*** 1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        f(cons(x,k),l) -> g(k,l,cons(x,k))
        f(empty(),l) -> l
        g(a,b,c) -> f(a,cons(b,c))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/2,g/3} / {cons/2,empty/0}
      Obligation:
        Innermost
        basic terms: {f,g}/{cons,empty}
    Applied Processor:
      DependencyPairs {dpKind_ = WIDP}
    Proof:
      We add the following weak innermost dependency pairs:
      
      Strict DPs
        f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
        f#(empty(),l) -> c_2()
        g#(a,b,c) -> c_3(f#(a,cons(b,c)))
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
        f#(empty(),l) -> c_2()
        g#(a,b,c) -> c_3(f#(a,cons(b,c)))
      Strict TRS Rules:
        f(cons(x,k),l) -> g(k,l,cons(x,k))
        f(empty(),l) -> l
        g(a,b,c) -> f(a,cons(b,c))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/2,g/3,f#/2,g#/3} / {cons/2,empty/0,c_1/1,c_2/0,c_3/1}
      Obligation:
        Innermost
        basic terms: {f#,g#}/{cons,empty}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
        f#(empty(),l) -> c_2()
        g#(a,b,c) -> c_3(f#(a,cons(b,c)))
*** 1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
        f#(empty(),l) -> c_2()
        g#(a,b,c) -> c_3(f#(a,cons(b,c)))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/2,g/3,f#/2,g#/3} / {cons/2,empty/0,c_1/1,c_2/0,c_3/1}
      Obligation:
        Innermost
        basic terms: {f#,g#}/{cons,empty}
    Applied Processor:
      Succeeding
    Proof:
      ()
*** 1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
        f#(empty(),l) -> c_2()
        g#(a,b,c) -> c_3(f#(a,cons(b,c)))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/2,g/3,f#/2,g#/3} / {cons/2,empty/0,c_1/1,c_2/0,c_3/1}
      Obligation:
        Innermost
        basic terms: {f#,g#}/{cons,empty}
    Applied Processor:
      PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    Proof:
      We estimate the number of application of
        {2}
      by application of
        Pre({2}) = {3}.
      Here rules are labelled as follows:
        1: f#(cons(x,k),l) -> c_1(g#(k          
                                    ,l          
                                    ,cons(x,k)))
        2: f#(empty(),l) -> c_2()               
        3: g#(a,b,c) -> c_3(f#(a                
                              ,cons(b,c)))      
*** 1.1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
        g#(a,b,c) -> c_3(f#(a,cons(b,c)))
      Strict TRS Rules:
        
      Weak DP Rules:
        f#(empty(),l) -> c_2()
      Weak TRS Rules:
        
      Signature:
        {f/2,g/3,f#/2,g#/3} / {cons/2,empty/0,c_1/1,c_2/0,c_3/1}
      Obligation:
        Innermost
        basic terms: {f#,g#}/{cons,empty}
    Applied Processor:
      RemoveWeakSuffixes
    Proof:
      Consider the dependency graph
        1:S:f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
           -->_1 g#(a,b,c) -> c_3(f#(a,cons(b,c))):2
        
        2:S:g#(a,b,c) -> c_3(f#(a,cons(b,c)))
           -->_1 f#(empty(),l) -> c_2():3
           -->_1 f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k))):1
        
        3:W:f#(empty(),l) -> c_2()
           
        
      The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
        3: f#(empty(),l) -> c_2()
*** 1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
        g#(a,b,c) -> c_3(f#(a,cons(b,c)))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {f/2,g/3,f#/2,g#/3} / {cons/2,empty/0,c_1/1,c_2/0,c_3/1}
      Obligation:
        Innermost
        basic terms: {f#,g#}/{cons,empty}
    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: f#(cons(x,k),l) -> c_1(g#(k          
                                    ,l          
                                    ,cons(x,k)))
        
