WORST_CASE(?,O(n^2))
* Step 1: DependencyPairs WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            gcd(0(),Y) -> 0()
            gcd(s(X),0()) -> s(X)
            gcd(s(X),s(Y)) -> if(le(Y,X),s(X),s(Y))
            if(false(),s(X),s(Y)) -> gcd(minus(Y,X),s(X))
            if(true(),s(X),s(Y)) -> gcd(minus(X,Y),s(Y))
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(X,0()) -> X
            minus(X,s(Y)) -> pred(minus(X,Y))
            pred(s(X)) -> X
        - Signature:
            {gcd/2,if/3,le/2,minus/2,pred/1} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {gcd,if,le,minus,pred} and constructors {0,false,s,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          gcd#(0(),Y) -> c_1()
          gcd#(s(X),0()) -> c_2()
          gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X))
          if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X))
          if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y))
          le#(0(),Y) -> c_6()
          le#(s(X),0()) -> c_7()
          le#(s(X),s(Y)) -> c_8(le#(X,Y))
          minus#(X,0()) -> c_9()
          minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y))
          pred#(s(X)) -> c_11()
        Weak DPs
          
        
        and mark the set of starting terms.
* Step 2: UsableRules WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            gcd#(0(),Y) -> c_1()
            gcd#(s(X),0()) -> c_2()
            gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X))
            if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X))
            if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y))
            le#(0(),Y) -> c_6()
            le#(s(X),0()) -> c_7()
            le#(s(X),s(Y)) -> c_8(le#(X,Y))
            minus#(X,0()) -> c_9()
            minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y))
            pred#(s(X)) -> c_11()
        - Weak TRS:
            gcd(0(),Y) -> 0()
            gcd(s(X),0()) -> s(X)
            gcd(s(X),s(Y)) -> if(le(Y,X),s(X),s(Y))
            if(false(),s(X),s(Y)) -> gcd(minus(Y,X),s(X))
            if(true(),s(X),s(Y)) -> gcd(minus(X,Y),s(Y))
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(X,0()) -> X
            minus(X,s(Y)) -> pred(minus(X,Y))
            pred(s(X)) -> X
        - Signature:
            {gcd/2,if/3,le/2,minus/2,pred/1,gcd#/2,if#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0
            ,c_3/2,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {gcd#,if#,le#,minus#,pred#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          le(0(),Y) -> true()
          le(s(X),0()) -> false()
          le(s(X),s(Y)) -> le(X,Y)
          minus(X,0()) -> X
          minus(X,s(Y)) -> pred(minus(X,Y))
          pred(s(X)) -> X
          gcd#(0(),Y) -> c_1()
          gcd#(s(X),0()) -> c_2()
          gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X))
          if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X))
          if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y))
          le#(0(),Y) -> c_6()
          le#(s(X),0()) -> c_7()
          le#(s(X),s(Y)) -> c_8(le#(X,Y))
          minus#(X,0()) -> c_9()
          minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y))
          pred#(s(X)) -> c_11()
* Step 3: PredecessorEstimation WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            gcd#(0(),Y) -> c_1()
            gcd#(s(X),0()) -> c_2()
            gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X))
            if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X))
            if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y))
            le#(0(),Y) -> c_6()
            le#(s(X),0()) -> c_7()
            le#(s(X),s(Y)) -> c_8(le#(X,Y))
            minus#(X,0()) -> c_9()
            minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y))
            pred#(s(X)) -> c_11()
        - Weak TRS:
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(X,0()) -> X
            minus(X,s(Y)) -> pred(minus(X,Y))
            pred(s(X)) -> X
        - Signature:
            {gcd/2,if/3,le/2,minus/2,pred/1,gcd#/2,if#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0
            ,c_3/2,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {gcd#,if#,le#,minus#,pred#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,2,6,7,9,11}
        by application of
          Pre({1,2,6,7,9,11}) = {3,4,5,8,10}.
        Here rules are labelled as follows:
          1: gcd#(0(),Y) -> c_1()
          2: gcd#(s(X),0()) -> c_2()
          3: gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X))
          4: if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X))
          5: if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y))
          6: le#(0(),Y) -> c_6()
          7: le#(s(X),0()) -> c_7()
          8: le#(s(X),s(Y)) -> c_8(le#(X,Y))
          9: minus#(X,0()) -> c_9()
          10: minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y))
          11: pred#(s(X)) -> c_11()
* Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X))
            if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X))
            if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y))
            le#(s(X),s(Y)) -> c_8(le#(X,Y))
            minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y))
        - Weak DPs:
            gcd#(0(),Y) -> c_1()
            gcd#(s(X),0()) -> c_2()
            le#(0(),Y) -> c_6()
            le#(s(X),0()) -> c_7()
            minus#(X,0()) -> c_9()
            pred#(s(X)) -> c_11()
        - Weak TRS:
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(X,0()) -> X
            minus(X,s(Y)) -> pred(minus(X,Y))
            pred(s(X)) -> X
        - Signature:
            {gcd/2,if/3,le/2,minus/2,pred/1,gcd#/2,if#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0
            ,c_3/2,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {gcd#,if#,le#,minus#,pred#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X))
             -->_2 le#(s(X),s(Y)) -> c_8(le#(X,Y)):4
             -->_1 if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y)):3
             -->_1 if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X)):2
             -->_2 le#(s(X),0()) -> c_7():9
             -->_2 le#(0(),Y) -> c_6():8
          
