/bin/sh: line 1: 24562 Quit                    (core dumped) z3 -T:4.6850000000000005 -smt2 /tmp/SMTP24411-27 > /tmp/SMTS24411-28
/bin/sh: line 1: 24573 Quit                    (core dumped) z3 -T:7.6 -smt2 /tmp/SMTP24411-76 > /tmp/SMTS24411-78
/bin/sh: line 1: 24567 Quit                    (core dumped) z3 -T:2.811 -smt2 /tmp/SMTP24411-37 > /tmp/SMTS24411-38
WORST_CASE(Omega(n^1),O(n^1))
* Step 1: Sum.  WORST_CASE(Omega(n^1),O(n^1))
    + Considered Problem:
        - Strict TRS:
            div(0(),s(Y)) -> 0()
            div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
            geq(X,0()) -> true()
            geq(0(),s(Y)) -> false()
            geq(s(X),s(Y)) -> geq(X,Y)
            if(false(),X,Y) -> Y
            if(true(),X,Y) -> X
            minus(0(),Y) -> 0()
            minus(s(X),s(Y)) -> minus(X,Y)
        - Signature:
            {div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div,geq,if,minus} and constructors {0,false,s,true}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
** Step 1.a:1: Sum.  WORST_CASE(Omega(n^1),?)
    + Considered Problem:
        - Strict TRS:
            div(0(),s(Y)) -> 0()
            div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
            geq(X,0()) -> true()
            geq(0(),s(Y)) -> false()
            geq(s(X),s(Y)) -> geq(X,Y)
            if(false(),X,Y) -> Y
            if(true(),X,Y) -> X
            minus(0(),Y) -> 0()
            minus(s(X),s(Y)) -> minus(X,Y)
        - Signature:
            {div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div,geq,if,minus} and constructors {0,false,s,true}
    + Applied Processor:
        Sum {left = someStrategy, right = someStrategy}
    + Details:
        ()
*** Step 1.a:1.a:1: Ara.  MAYBE
    + Considered Problem:
        - Strict TRS:
            div(0(),s(Y)) -> 0()
            div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
            geq(X,0()) -> true()
            geq(0(),s(Y)) -> false()
            geq(s(X),s(Y)) -> geq(X,Y)
            if(false(),X,Y) -> Y
            if(true(),X,Y) -> X
            minus(0(),Y) -> 0()
            minus(s(X),s(Y)) -> minus(X,Y)
        - Signature:
            {div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div,geq,if,minus} and constructors {0,false,s,true}
    + Applied Processor:
        Ara {minDegree = 1, maxDegree = 3, araTimeout = 15, araRuleShifting = Just 1, isBestCase = True, mkCompletelyDefined = False, verboseOutput = False}
    + Details:
        Signatures used:
        ----------------
          F (TrsFun "0") :: [] -(0)-> "A"(0)
          F (TrsFun "0") :: [] -(0)-> "A"(1)
          F (TrsFun "div") :: ["A"(0) x "A"(0)] -(1)-> "A"(0)
          F (TrsFun "false") :: [] -(0)-> "A"(0)
          F (TrsFun "geq") :: ["A"(0) x "A"(0)] -(1)-> "A"(0)
          F (TrsFun "if") :: ["A"(0) x "A"(0) x "A"(0)] -(1)-> "A"(0)
          F (TrsFun "main") :: ["A"(1) x "A"(0)] -(1)-> "A"(0)
          F (TrsFun "minus") :: ["A"(1) x "A"(0)] -(1)-> "A"(0)
          F (TrsFun "s") :: ["A"(0)] -(0)-> "A"(0)
          F (TrsFun "s") :: ["A"(1)] -(1)-> "A"(1)
          F (TrsFun "true") :: [] -(0)-> "A"(0)
        
        
        Cost-free Signatures used:
        --------------------------
        
        
        
        Base Constructor Signatures used:
        ---------------------------------
        
        
        
