* Step 1: Sum WORST_CASE(Omega(n^1),O(n^3))
    + Considered Problem:
        - Strict TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(true(),s(x),y) -> 0()
            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(0(),y) -> 0()
            minus(s(x),y) -> if_minus(le(s(x),y),s(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))
        - Signature:
            {if_minus/3,if_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus,if_mod,le,minus,mod} 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_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(true(),s(x),y) -> 0()
            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(0(),y) -> 0()
            minus(s(x),y) -> if_minus(le(s(x),y),s(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))
        - Signature:
            {if_minus/3,if_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus,if_mod,le,minus,mod} 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^3))
    + Considered Problem:
        - Strict TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(true(),s(x),y) -> 0()
            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(0(),y) -> 0()
            minus(s(x),y) -> if_minus(le(s(x),y),s(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))
        - Signature:
            {if_minus/3,if_mod/3,le/2,minus/2,mod/2} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus,if_mod,le,minus,mod} and constructors {0,false,s
            ,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
          if_minus#(true(),s(x),y) -> c_2()
          if_mod#(false(),s(x),s(y)) -> c_3()
          if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
          le#(0(),y) -> c_5()
          le#(s(x),0()) -> c_6()
          le#(s(x),s(y)) -> c_7(le#(x,y))
          minus#(0(),y) -> c_8()
          minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
          mod#(0(),y) -> c_10()
          mod#(s(x),0()) -> c_11()
          mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: UsableRules WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
            if_minus#(true(),s(x),y) -> c_2()
            if_mod#(false(),s(x),s(y)) -> c_3()
            if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
            le#(0(),y) -> c_5()
            le#(s(x),0()) -> c_6()
            le#(s(x),s(y)) -> c_7(le#(x,y))
            minus#(0(),y) -> c_8()
            minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
            mod#(0(),y) -> c_10()
            mod#(s(x),0()) -> c_11()
            mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(true(),s(x),y) -> 0()
            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(0(),y) -> 0()
            minus(s(x),y) -> if_minus(le(s(x),y),s(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))
        - Signature:
            {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1
            ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          if_minus(false(),s(x),y) -> s(minus(x,y))
          if_minus(true(),s(x),y) -> 0()
          le(0(),y) -> true()
          le(s(x),0()) -> false()
          le(s(x),s(y)) -> le(x,y)
          minus(0(),y) -> 0()
          minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
          if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
          if_minus#(true(),s(x),y) -> c_2()
          if_mod#(false(),s(x),s(y)) -> c_3()
          if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
          le#(0(),y) -> c_5()
          le#(s(x),0()) -> c_6()
          le#(s(x),s(y)) -> c_7(le#(x,y))
          minus#(0(),y) -> c_8()
          minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
          mod#(0(),y) -> c_10()
          mod#(s(x),0()) -> c_11()
          mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
** Step 1.b:3: PredecessorEstimation WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
            if_minus#(true(),s(x),y) -> c_2()
            if_mod#(false(),s(x),s(y)) -> c_3()
            if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
            le#(0(),y) -> c_5()
            le#(s(x),0()) -> c_6()
            le#(s(x),s(y)) -> c_7(le#(x,y))
            minus#(0(),y) -> c_8()
            minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
            mod#(0(),y) -> c_10()
            mod#(s(x),0()) -> c_11()
            mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(true(),s(x),y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(0(),y) -> 0()
            minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
        - Signature:
            {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1
            ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {2,3,5,6,8,10,11}
        by application of
          Pre({2,3,5,6,8,10,11}) = {1,4,7,9,12}.
        Here rules are labelled as follows:
          1: if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
          2: if_minus#(true(),s(x),y) -> c_2()
          3: if_mod#(false(),s(x),s(y)) -> c_3()
          4: if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
          5: le#(0(),y) -> c_5()
          6: le#(s(x),0()) -> c_6()
          7: le#(s(x),s(y)) -> c_7(le#(x,y))
          8: minus#(0(),y) -> c_8()
          9: minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
          10: mod#(0(),y) -> c_10()
          11: mod#(s(x),0()) -> c_11()
          12: mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
** Step 1.b:4: RemoveWeakSuffixes WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
            if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
            le#(s(x),s(y)) -> c_7(le#(x,y))
            minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
            mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
        - Weak DPs:
            if_minus#(true(),s(x),y) -> c_2()
            if_mod#(false(),s(x),s(y)) -> c_3()
            le#(0(),y) -> c_5()
            le#(s(x),0()) -> c_6()
            minus#(0(),y) -> c_8()
            mod#(0(),y) -> c_10()
            mod#(s(x),0()) -> c_11()
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(true(),s(x),y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(0(),y) -> 0()
            minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
        - Signature:
            {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1
            ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
             -->_1 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):4
             -->_1 minus#(0(),y) -> c_8():10
          
          2:S:if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
             -->_1 mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x)):5
             -->_2 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):4
             -->_1 mod#(0(),y) -> c_10():11
             -->_2 minus#(0(),y) -> c_8():10
          
          3:S:le#(s(x),s(y)) -> c_7(le#(x,y))
             -->_1 le#(s(x),0()) -> c_6():9
             -->_1 le#(0(),y) -> c_5():8
             -->_1 le#(s(x),s(y)) -> c_7(le#(x,y)):3
          
