*** 1 Progress [(?,O(n^3))]  ***
    Considered Problem:
      Strict DP Rules:
        
      Strict TRS Rules:
        *(0(),y) -> 0()
        *(p(x),y) -> +(*(x,y),minus(y))
        *(s(x),y) -> +(*(x,y),y)
        +(0(),y) -> y
        +(p(x),y) -> p(+(x,y))
        +(s(x),y) -> s(+(x,y))
        minus(0()) -> 0()
        minus(p(x)) -> s(minus(x))
        minus(s(x)) -> p(minus(x))
      Weak DP Rules:
        
      Weak TRS Rules:
        
      Signature:
        {*/2,+/2,minus/1} / {0/0,p/1,s/1}
      Obligation:
        Innermost
        basic terms: {*,+,minus}/{0,p,s}
    Applied Processor:
      DependencyPairs {dpKind_ = DT}
    Proof:
      We add the following dependency tuples:
      
      Strict DPs
        *#(0(),y) -> c_1()
        *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
        *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
        +#(0(),y) -> c_4()
        +#(p(x),y) -> c_5(+#(x,y))
        +#(s(x),y) -> c_6(+#(x,y))
        minus#(0()) -> c_7()
        minus#(p(x)) -> c_8(minus#(x))
        minus#(s(x)) -> c_9(minus#(x))
      Weak DPs
        
      
      and mark the set of starting terms.
*** 1.1 Progress [(?,O(n^3))]  ***
    Considered Problem:
      Strict DP Rules:
        *#(0(),y) -> c_1()
        *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
        *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
        +#(0(),y) -> c_4()
        +#(p(x),y) -> c_5(+#(x,y))
        +#(s(x),y) -> c_6(+#(x,y))
        minus#(0()) -> c_7()
        minus#(p(x)) -> c_8(minus#(x))
        minus#(s(x)) -> c_9(minus#(x))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        *(0(),y) -> 0()
        *(p(x),y) -> +(*(x,y),minus(y))
        *(s(x),y) -> +(*(x,y),y)
        +(0(),y) -> y
        +(p(x),y) -> p(+(x,y))
        +(s(x),y) -> s(+(x,y))
        minus(0()) -> 0()
        minus(p(x)) -> s(minus(x))
        minus(s(x)) -> p(minus(x))
      Signature:
        {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
      Obligation:
        Innermost
        basic terms: {*#,+#,minus#}/{0,p,s}
    Applied Processor:
      PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    Proof:
      We estimate the number of application of
        {1,4,7}
      by application of
        Pre({1,4,7}) = {2,3,5,6,8,9}.
      Here rules are labelled as follows:
        1: *#(0(),y) -> c_1()            
        2: *#(p(x),y) -> c_2(+#(*(x,y)   
                               ,minus(y))
                            ,*#(x,y)     
                            ,minus#(y))  
        3: *#(s(x),y) -> c_3(+#(*(x,y),y)
                            ,*#(x,y))    
        4: +#(0(),y) -> c_4()            
        5: +#(p(x),y) -> c_5(+#(x,y))    
        6: +#(s(x),y) -> c_6(+#(x,y))    
        7: minus#(0()) -> c_7()          
        8: minus#(p(x)) -> c_8(minus#(x))
        9: minus#(s(x)) -> c_9(minus#(x))
*** 1.1.1 Progress [(?,O(n^3))]  ***
    Considered Problem:
      Strict DP Rules:
        *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
        *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
        +#(p(x),y) -> c_5(+#(x,y))
        +#(s(x),y) -> c_6(+#(x,y))
        minus#(p(x)) -> c_8(minus#(x))
        minus#(s(x)) -> c_9(minus#(x))
      Strict TRS Rules:
        
      Weak DP Rules:
        *#(0(),y) -> c_1()
        +#(0(),y) -> c_4()
        minus#(0()) -> c_7()
      Weak TRS Rules:
        *(0(),y) -> 0()
        *(p(x),y) -> +(*(x,y),minus(y))
        *(s(x),y) -> +(*(x,y),y)
        +(0(),y) -> y
        +(p(x),y) -> p(+(x,y))
        +(s(x),y) -> s(+(x,y))
        minus(0()) -> 0()
        minus(p(x)) -> s(minus(x))
        minus(s(x)) -> p(minus(x))
      Signature:
        {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
      Obligation:
        Innermost
        basic terms: {*#,+#,minus#}/{0,p,s}
    Applied Processor:
      RemoveWeakSuffixes
    Proof:
      Consider the dependency graph
        1:S:*#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
           -->_3 minus#(s(x)) -> c_9(minus#(x)):6
           -->_3 minus#(p(x)) -> c_8(minus#(x)):5
           -->_1 +#(s(x),y) -> c_6(+#(x,y)):4
           -->_1 +#(p(x),y) -> c_5(+#(x,y)):3
           -->_2 *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)):2
           -->_3 minus#(0()) -> c_7():9
           -->_1 +#(0(),y) -> c_4():8
           -->_2 *#(0(),y) -> c_1():7
           -->_2 *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)):1
        
