KILLED
proof of input_BbcN10rF1P.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxRelTRS
(1) SInnermostTerminationProof [BOTH CONCRETE BOUNDS(ID, ID), 121 ms]
(2) CpxRelTRS
(3) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxRelTRS
(5) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
(6) CpxTRS
(7) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(8) CpxWeightedTrs
    (9) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxWeightedTrs
    (11) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxTypedWeightedTrs
    (13) CompletionProof [UPPER BOUND(ID), 0 ms]
    (14) CpxTypedWeightedCompleteTrs
        (15) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
        (16) CpxTypedWeightedCompleteTrs
        (17) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (18) CpxRNTS
        (19) InliningProof [UPPER BOUND(ID), 121 ms]
        (20) CpxRNTS
        (21) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
        (22) CpxRNTS
        (23) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
        (24) CpxRNTS
        (25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (26) CpxRNTS
        (27) IntTrsBoundProof [UPPER BOUND(ID), 190 ms]
        (28) CpxRNTS
        (29) IntTrsBoundProof [UPPER BOUND(ID), 9 ms]
        (30) CpxRNTS
        (31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (32) CpxRNTS
        (33) IntTrsBoundProof [UPPER BOUND(ID), 453 ms]
        (34) CpxRNTS
        (35) IntTrsBoundProof [UPPER BOUND(ID), 110 ms]
        (36) CpxRNTS
        (37) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (38) CpxRNTS
        (39) IntTrsBoundProof [UPPER BOUND(ID), 418 ms]
        (40) CpxRNTS
        (41) IntTrsBoundProof [UPPER BOUND(ID), 35 ms]
        (42) CpxRNTS
        (43) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
        (44) CpxRNTS
        (45) IntTrsBoundProof [UPPER BOUND(ID), 613 ms]
        (46) CpxRNTS
        (47) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
        (48) CpxRNTS
    (49) CompletionProof [UPPER BOUND(ID), 0 ms]
    (50) CpxTypedWeightedCompleteTrs
        (51) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
        (52) CpxRNTS
(53) CpxTrsToCdtProof [UPPER BOUND(ID), 0 ms]
(54) CdtProblem
    (55) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (56) CdtProblem
    (57) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CdtProblem
    (59) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 70 ms]
    (60) CdtProblem
    (61) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (62) CdtProblem
    (63) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (64) CdtProblem
    (65) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (66) CdtProblem
    (67) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (68) CdtProblem
    (69) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 18 ms]
    (70) CdtProblem
    (71) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (72) CdtProblem
    (73) CdtLeafRemovalProof [BOTH BOUNDS(ID, ID), 0 ms]
    (74) CdtProblem
    (75) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (76) CdtProblem
    (77) CdtNarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (78) CdtProblem
    (79) CdtRhsSimplificationProcessorProof [BOTH BOUNDS(ID, ID), 0 ms]
    (80) CdtProblem
    (81) CdtRuleRemovalProof [UPPER BOUND(ADD(n^1)), 19 ms]
    (82) CdtProblem
    (83) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (84) CdtProblem
    (85) CdtRewritingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (86) CdtProblem
    (87) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (88) CdtProblem
    (89) CdtForwardInstantiationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (90) CdtProblem


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   power(x', S(x)) -> mult(x', power(x', x))
   mult(x', S(x)) -> add0(x', mult(x', x))
   add0(x', S(x)) -> +(S(0), add0(x', x))
   power(x, 0) -> S(0)
   mult(x, 0) -> 0
   add0(x, 0) -> x

The (relative) TRS S consists of the following rules:

   +(x, S(0)) -> S(x)
   +(S(0), y) -> S(y)

Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(1) SInnermostTerminationProof (BOTH CONCRETE BOUNDS(ID, ID))
proved innermost termination of relative rules
----------------------------------------

(2)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   power(x', S(x)) -> mult(x', power(x', x))
   mult(x', S(x)) -> add0(x', mult(x', x))
   add0(x', S(x)) -> +(S(0), add0(x', x))
   power(x, 0) -> S(0)
   mult(x, 0) -> 0
   add0(x, 0) -> x

The (relative) TRS S consists of the following rules:

   +(x, S(0)) -> S(x)
   +(S(0), y) -> S(y)

Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(3) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(4)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxRelTRS could be proven to be BOUNDS(1, INF).


The TRS R consists of the following rules:

   power(x', S(x)) -> mult(x', power(x', x))
   mult(x', S(x)) -> add0(x', mult(x', x))
   add0(x', S(x)) -> +'(S(0'), add0(x', x))
   power(x, 0') -> S(0')
   mult(x, 0') -> 0'
   add0(x, 0') -> x

The (relative) TRS S consists of the following rules:

   +'(x, S(0')) -> S(x)
   +'(S(0'), y) -> S(y)

Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(5) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
----------------------------------------

(6)
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:

   power(x', S(x)) -> mult(x', power(x', x))
   mult(x', S(x)) -> add0(x', mult(x', x))
   add0(x', S(x)) -> +(S(0), add0(x', x))
   power(x, 0) -> S(0)
   mult(x, 0) -> 0
   add0(x, 0) -> x
   +(x, S(0)) -> S(x)
   +(S(0), y) -> S(y)

S is empty.
Rewrite Strategy: PARALLEL_INNERMOST
----------------------------------------

(7) RelTrsToWeightedTrsProof (UPPER BOUND(ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(8)
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:

   power(x', S(x)) -> mult(x', power(x', x)) [1]
   mult(x', S(x)) -> add0(x', mult(x', x)) [1]
   add0(x', S(x)) -> +(S(0), add0(x', x)) [1]
   power(x, 0) -> S(0) [1]
   mult(x, 0) -> 0 [1]
   add0(x, 0) -> x [1]
   +(x, S(0)) -> S(x) [0]
   +(S(0), y) -> S(y) [0]

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

(9) CpxWeightedTrsRenamingProof (BOTH BOUNDS(ID, ID))
Renamed defined symbols to avoid conflicts with arithmetic symbols:

+ => plus

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

(10)
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:

   power(x', S(x)) -> mult(x', power(x', x)) [1]
   mult(x', S(x)) -> add0(x', mult(x', x)) [1]
   add0(x', S(x)) -> plus(S(0), add0(x', x)) [1]
   power(x, 0) -> S(0) [1]
   mult(x, 0) -> 0 [1]
   add0(x, 0) -> x [1]
   plus(x, S(0)) -> S(x) [0]
   plus(S(0), y) -> S(y) [0]

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

(11) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Infered types.
----------------------------------------

(12)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   power(x', S(x)) -> mult(x', power(x', x)) [1]
   mult(x', S(x)) -> add0(x', mult(x', x)) [1]
   add0(x', S(x)) -> plus(S(0), add0(x', x)) [1]
   power(x, 0) -> S(0) [1]
   mult(x, 0) -> 0 [1]
   add0(x, 0) -> x [1]
   plus(x, S(0)) -> S(x) [0]
   plus(S(0), y) -> S(y) [0]

The TRS has the following type information:
power :: S:0 -> S:0 -> S:0
S :: S:0 -> S:0
mult :: S:0 -> S:0 -> S:0
add0 :: S:0 -> S:0 -> S:0
plus :: S:0 -> S:0 -> S:0
0 :: S:0

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

(13) 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:

   power_2
   mult_2
   add0_2
   plus_2

Due to the following rules being added:

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

And the following fresh constants: none

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

(14)
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:

   power(x', S(x)) -> mult(x', power(x', x)) [1]
   mult(x', S(x)) -> add0(x', mult(x', x)) [1]
   add0(x', S(x)) -> plus(S(0), add0(x', x)) [1]
   power(x, 0) -> S(0) [1]
   mult(x, 0) -> 0 [1]
   add0(x, 0) -> x [1]
   plus(x, S(0)) -> S(x) [0]
   plus(S(0), y) -> S(y) [0]
   plus(v0, v1) -> 0 [0]

The TRS has the following type information:
power :: S:0 -> S:0 -> S:0
S :: S:0 -> S:0
mult :: S:0 -> S:0 -> S:0
add0 :: S:0 -> S:0 -> S:0
plus :: S:0 -> S:0 -> S:0
0 :: S:0

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

(15) NarrowingProof (BOTH BOUNDS(ID, ID))
Narrowed the inner basic terms of all right-hand sides by a single narrowing step.
----------------------------------------

(16)
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:

   power(x', S(S(x1))) -> mult(x', mult(x', power(x', x1))) [2]
   power(x', S(0)) -> mult(x', S(0)) [2]
   mult(x', S(S(x2))) -> add0(x', add0(x', mult(x', x2))) [2]
   mult(x', S(0)) -> add0(x', 0) [2]
   add0(x', S(S(x3))) -> plus(S(0), plus(S(0), add0(x', x3))) [2]
   add0(x', S(0)) -> plus(S(0), x') [2]
   power(x, 0) -> S(0) [1]
   mult(x, 0) -> 0 [1]
   add0(x, 0) -> x [1]
   plus(x, S(0)) -> S(x) [0]
   plus(S(0), y) -> S(y) [0]
   plus(v0, v1) -> 0 [0]

