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) = [0]                            
              p(f) = [0]                            
              p(g) = [4] x2 + [2]                   
              p(h) = [1] x1 + [2]                   
              p(s) = [1] x1 + [4]                   
             p(f#) = [10] x1 + [2] x2 + [4] x3 + [0]
             p(g#) = [8] x1 + [1] x2 + [1]          
             p(h#) = [13] x1 + [4] x2 + [2]         
            p(c_1) = [1] x1 + [4]                   
            p(c_2) = [1] x1 + [3]                   
            p(c_3) = [2]                            
            p(c_4) = [1] x1 + [8]                   
          
          Following rules are strictly oriented:
          g#(X,s(Y)) = [8] X + [1] Y + [5]
                     > [8] X + [1] Y + [4]
                     = c_2(g#(X,Y))       
          
           g(X,s(Y)) = [4] Y + [18]       
                     > [4] Y + [2]        
                     = g(X,Y)             
          
            g(0(),Y) = [4] Y + [2]        
                     > [0]                
                     = 0()                
          
          
          Following rules are (at-least) weakly oriented:
          f#(X,Y,g(X,Y)) =  [10] X + [18] Y + [8]
                         >= [16] Y + [14]        
                         =  c_1(h#(0(),g(X,Y)))  
          
               g#(0(),Y) =  [1] Y + [1]          
                         >= [2]                  
                         =  c_3()                
          
                 h#(X,Z) =  [13] X + [4] Z + [2] 
                         >= [12] X + [4] Z + [16]
                         =  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)))
            g#(0(),Y) -> c_3()
            h#(X,Z) -> c_4(f#(X,s(X),Z))
        - Weak DPs:
            g#(X,s(Y)) -> c_2(g#(X,Y))
        - 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:
            g#(0(),Y) -> c_3()
            h#(X,Z) -> c_4(f#(X,s(X),Z))
        - Weak DPs:
            g#(X,s(Y)) -> c_2(g#(X,Y))
        - 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
          {2}
        by application of
          Pre({2}) = {}.
        Here rules are labelled as follows:
          1: g#(0(),Y) -> c_3()
          2: h#(X,Z) -> c_4(f#(X,s(X),Z))
          3: g#(X,s(Y)) -> c_2(g#(X,Y))
* Step 6: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            g#(0(),Y) -> c_3()
        - Weak DPs:
            g#(X,s(Y)) -> c_2(g#(X,Y))
            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:S:g#(0(),Y) -> c_3()
             
          
          2:W:g#(X,s(Y)) -> c_2(g#(X,Y))
             -->_1 g#(X,s(Y)) -> c_2(g#(X,Y)):2
             -->_1 g#(0(),Y) -> c_3():1
          
          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))
* Step 7: UsableRules WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            g#(0(),Y) -> c_3()
        - Weak DPs:
            g#(X,s(Y)) -> c_2(g#(X,Y))
        - 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:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          g#(X,s(Y)) -> c_2(g#(X,Y))
          g#(0(),Y) -> c_3()
* Step 8: PredecessorEstimationCP WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            g#(0(),Y) -> c_3()
        - 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:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: g#(0(),Y) -> c_3()
          
        The strictly oriented rules are moved into the weak component.
** Step 8.a:1: NaturalMI WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            g#(0(),Y) -> c_3()
        - 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:
        NaturalMI {miDimension = 1, miDegree = 0, 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 (containing no more than 0 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        The following argument positions are considered usable:
          uargs(c_2) = {1}
        
        Following symbols are considered usable:
          {f#,g#,h#}
        TcT has computed the following interpretation:
            p(0) = [0]         
            p(f) = [0]         
            p(g) = [0]         
            p(h) = [0]         
            p(s) = [0]         
           p(f#) = [0]         
           p(g#) = [5]         
           p(h#) = [0]         
          p(c_1) = [0]         
          p(c_2) = [1] x1 + [0]
          p(c_3) = [1]         
          p(c_4) = [0]         
        
        Following rules are strictly oriented:
        g#(0(),Y) = [5]  
                  > [1]  
                  = c_3()
        
        
        Following rules are (at-least) weakly oriented:
        g#(X,s(Y)) =  [5]         
                   >= [5]         
                   =  c_2(g#(X,Y))
        
** Step 8.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            g#(X,s(Y)) -> c_2(g#(X,Y))
            g#(0(),Y) -> c_3()
        - 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:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

** Step 8.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            g#(X,s(Y)) -> c_2(g#(X,Y))
            g#(0(),Y) -> c_3()
        - 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()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: g#(X,s(Y)) -> c_2(g#(X,Y))
          2: g#(0(),Y) -> c_3()
** Step 8.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - 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))