        2:S:*#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
           -->_1 +#(s(x),y) -> c_6(+#(x,y)):4
           -->_1 +#(p(x),y) -> c_5(+#(x,y)):3
           -->_1 +#(0(),y) -> c_4():8
           -->_2 *#(0(),y) -> c_1():7
           -->_2 *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)):2
           -->_2 *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)):1
        
        3:S:+#(p(x),y) -> c_5(+#(x,y))
           -->_1 +#(s(x),y) -> c_6(+#(x,y)):4
           -->_1 +#(0(),y) -> c_4():8
           -->_1 +#(p(x),y) -> c_5(+#(x,y)):3
        
        4:S:+#(s(x),y) -> c_6(+#(x,y))
           -->_1 +#(0(),y) -> c_4():8
           -->_1 +#(s(x),y) -> c_6(+#(x,y)):4
           -->_1 +#(p(x),y) -> c_5(+#(x,y)):3
        
        5:S:minus#(p(x)) -> c_8(minus#(x))
           -->_1 minus#(s(x)) -> c_9(minus#(x)):6
           -->_1 minus#(0()) -> c_7():9
           -->_1 minus#(p(x)) -> c_8(minus#(x)):5
        
        6:S:minus#(s(x)) -> c_9(minus#(x))
           -->_1 minus#(0()) -> c_7():9
           -->_1 minus#(s(x)) -> c_9(minus#(x)):6
           -->_1 minus#(p(x)) -> c_8(minus#(x)):5
        
        7:W:*#(0(),y) -> c_1()
           
        
        8:W:+#(0(),y) -> c_4()
           
        
        9:W:minus#(0()) -> c_7()
           
        
      The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
        7: *#(0(),y) -> c_1()  
        8: +#(0(),y) -> c_4()  
        9: minus#(0()) -> c_7()
*** 1.1.1.1 Progress [(?,O(n^3))]  ***
    Considered Problem:
      Strict DP Rules:
        *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
        *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
        +#(p(x),y) -> c_5(+#(x,y))
        +#(s(x),y) -> c_6(+#(x,y))
        minus#(p(x)) -> c_8(minus#(x))
        minus#(s(x)) -> c_9(minus#(x))
      Strict TRS Rules:
        
      Weak DP Rules:
        
      Weak TRS Rules:
        *(0(),y) -> 0()
        *(p(x),y) -> +(*(x,y),minus(y))
        *(s(x),y) -> +(*(x,y),y)
        +(0(),y) -> y
        +(p(x),y) -> p(+(x,y))
        +(s(x),y) -> s(+(x,y))
        minus(0()) -> 0()
        minus(p(x)) -> s(minus(x))
        minus(s(x)) -> p(minus(x))
      Signature:
        {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
      Obligation:
        Innermost
        basic terms: {*#,+#,minus#}/{0,p,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: *#(p(x),y) -> c_2(+#(*(x,y)   
                               ,minus(y))
                            ,*#(x,y)     
                            ,minus#(y))  
        2: *#(s(x),y) -> c_3(+#(*(x,y),y)
                            ,*#(x,y))    
        5: minus#(p(x)) -> c_8(minus#(x))
        6: minus#(s(x)) -> c_9(minus#(x))
        
      The strictly oriented rules are moved into the weak component.
  *** 1.1.1.1.1 Progress [(?,O(n^2))]  ***
      Considered Problem:
        Strict DP Rules:
          *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
          *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
          +#(p(x),y) -> c_5(+#(x,y))
          +#(s(x),y) -> c_6(+#(x,y))
          minus#(p(x)) -> c_8(minus#(x))
          minus#(s(x)) -> c_9(minus#(x))
        Strict TRS Rules:
          
        Weak DP Rules:
          
