/export/starexec/sandbox/solver/bin/starexec_run_tct_rci /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1),O(n^3)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^3)) + Considered Problem: - Strict TRS: 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: Sum. 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: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: 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^3)) + Considered Problem: - Strict TRS: div_active(x,y) -> div(x,y) div_active(0(),s(y)) -> 0() div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) -> ge(x,y) ge_active(x,0()) -> true() ge_active(0(),s(y)) -> false() ge_active(s(x),s(y)) -> ge_active(x,y) if_active(x,y,z) -> if(x,y,z) if_active(false(),x,y) -> mark(y) if_active(true(),x,y) -> mark(x) mark(0()) -> 0() mark(div(x,y)) -> div_active(mark(x),y) mark(ge(x,y)) -> ge_active(x,y) mark(if(x,y,z)) -> if_active(mark(x),y,z) mark(minus(x,y)) -> minus_active(x,y) mark(s(x)) -> s(mark(x)) minus_active(x,y) -> minus(x,y) minus_active(0(),y) -> 0() minus_active(s(x),s(y)) -> minus_active(x,y) - Signature: {div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2} / {0/0,div/2,false/0,ge/2,if/3,minus/2,s/1 ,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {div_active,ge_active,if_active,mark ,minus_active} and constructors {0,div,false,ge,if,minus,s,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs div_active#(x,y) -> c_1() div_active#(0(),s(y)) -> c_2() div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0()),ge_active#(x,y)) ge_active#(x,y) -> c_4() ge_active#(x,0()) -> c_5() ge_active#(0(),s(y)) -> c_6() ge_active#(s(x),s(y)) -> c_7(ge_active#(x,y)) if_active#(x,y,z) -> c_8() if_active#(false(),x,y) -> c_9(mark#(y)) if_active#(true(),x,y) -> c_10(mark#(x)) mark#(0()) -> c_11() mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)) mark#(ge(x,y)) -> c_13(ge_active#(x,y)) mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)) mark#(minus(x,y)) -> c_15(minus_active#(x,y)) mark#(s(x)) -> c_16(mark#(x)) minus_active#(x,y) -> c_17() minus_active#(0(),y) -> c_18() minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)) Weak DPs and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: div_active#(x,y) -> c_1() div_active#(0(),s(y)) -> c_2() div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0()),ge_active#(x,y)) ge_active#(x,y) -> c_4() ge_active#(x,0()) -> c_5() ge_active#(0(),s(y)) -> c_6() ge_active#(s(x),s(y)) -> c_7(ge_active#(x,y)) if_active#(x,y,z) -> c_8() if_active#(false(),x,y) -> c_9(mark#(y)) if_active#(true(),x,y) -> c_10(mark#(x)) mark#(0()) -> c_11() mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)) mark#(ge(x,y)) -> c_13(ge_active#(x,y)) mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)) mark#(minus(x,y)) -> c_15(minus_active#(x,y)) mark#(s(x)) -> c_16(mark#(x)) minus_active#(x,y) -> c_17() minus_active#(0(),y) -> c_18() minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)) - Weak TRS: div_active(x,y) -> div(x,y) div_active(0(),s(y)) -> 0() div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) -> ge(x,y) ge_active(x,0()) -> true() ge_active(0(),s(y)) -> false() ge_active(s(x),s(y)) -> ge_active(x,y) if_active(x,y,z) -> if(x,y,z) if_active(false(),x,y) -> mark(y) if_active(true(),x,y) -> mark(x) mark(0()) -> 0() mark(div(x,y)) -> div_active(mark(x),y) mark(ge(x,y)) -> ge_active(x,y) mark(if(x,y,z)) -> if_active(mark(x),y,z) mark(minus(x,y)) -> minus_active(x,y) mark(s(x)) -> s(mark(x)) minus_active(x,y) -> minus(x,y) minus_active(0(),y) -> 0() minus_active(s(x),s(y)) -> minus_active(x,y) - Signature: {div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2,div_active#/2,ge_active#/2,if_active#/3,mark#/1 ,minus_active#/2} / {0/0,div/2,false/0,ge/2,if/3,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0 ,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0,c_12/2,c_13/1,c_14/2,c_15/1,c_16/1,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {div_active#,ge_active#,if_active#,mark# ,minus_active#} and constructors {0,div,false,ge,if,minus,s,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,4,5,6,8,11,17,18} by application of Pre({1,2,4,5,6,8,11,17,18}) = {3,7,9,10,12,13,14,15,16,19}. Here rules are labelled as follows: 1: div_active#(x,y) -> c_1() 2: div_active#(0(),s(y)) -> c_2() 3: div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0()),ge_active#(x,y)) 4: ge_active#(x,y) -> c_4() 5: ge_active#(x,0()) -> c_5() 6: ge_active#(0(),s(y)) -> c_6() 7: ge_active#(s(x),s(y)) -> c_7(ge_active#(x,y)) 8: if_active#(x,y,z) -> c_8() 9: if_active#(false(),x,y) -> c_9(mark#(y)) 10: if_active#(true(),x,y) -> c_10(mark#(x)) 11: mark#(0()) -> c_11() 12: mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)) 13: mark#(ge(x,y)) -> c_13(ge_active#(x,y)) 14: mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)) 15: mark#(minus(x,y)) -> c_15(minus_active#(x,y)) 16: mark#(s(x)) -> c_16(mark#(x)) 17: minus_active#(x,y) -> c_17() 18: minus_active#(0(),y) -> c_18() 19: minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)) ** Step 1.b:3: RemoveWeakSuffixes. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0()),ge_active#(x,y)) ge_active#(s(x),s(y)) -> c_7(ge_active#(x,y)) if_active#(false(),x,y) -> c_9(mark#(y)) if_active#(true(),x,y) -> c_10(mark#(x)) mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)) mark#(ge(x,y)) -> c_13(ge_active#(x,y)) mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)) mark#(minus(x,y)) -> c_15(minus_active#(x,y)) mark#(s(x)) -> c_16(mark#(x)) minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)) - Weak DPs: div_active#(x,y) -> c_1() div_active#(0(),s(y)) -> c_2() ge_active#(x,y) -> c_4() ge_active#(x,0()) -> c_5() ge_active#(0(),s(y)) -> c_6() if_active#(x,y,z) -> c_8() mark#(0()) -> c_11() minus_active#(x,y) -> c_17() minus_active#(0(),y) -> c_18() - Weak TRS: div_active(x,y) -> div(x,y) div_active(0(),s(y)) -> 0() div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) -> ge(x,y) ge_active(x,0()) -> true() ge_active(0(),s(y)) -> false() ge_active(s(x),s(y)) -> ge_active(x,y) if_active(x,y,z) -> if(x,y,z) if_active(false(),x,y) -> mark(y) if_active(true(),x,y) -> mark(x) mark(0()) -> 0() mark(div(x,y)) -> div_active(mark(x),y) mark(ge(x,y)) -> ge_active(x,y) mark(if(x,y,z)) -> if_active(mark(x),y,z) mark(minus(x,y)) -> minus_active(x,y) mark(s(x)) -> s(mark(x)) minus_active(x,y) -> minus(x,y) minus_active(0(),y) -> 0() minus_active(s(x),s(y)) -> minus_active(x,y) - Signature: {div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2,div_active#/2,ge_active#/2,if_active#/3,mark#/1 ,minus_active#/2} / {0/0,div/2,false/0,ge/2,if/3,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0 ,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0,c_12/2,c_13/1,c_14/2,c_15/1,c_16/1,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {div_active#,ge_active#,if_active#,mark# ,minus_active#} and constructors {0,div,false,ge,if,minus,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0()),ge_active#(x,y)) -->_1 if_active#(true(),x,y) -> c_10(mark#(x)):4 -->_1 if_active#(false(),x,y) -> c_9(mark#(y)):3 -->_2 ge_active#(s(x),s(y)) -> c_7(ge_active#(x,y)):2 -->_1 if_active#(x,y,z) -> c_8():16 -->_2 ge_active#(0(),s(y)) -> c_6():15 -->_2 ge_active#(x,0()) -> c_5():14 -->_2 ge_active#(x,y) -> c_4():13 2:S:ge_active#(s(x),s(y)) -> c_7(ge_active#(x,y)) -->_1 ge_active#(0(),s(y)) -> c_6():15 -->_1 ge_active#(x,0()) -> c_5():14 -->_1 ge_active#(x,y) -> c_4():13 -->_1 ge_active#(s(x),s(y)) -> c_7(ge_active#(x,y)):2 3:S:if_active#(false(),x,y) -> c_9(mark#(y)) -->_1 mark#(s(x)) -> c_16(mark#(x)):9 -->_1 mark#(minus(x,y)) -> c_15(minus_active#(x,y)):8 -->_1 mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)):7 -->_1 mark#(ge(x,y)) -> c_13(ge_active#(x,y)):6 -->_1 mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)):5 -->_1 mark#(0()) -> c_11():17 4:S:if_active#(true(),x,y) -> c_10(mark#(x)) -->_1 mark#(s(x)) -> c_16(mark#(x)):9 -->_1 mark#(minus(x,y)) -> c_15(minus_active#(x,y)):8 -->_1 mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)):7 -->_1 mark#(ge(x,y)) -> c_13(ge_active#(x,y)):6 -->_1 mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)):5 -->_1 mark#(0()) -> c_11():17 5:S:mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)) -->_2 mark#(s(x)) -> c_16(mark#(x)):9 -->_2 mark#(minus(x,y)) -> c_15(minus_active#(x,y)):8 -->_2 mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)):7 -->_2 mark#(ge(x,y)) -> c_13(ge_active#(x,y)):6 -->_2 mark#(0()) -> c_11():17 -->_1 div_active#(0(),s(y)) -> c_2():12 -->_1 div_active#(x,y) -> c_1():11 -->_2 mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)):5 -->_1 div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ,ge_active#(x,y)):1 6:S:mark#(ge(x,y)) -> c_13(ge_active#(x,y)) -->_1 ge_active#(0(),s(y)) -> c_6():15 -->_1 ge_active#(x,0()) -> c_5():14 -->_1 ge_active#(x,y) -> c_4():13 -->_1 ge_active#(s(x),s(y)) -> c_7(ge_active#(x,y)):2 7:S:mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)) -->_2 mark#(s(x)) -> c_16(mark#(x)):9 -->_2 mark#(minus(x,y)) -> c_15(minus_active#(x,y)):8 -->_2 mark#(0()) -> c_11():17 -->_1 if_active#(x,y,z) -> c_8():16 -->_2 mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)):7 -->_2 mark#(ge(x,y)) -> c_13(ge_active#(x,y)):6 -->_2 mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)):5 -->_1 if_active#(true(),x,y) -> c_10(mark#(x)):4 -->_1 if_active#(false(),x,y) -> c_9(mark#(y)):3 8:S:mark#(minus(x,y)) -> c_15(minus_active#(x,y)) -->_1 minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)):10 -->_1 minus_active#(0(),y) -> c_18():19 -->_1 minus_active#(x,y) -> c_17():18 9:S:mark#(s(x)) -> c_16(mark#(x)) -->_1 mark#(0()) -> c_11():17 -->_1 mark#(s(x)) -> c_16(mark#(x)):9 -->_1 mark#(minus(x,y)) -> c_15(minus_active#(x,y)):8 -->_1 mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)):7 -->_1 mark#(ge(x,y)) -> c_13(ge_active#(x,y)):6 -->_1 mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)):5 10:S:minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)) -->_1 minus_active#(0(),y) -> c_18():19 -->_1 minus_active#(x,y) -> c_17():18 -->_1 minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)):10 11:W:div_active#(x,y) -> c_1() 12:W:div_active#(0(),s(y)) -> c_2() 13:W:ge_active#(x,y) -> c_4() 14:W:ge_active#(x,0()) -> c_5() 15:W:ge_active#(0(),s(y)) -> c_6() 16:W:if_active#(x,y,z) -> c_8() 17:W:mark#(0()) -> c_11() 18:W:minus_active#(x,y) -> c_17() 19:W:minus_active#(0(),y) -> c_18() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 11: div_active#(x,y) -> c_1() 12: div_active#(0(),s(y)) -> c_2() 13: ge_active#(x,y) -> c_4() 14: ge_active#(x,0()) -> c_5() 15: ge_active#(0(),s(y)) -> c_6() 16: if_active#(x,y,z) -> c_8() 18: minus_active#(x,y) -> c_17() 19: minus_active#(0(),y) -> c_18() 17: mark#(0()) -> c_11() ** Step 1.b:4: DecomposeDG. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0()),ge_active#(x,y)) ge_active#(s(x),s(y)) -> c_7(ge_active#(x,y)) if_active#(false(),x,y) -> c_9(mark#(y)) if_active#(true(),x,y) -> c_10(mark#(x)) mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)) mark#(ge(x,y)) -> c_13(ge_active#(x,y)) mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)) mark#(minus(x,y)) -> c_15(minus_active#(x,y)) mark#(s(x)) -> c_16(mark#(x)) minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)) - Weak TRS: div_active(x,y) -> div(x,y) div_active(0(),s(y)) -> 0() div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) -> ge(x,y) ge_active(x,0()) -> true() ge_active(0(),s(y)) -> false() ge_active(s(x),s(y)) -> ge_active(x,y) if_active(x,y,z) -> if(x,y,z) if_active(false(),x,y) -> mark(y) if_active(true(),x,y) -> mark(x) mark(0()) -> 0() mark(div(x,y)) -> div_active(mark(x),y) mark(ge(x,y)) -> ge_active(x,y) mark(if(x,y,z)) -> if_active(mark(x),y,z) mark(minus(x,y)) -> minus_active(x,y) mark(s(x)) -> s(mark(x)) minus_active(x,y) -> minus(x,y) minus_active(0(),y) -> 0() minus_active(s(x),s(y)) -> minus_active(x,y) - Signature: {div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2,div_active#/2,ge_active#/2,if_active#/3,mark#/1 ,minus_active#/2} / {0/0,div/2,false/0,ge/2,if/3,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0 ,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0,c_12/2,c_13/1,c_14/2,c_15/1,c_16/1,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {div_active#,ge_active#,if_active#,mark# ,minus_active#} and constructors {0,div,false,ge,if,minus,s,true} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0()),ge_active#(x,y)) if_active#(false(),x,y) -> c_9(mark#(y)) if_active#(true(),x,y) -> c_10(mark#(x)) mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)) mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)) mark#(s(x)) -> c_16(mark#(x)) and a lower component ge_active#(s(x),s(y)) -> c_7(ge_active#(x,y)) mark#(ge(x,y)) -> c_13(ge_active#(x,y)) mark#(minus(x,y)) -> c_15(minus_active#(x,y)) minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)) Further, following extension rules are added to the lower component. div_active#(s(x),s(y)) -> ge_active#(x,y) div_active#(s(x),s(y)) -> if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0()) if_active#(false(),x,y) -> mark#(y) if_active#(true(),x,y) -> mark#(x) mark#(div(x,y)) -> div_active#(mark(x),y) mark#(div(x,y)) -> mark#(x) mark#(if(x,y,z)) -> if_active#(mark(x),y,z) mark#(if(x,y,z)) -> mark#(x) mark#(s(x)) -> mark#(x) *** Step 1.b:4.a:1: SimplifyRHS. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0()),ge_active#(x,y)) if_active#(false(),x,y) -> c_9(mark#(y)) if_active#(true(),x,y) -> c_10(mark#(x)) mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)) mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)) mark#(s(x)) -> c_16(mark#(x)) - Weak TRS: div_active(x,y) -> div(x,y) div_active(0(),s(y)) -> 0() div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) -> ge(x,y) ge_active(x,0()) -> true() ge_active(0(),s(y)) -> false() ge_active(s(x),s(y)) -> ge_active(x,y) if_active(x,y,z) -> if(x,y,z) if_active(false(),x,y) -> mark(y) if_active(true(),x,y) -> mark(x) mark(0()) -> 0() mark(div(x,y)) -> div_active(mark(x),y) mark(ge(x,y)) -> ge_active(x,y) mark(if(x,y,z)) -> if_active(mark(x),y,z) mark(minus(x,y)) -> minus_active(x,y) mark(s(x)) -> s(mark(x)) minus_active(x,y) -> minus(x,y) minus_active(0(),y) -> 0() minus_active(s(x),s(y)) -> minus_active(x,y) - Signature: {div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2,div_active#/2,ge_active#/2,if_active#/3,mark#/1 ,minus_active#/2} / {0/0,div/2,false/0,ge/2,if/3,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0 ,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0,c_12/2,c_13/1,c_14/2,c_15/1,c_16/1,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {div_active#,ge_active#,if_active#,mark# ,minus_active#} and constructors {0,div,false,ge,if,minus,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0()),ge_active#(x,y)) -->_1 if_active#(true(),x,y) -> c_10(mark#(x)):3 -->_1 if_active#(false(),x,y) -> c_9(mark#(y)):2 2:S:if_active#(false(),x,y) -> c_9(mark#(y)) -->_1 mark#(s(x)) -> c_16(mark#(x)):6 -->_1 mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)):5 -->_1 mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)):4 3:S:if_active#(true(),x,y) -> c_10(mark#(x)) -->_1 mark#(s(x)) -> c_16(mark#(x)):6 -->_1 mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)):5 -->_1 mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)):4 4:S:mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)) -->_2 mark#(s(x)) -> c_16(mark#(x)):6 -->_2 mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)):5 -->_2 mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)):4 -->_1 div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ,ge_active#(x,y)):1 5:S:mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)) -->_2 mark#(s(x)) -> c_16(mark#(x)):6 -->_2 mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)):5 -->_2 mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)):4 -->_1 if_active#(true(),x,y) -> c_10(mark#(x)):3 -->_1 if_active#(false(),x,y) -> c_9(mark#(y)):2 6:S:mark#(s(x)) -> c_16(mark#(x)) -->_1 mark#(s(x)) -> c_16(mark#(x)):6 -->_1 mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)):5 -->_1 mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)):4 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())) *** Step 1.b:4.a:2: 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#(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 = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_3) = {1}, uargs(c_9) = {1}, uargs(c_10) = {1}, uargs(c_12) = {1,2}, uargs(c_14) = {1,2}, uargs(c_16) = {1} Following symbols are considered usable: {div_active#,ge_active#,if_active#,mark#,minus_active#} TcT has computed the following interpretation: p(0) = [0] p(div) = [1] x1 + [0] p(div_active) = [5] p(false) = [0] p(ge) = [1] x2 + [2] p(ge_active) = [0] p(if) = [1] x1 + [1] x2 + [1] x3 + [2] p(if_active) = [1] x1 + [0] p(mark) = [0] p(minus) = [0] p(minus_active) = [1] x2 + [0] p(s) = [1] x1 + [0] p(true) = [0] p(div_active#) = [0] p(ge_active#) = [1] x2 + [2] p(if_active#) = [4] x2 + [4] x3 + [0] p(mark#) = [4] x1 + [0] p(minus_active#) = [1] x1 + [1] p(c_1) = [4] p(c_2) = [0] p(c_3) = [4] x1 + [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [1] p(c_7) = [1] x1 + [1] p(c_8) = [4] p(c_9) = [1] x1 + [0] p(c_10) = [1] x1 + [0] p(c_11) = [4] p(c_12) = [2] x1 + [1] x2 + [0] p(c_13) = [2] p(c_14) = [1] x1 + [1] x2 + [6] p(c_15) = [0] p(c_16) = [1] x1 + [0] p(c_17) = [0] p(c_18) = [1] p(c_19) = [1] Following rules are strictly oriented: mark#(if(x,y,z)) = [4] x + [4] y + [4] z + [8] > [4] x + [4] y + [4] z + [6] = 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.a:3: WeightGap. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())) 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#(s(x)) -> c_16(mark#(x)) - Weak DPs: mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)) - Weak TRS: div_active(x,y) -> div(x,y) div_active(0(),s(y)) -> 0() div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) -> ge(x,y) ge_active(x,0()) -> true() ge_active(0(),s(y)) -> false() ge_active(s(x),s(y)) -> ge_active(x,y) if_active(x,y,z) -> if(x,y,z) if_active(false(),x,y) -> mark(y) if_active(true(),x,y) -> mark(x) mark(0()) -> 0() mark(div(x,y)) -> div_active(mark(x),y) mark(ge(x,y)) -> ge_active(x,y) mark(if(x,y,z)) -> if_active(mark(x),y,z) mark(minus(x,y)) -> minus_active(x,y) mark(s(x)) -> s(mark(x)) minus_active(x,y) -> minus(x,y) minus_active(0(),y) -> 0() minus_active(s(x),s(y)) -> minus_active(x,y) - Signature: {div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2,div_active#/2,ge_active#/2,if_active#/3,mark#/1 ,minus_active#/2} / {0/0,div/2,false/0,ge/2,if/3,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/0 ,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0,c_12/2,c_13/1,c_14/2,c_15/1,c_16/1,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {div_active#,ge_active#,if_active#,mark# ,minus_active#} and constructors {0,div,false,ge,if,minus,s,true} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(div_active) = {1}, uargs(if_active) = {1}, uargs(s) = {1}, uargs(div_active#) = {1}, uargs(if_active#) = {1}, uargs(c_3) = {1}, uargs(c_9) = {1}, uargs(c_10) = {1}, uargs(c_12) = {1,2}, uargs(c_14) = {1,2}, uargs(c_16) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(div) = [1] x1 + [0] p(div_active) = [1] x1 + [0] p(false) = [0] p(ge) = [0] p(ge_active) = [0] p(if) = [1] x1 + [0] p(if_active) = [1] x1 + [0] p(mark) = [0] p(minus) = [0] p(minus_active) = [0] p(s) = [1] x1 + [0] p(true) = [0] p(div_active#) = [1] x1 + [7] p(ge_active#) = [0] p(if_active#) = [1] x1 + [0] p(mark#) = [0] p(minus_active#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [1] x1 + [0] p(c_11) = [0] p(c_12) = [1] x1 + [1] x2 + [2] p(c_13) = [0] p(c_14) = [1] x1 + [1] x2 + [0] p(c_15) = [0] p(c_16) = [1] x1 + [0] p(c_17) = [0] p(c_18) = [2] p(c_19) = [2] x1 + [0] Following rules are strictly oriented: div_active#(s(x),s(y)) = [1] x + [7] > [0] = c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())) Following rules are (at-least) weakly oriented: if_active#(false(),x,y) = [0] >= [0] = c_9(mark#(y)) if_active#(true(),x,y) = [0] >= [0] = c_10(mark#(x)) mark#(div(x,y)) = [0] >= [9] = c_12(div_active#(mark(x),y),mark#(x)) mark#(if(x,y,z)) = [0] >= [0] = c_14(if_active#(mark(x),y,z),mark#(x)) mark#(s(x)) = [0] >= [0] = c_16(mark#(x)) div_active(x,y) = [1] x + [0] >= [1] x + [0] = div(x,y) div_active(0(),s(y)) = [0] >= [0] = 0() div_active(s(x),s(y)) = [1] x + [0] >= [0] = if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) = [0] >= [0] = ge(x,y) ge_active(x,0()) = [0] >= [0] = true() ge_active(0(),s(y)) = [0] >= [0] = false() ge_active(s(x),s(y)) = [0] >= [0] = ge_active(x,y) if_active(x,y,z) = [1] x + [0] >= [1] x + [0] = if(x,y,z) if_active(false(),x,y) = [0] >= [0] = mark(y) if_active(true(),x,y) = [0] >= [0] = mark(x) mark(0()) = [0] >= [0] = 0() mark(div(x,y)) = [0] >= [0] = div_active(mark(x),y) mark(ge(x,y)) = [0] >= [0] = ge_active(x,y) mark(if(x,y,z)) = [0] >= [0] = if_active(mark(x),y,z) mark(minus(x,y)) = [0] >= [0] = minus_active(x,y) mark(s(x)) = [0] >= [0] = s(mark(x)) minus_active(x,y) = [0] >= [0] = minus(x,y) minus_active(0(),y) = [0] >= [0] = 0() minus_active(s(x),s(y)) = [0] >= [0] = minus_active(x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 1.b:4.a:4: Ara. 