WORST_CASE(?,O(n^3))
proof of input_PzlPSpsDxr.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3).

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [UPPER BOUND(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), 0 ms]
(16) CpxRNTS
(17) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(18) CpxRNTS
(19) IntTrsBoundProof [UPPER BOUND(ID), 156 ms]
(20) CpxRNTS
(21) IntTrsBoundProof [UPPER BOUND(ID), 67 ms]
(22) CpxRNTS
(23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(24) CpxRNTS
(25) IntTrsBoundProof [UPPER BOUND(ID), 303 ms]
(26) CpxRNTS
(27) IntTrsBoundProof [UPPER BOUND(ID), 86 ms]
(28) CpxRNTS
(29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(30) CpxRNTS
(31) IntTrsBoundProof [UPPER BOUND(ID), 1256 ms]
(32) CpxRNTS
(33) IntTrsBoundProof [UPPER BOUND(ID), 132 ms]
(34) CpxRNTS
(35) FinalProof [FINISHED, 0 ms]
(36) BOUNDS(1, n^3)


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

(0)
Obligation:
The Runtime Complexity (parallel-innermost) of the given CpxTRS could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

   +(0, y) -> y
   +(s(x), y) -> s(+(x, y))
   +(p(x), y) -> p(+(x, y))
   minus(0) -> 0
   minus(s(x)) -> p(minus(x))
   minus(p(x)) -> s(minus(x))
   *(0, y) -> 0
   *(s(x), y) -> +(*(x, y), y)
   *(p(x), y) -> +(*(x, y), minus(y))

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

(1) RelTrsToWeightedTrsProof (UPPER BOUND(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^3).


The TRS R consists of the following rules:

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

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

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

+ => plus
* => times

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

(4)
Obligation:
The Runtime Complexity (innermost) of the given CpxWeightedTrs could be proven to be BOUNDS(1, n^3).


The TRS R consists of the following rules:

   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   plus(p(x), y) -> p(plus(x, y)) [1]
   minus(0) -> 0 [1]
   minus(s(x)) -> p(minus(x)) [1]
   minus(p(x)) -> s(minus(x)) [1]
   times(0, y) -> 0 [1]
   times(s(x), y) -> plus(times(x, y), y) [1]
   times(p(x), y) -> plus(times(x, y), minus(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:

   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   plus(p(x), y) -> p(plus(x, y)) [1]
   minus(0) -> 0 [1]
   minus(s(x)) -> p(minus(x)) [1]
   minus(p(x)) -> s(minus(x)) [1]
   times(0, y) -> 0 [1]
   times(s(x), y) -> plus(times(x, y), y) [1]
   times(p(x), y) -> plus(times(x, y), minus(y)) [1]

The TRS has the following type information:
plus :: 0:s:p -> 0:s:p -> 0:s:p
0 :: 0:s:p
s :: 0:s:p -> 0:s:p
p :: 0:s:p -> 0:s:p
minus :: 0:s:p -> 0:s:p
times :: 0:s:p -> 0:s:p -> 0:s:p

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:
none

(c) The following functions are completely defined:

   times_2
   minus_1
   plus_2

Due to the following rules being added:
none

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:

   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   plus(p(x), y) -> p(plus(x, y)) [1]
   minus(0) -> 0 [1]
   minus(s(x)) -> p(minus(x)) [1]
   minus(p(x)) -> s(minus(x)) [1]
   times(0, y) -> 0 [1]
   times(s(x), y) -> plus(times(x, y), y) [1]
   times(p(x), y) -> plus(times(x, y), minus(y)) [1]

The TRS has the following type information:
plus :: 0:s:p -> 0:s:p -> 0:s:p
0 :: 0:s:p
s :: 0:s:p -> 0:s:p
p :: 0:s:p -> 0:s:p
minus :: 0:s:p -> 0:s:p
times :: 0:s:p -> 0:s:p -> 0:s:p