The TRS has the following type information:
power :: S:0 -> S:0 -> S:0
S :: S:0 -> S:0
mult :: S:0 -> S:0 -> S:0
add0 :: S:0 -> S:0 -> S:0
plus :: S:0 -> S:0 -> S:0
0 :: S:0

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

(17) 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

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

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

   add0(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   add0(z, z') -{ 2 }-> plus(1 + 0, x') :|: x' >= 0, z' = 1 + 0, z = x'
   add0(z, z') -{ 2 }-> plus(1 + 0, plus(1 + 0, add0(x', x3))) :|: z' = 1 + (1 + x3), x' >= 0, x3 >= 0, z = x'
   mult(z, z') -{ 2 }-> add0(x', add0(x', mult(x', x2))) :|: z' = 1 + (1 + x2), x' >= 0, x2 >= 0, z = x'
   mult(z, z') -{ 2 }-> add0(x', 0) :|: x' >= 0, z' = 1 + 0, z = x'
   mult(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   plus(z, z') -{ 0 }-> 1 + x :|: x >= 0, z' = 1 + 0, z = x
   plus(z, z') -{ 0 }-> 1 + y :|: z = 1 + 0, y >= 0, z' = y
   power(z, z') -{ 2 }-> mult(x', mult(x', power(x', x1))) :|: x1 >= 0, z' = 1 + (1 + x1), x' >= 0, z = x'
   power(z, z') -{ 2 }-> mult(x', 1 + 0) :|: x' >= 0, z' = 1 + 0, z = x'
   power(z, z') -{ 1 }-> 1 + 0 :|: x >= 0, z = x, z' = 0


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

(19) InliningProof (UPPER BOUND(ID))
Inlined the following terminating rules on right-hand sides where appropriate:

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

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

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

   add0(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   add0(z, z') -{ 2 }-> plus(1 + 0, plus(1 + 0, add0(x', x3))) :|: z' = 1 + (1 + x3), x' >= 0, x3 >= 0, z = x'
   add0(z, z') -{ 2 }-> 0 :|: x' >= 0, z' = 1 + 0, z = x', v0 >= 0, v1 >= 0, 1 + 0 = v0, x' = v1
   add0(z, z') -{ 2 }-> 1 + x :|: x' >= 0, z' = 1 + 0, z = x', x >= 0, x' = 1 + 0, 1 + 0 = x
   add0(z, z') -{ 2 }-> 1 + y :|: x' >= 0, z' = 1 + 0, z = x', 1 + 0 = 1 + 0, y >= 0, x' = y
   mult(z, z') -{ 2 }-> add0(x', add0(x', mult(x', x2))) :|: z' = 1 + (1 + x2), x' >= 0, x2 >= 0, z = x'
   mult(z, z') -{ 2 }-> add0(x', 0) :|: x' >= 0, z' = 1 + 0, z = x'
   mult(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   plus(z, z') -{ 0 }-> 1 + x :|: x >= 0, z' = 1 + 0, z = x
   plus(z, z') -{ 0 }-> 1 + y :|: z = 1 + 0, y >= 0, z' = y
   power(z, z') -{ 2 }-> mult(x', mult(x', power(x', x1))) :|: x1 >= 0, z' = 1 + (1 + x1), x' >= 0, z = x'
   power(z, z') -{ 2 }-> mult(x', 1 + 0) :|: x' >= 0, z' = 1 + 0, z = x'
   power(z, z') -{ 1 }-> 1 + 0 :|: x >= 0, z = x, z' = 0


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

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

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

   add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   add0(z, z') -{ 2 }-> plus(1 + 0, plus(1 + 0, add0(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   add0(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1 + 0, v0 >= 0, 1 + 0 = v0
   add0(z, z') -{ 2 }-> 1 + x :|: z >= 0, z' = 1 + 0, x >= 0, z = 1 + 0, 1 + 0 = x
   add0(z, z') -{ 2 }-> 1 + z :|: z >= 0, z' = 1 + 0, 1 + 0 = 1 + 0
   mult(z, z') -{ 2 }-> add0(z, add0(z, mult(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   mult(z, z') -{ 2 }-> add0(z, 0) :|: z >= 0, z' = 1 + 0
   mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + z :|: z >= 0, z' = 1 + 0
   plus(z, z') -{ 0 }-> 1 + z' :|: z = 1 + 0, z' >= 0
   power(z, z') -{ 2 }-> mult(z, mult(z, power(z, z' - 2))) :|: z' - 2 >= 0, z >= 0
   power(z, z') -{ 2 }-> mult(z, 1 + 0) :|: z >= 0, z' = 1 + 0
   power(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0


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

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

   { plus }
   { add0 }
   { mult }
   { power }

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

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

   add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   add0(z, z') -{ 2 }-> plus(1 + 0, plus(1 + 0, add0(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   add0(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1 + 0, v0 >= 0, 1 + 0 = v0
   add0(z, z') -{ 2 }-> 1 + x :|: z >= 0, z' = 1 + 0, x >= 0, z = 1 + 0, 1 + 0 = x
   add0(z, z') -{ 2 }-> 1 + z :|: z >= 0, z' = 1 + 0, 1 + 0 = 1 + 0
   mult(z, z') -{ 2 }-> add0(z, add0(z, mult(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   mult(z, z') -{ 2 }-> add0(z, 0) :|: z >= 0, z' = 1 + 0
   mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + z :|: z >= 0, z' = 1 + 0
   plus(z, z') -{ 0 }-> 1 + z' :|: z = 1 + 0, z' >= 0
   power(z, z') -{ 2 }-> mult(z, mult(z, power(z, z' - 2))) :|: z' - 2 >= 0, z >= 0
   power(z, z') -{ 2 }-> mult(z, 1 + 0) :|: z >= 0, z' = 1 + 0
   power(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {plus}, {add0}, {mult}, {power}

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

(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:

   add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   add0(z, z') -{ 2 }-> plus(1 + 0, plus(1 + 0, add0(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   add0(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1 + 0, v0 >= 0, 1 + 0 = v0
   add0(z, z') -{ 2 }-> 1 + x :|: z >= 0, z' = 1 + 0, x >= 0, z = 1 + 0, 1 + 0 = x
   add0(z, z') -{ 2 }-> 1 + z :|: z >= 0, z' = 1 + 0, 1 + 0 = 1 + 0
   mult(z, z') -{ 2 }-> add0(z, add0(z, mult(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   mult(z, z') -{ 2 }-> add0(z, 0) :|: z >= 0, z' = 1 + 0
   mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + z :|: z >= 0, z' = 1 + 0
   plus(z, z') -{ 0 }-> 1 + z' :|: z = 1 + 0, z' >= 0
   power(z, z') -{ 2 }-> mult(z, mult(z, power(z, z' - 2))) :|: z' - 2 >= 0, z >= 0
   power(z, z') -{ 2 }-> mult(z, 1 + 0) :|: z >= 0, z' = 1 + 0
   power(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {plus}, {add0}, {mult}, {power}

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT 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:

   add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   add0(z, z') -{ 2 }-> plus(1 + 0, plus(1 + 0, add0(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   add0(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1 + 0, v0 >= 0, 1 + 0 = v0
   add0(z, z') -{ 2 }-> 1 + x :|: z >= 0, z' = 1 + 0, x >= 0, z = 1 + 0, 1 + 0 = x
   add0(z, z') -{ 2 }-> 1 + z :|: z >= 0, z' = 1 + 0, 1 + 0 = 1 + 0
   mult(z, z') -{ 2 }-> add0(z, add0(z, mult(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   mult(z, z') -{ 2 }-> add0(z, 0) :|: z >= 0, z' = 1 + 0
   mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + z :|: z >= 0, z' = 1 + 0
   plus(z, z') -{ 0 }-> 1 + z' :|: z = 1 + 0, z' >= 0
   power(z, z') -{ 2 }-> mult(z, mult(z, power(z, z' - 2))) :|: z' - 2 >= 0, z >= 0
   power(z, z') -{ 2 }-> mult(z, 1 + 0) :|: z >= 0, z' = 1 + 0
   power(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {plus}, {add0}, {mult}, {power}
Previous analysis results are:
  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(1) with polynomial bound: 0

