*** 1 Progress [(?,O(n^3))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        plus(x,0()) -> x
        plus(x,s(y)) -> s(plus(x,y))
        plus(0(),x) -> x
        plus(s(x),y) -> s(plus(x,y))
        times(x,0()) -> 0()
        times(x,s(y)) -> plus(times(x,y),x)
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {plus/2,times/2} / {0/0,s/1}
      Obligation:
        Innermost
        basic terms: {plus,times}/{0,s}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      We add the following dependency tuples:
      
      Strict DPs
        plus#(x,0()) -> c_1()
        plus#(x,s(y)) -> c_2(plus#(x,y))
        plus#(0(),x) -> c_3()
        plus#(s(x),y) -> c_4(plus#(x,y))
        times#(x,0()) -> c_5()
        times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1 Progress [(?,O(n^3))]  ***
    Considered Problem:
      Strict DP Rules:
        plus#(x,0()) -> c_1()
        plus#(x,s(y)) -> c_2(plus#(x,y))
        plus#(0(),x) -> c_3()
        plus#(s(x),y) -> c_4(plus#(x,y))
        times#(x,0()) -> c_5()
        times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        plus(x,0()) -> x
        plus(x,s(y)) -> s(plus(x,y))
        plus(0(),x) -> x
        plus(s(x),y) -> s(plus(x,y))
        times(x,0()) -> 0()
        times(x,s(y)) -> plus(times(x,y),x)
      Signature:
        {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2}
      Obligation:
        Innermost
        basic terms: {plus#,times#}/{0,s}
    Applied Processor:
      PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    Proof:
      We estimate the number of application of
        {1,3,5}
      by application of
        Pre({1,3,5}) = {2,4,6}.
      Here rules are labelled as follows:
        1: plus#(x,0()) -> c_1()           
        2: plus#(x,s(y)) -> c_2(plus#(x,y))
        3: plus#(0(),x) -> c_3()           
        4: plus#(s(x),y) -> c_4(plus#(x,y))
        5: times#(x,0()) -> c_5()          
        6: times#(x,s(y)) ->               
             c_6(plus#(times(x,y),x)       
                ,times#(x,y))              
*** 1.1.1 Progress [(?,O(n^3))]  ***
    Considered Problem:
      Strict DP Rules:
        plus#(x,s(y)) -> c_2(plus#(x,y))
        plus#(s(x),y) -> c_4(plus#(x,y))
        times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
      Strict TRS Rules:
        
      Weak DP Rules:
        plus#(x,0()) -> c_1()
        plus#(0(),x) -> c_3()
        times#(x,0()) -> c_5()
      Weak TRS Rules:
        plus(x,0()) -> x
        plus(x,s(y)) -> s(plus(x,y))
        plus(0(),x) -> x
        plus(s(x),y) -> s(plus(x,y))
        times(x,0()) -> 0()
        times(x,s(y)) -> plus(times(x,y),x)
      Signature:
        {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2}
      Obligation:
        Innermost
        basic terms: {plus#,times#}/{0,s}
    Applied Processor:
      RemoveWeakSuffixes
    Proof:
      Consider the dependency graph
        1:S:plus#(x,s(y)) -> c_2(plus#(x,y))
           -->_1 plus#(s(x),y) -> c_4(plus#(x,y)):2
           -->_1 plus#(0(),x) -> c_3():5
           -->_1 plus#(x,0()) -> c_1():4
           -->_1 plus#(x,s(y)) -> c_2(plus#(x,y)):1
        
        2:S:plus#(s(x),y) -> c_4(plus#(x,y))
           -->_1 plus#(0(),x) -> c_3():5
           -->_1 plus#(x,0()) -> c_1():4
           -->_1 plus#(s(x),y) -> c_4(plus#(x,y)):2
           -->_1 plus#(x,s(y)) -> c_2(plus#(x,y)):1
        
