KILLED
proof of input_L8ZEZcidUl.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 2 ms]
        (14) CpxRNTS
        (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (16) CpxRNTS
        (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 4 ms]
        (18) CpxRNTS
        (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (20) CpxRNTS
        (21) IntTrsBoundProof [UPPER BOUND(ID), 118 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 67 ms]
        (24) CpxRNTS
        (25) ResultPropagationProof [UPPER BOUND(ID), 1 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 198 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 84 ms]
        (30) CpxRNTS
        (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (32) CpxRNTS
        (33) IntTrsBoundProof [UPPER BOUND(ID), 932 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 17 ms]
        (36) CpxRNTS
    (37) CompletionProof [UPPER BOUND(ID), 0 ms]
    (38) CpxTypedWeightedCompleteTrs
        (39) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 9 ms]
        (40) CpxRNTS
(41) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(42) CdtProblem
    (43) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (44) CdtProblem
    (45) CdtUsableRulesProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) CdtProblem
    (47) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) CdtProblem
    (49) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) CdtProblem
    (51) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CdtProblem
    (53) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 386 ms]
    (54) CdtProblem
    (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CdtProblem
    (57) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CdtProblem
    (59) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
    (61) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (62) CdtProblem
    (63) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CdtProblem
    (65) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 314 ms]
    (66) CdtProblem
    (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) CdtProblem
    (87) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 2083 ms]
    (88) CdtProblem
    (89) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem
    (91) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (92) CdtProblem
    (93) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (94) CdtProblem
    (95) CdtKnowledgeProof [BOTH BOUNDS(ID, ID), 0 ms]
    (96) CdtProblem
    (97) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (98) CdtProblem
    (99) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (100) CdtProblem
    (101) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 1071 ms]
    (102) CdtProblem
    (103) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (104) CdtProblem
    (105) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (106) CdtProblem
    (107) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (108) CdtProblem
    (109) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (110) CdtProblem
    (111) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (112) CdtProblem
    (113) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (114) CdtProblem
    (115) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (116) CdtProblem
    (117) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (118) CdtProblem
    (119) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (120) CdtProblem
    (121) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (122) CdtProblem
    (123) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (124) CdtProblem
    (125) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (126) CdtProblem
    (127) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (128) CdtProblem
    (129) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (130) CdtProblem
    (131) CdtRewritingProof [BOTH BOUNDS(ID, ID), 3 ms]
    (132) CdtProblem
    (133) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (134) CdtProblem
    (135) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (136) CdtProblem
    (137) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (138) CdtProblem
    (139) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (140) CdtProblem
    (141) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (142) CdtProblem
    (143) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (144) CdtProblem
    (145) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (146) CdtProblem
    (147) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (148) CdtProblem
    (149) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (150) CdtProblem
    (151) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (152) CdtProblem
    (153) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (154) CdtProblem
    (155) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (156) CdtProblem
    (157) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (158) CdtProblem
    (159) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (160) CdtProblem
    (161) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (162) CdtProblem
    (163) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (164) CdtProblem
    (165) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (166) CdtProblem
    (167) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (168) CdtProblem
    (169) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 13 ms]
    (170) CdtProblem
    (171) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (172) CdtProblem
    (173) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 22 ms]
    (174) CdtProblem
    (175) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 2087 ms]
    (176) CdtProblem
    (177) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (178) CdtProblem
    (179) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (180) CdtProblem
    (181) CdtRuleRemovalProof [UPPER BOUND(ADD(n^2)), 2689 ms]
    (182) CdtProblem
    (183) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (184) CdtProblem
    (185) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (186) CdtProblem
    (187) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 82 ms]
    (188) CdtProblem
    (189) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (190) CdtProblem
    (191) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 37 ms]
    (192) CdtProblem
    (193) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (194) CdtProblem
    (195) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (196) CdtProblem
    (197) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (198) CdtProblem
    (199) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (200) CdtProblem
    (201) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 39 ms]
    (202) CdtProblem
    (203) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (204) CdtProblem
    (205) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (206) CdtProblem
    (207) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (208) CdtProblem
    (209) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (210) CdtProblem
    (211) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (212) CdtProblem
    (213) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (214) CdtProblem
    (215) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (216) CdtProblem
    (217) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (218) CdtProblem
    (219) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (220) CdtProblem
    (221) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (222) CdtProblem
    (223) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (224) CdtProblem
    (225) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (226) CdtProblem
    (227) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (228) CdtProblem
    (229) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (230) 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:

   tower(x) -> f(a, x, s(0))
   f(a, 0, y) -> y
   f(a, s(x), y) -> f(b, y, s(x))
   f(b, y, x) -> f(a, half(x), exp(y))
   exp(0) -> s(0)
   exp(s(x)) -> double(exp(x))
   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(x))) -> s(half(x))

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:

   tower(x) -> f(a, x, s(0'))
   f(a, 0', y) -> y
   f(a, s(x), y) -> f(b, y, s(x))
   f(b, y, x) -> f(a, half(x), exp(y))
   exp(0') -> s(0')
   exp(s(x)) -> double(exp(x))
   double(0') -> 0'
   double(s(x)) -> s(s(double(x)))
   half(0') -> double(0')
   half(s(0')) -> half(0')
   half(s(s(x))) -> s(half(x))

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:

   tower(x) -> f(a, x, s(0))
   f(a, 0, y) -> y
   f(a, s(x), y) -> f(b, y, s(x))
   f(b, y, x) -> f(a, half(x), exp(y))
   exp(0) -> s(0)
   exp(s(x)) -> double(exp(x))
   double(0) -> 0
   double(s(x)) -> s(s(double(x)))
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(x))) -> s(half(x))

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:

   tower(x) -> f(a, x, s(0)) [1]
   f(a, 0, y) -> y [1]
   f(a, s(x), y) -> f(b, y, s(x)) [1]
   f(b, y, x) -> f(a, half(x), exp(y)) [1]
   exp(0) -> s(0) [1]
   exp(s(x)) -> double(exp(x)) [1]
   double(0) -> 0 [1]
   double(s(x)) -> s(s(double(x))) [1]
   half(0) -> double(0) [1]
   half(s(0)) -> half(0) [1]
   half(s(s(x))) -> s(half(x)) [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:

   tower(x) -> f(a, x, s(0)) [1]
   f(a, 0, y) -> y [1]
   f(a, s(x), y) -> f(b, y, s(x)) [1]
   f(b, y, x) -> f(a, half(x), exp(y)) [1]
   exp(0) -> s(0) [1]
   exp(s(x)) -> double(exp(x)) [1]
   double(0) -> 0 [1]
   double(s(x)) -> s(s(double(x))) [1]
   half(0) -> double(0) [1]
   half(s(0)) -> half(0) [1]
   half(s(s(x))) -> s(half(x)) [1]

The TRS has the following type information:
tower :: 0:s -> 0:s
f :: a:b -> 0:s -> 0:s -> 0:s
a :: a:b
s :: 0:s -> 0:s
0 :: 0:s
b :: a:b
half :: 0:s -> 0:s
exp :: 0:s -> 0:s
double :: 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_1
   f_3

(c) The following functions are completely defined:

   exp_1
   half_1
   double_1

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:

   tower(x) -> f(a, x, s(0)) [1]
   f(a, 0, y) -> y [1]
   f(a, s(x), y) -> f(b, y, s(x)) [1]
   f(b, y, x) -> f(a, half(x), exp(y)) [1]
   exp(0) -> s(0) [1]
   exp(s(x)) -> double(exp(x)) [1]
   double(0) -> 0 [1]
   double(s(x)) -> s(s(double(x))) [1]
   half(0) -> double(0) [1]
   half(s(0)) -> half(0) [1]
   half(s(s(x))) -> s(half(x)) [1]

The TRS has the following type information:
tower :: 0:s -> 0:s
f :: a:b -> 0:s -> 0:s -> 0:s
a :: a:b
s :: 0:s -> 0:s
0 :: 0:s
b :: a:b
half :: 0:s -> 0:s
exp :: 0:s -> 0:s
double :: 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:

   tower(x) -> f(a, x, s(0)) [1]
   f(a, 0, y) -> y [1]
   f(a, s(x), y) -> f(b, y, s(x)) [1]
   f(b, 0, 0) -> f(a, double(0), s(0)) [3]
   f(b, s(x''), 0) -> f(a, double(0), double(exp(x''))) [3]
   f(b, 0, s(0)) -> f(a, half(0), s(0)) [3]
   f(b, s(x1), s(0)) -> f(a, half(0), double(exp(x1))) [3]
   f(b, 0, s(s(x'))) -> f(a, s(half(x')), s(0)) [3]
   f(b, s(x2), s(s(x'))) -> f(a, s(half(x')), double(exp(x2))) [3]
   exp(0) -> s(0) [1]
   exp(s(0)) -> double(s(0)) [2]
   exp(s(s(x3))) -> double(double(exp(x3))) [2]
   double(0) -> 0 [1]
   double(s(x)) -> s(s(double(x))) [1]
   half(0) -> double(0) [1]
   half(s(0)) -> half(0) [1]
   half(s(s(x))) -> s(half(x)) [1]

The TRS has the following type information:
tower :: 0:s -> 0:s
f :: a:b -> 0:s -> 0:s -> 0:s
a :: a:b
s :: 0:s -> 0:s
0 :: 0:s
b :: a:b
half :: 0:s -> 0:s
exp :: 0:s -> 0:s
double :: 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:

   a => 0
   0 => 0
   b => 1

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

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

   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 }-> 1 + (1 + double(x)) :|: x >= 0, z = 1 + x
   exp(z) -{ 2 }-> double(double(exp(x3))) :|: z = 1 + (1 + x3), x3 >= 0
   exp(z) -{ 2 }-> double(1 + 0) :|: z = 1 + 0
   exp(z) -{ 1 }-> 1 + 0 :|: z = 0
   f(z, z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z = 0, z' = 0
   f(z, z', z'') -{ 1 }-> f(1, y, 1 + x) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0, z = 0
   f(z, z', z'') -{ 3 }-> f(0, half(0), double(exp(x1))) :|: x1 >= 0, z = 1, z' = 1 + x1, z'' = 1 + 0
   f(z, z', z'') -{ 3 }-> f(0, half(0), 1 + 0) :|: z = 1, z'' = 1 + 0, z' = 0
   f(z, z', z'') -{ 3 }-> f(0, double(0), double(exp(x''))) :|: z'' = 0, z' = 1 + x'', z = 1, x'' >= 0
   f(z, z', z'') -{ 3 }-> f(0, double(0), 1 + 0) :|: z'' = 0, z = 1, z' = 0
   f(z, z', z'') -{ 3 }-> f(0, 1 + half(x'), double(exp(x2))) :|: z' = 1 + x2, z'' = 1 + (1 + x'), z = 1, x' >= 0, x2 >= 0
   f(z, z', z'') -{ 3 }-> f(0, 1 + half(x'), 1 + 0) :|: z'' = 1 + (1 + x'), z = 1, x' >= 0, z' = 0
   half(z) -{ 1 }-> half(0) :|: z = 1 + 0
   half(z) -{ 1 }-> double(0) :|: z = 0
   half(z) -{ 1 }-> 1 + half(x) :|: x >= 0, z = 1 + (1 + x)
   tower(z) -{ 1 }-> f(0, x, 1 + 0) :|: x >= 0, z = x


