WORST_CASE(Omega(n^1),O(n^2))
proof of input_A5t7ZFHiRX.trs
# AProVE Commit ID: 5b976082cb74a395683ed8cc7acf94bd611ab29f fuhs 20230524 unpublished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) InliningProof [UPPER BOUND(ID), 28 ms]
    (12) CpxRNTS
    (13) SimplificationProof [BOTH BOUNDS(ID, ID), 0 ms]
    (14) CpxRNTS
    (15) CpxRntsAnalysisOrderProof [BOTH BOUNDS(ID, ID), 2 ms]
    (16) CpxRNTS
    (17) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (18) CpxRNTS
    (19) IntTrsBoundProof [UPPER BOUND(ID), 254 ms]
    (20) CpxRNTS
    (21) IntTrsBoundProof [UPPER BOUND(ID), 70 ms]
    (22) CpxRNTS
    (23) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (24) CpxRNTS
    (25) IntTrsBoundProof [UPPER BOUND(ID), 151 ms]
    (26) CpxRNTS
    (27) IntTrsBoundProof [UPPER BOUND(ID), 23 ms]
    (28) CpxRNTS
    (29) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (30) CpxRNTS
    (31) IntTrsBoundProof [UPPER BOUND(ID), 131 ms]
    (32) CpxRNTS
    (33) IntTrsBoundProof [UPPER BOUND(ID), 75 ms]
    (34) CpxRNTS
    (35) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (36) CpxRNTS
    (37) IntTrsBoundProof [UPPER BOUND(ID), 100 ms]
    (38) CpxRNTS
    (39) IntTrsBoundProof [UPPER BOUND(ID), 12 ms]
    (40) CpxRNTS
    (41) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (42) CpxRNTS
    (43) IntTrsBoundProof [UPPER BOUND(ID), 1823 ms]
    (44) CpxRNTS
    (45) IntTrsBoundProof [UPPER BOUND(ID), 888 ms]
    (46) CpxRNTS
    (47) ResultPropagationProof [UPPER BOUND(ID), 0 ms]
    (48) CpxRNTS
    (49) IntTrsBoundProof [UPPER BOUND(ID), 168 ms]
    (50) CpxRNTS
    (51) IntTrsBoundProof [UPPER BOUND(ID), 0 ms]
    (52) CpxRNTS
    (53) FinalProof [FINISHED, 0 ms]
    (54) BOUNDS(1, n^2)
(55) CpxTrsToCdtProof [BOTH BOUNDS(ID, ID), 0 ms]
(56) CdtProblem
    (57) CdtToCpxRelTrsProof [BOTH BOUNDS(ID, ID), 0 ms]
    (58) CpxRelTRS
    (59) RenamingProof [BOTH BOUNDS(ID, ID), 0 ms]
    (60) CpxRelTRS
    (61) TypeInferenceProof [BOTH BOUNDS(ID, ID), 5 ms]
    (62) typed CpxTrs
    (63) OrderProof [LOWER BOUND(ID), 0 ms]
    (64) typed CpxTrs
    (65) RewriteLemmaProof [LOWER BOUND(ID), 314 ms]
    (66) BEST
        (67) proven lower bound
            (68) LowerBoundPropagationProof [FINISHED, 0 ms]
            (69) BOUNDS(n^1, INF)
        (70) typed CpxTrs
            (71) RewriteLemmaProof [LOWER BOUND(ID), 78 ms]
            (72) typed CpxTrs
            (73) RewriteLemmaProof [LOWER BOUND(ID), 53 ms]
            (74) typed CpxTrs
            (75) RewriteLemmaProof [LOWER BOUND(ID), 86 ms]
            (76) 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:

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

S is empty.
Rewrite Strategy: 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:

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

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

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

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

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

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

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

(5) CompletionProof (UPPER BOUND(ID))
The transformation into a RNTS is sound, since:

(a) The obligation is a constructor system where every type has a constant constructor,

(b) The following defined symbols do not have to be completely defined, as they can never occur inside other defined symbols:

   p_1
   div_2
   quot_3
   if_4

(c) The following functions are completely defined:

   minus_2
   zero_1
   plus_2

Due to the following rules being added:
none

And the following fresh constants: none

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

(6)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

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

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

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

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

(8)
Obligation:
Runtime Complexity Weighted TRS where critical functions are completely defined. The underlying TRS is:

Runtime Complexity Weighted TRS with Types.
The TRS R consists of the following rules:

   minus(0, y) -> 0 [1]
   minus(x, 0) -> x [1]
   minus(s(x), s(y)) -> minus(x, y) [1]
   plus(0, y) -> y [1]
   plus(s(x), y) -> plus(x, s(y)) [1]
   zero(s(x)) -> false [1]
   zero(0) -> true [1]
   p(s(x)) -> x [1]
   div(x, y) -> quot(x, y, 0) [1]
   quot(s(x'), y, 0) -> if(false, s(x'), y, s(0)) [3]
   quot(s(x'), y, s(x'')) -> if(false, s(x'), y, plus(x'', s(s(0)))) [3]
   quot(0, y, 0) -> if(true, 0, y, s(0)) [3]
   quot(0, y, s(x1)) -> if(true, 0, y, plus(x1, s(s(0)))) [3]
   if(true, x, y, z) -> p(z) [1]
   if(false, 0, s(y), z) -> quot(0, s(y), z) [2]
   if(false, s(x2), s(y), z) -> quot(minus(x2, y), s(y), z) [2]