        3:S:times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
           -->_2 times#(x,0()) -> c_5():6
           -->_1 plus#(0(),x) -> c_3():5
           -->_1 plus#(x,0()) -> c_1():4
           -->_2 times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y)):3
           -->_1 plus#(s(x),y) -> c_4(plus#(x,y)):2
           -->_1 plus#(x,s(y)) -> c_2(plus#(x,y)):1
        
        4:W:plus#(x,0()) -> c_1()
           
        
        5:W:plus#(0(),x) -> c_3()
           
        
        6:W:times#(x,0()) -> c_5()
           
        
      The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
        6: times#(x,0()) -> c_5()
        4: plus#(x,0()) -> c_1() 
        5: plus#(0(),x) -> c_3() 
*** 1.1.1.1 Progress [(?,O(n^3))]  ***
    Considered Problem:
      Strict DP Rules:
        plus#(x,s(y)) -> c_2(plus#(x,y))
        plus#(s(x),y) -> c_4(plus#(x,y))
        times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        plus(x,0()) -> x
        plus(x,s(y)) -> s(plus(x,y))
        plus(0(),x) -> x
        plus(s(x),y) -> s(plus(x,y))
        times(x,0()) -> 0()
        times(x,s(y)) -> plus(times(x,y),x)
      Signature:
        {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2}
      Obligation:
        Innermost
        basic terms: {plus#,times#}/{0,s}
    Applied Processor:
      Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
    Proof:
      We analyse the complexity of following sub-problems (R) and (S).
      Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
      
      Problem (R)
        Strict DP Rules:
          plus#(x,s(y)) -> c_2(plus#(x,y))
          plus#(s(x),y) -> c_4(plus#(x,y))
        Strict TRS Rules:
          
        Weak DP Rules:
          times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
        Weak TRS Rules:
          plus(x,0()) -> x
          plus(x,s(y)) -> s(plus(x,y))
          plus(0(),x) -> x
          plus(s(x),y) -> s(plus(x,y))
          times(x,0()) -> 0()
          times(x,s(y)) -> plus(times(x,y),x)
        Signature:
          {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2}
        Obligation:
          Innermost
          basic terms: {plus#,times#}/{0,s}
      
      Problem (S)
        Strict DP Rules:
          times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
        Strict TRS Rules:
          
        Weak DP Rules:
          plus#(x,s(y)) -> c_2(plus#(x,y))
          plus#(s(x),y) -> c_4(plus#(x,y))
        Weak TRS Rules:
          plus(x,0()) -> x
          plus(x,s(y)) -> s(plus(x,y))
          plus(0(),x) -> x
          plus(s(x),y) -> s(plus(x,y))
          times(x,0()) -> 0()
          times(x,s(y)) -> plus(times(x,y),x)
        Signature:
          {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2}
        Obligation:
          Innermost
          basic terms: {plus#,times#}/{0,s}
  *** 1.1.1.1.1 Progress [(?,O(n^3))]  ***
      Considered Problem:
        Strict DP Rules:
          plus#(x,s(y)) -> c_2(plus#(x,y))
          plus#(s(x),y) -> c_4(plus#(x,y))
        Strict TRS Rules:
          
        Weak DP Rules:
          times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
        Weak TRS Rules:
          plus(x,0()) -> x
          plus(x,s(y)) -> s(plus(x,y))
          plus(0(),x) -> x
          plus(s(x),y) -> s(plus(x,y))
          times(x,0()) -> 0()
          times(x,s(y)) -> plus(times(x,y),x)
        Signature:
          {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2}
        Obligation:
          Innermost
          basic terms: {plus#,times#}/{0,s}
      Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
      Proof:
        We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
          1: plus#(x,s(y)) -> c_2(plus#(x,y))
          
        The strictly oriented rules are moved into the weak component.
    *** 1.1.1.1.1.1 Progress [(?,O(n^2))]  ***
        Considered Problem:
          Strict DP Rules:
            plus#(x,s(y)) -> c_2(plus#(x,y))
            plus#(s(x),y) -> c_4(plus#(x,y))
          Strict TRS Rules:
            
