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

** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict TRS:
            +(x,+(y,z)) -> +(+(x,y),z)
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            f(g(f(x))) -> f(h(s(0()),x))
            f(g(h(x,y))) -> f(h(s(x),y))
            f(h(x,h(y,z))) -> f(h(+(x,y),z))
        - Signature:
            {+/2,f/1} / {0/0,g/1,h/2,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+,f} and constructors {0,g,h,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
          +#(x,0()) -> c_2()
          +#(x,s(y)) -> c_3(+#(x,y))
          +#(0(),y) -> c_4()
          +#(s(x),y) -> c_5(+#(x,y))
          f#(g(f(x))) -> c_6(f#(h(s(0()),x)))
          f#(g(h(x,y))) -> c_7(f#(h(s(x),y)))
          f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
            +#(x,0()) -> c_2()
            +#(x,s(y)) -> c_3(+#(x,y))
            +#(0(),y) -> c_4()
            +#(s(x),y) -> c_5(+#(x,y))
            f#(g(f(x))) -> c_6(f#(h(s(0()),x)))
            f#(g(h(x,y))) -> c_7(f#(h(s(x),y)))
            f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
        - Weak TRS:
            +(x,+(y,z)) -> +(+(x,y),z)
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            f(g(f(x))) -> f(h(s(0()),x))
            f(g(h(x,y))) -> f(h(s(x),y))
            f(h(x,h(y,z))) -> f(h(+(x,y),z))
        - Signature:
            {+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,4}
        by application of
          Pre({2,4}) = {1,3,5,8}.
        Here rules are labelled as follows:
          1: +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
          2: +#(x,0()) -> c_2()
          3: +#(x,s(y)) -> c_3(+#(x,y))
          4: +#(0(),y) -> c_4()
          5: +#(s(x),y) -> c_5(+#(x,y))
          6: f#(g(f(x))) -> c_6(f#(h(s(0()),x)))
          7: f#(g(h(x,y))) -> c_7(f#(h(s(x),y)))
          8: f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
** Step 1.b:3: RemoveWeakSuffixes WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
            +#(x,s(y)) -> c_3(+#(x,y))
            +#(s(x),y) -> c_5(+#(x,y))
            f#(g(f(x))) -> c_6(f#(h(s(0()),x)))
            f#(g(h(x,y))) -> c_7(f#(h(s(x),y)))
            f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
        - Weak DPs:
            +#(x,0()) -> c_2()
            +#(0(),y) -> c_4()
        - Weak TRS:
            +(x,+(y,z)) -> +(+(x,y),z)
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            f(g(f(x))) -> f(h(s(0()),x))
            f(g(h(x,y))) -> f(h(s(x),y))
            f(h(x,h(y,z))) -> f(h(+(x,y),z))
        - Signature:
            {+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
             -->_2 +#(s(x),y) -> c_5(+#(x,y)):3
             -->_1 +#(s(x),y) -> c_5(+#(x,y)):3
             -->_2 +#(x,s(y)) -> c_3(+#(x,y)):2
             -->_1 +#(x,s(y)) -> c_3(+#(x,y)):2
             -->_2 +#(0(),y) -> c_4():8
             -->_1 +#(0(),y) -> c_4():8
             -->_2 +#(x,0()) -> c_2():7
             -->_1 +#(x,0()) -> c_2():7
             -->_2 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
             -->_1 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
          
          2:S:+#(x,s(y)) -> c_3(+#(x,y))
             -->_1 +#(s(x),y) -> c_5(+#(x,y)):3
             -->_1 +#(0(),y) -> c_4():8
             -->_1 +#(x,0()) -> c_2():7
             -->_1 +#(x,s(y)) -> c_3(+#(x,y)):2
             -->_1 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
          
          3:S:+#(s(x),y) -> c_5(+#(x,y))
             -->_1 +#(0(),y) -> c_4():8
             -->_1 +#(x,0()) -> c_2():7
             -->_1 +#(s(x),y) -> c_5(+#(x,y)):3
             -->_1 +#(x,s(y)) -> c_3(+#(x,y)):2
             -->_1 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
          
          4:S:f#(g(f(x))) -> c_6(f#(h(s(0()),x)))
             -->_1 f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y)):6
          