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

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

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

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

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

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

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


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

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

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

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

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

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


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

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

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

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


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

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

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

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

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

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

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

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

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

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

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

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

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

(19) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

(21) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

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

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

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

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

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

(25) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

(27) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

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

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

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

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

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

(31) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

(33) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

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

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

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

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

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

(37) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

(39) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

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

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

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

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

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

(43) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

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

(45) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

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

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

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

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

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

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

(49) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

(51) IntTrsBoundProof (UPPER BOUND(ID))

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

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

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

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

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

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

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

(54)
BOUNDS(1, n^2)

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

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

(56)
Obligation:
Complexity Dependency Tuples Problem

Rules:
   minus(0, z0) -> 0
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   zero(s(z0)) -> false
   zero(0) -> true
   p(s(z0)) -> z0
   div(z0, z1) -> quot(z0, z1, 0)
   quot(z0, z1, z2) -> if(zero(z0), z0, z1, plus(z2, s(0)))
   if(true, z0, z1, z2) -> p(z2)
   if(false, z0, s(z1), z2) -> quot(minus(z0, s(z1)), s(z1), z2)
Tuples:
   MINUS(0, z0) -> c
   MINUS(z0, 0) -> c1
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   PLUS(0, z0) -> c3
   PLUS(s(z0), z1) -> c4(PLUS(z0, s(z1)))
   ZERO(s(z0)) -> c5
   ZERO(0) -> c6
   P(s(z0)) -> c7
   DIV(z0, z1) -> c8(QUOT(z0, z1, 0))
   QUOT(z0, z1, z2) -> c9(IF(zero(z0), z0, z1, plus(z2, s(0))), ZERO(z0))
   QUOT(z0, z1, z2) -> c10(IF(zero(z0), z0, z1, plus(z2, s(0))), PLUS(z2, s(0)))
   IF(true, z0, z1, z2) -> c11(P(z2))
   IF(false, z0, s(z1), z2) -> c12(QUOT(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
S tuples:
   MINUS(0, z0) -> c
   MINUS(z0, 0) -> c1
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   PLUS(0, z0) -> c3
   PLUS(s(z0), z1) -> c4(PLUS(z0, s(z1)))
   ZERO(s(z0)) -> c5
   ZERO(0) -> c6
   P(s(z0)) -> c7
   DIV(z0, z1) -> c8(QUOT(z0, z1, 0))
   QUOT(z0, z1, z2) -> c9(IF(zero(z0), z0, z1, plus(z2, s(0))), ZERO(z0))
   QUOT(z0, z1, z2) -> c10(IF(zero(z0), z0, z1, plus(z2, s(0))), PLUS(z2, s(0)))
   IF(true, z0, z1, z2) -> c11(P(z2))
   IF(false, z0, s(z1), z2) -> c12(QUOT(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
K tuples:none
Defined Rule Symbols:   minus_2, plus_2, zero_1, p_1, div_2, quot_3, if_4

Defined Pair Symbols:   MINUS_2, PLUS_2, ZERO_1, P_1, DIV_2, QUOT_3, IF_4

Compound Symbols:   c, c1, c2_1, c3, c4_1, c5, c6, c7, c8_1, c9_2, c10_2, c11_1, c12_2


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

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

(58)
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:

   MINUS(0, z0) -> c
   MINUS(z0, 0) -> c1
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   PLUS(0, z0) -> c3
   PLUS(s(z0), z1) -> c4(PLUS(z0, s(z1)))
   ZERO(s(z0)) -> c5
   ZERO(0) -> c6
   P(s(z0)) -> c7
   DIV(z0, z1) -> c8(QUOT(z0, z1, 0))
   QUOT(z0, z1, z2) -> c9(IF(zero(z0), z0, z1, plus(z2, s(0))), ZERO(z0))
   QUOT(z0, z1, z2) -> c10(IF(zero(z0), z0, z1, plus(z2, s(0))), PLUS(z2, s(0)))
   IF(true, z0, z1, z2) -> c11(P(z2))
   IF(false, z0, s(z1), z2) -> c12(QUOT(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))

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

   minus(0, z0) -> 0
   minus(z0, 0) -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   plus(0, z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   zero(s(z0)) -> false
   zero(0) -> true
   p(s(z0)) -> z0
   div(z0, z1) -> quot(z0, z1, 0)
   quot(z0, z1, z2) -> if(zero(z0), z0, z1, plus(z2, s(0)))
   if(true, z0, z1, z2) -> p(z2)
   if(false, z0, s(z1), z2) -> quot(minus(z0, s(z1)), s(z1), z2)

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

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

(60)
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:

   MINUS(0', z0) -> c
   MINUS(z0, 0') -> c1
   MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
   PLUS(0', z0) -> c3
   PLUS(s(z0), z1) -> c4(PLUS(z0, s(z1)))
   ZERO(s(z0)) -> c5
   ZERO(0') -> c6
   P(s(z0)) -> c7
   DIV(z0, z1) -> c8(QUOT(z0, z1, 0'))
   QUOT(z0, z1, z2) -> c9(IF(zero(z0), z0, z1, plus(z2, s(0'))), ZERO(z0))
   QUOT(z0, z1, z2) -> c10(IF(zero(z0), z0, z1, plus(z2, s(0'))), PLUS(z2, s(0')))
   IF(true, z0, z1, z2) -> c11(P(z2))
   IF(false, z0, s(z1), z2) -> c12(QUOT(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))

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

   minus(0', z0) -> 0'
   minus(z0, 0') -> z0
   minus(s(z0), s(z1)) -> minus(z0, z1)
   plus(0', z0) -> z0
   plus(s(z0), z1) -> plus(z0, s(z1))
   zero(s(z0)) -> false
   zero(0') -> true
   p(s(z0)) -> z0
   div(z0, z1) -> quot(z0, z1, 0')
   quot(z0, z1, z2) -> if(zero(z0), z0, z1, plus(z2, s(0')))
   if(true, z0, z1, z2) -> p(z2)
   if(false, z0, s(z1), z2) -> quot(minus(z0, s(z1)), s(z1), z2)

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

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

(62)
Obligation:
Innermost TRS:
Rules:
MINUS(0', z0) -> c
MINUS(z0, 0') -> c1
MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
PLUS(0', z0) -> c3
PLUS(s(z0), z1) -> c4(PLUS(z0, s(z1)))
ZERO(s(z0)) -> c5
ZERO(0') -> c6
P(s(z0)) -> c7
DIV(z0, z1) -> c8(QUOT(z0, z1, 0'))
QUOT(z0, z1, z2) -> c9(IF(zero(z0), z0, z1, plus(z2, s(0'))), ZERO(z0))
QUOT(z0, z1, z2) -> c10(IF(zero(z0), z0, z1, plus(z2, s(0'))), PLUS(z2, s(0')))
IF(true, z0, z1, z2) -> c11(P(z2))
IF(false, z0, s(z1), z2) -> c12(QUOT(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
minus(0', z0) -> 0'
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
plus(0', z0) -> z0
plus(s(z0), z1) -> plus(z0, s(z1))
zero(s(z0)) -> false
zero(0') -> true
p(s(z0)) -> z0
div(z0, z1) -> quot(z0, z1, 0')
quot(z0, z1, z2) -> if(zero(z0), z0, z1, plus(z2, s(0')))
if(true, z0, z1, z2) -> p(z2)
if(false, z0, s(z1), z2) -> quot(minus(z0, s(z1)), s(z1), z2)

Types:
MINUS :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
c1 :: c:c1:c2
s :: 0':s -> 0':s
c2 :: c:c1:c2 -> c:c1:c2
PLUS :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
ZERO :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
P :: 0':s -> c7
c7 :: c7
DIV :: 0':s -> 0':s -> c8
c8 :: c9:c10 -> c8
QUOT :: 0':s -> 0':s -> 0':s -> c9:c10
c9 :: c11:c12 -> c5:c6 -> c9:c10
IF :: true:false -> 0':s -> 0':s -> 0':s -> c11:c12
zero :: 0':s -> true:false
plus :: 0':s -> 0':s -> 0':s
c10 :: c11:c12 -> c3:c4 -> c9:c10
true :: true:false
c11 :: c7 -> c11:c12
false :: true:false
c12 :: c9:c10 -> c:c1:c2 -> c11:c12
minus :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c43_13 :: c3:c4
hole_c5:c64_13 :: c5:c6
hole_c75_13 :: c7
hole_c86_13 :: c8
hole_c9:c107_13 :: c9:c10
hole_c11:c128_13 :: c11:c12
hole_true:false9_13 :: true:false
gen_c:c1:c210_13 :: Nat -> c:c1:c2
gen_0':s11_13 :: Nat -> 0':s
gen_c3:c412_13 :: Nat -> c3:c4

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

(63) OrderProof (LOWER BOUND(ID))
Heuristically decided to analyse the following defined symbols:
MINUS, PLUS, QUOT, plus, minus, quot

They will be analysed ascendingly in the following order:
MINUS < QUOT
PLUS < QUOT
plus < QUOT
minus < QUOT
plus < quot
minus < quot

