WORST_CASE(Omega(n^1),O(n^2)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^2)) + Considered Problem: - Strict TRS: DEL(z0,.(z1,z2)) -> c7(DEL(z0,z2)) DEL(z0,nil()) -> c6() MIN(z0,.(z1,z2)) -> c4(MIN(z0,z2)) MIN(z0,.(z1,z2)) -> c5(MIN(z1,z2)) MIN(z0,nil()) -> c3() MSORT(.(z0,z1)) -> c1(MIN(z0,z1)) MSORT(.(z0,z1)) -> c2(MSORT(del(min(z0,z1),.(z0,z1))),DEL(min(z0,z1),.(z0,z1)),MIN(z0,z1)) MSORT(nil()) -> c() - Weak TRS: del(z0,.(z1,z2)) -> if(=(z0,z1),z2,.(z1,del(z0,z2))) del(z0,nil()) -> nil() min(z0,.(z1,z2)) -> if(<=(z0,z1),min(z0,z2),min(z1,z2)) min(z0,nil()) -> z0 msort(.(z0,z1)) -> .(min(z0,z1),msort(del(min(z0,z1),.(z0,z1)))) msort(nil()) -> nil() - Signature: {DEL/2,MIN/2,MSORT/1,del/2,min/2,msort/1} / {./2,<=/2,=/2,c/0,c1/1,c2/3,c3/0,c4/1,c5/1,c6/0,c7/1,if/3,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {DEL,MIN,MSORT,del,min,msort} and constructors {.,<=,=,c ,c1,c2,c3,c4,c5,c6,c7,if,nil} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: DEL(z0,.(z1,z2)) -> c7(DEL(z0,z2)) DEL(z0,nil()) -> c6() MIN(z0,.(z1,z2)) -> c4(MIN(z0,z2)) MIN(z0,.(z1,z2)) -> c5(MIN(z1,z2)) MIN(z0,nil()) -> c3() MSORT(.(z0,z1)) -> c1(MIN(z0,z1)) MSORT(.(z0,z1)) -> c2(MSORT(del(min(z0,z1),.(z0,z1))),DEL(min(z0,z1),.(z0,z1)),MIN(z0,z1)) MSORT(nil()) -> c() - Weak TRS: del(z0,.(z1,z2)) -> if(=(z0,z1),z2,.(z1,del(z0,z2))) del(z0,nil()) -> nil() min(z0,.(z1,z2)) -> if(<=(z0,z1),min(z0,z2),min(z1,z2)) min(z0,nil()) -> z0 msort(.(z0,z1)) -> .(min(z0,z1),msort(del(min(z0,z1),.(z0,z1)))) msort(nil()) -> nil() - Signature: {DEL/2,MIN/2,MSORT/1,del/2,min/2,msort/1} / {./2,<=/2,=/2,c/0,c1/1,c2/3,c3/0,c4/1,c5/1,c6/0,c7/1,if/3,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {DEL,MIN,MSORT,del,min,msort} and constructors {.,<=,=,c ,c1,c2,c3,c4,c5,c6,c7,if,nil} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: DEL(z0,.(z1,z2)) -> c7(DEL(z0,z2)) DEL(z0,nil()) -> c6() MIN(z0,.(z1,z2)) -> c4(MIN(z0,z2)) MIN(z0,.(z1,z2)) -> c5(MIN(z1,z2)) MIN(z0,nil()) -> c3() MSORT(.(z0,z1)) -> c1(MIN(z0,z1)) MSORT(.(z0,z1)) -> c2(MSORT(del(min(z0,z1),.(z0,z1))),DEL(min(z0,z1),.(z0,z1)),MIN(z0,z1)) MSORT(nil()) -> c() - Weak TRS: del(z0,.(z1,z2)) -> if(=(z0,z1),z2,.(z1,del(z0,z2))) del(z0,nil()) -> nil() min(z0,.(z1,z2)) -> if(<=(z0,z1),min(z0,z2),min(z1,z2)) min(z0,nil()) -> z0 msort(.(z0,z1)) -> .(min(z0,z1),msort(del(min(z0,z1),.(z0,z1)))) msort(nil()) -> nil() - Signature: {DEL/2,MIN/2,MSORT/1,del/2,min/2,msort/1} / {./2,<=/2,=/2,c/0,c1/1,c2/3,c3/0,c4/1,c5/1,c6/0,c7/1,if/3,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {DEL,MIN,MSORT,del,min,msort} and constructors {.,<=,=,c ,c1,c2,c3,c4,c5,c6,c7,if,nil} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: DEL(x,z){z -> .(y,z)} = DEL(x,.(y,z)) ->^+ c7(DEL(x,z)) = C[DEL(x,z) = DEL(x,z){}] ** Step 1.b:1: DependencyPairs. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: DEL(z0,.(z1,z2)) -> c7(DEL(z0,z2)) DEL(z0,nil()) -> c6() MIN(z0,.(z1,z2)) -> c4(MIN(z0,z2)) MIN(z0,.(z1,z2)) -> c5(MIN(z1,z2)) MIN(z0,nil()) -> c3() MSORT(.(z0,z1)) -> c1(MIN(z0,z1)) MSORT(.(z0,z1)) -> c2(MSORT(del(min(z0,z1),.(z0,z1))),DEL(min(z0,z1),.(z0,z1)),MIN(z0,z1)) MSORT(nil()) -> c() - Weak TRS: del(z0,.(z1,z2)) -> if(=(z0,z1),z2,.(z1,del(z0,z2))) del(z0,nil()) -> nil() min(z0,.(z1,z2)) -> if(<=(z0,z1),min(z0,z2),min(z1,z2)) min(z0,nil()) -> z0 msort(.(z0,z1)) -> .(min(z0,z1),msort(del(min(z0,z1),.