          Weak DP Rules:
            times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
          Weak TRS Rules:
            plus(x,0()) -> x
            plus(x,s(y)) -> s(plus(x,y))
            plus(0(),x) -> x
            plus(s(x),y) -> s(plus(x,y))
            times(x,0()) -> 0()
            times(x,s(y)) -> plus(times(x,y),x)
          Signature:
            {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2}
          Obligation:
            Innermost
            basic terms: {plus#,times#}/{0,s}
        Applied Processor:
          NaturalPI {shape = Mixed 2, restrict = Restrict, 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 polynomial interpretation of kind constructor-based(mixed(2)):
          The following argument positions are considered usable:
            uargs(c_2) = {1},
            uargs(c_4) = {1},
            uargs(c_6) = {1,2}
          
          Following symbols are considered usable:
            {plus#,times#}
          TcT has computed the following interpretation:
                 p(0) = 0                               
              p(plus) = 4 + x1 + x1*x2 + 2*x1^2 + 3*x2^2
                 p(s) = 1 + x1                          
             p(times) = 2 + x1*x2 + x2^2                
             p(plus#) = 2*x2                            
            p(times#) = 5 + 7*x1 + 6*x1*x2              
               p(c_1) = 1                               
               p(c_2) = 1 + x1                          
               p(c_3) = 0                               
               p(c_4) = x1                              
               p(c_5) = 1                               
               p(c_6) = x1 + x2                         
          
          Following rules are strictly oriented:
          plus#(x,s(y)) = 2 + 2*y        
                        > 1 + 2*y        
                        = c_2(plus#(x,y))
          
          
          Following rules are (at-least) weakly oriented:
           plus#(s(x),y) =  2*y                    
                         >= 2*y                    
                         =  c_4(plus#(x,y))        
          
          times#(x,s(y)) =  5 + 13*x + 6*x*y       
                         >= 5 + 9*x + 6*x*y        
                         =  c_6(plus#(times(x,y),x)
                               ,times#(x,y))       
          
    *** 1.1.1.1.1.1.1 Progress [(?,O(1))]  ***
        Considered Problem:
          Strict DP Rules:
            plus#(s(x),y) -> c_4(plus#(x,y))
          Strict TRS Rules:
            
          Weak DP Rules:
            plus#(x,s(y)) -> c_2(plus#(x,y))
            times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
          Weak TRS Rules:
            plus(x,0()) -> x
            plus(x,s(y)) -> s(plus(x,y))
            plus(0(),x) -> x
            plus(s(x),y) -> s(plus(x,y))
            times(x,0()) -> 0()
            times(x,s(y)) -> plus(times(x,y),x)
          Signature:
            {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2}
          Obligation:
            Innermost
            basic terms: {plus#,times#}/{0,s}
        Applied Processor:
          Assumption
        Proof:
          ()
    
    *** 1.1.1.1.1.2 Progress [(?,O(n^3))]  ***
        Considered Problem:
          Strict DP Rules:
            plus#(s(x),y) -> c_4(plus#(x,y))
          Strict TRS Rules:
            
          Weak DP Rules:
            plus#(x,s(y)) -> c_2(plus#(x,y))
            times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
          Weak TRS Rules:
            plus(x,0()) -> x
            plus(x,s(y)) -> s(plus(x,y))
            plus(0(),x) -> x
            plus(s(x),y) -> s(plus(x,y))
            times(x,0()) -> 0()
            times(x,s(y)) -> plus(times(x,y),x)
          Signature:
            {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2}
          Obligation:
            Innermost
            basic terms: {plus#,times#}/{0,s}
        Applied Processor:
          DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
        Proof:
          We decompose the input problem according to the dependency graph into the upper component
            times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
          and a lower component
            plus#(x,s(y)) -> c_2(plus#(x,y))
            plus#(s(x),y) -> c_4(plus#(x,y))
          Further, following extension rules are added to the lower component.
            times#(x,s(y)) -> plus#(times(x,y),x)
            times#(x,s(y)) -> times#(x,y)
      *** 1.1.1.1.1.2.1 Progress [(?,O(n^1))]  ***
          Considered Problem:
            Strict DP Rules:
              times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
            Strict TRS Rules:
              
            Weak DP Rules:
              