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

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

   add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   add0(z, z') -{ 2 }-> plus(1 + 0, plus(1 + 0, add0(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   add0(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1 + 0, v0 >= 0, 1 + 0 = v0
   add0(z, z') -{ 2 }-> 1 + x :|: z >= 0, z' = 1 + 0, x >= 0, z = 1 + 0, 1 + 0 = x
   add0(z, z') -{ 2 }-> 1 + z :|: z >= 0, z' = 1 + 0, 1 + 0 = 1 + 0
   mult(z, z') -{ 2 }-> add0(z, add0(z, mult(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   mult(z, z') -{ 2 }-> add0(z, 0) :|: z >= 0, z' = 1 + 0
   mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + z :|: z >= 0, z' = 1 + 0
   plus(z, z') -{ 0 }-> 1 + z' :|: z = 1 + 0, z' >= 0
   power(z, z') -{ 2 }-> mult(z, mult(z, power(z, z' - 2))) :|: z' - 2 >= 0, z >= 0
   power(z, z') -{ 2 }-> mult(z, 1 + 0) :|: z >= 0, z' = 1 + 0
   power(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {add0}, {mult}, {power}
Previous analysis results are:
  plus: runtime: O(1) [0], 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:

   add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   add0(z, z') -{ 2 }-> plus(1 + 0, plus(1 + 0, add0(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   add0(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1 + 0, v0 >= 0, 1 + 0 = v0
   add0(z, z') -{ 2 }-> 1 + x :|: z >= 0, z' = 1 + 0, x >= 0, z = 1 + 0, 1 + 0 = x
   add0(z, z') -{ 2 }-> 1 + z :|: z >= 0, z' = 1 + 0, 1 + 0 = 1 + 0
   mult(z, z') -{ 2 }-> add0(z, add0(z, mult(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   mult(z, z') -{ 2 }-> add0(z, 0) :|: z >= 0, z' = 1 + 0
   mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + z :|: z >= 0, z' = 1 + 0
   plus(z, z') -{ 0 }-> 1 + z' :|: z = 1 + 0, z' >= 0
   power(z, z') -{ 2 }-> mult(z, mult(z, power(z, z' - 2))) :|: z' - 2 >= 0, z >= 0
   power(z, z') -{ 2 }-> mult(z, 1 + 0) :|: z >= 0, z' = 1 + 0
   power(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {add0}, {mult}, {power}
Previous analysis results are:
  plus: runtime: O(1) [0], size: O(n^1) [z + z']

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using KoAT for: add0
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: z + z'

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

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

   add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   add0(z, z') -{ 2 }-> plus(1 + 0, plus(1 + 0, add0(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   add0(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1 + 0, v0 >= 0, 1 + 0 = v0
   add0(z, z') -{ 2 }-> 1 + x :|: z >= 0, z' = 1 + 0, x >= 0, z = 1 + 0, 1 + 0 = x
   add0(z, z') -{ 2 }-> 1 + z :|: z >= 0, z' = 1 + 0, 1 + 0 = 1 + 0
   mult(z, z') -{ 2 }-> add0(z, add0(z, mult(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   mult(z, z') -{ 2 }-> add0(z, 0) :|: z >= 0, z' = 1 + 0
   mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + z :|: z >= 0, z' = 1 + 0
   plus(z, z') -{ 0 }-> 1 + z' :|: z = 1 + 0, z' >= 0
   power(z, z') -{ 2 }-> mult(z, mult(z, power(z, z' - 2))) :|: z' - 2 >= 0, z >= 0
   power(z, z') -{ 2 }-> mult(z, 1 + 0) :|: z >= 0, z' = 1 + 0
   power(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {add0}, {mult}, {power}
Previous analysis results are:
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  add0: runtime: ?, size: O(n^1) [z + z']

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: add0
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 4 + 2*z'

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

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

   add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   add0(z, z') -{ 2 }-> plus(1 + 0, plus(1 + 0, add0(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   add0(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1 + 0, v0 >= 0, 1 + 0 = v0
   add0(z, z') -{ 2 }-> 1 + x :|: z >= 0, z' = 1 + 0, x >= 0, z = 1 + 0, 1 + 0 = x
   add0(z, z') -{ 2 }-> 1 + z :|: z >= 0, z' = 1 + 0, 1 + 0 = 1 + 0
   mult(z, z') -{ 2 }-> add0(z, add0(z, mult(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   mult(z, z') -{ 2 }-> add0(z, 0) :|: z >= 0, z' = 1 + 0
   mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + z :|: z >= 0, z' = 1 + 0
   plus(z, z') -{ 0 }-> 1 + z' :|: z = 1 + 0, z' >= 0
   power(z, z') -{ 2 }-> mult(z, mult(z, power(z, z' - 2))) :|: z' - 2 >= 0, z >= 0
   power(z, z') -{ 2 }-> mult(z, 1 + 0) :|: z >= 0, z' = 1 + 0
   power(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {mult}, {power}
Previous analysis results are:
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  add0: runtime: O(n^1) [4 + 2*z'], size: O(n^1) [z + z']

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

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

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

   add0(z, z') -{ 2 + 2*z' }-> s1 :|: s' >= 0, s' <= z + (z' - 2), s'' >= 0, s'' <= 1 + 0 + s', s1 >= 0, s1 <= 1 + 0 + s'', z >= 0, z' - 2 >= 0
   add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   add0(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1 + 0, v0 >= 0, 1 + 0 = v0
   add0(z, z') -{ 2 }-> 1 + x :|: z >= 0, z' = 1 + 0, x >= 0, z = 1 + 0, 1 + 0 = x
   add0(z, z') -{ 2 }-> 1 + z :|: z >= 0, z' = 1 + 0, 1 + 0 = 1 + 0
   mult(z, z') -{ 6 }-> s :|: s >= 0, s <= z + 0, z >= 0, z' = 1 + 0
   mult(z, z') -{ 2 }-> add0(z, add0(z, mult(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + z :|: z >= 0, z' = 1 + 0
   plus(z, z') -{ 0 }-> 1 + z' :|: z = 1 + 0, z' >= 0
   power(z, z') -{ 2 }-> mult(z, mult(z, power(z, z' - 2))) :|: z' - 2 >= 0, z >= 0
   power(z, z') -{ 2 }-> mult(z, 1 + 0) :|: z >= 0, z' = 1 + 0
   power(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {mult}, {power}
Previous analysis results are:
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  add0: runtime: O(n^1) [4 + 2*z'], size: O(n^1) [z + z']

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   add0(z, z') -{ 2 + 2*z' }-> s1 :|: s' >= 0, s' <= z + (z' - 2), s'' >= 0, s'' <= 1 + 0 + s', s1 >= 0, s1 <= 1 + 0 + s'', z >= 0, z' - 2 >= 0
   add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   add0(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1 + 0, v0 >= 0, 1 + 0 = v0
   add0(z, z') -{ 2 }-> 1 + x :|: z >= 0, z' = 1 + 0, x >= 0, z = 1 + 0, 1 + 0 = x
   add0(z, z') -{ 2 }-> 1 + z :|: z >= 0, z' = 1 + 0, 1 + 0 = 1 + 0
   mult(z, z') -{ 6 }-> s :|: s >= 0, s <= z + 0, z >= 0, z' = 1 + 0
   mult(z, z') -{ 2 }-> add0(z, add0(z, mult(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + z :|: z >= 0, z' = 1 + 0
   plus(z, z') -{ 0 }-> 1 + z' :|: z = 1 + 0, z' >= 0
   power(z, z') -{ 2 }-> mult(z, mult(z, power(z, z' - 2))) :|: z' - 2 >= 0, z >= 0
   power(z, z') -{ 2 }-> mult(z, 1 + 0) :|: z >= 0, z' = 1 + 0
   power(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {mult}, {power}
Previous analysis results are:
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  add0: runtime: O(n^1) [4 + 2*z'], size: O(n^1) [z + z']
  mult: runtime: ?, size: O(n^2) [z + 2*z*z']

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: mult
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 7 + 8*z*z'^2 + 10*z'

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

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

   add0(z, z') -{ 2 + 2*z' }-> s1 :|: s' >= 0, s' <= z + (z' - 2), s'' >= 0, s'' <= 1 + 0 + s', s1 >= 0, s1 <= 1 + 0 + s'', z >= 0, z' - 2 >= 0
   add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   add0(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1 + 0, v0 >= 0, 1 + 0 = v0
   add0(z, z') -{ 2 }-> 1 + x :|: z >= 0, z' = 1 + 0, x >= 0, z = 1 + 0, 1 + 0 = x
   add0(z, z') -{ 2 }-> 1 + z :|: z >= 0, z' = 1 + 0, 1 + 0 = 1 + 0
   mult(z, z') -{ 6 }-> s :|: s >= 0, s <= z + 0, z >= 0, z' = 1 + 0
   mult(z, z') -{ 2 }-> add0(z, add0(z, mult(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + z :|: z >= 0, z' = 1 + 0
   plus(z, z') -{ 0 }-> 1 + z' :|: z = 1 + 0, z' >= 0
   power(z, z') -{ 2 }-> mult(z, mult(z, power(z, z' - 2))) :|: z' - 2 >= 0, z >= 0
   power(z, z') -{ 2 }-> mult(z, 1 + 0) :|: z >= 0, z' = 1 + 0
   power(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {power}
Previous analysis results are:
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  add0: runtime: O(n^1) [4 + 2*z'], size: O(n^1) [z + z']
  mult: runtime: O(n^3) [7 + 8*z*z'^2 + 10*z'], size: O(n^2) [z + 2*z*z']

