/export/starexec/sandbox/solver/bin/starexec_run_tct_rci /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(x,0()) -> s(0())
            f(s(x),s(y)) -> s(f(x,y))
            g(0(),x) -> g(f(x,x),x)
        - Signature:
            {f/2,g/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g} 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(x,0()) -> s(0())
            f(s(x),s(y)) -> s(f(x,y))
            g(0(),x) -> g(f(x,x),x)
        - Signature:
            {f/2,g/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,s}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
*** Step 1.a:1.a:1: Ara.  MAYBE
    + Considered Problem:
        - Strict TRS:
            f(x,0()) -> s(0())
            f(s(x),s(y)) -> s(f(x,y))
            g(0(),x) -> g(f(x,x),x)
        - Signature:
            {f/2,g/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,s}
    + Applied Processor:
        Ara {minDegree = 1, maxDegree = 3, araTimeout = 15, araRuleShifting = Just 1, isBestCase = True, mkCompletelyDefined = False, verboseOutput = False}
    + Details:
        Signatures used:
        ----------------
          F (TrsFun "0") :: [] -(0)-> "A"(0, 0, 0)
          F (TrsFun "0") :: [] -(0)-> "A"(0, 0, 1)
          F (TrsFun "f") :: ["A"(0, 0, 0) x "A"(0, 0, 0)] -(1)-> "A"(0, 0, 0)
          F (TrsFun "g") :: ["A"(0, 0, 1) x "A"(0, 0, 0)] -(1)-> "A"(0, 0, 0)
          F (TrsFun "main") :: ["A"(0, 0, 1) x "A"(0, 0, 0)] -(1)-> "A"(0, 0, 0)
          F (TrsFun "s") :: ["A"(0, 0, 0)] -(0)-> "A"(0, 0, 0)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
        
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          f(x,0()) -> s(0())
          f(s(x),s(y)) -> s(f(x,y))
          g(0(),x) -> g(f(x,x),x)
          main(x1,x2) -> g(x1,x2)
        2. Weak:
          
        

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

** Step 1.b:1: DependencyPairs.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            f(x,0()) -> s(0())
            f(s(x),s(y)) -> s(f(x,y))
            g(0(),x) -> g(f(x,x),x)
        - Signature:
            {f/2,g/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f,g} and constructors {0,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          f#(x,0()) -> c_1()
          f#(s(x),s(y)) -> c_2(f#(x,y))
          g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
        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,0()) -> c_1()
            f#(s(x),s(y)) -> c_2(f#(x,y))
            g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
        - Weak TRS:
            f(x,0()) -> s(0())
            f(s(x),s(y)) -> s(f(x,y))
            g(0(),x) -> g(f(x,x),x)
        - Signature:
            {f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1}
        by application of
          Pre({1}) = {2,3}.
        Here rules are labelled as follows:
          1: f#(x,0()) -> c_1()
          2: f#(s(x),s(y)) -> c_2(f#(x,y))
          3: g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
** Step 1.b:3: RemoveWeakSuffixes.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(s(x),s(y)) -> c_2(f#(x,y))
            g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
        - Weak DPs:
            f#(x,0()) -> c_1()
        - Weak TRS:
            f(x,0()) -> s(0())
            f(s(x),s(y)) -> s(f(x,y))
            g(0(),x) -> g(f(x,x),x)
        - Signature:
            {f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:f#(s(x),s(y)) -> c_2(f#(x,y))
             -->_1 f#(x,0()) -> c_1():3
             -->_1 f#(s(x),s(y)) -> c_2(f#(x,y)):1
          
          2:S:g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
             -->_2 f#(x,0()) -> c_1():3
             -->_1 g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x)):2
             -->_2 f#(s(x),s(y)) -> c_2(f#(x,y)):1
          
