```* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict TRS:
divp(x,y) -> =(rem(x,y),0())
prime(0()) -> false()
prime(s(0())) -> false()
prime(s(s(x))) -> prime1(s(s(x)),s(x))
prime1(x,0()) -> false()
prime1(x,s(0())) -> true()
prime1(x,s(s(y))) -> and(not(divp(s(s(y)),x)),prime1(x,s(y)))
- Signature:
{divp/2,prime/1,prime1/2} / {0/0,=/2,and/2,false/0,not/1,rem/2,s/1,true/0}
- Obligation:
innermost runtime complexity wrt. defined symbols {divp,prime,prime1} and constructors {0,=,and,false,not
,rem,s,true}
+ Applied Processor:
DependencyPairs {dpKind_ = WIDP}
+ Details:
We add the following weak innermost dependency pairs:

Strict DPs
divp#(x,y) -> c_1()
prime#(0()) -> c_2()
prime#(s(0())) -> c_3()
prime#(s(s(x))) -> c_4(prime1#(s(s(x)),s(x)))
prime1#(x,0()) -> c_5()
prime1#(x,s(0())) -> c_6()
prime1#(x,s(s(y))) -> c_7(divp#(s(s(y)),x),prime1#(x,s(y)))
Weak DPs

and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
divp#(x,y) -> c_1()
prime#(0()) -> c_2()
prime#(s(0())) -> c_3()
prime#(s(s(x))) -> c_4(prime1#(s(s(x)),s(x)))
prime1#(x,0()) -> c_5()
prime1#(x,s(0())) -> c_6()
prime1#(x,s(s(y))) -> c_7(divp#(s(s(y)),x),prime1#(x,s(y)))
- Strict TRS:
divp(x,y) -> =(rem(x,y),0())
prime(0()) -> false()
prime(s(0())) -> false()
prime(s(s(x))) -> prime1(s(s(x)),s(x))
prime1(x,0()) -> false()
prime1(x,s(0())) -> true()
prime1(x,s(s(y))) -> and(not(divp(s(s(y)),x)),prime1(x,s(y)))
- Signature:
{divp/2,prime/1,prime1/2,divp#/2,prime#/1,prime1#/2} / {0/0,=/2,and/2,false/0,not/1,rem/2,s/1,true/0,c_1/0
,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {divp#,prime#,prime1#} and constructors {0,=,and,false,not
,rem,s,true}
+ Applied Processor:
UsableRules
+ Details:
We replace rewrite rules by usable rules:
divp#(x,y) -> c_1()
prime#(0()) -> c_2()
prime#(s(0())) -> c_3()
prime#(s(s(x))) -> c_4(prime1#(s(s(x)),s(x)))
prime1#(x,0()) -> c_5()
prime1#(x,s(0())) -> c_6()
prime1#(x,s(s(y))) -> c_7(divp#(s(s(y)),x),prime1#(x,s(y)))
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
divp#(x,y) -> c_1()
prime#(0()) -> c_2()
prime#(s(0())) -> c_3()
prime#(s(s(x))) -> c_4(prime1#(s(s(x)),s(x)))
prime1#(x,0()) -> c_5()
prime1#(x,s(0())) -> c_6()
prime1#(x,s(s(y))) -> c_7(divp#(s(s(y)),x),prime1#(x,s(y)))
- Signature:
{divp/2,prime/1,prime1/2,divp#/2,prime#/1,prime1#/2} / {0/0,=/2,and/2,false/0,not/1,rem/2,s/1,true/0,c_1/0
,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {divp#,prime#,prime1#} and constructors {0,=,and,false,not
,rem,s,true}
+ Applied Processor:
PredecessorEstimation {onSelection = all simple predecessor estimation selector}
+ Details:
We estimate the number of application of
{1,2,3,5,6}
by application of
Pre({1,2,3,5,6}) = {4,7}.
