WORST_CASE(Omega(n^1),O(n^2)) * Step 1: Sum WORST_CASE(Omega(n^1),O(n^2)) + 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: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?) + 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: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: a__geq(x,y){x -> s(x),y -> s(y)} = a__geq(s(x),s(y)) ->^+ a__geq(x,y) = C[a__geq(x,y) = a__geq(x,y){}] ** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^2)) + 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 1.b:2: PredecessorEstimation WORST_CASE(?,O(n^2)) + 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 1.b:3: RemoveWeakSuffixes 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__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 1.b:4: PredecessorEstimationCP 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__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: 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: 5: a__minus#(s(X),s(Y)) -> c_13(a__minus#(X,Y)) 9: mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)) The strictly oriented rules are moved into the weak component. *** Step 1.b:4.a:1: NaturalMI 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__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: 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_3) = {1,2}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_10) = {1}, uargs(c_13) = {1}, uargs(c_15) = {1,2}, uargs(c_17) = {1}, uargs(c_18) = {1,2}, uargs(c_19) = {1}, uargs(c_20) = {1} Following symbols are considered usable: {a__div,a__geq,a__if,a__minus,mark,a__div#,a__geq#,a__if#,a__minus#,mark#} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__div) = [0 0 0] [0 0 0] [0] [0 1 1] x1 + [0 0 0] x2 + [1] [0 0 1] [0 0 1] [0] p(a__geq) = [0] [0] [0] p(a__if) = [0 0 0] [0 0 0] [0 0 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0 1 1] x3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] p(a__minus) = [0 0 0] [0] [0 0 1] x1 + [1] [0 0 0] [0] p(div) = [0 0 0] [0 0 0] [0] [0 1 1] x1 + [0 0 0] x2 + [1] [0 0 1] [0 0 1] [0] p(false) = [0] [0] [0] p(geq) = [0] [0] [0] p(if) = [0 0 0] [0 0 0] [0 0 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0 1 1] x3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] p(mark) = [0 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(minus) = [0 0 0] [0] [0 0 1] x1 + [1] [0 0 0] [0] p(s) = [0 0 0] [0] [0 1 0] x1 + [0] [0 0 1] [1] p(true) = [0] [0] [0] p(a__div#) = [0 0 1] [0 0 0] [1] [0 0 0] x1 + [0 0 0] x2 + [1] [0 0 0] [0 0 1] [1] p(a__geq#) = [0 0 0] [0] [0 1 1] x2 + [0] [0 1 1] [0] p(a__if#) = [0 1 0] [0 1 1] [0] [0 0 0] x2 + [1 0 1] x3 + [1] [1 0 0] [1 0 1] [0] p(a__minus#) = [0 0 1] [0 0 0] [0] [0 0 0] x1 + [0 0 1] x2 + [0] [0 1 0] [0 0 1] [1] p(mark#) = [0 1 0] [0] [1 0 0] x1 + [0] [0 0 0] [0] p(c_1) = [0] [0] [0] p(c_2) = [0] [0] [0] p(c_3) = [1 0 0] [1 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 1 0] [0 0 0] [1] 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 1 0] x1 + [0] [0 1 0] [0] p(c_8) = [0] [0] [0] p(c_9) = [1 0 0] [0] [0 0 0] x1 + [1] [0 1 0] [0] p(c_10) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(c_11) = [0] [0] [0] p(c_12) = [0] [0] [0] p(c_13) = [1 0 0] [0] [0 1 0] x1 + [1] [0 0 0] [0] p(c_14) = [0] [0] [0] p(c_15) = [1 0 0] [1 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 0] [0 0 0] [0] p(c_16) = [0] [0] [0] p(c_17) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(c_18) = [1 0 0] [1 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 0] [0 0 0] [0] p(c_19) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(c_20) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(c_21) = [0] [0] [0] Following rules are strictly oriented: a__minus#(s(X),s(Y)) = [0 0 1] [0 0 0] [1] [0 0 0] X + [0 0 1] Y + [1] [0 1 0] [0 0 1] [2] > [0 0 1] [0 0 0] [0] [0 0 0] X + [0 0 1] Y + [1] [0 0 0] [0 0 0] [0] = c_13(a__minus#(X,Y)) mark#(minus(X1,X2)) = [0 0 1] [1] [0 0 0] X1 + [0] [0 0 0] [0] > [0 0 1] [0] [0 0 0] X1 + [0] [0 0 0] [0] = c_19(a__minus#(X1,X2)) Following rules are (at-least) weakly oriented: a__div#(s(X),s(Y)) = [0 0 1] [0 0 0] [2] [0 0 0] X + [0 0 0] Y + [1] [0 0 0] [0 0 1] [2] >= [0 0 1] [2] [0 0 0] X + [0] [0 0 0] [2] = 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)) = [0 0 0] [0] [0 1 1] Y + [1] [0 1 1] [1] >= [0 0 0] [0] [0 1 1] Y + [0] [0 1 1] [0] = c_7(a__geq#(X,Y)) a__if#(false(),X,Y) = [0 1 0] [0 1 1] [0] [0 0 0] X + [1 0 1] Y + [1] [1 0 0] [1 0 1] [0] >= [0 1 0] [0] [0 0 0] Y + [1] [1 0 0] [0] = c_9(mark#(Y)) a__if#(true(),X,Y) = [0 1 0] [0 1 1] [0] [0 0 0] X + [1 0 1] Y + [1] [1 0 0] [1 0 1] [0] >= [0 1 0] [0] [0 0 0] X + [1] [0 0 0] [0] = c_10(mark#(X)) mark#(div(X1,X2)) = [0 1 1] [1] [0 0 0] X1 + [0] [0 0 0] [0] >= [0 1 1] [1] [0 0 0] X1 + [0] [0 0 0] [0] = c_15(a__div#(mark(X1),X2),mark#(X1)) mark#(geq(X1,X2)) = [0] [0] [0] >= [0] [0] [0] = c_17(a__geq#(X1,X2)) mark#(if(X1,X2,X3)) = [0 1 0] [0 1 0] [0 1 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0 0 0] X3 + [0] [0 0 0] [0 0 0] [0 0 0] [0] >= [0 1 0] [0 1 0] [0 1 1] [0] [0 0 0] X1 + [0 0 0] X2 + [0 0 0] X3 + [0] [0 0 0] [0 0 0] [0 0 0] [0] = c_18(a__if#(mark(X1),X2,X3),mark#(X1)) mark#(s(X)) = [0 1 0] [0] [0 0 0] X + [0] [0 0 0] [0] >= [0 1 0] [0] [0 0 0] X + [0] [0 0 0] [0] = c_20(mark#(X)) a__div(X1,X2) = [0 0 0] [0 0 0] [0] [0 1 1] X1 + [0 0 0] X2 + [1] [0 0 1] [0 0 1] [0] >= [0 0 0] [0 0 0] [0] [0 1 1] X1 + [0 0 0] X2 + [1] [0 0 1] [0 0 1] [0] = div(X1,X2) a__div(0(),s(Y)) = [0 0 0] [0] [0 0 0] Y + [1] [0 0 1] [1] >= [0] [0] [0] = 0() a__div(s(X),s(Y)) = [0 0 0] [0 0 0] [0] [0 1 1] X + [0 0 0] Y + [2] [0 0 1] [0 0 1] [2] >= [0 0 0] [0 0 0] [0] [0 0 1] X + [0 0 0] Y + [2] [0 0 0] [0 0 1] [2] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) = [0] [0] [0] >= [0] [0] [0] = true() a__geq(X1,X2) = [0] [0] [0] >= [0] [0] [0] = geq(X1,X2) a__geq(0(),s(Y)) = [0] [0] [0] >= [0] [0] [0] = false() a__geq(s(X),s(Y)) = [0] [0] [0] >= [0] [0] [0] = a__geq(X,Y) a__if(X1,X2,X3) = [0 0 0] [0 0 0] [0 0 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 1] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] >= [0 0 0] [0 0 0] [0 0 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 1] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] = if(X1,X2,X3) a__if(false(),X,Y) = [0 0 0] [0 0 0] [0] [0 1 0] X + [0 1 1] Y + [0] [0 0 1] [0 0 1] [0] >= [0 0 0] [0] [0 1 0] Y + [0] [0 0 1] [0] = mark(Y) a__if(true(),X,Y) = [0 0 0] [0 0 0] [0] [0 1 0] X + [0 1 1] Y + [0] [0 0 1] [0 0 1] [0] >= [0 0 0] [0] [0 1 0] X + [0] [0 0 1] [0] = mark(X) a__minus(X1,X2) = [0 0 0] [0] [0 0 1] X1 + [1] [0 0 0] [0] >= [0 0 0] [0] [0 0 1] X1 + [1] [0 0 0] [0] = minus(X1,X2) a__minus(0(),Y) = [0] [1] [0] >= [0] [0] [0] = 0() a__minus(s(X),s(Y)) = [0 0 0] [0] [0 0 1] X + [2] [0 0 0] [0] >= [0 0 0] [0] [0 0 1] X + [1] [0 0 0] [0] = a__minus(X,Y) mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(div(X1,X2)) = [0 0 0] [0 0 0] [0] [0 1 1] X1 + [0 0 0] X2 + [1] [0 0 1] [0 0 1] [0] >= [0 0 0] [0 0 0] [0] [0 1 1] X1 + [0 0 0] X2 + [1] [0 0 1] [0 0 1] [0] = a__div(mark(X1),X2) mark(false()) = [0] [0] [0] >= [0] [0] [0] = false() mark(geq(X1,X2)) = [0] [0] [0] >= [0] [0] [0] = a__geq(X1,X2) mark(if(X1,X2,X3)) = [0 0 0] [0 0 0] [0 0 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 1] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] >= [0 0 0] [0 0 0] [0 0 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 1] X3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] = a__if(mark(X1),X2,X3) mark(minus(X1,X2)) = [0 0 0] [0] [0 0 1] X1 + [1] [0 0 0] [0] >= [0 0 0] [0] [0 0 1] X1 + [1] [0 0 0] [0] = a__minus(X1,X2) mark(s(X)) = [0 0 0] [0] [0 1 0] X + [0] [0 0 1] [1] >= [0 0 0] [0] [0 1 0] X + [0] [0 0 1] [1] = s(mark(X)) mark(true()) = [0] [0] [0] >= [0] [0] [0] = true() *** Step 1.b:4.a:2: Assumption WORST_CASE(?,O(1)) + 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)) 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#(s(X)) -> c_20(mark#(X)) - Weak DPs: a__minus#(s(X),s(Y)) -> c_13(a__minus#(X,Y)) mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)) - 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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 1.b:4.b:1: RemoveWeakSuffixes 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__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)) 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#(s(X)) -> c_20(mark#(X)) - Weak DPs: a__minus#(s(X),s(Y)) -> c_13(a__minus#(X,Y)) mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)) - 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 2:S:a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)) -->_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#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)):10 -->_1 mark#(s(X)) -> c_20(mark#(X)):8 -->_1 mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)):7 -->_1 mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)):6 -->_1 mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)):5 4:S:a__if#(true(),X,Y) -> c_10(mark#(X)) -->_1 mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)):10 -->_1 mark#(s(X)) -> c_20(mark#(X)):8 -->_1 mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)):7 -->_1 mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)):6 -->_1 mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)):5 5:S:mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)) -->_2 mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)):10 -->_2 mark#(s(X)) -> c_20(mark#(X)):8 -->_2 mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)):7 -->_2 mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)):6 -->_2 mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)):5 -->_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 6:S:mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)) -->_1 a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)):2 7:S:mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)) -->_2 mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)):10 -->_2 mark#(s(X)) -> c_20(mark#(X)):8 -->_2 mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)):7 -->_2 mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)):6 -->_2 mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)):5 -->_1 a__if#(true(),X,Y) -> c_10(mark#(X)):4 -->_1 a__if#(false(),X,Y) -> c_9(mark#(Y)):3 8:S:mark#(s(X)) -> c_20(mark#(X)) -->_1 mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)):10 -->_1 mark#(s(X)) -> c_20(mark#(X)):8 -->_1 mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)):7 -->_1 mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)):6 -->_1 mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)):5 9:W:a__minus#(s(X),s(Y)) -> c_13(a__minus#(X,Y)) -->_1 a__minus#(s(X),s(Y)) -> c_13(a__minus#(X,Y)):9 10:W:mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)) -->_1 a__minus#(s(X),s(Y)) -> c_13(a__minus#(X,Y)):9 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 10: mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)) 9: a__minus#(s(X),s(Y)) -> c_13(a__minus#(X,Y)) *** Step 1.b:4.b:2: PredecessorEstimationCP 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__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)) 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#(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: 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: a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)) The strictly oriented rules are moved into the weak component. **** Step 1.b:4.b:2.a:1: NaturalMI 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__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)) 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#(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: 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_3) = {1,2}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_10) = {1}, uargs(c_15) = {1,2}, uargs(c_17) = {1}, uargs(c_18) = {1,2}, uargs(c_20) = {1} Following symbols are considered usable: {a__div,a__geq,a__if,a__minus,mark,a__div#,a__geq#,a__if#,a__minus#,mark#} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(a__div) = [0 0 1] [0] [0 1 1] x1 + [0] [0 0 1] [0] p(a__geq) = [0 0 0] [0] [0 1 1] x1 + [0] [0 0 0] [0] p(a__if) = [0 0 1] [0 0 1] [0 0 1] [0] [0 1 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [1] [0 0 1] [0 0 1] [0 0 1] [0] p(a__minus) = [0] [0] [0] p(div) = [0 0 0] [0] [0 1 1] x1 + [0] [0 0 1] [0] p(false) = [0] [0] [0] p(geq) = [0 0 0] [0] [0 1 1] x1 + [0] [0 0 0] [0] p(if) = [0 0 0] [0 0 1] [0 0 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [1] [0 0 1] [0 0 1] [0 0 1] [0] p(mark) = [0 0 1] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(minus) = [0] [0] [0] p(s) = [0 0 1] [0] [0 1 0] x1 + [0] [0 0 1] [1] p(true) = [0] [0] [0] p(a__div#) = [1 0 0] [0 0 0] [0] [1 0 0] x1 + [0 0 1] x2 + [0] [1 0 0] [0 0 1] [1] p(a__geq#) = [0 0 1] [0] [0 1 0] x1 + [0] [0 0 