```* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
div(0(),s(Y)) -> 0()
div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
geq(X,0()) -> true()
geq(0(),s(Y)) -> false()
geq(s(X),s(Y)) -> geq(X,Y)
if(false(),X,Y) -> Y
if(true(),X,Y) -> X
minus(0(),Y) -> 0()
minus(s(X),s(Y)) -> minus(X,Y)
- Signature:
{div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {div,geq,if,minus} and constructors {0,false,s,true}
+ Applied Processor:
DependencyPairs {dpKind_ = DT}
+ Details:
We add the following dependency tuples:

Strict DPs
div#(0(),s(Y)) -> c_1()
div#(s(X),s(Y)) -> c_2(if#(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
,geq#(X,Y)
,div#(minus(X,Y),s(Y))
,minus#(X,Y))
geq#(X,0()) -> c_3()
geq#(0(),s(Y)) -> c_4()
geq#(s(X),s(Y)) -> c_5(geq#(X,Y))
if#(false(),X,Y) -> c_6()
if#(true(),X,Y) -> c_7()
minus#(0(),Y) -> c_8()
minus#(s(X),s(Y)) -> c_9(minus#(X,Y))
Weak DPs

and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
div#(0(),s(Y)) -> c_1()
div#(s(X),s(Y)) -> c_2(if#(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
,geq#(X,Y)
,div#(minus(X,Y),s(Y))
,minus#(X,Y))
geq#(X,0()) -> c_3()
geq#(0(),s(Y)) -> c_4()
geq#(s(X),s(Y)) -> c_5(geq#(X,Y))
if#(false(),X,Y) -> c_6()
if#(true(),X,Y) -> c_7()
minus#(0(),Y) -> c_8()
minus#(s(X),s(Y)) -> c_9(minus#(X,Y))
- Weak TRS:
div(0(),s(Y)) -> 0()
div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
geq(X,0()) -> true()
geq(0(),s(Y)) -> false()
geq(s(X),s(Y)) -> geq(X,Y)
if(false(),X,Y) -> Y
if(true(),X,Y) -> X
minus(0(),Y) -> 0()
minus(s(X),s(Y)) -> minus(X,Y)
- Signature:
{div/2,geq/2,if/3,minus/2,div#/2,geq#/2,if#/3,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/4,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,geq#,if#,minus#} and constructors {0,false,s,true}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,3,4,6,7,8}
by application of
Pre({1,3,4,6,7,8}) = {2,5,9}.
Here rules are labelled as follows:
1: div#(0(),s(Y)) -> c_1()
2: div#(s(X),s(Y)) -> c_2(if#(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
,geq#(X,Y)
,div#(minus(X,Y),s(Y))
,minus#(X,Y))
3: geq#(X,0()) -> c_3()
4: geq#(0(),s(Y)) -> c_4()
5: geq#(s(X),s(Y)) -> c_5(geq#(X,Y))
6: if#(false(),X,Y) -> c_6()
7: if#(true(),X,Y) -> c_7()
8: minus#(0(),Y) -> c_8()
9: minus#(s(X),s(Y)) -> c_9(minus#(X,Y))
* Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
div#(s(X),s(Y)) -> c_2(if#(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
,geq#(X,Y)
,div#(minus(X,Y),s(Y))
,minus#(X,Y))
geq#(s(X),s(Y)) -> c_5(geq#(X,Y))
minus#(s(X),s(Y)) -> c_9(minus#(X,Y))
- Weak DPs:
div#(0(),s(Y)) -> c_1()
geq#(X,0()) -> c_3()
geq#(0(),s(Y)) -> c_4()
if#(false(),X,Y) -> c_6()
if#(true(),X,Y) -> c_7()
minus#(0(),Y) -> c_8()
- Weak TRS:
div(0(),s(Y)) -> 0()
div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
geq(X,0()) -> true()
geq(0(),s(Y)) -> false()
geq(s(X),s(Y)) -> geq(X,Y)
if(false(),X,Y) -> Y
if(true(),X,Y) -> X
minus(0(),Y) -> 0()
minus(s(X),s(Y)) -> minus(X,Y)
- Signature:
{div/2,geq/2,if/3,minus/2,div#/2,geq#/2,if#/3,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/4,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,geq#,if#,minus#} and constructors {0,false,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:div#(s(X),s(Y)) -> c_2(if#(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
,geq#(X,Y)
,div#(minus(X,Y),s(Y))
,minus#(X,Y))
-->_4 minus#(s(X),s(Y)) -> c_9(minus#(X,Y)):3
-->_2 geq#(s(X),s(Y)) -> c_5(geq#(X,Y)):2
-->_4 minus#(0(),Y) -> c_8():9
-->_1 if#(true(),X,Y) -> c_7():8
-->_1 if#(false(),X,Y) -> c_6():7
-->_2 geq#(0(),s(Y)) -> c_4():6
-->_2 geq#(X,0()) -> c_3():5
-->_3 div#(0(),s(Y)) -> c_1():4
-->_3 div#(s(X),s(Y)) -> c_2(if#(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
,geq#(X,Y)
,div#(minus(X,Y),s(Y))
,minus#(X,Y)):1

2:S:geq#(s(X),s(Y)) -> c_5(geq#(X,Y))
-->_1 geq#(0(),s(Y)) -> c_4():6
-->_1 geq#(X,0()) -> c_3():5
-->_1 geq#(s(X),s(Y)) -> c_5(geq#(X,Y)):2

3:S:minus#(s(X),s(Y)) -> c_9(minus#(X,Y))
-->_1 minus#(0(),Y) -> c_8():9
-->_1 minus#(s(X),s(Y)) -> c_9(minus#(X,Y)):3

4:W:div#(0(),s(Y)) -> c_1()

5:W:geq#(X,0()) -> c_3()

6:W:geq#(0(),s(Y)) -> c_4()

7:W:if#(false(),X,Y) -> c_6()

8:W:if#(true(),X,Y) -> c_7()

9:W:minus#(0(),Y) -> c_8()

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
4: div#(0(),s(Y)) -> c_1()
7: if#(false(),X,Y) -> c_6()
8: if#(true(),X,Y) -> c_7()
5: geq#(X,0()) -> c_3()
6: geq#(0(),s(Y)) -> c_4()
9: minus#(0(),Y) -> c_8()
* Step 4: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
div#(s(X),s(Y)) -> c_2(if#(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
,geq#(X,Y)
,div#(minus(X,Y),s(Y))
,minus#(X,Y))
geq#(s(X),s(Y)) -> c_5(geq#(X,Y))
minus#(s(X),s(Y)) -> c_9(minus#(X,Y))
- Weak TRS:
div(0(),s(Y)) -> 0()
div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
geq(X,0()) -> true()
geq(0(),s(Y)) -> false()
geq(s(X),s(Y)) -> geq(X,Y)
if(false(),X,Y) -> Y
if(true(),X,Y) -> X
minus(0(),Y) -> 0()
minus(s(X),s(Y)) -> minus(X,Y)
- Signature:
{div/2,geq/2,if/3,minus/2,div#/2,geq#/2,if#/3,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/4,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,geq#,if#,minus#} and constructors {0,false,s,true}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:div#(s(X),s(Y)) -> c_2(if#(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
,geq#(X,Y)
,div#(minus(X,Y),s(Y))
,minus#(X,Y))
-->_4 minus#(s(X),s(Y)) -> c_9(minus#(X,Y)):3
-->_2 geq#(s(X),s(Y)) -> c_5(geq#(X,Y)):2
-->_3 div#(s(X),s(Y)) -> c_2(if#(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
,geq#(X,Y)
,div#(minus(X,Y),s(Y))
,minus#(X,Y)):1

2:S:geq#(s(X),s(Y)) -> c_5(geq#(X,Y))
-->_1 geq#(s(X),s(Y)) -> c_5(geq#(X,Y)):2

3:S:minus#(s(X),s(Y)) -> c_9(minus#(X,Y))
-->_1 minus#(s(X),s(Y)) -> c_9(minus#(X,Y)):3

Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
div#(s(X),s(Y)) -> c_2(geq#(X,Y),div#(minus(X,Y),s(Y)),minus#(X,Y))
* Step 5: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
div#(s(X),s(Y)) -> c_2(geq#(X,Y),div#(minus(X,Y),s(Y)),minus#(X,Y))
geq#(s(X),s(Y)) -> c_5(geq#(X,Y))
minus#(s(X),s(Y)) -> c_9(minus#(X,Y))
