KILLED
proof of input_dzthlO6myE.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, INF).

(0) CpxTRS
(1) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxTRS
(3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(4) CpxTRS
(5) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxWeightedTrs
    (7) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedTrs
    (9) CompletionProof [UPPER BOUND(ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
        (11) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (12) CpxTypedWeightedCompleteTrs
        (13) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (14) CpxRNTS
        (15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (16) CpxRNTS
        (17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (18) CpxRNTS
        (19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (20) CpxRNTS
        (21) IntTrsBoundProof [UPPER BOUND(ID), 276 ms]
        (22) CpxRNTS
        (23) IntTrsBoundProof [UPPER BOUND(ID), 41 ms]
        (24) CpxRNTS
        (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 234 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 57 ms]
        (30) CpxRNTS
        (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (32) CpxRNTS
        (33) IntTrsBoundProof [UPPER BOUND(ID), 8776 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 6141 ms]
        (36) CpxRNTS
    (37) CompletionProof [UPPER BOUND(ID), 0 ms]
    (38) CpxTypedWeightedCompleteTrs
        (39) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 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) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) CdtProblem
    (51) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CdtProblem
    (53) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 22 ms]
    (54) CdtProblem
    (55) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CdtProblem
    (57) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (58) CdtProblem
    (59) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CdtProblem
    (61) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CdtProblem
    (63) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 11 ms]
    (64) CdtProblem
    (65) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (70) CdtProblem
    (71) CdtLeafRemovalProof [ComplexityIfPolyImplication, 0 ms]
    (72) CdtProblem
    (73) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 15 ms]
    (74) CdtProblem
    (75) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (82) CdtProblem
    (83) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) CdtProblem
    (87) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (88) 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:

   div(x, s(y)) -> d(x, s(y), 0)
   d(x, s(y), z) -> cond(ge(x, z), x, y, z)
   cond(true, x, y, z) -> s(d(x, s(y), plus(s(y), z)))
   cond(false, x, y, z) -> 0
   ge(u, 0) -> true
   ge(0, s(v)) -> false
   ge(s(u), s(v)) -> ge(u, v)
   plus(n, 0) -> n
   plus(n, s(m)) -> s(plus(n, m))

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:

   div(x, s(y)) -> d(x, s(y), 0')
   d(x, s(y), z) -> cond(ge(x, z), x, y, z)
   cond(true, x, y, z) -> s(d(x, s(y), plus(s(y), z)))
   cond(false, x, y, z) -> 0'
   ge(u, 0') -> true
   ge(0', s(v)) -> false
   ge(s(u), s(v)) -> ge(u, v)
   plus(n, 0') -> n
   plus(n, s(m)) -> s(plus(n, m))

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:

   div(x, s(y)) -> d(x, s(y), 0)
   d(x, s(y), z) -> cond(ge(x, z), x, y, z)
   cond(true, x, y, z) -> s(d(x, s(y), plus(s(y), z)))
   cond(false, x, y, z) -> 0
   ge(u, 0) -> true
   ge(0, s(v)) -> false
   ge(s(u), s(v)) -> ge(u, v)
   plus(n, 0) -> n
   plus(n, s(m)) -> s(plus(n, m))

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:

   div(x, s(y)) -> d(x, s(y), 0) [1]
   d(x, s(y), z) -> cond(ge(x, z), x, y, z) [1]
   cond(true, x, y, z) -> s(d(x, s(y), plus(s(y), z))) [1]
   cond(false, x, y, z) -> 0 [1]
   ge(u, 0) -> true [1]
   ge(0, s(v)) -> false [1]
   ge(s(u), s(v)) -> ge(u, v) [1]
   plus(n, 0) -> n [1]
   plus(n, s(m)) -> s(plus(n, m)) [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:

   div(x, s(y)) -> d(x, s(y), 0) [1]
   d(x, s(y), z) -> cond(ge(x, z), x, y, z) [1]
   cond(true, x, y, z) -> s(d(x, s(y), plus(s(y), z))) [1]
   cond(false, x, y, z) -> 0 [1]
   ge(u, 0) -> true [1]
   ge(0, s(v)) -> false [1]
   ge(s(u), s(v)) -> ge(u, v) [1]
   plus(n, 0) -> n [1]
   plus(n, s(m)) -> s(plus(n, m)) [1]

The TRS has the following type information:
div :: s:0 -> s:0 -> s:0
s :: s:0 -> s:0
d :: s:0 -> s:0 -> s:0 -> s:0
0 :: s:0
cond :: true:false -> s:0 -> s:0 -> s:0 -> s:0
ge :: s:0 -> s:0 -> true:false
true :: true:false
plus :: s:0 -> s:0 -> s:0
false :: true:false

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:

   div_2
   d_3
   cond_4

(c) The following functions are completely defined:

   ge_2
   plus_2

Due to the following rules being added:
none

And the following fresh constants: none

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

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

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

   div(x, s(y)) -> d(x, s(y), 0) [1]
   d(x, s(y), z) -> cond(ge(x, z), x, y, z) [1]
   cond(true, x, y, z) -> s(d(x, s(y), plus(s(y), z))) [1]
   cond(false, x, y, z) -> 0 [1]
   ge(u, 0) -> true [1]
   ge(0, s(v)) -> false [1]
   ge(s(u), s(v)) -> ge(u, v) [1]
   plus(n, 0) -> n [1]
   plus(n, s(m)) -> s(plus(n, m)) [1]

The TRS has the following type information:
div :: s:0 -> s:0 -> s:0
s :: s:0 -> s:0
d :: s:0 -> s:0 -> s:0 -> s:0
0 :: s:0
cond :: true:false -> s:0 -> s:0 -> s:0 -> s:0
ge :: s:0 -> s:0 -> true:false
true :: true:false
plus :: s:0 -> s:0 -> s:0
false :: true:false

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:

   div(x, s(y)) -> d(x, s(y), 0) [1]
   d(x, s(y), 0) -> cond(true, x, y, 0) [2]
   d(0, s(y), s(v')) -> cond(false, 0, y, s(v')) [2]
   d(s(u'), s(y), s(v'')) -> cond(ge(u', v''), s(u'), y, s(v'')) [2]
   cond(true, x, y, 0) -> s(d(x, s(y), s(y))) [2]
   cond(true, x, y, s(m')) -> s(d(x, s(y), s(plus(s(y), m')))) [2]
   cond(false, x, y, z) -> 0 [1]
   ge(u, 0) -> true [1]
   ge(0, s(v)) -> false [1]
   ge(s(u), s(v)) -> ge(u, v) [1]
   plus(n, 0) -> n [1]
   plus(n, s(m)) -> s(plus(n, m)) [1]

The TRS has the following type information:
div :: s:0 -> s:0 -> s:0
s :: s:0 -> s:0
d :: s:0 -> s:0 -> s:0 -> s:0
0 :: s:0
cond :: true:false -> s:0 -> s:0 -> s:0 -> s:0
ge :: s:0 -> s:0 -> true:false
true :: true:false
plus :: s:0 -> s:0 -> s:0
false :: true:false

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

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

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

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

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

   cond(z', z'', z1, z2) -{ 1 }-> 0 :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 0
   cond(z', z'', z1, z2) -{ 2 }-> 1 + d(x, 1 + y, 1 + y) :|: z1 = y, z2 = 0, x >= 0, y >= 0, z'' = x, z' = 1
   cond(z', z'', z1, z2) -{ 2 }-> 1 + d(x, 1 + y, 1 + plus(1 + y, m')) :|: z2 = 1 + m', z1 = y, x >= 0, y >= 0, m' >= 0, z'' = x, z' = 1
   d(z', z'', z1) -{ 2 }-> cond(ge(u', v''), 1 + u', y, 1 + v'') :|: z1 = 1 + v'', y >= 0, z' = 1 + u', u' >= 0, z'' = 1 + y, v'' >= 0
   d(z', z'', z1) -{ 2 }-> cond(1, x, y, 0) :|: z1 = 0, z' = x, x >= 0, y >= 0, z'' = 1 + y
   d(z', z'', z1) -{ 2 }-> cond(0, 0, y, 1 + v') :|: y >= 0, z1 = 1 + v', v' >= 0, z'' = 1 + y, z' = 0
   div(z', z'') -{ 1 }-> d(x, 1 + y, 0) :|: z' = x, x >= 0, y >= 0, z'' = 1 + y
   ge(z', z'') -{ 1 }-> ge(u, v) :|: v >= 0, z' = 1 + u, z'' = 1 + v, u >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' = u, u >= 0
   ge(z', z'') -{ 1 }-> 0 :|: v >= 0, z'' = 1 + v, z' = 0
   plus(z', z'') -{ 1 }-> n :|: z'' = 0, n >= 0, z' = n
   plus(z', z'') -{ 1 }-> 1 + plus(n, m) :|: n >= 0, z'' = 1 + m, z' = n, m >= 0


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

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

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

   cond(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0
   cond(z', z'', z1, z2) -{ 2 }-> 1 + d(z'', 1 + z1, 1 + z1) :|: z2 = 0, z'' >= 0, z1 >= 0, z' = 1
   cond(z', z'', z1, z2) -{ 2 }-> 1 + d(z'', 1 + z1, 1 + plus(1 + z1, z2 - 1)) :|: z'' >= 0, z1 >= 0, z2 - 1 >= 0, z' = 1
   d(z', z'', z1) -{ 2 }-> cond(ge(z' - 1, z1 - 1), 1 + (z' - 1), z'' - 1, 1 + (z1 - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0, z1 - 1 >= 0
   d(z', z'', z1) -{ 2 }-> cond(1, z', z'' - 1, 0) :|: z1 = 0, z' >= 0, z'' - 1 >= 0
   d(z', z'', z1) -{ 2 }-> cond(0, 0, z'' - 1, 1 + (z1 - 1)) :|: z'' - 1 >= 0, z1 - 1 >= 0, z' = 0
   div(z', z'') -{ 1 }-> d(z', 1 + (z'' - 1), 0) :|: z' >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z'' - 1 >= 0, z' - 1 >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z', z'' - 1) :|: z' >= 0, z'' - 1 >= 0


