/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) 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), 56 ms]
    (12) CpxRNTS
    (13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRNTS
    (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 4 ms]
    (16) CpxRNTS
    (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 395 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 120 ms]
    (22) CpxRNTS
    (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 149 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 15 ms]
    (28) CpxRNTS
    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 299 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 95 ms]
    (34) CpxRNTS
    (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 128 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 3 ms]
    (40) CpxRNTS
    (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 1825 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 850 ms]
    (46) CpxRNTS
    (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (48) CpxRNTS
    (49) IntTrsBoundProof [UPPER BOUND(ID), 187 ms]
    (50) CpxRNTS
    (51) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (52) CpxRNTS
    (53) FinalProof [FINISHED, 0 ms]
    (54) BOUNDS(1, n^2)
(55) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(56) TRS for Loop Detection
    (57) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (58) BEST
        (59) proven lower bound
            (60) LowerBoundPropagationProof [FINISHED, 0 ms]
            (61) BOUNDS(n^1, INF)
        (62) 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:

   minus(0, y) -> 0
   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   plus(0, y) -> y
   plus(s(x), y) -> plus(x, s(y))
   zero(s(x)) -> false
   zero(0) -> true
   p(s(x)) -> x
   div(x, y) -> quot(x, y, 0)
   quot(x, y, z) -> if(zero(x), x, y, plus(z, s(0)))
   if(true, x, y, z) -> p(z)
   if(false, x, s(y), z) -> quot(minus(x, s(y)), s(y), z)

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:

   minus(0, y) -> 0 [1]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> plus(x, s(y)) [1]
   zero(s(x)) -> false [1]
   zero(0) -> true [1]
   p(s(x)) -> x [1]
   div(x, y) -> quot(x, y, 0) [1]
   quot(x, y, z) -> if(zero(x), x, y, plus(z, s(0))) [1]
   if(true, x, y, z) -> p(z) [1]
   if(false, x, s(y), z) -> quot(minus(x, s(y)), s(y), z) [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(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> plus(x, s(y)) [1]
   zero(s(x)) -> false [1]
   zero(0) -> true [1]
   p(s(x)) -> x [1]
   div(x, y) -> quot(x, y, 0) [1]
   quot(x, y, z) -> if(zero(x), x, y, plus(z, s(0))) [1]
   if(true, x, y, z) -> p(z) [1]
   if(false, x, s(y), z) -> quot(minus(x, s(y)), s(y), z) [1]

The TRS has the following type information:
minus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
plus :: 0:s -> 0:s -> 0:s
zero :: 0:s -> false:true
false :: false:true
true :: false:true
p :: 0:s -> 0:s
div :: 0:s -> 0:s -> 0:s
quot :: 0:s -> 0:s -> 0:s -> 0:s
if :: false:true -> 0:s -> 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:

   p_1
   div_2
   quot_3
   if_4

(c) The following functions are completely defined:

   minus_2
   zero_1
   plus_2

Due to the following rules being added:
none

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(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> plus(x, s(y)) [1]
   zero(s(x)) -> false [1]
   zero(0) -> true [1]
   p(s(x)) -> x [1]
   div(x, y) -> quot(x, y, 0) [1]
   quot(x, y, z) -> if(zero(x), x, y, plus(z, s(0))) [1]
   if(true, x, y, z) -> p(z) [1]
   if(false, x, s(y), z) -> quot(minus(x, s(y)), s(y), z) [1]

The TRS has the following type information:
minus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
plus :: 0:s -> 0:s -> 0:s
zero :: 0:s -> false:true
false :: false:true
true :: false:true
p :: 0:s -> 0:s
div :: 0:s -> 0:s -> 0:s
quot :: 0:s -> 0:s -> 0:s -> 0:s
if :: false:true -> 0:s -> 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(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> plus(x, s(y)) [1]
   zero(s(x)) -> false [1]
   zero(0) -> true [1]
   p(s(x)) -> x [1]
   div(x, y) -> quot(x, y, 0) [1]
   quot(s(x'), y, 0) -> if(false, s(x'), y, s(0)) [3]
   quot(s(x'), y, s(x'')) -> if(false, s(x'), y, plus(x'', s(s(0)))) [3]
   quot(0, y, 0) -> if(true, 0, y, s(0)) [3]
   quot(0, y, s(x1)) -> if(true, 0, y, plus(x1, s(s(0)))) [3]
   if(true, x, y, z) -> p(z) [1]
   if(false, 0, s(y), z) -> quot(0, s(y), z) [2]
   if(false, s(x2), s(y), z) -> quot(minus(x2, y), s(y), z) [2]

The TRS has the following type information:
minus :: 0:s -> 0:s -> 0:s
0 :: 0:s
s :: 0:s -> 0:s
plus :: 0:s -> 0:s -> 0:s
zero :: 0:s -> false:true
false :: false:true
true :: false:true
p :: 0:s -> 0:s
div :: 0:s -> 0:s -> 0:s
quot :: 0:s -> 0:s -> 0:s -> 0:s
if :: false:true -> 0:s -> 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
   false => 0
   true => 1

