*** 1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        circ(s,id()) -> s
        circ(circ(s,t),u) -> circ(s,circ(t,u))
        circ(cons(a,s),t) -> cons(msubst(a,t),circ(s,t))
        circ(cons(lift(),s),circ(cons(lift(),t),u)) -> circ(cons(lift(),circ(s,t)),u)
        circ(cons(lift(),s),cons(a,t)) -> cons(a,circ(s,t))
        circ(cons(lift(),s),cons(lift(),t)) -> cons(lift(),circ(s,t))
        circ(id(),s) -> s
        msubst(a,id()) -> a
        msubst(msubst(a,s),t) -> msubst(a,circ(s,t))
        subst(a,id()) -> a
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {circ/2,msubst/2,subst/2} / {cons/2,id/0,lift/0}
      Obligation:
        Innermost
        basic terms: {circ,msubst,subst}/{cons,id,lift}
    Applied Processor:
      InnermostRuleRemoval
    Proof:
      Arguments of following rules are not normal-forms.
        circ(cons(lift(),s),circ(cons(lift(),t),u)) -> circ(cons(lift(),circ(s,t)),u)
      All above mentioned rules can be savely removed.
*** 1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        circ(s,id()) -> s
        circ(circ(s,t),u) -> circ(s,circ(t,u))
        circ(cons(a,s),t) -> cons(msubst(a,t),circ(s,t))
        circ(cons(lift(),s),cons(a,t)) -> cons(a,circ(s,t))
        circ(cons(lift(),s),cons(lift(),t)) -> cons(lift(),circ(s,t))
        circ(id(),s) -> s
        msubst(a,id()) -> a
        msubst(msubst(a,s),t) -> msubst(a,circ(s,t))
        subst(a,id()) -> a
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {circ/2,msubst/2,subst/2} / {cons/2,id/0,lift/0}
      Obligation:
        Innermost
        basic terms: {circ,msubst,subst}/{cons,id,lift}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      We add the following dependency tuples:
      