        Weak TRS Rules:
          *(0(),y) -> 0()
          *(p(x),y) -> +(*(x,y),minus(y))
          *(s(x),y) -> +(*(x,y),y)
          +(0(),y) -> y
          +(p(x),y) -> p(+(x,y))
          +(s(x),y) -> s(+(x,y))
          minus(0()) -> 0()
          minus(p(x)) -> s(minus(x))
          minus(s(x)) -> p(minus(x))
        Signature:
          {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
        Obligation:
          Innermost
          basic terms: {*#,+#,minus#}/{0,p,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) = {1,2,3},
          uargs(c_3) = {1,2},
          uargs(c_5) = {1},
          uargs(c_6) = {1},
          uargs(c_8) = {1},
          uargs(c_9) = {1}
        
        Following symbols are considered usable:
          {minus,*#,+#,minus#}
        TcT has computed the following interpretation:
               p(*) = 0                                   
               p(+) = x2^2                                
               p(0) = 0                                   
           p(minus) = 2 + x1                              
               p(p) = 1 + x1                              
               p(s) = 1 + x1                              
              p(*#) = 2 + 2*x1 + 2*x1*x2 + 2*x1^2 + 3*x2^2
              p(+#) = x2                                  
          p(minus#) = x1                                  
             p(c_1) = 1                                   
             p(c_2) = x1 + x2 + x3                        
             p(c_3) = x1 + x2                             
             p(c_4) = 0                                   
             p(c_5) = x1                                  
             p(c_6) = x1                                  
             p(c_7) = 0                                   
             p(c_8) = x1                                  
             p(c_9) = x1                                  
        
        Following rules are strictly oriented:
          *#(p(x),y) = 6 + 6*x + 2*x*y + 2*x^2 + 2*y + 3*y^2
                     > 4 + 2*x + 2*x*y + 2*x^2 + 2*y + 3*y^2
                     = c_2(+#(*(x,y),minus(y))              
                          ,*#(x,y)                          
                          ,minus#(y))                       
        
          *#(s(x),y) = 6 + 6*x + 2*x*y + 2*x^2 + 2*y + 3*y^2
                     > 2 + 2*x + 2*x*y + 2*x^2 + y + 3*y^2  
                     = c_3(+#(*(x,y),y),*#(x,y))            
        
        minus#(p(x)) = 1 + x                                
                     > x                                    
                     = c_8(minus#(x))                       
        
        minus#(s(x)) = 1 + x                                
                     > x                                    
                     = c_9(minus#(x))                       
        
        
        Following rules are (at-least) weakly oriented:
         +#(p(x),y) =  y           
                    >= y           
                    =  c_5(+#(x,y))
        
         +#(s(x),y) =  y           
                    >= y           
                    =  c_6(+#(x,y))
        
         minus(0()) =  2           
                    >= 0           
                    =  0()         
        
        minus(p(x)) =  3 + x       
                    >= 3 + x       
                    =  s(minus(x)) 
        
        minus(s(x)) =  3 + x       
                    >= 3 + x       
                    =  p(minus(x)) 
        
  *** 1.1.1.1.1.1 Progress [(?,O(1))]  ***
      Considered Problem:
        Strict DP Rules:
          +#(p(x),y) -> c_5(+#(x,y))
          +#(s(x),y) -> c_6(+#(x,y))
        Strict TRS Rules:
          
        Weak DP Rules:
          *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
          *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
          minus#(p(x)) -> c_8(minus#(x))
          minus#(s(x)) -> c_9(minus#(x))
        Weak TRS Rules:
          *(0(),y) -> 0()
          *(p(x),y) -> +(*(x,y),minus(y))
          *(s(x),y) -> +(*(x,y),y)
          +(0(),y) -> y
          +(p(x),y) -> p(+(x,y))
          +(s(x),y) -> s(+(x,y))
          minus(0()) -> 0()
          minus(p(x)) -> s(minus(x))
          minus(s(x)) -> p(minus(x))
        Signature:
          {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
        Obligation:
          Innermost
          basic terms: {*#,+#,minus#}/{0,p,s}
      Applied Processor:
        Assumption
      Proof:
        ()
  
  *** 1.1.1.1.2 Progress [(?,O(n^3))]  ***
      Considered Problem:
        Strict DP Rules:
          +#(p(x),y) -> c_5(+#(x,y))
          +#(s(x),y) -> c_6(+#(x,y))
        Strict TRS Rules:
          