1] [1] p(a__if#) = [0 1 0] [0 1 0] [0] [0 0 0] x2 + [1 0 0] x3 + [1] [0 0 1] [0 0 0] [0] p(a__minus#) = [0] [0] [0] p(mark#) = [0 1 0] [0] [1 0 0] x1 + [1] [1 0 0] [0] p(c_1) = [0] [0] [0] p(c_2) = [0] [0] [0] p(c_3) = [1 0 0] [1 0 0] [0] [0 0 0] x1 + [0 0 1] x2 + [0] [0 0 1] [0 0 1] [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 + [0] [1 0 0] [1] p(c_8) = [0] [0] [0] p(c_9) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(c_10) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(c_11) = [0] [0] [0] p(c_12) = [0] [0] [0] p(c_13) = [0] [0] [0] p(c_14) = [0] [0] [0] p(c_15) = [1 0 0] [1 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [1] [0 0 0] [0 0 0] [0] p(c_16) = [0] [0] [0] p(c_17) = [1 1 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(c_18) = [1 0 0] [1 0 0] [1] [0 0 1] x1 + [0 0 0] x2 + [0] [0 0 0] [0 0 0] [0] p(c_19) = [0] [0] [0] p(c_20) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(c_21) = [0] [0] [0] Following rules are strictly oriented: a__geq#(s(X),s(Y)) = [0 0 1] [1] [0 1 0] X + [0] [0 0 1] [2] > [0 0 1] [0] [0 0 0] X + [0] [0 0 1] [1] = c_7(a__geq#(X,Y)) Following rules are (at-least) weakly oriented: a__div#(s(X),s(Y)) = [0 0 1] [0 0 0] [0] [0 0 1] X + [0 0 1] Y + [1] [0 0 1] [0 0 1] [2] >= [0 0 1] [0] [0 0 1] X + [1] [0 0 1] [2] = 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) = [0 1 0] [0 1 0] [0] [0 0 0] X + [1 0 0] Y + [1] [0 0 1] [0 0 0] [0] >= [0 1 0] [0] [1 0 0] Y + [1] [0 0 0] [0] = c_9(mark#(Y)) a__if#(true(),X,Y) = [0 1 0] [0 1 0] [0] [0 0 0] X + [1 0 0] Y + [1] [0 0 1] [0 0 0] [0] >= [0 1 0] [0] [0 0 0] X + [0] [0 0 0] [0] = c_10(mark#(X)) mark#(div(X1,X2)) = [0 1 1] [0] [0 0 0] X1 + [1] [0 0 0] [0] >= [0 1 1] [0] [0 0 0] X1 + [1] [0 0 0] [0] = c_15(a__div#(mark(X1),X2),mark#(X1)) mark#(geq(X1,X2)) = [0 1 1] [0] [0 0 0] X1 + [1] [0 0 0] [0] >= [0 1 1] [0] [0 0 0] X1 + [1] [0 0 0] [0] = c_17(a__geq#(X1,X2)) mark#(if(X1,X2,X3)) = [0 1 0] [0 1 0] [0 1 0] [1] [0 0 0] X1 + [0 0 1] X2 + [0 0 0] X3 + [1] [0 0 0] [0 0 1] [0 0 0] [0] >= [0 1 0] [0 1 0] [0 1 0] [1] [0 0 0] X1 + [0 0 1] X2 + [0 0 0] X3 + [0] [0 0 0] [0 0 0] [0 0 0] [0] = c_18(a__if#(mark(X1),X2,X3),mark#(X1)) mark#(s(X)) = [0 1 0] [0] [0 0 1] X + [1] [0 0 1] [0] >= [0 1 0] [0] [0 0 0] X + [1] [0 0 0] [0] = c_20(mark#(X)) a__div(X1,X2) = [0 0 1] [0] [0 1 1] X1 + [0] [0 0 1] [0] >= [0 0 0] [0] [0 1 1] X1 + [0] [0 0 1] [0] = div(X1,X2) a__div(0(),s(Y)) = [0] [0] [0] >= [0] [0] [0] = 0() a__div(s(X),s(Y)) = [0 0 1] [1] [0 1 1] X + [1] [0 0 1] [1] >= [0 0 0] [1] [0 1 1] X + [1] [0 0 0] [1] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) = [0 0 0] [0] [0 1 1] X + [0] [0 0 0] [0] >= [0] [0] [0] = true() a__geq(X1,X2) = [0 0 0] [0] [0 1 1] X1 + [0] [0 0 0] [0] >= [0 0 0] [0] [0 1 1] X1 + [0] [0 0 0] [0] = geq(X1,X2) a__geq(0(),s(Y)) = [0] [0] [0] >= [0] [0] [0] = false() a__geq(s(X),s(Y)) = [0 0 0] [0] [0 1 1] X + [1] [0 0 0] [0] >= [0 0 0] [0] [0 1 1] X + [0] [0 0 0] [0] = a__geq(X,Y) a__if(X1,X2,X3) = [0 0 1] [0 0 1] [0 0 1] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [1] [0 0 1] [0 0 1] [0 0 1] [0] >= [0 0 0] [0 0 1] [0 0 0] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [1] [0 0 1] [0 0 1] [0 0 1] [0] = if(X1,X2,X3) a__if(false(),X,Y) = [0 0 1] [0 0 1] [0] [0 1 0] X + [0 1 0] Y + [1] [0 0 1] [0 0 1] [0] >= [0 0 1] [0] [0 1 0] Y + [0] [0 0 1] [0] = mark(Y) a__if(true(),X,Y) = [0 0 1] [0 0 1] [0] [0 1 0] X + [0 1 0] Y + [1] [0 0 1] [0 0 1] [0] >= [0 0 1] [0] [0 1 0] X + [0] [0 0 1] [0] = mark(X) a__minus(X1,X2) = [0] [0] [0] >= [0] [0] [0] = minus(X1,X2) a__minus(0(),Y) = [0] [0] [0] >= [0] [0] [0] = 0() a__minus(s(X),s(Y)) = [0] [0] [0] >= [0] [0] [0] = a__minus(X,Y) mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(div(X1,X2)) = [0 0 1] [0] [0 1 1] X1 + [0] [0 0 1] [0] >= [0 0 1] [0] [0 1 1] X1 + [0] [0 0 1] [0] = a__div(mark(X1),X2) mark(false()) = [0] [0] [0] >= [0] [0] [0] = false() mark(geq(X1,X2)) = [0 0 0] [0] [0 1 1] X1 + [0] [0 0 0] [0] >= [0 0 0] [0] [0 1 1] X1 + [0] [0 0 0] [0] = a__geq(X1,X2) mark(if(X1,X2,X3)) = [0 0 1] [0 0 1] [0 0 1] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [1] [0 0 1] [0 0 1] [0 0 1] [0] >= [0 0 1] [0 0 1] [0 0 1] [0] [0 1 0] X1 + [0 1 0] X2 + [0 1 0] X3 + [1] [0 0 1] [0 0 1] [0 0 1] [0] = a__if(mark(X1),X2,X3) mark(minus(X1,X2)) = [0] [0] [0] >= [0] [0] [0] = a__minus(X1,X2) mark(s(X)) = [0 0 1] [1] [0 1 0] X + [0] [0 0 1] [1] >= [0 0 1] [0] [0 1 0] X + [0] [0 0 1] [1] = s(mark(X)) mark(true()) = [0] [0] [0] >= [0] [0] [0] = true() **** Step 1.b:4.b:2.a:2: Assumption WORST_CASE(?,O(1)) + 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#(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#(s(X)) -> c_20(mark#(X)) - Weak DPs: a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)) - 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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 1.b:4.b:2.b:1: RemoveWeakSuffixes 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#(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#(s(X)) -> c_20(mark#(X)) - Weak DPs: a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)) - 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)) -->_2 a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)):8 -->_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)):7 -->_1 mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)):6 -->_1 mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)):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)):7 -->_1 mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)):6 -->_1 mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)):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)):7 -->_2 mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)):6 -->_2 mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)):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#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)) -->_1 a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)):8 6:S:mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)) -->_2 mark#(s(X)) -> c_20(mark#(X)):7 -->_2 mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)):6 -->_2 mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)):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 7:S:mark#(s(X)) -> c_20(mark#(X)) -->_1 mark#(s(X)) -> c_20(mark#(X)):7 -->_1 mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)):6 -->_1 mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)):5 -->_1 mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)):4 8:W:a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)) -->_1 a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)):8 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 8: a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)) **** Step 1.b:4.b:2.b:2: 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#(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#(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)):7 -->_1 mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)):6 -->_1 mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)):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)):7 -->_1 mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)):6 -->_1 mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)):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)):7 -->_2 mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)):6 -->_2 mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)):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#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)) 6:S:mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)) -->_2 mark#(s(X)) -> c_20(mark#(X)):7 -->_2 mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)):6 -->_2 mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)):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 7:S:mark#(s(X)) -> c_20(mark#(X)) -->_1 mark#(s(X)) -> c_20(mark#(X)):7 -->_1 mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)):6 -->_1 mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)):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())) mark#(geq(X1,X2)) -> c_17() **** Step 1.b:4.b:2.b:3: PredecessorEstimationCP 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#(geq(X1,X2)) -> c_17() 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_17/0,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: 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: 5: mark#(geq(X1,X2)) -> c_17() The strictly oriented rules are moved into the weak component. ***** Step 1.b:4.b:2.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1)) + 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#(geq(X1,X2)) -> c_17() 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_17/0,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 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_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) = [0] p(a__geq) = [1] x1 + [2] p(a__if) = [1] x1 + [6] p(a__minus) = [0] p(div) = [1] x1 + [0] p(false) = [0] p(geq) = [1] x2 + [4] p(if) = [1] x1 + [1] x2 + [1] x3 + [0] p(mark) = [0] p(minus) = [0] p(s) = [1] x1 + [0] p(true) = [2] p(a__div#) = [0] p(a__geq#) = [4] p(a__if#) = [1] x2 + [1] x3 + [0] p(a__minus#) = [2] p(mark#) = [1] x1 + [0] p(c_1) = [1] p(c_2) = [1] p(c_3) = [4] x1 + [0] p(c_4) = [0] p(c_5) = [1] p(c_6) = [1] p(c_7) = [1] x1 + [2] p(c_8) = [1] 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) = [2] p(c_15) = [4] x1 + [1] x2 + [0] p(c_16) = [0] p(c_17) = [0] p(c_18) = [1] x1 + [1] x2 + [0] p(c_19) = [1] x1 + [1] p(c_20) = [1] x1 + [0] p(c_21) = [0] Following rules are strictly oriented: mark#(geq(X1,X2)) = [1] X2 + [4] > [0] = c_17() 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) = [1] X + [1] Y + [0] >= [1] Y + [0] = c_9(mark#(Y)) a__if#(true(),X,Y) = [1] X + [1] Y + [0] >= [1] X + [0] = c_10(mark#(X)) mark#(div(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = c_15(a__div#(mark(X1),X2),mark#(X1)) mark#(if(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = c_18(a__if#(mark(X1),X2,X3),mark#(X1)) mark#(s(X)) = [1] X + [0] >= [1] X + [0] = c_20(mark#(X)) ***** Step 1.b:4.b:2.b:3.a:2: Assumption WORST_CASE(?,O(1)) + 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 DPs: mark#(geq(X1,X2)) -> c_17() - 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_17/0,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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 1.b:4.b:2.b:3.b:1: RemoveWeakSuffixes 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 DPs: mark#(geq(X1,X2)) -> c_17() - 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_17/0,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())) -->_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 -->_1 mark#(geq(X1,X2)) -> c_17():7 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 -->_1 mark#(geq(X1,X2)) -> c_17():7 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#(geq(X1,X2)) -> c_17():7 -->_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())):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#(geq(X1,X2)) -> c_17():7 -->_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#(geq(X1,X2)) -> c_17():7 -->_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 7:W:mark#(geq(X1,X2)) -> c_17() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 7: mark#(geq(X1,X2)) -> c_17() ***** Step 1.b:4.b:2.b:3.b:2: PredecessorEstimationCP 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_17/0,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: 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: 5: mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)) The strictly oriented rules are moved into the weak component. ****** Step 1.b:4.b:2.b:3.b:2.a:1: NaturalMI WORST_CASE(?,O(n^1)) + 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_17/0,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 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_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) = [1] x1 + [4] p(a__geq) = [6] x1 + [4] x2 + [0] p(a__if) = [2] x1 + [2] x2 + [0] p(a__minus) = [4] x1 + [0] p(div) = [1] x1 + [4] p(false) = [0] p(geq) = [0] p(if) = [1] x1 + [1] x2 + [1] x3 + [1] p(mark) = [0] p(minus) = [0] p(s) = [1] x1 + [0] p(true) = [0] p(a__div#) = [4] p(a__geq#) = [4] x2 + [0] p(a__if#) = [1] x2 + [1] x3 + [0] p(a__minus#) = [1] x1 + [0] p(mark#) = [1] x1 + [0] p(c_1) = [0] p(c_2) = [1] p(c_3) = [1] x1 + [0] p(c_4) = [0] p(c_5) = [1] p(c_6) = [2] p(c_7) = [1] x1 + [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) = [4] x1 + [0] p(c_14) = [0] p(c_15) = [1] x1 + [1] x2 + [0] p(c_16) = [0] p(c_17) = [4] p(c_18) = [1] x1 + [1] x2 + [0] p(c_19) = [0] p(c_20) = [1] x1 + [0] p(c_21) = [0] Following rules are strictly oriented: mark#(if(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [1] > [1] X1 + [1] X2 + [1] 