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

** Step 1.b:1: DependencyPairs.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(0(),y) -> 0()
            f(s(x),y) -> f(f(x,y),y)
        - Signature:
            {f/2} / {0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f} and constructors {0,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak dependency pairs:
        
        Strict DPs
          f#(0(),y) -> c_1()
          f#(s(x),y) -> c_2(f#(f(x,y),y))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(0(),y) -> c_1()
            f#(s(x),y) -> c_2(f#(f(x,y),y))
        - Strict TRS:
            f(0(),y) -> 0()
            f(s(x),y) -> f(f(x,y),y)
        - Signature:
            {f/2,f#/2} / {0/0,s/1,c_1/0,c_2/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#} and constructors {0,s}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnTrs}
    + 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(f) = {1},
            uargs(f#) = {1},
            uargs(c_2) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
              p(0) = [2]         
              p(f) = [1] x1 + [2]
              p(s) = [1] x1 + [3]
             p(f#) = [1] x1 + [0]
            p(c_1) = [0]         
            p(c_2) = [1] x1 + [0]
          
          Following rules are strictly oriented:
           f#(0(),y) = [2]              
                     > [0]              
                     = c_1()            
          
          f#(s(x),y) = [1] x + [3]      
                     > [1] x + [2]      
                     = c_2(f#(f(x,y),y))
          
            f(0(),y) = [4]              
                     > [2]              
                     = 0()              
          
           f(s(x),y) = [1] x + [5]      
                     > [1] x + [4]      
                     = f(f(x,y),y)      
          
          
          Following rules are (at-least) weakly oriented:
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:3: RemoveWeakSuffixes.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f#(0(),y) -> c_1()
            f#(s(x),y) -> c_2(f#(f(x,y),y))
        - Weak TRS:
            f(0(),y) -> 0()
            f(s(x),y) -> f(f(x,y),y)
        - Signature:
            {f/2,f#/2} / {0/0,s/1,c_1/0,c_2/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:f#(0(),y) -> c_1()
             
          
          2:W:f#(s(x),y) -> c_2(f#(f(x,y),y))
             -->_1 f#(s(x),y) -> c_2(f#(f(x,y),y)):2
             -->_1 f#(0(),y) -> c_1():1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: f#(s(x),y) -> c_2(f#(f(x,y),y))
          1: f#(0(),y) -> c_1()
** Step 1.b:4: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            f(0(),y) -> 0()
            f(s(x),y) -> f(f(x,y),y)
        - Signature:
            {f/2,f#/2} / {0/0,s/1,c_1/0,c_2/1}
        - Obligation:
             runtime complexity wrt. defined symbols {f#} and constructors {0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

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