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

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

(44)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 2 + 2*z' }-> s1 :|: s' >= 0, s' <= z + (z' - 2), s'' >= 0, s'' <= 1 + 0 + s', s1 >= 0, s1 <= 1 + 0 + s'', z >= 0, z' - 2 >= 0
   add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   add0(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1 + 0, v0 >= 0, 1 + 0 = v0
   add0(z, z') -{ 2 }-> 1 + x :|: z >= 0, z' = 1 + 0, x >= 0, z = 1 + 0, 1 + 0 = x
   add0(z, z') -{ 2 }-> 1 + z :|: z >= 0, z' = 1 + 0, 1 + 0 = 1 + 0
   mult(z, z') -{ 6 }-> s :|: s >= 0, s <= z + 0, z >= 0, z' = 1 + 0
   mult(z, z') -{ -3 + 2*s3 + 2*s4 + 32*z + -32*z*z' + 8*z*z'^2 + 10*z' }-> s5 :|: s3 >= 0, s3 <= 2 * ((z' - 2) * z) + z, s4 >= 0, s4 <= z + s3, s5 >= 0, s5 <= z + s4, z >= 0, z' - 2 >= 0
   mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + z :|: z >= 0, z' = 1 + 0
   plus(z, z') -{ 0 }-> 1 + z' :|: z = 1 + 0, z' >= 0
   power(z, z') -{ 19 + 8*z }-> s2 :|: s2 >= 0, s2 <= 2 * ((1 + 0) * z) + z, z >= 0, z' = 1 + 0
   power(z, z') -{ 2 }-> mult(z, mult(z, power(z, z' - 2))) :|: z' - 2 >= 0, z >= 0
   power(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {power}
Previous analysis results are:
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  add0: runtime: O(n^1) [4 + 2*z'], size: O(n^1) [z + z']
  mult: runtime: O(n^3) [7 + 8*z*z'^2 + 10*z'], size: O(n^2) [z + 2*z*z']

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(46)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 2 + 2*z' }-> s1 :|: s' >= 0, s' <= z + (z' - 2), s'' >= 0, s'' <= 1 + 0 + s', s1 >= 0, s1 <= 1 + 0 + s'', z >= 0, z' - 2 >= 0
   add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   add0(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1 + 0, v0 >= 0, 1 + 0 = v0
   add0(z, z') -{ 2 }-> 1 + x :|: z >= 0, z' = 1 + 0, x >= 0, z = 1 + 0, 1 + 0 = x
   add0(z, z') -{ 2 }-> 1 + z :|: z >= 0, z' = 1 + 0, 1 + 0 = 1 + 0
   mult(z, z') -{ 6 }-> s :|: s >= 0, s <= z + 0, z >= 0, z' = 1 + 0
   mult(z, z') -{ -3 + 2*s3 + 2*s4 + 32*z + -32*z*z' + 8*z*z'^2 + 10*z' }-> s5 :|: s3 >= 0, s3 <= 2 * ((z' - 2) * z) + z, s4 >= 0, s4 <= z + s3, s5 >= 0, s5 <= z + s4, z >= 0, z' - 2 >= 0
   mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + z :|: z >= 0, z' = 1 + 0
   plus(z, z') -{ 0 }-> 1 + z' :|: z = 1 + 0, z' >= 0
   power(z, z') -{ 19 + 8*z }-> s2 :|: s2 >= 0, s2 <= 2 * ((1 + 0) * z) + z, z >= 0, z' = 1 + 0
   power(z, z') -{ 2 }-> mult(z, mult(z, power(z, z' - 2))) :|: z' - 2 >= 0, z >= 0
   power(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {power}
Previous analysis results are:
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  add0: runtime: O(n^1) [4 + 2*z'], size: O(n^1) [z + z']
  mult: runtime: O(n^3) [7 + 8*z*z'^2 + 10*z'], size: O(n^2) [z + 2*z*z']
  power: runtime: ?, size: INF

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

(47) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(48)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 2 + 2*z' }-> s1 :|: s' >= 0, s' <= z + (z' - 2), s'' >= 0, s'' <= 1 + 0 + s', s1 >= 0, s1 <= 1 + 0 + s'', z >= 0, z' - 2 >= 0
   add0(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   add0(z, z') -{ 2 }-> 0 :|: z >= 0, z' = 1 + 0, v0 >= 0, 1 + 0 = v0
   add0(z, z') -{ 2 }-> 1 + x :|: z >= 0, z' = 1 + 0, x >= 0, z = 1 + 0, 1 + 0 = x
   add0(z, z') -{ 2 }-> 1 + z :|: z >= 0, z' = 1 + 0, 1 + 0 = 1 + 0
   mult(z, z') -{ 6 }-> s :|: s >= 0, s <= z + 0, z >= 0, z' = 1 + 0
   mult(z, z') -{ -3 + 2*s3 + 2*s4 + 32*z + -32*z*z' + 8*z*z'^2 + 10*z' }-> s5 :|: s3 >= 0, s3 <= 2 * ((z' - 2) * z) + z, s4 >= 0, s4 <= z + s3, s5 >= 0, s5 <= z + s4, z >= 0, z' - 2 >= 0
   mult(z, z') -{ 1 }-> 0 :|: z >= 0, z' = 0
   plus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   plus(z, z') -{ 0 }-> 1 + z :|: z >= 0, z' = 1 + 0
   plus(z, z') -{ 0 }-> 1 + z' :|: z = 1 + 0, z' >= 0
   power(z, z') -{ 19 + 8*z }-> s2 :|: s2 >= 0, s2 <= 2 * ((1 + 0) * z) + z, z >= 0, z' = 1 + 0
   power(z, z') -{ 2 }-> mult(z, mult(z, power(z, z' - 2))) :|: z' - 2 >= 0, z >= 0
   power(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0

Function symbols to be analyzed: {power}
Previous analysis results are:
  plus: runtime: O(1) [0], size: O(n^1) [z + z']
  add0: runtime: O(n^1) [4 + 2*z'], size: O(n^1) [z + z']
  mult: runtime: O(n^3) [7 + 8*z*z'^2 + 10*z'], size: O(n^2) [z + 2*z*z']
  power: runtime: INF, size: INF

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

(49) 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:

   plus(v0, v1) -> null_plus [0]
   power(v0, v1) -> null_power [0]
   mult(v0, v1) -> null_mult [0]
   add0(v0, v1) -> null_add0 [0]

And the following fresh constants:    null_plus, null_power, null_mult, null_add0

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

(50)
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:

   power(x', S(x)) -> mult(x', power(x', x)) [1]
   mult(x', S(x)) -> add0(x', mult(x', x)) [1]
   add0(x', S(x)) -> plus(S(0), add0(x', x)) [1]
   power(x, 0) -> S(0) [1]
   mult(x, 0) -> 0 [1]
   add0(x, 0) -> x [1]
   plus(x, S(0)) -> S(x) [0]
   plus(S(0), y) -> S(y) [0]
   plus(v0, v1) -> null_plus [0]
   power(v0, v1) -> null_power [0]
   mult(v0, v1) -> null_mult [0]
   add0(v0, v1) -> null_add0 [0]

The TRS has the following type information:
power :: S:0:null_plus:null_power:null_mult:null_add0 -> S:0:null_plus:null_power:null_mult:null_add0 -> S:0:null_plus:null_power:null_mult:null_add0
S :: S:0:null_plus:null_power:null_mult:null_add0 -> S:0:null_plus:null_power:null_mult:null_add0
mult :: S:0:null_plus:null_power:null_mult:null_add0 -> S:0:null_plus:null_power:null_mult:null_add0 -> S:0:null_plus:null_power:null_mult:null_add0
add0 :: S:0:null_plus:null_power:null_mult:null_add0 -> S:0:null_plus:null_power:null_mult:null_add0 -> S:0:null_plus:null_power:null_mult:null_add0
plus :: S:0:null_plus:null_power:null_mult:null_add0 -> S:0:null_plus:null_power:null_mult:null_add0 -> S:0:null_plus:null_power:null_mult:null_add0
0 :: S:0:null_plus:null_power:null_mult:null_add0
null_plus :: S:0:null_plus:null_power:null_mult:null_add0
null_power :: S:0:null_plus:null_power:null_mult:null_add0
null_mult :: S:0:null_plus:null_power:null_mult:null_add0
null_add0 :: S:0:null_plus:null_power:null_mult:null_add0

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

(51) 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_plus => 0
   null_power => 0
   null_mult => 0
   null_add0 => 0

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

(52)
Obligation:
Complexity RNTS consisting of the following rules:

   add0(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   add0(z, z') -{ 1 }-> plus(1 + 0, add0(x', x)) :|: z' = 1 + x, x' >= 0, x >= 0, z = x'
   add0(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   mult(z, z') -{ 1 }-> add0(x', mult(x', x)) :|: z' = 1 + x, x' >= 0, x >= 0, z = x'
   mult(z, z') -{ 1 }-> 0 :|: x >= 0, z = x, z' = 0
   mult(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   plus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   plus(z, z') -{ 0 }-> 1 + x :|: x >= 0, z' = 1 + 0, z = x
   plus(z, z') -{ 0 }-> 1 + y :|: z = 1 + 0, y >= 0, z' = y
   power(z, z') -{ 1 }-> mult(x', power(x', x)) :|: z' = 1 + x, x' >= 0, x >= 0, z = x'
   power(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   power(z, z') -{ 1 }-> 1 + 0 :|: x >= 0, z = x, z' = 0

Only complete derivations are relevant for the runtime complexity.