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

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

   div(z', z'') -{ 1 }-> quot(x, y, 0) :|: z' = x, z'' = y, x >= 0, y >= 0
   if(z', z'', z1, z2) -{ 2 }-> quot(minus(x2, y), 1 + y, z) :|: z >= 0, z2 = z, y >= 0, z'' = 1 + x2, z1 = 1 + y, x2 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> quot(0, 1 + y, z) :|: z'' = 0, z >= 0, z2 = z, y >= 0, z1 = 1 + y, z' = 0
   if(z', z'', z1, z2) -{ 1 }-> p(z) :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1
   minus(z', z'') -{ 1 }-> x :|: z'' = 0, z' = x, x >= 0
   minus(z', z'') -{ 1 }-> minus(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   minus(z', z'') -{ 1 }-> 0 :|: z'' = y, y >= 0, z' = 0
   p(z') -{ 1 }-> x :|: z' = 1 + x, x >= 0
   plus(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 0
   plus(z', z'') -{ 1 }-> plus(x, 1 + y) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0
   quot(z', z'', z1) -{ 3 }-> if(1, 0, y, plus(x1, 1 + (1 + 0))) :|: x1 >= 0, z'' = y, y >= 0, z1 = 1 + x1, z' = 0
   quot(z', z'', z1) -{ 3 }-> if(1, 0, y, 1 + 0) :|: z1 = 0, z'' = y, y >= 0, z' = 0
   quot(z', z'', z1) -{ 3 }-> if(0, 1 + x', y, plus(x'', 1 + (1 + 0))) :|: z1 = 1 + x'', z' = 1 + x', z'' = y, x' >= 0, y >= 0, x'' >= 0
   quot(z', z'', z1) -{ 3 }-> if(0, 1 + x', y, 1 + 0) :|: z1 = 0, z' = 1 + x', z'' = y, x' >= 0, y >= 0
   zero(z') -{ 1 }-> 1 :|: z' = 0
   zero(z') -{ 1 }-> 0 :|: z' = 1 + x, x >= 0


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

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

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

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

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

   div(z', z'') -{ 1 }-> quot(x, y, 0) :|: z' = x, z'' = y, x >= 0, y >= 0
   if(z', z'', z1, z2) -{ 2 }-> x' :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1, z = 1 + x', x' >= 0
   if(z', z'', z1, z2) -{ 2 }-> quot(minus(x2, y), 1 + y, z) :|: z >= 0, z2 = z, y >= 0, z'' = 1 + x2, z1 = 1 + y, x2 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> quot(0, 1 + y, z) :|: z'' = 0, z >= 0, z2 = z, y >= 0, z1 = 1 + y, z' = 0
   minus(z', z'') -{ 1 }-> x :|: z'' = 0, z' = x, x >= 0
   minus(z', z'') -{ 1 }-> minus(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   minus(z', z'') -{ 1 }-> 0 :|: z'' = y, y >= 0, z' = 0
   p(z') -{ 1 }-> x :|: z' = 1 + x, x >= 0
   plus(z', z'') -{ 1 }-> y :|: z'' = y, y >= 0, z' = 0
   plus(z', z'') -{ 1 }-> plus(x, 1 + y) :|: z' = 1 + x, z'' = y, x >= 0, y >= 0
   quot(z', z'', z1) -{ 3 }-> if(1, 0, y, plus(x1, 1 + (1 + 0))) :|: x1 >= 0, z'' = y, y >= 0, z1 = 1 + x1, z' = 0
   quot(z', z'', z1) -{ 3 }-> if(1, 0, y, 1 + 0) :|: z1 = 0, z'' = y, y >= 0, z' = 0
   quot(z', z'', z1) -{ 3 }-> if(0, 1 + x', y, plus(x'', 1 + (1 + 0))) :|: z1 = 1 + x'', z' = 1 + x', z'' = y, x' >= 0, y >= 0, x'' >= 0
   quot(z', z'', z1) -{ 3 }-> if(0, 1 + x', y, 1 + 0) :|: z1 = 0, z' = 1 + x', z'' = y, x' >= 0, y >= 0
   zero(z') -{ 1 }-> 1 :|: z' = 0
   zero(z') -{ 1 }-> 0 :|: z' = 1 + x, x >= 0


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

(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'') -{ 1 }-> quot(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 2 }-> quot(minus(z'' - 1, z1 - 1), 1 + (z1 - 1), z2) :|: z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> quot(0, 1 + (z1 - 1), z2) :|: z'' = 0, z2 >= 0, z1 - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> z2 - 1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z2 - 1 >= 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
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> plus(z' - 1, 1 + z'') :|: z' - 1 >= 0, z'' >= 0
   quot(z', z'', z1) -{ 3 }-> if(1, 0, z'', plus(z1 - 1, 1 + (1 + 0))) :|: z1 - 1 >= 0, z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 }-> if(1, 0, z'', 1 + 0) :|: z1 = 0, z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 }-> if(0, 1 + (z' - 1), z'', plus(z1 - 1, 1 + (1 + 0))) :|: z' - 1 >= 0, z'' >= 0, z1 - 1 >= 0
   quot(z', z'', z1) -{ 3 }-> if(0, 1 + (z' - 1), z'', 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   zero(z') -{ 1 }-> 1 :|: z' = 0
   zero(z') -{ 1 }-> 0 :|: z' - 1 >= 0


