WORST_CASE(?,O(n^2))
proof of input_wLY1lpUp89.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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^2).

(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) InliningProof [UPPER BOUND(ID), 644 ms]
(14) CpxRNTS
(15) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
(16) CpxRNTS
(17) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 4 ms]
(18) CpxRNTS
(19) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(20) CpxRNTS
(21) IntTrsBoundProof [UPPER BOUND(ID), 384 ms]
(22) CpxRNTS
(23) IntTrsBoundProof [UPPER BOUND(ID), 101 ms]
(24) CpxRNTS
(25) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(26) CpxRNTS
(27) IntTrsBoundProof [UPPER BOUND(ID), 140 ms]
(28) CpxRNTS
(29) IntTrsBoundProof [UPPER BOUND(ID), 2 ms]
(30) CpxRNTS
(31) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(32) CpxRNTS
(33) IntTrsBoundProof [UPPER BOUND(ID), 774 ms]
(34) CpxRNTS
(35) IntTrsBoundProof [UPPER BOUND(ID), 1063 ms]
(36) CpxRNTS
(37) FinalProof [FINISHED, 0 ms]
(38) BOUNDS(1, n^2)


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

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


The TRS R consists of the following rules:

   -(x, 0) -> x
   -(s(x), s(y)) -> -(x, y)
   p(s(x)) -> x
   f(s(x), y) -> f(p(-(s(x), y)), p(-(y, s(x))))
   f(x, s(y)) -> f(p(-(x, s(y))), p(-(s(y), x)))

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^2).


The TRS R consists of the following rules:

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

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

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

- => 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:

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

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

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

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:

   f_2

(c) The following functions are completely defined:

   p_1
   minus_2

Due to the following rules being added:

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

And the following fresh constants:    const

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

(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:

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

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

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:

   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   p(s(x)) -> x [1]
   f(s(x), 0) -> f(p(s(x)), p(0)) [2]
   f(s(x), s(y')) -> f(p(minus(x, y')), p(minus(y', x))) [3]
   f(s(x), s(y')) -> f(p(minus(x, y')), p(0)) [2]
   f(s(x), s(x')) -> f(p(0), p(minus(x', x))) [2]
   f(s(x), y) -> f(p(0), p(0)) [1]
   f(s(x''), s(y)) -> f(p(minus(x'', y)), p(minus(y, x''))) [3]
   f(s(x''), s(y)) -> f(p(minus(x'', y)), p(0)) [2]
   f(0, s(y)) -> f(p(0), p(s(y))) [2]
   f(s(y''), s(y)) -> f(p(0), p(minus(y, y''))) [2]
   f(x, s(y)) -> f(p(0), p(0)) [1]
   p(v0) -> 0 [0]
   minus(v0, v1) -> 0 [0]

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

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
   const => 0

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

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

   f(z, z') -{ 3 }-> f(p(minus(x, y')), p(minus(y', x))) :|: x >= 0, y' >= 0, z = 1 + x, z' = 1 + y'
   f(z, z') -{ 2 }-> f(p(minus(x, y')), p(0)) :|: x >= 0, y' >= 0, z = 1 + x, z' = 1 + y'
   f(z, z') -{ 3 }-> f(p(minus(x'', y)), p(minus(y, x''))) :|: z = 1 + x'', z' = 1 + y, y >= 0, x'' >= 0
   f(z, z') -{ 2 }-> f(p(minus(x'', y)), p(0)) :|: z = 1 + x'', z' = 1 + y, y >= 0, x'' >= 0
   f(z, z') -{ 2 }-> f(p(0), p(minus(x', x))) :|: z' = 1 + x', x >= 0, x' >= 0, z = 1 + x
   f(z, z') -{ 2 }-> f(p(0), p(minus(y, y''))) :|: z' = 1 + y, z = 1 + y'', y >= 0, y'' >= 0
   f(z, z') -{ 1 }-> f(p(0), p(0)) :|: x >= 0, y >= 0, z = 1 + x, z' = y
   f(z, z') -{ 1 }-> f(p(0), p(0)) :|: z' = 1 + y, x >= 0, y >= 0, z = x
   f(z, z') -{ 2 }-> f(p(0), p(1 + y)) :|: z' = 1 + y, y >= 0, z = 0
   f(z, z') -{ 2 }-> f(p(1 + x), p(0)) :|: x >= 0, z = 1 + 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') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0