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)) = [4] >= [4] = c_3(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] = c_9(mark#(Y)) a__if#(true(),X,Y) = [1] X + [1] Y + [0] >= [1] X + [0] = c_10(mark#(X)) mark#(div(X1,X2)) = [1] X1 + [4] >= [1] X1 + [4] = c_15(a__div#(mark(X1),X2),mark#(X1)) mark#(s(X)) = [1] X + [0] >= [1] X + [0] = c_20(mark#(X)) ****** Step 1.b:4.b:2.b:3.b:2.a:2: Assumption WORST_CASE(?,O(1)) + 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#(s(X)) -> c_20(mark#(X)) - Weak DPs: 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_17/0,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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ****** Step 1.b:4.b:2.b:3.b:2.b:1: PredecessorEstimationCP 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#(s(X)) -> c_20(mark#(X)) - Weak DPs: 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_17/0,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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 3: a__if#(true(),X,Y) -> c_10(mark#(X)) The strictly oriented rules are moved into the weak component. ******* Step 1.b:4.b:2.b:3.b:2.b:1.a:1: NaturalMI 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#(s(X)) -> c_20(mark#(X)) - Weak DPs: 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_17/0,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 = 2, 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: 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,a__div#,a__geq#,a__if#,a__minus#,mark#} TcT has computed the following interpretation: p(0) = [0] [0] p(a__div) = [1 2] x1 + [0] [0 1] [0] p(a__geq) = [0] [2] p(a__if) = [1 1] x1 + [2 0] x2 + [2 0] x3 + [0] [0 0] [0 1] [0 1] [0] p(a__minus) = [0] [0] p(div) = [1 2] x1 + [0] [0 1] [0] p(false) = [0] [0] p(geq) = [0] [2] p(if) = [1 1] x1 + [1 0] x2 + [1 0] x3 + [0] [0 0] [0 1] [0 1] [0] p(mark) = [2 0] x1 + [0] [0 1] [0] p(minus) = [0] [0] p(s) = [1 0] x1 + [0] [0 0] [2] p(true) = [0] [2] p(a__div#) = [0 1] x1 + [0] [0 0] [0] p(a__geq#) = [0 0] x2 + [0] [1 1] [2] p(a__if#) = [0 1] x1 + [1 0] x2 + [1 0] x3 + [0] [1 0] [0 0] [0 1] [0] p(a__minus#) = [1 1] x1 + [1 0] x2 + [0] [0 0] [1 0] [2] p(mark#) = [1 0] x1 + [0] [0 0] [0] p(c_1) = [0] [0] p(c_2) = [2] [2] p(c_3) = [1 0] x1 + [0] [0 0] [0] p(c_4) = [0] [0] p(c_5) = [0] [0] p(c_6) = [0] [0] p(c_7) = [0] [0] p(c_8) = [2] [0] p(c_9) = [1 0] x1 + [0] [0 1] [0] p(c_10) = [1 1] x1 + [0] [0 0] [0] p(c_11) = [1] [0] p(c_12) = [0] [0] p(c_13) = [0] [0] p(c_14) = [0] [1] p(c_15) = [2 0] x1 + [1 2] x2 + [0] [0 0] [0 0] [0] p(c_16) = [0] [1] p(c_17) = [1] [0] p(c_18) = [1 0] x1 + [1 2] x2 + [0] [0 0] [0 0] [0] p(c_19) = [1] [2] p(c_20) = [1 0] x1 + [0] [0 0] [0] p(c_21) = [0] [2] Following rules are strictly oriented: a__if#(true(),X,Y) = [1 0] X + [1 0] Y + [2] [0 0] [0 1] [0] > [1 0] X + [0] [0 0] [0] = c_10(mark#(X)) Following rules are (at-least) weakly oriented: a__div#(s(X),s(Y)) = [2] [0] >= [2] [0] = c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())) a__if#(false(),X,Y) = [1 0] X + [1 0] Y + [0] [0 0] [0 1] [0] >= [1 0] Y + [0] [0 0] [0] = c_9(mark#(Y)) mark#(div(X1,X2)) = [1 2] X1 + [0] [0 0] [0] >= [1 2] X1 + [0] [0 0] [0] = c_15(a__div#(mark(X1),X2),mark#(X1)) mark#(if(X1,X2,X3)) = [1 1] X1 + [1 0] X2 + [1 0] X3 + [0] [0 0] [0 0] [0 0] [0] >= [1 1] X1 + [1 0] X2 + [1 0] X3 + [0] [0 0] [0 0] [0 0] [0] = c_18(a__if#(mark(X1),X2,X3),mark#(X1)) mark#(s(X)) = [1 0] X + [0] [0 0] [0] >= [1 0] X + [0] [0 0] [0] = c_20(mark#(X)) a__div(X1,X2) = [1 2] X1 + [0] [0 1] [0] >= [1 2] X1 + [0] [0 1] [0] = div(X1,X2) a__div(0(),s(Y)) = [0] [0] >= [0] [0] = 0() a__div(s(X),s(Y)) = [1 0] X + [4] [0 0] [2] >= [2] [2] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) = [0] [2] >= [0] [2] = true() a__geq(X1,X2) = [0] [2] >= [0] [2] = geq(X1,X2) a__geq(0(),s(Y)) = [0] [2] >= [0] [0] = false() a__geq(s(X),s(Y)) = [0] [2] >= [0] [2] = a__geq(X,Y) a__if(X1,X2,X3) = [1 1] X1 + [2 0] X2 + [2 0] X3 + [0] [0 0] [0 1] [0 1] [0] >= [1 1] X1 + [1 0] X2 + [1 0] X3 + [0] [0 0] [0 1] [0 1] [0] = if(X1,X2,X3) a__if(false(),X,Y) = [2 0] X + [2 0] Y + [0] [0 1] [0 1] [0] >= [2 0] Y + [0] [0 1] [0] = mark(Y) a__if(true(),X,Y) = [2 0] X + [2 0] Y + [2] [0 1] [0 1] [0] >= [2 0] X + [0] [0 1] [0] = mark(X) a__minus(X1,X2) = [0] [0] >= [0] [0] = minus(X1,X2) a__minus(0(),Y) = [0] [0] >= [0] [0] = 0() a__minus(s(X),s(Y)) = [0] [0] >= [0] [0] = a__minus(X,Y) mark(0()) = [0] [0] >= [0] [0] = 0() mark(div(X1,X2)) = [2 4] X1 + [0] [0 1] [0] >= [2 2] X1 + [0] [0 1] [0] = a__div(mark(X1),X2) mark(false()) = [0] [0] >= [0] [0] = false() mark(geq(X1,X2)) = [0] [2] >= [0] [2] = a__geq(X1,X2) mark(if(X1,X2,X3)) = [2 2] X1 + [2 0] X2 + [2 0] X3 + [0] [0 0] [0 1] [0 1] [0] >= [2 1] X1 + [2 0] X2 + [2 0] X3 + [0] [0 0] [0 1] [0 1] [0] = a__if(mark(X1),X2,X3) mark(minus(X1,X2)) = [0] [0] >= [0] [0] = a__minus(X1,X2) mark(s(X)) = [2 0] X + [0] [0 0] [2] >= [2 0] X + [0] [0 0] [2] = s(mark(X)) mark(true()) = [0] [2] >= [0] [2] = true() ******* Step 1.b:4.b:2.b:3.b:2.b:1.a:2: Assumption WORST_CASE(?,O(1)) + 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)) mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)) mark#(s(X)) -> c_20(mark#(X)) - Weak DPs: 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_17/0,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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ******* Step 1.b:4.b:2.b:3.b:2.b:1.b:1: PredecessorEstimationCP 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)) mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)) mark#(s(X)) -> c_20(mark#(X)) - Weak DPs: 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_17/0,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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: a__div#(s(X),s(Y)) -> c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())) The strictly oriented rules are moved into the weak component. ******** Step 1.b:4.b:2.b:3.b:2.b:1.b:1.a:1: NaturalMI 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)) mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)) mark#(s(X)) -> c_20(mark#(X)) - Weak DPs: 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_17/0,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 = 2, 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: 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,a__div#,a__geq#,a__if#,a__minus#,mark#} TcT has computed the following interpretation: p(0) = [0] [0] p(a__div) = [1 3] x1 + [0 0] x2 + [0] [0 0] [0 2] [2] p(a__geq) = [0] [1] p(a__if) = [1 