WORST_CASE(?,O(n^1))
* Step 1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            :(z,+(x,f(y))) -> :(g(z,y),+(x,a()))
            :(+(x,y),z) -> +(:(x,z),:(y,z))
            :(:(x,y),z) -> :(x,:(y,z))
        - Signature:
            {:/2} / {+/2,a/0,f/1,g/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {:} and constructors {+,a,f,g}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          :#(z,+(x,f(y))) -> c_1(:#(g(z,y),+(x,a())))
          :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z))
          :#(:(x,y),z) -> c_3(:#(x,:(y,z)),:#(y,z))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            :#(z,+(x,f(y))) -> c_1(:#(g(z,y),+(x,a())))
            :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z))
            :#(:(x,y),z) -> c_3(:#(x,:(y,z)),:#(y,z))
        - Weak TRS:
            :(z,+(x,f(y))) -> :(g(z,y),+(x,a()))
            :(+(x,y),z) -> +(:(x,z),:(y,z))
            :(:(x,y),z) -> :(x,:(y,z))
        - Signature:
            {:/2,:#/2} / {+/2,a/0,f/1,g/2,c_1/1,c_2/2,c_3/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {:#} and constructors {+,a,f,g}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1}
        by application of
          Pre({1}) = {2,3}.
        Here rules are labelled as follows:
          1: :#(z,+(x,f(y))) -> c_1(:#(g(z,y),+(x,a())))
          2: :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z))
          3: :#(:(x,y),z) -> c_3(:#(x,:(y,z)),:#(y,z))
* Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z))
            :#(:(x,y),z) -> c_3(:#(x,:(y,z)),:#(y,z))
        - Weak DPs:
            :#(z,+(x,f(y))) -> c_1(:#(g(z,y),+(x,a())))
        - Weak TRS:
            :(z,+(x,f(y))) -> :(g(z,y),+(x,a()))
            :(+(x,y),z) -> +(:(x,z),:(y,z))
            :(:(x,y),z) -> :(x,:(y,z))
        - Signature:
            {:/2,:#/2} / {+/2,a/0,f/1,g/2,c_1/1,c_2/2,c_3/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {:#} and constructors {+,a,f,g}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S::#(+(x,y),z) -> c_2(:#(x,z),:#(y,z))
             -->_2 :#(:(x,y),z) -> c_3(:#(x,:(y,z)),:#(y,z)):2
             -->_1 :#(:(x,y),z) -> c_3(:#(x,:(y,z)),:#(y,z)):2
             -->_2 :#(z,+(x,f(y))) -> c_1(:#(g(z,y),+(x,a()))):3
             -->_1 :#(z,+(x,f(y))) -> c_1(:#(g(z,y),+(x,a()))):3
             -->_2 :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z)):1
             -->_1 :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z)):1
          
          2:S::#(:(x,y),z) -> c_3(:#(x,:(y,z)),:#(y,z))
             -->_2 :#(z,+(x,f(y))) -> c_1(:#(g(z,y),+(x,a()))):3
             -->_1 :#(z,+(x,f(y))) -> c_1(:#(g(z,y),+(x,a()))):3
             -->_2 :#(:(x,y),z) -> c_3(:#(x,:(y,z)),:#(y,z)):2
             -->_1 :#(:(x,y),z) -> c_3(:#(x,:(y,z)),:#(y,z)):2
             -->_2 :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z)):1
             -->_1 :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z)):1
          
          3:W::#(z,+(x,f(y))) -> c_1(:#(g(z,y),+(x,a())))
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: :#(z,+(x,f(y))) -> c_1(:#(g(z,y),+(x,a())))
* Step 4: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z))
            :#(:(x,y),z) -> c_3(:#(x,:(y,z)),:#(y,z))
        - Weak TRS:
            :(z,+(x,f(y))) -> :(g(z,y),+(x,a()))
            :(+(x,y),z) -> +(:(x,z),:(y,z))
            :(:(x,y),z) -> :(x,:(y,z))
        - Signature:
            {:/2,:#/2} / {+/2,a/0,f/1,g/2,c_1/1,c_2/2,c_3/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {:#} and constructors {+,a,f,g}
    + 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: :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z))
          
        The strictly oriented rules are moved into the weak component.
** Step 4.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z))
            :#(:(x,y),z) -> c_3(:#(x,:(y,z)),:#(y,z))
        - Weak TRS:
            :(z,+(x,f(y))) -> :(g(z,y),+(x,a()))
            :(+(x,y),z) -> +(:(x,z),:(y,z))
            :(:(x,y),z) -> :(x,:(y,z))
        - Signature:
            {:/2,:#/2} / {+/2,a/0,f/1,g/2,c_1/1,c_2/2,c_3/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {:#} and constructors {+,a,f,g}
    + 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_2) = {1,2},
          uargs(c_3) = {1,2}
        
