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

** Step 1.b:1: DependencyPairs.  MAYBE
    + Considered Problem:
        - Strict TRS:
            help(false(),x,y) -> 0()
            help(true(),x,y) -> s(minus(x,s(y)))
            lt(x,0()) -> false()
            lt(0(),s(x)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
            minus(x,y) -> help(lt(y,x),x,y)
        - Signature:
            {help/3,lt/2,minus/2} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {help,lt,minus} and constructors {0,false,s,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          help#(false(),x,y) -> c_1()
          help#(true(),x,y) -> c_2(minus#(x,s(y)))
          lt#(x,0()) -> c_3()
          lt#(0(),s(x)) -> c_4()
          lt#(s(x),s(y)) -> c_5(lt#(x,y))
          minus#(x,y) -> c_6(help#(lt(y,x),x,y),lt#(y,x))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation.  MAYBE
    + Considered Problem:
        - Strict DPs:
            help#(false(),x,y) -> c_1()
            help#(true(),x,y) -> c_2(minus#(x,s(y)))
            lt#(x,0()) -> c_3()
            lt#(0(),s(x)) -> c_4()
            lt#(s(x),s(y)) -> c_5(lt#(x,y))
            minus#(x,y) -> c_6(help#(lt(y,x),x,y),lt#(y,x))
        - Weak TRS:
            help(false(),x,y) -> 0()
            help(true(),x,y) -> s(minus(x,s(y)))
            lt(x,0()) -> false()
            lt(0(),s(x)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
            minus(x,y) -> help(lt(y,x),x,y)
        - Signature:
            {help/3,lt/2,minus/2,help#/3,lt#/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 {help#,lt#,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: help#(false(),x,y) -> c_1()
          2: help#(true(),x,y) -> c_2(minus#(x,s(y)))
          3: lt#(x,0()) -> c_3()
          4: lt#(0(),s(x)) -> c_4()
          5: lt#(s(x),s(y)) -> c_5(lt#(x,y))
          6: minus#(x,y) -> c_6(help#(lt(y,x),x,y),lt#(y,x))
** Step 1.b:3: RemoveWeakSuffixes.  MAYBE
    + Considered Problem:
        - Strict DPs:
            help#(true(),x,y) -> c_2(minus#(x,s(y)))
            lt#(s(x),s(y)) -> c_5(lt#(x,y))
            minus#(x,y) -> c_6(help#(lt(y,x),x,y),lt#(y,x))
        - Weak DPs:
            help#(false(),x,y) -> c_1()
            lt#(x,0()) -> c_3()
            lt#(0(),s(x)) -> c_4()
        - Weak TRS:
            help(false(),x,y) -> 0()
            help(true(),x,y) -> s(minus(x,s(y)))
            lt(x,0()) -> false()
            lt(0(),s(x)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
            minus(x,y) -> help(lt(y,x),x,y)
        - Signature:
            {help/3,lt/2,minus/2,help#/3,lt#/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 {help#,lt#,minus#} and constructors {0,false,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:help#(true(),x,y) -> c_2(minus#(x,s(y)))
             -->_1 minus#(x,y) -> c_6(help#(lt(y,x),x,y),lt#(y,x)):3
          
          2:S:lt#(s(x),s(y)) -> c_5(lt#(x,y))
             -->_1 lt#(0(),s(x)) -> c_4():6
             -->_1 lt#(x,0()) -> c_3():5
             -->_1 lt#(s(x),s(y)) -> c_5(lt#(x,y)):2
          
          3:S:minus#(x,y) -> c_6(help#(lt(y,x),x,y),lt#(y,x))
             -->_2 lt#(0(),s(x)) -> c_4():6
             -->_2 lt#(x,0()) -> c_3():5
             -->_1 help#(false(),x,y) -> c_1():4
             -->_2 lt#(s(x),s(y)) -> c_5(lt#(x,y)):2
             -->_1 help#(true(),x,y) -> c_2(minus#(x,s(y))):1
          
