WORST_CASE(Omega(n^1),O(n^3))
* Step 1: Sum WORST_CASE(Omega(n^1),O(n^3))
    + Considered Problem:
        - Strict TRS:
            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} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {plus,times} and constructors {0,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            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} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {plus,times} and constructors {0,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          plus(x,y){y -> s(y)} =
            plus(x,s(y)) ->^+ s(plus(x,y))
              = C[plus(x,y) = plus(x,y){}]

** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict TRS:
            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} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {plus,times} and constructors {0,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        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.
** Step 1.b:2: PredecessorEstimation WORST_CASE(?,O(n^3))
    + Considered Problem:
        - 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 TRS:
            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 runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        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))
** Step 1.b:3: RemoveWeakSuffixes WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            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))
        - Weak DPs:
            plus#(x,0()) -> c_1()
            plus#(0(),x) -> c_3()
            times#(x,0()) -> c_5()
        - Weak TRS:
            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 runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        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()
** Step 1.b:4: DecomposeDG WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            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))
        - Weak TRS:
            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 runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s}
    + Applied Processor:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing}
    + Details:
        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)
*** Step 1.b:4.a:1: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y))
        - Weak TRS:
            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 runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        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))
*** Step 1.b:4.a:2: UsableRules WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            times#(x,s(y)) -> c_6(times#(x,y))
        - Weak TRS:
            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 runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          times#(x,s(y)) -> c_6(times#(x,y))
*** Step 1.b:4.a:3: WeightGap WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            times#(x,s(y)) -> c_6(times#(x,y))
        - 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 runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          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:
            all
          TcT has computed the following interpretation:
                 p(0) = [0]          
              p(plus) = [0]          
                 p(s) = [1] x1 + [11]
             p(times) = [0]          
             p(plus#) = [0]          
            p(times#) = [1] x2 + [0] 
               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 + [0] 
          
          Following rules are strictly oriented:
          times#(x,s(y)) = [1] y + [11]    
                         > [1] y + [0]     
                         = c_6(times#(x,y))
          
          
          Following rules are (at-least) weakly oriented:
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** Step 1.b:4.a:4: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            times#(x,s(y)) -> c_6(times#(x,y))
        - 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 runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

*** Step 1.b:4.b:1: NaturalMI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            plus#(x,s(y)) -> c_2(plus#(x,y))
            plus#(s(x),y) -> c_4(plus#(x,y))
        - Weak DPs:
            times#(x,s(y)) -> plus#(times(x,y),x)
            times#(x,s(y)) -> times#(x,y)
        - Weak TRS:
            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 runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_2) = {1},
          uargs(c_4) = {1}
        
        Following symbols are considered usable:
          {plus#,times#}
        TcT has computed the following interpretation:
               p(0) = [0]                  
            p(plus) = [1]                  
               p(s) = [1] x1 + [2]         
           p(times) = [2]                  
           p(plus#) = [8] x2 + [0]         
          p(times#) = [9] x1 + [8] x2 + [0]
             p(c_1) = [1]                  
             p(c_2) = [1] x1 + [9]         
             p(c_3) = [1]                  
             p(c_4) = [1] x1 + [0]         
             p(c_5) = [0]                  
             p(c_6) = [4] x1 + [1] x2 + [1]
        
        Following rules are strictly oriented:
        plus#(x,s(y)) = [8] y + [16]   
                      > [8] y + [9]    
                      = c_2(plus#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
         plus#(s(x),y) =  [8] y + [0]         
                       >= [8] y + [0]         
                       =  c_4(plus#(x,y))     
        
        times#(x,s(y)) =  [9] x + [8] y + [16]
                       >= [8] x + [0]         
                       =  plus#(times(x,y),x) 
        
        times#(x,s(y)) =  [9] x + [8] y + [16]
                       >= [9] x + [8] y + [0] 
                       =  times#(x,y)         
        
*** Step 1.b:4.b:2: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            plus#(s(x),y) -> c_4(plus#(x,y))
        - Weak DPs:
            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:
            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 runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s}
    + Applied Processor:
        NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        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) = x1*x2                                      
           p(plus#) = 3*x1 + 4*x2 + 3*x2^2                       
          p(times#) = 4 + 5*x1 + 3*x1*x2 + 6*x1^2 + 5*x2 + 4*x2^2
             p(c_1) = 0                                          
             p(c_2) = 7 + x1                                     
             p(c_3) = 1                                          
             p(c_4) = 2 + x1                                     
             p(c_5) = 4                                          
             p(c_6) = 1 + x1                                     
        
        Following rules are strictly oriented:
        plus#(s(x),y) = 3 + 3*x + 4*y + 3*y^2
                      > 2 + 3*x + 4*y + 3*y^2
                      = c_4(plus#(x,y))      
        
        
        Following rules are (at-least) weakly oriented:
         plus#(x,s(y)) =  7 + 3*x + 10*y + 3*y^2                 
                       >= 7 + 3*x + 4*y + 3*y^2                  
                       =  c_2(plus#(x,y))                        
        
        times#(x,s(y)) =  13 + 8*x + 3*x*y + 6*x^2 + 13*y + 4*y^2
                       >= 4*x + 3*x*y + 3*x^2                    
                       =  plus#(times(x,y),x)                    
        
        times#(x,s(y)) =  13 + 8*x + 3*x*y + 6*x^2 + 13*y + 4*y^2
                       >= 4 + 5*x + 3*x*y + 6*x^2 + 5*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()) =  0                                      
                       >= 0                                      
                       =  0()                                    
        
         times(x,s(y)) =  x + x*y                                
                       >= x + x*y                                
                       =  plus(times(x,y),x)                     
        
*** Step 1.b:4.b:3: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            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:
            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 runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),O(n^3))