/export/starexec/sandbox2/solver/bin/starexec_run_tct_rc /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum.  WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(x,y,s(z)) -> s(f(0(),1(),z))
            f(0(),1(),x) -> f(s(x),x,x)
            g(x,y) -> x
            g(x,y) -> y
        - Signature:
            {f/3,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:
            f(x,y,s(z)) -> s(f(0(),1(),z))
            f(0(),1(),x) -> f(s(x),x,x)
            g(x,y) -> x
            g(x,y) -> y
        - Signature:
            {f/3,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:
            f(x,y,s(z)) -> s(f(0(),1(),z))
            f(0(),1(),x) -> f(s(x),x,x)
            g(x,y) -> x
            g(x,y) -> y
        - Signature:
            {f/3,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:
          f(x,y,z){z -> s(z)} =
            f(x,y,s(z)) ->^+ s(f(0(),1(),z))
              = C[f(0(),1(),z) = f(x,y,z){x -> 0(),y -> 1()}]

** Step 1.b:1: DependencyPairs.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(x,y,s(z)) -> s(f(0(),1(),z))
            f(0(),1(),x) -> f(s(x),x,x)
            g(x,y) -> x
            g(x,y) -> y
        - Signature:
            {f/3,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
          f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
          f#(0(),1(),x) -> c_2(f#(s(x),x,x))
          g#(x,y) -> c_3(x)
          g#(x,y) -> c_4(y)
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: UsableRules.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
            f#(0(),1(),x) -> c_2(f#(s(x),x,x))
            g#(x,y) -> c_3(x)
            g#(x,y) -> c_4(y)
        - Strict TRS:
            f(x,y,s(z)) -> s(f(0(),1(),z))
            f(0(),1(),x) -> f(s(x),x,x)
            g(x,y) -> x
            g(x,y) -> y
        - Signature:
            {f/3,g/2,f#/3,g#/2} / {0/0,1/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,g#} and constructors {0,1,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
          f#(0(),1(),x) -> c_2(f#(s(x),x,x))
          g#(x,y) -> c_3(x)
          g#(x,y) -> c_4(y)
** Step 1.b:3: PredecessorEstimationCP.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
            f#(0(),1(),x) -> c_2(f#(s(x),x,x))
            g#(x,y) -> c_3(x)
            g#(x,y) -> c_4(y)
        - Signature:
            {f/3,g/2,f#/3,g#/2} / {0/0,1/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,g#} and constructors {0,1,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}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
          3: g#(x,y) -> c_3(x)
          
        The strictly oriented rules are moved into the weak component.
*** Step 1.b:3.a:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
            f#(0(),1(),x) -> c_2(f#(s(x),x,x))
            g#(x,y) -> c_3(x)
            g#(x,y) -> c_4(y)
        - Signature:
            {f/3,g/2,f#/3,g#/2} / {0/0,1/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,g#} and constructors {0,1,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_1) = {1},
          uargs(c_2) = {1}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
            p(0) = [0]                           
            p(1) = [0]                           
            p(f) = [2] x1 + [1] x2 + [1] x3 + [0]
            p(g) = [1]                           
            p(s) = [1] x1 + [1]                  
           p(f#) = [4] x3 + [8]                  
           p(g#) = [10] x1 + [5] x2 + [1]        
          p(c_1) = [1] x1 + [0]                  
          p(c_2) = [1] x1 + [0]                  
          p(c_3) = [0]                           
          p(c_4) = [2] x1 + [1]                  
        
        Following rules are strictly oriented:
        f#(x,y,s(z)) = [4] z + [12]        
                     > [4] z + [8]         
                     = c_1(f#(0(),1(),z))  
        
             g#(x,y) = [10] x + [5] y + [1]
                     > [0]                 
                     = c_3(x)              
        
        
        Following rules are (at-least) weakly oriented:
        f#(0(),1(),x) =  [4] x + [8]         
                      >= [4] x + [8]         
                      =  c_2(f#(s(x),x,x))   
        
              g#(x,y) =  [10] x + [5] y + [1]
                      >= [2] y + [1]         
                      =  c_4(y)              
        