          4:S:minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
             -->_2 le#(s(x),0()) -> c_6():9
             -->_1 if_minus#(true(),s(x),y) -> c_2():6
             -->_2 le#(s(x),s(y)) -> c_7(le#(x,y)):3
             -->_1 if_minus#(false(),s(x),y) -> c_1(minus#(x,y)):1
          
          5:S:mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
             -->_2 le#(s(x),0()) -> c_6():9
             -->_2 le#(0(),y) -> c_5():8
             -->_1 if_mod#(false(),s(x),s(y)) -> c_3():7
             -->_2 le#(s(x),s(y)) -> c_7(le#(x,y)):3
             -->_1 if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y)):2
          
          6:W:if_minus#(true(),s(x),y) -> c_2()
             
          
          7:W:if_mod#(false(),s(x),s(y)) -> c_3()
             
          
          8:W:le#(0(),y) -> c_5()
             
          
          9:W:le#(s(x),0()) -> c_6()
             
          
          10:W:minus#(0(),y) -> c_8()
             
          
          11:W:mod#(0(),y) -> c_10()
             
          
          12:W:mod#(s(x),0()) -> c_11()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          12: mod#(s(x),0()) -> c_11()
          11: mod#(0(),y) -> c_10()
          7: if_mod#(false(),s(x),s(y)) -> c_3()
          10: minus#(0(),y) -> c_8()
          8: le#(0(),y) -> c_5()
          6: if_minus#(true(),s(x),y) -> c_2()
          9: le#(s(x),0()) -> c_6()
** Step 1.b:5: Decompose WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
            if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
            le#(s(x),s(y)) -> c_7(le#(x,y))
            minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
            mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(true(),s(x),y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(0(),y) -> 0()
            minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
        - Signature:
            {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1
            ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd}
    + Details:
        We analyse the complexity of following sub-problems (R) and (S).
        Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component.
        
        Problem (R)
          - Strict DPs:
              if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
              le#(s(x),s(y)) -> c_7(le#(x,y))
              minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
          - Weak DPs:
              if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
              mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
          - Weak TRS:
              if_minus(false(),s(x),y) -> s(minus(x,y))
              if_minus(true(),s(x),y) -> 0()
              le(0(),y) -> true()
              le(s(x),0()) -> false()
              le(s(x),s(y)) -> le(x,y)
              minus(0(),y) -> 0()
              minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
          - Signature:
              {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1
              ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} and constructors {0
              ,false,s,true}
        
        Problem (S)
          - Strict DPs:
              if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
              mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
          - Weak DPs:
              if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
              le#(s(x),s(y)) -> c_7(le#(x,y))
              minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
          - Weak TRS:
              if_minus(false(),s(x),y) -> s(minus(x,y))
              if_minus(true(),s(x),y) -> 0()
              le(0(),y) -> true()
              le(s(x),0()) -> false()
              le(s(x),s(y)) -> le(x,y)
              minus(0(),y) -> 0()
              minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
          - Signature:
              {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1
              ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2}
          - Obligation:
              innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} and constructors {0
              ,false,s,true}
*** Step 1.b:5.a:1: DecomposeDG WORST_CASE(?,O(n^3))
    + Considered Problem:
        - Strict DPs:
            if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
            le#(s(x),s(y)) -> c_7(le#(x,y))
            minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
        - Weak DPs:
            if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
            mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(true(),s(x),y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(0(),y) -> 0()
            minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
        - Signature:
            {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1
            ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
          mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
        and a lower component
          if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
          le#(s(x),s(y)) -> c_7(le#(x,y))
          minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
        Further, following extension rules are added to the lower component.
          if_mod#(true(),s(x),s(y)) -> minus#(x,y)
          if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
          mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
          mod#(s(x),s(y)) -> le#(y,x)
**** Step 1.b:5.a:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
            mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(true(),s(x),y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(0(),y) -> 0()
            minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
        - Signature:
            {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1
            ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} 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:
          1: if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
          
        Consider the set of all dependency pairs
          1: if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
          2: mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
        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
          {1}
        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:5.a:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
            mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(true(),s(x),y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(0(),y) -> 0()
            minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
        - Signature:
            {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1
            ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} 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_4) = {1},
          uargs(c_12) = {1}
        
        Following symbols are considered usable:
          {if_minus,minus,if_minus#,if_mod#,le#,minus#,mod#}
        TcT has computed the following interpretation:
                  p(0) = [1]                    
              p(false) = [0]                    
           p(if_minus) = [1] x2 + [0]           
             p(if_mod) = [1] x1 + [4] x3 + [1]  
                 p(le) = [0]                    
              p(minus) = [1] x1 + [0]           
                p(mod) = [4] x1 + [1] x2 + [1]  
                  p(s) = [1] x1 + [1]           
               p(true) = [0]                    
          p(if_minus#) = [1] x2 + [0]           
            p(if_mod#) = [4] x2 + [11] x3 + [11]
                p(le#) = [1] x2 + [2]           
             p(minus#) = [3]                    
               p(mod#) = [4] x1 + [11] x2 + [11]
                p(c_1) = [4] x1 + [2]           
                p(c_2) = [4]                    
                p(c_3) = [0]                    
                p(c_4) = [1] x1 + [1] x2 + [0]  
                p(c_5) = [1]                    
                p(c_6) = [0]                    
                p(c_7) = [2] x1 + [1]           
                p(c_8) = [1]                    
                p(c_9) = [1] x1 + [1]           
               p(c_10) = [0]                    
               p(c_11) = [1]                    
               p(c_12) = [1] x1 + [0]           
        
