* Step 1: ToInnermost WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            U11(tt(),M,N) -> U12(tt(),activate(M),activate(N))
            U12(tt(),M,N) -> s(plus(activate(N),activate(M)))
            activate(X) -> X
            plus(N,0()) -> N
            plus(N,s(M)) -> U11(tt(),M,N)
        - Signature:
            {U11/3,U12/3,activate/1,plus/2} / {0/0,s/1,tt/0}
        - Obligation:
             runtime complexity wrt. defined symbols {U11,U12,activate,plus} and constructors {0,s,tt}
    + Applied Processor:
        ToInnermost
    + Details:
        switch to innermost, as the system is overlay and right linear and does not contain weak rules
* Step 2: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            U11(tt(),M,N) -> U12(tt(),activate(M),activate(N))
            U12(tt(),M,N) -> s(plus(activate(N),activate(M)))
            activate(X) -> X
            plus(N,0()) -> N
            plus(N,s(M)) -> U11(tt(),M,N)
        - Signature:
            {U11/3,U12/3,activate/1,plus/2} / {0/0,s/1,tt/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {U11,U12,activate,plus} and constructors {0,s,tt}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak innermost dependency pairs:
        
        Strict DPs
          U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)))
          U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)))
          activate#(X) -> c_3()
          plus#(N,0()) -> c_4()
          plus#(N,s(M)) -> c_5(U11#(tt(),M,N))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 3: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)))
            U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)))
            activate#(X) -> c_3()
            plus#(N,0()) -> c_4()
            plus#(N,s(M)) -> c_5(U11#(tt(),M,N))
        - Strict TRS:
            U11(tt(),M,N) -> U12(tt(),activate(M),activate(N))
            U12(tt(),M,N) -> s(plus(activate(N),activate(M)))
            activate(X) -> X
            plus(N,0()) -> N
            plus(N,s(M)) -> U11(tt(),M,N)
        - Signature:
            {U11/3,U12/3,activate/1,plus/2,U11#/3,U12#/3,activate#/1,plus#/2} / {0/0,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/0
            ,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,plus#} and constructors {0,s,tt}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          activate(X) -> X
          U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)))
          U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)))
          activate#(X) -> c_3()
          plus#(N,0()) -> c_4()
          plus#(N,s(M)) -> c_5(U11#(tt(),M,N))
* Step 4: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)))
            U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)))
            activate#(X) -> c_3()
            plus#(N,0()) -> c_4()
            plus#(N,s(M)) -> c_5(U11#(tt(),M,N))
        - Strict TRS:
            activate(X) -> X
        - Signature:
            {U11/3,U12/3,activate/1,plus/2,U11#/3,U12#/3,activate#/1,plus#/2} / {0/0,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/0
            ,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,plus#} and constructors {0,s,tt}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnTrs}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(U12#) = {2,3},
            uargs(plus#) = {1,2},
            uargs(c_1) = {1},
            uargs(c_2) = {1},
            uargs(c_5) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                    p(0) = [0]                           
                  p(U11) = [0]                           
                  p(U12) = [0]                           
             p(activate) = [1] x1 + [1]                  
                 p(plus) = [0]                           
                    p(s) = [1] x1 + [0]                  
                   p(tt) = [3]                           
                 p(U11#) = [8] x1 + [1] x2 + [1] x3 + [0]
                 p(U12#) = [1] x2 + [1] x3 + [0]         
            p(activate#) = [1] x1 + [8]                  
                p(plus#) = [1] x1 + [1] x2 + [0]         
                  p(c_1) = [1] x1 + [0]                  
                  p(c_2) = [1] x1 + [0]                  
                  p(c_3) = [0]                           
                  p(c_4) = [0]                           
                  p(c_5) = [1] x1 + [0]                  
          
          Following rules are strictly oriented:
          U11#(tt(),M,N) = [1] M + [1] N + [24]                   
                         > [1] M + [1] N + [2]                    
                         = c_1(U12#(tt(),activate(M),activate(N)))
          
            activate#(X) = [1] X + [8]                            
                         > [0]                                    
                         = c_3()                                  
          
             activate(X) = [1] X + [1]                            
                         > [1] X + [0]                            
                         = X                                      
          
          
          Following rules are (at-least) weakly oriented:
          U12#(tt(),M,N) =  [1] M + [1] N + [0]                
                         >= [1] M + [1] N + [2]                
                         =  c_2(plus#(activate(N),activate(M)))
          
            plus#(N,0()) =  [1] N + [0]                        
                         >= [0]                                
                         =  c_4()                              
          
           plus#(N,s(M)) =  [1] M + [1] N + [0]                
                         >= [1] M + [1] N + [24]               
                         =  c_5(U11#(tt(),M,N))                
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 5: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)))
            plus#(N,0()) -> c_4()
            plus#(N,s(M)) -> c_5(U11#(tt(),M,N))
        - Weak DPs:
            U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)))
            activate#(X) -> c_3()
        - Weak TRS:
            activate(X) -> X
        - Signature:
            {U11/3,U12/3,activate/1,plus/2,U11#/3,U12#/3,activate#/1,plus#/2} / {0/0,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/0
            ,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,plus#} and constructors {0,s,tt}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,3}
        by application of
          Pre({2,3}) = {1}.
        Here rules are labelled as follows:
          1: U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)))
          2: plus#(N,0()) -> c_4()
          3: plus#(N,s(M)) -> c_5(U11#(tt(),M,N))
          4: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)))
          5: activate#(X) -> c_3()
* Step 6: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)))
        - Weak DPs:
            U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)))
            activate#(X) -> c_3()
            plus#(N,0()) -> c_4()
            plus#(N,s(M)) -> c_5(U11#(tt(),M,N))
        - Weak TRS:
            activate(X) -> X
        - Signature:
            {U11/3,U12/3,activate/1,plus/2,U11#/3,U12#/3,activate#/1,plus#/2} / {0/0,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/0
            ,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,plus#} and constructors {0,s,tt}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)))
             -->_1 plus#(N,s(M)) -> c_5(U11#(tt(),M,N)):5
             -->_1 plus#(N,0()) -> c_4():4
          