        Weak DP Rules:
          *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
          *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
          minus#(p(x)) -> c_8(minus#(x))
          minus#(s(x)) -> c_9(minus#(x))
        Weak TRS Rules:
          *(0(),y) -> 0()
          *(p(x),y) -> +(*(x,y),minus(y))
          *(s(x),y) -> +(*(x,y),y)
          +(0(),y) -> y
          +(p(x),y) -> p(+(x,y))
          +(s(x),y) -> s(+(x,y))
          minus(0()) -> 0()
          minus(p(x)) -> s(minus(x))
          minus(s(x)) -> p(minus(x))
        Signature:
          {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
        Obligation:
          Innermost
          basic terms: {*#,+#,minus#}/{0,p,s}
      Applied Processor:
        RemoveWeakSuffixes
      Proof:
        Consider the dependency graph
          1:S:+#(p(x),y) -> c_5(+#(x,y))
             -->_1 +#(s(x),y) -> c_6(+#(x,y)):2
             -->_1 +#(p(x),y) -> c_5(+#(x,y)):1
          
          2:S:+#(s(x),y) -> c_6(+#(x,y))
             -->_1 +#(s(x),y) -> c_6(+#(x,y)):2
             -->_1 +#(p(x),y) -> c_5(+#(x,y)):1
          
          3:W:*#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
             -->_3 minus#(s(x)) -> c_9(minus#(x)):6
             -->_3 minus#(p(x)) -> c_8(minus#(x)):5
             -->_2 *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)):4
             -->_2 *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)):3
             -->_1 +#(s(x),y) -> c_6(+#(x,y)):2
             -->_1 +#(p(x),y) -> c_5(+#(x,y)):1
          
          4:W:*#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
             -->_2 *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)):4
             -->_2 *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)):3
             -->_1 +#(s(x),y) -> c_6(+#(x,y)):2
             -->_1 +#(p(x),y) -> c_5(+#(x,y)):1
          
          5:W:minus#(p(x)) -> c_8(minus#(x))
             -->_1 minus#(s(x)) -> c_9(minus#(x)):6
             -->_1 minus#(p(x)) -> c_8(minus#(x)):5
          
          6:W:minus#(s(x)) -> c_9(minus#(x))
             -->_1 minus#(s(x)) -> c_9(minus#(x)):6
             -->_1 minus#(p(x)) -> c_8(minus#(x)):5
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          6: minus#(s(x)) -> c_9(minus#(x))
          5: minus#(p(x)) -> c_8(minus#(x))
  *** 1.1.1.1.2.1 Progress [(?,O(n^3))]  ***
      Considered Problem:
        Strict DP Rules:
          +#(p(x),y) -> c_5(+#(x,y))
          +#(s(x),y) -> c_6(+#(x,y))
        Strict TRS Rules:
          
        Weak DP Rules:
          *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
          *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
        Weak TRS Rules:
          *(0(),y) -> 0()
          *(p(x),y) -> +(*(x,y),minus(y))
          *(s(x),y) -> +(*(x,y),y)
          +(0(),y) -> y
          +(p(x),y) -> p(+(x,y))
          +(s(x),y) -> s(+(x,y))
          minus(0()) -> 0()
          minus(p(x)) -> s(minus(x))
          minus(s(x)) -> p(minus(x))
        Signature:
          {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/3,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
        Obligation:
          Innermost
          basic terms: {*#,+#,minus#}/{0,p,s}
      Applied Processor:
        SimplifyRHS
      Proof:
        Consider the dependency graph
          1:S:+#(p(x),y) -> c_5(+#(x,y))
             -->_1 +#(s(x),y) -> c_6(+#(x,y)):2
             -->_1 +#(p(x),y) -> c_5(+#(x,y)):1
          
          2:S:+#(s(x),y) -> c_6(+#(x,y))
             -->_1 +#(s(x),y) -> c_6(+#(x,y)):2
             -->_1 +#(p(x),y) -> c_5(+#(x,y)):1
          
          3:W:*#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y))
             -->_2 *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)):4
             -->_2 *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)):3
             -->_1 +#(s(x),y) -> c_6(+#(x,y)):2
             -->_1 +#(p(x),y) -> c_5(+#(x,y)):1
          
          4:W:*#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
             -->_2 *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)):4
             -->_2 *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y),minus#(y)):3
             -->_1 +#(s(x),y) -> c_6(+#(x,y)):2
             -->_1 +#(p(x),y) -> c_5(+#(x,y)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y))
  *** 1.1.1.1.2.1.1 Progress [(?,O(n^3))]  ***
      Considered Problem:
        Strict DP Rules:
          +#(p(x),y) -> c_5(+#(x,y))
          +#(s(x),y) -> c_6(+#(x,y))
        Strict TRS Rules:
          
