/export/starexec/sandbox/solver/bin/starexec_run_tct_rc /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files


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WORST_CASE(Omega(n^1),O(n^2))
* Step 1: Sum.  WORST_CASE(Omega(n^1),O(n^2))
    + Considered Problem:
        - Strict TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            f(0()) -> 1()
            f(s(x)) -> g(x,s(x))
            g(0(),y) -> y
            g(s(x),y) -> g(x,+(y,s(x)))
            g(s(x),y) -> g(x,s(+(y,x)))
        - Signature:
            {+/2,f/1,g/2} / {0/0,1/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {+,f,g} and constructors {0,1,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            f(0()) -> 1()
            f(s(x)) -> g(x,s(x))
            g(0(),y) -> y
            g(s(x),y) -> g(x,+(y,s(x)))
            g(s(x),y) -> g(x,s(+(y,x)))
        - Signature:
            {+/2,f/1,g/2} / {0/0,1/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {+,f,g} and constructors {0,1,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            f(0()) -> 1()
            f(s(x)) -> g(x,s(x))
            g(0(),y) -> y
            g(s(x),y) -> g(x,+(y,s(x)))
            g(s(x),y) -> g(x,s(+(y,x)))
        - Signature:
            {+/2,f/1,g/2} / {0/0,1/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {+,f,g} and constructors {0,1,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          +(x,y){y -> s(y)} =
            +(x,s(y)) ->^+ s(+(x,y))
              = C[+(x,y) = +(x,y){}]

** Step 1.b:1: DependencyPairs.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            f(0()) -> 1()
            f(s(x)) -> g(x,s(x))
            g(0(),y) -> y
            g(s(x),y) -> g(x,+(y,s(x)))
            g(s(x),y) -> g(x,s(+(y,x)))
        - Signature:
            {+/2,f/1,g/2} / {0/0,1/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {+,f,g} and constructors {0,1,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak dependency pairs:
        
        Strict DPs
          +#(x,0()) -> c_1(x)
          +#(x,s(y)) -> c_2(+#(x,y))
          f#(0()) -> c_3()
          f#(s(x)) -> c_4(g#(x,s(x)))
          g#(0(),y) -> c_5(y)
          g#(s(x),y) -> c_6(g#(x,+(y,s(x))))
          g#(s(x),y) -> c_7(g#(x,s(+(y,x))))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: UsableRules.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            +#(x,0()) -> c_1(x)
            +#(x,s(y)) -> c_2(+#(x,y))
            f#(0()) -> c_3()
            f#(s(x)) -> c_4(g#(x,s(x)))
            g#(0(),y) -> c_5(y)
            g#(s(x),y) -> c_6(g#(x,+(y,s(x))))
            g#(s(x),y) -> c_7(g#(x,s(+(y,x))))
        - Strict TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
            f(0()) -> 1()
            f(s(x)) -> g(x,s(x))
            g(0(),y) -> y
            g(s(x),y) -> g(x,+(y,s(x)))
            g(s(x),y) -> g(x,s(+(y,x)))
        - Signature:
            {+/2,f/1,g/2,+#/2,f#/1,g#/2} / {0/0,1/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1}
        - Obligation:
             runtime complexity wrt. defined symbols {+#,f#,g#} and constructors {0,1,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          +(x,0()) -> x
          +(x,s(y)) -> s(+(x,y))
          +#(x,0()) -> c_1(x)
          +#(x,s(y)) -> c_2(+#(x,y))
          f#(0()) -> c_3()
          f#(s(x)) -> c_4(g#(x,s(x)))
          g#(0(),y) -> c_5(y)
          g#(s(x),y) -> c_6(g#(x,+(y,s(x))))
          g#(s(x),y) -> c_7(g#(x,s(+(y,x))))
** Step 1.b:3: PredecessorEstimation.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            +#(x,0()) -> c_1(x)
            +#(x,s(y)) -> c_2(+#(x,y))
            f#(0()) -> c_3()
            f#(s(x)) -> c_4(g#(x,s(x)))
            g#(0(),y) -> c_5(y)
            g#(s(x),y) -> c_6(g#(x,+(y,s(x))))
            g#(s(x),y) -> c_7(g#(x,s(+(y,x))))
        - Strict TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
        - Signature:
            {+/2,f/1,g/2,+#/2,f#/1,g#/2} / {0/0,1/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1}
        - Obligation:
             runtime complexity wrt. defined symbols {+#,f#,g#} and constructors {0,1,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {3}
        by application of
          Pre({3}) = {1,5}.
        Here rules are labelled as follows:
          1: +#(x,0()) -> c_1(x)
          2: +#(x,s(y)) -> c_2(+#(x,y))
          3: f#(0()) -> c_3()
          4: f#(s(x)) -> c_4(g#(x,s(x)))
          5: g#(0(),y) -> c_5(y)
          6: g#(s(x),y) -> c_6(g#(x,+(y,s(x))))
          7: g#(s(x),y) -> c_7(g#(x,s(+(y,x))))
** Step 1.b:4: WeightGap.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            +#(x,0()) -> c_1(x)
            +#(x,s(y)) -> c_2(+#(x,y))
            f#(s(x)) -> c_4(g#(x,s(x)))
            g#(0(),y) -> c_5(y)
            g#(s(x),y) -> c_6(g#(x,+(y,s(x))))
            g#(s(x),y) -> c_7(g#(x,s(+(y,x))))
        - Strict TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
        - Weak DPs:
            f#(0()) -> c_3()
        - Signature:
            {+/2,f/1,g/2,+#/2,f#/1,g#/2} / {0/0,1/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1}
        - Obligation:
             runtime complexity wrt. defined symbols {+#,f#,g#} and constructors {0,1,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(+) = {1},
            uargs(s) = {1},
            uargs(g#) = {2},
            uargs(c_2) = {1},
            uargs(c_4) = {1},
            uargs(c_5) = {1},
            uargs(c_6) = {1},
            uargs(c_7) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
              p(+) = [1] x1 + [1]         
              p(0) = [0]                  
              p(1) = [0]                  
              p(f) = [4] x1 + [0]         
              p(g) = [1] x2 + [1]         
              p(s) = [1] x1 + [2]         
             p(+#) = [0]                  
             p(f#) = [3] x1 + [0]         
             p(g#) = [2] x1 + [1] x2 + [0]
            p(c_1) = [0]                  
            p(c_2) = [1] x1 + [0]         
            p(c_3) = [0]                  
            p(c_4) = [1] x1 + [0]         
            p(c_5) = [1] x1 + [0]         
            p(c_6) = [1] x1 + [0]         
            p(c_7) = [1] x1 + [0]         
          
