* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(x,0()) -> x
            f(0(),y) -> y
            f(1(),g(x,y)) -> x
            f(2(),g(x,y)) -> y
            f(f(x,y),z) -> f(x,f(y,z))
            f(g(x,y),z) -> g(f(x,z),f(y,z))
            f(i(x),y) -> i(x)
        - Signature:
            {f/2} / {0/0,1/0,2/0,g/2,i/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f} and constructors {0,1,2,g,i}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            f(x,0()) -> x
            f(0(),y) -> y
            f(1(),g(x,y)) -> x
            f(2(),g(x,y)) -> y
            f(f(x,y),z) -> f(x,f(y,z))
            f(g(x,y),z) -> g(f(x,z),f(y,z))
            f(i(x),y) -> i(x)
        - Signature:
            {f/2} / {0/0,1/0,2/0,g/2,i/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f} and constructors {0,1,2,g,i}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          f(x,z){x -> g(x,y)} =
            f(g(x,y),z) ->^+ g(f(x,z),f(y,z))
              = C[f(x,z) = f(x,z){}]

** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(x,0()) -> x
            f(0(),y) -> y
            f(1(),g(x,y)) -> x
            f(2(),g(x,y)) -> y
            f(f(x,y),z) -> f(x,f(y,z))
            f(g(x,y),z) -> g(f(x,z),f(y,z))
            f(i(x),y) -> i(x)
        - Signature:
            {f/2} / {0/0,1/0,2/0,g/2,i/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f} and constructors {0,1,2,g,i}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          f#(x,0()) -> c_1()
          f#(0(),y) -> c_2()
          f#(1(),g(x,y)) -> c_3()
          f#(2(),g(x,y)) -> c_4()
          f#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z))
          f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z))
          f#(i(x),y) -> c_7()
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(x,0()) -> c_1()
            f#(0(),y) -> c_2()
            f#(1(),g(x,y)) -> c_3()
            f#(2(),g(x,y)) -> c_4()
            f#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z))
            f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z))
            f#(i(x),y) -> c_7()
        - Weak TRS:
            f(x,0()) -> x
            f(0(),y) -> y
            f(1(),g(x,y)) -> x
            f(2(),g(x,y)) -> y
            f(f(x,y),z) -> f(x,f(y,z))
            f(g(x,y),z) -> g(f(x,z),f(y,z))
            f(i(x),y) -> i(x)
        - Signature:
            {f/2,f#/2} / {0/0,1/0,2/0,g/2,i/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/2,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {0,1,2,g,i}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2,3,4,7}
        by application of
          Pre({1,2,3,4,7}) = {5,6}.
        Here rules are labelled as follows:
          1: f#(x,0()) -> c_1()
          2: f#(0(),y) -> c_2()
          3: f#(1(),g(x,y)) -> c_3()
          4: f#(2(),g(x,y)) -> c_4()
          5: f#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z))
          6: f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z))
          7: f#(i(x),y) -> c_7()
** Step 1.b:3: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z))
            f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z))
        - Weak DPs:
            f#(x,0()) -> c_1()
            f#(0(),y) -> c_2()
            f#(1(),g(x,y)) -> c_3()
            f#(2(),g(x,y)) -> c_4()
            f#(i(x),y) -> c_7()
        - Weak TRS:
            f(x,0()) -> x
            f(0(),y) -> y
            f(1(),g(x,y)) -> x
            f(2(),g(x,y)) -> y
            f(f(x,y),z) -> f(x,f(y,z))
            f(g(x,y),z) -> g(f(x,z),f(y,z))
            f(i(x),y) -> i(x)
        - Signature:
            {f/2,f#/2} / {0/0,1/0,2/0,g/2,i/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/2,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {0,1,2,g,i}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:f#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z))
             -->_2 f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z)):2
             -->_1 f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z)):2
             -->_2 f#(i(x),y) -> c_7():7
             -->_1 f#(i(x),y) -> c_7():7
             -->_2 f#(2(),g(x,y)) -> c_4():6
             -->_1 f#(2(),g(x,y)) -> c_4():6
             -->_2 f#(1(),g(x,y)) -> c_3():5
             -->_1 f#(1(),g(x,y)) -> c_3():5
             -->_2 f#(0(),y) -> c_2():4
             -->_1 f#(0(),y) -> c_2():4
             -->_2 f#(x,0()) -> c_1():3
             -->_1 f#(x,0()) -> c_1():3
             -->_2 f#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z)):1
             -->_1 f#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z)):1
          
          2:S:f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z))
             -->_2 f#(i(x),y) -> c_7():7
             -->_1 f#(i(x),y) -> c_7():7
             -->_2 f#(2(),g(x,y)) -> c_4():6
             -->_1 f#(2(),g(x,y)) -> c_4():6
             -->_2 f#(1(),g(x,y)) -> c_3():5
             -->_1 f#(1(),g(x,y)) -> c_3():5
             -->_2 f#(0(),y) -> c_2():4
             -->_1 f#(0(),y) -> c_2():4
             -->_2 f#(x,0()) -> c_1():3
             -->_1 f#(x,0()) -> c_1():3
             -->_2 f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z)):2
             -->_1 f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z)):2
             -->_2 f#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z)):1
             -->_1 f#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z)):1
          