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

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

   { ge }
   { plus }
   { d, cond }
   { div }

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

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

   cond(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0
   cond(z', z'', z1, z2) -{ 2 }-> 1 + d(z'', 1 + z1, 1 + z1) :|: z2 = 0, z'' >= 0, z1 >= 0, z' = 1
   cond(z', z'', z1, z2) -{ 2 }-> 1 + d(z'', 1 + z1, 1 + plus(1 + z1, z2 - 1)) :|: z'' >= 0, z1 >= 0, z2 - 1 >= 0, z' = 1
   d(z', z'', z1) -{ 2 }-> cond(ge(z' - 1, z1 - 1), 1 + (z' - 1), z'' - 1, 1 + (z1 - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0, z1 - 1 >= 0
   d(z', z'', z1) -{ 2 }-> cond(1, z', z'' - 1, 0) :|: z1 = 0, z' >= 0, z'' - 1 >= 0
   d(z', z'', z1) -{ 2 }-> cond(0, 0, z'' - 1, 1 + (z1 - 1)) :|: z'' - 1 >= 0, z1 - 1 >= 0, z' = 0
   div(z', z'') -{ 1 }-> d(z', 1 + (z'' - 1), 0) :|: z' >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z'' - 1 >= 0, z' - 1 >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z', z'' - 1) :|: z' >= 0, z'' - 1 >= 0

Function symbols to be analyzed: {ge}, {plus}, {d,cond}, {div}

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

(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:

   cond(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0
   cond(z', z'', z1, z2) -{ 2 }-> 1 + d(z'', 1 + z1, 1 + z1) :|: z2 = 0, z'' >= 0, z1 >= 0, z' = 1
   cond(z', z'', z1, z2) -{ 2 }-> 1 + d(z'', 1 + z1, 1 + plus(1 + z1, z2 - 1)) :|: z'' >= 0, z1 >= 0, z2 - 1 >= 0, z' = 1
   d(z', z'', z1) -{ 2 }-> cond(ge(z' - 1, z1 - 1), 1 + (z' - 1), z'' - 1, 1 + (z1 - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0, z1 - 1 >= 0
   d(z', z'', z1) -{ 2 }-> cond(1, z', z'' - 1, 0) :|: z1 = 0, z' >= 0, z'' - 1 >= 0
   d(z', z'', z1) -{ 2 }-> cond(0, 0, z'' - 1, 1 + (z1 - 1)) :|: z'' - 1 >= 0, z1 - 1 >= 0, z' = 0
   div(z', z'') -{ 1 }-> d(z', 1 + (z'' - 1), 0) :|: z' >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z'' - 1 >= 0, z' - 1 >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z', z'' - 1) :|: z' >= 0, z'' - 1 >= 0

Function symbols to be analyzed: {ge}, {plus}, {d,cond}, {div}

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   cond(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0
   cond(z', z'', z1, z2) -{ 2 }-> 1 + d(z'', 1 + z1, 1 + z1) :|: z2 = 0, z'' >= 0, z1 >= 0, z' = 1
   cond(z', z'', z1, z2) -{ 2 }-> 1 + d(z'', 1 + z1, 1 + plus(1 + z1, z2 - 1)) :|: z'' >= 0, z1 >= 0, z2 - 1 >= 0, z' = 1
   d(z', z'', z1) -{ 2 }-> cond(ge(z' - 1, z1 - 1), 1 + (z' - 1), z'' - 1, 1 + (z1 - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0, z1 - 1 >= 0
   d(z', z'', z1) -{ 2 }-> cond(1, z', z'' - 1, 0) :|: z1 = 0, z' >= 0, z'' - 1 >= 0
   d(z', z'', z1) -{ 2 }-> cond(0, 0, z'' - 1, 1 + (z1 - 1)) :|: z'' - 1 >= 0, z1 - 1 >= 0, z' = 0
   div(z', z'') -{ 1 }-> d(z', 1 + (z'' - 1), 0) :|: z' >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z'' - 1 >= 0, z' - 1 >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z', z'' - 1) :|: z' >= 0, z'' - 1 >= 0

Function symbols to be analyzed: {ge}, {plus}, {d,cond}, {div}
Previous analysis results are:
  ge: runtime: ?, size: O(1) [1]

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

(23) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: ge
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + z''

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

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

   cond(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0
   cond(z', z'', z1, z2) -{ 2 }-> 1 + d(z'', 1 + z1, 1 + z1) :|: z2 = 0, z'' >= 0, z1 >= 0, z' = 1
   cond(z', z'', z1, z2) -{ 2 }-> 1 + d(z'', 1 + z1, 1 + plus(1 + z1, z2 - 1)) :|: z'' >= 0, z1 >= 0, z2 - 1 >= 0, z' = 1
   d(z', z'', z1) -{ 2 }-> cond(ge(z' - 1, z1 - 1), 1 + (z' - 1), z'' - 1, 1 + (z1 - 1)) :|: z'' - 1 >= 0, z' - 1 >= 0, z1 - 1 >= 0
   d(z', z'', z1) -{ 2 }-> cond(1, z', z'' - 1, 0) :|: z1 = 0, z' >= 0, z'' - 1 >= 0
   d(z', z'', z1) -{ 2 }-> cond(0, 0, z'' - 1, 1 + (z1 - 1)) :|: z'' - 1 >= 0, z1 - 1 >= 0, z' = 0
   div(z', z'') -{ 1 }-> d(z', 1 + (z'' - 1), 0) :|: z' >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 1 }-> ge(z' - 1, z'' - 1) :|: z'' - 1 >= 0, z' - 1 >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z', z'' - 1) :|: z' >= 0, z'' - 1 >= 0

Function symbols to be analyzed: {plus}, {d,cond}, {div}
Previous analysis results are:
  ge: runtime: O(n^1) [2 + z''], size: O(1) [1]

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

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

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

   cond(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0
   cond(z', z'', z1, z2) -{ 2 }-> 1 + d(z'', 1 + z1, 1 + z1) :|: z2 = 0, z'' >= 0, z1 >= 0, z' = 1
   cond(z', z'', z1, z2) -{ 2 }-> 1 + d(z'', 1 + z1, 1 + plus(1 + z1, z2 - 1)) :|: z'' >= 0, z1 >= 0, z2 - 1 >= 0, z' = 1
   d(z', z'', z1) -{ 3 + z1 }-> cond(s, 1 + (z' - 1), z'' - 1, 1 + (z1 - 1)) :|: s >= 0, s <= 1, z'' - 1 >= 0, z' - 1 >= 0, z1 - 1 >= 0
   d(z', z'', z1) -{ 2 }-> cond(1, z', z'' - 1, 0) :|: z1 = 0, z' >= 0, z'' - 1 >= 0
   d(z', z'', z1) -{ 2 }-> cond(0, 0, z'' - 1, 1 + (z1 - 1)) :|: z'' - 1 >= 0, z1 - 1 >= 0, z' = 0
   div(z', z'') -{ 1 }-> d(z', 1 + (z'' - 1), 0) :|: z' >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 1, z'' - 1 >= 0, z' - 1 >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z', z'' - 1) :|: z' >= 0, z'' - 1 >= 0

Function symbols to be analyzed: {plus}, {d,cond}, {div}
Previous analysis results are:
  ge: runtime: O(n^1) [2 + z''], size: O(1) [1]

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   cond(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0
   cond(z', z'', z1, z2) -{ 2 }-> 1 + d(z'', 1 + z1, 1 + z1) :|: z2 = 0, z'' >= 0, z1 >= 0, z' = 1
   cond(z', z'', z1, z2) -{ 2 }-> 1 + d(z'', 1 + z1, 1 + plus(1 + z1, z2 - 1)) :|: z'' >= 0, z1 >= 0, z2 - 1 >= 0, z' = 1
   d(z', z'', z1) -{ 3 + z1 }-> cond(s, 1 + (z' - 1), z'' - 1, 1 + (z1 - 1)) :|: s >= 0, s <= 1, z'' - 1 >= 0, z' - 1 >= 0, z1 - 1 >= 0
   d(z', z'', z1) -{ 2 }-> cond(1, z', z'' - 1, 0) :|: z1 = 0, z' >= 0, z'' - 1 >= 0
   d(z', z'', z1) -{ 2 }-> cond(0, 0, z'' - 1, 1 + (z1 - 1)) :|: z'' - 1 >= 0, z1 - 1 >= 0, z' = 0
   div(z', z'') -{ 1 }-> d(z', 1 + (z'' - 1), 0) :|: z' >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 1, z'' - 1 >= 0, z' - 1 >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z', z'' - 1) :|: z' >= 0, z'' - 1 >= 0

Function symbols to be analyzed: {plus}, {d,cond}, {div}
Previous analysis results are:
  ge: runtime: O(n^1) [2 + z''], size: O(1) [1]
  plus: runtime: ?, size: O(n^1) [z' + z'']

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   cond(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0
   cond(z', z'', z1, z2) -{ 2 }-> 1 + d(z'', 1 + z1, 1 + z1) :|: z2 = 0, z'' >= 0, z1 >= 0, z' = 1
   cond(z', z'', z1, z2) -{ 2 }-> 1 + d(z'', 1 + z1, 1 + plus(1 + z1, z2 - 1)) :|: z'' >= 0, z1 >= 0, z2 - 1 >= 0, z' = 1
   d(z', z'', z1) -{ 3 + z1 }-> cond(s, 1 + (z' - 1), z'' - 1, 1 + (z1 - 1)) :|: s >= 0, s <= 1, z'' - 1 >= 0, z' - 1 >= 0, z1 - 1 >= 0
   d(z', z'', z1) -{ 2 }-> cond(1, z', z'' - 1, 0) :|: z1 = 0, z' >= 0, z'' - 1 >= 0
   d(z', z'', z1) -{ 2 }-> cond(0, 0, z'' - 1, 1 + (z1 - 1)) :|: z'' - 1 >= 0, z1 - 1 >= 0, z' = 0
   div(z', z'') -{ 1 }-> d(z', 1 + (z'' - 1), 0) :|: z' >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 1, z'' - 1 >= 0, z' - 1 >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 }-> 1 + plus(z', z'' - 1) :|: z' >= 0, z'' - 1 >= 0

