WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(0()) -> true()
            f(1()) -> false()
            f(s(x)) -> f(x)
            g(x,c(y)) -> g(x,g(s(c(y)),y))
            g(s(x),s(y)) -> if(f(x),s(x),s(y))
            if(false(),x,y) -> y
            if(true(),x,y) -> x
        - Signature:
            {f/1,g/2,if/3} / {0/0,1/0,c/1,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g,if} and constructors {0,1,c,false,s,true}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            f(0()) -> true()
            f(1()) -> false()
            f(s(x)) -> f(x)
            g(x,c(y)) -> g(x,g(s(c(y)),y))
            g(s(x),s(y)) -> if(f(x),s(x),s(y))
            if(false(),x,y) -> y
            if(true(),x,y) -> x
        - Signature:
            {f/1,g/2,if/3} / {0/0,1/0,c/1,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g,if} and constructors {0,1,c,false,s,true}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          f(x){x -> s(x)} =
            f(s(x)) ->^+ f(x)
              = C[f(x) = f(x){}]

** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(0()) -> true()
            f(1()) -> false()
            f(s(x)) -> f(x)
            g(x,c(y)) -> g(x,g(s(c(y)),y))
            g(s(x),s(y)) -> if(f(x),s(x),s(y))
            if(false(),x,y) -> y
            if(true(),x,y) -> x
        - Signature:
            {f/1,g/2,if/3} / {0/0,1/0,c/1,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g,if} and constructors {0,1,c,false,s,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          f#(0()) -> c_1()
          f#(1()) -> c_2()
          f#(s(x)) -> c_3(f#(x))
          g#(x,c(y)) -> c_4(g#(x,g(s(c(y)),y)),g#(s(c(y)),y))
          g#(s(x),s(y)) -> c_5(if#(f(x),s(x),s(y)),f#(x))
          if#(false(),x,y) -> c_6()
          if#(true(),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#(0()) -> c_1()
            f#(1()) -> c_2()
            f#(s(x)) -> c_3(f#(x))
            g#(x,c(y)) -> c_4(g#(x,g(s(c(y)),y)),g#(s(c(y)),y))
            g#(s(x),s(y)) -> c_5(if#(f(x),s(x),s(y)),f#(x))
            if#(false(),x,y) -> c_6()
            if#(true(),x,y) -> c_7()
        - Weak TRS:
            f(0()) -> true()
            f(1()) -> false()
            f(s(x)) -> f(x)
            g(x,c(y)) -> g(x,g(s(c(y)),y))
            g(s(x),s(y)) -> if(f(x),s(x),s(y))
            if(false(),x,y) -> y
            if(true(),x,y) -> x
        - Signature:
            {f/1,g/2,if/3,f#/1,g#/2,if#/3} / {0/0,1/0,c/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/2,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,if#} and constructors {0,1,c,false,s,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2,6,7}
        by application of
          Pre({1,2,6,7}) = {3,5}.
        Here rules are labelled as follows:
          1: f#(0()) -> c_1()
          2: f#(1()) -> c_2()
          3: f#(s(x)) -> c_3(f#(x))
          4: g#(x,c(y)) -> c_4(g#(x,g(s(c(y)),y)),g#(s(c(y)),y))
          5: g#(s(x),s(y)) -> c_5(if#(f(x),s(x),s(y)),f#(x))
          6: if#(false(),x,y) -> c_6()
          7: if#(true(),x,y) -> c_7()
** Step 1.b:3: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(s(x)) -> c_3(f#(x))
            g#(x,c(y)) -> c_4(g#(x,g(s(c(y)),y)),g#(s(c(y)),y))
            g#(s(x),s(y)) -> c_5(if#(f(x),s(x),s(y)),f#(x))
        - Weak DPs:
            f#(0()) -> c_1()
            f#(1()) -> c_2()
            if#(false(),x,y) -> c_6()
            if#(true(),x,y) -> c_7()
        - Weak TRS:
            f(0()) -> true()
            f(1()) -> false()
            f(s(x)) -> f(x)
            g(x,c(y)) -> g(x,g(s(c(y)),y))
            g(s(x),s(y)) -> if(f(x),s(x),s(y))
            if(false(),x,y) -> y
            if(true(),x,y) -> x
        - Signature:
            {f/1,g/2,if/3,f#/1,g#/2,if#/3} / {0/0,1/0,c/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/2,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,if#} and constructors {0,1,c,false,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:f#(s(x)) -> c_3(f#(x))
             -->_1 f#(1()) -> c_2():5
             -->_1 f#(0()) -> c_1():4
             -->_1 f#(s(x)) -> c_3(f#(x)):1
          