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

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

   { minus }
   { zero }
   { plus }
   { p }
   { if, quot }
   { div }

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

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

   div(z', z'') -{ 1 }-> quot(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 2 }-> quot(minus(z'' - 1, z1 - 1), 1 + (z1 - 1), z2) :|: z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> quot(0, 1 + (z1 - 1), z2) :|: z'' = 0, z2 >= 0, z1 - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> z2 - 1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z2 - 1 >= 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
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> plus(z' - 1, 1 + z'') :|: z' - 1 >= 0, z'' >= 0
   quot(z', z'', z1) -{ 3 }-> if(1, 0, z'', plus(z1 - 1, 1 + (1 + 0))) :|: z1 - 1 >= 0, z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 }-> if(1, 0, z'', 1 + 0) :|: z1 = 0, z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 }-> if(0, 1 + (z' - 1), z'', plus(z1 - 1, 1 + (1 + 0))) :|: z' - 1 >= 0, z'' >= 0, z1 - 1 >= 0
   quot(z', z'', z1) -{ 3 }-> if(0, 1 + (z' - 1), z'', 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   zero(z') -{ 1 }-> 1 :|: z' = 0
   zero(z') -{ 1 }-> 0 :|: z' - 1 >= 0

Function symbols to be analyzed: {minus}, {zero}, {plus}, {p}, {if,quot}, {div}

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

(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'') -{ 1 }-> quot(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 2 }-> quot(minus(z'' - 1, z1 - 1), 1 + (z1 - 1), z2) :|: z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> quot(0, 1 + (z1 - 1), z2) :|: z'' = 0, z2 >= 0, z1 - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> z2 - 1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z2 - 1 >= 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
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> plus(z' - 1, 1 + z'') :|: z' - 1 >= 0, z'' >= 0
   quot(z', z'', z1) -{ 3 }-> if(1, 0, z'', plus(z1 - 1, 1 + (1 + 0))) :|: z1 - 1 >= 0, z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 }-> if(1, 0, z'', 1 + 0) :|: z1 = 0, z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 }-> if(0, 1 + (z' - 1), z'', plus(z1 - 1, 1 + (1 + 0))) :|: z' - 1 >= 0, z'' >= 0, z1 - 1 >= 0
   quot(z', z'', z1) -{ 3 }-> if(0, 1 + (z' - 1), z'', 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   zero(z') -{ 1 }-> 1 :|: z' = 0
   zero(z') -{ 1 }-> 0 :|: z' - 1 >= 0

Function symbols to be analyzed: {minus}, {zero}, {plus}, {p}, {if,quot}, {div}

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

(19) 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'

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

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

   div(z', z'') -{ 1 }-> quot(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 2 }-> quot(minus(z'' - 1, z1 - 1), 1 + (z1 - 1), z2) :|: z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> quot(0, 1 + (z1 - 1), z2) :|: z'' = 0, z2 >= 0, z1 - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> z2 - 1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z2 - 1 >= 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
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> plus(z' - 1, 1 + z'') :|: z' - 1 >= 0, z'' >= 0
   quot(z', z'', z1) -{ 3 }-> if(1, 0, z'', plus(z1 - 1, 1 + (1 + 0))) :|: z1 - 1 >= 0, z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 }-> if(1, 0, z'', 1 + 0) :|: z1 = 0, z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 }-> if(0, 1 + (z' - 1), z'', plus(z1 - 1, 1 + (1 + 0))) :|: z' - 1 >= 0, z'' >= 0, z1 - 1 >= 0
   quot(z', z'', z1) -{ 3 }-> if(0, 1 + (z' - 1), z'', 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   zero(z') -{ 1 }-> 1 :|: z' = 0
   zero(z') -{ 1 }-> 0 :|: z' - 1 >= 0

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

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

(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: 2 + z''

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

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

   div(z', z'') -{ 1 }-> quot(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 2 }-> quot(minus(z'' - 1, z1 - 1), 1 + (z1 - 1), z2) :|: z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> quot(0, 1 + (z1 - 1), z2) :|: z'' = 0, z2 >= 0, z1 - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> z2 - 1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z2 - 1 >= 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
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> plus(z' - 1, 1 + z'') :|: z' - 1 >= 0, z'' >= 0
   quot(z', z'', z1) -{ 3 }-> if(1, 0, z'', plus(z1 - 1, 1 + (1 + 0))) :|: z1 - 1 >= 0, z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 }-> if(1, 0, z'', 1 + 0) :|: z1 = 0, z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 }-> if(0, 1 + (z' - 1), z'', plus(z1 - 1, 1 + (1 + 0))) :|: z' - 1 >= 0, z'' >= 0, z1 - 1 >= 0
   quot(z', z'', z1) -{ 3 }-> if(0, 1 + (z' - 1), z'', 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   zero(z') -{ 1 }-> 1 :|: z' = 0
   zero(z') -{ 1 }-> 0 :|: z' - 1 >= 0

