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

** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict TRS:
            if_mod(false(),s(x),s(y)) -> s(x)
            if_mod(true(),s(x),s(y)) -> mod(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))
            mod(0(),y) -> 0()
            mod(s(x),0()) -> 0()
            mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
            pred(s(x)) -> x
        - Signature:
            {if_mod/3,le/2,minus/2,mod/2,pred/1} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_mod,le,minus,mod,pred} and constructors {0,false,s
            ,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          if_mod#(false(),s(x),s(y)) -> c_1()
          if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
          le#(0(),y) -> c_3()
          le#(s(x),0()) -> c_4()
          le#(s(x),s(y)) -> c_5(le#(x,y))
          minus#(x,0()) -> c_6()
          minus#(x,s(y)) -> c_7(pred#(minus(x,y)),minus#(x,y))
          mod#(0(),y) -> c_8()
          mod#(s(x),0()) -> c_9()
          mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
          pred#(s(x)) -> c_11()
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: UsableRules WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            if_mod#(false(),s(x),s(y)) -> c_1()
            if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
            le#(0(),y) -> c_3()
            le#(s(x),0()) -> c_4()
            le#(s(x),s(y)) -> c_5(le#(x,y))
            minus#(x,0()) -> c_6()
            minus#(x,s(y)) -> c_7(pred#(minus(x,y)),minus#(x,y))
            mod#(0(),y) -> c_8()
            mod#(s(x),0()) -> c_9()
            mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
            pred#(s(x)) -> c_11()
        - Weak TRS:
            if_mod(false(),s(x),s(y)) -> s(x)
            if_mod(true(),s(x),s(y)) -> mod(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))
            mod(0(),y) -> 0()
            mod(s(x),0()) -> 0()
            mod(s(x),s(y)) -> if_mod(le(y,x),s(x),s(y))
            pred(s(x)) -> x
        - Signature:
            {if_mod/3,le/2,minus/2,mod/2,pred/1,if_mod#/3,le#/2,minus#/2,mod#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0
            ,c_2/2,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/0,c_10/2,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_mod#,le#,minus#,mod#,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
          if_mod#(false(),s(x),s(y)) -> c_1()
          if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
          le#(0(),y) -> c_3()
          le#(s(x),0()) -> c_4()
          le#(s(x),s(y)) -> c_5(le#(x,y))
          minus#(x,0()) -> c_6()
          minus#(x,s(y)) -> c_7(pred#(minus(x,y)),minus#(x,y))
          mod#(0(),y) -> c_8()
          mod#(s(x),0()) -> c_9()
          mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
          pred#(s(x)) -> c_11()
** Step 1.b:3: PredecessorEstimation WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            if_mod#(false(),s(x),s(y)) -> c_1()
            if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
            le#(0(),y) -> c_3()
            le#(s(x),0()) -> c_4()
            le#(s(x),s(y)) -> c_5(le#(x,y))
            minus#(x,0()) -> c_6()
            minus#(x,s(y)) -> c_7(pred#(minus(x,y)),minus#(x,y))
            mod#(0(),y) -> c_8()
            mod#(s(x),0()) -> c_9()
            mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
            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:
            {if_mod/3,le/2,minus/2,mod/2,pred/1,if_mod#/3,le#/2,minus#/2,mod#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0
            ,c_2/2,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/0,c_10/2,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_mod#,le#,minus#,mod#,pred#} and constructors {0,false
            ,s,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,3,4,6,8,9,11}
        by application of
          Pre({1,3,4,6,8,9,11}) = {2,5,7,10}.
        Here rules are labelled as follows:
          1: if_mod#(false(),s(x),s(y)) -> c_1()
          2: if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
          3: le#(0(),y) -> c_3()
          4: le#(s(x),0()) -> c_4()
          5: le#(s(x),s(y)) -> c_5(le#(x,y))
          6: minus#(x,0()) -> c_6()
          7: minus#(x,s(y)) -> c_7(pred#(minus(x,y)),minus#(x,y))
          8: mod#(0(),y) -> c_8()
          9: mod#(s(x),0()) -> c_9()
          10: mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
          11: pred#(s(x)) -> c_11()
** Step 1.b:4: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
            le#(s(x),s(y)) -> c_5(le#(x,y))
            minus#(x,s(y)) -> c_7(pred#(minus(x,y)),minus#(x,y))
            mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
        - Weak DPs:
            if_mod#(false(),s(x),s(y)) -> c_1()
            le#(0(),y) -> c_3()
            le#(s(x),0()) -> c_4()
            minus#(x,0()) -> c_6()
            mod#(0(),y) -> c_8()
            mod#(s(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:
            {if_mod/3,le/2,minus/2,mod/2,pred/1,if_mod#/3,le#/2,minus#/2,mod#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0
            ,c_2/2,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/0,c_10/2,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_mod#,le#,minus#,mod#,pred#} and constructors {0,false
            ,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
             -->_1 mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x)):4
             -->_2 minus#(x,s(y)) -> c_7(pred#(minus(x,y)),minus#(x,y)):3
             -->_1 mod#(0(),y) -> c_8():9
             -->_2 minus#(x,0()) -> c_6():8
          