          2:S:if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X))
             -->_2 minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y)):5
             -->_2 minus#(X,0()) -> c_9():10
             -->_1 gcd#(0(),Y) -> c_1():6
             -->_1 gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X)):1
          
          3:S:if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y))
             -->_2 minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y)):5
             -->_2 minus#(X,0()) -> c_9():10
             -->_1 gcd#(0(),Y) -> c_1():6
             -->_1 gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X)):1
          
          4:S:le#(s(X),s(Y)) -> c_8(le#(X,Y))
             -->_1 le#(s(X),0()) -> c_7():9
             -->_1 le#(0(),Y) -> c_6():8
             -->_1 le#(s(X),s(Y)) -> c_8(le#(X,Y)):4
          
          5:S:minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y))
             -->_1 pred#(s(X)) -> c_11():11
             -->_2 minus#(X,0()) -> c_9():10
             -->_2 minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y)):5
          
          6:W:gcd#(0(),Y) -> c_1()
             
          
          7:W:gcd#(s(X),0()) -> c_2()
             
          
          8:W:le#(0(),Y) -> c_6()
             
          
          9:W:le#(s(X),0()) -> c_7()
             
          
          10:W:minus#(X,0()) -> c_9()
             
          
          11:W:pred#(s(X)) -> c_11()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          7: gcd#(s(X),0()) -> c_2()
          6: gcd#(0(),Y) -> c_1()
          10: minus#(X,0()) -> c_9()
          11: pred#(s(X)) -> c_11()
          8: le#(0(),Y) -> c_6()
          9: le#(s(X),0()) -> c_7()
* Step 5: SimplifyRHS WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X))
            if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X))
            if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y))
            le#(s(X),s(Y)) -> c_8(le#(X,Y))
            minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y))
        - Weak TRS:
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(X,0()) -> X
            minus(X,s(Y)) -> pred(minus(X,Y))
            pred(s(X)) -> X
        - Signature:
            {gcd/2,if/3,le/2,minus/2,pred/1,gcd#/2,if#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0
            ,c_3/2,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {gcd#,if#,le#,minus#,pred#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X))
             -->_2 le#(s(X),s(Y)) -> c_8(le#(X,Y)):4
             -->_1 if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y)):3
             -->_1 if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X)):2
          
          2:S:if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X))
             -->_2 minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y)):5
             -->_1 gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X)):1
          
          3:S:if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y))
             -->_2 minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y)):5
             -->_1 gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X)):1
          
          4:S:le#(s(X),s(Y)) -> c_8(le#(X,Y))
             -->_1 le#(s(X),s(Y)) -> c_8(le#(X,Y)):4
          
          5:S:minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y))
             -->_2 minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y)):5
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          minus#(X,s(Y)) -> c_10(minus#(X,Y))
* Step 6: PredecessorEstimationCP WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X))
            if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X))
            if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y))
            le#(s(X),s(Y)) -> c_8(le#(X,Y))
            minus#(X,s(Y)) -> c_10(minus#(X,Y))
        - Weak TRS:
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(X,0()) -> X
            minus(X,s(Y)) -> pred(minus(X,Y))
            pred(s(X)) -> X
        - Signature:
            {gcd/2,if/3,le/2,minus/2,pred/1,gcd#/2,if#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0
            ,c_3/2,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {gcd#,if#,le#,minus#,pred#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          2: if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X))
          3: if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y))
          4: le#(s(X),s(Y)) -> c_8(le#(X,Y))
          
