WORST_CASE(?,O(1)) * Step 1: Sum WORST_CASE(?,O(1)) + Considered Problem: - Strict TRS: e1(x1,x1,x,y,z) -> e5(x1,x,y,z) e1(h1(),h2(),x,y,z) -> e2(x,x,y,z,z) e2(x,x,y,z,z) -> e6(x,y,z) e2(f1(),x,y,z,f2()) -> e3(x,y,x,y,y,z,y,z,x,y,z) e2(i(),x,y,z,i()) -> e6(x,y,z) e3(x,y,x,y,y,z,y,z,x,y,z) -> e6(x,y,z) e3(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z) -> e4(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z) e4(x,x,x,x,x,x,x,x,x,x,x) -> e6(x,x,x) e4(g1(),x1,g2(),x1,g1(),x1,g2(),x1,x,y,z) -> e1(x1,x1,x,y,z) e4(i(),x1,i(),x1,i(),x1,i(),x1,x,y,z) -> e5(x1,x,y,z) e5(i(),x,y,z) -> e6(x,y,z) f1() -> g1() f1() -> g2() f2() -> g1() f2() -> g2() g1() -> h1() g1() -> h2() g2() -> h1() g2() -> h2() h1() -> i() h2() -> i() - Signature: {e1/5,e2/5,e3/11,e4/11,e5/4,f1/0,f2/0,g1/0,g2/0,h1/0,h2/0} / {e6/3,i/0} - Obligation: innermost runtime complexity wrt. defined symbols {e1,e2,e3,e4,e5,f1,f2,g1,g2,h1,h2} and constructors {e6,i} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () * Step 2: DependencyPairs WORST_CASE(?,O(1)) + Considered Problem: - Strict TRS: e1(x1,x1,x,y,z) -> e5(x1,x,y,z) e1(h1(),h2(),x,y,z) -> e2(x,x,y,z,z) e2(x,x,y,z,z) -> e6(x,y,z) e2(f1(),x,y,z,f2()) -> e3(x,y,x,y,y,z,y,z,x,y,z) e2(i(),x,y,z,i()) -> e6(x,y,z) e3(x,y,x,y,y,z,y,z,x,y,z) -> e6(x,y,z) e3(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z) -> e4(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z) e4(x,x,x,x,x,x,x,x,x,x,x) -> e6(x,x,x) e4(g1(),x1,g2(),x1,g1(),x1,g2(),x1,x,y,z) -> e1(x1,x1,x,y,z) e4(i(),x1,i(),x1,i(),x1,i(),x1,x,y,z) -> e5(x1,x,y,z) e5(i(),x,y,z) -> e6(x,y,z) f1() -> g1() f1() -> g2() f2() -> g1() f2() -> g2() g1() -> h1() g1() -> h2() g2() -> h1() g2() -> h2() h1() -> i() h2() -> i() - Signature: {e1/5,e2/5,e3/11,e4/11,e5/4,f1/0,f2/0,g1/0,g2/0,h1/0,h2/0} / {e6/3,i/0} - Obligation: innermost runtime complexity wrt. defined symbols {e1,e2,e3,e4,e5,f1,f2,g1,g2,h1,h2} and constructors {e6,i} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs e1#(x1,x1,x,y,z) -> c_1(e5#(x1,x,y,z)) e1#(h1(),h2(),x,y,z) -> c_2(e2#(x,x,y,z,z)) e2#(x,x,y,z,z) -> c_3() e2#(f1(),x,y,z,f2()) -> c_4(e3#(x,y,x,y,y,z,y,z,x,y,z)) e2#(i(),x,y,z,i()) -> c_5() e3#(x,y,x,y,y,z,y,z,x,y,z) -> c_6() e3#(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z) -> c_7(e4#(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z)) e4#(x,x,x,x,x,x,x,x,x,x,x) -> c_8() e4#(g1(),x1,g2(),x1,g1(),x1,g2(),x1,x,y,z) -> c_9(e1#(x1,x1,x,y,z)) e4#(i(),x1,i(),x1,i(),x1,i(),x1,x,y,z) -> c_10(e5#(x1,x,y,z)) e5#(i(),x,y,z) -> c_11() f1#() -> c_12(g1#()) f1#() -> c_13(g2#()) f2#() -> c_14(g1#()) f2#() -> c_15(g2#()) g1#() -> c_16(h1#()) g1#() -> c_17(h2#()) g2#() -> c_18(h1#()) g2#() -> c_19(h2#()) h1#() -> c_20() h2#() -> c_21() Weak