      Strict DPs
        circ#(s,id()) -> c_1()
        circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u))
        circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t))
        circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t))
        circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t))
        circ#(id(),s) -> c_6()
        msubst#(a,id()) -> c_7()
        msubst#(msubst(a,s),t) -> c_8(msubst#(a,circ(s,t)),circ#(s,t))
        subst#(a,id()) -> c_9()
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        circ#(s,id()) -> c_1()
        circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u))
        circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t))
        circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t))
        circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t))
        circ#(id(),s) -> c_6()
        msubst#(a,id()) -> c_7()
        msubst#(msubst(a,s),t) -> c_8(msubst#(a,circ(s,t)),circ#(s,t))
        subst#(a,id()) -> c_9()
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        circ(s,id()) -> s
        circ(circ(s,t),u) -> circ(s,circ(t,u))
        circ(cons(a,s),t) -> cons(msubst(a,t),circ(s,t))
        circ(cons(lift(),s),cons(a,t)) -> cons(a,circ(s,t))
        circ(cons(lift(),s),cons(lift(),t)) -> cons(lift(),circ(s,t))
        circ(id(),s) -> s
        msubst(a,id()) -> a
        msubst(msubst(a,s),t) -> msubst(a,circ(s,t))
        subst(a,id()) -> a
      Signature:
        {circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/1,c_6/0,c_7/0,c_8/2,c_9/0}
      Obligation:
        Innermost
        basic terms: {circ#,msubst#,subst#}/{cons,id,lift}
    Applied Processor:
      UsableRules
    Proof:
      We replace rewrite rules by usable rules:
        circ(s,id()) -> s
        circ(circ(s,t),u) -> circ(s,circ(t,u))
        circ(cons(a,s),t) -> cons(msubst(a,t),circ(s,t))
        circ(cons(lift(),s),cons(a,t)) -> cons(a,circ(s,t))
        circ(cons(lift(),s),cons(lift(),t)) -> cons(lift(),circ(s,t))
        circ(id(),s) -> s
        msubst(a,id()) -> a
        msubst(msubst(a,s),t) -> msubst(a,circ(s,t))
        circ#(s,id()) -> c_1()
        circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u))
        circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t))
        circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t))
        circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t))
        circ#(id(),s) -> c_6()
        msubst#(a,id()) -> c_7()
        msubst#(msubst(a,s),t) -> c_8(msubst#(a,circ(s,t)),circ#(s,t))
        subst#(a,id()) -> c_9()
*** 1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        circ#(s,id()) -> c_1()
        circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u))
        circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t))
        circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t))
        circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t))
        circ#(id(),s) -> c_6()
        msubst#(a,id()) -> c_7()
        msubst#(msubst(a,s),t) -> c_8(msubst#(a,circ(s,t)),circ#(s,t))
        subst#(a,id()) -> c_9()
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        circ(s,id()) -> s
        circ(circ(s,t),u) -> circ(s,circ(t,u))
        circ(cons(a,s),t) -> cons(msubst(a,t),circ(s,t))
        circ(cons(lift(),s),cons(a,t)) -> cons(a,circ(s,t))
        circ(cons(lift(),s),cons(lift(),t)) -> cons(lift(),circ(s,t))
        circ(id(),s) -> s
        msubst(a,id()) -> a
        msubst(msubst(a,s),t) -> msubst(a,circ(s,t))
      Signature:
        {circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/1,c_6/0,c_7/0,c_8/2,c_9/0}
      Obligation:
        Innermost
        basic terms: {circ#,msubst#,subst#}/{cons,id,lift}
    Applied Processor:
      PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    Proof:
      We estimate the number of application of
        {1,6,7,9}
      by application of
        Pre({1,6,7,9}) = {2,3,4,5,8}.
      Here rules are labelled as follows:
        1: circ#(s,id()) -> c_1()                
        2: circ#(circ(s,t),u) ->                 
             c_2(circ#(s,circ(t,u))              
                ,circ#(t,u))                     
        3: circ#(cons(a,s),t) ->                 
             c_3(msubst#(a,t),circ#(s,t))        
        4: circ#(cons(lift(),s)                  
                ,cons(a,t)) -> c_4(circ#(s,t))   
        5: circ#(cons(lift(),s)                  
                ,cons(lift(),t)) -> c_5(circ#(s  
                                             ,t))
        6: circ#(id(),s) -> c_6()                
        7: msubst#(a,id()) -> c_7()              
        8: msubst#(msubst(a,s),t) ->             
             c_8(msubst#(a,circ(s,t))            
                ,circ#(s,t))                     
        9: subst#(a,id()) -> c_9()               
*** 1.1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u))
        circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t))
        circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t))
        circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t))
        msubst#(msubst(a,s),t) -> c_8(msubst#(a,circ(s,t)),circ#(s,t))
      Strict TRS Rules:
        
      Weak DP Rules:
        circ#(s,id()) -> c_1()
        circ#(id(),s) -> c_6()
        msubst#(a,id()) -> c_7()
        subst#(a,id()) -> c_9()
      Weak TRS Rules:
        circ(s,id()) -> s
        circ(circ(s,t),u) -> circ(s,circ(t,u))
        circ(cons(a,s),t) -> cons(msubst(a,t),circ(s,t))
        circ(cons(lift(),s),cons(a,t)) -> cons(a,circ(s,t))
        circ(cons(lift(),s),cons(lift(),t)) -> cons(lift(),circ(s,t))
        circ(id(),s) -> s
        msubst(a,id()) -> a
        msubst(msubst(a,s),t) -> msubst(a,circ(s,t))
      Signature:
        {circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/1,c_6/0,c_7/0,c_8/2,c_9/0}
      Obligation:
        Innermost
        basic terms: {circ#,msubst#,subst#}/{cons,id,lift}
    Applied Processor:
      RemoveWeakSuffixes
    Proof:
      Consider the dependency graph
        1:S:circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u))
           -->_2 circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t)):4
           -->_1 circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t)):4
           -->_2 circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t)):3
           -->_1 circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t)):3
           -->_2 circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t)):2
           -->_1 circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t)):2
           -->_2 circ#(id(),s) -> c_6():7
           -->_1 circ#(id(),s) -> c_6():7
           -->_2 circ#(s,id()) -> c_1():6
           -->_1 circ#(s,id()) -> c_1():6
           -->_2 circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u)):1
           -->_1 circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u)):1
        