          3:W:f#(x,0()) -> c_1()
             
          
          4:W:f#(0(),y) -> c_2()
             
          
          5:W:f#(1(),g(x,y)) -> c_3()
             
          
          6:W:f#(2(),g(x,y)) -> c_4()
             
          
          7:W:f#(i(x),y) -> c_7()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: f#(x,0()) -> c_1()
          4: f#(0(),y) -> c_2()
          5: f#(1(),g(x,y)) -> c_3()
          6: f#(2(),g(x,y)) -> c_4()
          7: f#(i(x),y) -> c_7()
** Step 1.b:4: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z))
            f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z))
        - Weak TRS:
            f(x,0()) -> x
            f(0(),y) -> y
            f(1(),g(x,y)) -> x
            f(2(),g(x,y)) -> y
            f(f(x,y),z) -> f(x,f(y,z))
            f(g(x,y),z) -> g(f(x,z),f(y,z))
            f(i(x),y) -> i(x)
        - Signature:
            {f/2,f#/2} / {0/0,1/0,2/0,g/2,i/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/2,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {0,1,2,g,i}
    + 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#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z))
          2: f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z))
          
        The strictly oriented rules are moved into the weak component.
*** Step 1.b:4.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z))
            f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z))
        - Weak TRS:
            f(x,0()) -> x
            f(0(),y) -> y
            f(1(),g(x,y)) -> x
            f(2(),g(x,y)) -> y
            f(f(x,y),z) -> f(x,f(y,z))
            f(g(x,y),z) -> g(f(x,z),f(y,z))
            f(i(x),y) -> i(x)
        - Signature:
            {f/2,f#/2} / {0/0,1/0,2/0,g/2,i/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/2,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {0,1,2,g,i}
    + 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_5) = {1,2},
          uargs(c_6) = {1,2}
        
        Following symbols are considered usable:
          {f#}
        TcT has computed the following interpretation:
            p(0) = [4]                  
            p(1) = [1]                  
            p(2) = [3]                  
            p(f) = [4] x1 + [1] x2 + [6]
            p(g) = [1] x1 + [1] x2 + [4]
            p(i) = [2]                  
           p(f#) = [4] x1 + [3]         
          p(c_1) = [0]                  
          p(c_2) = [1]                  
          p(c_3) = [0]                  
          p(c_4) = [0]                  
          p(c_5) = [4] x1 + [1] x2 + [7]
          p(c_6) = [1] x1 + [1] x2 + [4]
          p(c_7) = [2]                  
        
        Following rules are strictly oriented:
        f#(f(x,y),z) = [16] x + [4] y + [27]    
                     > [16] x + [4] y + [22]    
                     = c_5(f#(x,f(y,z)),f#(y,z))
        
        f#(g(x,y),z) = [4] x + [4] y + [19]     
                     > [4] x + [4] y + [10]     
                     = c_6(f#(x,z),f#(y,z))     
        
        
        Following rules are (at-least) weakly oriented:
        
*** Step 1.b:4.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z))
            f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z))
        - Weak TRS:
            f(x,0()) -> x
            f(0(),y) -> y
            f(1(),g(x,y)) -> x
            f(2(),g(x,y)) -> y
            f(f(x,y),z) -> f(x,f(y,z))
            f(g(x,y),z) -> g(f(x,z),f(y,z))
            f(i(x),y) -> i(x)
        - Signature:
            {f/2,f#/2} / {0/0,1/0,2/0,g/2,i/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/2,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {0,1,2,g,i}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

*** Step 1.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z))
            f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z))
        - Weak TRS:
            f(x,0()) -> x
            f(0(),y) -> y
            f(1(),g(x,y)) -> x
            f(2(),g(x,y)) -> y
            f(f(x,y),z) -> f(x,f(y,z))
            f(g(x,y),z) -> g(f(x,z),f(y,z))
            f(i(x),y) -> i(x)
        - Signature:
            {f/2,f#/2} / {0/0,1/0,2/0,g/2,i/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/2,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {0,1,2,g,i}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:f#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z))
             -->_2 f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z)):2
             -->_1 f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z)):2
             -->_2 f#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z)):1
             -->_1 f#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z)):1
          
          2:W:f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z))
             -->_2 f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z)):2
             -->_1 f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z)):2
             -->_2 f#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z)):1
             -->_1 f#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: f#(f(x,y),z) -> c_5(f#(x,f(y,z)),f#(y,z))
          2: f#(g(x,y),z) -> c_6(f#(x,z),f#(y,z))
*** Step 1.b:4.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(x,0()) -> x
            f(0(),y) -> y
            f(1(),g(x,y)) -> x
            f(2(),g(x,y)) -> y
            f(f(x,y),z) -> f(x,f(y,z))
            f(g(x,y),z) -> g(f(x,z),f(y,z))
            f(i(x),y) -> i(x)
        - Signature:
            {f/2,f#/2} / {0/0,1/0,2/0,g/2,i/1,c_1/0,c_2/0,c_3/0,c_4/0,c_5/2,c_6/2,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#} and constructors {0,1,2,g,i}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),O(n^1))