          2:S:le#(s(x),s(y)) -> c_5(le#(x,y))
             -->_1 le#(s(x),0()) -> c_4():7
             -->_1 le#(0(),y) -> c_3():6
             -->_1 le#(s(x),s(y)) -> c_5(le#(x,y)):2
          
          3:S:minus#(x,s(y)) -> c_7(pred#(minus(x,y)),minus#(x,y))
             -->_1 pred#(s(x)) -> c_11():11
             -->_2 minus#(x,0()) -> c_6():8
             -->_2 minus#(x,s(y)) -> c_7(pred#(minus(x,y)),minus#(x,y)):3
          
          4:S:mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
             -->_2 le#(s(x),0()) -> c_4():7
             -->_2 le#(0(),y) -> c_3():6
             -->_1 if_mod#(false(),s(x),s(y)) -> c_1():5
             -->_2 le#(s(x),s(y)) -> c_5(le#(x,y)):2
             -->_1 if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y)):1
          
          5:W:if_mod#(false(),s(x),s(y)) -> c_1()
             
          
          6:W:le#(0(),y) -> c_3()
             
          
          7:W:le#(s(x),0()) -> c_4()
             
          
          8:W:minus#(x,0()) -> c_6()
             
          
          9:W:mod#(0(),y) -> c_8()
             
          
          10:W:mod#(s(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.
          10: mod#(s(x),0()) -> c_9()
          9: mod#(0(),y) -> c_8()
          8: minus#(x,0()) -> c_6()
          11: pred#(s(x)) -> c_11()
          5: if_mod#(false(),s(x),s(y)) -> c_1()
          6: le#(0(),y) -> c_3()
          7: le#(s(x),0()) -> c_4()
** Step 1.b:5: SimplifyRHS WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
            le#(s(x),s(y)) -> c_5(le#(x,y))
            minus#(x,s(y)) -> c_7(pred#(minus(x,y)),minus#(x,y))
            mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
        - 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:
            {if_mod/3,le/2,minus/2,mod/2,pred/1,if_mod#/3,le#/2,minus#/2,mod#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0
            ,c_2/2,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/0,c_10/2,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_mod#,le#,minus#,mod#,pred#} and constructors {0,false
            ,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
             -->_1 mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x)):4
             -->_2 minus#(x,s(y)) -> c_7(pred#(minus(x,y)),minus#(x,y)):3
          
          2:S:le#(s(x),s(y)) -> c_5(le#(x,y))
             -->_1 le#(s(x),s(y)) -> c_5(le#(x,y)):2
          
