/export/starexec/sandbox2/solver/bin/starexec_run_tct_rci /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1),O(n^3)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^3)) + Considered Problem: - Strict TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0 ,div,false,geq,if,minus,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. 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: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: 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^3)) + Considered Problem: - Strict TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1} / {0/0,div/2,false/0,geq/2,if/3,minus/2,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div,a__geq,a__if,a__minus,mark} and constructors {0 ,div,false,geq,if,minus,s,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs a__div#(X1,X2) -> c_1() a__div#(0(),s(Y)) -> c_2() a__div#(s(X),s(Y)) -> c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()),a__geq#(X,Y)) a__geq#(X,0()) -> c_4() a__geq#(X1,X2) -> c_5() a__geq#(0(),s(Y)) -> c_6() a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)) a__if#(X1,X2,X3) -> c_8() a__if#(false(),X,Y) -> c_9(mark#(Y)) a__if#(true(),X,Y) -> c_10(mark#(X)) a__minus#(X1,X2) -> c_11() a__minus#(0(),Y) -> c_12() a__minus#(s(X),s(Y)) -> c_13(a__minus#(X,Y)) mark#(0()) -> c_14() mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)) mark#(false()) -> c_16() mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)) mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)) mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)) mark#(s(X)) -> c_20(mark#(X)) mark#(true()) -> c_21() Weak DPs and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: a__div#(X1,X2) -> c_1() a__div#(0(),s(Y)) -> c_2() a__div#(s(X),s(Y)) -> c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()),a__geq#(X,Y)) a__geq#(X,0()) -> c_4() a__geq#(X1,X2) -> c_5() a__geq#(0(),s(Y)) -> c_6() a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)) a__if#(X1,X2,X3) -> c_8() a__if#(false(),X,Y) -> c_9(mark#(Y)) a__if#(true(),X,Y) -> c_10(mark#(X)) a__minus#(X1,X2) -> c_11() a__minus#(0(),Y) -> c_12() a__minus#(s(X),s(Y)) -> c_13(a__minus#(X,Y)) mark#(0()) -> c_14() mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)) mark#(false()) -> c_16() mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)) mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)) mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)) mark#(s(X)) -> c_20(mark#(X)) mark#(true()) -> c_21() - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1,a__div#/2,a__geq#/2,a__if#/3,a__minus#/2,mark#/1} / {0/0,div/2 ,false/0,geq/2,if/3,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0,c_15/2,c_16/0,c_17/1,c_18/2,c_19/1,c_20/1,c_21/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div#,a__geq#,a__if#,a__minus# ,mark#} and constructors {0,div,false,geq,if,minus,s,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,4,5,6,8,11,12,14,16,21} by application of Pre({1,2,4,5,6,8,11,12,14,16,21}) = {3,7,9,10,13,15,17,18,19,20}. Here rules are labelled as follows: 1: a__div#(X1,X2) -> c_1() 2: a__div#(0(),s(Y)) -> c_2() 3: a__div#(s(X),s(Y)) -> c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()),a__geq#(X,Y)) 4: a__geq#(X,0()) -> c_4() 5: a__geq#(X1,X2) -> c_5() 6: a__geq#(0(),s(Y)) -> c_6() 7: a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)) 8: a__if#(X1,X2,X3) -> c_8() 9: a__if#(false(),X,Y) -> c_9(mark#(Y)) 10: a__if#(true(),X,Y) -> c_10(mark#(X)) 11: a__minus#(X1,X2) -> c_11() 12: a__minus#(0(),Y) -> c_12() 13: a__minus#(s(X),s(Y)) -> c_13(a__minus#(X,Y)) 14: mark#(0()) -> c_14() 15: mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)) 16: mark#(false()) -> c_16() 17: mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)) 18: mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)) 19: mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)) 20: mark#(s(X)) -> c_20(mark#(X)) 21: mark#(true()) -> c_21() ** Step 1.b:3: RemoveWeakSuffixes. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: a__div#(s(X),s(Y)) -> c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()),a__geq#(X,Y)) a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)) a__if#(false(),X,Y) -> c_9(mark#(Y)) a__if#(true(),X,Y) -> c_10(mark#(X)) a__minus#(s(X),s(Y)) -> c_13(a__minus#(X,Y)) mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)) mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)) mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)) mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)) mark#(s(X)) -> c_20(mark#(X)) - Weak DPs: a__div#(X1,X2) -> c_1() a__div#(0(),s(Y)) -> c_2() a__geq#(X,0()) -> c_4() a__geq#(X1,X2) -> c_5() a__geq#(0(),s(Y)) -> c_6() a__if#(X1,X2,X3) -> c_8() a__minus#(X1,X2) -> c_11() a__minus#(0(),Y) -> c_12() mark#(0()) -> c_14() mark#(false()) -> c_16() mark#(true()) -> c_21() - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1,a__div#/2,a__geq#/2,a__if#/3,a__minus#/2,mark#/1} / {0/0,div/2 ,false/0,geq/2,if/3,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0,c_15/2,c_16/0,c_17/1,c_18/2,c_19/1,c_20/1,c_21/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div#,a__geq#,a__if#,a__minus# ,mark#} and constructors {0,div,false,geq,if,minus,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:a__div#(s(X),s(Y)) -> c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()),a__geq#(X,Y)) -->_1 a__if#(true(),X,Y) -> c_10(mark#(X)):4 -->_1 a__if#(false(),X,Y) -> c_9(mark#(Y)):3 -->_2 a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)):2 -->_1 a__if#(X1,X2,X3) -> c_8():16 -->_2 a__geq#(0(),s(Y)) -> c_6():15 -->_2 a__geq#(X1,X2) -> c_5():14 -->_2 a__geq#(X,0()) -> c_4():13 2:S:a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)) -->_1 a__geq#(0(),s(Y)) -> c_6():15 -->_1 a__geq#(X1,X2) -> c_5():14 -->_1 a__geq#(X,0()) -> c_4():13 -->_1 a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)):2 3:S:a__if#(false(),X,Y) -> c_9(mark#(Y)) -->_1 mark#(s(X)) -> c_20(mark#(X)):10 -->_1 mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)):9 -->_1 mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)):8 -->_1 mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)):7 -->_1 mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)):6 -->_1 mark#(true()) -> c_21():21 -->_1 mark#(false()) -> c_16():20 -->_1 mark#(0()) -> c_14():19 4:S:a__if#(true(),X,Y) -> c_10(mark#(X)) -->_1 mark#(s(X)) -> c_20(mark#(X)):10 -->_1 mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)):9 -->_1 mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)):8 -->_1 mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)):7 -->_1 mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)):6 -->_1 mark#(true()) -> c_21():21 -->_1 mark#(false()) -> c_16():20 -->_1 mark#(0()) -> c_14():19 5:S:a__minus#(s(X),s(Y)) -> c_13(a__minus#(X,Y)) -->_1 a__minus#(0(),Y) -> c_12():18 -->_1 a__minus#(X1,X2) -> c_11():17 -->_1 a__minus#(s(X),s(Y)) -> c_13(a__minus#(X,Y)):5 6:S:mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)) -->_2 