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

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

   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0

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

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

   f(z, z') -{ 3 }-> f(x', 0) :|: x >= 0, z = 1 + x, z' = 0, x' >= 0, 1 + x = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(p(minus(x, y')), p(minus(y', x))) :|: x >= 0, y' >= 0, z = 1 + x, z' = 1 + y'
   f(z, z') -{ 2 }-> f(p(minus(x, y')), 0) :|: x >= 0, y' >= 0, z = 1 + x, z' = 1 + y', v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(p(minus(x'', y)), p(minus(y, x''))) :|: z = 1 + x'', z' = 1 + y, y >= 0, x'' >= 0
   f(z, z') -{ 2 }-> f(p(minus(x'', y)), 0) :|: z = 1 + x'', z' = 1 + y, y >= 0, x'' >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(0, x) :|: z' = 1 + y, y >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + y = 1 + x
   f(z, z') -{ 2 }-> f(0, p(minus(x', x))) :|: z' = 1 + x', x >= 0, x' >= 0, z = 1 + x, v0 >= 0, 0 = v0
   f(z, z') -{ 2 }-> f(0, p(minus(y, y''))) :|: z' = 1 + y, z = 1 + y'', y >= 0, y'' >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 2 }-> f(0, 0) :|: x >= 0, z = 1 + x, z' = 0, v0 >= 0, 1 + x = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: x >= 0, y >= 0, z = 1 + x, z' = y, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 2 }-> f(0, 0) :|: z' = 1 + y, y >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + y = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z' = 1 + y, x >= 0, y >= 0, z = x, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   minus(z, z') -{ 1 }-> x :|: x >= 0, z = x, z' = 0
   minus(z, z') -{ 1 }-> minus(x, y) :|: z' = 1 + y, x >= 0, y >= 0, z = 1 + x
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   p(z) -{ 1 }-> x :|: x >= 0, z = 1 + x
   p(z) -{ 0 }-> 0 :|: v0 >= 0, z = v0


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

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

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

   f(z, z') -{ 3 }-> f(x', 0) :|: z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(p(minus(z - 1, z' - 1)), p(minus(z' - 1, z - 1))) :|: z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 2 }-> f(p(minus(z - 1, z' - 1)), 0) :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(0, x) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 2 }-> f(0, p(minus(z' - 1, z - 1))) :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 2 }-> f(0, 0) :|: z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 2 }-> f(0, 0) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0


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

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

   { minus }
   { p }
   { f }

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

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

   f(z, z') -{ 3 }-> f(x', 0) :|: z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(p(minus(z - 1, z' - 1)), p(minus(z' - 1, z - 1))) :|: z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 2 }-> f(p(minus(z - 1, z' - 1)), 0) :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(0, x) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 2 }-> f(0, p(minus(z' - 1, z - 1))) :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 2 }-> f(0, 0) :|: z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 2 }-> f(0, 0) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {minus}, {p}, {f}

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

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

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

   f(z, z') -{ 3 }-> f(x', 0) :|: z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(p(minus(z - 1, z' - 1)), p(minus(z' - 1, z - 1))) :|: z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 2 }-> f(p(minus(z - 1, z' - 1)), 0) :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(0, x) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 2 }-> f(0, p(minus(z' - 1, z - 1))) :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 2 }-> f(0, 0) :|: z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 2 }-> f(0, 0) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: {minus}, {p}, {f}

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

(21) 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

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

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

   f(z, z') -{ 3 }-> f(x', 0) :|: z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(p(minus(z - 1, z' - 1)), p(minus(z' - 1, z - 1))) :|: z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 2 }-> f(p(minus(z - 1, z' - 1)), 0) :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(0, x) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 2 }-> f(0, p(minus(z' - 1, z - 1))) :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 2 }-> f(0, 0) :|: z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 2 }-> f(0, 0) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

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

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

(23) 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'

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

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

   f(z, z') -{ 3 }-> f(x', 0) :|: z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(p(minus(z - 1, z' - 1)), p(minus(z' - 1, z - 1))) :|: z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 2 }-> f(p(minus(z - 1, z' - 1)), 0) :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(0, x) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 2 }-> f(0, p(minus(z' - 1, z - 1))) :|: z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 2 }-> f(0, 0) :|: z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 2 }-> f(0, 0) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 1 }-> minus(z - 1, z' - 1) :|: z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

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

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

(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:

   f(z, z') -{ 3 }-> f(x', 0) :|: z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 3 + z + z' }-> f(p(s'), p(s'')) :|: s' >= 0, s' <= z - 1, s'' >= 0, s'' <= z' - 1, z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 2 + z' }-> f(p(s1), 0) :|: s1 >= 0, s1 <= z - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(0, x) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 2 + z }-> f(0, p(s2)) :|: s2 >= 0, s2 <= z' - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 2 }-> f(0, 0) :|: z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 2 }-> f(0, 0) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

