WORST_CASE(Omega(n^1),O(n^2))
proof of input_S3jpgRD1rr.trs
# AProVE Commit ID: aff8ecad908e01718a4c36e68d2e55d5e0f16e15 fuhs 20220216 unpublished


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

(0) CpxTRS
(1) RelTrsToWeightedTrsProof [UPPER BOUND(ID), 0 ms]
(2) CpxWeightedTrs
    (3) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (4) CpxTypedWeightedTrs
    (5) CompletionProof [UPPER BOUND(ID), 0 ms]
    (6) CpxTypedWeightedCompleteTrs
    (7) NarrowingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (8) CpxTypedWeightedCompleteTrs
    (9) CpxTypedWeightedTrsToRntsProof [UPPER BOUND(ID), 0 ms]
    (10) CpxRNTS
    (11) SimplificationProof [BOTH BOUNDS(ID, ID), 6 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), 251 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 71 ms]
    (20) CpxRNTS
    (21) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (22) CpxRNTS
    (23) IntTrsBoundProof [UPPER BOUND(ID), 284 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 106 ms]
    (26) CpxRNTS
    (27) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (28) CpxRNTS
    (29) IntTrsBoundProof [UPPER BOUND(ID), 241 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 75 ms]
    (32) CpxRNTS
    (33) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (34) CpxRNTS
    (35) IntTrsBoundProof [UPPER BOUND(ID), 1716 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 1420 ms]
    (38) CpxRNTS
    (39) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (40) CpxRNTS
    (41) IntTrsBoundProof [UPPER BOUND(ID), 190 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (44) CpxRNTS
    (45) FinalProof [FINISHED, 0 ms]
    (46) BOUNDS(1, n^2)
(47) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(48) CdtProblem
    (49) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (50) CpxRelTRS
    (51) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (52) CpxRelTRS
    (53) TypeInferenceProof [BOTH BOUNDS(ID, ID), 0 ms]
    (54) typed CpxTrs
    (55) OrderProof [LOWER BOUND(ID), 0 ms]
    (56) typed CpxTrs
    (57) RewriteLemmaProof [LOWER BOUND(ID), 338 ms]
    (58) typed CpxTrs
    (59) RewriteLemmaProof [LOWER BOUND(ID), 95 ms]
    (60) typed CpxTrs
    (61) RewriteLemmaProof [LOWER BOUND(ID), 146 ms]
    (62) BEST
        (63) proven lower bound
            (64) LowerBoundPropagationProof [FINISHED, 0 ms]
            (65) BOUNDS(n^1, INF)
        (66) typed CpxTrs
            (67) RewriteLemmaProof [LOWER BOUND(ID), 25 ms]
            (68) typed CpxTrs
            (69) RewriteLemmaProof [LOWER BOUND(ID), 0 ms]
            (70) typed CpxTrs
            (71) RewriteLemmaProof [LOWER BOUND(ID), 7 ms]
            (72) typed CpxTrs


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

(0)
Obligation:
The Runtime Complexity (parallel-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: PARALLEL_INNERMOST
----------------------------------------

(1) RelTrsToWeightedTrsProof (UPPER BOUND(ID))
Transformed relative TRS to weighted TRS
----------------------------------------

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


The TRS R consists of the following rules:

   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) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID))
Converted Cpx (relative) TRS with rewrite strategy PARALLEL_INNERMOST to CDT
----------------------------------------

(48)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   division(z0, z1) -> div(z0, z1, 0)
   div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2))
   if(true, z0, z1, z2) -> z2
   if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2)
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   inc(0) -> s(0)
   inc(s(z0)) -> s(inc(z0))
Tuples:
   DIVISION(z0, z1) -> c(DIV(z0, z1, 0))
   DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1))
   DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2))
   IF(true, z0, z1, z2) -> c3
   IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
   MINUS(z0, 0) -> c5
   MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1))
   LT(z0, 0) -> c7
   LT(0, s(z0)) -> c8
   LT(s(z0), s(z1)) -> c9(LT(z0, z1))
   INC(0) -> c10
   INC(s(z0)) -> c11(INC(z0))
S tuples:
   DIVISION(z0, z1) -> c(DIV(z0, z1, 0))
   DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1))
   DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2))
   IF(true, z0, z1, z2) -> c3
   IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
   MINUS(z0, 0) -> c5
   MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1))
   LT(z0, 0) -> c7
   LT(0, s(z0)) -> c8
   LT(s(z0), s(z1)) -> c9(LT(z0, z1))
   INC(0) -> c10
   INC(s(z0)) -> c11(INC(z0))
K tuples:none
Defined Rule Symbols:   division_2, div_3, if_4, minus_2, lt_2, inc_1

Defined Pair Symbols:   DIVISION_2, DIV_3, IF_4, MINUS_2, LT_2, INC_1

Compound Symbols:   c_1, c1_2, c2_2, c3, c4_2, c5, c6_1, c7, c8, c9_1, c10, c11_1


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

(49) CdtToCpxRelTrsProof (BOTH BOUNDS(ID, ID))
Converted S to standard rules, and D \ S as well as R to relative rules.
----------------------------------------

(50)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   DIVISION(z0, z1) -> c(DIV(z0, z1, 0))
   DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1))
   DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2))
   IF(true, z0, z1, z2) -> c3
   IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
   MINUS(z0, 0) -> c5
   MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1))
   LT(z0, 0) -> c7
   LT(0, s(z0)) -> c8
   LT(s(z0), s(z1)) -> c9(LT(z0, z1))
   INC(0) -> c10
   INC(s(z0)) -> c11(INC(z0))

The (relative) TRS S consists of the following rules:

   division(z0, z1) -> div(z0, z1, 0)
   div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2))
   if(true, z0, z1, z2) -> z2
   if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2)
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   lt(z0, 0) -> false
   lt(0, s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   inc(0) -> s(0)
   inc(s(z0)) -> s(inc(z0))

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

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

(52)
Obligation:
The Runtime Complexity (innermost) of the given CpxRelTRS could be proven to be BOUNDS(n^1, INF).


