/export/starexec/sandbox/solver/bin/starexec_run_complexity /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
(2) CpxWeightedTrs
    (3) CpxWeightedTrsRenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxWeightedTrs
    (5) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (6) CpxTypedWeightedTrs
    (7) CompletionProof [UPPER BOUND(ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (10) CpxTypedWeightedCompleteTrs
    (11) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (12) CpxRNTS
    (13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRNTS
    (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 5 ms]
    (16) CpxRNTS
    (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 312 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 117 ms]
    (22) CpxRNTS
    (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 383 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 165 ms]
    (28) CpxRNTS
    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 1463 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 175 ms]
    (34) CpxRNTS
    (35) FinalProof [FINISHED, 0 ms]
    (36) BOUNDS(1, n^2)
(37) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(38) TRS for Loop Detection
    (39) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (40) BEST
        (41) proven lower bound
            (42) LowerBoundPropagationProof [FINISHED, 0 ms]
            (43) BOUNDS(n^1, INF)
        (44) TRS for Loop Detection


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

(0)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   exp(x, 0) -> s(0)
   exp(x, s(y)) -> *(x, exp(x, y))
   *(0, y) -> 0
   *(s(x), y) -> +(y, *(x, y))
   -(0, y) -> 0
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)

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

(1) RelTrsToWeightedTrsProof (BOTH BOUNDS(ID, ID))
Transformed relative TRS to weighted TRS
----------------------------------------

(2)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   exp(x, 0) -> s(0) [1]
   exp(x, s(y)) -> *(x, exp(x, y)) [1]
   *(0, y) -> 0 [1]
   *(s(x), y) -> +(y, *(x, y)) [1]
   -(0, y) -> 0 [1]
   -(x, 0) -> x [1]
   -(s(x), s(y)) -> -(x, y) [1]

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

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

* => times
- => minus

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

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^2).


The TRS R consists of the following rules:

   exp(x, 0) -> s(0) [1]
   exp(x, s(y)) -> times(x, exp(x, y)) [1]
   times(0, y) -> 0 [1]
   times(s(x), y) -> +(y, times(x, y)) [1]
   minus(0, y) -> 0 [1]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]

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

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

(6)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   exp(x, 0) -> s(0) [1]
   exp(x, s(y)) -> times(x, exp(x, y)) [1]
   times(0, y) -> 0 [1]
   times(s(x), y) -> +(y, times(x, y)) [1]
   minus(0, y) -> 0 [1]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]

The TRS has the following type information:
exp :: 0:s:+ -> 0:s:+ -> 0:s:+
0 :: 0:s:+
s :: 0:s:+ -> 0:s:+
times :: 0:s:+ -> 0:s:+ -> 0:s:+
+ :: 0:s:+ -> 0:s:+ -> 0:s:+
minus :: 0:s:+ -> 0:s:+ -> 0:s:+

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

(7) 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:

   minus_2

(c) The following functions are completely defined:

   exp_2
   times_2

Due to the following rules being added:

   exp(v0, v1) -> 0 [0]
   times(v0, v1) -> 0 [0]

And the following fresh constants: none

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

(8)
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:

   exp(x, 0) -> s(0) [1]
   exp(x, s(y)) -> times(x, exp(x, y)) [1]
   times(0, y) -> 0 [1]
   times(s(x), y) -> +(y, times(x, y)) [1]
   minus(0, y) -> 0 [1]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   exp(v0, v1) -> 0 [0]
   times(v0, v1) -> 0 [0]

The TRS has the following type information:
exp :: 0:s:+ -> 0:s:+ -> 0:s:+
0 :: 0:s:+
s :: 0:s:+ -> 0:s:+
times :: 0:s:+ -> 0:s:+ -> 0:s:+
+ :: 0:s:+ -> 0:s:+ -> 0:s:+
minus :: 0:s:+ -> 0:s:+ -> 0:s:+

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

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

(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:

   exp(x, 0) -> s(0) [1]
   exp(x, s(0)) -> times(x, s(0)) [2]
   exp(x, s(s(y'))) -> times(x, times(x, exp(x, y'))) [2]
   exp(x, s(y)) -> times(x, 0) [1]
   times(0, y) -> 0 [1]
   times(s(x), y) -> +(y, times(x, y)) [1]
   minus(0, y) -> 0 [1]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   exp(v0, v1) -> 0 [0]
   times(v0, v1) -> 0 [0]

The TRS has the following type information:
exp :: 0:s:+ -> 0:s:+ -> 0:s:+
0 :: 0:s:+
s :: 0:s:+ -> 0:s:+
times :: 0:s:+ -> 0:s:+ -> 0:s:+
+ :: 0:s:+ -> 0:s:+ -> 0:s:+
minus :: 0:s:+ -> 0:s:+ -> 0:s:+

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

(11) 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

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

(12)
Obligation:
Complexity RNTS consisting of the following rules:

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


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

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

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

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


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

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

   { times }
   { minus }
   { exp }

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

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

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

Function symbols to be analyzed: {times}, {minus}, {exp}

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

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

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

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

Function symbols to be analyzed: {times}, {minus}, {exp}

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

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

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

   exp(z, z') -{ 3 + z }-> s :|: s >= 0, s <= (1 + 0) * z + z, z >= 0, z' = 1 + 0
   exp(z, z') -{ 2 + z }-> s' :|: s' >= 0, s' <= 0 * z + z, z >= 0, z' - 1 >= 0
   exp(z, z') -{ 2 }-> times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   exp(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   exp(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 1 + z }-> 1 + z' + s'' :|: s'' >= 0, s'' <= z' * (z - 1) + (z - 1), z - 1 >= 0, z' >= 0