          3:S:minus#(x,s(y)) -> c_7(pred#(minus(x,y)),minus#(x,y))
             -->_2 minus#(x,s(y)) -> c_7(pred#(minus(x,y)),minus#(x,y)):3
          
          4:S:mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
             -->_2 le#(s(x),s(y)) -> c_5(le#(x,y)):2
             -->_1 if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          minus#(x,s(y)) -> c_7(minus#(x,y))
** Step 1.b:6: PredecessorEstimationCP WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
            le#(s(x),s(y)) -> c_5(le#(x,y))
            minus#(x,s(y)) -> c_7(minus#(x,y))
            mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
        - 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:
            {if_mod/3,le/2,minus/2,mod/2,pred/1,if_mod#/3,le#/2,minus#/2,mod#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0
            ,c_2/2,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/2,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_mod#,le#,minus#,mod#,pred#} and constructors {0,false
            ,s,true}
    + Applied Processor:
        PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          2: le#(s(x),s(y)) -> c_5(le#(x,y))
          
        The strictly oriented rules are moved into the weak component.
*** Step 1.b:6.a:1: NaturalMI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
            le#(s(x),s(y)) -> c_5(le#(x,y))
            minus#(x,s(y)) -> c_7(minus#(x,y))
            mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
        - 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:
            {if_mod/3,le/2,minus/2,mod/2,pred/1,if_mod#/3,le#/2,minus#/2,mod#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0
            ,c_2/2,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/2,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_mod#,le#,minus#,mod#,pred#} and constructors {0,false
            ,s,true}
    + Applied Processor:
        NaturalMI {miDimension = 3, miDegree = 2, 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 (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        The following argument positions are considered usable:
          uargs(c_2) = {1,2},
          uargs(c_5) = {1},
          uargs(c_7) = {1},
          uargs(c_10) = {1,2}
        
        Following symbols are considered usable:
          {le,minus,pred,if_mod#,le#,minus#,mod#,pred#}
        TcT has computed the following interpretation:
                p(0) = [0]                                       
                       [0]                                       
                       [1]                                       
            p(false) = [1]                                       
                       [0]                                       
                       [0]                                       
           p(if_mod) = [0]                                       
                       [0]                                       
                       [0]                                       
               p(le) = [0 0 1]      [0 0 0]      [0]             
                       [0 0 1] x1 + [0 0 1] x2 + [0]             
                       [0 0 0]      [1 0 0]      [0]             
            p(minus) = [1 0 0]      [0 0 0]      [0]             
                       [1 1 1] x1 + [1 0 1] x2 + [0]             
                       [0 0 1]      [0 0 0]      [0]             
              p(mod) = [0]                                       
                       [0]                                       
                       [0]                                       
             p(pred) = [1 0 0]      [0]                          
                       [1 0 0] x1 + [0]                          
                       [0 0 1]      [0]                          
                p(s) = [1 1 1]      [0]                          
                       [0 0 1] x1 + [1]                          
                       [0 0 1]      [1]                          
             p(true) = [1]                                       
                       [1]                                       
                       [0]                                       
          p(if_mod#) = [0 0 0]      [1 0 0]      [0 1 1]      [1]
                       [1 0 0] x1 + [0 1 0] x2 + [0 0 0] x3 + [1]
                       [1 1 0]      [1 0 0]      [0 0 0]      [0]
              p(le#) = [0 0 1]      [0]                          
                       [1 0 0] x2 + [0]                          
                       [0 0 1]      [0]                          
           p(minus#) = [0 1 0]      [0]                          
                       [0 0 0] x1 + [1]                          
                       [0 0 1]      [1]                          
             p(mod#) = [1 0 1]      [0 1 1]      [0]             
                       [1 1 0] x1 + [0 0 1] x2 + [1]             
                       [0 0 0]      [1 0 1]      [0]             
            p(pred#) = [0]                                       
                       [0]                                       
                       [0]                                       
              p(c_1) = [0]                                       
                       [0]                                       
                       [0]                                       
              p(c_2) = [1 0 0]      [1 1 0]      [0]             
                       [0 0 0] x1 + [0 1 1] x2 + [1]             
                       [0 0 0]      [0 0 1]      [1]             
              p(c_3) = [0]                                       
                       [0]                                       
                       [0]                                       
              p(c_4) = [0]                                       
                       [0]                                       
                       [0]                                       
              p(c_5) = [1 0 0]      [0]                          
                       [1 1 0] x1 + [0]                          
                       [0 0 0]      [1]                          
              p(c_6) = [0]                                       
                       [0]                                       
                       [0]                                       
              p(c_7) = [1 0 0]      [0]                          
                       [0 0 0] x1 + [0]                          
                       [0 0 0]      [0]                          
              p(c_8) = [0]                                       
                       [0]                                       
                       [0]                                       
              p(c_9) = [0]                                       
                       [0]                                       
                       [0]                                       
             p(c_10) = [1 0 0]      [1 0 0]      [0]             
                       [0 1 0] x1 + [1 1 0] x2 + [1]             
                       [0 0 0]      [0 0 0]      [0]             
             p(c_11) = [0]                                       
                       [0]                                       
                       [0]                                       
        