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

(64)
Obligation:
Innermost TRS:
Rules:
MINUS(0', z0) -> c
MINUS(z0, 0') -> c1
MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
PLUS(0', z0) -> c3
PLUS(s(z0), z1) -> c4(PLUS(z0, s(z1)))
ZERO(s(z0)) -> c5
ZERO(0') -> c6
P(s(z0)) -> c7
DIV(z0, z1) -> c8(QUOT(z0, z1, 0'))
QUOT(z0, z1, z2) -> c9(IF(zero(z0), z0, z1, plus(z2, s(0'))), ZERO(z0))
QUOT(z0, z1, z2) -> c10(IF(zero(z0), z0, z1, plus(z2, s(0'))), PLUS(z2, s(0')))
IF(true, z0, z1, z2) -> c11(P(z2))
IF(false, z0, s(z1), z2) -> c12(QUOT(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
minus(0', z0) -> 0'
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
plus(0', z0) -> z0
plus(s(z0), z1) -> plus(z0, s(z1))
zero(s(z0)) -> false
zero(0') -> true
p(s(z0)) -> z0
div(z0, z1) -> quot(z0, z1, 0')
quot(z0, z1, z2) -> if(zero(z0), z0, z1, plus(z2, s(0')))
if(true, z0, z1, z2) -> p(z2)
if(false, z0, s(z1), z2) -> quot(minus(z0, s(z1)), s(z1), z2)

Types:
MINUS :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
c1 :: c:c1:c2
s :: 0':s -> 0':s
c2 :: c:c1:c2 -> c:c1:c2
PLUS :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
ZERO :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
P :: 0':s -> c7
c7 :: c7
DIV :: 0':s -> 0':s -> c8
c8 :: c9:c10 -> c8
QUOT :: 0':s -> 0':s -> 0':s -> c9:c10
c9 :: c11:c12 -> c5:c6 -> c9:c10
IF :: true:false -> 0':s -> 0':s -> 0':s -> c11:c12
zero :: 0':s -> true:false
plus :: 0':s -> 0':s -> 0':s
c10 :: c11:c12 -> c3:c4 -> c9:c10
true :: true:false
c11 :: c7 -> c11:c12
false :: true:false
c12 :: c9:c10 -> c:c1:c2 -> c11:c12
minus :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c43_13 :: c3:c4
hole_c5:c64_13 :: c5:c6
hole_c75_13 :: c7
hole_c86_13 :: c8
hole_c9:c107_13 :: c9:c10
hole_c11:c128_13 :: c11:c12
hole_true:false9_13 :: true:false
gen_c:c1:c210_13 :: Nat -> c:c1:c2
gen_0':s11_13 :: Nat -> 0':s
gen_c3:c412_13 :: Nat -> c3:c4


Generator Equations:
gen_c:c1:c210_13(0) <=> c
gen_c:c1:c210_13(+(x, 1)) <=> c2(gen_c:c1:c210_13(x))
gen_0':s11_13(0) <=> 0'
gen_0':s11_13(+(x, 1)) <=> s(gen_0':s11_13(x))
gen_c3:c412_13(0) <=> c3
gen_c3:c412_13(+(x, 1)) <=> c4(gen_c3:c412_13(x))


The following defined symbols remain to be analysed:
MINUS, PLUS, QUOT, plus, minus, quot

They will be analysed ascendingly in the following order:
MINUS < QUOT
PLUS < QUOT
plus < QUOT
minus < QUOT
plus < quot
minus < quot

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

(65) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
MINUS(gen_0':s11_13(n14_13), gen_0':s11_13(n14_13)) -> gen_c:c1:c210_13(n14_13), rt in Omega(1 + n14_13)

Induction Base:
MINUS(gen_0':s11_13(0), gen_0':s11_13(0)) ->_R^Omega(1)
c

Induction Step:
MINUS(gen_0':s11_13(+(n14_13, 1)), gen_0':s11_13(+(n14_13, 1))) ->_R^Omega(1)
c2(MINUS(gen_0':s11_13(n14_13), gen_0':s11_13(n14_13))) ->_IH
c2(gen_c:c1:c210_13(c15_13))

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

(66)
Complex Obligation (BEST)

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

(67)
Obligation:
Proved the lower bound n^1 for the following obligation:

Innermost TRS:
Rules:
MINUS(0', z0) -> c
MINUS(z0, 0') -> c1
MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
PLUS(0', z0) -> c3
PLUS(s(z0), z1) -> c4(PLUS(z0, s(z1)))
ZERO(s(z0)) -> c5
ZERO(0') -> c6
P(s(z0)) -> c7
DIV(z0, z1) -> c8(QUOT(z0, z1, 0'))
QUOT(z0, z1, z2) -> c9(IF(zero(z0), z0, z1, plus(z2, s(0'))), ZERO(z0))
QUOT(z0, z1, z2) -> c10(IF(zero(z0), z0, z1, plus(z2, s(0'))), PLUS(z2, s(0')))
IF(true, z0, z1, z2) -> c11(P(z2))
IF(false, z0, s(z1), z2) -> c12(QUOT(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
minus(0', z0) -> 0'
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
plus(0', z0) -> z0
plus(s(z0), z1) -> plus(z0, s(z1))
zero(s(z0)) -> false
zero(0') -> true
p(s(z0)) -> z0
div(z0, z1) -> quot(z0, z1, 0')
quot(z0, z1, z2) -> if(zero(z0), z0, z1, plus(z2, s(0')))
if(true, z0, z1, z2) -> p(z2)
if(false, z0, s(z1), z2) -> quot(minus(z0, s(z1)), s(z1), z2)

