WORST_CASE(?,O(n^3)) * Step 1: DependencyPairs WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0 ,div,false,geq,if,minus,s,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs a__div#(X1,X2) -> c_1() a__div#(0(),s(Y)) -> c_2() a__div#(s(X),s(Y)) -> c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()),a__geq#(X,Y)) a__geq#(X,0()) -> c_4() a__geq#(X1,X2) -> c_5() a__geq#(0(),s(Y)) -> c_6() a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)) a__if#(X1,X2,X3) -> c_8() a__if#(false(),X,Y) -> c_9(mark#(Y)) a__if#(true(),X,Y) -> c_10(mark#(X)) a__minus#(X1,X2) -> c_11() a__minus#(0(),Y) -> c_12() a__minus#(s(X),s(Y)) -> c_13(a__minus#(X,Y)) mark#(0()) -> c_14() mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)) mark#(false()) -> c_16() mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)) mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)) mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)) mark#(s(X)) -> c_20(mark#(X)) mark#(true()) -> c_21() Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: a__div#(X1,X2) -> c_1() a__div#(0(),s(Y)) -> c_2() a__div#(s(X),s(Y)) -> c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()),a__geq#(X,Y)) a__geq#(X,0()) -> c_4() a__geq#(X1,X2) -> c_5() a__geq#(0(),s(Y)) -> c_6() a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)) a__if#(X1,X2,X3) -> c_8() a__if#(false(),X,Y) -> c_9(mark#(Y)) a__if#(true(),X,Y) -> c_10(mark#(X)) a__minus#(X1,X2) -> c_11() a__minus#(0(),Y) -> c_12() a__minus#(s(X),s(Y)) -> c_13(a__minus#(X,Y)) mark#(0()) -> c_14() mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)) mark#(false()) -> c_16() mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)) mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)) mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)) mark#(s(X)) -> c_20(mark#(X)) mark#(true()) -> c_21() - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1,a__div#/2,a__geq#/2,a__if#/3,a__minus#/2,mark#/1} / {0/0,div/2 ,false/0,geq/2,if/3,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0,c_15/2,c_16/0,c_17/1,c_18/2,c_19/1,c_20/1,c_21/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div#,a__geq#,a__if#,a__minus# ,mark#} and constructors {0,div,false,geq,if,minus,s,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,4,5,6,8,11,12,14,16,21} by application of Pre({1,2,4,5,6,8,11,12,14,16,21}) = {3,7,9,10,13,15,17,18,19,20}. Here rules are labelled as follows: 1: a__div#(X1,X2) -> c_1() 2: a__div#(0(),s(Y)) -> c_2() 3: a__div#(s(X),s(Y)) -> c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()),a__geq#(X,Y)) 4: a__geq#(X,0()) -> c_4() 5: a__geq#(X1,X2) -> c_5() 6: a__geq#(0(),s(Y)) -> c_6() 7: a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)) 8: a__if#(X1,X2,X3) -> c_8() 9: a__if#(false(),X,Y) -> c_9(mark#(Y)) 10: a__if#(true(),X,Y) -> c_10(mark#(X)) 11: a__minus#(X1,X2) -> c_11() 12: a__minus#(0(),Y) -> c_12() 13: a__minus#(s(X),s(Y)) -> c_13(a__minus#(X,Y)) 14: mark#(0()) -> c_14() 15: mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)) 16: mark#(false()) -> c_16() 17: mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)) 18: mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)) 19: mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)) 20: mark#(s(X)) -> c_20(mark#(X)) 21: mark#(true()) -> c_21() * Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: a__div#(s(X),s(Y)) -> c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()),a__geq#(X,Y)) a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)) a__if#(false(),X,Y) -> c_9(mark#(Y)) a__if#(true(),X,Y) -> c_10(mark#(X)) a__minus#(s(X),s(Y)) -> c_13(a__minus#(X,Y)) mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)) mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)) mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)) mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)) mark#(s(X)) -> c_20(mark#(X)) - Weak DPs: a__div#(X1,X2) -> c_1() a__div#(0(),s(Y)) -> c_2() a__geq#(X,0()) -> c_4() a__geq#(X1,X2) -> c_5() a__geq#(0(),s(Y)) -> c_6() a__if#(X1,X2,X3) -> c_8() a__minus#(X1,X2) -> c_11() a__minus#(0(),Y) -> c_12() mark#(0()) -> c_14() mark#(false()) -> c_16() mark#(true()) -> c_21() - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1,a__div#/2,a__geq#/2,a__if#/3,a__minus#/2,mark#/1} / {0/0,div/2 ,false/0,geq/2,if/3,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0,c_15/2,c_16/0,c_17/1,c_18/2,c_19/1,c_20/1,c_21/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div#,a__geq#,a__if#,a__minus# ,mark#} and constructors {0,div,false,geq,if,minus,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:a__div#(s(X),s(Y)) -> c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()),a__geq#(X,Y)) -->_1 a__if#(true(),X,Y) -> c_10(mark#(X)):4 -->_1 a__if#(false(),X,Y) -> c_9(mark#(Y)):3 -->_2 a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)):2 -->_1 a__if#(X1,X2,X3) -> c_8():16 -->_2 a__geq#(0(),s(Y)) -> c_6():15 -->_2 a__geq#(X1,X2) -> c_5():14 -->_2 a__geq#(X,0()) -> c_4():13 2:S:a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)) -->_1 a__geq#(0(),s(Y)) -> c_6():15 -->_1 a__geq#(X1,X2) -> c_5():14 -->_1 a__geq#(X,0()) -> c_4():13 -->_1 a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)):2 3:S:a__if#(false(),X,Y) -> c_9(mark#(Y)) -->_1 mark#(s(X)) -> c_20(mark#(X)):10 -->_1 mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)):9 -->_1 mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)):8 -->_1 mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)):7 -->_1 mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)):6 -->_1 mark#(true()) -> c_21():21 -->_1 mark#(false()) -> c_16():20 -->_1 mark#(0()) -> c_14():19 4:S:a__if#(true(),X,Y) -> c_10(mark#(X)) -->_1 mark#(s(X)) -> c_20(mark#(X)):10 -->_1 mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)):9 -->_1 mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)):8 -->_1 mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)):7 -->_1 mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)):6 -->_1 mark#(true()) -> c_21():21 -->_1 mark#(false()) -> c_16():20 -->_1 mark#(0()) -> c_14():19 5:S:a__minus#(s(X),s(Y)) -> c_13(a__minus#(X,Y)) -->_1 a__minus#(0(),Y) -> c_12():18 -->_1 a__minus#(X1,X2) -> c_11():17 -->_1 a__minus#(s(X),s(Y)) -> c_13(a__minus#(X,Y)):5 6:S:mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)) -->_2 mark#(s(X)) -> c_20(mark#(X)):10 -->_2 mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)):9 -->_2 mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)):8 -->_2 mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)):7 -->_2 mark#(true()) -> c_21():21 -->_2 mark#(false()) -> c_16():20 -->_2 mark#(0()) -> c_14():19 -->_1 a__div#(0(),s(Y)) -> c_2():12 -->_1 a__div#(X1,X2) -> c_1():11 -->_2 mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)):6 -->_1 a__div#(s(X),s(Y)) -> c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()),a__geq#(X,Y)):1 7:S:mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)) -->_1 a__geq#(0(),s(Y)) -> c_6():15 -->_1 a__geq#(X1,X2) -> c_5():14 -->_1 a__geq#(X,0()) -> c_4():13 -->_1 a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)):2 8:S:mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)) -->_2 mark#(s(X)) -> c_20(mark#(X)):10 -->_2 mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)):9 -->_2 mark#(true()) -> c_21():21 -->_2 mark#(false()) -> c_16():20 -->_2 mark#(0()) -> c_14():19 -->_1 a__if#(X1,X2,X3) -> c_8():16 -->_2 mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)):8 -->_2 mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)):7 -->_2 mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)):6 -->_1 a__if#(true(),X,Y) -> c_10(mark#(X)):4 -->_1 a__if#(false(),X,Y) -> c_9(mark#(Y)):3 9:S:mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)) -->_1 a__minus#(0(),Y) -> c_12():18 -->_1 a__minus#(X1,X2) -> c_11():17 -->_1 a__minus#(s(X),s(Y)) -> c_13(a__minus#(X,Y)):5 10:S:mark#(s(X)) -> c_20(mark#(X)) -->_1 mark#(true()) -> c_21():21 -->_1 mark#(false()) -> c_16():20 -->_1 mark#(0()) -> c_14():19 -->_1 mark#(s(X)) -> c_20(mark#(X)):10 -->_1 mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)):9 -->_1 mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)):8 -->_1 mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)):7 -->_1 mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)):6 11:W:a__div#(X1,X2) -> c_1() 12:W:a__div#(0(),s(Y)) -> c_2() 13:W:a__geq#(X,0()) -> c_4() 14:W:a__geq#(X1,X2) -> c_5() 15:W:a__geq#(0(),s(Y)) -> c_6() 16:W:a__if#(X1,X2,X3) -> c_8() 17:W:a__minus#(X1,X2) -> c_11() 18:W:a__minus#(0(),Y) -> c_12() 19:W:mark#(0()) -> c_14() 20:W:mark#(false()) -> c_16() 21:W:mark#(true()) -> c_21() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 11: a__div#(X1,X2) -> c_1() 12: a__div#(0(),s(Y)) -> c_2() 13: a__geq#(X,0()) -> c_4() 14: a__geq#(X1,X2) -> c_5() 15: a__geq#(0(),s(Y)) -> c_6() 16: a__if#(X1,X2,X3) -> c_8() 17: a__minus#(X1,X2) -> c_11() 18: a__minus#(0(),Y) -> c_12() 19: mark#(0()) -> c_14() 20: mark#(false()) -> c_16() 21: mark#(true()) -> c_21() * Step 4: DecomposeDG WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: a__div#(s(X),s(Y)) -> c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()),a__geq#(X,Y)) a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)) a__if#(false(),X,Y) -> c_9(mark#(Y)) a__if#(true(),X,Y) -> c_10(mark#(X)) a__minus#(s(X),s(Y)) -> c_13(a__minus#(X,Y)) mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)) mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)) mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)) mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)) mark#(s(X)) -> c_20(mark#(X)) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1,a__div#/2,a__geq#/2,a__if#/3,a__minus#/2,mark#/1} / {0/0,div/2 ,false/0,geq/2,if/3,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0,c_15/2,c_16/0,c_17/1,c_18/2,c_19/1,c_20/1,c_21/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div#,a__geq#,a__if#,a__minus# ,mark#} and constructors {0,div,false,geq,if,minus,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 a__div#(s(X),s(Y)) -> c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()),a__geq#(X,Y)) a__if#(false(),X,Y) -> c_9(mark#(Y)) a__if#(true(),X,Y) -> c_10(mark#(X)) mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)) mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)) mark#(s(X)) -> c_20(mark#(X)) and a lower component a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)) a__minus#(s(X),s(Y)) -> c_13(a__minus#(X,Y)) mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)) mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)) Further, following extension rules are added to the lower component. a__div#(s(X),s(Y)) -> a__geq#(X,Y) a__div#(s(X),s(Y)) -> a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__if#(false(),X,Y) -> mark#(Y) a__if#(true(),X,Y) -> mark#(X) mark#(div(X1,X2)) -> a__div#(mark(X1),X2) mark#(div(X1,X2)) -> mark#(X1) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(if(X1,X2,X3)) -> mark#(X1) mark#(s(X)) -> mark#(X) ** Step 4.a:1: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: a__div#(s(X),s(Y)) -> c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()),a__geq#(X,Y)) a__if#(false(),X,Y) -> c_9(mark#(Y)) a__if#(true(),X,Y) -> c_10(mark#(X)) mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)) mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)) mark#(s(X)) -> c_20(mark#(X)) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1,a__div#/2,a__geq#/2,a__if#/3,a__minus#/2,mark#/1} / {0/0,div/2 ,false/0,geq/2,if/3,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0,c_15/2,c_16/0,c_17/1,c_18/2,c_19/1,c_20/1,c_21/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div#,a__geq#,a__if#,a__minus# ,mark#} and