The TRS R consists of the following rules:

   DIVISION(z0, z1) -> c(DIV(z0, z1, 0'))
   DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1))
   DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2))
   IF(true, z0, z1, z2) -> c3
   IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
   MINUS(z0, 0') -> c5
   MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1))
   LT(z0, 0') -> c7
   LT(0', s(z0)) -> c8
   LT(s(z0), s(z1)) -> c9(LT(z0, z1))
   INC(0') -> c10
   INC(s(z0)) -> c11(INC(z0))

The (relative) TRS S consists of the following rules:

   division(z0, z1) -> div(z0, z1, 0')
   div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2))
   if(true, z0, z1, z2) -> z2
   if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2)
   minus(z0, 0') -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   lt(z0, 0') -> false
   lt(0', s(z0)) -> true
   lt(s(z0), s(z1)) -> lt(z0, z1)
   inc(0') -> s(0')
   inc(s(z0)) -> s(inc(z0))

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

(53) TypeInferenceProof (BOTH BOUNDS(ID, ID))
Inferred types.
----------------------------------------

(54)
Obligation:
Innermost TRS:
Rules:
DIVISION(z0, z1) -> c(DIV(z0, z1, 0'))
DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1))
DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2))
IF(true, z0, z1, z2) -> c3
IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
MINUS(z0, 0') -> c5
MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1))
LT(z0, 0') -> c7
LT(0', s(z0)) -> c8
LT(s(z0), s(z1)) -> c9(LT(z0, z1))
INC(0') -> c10
INC(s(z0)) -> c11(INC(z0))
division(z0, z1) -> div(z0, z1, 0')
div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2))
if(true, z0, z1, z2) -> z2
if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2)
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
lt(z0, 0') -> false
lt(0', s(z0)) -> true
lt(s(z0), s(z1)) -> lt(z0, z1)
inc(0') -> s(0')
inc(s(z0)) -> s(inc(z0))

Types:
DIVISION :: 0':s -> 0':s -> c
c :: c1:c2 -> c
DIV :: 0':s -> 0':s -> 0':s -> c1:c2
0' :: 0':s
c1 :: c3:c4 -> c7:c8:c9 -> c1:c2
IF :: true:false -> 0':s -> 0':s -> 0':s -> c3:c4
lt :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
LT :: 0':s -> 0':s -> c7:c8:c9
c2 :: c3:c4 -> c10:c11 -> c1:c2
INC :: 0':s -> c10:c11
true :: true:false
c3 :: c3:c4
false :: true:false
s :: 0':s -> 0':s
c4 :: c1:c2 -> c5:c6 -> c3:c4
minus :: 0':s -> 0':s -> 0':s
MINUS :: 0':s -> 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c7:c8:c9 -> c7:c8:c9
c10 :: c10:c11
c11 :: c10:c11 -> c10:c11
division :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
hole_c1_12 :: c
hole_0':s2_12 :: 0':s
hole_c1:c23_12 :: c1:c2
hole_c3:c44_12 :: c3:c4
hole_c7:c8:c95_12 :: c7:c8:c9
hole_true:false6_12 :: true:false
hole_c10:c117_12 :: c10:c11
hole_c5:c68_12 :: c5:c6
gen_0':s9_12 :: Nat -> 0':s
gen_c7:c8:c910_12 :: Nat -> c7:c8:c9
gen_c10:c1111_12 :: Nat -> c10:c11
gen_c5:c612_12 :: Nat -> c5:c6

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

(55) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
DIV, lt, inc, LT, INC, minus, MINUS, div

They will be analysed ascendingly in the following order:
lt < DIV
inc < DIV
LT < DIV
INC < DIV
minus < DIV
MINUS < DIV
lt < div
inc < div
minus < div

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

(56)
Obligation:
Innermost TRS:
Rules:
DIVISION(z0, z1) -> c(DIV(z0, z1, 0'))
DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1))
DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2))
IF(true, z0, z1, z2) -> c3
IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
MINUS(z0, 0') -> c5
MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1))
LT(z0, 0') -> c7
LT(0', s(z0)) -> c8
LT(s(z0), s(z1)) -> c9(LT(z0, z1))
INC(0') -> c10
INC(s(z0)) -> c11(INC(z0))
division(z0, z1) -> div(z0, z1, 0')
div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2))
if(true, z0, z1, z2) -> z2
if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2)
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
lt(z0, 0') -> false
lt(0', s(z0)) -> true
lt(s(z0), s(z1)) -> lt(z0, z1)
inc(0') -> s(0')
inc(s(z0)) -> s(inc(z0))