          Following rules are strictly oriented:
            f#(s(x)) = [3] x + [6]         
                     > [3] x + [2]         
                     = c_4(g#(x,s(x)))     
          
          g#(s(x),y) = [2] x + [1] y + [4] 
                     > [2] x + [1] y + [1] 
                     = c_6(g#(x,+(y,s(x))))
          
          g#(s(x),y) = [2] x + [1] y + [4] 
                     > [2] x + [1] y + [3] 
                     = c_7(g#(x,s(+(y,x))))
          
            +(x,0()) = [1] x + [1]         
                     > [1] x + [0]         
                     = x                   
          
          
          Following rules are (at-least) weakly oriented:
           +#(x,0()) =  [0]         
                     >= [0]         
                     =  c_1(x)      
          
          +#(x,s(y)) =  [0]         
                     >= [0]         
                     =  c_2(+#(x,y))
          
             f#(0()) =  [0]         
                     >= [0]         
                     =  c_3()       
          
           g#(0(),y) =  [1] y + [0] 
                     >= [1] y + [0] 
                     =  c_5(y)      
          
           +(x,s(y)) =  [1] x + [1] 
                     >= [1] x + [3] 
                     =  s(+(x,y))   
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:5: WeightGap.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            +#(x,0()) -> c_1(x)
            +#(x,s(y)) -> c_2(+#(x,y))
            g#(0(),y) -> c_5(y)
        - Strict TRS:
            +(x,s(y)) -> s(+(x,y))
        - Weak DPs:
            f#(0()) -> c_3()
            f#(s(x)) -> c_4(g#(x,s(x)))
            g#(s(x),y) -> c_6(g#(x,+(y,s(x))))
            g#(s(x),y) -> c_7(g#(x,s(+(y,x))))
        - Weak TRS:
            +(x,0()) -> x
        - Signature:
            {+/2,f/1,g/2,+#/2,f#/1,g#/2} / {0/0,1/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1}
        - Obligation:
             runtime complexity wrt. defined symbols {+#,f#,g#} and constructors {0,1,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(+) = {1},
            uargs(s) = {1},
            uargs(g#) = {2},
            uargs(c_2) = {1},
            uargs(c_4) = {1},
            uargs(c_5) = {1},
            uargs(c_6) = {1},
            uargs(c_7) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
              p(+) = [1] x1 + [0]         
              p(0) = [0]                  
              p(1) = [1]                  
              p(f) = [1] x1 + [0]         
              p(g) = [8] x1 + [1] x2 + [1]
              p(s) = [1] x1 + [2]         
             p(+#) = [4] x2 + [2]         
             p(f#) = [10] x1 + [0]        
             p(g#) = [4] x1 + [1] x2 + [0]
            p(c_1) = [1]                  
            p(c_2) = [1] x1 + [7]         
            p(c_3) = [0]                  
            p(c_4) = [1] x1 + [2]         
            p(c_5) = [1] x1 + [4]         
            p(c_6) = [1] x1 + [0]         
            p(c_7) = [1] x1 + [0]         
          