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

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   f(z, z') -{ 3 }-> f(x', 0) :|: z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 3 + z + z' }-> f(p(s'), p(s'')) :|: s' >= 0, s' <= z - 1, s'' >= 0, s'' <= z' - 1, z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 2 + z' }-> f(p(s1), 0) :|: s1 >= 0, s1 <= z - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(0, x) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 2 + z }-> f(0, p(s2)) :|: s2 >= 0, s2 <= z' - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 2 }-> f(0, 0) :|: z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 2 }-> f(0, 0) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

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

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   f(z, z') -{ 3 }-> f(x', 0) :|: z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 3 + z + z' }-> f(p(s'), p(s'')) :|: s' >= 0, s' <= z - 1, s'' >= 0, s'' <= z' - 1, z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 2 + z' }-> f(p(s1), 0) :|: s1 >= 0, s1 <= z - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(0, x) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 2 + z }-> f(0, p(s2)) :|: s2 >= 0, s2 <= z' - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 2 }-> f(0, 0) :|: z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 2 }-> f(0, 0) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

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

   f(z, z') -{ 5 + z + z' }-> f(s3, s4) :|: s3 >= 0, s3 <= s', s4 >= 0, s4 <= s'', s' >= 0, s' <= z - 1, s'' >= 0, s'' <= z' - 1, z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 3 + z' }-> f(s5, 0) :|: s5 >= 0, s5 <= s1, s1 >= 0, s1 <= z - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(x', 0) :|: z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 3 + z }-> f(0, s6) :|: s6 >= 0, s6 <= s2, s2 >= 0, s2 <= z' - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(0, x) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 2 }-> f(0, 0) :|: z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 2 }-> f(0, 0) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

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

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: f
after applying outer abstraction to obtain an ITS,
resulting in: O(1) with polynomial bound: 0

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

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

   f(z, z') -{ 5 + z + z' }-> f(s3, s4) :|: s3 >= 0, s3 <= s', s4 >= 0, s4 <= s'', s' >= 0, s' <= z - 1, s'' >= 0, s'' <= z' - 1, z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 3 + z' }-> f(s5, 0) :|: s5 >= 0, s5 <= s1, s1 >= 0, s1 <= z - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(x', 0) :|: z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 3 + z }-> f(0, s6) :|: s6 >= 0, s6 <= s2, s2 >= 0, s2 <= z' - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(0, x) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 2 }-> f(0, 0) :|: z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 2 }-> f(0, 0) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

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

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

(35) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: f
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 17 + 8*z + z*z' + z^2 + 8*z' + z'^2

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

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

   f(z, z') -{ 5 + z + z' }-> f(s3, s4) :|: s3 >= 0, s3 <= s', s4 >= 0, s4 <= s'', s' >= 0, s' <= z - 1, s'' >= 0, s'' <= z' - 1, z - 1 >= 0, z' - 1 >= 0
   f(z, z') -{ 3 + z' }-> f(s5, 0) :|: s5 >= 0, s5 <= s1, s1 >= 0, s1 <= z - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(x', 0) :|: z - 1 >= 0, z' = 0, x' >= 0, 1 + (z - 1) = 1 + x', v0 >= 0, 0 = v0
   f(z, z') -{ 3 + z }-> f(0, s6) :|: s6 >= 0, s6 <= s2, s2 >= 0, s2 <= z' - 1, z - 1 >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0
   f(z, z') -{ 3 }-> f(0, x) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, x >= 0, 1 + (z' - 1) = 1 + x
   f(z, z') -{ 2 }-> f(0, 0) :|: z - 1 >= 0, z' = 0, v0 >= 0, 1 + (z - 1) = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z - 1 >= 0, z' >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   f(z, z') -{ 2 }-> f(0, 0) :|: z' - 1 >= 0, z = 0, v0 >= 0, 0 = v0, v0' >= 0, 1 + (z' - 1) = v0'
   f(z, z') -{ 1 }-> f(0, 0) :|: z >= 0, z' - 1 >= 0, v0 >= 0, 0 = v0, v0' >= 0, 0 = v0'
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= z - 1, z - 1 >= 0, z' - 1 >= 0
   minus(z, z') -{ 1 }-> z :|: z >= 0, z' = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   p(z) -{ 0 }-> 0 :|: z >= 0
   p(z) -{ 1 }-> z - 1 :|: z - 1 >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(n^1) [z]
  p: runtime: O(1) [1], size: O(n^1) [z]
  f: runtime: O(n^2) [17 + 8*z + z*z' + z^2 + 8*z' + z'^2], size: O(1) [0]

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

(37) FinalProof (FINISHED)
Computed overall runtime complexity
----------------------------------------

(38)
BOUNDS(1, n^2)