          2:S:g#(x,c(y)) -> c_4(g#(x,g(s(c(y)),y)),g#(s(c(y)),y))
             -->_2 g#(s(x),s(y)) -> c_5(if#(f(x),s(x),s(y)),f#(x)):3
             -->_1 g#(s(x),s(y)) -> c_5(if#(f(x),s(x),s(y)),f#(x)):3
             -->_2 g#(x,c(y)) -> c_4(g#(x,g(s(c(y)),y)),g#(s(c(y)),y)):2
             -->_1 g#(x,c(y)) -> c_4(g#(x,g(s(c(y)),y)),g#(s(c(y)),y)):2
          
          3:S:g#(s(x),s(y)) -> c_5(if#(f(x),s(x),s(y)),f#(x))
             -->_1 if#(true(),x,y) -> c_7():7
             -->_1 if#(false(),x,y) -> c_6():6
             -->_2 f#(1()) -> c_2():5
             -->_2 f#(0()) -> c_1():4
             -->_2 f#(s(x)) -> c_3(f#(x)):1
          
          4:W:f#(0()) -> c_1()
             
          
          5:W:f#(1()) -> c_2()
             
          
          6:W:if#(false(),x,y) -> c_6()
             
          
          7:W:if#(true(),x,y) -> c_7()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          6: if#(false(),x,y) -> c_6()
          7: if#(true(),x,y) -> c_7()
          4: f#(0()) -> c_1()
          5: f#(1()) -> c_2()
** Step 1.b:4: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(s(x)) -> c_3(f#(x))
            g#(x,c(y)) -> c_4(g#(x,g(s(c(y)),y)),g#(s(c(y)),y))
            g#(s(x),s(y)) -> c_5(if#(f(x),s(x),s(y)),f#(x))
        - Weak TRS:
            f(0()) -> true()
            f(1()) -> false()
            f(s(x)) -> f(x)
            g(x,c(y)) -> g(x,g(s(c(y)),y))
            g(s(x),s(y)) -> if(f(x),s(x),s(y))
            if(false(),x,y) -> y
            if(true(),x,y) -> x
        - Signature:
            {f/1,g/2,if/3,f#/1,g#/2,if#/3} / {0/0,1/0,c/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/2,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,if#} and constructors {0,1,c,false,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:f#(s(x)) -> c_3(f#(x))
             -->_1 f#(s(x)) -> c_3(f#(x)):1
          
          2:S:g#(x,c(y)) -> c_4(g#(x,g(s(c(y)),y)),g#(s(c(y)),y))
             -->_2 g#(s(x),s(y)) -> c_5(if#(f(x),s(x),s(y)),f#(x)):3
             -->_1 g#(s(x),s(y)) -> c_5(if#(f(x),s(x),s(y)),f#(x)):3
             -->_2 g#(x,c(y)) -> c_4(g#(x,g(s(c(y)),y)),g#(s(c(y)),y)):2
             -->_1 g#(x,c(y)) -> c_4(g#(x,g(s(c(y)),y)),g#(s(c(y)),y)):2
          
          3:S:g#(s(x),s(y)) -> c_5(if#(f(x),s(x),s(y)),f#(x))
             -->_2 f#(s(x)) -> c_3(f#(x)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          g#(s(x),s(y)) -> c_5(f#(x))
** Step 1.b:5: Decompose WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(s(x)) -> c_3(f#(x))
            g#(x,c(y)) -> c_4(g#(x,g(s(c(y)),y)),g#(s(c(y)),y))
            g#(s(x),s(y)) -> c_5(f#(x))
        - Weak TRS:
            f(0()) -> true()
            f(1()) -> false()
            f(s(x)) -> f(x)
            g(x,c(y)) -> g(x,g(s(c(y)),y))
            g(s(x),s(y)) -> if(f(x),s(x),s(y))
            if(false(),x,y) -> y
            if(true(),x,y) -> x
        - Signature:
            {f/1,g/2,if/3,f#/1,g#/2,if#/3} / {0/0,1/0,c/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/1,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,if#} and constructors {0,1,c,false,s,true}
    + Applied Processor:
        Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
    + Details:
        We analyse the complexity of following sub-problems (R) and (S).
        Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
        