2] x1 + [2 0] x2 + [2 0] x3 + [0] [0 0] [0 2] [0 2] [1] p(a__minus) = [0] [0] p(div) = [1 3] x1 + [0 0] x2 + [0] [0 0] [0 1] [2] p(false) = [0] [0] p(geq) = [0] [1] p(if) = [1 2] x1 + [1 0] x2 + [1 0] x3 + [0] [0 0] [0 1] [0 1] [1] p(mark) = [2 0] x1 + [0] [0 2] [0] p(minus) = [0] [0] p(s) = [1 0] x1 + [0] [0 0] [1] p(true) = [0] [0] p(a__div#) = [0 3] x1 + [0] [0 0] [1] p(a__geq#) = [0 1] x2 + [0] [0 1] [0] p(a__if#) = [0 2] x1 + [2 0] x2 + [2 0] x3 + [0] [0 1] [0 0] [0 0] [2] p(a__minus#) = [0 2] x1 + [0 0] x2 + [0] [0 2] [0 1] [0] p(mark#) = [2 0] x1 + [0] [0 1] [0] p(c_1) = [0] [0] p(c_2) = [0] [1] p(c_3) = [1 0] x1 + [0] [0 0] [1] p(c_4) = [1] [0] p(c_5) = [0] [0] p(c_6) = [0] [2] p(c_7) = [1 2] x1 + [2] [0 0] [0] p(c_8) = [2] [2] p(c_9) = [1 0] x1 + [0] [0 0] [0] p(c_10) = [1 0] x1 + [0] [0 0] [2] p(c_11) = [1] [0] p(c_12) = [1] [2] p(c_13) = [0] [0] p(c_14) = [0] [0] p(c_15) = [1 0] x1 + [1 0] x2 + [0] [0 1] [0 0] [1] p(c_16) = [0] [0] p(c_17) = [0] [0] p(c_18) = [1 0] x1 + [1 0] x2 + [0] [0 0] [0 0] [1] p(c_19) = [0 0] x1 + [1] [2 1] [2] p(c_20) = [1 0] x1 + [0] [0 0] [0] p(c_21) = [0] [1] Following rules are strictly oriented: a__div#(s(X),s(Y)) = [3] [1] > [2] [1] = 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) = [2 0] X + [2 0] Y + [0] [0 0] [0 0] [2] >= [2 0] Y + [0] [0 0] [0] = c_9(mark#(Y)) a__if#(true(),X,Y) = [2 0] X + [2 0] Y + [0] [0 0] [0 0] [2] >= [2 0] X + [0] [0 0] [2] = c_10(mark#(X)) mark#(div(X1,X2)) = [2 6] X1 + [0 0] X2 + [0] [0 0] [0 1] [2] >= [2 6] X1 + [0] [0 0] [2] = c_15(a__div#(mark(X1),X2),mark#(X1)) mark#(if(X1,X2,X3)) = [2 4] X1 + [2 0] X2 + [2 0] X3 + [0] [0 0] [0 1] [0 1] [1] >= [2 4] X1 + [2 0] X2 + [2 0] X3 + [0] [0 0] [0 0] [0 0] [1] = c_18(a__if#(mark(X1),X2,X3),mark#(X1)) mark#(s(X)) = [2 0] X + [0] [0 0] [1] >= [2 0] X + [0] [0 0] [0] = c_20(mark#(X)) a__div(X1,X2) = [1 3] X1 + [0 0] X2 + [0] [0 0] [0 2] [2] >= [1 3] X1 + [0 0] X2 + [0] [0 0] [0 1] [2] = div(X1,X2) a__div(0(),s(Y)) = [0] [4] >= [0] [0] = 0() a__div(s(X),s(Y)) = [1 0] X + [3] [0 0] [4] >= [2] [3] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) = [0] [1] >= [0] [0] = true() a__geq(X1,X2) = [0] [1] >= [0] [1] = geq(X1,X2) a__geq(0(),s(Y)) = [0] [1] >= [0] [0] = false() a__geq(s(X),s(Y)) = [0] [1] >= [0] [1] = a__geq(X,Y) a__if(X1,X2,X3) = [1 2] X1 + [2 0] X2 + [2 0] X3 + [0] [0 0] [0 2] [0 2] [1] >= [1 2] X1 + [1 0] X2 + [1 0] X3 + [0] [0 0] [0 1] [0 1] [1] = if(X1,X2,X3) a__if(false(),X,Y) = [2 0] X + [2 0] Y + [0] [0 2] [0 2] [1] >= [2 0] Y + [0] [0 2] [0] = mark(Y) a__if(true(),X,Y) = [2 0] X + [2 0] Y + [0] [0 2] [0 2] [1] >= [2 0] X + [0] [0 2] [0] = mark(X) a__minus(X1,X2) = [0] [0] >= [0] [0] = minus(X1,X2) a__minus(0(),Y) = [0] [0] >= [0] [0] = 0() a__minus(s(X),s(Y)) = [0] [0] >= [0] [0] = a__minus(X,Y) mark(0()) = [0] [0] >= [0] [0] = 0() mark(div(X1,X2)) = [2 6] X1 + [0 0] X2 + [0] [0 0] [0 2] [4] >= [2 6] X1 + [0 0] X2 + [0] [0 0] [0 2] [2] = a__div(mark(X1),X2) mark(false()) = [0] [0] >= [0] [0] = false() mark(geq(X1,X2)) = [0] [2] >= [0] [1] = a__geq(X1,X2) mark(if(X1,X2,X3)) = [2 4] X1 + [2 0] X2 + [2 0] X3 + [0] [0 0] [0 2] [0 2] [2] >= [2 4] X1 + [2 0] X2 + [2 0] X3 + [0] [0 0] [0 2] [0 2] [1] = a__if(mark(X1),X2,X3) mark(minus(X1,X2)) = [0] [0] >= [0] [0] = a__minus(X1,X2) mark(s(X)) = [2 0] X + [0] [0 0] [2] >= [2 0] X + [0] [0 0] [1] = s(mark(X)) mark(true()) = [0] [0] >= [0] [0] = true() ******** Step 1.b:4.b:2.b:3.b:2.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: a__if#(false(),X,Y) -> c_9(mark#(Y)) 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#(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_17/0,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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ******** Step 1.b:4.b:2.b:3.b:2.b:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: a__if#(false(),X,Y) -> c_9(mark#(Y)) 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#(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_17/0,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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: a__if#(false(),X,Y) -> c_9(mark#(Y)) The strictly oriented rules are moved into the weak component. ********* Step 1.b:4.b:2.b:3.b:2.b:1.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: a__if#(false(),X,Y) -> c_9(mark#(Y)) 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#(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_17/0,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 = 2, 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: 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,a__div#,a__geq#,a__if#,a__minus#,mark#} TcT has computed the following interpretation: p(0) = [0] [0] p(a__div) = [1 2] x1 + [2] [0 1] [1] p(a__geq) = [0] [0] p(a__if) = [1 1] x1 + [2 0] x2 + [2 0] x3 + [2] [0 0] [0 1] [0 1] [1] p(a__minus) = [0] [1] p(div) = [1 2] x1 + [2] [0 1] [1] p(false) = [0] [0] p(geq) = [0] [0] p(if) = [1 1] x1 + [1 0] x2 + [1 0] x3 + [2] [0 0] [0 1] [0 1] [0] p(mark) = [2 0] x1 + [0] [0 1] [1] p(minus) = [0] [0] p(s) = [1 0] x1 + [0] [0 0] [2] p(true) = [0] [0] p(a__div#) = [0 3] x1 + [0 0] x2 + [0] [0 0] [0 1] [1] p(a__geq#) = [0] [0] p(a__if#) = [0 2] x1 + [2 0] x2 + [2 0] x3 + [2] [0 0] [0 0] [0 0] [0] p(a__minus#) = [2] [0] p(mark#) = [2 0] x1 + [0] [0 1] [0] p(c_1) = [1] [2] p(c_2) = [0] [1] p(c_3) = [1 0] x1 + [0] [0 1] [0] p(c_4) = [0] [0] p(c_5) = [0] [0] p(c_6) = [0] [0] p(c_7) = [2 2] x1 + [0] [0 0] [2] p(c_8) = [0] [1] p(c_9) = [1 0] x1 + [0] [0 0] [0] p(c_10) = [1 0] x1 + [2] [0 0] [0] p(c_11) = [0] [0] p(c_12) = [1] [0] p(c_13) = [1] [0] p(c_14) = [0] [0] p(c_15) = [1 0] x1 + [1 1] x2 + [1] [0 0] [0 0] [1] p(c_16) = [1] [1] p(c_17) = [0] [0] p(c_18) = [1 0] x1 + [1 0] x2 + [0] [0 0] [0 0] [0] p(c_19) = [0] [2] p(c_20) = [1 0] x1 + [0] [0 0] [2] p(c_21) = [0] [0] Following rules are strictly oriented: a__if#(false(),X,Y) = [2 0] X + [2 0] Y + [2] [0 0] [0 0] [0] > [2 0] Y + [0] [0 0] [0] = c_9(mark#(Y)) Following rules are (at-least) weakly oriented: a__div#(s(X),s(Y)) = [6] [3] >= [6] [0] = c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())) a__if#(true(),X,Y) = [2 0] X + [2 0] Y + [2] [0 0] [0 0] [0] >= [2 0] X + [2] [0 0] [0] = c_10(mark#(X)) mark#(div(X1,X2)) = [2 4] X1 + [4] [0 1] [1] >= [2 4] X1 + [4] [0 0] [1] = c_15(a__div#(mark(X1),X2),mark#(X1)) mark#(if(X1,X2,X3)) = [2 2] X1 + [2 0] X2 + [2 0] X3 + [4] [0 0] [0 1] [0 1] [0] >= [2 2] X1 + [2 0] X2 + [2 