WORST_CASE(?,O(n^3)) * Step 1: DependencyPairs WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: div_active(x,y) -> div(x,y) div_active(0(),s(y)) -> 0() div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) -> ge(x,y) ge_active(x,0()) -> true() ge_active(0(),s(y)) -> false() ge_active(s(x),s(y)) -> ge_active(x,y) if_active(x,y,z) -> if(x,y,z) if_active(false(),x,y) -> mark(y) if_active(true(),x,y) -> mark(x) mark(0()) -> 0() mark(div(x,y)) -> div_active(mark(x),y) mark(ge(x,y)) -> ge_active(x,y) mark(if(x,y,z)) -> if_active(mark(x),y,z) mark(minus(x,y)) -> minus_active(x,y) mark(s(x)) -> s(mark(x)) minus_active(x,y) -> minus(x,y) minus_active(0(),y) -> 0() minus_active(s(x),s(y)) -> minus_active(x,y) - Signature: {div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2} / {0/0,div/2,false/0,ge/2,if/3,minus/2,s/1 ,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div_active,ge_active,if_active,mark ,minus_active} and constructors {0,div,false,ge,if,minus,s,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs div_active#(x,y) -> c_1() div_active#(0(),s(y)) -> c_2() div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0()),ge_active#(x,y)) ge_active#(x,y) -> c_4() ge_active#(x,0()) -> c_5() ge_active#(0(),s(y)) -> c_6() ge_active#(s(x),s(y)) -> c_7(ge_active#(x,y)) if_active#(x,y,z) -> c_8() if_active#(false(),x,y) -> c_9(mark#(y)) if_active#(true(),x,y) -> c_10(mark#(x)) mark#(0()) -> c_11() mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)) mark#(ge(x,y)) -> c_13(ge_active#(x,y)) mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)) mark#(minus(x,y)) -> c_15(minus_active#(x,y)) mark#(s(x)) -> c_16(mark#(x)) minus_active#(x,y) -> c_17() minus_active#(0(),y) -> c_18() minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)) Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: div_active#(x,y) -> c_1() div_active#(0(),s(y)) -> c_2() div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0()),ge_active#(x,y)) ge_active#(x,y) -> c_4() ge_active#(x,0()) -> c_5() ge_active#(0(),s(y)) -> c_6() ge_active#(s(x),s(y)) -> c_7(ge_active#(x,y)) if_active#(x,y,z) -> c_8() if_active#(false(),x,y) -> c_9(mark#(y)) if_active#(true(),x,y) -> c_10(mark#(x)) mark#(0()) -> c_11() mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)) mark#(ge(x,y)) -> c_13(ge_active#(x,y)) mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)) mark#(minus(x,y)) -> c_15(minus_active#(x,y)) mark#(s(x)) -> c_16(mark#(x)) minus_active#(x,y) -> c_17() minus_active#(0(),y) -> c_18() minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)) - Weak TRS: div_active(x,y) -> div(x,y) div_active(0(),s(y)) -> 0() div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) -> ge(x,y) ge_active(x,0()) -> true() ge_active(0(),s(y)) -> false() ge_active(s(x),s(y)) -> ge_active(x,y) if_active(x,y,z) -> if(x,y,z) if_active(false(),x,y) -> mark(y) if_active(true(),x,y) -> mark(x) mark(0()) -> 0() mark(div(x,y)) -> div_active(mark(x),y) mark(ge(x,y)) -> ge_active(x,y) mark(if(x,y,z)) -> if_active(mark(x),y,z) mark(minus(x,y)) -> minus_active(x,y) mark(s(x)) -> s(mark(x)) minus_active(x,y) -> minus(x,y) minus_active(0(),y) -> 0() minus_active(s(x),s(y)) -> minus_active(x,y) - Signature: {div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2,div_active#/2,ge_active#/2,if_active#/3,mark#/1 ,minus_active#/2} / {0/0,div/2,false/0,ge/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_12/2,c_13/1,c_14/2,c_15/1,c_16/1,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {div_active#,ge_active#,if_active#,mark# ,minus_active#} and constructors {0,div,false,ge,if,minus,s,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,4,5,6,8,11,17,18} by application of Pre({1,2,4,5,6,8,11,17,18}) = {3,7,9,10,12,13,14,15,16,19}. Here rules are labelled as follows: 1: div_active#(x,y) -> c_1() 2: div_active#(0(),s(y)) -> c_2() 3: div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0()),ge_active#(x,y)) 4: ge_active#(x,y) -> c_4() 5: ge_active#(x,0()) -> c_5() 6: ge_active#(0(),s(y)) -> c_6() 7: ge_active#(s(x),s(y)) -> c_7(ge_active#(x,y)) 8: if_active#(x,y,z) -> c_8() 9: if_active#(false(),x,y) -> c_9(mark#(y)) 10: if_active#(true(),x,y) -> c_10(mark#(x)) 11: mark#(0()) -> c_11() 12: mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)) 13: mark#(ge(x,y)) -> c_13(ge_active#(x,y)) 14: mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)) 15: mark#(minus(x,y)) -> c_15(minus_active#(x,y)) 16: mark#(s(x)) -> c_16(mark#(x)) 17: minus_active#(x,y) -> c_17() 18: minus_active#(0(),y) -> c_18() 19: minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)) * Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0()),ge_active#(x,y)) ge_active#(s(x),s(y)) -> c_7(ge_active#(x,y)) if_active#(false(),x,y) -> c_9(mark#(y)) if_active#(true(),x,y) -> c_10(mark#(x)) mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)) mark#(ge(x,y)) -> c_13(ge_active#(x,y)) mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)) mark#(minus(x,y)) -> c_15(minus_active#(x,y)) mark#(s(x)) -> c_16(mark#(x)) minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)) - Weak DPs: div_active#(x,y) -> c_1() div_active#(0(),s(y)) -> c_2() ge_active#(x,y) -> c_4() ge_active#(x,0()) -> c_5() ge_active#(0(),s(y)) -> c_6() if_active#(x,y,z) -> c_8() mark#(0()) -> c_11() minus_active#(x,y) -> c_17() minus_active#(0(),y) -> c_18() - Weak TRS: div_active(x,y) -> div(x,y) div_active(0(),s(y)) -> 0() div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) -> ge(x,y) ge_active(x,0()) -> true() ge_active(0(),s(y)) -> false() ge_active(s(x),s(y)) -> ge_active(x,y) if_active(x,y,z) -> if(x,y,z) if_active(false(),x,y) -> mark(y) if_active(true(),x,y) -> mark(x) mark(0()) -> 0() mark(div(x,y)) -> div_active(mark(x),y) mark(ge(x,y)) -> ge_active(x,y) mark(if(x,y,z)) -> if_active(mark(x),y,z) mark(minus(x,y)) -> minus_active(x,y) mark(s(x)) -> s(mark(x)) minus_active(x,y) -> minus(x,y) minus_active(0(),y) -> 0() minus_active(s(x),s(y)) -> minus_active(x,y) - Signature: {div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2,div_active#/2,ge_active#/2,if_active#/3,mark#/1 ,minus_active#/2} / {0/0,div/2,false/0,ge/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_12/2,c_13/1,c_14/2,c_15/1,c_16/1,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {div_active#,ge_active#,if_active#,mark# ,minus_active#} and constructors {0,div,false,ge,if,minus,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0()),ge_active#(x,y)) -->_1 if_active#(true(),x,y) -> c_10(mark#(x)):4 -->_1 if_active#(false(),x,y) -> c_9(mark#(y)):3 -->_2 ge_active#(s(x),s(y)) -> c_7(ge_active#(x,y)):2 -->_1 if_active#(x,y,z) -> c_8():16 -->_2 ge_active#(0(),s(y)) -> c_6():15 -->_2 ge_active#(x,0()) -> c_5():14 -->_2 ge_active#(x,y) -> c_4():13 2:S:ge_active#(s(x),s(y)) -> c_7(ge_active#(x,y)) -->_1 ge_active#(0(),s(y)) -> c_6():15 -->_1 ge_active#(x,0()) -> c_5():14 -->_1 ge_active#(x,y) -> c_4():13 -->_1 ge_active#(s(x),s(y)) -> c_7(ge_active#(x,y)):2 3:S:if_active#(false(),x,y) -> c_9(mark#(y)) -->_1 mark#(s(x)) -> c_16(mark#(x)):9 -->_1 