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:

   plus(0, y) -> y [1]
   plus(s(x), y) -> s(plus(x, y)) [1]
   plus(p(x), y) -> p(plus(x, y)) [1]
   minus(0) -> 0 [1]
   minus(s(x)) -> p(minus(x)) [1]
   minus(p(x)) -> s(minus(x)) [1]
   times(0, y) -> 0 [1]
   times(s(0), y) -> plus(0, y) [2]
   times(s(s(x')), y) -> plus(plus(times(x', y), y), y) [2]
   times(s(p(x'')), y) -> plus(plus(times(x'', y), minus(y)), y) [2]
   times(p(0), 0) -> plus(0, 0) [3]
   times(p(0), s(x3)) -> plus(0, p(minus(x3))) [3]
   times(p(0), p(x4)) -> plus(0, s(minus(x4))) [3]
   times(p(s(x1)), 0) -> plus(plus(times(x1, 0), 0), 0) [3]
   times(p(s(x1)), s(x5)) -> plus(plus(times(x1, s(x5)), s(x5)), p(minus(x5))) [3]
   times(p(s(x1)), p(x6)) -> plus(plus(times(x1, p(x6)), p(x6)), s(minus(x6))) [3]
   times(p(p(x2)), 0) -> plus(plus(times(x2, 0), minus(0)), 0) [3]
   times(p(p(x2)), s(x7)) -> plus(plus(times(x2, s(x7)), minus(s(x7))), p(minus(x7))) [3]
   times(p(p(x2)), p(x8)) -> plus(plus(times(x2, p(x8)), minus(p(x8))), s(minus(x8))) [3]

The TRS has the following type information:
plus :: 0:s:p -> 0:s:p -> 0:s:p
0 :: 0:s:p
s :: 0:s:p -> 0:s:p
p :: 0:s:p -> 0:s:p
minus :: 0:s:p -> 0:s:p
times :: 0:s:p -> 0:s:p -> 0:s:p

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:

   minus(z) -{ 1 }-> 0 :|: z = 0
   minus(z) -{ 1 }-> 1 + minus(x) :|: x >= 0, z = 1 + x
   plus(z, z') -{ 1 }-> y :|: y >= 0, z = 0, z' = y
   plus(z, z') -{ 1 }-> 1 + plus(x, y) :|: x >= 0, y >= 0, z = 1 + x, z' = y
   times(z, z') -{ 2 }-> plus(plus(times(x', y), y), y) :|: x' >= 0, y >= 0, z = 1 + (1 + x'), z' = y
   times(z, z') -{ 2 }-> plus(plus(times(x'', y), minus(y)), y) :|: y >= 0, x'' >= 0, z = 1 + (1 + x''), z' = y
   times(z, z') -{ 3 }-> plus(plus(times(x1, 0), 0), 0) :|: z = 1 + (1 + x1), x1 >= 0, z' = 0
   times(z, z') -{ 3 }-> plus(plus(times(x1, 1 + x5), 1 + x5), 1 + minus(x5)) :|: z = 1 + (1 + x1), x1 >= 0, x5 >= 0, z' = 1 + x5
   times(z, z') -{ 3 }-> plus(plus(times(x1, 1 + x6), 1 + x6), 1 + minus(x6)) :|: z' = 1 + x6, z = 1 + (1 + x1), x1 >= 0, x6 >= 0
   times(z, z') -{ 3 }-> plus(plus(times(x2, 0), minus(0)), 0) :|: z = 1 + (1 + x2), x2 >= 0, z' = 0
   times(z, z') -{ 3 }-> plus(plus(times(x2, 1 + x7), minus(1 + x7)), 1 + minus(x7)) :|: z' = 1 + x7, z = 1 + (1 + x2), x7 >= 0, x2 >= 0
   times(z, z') -{ 3 }-> plus(plus(times(x2, 1 + x8), minus(1 + x8)), 1 + minus(x8)) :|: z = 1 + (1 + x2), z' = 1 + x8, x8 >= 0, x2 >= 0
   times(z, z') -{ 2 }-> plus(0, y) :|: z = 1 + 0, y >= 0, z' = y
   times(z, z') -{ 3 }-> plus(0, 0) :|: z = 1 + 0, z' = 0
   times(z, z') -{ 3 }-> plus(0, 1 + minus(x3)) :|: z' = 1 + x3, z = 1 + 0, x3 >= 0
   times(z, z') -{ 3 }-> plus(0, 1 + minus(x4)) :|: x4 >= 0, z = 1 + 0, z' = 1 + x4
   times(z, z') -{ 1 }-> 0 :|: y >= 0, z = 0, 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:

   minus(z) -{ 1 }-> 0 :|: z = 0
   minus(z) -{ 1 }-> 1 + minus(z - 1) :|: z - 1 >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), minus(z')), z') :|: z' >= 0, z - 2 >= 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), minus(0)), 0) :|: z - 2 >= 0, z' = 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), 0), 0) :|: z - 2 >= 0, z' = 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 1 + (z' - 1)), minus(1 + (z' - 1))), 1 + minus(z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 1 + (z' - 1)), 1 + (z' - 1)), 1 + minus(z' - 1)) :|: z - 2 >= 0, z' - 1 >= 0
   times(z, z') -{ 2 }-> plus(0, z') :|: z = 1 + 0, z' >= 0
   times(z, z') -{ 3 }-> plus(0, 0) :|: z = 1 + 0, z' = 0
   times(z, z') -{ 3 }-> plus(0, 1 + minus(z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0


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

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

   { minus }
   { plus }
   { times }

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

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

   minus(z) -{ 1 }-> 0 :|: z = 0
   minus(z) -{ 1 }-> 1 + minus(z - 1) :|: z - 1 >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), minus(z')), z') :|: z' >= 0, z - 2 >= 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), minus(0)), 0) :|: z - 2 >= 0, z' = 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), 0), 0) :|: z - 2 >= 0, z' = 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 1 + (z' - 1)), minus(1 + (z' - 1))), 1 + minus(z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 1 + (z' - 1)), 1 + (z' - 1)), 1 + minus(z' - 1)) :|: z - 2 >= 0, z' - 1 >= 0
   times(z, z') -{ 2 }-> plus(0, z') :|: z = 1 + 0, z' >= 0
   times(z, z') -{ 3 }-> plus(0, 0) :|: z = 1 + 0, z' = 0
   times(z, z') -{ 3 }-> plus(0, 1 + minus(z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

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

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

(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:

   minus(z) -{ 1 }-> 0 :|: z = 0
   minus(z) -{ 1 }-> 1 + minus(z - 1) :|: z - 1 >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), minus(z')), z') :|: z' >= 0, z - 2 >= 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), minus(0)), 0) :|: z - 2 >= 0, z' = 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), 0), 0) :|: z - 2 >= 0, z' = 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 1 + (z' - 1)), minus(1 + (z' - 1))), 1 + minus(z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 1 + (z' - 1)), 1 + (z' - 1)), 1 + minus(z' - 1)) :|: z - 2 >= 0, z' - 1 >= 0
   times(z, z') -{ 2 }-> plus(0, z') :|: z = 1 + 0, z' >= 0
   times(z, z') -{ 3 }-> plus(0, 0) :|: z = 1 + 0, z' = 0
   times(z, z') -{ 3 }-> plus(0, 1 + minus(z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

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

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   minus(z) -{ 1 }-> 0 :|: z = 0
   minus(z) -{ 1 }-> 1 + minus(z - 1) :|: z - 1 >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), minus(z')), z') :|: z' >= 0, z - 2 >= 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), minus(0)), 0) :|: z - 2 >= 0, z' = 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), 0), 0) :|: z - 2 >= 0, z' = 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 1 + (z' - 1)), minus(1 + (z' - 1))), 1 + minus(z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 1 + (z' - 1)), 1 + (z' - 1)), 1 + minus(z' - 1)) :|: z - 2 >= 0, z' - 1 >= 0
   times(z, z') -{ 2 }-> plus(0, z') :|: z = 1 + 0, z' >= 0
   times(z, z') -{ 3 }-> plus(0, 0) :|: z = 1 + 0, z' = 0
   times(z, z') -{ 3 }-> plus(0, 1 + minus(z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

