/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^1))
* Step 1: Sum.  WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(cons(x,k),l) -> g(k,l,cons(x,k))
            f(empty(),l) -> l
            g(a,b,c) -> f(a,cons(b,c))
        - Signature:
            {f/2,g/3} / {cons/2,empty/0}
        - Obligation:
             runtime complexity wrt. defined symbols {f,g} and constructors {cons,empty}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            f(cons(x,k),l) -> g(k,l,cons(x,k))
            f(empty(),l) -> l
            g(a,b,c) -> f(a,cons(b,c))
        - Signature:
            {f/2,g/3} / {cons/2,empty/0}
        - Obligation:
             runtime complexity wrt. defined symbols {f,g} and constructors {cons,empty}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            f(cons(x,k),l) -> g(k,l,cons(x,k))
            f(empty(),l) -> l
            g(a,b,c) -> f(a,cons(b,c))
        - Signature:
            {f/2,g/3} / {cons/2,empty/0}
        - Obligation:
             runtime complexity wrt. defined symbols {f,g} and constructors {cons,empty}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          f(x,z){x -> cons(y,x)} =
            f(cons(y,x),z) ->^+ f(x,cons(z,cons(y,x)))
              = C[f(x,cons(z,cons(y,x))) = f(x,z){z -> cons(z,cons(y,x))}]

** Step 1.b:1: DependencyPairs.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(cons(x,k),l) -> g(k,l,cons(x,k))
            f(empty(),l) -> l
            g(a,b,c) -> f(a,cons(b,c))
        - Signature:
            {f/2,g/3} / {cons/2,empty/0}
        - Obligation:
             runtime complexity wrt. defined symbols {f,g} and constructors {cons,empty}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak dependency pairs:
        
        Strict DPs
          f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
          f#(empty(),l) -> c_2(l)
          g#(a,b,c) -> c_3(f#(a,cons(b,c)))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: UsableRules.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
            f#(empty(),l) -> c_2(l)
            g#(a,b,c) -> c_3(f#(a,cons(b,c)))
        - Strict TRS:
            f(cons(x,k),l) -> g(k,l,cons(x,k))
            f(empty(),l) -> l
            g(a,b,c) -> f(a,cons(b,c))
        - Signature:
            {f/2,g/3,f#/2,g#/3} / {cons/2,empty/0,c_1/1,c_2/1,c_3/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,g#} and constructors {cons,empty}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
          f#(empty(),l) -> c_2(l)
          g#(a,b,c) -> c_3(f#(a,cons(b,c)))
** Step 1.b:3: PredecessorEstimationCP.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
            f#(empty(),l) -> c_2(l)
            g#(a,b,c) -> c_3(f#(a,cons(b,c)))
        - Signature:
            {f/2,g/3,f#/2,g#/3} / {cons/2,empty/0,c_1/1,c_2/1,c_3/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,g#} and constructors {cons,empty}
    + 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#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
          2: f#(empty(),l) -> c_2(l)
          3: g#(a,b,c) -> c_3(f#(a,cons(b,c)))
          
        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#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
            f#(empty(),l) -> c_2(l)
            g#(a,b,c) -> c_3(f#(a,cons(b,c)))
        - Signature:
            {f/2,g/3,f#/2,g#/3} / {cons/2,empty/0,c_1/1,c_2/1,c_3/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,g#} and constructors {cons,empty}
    + 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_3) = {1}
        
        Following symbols are considered usable:
          all
        TcT has computed the following interpretation:
           p(cons) = [1] x2 + [8]         
          p(empty) = [6]                  
              p(f) = [2] x1 + [1] x2 + [1]
              p(g) = [4] x2 + [1] x3 + [1]
             p(f#) = [3] x1 + [0]         
             p(g#) = [3] x1 + [1]         
            p(c_1) = [1] x1 + [15]        
            p(c_2) = [9]                  
            p(c_3) = [1] x1 + [0]         
        
        Following rules are strictly oriented:
        f#(cons(x,k),l) = [3] k + [24]          
                        > [3] k + [16]          
                        = c_1(g#(k,l,cons(x,k)))
        
          f#(empty(),l) = [18]                  
                        > [9]                   
                        = c_2(l)                
        
              g#(a,b,c) = [3] a + [1]           
                        > [3] a + [0]           
                        = c_3(f#(a,cons(b,c)))  
        
        
        Following rules are (at-least) weakly oriented:
        
*** Step 1.b:3.a:2: Assumption.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
            f#(empty(),l) -> c_2(l)
            g#(a,b,c) -> c_3(f#(a,cons(b,c)))
        - Signature:
            {f/2,g/3,f#/2,g#/3} / {cons/2,empty/0,c_1/1,c_2/1,c_3/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,g#} and constructors {cons,empty}
    + 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: RemoveWeakSuffixes.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
            f#(empty(),l) -> c_2(l)
            g#(a,b,c) -> c_3(f#(a,cons(b,c)))
        - Signature:
            {f/2,g/3,f#/2,g#/3} / {cons/2,empty/0,c_1/1,c_2/1,c_3/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,g#} and constructors {cons,empty}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
             -->_1 g#(a,b,c) -> c_3(f#(a,cons(b,c))):3
          
          2:W:f#(empty(),l) -> c_2(l)
             -->_1 g#(a,b,c) -> c_3(f#(a,cons(b,c))):3
             -->_1 f#(empty(),l) -> c_2(l):2
             -->_1 f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k))):1
          
          3:W:g#(a,b,c) -> c_3(f#(a,cons(b,c)))
             -->_1 f#(empty(),l) -> c_2(l):2
             -->_1 f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k))):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: f#(cons(x,k),l) -> c_1(g#(k,l,cons(x,k)))
          3: g#(a,b,c) -> c_3(f#(a,cons(b,c)))
          2: f#(empty(),l) -> c_2(l)
*** Step 1.b:3.b:2: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        
        - Signature:
            {f/2,g/3,f#/2,g#/3} / {cons/2,empty/0,c_1/1,c_2/1,c_3/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#,g#} and constructors {cons,empty}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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