        Following rules are strictly oriented:
        if_mod#(true(),s(x),s(y)) = [4] x + [11] y + [26]                 
                                  > [4] x + [11] y + [25]                 
                                  = c_4(mod#(minus(x,y),s(y)),minus#(x,y))
        
        
        Following rules are (at-least) weakly oriented:
                 mod#(s(x),s(y)) =  [4] x + [11] y + [26]                    
                                 >= [4] x + [11] y + [26]                    
                                 =  c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
        
        if_minus(false(),s(x),y) =  [1] x + [1]                              
                                 >= [1] x + [1]                              
                                 =  s(minus(x,y))                            
        
         if_minus(true(),s(x),y) =  [1] x + [1]                              
                                 >= [1]                                      
                                 =  0()                                      
        
                    minus(0(),y) =  [1]                                      
                                 >= [1]                                      
                                 =  0()                                      
        
                   minus(s(x),y) =  [1] x + [1]                              
                                 >= [1] x + [1]                              
                                 =  if_minus(le(s(x),y),s(x),y)              
        
***** Step 1.b:5.a:1.a:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
        - Weak DPs:
            if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(true(),s(x),y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(0(),y) -> 0()
            minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
        - Signature:
            {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1
            ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} 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:5.a:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
            mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(true(),s(x),y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(0(),y) -> 0()
            minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
        - Signature:
            {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1
            ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
             -->_1 mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x)):2
          
          2:W:mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
             -->_1 if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,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_4(mod#(minus(x,y),s(y)),minus#(x,y))
          2: mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
***** Step 1.b:5.a:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(true(),s(x),y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(0(),y) -> 0()
            minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
        - Signature:
            {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1
            ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

**** Step 1.b:5.a:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
            le#(s(x),s(y)) -> c_7(le#(x,y))
            minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
        - Weak DPs:
            if_mod#(true(),s(x),s(y)) -> minus#(x,y)
            if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
            mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
            mod#(s(x),s(y)) -> le#(y,x)
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(true(),s(x),y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(0(),y) -> 0()
            minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
        - Signature:
            {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1
            ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} 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_7(le#(x,y))
          
        The strictly oriented rules are moved into the weak component.
***** Step 1.b:5.a:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^2))
    + Considered Problem:
        - Strict DPs:
            if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
            le#(s(x),s(y)) -> c_7(le#(x,y))
            minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
        - Weak DPs:
            if_mod#(true(),s(x),s(y)) -> minus#(x,y)
            if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
            mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
            mod#(s(x),s(y)) -> le#(y,x)
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(true(),s(x),y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(0(),y) -> 0()
            minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
        - Signature:
            {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1
            ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} 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_1) = {1},
          uargs(c_7) = {1},
          uargs(c_9) = {1,2}
        
        Following symbols are considered usable:
          {if_minus,minus,if_minus#,if_mod#,le#,minus#,mod#}
        TcT has computed the following interpretation:
                  p(0) = [1]                          
                         [0]                          
                         [0]                          
              p(false) = [0]                          
                         [0]                          
                         [0]                          
           p(if_minus) = [1 0 0]      [1]             
                         [1 1 0] x2 + [0]             
                         [1 0 0]      [1]             
             p(if_mod) = [0]                          
                         [0]                          
                         [0]                          
                 p(le) = [1 0 0]      [0]             
                         [0 0 1] x1 + [1]             
                         [0 0 1]      [1]             
              p(minus) = [0 0 1]      [1]             
                         [0 1 1] x1 + [0]             
                         [0 0 1]      [0]             
                p(mod) = [0]                          
                         [0]                          
                         [0]                          
                  p(s) = [0 0 1]      [0]             
                         [0 1 1] x1 + [0]             
                         [0 0 1]      [1]             
               p(true) = [0]                          
                         [0]                          
                         [0]                          
          p(if_minus#) = [0 1 0]      [0 0 0]      [0]
                         [0 0 0] x2 + [0 0 1] x3 + [0]
                         [1 0 1]      [1 0 0]      [0]
            p(if_mod#) = [1 1 0]      [1 0 0]      [1]
                         [1 0 1] x2 + [0 0 0] x3 + [1]
                         [0 0 0]      [0 0 1]      [0]
                p(le#) = [0 0 1]      [0 0 0]      [0]
                         [0 0 0] x1 + [0 0 1] x2 + [1]
                         [0 0 1]      [0 0 0]      [1]
             p(minus#) = [0 1 1]      [0 0 0]      [0]
                         [0 0 1] x1 + [0 0 0] x2 + [0]
                         [0 0 0]      [0 0 1]      [1]
               p(mod#) = [0 1 1]      [1 0 0]      [0]
                         [1 0 1] x1 + [0 0 0] x2 + [1]
                         [0 0 0]      [1 0 0]      [1]
                p(c_1) = [1 0 0]      [0]             
                         [0 0 0] x1 + [0]             
                         [0 0 0]      [0]             
                p(c_2) = [0]                          
                         [0]                          
                         [0]                          
                p(c_3) = [0]                          
                         [0]                          
                         [0]                          
                p(c_4) = [0]                          
                         [0]                          
                         [0]                          
                p(c_5) = [0]                          
                         [0]                          
                         [0]                          
                p(c_6) = [0]                          
                         [0]                          
                         [0]                          
                p(c_7) = [1 0 0]      [0]             
                         [0 0 0] x1 + [1]             
                         [0 0 1]      [1]             
                p(c_8) = [0]                          
                         [0]                          
                         [0]                          
                p(c_9) = [1 0 0]      [1 0 0]      [0]
                         [0 0 0] x1 + [1 0 0] x2 + [0]
                         [0 1 0]      [0 0 0]      [0]
               p(c_10) = [0]                          
                         [0]                          
                         [0]                          
               p(c_11) = [0]                          
                         [0]                          
                         [0]                          
               p(c_12) = [0]                          
                         [0]                          
                         [0]                          
        