            Weak TRS Rules:
              plus(x,0()) -> x
              plus(x,s(y)) -> s(plus(x,y))
              plus(0(),x) -> x
              plus(s(x),y) -> s(plus(x,y))
              times(x,0()) -> 0()
              times(x,s(y)) -> plus(times(x,y),x)
            Signature:
              {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2}
            Obligation:
              Innermost
              basic terms: {plus#,times#}/{0,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: times#(x,s(y)) ->        
                   c_6(plus#(times(x,y),x)
                      ,times#(x,y))       
              
            The strictly oriented rules are moved into the weak component.
        *** 1.1.1.1.1.2.1.1 Progress [(?,O(n^1))]  ***
            Considered Problem:
              Strict DP Rules:
                times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
              Strict TRS Rules:
                
              Weak DP Rules:
                
              Weak TRS Rules:
                plus(x,0()) -> x
                plus(x,s(y)) -> s(plus(x,y))
                plus(0(),x) -> x
                plus(s(x),y) -> s(plus(x,y))
                times(x,0()) -> 0()
                times(x,s(y)) -> plus(times(x,y),x)
              Signature:
                {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2}
              Obligation:
                Innermost
                basic terms: {plus#,times#}/{0,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_6) = {1,2}
              
              Following symbols are considered usable:
                {plus#,times#}
              TcT has computed the following interpretation:
                     p(0) = [0]                  
                  p(plus) = [0]                  
                     p(s) = [1] x1 + [15]        
                 p(times) = [1] x2 + [1]         
                 p(plus#) = [2]                  
                p(times#) = [1] x2 + [8]         
                   p(c_1) = [2]                  
                   p(c_2) = [1] x1 + [0]         
                   p(c_3) = [0]                  
                   p(c_4) = [0]                  
                   p(c_5) = [0]                  
                   p(c_6) = [4] x1 + [1] x2 + [5]
              
              Following rules are strictly oriented:
              times#(x,s(y)) = [1] y + [23]           
                             > [1] y + [21]           
                             = c_6(plus#(times(x,y),x)
                                  ,times#(x,y))       
              
              
              Following rules are (at-least) weakly oriented:
              
        *** 1.1.1.1.1.2.1.1.1 Progress [(?,O(1))]  ***
            Considered Problem:
              Strict DP Rules:
                
              Strict TRS Rules:
                
              Weak DP Rules:
                times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
              Weak TRS Rules:
                plus(x,0()) -> x
                plus(x,s(y)) -> s(plus(x,y))
                plus(0(),x) -> x
                plus(s(x),y) -> s(plus(x,y))
                times(x,0()) -> 0()
                times(x,s(y)) -> plus(times(x,y),x)
              Signature:
                {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2}
              Obligation:
                Innermost
                basic terms: {plus#,times#}/{0,s}
            Applied Processor:
              Assumption
            Proof:
              ()
        
        *** 1.1.1.1.1.2.1.2 Progress [(O(1),O(1))]  ***
            Considered Problem:
              Strict DP Rules:
                
              Strict TRS Rules:
                
              Weak DP Rules:
                times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
              Weak TRS Rules:
                plus(x,0()) -> x
                plus(x,s(y)) -> s(plus(x,y))
                plus(0(),x) -> x
                plus(s(x),y) -> s(plus(x,y))
                times(x,0()) -> 0()
                times(x,s(y)) -> plus(times(x,y),x)
              Signature:
                {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2}
              Obligation:
                Innermost
                basic terms: {plus#,times#}/{0,s}
            Applied Processor:
              RemoveWeakSuffixes
            Proof:
              Consider the dependency graph
                1:W:times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
                   -->_2 times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y)):1
                
              The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
                1: times#(x,s(y)) ->        
                     c_6(plus#(times(x,y),x)
                        ,times#(x,y))       
        *** 1.1.1.1.1.2.1.2.1 Progress [(O(1),O(1))]  ***
            Considered Problem:
              Strict DP Rules:
                
              Strict TRS Rules:
                
              Weak DP Rules:
                
              Weak TRS Rules:
                plus(x,0()) -> x
                plus(x,s(y)) -> s(plus(x,y))
                plus(0(),x) -> x
                plus(s(x),y) -> s(plus(x,y))
                times(x,0()) -> 0()
                times(x,s(y)) -> plus(times(x,y),x)
              Signature:
                {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2}
              Obligation:
                Innermost
                basic terms: {plus#,times#}/{0,s}
            Applied Processor:
              EmptyProcessor
            Proof:
              The problem is already closed. The intended complexity is O(1).
        