        Problem (R)
          - Strict DPs:
              f#(s(x)) -> c_3(f#(x))
          - Weak DPs:
              g#(x,c(y)) -> c_4(g#(x,g(s(c(y)),y)),g#(s(c(y)),y))
              g#(s(x),s(y)) -> c_5(f#(x))
          - Weak TRS:
              f(0()) -> true()
              f(1()) -> false()
              f(s(x)) -> f(x)
              g(x,c(y)) -> g(x,g(s(c(y)),y))
              g(s(x),s(y)) -> if(f(x),s(x),s(y))
              if(false(),x,y) -> y
              if(true(),x,y) -> x
          - Signature:
              {f/1,g/2,if/3,f#/1,g#/2,if#/3} / {0/0,1/0,c/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/1,c_6/0
              ,c_7/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {f#,g#,if#} and constructors {0,1,c,false,s,true}
        
        Problem (S)
          - Strict DPs:
              g#(x,c(y)) -> c_4(g#(x,g(s(c(y)),y)),g#(s(c(y)),y))
              g#(s(x),s(y)) -> c_5(f#(x))
          - Weak DPs:
              f#(s(x)) -> c_3(f#(x))
          - Weak TRS:
              f(0()) -> true()
              f(1()) -> false()
              f(s(x)) -> f(x)
              g(x,c(y)) -> g(x,g(s(c(y)),y))
              g(s(x),s(y)) -> if(f(x),s(x),s(y))
              if(false(),x,y) -> y
              if(true(),x,y) -> x
          - Signature:
              {f/1,g/2,if/3,f#/1,g#/2,if#/3} / {0/0,1/0,c/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/1,c_6/0
              ,c_7/0}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {f#,g#,if#} and constructors {0,1,c,false,s,true}
*** Step 1.b:5.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(s(x)) -> c_3(f#(x))
        - Weak DPs:
            g#(x,c(y)) -> c_4(g#(x,g(s(c(y)),y)),g#(s(c(y)),y))
            g#(s(x),s(y)) -> c_5(f#(x))
        - Weak TRS:
            f(0()) -> true()
            f(1()) -> false()
            f(s(x)) -> f(x)
            g(x,c(y)) -> g(x,g(s(c(y)),y))
            g(s(x),s(y)) -> if(f(x),s(x),s(y))
            if(false(),x,y) -> y
            if(true(),x,y) -> x
        - Signature:
            {f/1,g/2,if/3,f#/1,g#/2,if#/3} / {0/0,1/0,c/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/1,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,if#} and constructors {0,1,c,false,s,true}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: f#(s(x)) -> c_3(f#(x))
          
        The strictly oriented rules are moved into the weak component.
**** Step 1.b:5.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(s(x)) -> c_3(f#(x))
        - Weak DPs:
            g#(x,c(y)) -> c_4(g#(x,g(s(c(y)),y)),g#(s(c(y)),y))
            g#(s(x),s(y)) -> c_5(f#(x))
        - Weak TRS:
            f(0()) -> true()
            f(1()) -> false()
            f(s(x)) -> f(x)
            g(x,c(y)) -> g(x,g(s(c(y)),y))
            g(s(x),s(y)) -> if(f(x),s(x),s(y))
            if(false(),x,y) -> y
            if(true(),x,y) -> x
        - Signature:
            {f/1,g/2,if/3,f#/1,g#/2,if#/3} / {0/0,1/0,c/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/1,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,if#} and constructors {0,1,c,false,s,true}
    + Applied Processor:
        NaturalMI {miDimension = 2, 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 (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        The following argument positions are considered usable:
          uargs(c_3) = {1},
          uargs(c_4) = {1,2},
          uargs(c_5) = {1}
        
