/export/starexec/sandbox2/solver/bin/starexec_run_tct_rci /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


--------------------------------------------------------------------------------


WORST_CASE(Omega(n^1),?)
* Step 1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            cond(false(),x,y) -> 0()
            cond(true(),x,y) -> s(minus(x,s(y)))
            ge(u,0()) -> true()
            ge(0(),s(v)) -> false()
            ge(s(u),s(v)) -> ge(u,v)
            minus(x,y) -> cond(ge(x,s(y)),x,y)
        - Signature:
            {cond/3,ge/2,minus/2} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {cond,ge,minus} and constructors {0,false,s,true}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            cond(false(),x,y) -> 0()
            cond(true(),x,y) -> s(minus(x,s(y)))
            ge(u,0()) -> true()
            ge(0(),s(v)) -> false()
            ge(s(u),s(v)) -> ge(u,v)
            minus(x,y) -> cond(ge(x,s(y)),x,y)
        - Signature:
            {cond/3,ge/2,minus/2} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {cond,ge,minus} and constructors {0,false,s,true}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:2: DecreasingLoops.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            cond(false(),x,y) -> 0()
            cond(true(),x,y) -> s(minus(x,s(y)))
            ge(u,0()) -> true()
            ge(0(),s(v)) -> false()
            ge(s(u),s(v)) -> ge(u,v)
            minus(x,y) -> cond(ge(x,s(y)),x,y)
        - Signature:
            {cond/3,ge/2,minus/2} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {cond,ge,minus} and constructors {0,false,s,true}
    + Applied Processor:
        DecreasingLoops {bound = AnyLoop, narrow = 10}
    + Details:
        The system has following decreasing Loops:
          ge(x,y){x -> s(x),y -> s(y)} =
            ge(s(x),s(y)) ->^+ ge(x,y)
              = C[ge(x,y) = ge(x,y){}]

** Step 1.b:1: DependencyPairs.  MAYBE
    + Considered Problem:
        - Strict TRS:
            cond(false(),x,y) -> 0()
            cond(true(),x,y) -> s(minus(x,s(y)))
            ge(u,0()) -> true()
            ge(0(),s(v)) -> false()
            ge(s(u),s(v)) -> ge(u,v)
            minus(x,y) -> cond(ge(x,s(y)),x,y)
        - Signature:
            {cond/3,ge/2,minus/2} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {cond,ge,minus} and constructors {0,false,s,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          cond#(false(),x,y) -> c_1()
          cond#(true(),x,y) -> c_2(minus#(x,s(y)))
          ge#(u,0()) -> c_3()
          ge#(0(),s(v)) -> c_4()
          ge#(s(u),s(v)) -> c_5(ge#(u,v))
          minus#(x,y) -> c_6(cond#(ge(x,s(y)),x,y),ge#(x,s(y)))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation.  MAYBE
    + Considered Problem:
        - Strict DPs:
            cond#(false(),x,y) -> c_1()
            cond#(true(),x,y) -> c_2(minus#(x,s(y)))
            ge#(u,0()) -> c_3()
            ge#(0(),s(v)) -> c_4()
            ge#(s(u),s(v)) -> c_5(ge#(u,v))
            minus#(x,y) -> c_6(cond#(ge(x,s(y)),x,y),ge#(x,s(y)))
        - Weak TRS:
            cond(false(),x,y) -> 0()
            cond(true(),x,y) -> s(minus(x,s(y)))
            ge(u,0()) -> true()
            ge(0(),s(v)) -> false()
            ge(s(u),s(v)) -> ge(u,v)
            minus(x,y) -> cond(ge(x,s(y)),x,y)
        - Signature:
            {cond/3,ge/2,minus/2,cond#/3,ge#/2,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {cond#,ge#,minus#} and constructors {0,false,s,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,3,4}
        by application of
          Pre({1,3,4}) = {5,6}.
        Here rules are labelled as follows:
          1: cond#(false(),x,y) -> c_1()
          2: cond#(true(),x,y) -> c_2(minus#(x,s(y)))
          3: ge#(u,0()) -> c_3()
          4: ge#(0(),s(v)) -> c_4()
          5: ge#(s(u),s(v)) -> c_5(ge#(u,v))
          6: minus#(x,y) -> c_6(cond#(ge(x,s(y)),x,y),ge#(x,s(y)))
** Step 1.b:3: RemoveWeakSuffixes.  MAYBE
    + Considered Problem:
        - Strict DPs:
            cond#(true(),x,y) -> c_2(minus#(x,s(y)))
            ge#(s(u),s(v)) -> c_5(ge#(u,v))
            minus#(x,y) -> c_6(cond#(ge(x,s(y)),x,y),ge#(x,s(y)))
        - Weak DPs:
            cond#(false(),x,y) -> c_1()
            ge#(u,0()) -> c_3()
            ge#(0(),s(v)) -> c_4()
        - Weak TRS:
            cond(false(),x,y) -> 0()
            cond(true(),x,y) -> s(minus(x,s(y)))
            ge(u,0()) -> true()
            ge(0(),s(v)) -> false()
            ge(s(u),s(v)) -> ge(u,v)
            minus(x,y) -> cond(ge(x,s(y)),x,y)
        - Signature:
            {cond/3,ge/2,minus/2,cond#/3,ge#/2,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {cond#,ge#,minus#} and constructors {0,false,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:cond#(true(),x,y) -> c_2(minus#(x,s(y)))
             -->_1 minus#(x,y) -> c_6(cond#(ge(x,s(y)),x,y),ge#(x,s(y))):3
          