DPs and mark the set of starting terms. * Step 3: PredecessorEstimation WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: e1#(x1,x1,x,y,z) -> c_1(e5#(x1,x,y,z)) e1#(h1(),h2(),x,y,z) -> c_2(e2#(x,x,y,z,z)) e2#(x,x,y,z,z) -> c_3() e2#(f1(),x,y,z,f2()) -> c_4(e3#(x,y,x,y,y,z,y,z,x,y,z)) e2#(i(),x,y,z,i()) -> c_5() e3#(x,y,x,y,y,z,y,z,x,y,z) -> c_6() e3#(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z) -> c_7(e4#(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z)) e4#(x,x,x,x,x,x,x,x,x,x,x) -> c_8() e4#(g1(),x1,g2(),x1,g1(),x1,g2(),x1,x,y,z) -> c_9(e1#(x1,x1,x,y,z)) e4#(i(),x1,i(),x1,i(),x1,i(),x1,x,y,z) -> c_10(e5#(x1,x,y,z)) e5#(i(),x,y,z) -> c_11() f1#() -> c_12(g1#()) f1#() -> c_13(g2#()) f2#() -> c_14(g1#()) f2#() -> c_15(g2#()) g1#() -> c_16(h1#()) g1#() -> c_17(h2#()) g2#() -> c_18(h1#()) g2#() -> c_19(h2#()) h1#() -> c_20() h2#() -> c_21() - Weak TRS: e1(x1,x1,x,y,z) -> e5(x1,x,y,z) e1(h1(),h2(),x,y,z) -> e2(x,x,y,z,z) e2(x,x,y,z,z) -> e6(x,y,z) e2(f1(),x,y,z,f2()) -> e3(x,y,x,y,y,z,y,z,x,y,z) e2(i(),x,y,z,i()) -> e6(x,y,z) e3(x,y,x,y,y,z,y,z,x,y,z) -> e6(x,y,z) e3(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z) -> e4(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z) e4(x,x,x,x,x,x,x,x,x,x,x) -> e6(x,x,x) e4(g1(),x1,g2(),x1,g1(),x1,g2(),x1,x,y,z) -> e1(x1,x1,x,y,z) e4(i(),x1,i(),x1,i(),x1,i(),x1,x,y,z) -> e5(x1,x,y,z) e5(i(),x,y,z) -> e6(x,y,z) f1() -> g1() f1() -> g2() f2() -> g1() f2() -> g2() g1() -> h1() g1() -> h2() g2() -> h1() g2() -> h2() h1() -> i() h2() -> i() - Signature: {e1/5,e2/5,e3/11,e4/11,e5/4,f1/0,f2/0,g1/0,g2/0,h1/0,h2/0,e1#/5,e2#/5,e3#/11,e4#/11,e5#/4,f1#/0,f2#/0,g1#/0 ,g2#/0,h1#/0,h2#/0} / {e6/3,i/0,c_1/1,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0,c_12/1 ,c_13/1,c_14/1,c_15/1,c_16/1,c_17/1,c_18/1,c_19/1,c_20/0,c_21/0} - Obligation: innermost runtime complexity wrt. defined symbols {e1#,e2#,e3#,e4#,e5#,f1#,f2#,g1#,g2#,h1# ,h2#} and constructors {e6,i} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2,3,4,5,6,8,9,11,20,21} by application of Pre({2,3,4,5,6,8,9,11,20,21}) = {1,7,10,16,17,18,19}. Here