Function symbols to be analyzed: {d,cond}, {div}
Previous analysis results are:
  ge: runtime: O(n^1) [2 + z''], size: O(1) [1]
  plus: runtime: O(n^1) [1 + z''], size: O(n^1) [z' + z'']

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

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

(32)
Obligation:
Complexity RNTS consisting of the following rules:

   cond(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0
   cond(z', z'', z1, z2) -{ 2 + z2 }-> 1 + d(z'', 1 + z1, 1 + s'') :|: s'' >= 0, s'' <= 1 + z1 + (z2 - 1), z'' >= 0, z1 >= 0, z2 - 1 >= 0, z' = 1
   cond(z', z'', z1, z2) -{ 2 }-> 1 + d(z'', 1 + z1, 1 + z1) :|: z2 = 0, z'' >= 0, z1 >= 0, z' = 1
   d(z', z'', z1) -{ 3 + z1 }-> cond(s, 1 + (z' - 1), z'' - 1, 1 + (z1 - 1)) :|: s >= 0, s <= 1, z'' - 1 >= 0, z' - 1 >= 0, z1 - 1 >= 0
   d(z', z'', z1) -{ 2 }-> cond(1, z', z'' - 1, 0) :|: z1 = 0, z' >= 0, z'' - 1 >= 0
   d(z', z'', z1) -{ 2 }-> cond(0, 0, z'' - 1, 1 + (z1 - 1)) :|: z'' - 1 >= 0, z1 - 1 >= 0, z' = 0
   div(z', z'') -{ 1 }-> d(z', 1 + (z'' - 1), 0) :|: z' >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 1, z'' - 1 >= 0, z' - 1 >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 + z'' }-> 1 + s1 :|: s1 >= 0, s1 <= z' + (z'' - 1), z' >= 0, z'' - 1 >= 0

Function symbols to be analyzed: {d,cond}, {div}
Previous analysis results are:
  ge: runtime: O(n^1) [2 + z''], size: O(1) [1]
  plus: runtime: O(n^1) [1 + z''], size: O(n^1) [z' + z'']

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

   cond(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0
   cond(z', z'', z1, z2) -{ 2 + z2 }-> 1 + d(z'', 1 + z1, 1 + s'') :|: s'' >= 0, s'' <= 1 + z1 + (z2 - 1), z'' >= 0, z1 >= 0, z2 - 1 >= 0, z' = 1
   cond(z', z'', z1, z2) -{ 2 }-> 1 + d(z'', 1 + z1, 1 + z1) :|: z2 = 0, z'' >= 0, z1 >= 0, z' = 1
   d(z', z'', z1) -{ 3 + z1 }-> cond(s, 1 + (z' - 1), z'' - 1, 1 + (z1 - 1)) :|: s >= 0, s <= 1, z'' - 1 >= 0, z' - 1 >= 0, z1 - 1 >= 0
   d(z', z'', z1) -{ 2 }-> cond(1, z', z'' - 1, 0) :|: z1 = 0, z' >= 0, z'' - 1 >= 0
   d(z', z'', z1) -{ 2 }-> cond(0, 0, z'' - 1, 1 + (z1 - 1)) :|: z'' - 1 >= 0, z1 - 1 >= 0, z' = 0
   div(z', z'') -{ 1 }-> d(z', 1 + (z'' - 1), 0) :|: z' >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 1, z'' - 1 >= 0, z' - 1 >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 + z'' }-> 1 + s1 :|: s1 >= 0, s1 <= z' + (z'' - 1), z' >= 0, z'' - 1 >= 0

Function symbols to be analyzed: {d,cond}, {div}
Previous analysis results are:
  ge: runtime: O(n^1) [2 + z''], size: O(1) [1]
  plus: runtime: O(n^1) [1 + z''], size: O(n^1) [z' + z'']
  d: runtime: ?, size: INF
  cond: runtime: ?, size: INF

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   cond(z', z'', z1, z2) -{ 1 }-> 0 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 0
   cond(z', z'', z1, z2) -{ 2 + z2 }-> 1 + d(z'', 1 + z1, 1 + s'') :|: s'' >= 0, s'' <= 1 + z1 + (z2 - 1), z'' >= 0, z1 >= 0, z2 - 1 >= 0, z' = 1
   cond(z', z'', z1, z2) -{ 2 }-> 1 + d(z'', 1 + z1, 1 + z1) :|: z2 = 0, z'' >= 0, z1 >= 0, z' = 1
   d(z', z'', z1) -{ 3 + z1 }-> cond(s, 1 + (z' - 1), z'' - 1, 1 + (z1 - 1)) :|: s >= 0, s <= 1, z'' - 1 >= 0, z' - 1 >= 0, z1 - 1 >= 0
   d(z', z'', z1) -{ 2 }-> cond(1, z', z'' - 1, 0) :|: z1 = 0, z' >= 0, z'' - 1 >= 0
   d(z', z'', z1) -{ 2 }-> cond(0, 0, z'' - 1, 1 + (z1 - 1)) :|: z'' - 1 >= 0, z1 - 1 >= 0, z' = 0
   div(z', z'') -{ 1 }-> d(z', 1 + (z'' - 1), 0) :|: z' >= 0, z'' - 1 >= 0
   ge(z', z'') -{ 2 + z'' }-> s' :|: s' >= 0, s' <= 1, z'' - 1 >= 0, z' - 1 >= 0
   ge(z', z'') -{ 1 }-> 1 :|: z'' = 0, z' >= 0
   ge(z', z'') -{ 1 }-> 0 :|: z'' - 1 >= 0, z' = 0
   plus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   plus(z', z'') -{ 1 + z'' }-> 1 + s1 :|: s1 >= 0, s1 <= z' + (z'' - 1), z' >= 0, z'' - 1 >= 0

Function symbols to be analyzed: {d,cond}, {div}
Previous analysis results are:
  ge: runtime: O(n^1) [2 + z''], size: O(1) [1]
  plus: runtime: O(n^1) [1 + z''], size: O(n^1) [z' + z'']
  d: runtime: INF, size: INF
  cond: runtime: ?, size: INF

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

(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:

   div(v0, v1) -> null_div [0]
   d(v0, v1, v2) -> null_d [0]
   ge(v0, v1) -> null_ge [0]
   plus(v0, v1) -> null_plus [0]
   cond(v0, v1, v2, v3) -> null_cond [0]

And the following fresh constants:    null_div, null_d, null_ge, null_plus, null_cond

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

(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:

   div(x, s(y)) -> d(x, s(y), 0) [1]
   d(x, s(y), z) -> cond(ge(x, z), x, y, z) [1]
   cond(true, x, y, z) -> s(d(x, s(y), plus(s(y), z))) [1]
   cond(false, x, y, z) -> 0 [1]
   ge(u, 0) -> true [1]
   ge(0, s(v)) -> false [1]
   ge(s(u), s(v)) -> ge(u, v) [1]
   plus(n, 0) -> n [1]
   plus(n, s(m)) -> s(plus(n, m)) [1]
   div(v0, v1) -> null_div [0]
   d(v0, v1, v2) -> null_d [0]
   ge(v0, v1) -> null_ge [0]
   plus(v0, v1) -> null_plus [0]
   cond(v0, v1, v2, v3) -> null_cond [0]

The TRS has the following type information:
div :: s:0:null_div:null_d:null_plus:null_cond -> s:0:null_div:null_d:null_plus:null_cond -> s:0:null_div:null_d:null_plus:null_cond
s :: s:0:null_div:null_d:null_plus:null_cond -> s:0:null_div:null_d:null_plus:null_cond
d :: s:0:null_div:null_d:null_plus:null_cond -> s:0:null_div:null_d:null_plus:null_cond -> s:0:null_div:null_d:null_plus:null_cond -> s:0:null_div:null_d:null_plus:null_cond
0 :: s:0:null_div:null_d:null_plus:null_cond
cond :: true:false:null_ge -> s:0:null_div:null_d:null_plus:null_cond -> s:0:null_div:null_d:null_plus:null_cond -> s:0:null_div:null_d:null_plus:null_cond -> s:0:null_div:null_d:null_plus:null_cond
ge :: s:0:null_div:null_d:null_plus:null_cond -> s:0:null_div:null_d:null_plus:null_cond -> true:false:null_ge
true :: true:false:null_ge
plus :: s:0:null_div:null_d:null_plus:null_cond -> s:0:null_div:null_d:null_plus:null_cond -> s:0:null_div:null_d:null_plus:null_cond
false :: true:false:null_ge
null_div :: s:0:null_div:null_d:null_plus:null_cond
null_d :: s:0:null_div:null_d:null_plus:null_cond
null_ge :: true:false:null_ge
null_plus :: s:0:null_div:null_d:null_plus:null_cond
null_cond :: s:0:null_div:null_d:null_plus:null_cond

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:

   0 => 0
   true => 2
   false => 1
   null_div => 0
   null_d => 0
   null_ge => 0
   null_plus => 0
   null_cond => 0