        Following Still Strict Rules were Typed as:
        -------------------------------------------
        1. Strict:
          div(0(),s(Y)) -> 0()
          div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
          geq(X,0()) -> true()
          geq(0(),s(Y)) -> false()
          geq(s(X),s(Y)) -> geq(X,Y)
          if(false(),X,Y) -> Y
          if(true(),X,Y) -> X
          minus(0(),Y) -> 0()
          minus(s(X),s(Y)) -> minus(X,Y)
          main(x1,x2) -> minus(x1,x2)
        2. Weak:
          
        

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

** Step 1.b:1: DependencyPairs.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict TRS:
            div(0(),s(Y)) -> 0()
            div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
            geq(X,0()) -> true()
            geq(0(),s(Y)) -> false()
            geq(s(X),s(Y)) -> geq(X,Y)
            if(false(),X,Y) -> Y
            if(true(),X,Y) -> X
            minus(0(),Y) -> 0()
            minus(s(X),s(Y)) -> minus(X,Y)
        - Signature:
            {div/2,geq/2,if/3,minus/2} / {0/0,false/0,s/1,true/0}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div,geq,if,minus} and constructors {0,false,s,true}
    + Applied Processor:
        DependencyPairs {dpKind_ = DT}
    + Details:
        We add the following dependency tuples:
        
        Strict DPs
          div#(0(),s(Y)) -> c_1()
          div#(s(X),s(Y)) -> c_2(if#(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
                                ,geq#(X,Y)
                                ,div#(minus(X,Y),s(Y))
                                ,minus#(X,Y))
          geq#(X,0()) -> c_3()
          geq#(0(),s(Y)) -> c_4()
          geq#(s(X),s(Y)) -> c_5(geq#(X,Y))
          if#(false(),X,Y) -> c_6()
          if#(true(),X,Y) -> c_7()
          minus#(0(),Y) -> c_8()
          minus#(s(X),s(Y)) -> c_9(minus#(X,Y))
        Weak DPs
          