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

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

(54)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, S(0)) -> S(z0)
   +(S(0), z0) -> S(z0)
   power(z0, S(z1)) -> mult(z0, power(z0, z1))
   power(z0, 0) -> S(0)
   mult(z0, S(z1)) -> add0(z0, mult(z0, z1))
   mult(z0, 0) -> 0
   add0(z0, S(z1)) -> +(S(0), add0(z0, z1))
   add0(z0, 0) -> z0
Tuples:
   +'(z0, S(0)) -> c
   +'(S(0), z0) -> c1
   POWER(z0, S(z1)) -> c2(MULT(z0, power(z0, z1)), POWER(z0, z1))
   POWER(z0, 0) -> c3
   MULT(z0, S(z1)) -> c4(ADD0(z0, mult(z0, z1)), MULT(z0, z1))
   MULT(z0, 0) -> c5
   ADD0(z0, S(z1)) -> c6(+'(S(0), add0(z0, z1)), ADD0(z0, z1))
   ADD0(z0, 0) -> c7
S tuples:
   POWER(z0, S(z1)) -> c2(MULT(z0, power(z0, z1)), POWER(z0, z1))
   POWER(z0, 0) -> c3
   MULT(z0, S(z1)) -> c4(ADD0(z0, mult(z0, z1)), MULT(z0, z1))
   MULT(z0, 0) -> c5
   ADD0(z0, S(z1)) -> c6(+'(S(0), add0(z0, z1)), ADD0(z0, z1))
   ADD0(z0, 0) -> c7
K tuples:none
Defined Rule Symbols:   power_2, mult_2, add0_2, +_2

Defined Pair Symbols:   +'_2, POWER_2, MULT_2, ADD0_2

Compound Symbols:   c, c1, c2_2, c3, c4_2, c5, c6_2, c7


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

(55) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 5 trailing nodes:
   +'(S(0), z0) -> c1
   +'(z0, S(0)) -> c
   POWER(z0, 0) -> c3
   ADD0(z0, 0) -> c7
   MULT(z0, 0) -> c5

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, S(0)) -> S(z0)
   +(S(0), z0) -> S(z0)
   power(z0, S(z1)) -> mult(z0, power(z0, z1))
   power(z0, 0) -> S(0)
   mult(z0, S(z1)) -> add0(z0, mult(z0, z1))
   mult(z0, 0) -> 0
   add0(z0, S(z1)) -> +(S(0), add0(z0, z1))
   add0(z0, 0) -> z0
Tuples:
   POWER(z0, S(z1)) -> c2(MULT(z0, power(z0, z1)), POWER(z0, z1))
   MULT(z0, S(z1)) -> c4(ADD0(z0, mult(z0, z1)), MULT(z0, z1))
   ADD0(z0, S(z1)) -> c6(+'(S(0), add0(z0, z1)), ADD0(z0, z1))
S tuples:
   POWER(z0, S(z1)) -> c2(MULT(z0, power(z0, z1)), POWER(z0, z1))
   MULT(z0, S(z1)) -> c4(ADD0(z0, mult(z0, z1)), MULT(z0, z1))
   ADD0(z0, S(z1)) -> c6(+'(S(0), add0(z0, z1)), ADD0(z0, z1))
K tuples:none
Defined Rule Symbols:   power_2, mult_2, add0_2, +_2

Defined Pair Symbols:   POWER_2, MULT_2, ADD0_2

Compound Symbols:   c2_2, c4_2, c6_2


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

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

(58)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, S(0)) -> S(z0)
   +(S(0), z0) -> S(z0)
   power(z0, S(z1)) -> mult(z0, power(z0, z1))
   power(z0, 0) -> S(0)
   mult(z0, S(z1)) -> add0(z0, mult(z0, z1))
   mult(z0, 0) -> 0
   add0(z0, S(z1)) -> +(S(0), add0(z0, z1))
   add0(z0, 0) -> z0
Tuples:
   POWER(z0, S(z1)) -> c2(MULT(z0, power(z0, z1)), POWER(z0, z1))
   MULT(z0, S(z1)) -> c4(ADD0(z0, mult(z0, z1)), MULT(z0, z1))
   ADD0(z0, S(z1)) -> c6(ADD0(z0, z1))
S tuples:
   POWER(z0, S(z1)) -> c2(MULT(z0, power(z0, z1)), POWER(z0, z1))
   MULT(z0, S(z1)) -> c4(ADD0(z0, mult(z0, z1)), MULT(z0, z1))
   ADD0(z0, S(z1)) -> c6(ADD0(z0, z1))
K tuples:none
Defined Rule Symbols:   power_2, mult_2, add0_2, +_2

Defined Pair Symbols:   POWER_2, MULT_2, ADD0_2

Compound Symbols:   c2_2, c4_2, c6_1


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

(59) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   POWER(z0, S(z1)) -> c2(MULT(z0, power(z0, z1)), POWER(z0, z1))
We considered the (Usable) Rules:none
And the Tuples:
   POWER(z0, S(z1)) -> c2(MULT(z0, power(z0, z1)), POWER(z0, z1))
   MULT(z0, S(z1)) -> c4(ADD0(z0, mult(z0, z1)), MULT(z0, z1))
   ADD0(z0, S(z1)) -> c6(ADD0(z0, z1))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(+(x_1, x_2)) = [1] + x_1 + x_2
   POL(0) = [1]
   POL(ADD0(x_1, x_2)) = 0
   POL(MULT(x_1, x_2)) = 0
   POL(POWER(x_1, x_2)) = x_2
   POL(S(x_1)) = [1] + x_1
   POL(add0(x_1, x_2)) = [1] + x_1 + x_2
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(mult(x_1, x_2)) = [1] + x_2
   POL(power(x_1, x_2)) = [1] + x_1 + x_2

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

(60)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, S(0)) -> S(z0)
   +(S(0), z0) -> S(z0)
   power(z0, S(z1)) -> mult(z0, power(z0, z1))
   power(z0, 0) -> S(0)
   mult(z0, S(z1)) -> add0(z0, mult(z0, z1))
   mult(z0, 0) -> 0
   add0(z0, S(z1)) -> +(S(0), add0(z0, z1))
   add0(z0, 0) -> z0
Tuples:
   POWER(z0, S(z1)) -> c2(MULT(z0, power(z0, z1)), POWER(z0, z1))
   MULT(z0, S(z1)) -> c4(ADD0(z0, mult(z0, z1)), MULT(z0, z1))
   ADD0(z0, S(z1)) -> c6(ADD0(z0, z1))
S tuples:
   MULT(z0, S(z1)) -> c4(ADD0(z0, mult(z0, z1)), MULT(z0, z1))
   ADD0(z0, S(z1)) -> c6(ADD0(z0, z1))
K tuples:
   POWER(z0, S(z1)) -> c2(MULT(z0, power(z0, z1)), POWER(z0, z1))
Defined Rule Symbols:   power_2, mult_2, add0_2, +_2

Defined Pair Symbols:   POWER_2, MULT_2, ADD0_2

Compound Symbols:   c2_2, c4_2, c6_1


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

(61) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace POWER(z0, S(z1)) -> c2(MULT(z0, power(z0, z1)), POWER(z0, z1)) by 
   POWER(z0, S(S(z1))) -> c2(MULT(z0, mult(z0, power(z0, z1))), POWER(z0, S(z1)))
   POWER(z0, S(0)) -> c2(MULT(z0, S(0)), POWER(z0, 0))

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

(62)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, S(0)) -> S(z0)
   +(S(0), z0) -> S(z0)
   power(z0, S(z1)) -> mult(z0, power(z0, z1))
   power(z0, 0) -> S(0)
   mult(z0, S(z1)) -> add0(z0, mult(z0, z1))
   mult(z0, 0) -> 0
   add0(z0, S(z1)) -> +(S(0), add0(z0, z1))
   add0(z0, 0) -> z0
Tuples:
   MULT(z0, S(z1)) -> c4(ADD0(z0, mult(z0, z1)), MULT(z0, z1))
   ADD0(z0, S(z1)) -> c6(ADD0(z0, z1))
   POWER(z0, S(S(z1))) -> c2(MULT(z0, mult(z0, power(z0, z1))), POWER(z0, S(z1)))
   POWER(z0, S(0)) -> c2(MULT(z0, S(0)), POWER(z0, 0))
S tuples:
   MULT(z0, S(z1)) -> c4(ADD0(z0, mult(z0, z1)), MULT(z0, z1))
   ADD0(z0, S(z1)) -> c6(ADD0(z0, z1))
K tuples:
   POWER(z0, S(z1)) -> c2(MULT(z0, power(z0, z1)), POWER(z0, z1))
Defined Rule Symbols:   power_2, mult_2, add0_2, +_2

Defined Pair Symbols:   MULT_2, ADD0_2, POWER_2

Compound Symbols:   c4_2, c6_1, c2_2


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

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

(64)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, S(0)) -> S(z0)
   +(S(0), z0) -> S(z0)
   power(z0, S(z1)) -> mult(z0, power(z0, z1))
   power(z0, 0) -> S(0)
   mult(z0, S(z1)) -> add0(z0, mult(z0, z1))
   mult(z0, 0) -> 0
   add0(z0, S(z1)) -> +(S(0), add0(z0, z1))
   add0(z0, 0) -> z0
Tuples:
   MULT(z0, S(z1)) -> c4(ADD0(z0, mult(z0, z1)), MULT(z0, z1))
   ADD0(z0, S(z1)) -> c6(ADD0(z0, z1))
   POWER(z0, S(S(z1))) -> c2(MULT(z0, mult(z0, power(z0, z1))), POWER(z0, S(z1)))
   POWER(z0, S(0)) -> c2(MULT(z0, S(0)))
S tuples:
   MULT(z0, S(z1)) -> c4(ADD0(z0, mult(z0, z1)), MULT(z0, z1))
   ADD0(z0, S(z1)) -> c6(ADD0(z0, z1))
K tuples:
   POWER(z0, S(z1)) -> c2(MULT(z0, power(z0, z1)), POWER(z0, z1))
Defined Rule Symbols:   power_2, mult_2, add0_2, +_2