          5:S:f#(g(h(x,y))) -> c_7(f#(h(s(x),y)))
             -->_1 f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y)):6
          
          6:S:f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
             -->_2 +#(0(),y) -> c_4():8
             -->_2 +#(x,0()) -> c_2():7
             -->_1 f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y)):6
             -->_2 +#(s(x),y) -> c_5(+#(x,y)):3
             -->_2 +#(x,s(y)) -> c_3(+#(x,y)):2
             -->_2 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
          
          7:W:+#(x,0()) -> c_2()
             
          
          8:W:+#(0(),y) -> c_4()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          7: +#(x,0()) -> c_2()
          8: +#(0(),y) -> c_4()
** Step 1.b:4: RemoveHeads WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
            +#(x,s(y)) -> c_3(+#(x,y))
            +#(s(x),y) -> c_5(+#(x,y))
            f#(g(f(x))) -> c_6(f#(h(s(0()),x)))
            f#(g(h(x,y))) -> c_7(f#(h(s(x),y)))
            f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
        - Weak TRS:
            +(x,+(y,z)) -> +(+(x,y),z)
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            f(g(f(x))) -> f(h(s(0()),x))
            f(g(h(x,y))) -> f(h(s(x),y))
            f(h(x,h(y,z))) -> f(h(+(x,y),z))
        - Signature:
            {+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,s}
    + Applied Processor:
        RemoveHeads
    + Details:
        Consider the dependency graph
        
        1:S:+#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
           -->_2 +#(s(x),y) -> c_5(+#(x,y)):3
           -->_1 +#(s(x),y) -> c_5(+#(x,y)):3
           -->_2 +#(x,s(y)) -> c_3(+#(x,y)):2
           -->_1 +#(x,s(y)) -> c_3(+#(x,y)):2
           -->_2 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
           -->_1 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
        
        2:S:+#(x,s(y)) -> c_3(+#(x,y))
           -->_1 +#(s(x),y) -> c_5(+#(x,y)):3
           -->_1 +#(x,s(y)) -> c_3(+#(x,y)):2
           -->_1 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
        
        3:S:+#(s(x),y) -> c_5(+#(x,y))
           -->_1 +#(s(x),y) -> c_5(+#(x,y)):3
           -->_1 +#(x,s(y)) -> c_3(+#(x,y)):2
           -->_1 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
        
        4:S:f#(g(f(x))) -> c_6(f#(h(s(0()),x)))
           -->_1 f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y)):6
        
        5:S:f#(g(h(x,y))) -> c_7(f#(h(s(x),y)))
           -->_1 f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y)):6
        
        6:S:f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
           -->_1 f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y)):6
           -->_2 +#(s(x),y) -> c_5(+#(x,y)):3
           -->_2 +#(x,s(y)) -> c_3(+#(x,y)):2
           -->_2 +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y)):1
        
        
        Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts).
        