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

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

   cond(z', z'', z1, z2) -{ 1 }-> 0 :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1
   cond(z', z'', z1, z2) -{ 0 }-> 0 :|: z2 = v3, v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, v3 >= 0, z' = v0
   cond(z', z'', z1, z2) -{ 1 }-> 1 + d(x, 1 + y, plus(1 + y, z)) :|: z1 = y, z >= 0, z' = 2, z2 = z, x >= 0, y >= 0, z'' = x
   d(z', z'', z1) -{ 1 }-> cond(ge(x, z), x, y, z) :|: z1 = z, z >= 0, z' = x, x >= 0, y >= 0, z'' = 1 + y
   d(z', z'', z1) -{ 0 }-> 0 :|: v0 >= 0, z1 = v2, v1 >= 0, z'' = v1, v2 >= 0, z' = v0
   div(z', z'') -{ 1 }-> d(x, 1 + y, 0) :|: z' = x, x >= 0, y >= 0, z'' = 1 + y
   div(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   ge(z', z'') -{ 1 }-> ge(u, v) :|: v >= 0, z' = 1 + u, z'' = 1 + v, u >= 0
   ge(z', z'') -{ 1 }-> 2 :|: z'' = 0, z' = u, u >= 0
   ge(z', z'') -{ 1 }-> 1 :|: v >= 0, z'' = 1 + v, z' = 0
   ge(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   plus(z', z'') -{ 1 }-> n :|: z'' = 0, n >= 0, z' = n
   plus(z', z'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0
   plus(z', z'') -{ 1 }-> 1 + plus(n, m) :|: n >= 0, z'' = 1 + m, z' = n, m >= 0

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:
   div(z0, s(z1)) -> d(z0, s(z1), 0)
   d(z0, s(z1), z2) -> cond(ge(z0, z2), z0, z1, z2)
   cond(true, z0, z1, z2) -> s(d(z0, s(z1), plus(s(z1), z2)))
   cond(false, z0, z1, z2) -> 0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
Tuples:
   DIV(z0, s(z1)) -> c(D(z0, s(z1), 0))
   D(z0, s(z1), z2) -> c1(COND(ge(z0, z2), z0, z1, z2), GE(z0, z2))
   COND(true, z0, z1, z2) -> c2(D(z0, s(z1), plus(s(z1), z2)), PLUS(s(z1), z2))
   COND(false, z0, z1, z2) -> c3
   GE(z0, 0) -> c4
   GE(0, s(z0)) -> c5
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   PLUS(z0, 0) -> c7
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
S tuples:
   DIV(z0, s(z1)) -> c(D(z0, s(z1), 0))
   D(z0, s(z1), z2) -> c1(COND(ge(z0, z2), z0, z1, z2), GE(z0, z2))
   COND(true, z0, z1, z2) -> c2(D(z0, s(z1), plus(s(z1), z2)), PLUS(s(z1), z2))
   COND(false, z0, z1, z2) -> c3
   GE(z0, 0) -> c4
   GE(0, s(z0)) -> c5
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   PLUS(z0, 0) -> c7
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
K tuples:none
Defined Rule Symbols:   div_2, d_3, cond_4, ge_2, plus_2

Defined Pair Symbols:   DIV_2, D_3, COND_4, GE_2, PLUS_2

Compound Symbols:   c_1, c1_2, c2_2, c3, c4, c5, c6_1, c7, c8_1


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

(43) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 1 leading nodes:
   DIV(z0, s(z1)) -> c(D(z0, s(z1), 0))
Removed 4 trailing nodes:
   PLUS(z0, 0) -> c7
   GE(z0, 0) -> c4
   GE(0, s(z0)) -> c5
   COND(false, z0, z1, z2) -> c3

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

(44)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   div(z0, s(z1)) -> d(z0, s(z1), 0)
   d(z0, s(z1), z2) -> cond(ge(z0, z2), z0, z1, z2)
   cond(true, z0, z1, z2) -> s(d(z0, s(z1), plus(s(z1), z2)))
   cond(false, z0, z1, z2) -> 0
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
Tuples:
   D(z0, s(z1), z2) -> c1(COND(ge(z0, z2), z0, z1, z2), GE(z0, z2))
   COND(true, z0, z1, z2) -> c2(D(z0, s(z1), plus(s(z1), z2)), PLUS(s(z1), z2))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
S tuples:
   D(z0, s(z1), z2) -> c1(COND(ge(z0, z2), z0, z1, z2), GE(z0, z2))
   COND(true, z0, z1, z2) -> c2(D(z0, s(z1), plus(s(z1), z2)), PLUS(s(z1), z2))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
K tuples:none
Defined Rule Symbols:   div_2, d_3, cond_4, ge_2, plus_2

Defined Pair Symbols:   D_3, COND_4, GE_2, PLUS_2

Compound Symbols:   c1_2, c2_2, c6_1, c8_1


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

(45) CdtUsableRulesProof (BOTH BOUNDS(ID, ID))
The following rules are not usable and were removed:
   div(z0, s(z1)) -> d(z0, s(z1), 0)
   d(z0, s(z1), z2) -> cond(ge(z0, z2), z0, z1, z2)
   cond(true, z0, z1, z2) -> s(d(z0, s(z1), plus(s(z1), z2)))
   cond(false, z0, z1, z2) -> 0

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

(46)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
Tuples:
   D(z0, s(z1), z2) -> c1(COND(ge(z0, z2), z0, z1, z2), GE(z0, z2))
   COND(true, z0, z1, z2) -> c2(D(z0, s(z1), plus(s(z1), z2)), PLUS(s(z1), z2))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
S tuples:
   D(z0, s(z1), z2) -> c1(COND(ge(z0, z2), z0, z1, z2), GE(z0, z2))
   COND(true, z0, z1, z2) -> c2(D(z0, s(z1), plus(s(z1), z2)), PLUS(s(z1), z2))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
K tuples:none
Defined Rule Symbols:   ge_2, plus_2

Defined Pair Symbols:   D_3, COND_4, GE_2, PLUS_2

Compound Symbols:   c1_2, c2_2, c6_1, c8_1


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

(47) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace D(z0, s(z1), z2) -> c1(COND(ge(z0, z2), z0, z1, z2), GE(z0, z2)) by 
   D(z0, s(x1), 0) -> c1(COND(true, z0, x1, 0), GE(z0, 0))
   D(0, s(x1), s(z0)) -> c1(COND(false, 0, x1, s(z0)), GE(0, s(z0)))
   D(s(z0), s(x1), s(z1)) -> c1(COND(ge(z0, z1), s(z0), x1, s(z1)), GE(s(z0), s(z1)))

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

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
Tuples:
   COND(true, z0, z1, z2) -> c2(D(z0, s(z1), plus(s(z1), z2)), PLUS(s(z1), z2))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
   D(z0, s(x1), 0) -> c1(COND(true, z0, x1, 0), GE(z0, 0))
   D(0, s(x1), s(z0)) -> c1(COND(false, 0, x1, s(z0)), GE(0, s(z0)))
   D(s(z0), s(x1), s(z1)) -> c1(COND(ge(z0, z1), s(z0), x1, s(z1)), GE(s(z0), s(z1)))
S tuples:
   COND(true, z0, z1, z2) -> c2(D(z0, s(z1), plus(s(z1), z2)), PLUS(s(z1), z2))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
   D(z0, s(x1), 0) -> c1(COND(true, z0, x1, 0), GE(z0, 0))
   D(0, s(x1), s(z0)) -> c1(COND(false, 0, x1, s(z0)), GE(0, s(z0)))
   D(s(z0), s(x1), s(z1)) -> c1(COND(ge(z0, z1), s(z0), x1, s(z1)), GE(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   ge_2, plus_2

Defined Pair Symbols:   COND_4, GE_2, PLUS_2, D_3

Compound Symbols:   c2_2, c6_1, c8_1, c1_2


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

(49) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 1 trailing nodes:
   D(0, s(x1), s(z0)) -> c1(COND(false, 0, x1, s(z0)), GE(0, s(z0)))

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

(50)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
Tuples:
   COND(true, z0, z1, z2) -> c2(D(z0, s(z1), plus(s(z1), z2)), PLUS(s(z1), z2))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
   D(z0, s(x1), 0) -> c1(COND(true, z0, x1, 0), GE(z0, 0))
   D(s(z0), s(x1), s(z1)) -> c1(COND(ge(z0, z1), s(z0), x1, s(z1)), GE(s(z0), s(z1)))
S tuples:
   COND(true, z0, z1, z2) -> c2(D(z0, s(z1), plus(s(z1), z2)), PLUS(s(z1), z2))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
   D(z0, s(x1), 0) -> c1(COND(true, z0, x1, 0), GE(z0, 0))
   D(s(z0), s(x1), s(z1)) -> c1(COND(ge(z0, z1), s(z0), x1, s(z1)), GE(s(z0), s(z1)))
K tuples:none
Defined Rule Symbols:   ge_2, plus_2

Defined Pair Symbols:   COND_4, GE_2, PLUS_2, D_3

Compound Symbols:   c2_2, c6_1, c8_1, c1_2


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

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

(52)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
Tuples:
   COND(true, z0, z1, z2) -> c2(D(z0, s(z1), plus(s(z1), z2)), PLUS(s(z1), z2))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
   D(s(z0), s(x1), s(z1)) -> c1(COND(ge(z0, z1), s(z0), x1, s(z1)), GE(s(z0), s(z1)))
   D(z0, s(x1), 0) -> c1(COND(true, z0, x1, 0))
S tuples:
   COND(true, z0, z1, z2) -> c2(D(z0, s(z1), plus(s(z1), z2)), PLUS(s(z1), z2))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
   D(s(z0), s(x1), s(z1)) -> c1(COND(ge(z0, z1), s(z0), x1, s(z1)), GE(s(z0), s(z1)))
   D(z0, s(x1), 0) -> c1(COND(true, z0, x1, 0))
K tuples:none
Defined Rule Symbols:   ge_2, plus_2

