*** 1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        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
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {copy/3,f/1} / {0/0,cons/2,n/0,nil/0,s/1}
      Obligation:
        Innermost
        basic terms: {copy,f}/{0,cons,n,nil,s}
    Applied Processor:
      InnermostRuleRemoval
    Proof:
      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.
*** 1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        copy(0(),y,z) -> f(z)
        copy(s(x),y,z) -> copy(x,y,cons(f(y),z))
        f(cons(nil(),y)) -> y
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {copy/3,f/1} / {0/0,cons/2,n/0,nil/0,s/1}
      Obligation:
        Innermost
        basic terms: {copy,f}/{0,cons,n,nil,s}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      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.
*** 1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        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()
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        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
        basic terms: {copy#,f#}/{0,cons,n,nil,s}
    Applied Processor:
      UsableRules
    Proof:
      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()
*** 1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        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()
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        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
        basic terms: {copy#,f#}/{0,cons,n,nil,s}
    Applied Processor:
      PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    Proof:
      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()                
*** 1.1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        copy#(0(),y,z) -> c_1(f#(z))
        copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)),f#(y))
      Strict TRS Rules:
        
      Weak DP Rules:
        f#(cons(nil(),y)) -> c_3()
      Weak TRS Rules:
        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
        basic terms: {copy#,f#}/{0,cons,n,nil,s}
    Applied Processor:
      PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    Proof:
      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()                
*** 1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)),f#(y))
      Strict TRS Rules:
        
      Weak DP Rules:
        copy#(0(),y,z) -> c_1(f#(z))
        f#(cons(nil(),y)) -> c_3()
      Weak TRS Rules:
        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
        basic terms: {copy#,f#}/{0,cons,n,nil,s}
    Applied Processor:
      RemoveWeakSuffixes
    Proof:
      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()  
*** 1.1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)),f#(y))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        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
        basic terms: {copy#,f#}/{0,cons,n,nil,s}
    Applied Processor:
      SimplifyRHS
    Proof:
      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)))
*** 1.1.1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        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
        basic terms: {copy#,f#}/{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, greedy = NoGreedy}}
    Proof:
      We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} 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.
  *** 1.1.1.1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
      Considered Problem:
        Strict DP Rules:
          copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)))
        Strict TRS Rules:
          
        Weak DP Rules:
          
        Weak TRS Rules:
          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
          basic terms: {copy#,f#}/{0,cons,n,nil,s}
      Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation any intersect of rules of CDG leaf and strict-rules, greedy = NoGreedy}
      Proof:
        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) = [2]                           
           p(copy) = [1] x3 + [2]                  
              p(f) = [2] x1 + [12]                 
              p(n) = [0]                           
            p(nil) = [1]                           
              p(s) = [1] x1 + [5]                  
          p(copy#) = [1] x1 + [8] x2 + [2] x3 + [0]
             p(f#) = [1] x1 + [2]                  
            p(c_1) = [2]                           
            p(c_2) = [1] x1 + [0]                  
            p(c_3) = [0]                           
        
        Following rules are strictly oriented:
        copy#(s(x),y,z) = [1] x + [8] y + [2] z + [5] 
                        > [1] x + [8] y + [4]         
                        = c_2(copy#(x,y,cons(f(y),z)))
        
        
        Following rules are (at-least) weakly oriented:
        
  *** 1.1.1.1.1.1.1.1.1.1 Progress [(?,O(1))]  ***
      Considered Problem:
        Strict DP Rules:
          
        Strict TRS Rules:
          
        Weak DP Rules:
          copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)))
        Weak TRS Rules:
          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
          basic terms: {copy#,f#}/{0,cons,n,nil,s}
      Applied Processor:
        Assumption
      Proof:
        ()
  
  *** 1.1.1.1.1.1.1.1.2 Progress [(O(1),O(1))]  ***
      Considered Problem:
        Strict DP Rules:
          
        Strict TRS Rules:
          
        Weak DP Rules:
          copy#(s(x),y,z) -> c_2(copy#(x,y,cons(f(y),z)))
        Weak TRS Rules:
          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
          basic terms: {copy#,f#}/{0,cons,n,nil,s}
      Applied Processor:
        RemoveWeakSuffixes
      Proof:
        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)))
  *** 1.1.1.1.1.1.1.1.2.1 Progress [(O(1),O(1))]  ***
      Considered Problem:
        Strict DP Rules:
          
        Strict TRS Rules:
          
        Weak DP Rules:
          
        Weak TRS Rules:
          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
          basic terms: {copy#,f#}/{0,cons,n,nil,s}
      Applied Processor:
        EmptyProcessor
      Proof:
        The problem is already closed. The intended complexity is O(1).