*** Step 1.b:3.a:2: Assumption.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(0(),1(),x) -> c_2(f#(s(x),x,x))
            g#(x,y) -> c_4(y)
        - Weak DPs:
            f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
            g#(x,y) -> c_3(x)
        - Signature:
            {f/3,g/2,f#/3,g#/2} / {0/0,1/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,g#} and constructors {0,1,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}}
    + Details:
        ()

*** Step 1.b:3.b:1: PredecessorEstimationCP.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(0(),1(),x) -> c_2(f#(s(x),x,x))
            g#(x,y) -> c_4(y)
        - Weak DPs:
            f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
            g#(x,y) -> c_3(x)
        - Signature:
            {f/3,g/2,f#/3,g#/2} / {0/0,1/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,g#} and constructors {0,1,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}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          2: g#(x,y) -> c_4(y)
          
        The strictly oriented rules are moved into the weak component.
**** Step 1.b:3.b:1.a:1: NaturalMI.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(0(),1(),x) -> c_2(f#(s(x),x,x))
            g#(x,y) -> c_4(y)
        - Weak DPs:
            f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
            g#(x,y) -> c_3(x)
        - Signature:
            {f/3,g/2,f#/3,g#/2} / {0/0,1/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,g#} and constructors {0,1,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_1) = {1},
          uargs(c_2) = {1}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
            p(0) = [0]         
            p(1) = [1]         
            p(f) = [1] x1 + [1]
            p(g) = [4] x1 + [1]
            p(s) = [1]         
           p(f#) = [0]         
           p(g#) = [1] x1 + [7]
          p(c_1) = [1] x1 + [0]
          p(c_2) = [4] x1 + [0]
          p(c_3) = [4]         
          p(c_4) = [0]         
        
        Following rules are strictly oriented:
        g#(x,y) = [1] x + [7]
                > [0]        
                = c_4(y)     
        
        
        Following rules are (at-least) weakly oriented:
         f#(x,y,s(z)) =  [0]               
                      >= [0]               
                      =  c_1(f#(0(),1(),z))
        
        f#(0(),1(),x) =  [0]               
                      >= [0]               
                      =  c_2(f#(s(x),x,x)) 
        
              g#(x,y) =  [1] x + [7]       
                      >= [4]               
                      =  c_3(x)            
        
**** Step 1.b:3.b:1.a:2: Assumption.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            f#(0(),1(),x) -> c_2(f#(s(x),x,x))
        - Weak DPs:
            f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
            g#(x,y) -> c_3(x)
            g#(x,y) -> c_4(y)
        - Signature:
            {f/3,g/2,f#/3,g#/2} / {0/0,1/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,g#} and constructors {0,1,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}}
    + Details:
        ()

**** Step 1.b:3.b:1.b:1: PredecessorEstimationCP.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(0(),1(),x) -> c_2(f#(s(x),x,x))
        - Weak DPs:
            f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
            g#(x,y) -> c_3(x)
            g#(x,y) -> c_4(y)
        - Signature:
            {f/3,g/2,f#/3,g#/2} / {0/0,1/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,g#} and constructors {0,1,s}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: f#(0(),1(),x) -> c_2(f#(s(x),x,x))
          
        The strictly oriented rules are moved into the weak component.
***** Step 1.b:3.b:1.b:1.a:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(0(),1(),x) -> c_2(f#(s(x),x,x))
        - Weak DPs:
            f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
            g#(x,y) -> c_3(x)
            g#(x,y) -> c_4(y)
        - Signature:
            {f/3,g/2,f#/3,g#/2} / {0/0,1/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,g#} and constructors {0,1,s}
    + Applied Processor:
        NaturalMI {miDimension = 2, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        The following argument positions are considered usable:
          uargs(c_1) = {1},
          uargs(c_2) = {1}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
            p(0) = [0]                                 
                   [5]                                 
            p(1) = [1]                                 
                   [7]                                 
            p(f) = [4 4] x1 + [0 4] x2 + [1 1] x3 + [0]
                   [4 1]      [1 1]      [0 2]      [4]
            p(g) = [0 4] x1 + [2 1] x2 + [0]           
                   [4 1]      [1 0]      [2]           
            p(s) = [1 0] x1 + [2]                      
                   [0 0]      [4]                      
           p(f#) = [0 1] x1 + [0 0] x2 + [4 0] x3 + [2]
                   [4 0]      [0 1]      [0 0]      [1]
           p(g#) = [0 1] x1 + [1 1] x2 + [2]           
                   [2 1]      [0 2]      [0]           
          p(c_1) = [1 0] x1 + [3]                      
                   [0 0]      [1]                      
          p(c_2) = [1 0] x1 + [0]                      
                   [0 0]      [4]                      
          p(c_3) = [0 1] x1 + [0]                      
                   [1 0]      [0]                      
          p(c_4) = [0 1] x1 + [1]                      
                   [0 2]      [0]                      
        