0] X3 + [4] [0 0] [0 0] [0 0] [0] = c_18(a__if#(mark(X1),X2,X3),mark#(X1)) mark#(s(X)) = [2 0] X + [0] [0 0] [2] >= [2 0] X + [0] [0 0] [2] = c_20(mark#(X)) a__div(X1,X2) = [1 2] X1 + [2] [0 1] [1] >= [1 2] X1 + [2] [0 1] [1] = div(X1,X2) a__div(0(),s(Y)) = [2] [1] >= [0] [0] = 0() a__div(s(X),s(Y)) = [1 0] X + [6] [0 0] [3] >= [6] [3] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) = [0] [0] >= [0] [0] = true() a__geq(X1,X2) = [0] [0] >= [0] [0] = geq(X1,X2) a__geq(0(),s(Y)) = [0] [0] >= [0] [0] = false() a__geq(s(X),s(Y)) = [0] [0] >= [0] [0] = a__geq(X,Y) a__if(X1,X2,X3) = [1 1] X1 + [2 0] X2 + [2 0] X3 + [2] [0 0] [0 1] [0 1] [1] >= [1 1] X1 + [1 0] X2 + [1 0] X3 + [2] [0 0] [0 1] [0 1] [0] = if(X1,X2,X3) a__if(false(),X,Y) = [2 0] X + [2 0] Y + [2] [0 1] [0 1] [1] >= [2 0] Y + [0] [0 1] [1] = mark(Y) a__if(true(),X,Y) = [2 0] X + [2 0] Y + [2] [0 1] [0 1] [1] >= [2 0] X + [0] [0 1] [1] = mark(X) a__minus(X1,X2) = [0] [1] >= [0] [0] = minus(X1,X2) a__minus(0(),Y) = [0] [1] >= [0] [0] = 0() a__minus(s(X),s(Y)) = [0] [1] >= [0] [1] = a__minus(X,Y) mark(0()) = [0] [1] >= [0] [0] = 0() mark(div(X1,X2)) = [2 4] X1 + [4] [0 1] [2] >= [2 2] X1 + [4] [0 1] [2] = a__div(mark(X1),X2) mark(false()) = [0] [1] >= [0] [0] = false() mark(geq(X1,X2)) = [0] [1] >= [0] [0] = a__geq(X1,X2) mark(if(X1,X2,X3)) = [2 2] X1 + [2 0] X2 + [2 0] X3 + [4] [0 0] [0 1] [0 1] [1] >= [2 1] X1 + [2 0] X2 + [2 0] X3 + [3] [0 0] [0 1] [0 1] [1] = a__if(mark(X1),X2,X3) mark(minus(X1,X2)) = [0] [1] >= [0] [1] = a__minus(X1,X2) mark(s(X)) = [2 0] X + [0] [0 0] [3] >= [2 0] X + [0] [0 0] [2] = s(mark(X)) mark(true()) = [0] [1] >= [0] [0] = true() ********* Step 1.b:4.b:2.b:3.b:2.b:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + 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_17/0,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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ********* Step 1.b:4.b:2.b:3.b:2.b:1.b:1.b:1.b:1: PredecessorEstimationCP 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_17/0,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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 2: mark#(s(X)) -> c_20(mark#(X)) The strictly oriented rules are moved into the weak component. ********** Step 1.b:4.b:2.b:3.b:2.b:1.b:1.b:1.b:1.a:1: NaturalMI 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_17/0,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 = 2, 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: 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,a__div#,a__geq#,a__if#,a__minus#,mark#} TcT has computed the following interpretation: p(0) = [0] [0] p(a__div) = [1 2] x1 + [2 0] x2 + [0] [0 0] [0 1] [2] p(a__geq) = [1 0] x1 + [0] [0 0] [0] p(a__if) = [1 2] x1 + [2 0] x2 + [2 0] x3 + [1] [0 1] [0 1] [0 1] [2] p(a__minus) = [1] [0] p(div) = [1 1] x1 + [1 0] x2 + [0] [0 0] [0 1] [2] p(false) = [0] [0] p(geq) = [1 0] x1 + [0] [0 0] [0] p(if) = [1 1] x1 + [1 0] x2 + [1 0] x3 + [1] [0 1] [0 1] [0 1] [2] p(mark) = [2 0] x1 + [0] [0 1] [0] p(minus) = [1] [0] p(s) = [1 0] x1 + [1] [0 0] [2] p(true) = [0] [0] p(a__div#) = [0 2] x1 + [2 0] x2 + [0] [0 0] [2 0] [0] p(a__geq#) = [0 2] x2 + [1] [2 1] [0] p(a__if#) = [2 0] x2 + [2 0] x3 + [0] [0 0] [0 0] [1] p(a__minus#) = [0] [2] p(mark#) = [2 0] x1 + [0] [0 0] [0] p(c_1) = [2] [1] p(c_2) = [0] [0] p(c_3) = [1 0] x1 + [0] [0 2] [0] p(c_4) = [0] [0] p(c_5) = [0] [0] p(c_6) = [0] [0] p(c_7) = [1 2] x1 + [0] [0 1] [2] p(c_8) = [2] [0] p(c_9) = [1 0] x1 + [0] [0 0] [0] p(c_10) = [1 2] x1 + [0] [0 0] [0] p(c_11) = [0] [0] p(c_12) = [0] [0] p(c_13) = [2 1] x1 + [0] [0 0] [0] p(c_14) = [0] [1] p(c_15) = [1 0] x1 + [1 0] x2 + [0] [0 0] [0 1] [0] p(c_16) = [1] [1] p(c_17) = [0] [0] p(c_18) = [1 1] x1 + [1 0] x2 + [1] [0 0] [0 1] [0] p(c_19) = [1 0] x1 + [2] [2 0] [0] p(c_20) = [1 0] x1 + [0] [0 1] [0] p(c_21) = [0] [1] Following rules are strictly oriented: mark#(s(X)) = [2 0] X + [2] [0 0] [0] > [2 0] X + [0] [0 0] [0] = c_20(mark#(X)) Following rules are (at-least) weakly oriented: a__div#(s(X),s(Y)) = [2 0] Y + [6] [2 0] [2] >= [2 0] Y + [6] [0 0] [2] = c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())) a__if#(false(),X,Y) = [2 0] X + [2 0] Y + [0] [0 0] [0 0] [1] >= [2 0] Y + [0] [0 0] [0] = c_9(mark#(Y)) a__if#(true(),X,Y) = [2 0] X + [2 0] Y + [0] [0 0] [0 0] [1] >= [2 0] X + [0] [0 0] [0] = c_10(mark#(X)) mark#(div(X1,X2)) = [2 2] X1 + [2 0] X2 + [0] [0 0] [0 0] [0] >= [2 2] X1 + [2 0] X2 + [0] [0 0] [0 0] [0] = c_15(a__div#(mark(X1),X2),mark#(X1)) mark#(if(X1,X2,X3)) = [2 2] X1 + [2 0] X2 + [2 0] X3 + [2] [0 0] [0 0] [0 0] [0] >= [2 0] X1 + [2 0] X2 + [2 0] X3 + [2] [0 0] [0 0] [0 0] [0] = c_18(a__if#(mark(X1),X2,X3),mark#(X1)) a__div(X1,X2) = [1 2] X1 + [2 0] X2 + [0] [0 0] [0 1] [2] >= [1 1] X1 + [1 0] X2 + [0] [0 0] [0 1] [2] = div(X1,X2) a__div(0(),s(Y)) = [2 0] Y + [2] [0 0] [4] >= [0] [0] = 0() a__div(s(X),s(Y)) = [1 0] X + [2 0] Y + [7] [0 0] [0 0] [4] >= [1 0] X + [2 0] Y + [7] [0 0] [0 0] [4] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) = [1 0] X + [0] [0 0] [0] >= [0] [0] = true() a__geq(X1,X2) = [1 0] X1 + [0] [0 0] [0] >= [1 0] X1 + [0] [0 0] [0] = geq(X1,X2) a__geq(0(),s(Y)) = [0] [0] >= [0] [0] = false() a__geq(s(X),s(Y)) = [1 0] X + [1] [0 0] [0] >= [1 0] X + [0] [0 0] [0] = a__geq(X,Y) a__if(X1,X2,X3) = [1 2] X1 + [2 0] X2 + [2 0] X3 + [1] [0 1] [0 1] [0 1] [2] >= [1 1] X1 + [1 0] X2 + [1 0] X3 + [1] [0 1] [0 1] [0 1] [2] = if(X1,X2,X3) a__if(false(),X,Y) = [2 0] X + [2 0] Y + [1] [0 1] [0 1] [2] >= [2 0] Y + [0] [0 1] [0] = mark(Y) a__if(true(),X,Y) = [2 0] X + [2 0] Y + [1] [0 1] [0 1] [2] >= [2 0] X + [0] [0 1] [0] = mark(X) a__minus(X1,X2) = [1] [0] >= [1] [0] = minus(X1,X2) a__minus(0(),Y) = [1] [0] >= [0] [0] = 0() a__minus(s(X),s(Y)) = [1] [0] >= [1] [0] = a__minus(X,Y) mark(0()) = [0] [0] >= [0] [0] = 0() mark(div(X1,X2)) = [2 2] X1 + [2 0] X2 + [0] [0 0] [0 1] [2] >= [2 2] X1 + [2 0] X2 + [0] [0 0] [0 1] [2] = a__div(mark(X1),X2) mark(false()) = [0] [0] >= [0] [0] = false() mark(geq(X1,X2)) = [2 0] X1 + [0] [0 0] [0] >= [1 0] X1 + [0] [0 0] [0] = a__geq(X1,X2) mark(if(X1,X2,X3)) = [2 2] X1 + [2 0] X2 + [2 0] X3 + [2] [0 1] [0 1] [0 1] [2] >= [2 2] X1 + [2 0] X2 + [2 0] X3 + [1] [0 1] [0 1] [0 1] [2] = a__if(mark(X1),X2,X3) mark(minus(X1,X2)) = [2] [0] >= [1] [0] = a__minus(X1,X2) mark(s(X)) = [2 0] X + [2] [0 0] [2] >= [2 0] X + [1] [0 0] [2] = s(mark(X)) mark(true()) = [0] [0] >= [0] [0] = true() ********** Step 