        
        and mark the set of starting terms.
** Step 1.b:2: PredecessorEstimation.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            div#(0(),s(Y)) -> c_1()
            div#(s(X),s(Y)) -> c_2(if#(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
                                  ,geq#(X,Y)
                                  ,div#(minus(X,Y),s(Y))
                                  ,minus#(X,Y))
            geq#(X,0()) -> c_3()
            geq#(0(),s(Y)) -> c_4()
            geq#(s(X),s(Y)) -> c_5(geq#(X,Y))
            if#(false(),X,Y) -> c_6()
            if#(true(),X,Y) -> c_7()
            minus#(0(),Y) -> c_8()
            minus#(s(X),s(Y)) -> c_9(minus#(X,Y))
        - Weak TRS:
            div(0(),s(Y)) -> 0()
            div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
            geq(X,0()) -> true()
            geq(0(),s(Y)) -> false()
            geq(s(X),s(Y)) -> geq(X,Y)
            if(false(),X,Y) -> Y
            if(true(),X,Y) -> X
            minus(0(),Y) -> 0()
            minus(s(X),s(Y)) -> minus(X,Y)
        - Signature:
            {div/2,geq/2,if/3,minus/2,div#/2,geq#/2,if#/3,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/4,c_3/0,c_4/0
            ,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,geq#,if#,minus#} and constructors {0,false,s,true}
    + Applied Processor:
        PredecessorEstimation {onSelection = all simple predecessor estimation selector}
    + Details:
        We estimate the number of application of
          {1,3,4,6,7,8}
        by application of
          Pre({1,3,4,6,7,8}) = {2,5,9}.
        Here rules are labelled as follows:
          1: div#(0(),s(Y)) -> c_1()
          2: div#(s(X),s(Y)) -> c_2(if#(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
                                   ,geq#(X,Y)
                                   ,div#(minus(X,Y),s(Y))
                                   ,minus#(X,Y))
          3: geq#(X,0()) -> c_3()
          4: geq#(0(),s(Y)) -> c_4()
          5: geq#(s(X),s(Y)) -> c_5(geq#(X,Y))
          6: if#(false(),X,Y) -> c_6()
          7: if#(true(),X,Y) -> c_7()
          8: minus#(0(),Y) -> c_8()
          9: minus#(s(X),s(Y)) -> c_9(minus#(X,Y))
** Step 1.b:3: RemoveWeakSuffixes.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            div#(s(X),s(Y)) -> c_2(if#(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
                                  ,geq#(X,Y)
                                  ,div#(minus(X,Y),s(Y))
                                  ,minus#(X,Y))
            geq#(s(X),s(Y)) -> c_5(geq#(X,Y))
            minus#(s(X),s(Y)) -> c_9(minus#(X,Y))
        - Weak DPs:
            div#(0(),s(Y)) -> c_1()
            geq#(X,0()) -> c_3()
            geq#(0(),s(Y)) -> c_4()
            if#(false(),X,Y) -> c_6()
            if#(true(),X,Y) -> c_7()
            minus#(0(),Y) -> c_8()
        - Weak TRS:
            div(0(),s(Y)) -> 0()
            div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
            geq(X,0()) -> true()
            geq(0(),s(Y)) -> false()
            geq(s(X),s(Y)) -> geq(X,Y)
            if(false(),X,Y) -> Y
            if(true(),X,Y) -> X
            minus(0(),Y) -> 0()
            minus(s(X),s(Y)) -> minus(X,Y)
        - Signature:
            {div/2,geq/2,if/3,minus/2,div#/2,geq#/2,if#/3,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/4,c_3/0,c_4/0
            ,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,geq#,if#,minus#} and constructors {0,false,s,true}
    + Applied Processor:
        RemoveWeakSuffixes
    + Details:
        Consider the dependency graph
          1:S:div#(s(X),s(Y)) -> c_2(if#(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
                                    ,geq#(X,Y)
                                    ,div#(minus(X,Y),s(Y))
                                    ,minus#(X,Y))
             -->_4 minus#(s(X),s(Y)) -> c_9(minus#(X,Y)):3
             -->_2 geq#(s(X),s(Y)) -> c_5(geq#(X,Y)):2
             -->_4 minus#(0(),Y) -> c_8():9
             -->_1 if#(true(),X,Y) -> c_7():8
             -->_1 if#(false(),X,Y) -> c_6():7
             -->_2 geq#(0(),s(Y)) -> c_4():6
             -->_2 geq#(X,0()) -> c_3():5
             -->_3 div#(0(),s(Y)) -> c_1():4
             -->_3 div#(s(X),s(Y)) -> c_2(if#(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
                                         ,geq#(X,Y)
                                         ,div#(minus(X,Y),s(Y))
                                         ,minus#(X,Y)):1
          
          2:S:geq#(s(X),s(Y)) -> c_5(geq#(X,Y))
             -->_1 geq#(0(),s(Y)) -> c_4():6
             -->_1 geq#(X,0()) -> c_3():5
             -->_1 geq#(s(X),s(Y)) -> c_5(geq#(X,Y)):2
          
          3:S:minus#(s(X),s(Y)) -> c_9(minus#(X,Y))
             -->_1 minus#(0(),Y) -> c_8():9
             -->_1 minus#(s(X),s(Y)) -> c_9(minus#(X,Y)):3
          
          4:W:div#(0(),s(Y)) -> c_1()
             
          
          5:W:geq#(X,0()) -> c_3()
             
          
          6:W:geq#(0(),s(Y)) -> c_4()
             
          
          7:W:if#(false(),X,Y) -> c_6()
             
          
          8:W:if#(true(),X,Y) -> c_7()
             
          
          9:W:minus#(0(),Y) -> c_8()
             