Function symbols to be analyzed: {zero}, {plus}, {p}, {if,quot}, {div}
Previous analysis results are:
  minus: runtime: O(n^1) [2 + 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:

   div(z', z'') -{ 1 }-> quot(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 3 + z1 }-> quot(s', 1 + (z1 - 1), z2) :|: s' >= 0, s' <= z'' - 1, z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> quot(0, 1 + (z1 - 1), z2) :|: z'' = 0, z2 >= 0, z1 - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> z2 - 1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z2 - 1 >= 0
   minus(z', z'') -{ 2 + 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'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> plus(z' - 1, 1 + z'') :|: z' - 1 >= 0, z'' >= 0
   quot(z', z'', z1) -{ 3 }-> if(1, 0, z'', plus(z1 - 1, 1 + (1 + 0))) :|: z1 - 1 >= 0, z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 }-> if(1, 0, z'', 1 + 0) :|: z1 = 0, z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 }-> if(0, 1 + (z' - 1), z'', plus(z1 - 1, 1 + (1 + 0))) :|: z' - 1 >= 0, z'' >= 0, z1 - 1 >= 0
   quot(z', z'', z1) -{ 3 }-> if(0, 1 + (z' - 1), z'', 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   zero(z') -{ 1 }-> 1 :|: z' = 0
   zero(z') -{ 1 }-> 0 :|: z' - 1 >= 0

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

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   div(z', z'') -{ 1 }-> quot(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 3 + z1 }-> quot(s', 1 + (z1 - 1), z2) :|: s' >= 0, s' <= z'' - 1, z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> quot(0, 1 + (z1 - 1), z2) :|: z'' = 0, z2 >= 0, z1 - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> z2 - 1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z2 - 1 >= 0
   minus(z', z'') -{ 2 + 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'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> plus(z' - 1, 1 + z'') :|: z' - 1 >= 0, z'' >= 0
   quot(z', z'', z1) -{ 3 }-> if(1, 0, z'', plus(z1 - 1, 1 + (1 + 0))) :|: z1 - 1 >= 0, z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 }-> if(1, 0, z'', 1 + 0) :|: z1 = 0, z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 }-> if(0, 1 + (z' - 1), z'', plus(z1 - 1, 1 + (1 + 0))) :|: z' - 1 >= 0, z'' >= 0, z1 - 1 >= 0
   quot(z', z'', z1) -{ 3 }-> if(0, 1 + (z' - 1), z'', 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   zero(z') -{ 1 }-> 1 :|: z' = 0
   zero(z') -{ 1 }-> 0 :|: z' - 1 >= 0

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

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: zero
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'') -{ 1 }-> quot(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 3 + z1 }-> quot(s', 1 + (z1 - 1), z2) :|: s' >= 0, s' <= z'' - 1, z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> quot(0, 1 + (z1 - 1), z2) :|: z'' = 0, z2 >= 0, z1 - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> z2 - 1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z2 - 1 >= 0
   minus(z', z'') -{ 2 + 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'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> plus(z' - 1, 1 + z'') :|: z' - 1 >= 0, z'' >= 0
   quot(z', z'', z1) -{ 3 }-> if(1, 0, z'', plus(z1 - 1, 1 + (1 + 0))) :|: z1 - 1 >= 0, z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 }-> if(1, 0, z'', 1 + 0) :|: z1 = 0, z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 }-> if(0, 1 + (z' - 1), z'', plus(z1 - 1, 1 + (1 + 0))) :|: z' - 1 >= 0, z'' >= 0, z1 - 1 >= 0
   quot(z', z'', z1) -{ 3 }-> if(0, 1 + (z' - 1), z'', 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   zero(z') -{ 1 }-> 1 :|: z' = 0
   zero(z') -{ 1 }-> 0 :|: z' - 1 >= 0

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

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

(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'') -{ 1 }-> quot(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 3 + z1 }-> quot(s', 1 + (z1 - 1), z2) :|: s' >= 0, s' <= z'' - 1, z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> quot(0, 1 + (z1 - 1), z2) :|: z'' = 0, z2 >= 0, z1 - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> z2 - 1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z2 - 1 >= 0
   minus(z', z'') -{ 2 + 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'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> plus(z' - 1, 1 + z'') :|: z' - 1 >= 0, z'' >= 0
   quot(z', z'', z1) -{ 3 }-> if(1, 0, z'', plus(z1 - 1, 1 + (1 + 0))) :|: z1 - 1 >= 0, z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 }-> if(1, 0, z'', 1 + 0) :|: z1 = 0, z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 }-> if(0, 1 + (z' - 1), z'', plus(z1 - 1, 1 + (1 + 0))) :|: z' - 1 >= 0, z'' >= 0, z1 - 1 >= 0
   quot(z', z'', z1) -{ 3 }-> if(0, 1 + (z' - 1), z'', 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   zero(z') -{ 1 }-> 1 :|: z' = 0
   zero(z') -{ 1 }-> 0 :|: z' - 1 >= 0

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

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

(31) 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''

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

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

   div(z', z'') -{ 1 }-> quot(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 3 + z1 }-> quot(s', 1 + (z1 - 1), z2) :|: s' >= 0, s' <= z'' - 1, z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> quot(0, 1 + (z1 - 1), z2) :|: z'' = 0, z2 >= 0, z1 - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> z2 - 1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z2 - 1 >= 0
   minus(z', z'') -{ 2 + 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'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> plus(z' - 1, 1 + z'') :|: z' - 1 >= 0, z'' >= 0
   quot(z', z'', z1) -{ 3 }-> if(1, 0, z'', plus(z1 - 1, 1 + (1 + 0))) :|: z1 - 1 >= 0, z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 }-> if(1, 0, z'', 1 + 0) :|: z1 = 0, z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 }-> if(0, 1 + (z' - 1), z'', plus(z1 - 1, 1 + (1 + 0))) :|: z' - 1 >= 0, z'' >= 0, z1 - 1 >= 0
   quot(z', z'', z1) -{ 3 }-> if(0, 1 + (z' - 1), z'', 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   zero(z') -{ 1 }-> 1 :|: z' = 0
   zero(z') -{ 1 }-> 0 :|: z' - 1 >= 0

