/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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


WORST_CASE(Omega(n^1), O(n^2))
proof of /export/starexec/sandbox2/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# AProVE Commit ID: 794c25de1cacf0d048858bcd21c9a779e1221865 marcel 20200619 unpublished dirty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) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (12) CpxRNTS
    (13) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRNTS
    (15) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (16) CpxRNTS
    (17) IntTrsBoundProof [UPPER BOUND(ID), 366 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 130 ms]
    (20) CpxRNTS
    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 312 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 115 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 204 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 84 ms]
    (32) CpxRNTS
    (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 1818 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 1455 ms]
    (38) CpxRNTS
    (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 157 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 2 ms]
    (44) CpxRNTS
    (45) FinalProof [FINISHED, 0 ms]
    (46) BOUNDS(1, n^2)
(47) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
(48) CpxTRS
    (49) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) typed CpxTrs
    (51) OrderProof [LOWER BOUND(ID), 0 ms]
    (52) typed CpxTrs
    (53) RewriteLemmaProof [LOWER BOUND(ID), 295 ms]
    (54) BEST
        (55) proven lower bound
            (56) LowerBoundPropagationProof [FINISHED, 0 ms]
            (57) BOUNDS(n^1, INF)
        (58) typed CpxTrs
            (59) RewriteLemmaProof [LOWER BOUND(ID), 67 ms]
            (60) typed CpxTrs
            (61) RewriteLemmaProof [LOWER BOUND(ID), 33 ms]
            (62) typed CpxTrs


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

(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:

   division(x, y) -> div(x, y, 0)
   div(x, y, z) -> if(lt(x, y), x, y, inc(z))
   if(true, x, y, z) -> z
   if(false, x, s(y), z) -> div(minus(x, s(y)), s(y), z)
   minus(x, 0) -> x
   minus(s(x), s(y)) -> minus(x, y)
   lt(x, 0) -> false
   lt(0, s(y)) -> true
   lt(s(x), s(y)) -> lt(x, y)
   inc(0) -> s(0)
   inc(s(x)) -> s(inc(x))

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:

   division(x, y) -> div(x, y, 0) [1]
   div(x, y, z) -> if(lt(x, y), x, y, inc(z)) [1]
   if(true, x, y, z) -> z [1]
   if(false, x, s(y), z) -> div(minus(x, s(y)), s(y), z) [1]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   lt(x, 0) -> false [1]
   lt(0, s(y)) -> true [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   inc(0) -> s(0) [1]
   inc(s(x)) -> s(inc(x)) [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:

   division(x, y) -> div(x, y, 0) [1]
   div(x, y, z) -> if(lt(x, y), x, y, inc(z)) [1]
   if(true, x, y, z) -> z [1]
   if(false, x, s(y), z) -> div(minus(x, s(y)), s(y), z) [1]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   lt(x, 0) -> false [1]
   lt(0, s(y)) -> true [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   inc(0) -> s(0) [1]
   inc(s(x)) -> s(inc(x)) [1]

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

   division_2
   div_3
   if_4

(c) The following functions are completely defined:

   minus_2
   lt_2
   inc_1

Due to the following rules being added:

   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:

   division(x, y) -> div(x, y, 0) [1]
   div(x, y, z) -> if(lt(x, y), x, y, inc(z)) [1]
   if(true, x, y, z) -> z [1]
   if(false, x, s(y), z) -> div(minus(x, s(y)), s(y), z) [1]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   lt(x, 0) -> false [1]
   lt(0, s(y)) -> true [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   inc(0) -> s(0) [1]
   inc(s(x)) -> s(inc(x)) [1]
   minus(v0, v1) -> 0 [0]

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

   division(x, y) -> div(x, y, 0) [1]
   div(x, 0, 0) -> if(false, x, 0, s(0)) [3]
   div(x, 0, s(x'')) -> if(false, x, 0, s(inc(x''))) [3]
   div(0, s(y'), 0) -> if(true, 0, s(y'), s(0)) [3]
   div(0, s(y'), s(x1)) -> if(true, 0, s(y'), s(inc(x1))) [3]
   div(s(x'), s(y''), 0) -> if(lt(x', y''), s(x'), s(y''), s(0)) [3]
   div(s(x'), s(y''), s(x2)) -> if(lt(x', y''), s(x'), s(y''), s(inc(x2))) [3]
   if(true, x, y, z) -> z [1]
   if(false, s(x3), s(y), z) -> div(minus(x3, y), s(y), z) [2]
   if(false, x, s(y), z) -> div(0, s(y), z) [1]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   lt(x, 0) -> false [1]
   lt(0, s(y)) -> true [1]
   lt(s(x), s(y)) -> lt(x, y) [1]
   inc(0) -> s(0) [1]
   inc(s(x)) -> s(inc(x)) [1]
   minus(v0, v1) -> 0 [0]

