WORST_CASE(Omega(n^1),O(n^2)) * Step 1: Sum WORST_CASE(Omega(n^1),O(n^2)) + 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: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?) + 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: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: ge_active(x,y){x -> s(x),y -> s(y)} = ge_active(s(x),s(y)) ->^+ ge_active(x,y) = C[ge_active(x,y) = ge_active(x,y){}] ** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^2)) + 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 1.b:2: PredecessorEstimation WORST_CASE(?,O(n^2)) + 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 1.b:3: RemoveWeakSuffixes 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)) 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 1.b:4: PredecessorEstimationCP 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)) 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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 2: ge_active#(s(x),s(y)) -> c_7(ge_active#(x,y)) 6: mark#(ge(x,y)) -> c_13(ge_active#(x,y)) 9: mark#(s(x)) -> c_16(mark#(x)) The strictly oriented rules are moved into the weak component. *** Step 1.b:4.a:1: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 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: NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_3) = {1,2}, uargs(c_7) = {1}, uargs(c_9) = {1}, uargs(c_10) = {1}, uargs(c_12) = {1,2}, uargs(c_13) = {1}, uargs(c_14) = {1,2}, uargs(c_15) = {1}, uargs(c_16) = {1}, uargs(c_19) = {1} Following symbols are considered usable: {div_active,ge_active,if_active,mark,minus_active,div_active#,ge_active#,if_active#,mark#,minus_active#} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(div) = [0 0 0] [0 0 0] [0] [0 1 1] x1 + [0 0 0] x2 + [1] [0 0 1] [0 0 1] [1] p(div_active) = [1 0 0] [0 0 1] [1] [0 1 1] x1 + [0 0 0] x2 + [1] [0 0 1] [0 0 1] [1] p(false) = [0] [0] [1] p(ge) = [0 0 0] [0] [0 0 1] x1 + [1] [0 0 0] [1] p(ge_active) = [0 0 0] [0] [0 0 1] x1 + [1] [0 0 0] [1] p(if) = [0 0 0] [0 0 0] [0 0 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] p(if_active) = [0 0 0] [0 0 1] [0 0 1] [0] [0 1 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] p(mark) = [0 0 1] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(minus) = [0] [0] [0] p(minus_active) = [0] [0] [0] p(s) = [0 0 0] [1] [0 1 0] x1 + [1] [0 0 1] [1] p(true) = [0] [0] [0] p(div_active#) = [0 0 1] [0 0 0] [1] [0 0 0] x1 + [1 0 0] x2 + [1] [0 0 0] [1 1 0] [1] p(ge_active#) = [0 0 1] [0 0 0] [0] [1 0 0] x1 + [0 1 0] x2 + [0] [0 0 0] [0 1 0] [0] p(if_active#) = [0 1 0] [0 1 0] [0] [0 0 0] x2 + [1 0 0] x3 + [1] [0 1 0] [1 0 1] [0] p(mark#) = [0 1 0] [0] [0 0 0] x1 + [0] [0 1 1] [0] p(minus_active#) = [0 0 0] [0 0 0] [0] [0 0 1] x1 + [1 0 0] x2 + [0] [0 0 0] [0 0 0] [0] p(c_1) = [0] [0] [0] p(c_2) = [0] [0] [0] p(c_3) = [1 0 0] [1 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 1 1] [0 0 0] [0] p(c_4) = [0] [0] [0] p(c_5) = [0] [0] [0] p(c_6) = [0] [0] [0] p(c_7) = [1 0 0] [0] [0 0 1] x1 + [0] [0 0 0] [0] p(c_8) = [0] [0] [0] p(c_9) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(c_10) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(c_11) = [0] [0] [0] p(c_12) = [1 0 0] [1 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 0] [0 0 0] [0] p(c_13) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(c_14) = [1 0 0] [1 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 0] [0 0 0] [0] p(c_15) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(c_16) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(c_17) = [0] [0] [0] p(c_18) = [0] [0] [0] p(c_19) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] Following rules are strictly oriented: ge_active#(s(x),s(y)) = [0 0 1] [0 0 0] [1] [0 0 0] x + [0 1 0] y + [2] [0 0 0] [0 1 0] [1] > [0 0 1] [0 0 0] [0] [0 0 0] x + [0 1 0] y + [0] [0 0 0] [0 0 0] [0] = c_7(ge_active#(x,y)) mark#(ge(x,y)) = [0 0 1] [1] [0 0 0] x + [0] [0 0 1] [2] > [0 0 1] [0] [0 0 0] x + [0] [0 0 0] [0] = c_13(ge_active#(x,y)) mark#(s(x)) = [0 1 0] [1] [0 0 0] x + [0] [0 1 1] [2] > [0 1 0] [0] [0 0 0] x + [0] [0 0 0] [0] = c_16(mark#(x)) Following rules are (at-least) weakly oriented: div_active#(s(x),s(y)) = [0 0 1] [0 0 0] [2] [0 0 0] x + [0 0 0] y + [2] [0 0 0] [0 1 0] [3] >= [0 0 1] [2] [0 0 0] x + [0] [0 0 0] [3] = 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) = [0 1 0] [0 1 0] [0] [0 0 0] x + [1 0 0] y + [1] [0 1 0] [1 0 1] [0] >= [0 1 0] [0] [0 0 0] y + [0] [0 0 0] [0] = c_9(mark#(y)) if_active#(true(),x,y) = [0 1 0] [0 1 0] [0] [0 0 0] x + [1 0 0] y + [1] [0 1 0] [1 0 1] [0] >= [0 1 0] [0] [0 0 0] x + [1] [0 0 0] [0] = c_10(mark#(x)) mark#(div(x,y)) = [0 1 1] [0 0 0] [1] [0 0 0] x + [0 0 0] y + [0] [0 1 2] [0 0 1] [2] >= [0 1 1] [1] [0 0 0] x + [0] [0 0 0] [0] = c_12(div_active#(mark(x),y),mark#(x)) mark#(if(x,y,z)) = [0 1 0] [0 1 0] [0 1 0] [0] [0 0 0] x + [0 0 0] y + [0 0 0] z + [0] [0 1 0] [0 1 1] [0 1 1] [0] >= [0 1 0] [0 1 0] [0 1 0] [0] [0 0 0] x + [0 0 0] y + [0 0 0] z + [0] [0 0 0] [0 0 0] [0 0 0] [0] = c_14(if_active#(mark(x),y,z),mark#(x)) mark#(minus(x,y)) = [0] [0] [0] >= [0] [0] [0] = c_15(minus_active#(x,y)) minus_active#(s(x),s(y)) = [0 0 0] [0] [0 0 1] x + [2] [0 0 0] [0] >= [0] [0] [0] = c_19(minus_active#(x,y)) div_active(x,y) = [1 0 0] [0 0 1] [1] [0 1 1] x + [0 0 0] y + [1] [0 0 1] [0 0 1] [1] >= [0 0 0] [0 0 0] [0] [0 1 1] x + [0 0 0] y + [1] [0 0 1] [0 0 1] [1] = div(x,y) div_active(0(),s(y)) = [0 0 1] [2] [0 0 0] y + [1] [0 0 1] [2] >= [0] [0] [0] = 0() div_active(s(x),s(y)) = [0 0 0] [0 0 1] [3] [0 1 1] x + [0 0 0] y + [3] [0 0 1] [0 0 1] [3] >= [0 0 0] [0 0 1] [3] [0 0 1] x + [0 0 0] y + [3] [0 0 0] [0 0 1] [3] = if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) = [0 0 0] [0] [0 0 1] x + [1] [0 0 0] [1] >= [0 0 0] [0] [0 0 1] x + [1] [0 0 0] [1] = ge(x,y) ge_active(x,0()) = [0 0 0] [0] [0 0 1] x + [1] [0 0 0] [1] >= [0] [0] [0] = true() ge_active(0(),s(y)) = [0] [1] [1] >= [0] [0] [1] = false() ge_active(s(x),s(y)) = [0 0 0] [0] [0 0 1] x + [2] [0 0 0] [1] >= [0 0 0] [0] [0 0 1] x + [1] [0 0 0] [1] = ge_active(x,y) if_active(x,y,z) = [0 0 