        [(4,f#(g(f(x))) -> c_6(f#(h(s(0()),x)))),(5,f#(g(h(x,y))) -> c_7(f#(h(s(x),y))))]
** Step 1.b:5: UsableRules WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
            +#(x,s(y)) -> c_3(+#(x,y))
            +#(s(x),y) -> c_5(+#(x,y))
            f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
        - Weak TRS:
            +(x,+(y,z)) -> +(+(x,y),z)
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
            f(g(f(x))) -> f(h(s(0()),x))
            f(g(h(x,y))) -> f(h(s(x),y))
            f(h(x,h(y,z))) -> f(h(+(x,y),z))
        - Signature:
            {+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          +(x,+(y,z)) -> +(+(x,y),z)
          +(x,0()) -> x
          +(x,s(y)) -> s(+(x,y))
          +(0(),y) -> y
          +(s(x),y) -> s(+(x,y))
          +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
          +#(x,s(y)) -> c_3(+#(x,y))
          +#(s(x),y) -> c_5(+#(x,y))
          f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
** Step 1.b:6: DecomposeDG WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
            +#(x,s(y)) -> c_3(+#(x,y))
            +#(s(x),y) -> c_5(+#(x,y))
            f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
        - Weak TRS:
            +(x,+(y,z)) -> +(+(x,y),z)
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
        - Signature:
            {+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,s}
    + Applied Processor:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
        and a lower component
          +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
          +#(x,s(y)) -> c_3(+#(x,y))
          +#(s(x),y) -> c_5(+#(x,y))
        Further, following extension rules are added to the lower component.
          f#(h(x,h(y,z))) -> +#(x,y)
          f#(h(x,h(y,z))) -> f#(h(+(x,y),z))
*** Step 1.b:6.a:1: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
        - Weak TRS:
            +(x,+(y,z)) -> +(+(x,y),z)
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
        - Signature:
            {+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y))
             -->_1 f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)),+#(x,y)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)))
*** Step 1.b:6.a:2: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)))
        - Weak TRS:
            +(x,+(y,z)) -> +(+(x,y),z)
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
        - Signature:
            {+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,s}
    + 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(+) = {1},
            uargs(h) = {1},
            uargs(s) = {1},
            uargs(f#) = {1},
            uargs(c_8) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
              p(+) = [1] x1 + [1] x2 + [0]
              p(0) = [0]                  
              p(f) = [0]                  
              p(g) = [1] x1 + [0]         
              p(h) = [1] x1 + [1] x2 + [6]
              p(s) = [1] x1 + [13]        
             p(+#) = [0]                  
             p(f#) = [1] x1 + [4]         
            p(c_1) = [0]                  
            p(c_2) = [0]                  
            p(c_3) = [0]                  
            p(c_4) = [0]                  
            p(c_5) = [0]                  
            p(c_6) = [0]                  
            p(c_7) = [0]                  
            p(c_8) = [1] x1 + [2]         
          
          Following rules are strictly oriented:
          f#(h(x,h(y,z))) = [1] x + [1] y + [1] z + [16]
                          > [1] x + [1] y + [1] z + [12]
                          = c_8(f#(h(+(x,y),z)))        
          
          
          Following rules are (at-least) weakly oriented:
          +(x,+(y,z)) =  [1] x + [1] y + [1] z + [0]
                      >= [1] x + [1] y + [1] z + [0]
                      =  +(+(x,y),z)                
          
             +(x,0()) =  [1] x + [0]                
                      >= [1] x + [0]                
                      =  x                          
          
            +(x,s(y)) =  [1] x + [1] y + [13]       
                      >= [1] x + [1] y + [13]       
                      =  s(+(x,y))                  
          
             +(0(),y) =  [1] y + [0]                
                      >= [1] y + [0]                
                      =  y                          
          
            +(s(x),y) =  [1] x + [1] y + [13]       
                      >= [1] x + [1] y + [13]       
                      =  s(+(x,y))                  
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** Step 1.b:6.a:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f#(h(x,h(y,z))) -> c_8(f#(h(+(x,y),z)))
        - Weak TRS:
            +(x,+(y,z)) -> +(+(x,y),z)
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
        - Signature:
            {+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

*** Step 1.b:6.b:1: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
            +#(x,s(y)) -> c_3(+#(x,y))
            +#(s(x),y) -> c_5(+#(x,y))
        - Weak DPs:
            f#(h(x,h(y,z))) -> +#(x,y)
            f#(h(x,h(y,z))) -> f#(h(+(x,y),z))
        - Weak TRS:
            +(x,+(y,z)) -> +(+(x,y),z)
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
        - Signature:
            {+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,s}
    + Applied Processor:
        NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(linear):
        The following argument positions are considered usable:
          uargs(c_1) = {1,2},
          uargs(c_3) = {1},
          uargs(c_5) = {1}
        
        Following symbols are considered usable:
          {+,+#,f#}
        TcT has computed the following interpretation:
            p(+) = x1 + x2
            p(0) = 2      
            p(f) = 0      
            p(g) = 1 + x1 
            p(h) = x1 + x2
            p(s) = 3 + x1 
           p(+#) = 4*x2   
           p(f#) = 4*x1   
          p(c_1) = x1 + x2
          p(c_2) = 4      
          p(c_3) = 5 + x1 
          p(c_4) = 0      
          p(c_5) = x1     
          p(c_6) = 0      
          p(c_7) = 4      
          p(c_8) = x1     
        