The TRS has the following type information:
division :: 0:s -> 0:s -> 0:s
div :: 0:s -> 0:s -> 0:s -> 0:s
0 :: 0:s
if :: true:false -> 0:s -> 0:s -> 0:s -> 0:s
lt :: 0:s -> 0:s -> true:false
inc :: 0:s -> 0:s
true :: true:false
false :: true:false
s :: 0:s -> 0:s
minus :: 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'', z1) -{ 3 }-> if(lt(x', y''), 1 + x', 1 + y'', 1 + inc(x2)) :|: z' = 1 + x', x' >= 0, y'' >= 0, x2 >= 0, z'' = 1 + y'', z1 = 1 + x2
   div(z', z'', z1) -{ 3 }-> if(lt(x', y''), 1 + x', 1 + y'', 1 + 0) :|: z1 = 0, z' = 1 + x', x' >= 0, y'' >= 0, z'' = 1 + y''
   div(z', z'', z1) -{ 3 }-> if(1, 0, 1 + y', 1 + inc(x1)) :|: x1 >= 0, y' >= 0, z1 = 1 + x1, z' = 0, z'' = 1 + y'
   div(z', z'', z1) -{ 3 }-> if(1, 0, 1 + y', 1 + 0) :|: z1 = 0, y' >= 0, z' = 0, z'' = 1 + y'
   div(z', z'', z1) -{ 3 }-> if(0, x, 0, 1 + inc(x'')) :|: z'' = 0, z' = x, z1 = 1 + x'', x >= 0, x'' >= 0
   div(z', z'', z1) -{ 3 }-> if(0, x, 0, 1 + 0) :|: z'' = 0, z1 = 0, z' = x, x >= 0
   division(z', z'') -{ 1 }-> div(x, y, 0) :|: z' = x, z'' = y, x >= 0, y >= 0
   if(z', z'', z1, z2) -{ 1 }-> z :|: z1 = y, z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> div(minus(x3, y), 1 + y, z) :|: z >= 0, z'' = 1 + x3, z2 = z, y >= 0, z1 = 1 + y, x3 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 1 }-> div(0, 1 + y, z) :|: z >= 0, z2 = z, x >= 0, y >= 0, z'' = x, z1 = 1 + y, z' = 0
   inc(z') -{ 1 }-> 1 + inc(x) :|: z' = 1 + x, x >= 0
   inc(z') -{ 1 }-> 1 + 0 :|: z' = 0
   lt(z', z'') -{ 1 }-> lt(x, y) :|: z' = 1 + x, x >= 0, y >= 0, z'' = 1 + y
   lt(z', z'') -{ 1 }-> 1 :|: y >= 0, z'' = 1 + y, z' = 0
   lt(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' = x, x >= 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'') -{ 0 }-> 0 :|: v0 >= 0, v1 >= 0, z'' = v1, z' = v0


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

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

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

   div(z', z'', z1) -{ 3 }-> if(lt(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0
   div(z', z'', z1) -{ 3 }-> if(lt(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' - 1 >= 0
   div(z', z'', z1) -{ 3 }-> if(1, 0, 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   div(z', z'', z1) -{ 3 }-> if(1, 0, 1 + (z'' - 1), 1 + 0) :|: z1 = 0, z'' - 1 >= 0, z' = 0
   div(z', z'', z1) -{ 3 }-> if(0, z', 0, 1 + inc(z1 - 1)) :|: z'' = 0, z' >= 0, z1 - 1 >= 0
   div(z', z'', z1) -{ 3 }-> if(0, z', 0, 1 + 0) :|: z'' = 0, z1 = 0, z' >= 0
   division(z', z'') -{ 1 }-> div(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> div(minus(z'' - 1, z1 - 1), 1 + (z1 - 1), z2) :|: z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 1 }-> div(0, 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   inc(z') -{ 1 }-> 1 + inc(z' - 1) :|: z' - 1 >= 0
   inc(z') -{ 1 }-> 1 + 0 :|: z' = 0
   lt(z', z'') -{ 1 }-> lt(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   lt(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   lt(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 }-> minus(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0


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

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

   { lt }
   { minus }
   { inc }
   { div, if }
   { division }

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

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

   div(z', z'', z1) -{ 3 }-> if(lt(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0
   div(z', z'', z1) -{ 3 }-> if(lt(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' - 1 >= 0
   div(z', z'', z1) -{ 3 }-> if(1, 0, 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   div(z', z'', z1) -{ 3 }-> if(1, 0, 1 + (z'' - 1), 1 + 0) :|: z1 = 0, z'' - 1 >= 0, z' = 0
   div(z', z'', z1) -{ 3 }-> if(0, z', 0, 1 + inc(z1 - 1)) :|: z'' = 0, z' >= 0, z1 - 1 >= 0
   div(z', z'', z1) -{ 3 }-> if(0, z', 0, 1 + 0) :|: z'' = 0, z1 = 0, z' >= 0
   division(z', z'') -{ 1 }-> div(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> div(minus(z'' - 1, z1 - 1), 1 + (z1 - 1), z2) :|: z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 1 }-> div(0, 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   inc(z') -{ 1 }-> 1 + inc(z' - 1) :|: z' - 1 >= 0
   inc(z') -{ 1 }-> 1 + 0 :|: z' = 0
   lt(z', z'') -{ 1 }-> lt(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   lt(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   lt(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 }-> minus(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {lt}, {minus}, {inc}, {div,if}, {division}

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

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

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

   div(z', z'', z1) -{ 3 }-> if(lt(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0
   div(z', z'', z1) -{ 3 }-> if(lt(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' - 1 >= 0
   div(z', z'', z1) -{ 3 }-> if(1, 0, 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   div(z', z'', z1) -{ 3 }-> if(1, 0, 1 + (z'' - 1), 1 + 0) :|: z1 = 0, z'' - 1 >= 0, z' = 0
   div(z', z'', z1) -{ 3 }-> if(0, z', 0, 1 + inc(z1 - 1)) :|: z'' = 0, z' >= 0, z1 - 1 >= 0
   div(z', z'', z1) -{ 3 }-> if(0, z', 0, 1 + 0) :|: z'' = 0, z1 = 0, z' >= 0
   division(z', z'') -{ 1 }-> div(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> div(minus(z'' - 1, z1 - 1), 1 + (z1 - 1), z2) :|: z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 1 }-> div(0, 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   inc(z') -{ 1 }-> 1 + inc(z' - 1) :|: z' - 1 >= 0
   inc(z') -{ 1 }-> 1 + 0 :|: z' = 0
   lt(z', z'') -{ 1 }-> lt(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   lt(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   lt(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 }-> minus(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0

Function symbols to be analyzed: {lt}, {minus}, {inc}, {div,if}, {division}

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

(17) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   div(z', z'', z1) -{ 3 }-> if(lt(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0
   div(z', z'', z1) -{ 3 }-> if(lt(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' - 1 >= 0
   div(z', z'', z1) -{ 3 }-> if(1, 0, 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   div(z', z'', z1) -{ 3 }-> if(1, 0, 1 + (z'' - 1), 1 + 0) :|: z1 = 0, z'' - 1 >= 0, z' = 0
   div(z', z'', z1) -{ 3 }-> if(0, z', 0, 1 + inc(z1 - 1)) :|: z'' = 0, z' >= 0, z1 - 1 >= 0
   div(z', z'', z1) -{ 3 }-> if(0, z', 0, 1 + 0) :|: z'' = 0, z1 = 0, z' >= 0
   division(z', z'') -{ 1 }-> div(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> div(minus(z'' - 1, z1 - 1), 1 + (z1 - 1), z2) :|: z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 1 }-> div(0, 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   inc(z') -{ 1 }-> 1 + inc(z' - 1) :|: z' - 1 >= 0
   inc(z') -{ 1 }-> 1 + 0 :|: z' = 0
   lt(z', z'') -{ 1 }-> lt(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   lt(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   lt(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 }-> minus(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0