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

(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:

   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0
   exp(z) -{ 2 }-> double(double(exp(z - 2))) :|: z - 2 >= 0
   exp(z) -{ 2 }-> double(1 + 0) :|: z = 1 + 0
   exp(z) -{ 1 }-> 1 + 0 :|: z = 0
   f(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' = 0
   f(z, z', z'') -{ 1 }-> f(1, z'', 1 + (z' - 1)) :|: z' - 1 >= 0, z'' >= 0, z = 0
   f(z, z', z'') -{ 3 }-> f(0, half(0), double(exp(z' - 1))) :|: z' - 1 >= 0, z = 1, z'' = 1 + 0
   f(z, z', z'') -{ 3 }-> f(0, half(0), 1 + 0) :|: z = 1, z'' = 1 + 0, z' = 0
   f(z, z', z'') -{ 3 }-> f(0, double(0), double(exp(z' - 1))) :|: z'' = 0, z = 1, z' - 1 >= 0
   f(z, z', z'') -{ 3 }-> f(0, double(0), 1 + 0) :|: z'' = 0, z = 1, z' = 0
   f(z, z', z'') -{ 3 }-> f(0, 1 + half(z'' - 2), double(exp(z' - 1))) :|: z = 1, z'' - 2 >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 3 }-> f(0, 1 + half(z'' - 2), 1 + 0) :|: z = 1, z'' - 2 >= 0, z' = 0
   half(z) -{ 1 }-> half(0) :|: z = 1 + 0
   half(z) -{ 1 }-> double(0) :|: z = 0
   half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 0
   tower(z) -{ 1 }-> f(0, z, 1 + 0) :|: z >= 0


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

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

   { double }
   { half }
   { exp }
   { f }
   { tower }

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

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

   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0
   exp(z) -{ 2 }-> double(double(exp(z - 2))) :|: z - 2 >= 0
   exp(z) -{ 2 }-> double(1 + 0) :|: z = 1 + 0
   exp(z) -{ 1 }-> 1 + 0 :|: z = 0
   f(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' = 0
   f(z, z', z'') -{ 1 }-> f(1, z'', 1 + (z' - 1)) :|: z' - 1 >= 0, z'' >= 0, z = 0
   f(z, z', z'') -{ 3 }-> f(0, half(0), double(exp(z' - 1))) :|: z' - 1 >= 0, z = 1, z'' = 1 + 0
   f(z, z', z'') -{ 3 }-> f(0, half(0), 1 + 0) :|: z = 1, z'' = 1 + 0, z' = 0
   f(z, z', z'') -{ 3 }-> f(0, double(0), double(exp(z' - 1))) :|: z'' = 0, z = 1, z' - 1 >= 0
   f(z, z', z'') -{ 3 }-> f(0, double(0), 1 + 0) :|: z'' = 0, z = 1, z' = 0
   f(z, z', z'') -{ 3 }-> f(0, 1 + half(z'' - 2), double(exp(z' - 1))) :|: z = 1, z'' - 2 >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 3 }-> f(0, 1 + half(z'' - 2), 1 + 0) :|: z = 1, z'' - 2 >= 0, z' = 0
   half(z) -{ 1 }-> half(0) :|: z = 1 + 0
   half(z) -{ 1 }-> double(0) :|: z = 0
   half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 0
   tower(z) -{ 1 }-> f(0, z, 1 + 0) :|: z >= 0

Function symbols to be analyzed: {double}, {half}, {exp}, {f}, {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:

   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0
   exp(z) -{ 2 }-> double(double(exp(z - 2))) :|: z - 2 >= 0
   exp(z) -{ 2 }-> double(1 + 0) :|: z = 1 + 0
   exp(z) -{ 1 }-> 1 + 0 :|: z = 0
   f(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' = 0
   f(z, z', z'') -{ 1 }-> f(1, z'', 1 + (z' - 1)) :|: z' - 1 >= 0, z'' >= 0, z = 0
   f(z, z', z'') -{ 3 }-> f(0, half(0), double(exp(z' - 1))) :|: z' - 1 >= 0, z = 1, z'' = 1 + 0
   f(z, z', z'') -{ 3 }-> f(0, half(0), 1 + 0) :|: z = 1, z'' = 1 + 0, z' = 0
   f(z, z', z'') -{ 3 }-> f(0, double(0), double(exp(z' - 1))) :|: z'' = 0, z = 1, z' - 1 >= 0
   f(z, z', z'') -{ 3 }-> f(0, double(0), 1 + 0) :|: z'' = 0, z = 1, z' = 0
   f(z, z', z'') -{ 3 }-> f(0, 1 + half(z'' - 2), double(exp(z' - 1))) :|: z = 1, z'' - 2 >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 3 }-> f(0, 1 + half(z'' - 2), 1 + 0) :|: z = 1, z'' - 2 >= 0, z' = 0
   half(z) -{ 1 }-> half(0) :|: z = 1 + 0
   half(z) -{ 1 }-> double(0) :|: z = 0
   half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 0
   tower(z) -{ 1 }-> f(0, z, 1 + 0) :|: z >= 0

Function symbols to be analyzed: {double}, {half}, {exp}, {f}, {tower}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: double
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2*z

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

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

   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0
   exp(z) -{ 2 }-> double(double(exp(z - 2))) :|: z - 2 >= 0
   exp(z) -{ 2 }-> double(1 + 0) :|: z = 1 + 0
   exp(z) -{ 1 }-> 1 + 0 :|: z = 0
   f(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' = 0
   f(z, z', z'') -{ 1 }-> f(1, z'', 1 + (z' - 1)) :|: z' - 1 >= 0, z'' >= 0, z = 0
   f(z, z', z'') -{ 3 }-> f(0, half(0), double(exp(z' - 1))) :|: z' - 1 >= 0, z = 1, z'' = 1 + 0
   f(z, z', z'') -{ 3 }-> f(0, half(0), 1 + 0) :|: z = 1, z'' = 1 + 0, z' = 0
   f(z, z', z'') -{ 3 }-> f(0, double(0), double(exp(z' - 1))) :|: z'' = 0, z = 1, z' - 1 >= 0
   f(z, z', z'') -{ 3 }-> f(0, double(0), 1 + 0) :|: z'' = 0, z = 1, z' = 0
   f(z, z', z'') -{ 3 }-> f(0, 1 + half(z'' - 2), double(exp(z' - 1))) :|: z = 1, z'' - 2 >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 3 }-> f(0, 1 + half(z'' - 2), 1 + 0) :|: z = 1, z'' - 2 >= 0, z' = 0
   half(z) -{ 1 }-> half(0) :|: z = 1 + 0
   half(z) -{ 1 }-> double(0) :|: z = 0
   half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 0
   tower(z) -{ 1 }-> f(0, z, 1 + 0) :|: z >= 0

Function symbols to be analyzed: {double}, {half}, {exp}, {f}, {tower}
Previous analysis results are:
  double: runtime: ?, size: O(n^1) [2*z]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 }-> 1 + (1 + double(z - 1)) :|: z - 1 >= 0
   exp(z) -{ 2 }-> double(double(exp(z - 2))) :|: z - 2 >= 0
   exp(z) -{ 2 }-> double(1 + 0) :|: z = 1 + 0
   exp(z) -{ 1 }-> 1 + 0 :|: z = 0
   f(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' = 0
   f(z, z', z'') -{ 1 }-> f(1, z'', 1 + (z' - 1)) :|: z' - 1 >= 0, z'' >= 0, z = 0
   f(z, z', z'') -{ 3 }-> f(0, half(0), double(exp(z' - 1))) :|: z' - 1 >= 0, z = 1, z'' = 1 + 0
   f(z, z', z'') -{ 3 }-> f(0, half(0), 1 + 0) :|: z = 1, z'' = 1 + 0, z' = 0
   f(z, z', z'') -{ 3 }-> f(0, double(0), double(exp(z' - 1))) :|: z'' = 0, z = 1, z' - 1 >= 0
   f(z, z', z'') -{ 3 }-> f(0, double(0), 1 + 0) :|: z'' = 0, z = 1, z' = 0
   f(z, z', z'') -{ 3 }-> f(0, 1 + half(z'' - 2), double(exp(z' - 1))) :|: z = 1, z'' - 2 >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 3 }-> f(0, 1 + half(z'' - 2), 1 + 0) :|: z = 1, z'' - 2 >= 0, z' = 0
   half(z) -{ 1 }-> half(0) :|: z = 1 + 0
   half(z) -{ 1 }-> double(0) :|: z = 0
   half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 0
   tower(z) -{ 1 }-> f(0, z, 1 + 0) :|: z >= 0

Function symbols to be analyzed: {half}, {exp}, {f}, {tower}
Previous analysis results are:
  double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]

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

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

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

   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 + z }-> 1 + (1 + s1) :|: s1 >= 0, s1 <= 2 * (z - 1), z - 1 >= 0
   exp(z) -{ 4 }-> s'' :|: s'' >= 0, s'' <= 2 * (1 + 0), z = 1 + 0
   exp(z) -{ 2 }-> double(double(exp(z - 2))) :|: z - 2 >= 0
   exp(z) -{ 1 }-> 1 + 0 :|: z = 0
   f(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' = 0
   f(z, z', z'') -{ 1 }-> f(1, z'', 1 + (z' - 1)) :|: z' - 1 >= 0, z'' >= 0, z = 0
   f(z, z', z'') -{ 4 }-> f(0, s, 1 + 0) :|: s >= 0, s <= 2 * 0, z'' = 0, z = 1, z' = 0
   f(z, z', z'') -{ 4 }-> f(0, s', double(exp(z' - 1))) :|: s' >= 0, s' <= 2 * 0, z'' = 0, z = 1, z' - 1 >= 0
   f(z, z', z'') -{ 3 }-> f(0, half(0), double(exp(z' - 1))) :|: z' - 1 >= 0, z = 1, z'' = 1 + 0
   f(z, z', z'') -{ 3 }-> f(0, half(0), 1 + 0) :|: z = 1, z'' = 1 + 0, z' = 0
   f(z, z', z'') -{ 3 }-> f(0, 1 + half(z'' - 2), double(exp(z' - 1))) :|: z = 1, z'' - 2 >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 3 }-> f(0, 1 + half(z'' - 2), 1 + 0) :|: z = 1, z'' - 2 >= 0, z' = 0
   half(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 2 * 0, z = 0
   half(z) -{ 1 }-> half(0) :|: z = 1 + 0
   half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 0
   tower(z) -{ 1 }-> f(0, z, 1 + 0) :|: z >= 0