0] [0 0 1] [0 0 1] [0] [0 1 0] x + [0 1 0] y + [0 1 0] z + [0] [0 0 0] [0 0 1] [0 0 1] [0] >= [0 0 0] [0 0 0] [0 0 0] [0] [0 1 0] x + [0 1 0] y + [0 1 0] z + [0] [0 0 0] [0 0 1] [0 0 1] [0] = if(x,y,z) if_active(false(),x,y) = [0 0 1] [0 0 1] [0] [0 1 0] x + [0 1 0] y + [0] [0 0 1] [0 0 1] [0] >= [0 0 1] [0] [0 1 0] y + [0] [0 0 1] [0] = mark(y) if_active(true(),x,y) = [0 0 1] [0 0 1] [0] [0 1 0] x + [0 1 0] y + [0] [0 0 1] [0 0 1] [0] >= [0 0 1] [0] [0 1 0] x + [0] [0 0 1] [0] = mark(x) mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(div(x,y)) = [0 0 1] [0 0 1] [1] [0 1 1] x + [0 0 0] y + [1] [0 0 1] [0 0 1] [1] >= [0 0 1] [0 0 1] [1] [0 1 1] x + [0 0 0] y + [1] [0 0 1] [0 0 1] [1] = div_active(mark(x),y) mark(ge(x,y)) = [0 0 0] [1] [0 0 1] x + [1] [0 0 0] [1] >= [0 0 0] [0] [0 0 1] x + [1] [0 0 0] [1] = ge_active(x,y) mark(if(x,y,z)) = [0 0 0] [0 0 1] [0 0 1] [0] [0 1 0] x + [0 1 0] y + [0 1 0] z + [0] [0 0 0] [0 0 1] [0 0 1] [0] >= [0 0 0] [0 0 1] [0 0 1] [0] [0 1 0] x + [0 1 0] y + [0 1 0] z + [0] [0 0 0] [0 0 1] [0 0 1] [0] = if_active(mark(x),y,z) mark(minus(x,y)) = [0] [0] [0] >= [0] [0] [0] = minus_active(x,y) mark(s(x)) = [0 0 1] [1] [0 1 0] x + [1] [0 0 1] [1] >= [0 0 0] [1] [0 1 0] x + [1] [0 0 1] [1] = s(mark(x)) minus_active(x,y) = [0] [0] [0] >= [0] [0] [0] = minus(x,y) minus_active(0(),y) = [0] [0] [0] >= [0] [0] [0] = 0() minus_active(s(x),s(y)) = [0] [0] [0] >= [0] [0] [0] = minus_active(x,y) *** Step 1.b:4.a:2: Assumption WORST_CASE(?,O(1)) + 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#(minus(x,y)) -> c_15(minus_active#(x,y)) minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)) - Weak DPs: ge_active#(s(x),s(y)) -> c_7(ge_active#(x,y)) mark#(ge(x,y)) -> c_13(ge_active#(x,y)) 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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 1.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 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#(minus(x,y)) -> c_15(minus_active#(x,y)) minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)) - Weak DPs: ge_active#(s(x),s(y)) -> c_7(ge_active#(x,y)) mark#(ge(x,y)) -> c_13(ge_active#(x,y)) 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: 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)) -->_2 ge_active#(s(x),s(y)) -> c_7(ge_active#(x,y)):8 -->_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)):10 -->_1 mark#(ge(x,y)) -> c_13(ge_active#(x,y)):9 -->_1 mark#(minus(x,y)) -> c_15(minus_active#(x,y)):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)):10 -->_1 mark#(ge(x,y)) -> c_13(ge_active#(x,y)):9 -->_1 mark#(minus(x,y)) -> c_15(minus_active#(x,y)):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)):10 -->_2 mark#(ge(x,y)) -> c_13(ge_active#(x,y)):9 -->_2 mark#(minus(x,y)) -> c_15(minus_active#(x,y)):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)):10 -->_2 mark#(ge(x,y)) -> c_13(ge_active#(x,y)):9 -->_2 mark#(minus(x,y)) -> c_15(minus_active#(x,y)):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#(minus(x,y)) -> c_15(minus_active#(x,y)) -->_1 minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)):7 7:S:minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)) -->_1 minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)):7 8:W:ge_active#(s(x),s(y)) -> c_7(ge_active#(x,y)) -->_1 ge_active#(s(x),s(y)) -> c_7(ge_active#(x,y)):8 9:W:mark#(ge(x,y)) -> c_13(ge_active#(x,y)) -->_1 ge_active#(s(x),s(y)) -> c_7(ge_active#(x,y)):8 10:W:mark#(s(x)) -> c_16(mark#(x)) -->_1 mark#(s(x)) -> c_16(mark#(x)):10 -->_1 mark#(ge(x,y)) -> c_13(ge_active#(x,y)):9 -->_1 mark#(minus(x,y)) -> c_15(minus_active#(x,y)):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 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 9: mark#(ge(x,y)) -> c_13(ge_active#(x,y)) 8: ge_active#(s(x),s(y)) -> c_7(ge_active#(x,y)) *** Step 1.b:4.b:2: 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#(minus(x,y)) -> c_15(minus_active#(x,y)) minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)) - Weak DPs: 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)):10 -->_1 mark#(minus(x,y)) -> c_15(minus_active#(x,y)):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)):10 -->_1 mark#(minus(x,y)) -> c_15(minus_active#(x,y)):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)):10 -->_2 mark#(minus(x,y)) -> c_15(minus_active#(x,y)):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)):10 -->_2 mark#(minus(x,y)) -> c_15(minus_active#(x,y)):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#(minus(x,y)) -> c_15(minus_active#(x,y)) -->_1 minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)):7 7:S:minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)) -->_1 minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)):7 10:W:mark#(s(x)) -> c_16(mark#(x)) -->_1 mark#(s(x)) -> c_16(mark#(x)):10 -->_1 mark#(minus(x,y)) -> c_15(minus_active#(x,y)):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 1.b:4.b:3: PredecessorEstimationCP 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#(minus(x,y)) -> c_15(minus_active#(x,y)) minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)) - Weak DPs: 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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())) 7: minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)) The strictly oriented rules are moved into the weak component. **** Step 1.b:4.b:3.a:1: NaturalMI 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#(minus(x,y)) -> c_15(minus_active#(x,y)) minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)) - Weak DPs: 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: NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_3) = {1}, uargs(c_9) = {1}, uargs(c_10) = {1}, uargs(c_12) = {1,2}, uargs(c_14) = {1,2}, uargs(c_15) = {1}, uargs(c_16) = {1}, uargs(c_19) = {1} Following symbols are considered usable: {div_active,ge_active,if_active,mark,minus_active,div_active#,ge_active#,if_active#,mark#,minus_active#} TcT has computed the following interpretation: p(0) = [0] [0] p(div) = [1 2] x1 + [0] [0 1] [0] p(div_active) = [1 2] x1 + [0] [0 