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

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   div(z', z'', z1) -{ 3 }-> if(lt(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: z' - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0
   div(z', z'', z1) -{ 3 }-> if(lt(z' - 1, z'' - 1), 1 + (z' - 1), 1 + (z'' - 1), 1 + 0) :|: z1 = 0, z' - 1 >= 0, z'' - 1 >= 0
   div(z', z'', z1) -{ 3 }-> if(1, 0, 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   div(z', z'', z1) -{ 3 }-> if(1, 0, 1 + (z'' - 1), 1 + 0) :|: z1 = 0, z'' - 1 >= 0, z' = 0
   div(z', z'', z1) -{ 3 }-> if(0, z', 0, 1 + inc(z1 - 1)) :|: z'' = 0, z' >= 0, z1 - 1 >= 0
   div(z', z'', z1) -{ 3 }-> if(0, z', 0, 1 + 0) :|: z'' = 0, z1 = 0, z' >= 0
   division(z', z'') -{ 1 }-> div(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> div(minus(z'' - 1, z1 - 1), 1 + (z1 - 1), z2) :|: z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 1 }-> div(0, 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   inc(z') -{ 1 }-> 1 + inc(z' - 1) :|: z' - 1 >= 0
   inc(z') -{ 1 }-> 1 + 0 :|: z' = 0
   lt(z', z'') -{ 1 }-> lt(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   lt(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   lt(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 }-> minus(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0

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

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

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

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

   div(z', z'', z1) -{ 4 + z'' }-> if(s, 1 + (z' - 1), 1 + (z'' - 1), 1 + 0) :|: s >= 0, s <= 1, z1 = 0, z' - 1 >= 0, z'' - 1 >= 0
   div(z', z'', z1) -{ 4 + z'' }-> if(s', 1 + (z' - 1), 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0
   div(z', z'', z1) -{ 3 }-> if(1, 0, 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   div(z', z'', z1) -{ 3 }-> if(1, 0, 1 + (z'' - 1), 1 + 0) :|: z1 = 0, z'' - 1 >= 0, z' = 0
   div(z', z'', z1) -{ 3 }-> if(0, z', 0, 1 + inc(z1 - 1)) :|: z'' = 0, z' >= 0, z1 - 1 >= 0
   div(z', z'', z1) -{ 3 }-> if(0, z', 0, 1 + 0) :|: z'' = 0, z1 = 0, z' >= 0
   division(z', z'') -{ 1 }-> div(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> div(minus(z'' - 1, z1 - 1), 1 + (z1 - 1), z2) :|: z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 1 }-> div(0, 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   inc(z') -{ 1 }-> 1 + inc(z' - 1) :|: z' - 1 >= 0
   inc(z') -{ 1 }-> 1 + 0 :|: z' = 0
   lt(z', z'') -{ 2 + z'' }-> s'' :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z'' - 1 >= 0
   lt(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   lt(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 }-> minus(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0

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

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

(23) 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'

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

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

   div(z', z'', z1) -{ 4 + z'' }-> if(s, 1 + (z' - 1), 1 + (z'' - 1), 1 + 0) :|: s >= 0, s <= 1, z1 = 0, z' - 1 >= 0, z'' - 1 >= 0
   div(z', z'', z1) -{ 4 + z'' }-> if(s', 1 + (z' - 1), 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0
   div(z', z'', z1) -{ 3 }-> if(1, 0, 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   div(z', z'', z1) -{ 3 }-> if(1, 0, 1 + (z'' - 1), 1 + 0) :|: z1 = 0, z'' - 1 >= 0, z' = 0
   div(z', z'', z1) -{ 3 }-> if(0, z', 0, 1 + inc(z1 - 1)) :|: z'' = 0, z' >= 0, z1 - 1 >= 0
   div(z', z'', z1) -{ 3 }-> if(0, z', 0, 1 + 0) :|: z'' = 0, z1 = 0, z' >= 0
   division(z', z'') -{ 1 }-> div(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> div(minus(z'' - 1, z1 - 1), 1 + (z1 - 1), z2) :|: z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 1 }-> div(0, 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   inc(z') -{ 1 }-> 1 + inc(z' - 1) :|: z' - 1 >= 0
   inc(z') -{ 1 }-> 1 + 0 :|: z' = 0
   lt(z', z'') -{ 2 + z'' }-> s'' :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z'' - 1 >= 0
   lt(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   lt(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 }-> minus(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0

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

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

(25) 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''

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

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

   div(z', z'', z1) -{ 4 + z'' }-> if(s, 1 + (z' - 1), 1 + (z'' - 1), 1 + 0) :|: s >= 0, s <= 1, z1 = 0, z' - 1 >= 0, z'' - 1 >= 0
   div(z', z'', z1) -{ 4 + z'' }-> if(s', 1 + (z' - 1), 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0
   div(z', z'', z1) -{ 3 }-> if(1, 0, 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   div(z', z'', z1) -{ 3 }-> if(1, 0, 1 + (z'' - 1), 1 + 0) :|: z1 = 0, z'' - 1 >= 0, z' = 0
   div(z', z'', z1) -{ 3 }-> if(0, z', 0, 1 + inc(z1 - 1)) :|: z'' = 0, z' >= 0, z1 - 1 >= 0
   div(z', z'', z1) -{ 3 }-> if(0, z', 0, 1 + 0) :|: z'' = 0, z1 = 0, z' >= 0
   division(z', z'') -{ 1 }-> div(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 }-> div(minus(z'' - 1, z1 - 1), 1 + (z1 - 1), z2) :|: z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 1 }-> div(0, 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   inc(z') -{ 1 }-> 1 + inc(z' - 1) :|: z' - 1 >= 0
   inc(z') -{ 1 }-> 1 + 0 :|: z' = 0
   lt(z', z'') -{ 2 + z'' }-> s'' :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z'' - 1 >= 0
   lt(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   lt(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 }-> z' :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 }-> minus(z' - 1, z'' - 1) :|: z' - 1 >= 0, z'' - 1 >= 0
   minus(z', z'') -{ 0 }-> 0 :|: z' >= 0, z'' >= 0