        Following symbols are considered usable:
          {f#,g#,if#}
        TcT has computed the following interpretation:
              p(0) = [1]                                 
                     [3]                                 
              p(1) = [3]                                 
                     [3]                                 
              p(c) = [0]                                 
                     [0]                                 
              p(f) = [2 0] x1 + [4]                      
                     [1 3]      [0]                      
          p(false) = [0]                                 
                     [2]                                 
              p(g) = [0 2] x1 + [7]                      
                     [0 0]      [0]                      
             p(if) = [0 0] x1 + [2 2] x3 + [2]           
                     [2 4]      [0 0]      [0]           
              p(s) = [0 4] x1 + [0]                      
                     [0 1]      [1]                      
           p(true) = [0]                                 
                     [0]                                 
             p(f#) = [0 4] x1 + [0]                      
                     [0 0]      [2]                      
             p(g#) = [2 0] x1 + [0]                      
                     [2 1]      [7]                      
            p(if#) = [0 0] x1 + [0 0] x2 + [0 0] x3 + [1]
                     [0 1]      [2 4]      [0 1]      [0]
            p(c_1) = [0]                                 
                     [4]                                 
            p(c_2) = [0]                                 
                     [1]                                 
            p(c_3) = [1 0] x1 + [0]                      
                     [0 0]      [0]                      
            p(c_4) = [1 0] x1 + [4 0] x2 + [0]           
                     [0 1]      [2 0]      [0]           
            p(c_5) = [2 0] x1 + [0]                      
                     [0 4]      [0]                      
            p(c_6) = [0]                                 
                     [0]                                 
            p(c_7) = [1]                                 
                     [1]                                 
        
        Following rules are strictly oriented:
        f#(s(x)) = [0 4] x + [4]
                   [0 0]     [2]
                 > [0 4] x + [0]
                   [0 0]     [0]
                 = c_3(f#(x))   
        
        
        Following rules are (at-least) weakly oriented:
           g#(x,c(y)) =  [2 0] x + [0]                        
                         [2 1]     [7]                        
                      >= [2 0] x + [0]                        
                         [2 1]     [7]                        
                      =  c_4(g#(x,g(s(c(y)),y)),g#(s(c(y)),y))
        
        g#(s(x),s(y)) =  [0 8] x + [0]                        
                         [0 9]     [8]                        
                      >= [0 8] x + [0]                        
                         [0 0]     [8]                        
                      =  c_5(f#(x))                           
        
**** Step 1.b:5.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f#(s(x)) -> c_3(f#(x))
            g#(x,c(y)) -> c_4(g#(x,g(s(c(y)),y)),g#(s(c(y)),y))
            g#(s(x),s(y)) -> c_5(f#(x))
        - Weak TRS:
            f(0()) -> true()
            f(1()) -> false()
            f(s(x)) -> f(x)
            g(x,c(y)) -> g(x,g(s(c(y)),y))
            g(s(x),s(y)) -> if(f(x),s(x),s(y))
            if(false(),x,y) -> y
            if(true(),x,y) -> x
        - Signature:
            {f/1,g/2,if/3,f#/1,g#/2,if#/3} / {0/0,1/0,c/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/1,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,if#} and constructors {0,1,c,false,s,true}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

**** Step 1.b:5.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f#(s(x)) -> c_3(f#(x))
            g#(x,c(y)) -> c_4(g#(x,g(s(c(y)),y)),g#(s(c(y)),y))
            g#(s(x),s(y)) -> c_5(f#(x))
        - Weak TRS:
            f(0()) -> true()
            f(1()) -> false()
            f(s(x)) -> f(x)
            g(x,c(y)) -> g(x,g(s(c(y)),y))
            g(s(x),s(y)) -> if(f(x),s(x),s(y))
            if(false(),x,y) -> y
            if(true(),x,y) -> x
        - Signature:
            {f/1,g/2,if/3,f#/1,g#/2,if#/3} / {0/0,1/0,c/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/1,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,if#} and constructors {0,1,c,false,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:f#(s(x)) -> c_3(f#(x))
             -->_1 f#(s(x)) -> c_3(f#(x)):1
          
          2:W:g#(x,c(y)) -> c_4(g#(x,g(s(c(y)),y)),g#(s(c(y)),y))
             -->_2 g#(s(x),s(y)) -> c_5(f#(x)):3
             -->_1 g#(s(x),s(y)) -> c_5(f#(x)):3
             -->_2 g#(x,c(y)) -> c_4(g#(x,g(s(c(y)),y)),g#(s(c(y)),y)):2
             -->_1 g#(x,c(y)) -> c_4(g#(x,g(s(c(y)),y)),g#(s(c(y)),y)):2
          
          3:W:g#(s(x),s(y)) -> c_5(f#(x))
             -->_1 f#(s(x)) -> c_3(f#(x)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: g#(x,c(y)) -> c_4(g#(x,g(s(c(y)),y)),g#(s(c(y)),y))
          3: g#(s(x),s(y)) -> c_5(f#(x))
          1: f#(s(x)) -> c_3(f#(x))
**** Step 1.b:5.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(0()) -> true()
            f(1()) -> false()
            f(s(x)) -> f(x)
            g(x,c(y)) -> g(x,g(s(c(y)),y))
            g(s(x),s(y)) -> if(f(x),s(x),s(y))
            if(false(),x,y) -> y
            if(true(),x,y) -> x
        - Signature:
            {f/1,g/2,if/3,f#/1,g#/2,if#/3} / {0/0,1/0,c/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/1,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,if#} and constructors {0,1,c,false,s,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