        2:S:circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t))
           -->_1 msubst#(msubst(a,s),t) -> c_8(msubst#(a,circ(s,t)),circ#(s,t)):5
           -->_2 circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t)):4
           -->_2 circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t)):3
           -->_1 msubst#(a,id()) -> c_7():8
           -->_2 circ#(id(),s) -> c_6():7
           -->_2 circ#(s,id()) -> c_1():6
           -->_2 circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t)):2
           -->_2 circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u)):1
        
        3:S:circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t))
           -->_1 circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t)):4
           -->_1 circ#(id(),s) -> c_6():7
           -->_1 circ#(s,id()) -> c_1():6
           -->_1 circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t)):3
           -->_1 circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t)):2
           -->_1 circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u)):1
        
        4:S:circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t))
           -->_1 circ#(id(),s) -> c_6():7
           -->_1 circ#(s,id()) -> c_1():6
           -->_1 circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t)):4
           -->_1 circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t)):3
           -->_1 circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t)):2
           -->_1 circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u)):1
        
        5:S:msubst#(msubst(a,s),t) -> c_8(msubst#(a,circ(s,t)),circ#(s,t))
           -->_1 msubst#(a,id()) -> c_7():8
           -->_2 circ#(id(),s) -> c_6():7
           -->_2 circ#(s,id()) -> c_1():6
           -->_1 msubst#(msubst(a,s),t) -> c_8(msubst#(a,circ(s,t)),circ#(s,t)):5
           -->_2 circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t)):4
           -->_2 circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t)):3
           -->_2 circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t)):2
           -->_2 circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u)):1
        
        6:W:circ#(s,id()) -> c_1()
           
        
        7:W:circ#(id(),s) -> c_6()
           
        
        8:W:msubst#(a,id()) -> c_7()
           
        
        9:W:subst#(a,id()) -> c_9()
           
        
      The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
        9: subst#(a,id()) -> c_9() 
        8: msubst#(a,id()) -> c_7()
        6: circ#(s,id()) -> c_1()  
        7: circ#(id(),s) -> c_6()  
*** 1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
    Considered Problem:
      Strict DP Rules:
        circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u))
        circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t))
        circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t))
        circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t))
        msubst#(msubst(a,s),t) -> c_8(msubst#(a,circ(s,t)),circ#(s,t))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        circ(s,id()) -> s
        circ(circ(s,t),u) -> circ(s,circ(t,u))
        circ(cons(a,s),t) -> cons(msubst(a,t),circ(s,t))
        circ(cons(lift(),s),cons(a,t)) -> cons(a,circ(s,t))
        circ(cons(lift(),s),cons(lift(),t)) -> cons(lift(),circ(s,t))
        circ(id(),s) -> s
        msubst(a,id()) -> a
        msubst(msubst(a,s),t) -> msubst(a,circ(s,t))
      Signature:
        {circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/1,c_6/0,c_7/0,c_8/2,c_9/0}
      Obligation:
        Innermost
        basic terms: {circ#,msubst#,subst#}/{cons,id,lift}
    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: circ#(circ(s,t),u) ->                 
             c_2(circ#(s,circ(t,u))              
                ,circ#(t,u))                     
        2: circ#(cons(a,s),t) ->                 
             c_3(msubst#(a,t),circ#(s,t))        
        3: circ#(cons(lift(),s)                  
                ,cons(a,t)) -> c_4(circ#(s,t))   
        4: circ#(cons(lift(),s)                  
                ,cons(lift(),t)) -> c_5(circ#(s  
                                             ,t))
        