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

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

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

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

   div(z', z'', z1) -{ 4 + z'' }-> if(s, 1 + (z' - 1), 1 + (z'' - 1), 1 + 0) :|: s >= 0, s <= 1, z1 = 0, z' - 1 >= 0, z'' - 1 >= 0
   div(z', z'', z1) -{ 4 + z'' }-> if(s', 1 + (z' - 1), 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0
   div(z', z'', z1) -{ 3 }-> if(1, 0, 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   div(z', z'', z1) -{ 3 }-> if(1, 0, 1 + (z'' - 1), 1 + 0) :|: z1 = 0, z'' - 1 >= 0, z' = 0
   div(z', z'', z1) -{ 3 }-> if(0, z', 0, 1 + inc(z1 - 1)) :|: z'' = 0, z' >= 0, z1 - 1 >= 0
   div(z', z'', z1) -{ 3 }-> if(0, z', 0, 1 + 0) :|: z'' = 0, z1 = 0, z' >= 0
   division(z', z'') -{ 1 }-> div(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 + z1 }-> div(s1, 1 + (z1 - 1), z2) :|: s1 >= 0, s1 <= z'' - 1, z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 1 }-> div(0, 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   inc(z') -{ 1 }-> 1 + inc(z' - 1) :|: z' - 1 >= 0
   inc(z') -{ 1 }-> 1 + 0 :|: z' = 0
   lt(z', z'') -{ 2 + z'' }-> s'' :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z'' - 1 >= 0
   lt(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   lt(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 + z'' }-> s2 :|: s2 >= 0, s2 <= 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

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

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

(29) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   div(z', z'', z1) -{ 4 + z'' }-> if(s, 1 + (z' - 1), 1 + (z'' - 1), 1 + 0) :|: s >= 0, s <= 1, z1 = 0, z' - 1 >= 0, z'' - 1 >= 0
   div(z', z'', z1) -{ 4 + z'' }-> if(s', 1 + (z' - 1), 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0
   div(z', z'', z1) -{ 3 }-> if(1, 0, 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   div(z', z'', z1) -{ 3 }-> if(1, 0, 1 + (z'' - 1), 1 + 0) :|: z1 = 0, z'' - 1 >= 0, z' = 0
   div(z', z'', z1) -{ 3 }-> if(0, z', 0, 1 + inc(z1 - 1)) :|: z'' = 0, z' >= 0, z1 - 1 >= 0
   div(z', z'', z1) -{ 3 }-> if(0, z', 0, 1 + 0) :|: z'' = 0, z1 = 0, z' >= 0
   division(z', z'') -{ 1 }-> div(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 + z1 }-> div(s1, 1 + (z1 - 1), z2) :|: s1 >= 0, s1 <= z'' - 1, z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 1 }-> div(0, 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   inc(z') -{ 1 }-> 1 + inc(z' - 1) :|: z' - 1 >= 0
   inc(z') -{ 1 }-> 1 + 0 :|: z' = 0
   lt(z', z'') -{ 2 + z'' }-> s'' :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z'' - 1 >= 0
   lt(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   lt(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 + z'' }-> s2 :|: s2 >= 0, s2 <= 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

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

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   div(z', z'', z1) -{ 4 + z'' }-> if(s, 1 + (z' - 1), 1 + (z'' - 1), 1 + 0) :|: s >= 0, s <= 1, z1 = 0, z' - 1 >= 0, z'' - 1 >= 0
   div(z', z'', z1) -{ 4 + z'' }-> if(s', 1 + (z' - 1), 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0
   div(z', z'', z1) -{ 3 }-> if(1, 0, 1 + (z'' - 1), 1 + inc(z1 - 1)) :|: z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   div(z', z'', z1) -{ 3 }-> if(1, 0, 1 + (z'' - 1), 1 + 0) :|: z1 = 0, z'' - 1 >= 0, z' = 0
   div(z', z'', z1) -{ 3 }-> if(0, z', 0, 1 + inc(z1 - 1)) :|: z'' = 0, z' >= 0, z1 - 1 >= 0
   div(z', z'', z1) -{ 3 }-> if(0, z', 0, 1 + 0) :|: z'' = 0, z1 = 0, z' >= 0
   division(z', z'') -{ 1 }-> div(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 + z1 }-> div(s1, 1 + (z1 - 1), z2) :|: s1 >= 0, s1 <= z'' - 1, z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 1 }-> div(0, 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   inc(z') -{ 1 }-> 1 + inc(z' - 1) :|: z' - 1 >= 0
   inc(z') -{ 1 }-> 1 + 0 :|: z' = 0
   lt(z', z'') -{ 2 + z'' }-> s'' :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z'' - 1 >= 0
   lt(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   lt(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 + z'' }-> s2 :|: s2 >= 0, s2 <= 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

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

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

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

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

   div(z', z'', z1) -{ 4 + z'' }-> if(s, 1 + (z' - 1), 1 + (z'' - 1), 1 + 0) :|: s >= 0, s <= 1, z1 = 0, z' - 1 >= 0, z'' - 1 >= 0
   div(z', z'', z1) -{ 4 + z'' + z1 }-> if(s', 1 + (z' - 1), 1 + (z'' - 1), 1 + s5) :|: s5 >= 0, s5 <= z1 - 1 + 1, s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0
   div(z', z'', z1) -{ 3 + z1 }-> if(1, 0, 1 + (z'' - 1), 1 + s4) :|: s4 >= 0, s4 <= z1 - 1 + 1, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   div(z', z'', z1) -{ 3 }-> if(1, 0, 1 + (z'' - 1), 1 + 0) :|: z1 = 0, z'' - 1 >= 0, z' = 0
   div(z', z'', z1) -{ 3 + z1 }-> if(0, z', 0, 1 + s3) :|: s3 >= 0, s3 <= z1 - 1 + 1, z'' = 0, z' >= 0, z1 - 1 >= 0
   div(z', z'', z1) -{ 3 }-> if(0, z', 0, 1 + 0) :|: z'' = 0, z1 = 0, z' >= 0
   division(z', z'') -{ 1 }-> div(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 + z1 }-> div(s1, 1 + (z1 - 1), z2) :|: s1 >= 0, s1 <= z'' - 1, z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 1 }-> div(0, 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   inc(z') -{ 1 + z' }-> 1 + s6 :|: s6 >= 0, s6 <= z' - 1 + 1, z' - 1 >= 0
   inc(z') -{ 1 }-> 1 + 0 :|: z' = 0
   lt(z', z'') -{ 2 + z'' }-> s'' :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z'' - 1 >= 0
   lt(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   lt(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 + z'' }-> s2 :|: s2 >= 0, s2 <= 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