Types:
DIVISION :: 0':s -> 0':s -> c
c :: c1:c2 -> c
DIV :: 0':s -> 0':s -> 0':s -> c1:c2
0' :: 0':s
c1 :: c3:c4 -> c7:c8:c9 -> c1:c2
IF :: true:false -> 0':s -> 0':s -> 0':s -> c3:c4
lt :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
LT :: 0':s -> 0':s -> c7:c8:c9
c2 :: c3:c4 -> c10:c11 -> c1:c2
INC :: 0':s -> c10:c11
true :: true:false
c3 :: c3:c4
false :: true:false
s :: 0':s -> 0':s
c4 :: c1:c2 -> c5:c6 -> c3:c4
minus :: 0':s -> 0':s -> 0':s
MINUS :: 0':s -> 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c7:c8:c9 -> c7:c8:c9
c10 :: c10:c11
c11 :: c10:c11 -> c10:c11
division :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
hole_c1_12 :: c
hole_0':s2_12 :: 0':s
hole_c1:c23_12 :: c1:c2
hole_c3:c44_12 :: c3:c4
hole_c7:c8:c95_12 :: c7:c8:c9
hole_true:false6_12 :: true:false
hole_c10:c117_12 :: c10:c11
hole_c5:c68_12 :: c5:c6
gen_0':s9_12 :: Nat -> 0':s
gen_c7:c8:c910_12 :: Nat -> c7:c8:c9
gen_c10:c1111_12 :: Nat -> c10:c11
gen_c5:c612_12 :: Nat -> c5:c6


Generator Equations:
gen_0':s9_12(0) <=> 0'
gen_0':s9_12(+(x, 1)) <=> s(gen_0':s9_12(x))
gen_c7:c8:c910_12(0) <=> c7
gen_c7:c8:c910_12(+(x, 1)) <=> c9(gen_c7:c8:c910_12(x))
gen_c10:c1111_12(0) <=> c10
gen_c10:c1111_12(+(x, 1)) <=> c11(gen_c10:c1111_12(x))
gen_c5:c612_12(0) <=> c5
gen_c5:c612_12(+(x, 1)) <=> c6(gen_c5:c612_12(x))


The following defined symbols remain to be analysed:
lt, DIV, inc, LT, INC, minus, MINUS, div

They will be analysed ascendingly in the following order:
lt < DIV
inc < DIV
LT < DIV
INC < DIV
minus < DIV
MINUS < DIV
lt < div
inc < div
minus < div

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

(57) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
lt(gen_0':s9_12(n14_12), gen_0':s9_12(n14_12)) -> false, rt in Omega(0)

Induction Base:
lt(gen_0':s9_12(0), gen_0':s9_12(0)) ->_R^Omega(0)
false

Induction Step:
lt(gen_0':s9_12(+(n14_12, 1)), gen_0':s9_12(+(n14_12, 1))) ->_R^Omega(0)
lt(gen_0':s9_12(n14_12), gen_0':s9_12(n14_12)) ->_IH
false

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(58)
Obligation:
Innermost TRS:
Rules:
DIVISION(z0, z1) -> c(DIV(z0, z1, 0'))
DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1))
DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2))
IF(true, z0, z1, z2) -> c3
IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
MINUS(z0, 0') -> c5
MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1))
LT(z0, 0') -> c7
LT(0', s(z0)) -> c8
LT(s(z0), s(z1)) -> c9(LT(z0, z1))
INC(0') -> c10
INC(s(z0)) -> c11(INC(z0))
division(z0, z1) -> div(z0, z1, 0')
div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2))
if(true, z0, z1, z2) -> z2
if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2)
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
lt(z0, 0') -> false
lt(0', s(z0)) -> true
lt(s(z0), s(z1)) -> lt(z0, z1)
inc(0') -> s(0')
inc(s(z0)) -> s(inc(z0))

Types:
DIVISION :: 0':s -> 0':s -> c
c :: c1:c2 -> c
DIV :: 0':s -> 0':s -> 0':s -> c1:c2
0' :: 0':s
c1 :: c3:c4 -> c7:c8:c9 -> c1:c2
IF :: true:false -> 0':s -> 0':s -> 0':s -> c3:c4
lt :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
LT :: 0':s -> 0':s -> c7:c8:c9
c2 :: c3:c4 -> c10:c11 -> c1:c2
INC :: 0':s -> c10:c11
true :: true:false
c3 :: c3:c4
false :: true:false
s :: 0':s -> 0':s
c4 :: c1:c2 -> c5:c6 -> c3:c4
minus :: 0':s -> 0':s -> 0':s
MINUS :: 0':s -> 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c7:c8:c9 -> c7:c8:c9
c10 :: c10:c11
c11 :: c10:c11 -> c10:c11
division :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
hole_c1_12 :: c
hole_0':s2_12 :: 0':s
hole_c1:c23_12 :: c1:c2
hole_c3:c44_12 :: c3:c4
hole_c7:c8:c95_12 :: c7:c8:c9
hole_true:false6_12 :: true:false
hole_c10:c117_12 :: c10:c11
hole_c5:c68_12 :: c5:c6
gen_0':s9_12 :: Nat -> 0':s
gen_c7:c8:c910_12 :: Nat -> c7:c8:c9
gen_c10:c1111_12 :: Nat -> c10:c11
gen_c5:c612_12 :: Nat -> c5:c6


