KILLED
proof of input_NdOCBnLBCd.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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), 10 ms]
        (14) CpxRNTS
        (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (16) CpxRNTS
    (17) CompletionProof [UPPER BOUND(ID), 0 ms]
    (18) CpxTypedWeightedCompleteTrs
        (19) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 2 ms]
        (20) CpxRNTS
(21) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(22) CdtProblem
    (23) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (24) CdtProblem
    (25) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (26) CdtProblem
    (27) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (28) CdtProblem
    (29) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (30) CdtProblem
    (31) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 1 ms]
    (32) CdtProblem
    (33) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (34) CdtProblem
    (35) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (36) CdtProblem
    (37) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (38) CdtProblem
    (39) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (40) CdtProblem
    (41) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (42) CdtProblem
    (43) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (44) CdtProblem
    (45) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (46) CdtProblem
    (47) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (48) CdtProblem
    (49) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) CdtProblem
    (51) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CdtProblem
    (53) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) CdtProblem
    (55) CdtGraphSplitRhsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CdtProblem
    (57) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CdtProblem
    (59) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
    (61) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CdtProblem
    (63) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CdtProblem
    (65) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem


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

(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:

   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   f(0) -> s(0)
   f(s(x)) -> minus(s(x), g(f(x)))
   g(0) -> 0
   g(s(x)) -> minus(s(x), f(g(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:

   minus(x, 0') -> x
   minus(s(x), s(y)) -> minus(x, y)
   f(0') -> s(0')
   f(s(x)) -> minus(s(x), g(f(x)))
   g(0') -> 0'
   g(s(x)) -> minus(s(x), f(g(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:

   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   f(0) -> s(0)
   f(s(x)) -> minus(s(x), g(f(x)))
   g(0) -> 0
   g(s(x)) -> minus(s(x), f(g(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:

   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   f(0) -> s(0) [1]
   f(s(x)) -> minus(s(x), g(f(x))) [1]
   g(0) -> 0 [1]
   g(s(x)) -> minus(s(x), f(g(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:

   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   f(0) -> s(0) [1]
   f(s(x)) -> minus(s(x), g(f(x))) [1]
   g(0) -> 0 [1]
   g(s(x)) -> minus(s(x), f(g(x))) [1]

The TRS has the following type information:
minus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
f :: 0:s -> 0:s
g :: 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:
none

(c) The following functions are completely defined:

   f_1
   g_1
   minus_2

Due to the following rules being added:

   minus(v0, v1) -> 0 [0]

And the following fresh constants: none

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

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

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

   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   f(0) -> s(0) [1]
   f(s(x)) -> minus(s(x), g(f(x))) [1]
   g(0) -> 0 [1]
   g(s(x)) -> minus(s(x), f(g(x))) [1]
   minus(v0, v1) -> 0 [0]

The TRS has the following type information:
minus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
f :: 0:s -> 0:s
g :: 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:

   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   f(0) -> s(0) [1]
   f(s(0)) -> minus(s(0), g(s(0))) [2]
   f(s(s(x'))) -> minus(s(s(x')), g(minus(s(x'), g(f(x'))))) [2]
   g(0) -> 0 [1]
   g(s(0)) -> minus(s(0), f(0)) [2]
   g(s(s(x''))) -> minus(s(s(x'')), f(minus(s(x''), f(g(x''))))) [2]
   minus(v0, v1) -> 0 [0]

The TRS has the following type information:
minus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
f :: 0:s -> 0:s
g :: 0:s -> 0:s

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

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

   0 => 0

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

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

   f(z) -{ 2 }-> minus(1 + 0, g(1 + 0)) :|: z = 1 + 0
   f(z) -{ 2 }-> minus(1 + (1 + x'), g(minus(1 + x', g(f(x'))))) :|: x' >= 0, z = 1 + (1 + x')
   f(z) -{ 1 }-> 1 + 0 :|: z = 0
   g(z) -{ 2 }-> minus(1 + 0, f(0)) :|: z = 1 + 0
   g(z) -{ 2 }-> minus(1 + (1 + x''), f(minus(1 + x'', f(g(x''))))) :|: x'' >= 0, z = 1 + (1 + x'')
   g(z) -{ 1 }-> 0 :|: z = 0
   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1


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

(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:

   f(z) -{ 2 }-> minus(1 + 0, g(1 + 0)) :|: z = 1 + 0
   f(z) -{ 2 }-> minus(1 + (1 + (z - 2)), g(minus(1 + (z - 2), g(f(z - 2))))) :|: z - 2 >= 0
   f(z) -{ 1 }-> 1 + 0 :|: z = 0
   g(z) -{ 2 }-> minus(1 + 0, f(0)) :|: z = 1 + 0
   g(z) -{ 2 }-> minus(1 + (1 + (z - 2)), f(minus(1 + (z - 2), f(g(z - 2))))) :|: z - 2 >= 0
   g(z) -{ 1 }-> 0 :|: z = 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0


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

(17) CompletionProof (UPPER BOUND(ID))
The TRS is a completely defined constructor system, as every type has a constant constructor and the following rules were added:

   minus(v0, v1) -> null_minus [0]
   f(v0) -> null_f [0]
   g(v0) -> null_g [0]

And the following fresh constants:    null_minus, null_f, null_g

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

(18)
Obligation:
Runtime Complexity Weighted TRS where all functions are completely defined. The underlying TRS is:

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

   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   f(0) -> s(0) [1]
   f(s(x)) -> minus(s(x), g(f(x))) [1]
   g(0) -> 0 [1]
   g(s(x)) -> minus(s(x), f(g(x))) [1]
   minus(v0, v1) -> null_minus [0]
   f(v0) -> null_f [0]
   g(v0) -> null_g [0]

