WORST_CASE(?,O(n^1))
proof of input_EHGoiqnEkG.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# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished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^1).

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxWeightedTrs
(3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
(4) CpxTypedWeightedTrs
(5) CompletionProof [UPPER BOUND(ID), 0 ms]
(6) CpxTypedWeightedCompleteTrs
(7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
(8) CpxTypedWeightedCompleteTrs
(9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
(10) CpxRNTS
(11) InliningProof [UPPER BOUND(ID), 2 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), 163 ms]
(20) CpxRNTS
(21) IntTrsBoundProof [UPPER BOUND(ID), 130 ms]
(22) CpxRNTS
(23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(24) CpxRNTS
(25) IntTrsBoundProof [UPPER BOUND(ID), 253 ms]
(26) CpxRNTS
(27) IntTrsBoundProof [UPPER BOUND(ID), 63 ms]
(28) CpxRNTS
(29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(30) CpxRNTS
(31) IntTrsBoundProof [UPPER BOUND(ID), 312 ms]
(32) CpxRNTS
(33) IntTrsBoundProof [UPPER BOUND(ID), 114 ms]
(34) CpxRNTS
(35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
(36) CpxRNTS
(37) IntTrsBoundProof [UPPER BOUND(ID), 412 ms]
(38) CpxRNTS
(39) IntTrsBoundProof [UPPER BOUND(ID), 220 ms]
(40) CpxRNTS
(41) FinalProof [FINISHED, 0 ms]
(42) BOUNDS(1, n^1)


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

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


The TRS R consists of the following rules:

   minus(0, Y) -> 0
   minus(s(X), s(Y)) -> minus(X, Y)
   geq(X, 0) -> true
   geq(0, s(Y)) -> false
   geq(s(X), s(Y)) -> geq(X, Y)
   div(0, s(Y)) -> 0
   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0)
   if(true, X, Y) -> X
   if(false, X, Y) -> 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^1).


The TRS R consists of the following rules:

   minus(0, Y) -> 0 [1]
   minus(s(X), s(Y)) -> minus(X, Y) [1]
   geq(X, 0) -> true [1]
   geq(0, s(Y)) -> false [1]
   geq(s(X), s(Y)) -> geq(X, Y) [1]
   div(0, s(Y)) -> 0 [1]
   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0) [1]
   if(true, X, Y) -> X [1]
   if(false, X, Y) -> Y [1]

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

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

(4)
Obligation:
Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   minus(0, Y) -> 0 [1]
   minus(s(X), s(Y)) -> minus(X, Y) [1]
   geq(X, 0) -> true [1]
   geq(0, s(Y)) -> false [1]
   geq(s(X), s(Y)) -> geq(X, Y) [1]
   div(0, s(Y)) -> 0 [1]
   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0) [1]
   if(true, X, Y) -> X [1]
   if(false, X, Y) -> Y [1]

The TRS has the following type information:
minus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
geq :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
div :: 0:s -> 0:s -> 0:s
if :: true:false -> 0:s -> 0:s -> 0:s

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

(5) 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:

   geq_2
   div_2
   minus_2
   if_3

Due to the following rules being added:

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

And the following fresh constants: none

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

(6)
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(0, Y) -> 0 [1]
   minus(s(X), s(Y)) -> minus(X, Y) [1]
   geq(X, 0) -> true [1]
   geq(0, s(Y)) -> false [1]
   geq(s(X), s(Y)) -> geq(X, Y) [1]
   div(0, s(Y)) -> 0 [1]
   div(s(X), s(Y)) -> if(geq(X, Y), s(div(minus(X, Y), s(Y))), 0) [1]
   if(true, X, Y) -> X [1]
   if(false, X, Y) -> Y [1]
   div(v0, v1) -> 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
geq :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
div :: 0:s -> 0:s -> 0:s
if :: true:false -> 0:s -> 0:s -> 0:s