(z0,z1)))) msort(nil()) -> nil() - Signature: {DEL/2,MIN/2,MSORT/1,del/2,min/2,msort/1} / {./2,<=/2,=/2,c/0,c1/1,c2/3,c3/0,c4/1,c5/1,c6/0,c7/1,if/3,nil/0} - Obligation: innermost runtime complexity wrt. defined symbols {DEL,MIN,MSORT,del,min,msort} and constructors {.,<=,=,c ,c1,c2,c3,c4,c5,c6,c7,if,nil} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs DEL#(z0,.(z1,z2)) -> c_1(DEL#(z0,z2)) DEL#(z0,nil()) -> c_2() MIN#(z0,.(z1,z2)) -> c_3(MIN#(z0,z2)) MIN#(z0,.(z1,z2)) -> c_4(MIN#(z1,z2)) MIN#(z0,nil()) -> c_5() MSORT#(.(z0,z1)) -> c_6(MIN#(z0,z1)) MSORT#(.(z0,z1)) -> c_7(MSORT#(del(min(z0,z1),.(z0,z1))) ,del#(min(z0,z1),.(z0,z1)) ,min#(z0,z1) ,DEL#(min(z0,z1),.(z0,z1)) ,min#(z0,z1) ,MIN#(z0,z1)) MSORT#(nil()) -> c_8() Weak DPs del#(z0,.(z1,z2)) -> c_9(del#(z0,z2)) del#(z0,nil()) -> c_10() min#(z0,.(z1,z2)) -> c_11(min#(z0,z2),min#(z1,z2)) min#(z0,nil()) -> c_12() msort#(.(z0,z1)) -> c_13(min#(z0,z1),msort#(del(min(z0,z1),.(z0,z1))),del#(min(z0,z1),.(z0,z1)),min#(z0,z1)) msort#(nil()) -> c_14() and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: DEL#(z0,.(z1,z2)) -> c_1(DEL#(z0,z2)) DEL#(z0,nil()) -> c_2() MIN#(z0,.(z1,z2)) -> c_3(MIN#(z0,z2)) MIN#(z0,.(z1,z2)) -> c_4(MIN#(z1,z2)) MIN#(z0,nil()) -> c_5() MSORT#(.(z0,z1)) -> c_6(MIN#(z0,z1)) MSORT#(.(z0,z1)) -> c_7(MSORT#(del(min(z0,z1),.(z0,z1))) ,del#(min(z0,z1),.(z0,z1)) ,min#(z0,z1) ,DEL#(min(z0,z1),.(z0,z1)) ,min#(z0,z1) ,MIN#(z0,z1)) MSORT#(nil()) -> c_8() - Weak DPs: del#(z0,.(z1,z2)) -> c_9(del#(z0,z2)) del#(z0,nil()) -> c_10() min#(z0,.(z1,z2)) -> c_11(min#(z0,z2),min#(z1,z2)) min#(z0,nil()) -> c_12() msort#(.(z0,z1)) -> c_13(min#(z0,z1),msort#(del(min(z0,z1),.(z0,z1))),del#(min(z0,z1),.(z0,z1)),min#(z0,z1)) msort#(nil()) -> c_14() - Weak TRS: DEL(z0,.(z1,z2)) -> c7(DEL(z0,z2)) DEL(z0,nil()) -> c6() MIN(z0,.(z1,z2)) -> c4(MIN(z0,z2)) MIN(z0,.(z1,z2)) -> c5(MIN(z1,z2)) MIN(z0,nil()) -> c3() MSORT(.(z0,z1)) -> c1(MIN(z0,z1)) MSORT(.(z0,z1)) -> c2(MSORT(del(min(z0,z1),.(z0,z1))),DEL(min(z0,z1),.(z0,z1)),MIN(z0,z1)) MSORT(nil()) -> c() del(z0,.(z1,z2)) -> if(=(z0,z1),z2,.(z1,del(z0,z2))) del(z0,nil()) -> nil() min(z0,.(z1,z2)) -> if(<=(z0,z1),min(z0,z2),min(z1,z2)) min(z0,nil()) -> z0 msort(.(z0,z1)) -> .(min(z0,z1),msort(del(min(z0,z1),.(z0,z1)))) msort(nil()) -> nil() - Signature: {DEL/2,MIN/2,MSORT/1,del/2,min/2,msort/1,DEL#/2,MIN#/2,MSORT#/1,del#/2,min#/2,msort#/1} / {./2,<=/2,=/2,c/0 ,c1/1,c2/3,c3/0,c4/1,c5/1,c6/0,c7/1,if/3,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/1,c_7/6,c_8/0,c_9/1,c_10/0 ,c_11/2,c_12/0,c_13/4,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {DEL#,MIN#,MSORT#,del#,min#,msort#} and constructors {.,<= ,=,c,c1,c2,c3,c4,c5,c6,c7,if,nil} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2,5,8} by application of Pre({2,5,8}) = {1,3,4,6,7}. Here rules are labelled as follows: 1: DEL#(z0,.(z1,z2)) -> c_1(DEL#(z0,z2)) 2: DEL#(z0,nil()) -> c_2() 3: MIN#(z0,.(z1,z2)) -> c_3(MIN#(z0,z2)) 4: MIN#(z0,.(z1,z2)) -> c_4(MIN#(z1,z2)) 5: MIN#(z0,nil()) -> c_5() 6: MSORT#(.(z0,z1)) -> c_6(MIN#(z0,z1)) 7: MSORT#(.(z0,z1)) -> c_7(MSORT#(del(min(z0,z1),.(z0,z1))) ,del#(min(z0,z1),.(z0,z1)) ,min#(z0,z1) ,DEL#(min(z0,z1),.(z0,z1)) ,min#(z0,z1) ,MIN#(z0,z1)) 8: MSORT#(nil()) -> c_8() 9: del#(z0,.(z1,z2)) -> c_9(del#(z0,z2)) 10: del#(z0,nil()) -> c_10() 11: min#(z0,.(z1,z2)) -> c_11(min#(z0,z2),min#(z1,z2)) 12: min#(z0,nil()) -> c_12() 13: msort#(.