mark#(minus(x,y)) -> c_15(minus_active#(x,y)):8 -->_1 mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)):7 -->_1 mark#(ge(x,y)) -> c_13(ge_active#(x,y)):6 -->_1 mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)):5 -->_1 mark#(0()) -> c_11():17 4:S:if_active#(true(),x,y) -> c_10(mark#(x)) -->_1 mark#(s(x)) -> c_16(mark#(x)):9 -->_1 mark#(minus(x,y)) -> c_15(minus_active#(x,y)):8 -->_1 mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)):7 -->_1 mark#(ge(x,y)) -> c_13(ge_active#(x,y)):6 -->_1 mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)):5 -->_1 mark#(0()) -> c_11():17 5:S:mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)) -->_2 mark#(s(x)) -> c_16(mark#(x)):9 -->_2 mark#(minus(x,y)) -> c_15(minus_active#(x,y)):8 -->_2 mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)):7 -->_2 mark#(ge(x,y)) -> c_13(ge_active#(x,y)):6 -->_2 mark#(0()) -> c_11():17 -->_1 div_active#(0(),s(y)) -> c_2():12 -->_1 div_active#(x,y) -> c_1():11 -->_2 mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)):5 -->_1 div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ,ge_active#(x,y)):1 6:S:mark#(ge(x,y)) -> c_13(ge_active#(x,y)) -->_1 ge_active#(0(),s(y)) -> c_6():15 -->_1 ge_active#(x,0()) -> c_5():14 -->_1 ge_active#(x,y) -> c_4():13 -->_1 ge_active#(s(x),s(y)) -> c_7(ge_active#(x,y)):2 7:S:mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)) -->_2 mark#(s(x)) -> c_16(mark#(x)):9 -->_2 mark#(minus(x,y)) -> c_15(minus_active#(x,y)):8 -->_2 mark#(0()) -> c_11():17 -->_1 if_active#(x,y,z) -> c_8():16 -->_2 mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)):7 -->_2 mark#(ge(x,y)) -> c_13(ge_active#(x,y)):6 -->_2 mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)):5 -->_1 if_active#(true(),x,y) -> c_10(mark#(x)):4 -->_1 if_active#(false(),x,y) -> c_9(mark#(y)):3 8:S:mark#(minus(x,y)) -> c_15(minus_active#(x,y)) -->_1 minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)):10 -->_1 minus_active#(0(),y) -> c_18():19 -->_1 minus_active#(x,y) -> c_17():18 9:S:mark#(s(x)) -> c_16(mark#(x)) -->_1 mark#(0()) -> c_11():17 -->_1 mark#(s(x)) -> c_16(mark#(x)):9 -->_1 mark#(minus(x,y)) -> c_15(minus_active#(x,y)):8 -->_1 mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)):7 -->_1 mark#(ge(x,y)) -> c_13(ge_active#(x,y)):6 -->_1 mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)):5 10:S:minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)) -->_1 minus_active#(0(),y) -> c_18():19 -->_1 minus_active#(x,y) -> c_17():18 -->_1 minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)):10 11:W:div_active#(x,y) -> c_1() 12:W:div_active#(0(),s(y)) -> c_2() 13:W:ge_active#(x,y) -> c_4() 14:W:ge_active#(x,0()) -> c_5() 15:W:ge_active#(0(),s(y)) -> c_6() 16:W:if_active#(x,y,z) -> c_8() 17:W:mark#(0()) -> c_11() 18:W:minus_active#(x,y) -> c_17() 19:W:minus_active#(0(),y) -> c_18() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 11: div_active#(x,y) -> c_1() 12: div_active#(0(),s(y)) -> c_2() 13: ge_active#(x,y) -> c_4() 14: ge_active#(x,0()) -> c_5() 15: ge_active#(0(),s(y)) -> c_6() 16: if_active#(x,y,z) -> c_8() 18: minus_active#(x,y) -> c_17() 19: minus_active#(0(),y) -> c_18() 17: mark#(0()) -> c_11() * Step 4: DecomposeDG WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0()),ge_active#(x,y)) ge_active#(s(x),s(y)) -> c_7(ge_active#(x,y)) if_active#(false(),x,y) -> c_9(mark#(y)) if_active#(true(),x,y) -> c_10(mark#(x)) mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)) mark#(ge(x,y)) -> c_13(ge_active#(x,y)) mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)) mark#(minus(x,y)) -> c_15(minus_active#(x,y)) mark#(s(x)) -> c_16(mark#(x)) minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)) - Weak TRS: div_active(x,y) -> div(x,y) div_active(0(),s(y)) -> 0() div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) -> ge(x,y) ge_active(x,0()) -> true() ge_active(0(),s(y)) -> false() ge_active(s(x),s(y)) -> ge_active(x,y) if_active(x,y,z) -> if(x,y,z) if_active(false(),x,y) -> mark(y) if_active(true(),x,y) -> mark(x) mark(0()) -> 0() mark(div(x,y)) -> div_active(mark(x),y) mark(ge(x,y)) -> ge_active(x,y) mark(if(x,y,z)) -> if_active(mark(x),y,z) mark(minus(x,y)) -> minus_active(x,y) mark(s(x)) -> s(mark(x)) minus_active(x,y) -> minus(x,y) minus_active(0(),y) -> 0() minus_active(s(x),s(y)) -> minus_active(x,y) - Signature: {div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2,div_active#/2,ge_active#/2,if_active#/3,mark#/1 ,minus_active#/2} / {0/0,div/2,false/0,ge/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_12/2,c_13/1,c_14/2,c_15/1,c_16/1,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {div_active#,ge_active#,if_active#,mark# ,minus_active#} and constructors {0,div,false,ge,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 div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0()),ge_active#(x,y)) if_active#(false(),x,y) -> c_9(mark#(y)) if_active#(true(),x,y) -> c_10(mark#(x)) mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)) mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)) mark#(s(x)) -> c_16(mark#(x)) and a lower component ge_active#(s(x),s(y)) -> c_7(ge_active#(x,y)) mark#(ge(x,y)) -> c_13(ge_active#(x,y)) mark#(minus(x,y)) -> c_15(minus_active#(x,y)) minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)) Further, following extension rules are added to the lower component. div_active#(s(x),s(y)) -> ge_active#(x,y) div_active#(s(x),s(y)) -> if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0()) if_active#(false(),x,y) -> mark#(y) if_active#(true(),x,y) -> mark#(x) mark#(div(x,y)) -> div_active#(mark(x),y) mark#(div(x,y)) -> mark#(x) mark#(if(x,y,z)) -> if_active#(mark(x),y,z) mark#(if(x,y,z)) -> mark#(x) mark#(s(x)) -> mark#(x) ** Step 4.a:1: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0()),ge_active#(x,y)) if_active#(false(),x,y) -> c_9(mark#(y)) if_active#(true(),x,y) -> c_10(mark#(x)) mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)) mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)) mark#(s(x)) -> c_16(mark#(x)) - Weak TRS: div_active(x,y) -> div(x,y) div_active(0(),s(y)) -> 0() div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) -> ge(x,y) ge_active(x,0()) -> true() ge_active(0(),s(y)) -> false() ge_active(s(x),s(y)) -> ge_active(x,y) if_active(x,y,z) -> if(x,y,z) if_active(false(),x,y) -> mark(y) if_active(true(),x,y) -> mark(x) mark(0()) -> 0() mark(div(x,y)) -> div_active(mark(x),y) mark(ge(x,y)) -> ge_active(x,y) mark(if(x,y,z)) -> if_active(mark(x),y,z) mark(minus(x,y)) -> minus_active(x,y) mark(s(x)) -> s(mark(x)) minus_active(x,y) -> minus(x,y) minus_active(0(),y) -> 0() minus_active(s(x),s(y)) -> minus_active(x,y) - Signature: {div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2,div_active#/2,ge_active#/2,if_active#/3,mark#/1 ,minus_active#/2} / {0/0,div/2,false/0,ge/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_12/2,c_13/1,c_14/2,c_15/1,c_16/1,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {div_active#,ge_active#,if_active#,mark# ,minus_active#} and constructors {0,div,false,ge,if,minus,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0()),ge_active#(x,y)) -->_1 if_active#(true(),x,y) -> c_10(mark#(x)):3 -->_1 if_active#(false(),x,y) -> c_9(mark#(y)):2 2:S:if_active#(false(),x,y) -> c_9(mark#(y)) -->_1 mark#(s(x)) -> c_16(mark#(x)):6 -->_1 mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)):5 -->_1 mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)):4 3:S:if_active#(true(),x,y) -> c_10(mark#(x)) -->_1 mark#(s(x)) -> c_16(mark#(x)):6 -->_1 mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)):5 -->_1 mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)):4 4:S:mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)) -->_2 mark#(s(x)) -> c_16(mark#(x)):6 -->_2 mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)):5 -->_2 mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)):4 -->_1 div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ,ge_active#(x,y)):1 5:S:mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)) -->_2 mark#(s(x)) -> c_16(mark#(x)):6 -->_2 mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)):5 -->_2 mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)):4 -->_1 if_active#(true(),x,y) -> c_10(mark#(x)):3 -->_1 if_active#(false(),x,y) -> c_9(mark#(y)):2 6:S:mark#(s(x)) -> c_16(mark#(x)) -->_1 mark#(s(x)) -> c_16(mark#(x)):6 -->_1 mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)):5 -->_1 mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)):4 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())) ** Step 4.a:2: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())) if_active#(false(),x,y) -> c_9(mark#(y)) if_active#(true(),x,y) -> c_10(mark#(x)) mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)) mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)) mark#(s(x)) -> c_16(mark#(x)) - Weak TRS: div_active(x,y) -> div(x,y) div_active(0(),s(y)) -> 0() div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) -> ge(x,y) ge_active(x,0()) -> true() ge_active(0(),s(y)) -> false() ge_active(s(x),s(y)) -> ge_active(x,y) if_active(x,y,z) -> if(x,y,z) if_active(false(),x,y) -> mark(y) if_active(true(),x,y) -> mark(x) mark(0()) -> 0() mark(div(x,y)) -> div_active(mark(x),y) mark(ge(x,y)) -> ge_active(x,y) mark(if(x,y,z)) -> if_active(mark(x),y,z) mark(minus(x,y)) -> minus_active(x,y) mark(s(x)) -> s(mark(x)) minus_active(x,y) -> minus(x,y) minus_active(0(),y) -> 0() minus_active(s(x),s(y)) -> minus_active(x,y) - Signature: {div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2,div_active#/2,ge_active#/2,if_active#/3,mark#/1 ,minus_active#/2} / {0/0,div/2,false/0,ge/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_12/2,c_13/1,c_14/2,c_15/1,c_16/1,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {div_active#,ge_active#,if_active#,mark# ,minus_active#} and constructors {0,div,false,ge,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(div_active) = {1}, uargs(if_active) = {1}, uargs(s) = {1}, uargs(div_active#) = {1}, uargs(if_active#) = {1}, uargs(c_3) = {1}, uargs(c_9) = {1}, uargs(c_10) = {1}, uargs(c_12) = {1,2}, uargs(c_14) = {1,2}, uargs(c_16) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(div) = [0] p(div_active) = [1] x1 + [0] p(false) = [0] p(ge) = [0] p(ge_active) = [0] p(if) = [0] p(if_active) = [1] x1 + [0] p(mark) = [0] p(minus) = [0] p(minus_active) = [0] p(s) = [1] x1 + [0] p(true) = [0] p(div_active#) = [1] x1 + [5] p(ge_active#) = [0] p(if_active#) = [1] x1 + [5] p(mark#) = [0] p(minus_active#) = [0] 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 + [4] p(c_10) = [1] x1 + [0] p(c_11) = [0] p(c_12) = [1] x1 + [1] x2 + [1] p(c_13) = [1] x1 + [4] p(c_14) = [1] x1 + [1] x2 + [2] p(c_15) = [1] x1 + [0] p(c_16) = [1] x1 + [4] p(c_17) = [0] p(c_18) = [2] p(c_19) = [1] x1 + [0] Following rules are strictly oriented: if_active#(false(),x,y) = [5] > [4] = c_9(mark#(y)) if_active#(true(),x,y) = [5] > [0] = c_10(mark#(x)) Following rules are (at-least) weakly oriented: div_active#(s(x),s(y)) = [1] x + [5] >= [5] = c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())) mark#(div(x,y)) = [0] >= [6] = c_12(div_active#(mark(x),y),mark#(x)) mark#(if(x,y,z)) = [0] >= [7] = c_14(if_active#(mark(x),y,z),mark#(x)) mark#(s(x)) = [0] >= [4] = c_16(mark#(x)) div_active(x,y) = [1] x + [0] >= [0] = div(x,y) div_active(0(),s(y)) = [0] >= [0] = 0() div_active(s(x),s(y)) = [1] x + [0] >= [0] = if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) = [0] >= [0] = ge(x,y) ge_active(x,0()) = [0] >= [0] = true() ge_active(0(),s(y)) = [0] >= [0] = false() ge_active(s(x),s(y)) = [0] >= [0] = ge_active(x,y) if_active(x,y,z) = [1] x + [0] >= [0] = if(x,y,z) if_active(false(),x,y) = [0] >= [0] = mark(y) if_active(true(),x,y) = [0] >= [0] = mark(x) mark(0()) = [0] >= [0] = 0() mark(div(x,y)) = [0] >= [0] = div_active(mark(x),y) mark(ge(x,y)) = [0] >= [0] = ge_active(x,y) mark(if(x,y,z)) = [0] >= [0] = if_active(mark(x),y,z) mark(minus(x,y)) = [0] >= [0] = minus_active(x,y) mark(s(x)) = [0] >= [0] = s(mark(x)) minus_active(x,y) = [0] >= [0] = minus(x,y) minus_active(0(),y) = [0] >= [0] = 0() minus_active(s(x),s(y)) = [0] >= [0] = minus_active(x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 4.a:3: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())) mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)) mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)) mark#(s(x)) -> c_16(mark#(x)) - Weak DPs: if_active#(false(),x,y) -> c_9(mark#(y)) if_active#(true(),x,y) -> c_10(mark#(x)) - Weak TRS: div_active(x,y) -> div(x,y) div_active(0(),s(y)) -> 0() div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) -> ge(x,y) ge_active(x,0()) -> true() ge_active(0(),s(y)) -> false() ge_active(s(x),s(y)) -> ge_active(x,y) if_active(x,y,z) -> if(x,y,z) if_active(false(),x,y) -> mark(y) if_active(true(),x,y) -> mark(x) mark(0()) -> 0() mark(div(x,y)) -> div_active(mark(x),y) mark(ge(x,y)) -> ge_active(x,y) mark(if(x,y,z)) -> if_active(mark(x),y,z) mark(minus(x,y)) -> minus_active(x,y) mark(s(x)) -> s(mark(x)) minus_active(x,y) -> minus(x,y) minus_active(0(),y) -> 0() minus_active(s(x),s(y)) -> minus_active(x,y) - Signature: {div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2,div_active#/2,ge_active#/2,if_active#/3,mark#/1 ,minus_active#/2} / {0/0,div/2,false/0,ge/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_12/2,c_13/1,c_14/2,c_15/1,c_16/1,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {div_active#,ge_active#,if_active#,mark# ,minus_active#} and constructors {0,div,false,ge,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(div_active) = {1}, uargs(if_active) = {1}, uargs(s) = {1}, uargs(div_active#) = {1}, uargs(if_active#) = {1}, uargs(c_3) = {1}, uargs(c_9) = {1}, uargs(c_10) = {1}, uargs(c_12) = {1,2}, uargs(c_14) = {1,2}, uargs(c_16) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(div) = [0] p(div_active) = [1] x1 + [0] p(false) = [0] p(ge) = [0] p(ge_active) = [0] p(if) = [0] p(if_active) = [1] x1 + [0] p(mark) = [0] p(minus) = [0] p(minus_active) = [0] p(s) = [1] x1 + [0] p(true) = [0] p(div_active#) = [1] x1 + [4] p(ge_active#) = [1] x1 + [0] p(if_active#) = [1] x1 + [1] p(mark#) = [1] p(minus_active#) = [1] x1 + [1] x2 + [4] p(c_1) = [1] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] p(c_5) = [1] p(c_6) = [1] p(c_7) = [1] p(c_8) = [1] p(c_9) = [1] x1 + [0] p(c_10) = [1] x1 + [0] p(c_11) = [0] p(c_12) = [1] x1 + [1] x2 + [7] p(c_13) = [1] x1 + [1] p(c_14) = [1] x1 + [1] x2 + [4] p(c_15) = [1] x1 + [1] p(c_16) = [1] x1 + [7] p(c_17) = [1] p(c_18) = [0] p(c_19) = [1] x1 + [2] Following rules are strictly oriented: div_active#(s(x),s(y)) = [1] x + [4] > [1] = c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())) Following rules are (at-least) weakly oriented: if_active#(false(),x,y) = [1] >= [1] = c_9(mark#(y)) if_active#(true(),x,y) = [1] >= [1] = c_10(mark#(x)) mark#(div(x,y)) = [1] >= [12] = c_12(div_active#(mark(x),y),mark#(x)) mark#(if(x,y,z)) = [1] >= [6] = c_14(if_active#(mark(x),y,z),mark#(x)) mark#(s(x)) = [1] >= [8] = c_16(mark#(x)) div_active(x,y) = [1] x + [0] >= [0] = div(x,y) div_active(0(),s(y)) = [0] >= [0] = 0() div_active(s(x),s(y)) = [1] x + [0] >= [0] = if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) = [0] >= [0] = ge(x,y) ge_active(x,0()) = [0] >= [0] = true() ge_active(0(),s(y)) = [0] >= [0] = false() ge_active(s(x),s(y)) = [0] >= [0] = ge_active(x,y) if_active(x,y,z) = [1] x + [0] >= [0] = if(x,y,z) if_active(false(),x,y) = [0] >= [0] = mark(y) if_active(true(),x,y) = [0] >= [0] = mark(x) mark(0()) = [0] >= [0] = 0() mark(div(x,y)) = [0] >= [0] = div_active(mark(x),y) mark(ge(x,y)) = [0] >= [0] = ge_active(x,y) mark(if(x,y,z)) = [0] >= [0] = if_active(mark(x),y,z) mark(minus(x,y)) = [0] >= [0] = minus_active(x,y) mark(s(x)) = [0] >= [0] = s(mark(x)) minus_active(x,y) = [0] >= [0] = minus(x,y) minus_active(0(),y) = [0] >= [0] = 0() minus_active(s(x),s(y)) = [0] >= [0] = minus_active(x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 4.a:4: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)) mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)) mark#(s(x)) -> c_16(mark#(x)) - Weak DPs: div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())) if_active#(false(),x,y) -> c_9(mark#(y)) if_active#(true(),x,y) -> c_10(mark#(x)) - Weak TRS: div_active(x,y) -> div(x,y) div_active(0(),s(y)) -> 0() div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) -> ge(x,y) ge_active(x,0()) -> true() ge_active(0(),s(y)) -> false() ge_active(s(x),s(y)) -> ge_active(x,y) if_active(x,y,z) -> if(x,y,z) if_active(false(),x,y) -> mark(y) if_active(true(),x,y) -> mark(x) mark(0()) -> 0() mark(div(x,y)) -> div_active(mark(x),y) mark(ge(x,y)) -> ge_active(x,y) mark(if(x,y,z)) -> if_active(mark(x),y,z) mark(minus(x,y)) -> minus_active(x,y) mark(s(x)) -> s(mark(x)) minus_active(x,y) -> minus(x,y) minus_active(0(),y) -> 0() minus_active(s(x),s(y)) -> minus_active(x,y) - Signature: {div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2,div_active#/2,ge_active#/2,if_active#/3,mark#/1 ,minus_active#/2} / {0/0,div/2,false/0,ge/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_12/2,c_13/1,c_14/2,c_15/1,c_16/1,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {div_active#,ge_active#,if_active#,mark# ,minus_active#} and constructors {0,div,false,ge,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_12) = {1,2}, uargs(c_14) = {1,2}, uargs(c_16) = {1} Following symbols are considered usable: {div_active#,ge_active#,if_active#,mark#,minus_active#} TcT has computed the following interpretation: p(0) = [0] p(div) = [1] x1 + [0] p(div_active) = [0] p(false) = [6] p(ge) = [2] p(ge_active) = [4] x1 + [6] x2 + [4] p(if) = [1] x1 + [1] x2 + [1] x3 + [4] p(if_active) = [1] x1 + [2] p(mark) = [0] p(minus) = [0] p(minus_active) = [4] x2 + [0] p(s) = [1] x1 + [0] p(true) = [4] p(div_active#) = [0] p(ge_active#) = [1] x1 + [1] p(if_active#) = [1] x2 + [1] x3 + [0] p(mark#) = [1] x1 + [0] p(minus_active#) = [2] x2 + [0] p(c_1) = [0] p(c_2) = [1] p(c_3) = [4] x1 + [0] p(c_4) = [1] p(c_5) = [4] p(c_6) = [4] p(c_7) = [1] x1 + [1] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [1] x1 + [0] p(c_11) = [1] p(c_12) = [4] x1 + [1] x2 + [0] p(c_13) = [0] p(c_14) = [1] x1 + [1] x2 + [3] p(c_15) = [1] x1 + [0] p(c_16) = [1] x1 + [0] p(c_17) = [0] p(c_18) = [1] p(c_19) = [4] x1 + [0] Following rules are strictly oriented: mark#(if(x,y,z)) = [1] x + [1] y + [1] z + [4] > [1] x + [1] y + [1] z + [3] = c_14(if_active#(mark(x),y,z),mark#(x)) Following rules are (at-least) weakly oriented: div_active#(s(x),s(y)) = [0] >= [0] = c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())) if_active#(false(),x,y) = [1] x + [1] y + [0] >= [1] y + [0] = c_9(mark#(y)) if_active#(true(),x,y) = [1] x + [1] y + [0] >= [1] x + [0] = c_10(mark#(x)) mark#(div(x,y)) = [1] x + [0] >= [1] x + [0] = c_12(div_active#(mark(x),y),mark#(x)) mark#(s(x)) = [1] x + [0] >= [1] x + [0] = c_16(mark#(x)) ** Step 4.a:5: Ara WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)) mark#(s(x)) -> c_16(mark#(x)) - Weak DPs: div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())) if_active#(false(),x,y) -> c_9(mark#(y)) if_active#(true(),x,y) -> c_10(mark#(x)) mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)) - Weak TRS: div_active(x,y) -> div(x,y) div_active(0(),s(y)) -> 0() div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) -> ge(x,y) ge_active(x,0()) -> true() ge_active(0(),s(y)) -> false() ge_active(s(x),s(y)) -> ge_active(x,y) if_active(x,y,z) -> if(x,y,z) if_active(false(),x,y) -> mark(y) if_active(true(),x,y) -> mark(x) mark(0()) -> 0() mark(div(x,y)) -> div_active(mark(x),y) mark(ge(x,y)) -> ge_active(x,y) mark(if(x,y,z)) -> if_active(mark(x),y,z) mark(minus(x,y)) -> minus_active(x,y) mark(s(x)) -> s(mark(x)) minus_active(x,y) -> minus(x,y) minus_active(0(),y) -> 0() minus_active(s(x),s(y)) -> minus_active(x,y) - Signature: {div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2,div_active#/2,ge_active#/2,if_active#/3,mark#/1 ,minus_active#/2} / {0/0,div/2,false/0,ge/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_12/2,c_13/1,c_14/2,c_15/1,c_16/1,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {div_active#,ge_active#,if_active#,mark# ,minus_active#} and constructors {0,div,false,ge,if,minus,s,true} + Applied Processor: Ara {araHeuristics = NoHeuristics, minDegree = 2, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1} + Details: Signatures used: ---------------- 0 :: [] -(0)-> "A"(8, 0) 0 :: [] -(0)-> "A"(0, 0) 0 :: [] -(0)-> "A"(15, 7) 0 :: [] -(0)-> "A"(13, 5) 0 :: [] -(0)-> "A"(14, 14) div :: ["A"(8, 8) x "A"(0, 0)] -(8)-> "A"(0, 8) div :: ["A"(8, 0) x "A"(0, 0)] -(0)-> "A"(8, 0) div :: ["A"(10, 0) x "A"(0, 0)] -(0)-> "A"(10, 0) div :: ["A"(12, 8) x "A"(0, 0)] -(8)-> "A"(4, 8) div_active :: ["A"(8, 0) x "A"(0, 0)] -(0)-> "A"(8, 0) false :: [] -(0)-> "A"(8, 0) false :: [] -(0)-> "A"(5, 0) false :: [] -(0)-> "A"(15, 14) ge :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(8, 0) ge :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(14, 14) ge_active :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(14, 12) if :: ["A"(8, 0) x "A"(8, 0) x "A"(8, 0)] -(0)-> "A"(8, 0) if :: ["A"(8, 8) x "A"(8, 8) x "A"(8, 8)] -(0)-> "A"(0, 8) if_active :: ["A"(8, 0) x "A"(8, 0) x "A"(8, 0)] -(0)-> "A"(8, 0) mark :: ["A"(8, 0)] -(0)-> "A"(8, 0) minus :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(8, 0) minus :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(15, 11) minus :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(14, 6) minus :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(14, 13) minus_active :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(11, 3) s :: ["A"(0, 8)] -(0)-> "A"(0, 8) s :: ["A"(0, 0)] -(0)-> "A"(0, 0) s :: ["A"(8, 0)] -(8)-> "A"(8, 0) true :: [] -(0)-> "A"(8, 0) true :: [] -(0)-> "A"(5, 0) true :: [] -(0)-> "A"(15, 14) div_active# :: ["A"(8, 0) x "A"(0, 0)] -(0)-> "A"(1, 1) if_active# :: ["A"(5, 0) x "A"(0, 8) x "A"(8, 8)] -(0)-> "A"(3, 11) mark# :: ["A"(0, 8)] -(0)-> "A"(8, 1) c_3 :: ["A"(0, 8)] -(0)-> "A"(1, 8) c_9 :: ["A"(0, 0)] -(0)-> "A"(3, 15) c_10 :: ["A"(0, 0)] -(0)-> "A"(7, 14) c_12 :: ["A"(0, 0) x "A"(2, 0)] -(2)-> "A"(9, 2) c_14 :: ["A"(0, 8) x "A"(0, 0)] -(0)-> "A"(15, 8) c_16 :: ["A"(2, 0)] -(0)-> "A"(12, 2) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "0_A" :: [] -(0)-> "A"(1, 0) "0_A" :: [] -(0)-> "A"(0, 1) "c_10_A" :: ["A"(0)] -(0)-> "A"(1, 0) "c_10_A" :: ["A"(0)] -(0)-> "A"(0, 1) "c_12_A" :: ["A"(0) x "A"(0)] -(0)-> "A"(1, 0) "c_12_A" :: ["A"(0) x "A"(0)] -(1)-> "A"(0, 1) "c_14_A" :: ["A"(0) x "A"(0)] -(0)-> "A"(1, 0) "c_14_A" :: ["A"(0) x "A"(0)] -(0)-> "A"(0, 1) "c_16_A" :: ["A"(0)] -(0)-> "A"(1, 0) "c_16_A" :: ["A"(0)] -(0)-> "A"(0, 1) "c_3_A" :: ["A"(0)] -(0)-> "A"(1, 0) "c_3_A" :: ["A"(0)] -(0)-> "A"(0, 1) "c_9_A" :: ["A"(0)] -(0)-> "A"(1, 0) "c_9_A" :: ["A"(0)] -(0)-> "A"(0, 1) "div_A" :: ["A"(1, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "div_A" :: ["A"(1, 1) x "A"(0, 0)] -(1)-> "A"(0, 1) "false_A" :: [] -(0)-> "A"(1, 0) "false_A" :: [] -(0)-> "A"(0, 1) "ge_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "ge_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(0, 1) "if_A" :: ["A"(1, 0) x "A"(1, 0) x "A"(1, 0)] -(0)-> "A"(1, 0) "if_A" :: ["A"(1, 1) x "A"(1, 1) x "A"(1, 1)] -(0)-> "A"(0, 1) "minus_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "minus_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(0, 1) "s_A" :: ["A"(1, 0)] -(1)-> "A"(1, 0) "s_A" :: ["A"(0, 1)] -(0)-> "A"(0, 1) "true_A" :: [] -(0)-> "A"(1, 0) "true_A" :: [] -(0)-> "A"(0, 1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)) 2. Weak: mark#(s(x)) -> c_16(mark#(x)) ** Step 4.a:6: Ara WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: mark#(s(x)) -> c_16(mark#(x)) - Weak DPs: div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())) if_active#(false(),x,y) -> c_9(mark#(y)) if_active#(true(),x,y) -> c_10(mark#(x)) mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)) mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)) - Weak TRS: div_active(x,y) -> div(x,y) div_active(0(),s(y)) -> 0() div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) -> ge(x,y) ge_active(x,0()) -> true() ge_active(0(),s(y)) -> false() ge_active(s(x),s(y)) -> ge_active(x,y) if_active(x,y,z) -> if(x,y,z) if_active(false(),x,y) -> mark(y) if_active(true(),x,y) -> mark(x) mark(0()) -> 0() mark(div(x,y)) -> div_active(mark(x),y) mark(ge(x,y)) -> ge_active(x,y) mark(if(x,y,z)) -> if_active(mark(x),y,z) mark(minus(x,y)) -> minus_active(x,y) mark(s(x)) -> s(mark(x)) minus_active(x,y) -> minus(x,y) minus_active(0(),y) -> 0() minus_active(s(x),s(y)) -> minus_active(x,y) - Signature: {div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2,div_active#/2,ge_active#/2,if_active#/3,mark#/1 ,minus_active#/2} / {0/0,div/2,false/0,ge/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_12/2,c_13/1,c_14/2,c_15/1,c_16/1,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {div_active#,ge_active#,if_active#,mark# ,minus_active#} and constructors {0,div,false,ge,if,minus,s,true} + Applied Processor: Ara {araHeuristics = NoHeuristics, minDegree = 2, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1} + Details: Signatures used: ---------------- 0 :: [] -(0)-> "A"(1, 0) 0 :: [] -(0)-> "A"(0, 0) 0 :: [] -(0)-> "A"(7, 7) 0 :: [] -(0)-> "A"(13, 13) 0 :: [] -(0)-> "A"(6, 6) div :: ["A"(1, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) div :: ["A"(1, 1) x "A"(0, 1)] -(1)-> "A"(0, 1) div :: ["A"(2, 0) x "A"(0, 0)] -(0)-> "A"(2, 0) div_active :: ["A"(1, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) false :: [] -(0)-> "A"(1, 0) false :: [] -(0)-> "A"(0, 0) false :: [] -(0)-> "A"(15, 14) ge :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) ge :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(12, 13) ge_active :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(9, 12) if :: ["A"(1, 0) x "A"(1, 0) x "A"(1, 0)] -(0)-> "A"(1, 0) if :: ["A"(1, 1) x "A"(0, 1) x "A"(0, 1)] -(1)-> "A"(0, 1) if_active :: ["A"(1, 0) x "A"(1, 0) x "A"(1, 0)] -(0)-> "A"(1, 0) mark :: ["A"(1, 0)] -(0)-> "A"(1, 0) minus :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) minus :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(9, 4) minus :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(9, 8) minus :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(3, 2) minus_active :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(7, 7) s :: ["A"(0, 1)] -(1)-> "A"(0, 1) s :: ["A"(0, 0)] -(0)-> "A"(0, 0) s :: ["A"(1, 0)] -(1)-> "A"(1, 0) true :: [] -(0)-> "A"(1, 0) true :: [] -(0)-> "A"(0, 0) true :: [] -(0)-> "A"(15, 14) div_active# :: ["A"(1, 0) x "A"(0, 1)] -(1)-> "A"(1, 1) if_active# :: ["A"(0, 0) x "A"(0, 1) x "A"(0, 1)] -(0)-> "A"(5, 13) mark# :: ["A"(0, 1)] -(0)-> "A"(1, 0) c_3 :: ["A"(4, 3)] -(0)-> "A"(1, 3) c_9 :: ["A"(0, 0)] -(0)-> "A"(5, 15) c_10 :: ["A"(0, 0)] -(0)-> "A"(14, 13) c_12 :: ["A"(1, 0) x "A"(0, 0)] -(0)-> "A"(7, 1) c_14 :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(3, 0) c_16 :: ["A"(1, 0)] -(0)-> "A"(1, 4) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "0_A" :: [] -(0)-> "A"(1, 0) "0_A" :: [] -(0)-> "A"(0, 1) "c_10_A" :: ["A"(0)] -(0)-> "A"(1, 0) "c_10_A" :: ["A"(0)] -(0)-> "A"(0, 1) "c_12_A" :: ["A"(0) x "A"(0)] -(0)-> "A"(1, 0) "c_12_A" :: ["A"(0) x "A"(0)] -(0)-> "A"(0, 1) "c_14_A" :: ["A"(0) x "A"(0)] -(0)-> "A"(1, 0) "c_14_A" :: ["A"(0) x "A"(0)] -(1)-> "A"(0, 1) "c_16_A" :: ["A"(0)] -(0)-> "A"(1, 0) "c_16_A" :: ["A"(0)] -(0)-> "A"(0, 1) "c_3_A" :: ["A"(0)] -(0)-> "A"(1, 0) "c_3_A" :: ["A"(0)] -(0)-> "A"(0, 1) "c_9_A" :: ["A"(0)] -(0)-> "A"(1, 0) "c_9_A" :: ["A"(0)] -(0)-> "A"(0, 1) "div_A" :: ["A"(1, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "div_A" :: ["A"(1, 1) x "A"(0, 1)] -(1)-> "A"(0, 1) "false_A" :: [] -(0)-> "A"(1, 0) "false_A" :: [] -(0)-> "A"(0, 1) "ge_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "ge_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(0, 1) "if_A" :: ["A"(1, 0) x "A"(1, 0) x "A"(1, 0)] -(0)-> "A"(1, 0) "if_A" :: ["A"(1, 1) x "A"(0, 1) x "A"(0, 1)] -(1)-> "A"(0, 1) "minus_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "minus_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(0, 