Function symbols to be analyzed: {half}, {exp}, {f}, {tower}
Previous analysis results are:
  double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 + z }-> 1 + (1 + s1) :|: s1 >= 0, s1 <= 2 * (z - 1), z - 1 >= 0
   exp(z) -{ 4 }-> s'' :|: s'' >= 0, s'' <= 2 * (1 + 0), z = 1 + 0
   exp(z) -{ 2 }-> double(double(exp(z - 2))) :|: z - 2 >= 0
   exp(z) -{ 1 }-> 1 + 0 :|: z = 0
   f(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' = 0
   f(z, z', z'') -{ 1 }-> f(1, z'', 1 + (z' - 1)) :|: z' - 1 >= 0, z'' >= 0, z = 0
   f(z, z', z'') -{ 4 }-> f(0, s, 1 + 0) :|: s >= 0, s <= 2 * 0, z'' = 0, z = 1, z' = 0
   f(z, z', z'') -{ 4 }-> f(0, s', double(exp(z' - 1))) :|: s' >= 0, s' <= 2 * 0, z'' = 0, z = 1, z' - 1 >= 0
   f(z, z', z'') -{ 3 }-> f(0, half(0), double(exp(z' - 1))) :|: z' - 1 >= 0, z = 1, z'' = 1 + 0
   f(z, z', z'') -{ 3 }-> f(0, half(0), 1 + 0) :|: z = 1, z'' = 1 + 0, z' = 0
   f(z, z', z'') -{ 3 }-> f(0, 1 + half(z'' - 2), double(exp(z' - 1))) :|: z = 1, z'' - 2 >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 3 }-> f(0, 1 + half(z'' - 2), 1 + 0) :|: z = 1, z'' - 2 >= 0, z' = 0
   half(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 2 * 0, z = 0
   half(z) -{ 1 }-> half(0) :|: z = 1 + 0
   half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 0
   tower(z) -{ 1 }-> f(0, z, 1 + 0) :|: z >= 0

Function symbols to be analyzed: {half}, {exp}, {f}, {tower}
Previous analysis results are:
  double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  half: runtime: ?, size: O(n^1) [z]

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: half
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 4 + z

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

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

   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 + z }-> 1 + (1 + s1) :|: s1 >= 0, s1 <= 2 * (z - 1), z - 1 >= 0
   exp(z) -{ 4 }-> s'' :|: s'' >= 0, s'' <= 2 * (1 + 0), z = 1 + 0
   exp(z) -{ 2 }-> double(double(exp(z - 2))) :|: z - 2 >= 0
   exp(z) -{ 1 }-> 1 + 0 :|: z = 0
   f(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' = 0
   f(z, z', z'') -{ 1 }-> f(1, z'', 1 + (z' - 1)) :|: z' - 1 >= 0, z'' >= 0, z = 0
   f(z, z', z'') -{ 4 }-> f(0, s, 1 + 0) :|: s >= 0, s <= 2 * 0, z'' = 0, z = 1, z' = 0
   f(z, z', z'') -{ 4 }-> f(0, s', double(exp(z' - 1))) :|: s' >= 0, s' <= 2 * 0, z'' = 0, z = 1, z' - 1 >= 0
   f(z, z', z'') -{ 3 }-> f(0, half(0), double(exp(z' - 1))) :|: z' - 1 >= 0, z = 1, z'' = 1 + 0
   f(z, z', z'') -{ 3 }-> f(0, half(0), 1 + 0) :|: z = 1, z'' = 1 + 0, z' = 0
   f(z, z', z'') -{ 3 }-> f(0, 1 + half(z'' - 2), double(exp(z' - 1))) :|: z = 1, z'' - 2 >= 0, z' - 1 >= 0
   f(z, z', z'') -{ 3 }-> f(0, 1 + half(z'' - 2), 1 + 0) :|: z = 1, z'' - 2 >= 0, z' = 0
   half(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 2 * 0, z = 0
   half(z) -{ 1 }-> half(0) :|: z = 1 + 0
   half(z) -{ 1 }-> 1 + half(z - 2) :|: z - 2 >= 0
   tower(z) -{ 1 }-> f(0, z, 1 + 0) :|: z >= 0

Function symbols to be analyzed: {exp}, {f}, {tower}
Previous analysis results are:
  double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  half: runtime: O(n^1) [4 + z], size: O(n^1) [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:

   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 + z }-> 1 + (1 + s1) :|: s1 >= 0, s1 <= 2 * (z - 1), z - 1 >= 0
   exp(z) -{ 4 }-> s'' :|: s'' >= 0, s'' <= 2 * (1 + 0), z = 1 + 0
   exp(z) -{ 2 }-> double(double(exp(z - 2))) :|: z - 2 >= 0
   exp(z) -{ 1 }-> 1 + 0 :|: z = 0
   f(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' = 0
   f(z, z', z'') -{ 1 }-> f(1, z'', 1 + (z' - 1)) :|: z' - 1 >= 0, z'' >= 0, z = 0
   f(z, z', z'') -{ 4 }-> f(0, s, 1 + 0) :|: s >= 0, s <= 2 * 0, z'' = 0, z = 1, z' = 0
   f(z, z', z'') -{ 4 }-> f(0, s', double(exp(z' - 1))) :|: s' >= 0, s' <= 2 * 0, z'' = 0, z = 1, z' - 1 >= 0
   f(z, z', z'') -{ 7 }-> f(0, s3, 1 + 0) :|: s3 >= 0, s3 <= 0, z = 1, z'' = 1 + 0, z' = 0
   f(z, z', z'') -{ 7 }-> f(0, s4, double(exp(z' - 1))) :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z = 1, z'' = 1 + 0
   f(z, z', z'') -{ 5 + z'' }-> f(0, 1 + s5, 1 + 0) :|: s5 >= 0, s5 <= z'' - 2, z = 1, z'' - 2 >= 0, z' = 0
   f(z, z', z'') -{ 5 + z'' }-> f(0, 1 + s6, double(exp(z' - 1))) :|: s6 >= 0, s6 <= z'' - 2, z = 1, z'' - 2 >= 0, z' - 1 >= 0
   half(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 2 * 0, z = 0
   half(z) -{ 5 }-> s7 :|: s7 >= 0, s7 <= 0, z = 1 + 0
   half(z) -{ 3 + z }-> 1 + s8 :|: s8 >= 0, s8 <= z - 2, z - 2 >= 0
   tower(z) -{ 1 }-> f(0, z, 1 + 0) :|: z >= 0

Function symbols to be analyzed: {exp}, {f}, {tower}
Previous analysis results are:
  double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  half: runtime: O(n^1) [4 + z], size: O(n^1) [z]

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 + z }-> 1 + (1 + s1) :|: s1 >= 0, s1 <= 2 * (z - 1), z - 1 >= 0
   exp(z) -{ 4 }-> s'' :|: s'' >= 0, s'' <= 2 * (1 + 0), z = 1 + 0
   exp(z) -{ 2 }-> double(double(exp(z - 2))) :|: z - 2 >= 0
   exp(z) -{ 1 }-> 1 + 0 :|: z = 0
   f(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' = 0
   f(z, z', z'') -{ 1 }-> f(1, z'', 1 + (z' - 1)) :|: z' - 1 >= 0, z'' >= 0, z = 0
   f(z, z', z'') -{ 4 }-> f(0, s, 1 + 0) :|: s >= 0, s <= 2 * 0, z'' = 0, z = 1, z' = 0
   f(z, z', z'') -{ 4 }-> f(0, s', double(exp(z' - 1))) :|: s' >= 0, s' <= 2 * 0, z'' = 0, z = 1, z' - 1 >= 0
   f(z, z', z'') -{ 7 }-> f(0, s3, 1 + 0) :|: s3 >= 0, s3 <= 0, z = 1, z'' = 1 + 0, z' = 0
   f(z, z', z'') -{ 7 }-> f(0, s4, double(exp(z' - 1))) :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z = 1, z'' = 1 + 0
   f(z, z', z'') -{ 5 + z'' }-> f(0, 1 + s5, 1 + 0) :|: s5 >= 0, s5 <= z'' - 2, z = 1, z'' - 2 >= 0, z' = 0
   f(z, z', z'') -{ 5 + z'' }-> f(0, 1 + s6, double(exp(z' - 1))) :|: s6 >= 0, s6 <= z'' - 2, z = 1, z'' - 2 >= 0, z' - 1 >= 0
   half(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 2 * 0, z = 0
   half(z) -{ 5 }-> s7 :|: s7 >= 0, s7 <= 0, z = 1 + 0
   half(z) -{ 3 + z }-> 1 + s8 :|: s8 >= 0, s8 <= z - 2, z - 2 >= 0
   tower(z) -{ 1 }-> f(0, z, 1 + 0) :|: z >= 0

Function symbols to be analyzed: {exp}, {f}, {tower}
Previous analysis results are:
  double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  half: runtime: O(n^1) [4 + z], size: O(n^1) [z]
  exp: runtime: ?, size: EXP

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

(35) 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: ?

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

(36)
Obligation:
Complexity RNTS consisting of the following rules:

   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 + z }-> 1 + (1 + s1) :|: s1 >= 0, s1 <= 2 * (z - 1), z - 1 >= 0
   exp(z) -{ 4 }-> s'' :|: s'' >= 0, s'' <= 2 * (1 + 0), z = 1 + 0
   exp(z) -{ 2 }-> double(double(exp(z - 2))) :|: z - 2 >= 0
   exp(z) -{ 1 }-> 1 + 0 :|: z = 0
   f(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z = 0, z' = 0
   f(z, z', z'') -{ 1 }-> f(1, z'', 1 + (z' - 1)) :|: z' - 1 >= 0, z'' >= 0, z = 0
   f(z, z', z'') -{ 4 }-> f(0, s, 1 + 0) :|: s >= 0, s <= 2 * 0, z'' = 0, z = 1, z' = 0
   f(z, z', z'') -{ 4 }-> f(0, s', double(exp(z' - 1))) :|: s' >= 0, s' <= 2 * 0, z'' = 0, z = 1, z' - 1 >= 0
   f(z, z', z'') -{ 7 }-> f(0, s3, 1 + 0) :|: s3 >= 0, s3 <= 0, z = 1, z'' = 1 + 0, z' = 0
   f(z, z', z'') -{ 7 }-> f(0, s4, double(exp(z' - 1))) :|: s4 >= 0, s4 <= 0, z' - 1 >= 0, z = 1, z'' = 1 + 0
   f(z, z', z'') -{ 5 + z'' }-> f(0, 1 + s5, 1 + 0) :|: s5 >= 0, s5 <= z'' - 2, z = 1, z'' - 2 >= 0, z' = 0
   f(z, z', z'') -{ 5 + z'' }-> f(0, 1 + s6, double(exp(z' - 1))) :|: s6 >= 0, s6 <= z'' - 2, z = 1, z'' - 2 >= 0, z' - 1 >= 0
   half(z) -{ 2 }-> s2 :|: s2 >= 0, s2 <= 2 * 0, z = 0
   half(z) -{ 5 }-> s7 :|: s7 >= 0, s7 <= 0, z = 1 + 0
   half(z) -{ 3 + z }-> 1 + s8 :|: s8 >= 0, s8 <= z - 2, z - 2 >= 0
   tower(z) -{ 1 }-> f(0, z, 1 + 0) :|: z >= 0

Function symbols to be analyzed: {exp}, {f}, {tower}
Previous analysis results are:
  double: runtime: O(n^1) [1 + z], size: O(n^1) [2*z]
  half: runtime: O(n^1) [4 + z], size: O(n^1) [z]
  exp: runtime: INF, size: EXP