        Consider the set of all dependency pairs
          1: gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X))
          2: if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X))
          3: if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y))
          4: le#(s(X),s(Y)) -> c_8(le#(X,Y))
          5: minus#(X,s(Y)) -> c_10(minus#(X,Y))
        Processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^2))
        SPACE(?,?)on application of the dependency pairs
          {2,3,4}
        These cover all (indirect) predecessors of dependency pairs
          {1,2,3,4}
        their number of applications is equally bounded.
        The dependency pairs are shifted into the weak component.
** Step 6.a:1: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X))
            if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X))
            if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y))
            le#(s(X),s(Y)) -> c_8(le#(X,Y))
            minus#(X,s(Y)) -> c_10(minus#(X,Y))
        - Weak TRS:
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(X,0()) -> X
            minus(X,s(Y)) -> pred(minus(X,Y))
            pred(s(X)) -> X
        - Signature:
            {gcd/2,if/3,le/2,minus/2,pred/1,gcd#/2,if#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0
            ,c_3/2,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {gcd#,if#,le#,minus#,pred#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        NaturalPI {shape = Mixed 2, restrict = Restrict, 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 polynomial interpretation of kind constructor-based(mixed(2)):
        The following argument positions are considered usable:
          uargs(c_3) = {1,2},
          uargs(c_4) = {1,2},
          uargs(c_5) = {1,2},
          uargs(c_8) = {1},
          uargs(c_10) = {1}
        
        Following symbols are considered usable:
          {minus,pred,gcd#,if#,le#,minus#,pred#}
        TcT has computed the following interpretation:
               p(0) = 0                    
           p(false) = 2                    
             p(gcd) = x1^2 + x2^2          
              p(if) = 2*x3                 
              p(le) = 1 + 2*x2             
           p(minus) = x1                   
            p(pred) = x1                   
               p(s) = 1 + x1               
            p(true) = 0                    
            p(gcd#) = 2*x1 + 2*x1*x2 + 3*x2
             p(if#) = 2*x2 + 2*x2*x3 + 2*x3
             p(le#) = 1 + x1               
          p(minus#) = 2                    
           p(pred#) = 1                    
             p(c_1) = 1                    
             p(c_2) = 0                    
             p(c_3) = x1 + x2              
             p(c_4) = x1 + x2              
             p(c_5) = x1 + x2              
             p(c_6) = 0                    
             p(c_7) = 0                    
             p(c_8) = x1                   
             p(c_9) = 0                    
            p(c_10) = x1                   
            p(c_11) = 1                    
        
        Following rules are strictly oriented:
        if#(false(),s(X),s(Y)) = 6 + 4*X + 2*X*Y + 4*Y                 
                               > 5 + 3*X + 2*X*Y + 4*Y                 
                               = c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X))
        
         if#(true(),s(X),s(Y)) = 6 + 4*X + 2*X*Y + 4*Y                 
                               > 5 + 4*X + 2*X*Y + 3*Y                 
                               = c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y))
        
                le#(s(X),s(Y)) = 2 + X                                 
                               > 1 + X                                 
                               = c_8(le#(X,Y))                         
        
        
        Following rules are (at-least) weakly oriented:
        gcd#(s(X),s(Y)) =  7 + 4*X + 2*X*Y + 5*Y               
                        >= 7 + 4*X + 2*X*Y + 5*Y               
                        =  c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X))
        
         minus#(X,s(Y)) =  2                                   
                        >= 2                                   
                        =  c_10(minus#(X,Y))                   
        
           minus(X,0()) =  X                                   
                        >= X                                   
                        =  X                                   
        
          minus(X,s(Y)) =  X                                   
                        >= X                                   
                        =  pred(minus(X,Y))                    
        
             pred(s(X)) =  1 + X                               
                        >= X                                   
                        =  X                                   
        
** Step 6.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X))
            minus#(X,s(Y)) -> c_10(minus#(X,Y))
        - Weak DPs:
            if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X))
            if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y))
            le#(s(X),s(Y)) -> c_8(le#(X,Y))
        - Weak TRS:
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(X,0()) -> X
            minus(X,s(Y)) -> pred(minus(X,Y))
            pred(s(X)) -> X
        - Signature:
            {gcd/2,if/3,le/2,minus/2,pred/1,gcd#/2,if#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0
            ,c_3/2,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {gcd#,if#,le#,minus#,pred#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