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

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: minus
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:

   minus(z) -{ 1 }-> 0 :|: z = 0
   minus(z) -{ 1 }-> 1 + minus(z - 1) :|: z - 1 >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), minus(z')), z') :|: z' >= 0, z - 2 >= 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), minus(0)), 0) :|: z - 2 >= 0, z' = 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), 0), 0) :|: z - 2 >= 0, z' = 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 1 + (z' - 1)), minus(1 + (z' - 1))), 1 + minus(z' - 1)) :|: z' - 1 >= 0, z - 2 >= 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 1 + (z' - 1)), 1 + (z' - 1)), 1 + minus(z' - 1)) :|: z - 2 >= 0, z' - 1 >= 0
   times(z, z') -{ 2 }-> plus(0, z') :|: z = 1 + 0, z' >= 0
   times(z, z') -{ 3 }-> plus(0, 0) :|: z = 1 + 0, z' = 0
   times(z, z') -{ 3 }-> plus(0, 1 + minus(z' - 1)) :|: z = 1 + 0, z' - 1 >= 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

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

   minus(z) -{ 1 }-> 0 :|: z = 0
   minus(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   times(z, z') -{ 3 + z' }-> plus(plus(times(z - 2, z'), s'), z') :|: s' >= 0, s' <= z', z' >= 0, z - 2 >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0
   times(z, z') -{ 4 }-> plus(plus(times(z - 2, 0), s2), 0) :|: s2 >= 0, s2 <= 0, z - 2 >= 0, z' = 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), 0), 0) :|: z - 2 >= 0, z' = 0
   times(z, z') -{ 4 + 2*z' }-> plus(plus(times(z - 2, 1 + (z' - 1)), s3), 1 + s4) :|: s3 >= 0, s3 <= 1 + (z' - 1), s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 2 >= 0
   times(z, z') -{ 3 + z' }-> plus(plus(times(z - 2, 1 + (z' - 1)), 1 + (z' - 1)), 1 + s1) :|: s1 >= 0, s1 <= z' - 1, z - 2 >= 0, z' - 1 >= 0
   times(z, z') -{ 2 }-> plus(0, z') :|: z = 1 + 0, z' >= 0
   times(z, z') -{ 3 }-> plus(0, 0) :|: z = 1 + 0, z' = 0
   times(z, z') -{ 3 + z' }-> plus(0, 1 + s'') :|: s'' >= 0, s'' <= z' - 1, z = 1 + 0, z' - 1 >= 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

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

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

(25) 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'

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

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

   minus(z) -{ 1 }-> 0 :|: z = 0
   minus(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   times(z, z') -{ 3 + z' }-> plus(plus(times(z - 2, z'), s'), z') :|: s' >= 0, s' <= z', z' >= 0, z - 2 >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0
   times(z, z') -{ 4 }-> plus(plus(times(z - 2, 0), s2), 0) :|: s2 >= 0, s2 <= 0, z - 2 >= 0, z' = 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), 0), 0) :|: z - 2 >= 0, z' = 0
   times(z, z') -{ 4 + 2*z' }-> plus(plus(times(z - 2, 1 + (z' - 1)), s3), 1 + s4) :|: s3 >= 0, s3 <= 1 + (z' - 1), s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 2 >= 0
   times(z, z') -{ 3 + z' }-> plus(plus(times(z - 2, 1 + (z' - 1)), 1 + (z' - 1)), 1 + s1) :|: s1 >= 0, s1 <= z' - 1, z - 2 >= 0, z' - 1 >= 0
   times(z, z') -{ 2 }-> plus(0, z') :|: z = 1 + 0, z' >= 0
   times(z, z') -{ 3 }-> plus(0, 0) :|: z = 1 + 0, z' = 0
   times(z, z') -{ 3 + z' }-> plus(0, 1 + s'') :|: s'' >= 0, s'' <= z' - 1, z = 1 + 0, z' - 1 >= 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