Types:
MINUS :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
c1 :: c:c1:c2
s :: 0':s -> 0':s
c2 :: c:c1:c2 -> c:c1:c2
PLUS :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
ZERO :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
P :: 0':s -> c7
c7 :: c7
DIV :: 0':s -> 0':s -> c8
c8 :: c9:c10 -> c8
QUOT :: 0':s -> 0':s -> 0':s -> c9:c10
c9 :: c11:c12 -> c5:c6 -> c9:c10
IF :: true:false -> 0':s -> 0':s -> 0':s -> c11:c12
zero :: 0':s -> true:false
plus :: 0':s -> 0':s -> 0':s
c10 :: c11:c12 -> c3:c4 -> c9:c10
true :: true:false
c11 :: c7 -> c11:c12
false :: true:false
c12 :: c9:c10 -> c:c1:c2 -> c11:c12
minus :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c43_13 :: c3:c4
hole_c5:c64_13 :: c5:c6
hole_c75_13 :: c7
hole_c86_13 :: c8
hole_c9:c107_13 :: c9:c10
hole_c11:c128_13 :: c11:c12
hole_true:false9_13 :: true:false
gen_c:c1:c210_13 :: Nat -> c:c1:c2
gen_0':s11_13 :: Nat -> 0':s
gen_c3:c412_13 :: Nat -> c3:c4


Generator Equations:
gen_c:c1:c210_13(0) <=> c
gen_c:c1:c210_13(+(x, 1)) <=> c2(gen_c:c1:c210_13(x))
gen_0':s11_13(0) <=> 0'
gen_0':s11_13(+(x, 1)) <=> s(gen_0':s11_13(x))
gen_c3:c412_13(0) <=> c3
gen_c3:c412_13(+(x, 1)) <=> c4(gen_c3:c412_13(x))


The following defined symbols remain to be analysed:
MINUS, PLUS, QUOT, plus, minus, quot

They will be analysed ascendingly in the following order:
MINUS < QUOT
PLUS < QUOT
plus < QUOT
minus < QUOT
plus < quot
minus < quot

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

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

(69)
BOUNDS(n^1, INF)

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

(70)
Obligation:
Innermost TRS:
Rules:
MINUS(0', z0) -> c
MINUS(z0, 0') -> c1
MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
PLUS(0', z0) -> c3
PLUS(s(z0), z1) -> c4(PLUS(z0, s(z1)))
ZERO(s(z0)) -> c5
ZERO(0') -> c6
P(s(z0)) -> c7
DIV(z0, z1) -> c8(QUOT(z0, z1, 0'))
QUOT(z0, z1, z2) -> c9(IF(zero(z0), z0, z1, plus(z2, s(0'))), ZERO(z0))
QUOT(z0, z1, z2) -> c10(IF(zero(z0), z0, z1, plus(z2, s(0'))), PLUS(z2, s(0')))
IF(true, z0, z1, z2) -> c11(P(z2))
IF(false, z0, s(z1), z2) -> c12(QUOT(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
minus(0', z0) -> 0'
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
plus(0', z0) -> z0
plus(s(z0), z1) -> plus(z0, s(z1))
zero(s(z0)) -> false
zero(0') -> true
p(s(z0)) -> z0
div(z0, z1) -> quot(z0, z1, 0')
quot(z0, z1, z2) -> if(zero(z0), z0, z1, plus(z2, s(0')))
if(true, z0, z1, z2) -> p(z2)
if(false, z0, s(z1), z2) -> quot(minus(z0, s(z1)), s(z1), z2)

Types:
MINUS :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
c1 :: c:c1:c2
s :: 0':s -> 0':s
c2 :: c:c1:c2 -> c:c1:c2
PLUS :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
ZERO :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
P :: 0':s -> c7
c7 :: c7
DIV :: 0':s -> 0':s -> c8
c8 :: c9:c10 -> c8
QUOT :: 0':s -> 0':s -> 0':s -> c9:c10
c9 :: c11:c12 -> c5:c6 -> c9:c10
IF :: true:false -> 0':s -> 0':s -> 0':s -> c11:c12
zero :: 0':s -> true:false
plus :: 0':s -> 0':s -> 0':s
c10 :: c11:c12 -> c3:c4 -> c9:c10
true :: true:false
c11 :: c7 -> c11:c12
false :: true:false
c12 :: c9:c10 -> c:c1:c2 -> c11:c12
minus :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c43_13 :: c3:c4
hole_c5:c64_13 :: c5:c6
hole_c75_13 :: c7
hole_c86_13 :: c8
hole_c9:c107_13 :: c9:c10
hole_c11:c128_13 :: c11:c12
hole_true:false9_13 :: true:false
gen_c:c1:c210_13 :: Nat -> c:c1:c2
gen_0':s11_13 :: Nat -> 0':s
gen_c3:c412_13 :: Nat -> c3:c4