Defined Pair Symbols:   COND_4, GE_2, PLUS_2, D_3

Compound Symbols:   c2_2, c6_1, c8_1, c1_2, c1_1


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

(53) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   D(z0, s(x1), 0) -> c1(COND(true, z0, x1, 0))
We considered the (Usable) Rules:
   plus(z0, s(z1)) -> s(plus(z0, z1))
   plus(z0, 0) -> z0
And the Tuples:
   COND(true, z0, z1, z2) -> c2(D(z0, s(z1), plus(s(z1), z2)), PLUS(s(z1), z2))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
   D(s(z0), s(x1), s(z1)) -> c1(COND(ge(z0, z1), s(z0), x1, s(z1)), GE(s(z0), s(z1)))
   D(z0, s(x1), 0) -> c1(COND(true, z0, x1, 0))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [3]
   POL(COND(x_1, x_2, x_3, x_4)) = x_4
   POL(D(x_1, x_2, x_3)) = [3]x_2 + [2]x_3
   POL(GE(x_1, x_2)) = 0
   POL(PLUS(x_1, x_2)) = [3]x_1
   POL(c1(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(false) = [3]
   POL(ge(x_1, x_2)) = [3] + [3]x_1
   POL(plus(x_1, x_2)) = x_1
   POL(s(x_1)) = 0
   POL(true) = [3]

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
Tuples:
   COND(true, z0, z1, z2) -> c2(D(z0, s(z1), plus(s(z1), z2)), PLUS(s(z1), z2))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
   D(s(z0), s(x1), s(z1)) -> c1(COND(ge(z0, z1), s(z0), x1, s(z1)), GE(s(z0), s(z1)))
   D(z0, s(x1), 0) -> c1(COND(true, z0, x1, 0))
S tuples:
   COND(true, z0, z1, z2) -> c2(D(z0, s(z1), plus(s(z1), z2)), PLUS(s(z1), z2))
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
   D(s(z0), s(x1), s(z1)) -> c1(COND(ge(z0, z1), s(z0), x1, s(z1)), GE(s(z0), s(z1)))
K tuples:
   D(z0, s(x1), 0) -> c1(COND(true, z0, x1, 0))
Defined Rule Symbols:   ge_2, plus_2

Defined Pair Symbols:   COND_4, GE_2, PLUS_2, D_3

Compound Symbols:   c2_2, c6_1, c8_1, c1_2, c1_1


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

(55) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace COND(true, z0, z1, z2) -> c2(D(z0, s(z1), plus(s(z1), z2)), PLUS(s(z1), z2)) by 
   COND(true, x0, x1, 0) -> c2(D(x0, s(x1), s(x1)), PLUS(s(x1), 0))
   COND(true, x0, x1, s(z1)) -> c2(D(x0, s(x1), s(plus(s(x1), z1))), PLUS(s(x1), s(z1)))

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
Tuples:
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
   D(s(z0), s(x1), s(z1)) -> c1(COND(ge(z0, z1), s(z0), x1, s(z1)), GE(s(z0), s(z1)))
   D(z0, s(x1), 0) -> c1(COND(true, z0, x1, 0))
   COND(true, x0, x1, 0) -> c2(D(x0, s(x1), s(x1)), PLUS(s(x1), 0))
   COND(true, x0, x1, s(z1)) -> c2(D(x0, s(x1), s(plus(s(x1), z1))), PLUS(s(x1), s(z1)))
S tuples:
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
   D(s(z0), s(x1), s(z1)) -> c1(COND(ge(z0, z1), s(z0), x1, s(z1)), GE(s(z0), s(z1)))
   COND(true, x0, x1, 0) -> c2(D(x0, s(x1), s(x1)), PLUS(s(x1), 0))
   COND(true, x0, x1, s(z1)) -> c2(D(x0, s(x1), s(plus(s(x1), z1))), PLUS(s(x1), s(z1)))
K tuples:
   D(z0, s(x1), 0) -> c1(COND(true, z0, x1, 0))
Defined Rule Symbols:   ge_2, plus_2

Defined Pair Symbols:   GE_2, PLUS_2, D_3, COND_4

Compound Symbols:   c6_1, c8_1, c1_2, c1_1, c2_2


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

(57) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 2 leading nodes:
   D(z0, s(x1), 0) -> c1(COND(true, z0, x1, 0))
   COND(true, x0, x1, 0) -> c2(D(x0, s(x1), s(x1)), PLUS(s(x1), 0))

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
Tuples:
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
   D(s(z0), s(x1), s(z1)) -> c1(COND(ge(z0, z1), s(z0), x1, s(z1)), GE(s(z0), s(z1)))
   COND(true, x0, x1, s(z1)) -> c2(D(x0, s(x1), s(plus(s(x1), z1))), PLUS(s(x1), s(z1)))
S tuples:
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
   D(s(z0), s(x1), s(z1)) -> c1(COND(ge(z0, z1), s(z0), x1, s(z1)), GE(s(z0), s(z1)))
   COND(true, x0, x1, s(z1)) -> c2(D(x0, s(x1), s(plus(s(x1), z1))), PLUS(s(x1), s(z1)))
K tuples:none
Defined Rule Symbols:   ge_2, plus_2

Defined Pair Symbols:   GE_2, PLUS_2, D_3, COND_4

Compound Symbols:   c6_1, c8_1, c1_2, c2_2


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

(59) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace D(s(z0), s(x1), s(z1)) -> c1(COND(ge(z0, z1), s(z0), x1, s(z1)), GE(s(z0), s(z1))) by 
   D(s(z0), s(x1), s(0)) -> c1(COND(true, s(z0), x1, s(0)), GE(s(z0), s(0)))
   D(s(0), s(x1), s(s(z0))) -> c1(COND(false, s(0), x1, s(s(z0))), GE(s(0), s(s(z0))))
   D(s(s(z0)), s(x1), s(s(z1))) -> c1(COND(ge(z0, z1), s(s(z0)), x1, s(s(z1))), GE(s(s(z0)), s(s(z1))))
   D(s(x0), s(x1), s(x2)) -> c1(GE(s(x0), s(x2)))

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
Tuples:
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
   COND(true, x0, x1, s(z1)) -> c2(D(x0, s(x1), s(plus(s(x1), z1))), PLUS(s(x1), s(z1)))
   D(s(z0), s(x1), s(0)) -> c1(COND(true, s(z0), x1, s(0)), GE(s(z0), s(0)))
   D(s(0), s(x1), s(s(z0))) -> c1(COND(false, s(0), x1, s(s(z0))), GE(s(0), s(s(z0))))
   D(s(s(z0)), s(x1), s(s(z1))) -> c1(COND(ge(z0, z1), s(s(z0)), x1, s(s(z1))), GE(s(s(z0)), s(s(z1))))
   D(s(x0), s(x1), s(x2)) -> c1(GE(s(x0), s(x2)))
S tuples:
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
   COND(true, x0, x1, s(z1)) -> c2(D(x0, s(x1), s(plus(s(x1), z1))), PLUS(s(x1), s(z1)))
   D(s(z0), s(x1), s(0)) -> c1(COND(true, s(z0), x1, s(0)), GE(s(z0), s(0)))
   D(s(0), s(x1), s(s(z0))) -> c1(COND(false, s(0), x1, s(s(z0))), GE(s(0), s(s(z0))))
   D(s(s(z0)), s(x1), s(s(z1))) -> c1(COND(ge(z0, z1), s(s(z0)), x1, s(s(z1))), GE(s(s(z0)), s(s(z1))))
   D(s(x0), s(x1), s(x2)) -> c1(GE(s(x0), s(x2)))
K tuples:none
Defined Rule Symbols:   ge_2, plus_2

Defined Pair Symbols:   GE_2, PLUS_2, COND_4, D_3

Compound Symbols:   c6_1, c8_1, c2_2, c1_2, c1_1


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

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
Tuples:
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
   COND(true, x0, x1, s(z1)) -> c2(D(x0, s(x1), s(plus(s(x1), z1))), PLUS(s(x1), s(z1)))
   D(s(z0), s(x1), s(0)) -> c1(COND(true, s(z0), x1, s(0)), GE(s(z0), s(0)))
   D(s(s(z0)), s(x1), s(s(z1))) -> c1(COND(ge(z0, z1), s(s(z0)), x1, s(s(z1))), GE(s(s(z0)), s(s(z1))))
   D(s(x0), s(x1), s(x2)) -> c1(GE(s(x0), s(x2)))
   D(s(0), s(x1), s(s(z0))) -> c1(GE(s(0), s(s(z0))))
S tuples:
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
   COND(true, x0, x1, s(z1)) -> c2(D(x0, s(x1), s(plus(s(x1), z1))), PLUS(s(x1), s(z1)))
   D(s(z0), s(x1), s(0)) -> c1(COND(true, s(z0), x1, s(0)), GE(s(z0), s(0)))
   D(s(s(z0)), s(x1), s(s(z1))) -> c1(COND(ge(z0, z1), s(s(z0)), x1, s(s(z1))), GE(s(s(z0)), s(s(z1))))
   D(s(x0), s(x1), s(x2)) -> c1(GE(s(x0), s(x2)))
   D(s(0), s(x1), s(s(z0))) -> c1(GE(s(0), s(s(z0))))
K tuples:none
Defined Rule Symbols:   ge_2, plus_2

Defined Pair Symbols:   GE_2, PLUS_2, COND_4, D_3

Compound Symbols:   c6_1, c8_1, c2_2, c1_2, c1_1


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

(63) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   D(s(x0), s(x1), s(x2)) -> c1(GE(s(x0), s(x2)))
   D(s(0), s(x1), s(s(z0))) -> c1(GE(s(0), s(s(z0))))
We considered the (Usable) Rules:none
And the Tuples:
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
   COND(true, x0, x1, s(z1)) -> c2(D(x0, s(x1), s(plus(s(x1), z1))), PLUS(s(x1), s(z1)))
   D(s(z0), s(x1), s(0)) -> c1(COND(true, s(z0), x1, s(0)), GE(s(z0), s(0)))
   D(s(s(z0)), s(x1), s(s(z1))) -> c1(COND(ge(z0, z1), s(s(z0)), x1, s(s(z1))), GE(s(s(z0)), s(s(z1))))
   D(s(x0), s(x1), s(x2)) -> c1(GE(s(x0), s(x2)))
   D(s(0), s(x1), s(s(z0))) -> c1(GE(s(0), s(s(z0))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(COND(x_1, x_2, x_3, x_4)) = x_2
   POL(D(x_1, x_2, x_3)) = x_1
   POL(GE(x_1, x_2)) = 0
   POL(PLUS(x_1, x_2)) = 0
   POL(c1(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(false) = [1]
   POL(ge(x_1, x_2)) = [1]
   POL(plus(x_1, x_2)) = [1] + x_1 + x_2
   POL(s(x_1)) = [1]
   POL(true) = [1]