      The strictly oriented rules are moved into the weak component.
  *** 1.1.1.1.1.1.1 Progress [(?,O(n^1))]  ***
      Considered Problem:
        Strict DP Rules:
          circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u))
          circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t))
          circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t))
          circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t))
          msubst#(msubst(a,s),t) -> c_8(msubst#(a,circ(s,t)),circ#(s,t))
        Strict TRS Rules:
          
        Weak DP Rules:
          
        Weak TRS Rules:
          circ(s,id()) -> s
          circ(circ(s,t),u) -> circ(s,circ(t,u))
          circ(cons(a,s),t) -> cons(msubst(a,t),circ(s,t))
          circ(cons(lift(),s),cons(a,t)) -> cons(a,circ(s,t))
          circ(cons(lift(),s),cons(lift(),t)) -> cons(lift(),circ(s,t))
          circ(id(),s) -> s
          msubst(a,id()) -> a
          msubst(msubst(a,s),t) -> msubst(a,circ(s,t))
        Signature:
          {circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/1,c_6/0,c_7/0,c_8/2,c_9/0}
        Obligation:
          Innermost
          basic terms: {circ#,msubst#,subst#}/{cons,id,lift}
      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) = {1,2},
          uargs(c_3) = {1,2},
          uargs(c_4) = {1},
          uargs(c_5) = {1},
          uargs(c_8) = {1,2}
        
        Following symbols are considered usable:
          {circ#,msubst#,subst#}
        TcT has computed the following interpretation:
             p(circ) = [2] x1 + [1] x2 + [3]
             p(cons) = [1] x1 + [1] x2 + [2]
               p(id) = [0]                  
             p(lift) = [0]                  
           p(msubst) = [2] x1 + [2] x2 + [1]
            p(subst) = [0]                  
            p(circ#) = [4] x1 + [2]         
          p(msubst#) = [4] x1 + [5]         
           p(subst#) = [1] x1 + [0]         
              p(c_1) = [0]                  
              p(c_2) = [1] x1 + [1] x2 + [5]
              p(c_3) = [1] x1 + [1] x2 + [1]
              p(c_4) = [1] x1 + [5]         
              p(c_5) = [1] x1 + [5]         
              p(c_6) = [0]                  
              p(c_7) = [1]                  
              p(c_8) = [1] x1 + [1] x2 + [2]
              p(c_9) = [0]                  
        
        Following rules are strictly oriented:
                     circ#(circ(s,t),u) = [8] s + [4] t + [14]        
                                        > [4] s + [4] t + [9]         
                                        = c_2(circ#(s,circ(t,u))      
                                             ,circ#(t,u))             
        
                     circ#(cons(a,s),t) = [4] a + [4] s + [10]        
                                        > [4] a + [4] s + [8]         
                                        = c_3(msubst#(a,t),circ#(s,t))
        
        circ#(cons(lift(),s),cons(a,t)) = [4] s + [10]                
                                        > [4] s + [7]                 
                                        = c_4(circ#(s,t))             
        
                   circ#(cons(lift(),s) = [4] s + [10]                
                       ,cons(lift(),t))                               
                                        > [4] s + [7]                 
                                        = c_5(circ#(s,t))             
        
        
        Following rules are (at-least) weakly oriented:
        msubst#(msubst(a,s),t) =  [8] a + [8] s + [9]     
                               >= [4] a + [4] s + [9]     
                               =  c_8(msubst#(a,circ(s,t))
                                     ,circ#(s,t))         
        
  *** 1.1.1.1.1.1.1.1 Progress [(?,O(1))]  ***
      Considered Problem:
        Strict DP Rules:
          msubst#(msubst(a,s),t) -> c_8(msubst#(a,circ(s,t)),circ#(s,t))
        Strict TRS Rules:
          