          
        The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed.
          4: div#(0(),s(Y)) -> c_1()
          7: if#(false(),X,Y) -> c_6()
          8: if#(true(),X,Y) -> c_7()
          5: geq#(X,0()) -> c_3()
          6: geq#(0(),s(Y)) -> c_4()
          9: minus#(0(),Y) -> c_8()
** Step 1.b:4: SimplifyRHS.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            div#(s(X),s(Y)) -> c_2(if#(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
                                  ,geq#(X,Y)
                                  ,div#(minus(X,Y),s(Y))
                                  ,minus#(X,Y))
            geq#(s(X),s(Y)) -> c_5(geq#(X,Y))
            minus#(s(X),s(Y)) -> c_9(minus#(X,Y))
        - Weak TRS:
            div(0(),s(Y)) -> 0()
            div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
            geq(X,0()) -> true()
            geq(0(),s(Y)) -> false()
            geq(s(X),s(Y)) -> geq(X,Y)
            if(false(),X,Y) -> Y
            if(true(),X,Y) -> X
            minus(0(),Y) -> 0()
            minus(s(X),s(Y)) -> minus(X,Y)
        - Signature:
            {div/2,geq/2,if/3,minus/2,div#/2,geq#/2,if#/3,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/4,c_3/0,c_4/0
            ,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,geq#,if#,minus#} and constructors {0,false,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:div#(s(X),s(Y)) -> c_2(if#(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
                                    ,geq#(X,Y)
                                    ,div#(minus(X,Y),s(Y))
                                    ,minus#(X,Y))
             -->_4 minus#(s(X),s(Y)) -> c_9(minus#(X,Y)):3
             -->_2 geq#(s(X),s(Y)) -> c_5(geq#(X,Y)):2
             -->_3 div#(s(X),s(Y)) -> c_2(if#(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
                                         ,geq#(X,Y)
                                         ,div#(minus(X,Y),s(Y))
                                         ,minus#(X,Y)):1
          
          2:S:geq#(s(X),s(Y)) -> c_5(geq#(X,Y))
             -->_1 geq#(s(X),s(Y)) -> c_5(geq#(X,Y)):2
          
          3:S:minus#(s(X),s(Y)) -> c_9(minus#(X,Y))
             -->_1 minus#(s(X),s(Y)) -> c_9(minus#(X,Y)):3
          
        Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified:
          div#(s(X),s(Y)) -> c_2(geq#(X,Y),div#(minus(X,Y),s(Y)),minus#(X,Y))
** Step 1.b:5: UsableRules.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            div#(s(X),s(Y)) -> c_2(geq#(X,Y),div#(minus(X,Y),s(Y)),minus#(X,Y))
            geq#(s(X),s(Y)) -> c_5(geq#(X,Y))
            minus#(s(X),s(Y)) -> c_9(minus#(X,Y))
        - Weak TRS:
            div(0(),s(Y)) -> 0()
            div(s(X),s(Y)) -> if(geq(X,Y),s(div(minus(X,Y),s(Y))),0())
            geq(X,0()) -> true()
            geq(0(),s(Y)) -> false()
            geq(s(X),s(Y)) -> geq(X,Y)
            if(false(),X,Y) -> Y
            if(true(),X,Y) -> X
            minus(0(),Y) -> 0()
            minus(s(X),s(Y)) -> minus(X,Y)
        - Signature:
            {div/2,geq/2,if/3,minus/2,div#/2,geq#/2,if#/3,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/3,c_3/0,c_4/0
            ,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,geq#,if#,minus#} and constructors {0,false,s,true}
    + Applied Processor:
        UsableRules
    + Details:
        We replace rewrite rules by usable rules:
          minus(0(),Y) -> 0()
          minus(s(X),s(Y)) -> minus(X,Y)
          div#(s(X),s(Y)) -> c_2(geq#(X,Y),div#(minus(X,Y),s(Y)),minus#(X,Y))
          geq#(s(X),s(Y)) -> c_5(geq#(X,Y))
          minus#(s(X),s(Y)) -> c_9(minus#(X,Y))
** Step 1.b:6: DecomposeDG.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            div#(s(X),s(Y)) -> c_2(geq#(X,Y),div#(minus(X,Y),s(Y)),minus#(X,Y))
            geq#(s(X),s(Y)) -> c_5(geq#(X,Y))
            minus#(s(X),s(Y)) -> c_9(minus#(X,Y))
        - Weak TRS:
            minus(0(),Y) -> 0()
            minus(s(X),s(Y)) -> minus(X,Y)
        - Signature:
            {div/2,geq/2,if/3,minus/2,div#/2,geq#/2,if#/3,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/3,c_3/0,c_4/0
            ,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,geq#,if#,minus#} and constructors {0,false,s,true}
    + Applied Processor:
        DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing}
    + Details:
        We decompose the input problem according to the dependency graph into the upper component
          div#(s(X),s(Y)) -> c_2(geq#(X,Y),div#(minus(X,Y),s(Y)),minus#(X,Y))
        and a lower component
          geq#(s(X),s(Y)) -> c_5(geq#(X,Y))
          minus#(s(X),s(Y)) -> c_9(minus#(X,Y))
        Further, following extension rules are added to the lower component.
          div#(s(X),s(Y)) -> div#(minus(X,Y),s(Y))
          div#(s(X),s(Y)) -> geq#(X,Y)
          div#(s(X),s(Y)) -> minus#(X,Y)
*** Step 1.b:6.a:1: SimplifyRHS.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            div#(s(X),s(Y)) -> c_2(geq#(X,Y),div#(minus(X,Y),s(Y)),minus#(X,Y))
        - Weak TRS:
            minus(0(),Y) -> 0()
            minus(s(X),s(Y)) -> minus(X,Y)
        - Signature:
            {div/2,geq/2,if/3,minus/2,div#/2,geq#/2,if#/3,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/3,c_3/0,c_4/0
            ,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,geq#,if#,minus#} and constructors {0,false,s,true}
    + Applied Processor:
        SimplifyRHS
    + Details:
        Consider the dependency graph
          1:S:div#(s(X),s(Y)) -> c_2(geq#(X,Y),div#(minus(X,Y),s(Y)),minus#(X,Y))
             -->_2 div#(s(X),s(Y)) -> c_2(geq#(X,Y),div#(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:
          div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y)))
*** Step 1.b:6.a:2: NaturalMI.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Strict DPs:
            div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y)))
        - Weak TRS:
            minus(0(),Y) -> 0()
            minus(s(X),s(Y)) -> minus(X,Y)
        - Signature:
            {div/2,geq/2,if/3,minus/2,div#/2,geq#/2,if#/3,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/0,c_4/0
            ,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,geq#,if#,minus#} and constructors {0,false,s,true}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_2) = {1}
        
        Following symbols are considered usable:
          {minus,div#,geq#,if#,minus#}
        TcT has computed the following interpretation:
               p(0) = [0]         
             p(div) = [0]         
           p(false) = [8]         
             p(geq) = [1]         
              p(if) = [1]         
           p(minus) = [0]         
               p(s) = [5]         
            p(true) = [1]         
            p(div#) = [1] x1 + [0]
            p(geq#) = [0]         
             p(if#) = [0]         
          p(minus#) = [0]         
             p(c_1) = [0]         
             p(c_2) = [2] x1 + [0]
             p(c_3) = [0]         
             p(c_4) = [0]         
             p(c_5) = [0]         
             p(c_6) = [0]         
             p(c_7) = [0]         
             p(c_8) = [0]         
             p(c_9) = [0]         
        
        Following rules are strictly oriented:
        div#(s(X),s(Y)) = [5]                       
                        > [0]                       
                        = c_2(div#(minus(X,Y),s(Y)))
        
        
        Following rules are (at-least) weakly oriented:
            minus(0(),Y) =  [0]       
                         >= [0]       
                         =  0()       
        
        minus(s(X),s(Y)) =  [0]       
                         >= [0]       
                         =  minus(X,Y)
        
*** Step 1.b:6.a:3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            div#(s(X),s(Y)) -> c_2(div#(minus(X,Y),s(Y)))
        - Weak TRS:
            minus(0(),Y) -> 0()
            minus(s(X),s(Y)) -> minus(X,Y)
        - Signature:
            {div/2,geq/2,if/3,minus/2,div#/2,geq#/2,if#/3,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/1,c_3/0,c_4/0
            ,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,geq#,if#,minus#} and constructors {0,false,s,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

