/export/starexec/sandbox2/solver/bin/starexec_run_tct_rc /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:
            +(0(),y) -> y
            +(s(x),0()) -> s(x)
            +(s(x),s(y)) -> s(+(s(x),+(y,0())))
        - Signature:
            {+/2} / {0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {+} 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:
            +(0(),y) -> y
            +(s(x),0()) -> s(x)
            +(s(x),s(y)) -> s(+(s(x),+(y,0())))
        - Signature:
            {+/2} / {0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {+} 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:
            +(0(),y) -> y
            +(s(x),0()) -> s(x)
            +(s(x),s(y)) -> s(+(s(x),+(y,0())))
        - Signature:
            {+/2} / {0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {+} and constructors {0,s}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          +(s(y),s(x)){x -> s(x)} =
            +(s(y),s(s(x))) ->^+ s(+(s(y),s(x)))
              = C[+(s(y),s(x)) = +(s(y),s(x)){}]

** Step 1.b:1: ToInnermost.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            +(0(),y) -> y
            +(s(x),0()) -> s(x)
            +(s(x),s(y)) -> s(+(s(x),+(y,0())))
        - Signature:
            {+/2} / {0/0,s/1}
        - Obligation:
             runtime complexity wrt. defined symbols {+} and constructors {0,s}
    + Applied Processor:
        ToInnermost
    + Details:
        switch to innermost, as the system is overlay and right linear and does not contain weak rules
** Step 1.b:2: DependencyPairs.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            +(0(),y) -> y
            +(s(x),0()) -> s(x)
            +(s(x),s(y)) -> s(+(s(x),+(y,0())))
        - Signature:
            {+/2} / {0/0,s/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+} and constructors {0,s}
    + Applied Processor:
        DependencyPairs {dpKind_ = WIDP}
    + Details:
        We add the following weak innermost dependency pairs:
        
        Strict DPs
          +#(0(),y) -> c_1()
          +#(s(x),0()) -> c_2()
          +#(s(x),s(y)) -> c_3(+#(s(x),+(y,0())))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:3: UsableRules.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            +#(0(),y) -> c_1()
            +#(s(x),0()) -> c_2()
            +#(s(x),s(y)) -> c_3(+#(s(x),+(y,0())))
        - Strict TRS:
            +(0(),y) -> y
            +(s(x),0()) -> s(x)
            +(s(x),s(y)) -> s(+(s(x),+(y,0())))
        - Signature:
            {+/2,+#/2} / {0/0,s/1,c_1/0,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#} and constructors {0,s}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          +(0(),y) -> y
          +(s(x),0()) -> s(x)
          +#(0(),y) -> c_1()
          +#(s(x),0()) -> c_2()
          +#(s(x),s(y)) -> c_3(+#(s(x),+(y,0())))
** Step 1.b:4: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            +#(0(),y) -> c_1()
            +#(s(x),0()) -> c_2()
            +#(s(x),s(y)) -> c_3(+#(s(x),+(y,0())))
        - Strict TRS:
            +(0(),y) -> y
            +(s(x),0()) -> s(x)
        - Signature:
            {+/2,+#/2} / {0/0,s/1,c_1/0,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#} 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(+#) = {2},
            uargs(c_3) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
              p(+) = [1] x1 + [2] x2 + [5]
              p(0) = [0]                  
              p(s) = [1] x1 + [0]         
             p(+#) = [1] x2 + [0]         
            p(c_1) = [0]                  
            p(c_2) = [0]                  
            p(c_3) = [1] x1 + [0]         
          
          Following rules are strictly oriented:
             +(0(),y) = [2] y + [5]
                      > [1] y + [0]
                      = y          
          
          +(s(x),0()) = [1] x + [5]
                      > [1] x + [0]
                      = s(x)       
          
          
          Following rules are (at-least) weakly oriented:
              +#(0(),y) =  [1] y + [0]           
                        >= [0]                   
                        =  c_1()                 
          
           +#(s(x),0()) =  [0]                   
                        >= [0]                   
                        =  c_2()                 
          