          3:W:f#(x,0()) -> c_1()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: f#(x,0()) -> c_1()
** Step 1.b:4: UsableRules.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(s(x),s(y)) -> c_2(f#(x,y))
            g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
        - Weak TRS:
            f(x,0()) -> s(0())
            f(s(x),s(y)) -> s(f(x,y))
            g(0(),x) -> g(f(x,x),x)
        - Signature:
            {f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          f(x,0()) -> s(0())
          f(s(x),s(y)) -> s(f(x,y))
          f#(s(x),s(y)) -> c_2(f#(x,y))
          g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
** Step 1.b:5: DecomposeDG.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(s(x),s(y)) -> c_2(f#(x,y))
            g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
        - Weak TRS:
            f(x,0()) -> s(0())
            f(s(x),s(y)) -> s(f(x,y))
        - Signature:
            {f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
    + Applied Processor:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
        and a lower component
          f#(s(x),s(y)) -> c_2(f#(x,y))
        Further, following extension rules are added to the lower component.
          g#(0(),x) -> f#(x,x)
          g#(0(),x) -> g#(f(x,x),x)
*** Step 1.b:5.a:1: SimplifyRHS.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
        - Weak TRS:
            f(x,0()) -> s(0())
            f(s(x),s(y)) -> s(f(x,y))
        - Signature:
            {f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x))
             -->_1 g#(0(),x) -> c_3(g#(f(x,x),x),f#(x,x)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          g#(0(),x) -> c_3(g#(f(x,x),x))
*** Step 1.b:5.a:2: NaturalMI.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            g#(0(),x) -> c_3(g#(f(x,x),x))
        - Weak TRS:
            f(x,0()) -> s(0())
            f(s(x),s(y)) -> s(f(x,y))
        - Signature:
            {f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,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_3) = {1}
        
        Following symbols are considered usable:
          {f,f#,g#}
        TcT has computed the following interpretation:
            p(0) = [7]         
            p(f) = [0]         
            p(g) = [4] x2 + [2]
            p(s) = [0]         
           p(f#) = [2] x2 + [1]
           p(g#) = [3] x1 + [4]
          p(c_1) = [1]         
          p(c_2) = [2] x1 + [1]
          p(c_3) = [2] x1 + [4]
        
        Following rules are strictly oriented:
        g#(0(),x) = [25]             
                  > [12]             
                  = c_3(g#(f(x,x),x))
        
        
        Following rules are (at-least) weakly oriented:
            f(x,0()) =  [0]      
                     >= [0]      
                     =  s(0())   
        
        f(s(x),s(y)) =  [0]      
                     >= [0]      
                     =  s(f(x,y))
        
*** Step 1.b:5.a:3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            g#(0(),x) -> c_3(g#(f(x,x),x))
        - Weak TRS:
            f(x,0()) -> s(0())
            f(s(x),s(y)) -> s(f(x,y))
        - Signature:
            {f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,s}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

*** Step 1.b:5.b:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            f#(s(x),s(y)) -> c_2(f#(x,y))
        - Weak DPs:
            g#(0(),x) -> f#(x,x)
            g#(0(),x) -> g#(f(x,x),x)
        - Weak TRS:
            f(x,0()) -> s(0())
            f(s(x),s(y)) -> s(f(x,y))
        - Signature:
            {f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} and constructors {0,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_2) = {1}
        
        Following symbols are considered usable:
          {f#,g#}
        TcT has computed the following interpretation:
            p(0) = [1]                  
            p(f) = [2] x1 + [4] x2 + [7]
            p(g) = [1] x1 + [0]         
            p(s) = [1] x1 + [1]         
           p(f#) = [2] x1 + [0]         
           p(g#) = [6] x2 + [0]         
          p(c_1) = [1]                  
          p(c_2) = [1] x1 + [0]         
          p(c_3) = [1] x2 + [1]         
        
        Following rules are strictly oriented:
        f#(s(x),s(y)) = [2] x + [2] 
                      > [2] x + [0] 
                      = c_2(f#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
        g#(0(),x) =  [6] x + [0] 
                  >= [2] x + [0] 
                  =  f#(x,x)     
        
        g#(0(),x) =  [6] x + [0] 
                  >= [6] x + [0] 
                  =  g#(f(x,x),x)
        
*** Step 1.b:5.b:2: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            f#(s(x),s(y)) -> c_2(f#(x,y))
            g#(0(),x) -> f#(x,x)
            g#(0(),x) -> g#(f(x,x),x)
        - Weak TRS:
            f(x,0()) -> s(0())
            f(s(x),s(y)) -> s(f(x,y))
        - Signature:
            {f/2,g/2,f#/2,g#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {f#,g#} 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))