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
Tuples:
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
   COND(true, x0, x1, s(z1)) -> c2(D(x0, s(x1), s(plus(s(x1), z1))), PLUS(s(x1), s(z1)))
   D(s(z0), s(x1), s(0)) -> c1(COND(true, s(z0), x1, s(0)), GE(s(z0), s(0)))
   D(s(s(z0)), s(x1), s(s(z1))) -> c1(COND(ge(z0, z1), s(s(z0)), x1, s(s(z1))), GE(s(s(z0)), s(s(z1))))
   D(s(x0), s(x1), s(x2)) -> c1(GE(s(x0), s(x2)))
   D(s(0), s(x1), s(s(z0))) -> c1(GE(s(0), s(s(z0))))
S tuples:
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
   COND(true, x0, x1, s(z1)) -> c2(D(x0, s(x1), s(plus(s(x1), z1))), PLUS(s(x1), s(z1)))
   D(s(z0), s(x1), s(0)) -> c1(COND(true, s(z0), x1, s(0)), GE(s(z0), s(0)))
   D(s(s(z0)), s(x1), s(s(z1))) -> c1(COND(ge(z0, z1), s(s(z0)), x1, s(s(z1))), GE(s(s(z0)), s(s(z1))))
K tuples:
   D(s(x0), s(x1), s(x2)) -> c1(GE(s(x0), s(x2)))
   D(s(0), s(x1), s(s(z0))) -> c1(GE(s(0), s(s(z0))))
Defined Rule Symbols:   ge_2, plus_2

Defined Pair Symbols:   GE_2, PLUS_2, COND_4, D_3

Compound Symbols:   c6_1, c8_1, c2_2, c1_2, c1_1


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

(65) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace COND(true, x0, x1, s(z1)) -> c2(D(x0, s(x1), s(plus(s(x1), z1))), PLUS(s(x1), s(z1))) by 
   COND(true, s(x0), x1, s(0)) -> c2(D(s(x0), s(x1), s(plus(s(x1), 0))), PLUS(s(x1), s(0)))
   COND(true, s(s(x0)), x1, s(s(x2))) -> c2(D(s(s(x0)), s(x1), s(plus(s(x1), s(x2)))), PLUS(s(x1), s(s(x2))))

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
Tuples:
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
   D(s(z0), s(x1), s(0)) -> c1(COND(true, s(z0), x1, s(0)), GE(s(z0), s(0)))
   D(s(s(z0)), s(x1), s(s(z1))) -> c1(COND(ge(z0, z1), s(s(z0)), x1, s(s(z1))), GE(s(s(z0)), s(s(z1))))
   D(s(x0), s(x1), s(x2)) -> c1(GE(s(x0), s(x2)))
   D(s(0), s(x1), s(s(z0))) -> c1(GE(s(0), s(s(z0))))
   COND(true, s(x0), x1, s(0)) -> c2(D(s(x0), s(x1), s(plus(s(x1), 0))), PLUS(s(x1), s(0)))
   COND(true, s(s(x0)), x1, s(s(x2))) -> c2(D(s(s(x0)), s(x1), s(plus(s(x1), s(x2)))), PLUS(s(x1), s(s(x2))))
S tuples:
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
   D(s(z0), s(x1), s(0)) -> c1(COND(true, s(z0), x1, s(0)), GE(s(z0), s(0)))
   D(s(s(z0)), s(x1), s(s(z1))) -> c1(COND(ge(z0, z1), s(s(z0)), x1, s(s(z1))), GE(s(s(z0)), s(s(z1))))
   COND(true, s(x0), x1, s(0)) -> c2(D(s(x0), s(x1), s(plus(s(x1), 0))), PLUS(s(x1), s(0)))
   COND(true, s(s(x0)), x1, s(s(x2))) -> c2(D(s(s(x0)), s(x1), s(plus(s(x1), s(x2)))), PLUS(s(x1), s(s(x2))))
K tuples:
   D(s(x0), s(x1), s(x2)) -> c1(GE(s(x0), s(x2)))
   D(s(0), s(x1), s(s(z0))) -> c1(GE(s(0), s(s(z0))))
Defined Rule Symbols:   ge_2, plus_2

Defined Pair Symbols:   GE_2, PLUS_2, D_3, COND_4

Compound Symbols:   c6_1, c8_1, c1_2, c1_1, c2_2


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

(67) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace COND(true, s(x0), x1, s(0)) -> c2(D(s(x0), s(x1), s(plus(s(x1), 0))), PLUS(s(x1), s(0))) by 
   COND(true, s(x0), x1, s(0)) -> c2(D(s(x0), s(x1), s(s(x1))), PLUS(s(x1), s(0)))

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
Tuples:
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
   D(s(z0), s(x1), s(0)) -> c1(COND(true, s(z0), x1, s(0)), GE(s(z0), s(0)))
   D(s(s(z0)), s(x1), s(s(z1))) -> c1(COND(ge(z0, z1), s(s(z0)), x1, s(s(z1))), GE(s(s(z0)), s(s(z1))))
   D(s(x0), s(x1), s(x2)) -> c1(GE(s(x0), s(x2)))
   D(s(0), s(x1), s(s(z0))) -> c1(GE(s(0), s(s(z0))))
   COND(true, s(s(x0)), x1, s(s(x2))) -> c2(D(s(s(x0)), s(x1), s(plus(s(x1), s(x2)))), PLUS(s(x1), s(s(x2))))
   COND(true, s(x0), x1, s(0)) -> c2(D(s(x0), s(x1), s(s(x1))), PLUS(s(x1), s(0)))
S tuples:
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
   D(s(z0), s(x1), s(0)) -> c1(COND(true, s(z0), x1, s(0)), GE(s(z0), s(0)))
   D(s(s(z0)), s(x1), s(s(z1))) -> c1(COND(ge(z0, z1), s(s(z0)), x1, s(s(z1))), GE(s(s(z0)), s(s(z1))))
   COND(true, s(s(x0)), x1, s(s(x2))) -> c2(D(s(s(x0)), s(x1), s(plus(s(x1), s(x2)))), PLUS(s(x1), s(s(x2))))
   COND(true, s(x0), x1, s(0)) -> c2(D(s(x0), s(x1), s(s(x1))), PLUS(s(x1), s(0)))
K tuples:
   D(s(x0), s(x1), s(x2)) -> c1(GE(s(x0), s(x2)))
   D(s(0), s(x1), s(s(z0))) -> c1(GE(s(0), s(s(z0))))
Defined Rule Symbols:   ge_2, plus_2

Defined Pair Symbols:   GE_2, PLUS_2, D_3, COND_4

Compound Symbols:   c6_1, c8_1, c1_2, c1_1, c2_2


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

(69) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace COND(true, s(s(x0)), x1, s(s(x2))) -> c2(D(s(s(x0)), s(x1), s(plus(s(x1), s(x2)))), PLUS(s(x1), s(s(x2)))) by 
   COND(true, s(s(x0)), x1, s(s(z1))) -> c2(D(s(s(x0)), s(x1), s(s(plus(s(x1), z1)))), PLUS(s(x1), s(s(z1))))
   COND(true, s(s(x0)), x1, s(s(x2))) -> c2(PLUS(s(x1), s(s(x2))))

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
Tuples:
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
   D(s(z0), s(x1), s(0)) -> c1(COND(true, s(z0), x1, s(0)), GE(s(z0), s(0)))
   D(s(s(z0)), s(x1), s(s(z1))) -> c1(COND(ge(z0, z1), s(s(z0)), x1, s(s(z1))), GE(s(s(z0)), s(s(z1))))
   D(s(x0), s(x1), s(x2)) -> c1(GE(s(x0), s(x2)))
   D(s(0), s(x1), s(s(z0))) -> c1(GE(s(0), s(s(z0))))
   COND(true, s(x0), x1, s(0)) -> c2(D(s(x0), s(x1), s(s(x1))), PLUS(s(x1), s(0)))
   COND(true, s(s(x0)), x1, s(s(z1))) -> c2(D(s(s(x0)), s(x1), s(s(plus(s(x1), z1)))), PLUS(s(x1), s(s(z1))))
   COND(true, s(s(x0)), x1, s(s(x2))) -> c2(PLUS(s(x1), s(s(x2))))
S tuples:
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
   D(s(z0), s(x1), s(0)) -> c1(COND(true, s(z0), x1, s(0)), GE(s(z0), s(0)))
   D(s(s(z0)), s(x1), s(s(z1))) -> c1(COND(ge(z0, z1), s(s(z0)), x1, s(s(z1))), GE(s(s(z0)), s(s(z1))))
   COND(true, s(x0), x1, s(0)) -> c2(D(s(x0), s(x1), s(s(x1))), PLUS(s(x1), s(0)))
   COND(true, s(s(x0)), x1, s(s(z1))) -> c2(D(s(s(x0)), s(x1), s(s(plus(s(x1), z1)))), PLUS(s(x1), s(s(z1))))
   COND(true, s(s(x0)), x1, s(s(x2))) -> c2(PLUS(s(x1), s(s(x2))))
K tuples:
   D(s(x0), s(x1), s(x2)) -> c1(GE(s(x0), s(x2)))
   D(s(0), s(x1), s(s(z0))) -> c1(GE(s(0), s(s(z0))))
Defined Rule Symbols:   ge_2, plus_2

