YES(O(1),O(n^2)) We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict Trs: { #less(@x, @y) -> #cklt(#compare(@x, @y)) , merge(@l1, @l2) -> merge#1(@l1, @l2) , merge#1(::(@x, @xs), @l2) -> merge#2(@l2, @x, @xs) , merge#1(nil(), @l2) -> @l2 , merge#2(::(@y, @ys), @x, @xs) -> merge#3(#less(@x, @y), @x, @xs, @y, @ys) , merge#2(nil(), @x, @xs) -> ::(@x, @xs) , merge#3(#false(), @x, @xs, @y, @ys) -> ::(@y, merge(::(@x, @xs), @ys)) , merge#3(#true(), @x, @xs, @y, @ys) -> ::(@x, merge(@xs, ::(@y, @ys))) , mergesort(@l) -> mergesort#1(@l) , mergesort#1(::(@x1, @xs)) -> mergesort#2(@xs, @x1) , mergesort#1(nil()) -> nil() , mergesort#2(::(@x2, @xs'), @x1) -> mergesort#3(msplit(::(@x1, ::(@x2, @xs')))) , mergesort#2(nil(), @x1) -> ::(@x1, nil()) , msplit(@l) -> msplit#1(@l) , mergesort#3(tuple#2(@l1, @l2)) -> merge(mergesort(@l1), mergesort(@l2)) , msplit#1(::(@x1, @xs)) -> msplit#2(@xs, @x1) , msplit#1(nil()) -> tuple#2(nil(), nil()) , msplit#2(::(@x2, @xs'), @x1) -> msplit#3(msplit(@xs'), @x1, @x2) , msplit#2(nil(), @x1) -> tuple#2(::(@x1, nil()), nil()) , msplit#3(tuple#2(@l1, @l2), @x1, @x2) -> tuple#2(::(@x1, @l1), ::(@x2, @l2)) } Weak Trs: { #compare(#0(), #0()) -> #EQ() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #s(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#pos(@x), #0()) -> #GT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , #cklt(#EQ()) -> #false() , #cklt(#GT()) -> #false() , #cklt(#LT()) -> #true() } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We add following dependency tuples: Strict DPs: { #less^#(@x, @y) -> c_1(#cklt^#(#compare(@x, @y)), #compare^#(@x, @y)) , merge^#(@l1, @l2) -> c_2(merge#1^#(@l1, @l2)) , merge#1^#(::(@x, @xs), @l2) -> c_3(merge#2^#(@l2, @x, @xs)) , merge#1^#(nil(), @l2) -> c_4() , merge#2^#(::(@y, @ys), @x, @xs) -> c_5(merge#3^#(#less(@x, @y), @x, @xs, @y, @ys), #less^#(@x, @y)) , merge#2^#(nil(), @x, @xs) -> c_6() , merge#3^#(#false(), @x, @xs, @y, @ys) -> c_7(merge^#(::(@x, @xs), @ys)) , merge#3^#(#true(), @x, @xs, @y, @ys) -> c_8(merge^#(@xs, ::(@y, @ys))) , mergesort^#(@l) -> c_9(mergesort#1^#(@l)) , mergesort#1^#(::(@x1, @xs)) -> c_10(mergesort#2^#(@xs, @x1)) , mergesort#1^#(nil()) -> c_11() , mergesort#2^#(::(@x2, @xs'), @x1) -> c_12(mergesort#3^#(msplit(::(@x1, ::(@x2, @xs')))), msplit^#(::(@x1, ::(@x2, @xs')))) , mergesort#2^#(nil(), @x1) -> c_13() , mergesort#3^#(tuple#2(@l1, @l2)) -> c_15(merge^#(mergesort(@l1), mergesort(@l2)), mergesort^#(@l1), mergesort^#(@l2)) , msplit^#(@l) -> c_14(msplit#1^#(@l)) , msplit#1^#(::(@x1, @xs)) -> c_16(msplit#2^#(@xs, @x1)) , msplit#1^#(nil()) -> c_17() , msplit#2^#(::(@x2, @xs'), @x1) -> c_18(msplit#3^#(msplit(@xs'), @x1, @x2), msplit^#(@xs')) , msplit#2^#(nil(), @x1) -> c_19() , msplit#3^#(tuple#2(@l1, @l2), @x1, @x2) -> c_20() } Weak DPs: { #cklt^#(#EQ()) -> c_33() , #cklt^#(#GT()) -> c_34() , #cklt^#(#LT()) -> c_35() , #compare^#(#0(), #0()) -> c_21() , #compare^#(#0(), #neg(@y)) -> c_22() , #compare^#(#0(), #pos(@y)) -> c_23() , #compare^#(#0(), #s(@y)) -> c_24() , #compare^#(#neg(@x), #0()) -> c_25() , #compare^#(#neg(@x), #neg(@y)) -> c_26(#compare^#(@y, @x)) , #compare^#(#neg(@x), #pos(@y)) -> c_27() , #compare^#(#pos(@x), #0()) -> c_28() , #compare^#(#pos(@x), #neg(@y)) -> c_29() , #compare^#(#pos(@x), #pos(@y)) -> c_30(#compare^#(@x, @y)) , #compare^#(#s(@x), #0()) -> c_31() , #compare^#(#s(@x), #s(@y)) -> c_32(#compare^#(@x, @y)) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { #less^#(@x, @y) -> c_1(#cklt^#(#compare(@x, @y)), #compare^#(@x, @y)) , merge^#(@l1, @l2) -> c_2(merge#1^#(@l1, @l2)) , merge#1^#(::(@x, @xs), @l2) -> c_3(merge#2^#(@l2, @x, @xs)) , merge#1^#(nil(), @l2) -> c_4() , merge#2^#(::(@y, @ys), @x, @xs) -> c_5(merge#3^#(#less(@x, @y), @x, @xs, @y, @ys), #less^#(@x, @y)) , merge#2^#(nil(), @x, @xs) -> c_6() , merge#3^#(#false(), @x, @xs, @y, @ys) -> c_7(merge^#(::(@x, @xs), @ys)) , merge#3^#(#true(), @x, @xs, @y, @ys) -> c_8(merge^#(@xs, ::(@y, @ys))) , mergesort^#(@l) -> c_9(mergesort#1^#(@l)) , mergesort#1^#(::(@x1, @xs)) -> c_10(mergesort#2^#(@xs, @x1)) , mergesort#1^#(nil()) -> c_11() , mergesort#2^#(::(@x2, @xs'), @x1) -> c_12(mergesort#3^#(msplit(::(@x1, ::(@x2, @xs')))), msplit^#(::(@x1, ::(@x2, @xs')))) , mergesort#2^#(nil(), @x1) -> c_13() , mergesort#3^#(tuple#2(@l1, @l2)) -> c_15(merge^#(mergesort(@l1), mergesort(@l2)), mergesort^#(@l1), mergesort^#(@l2)) , msplit^#(@l) -> c_14(msplit#1^#(@l)) , msplit#1^#(::(@x1, @xs)) -> c_16(msplit#2^#(@xs, @x1)) , msplit#1^#(nil()) -> c_17() , msplit#2^#(::(@x2, @xs'), @x1) -> c_18(msplit#3^#(msplit(@xs'), @x1, @x2), msplit^#(@xs')) , msplit#2^#(nil(), @x1) -> c_19() , msplit#3^#(tuple#2(@l1, @l2), @x1, @x2) -> c_20() } Weak DPs: { #cklt^#(#EQ()) -> c_33() , #cklt^#(#GT()) -> c_34() , #cklt^#(#LT()) -> c_35() , #compare^#(#0(), #0()) -> c_21() , #compare^#(#0(), #neg(@y)) -> c_22() , #compare^#(#0(), #pos(@y)) -> c_23() , #compare^#(#0(), #s(@y)) -> c_24() , #compare^#(#neg(@x), #0()) -> c_25() , #compare^#(#neg(@x), #neg(@y)) -> c_26(#compare^#(@y, @x)) , #compare^#(#neg(@x), #pos(@y)) -> c_27() , #compare^#(#pos(@x), #0()) -> c_28() , #compare^#(#pos(@x), #neg(@y)) -> c_29() , #compare^#(#pos(@x), #pos(@y)) -> c_30(#compare^#(@x, @y)) , #compare^#(#s(@x), #0()) -> c_31() , #compare^#(#s(@x), #s(@y)) -> c_32(#compare^#(@x, @y)) } Weak Trs: { #less(@x, @y) -> #cklt(#compare(@x, @y)) , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #s(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#pos(@x), #0()) -> #GT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , #cklt(#EQ()) -> #false() , #cklt(#GT()) -> #false() , #cklt(#LT()) -> #true() , merge(@l1, @l2) -> merge#1(@l1, @l2) , merge#1(::(@x, @xs), @l2) -> merge#2(@l2, @x, @xs) , merge#1(nil(), @l2) -> @l2 , merge#2(::(@y, @ys), @x, @xs) -> merge#3(#less(@x, @y), @x, @xs, @y, @ys) , merge#2(nil(), @x, @xs) -> ::(@x, @xs) , merge#3(#false(), @x, @xs, @y, @ys) -> ::(@y, merge(::(@x, @xs), @ys)) , merge#3(#true(), @x, @xs, @y, @ys) -> ::(@x, merge(@xs, ::(@y, @ys))) , mergesort(@l) -> mergesort#1(@l) , mergesort#1(::(@x1, @xs)) -> mergesort#2(@xs, @x1) , mergesort#1(nil()) -> nil() , mergesort#2(::(@x2, @xs'), @x1) -> mergesort#3(msplit(::(@x1, ::(@x2, @xs')))) , mergesort#2(nil(), @x1) -> ::(@x1, nil()) , msplit(@l) -> msplit#1(@l) , mergesort#3(tuple#2(@l1, @l2)) -> merge(mergesort(@l1), mergesort(@l2)) , msplit#1(::(@x1, @xs)) -> msplit#2(@xs, @x1) , msplit#1(nil()) -> tuple#2(nil(), nil()) , msplit#2(::(@x2, @xs'), @x1) -> msplit#3(msplit(@xs'), @x1, @x2) , msplit#2(nil(), @x1) -> tuple#2(::(@x1, nil()), nil()) , msplit#3(tuple#2(@l1, @l2), @x1, @x2) -> tuple#2(::(@x1, @l1), ::(@x2, @l2)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We estimate the number of application of {1,4,6,11,13,17,19,20} by applications of Pre({1,4,6,11,13,17,19,20}) = {2,3,5,9,10,15,16,18}. Here rules are labeled as follows: DPs: { 1: #less^#(@x, @y) -> c_1(#cklt^#(#compare(@x, @y)), #compare^#(@x, @y)) , 2: merge^#(@l1, @l2) -> c_2(merge#1^#(@l1, @l2)) , 3: merge#1^#(::(@x, @xs), @l2) -> c_3(merge#2^#(@l2, @x, @xs)) , 4: merge#1^#(nil(), @l2) -> c_4() , 5: merge#2^#(::(@y, @ys), @x, @xs) -> c_5(merge#3^#(#less(@x, @y), @x, @xs, @y, @ys), #less^#(@x, @y)) , 6: merge#2^#(nil(), @x, @xs) -> c_6() , 7: merge#3^#(#false(), @x, @xs, @y, @ys) -> c_7(merge^#(::(@x, @xs), @ys)) , 8: merge#3^#(#true(), @x, @xs, @y, @ys) -> c_8(merge^#(@xs, ::(@y, @ys))) , 9: mergesort^#(@l) -> c_9(mergesort#1^#(@l)) , 10: mergesort#1^#(::(@x1, @xs)) -> c_10(mergesort#2^#(@xs, @x1)) , 11: mergesort#1^#(nil()) -> c_11() , 12: mergesort#2^#(::(@x2, @xs'), @x1) -> c_12(mergesort#3^#(msplit(::(@x1, ::(@x2, @xs')))), msplit^#(::(@x1, ::(@x2, @xs')))) , 13: mergesort#2^#(nil(), @x1) -> c_13() , 14: mergesort#3^#(tuple#2(@l1, @l2)) -> c_15(merge^#(mergesort(@l1), mergesort(@l2)), mergesort^#(@l1), mergesort^#(@l2)) , 15: msplit^#(@l) -> c_14(msplit#1^#(@l)) , 16: msplit#1^#(::(@x1, @xs)) -> c_16(msplit#2^#(@xs, @x1)) , 17: msplit#1^#(nil()) -> c_17() , 18: msplit#2^#(::(@x2, @xs'), @x1) -> c_18(msplit#3^#(msplit(@xs'), @x1, @x2), msplit^#(@xs')) , 19: msplit#2^#(nil(), @x1) -> c_19() , 20: msplit#3^#(tuple#2(@l1, @l2), @x1, @x2) -> c_20() , 21: #cklt^#(#EQ()) -> c_33() , 22: #cklt^#(#GT()) -> c_34() , 23: #cklt^#(#LT()) -> c_35() , 24: #compare^#(#0(), #0()) -> c_21() , 25: #compare^#(#0(), #neg(@y)) -> c_22() , 26: #compare^#(#0(), #pos(@y)) -> c_23() , 27: #compare^#(#0(), #s(@y)) -> c_24() , 28: #compare^#(#neg(@x), #0()) -> c_25() , 29: #compare^#(#neg(@x), #neg(@y)) -> c_26(#compare^#(@y, @x)) , 30: #compare^#(#neg(@x), #pos(@y)) -> c_27() , 31: #compare^#(#pos(@x), #0()) -> c_28() , 32: #compare^#(#pos(@x), #neg(@y)) -> c_29() , 33: #compare^#(#pos(@x), #pos(@y)) -> c_30(#compare^#(@x, @y)) , 34: #compare^#(#s(@x), #0()) -> c_31() , 35: #compare^#(#s(@x), #s(@y)) -> c_32(#compare^#(@x, @y)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { merge^#(@l1, @l2) -> c_2(merge#1^#(@l1, @l2)) , merge#1^#(::(@x, @xs), @l2) -> c_3(merge#2^#(@l2, @x, @xs)) , merge#2^#(::(@y, @ys), @x, @xs) -> c_5(merge#3^#(#less(@x, @y), @x, @xs, @y, @ys), #less^#(@x, @y)) , merge#3^#(#false(), @x, @xs, @y, @ys) -> c_7(merge^#(::(@x, @xs), @ys)) , merge#3^#(#true(), @x, @xs, @y, @ys) -> c_8(merge^#(@xs, ::(@y, @ys))) , mergesort^#(@l) -> c_9(mergesort#1^#(@l)) , mergesort#1^#(::(@x1, @xs)) -> c_10(mergesort#2^#(@xs, @x1)) , mergesort#2^#(::(@x2, @xs'), @x1) -> c_12(mergesort#3^#(msplit(::(@x1, ::(@x2, @xs')))), msplit^#(::(@x1, ::(@x2, @xs')))) , mergesort#3^#(tuple#2(@l1, @l2)) -> c_15(merge^#(mergesort(@l1), mergesort(@l2)), mergesort^#(@l1), mergesort^#(@l2)) , msplit^#(@l) -> c_14(msplit#1^#(@l)) , msplit#1^#(::(@x1, @xs)) -> c_16(msplit#2^#(@xs, @x1)) , msplit#2^#(::(@x2, @xs'), @x1) -> c_18(msplit#3^#(msplit(@xs'), @x1, @x2), msplit^#(@xs')) } Weak DPs: { #less^#(@x, @y) -> c_1(#cklt^#(#compare(@x, @y)), #compare^#(@x, @y)) , #cklt^#(#EQ()) -> c_33() , #cklt^#(#GT()) -> c_34() , #cklt^#(#LT()) -> c_35() , #compare^#(#0(), #0()) -> c_21() , #compare^#(#0(), #neg(@y)) -> c_22() , #compare^#(#0(), #pos(@y)) -> c_23() , #compare^#(#0(), #s(@y)) -> c_24() , #compare^#(#neg(@x), #0()) -> c_25() , #compare^#(#neg(@x), #neg(@y)) -> c_26(#compare^#(@y, @x)) , #compare^#(#neg(@x), #pos(@y)) -> c_27() , #compare^#(#pos(@x), #0()) -> c_28() , #compare^#(#pos(@x), #neg(@y)) -> c_29() , #compare^#(#pos(@x), #pos(@y)) -> c_30(#compare^#(@x, @y)) , #compare^#(#s(@x), #0()) -> c_31() , #compare^#(#s(@x), #s(@y)) -> c_32(#compare^#(@x, @y)) , merge#1^#(nil(), @l2) -> c_4() , merge#2^#(nil(), @x, @xs) -> c_6() , mergesort#1^#(nil()) -> c_11() , mergesort#2^#(nil(), @x1) -> c_13() , msplit#1^#(nil()) -> c_17() , msplit#2^#(nil(), @x1) -> c_19() , msplit#3^#(tuple#2(@l1, @l2), @x1, @x2) -> c_20() } Weak Trs: { #less(@x, @y) -> #cklt(#compare(@x, @y)) , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #s(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#pos(@x), #0()) -> #GT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , #cklt(#EQ()) -> #false() , #cklt(#GT()) -> #false() , #cklt(#LT()) -> #true() , merge(@l1, @l2) -> merge#1(@l1, @l2) , merge#1(::(@x, @xs), @l2) -> merge#2(@l2, @x, @xs) , merge#1(nil(), @l2) -> @l2 , merge#2(::(@y, @ys), @x, @xs) -> merge#3(#less(@x, @y), @x, @xs, @y, @ys) , merge#2(nil(), @x, @xs) -> ::(@x, @xs) , merge#3(#false(), @x, @xs, @y, @ys) -> ::(@y, merge(::(@x, @xs), @ys)) , merge#3(#true(), @x, @xs, @y, @ys) -> ::(@x, merge(@xs, ::(@y, @ys))) , mergesort(@l) -> mergesort#1(@l) , mergesort#1(::(@x1, @xs)) -> mergesort#2(@xs, @x1) , mergesort#1(nil()) -> nil() , mergesort#2(::(@x2, @xs'), @x1) -> mergesort#3(msplit(::(@x1, ::(@x2, @xs')))) , mergesort#2(nil(), @x1) -> ::(@x1, nil()) , msplit(@l) -> msplit#1(@l) , mergesort#3(tuple#2(@l1, @l2)) -> merge(mergesort(@l1), mergesort(@l2)) , msplit#1(::(@x1, @xs)) -> msplit#2(@xs, @x1) , msplit#1(nil()) -> tuple#2(nil(), nil()) , msplit#2(::(@x2, @xs'), @x1) -> msplit#3(msplit(@xs'), @x1, @x2) , msplit#2(nil(), @x1) -> tuple#2(::(@x1, nil()), nil()) , msplit#3(tuple#2(@l1, @l2), @x1, @x2) -> tuple#2(::(@x1, @l1), ::(@x2, @l2)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { #less^#(@x, @y) -> c_1(#cklt^#(#compare(@x, @y)), #compare^#(@x, @y)) , #cklt^#(#EQ()) -> c_33() , #cklt^#(#GT()) -> c_34() , #cklt^#(#LT()) -> c_35() , #compare^#(#0(), #0()) -> c_21() , #compare^#(#0(), #neg(@y)) -> c_22() , #compare^#(#0(), #pos(@y)) -> c_23() , #compare^#(#0(), #s(@y)) -> c_24() , #compare^#(#neg(@x), #0()) -> c_25() , #compare^#(#neg(@x), #neg(@y)) -> c_26(#compare^#(@y, @x)) , #compare^#(#neg(@x), #pos(@y)) -> c_27() , #compare^#(#pos(@x), #0()) -> c_28() , #compare^#(#pos(@x), #neg(@y)) -> c_29() , #compare^#(#pos(@x), #pos(@y)) -> c_30(#compare^#(@x, @y)) , #compare^#(#s(@x), #0()) -> c_31() , #compare^#(#s(@x), #s(@y)) -> c_32(#compare^#(@x, @y)) , merge#1^#(nil(), @l2) -> c_4() , merge#2^#(nil(), @x, @xs) -> c_6() , mergesort#1^#(nil()) -> c_11() , mergesort#2^#(nil(), @x1) -> c_13() , msplit#1^#(nil()) -> c_17() , msplit#2^#(nil(), @x1) -> c_19() , msplit#3^#(tuple#2(@l1, @l2), @x1, @x2) -> c_20() } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { merge^#(@l1, @l2) -> c_2(merge#1^#(@l1, @l2)) , merge#1^#(::(@x, @xs), @l2) -> c_3(merge#2^#(@l2, @x, @xs)) , merge#2^#(::(@y, @ys), @x, @xs) -> c_5(merge#3^#(#less(@x, @y), @x, @xs, @y, @ys), #less^#(@x, @y)) , merge#3^#(#false(), @x, @xs, @y, @ys) -> c_7(merge^#(::(@x, @xs), @ys)) , merge#3^#(#true(), @x, @xs, @y, @ys) -> c_8(merge^#(@xs, ::(@y, @ys))) , mergesort^#(@l) -> c_9(mergesort#1^#(@l)) , mergesort#1^#(::(@x1, @xs)) -> c_10(mergesort#2^#(@xs, @x1)) , mergesort#2^#(::(@x2, @xs'), @x1) -> c_12(mergesort#3^#(msplit(::(@x1, ::(@x2, @xs')))), msplit^#(::(@x1, ::(@x2, @xs')))) , mergesort#3^#(tuple#2(@l1, @l2)) -> c_15(merge^#(mergesort(@l1), mergesort(@l2)), mergesort^#(@l1), mergesort^#(@l2)) , msplit^#(@l) -> c_14(msplit#1^#(@l)) , msplit#1^#(::(@x1, @xs)) -> c_16(msplit#2^#(@xs, @x1)) , msplit#2^#(::(@x2, @xs'), @x1) -> c_18(msplit#3^#(msplit(@xs'), @x1, @x2), msplit^#(@xs')) } Weak Trs: { #less(@x, @y) -> #cklt(#compare(@x, @y)) , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #s(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#pos(@x), #0()) -> #GT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , #cklt(#EQ()) -> #false() , #cklt(#GT()) -> #false() , #cklt(#LT()) -> #true() , merge(@l1, @l2) -> merge#1(@l1, @l2) , merge#1(::(@x, @xs), @l2) -> merge#2(@l2, @x, @xs) , merge#1(nil(), @l2) -> @l2 , merge#2(::(@y, @ys), @x, @xs) -> merge#3(#less(@x, @y), @x, @xs, @y, @ys) , merge#2(nil(), @x, @xs) -> ::(@x, @xs) , merge#3(#false(), @x, @xs, @y, @ys) -> ::(@y, merge(::(@x, @xs), @ys)) , merge#3(#true(), @x, @xs, @y, @ys) -> ::(@x, merge(@xs, ::(@y, @ys))) , mergesort(@l) -> mergesort#1(@l) , mergesort#1(::(@x1, @xs)) -> mergesort#2(@xs, @x1) , mergesort#1(nil()) -> nil() , mergesort#2(::(@x2, @xs'), @x1) -> mergesort#3(msplit(::(@x1, ::(@x2, @xs')))) , mergesort#2(nil(), @x1) -> ::(@x1, nil()) , msplit(@l) -> msplit#1(@l) , mergesort#3(tuple#2(@l1, @l2)) -> merge(mergesort(@l1), mergesort(@l2)) , msplit#1(::(@x1, @xs)) -> msplit#2(@xs, @x1) , msplit#1(nil()) -> tuple#2(nil(), nil()) , msplit#2(::(@x2, @xs'), @x1) -> msplit#3(msplit(@xs'), @x1, @x2) , msplit#2(nil(), @x1) -> tuple#2(::(@x1, nil()), nil()) , msplit#3(tuple#2(@l1, @l2), @x1, @x2) -> tuple#2(::(@x1, @l1), ::(@x2, @l2)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) Due to missing edges in the dependency-graph, the right-hand sides of following rules could be simplified: { merge#2^#(::(@y, @ys), @x, @xs) -> c_5(merge#3^#(#less(@x, @y), @x, @xs, @y, @ys), #less^#(@x, @y)) , msplit#2^#(::(@x2, @xs'), @x1) -> c_18(msplit#3^#(msplit(@xs'), @x1, @x2), msplit^#(@xs')) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { merge^#(@l1, @l2) -> c_1(merge#1^#(@l1, @l2)) , merge#1^#(::(@x, @xs), @l2) -> c_2(merge#2^#(@l2, @x, @xs)) , merge#2^#(::(@y, @ys), @x, @xs) -> c_3(merge#3^#(#less(@x, @y), @x, @xs, @y, @ys)) , merge#3^#(#false(), @x, @xs, @y, @ys) -> c_4(merge^#(::(@x, @xs), @ys)) , merge#3^#(#true(), @x, @xs, @y, @ys) -> c_5(merge^#(@xs, ::(@y, @ys))) , mergesort^#(@l) -> c_6(mergesort#1^#(@l)) , mergesort#1^#(::(@x1, @xs)) -> c_7(mergesort#2^#(@xs, @x1)) , mergesort#2^#(::(@x2, @xs'), @x1) -> c_8(mergesort#3^#(msplit(::(@x1, ::(@x2, @xs')))), msplit^#(::(@x1, ::(@x2, @xs')))) , mergesort#3^#(tuple#2(@l1, @l2)) -> c_9(merge^#(mergesort(@l1), mergesort(@l2)), mergesort^#(@l1), mergesort^#(@l2)) , msplit^#(@l) -> c_10(msplit#1^#(@l)) , msplit#1^#(::(@x1, @xs)) -> c_11(msplit#2^#(@xs, @x1)) , msplit#2^#(::(@x2, @xs'), @x1) -> c_12(msplit^#(@xs')) } Weak Trs: { #less(@x, @y) -> #cklt(#compare(@x, @y)) , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #s(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#pos(@x), #0()) -> #GT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , #cklt(#EQ()) -> #false() , #cklt(#GT()) -> #false() , #cklt(#LT()) -> #true() , merge(@l1, @l2) -> merge#1(@l1, @l2) , merge#1(::(@x, @xs), @l2) -> merge#2(@l2, @x, @xs) , merge#1(nil(), @l2) -> @l2 , merge#2(::(@y, @ys), @x, @xs) -> merge#3(#less(@x, @y), @x, @xs, @y, @ys) , merge#2(nil(), @x, @xs) -> ::(@x, @xs) , merge#3(#false(), @x, @xs, @y, @ys) -> ::(@y, merge(::(@x, @xs), @ys)) , merge#3(#true(), @x, @xs, @y, @ys) -> ::(@x, merge(@xs, ::(@y, @ys))) , mergesort(@l) -> mergesort#1(@l) , mergesort#1(::(@x1, @xs)) -> mergesort#2(@xs, @x1) , mergesort#1(nil()) -> nil() , mergesort#2(::(@x2, @xs'), @x1) -> mergesort#3(msplit(::(@x1, ::(@x2, @xs')))) , mergesort#2(nil(), @x1) -> ::(@x1, nil()) , msplit(@l) -> msplit#1(@l) , mergesort#3(tuple#2(@l1, @l2)) -> merge(mergesort(@l1), mergesort(@l2)) , msplit#1(::(@x1, @xs)) -> msplit#2(@xs, @x1) , msplit#1(nil()) -> tuple#2(nil(), nil()) , msplit#2(::(@x2, @xs'), @x1) -> msplit#3(msplit(@xs'), @x1, @x2) , msplit#2(nil(), @x1) -> tuple#2(::(@x1, nil()), nil()) , msplit#3(tuple#2(@l1, @l2), @x1, @x2) -> tuple#2(::(@x1, @l1), ::(@x2, @l2)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We decompose the input problem according to the dependency graph into the upper component { mergesort^#(@l) -> c_6(mergesort#1^#(@l)) , mergesort#1^#(::(@x1, @xs)) -> c_7(mergesort#2^#(@xs, @x1)) , mergesort#2^#(::(@x2, @xs'), @x1) -> c_8(mergesort#3^#(msplit(::(@x1, ::(@x2, @xs')))), msplit^#(::(@x1, ::(@x2, @xs')))) , mergesort#3^#(tuple#2(@l1, @l2)) -> c_9(merge^#(mergesort(@l1), mergesort(@l2)), mergesort^#(@l1), mergesort^#(@l2)) } and lower component { merge^#(@l1, @l2) -> c_1(merge#1^#(@l1, @l2)) , merge#1^#(::(@x, @xs), @l2) -> c_2(merge#2^#(@l2, @x, @xs)) , merge#2^#(::(@y, @ys), @x, @xs) -> c_3(merge#3^#(#less(@x, @y), @x, @xs, @y, @ys)) , merge#3^#(#false(), @x, @xs, @y, @ys) -> c_4(merge^#(::(@x, @xs), @ys)) , merge#3^#(#true(), @x, @xs, @y, @ys) -> c_5(merge^#(@xs, ::(@y, @ys))) , msplit^#(@l) -> c_10(msplit#1^#(@l)) , msplit#1^#(::(@x1, @xs)) -> c_11(msplit#2^#(@xs, @x1)) , msplit#2^#(::(@x2, @xs'), @x1) -> c_12(msplit^#(@xs')) } Further, following extension rules are added to the lower component. { mergesort^#(@l) -> mergesort#1^#(@l) , mergesort#1^#(::(@x1, @xs)) -> mergesort#2^#(@xs, @x1) , mergesort#2^#(::(@x2, @xs'), @x1) -> mergesort#3^#(msplit(::(@x1, ::(@x2, @xs')))) , mergesort#2^#(::(@x2, @xs'), @x1) -> msplit^#(::(@x1, ::(@x2, @xs'))) , mergesort#3^#(tuple#2(@l1, @l2)) -> merge^#(mergesort(@l1), mergesort(@l2)) , mergesort#3^#(tuple#2(@l1, @l2)) -> mergesort^#(@l1) , mergesort#3^#(tuple#2(@l1, @l2)) -> mergesort^#(@l2) } TcT solves the upper component with certificate YES(O(1),O(n^1)). Sub-proof: ---------- We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { mergesort^#(@l) -> c_6(mergesort#1^#(@l)) , mergesort#1^#(::(@x1, @xs)) -> c_7(mergesort#2^#(@xs, @x1)) , mergesort#2^#(::(@x2, @xs'), @x1) -> c_8(mergesort#3^#(msplit(::(@x1, ::(@x2, @xs')))), msplit^#(::(@x1, ::(@x2, @xs')))) , mergesort#3^#(tuple#2(@l1, @l2)) -> c_9(merge^#(mergesort(@l1), mergesort(@l2)), mergesort^#(@l1), mergesort^#(@l2)) } Weak Trs: { #less(@x, @y) -> #cklt(#compare(@x, @y)) , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #s(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#pos(@x), #0()) -> #GT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , #cklt(#EQ()) -> #false() , #cklt(#GT()) -> #false() , #cklt(#LT()) -> #true() , merge(@l1, @l2) -> merge#1(@l1, @l2) , merge#1(::(@x, @xs), @l2) -> merge#2(@l2, @x, @xs) , merge#1(nil(), @l2) -> @l2 , merge#2(::(@y, @ys), @x, @xs) -> merge#3(#less(@x, @y), @x, @xs, @y, @ys) , merge#2(nil(), @x, @xs) -> ::(@x, @xs) , merge#3(#false(), @x, @xs, @y, @ys) -> ::(@y, merge(::(@x, @xs), @ys)) , merge#3(#true(), @x, @xs, @y, @ys) -> ::(@x, merge(@xs, ::(@y, @ys))) , mergesort(@l) -> mergesort#1(@l) , mergesort#1(::(@x1, @xs)) -> mergesort#2(@xs, @x1) , mergesort#1(nil()) -> nil() , mergesort#2(::(@x2, @xs'), @x1) -> mergesort#3(msplit(::(@x1, ::(@x2, @xs')))) , mergesort#2(nil(), @x1) -> ::(@x1, nil()) , msplit(@l) -> msplit#1(@l) , mergesort#3(tuple#2(@l1, @l2)) -> merge(mergesort(@l1), mergesort(@l2)) , msplit#1(::(@x1, @xs)) -> msplit#2(@xs, @x1) , msplit#1(nil()) -> tuple#2(nil(), nil()) , msplit#2(::(@x2, @xs'), @x1) -> msplit#3(msplit(@xs'), @x1, @x2) , msplit#2(nil(), @x1) -> tuple#2(::(@x1, nil()), nil()) , msplit#3(tuple#2(@l1, @l2), @x1, @x2) -> tuple#2(::(@x1, @l1), ::(@x2, @l2)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) Due to missing edges in the dependency-graph, the right-hand sides of following rules could be simplified: { mergesort#2^#(::(@x2, @xs'), @x1) -> c_8(mergesort#3^#(msplit(::(@x1, ::(@x2, @xs')))), msplit^#(::(@x1, ::(@x2, @xs')))) , mergesort#3^#(tuple#2(@l1, @l2)) -> c_9(merge^#(mergesort(@l1), mergesort(@l2)), mergesort^#(@l1), mergesort^#(@l2)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { mergesort^#(@l) -> c_1(mergesort#1^#(@l)) , mergesort#1^#(::(@x1, @xs)) -> c_2(mergesort#2^#(@xs, @x1)) , mergesort#2^#(::(@x2, @xs'), @x1) -> c_3(mergesort#3^#(msplit(::(@x1, ::(@x2, @xs'))))) , mergesort#3^#(tuple#2(@l1, @l2)) -> c_4(mergesort^#(@l1), mergesort^#(@l2)) } Weak Trs: { #less(@x, @y) -> #cklt(#compare(@x, @y)) , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #s(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#pos(@x), #0()) -> #GT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , #cklt(#EQ()) -> #false() , #cklt(#GT()) -> #false() , #cklt(#LT()) -> #true() , merge(@l1, @l2) -> merge#1(@l1, @l2) , merge#1(::(@x, @xs), @l2) -> merge#2(@l2, @x, @xs) , merge#1(nil(), @l2) -> @l2 , merge#2(::(@y, @ys), @x, @xs) -> merge#3(#less(@x, @y), @x, @xs, @y, @ys) , merge#2(nil(), @x, @xs) -> ::(@x, @xs) , merge#3(#false(), @x, @xs, @y, @ys) -> ::(@y, merge(::(@x, @xs), @ys)) , merge#3(#true(), @x, @xs, @y, @ys) -> ::(@x, merge(@xs, ::(@y, @ys))) , mergesort(@l) -> mergesort#1(@l) , mergesort#1(::(@x1, @xs)) -> mergesort#2(@xs, @x1) , mergesort#1(nil()) -> nil() , mergesort#2(::(@x2, @xs'), @x1) -> mergesort#3(msplit(::(@x1, ::(@x2, @xs')))) , mergesort#2(nil(), @x1) -> ::(@x1, nil()) , msplit(@l) -> msplit#1(@l) , mergesort#3(tuple#2(@l1, @l2)) -> merge(mergesort(@l1), mergesort(@l2)) , msplit#1(::(@x1, @xs)) -> msplit#2(@xs, @x1) , msplit#1(nil()) -> tuple#2(nil(), nil()) , msplit#2(::(@x2, @xs'), @x1) -> msplit#3(msplit(@xs'), @x1, @x2) , msplit#2(nil(), @x1) -> tuple#2(::(@x1, nil()), nil()) , msplit#3(tuple#2(@l1, @l2), @x1, @x2) -> tuple#2(::(@x1, @l1), ::(@x2, @l2)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We replace rewrite rules by usable rules: Weak Usable Rules: { msplit(@l) -> msplit#1(@l) , msplit#1(::(@x1, @xs)) -> msplit#2(@xs, @x1) , msplit#1(nil()) -> tuple#2(nil(), nil()) , msplit#2(::(@x2, @xs'), @x1) -> msplit#3(msplit(@xs'), @x1, @x2) , msplit#2(nil(), @x1) -> tuple#2(::(@x1, nil()), nil()) , msplit#3(tuple#2(@l1, @l2), @x1, @x2) -> tuple#2(::(@x1, @l1), ::(@x2, @l2)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { mergesort^#(@l) -> c_1(mergesort#1^#(@l)) , mergesort#1^#(::(@x1, @xs)) -> c_2(mergesort#2^#(@xs, @x1)) , mergesort#2^#(::(@x2, @xs'), @x1) -> c_3(mergesort#3^#(msplit(::(@x1, ::(@x2, @xs'))))) , mergesort#3^#(tuple#2(@l1, @l2)) -> c_4(mergesort^#(@l1), mergesort^#(@l2)) } Weak Trs: { msplit(@l) -> msplit#1(@l) , msplit#1(::(@x1, @xs)) -> msplit#2(@xs, @x1) , msplit#1(nil()) -> tuple#2(nil(), nil()) , msplit#2(::(@x2, @xs'), @x1) -> msplit#3(msplit(@xs'), @x1, @x2) , msplit#2(nil(), @x1) -> tuple#2(::(@x1, nil()), nil()) , msplit#3(tuple#2(@l1, @l2), @x1, @x2) -> tuple#2(::(@x1, @l1), ::(@x2, @l2)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We use the processor 'matrix interpretation of dimension 3' to orient following rules strictly. DPs: { 4: mergesort#3^#(tuple#2(@l1, @l2)) -> c_4(mergesort^#(@l1), mergesort^#(@l2)) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_3) = {1}, Uargs(c_4) = {1, 2} TcT has computed following constructor-based matrix interpretation satisfying not(EDA) and not(IDA(1)). [0] [#less](x1, x2) = [0] [0] [0] [#compare](x1, x2) = [0] [0] [0] [#cklt](x1) = [0] [0] [0] [merge](x1, x2) = [0] [0] [0] [merge#1](x1, x2) = [0] [0] [0 