          +#(s(x),s(y)) =  [1] y + [0]           
                        >= [1] y + [5]           
                        =  c_3(+#(s(x),+(y,0())))
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
** Step 1.b:5: PredecessorEstimation.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            +#(0(),y) -> c_1()
            +#(s(x),0()) -> c_2()
            +#(s(x),s(y)) -> c_3(+#(s(x),+(y,0())))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),0()) -> s(x)
        - Signature:
            {+/2,+#/2} / {0/0,s/1,c_1/0,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#} and constructors {0,s}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2}
        by application of
          Pre({1,2}) = {3}.
        Here rules are labelled as follows:
          1: +#(0(),y) -> c_1()
          2: +#(s(x),0()) -> c_2()
          3: +#(s(x),s(y)) -> c_3(+#(s(x),+(y,0())))
** Step 1.b:6: RemoveWeakSuffixes.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            +#(s(x),s(y)) -> c_3(+#(s(x),+(y,0())))
        - Weak DPs:
            +#(0(),y) -> c_1()
            +#(s(x),0()) -> c_2()
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),0()) -> s(x)
        - Signature:
            {+/2,+#/2} / {0/0,s/1,c_1/0,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:+#(s(x),s(y)) -> c_3(+#(s(x),+(y,0())))
             -->_1 +#(s(x),0()) -> c_2():3
             -->_1 +#(s(x),s(y)) -> c_3(+#(s(x),+(y,0()))):1
          
          2:W:+#(0(),y) -> c_1()
             
          
          3:W:+#(s(x),0()) -> c_2()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          2: +#(0(),y) -> c_1()
          3: +#(s(x),0()) -> c_2()
** Step 1.b:7: PredecessorEstimationCP.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            +#(s(x),s(y)) -> c_3(+#(s(x),+(y,0())))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),0()) -> s(x)
        - Signature:
            {+/2,+#/2} / {0/0,s/1,c_1/0,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#} and constructors {0,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:
          1: +#(s(x),s(y)) -> c_3(+#(s(x),+(y,0())))
          
        The strictly oriented rules are moved into the weak component.
*** Step 1.b:7.a:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            +#(s(x),s(y)) -> c_3(+#(s(x),+(y,0())))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),0()) -> s(x)
        - Signature:
            {+/2,+#/2} / {0/0,s/1,c_1/0,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#} and constructors {0,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:
          uargs(c_3) = {1}
        
        Following symbols are considered usable:
          {+,+#}
        TcT has computed the following interpretation:
            p(+) = [1] x1 + [4] x2 + [0]
            p(0) = [0]                  
            p(s) = [1] x1 + [2]         
           p(+#) = [8] x2 + [0]         
          p(c_1) = [0]                  
          p(c_2) = [0]                  
          p(c_3) = [1] x1 + [15]        
        
        Following rules are strictly oriented:
        +#(s(x),s(y)) = [8] y + [16]          
                      > [8] y + [15]          
                      = c_3(+#(s(x),+(y,0())))
        
        
        Following rules are (at-least) weakly oriented:
           +(0(),y) =  [4] y + [0]
                    >= [1] y + [0]
                    =  y          
        
        +(s(x),0()) =  [1] x + [2]
                    >= [1] x + [2]
                    =  s(x)       
        
*** Step 1.b:7.a:2: Assumption.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            +#(s(x),s(y)) -> c_3(+#(s(x),+(y,0())))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),0()) -> s(x)
        - Signature:
            {+/2,+#/2} / {0/0,s/1,c_1/0,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#} and constructors {0,s}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}}
    + Details:
        ()

*** Step 1.b:7.b:1: RemoveWeakSuffixes.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            +#(s(x),s(y)) -> c_3(+#(s(x),+(y,0())))
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),0()) -> s(x)
        - Signature:
            {+/2,+#/2} / {0/0,s/1,c_1/0,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#} and constructors {0,s}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:+#(s(x),s(y)) -> c_3(+#(s(x),+(y,0())))
             -->_1 +#(s(x),s(y)) -> c_3(+#(s(x),+(y,0()))):1
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: +#(s(x),s(y)) -> c_3(+#(s(x),+(y,0())))
*** Step 1.b:7.b:2: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            +(0(),y) -> y
            +(s(x),0()) -> s(x)
        - Signature:
            {+/2,+#/2} / {0/0,s/1,c_1/0,c_2/0,c_3/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {+#} 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))