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

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   exp(z, z') -{ 3 + z }-> s :|: s >= 0, s <= (1 + 0) * z + z, z >= 0, z' = 1 + 0
   exp(z, z') -{ 2 + z }-> s' :|: s' >= 0, s' <= 0 * z + z, z >= 0, z' - 1 >= 0
   exp(z, z') -{ 2 }-> times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   exp(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   exp(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 1 + z }-> 1 + z' + s'' :|: s'' >= 0, s'' <= z' * (z - 1) + (z - 1), z - 1 >= 0, z' >= 0

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

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   exp(z, z') -{ 3 + z }-> s :|: s >= 0, s <= (1 + 0) * z + z, z >= 0, z' = 1 + 0
   exp(z, z') -{ 2 + z }-> s' :|: s' >= 0, s' <= 0 * z + z, z >= 0, z' - 1 >= 0
   exp(z, z') -{ 2 }-> times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   exp(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   exp(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 1 + z }-> 1 + z' + s'' :|: s'' >= 0, s'' <= z' * (z - 1) + (z - 1), z - 1 >= 0, z' >= 0

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

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

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

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

   exp(z, z') -{ 3 + z }-> s :|: s >= 0, s <= (1 + 0) * z + z, z >= 0, z' = 1 + 0
   exp(z, z') -{ 2 + z }-> s' :|: s' >= 0, s' <= 0 * z + z, z >= 0, z' - 1 >= 0
   exp(z, z') -{ 2 }-> times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   exp(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   exp(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 2 + z' }-> s1 :|: s1 >= 0, s1 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 1 + z }-> 1 + z' + s'' :|: s'' >= 0, s'' <= z' * (z - 1) + (z - 1), z - 1 >= 0, z' >= 0

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

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   exp(z, z') -{ 3 + z }-> s :|: s >= 0, s <= (1 + 0) * z + z, z >= 0, z' = 1 + 0
   exp(z, z') -{ 2 + z }-> s' :|: s' >= 0, s' <= 0 * z + z, z >= 0, z' - 1 >= 0
   exp(z, z') -{ 2 }-> times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   exp(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   exp(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 2 + z' }-> s1 :|: s1 >= 0, s1 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 1 + z }-> 1 + z' + s'' :|: s'' >= 0, s'' <= z' * (z - 1) + (z - 1), z - 1 >= 0, z' >= 0

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

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(33) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: exp
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 6 + 2*z + 2*z*z' + 4*z'

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

   exp(z, z') -{ 3 + z }-> s :|: s >= 0, s <= (1 + 0) * z + z, z >= 0, z' = 1 + 0
   exp(z, z') -{ 2 + z }-> s' :|: s' >= 0, s' <= 0 * z + z, z >= 0, z' - 1 >= 0
   exp(z, z') -{ 2 }-> times(z, times(z, exp(z, z' - 2))) :|: z >= 0, z' - 2 >= 0
   exp(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   exp(z, z') -{ 1 }-> 1 + 0 :|: z >= 0, z' = 0
   minus(z, z') -{ 2 + z' }-> s1 :|: s1 >= 0, s1 <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   times(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   times(z, z') -{ 1 + z }-> 1 + z' + s'' :|: s'' >= 0, s'' <= z' * (z - 1) + (z - 1), z - 1 >= 0, z' >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  times: runtime: O(n^1) [1 + z], size: O(n^2) [z + z*z']
  minus: runtime: O(n^1) [2 + z'], size: O(n^1) [z]
  exp: runtime: O(n^2) [6 + 2*z + 2*z*z' + 4*z'], size: INF

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(35) FinalProof (FINISHED)
Computed overall runtime complexity
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(36)
BOUNDS(1, n^2)

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(37) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(38)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   exp(x, 0) -> s(0)
   exp(x, s(y)) -> *(x, exp(x, y))
   *(0, y) -> 0
   *(s(x), y) -> +(y, *(x, y))
   -(0, y) -> 0
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)

S is empty.
Rewrite Strategy: INNERMOST
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(39) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

exp(x, s(y)) ->^+ *(x, exp(x, y))

gives rise to a decreasing loop by considering the right hand sides subterm at position [1].

The pumping substitution is [y / s(y)].

The result substitution is [ ].




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(40)
Complex Obligation (BEST)

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(41)
Obligation:
Proved the lower bound n^1 for the following obligation:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   exp(x, 0) -> s(0)
   exp(x, s(y)) -> *(x, exp(x, y))
   *(0, y) -> 0
   *(s(x), y) -> +(y, *(x, y))
   -(0, y) -> 0
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)

S is empty.
Rewrite Strategy: INNERMOST
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(42) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(43)
BOUNDS(n^1, INF)

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(44)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, n^2).


The TRS R consists of the following rules:

   exp(x, 0) -> s(0)
   exp(x, s(y)) -> *(x, exp(x, y))
   *(0, y) -> 0
   *(s(x), y) -> +(y, *(x, y))
   -(0, y) -> 0
   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)

S is empty.
Rewrite Strategy: INNERMOST