mark#(s(X)) -> c_20(mark#(X)):10 -->_2 mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)):9 -->_2 mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)):8 -->_2 mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)):7 -->_2 mark#(true()) -> c_21():21 -->_2 mark#(false()) -> c_16():20 -->_2 mark#(0()) -> c_14():19 -->_1 a__div#(0(),s(Y)) -> c_2():12 -->_1 a__div#(X1,X2) -> c_1():11 -->_2 mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)):6 -->_1 a__div#(s(X),s(Y)) -> c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()),a__geq#(X,Y)):1 7:S:mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)) -->_1 a__geq#(0(),s(Y)) -> c_6():15 -->_1 a__geq#(X1,X2) -> c_5():14 -->_1 a__geq#(X,0()) -> c_4():13 -->_1 a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)):2 8:S:mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)) -->_2 mark#(s(X)) -> c_20(mark#(X)):10 -->_2 mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)):9 -->_2 mark#(true()) -> c_21():21 -->_2 mark#(false()) -> c_16():20 -->_2 mark#(0()) -> c_14():19 -->_1 a__if#(X1,X2,X3) -> c_8():16 -->_2 mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)):8 -->_2 mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)):7 -->_2 mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)):6 -->_1 a__if#(true(),X,Y) -> c_10(mark#(X)):4 -->_1 a__if#(false(),X,Y) -> c_9(mark#(Y)):3 9:S:mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)) -->_1 a__minus#(0(),Y) -> c_12():18 -->_1 a__minus#(X1,X2) -> c_11():17 -->_1 a__minus#(s(X),s(Y)) -> c_13(a__minus#(X,Y)):5 10:S:mark#(s(X)) -> c_20(mark#(X)) -->_1 mark#(true()) -> c_21():21 -->_1 mark#(false()) -> c_16():20 -->_1 mark#(0()) -> c_14():19 -->_1 mark#(s(X)) -> c_20(mark#(X)):10 -->_1 mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)):9 -->_1 mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)):8 -->_1 mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)):7 -->_1 mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)):6 11:W:a__div#(X1,X2) -> c_1() 12:W:a__div#(0(),s(Y)) -> c_2() 13:W:a__geq#(X,0()) -> c_4() 14:W:a__geq#(X1,X2) -> c_5() 15:W:a__geq#(0(),s(Y)) -> c_6() 16:W:a__if#(X1,X2,X3) -> c_8() 17:W:a__minus#(X1,X2) -> c_11() 18:W:a__minus#(0(),Y) -> c_12() 19:W:mark#(0()) -> c_14() 20:W:mark#(false()) -> c_16() 21:W:mark#(true()) -> c_21() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 11: a__div#(X1,X2) -> c_1() 12: a__div#(0(),s(Y)) -> c_2() 13: a__geq#(X,0()) -> c_4() 14: a__geq#(X1,X2) -> c_5() 15: a__geq#(0(),s(Y)) -> c_6() 16: a__if#(X1,X2,X3) -> c_8() 17: a__minus#(X1,X2) -> c_11() 18: a__minus#(0(),Y) -> c_12() 19: mark#(0()) -> c_14() 20: mark#(false()) -> c_16() 21: mark#(true()) -> c_21() ** Step 1.b:4: DecomposeDG. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: a__div#(s(X),s(Y)) -> c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()),a__geq#(X,Y)) a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)) a__if#(false(),X,Y) -> c_9(mark#(Y)) a__if#(true(),X,Y) -> c_10(mark#(X)) a__minus#(s(X),s(Y)) -> c_13(a__minus#(X,Y)) mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)) mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)) mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)) mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)) mark#(s(X)) -> c_20(mark#(X)) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1,a__div#/2,a__geq#/2,a__if#/3,a__minus#/2,mark#/1} / {0/0,div/2 ,false/0,geq/2,if/3,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0,c_15/2,c_16/0,c_17/1,c_18/2,c_19/1,c_20/1,c_21/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div#,a__geq#,a__if#,a__minus# ,mark#} and constructors {0,div,false,geq,if,minus,s,true} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component a__div#(s(X),s(Y)) -> c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()),a__geq#(X,Y)) a__if#(false(),X,Y) -> c_9(mark#(Y)) a__if#(true(),X,Y) -> c_10(mark#(X)) mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)) mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)) mark#(s(X)) -> c_20(mark#(X)) and a lower component a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)) a__minus#(s(X),s(Y)) -> c_13(a__minus#(X,Y)) mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)) mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)) Further, following extension rules are added to the lower component. a__div#(s(X),s(Y)) -> a__geq#(X,Y) a__div#(s(X),s(Y)) -> a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__if#(false(),X,Y) -> mark#(Y) a__if#(true(),X,Y) -> mark#(X) mark#(div(X1,X2)) -> a__div#(mark(X1),X2) mark#(div(X1,X2)) -> mark#(X1) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(if(X1,X2,X3)) -> mark#(X1) mark#(s(X)) -> mark#(X) *** Step 1.b:4.a:1: SimplifyRHS. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: a__div#(s(X),s(Y)) -> c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()),a__geq#(X,Y)) a__if#(false(),X,Y) -> c_9(mark#(Y)) a__if#(true(),X,Y) -> c_10(mark#(X)) mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)) mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)) mark#(s(X)) -> c_20(mark#(X)) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1,a__div#/2,a__geq#/2,a__if#/3,a__minus#/2,mark#/1} / {0/0,div/2 ,false/0,geq/2,if/3,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0,c_15/2,c_16/0,c_17/1,c_18/2,c_19/1,c_20/1,c_21/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div#,a__geq#,a__if#,a__minus# ,mark#} and constructors {0,div,false,geq,if,minus,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:a__div#(s(X),s(Y)) -> c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()),a__geq#(X,Y)) -->_1 a__if#(true(),X,Y) -> c_10(mark#(X)):3 -->_1 a__if#(false(),X,Y) -> c_9(mark#(Y)):2 2:S:a__if#(false(),X,Y) -> c_9(mark#(Y)) -->_1 mark#(s(X)) -> c_20(mark#(X)):6 -->_1 mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)):5 -->_1 mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)):4 3:S:a__if#(true(),X,Y) -> c_10(mark#(X)) -->_1 mark#(s(X)) -> c_20(mark#(X)):6 -->_1 mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)):5 -->_1 mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)):4 4:S:mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)) -->_2 mark#(s(X)) -> c_20(mark#(X)):6 -->_2 mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)):5 -->_2 mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)):4 -->_1 a__div#(s(X),s(Y)) -> c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()),a__geq#(X,Y)):1 5:S:mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)) -->_2 mark#(s(X)) -> c_20(mark#(X)):6 -->_2 mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)):5 -->_2 mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)):4 -->_1 a__if#(true(),X,Y) -> c_10(mark#(X)):3 -->_1 a__if#(false(),X,Y) -> c_9(mark#(Y)):2 6:S:mark#(s(X)) -> c_20(mark#(X)) -->_1 mark#(s(X)) -> c_20(mark#(X)):6 -->_1 mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)):5 -->_1 mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)):4 