Defined Pair Symbols:   MULT_2, ADD0_2, POWER_2

Compound Symbols:   c4_2, c6_1, c2_2, c2_1


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

(65) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace MULT(z0, S(z1)) -> c4(ADD0(z0, mult(z0, z1)), MULT(z0, z1)) by 
   MULT(z0, S(S(z1))) -> c4(ADD0(z0, add0(z0, mult(z0, z1))), MULT(z0, S(z1)))
   MULT(z0, S(0)) -> c4(ADD0(z0, 0), MULT(z0, 0))

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

(66)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, S(0)) -> S(z0)
   +(S(0), z0) -> S(z0)
   power(z0, S(z1)) -> mult(z0, power(z0, z1))
   power(z0, 0) -> S(0)
   mult(z0, S(z1)) -> add0(z0, mult(z0, z1))
   mult(z0, 0) -> 0
   add0(z0, S(z1)) -> +(S(0), add0(z0, z1))
   add0(z0, 0) -> z0
Tuples:
   ADD0(z0, S(z1)) -> c6(ADD0(z0, z1))
   POWER(z0, S(S(z1))) -> c2(MULT(z0, mult(z0, power(z0, z1))), POWER(z0, S(z1)))
   POWER(z0, S(0)) -> c2(MULT(z0, S(0)))
   MULT(z0, S(S(z1))) -> c4(ADD0(z0, add0(z0, mult(z0, z1))), MULT(z0, S(z1)))
   MULT(z0, S(0)) -> c4(ADD0(z0, 0), MULT(z0, 0))
S tuples:
   ADD0(z0, S(z1)) -> c6(ADD0(z0, z1))
   MULT(z0, S(S(z1))) -> c4(ADD0(z0, add0(z0, mult(z0, z1))), MULT(z0, S(z1)))
   MULT(z0, S(0)) -> c4(ADD0(z0, 0), MULT(z0, 0))
K tuples:
   POWER(z0, S(z1)) -> c2(MULT(z0, power(z0, z1)), POWER(z0, z1))
Defined Rule Symbols:   power_2, mult_2, add0_2, +_2

Defined Pair Symbols:   ADD0_2, POWER_2, MULT_2

Compound Symbols:   c6_1, c2_2, c2_1, c4_2


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

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

(68)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, S(0)) -> S(z0)
   +(S(0), z0) -> S(z0)
   power(z0, S(z1)) -> mult(z0, power(z0, z1))
   power(z0, 0) -> S(0)
   mult(z0, S(z1)) -> add0(z0, mult(z0, z1))
   mult(z0, 0) -> 0
   add0(z0, S(z1)) -> +(S(0), add0(z0, z1))
   add0(z0, 0) -> z0
Tuples:
   ADD0(z0, S(z1)) -> c6(ADD0(z0, z1))
   POWER(z0, S(S(z1))) -> c2(MULT(z0, mult(z0, power(z0, z1))), POWER(z0, S(z1)))
   POWER(z0, S(0)) -> c2(MULT(z0, S(0)))
   MULT(z0, S(S(z1))) -> c4(ADD0(z0, add0(z0, mult(z0, z1))), MULT(z0, S(z1)))
   MULT(z0, S(0)) -> c4
S tuples:
   ADD0(z0, S(z1)) -> c6(ADD0(z0, z1))
   MULT(z0, S(S(z1))) -> c4(ADD0(z0, add0(z0, mult(z0, z1))), MULT(z0, S(z1)))
   MULT(z0, S(0)) -> c4
K tuples:
   POWER(z0, S(z1)) -> c2(MULT(z0, power(z0, z1)), POWER(z0, z1))
Defined Rule Symbols:   power_2, mult_2, add0_2, +_2

Defined Pair Symbols:   ADD0_2, POWER_2, MULT_2

Compound Symbols:   c6_1, c2_2, c2_1, c4_2, c4


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

(69) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   MULT(z0, S(0)) -> c4
We considered the (Usable) Rules:none
And the Tuples:
   ADD0(z0, S(z1)) -> c6(ADD0(z0, z1))
   POWER(z0, S(S(z1))) -> c2(MULT(z0, mult(z0, power(z0, z1))), POWER(z0, S(z1)))
   POWER(z0, S(0)) -> c2(MULT(z0, S(0)))
   MULT(z0, S(S(z1))) -> c4(ADD0(z0, add0(z0, mult(z0, z1))), MULT(z0, S(z1)))
   MULT(z0, S(0)) -> c4
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(+(x_1, x_2)) = x_1 + x_2
   POL(0) = 0
   POL(ADD0(x_1, x_2)) = 0
   POL(MULT(x_1, x_2)) = [1]
   POL(POWER(x_1, x_2)) = [1] + x_2
   POL(S(x_1)) = [1] + x_1
   POL(add0(x_1, x_2)) = [1] + x_1 + x_2
   POL(c2(x_1)) = x_1
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c4) = 0
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(mult(x_1, x_2)) = [1] + x_2
   POL(power(x_1, x_2)) = [1] + x_1 + x_2

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

(70)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, S(0)) -> S(z0)
   +(S(0), z0) -> S(z0)
   power(z0, S(z1)) -> mult(z0, power(z0, z1))
   power(z0, 0) -> S(0)
   mult(z0, S(z1)) -> add0(z0, mult(z0, z1))
   mult(z0, 0) -> 0
   add0(z0, S(z1)) -> +(S(0), add0(z0, z1))
   add0(z0, 0) -> z0
Tuples:
   ADD0(z0, S(z1)) -> c6(ADD0(z0, z1))
   POWER(z0, S(S(z1))) -> c2(MULT(z0, mult(z0, power(z0, z1))), POWER(z0, S(z1)))
   POWER(z0, S(0)) -> c2(MULT(z0, S(0)))
   MULT(z0, S(S(z1))) -> c4(ADD0(z0, add0(z0, mult(z0, z1))), MULT(z0, S(z1)))
   MULT(z0, S(0)) -> c4
S tuples:
   ADD0(z0, S(z1)) -> c6(ADD0(z0, z1))
   MULT(z0, S(S(z1))) -> c4(ADD0(z0, add0(z0, mult(z0, z1))), MULT(z0, S(z1)))
K tuples:
   POWER(z0, S(z1)) -> c2(MULT(z0, power(z0, z1)), POWER(z0, z1))
   MULT(z0, S(0)) -> c4
Defined Rule Symbols:   power_2, mult_2, add0_2, +_2

Defined Pair Symbols:   ADD0_2, POWER_2, MULT_2

Compound Symbols:   c6_1, c2_2, c2_1, c4_2, c4


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

(71) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace POWER(z0, S(S(z1))) -> c2(MULT(z0, mult(z0, power(z0, z1))), POWER(z0, S(z1))) by 
   POWER(z0, S(S(S(z1)))) -> c2(MULT(z0, mult(z0, mult(z0, power(z0, z1)))), POWER(z0, S(S(z1))))
   POWER(z0, S(S(0))) -> c2(MULT(z0, mult(z0, S(0))), POWER(z0, S(0)))

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

(72)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, S(0)) -> S(z0)
   +(S(0), z0) -> S(z0)
   power(z0, S(z1)) -> mult(z0, power(z0, z1))
   power(z0, 0) -> S(0)
   mult(z0, S(z1)) -> add0(z0, mult(z0, z1))
   mult(z0, 0) -> 0
   add0(z0, S(z1)) -> +(S(0), add0(z0, z1))
   add0(z0, 0) -> z0
Tuples:
   ADD0(z0, S(z1)) -> c6(ADD0(z0, z1))
   POWER(z0, S(0)) -> c2(MULT(z0, S(0)))
   MULT(z0, S(S(z1))) -> c4(ADD0(z0, add0(z0, mult(z0, z1))), MULT(z0, S(z1)))
   MULT(z0, S(0)) -> c4
   POWER(z0, S(S(S(z1)))) -> c2(MULT(z0, mult(z0, mult(z0, power(z0, z1)))), POWER(z0, S(S(z1))))
   POWER(z0, S(S(0))) -> c2(MULT(z0, mult(z0, S(0))), POWER(z0, S(0)))
S tuples:
   ADD0(z0, S(z1)) -> c6(ADD0(z0, z1))
   MULT(z0, S(S(z1))) -> c4(ADD0(z0, add0(z0, mult(z0, z1))), MULT(z0, S(z1)))
K tuples:
   POWER(z0, S(z1)) -> c2(MULT(z0, power(z0, z1)), POWER(z0, z1))
   MULT(z0, S(0)) -> c4
Defined Rule Symbols:   power_2, mult_2, add0_2, +_2

