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: PredecessorEstimationCP 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:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: f#(g(x,y)) -> c_3(f#(x),f#(y))
          2: f#(h(x,y)) -> c_4(f#(x),f#(y))
          
        The strictly oriented rules are moved into the weak component.
** Step 6.a:1: NaturalMI 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:
        NaturalMI {miDimension = 1, miDegree = 1, 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:
        The following argument positions are considered usable:
          uargs(c_3) = {1,2},
          uargs(c_4) = {1,2}
        
        Following symbols are considered usable:
          {f#,g#}
        TcT has computed the following interpretation:
            p(a) = [1]                  
            p(b) = [2]                  
            p(c) = [2]                  
            p(d) = [0]                  
            p(e) = [2]                  
            p(f) = [4] x1 + [1]         
            p(g) = [2] x1 + [3] x2 + [2]
            p(h) = [1] x1 + [1] x2 + [2]
           p(f#) = [8] x1 + [3]         
           p(g#) = [4] x1 + [1]         
          p(c_1) = [0]                  
          p(c_2) = [1]                  
          p(c_3) = [2] x1 + [3] x2 + [1]
          p(c_4) = [1] x1 + [1] x2 + [3]
          p(c_5) = [1]                  
        
        Following rules are strictly oriented:
        f#(g(x,y)) = [16] x + [24] y + [19]
                   > [16] x + [24] y + [16]
                   = c_3(f#(x),f#(y))      
        
        f#(h(x,y)) = [8] x + [8] y + [19]  
                   > [8] x + [8] y + [9]   
                   = c_4(f#(x),f#(y))      
        
        
        Following rules are (at-least) weakly oriented:
        
** Step 6.a:2: Assumption 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:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

** Step 6.b:1: RemoveWeakSuffixes 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:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:f#(g(x,y)) -> c_3(f#(x),f#(y))
             -->_2 f#(h(x,y)) -> c_4(f#(x),f#(y)):2
             -->_1 f#(h(x,y)) -> c_4(f#(x),f#(y)):2
             -->_2 f#(g(x,y)) -> c_3(f#(x),f#(y)):1
             -->_1 f#(g(x,y)) -> c_3(f#(x),f#(y)):1
          
          2:W:f#(h(x,y)) -> c_4(f#(x),f#(y))
             -->_2 f#(h(x,y)) -> c_4(f#(x),f#(y)):2
             -->_1 f#(h(x,y)) -> c_4(f#(x),f#(y)):2
             -->_2 f#(g(x,y)) -> c_3(f#(x),f#(y)):1
             -->_1 f#(g(x,y)) -> c_3(f#(x),f#(y)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: f#(g(x,y)) -> c_3(f#(x),f#(y))
          2: f#(h(x,y)) -> c_4(f#(x),f#(y))
** Step 6.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        
        - 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))