        Weak DP Rules:
          *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y))
          *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
        Weak TRS Rules:
          *(0(),y) -> 0()
          *(p(x),y) -> +(*(x,y),minus(y))
          *(s(x),y) -> +(*(x,y),y)
          +(0(),y) -> y
          +(p(x),y) -> p(+(x,y))
          +(s(x),y) -> s(+(x,y))
          minus(0()) -> 0()
          minus(p(x)) -> s(minus(x))
          minus(s(x)) -> p(minus(x))
        Signature:
          {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
        Obligation:
          Innermost
          basic terms: {*#,+#,minus#}/{0,p,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
          *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y))
          *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
        and a lower component
          +#(p(x),y) -> c_5(+#(x,y))
          +#(s(x),y) -> c_6(+#(x,y))
        Further, following extension rules are added to the lower component.
          *#(p(x),y) -> *#(x,y)
          *#(p(x),y) -> +#(*(x,y),minus(y))
          *#(s(x),y) -> *#(x,y)
          *#(s(x),y) -> +#(*(x,y),y)
    *** 1.1.1.1.2.1.1.1 Progress [(?,O(n^1))]  ***
        Considered Problem:
          Strict DP Rules:
            *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y))
            *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
          Strict TRS Rules:
            
          Weak DP Rules:
            
          Weak TRS Rules:
            *(0(),y) -> 0()
            *(p(x),y) -> +(*(x,y),minus(y))
            *(s(x),y) -> +(*(x,y),y)
            +(0(),y) -> y
            +(p(x),y) -> p(+(x,y))
            +(s(x),y) -> s(+(x,y))
            minus(0()) -> 0()
            minus(p(x)) -> s(minus(x))
            minus(s(x)) -> p(minus(x))
          Signature:
            {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
          Obligation:
            Innermost
            basic terms: {*#,+#,minus#}/{0,p,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: *#(p(x),y) -> c_2(+#(*(x,y)   
                                   ,minus(y))
                                ,*#(x,y))    
            2: *#(s(x),y) -> c_3(+#(*(x,y),y)
                                ,*#(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:
              *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y))
              *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
            Strict TRS Rules:
              
            Weak DP Rules:
              
            Weak TRS Rules:
              *(0(),y) -> 0()
              *(p(x),y) -> +(*(x,y),minus(y))
              *(s(x),y) -> +(*(x,y),y)
              +(0(),y) -> y
              +(p(x),y) -> p(+(x,y))
              +(s(x),y) -> s(+(x,y))
              minus(0()) -> 0()
              minus(p(x)) -> s(minus(x))
              minus(s(x)) -> p(minus(x))
            Signature:
              {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
            Obligation:
              Innermost
              basic terms: {*#,+#,minus#}/{0,p,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) = {1,2},
              uargs(c_3) = {1,2}
            
            Following symbols are considered usable:
              {*#,+#,minus#}
            TcT has computed the following interpretation:
                   p(*) = [5] x1 + [1]         
                   p(+) = [4] x1 + [8]         
                   p(0) = [0]                  
               p(minus) = [1] x1 + [3]         
                   p(p) = [1] x1 + [3]         
                   p(s) = [1] x1 + [2]         
                  p(*#) = [8] x1 + [0]         
                  p(+#) = [0]                  
              p(minus#) = [0]                  
                 p(c_1) = [0]                  
                 p(c_2) = [1] x1 + [1] x2 + [9]
                 p(c_3) = [4] x1 + [1] x2 + [0]
                 p(c_4) = [0]                  
                 p(c_5) = [2]                  
                 p(c_6) = [0]                  
                 p(c_7) = [1]                  
                 p(c_8) = [1] x1 + [4]         
                 p(c_9) = [1]                  
            
            Following rules are strictly oriented:
            *#(p(x),y) = [8] x + [24]                    
                       > [8] x + [9]                     
                       = c_2(+#(*(x,y),minus(y)),*#(x,y))
            