** Step 6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            minus#(X,s(Y)) -> c_10(minus#(X,Y))
        - Weak DPs:
            gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X))
            if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X))
            if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y))
            le#(s(X),s(Y)) -> c_8(le#(X,Y))
        - Weak TRS:
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(X,0()) -> X
            minus(X,s(Y)) -> pred(minus(X,Y))
            pred(s(X)) -> X
        - Signature:
            {gcd/2,if/3,le/2,minus/2,pred/1,gcd#/2,if#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0
            ,c_3/2,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {gcd#,if#,le#,minus#,pred#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:minus#(X,s(Y)) -> c_10(minus#(X,Y))
             -->_1 minus#(X,s(Y)) -> c_10(minus#(X,Y)):1
          
          2:W:gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X))
             -->_2 le#(s(X),s(Y)) -> c_8(le#(X,Y)):5
             -->_1 if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y)):4
             -->_1 if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X)):3
          
          3:W:if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X))
             -->_1 gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X)):2
             -->_2 minus#(X,s(Y)) -> c_10(minus#(X,Y)):1
          
          4:W:if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y))
             -->_1 gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X)):2
             -->_2 minus#(X,s(Y)) -> c_10(minus#(X,Y)):1
          
          5:W:le#(s(X),s(Y)) -> c_8(le#(X,Y))
             -->_1 le#(s(X),s(Y)) -> c_8(le#(X,Y)):5
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: le#(s(X),s(Y)) -> c_8(le#(X,Y))
** Step 6.b:2: SimplifyRHS WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            minus#(X,s(Y)) -> c_10(minus#(X,Y))
        - Weak DPs:
            gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X))
            if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X))
            if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y))
        - Weak TRS:
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(X,0()) -> X
            minus(X,s(Y)) -> pred(minus(X,Y))
            pred(s(X)) -> X
        - Signature:
            {gcd/2,if/3,le/2,minus/2,pred/1,gcd#/2,if#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0
            ,c_3/2,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {gcd#,if#,le#,minus#,pred#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:minus#(X,s(Y)) -> c_10(minus#(X,Y))
             -->_1 minus#(X,s(Y)) -> c_10(minus#(X,Y)):1
          
          2:W:gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X))
             -->_1 if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y)):4
             -->_1 if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X)):3
          
          3:W:if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X))
             -->_1 gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X)):2
             -->_2 minus#(X,s(Y)) -> c_10(minus#(X,Y)):1
          
          4:W:if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y))
             -->_1 gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X)):2
             -->_2 minus#(X,s(Y)) -> c_10(minus#(X,Y)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)))
** Step 6.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            minus#(X,s(Y)) -> c_10(minus#(X,Y))
        - Weak DPs:
            gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)))
            if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X))
            if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y))
        - Weak TRS:
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(X,0()) -> X
            minus(X,s(Y)) -> pred(minus(X,Y))
            pred(s(X)) -> X
        - Signature:
            {gcd/2,if/3,le/2,minus/2,pred/1,gcd#/2,if#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0
            ,c_3/1,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {gcd#,if#,le#,minus#,pred#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: minus#(X,s(Y)) -> c_10(minus#(X,Y))
          
        The strictly oriented rules are moved into the weak component.
*** Step 6.b:3.a:1: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            minus#(X,s(Y)) -> c_10(minus#(X,Y))
        - Weak DPs:
            gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)))
            if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X))
            if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y))
        - Weak TRS:
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(X,0()) -> X
            minus(X,s(Y)) -> pred(minus(X,Y))
            pred(s(X)) -> X
        - Signature:
            {gcd/2,if/3,le/2,minus/2,pred/1,gcd#/2,if#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0
            ,c_3/1,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {gcd#,if#,le#,minus#,pred#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        NaturalPI {shape = Mixed 2, restrict = Restrict, 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 polynomial interpretation of kind constructor-based(mixed(2)):
        The following argument positions are considered usable:
          uargs(c_3) = {1},
          uargs(c_4) = {1,2},
          uargs(c_5) = {1,2},
          uargs(c_10) = {1}
        
