WORST_CASE(?,O(n^1))
* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(a()) -> b()
            f(c()) -> d()
            f(g(x,y)) -> g(f(x),f(y))
            f(h(x,y)) -> g(h(y,f(x)),h(x,f(y)))
            g(x,x) -> h(e(),x)
        - Signature:
            {f/1,g/2} / {a/0,b/0,c/0,d/0,e/0,h/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g} and constructors {a,b,c,d,e,h}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          f#(a()) -> c_1()
          f#(c()) -> c_2()
          f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y))
          f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y))
          g#(x,x) -> c_5()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(a()) -> c_1()
            f#(c()) -> c_2()
            f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y))
            f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y))
            g#(x,x) -> c_5()
        - Weak TRS:
            f(a()) -> b()
            f(c()) -> d()
            f(g(x,y)) -> g(f(x),f(y))
            f(h(x,y)) -> g(h(y,f(x)),h(x,f(y)))
            g(x,x) -> h(e(),x)
        - Signature:
            {f/1,g/2,f#/1,g#/2} / {a/0,b/0,c/0,d/0,e/0,h/2,c_1/0,c_2/0,c_3/3,c_4/3,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {a,b,c,d,e,h}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2,5}
        by application of
          Pre({1,2,5}) = {3,4}.
        Here rules are labelled as follows:
          1: f#(a()) -> c_1()
          2: f#(c()) -> c_2()
          3: f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y))
          4: f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y))
          5: g#(x,x) -> c_5()
* Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y))
            f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y))
        - Weak DPs:
            f#(a()) -> c_1()
            f#(c()) -> c_2()
            g#(x,x) -> c_5()
        - Weak TRS:
            f(a()) -> b()
            f(c()) -> d()
            f(g(x,y)) -> g(f(x),f(y))
            f(h(x,y)) -> g(h(y,f(x)),h(x,f(y)))
            g(x,x) -> h(e(),x)
        - Signature:
            {f/1,g/2,f#/1,g#/2} / {a/0,b/0,c/0,d/0,e/0,h/2,c_1/0,c_2/0,c_3/3,c_4/3,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {a,b,c,d,e,h}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y))
             -->_3 f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y)):2
             -->_2 f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y)):2
             -->_1 g#(x,x) -> c_5():5
             -->_3 f#(c()) -> c_2():4
             -->_2 f#(c()) -> c_2():4
             -->_3 f#(a()) -> c_1():3
             -->_2 f#(a()) -> c_1():3
             -->_3 f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y)):1
             -->_2 f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y)):1
          
          2:S:f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y))
             -->_1 g#(x,x) -> c_5():5
             -->_3 f#(c()) -> c_2():4
             -->_2 f#(c()) -> c_2():4
             -->_3 f#(a()) -> c_1():3
             -->_2 f#(a()) -> c_1():3
             -->_3 f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y)):2
             -->_2 f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y)):2
             -->_3 f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y)):1
             -->_2 f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y)):1
          
          3:W:f#(a()) -> c_1()
             
          
          4:W:f#(c()) -> c_2()
             
          
          5:W:g#(x,x) -> c_5()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: f#(a()) -> c_1()
          4: f#(c()) -> c_2()
          5: g#(x,x) -> c_5()
* Step 4: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y))
            f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y))
        - Weak TRS:
            f(a()) -> b()
            f(c()) -> d()
            f(g(x,y)) -> g(f(x),f(y))
            f(h(x,y)) -> g(h(y,f(x)),h(x,f(y)))
            g(x,x) -> h(e(),x)
        - Signature:
            {f/1,g/2,f#/1,g#/2} / {a/0,b/0,c/0,d/0,e/0,h/2,c_1/0,c_2/0,c_3/3,c_4/3,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {a,b,c,d,e,h}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y))
             -->_3 f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y)):2
             -->_2 f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y)):2
             -->_3 f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y)):1
             -->_2 f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y)):1
          
          2:S:f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y))
             -->_3 f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y)):2
             -->_2 f#(h(x,y)) -> c_4(g#(h(y,f(x)),h(x,f(y))),f#(x),f#(y)):2
             -->_3 f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y)):1
             -->_2 f#(g(x,y)) -> c_3(g#(f(x),f(y)),f#(x),f#(y)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          f#(g(x,y)) -> c_3(f#(x),f#(y))
          f#(h(x,y)) -> c_4(f#(x),f#(y))
* Step 5: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(g(x,y)) -> c_3(f#(x),f#(y))
            f#(h(x,y)) -> c_4(f#(x),f#(y))
        - Weak TRS:
            f(a()) -> b()
            f(c()) -> d()
            f(g(x,y)) -> g(f(x),f(y))
            f(h(x,y)) -> g(h(y,f(x)),h(x,f(y)))
            g(x,x) -> h(e(),x)
        - Signature:
            {f/1,g/2,f#/1,g#/2} / {a/0,b/0,c/0,d/0,e/0,h/2,c_1/0,c_2/0,c_3/2,c_4/2,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {a,b,c,d,e,h}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          f#(g(x,y)) -> c_3(f#(x),f#(y))
          f#(h(x,y)) -> c_4(f#(x),f#(y))
* Step 6: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(g(x,y)) -> c_3(f#(x),f#(y))
            f#(h(x,y)) -> c_4(f#(x),f#(y))
        - Signature:
            {f/1,g/2,f#/1,g#/2} / {a/0,b/0,c/0,d/0,e/0,h/2,c_1/0,c_2/0,c_3/2,c_4/2,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {a,b,c,d,e,h}
    + 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_3) = {1,2},
            uargs(c_4) = {1,2}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
              p(a) = [0]                  
              p(b) = [0]                  
              p(c) = [0]                  
              p(d) = [0]                  
              p(e) = [0]                  
              p(f) = [0]                  
              p(g) = [1] x1 + [1] x2 + [5]
              p(h) = [1] x1 + [1] x2 + [1]
             p(f#) = [5] x1 + [0]         
             p(g#) = [1] x2 + [4]         
            p(c_1) = [1]                  
            p(c_2) = [1]                  
            p(c_3) = [1] x1 + [1] x2 + [8]
            p(c_4) = [1] x1 + [1] x2 + [0]
            p(c_5) = [0]                  
          
          Following rules are strictly oriented:
          f#(g(x,y)) = [5] x + [5] y + [25]
                     > [5] x + [5] y + [8] 
                     = c_3(f#(x),f#(y))    
          
          f#(h(x,y)) = [5] x + [5] y + [5] 
                     > [5] x + [5] y + [0] 
                     = c_4(f#(x),f#(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:
            f#(g(x,y)) -> c_3(f#(x),f#(y))
            f#(h(x,y)) -> c_4(f#(x),f#(y))
        - Signature:
            {f/1,g/2,f#/1,g#/2} / {a/0,b/0,c/0,d/0,e/0,h/2,c_1/0,c_2/0,c_3/2,c_4/2,c_5/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {a,b,c,d,e,h}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^1))