            *#(s(x),y) = [8] x + [16]                    
                       > [8] x + [0]                     
                       = c_3(+#(*(x,y),y),*#(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:
              *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y))
              *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
            Weak TRS Rules:
              *(0(),y) -> 0()
              *(p(x),y) -> +(*(x,y),minus(y))
              *(s(x),y) -> +(*(x,y),y)
              +(0(),y) -> y
              +(p(x),y) -> p(+(x,y))
              +(s(x),y) -> s(+(x,y))
              minus(0()) -> 0()
              minus(p(x)) -> s(minus(x))
              minus(s(x)) -> p(minus(x))
            Signature:
              {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
            Obligation:
              Innermost
              basic terms: {*#,+#,minus#}/{0,p,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:
              *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y))
              *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
            Weak TRS Rules:
              *(0(),y) -> 0()
              *(p(x),y) -> +(*(x,y),minus(y))
              *(s(x),y) -> +(*(x,y),y)
              +(0(),y) -> y
              +(p(x),y) -> p(+(x,y))
              +(s(x),y) -> s(+(x,y))
              minus(0()) -> 0()
              minus(p(x)) -> s(minus(x))
              minus(s(x)) -> p(minus(x))
            Signature:
              {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
            Obligation:
              Innermost
              basic terms: {*#,+#,minus#}/{0,p,s}
          Applied Processor:
            RemoveWeakSuffixes
          Proof:
            Consider the dependency graph
              1:W:*#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y))
                 -->_2 *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)):2
                 -->_2 *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y)):1
              
              2:W:*#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y))
                 -->_2 *#(s(x),y) -> c_3(+#(*(x,y),y),*#(x,y)):2
                 -->_2 *#(p(x),y) -> c_2(+#(*(x,y),minus(y)),*#(x,y)):1
              
            The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
              1: *#(p(x),y) -> c_2(+#(*(x,y)   
                                     ,minus(y))
                                  ,*#(x,y))    
              2: *#(s(x),y) -> c_3(+#(*(x,y),y)
                                  ,*#(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:
              *(0(),y) -> 0()
              *(p(x),y) -> +(*(x,y),minus(y))
              *(s(x),y) -> +(*(x,y),y)
              +(0(),y) -> y
              +(p(x),y) -> p(+(x,y))
              +(s(x),y) -> s(+(x,y))
              minus(0()) -> 0()
              minus(p(x)) -> s(minus(x))
              minus(s(x)) -> p(minus(x))
            Signature:
              {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
            Obligation:
              Innermost
              basic terms: {*#,+#,minus#}/{0,p,s}
          Applied Processor:
            EmptyProcessor
          Proof:
            The problem is already closed. The intended complexity is O(1).
      
    *** 1.1.1.1.2.1.1.2 Progress [(?,O(n^2))]  ***
        Considered Problem:
          Strict DP Rules:
            +#(p(x),y) -> c_5(+#(x,y))
            +#(s(x),y) -> c_6(+#(x,y))
          Strict TRS Rules:
            
          Weak DP Rules:
            *#(p(x),y) -> *#(x,y)
            *#(p(x),y) -> +#(*(x,y),minus(y))
            *#(s(x),y) -> *#(x,y)
            *#(s(x),y) -> +#(*(x,y),y)
          Weak TRS Rules:
            *(0(),y) -> 0()
            *(p(x),y) -> +(*(x,y),minus(y))
            *(s(x),y) -> +(*(x,y),y)
            +(0(),y) -> y
            +(p(x),y) -> p(+(x,y))
            +(s(x),y) -> s(+(x,y))
            minus(0()) -> 0()
            minus(p(x)) -> s(minus(x))
            minus(s(x)) -> p(minus(x))
          Signature:
            {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
          Obligation:
            Innermost
            basic terms: {*#,+#,minus#}/{0,p,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: +#(p(x),y) -> c_5(+#(x,y))
            2: +#(s(x),y) -> c_6(+#(x,y))
            
          The strictly oriented rules are moved into the weak component.
      *** 1.1.1.1.2.1.1.2.1 Progress [(?,O(n^2))]  ***
          Considered Problem:
            Strict DP Rules:
              +#(p(x),y) -> c_5(+#(x,y))
              +#(s(x),y) -> c_6(+#(x,y))
            Strict TRS Rules:
              
            Weak DP Rules:
              *#(p(x),y) -> *#(x,y)
              *#(p(x),y) -> +#(*(x,y),minus(y))
              *#(s(x),y) -> *#(x,y)
              *#(s(x),y) -> +#(*(x,y),y)
            Weak TRS Rules:
              *(0(),y) -> 0()
              *(p(x),y) -> +(*(x,y),minus(y))
              *(s(x),y) -> +(*(x,y),y)
              +(0(),y) -> y
              +(p(x),y) -> p(+(x,y))
              +(s(x),y) -> s(+(x,y))
              minus(0()) -> 0()
              minus(p(x)) -> s(minus(x))
              minus(s(x)) -> p(minus(x))
            Signature:
              {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
            Obligation:
              Innermost
              basic terms: {*#,+#,minus#}/{0,p,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_5) = {1},
              uargs(c_6) = {1}
            