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

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

(33) 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'

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

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

   div(z', z'') -{ 1 }-> quot(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 3 + z1 }-> quot(s', 1 + (z1 - 1), z2) :|: s' >= 0, s' <= z'' - 1, z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> quot(0, 1 + (z1 - 1), z2) :|: z'' = 0, z2 >= 0, z1 - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> z2 - 1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z2 - 1 >= 0
   minus(z', z'') -{ 2 + 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'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   plus(z', z'') -{ 1 }-> plus(z' - 1, 1 + z'') :|: z' - 1 >= 0, z'' >= 0
   quot(z', z'', z1) -{ 3 }-> if(1, 0, z'', plus(z1 - 1, 1 + (1 + 0))) :|: z1 - 1 >= 0, z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 }-> if(1, 0, z'', 1 + 0) :|: z1 = 0, z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 }-> if(0, 1 + (z' - 1), z'', plus(z1 - 1, 1 + (1 + 0))) :|: z' - 1 >= 0, z'' >= 0, z1 - 1 >= 0
   quot(z', z'', z1) -{ 3 }-> if(0, 1 + (z' - 1), z'', 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   zero(z') -{ 1 }-> 1 :|: z' = 0
   zero(z') -{ 1 }-> 0 :|: z' - 1 >= 0

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

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

(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'') -{ 1 }-> quot(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 3 + z1 }-> quot(s', 1 + (z1 - 1), z2) :|: s' >= 0, s' <= z'' - 1, z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> quot(0, 1 + (z1 - 1), z2) :|: z'' = 0, z2 >= 0, z1 - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> z2 - 1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z2 - 1 >= 0
   minus(z', z'') -{ 2 + 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'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 + z' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + (1 + z''), z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 + z1 }-> if(1, 0, z'', s2) :|: s2 >= 0, s2 <= z1 - 1 + (1 + (1 + 0)), z1 - 1 >= 0, z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 }-> if(1, 0, z'', 1 + 0) :|: z1 = 0, z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 + z1 }-> if(0, 1 + (z' - 1), z'', s1) :|: s1 >= 0, s1 <= z1 - 1 + (1 + (1 + 0)), z' - 1 >= 0, z'' >= 0, z1 - 1 >= 0
   quot(z', z'', z1) -{ 3 }-> if(0, 1 + (z' - 1), z'', 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   zero(z') -{ 1 }-> 1 :|: z' = 0
   zero(z') -{ 1 }-> 0 :|: z' - 1 >= 0

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

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

(37) 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'

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

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

   div(z', z'') -{ 1 }-> quot(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 3 + z1 }-> quot(s', 1 + (z1 - 1), z2) :|: s' >= 0, s' <= z'' - 1, z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> quot(0, 1 + (z1 - 1), z2) :|: z'' = 0, z2 >= 0, z1 - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> z2 - 1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z2 - 1 >= 0
   minus(z', z'') -{ 2 + 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'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 + z' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + (1 + z''), z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 + z1 }-> if(1, 0, z'', s2) :|: s2 >= 0, s2 <= z1 - 1 + (1 + (1 + 0)), z1 - 1 >= 0, z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 }-> if(1, 0, z'', 1 + 0) :|: z1 = 0, z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 + z1 }-> if(0, 1 + (z' - 1), z'', s1) :|: s1 >= 0, s1 <= z1 - 1 + (1 + (1 + 0)), z' - 1 >= 0, z'' >= 0, z1 - 1 >= 0
   quot(z', z'', z1) -{ 3 }-> if(0, 1 + (z' - 1), z'', 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   zero(z') -{ 1 }-> 1 :|: z' = 0
   zero(z') -{ 1 }-> 0 :|: z' - 1 >= 0

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

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

(39) 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

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

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

   div(z', z'') -{ 1 }-> quot(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 3 + z1 }-> quot(s', 1 + (z1 - 1), z2) :|: s' >= 0, s' <= z'' - 1, z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> quot(0, 1 + (z1 - 1), z2) :|: z'' = 0, z2 >= 0, z1 - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> z2 - 1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z2 - 1 >= 0
   minus(z', z'') -{ 2 + 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'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 + z' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + (1 + z''), z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 + z1 }-> if(1, 0, z'', s2) :|: s2 >= 0, s2 <= z1 - 1 + (1 + (1 + 0)), z1 - 1 >= 0, z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 }-> if(1, 0, z'', 1 + 0) :|: z1 = 0, z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 + z1 }-> if(0, 1 + (z' - 1), z'', s1) :|: s1 >= 0, s1 <= z1 - 1 + (1 + (1 + 0)), z' - 1 >= 0, z'' >= 0, z1 - 1 >= 0
   quot(z', z'', z1) -{ 3 }-> if(0, 1 + (z' - 1), z'', 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   zero(z') -{ 1 }-> 1 :|: z' = 0
   zero(z') -{ 1 }-> 0 :|: z' - 1 >= 0