        Following rules are strictly oriented:
        le#(s(x),s(y)) = [0 0 1]     [0 0 0]     [1]
                         [0 0 0] x + [0 0 1] y + [2]
                         [0 0 1]     [0 0 0]     [2]
                       > [0 0 1]     [0]            
                         [0 0 0] x + [1]            
                         [0 0 1]     [2]            
                       = c_7(le#(x,y))              
        
        
        Following rules are (at-least) weakly oriented:
        if_minus#(false(),s(x),y) =  [0 1 1]     [0 0 0]     [0]                  
                                     [0 0 0] x + [0 0 1] y + [0]                  
                                     [0 0 2]     [1 0 0]     [1]                  
                                  >= [0 1 1]     [0]                              
                                     [0 0 0] x + [0]                              
                                     [0 0 0]     [0]                              
                                  =  c_1(minus#(x,y))                             
        
        if_mod#(true(),s(x),s(y)) =  [0 1 2]     [0 0 1]     [1]                  
                                     [0 0 2] x + [0 0 0] y + [2]                  
                                     [0 0 0]     [0 0 1]     [1]                  
                                  >= [0 1 1]     [0 0 0]     [0]                  
                                     [0 0 1] x + [0 0 0] y + [0]                  
                                     [0 0 0]     [0 0 1]     [1]                  
                                  =  minus#(x,y)                                  
        
        if_mod#(true(),s(x),s(y)) =  [0 1 2]     [0 0 1]     [1]                  
                                     [0 0 2] x + [0 0 0] y + [2]                  
                                     [0 0 0]     [0 0 1]     [1]                  
                                  >= [0 1 2]     [0 0 1]     [0]                  
                                     [0 0 2] x + [0 0 0] y + [2]                  
                                     [0 0 0]     [0 0 1]     [1]                  
                                  =  mod#(minus(x,y),s(y))                        
        
                   minus#(s(x),y) =  [0 1 2]     [0 0 0]     [1]                  
                                     [0 0 1] x + [0 0 0] y + [1]                  
                                     [0 0 0]     [0 0 1]     [1]                  
                                  >= [0 1 2]     [0 0 0]     [1]                  
                                     [0 0 1] x + [0 0 0] y + [1]                  
                                     [0 0 0]     [0 0 1]     [0]                  
                                  =  c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
        
                  mod#(s(x),s(y)) =  [0 1 2]     [0 0 1]     [1]                  
                                     [0 0 2] x + [0 0 0] y + [2]                  
                                     [0 0 0]     [0 0 1]     [1]                  
                                  >= [0 1 2]     [0 0 1]     [1]                  
                                     [0 0 2] x + [0 0 0] y + [2]                  
                                     [0 0 0]     [0 0 1]     [1]                  
                                  =  if_mod#(le(y,x),s(x),s(y))                   
        
                  mod#(s(x),s(y)) =  [0 1 2]     [0 0 1]     [1]                  
                                     [0 0 2] x + [0 0 0] y + [2]                  
                                     [0 0 0]     [0 0 1]     [1]                  
                                  >= [0 0 0]     [0 0 1]     [0]                  
                                     [0 0 1] x + [0 0 0] y + [1]                  
                                     [0 0 0]     [0 0 1]     [1]                  
                                  =  le#(y,x)                                     
        
         if_minus(false(),s(x),y) =  [0 0 1]     [1]                              
                                     [0 1 2] x + [0]                              
                                     [0 0 1]     [1]                              
                                  >= [0 0 1]     [0]                              
                                     [0 1 2] x + [0]                              
                                     [0 0 1]     [1]                              
                                  =  s(minus(x,y))                                
        
          if_minus(true(),s(x),y) =  [0 0 1]     [1]                              
                                     [0 1 2] x + [0]                              
                                     [0 0 1]     [1]                              
                                  >= [1]                                          
                                     [0]                                          
                                     [0]                                          
                                  =  0()                                          
        
                     minus(0(),y) =  [1]                                          
                                     [0]                                          
                                     [0]                                          
                                  >= [1]                                          
                                     [0]                                          
                                     [0]                                          
                                  =  0()                                          
        
                    minus(s(x),y) =  [0 0 1]     [2]                              
                                     [0 1 2] x + [1]                              
                                     [0 0 1]     [1]                              
                                  >= [0 0 1]     [1]                              
                                     [0 1 2] x + [0]                              
                                     [0 0 1]     [1]                              
                                  =  if_minus(le(s(x),y),s(x),y)                  
        