      *** 1.1.1.1.1.2.2 Progress [(?,O(n^2))]  ***
          Considered Problem:
            Strict DP Rules:
              plus#(s(x),y) -> c_4(plus#(x,y))
            Strict TRS Rules:
              
            Weak DP Rules:
              plus#(x,s(y)) -> c_2(plus#(x,y))
              times#(x,s(y)) -> plus#(times(x,y),x)
              times#(x,s(y)) -> times#(x,y)
            Weak TRS Rules:
              plus(x,0()) -> x
              plus(x,s(y)) -> s(plus(x,y))
              plus(0(),x) -> x
              plus(s(x),y) -> s(plus(x,y))
              times(x,0()) -> 0()
              times(x,s(y)) -> plus(times(x,y),x)
            Signature:
              {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2}
            Obligation:
              Innermost
              basic terms: {plus#,times#}/{0,s}
          Applied Processor:
            PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy}}
          Proof:
            We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing, greedy = NoGreedy} to orient following rules strictly:
              1: plus#(s(x),y) -> c_4(plus#(x,y))
              
            The strictly oriented rules are moved into the weak component.
        *** 1.1.1.1.1.2.2.1 Progress [(?,O(n^2))]  ***
            Considered Problem:
              Strict DP Rules:
                plus#(s(x),y) -> c_4(plus#(x,y))
              Strict TRS Rules:
                
              Weak DP Rules:
                plus#(x,s(y)) -> c_2(plus#(x,y))
                times#(x,s(y)) -> plus#(times(x,y),x)
                times#(x,s(y)) -> times#(x,y)
              Weak TRS Rules:
                plus(x,0()) -> x
                plus(x,s(y)) -> s(plus(x,y))
                plus(0(),x) -> x
                plus(s(x),y) -> s(plus(x,y))
                times(x,0()) -> 0()
                times(x,s(y)) -> plus(times(x,y),x)
              Signature:
                {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2}
              Obligation:
                Innermost
                basic terms: {plus#,times#}/{0,s}
            Applied Processor:
              NaturalPI {shape = Mixed 2, restrict = Restrict, 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 polynomial interpretation of kind constructor-based(mixed(2)):
              The following argument positions are considered usable:
                uargs(c_2) = {1},
                uargs(c_4) = {1}
              
              Following symbols are considered usable:
                {plus,times,plus#,times#}
              TcT has computed the following interpretation:
                     p(0) = 0                                          
                  p(plus) = x1 + x2                                    
                     p(s) = 1 + x1                                     
                 p(times) = 2*x1 + 2*x1*x2 + x2 + x2^2                 
                 p(plus#) = 2*x1 + 6*x2^2                              
                p(times#) = 2 + 2*x1 + 4*x1*x2 + 6*x1^2 + 6*x2 + 4*x2^2
                   p(c_1) = 1                                          
                   p(c_2) = x1                                         
                   p(c_3) = 0                                          
                   p(c_4) = x1                                         
                   p(c_5) = 1                                          
                   p(c_6) = x2                                         
              
              Following rules are strictly oriented:
              plus#(s(x),y) = 2 + 2*x + 6*y^2
                            > 2*x + 6*y^2    
                            = c_4(plus#(x,y))
              