rules are labelled as follows: 1: e1#(x1,x1,x,y,z) -> c_1(e5#(x1,x,y,z)) 2: e1#(h1(),h2(),x,y,z) -> c_2(e2#(x,x,y,z,z)) 3: e2#(x,x,y,z,z) -> c_3() 4: e2#(f1(),x,y,z,f2()) -> c_4(e3#(x,y,x,y,y,z,y,z,x,y,z)) 5: e2#(i(),x,y,z,i()) -> c_5() 6: e3#(x,y,x,y,y,z,y,z,x,y,z) -> c_6() 7: e3#(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z) -> c_7(e4#(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z)) 8: e4#(x,x,x,x,x,x,x,x,x,x,x) -> c_8() 9: e4#(g1(),x1,g2(),x1,g1(),x1,g2(),x1,x,y,z) -> c_9(e1#(x1,x1,x,y,z)) 10: e4#(i(),x1,i(),x1,i(),x1,i(),x1,x,y,z) -> c_10(e5#(x1,x,y,z)) 11: e5#(i(),x,y,z) -> c_11() 12: f1#() -> c_12(g1#()) 13: f1#() -> c_13(g2#()) 14: f2#() -> c_14(g1#()) 15: f2#() -> c_15(g2#()) 16: g1#() -> c_16(h1#()) 17: g1#() -> c_17(h2#()) 18: g2#() -> c_18(h1#()) 19: g2#() -> c_19(h2#()) 20: h1#() -> c_20() 21: h2#() -> c_21() * Step 4: PredecessorEstimation WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: e1#(x1,x1,x,y,z) -> c_1(e5#(x1,x,y,z)) e3#(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z) -> c_7(e4#(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z)) e4#(i(),x1,i(),x1,i(),x1,i(),x1,x,y,z) -> c_10(e5#(x1,x,y,z)) f1#() -> c_12(g1#()) f1#() -> c_13(g2#()) f2#() -> c_14(g1#()) f2#() -> c_15(g2#()) g1#() -> c_16(h1#()) g1#() -> c_17(h2#()) g2#() -> c_18(h1#()) g2#() -> c_19(h2#()) - Weak DPs: e1#(h1(),h2(),x,y,z) -> c_2(e2#(x,x,y,z,z)) e2#(x,x,y,z,z) -> c_3() e2#(f1(),x,y,z,f2()) -> c_4(e3#(x,y,x,y,y,z,y,z,x,y,z)) e2#(i(),x,y,z,i()) -> c_5() e3#(x,y,x,y,y,z,y,z,x,y,z) -> c_6() e4#(x,x,x,x,x,x,x,x,x,x,x) -> c_8() e4#(g1(),x1,g2(),x1,g1(),x1,g2(),x1,x,y,z) -> c_9(e1#(x1,x1,x,y,z)) e5#(i(),x,y,z) -> c_11() h1#() -> c_20() h2#() -> c_21() - Weak TRS: e1(x1,x1,x,y,z) -> e5(x1,x,y,z) e1(h1(),h2(),x,y,z) -> e2(x,x,y,z,z) e2(x,x,y,z,z) -> e6(x,y,z) e2(f1(),x,y,z,f2()) -> e3(x,y,x,y,y,z,y,z,x,y,z) e2(i(),x,y,z,i()) -> e6(x,y,z) e3(x,y,x,y,y,z,y,z,x,y,z) -> e6(x,y,z) e3(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z) -> e4(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z) e4(x,x,x,x,x,x,x,x,x,x,x) -> e6(x,x,x) e4(g1(),x1,g2(),x1,g1(),x1,g2(),x1,x,y,z) -> e1(x1,x1,x,y,z) e4(i(),x1,i(),x1,i(),x1,i(),x1,x,y,z) -> e5(x1,x,y,z) e5(i(),x,y,z) -> e6(x,y,z) f1() -> g1() f1() -> g2() f2() -> g1() f2() -> g2() g1() -> h1() g1() -> h2() g2() -> h1() g2() -> h2() h1() -> i() h2() -> i() - Signature: {e1/5,e2/5,e3/11,e4/11,e5/4,f1/0,f2/0,g1/0,g2/0,h1/0,h2/0,e1#/5,e2#/5,e3#/11,e4#/11,e5#/4,f1#/0,f2#/0,g1#/0 ,g2#/0,h1#/0,h2#/0} / {e6/3,i/0,c_1/1,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0,c_12/1 ,c_13/1,c_14/1,c_15/1,c_16/1,c_17/1,c_18/1,c_19/1,c_20/0,c_21/0} - Obligation: innermost runtime complexity wrt. defined symbols {e1#,e2#,e3#,e4#,e5#,f1#,f2#,g1#,g2#,h1# ,h2#} and constructors {e6,i} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,3,8,9,10,11} by application of Pre({1,3,8,9,10,11}) = {2,4,5,6,7}. Here rules are labelled as follows: 1: e1#(x1,x1,x,y,z) -> c_1(e5#(x1,x,y,z)) 2: e3#(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z) -> c_7(e4#(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z)) 3: e4#(i(),x1,i(),x1,i(),x1,i(),x1,x,y,z) -> c_10(e5#(x1,x,y,z)) 4: f1#() -> c_12(g1#()) 5: f1#() -> c_13(g2#()) 6: f2#() -> c_14(g1#()) 7: f2#() -> c_15(g2#()) 8: g1#() -> c_16(h1#()) 9: g1#() -> c_17(h2#()) 10: g2#() -> c_18(h1#()) 11: g2#() -> c_19(h2#()) 12: e1#(h1(),h2(),x,y,z) -> c_2(e2#(x,x,y,z,z)) 13: e2#(x,x,y,z,z) -> c_3() 14: e2#(f1(),x,y,z,f2()) -> c_4(e3#(x,y,x,y,y,z,y,z,x,y,z)) 15: e2#(i(),x,y,z,i()) -> c_5() 16: e3#(x,y,x,y,y,z,y,z,x,y,z) -> c_6() 17: e4#(x,x,x,x,x,x,x,x,x,x,x) -> c_8() 18: e4#(g1(),x1,g2(),x1,g1(),x1,g2(),x1,x,y,z) -> c_9(e1#(x1,x1,x,y,z)) 19: e5#(i(),x,y,z) -> c_11() 20: h1#() -> c_20() 21: h2#() -> c_21() * Step 5: PredecessorEstimation WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: e3#(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z) -> c_7(e4#(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z)) f1#() -> c_12(g1#()) f1#() -> c_13(g2#()) f2#() -> c_14(g1#()) f2#() -> c_15(g2#()) - Weak DPs: e1#(x1,x1,x,y,z) -> c_1(e5#(x1,x,y,z)) e1#(h1(),h2(),x,y,z) -> c_2(e2#(x,x,y,z,z)) e2#(x,x,y,z,z) -> c_3() e2#(f1(),x,y,z,f2()) -> c_4(e3#(x,y,x,y,y,z,y,z,x,y,z)) e2#(i(),x,y,z,i()) -> c_5() e3#(x,y,x,y,y,z,y,z,x,y,z) -> c_6() e4#(x,x,x,x,x,x,x,x,x,x,x) -> c_8() e4#(g1(),x1,g2(),x1,g1(),x1,g2(),x1,x,y,z) -> c_9(e1#(x1,x1,x,y,z)) e4#(i(),x1,i(),x1,i(),x1,i(),x1,x,y,z) -> c_10(e5#(x1,x,y,z)) e5#(i(),x,y,z) -> c_11() g1#() -> c_16(h1#()) g1#() -> c_17(h2#()) g2#() -> c_18(h1#()) g2#() -> c_19(h2#()) h1#() -> c_20() h2#() -> c_21() - Weak TRS: e1(x1,x1,x,y,z) -> e5(x1,x,y,z) e1(h1(),h2(),x,y,z) -> e2(x,x,y,z,z) e2(x,x,y,z,z) -> e6(x,y,z) e2(f1(),x,y,z,f2()) -> e3(x,y,x,y,y,z,y,z,x,y,z) e2(i(),x,y,z,i()) -> e6(x,y,z) e3(x,y,x,y,y,z,y,z,x,y,z) -> e6(x,y,z) e3(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z) -> e4(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z) e4(x,x,x,x,x,x,x,x,x,x,x) -> e6(x,x,x) e4(g1(),x1,g2(),x1,g1(),x1,g2(),x1,x,y,z) -> e1(x1,x1,x,y,z) e4(i(),x1,i(),x1,i(),x1,i(),x1,x,y,z) -> e5(x1,x,y,z) e5(i(),x,y,z) -> e6(x,y,z) f1() -> g1() f1() -> g2() f2() -> g1() f2() -> g2() g1() -> h1() g1() -> h2() g2() -> h1() g2() -> h2() h1() -> i() h2() -> i() - Signature: {e1/5,e2/5,e3/11,e4/11,e5/4,f1/0,f2/0,g1/0,g2/0,h1/0,h2/0,e1#/5,e2#/5,e3#/11,e4#/11,e5#/4,f1#/0,f2#/0,g1#/0 ,g2#/0,h1#/0,h2#/0} / {e6/3,i/0,c_1/1,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0,c_12/1 ,c_13/1,c_14/1,c_15/1,c_16/1,c_17/1,c_18/1,c_19/1,c_20/0,c_21/0} - Obligation: innermost runtime complexity wrt. defined symbols {e1#,e2#,e3#,e4#,e5#,f1#,f2#,g1#,g2#,h1# ,h2#} and constructors {e6,i} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,3,4,5} by application of Pre({1,2,3,4,5}) = {}. Here rules are labelled as follows: 1: e3#(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z) -> c_7(e4#(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z)) 2: f1#() -> c_12(g1#()) 3: f1#() -> c_13(g2#()) 4: f2#() -> c_14(g1#()) 5: f2#() -> c_15(g2#()) 6: e1#(x1,x1,x,y,z) -> c_1(e5#(x1,x,y,z)) 7: e1#(h1(),h2(),x,y,z) -> c_2(e2#(x,x,y,z,z)) 8: e2#(x,x,y,z,z) -> c_3() 9: e2#(f1(),x,y,z,f2()) -> c_4(e3#(x,y,x,y,y,z,y,z,x,y,z)) 10: e2#(i(),x,y,z,i()) -> c_5() 11: e3#(x,y,x,y,y,z,y,z,x,y,z) -> c_6() 12: e4#(x,x,x,x,x,x,x,x,x,x,x) -> c_8() 13: e4#(g1(),x1,g2(),x1,g1(),x1,g2(),x1,x,y,z) -> c_9(e1#(x1,x1,x,y,z)) 14: e4#(i(),x1,i(),x1,i(),x1,i(),x1,x,y,z) -> c_10(e5#(x1,x,y,z)) 15: e5#(i(),x,y,z) -> c_11() 16: g1#() -> c_16(h1#()) 17: g1#() -> c_17(h2#()) 18: g2#() -> c_18(h1#()) 19: g2#() -> c_19(h2#()) 20: h1#() -> c_20() 21: h2#() -> c_21() * Step 6: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: e1#(x1,x1,x,y,z) -> c_1(e5#(x1,x,y,z)) e1#(h1(),h2(),x,y,z) -> c_2(e2#(x,x,y,z,z)) e2#(x,x,y,z,z) -> c_3() e2#(f1(),x,y,z,f2()) -> c_4(e3#(x,y,x,y,y,z,y,z,x,y,z)) e2#(i(),x,y,z,i()) -> c_5() e3#(x,y,x,y,y,z,y,z,x,y,z) -> c_6() e3#(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z) -> c_7(e4#(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z)) e4#(x,x,x,x,x,x,x,x,x,x,x) -> c_8() e4#(g1(),x1,g2(),x1,g1(),x1,g2(),x1,x,y,z) -> c_9(e1#(x1,x1,x,y,z)) e4#(i(),x1,i(),x1,i(),x1,i(),x1,x,y,z) -> c_10(e5#(x1,x,y,z)) e5#(i(),x,y,z) -> c_11() f1#() -> c_12(g1#()) f1#() -> c_13(g2#()) f2#() -> c_14(g1#()) f2#() -> c_15(g2#()) g1#() -> c_16(h1#()) g1#() -> c_17(h2#()) g2#() -> c_18(h1#()) g2#() -> c_19(h2#()) h1#() -> c_20() h2#() -> c_21() - Weak