        Weak DP Rules:
          circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u))
          circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t))
          circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t))
          circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t))
        Weak TRS Rules:
          circ(s,id()) -> s
          circ(circ(s,t),u) -> circ(s,circ(t,u))
          circ(cons(a,s),t) -> cons(msubst(a,t),circ(s,t))
          circ(cons(lift(),s),cons(a,t)) -> cons(a,circ(s,t))
          circ(cons(lift(),s),cons(lift(),t)) -> cons(lift(),circ(s,t))
          circ(id(),s) -> s
          msubst(a,id()) -> a
          msubst(msubst(a,s),t) -> msubst(a,circ(s,t))
        Signature:
          {circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/1,c_6/0,c_7/0,c_8/2,c_9/0}
        Obligation:
          Innermost
          basic terms: {circ#,msubst#,subst#}/{cons,id,lift}
      Applied Processor:
        Assumption
      Proof:
        ()
  
  *** 1.1.1.1.1.1.2 Progress [(?,O(n^1))]  ***
      Considered Problem:
        Strict DP Rules:
          msubst#(msubst(a,s),t) -> c_8(msubst#(a,circ(s,t)),circ#(s,t))
        Strict TRS Rules:
          
        Weak DP Rules:
          circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u))
          circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t))
          circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t))
          circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t))
        Weak TRS Rules:
          circ(s,id()) -> s
          circ(circ(s,t),u) -> circ(s,circ(t,u))
          circ(cons(a,s),t) -> cons(msubst(a,t),circ(s,t))
          circ(cons(lift(),s),cons(a,t)) -> cons(a,circ(s,t))
          circ(cons(lift(),s),cons(lift(),t)) -> cons(lift(),circ(s,t))
          circ(id(),s) -> s
          msubst(a,id()) -> a
          msubst(msubst(a,s),t) -> msubst(a,circ(s,t))
        Signature:
          {circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/1,c_6/0,c_7/0,c_8/2,c_9/0}
        Obligation:
          Innermost
          basic terms: {circ#,msubst#,subst#}/{cons,id,lift}
      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: msubst#(msubst(a,s),t) -> 
               c_8(msubst#(a,circ(s,t))
                  ,circ#(s,t))         
          
        The strictly oriented rules are moved into the weak component.
    *** 1.1.1.1.1.1.2.1 Progress [(?,O(n^1))]  ***
        Considered Problem:
          Strict DP Rules:
            msubst#(msubst(a,s),t) -> c_8(msubst#(a,circ(s,t)),circ#(s,t))
          Strict TRS Rules:
            
          Weak DP Rules:
            circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u))
            circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t))
            circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t))
            circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t))
          Weak TRS Rules:
            circ(s,id()) -> s
            circ(circ(s,t),u) -> circ(s,circ(t,u))
            circ(cons(a,s),t) -> cons(msubst(a,t),circ(s,t))
            circ(cons(lift(),s),cons(a,t)) -> cons(a,circ(s,t))
            circ(cons(lift(),s),cons(lift(),t)) -> cons(lift(),circ(s,t))
            circ(id(),s) -> s
            msubst(a,id()) -> a
            msubst(msubst(a,s),t) -> msubst(a,circ(s,t))
          Signature:
            {circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/1,c_6/0,c_7/0,c_8/2,c_9/0}
          Obligation:
            Innermost
            basic terms: {circ#,msubst#,subst#}/{cons,id,lift}
        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) = {1,2},
            uargs(c_3) = {1,2},
            uargs(c_4) = {1},
            uargs(c_5) = {1},
            uargs(c_8) = {1,2}
          