        Following rules are strictly oriented:
        le#(s(x),s(y)) = [0 0 1]     [1]
                         [1 1 1] y + [0]
                         [0 0 1]     [1]
                       > [0 0 1]     [0]
                         [1 0 1] y + [0]
                         [0 0 0]     [1]
                       = c_5(le#(x,y))  
        
        
        Following rules are (at-least) weakly oriented:
        if_mod#(true(),s(x),s(y)) =  [1 1 1]     [0 0 2]     [3]              
                                     [0 0 1] x + [0 0 0] y + [3]              
                                     [1 1 1]     [0 0 0]     [2]              
                                  >= [1 1 1]     [0 0 2]     [3]              
                                     [0 0 1] x + [0 0 0] y + [3]              
                                     [0 0 1]     [0 0 0]     [2]              
                                  =  c_2(mod#(minus(x,y),s(y)),minus#(x,y))   
        
                   minus#(x,s(y)) =  [0 1 0]     [0]                          
                                     [0 0 0] x + [1]                          
                                     [0 0 1]     [1]                          
                                  >= [0 1 0]     [0]                          
                                     [0 0 0] x + [0]                          
                                     [0 0 0]     [0]                          
                                  =  c_7(minus#(x,y))                         
        
                  mod#(s(x),s(y)) =  [1 1 2]     [0 0 2]     [3]              
                                     [1 1 2] x + [0 0 1] y + [3]              
                                     [0 0 0]     [1 1 2]     [1]              
                                  >= [1 1 2]     [0 0 2]     [3]              
                                     [1 0 2] x + [0 0 1] y + [3]              
                                     [0 0 0]     [0 0 0]     [0]              
                                  =  c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
        
                        le(0(),y) =  [0 0 0]     [1]                          
                                     [0 0 1] y + [1]                          
                                     [1 0 0]     [0]                          
                                  >= [1]                                      
                                     [1]                                      
                                     [0]                                      
                                  =  true()                                   
        
                     le(s(x),0()) =  [0 0 1]     [1]                          
                                     [0 0 1] x + [2]                          
                                     [0 0 0]     [0]                          
                                  >= [1]                                      
                                     [0]                                      
                                     [0]                                      
                                  =  false()                                  
        
                    le(s(x),s(y)) =  [0 0 1]     [0 0 0]     [1]              
                                     [0 0 1] x + [0 0 1] y + [2]              
                                     [0 0 0]     [1 1 1]     [0]              
                                  >= [0 0 1]     [0 0 0]     [0]              
                                     [0 0 1] x + [0 0 1] y + [0]              
                                     [0 0 0]     [1 0 0]     [0]              
                                  =  le(x,y)                                  
        