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#(s(x)) -> c_16(mark#(x)) - Weak DPs: div_active#(s(x),s(y)) -> c_3(if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0())) mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)) - Weak TRS: div_active(x,y) -> div(x,y) div_active(0(),s(y)) -> 0() div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) -> ge(x,y) ge_active(x,0()) -> true() ge_active(0(),s(y)) -> false() ge_active(s(x),s(y)) -> ge_active(x,y) if_active(x,y,z) -> if(x,y,z) if_active(false(),x,y) -> mark(y) if_active(true(),x,y) -> mark(x) mark(0()) -> 0() mark(div(x,y)) -> div_active(mark(x),y) mark(ge(x,y)) -> ge_active(x,y) mark(if(x,y,z)) -> if_active(mark(x),y,z) mark(minus(x,y)) -> minus_active(x,y) mark(s(x)) -> s(mark(x)) minus_active(x,y) -> minus(x,y) minus_active(0(),y) -> 0() minus_active(s(x),s(y)) -> minus_active(x,y) - Signature: {div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2,div_active#/2,ge_active#/2,if_active#/3,mark#/1 ,minus_active#/2} / {0/0,div/2,false/0,ge/2,if/3,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/0 ,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0,c_12/2,c_13/1,c_14/2,c_15/1,c_16/1,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {div_active#,ge_active#,if_active#,mark# ,minus_active#} and constructors {0,div,false,ge,if,minus,s,true} + Applied Processor: Ara {minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1, isBestCase = False, mkCompletelyDefined = False, verboseOutput = False} + Details: Signatures used: ---------------- F (TrsFun "0") :: [] -(0)-> "A"(12, 0) F (TrsFun "0") :: [] -(0)-> "A"(0, 0) F (TrsFun "0") :: [] -(0)-> "A"(12, 6) F (TrsFun "0") :: [] -(0)-> "A"(14, 0) F (TrsFun "0") :: [] -(0)-> "A"(12, 2) F (TrsFun "0") :: [] -(0)-> "A"(14, 6) F (TrsFun "0") :: [] -(0)-> "A"(12, 14) F (TrsFun "div") :: ["A"(14, 13) x "A"(0, 0)] -(13)-> "A"(1, 13) F (TrsFun "div") :: ["A"(12, 0) x "A"(0, 0)] -(0)-> "A"(12, 0) F (TrsFun "div") :: ["A"(13, 0) x "A"(0, 0)] -(0)-> "A"(13, 0) F (TrsFun "div") :: ["A"(15, 13) x "A"(0, 0)] -(13)-> "A"(2, 13) F (TrsFun "div_active") :: ["A"(12, 0) x "A"(0, 0)] -(0)-> "A"(12, 0) F (TrsFun "false") :: [] -(0)-> "A"(1, 0) F (TrsFun "false") :: [] -(0)-> "A"(12, 0) F (TrsFun "false") :: [] -(0)-> "A"(15, 6) F (TrsFun "ge") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(12, 0) F (TrsFun "ge") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(14, 8) F (TrsFun "ge_active") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(13, 0) F (TrsFun "if") :: ["A"(12, 0) x "A"(12, 0) x "A"(12, 0)] -(0)-> "A"(12, 0) F (TrsFun "if") :: ["A"(14, 13) x "A"(1, 13) x "A"(1, 13)] -(13)-> "A"(1, 13) F (TrsFun "if_active") :: ["A"(12, 0) x "A"(12, 0) x "A"(12, 0)] -(0)-> "A"(12, 0) F (TrsFun "mark") :: ["A"(12, 0)] -(0)-> "A"(12, 0) F (TrsFun "minus") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(12, 0) F (TrsFun "minus") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(15, 14) F (TrsFun "minus") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(14, 0) F (TrsFun "minus") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(15, 13) F (TrsFun "minus_active") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(14, 0) F (TrsFun "s") :: ["A"(1, 13)] -(1)-> "A"(1, 13) F (TrsFun "s") :: ["A"(0, 0)] -(0)-> "A"(0, 0) F (TrsFun "s") :: ["A"(12, 0)] -(12)-> "A"(12, 0) F (TrsFun "true") :: [] -(0)-> "A"(1, 0) F (TrsFun "true") :: [] -(0)-> "A"(12, 0) F (TrsFun "true") :: [] -(0)-> "A"(15, 6) F (DpFun "div_active") :: ["A"(12, 0) x "A"(0, 0)] -(3)-> "A"(15, 1) F (DpFun "if_active") :: ["A"(1, 0) x "A"(1, 13) x "A"(1, 13)] -(0)-> "A"(2, 0) F (DpFun "mark") :: ["A"(1, 13)] -(0)-> "A"(9, 4) F (ComFun 3) :: ["A"(0, 0)] -(0)-> "A"(15, 11) F (ComFun 9) :: ["A"(0, 0)] -(0)-> "A"(2, 1) F (ComFun 10) :: ["A"(0, 0)] -(0)-> "A"(12, 12) F (ComFun 12) :: ["A"(9, 0) x "A"(9, 0)] -(0)-> "A"(9, 15) F (ComFun 14) :: ["A"(0, 0) x "A"(9, 0)] -(0)-> "A"(9, 10) F (ComFun 16) :: ["A"(0, 0)] -(0)-> "A"(11, 11) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "F (ComFun 10)_A" :: ["A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 10)_A" :: ["A"(0, 0)] -(0)-> "A"(0, 1) "F (ComFun 12)_A" :: ["A"(1, 0) x "A"(1, 0)] -(0)-> "A"(1, 0) "F (ComFun 12)_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(0, 1) "F (ComFun 14)_A" :: ["A"(0, 0) x "A"(1, 0)] -(0)-> "A"(1, 0) "F (ComFun 14)_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(0, 1) "F (ComFun 16)_A" :: ["A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 16)_A" :: ["A"(0, 0)] -(0)-> "A"(0, 1) "F (ComFun 3)_A" :: ["A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 3)_A" :: ["A"(0, 0)] -(0)-> "A"(0, 1) "F (ComFun 9)_A" :: ["A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 9)_A" :: ["A"(0, 0)] -(0)-> "A"(0, 1) "F (TrsFun \"0\")_A" :: [] -(0)-> "A"(1, 0) "F (TrsFun \"0\")_A" :: [] -(0)-> "A"(0, 1) "F (TrsFun \"div\")_A" :: ["A"(1, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "F (TrsFun \"div\")_A" :: ["A"(1, 1) x "A"(0, 0)] -(1)-> "A"(0, 1) "F (TrsFun \"false\")_A" :: [] -(0)-> "A"(1, 0) "F (TrsFun \"false\")_A" :: [] -(0)-> "A"(0, 1) "F (TrsFun \"ge\")_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "F (TrsFun \"ge\")_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(0, 1) "F (TrsFun \"if\")_A" :: ["A"(1, 0) x "A"(1, 0) x "A"(1, 0)] -(0)-> "A"(1, 0) "F (TrsFun \"if\")_A" :: ["A"(1, 1) x "A"(0, 1) x "A"(0, 1)] -(1)-> "A"(0, 1) "F (TrsFun \"minus\")_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "F (TrsFun \"minus\")_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(0, 1) "F (TrsFun \"s\")_A" :: ["A"(1, 0)] -(1)-> "A"(1, 0) "F (TrsFun \"s\")_A" :: ["A"(0, 1)] -(0)-> "A"(0, 1) "F (TrsFun \"true\")_A" :: [] -(0)-> "A"(1, 0) "F (TrsFun \"true\")_A" :: [] -(0)-> "A"(0, 1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: mark#(s(x)) -> c_16(mark#(x)) 2. Weak: 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)) *** Step 1.b:4.a:5: Ara. 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_15/1,c_16/1,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {div_active#,ge_active#,if_active#,mark# ,minus_active#} and constructors {0,div,false,ge,if,minus,s,true} + Applied Processor: Ara {minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1, isBestCase = False, mkCompletelyDefined = False, verboseOutput = False} + Details: Signatures used: ---------------- F (TrsFun "0") :: [] -(0)-> "A"(7, 0) F (TrsFun "0") :: [] -(0)-> "A"(0, 0) F (TrsFun "0") :: [] -(0)-> "A"(7, 8) F (TrsFun "0") :: [] -(0)-> "A"(7, 10) F (TrsFun "0") :: [] -(0)-> "A"(14, 10) F (TrsFun "0") :: [] -(0)-> "A"(10, 11) F (TrsFun "div") :: ["A"(8, 7) x "A"(7, 7)] -(1)-> "A"(1, 7) F (TrsFun "div") :: ["A"(7, 0) x "A"(0, 0)] -(7)-> "A"(7, 0) F (TrsFun "div") :: ["A"(9, 7) x "A"(7, 7)] -(2)-> "A"(2, 7) F (TrsFun "div_active") :: ["A"(7, 0) x "A"(0, 0)] -(7)-> "A"(7, 0) F (TrsFun "false") :: [] -(0)-> "A"(0, 0) F (TrsFun "false") :: [] -(0)-> "A"(7, 0) F (TrsFun "false") :: [] -(0)-> "A"(7, 8) F (TrsFun "ge") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(7, 0) F (TrsFun "ge") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(9, 14) F (TrsFun "ge_active") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(7, 0) F (TrsFun "if") :: ["A"(7, 0) x "A"(7, 0) x "A"(7, 