1] [0] p(false) = [2] [0] p(ge) = [1] [0] p(ge_active) = [2] [0] p(if) = [1 2] x1 + [1 0] x2 + [1 2] x3 + [0] [0 0] [0 1] [0 1] [0] p(if_active) = [1 2] x1 + [2 0] x2 + [2 2] x3 + [0] [0 0] [0 1] [0 1] [0] p(mark) = [2 0] x1 + [0] [0 1] [0] p(minus) = [0 1] x1 + [0] [0 0] [0] p(minus_active) = [0 2] x1 + [0] [0 0] [0] p(s) = [1 2] x1 + [0] [0 1] [1] p(true) = [0] [0] p(div_active#) = [0 2] x1 + [0] [0 0] [3] p(ge_active#) = [1 2] x1 + [0 0] x2 + [1] [0 2] [1 2] [2] p(if_active#) = [0 0] x1 + [1 0] x2 + [1 2] x3 + [0] [0 1] [0 0] [0 0] [0] p(mark#) = [1 0] x1 + [0] [0 1] [0] p(minus_active#) = [0 1] x1 + [0 0] x2 + [0] [2 1] [2 0] [0] p(c_1) = [0] [0] p(c_2) = [0] [0] p(c_3) = [2 1] x1 + [1] [0 0] [3] p(c_4) = [0] [0] p(c_5) = [0] [0] p(c_6) = [0] [2] p(c_7) = [0] [0] p(c_8) = [0] [2] p(c_9) = [1 0] x1 + [0] [0 0] [0] p(c_10) = [1 0] x1 + [0] [0 0] [0] p(c_11) = [0] [0] p(c_12) = [1 0] x1 + [1 0] x2 + [0] [0 0] [0 0] [0] p(c_13) = [0] [0] p(c_14) = [1 1] x1 + [1 1] x2 + [0] [0 0] [0 0] [0] p(c_15) = [1 0] x1 + [0] [0 0] [0] p(c_16) = [1 2] x1 + [0] [0 0] [0] p(c_17) = [1] [0] p(c_18) = [1] [2] p(c_19) = [1 0] x1 + [0] [3 1] [1] Following rules are strictly oriented: div_active#(s(x),s(y)) = [0 2] x + [2] [0 0] [3] > [0 2] x + [1] [0 0] [3] = c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())) minus_active#(s(x),s(y)) = [0 1] x + [0 0] y + [1] [2 5] [2 4] [1] > [0 1] x + [0 0] y + [0] [2 4] [2 0] [1] = c_19(minus_active#(x,y)) Following rules are (at-least) weakly oriented: if_active#(false(),x,y) = [1 0] x + [1 2] y + [0] [0 0] [0 0] [0] >= [1 0] y + [0] [0 0] [0] = c_9(mark#(y)) if_active#(true(),x,y) = [1 0] x + [1 2] y + [0] [0 0] [0 0] [0] >= [1 0] x + [0] [0 0] [0] = c_10(mark#(x)) mark#(div(x,y)) = [1 2] x + [0] [0 1] [0] >= [1 2] x + [0] [0 0] [0] = c_12(div_active#(mark(x),y),mark#(x)) mark#(if(x,y,z)) = [1 2] x + [1 0] y + [1 2] z + [0] [0 0] [0 1] [0 1] [0] >= [1 2] x + [1 0] y + [1 2] z + [0] [0 0] [0 0] [0 0] [0] = c_14(if_active#(mark(x),y,z),mark#(x)) mark#(minus(x,y)) = [0 1] x + [0] [0 0] [0] >= [0 1] x + [0] [0 0] [0] = c_15(minus_active#(x,y)) mark#(s(x)) = [1 2] x + [0] [0 1] [1] >= [1 2] x + [0] [0 0] [0] = c_16(mark#(x)) div_active(x,y) = [1 2] x + [0] [0 1] [0] >= [1 2] x + [0] [0 1] [0] = div(x,y) div_active(0(),s(y)) = [0] [0] >= [0] [0] = 0() div_active(s(x),s(y)) = [1 4] x + [2] [0 1] [1] >= [0 2] x + [2] [0 0] [1] = if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) = [2] [0] >= [1] [0] = ge(x,y) ge_active(x,0()) = [2] [0] >= [0] [0] = true() ge_active(0(),s(y)) = [2] [0] >= [2] [0] = false() ge_active(s(x),s(y)) = [2] [0] >= [2] [0] = ge_active(x,y) if_active(x,y,z) = [1 2] x + [2 0] y + [2 2] z + [0] [0 0] [0 1] [0 1] [0] >= [1 2] x + [1 0] y + [1 2] z + [0] [0 0] [0 1] [0 1] [0] = if(x,y,z) if_active(false(),x,y) = [2 0] x + [2 2] y + [2] [0 1] [0 1] [0] >= [2 0] y + [0] [0 1] [0] = mark(y) if_active(true(),x,y) = [2 0] x + [2 2] y + [0] [0 1] [0 1] [0] >= [2 0] x + [0] [0 1] [0] = mark(x) mark(0()) = [0] [0] >= [0] [0] = 0() mark(div(x,y)) = [2 4] x + [0] [0 1] [0] >= [2 2] x + [0] [0 1] [0] = div_active(mark(x),y) mark(ge(x,y)) = [2] [0] >= [2] [0] = ge_active(x,y) mark(if(x,y,z)) = [2 4] x + [2 0] y + [2 4] z + [0] [0 0] [0 1] [0 1] [0] >= [2 2] x + [2 0] y + [2 2] z + [0] [0 0] [0 1] [0 1] [0] = if_active(mark(x),y,z) mark(minus(x,y)) = [0 2] x + [0] [0 0] [0] >= [0 2] x + [0] [0 0] [0] = minus_active(x,y) mark(s(x)) = [2 4] x + [0] [0 1] [1] >= [2 2] x + [0] [0 1] [1] = s(mark(x)) minus_active(x,y) = [0 2] x + [0] [0 0] [0] >= [0 1] x + [0] [0 0] [0] = minus(x,y) minus_active(0(),y) = [0] [0] >= [0] [0] = 0() minus_active(s(x),s(y)) = [0 2] x + [2] [0 0] [0] >= [0 2] x + [0] [0 0] [0] = minus_active(x,y) **** Step 1.b:4.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: 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#(minus(x,y)) -> c_15(minus_active#(x,y)) - Weak DPs: div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())) 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_3/1,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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 1.b:4.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 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#(minus(x,y)) -> c_15(minus_active#(x,y)) - Weak DPs: div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())) 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_3/1,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: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:if_active#(false(),x,y) -> c_9(mark#(y)) -->_1 mark#(s(x)) -> c_16(mark#(x)):7 -->_1 mark#(minus(x,y)) -> c_15(minus_active#(x,y)):5 -->_1 mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)):4 -->_1 mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)):3 2:S:if_active#(true(),x,y) -> c_10(mark#(x)) -->_1 mark#(s(x)) -> c_16(mark#(x)):7 -->_1 mark#(minus(x,y)) -> c_15(minus_active#(x,y)):5 -->_1 mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)):4 -->_1 mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)):3 3:S:mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)) -->_2 mark#(s(x)) -> c_16(mark#(x)):7 -->_1 div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())):6 -->_2 mark#(minus(x,y)) -> c_15(minus_active#(x,y)):5 -->_2 mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)):4 -->_2 mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)):3 4:S:mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)) -->_2 mark#(s(x)) -> c_16(mark#(x)):7 -->_2 mark#(minus(x,y)) -> c_15(minus_active#(x,y)):5 -->_2 mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)):4 -->_2 mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)):3 -->_1 if_active#(true(),x,y) -> c_10(mark#(x)):2 -->_1 if_active#(false(),x,y) -> c_9(mark#(y)):1 5:S:mark#(minus(x,y)) -> c_15(minus_active#(x,y)) -->_1 minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)):8 6:W:div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())) -->_1 if_active#(true(),x,y) -> c_10(mark#(x)):2 -->_1 if_active#(false(),x,y) -> c_9(mark#(y)):1 7:W:mark#(s(x)) -> c_16(mark#(x)) -->_1 mark#(s(x)) -> c_16(mark#(x)):7 -->_1 mark#(minus(x,y)) -> c_15(minus_active#(x,y)):5 -->_1 mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)):4 -->_1 mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)):3 8:W:minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)) -->_1 minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)):8 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 8: minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)) **** Step 1.b:4.b:3.b:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 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#(minus(x,y)) -> c_15(minus_active#(x,y)) - Weak DPs: div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())) 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: SimplifyRHS + Details: Consider the dependency graph 1:S:if_active#(false(),x,y) -> c_9(mark#(y)) -->_1 mark#(s(x)) -> c_16(mark#(x)):7 -->_1 mark#(minus(x,y)) -> c_15(minus_active#(x,y)):5 -->_1 mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)):4 -->_1 mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)):3 2:S:if_active#(true(),x,y) -> c_10(mark#(x)) -->_1 mark#(s(x)) -> c_16(mark#(x)):7 -->_1 mark#(minus(x,y)) -> c_15(minus_active#(x,y)):5 -->_1 mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)):4 -->_1 mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)):3 3:S:mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)) -->_2 mark#(s(x)) -> c_16(mark#(x)):7 -->_1 div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())):6 -->_2 mark#(minus(x,y)) -> c_15(minus_active#(x,y)):5 -->_2 mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)):4 -->_2 mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)):3 4:S:mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)) -->_2 mark#(s(x)) -> c_16(mark#(x)):7 -->_2 mark#(minus(x,y)) -> c_15(minus_active#(x,y)):5 -->_2 mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)):4 -->_2 mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)):3 -->_1 if_active#(true(),x,y) -> c_10(mark#(x)):2 -->_1 if_active#(false(),x,y) -> c_9(mark#(y)):1 5:S:mark#(minus(x,y)) -> c_15(minus_active#(x,y)) 6:W:div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())) -->_1 if_active#(true(),x,y) -> c_10(mark#(x)):2 -->_1 if_active#(false(),x,y) -> c_9(mark#(y)):1 7:W:mark#(s(x)) -> c_16(mark#(x)) -->_1 mark#(s(x)) -> c_16(mark#(x)):7 -->_1 mark#(minus(x,y)) -> c_15(minus_active#(x,y)):5 -->_1 mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)):4 -->_1 mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)):3 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: mark#(minus(x,y)) -> c_15() **** Step 1.b:4.b:3.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 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#(minus(x,y)) -> c_15() - Weak DPs: div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())) 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_15/0,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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 5: mark#(minus(x,y)) -> c_15() The strictly oriented rules are moved into the weak component. ***** Step 1.b:4.b:3.b:3.a:1: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 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#(minus(x,y)) -> c_15() - Weak DPs: div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())) 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_15/0,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 = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_3) = {1}, 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,div_active#,ge_active#,if_active#,mark#,minus_active#} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(div) = [1 1 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(div_active) = [1 1 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(false) = [0] [0] [0] p(ge) = [0] [0] [0] p(ge_active) = [0] [0] [0] p(if) = [1 0 0] [1 0 0] [1 0 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [0] [0 0 0] [0 0 0] [0 0 0] [0] p(if_active) = [1 0 0] [1 0 0] [1 0 0] [0] [0 1 0] x1 + [0 1 0] x2 + [0 1 0] x3 + [0] [0 1 0] [0 1 0] [0 1 0] [0] p(mark) = [1 0 0] [0] [0 1 0] x1 + [0] [0 1 0] [0] p(minus) = [1] [0] [0] p(minus_active) = [1] [0] [0] p(s) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [1] p(true) = [0] [0] [0] p(div_active#) = [0 0 1] [0 0 0] [0] [0 0 1] x1 + [1 0 1] x2 + [0] [0 0 0] [1 0 1] [0] p(ge_active#) = [0] [0] [0] p(if_active#) = [1 0 0] [1 0 0] [0] [1 1 1] x2 + [1 0 1] x3 + [0] [0 0 0] [1 0 0] [0] p(mark#) = [1 0 0] [0] [1 0 0] x1 + [1] [0 1 0] [0] p(minus_active#) = [0] [0] [0] p(c_1) = [0] [0] [0] p(c_2) = [0] [0] [0] p(c_3) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(c_4) = [0] [0] [0] p(c_5) = [0] [0] [0] p(c_6) = [0] [0] [0] p(c_7) = [0] [0] [0] p(c_8) = [0] [0] [0] p(c_9) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(c_10) = [1 0 0] [0] [1 0 0] x1 + [0] [0 0 0] [0] p(c_11) = [0] [0] [0] p(c_12) = [1 0 0] [1 0 0] [0] [0 0 0] x1 + [1 0 1] x2 + [0] [0 0 0] [0 0 0] [0] p(c_13) = [0] [0] [0] p(c_14) = [1 0 0] [1 0 0] [0] [0 0 1] x1 + [0 0 0] x2 + [0] [0 0 0] [0 0 0] [0] p(c_15) = [0] [0] [0] p(c_16) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] p(c_17) = [0] [0] [0] p(c_18) = [0] [0] [0] p(c_19) = [0] [0] [0] Following rules are strictly oriented: mark#(minus(x,y)) = [1] [2] [0] > [0] [0] [0] = c_15() Following rules are (at-least) weakly oriented: div_active#(s(x),s(y)) = [0 0 0] [1] [1 0 0] y + [2] [1 0 0] [1] >= [1] [0] [0] = c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())) if_active#(false(),x,y) = [1 0 0] [1 0 0] [0] [1 1 1] x + [1 0 1] y + [0] [0 0 0] [1 0 0] [0] >= [1 0 0] [0] [0 0 0] y + [0] [0 0 0] [0] = c_9(mark#(y)) if_active#(true(),x,y) = [1 0 0] [1 0 0] [0] [1 1 1] x + [1 0 1] y + [0] [0 0 0] [1 0 0] [0] >= [1 0 0] [0] [1 0 0] x + [0] [0 0 0] [0] = c_10(mark#(x)) mark#(div(x,y)) = [1 1 0] [0] [1 1 0] x + [1] [0 1 0] [0] >= [1 1 0] [0] [1 1 0] x + [0] [0 0 0] [0] = c_12(div_active#(mark(x),y),mark#(x)) mark#(if(x,y,z)) = [1 0 0] [1 0 0] [1 0 0] [0] [1 0 0] x + [1 0 0] y + [1 0 0] z + [1] [0 1 0] [0 1 0] [0 1 0] [0] >= [1 0 0] [1 0 0] [1 