***** Step 1.b:5.a:1.b:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
            minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
        - Weak DPs:
            if_mod#(true(),s(x),s(y)) -> minus#(x,y)
            if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
            le#(s(x),s(y)) -> c_7(le#(x,y))
            mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
            mod#(s(x),s(y)) -> le#(y,x)
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(true(),s(x),y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(0(),y) -> 0()
            minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
        - Signature:
            {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1
            ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} 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:5.a:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
            minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
        - Weak DPs:
            if_mod#(true(),s(x),s(y)) -> minus#(x,y)
            if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
            le#(s(x),s(y)) -> c_7(le#(x,y))
            mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
            mod#(s(x),s(y)) -> le#(y,x)
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(true(),s(x),y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(0(),y) -> 0()
            minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
        - Signature:
            {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1
            ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
             -->_1 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):2
          
          2:S:minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
             -->_2 le#(s(x),s(y)) -> c_7(le#(x,y)):5
             -->_1 if_minus#(false(),s(x),y) -> c_1(minus#(x,y)):1
          
          3:W:if_mod#(true(),s(x),s(y)) -> minus#(x,y)
             -->_1 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):2
          
          4:W:if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
             -->_1 mod#(s(x),s(y)) -> le#(y,x):7
             -->_1 mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y)):6
          
          5:W:le#(s(x),s(y)) -> c_7(le#(x,y))
             -->_1 le#(s(x),s(y)) -> c_7(le#(x,y)):5
          
          6:W:mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
             -->_1 if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y)):4
             -->_1 if_mod#(true(),s(x),s(y)) -> minus#(x,y):3
          
          7:W:mod#(s(x),s(y)) -> le#(y,x)
             -->_1 le#(s(x),s(y)) -> c_7(le#(x,y)):5
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          7: mod#(s(x),s(y)) -> le#(y,x)
          5: le#(s(x),s(y)) -> c_7(le#(x,y))
***** Step 1.b:5.a:1.b:1.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
            minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
        - Weak DPs:
            if_mod#(true(),s(x),s(y)) -> minus#(x,y)
            if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
            mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(true(),s(x),y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(0(),y) -> 0()
            minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
        - Signature:
            {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1
            ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
             -->_1 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):2
          
          2:S:minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
             -->_1 if_minus#(false(),s(x),y) -> c_1(minus#(x,y)):1
          
          3:W:if_mod#(true(),s(x),s(y)) -> minus#(x,y)
             -->_1 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):2
          
          4:W:if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
             -->_1 mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y)):6
          
          6:W:mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
             -->_1 if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y)):4
             -->_1 if_mod#(true(),s(x),s(y)) -> minus#(x,y):3
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y))
***** Step 1.b:5.a:1.b:1.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
            minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y))
        - Weak DPs:
            if_mod#(true(),s(x),s(y)) -> minus#(x,y)
            if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
            mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(true(),s(x),y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(0(),y) -> 0()
            minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
        - Signature:
            {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1
            ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} 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:
          1: if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
          
        The strictly oriented rules are moved into the weak component.
****** Step 1.b:5.a:1.b:1.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
            minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y))
        - Weak DPs:
            if_mod#(true(),s(x),s(y)) -> minus#(x,y)
            if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
            mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(true(),s(x),y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(0(),y) -> 0()
            minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
        - Signature:
            {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1
            ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} 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_1) = {1},
          uargs(c_9) = {1}
        
        Following symbols are considered usable:
          {if_minus,minus,if_minus#,if_mod#,le#,minus#,mod#}
        TcT has computed the following interpretation:
                  p(0) = [0]                  
              p(false) = [0]                  
           p(if_minus) = [1] x2 + [2]         
             p(if_mod) = [1] x1 + [1]         
                 p(le) = [0]                  
              p(minus) = [1] x1 + [2]         
                p(mod) = [1] x1 + [2] x2 + [1]
                  p(s) = [1] x1 + [2]         
               p(true) = [0]                  
          p(if_minus#) = [8] x2 + [7] x3 + [0]
            p(if_mod#) = [8] x2 + [7] x3 + [0]
                p(le#) = [4] x2 + [1]         
             p(minus#) = [8] x1 + [7] x2 + [0]
               p(mod#) = [8] x1 + [7] x2 + [0]
                p(c_1) = [1] x1 + [15]        
                p(c_2) = [2]                  
                p(c_3) = [1]                  
                p(c_4) = [1] x1 + [0]         
                p(c_5) = [2]                  
                p(c_6) = [8]                  
                p(c_7) = [0]                  
                p(c_8) = [4]                  
                p(c_9) = [1] x1 + [0]         
               p(c_10) = [0]                  
               p(c_11) = [8]                  
               p(c_12) = [2] x2 + [1]         
        
        Following rules are strictly oriented:
        if_minus#(false(),s(x),y) = [8] x + [7] y + [16]
                                  > [8] x + [7] y + [15]
                                  = c_1(minus#(x,y))    
        