        Following symbols are considered usable:
          {minus,pred,gcd#,if#,le#,minus#,pred#}
        TcT has computed the following interpretation:
               p(0) = 0                          
           p(false) = 0                          
             p(gcd) = 2*x1 + x1^2 + 2*x2 + 2*x2^2
              p(if) = 1 + x1*x2 + 2*x1*x3        
              p(le) = 2*x1                       
           p(minus) = x1                         
            p(pred) = x1                         
               p(s) = 1 + x1                     
            p(true) = 0                          
            p(gcd#) = x1 + x1*x2 + x2            
             p(if#) = x2 + x2*x3 + x3            
             p(le#) = 2*x1                       
          p(minus#) = 2 + x2                     
           p(pred#) = 0                          
             p(c_1) = 1                          
             p(c_2) = 1                          
             p(c_3) = x1                         
             p(c_4) = x1 + x2                    
             p(c_5) = x1 + x2                    
             p(c_6) = 1                          
             p(c_7) = 0                          
             p(c_8) = x1                         
             p(c_9) = 1                          
            p(c_10) = x1                         
            p(c_11) = 1                          
        
        Following rules are strictly oriented:
        minus#(X,s(Y)) = 3 + Y            
                       > 2 + Y            
                       = c_10(minus#(X,Y))
        
        
        Following rules are (at-least) weakly oriented:
               gcd#(s(X),s(Y)) =  3 + 2*X + X*Y + 2*Y                   
                               >= 3 + 2*X + X*Y + 2*Y                   
                               =  c_3(if#(le(Y,X),s(X),s(Y)))           
        
        if#(false(),s(X),s(Y)) =  3 + 2*X + X*Y + 2*Y                   
                               >= 3 + 2*X + X*Y + 2*Y                   
                               =  c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X))
        
         if#(true(),s(X),s(Y)) =  3 + 2*X + X*Y + 2*Y                   
                               >= 3 + 2*X + X*Y + 2*Y                   
                               =  c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y))
        
                  minus(X,0()) =  X                                     
                               >= X                                     
                               =  X                                     
        
                 minus(X,s(Y)) =  X                                     
                               >= X                                     
                               =  pred(minus(X,Y))                      
        
                    pred(s(X)) =  1 + X                                 
                               >= X                                     
                               =  X                                     
        
*** Step 6.b:3.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)))
            if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X))
            if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y))
            minus#(X,s(Y)) -> c_10(minus#(X,Y))
        - Weak TRS:
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(X,0()) -> X
            minus(X,s(Y)) -> pred(minus(X,Y))
            pred(s(X)) -> X
        - Signature:
            {gcd/2,if/3,le/2,minus/2,pred/1,gcd#/2,if#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0
            ,c_3/1,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {gcd#,if#,le#,minus#,pred#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

*** Step 6.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)))
            if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X))
            if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y))
            minus#(X,s(Y)) -> c_10(minus#(X,Y))
        - Weak TRS:
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(X,0()) -> X
            minus(X,s(Y)) -> pred(minus(X,Y))
            pred(s(X)) -> X
        - Signature:
            {gcd/2,if/3,le/2,minus/2,pred/1,gcd#/2,if#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0
            ,c_3/1,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {gcd#,if#,le#,minus#,pred#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)))
             -->_1 if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y)):3
             -->_1 if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X)):2
          
          2:W:if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X))
             -->_2 minus#(X,s(Y)) -> c_10(minus#(X,Y)):4
             -->_1 gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y))):1
          
          3:W:if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y))
             -->_2 minus#(X,s(Y)) -> c_10(minus#(X,Y)):4
             -->_1 gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y))):1
          
          4:W:minus#(X,s(Y)) -> c_10(minus#(X,Y))
             -->_1 minus#(X,s(Y)) -> c_10(minus#(X,Y)):4
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          1: gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)))
          3: if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y))
          2: if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X))
          4: minus#(X,s(Y)) -> c_10(minus#(X,Y))
*** Step 6.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            le(0(),Y) -> true()
            le(s(X),0()) -> false()
            le(s(X),s(Y)) -> le(X,Y)
            minus(X,0()) -> X
            minus(X,s(Y)) -> pred(minus(X,Y))
            pred(s(X)) -> X
        - Signature:
            {gcd/2,if/3,le/2,minus/2,pred/1,gcd#/2,if#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0
            ,c_3/1,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {gcd#,if#,le#,minus#,pred#} and constructors {0,false,s
            ,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(?,O(n^2))