The TRS has the following type information:
minus :: 0:s:null_minus:null_f:null_g -> 0:s:null_minus:null_f:null_g -> 0:s:null_minus:null_f:null_g
0 :: 0:s:null_minus:null_f:null_g
s :: 0:s:null_minus:null_f:null_g -> 0:s:null_minus:null_f:null_g
f :: 0:s:null_minus:null_f:null_g -> 0:s:null_minus:null_f:null_g
g :: 0:s:null_minus:null_f:null_g -> 0:s:null_minus:null_f:null_g
null_minus :: 0:s:null_minus:null_f:null_g
null_f :: 0:s:null_minus:null_f:null_g
null_g :: 0:s:null_minus:null_f:null_g

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

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

   0 => 0
   null_minus => 0
   null_f => 0
   null_g => 0

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

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

   f(z) -{ 1 }-> minus(1 + x, g(f(x))) :|: x >= 0, z = 1 + x
   f(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   f(z) -{ 1 }-> 1 + 0 :|: z = 0
   g(z) -{ 1 }-> minus(1 + x, f(g(x))) :|: x >= 0, z = 1 + x
   g(z) -{ 1 }-> 0 :|: z = 0
   g(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0
   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1

Only complete derivations are relevant for the runtime complexity.

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

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

(22)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   f(0) -> s(0)
   f(s(z0)) -> minus(s(z0), g(f(z0)))
   g(0) -> 0
   g(s(z0)) -> minus(s(z0), f(g(z0)))
Tuples:
   MINUS(z0, 0) -> c
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   F(0) -> c2
   F(s(z0)) -> c3(MINUS(s(z0), g(f(z0))), G(f(z0)), F(z0))
   G(0) -> c4
   G(s(z0)) -> c5(MINUS(s(z0), f(g(z0))), F(g(z0)), G(z0))
S tuples:
   MINUS(z0, 0) -> c
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   F(0) -> c2
   F(s(z0)) -> c3(MINUS(s(z0), g(f(z0))), G(f(z0)), F(z0))
   G(0) -> c4
   G(s(z0)) -> c5(MINUS(s(z0), f(g(z0))), F(g(z0)), G(z0))
K tuples:none
Defined Rule Symbols:   minus_2, f_1, g_1

Defined Pair Symbols:   MINUS_2, F_1, G_1

Compound Symbols:   c, c1_1, c2, c3_3, c4, c5_3


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

(23) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 3 trailing nodes:
   F(0) -> c2
   G(0) -> c4
   MINUS(z0, 0) -> c

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

(24)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   f(0) -> s(0)
   f(s(z0)) -> minus(s(z0), g(f(z0)))
   g(0) -> 0
   g(s(z0)) -> minus(s(z0), f(g(z0)))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   F(s(z0)) -> c3(MINUS(s(z0), g(f(z0))), G(f(z0)), F(z0))
   G(s(z0)) -> c5(MINUS(s(z0), f(g(z0))), F(g(z0)), G(z0))
S tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   F(s(z0)) -> c3(MINUS(s(z0), g(f(z0))), G(f(z0)), F(z0))
   G(s(z0)) -> c5(MINUS(s(z0), f(g(z0))), F(g(z0)), G(z0))
K tuples:none
Defined Rule Symbols:   minus_2, f_1, g_1

Defined Pair Symbols:   MINUS_2, F_1, G_1

Compound Symbols:   c1_1, c3_3, c5_3


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

(25) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(z0)) -> c3(MINUS(s(z0), g(f(z0))), G(f(z0)), F(z0)) by 
   F(s(0)) -> c3(MINUS(s(0), g(s(0))), G(f(0)), F(0))
   F(s(s(z0))) -> c3(MINUS(s(s(z0)), g(minus(s(z0), g(f(z0))))), G(f(s(z0))), F(s(z0)))

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

(26)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   f(0) -> s(0)
   f(s(z0)) -> minus(s(z0), g(f(z0)))
   g(0) -> 0
   g(s(z0)) -> minus(s(z0), f(g(z0)))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   G(s(z0)) -> c5(MINUS(s(z0), f(g(z0))), F(g(z0)), G(z0))
   F(s(0)) -> c3(MINUS(s(0), g(s(0))), G(f(0)), F(0))
   F(s(s(z0))) -> c3(MINUS(s(s(z0)), g(minus(s(z0), g(f(z0))))), G(f(s(z0))), F(s(z0)))
S tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   G(s(z0)) -> c5(MINUS(s(z0), f(g(z0))), F(g(z0)), G(z0))
   F(s(0)) -> c3(MINUS(s(0), g(s(0))), G(f(0)), F(0))
   F(s(s(z0))) -> c3(MINUS(s(s(z0)), g(minus(s(z0), g(f(z0))))), G(f(s(z0))), F(s(z0)))
K tuples:none
Defined Rule Symbols:   minus_2, f_1, g_1

Defined Pair Symbols:   MINUS_2, G_1, F_1

Compound Symbols:   c1_1, c5_3, c3_3


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

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

(28)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   f(0) -> s(0)
   f(s(z0)) -> minus(s(z0), g(f(z0)))
   g(0) -> 0
   g(s(z0)) -> minus(s(z0), f(g(z0)))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   G(s(z0)) -> c5(MINUS(s(z0), f(g(z0))), F(g(z0)), G(z0))
   F(s(s(z0))) -> c3(MINUS(s(s(z0)), g(minus(s(z0), g(f(z0))))), G(f(s(z0))), F(s(z0)))
   F(s(0)) -> c3(MINUS(s(0), g(s(0))), G(f(0)))
S tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   G(s(z0)) -> c5(MINUS(s(z0), f(g(z0))), F(g(z0)), G(z0))
   F(s(s(z0))) -> c3(MINUS(s(s(z0)), g(minus(s(z0), g(f(z0))))), G(f(s(z0))), F(s(z0)))
   F(s(0)) -> c3(MINUS(s(0), g(s(0))), G(f(0)))
K tuples:none
Defined Rule Symbols:   minus_2, f_1, g_1

