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

** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            copy(0(),y,z) -> f(z)
            copy(s(x),y,z) -> copy(x,y,cons(f(y),z))
            f(cons(f(cons(nil(),y)),z)) -> copy(n(),y,z)
            f(cons(nil(),y)) -> y
        - Signature:
            {copy/3,f/1} / {0/0,cons/2,n/0,nil/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {copy,f} and constructors {0,cons,n,nil,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          copy#(0(),y,z) -> c_1(f#(z))
          copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)),f#(y))
          f#(cons(f(cons(nil(),y)),z)) -> c_3(copy#(n(),y,z))
          f#(cons(nil(),y)) -> c_4()
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            copy#(0(),y,z) -> c_1(f#(z))
            copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)),f#(y))
            f#(cons(f(cons(nil(),y)),z)) -> c_3(copy#(n(),y,z))
            f#(cons(nil(),y)) -> c_4()
        - Weak TRS:
            copy(0(),y,z) -> f(z)
            copy(s(x),y,z) -> copy(x,y,cons(f(y),z))
            f(cons(f(cons(nil(),y)),z)) -> copy(n(),y,z)
            f(cons(nil(),y)) -> y
        - Signature:
            {copy/3,f/1,copy#/3,f#/1} / {0/0,cons/2,n/0,nil/0,s/1,c_1/1,c_2/2,c_3/1,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {copy#,f#} and constructors {0,cons,n,nil,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {3,4}
        by application of
          Pre({3,4}) = {1,2}.
        Here rules are labelled as follows:
          1: copy#(0(),y,z) -> c_1(f#(z))
          2: copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)),f#(y))
          3: f#(cons(f(cons(nil(),y)),z)) -> c_3(copy#(n(),y,z))
          4: f#(cons(nil(),y)) -> c_4()
** Step 1.b:3: PredecessorEstimation WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            copy#(0(),y,z) -> c_1(f#(z))
            copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)),f#(y))
        - Weak DPs:
            f#(cons(f(cons(nil(),y)),z)) -> c_3(copy#(n(),y,z))
            f#(cons(nil(),y)) -> c_4()
        - Weak TRS:
            copy(0(),y,z) -> f(z)
            copy(s(x),y,z) -> copy(x,y,cons(f(y),z))
            f(cons(f(cons(nil(),y)),z)) -> copy(n(),y,z)
            f(cons(nil(),y)) -> y
        - Signature:
            {copy/3,f/1,copy#/3,f#/1} / {0/0,cons/2,n/0,nil/0,s/1,c_1/1,c_2/2,c_3/1,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {copy#,f#} and constructors {0,cons,n,nil,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1}
        by application of
          Pre({1}) = {2}.
        Here rules are labelled as follows:
          1: copy#(0(),y,z) -> c_1(f#(z))
          2: copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)),f#(y))
          3: f#(cons(f(cons(nil(),y)),z)) -> c_3(copy#(n(),y,z))
          4: f#(cons(nil(),y)) -> c_4()
** Step 1.b:4: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)),f#(y))
        - Weak DPs:
            copy#(0(),y,z) -> c_1(f#(z))
            f#(cons(f(cons(nil(),y)),z)) -> c_3(copy#(n(),y,z))
            f#(cons(nil(),y)) -> c_4()
        - Weak TRS:
            copy(0(),y,z) -> f(z)
            copy(s(x),y,z) -> copy(x,y,cons(f(y),z))
            f(cons(f(cons(nil(),y)),z)) -> copy(n(),y,z)
            f(cons(nil(),y)) -> y
        - Signature:
            {copy/3,f/1,copy#/3,f#/1} / {0/0,cons/2,n/0,nil/0,s/1,c_1/1,c_2/2,c_3/1,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {copy#,f#} and constructors {0,cons,n,nil,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)),f#(y))
             -->_1 copy#(0(),y,z) -> c_1(f#(z)):2
             -->_2 f#(cons(nil(),y)) -> c_4():4
             -->_1 copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)),f#(y)):1
          
          2:W:copy#(0(),y,z) -> c_1(f#(z))
             -->_1 f#(cons(nil(),y)) -> c_4():4
          
          3:W:f#(cons(f(cons(nil(),y)),z)) -> c_3(copy#(n(),y,z))
             
          
          4:W:f#(cons(nil(),y)) -> c_4()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: f#(cons(f(cons(nil(),y)),z)) -> c_3(copy#(n(),y,z))
          2: copy#(0(),y,z) -> c_1(f#(z))
          4: f#(cons(nil(),y)) -> c_4()
** Step 1.b:5: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)),f#(y))
        - Weak TRS:
            copy(0(),y,z) -> f(z)
            copy(s(x),y,z) -> copy(x,y,cons(f(y),z))
            f(cons(f(cons(nil(),y)),z)) -> copy(n(),y,z)
            f(cons(nil(),y)) -> y
        - Signature:
            {copy/3,f/1,copy#/3,f#/1} / {0/0,cons/2,n/0,nil/0,s/1,c_1/1,c_2/2,c_3/1,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {copy#,f#} and constructors {0,cons,n,nil,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)),f#(y))
             -->_1 copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)),f#(y)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)))
** Step 1.b:6: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)))
        - Weak TRS:
            copy(0(),y,z) -> f(z)
            copy(s(x),y,z) -> copy(x,y,cons(f(y),z))
            f(cons(f(cons(nil(),y)),z)) -> copy(n(),y,z)
            f(cons(nil(),y)) -> y
        - Signature:
            {copy/3,f/1,copy#/3,f#/1} / {0/0,cons/2,n/0,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {copy#,f#} and constructors {0,cons,n,nil,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          f(cons(f(cons(nil(),y)),z)) -> copy(n(),y,z)
          f(cons(nil(),y)) -> y
          copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)))
** Step 1.b:7: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)))
        - Weak TRS:
            f(cons(f(cons(nil(),y)),z)) -> copy(n(),y,z)
            f(cons(nil(),y)) -> y
        - Signature:
            {copy/3,f/1,copy#/3,f#/1} / {0/0,cons/2,n/0,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {copy#,f#} and constructors {0,cons,n,nil,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any 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}
        
        Following symbols are considered usable:
          {copy#,f#}
        TcT has computed the following interpretation:
              p(0) = [0]                  
           p(cons) = [1]                  
           p(copy) = [2] x1 + [1] x3 + [2]
              p(f) = [12]                 
              p(n) = [2]                  
            p(nil) = [2]                  
              p(s) = [1] x1 + [2]         
          p(copy#) = [4] x1 + [1] x3 + [0]
             p(f#) = [1] x1 + [0]         
            p(c_1) = [1] x1 + [1]         
            p(c_2) = [1] x1 + [5]         
            p(c_3) = [1] x1 + [1]         
            p(c_4) = [1]                  
        
        Following rules are strictly oriented:
        copy#(s(x),y,z) = [4] x + [1] z + [8]         
                        > [4] x + [6]                 
                        = c_2(copy#(x,y,cons(f(y),z)))
        
        
        Following rules are (at-least) weakly oriented:
        
** Step 1.b:8: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)))
        - Weak TRS:
            f(cons(f(cons(nil(),y)),z)) -> copy(n(),y,z)
            f(cons(nil(),y)) -> y
        - Signature:
            {copy/3,f/1,copy#/3,f#/1} / {0/0,cons/2,n/0,nil/0,s/1,c_1/1,c_2/1,c_3/1,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {copy#,f#} and constructors {0,cons,n,nil,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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