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

(37) 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

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

(38)
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:

   tower(x) -> f(a, x, s(0)) [1]
   f(a, 0, y) -> y [1]
   f(a, s(x), y) -> f(b, y, s(x)) [1]
   f(b, y, x) -> f(a, half(x), exp(y)) [1]
   exp(0) -> s(0) [1]
   exp(s(x)) -> double(exp(x)) [1]
   double(0) -> 0 [1]
   double(s(x)) -> s(s(double(x))) [1]
   half(0) -> double(0) [1]
   half(s(0)) -> half(0) [1]
   half(s(s(x))) -> s(half(x)) [1]

The TRS has the following type information:
tower :: 0:s -> 0:s
f :: a:b -> 0:s -> 0:s -> 0:s
a :: a:b
s :: 0:s -> 0:s
0 :: 0:s
b :: a:b
half :: 0:s -> 0:s
exp :: 0:s -> 0:s
double :: 0:s -> 0:s

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

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

   a => 0
   0 => 0
   b => 1

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

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

   double(z) -{ 1 }-> 0 :|: z = 0
   double(z) -{ 1 }-> 1 + (1 + double(x)) :|: x >= 0, z = 1 + x
   exp(z) -{ 1 }-> double(exp(x)) :|: x >= 0, z = 1 + x
   exp(z) -{ 1 }-> 1 + 0 :|: z = 0
   f(z, z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z = 0, z' = 0
   f(z, z', z'') -{ 1 }-> f(1, y, 1 + x) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0, z = 0
   f(z, z', z'') -{ 1 }-> f(0, half(x), exp(y)) :|: z = 1, y >= 0, x >= 0, z'' = x, z' = y
   half(z) -{ 1 }-> half(0) :|: z = 1 + 0
   half(z) -{ 1 }-> double(0) :|: z = 0
   half(z) -{ 1 }-> 1 + half(x) :|: x >= 0, z = 1 + (1 + x)
   tower(z) -{ 1 }-> f(0, x, 1 + 0) :|: x >= 0, z = x

Only complete derivations are relevant for the runtime complexity.

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

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

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   tower(z0) -> f(a, z0, s(0))
   f(a, 0, z0) -> z0
   f(a, s(z0), z1) -> f(b, z1, s(z0))
   f(b, z0, z1) -> f(a, half(z1), exp(z0))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
Tuples:
   TOWER(z0) -> c(F(a, z0, s(0)))
   F(a, 0, z0) -> c1
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   F(b, z0, z1) -> c3(F(a, half(z1), exp(z0)), HALF(z1))
   F(b, z0, z1) -> c4(F(a, half(z1), exp(z0)), EXP(z0))
   EXP(0) -> c5
   EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0))
   DOUBLE(0) -> c7
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(0) -> c9(DOUBLE(0))
   HALF(s(0)) -> c10(HALF(0))
   HALF(s(s(z0))) -> c11(HALF(z0))
S tuples:
   TOWER(z0) -> c(F(a, z0, s(0)))
   F(a, 0, z0) -> c1
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   F(b, z0, z1) -> c3(F(a, half(z1), exp(z0)), HALF(z1))
   F(b, z0, z1) -> c4(F(a, half(z1), exp(z0)), EXP(z0))
   EXP(0) -> c5
   EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0))
   DOUBLE(0) -> c7
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(0) -> c9(DOUBLE(0))
   HALF(s(0)) -> c10(HALF(0))
   HALF(s(s(z0))) -> c11(HALF(z0))
K tuples:none
Defined Rule Symbols:   tower_1, f_3, exp_1, double_1, half_1

Defined Pair Symbols:   TOWER_1, F_3, EXP_1, DOUBLE_1, HALF_1

Compound Symbols:   c_1, c1, c2_1, c3_2, c4_2, c5, c6_2, c7, c8_1, c9_1, c10_1, c11_1


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

(43) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   TOWER(z0) -> c(F(a, z0, s(0)))
Removed 5 trailing nodes:
   EXP(0) -> c5
   DOUBLE(0) -> c7
   HALF(0) -> c9(DOUBLE(0))
   HALF(s(0)) -> c10(HALF(0))
   F(a, 0, z0) -> c1

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   tower(z0) -> f(a, z0, s(0))
   f(a, 0, z0) -> z0
   f(a, s(z0), z1) -> f(b, z1, s(z0))
   f(b, z0, z1) -> f(a, half(z1), exp(z0))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   F(b, z0, z1) -> c3(F(a, half(z1), exp(z0)), HALF(z1))
   F(b, z0, z1) -> c4(F(a, half(z1), exp(z0)), EXP(z0))
   EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   F(b, z0, z1) -> c3(F(a, half(z1), exp(z0)), HALF(z1))
   F(b, z0, z1) -> c4(F(a, half(z1), exp(z0)), EXP(z0))
   EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
K tuples:none
Defined Rule Symbols:   tower_1, f_3, exp_1, double_1, half_1

Defined Pair Symbols:   F_3, EXP_1, DOUBLE_1, HALF_1

Compound Symbols:   c2_1, c3_2, c4_2, c6_2, c8_1, c11_1


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

(45) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   tower(z0) -> f(a, z0, s(0))
   f(a, 0, z0) -> z0
   f(a, s(z0), z1) -> f(b, z1, s(z0))
   f(b, z0, z1) -> f(a, half(z1), exp(z0))

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   F(b, z0, z1) -> c3(F(a, half(z1), exp(z0)), HALF(z1))
   F(b, z0, z1) -> c4(F(a, half(z1), exp(z0)), EXP(z0))
   EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   F(b, z0, z1) -> c3(F(a, half(z1), exp(z0)), HALF(z1))
   F(b, z0, z1) -> c4(F(a, half(z1), exp(z0)), EXP(z0))
   EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
K tuples:none
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, EXP_1, DOUBLE_1, HALF_1

Compound Symbols:   c2_1, c3_2, c4_2, c6_2, c8_1, c11_1


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

(47) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(b, z0, z1) -> c3(F(a, half(z1), exp(z0)), HALF(z1)) by 
   F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1))
   F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1))
   F(b, x0, 0) -> c3(F(a, double(0), exp(x0)), HALF(0))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)), HALF(s(0)))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   F(b, z0, z1) -> c4(F(a, half(z1), exp(z0)), EXP(z0))
   EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1))
   F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1))
   F(b, x0, 0) -> c3(F(a, double(0), exp(x0)), HALF(0))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)), HALF(s(0)))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   F(b, z0, z1) -> c4(F(a, half(z1), exp(z0)), EXP(z0))
   EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1))
   F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1))
   F(b, x0, 0) -> c3(F(a, double(0), exp(x0)), HALF(0))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)), HALF(s(0)))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
K tuples:none
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, EXP_1, DOUBLE_1, HALF_1

Compound Symbols:   c2_1, c4_2, c6_2, c8_1, c11_1, c3_2


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

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   F(b, z0, z1) -> c4(F(a, half(z1), exp(z0)), EXP(z0))
   EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1))
   F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, 0) -> c3(F(a, double(0), exp(x0)))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   F(b, z0, z1) -> c4(F(a, half(z1), exp(z0)), EXP(z0))
   EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1))
   F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, 0) -> c3(F(a, double(0), exp(x0)))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)))
K tuples:none
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, EXP_1, DOUBLE_1, HALF_1

Compound Symbols:   c2_1, c4_2, c6_2, c8_1, c11_1, c3_2, c3_1


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

(51) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   F(b, x0, 0) -> c3(F(a, double(0), exp(x0)))

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   F(b, z0, z1) -> c4(F(a, half(z1), exp(z0)), EXP(z0))
   EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1))
   F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, 0) -> c3(F(a, double(0), exp(x0)))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   F(b, z0, z1) -> c4(F(a, half(z1), exp(z0)), EXP(z0))
   EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1))
   F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)))
K tuples:
   F(b, x0, 0) -> c3(F(a, double(0), exp(x0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, EXP_1, DOUBLE_1, HALF_1

Compound Symbols:   c2_1, c4_2, c6_2, c8_1, c11_1, c3_2, c3_1


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

(53) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1))
We considered the (Usable) Rules:
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   half(s(s(z0))) -> s(half(z0))
   exp(s(z0)) -> double(exp(z0))
   half(s(0)) -> half(0)
   exp(0) -> s(0)
   half(0) -> double(0)
And the Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   F(b, z0, z1) -> c4(F(a, half(z1), exp(z0)), EXP(z0))
   EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1))
   F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, 0) -> c3(F(a, double(0), exp(x0)))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [2]
   POL(DOUBLE(x_1)) = 0
   POL(EXP(x_1)) = 0
   POL(F(x_1, x_2, x_3)) = [2]x_3 + [2]x_2*x_3 + x_1*x_2
   POL(HALF(x_1)) = 0
   POL(a) = 0
   POL(b) = [1]
   POL(c11(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1)) = x_1
   POL(double(x_1)) = x_1
   POL(exp(x_1)) = 0
   POL(half(x_1)) = [2]
   POL(s(x_1)) = 0

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   F(b, z0, z1) -> c4(F(a, half(z1), exp(z0)), EXP(z0))
   EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1))
   F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, 0) -> c3(F(a, double(0), exp(x0)))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   F(b, z0, z1) -> c4(F(a, half(z1), exp(z0)), EXP(z0))
   EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)))
K tuples:
   F(b, x0, 0) -> c3(F(a, double(0), exp(x0)))
   F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, EXP_1, DOUBLE_1, HALF_1

Compound Symbols:   c2_1, c4_2, c6_2, c8_1, c11_1, c3_2, c3_1


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

(55) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(b, z0, z1) -> c4(F(a, half(z1), exp(z0)), EXP(z0)) by 
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)), EXP(0))
   F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0)))
   F(b, x0, 0) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1))
   F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, 0) -> c3(F(a, double(0), exp(x0)))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)), EXP(0))
   F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0)))
   F(b, x0, 0) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)), EXP(0))
   F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0)))
   F(b, x0, 0) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
K tuples:
   F(b, x0, 0) -> c3(F(a, double(0), exp(x0)))
   F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, EXP_1, DOUBLE_1, HALF_1

Compound Symbols:   c2_1, c6_2, c8_1, c11_1, c3_2, c3_1, c4_2


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

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1))
   F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, 0) -> c3(F(a, double(0), exp(x0)))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)))
   F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0)))
   F(b, x0, 0) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)))
   F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0)))
   F(b, x0, 0) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
K tuples:
   F(b, x0, 0) -> c3(F(a, double(0), exp(x0)))
   F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, EXP_1, DOUBLE_1, HALF_1

Compound Symbols:   c2_1, c6_2, c8_1, c11_1, c3_2, c3_1, c4_2, c4_1


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

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1))
   F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, 0) -> c3(F(a, double(0), exp(x0)))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)))
   F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0)))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   F(b, x0, 0) -> c(EXP(x0))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)))
   F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0)))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   F(b, x0, 0) -> c(EXP(x0))
K tuples:
   F(b, x0, 0) -> c3(F(a, double(0), exp(x0)))
   F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, EXP_1, DOUBLE_1, HALF_1