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

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(27) 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

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

   minus(z) -{ 1 }-> 0 :|: z = 0
   minus(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 }-> 1 + plus(z - 1, z') :|: z - 1 >= 0, z' >= 0
   times(z, z') -{ 3 + z' }-> plus(plus(times(z - 2, z'), s'), z') :|: s' >= 0, s' <= z', z' >= 0, z - 2 >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0
   times(z, z') -{ 4 }-> plus(plus(times(z - 2, 0), s2), 0) :|: s2 >= 0, s2 <= 0, z - 2 >= 0, z' = 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), 0), 0) :|: z - 2 >= 0, z' = 0
   times(z, z') -{ 4 + 2*z' }-> plus(plus(times(z - 2, 1 + (z' - 1)), s3), 1 + s4) :|: s3 >= 0, s3 <= 1 + (z' - 1), s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 2 >= 0
   times(z, z') -{ 3 + z' }-> plus(plus(times(z - 2, 1 + (z' - 1)), 1 + (z' - 1)), 1 + s1) :|: s1 >= 0, s1 <= z' - 1, z - 2 >= 0, z' - 1 >= 0
   times(z, z') -{ 2 }-> plus(0, z') :|: z = 1 + 0, z' >= 0
   times(z, z') -{ 3 }-> plus(0, 0) :|: z = 1 + 0, z' = 0
   times(z, z') -{ 3 + z' }-> plus(0, 1 + s'') :|: s'' >= 0, s'' <= z' - 1, z = 1 + 0, z' - 1 >= 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

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

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(29) ResultPropagationProof (UPPER BOUND(ID))
Applied inner abstraction using the recently inferred runtime/size bounds where possible.
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(30)
Obligation:
Complexity RNTS consisting of the following rules:

   minus(z) -{ 1 }-> 0 :|: z = 0
   minus(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 + z }-> 1 + s5 :|: s5 >= 0, s5 <= z - 1 + z', z - 1 >= 0, z' >= 0
   times(z, z') -{ 3 }-> s6 :|: s6 >= 0, s6 <= 0 + z', z = 1 + 0, z' >= 0
   times(z, z') -{ 4 }-> s7 :|: s7 >= 0, s7 <= 0 + 0, z = 1 + 0, z' = 0
   times(z, z') -{ 4 + z' }-> s8 :|: s8 >= 0, s8 <= 0 + (1 + s''), s'' >= 0, s'' <= z' - 1, z = 1 + 0, z' - 1 >= 0
   times(z, z') -{ 3 + z' }-> plus(plus(times(z - 2, z'), s'), z') :|: s' >= 0, s' <= z', z' >= 0, z - 2 >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0
   times(z, z') -{ 4 }-> plus(plus(times(z - 2, 0), s2), 0) :|: s2 >= 0, s2 <= 0, z - 2 >= 0, z' = 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), 0), 0) :|: z - 2 >= 0, z' = 0
   times(z, z') -{ 4 + 2*z' }-> plus(plus(times(z - 2, 1 + (z' - 1)), s3), 1 + s4) :|: s3 >= 0, s3 <= 1 + (z' - 1), s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 2 >= 0
   times(z, z') -{ 3 + z' }-> plus(plus(times(z - 2, 1 + (z' - 1)), 1 + (z' - 1)), 1 + s1) :|: s1 >= 0, s1 <= z' - 1, z - 2 >= 0, z' - 1 >= 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

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

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(31) 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: 2*z*z' + 3*z'