Defined Pair Symbols:   MINUS_2, G_1, F_1

Compound Symbols:   c1_1, c5_3, c3_3, c3_2


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

(29) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(s(z0)) -> c5(MINUS(s(z0), f(g(z0))), F(g(z0)), G(z0)) by 
   G(s(0)) -> c5(MINUS(s(0), f(0)), F(g(0)), G(0))
   G(s(s(z0))) -> c5(MINUS(s(s(z0)), f(minus(s(z0), f(g(z0))))), F(g(s(z0))), G(s(z0)))

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

(30)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   f(0) -> s(0)
   f(s(z0)) -> minus(s(z0), g(f(z0)))
   g(0) -> 0
   g(s(z0)) -> minus(s(z0), f(g(z0)))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   F(s(s(z0))) -> c3(MINUS(s(s(z0)), g(minus(s(z0), g(f(z0))))), G(f(s(z0))), F(s(z0)))
   F(s(0)) -> c3(MINUS(s(0), g(s(0))), G(f(0)))
   G(s(0)) -> c5(MINUS(s(0), f(0)), F(g(0)), G(0))
   G(s(s(z0))) -> c5(MINUS(s(s(z0)), f(minus(s(z0), f(g(z0))))), F(g(s(z0))), G(s(z0)))
S tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   F(s(s(z0))) -> c3(MINUS(s(s(z0)), g(minus(s(z0), g(f(z0))))), G(f(s(z0))), F(s(z0)))
   F(s(0)) -> c3(MINUS(s(0), g(s(0))), G(f(0)))
   G(s(0)) -> c5(MINUS(s(0), f(0)), F(g(0)), G(0))
   G(s(s(z0))) -> c5(MINUS(s(s(z0)), f(minus(s(z0), f(g(z0))))), F(g(s(z0))), G(s(z0)))
K tuples:none
Defined Rule Symbols:   minus_2, f_1, g_1

Defined Pair Symbols:   MINUS_2, F_1, G_1

Compound Symbols:   c1_1, c3_3, c3_2, c5_3


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

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

(32)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   f(0) -> s(0)
   f(s(z0)) -> minus(s(z0), g(f(z0)))
   g(0) -> 0
   g(s(z0)) -> minus(s(z0), f(g(z0)))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   F(s(s(z0))) -> c3(MINUS(s(s(z0)), g(minus(s(z0), g(f(z0))))), G(f(s(z0))), F(s(z0)))
   F(s(0)) -> c3(MINUS(s(0), g(s(0))), G(f(0)))
   G(s(s(z0))) -> c5(MINUS(s(s(z0)), f(minus(s(z0), f(g(z0))))), F(g(s(z0))), G(s(z0)))
   G(s(0)) -> c5(MINUS(s(0), f(0)), F(g(0)))
S tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   F(s(s(z0))) -> c3(MINUS(s(s(z0)), g(minus(s(z0), g(f(z0))))), G(f(s(z0))), F(s(z0)))
   F(s(0)) -> c3(MINUS(s(0), g(s(0))), G(f(0)))
   G(s(s(z0))) -> c5(MINUS(s(s(z0)), f(minus(s(z0), f(g(z0))))), F(g(s(z0))), G(s(z0)))
   G(s(0)) -> c5(MINUS(s(0), f(0)), F(g(0)))
K tuples:none
Defined Rule Symbols:   minus_2, f_1, g_1

Defined Pair Symbols:   MINUS_2, F_1, G_1

Compound Symbols:   c1_1, c3_3, c3_2, c5_3, c5_2


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

(33) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(s(z0))) -> c3(MINUS(s(s(z0)), g(minus(s(z0), g(f(z0))))), G(f(s(z0))), F(s(z0))) by 
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(f(s(0))), F(s(0)))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0))))
   F(s(s(x0))) -> c3(G(f(s(x0))))

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

(34)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   f(0) -> s(0)
   f(s(z0)) -> minus(s(z0), g(f(z0)))
   g(0) -> 0
   g(s(z0)) -> minus(s(z0), f(g(z0)))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   F(s(0)) -> c3(MINUS(s(0), g(s(0))), G(f(0)))
   G(s(s(z0))) -> c5(MINUS(s(s(z0)), f(minus(s(z0), f(g(z0))))), F(g(s(z0))), G(s(z0)))
   G(s(0)) -> c5(MINUS(s(0), f(0)), F(g(0)))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(f(s(0))), F(s(0)))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0))))
   F(s(s(x0))) -> c3(G(f(s(x0))))
S tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   F(s(0)) -> c3(MINUS(s(0), g(s(0))), G(f(0)))
   G(s(s(z0))) -> c5(MINUS(s(s(z0)), f(minus(s(z0), f(g(z0))))), F(g(s(z0))), G(s(z0)))
   G(s(0)) -> c5(MINUS(s(0), f(0)), F(g(0)))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(f(s(0))), F(s(0)))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0))))
   F(s(s(x0))) -> c3(G(f(s(x0))))
K tuples:none
Defined Rule Symbols:   minus_2, f_1, g_1

Defined Pair Symbols:   MINUS_2, F_1, G_1

Compound Symbols:   c1_1, c3_2, c5_3, c5_2, c3_3, c3_1


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

(35) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(0)) -> c3(MINUS(s(0), g(s(0))), G(f(0))) by 
   F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0)))
   F(s(0)) -> c3(G(f(0)))

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

(36)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   f(0) -> s(0)
   f(s(z0)) -> minus(s(z0), g(f(z0)))
   g(0) -> 0
   g(s(z0)) -> minus(s(z0), f(g(z0)))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   G(s(s(z0))) -> c5(MINUS(s(s(z0)), f(minus(s(z0), f(g(z0))))), F(g(s(z0))), G(s(z0)))
   G(s(0)) -> c5(MINUS(s(0), f(0)), F(g(0)))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(f(s(0))), F(s(0)))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0))))
   F(s(s(x0))) -> c3(G(f(s(x0))))
   F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0)))
   F(s(0)) -> c3(G(f(0)))
S tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   G(s(s(z0))) -> c5(MINUS(s(s(z0)), f(minus(s(z0), f(g(z0))))), F(g(s(z0))), G(s(z0)))
   G(s(0)) -> c5(MINUS(s(0), f(0)), F(g(0)))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(f(s(0))), F(s(0)))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0))))
   F(s(s(x0))) -> c3(G(f(s(x0))))
   F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0)))
   F(s(0)) -> c3(G(f(0)))
K tuples:none
Defined Rule Symbols:   minus_2, f_1, g_1