Here rules are labelled as follows:
1: divp#(x,y) -> c_1()
2: prime#(0()) -> c_2()
3: prime#(s(0())) -> c_3()
4: prime#(s(s(x))) -> c_4(prime1#(s(s(x)),s(x)))
5: prime1#(x,0()) -> c_5()
6: prime1#(x,s(0())) -> c_6()
7: prime1#(x,s(s(y))) -> c_7(divp#(s(s(y)),x),prime1#(x,s(y)))
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
prime#(s(s(x))) -> c_4(prime1#(s(s(x)),s(x)))
prime1#(x,s(s(y))) -> c_7(divp#(s(s(y)),x),prime1#(x,s(y)))
- Weak DPs:
divp#(x,y) -> c_1()
prime#(0()) -> c_2()
prime#(s(0())) -> c_3()
prime1#(x,0()) -> c_5()
prime1#(x,s(0())) -> c_6()
- Signature:
{divp/2,prime/1,prime1/2,divp#/2,prime#/1,prime1#/2} / {0/0,=/2,and/2,false/0,not/1,rem/2,s/1,true/0,c_1/0
,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {divp#,prime#,prime1#} and constructors {0,=,and,false,not
,rem,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:S:prime#(s(s(x))) -> c_4(prime1#(s(s(x)),s(x)))
-->_1 prime1#(x,s(s(y))) -> c_7(divp#(s(s(y)),x),prime1#(x,s(y))):2
-->_1 prime1#(x,s(0())) -> c_6():7

2:S:prime1#(x,s(s(y))) -> c_7(divp#(s(s(y)),x),prime1#(x,s(y)))
-->_2 prime1#(x,s(0())) -> c_6():7
-->_1 divp#(x,y) -> c_1():3
-->_2 prime1#(x,s(s(y))) -> c_7(divp#(s(s(y)),x),prime1#(x,s(y))):2

3:W:divp#(x,y) -> c_1()

4:W:prime#(0()) -> c_2()

5:W:prime#(s(0())) -> c_3()

6:W:prime1#(x,0()) -> c_5()

7:W:prime1#(x,s(0())) -> c_6()

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
6: prime1#(x,0()) -> c_5()
5: prime#(s(0())) -> c_3()
4: prime#(0()) -> c_2()
3: divp#(x,y) -> c_1()
7: prime1#(x,s(0())) -> c_6()
* Step 5: SimplifyRHS WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
prime#(s(s(x))) -> c_4(prime1#(s(s(x)),s(x)))
prime1#(x,s(s(y))) -> c_7(divp#(s(s(y)),x),prime1#(x,s(y)))
- Signature:
{divp/2,prime/1,prime1/2,divp#/2,prime#/1,prime1#/2} / {0/0,=/2,and/2,false/0,not/1,rem/2,s/1,true/0,c_1/0
,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/2}
- Obligation:
innermost runtime complexity wrt. defined symbols {divp#,prime#,prime1#} and constructors {0,=,and,false,not
,rem,s,true}
+ Applied Processor:
SimplifyRHS
+ Details:
Consider the dependency graph
1:S:prime#(s(s(x))) -> c_4(prime1#(s(s(x)),s(x)))
-->_1 prime1#(x,s(s(y))) -> c_7(divp#(s(s(y)),x),prime1#(x,s(y))):2

2:S:prime1#(x,s(s(y))) -> c_7(divp#(s(s(y)),x),prime1#(x,s(y)))
-->_2 prime1#(x,s(s(y))) -> c_7(divp#(s(s(y)),x),prime1#(x,s(y))):2

Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
prime1#(x,s(s(y))) -> c_7(prime1#(x,s(y)))
+ Considered Problem:
- Strict DPs:
prime#(s(s(x))) -> c_4(prime1#(s(s(x)),s(x)))
prime1#(x,s(s(y))) -> c_7(prime1#(x,s(y)))
- Signature:
{divp/2,prime/1,prime1/2,divp#/2,prime#/1,prime1#/2} / {0/0,=/2,and/2,false/0,not/1,rem/2,s/1,true/0,c_1/0
,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {divp#,prime#,prime1#} and constructors {0,=,and,false,not
,rem,s,true}
+ Applied Processor:
+ Details:
Consider the dependency graph

1:S:prime#(s(s(x))) -> c_4(prime1#(s(s(x)),s(x)))
-->_1 prime1#(x,s(s(y))) -> c_7(prime1#(x,s(y))):2

2:S:prime1#(x,s(s(y))) -> c_7(prime1#(x,s(y)))
-->_1 prime1#(x,s(s(y))) -> c_7(prime1#(x,s(y))):2

Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).