        Following rules are strictly oriented:
        +#(x,s(y)) = 12 + 4*y    
                   > 5 + 4*y     
                   = c_3(+#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
           +#(x,+(y,z)) =  4*y + 4*z                
                        >= 4*y + 4*z                
                        =  c_1(+#(+(x,y),z),+#(x,y))
        
             +#(s(x),y) =  4*y                      
                        >= 4*y                      
                        =  c_5(+#(x,y))             
        
        f#(h(x,h(y,z))) =  4*x + 4*y + 4*z          
                        >= 4*y                      
                        =  +#(x,y)                  
        
        f#(h(x,h(y,z))) =  4*x + 4*y + 4*z          
                        >= 4*x + 4*y + 4*z          
                        =  f#(h(+(x,y),z))          
        
            +(x,+(y,z)) =  x + y + z                
                        >= x + y + z                
                        =  +(+(x,y),z)              
        
               +(x,0()) =  2 + x                    
                        >= x                        
                        =  x                        
        
              +(x,s(y)) =  3 + x + y                
                        >= 3 + x + y                
                        =  s(+(x,y))                
        
               +(0(),y) =  2 + y                    
                        >= y                        
                        =  y                        
        
              +(s(x),y) =  3 + x + y                
                        >= 3 + x + y                
                        =  s(+(x,y))                
        
*** Step 1.b:6.b:2: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
            +#(s(x),y) -> c_5(+#(x,y))
        - Weak DPs:
            +#(x,s(y)) -> c_3(+#(x,y))
            f#(h(x,h(y,z))) -> +#(x,y)
            f#(h(x,h(y,z))) -> f#(h(+(x,y),z))
        - Weak TRS:
            +(x,+(y,z)) -> +(+(x,y),z)
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
        - Signature:
            {+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,s}
    + Applied Processor:
        NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(linear):
        The following argument positions are considered usable:
          uargs(c_1) = {1,2},
          uargs(c_3) = {1},
          uargs(c_5) = {1}
        
        Following symbols are considered usable:
          {+,+#,f#}
        TcT has computed the following interpretation:
            p(+) = 1 + x1 + x2
            p(0) = 2          
            p(f) = 4 + x1     
            p(g) = 1 + x1     
            p(h) = 1 + x1 + x2
            p(s) = x1         
           p(+#) = x2         
           p(f#) = 1 + 4*x1   
          p(c_1) = x1 + x2    
          p(c_2) = 0          
          p(c_3) = x1         
          p(c_4) = 0          
          p(c_5) = x1         
          p(c_6) = 1          
          p(c_7) = 1          
          p(c_8) = 2          
        
        Following rules are strictly oriented:
        +#(x,+(y,z)) = 1 + y + z                
                     > y + z                    
                     = c_1(+#(+(x,y),z),+#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
             +#(x,s(y)) =  y                  
                        >= y                  
                        =  c_3(+#(x,y))       
        
             +#(s(x),y) =  y                  
                        >= y                  
                        =  c_5(+#(x,y))       
        
        f#(h(x,h(y,z))) =  9 + 4*x + 4*y + 4*z
                        >= y                  
                        =  +#(x,y)            
        
        f#(h(x,h(y,z))) =  9 + 4*x + 4*y + 4*z
                        >= 9 + 4*x + 4*y + 4*z
                        =  f#(h(+(x,y),z))    
        
            +(x,+(y,z)) =  2 + x + y + z      
                        >= 2 + x + y + z      
                        =  +(+(x,y),z)        
        
               +(x,0()) =  3 + x              
                        >= x                  
                        =  x                  
        
              +(x,s(y)) =  1 + x + y          
                        >= 1 + x + y          
                        =  s(+(x,y))          
        
               +(0(),y) =  3 + y              
                        >= y                  
                        =  y                  
        
              +(s(x),y) =  1 + x + y          
                        >= 1 + x + y          
                        =  s(+(x,y))          
        