Defined Pair Symbols:   MINUS_2, G_1, F_1

Compound Symbols:   c1_1, c5_3, c5_2, c3_3, c3_1, c3_2


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

(37) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(s(s(z0))) -> c5(MINUS(s(s(z0)), f(minus(s(z0), f(g(z0))))), F(g(s(z0))), G(s(z0))) by 
   G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0)))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0))))
   G(s(s(x0))) -> c5(F(g(s(x0))))

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

(38)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   f(0) -> s(0)
   f(s(z0)) -> minus(s(z0), g(f(z0)))
   g(0) -> 0
   g(s(z0)) -> minus(s(z0), f(g(z0)))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   G(s(0)) -> c5(MINUS(s(0), f(0)), F(g(0)))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(f(s(0))), F(s(0)))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0))))
   F(s(s(x0))) -> c3(G(f(s(x0))))
   F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0)))
   F(s(0)) -> c3(G(f(0)))
   G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0)))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0))))
   G(s(s(x0))) -> c5(F(g(s(x0))))
S tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   G(s(0)) -> c5(MINUS(s(0), f(0)), F(g(0)))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(f(s(0))), F(s(0)))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0))))
   F(s(s(x0))) -> c3(G(f(s(x0))))
   F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0)))
   F(s(0)) -> c3(G(f(0)))
   G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0)))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0))))
   G(s(s(x0))) -> c5(F(g(s(x0))))
K tuples:none
Defined Rule Symbols:   minus_2, f_1, g_1

Defined Pair Symbols:   MINUS_2, G_1, F_1

Compound Symbols:   c1_1, c5_2, c3_3, c3_1, c3_2, c5_3, c5_1


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

(39) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(s(0)) -> c5(MINUS(s(0), f(0)), F(g(0))) by 
   G(s(0)) -> c5(MINUS(s(0), s(0)), F(g(0)))

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

(40)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   f(0) -> s(0)
   f(s(z0)) -> minus(s(z0), g(f(z0)))
   g(0) -> 0
   g(s(z0)) -> minus(s(z0), f(g(z0)))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(f(s(0))), F(s(0)))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0))))
   F(s(s(x0))) -> c3(G(f(s(x0))))
   F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0)))
   F(s(0)) -> c3(G(f(0)))
   G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0)))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0))))
   G(s(s(x0))) -> c5(F(g(s(x0))))
   G(s(0)) -> c5(MINUS(s(0), s(0)), F(g(0)))
S tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(f(s(0))), F(s(0)))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0))))
   F(s(s(x0))) -> c3(G(f(s(x0))))
   F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0)))
   F(s(0)) -> c3(G(f(0)))
   G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0)))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0))))
   G(s(s(x0))) -> c5(F(g(s(x0))))
   G(s(0)) -> c5(MINUS(s(0), s(0)), F(g(0)))
K tuples:none
Defined Rule Symbols:   minus_2, f_1, g_1

Defined Pair Symbols:   MINUS_2, F_1, G_1

Compound Symbols:   c1_1, c3_3, c3_1, c3_2, c5_3, c5_1, c5_2


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

(41) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace G(s(0)) -> c5(MINUS(s(0), s(0)), F(g(0))) by 
   G(s(0)) -> c5(MINUS(s(0), s(0)), F(0))

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

(42)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   f(0) -> s(0)
   f(s(z0)) -> minus(s(z0), g(f(z0)))
   g(0) -> 0
   g(s(z0)) -> minus(s(z0), f(g(z0)))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(f(s(0))), F(s(0)))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0))))
   F(s(s(x0))) -> c3(G(f(s(x0))))
   F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0)))
   F(s(0)) -> c3(G(f(0)))
   G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0)))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0))))
   G(s(s(x0))) -> c5(F(g(s(x0))))
   G(s(0)) -> c5(MINUS(s(0), s(0)), F(0))
S tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(f(s(0))), F(s(0)))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0))))
   F(s(s(x0))) -> c3(G(f(s(x0))))
   F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0)))
   F(s(0)) -> c3(G(f(0)))
   G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0)))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0))))
   G(s(s(x0))) -> c5(F(g(s(x0))))
   G(s(0)) -> c5(MINUS(s(0), s(0)), F(0))
K tuples:none
Defined Rule Symbols:   minus_2, f_1, g_1

Defined Pair Symbols:   MINUS_2, F_1, G_1

Compound Symbols:   c1_1, c3_3, c3_1, c3_2, c5_3, c5_1, c5_2


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

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   f(0) -> s(0)
   f(s(z0)) -> minus(s(z0), g(f(z0)))
   g(0) -> 0
   g(s(z0)) -> minus(s(z0), f(g(z0)))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(f(s(0))), F(s(0)))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0))))
   F(s(s(x0))) -> c3(G(f(s(x0))))
   F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0)))
   F(s(0)) -> c3(G(f(0)))
   G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0)))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0))))
   G(s(s(x0))) -> c5(F(g(s(x0))))
   G(s(0)) -> c5(MINUS(s(0), s(0)))
S tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(f(s(0))), F(s(0)))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0))))
   F(s(s(x0))) -> c3(G(f(s(x0))))
   F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0)))
   F(s(0)) -> c3(G(f(0)))
   G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0)))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0))))
   G(s(s(x0))) -> c5(F(g(s(x0))))
   G(s(0)) -> c5(MINUS(s(0), s(0)))
K tuples:none
Defined Rule Symbols:   minus_2, f_1, g_1

Defined Pair Symbols:   MINUS_2, F_1, G_1

Compound Symbols:   c1_1, c3_3, c3_1, c3_2, c5_3, c5_1


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

(45) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace F(s(0)) -> c3(G(f(0))) by 
   F(s(0)) -> c3(G(s(0)))