        
        Following rules are (at-least) weakly oriented:
        if_mod#(true(),s(x),s(y)) =  [8] x + [7] y + [30]             
                                  >= [8] x + [7] y + [0]              
                                  =  minus#(x,y)                      
        
        if_mod#(true(),s(x),s(y)) =  [8] x + [7] y + [30]             
                                  >= [8] x + [7] y + [30]             
                                  =  mod#(minus(x,y),s(y))            
        
                   minus#(s(x),y) =  [8] x + [7] y + [16]             
                                  >= [8] x + [7] y + [16]             
                                  =  c_9(if_minus#(le(s(x),y),s(x),y))
        
                  mod#(s(x),s(y)) =  [8] x + [7] y + [30]             
                                  >= [8] x + [7] y + [30]             
                                  =  if_mod#(le(y,x),s(x),s(y))       
        
         if_minus(false(),s(x),y) =  [1] x + [4]                      
                                  >= [1] x + [4]                      
                                  =  s(minus(x,y))                    
        
          if_minus(true(),s(x),y) =  [1] x + [4]                      
                                  >= [0]                              
                                  =  0()                              
        
                     minus(0(),y) =  [2]                              
                                  >= [0]                              
                                  =  0()                              
        
                    minus(s(x),y) =  [1] x + [4]                      
                                  >= [1] x + [4]                      
                                  =  if_minus(le(s(x),y),s(x),y)      
        
****** Step 1.b:5.a:1.b:1.b:3.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y))
        - Weak DPs:
            if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
            if_mod#(true(),s(x),s(y)) -> minus#(x,y)
            if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
            mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(true(),s(x),y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(0(),y) -> 0()
            minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
        - Signature:
            {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1
            ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} 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:5.a:1.b:1.b:3.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y))
        - Weak DPs:
            if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
            if_mod#(true(),s(x),s(y)) -> minus#(x,y)
            if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
            mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(true(),s(x),y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(0(),y) -> 0()
            minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
        - Signature:
            {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1
            ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} 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:
          1: minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y))
          
        Consider the set of all dependency pairs
          1: minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y))
          2: if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
          3: if_mod#(true(),s(x),s(y)) -> minus#(x,y)
          4: if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
          5: mod#(s(x),s(y)) -> 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
          {1}
        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:5.a:1.b:1.b:3.b:1.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y))
        - Weak DPs:
            if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
            if_mod#(true(),s(x),s(y)) -> minus#(x,y)
            if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
            mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(true(),s(x),y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(0(),y) -> 0()
            minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
        - Signature:
            {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1
            ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} 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_1) = {1},
          uargs(c_9) = {1}
        
        Following symbols are considered usable:
          {if_minus,minus,if_minus#,if_mod#,le#,minus#,mod#}
        TcT has computed the following interpretation:
                  p(0) = [2]                  
              p(false) = [0]                  
           p(if_minus) = [1] x2 + [0]         
             p(if_mod) = [1] x1 + [2] x3 + [2]
                 p(le) = [1] x1 + [2] x2 + [0]
              p(minus) = [1] x1 + [0]         
                p(mod) = [1] x2 + [0]         
                  p(s) = [1] x1 + [4]         
               p(true) = [0]                  
          p(if_minus#) = [4] x2 + [1]         
            p(if_mod#) = [6] x2 + [5]         
                p(le#) = [4] x1 + [1]         
             p(minus#) = [4] x1 + [15]        
               p(mod#) = [6] x1 + [5]         
                p(c_1) = [1] x1 + [0]         
                p(c_2) = [8]                  
                p(c_3) = [0]                  
                p(c_4) = [4] x2 + [1]         
                p(c_5) = [2]                  
                p(c_6) = [0]                  
                p(c_7) = [1]                  
                p(c_8) = [1]                  
                p(c_9) = [1] x1 + [10]        
               p(c_10) = [1]                  
               p(c_11) = [1]                  
               p(c_12) = [1] x1 + [8]         
        
        Following rules are strictly oriented:
        minus#(s(x),y) = [4] x + [31]                     
                       > [4] x + [27]                     
                       = c_9(if_minus#(le(s(x),y),s(x),y))
        
        
        Following rules are (at-least) weakly oriented:
        if_minus#(false(),s(x),y) =  [4] x + [17]               
                                  >= [4] x + [15]               
                                  =  c_1(minus#(x,y))           
        
        if_mod#(true(),s(x),s(y)) =  [6] x + [29]               
                                  >= [4] x + [15]               
                                  =  minus#(x,y)                
        
        if_mod#(true(),s(x),s(y)) =  [6] x + [29]               
                                  >= [6] x + [5]                
                                  =  mod#(minus(x,y),s(y))      
        
                  mod#(s(x),s(y)) =  [6] x + [29]               
                                  >= [6] x + [29]               
                                  =  if_mod#(le(y,x),s(x),s(y)) 
        
         if_minus(false(),s(x),y) =  [1] x + [4]                
                                  >= [1] x + [4]                
                                  =  s(minus(x,y))              
        
          if_minus(true(),s(x),y) =  [1] x + [4]                
                                  >= [2]                        
                                  =  0()                        
        
                     minus(0(),y) =  [2]                        
                                  >= [2]                        
                                  =  0()                        
        
                    minus(s(x),y) =  [1] x + [4]                
                                  >= [1] x + [4]                
                                  =  if_minus(le(s(x),y),s(x),y)
        