0 0] [0] [0 0 0] x + [0 0 0] y + [1 0 0] z + [0] [0 0 0] [0 0 0] [0 0 0] [0] = c_14(if_active#(mark(x),y,z),mark#(x)) mark#(s(x)) = [1 0 0] [0] [1 0 0] x + [1] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x + [0] [0 0 0] [1] = c_16(mark#(x)) div_active(x,y) = [1 1 0] [0] [0 1 0] x + [0] [0 0 1] [0] >= [1 1 0] [0] [0 1 0] x + [0] [0 0 0] [0] = div(x,y) div_active(0(),s(y)) = [0] [0] [0] >= [0] [0] [0] = 0() div_active(s(x),s(y)) = [1 0 0] [1] [0 0 0] x + [1] [0 0 0] [1] >= [1] [1] [1] = if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) = [0] [0] [0] >= [0] [0] [0] = ge(x,y) ge_active(x,0()) = [0] [0] [0] >= [0] [0] [0] = true() ge_active(0(),s(y)) = [0] [0] [0] >= [0] [0] [0] = false() ge_active(s(x),s(y)) = [0] [0] [0] >= [0] [0] [0] = ge_active(x,y) if_active(x,y,z) = [1 0 0] [1 0 0] [1 0 0] [0] [0 1 0] x + [0 1 0] y + [0 1 0] z + [0] [0 1 0] [0 1 0] [0 1 0] [0] >= [1 0 0] [1 0 0] [1 0 0] [0] [0 1 0] x + [0 1 0] y + [0 1 0] z + [0] [0 0 0] [0 0 0] [0 0 0] [0] = if(x,y,z) if_active(false(),x,y) = [1 0 0] [1 0 0] [0] [0 1 0] x + [0 1 0] y + [0] [0 1 0] [0 1 0] [0] >= [1 0 0] [0] [0 1 0] y + [0] [0 1 0] [0] = mark(y) if_active(true(),x,y) = [1 0 0] [1 0 0] [0] [0 1 0] x + [0 1 0] y + [0] [0 1 0] [0 1 0] [0] >= [1 0 0] [0] [0 1 0] x + [0] [0 1 0] [0] = mark(x) mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(div(x,y)) = [1 1 0] [0] [0 1 0] x + [0] [0 1 0] [0] >= [1 1 0] [0] [0 1 0] x + [0] [0 1 0] [0] = div_active(mark(x),y) mark(ge(x,y)) = [0] [0] [0] >= [0] [0] [0] = ge_active(x,y) mark(if(x,y,z)) = [1 0 0] [1 0 0] [1 0 0] [0] [0 1 0] x + [0 1 0] y + [0 1 0] z + [0] [0 1 0] [0 1 0] [0 1 0] [0] >= [1 0 0] [1 0 0] [1 0 0] [0] [0 1 0] x + [0 1 0] y + [0 1 0] z + [0] [0 1 0] [0 1 0] [0 1 0] [0] = if_active(mark(x),y,z) mark(minus(x,y)) = [1] [0] [0] >= [1] [0] [0] = minus_active(x,y) mark(s(x)) = [1 0 0] [0] [0 0 0] x + [1] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x + [1] [0 0 0] [1] = s(mark(x)) minus_active(x,y) = [1] [0] [0] >= [1] [0] [0] = minus(x,y) minus_active(0(),y) = [1] [0] [0] >= [0] [0] [0] = 0() minus_active(s(x),s(y)) = [1] [0] [0] >= [1] [0] [0] = minus_active(x,y) ***** Step 1.b:4.b:3.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: 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 DPs: div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())) mark#(minus(x,y)) -> c_15() 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_15/0,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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 1.b:4.b:3.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 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 DPs: div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())) mark#(minus(x,y)) -> c_15() 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_15/0,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:if_active#(false(),x,y) -> c_9(mark#(y)) -->_1 mark#(s(x)) -> c_16(mark#(x)):7 -->_1 mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)):4 -->_1 mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)):3 -->_1 mark#(minus(x,y)) -> c_15():6 2:S:if_active#(true(),x,y) -> c_10(mark#(x)) -->_1 mark#(s(x)) -> c_16(mark#(x)):7 -->_1 mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)):4 -->_1 mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)):3 -->_1 mark#(minus(x,y)) -> c_15():6 3:S:mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)) -->_2 mark#(s(x)) -> c_16(mark#(x)):7 -->_1 div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())):5 -->_2 mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)):4 -->_2 mark#(minus(x,y)) -> c_15():6 -->_2 mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)):3 4:S:mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)) -->_2 mark#(s(x)) -> c_16(mark#(x)):7 -->_2 mark#(minus(x,y)) -> c_15():6 -->_2 mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)):4 -->_2 mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)):3 -->_1 if_active#(true(),x,y) -> c_10(mark#(x)):2 -->_1 if_active#(false(),x,y) -> c_9(mark#(y)):1 5:W:div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())) -->_1 if_active#(true(),x,y) -> c_10(mark#(x)):2 -->_1 if_active#(false(),x,y) -> c_9(mark#(y)):1 6:W:mark#(minus(x,y)) -> c_15() 7:W:mark#(s(x)) -> c_16(mark#(x)) -->_1 mark#(s(x)) -> c_16(mark#(x)):7 -->_1 mark#(minus(x,y)) -> c_15():6 -->_1 mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)):4 -->_1 mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 6: mark#(minus(x,y)) -> c_15() ***** Step 1.b:4.b:3.b:3.b:2: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 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 DPs: div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())) 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_15/0,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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 4: mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)) The strictly oriented rules are moved into the weak component. ****** Step 1.b:4.b:3.b:3.b:2.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: 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 DPs: div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())) 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_15/0,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 first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_3) = {1}, uargs(c_9) = {1}, uargs(c_10) = {1}, uargs(c_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) = [6] p(false) = [0] p(ge) = [1] x1 + [0] p(ge_active) = [0] p(if) = [1] x1 + [1] x2 + [1] x3 + [2] p(if_active) = [4] x1 + [6] p(mark) = [0] p(minus) = [0] p(minus_active) = [2] p(s) = [1] x1 + [0] p(true) = [0] p(div_active#) = [0] p(ge_active#) = [1] p(if_active#) = [4] x2 + [4] x3 + [0] p(mark#) = [4] x1 + [0] p(minus_active#) = [1] x1 + [0] p(c_1) = [0] p(c_2) = [2] p(c_3) = [4] x1 + [0] p(c_4) = [4] p(c_5) = [0] p(c_6) = [2] 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) = [1] x1 + [1] x2 + [0] p(c_13) = [2] x1 + [4] p(c_14) = [1] x1 + [1] x2 + [7] p(c_15) = [0] p(c_16) = [1] x1 + [0] p(c_17) = [1] p(c_18) = [0] p(c_19) = [2] x1 + [1] Following rules are strictly oriented: mark#(if(x,y,z)) = [4] x + [4] y + [4] z + [8] > [4] x + [4] y + [4] z + [7] = 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) = [4] x + [4] y + [0] >= [4] y + [0] = c_9(mark#(y)) if_active#(true(),x,y) = [4] x + [4] y + [0] >= [4] x + [0] = c_10(mark#(x)) mark#(div(x,y)) = [4] x + [0] >= [4] x + [0] = c_12(div_active#(mark(x),y),mark#(x)) mark#(s(x)) = [4] x + [0] >= [4] x + [0] = c_16(mark#(x)) ****** Step 1.b:4.b:3.b:3.b:2.