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

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

(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(0, Y) -> 0 [1]
   minus(s(X), s(Y)) -> minus(X, Y) [1]
   geq(X, 0) -> true [1]
   geq(0, s(Y)) -> false [1]
   geq(s(X), s(Y)) -> geq(X, Y) [1]
   div(0, s(Y)) -> 0 [1]
   div(s(0), s(0)) -> if(true, s(div(0, s(0))), 0) [3]
   div(s(X), s(0)) -> if(true, s(div(0, s(0))), 0) [2]
   div(s(0), s(s(Y'))) -> if(false, s(div(0, s(s(Y')))), 0) [3]
   div(s(0), s(s(Y'))) -> if(false, s(div(0, s(s(Y')))), 0) [2]
   div(s(s(X')), s(s(Y''))) -> if(geq(X', Y''), s(div(minus(X', Y''), s(s(Y'')))), 0) [3]
   div(s(s(X')), s(s(Y''))) -> if(geq(X', Y''), s(div(0, s(s(Y'')))), 0) [2]
   if(true, X, Y) -> X [1]
   if(false, X, Y) -> Y [1]
   div(v0, v1) -> 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
geq :: 0:s -> 0:s -> true:false
true :: true:false
false :: true:false
div :: 0:s -> 0:s -> 0:s
if :: true:false -> 0:s -> 0:s -> 0:s

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

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

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

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

(10)
Obligation:
Complexity RNTS consisting of the following rules:

   div(z, z') -{ 3 }-> if(geq(X', Y''), 1 + div(minus(X', Y''), 1 + (1 + Y'')), 0) :|: Y'' >= 0, z' = 1 + (1 + Y''), X' >= 0, z = 1 + (1 + X')
   div(z, z') -{ 2 }-> if(geq(X', Y''), 1 + div(0, 1 + (1 + Y'')), 0) :|: Y'' >= 0, z' = 1 + (1 + Y''), X' >= 0, z = 1 + (1 + X')
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + X, X >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + Y')), 0) :|: z' = 1 + (1 + Y'), Y' >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + Y')), 0) :|: z' = 1 + (1 + Y'), Y' >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: Y >= 0, z' = 1 + Y, z = 0
   div(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   geq(z, z') -{ 1 }-> geq(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
   geq(z, z') -{ 1 }-> 1 :|: X >= 0, z = X, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: Y >= 0, z' = 1 + Y, z = 0
   if(z, z', z'') -{ 1 }-> X :|: z' = X, Y >= 0, z = 1, z'' = Y, X >= 0
   if(z, z', z'') -{ 1 }-> Y :|: z' = X, Y >= 0, z'' = Y, X >= 0, z = 0
   minus(z, z') -{ 1 }-> minus(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' = Y, Y >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1


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

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

   if(z, z', z'') -{ 1 }-> X :|: z' = X, Y >= 0, z = 1, z'' = Y, X >= 0
   if(z, z', z'') -{ 1 }-> Y :|: z' = X, Y >= 0, z'' = Y, X >= 0, z = 0

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

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

   div(z, z') -{ 3 }-> if(geq(X', Y''), 1 + div(minus(X', Y''), 1 + (1 + Y'')), 0) :|: Y'' >= 0, z' = 1 + (1 + Y''), X' >= 0, z = 1 + (1 + X')
   div(z, z') -{ 2 }-> if(geq(X', Y''), 1 + div(0, 1 + (1 + Y'')), 0) :|: Y'' >= 0, z' = 1 + (1 + Y''), X' >= 0, z = 1 + (1 + X')
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + X, X >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + Y')), 0) :|: z' = 1 + (1 + Y'), Y' >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + Y')), 0) :|: z' = 1 + (1 + Y'), Y' >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: Y >= 0, z' = 1 + Y, z = 0
   div(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1
   geq(z, z') -{ 1 }-> geq(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
   geq(z, z') -{ 1 }-> 1 :|: X >= 0, z = X, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: Y >= 0, z' = 1 + Y, z = 0
   if(z, z', z'') -{ 1 }-> X :|: z' = X, Y >= 0, z = 1, z'' = Y, X >= 0
   if(z, z', z'') -{ 1 }-> Y :|: z' = X, Y >= 0, z'' = Y, X >= 0, z = 0
   minus(z, z') -{ 1 }-> minus(X, Y) :|: z = 1 + X, Y >= 0, z' = 1 + Y, X >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' = Y, Y >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z = v0, z' = v1


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(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:

   div(z, z') -{ 3 }-> if(geq(z - 2, z' - 2), 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, 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
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0