(z0,z1)) -> c_13(min#(z0,z1) ,msort#(del(min(z0,z1),.(z0,z1))) ,del#(min(z0,z1),.(z0,z1)) ,min#(z0,z1)) 14: msort#(nil()) -> c_14() ** Step 1.b:3: RemoveWeakSuffixes. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: DEL#(z0,.(z1,z2)) -> c_1(DEL#(z0,z2)) MIN#(z0,.(z1,z2)) -> c_3(MIN#(z0,z2)) MIN#(z0,.(z1,z2)) -> c_4(MIN#(z1,z2)) MSORT#(.(z0,z1)) -> c_6(MIN#(z0,z1)) MSORT#(.(z0,z1)) -> c_7(MSORT#(del(min(z0,z1),.(z0,z1))) ,del#(min(z0,z1),.(z0,z1)) ,min#(z0,z1) ,DEL#(min(z0,z1),.(z0,z1)) ,min#(z0,z1) ,MIN#(z0,z1)) - Weak DPs: DEL#(z0,nil()) -> c_2() MIN#(z0,nil()) -> c_5() MSORT#(nil()) -> c_8() del#(z0,.(z1,z2)) -> c_9(del#(z0,z2)) del#(z0,nil()) -> c_10() min#(z0,.(z1,z2)) -> c_11(min#(z0,z2),min#(z1,z2)) min#(z0,nil()) -> c_12() msort#(.(z0,z1)) -> c_13(min#(z0,z1),msort#(del(min(z0,z1),.(z0,z1))),del#(min(z0,z1),.(z0,z1)),min#(z0,z1)) msort#(nil()) -> c_14() - Weak TRS: DEL(z0,.(z1,z2)) -> c7(DEL(z0,z2)) DEL(z0,nil()) -> c6() MIN(z0,.(z1,z2)) -> c4(MIN(z0,z2)) MIN(z0,.(z1,z2)) -> c5(MIN(z1,z2)) MIN(z0,nil()) -> c3() MSORT(.(z0,z1)) -> c1(MIN(z0,z1)) MSORT(.(z0,z1)) -> c2(MSORT(del(min(z0,z1),.(z0,z1))),DEL(min(z0,z1),.(z0,z1)),MIN(z0,z1)) MSORT(nil()) -> c() del(z0,.(z1,z2)) -> if(=(z0,z1),z2,.(z1,del(z0,z2))) del(z0,nil()) -> nil() min(z0,.(z1,z2)) -> if(<=(z0,z1),min(z0,z2),min(z1,z2)) min(z0,nil()) -> z0 msort(.(z0,z1)) -> .(min(z0,z1),msort(del(min(z0,z1),.(z0,z1)))) msort(nil()) -> nil() - Signature: {DEL/2,MIN/2,MSORT/1,del/2,min/2,msort/1,DEL#/2,MIN#/2,MSORT#/1,del#/2,min#/2,msort#/1} / {./2,<=/2,=/2,c/0 ,c1/1,c2/3,c3/0,c4/1,c5/1,c6/0,c7/1,if/3,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/1,c_7/6,c_8/0,c_9/1,c_10/0 ,c_11/2,c_12/0,c_13/4,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {DEL#,MIN#,MSORT#,del#,min#,msort#} and constructors {.,<= ,=,c,c1,c2,c3,c4,c5,c6,c7,if,nil} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:DEL#(z0,.(z1,z2)) -> c_1(DEL#(z0,z2)) -->_1 DEL#(z0,nil()) -> c_2():6 -->_1 DEL#(z0,.(z1,z2)) -> c_1(DEL#(z0,z2)):1 2:S:MIN#(z0,.(z1,z2)) -> c_3(MIN#(z0,z2)) -->_1 MIN#(z0,.(z1,z2)) -> c_4(MIN#(z1,z2)):3 -->_1 MIN#(z0,nil()) -> c_5():7 -->_1 MIN#(z0,.(z1,z2)) -> c_3(MIN#(z0,z2)):2 3:S:MIN#(z0,.(z1,z2)) -> c_4(MIN#(z1,z2)) -->_1 MIN#(z0,nil()) -> c_5():7 -->_1 MIN#(z0,.(z1,z2)) -> c_4(MIN#(z1,z2)):3 -->_1 MIN#(z0,.(z1,z2)) -> c_3(MIN#(z0,z2)):2 4:S:MSORT#(.(z0,z1)) -> c_6(MIN#(z0,z1)) -->_1 MIN#(z0,nil()) -> c_5():7 -->_1 MIN#(z0,.(z1,z2)) -> c_4(MIN#(z1,z2)):3 -->_1 MIN#(z0,.(z1,z2)) -> c_3(MIN#(z0,z2)):2 5:S:MSORT#(.(z0,z1)) -> c_7(MSORT#(del(min(z0,z1),.(z0,z1))) ,del#(min(z0,z1),.(z0,z1)) ,min#(z0,z1) ,DEL#(min(z0,z1),.(z0,z1)) ,min#(z0,z1) ,MIN#(z0,z1)) -->_5 min#(z0,.(z1,z2)) -> c_11(min#(z0,z2),min#(z1,z2)):11 -->_3 min#(z0,.(z1,z2)) -> c_11(min#(z0,z2),min#(z1,z2)):11 -->_2 del#(z0,.(z1,z2)) -> c_9(del#(z0,z2)):9 -->_5 min#(z0,nil()) -> c_12():12 -->_3 min#(z0,nil()) -> c_12():12 -->_1 MSORT#(nil()) -> c_8():8 -->_6 MIN#(z0,nil()) -> c_5():7 -->_1 MSORT#(.(z0,z1)) -> c_7(MSORT#(del(min(z0,z1),.(z0,z1))) ,del#(min(z0,z1),.(z0,z1)) ,min#(z0,z1) ,DEL#(min(z0,z1),.(z0,z1)) ,min#(z0,z1) ,MIN#(z0,z1)):5 -->_1 MSORT#(.(z0,z1)) -> c_6(MIN#(z0,z1)):4 -->_6 MIN#(z0,.(z1,z2)) -> c_4(MIN#(z1,z2)):3 -->_6 MIN#(z0,.(z1,z2)) -> c_3(MIN#(z0,z2)):2 -->_4 DEL#(z0,.(z1,z2)) -> c_1(DEL#(z0,z2)):1 6:W:DEL#(z0,nil()) -> c_2() 7:W:MIN#(z0,nil()) -> c_5() 8:W:MSORT#(nil()) -> c_8() 9:W:del#(z0,.