0)] -(0)-> "A"(7, 0) F (TrsFun "if") :: ["A"(8, 7) x "A"(1, 7) x "A"(8, 7)] -(0)-> "A"(1, 7) F (TrsFun "if_active") :: ["A"(7, 0) x "A"(7, 0) x "A"(7, 0)] -(0)-> "A"(7, 0) F (TrsFun "mark") :: ["A"(7, 0)] -(0)-> "A"(7, 0) F (TrsFun "minus") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(7, 0) F (TrsFun "minus") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(11, 6) F (TrsFun "minus") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(8, 2) F (TrsFun "minus") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(15, 15) F (TrsFun "minus_active") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(7, 0) F (TrsFun "s") :: ["A"(0, 0)] -(0)-> "A"(0, 0) F (TrsFun "s") :: ["A"(7, 0)] -(7)-> "A"(7, 0) F (TrsFun "s") :: ["A"(7, 7)] -(7)-> "A"(7, 7) F (TrsFun "s") :: ["A"(1, 7)] -(1)-> "A"(1, 7) F (TrsFun "true") :: [] -(0)-> "A"(0, 0) F (TrsFun "true") :: [] -(0)-> "A"(7, 0) F (TrsFun "true") :: [] -(0)-> "A"(7, 8) F (DpFun "div_active") :: ["A"(7, 0) x "A"(7, 7)] -(0)-> "A"(6, 1) F (DpFun "if_active") :: ["A"(0, 0) x "A"(1, 7) x "A"(8, 7)] -(0)-> "A"(14, 12) F (DpFun "mark") :: ["A"(1, 7)] -(0)-> "A"(2, 1) F (ComFun 3) :: ["A"(0, 0)] -(0)-> "A"(6, 6) F (ComFun 9) :: ["A"(0, 0)] -(0)-> "A"(14, 14) F (ComFun 10) :: ["A"(0, 0)] -(0)-> "A"(14, 14) F (ComFun 12) :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(12, 12) F (ComFun 14) :: ["A"(3, 0) x "A"(0, 0)] -(0)-> "A"(8, 3) F (ComFun 16) :: ["A"(0, 0)] -(0)-> "A"(14, 2) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "F (ComFun 10)_A" :: ["A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 10)_A" :: ["A"(0, 0)] -(0)-> "A"(0, 1) "F (ComFun 12)_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 12)_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(0, 1) "F (ComFun 14)_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 14)_A" :: ["A"(1, 0) x "A"(0, 0)] -(0)-> "A"(0, 1) "F (ComFun 16)_A" :: ["A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 16)_A" :: ["A"(0, 0)] -(0)-> "A"(0, 1) "F (ComFun 3)_A" :: ["A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 3)_A" :: ["A"(0, 0)] -(0)-> "A"(0, 1) "F (ComFun 9)_A" :: ["A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 9)_A" :: ["A"(0, 0)] -(0)-> "A"(0, 1) "F (TrsFun \"0\")_A" :: [] -(0)-> "A"(1, 0) "F (TrsFun \"0\")_A" :: [] -(0)-> "A"(0, 1) "F (TrsFun \"div\")_A" :: ["A"(1, 0) x "A"(0, 0)] -(1)-> "A"(1, 0) "F (TrsFun \"div\")_A" :: ["A"(1, 1) x "A"(1, 1)] -(0)-> "A"(0, 1) "F (TrsFun \"false\")_A" :: [] -(0)-> "A"(1, 0) "F (TrsFun \"false\")_A" :: [] -(0)-> "A"(0, 1) "F (TrsFun \"ge\")_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "F (TrsFun \"ge\")_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(0, 1) "F (TrsFun \"if\")_A" :: ["A"(1, 0) x "A"(1, 0) x "A"(1, 0)] -(0)-> "A"(1, 0) "F (TrsFun \"if\")_A" :: ["A"(1, 1) x "A"(0, 1) x "A"(1, 1)] -(0)-> "A"(0, 1) "F (TrsFun \"minus\")_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "F (TrsFun \"minus\")_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(0, 1) "F (TrsFun \"s\")_A" :: ["A"(1, 0)] -(1)-> "A"(1, 0) "F (TrsFun \"s\")_A" :: ["A"(0, 1)] -(0)-> "A"(0, 1) "F (TrsFun \"true\")_A" :: [] -(0)-> "A"(1, 0) "F (TrsFun \"true\")_A" :: [] -(0)-> "A"(0, 1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: mark#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)) 2. Weak: if_active#(false(),x,y) -> c_9(mark#(y)) if_active#(true(),x,y) -> c_10(mark#(x)) *** Step 1.b:4.a:6: Ara. 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)) - 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#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)) mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)) mark#(s(x)) -> c_16(mark#(x)) - Weak TRS: div_active(x,y) -> div(x,y) div_active(0(),s(y)) -> 0() div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) -> ge(x,y) ge_active(x,0()) -> true() ge_active(0(),s(y)) -> false() ge_active(s(x),s(y)) -> ge_active(x,y) if_active(x,y,z) -> if(x,y,z) if_active(false(),x,y) -> mark(y) if_active(true(),x,y) -> mark(x) mark(0()) -> 0() mark(div(x,y)) -> div_active(mark(x),y) mark(ge(x,y)) -> ge_active(x,y) mark(if(x,y,z)) -> if_active(mark(x),y,z) mark(minus(x,y)) -> minus_active(x,y) mark(s(x)) -> s(mark(x)) minus_active(x,y) -> minus(x,y) minus_active(0(),y) -> 0() minus_active(s(x),s(y)) -> minus_active(x,y) - Signature: {div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2,div_active#/2,ge_active#/2,if_active#/3,mark#/1 ,minus_active#/2} / {0/0,div/2,false/0,ge/2,if/3,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/0 ,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0,c_12/2,c_13/1,c_14/2,c_15/1,c_16/1,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {div_active#,ge_active#,if_active#,mark# ,minus_active#} and constructors {0,div,false,ge,if,minus,s,true} + Applied Processor: Ara {minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1, isBestCase = False, mkCompletelyDefined = False, verboseOutput = False} + Details: Signatures used: ---------------- F (TrsFun "0") :: [] -(0)-> "A"(4, 0) F (TrsFun "0") :: [] -(0)-> "A"(0, 0) F (TrsFun "0") :: [] -(0)-> "A"(8, 7) F (TrsFun "0") :: [] -(0)-> "A"(4, 3) F (TrsFun "0") :: [] -(0)-> "A"(15, 15) F (TrsFun "div") :: ["A"(4, 0) x "A"(0, 0)] -(0)-> "A"(4, 0) F (TrsFun "div") :: ["A"(5, 5) x "A"(0, 0)] -(5)-> "A"(0, 5) F (TrsFun "div") :: ["A"(14, 0) x "A"(0, 0)] -(0)-> "A"(14, 0) F (TrsFun "div") :: ["A"(13, 5) x "A"(0, 0)] -(5)-> "A"(8, 5) F (TrsFun "div_active") :: ["A"(4, 0) x "A"(0, 0)] -(0)-> "A"(4, 0) F (TrsFun "false") :: [] -(0)-> "A"(0, 0) F (TrsFun "false") :: [] -(0)-> "A"(4, 0) F (TrsFun "false") :: [] -(0)-> "A"(8, 7) F (TrsFun "ge") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(4, 0) F (TrsFun "ge") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(7, 15) F (TrsFun "ge_active") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(6, 7) F (TrsFun "if") :: ["A"(4, 0) x "A"(4, 0) x "A"(4, 0)] -(0)-> "A"(4, 0) F (TrsFun "if") :: ["A"(5, 5) x "A"(0, 5) x "A"(0, 5)] -(5)-> "A"(0, 5) F (TrsFun "if_active") :: ["A"(4, 0) x "A"(4, 0) x "A"(4, 0)] -(0)-> "A"(4, 0) F (TrsFun "mark") :: ["A"(4, 0)] -(0)-> "A"(4, 0) F (TrsFun "minus") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(4, 0) F (TrsFun "minus") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(14, 12) F (TrsFun "minus") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(7, 14) F (TrsFun "minus") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(15, 15) F (TrsFun "minus_active") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(7, 7) F (TrsFun "s") :: ["A"(0, 0)] -(0)-> "A"(0, 0) F (TrsFun "s") :: ["A"(4, 0)] -(4)-> "A"(4, 0) F (TrsFun "s") :: ["A"(3, 0)] -(3)-> "A"(3, 0) F (TrsFun "s") :: ["A"(0, 5)] -(0)-> "A"(0, 5) F (TrsFun "s") :: ["A"(2, 5)] -(2)-> "A"(2, 5) F (TrsFun "true") :: [] -(0)-> "A"(0, 0) F (TrsFun "true") :: [] -(0)-> "A"(4, 0) F (TrsFun "true") :: [] -(0)-> "A"(8, 7) F (DpFun "div_active") :: ["A"(3, 0) x "A"(0, 0)] -(5)-> "A"(8, 0) F (DpFun "if_active") :: ["A"(0, 0) x "A"(0, 5) x "A"(0, 5)] -(1)-> "A"(14, 12) F (DpFun "mark") :: ["A"(0, 5)] -(0)-> "A"(11, 10) F (ComFun 3) :: ["A"(10, 0)] -(0)-> "A"(9, 1) F (ComFun 9) :: ["A"(0, 