*** Step 1.b:6.b:3: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            +#(s(x),y) -> c_5(+#(x,y))
        - Weak DPs:
            +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
            +#(x,s(y)) -> c_3(+#(x,y))
            f#(h(x,h(y,z))) -> +#(x,y)
            f#(h(x,h(y,z))) -> f#(h(+(x,y),z))
        - Weak TRS:
            +(x,+(y,z)) -> +(+(x,y),z)
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
        - Signature:
            {+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,s}
    + Applied Processor:
        NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(mixed(2)):
        The following argument positions are considered usable:
          uargs(c_1) = {1,2},
          uargs(c_3) = {1},
          uargs(c_5) = {1}
        
        Following symbols are considered usable:
          {+,+#,f#}
        TcT has computed the following interpretation:
            p(+) = 1 + x1 + x2               
            p(0) = 0                         
            p(f) = x1                        
            p(g) = 0                         
            p(h) = 1 + x1 + x2               
            p(s) = 2 + x1                    
           p(+#) = 2*x1 + 2*x1*x2 + x2 + x2^2
           p(f#) = x1 + x1^2                 
          p(c_1) = x1 + x2                   
          p(c_2) = 2                         
          p(c_3) = 1 + x1                    
          p(c_4) = 0                         
          p(c_5) = x1                        
          p(c_6) = 2 + x1                    
          p(c_7) = 0                         
          p(c_8) = 1                         
        
        Following rules are strictly oriented:
        +#(s(x),y) = 4 + 2*x + 2*x*y + 5*y + y^2
                   > 2*x + 2*x*y + y + y^2      
                   = c_5(+#(x,y))               
        
        
        Following rules are (at-least) weakly oriented:
           +#(x,+(y,z)) =  2 + 4*x + 2*x*y + 2*x*z + 3*y + 2*y*z + y^2 + 3*z + z^2      
                        >= 2 + 4*x + 2*x*y + 2*x*z + 3*y + 2*y*z + y^2 + 3*z + z^2      
                        =  c_1(+#(+(x,y),z),+#(x,y))                                    
        
             +#(x,s(y)) =  6 + 6*x + 2*x*y + 5*y + y^2                                  
                        >= 1 + 2*x + 2*x*y + y + y^2                                    
                        =  c_3(+#(x,y))                                                 
        
        f#(h(x,h(y,z))) =  6 + 5*x + 2*x*y + 2*x*z + x^2 + 5*y + 2*y*z + y^2 + 5*z + z^2
                        >= 2*x + 2*x*y + y + y^2                                        
                        =  +#(x,y)                                                      
        
        f#(h(x,h(y,z))) =  6 + 5*x + 2*x*y + 2*x*z + x^2 + 5*y + 2*y*z + y^2 + 5*z + z^2
                        >= 6 + 5*x + 2*x*y + 2*x*z + x^2 + 5*y + 2*y*z + y^2 + 5*z + z^2
                        =  f#(h(+(x,y),z))                                              
        
            +(x,+(y,z)) =  2 + x + y + z                                                
                        >= 2 + x + y + z                                                
                        =  +(+(x,y),z)                                                  
        
               +(x,0()) =  1 + x                                                        
                        >= x                                                            
                        =  x                                                            
        
              +(x,s(y)) =  3 + x + y                                                    
                        >= 3 + x + y                                                    
                        =  s(+(x,y))                                                    
        
               +(0(),y) =  1 + y                                                        
                        >= y                                                            
                        =  y                                                            
        
              +(s(x),y) =  3 + x + y                                                    
                        >= 3 + x + y                                                    
                        =  s(+(x,y))                                                    
        
*** Step 1.b:6.b:4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            +#(x,+(y,z)) -> c_1(+#(+(x,y),z),+#(x,y))
            +#(x,s(y)) -> c_3(+#(x,y))
            +#(s(x),y) -> c_5(+#(x,y))
            f#(h(x,h(y,z))) -> +#(x,y)
            f#(h(x,h(y,z))) -> f#(h(+(x,y),z))
        - Weak TRS:
            +(x,+(y,z)) -> +(+(x,y),z)
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            +(0(),y) -> y
            +(s(x),y) -> s(+(x,y))
        - Signature:
            {+/2,f/1,+#/2,f#/1} / {0/0,g/1,h/2,s/1,c_1/2,c_2/0,c_3/1,c_4/0,c_5/1,c_6/1,c_7/1,c_8/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#,f#} and constructors {0,g,h,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),O(n^3))