Compound Symbols:   c2_1, c6_2, c8_1, c11_1, c3_2, c3_1, c4_2, c4_1, c_1


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

(61) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   F(b, x0, 0) -> c(EXP(x0))

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1))
   F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, 0) -> c3(F(a, double(0), exp(x0)))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)))
   F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0)))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)))
   F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0)))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
K tuples:
   F(b, x0, 0) -> c3(F(a, double(0), exp(x0)))
   F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, EXP_1, DOUBLE_1, HALF_1

Compound Symbols:   c2_1, c6_2, c8_1, c11_1, c3_2, c3_1, c4_2, c4_1, c_1


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

(63) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1))
   F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, 0) -> c3(F(a, double(0), exp(x0)))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)))
   F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0)))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)))
   F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0)))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
K tuples:
   F(b, x0, 0) -> c3(F(a, double(0), exp(x0)))
   F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1))
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, EXP_1, DOUBLE_1, HALF_1

Compound Symbols:   c2_1, c6_2, c8_1, c11_1, c3_2, c3_1, c4_2, c4_1, c_1


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

(65) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
We considered the (Usable) Rules:
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   half(s(s(z0))) -> s(half(z0))
   exp(s(z0)) -> double(exp(z0))
   half(s(0)) -> half(0)
   exp(0) -> s(0)
   half(0) -> double(0)
And the Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1))
   F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, 0) -> c3(F(a, double(0), exp(x0)))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)))
   F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0)))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(DOUBLE(x_1)) = 0
   POL(EXP(x_1)) = 0
   POL(F(x_1, x_2, x_3)) = x_3 + x_1*x_2
   POL(HALF(x_1)) = 0
   POL(a) = 0
   POL(b) = [1]
   POL(c(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1)) = x_1
   POL(double(x_1)) = x_1^2
   POL(exp(x_1)) = 0
   POL(half(x_1)) = [1]
   POL(s(x_1)) = 0

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1))
   F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, 0) -> c3(F(a, double(0), exp(x0)))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)))
   F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0)))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)))
   F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0)))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
K tuples:
   F(b, x0, 0) -> c3(F(a, double(0), exp(x0)))
   F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1))
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, EXP_1, DOUBLE_1, HALF_1

Compound Symbols:   c2_1, c6_2, c8_1, c11_1, c3_2, c3_1, c4_2, c4_1, c_1


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

(67) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace EXP(s(z0)) -> c6(DOUBLE(exp(z0)), EXP(z0)) by 
   EXP(s(0)) -> c6(DOUBLE(s(0)), EXP(0))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1))
   F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, 0) -> c3(F(a, double(0), exp(x0)))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)))
   F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0)))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)), EXP(0))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)))
   F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0)))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)), EXP(0))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
K tuples:
   F(b, x0, 0) -> c3(F(a, double(0), exp(x0)))
   F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1))
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c3_1, c4_2, c4_1, c_1, c6_2


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

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1))
   F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, 0) -> c3(F(a, double(0), exp(x0)))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)))
   F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0)))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)))
   F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0)))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
K tuples:
   F(b, x0, 0) -> c3(F(a, double(0), exp(x0)))
   F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1))
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c3_1, c4_2, c4_1, c_1, c6_2, c6_1


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

(71) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1)) by 
   F(b, 0, 0) -> c3(F(a, double(0), s(0)), HALF(0))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)), HALF(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, 0) -> c3(F(a, double(0), exp(x0)))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)))
   F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0)))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, 0) -> c3(F(a, double(0), s(0)), HALF(0))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)), HALF(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)))
   F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0)))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
K tuples:
   F(b, x0, 0) -> c3(F(a, double(0), exp(x0)))
   F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1))
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c3_1, c4_2, c4_1, c_1, c6_2, c6_1


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

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, 0) -> c3(F(a, double(0), exp(x0)))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)))
   F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0)))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, 0) -> c3(F(a, double(0), s(0)))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)))
   F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0)))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
K tuples:
   F(b, x0, 0) -> c3(F(a, double(0), exp(x0)))
   F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1))
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c3_1, c4_2, c4_1, c_1, c6_2, c6_1


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

(75) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(b, s(z0), x1) -> c3(F(a, half(x1), double(exp(z0))), HALF(x1)) by 
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))), HALF(0))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))), HALF(s(0)))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, 0) -> c3(F(a, double(0), exp(x0)))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)))
   F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0)))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, 0) -> c3(F(a, double(0), s(0)))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))), HALF(0))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))), HALF(s(0)))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)))
   F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0)))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))), HALF(0))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))), HALF(s(0)))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
K tuples:
   F(b, x0, 0) -> c3(F(a, double(0), exp(x0)))
   F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1))
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c3_1, c4_2, c4_1, c_1, c6_2, c6_1


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

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, 0) -> c3(F(a, double(0), exp(x0)))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)))
   F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0)))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, 0) -> c3(F(a, double(0), s(0)))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)))
   F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0)))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
K tuples:
   F(b, x0, 0) -> c3(F(a, double(0), exp(x0)))
   F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1))
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c3_1, c4_2, c4_1, c_1, c6_2, c6_1


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

(79) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, 0) -> c3(F(a, double(0), exp(x0)))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)))
   F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0)))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, 0) -> c3(F(a, double(0), s(0)))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)))
   F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0)))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
K tuples:
   F(b, x0, 0) -> c3(F(a, double(0), exp(x0)))
   F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1))
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c3_1, c4_2, c4_1, c_1, c6_2, c6_1


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

(81) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(b, x0, 0) -> c3(F(a, double(0), exp(x0))) by 
   F(b, 0, 0) -> c3(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c3(F(a, double(0), double(exp(z0))))
   F(b, x0, 0) -> c3(F(a, 0, exp(x0)))

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)))
   F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0)))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, 0) -> c3(F(a, double(0), s(0)))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, x0, 0) -> c3(F(a, 0, exp(x0)))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)))
   F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0)))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
K tuples:
   F(b, x0, 0) -> c3(F(a, double(0), exp(x0)))
   F(b, 0, x1) -> c3(F(a, half(x1), s(0)), HALF(x1))
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c3_1, c4_2, c4_1, c_1, c6_2, c6_1


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

(83) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(b, x0, 0) -> c3(F(a, 0, exp(x0)))

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)))
   F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0)))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, 0) -> c3(F(a, double(0), s(0)))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0)))
   F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0)))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
K tuples:
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c3_1, c4_2, c4_1, c_1, c6_2, c6_1


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

(85) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(b, x0, s(0)) -> c3(F(a, half(0), exp(x0))) by 
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, half(0), double(exp(z0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0)))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, 0) -> c3(F(a, double(0), s(0)))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0)))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
K tuples:
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c4_1, c_1, c6_2, c6_1, c3_1


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

(87) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
We considered the (Usable) Rules:
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   half(s(s(z0))) -> s(half(z0))
   exp(s(z0)) -> double(exp(z0))
   half(s(0)) -> half(0)
   exp(0) -> s(0)
   half(0) -> double(0)
And the Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0)))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, 0) -> c3(F(a, double(0), s(0)))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [2]
   POL(DOUBLE(x_1)) = 0
   POL(EXP(x_1)) = 0
   POL(F(x_1, x_2, x_3)) = [2]x_3 + x_1*x_2
   POL(HALF(x_1)) = 0
   POL(a) = 0
   POL(b) = [2]
   POL(c(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1)) = x_1
   POL(double(x_1)) = x_1
   POL(exp(x_1)) = 0
   POL(half(x_1)) = [2]
   POL(s(x_1)) = 0

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0)))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, 0) -> c3(F(a, double(0), s(0)))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0)))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
K tuples:
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c4_1, c_1, c6_2, c6_1, c3_1


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

(89) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(b, s(z0), x1) -> c4(F(a, half(x1), double(exp(z0))), EXP(s(z0))) by 
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), 0) -> c4(F(a, double(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, 0) -> c3(F(a, double(0), s(0)))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), 0) -> c4(F(a, double(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), 0) -> c4(F(a, double(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
K tuples:
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c4_1, c_1, c6_2, c6_1, c3_1


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

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

(92)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, 0) -> c3(F(a, double(0), s(0)))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, s(x0), 0) -> c1(EXP(s(x0)))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, s(x0), 0) -> c1(EXP(s(x0)))
K tuples:
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c4_1, c_1, c6_2, c6_1, c3_1, c1_1


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

(93) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   F(b, s(x0), 0) -> c1(EXP(s(x0)))

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

(94)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, 0) -> c3(F(a, double(0), s(0)))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
K tuples:
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c4_1, c_1, c6_2, c6_1, c3_1, c1_1


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

(95) CdtKnowledgeProof (BOTH BOUNDS(ID, ID))
The following tuples could be moved from S to K by knowledge propagation:
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))

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

(96)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, 0) -> c3(F(a, double(0), s(0)))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
K tuples:
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c4_1, c_1, c6_2, c6_1, c3_1, c1_1


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

(97) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(b, x0, s(0)) -> c4(F(a, half(0), exp(x0)), EXP(x0)) by 
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)), EXP(0))
   F(b, s(z0), s(0)) -> c4(F(a, half(0), double(exp(z0))), EXP(s(z0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))

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

(98)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, 0) -> c3(F(a, double(0), s(0)))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)), EXP(0))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)), EXP(0))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
K tuples:
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c4_1, c_1, c6_2, c6_1, c3_1, c1_1


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

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

(100)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, 0) -> c3(F(a, double(0), s(0)))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
K tuples:
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c4_1, c_1, c6_2, c6_1, c3_1, c1_1


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

(101) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
We considered the (Usable) Rules:
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   half(s(s(z0))) -> s(half(z0))
   exp(s(z0)) -> double(exp(z0))
   half(s(0)) -> half(0)
   exp(0) -> s(0)
   half(0) -> double(0)
And the Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, 0) -> c3(F(a, double(0), s(0)))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [2]
   POL(DOUBLE(x_1)) = 0
   POL(EXP(x_1)) = 0
   POL(F(x_1, x_2, x_3)) = [2]x_3 + x_1*x_2
   POL(HALF(x_1)) = 0
   POL(a) = 0
   POL(b) = [1]
   POL(c(x_1)) = x_1
   POL(c1(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1)) = x_1
   POL(double(x_1)) = x_1
   POL(exp(x_1)) = 0
   POL(half(x_1)) = [2]
   POL(s(x_1)) = 0

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

(102)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, 0) -> c3(F(a, double(0), s(0)))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
K tuples:
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c4_1, c_1, c6_2, c6_1, c3_1, c1_1


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

(103) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(b, 0, x1) -> c4(F(a, half(x1), s(0))) by 
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))

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

(104)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, 0) -> c3(F(a, double(0), s(0)))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
K tuples:
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c_1, c6_2, c6_1, c3_1, c1_1, c4_1


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

(105) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(b, x0, 0) -> c(F(a, double(0), exp(x0))) by 
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   F(b, x0, 0) -> c(F(a, 0, exp(x0)))