constructors {0,div,false,geq,if,minus,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:a__div#(s(X),s(Y)) -> c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()),a__geq#(X,Y)) -->_1 a__if#(true(),X,Y) -> c_10(mark#(X)):3 -->_1 a__if#(false(),X,Y) -> c_9(mark#(Y)):2 2:S:a__if#(false(),X,Y) -> c_9(mark#(Y)) -->_1 mark#(s(X)) -> c_20(mark#(X)):6 -->_1 mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)):5 -->_1 mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)):4 3:S:a__if#(true(),X,Y) -> c_10(mark#(X)) -->_1 mark#(s(X)) -> c_20(mark#(X)):6 -->_1 mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)):5 -->_1 mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)):4 4:S:mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)) -->_2 mark#(s(X)) -> c_20(mark#(X)):6 -->_2 mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)):5 -->_2 mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)):4 -->_1 a__div#(s(X),s(Y)) -> c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()),a__geq#(X,Y)):1 5:S:mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)) -->_2 mark#(s(X)) -> c_20(mark#(X)):6 -->_2 mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)):5 -->_2 mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)):4 -->_1 a__if#(true(),X,Y) -> c_10(mark#(X)):3 -->_1 a__if#(false(),X,Y) -> c_9(mark#(Y)):2 6:S:mark#(s(X)) -> c_20(mark#(X)) -->_1 mark#(s(X)) -> c_20(mark#(X)):6 -->_1 mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)):5 -->_1 mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)):4 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: a__div#(s(X),s(Y)) -> c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())) ** Step 4.a:2: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: a__div#(s(X),s(Y)) -> c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())) a__if#(false(),X,Y) -> c_9(mark#(Y)) a__if#(true(),X,Y) -> c_10(mark#(X)) mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)) mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)) mark#(s(X)) -> c_20(mark#(X)) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1,a__div#/2,a__geq#/2,a__if#/3,a__minus#/2,mark#/1} / {0/0,div/2 ,false/0,geq/2,if/3,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0,c_15/2,c_16/0,c_17/1,c_18/2,c_19/1,c_20/1,c_21/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div#,a__geq#,a__if#,a__minus# ,mark#} and constructors {0,div,false,geq,if,minus,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(a__div) = {1}, uargs(a__if) = {1}, uargs(s) = {1}, uargs(a__div#) = {1}, uargs(a__if#) = {1}, uargs(c_3) = {1}, uargs(c_9) = {1}, uargs(c_10) = {1}, uargs(c_15) = {1,2}, uargs(c_18) = {1,2}, uargs(c_20) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__div) = [1] x1 + [0] p(a__geq) = [0] p(a__if) = [1] x1 + [0] p(a__minus) = [0] p(div) = [1] x1 + [0] p(false) = [0] p(geq) = [0] p(if) = [0] p(mark) = [0] p(minus) = [0] p(s) = [1] x1 + [0] p(true) = [0] p(a__div#) = [1] x1 + [4] p(a__geq#) = [4] x1 + [0] p(a__if#) = [1] x1 + [0] p(a__minus#) = [0] p(mark#) = [3] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [1] x1 + [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [1] x1 + [1] x2 + [0] p(c_16) = [0] p(c_17) = [0] p(c_18) = [1] x1 + [1] x2 + [0] p(c_19) = [0] p(c_20) = [1] x1 + [5] p(c_21) = [0] Following rules are strictly oriented: a__div#(s(X),s(Y)) = [1] X + [4] > [0] = c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())) Following rules are (at-least) weakly oriented: a__if#(false(),X,Y) = [0] >= [3] = c_9(mark#(Y)) a__if#(true(),X,Y) = [0] >= [3] = c_10(mark#(X)) mark#(div(X1,X2)) = [3] >= [7] = c_15(a__div#(mark(X1),X2),mark#(X1)) mark#(if(X1,X2,X3)) = [3] >= [3] = c_18(a__if#(mark(X1),X2,X3),mark#(X1)) mark#(s(X)) = [3] >= [8] = c_20(mark#(X)) a__div(X1,X2) = [1] X1 + [0] >= [1] X1 + [0] = div(X1,X2) a__div(0(),s(Y)) = [0] >= [0] = 0() a__div(s(X),s(Y)) = [1] X + [0] >= [0] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) = [0] >= [0] = true() a__geq(X1,X2) = [0] >= [0] = geq(X1,X2) a__geq(0(),s(Y)) = [0] >= [0] = false() a__geq(s(X),s(Y)) = [0] >= [0] = a__geq(X,Y) a__if(X1,X2,X3) = [1] X1 + [0] >= [0] = if(X1,X2,X3) a__if(false(),X,Y) = [0] >= [0] = mark(Y) a__if(true(),X,Y) = [0] >= [0] = mark(X) a__minus(X1,X2) = [0] >= [0] = minus(X1,X2) a__minus(0(),Y) = [0] >= [0] = 0() a__minus(s(X),s(Y)) = [0] >= [0] = a__minus(X,Y) mark(0()) = [0] >= [0] = 0() mark(div(X1,X2)) = [0] >= [0] = a__div(mark(X1),X2) mark(false()) = [0] >= [0] = false() mark(geq(X1,X2)) = [0] >= [0] = a__geq(X1,X2) mark(if(X1,X2,X3)) = [0] >= [0] = a__if(mark(X1),X2,X3) mark(minus(X1,X2)) = [0] >= [0] = a__minus(X1,X2) mark(s(X)) = [0] >= [0] = s(mark(X)) mark(true()) = [0] >= [0] = true() Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 4.a:3: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: a__if#(false(),X,Y) -> c_9(mark#(Y)) a__if#(true(),X,Y) -> c_10(mark#(X)) mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)) mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)) mark#(s(X)) -> c_20(mark#(X)) - Weak DPs: a__div#(s(X),s(Y)) -> c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1,a__div#/2,a__geq#/2,a__if#/3,a__minus#/2,mark#/1} / {0/0,div/2 ,false/0,geq/2,if/3,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0,c_15/2,c_16/0,c_17/1,c_18/2,c_19/1,c_20/1,c_21/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div#,a__geq#,a__if#,a__minus# ,mark#} and constructors {0,div,false,geq,if,minus,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(a__div) = {1}, uargs(a__if) = {1}, uargs(s) = {1}, uargs(a__div#) = {1}, uargs(a__if#) = {1}, uargs(c_3) = {1}, uargs(c_9) = {1}, uargs(c_10) = {1}, uargs(c_15) = {1,2}, uargs(c_18) = {1,2}, uargs(c_20) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__div) = [1] x1 + [0] p(a__geq) = [0] p(a__if) = [1] x1 + [0] p(a__minus) = [0] p(div) = [0] p(false) = [0] p(geq) = [0] p(if) = [0] p(mark) = [0] p(minus) = [0] p(s) = [1] x1 + [0] p(true) = [0] p(a__div#) = [1] x1 + [5] p(a__geq#) = [1] x1 + [0] p(a__if#) = [1] x1 + [5] p(a__minus#) = [0] p(mark#) = [1] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [1] x1 + [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [1] x1 + [1] x2 + [0] p(c_16) = [2] p(c_17) = [0] p(c_18) = [1] x1 + [1] x2 + [0] p(c_19) = [0] p(c_20) = [1] x1 + [7] p(c_21) = [0] Following rules are strictly oriented: a__if#(false(),X,Y) = [5] > [1] = c_9(mark#(Y)) a__if#(true(),X,Y) = [5] > [1] = c_10(mark#(X)) Following rules are (at-least) weakly oriented: a__div#(s(X),s(Y)) = [1] X + [5] >= [5] = c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())) mark#(div(X1,X2)) = [1] >= [6] = c_15(a__div#(mark(X1),X2),mark#(X1)) mark#(if(X1,X2,X3)) = [1] >= [6] = c_18(a__if#(mark(X1),X2,X3),mark#(X1)) mark#(s(X)) = [1] >= [8] = c_20(mark#(X)) a__div(X1,X2) = [1] X1 + [0] >= [0] = div(X1,X2) a__div(0(),s(Y)) = [0] >= [0] = 0() a__div(s(X),s(Y)) = [1] X + [0] >= [0] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) = [0] >= [0] = true() a__geq(X1,X2) = [0] >= [0] = geq(X1,X2) a__geq(0(),s(Y)) = [0] >= [0] = false() a__geq(s(X),s(Y)) = [0] >= [0] = a__geq(X,Y) a__if(X1,X2,X3) = [1] X1 + [0] >= [0] = if(X1,X2,X3) a__if(false(),X,Y) = [0] >= [0] = mark(Y) a__if(true(),X,Y) = [0] >= [0] = mark(X) a__minus(X1,X2) = [0] >= [0] = minus(X1,X2) a__minus(0(),Y) = [0] >= [0] = 0() a__minus(s(X),s(Y)) = [0] >= [0] = a__minus(X,Y) mark(0()) = [0] >= [0] = 0() mark(div(X1,X2)) = [0] >= [0] = a__div(mark(X1),X2) mark(false()) = [0] >= [0] = false() mark(geq(X1,X2)) = [0] >= [0] = a__geq(X1,X2) mark(if(X1,X2,X3)) = [0] >= [0] = a__if(mark(X1),X2,X3) mark(minus(X1,X2)) = [0] >= [0] = a__minus(X1,X2) mark(s(X)) = [0] >= [0] = s(mark(X)) mark(true()) = [0] >= [0] = true() Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 4.a:4: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)) mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)) mark#(s(X)) -> c_20(mark#(X)) - Weak DPs: a__div#(s(X),s(Y)) -> c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())) a__if#(false(),X,Y) -> c_9(mark#(Y)) a__if#(true(),X,Y) -> c_10(mark#(X)) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1,a__div#/2,a__geq#/2,a__if#/3,a__minus#/2,mark#/1} / {0/0,div/2 ,false/0,geq/2,if/3,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0,c_15/2,c_16/0,c_17/1,c_18/2,c_19/1,c_20/1,c_21/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div#,a__geq#,a__if#,a__minus# ,mark#} and constructors {0,div,false,geq,if,minus,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_3) = {1}, uargs(c_9) = {1}, uargs(c_10) = {1}, uargs(c_15) = {1,2}, uargs(c_18) = {1,2}, uargs(c_20) = {1} Following symbols are considered usable: {a__div#,a__geq#,a__if#,a__minus#,mark#} TcT has computed the following interpretation: p(0) = [0] p(a__div) = [3] x1 + [0] p(a__geq) = [2] p(a__if) = [4] x1 + [2] p(a__minus) = [1] x1 + [2] x2 + [0] p(div) = [1] x1 + [0] p(false) = [0] p(geq) = [0] p(if) = [1] x1 + [1] x2 + [1] x3 + [2] p(mark) = [0] p(minus) = [0] p(s) = [1] x1 + [0] p(true) = [0] p(a__div#) = [0] p(a__geq#) = [4] x1 + [1] x2 + [1] p(a__if#) = [4] x2 + [4] x3 + [0] p(a__minus#) = [0] p(mark#) = [4] x1 + [0] p(c_1) = [1] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] p(c_5) = [2] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [1] x1 + [0] p(c_11) = [1] p(c_12) = [0] p(c_13) = [1] x1 + [2] p(c_14) = [4] p(c_15) = [2] x1 + [1] x2 + [0] p(c_16) = [2] p(c_17) = [1] p(c_18) = [1] x1 + [1] x2 + [0] p(c_19) = [0] p(c_20) = [1] x1 + [0] p(c_21) = [4] Following rules are strictly oriented: mark#(if(X1,X2,X3)) = [4] X1 + [4] X2 + [4] X3 + [8] > [4] X1 + [4] X2 + [4] X3 + [0] = c_18(a__if#(mark(X1),X2,X3),mark#(X1)) Following rules are (at-least) weakly oriented: a__div#(s(X),s(Y)) = [0] >= [0] = c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())) a__if#(false(),X,Y) = [4] X + [4] Y + [0] >= [4] Y + [0] = c_9(mark#(Y)) a__if#(true(),X,Y) = [4] X + [4] Y + [0] >= [4] X + [0] = c_10(mark#(X)) mark#(div(X1,X2)) = [4] X1 + [0] >= [4] X1 + [0] = c_15(a__div#(mark(X1),X2),mark#(X1)) mark#(s(X)) = [4] X + [0] >= [4] X + [0] = c_20(mark#(X)) ** Step 4.a:5: Ara WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)) mark#(s(X)) -> c_20(mark#(X)) - Weak DPs: a__div#(s(X),s(Y)) -> c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())) a__if#(false(),X,Y) -> c_9(mark#(Y)) a__if#(true(),X,Y) -> c_10(mark#(X)) mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1,a__div#/2,a__geq#/2,a__if#/3,a__minus#/2,mark#/1} / {0/0,div/2 ,false/0,geq/2,if/3,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0,c_15/2,c_16/0,c_17/1,c_18/2,c_19/1,c_20/1,c_21/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div#,a__geq#,a__if#,a__minus# ,mark#} and constructors {0,div,false,geq,if,minus,s,true} + Applied Processor: Ara {araHeuristics = NoHeuristics, minDegree = 2, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1} + Details: Signatures used: ---------------- 0 :: [] -(0)-> "A"(11, 0) 0 :: [] -(0)-> "A"(0, 0) 0 :: [] -(0)-> "A"(13, 7) 0 :: [] -(0)-> "A"(15, 7) 0 :: [] -(0)-> "A"(13, 6) 0 :: [] -(0)-> "A"(6, 14) a__div :: ["A"(11, 0) x "A"(0, 0)] -(0)-> "A"(11, 0) a__geq :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(13, 0) a__if :: ["A"(11, 0) x "A"(11, 0) x "A"(11, 0)] -(0)-> "A"(11, 0) a__minus :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(11, 1) div :: ["A"(15, 14) x "A"(0, 0)] -(0)-> "A"(1, 14) div :: ["A"(11, 0) x "A"(0, 0)] -(0)-> "A"(11, 0) div :: ["A"(14, 3) x "A"(0, 0)] -(0)-> "A"(11, 3) false :: [] -(0)-> "A"(11, 0) false :: [] -(0)-> "A"(3, 0) false :: [] -(0)-> "A"(15, 6) false :: [] -(0)-> "A"(13, 6) geq :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(11, 0) geq :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(13, 5) if :: ["A"(11, 0) x "A"(11, 0) x "A"(11, 0)] -(0)-> "A"(11, 0) if :: ["A"(15, 14) x "A"(1, 14) x "A"(1, 14)] -(14)-> "A"(1, 14) mark :: ["A"(11, 0)] -(0)-> "A"(11, 0) minus :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(11, 0) minus :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(14, 12) minus :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(15, 1) minus :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(15, 15) s :: ["A"(1, 14)] -(1)-> "A"(1, 14) s :: ["A"(0, 0)] -(0)-> "A"(0, 0) s :: ["A"(11, 0)] -(11)-> "A"(11, 0) s :: ["A"(11, 1)] -(11)-> "A"(11, 1) true :: [] -(0)-> "A"(11, 0) true :: [] -(0)-> "A"(3, 0) true :: [] -(0)-> "A"(15, 6) true :: [] -(0)-> "A"(13, 6) a__div# :: ["A"(11, 0) x "A"(0, 0)] -(0)-> "A"(1, 1) a__if# :: ["A"(3, 0) x "A"(1, 14) x "A"(1, 14)] -(6)-> "A"(3, 3) mark# :: ["A"(1, 14)] -(0)-> "A"(7, 1) c_3 :: ["A"(2, 1)] -(0)-> "A"(1, 1) c_9 :: ["A"(6, 0)] -(3)-> "A"(3, 3) c_10 :: ["A"(3, 0)] -(0)-> "A"(7, 3) c_15 :: ["A"(1, 1) x "A"(0, 0)] -(0)-> "A"(7, 1) c_18 :: ["A"(0, 2) x "A"(0, 0)] -(0)-> "A"(7, 2) c_20 :: ["A"(0, 0)] -(0)-> "A"(7, 12) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "0_A" :: [] -(0)-> "A"(1, 0) "0_A" :: [] -(0)-> "A"(0, 1) "c_10_A" :: ["A"(0)] -(0)-> "A"(1, 0) "c_10_A" :: ["A"(0)] -(0)-> "A"(0, 1) "c_15_A" :: ["A"(0) x "A"(0)] -(0)-> "A"(1, 0) "c_15_A" :: ["A"(0) x "A"(0)] -(0)-> "A"(0, 1) "c_18_A" :: ["A"(0) x "A"(0)] -(0)-> "A"(1, 0) "c_18_A" :: ["A"(0) x "A"(0)] -(0)-> "A"(0, 1) "c_20_A" :: ["A"(0)] -(0)-> "A"(1, 0) "c_20_A" :: ["A"(0)] -(0)-> "A"(0, 1) "c_3_A" :: ["A"(0)] -(0)-> "A"(1, 0) "c_3_A" :: ["A"(0)] -(0)-> "A"(0, 1) "c_9_A" :: ["A"(0)] -(1)-> "A"(1, 0) "c_9_A" :: ["A"(0)] -(0)-> "A"(0, 1) "div_A" :: ["A"(1, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "div_A" :: ["A"(1, 1) x "A"(0, 0)] -(0)-> "A"(0, 1) "false_A" :: [] -(0)-> "A"(1, 0) "false_A" :: [] -(0)-> "A"(0, 1) "geq_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "geq_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(0, 1) "if_A" :: ["A"(1, 0) x "A"(1, 0) x "A"(1, 0)] -(0)-> "A"(1, 0) "if_A" :: ["A"(1, 1) x "A"(0, 1) x "A"(0, 1)] -(1)-> "A"(0, 1) "minus_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "minus_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(0, 1) "s_A" :: ["A"(1, 0)] -(1)-> "A"(1, 0) "s_A" :: ["A"(0, 1)] -(0)-> "A"(0, 1) "true_A" :: [] -(0)-> "A"(1, 0) "true_A" :: [] -(0)-> "A"(0, 1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: mark#(s(X)) -> c_20(mark#(X)) 2. Weak: mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)) ** Step 4.a:6: Ara WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)) - Weak DPs: a__div#(s(X),s(Y)) -> c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())) a__if#(false(),X,Y) -> c_9(mark#(Y)) a__if#(true(),X,Y) -> c_10(mark#(X)) mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)) mark#(s(X)) -> c_20(mark#(X)) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1,a__div#/2,a__geq#/2,a__if#/3,a__minus#/2,mark#/1} / {0/0,div/2 ,false/0,geq/2,if/3,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0,c_15/2,c_16/0,c_17/1,c_18/2,c_19/1,c_20/1,c_21/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div#,a__geq#,a__if#,a__minus# ,mark#} and constructors {0,div,false,geq,if,minus,s,true} + Applied Processor: Ara {araHeuristics = NoHeuristics, minDegree = 2, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1} + Details: Signatures used: ---------------- 0 :: [] -(0)-> "A"(4, 0) 0 :: [] -(0)-> "A"(0, 0) 0 :: [] -(0)-> "A"(11, 7) 0 :: [] -(0)-> "A"(13, 5) 0 :: [] -(0)-> "A"(10, 6) 0 :: [] -(0)-> "A"(10, 14) a__div :: ["A"(4, 0) x "A"(0, 0)] -(4)-> "A"(4, 0) a__geq :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(9, 3) a__if :: ["A"(4, 0) x "A"(4, 0) x "A"(4, 0)] -(0)-> "A"(4, 0) a__minus :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(7, 3) div :: ["A"(14, 13) x "A"(13, 13)] -(1)-> "A"(1, 13) div :: ["A"(4, 0) x "A"(0, 0)] -(4)-> "A"(4, 0) false :: [] -(0)-> "A"(4, 0) false :: [] -(0)-> "A"(1, 0) false :: [] -(0)-> "A"(15, 5) false :: [] -(0)-> "A"(11, 6) geq :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(4, 0) geq :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(9, 4) if :: ["A"(4, 0) x "A"(4, 0) x "A"(4, 0)] -(0)-> "A"(4, 0) if :: ["A"(14, 13) x "A"(1, 13) x "A"(14, 13)] -(0)-> "A"(1, 13) mark :: ["A"(4, 0)] -(0)-> "A"(4, 0) minus :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(4, 0) minus :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(15, 8) minus :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(7, 13) minus :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(14, 15) s :: ["A"(0, 0)] -(0)-> "A"(0, 0) s :: ["A"(4, 0)] -(4)-> "A"(4, 0) s :: ["A"(3, 0)] -(3)-> "A"(3, 0) s :: ["A"(13, 13)] -(13)-> "A"(13, 13) s :: ["A"(1, 13)] -(1)-> "A"(1, 13) true :: [] -(0)-> "A"(4, 0) true :: [] -(0)-> "A"(1, 0) true :: [] -(0)-> "A"(15, 5) true :: [] -(0)-> "A"(11, 6) a__div# :: ["A"(3, 0) x "A"(13, 13)] -(0)-> "A"(14, 9) a__if# :: ["A"(1, 0) x "A"(1, 13) x "A"(4, 13)] -(0)-> "A"(1, 3) mark# :: ["A"(1, 13)] -(0)-> "A"(5, 0) c_3 :: ["A"(0, 0)] -(0)-> "A"(14, 14) c_9 :: ["A"(0, 0)] -(0)-> "A"(5, 6) c_10 :: ["A"(0, 0)] -(0)-> "A"(2, 7) c_15 :: ["A"(6, 1) x "A"(5, 0)] -(0)-> "A"(5, 1) c_18 :: ["A"(0, 3) x "A"(3, 0)] -(0)-> "A"(12, 3) c_20 :: ["A"(0, 0)] -(0)-> "A"(11, 2) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "0_A" :: [] -(0)-> "A"(1, 0) "0_A" :: [] -(0)-> "A"(0, 1) "c_10_A" :: ["A"(0)] -(0)-> "A"(1, 0) "c_10_A" :: ["A"(0)] -(0)-> "A"(0, 1) "c_15_A" :: ["A"(0) x "A"(0)] -(0)-> "A"(1, 0) "c_15_A" :: ["A"(0) x "A"(0)] -(0)-> "A"(0, 1) "c_18_A" :: ["A"(0) x "A"(0)] -(0)-> "A"(1, 