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: 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)) - Weak DPs: div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())) 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_15/0,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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ****** Step 1.b:4.b:3.b:3.b:2.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 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)) - Weak DPs: div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())) 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_15/0,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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: if_active#(false(),x,y) -> c_9(mark#(y)) The strictly oriented rules are moved into the weak component. ******* Step 1.b:4.b:3.b:3.b:2.b:1.a:1: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: 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)) - Weak DPs: div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())) 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_15/0,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 = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_3) = {1}, 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,div_active#,ge_active#,if_active#,mark#,minus_active#} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(div) = [1 0 1] [0 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(div_active) = [1 0 1] [0 0 0] [0] [0 1 0] x1 + [0 0 0] x2 + [0] [0 0 1] [0 0 1] [1] p(false) = [0] [0] [1] p(ge) = [0] [0] [1] p(ge_active) = [0] [0] [1] p(if) = [1 0 1] [1 0 0] [1 1 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] p(if_active) = [1 0 1] [1 0 0] [1 1 0] [0] [0 0 0] x1 + [0 0 1] x2 + [0 0 1] x3 + [0] [0 0 0] [0 0 1] [0 0 1] [0] p(mark) = [1 0 0] [0] [0 0 1] x1 + [0] [0 0 1] [0] p(minus) = [0] [0] [0] p(minus_active) = [0] [0] [0] p(s) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [1] p(true) = [0] [0] [0] p(div_active#) = [0 1 0] [0 0 0] [0] [0 0 0] x1 + [0 0 1] x2 + [1] [0 0 0] [0 0 0] [0] p(ge_active#) = [0] [0] [0] p(if_active#) = [0 0 1] [1 0 0] [1 0 0] [0] [0 0 0] x1 + [1 0 0] x2 + [0 0 1] x3 + [1] [0 0 0] [0 0 0] [0 1 0] [0] p(mark#) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(minus_active#) = [0] [0] [0] p(c_1) = [0] [0] [0] p(c_2) = [0] [0] [0] p(c_3) = [1 0 0] [0] [1 0 0] x1 + [1] [0 0 0] [0] p(c_4) = [0] [0] [0] p(c_5) = [0] [0] [0] p(c_6) = [0] [0] [0] p(c_7) = [0] [0] [0] p(c_8) = [0] [0] [0] p(c_9) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(c_10) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(c_11) = [0] [0] [0] p(c_12) = [1 0 0] [1 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 0] [0 0 0] [0] p(c_13) = [0] [0] [0] p(c_14) = [1 0 1] [1 0 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 0] [0 0 0] [0] p(c_15) = [0] [0] [0] p(c_16) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(c_17) = [0] [0] [0] p(c_18) = [0] [0] [0] p(c_19) = [0] [0] [0] Following rules are strictly oriented: if_active#(false(),x,y) = [1 0 0] [1 0 0] [1] [1 0 0] x + [0 0 1] y + [1] [0 0 0] [0 1 0] [0] > [1 0 0] [0] [0 0 0] y + [0] [0 0 0] [0] = c_9(mark#(y)) Following rules are (at-least) weakly oriented: div_active#(s(x),s(y)) = [1] [2] [0] >= [1] [2] [0] = c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())) if_active#(true(),x,y) = [1 0 0] [1 0 0] [0] [1 0 0] x + [0 0 1] y + [1] [0 0 0] [0 1 0] [0] >= [1 0 0] [0] [0 0 0] x + [1] [0 0 0] [0] = c_10(mark#(x)) mark#(div(x,y)) = [1 0 1] [0] [0 0 0] x + [0] [0 0 0] [0] >= [1 0 1] [0] [0 0 0] x + [0] [0 0 0] [0] = c_12(div_active#(mark(x),y),mark#(x)) mark#(if(x,y,z)) = [1 0 1] [1 0 0] [1 1 0] [0] [0 0 0] x + [0 0 0] y + [0 0 0] z + [0] [0 0 0] [0 0 0] [0 0 0] [0] >= [1 0 1] [1 0 0] [1 1 0] [0] [0 0 0] x + [0 0 0] y + [0 0 0] z + [0] [0 0 0] [0 0 0] [0 0 0] [0] = c_14(if_active#(mark(x),y,z),mark#(x)) mark#(s(x)) = [1 0 0] [0] [0 0 0] x + [0] [0 0 0] [0] >= [1 0 0] [0] [0 0 0] x + [0] [0 0 0] [0] = c_16(mark#(x)) div_active(x,y) = [1 0 1] [0 0 0] [0] [0 1 0] x + [0 0 0] y + [0] [0 0 1] [0 0 1] [1] >= [1 0 1] [0 0 0] [0] [0 0 0] x + [0 0 0] y + [0] [0 0 1] [0 0 1] [1] = div(x,y) div_active(0(),s(y)) = [0] [0] [2] >= [0] [0] [0] = 0() div_active(s(x),s(y)) = [1 0 0] [1] [0 0 0] x + [1] [0 0 0] [3] >= [1] [1] [1] = if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) = [0] [0] [1] >= [0] [0] [1] = ge(x,y) ge_active(x,0()) = [0] [0] [1] >= [0] [0] [0] = true() ge_active(0(),s(y)) = [0] [0] [1] >= [0] [0] [1] = false() ge_active(s(x),s(y)) = [0] [0] [1] >= [0] [0] [1] = ge_active(x,y) if_active(x,y,z) = [1 0 1] [1 0 0] [1 1 0] [0] [0 0 0] x + [0 0 1] y + [0 0 1] z + [0] [0 0 0] [0 0 1] [0 0 1] [0] >= [1 0 1] [1 0 0] [1 1 0] [0] [0 0 0] x + [0 0 0] y + [0 0 0] z + [0] [0 0 0] [0 0 1] [0 0 1] [0] = if(x,y,z) if_active(false(),x,y) = [1 0 0] [1 1 0] [1] [0 0 1] x + [0 0 1] y + [0] [0 0 1] [0 0 1] [0] >= [1 0 0] [0] [0 0 1] y + [0] [0 0 1] [0] = mark(y) if_active(true(),x,y) = [1 0 0] [1 1 0] [0] [0 0 1] x + [0 0 1] y + [0] [0 0 1] [0 0 1] [0] >= [1 0 0] [0] [0 0 1] x + [0] [0 0 1] [0] = mark(x) mark(0()) = [0] [0] [0] >= [0] [0] [0] = 0() mark(div(x,y)) = [1 0 1] [0 0 0] [0] [0 0 1] x + [0 0 1] y + [1] [0 0 1] [0 0 1] [1] >= [1 0 1] [0 0 0] [0] [0 0 1] x + [0 0 0] y + [0] [0 0 1] [0 0 1] [1] = div_active(mark(x),y) mark(ge(x,y)) = [0] [1] [1] >= [0] [0] [1] = ge_active(x,y) mark(if(x,y,z)) = [1 0 1] [1 0 0] [1 1 0] [0] [0 0 0] x + [0 0 1] y + [0 0 1] z + [0] [0 0 0] [0 0 1] [0 0 1] [0] >= [1 0 1] [1 0 0] [1 1 0] [0] [0 0 0] x + [0 0 1] y + [0 0 1] z + [0] [0 0 0] [0 0 1] [0 0 1] [0] = if_active(mark(x),y,z) mark(minus(x,y)) = [0] [0] [0] >= [0] [0] [0] = minus_active(x,y) mark(s(x)) = [1 0 0] [0] [0 0 0] x + [1] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] x + [1] [0 0 0] [1] = s(mark(x)) minus_active(x,y) = [0] [0] [0] >= [0] [0] [0] = minus(x,y) minus_active(0(),y) = [0] [0] [0] >= [0] [0] [0] = 0() minus_active(s(x),s(y)) = [0] [0] [0] >= [0] [0] [0] = minus_active(x,y) ******* Step 1.b:4.b:3.b:3.b:2.