          2:S:ge#(s(u),s(v)) -> c_5(ge#(u,v))
             -->_1 ge#(0(),s(v)) -> c_4():6
             -->_1 ge#(u,0()) -> c_3():5
             -->_1 ge#(s(u),s(v)) -> c_5(ge#(u,v)):2
          
          3:S:minus#(x,y) -> c_6(cond#(ge(x,s(y)),x,y),ge#(x,s(y)))
             -->_2 ge#(0(),s(v)) -> c_4():6
             -->_1 cond#(false(),x,y) -> c_1():4
             -->_2 ge#(s(u),s(v)) -> c_5(ge#(u,v)):2
             -->_1 cond#(true(),x,y) -> c_2(minus#(x,s(y))):1
          
          4:W:cond#(false(),x,y) -> c_1()
             
          
          5:W:ge#(u,0()) -> c_3()
             
          
          6:W:ge#(0(),s(v)) -> c_4()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: ge#(u,0()) -> c_3()
          4: cond#(false(),x,y) -> c_1()
          6: ge#(0(),s(v)) -> c_4()
** Step 1.b:4: UsableRules.  MAYBE
    + Considered Problem:
        - Strict DPs:
            cond#(true(),x,y) -> c_2(minus#(x,s(y)))
            ge#(s(u),s(v)) -> c_5(ge#(u,v))
            minus#(x,y) -> c_6(cond#(ge(x,s(y)),x,y),ge#(x,s(y)))
        - Weak TRS:
            cond(false(),x,y) -> 0()
            cond(true(),x,y) -> s(minus(x,s(y)))
            ge(u,0()) -> true()
            ge(0(),s(v)) -> false()
            ge(s(u),s(v)) -> ge(u,v)
            minus(x,y) -> cond(ge(x,s(y)),x,y)
        - Signature:
            {cond/3,ge/2,minus/2,cond#/3,ge#/2,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {cond#,ge#,minus#} and constructors {0,false,s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          ge(u,0()) -> true()
          ge(0(),s(v)) -> false()
          ge(s(u),s(v)) -> ge(u,v)
          cond#(true(),x,y) -> c_2(minus#(x,s(y)))
          ge#(s(u),s(v)) -> c_5(ge#(u,v))
          minus#(x,y) -> c_6(cond#(ge(x,s(y)),x,y),ge#(x,s(y)))
** Step 1.b:5: DecomposeDG.  MAYBE
    + Considered Problem:
        - Strict DPs:
            cond#(true(),x,y) -> c_2(minus#(x,s(y)))
            ge#(s(u),s(v)) -> c_5(ge#(u,v))
            minus#(x,y) -> c_6(cond#(ge(x,s(y)),x,y),ge#(x,s(y)))
        - Weak TRS:
            ge(u,0()) -> true()
            ge(0(),s(v)) -> false()
            ge(s(u),s(v)) -> ge(u,v)
        - Signature:
            {cond/3,ge/2,minus/2,cond#/3,ge#/2,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {cond#,ge#,minus#} and constructors {0,false,s,true}
    + 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
          cond#(true(),x,y) -> c_2(minus#(x,s(y)))
          minus#(x,y) -> c_6(cond#(ge(x,s(y)),x,y),ge#(x,s(y)))
        and a lower component
          ge#(s(u),s(v)) -> c_5(ge#(u,v))
        Further, following extension rules are added to the lower component.
          cond#(true(),x,y) -> minus#(x,s(y))
