WORST_CASE(?,O(n^1))
* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            plus(plus(X,Y),Z) -> plus(X,plus(Y,Z))
            times(X,s(Y)) -> plus(X,times(Y,X))
        - Signature:
            {plus/2,times/2} / {s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {plus,times} and constructors {s}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak innermost dependency pairs:
        
        Strict DPs
          plus#(plus(X,Y),Z) -> c_1(plus#(X,plus(Y,Z)))
          times#(X,s(Y)) -> c_2(plus#(X,times(Y,X)))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            plus#(plus(X,Y),Z) -> c_1(plus#(X,plus(Y,Z)))
            times#(X,s(Y)) -> c_2(plus#(X,times(Y,X)))
        - Strict TRS:
            plus(plus(X,Y),Z) -> plus(X,plus(Y,Z))
            times(X,s(Y)) -> plus(X,times(Y,X))
        - Signature:
            {plus/2,times/2,plus#/2,times#/2} / {s/1,c_1/1,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {plus#,times#} and constructors {s}
    + Applied Processor:
        WeightGap {wgDimension = 2, 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 (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima):
          The following argument positions are considered usable:
            uargs(plus) = {2},
            uargs(plus#) = {2},
            uargs(c_1) = {1},
            uargs(c_2) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
              p(plus) = [0 1] x1 + [1 0] x2 + [0]
                        [0 2]      [0 1]      [2]
                 p(s) = [1 1] x1 + [4]           
                        [0 0]      [0]           
             p(times) = [1 1] x1 + [1 0] x2 + [5]
                        [2 2]      [2 0]      [0]
             p(plus#) = [0 4] x1 + [1 0] x2 + [5]
                        [1 7]      [0 1]      [0]
            p(times#) = [1 6] x1 + [2 0] x2 + [0]
                        [0 0]      [0 0]      [0]
               p(c_1) = [1 0] x1 + [0]           
                        [0 1]      [0]           
               p(c_2) = [1 0] x1 + [0]           
                        [0 1]      [0]           
          
          Following rules are strictly oriented:
          plus#(plus(X,Y),Z) = [0  8] X + [0 4] Y + [1 0] Z + [13]
                               [0 15]     [1 7]     [0 1]     [14]
                             > [0 4] X + [0 1] Y + [1 0] Z + [5]  
                               [1 7]     [0 2]     [0 1]     [2]  
                             = c_1(plus#(X,plus(Y,Z)))            
          
           plus(plus(X,Y),Z) = [0 2] X + [0 1] Y + [1 0] Z + [2]  
                               [0 4]     [0 2]     [0 1]     [6]  
                             > [0 1] X + [0 1] Y + [1 0] Z + [0]  
                               [0 2]     [0 2]     [0 1]     [4]  
                             = plus(X,plus(Y,Z))                  
          
               times(X,s(Y)) = [1 1] X + [1 1] Y + [9]            
                               [2 2]     [2 2]     [8]            
                             > [1 1] X + [1 1] Y + [5]            
                               [2 2]     [2 2]     [2]            
                             = plus(X,times(Y,X))                 
          
          
          Following rules are (at-least) weakly oriented:
          times#(X,s(Y)) =  [1 6] X + [2 2] Y + [8] 
                            [0 0]     [0 0]     [0] 
                         >= [1 4] X + [1 1] Y + [10]
                            [3 7]     [2 2]     [0] 
                         =  c_2(plus#(X,times(Y,X)))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 3: PredecessorEstimation WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            times#(X,s(Y)) -> c_2(plus#(X,times(Y,X)))
        - Weak DPs:
            plus#(plus(X,Y),Z) -> c_1(plus#(X,plus(Y,Z)))
        - Weak TRS:
            plus(plus(X,Y),Z) -> plus(X,plus(Y,Z))
            times(X,s(Y)) -> plus(X,times(Y,X))
        - Signature:
            {plus/2,times/2,plus#/2,times#/2} / {s/1,c_1/1,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {plus#,times#} and constructors {s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1}
        by application of
          Pre({1}) = {}.
        Here rules are labelled as follows:
          1: times#(X,s(Y)) -> c_2(plus#(X,times(Y,X)))
          2: plus#(plus(X,Y),Z) -> c_1(plus#(X,plus(Y,Z)))
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            plus#(plus(X,Y),Z) -> c_1(plus#(X,plus(Y,Z)))
            times#(X,s(Y)) -> c_2(plus#(X,times(Y,X)))
        - Weak TRS:
            plus(plus(X,Y),Z) -> plus(X,plus(Y,Z))
            times(X,s(Y)) -> plus(X,times(Y,X))
        - Signature:
            {plus/2,times/2,plus#/2,times#/2} / {s/1,c_1/1,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {plus#,times#} and constructors {s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:plus#(plus(X,Y),Z) -> c_1(plus#(X,plus(Y,Z)))
             -->_1 plus#(plus(X,Y),Z) -> c_1(plus#(X,plus(Y,Z))):1
          
          2:W:times#(X,s(Y)) -> c_2(plus#(X,times(Y,X)))
             -->_1 plus#(plus(X,Y),Z) -> c_1(plus#(X,plus(Y,Z))):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: times#(X,s(Y)) -> c_2(plus#(X,times(Y,X)))
          1: plus#(plus(X,Y),Z) -> c_1(plus#(X,plus(Y,Z)))
* Step 5: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            plus(plus(X,Y),Z) -> plus(X,plus(Y,Z))
            times(X,s(Y)) -> plus(X,times(Y,X))
        - Signature:
            {plus/2,times/2,plus#/2,times#/2} / {s/1,c_1/1,c_2/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {plus#,times#} and constructors {s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))