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   f(0) -> s(0)
   f(s(z0)) -> minus(s(z0), g(f(z0)))
   g(0) -> 0
   g(s(z0)) -> minus(s(z0), f(g(z0)))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(f(s(0))), F(s(0)))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0))))
   F(s(s(x0))) -> c3(G(f(s(x0))))
   F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0)))
   G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0)))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0))))
   G(s(s(x0))) -> c5(F(g(s(x0))))
   G(s(0)) -> c5(MINUS(s(0), s(0)))
   F(s(0)) -> c3(G(s(0)))
S tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(f(s(0))), F(s(0)))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0))))
   F(s(s(x0))) -> c3(G(f(s(x0))))
   F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0)))
   G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0)))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0))))
   G(s(s(x0))) -> c5(F(g(s(x0))))
   G(s(0)) -> c5(MINUS(s(0), s(0)))
   F(s(0)) -> c3(G(s(0)))
K tuples:none
Defined Rule Symbols:   minus_2, f_1, g_1

Defined Pair Symbols:   MINUS_2, F_1, G_1

Compound Symbols:   c1_1, c3_3, c3_1, c3_2, c5_3, c5_1


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

(47) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(f(s(0))), F(s(0))) by F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))), F(s(0)))
----------------------------------------

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   f(0) -> s(0)
   f(s(z0)) -> minus(s(z0), g(f(z0)))
   g(0) -> 0
   g(s(z0)) -> minus(s(z0), f(g(z0)))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0))))
   F(s(s(x0))) -> c3(G(f(s(x0))))
   F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0)))
   G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0)))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0))))
   G(s(s(x0))) -> c5(F(g(s(x0))))
   G(s(0)) -> c5(MINUS(s(0), s(0)))
   F(s(0)) -> c3(G(s(0)))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))), F(s(0)))
S tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0))))
   F(s(s(x0))) -> c3(G(f(s(x0))))
   F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0)))
   G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0)))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0))))
   G(s(s(x0))) -> c5(F(g(s(x0))))
   G(s(0)) -> c5(MINUS(s(0), s(0)))
   F(s(0)) -> c3(G(s(0)))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))), F(s(0)))
K tuples:none
Defined Rule Symbols:   minus_2, f_1, g_1

Defined Pair Symbols:   MINUS_2, F_1, G_1

Compound Symbols:   c1_1, c3_3, c3_1, c3_2, c5_3, c5_1


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

(49) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(f(s(s(z0)))), F(s(s(z0)))) by F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0))))
----------------------------------------

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   f(0) -> s(0)
   f(s(z0)) -> minus(s(z0), g(f(z0)))
   g(0) -> 0
   g(s(z0)) -> minus(s(z0), f(g(z0)))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   F(s(s(x0))) -> c3(G(f(s(x0))))
   F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0)))
   G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0)))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0))))
   G(s(s(x0))) -> c5(F(g(s(x0))))
   G(s(0)) -> c5(MINUS(s(0), s(0)))
   F(s(0)) -> c3(G(s(0)))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))), F(s(0)))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0))))
S tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   F(s(s(x0))) -> c3(G(f(s(x0))))
   F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0)))
   G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0)))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0))))
   G(s(s(x0))) -> c5(F(g(s(x0))))
   G(s(0)) -> c5(MINUS(s(0), s(0)))
   F(s(0)) -> c3(G(s(0)))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))), F(s(0)))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0))))
K tuples:none
Defined Rule Symbols:   minus_2, f_1, g_1

Defined Pair Symbols:   MINUS_2, F_1, G_1

Compound Symbols:   c1_1, c3_1, c3_2, c5_3, c5_1, c3_3


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

(51) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(s(x0))) -> c3(G(f(s(x0)))) by F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0)))))
----------------------------------------

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   f(0) -> s(0)
   f(s(z0)) -> minus(s(z0), g(f(z0)))
   g(0) -> 0
   g(s(z0)) -> minus(s(z0), f(g(z0)))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0)))
   G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0)))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0))))
   G(s(s(x0))) -> c5(F(g(s(x0))))
   G(s(0)) -> c5(MINUS(s(0), s(0)))
   F(s(0)) -> c3(G(s(0)))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))), F(s(0)))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0))))
   F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0)))))
S tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0)))
   G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0)))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0))))
   G(s(s(x0))) -> c5(F(g(s(x0))))
   G(s(0)) -> c5(MINUS(s(0), s(0)))
   F(s(0)) -> c3(G(s(0)))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))), F(s(0)))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0))))
   F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0)))))
K tuples:none
Defined Rule Symbols:   minus_2, f_1, g_1

Defined Pair Symbols:   MINUS_2, F_1, G_1

Compound Symbols:   c1_1, c3_2, c5_3, c5_1, c3_1, c3_3


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

(53) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(f(0))) by F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(s(0)))
----------------------------------------

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   f(0) -> s(0)
   f(s(z0)) -> minus(s(z0), g(f(z0)))
   g(0) -> 0
   g(s(z0)) -> minus(s(z0), f(g(z0)))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0)))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0))))
   G(s(s(x0))) -> c5(F(g(s(x0))))
   G(s(0)) -> c5(MINUS(s(0), s(0)))
   F(s(0)) -> c3(G(s(0)))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))), F(s(0)))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0))))
   F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0)))))
   F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(s(0)))
S tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0)))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0))))
   G(s(s(x0))) -> c5(F(g(s(x0))))
   G(s(0)) -> c5(MINUS(s(0), s(0)))
   F(s(0)) -> c3(G(s(0)))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))), F(s(0)))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0))))
   F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0)))))
   F(s(0)) -> c3(MINUS(s(0), minus(s(0), f(g(0)))), G(s(0)))
K tuples:none
Defined Rule Symbols:   minus_2, f_1, g_1