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: a__div#(s(X),s(Y)) -> c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())) *** Step 1.b:4.a:2: 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#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)) mark#(s(X)) -> c_20(mark#(X)) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1,a__div#/2,a__geq#/2,a__if#/3,a__minus#/2,mark#/1} / {0/0,div/2 ,false/0,geq/2,if/3,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0,c_15/2,c_16/0,c_17/1,c_18/2,c_19/1,c_20/1,c_21/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div#,a__geq#,a__if#,a__minus# ,mark#} and constructors {0,div,false,geq,if,minus,s,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_3) = {1}, uargs(c_9) = {1}, uargs(c_10) = {1}, uargs(c_15) = {1,2}, uargs(c_18) = {1,2}, uargs(c_20) = {1} Following symbols are considered usable: {a__div#,a__geq#,a__if#,a__minus#,mark#} TcT has computed the following interpretation: p(0) = [0] p(a__div) = [4] x1 + [6] p(a__geq) = [1] x1 + [0] p(a__if) = [2] x1 + [1] p(a__minus) = [4] p(div) = [1] x1 + [1] x2 + [1] p(false) = [0] p(geq) = [1] x2 + [0] p(if) = [1] x1 + [1] x2 + [1] x3 + [4] p(mark) = [0] p(minus) = [0] p(s) = [1] x1 + [0] p(true) = [0] p(a__div#) = [2] x2 + [2] p(a__geq#) = [2] x1 + [1] p(a__if#) = [2] x2 + [2] x3 + [0] p(a__minus#) = [2] x2 + [1] p(mark#) = [2] x1 + [0] p(c_1) = [1] p(c_2) = [1] p(c_3) = [1] x1 + [0] p(c_4) = [0] p(c_5) = [4] p(c_6) = [2] p(c_7) = [1] p(c_8) = [2] p(c_9) = [1] x1 + [0] p(c_10) = [1] x1 + [0] p(c_11) = [2] p(c_12) = [1] p(c_13) = [0] p(c_14) = [2] p(c_15) = [1] x1 + [1] x2 + [0] p(c_16) = [2] p(c_17) = [2] x1 + [4] p(c_18) = [1] x1 + [1] x2 + [7] p(c_19) = [1] x1 + [0] p(c_20) = [1] x1 + [0] p(c_21) = [1] Following rules are strictly oriented: mark#(if(X1,X2,X3)) = [2] X1 + [2] X2 + [2] X3 + [8] > [2] X1 + [2] X2 + [2] X3 + [7] = c_18(a__if#(mark(X1),X2,X3),mark#(X1)) Following rules are (at-least) weakly oriented: a__div#(s(X),s(Y)) = [2] Y + [2] >= [2] Y + [2] = c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())) a__if#(false(),X,Y) = [2] X + [2] Y + [0] >= [2] Y + [0] = c_9(mark#(Y)) a__if#(true(),X,Y) = [2] X + [2] Y + [0] >= [2] X + [0] = c_10(mark#(X)) mark#(div(X1,X2)) = [2] X1 + [2] X2 + [2] >= [2] X1 + [2] X2 + [2] = c_15(a__div#(mark(X1),X2),mark#(X1)) mark#(s(X)) = [2] X + [0] >= [2] X + [0] = c_20(mark#(X)) *** Step 1.b:4.a:3: WeightGap. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: a__div#(s(X),s(Y)) -> c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())) a__if#(false(),X,Y) -> c_9(mark#(Y)) a__if#(true(),X,Y) -> c_10(mark#(X)) mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)) mark#(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_17/1,c_18/2,c_19/1,c_20/1,c_21/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div#,a__geq#,a__if#,a__minus# ,mark#} and constructors {0,div,false,geq,if,minus,s,true} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(a__div) = {1}, uargs(a__if) = {1}, uargs(s) = {1}, uargs(a__div#) = {1}, uargs(a__if#) = {1}, uargs(c_3) = {1}, uargs(c_9) = {1}, uargs(c_10) = {1}, uargs(c_15) = {1,2}, uargs(c_18) = {1,2}, uargs(c_20) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(a__div) = [1] x1 + [0] p(a__geq) = [0] p(a__if) = [1] x1 + [0] p(a__minus) = [0] p(div) = [0] p(false) = [0] p(geq) = [0] p(if) = [0] p(mark) = [0] p(minus) = [0] p(s) = [1] x1 + [0] p(true) = [0] p(a__div#) = [1] x1 + [3] p(a__geq#) = [0] p(a__if#) = [1] x1 + [0] p(a__minus#) = [0] p(mark#) = [2] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [1] x1 + [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [0] p(c_14) = [0] p(c_15) = [1] x1 + [1] x2 + [0] p(c_16) = [0] p(c_17) = [0] p(c_18) = [1] x1 + [1] x2 + [0] p(c_19) = [0] p(c_20) = [1] x1 + [7] p(c_21) = [0] Following rules are strictly oriented: a__div#(s(X),s(Y)) = [1] X + [3] > [0] = c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())) Following rules are (at-least) weakly oriented: a__if#(false(),X,Y) = [0] >= [2] = c_9(mark#(Y)) a__if#(true(),X,Y) = [0] >= [2] = c_10(mark#(X)) mark#(div(X1,X2)) = [2] >= [5] = c_15(a__div#(mark(X1),X2),mark#(X1)) mark#(if(X1,X2,X3)) = [2] >= [2] = c_18(a__if#(mark(X1),X2,X3),mark#(X1)) mark#(s(X)) = [2] >= [9] = c_20(mark#(X)) a__div(X1,X2) = [1] X1 + [0] >= [0] = div(X1,X2) a__div(0(),s(Y)) = [0] >= [0] = 0() a__div(s(X),s(Y)) = [1] X + [0] >= [0] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) = [0] >= [0] = true() a__geq(X1,X2) = [0] >= [0] = geq(X1,X2) a__geq(0(),s(Y)) = [0] >= [0] = false() a__geq(s(X),s(Y)) = [0] >= [0] = a__geq(X,Y) a__if(X1,X2,X3) = [1] X1 + [0] >= [0] = if(X1,X2,X3) a__if(false(),X,Y) = [0] >= [0] = mark(Y) a__if(true(),X,Y) = [0] >= [0] = mark(X) a__minus(X1,X2) = [0] >= [0] = minus(X1,X2) a__minus(0(),Y) = [0] >= [0] = 0() a__minus(s(X),s(Y)) = [0] >= [0] = a__minus(X,Y) mark(0()) = [0] >= [0] = 0() mark(div(X1,X2)) = [0] >= [0] = a__div(mark(X1),X2) mark(false()) = [0] >= [0] = false() mark(geq(X1,X2)) = [0] >= [0] = a__geq(X1,X2) mark(if(X1,X2,X3)) = [0] >= [0] = a__if(mark(X1),X2,X3) mark(minus(X1,X2)) = [0] >= [0] = a__minus(X1,X2) mark(s(X)) = [0] >= [0] = s(mark(X)) mark(true()) = [0] >= [0] = true() Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 1.b:4.a:4: Ara. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: a__if#(false(),X,Y) -> c_9(mark#(Y)) a__if#(true(),X,Y) -> c_10(mark#(X)) mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)) mark#(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())) mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1,a__div#/2,a__geq#/2,a__if#/3,a__minus#/2,mark#/1} / {0/0,div/2 ,false/0,geq/2,if/3,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0,c_15/2,c_16/0,c_17/1,c_18/2,c_19/1,c_20/1,c_21/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div#,a__geq#,a__if#,a__minus# ,mark#} and constructors {0,div,false,geq,if,minus,s,true} + Applied Processor: Ara {minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1, isBestCase = False, mkCompletelyDefined = False, verboseOutput = False} + Details: Signatures used: ---------------- F (TrsFun "0") :: [] -(0)-> "A"(8, 0) F (TrsFun "0") :: [] -(0)-> "A"(0, 0) F (TrsFun "0") :: [] -(0)-> "A"(14, 4) F (TrsFun "0") :: [] -(0)-> "A"(15, 0) F (TrsFun "0") :: [] -(0)-> "A"(15, 7) F (TrsFun "0") :: [] -(0)-> "A"(11, 6) F (TrsFun "0") :: [] -(0)-> "A"(10, 14) F (TrsFun "a__div") :: ["A"(8, 0) x "A"(0, 0)] -(0)-> "A"(8, 0) F (TrsFun "a__geq") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(9, 8) F (TrsFun "a__if") :: ["A"(8, 0) x "A"(8, 0) x "A"(8, 0)] -(0)-> "A"(8, 0) F (TrsFun "a__minus") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(12, 3) F (TrsFun "div") :: ["A"(15, 8) x "A"(8, 0)] -(0)-> "A"(7, 8) F (TrsFun "div") :: ["A"(8, 0) x "A"(0, 0)] -(0)-> "A"(8, 0) F (TrsFun "div") :: ["A"(13, 0) x "A"(0, 0)] -(0)-> "A"(13, 0) F (TrsFun "false") :: [] -(0)-> "A"(0, 0) F (TrsFun "false") :: [] -(0)-> "A"(8, 0) F (TrsFun "false") :: [] -(0)-> "A"(15, 12) F (TrsFun "false") :: [] -(0)-> "A"(11, 6) F (TrsFun "geq") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(8, 0) F (TrsFun "geq") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(10, 10) F (TrsFun "if") :: ["A"(8, 0) x "A"(8, 0) x "A"(8, 0)] -(0)-> "A"(8, 0) F (TrsFun "if") :: ["A"(15, 8) x "A"(15, 8) x "A"(7, 8)] -(8)-> "A"(7, 8) F (TrsFun "mark") :: ["A"(8, 0)] -(0)-> "A"(8, 0) F (TrsFun "minus") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(8, 0) F (TrsFun "minus") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(14, 6) F (TrsFun "minus") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(14, 14) F (TrsFun "minus") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(15, 14) F (TrsFun "s") :: ["A"(7, 8)] -(7)-> "A"(7, 8) F (TrsFun "s") :: ["A"(0, 0)] -(0)-> "A"(0, 0) F (TrsFun "s") :: ["A"(8, 0)] -(8)-> "A"(8, 0) F (TrsFun "s") :: ["A"(7, 0)] -(7)-> "A"(7, 0) F (TrsFun "true") :: [] -(0)-> "A"(0, 0) F (TrsFun "true") :: [] -(0)-> "A"(8, 0) F (TrsFun "true") :: [] -(0)-> "A"(15, 12) F (TrsFun "true") :: [] -(0)-> "A"(11, 6) F (DpFun "a__div") :: ["A"(7, 0) x "A"(8, 0)] -(0)-> "A"(6, 1) F (DpFun "a__if") :: ["A"(0, 0) x "A"(7, 8) x "A"(7, 8)] -(0)-> "A"(11, 12) F (DpFun "mark") :: ["A"(7, 8)] -(0)-> "A"(1, 1) F (ComFun 3) :: ["A"(0, 0)] -(0)-> "A"(10, 10) F (ComFun 9) :: ["A"(0, 0)] -(0)-> "A"(11, 15) F (ComFun 10) :: ["A"(0, 0)] -(0)-> "A"(12, 12) F (ComFun 15) :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(7, 15) F (ComFun 18) :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(6, 3) F (ComFun 20) :: ["A"(1, 0)] -(0)-> "A"(15, 1) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "F (ComFun 10)_A" :: ["A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 10)_A" :: ["A"(0, 0)] -(0)-> "A"(0, 1) "F (ComFun 15)_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 15)_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(0, 1) "F (ComFun 18)_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 18)_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(0, 1) "F (ComFun 20)_A" :: ["A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 20)_A" :: ["A"(1, 0)] -(0)-> "A"(0, 1) "F (ComFun 3)_A" :: ["A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 3)_A" :: ["A"(0, 0)] -(0)-> "A"(0, 1) "F (ComFun 9)_A" :: ["A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 9)_A" :: ["A"(0, 0)] -(0)-> "A"(0, 1) "F (TrsFun \"0\")_A" :: [] -(0)-> "A"(1, 0) "F (TrsFun \"0\")_A" :: [] -(0)-> "A"(0, 1) "F (TrsFun \"div\")_A" :: ["A"(1, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "F (TrsFun \"div\")_A" :: ["A"(1, 1) x "A"(1, 0)] -(0)-> "A"(0, 1) "F (TrsFun \"false\")_A" :: [] -(0)-> "A"(1, 0) "F (TrsFun \"false\")_A" :: [] -(0)-> "A"(0, 1) "F (TrsFun \"geq\")_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "F (TrsFun \"geq\")_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(0, 1) "F (TrsFun \"if\")_A" :: ["A"(1, 0) x "A"(1, 0) x "A"(1, 0)] -(0)-> "A"(1, 0) "F (TrsFun \"if\")_A" :: ["A"(1, 1) x "A"(1, 1) x "A"(0, 1)] -(1)-> "A"(0, 1) "F (TrsFun \"minus\")_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "F (TrsFun \"minus\")_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(0, 1) "F (TrsFun \"s\")_A" :: ["A"(1, 0)] -(1)-> "A"(1, 0) "F (TrsFun \"s\")_A" :: ["A"(0, 1)] -(0)-> "A"(0, 1) "F (TrsFun \"true\")_A" :: [] -(0)-> "A"(1, 0) "F (TrsFun \"true\")_A" :: [] -(0)-> "A"(0, 1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: mark#(s(X)) -> c_20(mark#(X)) 2. Weak: 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)) *** Step 1.b:4.a:5: Ara. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: a__if#(false(),X,Y) -> c_9(mark#(Y)) a__if#(true(),X,Y) -> c_10(mark#(X)) mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)) - Weak DPs: a__div#(s(X),s(Y)) -> c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())) mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)) mark#(s(X)) -> c_20(mark#(X)) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1,a__div#/2,a__geq#/2,a__if#/3,a__minus#/2,mark#/1} / {0/0,div/2 ,false/0,geq/2,if/3,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0,c_15/2,c_16/0,c_17/1,c_18/2,c_19/1,c_20/1,c_21/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div#,a__geq#,a__if#,a__minus# ,mark#} and constructors {0,div,false,geq,if,minus,s,true} + Applied Processor: Ara {minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1, isBestCase = False, mkCompletelyDefined = False, verboseOutput = False} + Details: Signatures used: ---------------- F (TrsFun "0") :: [] -(0)-> "A"(12, 0) F (TrsFun "0") :: [] -(0)-> "A"(0, 0) F (TrsFun "0") :: [] -(0)-> "A"(12, 8) F (TrsFun "0") :: [] -(0)-> "A"(12, 14) F (TrsFun "0") :: [] -(0)-> "A"(14, 12) F (TrsFun "a__div") :: ["A"(12, 0) x "A"(0, 0)] -(0)-> "A"(12, 0) F (TrsFun "a__geq") :: ["A"(12, 0) x "A"(0, 0)] -(0)-> "A"(12, 8) F (TrsFun "a__if") :: ["A"(12, 0) x "A"(12, 0) x "A"(12, 0)] -(0)-> "A"(12, 0) F (TrsFun "a__minus") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(12, 11) F (TrsFun "div") :: ["A"(12, 12) x "A"(0, 12)] -(12)-> "A"(0, 12) F (TrsFun "div") :: ["A"(12, 0) x "A"(0, 0)] -(0)-> "A"(12, 0) F (TrsFun "div") :: ["A"(15, 12) x "A"(0, 12)] -(12)-> "A"(3, 12) F (TrsFun "false") :: [] -(0)-> "A"(4, 0) F (TrsFun "false") :: [] -(0)-> "A"(12, 0) F (TrsFun "false") :: [] -(0)-> "A"(12, 8) F (TrsFun "false") :: [] -(0)-> "A"(12, 14) F (TrsFun "geq") :: ["A"(12, 0) x "A"(0, 0)] -(0)-> "A"(12, 0) F (TrsFun "geq") :: ["A"(12, 0) x "A"(0, 0)] -(0)-> "A"(12, 14) F (TrsFun "if") :: ["A"(12, 0) x "A"(12, 0) x "A"(12, 0)] -(0)-> "A"(12, 0) F (TrsFun "if") :: ["A"(12, 12) x "A"(0, 12) x "A"(12, 12)] -(12)-> "A"(0, 12) F (TrsFun "mark") :: ["A"(12, 0)] -(0)-> "A"(12, 0) F (TrsFun "minus") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(12, 0) F (TrsFun "minus") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(14, 12) F (TrsFun "minus") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(13, 13) F (TrsFun "minus") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(15, 15) F (TrsFun "s") :: ["A"(0, 0)] -(0)-> "A"(0, 0) F (TrsFun "s") :: ["A"(12, 0)] -(12)-> "A"(12, 0) F (TrsFun "s") :: ["A"(0, 12)] -(0)-> "A"(0, 12) F (TrsFun "s") :: ["A"(1, 12)] -(1)-> "A"(1, 12) F (TrsFun "true") :: [] -(0)-> "A"(4, 0) F (TrsFun "true") :: [] -(0)-> "A"(12, 0) F (TrsFun "true") :: [] -(0)-> "A"(12, 8) F (TrsFun "true") :: [] -(0)-> "A"(12, 14) F (DpFun "a__div") :: ["A"(12, 0) x "A"(0, 12)] -(3)-> "A"(1, 1) F (DpFun "a__if") :: ["A"(4, 0) x "A"(0, 12) x "A"(0, 12)] -(0)-> "A"(12, 10) F (DpFun "mark") :: ["A"(0, 12)] -(0)-> "A"(7, 0) F (ComFun 3) :: ["A"(0, 0)] -(0)-> "A"(12, 12) F (ComFun 9) :: ["A"(0, 0)] -(0)-> "A"(12, 10) F (ComFun 10) :: ["A"(0, 0)] -(0)-> "A"(13, 14) F (ComFun 15) :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(9, 13) F (ComFun 18) :: ["A"(0, 5) x "A"(0, 0)] -(0)-> "A"(7, 5) F (ComFun 20) :: ["A"(7, 0)] -(0)-> "A"(7, 13) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "F (ComFun 10)_A" :: ["A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 