Lemmas:
MINUS(gen_0':s11_13(n14_13), gen_0':s11_13(n14_13)) -> gen_c:c1:c210_13(n14_13), rt in Omega(1 + n14_13)


Generator Equations:
gen_c:c1:c210_13(0) <=> c
gen_c:c1:c210_13(+(x, 1)) <=> c2(gen_c:c1:c210_13(x))
gen_0':s11_13(0) <=> 0'
gen_0':s11_13(+(x, 1)) <=> s(gen_0':s11_13(x))
gen_c3:c412_13(0) <=> c3
gen_c3:c412_13(+(x, 1)) <=> c4(gen_c3:c412_13(x))


The following defined symbols remain to be analysed:
PLUS, QUOT, plus, minus, quot

They will be analysed ascendingly in the following order:
PLUS < QUOT
plus < QUOT
minus < QUOT
plus < quot
minus < quot

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

(71) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
PLUS(gen_0':s11_13(n491_13), gen_0':s11_13(b)) -> gen_c3:c412_13(n491_13), rt in Omega(1 + n491_13)

Induction Base:
PLUS(gen_0':s11_13(0), gen_0':s11_13(b)) ->_R^Omega(1)
c3

Induction Step:
PLUS(gen_0':s11_13(+(n491_13, 1)), gen_0':s11_13(b)) ->_R^Omega(1)
c4(PLUS(gen_0':s11_13(n491_13), s(gen_0':s11_13(b)))) ->_IH
c4(gen_c3:c412_13(c492_13))

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:
MINUS(0', z0) -> c
MINUS(z0, 0') -> c1
MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
PLUS(0', z0) -> c3
PLUS(s(z0), z1) -> c4(PLUS(z0, s(z1)))
ZERO(s(z0)) -> c5
ZERO(0') -> c6
P(s(z0)) -> c7
DIV(z0, z1) -> c8(QUOT(z0, z1, 0'))
QUOT(z0, z1, z2) -> c9(IF(zero(z0), z0, z1, plus(z2, s(0'))), ZERO(z0))
QUOT(z0, z1, z2) -> c10(IF(zero(z0), z0, z1, plus(z2, s(0'))), PLUS(z2, s(0')))
IF(true, z0, z1, z2) -> c11(P(z2))
IF(false, z0, s(z1), z2) -> c12(QUOT(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
minus(0', z0) -> 0'
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
plus(0', z0) -> z0
plus(s(z0), z1) -> plus(z0, s(z1))
zero(s(z0)) -> false
zero(0') -> true
p(s(z0)) -> z0
div(z0, z1) -> quot(z0, z1, 0')
quot(z0, z1, z2) -> if(zero(z0), z0, z1, plus(z2, s(0')))
if(true, z0, z1, z2) -> p(z2)
if(false, z0, s(z1), z2) -> quot(minus(z0, s(z1)), s(z1), z2)

Types:
MINUS :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
c1 :: c:c1:c2
s :: 0':s -> 0':s
c2 :: c:c1:c2 -> c:c1:c2
PLUS :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
ZERO :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
P :: 0':s -> c7
c7 :: c7
DIV :: 0':s -> 0':s -> c8
c8 :: c9:c10 -> c8
QUOT :: 0':s -> 0':s -> 0':s -> c9:c10
c9 :: c11:c12 -> c5:c6 -> c9:c10
IF :: true:false -> 0':s -> 0':s -> 0':s -> c11:c12
zero :: 0':s -> true:false
plus :: 0':s -> 0':s -> 0':s
c10 :: c11:c12 -> c3:c4 -> c9:c10
true :: true:false
c11 :: c7 -> c11:c12
false :: true:false
c12 :: c9:c10 -> c:c1:c2 -> c11:c12
minus :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c43_13 :: c3:c4
hole_c5:c64_13 :: c5:c6
hole_c75_13 :: c7
hole_c86_13 :: c8
hole_c9:c107_13 :: c9:c10
hole_c11:c128_13 :: c11:c12
hole_true:false9_13 :: true:false
gen_c:c1:c210_13 :: Nat -> c:c1:c2
gen_0':s11_13 :: Nat -> 0':s
gen_c3:c412_13 :: Nat -> c3:c4


Lemmas:
MINUS(gen_0':s11_13(n14_13), gen_0':s11_13(n14_13)) -> gen_c:c1:c210_13(n14_13), rt in Omega(1 + n14_13)
PLUS(gen_0':s11_13(n491_13), gen_0':s11_13(b)) -> gen_c3:c412_13(n491_13), rt in Omega(1 + n491_13)


Generator Equations:
gen_c:c1:c210_13(0) <=> c
gen_c:c1:c210_13(+(x, 1)) <=> c2(gen_c:c1:c210_13(x))
gen_0':s11_13(0) <=> 0'
gen_0':s11_13(+(x, 1)) <=> s(gen_0':s11_13(x))
gen_c3:c412_13(0) <=> c3
gen_c3:c412_13(+(x, 1)) <=> c4(gen_c3:c412_13(x))


The following defined symbols remain to be analysed:
plus, QUOT, minus, quot

They will be analysed ascendingly in the following order:
plus < QUOT
minus < QUOT
plus < quot
minus < quot

