/export/starexec/sandbox2/solver/bin/starexec_run_tct_rci /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:
            innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,1,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: 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:
            innermost 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:
            innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,1,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        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()
          g#(x,y) -> c_4()
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation 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()
            g#(x,y) -> c_4()
        - Weak 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_3/0,c_4/0}
        - Obligation:
            innermost 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
          {3,4}
        by application of
          Pre({3,4}) = {}.
        Here rules are labelled as follows:
          1: f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
          2: f#(0(),1(),x) -> c_2(f#(s(x),x,x))
          3: g#(x,y) -> c_3()
          4: g#(x,y) -> c_4()
** Step 1.b:3: RemoveWeakSuffixes 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))
        - Weak DPs:
            g#(x,y) -> c_3()
            g#(x,y) -> c_4()
        - Weak 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_3/0,c_4/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,1,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S: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:S: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()
             
          
          4:W:g#(x,y) -> c_4()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: g#(x,y) -> c_4()
          3: g#(x,y) -> c_3()
** Step 1.b:4: 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))
        - Weak 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_3/0,c_4/0}
        - Obligation:
            innermost 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))
** Step 1.b:5: WeightGap 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))
        - Signature:
            {f/3,g/2,f#/3,g#/2} / {0/0,1/0,s/1,c_1/1,c_2/1,c_3/0,c_4/0}
        - Obligation:
            innermost 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 constant growth matrix-interpretation:
          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) = [1]         
              p(1) = [0]         
              p(f) = [0]         
              p(g) = [0]         
              p(s) = [0]         
             p(f#) = [4] x1 + [3]
             p(g#) = [0]         
            p(c_1) = [1] x1 + [2]
            p(c_2) = [1] x1 + [1]
            p(c_3) = [4]         
            p(c_4) = [2]         
          
          Following rules are strictly oriented:
          f#(0(),1(),x) = [7]              
                        > [4]              
                        = c_2(f#(s(x),x,x))
          
          
          Following rules are (at-least) weakly oriented:
          f#(x,y,s(z)) =  [4] x + [3]       
                       >= [9]               
                       =  c_1(f#(0(),1(),z))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:6: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(x,y,s(z)) -> c_1(f#(0(),1(),z))
        - Weak DPs:
            f#(0(),1(),x) -> c_2(f#(s(x),x,x))
        - Signature:
            {f/3,g/2,f#/3,g#/2} / {0/0,1/0,s/1,c_1/1,c_2/1,c_3/0,c_4/0}
        - Obligation:
            innermost 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 any 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:
          {f#,g#}
        TcT has computed the following interpretation:
            p(0) = [4]                  
            p(1) = [1]                  
            p(f) = [1] x1 + [2] x2 + [1]
            p(g) = [1] x1 + [2] x2 + [0]
            p(s) = [1] x1 + [8]         
           p(f#) = [1] x3 + [0]         
           p(g#) = [1] x2 + [4]         
          p(c_1) = [1] x1 + [7]         
          p(c_2) = [1] x1 + [0]         
          p(c_3) = [1]                  
          p(c_4) = [0]                  
        
        Following rules are strictly oriented:
        f#(x,y,s(z)) = [1] z + [8]       
                     > [1] z + [7]       
                     = c_1(f#(0(),1(),z))
        
        
        Following rules are (at-least) weakly oriented:
        f#(0(),1(),x) =  [1] x + [0]      
                      >= [1] x + [0]      
                      =  c_2(f#(s(x),x,x))
        
** Step 1.b:7: EmptyProcessor 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))
        - Signature:
            {f/3,g/2,f#/3,g#/2} / {0/0,1/0,s/1,c_1/1,c_2/1,c_3/0,c_4/0}
        - Obligation:
            innermost 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))