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: if_active#(true(),x,y) -> c_10(mark#(x)) mark#(div(x,y)) -> c_12(div_active#(mark(x),y),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)) 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_15/0,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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ******* Step 1.b:4.b:3.b:3.b:2.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: if_active#(true(),x,y) -> c_10(mark#(x)) mark#(div(x,y)) -> c_12(div_active#(mark(x),y),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)) 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_15/0,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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: if_active#(true(),x,y) -> c_10(mark#(x)) The strictly oriented rules are moved into the weak component. ******** Step 1.b:4.b:3.b:3.b:2.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: if_active#(true(),x,y) -> c_10(mark#(x)) mark#(div(x,y)) -> c_12(div_active#(mark(x),y),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)) 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_15/0,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 = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_3) = {1}, uargs(c_9) = {1}, uargs(c_10) = {1}, uargs(c_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,div_active#,ge_active#,if_active#,mark#,minus_active#} TcT has computed the following interpretation: p(0) = [0] [0] p(div) = [1 2] x1 + [0] [0 1] [1] p(div_active) = [1 2] x1 + [0] [0 1] [1] p(false) = [1] [0] p(ge) = [1] [0] p(ge_active) = [1] [0] p(if) = [1 2] x1 + [1 0] x2 + [1 0] x3 + [2] [0 1] [0 1] [0 1] [1] p(if_active) = [1 2] x1 + [2 0] x2 + [2 0] x3 + [3] [0 1] [0 1] [0 1] [1] p(mark) = [2 0] x1 + [0] [0 1] [0] p(minus) = [0] [0] p(minus_active) = [0] [0] p(s) = [1 0] x1 + [0] [0 0] [2] p(true) = [0] [0] p(div_active#) = [0 2] x1 + [0 0] x2 + [0] [0 0] [0 2] [2] p(ge_active#) = [2] [0] p(if_active#) = [2 0] x2 + [2 0] x3 + [2] [0 0] [0 0] [1] p(mark#) = [2 0] x1 + [0] [0 3] [1] p(minus_active#) = [0 1] x1 + [2] [0 2] [0] p(c_1) = [2] [0] p(c_2) = [0] [0] p(c_3) = [2 0] x1 + [0] [2 2] [0] p(c_4) = [0] [0] p(c_5) = [0] [0] p(c_6) = [0] [0] p(c_7) = [0 1] x1 + [0] [0 2] [0] p(c_8) = [0] [0] p(c_9) = [1 0] x1 + [0] [0 0] [1] p(c_10) = [1 0] x1 + [1] [0 0] [1] p(c_11) = [2] [0] p(c_12) = [2 0] x1 + [1 0] x2 + [0] [1 0] [0 0] [0] p(c_13) = [0 2] x1 + [0] [0 0] [1] p(c_14) = [1 0] x1 + [1 1] x2 + [1] [0 1] [0 1] [2] p(c_15) = [0] [1] p(c_16) = [1 0] x1 + [0] [0 0] [0] p(c_17) = [2] [2] p(c_18) = [0] [0] p(c_19) = [2] [1] Following rules are strictly oriented: if_active#(true(),x,y) = [2 0] x + [2 0] y + [2] [0 0] [0 0] [1] > [2 0] x + [1] [0 0] [1] = c_10(mark#(x)) Following rules are (at-least) weakly oriented: div_active#(s(x),s(y)) = [4] [6] >= [4] [6] = c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())) if_active#(false(),x,y) = [2 0] x + [2 0] y + [2] [0 0] [0 0] [1] >= [2 0] y + [0] [0 0] [1] = c_9(mark#(y)) mark#(div(x,y)) = [2 4] x + [0] [0 3] [4] >= [2 4] x + [0] [0 2] [0] = c_12(div_active#(mark(x),y),mark#(x)) mark#(if(x,y,z)) = [2 4] x + [2 0] y + [2 0] z + [4] [0 3] [0 3] [0 3] [4] >= [2 3] x + [2 0] y + [2 0] z + [4] [0 3] [0 0] [0 0] [4] = c_14(if_active#(mark(x),y,z),mark#(x)) mark#(s(x)) = [2 0] x + [0] [0 0] [7] >= [2 0] x + [0] [0 0] [0] = c_16(mark#(x)) div_active(x,y) = [1 2] x + [0] [0 1] [1] >= [1 2] x + [0] [0 1] [1] = div(x,y) div_active(0(),s(y)) = [0] [1] >= [0] [0] = 0() div_active(s(x),s(y)) = [1 0] x + [4] [0 0] [3] >= [4] [3] = if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) = [1] [0] >= [1] [0] = ge(x,y) ge_active(x,0()) = [1] [0] >= [0] [0] = true() ge_active(0(),s(y)) = [1] [0] >= [1] [0] = false() ge_active(s(x),s(y)) = [1] [0] >= [1] [0] = ge_active(x,y) if_active(x,y,z) = [1 2] x + [2 0] y + [2 0] z + [3] [0 1] [0 1] [0 1] [1] >= [1 2] x + [1 0] y + [1 0] z + [2] [0 1] [0 1] [0 1] [1] = if(x,y,z) if_active(false(),x,y) = [2 0] x + [2 0] y + [4] [0 1] [0 1] [1] >= [2 0] y + [0] [0 1] [0] = mark(y) if_active(true(),x,y) = [2 0] x + [2 0] y + [3] [0 1] [0 1] [1] >= [2 0] x + [0] [0 1] [0] = mark(x) mark(0()) = [0] [0] >= [0] [0] = 0() mark(div(x,y)) = [2 4] x + [0] [0 1] [1] >= [2 2] x + [0] [0 1] [1] = div_active(mark(x),y) mark(ge(x,y)) = [2] [0] >= [1] [0] = ge_active(x,y) mark(if(x,y,z)) = [2 4] x + [2 0] y + [2 0] z + [4] [0 1] [0 1] [0 1] [1] >= [2 2] x + [2 0] y + [2 0] z + [3] [0 1] [0 1] [0 1] [1] = if_active(mark(x),y,z) mark(minus(x,y)) = [0] [0] >= [0] [0] = minus_active(x,y) mark(s(x)) = [2 0] x + [0] [0 0] [2] >= [2 0] x + [0] [0 0] [2] = s(mark(x)) minus_active(x,y) = [0] [0] >= [0] [0] = minus(x,y) minus_active(0(),y) = [0] [0] >= [0] [0] = 0() minus_active(s(x),s(y)) = [0] [0] >= [0] [0] = minus_active(x,y) ******** Step 1.b:4.b:3.b:3.b:2.