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_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          f#(X,Y,g(X,Y)) -> c_1(h#(0(),g(X,Y)),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: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(X,Y,g(X,Y)) -> c_1(h#(0(),g(X,Y)),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 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_1/2,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
          {3,4}
        by application of
          Pre({3,4}) = {1,2}.
        Here rules are labelled as follows:
          1: f#(X,Y,g(X,Y)) -> c_1(h#(0(),g(X,Y)),g#(X,Y))
          2: g#(X,s(Y)) -> c_2(g#(X,Y))
          3: g#(0(),Y) -> c_3()
          4: h#(X,Z) -> c_4(f#(X,s(X),Z))
* Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(X,Y,g(X,Y)) -> c_1(h#(0(),g(X,Y)),g#(X,Y))
            g#(X,s(Y)) -> c_2(g#(X,Y))
        - Weak DPs:
            g#(0(),Y) -> c_3()
            h#(X,Z) -> c_4(f#(X,s(X),Z))
        - Weak 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_1/2,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:S:f#(X,Y,g(X,Y)) -> c_1(h#(0(),g(X,Y)),g#(X,Y))
             -->_2 g#(X,s(Y)) -> c_2(g#(X,Y)):2
             -->_2 g#(0(),Y) -> c_3():3
          
          2:S:g#(X,s(Y)) -> c_2(g#(X,Y))
             -->_1 g#(0(),Y) -> c_3():3
             -->_1 g#(X,s(Y)) -> c_2(g#(X,Y)):2
          
          3:W:g#(0(),Y) -> c_3()
             
          
          4: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.
          4: h#(X,Z) -> c_4(f#(X,s(X),Z))
          3: g#(0(),Y) -> c_3()
* Step 4: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(X,Y,g(X,Y)) -> c_1(h#(0(),g(X,Y)),g#(X,Y))
            g#(X,s(Y)) -> c_2(g#(X,Y))
        - Weak 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_1/2,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:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:f#(X,Y,g(X,Y)) -> c_1(h#(0(),g(X,Y)),g#(X,Y))
             -->_2 g#(X,s(Y)) -> c_2(g#(X,Y)):2
          
          2:S:g#(X,s(Y)) -> c_2(g#(X,Y))
             -->_1 g#(X,s(Y)) -> c_2(g#(X,Y)):2
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          f#(X,Y,g(X,Y)) -> c_1(g#(X,Y))
* Step 5: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(X,Y,g(X,Y)) -> c_1(g#(X,Y))
            g#(X,s(Y)) -> c_2(g#(X,Y))
        - Weak 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)) -> c_2(g#(X,Y))
* Step 6: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g#(X,s(Y)) -> c_2(g#(X,Y))
        - 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 = WgOnAny}
    + 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_2) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
              p(0) = [0]          
              p(f) = [0]          
              p(g) = [0]          
              p(h) = [0]          
              p(s) = [1] x1 + [1] 
             p(f#) = [0]          
             p(g#) = [8] x2 + [15]
             p(h#) = [0]          
            p(c_1) = [0]          
            p(c_2) = [1] x1 + [5] 
            p(c_3) = [0]          
            p(c_4) = [8] x1 + [0] 
          
          Following rules are strictly oriented:
          g#(X,s(Y)) = [8] Y + [23]
                     > [8] Y + [20]
                     = c_2(g#(X,Y))
          
          
          Following rules are (at-least) weakly oriented:
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
* Step 7: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            g#(X,s(Y)) -> c_2(g#(X,Y))
        - 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))