* 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:
             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 dependency pairs:
        
        Strict DPs
          divp#(x,y) -> c_1(x,y)
          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(x,y)
            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_1/2
            ,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/2}
        - Obligation:
             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(x,y)
          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(x,y)
            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_1/2
            ,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/2}
        - Obligation:
             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
          {2,3,5,6}
        by application of
          Pre({2,3,5,6}) = {1,4,7}.
        Here rules are labelled as follows:
          1: divp#(x,y) -> c_1(x,y)
          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:
            divp#(x,y) -> c_1(x,y)
            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:
            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_1/2
            ,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/2}
        - Obligation:
             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:divp#(x,y) -> c_1(x,y)
             -->_2 prime1#(x,s(s(y))) -> c_7(divp#(s(s(y)),x),prime1#(x,s(y))):3
             -->_1 prime1#(x,s(s(y))) -> c_7(divp#(s(s(y)),x),prime1#(x,s(y))):3
             -->_2 prime#(s(s(x))) -> c_4(prime1#(s(s(x)),s(x))):2
             -->_1 prime#(s(s(x))) -> c_4(prime1#(s(s(x)),s(x))):2
             -->_2 prime1#(x,s(0())) -> c_6():7
             -->_1 prime1#(x,s(0())) -> c_6():7
             -->_2 prime1#(x,0()) -> c_5():6
             -->_1 prime1#(x,0()) -> c_5():6
             -->_2 prime#(s(0())) -> c_3():5
             -->_1 prime#(s(0())) -> c_3():5
             -->_2 prime#(0()) -> c_2():4
             -->_1 prime#(0()) -> c_2():4
             -->_2 divp#(x,y) -> c_1(x,y):1
             -->_1 divp#(x,y) -> c_1(x,y):1
          
          2: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))):3
             -->_1 prime1#(x,s(0())) -> c_6():7
          
          3: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
             -->_2 prime1#(x,s(s(y))) -> c_7(divp#(s(s(y)),x),prime1#(x,s(y))):3
             -->_1 divp#(x,y) -> c_1(x,y):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.
          4: prime#(0()) -> c_2()
          5: prime#(s(0())) -> c_3()
          6: prime1#(x,0()) -> c_5()
          7: prime1#(x,s(0())) -> c_6()
* Step 5: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            divp#(x,y) -> c_1(x,y)
            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_1/2
            ,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/2}
        - Obligation:
             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: prime#(s(s(x))) -> c_4(prime1#(s(s(x)),s(x)))
          
        The strictly oriented rules are moved into the weak component.
** Step 5.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            divp#(x,y) -> c_1(x,y)
            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_1/2
            ,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/2}
        - Obligation:
             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_4) = {1},
          uargs(c_7) = {1,2}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
                p(0) = [0]                  
                p(=) = [1] x2 + [0]         
              p(and) = [1] x1 + [1] x2 + [0]
             p(divp) = [1] x1 + [4] x2 + [1]
            p(false) = [1]                  
              p(not) = [1]                  
            p(prime) = [2] x1 + [4]         
           p(prime1) = [1] x1 + [1] x2 + [1]
              p(rem) = [1] x1 + [1]         
                p(s) = [0]                  
             p(true) = [2]                  
            p(divp#) = [9] x1 + [0]         
           p(prime#) = [1] x1 + [5]         
          p(prime1#) = [0]                  
              p(c_1) = [0]                  
              p(c_2) = [1]                  
              p(c_3) = [2]                  
              p(c_4) = [1] x1 + [3]         
              p(c_5) = [0]                  
              p(c_6) = [1]                  
              p(c_7) = [8] x1 + [8] x2 + [0]
        
        Following rules are strictly oriented:
        prime#(s(s(x))) = [5]                       
                        > [3]                       
                        = c_4(prime1#(s(s(x)),s(x)))
        
        
        Following rules are (at-least) weakly oriented:
                divp#(x,y) =  [9] x + [0]                          
                           >= [0]                                  
                           =  c_1(x,y)                             
        
        prime1#(x,s(s(y))) =  [0]                                  
                           >= [0]                                  
                           =  c_7(divp#(s(s(y)),x),prime1#(x,s(y)))
        
** Step 5.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            divp#(x,y) -> c_1(x,y)
            prime1#(x,s(s(y))) -> c_7(divp#(s(s(y)),x),prime1#(x,s(y)))
        - Weak DPs:
            prime#(s(s(x))) -> c_4(prime1#(s(s(x)),s(x)))
        - 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_1/2
            ,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/2}
        - Obligation:
             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 5.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            divp#(x,y) -> c_1(x,y)
            prime1#(x,s(s(y))) -> c_7(divp#(s(s(y)),x),prime1#(x,s(y)))
        - Weak DPs:
            prime#(s(s(x))) -> c_4(prime1#(s(s(x)),s(x)))
        - 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_1/2
            ,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/2}
        - Obligation:
             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(divp#(s(s(y)),x),prime1#(x,s(y)))
          