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: mark#(div(x,y)) -> c_12(div_active#(mark(x),y),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)) 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_15/0,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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ******** Step 1.b:4.b:3.b:3.b:2.b:1.b:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: mark#(div(x,y)) -> c_12(div_active#(mark(x),y),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)) 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_15/0,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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)) Consider the set of all dependency pairs 1: mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)) 2: div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())) 3: if_active#(false(),x,y) -> c_9(mark#(y)) 4: if_active#(true(),x,y) -> c_10(mark#(x)) 5: mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)) 6: mark#(s(x)) -> c_16(mark#(x)) Processor NaturalMI {miDimension = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^2)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,2} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. ********* Step 1.b:4.b:3.b:3.b:2.b:1.b:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: mark#(div(x,y)) -> c_12(div_active#(mark(x),y),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)) 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_15/0,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 = 2, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_3) = {1}, uargs(c_9) = {1}, uargs(c_10) = {1}, uargs(c_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,div_active#,ge_active#,if_active#,mark#,minus_active#} TcT has computed the following interpretation: p(0) = [0] [0] p(div) = [1 2] x1 + [0] [0 1] [2] p(div_active) = [1 2] x1 + [0] [0 1] [2] p(false) = [0] [0] p(ge) = [0 0] x1 + [0] [0 1] [2] p(ge_active) = [0 0] x1 + [0] [0 1] [2] p(if) = [1 2] x1 + [1 0] x2 + [1 0] x3 + [0] [0 0] [0 1] [0 1] [0] p(if_active) = [1 2] x1 + [1 0] x2 + [1 0] x3 + [0] [0 0] [0 1] [0 1] [0] p(mark) = [1 0] x1 + [0] [0 1] [0] p(minus) = [0] [0] p(minus_active) = [0] [0] p(s) = [1 0] x1 + [0] [0 1] [2] p(true) = [0] [0] p(div_active#) = [0 2] x1 + [0] [0 0] [2] p(ge_active#) = [0 0] x1 + [2 1] x2 + [0] [0 1] [0 0] [0] p(if_active#) = [0 0] x1 + [2 1] x2 + [2 1] x3 + [0] [2 0] [0 0] [2 2] [0] p(mark#) = [2 1] x1 + [0] [0 0] [0] p(minus_active#) = [0] [0] p(c_1) = [0] [0] p(c_2) = [0] [0] p(c_3) = [1 0] x1 + [0] [0 0] [2] p(c_4) = [0] [0] p(c_5) = [0] [0] p(c_6) = [0] [0] p(c_7) = [0 2] x1 + [0] [0 0] [0] p(c_8) = [0] [0] p(c_9) = [1 0] x1 + [0] [1 0] [0] p(c_10) = [1 0] x1 + [0] [0 0] [0] p(c_11) = [2] [0] p(c_12) = [2 0] x1 + [1 0] x2 + [0] [0 0] [0 0] [0] p(c_13) = [2 0] x1 + [0] [2 0] [0] p(c_14) = [1 0] x1 + [1 0] x2 + [0] [0 0] [0 0] [0] p(c_15) = [2] [2] p(c_16) = [1 0] x1 + [2] [0 0] [0] p(c_17) = [0] [0] p(c_18) = [0] [0] p(c_19) = [0 0] x1 + [0] [0 2] [0] Following rules are strictly oriented: mark#(div(x,y)) = [2 5] x + [2] [0 0] [0] > [2 5] x + [0] [0 0] [0] = c_12(div_active#(mark(x),y),mark#(x)) Following rules are (at-least) weakly oriented: div_active#(s(x),s(y)) = [0 2] x + [4] [0 0] [2] >= [4] [2] = c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())) if_active#(false(),x,y) = [2 1] x + [2 1] y + [0] [0 0] [2 2] [0] >= [2 1] y + [0] [2 1] [0] = c_9(mark#(y)) if_active#(true(),x,y) = [2 1] x + [2 1] y + [0] [0 0] [2 2] [0] >= [2 1] x + [0] [0 0] [0] = c_10(mark#(x)) mark#(if(x,y,z)) = [2 4] x + [2 1] y + [2 1] z + [0] [0 0] [0 0] [0 0] [0] >= [2 1] x + [2 1] y + [2 1] z + [0] [0 0] [0 0] [0 0] [0] = c_14(if_active#(mark(x),y,z),mark#(x)) mark#(s(x)) = [2 1] x + [2] [0 0] [0] >= [2 1] x + [2] [0 0] [0] = c_16(mark#(x)) div_active(x,y) = [1 2] x + [0] [0 1] [2] >= [1 2] x + [0] [0 1] [2] = div(x,y) div_active(0(),s(y)) = [0] [2] >= [0] [0] = 0() div_active(s(x),s(y)) = [1 2] x + [4] [0 1] [4] >= [0 2] x + [4] [0 0] [4] = if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) = [0 0] x + [0] [0 1] [2] >= [0 0] x + [0] [0 1] [2] = ge(x,y) ge_active(x,0()) = [0 0] x + [0] [0 1] [2] >= [0] [0] = true() ge_active(0(),s(y)) = [0] [2] >= [0] [0] = false() ge_active(s(x),s(y)) = [0 0] x + [0] [0 1] [4] >= [0 0] x + [0] [0 1] [2] = ge_active(x,y) if_active(x,y,z) = [1 2] x + [1 0] y + [1 0] z + [0] [0 0] [0 1] [0 1] [0] >= [1 2] x + [1 0] y + [1 0] z + [0] [0 0] [0 1] [0 1] [0] = if(x,y,z) if_active(false(),x,y) = [1 0] x + [1 0] y + [0] [0 1] [0 1] [0] >= [1 0] y + [0] [0 1] [0] = mark(y) if_active(true(),x,y) = [1 0] x + [1 0] y + [0] [0 1] [0 1] [0] >= [1 0] x + [0] [0 1] [0] = mark(x) mark(0()) = [0] [0] >= [0] [0] = 0() mark(div(x,y)) = [1 2] x + [0] [0 1] [2] >= [1 2] x + [0] [0 1] [2] = div_active(mark(x),y) mark(ge(x,y)) = [0 0] x + [0] [0 1] [2] >= [0 0] x + [0] [0 1] [2] = ge_active(x,y) mark(if(x,y,z)) = [1 2] x + [1 0] y + [1 0] z + [0] [0 0] [0 1] [0 1] [0] >= [1 2] x + [1 0] y + [1 0] z + [0] [0 0] [0 1] [0 1] [0] = if_active(mark(x),y,z) mark(minus(x,y)) = [0] [0] >= [0] [0] = minus_active(x,y) mark(s(x)) = [1 0] x + [0] [0 1] [2] >= [1 0] x + [0] [0 1] [2] = s(mark(x)) minus_active(x,y) = [0] [0] >= [0] [0] = minus(x,y) minus_active(0(),y) = [0] [0] >= [0] [0] = 0() minus_active(s(x),s(y)) = [0] [0] >= [0] [0] = minus_active(x,y) ********* Step 1.b:4.b:3.b:3.b:2.b:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - 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)) 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_15/0,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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ********* Step 1.b:4.b:3.b:3.b:2.b:1.b:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - 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)) 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_15/0,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:W:div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())) -->_1 if_active#(true(),x,y) -> c_10(mark#(x)):3 -->_1 if_active#(false(),x,y) -> c_9(mark#(y)):2 2:W: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:W: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:W: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())):1 5:W: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:W: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 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())) 4: mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)) 6: mark#(s(x)) -> c_16(mark#(x)) 5: mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)) 3: if_active#(true(),x,y) -> c_10(mark#(x)) 2: if_active#(false(),x,y) -> c_9(mark#(y)) ********* Step 1.b:4.b:3.b:3.b:2.b:1.b:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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_15/0,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(Omega(n^1),O(n^2))