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

(106)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, 0) -> c3(F(a, double(0), s(0)))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   F(b, x0, 0) -> c(F(a, 0, exp(x0)))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
K tuples:
   F(b, x0, 0) -> c(F(a, double(0), exp(x0)))
   F(b, 0, x1) -> c4(F(a, half(x1), s(0)))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c6_2, c6_1, c3_1, c1_1, c4_1, c_1


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

(107) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(b, x0, 0) -> c(F(a, 0, exp(x0)))

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

(108)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, 0) -> c3(F(a, double(0), s(0)))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
K tuples:
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c6_2, c6_1, c3_1, c1_1, c4_1, c_1


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

(109) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace EXP(s(s(z0))) -> c6(DOUBLE(double(exp(z0))), EXP(s(z0))) by 
   EXP(s(s(0))) -> c6(DOUBLE(double(s(0))), EXP(s(0)))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))

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

(110)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, 0) -> c3(F(a, double(0), s(0)))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(0))) -> c6(DOUBLE(double(s(0))), EXP(s(0)))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   EXP(s(s(0))) -> c6(DOUBLE(double(s(0))), EXP(s(0)))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
K tuples:
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2


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

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

(112)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, 0) -> c3(F(a, double(0), s(0)))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
K tuples:
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2, c5_1


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

(113) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(b, 0, 0) -> c3(F(a, double(0), s(0))) by F(b, 0, 0) -> c3(F(a, 0, s(0)))
----------------------------------------

(114)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, 0) -> c3(F(a, double(0), s(0)))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, 0, 0) -> c3(F(a, 0, s(0)))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
K tuples:
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2, c5_1


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

(115) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(b, 0, 0) -> c3(F(a, 0, s(0)))

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

(116)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, 0, 0) -> c3(F(a, double(0), s(0)))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
K tuples:
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2, c5_1


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

(117) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) by F(b, 0, s(0)) -> c3(F(a, double(0), s(0)))
----------------------------------------

(118)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, 0, 0) -> c3(F(a, double(0), s(0)))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, 0, s(0)) -> c3(F(a, double(0), s(0)))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
K tuples:
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2, c5_1


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

(119) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(b, 0, s(0)) -> c3(F(a, double(0), s(0))) by 
   F(b, 0, s(0)) -> c3(F(a, 0, s(0)))

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

(120)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, 0, 0) -> c3(F(a, double(0), s(0)))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, 0, s(0)) -> c3(F(a, 0, s(0)))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
K tuples:
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2, c5_1


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

(121) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(b, 0, s(0)) -> c3(F(a, 0, s(0)))

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

(122)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, 0, 0) -> c3(F(a, double(0), s(0)))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
K tuples:
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2, c5_1


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

(123) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(b, s(0), x1) -> c3(F(a, half(x1), double(s(0))), HALF(x1)) by F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0))
----------------------------------------

(124)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, 0, 0) -> c3(F(a, double(0), s(0)))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0))
K tuples:
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2, c5_1


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

(125) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0)))) by F(b, s(z0), 0) -> c3(F(a, 0, double(exp(z0))))
----------------------------------------

(126)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, 0, 0) -> c3(F(a, double(0), s(0)))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0))
   F(b, s(z0), 0) -> c3(F(a, 0, double(exp(z0))))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0))
K tuples:
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2, c5_1


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

(127) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(b, s(z0), 0) -> c3(F(a, 0, double(exp(z0))))

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

(128)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, 0, 0) -> c3(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c3(F(a, double(0), double(exp(z0))))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0))
K tuples:
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2, c5_1


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

(129) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0)))) by F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
----------------------------------------

(130)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, 0, 0) -> c3(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c3(F(a, double(0), double(exp(z0))))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
K tuples:
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2, c5_1


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

(131) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(b, 0, 0) -> c3(F(a, double(0), s(0))) by F(b, 0, 0) -> c3(F(a, 0, s(0)))
----------------------------------------

(132)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(0)) -> c3(F(a, half(0), double(exp(x0))))
   F(b, s(z0), 0) -> c3(F(a, double(0), double(exp(z0))))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, 0, 0) -> c3(F(a, 0, s(0)))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
K tuples:
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2, c5_1


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

(133) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(b, 0, 0) -> c3(F(a, 0, s(0)))

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

(134)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(z0), 0) -> c3(F(a, double(0), double(exp(z0))))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, half(0), double(exp(z0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
K tuples:
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2, c5_1


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

(135) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(b, s(z0), 0) -> c3(F(a, double(0), double(exp(z0)))) by F(b, s(z0), 0) -> c3(F(a, 0, double(exp(z0))))
----------------------------------------

(136)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, half(0), double(exp(z0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(z0), 0) -> c3(F(a, 0, double(exp(z0))))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
K tuples:
   F(b, s(x0), 0) -> c3(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2, c5_1


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

(137) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(b, s(z0), 0) -> c3(F(a, 0, double(exp(z0))))

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

(138)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, half(0), double(exp(z0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
K tuples:
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2, c5_1


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

(139) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace F(b, s(s(z0)), x1) -> c3(F(a, half(x1), double(double(exp(z0)))), HALF(x1)) by 
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))

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

(140)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, half(0), double(exp(z0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
K tuples:
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2, c5_1


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

(141) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(b, 0, s(0)) -> c3(F(a, half(0), s(0))) by F(b, 0, s(0)) -> c3(F(a, double(0), s(0)))
----------------------------------------

(142)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(z0), s(0)) -> c3(F(a, half(0), double(exp(z0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, 0, s(0)) -> c3(F(a, double(0), s(0)))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
K tuples:
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2, c5_1


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

(143) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(b, 0, s(0)) -> c3(F(a, double(0), s(0))) by 
   F(b, 0, s(0)) -> c3(F(a, 0, s(0)))

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

(144)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(z0), s(0)) -> c3(F(a, half(0), double(exp(z0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, 0, s(0)) -> c3(F(a, 0, s(0)))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
K tuples:
   F(b, 0, s(0)) -> c3(F(a, half(0), s(0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2, c5_1


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

(145) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(b, 0, s(0)) -> c3(F(a, 0, s(0)))

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

(146)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(z0), s(0)) -> c3(F(a, half(0), double(exp(z0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
K tuples:
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2, c5_1


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

(147) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(b, s(z0), s(0)) -> c3(F(a, half(0), double(exp(z0)))) by F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
----------------------------------------

(148)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0)))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
K tuples:
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c3_1, c1_1, c4_1, c_1, c6_2, c5_1


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

(149) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(b, x0, s(0)) -> c3(F(a, double(0), exp(x0))) by F(b, z0, s(0)) -> c3(F(a, 0, exp(z0)))
----------------------------------------

(150)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, z0, s(0)) -> c3(F(a, 0, exp(z0)))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, z0, s(0)) -> c3(F(a, 0, exp(z0)))
K tuples:
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c1_1, c4_1, c_1, c6_2, c5_1, c3_1


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

(151) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(b, z0, s(0)) -> c3(F(a, 0, exp(z0)))

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

(152)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
K tuples:
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c1_1, c4_1, c_1, c6_2, c5_1, c3_1


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

(153) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace F(b, s(0), x1) -> c4(F(a, half(x1), double(s(0))), EXP(s(0))) by 
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))

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

(154)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
K tuples:
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c1_1, c4_1, c_1, c6_2, c5_1, c3_1


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

(155) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace F(b, s(s(z0)), x1) -> c4(F(a, half(x1), double(double(exp(z0)))), EXP(s(s(z0)))) by 
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))

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

(156)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
K tuples:
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c1_1, c4_1, c_1, c6_2, c5_1, c3_1


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

(157) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace F(b, s(0), z0) -> c3(F(a, half(z0), s(s(double(0)))), HALF(z0)) by 
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))

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

(158)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
K tuples:
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c1_1, c4_1, c_1, c6_2, c5_1, c3_1


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

(159) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0))) by F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
----------------------------------------

(160)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
K tuples:
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c1_1, c4_1, c_1, c6_2, c5_1, c3_1


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

(161) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0)))) by F(b, s(z0), 0) -> c1(F(a, 0, double(exp(z0))))
----------------------------------------

(162)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(0)) -> c4(F(a, half(0), double(exp(x0))), EXP(s(x0)))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(b, s(z0), 0) -> c1(F(a, 0, double(exp(z0))))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
K tuples:
   F(b, s(x0), 0) -> c1(F(a, double(0), double(exp(x0))))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c4_1, c_1, c6_2, c5_1, c3_1, c1_1


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

(163) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(b, s(z0), 0) -> c1(F(a, 0, double(exp(z0))))

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

(164)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, half(0), double(exp(z0))), EXP(s(z0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
K tuples:
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c4_1, c_1, c6_2, c5_1, c3_1


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

(165) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(b, s(z0), s(0)) -> c4(F(a, half(0), double(exp(z0))), EXP(s(z0))) by F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
----------------------------------------

(166)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
S tuples:
   F(a, s(z0), z1) -> c2(F(b, z1, s(z0)))
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
K tuples:
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   F_3, DOUBLE_1, HALF_1, EXP_1

Compound Symbols:   c2_1, c8_1, c11_1, c3_2, c4_2, c6_1, c4_1, c_1, c6_2, c5_1, c3_1


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

(167) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace F(a, s(z0), z1) -> c2(F(b, z1, s(z0))) by 
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))

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

(168)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
S tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
K tuples:
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   DOUBLE_1, HALF_1, F_3, EXP_1

Compound Symbols:   c8_1, c11_1, c3_2, c4_2, c6_1, c4_1, c_1, c6_2, c5_1, c3_1, c2_1


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

(169) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
We considered the (Usable) Rules:
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(s(z0)) -> double(exp(z0))
   exp(0) -> s(0)
And the Tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(DOUBLE(x_1)) = 0
   POL(EXP(x_1)) = 0
   POL(F(x_1, x_2, x_3)) = x_1 + x_3
   POL(HALF(x_1)) = 0
   POL(a) = 0
   POL(b) = 0
   POL(c(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1)) = x_1
   POL(double(x_1)) = x_1
   POL(exp(x_1)) = 0
   POL(half(x_1)) = [1]
   POL(s(x_1)) = 0

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

(170)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
S tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0))))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
K tuples:
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   DOUBLE_1, HALF_1, F_3, EXP_1

Compound Symbols:   c8_1, c11_1, c3_2, c4_2, c6_1, c4_1, c_1, c6_2, c5_1, c3_1, c2_1


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

(171) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(b, x0, s(s(z0))) -> c3(F(a, s(half(z0)), exp(x0)), HALF(s(s(z0)))) by 
   F(b, 0, s(s(x1))) -> c3(F(a, s(half(x1)), s(0)), HALF(s(s(x1))))
   F(b, s(z0), s(s(x1))) -> c3(F(a, s(half(x1)), double(exp(z0))), HALF(s(s(x1))))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))

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

(172)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
S tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, 0, s(s(x1))) -> c3(F(a, s(half(x1)), s(0)), HALF(s(s(x1))))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
K tuples:
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   DOUBLE_1, HALF_1, F_3, EXP_1

Compound Symbols:   c8_1, c11_1, c4_2, c6_1, c3_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1