Defined Pair Symbols:   GE_2, PLUS_2, D_3, COND_4

Compound Symbols:   c6_1, c8_1, c1_2, c1_1, c2_2, c2_1


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

(71) CdtLeafRemovalProof (ComplexityIfPolyImplication)
Removed 3 leading nodes:
   D(s(z0), s(x1), s(0)) -> c1(COND(true, s(z0), x1, s(0)), GE(s(z0), s(0)))
   COND(true, s(x0), x1, s(0)) -> c2(D(s(x0), s(x1), s(s(x1))), PLUS(s(x1), s(0)))
   D(s(0), s(x1), s(s(z0))) -> c1(GE(s(0), s(s(z0))))

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
Tuples:
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
   D(s(s(z0)), s(x1), s(s(z1))) -> c1(COND(ge(z0, z1), s(s(z0)), x1, s(s(z1))), GE(s(s(z0)), s(s(z1))))
   D(s(x0), s(x1), s(x2)) -> c1(GE(s(x0), s(x2)))
   COND(true, s(s(x0)), x1, s(s(z1))) -> c2(D(s(s(x0)), s(x1), s(s(plus(s(x1), z1)))), PLUS(s(x1), s(s(z1))))
   COND(true, s(s(x0)), x1, s(s(x2))) -> c2(PLUS(s(x1), s(s(x2))))
S tuples:
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
   D(s(s(z0)), s(x1), s(s(z1))) -> c1(COND(ge(z0, z1), s(s(z0)), x1, s(s(z1))), GE(s(s(z0)), s(s(z1))))
   COND(true, s(s(x0)), x1, s(s(z1))) -> c2(D(s(s(x0)), s(x1), s(s(plus(s(x1), z1)))), PLUS(s(x1), s(s(z1))))
   COND(true, s(s(x0)), x1, s(s(x2))) -> c2(PLUS(s(x1), s(s(x2))))
K tuples:
   D(s(x0), s(x1), s(x2)) -> c1(GE(s(x0), s(x2)))
Defined Rule Symbols:   ge_2, plus_2

Defined Pair Symbols:   GE_2, PLUS_2, D_3, COND_4

Compound Symbols:   c6_1, c8_1, c1_2, c1_1, c2_2, c2_1


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

(73) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   COND(true, s(s(x0)), x1, s(s(x2))) -> c2(PLUS(s(x1), s(s(x2))))
We considered the (Usable) Rules:
   ge(s(z0), s(z1)) -> ge(z0, z1)
   ge(0, s(z0)) -> false
   ge(z0, 0) -> true
And the Tuples:
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
   D(s(s(z0)), s(x1), s(s(z1))) -> c1(COND(ge(z0, z1), s(s(z0)), x1, s(s(z1))), GE(s(s(z0)), s(s(z1))))
   D(s(x0), s(x1), s(x2)) -> c1(GE(s(x0), s(x2)))
   COND(true, s(s(x0)), x1, s(s(z1))) -> c2(D(s(s(x0)), s(x1), s(s(plus(s(x1), z1)))), PLUS(s(x1), s(s(z1))))
   COND(true, s(s(x0)), x1, s(s(x2))) -> c2(PLUS(s(x1), s(s(x2))))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(0) = [1]
   POL(COND(x_1, x_2, x_3, x_4)) = [1] + x_1 + x_2 + x_4
   POL(D(x_1, x_2, x_3)) = [1] + x_1 + x_2 + x_3
   POL(GE(x_1, x_2)) = 0
   POL(PLUS(x_1, x_2)) = 0
   POL(c1(x_1)) = x_1
   POL(c1(x_1, x_2)) = x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(c8(x_1)) = x_1
   POL(false) = [1]
   POL(ge(x_1, x_2)) = [1]
   POL(plus(x_1, x_2)) = [1] + x_1 + x_2
   POL(s(x_1)) = [1]
   POL(true) = [1]

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
Tuples:
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
   D(s(s(z0)), s(x1), s(s(z1))) -> c1(COND(ge(z0, z1), s(s(z0)), x1, s(s(z1))), GE(s(s(z0)), s(s(z1))))
   D(s(x0), s(x1), s(x2)) -> c1(GE(s(x0), s(x2)))
   COND(true, s(s(x0)), x1, s(s(z1))) -> c2(D(s(s(x0)), s(x1), s(s(plus(s(x1), z1)))), PLUS(s(x1), s(s(z1))))
   COND(true, s(s(x0)), x1, s(s(x2))) -> c2(PLUS(s(x1), s(s(x2))))
S tuples:
   GE(s(z0), s(z1)) -> c6(GE(z0, z1))
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
   D(s(s(z0)), s(x1), s(s(z1))) -> c1(COND(ge(z0, z1), s(s(z0)), x1, s(s(z1))), GE(s(s(z0)), s(s(z1))))
   COND(true, s(s(x0)), x1, s(s(z1))) -> c2(D(s(s(x0)), s(x1), s(s(plus(s(x1), z1)))), PLUS(s(x1), s(s(z1))))
K tuples:
   D(s(x0), s(x1), s(x2)) -> c1(GE(s(x0), s(x2)))
   COND(true, s(s(x0)), x1, s(s(x2))) -> c2(PLUS(s(x1), s(s(x2))))
Defined Rule Symbols:   ge_2, plus_2

Defined Pair Symbols:   GE_2, PLUS_2, D_3, COND_4

Compound Symbols:   c6_1, c8_1, c1_2, c1_1, c2_2, c2_1


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

(75) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace GE(s(z0), s(z1)) -> c6(GE(z0, z1)) by 
   GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1)))

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
Tuples:
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
   D(s(s(z0)), s(x1), s(s(z1))) -> c1(COND(ge(z0, z1), s(s(z0)), x1, s(s(z1))), GE(s(s(z0)), s(s(z1))))
   D(s(x0), s(x1), s(x2)) -> c1(GE(s(x0), s(x2)))
   COND(true, s(s(x0)), x1, s(s(z1))) -> c2(D(s(s(x0)), s(x1), s(s(plus(s(x1), z1)))), PLUS(s(x1), s(s(z1))))
   COND(true, s(s(x0)), x1, s(s(x2))) -> c2(PLUS(s(x1), s(s(x2))))
   GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1)))
S tuples:
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
   D(s(s(z0)), s(x1), s(s(z1))) -> c1(COND(ge(z0, z1), s(s(z0)), x1, s(s(z1))), GE(s(s(z0)), s(s(z1))))
   COND(true, s(s(x0)), x1, s(s(z1))) -> c2(D(s(s(x0)), s(x1), s(s(plus(s(x1), z1)))), PLUS(s(x1), s(s(z1))))
   GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1)))
K tuples:
   D(s(x0), s(x1), s(x2)) -> c1(GE(s(x0), s(x2)))
   COND(true, s(s(x0)), x1, s(s(x2))) -> c2(PLUS(s(x1), s(s(x2))))
Defined Rule Symbols:   ge_2, plus_2

Defined Pair Symbols:   PLUS_2, D_3, COND_4, GE_2

Compound Symbols:   c8_1, c1_2, c1_1, c2_2, c2_1, c6_1


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

(77) CdtInstantiationProof (BOTH BOUNDS(ID, ID))
Use instantiation to replace D(s(x0), s(x1), s(x2)) -> c1(GE(s(x0), s(x2))) by 
   D(s(s(x0)), s(x1), s(s(y0))) -> c1(GE(s(s(x0)), s(s(y0))))

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
Tuples:
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
   D(s(s(z0)), s(x1), s(s(z1))) -> c1(COND(ge(z0, z1), s(s(z0)), x1, s(s(z1))), GE(s(s(z0)), s(s(z1))))
   COND(true, s(s(x0)), x1, s(s(z1))) -> c2(D(s(s(x0)), s(x1), s(s(plus(s(x1), z1)))), PLUS(s(x1), s(s(z1))))
   COND(true, s(s(x0)), x1, s(s(x2))) -> c2(PLUS(s(x1), s(s(x2))))
   GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1)))
   D(s(s(x0)), s(x1), s(s(y0))) -> c1(GE(s(s(x0)), s(s(y0))))
S tuples:
   PLUS(z0, s(z1)) -> c8(PLUS(z0, z1))
   D(s(s(z0)), s(x1), s(s(z1))) -> c1(COND(ge(z0, z1), s(s(z0)), x1, s(s(z1))), GE(s(s(z0)), s(s(z1))))
   COND(true, s(s(x0)), x1, s(s(z1))) -> c2(D(s(s(x0)), s(x1), s(s(plus(s(x1), z1)))), PLUS(s(x1), s(s(z1))))
   GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1)))
K tuples:
   COND(true, s(s(x0)), x1, s(s(x2))) -> c2(PLUS(s(x1), s(s(x2))))
   D(s(s(x0)), s(x1), s(s(y0))) -> c1(GE(s(s(x0)), s(s(y0))))
Defined Rule Symbols:   ge_2, plus_2

Defined Pair Symbols:   PLUS_2, D_3, COND_4, GE_2

Compound Symbols:   c8_1, c1_2, c2_2, c2_1, c6_1, c1_1


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

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

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
Tuples:
   D(s(s(z0)), s(x1), s(s(z1))) -> c1(COND(ge(z0, z1), s(s(z0)), x1, s(s(z1))), GE(s(s(z0)), s(s(z1))))
   COND(true, s(s(x0)), x1, s(s(z1))) -> c2(D(s(s(x0)), s(x1), s(s(plus(s(x1), z1)))), PLUS(s(x1), s(s(z1))))
   COND(true, s(s(x0)), x1, s(s(x2))) -> c2(PLUS(s(x1), s(s(x2))))
   GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1)))
   D(s(s(x0)), s(x1), s(s(y0))) -> c1(GE(s(s(x0)), s(s(y0))))
   PLUS(z0, s(s(y1))) -> c8(PLUS(z0, s(y1)))
S tuples:
   D(s(s(z0)), s(x1), s(s(z1))) -> c1(COND(ge(z0, z1), s(s(z0)), x1, s(s(z1))), GE(s(s(z0)), s(s(z1))))
   COND(true, s(s(x0)), x1, s(s(z1))) -> c2(D(s(s(x0)), s(x1), s(s(plus(s(x1), z1)))), PLUS(s(x1), s(s(z1))))
   GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1)))
   PLUS(z0, s(s(y1))) -> c8(PLUS(z0, s(y1)))
K tuples:
   COND(true, s(s(x0)), x1, s(s(x2))) -> c2(PLUS(s(x1), s(s(x2))))
   D(s(s(x0)), s(x1), s(s(y0))) -> c1(GE(s(s(x0)), s(s(y0))))
Defined Rule Symbols:   ge_2, plus_2