*** Step 1.b:6.b:1: NaturalMI.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            geq#(s(X),s(Y)) -> c_5(geq#(X,Y))
            minus#(s(X),s(Y)) -> c_9(minus#(X,Y))
        - Weak DPs:
            div#(s(X),s(Y)) -> div#(minus(X,Y),s(Y))
            div#(s(X),s(Y)) -> geq#(X,Y)
            div#(s(X),s(Y)) -> minus#(X,Y)
        - Weak TRS:
            minus(0(),Y) -> 0()
            minus(s(X),s(Y)) -> minus(X,Y)
        - Signature:
            {div/2,geq/2,if/3,minus/2,div#/2,geq#/2,if#/3,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/3,c_3/0,c_4/0
            ,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,geq#,if#,minus#} and constructors {0,false,s,true}
    + Applied Processor:
        NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules}
    + Details:
        We apply a matrix interpretation of kind constructor based matrix interpretation:
        The following argument positions are considered usable:
          uargs(c_5) = {1},
          uargs(c_9) = {1}
        
        Following symbols are considered usable:
          {minus,div#,geq#,if#,minus#}
        TcT has computed the following interpretation:
               p(0) = [0]                           
             p(div) = [1] x1 + [2]                  
           p(false) = [1]                           
             p(geq) = [1] x1 + [2]                  
              p(if) = [0]                           
           p(minus) = [1] x1 + [0]                  
               p(s) = [1] x1 + [2]                  
            p(true) = [0]                           
            p(div#) = [2] x1 + [5]                  
            p(geq#) = [1] x1 + [0]                  
             p(if#) = [2] x2 + [1] x3 + [8]         
          p(minus#) = [0]                           
             p(c_1) = [0]                           
             p(c_2) = [1] x1 + [1] x2 + [1] x3 + [2]
             p(c_3) = [1]                           
             p(c_4) = [1]                           
             p(c_5) = [1] x1 + [0]                  
             p(c_6) = [8]                           
             p(c_7) = [1]                           
             p(c_8) = [1]                           
             p(c_9) = [8] x1 + [0]                  
        
        Following rules are strictly oriented:
        geq#(s(X),s(Y)) = [1] X + [2]   
                        > [1] X + [0]   
                        = c_5(geq#(X,Y))
        
        
        Following rules are (at-least) weakly oriented:
          div#(s(X),s(Y)) =  [2] X + [9]          
                          >= [2] X + [5]          
                          =  div#(minus(X,Y),s(Y))
        
          div#(s(X),s(Y)) =  [2] X + [9]          
                          >= [1] X + [0]          
                          =  geq#(X,Y)            
        
          div#(s(X),s(Y)) =  [2] X + [9]          
                          >= [0]                  
                          =  minus#(X,Y)          
        
        minus#(s(X),s(Y)) =  [0]                  
                          >= [0]                  
                          =  c_9(minus#(X,Y))     
        
             minus(0(),Y) =  [0]                  
                          >= [0]                  
                          =  0()                  
        
         minus(s(X),s(Y)) =  [1] X + [2]          
                          >= [1] X + [0]          
                          =  minus(X,Y)           
        
*** Step 1.b:6.b:2: WeightGap.  WORST_CASE(?,O(n^1))
    + Considered Problem:
        - Strict DPs:
            minus#(s(X),s(Y)) -> c_9(minus#(X,Y))
        - Weak DPs:
            div#(s(X),s(Y)) -> div#(minus(X,Y),s(Y))
            div#(s(X),s(Y)) -> geq#(X,Y)
            div#(s(X),s(Y)) -> minus#(X,Y)
            geq#(s(X),s(Y)) -> c_5(geq#(X,Y))
        - Weak TRS:
            minus(0(),Y) -> 0()
            minus(s(X),s(Y)) -> minus(X,Y)
        - Signature:
            {div/2,geq/2,if/3,minus/2,div#/2,geq#/2,if#/3,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/3,c_3/0,c_4/0
            ,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,geq#,if#,minus#} and constructors {0,false,s,true}
    + Applied Processor:
        WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny}
    + Details:
        The weightgap principle applies using the following constant growth matrix-interpretation:
          We apply a matrix interpretation of kind constructor based matrix interpretation:
          The following argument positions are considered usable:
            uargs(div#) = {1},
            uargs(c_5) = {1},
            uargs(c_9) = {1}
          