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

   minus(z) -{ 1 }-> 0 :|: z = 0
   minus(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 + z }-> 1 + s5 :|: s5 >= 0, s5 <= z - 1 + z', z - 1 >= 0, z' >= 0
   times(z, z') -{ 3 }-> s6 :|: s6 >= 0, s6 <= 0 + z', z = 1 + 0, z' >= 0
   times(z, z') -{ 4 }-> s7 :|: s7 >= 0, s7 <= 0 + 0, z = 1 + 0, z' = 0
   times(z, z') -{ 4 + z' }-> s8 :|: s8 >= 0, s8 <= 0 + (1 + s''), s'' >= 0, s'' <= z' - 1, z = 1 + 0, z' - 1 >= 0
   times(z, z') -{ 3 + z' }-> plus(plus(times(z - 2, z'), s'), z') :|: s' >= 0, s' <= z', z' >= 0, z - 2 >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0
   times(z, z') -{ 4 }-> plus(plus(times(z - 2, 0), s2), 0) :|: s2 >= 0, s2 <= 0, z - 2 >= 0, z' = 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), 0), 0) :|: z - 2 >= 0, z' = 0
   times(z, z') -{ 4 + 2*z' }-> plus(plus(times(z - 2, 1 + (z' - 1)), s3), 1 + s4) :|: s3 >= 0, s3 <= 1 + (z' - 1), s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 2 >= 0
   times(z, z') -{ 3 + z' }-> plus(plus(times(z - 2, 1 + (z' - 1)), 1 + (z' - 1)), 1 + s1) :|: s1 >= 0, s1 <= z' - 1, z - 2 >= 0, z' - 1 >= 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: {times}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  times: runtime: ?, size: O(n^2) [2*z*z' + 3*z']

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(33) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: times
after applying outer abstraction to obtain an ITS,
resulting in: O(n^3) with polynomial bound: 12 + 31*z + z*z' + 16*z^2*z' + z'

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

   minus(z) -{ 1 }-> 0 :|: z = 0
   minus(z) -{ 1 + z }-> 1 + s :|: s >= 0, s <= z - 1, z - 1 >= 0
   plus(z, z') -{ 1 }-> z' :|: z' >= 0, z = 0
   plus(z, z') -{ 1 + z }-> 1 + s5 :|: s5 >= 0, s5 <= z - 1 + z', z - 1 >= 0, z' >= 0
   times(z, z') -{ 3 }-> s6 :|: s6 >= 0, s6 <= 0 + z', z = 1 + 0, z' >= 0
   times(z, z') -{ 4 }-> s7 :|: s7 >= 0, s7 <= 0 + 0, z = 1 + 0, z' = 0
   times(z, z') -{ 4 + z' }-> s8 :|: s8 >= 0, s8 <= 0 + (1 + s''), s'' >= 0, s'' <= z' - 1, z = 1 + 0, z' - 1 >= 0
   times(z, z') -{ 3 + z' }-> plus(plus(times(z - 2, z'), s'), z') :|: s' >= 0, s' <= z', z' >= 0, z - 2 >= 0
   times(z, z') -{ 2 }-> plus(plus(times(z - 2, z'), z'), z') :|: z - 2 >= 0, z' >= 0
   times(z, z') -{ 4 }-> plus(plus(times(z - 2, 0), s2), 0) :|: s2 >= 0, s2 <= 0, z - 2 >= 0, z' = 0
   times(z, z') -{ 3 }-> plus(plus(times(z - 2, 0), 0), 0) :|: z - 2 >= 0, z' = 0
   times(z, z') -{ 4 + 2*z' }-> plus(plus(times(z - 2, 1 + (z' - 1)), s3), 1 + s4) :|: s3 >= 0, s3 <= 1 + (z' - 1), s4 >= 0, s4 <= z' - 1, z' - 1 >= 0, z - 2 >= 0
   times(z, z') -{ 3 + z' }-> plus(plus(times(z - 2, 1 + (z' - 1)), 1 + (z' - 1)), 1 + s1) :|: s1 >= 0, s1 <= z' - 1, z - 2 >= 0, z' - 1 >= 0
   times(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0

Function symbols to be analyzed: 
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z], size: O(n^1) [z]
  plus: runtime: O(n^1) [1 + z], size: O(n^1) [z + z']
  times: runtime: O(n^3) [12 + 31*z + z*z' + 16*z^2*z' + z'], size: O(n^2) [2*z*z' + 3*z']

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(35) FinalProof (FINISHED)
Computed overall runtime complexity
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(36)
BOUNDS(1, n^3)