******* Step 1.b:5.a:1.b:1.b:3.b:1.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
            if_mod#(true(),s(x),s(y)) -> minus#(x,y)
            if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
            minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y))
            mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(true(),s(x),y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(0(),y) -> 0()
            minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
        - Signature:
            {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1
            ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} 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:5.a:1.b:1.b:3.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
            if_mod#(true(),s(x),s(y)) -> minus#(x,y)
            if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
            minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y))
            mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(true(),s(x),y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(0(),y) -> 0()
            minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
        - Signature:
            {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1
            ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
             -->_1 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y)):4
          
          2:W:if_mod#(true(),s(x),s(y)) -> minus#(x,y)
             -->_1 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y)):4
          
          3:W:if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
             -->_1 mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y)):5
          
          4:W:minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y))
             -->_1 if_minus#(false(),s(x),y) -> c_1(minus#(x,y)):1
          
          5:W:mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
             -->_1 if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y)):3
             -->_1 if_mod#(true(),s(x),s(y)) -> minus#(x,y):2
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          3: if_mod#(true(),s(x),s(y)) -> mod#(minus(x,y),s(y))
          5: mod#(s(x),s(y)) -> if_mod#(le(y,x),s(x),s(y))
          2: if_mod#(true(),s(x),s(y)) -> minus#(x,y)
          1: if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
          4: minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y))
******* Step 1.b:5.a:1.b:1.b:3.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(true(),s(x),y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(0(),y) -> 0()
            minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
        - Signature:
            {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1
            ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/0,c_11/0,c_12/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

*** Step 1.b:5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
            mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
        - Weak DPs:
            if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
            le#(s(x),s(y)) -> c_7(le#(x,y))
            minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(true(),s(x),y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(0(),y) -> 0()
            minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
        - Signature:
            {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1
            ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
             -->_2 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):5
             -->_1 mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x)):2
          
          2:S:mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
             -->_2 le#(s(x),s(y)) -> c_7(le#(x,y)):4
             -->_1 if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y)):1
          
          3:W:if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
             -->_1 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):5
          
          4:W:le#(s(x),s(y)) -> c_7(le#(x,y))
             -->_1 le#(s(x),s(y)) -> c_7(le#(x,y)):4
          
          5:W:minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
             -->_2 le#(s(x),s(y)) -> c_7(le#(x,y)):4
             -->_1 if_minus#(false(),s(x),y) -> c_1(minus#(x,y)):3
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          5: minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y))
          3: if_minus#(false(),s(x),y) -> c_1(minus#(x,y))
          4: le#(s(x),s(y)) -> c_7(le#(x,y))
*** Step 1.b:5.b:2: SimplifyRHS WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
            mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(true(),s(x),y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(0(),y) -> 0()
            minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
        - Signature:
            {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1
            ,true/0,c_1/1,c_2/0,c_3/0,c_4/2,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/2}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)),minus#(x,y))
             -->_1 mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x)):2
          
          2:S:mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)),le#(y,x))
             -->_1 if_mod#(true(),s(x),s(y)) -> c_4(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_4(mod#(minus(x,y),s(y)))
          mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)))
*** Step 1.b:5.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)))
            mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(true(),s(x),y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(0(),y) -> 0()
            minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
        - Signature:
            {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1
            ,true/0,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} 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 = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}}
    + Details:
        We first use the processor NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly:
          1: if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)))
          
        Consider the set of all dependency pairs
          1: if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)))
          2: mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)))
        Processor NaturalMI {miDimension = 3, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1))
        SPACE(?,?)on application of the dependency pairs
          {1}
        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:5.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)))
            mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(true(),s(x),y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(0(),y) -> 0()
            minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
        - Signature:
            {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1
            ,true/0,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        NaturalMI {miDimension = 3, 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 (containing no more than 1 non-zero interpretation-entries in the diagonal of the component-wise maxima):
        The following argument positions are considered usable:
          uargs(c_4) = {1},
          uargs(c_12) = {1}
        