          Following symbols are considered usable:
            all
          TcT has computed the following interpretation:
                 p(0) = [0]                           
               p(div) = [2] x1 + [1] x2 + [1]         
             p(false) = [1]                           
               p(geq) = [2] x2 + [0]                  
                p(if) = [1] x1 + [2] x2 + [0]         
             p(minus) = [0]                           
                 p(s) = [1] x1 + [2]                  
              p(true) = [1]                           
              p(div#) = [1] x1 + [4] x2 + [2]         
              p(geq#) = [2] x2 + [4]                  
               p(if#) = [1] x1 + [1] x3 + [2]         
            p(minus#) = [4] x2 + [4]                  
               p(c_1) = [0]                           
               p(c_2) = [2] x1 + [1] x2 + [1] x3 + [4]
               p(c_3) = [2]                           
               p(c_4) = [0]                           
               p(c_5) = [1] x1 + [1]                  
               p(c_6) = [1]                           
               p(c_7) = [0]                           
               p(c_8) = [2]                           
               p(c_9) = [1] x1 + [1]                  
          
          Following rules are strictly oriented:
          minus#(s(X),s(Y)) = [4] Y + [12]    
                            > [4] Y + [5]     
                            = c_9(minus#(X,Y))
          
          
          Following rules are (at-least) weakly oriented:
           div#(s(X),s(Y)) =  [1] X + [4] Y + [12] 
                           >= [4] Y + [10]         
                           =  div#(minus(X,Y),s(Y))
          
           div#(s(X),s(Y)) =  [1] X + [4] Y + [12] 
                           >= [2] Y + [4]          
                           =  geq#(X,Y)            
          
           div#(s(X),s(Y)) =  [1] X + [4] Y + [12] 
                           >= [4] Y + [4]          
                           =  minus#(X,Y)          
          
           geq#(s(X),s(Y)) =  [2] Y + [8]          
                           >= [2] Y + [5]          
                           =  c_5(geq#(X,Y))       
          
              minus(0(),Y) =  [0]                  
                           >= [0]                  
                           =  0()                  
          
          minus(s(X),s(Y)) =  [0]                  
                           >= [0]                  
                           =  minus(X,Y)           
          
        Further, it can be verified that all rules not oriented are covered by the weightgap condition.
*** Step 1.b:6.b:3: EmptyProcessor.  WORST_CASE(?,O(1))
    + Considered Problem:
        - Weak DPs:
            div#(s(X),s(Y)) -> div#(minus(X,Y),s(Y))
            div#(s(X),s(Y)) -> geq#(X,Y)
            div#(s(X),s(Y)) -> minus#(X,Y)
            geq#(s(X),s(Y)) -> c_5(geq#(X,Y))
            minus#(s(X),s(Y)) -> c_9(minus#(X,Y))
        - Weak TRS:
            minus(0(),Y) -> 0()
            minus(s(X),s(Y)) -> minus(X,Y)
        - Signature:
            {div/2,geq/2,if/3,minus/2,div#/2,geq#/2,if#/3,minus#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/3,c_3/0,c_4/0
            ,c_5/1,c_6/0,c_7/0,c_8/0,c_9/1}
        - Obligation:
            innermost runtime complexity wrt. defined symbols {div#,geq#,if#,minus#} and constructors {0,false,s,true}
    + Applied Processor:
        EmptyProcessor
    + Details:
        The problem is already closed. The intended complexity is O(1).

WORST_CASE(Omega(n^1),O(n^1))