(z1,z2)) -> c_9(del#(z0,z2)) -->_1 del#(z0,nil()) -> c_10():10 -->_1 del#(z0,.(z1,z2)) -> c_9(del#(z0,z2)):9 10:W:del#(z0,nil()) -> c_10() 11:W:min#(z0,.(z1,z2)) -> c_11(min#(z0,z2),min#(z1,z2)) -->_2 min#(z0,nil()) -> c_12():12 -->_1 min#(z0,nil()) -> c_12():12 -->_2 min#(z0,.(z1,z2)) -> c_11(min#(z0,z2),min#(z1,z2)):11 -->_1 min#(z0,.(z1,z2)) -> c_11(min#(z0,z2),min#(z1,z2)):11 12:W:min#(z0,nil()) -> c_12() 13:W:msort#(.(z0,z1)) -> c_13(min#(z0,z1) ,msort#(del(min(z0,z1),.(z0,z1))) ,del#(min(z0,z1),.(z0,z1)) ,min#(z0,z1)) -->_2 msort#(nil()) -> c_14():14 -->_2 msort#(.(z0,z1)) -> c_13(min#(z0,z1) ,msort#(del(min(z0,z1),.(z0,z1))) ,del#(min(z0,z1),.(z0,z1)) ,min#(z0,z1)):13 -->_4 min#(z0,nil()) -> c_12():12 -->_1 min#(z0,nil()) -> c_12():12 -->_4 min#(z0,.(z1,z2)) -> c_11(min#(z0,z2),min#(z1,z2)):11 -->_1 min#(z0,.(z1,z2)) -> c_11(min#(z0,z2),min#(z1,z2)):11 -->_3 del#(z0,.(z1,z2)) -> c_9(del#(z0,z2)):9 14:W:msort#(nil()) -> c_14() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 13: msort#(.(z0,z1)) -> c_13(min#(z0,z1) ,msort#(del(min(z0,z1),.(z0,z1))) ,del#(min(z0,z1),.(z0,z1)) ,min#(z0,z1)) 14: msort#(nil()) -> c_14() 8: MSORT#(nil()) -> c_8() 9: del#(z0,.(z1,z2)) -> c_9(del#(z0,z2)) 10: del#(z0,nil()) -> c_10() 11: min#(z0,.(z1,z2)) -> c_11(min#(z0,z2),min#(z1,z2)) 12: min#(z0,nil()) -> c_12() 7: MIN#(z0,nil()) -> c_5() 6: DEL#(z0,nil()) -> c_2() ** Step 1.b:4: SimplifyRHS. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: DEL#(z0,.(z1,z2)) -> c_1(DEL#(z0,z2)) MIN#(z0,.(z1,z2)) -> c_3(MIN#(z0,z2)) MIN#(z0,.(z1,z2)) -> c_4(MIN#(z1,z2)) MSORT#(.(z0,z1)) -> c_6(MIN#(z0,z1)) MSORT#(.(z0,z1)) -> c_7(MSORT#(del(min(z0,z1),.(z0,z1))) ,del#(min(z0,z1),.(z0,z1)) ,min#(z0,z1) ,DEL#(min(z0,z1),.(z0,z1)) ,min#(z0,z1) ,MIN#(z0,z1)) - Weak TRS: DEL(z0,.(z1,z2)) -> c7(DEL(z0,z2)) DEL(z0,nil()) -> c6() MIN(z0,.(z1,z2)) -> c4(MIN(z0,z2)) MIN(z0,.(z1,z2)) -> c5(MIN(z1,z2)) MIN(z0,nil()) -> c3() MSORT(.(z0,z1)) -> c1(MIN(z0,z1)) MSORT(.(z0,z1)) -> c2(MSORT(del(min(z0,z1),.(z0,z1))),DEL(min(z0,z1),.(z0,z1)),MIN(z0,z1)) MSORT(nil()) -> c() del(z0,.(z1,z2)) -> if(=(z0,z1),z2,.(z1,del(z0,z2))) del(z0,nil()) -> nil() min(z0,.(z1,z2)) -> if(<=(z0,z1),min(z0,z2),min(z1,z2)) min(z0,nil()) -> z0 msort(.(z0,z1)) -> .(min(z0,z1),msort(del(min(z0,z1),.(z0,z1)))) msort(nil()) -> nil() - Signature: {DEL/2,MIN/2,MSORT/1,del/2,min/2,msort/1,DEL#/2,MIN#/2,MSORT#/1,del#/2,min#/2,msort#/1} / {./2,<=/2,=/2,c/0 ,c1/1,c2/3,c3/0,c4/1,c5/1,c6/0,c7/1,if/3,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/1,c_7/6,c_8/0,c_9/1,c_10/0 ,c_11/2,c_12/0,c_13/4,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {DEL#,MIN#,MSORT#,del#,min#,msort#} and constructors {.,<= ,=,c,c1,c2,c3,c4,c5,c6,c7,if,nil} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:DEL#(z0,.(z1,z2)) -> c_1(DEL#(z0,z2)) -->_1 DEL#(z0,.(z1,z2)) -> c_1(DEL#(z0,z2)):1 2:S:MIN#(z0,.(z1,z2)) -> c_3(MIN#(z0,z2)) -->_1 MIN#(z0,.(z1,z2)) -> c_4(MIN#(z1,z2)):3 -->_1 MIN#(z0,.(z1,z2)) -> c_3(MIN#(z0,z2)):2 3:S:MIN#(z0,.(z1,z2)) -> c_4(MIN#(z1,z2)) -->_1 MIN#(z0,.(z1,z2)) -> c_4(MIN#(z1,z2)):3 -->_1 MIN#(z0,.(z1,z2)) -> c_3(MIN#(z0,z2)):2 4:S:MSORT#(.(z0,z1)) -> c_6(MIN#(z0,z1)) -->_1 MIN#(z0,.(z1,z2)) -> c_4(MIN#(z1,z2)):3 -->_1 MIN#(z0,.(z1,z2)) -> c_3(MIN#(z0,z2)):2 5:S:MSORT#(.(z0,z1)) -> c_7(MSORT#(del(min(z0,z1),.(z0,z1))) ,del#(min(z0,z1),.(z0,z1)) ,min#(z0,z1) ,DEL#(min(z0,z1),.