WORST_CASE(?,O(n^1))
* 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)))
* Step 6: RemoveHeads 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(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:
        RemoveHeads
    + 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))