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

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

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

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

   div(z', z'') -{ 1 }-> quot(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 3 + z1 }-> quot(s', 1 + (z1 - 1), z2) :|: s' >= 0, s' <= z'' - 1, z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> quot(0, 1 + (z1 - 1), z2) :|: z'' = 0, z2 >= 0, z1 - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> z2 - 1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z2 - 1 >= 0
   minus(z', z'') -{ 2 + 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'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 + z' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + (1 + z''), z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 + z1 }-> if(1, 0, z'', s2) :|: s2 >= 0, s2 <= z1 - 1 + (1 + (1 + 0)), z1 - 1 >= 0, z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 }-> if(1, 0, z'', 1 + 0) :|: z1 = 0, z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 + z1 }-> if(0, 1 + (z' - 1), z'', s1) :|: s1 >= 0, s1 <= z1 - 1 + (1 + (1 + 0)), z' - 1 >= 0, z'' >= 0, z1 - 1 >= 0
   quot(z', z'', z1) -{ 3 }-> if(0, 1 + (z' - 1), z'', 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   zero(z') -{ 1 }-> 1 :|: z' = 0
   zero(z') -{ 1 }-> 0 :|: z' - 1 >= 0

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

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

(43) 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: 1 + z'' + z2

Computed SIZE bound using CoFloCo for: quot
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + z' + z1

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

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

   div(z', z'') -{ 1 }-> quot(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 3 + z1 }-> quot(s', 1 + (z1 - 1), z2) :|: s' >= 0, s' <= z'' - 1, z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> quot(0, 1 + (z1 - 1), z2) :|: z'' = 0, z2 >= 0, z1 - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> z2 - 1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z2 - 1 >= 0
   minus(z', z'') -{ 2 + 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'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 + z' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + (1 + z''), z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 + z1 }-> if(1, 0, z'', s2) :|: s2 >= 0, s2 <= z1 - 1 + (1 + (1 + 0)), z1 - 1 >= 0, z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 }-> if(1, 0, z'', 1 + 0) :|: z1 = 0, z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 + z1 }-> if(0, 1 + (z' - 1), z'', s1) :|: s1 >= 0, s1 <= z1 - 1 + (1 + (1 + 0)), z' - 1 >= 0, z'' >= 0, z1 - 1 >= 0
   quot(z', z'', z1) -{ 3 }-> if(0, 1 + (z' - 1), z'', 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   zero(z') -{ 1 }-> 1 :|: z' = 0
   zero(z') -{ 1 }-> 0 :|: z' - 1 >= 0

Function symbols to be analyzed: {if,quot}, {div}
Previous analysis results are:
  minus: runtime: O(n^1) [2 + z''], size: O(n^1) [z']
  zero: runtime: O(1) [1], size: O(1) [1]
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [z']
  if: runtime: ?, size: O(n^1) [1 + z'' + z2]
  quot: runtime: ?, size: O(n^1) [2 + z' + z1]

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: if
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 18 + 12*z'' + 2*z''*z1 + z''*z2 + z''^2 + 3*z1 + 2*z2

Computed RUNTIME bound using KoAT for: quot
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 92 + 26*z' + 4*z'*z'' + z'*z1 + 2*z'^2 + 12*z'' + 6*z1

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

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

   div(z', z'') -{ 1 }-> quot(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 3 + z1 }-> quot(s', 1 + (z1 - 1), z2) :|: s' >= 0, s' <= z'' - 1, z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> quot(0, 1 + (z1 - 1), z2) :|: z'' = 0, z2 >= 0, z1 - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> z2 - 1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z2 - 1 >= 0
   minus(z', z'') -{ 2 + 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'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 + z' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + (1 + z''), z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 + z1 }-> if(1, 0, z'', s2) :|: s2 >= 0, s2 <= z1 - 1 + (1 + (1 + 0)), z1 - 1 >= 0, z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 }-> if(1, 0, z'', 1 + 0) :|: z1 = 0, z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 3 + z1 }-> if(0, 1 + (z' - 1), z'', s1) :|: s1 >= 0, s1 <= z1 - 1 + (1 + (1 + 0)), z' - 1 >= 0, z'' >= 0, z1 - 1 >= 0
   quot(z', z'', z1) -{ 3 }-> if(0, 1 + (z' - 1), z'', 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' >= 0
   zero(z') -{ 1 }-> 1 :|: z' = 0
   zero(z') -{ 1 }-> 0 :|: z' - 1 >= 0

Function symbols to be analyzed: {div}
Previous analysis results are:
  minus: runtime: O(n^1) [2 + z''], size: O(n^1) [z']
  zero: runtime: O(1) [1], size: O(1) [1]
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [z']
  if: runtime: O(n^2) [18 + 12*z'' + 2*z''*z1 + z''*z2 + z''^2 + 3*z1 + 2*z2], size: O(n^1) [1 + z'' + z2]
  quot: runtime: O(n^2) [92 + 26*z' + 4*z'*z'' + z'*z1 + 2*z'^2 + 12*z'' + 6*z1], size: O(n^1) [2 + z' + z1]

