/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(?,O(n^1))
* Step 1: Sum.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            cons(x,y) -> x
            cons(x,y) -> y
            f(s(a()),s(b()),x) -> f(x,x,x)
            g(f(s(x),s(y),z)) -> g(f(x,y,z))
        - Signature:
            {cons/2,f/3,g/1} / {a/0,b/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {cons,f,g} and constructors {a,b,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
* Step 2: DependencyPairs.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            cons(x,y) -> x
            cons(x,y) -> y
            f(s(a()),s(b()),x) -> f(x,x,x)
            g(f(s(x),s(y),z)) -> g(f(x,y,z))
        - Signature:
            {cons/2,f/3,g/1} / {a/0,b/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {cons,f,g} and constructors {a,b,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak dependency pairs:
        
        Strict DPs
          cons#(x,y) -> c_1(x)
          cons#(x,y) -> c_2(y)
          f#(s(a()),s(b()),x) -> c_3(f#(x,x,x))
          g#(f(s(x),s(y),z)) -> c_4(g#(f(x,y,z)))
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 3: UsableRules.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            cons#(x,y) -> c_1(x)
            cons#(x,y) -> c_2(y)
            f#(s(a()),s(b()),x) -> c_3(f#(x,x,x))
            g#(f(s(x),s(y),z)) -> c_4(g#(f(x,y,z)))
        - Strict TRS:
            cons(x,y) -> x
            cons(x,y) -> y
            f(s(a()),s(b()),x) -> f(x,x,x)
            g(f(s(x),s(y),z)) -> g(f(x,y,z))
        - Signature:
            {cons/2,f/3,g/1,cons#/2,f#/3,g#/1} / {a/0,b/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1}
        - Obligation:
             runtime complexity wrt. defined symbols {cons#,f#,g#} and constructors {a,b,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          cons#(x,y) -> c_1(x)
          cons#(x,y) -> c_2(y)
          f#(s(a()),s(b()),x) -> c_3(f#(x,x,x))
* Step 4: PredecessorEstimationCP.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            cons#(x,y) -> c_1(x)
            cons#(x,y) -> c_2(y)
            f#(s(a()),s(b()),x) -> c_3(f#(x,x,x))
        - Signature:
            {cons/2,f/3,g/1,cons#/2,f#/3,g#/1} / {a/0,b/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1}
        - Obligation:
             runtime complexity wrt. defined symbols {cons#,f#,g#} and constructors {a,b,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:
          3: f#(s(a()),s(b()),x) -> c_3(f#(x,x,x))
          
        The strictly oriented rules are moved into the weak component.
** Step 4.a:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            cons#(x,y) -> c_1(x)
            cons#(x,y) -> c_2(y)
            f#(s(a()),s(b()),x) -> c_3(f#(x,x,x))
        - Signature:
            {cons/2,f/3,g/1,cons#/2,f#/3,g#/1} / {a/0,b/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1}
        - Obligation:
             runtime complexity wrt. defined symbols {cons#,f#,g#} and constructors {a,b,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:
          none
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
              p(a) = [1]                  
              p(b) = [0]                  
           p(cons) = [0]                  
              p(f) = [1] x1 + [2] x3 + [0]
              p(g) = [0]                  
              p(s) = [1] x1 + [0]         
          p(cons#) = [0]                  
             p(f#) = [8] x1 + [9]         
             p(g#) = [0]                  
            p(c_1) = [0]                  
            p(c_2) = [0]                  
            p(c_3) = [8]                  
            p(c_4) = [0]                  
        
        Following rules are strictly oriented:
        f#(s(a()),s(b()),x) = [17]          
                            > [8]           
                            = c_3(f#(x,x,x))
        
        
        Following rules are (at-least) weakly oriented:
        cons#(x,y) =  [0]   
                   >= [0]   
                   =  c_1(x)
        
        cons#(x,y) =  [0]   
                   >= [0]   
                   =  c_2(y)
        