                     minus(x,0()) =  [1 0 0]     [0]                          
                                     [1 1 1] x + [1]                          
                                     [0 0 1]     [0]                          
                                  >= [1 0 0]     [0]                          
                                     [0 1 0] x + [0]                          
                                     [0 0 1]     [0]                          
                                  =  x                                        
        
                    minus(x,s(y)) =  [1 0 0]     [0 0 0]     [0]              
                                     [1 1 1] x + [1 1 2] y + [1]              
                                     [0 0 1]     [0 0 0]     [0]              
                                  >= [1 0 0]     [0]                          
                                     [1 0 0] x + [0]                          
                                     [0 0 1]     [0]                          
                                  =  pred(minus(x,y))                         
        
                       pred(s(x)) =  [1 1 1]     [0]                          
                                     [1 1 1] x + [0]                          
                                     [0 0 1]     [1]                          
                                  >= [1 0 0]     [0]                          
                                     [0 1 0] x + [0]                          
                                     [0 0 1]     [0]                          
                                  =  x                                        
        
*** Step 1.b:6.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
            minus#(x,s(y)) -> c_7(minus#(x,y))
            mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
        - Weak DPs:
            le#(s(x),s(y)) -> c_5(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:
            {if_mod/3,le/2,minus/2,mod/2,pred/1,if_mod#/3,le#/2,minus#/2,mod#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0
            ,c_2/2,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/2,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_mod#,le#,minus#,mod#,pred#} and constructors {0,false
            ,s,true}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

*** Step 1.b:6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
            minus#(x,s(y)) -> c_7(minus#(x,y))
            mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
        - Weak DPs:
            le#(s(x),s(y)) -> c_5(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:
            {if_mod/3,le/2,minus/2,mod/2,pred/1,if_mod#/3,le#/2,minus#/2,mod#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0
            ,c_2/2,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/2,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_mod#,le#,minus#,mod#,pred#} and constructors {0,false
            ,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
             -->_1 mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x)):3
             -->_2 minus#(x,s(y)) -> c_7(minus#(x,y)):2
          
          2:S:minus#(x,s(y)) -> c_7(minus#(x,y))
             -->_1 minus#(x,s(y)) -> c_7(minus#(x,y)):2
          
          3:S:mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
             -->_2 le#(s(x),s(y)) -> c_5(le#(x,y)):4
             -->_1 if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y)):1
          
          4:W:le#(s(x),s(y)) -> c_5(le#(x,y))
             -->_1 le#(s(x),s(y)) -> c_5(le#(x,y)):4
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: le#(s(x),s(y)) -> c_5(le#(x,y))
*** Step 1.b:6.b:2: SimplifyRHS WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
            minus#(x,s(y)) -> c_7(minus#(x,y))
            mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
        - 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:
            {if_mod/3,le/2,minus/2,mod/2,pred/1,if_mod#/3,le#/2,minus#/2,mod#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0
            ,c_2/2,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/2,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_mod#,le#,minus#,mod#,pred#} and constructors {0,false
            ,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
             -->_1 mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x)):3
             -->_2 minus#(x,s(y)) -> c_7(minus#(x,y)):2
          
          2:S:minus#(x,s(y)) -> c_7(minus#(x,y))
             -->_1 minus#(x,s(y)) -> c_7(minus#(x,y)):2
          
          3:S:mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
             -->_1 if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)))
*** Step 1.b:6.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
            minus#(x,s(y)) -> c_7(minus#(x,y))
            mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(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:
            {if_mod/3,le/2,minus/2,mod/2,pred/1,if_mod#/3,le#/2,minus#/2,mod#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0
            ,c_2/2,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_mod#,le#,minus#,mod#,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: minus#(x,s(y)) -> c_7(minus#(x,y))
          