            Following symbols are considered usable:
              {*,+,minus,*#,+#,minus#}
            TcT has computed the following interpretation:
                   p(*) = x1*x2 + 2*x1^2               
                   p(+) = 1 + x1 + x2                  
                   p(0) = 1                            
               p(minus) = 1 + x1                       
                   p(p) = 1 + x1                       
                   p(s) = 1 + x1                       
                  p(*#) = 3 + 2*x1*x2 + 2*x1^2 + 2*x2^2
                  p(+#) = x1 + 2*x2                    
              p(minus#) = x1                           
                 p(c_1) = 0                            
                 p(c_2) = 1 + x2                       
                 p(c_3) = x1                           
                 p(c_4) = 0                            
                 p(c_5) = x1                           
                 p(c_6) = x1                           
                 p(c_7) = 1                            
                 p(c_8) = 0                            
                 p(c_9) = 0                            
            
            Following rules are strictly oriented:
            +#(p(x),y) = 1 + x + 2*y 
                       > x + 2*y     
                       = c_5(+#(x,y))
            
            +#(s(x),y) = 1 + x + 2*y 
                       > x + 2*y     
                       = c_6(+#(x,y))
            
            
            Following rules are (at-least) weakly oriented:
             *#(p(x),y) =  5 + 4*x + 2*x*y + 2*x^2 + 2*y + 2*y^2
                        >= 3 + 2*x*y + 2*x^2 + 2*y^2            
                        =  *#(x,y)                              
            
             *#(p(x),y) =  5 + 4*x + 2*x*y + 2*x^2 + 2*y + 2*y^2
                        >= 2 + x*y + 2*x^2 + 2*y                
                        =  +#(*(x,y),minus(y))                  
            
             *#(s(x),y) =  5 + 4*x + 2*x*y + 2*x^2 + 2*y + 2*y^2
                        >= 3 + 2*x*y + 2*x^2 + 2*y^2            
                        =  *#(x,y)                              
            
             *#(s(x),y) =  5 + 4*x + 2*x*y + 2*x^2 + 2*y + 2*y^2
                        >= x*y + 2*x^2 + 2*y                    
                        =  +#(*(x,y),y)                         
            
               *(0(),y) =  2 + y                                
                        >= 1                                    
                        =  0()                                  
            
              *(p(x),y) =  2 + 4*x + x*y + 2*x^2 + y            
                        >= 2 + x*y + 2*x^2 + y                  
                        =  +(*(x,y),minus(y))                   
            
              *(s(x),y) =  2 + 4*x + x*y + 2*x^2 + y            
                        >= 1 + x*y + 2*x^2 + y                  
                        =  +(*(x,y),y)                          
            
               +(0(),y) =  2 + y                                
                        >= y                                    
                        =  y                                    
            
              +(p(x),y) =  2 + x + y                            
                        >= 2 + x + y                            
                        =  p(+(x,y))                            
            
              +(s(x),y) =  2 + x + y                            
                        >= 2 + x + y                            
                        =  s(+(x,y))                            
            
             minus(0()) =  2                                    
                        >= 1                                    
                        =  0()                                  
            
            minus(p(x)) =  2 + x                                
                        >= 2 + x                                
                        =  s(minus(x))                          
            
            minus(s(x)) =  2 + x                                
                        >= 2 + x                                
                        =  p(minus(x))                          
            
      *** 1.1.1.1.2.1.1.2.1.1 Progress [(?,O(1))]  ***
          Considered Problem:
            Strict DP Rules:
              
            Strict TRS Rules:
              
            Weak DP Rules:
              *#(p(x),y) -> *#(x,y)
              *#(p(x),y) -> +#(*(x,y),minus(y))
              *#(s(x),y) -> *#(x,y)
              *#(s(x),y) -> +#(*(x,y),y)
              +#(p(x),y) -> c_5(+#(x,y))
              +#(s(x),y) -> c_6(+#(x,y))
            Weak TRS Rules:
              *(0(),y) -> 0()
              *(p(x),y) -> +(*(x,y),minus(y))
              *(s(x),y) -> +(*(x,y),y)
              +(0(),y) -> y
              +(p(x),y) -> p(+(x,y))
              +(s(x),y) -> s(+(x,y))
              minus(0()) -> 0()
              minus(p(x)) -> s(minus(x))
              minus(s(x)) -> p(minus(x))
            Signature:
              {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
            Obligation:
              Innermost
              basic terms: {*#,+#,minus#}/{0,p,s}
          Applied Processor:
            Assumption
          Proof:
            ()
      