        The strictly oriented rules are moved into the weak component.
*** Step 5.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            divp#(x,y) -> c_1(x,y)
            prime1#(x,s(s(y))) -> c_7(divp#(s(s(y)),x),prime1#(x,s(y)))
        - Weak DPs:
            prime#(s(s(x))) -> c_4(prime1#(s(s(x)),s(x)))
        - 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_1/2
            ,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/2}
        - Obligation:
             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_4) = {1},
          uargs(c_7) = {1,2}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
                p(0) = [2]                  
                p(=) = [2]                  
              p(and) = [1] x2 + [1]         
             p(divp) = [1] x1 + [2] x2 + [0]
            p(false) = [0]                  
              p(not) = [0]                  
            p(prime) = [4] x1 + [8]         
           p(prime1) = [1] x1 + [1]         
              p(rem) = [1] x1 + [1]         
                p(s) = [1] x1 + [4]         
             p(true) = [0]                  
            p(divp#) = [0]                  
           p(prime#) = [2] x1 + [0]         
          p(prime1#) = [1] x2 + [4]         
              p(c_1) = [0]                  
              p(c_2) = [2]                  
              p(c_3) = [0]                  
              p(c_4) = [2] x1 + [0]         
              p(c_5) = [1]                  
              p(c_6) = [1]                  
              p(c_7) = [8] x1 + [1] x2 + [1]
        
        Following rules are strictly oriented:
        prime1#(x,s(s(y))) = [1] y + [12]                         
                           > [1] y + [9]                          
                           = c_7(divp#(s(s(y)),x),prime1#(x,s(y)))
        
        
        Following rules are (at-least) weakly oriented:
             divp#(x,y) =  [0]                       
                        >= [0]                       
                        =  c_1(x,y)                  
        
        prime#(s(s(x))) =  [2] x + [16]              
                        >= [2] x + [16]              
                        =  c_4(prime1#(s(s(x)),s(x)))
        
*** Step 5.b:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            divp#(x,y) -> c_1(x,y)
        - Weak 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_1/2
            ,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/2}
        - Obligation:
             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 5.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            divp#(x,y) -> c_1(x,y)
        - Weak 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_1/2
            ,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/2}
        - Obligation:
             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:
          1: divp#(x,y) -> c_1(x,y)
          
        Consider the set of all dependency pairs
          1: divp#(x,y) -> c_1(x,y)
          2: prime#(s(s(x))) -> c_4(prime1#(s(s(x)),s(x)))
          3: prime1#(x,s(s(y))) -> c_7(divp#(s(s(y)),x),prime1#(x,s(y)))
        Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1))
        SPACE(?,?)on application of the dependency pairs
          {1}
        These cover all (indirect) predecessors of dependency pairs
          {1,2}
        their number of applications is equally bounded.
        The dependency pairs are shifted into the weak component.
**** Step 5.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            divp#(x,y) -> c_1(x,y)
        - Weak 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_1/2
            ,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/2}
        - Obligation:
             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_4) = {1},
          uargs(c_7) = {1,2}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
                p(0) = [1]                  
                p(=) = [1] x2 + [0]         
              p(and) = [1]                  
             p(divp) = [1] x1 + [2]         
            p(false) = [2]                  
              p(not) = [1]                  
            p(prime) = [1]                  
           p(prime1) = [2] x1 + [8] x2 + [0]
              p(rem) = [8]                  
                p(s) = [1] x1 + [1]         
             p(true) = [4]                  
            p(divp#) = [1]                  
           p(prime#) = [8] x1 + [13]        
          p(prime1#) = [2] x2 + [7]         
              p(c_1) = [0]                  
              p(c_2) = [2]                  
              p(c_3) = [0]                  
              p(c_4) = [1] x1 + [9]         
              p(c_5) = [1]                  
              p(c_6) = [0]                  
              p(c_7) = [1] x1 + [1] x2 + [1]
        
        Following rules are strictly oriented:
        divp#(x,y) = [1]     
                   > [0]     
                   = c_1(x,y)
        
        
        Following rules are (at-least) weakly oriented:
           prime#(s(s(x))) =  [8] x + [29]                         
                           >= [2] x + [18]                         
                           =  c_4(prime1#(s(s(x)),s(x)))           
        
        prime1#(x,s(s(y))) =  [2] y + [11]                         
                           >= [2] y + [11]                         
                           =  c_7(divp#(s(s(y)),x),prime1#(x,s(y)))
        
**** Step 5.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            divp#(x,y) -> c_1(x,y)
            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_1/2
            ,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/2}
        - Obligation:
             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 5.b:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            divp#(x,y) -> c_1(x,y)
            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_1/2
            ,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/2}
        - Obligation:
             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:divp#(x,y) -> c_1(x,y)
             -->_2 prime1#(x,s(s(y))) -> c_7(divp#(s(s(y)),x),prime1#(x,s(y))):3
             -->_1 prime1#(x,s(s(y))) -> c_7(divp#(s(s(y)),x),prime1#(x,s(y))):3
             -->_2 prime#(s(s(x))) -> c_4(prime1#(s(s(x)),s(x))):2
             -->_1 prime#(s(s(x))) -> c_4(prime1#(s(s(x)),s(x))):2
             -->_2 divp#(x,y) -> c_1(x,y):1
             -->_1 divp#(x,y) -> c_1(x,y):1
          
          2:W: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))):3
          
          3:W: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))):3
             -->_1 divp#(x,y) -> c_1(x,y):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: divp#(x,y) -> c_1(x,y)
          3: prime1#(x,s(s(y))) -> c_7(divp#(s(s(y)),x),prime1#(x,s(y)))
          2: prime#(s(s(x))) -> c_4(prime1#(s(s(x)),s(x)))
**** Step 5.b:1.b:1.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_1/2
            ,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/2}
        - Obligation:
             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))