Lemmas:
lt(gen_0':s9_12(n14_12), gen_0':s9_12(n14_12)) -> false, rt in Omega(0)


Generator Equations:
gen_0':s9_12(0) <=> 0'
gen_0':s9_12(+(x, 1)) <=> s(gen_0':s9_12(x))
gen_c7:c8:c910_12(0) <=> c7
gen_c7:c8:c910_12(+(x, 1)) <=> c9(gen_c7:c8:c910_12(x))
gen_c10:c1111_12(0) <=> c10
gen_c10:c1111_12(+(x, 1)) <=> c11(gen_c10:c1111_12(x))
gen_c5:c612_12(0) <=> c5
gen_c5:c612_12(+(x, 1)) <=> c6(gen_c5:c612_12(x))


The following defined symbols remain to be analysed:
inc, DIV, LT, INC, minus, MINUS, div

They will be analysed ascendingly in the following order:
inc < DIV
LT < DIV
INC < DIV
minus < DIV
MINUS < DIV
inc < div
minus < div

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

(59) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
inc(gen_0':s9_12(n398_12)) -> gen_0':s9_12(+(1, n398_12)), rt in Omega(0)

Induction Base:
inc(gen_0':s9_12(0)) ->_R^Omega(0)
s(0')

Induction Step:
inc(gen_0':s9_12(+(n398_12, 1))) ->_R^Omega(0)
s(inc(gen_0':s9_12(n398_12))) ->_IH
s(gen_0':s9_12(+(1, c399_12)))

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(60)
Obligation:
Innermost TRS:
Rules:
DIVISION(z0, z1) -> c(DIV(z0, z1, 0'))
DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1))
DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2))
IF(true, z0, z1, z2) -> c3
IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
MINUS(z0, 0') -> c5
MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1))
LT(z0, 0') -> c7
LT(0', s(z0)) -> c8
LT(s(z0), s(z1)) -> c9(LT(z0, z1))
INC(0') -> c10
INC(s(z0)) -> c11(INC(z0))
division(z0, z1) -> div(z0, z1, 0')
div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2))
if(true, z0, z1, z2) -> z2
if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2)
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
lt(z0, 0') -> false
lt(0', s(z0)) -> true
lt(s(z0), s(z1)) -> lt(z0, z1)
inc(0') -> s(0')
inc(s(z0)) -> s(inc(z0))

Types:
DIVISION :: 0':s -> 0':s -> c
c :: c1:c2 -> c
DIV :: 0':s -> 0':s -> 0':s -> c1:c2
0' :: 0':s
c1 :: c3:c4 -> c7:c8:c9 -> c1:c2
IF :: true:false -> 0':s -> 0':s -> 0':s -> c3:c4
lt :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
LT :: 0':s -> 0':s -> c7:c8:c9
c2 :: c3:c4 -> c10:c11 -> c1:c2
INC :: 0':s -> c10:c11
true :: true:false
c3 :: c3:c4
false :: true:false
s :: 0':s -> 0':s
c4 :: c1:c2 -> c5:c6 -> c3:c4
minus :: 0':s -> 0':s -> 0':s
MINUS :: 0':s -> 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c7:c8:c9 -> c7:c8:c9
c10 :: c10:c11
c11 :: c10:c11 -> c10:c11
division :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
hole_c1_12 :: c
hole_0':s2_12 :: 0':s
hole_c1:c23_12 :: c1:c2
hole_c3:c44_12 :: c3:c4
hole_c7:c8:c95_12 :: c7:c8:c9
hole_true:false6_12 :: true:false
hole_c10:c117_12 :: c10:c11
hole_c5:c68_12 :: c5:c6
gen_0':s9_12 :: Nat -> 0':s
gen_c7:c8:c910_12 :: Nat -> c7:c8:c9
gen_c10:c1111_12 :: Nat -> c10:c11
gen_c5:c612_12 :: Nat -> c5:c6


Lemmas:
lt(gen_0':s9_12(n14_12), gen_0':s9_12(n14_12)) -> false, rt in Omega(0)
inc(gen_0':s9_12(n398_12)) -> gen_0':s9_12(+(1, n398_12)), rt in Omega(0)


Generator Equations:
gen_0':s9_12(0) <=> 0'
gen_0':s9_12(+(x, 1)) <=> s(gen_0':s9_12(x))
gen_c7:c8:c910_12(0) <=> c7
gen_c7:c8:c910_12(+(x, 1)) <=> c9(gen_c7:c8:c910_12(x))
gen_c10:c1111_12(0) <=> c10
gen_c10:c1111_12(+(x, 1)) <=> c11(gen_c10:c1111_12(x))
gen_c5:c612_12(0) <=> c5
gen_c5:c612_12(+(x, 1)) <=> c6(gen_c5:c612_12(x))


The following defined symbols remain to be analysed:
LT, DIV, INC, minus, MINUS, div

They will be analysed ascendingly in the following order:
LT < DIV
INC < DIV
minus < DIV
MINUS < DIV
minus < div

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

(61) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
LT(gen_0':s9_12(n694_12), gen_0':s9_12(n694_12)) -> gen_c7:c8:c910_12(n694_12), rt in Omega(1 + n694_12)