(z0,z1)) ,min#(z0,z1) ,MIN#(z0,z1)) -->_1 MSORT#(.(z0,z1)) -> c_7(MSORT#(del(min(z0,z1),.(z0,z1))) ,del#(min(z0,z1),.(z0,z1)) ,min#(z0,z1) ,DEL#(min(z0,z1),.(z0,z1)) ,min#(z0,z1) ,MIN#(z0,z1)):5 -->_1 MSORT#(.(z0,z1)) -> c_6(MIN#(z0,z1)):4 -->_6 MIN#(z0,.(z1,z2)) -> c_4(MIN#(z1,z2)):3 -->_6 MIN#(z0,.(z1,z2)) -> c_3(MIN#(z0,z2)):2 -->_4 DEL#(z0,.(z1,z2)) -> c_1(DEL#(z0,z2)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: MSORT#(.(z0,z1)) -> c_7(MSORT#(del(min(z0,z1),.(z0,z1))),DEL#(min(z0,z1),.(z0,z1)),MIN#(z0,z1)) ** Step 1.b:5: UsableRules. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: DEL#(z0,.(z1,z2)) -> c_1(DEL#(z0,z2)) MIN#(z0,.(z1,z2)) -> c_3(MIN#(z0,z2)) MIN#(z0,.(z1,z2)) -> c_4(MIN#(z1,z2)) MSORT#(.(z0,z1)) -> c_6(MIN#(z0,z1)) MSORT#(.(z0,z1)) -> c_7(MSORT#(del(min(z0,z1),.(z0,z1))),DEL#(min(z0,z1),.(z0,z1)),MIN#(z0,z1)) - Weak TRS: DEL(z0,.(z1,z2)) -> c7(DEL(z0,z2)) DEL(z0,nil()) -> c6() MIN(z0,.(z1,z2)) -> c4(MIN(z0,z2)) MIN(z0,.(z1,z2)) -> c5(MIN(z1,z2)) MIN(z0,nil()) -> c3() MSORT(.(z0,z1)) -> c1(MIN(z0,z1)) MSORT(.(z0,z1)) -> c2(MSORT(del(min(z0,z1),.(z0,z1))),DEL(min(z0,z1),.(z0,z1)),MIN(z0,z1)) MSORT(nil()) -> c() del(z0,.(z1,z2)) -> if(=(z0,z1),z2,.(z1,del(z0,z2))) del(z0,nil()) -> nil() min(z0,.(z1,z2)) -> if(<=(z0,z1),min(z0,z2),min(z1,z2)) min(z0,nil()) -> z0 msort(.(z0,z1)) -> .(min(z0,z1),msort(del(min(z0,z1),.(z0,z1)))) msort(nil()) -> nil() - Signature: {DEL/2,MIN/2,MSORT/1,del/2,min/2,msort/1,DEL#/2,MIN#/2,MSORT#/1,del#/2,min#/2,msort#/1} / {./2,<=/2,=/2,c/0 ,c1/1,c2/3,c3/0,c4/1,c5/1,c6/0,c7/1,if/3,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/1,c_7/3,c_8/0,c_9/1,c_10/0 ,c_11/2,c_12/0,c_13/4,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {DEL#,MIN#,MSORT#,del#,min#,msort#} and constructors {.,<= ,=,c,c1,c2,c3,c4,c5,c6,c7,if,nil} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: del(z0,.(z1,z2)) -> if(=(z0,z1),z2,.(z1,del(z0,z2))) del(z0,nil()) -> nil() min(z0,.(z1,z2)) -> if(<=(z0,z1),min(z0,z2),min(z1,z2)) min(z0,nil()) -> z0 DEL#(z0,.(z1,z2)) -> c_1(DEL#(z0,z2)) MIN#(z0,.(z1,z2)) -> c_3(MIN#(z0,z2)) MIN#(z0,.(z1,z2)) -> c_4(MIN#(z1,z2)) MSORT#(.(z0,z1)) -> c_6(MIN#(z0,z1)) MSORT#(.(z0,z1)) -> c_7(MSORT#(del(min(z0,z1),.(z0,z1))),DEL#(min(z0,z1),.(z0,z1)),MIN#(z0,z1)) ** Step 1.b:6: DecomposeDG. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: DEL#(z0,.(z1,z2)) -> c_1(DEL#(z0,z2)) MIN#(z0,.(z1,z2)) -> c_3(MIN#(z0,z2)) MIN#(z0,.(z1,z2)) -> c_4(MIN#(z1,z2)) MSORT#(.(z0,z1)) -> c_6(MIN#(z0,z1)) MSORT#(.(z0,z1)) -> c_7(MSORT#(del(min(z0,z1),.(z0,z1))),DEL#(min(z0,z1),.(z0,z1)),MIN#(z0,z1)) - Weak TRS: del(z0,.(z1,z2)) -> if(=(z0,z1),z2,.(z1,del(z0,z2))) del(z0,nil()) -> nil() min(z0,.(z1,z2)) -> if(<=(z0,z1),min(z0,z2),min(z1,z2)) min(z0,nil()) -> z0 - Signature: {DEL/2,MIN/2,MSORT/1,del/2,min/2,msort/1,DEL#/2,MIN#/2,MSORT#/1,del#/2,min#/2,msort#/1} / {./2,<=/2,=/2,c/0 ,c1/1,c2/3,c3/0,c4/1,c5/1,c6/0,c7/1,if/3,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/1,c_7/3,c_8/0,c_9/1,c_10/0 ,c_11/2,c_12/0,c_13/4,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {DEL#,MIN#,MSORT#,del#,min#,msort#} and constructors {.,<= ,=,c,c1,c2,c3,c4,c5,c6,c7,if,nil} + 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 MSORT#(.(z0,z1)) -> c_7(MSORT#(del(min(z0,z1),.(z0,z1))),DEL#(min(z0,z1),.(z0,z1)),MIN#(z0,z1)) and a lower component DEL#(z0,.