          minus#(x,y) -> cond#(ge(x,s(y)),x,y)
          minus#(x,y) -> ge#(x,s(y))
*** Step 1.b:5.a:1: SimplifyRHS.  MAYBE
    + Considered Problem:
        - Strict DPs:
            cond#(true(),x,y) -> c_2(minus#(x,s(y)))
            minus#(x,y) -> c_6(cond#(ge(x,s(y)),x,y),ge#(x,s(y)))
        - Weak TRS:
            ge(u,0()) -> true()
            ge(0(),s(v)) -> false()
            ge(s(u),s(v)) -> ge(u,v)
        - Signature:
            {cond/3,ge/2,minus/2,cond#/3,ge#/2,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {cond#,ge#,minus#} and constructors {0,false,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:cond#(true(),x,y) -> c_2(minus#(x,s(y)))
             -->_1 minus#(x,y) -> c_6(cond#(ge(x,s(y)),x,y),ge#(x,s(y))):2
          
          2:S:minus#(x,y) -> c_6(cond#(ge(x,s(y)),x,y),ge#(x,s(y)))
             -->_1 cond#(true(),x,y) -> c_2(minus#(x,s(y))):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          minus#(x,y) -> c_6(cond#(ge(x,s(y)),x,y))
*** Step 1.b:5.a:2: WeightGap.  MAYBE
    + Considered Problem:
        - Strict DPs:
            cond#(true(),x,y) -> c_2(minus#(x,s(y)))
            minus#(x,y) -> c_6(cond#(ge(x,s(y)),x,y))
        - Weak TRS:
            ge(u,0()) -> true()
            ge(0(),s(v)) -> false()
            ge(s(u),s(v)) -> ge(u,v)
        - Signature:
            {cond/3,ge/2,minus/2,cond#/3,ge#/2,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {cond#,ge#,minus#} and constructors {0,false,s,true}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + 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(cond#) = {1},
            uargs(c_2) = {1},
            uargs(c_6) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                 p(0) = [0]                  
              p(cond) = [0]                  
             p(false) = [0]                  
                p(ge) = [1]                  
             p(minus) = [0]                  
                 p(s) = [1] x1 + [1]         
              p(true) = [1]                  
             p(cond#) = [1] x1 + [3] x3 + [7]
               p(ge#) = [1] x1 + [0]         
            p(minus#) = [3] x2 + [4]         
               p(c_1) = [0]                  
               p(c_2) = [1] x1 + [0]         
               p(c_3) = [0]                  
               p(c_4) = [0]                  
               p(c_5) = [0]                  
               p(c_6) = [1] x1 + [0]         
          
          Following rules are strictly oriented:
          cond#(true(),x,y) = [3] y + [8]        
                            > [3] y + [7]        
                            = c_2(minus#(x,s(y)))
          