Defined Pair Symbols:   MINUS_2, G_1, F_1

Compound Symbols:   c1_1, c5_3, c5_1, c3_1, c3_3, c3_2


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

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   f(0) -> s(0)
   f(s(z0)) -> minus(s(z0), g(f(z0)))
   g(0) -> 0
   g(s(z0)) -> minus(s(z0), f(g(z0)))
Tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0)))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0))))
   G(s(s(x0))) -> c5(F(g(s(x0))))
   G(s(0)) -> c5(MINUS(s(0), s(0)))
   F(s(0)) -> c3(G(s(0)))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))), F(s(0)))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0))))
   F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0)))))
   F(s(0)) -> c(MINUS(s(0), minus(s(0), f(g(0)))))
   F(s(0)) -> c(G(s(0)))
S tuples:
   MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1))
   G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0)))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0))))
   G(s(s(x0))) -> c5(F(g(s(x0))))
   G(s(0)) -> c5(MINUS(s(0), s(0)))
   F(s(0)) -> c3(G(s(0)))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))), F(s(0)))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0))))
   F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0)))))
   F(s(0)) -> c(MINUS(s(0), minus(s(0), f(g(0)))))
   F(s(0)) -> c(G(s(0)))
K tuples:none
Defined Rule Symbols:   minus_2, f_1, g_1

Defined Pair Symbols:   MINUS_2, G_1, F_1

Compound Symbols:   c1_1, c5_3, c5_1, c3_1, c3_3, c_1


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

(57) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace MINUS(s(z0), s(z1)) -> c1(MINUS(z0, z1)) by 
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   f(0) -> s(0)
   f(s(z0)) -> minus(s(z0), g(f(z0)))
   g(0) -> 0
   g(s(z0)) -> minus(s(z0), f(g(z0)))
Tuples:
   G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0)))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0))))
   G(s(s(x0))) -> c5(F(g(s(x0))))
   G(s(0)) -> c5(MINUS(s(0), s(0)))
   F(s(0)) -> c3(G(s(0)))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))), F(s(0)))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0))))
   F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0)))))
   F(s(0)) -> c(MINUS(s(0), minus(s(0), f(g(0)))))
   F(s(0)) -> c(G(s(0)))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
S tuples:
   G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0)))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0))))
   G(s(s(x0))) -> c5(F(g(s(x0))))
   G(s(0)) -> c5(MINUS(s(0), s(0)))
   F(s(0)) -> c3(G(s(0)))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))), F(s(0)))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0))))
   F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0)))))
   F(s(0)) -> c(MINUS(s(0), minus(s(0), f(g(0)))))
   F(s(0)) -> c(G(s(0)))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
K tuples:none
Defined Rule Symbols:   minus_2, f_1, g_1

Defined Pair Symbols:   G_1, F_1, MINUS_2

Compound Symbols:   c5_3, c5_1, c3_1, c3_3, c_1, c1_1


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

(59) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 4 trailing nodes:
   F(s(0)) -> c(MINUS(s(0), minus(s(0), f(g(0)))))
   G(s(0)) -> c5(MINUS(s(0), s(0)))
   F(s(0)) -> c3(G(s(0)))
   F(s(0)) -> c(G(s(0)))

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   f(0) -> s(0)
   f(s(z0)) -> minus(s(z0), g(f(z0)))
   g(0) -> 0
   g(s(z0)) -> minus(s(z0), f(g(z0)))
Tuples:
   G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0)))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0))))
   G(s(s(x0))) -> c5(F(g(s(x0))))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))), F(s(0)))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0))))
   F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0)))))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
S tuples:
   G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))), G(s(0)))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0))))
   G(s(s(x0))) -> c5(F(g(s(x0))))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))), F(s(0)))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0))))
   F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0)))))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
K tuples:none
Defined Rule Symbols:   minus_2, f_1, g_1

Defined Pair Symbols:   G_1, F_1, MINUS_2

Compound Symbols:   c5_3, c5_1, c3_3, c3_1, c1_1


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

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   f(0) -> s(0)
   f(s(z0)) -> minus(s(z0), g(f(z0)))
   g(0) -> 0
   g(s(z0)) -> minus(s(z0), f(g(z0)))
Tuples:
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0))))
   G(s(s(x0))) -> c5(F(g(s(x0))))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0))))
   F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0)))))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))))
S tuples:
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0))))
   G(s(s(x0))) -> c5(F(g(s(x0))))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0))))
   F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0)))))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))))
K tuples:none
Defined Rule Symbols:   minus_2, f_1, g_1

Defined Pair Symbols:   G_1, F_1, MINUS_2

Compound Symbols:   c5_3, c5_1, c3_3, c3_1, c1_1, c5_2, c3_2


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

(63) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(g(s(s(z0)))), G(s(s(z0)))) by G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(g(s(z0))))), G(s(s(z0))))
----------------------------------------

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   f(0) -> s(0)
   f(s(z0)) -> minus(s(z0), g(f(z0)))
   g(0) -> 0
   g(s(z0)) -> minus(s(z0), f(g(z0)))
Tuples:
   G(s(s(x0))) -> c5(F(g(s(x0))))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0))))
   F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0)))))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(g(s(z0))))), G(s(s(z0))))
S tuples:
   G(s(s(x0))) -> c5(F(g(s(x0))))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0))))
   F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0)))))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(g(s(z0))))), G(s(s(z0))))
K tuples:none
Defined Rule Symbols:   minus_2, f_1, g_1

Defined Pair Symbols:   G_1, F_1, MINUS_2

Compound Symbols:   c5_1, c3_3, c3_1, c1_1, c5_2, c3_2, c5_3


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

(65) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace G(s(s(x0))) -> c5(F(g(s(x0)))) by G(s(s(z0))) -> c5(F(minus(s(z0), f(g(z0)))))
----------------------------------------