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

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

(35) 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' + z1

Computed SIZE bound using CoFloCo for: if
after applying outer abstraction to obtain an ITS,
resulting in: O(n^1) with polynomial bound: 2 + z'' + z2

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

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

   div(z', z'', z1) -{ 4 + z'' }-> if(s, 1 + (z' - 1), 1 + (z'' - 1), 1 + 0) :|: s >= 0, s <= 1, z1 = 0, z' - 1 >= 0, z'' - 1 >= 0
   div(z', z'', z1) -{ 4 + z'' + z1 }-> if(s', 1 + (z' - 1), 1 + (z'' - 1), 1 + s5) :|: s5 >= 0, s5 <= z1 - 1 + 1, s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0
   div(z', z'', z1) -{ 3 + z1 }-> if(1, 0, 1 + (z'' - 1), 1 + s4) :|: s4 >= 0, s4 <= z1 - 1 + 1, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   div(z', z'', z1) -{ 3 }-> if(1, 0, 1 + (z'' - 1), 1 + 0) :|: z1 = 0, z'' - 1 >= 0, z' = 0
   div(z', z'', z1) -{ 3 + z1 }-> if(0, z', 0, 1 + s3) :|: s3 >= 0, s3 <= z1 - 1 + 1, z'' = 0, z' >= 0, z1 - 1 >= 0
   div(z', z'', z1) -{ 3 }-> if(0, z', 0, 1 + 0) :|: z'' = 0, z1 = 0, z' >= 0
   division(z', z'') -{ 1 }-> div(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 + z1 }-> div(s1, 1 + (z1 - 1), z2) :|: s1 >= 0, s1 <= z'' - 1, z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 1 }-> div(0, 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   inc(z') -{ 1 + z' }-> 1 + s6 :|: s6 >= 0, s6 <= z' - 1 + 1, z' - 1 >= 0
   inc(z') -{ 1 }-> 1 + 0 :|: z' = 0
   lt(z', z'') -{ 2 + z'' }-> s'' :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z'' - 1 >= 0
   lt(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   lt(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 + z'' }-> s2 :|: s2 >= 0, s2 <= 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

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

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using CoFloCo for: div
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 24 + 9*z' + 2*z'*z'' + z'*z1 + z'^2 + 7*z'' + 3*z1

Computed RUNTIME bound using KoAT for: if
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 44 + 7*z'' + 2*z''*z1 + z''*z2 + z''^2 + 13*z1 + 5*z2

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

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

   div(z', z'', z1) -{ 4 + z'' }-> if(s, 1 + (z' - 1), 1 + (z'' - 1), 1 + 0) :|: s >= 0, s <= 1, z1 = 0, z' - 1 >= 0, z'' - 1 >= 0
   div(z', z'', z1) -{ 4 + z'' + z1 }-> if(s', 1 + (z' - 1), 1 + (z'' - 1), 1 + s5) :|: s5 >= 0, s5 <= z1 - 1 + 1, s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0
   div(z', z'', z1) -{ 3 + z1 }-> if(1, 0, 1 + (z'' - 1), 1 + s4) :|: s4 >= 0, s4 <= z1 - 1 + 1, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   div(z', z'', z1) -{ 3 }-> if(1, 0, 1 + (z'' - 1), 1 + 0) :|: z1 = 0, z'' - 1 >= 0, z' = 0
   div(z', z'', z1) -{ 3 + z1 }-> if(0, z', 0, 1 + s3) :|: s3 >= 0, s3 <= z1 - 1 + 1, z'' = 0, z' >= 0, z1 - 1 >= 0
   div(z', z'', z1) -{ 3 }-> if(0, z', 0, 1 + 0) :|: z'' = 0, z1 = 0, z' >= 0
   division(z', z'') -{ 1 }-> div(z', z'', 0) :|: z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   if(z', z'', z1, z2) -{ 2 + z1 }-> div(s1, 1 + (z1 - 1), z2) :|: s1 >= 0, s1 <= z'' - 1, z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 1 }-> div(0, 1 + (z1 - 1), z2) :|: z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   inc(z') -{ 1 + z' }-> 1 + s6 :|: s6 >= 0, s6 <= z' - 1 + 1, z' - 1 >= 0
   inc(z') -{ 1 }-> 1 + 0 :|: z' = 0
   lt(z', z'') -{ 2 + z'' }-> s'' :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z'' - 1 >= 0
   lt(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   lt(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 + z'' }-> s2 :|: s2 >= 0, s2 <= 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

Function symbols to be analyzed: {division}
Previous analysis results are:
  lt: runtime: O(n^1) [2 + z''], size: O(1) [1]
  minus: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  inc: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z']
  div: runtime: O(n^2) [24 + 9*z' + 2*z'*z'' + z'*z1 + z'^2 + 7*z'' + 3*z1], size: O(n^1) [2 + z' + z1]
  if: runtime: O(n^2) [44 + 7*z'' + 2*z''*z1 + z''*z2 + z''^2 + 13*z1 + 5*z2], size: O(n^1) [2 + z'' + z2]