- Weak TRS:
div(0(),s(Y)) -> 0()
div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
geq(X,0()) -> true()
geq(0(),s(Y)) -> false()
geq(s(X),s(Y)) -> geq(X,Y)
if(false(),X,Y) -> Y
if(true(),X,Y) -> X
minus(0(),Y) -> 0()
minus(s(X),s(Y)) -> minus(X,Y)
- Signature:
{div/2,geq/2,if/3,minus/2,div#/2,geq#/2,if#/3,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/3,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,geq#,if#,minus#} and constructors {0,false,s,true}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
minus(0(),Y) -> 0()
minus(s(X),s(Y)) -> minus(X,Y)
div#(s(X),s(Y)) -> c_2(geq#(X,Y),div#(minus(X,Y),s(Y)),minus#(X,Y))
geq#(s(X),s(Y)) -> c_5(geq#(X,Y))
minus#(s(X),s(Y)) -> c_9(minus#(X,Y))
* Step 6: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
div#(s(X),s(Y)) -> c_2(geq#(X,Y),div#(minus(X,Y),s(Y)),minus#(X,Y))
geq#(s(X),s(Y)) -> c_5(geq#(X,Y))
minus#(s(X),s(Y)) -> c_9(minus#(X,Y))
- Weak TRS:
minus(0(),Y) -> 0()
minus(s(X),s(Y)) -> minus(X,Y)
- Signature:
{div/2,geq/2,if/3,minus/2,div#/2,geq#/2,if#/3,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/3,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,geq#,if#,minus#} and constructors {0,false,s,true}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
2: geq#(s(X),s(Y)) -> c_5(geq#(X,Y))

The strictly oriented rules are moved into the weak component.
** Step 6.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
div#(s(X),s(Y)) -> c_2(geq#(X,Y),div#(minus(X,Y),s(Y)),minus#(X,Y))
geq#(s(X),s(Y)) -> c_5(geq#(X,Y))
minus#(s(X),s(Y)) -> c_9(minus#(X,Y))
- Weak TRS:
minus(0(),Y) -> 0()
minus(s(X),s(Y)) -> minus(X,Y)
- Signature:
{div/2,geq/2,if/3,minus/2,div#/2,geq#/2,if#/3,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/3,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,geq#,if#,minus#} and constructors {0,false,s,true}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_2) = {1,2,3},
uargs(c_5) = {1},
uargs(c_9) = {1}

Following symbols are considered usable:
{minus,div#,geq#,if#,minus#}
TcT has computed the following interpretation:
p(0) = [0]
p(div) = [1] x1 + [0]
p(false) = [1]
p(geq) = [8] x1 + [2] x2 + [0]
p(if) = [1] x1 + [1] x2 + [0]
p(minus) = [0]
p(s) = [1] x1 + [2]
p(true) = [2]
p(div#) = [11] x1 + [4]
p(geq#) = [4] x1 + [0]
p(if#) = [0]
p(minus#) = [0]
p(c_1) = [0]
p(c_2) = [2] x1 + [4] x2 + [1] x3 + [10]
p(c_3) = [1]
p(c_4) = [1]
p(c_5) = [1] x1 + [0]
p(c_6) = [1]
p(c_7) = [4]
p(c_8) = [2]
p(c_9) = [8] x1 + [0]

Following rules are strictly oriented:
geq#(s(X),s(Y)) = [4] X + [8]
> [4] X + [0]
= c_5(geq#(X,Y))

Following rules are (at-least) weakly oriented:
div#(s(X),s(Y)) =  [11] X + [26]
>= [8] X + [26]
=  c_2(geq#(X,Y),div#(minus(X,Y),s(Y)),minus#(X,Y))

minus#(s(X),s(Y)) =  [0]
>= [0]
=  c_9(minus#(X,Y))

minus(0(),Y) =  [0]
>= [0]
=  0()

minus(s(X),s(Y)) =  [0]
>= [0]
=  minus(X,Y)

** Step 6.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Strict DPs:
div#(s(X),s(Y)) -> c_2(geq#(X,Y),div#(minus(X,Y),s(Y)),minus#(X,Y))