          
          Following rules are (at-least) weakly oriented:
            minus#(x,y) =  [3] y + [4]               
                        >= [3] y + [8]               
                        =  c_6(cond#(ge(x,s(y)),x,y))
          
              ge(u,0()) =  [1]                       
                        >= [1]                       
                        =  true()                    
          
           ge(0(),s(v)) =  [1]                       
                        >= [0]                       
                        =  false()                   
          
          ge(s(u),s(v)) =  [1]                       
                        >= [1]                       
                        =  ge(u,v)                   
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** Step 1.b:5.a:3: Failure MAYBE
  + Considered Problem:
      - Strict DPs:
          minus#(x,y) -> c_6(cond#(ge(x,s(y)),x,y))
      - Weak DPs:
          cond#(true(),x,y) -> c_2(minus#(x,s(y)))
      - Weak TRS:
          ge(u,0()) -> true()
          ge(0(),s(v)) -> false()
          ge(s(u),s(v)) -> ge(u,v)
      - Signature:
          {cond/3,ge/2,minus/2,cond#/3,ge#/2,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/1}
      - Obligation:
          innermost runtime complexity wrt. defined symbols {cond#,ge#,minus#} and constructors {0,false,s,true}
  + Applied Processor:
      EmptyProcessor
  + Details:
      The problem is still open.
*** Step 1.b:5.b:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            ge#(s(u),s(v)) -> c_5(ge#(u,v))
        - Weak DPs:
            cond#(true(),x,y) -> minus#(x,s(y))
            minus#(x,y) -> cond#(ge(x,s(y)),x,y)
            minus#(x,y) -> ge#(x,s(y))
        - Weak TRS:
            ge(u,0()) -> true()
            ge(0(),s(v)) -> false()
            ge(s(u),s(v)) -> ge(u,v)
        - Signature:
            {cond/3,ge/2,minus/2,cond#/3,ge#/2,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {cond#,ge#,minus#} and constructors {0,false,s,true}
    + 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_5) = {1}
        
        Following symbols are considered usable:
          {cond#,ge#,minus#}
        TcT has computed the following interpretation:
               p(0) = [5]                  
            p(cond) = [1] x3 + [2]         
           p(false) = [0]                  
              p(ge) = [2] x1 + [2] x2 + [0]
           p(minus) = [2] x1 + [2]         
               p(s) = [1] x1 + [4]         
            p(true) = [8]                  
           p(cond#) = [4] x2 + [11]        
             p(ge#) = [4] x1 + [0]         
          p(minus#) = [4] x1 + [11]        
             p(c_1) = [0]                  
             p(c_2) = [2] x1 + [1]         
             p(c_3) = [1]                  
             p(c_4) = [1]                  
             p(c_5) = [1] x1 + [14]        
             p(c_6) = [1] x1 + [2] x2 + [0]
        
        Following rules are strictly oriented:
        ge#(s(u),s(v)) = [4] u + [16] 
                       > [4] u + [14] 
                       = c_5(ge#(u,v))
        
        
        Following rules are (at-least) weakly oriented:
        cond#(true(),x,y) =  [4] x + [11]         
                          >= [4] x + [11]         
                          =  minus#(x,s(y))       
        
              minus#(x,y) =  [4] x + [11]         
                          >= [4] x + [11]         
                          =  cond#(ge(x,s(y)),x,y)
        
              minus#(x,y) =  [4] x + [11]         
                          >= [4] x + [0]          
                          =  ge#(x,s(y))          
        
*** Step 1.b:5.b:2: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            cond#(true(),x,y) -> minus#(x,s(y))
            ge#(s(u),s(v)) -> c_5(ge#(u,v))
            minus#(x,y) -> cond#(ge(x,s(y)),x,y)
            minus#(x,y) -> ge#(x,s(y))
        - Weak TRS:
            ge(u,0()) -> true()
            ge(0(),s(v)) -> false()
            ge(s(u),s(v)) -> ge(u,v)
        - Signature:
            {cond/3,ge/2,minus/2,cond#/3,ge#/2,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/0,c_4/0,c_5/1,c_6/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {cond#,ge#,minus#} and constructors {0,false,s,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),?)