              
              Following rules are (at-least) weakly oriented:
               plus#(x,s(y)) =  6 + 2*x + 12*y + 6*y^2                 
                             >= 2*x + 6*y^2                            
                             =  c_2(plus#(x,y))                        
              
              times#(x,s(y)) =  12 + 6*x + 4*x*y + 6*x^2 + 14*y + 4*y^2
                             >= 4*x + 4*x*y + 6*x^2 + 2*y + 2*y^2      
                             =  plus#(times(x,y),x)                    
              
              times#(x,s(y)) =  12 + 6*x + 4*x*y + 6*x^2 + 14*y + 4*y^2
                             >= 2 + 2*x + 4*x*y + 6*x^2 + 6*y + 4*y^2  
                             =  times#(x,y)                            
              
                 plus(x,0()) =  x                                      
                             >= x                                      
                             =  x                                      
              
                plus(x,s(y)) =  1 + x + y                              
                             >= 1 + x + y                              
                             =  s(plus(x,y))                           
              
                 plus(0(),x) =  x                                      
                             >= x                                      
                             =  x                                      
              
                plus(s(x),y) =  1 + x + y                              
                             >= 1 + x + y                              
                             =  s(plus(x,y))                           
              
                times(x,0()) =  2*x                                    
                             >= 0                                      
                             =  0()                                    
              
               times(x,s(y)) =  2 + 4*x + 2*x*y + 3*y + y^2            
                             >= 3*x + 2*x*y + y + y^2                  
                             =  plus(times(x,y),x)                     
              
        *** 1.1.1.1.1.2.2.1.1 Progress [(?,O(1))]  ***
            Considered Problem:
              Strict DP Rules:
                
              Strict TRS Rules:
                
              Weak DP Rules:
                plus#(x,s(y)) -> c_2(plus#(x,y))
                plus#(s(x),y) -> c_4(plus#(x,y))
                times#(x,s(y)) -> plus#(times(x,y),x)
                times#(x,s(y)) -> times#(x,y)
              Weak TRS Rules:
                plus(x,0()) -> x
                plus(x,s(y)) -> s(plus(x,y))
                plus(0(),x) -> x
                plus(s(x),y) -> s(plus(x,y))
                times(x,0()) -> 0()
                times(x,s(y)) -> plus(times(x,y),x)
              Signature:
                {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2}
              Obligation:
                Innermost
                basic terms: {plus#,times#}/{0,s}
            Applied Processor:
              Assumption
            Proof:
              ()
        
        *** 1.1.1.1.1.2.2.2 Progress [(O(1),O(1))]  ***
            Considered Problem:
              Strict DP Rules:
                
              Strict TRS Rules:
                
              Weak DP Rules:
                plus#(x,s(y)) -> c_2(plus#(x,y))
                plus#(s(x),y) -> c_4(plus#(x,y))
                times#(x,s(y)) -> plus#(times(x,y),x)
                times#(x,s(y)) -> times#(x,y)
              Weak TRS Rules:
                plus(x,0()) -> x
                plus(x,s(y)) -> s(plus(x,y))
                plus(0(),x) -> x
                plus(s(x),y) -> s(plus(x,y))
                times(x,0()) -> 0()
                times(x,s(y)) -> plus(times(x,y),x)
              Signature:
                {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2}
              Obligation:
                Innermost
                basic terms: {plus#,times#}/{0,s}
            Applied Processor:
              RemoveWeakSuffixes
            Proof:
              Consider the dependency graph
                1:W:plus#(x,s(y)) -> c_2(plus#(x,y))
                   -->_1 plus#(s(x),y) -> c_4(plus#(x,y)):2
                   -->_1 plus#(x,s(y)) -> c_2(plus#(x,y)):1
                
                2:W:plus#(s(x),y) -> c_4(plus#(x,y))
                   -->_1 plus#(s(x),y) -> c_4(plus#(x,y)):2
                   -->_1 plus#(x,s(y)) -> c_2(plus#(x,y)):1
                
                3:W:times#(x,s(y)) -> plus#(times(x,y),x)
                   -->_1 plus#(s(x),y) -> c_4(plus#(x,y)):2
                   -->_1 plus#(x,s(y)) -> c_2(plus#(x,y)):1
                
                4:W:times#(x,s(y)) -> times#(x,y)
                   -->_1 times#(x,s(y)) -> times#(x,y):4
                   -->_1 times#(x,s(y)) -> plus#(times(x,y),x):3
                