      *** 1.1.1.1.2.1.1.2.2 Progress [(O(1),O(1))]  ***
          Considered Problem:
            Strict DP Rules:
              
            Strict TRS Rules:
              
            Weak DP Rules:
              *#(p(x),y) -> *#(x,y)
              *#(p(x),y) -> +#(*(x,y),minus(y))
              *#(s(x),y) -> *#(x,y)
              *#(s(x),y) -> +#(*(x,y),y)
              +#(p(x),y) -> c_5(+#(x,y))
              +#(s(x),y) -> c_6(+#(x,y))
            Weak TRS Rules:
              *(0(),y) -> 0()
              *(p(x),y) -> +(*(x,y),minus(y))
              *(s(x),y) -> +(*(x,y),y)
              +(0(),y) -> y
              +(p(x),y) -> p(+(x,y))
              +(s(x),y) -> s(+(x,y))
              minus(0()) -> 0()
              minus(p(x)) -> s(minus(x))
              minus(s(x)) -> p(minus(x))
            Signature:
              {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
            Obligation:
              Innermost
              basic terms: {*#,+#,minus#}/{0,p,s}
          Applied Processor:
            RemoveWeakSuffixes
          Proof:
            Consider the dependency graph
              1:W:*#(p(x),y) -> *#(x,y)
                 -->_1 *#(s(x),y) -> +#(*(x,y),y):4
                 -->_1 *#(s(x),y) -> *#(x,y):3
                 -->_1 *#(p(x),y) -> +#(*(x,y),minus(y)):2
                 -->_1 *#(p(x),y) -> *#(x,y):1
              
              2:W:*#(p(x),y) -> +#(*(x,y),minus(y))
                 -->_1 +#(s(x),y) -> c_6(+#(x,y)):6
                 -->_1 +#(p(x),y) -> c_5(+#(x,y)):5
              
              3:W:*#(s(x),y) -> *#(x,y)
                 -->_1 *#(s(x),y) -> +#(*(x,y),y):4
                 -->_1 *#(s(x),y) -> *#(x,y):3
                 -->_1 *#(p(x),y) -> +#(*(x,y),minus(y)):2
                 -->_1 *#(p(x),y) -> *#(x,y):1
              
              4:W:*#(s(x),y) -> +#(*(x,y),y)
                 -->_1 +#(s(x),y) -> c_6(+#(x,y)):6
                 -->_1 +#(p(x),y) -> c_5(+#(x,y)):5
              
              5:W:+#(p(x),y) -> c_5(+#(x,y))
                 -->_1 +#(s(x),y) -> c_6(+#(x,y)):6
                 -->_1 +#(p(x),y) -> c_5(+#(x,y)):5
              
              6:W:+#(s(x),y) -> c_6(+#(x,y))
                 -->_1 +#(s(x),y) -> c_6(+#(x,y)):6
                 -->_1 +#(p(x),y) -> c_5(+#(x,y)):5
              
            The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
              1: *#(p(x),y) -> *#(x,y)     
              3: *#(s(x),y) -> *#(x,y)     
              2: *#(p(x),y) -> +#(*(x,y)   
                                 ,minus(y))
              4: *#(s(x),y) -> +#(*(x,y),y)
              6: +#(s(x),y) -> c_6(+#(x,y))
              5: +#(p(x),y) -> c_5(+#(x,y))
      *** 1.1.1.1.2.1.1.2.2.1 Progress [(O(1),O(1))]  ***
          Considered Problem:
            Strict DP Rules:
              
            Strict TRS Rules:
              
            Weak DP Rules:
              
            Weak TRS Rules:
              *(0(),y) -> 0()
              *(p(x),y) -> +(*(x,y),minus(y))
              *(s(x),y) -> +(*(x,y),y)
              +(0(),y) -> y
              +(p(x),y) -> p(+(x,y))
              +(s(x),y) -> s(+(x,y))
              minus(0()) -> 0()
              minus(p(x)) -> s(minus(x))
              minus(s(x)) -> p(minus(x))
            Signature:
              {*/2,+/2,minus/1,*#/2,+#/2,minus#/1} / {0/0,p/1,s/1,c_1/0,c_2/2,c_3/2,c_4/0,c_5/1,c_6/1,c_7/0,c_8/1,c_9/1}
            Obligation:
              Innermost
              basic terms: {*#,+#,minus#}/{0,p,s}
          Applied Processor:
            EmptyProcessor
          Proof:
            The problem is already closed. The intended complexity is O(1).