minus#(s(X),s(Y)) -> c_9(minus#(X,Y))
- Weak DPs:
geq#(s(X),s(Y)) -> c_5(geq#(X,Y))
- Weak TRS:
minus(0(),Y) -> 0()
minus(s(X),s(Y)) -> minus(X,Y)
- Signature:
{div/2,geq/2,if/3,minus/2,div#/2,geq#/2,if#/3,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/3,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,geq#,if#,minus#} and constructors {0,false,s,true}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

** Step 6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
div#(s(X),s(Y)) -> c_2(geq#(X,Y),div#(minus(X,Y),s(Y)),minus#(X,Y))
minus#(s(X),s(Y)) -> c_9(minus#(X,Y))
- Weak DPs:
geq#(s(X),s(Y)) -> c_5(geq#(X,Y))
- Weak TRS:
minus(0(),Y) -> 0()
minus(s(X),s(Y)) -> minus(X,Y)
- Signature:
{div/2,geq/2,if/3,minus/2,div#/2,geq#/2,if#/3,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/3,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,geq#,if#,minus#} and constructors {0,false,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:div#(s(X),s(Y)) -> c_2(geq#(X,Y),div#(minus(X,Y),s(Y)),minus#(X,Y))
-->_1 geq#(s(X),s(Y)) -> c_5(geq#(X,Y)):3
-->_3 minus#(s(X),s(Y)) -> c_9(minus#(X,Y)):2
-->_2 div#(s(X),s(Y)) -> c_2(geq#(X,Y),div#(minus(X,Y),s(Y)),minus#(X,Y)):1

2:S:minus#(s(X),s(Y)) -> c_9(minus#(X,Y))
-->_1 minus#(s(X),s(Y)) -> c_9(minus#(X,Y)):2

3:W:geq#(s(X),s(Y)) -> c_5(geq#(X,Y))
-->_1 geq#(s(X),s(Y)) -> c_5(geq#(X,Y)):3

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
3: geq#(s(X),s(Y)) -> c_5(geq#(X,Y))
** Step 6.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
div#(s(X),s(Y)) -> c_2(geq#(X,Y),div#(minus(X,Y),s(Y)),minus#(X,Y))
minus#(s(X),s(Y)) -> c_9(minus#(X,Y))
- Weak TRS:
minus(0(),Y) -> 0()
minus(s(X),s(Y)) -> minus(X,Y)
- Signature:
{div/2,geq/2,if/3,minus/2,div#/2,geq#/2,if#/3,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/3,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,geq#,if#,minus#} and constructors {0,false,s,true}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:div#(s(X),s(Y)) -> c_2(geq#(X,Y),div#(minus(X,Y),s(Y)),minus#(X,Y))
-->_3 minus#(s(X),s(Y)) -> c_9(minus#(X,Y)):2
-->_2 div#(s(X),s(Y)) -> c_2(geq#(X,Y),div#(minus(X,Y),s(Y)),minus#(X,Y)):1

2:S:minus#(s(X),s(Y)) -> c_9(minus#(X,Y))
-->_1 minus#(s(X),s(Y)) -> c_9(minus#(X,Y)):2

Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y)),minus#(X,Y))
** Step 6.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y)),minus#(X,Y))
minus#(s(X),s(Y)) -> c_9(minus#(X,Y))
- Weak TRS:
minus(0(),Y) -> 0()
minus(s(X),s(Y)) -> minus(X,Y)
- Signature:
{div/2,geq/2,if/3,minus/2,div#/2,geq#/2,if#/3,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/2,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,geq#,if#,minus#} and constructors {0,false,s,true}
+ Applied Processor:
PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
+ Details:
We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
1: div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y)),minus#(X,Y))
2: minus#(s(X),s(Y)) -> c_9(minus#(X,Y))

The strictly oriented rules are moved into the weak component.