(z1,z2)) -> c_1(DEL#(z0,z2)) MIN#(z0,.(z1,z2)) -> c_3(MIN#(z0,z2)) MIN#(z0,.(z1,z2)) -> c_4(MIN#(z1,z2)) MSORT#(.(z0,z1)) -> c_6(MIN#(z0,z1)) Further, following extension rules are added to the lower component. MSORT#(.(z0,z1)) -> DEL#(min(z0,z1),.(z0,z1)) MSORT#(.(z0,z1)) -> MIN#(z0,z1) MSORT#(.(z0,z1)) -> MSORT#(del(min(z0,z1),.(z0,z1))) *** Step 1.b:6.a:1: SimplifyRHS. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: MSORT#(.(z0,z1)) -> c_7(MSORT#(del(min(z0,z1),.(z0,z1))),DEL#(min(z0,z1),.(z0,z1)),MIN#(z0,z1)) - Weak TRS: del(z0,.(z1,z2)) -> if(=(z0,z1),z2,.(z1,del(z0,z2))) del(z0,nil()) -> nil() min(z0,.(z1,z2)) -> if(<=(z0,z1),min(z0,z2),min(z1,z2)) min(z0,nil()) -> z0 - Signature: {DEL/2,MIN/2,MSORT/1,del/2,min/2,msort/1,DEL#/2,MIN#/2,MSORT#/1,del#/2,min#/2,msort#/1} / {./2,<=/2,=/2,c/0 ,c1/1,c2/3,c3/0,c4/1,c5/1,c6/0,c7/1,if/3,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/1,c_7/3,c_8/0,c_9/1,c_10/0 ,c_11/2,c_12/0,c_13/4,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {DEL#,MIN#,MSORT#,del#,min#,msort#} and constructors {.,<= ,=,c,c1,c2,c3,c4,c5,c6,c7,if,nil} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:MSORT#(.(z0,z1)) -> c_7(MSORT#(del(min(z0,z1),.(z0,z1))),DEL#(min(z0,z1),.(z0,z1)),MIN#(z0,z1)) -->_1 MSORT#(.(z0,z1)) -> c_7(MSORT#(del(min(z0,z1),.(z0,z1))),DEL#(min(z0,z1),.(z0,z1)),MIN#(z0,z1)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: MSORT#(.(z0,z1)) -> c_7(MSORT#(del(min(z0,z1),.(z0,z1)))) *** Step 1.b:6.a:2: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: MSORT#(.(z0,z1)) -> c_7(MSORT#(del(min(z0,z1),.(z0,z1)))) - Weak TRS: del(z0,.(z1,z2)) -> if(=(z0,z1),z2,.(z1,del(z0,z2))) del(z0,nil()) -> nil() min(z0,.(z1,z2)) -> if(<=(z0,z1),min(z0,z2),min(z1,z2)) min(z0,nil()) -> z0 - Signature: {DEL/2,MIN/2,MSORT/1,del/2,min/2,msort/1,DEL#/2,MIN#/2,MSORT#/1,del#/2,min#/2,msort#/1} / {./2,<=/2,=/2,c/0 ,c1/1,c2/3,c3/0,c4/1,c5/1,c6/0,c7/1,if/3,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,c_10/0 ,c_11/2,c_12/0,c_13/4,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {DEL#,MIN#,MSORT#,del#,min#,msort#} and constructors {.,<= ,=,c,c1,c2,c3,c4,c5,c6,c7,if,nil} + 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} Following symbols are considered usable: {del,DEL#,MIN#,MSORT#,del#,min#,msort#} TcT has computed the following interpretation: p(.) = [2] p(<=) = [1] x1 + [1] x2 + [0] p(=) = [0] p(DEL) = [0] p(MIN) = [0] p(MSORT) = [0] p(c) = [0] p(c1) = [1] x1 + [0] p(c2) = [1] x1 + [1] x2 + [1] x3 + [0] p(c3) = [0] p(c4) = [1] x1 + [0] p(c5) = [1] x1 + [0] p(c6) = [0] p(c7) = [1] x1 + [0] p(del) = [0] p(if) = [1] x1 + [0] p(min) = [0] p(msort) = [0] p(nil) = [0] p(DEL#) = [0] p(MIN#) = [0] p(MSORT#) = [8] x1 + [0] p(del#) = [1] x2 + [0] p(min#) = [1] x2 + [0] p(msort#) = [2] x1 + [0] p(c_1) = [2] x1 + [0] p(c_2) = [0] p(c_3) = [2] x1 + [0] p(c_4) = [2] p(c_5) = [1] p(c_6) = [1] x1 + [0] p(c_7) = [8] x1 + [8] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [8] p(c_11) = [1] x1 + [0] p(c_12) = [1] p(c_13) = [1] x2 + [8] x3 + [1] p(c_14) = [0] Following rules are strictly oriented: MSORT#(.(z0,z1)) = [16] > [8] = c_7(MSORT#(del(min(z0,z1),.(z0,z1)))) Following rules are (at-least) weakly oriented: del(z0,.(z1,z2)) = [0] >= [0] = if(=(z0,z1),z2,.(z1,del(z0,z2))) del(z0,nil()) = [0] >= [0] = nil() *** Step 1.b:6.a:3: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: MSORT#(.