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

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

   { minus }
   { if }
   { geq }
   { div }

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

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

   div(z, z') -{ 3 }-> if(geq(z - 2, z' - 2), 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, 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
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {minus}, {if}, {geq}, {div}

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(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:

   div(z, z') -{ 3 }-> if(geq(z - 2, z' - 2), 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, 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
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {minus}, {if}, {geq}, {div}

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(19) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   div(z, z') -{ 3 }-> if(geq(z - 2, z' - 2), 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, 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
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {minus}, {if}, {geq}, {div}
Previous analysis results are:
  minus: runtime: ?, size: O(1) [0]

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(21) 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: 1 + z'

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

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

   div(z, z') -{ 3 }-> if(geq(z - 2, z' - 2), 1 + div(minus(z - 2, z' - 2), 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, 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
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

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

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

(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:

   div(z, z') -{ 2 + z' }-> if(geq(z - 2, z' - 2), 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

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

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

Computed SIZE bound using CoFloCo for: if
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:

   div(z, z') -{ 2 + z' }-> if(geq(z - 2, z' - 2), 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

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

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   div(z, z') -{ 2 + z' }-> if(geq(z - 2, z' - 2), 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

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

   div(z, z') -{ 2 + z' }-> if(geq(z - 2, z' - 2), 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

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

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   div(z, z') -{ 2 + z' }-> if(geq(z - 2, z' - 2), 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

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

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   div(z, z') -{ 2 + z' }-> if(geq(z - 2, z' - 2), 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 }-> if(geq(z - 2, z' - 2), 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 1 }-> geq(z - 1, z' - 1) :|: z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {div}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(1) [0]
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
  geq: runtime: O(n^1) [2 + z'], size: O(1) [1]

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

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

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

   div(z, z') -{ 2 + 2*z' }-> if(s1, 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s1 >= 0, s1 <= 1, s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 + z' }-> if(s2, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: s2 >= 0, s2 <= 1, z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {div}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(1) [0]
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
  geq: runtime: O(n^1) [2 + z'], size: O(1) [1]

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   div(z, z') -{ 2 + 2*z' }-> if(s1, 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s1 >= 0, s1 <= 1, s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 + z' }-> if(s2, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: s2 >= 0, s2 <= 1, z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: {div}
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(1) [0]
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
  geq: runtime: O(n^1) [2 + z'], size: O(1) [1]
  div: runtime: ?, size: O(1) [1]

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: div
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 5 + 2*z'

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

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

   div(z, z') -{ 2 + 2*z' }-> if(s1, 1 + div(s', 1 + (1 + (z' - 2))), 0) :|: s1 >= 0, s1 <= 1, s' >= 0, s' <= 0, z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 2 + z' }-> if(s2, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: s2 >= 0, s2 <= 1, z' - 2 >= 0, z - 2 >= 0
   div(z, z') -{ 3 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z = 1 + 0, z' = 1 + 0
   div(z, z') -{ 2 }-> if(1, 1 + div(0, 1 + 0), 0) :|: z - 1 >= 0, z' = 1 + 0
   div(z, z') -{ 3 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 2 }-> if(0, 1 + div(0, 1 + (1 + (z' - 2))), 0) :|: z' - 2 >= 0, z = 1 + 0
   div(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   div(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0
   geq(z, z') -{ 2 + z' }-> s'' :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z - 1 >= 0
   geq(z, z') -{ 1 }-> 1 :|: z >= 0, z' = 0
   geq(z, z') -{ 1 }-> 0 :|: z' - 1 >= 0, z = 0
   if(z, z', z'') -{ 1 }-> z' :|: z'' >= 0, z = 1, z' >= 0
   if(z, z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' >= 0, z = 0
   minus(z, z') -{ 1 + z' }-> s :|: s >= 0, s <= 0, z' - 1 >= 0, z - 1 >= 0
   minus(z, z') -{ 1 }-> 0 :|: z' >= 0, z = 0
   minus(z, z') -{ 0 }-> 0 :|: z >= 0, z' >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  minus: runtime: O(n^1) [1 + z'], size: O(1) [0]
  if: runtime: O(1) [1], size: O(n^1) [z' + z'']
  geq: runtime: O(n^1) [2 + z'], size: O(1) [1]
  div: runtime: O(n^1) [5 + 2*z'], size: O(1) [1]

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

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

(42)
BOUNDS(1, n^1)