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

(73) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
plus(gen_0':s11_13(n1206_13), gen_0':s11_13(b)) -> gen_0':s11_13(+(n1206_13, b)), rt in Omega(0)

Induction Base:
plus(gen_0':s11_13(0), gen_0':s11_13(b)) ->_R^Omega(0)
gen_0':s11_13(b)

Induction Step:
plus(gen_0':s11_13(+(n1206_13, 1)), gen_0':s11_13(b)) ->_R^Omega(0)
plus(gen_0':s11_13(n1206_13), s(gen_0':s11_13(b))) ->_IH
gen_0':s11_13(+(+(b, 1), c1207_13))

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

(74)
Obligation:
Innermost TRS:
Rules:
MINUS(0', z0) -> c
MINUS(z0, 0') -> c1
MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
PLUS(0', z0) -> c3
PLUS(s(z0), z1) -> c4(PLUS(z0, s(z1)))
ZERO(s(z0)) -> c5
ZERO(0') -> c6
P(s(z0)) -> c7
DIV(z0, z1) -> c8(QUOT(z0, z1, 0'))
QUOT(z0, z1, z2) -> c9(IF(zero(z0), z0, z1, plus(z2, s(0'))), ZERO(z0))
QUOT(z0, z1, z2) -> c10(IF(zero(z0), z0, z1, plus(z2, s(0'))), PLUS(z2, s(0')))
IF(true, z0, z1, z2) -> c11(P(z2))
IF(false, z0, s(z1), z2) -> c12(QUOT(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
minus(0', z0) -> 0'
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
plus(0', z0) -> z0
plus(s(z0), z1) -> plus(z0, s(z1))
zero(s(z0)) -> false
zero(0') -> true
p(s(z0)) -> z0
div(z0, z1) -> quot(z0, z1, 0')
quot(z0, z1, z2) -> if(zero(z0), z0, z1, plus(z2, s(0')))
if(true, z0, z1, z2) -> p(z2)
if(false, z0, s(z1), z2) -> quot(minus(z0, s(z1)), s(z1), z2)

Types:
MINUS :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
c1 :: c:c1:c2
s :: 0':s -> 0':s
c2 :: c:c1:c2 -> c:c1:c2
PLUS :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
ZERO :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
P :: 0':s -> c7
c7 :: c7
DIV :: 0':s -> 0':s -> c8
c8 :: c9:c10 -> c8
QUOT :: 0':s -> 0':s -> 0':s -> c9:c10
c9 :: c11:c12 -> c5:c6 -> c9:c10
IF :: true:false -> 0':s -> 0':s -> 0':s -> c11:c12
zero :: 0':s -> true:false
plus :: 0':s -> 0':s -> 0':s
c10 :: c11:c12 -> c3:c4 -> c9:c10
true :: true:false
c11 :: c7 -> c11:c12
false :: true:false
c12 :: c9:c10 -> c:c1:c2 -> c11:c12
minus :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c43_13 :: c3:c4
hole_c5:c64_13 :: c5:c6
hole_c75_13 :: c7
hole_c86_13 :: c8
hole_c9:c107_13 :: c9:c10
hole_c11:c128_13 :: c11:c12
hole_true:false9_13 :: true:false
gen_c:c1:c210_13 :: Nat -> c:c1:c2
gen_0':s11_13 :: Nat -> 0':s
gen_c3:c412_13 :: Nat -> c3:c4


Lemmas:
MINUS(gen_0':s11_13(n14_13), gen_0':s11_13(n14_13)) -> gen_c:c1:c210_13(n14_13), rt in Omega(1 + n14_13)
PLUS(gen_0':s11_13(n491_13), gen_0':s11_13(b)) -> gen_c3:c412_13(n491_13), rt in Omega(1 + n491_13)
plus(gen_0':s11_13(n1206_13), gen_0':s11_13(b)) -> gen_0':s11_13(+(n1206_13, b)), rt in Omega(0)


Generator Equations:
gen_c:c1:c210_13(0) <=> c
gen_c:c1:c210_13(+(x, 1)) <=> c2(gen_c:c1:c210_13(x))
gen_0':s11_13(0) <=> 0'
gen_0':s11_13(+(x, 1)) <=> s(gen_0':s11_13(x))
gen_c3:c412_13(0) <=> c3
gen_c3:c412_13(+(x, 1)) <=> c4(gen_c3:c412_13(x))


The following defined symbols remain to be analysed:
minus, QUOT, quot

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

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

(75) RewriteLemmaProof (LOWER BOUND(ID))
Proved the following rewrite lemma:
minus(gen_0':s11_13(n2428_13), gen_0':s11_13(n2428_13)) -> gen_0':s11_13(0), rt in Omega(0)

Induction Base:
minus(gen_0':s11_13(0), gen_0':s11_13(0)) ->_R^Omega(0)
0'