*** Step 6.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y)),minus#(X,Y))
minus#(s(X),s(Y)) -> c_9(minus#(X,Y))
- Weak TRS:
minus(0(),Y) -> 0()
minus(s(X),s(Y)) -> minus(X,Y)
- Signature:
{div/2,geq/2,if/3,minus/2,div#/2,geq#/2,if#/3,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/2,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,geq#,if#,minus#} and constructors {0,false,s,true}
+ Applied Processor:
NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
+ Details:
We apply a matrix interpretation of kind constructor based matrix interpretation:
The following argument positions are considered usable:
uargs(c_2) = {1,2},
uargs(c_9) = {1}

Following symbols are considered usable:
{minus,div#,geq#,if#,minus#}
TcT has computed the following interpretation:
p(0) = [0]
p(div) = [1] x1 + [0]
p(false) = [2]
p(geq) = [1] x1 + [2]
p(if) = [1] x1 + [0]
p(minus) = [0]
p(s) = [1] x1 + [6]
p(true) = [0]
p(div#) = [5] x1 + [0]
p(geq#) = [1] x1 + [1] x2 + [2]
p(if#) = [1] x1 + [2] x2 + [8] x3 + [2]
p(minus#) = [2] x1 + [1]
p(c_1) = [8]
p(c_2) = [8] x1 + [2] x2 + [14]
p(c_3) = [4]
p(c_4) = [0]
p(c_5) = [1] x1 + [4]
p(c_6) = [0]
p(c_7) = [1]
p(c_8) = [1]
p(c_9) = [1] x1 + [10]

Following rules are strictly oriented:
div#(s(X),s(Y)) = [5] X + [30]
> [4] X + [16]
= c_2(div#(minus(X,Y),s(Y)),minus#(X,Y))

minus#(s(X),s(Y)) = [2] X + [13]
> [2] X + [11]
= c_9(minus#(X,Y))

Following rules are (at-least) weakly oriented:
minus(0(),Y) =  [0]
>= [0]
=  0()

minus(s(X),s(Y)) =  [0]
>= [0]
=  minus(X,Y)

*** Step 6.b:3.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y)),minus#(X,Y))
minus#(s(X),s(Y)) -> c_9(minus#(X,Y))
- Weak TRS:
minus(0(),Y) -> 0()
minus(s(X),s(Y)) -> minus(X,Y)
- Signature:
{div/2,geq/2,if/3,minus/2,div#/2,geq#/2,if#/3,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/2,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,geq#,if#,minus#} and constructors {0,false,s,true}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

*** Step 6.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y)),minus#(X,Y))
minus#(s(X),s(Y)) -> c_9(minus#(X,Y))
- Weak TRS:
minus(0(),Y) -> 0()
minus(s(X),s(Y)) -> minus(X,Y)
- Signature:
{div/2,geq/2,if/3,minus/2,div#/2,geq#/2,if#/3,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/2,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,geq#,if#,minus#} and constructors {0,false,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y)),minus#(X,Y))
-->_2 minus#(s(X),s(Y)) -> c_9(minus#(X,Y)):2
-->_1 div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y)),minus#(X,Y)):1

2:W:minus#(s(X),s(Y)) -> c_9(minus#(X,Y))
-->_1 minus#(s(X),s(Y)) -> c_9(minus#(X,Y)):2

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y)),minus#(X,Y))
2: minus#(s(X),s(Y)) -> c_9(minus#(X,Y))
*** Step 6.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:
- Weak TRS:
minus(0(),Y) -> 0()
minus(s(X),s(Y)) -> minus(X,Y)
- Signature:
{div/2,geq/2,if/3,minus/2,div#/2,geq#/2,if#/3,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/2,c_3/0,c_4/0
,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {div#,geq#,if#,minus#} and constructors {0,false,s,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))
```