TRS: e1(x1,x1,x,y,z) -> e5(x1,x,y,z) e1(h1(),h2(),x,y,z) -> e2(x,x,y,z,z) e2(x,x,y,z,z) -> e6(x,y,z) e2(f1(),x,y,z,f2()) -> e3(x,y,x,y,y,z,y,z,x,y,z) e2(i(),x,y,z,i()) -> e6(x,y,z) e3(x,y,x,y,y,z,y,z,x,y,z) -> e6(x,y,z) e3(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z) -> e4(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z) e4(x,x,x,x,x,x,x,x,x,x,x) -> e6(x,x,x) e4(g1(),x1,g2(),x1,g1(),x1,g2(),x1,x,y,z) -> e1(x1,x1,x,y,z) e4(i(),x1,i(),x1,i(),x1,i(),x1,x,y,z) -> e5(x1,x,y,z) e5(i(),x,y,z) -> e6(x,y,z) f1() -> g1() f1() -> g2() f2() -> g1() f2() -> g2() g1() -> h1() g1() -> h2() g2() -> h1() g2() -> h2() h1() -> i() h2() -> i() - Signature: {e1/5,e2/5,e3/11,e4/11,e5/4,f1/0,f2/0,g1/0,g2/0,h1/0,h2/0,e1#/5,e2#/5,e3#/11,e4#/11,e5#/4,f1#/0,f2#/0,g1#/0 ,g2#/0,h1#/0,h2#/0} / {e6/3,i/0,c_1/1,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0,c_12/1 ,c_13/1,c_14/1,c_15/1,c_16/1,c_17/1,c_18/1,c_19/1,c_20/0,c_21/0} - Obligation: innermost runtime complexity wrt. defined symbols {e1#,e2#,e3#,e4#,e5#,f1#,f2#,g1#,g2#,h1# ,h2#} and constructors {e6,i} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:e1#(x1,x1,x,y,z) -> c_1(e5#(x1,x,y,z)) -->_1 e5#(i(),x,y,z) -> c_11():11 2:W:e1#(h1(),h2(),x,y,z) -> c_2(e2#(x,x,y,z,z)) 3:W:e2#(x,x,y,z,z) -> c_3() 4:W:e2#(f1(),x,y,z,f2()) -> c_4(e3#(x,y,x,y,y,z,y,z,x,y,z)) 5:W:e2#(i(),x,y,z,i()) -> c_5() 6:W:e3#(x,y,x,y,y,z,y,z,x,y,z) -> c_6() 7:W:e3#(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z) -> c_7(e4#(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z)) -->_1 e4#(i(),x1,i(),x1,i(),x1,i(),x1,x,y,z) -> c_10(e5#(x1,x,y,z)):10 -->_1 e4#(x,x,x,x,x,x,x,x,x,x,x) -> c_8():8 8:W:e4#(x,x,x,x,x,x,x,x,x,x,x) -> c_8() 9:W:e4#(g1(),x1,g2(),x1,g1(),x1,g2(),x1,x,y,z) -> c_9(e1#(x1,x1,x,y,z)) 10:W:e4#(i(),x1,i(),x1,i(),x1,i(),x1,x,y,z) -> c_10(e5#(x1,x,y,z)) -->_1 e5#(i(),x,y,z) -> c_11():11 11:W:e5#(i(),x,y,z) -> c_11() 12:W:f1#() -> c_12(g1#()) -->_1 g1#() -> c_17(h2#()):17 -->_1 g1#() -> c_16(h1#()):16 13:W:f1#() -> c_13(g2#()) -->_1 g2#() -> c_19(h2#()):19 -->_1 g2#() -> c_18(h1#()):18 14:W:f2#() -> c_14(g1#()) -->_1 g1#() -> c_17(h2#()):17 -->_1 g1#() -> c_16(h1#()):16 