        Following symbols are considered usable:
          {:#}
        TcT has computed the following interpretation:
            p(+) = [1] x1 + [1] x2 + [2]
            p(:) = [2] x1 + [2] x2 + [0]
            p(a) = [0]                  
            p(f) = [1] x1 + [0]         
            p(g) = [1] x1 + [1] x2 + [0]
           p(:#) = [8] x1 + [0]         
          p(c_1) = [0]                  
          p(c_2) = [1] x1 + [1] x2 + [3]
          p(c_3) = [2] x1 + [1] x2 + [0]
        
        Following rules are strictly oriented:
        :#(+(x,y),z) = [8] x + [8] y + [16]
                     > [8] x + [8] y + [3] 
                     = c_2(:#(x,z),:#(y,z))
        
        
        Following rules are (at-least) weakly oriented:
        :#(:(x,y),z) =  [16] x + [16] y + [0]    
                     >= [16] x + [8] y + [0]     
                     =  c_3(:#(x,:(y,z)),:#(y,z))
        
** Step 4.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            :#(:(x,y),z) -> c_3(:#(x,:(y,z)),:#(y,z))
        - Weak DPs:
            :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z))
        - Weak TRS:
            :(z,+(x,f(y))) -> :(g(z,y),+(x,a()))
            :(+(x,y),z) -> +(:(x,z),:(y,z))
            :(:(x,y),z) -> :(x,:(y,z))
        - Signature:
            {:/2,:#/2} / {+/2,a/0,f/1,g/2,c_1/1,c_2/2,c_3/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {:#} and constructors {+,a,f,g}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

** Step 4.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            :#(:(x,y),z) -> c_3(:#(x,:(y,z)),:#(y,z))
        - Weak DPs:
            :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z))
        - Weak TRS:
            :(z,+(x,f(y))) -> :(g(z,y),+(x,a()))
            :(+(x,y),z) -> +(:(x,z),:(y,z))
            :(:(x,y),z) -> :(x,:(y,z))
        - Signature:
            {:/2,:#/2} / {+/2,a/0,f/1,g/2,c_1/1,c_2/2,c_3/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {:#} and constructors {+,a,f,g}
    + 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: :#(:(x,y),z) -> c_3(:#(x,:(y,z)),:#(y,z))
          
        The strictly oriented rules are moved into the weak component.
*** Step 4.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            :#(:(x,y),z) -> c_3(:#(x,:(y,z)),:#(y,z))
        - Weak DPs:
            :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z))
        - Weak TRS:
            :(z,+(x,f(y))) -> :(g(z,y),+(x,a()))
            :(+(x,y),z) -> +(:(x,z),:(y,z))
            :(:(x,y),z) -> :(x,:(y,z))
        - Signature:
            {:/2,:#/2} / {+/2,a/0,f/1,g/2,c_1/1,c_2/2,c_3/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {:#} and constructors {+,a,f,g}
    + 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_2) = {1,2},
          uargs(c_3) = {1,2}
        
        Following symbols are considered usable:
          {:#}
        TcT has computed the following interpretation:
            p(+) = [1] x1 + [1] x2 + [1]
            p(:) = [4] x1 + [2] x2 + [1]
            p(a) = [8]                  
            p(f) = [4]                  
            p(g) = [3]                  
           p(:#) = [1] x1 + [0]         
          p(c_1) = [0]                  
          p(c_2) = [1] x1 + [1] x2 + [1]
          p(c_3) = [4] x1 + [1] x2 + [0]
        
        Following rules are strictly oriented:
        :#(:(x,y),z) = [4] x + [2] y + [1]      
                     > [4] x + [1] y + [0]      
                     = c_3(:#(x,:(y,z)),:#(y,z))
        
        
        Following rules are (at-least) weakly oriented:
        :#(+(x,y),z) =  [1] x + [1] y + [1] 
                     >= [1] x + [1] y + [1] 
                     =  c_2(:#(x,z),:#(y,z))
        
*** Step 4.b:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            :#(+(x,y),z) -> c_2(:#(x,z),:#(y,z))
            :#(:(x,y),z) -> c_3(:#(x,:(y,z)),:#(y,z))
        - Weak TRS:
            :(z,+(x,f(y))) -> :(g(z,y),+(x,a()))
            :(+(x,y),z) -> +(:(x,z),:(y,z))
            :(:(x,y),z) -> :(x,:(y,z))
        - Signature:
            {:/2,:#/2} / {+/2,a/0,f/1,g/2,c_1/1,c_2/2,c_3/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {:#} and constructors {+,a,f,g}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

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

WORST_CASE(?,O(n^1))