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

(173) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
We considered the (Usable) Rules:none
And the Tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = 0
   POL(DOUBLE(x_1)) = 0
   POL(EXP(x_1)) = 0
   POL(F(x_1, x_2, x_3)) = [1] + x_1
   POL(HALF(x_1)) = 0
   POL(a) = 0
   POL(b) = 0
   POL(c(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1)) = x_1
   POL(double(x_1)) = 0
   POL(exp(x_1)) = 0
   POL(half(x_1)) = [1]
   POL(s(x_1)) = 0

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

(174)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
S tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, 0, s(s(x1))) -> c3(F(a, s(half(x1)), s(0)), HALF(s(s(x1))))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
K tuples:
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   DOUBLE_1, HALF_1, F_3, EXP_1

Compound Symbols:   c8_1, c11_1, c4_2, c6_1, c3_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1


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

(175) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
We considered the (Usable) Rules:
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   half(s(s(z0))) -> s(half(z0))
   exp(s(z0)) -> double(exp(z0))
   half(s(0)) -> half(0)
   exp(0) -> s(0)
   half(0) -> double(0)
And the Tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(DOUBLE(x_1)) = 0
   POL(EXP(x_1)) = 0
   POL(F(x_1, x_2, x_3)) = [2]x_3 + [2]x_1*x_3 + x_1*x_2
   POL(HALF(x_1)) = 0
   POL(a) = 0
   POL(b) = [1]
   POL(c(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1)) = x_1
   POL(double(x_1)) = x_1^2
   POL(exp(x_1)) = 0
   POL(half(x_1)) = [1]
   POL(s(x_1)) = 0

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

(176)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
S tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
K tuples:
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   DOUBLE_1, HALF_1, F_3, EXP_1

Compound Symbols:   c8_1, c11_1, c4_2, c6_1, c3_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1


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

(177) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(b, x0, s(s(z0))) -> c4(F(a, s(half(z0)), exp(x0)), EXP(x0)) by 
   F(b, 0, s(s(x1))) -> c4(F(a, s(half(x1)), s(0)), EXP(0))
   F(b, s(z0), s(s(x1))) -> c4(F(a, s(half(x1)), double(exp(z0))), EXP(s(z0)))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))

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

(178)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, 0, s(s(x1))) -> c4(F(a, s(half(x1)), s(0)), EXP(0))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
S tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, 0, s(s(x1))) -> c4(F(a, s(half(x1)), s(0)), EXP(0))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
K tuples:
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   DOUBLE_1, HALF_1, EXP_1, F_3

Compound Symbols:   c8_1, c11_1, c6_1, c3_2, c4_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1


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

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

(180)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
S tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
   F(b, 0, s(s(x1))) -> c4(F(a, s(half(x1)), s(0)))
K tuples:
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   DOUBLE_1, HALF_1, EXP_1, F_3

Compound Symbols:   c8_1, c11_1, c6_1, c3_2, c4_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1


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

(181) CdtRuleRemovalProof (UPPER BOUND(ADD(n^2)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
We considered the (Usable) Rules:
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   half(s(s(z0))) -> s(half(z0))
   exp(s(z0)) -> double(exp(z0))
   half(s(0)) -> half(0)
   exp(0) -> s(0)
   half(0) -> double(0)
And the Tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(DOUBLE(x_1)) = 0
   POL(EXP(x_1)) = 0
   POL(F(x_1, x_2, x_3)) = [2] + [2]x_3 + x_1*x_2
   POL(HALF(x_1)) = 0
   POL(a) = 0
   POL(b) = [1]
   POL(c(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1)) = x_1
   POL(double(x_1)) = [2]x_1
   POL(exp(x_1)) = 0
   POL(half(x_1)) = [2]
   POL(s(x_1)) = 0

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

(182)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
S tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
K tuples:
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   DOUBLE_1, HALF_1, EXP_1, F_3

Compound Symbols:   c8_1, c11_1, c6_1, c3_2, c4_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1


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

(183) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(b, x0, s(0)) -> c4(F(a, double(0), exp(x0)), EXP(x0)) by F(b, z0, s(0)) -> c4(F(a, 0, exp(z0)), EXP(z0))
----------------------------------------

(184)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
   F(b, z0, s(0)) -> c4(F(a, 0, exp(z0)), EXP(z0))
S tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
   F(b, z0, s(0)) -> c4(F(a, 0, exp(z0)), EXP(z0))
K tuples:
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   DOUBLE_1, HALF_1, EXP_1, F_3

Compound Symbols:   c8_1, c11_1, c6_1, c3_2, c4_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1


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

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

(186)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
   F(b, z0, s(0)) -> c4(EXP(z0))
S tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
   F(b, z0, s(0)) -> c4(EXP(z0))
K tuples:
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   DOUBLE_1, HALF_1, EXP_1, F_3

Compound Symbols:   c8_1, c11_1, c6_1, c3_2, c4_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1


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

(187) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   F(b, z0, s(0)) -> c4(EXP(z0))
We considered the (Usable) Rules:none
And the Tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
   F(b, z0, s(0)) -> c4(EXP(z0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [2]
   POL(DOUBLE(x_1)) = 0
   POL(EXP(x_1)) = 0
   POL(F(x_1, x_2, x_3)) = [1]
   POL(HALF(x_1)) = 0
   POL(a) = [3]
   POL(b) = [3]
   POL(c(x_1)) = x_1
   POL(c11(x_1)) = x_1
   POL(c2(x_1)) = x_1
   POL(c3(x_1)) = x_1
   POL(c3(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c5(x_1)) = x_1
   POL(c6(x_1)) = x_1
   POL(c6(x_1, x_2)) = x_1 + x_2
   POL(c8(x_1)) = x_1
   POL(double(x_1)) = [2]
   POL(exp(x_1)) = 0
   POL(half(x_1)) = [2]x_1
   POL(s(x_1)) = 0

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

(188)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
   F(b, z0, s(0)) -> c4(EXP(z0))
S tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
K tuples:
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, z0, s(0)) -> c4(EXP(z0))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   DOUBLE_1, HALF_1, EXP_1, F_3

Compound Symbols:   c8_1, c11_1, c6_1, c3_2, c4_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1


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

(189) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) by F(b, 0, s(0)) -> c4(F(a, double(0), s(0)))
----------------------------------------

(190)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
   F(b, z0, s(0)) -> c4(EXP(z0))
   F(b, 0, s(0)) -> c4(F(a, double(0), s(0)))
S tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
K tuples:
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, z0, s(0)) -> c4(EXP(z0))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   DOUBLE_1, HALF_1, EXP_1, F_3

Compound Symbols:   c8_1, c11_1, c6_1, c3_2, c4_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1


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

(191) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(b, 0, s(0)) -> c4(F(a, double(0), s(0))) by 
   F(b, 0, s(0)) -> c4(F(a, 0, s(0)))

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

(192)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
   F(b, z0, s(0)) -> c4(EXP(z0))
   F(b, 0, s(0)) -> c4(F(a, 0, s(0)))
S tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
K tuples:
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, z0, s(0)) -> c4(EXP(z0))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   DOUBLE_1, HALF_1, EXP_1, F_3

Compound Symbols:   c8_1, c11_1, c6_1, c3_2, c4_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1


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

(193) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(b, 0, s(0)) -> c4(F(a, 0, s(0)))

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

(194)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, 0, 0) -> c4(F(a, double(0), s(0)))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
   F(b, z0, s(0)) -> c4(EXP(z0))
S tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
K tuples:
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, z0, s(0)) -> c4(EXP(z0))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   DOUBLE_1, HALF_1, EXP_1, F_3

Compound Symbols:   c8_1, c11_1, c6_1, c3_2, c4_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1


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

(195) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(b, 0, 0) -> c4(F(a, double(0), s(0))) by F(b, 0, 0) -> c4(F(a, 0, s(0)))
----------------------------------------

(196)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
   F(b, z0, s(0)) -> c4(EXP(z0))
   F(b, 0, 0) -> c4(F(a, 0, s(0)))
S tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
K tuples:
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, z0, s(0)) -> c4(EXP(z0))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   DOUBLE_1, HALF_1, EXP_1, F_3

Compound Symbols:   c8_1, c11_1, c6_1, c3_2, c4_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1


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

(197) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(b, 0, 0) -> c4(F(a, 0, s(0)))

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

(198)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
   F(b, z0, s(0)) -> c4(EXP(z0))
S tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
K tuples:
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, z0, s(0)) -> c4(EXP(z0))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   DOUBLE_1, HALF_1, EXP_1, F_3

Compound Symbols:   c8_1, c11_1, c6_1, c3_2, c4_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1


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

(199) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(b, 0, s(0)) -> c4(F(a, half(0), s(0))) by F(b, 0, s(0)) -> c4(F(a, double(0), s(0)))
----------------------------------------

(200)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
   F(b, z0, s(0)) -> c4(EXP(z0))
   F(b, 0, s(0)) -> c4(F(a, double(0), s(0)))
S tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
K tuples:
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, z0, s(0)) -> c4(EXP(z0))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   DOUBLE_1, HALF_1, EXP_1, F_3

Compound Symbols:   c8_1, c11_1, c6_1, c3_2, c4_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1


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

(201) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(b, 0, s(0)) -> c4(F(a, double(0), s(0))) by 
   F(b, 0, s(0)) -> c4(F(a, 0, s(0)))

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

(202)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
   F(b, z0, s(0)) -> c4(EXP(z0))
   F(b, 0, s(0)) -> c4(F(a, 0, s(0)))
S tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
K tuples:
   F(b, 0, s(0)) -> c4(F(a, half(0), s(0)))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, z0, s(0)) -> c4(EXP(z0))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   DOUBLE_1, HALF_1, EXP_1, F_3

Compound Symbols:   c8_1, c11_1, c6_1, c3_2, c4_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1


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

(203) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(b, 0, s(0)) -> c4(F(a, 0, s(0)))

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

(204)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, 0, 0) -> c(F(a, double(0), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
   F(b, z0, s(0)) -> c4(EXP(z0))
S tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
K tuples:
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, z0, s(0)) -> c4(EXP(z0))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   DOUBLE_1, HALF_1, EXP_1, F_3

Compound Symbols:   c8_1, c11_1, c6_1, c3_2, c4_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1


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

(205) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(b, 0, 0) -> c(F(a, double(0), s(0))) by F(b, 0, 0) -> c(F(a, 0, s(0)))
----------------------------------------

(206)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
   F(b, z0, s(0)) -> c4(EXP(z0))
   F(b, 0, 0) -> c(F(a, 0, s(0)))
S tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
K tuples:
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, z0, s(0)) -> c4(EXP(z0))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   DOUBLE_1, HALF_1, EXP_1, F_3

Compound Symbols:   c8_1, c11_1, c6_1, c3_2, c4_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1


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

(207) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(b, 0, 0) -> c(F(a, 0, s(0)))

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

(208)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
   F(b, z0, s(0)) -> c4(EXP(z0))
S tuples:
   DOUBLE(s(z0)) -> c8(DOUBLE(z0))
   HALF(s(s(z0))) -> c11(HALF(z0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
K tuples:
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, z0, s(0)) -> c4(EXP(z0))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   DOUBLE_1, HALF_1, EXP_1, F_3