          2:W:U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)))
             -->_1 U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))):1
          
          3:W:activate#(X) -> c_3()
             
          
          4:W:plus#(N,0()) -> c_4()
             
          
          5:W:plus#(N,s(M)) -> c_5(U11#(tt(),M,N))
             -->_1 U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: activate#(X) -> c_3()
          4: plus#(N,0()) -> c_4()
* Step 7: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)))
        - Weak DPs:
            U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)))
            plus#(N,s(M)) -> c_5(U11#(tt(),M,N))
        - Weak TRS:
            activate(X) -> X
        - Signature:
            {U11/3,U12/3,activate/1,plus/2,U11#/3,U12#/3,activate#/1,plus#/2} / {0/0,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/0
            ,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,plus#} and constructors {0,s,tt}
    + 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: U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)))
          
        Consider the set of all dependency pairs
          1: U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)))
          2: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)))
          5: plus#(N,s(M)) -> c_5(U11#(tt(),M,N))
        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
          {1,2,5}
        their number of applications is equally bounded.
        The dependency pairs are shifted into the weak component.
** Step 7.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)))
        - Weak DPs:
            U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)))
            plus#(N,s(M)) -> c_5(U11#(tt(),M,N))
        - Weak TRS:
            activate(X) -> X
        - Signature:
            {U11/3,U12/3,activate/1,plus/2,U11#/3,U12#/3,activate#/1,plus#/2} / {0/0,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/0
            ,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,plus#} and constructors {0,s,tt}
    + 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_1) = {1},
          uargs(c_2) = {1},
          uargs(c_5) = {1}
        
        Following symbols are considered usable:
          {activate,U11#,U12#,activate#,plus#}
        TcT has computed the following interpretation:
                  p(0) = [4]                           
                p(U11) = [1] x1 + [1]                  
                p(U12) = [1] x1 + [1] x3 + [2]         
           p(activate) = [1] x1 + [1]                  
               p(plus) = [1] x1 + [1] x2 + [2]         
                  p(s) = [1] x1 + [5]                  
                 p(tt) = [3]                           
               p(U11#) = [4] x1 + [4] x2 + [1] x3 + [4]
               p(U12#) = [2] x1 + [4] x2 + [1] x3 + [5]
          p(activate#) = [0]                           
              p(plus#) = [1] x1 + [4] x2 + [4]         
                p(c_1) = [1] x1 + [0]                  
                p(c_2) = [1] x1 + [0]                  
                p(c_3) = [1]                           
                p(c_4) = [1]                           
                p(c_5) = [1] x1 + [3]                  
        
        Following rules are strictly oriented:
        U12#(tt(),M,N) = [4] M + [1] N + [11]               
                       > [4] M + [1] N + [9]                
                       = c_2(plus#(activate(N),activate(M)))
        
        
        Following rules are (at-least) weakly oriented:
        U11#(tt(),M,N) =  [4] M + [1] N + [16]                   
                       >= [4] M + [1] N + [16]                   
                       =  c_1(U12#(tt(),activate(M),activate(N)))
        
         plus#(N,s(M)) =  [4] M + [1] N + [24]                   
                       >= [4] M + [1] N + [19]                   
                       =  c_5(U11#(tt(),M,N))                    
        
           activate(X) =  [1] X + [1]                            
                       >= [1] X + [0]                            
                       =  X                                      
        
** Step 7.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)))
            U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)))
            plus#(N,s(M)) -> c_5(U11#(tt(),M,N))
        - Weak TRS:
            activate(X) -> X
        - Signature:
            {U11/3,U12/3,activate/1,plus/2,U11#/3,U12#/3,activate#/1,plus#/2} / {0/0,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/0
            ,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,plus#} and constructors {0,s,tt}
    + 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:
            U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)))
            U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)))
            plus#(N,s(M)) -> c_5(U11#(tt(),M,N))
        - Weak TRS:
            activate(X) -> X
        - Signature:
            {U11/3,U12/3,activate/1,plus/2,U11#/3,U12#/3,activate#/1,plus#/2} / {0/0,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/0
            ,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,plus#} and constructors {0,s,tt}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)))
             -->_1 U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M))):2
          
          2:W:U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)))
             -->_1 plus#(N,s(M)) -> c_5(U11#(tt(),M,N)):3
          
          3:W:plus#(N,s(M)) -> c_5(U11#(tt(),M,N))
             -->_1 U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N))):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: U11#(tt(),M,N) -> c_1(U12#(tt(),activate(M),activate(N)))
          3: plus#(N,s(M)) -> c_5(U11#(tt(),M,N))
          2: U12#(tt(),M,N) -> c_2(plus#(activate(N),activate(M)))
** Step 7.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            activate(X) -> X
        - Signature:
            {U11/3,U12/3,activate/1,plus/2,U11#/3,U12#/3,activate#/1,plus#/2} / {0/0,s/1,tt/0,c_1/1,c_2/1,c_3/0,c_4/0
            ,c_5/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {U11#,U12#,activate#,plus#} and constructors {0,s,tt}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))