          Following symbols are considered usable:
            {circ#,msubst#,subst#}
          TcT has computed the following interpretation:
               p(circ) = [2] x1 + [1] x2 + [0]
               p(cons) = [1] x1 + [1] x2 + [0]
                 p(id) = [0]                  
               p(lift) = [0]                  
             p(msubst) = [2] x1 + [7] x2 + [2]
              p(subst) = [1] x1 + [4]         
              p(circ#) = [6] x1 + [0]         
            p(msubst#) = [1] x1 + [0]         
             p(subst#) = [1] x1 + [1] x2 + [0]
                p(c_1) = [4]                  
                p(c_2) = [1] x1 + [1] x2 + [0]
                p(c_3) = [6] x1 + [1] x2 + [0]
                p(c_4) = [1] x1 + [0]         
                p(c_5) = [1] x1 + [0]         
                p(c_6) = [0]                  
                p(c_7) = [0]                  
                p(c_8) = [2] x1 + [1] x2 + [1]
                p(c_9) = [1]                  
          
          Following rules are strictly oriented:
          msubst#(msubst(a,s),t) = [2] a + [7] s + [2]     
                                 > [2] a + [6] s + [1]     
                                 = c_8(msubst#(a,circ(s,t))
                                      ,circ#(s,t))         
          
          
          Following rules are (at-least) weakly oriented:
                       circ#(circ(s,t),u) =  [12] s + [6] t + [0]        
                                          >= [6] s + [6] t + [0]         
                                          =  c_2(circ#(s,circ(t,u))      
                                                ,circ#(t,u))             
          
                       circ#(cons(a,s),t) =  [6] a + [6] s + [0]         
                                          >= [6] a + [6] s + [0]         
                                          =  c_3(msubst#(a,t),circ#(s,t))
          
          circ#(cons(lift(),s),cons(a,t)) =  [6] s + [0]                 
                                          >= [6] s + [0]                 
                                          =  c_4(circ#(s,t))             
          
                     circ#(cons(lift(),s) =  [6] s + [0]                 
                         ,cons(lift(),t))                                
                                          >= [6] s + [0]                 
                                          =  c_5(circ#(s,t))             
          
    *** 1.1.1.1.1.1.2.1.1 Progress [(?,O(1))]  ***
        Considered Problem:
          Strict DP Rules:
            
          Strict TRS Rules:
            
          Weak DP Rules:
            circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u))
            circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t))
            circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t))
            circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t))
            msubst#(msubst(a,s),t) -> c_8(msubst#(a,circ(s,t)),circ#(s,t))
          Weak TRS Rules:
            circ(s,id()) -> s
            circ(circ(s,t),u) -> circ(s,circ(t,u))
            circ(cons(a,s),t) -> cons(msubst(a,t),circ(s,t))
            circ(cons(lift(),s),cons(a,t)) -> cons(a,circ(s,t))
            circ(cons(lift(),s),cons(lift(),t)) -> cons(lift(),circ(s,t))
            circ(id(),s) -> s
            msubst(a,id()) -> a
            msubst(msubst(a,s),t) -> msubst(a,circ(s,t))
          Signature:
            {circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/1,c_6/0,c_7/0,c_8/2,c_9/0}
          Obligation:
            Innermost
            basic terms: {circ#,msubst#,subst#}/{cons,id,lift}
        Applied Processor:
          Assumption
        Proof:
          ()
    
    *** 1.1.1.1.1.1.2.2 Progress [(O(1),O(1))]  ***
        Considered Problem:
          Strict DP Rules:
            
          Strict TRS Rules:
            
          Weak DP Rules:
            circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u))
            circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t))
            circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t))
            circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t))
            msubst#(msubst(a,s),t) -> c_8(msubst#(a,circ(s,t)),circ#(s,t))
          Weak TRS Rules:
            circ(s,id()) -> s
            circ(circ(s,t),u) -> circ(s,circ(t,u))
            circ(cons(a,s),t) -> cons(msubst(a,t),circ(s,t))
            circ(cons(lift(),s),cons(a,t)) -> cons(a,circ(s,t))
            circ(cons(lift(),s),cons(lift(),t)) -> cons(lift(),circ(s,t))
            circ(id(),s) -> s
            msubst(a,id()) -> a
            msubst(msubst(a,s),t) -> msubst(a,circ(s,t))
          Signature:
            {circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/1,c_6/0,c_7/0,c_8/2,c_9/0}
          Obligation:
            Innermost
            basic terms: {circ#,msubst#,subst#}/{cons,id,lift}
        Applied Processor:
          RemoveWeakSuffixes
        Proof:
          Consider the dependency graph
            1:W:circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u))
               -->_2 circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t)):4
               -->_1 circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t)):4
               -->_2 circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t)):3
               -->_1 circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t)):3
               -->_2 circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t)):2
               -->_1 circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t)):2
               -->_2 circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u)):1
               -->_1 circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u)):1
            