Induction Base:
LT(gen_0':s9_12(0), gen_0':s9_12(0)) ->_R^Omega(1)
c7

Induction Step:
LT(gen_0':s9_12(+(n694_12, 1)), gen_0':s9_12(+(n694_12, 1))) ->_R^Omega(1)
c9(LT(gen_0':s9_12(n694_12), gen_0':s9_12(n694_12))) ->_IH
c9(gen_c7:c8:c910_12(c695_12))

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

(62)
Complex Obligation (BEST)

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

(63)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
DIVISION(z0, z1) -> c(DIV(z0, z1, 0'))
DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1))
DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2))
IF(true, z0, z1, z2) -> c3
IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
MINUS(z0, 0') -> c5
MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1))
LT(z0, 0') -> c7
LT(0', s(z0)) -> c8
LT(s(z0), s(z1)) -> c9(LT(z0, z1))
INC(0') -> c10
INC(s(z0)) -> c11(INC(z0))
division(z0, z1) -> div(z0, z1, 0')
div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2))
if(true, z0, z1, z2) -> z2
if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2)
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
lt(z0, 0') -> false
lt(0', s(z0)) -> true
lt(s(z0), s(z1)) -> lt(z0, z1)
inc(0') -> s(0')
inc(s(z0)) -> s(inc(z0))

Types:
DIVISION :: 0':s -> 0':s -> c
c :: c1:c2 -> c
DIV :: 0':s -> 0':s -> 0':s -> c1:c2
0' :: 0':s
c1 :: c3:c4 -> c7:c8:c9 -> c1:c2
IF :: true:false -> 0':s -> 0':s -> 0':s -> c3:c4
lt :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
LT :: 0':s -> 0':s -> c7:c8:c9
c2 :: c3:c4 -> c10:c11 -> c1:c2
INC :: 0':s -> c10:c11
true :: true:false
c3 :: c3:c4
false :: true:false
s :: 0':s -> 0':s
c4 :: c1:c2 -> c5:c6 -> c3:c4
minus :: 0':s -> 0':s -> 0':s
MINUS :: 0':s -> 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c7:c8:c9 -> c7:c8:c9
c10 :: c10:c11
c11 :: c10:c11 -> c10:c11
division :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
hole_c1_12 :: c
hole_0':s2_12 :: 0':s
hole_c1:c23_12 :: c1:c2
hole_c3:c44_12 :: c3:c4
hole_c7:c8:c95_12 :: c7:c8:c9
hole_true:false6_12 :: true:false
hole_c10:c117_12 :: c10:c11
hole_c5:c68_12 :: c5:c6
gen_0':s9_12 :: Nat -> 0':s
gen_c7:c8:c910_12 :: Nat -> c7:c8:c9
gen_c10:c1111_12 :: Nat -> c10:c11
gen_c5:c612_12 :: Nat -> c5:c6


Lemmas:
lt(gen_0':s9_12(n14_12), gen_0':s9_12(n14_12)) -> false, rt in Omega(0)
inc(gen_0':s9_12(n398_12)) -> gen_0':s9_12(+(1, n398_12)), rt in Omega(0)


Generator Equations:
gen_0':s9_12(0) <=> 0'
gen_0':s9_12(+(x, 1)) <=> s(gen_0':s9_12(x))
gen_c7:c8:c910_12(0) <=> c7
gen_c7:c8:c910_12(+(x, 1)) <=> c9(gen_c7:c8:c910_12(x))
gen_c10:c1111_12(0) <=> c10
gen_c10:c1111_12(+(x, 1)) <=> c11(gen_c10:c1111_12(x))
gen_c5:c612_12(0) <=> c5
gen_c5:c612_12(+(x, 1)) <=> c6(gen_c5:c612_12(x))


The following defined symbols remain to be analysed:
LT, DIV, INC, minus, MINUS, div

They will be analysed ascendingly in the following order:
LT < DIV
INC < DIV
minus < DIV
MINUS < DIV
minus < div

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

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

(65)
BOUNDS(n^1, INF)

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

(66)
Obligation:
Innermost TRS:
Rules:
DIVISION(z0, z1) -> c(DIV(z0, z1, 0'))
DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1))
DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2))
IF(true, z0, z1, z2) -> c3
IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
MINUS(z0, 0') -> c5
MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1))
LT(z0, 0') -> c7
LT(0', s(z0)) -> c8
LT(s(z0), s(z1)) -> c9(LT(z0, z1))
INC(0') -> c10
INC(s(z0)) -> c11(INC(z0))
division(z0, z1) -> div(z0, z1, 0')
div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2))
if(true, z0, z1, z2) -> z2
if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2)
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
lt(z0, 0') -> false
lt(0', s(z0)) -> true
lt(s(z0), s(z1)) -> lt(z0, z1)
inc(0') -> s(0')
inc(s(z0)) -> s(inc(z0))