(z0,z1)) -> c_7(MSORT#(del(min(z0,z1),.(z0,z1)))) - Weak TRS: del(z0,.(z1,z2)) -> if(=(z0,z1),z2,.(z1,del(z0,z2))) del(z0,nil()) -> nil() min(z0,.(z1,z2)) -> if(<=(z0,z1),min(z0,z2),min(z1,z2)) min(z0,nil()) -> z0 - Signature: {DEL/2,MIN/2,MSORT/1,del/2,min/2,msort/1,DEL#/2,MIN#/2,MSORT#/1,del#/2,min#/2,msort#/1} / {./2,<=/2,=/2,c/0 ,c1/1,c2/3,c3/0,c4/1,c5/1,c6/0,c7/1,if/3,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/1,c_7/1,c_8/0,c_9/1,c_10/0 ,c_11/2,c_12/0,c_13/4,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {DEL#,MIN#,MSORT#,del#,min#,msort#} and constructors {.,<= ,=,c,c1,c2,c3,c4,c5,c6,c7,if,nil} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 1.b:6.b:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: DEL#(z0,.(z1,z2)) -> c_1(DEL#(z0,z2)) MIN#(z0,.(z1,z2)) -> c_3(MIN#(z0,z2)) MIN#(z0,.(z1,z2)) -> c_4(MIN#(z1,z2)) MSORT#(.(z0,z1)) -> c_6(MIN#(z0,z1)) - Weak DPs: MSORT#(.(z0,z1)) -> DEL#(min(z0,z1),.(z0,z1)) MSORT#(.(z0,z1)) -> MIN#(z0,z1) MSORT#(.(z0,z1)) -> MSORT#(del(min(z0,z1),.(z0,z1))) - Weak TRS: del(z0,.(z1,z2)) -> if(=(z0,z1),z2,.(z1,del(z0,z2))) del(z0,nil()) -> nil() min(z0,.(z1,z2)) -> if(<=(z0,z1),min(z0,z2),min(z1,z2)) min(z0,nil()) -> z0 - Signature: {DEL/2,MIN/2,MSORT/1,del/2,min/2,msort/1,DEL#/2,MIN#/2,MSORT#/1,del#/2,min#/2,msort#/1} / {./2,<=/2,=/2,c/0 ,c1/1,c2/3,c3/0,c4/1,c5/1,c6/0,c7/1,if/3,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/1,c_7/3,c_8/0,c_9/1,c_10/0 ,c_11/2,c_12/0,c_13/4,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {DEL#,MIN#,MSORT#,del#,min#,msort#} and constructors {.,<= ,=,c,c1,c2,c3,c4,c5,c6,c7,if,nil} + 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_1) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_6) = {1} Following symbols are considered usable: {del,DEL#,MIN#,MSORT#,del#,min#,msort#} TcT has computed the following interpretation: p(.) = [1] x2 + [3] p(<=) = [1] x1 + [1] p(=) = [0] p(DEL) = [0] p(MIN) = [2] x2 + [2] p(MSORT) = [1] x1 + [0] p(c) = [1] p(c1) = [1] p(c2) = [1] x2 + [1] x3 + [2] p(c3) = [0] p(c4) = [1] x1 + [2] p(c5) = [4] p(c6) = [1] p(c7) = [2] p(del) = [1] x2 + [0] p(if) = [3] p(min) = [0] p(msort) = [2] x1 + [1] p(nil) = [1] p(DEL#) = [0] p(MIN#) = [3] x2 + [8] p(MSORT#) = [4] x1 + [14] p(del#) = [1] x1 + [1] x2 + [8] p(min#) = [2] x1 + [0] p(msort#) = [1] p(c_1) = [8] x1 + [0] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [5] p(c_5) = [0] p(c_6) = [1] x1 + [10] p(c_7) = [2] x1 + [2] p(c_8) = [0] p(c_9) = [1] x1 + [1] p(c_10) = [2] p(c_11) = [1] p(c_12) = [2] p(c_13) = [1] x1 + [2] x2 + [1] x4 + [0] p(c_14) = [0] Following rules are strictly oriented: MIN#(z0,.(z1,z2)) = [3] z2 + [17] > [3] z2 + [8] = c_3(MIN#(z0,z2)) MIN#(z0,.(z1,z2)) = [3] z2 + [17] > [3] z2 + [13] = c_4(MIN#(z1,z2)) MSORT#(.(z0,z1)) = [4] z1 + [26] > [3] z1 + [18] = c_6(MIN#(z0,z1)) Following rules are (at-least) weakly oriented: DEL#(z0,.(z1,z2)) = [0] >= [0] = c_1(DEL#(z0,z2)) MSORT#(.(z0,z1)) = [4] z1 + [26] >= [0] = DEL#(min(z0,z1),.(z0,z1)) MSORT#(.(z0,z1)) = [4] z1 + [26] >= [3] z1 + [8] = MIN#(z0,z1) MSORT#(.(z0,z1)) = [4] z1 + [26] >= [4] z1 + [26] = MSORT#(del(min(z0,z1),.(z0,z1))) del(z0,.(z1,z2)) = [1] z2 + [3] >= [3] = if(=(z0,z1),z2,.(z1,del(z0,z2))) del(z0,nil()) = [1] >= [1] = nil() *** Step 1.b:6.b:2: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: DEL#(z0,.(z1,z2)) -> c_1(DEL#(z0,z2)) - Weak DPs: MIN#(z0,.