            2:W:circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t))
               -->_1 msubst#(msubst(a,s),t) -> c_8(msubst#(a,circ(s,t)),circ#(s,t)):5
               -->_2 circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t)):4
               -->_2 circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t)):3
               -->_2 circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t)):2
               -->_2 circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u)):1
            
            3:W:circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t))
               -->_1 circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t)):4
               -->_1 circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t)):3
               -->_1 circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t)):2
               -->_1 circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u)):1
            
            4:W:circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t))
               -->_1 circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t)):4
               -->_1 circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t)):3
               -->_1 circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t)):2
               -->_1 circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u)):1
            
            5:W:msubst#(msubst(a,s),t) -> c_8(msubst#(a,circ(s,t)),circ#(s,t))
               -->_1 msubst#(msubst(a,s),t) -> c_8(msubst#(a,circ(s,t)),circ#(s,t)):5
               -->_2 circ#(cons(lift(),s),cons(lift(),t)) -> c_5(circ#(s,t)):4
               -->_2 circ#(cons(lift(),s),cons(a,t)) -> c_4(circ#(s,t)):3
               -->_2 circ#(cons(a,s),t) -> c_3(msubst#(a,t),circ#(s,t)):2
               -->_2 circ#(circ(s,t),u) -> c_2(circ#(s,circ(t,u)),circ#(t,u)):1
            
          The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
            1: circ#(circ(s,t),u) ->                 
                 c_2(circ#(s,circ(t,u))              
                    ,circ#(t,u))                     
            5: msubst#(msubst(a,s),t) ->             
                 c_8(msubst#(a,circ(s,t))            
                    ,circ#(s,t))                     
            2: circ#(cons(a,s),t) ->                 
                 c_3(msubst#(a,t),circ#(s,t))        
            4: circ#(cons(lift(),s)                  
                    ,cons(lift(),t)) -> c_5(circ#(s  
                                                 ,t))
            3: circ#(cons(lift(),s)                  
                    ,cons(a,t)) -> c_4(circ#(s,t))   
    *** 1.1.1.1.1.1.2.2.1 Progress [(O(1),O(1))]  ***
        Considered Problem:
          Strict DP Rules:
            
          Strict TRS Rules:
            
          Weak DP Rules:
            
          Weak TRS Rules:
            circ(s,id()) -> s
            circ(circ(s,t),u) -> circ(s,circ(t,u))
            circ(cons(a,s),t) -> cons(msubst(a,t),circ(s,t))
            circ(cons(lift(),s),cons(a,t)) -> cons(a,circ(s,t))
            circ(cons(lift(),s),cons(lift(),t)) -> cons(lift(),circ(s,t))
            circ(id(),s) -> s
            msubst(a,id()) -> a
            msubst(msubst(a,s),t) -> msubst(a,circ(s,t))
          Signature:
            {circ/2,msubst/2,subst/2,circ#/2,msubst#/2,subst#/2} / {cons/2,id/0,lift/0,c_1/0,c_2/2,c_3/2,c_4/1,c_5/1,c_6/0,c_7/0,c_8/2,c_9/0}
          Obligation:
            Innermost
            basic terms: {circ#,msubst#,subst#}/{cons,id,lift}
        Applied Processor:
          EmptyProcessor
        Proof:
          The problem is already closed. The intended complexity is O(1).