        Following rules are strictly oriented:
        f#(0(),1(),x) = [4 0] x + [7]    
                        [0 0]     [8]    
                      > [4 0] x + [6]    
                        [0 0]     [4]    
                      = c_2(f#(s(x),x,x))
        
        
        Following rules are (at-least) weakly oriented:
        f#(x,y,s(z)) =  [0 1] x + [0 0] y + [4 0] z + [10]
                        [4 0]     [0 1]     [0 0]     [1] 
                     >= [4 0] z + [10]                    
                        [0 0]     [1]                     
                     =  c_1(f#(0(),1(),z))                
        
             g#(x,y) =  [0 1] x + [1 1] y + [2]           
                        [2 1]     [0 2]     [0]           
                     >= [0 1] x + [0]                     
                        [1 0]     [0]                     
                     =  c_3(x)                            
        
             g#(x,y) =  [0 1] x + [1 1] y + [2]           
                        [2 1]     [0 2]     [0]           
                     >= [0 1] y + [1]                     
                        [0 2]     [0]                     
                     =  c_4(y)                            
        
***** Step 1.b:3.b:1.b:1.a:2: Assumption.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
            f#(0(),1(),x) -> c_2(f#(s(x),x,x))
            g#(x,y) -> c_3(x)
            g#(x,y) -> c_4(y)
        - Signature:
            {f/3,g/2,f#/3,g#/2} / {0/0,1/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,g#} and constructors {0,1,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}}
    + Details:
        ()

***** Step 1.b:3.b:1.b:1.b:1: RemoveWeakSuffixes.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
            f#(0(),1(),x) -> c_2(f#(s(x),x,x))
            g#(x,y) -> c_3(x)
            g#(x,y) -> c_4(y)
        - Signature:
            {f/3,g/2,f#/3,g#/2} / {0/0,1/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,g#} and constructors {0,1,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
             -->_1 f#(0(),1(),x) -> c_2(f#(s(x),x,x)):2
             -->_1 f#(x,y,s(z)) -> c_1(f#(0(),1(),z)):1
          
          2:W:f#(0(),1(),x) -> c_2(f#(s(x),x,x))
             -->_1 f#(x,y,s(z)) -> c_1(f#(0(),1(),z)):1
          
          3:W:g#(x,y) -> c_3(x)
             -->_1 g#(x,y) -> c_4(y):4
             -->_1 g#(x,y) -> c_3(x):3
             -->_1 f#(0(),1(),x) -> c_2(f#(s(x),x,x)):2
             -->_1 f#(x,y,s(z)) -> c_1(f#(0(),1(),z)):1
          
          4:W:g#(x,y) -> c_4(y)
             -->_1 g#(x,y) -> c_4(y):4
             -->_1 g#(x,y) -> c_3(x):3
             -->_1 f#(0(),1(),x) -> c_2(f#(s(x),x,x)):2
             -->_1 f#(x,y,s(z)) -> c_1(f#(0(),1(),z)):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: g#(x,y) -> c_3(x)
          4: g#(x,y) -> c_4(y)
          1: f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
          2: f#(0(),1(),x) -> c_2(f#(s(x),x,x))
***** Step 1.b:3.b:1.b:1.b:2: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {f/3,g/2,f#/3,g#/2} / {0/0,1/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1}
        - 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^1))