          Following rules are strictly oriented:
           +#(x,0()) = [2]         
                     > [1]         
                     = c_1(x)      
          
          +#(x,s(y)) = [4] y + [10]
                     > [4] y + [9] 
                     = c_2(+#(x,y))
          
          
          Following rules are (at-least) weakly oriented:
             f#(0()) =  [0]                 
                     >= [0]                 
                     =  c_3()               
          
            f#(s(x)) =  [10] x + [20]       
                     >= [5] x + [4]         
                     =  c_4(g#(x,s(x)))     
          
           g#(0(),y) =  [1] y + [0]         
                     >= [1] y + [4]         
                     =  c_5(y)              
          
          g#(s(x),y) =  [4] x + [1] y + [8] 
                     >= [4] x + [1] y + [0] 
                     =  c_6(g#(x,+(y,s(x))))
          
          g#(s(x),y) =  [4] x + [1] y + [8] 
                     >= [4] x + [1] y + [2] 
                     =  c_7(g#(x,s(+(y,x))))
          
            +(x,0()) =  [1] x + [0]         
                     >= [1] x + [0]         
                     =  x                   
          
           +(x,s(y)) =  [1] x + [0]         
                     >= [1] x + [2]         
                     =  s(+(x,y))           
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:6: WeightGap.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            g#(0(),y) -> c_5(y)
        - Strict TRS:
            +(x,s(y)) -> s(+(x,y))
        - Weak DPs:
            +#(x,0()) -> c_1(x)
            +#(x,s(y)) -> c_2(+#(x,y))
            f#(0()) -> c_3()
            f#(s(x)) -> c_4(g#(x,s(x)))
            g#(s(x),y) -> c_6(g#(x,+(y,s(x))))
            g#(s(x),y) -> c_7(g#(x,s(+(y,x))))
        - Weak TRS:
            +(x,0()) -> x
        - Signature:
            {+/2,f/1,g/2,+#/2,f#/1,g#/2} / {0/0,1/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1}
        - Obligation:
             runtime complexity wrt. defined symbols {+#,f#,g#} and constructors {0,1,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following nonconstant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(+) = {1},
            uargs(s) = {1},
            uargs(g#) = {2},
            uargs(c_2) = {1},
            uargs(c_4) = {1},
            uargs(c_5) = {1},
            uargs(c_6) = {1},
            uargs(c_7) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
              p(+) = [1] x1 + [0]         
              p(0) = [0]                  
              p(1) = [0]                  
              p(f) = [1]                  
              p(g) = [1]                  
              p(s) = [1] x1 + [2]         
             p(+#) = [2] x1 + [1]         
             p(f#) = [6] x1 + [2]         
             p(g#) = [4] x1 + [1] x2 + [2]
            p(c_1) = [1] x1 + [0]         
            p(c_2) = [1] x1 + [0]         
            p(c_3) = [2]                  
            p(c_4) = [1] x1 + [0]         
            p(c_5) = [1] x1 + [1]         
            p(c_6) = [1] x1 + [0]         
            p(c_7) = [1] x1 + [1]         
          
          Following rules are strictly oriented:
          g#(0(),y) = [1] y + [2]
                    > [1] y + [1]
                    = c_5(y)     
          
          
          Following rules are (at-least) weakly oriented:
           +#(x,0()) =  [2] x + [1]         
                     >= [1] x + [0]         
                     =  c_1(x)              
          
          +#(x,s(y)) =  [2] x + [1]         
                     >= [2] x + [1]         
                     =  c_2(+#(x,y))        
          
             f#(0()) =  [2]                 
                     >= [2]                 
                     =  c_3()               
          
            f#(s(x)) =  [6] x + [14]        
                     >= [5] x + [4]         
                     =  c_4(g#(x,s(x)))     
          
          g#(s(x),y) =  [4] x + [1] y + [10]
                     >= [4] x + [1] y + [2] 
                     =  c_6(g#(x,+(y,s(x))))
          
          g#(s(x),y) =  [4] x + [1] y + [10]
                     >= [4] x + [1] y + [5] 
                     =  c_7(g#(x,s(+(y,x))))
          