Defined Pair Symbols:   D_3, COND_4, GE_2, PLUS_2

Compound Symbols:   c1_2, c2_2, c2_1, c6_1, c1_1, c8_1


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

(81) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace GE(s(s(y0)), s(s(y1))) -> c6(GE(s(y0), s(y1))) by 
   GE(s(s(s(y0))), s(s(s(y1)))) -> c6(GE(s(s(y0)), s(s(y1))))

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
Tuples:
   D(s(s(z0)), s(x1), s(s(z1))) -> c1(COND(ge(z0, z1), s(s(z0)), x1, s(s(z1))), GE(s(s(z0)), s(s(z1))))
   COND(true, s(s(x0)), x1, s(s(z1))) -> c2(D(s(s(x0)), s(x1), s(s(plus(s(x1), z1)))), PLUS(s(x1), s(s(z1))))
   COND(true, s(s(x0)), x1, s(s(x2))) -> c2(PLUS(s(x1), s(s(x2))))
   D(s(s(x0)), s(x1), s(s(y0))) -> c1(GE(s(s(x0)), s(s(y0))))
   PLUS(z0, s(s(y1))) -> c8(PLUS(z0, s(y1)))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c6(GE(s(s(y0)), s(s(y1))))
S tuples:
   D(s(s(z0)), s(x1), s(s(z1))) -> c1(COND(ge(z0, z1), s(s(z0)), x1, s(s(z1))), GE(s(s(z0)), s(s(z1))))
   COND(true, s(s(x0)), x1, s(s(z1))) -> c2(D(s(s(x0)), s(x1), s(s(plus(s(x1), z1)))), PLUS(s(x1), s(s(z1))))
   PLUS(z0, s(s(y1))) -> c8(PLUS(z0, s(y1)))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c6(GE(s(s(y0)), s(s(y1))))
K tuples:
   COND(true, s(s(x0)), x1, s(s(x2))) -> c2(PLUS(s(x1), s(s(x2))))
   D(s(s(x0)), s(x1), s(s(y0))) -> c1(GE(s(s(x0)), s(s(y0))))
Defined Rule Symbols:   ge_2, plus_2

Defined Pair Symbols:   D_3, COND_4, PLUS_2, GE_2

Compound Symbols:   c1_2, c2_2, c2_1, c1_1, c8_1, c6_1


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

(83) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace D(s(s(x0)), s(x1), s(s(y0))) -> c1(GE(s(s(x0)), s(s(y0)))) by 
   D(s(s(s(y0))), s(z1), s(s(s(y1)))) -> c1(GE(s(s(s(y0))), s(s(s(y1)))))

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
Tuples:
   D(s(s(z0)), s(x1), s(s(z1))) -> c1(COND(ge(z0, z1), s(s(z0)), x1, s(s(z1))), GE(s(s(z0)), s(s(z1))))
   COND(true, s(s(x0)), x1, s(s(z1))) -> c2(D(s(s(x0)), s(x1), s(s(plus(s(x1), z1)))), PLUS(s(x1), s(s(z1))))
   COND(true, s(s(x0)), x1, s(s(x2))) -> c2(PLUS(s(x1), s(s(x2))))
   PLUS(z0, s(s(y1))) -> c8(PLUS(z0, s(y1)))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c6(GE(s(s(y0)), s(s(y1))))
   D(s(s(s(y0))), s(z1), s(s(s(y1)))) -> c1(GE(s(s(s(y0))), s(s(s(y1)))))
S tuples:
   D(s(s(z0)), s(x1), s(s(z1))) -> c1(COND(ge(z0, z1), s(s(z0)), x1, s(s(z1))), GE(s(s(z0)), s(s(z1))))
   COND(true, s(s(x0)), x1, s(s(z1))) -> c2(D(s(s(x0)), s(x1), s(s(plus(s(x1), z1)))), PLUS(s(x1), s(s(z1))))
   PLUS(z0, s(s(y1))) -> c8(PLUS(z0, s(y1)))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c6(GE(s(s(y0)), s(s(y1))))
K tuples:
   COND(true, s(s(x0)), x1, s(s(x2))) -> c2(PLUS(s(x1), s(s(x2))))
   D(s(s(s(y0))), s(z1), s(s(s(y1)))) -> c1(GE(s(s(s(y0))), s(s(s(y1)))))
Defined Rule Symbols:   ge_2, plus_2

Defined Pair Symbols:   D_3, COND_4, PLUS_2, GE_2

Compound Symbols:   c1_2, c2_2, c2_1, c8_1, c6_1, c1_1


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

(85) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace PLUS(z0, s(s(y1))) -> c8(PLUS(z0, s(y1))) by 
   PLUS(z0, s(s(s(y1)))) -> c8(PLUS(z0, s(s(y1))))

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

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
Tuples:
   D(s(s(z0)), s(x1), s(s(z1))) -> c1(COND(ge(z0, z1), s(s(z0)), x1, s(s(z1))), GE(s(s(z0)), s(s(z1))))
   COND(true, s(s(x0)), x1, s(s(z1))) -> c2(D(s(s(x0)), s(x1), s(s(plus(s(x1), z1)))), PLUS(s(x1), s(s(z1))))
   COND(true, s(s(x0)), x1, s(s(x2))) -> c2(PLUS(s(x1), s(s(x2))))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c6(GE(s(s(y0)), s(s(y1))))
   D(s(s(s(y0))), s(z1), s(s(s(y1)))) -> c1(GE(s(s(s(y0))), s(s(s(y1)))))
   PLUS(z0, s(s(s(y1)))) -> c8(PLUS(z0, s(s(y1))))
S tuples:
   D(s(s(z0)), s(x1), s(s(z1))) -> c1(COND(ge(z0, z1), s(s(z0)), x1, s(s(z1))), GE(s(s(z0)), s(s(z1))))
   COND(true, s(s(x0)), x1, s(s(z1))) -> c2(D(s(s(x0)), s(x1), s(s(plus(s(x1), z1)))), PLUS(s(x1), s(s(z1))))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c6(GE(s(s(y0)), s(s(y1))))
   PLUS(z0, s(s(s(y1)))) -> c8(PLUS(z0, s(s(y1))))
K tuples:
   COND(true, s(s(x0)), x1, s(s(x2))) -> c2(PLUS(s(x1), s(s(x2))))
   D(s(s(s(y0))), s(z1), s(s(s(y1)))) -> c1(GE(s(s(s(y0))), s(s(s(y1)))))
Defined Rule Symbols:   ge_2, plus_2

Defined Pair Symbols:   D_3, COND_4, GE_2, PLUS_2

Compound Symbols:   c1_2, c2_2, c2_1, c6_1, c1_1, c8_1


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

(87) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace COND(true, s(s(x0)), x1, s(s(x2))) -> c2(PLUS(s(x1), s(s(x2)))) by 
   COND(true, s(s(z0)), z1, s(s(s(y1)))) -> c2(PLUS(s(z1), s(s(s(y1)))))

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   ge(z0, 0) -> true
   ge(0, s(z0)) -> false
   ge(s(z0), s(z1)) -> ge(z0, z1)
   plus(z0, 0) -> z0
   plus(z0, s(z1)) -> s(plus(z0, z1))
Tuples:
   D(s(s(z0)), s(x1), s(s(z1))) -> c1(COND(ge(z0, z1), s(s(z0)), x1, s(s(z1))), GE(s(s(z0)), s(s(z1))))
   COND(true, s(s(x0)), x1, s(s(z1))) -> c2(D(s(s(x0)), s(x1), s(s(plus(s(x1), z1)))), PLUS(s(x1), s(s(z1))))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c6(GE(s(s(y0)), s(s(y1))))
   D(s(s(s(y0))), s(z1), s(s(s(y1)))) -> c1(GE(s(s(s(y0))), s(s(s(y1)))))
   PLUS(z0, s(s(s(y1)))) -> c8(PLUS(z0, s(s(y1))))
   COND(true, s(s(z0)), z1, s(s(s(y1)))) -> c2(PLUS(s(z1), s(s(s(y1)))))
S tuples:
   D(s(s(z0)), s(x1), s(s(z1))) -> c1(COND(ge(z0, z1), s(s(z0)), x1, s(s(z1))), GE(s(s(z0)), s(s(z1))))
   COND(true, s(s(x0)), x1, s(s(z1))) -> c2(D(s(s(x0)), s(x1), s(s(plus(s(x1), z1)))), PLUS(s(x1), s(s(z1))))
   GE(s(s(s(y0))), s(s(s(y1)))) -> c6(GE(s(s(y0)), s(s(y1))))
   PLUS(z0, s(s(s(y1)))) -> c8(PLUS(z0, s(s(y1))))
K tuples:
   D(s(s(s(y0))), s(z1), s(s(s(y1)))) -> c1(GE(s(s(s(y0))), s(s(s(y1)))))
   COND(true, s(s(z0)), z1, s(s(s(y1)))) -> c2(PLUS(s(z1), s(s(s(y1)))))
Defined Rule Symbols:   ge_2, plus_2

Defined Pair Symbols:   D_3, COND_4, GE_2, PLUS_2

Compound Symbols:   c1_2, c2_2, c6_1, c1_1, c8_1, c2_1