Defined Pair Symbols:   ADD0_2, POWER_2, MULT_2

Compound Symbols:   c6_1, c2_1, c4_2, c4, c2_2


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

(73) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID))
Removed 2 trailing nodes:
   POWER(z0, S(0)) -> c2(MULT(z0, S(0)))
   MULT(z0, S(0)) -> c4

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

(74)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, S(0)) -> S(z0)
   +(S(0), z0) -> S(z0)
   power(z0, S(z1)) -> mult(z0, power(z0, z1))
   power(z0, 0) -> S(0)
   mult(z0, S(z1)) -> add0(z0, mult(z0, z1))
   mult(z0, 0) -> 0
   add0(z0, S(z1)) -> +(S(0), add0(z0, z1))
   add0(z0, 0) -> z0
Tuples:
   ADD0(z0, S(z1)) -> c6(ADD0(z0, z1))
   MULT(z0, S(S(z1))) -> c4(ADD0(z0, add0(z0, mult(z0, z1))), MULT(z0, S(z1)))
   POWER(z0, S(S(S(z1)))) -> c2(MULT(z0, mult(z0, mult(z0, power(z0, z1)))), POWER(z0, S(S(z1))))
   POWER(z0, S(S(0))) -> c2(MULT(z0, mult(z0, S(0))), POWER(z0, S(0)))
S tuples:
   ADD0(z0, S(z1)) -> c6(ADD0(z0, z1))
   MULT(z0, S(S(z1))) -> c4(ADD0(z0, add0(z0, mult(z0, z1))), MULT(z0, S(z1)))
K tuples:none
Defined Rule Symbols:   power_2, mult_2, add0_2, +_2

Defined Pair Symbols:   ADD0_2, MULT_2, POWER_2

Compound Symbols:   c6_1, c4_2, c2_2


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

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

(76)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, S(0)) -> S(z0)
   +(S(0), z0) -> S(z0)
   power(z0, S(z1)) -> mult(z0, power(z0, z1))
   power(z0, 0) -> S(0)
   mult(z0, S(z1)) -> add0(z0, mult(z0, z1))
   mult(z0, 0) -> 0
   add0(z0, S(z1)) -> +(S(0), add0(z0, z1))
   add0(z0, 0) -> z0
Tuples:
   ADD0(z0, S(z1)) -> c6(ADD0(z0, z1))
   MULT(z0, S(S(z1))) -> c4(ADD0(z0, add0(z0, mult(z0, z1))), MULT(z0, S(z1)))
   POWER(z0, S(S(S(z1)))) -> c2(MULT(z0, mult(z0, mult(z0, power(z0, z1)))), POWER(z0, S(S(z1))))
   POWER(z0, S(S(0))) -> c2(MULT(z0, mult(z0, S(0))))
S tuples:
   ADD0(z0, S(z1)) -> c6(ADD0(z0, z1))
   MULT(z0, S(S(z1))) -> c4(ADD0(z0, add0(z0, mult(z0, z1))), MULT(z0, S(z1)))
K tuples:none
Defined Rule Symbols:   power_2, mult_2, add0_2, +_2

Defined Pair Symbols:   ADD0_2, MULT_2, POWER_2

Compound Symbols:   c6_1, c4_2, c2_2, c2_1


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

(77) CdtNarrowingProof (BOTH BOUNDS(ID, ID))
Use narrowing to replace MULT(z0, S(S(z1))) -> c4(ADD0(z0, add0(z0, mult(z0, z1))), MULT(z0, S(z1))) by 
   MULT(z0, S(S(S(z1)))) -> c4(ADD0(z0, add0(z0, add0(z0, mult(z0, z1)))), MULT(z0, S(S(z1))))
   MULT(z0, S(S(0))) -> c4(ADD0(z0, add0(z0, 0)), MULT(z0, S(0)))

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

(78)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, S(0)) -> S(z0)
   +(S(0), z0) -> S(z0)
   power(z0, S(z1)) -> mult(z0, power(z0, z1))
   power(z0, 0) -> S(0)
   mult(z0, S(z1)) -> add0(z0, mult(z0, z1))
   mult(z0, 0) -> 0
   add0(z0, S(z1)) -> +(S(0), add0(z0, z1))
   add0(z0, 0) -> z0
Tuples:
   ADD0(z0, S(z1)) -> c6(ADD0(z0, z1))
   POWER(z0, S(S(S(z1)))) -> c2(MULT(z0, mult(z0, mult(z0, power(z0, z1)))), POWER(z0, S(S(z1))))
   POWER(z0, S(S(0))) -> c2(MULT(z0, mult(z0, S(0))))
   MULT(z0, S(S(S(z1)))) -> c4(ADD0(z0, add0(z0, add0(z0, mult(z0, z1)))), MULT(z0, S(S(z1))))
   MULT(z0, S(S(0))) -> c4(ADD0(z0, add0(z0, 0)), MULT(z0, S(0)))
S tuples:
   ADD0(z0, S(z1)) -> c6(ADD0(z0, z1))
   MULT(z0, S(S(S(z1)))) -> c4(ADD0(z0, add0(z0, add0(z0, mult(z0, z1)))), MULT(z0, S(S(z1))))
   MULT(z0, S(S(0))) -> c4(ADD0(z0, add0(z0, 0)), MULT(z0, S(0)))
K tuples:none
Defined Rule Symbols:   power_2, mult_2, add0_2, +_2

Defined Pair Symbols:   ADD0_2, POWER_2, MULT_2

Compound Symbols:   c6_1, c2_2, c2_1, c4_2


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

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

(80)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, S(0)) -> S(z0)
   +(S(0), z0) -> S(z0)
   power(z0, S(z1)) -> mult(z0, power(z0, z1))
   power(z0, 0) -> S(0)
   mult(z0, S(z1)) -> add0(z0, mult(z0, z1))
   mult(z0, 0) -> 0
   add0(z0, S(z1)) -> +(S(0), add0(z0, z1))
   add0(z0, 0) -> z0
Tuples:
   ADD0(z0, S(z1)) -> c6(ADD0(z0, z1))
   POWER(z0, S(S(S(z1)))) -> c2(MULT(z0, mult(z0, mult(z0, power(z0, z1)))), POWER(z0, S(S(z1))))
   POWER(z0, S(S(0))) -> c2(MULT(z0, mult(z0, S(0))))
   MULT(z0, S(S(S(z1)))) -> c4(ADD0(z0, add0(z0, add0(z0, mult(z0, z1)))), MULT(z0, S(S(z1))))
   MULT(z0, S(S(0))) -> c4(ADD0(z0, add0(z0, 0)))
S tuples:
   ADD0(z0, S(z1)) -> c6(ADD0(z0, z1))
   MULT(z0, S(S(S(z1)))) -> c4(ADD0(z0, add0(z0, add0(z0, mult(z0, z1)))), MULT(z0, S(S(z1))))
   MULT(z0, S(S(0))) -> c4(ADD0(z0, add0(z0, 0)))
K tuples:none
Defined Rule Symbols:   power_2, mult_2, add0_2, +_2

Defined Pair Symbols:   ADD0_2, POWER_2, MULT_2

Compound Symbols:   c6_1, c2_2, c2_1, c4_2, c4_1


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

(81) CdtRuleRemovalProof (UPPER BOUND(ADD(n^1)))
Found a reduction pair which oriented the following tuples strictly. Hence they can be removed from S.
   MULT(z0, S(S(0))) -> c4(ADD0(z0, add0(z0, 0)))
We considered the (Usable) Rules:none
And the Tuples:
   ADD0(z0, S(z1)) -> c6(ADD0(z0, z1))
   POWER(z0, S(S(S(z1)))) -> c2(MULT(z0, mult(z0, mult(z0, power(z0, z1)))), POWER(z0, S(S(z1))))
   POWER(z0, S(S(0))) -> c2(MULT(z0, mult(z0, S(0))))
   MULT(z0, S(S(S(z1)))) -> c4(ADD0(z0, add0(z0, add0(z0, mult(z0, z1)))), MULT(z0, S(S(z1))))
   MULT(z0, S(S(0))) -> c4(ADD0(z0, add0(z0, 0)))
The order we found is given by the following interpretation:

Polynomial interpretation :

   POL(+(x_1, x_2)) = x_1 + x_2
   POL(0) = 0
   POL(ADD0(x_1, x_2)) = 0
   POL(MULT(x_1, x_2)) = [1]
   POL(POWER(x_1, x_2)) = [1] + x_2
   POL(S(x_1)) = [1] + x_1
   POL(add0(x_1, x_2)) = [1] + x_2
   POL(c2(x_1)) = x_1
   POL(c2(x_1, x_2)) = x_1 + x_2
   POL(c4(x_1)) = x_1
   POL(c4(x_1, x_2)) = x_1 + x_2
   POL(c6(x_1)) = x_1
   POL(mult(x_1, x_2)) = [1] + x_2
   POL(power(x_1, x_2)) = [1] + x_1 + x_2