          4:W:help#(false(),x,y) -> c_1()
             
          
          5:W:lt#(x,0()) -> c_3()
             
          
          6:W:lt#(0(),s(x)) -> c_4()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: help#(false(),x,y) -> c_1()
          5: lt#(x,0()) -> c_3()
          6: lt#(0(),s(x)) -> c_4()
** Step 1.b:4: UsableRules.  MAYBE
    + Considered Problem:
        - Strict DPs:
            help#(true(),x,y) -> c_2(minus#(x,s(y)))
            lt#(s(x),s(y)) -> c_5(lt#(x,y))
            minus#(x,y) -> c_6(help#(lt(y,x),x,y),lt#(y,x))
        - Weak TRS:
            help(false(),x,y) -> 0()
            help(true(),x,y) -> s(minus(x,s(y)))
            lt(x,0()) -> false()
            lt(0(),s(x)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
            minus(x,y) -> help(lt(y,x),x,y)
        - Signature:
            {help/3,lt/2,minus/2,help#/3,lt#/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 {help#,lt#,minus#} and constructors {0,false,s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          lt(x,0()) -> false()
          lt(0(),s(x)) -> true()
          lt(s(x),s(y)) -> lt(x,y)
          help#(true(),x,y) -> c_2(minus#(x,s(y)))
          lt#(s(x),s(y)) -> c_5(lt#(x,y))
          minus#(x,y) -> c_6(help#(lt(y,x),x,y),lt#(y,x))
** Step 1.b:5: DecomposeDG.  MAYBE
    + Considered Problem:
        - Strict DPs:
            help#(true(),x,y) -> c_2(minus#(x,s(y)))
            lt#(s(x),s(y)) -> c_5(lt#(x,y))
            minus#(x,y) -> c_6(help#(lt(y,x),x,y),lt#(y,x))
        - Weak TRS:
            lt(x,0()) -> false()
            lt(0(),s(x)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
        - Signature:
            {help/3,lt/2,minus/2,help#/3,lt#/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 {help#,lt#,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
          help#(true(),x,y) -> c_2(minus#(x,s(y)))
          minus#(x,y) -> c_6(help#(lt(y,x),x,y),lt#(y,x))
        and a lower component
          lt#(s(x),s(y)) -> c_5(lt#(x,y))
        Further, following extension rules are added to the lower component.
          help#(true(),x,y) -> minus#(x,s(y))
          minus#(x,y) -> help#(lt(y,x),x,y)
          minus#(x,y) -> lt#(y,x)
*** Step 1.b:5.a:1: SimplifyRHS.  MAYBE
    + Considered Problem:
        - Strict DPs:
            help#(true(),x,y) -> c_2(minus#(x,s(y)))
            minus#(x,y) -> c_6(help#(lt(y,x),x,y),lt#(y,x))
        - Weak TRS:
            lt(x,0()) -> false()
            lt(0(),s(x)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
        - Signature:
            {help/3,lt/2,minus/2,help#/3,lt#/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 {help#,lt#,minus#} and constructors {0,false,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:help#(true(),x,y) -> c_2(minus#(x,s(y)))
             -->_1 minus#(x,y) -> c_6(help#(lt(y,x),x,y),lt#(y,x)):2
          
          2:S:minus#(x,y) -> c_6(help#(lt(y,x),x,y),lt#(y,x))
             -->_1 help#(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(help#(lt(y,x),x,y))
*** Step 1.b:5.a:2: WeightGap.  MAYBE
    + Considered Problem:
        - Strict DPs:
            help#(true(),x,y) -> c_2(minus#(x,s(y)))
            minus#(x,y) -> c_6(help#(lt(y,x),x,y))
        - Weak TRS:
            lt(x,0()) -> false()
            lt(0(),s(x)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
        - Signature:
            {help/3,lt/2,minus/2,help#/3,lt#/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 {help#,lt#,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(help#) = {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(false) = [0]         
              p(help) = [0]         
                p(lt) = [0]         
             p(minus) = [0]         
                 p(s) = [0]         
              p(true) = [0]         
             p(help#) = [1] x1 + [2]
               p(lt#) = [4] x2 + [0]
            p(minus#) = [5]         
               p(c_1) = [1]         
               p(c_2) = [1] x1 + [7]
               p(c_3) = [1]         
               p(c_4) = [1]         
               p(c_5) = [1]         
               p(c_6) = [1] x1 + [0]
          