15:W:f2#() -> c_15(g2#()) -->_1 g2#() -> c_19(h2#()):19 -->_1 g2#() -> c_18(h1#()):18 16:W:g1#() -> c_16(h1#()) -->_1 h1#() -> c_20():20 17:W:g1#() -> c_17(h2#()) -->_1 h2#() -> c_21():21 18:W:g2#() -> c_18(h1#()) -->_1 h1#() -> c_20():20 19:W:g2#() -> c_19(h2#()) -->_1 h2#() -> c_21():21 20:W:h1#() -> c_20() 21:W:h2#() -> c_21() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 15: f2#() -> c_15(g2#()) 14: f2#() -> c_14(g1#()) 13: f1#() -> c_13(g2#()) 18: g2#() -> c_18(h1#()) 19: g2#() -> c_19(h2#()) 12: f1#() -> c_12(g1#()) 16: g1#() -> c_16(h1#()) 20: h1#() -> c_20() 17: g1#() -> c_17(h2#()) 21: h2#() -> c_21() 9: e4#(g1(),x1,g2(),x1,g1(),x1,g2(),x1,x,y,z) -> c_9(e1#(x1,x1,x,y,z)) 7: e3#(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z) -> c_7(e4#(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z)) 8: e4#(x,x,x,x,x,x,x,x,x,x,x) -> c_8() 10: e4#(i(),x1,i(),x1,i(),x1,i(),x1,x,y,z) -> c_10(e5#(x1,x,y,z)) 6: e3#(x,y,x,y,y,z,y,z,x,y,z) -> c_6() 5: e2#(i(),x,y,z,i()) -> c_5() 4: e2#(f1(),x,y,z,f2()) -> c_4(e3#(x,y,x,y,y,z,y,z,x,y,z)) 3: e2#(x,x,y,z,z) -> c_3() 2: e1#(h1(),h2(),x,y,z) -> c_2(e2#(x,x,y,z,z)) 1: e1#(x1,x1,x,y,z) -> c_1(e5#(x1,x,y,z)) 11: e5#(i(),x,y,z) -> c_11() * Step 7: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: e1(x1,x1,x,y,z) -> e5(x1,x,y,z) e1(h1(),h2(),x,y,z) -> e2(x,x,y,z,z) e2(x,x,y,z,z) -> e6(x,y,z) e2(f1(),x,y,z,f2()) -> e3(x,y,x,y,y,z,y,z,x,y,z) e2(i(),x,y,z,i()) -> e6(x,y,z) e3(x,y,x,y,y,z,y,z,x,y,z) -> e6(x,y,z) e3(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z) -> e4(x1,x1,x2,x2,x3,x3,x4,x4,x,y,z) e4(x,x,x,x,x,x,x,x,x,x,x) -> e6(x,x,x) e4(g1(),x1,g2(),x1,g1(),x1,g2(),x1,x,y,z) -> e1(x1,x1,x,y,z) e4(i(),x1,i(),x1,i(),x1,i(),x1,x,y,z) -> e5(x1,x,y,z) e5(i(),x,y,z) -> e6(x,y,z) f1() -> g1() f1() -> g2() f2() -> g1() f2() -> g2() g1() -> h1() g1() -> h2() g2() -> h1() g2() -> h2() h1() -> i() h2() -> i() - Signature: {e1/5,e2/5,e3/11,e4/11,e5/4,f1/0,f2/0,g1/0,g2/0,h1/0,h2/0,e1#/5,e2#/5,e3#/11,e4#/11,e5#/4,f1#/0,f2#/0,g1#/0 ,g2#/0,h1#/0,h2#/0} / {e6/3,i/0,c_1/1,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0,c_7/1,c_8/0,c_9/1,c_10/1,c_11/0,c_12/1 ,c_13/1,c_14/1,c_15/1,c_16/1,c_17/1,c_18/1,c_19/1,c_20/0,c_21/0} - Obligation: innermost runtime complexity wrt. defined symbols {e1#,e2#,e3#,e4#,e5#,f1#,f2#,g1#,g2#,h1# ,h2#} and constructors {e6,i} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(1))