Types:
DIVISION :: 0':s -> 0':s -> c
c :: c1:c2 -> c
DIV :: 0':s -> 0':s -> 0':s -> c1:c2
0' :: 0':s
c1 :: c3:c4 -> c7:c8:c9 -> c1:c2
IF :: true:false -> 0':s -> 0':s -> 0':s -> c3:c4
lt :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
LT :: 0':s -> 0':s -> c7:c8:c9
c2 :: c3:c4 -> c10:c11 -> c1:c2
INC :: 0':s -> c10:c11
true :: true:false
c3 :: c3:c4
false :: true:false
s :: 0':s -> 0':s
c4 :: c1:c2 -> c5:c6 -> c3:c4
minus :: 0':s -> 0':s -> 0':s
MINUS :: 0':s -> 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c7:c8:c9 -> c7:c8:c9
c10 :: c10:c11
c11 :: c10:c11 -> c10:c11
division :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
hole_c1_12 :: c
hole_0':s2_12 :: 0':s
hole_c1:c23_12 :: c1:c2
hole_c3:c44_12 :: c3:c4
hole_c7:c8:c95_12 :: c7:c8:c9
hole_true:false6_12 :: true:false
hole_c10:c117_12 :: c10:c11
hole_c5:c68_12 :: c5:c6
gen_0':s9_12 :: Nat -> 0':s
gen_c7:c8:c910_12 :: Nat -> c7:c8:c9
gen_c10:c1111_12 :: Nat -> c10:c11
gen_c5:c612_12 :: Nat -> c5:c6


Lemmas:
lt(gen_0':s9_12(n14_12), gen_0':s9_12(n14_12)) -> false, rt in Omega(0)
inc(gen_0':s9_12(n398_12)) -> gen_0':s9_12(+(1, n398_12)), rt in Omega(0)
LT(gen_0':s9_12(n694_12), gen_0':s9_12(n694_12)) -> gen_c7:c8:c910_12(n694_12), rt in Omega(1 + n694_12)


Generator Equations:
gen_0':s9_12(0) <=> 0'
gen_0':s9_12(+(x, 1)) <=> s(gen_0':s9_12(x))
gen_c7:c8:c910_12(0) <=> c7
gen_c7:c8:c910_12(+(x, 1)) <=> c9(gen_c7:c8:c910_12(x))
gen_c10:c1111_12(0) <=> c10
gen_c10:c1111_12(+(x, 1)) <=> c11(gen_c10:c1111_12(x))
gen_c5:c612_12(0) <=> c5
gen_c5:c612_12(+(x, 1)) <=> c6(gen_c5:c612_12(x))


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

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

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

(67) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
INC(gen_0':s9_12(n1356_12)) -> gen_c10:c1111_12(n1356_12), rt in Omega(1 + n1356_12)

Induction Base:
INC(gen_0':s9_12(0)) ->_R^Omega(1)
c10

Induction Step:
INC(gen_0':s9_12(+(n1356_12, 1))) ->_R^Omega(1)
c11(INC(gen_0':s9_12(n1356_12))) ->_IH
c11(gen_c10:c1111_12(c1357_12))

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

(68)
Obligation:
Innermost TRS:
Rules:
DIVISION(z0, z1) -> c(DIV(z0, z1, 0'))
DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1))
DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2))
IF(true, z0, z1, z2) -> c3
IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
MINUS(z0, 0') -> c5
MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1))
LT(z0, 0') -> c7
LT(0', s(z0)) -> c8
LT(s(z0), s(z1)) -> c9(LT(z0, z1))
INC(0') -> c10
INC(s(z0)) -> c11(INC(z0))
division(z0, z1) -> div(z0, z1, 0')
div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2))
if(true, z0, z1, z2) -> z2
if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2)
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
lt(z0, 0') -> false
lt(0', s(z0)) -> true
lt(s(z0), s(z1)) -> lt(z0, z1)
inc(0') -> s(0')
inc(s(z0)) -> s(inc(z0))

Types:
DIVISION :: 0':s -> 0':s -> c
c :: c1:c2 -> c
DIV :: 0':s -> 0':s -> 0':s -> c1:c2
0' :: 0':s
c1 :: c3:c4 -> c7:c8:c9 -> c1:c2
IF :: true:false -> 0':s -> 0':s -> 0':s -> c3:c4
lt :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
LT :: 0':s -> 0':s -> c7:c8:c9
c2 :: c3:c4 -> c10:c11 -> c1:c2
INC :: 0':s -> c10:c11
true :: true:false
c3 :: c3:c4
false :: true:false
s :: 0':s -> 0':s
c4 :: c1:c2 -> c5:c6 -> c3:c4
minus :: 0':s -> 0':s -> 0':s
MINUS :: 0':s -> 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c7:c8:c9 -> c7:c8:c9
c10 :: c10:c11
c11 :: c10:c11 -> c10:c11
division :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
hole_c1_12 :: c
hole_0':s2_12 :: 0':s
hole_c1:c23_12 :: c1:c2
hole_c3:c44_12 :: c3:c4
hole_c7:c8:c95_12 :: c7:c8:c9
hole_true:false6_12 :: true:false
hole_c10:c117_12 :: c10:c11
hole_c5:c68_12 :: c5:c6
gen_0':s9_12 :: Nat -> 0':s
gen_c7:c8:c910_12 :: Nat -> c7:c8:c9
gen_c10:c1111_12 :: Nat -> c10:c11
gen_c5:c612_12 :: Nat -> c5:c6


Lemmas:
lt(gen_0':s9_12(n14_12), gen_0':s9_12(n14_12)) -> false, rt in Omega(0)
inc(gen_0':s9_12(n398_12)) -> gen_0':s9_12(+(1, n398_12)), rt in Omega(0)
LT(gen_0':s9_12(n694_12), gen_0':s9_12(n694_12)) -> gen_c7:c8:c910_12(n694_12), rt in Omega(1 + n694_12)
INC(gen_0':s9_12(n1356_12)) -> gen_c10:c1111_12(n1356_12), rt in Omega(1 + n1356_12)