Induction Step:
minus(gen_0':s11_13(+(n2428_13, 1)), gen_0':s11_13(+(n2428_13, 1))) ->_R^Omega(0)
minus(gen_0':s11_13(n2428_13), gen_0':s11_13(n2428_13)) ->_IH
gen_0':s11_13(0)

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

(76)
Obligation:
Innermost TRS:
Rules:
MINUS(0', z0) -> c
MINUS(z0, 0') -> c1
MINUS(s(z0), s(z1)) -> c2(MINUS(z0, z1))
PLUS(0', z0) -> c3
PLUS(s(z0), z1) -> c4(PLUS(z0, s(z1)))
ZERO(s(z0)) -> c5
ZERO(0') -> c6
P(s(z0)) -> c7
DIV(z0, z1) -> c8(QUOT(z0, z1, 0'))
QUOT(z0, z1, z2) -> c9(IF(zero(z0), z0, z1, plus(z2, s(0'))), ZERO(z0))
QUOT(z0, z1, z2) -> c10(IF(zero(z0), z0, z1, plus(z2, s(0'))), PLUS(z2, s(0')))
IF(true, z0, z1, z2) -> c11(P(z2))
IF(false, z0, s(z1), z2) -> c12(QUOT(minus(z0, s(z1)), s(z1), z2), MINUS(z0, s(z1)))
minus(0', z0) -> 0'
minus(z0, 0') -> z0
minus(s(z0), s(z1)) -> minus(z0, z1)
plus(0', z0) -> z0
plus(s(z0), z1) -> plus(z0, s(z1))
zero(s(z0)) -> false
zero(0') -> true
p(s(z0)) -> z0
div(z0, z1) -> quot(z0, z1, 0')
quot(z0, z1, z2) -> if(zero(z0), z0, z1, plus(z2, s(0')))
if(true, z0, z1, z2) -> p(z2)
if(false, z0, s(z1), z2) -> quot(minus(z0, s(z1)), s(z1), z2)

Types:
MINUS :: 0':s -> 0':s -> c:c1:c2
0' :: 0':s
c :: c:c1:c2
c1 :: c:c1:c2
s :: 0':s -> 0':s
c2 :: c:c1:c2 -> c:c1:c2
PLUS :: 0':s -> 0':s -> c3:c4
c3 :: c3:c4
c4 :: c3:c4 -> c3:c4
ZERO :: 0':s -> c5:c6
c5 :: c5:c6
c6 :: c5:c6
P :: 0':s -> c7
c7 :: c7
DIV :: 0':s -> 0':s -> c8
c8 :: c9:c10 -> c8
QUOT :: 0':s -> 0':s -> 0':s -> c9:c10
c9 :: c11:c12 -> c5:c6 -> c9:c10
IF :: true:false -> 0':s -> 0':s -> 0':s -> c11:c12
zero :: 0':s -> true:false
plus :: 0':s -> 0':s -> 0':s
c10 :: c11:c12 -> c3:c4 -> c9:c10
true :: true:false
c11 :: c7 -> c11:c12
false :: true:false
c12 :: c9:c10 -> c:c1:c2 -> c11:c12
minus :: 0':s -> 0':s -> 0':s
p :: 0':s -> 0':s
div :: 0':s -> 0':s -> 0':s
quot :: 0':s -> 0':s -> 0':s -> 0':s
if :: true:false -> 0':s -> 0':s -> 0':s -> 0':s
hole_c:c1:c21_13 :: c:c1:c2
hole_0':s2_13 :: 0':s
hole_c3:c43_13 :: c3:c4
hole_c5:c64_13 :: c5:c6
hole_c75_13 :: c7
hole_c86_13 :: c8
hole_c9:c107_13 :: c9:c10
hole_c11:c128_13 :: c11:c12
hole_true:false9_13 :: true:false
gen_c:c1:c210_13 :: Nat -> c:c1:c2
gen_0':s11_13 :: Nat -> 0':s
gen_c3:c412_13 :: Nat -> c3:c4


Lemmas:
MINUS(gen_0':s11_13(n14_13), gen_0':s11_13(n14_13)) -> gen_c:c1:c210_13(n14_13), rt in Omega(1 + n14_13)
PLUS(gen_0':s11_13(n491_13), gen_0':s11_13(b)) -> gen_c3:c412_13(n491_13), rt in Omega(1 + n491_13)
plus(gen_0':s11_13(n1206_13), gen_0':s11_13(b)) -> gen_0':s11_13(+(n1206_13, b)), rt in Omega(0)
minus(gen_0':s11_13(n2428_13), gen_0':s11_13(n2428_13)) -> gen_0':s11_13(0), rt in Omega(0)


Generator Equations:
gen_c:c1:c210_13(0) <=> c
gen_c:c1:c210_13(+(x, 1)) <=> c2(gen_c:c1:c210_13(x))
gen_0':s11_13(0) <=> 0'
gen_0':s11_13(+(x, 1)) <=> s(gen_0':s11_13(x))
gen_c3:c412_13(0) <=> c3
gen_c3:c412_13(+(x, 1)) <=> c4(gen_c3:c412_13(x))


The following defined symbols remain to be analysed:
QUOT, quot