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   f(0) -> s(0)
   f(s(z0)) -> minus(s(z0), g(f(z0)))
   g(0) -> 0
   g(s(z0)) -> minus(s(z0), f(g(z0)))
Tuples:
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0))))
   F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0)))))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(g(s(z0))))), G(s(s(z0))))
   G(s(s(z0))) -> c5(F(minus(s(z0), f(g(z0)))))
S tuples:
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0))))
   F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0)))))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(g(s(z0))))), G(s(s(z0))))
   G(s(s(z0))) -> c5(F(minus(s(z0), f(g(z0)))))
K tuples:none
Defined Rule Symbols:   minus_2, f_1, g_1

Defined Pair Symbols:   F_1, MINUS_2, G_1

Compound Symbols:   c3_3, c3_1, c1_1, c5_2, c3_2, c5_3, c5_1


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

(67) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(f(s(z0))))), F(s(s(z0)))) by F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(minus(s(z0), g(f(z0)))))), F(s(s(z0))))
----------------------------------------

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   f(0) -> s(0)
   f(s(z0)) -> minus(s(z0), g(f(z0)))
   g(0) -> 0
   g(s(z0)) -> minus(s(z0), f(g(z0)))
Tuples:
   F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0)))))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(g(s(z0))))), G(s(s(z0))))
   G(s(s(z0))) -> c5(F(minus(s(z0), f(g(z0)))))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(minus(s(z0), g(f(z0)))))), F(s(s(z0))))
S tuples:
   F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0)))))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0))))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(g(s(z0))))), G(s(s(z0))))
   G(s(s(z0))) -> c5(F(minus(s(z0), f(g(z0)))))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(minus(s(z0), g(f(z0)))))), F(s(s(z0))))
K tuples:none
Defined Rule Symbols:   minus_2, f_1, g_1

Defined Pair Symbols:   F_1, MINUS_2, G_1

Compound Symbols:   c3_1, c1_1, c5_2, c3_2, c5_3, c5_1, c3_3


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

(69) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(g(s(0)))) by G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(minus(s(0), f(g(0)))))
----------------------------------------

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   f(0) -> s(0)
   f(s(z0)) -> minus(s(z0), g(f(z0)))
   g(0) -> 0
   g(s(z0)) -> minus(s(z0), f(g(z0)))
Tuples:
   F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0)))))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(g(s(z0))))), G(s(s(z0))))
   G(s(s(z0))) -> c5(F(minus(s(z0), f(g(z0)))))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(minus(s(z0), g(f(z0)))))), F(s(s(z0))))
   G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(minus(s(0), f(g(0)))))
S tuples:
   F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0)))))
   MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1)))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(g(s(z0))))), G(s(s(z0))))
   G(s(s(z0))) -> c5(F(minus(s(z0), f(g(z0)))))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(minus(s(z0), g(f(z0)))))), F(s(s(z0))))
   G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(minus(s(0), f(g(0)))))
K tuples:none
Defined Rule Symbols:   minus_2, f_1, g_1

Defined Pair Symbols:   F_1, MINUS_2, G_1

Compound Symbols:   c3_1, c1_1, c3_2, c5_3, c5_1, c3_3, c5_2


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

(71) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace MINUS(s(s(y0)), s(s(y1))) -> c1(MINUS(s(y0), s(y1))) by 
   MINUS(s(s(s(y0))), s(s(s(y1)))) -> c1(MINUS(s(s(y0)), s(s(y1))))

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   f(0) -> s(0)
   f(s(z0)) -> minus(s(z0), g(f(z0)))
   g(0) -> 0
   g(s(z0)) -> minus(s(z0), f(g(z0)))
Tuples:
   F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0)))))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(g(s(z0))))), G(s(s(z0))))
   G(s(s(z0))) -> c5(F(minus(s(z0), f(g(z0)))))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(minus(s(z0), g(f(z0)))))), F(s(s(z0))))
   G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(minus(s(0), f(g(0)))))
   MINUS(s(s(s(y0))), s(s(s(y1)))) -> c1(MINUS(s(s(y0)), s(s(y1))))
S tuples:
   F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0)))))
   F(s(s(0))) -> c3(MINUS(s(s(0)), g(minus(s(0), g(s(0))))), G(minus(s(0), g(f(0)))))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(g(s(z0))))), G(s(s(z0))))
   G(s(s(z0))) -> c5(F(minus(s(z0), f(g(z0)))))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(minus(s(z0), g(f(z0)))))), F(s(s(z0))))
   G(s(s(0))) -> c5(MINUS(s(s(0)), f(minus(s(0), f(0)))), F(minus(s(0), f(g(0)))))
   MINUS(s(s(s(y0))), s(s(s(y1)))) -> c1(MINUS(s(s(y0)), s(s(y1))))
K tuples:none
Defined Rule Symbols:   minus_2, f_1, g_1

Defined Pair Symbols:   F_1, G_1, MINUS_2

Compound Symbols:   c3_1, c3_2, c5_3, c5_1, c3_3, c5_2, c1_1


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

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   f(0) -> s(0)
   f(s(z0)) -> minus(s(z0), g(f(z0)))
   g(0) -> 0
   g(s(z0)) -> minus(s(z0), f(g(z0)))
Tuples:
   F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0)))))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(g(s(z0))))), G(s(s(z0))))
   G(s(s(z0))) -> c5(F(minus(s(z0), f(g(z0)))))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(minus(s(z0), g(f(z0)))))), F(s(s(z0))))
   MINUS(s(s(s(y0))), s(s(s(y1)))) -> c1(MINUS(s(s(y0)), s(s(y1))))
   F(s(s(0))) -> c3(G(minus(s(0), g(f(0)))))
   G(s(s(0))) -> c5(F(minus(s(0), f(g(0)))))
S tuples:
   F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0)))))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(g(s(z0))))), G(s(s(z0))))
   G(s(s(z0))) -> c5(F(minus(s(z0), f(g(z0)))))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(minus(s(z0), g(f(z0)))))), F(s(s(z0))))
   MINUS(s(s(s(y0))), s(s(s(y1)))) -> c1(MINUS(s(s(y0)), s(s(y1))))
   F(s(s(0))) -> c3(G(minus(s(0), g(f(0)))))
   G(s(s(0))) -> c5(F(minus(s(0), f(g(0)))))
K tuples:none
Defined Rule Symbols:   minus_2, f_1, g_1