Compound Symbols:   c8_1, c11_1, c6_1, c3_2, c4_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1


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

(209) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace DOUBLE(s(z0)) -> c8(DOUBLE(z0)) by 
   DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0)))

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

(210)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   HALF(s(s(z0))) -> c11(HALF(z0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
   F(b, z0, s(0)) -> c4(EXP(z0))
   DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0)))
S tuples:
   HALF(s(s(z0))) -> c11(HALF(z0))
   EXP(s(0)) -> c6(DOUBLE(s(0)))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   EXP(s(s(0))) -> c5(EXP(s(0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
   DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0)))
K tuples:
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, z0, s(0)) -> c4(EXP(z0))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   HALF_1, EXP_1, F_3, DOUBLE_1

Compound Symbols:   c11_1, c6_1, c3_2, c4_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1, c8_1


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

(211) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   EXP(s(s(0))) -> c5(EXP(s(0)))
   EXP(s(0)) -> c6(DOUBLE(s(0)))

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

(212)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
   F(b, z0, s(0)) -> c4(EXP(z0))
   DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0)))
S tuples:
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))), EXP(s(0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
   DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0)))
K tuples:
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, z0, s(0)) -> c4(EXP(z0))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   HALF_1, F_3, EXP_1, DOUBLE_1

Compound Symbols:   c11_1, c3_2, c4_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1, c8_1


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

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

(214)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0))))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
   F(b, z0, s(0)) -> c4(EXP(z0))
   DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))))
S tuples:
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
   DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))))
K tuples:
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, z0, s(0)) -> c4(EXP(z0))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   HALF_1, F_3, EXP_1, DOUBLE_1

Compound Symbols:   c11_1, c3_2, c4_2, c4_1, c_1, c6_2, c5_1, c3_1, c2_1, c8_1


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

(215) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(b, s(z0), 0) -> c(F(a, double(0), double(exp(z0)))) by F(b, s(z0), 0) -> c(F(a, 0, double(exp(z0))))
----------------------------------------

(216)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
   F(b, z0, s(0)) -> c4(EXP(z0))
   DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))))
   F(b, s(z0), 0) -> c(F(a, 0, double(exp(z0))))
S tuples:
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
   DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))))
K tuples:
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, z0, s(0)) -> c4(EXP(z0))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   HALF_1, F_3, EXP_1, DOUBLE_1

Compound Symbols:   c11_1, c3_2, c4_2, c4_1, c6_2, c5_1, c3_1, c2_1, c8_1, c_1


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

(217) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(b, s(z0), 0) -> c(F(a, 0, double(exp(z0))))

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

(218)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
   F(b, z0, s(0)) -> c4(EXP(z0))
   DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))))
S tuples:
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   EXP(s(s(0))) -> c5(DOUBLE(double(s(0))))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
   DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))))
K tuples:
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, z0, s(0)) -> c4(EXP(z0))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   HALF_1, F_3, EXP_1, DOUBLE_1

Compound Symbols:   c11_1, c3_2, c4_2, c4_1, c6_2, c5_1, c3_1, c2_1, c8_1


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

(219) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace EXP(s(s(0))) -> c5(DOUBLE(double(s(0)))) by EXP(s(s(0))) -> c5(DOUBLE(s(s(double(0)))))
----------------------------------------

(220)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
   F(b, z0, s(0)) -> c4(EXP(z0))
   DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))))
   EXP(s(s(0))) -> c5(DOUBLE(s(s(double(0)))))
S tuples:
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
   DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))))
   EXP(s(s(0))) -> c5(DOUBLE(s(s(double(0)))))
K tuples:
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, z0, s(0)) -> c4(EXP(z0))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   HALF_1, F_3, EXP_1, DOUBLE_1

Compound Symbols:   c11_1, c3_2, c4_2, c4_1, c6_2, c3_1, c2_1, c8_1, c5_1


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

(221) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) by F(b, s(z0), s(0)) -> c3(F(a, 0, double(exp(z0))))
----------------------------------------

(222)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
   F(b, z0, s(0)) -> c4(EXP(z0))
   DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))))
   EXP(s(s(0))) -> c5(DOUBLE(s(s(double(0)))))
   F(b, s(z0), s(0)) -> c3(F(a, 0, double(exp(z0))))
S tuples:
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
   DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))))
   EXP(s(s(0))) -> c5(DOUBLE(s(s(double(0)))))
   F(b, s(z0), s(0)) -> c3(F(a, 0, double(exp(z0))))
K tuples:
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, z0, s(0)) -> c4(EXP(z0))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   HALF_1, F_3, EXP_1, DOUBLE_1

Compound Symbols:   c11_1, c3_2, c4_2, c4_1, c6_2, c3_1, c2_1, c8_1, c5_1


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

(223) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(b, s(z0), s(0)) -> c3(F(a, 0, double(exp(z0))))

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

(224)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0))))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
   F(b, z0, s(0)) -> c4(EXP(z0))
   DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))))
   EXP(s(s(0))) -> c5(DOUBLE(s(s(double(0)))))
S tuples:
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
   DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))))
   EXP(s(s(0))) -> c5(DOUBLE(s(s(double(0)))))
K tuples:
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, z0, s(0)) -> c4(EXP(z0))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   HALF_1, F_3, EXP_1, DOUBLE_1

Compound Symbols:   c11_1, c3_2, c4_2, c4_1, c6_2, c3_1, c2_1, c8_1, c5_1


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

(225) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(b, s(z0), s(0)) -> c3(F(a, double(0), double(exp(z0)))) by F(b, s(z0), s(0)) -> c3(F(a, 0, double(exp(z0))))
----------------------------------------

(226)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
   F(b, z0, s(0)) -> c4(EXP(z0))
   DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))))
   EXP(s(s(0))) -> c5(DOUBLE(s(s(double(0)))))
   F(b, s(z0), s(0)) -> c3(F(a, 0, double(exp(z0))))
S tuples:
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
   DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))))
   EXP(s(s(0))) -> c5(DOUBLE(s(s(double(0)))))
K tuples:
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, z0, s(0)) -> c4(EXP(z0))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   HALF_1, F_3, EXP_1, DOUBLE_1

Compound Symbols:   c11_1, c3_2, c4_2, c4_1, c6_2, c2_1, c3_1, c8_1, c5_1


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

(227) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   F(b, s(z0), s(0)) -> c3(F(a, 0, double(exp(z0))))

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

(228)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
   F(b, z0, s(0)) -> c4(EXP(z0))
   DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))))
   EXP(s(s(0))) -> c5(DOUBLE(s(s(double(0)))))
S tuples:
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0)))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
   DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))))
   EXP(s(s(0))) -> c5(DOUBLE(s(s(double(0)))))
K tuples:
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, z0, s(0)) -> c4(EXP(z0))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   HALF_1, F_3, EXP_1, DOUBLE_1

Compound Symbols:   c11_1, c3_2, c4_2, c4_1, c6_2, c2_1, c3_1, c8_1, c5_1


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

(229) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(b, s(0), s(x0)) -> c3(F(a, half(s(x0)), s(s(double(0)))), HALF(s(x0))) by F(b, s(0), s(z0)) -> c3(F(a, half(s(z0)), s(s(0))), HALF(s(z0)))
----------------------------------------

(230)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   half(0) -> double(0)
   half(s(0)) -> half(0)
   half(s(s(z0))) -> s(half(z0))
   double(0) -> 0
   double(s(z0)) -> s(s(double(z0)))
   exp(0) -> s(0)
   exp(s(z0)) -> double(exp(z0))
Tuples:
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
   F(b, z0, s(0)) -> c4(EXP(z0))
   DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))))
   EXP(s(s(0))) -> c5(DOUBLE(s(s(double(0)))))
   F(b, s(0), s(z0)) -> c3(F(a, half(s(z0)), s(s(0))), HALF(s(z0)))
S tuples:
   HALF(s(s(z0))) -> c11(HALF(z0))
   F(b, s(x0), s(s(z0))) -> c3(F(a, s(half(z0)), double(exp(x0))), HALF(s(s(z0))))
   F(b, s(x0), s(s(z0))) -> c4(F(a, s(half(z0)), double(exp(x0))), EXP(s(x0)))
   EXP(s(s(s(z0)))) -> c6(DOUBLE(double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(s(z0)), s(x0)) -> c3(F(a, half(s(x0)), double(double(exp(z0)))), HALF(s(x0)))
   F(b, s(s(z0)), s(x0)) -> c4(F(a, half(s(x0)), double(double(exp(z0)))), EXP(s(s(z0))))
   F(b, s(z0), s(0)) -> c4(F(a, double(0), double(exp(z0))), EXP(s(z0)))
   F(a, s(s(y1)), z1) -> c2(F(b, z1, s(s(y1))))
   F(a, s(s(y1)), s(y0)) -> c2(F(b, s(y0), s(s(y1))))
   F(a, s(0), z1) -> c2(F(b, z1, s(0)))
   F(a, s(0), s(y0)) -> c2(F(b, s(y0), s(0)))
   F(a, s(z0), s(s(y0))) -> c2(F(b, s(s(y0)), s(z0)))
   F(a, s(z0), s(0)) -> c2(F(b, s(0), s(z0)))
   F(b, x0, s(s(0))) -> c3(F(a, s(double(0)), exp(x0)), HALF(s(s(0))))
   F(b, x0, s(s(s(0)))) -> c3(F(a, s(half(0)), exp(x0)), HALF(s(s(s(0)))))
   F(b, x0, s(s(s(s(z0))))) -> c3(F(a, s(s(half(z0))), exp(x0)), HALF(s(s(s(s(z0))))))
   F(b, x0, s(s(0))) -> c4(F(a, s(double(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(0)))) -> c4(F(a, s(half(0)), exp(x0)), EXP(x0))
   F(b, x0, s(s(s(s(z0))))) -> c4(F(a, s(s(half(z0))), exp(x0)), EXP(x0))
   DOUBLE(s(s(y0))) -> c8(DOUBLE(s(y0)))
   F(b, s(0), s(x0)) -> c4(F(a, half(s(x0)), double(s(0))))
   EXP(s(s(0))) -> c5(DOUBLE(s(s(double(0)))))
   F(b, s(0), s(z0)) -> c3(F(a, half(s(z0)), s(s(0))), HALF(s(z0)))
K tuples:
   F(a, s(s(y0)), 0) -> c2(F(b, 0, s(s(y0))))
   F(a, s(0), 0) -> c2(F(b, 0, s(0)))
   F(b, x0, s(s(x1))) -> c3(HALF(s(s(x1))))
   F(b, 0, s(s(z0))) -> c3(F(a, s(half(z0)), s(0)), HALF(s(s(z0))))
   F(b, 0, s(s(z0))) -> c4(F(a, s(half(z0)), s(0)))
   F(b, z0, s(0)) -> c4(EXP(z0))
Defined Rule Symbols:   half_1, double_1, exp_1

Defined Pair Symbols:   HALF_1, F_3, EXP_1, DOUBLE_1

Compound Symbols:   c11_1, c3_2, c4_2, c4_1, c6_2, c2_1, c3_1, c8_1, c5_1