          Following rules are strictly oriented:
          minus#(x,y) = [5]                    
                      > [2]                    
                      = c_6(help#(lt(y,x),x,y))
          
          
          Following rules are (at-least) weakly oriented:
          help#(true(),x,y) =  [2]                
                            >= [12]               
                            =  c_2(minus#(x,s(y)))
          
                  lt(x,0()) =  [0]                
                            >= [0]                
                            =  false()            
          
               lt(0(),s(x)) =  [0]                
                            >= [0]                
                            =  true()             
          
              lt(s(x),s(y)) =  [0]                
                            >= [0]                
                            =  lt(x,y)            
          
        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:
          help#(true(),x,y) -> c_2(minus#(x,s(y)))
      - Weak DPs:
          minus#(x,y) -> c_6(help#(lt(y,x),x,y))
      - Weak TRS:
          lt(x,0()) -> false()
          lt(0(),s(x)) -> true()
          lt(s(x),s(y)) -> lt(x,y)
      - Signature:
          {help/3,lt/2,minus/2,help#/3,lt#/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 {help#,lt#,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:
            lt#(s(x),s(y)) -> c_5(lt#(x,y))
        - Weak DPs:
            help#(true(),x,y) -> minus#(x,s(y))
            minus#(x,y) -> help#(lt(y,x),x,y)
            minus#(x,y) -> lt#(y,x)
        - Weak TRS:
            lt(x,0()) -> false()
            lt(0(),s(x)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
        - Signature:
            {help/3,lt/2,minus/2,help#/3,lt#/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 {help#,lt#,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:
          {help#,lt#,minus#}
        TcT has computed the following interpretation:
               p(0) = [0]          
           p(false) = [0]          
            p(help) = [1] x2 + [8] 
              p(lt) = [0]          
           p(minus) = [0]          
               p(s) = [1] x1 + [8] 
            p(true) = [0]          
           p(help#) = [3] x2 + [0] 
             p(lt#) = [2] x2 + [0] 
          p(minus#) = [3] x1 + [0] 
             p(c_1) = [1]          
             p(c_2) = [1] x1 + [1] 
             p(c_3) = [0]          
             p(c_4) = [1]          
             p(c_5) = [1] x1 + [12]
             p(c_6) = [1] x2 + [1] 
        
        Following rules are strictly oriented:
        lt#(s(x),s(y)) = [2] y + [16] 
                       > [2] y + [12] 
                       = c_5(lt#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
        help#(true(),x,y) =  [3] x + [0]       
                          >= [3] x + [0]       
                          =  minus#(x,s(y))    
        
              minus#(x,y) =  [3] x + [0]       
                          >= [3] x + [0]       
                          =  help#(lt(y,x),x,y)
        
              minus#(x,y) =  [3] x + [0]       
                          >= [2] x + [0]       
                          =  lt#(y,x)          
        
*** Step 1.b:5.b:2: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            help#(true(),x,y) -> minus#(x,s(y))
            lt#(s(x),s(y)) -> c_5(lt#(x,y))
            minus#(x,y) -> help#(lt(y,x),x,y)
            minus#(x,y) -> lt#(y,x)
        - Weak TRS:
            lt(x,0()) -> false()
            lt(0(),s(x)) -> true()
            lt(s(x),s(y)) -> lt(x,y)
        - Signature:
            {help/3,lt/2,minus/2,help#/3,lt#/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 {help#,lt#,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),?)