Defined Pair Symbols:   F_1, G_1, MINUS_2

Compound Symbols:   c3_1, c5_3, c5_1, c3_3, c1_1


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

(75) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(g(s(z0))))), G(s(s(z0)))) by G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(minus(s(z0), f(g(z0)))))), G(s(s(z0))))
----------------------------------------

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   f(0) -> s(0)
   f(s(z0)) -> minus(s(z0), g(f(z0)))
   g(0) -> 0
   g(s(z0)) -> minus(s(z0), f(g(z0)))
Tuples:
   F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0)))))
   G(s(s(z0))) -> c5(F(minus(s(z0), f(g(z0)))))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(minus(s(z0), g(f(z0)))))), F(s(s(z0))))
   MINUS(s(s(s(y0))), s(s(s(y1)))) -> c1(MINUS(s(s(y0)), s(s(y1))))
   F(s(s(0))) -> c3(G(minus(s(0), g(f(0)))))
   G(s(s(0))) -> c5(F(minus(s(0), f(g(0)))))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(minus(s(z0), f(g(z0)))))), G(s(s(z0))))
S tuples:
   F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0)))))
   G(s(s(z0))) -> c5(F(minus(s(z0), f(g(z0)))))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(minus(s(z0), g(f(z0)))))), F(s(s(z0))))
   MINUS(s(s(s(y0))), s(s(s(y1)))) -> c1(MINUS(s(s(y0)), s(s(y1))))
   F(s(s(0))) -> c3(G(minus(s(0), g(f(0)))))
   G(s(s(0))) -> c5(F(minus(s(0), f(g(0)))))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(minus(s(z0), f(g(z0)))))), G(s(s(z0))))
K tuples:none
Defined Rule Symbols:   minus_2, f_1, g_1

Defined Pair Symbols:   F_1, G_1, MINUS_2

Compound Symbols:   c3_1, c5_1, c3_3, c1_1, c5_3


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

(77) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace F(s(s(0))) -> c3(G(minus(s(0), g(f(0))))) by F(s(s(0))) -> c3(G(minus(s(0), g(s(0)))))
----------------------------------------

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   f(0) -> s(0)
   f(s(z0)) -> minus(s(z0), g(f(z0)))
   g(0) -> 0
   g(s(z0)) -> minus(s(z0), f(g(z0)))
Tuples:
   F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0)))))
   G(s(s(z0))) -> c5(F(minus(s(z0), f(g(z0)))))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(minus(s(z0), g(f(z0)))))), F(s(s(z0))))
   MINUS(s(s(s(y0))), s(s(s(y1)))) -> c1(MINUS(s(s(y0)), s(s(y1))))
   G(s(s(0))) -> c5(F(minus(s(0), f(g(0)))))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(minus(s(z0), f(g(z0)))))), G(s(s(z0))))
   F(s(s(0))) -> c3(G(minus(s(0), g(s(0)))))
S tuples:
   F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0)))))
   G(s(s(z0))) -> c5(F(minus(s(z0), f(g(z0)))))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(minus(s(z0), g(f(z0)))))), F(s(s(z0))))
   MINUS(s(s(s(y0))), s(s(s(y1)))) -> c1(MINUS(s(s(y0)), s(s(y1))))
   G(s(s(0))) -> c5(F(minus(s(0), f(g(0)))))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(minus(s(z0), f(g(z0)))))), G(s(s(z0))))
   F(s(s(0))) -> c3(G(minus(s(0), g(s(0)))))
K tuples:none
Defined Rule Symbols:   minus_2, f_1, g_1

Defined Pair Symbols:   F_1, G_1, MINUS_2

Compound Symbols:   c3_1, c5_1, c3_3, c1_1, c5_3


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

(79) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace G(s(s(0))) -> c5(F(minus(s(0), f(g(0))))) by G(s(s(0))) -> c5(F(minus(s(0), f(0))))
----------------------------------------

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   f(0) -> s(0)
   f(s(z0)) -> minus(s(z0), g(f(z0)))
   g(0) -> 0
   g(s(z0)) -> minus(s(z0), f(g(z0)))
Tuples:
   F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0)))))
   G(s(s(z0))) -> c5(F(minus(s(z0), f(g(z0)))))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(minus(s(z0), g(f(z0)))))), F(s(s(z0))))
   MINUS(s(s(s(y0))), s(s(s(y1)))) -> c1(MINUS(s(s(y0)), s(s(y1))))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(minus(s(z0), f(g(z0)))))), G(s(s(z0))))
   F(s(s(0))) -> c3(G(minus(s(0), g(s(0)))))
   G(s(s(0))) -> c5(F(minus(s(0), f(0))))
S tuples:
   F(s(s(z0))) -> c3(G(minus(s(z0), g(f(z0)))))
   G(s(s(z0))) -> c5(F(minus(s(z0), f(g(z0)))))
   F(s(s(s(z0)))) -> c3(MINUS(s(s(s(z0))), g(minus(s(s(z0)), g(minus(s(z0), g(f(z0))))))), G(minus(s(s(z0)), g(minus(s(z0), g(f(z0)))))), F(s(s(z0))))
   MINUS(s(s(s(y0))), s(s(s(y1)))) -> c1(MINUS(s(s(y0)), s(s(y1))))
   G(s(s(s(z0)))) -> c5(MINUS(s(s(s(z0))), f(minus(s(s(z0)), f(minus(s(z0), f(g(z0))))))), F(minus(s(s(z0)), f(minus(s(z0), f(g(z0)))))), G(s(s(z0))))
   F(s(s(0))) -> c3(G(minus(s(0), g(s(0)))))
   G(s(s(0))) -> c5(F(minus(s(0), f(0))))
K tuples:none
Defined Rule Symbols:   minus_2, f_1, g_1

Defined Pair Symbols:   F_1, G_1, MINUS_2

Compound Symbols:   c3_1, c5_1, c3_3, c1_1, c5_3