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

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

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

   div(z', z'') -{ 93 + 26*z' + 4*z'*z'' + 2*z'^2 + 12*z'' }-> s3 :|: s3 >= 0, s3 <= z' + 0 + 2, z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 94 + 12*z1 + 6*z2 }-> s8 :|: s8 >= 0, s8 <= 0 + z2 + 2, z'' = 0, z2 >= 0, z1 - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 95 + 26*s' + 4*s'*z1 + s'*z2 + 2*s'^2 + 13*z1 + 6*z2 }-> s9 :|: s9 >= 0, s9 <= s' + z2 + 2, s' >= 0, s' <= z'' - 1, z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> z2 - 1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z2 - 1 >= 0
   minus(z', z'') -{ 2 + 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'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 + z' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + (1 + z''), z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 23 + 13*z' + 2*z'*z'' + z'^2 + 3*z'' }-> s4 :|: s4 >= 0, s4 <= 1 + 0 + 1 + (1 + (z' - 1)), z1 = 0, z' - 1 >= 0, z'' >= 0
   quot(z', z'', z1) -{ 21 + 2*s1 + s1*z' + 12*z' + 2*z'*z'' + z'^2 + 3*z'' + z1 }-> s5 :|: s5 >= 0, s5 <= s1 + 1 + (1 + (z' - 1)), s1 >= 0, s1 <= z1 - 1 + (1 + (1 + 0)), z' - 1 >= 0, z'' >= 0, z1 - 1 >= 0
   quot(z', z'', z1) -{ 23 + 3*z'' }-> s6 :|: s6 >= 0, s6 <= 1 + 0 + 1 + 0, z1 = 0, z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 21 + 2*s2 + 3*z'' + z1 }-> s7 :|: s7 >= 0, s7 <= s2 + 1 + 0, s2 >= 0, s2 <= z1 - 1 + (1 + (1 + 0)), z1 - 1 >= 0, z'' >= 0, z' = 0
   zero(z') -{ 1 }-> 1 :|: z' = 0
   zero(z') -{ 1 }-> 0 :|: z' - 1 >= 0

Function symbols to be analyzed: {div}
Previous analysis results are:
  minus: runtime: O(n^1) [2 + z''], size: O(n^1) [z']
  zero: runtime: O(1) [1], size: O(1) [1]
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [z']
  if: runtime: O(n^2) [18 + 12*z'' + 2*z''*z1 + z''*z2 + z''^2 + 3*z1 + 2*z2], size: O(n^1) [1 + z'' + z2]
  quot: runtime: O(n^2) [92 + 26*z' + 4*z'*z'' + z'*z1 + 2*z'^2 + 12*z'' + 6*z1], size: O(n^1) [2 + z' + z1]

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

(49) IntTrsBoundProof (UPPER BOUND(ID))

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

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

(50)
Obligation:
Complexity RNTS consisting of the following rules:

   div(z', z'') -{ 93 + 26*z' + 4*z'*z'' + 2*z'^2 + 12*z'' }-> s3 :|: s3 >= 0, s3 <= z' + 0 + 2, z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 94 + 12*z1 + 6*z2 }-> s8 :|: s8 >= 0, s8 <= 0 + z2 + 2, z'' = 0, z2 >= 0, z1 - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 95 + 26*s' + 4*s'*z1 + s'*z2 + 2*s'^2 + 13*z1 + 6*z2 }-> s9 :|: s9 >= 0, s9 <= s' + z2 + 2, s' >= 0, s' <= z'' - 1, z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> z2 - 1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z2 - 1 >= 0
   minus(z', z'') -{ 2 + 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'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 + z' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + (1 + z''), z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 23 + 13*z' + 2*z'*z'' + z'^2 + 3*z'' }-> s4 :|: s4 >= 0, s4 <= 1 + 0 + 1 + (1 + (z' - 1)), z1 = 0, z' - 1 >= 0, z'' >= 0
   quot(z', z'', z1) -{ 21 + 2*s1 + s1*z' + 12*z' + 2*z'*z'' + z'^2 + 3*z'' + z1 }-> s5 :|: s5 >= 0, s5 <= s1 + 1 + (1 + (z' - 1)), s1 >= 0, s1 <= z1 - 1 + (1 + (1 + 0)), z' - 1 >= 0, z'' >= 0, z1 - 1 >= 0
   quot(z', z'', z1) -{ 23 + 3*z'' }-> s6 :|: s6 >= 0, s6 <= 1 + 0 + 1 + 0, z1 = 0, z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 21 + 2*s2 + 3*z'' + z1 }-> s7 :|: s7 >= 0, s7 <= s2 + 1 + 0, s2 >= 0, s2 <= z1 - 1 + (1 + (1 + 0)), z1 - 1 >= 0, z'' >= 0, z' = 0
   zero(z') -{ 1 }-> 1 :|: z' = 0
   zero(z') -{ 1 }-> 0 :|: z' - 1 >= 0

Function symbols to be analyzed: {div}
Previous analysis results are:
  minus: runtime: O(n^1) [2 + z''], size: O(n^1) [z']
  zero: runtime: O(1) [1], size: O(1) [1]
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [z']
  if: runtime: O(n^2) [18 + 12*z'' + 2*z''*z1 + z''*z2 + z''^2 + 3*z1 + 2*z2], size: O(n^1) [1 + z'' + z2]
  quot: runtime: O(n^2) [92 + 26*z' + 4*z'*z'' + z'*z1 + 2*z'^2 + 12*z'' + 6*z1], size: O(n^1) [2 + z' + z1]
  div: runtime: ?, size: O(n^1) [2 + z']

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

(51) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: div
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 93 + 26*z' + 4*z'*z'' + 2*z'^2 + 12*z''