1.b:4.b:2.b:3.b:2.b:1.b:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + 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_17/0,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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ********** Step 1.b:4.b:2.b:3.b:2.b:1.b:1.b:1.b:1.b:1: PredecessorEstimationCP 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_17/0,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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)) Consider the set of all dependency pairs 1: mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)) 2: a__div#(s(X),s(Y)) -> c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())) 3: a__if#(false(),X,Y) -> c_9(mark#(Y)) 4: a__if#(true(),X,Y) -> c_10(mark#(X)) 5: mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)) 6: mark#(s(X)) -> c_20(mark#(X)) Processor NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^2)) 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:4.b:2.b:3.b:2.b:1.b:1.b:1.b:1.b:1.a:1: NaturalMI 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_17/0,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 = 2, 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: 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,a__div#,a__geq#,a__if#,a__minus#,mark#} TcT has computed the following interpretation: p(0) = [0] [0] p(a__div) = [1 3] x1 + [0] [0 1] [1] p(a__geq) = [0] [0] p(a__if) = [1 3] x1 + [2 0] x2 + [2 0] x3 + [0] [0 1] [0 1] [0 1] [1] p(a__minus) = [0] [0] p(div) = [1 3] x1 + [0] [0 1] [1] p(false) = [0] [0] p(geq) = [0] [0] p(if) = [1 3] x1 + [1 0] x2 + [1 0] x3 + [0] [0 1] [0 1] [0 1] [1] p(mark) = [2 0] x1 + [0] [0 1] [0] p(minus) = [0] [0] p(s) = [1 2] x1 + [0] [0 0] [2] p(true) = [0] [0] p(a__div#) = [0 3] x1 + [0 0] x2 + [0] [0 0] [0 2] [2] p(a__geq#) = [0 0] x1 + [0 2] x2 + [0] [2 0] [2 2] [0] p(a__if#) = [0 0] x1 + [1 2] x2 + [1 2] x3 + [0] [0 2] [0 0] [0 0] [1] p(a__minus#) = [0] [2] p(mark#) = [1 2] x1 + [0] [0 1] [1] p(c_1) = [1] [0] p(c_2) = [2] [0] p(c_3) = [1 0] x1 + [0] [0 0] [0] p(c_4) = [1] [2] p(c_5) = [2] [0] p(c_6) = [2] [2] p(c_7) = [2] [0] p(c_8) = [0] [0] p(c_9) = [1 0] x1 + [0] [0 0] [0] p(c_10) = [1 0] x1 + [0] [0 0] [0] p(c_11) = [2] [0] p(c_12) = [0] [1] p(c_13) = [1 1] x1 + [2] [0 1] [0] p(c_14) = [0] [0] p(c_15) = [1 0] x1 + [1 0] x2 + [0] [0 0] [0 1] [1] p(c_16) = [0] [1] p(c_17) = [2] [2] p(c_18) = [1 1] x1 + [1 1] x2 + [0] [0 0] [0 1] [1] p(c_19) = [0 1] x1 + [0] [1 0] [0] p(c_20) = [1 0] x1 + [2] [0 0] [0] p(c_21) = [1] [1] Following rules are strictly oriented: mark#(div(X1,X2)) = [1 5] X1 + [2] [0 1] [2] > [1 5] X1 + [0] [0 1] [2] = c_15(a__div#(mark(X1),X2),mark#(X1)) Following rules are (at-least) weakly oriented: a__div#(s(X),s(Y)) = [6] [6] >= [6] [0] = c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())) a__if#(false(),X,Y) = [1 2] X + [1 2] Y + [0] [0 0] [0 0] [1] >= [1 2] Y + [0] [0 0] [0] = c_9(mark#(Y)) a__if#(true(),X,Y) = [1 2] X + [1 2] Y + [0] [0 0] [0 0] [1] >= [1 2] X + [0] [0 0] [0] = c_10(mark#(X)) mark#(if(X1,X2,X3)) = [1 5] X1 + [1 2] X2 + [1 2] X3 + [2] [0 1] [0 1] [0 1] [2] >= [1 5] X1 + [1 2] X2 + [1 2] X3 + [2] [0 1] [0 0] [0 0] [2] = c_18(a__if#(mark(X1),X2,X3),mark#(X1)) mark#(s(X)) = [1 2] X + [4] [0 0] [3] >= [1 2] X + [2] [0 0] [0] = c_20(mark#(X)) a__div(X1,X2) = [1 3] X1 + [0] [0 1] [1] >= [1 3] X1 + [0] [0 1] [1] = div(X1,X2) a__div(0(),s(Y)) = [0] [1] >= [0] [0] = 0() a__div(s(X),s(Y)) = [1 2] X + [6] [0 0] [3] >= [4] [3] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) = [0] [0] >= [0] [0] = true() a__geq(X1,X2) = [0] [0] >= [0] [0] = geq(X1,X2) a__geq(0(),s(Y)) = [0] [0] >= [0] [0] = false() a__geq(s(X),s(Y)) = [0] [0] >= [0] [0] = a__geq(X,Y) a__if(X1,X2,X3) = [1 3] X1 + [2 0] X2 + [2 0] X3 + [0] [0 1] [0 1] [0 1] [1] >= [1 3] X1 + [1 0] X2 + [1 0] X3 + [0] [0 1] [0 1] [0 1] [1] = if(X1,X2,X3) a__if(false(),X,Y) = [2 0] X + [2 0] Y + [0] [0 1] [0 1] [1] >= [2 0] Y + [0] [0 1] [0] = mark(Y) a__if(true(),X,Y) = [2 0] X + [2 0] Y + [0] [0 1] [0 1] [1] >= [2 0] X + [0] [0 1] [0] = mark(X) a__minus(X1,X2) = [0] [0] >= [0] [0] = minus(X1,X2) a__minus(0(),Y) = [0] [0] >= [0] [0] = 0() a__minus(s(X),s(Y)) = [0] [0] >= [0] [0] = a__minus(X,Y) mark(0()) = [0] [0] >= [0] [0] = 0() mark(div(X1,X2)) = [2 6] X1 + [0] [0 1] [1] >= [2 3] X1 + [0] [0 1] [1] = a__div(mark(X1),X2) mark(false()) = [0] [0] >= [0] [0] = false() mark(geq(X1,X2)) = [0] [0] >= [0] [0] = a__geq(X1,X2) mark(if(X1,X2,X3)) = [2 6] X1 + [2 0] X2 + [2 0] X3 + [0] [0 1] [0 1] [0 1] [1] >= [2 3] X1 + [2 0] X2 + [2 0] X3 + [0] [0 1] [0 1] [0 1] [1] = a__if(mark(X1),X2,X3) mark(minus(X1,X2)) = [0] [0] >= [0] [0] = a__minus(X1,X2) mark(s(X)) = [2 4] X + [0] [0 0] [2] >= [2 2] X + [0] [0 0] [2] = s(mark(X)) mark(true()) = [0] [0] >= [0] [0] = true() *********** Step 1.b:4.b:2.b:3.b:2.b:1.b:1.b:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - 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#(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_17/0,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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *********** Step 1.b:4.b:2.b:3.b:2.b:1.b:1.b:1.b:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - 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#(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_17/0,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:W:a__div#(s(X),s(Y)) -> c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())) -->_1 a__if#(true(),X,Y) -> c_10(mark#(X)):3 -->_1 a__if#(false(),X,Y) -> c_9(mark#(Y)):2 2:W: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:W: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:W: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())):1 5:W: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:W: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 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: a__div#(s(X),s(Y)) -> c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())) 4: mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)) 6: mark#(s(X)) -> c_20(mark#(X)) 5: mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)) 3: a__if#(true(),X,Y) -> c_10(mark#(X)) 2: a__if#(false(),X,Y) -> c_9(mark#(Y)) *********** Step 1.b:4.b:2.b:3.b:2.b:1.b:1.b:1.b:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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_17/0,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(Omega(n^1),O(n^2))