WORST_CASE(?,O(n^1))
* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(X,Y,g(X,Y)) -> h(0(),g(X,Y))
            g(X,s(Y)) -> g(X,Y)
            g(0(),Y) -> 0()
            h(X,Z) -> f(X,s(X),Z)
        - Signature:
            {f/3,g/2,h/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g,h} and constructors {0,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak innermost dependency pairs:
        
        Strict DPs
          f#(X,Y,g(X,Y)) -> c_1(h#(0(),g(X,Y)))
          g#(X,s(Y)) -> c_2(g#(X,Y))
          g#(0(),Y) -> c_3()
          h#(X,Z) -> c_4(f#(X,s(X),Z))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(X,Y,g(X,Y)) -> c_1(h#(0(),g(X,Y)))
            g#(X,s(Y)) -> c_2(g#(X,Y))
            g#(0(),Y) -> c_3()
            h#(X,Z) -> c_4(f#(X,s(X),Z))
        - Strict TRS:
            f(X,Y,g(X,Y)) -> h(0(),g(X,Y))
            g(X,s(Y)) -> g(X,Y)
            g(0(),Y) -> 0()
            h(X,Z) -> f(X,s(X),Z)
        - Signature:
            {f/3,g/2,h/2,f#/3,g#/2,h#/2} / {0/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,h#} and constructors {0,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          g(X,s(Y)) -> g(X,Y)
          g(0(),Y) -> 0()
          f#(X,Y,g(X,Y)) -> c_1(h#(0(),g(X,Y)))
          g#(X,s(Y)) -> c_2(g#(X,Y))
          g#(0(),Y) -> c_3()
          h#(X,Z) -> c_4(f#(X,s(X),Z))
* Step 3: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(X,Y,g(X,Y)) -> c_1(h#(0(),g(X,Y)))
            g#(X,s(Y)) -> c_2(g#(X,Y))
            g#(0(),Y) -> c_3()
            h#(X,Z) -> c_4(f#(X,s(X),Z))
        - Strict TRS:
            g(X,s(Y)) -> g(X,Y)
            g(0(),Y) -> 0()
        - Signature:
            {f/3,g/2,h/2,f#/3,g#/2,h#/2} / {0/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,h#} and constructors {0,s}
    + 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(c_1) = {1},
            uargs(c_2) = {1},
            uargs(c_4) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
              p(0) = [8]                   
              p(f) = [0]                   
              p(g) = [2] x1 + [3] x2 + [10]
              p(h) = [0]                   
              p(s) = [1] x1 + [7]          
             p(f#) = [2] x3 + [0]          
             p(g#) = [2] x1 + [1] x2 + [0] 
             p(h#) = [2] x2 + [0]          
            p(c_1) = [1] x1 + [0]          
            p(c_2) = [1] x1 + [3]          
            p(c_3) = [8]                   
            p(c_4) = [1] x1 + [2]          
          
          Following rules are strictly oriented:
          g#(X,s(Y)) = [2] X + [1] Y + [7] 
                     > [2] X + [1] Y + [3] 
                     = c_2(g#(X,Y))        
          
           g#(0(),Y) = [1] Y + [16]        
                     > [8]                 
                     = c_3()               
          
           g(X,s(Y)) = [2] X + [3] Y + [31]
                     > [2] X + [3] Y + [10]
                     = g(X,Y)              
          
            g(0(),Y) = [3] Y + [26]        
                     > [8]                 
                     = 0()                 
          
          
          Following rules are (at-least) weakly oriented:
          f#(X,Y,g(X,Y)) =  [4] X + [6] Y + [20]
                         >= [4] X + [6] Y + [20]
                         =  c_1(h#(0(),g(X,Y))) 
          
                 h#(X,Z) =  [2] Z + [0]         
                         >= [2] Z + [2]         
                         =  c_4(f#(X,s(X),Z))   
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 4: RemoveInapplicable WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(X,Y,g(X,Y)) -> c_1(h#(0(),g(X,Y)))
            h#(X,Z) -> c_4(f#(X,s(X),Z))
        - Weak DPs:
            g#(X,s(Y)) -> c_2(g#(X,Y))
            g#(0(),Y) -> c_3()
        - Weak TRS:
            g(X,s(Y)) -> g(X,Y)
            g(0(),Y) -> 0()
        - Signature:
            {f/3,g/2,h/2,f#/3,g#/2,h#/2} / {0/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,h#} and constructors {0,s}
    + Applied Processor:
        RemoveInapplicable
    + Details:
        Only the nodes
          {2,3,4}
        are reachable from nodes
          {2,3,4}
        that start derivation from marked basic terms.
        The nodes not reachable are removed from the problem.
* Step 5: PredecessorEstimation WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            h#(X,Z) -> c_4(f#(X,s(X),Z))
        - Weak DPs:
            g#(X,s(Y)) -> c_2(g#(X,Y))
            g#(0(),Y) -> c_3()
        - Weak TRS:
            g(X,s(Y)) -> g(X,Y)
            g(0(),Y) -> 0()
        - Signature:
            {f/3,g/2,h/2,f#/3,g#/2,h#/2} / {0/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,h#} and constructors {0,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: h#(X,Z) -> c_4(f#(X,s(X),Z))
          2: g#(X,s(Y)) -> c_2(g#(X,Y))
          3: g#(0(),Y) -> c_3()
* Step 6: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            g#(X,s(Y)) -> c_2(g#(X,Y))
            g#(0(),Y) -> c_3()
            h#(X,Z) -> c_4(f#(X,s(X),Z))
        - Weak TRS:
            g(X,s(Y)) -> g(X,Y)
            g(0(),Y) -> 0()
        - Signature:
            {f/3,g/2,h/2,f#/3,g#/2,h#/2} / {0/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,h#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:g#(X,s(Y)) -> c_2(g#(X,Y))
             -->_1 g#(0(),Y) -> c_3():2
             -->_1 g#(X,s(Y)) -> c_2(g#(X,Y)):1
          
          2:W:g#(0(),Y) -> c_3()
             
          
          3:W:h#(X,Z) -> c_4(f#(X,s(X),Z))
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: h#(X,Z) -> c_4(f#(X,s(X),Z))
          1: g#(X,s(Y)) -> c_2(g#(X,Y))
          2: g#(0(),Y) -> c_3()
* Step 7: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            g(X,s(Y)) -> g(X,Y)
            g(0(),Y) -> 0()
        - Signature:
            {f/3,g/2,h/2,f#/3,g#/2,h#/2} / {0/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,h#} and constructors {0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))