0)] -(0)-> "A"(14, 12) F (ComFun 10) :: ["A"(0, 0)] -(0)-> "A"(14, 14) F (ComFun 12) :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(14, 15) F (ComFun 14) :: ["A"(11, 0) x "A"(0, 0)] -(0)-> "A"(11, 13) F (ComFun 16) :: ["A"(11, 10)] -(0)-> "A"(11, 10) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "F (ComFun 10)_A" :: ["A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 10)_A" :: ["A"(0, 0)] -(0)-> "A"(0, 1) "F (ComFun 12)_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 12)_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(0, 1) "F (ComFun 14)_A" :: ["A"(1, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 14)_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(0, 1) "F (ComFun 16)_A" :: ["A"(1, 0)] -(0)-> "A"(1, 0) "F (ComFun 16)_A" :: ["A"(0, 1)] -(0)-> "A"(0, 1) "F (ComFun 3)_A" :: ["A"(1, 0)] -(0)-> "A"(1, 0) "F (ComFun 3)_A" :: ["A"(1, 0)] -(0)-> "A"(0, 1) "F (ComFun 9)_A" :: ["A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 9)_A" :: ["A"(0, 0)] -(0)-> "A"(0, 1) "F (TrsFun \"0\")_A" :: [] -(0)-> "A"(1, 0) "F (TrsFun \"0\")_A" :: [] -(0)-> "A"(0, 1) "F (TrsFun \"div\")_A" :: ["A"(1, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "F (TrsFun \"div\")_A" :: ["A"(1, 1) x "A"(0, 0)] -(1)-> "A"(0, 1) "F (TrsFun \"false\")_A" :: [] -(0)-> "A"(1, 0) "F (TrsFun \"false\")_A" :: [] -(0)-> "A"(0, 1) "F (TrsFun \"ge\")_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "F (TrsFun \"ge\")_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(0, 1) "F (TrsFun \"if\")_A" :: ["A"(1, 0) x "A"(1, 0) x "A"(1, 0)] -(0)-> "A"(1, 0) "F (TrsFun \"if\")_A" :: ["A"(1, 1) x "A"(0, 1) x "A"(0, 1)] -(1)-> "A"(0, 1) "F (TrsFun \"minus\")_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "F (TrsFun \"minus\")_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(0, 1) "F (TrsFun \"s\")_A" :: ["A"(1, 0)] -(1)-> "A"(1, 0) "F (TrsFun \"s\")_A" :: ["A"(0, 1)] -(0)-> "A"(0, 1) "F (TrsFun \"true\")_A" :: [] -(0)-> "A"(1, 0) "F (TrsFun \"true\")_A" :: [] -(0)-> "A"(0, 1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: if_active#(false(),x,y) -> c_9(mark#(y)) 2. Weak: if_active#(true(),x,y) -> c_10(mark#(x)) *** Step 1.b:4.a:7: Ara. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: if_active#(true(),x,y) -> c_10(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#(div(x,y)) -> c_12(div_active#(mark(x),y),mark#(x)) mark#(if(x,y,z)) -> c_14(if_active#(mark(x),y,z),mark#(x)) mark#(s(x)) -> c_16(mark#(x)) - Weak TRS: div_active(x,y) -> div(x,y) div_active(0(),s(y)) -> 0() div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) -> ge(x,y) ge_active(x,0()) -> true() ge_active(0(),s(y)) -> false() ge_active(s(x),s(y)) -> ge_active(x,y) if_active(x,y,z) -> if(x,y,z) if_active(false(),x,y) -> mark(y) if_active(true(),x,y) -> mark(x) mark(0()) -> 0() mark(div(x,y)) -> div_active(mark(x),y) mark(ge(x,y)) -> ge_active(x,y) mark(if(x,y,z)) -> if_active(mark(x),y,z) mark(minus(x,y)) -> minus_active(x,y) mark(s(x)) -> s(mark(x)) minus_active(x,y) -> minus(x,y) minus_active(0(),y) -> 0() minus_active(s(x),s(y)) -> minus_active(x,y) - Signature: {div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2,div_active#/2,ge_active#/2,if_active#/3,mark#/1 ,minus_active#/2} / {0/0,div/2,false/0,ge/2,if/3,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/0 ,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0,c_12/2,c_13/1,c_14/2,c_15/1,c_16/1,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {div_active#,ge_active#,if_active#,mark# ,minus_active#} and constructors {0,div,false,ge,if,minus,s,true} + Applied Processor: Ara {minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1, isBestCase = False, mkCompletelyDefined = False, verboseOutput = False} + Details: Signatures used: ---------------- F (TrsFun "0") :: [] -(0)-> "A"(4, 0) F (TrsFun "0") :: [] -(0)-> "A"(0, 0) F (TrsFun "0") :: [] -(0)-> "A"(12, 0) F (TrsFun "0") :: [] -(0)-> "A"(8, 0) F (TrsFun "0") :: [] -(0)-> "A"(7, 0) F (TrsFun "0") :: [] -(0)-> "A"(11, 7) F (TrsFun "div") :: ["A"(4, 0) x "A"(0, 0)] -(4)-> "A"(4, 0) F (TrsFun "div") :: ["A"(4, 4) x "A"(0, 4)] -(4)-> "A"(0, 4) F (TrsFun "div_active") :: ["A"(4, 0) x "A"(0, 0)] -(4)-> "A"(4, 0) F (TrsFun "false") :: [] -(0)-> "A"(4, 0) F (TrsFun "false") :: [] -(0)-> "A"(0, 0) F (TrsFun "false") :: [] -(0)-> "A"(7, 0) F (TrsFun "ge") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(4, 0) F (TrsFun "ge") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(14, 6) F (TrsFun "ge_active") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(4, 0) F (TrsFun "if") :: ["A"(4, 0) x "A"(4, 0) x "A"(4, 0)] -(0)-> "A"(4, 0) F (TrsFun "if") :: ["A"(4, 4) x "A"(4, 4) x "A"(0, 4)] -(4)-> "A"(0, 4) F (TrsFun "if_active") :: ["A"(4, 0) x "A"(4, 0) x "A"(4, 0)] -(0)-> "A"(4, 0) F (TrsFun "mark") :: ["A"(4, 0)] -(0)-> "A"(4, 0) F (TrsFun "minus") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(4, 0) F (TrsFun "minus") :: ["A"(2, 0) x "A"(0, 0)] -(0)-> "A"(6, 2) F (TrsFun "minus") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(8, 0) F (TrsFun "minus") :: ["A"(4, 0) x "A"(0, 0)] -(0)-> "A"(11, 4) F (TrsFun "minus_active") :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(6, 0) F (TrsFun "s") :: ["A"(0, 0)] -(0)-> "A"(0, 0) F (TrsFun "s") :: ["A"(4, 0)] -(4)-> "A"(4, 0) F (TrsFun "s") :: ["A"(0, 4)] -(0)-> "A"(0, 4) F (TrsFun "true") :: [] -(0)-> "A"(0, 0) F (TrsFun "true") :: [] -(0)-> "A"(4, 0) F (DpFun "div_active") :: ["A"(4, 0) x "A"(0, 4)] -(3)-> "A"(0, 0) F (DpFun "if_active") :: ["A"(0, 0) x "A"(0, 4) x "A"(0, 4)] -(1)-> "A"(0, 0) F (DpFun "mark") :: ["A"(0, 4)] -(0)-> "A"(1, 4) F (ComFun 3) :: ["A"(0, 0)] -(0)-> "A"(8, 0) F (ComFun 9) :: ["A"(1, 1)] -(0)-> "A"(9, 1) F (ComFun 10) :: ["A"(0, 0)] -(0)-> "A"(0, 0) F (ComFun 12) :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(13, 6) F (ComFun 14) :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(8, 14) F (ComFun 16) :: ["A"(0, 0)] -(0)-> "A"(12, 12) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "F (ComFun 10)_A" :: ["A"(1, 0)] -(1)-> "A"(1, 0) "F (ComFun 10)_A" :: ["A"(0, 1)] -(1)-> "A"(0, 1) "F (ComFun 12)_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 12)_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(0, 1) "F (ComFun 14)_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 14)_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(0, 1) "F (ComFun 16)_A" :: ["A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 16)_A" :: ["A"(0, 0)] -(0)-> "A"(0, 1) "F (ComFun 3)_A" :: ["A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 3)_A" :: ["A"(0, 1)] -(1)-> "A"(0, 1) "F (ComFun 9)_A" :: ["A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 9)_A" :: ["A"(1, 1)] -(0)-> "A"(0, 1) "F (TrsFun \"0\")_A" :: [] -(0)-> "A"(1, 0) "F (TrsFun \"0\")_A" :: [] -(0)-> "A"(0, 1) "F (TrsFun \"div\")_A" :: ["A"(1, 0) x "A"(0, 0)] -(1)-> "A"(1, 0) "F (TrsFun \"div\")_A" :: ["A"(1, 1) x "A"(0, 1)] -(1)-> "A"(0, 1) "F (TrsFun \"false\")_A" :: [] -(0)-> "A"(1, 0) "F (TrsFun \"false\")_A" :: [] -(0)-> "A"(0, 1) "F (TrsFun \"ge\")_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "F (TrsFun \"ge\")_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(0, 