WORST_CASE(?,O(n^1))
* Step 1: InnermostRuleRemoval 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:
        InnermostRuleRemoval
    + Details:
        Arguments of following rules are not normal-forms.
          f(cons(f(cons(nil(),y)),z)) -> copy(n(),y,z)
        All above mentioned rules can be savely removed.
* Step 2: 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(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(nil(),y)) -> c_3()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 3: UsableRules 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(nil(),y)) -> c_3()
        - Weak TRS:
            copy(0(),y,z) -> f(z)
            copy(s(x),y,z) -> copy(x,y,cons(f(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_3/0}
        - 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(nil(),y)) -> y
          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(nil(),y)) -> c_3()
* Step 4: 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(nil(),y)) -> c_3()
        - Weak TRS:
            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_3/0}
        - 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}
        by application of
          Pre({3}) = {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(nil(),y)) -> c_3()
* Step 5: 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(nil(),y)) -> c_3()
        - Weak TRS:
            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_3/0}
        - 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(nil(),y)) -> c_3()
* Step 6: 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(nil(),y)) -> c_3()
        - Weak TRS:
            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_3/0}
        - 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_3():3
             -->_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_3():3
          
          3:W:f#(cons(nil(),y)) -> c_3()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: copy#(0(),y,z) -> c_1(f#(z))
          3: f#(cons(nil(),y)) -> c_3()
* Step 7: 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:
            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_3/0}
        - 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 8: PredecessorEstimationCP 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(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_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {copy#,f#} and constructors {0,cons,n,nil,s}
    + 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: copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)))
          
        The strictly oriented rules are moved into the weak component.
** Step 8.a:1: 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(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_3/0}
        - 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 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}
        
        Following symbols are considered usable:
          {copy#,f#}
        TcT has computed the following interpretation:
              p(0) = [1]                  
           p(cons) = [1] x1 + [2]         
           p(copy) = [1] x1 + [1]         
              p(f) = [2] x1 + [0]         
              p(n) = [1]                  
            p(nil) = [6]                  
              p(s) = [1] x1 + [2]         
          p(copy#) = [8] x1 + [1] x2 + [0]
             p(f#) = [1] x1 + [1]         
            p(c_1) = [2]                  
            p(c_2) = [1] x1 + [12]        
            p(c_3) = [1]                  
        
        Following rules are strictly oriented:
        copy#(s(x),y,z) = [8] x + [1] y + [16]        
                        > [8] x + [1] y + [12]        
                        = c_2(copy#(x,y,cons(f(y),z)))
        
        
        Following rules are (at-least) weakly oriented:
        
** Step 8.a:2: Assumption 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(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_3/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {copy#,f#} and constructors {0,cons,n,nil,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

** Step 8.b:1: RemoveWeakSuffixes 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(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_3/0}
        - 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:W:copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)))
             -->_1 copy#(s(x),y,z) -> c_2(copy#(x,y,cons(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: copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)))
** Step 8.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            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_3/0}
        - 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(?,O(n^1))