*** Step 1.b:5.b:1: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g#(x,c(y)) -> c_4(g#(x,g(s(c(y)),y)),g#(s(c(y)),y))
            g#(s(x),s(y)) -> c_5(f#(x))
        - Weak DPs:
            f#(s(x)) -> c_3(f#(x))
        - Weak TRS:
            f(0()) -> true()
            f(1()) -> false()
            f(s(x)) -> f(x)
            g(x,c(y)) -> g(x,g(s(c(y)),y))
            g(s(x),s(y)) -> if(f(x),s(x),s(y))
            if(false(),x,y) -> y
            if(true(),x,y) -> x
        - Signature:
            {f/1,g/2,if/3,f#/1,g#/2,if#/3} / {0/0,1/0,c/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/1,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,if#} and constructors {0,1,c,false,s,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2}
        by application of
          Pre({2}) = {1}.
        Here rules are labelled as follows:
          1: g#(x,c(y)) -> c_4(g#(x,g(s(c(y)),y)),g#(s(c(y)),y))
          2: g#(s(x),s(y)) -> c_5(f#(x))
          3: f#(s(x)) -> c_3(f#(x))
*** Step 1.b:5.b:2: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g#(x,c(y)) -> c_4(g#(x,g(s(c(y)),y)),g#(s(c(y)),y))
        - Weak DPs:
            f#(s(x)) -> c_3(f#(x))
            g#(s(x),s(y)) -> c_5(f#(x))
        - Weak TRS:
            f(0()) -> true()
            f(1()) -> false()
            f(s(x)) -> f(x)
            g(x,c(y)) -> g(x,g(s(c(y)),y))
            g(s(x),s(y)) -> if(f(x),s(x),s(y))
            if(false(),x,y) -> y
            if(true(),x,y) -> x
        - Signature:
            {f/1,g/2,if/3,f#/1,g#/2,if#/3} / {0/0,1/0,c/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/1,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,if#} and constructors {0,1,c,false,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:g#(x,c(y)) -> c_4(g#(x,g(s(c(y)),y)),g#(s(c(y)),y))
             -->_2 g#(s(x),s(y)) -> c_5(f#(x)):3
             -->_1 g#(s(x),s(y)) -> c_5(f#(x)):3
             -->_2 g#(x,c(y)) -> c_4(g#(x,g(s(c(y)),y)),g#(s(c(y)),y)):1
             -->_1 g#(x,c(y)) -> c_4(g#(x,g(s(c(y)),y)),g#(s(c(y)),y)):1
          
          2:W:f#(s(x)) -> c_3(f#(x))
             -->_1 f#(s(x)) -> c_3(f#(x)):2
          
          3:W:g#(s(x),s(y)) -> c_5(f#(x))
             -->_1 f#(s(x)) -> c_3(f#(x)):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: g#(s(x),s(y)) -> c_5(f#(x))
          2: f#(s(x)) -> c_3(f#(x))
*** Step 1.b:5.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g#(x,c(y)) -> c_4(g#(x,g(s(c(y)),y)),g#(s(c(y)),y))
        - Weak TRS:
            f(0()) -> true()
            f(1()) -> false()
            f(s(x)) -> f(x)
            g(x,c(y)) -> g(x,g(s(c(y)),y))
            g(s(x),s(y)) -> if(f(x),s(x),s(y))
            if(false(),x,y) -> y
            if(true(),x,y) -> x
        - Signature:
            {f/1,g/2,if/3,f#/1,g#/2,if#/3} / {0/0,1/0,c/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/1,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,if#} and constructors {0,1,c,false,s,true}
    + 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: g#(x,c(y)) -> c_4(g#(x,g(s(c(y)),y)),g#(s(c(y)),y))
          