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

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

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

   div(z', z'', z1) -{ 52 + 13*z'' }-> s10 :|: s10 >= 0, s10 <= 0 + (1 + 0) + 2, z1 = 0, z'' - 1 >= 0, z' = 0
   div(z', z'', z1) -{ 52 + 5*s4 + 13*z'' + z1 }-> s11 :|: s11 >= 0, s11 <= 0 + (1 + s4) + 2, s4 >= 0, s4 <= z1 - 1 + 1, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   div(z', z'', z1) -{ 53 + 8*z' + 2*z'*z'' + z'^2 + 14*z'' }-> s12 :|: s12 >= 0, s12 <= 1 + (z' - 1) + (1 + 0) + 2, s >= 0, s <= 1, z1 = 0, z' - 1 >= 0, z'' - 1 >= 0
   div(z', z'', z1) -{ 53 + 5*s5 + s5*z' + 8*z' + 2*z'*z'' + z'^2 + 14*z'' + z1 }-> s13 :|: s13 >= 0, s13 <= 1 + (z' - 1) + (1 + s5) + 2, s5 >= 0, s5 <= z1 - 1 + 1, s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0
   div(z', z'', z1) -{ 52 + 8*z' + z'^2 }-> s8 :|: s8 >= 0, s8 <= z' + (1 + 0) + 2, z'' = 0, z1 = 0, z' >= 0
   div(z', z'', z1) -{ 52 + 5*s3 + s3*z' + 8*z' + z'^2 + z1 }-> s9 :|: s9 >= 0, s9 <= z' + (1 + s3) + 2, s3 >= 0, s3 <= z1 - 1 + 1, z'' = 0, z' >= 0, z1 - 1 >= 0
   division(z', z'') -{ 25 + 9*z' + 2*z'*z'' + z'^2 + 7*z'' }-> s7 :|: s7 >= 0, s7 <= z' + 0 + 2, z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 26 + 9*s1 + 2*s1*z1 + s1*z2 + s1^2 + 8*z1 + 3*z2 }-> s14 :|: s14 >= 0, s14 <= s1 + z2 + 2, s1 >= 0, s1 <= z'' - 1, z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 25 + 7*z1 + 3*z2 }-> s15 :|: s15 >= 0, s15 <= 0 + z2 + 2, z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   inc(z') -{ 1 + z' }-> 1 + s6 :|: s6 >= 0, s6 <= z' - 1 + 1, z' - 1 >= 0
   inc(z') -{ 1 }-> 1 + 0 :|: z' = 0
   lt(z', z'') -{ 2 + z'' }-> s'' :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z'' - 1 >= 0
   lt(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   lt(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 + z'' }-> s2 :|: s2 >= 0, s2 <= 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

Function symbols to be analyzed: {division}
Previous analysis results are:
  lt: runtime: O(n^1) [2 + z''], size: O(1) [1]
  minus: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  inc: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z']
  div: runtime: O(n^2) [24 + 9*z' + 2*z'*z'' + z'*z1 + z'^2 + 7*z'' + 3*z1], size: O(n^1) [2 + z' + z1]
  if: runtime: O(n^2) [44 + 7*z'' + 2*z''*z1 + z''*z2 + z''^2 + 13*z1 + 5*z2], size: O(n^1) [2 + z'' + z2]

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

(41) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

   div(z', z'', z1) -{ 52 + 13*z'' }-> s10 :|: s10 >= 0, s10 <= 0 + (1 + 0) + 2, z1 = 0, z'' - 1 >= 0, z' = 0
   div(z', z'', z1) -{ 52 + 5*s4 + 13*z'' + z1 }-> s11 :|: s11 >= 0, s11 <= 0 + (1 + s4) + 2, s4 >= 0, s4 <= z1 - 1 + 1, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   div(z', z'', z1) -{ 53 + 8*z' + 2*z'*z'' + z'^2 + 14*z'' }-> s12 :|: s12 >= 0, s12 <= 1 + (z' - 1) + (1 + 0) + 2, s >= 0, s <= 1, z1 = 0, z' - 1 >= 0, z'' - 1 >= 0
   div(z', z'', z1) -{ 53 + 5*s5 + s5*z' + 8*z' + 2*z'*z'' + z'^2 + 14*z'' + z1 }-> s13 :|: s13 >= 0, s13 <= 1 + (z' - 1) + (1 + s5) + 2, s5 >= 0, s5 <= z1 - 1 + 1, s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0
   div(z', z'', z1) -{ 52 + 8*z' + z'^2 }-> s8 :|: s8 >= 0, s8 <= z' + (1 + 0) + 2, z'' = 0, z1 = 0, z' >= 0
   div(z', z'', z1) -{ 52 + 5*s3 + s3*z' + 8*z' + z'^2 + z1 }-> s9 :|: s9 >= 0, s9 <= z' + (1 + s3) + 2, s3 >= 0, s3 <= z1 - 1 + 1, z'' = 0, z' >= 0, z1 - 1 >= 0
   division(z', z'') -{ 25 + 9*z' + 2*z'*z'' + z'^2 + 7*z'' }-> s7 :|: s7 >= 0, s7 <= z' + 0 + 2, z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 26 + 9*s1 + 2*s1*z1 + s1*z2 + s1^2 + 8*z1 + 3*z2 }-> s14 :|: s14 >= 0, s14 <= s1 + z2 + 2, s1 >= 0, s1 <= z'' - 1, z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 25 + 7*z1 + 3*z2 }-> s15 :|: s15 >= 0, s15 <= 0 + z2 + 2, z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   inc(z') -{ 1 + z' }-> 1 + s6 :|: s6 >= 0, s6 <= z' - 1 + 1, z' - 1 >= 0
   inc(z') -{ 1 }-> 1 + 0 :|: z' = 0
   lt(z', z'') -{ 2 + z'' }-> s'' :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z'' - 1 >= 0
   lt(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   lt(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 + z'' }-> s2 :|: s2 >= 0, s2 <= 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

Function symbols to be analyzed: {division}
Previous analysis results are:
  lt: runtime: O(n^1) [2 + z''], size: O(1) [1]
  minus: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  inc: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z']
  div: runtime: O(n^2) [24 + 9*z' + 2*z'*z'' + z'*z1 + z'^2 + 7*z'' + 3*z1], size: O(n^1) [2 + z' + z1]
  if: runtime: O(n^2) [44 + 7*z'' + 2*z''*z1 + z''*z2 + z''^2 + 13*z1 + 5*z2], size: O(n^1) [2 + z'' + z2]
  division: runtime: ?, size: O(n^1) [2 + z']