              The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
                4: times#(x,s(y)) -> times#(x,y)   
                3: times#(x,s(y)) -> plus#(times(x 
                                                ,y)
                                          ,x)      
                1: plus#(x,s(y)) -> c_2(plus#(x,y))
                2: plus#(s(x),y) -> c_4(plus#(x,y))
        *** 1.1.1.1.1.2.2.2.1 Progress [(O(1),O(1))]  ***
            Considered Problem:
              Strict DP Rules:
                
              Strict TRS Rules:
                
              Weak DP Rules:
                
              Weak TRS Rules:
                plus(x,0()) -> x
                plus(x,s(y)) -> s(plus(x,y))
                plus(0(),x) -> x
                plus(s(x),y) -> s(plus(x,y))
                times(x,0()) -> 0()
                times(x,s(y)) -> plus(times(x,y),x)
              Signature:
                {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2}
              Obligation:
                Innermost
                basic terms: {plus#,times#}/{0,s}
            Applied Processor:
              EmptyProcessor
            Proof:
              The problem is already closed. The intended complexity is O(1).
        
  *** 1.1.1.1.2 Progress [(?,O(n^1))]  ***
      Considered Problem:
        Strict DP Rules:
          times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
        Strict TRS Rules:
          
        Weak DP Rules:
          plus#(x,s(y)) -> c_2(plus#(x,y))
          plus#(s(x),y) -> c_4(plus#(x,y))
        Weak TRS Rules:
          plus(x,0()) -> x
          plus(x,s(y)) -> s(plus(x,y))
          plus(0(),x) -> x
          plus(s(x),y) -> s(plus(x,y))
          times(x,0()) -> 0()
          times(x,s(y)) -> plus(times(x,y),x)
        Signature:
          {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2}
        Obligation:
          Innermost
          basic terms: {plus#,times#}/{0,s}
      Applied Processor:
        RemoveWeakSuffixes
      Proof:
        Consider the dependency graph
          1:S:times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
             -->_1 plus#(s(x),y) -> c_4(plus#(x,y)):3
             -->_1 plus#(x,s(y)) -> c_2(plus#(x,y)):2
             -->_2 times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y)):1
          
          2:W:plus#(x,s(y)) -> c_2(plus#(x,y))
             -->_1 plus#(s(x),y) -> c_4(plus#(x,y)):3
             -->_1 plus#(x,s(y)) -> c_2(plus#(x,y)):2
          
          3:W:plus#(s(x),y) -> c_4(plus#(x,y))
             -->_1 plus#(s(x),y) -> c_4(plus#(x,y)):3
             -->_1 plus#(x,s(y)) -> c_2(plus#(x,y)):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: plus#(s(x),y) -> c_4(plus#(x,y))
          2: plus#(x,s(y)) -> c_2(plus#(x,y))
  *** 1.1.1.1.2.1 Progress [(?,O(n^1))]  ***
      Considered Problem:
        Strict DP Rules:
          times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
        Strict TRS Rules:
          
        Weak DP Rules:
          
        Weak TRS Rules:
          plus(x,0()) -> x
          plus(x,s(y)) -> s(plus(x,y))
          plus(0(),x) -> x
          plus(s(x),y) -> s(plus(x,y))
          times(x,0()) -> 0()
          times(x,s(y)) -> plus(times(x,y),x)
        Signature:
          {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2}
        Obligation:
          Innermost
          basic terms: {plus#,times#}/{0,s}
      Applied Processor:
        SimplifyRHS
      Proof:
        Consider the dependency graph
          1:S:times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
             -->_2 times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          times#(x,s(y)) -> c_6(times#(x,y))
  *** 1.1.1.1.2.1.1 Progress [(?,O(n^1))]  ***
      Considered Problem:
        Strict DP Rules:
          times#(x,s(y)) -> c_6(times#(x,y))
        Strict TRS Rules:
          
        Weak DP Rules:
          