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

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

   div(z', z'') -{ 93 + 26*z' + 4*z'*z'' + 2*z'^2 + 12*z'' }-> s3 :|: s3 >= 0, s3 <= z' + 0 + 2, z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 94 + 12*z1 + 6*z2 }-> s8 :|: s8 >= 0, s8 <= 0 + z2 + 2, z'' = 0, z2 >= 0, z1 - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 95 + 26*s' + 4*s'*z1 + s'*z2 + 2*s'^2 + 13*z1 + 6*z2 }-> s9 :|: s9 >= 0, s9 <= s' + z2 + 2, s' >= 0, s' <= z'' - 1, z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 2 }-> z2 - 1 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1, z2 - 1 >= 0
   minus(z', z'') -{ 2 + 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'') -{ 1 }-> 0 :|: z'' >= 0, z' = 0
   p(z') -{ 1 }-> z' - 1 :|: z' - 1 >= 0
   plus(z', z'') -{ 1 + z' }-> s'' :|: s'' >= 0, s'' <= z' - 1 + (1 + z''), z' - 1 >= 0, z'' >= 0
   plus(z', z'') -{ 1 }-> z'' :|: z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 23 + 13*z' + 2*z'*z'' + z'^2 + 3*z'' }-> s4 :|: s4 >= 0, s4 <= 1 + 0 + 1 + (1 + (z' - 1)), z1 = 0, z' - 1 >= 0, z'' >= 0
   quot(z', z'', z1) -{ 21 + 2*s1 + s1*z' + 12*z' + 2*z'*z'' + z'^2 + 3*z'' + z1 }-> s5 :|: s5 >= 0, s5 <= s1 + 1 + (1 + (z' - 1)), s1 >= 0, s1 <= z1 - 1 + (1 + (1 + 0)), z' - 1 >= 0, z'' >= 0, z1 - 1 >= 0
   quot(z', z'', z1) -{ 23 + 3*z'' }-> s6 :|: s6 >= 0, s6 <= 1 + 0 + 1 + 0, z1 = 0, z'' >= 0, z' = 0
   quot(z', z'', z1) -{ 21 + 2*s2 + 3*z'' + z1 }-> s7 :|: s7 >= 0, s7 <= s2 + 1 + 0, s2 >= 0, s2 <= z1 - 1 + (1 + (1 + 0)), z1 - 1 >= 0, z'' >= 0, z' = 0
   zero(z') -{ 1 }-> 1 :|: z' = 0
   zero(z') -{ 1 }-> 0 :|: z' - 1 >= 0

Function symbols to be analyzed: 
Previous analysis results are:
  minus: runtime: O(n^1) [2 + z''], size: O(n^1) [z']
  zero: runtime: O(1) [1], size: O(1) [1]
  plus: runtime: O(n^1) [1 + z'], size: O(n^1) [z' + z'']
  p: runtime: O(1) [1], size: O(n^1) [z']
  if: runtime: O(n^2) [18 + 12*z'' + 2*z''*z1 + z''*z2 + z''^2 + 3*z1 + 2*z2], size: O(n^1) [1 + z'' + z2]
  quot: runtime: O(n^2) [92 + 26*z' + 4*z'*z'' + z'*z1 + 2*z'^2 + 12*z'' + 6*z1], size: O(n^1) [2 + z' + z1]
  div: runtime: O(n^2) [93 + 26*z' + 4*z'*z'' + 2*z'^2 + 12*z''], size: O(n^1) [2 + z']

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(53) FinalProof (FINISHED)
Computed overall runtime complexity
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(54)
BOUNDS(1, n^2)

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(55) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(56)
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:

   minus(0, y) -> 0
   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   plus(0, y) -> y
   plus(s(x), y) -> plus(x, s(y))
   zero(s(x)) -> false
   zero(0) -> true
   p(s(x)) -> x
   div(x, y) -> quot(x, y, 0)
   quot(x, y, z) -> if(zero(x), x, y, plus(z, s(0)))
   if(true, x, y, z) -> p(z)
   if(false, x, s(y), z) -> quot(minus(x, s(y)), s(y), z)

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

The rewrite sequence

minus(s(x), s(y)) ->^+ minus(x, y)

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

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

The result substitution is [ ].




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(58)
Complex Obligation (BEST)

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(59)
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:

   minus(0, y) -> 0
   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   plus(0, y) -> y
   plus(s(x), y) -> plus(x, s(y))
   zero(s(x)) -> false
   zero(0) -> true
   p(s(x)) -> x
   div(x, y) -> quot(x, y, 0)
   quot(x, y, z) -> if(zero(x), x, y, plus(z, s(0)))
   if(true, x, y, z) -> p(z)
   if(false, x, s(y), z) -> quot(minus(x, s(y)), s(y), z)

S is empty.
Rewrite Strategy: INNERMOST
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(60) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(61)
BOUNDS(n^1, INF)

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(62)
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:

   minus(0, y) -> 0
   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   plus(0, y) -> y
   plus(s(x), y) -> plus(x, s(y))
   zero(s(x)) -> false
   zero(0) -> true
   p(s(x)) -> x
   div(x, y) -> quot(x, y, 0)
   quot(x, y, z) -> if(zero(x), x, y, plus(z, s(0)))
   if(true, x, y, z) -> p(z)
   if(false, x, s(y), z) -> quot(minus(x, s(y)), s(y), z)

S is empty.
Rewrite Strategy: INNERMOST