1) "F (TrsFun \"if\")_A" :: ["A"(1, 0) x "A"(1, 0) x "A"(1, 0)] -(0)-> "A"(1, 0) "F (TrsFun \"if\")_A" :: ["A"(1, 1) x "A"(1, 1) x "A"(0, 1)] -(1)-> "A"(0, 1) "F (TrsFun \"minus\")_A" :: ["A"(0, 0) x "A"(0, 0)] -(0)-> "A"(1, 0) "F (TrsFun \"minus\")_A" :: ["A"(1, 0) x "A"(0, 0)] -(0)-> "A"(0, 1) "F (TrsFun \"s\")_A" :: ["A"(1, 0)] -(1)-> "A"(1, 0) "F (TrsFun \"s\")_A" :: ["A"(0, 1)] -(0)-> "A"(0, 1) "F (TrsFun \"true\")_A" :: [] -(0)-> "A"(1, 0) "F (TrsFun \"true\")_A" :: [] -(0)-> "A"(0, 1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: if_active#(true(),x,y) -> c_10(mark#(x)) 2. Weak: *** Step 1.b:4.b:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: ge_active#(s(x),s(y)) -> c_7(ge_active#(x,y)) mark#(ge(x,y)) -> c_13(ge_active#(x,y)) mark#(minus(x,y)) -> c_15(minus_active#(x,y)) minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)) - Weak DPs: div_active#(s(x),s(y)) -> ge_active#(x,y) div_active#(s(x),s(y)) -> if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0()) if_active#(false(),x,y) -> mark#(y) if_active#(true(),x,y) -> mark#(x) mark#(div(x,y)) -> div_active#(mark(x),y) mark#(div(x,y)) -> mark#(x) mark#(if(x,y,z)) -> if_active#(mark(x),y,z) mark#(if(x,y,z)) -> mark#(x) mark#(s(x)) -> mark#(x) - Weak TRS: div_active(x,y) -> div(x,y) div_active(0(),s(y)) -> 0() div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) -> ge(x,y) ge_active(x,0()) -> true() ge_active(0(),s(y)) -> false() ge_active(s(x),s(y)) -> ge_active(x,y) if_active(x,y,z) -> if(x,y,z) if_active(false(),x,y) -> mark(y) if_active(true(),x,y) -> mark(x) mark(0()) -> 0() mark(div(x,y)) -> div_active(mark(x),y) mark(ge(x,y)) -> ge_active(x,y) mark(if(x,y,z)) -> if_active(mark(x),y,z) mark(minus(x,y)) -> minus_active(x,y) mark(s(x)) -> s(mark(x)) minus_active(x,y) -> minus(x,y) minus_active(0(),y) -> 0() minus_active(s(x),s(y)) -> minus_active(x,y) - Signature: {div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2,div_active#/2,ge_active#/2,if_active#/3,mark#/1 ,minus_active#/2} / {0/0,div/2,false/0,ge/2,if/3,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0 ,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0,c_12/2,c_13/1,c_14/2,c_15/1,c_16/1,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {div_active#,ge_active#,if_active#,mark# ,minus_active#} and constructors {0,div,false,ge,if,minus,s,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_7) = {1}, uargs(c_13) = {1}, uargs(c_15) = {1}, uargs(c_19) = {1} Following symbols are considered usable: {div_active#,ge_active#,if_active#,mark#,minus_active#} TcT has computed the following interpretation: p(0) = [1] p(div) = [1] x1 + [1] x2 + [0] p(div_active) = [0] p(false) = [0] p(ge) = [1] x1 + [1] x2 + [0] p(ge_active) = [1] x2 + [0] p(if) = [1] x3 + [0] p(if_active) = [0] p(mark) = [0] p(minus) = [1] x2 + [0] p(minus_active) = [0] p(s) = [0] p(true) = [0] p(div_active#) = [2] p(ge_active#) = [0] p(if_active#) = [2] p(mark#) = [2] p(minus_active#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [1] x1 + [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [0] p(c_12) = [0] p(c_13) = [2] x1 + [0] p(c_14) = [0] p(c_15) = [1] x1 + [0] p(c_16) = [4] x1 + [0] p(c_17) = [0] p(c_18) = [0] p(c_19) = [4] x1 + [0] Following rules are strictly oriented: mark#(ge(x,y)) = [2] > [0] = c_13(ge_active#(x,y)) mark#(minus(x,y)) = [2] > [0] = c_15(minus_active#(x,y)) Following rules are (at-least) weakly oriented: div_active#(s(x),s(y)) = [2] >= [0] = ge_active#(x,y) div_active#(s(x),s(y)) = [2] >= [2] = if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active#(s(x),s(y)) = [0] >= [0] = c_7(ge_active#(x,y)) if_active#(false(),x,y) = [2] >= [2] = mark#(y) if_active#(true(),x,y) = [2] >= [2] = mark#(x) mark#(div(x,y)) = [2] >= [2] = div_active#(mark(x),y) mark#(div(x,y)) = [2] >= [2] = mark#(x) mark#(if(x,y,z)) = [2] >= [2] = if_active#(mark(x),y,z) mark#(if(x,y,z)) = [2] >= [2] = mark#(x) mark#(s(x)) = [2] >= [2] = mark#(x) minus_active#(s(x),s(y)) = [0] >= [0] = c_19(minus_active#(x,y)) *** Step 1.b:4.b:2: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: ge_active#(s(x),s(y)) -> c_7(ge_active#(x,y)) minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)) - Weak DPs: div_active#(s(x),s(y)) -> ge_active#(x,y) div_active#(s(x),s(y)) -> if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0()) if_active#(false(),x,y) -> mark#(y) if_active#(true(),x,y) -> mark#(x) mark#(div(x,y)) -> div_active#(mark(x),y) mark#(div(x,y)) -> mark#(x) mark#(ge(x,y)) -> c_13(ge_active#(x,y)) mark#(if(x,y,z)) -> if_active#(mark(x),y,z) mark#(if(x,y,z)) -> mark#(x) mark#(minus(x,y)) -> c_15(minus_active#(x,y)) mark#(s(x)) -> mark#(x) - Weak TRS: div_active(x,y) -> div(x,y) div_active(0(),s(y)) -> 0() div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) -> ge(x,y) ge_active(x,0()) -> true() ge_active(0(),s(y)) -> false() ge_active(s(x),s(y)) -> ge_active(x,y) if_active(x,y,z) -> if(x,y,z) if_active(false(),x,y) -> mark(y) if_active(true(),x,y) -> mark(x) mark(0()) -> 0() mark(div(x,y)) -> div_active(mark(x),y) mark(ge(x,y)) -> ge_active(x,y) mark(if(x,y,z)) -> if_active(mark(x),y,z) mark(minus(x,y)) -> minus_active(x,y) mark(s(x)) -> s(mark(x)) minus_active(x,y) -> minus(x,y) minus_active(0(),y) -> 0() minus_active(s(x),s(y)) -> minus_active(x,y) - Signature: {div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2,div_active#/2,ge_active#/2,if_active#/3,mark#/1 ,minus_active#/2} / {0/0,div/2,false/0,ge/2,if/3,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0 ,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0,c_12/2,c_13/1,c_14/2,c_15/1,c_16/1,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {div_active#,ge_active#,if_active#,mark# ,minus_active#} and constructors {0,div,false,ge,if,minus,s,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_7) = {1}, uargs(c_13) = {1}, uargs(c_15) = {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] p(div) = [1] x1 + [0] p(div_active) = [1] x1 + [0] p(false) = [0] p(ge) = [1] x1 + [0] p(ge_active) = [1] x1 + [0] p(if) = [1] x1 + [1] x2 + [1] x3 + [0] p(if_active) = [1] x1 + [1] x2 + [1] x3 + [0] p(mark) = [1] x1 + [0] p(minus) = [0] p(minus_active) = [0] p(s) = [1] x1 + [2] p(true) = [0] p(div_active#) = [4] x1 + [0] p(ge_active#) = [2] x1 + [0] p(if_active#) = [4] x2 + [4] x3 + [0] p(mark#) = [4] x1 + [0] p(minus_active#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] p(c_4) = [4] p(c_5) = [4] p(c_6) = [0] p(c_7) = [1] x1 + [2] p(c_8) = [0] p(c_9) = [2] p(c_10) = [1] p(c_11) = [0] p(c_12) = [1] x1 + [0] p(c_13) = [2] x1 + [0] p(c_14) = [1] x1 + [4] x2 + [0] p(c_15) = [1] x1 + [0] p(c_16) = [4] x1 + [0] p(c_17) = [0] p(c_18) = [0] p(c_19) = [4] x1 + [0] Following rules are strictly oriented: ge_active#(s(x),s(y)) = [2] x + [4] > [2] x + [2] = c_7(ge_active#(x,y)) Following rules are (at-least) weakly oriented: div_active#(s(x),s(y)) = [4] x + [8] >= [2] x + [0] = ge_active#(x,y) div_active#(s(x),s(y)) = [4] x + [8] >= [8] = 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] = mark#(y) if_active#(true(),x,y) = [4] x + [4] y + [0] >= [4] x + [0] = mark#(x) mark#(div(x,y)) = [4] x + [0] >= [4] x + [0] = div_active#(mark(x),y) mark#(div(x,y)) = [4] x + [0] >= [4] x + [0] = mark#(x) mark#(ge(x,y)) = [4] x + [0] >= [4] x + [0] = c_13(ge_active#(x,y)) mark#(if(x,y,z)) = [4] x + [4] y + [4] z + [0] >= [4] y + [4] z + [0] = if_active#(mark(x),y,z) mark#(if(x,y,z)) = [4] x + [4] y + [4] z + [0] >= [4] x + [0] = mark#(x) mark#(minus(x,y)) = [0] >= [0] = c_15(minus_active#(x,y)) mark#(s(x)) = [4] x + [8] >= [4] x + [0] = mark#(x) minus_active#(s(x),s(y)) = [0] >= [0] = c_19(minus_active#(x,y)) div_active(x,y) = [1] x + [0] >= [1] x + [0] = div(x,y) div_active(0(),s(y)) = [0] >= [0] = 0() div_active(s(x),s(y)) = [1] x + [2] >= [1] x + [2] = if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) = [1] x + [0] >= [1] x + [0] = ge(x,y) ge_active(x,0()) = [1] x + [0] >= [0] = true() ge_active(0(),s(y)) = [0] >= [0] = false() ge_active(s(x),s(y)) = [1] x + [2] >= [1] x + [0] = ge_active(x,y) if_active(x,y,z) = [1] x + [1] y + [1] z + [0] >= [1] x + [1] y + [1] z + [0] = if(x,y,z) if_active(false(),x,y) = [1] x + [1] y + [0] >= [1] y + [0] = mark(y) if_active(true(),x,y) = [1] x + [1] y + [0] >= [1] x + [0] = mark(x) mark(0()) = [0] >= [0] = 0() mark(div(x,y)) = [1] x + [0] >= [1] x + [0] = div_active(mark(x),y) mark(ge(x,y)) = [1] x + [0] >= [1] x + [0] = ge_active(x,y) mark(if(x,y,z)) = [1] x + [1] y + [1] z + [0] >= [1] x + [1] y + [1] z + [0] = if_active(mark(x),y,z) mark(minus(x,y)) = [0] >= [0] = minus_active(x,y) mark(s(x)) = [1] x + [2] >= [1] x + [2] = s(mark(x)) minus_active(x,y) = [0] >= [0] = minus(x,y) minus_active(0(),y) = [0] >= [0] = 0() minus_active(s(x),s(y)) = [0] >= [0] = minus_active(x,y) *** Step 1.b:4.b:3: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)) - Weak DPs: div_active#(s(x),s(y)) -> ge_active#(x,y) div_active#(s(x),s(y)) -> if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active#(s(x),s(y)) -> c_7(ge_active#(x,y)) if_active#(false(),x,y) -> mark#(y) if_active#(true(),x,y) -> mark#(x) mark#(div(x,y)) -> div_active#(mark(x),y) mark#(div(x,y)) -> mark#(x) mark#(ge(x,y)) -> c_13(ge_active#(x,y)) mark#(if(x,y,z)) -> if_active#(mark(x),y,z) mark#(if(x,y,z)) -> mark#(x) mark#(minus(x,y)) -> c_15(minus_active#(x,y)) mark#(s(x)) -> mark#(x) - 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 = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_7) = {1}, uargs(c_13) = {1}, uargs(c_15) = {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] p(div) = [1] x1 + [0] p(div_active) = [1] x1 + [0] p(false) = [0] p(ge) = [0] p(ge_active) = [0] p(if) = [1] x1 + [1] x2 + [1] x3 + [0] p(if_active) = [1] x1 + [1] x2 + [1] x3 + [0] p(mark) = [1] x1 + [0] p(minus) = [1] x1 + [0] p(minus_active) = [1] x1 + [0] p(s) = [1] x1 + [1] p(true) = [0] p(div_active#) = [4] x1 + [0] p(ge_active#) = [0] p(if_active#) = [4] x2 + [4] x3 + [0] p(mark#) = [4] x1 + [0] p(minus_active#) = [4] x1 + [0] p(c_1) = [4] p(c_2) = [1] p(c_3) = [1] p(c_4) = [1] p(c_5) = [0] p(c_6) = [0] p(c_7) = [1] x1 + [0] p(c_8) = [1] p(c_9) = [0] p(c_10) = [1] x1 + [0] p(c_11) = [2] p(c_12) = [1] x1 + [2] p(c_13) = [4] x1 + [0] p(c_14) = [1] x1 + [2] x2 + [1] p(c_15) = [1] x1 + [0] p(c_16) = [0] p(c_17) = [1] p(c_18) = [2] p(c_19) = [1] x1 + [3] Following rules are strictly oriented: minus_active#(s(x),s(y)) = [4] x + [4] > [4] x + [3] = c_19(minus_active#(x,y)) Following rules are (at-least) weakly oriented: div_active#(s(x),s(y)) = [4] x + [4] >= [0] = ge_active#(x,y) div_active#(s(x),s(y)) = [4] x + [4] >= [4] x + [4] = if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active#(s(x),s(y)) = [0] >= [0] = c_7(ge_active#(x,y)) if_active#(false(),x,y) = [4] x + [4] y + [0] >= [4] y + [0] = mark#(y) if_active#(true(),x,y) = [4] x + [4] y + [0] >= [4] x + [0] = mark#(x) mark#(div(x,y)) = [4] x + [0] >= [4] x + [0] = div_active#(mark(x),y) mark#(div(x,y)) = [4] x + [0] >= [4] x + [0] = mark#(x) mark#(ge(x,y)) = [0] >= [0] = c_13(ge_active#(x,y)) mark#(if(x,y,z)) = [4] x + [4] y + [4] z + [0] >= [4] y + [4] z + [0] = if_active#(mark(x),y,z) mark#(if(x,y,z)) = [4] x + [4] y + [4] z + [0] >= [4] x + [0] = mark#(x) mark#(minus(x,y)) = [4] x + [0] >= [4] x + [0] = c_15(minus_active#(x,y)) mark#(s(x)) = [4] x + [4] >= [4] x + [0] = mark#(x) div_active(x,y) = [1] x + [0] >= [1] x + [0] = div(x,y) div_active(0(),s(y)) = [0] >= [0] = 0() div_active(s(x),s(y)) = [1] x + [1] >= [1] x + [1] = if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) = [0] >= [0] = ge(x,y) ge_active(x,0()) = [0] >= [0] = true() ge_active(0(),s(y)) = [0] >= [0] = false() ge_active(s(x),s(y)) = [0] >= [0] = ge_active(x,y) if_active(x,y,z) = [1] x + [1] y + [1] z + [0] >= [1] x + [1] y + [1] z + [0] = if(x,y,z) if_active(false(),x,y) = [1] x + [1] y + [0] >= [1] y + [0] = mark(y) if_active(true(),x,y) = [1] x + [1] y + [0] >= [1] x + [0] = mark(x) mark(0()) = [0] >= [0] = 0() mark(div(x,y)) = [1] x + [0] >= [1] x + [0] = div_active(mark(x),y) mark(ge(x,y)) = [0] >= [0] = ge_active(x,y) mark(if(x,y,z)) = [1] x + [1] y + [1] z + [0] >= [1] x + [1] y + [1] z + [0] = if_active(mark(x),y,z) mark(minus(x,y)) = [1] x + [0] >= [1] x + [0] = minus_active(x,y) mark(s(x)) = [1] x + [1] >= [1] x + [1] = s(mark(x)) minus_active(x,y) = [1] x + [0] >= [1] x + [0] = minus(x,y) minus_active(0(),y) = [0] >= [0] = 0() minus_active(s(x),s(y)) = [1] x + [1] >= [1] x + [0] = minus_active(x,y) *** Step 1.b:4.b:4: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: div_active#(s(x),s(y)) -> ge_active#(x,y) div_active#(s(x),s(y)) -> if_active#(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active#(s(x),s(y)) -> c_7(ge_active#(x,y)) if_active#(false(),x,y) -> mark#(y) if_active#(true(),x,y) -> mark#(x) mark#(div(x,y)) -> div_active#(mark(x),y) mark#(div(x,y)) -> mark#(x) mark#(ge(x,y)) -> c_13(ge_active#(x,y)) mark#(if(x,y,z)) -> if_active#(mark(x),y,z) mark#(if(x,y,z)) -> mark#(x) mark#(minus(x,y)) -> c_15(minus_active#(x,y)) mark#(s(x)) -> mark#(x) minus_active#(s(x),s(y)) -> c_19(minus_active#(x,y)) - Weak TRS: div_active(x,y) -> div(x,y) div_active(0(),s(y)) -> 0() div_active(s(x),s(y)) -> if_active(ge_active(x,y),s(div(minus(x,y),s(y))),0()) ge_active(x,y) -> ge(x,y) ge_active(x,0()) -> true() ge_active(0(),s(y)) -> false() ge_active(s(x),s(y)) -> ge_active(x,y) if_active(x,y,z) -> if(x,y,z) if_active(false(),x,y) -> mark(y) if_active(true(),x,y) -> mark(x) mark(0()) -> 0() mark(div(x,y)) -> div_active(mark(x),y) mark(ge(x,y)) -> ge_active(x,y) mark(if(x,y,z)) -> if_active(mark(x),y,z) mark(minus(x,y)) -> minus_active(x,y) mark(s(x)) -> s(mark(x)) minus_active(x,y) -> minus(x,y) minus_active(0(),y) -> 0() minus_active(s(x),s(y)) -> minus_active(x,y) - Signature: {div_active/2,ge_active/2,if_active/3,mark/1,minus_active/2,div_active#/2,ge_active#/2,if_active#/3,mark#/1 ,minus_active#/2} / {0/0,div/2,false/0,ge/2,if/3,minus/2,s/1,true/0,c_1/0,c_2/0,c_3/2,c_4/0,c_5/0,c_6/0 ,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0,c_12/2,c_13/1,c_14/2,c_15/1,c_16/1,c_17/0,c_18/0,c_19/1} - Obligation: innermost runtime complexity wrt. defined symbols {div_active#,ge_active#,if_active#,mark# ,minus_active#} and constructors {0,div,false,ge,if,minus,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^3))