(z1,z2)) -> c_3(MIN#(z0,z2)) MIN#(z0,.(z1,z2)) -> c_4(MIN#(z1,z2)) MSORT#(.(z0,z1)) -> DEL#(min(z0,z1),.(z0,z1)) MSORT#(.(z0,z1)) -> MIN#(z0,z1) MSORT#(.(z0,z1)) -> MSORT#(del(min(z0,z1),.(z0,z1))) MSORT#(.(z0,z1)) -> c_6(MIN#(z0,z1)) - Weak TRS: del(z0,.(z1,z2)) -> if(=(z0,z1),z2,.(z1,del(z0,z2))) del(z0,nil()) -> nil() min(z0,.(z1,z2)) -> if(<=(z0,z1),min(z0,z2),min(z1,z2)) min(z0,nil()) -> z0 - Signature: {DEL/2,MIN/2,MSORT/1,del/2,min/2,msort/1,DEL#/2,MIN#/2,MSORT#/1,del#/2,min#/2,msort#/1} / {./2,<=/2,=/2,c/0 ,c1/1,c2/3,c3/0,c4/1,c5/1,c6/0,c7/1,if/3,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/1,c_7/3,c_8/0,c_9/1,c_10/0 ,c_11/2,c_12/0,c_13/4,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {DEL#,MIN#,MSORT#,del#,min#,msort#} and constructors {.,<= ,=,c,c1,c2,c3,c4,c5,c6,c7,if,nil} + 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_1) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_6) = {1} Following symbols are considered usable: {del,DEL#,MIN#,MSORT#,del#,min#,msort#} TcT has computed the following interpretation: p(.) = [1] x2 + [12] p(<=) = [0] p(=) = [1] x1 + [1] x2 + [2] p(DEL) = [1] x1 + [1] x2 + [2] p(MIN) = [1] x1 + [0] p(MSORT) = [1] p(c) = [1] p(c1) = [4] p(c2) = [1] x3 + [1] p(c3) = [1] p(c4) = [1] x1 + [1] p(c5) = [0] p(c6) = [1] p(c7) = [1] p(del) = [5] p(if) = [0] p(min) = [9] p(msort) = [8] x1 + [1] p(nil) = [5] p(DEL#) = [1] x2 + [4] p(MIN#) = [4] p(MSORT#) = [1] x1 + [8] p(del#) = [2] x1 + [1] p(min#) = [1] x1 + [1] x2 + [1] p(msort#) = [4] x1 + [2] p(c_1) = [1] x1 + [10] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [2] p(c_6) = [4] x1 + [0] p(c_7) = [1] x3 + [1] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [0] p(c_11) = [1] x2 + [0] p(c_12) = [1] p(c_13) = [1] p(c_14) = [0] Following rules are strictly oriented: DEL#(z0,.(z1,z2)) = [1] z2 + [16] > [1] z2 + [14] = c_1(DEL#(z0,z2)) Following rules are (at-least) weakly oriented: MIN#(z0,.(z1,z2)) = [4] >= [4] = c_3(MIN#(z0,z2)) MIN#(z0,.(z1,z2)) = [4] >= [4] = c_4(MIN#(z1,z2)) MSORT#(.(z0,z1)) = [1] z1 + [20] >= [1] z1 + [16] = DEL#(min(z0,z1),.(z0,z1)) MSORT#(.(z0,z1)) = [1] z1 + [20] >= [4] = MIN#(z0,z1) MSORT#(.(z0,z1)) = [1] z1 + [20] >= [13] = MSORT#(del(min(z0,z1),.(z0,z1))) MSORT#(.(z0,z1)) = [1] z1 + [20] >= [16] = c_6(MIN#(z0,z1)) del(z0,.(z1,z2)) = [5] >= [0] = if(=(z0,z1),z2,.(z1,del(z0,z2))) del(z0,nil()) = [5] >= [5] = nil() *** Step 1.b:6.b:3: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: DEL#(z0,.(z1,z2)) -> c_1(DEL#(z0,z2)) MIN#(z0,.(z1,z2)) -> c_3(MIN#(z0,z2)) MIN#(z0,.(z1,z2)) -> c_4(MIN#(z1,z2)) MSORT#(.(z0,z1)) -> DEL#(min(z0,z1),.(z0,z1)) MSORT#(.(z0,z1)) -> MIN#(z0,z1) MSORT#(.(z0,z1)) -> MSORT#(del(min(z0,z1),.(z0,z1))) MSORT#(.(z0,z1)) -> c_6(MIN#(z0,z1)) - Weak TRS: del(z0,.(z1,z2)) -> if(=(z0,z1),z2,.(z1,del(z0,z2))) del(z0,nil()) -> nil() min(z0,.(z1,z2)) -> if(<=(z0,z1),min(z0,z2),min(z1,z2)) min(z0,nil()) -> z0 - Signature: {DEL/2,MIN/2,MSORT/1,del/2,min/2,msort/1,DEL#/2,MIN#/2,MSORT#/1,del#/2,min#/2,msort#/1} / {./2,<=/2,=/2,c/0 ,c1/1,c2/3,c3/0,c4/1,c5/1,c6/0,c7/1,if/3,nil/0,c_1/1,c_2/0,c_3/1,c_4/1,c_5/0,c_6/1,c_7/3,c_8/0,c_9/1,c_10/0 ,c_11/2,c_12/0,c_13/4,c_14/0} - Obligation: innermost runtime complexity wrt. defined symbols {DEL#,MIN#,MSORT#,del#,min#,msort#} and constructors {.,<= ,=,c,c1,c2,c3,c4,c5,c6,c7,if,nil} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^2))