[(1,prime#(s(s(x))) -> c_4(prime1#(s(s(x)),s(x))))]
* Step 7: PredecessorEstimationCP WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
prime1#(x,s(s(y))) -> c_7(prime1#(x,s(y)))
- Signature:
{divp/2,prime/1,prime1/2,divp#/2,prime#/1,prime1#/2} / {0/0,=/2,and/2,false/0,not/1,rem/2,s/1,true/0,c_1/0
,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {divp#,prime#,prime1#} and constructors {0,=,and,false,not
,rem,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: prime1#(x,s(s(y))) -> c_7(prime1#(x,s(y)))

The strictly oriented rules are moved into the weak component.
** Step 7.a:1: NaturalMI WORST_CASE(?,O(n^1))
+ Considered Problem:
- Strict DPs:
prime1#(x,s(s(y))) -> c_7(prime1#(x,s(y)))
- Signature:
{divp/2,prime/1,prime1/2,divp#/2,prime#/1,prime1#/2} / {0/0,=/2,and/2,false/0,not/1,rem/2,s/1,true/0,c_1/0
,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {divp#,prime#,prime1#} and constructors {0,=,and,false,not
,rem,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_7) = {1}

Following symbols are considered usable:
{divp#,prime#,prime1#}
TcT has computed the following interpretation:
p(0) = [2]
p(=) = [1]
p(and) = [1]
p(divp) = [2] x2 + [0]
p(false) = [2]
p(not) = [1]
p(prime) = [1] x1 + [1]
p(prime1) = [2] x1 + [1] x2 + [0]
p(rem) = [1]
p(s) = [1] x1 + [1]
p(true) = [1]
p(divp#) = [1] x1 + [1] x2 + [8]
p(prime#) = [8]
p(prime1#) = [4] x1 + [8] x2 + [8]
p(c_1) = [1]
p(c_2) = [1]
p(c_3) = [4]
p(c_4) = [0]
p(c_5) = [1]
p(c_6) = [4]
p(c_7) = [1] x1 + [6]

Following rules are strictly oriented:
prime1#(x,s(s(y))) = [4] x + [8] y + [24]
> [4] x + [8] y + [22]
= c_7(prime1#(x,s(y)))

Following rules are (at-least) weakly oriented:

** Step 7.a:2: Assumption WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
prime1#(x,s(s(y))) -> c_7(prime1#(x,s(y)))
- Signature:
{divp/2,prime/1,prime1/2,divp#/2,prime#/1,prime1#/2} / {0/0,=/2,and/2,false/0,not/1,rem/2,s/1,true/0,c_1/0
,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {divp#,prime#,prime1#} and constructors {0,=,and,false,not
,rem,s,true}
+ Applied Processor:
Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
+ Details:
()

** Step 7.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
+ Considered Problem:
- Weak DPs:
prime1#(x,s(s(y))) -> c_7(prime1#(x,s(y)))
- Signature:
{divp/2,prime/1,prime1/2,divp#/2,prime#/1,prime1#/2} / {0/0,=/2,and/2,false/0,not/1,rem/2,s/1,true/0,c_1/0
,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {divp#,prime#,prime1#} and constructors {0,=,and,false,not
,rem,s,true}
+ Applied Processor:
RemoveWeakSuffixes
+ Details:
Consider the dependency graph
1:W:prime1#(x,s(s(y))) -> c_7(prime1#(x,s(y)))
-->_1 prime1#(x,s(s(y))) -> c_7(prime1#(x,s(y))):1

The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
1: prime1#(x,s(s(y))) -> c_7(prime1#(x,s(y)))
** Step 7.b:2: EmptyProcessor WORST_CASE(?,O(1))
+ Considered Problem:

- Signature:
{divp/2,prime/1,prime1/2,divp#/2,prime#/1,prime1#/2} / {0/0,=/2,and/2,false/0,not/1,rem/2,s/1,true/0,c_1/0
,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1}
- Obligation:
innermost runtime complexity wrt. defined symbols {divp#,prime#,prime1#} and constructors {0,=,and,false,not
,rem,s,true}
+ Applied Processor:
EmptyProcessor
+ Details:
The problem is already closed. The intended complexity is O(1).

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