Generator Equations:
gen_0':s9_12(0) <=> 0'
gen_0':s9_12(+(x, 1)) <=> s(gen_0':s9_12(x))
gen_c7:c8:c910_12(0) <=> c7
gen_c7:c8:c910_12(+(x, 1)) <=> c9(gen_c7:c8:c910_12(x))
gen_c10:c1111_12(0) <=> c10
gen_c10:c1111_12(+(x, 1)) <=> c11(gen_c10:c1111_12(x))
gen_c5:c612_12(0) <=> c5
gen_c5:c612_12(+(x, 1)) <=> c6(gen_c5:c612_12(x))


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

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

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

(69) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s9_12(n1724_12), gen_0':s9_12(n1724_12)) -> gen_0':s9_12(0), rt in Omega(0)

Induction Base:
minus(gen_0':s9_12(0), gen_0':s9_12(0)) ->_R^Omega(0)
gen_0':s9_12(0)

Induction Step:
minus(gen_0':s9_12(+(n1724_12, 1)), gen_0':s9_12(+(n1724_12, 1))) ->_R^Omega(0)
minus(gen_0':s9_12(n1724_12), gen_0':s9_12(n1724_12)) ->_IH
gen_0':s9_12(0)

We have rt in Omega(1) and sz in O(n). Thus, we have irc_R in Omega(n^0).
----------------------------------------

(70)
Obligation:
Innermost TRS:
Rules:
DIVISION(z0, z1) -> c(DIV(z0, z1, 0'))
DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1))
DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2))
IF(true, z0, z1, z2) -> c3
IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
MINUS(z0, 0') -> c5
MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1))
LT(z0, 0') -> c7
LT(0', s(z0)) -> c8
LT(s(z0), s(z1)) -> c9(LT(z0, z1))
INC(0') -> c10
INC(s(z0)) -> c11(INC(z0))
division(z0, z1) -> div(z0, z1, 0')
div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2))
if(true, z0, z1, z2) -> z2
if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2)
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
lt(z0, 0') -> false
lt(0', s(z0)) -> true
lt(s(z0), s(z1)) -> lt(z0, z1)
inc(0') -> s(0')
inc(s(z0)) -> s(inc(z0))

Types:
DIVISION :: 0':s -> 0':s -> c
c :: c1:c2 -> c
DIV :: 0':s -> 0':s -> 0':s -> c1:c2
0' :: 0':s
c1 :: c3:c4 -> c7:c8:c9 -> c1:c2
IF :: true:false -> 0':s -> 0':s -> 0':s -> c3:c4
lt :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
LT :: 0':s -> 0':s -> c7:c8:c9
c2 :: c3:c4 -> c10:c11 -> c1:c2
INC :: 0':s -> c10:c11
true :: true:false
c3 :: c3:c4
false :: true:false
s :: 0':s -> 0':s
c4 :: c1:c2 -> c5:c6 -> c3:c4
minus :: 0':s -> 0':s -> 0':s
MINUS :: 0':s -> 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c7:c8:c9 -> c7:c8:c9
c10 :: c10:c11
c11 :: c10:c11 -> c10:c11
division :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
hole_c1_12 :: c
hole_0':s2_12 :: 0':s
hole_c1:c23_12 :: c1:c2
hole_c3:c44_12 :: c3:c4
hole_c7:c8:c95_12 :: c7:c8:c9
hole_true:false6_12 :: true:false
hole_c10:c117_12 :: c10:c11
hole_c5:c68_12 :: c5:c6
gen_0':s9_12 :: Nat -> 0':s
gen_c7:c8:c910_12 :: Nat -> c7:c8:c9
gen_c10:c1111_12 :: Nat -> c10:c11
gen_c5:c612_12 :: Nat -> c5:c6


Lemmas:
lt(gen_0':s9_12(n14_12), gen_0':s9_12(n14_12)) -> false, rt in Omega(0)
inc(gen_0':s9_12(n398_12)) -> gen_0':s9_12(+(1, n398_12)), rt in Omega(0)
LT(gen_0':s9_12(n694_12), gen_0':s9_12(n694_12)) -> gen_c7:c8:c910_12(n694_12), rt in Omega(1 + n694_12)
INC(gen_0':s9_12(n1356_12)) -> gen_c10:c1111_12(n1356_12), rt in Omega(1 + n1356_12)
minus(gen_0':s9_12(n1724_12), gen_0':s9_12(n1724_12)) -> gen_0':s9_12(0), rt in Omega(0)


Generator Equations:
gen_0':s9_12(0) <=> 0'
gen_0':s9_12(+(x, 1)) <=> s(gen_0':s9_12(x))
gen_c7:c8:c910_12(0) <=> c7
gen_c7:c8:c910_12(+(x, 1)) <=> c9(gen_c7:c8:c910_12(x))
gen_c10:c1111_12(0) <=> c10
gen_c10:c1111_12(+(x, 1)) <=> c11(gen_c10:c1111_12(x))
gen_c5:c612_12(0) <=> c5
gen_c5:c612_12(+(x, 1)) <=> c6(gen_c5:c612_12(x))