      Consider the set of all dependency pairs
        1: f#(cons(x,k),l) -> c_1(g#(k          
                                    ,l          
                                    ,cons(x,k)))
        2: g#(a,b,c) -> c_3(f#(a                
                              ,cons(b,c)))      
      Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}induces the complexity certificateTIME (?,O(n^1))
      SPACE(?,?)on application of the dependency pairs
        {1}
      These cover all (indirect) predecessors of dependency pairs
        {1,2}
      their number of applications is equally bounded.
      The dependency pairs are shifted into the weak component.
  *** 1.1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
      Considered Problem:
        Strict DP Rules:
          f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
          g#(a,b,c) -> c_3(f#(a,cons(b,c)))
        Strict TRS Rules:
          
        Weak DP Rules:
          
        Weak TRS Rules:
          
        Signature:
          {f/2,g/3,f#/2,g#/3} / {cons/2,empty/0,c_1/1,c_2/0,c_3/1}
        Obligation:
          Innermost
          basic terms: {f#,g#}/{cons,empty}
      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_1) = {1},
          uargs(c_3) = {1}
        
        Following symbols are considered usable:
          {f#,g#}
        TcT has computed the following interpretation:
           p(cons) = [1] x1 + [1] x2 + [4]
          p(empty) = [1]                  
              p(f) = [1] x1 + [0]         
              p(g) = [1] x1 + [1]         
             p(f#) = [4] x1 + [0]         
             p(g#) = [4] x1 + [0]         
            p(c_1) = [1] x1 + [15]        
            p(c_2) = [1]                  
            p(c_3) = [1] x1 + [0]         
        
        Following rules are strictly oriented:
        f#(cons(x,k),l) = [4] k + [4] x + [16]  
                        > [4] k + [15]          
                        = c_1(g#(k,l,cons(x,k)))
        
        
        Following rules are (at-least) weakly oriented:
        g#(a,b,c) =  [4] a + [0]         
                  >= [4] a + [0]         
                  =  c_3(f#(a,cons(b,c)))
        
  *** 1.1.1.1.1.1.1.1 Progress [(?,O(1))]  ***
      Considered Problem:
        Strict DP Rules:
          g#(a,b,c) -> c_3(f#(a,cons(b,c)))
        Strict TRS Rules:
          
        Weak DP Rules:
          f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
        Weak TRS Rules:
          
        Signature:
          {f/2,g/3,f#/2,g#/3} / {cons/2,empty/0,c_1/1,c_2/0,c_3/1}
        Obligation:
          Innermost
          basic terms: {f#,g#}/{cons,empty}
      Applied Processor:
        Assumption
      Proof:
        ()
  
  *** 1.1.1.1.1.1.2 Progress [(O(1),O(1))]  ***
      Considered Problem:
        Strict DP Rules:
          
        Strict TRS Rules:
          
        Weak DP Rules:
          f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
          g#(a,b,c) -> c_3(f#(a,cons(b,c)))
        Weak TRS Rules:
          
        Signature:
          {f/2,g/3,f#/2,g#/3} / {cons/2,empty/0,c_1/1,c_2/0,c_3/1}
        Obligation:
          Innermost
          basic terms: {f#,g#}/{cons,empty}
      Applied Processor:
        RemoveWeakSuffixes
      Proof:
        Consider the dependency graph
          1:W:f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
             -->_1 g#(a,b,c) -> c_3(f#(a,cons(b,c))):2
          
          2:W:g#(a,b,c) -> c_3(f#(a,cons(b,c)))
             -->_1 f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k))):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: f#(cons(x,k),l) -> c_1(g#(k          
                                      ,l          
                                      ,cons(x,k)))
          2: g#(a,b,c) -> c_3(f#(a                
                                ,cons(b,c)))      
  *** 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:
          
        Signature:
          {f/2,g/3,f#/2,g#/3} / {cons/2,empty/0,c_1/1,c_2/0,c_3/1}
        Obligation:
          Innermost
          basic terms: {f#,g#}/{cons,empty}
      Applied Processor:
        EmptyProcessor
      Proof:
        The problem is already closed. The intended complexity is O(1).