** Step 4.a:2: Assumption.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            cons#(x,y) -> c_1(x)
            cons#(x,y) -> c_2(y)
        - Weak DPs:
            f#(s(a()),s(b()),x) -> c_3(f#(x,x,x))
        - Signature:
            {cons/2,f/3,g/1,cons#/2,f#/3,g#/1} / {a/0,b/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1}
        - Obligation:
             runtime complexity wrt. defined symbols {cons#,f#,g#} and constructors {a,b,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}}
    + Details:
        ()

** Step 4.b:1: RemoveWeakSuffixes.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            cons#(x,y) -> c_1(x)
            cons#(x,y) -> c_2(y)
        - Weak DPs:
            f#(s(a()),s(b()),x) -> c_3(f#(x,x,x))
        - Signature:
            {cons/2,f/3,g/1,cons#/2,f#/3,g#/1} / {a/0,b/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1}
        - Obligation:
             runtime complexity wrt. defined symbols {cons#,f#,g#} and constructors {a,b,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:cons#(x,y) -> c_1(x)
             -->_1 f#(s(a()),s(b()),x) -> c_3(f#(x,x,x)):3
             -->_1 cons#(x,y) -> c_2(y):2
             -->_1 cons#(x,y) -> c_1(x):1
          
          2:S:cons#(x,y) -> c_2(y)
             -->_1 f#(s(a()),s(b()),x) -> c_3(f#(x,x,x)):3
             -->_1 cons#(x,y) -> c_2(y):2
             -->_1 cons#(x,y) -> c_1(x):1
          
          3:W:f#(s(a()),s(b()),x) -> c_3(f#(x,x,x))
             -->_1 f#(s(a()),s(b()),x) -> c_3(f#(x,x,x)):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: f#(s(a()),s(b()),x) -> c_3(f#(x,x,x))
** Step 4.b:2: PredecessorEstimationCP.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            cons#(x,y) -> c_1(x)
            cons#(x,y) -> c_2(y)
        - Signature:
            {cons/2,f/3,g/1,cons#/2,f#/3,g#/1} / {a/0,b/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1}
        - Obligation:
             runtime complexity wrt. defined symbols {cons#,f#,g#} and constructors {a,b,s}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 1, miDegree = 0, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: cons#(x,y) -> c_1(x)
          2: cons#(x,y) -> c_2(y)
          
        The strictly oriented rules are moved into the weak component.
*** Step 4.b:2.a:1: NaturalMI.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            cons#(x,y) -> c_1(x)
            cons#(x,y) -> c_2(y)
        - Signature:
            {cons/2,f/3,g/1,cons#/2,f#/3,g#/1} / {a/0,b/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1}
        - Obligation:
             runtime complexity wrt. defined symbols {cons#,f#,g#} and constructors {a,b,s}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 0, 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 0 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        The following argument positions are considered usable:
          none
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
              p(a) = [0]
              p(b) = [0]
           p(cons) = [0]
              p(f) = [0]
              p(g) = [0]
              p(s) = [0]
          p(cons#) = [5]
             p(f#) = [0]
             p(g#) = [0]
            p(c_1) = [0]
            p(c_2) = [0]
            p(c_3) = [0]
            p(c_4) = [0]
        
        Following rules are strictly oriented:
        cons#(x,y) = [5]   
                   > [0]   
                   = c_1(x)
        
        cons#(x,y) = [5]   
                   > [0]   
                   = c_2(y)
        
        
        Following rules are (at-least) weakly oriented:
        
*** Step 4.b:2.a:2: Assumption.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            cons#(x,y) -> c_1(x)
            cons#(x,y) -> c_2(y)
        - Signature:
            {cons/2,f/3,g/1,cons#/2,f#/3,g#/1} / {a/0,b/0,s/1,c_1/1,c_2/1,c_3/1,c_4/1}
        - Obligation:
             runtime complexity wrt. defined symbols {cons#,f#,g#} and constructors {a,b,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}}
    + Details:
        ()

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

WORST_CASE(?,O(n^1))