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

Computed RUNTIME bound using KoAT for: division
after applying outer abstraction to obtain an ITS,
resulting in: O(n^2) with polynomial bound: 25 + 9*z' + 2*z'*z'' + z'^2 + 7*z''

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

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

   div(z', z'', z1) -{ 52 + 13*z'' }-> s10 :|: s10 >= 0, s10 <= 0 + (1 + 0) + 2, z1 = 0, z'' - 1 >= 0, z' = 0
   div(z', z'', z1) -{ 52 + 5*s4 + 13*z'' + z1 }-> s11 :|: s11 >= 0, s11 <= 0 + (1 + s4) + 2, s4 >= 0, s4 <= z1 - 1 + 1, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   div(z', z'', z1) -{ 53 + 8*z' + 2*z'*z'' + z'^2 + 14*z'' }-> s12 :|: s12 >= 0, s12 <= 1 + (z' - 1) + (1 + 0) + 2, s >= 0, s <= 1, z1 = 0, z' - 1 >= 0, z'' - 1 >= 0
   div(z', z'', z1) -{ 53 + 5*s5 + s5*z' + 8*z' + 2*z'*z'' + z'^2 + 14*z'' + z1 }-> s13 :|: s13 >= 0, s13 <= 1 + (z' - 1) + (1 + s5) + 2, s5 >= 0, s5 <= z1 - 1 + 1, s' >= 0, s' <= 1, z' - 1 >= 0, z'' - 1 >= 0, z1 - 1 >= 0
   div(z', z'', z1) -{ 52 + 8*z' + z'^2 }-> s8 :|: s8 >= 0, s8 <= z' + (1 + 0) + 2, z'' = 0, z1 = 0, z' >= 0
   div(z', z'', z1) -{ 52 + 5*s3 + s3*z' + 8*z' + z'^2 + z1 }-> s9 :|: s9 >= 0, s9 <= z' + (1 + s3) + 2, s3 >= 0, s3 <= z1 - 1 + 1, z'' = 0, z' >= 0, z1 - 1 >= 0
   division(z', z'') -{ 25 + 9*z' + 2*z'*z'' + z'^2 + 7*z'' }-> s7 :|: s7 >= 0, s7 <= z' + 0 + 2, z' >= 0, z'' >= 0
   if(z', z'', z1, z2) -{ 26 + 9*s1 + 2*s1*z1 + s1*z2 + s1^2 + 8*z1 + 3*z2 }-> s14 :|: s14 >= 0, s14 <= s1 + z2 + 2, s1 >= 0, s1 <= z'' - 1, z2 >= 0, z1 - 1 >= 0, z'' - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 25 + 7*z1 + 3*z2 }-> s15 :|: s15 >= 0, s15 <= 0 + z2 + 2, z2 >= 0, z'' >= 0, z1 - 1 >= 0, z' = 0
   if(z', z'', z1, z2) -{ 1 }-> z2 :|: z2 >= 0, z'' >= 0, z1 >= 0, z' = 1
   inc(z') -{ 1 + z' }-> 1 + s6 :|: s6 >= 0, s6 <= z' - 1 + 1, z' - 1 >= 0
   inc(z') -{ 1 }-> 1 + 0 :|: z' = 0
   lt(z', z'') -{ 2 + z'' }-> s'' :|: s'' >= 0, s'' <= 1, z' - 1 >= 0, z'' - 1 >= 0
   lt(z', z'') -{ 1 }-> 1 :|: z'' - 1 >= 0, z' = 0
   lt(z', z'') -{ 1 }-> 0 :|: z'' = 0, z' >= 0
   minus(z', z'') -{ 1 + z'' }-> s2 :|: s2 >= 0, s2 <= 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

Function symbols to be analyzed: 
Previous analysis results are:
  lt: runtime: O(n^1) [2 + z''], size: O(1) [1]
  minus: runtime: O(n^1) [1 + z''], size: O(n^1) [z']
  inc: runtime: O(n^1) [1 + z'], size: O(n^1) [1 + z']
  div: runtime: O(n^2) [24 + 9*z' + 2*z'*z'' + z'*z1 + z'^2 + 7*z'' + 3*z1], size: O(n^1) [2 + z' + z1]
  if: runtime: O(n^2) [44 + 7*z'' + 2*z''*z1 + z''*z2 + z''^2 + 13*z1 + 5*z2], size: O(n^1) [2 + z'' + z2]
  division: runtime: O(n^2) [25 + 9*z' + 2*z'*z'' + z'^2 + 7*z''], size: O(n^1) [2 + z']

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

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

(46)
BOUNDS(1, n^2)

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

(47) RenamingProof (BOTH BOUNDS(ID, ID))
Renamed function symbols to avoid clashes with predefined symbol.
----------------------------------------

(48)
Obligation:
The Runtime Complexity (innermost) of the given CpxTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   division(x, y) -> div(x, y, 0')
   div(x, y, z) -> if(lt(x, y), x, y, inc(z))
   if(true, x, y, z) -> z
   if(false, x, s(y), z) -> div(minus(x, s(y)), s(y), z)
   minus(x, 0') -> x
   minus(s(x), s(y)) -> minus(x, y)
   lt(x, 0') -> false
   lt(0', s(y)) -> true
   lt(s(x), s(y)) -> lt(x, y)
   inc(0') -> s(0')
   inc(s(x)) -> s(inc(x))

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

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

(50)
Obligation:
Innermost TRS:
Rules:
division(x, y) -> div(x, y, 0')
div(x, y, z) -> if(lt(x, y), x, y, inc(z))
if(true, x, y, z) -> z
if(false, x, s(y), z) -> div(minus(x, s(y)), s(y), z)
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
lt(x, 0') -> false
lt(0', s(y)) -> true
lt(s(x), s(y)) -> lt(x, y)
inc(0') -> s(0')
inc(s(x)) -> s(inc(x))

Types:
division :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s -> 0':s
0' :: 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
lt :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
true :: true:false
false :: true:false
s :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s

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

(51) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
div, lt, inc, minus

They will be analysed ascendingly in the following order:
lt < div
inc < div
minus < div