        Weak TRS Rules:
          plus(x,0()) -> x
          plus(x,s(y)) -> s(plus(x,y))
          plus(0(),x) -> x
          plus(s(x),y) -> s(plus(x,y))
          times(x,0()) -> 0()
          times(x,s(y)) -> plus(times(x,y),x)
        Signature:
          {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1}
        Obligation:
          Innermost
          basic terms: {plus#,times#}/{0,s}
      Applied Processor:
        UsableRules
      Proof:
        We replace rewrite rules by usable rules:
          times#(x,s(y)) -> c_6(times#(x,y))
  *** 1.1.1.1.2.1.1.1 Progress [(?,O(n^1))]  ***
      Considered Problem:
        Strict DP Rules:
          times#(x,s(y)) -> c_6(times#(x,y))
        Strict TRS Rules:
          
        Weak DP Rules:
          
        Weak TRS Rules:
          
        Signature:
          {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1}
        Obligation:
          Innermost
          basic terms: {plus#,times#}/{0,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: times#(x,s(y)) -> c_6(times#(x  
                                         ,y))
          
        The strictly oriented rules are moved into the weak component.
    *** 1.1.1.1.2.1.1.1.1 Progress [(?,O(n^1))]  ***
        Considered Problem:
          Strict DP Rules:
            times#(x,s(y)) -> c_6(times#(x,y))
          Strict TRS Rules:
            
          Weak DP Rules:
            
          Weak TRS Rules:
            
          Signature:
            {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1}
          Obligation:
            Innermost
            basic terms: {plus#,times#}/{0,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_6) = {1}
          
          Following symbols are considered usable:
            {plus#,times#}
          TcT has computed the following interpretation:
                 p(0) = [0]         
              p(plus) = [0]         
                 p(s) = [1] x1 + [9]
             p(times) = [0]         
             p(plus#) = [0]         
            p(times#) = [1] x2 + [3]
               p(c_1) = [0]         
               p(c_2) = [0]         
               p(c_3) = [0]         
               p(c_4) = [0]         
               p(c_5) = [0]         
               p(c_6) = [1] x1 + [1]
          
          Following rules are strictly oriented:
          times#(x,s(y)) = [1] y + [12]    
                         > [1] y + [4]     
                         = c_6(times#(x,y))
          
          
          Following rules are (at-least) weakly oriented:
          
    *** 1.1.1.1.2.1.1.1.1.1 Progress [(?,O(1))]  ***
        Considered Problem:
          Strict DP Rules:
            
          Strict TRS Rules:
            
          Weak DP Rules:
            times#(x,s(y)) -> c_6(times#(x,y))
          Weak TRS Rules:
            
          Signature:
            {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1}
          Obligation:
            Innermost
            basic terms: {plus#,times#}/{0,s}
        Applied Processor:
          Assumption
        Proof:
          ()
    
    *** 1.1.1.1.2.1.1.1.2 Progress [(O(1),O(1))]  ***
        Considered Problem:
          Strict DP Rules:
            
          Strict TRS Rules:
            
          Weak DP Rules:
            times#(x,s(y)) -> c_6(times#(x,y))
          Weak TRS Rules:
            
          Signature:
            {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1}
          Obligation:
            Innermost
            basic terms: {plus#,times#}/{0,s}
        Applied Processor:
          RemoveWeakSuffixes
        Proof:
          Consider the dependency graph
            1:W:times#(x,s(y)) -> c_6(times#(x,y))
               -->_1 times#(x,s(y)) -> c_6(times#(x,y)):1
            
          The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
            1: times#(x,s(y)) -> c_6(times#(x  
                                           ,y))
    *** 1.1.1.1.2.1.1.1.2.1 Progress [(O(1),O(1))]  ***
        Considered Problem:
          Strict DP Rules:
            
          Strict TRS Rules:
            
          Weak DP Rules:
            
          Weak TRS Rules:
            
          Signature:
            {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1}
          Obligation:
            Innermost
            basic terms: {plus#,times#}/{0,s}
        Applied Processor:
          EmptyProcessor
        Proof:
          The problem is already closed. The intended complexity is O(1).