            +(x,0()) =  [1] x + [0]         
                     >= [1] x + [0]         
                     =  x                   
          
           +(x,s(y)) =  [1] x + [0]         
                     >= [1] x + [2]         
                     =  s(+(x,y))           
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:7: NaturalPI.  WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            +(x,s(y)) -> s(+(x,y))
        - Weak DPs:
            +#(x,0()) -> c_1(x)
            +#(x,s(y)) -> c_2(+#(x,y))
            f#(0()) -> c_3()
            f#(s(x)) -> c_4(g#(x,s(x)))
            g#(0(),y) -> c_5(y)
            g#(s(x),y) -> c_6(g#(x,+(y,s(x))))
            g#(s(x),y) -> c_7(g#(x,s(+(y,x))))
        - Weak TRS:
            +(x,0()) -> x
        - Signature:
            {+/2,f/1,g/2,+#/2,f#/1,g#/2} / {0/0,1/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1}
        - Obligation:
             runtime complexity wrt. defined symbols {+#,f#,g#} and constructors {0,1,s}
    + Applied Processor:
        NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any intersect of strict-rules and rewrite-rules}
    + Details:
        We apply a polynomial interpretation of kind constructor-based(mixed(2)):
        The following argument positions are considered usable:
          uargs(+) = {1},
          uargs(s) = {1},
          uargs(g#) = {2},
          uargs(c_2) = {1},
          uargs(c_4) = {1},
          uargs(c_5) = {1},
          uargs(c_6) = {1},
          uargs(c_7) = {1}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
            p(+) = x1 + 2*x2               
            p(0) = 0                       
            p(1) = 1                       
            p(f) = 1 + 4*x1^2              
            p(g) = 1 + x1 + 4*x1^2 + 4*x2^2
            p(s) = 1 + x1                  
           p(+#) = 2 + 4*x1                
           p(f#) = 2 + 2*x1 + 4*x1^2       
           p(g#) = 2*x1 + 3*x1^2 + 2*x2    
          p(c_1) = x1                      
          p(c_2) = x1                      
          p(c_3) = 0                       
          p(c_4) = 5 + x1                  
          p(c_5) = x1                      
          p(c_6) = x1                      
          p(c_7) = 1 + x1                  
        
        Following rules are strictly oriented:
        +(x,s(y)) = 2 + x + 2*y
                  > 1 + x + 2*y
                  = s(+(x,y))  
        
        
        Following rules are (at-least) weakly oriented:
         +#(x,0()) =  2 + 4*x              
                   >= x                    
                   =  c_1(x)               
        
        +#(x,s(y)) =  2 + 4*x              
                   >= 2 + 4*x              
                   =  c_2(+#(x,y))         
        
           f#(0()) =  2                    
                   >= 0                    
                   =  c_3()                
        
          f#(s(x)) =  8 + 10*x + 4*x^2     
                   >= 7 + 4*x + 3*x^2      
                   =  c_4(g#(x,s(x)))      
        
         g#(0(),y) =  2*y                  
                   >= y                    
                   =  c_5(y)               
        
        g#(s(x),y) =  5 + 8*x + 3*x^2 + 2*y
                   >= 4 + 6*x + 3*x^2 + 2*y
                   =  c_6(g#(x,+(y,s(x)))) 
        
        g#(s(x),y) =  5 + 8*x + 3*x^2 + 2*y
                   >= 3 + 6*x + 3*x^2 + 2*y
                   =  c_7(g#(x,s(+(y,x)))) 
        
          +(x,0()) =  x                    
                   >= x                    
                   =  x                    
        
** Step 1.b:8: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            +#(x,0()) -> c_1(x)
            +#(x,s(y)) -> c_2(+#(x,y))
            f#(0()) -> c_3()
            f#(s(x)) -> c_4(g#(x,s(x)))
            g#(0(),y) -> c_5(y)
            g#(s(x),y) -> c_6(g#(x,+(y,s(x))))
            g#(s(x),y) -> c_7(g#(x,s(+(y,x))))
        - Weak TRS:
            +(x,0()) -> x
            +(x,s(y)) -> s(+(x,y))
        - Signature:
            {+/2,f/1,g/2,+#/2,f#/1,g#/2} / {0/0,1/0,s/1,c_1/1,c_2/1,c_3/0,c_4/1,c_5/1,c_6/1,c_7/1}
        - Obligation:
             runtime complexity wrt. defined symbols {+#,f#,g#} and constructors {0,1,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),O(n^2))