        Following symbols are considered usable:
          {if_minus,minus,if_minus#,if_mod#,le#,minus#,mod#}
        TcT has computed the following interpretation:
                  p(0) = [0]                          
                         [0]                          
                         [0]                          
              p(false) = [0]                          
                         [0]                          
                         [0]                          
           p(if_minus) = [0 1 0]      [1]             
                         [0 1 0] x2 + [0]             
                         [0 0 0]      [1]             
             p(if_mod) = [0]                          
                         [0]                          
                         [0]                          
                 p(le) = [0]                          
                         [0]                          
                         [0]                          
              p(minus) = [0 1 1]      [0]             
                         [0 1 0] x1 + [0]             
                         [0 0 1]      [0]             
                p(mod) = [0]                          
                         [0]                          
                         [0]                          
                  p(s) = [0 1 0]      [1]             
                         [0 1 1] x1 + [0]             
                         [0 0 0]      [1]             
               p(true) = [0]                          
                         [0]                          
                         [0]                          
          p(if_minus#) = [0]                          
                         [0]                          
                         [0]                          
            p(if_mod#) = [1 1 1]      [0 0 0]      [0]
                         [1 0 0] x2 + [0 0 0] x3 + [0]
                         [0 0 1]      [1 0 0]      [0]
                p(le#) = [0]                          
                         [0]                          
                         [0]                          
             p(minus#) = [0]                          
                         [0]                          
                         [0]                          
               p(mod#) = [1 1 0]      [0 0 0]      [1]
                         [1 1 0] x1 + [0 0 1] x2 + [0]
                         [0 0 0]      [0 1 0]      [1]
                p(c_1) = [0]                          
                         [0]                          
                         [0]                          
                p(c_2) = [0]                          
                         [0]                          
                         [0]                          
                p(c_3) = [0]                          
                         [0]                          
                         [0]                          
                p(c_4) = [1 0 0]      [0]             
                         [0 0 0] x1 + [1]             
                         [0 0 0]      [0]             
                p(c_5) = [0]                          
                         [0]                          
                         [0]                          
                p(c_6) = [0]                          
                         [0]                          
                         [0]                          
                p(c_7) = [0]                          
                         [0]                          
                         [0]                          
                p(c_8) = [0]                          
                         [0]                          
                         [0]                          
                p(c_9) = [0]                          
                         [0]                          
                         [0]                          
               p(c_10) = [0]                          
                         [0]                          
                         [0]                          
               p(c_11) = [0]                          
                         [0]                          
                         [0]                          
               p(c_12) = [1 0 0]      [0]             
                         [1 0 0] x1 + [0]             
                         [0 0 0]      [1]             
        
        Following rules are strictly oriented:
        if_mod#(true(),s(x),s(y)) = [0 2 1]     [0 0 0]     [2]
                                    [0 1 0] x + [0 0 0] y + [1]
                                    [0 0 0]     [0 1 0]     [2]
                                  > [0 2 1]     [1]            
                                    [0 0 0] x + [1]            
                                    [0 0 0]     [0]            
                                  = c_4(mod#(minus(x,y),s(y))) 
        
        
        Following rules are (at-least) weakly oriented:
                 mod#(s(x),s(y)) =  [0 2 1]     [0 0 0]     [2]     
                                    [0 2 1] x + [0 0 0] y + [2]     
                                    [0 0 0]     [0 1 1]     [1]     
                                 >= [0 2 1]     [2]                 
                                    [0 2 1] x + [2]                 
                                    [0 0 0]     [1]                 
                                 =  c_12(if_mod#(le(y,x),s(x),s(y)))
        
        if_minus(false(),s(x),y) =  [0 1 1]     [1]                 
                                    [0 1 1] x + [0]                 
                                    [0 0 0]     [1]                 
                                 >= [0 1 0]     [1]                 
                                    [0 1 1] x + [0]                 
                                    [0 0 0]     [1]                 
                                 =  s(minus(x,y))                   
        
         if_minus(true(),s(x),y) =  [0 1 1]     [1]                 
                                    [0 1 1] x + [0]                 
                                    [0 0 0]     [1]                 
                                 >= [0]                             
                                    [0]                             
                                    [0]                             
                                 =  0()                             
        
                    minus(0(),y) =  [0]                             
                                    [0]                             
                                    [0]                             
                                 >= [0]                             
                                    [0]                             
                                    [0]                             
                                 =  0()                             
        
                   minus(s(x),y) =  [0 1 1]     [1]                 
                                    [0 1 1] x + [0]                 
                                    [0 0 0]     [1]                 
                                 >= [0 1 1]     [1]                 
                                    [0 1 1] x + [0]                 
                                    [0 0 0]     [1]                 
                                 =  if_minus(le(s(x),y),s(x),y)     
        
**** Step 1.b:5.b:3.a:2: Assumption WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)))
        - Weak DPs:
            if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(true(),s(x),y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(0(),y) -> 0()
            minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
        - Signature:
            {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1
            ,true/0,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} 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:5.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)))
            mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)))
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(true(),s(x),y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(0(),y) -> 0()
            minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
        - Signature:
            {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1
            ,true/0,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} and constructors {0
            ,false,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:W:if_mod#(true(),s(x),s(y)) -> c_4(mod#(minus(x,y),s(y)))
             -->_1 mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y))):2
          
          2:W:mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)))
             -->_1 if_mod#(true(),s(x),s(y)) -> c_4(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_4(mod#(minus(x,y),s(y)))
          2: mod#(s(x),s(y)) -> c_12(if_mod#(le(y,x),s(x),s(y)))
**** Step 1.b:5.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak TRS:
            if_minus(false(),s(x),y) -> s(minus(x,y))
            if_minus(true(),s(x),y) -> 0()
            le(0(),y) -> true()
            le(s(x),0()) -> false()
            le(s(x),s(y)) -> le(x,y)
            minus(0(),y) -> 0()
            minus(s(x),y) -> if_minus(le(s(x),y),s(x),y)
        - Signature:
            {if_minus/3,if_mod/3,le/2,minus/2,mod/2,if_minus#/3,if_mod#/3,le#/2,minus#/2,mod#/2} / {0/0,false/0,s/1
            ,true/0,c_1/1,c_2/0,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/0,c_12/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {if_minus#,if_mod#,le#,minus#,mod#} 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^3))