        The strictly oriented rules are moved into the weak component.
**** Step 1.b:6.b:3.a:1: NaturalPI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
            minus#(x,s(y)) -> c_7(minus#(x,y))
            mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(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:
            {if_mod/3,le/2,minus/2,mod/2,pred/1,if_mod#/3,le#/2,minus#/2,mod#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0
            ,c_2/2,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_mod#,le#,minus#,mod#,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_2) = {1,2},
          uargs(c_7) = {1},
          uargs(c_10) = {1}
        
        Following symbols are considered usable:
          {minus,pred,if_mod#,le#,minus#,mod#,pred#}
        TcT has computed the following interpretation:
                p(0) = 0                        
            p(false) = 1                        
           p(if_mod) = 2 + 4*x1*x3 + x1^2 + 4*x3
               p(le) = 0                        
            p(minus) = x1                       
              p(mod) = 1 + 2*x1 + x1^2 + 4*x2^2 
             p(pred) = x1                       
                p(s) = 1 + x1                   
             p(true) = 0                        
          p(if_mod#) = 6 + 2*x2*x3              
              p(le#) = 2 + x1 + 4*x1*x2         
           p(minus#) = 1 + x2                   
             p(mod#) = 6 + 2*x1*x2              
            p(pred#) = 1 + 2*x1^2               
              p(c_1) = 0                        
              p(c_2) = 1 + x1 + x2              
              p(c_3) = 0                        
              p(c_4) = 1                        
              p(c_5) = 1                        
              p(c_6) = 1                        
              p(c_7) = x1                       
              p(c_8) = 0                        
              p(c_9) = 0                        
             p(c_10) = x1                       
             p(c_11) = 1                        
        
        Following rules are strictly oriented:
        minus#(x,s(y)) = 2 + y           
                       > 1 + y           
                       = c_7(minus#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
        if_mod#(true(),s(x),s(y)) =  8 + 2*x + 2*x*y + 2*y                 
                                  >= 8 + 2*x + 2*x*y + y                   
                                  =  c_2(mod#(minus(x,y),s(y)),minus#(x,y))
        
                  mod#(s(x),s(y)) =  8 + 2*x + 2*x*y + 2*y                 
                                  >= 8 + 2*x + 2*x*y + 2*y                 
                                  =  c_10(if_mod#(le(y,x),s(x),s(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 1.b:6.b:3.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
            mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)))
        - Weak DPs:
            minus#(x,s(y)) -> c_7(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:
            {if_mod/3,le/2,minus/2,mod/2,pred/1,if_mod#/3,le#/2,minus#/2,mod#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0
            ,c_2/2,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_mod#,le#,minus#,mod#,pred#} and constructors {0,false
            ,s,true}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

**** Step 1.b:6.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
            mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)))
        - Weak DPs:
            minus#(x,s(y)) -> c_7(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:
            {if_mod/3,le/2,minus/2,mod/2,pred/1,if_mod#/3,le#/2,minus#/2,mod#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0
            ,c_2/2,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_mod#,le#,minus#,mod#,pred#} and constructors {0,false
            ,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
             -->_2 minus#(x,s(y)) -> c_7(minus#(x,y)):3
             -->_1 mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y))):2
          
          2:S:mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)))
             -->_1 if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y)):1
          
          3:W:minus#(x,s(y)) -> c_7(minus#(x,y))
             -->_1 minus#(x,s(y)) -> c_7(minus#(x,y)):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: minus#(x,s(y)) -> c_7(minus#(x,y))
**** Step 1.b:6.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
            mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(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:
            {if_mod/3,le/2,minus/2,mod/2,pred/1,if_mod#/3,le#/2,minus#/2,mod#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0
            ,c_2/2,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_mod#,le#,minus#,mod#,pred#} and constructors {0,false
            ,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y))
             -->_1 mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y))):2
          
          2:S:mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)))
             -->_1 if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)),minus#(x,y)):1
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)))
**** Step 1.b:6.b:3.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)))
            mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(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:
            {if_mod/3,le/2,minus/2,mod/2,pred/1,if_mod#/3,le#/2,minus#/2,mod#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0
            ,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_mod#,le#,minus#,mod#,pred#} and constructors {0,false
            ,s,true}
    + 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:
          2: mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)))
          