0) "c_18_A" :: ["A"(0) x "A"(0)] -(0)-> "A"(0, 1) "c_20_A" :: ["A"(0)] -(0)-> "A"(1, 0) "c_20_A" :: ["A"(0)] -(0)-> "A"(0, 1) "c_3_A" :: ["A"(0)] -(0)-> "A"(1, 0) "c_3_A" :: ["A"(0)] -(0)-> "A"(0, 1) "c_9_A" :: ["A"(0)] -(0)-> "A"(1, 0) "c_9_A" :: ["A"(0)] -(0)-> "A"(0, 1) "div_A" :: ["A"(1, 0) x "A"(0, 0)] -(1)-> "A"(1, 0) "div_A" :: ["A"(1, 1) x "A"(1, 1)] -(0)-> "A"(0, 1) "false_A" :: [] -(0)-> "A"(1, 0) "false_A" :: [] -(0)-> "A"(0, 1) "geq_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "geq_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(0, 1) "if_A" :: ["A"(1, 0) x "A"(1, 0) x "A"(1, 0)] -(0)-> "A"(1, 0) "if_A" :: ["A"(1, 1) x "A"(0, 1) x "A"(1, 1)] -(0)-> "A"(0, 1) "minus_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "minus_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(0, 1) "s_A" :: ["A"(1, 0)] -(1)-> "A"(1, 0) "s_A" :: ["A"(0, 1)] -(0)-> "A"(0, 1) "true_A" :: [] -(0)-> "A"(1, 0) "true_A" :: [] -(0)-> "A"(0, 1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)) 2. Weak: ** Step 4.b:1: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)) a__minus#(s(X),s(Y)) -> c_13(a__minus#(X,Y)) mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)) mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)) - Weak DPs: a__div#(s(X),s(Y)) -> a__geq#(X,Y) a__div#(s(X),s(Y)) -> a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__if#(false(),X,Y) -> mark#(Y) a__if#(true(),X,Y) -> mark#(X) mark#(div(X1,X2)) -> a__div#(mark(X1),X2) mark#(div(X1,X2)) -> mark#(X1) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(if(X1,X2,X3)) -> mark#(X1) mark#(s(X)) -> mark#(X) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1,a__div#/2,a__geq#/2,a__if#/3,a__minus#/2,mark#/1} / {0/0,div/2 ,false/0,geq/2,if/3,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0,c_15/2,c_16/0,c_17/1,c_18/2,c_19/1,c_20/1,c_21/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div#,a__geq#,a__if#,a__minus# ,mark#} and constructors {0,div,false,geq,if,minus,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(a__div) = {1}, uargs(a__if) = {1}, uargs(s) = {1}, uargs(a__div#) = {1}, uargs(a__if#) = {1}, uargs(c_7) = {1}, uargs(c_13) = {1}, uargs(c_17) = {1}, uargs(c_19) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__div) = [1] x1 + [0] p(a__geq) = [0] p(a__if) = [1] x1 + [0] p(a__minus) = [0] p(div) = [0] p(false) = [0] p(geq) = [0] p(if) = [0] p(mark) = [0] p(minus) = [0] p(s) = [1] x1 + [0] p(true) = [0] p(a__div#) = [1] x1 + [2] p(a__geq#) = [0] p(a__if#) = [1] x1 + [2] p(a__minus#) = [7] p(mark#) = [2] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [1] x1 + [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [1] x1 + [2] p(c_14) = [0] p(c_15) = [0] p(c_16) = [0] p(c_17) = [1] x1 + [0] p(c_18) = [0] p(c_19) = [1] x1 + [0] p(c_20) = [0] p(c_21) = [0] Following rules are strictly oriented: mark#(geq(X1,X2)) = [2] > [0] = c_17(a__geq#(X1,X2)) Following rules are (at-least) weakly oriented: a__div#(s(X),s(Y)) = [1] X + [2] >= [0] = a__geq#(X,Y) a__div#(s(X),s(Y)) = [1] X + [2] >= [2] = a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq#(s(X),s(Y)) = [0] >= [0] = c_7(a__geq#(X,Y)) a__if#(false(),X,Y) = [2] >= [2] = mark#(Y) a__if#(true(),X,Y) = [2] >= [2] = mark#(X) a__minus#(s(X),s(Y)) = [7] >= [9] = c_13(a__minus#(X,Y)) mark#(div(X1,X2)) = [2] >= [2] = a__div#(mark(X1),X2) mark#(div(X1,X2)) = [2] >= [2] = mark#(X1) mark#(if(X1,X2,X3)) = [2] >= [2] = a__if#(mark(X1),X2,X3) mark#(if(X1,X2,X3)) = [2] >= [2] = mark#(X1) mark#(minus(X1,X2)) = [2] >= [7] = c_19(a__minus#(X1,X2)) mark#(s(X)) = [2] >= [2] = mark#(X) a__div(X1,X2) = [1] X1 + [0] >= [0] = div(X1,X2) a__div(0(),s(Y)) = [0] >= [0] = 0() a__div(s(X),s(Y)) = [1] X + [0] >= [0] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) = [0] >= [0] = true() a__geq(X1,X2) = [0] >= [0] = geq(X1,X2) a__geq(0(),s(Y)) = [0] >= [0] = false() a__geq(s(X),s(Y)) = [0] >= [0] = a__geq(X,Y) a__if(X1,X2,X3) = [1] X1 + [0] >= [0] = if(X1,X2,X3) a__if(false(),X,Y) = [0] >= [0] = mark(Y) a__if(true(),X,Y) = [0] >= [0] = mark(X) a__minus(X1,X2) = [0] >= [0] = minus(X1,X2) a__minus(0(),Y) = [0] >= [0] = 0() a__minus(s(X),s(Y)) = [0] >= [0] = a__minus(X,Y) mark(0()) = [0] >= [0] = 0() mark(div(X1,X2)) = [0] >= [0] = a__div(mark(X1),X2) mark(false()) = [0] >= [0] = false() mark(geq(X1,X2)) = [0] >= [0] = a__geq(X1,X2) mark(if(X1,X2,X3)) = [0] >= [0] = a__if(mark(X1),X2,X3) mark(minus(X1,X2)) = [0] >= [0] = a__minus(X1,X2) mark(s(X)) = [0] >= [0] = s(mark(X)) mark(true()) = [0] >= [0] = true() Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 4.b:2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)) a__minus#(s(X),s(Y)) -> c_13(a__minus#(X,Y)) mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)) - Weak DPs: a__div#(s(X),s(Y)) -> a__geq#(X,Y) a__div#(s(X),s(Y)) -> a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__if#(false(),X,Y) -> mark#(Y) a__if#(true(),X,Y) -> mark#(X) mark#(div(X1,X2)) -> a__div#(mark(X1),X2) mark#(div(X1,X2)) -> mark#(X1) mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(if(X1,X2,X3)) -> mark#(X1) mark#(s(X)) -> mark#(X) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1,a__div#/2,a__geq#/2,a__if#/3,a__minus#/2,mark#/1} / {0/0,div/2 ,false/0,geq/2,if/3,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0,c_15/2,c_16/0,c_17/1,c_18/2,c_19/1,c_20/1,c_21/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div#,a__geq#,a__if#,a__minus# ,mark#} and constructors {0,div,false,geq,if,minus,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(a__div) = {1}, uargs(a__if) = {1}, uargs(s) = {1}, uargs(a__div#) = {1}, uargs(a__if#) = {1}, uargs(c_7) = {1}, uargs(c_13) = {1}, uargs(c_17) = {1}, uargs(c_19) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__div) = [1] x1 + [0] p(a__geq) = [0] p(a__if) = [1] x1 + [0] p(a__minus) = [0] p(div) = [0] p(false) = [0] p(geq) = [0] p(if) = [0] p(mark) = [0] p(minus) = [0] p(s) = [1] x1 + [0] p(true) = [0] p(a__div#) = [1] x1 + [6] p(a__geq#) = [0] p(a__if#) = [1] x1 + [6] p(a__minus#) = [0] p(mark#) = [6] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [1] x1 + [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [1] x1 + [3] p(c_14) = [0] p(c_15) = [0] p(c_16) = [0] p(c_17) = [1] x1 + [6] p(c_18) = [0] p(c_19) = [1] x1 + [0] p(c_20) = [1] x1 + [0] p(c_21) = [1] Following rules are strictly oriented: mark#(minus(X1,X2)) = [6] > [0] = c_19(a__minus#(X1,X2)) Following rules are (at-least) weakly oriented: a__div#(s(X),s(Y)) = [1] X + [6] >= [0] = a__geq#(X,Y) a__div#(s(X),s(Y)) = [1] X + [6] >= [6] = a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq#(s(X),s(Y)) = [0] >= [0] = c_7(a__geq#(X,Y)) a__if#(false(),X,Y) = [6] >= [6] = mark#(Y) a__if#(true(),X,Y) = [6] >= [6] = mark#(X) a__minus#(s(X),s(Y)) = [0] >= [3] = c_13(a__minus#(X,Y)) mark#(div(X1,X2)) = [6] >= [6] = a__div#(mark(X1),X2) mark#(div(X1,X2)) = [6] >= [6] = mark#(X1) mark#(geq(X1,X2)) = [6] >= [6] = c_17(a__geq#(X1,X2)) mark#(if(X1,X2,X3)) = [6] >= [6] = a__if#(mark(X1),X2,X3) mark#(if(X1,X2,X3)) = [6] >= [6] = mark#(X1) mark#(s(X)) = [6] >= [6] = mark#(X) a__div(X1,X2) = [1] X1 + [0] >= [0] = div(X1,X2) a__div(0(),s(Y)) = [0] >= [0] = 0() a__div(s(X),s(Y)) = [1] X + [0] >= [0] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) = [0] >= [0] = true() a__geq(X1,X2) = [0] >= [0] = geq(X1,X2) a__geq(0(),s(Y)) = [0] >= [0] = false() a__geq(s(X),s(Y)) = [0] >= [0] = a__geq(X,Y) a__if(X1,X2,X3) = [1] X1 + [0] >= [0] = if(X1,X2,X3) a__if(false(),X,Y) = [0] >= [0] = mark(Y) a__if(true(),X,Y) = [0] >= [0] = mark(X) a__minus(X1,X2) = [0] >= [0] = minus(X1,X2) a__minus(0(),Y) = [0] >= [0] = 0() a__minus(s(X),s(Y)) = [0] >= [0] = a__minus(X,Y) mark(0()) = [0] >= [0] = 0() mark(div(X1,X2)) = [0] >= [0] = a__div(mark(X1),X2) mark(false()) = [0] >= [0] = false() mark(geq(X1,X2)) = [0] >= [0] = a__geq(X1,X2) mark(if(X1,X2,X3)) = [0] >= [0] = a__if(mark(X1),X2,X3) mark(minus(X1,X2)) = [0] >= [0] = a__minus(X1,X2) mark(s(X)) = [0] >= [0] = s(mark(X)) mark(true()) = [0] >= [0] = true() Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 4.b:3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)) a__minus#(s(X),s(Y)) -> c_13(a__minus#(X,Y)) - Weak DPs: a__div#(s(X),s(Y)) -> a__geq#(X,Y) a__div#(s(X),s(Y)) -> a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__if#(false(),X,Y) -> mark#(Y) a__if#(true(),X,Y) -> mark#(X) mark#(div(X1,X2)) -> a__div#(mark(X1),X2) mark#(div(X1,X2)) -> mark#(X1) mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(if(X1,X2,X3)) -> mark#(X1) mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)) mark#(s(X)) -> mark#(X) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1,a__div#/2,a__geq#/2,a__if#/3,a__minus#/2,mark#/1} / {0/0,div/2 ,false/0,geq/2,if/3,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0,c_15/2,c_16/0,c_17/1,c_18/2,c_19/1,c_20/1,c_21/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div#,a__geq#,a__if#,a__minus# ,mark#} and constructors {0,div,false,geq,if,minus,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(a__div) = {1}, uargs(a__if) = {1}, uargs(s) = {1}, uargs(a__div#) = {1}, uargs(a__if#) = {1}, uargs(c_7) = {1}, uargs(c_13) = {1}, uargs(c_17) = {1}, uargs(c_19) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__div) = [1] x1 + [0] p(a__geq) = [0] p(a__if) = [1] x1 + [1] x2 + [1] x3 + [0] p(a__minus) = [1] x1 + [0] p(div) = [1] x1 + [0] p(false) = [0] p(geq) = [0] p(if) = [1] x1 + [1] x2 + [1] x3 + [0] p(mark) = [1] x1 + [0] p(minus) = [1] x1 + [0] p(s) = [1] x1 + [4] p(true) = [0] p(a__div#) = [1] x1 + [3] p(a__geq#) = [0] p(a__if#) = [1] x1 + [1] x2 + [1] x3 + [3] p(a__minus#) = [1] x1 + [2] p(mark#) = [1] x1 + [3] p(c_1) = [0] p(c_2) = [1] p(c_3) = [4] x1 + [1] x2 + [1] p(c_4) = [1] p(c_5) = [0] p(c_6) = [2] p(c_7) = [1] x1 + [5] p(c_8) = [1] p(c_9) = [1] x1 + [1] p(c_10) = [4] p(c_11) = [0] p(c_12) = [1] p(c_13) = [1] x1 + [2] p(c_14) = [1] p(c_15) = [1] x1 + [1] x2 + [0] p(c_16) = [2] p(c_17) = [1] x1 + [3] p(c_18) = [2] x1 + [1] x2 + [0] p(c_19) = [1] x1 + [1] p(c_20) = [1] x1 + [4] p(c_21) = [1] Following rules are strictly oriented: a__minus#(s(X),s(Y)) = [1] X + [6] > [1] X + [4] = c_13(a__minus#(X,Y)) Following rules are (at-least) weakly oriented: a__div#(s(X),s(Y)) = [1] X + [7] >= [0] = a__geq#(X,Y) a__div#(s(X),s(Y)) = [1] X + [7] >= [1] X + [7] = a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq#(s(X),s(Y)) = [0] >= [5] = c_7(a__geq#(X,Y)) a__if#(false(),X,Y) = [1] X + [1] Y + [3] >= [1] Y + [3] = mark#(Y) a__if#(true(),X,Y) = [1] X + [1] Y + [3] >= [1] X + [3] = mark#(X) mark#(div(X1,X2)) = [1] X1 + [3] >= [1] X1 + [3] = a__div#(mark(X1),X2) mark#(div(X1,X2)) = [1] X1 + [3] >= [1] X1 + [3] = mark#(X1) mark#(geq(X1,X2)) = [3] >= [3] = c_17(a__geq#(X1,X2)) mark#(if(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [3] >= [1] X1 + [1] X2 + [1] X3 + [3] = a__if#(mark(X1),X2,X3) mark#(if(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [3] >= [1] X1 + [3] = mark#(X1) mark#(minus(X1,X2)) = [1] X1 + [3] >= [1] X1 + [3] = c_19(a__minus#(X1,X2)) mark#(s(X)) = [1] X + [7] >= [1] X + [3] = mark#(X) a__div(X1,X2) = [1] X1 + [0] >= [1] X1 + [0] = div(X1,X2) a__div(0(),s(Y)) = [0] >= [0] = 0() a__div(s(X),s(Y)) = [1] X + [4] >= [1] X + [4] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) = [0] >= [0] = true() a__geq(X1,X2) = [0] >= [0] = geq(X1,X2) a__geq(0(),s(Y)) = [0] >= [0] = false() a__geq(s(X),s(Y)) = [0] >= [0] = a__geq(X,Y) a__if(X1,X2,X3) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = if(X1,X2,X3) a__if(false(),X,Y) = [1] X + [1] Y + [0] >= [1] Y + [0] = mark(Y) a__if(true(),X,Y) = [1] X + [1] Y + [0] >= [1] X + [0] = mark(X) a__minus(X1,X2) = [1] X1 + [0] >= [1] X1 + [0] = minus(X1,X2) a__minus(0(),Y) = [0] >= [0] = 0() a__minus(s(X),s(Y)) = [1] X + [4] >= [1] X + [0] = a__minus(X,Y) mark(0()) = [0] >= [0] = 0() mark(div(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = a__div(mark(X1),X2) mark(false()) = [0] >= [0] = false() mark(geq(X1,X2)) = [0] >= [0] = a__geq(X1,X2) mark(if(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = a__if(mark(X1),X2,X3) mark(minus(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = a__minus(X1,X2) mark(s(X)) = [1] X + [4] >= [1] X + [4] = s(mark(X)) mark(true()) = [0] >= [0] = true() Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 4.b:4: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)) - Weak DPs: a__div#(s(X),s(Y)) -> a__geq#(X,Y) a__div#(s(X),s(Y)) -> a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__if#(false(),X,Y) -> mark#(Y) a__if#(true(),X,Y) -> mark#(X) a__minus#(s(X),s(Y)) -> c_13(a__minus#(X,Y)) mark#(div(X1,X2)) -> a__div#(mark(X1),X2) mark#(div(X1,X2)) -> mark#(X1) mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(if(X1,X2,X3)) -> mark#(X1) mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)) mark#(s(X)) -> mark#(X) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1,a__div#/2,a__geq#/2,a__if#/3,a__minus#/2,mark#/1} / {0/0,div/2 ,false/0,geq/2,if/3,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0,c_15/2,c_16/0,c_17/1,c_18/2,c_19/1,c_20/1,c_21/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div#,a__geq#,a__if#,a__minus# ,mark#} and constructors {0,div,false,geq,if,minus,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(a__div) = {1}, uargs(a__if) = {1}, uargs(s) = {1}, uargs(a__div#) = {1}, uargs(a__if#) = {1}, uargs(c_7) = {1}, uargs(c_13) = {1}, uargs(c_17) = {1}, uargs(c_19) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__div) = [1] x1 + [0] p(a__geq) = [1] x1 + [0] p(a__if) = [1] x1 + [1] x2 + [1] x3 + [0] p(a__minus) = [0] p(div) = [1] x1 + [0] p(false) = [0] p(geq) = [1] x1 + [0] p(if) = [1] x1 + [1] x2 + [1] x3 + [0] p(mark) = [1] x1 + [0] p(minus) = [0] p(s) = [1] x1 + [1] p(true) = [0] p(a__div#) = [1] x1 + [0] p(a__geq#) = [1] x1 + [0] p(a__if#) = [1] x1 + [1] x2 + [1] x3 + [0] p(a__minus#) = [0] p(mark#) = [1] x1 + [0] p(c_1) = [0] p(c_2) = [1] p(c_3) = [1] x1 + [1] x2 + [4] p(c_4) = [0] p(c_5) = [2] p(c_6) = [1] p(c_7) = [1] x1 + [0] p(c_8) = [4] p(c_9) = [1] x1 + [1] p(c_10) = [1] p(c_11) = [1] p(c_12) = [0] p(c_13) = [1] x1 + [0] p(c_14) = [2] p(c_15) = [2] x1 + [4] p(c_16) = [2] p(c_17) = [1] x1 + [0] p(c_18) = [1] x1 + [0] p(c_19) = [1] x1 + [0] p(c_20) = [1] x1 + [2] p(c_21) = [0] Following rules are strictly oriented: a__geq#(s(X),s(Y)) = [1] X + [1] > [1] X + [0] = c_7(a__geq#(X,Y)) Following rules are (at-least) weakly oriented: a__div#(s(X),s(Y)) = [1] X + [1] >= [1] X + [0] = a__geq#(X,Y) a__div#(s(X),s(Y)) = [1] X + [1] >= [1] X + [1] = a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__if#(false(),X,Y) = [1] X + [1] Y + [0] >= [1] Y + [0] = mark#(Y) a__if#(true(),X,Y) = [1] X + [1] Y + [0] >= [1] X + [0] = mark#(X) a__minus#(s(X),s(Y)) = [0] >= [0] = c_13(a__minus#(X,Y)) mark#(div(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = a__div#(mark(X1),X2) mark#(div(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = mark#(X1) mark#(geq(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = c_17(a__geq#(X1,X2)) mark#(if(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = a__if#(mark(X1),X2,X3) mark#(if(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [0] = mark#(X1) mark#(minus(X1,X2)) = [0] >= [0] = c_19(a__minus#(X1,X2)) mark#(s(X)) = [1] X + [1] >= [1] X + [0] = mark#(X) a__div(X1,X2) = [1] X1 + [0] >= [1] X1 + [0] = div(X1,X2) a__div(0(),s(Y)) = [0] >= [0] = 0() a__div(s(X),s(Y)) = [1] X + [1] >= [1] X + [1] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) = [1] X + [0] >= [0] = true() a__geq(X1,X2) = [1] X1 + [0] >= [1] X1 + [0] = geq(X1,X2) a__geq(0(),s(Y)) = [0] >= [0] = false() a__geq(s(X),s(Y)) = [1] X + [1] >= [1] X + [0] = a__geq(X,Y) a__if(X1,X2,X3) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = if(X1,X2,X3) a__if(false(),X,Y) = [1] X + [1] Y + [0] >= [1] Y + [0] = mark(Y) a__if(true(),X,Y) = [1] X + [1] Y + [0] >= [1] X + [0] = mark(X) a__minus(X1,X2) = [0] >= [0] = minus(X1,X2) a__minus(0(),Y) = [0] >= [0] = 0() a__minus(s(X),s(Y)) = [0] >= [0] = a__minus(X,Y) mark(0()) = [0] >= [0] = 0() mark(div(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = a__div(mark(X1),X2) mark(false()) = [0] >= [0] = false() mark(geq(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = a__geq(X1,X2) mark(if(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = a__if(mark(X1),X2,X3) mark(minus(X1,X2)) = [0] >= [0] = a__minus(X1,X2) mark(s(X)) = [1] X + [1] >= [1] X + [1] = s(mark(X)) mark(true()) = [0] >= [0] = true() Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 4.b:5: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: a__div#(s(X),s(Y)) -> a__geq#(X,Y) a__div#(s(X),s(Y)) -> a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)) a__if#(false(),X,Y) -> mark#(Y) a__if#(true(),X,Y) -> mark#(X) a__minus#(s(X),s(Y)) -> c_13(a__minus#(X,Y)) mark#(div(X1,X2)) -> a__div#(mark(X1),X2) mark#(div(X1,X2)) -> mark#(X1) mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(if(X1,X2,X3)) -> mark#(X1) mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)) mark#(s(X)) -> mark#(X) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1,a__div#/2,a__geq#/2,a__if#/3,a__minus#/2,mark#/1} / {0/0,div/2 ,false/0,geq/2,if/3,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0,c_15/2,c_16/0,c_17/1,c_18/2,c_19/1,c_20/1,c_21/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div#,a__geq#,a__if#,a__minus# ,mark#} and constructors {0,div,false,geq,if,minus,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^3))