The following defined symbols remain to be analysed:
MINUS, DIV, div

They will be analysed ascendingly in the following order:
MINUS < DIV

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

(71) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
MINUS(gen_0':s9_12(n2396_12), gen_0':s9_12(n2396_12)) -> gen_c5:c612_12(n2396_12), rt in Omega(1 + n2396_12)

Induction Base:
MINUS(gen_0':s9_12(0), gen_0':s9_12(0)) ->_R^Omega(1)
c5

Induction Step:
MINUS(gen_0':s9_12(+(n2396_12, 1)), gen_0':s9_12(+(n2396_12, 1))) ->_R^Omega(1)
c6(MINUS(gen_0':s9_12(n2396_12), gen_0':s9_12(n2396_12))) ->_IH
c6(gen_c5:c612_12(c2397_12))

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

(72)
Obligation:
Innermost TRS:
Rules:
DIVISION(z0, z1) -> c(DIV(z0, z1, 0'))
DIV(z0, z1, z2) -> c1(IF(lt(z0, z1), z0, z1, inc(z2)), LT(z0, z1))
DIV(z0, z1, z2) -> c2(IF(lt(z0, z1), z0, z1, inc(z2)), INC(z2))
IF(true, z0, z1, z2) -> c3
IF(false, z0, s(z1), z2) -> c4(DIV(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
MINUS(z0, 0') -> c5
MINUS(s(z0), s(z1)) -> c6(MINUS(z0, z1))
LT(z0, 0') -> c7
LT(0', s(z0)) -> c8
LT(s(z0), s(z1)) -> c9(LT(z0, z1))
INC(0') -> c10
INC(s(z0)) -> c11(INC(z0))
division(z0, z1) -> div(z0, z1, 0')
div(z0, z1, z2) -> if(lt(z0, z1), z0, z1, inc(z2))
if(true, z0, z1, z2) -> z2
if(false, z0, s(z1), z2) -> div(minus(z0, s(z1)), s(z1), z2)
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
lt(z0, 0') -> false
lt(0', s(z0)) -> true
lt(s(z0), s(z1)) -> lt(z0, z1)
inc(0') -> s(0')
inc(s(z0)) -> s(inc(z0))

Types:
DIVISION :: 0':s -> 0':s -> c
c :: c1:c2 -> c
DIV :: 0':s -> 0':s -> 0':s -> c1:c2
0' :: 0':s
c1 :: c3:c4 -> c7:c8:c9 -> c1:c2
IF :: true:false -> 0':s -> 0':s -> 0':s -> c3:c4
lt :: 0':s -> 0':s -> true:false
inc :: 0':s -> 0':s
LT :: 0':s -> 0':s -> c7:c8:c9
c2 :: c3:c4 -> c10:c11 -> c1:c2
INC :: 0':s -> c10:c11
true :: true:false
c3 :: c3:c4
false :: true:false
s :: 0':s -> 0':s
c4 :: c1:c2 -> c5:c6 -> c3:c4
minus :: 0':s -> 0':s -> 0':s
MINUS :: 0':s -> 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6 -> c5:c6
c7 :: c7:c8:c9
c8 :: c7:c8:c9
c9 :: c7:c8:c9 -> c7:c8:c9
c10 :: c10:c11
c11 :: c10:c11 -> c10:c11
division :: 0':s -> 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
hole_c1_12 :: c
hole_0':s2_12 :: 0':s
hole_c1:c23_12 :: c1:c2
hole_c3:c44_12 :: c3:c4
hole_c7:c8:c95_12 :: c7:c8:c9
hole_true:false6_12 :: true:false
hole_c10:c117_12 :: c10:c11
hole_c5:c68_12 :: c5:c6
gen_0':s9_12 :: Nat -> 0':s
gen_c7:c8:c910_12 :: Nat -> c7:c8:c9
gen_c10:c1111_12 :: Nat -> c10:c11
gen_c5:c612_12 :: Nat -> c5:c6


Lemmas:
lt(gen_0':s9_12(n14_12), gen_0':s9_12(n14_12)) -> false, rt in Omega(0)
inc(gen_0':s9_12(n398_12)) -> gen_0':s9_12(+(1, n398_12)), rt in Omega(0)
LT(gen_0':s9_12(n694_12), gen_0':s9_12(n694_12)) -> gen_c7:c8:c910_12(n694_12), rt in Omega(1 + n694_12)
INC(gen_0':s9_12(n1356_12)) -> gen_c10:c1111_12(n1356_12), rt in Omega(1 + n1356_12)
minus(gen_0':s9_12(n1724_12), gen_0':s9_12(n1724_12)) -> gen_0':s9_12(0), rt in Omega(0)
MINUS(gen_0':s9_12(n2396_12), gen_0':s9_12(n2396_12)) -> gen_c5:c612_12(n2396_12), rt in Omega(1 + n2396_12)


Generator Equations:
gen_0':s9_12(0) <=> 0'
gen_0':s9_12(+(x, 1)) <=> s(gen_0':s9_12(x))
gen_c7:c8:c910_12(0) <=> c7
gen_c7:c8:c910_12(+(x, 1)) <=> c9(gen_c7:c8:c910_12(x))
gen_c10:c1111_12(0) <=> c10
gen_c10:c1111_12(+(x, 1)) <=> c11(gen_c10:c1111_12(x))
gen_c5:c612_12(0) <=> c5
gen_c5:c612_12(+(x, 1)) <=> c6(gen_c5:c612_12(x))


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