10)_A" :: ["A"(0, 0)] -(0)-> "A"(0, 1) "F (ComFun 15)_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 15)_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(0, 1) "F (ComFun 18)_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 18)_A" :: ["A"(0, 1) x "A"(0, 0)] -(0)-> "A"(0, 1) "F (ComFun 20)_A" :: ["A"(1, 0)] -(0)-> "A"(1, 0) "F (ComFun 20)_A" :: ["A"(0, 0)] -(0)-> "A"(0, 1) "F (ComFun 3)_A" :: ["A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 3)_A" :: ["A"(0, 0)] -(0)-> "A"(0, 1) "F (ComFun 9)_A" :: ["A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 9)_A" :: ["A"(0, 0)] -(0)-> "A"(0, 1) "F (TrsFun \"0\")_A" :: [] -(0)-> "A"(1, 0) "F (TrsFun \"0\")_A" :: [] -(0)-> "A"(0, 1) "F (TrsFun \"div\")_A" :: ["A"(1, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "F (TrsFun \"div\")_A" :: ["A"(1, 1) x "A"(0, 1)] -(1)-> "A"(0, 1) "F (TrsFun \"false\")_A" :: [] -(0)-> "A"(1, 0) "F (TrsFun \"false\")_A" :: [] -(0)-> "A"(0, 1) "F (TrsFun \"geq\")_A" :: ["A"(1, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "F (TrsFun \"geq\")_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(0, 1) "F (TrsFun \"if\")_A" :: ["A"(1, 0) x "A"(1, 0) x "A"(1, 0)] -(0)-> "A"(1, 0) "F (TrsFun \"if\")_A" :: ["A"(1, 1) x "A"(0, 1) x "A"(1, 1)] -(1)-> "A"(0, 1) "F (TrsFun \"minus\")_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "F (TrsFun \"minus\")_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(0, 1) "F (TrsFun \"s\")_A" :: ["A"(1, 0)] -(1)-> "A"(1, 0) "F (TrsFun \"s\")_A" :: ["A"(0, 1)] -(0)-> "A"(0, 1) "F (TrsFun \"true\")_A" :: [] -(0)-> "A"(1, 0) "F (TrsFun \"true\")_A" :: [] -(0)-> "A"(0, 1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)) 2. Weak: a__if#(false(),X,Y) -> c_9(mark#(Y)) a__if#(true(),X,Y) -> c_10(mark#(X)) *** Step 1.b:4.a:6: Ara. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: a__if#(false(),X,Y) -> c_9(mark#(Y)) a__if#(true(),X,Y) -> c_10(mark#(X)) - Weak DPs: a__div#(s(X),s(Y)) -> c_3(a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0())) mark#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)) mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)) mark#(s(X)) -> c_20(mark#(X)) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1,a__div#/2,a__geq#/2,a__if#/3,a__minus#/2,mark#/1} / {0/0,div/2 ,false/0,geq/2,if/3,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0,c_15/2,c_16/0,c_17/1,c_18/2,c_19/1,c_20/1,c_21/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div#,a__geq#,a__if#,a__minus# ,mark#} and constructors {0,div,false,geq,if,minus,s,true} + Applied Processor: Ara {minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1, isBestCase = False, mkCompletelyDefined = False, verboseOutput = False} + Details: Signatures used: ---------------- F (TrsFun "0") :: [] -(0)-> "A"(8, 0) F (TrsFun "0") :: [] -(0)-> "A"(0, 0) F (TrsFun "0") :: [] -(0)-> "A"(9, 14) F (TrsFun "a__div") :: ["A"(8, 0) x "A"(0, 0)] -(0)-> "A"(8, 0) F (TrsFun "a__geq") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(9, 0) F (TrsFun "a__if") :: ["A"(8, 0) x "A"(8, 0) x "A"(8, 0)] -(0)-> "A"(8, 0) F (TrsFun "a__minus") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(8, 0) F (TrsFun "div") :: ["A"(8, 0) x "A"(0, 0)] -(0)-> "A"(8, 0) F (TrsFun "div") :: ["A"(10, 10) x "A"(0, 10)] -(0)-> "A"(0, 10) F (TrsFun "div") :: ["A"(11, 0) x "A"(0, 0)] -(0)-> "A"(11, 0) F (TrsFun "false") :: [] -(0)-> "A"(0, 0) F (TrsFun "false") :: [] -(0)-> "A"(8, 0) F (TrsFun "false") :: [] -(0)-> "A"(9, 0) F (TrsFun "false") :: [] -(0)-> "A"(10, 6) F (TrsFun "geq") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(8, 0) F (TrsFun "geq") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(13, 8) F (TrsFun "if") :: ["A"(8, 0) x "A"(8, 0) x "A"(8, 0)] -(0)-> "A"(8, 0) F (TrsFun "if") :: ["A"(10, 10) x "A"(10, 10) x "A"(0, 10)] -(10)-> "A"(0, 10) F (TrsFun "mark") :: ["A"(8, 0)] -(0)-> "A"(8, 0) F (TrsFun "minus") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(8, 0) F (TrsFun "minus") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(11, 15) F (TrsFun "minus") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(9, 0) F (TrsFun "minus") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(13, 10) F (TrsFun "s") :: ["A"(0, 0)] -(0)-> "A"(0, 0) F (TrsFun "s") :: ["A"(8, 0)] -(8)-> "A"(8, 0) F (TrsFun "s") :: ["A"(0, 10)] -(0)-> "A"(0, 10) F (TrsFun "true") :: [] -(0)-> "A"(0, 0) F (TrsFun "true") :: [] -(0)-> "A"(8, 0) F (TrsFun "true") :: [] -(0)-> "A"(9, 0) F (TrsFun "true") :: [] -(0)-> "A"(8, 6) F (DpFun "a__div") :: ["A"(8, 0) x "A"(0, 10)] -(0)-> "A"(2, 1) F (DpFun "a__if") :: ["A"(0, 0) x "A"(0, 10) x "A"(0, 10)] -(3)-> "A"(14, 14) F (DpFun "mark") :: ["A"(0, 10)] -(0)-> "A"(6, 8) F (ComFun 3) :: ["A"(10, 0)] -(0)-> "A"(10, 2) F (ComFun 9) :: ["A"(0, 0)] -(0)-> "A"(14, 14) F (ComFun 10) :: ["A"(0, 0)] -(0)-> "A"(14, 14) F (ComFun 15) :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(10, 15) F (ComFun 18) :: ["A"(14, 14) x "A"(0, 0)] -(0)-> "A"(14, 14) F (ComFun 20) :: ["A"(0, 0)] -(0)-> "A"(8, 8) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "F (ComFun 10)_A" :: ["A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 10)_A" :: ["A"(0, 0)] -(0)-> "A"(0, 1) "F (ComFun 15)_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 15)_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(0, 1) "F (ComFun 18)_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 18)_A" :: ["A"(1, 1) x "A"(0, 0)] -(0)-> "A"(0, 1) "F (ComFun 20)_A" :: ["A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 20)_A" :: ["A"(0, 0)] -(0)-> "A"(0, 1) "F (ComFun 3)_A" :: ["A"(1, 0)] -(0)-> "A"(1, 0) "F (ComFun 3)_A" :: ["A"(0, 0)] -(0)-> "A"(0, 1) "F (ComFun 9)_A" :: ["A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 9)_A" :: ["A"(0, 0)] -(0)-> "A"(0, 1) "F (TrsFun \"0\")_A" :: [] -(0)-> "A"(1, 0) "F (TrsFun \"0\")_A" :: [] -(0)-> "A"(0, 1) "F (TrsFun \"div\")_A" :: ["A"(1, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "F (TrsFun \"div\")_A" :: ["A"(1, 1) x "A"(0, 1)] -(0)-> "A"(0, 1) "F (TrsFun \"false\")_A" :: [] -(0)-> "A"(1, 0) "F (TrsFun \"false\")_A" :: [] -(0)-> "A"(0, 1) "F (TrsFun \"geq\")_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "F (TrsFun \"geq\")_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(0, 1) "F (TrsFun \"if\")_A" :: ["A"(1, 0) x "A"(1, 0) x "A"(1, 0)] -(0)-> "A"(1, 0) "F (TrsFun \"if\")_A" :: ["A"(1, 1) x "A"(1, 1) x "A"(0, 1)] -(1)-> "A"(0, 1) "F (TrsFun \"minus\")_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "F (TrsFun \"minus\")_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(0, 1) "F (TrsFun \"s\")_A" :: ["A"(1, 0)] -(1)-> "A"(1, 0) "F (TrsFun \"s\")_A" :: ["A"(0, 1)] -(0)-> "A"(0, 1) "F (TrsFun \"true\")_A" :: [] -(0)-> "A"(1, 0) "F (TrsFun \"true\")_A" :: [] -(0)-> "A"(0, 1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: a__if#(true(),X,Y) -> c_10(mark#(X)) 2. Weak: a__if#(false(),X,Y) -> c_9(mark#(Y)) *** Step 1.b:4.a:7: Ara. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: a__if#(false(),X,Y) -> c_9(mark#(Y)) - 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#(div(X1,X2)) -> c_15(a__div#(mark(X1),X2),mark#(X1)) mark#(if(X1,X2,X3)) -> c_18(a__if#(mark(X1),X2,X3),mark#(X1)) mark#(s(X)) -> c_20(mark#(X)) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1,a__div#/2,a__geq#/2,a__if#/3,a__minus#/2,mark#/1} / {0/0,div/2 ,false/0,geq/2,if/3,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0,c_15/2,c_16/0,c_17/1,c_18/2,c_19/1,c_20/1,c_21/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div#,a__geq#,a__if#,a__minus# ,mark#} and constructors {0,div,false,geq,if,minus,s,true} + Applied Processor: Ara {minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1, isBestCase = False, mkCompletelyDefined = False, verboseOutput = False} + Details: Signatures used: ---------------- F (TrsFun "0") :: [] -(0)-> "A"(12, 0) F (TrsFun "0") :: [] -(0)-> "A"(0, 0) F (TrsFun "0") :: [] -(0)-> "A"(13, 0) F (TrsFun "0") :: [] -(0)-> "A"(15, 0) F (TrsFun "0") :: [] -(0)-> "A"(14, 0) F (TrsFun "0") :: [] -(0)-> "A"(10, 14) F (TrsFun "a__div") :: ["A"(12, 0) x "A"(0, 0)] -(0)-> "A"(12, 0) F (TrsFun "a__geq") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(13, 0) F (TrsFun "a__if") :: ["A"(12, 0) x "A"(12, 0) x "A"(12, 0)] -(0)-> "A"(12, 0) F (TrsFun "a__minus") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(12, 0) F (TrsFun "div") :: ["A"(12, 0) x "A"(0, 0)] -(0)-> "A"(12, 0) F (TrsFun "div") :: ["A"(12, 12) x "A"(0, 12)] -(12)-> "A"(0, 12) F (TrsFun "false") :: [] -(0)-> "A"(5, 0) F (TrsFun "false") :: [] -(0)-> "A"(12, 0) F (TrsFun "false") :: [] -(0)-> "A"(14, 1) F (TrsFun "false") :: [] -(0)-> "A"(15, 6) F (TrsFun "geq") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(12, 0) F (TrsFun "geq") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(13, 8) F (TrsFun "if") :: ["A"(12, 0) x "A"(12, 0) x "A"(12, 0)] -(0)-> "A"(12, 0) F (TrsFun "if") :: ["A"(12, 12) x "A"(0, 12) x "A"(12, 12)] -(12)-> "A"(0, 12) F (TrsFun "mark") :: ["A"(12, 0)] -(0)-> "A"(12, 0) F (TrsFun "minus") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(12, 0) F (TrsFun "minus") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(14, 14) F (TrsFun "minus") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(12, 1) F (TrsFun "minus") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(13, 15) F (TrsFun "s") :: ["A"(0, 0)] -(0)-> "A"(0, 0) F (TrsFun "s") :: ["A"(12, 0)] -(12)-> "A"(12, 0) F (TrsFun "s") :: ["A"(0, 12)] -(0)-> "A"(0, 12) F (TrsFun "true") :: [] -(0)-> "A"(12, 0) F (TrsFun "true") :: [] -(0)-> "A"(5, 0) F (TrsFun "true") :: [] -(0)-> "A"(15, 1) F (TrsFun "true") :: [] -(0)-> "A"(15, 6) F (DpFun "a__div") :: ["A"(12, 0) x "A"(0, 12)] -(4)-> "A"(8, 1) F (DpFun "a__if") :: ["A"(5, 0) x "A"(0, 12) x "A"(2, 12)] -(1)-> "A"(6, 6) F (DpFun "mark") :: ["A"(0, 12)] -(0)-> "A"(0, 14) F (ComFun 3) :: ["A"(0, 0)] -(0)-> "A"(11, 1) F (ComFun 9) :: ["A"(0, 6)] -(0)-> "A"(7, 6) F (ComFun 10) :: ["A"(0, 13)] -(0)-> "A"(11, 13) F (ComFun 15) :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(6, 14) F (ComFun 18) :: ["A"(1, 0) x "A"(0, 0)] -(1)-> "A"(1, 15) F (ComFun 20) :: ["A"(0, 0)] -(0)-> "A"(15, 15) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "F (ComFun 10)_A" :: ["A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 10)_A" :: ["A"(0, 1)] -(0)-> "A"(0, 1) "F (ComFun 15)_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 15)_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(0, 1) "F (ComFun 18)_A" :: ["A"(1, 0) x "A"(0, 0)] -(1)-> "A"(1, 0) "F (ComFun 18)_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(0, 1) "F (ComFun 20)_A" :: ["A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 20)_A" :: ["A"(0, 0)] -(0)-> "A"(0, 1) "F (ComFun 3)_A" :: ["A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 3)_A" :: ["A"(0, 0)] -(0)-> "A"(0, 1) "F (ComFun 9)_A" :: ["A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 9)_A" :: ["A"(0, 1)] -(0)-> "A"(0, 1) "F (TrsFun \"0\")_A" :: [] -(0)-> "A"(1, 0) "F (TrsFun \"0\")_A" :: [] -(0)-> "A"(0, 1) "F (TrsFun \"div\")_A" :: ["A"(1, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "F (TrsFun \"div\")_A" :: ["A"(1, 1) x "A"(0, 1)] -(1)-> "A"(0, 1) "F (TrsFun \"false\")_A" :: [] -(0)-> "A"(1, 0) "F (TrsFun \"false\")_A" :: [] -(0)-> "A"(0, 1) "F (TrsFun \"geq\")_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "F (TrsFun \"geq\")_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(0, 1) "F (TrsFun \"if\")_A" :: ["A"(1, 0) x "A"(1, 0) x "A"(1, 0)] -(0)-> "A"(1, 0) "F (TrsFun \"if\")_A" :: ["A"(1, 1) x "A"(0, 1) x "A"(1, 1)] -(1)-> "A"(0, 1) "F (TrsFun \"minus\")_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "F (TrsFun \"minus\")_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(0, 1) "F (TrsFun \"s\")_A" :: ["A"(1, 0)] -(1)-> "A"(1, 0) "F (TrsFun \"s\")_A" :: ["A"(0, 1)] -(0)-> "A"(0, 1) "F (TrsFun \"true\")_A" :: [] -(0)-> "A"(1, 0) "F (TrsFun \"true\")_A" :: [] -(0)-> "A"(0, 1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: a__if#(false(),X,Y) -> c_9(mark#(Y)) 2. Weak: *** Step 1.b:4.b:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)) a__minus#(s(X),s(Y)) -> c_13(a__minus#(X,Y)) mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)) mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)) - Weak DPs: a__div#(s(X),s(Y)) -> a__geq#(X,Y) a__div#(s(X),s(Y)) -> a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__if#(false(),X,Y) -> mark#(Y) a__if#(true(),X,Y) -> mark#(X) mark#(div(X1,X2)) -> a__div#(mark(X1),X2) mark#(div(X1,X2)) -> mark#(X1) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(if(X1,X2,X3)) -> mark#(X1) mark#(s(X)) -> mark#(X) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1,a__div#/2,a__geq#/2,a__if#/3,a__minus#/2,mark#/1} / {0/0,div/2 ,false/0,geq/2,if/3,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0,c_15/2,c_16/0,c_17/1,c_18/2,c_19/1,c_20/1,c_21/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div#,a__geq#,a__if#,a__minus# ,mark#} and constructors {0,div,false,geq,if,minus,s,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_7) = {1}, uargs(c_13) = {1}, uargs(c_17) = {1}, uargs(c_19) = {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] p(a__div) = [0] p(a__geq) = [0] p(a__if) = [4] x1 + [0] p(a__minus) = [0] p(div) = [0] p(false) = [0] p(geq) = [0] p(if) = [0] p(mark) = [0] p(minus) = [0] p(s) = [1] x1 + [0] p(true) = [0] p(a__div#) = [1] x1 + [1] p(a__geq#) = [0] p(a__if#) = [1] p(a__minus#) = [0] p(mark#) = [1] p(c_1) = [1] p(c_2) = [2] p(c_3) = [1] p(c_4) = [4] p(c_5) = [4] p(c_6) = [0] p(c_7) = [2] x1 + [0] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [2] x1 + [2] p(c_11) = [1] p(c_12) = [4] p(c_13) = [4] x1 + [0] p(c_14) = [1] p(c_15) = [1] x2 + [0] p(c_16) = [2] p(c_17) = [1] x1 + [0] p(c_18) = [1] x1 + [2] x2 + [1] p(c_19) = [4] x1 + [1] p(c_20) = [0] p(c_21) = [0] Following rules are strictly oriented: mark#(geq(X1,X2)) = [1] > [0] = c_17(a__geq#(X1,X2)) Following rules are (at-least) weakly oriented: a__div#(s(X),s(Y)) = [1] X + [1] >= [0] = a__geq#(X,Y) a__div#(s(X),s(Y)) = [1] X + [1] >= [1] = a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq#(s(X),s(Y)) = [0] >= [0] = c_7(a__geq#(X,Y)) a__if#(false(),X,Y) = [1] >= [1] = mark#(Y) a__if#(true(),X,Y) = [1] >= [1] = mark#(X) a__minus#(s(X),s(Y)) = [0] >= [0] = c_13(a__minus#(X,Y)) mark#(div(X1,X2)) = [1] >= [1] = a__div#(mark(X1),X2) mark#(div(X1,X2)) = [1] >= [1] = mark#(X1) mark#(if(X1,X2,X3)) = [1] >= [1] = a__if#(mark(X1),X2,X3) mark#(if(X1,X2,X3)) = [1] >= [1] = mark#(X1) mark#(minus(X1,X2)) = [1] >= [1] = c_19(a__minus#(X1,X2)) mark#(s(X)) = [1] >= [1] = mark#(X) a__div(X1,X2) = [0] >= [0] = div(X1,X2) a__div(0(),s(Y)) = [0] >= [0] = 0() a__div(s(X),s(Y)) = [0] >= [0] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) = [0] >= [0] = true() a__geq(X1,X2) = [0] >= [0] = geq(X1,X2) a__geq(0(),s(Y)) = [0] >= [0] = false() a__geq(s(X),s(Y)) = [0] >= [0] = a__geq(X,Y) a__if(X1,X2,X3) = [4] X1 + [0] >= [0] = if(X1,X2,X3) a__if(false(),X,Y) = [0] >= [0] = mark(Y) a__if(true(),X,Y) = [0] >= [0] = mark(X) a__minus(X1,X2) = [0] >= [0] = minus(X1,X2) a__minus(0(),Y) = [0] >= [0] = 0() a__minus(s(X),s(Y)) = [0] >= [0] = a__minus(X,Y) mark(0()) = [0] >= [0] = 0() mark(div(X1,X2)) = [0] >= [0] = a__div(mark(X1),X2) mark(false()) = [0] >= [0] = false() mark(geq(X1,X2)) = [0] >= [0] = a__geq(X1,X2) mark(if(X1,X2,X3)) = [0] >= [0] = a__if(mark(X1),X2,X3) mark(minus(X1,X2)) = [0] >= [0] = a__minus(X1,X2) mark(s(X)) = [0] >= [0] = s(mark(X)) mark(true()) = [0] >= [0] = true() *** Step 1.b:4.b:2: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)) a__minus#(s(X),s(Y)) -> c_13(a__minus#(X,Y)) mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)) - Weak DPs: a__div#(s(X),s(Y)) -> a__geq#(X,Y) a__div#(s(X),s(Y)) -> a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__if#(false(),X,Y) -> mark#(Y) a__if#(true(),X,Y) -> mark#(X) mark#(div(X1,X2)) -> a__div#(mark(X1),X2) mark#(div(X1,X2)) -> mark#(X1) mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(if(X1,X2,X3)) -> mark#(X1) mark#(s(X)) -> mark#(X) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1,a__div#/2,a__geq#/2,a__if#/3,a__minus#/2,mark#/1} / {0/0,div/2 ,false/0,geq/2,if/3,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0,c_15/2,c_16/0,c_17/1,c_18/2,c_19/1,c_20/1,c_21/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div#,a__geq#,a__if#,a__minus# ,mark#} and constructors {0,div,false,geq,if,minus,s,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_7) = {1}, uargs(c_13) = {1}, uargs(c_17) = {1}, uargs(c_19) = {1} Following symbols are considered usable: {a__div#,a__geq#,a__if#,a__minus#,mark#} TcT has computed the following interpretation: p(0) = [2] p(a__div) = [0] p(a__geq) = [4] x1 + [0] p(a__if) = [1] x1 + [7] x3 + [0] p(a__minus) = [0] p(div) = [1] x2 + [0] p(false) = [1] p(geq) = [1] x1 + [0] p(if) = [1] x1 + [1] x2 + [1] x3 + [0] p(mark) = [4] x1 + [0] p(minus) = [1] x1 + [1] x2 + [0] p(s) = [0] p(true) = [0] p(a__div#) = [5] p(a__geq#) = [0] p(a__if#) = [5] p(a__minus#) = [1] p(mark#) = [5] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [2] x1 + [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [1] x1 + [0] p(c_14) = [0] p(c_15) = [0] p(c_16) = [0] p(c_17) = [2] x1 + [5] p(c_18) = [2] x1 + [0] p(c_19) = [4] x1 + [0] p(c_20) = [0] p(c_21) = [0] Following rules are strictly oriented: mark#(minus(X1,X2)) = [5] > [4] = c_19(a__minus#(X1,X2)) Following rules are (at-least) weakly oriented: a__div#(s(X),s(Y)) = [5] >= [0] = a__geq#(X,Y) a__div#(s(X),s(Y)) = [5] >= [5] = a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq#(s(X),s(Y)) = [0] >= [0] = c_7(a__geq#(X,Y)) a__if#(false(),X,Y) = [5] >= [5] = mark#(Y) a__if#(true(),X,Y) = [5] >= [5] = mark#(X) a__minus#(s(X),s(Y)) = [1] >= [1] = c_13(a__minus#(X,Y)) mark#(div(X1,X2)) = [5] >= [5] = a__div#(mark(X1),X2) mark#(div(X1,X2)) = [5] >= [5] = mark#(X1) mark#(geq(X1,X2)) = [5] >= [5] = c_17(a__geq#(X1,X2)) mark#(if(X1,X2,X3)) = [5] >= [5] = a__if#(mark(X1),X2,X3) mark#(if(X1,X2,X3)) = [5] >= [5] = mark#(X1) mark#(s(X)) = [5] >= [5] = mark#(X) *** Step 1.b:4.b:3: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)) a__minus#(s(X),s(Y)) -> c_13(a__minus#(X,Y)) - Weak DPs: a__div#(s(X),s(Y)) -> a__geq#(X,Y) a__div#(s(X),s(Y)) -> a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__if#(false(),X,Y) -> mark#(Y) a__if#(true(),X,Y) -> mark#(X) mark#(div(X1,X2)) -> a__div#(mark(X1),X2) mark#(div(X1,X2)) -> mark#(X1) mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(if(X1,X2,X3)) -> mark#(X1) mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)) mark#(s(X)) -> mark#(X) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1,a__div#/2,a__geq#/2,a__if#/3,a__minus#/2,mark#/1} / {0/0,div/2 ,false/0,geq/2,if/3,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0,c_15/2,c_16/0,c_17/1,c_18/2,c_19/1,c_20/1,c_21/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div#,a__geq#,a__if#,a__minus# ,mark#} and constructors {0,div,false,geq,if,minus,s,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_7) = {1}, uargs(c_13) = {1}, uargs(c_17) = {1}, uargs(c_19) = {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] p(a__div) = [1] x1 + [0] p(a__geq) = [0] p(a__if) = [1] x1 + [1] x2 + [1] x3 + [0] p(a__minus) = [1] x1 + [0] p(div) = [1] x1 + [0] p(false) = [0] p(geq) = [0] p(if) = [1] x1 + [1] x2 + [1] x3 + [0] p(mark) = [1] x1 + [0] p(minus) = [1] x1 + [0] p(s) = [1] x1 + [1] p(true) = [0] p(a__div#) = [1] x1 + [0] p(a__geq#) = [0] p(a__if#) = [1] x1 + [1] x2 + [1] x3 + [0] p(a__minus#) = [1] x1 + [0] p(mark#) = [1] x1 + [0] p(c_1) = [1] p(c_2) = [1] p(c_3) = [1] x1 + [0] p(c_4) = [1] p(c_5) = [2] p(c_6) = [0] p(c_7) = [1] x1 + [0] p(c_8) = [1] p(c_9) = [1] x1 + [2] p(c_10) = [4] p(c_11) = [0] p(c_12) = [1] p(c_13) = [1] x1 + [0] p(c_14) = [2] p(c_15) = [2] x1 + [1] p(c_16) = [4] p(c_17) = [4] x1 + [0] p(c_18) = [2] x1 + [1] x2 + [1] p(c_19) = [1] x1 + [0] p(c_20) = [4] x1 + [1] p(c_21) = [1] Following rules are strictly oriented: a__minus#(s(X),s(Y)) = [1] X + [1] > [1] X + [0] = c_13(a__minus#(X,Y)) Following rules are (at-least) weakly oriented: a__div#(s(X),s(Y)) = [1] X + [1] >= [0] = a__geq#(X,Y) a__div#(s(X),s(Y)) = [1] X + [1] >= [1] X + [1] = a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq#(s(X),s(Y)) = [0] >= [0] = c_7(a__geq#(X,Y)) a__if#(false(),X,Y) = [1] X + [1] Y + [0] >= [1] Y + [0] = mark#(Y) a__if#(true(),X,Y) = [1] X + [1] Y + [0] >= [1] X + [0] = mark#(X) mark#(div(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = a__div#(mark(X1),X2) mark#(div(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = mark#(X1) mark#(geq(X1,X2)) = [0] >= [0] = c_17(a__geq#(X1,X2)) mark#(if(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = a__if#(mark(X1),X2,X3) mark#(if(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [0] = mark#(X1) mark#(minus(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = c_19(a__minus#(X1,X2)) mark#(s(X)) = [1] X + [1] >= [1] X + [0] = mark#(X) a__div(X1,X2) = [1] X1 + [0] >= [1] X1 + [0] = div(X1,X2) a__div(0(),s(Y)) = [0] >= [0] = 0() a__div(s(X),s(Y)) = [1] X + [1] >= [1] X + [1] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) = [0] >= [0] = true() a__geq(X1,X2) = [0] >= [0] = geq(X1,X2) a__geq(0(),s(Y)) = [0] >= [0] = false() a__geq(s(X),s(Y)) = [0] >= [0] = a__geq(X,Y) a__if(X1,X2,X3) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = if(X1,X2,X3) a__if(false(),X,Y) = [1] X + [1] Y + [0] >= [1] Y + [0] = mark(Y) a__if(true(),X,Y) = [1] X + [1] Y + [0] >= [1] X + [0] = mark(X) a__minus(X1,X2) = [1] X1 + [0] >= [1] X1 + [0] = minus(X1,X2) a__minus(0(),Y) = [0] >= [0] = 0() a__minus(s(X),s(Y)) = [1] X + [1] >= [1] X + [0] = a__minus(X,Y) mark(0()) = [0] >= [0] = 0() mark(div(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = a__div(mark(X1),X2) mark(false()) = [0] >= [0] = false() mark(geq(X1,X2)) = [0] >= [0] = a__geq(X1,X2) mark(if(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = a__if(mark(X1),X2,X3) mark(minus(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = a__minus(X1,X2) mark(s(X)) = [1] X + [1] >= [1] X + [1] = s(mark(X)) mark(true()) = [0] >= [0] = true() *** Step 1.b:4.b:4: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)) - Weak DPs: a__div#(s(X),s(Y)) -> a__geq#(X,Y) a__div#(s(X),s(Y)) -> a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__if#(false(),X,Y) -> mark#(Y) a__if#(true(),X,Y) -> mark#(X) a__minus#(s(X),s(Y)) -> c_13(a__minus#(X,Y)) mark#(div(X1,X2)) -> a__div#(mark(X1),X2) mark#(div(X1,X2)) -> mark#(X1) mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(if(X1,X2,X3)) -> mark#(X1) mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)) mark#(s(X)) -> mark#(X) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1,a__div#/2,a__geq#/2,a__if#/3,a__minus#/2,mark#/1} / {0/0,div/2 ,false/0,geq/2,if/3,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0,c_15/2,c_16/0,c_17/1,c_18/2,c_19/1,c_20/1,c_21/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div#,a__geq#,a__if#,a__minus# ,mark#} and constructors {0,div,false,geq,if,minus,s,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_7) = {1}, uargs(c_13) = {1}, uargs(c_17) = {1}, uargs(c_19) = {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] p(a__div) = [1] x1 + [0] p(a__geq) = [1] x1 + [0] p(a__if) = [1] x1 + [1] x2 + [1] x3 + [0] p(a__minus) = [0] p(div) = [1] x1 + [0] p(false) = [0] p(geq) = [1] x1 + [0] p(if) = [1] x1 + [1] x2 + [1] x3 + [0] p(mark) = [1] x1 + [0] p(minus) = [0] p(s) = [1] x1 + [4] p(true) = [0] p(a__div#) = [1] x1 + [0] p(a__geq#) = [1] x1 + [0] p(a__if#) = [1] x2 + [1] x3 + [0] p(a__minus#) = [0] p(mark#) = [1] x1 + [0] p(c_1) = [0] p(c_2) = [4] p(c_3) = [1] p(c_4) = [1] p(c_5) = [1] p(c_6) = [0] p(c_7) = [1] x1 + [0] p(c_8) = [1] p(c_9) = [1] p(c_10) = [2] x1 + [4] p(c_11) = [1] p(c_12) = [4] p(c_13) = [1] x1 + [0] p(c_14) = [4] p(c_15) = [1] p(c_16) = [0] p(c_17) = [1] x1 + [0] p(c_18) = [4] x1 + [1] p(c_19) = [1] x1 + [0] p(c_20) = [1] x1 + [0] p(c_21) = [1] Following rules are strictly oriented: a__geq#(s(X),s(Y)) = [1] X + [4] > [1] X + [0] = c_7(a__geq#(X,Y)) Following rules are (at-least) weakly oriented: a__div#(s(X),s(Y)) = [1] X + [4] >= [1] X + [0] = a__geq#(X,Y) a__div#(s(X),s(Y)) = [1] X + [4] >= [4] = a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__if#(false(),X,Y) = [1] X + [1] Y + [0] >= [1] Y + [0] = mark#(Y) a__if#(true(),X,Y) = [1] X + [1] Y + [0] >= [1] X + [0] = mark#(X) a__minus#(s(X),s(Y)) = [0] >= [0] = c_13(a__minus#(X,Y)) mark#(div(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = a__div#(mark(X1),X2) mark#(div(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = mark#(X1) mark#(geq(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = c_17(a__geq#(X1,X2)) mark#(if(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X2 + [1] X3 + [0] = a__if#(mark(X1),X2,X3) mark#(if(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [0] = mark#(X1) mark#(minus(X1,X2)) = [0] >= [0] = c_19(a__minus#(X1,X2)) mark#(s(X)) = [1] X + [4] >= [1] X + [0] = mark#(X) a__div(X1,X2) = [1] X1 + [0] >= [1] X1 + [0] = div(X1,X2) a__div(0(),s(Y)) = [0] >= [0] = 0() a__div(s(X),s(Y)) = [1] X + [4] >= [1] X + [4] = a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) = [1] X + [0] >= [0] = true() a__geq(X1,X2) = [1] X1 + [0] >= [1] X1 + [0] = geq(X1,X2) a__geq(0(),s(Y)) = [0] >= [0] = false() a__geq(s(X),s(Y)) = [1] X + [4] >= [1] X + [0] = a__geq(X,Y) a__if(X1,X2,X3) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = if(X1,X2,X3) a__if(false(),X,Y) = [1] X + [1] Y + [0] >= [1] Y + [0] = mark(Y) a__if(true(),X,Y) = [1] X + [1] Y + [0] >= [1] X + [0] = mark(X) a__minus(X1,X2) = [0] >= [0] = minus(X1,X2) a__minus(0(),Y) = [0] >= [0] = 0() a__minus(s(X),s(Y)) = [0] >= [0] = a__minus(X,Y) mark(0()) = [0] >= [0] = 0() mark(div(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = a__div(mark(X1),X2) mark(false()) = [0] >= [0] = false() mark(geq(X1,X2)) = [1] X1 + [0] >= [1] X1 + [0] = a__geq(X1,X2) mark(if(X1,X2,X3)) = [1] X1 + [1] X2 + [1] X3 + [0] >= [1] X1 + [1] X2 + [1] X3 + [0] = a__if(mark(X1),X2,X3) mark(minus(X1,X2)) = [0] >= [0] = a__minus(X1,X2) mark(s(X)) = [1] X + [4] >= [1] X + [4] = s(mark(X)) mark(true()) = [0] >= [0] = true() *** Step 1.b:4.b:5: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: a__div#(s(X),s(Y)) -> a__geq#(X,Y) a__div#(s(X),s(Y)) -> a__if#(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq#(s(X),s(Y)) -> c_7(a__geq#(X,Y)) a__if#(false(),X,Y) -> mark#(Y) a__if#(true(),X,Y) -> mark#(X) a__minus#(s(X),s(Y)) -> c_13(a__minus#(X,Y)) mark#(div(X1,X2)) -> a__div#(mark(X1),X2) mark#(div(X1,X2)) -> mark#(X1) mark#(geq(X1,X2)) -> c_17(a__geq#(X1,X2)) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(if(X1,X2,X3)) -> mark#(X1) mark#(minus(X1,X2)) -> c_19(a__minus#(X1,X2)) mark#(s(X)) -> mark#(X) - Weak TRS: a__div(X1,X2) -> div(X1,X2) a__div(0(),s(Y)) -> 0() a__div(s(X),s(Y)) -> a__if(a__geq(X,Y),s(div(minus(X,Y),s(Y))),0()) a__geq(X,0()) -> true() a__geq(X1,X2) -> geq(X1,X2) a__geq(0(),s(Y)) -> false() a__geq(s(X),s(Y)) -> a__geq(X,Y) a__if(X1,X2,X3) -> if(X1,X2,X3) a__if(false(),X,Y) -> mark(Y) a__if(true(),X,Y) -> mark(X) a__minus(X1,X2) -> minus(X1,X2) a__minus(0(),Y) -> 0() a__minus(s(X),s(Y)) -> a__minus(X,Y) mark(0()) -> 0() mark(div(X1,X2)) -> a__div(mark(X1),X2) mark(false()) -> false() mark(geq(X1,X2)) -> a__geq(X1,X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(minus(X1,X2)) -> a__minus(X1,X2) mark(s(X)) -> s(mark(X)) mark(true()) -> true() - Signature: {a__div/2,a__geq/2,a__if/3,a__minus/2,mark/1,a__div#/2,a__geq#/2,a__if#/3,a__minus#/2,mark#/1} / {0/0,div/2 ,false/0,geq/2,if/3,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0 ,c_12/0,c_13/1,c_14/0,c_15/2,c_16/0,c_17/1,c_18/2,c_19/1,c_20/1,c_21/0} - Obligation: innermost runtime complexity wrt. defined symbols {a__div#,a__geq#,a__if#,a__minus# ,mark#} and constructors {0,div,false,geq,if,minus,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^3))