1) "s_A" :: ["A"(1, 0)] -(1)-> "A"(1, 0) "s_A" :: ["A"(0, 1)] -(1)-> "A"(0, 1) "true_A" :: [] -(0)-> "A"(1, 0) "true_A" :: [] -(0)-> "A"(0, 1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: mark#(s(x)) -> c_16(mark#(x)) 2. Weak: ** Step 4.b:1: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: ge_active#(s(x),s(y)) -> c_7(ge_active#(x,y)) mark#(ge(x,y)) -> c_13(ge_active#(x,y)) mark#(minus(x,y)) -> c_15(minus_active#(x,y)) minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)) - Weak DPs: div_active#(s(x),s(y)) -> ge_active#(x,y) div_active#(s(x),s(y)) -> if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0()) if_active#(false(),x,y) -> mark#(y) if_active#(true(),x,y) -> mark#(x) mark#(div(x,y)) -> div_active#(mark(x),y) mark#(div(x,y)) -> mark#(x) mark#(if(x,y,z)) -> if_active#(mark(x),y,z) mark#(if(x,y,z)) -> mark#(x) mark#(s(x)) -> mark#(x) - Weak TRS: div_active(x,y) -> div(x,y) div_active(0(),s(y)) -> 0() div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) -> ge(x,y) ge_active(x,0()) -> true() ge_active(0(),s(y)) -> false() ge_active(s(x),s(y)) -> ge_active(x,y) if_active(x,y,z) -> if(x,y,z) if_active(false(),x,y) -> mark(y) if_active(true(),x,y) -> mark(x) mark(0()) -> 0() mark(div(x,y)) -> div_active(mark(x),y) mark(ge(x,y)) -> ge_active(x,y) mark(if(x,y,z)) -> if_active(mark(x),y,z) mark(minus(x,y)) -> minus_active(x,y) mark(s(x)) -> s(mark(x)) minus_active(x,y) -> minus(x,y) minus_active(0(),y) -> 0() minus_active(s(x),s(y)) -> minus_active(x,y) - Signature: {div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2,div_active#/2,ge_active#/2,if_active#/3,mark#/1 ,minus_active#/2} / {0/0,div/2,false/0,ge/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_12/2,c_13/1,c_14/2,c_15/1,c_16/1,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {div_active#,ge_active#,if_active#,mark# ,minus_active#} and constructors {0,div,false,ge,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(div_active) = {1}, uargs(if_active) = {1}, uargs(s) = {1}, uargs(div_active#) = {1}, uargs(if_active#) = {1}, uargs(c_7) = {1}, uargs(c_13) = {1}, uargs(c_15) = {1}, uargs(c_19) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(div) = [0] p(div_active) = [1] x1 + [0] p(false) = [0] p(ge) = [0] p(ge_active) = [0] p(if) = [0] p(if_active) = [1] x1 + [0] p(mark) = [0] p(minus) = [0] p(minus_active) = [0] p(s) = [1] x1 + [0] p(true) = [0] p(div_active#) = [1] x1 + [1] p(ge_active#) = [0] p(if_active#) = [1] x1 + [1] p(mark#) = [1] p(minus_active#) = [4] p(c_1) = [2] p(c_2) = [0] p(c_3) = [2] x1 + [1] x2 + [0] p(c_4) = [2] p(c_5) = [2] p(c_6) = [0] p(c_7) = [1] x1 + [0] p(c_8) = [1] p(c_9) = [1] x1 + [4] p(c_10) = [1] x1 + [0] p(c_11) = [1] p(c_12) = [2] x2 + [0] p(c_13) = [1] x1 + [0] p(c_14) = [4] x1 + [2] x2 + [0] p(c_15) = [1] x1 + [4] p(c_16) = [1] x1 + [4] p(c_17) = [0] p(c_18) = [1] p(c_19) = [1] x1 + [0] Following rules are strictly oriented: mark#(ge(x,y)) = [1] > [0] = c_13(ge_active#(x,y)) Following rules are (at-least) weakly oriented: div_active#(s(x),s(y)) = [1] x + [1] >= [0] = ge_active#(x,y) div_active#(s(x),s(y)) = [1] x + [1] >= [1] = if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active#(s(x),s(y)) = [0] >= [0] = c_7(ge_active#(x,y)) if_active#(false(),x,y) = [1] >= [1] = mark#(y) if_active#(true(),x,y) = [1] >= [1] = mark#(x) mark#(div(x,y)) = [1] >= [1] = div_active#(mark(x),y) mark#(div(x,y)) = [1] >= [1] = mark#(x) mark#(if(x,y,z)) = [1] >= [1] = if_active#(mark(x),y,z) mark#(if(x,y,z)) = [1] >= [1] = mark#(x) mark#(minus(x,y)) = [1] >= [8] = c_15(minus_active#(x,y)) mark#(s(x)) = [1] >= [1] = mark#(x) minus_active#(s(x),s(y)) = [4] >= [4] = c_19(minus_active#(x,y)) div_active(x,y) = [1] x + [0] >= [0] = div(x,y) div_active(0(),s(y)) = [0] >= [0] = 0() div_active(s(x),s(y)) = [1] x + [0] >= [0] = if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) = [0] >= [0] = ge(x,y) ge_active(x,0()) = [0] >= [0] = true() ge_active(0(),s(y)) = [0] >= [0] = false() ge_active(s(x),s(y)) = [0] >= [0] = ge_active(x,y) if_active(x,y,z) = [1] x + [0] >= [0] = if(x,y,z) if_active(false(),x,y) = [0] >= [0] = mark(y) if_active(true(),x,y) = [0] >= [0] = mark(x) mark(0()) = [0] >= [0] = 0() mark(div(x,y)) = [0] >= [0] = div_active(mark(x),y) mark(ge(x,y)) = [0] >= [0] = ge_active(x,y) mark(if(x,y,z)) = [0] >= [0] = if_active(mark(x),y,z) mark(minus(x,y)) = [0] >= [0] = minus_active(x,y) mark(s(x)) = [0] >= [0] = s(mark(x)) minus_active(x,y) = [0] >= [0] = minus(x,y) minus_active(0(),y) = [0] >= [0] = 0() minus_active(s(x),s(y)) = [0] >= [0] = minus_active(x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 4.b:2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: ge_active#(s(x),s(y)) -> c_7(ge_active#(x,y)) mark#(minus(x,y)) -> c_15(minus_active#(x,y)) minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)) - Weak DPs: div_active#(s(x),s(y)) -> ge_active#(x,y) div_active#(s(x),s(y)) -> if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0()) if_active#(false(),x,y) -> mark#(y) if_active#(true(),x,y) -> mark#(x) mark#(div(x,y)) -> div_active#(mark(x),y) mark#(div(x,y)) -> mark#(x) mark#(ge(x,y)) -> c_13(ge_active#(x,y)) mark#(if(x,y,z)) -> if_active#(mark(x),y,z) mark#(if(x,y,z)) -> mark#(x) mark#(s(x)) -> mark#(x) - Weak TRS: div_active(x,y) -> div(x,y) div_active(0(),s(y)) -> 0() div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) -> ge(x,y) ge_active(x,0()) -> true() ge_active(0(),s(y)) -> false() ge_active(s(x),s(y)) -> ge_active(x,y) if_active(x,y,z) -> if(x,y,z) if_active(false(),x,y) -> mark(y) if_active(true(),x,y) -> mark(x) mark(0()) -> 0() mark(div(x,y)) -> div_active(mark(x),y) mark(ge(x,y)) -> ge_active(x,y) mark(if(x,y,z)) -> if_active(mark(x),y,z) mark(minus(x,y)) -> minus_active(x,y) mark(s(x)) -> s(mark(x)) minus_active(x,y) -> minus(x,y) minus_active(0(),y) -> 0() minus_active(s(x),s(y)) -> minus_active(x,y) - Signature: {div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2,div_active#/2,ge_active#/2,if_active#/3,mark#/1 ,minus_active#/2} / {0/0,div/2,false/0,ge/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_12/2,c_13/1,c_14/2,c_15/1,c_16/1,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {div_active#,ge_active#,if_active#,mark# ,minus_active#} and constructors {0,div,false,ge,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(div_active) = {1}, uargs(if_active) = {1}, uargs(s) = {1}, uargs(div_active#) = {1}, uargs(if_active#) = {1}, uargs(c_7) = {1}, uargs(c_13) = {1}, uargs(c_15) = {1}, uargs(c_19) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(div) = [1] x1 + [0] p(div_active) = [1] x1 + [0] p(false) = [0] p(ge) = [1] x1 + [0] p(ge_active) = [1] x1 + [0] p(if) = [1] x1 + [1] x2 + [1] x3 + [0] p(if_active) = [1] x1 + [1] x2 + [1] x3 + [0] p(mark) = [1] x1 + [0] p(minus) = [0] p(minus_active) = [0] p(s) = [1] x1 + [0] p(true) = [0] p(div_active#) = [1] x1 + [2] p(ge_active#) = [1] x1 + [2] p(if_active#) = [1] x1 + [1] x2 + [1] x3 + [2] p(mark#) = [1] x1 + [2] p(minus_active#) = [0] p(c_1) = [1] p(c_2) = [1] p(c_3) = [2] x1 + [2] x2 + [0] p(c_4) = [1] p(c_5) = [1] p(c_6) = [1] p(c_7) = [1] x1 + [6] p(c_8) = [0] p(c_9) = [4] x1 + [0] p(c_10) = [4] x1 + [0] p(c_11) = [0] p(c_12) = [1] x1 + [1] x2 + [2] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [2] x2 + [1] p(c_15) = [1] x1 + [0] p(c_16) = [1] x1 + [0] p(c_17) = [0] p(c_18) = [0] p(c_19) = [1] x1 + [2] Following rules are strictly oriented: mark#(minus(x,y)) = [2] > [0] = c_15(minus_active#(x,y)) Following rules are (at-least) weakly oriented: div_active#(s(x),s(y)) = [1] x + [2] >= [1] x + [2] = ge_active#(x,y) div_active#(s(x),s(y)) = [1] x + [2] >= [1] x + [2] = if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active#(s(x),s(y)) = [1] x + [2] >= [1] x + [8] = c_7(ge_active#(x,y)) if_active#(false(),x,y) = [1] x + [1] y + [2] >= [1] y + [2] = mark#(y) if_active#(true(),x,y) = [1] x + [1] y + [2] >= [1] x + [2] = mark#(x) mark#(div(x,y)) = [1] x + [2] >= [1] x + [2] = div_active#(mark(x),y) mark#(div(x,y)) = [1] x + [2] >= [1] x + [2] = mark#(x) mark#(ge(x,y)) = [1] x + [2] >= [1] x + [2] = c_13(ge_active#(x,y)) mark#(if(x,y,z)) = [1] x + [1] y + [1] z + [2] >= [1] x + [1] y + [1] z + [2] = if_active#(mark(x),y,z) mark#(if(x,y,z)) = [1] x + [1] y + [1] z + [2] >= [1] x + [2] = mark#(x) mark#(s(x)) = [1] x + [2] >= [1] x + [2] = mark#(x) minus_active#(s(x),s(y)) = [0] >= [2] = c_19(minus_active#(x,y)) div_active(x,y) = [1] x + [0] >= [1] x + [0] = div(x,y) div_active(0(),s(y)) = [0] >= [0] = 0() div_active(s(x),s(y)) = [1] x + [0] >= [1] x + [0] = if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) = [1] x + [0] >= [1] x + [0] = ge(x,y) ge_active(x,0()) = [1] x + [0] >= [0] = true() ge_active(0(),s(y)) = [0] >= [0] = false() ge_active(s(x),s(y)) = [1] x + [0] >= [1] x + [0] = ge_active(x,y) if_active(x,y,z) = [1] x + [1] y + [1] z + [0] >= [1] x + [1] y + [1] z + [0] = if(x,y,z) if_active(false(),x,y) = [1] x + [1] y + [0] >= [1] y + [0] = mark(y) if_active(true(),x,y) = [1] x + [1] y + [0] >= [1] x + [0] = mark(x) mark(0()) = [0] >= [0] = 0() mark(div(x,y)) = [1] x + [0] >= [1] x + [0] = div_active(mark(x),y) mark(ge(x,y)) = [1] x + [0] >= [1] x + [0] = ge_active(x,y) mark(if(x,y,z)) = [1] x + [1] y + [1] z + [0] >= [1] x + [1] y + [1] z + [0] = if_active(mark(x),y,z) mark(minus(x,y)) = [0] >= [0] = minus_active(x,y) mark(s(x)) = [1] x + [0] >= [1] x + [0] = s(mark(x)) minus_active(x,y) = [0] >= [0] = minus(x,y) minus_active(0(),y) = [0] >= [0] = 0() minus_active(s(x),s(y)) = [0] >= [0] = minus_active(x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 4.b:3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: ge_active#(s(x),s(y)) -> c_7(ge_active#(x,y)) minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)) - Weak DPs: div_active#(s(x),s(y)) -> ge_active#(x,y) div_active#(s(x),s(y)) -> if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0()) if_active#(false(),x,y) -> mark#(y) if_active#(true(),x,y) -> mark#(x) mark#(div(x,y)) -> div_active#(mark(x),y) mark#(div(x,y)) -> mark#(x) mark#(ge(x,y)) -> c_13(ge_active#(x,y)) mark#(if(x,y,z)) -> if_active#(mark(x),y,z) mark#(if(x,y,z)) -> mark#(x) mark#(minus(x,y)) -> c_15(minus_active#(x,y)) mark#(s(x)) -> mark#(x) - Weak TRS: div_active(x,y) -> div(x,y) div_active(0(),s(y)) -> 0() div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) -> ge(x,y) ge_active(x,0()) -> true() ge_active(0(),s(y)) -> false() ge_active(s(x),s(y)) -> ge_active(x,y) if_active(x,y,z) -> if(x,y,z) if_active(false(),x,y) -> mark(y) if_active(true(),x,y) -> mark(x) mark(0()) -> 0() mark(div(x,y)) -> div_active(mark(x),y) mark(ge(x,y)) -> ge_active(x,y) mark(if(x,y,z)) -> if_active(mark(x),y,z) mark(minus(x,y)) -> minus_active(x,y) mark(s(x)) -> s(mark(x)) minus_active(x,y) -> minus(x,y) minus_active(0(),y) -> 0() minus_active(s(x),s(y)) -> minus_active(x,y) - Signature: {div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2,div_active#/2,ge_active#/2,if_active#/3,mark#/1 ,minus_active#/2} / {0/0,div/2,false/0,ge/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_12/2,c_13/1,c_14/2,c_15/1,c_16/1,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {div_active#,ge_active#,if_active#,mark# ,minus_active#} and constructors {0,div,false,ge,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(div_active) = {1}, uargs(if_active) = {1}, uargs(s) = {1}, uargs(div_active#) = {1}, uargs(if_active#) = {1}, uargs(c_7) = {1}, uargs(c_13) = {1}, uargs(c_15) = {1}, uargs(c_19) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(div) = [1] x1 + [0] p(div_active) = [1] x1 + [0] p(false) = [0] p(ge) = [0] p(ge_active) = [0] p(if) = [1] x1 + [1] x2 + [1] x3 + [0] p(if_active) = [1] x1 + [1] x2 + [1] x3 + [0] p(mark) = [1] x1 + [0] p(minus) = [1] x1 + [0] p(minus_active) = [1] x1 + [0] p(s) = [1] x1 + [1] p(true) = [0] p(div_active#) = [1] x1 + [0] p(ge_active#) = [0] p(if_active#) = [1] x1 + [1] x2 + [1] x3 + [0] p(mark#) = [1] x1 + [0] p(minus_active#) = [1] x1 + [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [2] p(c_4) = [1] p(c_5) = [0] p(c_6) = [1] p(c_7) = [1] x1 + [2] p(c_8) = [1] p(c_9) = [2] x1 + [1] p(c_10) = [1] p(c_11) = [1] p(c_12) = [2] x1 + [1] x2 + [1] p(c_13) = [1] x1 + [0] p(c_14) = [1] x2 + [1] p(c_15) = [1] x1 + [0] p(c_16) = [2] p(c_17) = [0] p(c_18) = [0] p(c_19) = [1] x1 + [0] Following rules are strictly oriented: minus_active#(s(x),s(y)) = [1] x + [1] > [1] x + [0] = c_19(minus_active#(x,y)) Following rules are (at-least) weakly oriented: div_active#(s(x),s(y)) = [1] x + [1] >= [0] = ge_active#(x,y) div_active#(s(x),s(y)) = [1] x + [1] >= [1] x + [1] = if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active#(s(x),s(y)) = [0] >= [2] = c_7(ge_active#(x,y)) if_active#(false(),x,y) = [1] x + [1] y + [0] >= [1] y + [0] = mark#(y) if_active#(true(),x,y) = [1] x + [1] y + [0] >= [1] x + [0] = mark#(x) mark#(div(x,y)) = [1] x + [0] >= [1] x + [0] = div_active#(mark(x),y) mark#(div(x,y)) = [1] x + [0] >= [1] x + [0] = mark#(x) mark#(ge(x,y)) = [0] >= [0] = c_13(ge_active#(x,y)) mark#(if(x,y,z)) = [1] x + [1] y + [1] z + [0] >= [1] x + [1] y + [1] z + [0] = if_active#(mark(x),y,z) mark#(if(x,y,z)) = [1] x + [1] y + [1] z + [0] >= [1] x + [0] = mark#(x) mark#(minus(x,y)) = [1] x + [0] >= [1] x + [0] = c_15(minus_active#(x,y)) mark#(s(x)) = [1] x + [1] >= [1] x + [0] = mark#(x) div_active(x,y) = [1] x + [0] >= [1] x + [0] = div(x,y) div_active(0(),s(y)) = [0] >= [0] = 0() div_active(s(x),s(y)) = [1] x + [1] >= [1] x + [1] = if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) = [0] >= [0] = ge(x,y) ge_active(x,0()) = [0] >= [0] = true() ge_active(0(),s(y)) = [0] >= [0] = false() ge_active(s(x),s(y)) = [0] >= [0] = ge_active(x,y) if_active(x,y,z) = [1] x + [1] y + [1] z + [0] >= [1] x + [1] y + [1] z + [0] = if(x,y,z) if_active(false(),x,y) = [1] x + [1] y + [0] >= [1] y + [0] = mark(y) if_active(true(),x,y) = [1] x + [1] y + [0] >= [1] x + [0] = mark(x) mark(0()) = [0] >= [0] = 0() mark(div(x,y)) = [1] x + [0] >= [1] x + [0] = div_active(mark(x),y) mark(ge(x,y)) = [0] >= [0] = ge_active(x,y) mark(if(x,y,z)) = [1] x + [1] y + [1] z + [0] >= [1] x + [1] y + [1] z + [0] = if_active(mark(x),y,z) mark(minus(x,y)) = [1] x + [0] >= [1] x + [0] = minus_active(x,y) mark(s(x)) = [1] x + [1] >= [1] x + [1] = s(mark(x)) minus_active(x,y) = [1] x + [0] >= [1] x + [0] = minus(x,y) minus_active(0(),y) = [0] >= [0] = 0() minus_active(s(x),s(y)) = [1] x + [1] >= [1] x + [0] = minus_active(x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 4.b:4: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: ge_active#(s(x),s(y)) -> c_7(ge_active#(x,y)) - Weak DPs: div_active#(s(x),s(y)) -> ge_active#(x,y) div_active#(s(x),s(y)) -> if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0()) if_active#(false(),x,y) -> mark#(y) if_active#(true(),x,y) -> mark#(x) mark#(div(x,y)) -> div_active#(mark(x),y) mark#(div(x,y)) -> mark#(x) mark#(ge(x,y)) -> c_13(ge_active#(x,y)) mark#(if(x,y,z)) -> if_active#(mark(x),y,z) mark#(if(x,y,z)) -> mark#(x) mark#(minus(x,y)) -> c_15(minus_active#(x,y)) mark#(s(x)) -> mark#(x) minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)) - Weak TRS: div_active(x,y) -> div(x,y) div_active(0(),s(y)) -> 0() div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) -> ge(x,y) ge_active(x,0()) -> true() ge_active(0(),s(y)) -> false() ge_active(s(x),s(y)) -> ge_active(x,y) if_active(x,y,z) -> if(x,y,z) if_active(false(),x,y) -> mark(y) if_active(true(),x,y) -> mark(x) mark(0()) -> 0() mark(div(x,y)) -> div_active(mark(x),y) mark(ge(x,y)) -> ge_active(x,y) mark(if(x,y,z)) -> if_active(mark(x),y,z) mark(minus(x,y)) -> minus_active(x,y) mark(s(x)) -> s(mark(x)) minus_active(x,y) -> minus(x,y) minus_active(0(),y) -> 0() minus_active(s(x),s(y)) -> minus_active(x,y) - Signature: {div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2,div_active#/2,ge_active#/2,if_active#/3,mark#/1 ,minus_active#/2} / {0/0,div/2,false/0,ge/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_12/2,c_13/1,c_14/2,c_15/1,c_16/1,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {div_active#,ge_active#,if_active#,mark# ,minus_active#} and constructors {0,div,false,ge,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(div_active) = {1}, uargs(if_active) = {1}, uargs(s) = {1}, uargs(div_active#) = {1}, uargs(if_active#) = {1}, uargs(c_7) = {1}, uargs(c_13) = {1}, uargs(c_15) = {1}, uargs(c_19) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(div) = [1] x1 + [0] p(div_active) = [1] x1 + [0] p(false) = [0] p(ge) = [1] x1 + [0] p(ge_active) = [1] x1 + [0] p(if) = [1] x1 + [1] x2 + [1] x3 + [0] p(if_active) = [1] x1 + [1] x2 + [1] x3 + [0] p(mark) = [1] x1 + [0] p(minus) = [0] p(minus_active) = [0] p(s) = [1] x1 + [2] p(true) = [0] p(div_active#) = [1] x1 + [0] p(ge_active#) = [1] x1 + [0] p(if_active#) = [1] x1 + [1] x2 + [1] x3 + [0] p(mark#) = [1] x1 + [0] p(minus_active#) = [0] p(c_1) = [1] p(c_2) = [0] p(c_3) = [1] x1 + [2] x2 + [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [2] p(c_7) = [1] x1 + [1] p(c_8) = [0] p(c_9) = [1] p(c_10) = [0] p(c_11) = [1] p(c_12) = [1] x2 + [4] p(c_13) = [1] x1 + [0] p(c_14) = [1] x1 + [0] p(c_15) = [1] x1 + [0] p(c_16) = [1] p(c_17) = [0] p(c_18) = [2] p(c_19) = [1] x1 + [0] Following rules are strictly oriented: ge_active#(s(x),s(y)) = [1] x + [2] > [1] x + [1] = c_7(ge_active#(x,y)) Following rules are (at-least) weakly oriented: div_active#(s(x),s(y)) = [1] x + [2] >= [1] x + [0] = ge_active#(x,y) div_active#(s(x),s(y)) = [1] x + [2] >= [1] x + [2] = if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0()) if_active#(false(),x,y) = [1] x + [1] y + [0] >= [1] y + [0] = mark#(y) if_active#(true(),x,y) = [1] x + [1] y + [0] >= [1] x + [0] = mark#(x) mark#(div(x,y)) = [1] x + [0] >= [1] x + [0] = div_active#(mark(x),y) mark#(div(x,y)) = [1] x + [0] >= [1] x + [0] = mark#(x) mark#(ge(x,y)) = [1] x + [0] >= [1] x + [0] = c_13(ge_active#(x,y)) mark#(if(x,y,z)) = [1] x + [1] y + [1] z + [0] >= [1] x + [1] y + [1] z + [0] = if_active#(mark(x),y,z) mark#(if(x,y,z)) = [1] x + [1] y + [1] z + [0] >= [1] x + [0] = mark#(x) mark#(minus(x,y)) = [0] >= [0] = c_15(minus_active#(x,y)) mark#(s(x)) = [1] x + [2] >= [1] x + [0] = mark#(x) minus_active#(s(x),s(y)) = [0] >= [0] = c_19(minus_active#(x,y)) div_active(x,y) = [1] x + [0] >= [1] x + [0] = div(x,y) div_active(0(),s(y)) = [0] >= [0] = 0() div_active(s(x),s(y)) = [1] x + [2] >= [1] x + [2] = if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) = [1] x + [0] >= [1] x + [0] = ge(x,y) ge_active(x,0()) = [1] x + [0] >= [0] = true() ge_active(0(),s(y)) = [0] >= [0] = false() ge_active(s(x),s(y)) = [1] x + [2] >= [1] x + [0] = ge_active(x,y) if_active(x,y,z) = [1] x + [1] y + [1] z + [0] >= [1] x + [1] y + [1] z + [0] = if(x,y,z) if_active(false(),x,y) = [1] x + [1] y + [0] >= [1] y + [0] = mark(y) if_active(true(),x,y) = [1] x + [1] y + [0] >= [1] x + [0] = mark(x) mark(0()) = [0] >= [0] = 0() mark(div(x,y)) = [1] x + [0] >= [1] x + [0] = div_active(mark(x),y) mark(ge(x,y)) = [1] x + [0] >= [1] x + [0] = ge_active(x,y) mark(if(x,y,z)) = [1] x + [1] y + [1] z + [0] >= [1] x + [1] y + [1] z + [0] = if_active(mark(x),y,z) mark(minus(x,y)) = [0] >= [0] = minus_active(x,y) mark(s(x)) = [1] x + [2] >= [1] x + [2] = s(mark(x)) minus_active(x,y) = [0] >= [0] = minus(x,y) minus_active(0(),y) = [0] >= [0] = 0() minus_active(s(x),s(y)) = [0] >= [0] = minus_active(x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 4.b:5: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: div_active#(s(x),s(y)) -> ge_active#(x,y) div_active#(s(x),s(y)) -> if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active#(s(x),s(y)) -> c_7(ge_active#(x,y)) if_active#(false(),x,y) -> mark#(y) if_active#(true(),x,y) -> mark#(x) mark#(div(x,y)) -> div_active#(mark(x),y) mark#(div(x,y)) -> mark#(x) mark#(ge(x,y)) -> c_13(ge_active#(x,y)) mark#(if(x,y,z)) -> if_active#(mark(x),y,z) mark#(if(x,y,z)) -> mark#(x) mark#(minus(x,y)) -> c_15(minus_active#(x,y)) mark#(s(x)) -> mark#(x) minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)) - Weak TRS: div_active(x,y) -> div(x,y) div_active(0(),s(y)) -> 0() div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) -> ge(x,y) ge_active(x,0()) -> true() ge_active(0(),s(y)) -> false() ge_active(s(x),s(y)) -> ge_active(x,y) if_active(x,y,z) -> if(x,y,z) if_active(false(),x,y) -> mark(y) if_active(true(),x,y) -> mark(x) mark(0()) -> 0() mark(div(x,y)) -> div_active(mark(x),y) mark(ge(x,y)) -> ge_active(x,y) mark(if(x,y,z)) -> if_active(mark(x),y,z) mark(minus(x,y)) -> minus_active(x,y) mark(s(x)) -> s(mark(x)) minus_active(x,y) -> minus(x,y) minus_active(0(),y) -> 0() minus_active(s(x),s(y)) -> minus_active(x,y) - Signature: {div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2,div_active#/2,ge_active#/2,if_active#/3,mark#/1 ,minus_active#/2} / {0/0,div/2,false/0,ge/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_12/2,c_13/1,c_14/2,c_15/1,c_16/1,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {div_active#,ge_active#,if_active#,mark# ,minus_active#} and constructors {0,div,false,ge,if,minus,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^3))