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

(52)
Obligation:
Innermost TRS:
Rules:
division(x, y) -> div(x, y, 0')
div(x, y, z) -> if(lt(x, y), x, y, inc(z))
if(true, x, y, z) -> z
if(false, x, s(y), z) -> div(minus(x, s(y)), s(y), z)
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
lt(x, 0') -> false
lt(0', s(y)) -> true
lt(s(x), s(y)) -> lt(x, y)
inc(0') -> s(0')
inc(s(x)) -> s(inc(x))

Types:
division :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s -> 0':s
0' :: 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
lt :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
true :: true:false
false :: true:false
s :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
lt, div, inc, minus

They will be analysed ascendingly in the following order:
lt < div
inc < div
minus < div

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

(53) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> false, rt in Omega(1 + n5_0)

Induction Base:
lt(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
false

Induction Step:
lt(gen_0':s3_0(+(n5_0, 1)), gen_0':s3_0(+(n5_0, 1))) ->_R^Omega(1)
lt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) ->_IH
false

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(54)
Complex Obligation (BEST)

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

(55)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
division(x, y) -> div(x, y, 0')
div(x, y, z) -> if(lt(x, y), x, y, inc(z))
if(true, x, y, z) -> z
if(false, x, s(y), z) -> div(minus(x, s(y)), s(y), z)
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
lt(x, 0') -> false
lt(0', s(y)) -> true
lt(s(x), s(y)) -> lt(x, y)
inc(0') -> s(0')
inc(s(x)) -> s(inc(x))

Types:
division :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s -> 0':s
0' :: 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
lt :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
true :: true:false
false :: true:false
s :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
lt, div, inc, minus

They will be analysed ascendingly in the following order:
lt < div
inc < div
minus < div

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

(56) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
----------------------------------------

(57)
BOUNDS(n^1, INF)

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

(58)
Obligation:
Innermost TRS:
Rules:
division(x, y) -> div(x, y, 0')
div(x, y, z) -> if(lt(x, y), x, y, inc(z))
if(true, x, y, z) -> z
if(false, x, s(y), z) -> div(minus(x, s(y)), s(y), z)
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
lt(x, 0') -> false
lt(0', s(y)) -> true
lt(s(x), s(y)) -> lt(x, y)
inc(0') -> s(0')
inc(s(x)) -> s(inc(x))

Types:
division :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s -> 0':s
0' :: 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
lt :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
true :: true:false
false :: true:false
s :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> false, rt in Omega(1 + n5_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
inc, div, minus

They will be analysed ascendingly in the following order:
inc < div
minus < div

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

(59) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
inc(gen_0':s3_0(n305_0)) -> gen_0':s3_0(+(1, n305_0)), rt in Omega(1 + n305_0)

Induction Base:
inc(gen_0':s3_0(0)) ->_R^Omega(1)
s(0')

Induction Step:
inc(gen_0':s3_0(+(n305_0, 1))) ->_R^Omega(1)
s(inc(gen_0':s3_0(n305_0))) ->_IH
s(gen_0':s3_0(+(1, c306_0)))

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(60)
Obligation:
Innermost TRS:
Rules:
division(x, y) -> div(x, y, 0')
div(x, y, z) -> if(lt(x, y), x, y, inc(z))
if(true, x, y, z) -> z
if(false, x, s(y), z) -> div(minus(x, s(y)), s(y), z)
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
lt(x, 0') -> false
lt(0', s(y)) -> true
lt(s(x), s(y)) -> lt(x, y)
inc(0') -> s(0')
inc(s(x)) -> s(inc(x))

Types:
division :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s -> 0':s
0' :: 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
lt :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
true :: true:false
false :: true:false
s :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> false, rt in Omega(1 + n5_0)
inc(gen_0':s3_0(n305_0)) -> gen_0':s3_0(+(1, n305_0)), rt in Omega(1 + n305_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
minus, div

They will be analysed ascendingly in the following order:
minus < div

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

(61) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s3_0(n547_0), gen_0':s3_0(n547_0)) -> gen_0':s3_0(0), rt in Omega(1 + n547_0)

Induction Base:
minus(gen_0':s3_0(0), gen_0':s3_0(0)) ->_R^Omega(1)
gen_0':s3_0(0)

Induction Step:
minus(gen_0':s3_0(+(n547_0, 1)), gen_0':s3_0(+(n547_0, 1))) ->_R^Omega(1)
minus(gen_0':s3_0(n547_0), gen_0':s3_0(n547_0)) ->_IH
gen_0':s3_0(0)

We have rt in Omega(n^1) and sz in O(n). Thus, we have irc_R in Omega(n).
----------------------------------------

(62)
Obligation:
Innermost TRS:
Rules:
division(x, y) -> div(x, y, 0')
div(x, y, z) -> if(lt(x, y), x, y, inc(z))
if(true, x, y, z) -> z
if(false, x, s(y), z) -> div(minus(x, s(y)), s(y), z)
minus(x, 0') -> x
minus(s(x), s(y)) -> minus(x, y)
lt(x, 0') -> false
lt(0', s(y)) -> true
lt(s(x), s(y)) -> lt(x, y)
inc(0') -> s(0')
inc(s(x)) -> s(inc(x))

Types:
division :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s -> 0':s
0' :: 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
lt :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
true :: true:false
false :: true:false
s :: 0':s -> 0':s
minus :: 0':s -> 0':s -> 0':s
hole_0':s1_0 :: 0':s
hole_true:false2_0 :: true:false
gen_0':s3_0 :: Nat -> 0':s


Lemmas:
lt(gen_0':s3_0(n5_0), gen_0':s3_0(n5_0)) -> false, rt in Omega(1 + n5_0)
inc(gen_0':s3_0(n305_0)) -> gen_0':s3_0(+(1, n305_0)), rt in Omega(1 + n305_0)
minus(gen_0':s3_0(n547_0), gen_0':s3_0(n547_0)) -> gen_0':s3_0(0), rt in Omega(1 + n547_0)


Generator Equations:
gen_0':s3_0(0) <=> 0'
gen_0':s3_0(+(x, 1)) <=> s(gen_0':s3_0(x))


The following defined symbols remain to be analysed:
div