        The strictly oriented rules are moved into the weak component.
**** Step 1.b:5.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            g#(x,c(y)) -> c_4(g#(x,g(s(c(y)),y)),g#(s(c(y)),y))
        - Weak TRS:
            f(0()) -> true()
            f(1()) -> false()
            f(s(x)) -> f(x)
            g(x,c(y)) -> g(x,g(s(c(y)),y))
            g(s(x),s(y)) -> if(f(x),s(x),s(y))
            if(false(),x,y) -> y
            if(true(),x,y) -> x
        - Signature:
            {f/1,g/2,if/3,f#/1,g#/2,if#/3} / {0/0,1/0,c/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/1,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,if#} and constructors {0,1,c,false,s,true}
    + 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_4) = {1,2}
        
        Following symbols are considered usable:
          {g,if,f#,g#,if#}
        TcT has computed the following interpretation:
              p(0) = [0]                  
              p(1) = [4]                  
              p(c) = [1] x1 + [7]         
              p(f) = [0]                  
          p(false) = [0]                  
              p(g) = [0]                  
             p(if) = [2] x2 + [1] x3 + [0]
              p(s) = [0]                  
           p(true) = [1]                  
             p(f#) = [1] x1 + [1]         
             p(g#) = [1] x2 + [0]         
            p(if#) = [1] x1 + [2] x3 + [2]
            p(c_1) = [2]                  
            p(c_2) = [0]                  
            p(c_3) = [1]                  
            p(c_4) = [2] x1 + [1] x2 + [0]
            p(c_5) = [1]                  
            p(c_6) = [1]                  
            p(c_7) = [4]                  
        
        Following rules are strictly oriented:
        g#(x,c(y)) = [1] y + [7]                          
                   > [1] y + [0]                          
                   = c_4(g#(x,g(s(c(y)),y)),g#(s(c(y)),y))
        
        
        Following rules are (at-least) weakly oriented:
              g(x,c(y)) =  [0]                
                        >= [0]                
                        =  g(x,g(s(c(y)),y))  
        
           g(s(x),s(y)) =  [0]                
                        >= [0]                
                        =  if(f(x),s(x),s(y)) 
        
        if(false(),x,y) =  [2] x + [1] y + [0]
                        >= [1] y + [0]        
                        =  y                  
        
         if(true(),x,y) =  [2] x + [1] y + [0]
                        >= [1] x + [0]        
                        =  x                  
        
**** Step 1.b:5.b:3.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            g#(x,c(y)) -> c_4(g#(x,g(s(c(y)),y)),g#(s(c(y)),y))
        - Weak TRS:
            f(0()) -> true()
            f(1()) -> false()
            f(s(x)) -> f(x)
            g(x,c(y)) -> g(x,g(s(c(y)),y))
            g(s(x),s(y)) -> if(f(x),s(x),s(y))
            if(false(),x,y) -> y
            if(true(),x,y) -> x
        - Signature:
            {f/1,g/2,if/3,f#/1,g#/2,if#/3} / {0/0,1/0,c/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/1,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,if#} and constructors {0,1,c,false,s,true}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

**** Step 1.b:5.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            g#(x,c(y)) -> c_4(g#(x,g(s(c(y)),y)),g#(s(c(y)),y))
        - Weak TRS:
            f(0()) -> true()
            f(1()) -> false()
            f(s(x)) -> f(x)
            g(x,c(y)) -> g(x,g(s(c(y)),y))
            g(s(x),s(y)) -> if(f(x),s(x),s(y))
            if(false(),x,y) -> y
            if(true(),x,y) -> x
        - Signature:
            {f/1,g/2,if/3,f#/1,g#/2,if#/3} / {0/0,1/0,c/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/1,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,if#} and constructors {0,1,c,false,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:g#(x,c(y)) -> c_4(g#(x,g(s(c(y)),y)),g#(s(c(y)),y))
             -->_2 g#(x,c(y)) -> c_4(g#(x,g(s(c(y)),y)),g#(s(c(y)),y)):1
             -->_1 g#(x,c(y)) -> c_4(g#(x,g(s(c(y)),y)),g#(s(c(y)),y)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: g#(x,c(y)) -> c_4(g#(x,g(s(c(y)),y)),g#(s(c(y)),y))
**** Step 1.b:5.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(0()) -> true()
            f(1()) -> false()
            f(s(x)) -> f(x)
            g(x,c(y)) -> g(x,g(s(c(y)),y))
            g(s(x),s(y)) -> if(f(x),s(x),s(y))
            if(false(),x,y) -> y
            if(true(),x,y) -> x
        - Signature:
            {f/1,g/2,if/3,f#/1,g#/2,if#/3} / {0/0,1/0,c/1,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/2,c_5/1,c_6/0,c_7/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#,if#} and constructors {0,1,c,false,s,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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