        Consider the set of all dependency pairs
          1: if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)))
          2: mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(y)))
        Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1))
        SPACE(?,?)on application of the dependency pairs
          {2}
        These cover all (indirect) predecessors of dependency pairs
          {1,2}
        their number of applications is equally bounded.
        The dependency pairs are shifted into the weak component.
***** Step 1.b:6.b:3.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)))
            mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(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:
            {if_mod/3,le/2,minus/2,mod/2,pred/1,if_mod#/3,le#/2,minus#/2,mod#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0
            ,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_mod#,le#,minus#,mod#,pred#} and constructors {0,false
            ,s,true}
    + 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_2) = {1},
          uargs(c_10) = {1}
        
        Following symbols are considered usable:
          {le,minus,pred,if_mod#,le#,minus#,mod#,pred#}
        TcT has computed the following interpretation:
                p(0) = [0]                           
            p(false) = [1]                           
           p(if_mod) = [8] x2 + [1] x3 + [4]         
               p(le) = [1]                           
            p(minus) = [1] x1 + [0]                  
              p(mod) = [1] x1 + [1] x2 + [2]         
             p(pred) = [1] x1 + [0]                  
                p(s) = [1] x1 + [2]                  
             p(true) = [1]                           
          p(if_mod#) = [4] x1 + [1] x2 + [9] x3 + [1]
              p(le#) = [0]                           
           p(minus#) = [2] x2 + [1]                  
             p(mod#) = [1] x1 + [9] x2 + [7]         
            p(pred#) = [1] x1 + [0]                  
              p(c_1) = [1]                           
              p(c_2) = [1] x1 + [0]                  
              p(c_3) = [1]                           
              p(c_4) = [1]                           
              p(c_5) = [1]                           
              p(c_6) = [1]                           
              p(c_7) = [8]                           
              p(c_8) = [0]                           
              p(c_9) = [0]                           
             p(c_10) = [1] x1 + [0]                  
             p(c_11) = [0]                           
        
        Following rules are strictly oriented:
        mod#(s(x),s(y)) = [1] x + [9] y + [27]            
                        > [1] x + [9] y + [25]            
                        = c_10(if_mod#(le(y,x),s(x),s(y)))
        
        
        Following rules are (at-least) weakly oriented:
        if_mod#(true(),s(x),s(y)) =  [1] x + [9] y + [25]      
                                  >= [1] x + [9] y + [25]      
                                  =  c_2(mod#(minus(x,y),s(y)))
        
                        le(0(),y) =  [1]                       
                                  >= [1]                       
                                  =  true()                    
        
                     le(s(x),0()) =  [1]                       
                                  >= [1]                       
                                  =  false()                   
        
                    le(s(x),s(y)) =  [1]                       
                                  >= [1]                       
                                  =  le(x,y)                   
        
                     minus(x,0()) =  [1] x + [0]               
                                  >= [1] x + [0]               
                                  =  x                         
        
                    minus(x,s(y)) =  [1] x + [0]               
                                  >= [1] x + [0]               
                                  =  pred(minus(x,y))          
        
                       pred(s(x)) =  [1] x + [2]               
                                  >= [1] x + [0]               
                                  =  x                         
        
***** Step 1.b:6.b:3.b:3.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            if_mod#(true(),s(x),s(y)) -> c_2(mod#(minus(x,y),s(y)))
        - Weak DPs:
            mod#(s(x),s(y)) -> c_10(if_mod#(le(y,x),s(x),s(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:
            {if_mod/3,le/2,minus/2,mod/2,pred/1,if_mod#/3,le#/2,minus#/2,mod#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0
            ,c_2/1,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/0,c_10/1,c_11/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_mod#,le#,minus#,mod#,pred#} and constructors {0,false
            ,s,true}
    + Applied Processor:
        Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}}
    + Details:
        ()

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