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

(82)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, S(0)) -> S(z0)
   +(S(0), z0) -> S(z0)
   power(z0, S(z1)) -> mult(z0, power(z0, z1))
   power(z0, 0) -> S(0)
   mult(z0, S(z1)) -> add0(z0, mult(z0, z1))
   mult(z0, 0) -> 0
   add0(z0, S(z1)) -> +(S(0), add0(z0, z1))
   add0(z0, 0) -> z0
Tuples:
   ADD0(z0, S(z1)) -> c6(ADD0(z0, z1))
   POWER(z0, S(S(S(z1)))) -> c2(MULT(z0, mult(z0, mult(z0, power(z0, z1)))), POWER(z0, S(S(z1))))
   POWER(z0, S(S(0))) -> c2(MULT(z0, mult(z0, S(0))))
   MULT(z0, S(S(S(z1)))) -> c4(ADD0(z0, add0(z0, add0(z0, mult(z0, z1)))), MULT(z0, S(S(z1))))
   MULT(z0, S(S(0))) -> c4(ADD0(z0, add0(z0, 0)))
S tuples:
   ADD0(z0, S(z1)) -> c6(ADD0(z0, z1))
   MULT(z0, S(S(S(z1)))) -> c4(ADD0(z0, add0(z0, add0(z0, mult(z0, z1)))), MULT(z0, S(S(z1))))
K tuples:
   MULT(z0, S(S(0))) -> c4(ADD0(z0, add0(z0, 0)))
Defined Rule Symbols:   power_2, mult_2, add0_2, +_2

Defined Pair Symbols:   ADD0_2, POWER_2, MULT_2

Compound Symbols:   c6_1, c2_2, c2_1, c4_2, c4_1


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

(83) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ADD0(z0, S(z1)) -> c6(ADD0(z0, z1)) by 
   ADD0(z0, S(S(y1))) -> c6(ADD0(z0, S(y1)))

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

(84)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, S(0)) -> S(z0)
   +(S(0), z0) -> S(z0)
   power(z0, S(z1)) -> mult(z0, power(z0, z1))
   power(z0, 0) -> S(0)
   mult(z0, S(z1)) -> add0(z0, mult(z0, z1))
   mult(z0, 0) -> 0
   add0(z0, S(z1)) -> +(S(0), add0(z0, z1))
   add0(z0, 0) -> z0
Tuples:
   POWER(z0, S(S(S(z1)))) -> c2(MULT(z0, mult(z0, mult(z0, power(z0, z1)))), POWER(z0, S(S(z1))))
   POWER(z0, S(S(0))) -> c2(MULT(z0, mult(z0, S(0))))
   MULT(z0, S(S(S(z1)))) -> c4(ADD0(z0, add0(z0, add0(z0, mult(z0, z1)))), MULT(z0, S(S(z1))))
   MULT(z0, S(S(0))) -> c4(ADD0(z0, add0(z0, 0)))
   ADD0(z0, S(S(y1))) -> c6(ADD0(z0, S(y1)))
S tuples:
   MULT(z0, S(S(S(z1)))) -> c4(ADD0(z0, add0(z0, add0(z0, mult(z0, z1)))), MULT(z0, S(S(z1))))
   ADD0(z0, S(S(y1))) -> c6(ADD0(z0, S(y1)))
K tuples:
   MULT(z0, S(S(0))) -> c4(ADD0(z0, add0(z0, 0)))
Defined Rule Symbols:   power_2, mult_2, add0_2, +_2

Defined Pair Symbols:   POWER_2, MULT_2, ADD0_2

Compound Symbols:   c2_2, c2_1, c4_2, c4_1, c6_1


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

(85) CdtRewritingProof (BOTH BOUNDS(ID, ID))
Used rewriting to replace MULT(z0, S(S(0))) -> c4(ADD0(z0, add0(z0, 0))) by MULT(z0, S(S(0))) -> c4(ADD0(z0, z0))
----------------------------------------

(86)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, S(0)) -> S(z0)
   +(S(0), z0) -> S(z0)
   power(z0, S(z1)) -> mult(z0, power(z0, z1))
   power(z0, 0) -> S(0)
   mult(z0, S(z1)) -> add0(z0, mult(z0, z1))
   mult(z0, 0) -> 0
   add0(z0, S(z1)) -> +(S(0), add0(z0, z1))
   add0(z0, 0) -> z0
Tuples:
   POWER(z0, S(S(S(z1)))) -> c2(MULT(z0, mult(z0, mult(z0, power(z0, z1)))), POWER(z0, S(S(z1))))
   POWER(z0, S(S(0))) -> c2(MULT(z0, mult(z0, S(0))))
   MULT(z0, S(S(S(z1)))) -> c4(ADD0(z0, add0(z0, add0(z0, mult(z0, z1)))), MULT(z0, S(S(z1))))
   ADD0(z0, S(S(y1))) -> c6(ADD0(z0, S(y1)))
   MULT(z0, S(S(0))) -> c4(ADD0(z0, z0))
S tuples:
   MULT(z0, S(S(S(z1)))) -> c4(ADD0(z0, add0(z0, add0(z0, mult(z0, z1)))), MULT(z0, S(S(z1))))
   ADD0(z0, S(S(y1))) -> c6(ADD0(z0, S(y1)))
K tuples:
   MULT(z0, S(S(0))) -> c4(ADD0(z0, add0(z0, 0)))
Defined Rule Symbols:   power_2, mult_2, add0_2, +_2

Defined Pair Symbols:   POWER_2, MULT_2, ADD0_2

Compound Symbols:   c2_2, c2_1, c4_2, c6_1, c4_1


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

(87) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace ADD0(z0, S(S(y1))) -> c6(ADD0(z0, S(y1))) by 
   ADD0(z0, S(S(S(y1)))) -> c6(ADD0(z0, S(S(y1))))

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

(88)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, S(0)) -> S(z0)
   +(S(0), z0) -> S(z0)
   power(z0, S(z1)) -> mult(z0, power(z0, z1))
   power(z0, 0) -> S(0)
   mult(z0, S(z1)) -> add0(z0, mult(z0, z1))
   mult(z0, 0) -> 0
   add0(z0, S(z1)) -> +(S(0), add0(z0, z1))
   add0(z0, 0) -> z0
Tuples:
   POWER(z0, S(S(S(z1)))) -> c2(MULT(z0, mult(z0, mult(z0, power(z0, z1)))), POWER(z0, S(S(z1))))
   POWER(z0, S(S(0))) -> c2(MULT(z0, mult(z0, S(0))))
   MULT(z0, S(S(S(z1)))) -> c4(ADD0(z0, add0(z0, add0(z0, mult(z0, z1)))), MULT(z0, S(S(z1))))
   MULT(z0, S(S(0))) -> c4(ADD0(z0, z0))
   ADD0(z0, S(S(S(y1)))) -> c6(ADD0(z0, S(S(y1))))
S tuples:
   MULT(z0, S(S(S(z1)))) -> c4(ADD0(z0, add0(z0, add0(z0, mult(z0, z1)))), MULT(z0, S(S(z1))))
   ADD0(z0, S(S(S(y1)))) -> c6(ADD0(z0, S(S(y1))))
K tuples:
   MULT(z0, S(S(0))) -> c4(ADD0(z0, add0(z0, 0)))
Defined Rule Symbols:   power_2, mult_2, add0_2, +_2

Defined Pair Symbols:   POWER_2, MULT_2, ADD0_2

Compound Symbols:   c2_2, c2_1, c4_2, c4_1, c6_1


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

(89) CdtForwardInstantiationProof (BOTH BOUNDS(ID, ID))
Use forward instantiation to replace MULT(z0, S(S(0))) -> c4(ADD0(z0, z0)) by 
   MULT(S(S(S(y1))), S(S(0))) -> c4(ADD0(S(S(S(y1))), S(S(S(y1)))))

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

(90)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   +(z0, S(0)) -> S(z0)
   +(S(0), z0) -> S(z0)
   power(z0, S(z1)) -> mult(z0, power(z0, z1))
   power(z0, 0) -> S(0)
   mult(z0, S(z1)) -> add0(z0, mult(z0, z1))
   mult(z0, 0) -> 0
   add0(z0, S(z1)) -> +(S(0), add0(z0, z1))
   add0(z0, 0) -> z0
Tuples:
   POWER(z0, S(S(S(z1)))) -> c2(MULT(z0, mult(z0, mult(z0, power(z0, z1)))), POWER(z0, S(S(z1))))
   POWER(z0, S(S(0))) -> c2(MULT(z0, mult(z0, S(0))))
   MULT(z0, S(S(S(z1)))) -> c4(ADD0(z0, add0(z0, add0(z0, mult(z0, z1)))), MULT(z0, S(S(z1))))
   ADD0(z0, S(S(S(y1)))) -> c6(ADD0(z0, S(S(y1))))
   MULT(S(S(S(y1))), S(S(0))) -> c4(ADD0(S(S(S(y1))), S(S(S(y1)))))
S tuples:
   MULT(z0, S(S(S(z1)))) -> c4(ADD0(z0, add0(z0, add0(z0, mult(z0, z1)))), MULT(z0, S(S(z1))))
   ADD0(z0, S(S(S(y1)))) -> c6(ADD0(z0, S(S(y1))))
K tuples:
   MULT(z0, S(S(0))) -> c4(ADD0(z0, add0(z0, 0)))
Defined Rule Symbols:   power_2, mult_2, add0_2, +_2

Defined Pair Symbols:   POWER_2, MULT_2, ADD0_2

Compound Symbols:   c2_2, c2_1, c4_2, c6_1, c4_1