0 0] [0 0 1] [0] [::](x1, x2) = [0 0 0] x1 + [1 0 0] x2 + [1] [0 1 0] [0 0 1] [1] [0] [merge#2](x1, x2, x3) = [0] [0] [0] [nil] = [1] [0] [0] [merge#3](x1, x2, x3, x4, x5) = [0] [0] [0] [#false] = [0] [0] [0] [#true] = [0] [0] [0] [mergesort](x1) = [0] [0] [0] [mergesort#1](x1) = [0] [0] [0] [mergesort#2](x1, x2) = [0] [0] [0 0 1] [0] [msplit](x1) = [0 0 0] x1 + [1] [0 1 0] [0] [0] [mergesort#3](x1) = [0] [0] [0 0 1] [0 0 1] [0] [tuple#2](x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [1] [1 0 0] [1 0 0] [1] [0 0 1] [0] [msplit#1](x1) = [0 0 0] x1 + [1] [0 1 0] [0] [0 0 1] [0 1 0] [1] [msplit#2](x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [1] [1 0 0] [0 0 0] [1] [1 1 0] [0 1 0] [0 1 0] [1] [msplit#3](x1, x2, x3) = [0 1 0] x1 + [0 0 0] x2 + [0 0 0] x3 + [0] [1 1 0] [0 0 0] [0 0 0] [0] [0] [#EQ] = [0] [0] [0] [#GT] = [0] [0] [0] [#LT] = [0] [0] [0] [#0] = [0] [0] [0 0 1] [0] [#neg](x1) = [0 1 0] x1 + [0] [1 0 0] [0] [0 0 1] [0] [#pos](x1) = [0 0 1] x1 + [0] [1 0 0] [0] [0 0 1] [0] [#s](x1) = [0 0 0] x1 + [0] [0 1 0] [0] [0] [#less^#](x1, x2) = [0] [0] [0] [#cklt^#](x1) = [0] [0] [0] [#compare^#](x1, x2) = [0] [0] [0] [merge^#](x1, x2) = [0] [0] [0] [merge#1^#](x1, x2) = [0] [0] [0] [merge#2^#](x1, x2, x3) = [0] [0] [0] [merge#3^#](x1, x2, x3, x4, x5) = [0] [0] [1 0 0] [0] [mergesort^#](x1) = [0 0 0] x1 + [1] [0 0 0] [1] [1 0 0] [0] [mergesort#1^#](x1) = [0 0 0] x1 + [0] [0 0 0] [0] [0 0 1] [0] [mergesort#2^#](x1, x2) = [0 0 0] x1 + [0] [0 0 0] [1] [0 0 1] [0] [mergesort#3^#](x1) = [0 1 0] x1 + [0] [0 0 0] [0] [0] [msplit^#](x1) = [0] [0] [0] [msplit#1^#](x1) = [0] [0] [0] [msplit#2^#](x1, x2) = [0] [0] [0] [msplit#3^#](x1, x2, x3) = [0] [0] [0] [c_6](x1) = [0] [0] [0] [c_7](x1) = [0] [0] [0] [c_8](x1, x2) = [0] [0] [0] [c_9](x1, x2, x3) = [0] [0] [0] [c] = [0] [0] [1 1 1] [0] [c_1](x1) = [0 0 0] x1 + [0] [0 0 0] [0] [1 0 0] [0] [c_2](x1) = [0 0 0] x1 + [0] [0 0 0] [0] [1 0 1] [0] [c_3](x1) = [0 0 0] x1 + [0] [0 0 0] [0] [1 0 0] [1 0 0] [0] [c_4](x1, x2) = [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 0] [0 0 0] [0] This order satisfies following ordering constraints [msplit(@l)] = [0 0 1] [0] [0 0 0] @l + [1] [0 1 0] [0] >= [0 0 1] [0] [0 0 0] @l + [1] [0 1 0] [0] = [msplit#1(@l)] [msplit#1(::(@x1, @xs))] = [0 1 0] [0 0 1] [1] [0 0 0] @x1 + [0 0 0] @xs + [1] [0 0 0] [1 0 0] [1] >= [0 1 0] [0 0 1] [1] [0 0 0] @x1 + [0 0 0] @xs + [1] [0 0 0] [1 0 0] [1] = [msplit#2(@xs, @x1)] [msplit#1(nil())] = [0] [1] [1] >= [0] [1] [1] = [tuple#2(nil(), nil())] [msplit#2(::(@x2, @xs'), @x1)] = [0 1 0] [0 1 0] [0 0 1] [2] [0 0 0] @x1 + [0 0 0] @x2 + [0 0 0] @xs' + [1] [0 0 0] [0 0 0] [0 0 1] [1] >= [0 1 0] [0 1 0] [0 0 1] [2] [0 0 0] @x1 + [0 0 0] @x2 + [0 0 0] @xs' + [1] [0 0 0] [0 0 0] [0 0 1] [1] = [msplit#3(msplit(@xs'), @x1, @x2)] [msplit#2(nil(), @x1)] = [0 1 0] [1] [0 0 0] @x1 + [1] [0 0 0] [1] >= [0 1 0] [1] [0 0 0] @x1 + [1] [0 0 0] [1] = [tuple#2(::(@x1, nil()), nil())] [msplit#3(tuple#2(@l1, @l2), @x1, @x2)] = [0 0 1] [0 0 1] [0 1 0] [0 1 0] [2] [0 0 0] @l1 + [0 0 0] @l2 + [0 0 0] @x1 + [0 0 0] @x2 + [1] [0 0 1] [0 0 1] [0 0 0] [0 0 0] [1] >= [0 0 1] [0 0 1] [0 1 0] [0 1 0] [2] [0 0 0] @l1 + [0 0 0] @l2 + [0 0 0] @x1 + [0 0 0] @x2 + [1] [0 0 1] [0 0 1] [0 0 0] [0 0 0] [1] = [tuple#2(::(@x1, @l1), ::(@x2, @l2))] [mergesort^#(@l)] = [1 0 0] [0] [0 0 0] @l + [1] [0 0 0] [1] >= [1 0 0] [0] [0 0 0] @l + [0] [0 0 0] [0] = [c_1(mergesort#1^#(@l))] [mergesort#1^#(::(@x1, @xs))] = [0 0 1] [0] [0 0 0] @xs + [0] [0 0 0] [0] >= [0 0 1] [0] [0 0 0] @xs + [0] [0 0 0] [0] = [c_2(mergesort#2^#(@xs, @x1))] [mergesort#2^#(::(@x2, @xs'), @x1)] = [0 1 0] [0 0 1] [1] [0 0 0] @x2 + [0 0 0] @xs' + [0] [0 0 0] [0 0 0] [1] >= [0 0 1] [1] [0 0 0] @xs' + [0] [0 0 0] [0] = [c_3(mergesort#3^#(msplit(::(@x1, ::(@x2, @xs')))))] [mergesort#3^#(tuple#2(@l1, @l2))] = [1 0 0] [1 0 0] [1] [0 0 0] @l1 + [0 0 0] @l2 + [1] [0 0 0] [0 0 0] [0] > [1 0 0] [1 0 0] [0] [0 0 0] @l1 + [0 0 0] @l2 + [0] [0 0 0] [0 0 0] [0] = [c_4(mergesort^#(@l1), mergesort^#(@l2))] Consider the set of all dependency pairs DPs: { 1: mergesort^#(@l) -> c_1(mergesort#1^#(@l)) , 2: mergesort#1^#(::(@x1, @xs)) -> c_2(mergesort#2^#(@xs, @x1)) , 3: mergesort#2^#(::(@x2, @xs'), @x1) -> c_3(mergesort#3^#(msplit(::(@x1, ::(@x2, @xs'))))) , 4: mergesort#3^#(tuple#2(@l1, @l2)) -> c_4(mergesort^#(@l1), mergesort^#(@l2)) } Processor 'matrix interpretation of dimension 3' induces the complexity certificate YES(?,O(n^1)) on application of dependency pairs {4}. These cover all (indirect) predecessors of dependency pairs {1,2,3,4}, their number of application is equally bounded. The dependency pairs are shifted into the corresponding weak component(s). We apply the transformation 'removetails' on the sub-problem: Weak DPs: { mergesort^#(@l) -> c_1(mergesort#1^#(@l)) , mergesort#1^#(::(@x1, @xs)) -> c_2(mergesort#2^#(@xs, @x1)) , mergesort#2^#(::(@x2, @xs'), @x1) -> c_3(mergesort#3^#(msplit(::(@x1, ::(@x2, @xs'))))) , mergesort#3^#(tuple#2(@l1, @l2)) -> c_4(mergesort^#(@l1), mergesort^#(@l2)) } Weak Trs: { msplit(@l) -> msplit#1(@l) , msplit#1(::(@x1, @xs)) -> msplit#2(@xs, @x1) , msplit#1(nil()) -> tuple#2(nil(), nil()) , msplit#2(::(@x2, @xs'), @x1) -> msplit#3(msplit(@xs'), @x1, @x2) , msplit#2(nil(), @x1) -> tuple#2(::(@x1, nil()), nil()) , msplit#3(tuple#2(@l1, @l2), @x1, @x2) -> tuple#2(::(@x1, @l1), ::(@x2, @l2)) } StartTerms: basic terms Strategy: innermost The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { mergesort^#(@l) -> c_1(mergesort#1^#(@l)) , mergesort#1^#(::(@x1, @xs)) -> c_2(mergesort#2^#(@xs, @x1)) , mergesort#2^#(::(@x2, @xs'), @x1) -> c_3(mergesort#3^#(msplit(::(@x1, ::(@x2, @xs'))))) , mergesort#3^#(tuple#2(@l1, @l2)) -> c_4(mergesort^#(@l1), mergesort^#(@l2)) } We apply the transformation 'usablerules' on the sub-problem: Weak Trs: { msplit(@l) -> msplit#1(@l) , msplit#1(::(@x1, @xs)) -> msplit#2(@xs, @x1) , msplit#1(nil()) -> tuple#2(nil(), nil()) , msplit#2(::(@x2, @xs'), @x1) -> msplit#3(msplit(@xs'), @x1, @x2) , msplit#2(nil(), @x1) -> tuple#2(::(@x1, nil()), nil()) , msplit#3(tuple#2(@l1, @l2), @x1, @x2) -> tuple#2(::(@x1, @l1), ::(@x2, @l2)) } StartTerms: basic terms Strategy: innermost No rule is usable, rules are removed from the input problem. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Rules: Empty Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) Empty rules are trivially bounded We return to the main proof. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { merge^#(@l1, @l2) -> c_1(merge#1^#(@l1, @l2)) , merge#1^#(::(@x, @xs), @l2) -> c_2(merge#2^#(@l2, @x, @xs)) , merge#2^#(::(@y, @ys), @x, @xs) -> c_3(merge#3^#(#less(@x, @y), @x, @xs, @y, @ys)) , merge#3^#(#false(), @x, @xs, @y, @ys) -> c_4(merge^#(::(@x, @xs), @ys)) , merge#3^#(#true(), @x, @xs, @y, @ys) -> c_5(merge^#(@xs, ::(@y, @ys))) , msplit^#(@l) -> c_10(msplit#1^#(@l)) , msplit#1^#(::(@x1, @xs)) -> c_11(msplit#2^#(@xs, @x1)) , msplit#2^#(::(@x2, @xs'), @x1) -> c_12(msplit^#(@xs')) } Weak DPs: { mergesort^#(@l) -> mergesort#1^#(@l) , mergesort#1^#(::(@x1, @xs)) -> mergesort#2^#(@xs, @x1) , mergesort#2^#(::(@x2, @xs'), @x1) -> mergesort#3^#(msplit(::(@x1, ::(@x2, @xs')))) , mergesort#2^#(::(@x2, @xs'), @x1) -> msplit^#(::(@x1, ::(@x2, @xs'))) , mergesort#3^#(tuple#2(@l1, @l2)) -> merge^#(mergesort(@l1), mergesort(@l2)) , mergesort#3^#(tuple#2(@l1, @l2)) -> mergesort^#(@l1) , mergesort#3^#(tuple#2(@l1, @l2)) -> mergesort^#(@l2) } Weak Trs: { #less(@x, @y) -> #cklt(#compare(@x, @y)) , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #s(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#pos(@x), #0()) -> #GT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , #cklt(#EQ()) -> #false() , #cklt(#GT()) -> #false() , #cklt(#LT()) -> #true() , merge(@l1, @l2) -> merge#1(@l1, @l2) , merge#1(::(@x, @xs), @l2) -> merge#2(@l2, @x, @xs) , merge#1(nil(), @l2) -> @l2 , merge#2(::(@y, @ys), @x, @xs) -> merge#3(#less(@x, @y), @x, @xs, @y, @ys) , merge#2(nil(), @x, @xs) -> ::(@x, @xs) , merge#3(#false(), @x, @xs, @y, @ys) -> ::(@y, merge(::(@x, @xs), @ys)) , merge#3(#true(), @x, @xs, @y, @ys) -> ::(@x, merge(@xs, ::(@y, @ys))) , mergesort(@l) -> mergesort#1(@l) , mergesort#1(::(@x1, @xs)) -> mergesort#2(@xs, @x1) , mergesort#1(nil()) -> nil() , mergesort#2(::(@x2, @xs'), @x1) -> mergesort#3(msplit(::(@x1, ::(@x2, @xs')))) , mergesort#2(nil(), @x1) -> ::(@x1, nil()) , msplit(@l) -> msplit#1(@l) , mergesort#3(tuple#2(@l1, @l2)) -> merge(mergesort(@l1), mergesort(@l2)) , msplit#1(::(@x1, @xs)) -> msplit#2(@xs, @x1) , msplit#1(nil()) -> tuple#2(nil(), nil()) , msplit#2(::(@x2, @xs'), @x1) -> msplit#3(msplit(@xs'), @x1, @x2) , msplit#2(nil(), @x1) -> tuple#2(::(@x1, nil()), nil()) , msplit#3(tuple#2(@l1, @l2), @x1, @x2) -> tuple#2(::(@x1, @l1), ::(@x2, @l2)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 5: merge#3^#(#true(), @x, @xs, @y, @ys) -> c_5(merge^#(@xs, ::(@y, @ys))) , 12: mergesort#2^#(::(@x2, @xs'), @x1) -> msplit^#(::(@x1, ::(@x2, @xs'))) } Trs: { mergesort#2(nil(), @x1) -> ::(@x1, nil()) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_3) = {1}, Uargs(c_4) = {1}, Uargs(c_5) = {1}, Uargs(c_10) = {1}, Uargs(c_11) = {1}, Uargs(c_12) = {1} TcT has computed following constructor-based matrix interpretation satisfying not(EDA). [#less](x1, x2) = [0] [#compare](x1, x2) = [0] [#cklt](x1) = [0] [merge](x1, x2) = [1] x1 + [1] x2 + [0] [merge#1](x1, x2) = [1] x1 + [1] x2 + [0] [::](x1, x2) = [1] x2 + [1] [merge#2](x1, x2, x3) = [1] x1 + [1] x3 + [1] [nil] = [0] [merge#3](x1, x2, x3, x4, x5) = [1] x3 + [1] x5 + [2] [#false] = [0] [#true] = [0] [mergesort](x1) = [2] x1 + [0] [mergesort#1](x1) = [2] x1 + [0] [mergesort#2](x1, x2) = [2] x1 + [2] [msplit](x1) = [1] x1 + [0] [mergesort#3](x1) = [2] x1 + [0] [tuple#2](x1, x2) = [1] x1 + [1] x2 + [0] [msplit#1](x1) = [1] x1 + [0] [msplit#2](x1, x2) = [1] x1 + [1] [msplit#3](x1, x2, x3) = [1] x1 + [2] [#EQ] = [0] [#GT] = [0] [#LT] = [0] [#0] = [2] [#neg](x1) = [1] x1 + [0] [#pos](x1) = [1] x1 + [0] [#s](x1) = [1] x1 + [0] [#less^#](x1, x2) = [0] [#cklt^#](x1) = [0] [#compare^#](x1, x2) = [0] [merge^#](x1, x2) = [1] x1 + [0] [merge#1^#](x1, x2) = [1] x1 + [0] [merge#2^#](x1, x2, x3) = [1] x3 + [1] [merge#3^#](x1, x2, x3, x4, x5) = [1] x1 + [1] x3 + [1] [mergesort^#](x1) = [2] x1 + [0] [mergesort#1^#](x1) = [2] x1 + [0] [mergesort#2^#](x1, x2) = [2] x1 + [2] [mergesort#3^#](x1) = [2] x1 + [0] [msplit^#](x1) = [0] [msplit#1^#](x1) = [0] [msplit#2^#](x1, x2) = [0] [msplit#3^#](x1, x2, x3) = [0] [c_1](x1) = [1] x1 + [0] [c_2](x1) = [1] x1 + [0] [c_3](x1) = [1] x1 + [0] [c_4](x1) = [1] x1 + [0] [c_5](x1) = [1] x1 + [0] [c_10](x1) = [1] x1 + [0] [c_11](x1) = [1] x1 + [0] [c_12](x1) = [1] x1 + [0] This order satisfies following ordering constraints [#less(@x, @y)] = [0] >= [0] = [#cklt(#compare(@x, @y))] [#compare(#0(), #0())] = [0] >= [0] = [#EQ()] [#compare(#0(), #neg(@y))] = [0] >= [0] = [#GT()] [#compare(#0(), #pos(@y))] = [0] >= [0] = [#LT()] [#compare(#0(), #s(@y))] = [0] >= [0] = [#LT()] [#compare(#neg(@x), #0())] = [0] >= [0] = [#LT()] [#compare(#neg(@x), #neg(@y))] = [0] >= [0] = [#compare(@y, @x)] [#compare(#neg(@x), #pos(@y))] = [0] >= [0] = [#LT()] [#compare(#pos(@x), #0())] = [0] >= [0] = [#GT()] [#compare(#pos(@x), #neg(@y))] = [0] >= [0] = [#GT()] [#compare(#pos(@x), #pos(@y))] = [0] >= [0] = [#compare(@x, @y)] [#compare(#s(@x), #0())] = [0] >= [0] = [#GT()] [#compare(#s(@x), #s(@y))] = [0] >= [0] = [#compare(@x, @y)] [#cklt(#EQ())] = [0] >= [0] = [#false()] [#cklt(#GT())] = [0] >= [0] = [#false()] [#cklt(#LT())] = [0] >= [0] = [#true()] [merge(@l1, @l2)] = [1] @l1 + [1] @l2 + [0] >= [1] @l1 + [1] @l2 + [0] = [merge#1(@l1, @l2)] [merge#1(::(@x, @xs), @l2)] = [1] @l2 + [1] @xs + [1] >= [1] @l2 + [1] @xs + [1] = [merge#2(@l2, @x, @xs)] [merge#1(nil(), @l2)] = [1] @l2 + [0] >= [1] @l2 + [0] = [@l2] [merge#2(::(@y, @ys), @x, @xs)] = [1] @xs + [1] @ys + [2] >= [1] @xs + [1] @ys + [2] = [merge#3(#less(@x, @y), @x, @xs, @y, @ys)] [merge#2(nil(), @x, @xs)] = [1] @xs + [1] >= [1] @xs + [1] = [::(@x, @xs)] [merge#3(#false(), @x, @xs, @y, @ys)] = [1] @xs + [1] @ys + [2] >= [1] @xs + [1] @ys + [2] = [::(@y, merge(::(@x, @xs), @ys))] [merge#3(#true(), @x, @xs, @y, @ys)] = [1] @xs + [1] @ys + [2] >= [1] @xs + [1] @ys + [2] = [::(@x, merge(@xs, ::(@y, @ys)))] [mergesort(@l)] = [2] @l + [0] >= [2] @l + [0] = [mergesort#1(@l)] [mergesort#1(::(@x1, @xs))] = [2] @xs + [2] >= [2] @xs + [2] = [mergesort#2(@xs, @x1)] [mergesort#1(nil())] = [0] >= [0] = [nil()] [mergesort#2(::(@x2, @xs'), @x1)] = [2] @xs' + [4] >= [2] @xs' + [4] = [mergesort#3(msplit(::(@x1, ::(@x2, @xs'))))] [mergesort#2(nil(), @x1)] = [2] > [1] = [::(@x1, nil())] [msplit(@l)] = [1] @l + [0] >= [1] @l + [0] = [msplit#1(@l)] [mergesort#3(tuple#2(@l1, @l2))] = [2] @l1 + [2] @l2 + [0] >= [2] @l1 + [2] @l2 + [0] = [merge(mergesort(@l1), mergesort(@l2))] [msplit#1(::(@x1, @xs))] = [1] @xs + [1] >= [1] @xs + [1] = [msplit#2(@xs, @x1)] [msplit#1(nil())] = [0] >= [0] = [tuple#2(nil(), nil())] [msplit#2(::(@x2, @xs'), @x1)] = [1] @xs' + [2] >= [1] @xs' + [2] = [msplit#3(msplit(@xs'), @x1, @x2)] [msplit#2(nil(), @x1)] = [1] >= [1] = [tuple#2(::(@x1, nil()), nil())] [msplit#3(tuple#2(@l1, @l2), @x1, @x2)] = [1] @l1 + [1] @l2 + [2] >= [1] @l1 + [1] @l2 + [2] = [tuple#2(::(@x1, @l1), ::(@x2, @l2))] [merge^#(@l1, @l2)] = [1] @l1 + [0] >= [1] @l1 + [0] = [c_1(merge#1^#(@l1, @l2))] [merge#1^#(::(@x, @xs), @l2)] = [1] @xs + [1] >= [1] @xs + [1] = [c_2(merge#2^#(@l2, @x, @xs))] [merge#2^#(::(@y, @ys), @x, @xs)] = [1] @xs + [1] >= [1] @xs + [1] = [c_3(merge#3^#(#less(@x, @y), @x, @xs, @y, @ys))] [merge#3^#(#false(), @x, @xs, @y, @ys)] = [1] @xs + [1] >= [1] @xs + [1] = [c_4(merge^#(::(@x, @xs), @ys))] [merge#3^#(#true(), @x, @xs, @y, @ys)] = [1] @xs + [1] > [1] @xs + [0] = [c_5(merge^#(@xs, ::(@y, @ys)))] [mergesort^#(@l)] = [2] @l + [0] >= [2] @l + [0] = [mergesort#1^#(@l)] [mergesort#1^#(::(@x1, @xs))] = [2] @xs + [2] >= [2] @xs + [2] = [mergesort#2^#(@xs, @x1)] [mergesort#2^#(::(@x2, @xs'), @x1)] = [2] @xs' + [4] >= [2] @xs' + [4] = [mergesort#3^#(msplit(::(@x1, ::(@x2, @xs'))))] [mergesort#2^#(::(@x2, @xs'), @x1)] = [2] @xs' + [4] > [0] = [msplit^#(::(@x1, ::(@x2, @xs')))] [mergesort#3^#(tuple#2(@l1, @l2))] = [2] @l1 + [2] @l2 + [0] >= [2] @l1 + [0] = [merge^#(mergesort(@l1), mergesort(@l2))] [mergesort#3^#(tuple#2(@l1, @l2))] = [2] @l1 + [2] @l2 + [0] >= [2] @l1 + [0] = [mergesort^#(@l1)] [mergesort#3^#(tuple#2(@l1, @l2))] = [2] @l1 + [2] @l2 + [0] >= [2] @l2 + [0] = [mergesort^#(@l2)] [msplit^#(@l)] = [0] >= [0] = [c_10(msplit#1^#(@l))] [msplit#1^#(::(@x1, @xs))] = [0] >= [0] = [c_11(msplit#2^#(@xs, @x1))] [msplit#2^#(::(@x2, @xs'), @x1)] = [0] >= [0] = [c_12(msplit^#(@xs'))] The strictly oriented rules are moved into the corresponding weak component(s). We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { merge^#(@l1, @l2) -> c_1(merge#1^#(@l1, @l2)) , merge#1^#(::(@x, @xs), @l2) -> c_2(merge#2^#(@l2, @x, @xs)) , merge#2^#(::(@y, @ys), @x, @xs) -> c_3(merge#3^#(#less(@x, @y), @x, @xs, @y, @ys)) , merge#3^#(#false(), @x, @xs, @y, @ys) -> c_4(merge^#(::(@x, @xs), @ys)) , msplit^#(@l) -> c_10(msplit#1^#(@l)) , msplit#1^#(::(@x1, @xs)) -> c_11(msplit#2^#(@xs, @x1)) , msplit#2^#(::(@x2, @xs'), @x1) -> c_12(msplit^#(@xs')) } Weak DPs: { merge#3^#(#true(), @x, @xs, @y, @ys) -> c_5(merge^#(@xs, ::(@y, @ys))) , mergesort^#(@l) -> mergesort#1^#(@l) , mergesort#1^#(::(@x1, @xs)) -> mergesort#2^#(@xs, @x1) , mergesort#2^#(::(@x2, @xs'), @x1) -> mergesort#3^#(msplit(::(@x1, ::(@x2, @xs')))) , mergesort#2^#(::(@x2, @xs'), @x1) -> msplit^#(::(@x1, ::(@x2, @xs'))) , mergesort#3^#(tuple#2(@l1, @l2)) -> merge^#(mergesort(@l1), mergesort(@l2)) , mergesort#3^#(tuple#2(@l1, @l2)) -> mergesort^#(@l1) , mergesort#3^#(tuple#2(@l1, @l2)) -> mergesort^#(@l2) } Weak Trs: { #less(@x, @y) -> #cklt(#compare(@x, @y)) , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #s(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#pos(@x), #0()) -> #GT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , #cklt(#EQ()) -> #false() , #cklt(#GT()) -> #false() , #cklt(#LT()) -> #true() , merge(@l1, @l2) -> merge#1(@l1, @l2) , merge#1(::(@x, @xs), @l2) -> merge#2(@l2, @x, @xs) , merge#1(nil(), @l2) -> @l2 , merge#2(::(@y, @ys), @x, @xs) -> merge#3(#less(@x, @y), @x, @xs, @y, @ys) , merge#2(nil(), @x, @xs) -> ::(@x, @xs) , merge#3(#false(), @x, @xs, @y, @ys) -> ::(@y, merge(::(@x, @xs), @ys)) , merge#3(#true(), @x, @xs, @y, @ys) -> ::(@x, merge(@xs, ::(@y, @ys))) , mergesort(@l) -> mergesort#1(@l) , mergesort#1(::(@x1, @xs)) -> mergesort#2(@xs, @x1) , mergesort#1(nil()) -> nil() , mergesort#2(::(@x2, @xs'), @x1) -> mergesort#3(msplit(::(@x1, ::(@x2, @xs')))) , mergesort#2(nil(), @x1) -> ::(@x1, nil()) , msplit(@l) -> msplit#1(@l) , mergesort#3(tuple#2(@l1, @l2)) -> merge(mergesort(@l1), mergesort(@l2)) , msplit#1(::(@x1, @xs)) -> msplit#2(@xs, @x1) , msplit#1(nil()) -> tuple#2(nil(), nil()) , msplit#2(::(@x2, @xs'), @x1) -> msplit#3(msplit(@xs'), @x1, @x2) , msplit#2(nil(), @x1) -> tuple#2(::(@x1, nil()), nil()) , msplit#3(tuple#2(@l1, @l2), @x1, @x2) -> tuple#2(::(@x1, @l1), ::(@x2, @l2)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 5: msplit^#(@l) -> c_10(msplit#1^#(@l)) , 6: msplit#1^#(::(@x1, @xs)) -> c_11(msplit#2^#(@xs, @x1)) } Trs: { mergesort#2(::(@x2, @xs'), @x1) -> mergesort#3(msplit(::(@x1, ::(@x2, @xs')))) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_3) = {1}, Uargs(c_4) = {1}, Uargs(c_5) = {1}, Uargs(c_10) = {1}, Uargs(c_11) = {1}, Uargs(c_12) = {1} TcT has computed following constructor-based matrix interpretation satisfying not(EDA). [#less](x1, x2) = [0] [#compare](x1, x2) = [0] [#cklt](x1) = [0] [merge](x1, x2) = [1] x2 + [2] [merge#1](x1, x2) = [1] x1 + [2] x2 + [0] [::](x1, x2) = [1] x2 + [1] [merge#2](x1, x2, x3) = [2] x2 + [2] x3 + [0] [nil] = [0] [merge#3](x1, x2, x3, x4, x5) = [2] x3 + [1] x5 + [2] [#false] = [0] [#true] = [0] [mergesort](x1) = [0] [mergesort#1](x1) = [0] [mergesort#2](x1, x2) = [2] x2 + [1] [msplit](x1) = [1] x1 + [1] [mergesort#3](x1) = [0] [tuple#2](x1, x2) = [1] x1 + [1] x2 + [1] [msplit#1](x1) = [1] x1 + [1] [msplit#2](x1, x2) = [1] x1 + [2] [msplit#3](x1, x2, x3) = [1] x1 + [2] [#EQ] = [0] [#GT] = [0] [#LT] = [0] [#0] = [0] [#neg](x1) = [1] x1 + [0] [#pos](x1) = [1] x1 + [0] [#s](x1) = [1] x1 + [0] [#less^#](x1, x2) = [0] [#cklt^#](x1) = [0] [#compare^#](x1, x2) = [0] [merge^#](x1, x2) = [1] [merge#1^#](x1, x2) = [1] [merge#2^#](x1, x2, x3) = [1] [merge#3^#](x1, x2, x3, x4, x5) = [1] [mergesort^#](x1) = [1] x1 + [1] [mergesort#1^#](x1) = [1] x1 + [1] [mergesort#2^#](x1, x2) = [1] x1 + [2] [mergesort#3^#](x1) = [1] x1 + [0] [msplit^#](x1) = [1] x1 + [1] [msplit#1^#](x1) = [1] x1 + [0] [msplit#2^#](x1, x2) = [1] x1 + [0] [msplit#3^#](x1, x2, x3) = [0] [c_1](x1) = [1] x1 + [0] [c_2](x1) = [1] x1 + [0] [c_3](x1) = [1] x1 + [0] [c_4](x1) = [1] x1 + [0] [c_5](x1) = [1] x1 + [0] [c_10](x1) = [1] x1 + [0] [c_11](x1) = [1] x1 + [0] [c_12](x1) = [1] x1 + [0] This order satisfies following ordering constraints [msplit(@l)] = [1] @l + [1] >= [1] @l + [1] = [msplit#1(@l)] [msplit#1(::(@x1, @xs))] = [1] @xs + [2] >= [1] @xs + [2] = [msplit#2(@xs, @x1)] [msplit#1(nil())] = [1] >= [1] = [tuple#2(nil(), nil())] [msplit#2(::(@x2, @xs'), @x1)] = [1] @xs' + [3] >= [1] @xs' + [3] = [msplit#3(msplit(@xs'), @x1, @x2)] [msplit#2(nil(), @x1)] = [2] >= [2] = [tuple#2(::(@x1, nil()), nil())] [msplit#3(tuple#2(@l1, @l2), @x1, @x2)] = [1] @l1 + [1] @l2 + [3] >= [1] @l1 + [1] @l2 + [3] = [tuple#2(::(@x1, @l1), ::(@x2, @l2))] [merge^#(@l1, @l2)] = [1] >= [1] = [c_1(merge#1^#(@l1, @l2))] [merge#1^#(::(@x, @xs), @l2)] = [1] >= [1] = [c_2(merge#2^#(@l2, @x, @xs))] [merge#2^#(::(@y, @ys), @x, @xs)] = [1] >= [1] = [c_3(merge#3^#(#less(@x, @y), @x, @xs, @y, @ys))] [merge#3^#(#false(), @x, @xs, @y, @ys)] = [1] >= [1] = [c_4(merge^#(::(@x, @xs), @ys))] [merge#3^#(#true(), @x, @xs, @y, @ys)] = [1] >= [1] = [c_5(merge^#(@xs, ::(@y, @ys)))] [mergesort^#(@l)] = [1] @l + [1] >= [1] @l + [1] = [mergesort#1^#(@l)] [mergesort#1^#(::(@x1, @xs))] = [1] @xs + [2] >= [1] @xs + [2] = [mergesort#2^#(@xs, @x1)] [mergesort#2^#(::(@x2, @xs'), @x1)] = [1] @xs' + [3] >= [1] @xs' + [3] = [mergesort#3^#(msplit(::(@x1, ::(@x2, @xs'))))] [mergesort#2^#(::(@x2, @xs'), @x1)] = [1] @xs' + [3] >= [1] @xs' + [3] = [msplit^#(::(@x1, ::(@x2, @xs')))] [mergesort#3^#(tuple#2(@l1, @l2))] = [1] @l1 + [1] @l2 + [1] >= [1] = [merge^#(mergesort(@l1), mergesort(@l2))] [mergesort#3^#(tuple#2(@l1, @l2))] = [1] @l1 + [1] @l2 + [1] >= [1] @l1 + [1] = [mergesort^#(@l1)] [mergesort#3^#(tuple#2(@l1, @l2))] = [1] @l1 + [1] @l2 + [1] >= [1] @l2 + [1] = [mergesort^#(@l2)] [msplit^#(@l)] = [1] @l + [1] > [1] @l + [0] = [c_10(msplit#1^#(@l))] [msplit#1^#(::(@x1, @xs))] = [1] @xs + [1] > [1] @xs + [0] = [c_11(msplit#2^#(@xs, @x1))] [msplit#2^#(::(@x2, @xs'), @x1)] = [1] @xs' + [1] >= [1] @xs' + [1] = [c_12(msplit^#(@xs'))] Consider the set of all dependency pairs DPs: { 1: merge^#(@l1, @l2) -> c_1(merge#1^#(@l1, @l2)) , 2: merge#1^#(::(@x, @xs), @l2) -> c_2(merge#2^#(@l2, @x, @xs)) , 3: merge#2^#(::(@y, @ys), @x, @xs) -> c_3(merge#3^#(#less(@x, @y), @x, @xs, @y, @ys)) , 4: merge#3^#(#false(), @x, @xs, @y, @ys) -> c_4(merge^#(::(@x, @xs), @ys)) , 5: msplit^#(@l) -> c_10(msplit#1^#(@l)) , 6: msplit#1^#(::(@x1, @xs)) -> c_11(msplit#2^#(@xs, @x1)) , 7: msplit#2^#(::(@x2, @xs'), @x1) -> c_12(msplit^#(@xs')) , 8: merge#3^#(#true(), @x, @xs, @y, @ys) -> c_5(merge^#(@xs, ::(@y, @ys))) , 9: mergesort^#(@l) -> mergesort#1^#(@l) , 10: mergesort#1^#(::(@x1, @xs)) -> mergesort#2^#(@xs, @x1) , 11: mergesort#2^#(::(@x2, @xs'), @x1) -> mergesort#3^#(msplit(::(@x1, ::(@x2, @xs')))) , 12: mergesort#2^#(::(@x2, @xs'), @x1) -> msplit^#(::(@x1, ::(@x2, @xs'))) , 13: mergesort#3^#(tuple#2(@l1, @l2)) -> merge^#(mergesort(@l1), mergesort(@l2)) , 14: mergesort#3^#(tuple#2(@l1, @l2)) -> mergesort^#(@l1) , 15: mergesort#3^#(tuple#2(@l1, @l2)) -> mergesort^#(@l2) } Processor 'matrix interpretation of dimension 1' induces the complexity certificate YES(?,O(n^1)) on application of dependency pairs {5,6}. These cover all (indirect) predecessors of dependency pairs {5,6,7}, their number of application is equally bounded. The dependency pairs are shifted into the corresponding weak component(s). We apply the transformation 'removetails' on the sub-problem: Strict DPs: { merge^#(@l1, @l2) -> c_1(merge#1^#(@l1, @l2)) , merge#1^#(::(@x, @xs), @l2) -> c_2(merge#2^#(@l2, @x, @xs)) , merge#2^#(::(@y, @ys), @x, @xs) -> c_3(merge#3^#(#less(@x, @y), @x, @xs, @y, @ys)) , merge#3^#(#false(), @x, @xs, @y, @ys) -> c_4(merge^#(::(@x, @xs), @ys)) } Weak DPs: { merge#3^#(#true(), @x, @xs, @y, @ys) -> c_5(merge^#(@xs, ::(@y, @ys))) , mergesort^#(@l) -> mergesort#1^#(@l) , mergesort#1^#(::(@x1, @xs)) -> mergesort#2^#(@xs, @x1) , mergesort#2^#(::(@x2, @xs'), @x1) -> mergesort#3^#(msplit(::(@x1, ::(@x2, @xs')))) , mergesort#2^#(::(@x2, @xs'), @x1) -> msplit^#(::(@x1, ::(@x2, @xs'))) , mergesort#3^#(tuple#2(@l1, @l2)) -> merge^#(mergesort(@l1), mergesort(@l2)) , mergesort#3^#(tuple#2(@l1, @l2)) -> mergesort^#(@l1) , mergesort#3^#(tuple#2(@l1, @l2)) -> mergesort^#(@l2) , msplit^#(@l) -> c_10(msplit#1^#(@l)) , msplit#1^#(::(@x1, @xs)) -> c_11(msplit#2^#(@xs, @x1)) , msplit#2^#(::(@x2, @xs'), @x1) -> c_12(msplit^#(@xs')) } Weak Trs: { #less(@x, @y) -> #cklt(#compare(@x, @y)) , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #s(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#pos(@x), #0()) -> #GT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , #cklt(#EQ()) -> #false() , #cklt(#GT()) -> #false() , #cklt(#LT()) -> #true() , merge(@l1, @l2) -> merge#1(@l1, @l2) , merge#1(::(@x, @xs), @l2) -> merge#2(@l2, @x, @xs) , merge#1(nil(), @l2) -> @l2 , merge#2(::(@y, @ys), @x, @xs) -> merge#3(#less(@x, @y), @x, @xs, @y, @ys) , merge#2(nil(), @x, @xs) -> ::(@x, @xs) , merge#3(#false(), @x, @xs, @y, @ys) -> ::(@y, merge(::(@x, @xs), @ys)) , merge#3(#true(), @x, @xs, @y, @ys) -> ::(@x, merge(@xs, ::(@y, @ys))) , mergesort(@l) -> mergesort#1(@l) , mergesort#1(::(@x1, @xs)) -> mergesort#2(@xs, @x1) , mergesort#1(nil()) -> nil() , mergesort#2(::(@x2, @xs'), @x1) -> mergesort#3(msplit(::(@x1, ::(@x2, @xs')))) , mergesort#2(nil(), @x1) -> ::(@x1, nil()) , msplit(@l) -> msplit#1(@l) , mergesort#3(tuple#2(@l1, @l2)) -> merge(mergesort(@l1), mergesort(@l2)) , msplit#1(::(@x1, @xs)) -> msplit#2(@xs, @x1) , msplit#1(nil()) -> tuple#2(nil(), nil()) , msplit#2(::(@x2, @xs'), @x1) -> msplit#3(msplit(@xs'), @x1, @x2) , msplit#2(nil(), @x1) -> tuple#2(::(@x1, nil()), nil()) , msplit#3(tuple#2(@l1, @l2), @x1, @x2) -> tuple#2(::(@x1, @l1), ::(@x2, @l2)) } StartTerms: basic terms Strategy: innermost The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { mergesort#2^#(::(@x2, @xs'), @x1) -> msplit^#(::(@x1, ::(@x2, @xs'))) , msplit^#(@l) -> c_10(msplit#1^#(@l)) , msplit#1^#(::(@x1, @xs)) -> c_11(msplit#2^#(@xs, @x1)) , msplit#2^#(::(@x2, @xs'), @x1) -> c_12(msplit^#(@xs')) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { merge^#(@l1, @l2) -> c_1(merge#1^#(@l1, @l2)) , merge#1^#(::(@x, @xs), @l2) -> c_2(merge#2^#(@l2, @x, @xs)) , merge#2^#(::(@y, @ys), @x, @xs) -> c_3(merge#3^#(#less(@x, @y), @x, @xs, @y, @ys)) , merge#3^#(#false(), @x, @xs, @y, @ys) -> c_4(merge^#(::(@x, @xs), @ys)) } Weak DPs: { merge#3^#(#true(), @x, @xs, @y, @ys) -> c_5(merge^#(@xs, ::(@y, @ys))) , mergesort^#(@l) -> mergesort#1^#(@l) , mergesort#1^#(::(@x1, @xs)) -> mergesort#2^#(@xs, @x1) , mergesort#2^#(::(@x2, @xs'), @x1) -> mergesort#3^#(msplit(::(@x1, ::(@x2, @xs')))) , mergesort#3^#(tuple#2(@l1, @l2)) -> merge^#(mergesort(@l1), mergesort(@l2)) , mergesort#3^#(tuple#2(@l1, @l2)) -> mergesort^#(@l1) , mergesort#3^#(tuple#2(@l1, @l2)) -> mergesort^#(@l2) } Weak Trs: { #less(@x, @y) -> #cklt(#compare(@x, @y)) , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #s(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#pos(@x), #0()) -> #GT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , #cklt(#EQ()) -> #false() , #cklt(#GT()) -> #false() , #cklt(#LT()) -> #true() , merge(@l1, @l2) -> merge#1(@l1, @l2) , merge#1(::(@x, @xs), @l2) -> merge#2(@l2, @x, @xs) , merge#1(nil(), @l2) -> @l2 , merge#2(::(@y, @ys), @x, @xs) -> merge#3(#less(@x, @y), @x, @xs, @y, @ys) , merge#2(nil(), @x, @xs) -> ::(@x, @xs) , merge#3(#false(), @x, @xs, @y, @ys) -> ::(@y, merge(::(@x, @xs), @ys)) , merge#3(#true(), @x, @xs, @y, @ys) -> ::(@x, merge(@xs, ::(@y, @ys))) , mergesort(@l) -> mergesort#1(@l) , mergesort#1(::(@x1, @xs)) -> mergesort#2(@xs, @x1) , mergesort#1(nil()) -> nil() , mergesort#2(::(@x2, @xs'), @x1) -> mergesort#3(msplit(::(@x1, ::(@x2, @xs')))) , mergesort#2(nil(), @x1) -> ::(@x1, nil()) , msplit(@l) -> msplit#1(@l) , mergesort#3(tuple#2(@l1, @l2)) -> merge(mergesort(@l1), mergesort(@l2)) , msplit#1(::(@x1, @xs)) -> msplit#2(@xs, @x1) , msplit#1(nil()) -> tuple#2(nil(), nil()) , msplit#2(::(@x2, @xs'), @x1) -> msplit#3(msplit(@xs'), @x1, @x2) , msplit#2(nil(), @x1) -> tuple#2(::(@x1, nil()), nil()) , msplit#3(tuple#2(@l1, @l2), @x1, @x2) -> tuple#2(::(@x1, @l1), ::(@x2, @l2)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 3: merge#2^#(::(@y, @ys), @x, @xs) -> c_3(merge#3^#(#less(@x, @y), @x, @xs, @y, @ys)) , 5: merge#3^#(#true(), @x, @xs, @y, @ys) -> c_5(merge^#(@xs, ::(@y, @ys))) } Trs: { #less(@x, @y) -> #cklt(#compare(@x, @y)) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_3) = {1}, Uargs(c_4) = {1}, Uargs(c_5) = {1} TcT has computed following constructor-based matrix interpretation satisfying not(EDA). [#less](x1, x2) = [1] [#compare](x1, x2) = [0] [#cklt](x1) = [0] [merge](x1, x2) = [1] x1 + [1] x2 + [0] [merge#1](x1, x2) = [1] x1 + [1] x2 + [0] [::](x1, x2) = [1] x2 + [1] [merge#2](x1, x2, x3) = [1] x1 + [1] x3 + [1] [nil] = [0] [merge#3](x1, x2, x3, x4, x5) = [1] x3 + [1] x5 + [2] [#false] = [0] [#true] = [0] [mergesort](x1) = [1] x1 + [0] [mergesort#1](x1) = [1] x1 + [0] [mergesort#2](x1, x2) = [1] x1 + [1] [msplit](x1) = [1] x1 + [0] [mergesort#3](x1) = [1] x1 + [0] [tuple#2](x1, x2) = [1] x1 + [1] x2 + [0] [msplit#1](x1) = [1] x1 + [0] [msplit#2](x1, x2) = [1] x1 + [1] [msplit#3](x1, x2, x3) = [1] x1 + [2] [#EQ] = [0] [#GT] = [0] [#LT] = [0] [#0] = [0] [#neg](x1) = [1] x1 + [0] [#pos](x1) = [1] x1 + [0] [#s](x1) = [1] x1 + [0] [#less^#](x1, x2) = [0] [#cklt^#](x1) = [0] [#compare^#](x1, x2) = [0] [merge^#](x1, x2) = [2] x1 + [1] x2 + [0] [merge#1^#](x1, x2) = [2] x1 + [1] x2 + [0] [merge#2^#](x1, x2, x3) = [1] x1 + [2] x3 + [2] [merge#3^#](x1, x2, x3, x4, x5) = [2] x3 + [1] x5 + [2] [mergesort^#](x1) = [2] x1 + [0] [mergesort#1^#](x1) = [2] x1 + [0] [mergesort#2^#](x1, x2) = [2] x1 + [2] [mergesort#3^#](x1) = [2] x1 + [0] [msplit^#](x1) = [0] [msplit#1^#](x1) = [0] [msplit#2^#](x1, x2) = [0] [msplit#3^#](x1, x2, x3) = [0] [c_1](x1) = [1] x1 + [0] [c_2](x1) = [1] x1 + [0] [c_3](x1) = [1] x1 + [0] [c_4](x1) = [1] x1 + [0] [c_5](x1) = [1] x1 + [0] [c_10](x1) = [0] [c_11](x1) = [0] [c_12](x1) = [0] This order satisfies following ordering constraints [merge(@l1, @l2)] = [1] @l1 + [1] @l2 + [0] >= [1] @l1 + [1] @l2 + [0] = [merge#1(@l1, @l2)] [merge#1(::(@x, @xs), @l2)] = [1] @l2 + [1] @xs + [1] >= [1] @l2 + [1] @xs + [1] = [merge#2(@l2, @x, @xs)] [merge#1(nil(), @l2)] = [1] @l2 + [0] >= [1] @l2 + [0] = [@l2] [merge#2(::(@y, @ys), @x, @xs)] = [1] @xs + [1] @ys + [2] >= [1] @xs + [1] @ys + [2] = [merge#3(#less(@x, @y), @x, @xs, @y, @ys)] [merge#2(nil(), @x, @xs)] = [1] @xs + [1] >= [1] @xs + [1] = [::(@x, @xs)] [merge#3(#false(), @x, @xs, @y, @ys)] = [1] @xs + [1] @ys + [2] >= [1] @xs + [1] @ys + [2] = [::(@y, merge(::(@x, @xs), @ys))] [merge#3(#true(), @x, @xs, @y, @ys)] = [1] @xs + [1] @ys + [2] >= [1] @xs + [1] @ys + [2] = [::(@x, merge(@xs, ::(@y, @ys)))] [mergesort(@l)] = [1] @l + [0] >= [1] @l + [0] = [mergesort#1(@l)] [mergesort#1(::(@x1, @xs))] = [1] @xs + [1] >= [1] @xs + [1] = [mergesort#2(@xs, @x1)] [mergesort#1(nil())] = [0] >= [0] = [nil()] [mergesort#2(::(@x2, @xs'), @x1)] = [1] @xs' + [2] >= [1] @xs' + [2] = [mergesort#3(msplit(::(@x1, ::(@x2, @xs'))))] [mergesort#2(nil(), @x1)] = [1] >= [1] = [::(@x1, nil())] [msplit(@l)] = [1] @l + [0] >= [1] @l + [0] = [msplit#1(@l)] [mergesort#3(tuple#2(@l1, @l2))] = [1] @l1 + [1] @l2 + [0] >= [1] @l1 + [1] @l2 + [0] = [merge(mergesort(@l1), mergesort(@l2))] [msplit#1(::(@x1, @xs))] = [1] @xs + [1] >= [1] @xs + [1] = [msplit#2(@xs, @x1)] [msplit#1(nil())] = [0] >= [0] = [tuple#2(nil(), nil())] [msplit#2(::(@x2, @xs'), @x1)] = [1] @xs' + [2] >= [1] @xs' + [2] = [msplit#3(msplit(@xs'), @x1, @x2)] [msplit#2(nil(), @x1)] = [1] >= [1] = [tuple#2(::(@x1, nil()), nil())] [msplit#3(tuple#2(@l1, @l2), @x1, @x2)] = [1] @l1 + [1] @l2 + [2] >= [1] @l1 + [1] @l2 + [2] = [tuple#2(::(@x1, @l1), ::(@x2, @l2))] [merge^#(@l1, @l2)] = [2] @l1 + [1] @l2 + [0] >= [2] @l1 + [1] @l2 + [0] = [c_1(merge#1^#(@l1, @l2))] [merge#1^#(::(@x, @xs), @l2)] = [1] @l2 + [2] @xs + [2] >= [1] @l2 + [2] @xs + [2] = [c_2(merge#2^#(@l2, @x, @xs))] [merge#2^#(::(@y, @ys), @x, @xs)] = [2] @xs + [1] @ys + [3] > [2] @xs + [1] @ys + [2] = [c_3(merge#3^#(#less(@x, @y), @x, @xs, @y, @ys))] [merge#3^#(#false(), @x, @xs, @y, @ys)] = [2] @xs + [1] @ys + [2] >= [2] @xs + [1] @ys + [2] = [c_4(merge^#(::(@x, @xs), @ys))] [merge#3^#(#true(), @x, @xs, @y, @ys)] = [2] @xs + [1] @ys + [2] > [2] @xs + [1] @ys + [1] = [c_5(merge^#(@xs, ::(@y, @ys)))] [mergesort^#(@l)] = [2] @l + [0] >= [2] @l + [0] = [mergesort#1^#(@l)] [mergesort#1^#(::(@x1, @xs))] = [2] @xs + [2] >= [2] @xs + [2] = [mergesort#2^#(@xs, @x1)] [mergesort#2^#(::(@x2, @xs'), @x1)] = [2] @xs' + [4] >= [2] @xs' + [4] = [mergesort#3^#(msplit(::(@x1, ::(@x2, @xs'))))] [mergesort#3^#(tuple#2(@l1, @l2))] = [2] @l1 + [2] @l2 + [0] >= [2] @l1 + [1] @l2 + [0] = [merge^#(mergesort(@l1), mergesort(@l2))] [mergesort#3^#(tuple#2(@l1, @l2))] = [2] @l1 + [2] @l2 + [0] >= [2] @l1 + [0] = [mergesort^#(@l1)] [mergesort#3^#(tuple#2(@l1, @l2))] = [2] @l1 + [2] @l2 + [0] >= [2] @l2 + [0] = [mergesort^#(@l2)] Consider the set of all dependency pairs DPs: { 1: merge^#(@l1, @l2) -> c_1(merge#1^#(@l1, @l2)) , 2: merge#1^#(::(@x, @xs), @l2) -> c_2(merge#2^#(@l2, @x, @xs)) , 3: merge#2^#(::(@y, @ys), @x, @xs) -> c_3(merge#3^#(#less(@x, @y), @x, @xs, @y, @ys)) , 4: merge#3^#(#false(), @x, @xs, @y, @ys) -> c_4(merge^#(::(@x, @xs), @ys)) , 5: merge#3^#(#true(), @x, @xs, @y, @ys) -> c_5(merge^#(@xs, ::(@y, @ys))) , 6: mergesort^#(@l) -> mergesort#1^#(@l) , 7: mergesort#1^#(::(@x1, @xs)) -> mergesort#2^#(@xs, @x1) , 8: mergesort#2^#(::(@x2, @xs'), @x1) -> mergesort#3^#(msplit(::(@x1, ::(@x2, @xs')))) , 9: mergesort#3^#(tuple#2(@l1, @l2)) -> merge^#(mergesort(@l1), mergesort(@l2)) , 10: mergesort#3^#(tuple#2(@l1, @l2)) -> mergesort^#(@l1) , 11: mergesort#3^#(tuple#2(@l1, @l2)) -> mergesort^#(@l2) } Processor 'matrix interpretation of dimension 1' induces the complexity certificate YES(?,O(n^1)) on application of dependency pairs {3,5}. These cover all (indirect) predecessors of dependency pairs {3,4,5}, their number of application is equally bounded. The dependency pairs are shifted into the corresponding weak component(s). We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { merge^#(@l1, @l2) -> c_1(merge#1^#(@l1, @l2)) , merge#1^#(::(@x, @xs), @l2) -> c_2(merge#2^#(@l2, @x, @xs)) } Weak DPs: { merge#2^#(::(@y, @ys), @x, @xs) -> c_3(merge#3^#(#less(@x, @y), @x, @xs, @y, @ys)) , merge#3^#(#false(), @x, @xs, @y, @ys) -> c_4(merge^#(::(@x, @xs), @ys)) , merge#3^#(#true(), @x, @xs, @y, @ys) -> c_5(merge^#(@xs, ::(@y, @ys))) , mergesort^#(@l) -> mergesort#1^#(@l) , mergesort#1^#(::(@x1, @xs)) -> mergesort#2^#(@xs, @x1) , mergesort#2^#(::(@x2, @xs'), @x1) -> mergesort#3^#(msplit(::(@x1, ::(@x2, @xs')))) , mergesort#3^#(tuple#2(@l1, @l2)) -> merge^#(mergesort(@l1), mergesort(@l2)) , mergesort#3^#(tuple#2(@l1, @l2)) -> mergesort^#(@l1) , mergesort#3^#(tuple#2(@l1, @l2)) -> mergesort^#(@l2) } Weak Trs: { #less(@x, @y) -> #cklt(#compare(@x, @y)) , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #s(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#pos(@x), #0()) -> #GT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , #cklt(#EQ()) -> #false() , #cklt(#GT()) -> #false() , #cklt(#LT()) -> #true() , merge(@l1, @l2) -> merge#1(@l1, @l2) , merge#1(::(@x, @xs), @l2) -> merge#2(@l2, @x, @xs) , merge#1(nil(), @l2) -> @l2 , merge#2(::(@y, @ys), @x, @xs) -> merge#3(#less(@x, @y), @x, @xs, @y, @ys) , merge#2(nil(), @x, @xs) -> ::(@x, @xs) , merge#3(#false(), @x, @xs, @y, @ys) -> ::(@y, merge(::(@x, @xs), @ys)) , merge#3(#true(), @x, @xs, @y, @ys) -> ::(@x, merge(@xs, ::(@y, @ys))) , mergesort(@l) -> mergesort#1(@l) , mergesort#1(::(@x1, @xs)) -> mergesort#2(@xs, @x1) , mergesort#1(nil()) -> nil() , mergesort#2(::(@x2, @xs'), @x1) -> mergesort#3(msplit(::(@x1, ::(@x2, @xs')))) , mergesort#2(nil(), @x1) -> ::(@x1, nil()) , msplit(@l) -> msplit#1(@l) , mergesort#3(tuple#2(@l1, @l2)) -> merge(mergesort(@l1), mergesort(@l2)) , msplit#1(::(@x1, @xs)) -> msplit#2(@xs, @x1) , msplit#1(nil()) -> tuple#2(nil(), nil()) , msplit#2(::(@x2, @xs'), @x1) -> msplit#3(msplit(@xs'), @x1, @x2) , msplit#2(nil(), @x1) -> tuple#2(::(@x1, nil()), nil()) , msplit#3(tuple#2(@l1, @l2), @x1, @x2) -> tuple#2(::(@x1, @l1), ::(@x2, @l2)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 2: merge#1^#(::(@x, @xs), @l2) -> c_2(merge#2^#(@l2, @x, @xs)) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_3) = {1}, Uargs(c_4) = {1}, Uargs(c_5) = {1} TcT has computed following constructor-based matrix interpretation satisfying not(EDA). [#less](x1, x2) = [1] [#compare](x1, x2) = [1] [#cklt](x1) = [1] x1 + [0] [merge](x1, x2) = [1] x1 + [1] x2 + [0] [merge#1](x1, x2) = [1] x1 + [1] x2 + [0] [::](x1, x2) = [1] x2 + [1] [merge#2](x1, x2, x3) = [1] x1 + [1] x3 + [1] [nil] = [0] [merge#3](x1, x2, x3, x4, x5) = [1] x3 + [1] x5 + [2] [#false] = [1] [#true] = [1] [mergesort](x1) = [1] x1 + [0] [mergesort#1](x1) = [1] x1 + [0] [mergesort#2](x1, x2) = [1] x1 + [1] [msplit](x1) = [1] x1 + [0] [mergesort#3](x1) = [1] x1 + [0] [tuple#2](x1, x2) = [1] x1 + [1] x2 + [0] [msplit#1](x1) = [1] x1 + [0] [msplit#2](x1, x2) = [1] x1 + [1] [msplit#3](x1, x2, x3) = [1] x1 + [2] [#EQ] = [1] [#GT] = [1] [#LT] = [1] [#0] = [1] [#neg](x1) = [1] x1 + [0] [#pos](x1) = [1] x1 + [0] [#s](x1) = [1] x1 + [0] [#less^#](x1, x2) = [0] [#cklt^#](x1) = [0] [#compare^#](x1, x2) = [0] [merge^#](x1, x2) = [1] x1 + [2] x2 + [0] [merge#1^#](x1, x2) = [1] x1 + [2] x2 + [0] [merge#2^#](x1, x2, x3) = [2] x1 + [1] x3 + [0] [merge#3^#](x1, x2, x3, x4, x5) = [1] x1 + [1] x3 + [2] x5 + [1] [mergesort^#](x1) = [2] x1 + [0] [mergesort#1^#](x1) = [2] x1 + [0] [mergesort#2^#](x1, x2) = [2] x1 + [2] [mergesort#3^#](x1) = [2] x1 + [0] [msplit^#](x1) = [0] [msplit#1^#](x1) = [0] [msplit#2^#](x1, x2) = [0] [msplit#3^#](x1, x2, x3) = [0] [c_1](x1) = [1] x1 + [0] [c_2](x1) = [1] x1 + [0] [c_3](x1) = [1] x1 + [0] [c_4](x1) = [1] x1 + [1] [c_5](x1) = [1] x1 + [0] [c_10](x1) = [0] [c_11](x1) = [0] [c_12](x1) = [0] This order satisfies following ordering constraints [#less(@x, @y)] = [1] >= [1] = [#cklt(#compare(@x, @y))] [#compare(#0(), #0())] = [1] >= [1] = [#EQ()] [#compare(#0(), #neg(@y))] = [1] >= [1] = [#GT()] [#compare(#0(), #pos(@y))] = [1] >= [1] = [#LT()] [#compare(#0(), #s(@y))] = [1] >= [1] = [#LT()] [#compare(#neg(@x), #0())] = [1] >= [1] = [#LT()] [#compare(#neg(@x), #neg(@y))] = [1] >= [1] = [#compare(@y, @x)] [#compare(#neg(@x), #pos(@y))] = [1] >= [1] = [#LT()] [#compare(#pos(@x), #0())] = [1] >= [1] = [#GT()] [#compare(#pos(@x), #neg(@y))] = [1] >= [1] = [#GT()] [#compare(#pos(@x), #pos(@y))] = [1] >= [1] = [#compare(@x, @y)] [#compare(#s(@x), #0())] = [1] >= [1] = [#GT()] [#compare(#s(@x), #s(@y))] = [1] >= [1] = [#compare(@x, @y)] [#cklt(#EQ())] = [1] >= [1] = [#false()] [#cklt(#GT())] = [1] >= [1] = [#false()] [#cklt(#LT())] = [1] >= [1] = [#true()] [merge(@l1, @l2)] = [1] @l1 + [1] @l2 + [0] >= [1] @l1 + [1] @l2 + [0] = [merge#1(@l1, @l2)] [merge#1(::(@x, @xs), @l2)] = [1] @l2 + [1] @xs + [1] >= [1] @l2 + [1] @xs + [1] = [merge#2(@l2, @x, @xs)] [merge#1(nil(), @l2)] = [1] @l2 + [0] >= [1] @l2 + [0] = [@l2] [merge#2(::(@y, @ys), @x, @xs)] = [1] @xs + [1] @ys + [2] >= [1] @xs + [1] @ys + [2] = [merge#3(#less(@x, @y), @x, @xs, @y, @ys)] [merge#2(nil(), @x, @xs)] = [1] @xs + [1] >= [1] @xs + [1] = [::(@x, @xs)] [merge#3(#false(), @x, @xs, @y, @ys)] = [1] @xs + [1] @ys + [2] >= [1] @xs + [1] @ys + [2] = [::(@y, merge(::(@x, @xs), @ys))] [merge#3(#true(), @x, @xs, @y, @ys)] = [1] @xs + [1] @ys + [2] >= [1] @xs + [1] @ys + [2] = [::(@x, merge(@xs, ::(@y, @ys)))] [mergesort(@l)] = [1] @l + [0] >= [1] @l + [0] = [mergesort#1(@l)] [mergesort#1(::(@x1, @xs))] = [1] @xs + [1] >= [1] @xs + [1] = [mergesort#2(@xs, @x1)] [mergesort#1(nil())] = [0] >= [0] = [nil()] [mergesort#2(::(@x2, @xs'), @x1)] = [1] @xs' + [2] >= [1] @xs' + [2] = [mergesort#3(msplit(::(@x1, ::(@x2, @xs'))))] [mergesort#2(nil(), @x1)] = [1] >= [1] = [::(@x1, nil())] [msplit(@l)] = [1] @l + [0] >= [1] @l + [0] = [msplit#1(@l)] [mergesort#3(tuple#2(@l1, @l2))] = [1] @l1 + [1] @l2 + [0] >= [1] @l1 + [1] @l2 + [0] = [merge(mergesort(@l1), mergesort(@l2))] [msplit#1(::(@x1, @xs))] = [1] @xs + [1] >= [1] @xs + [1] = [msplit#2(@xs, @x1)] [msplit#1(nil())] = [0] >= [0] = [tuple#2(nil(), nil())] [msplit#2(::(@x2, @xs'), @x1)] = [1] @xs' + [2] >= [1] @xs' + [2] = [msplit#3(msplit(@xs'), @x1, @x2)] [msplit#2(nil(), @x1)] = [1] >= [1] = [tuple#2(::(@x1, nil()), nil())] [msplit#3(tuple#2(@l1, @l2), @x1, @x2)] = [1] @l1 + [1] @l2 + [2] >= [1] @l1 + [1] @l2 + [2] = [tuple#2(::(@x1, @l1), ::(@x2, @l2))] [merge^#(@l1, @l2)] = [1] @l1 + [2] @l2 + [0] >= [1] @l1 + [2] @l2 + [0] = [c_1(merge#1^#(@l1, @l2))] [merge#1^#(::(@x, @xs), @l2)] = [2] @l2 + [1] @xs + [1] > [2] @l2 + [1] @xs + [0] = [c_2(merge#2^#(@l2, @x, @xs))] [merge#2^#(::(@y, @ys), @x, @xs)] = [1] @xs + [2] @ys + [2] >= [1] @xs + [2] @ys + [2] = [c_3(merge#3^#(#less(@x, @y), @x, @xs, @y, @ys))] [merge#3^#(#false(), @x, @xs, @y, @ys)] = [1] @xs + [2] @ys + [2] >= [1] @xs + [2] @ys + [2] = [c_4(merge^#(::(@x, @xs), @ys))] [merge#3^#(#true(), @x, @xs, @y, @ys)] = [1] @xs + [2] @ys + [2] >= [1] @xs + [2] @ys + [2] = [c_5(merge^#(@xs, ::(@y, @ys)))] [mergesort^#(@l)] = [2] @l + [0] >= [2] @l + [0] = [mergesort#1^#(@l)] [mergesort#1^#(::(@x1, @xs))] = [2] @xs + [2] >= [2] @xs + [2] = [mergesort#2^#(@xs, @x1)] [mergesort#2^#(::(@x2, @xs'), @x1)] = [2] @xs' + [4] >= [2] @xs' + [4] = [mergesort#3^#(msplit(::(@x1, ::(@x2, @xs'))))] [mergesort#3^#(tuple#2(@l1, @l2))] = [2] @l1 + [2] @l2 + [0] >= [1] @l1 + [2] @l2 + [0] = [merge^#(mergesort(@l1), mergesort(@l2))] [mergesort#3^#(tuple#2(@l1, @l2))] = [2] @l1 + [2] @l2 + [0] >= [2] @l1 + [0] = [mergesort^#(@l1)] [mergesort#3^#(tuple#2(@l1, @l2))] = [2] @l1 + [2] @l2 + [0] >= [2] @l2 + [0] = [mergesort^#(@l2)] Consider the set of all dependency pairs DPs: { 1: merge^#(@l1, @l2) -> c_1(merge#1^#(@l1, @l2)) , 2: merge#1^#(::(@x, @xs), @l2) -> c_2(merge#2^#(@l2, @x, @xs)) , 3: merge#2^#(::(@y, @ys), @x, @xs) -> c_3(merge#3^#(#less(@x, @y), @x, @xs, @y, @ys)) , 4: merge#3^#(#false(), @x, @xs, @y, @ys) -> c_4(merge^#(::(@x, @xs), @ys)) , 5: merge#3^#(#true(), @x, @xs, @y, @ys) -> c_5(merge^#(@xs, ::(@y, @ys))) , 6: mergesort^#(@l) -> mergesort#1^#(@l) , 7: mergesort#1^#(::(@x1, @xs)) -> mergesort#2^#(@xs, @x1) , 8: mergesort#2^#(::(@x2, @xs'), @x1) -> mergesort#3^#(msplit(::(@x1, ::(@x2, @xs')))) , 9: mergesort#3^#(tuple#2(@l1, @l2)) -> merge^#(mergesort(@l1), mergesort(@l2)) , 10: mergesort#3^#(tuple#2(@l1, @l2)) -> mergesort^#(@l1) , 11: mergesort#3^#(tuple#2(@l1, @l2)) -> mergesort^#(@l2) } Processor 'matrix interpretation of dimension 1' induces the complexity certificate YES(?,O(n^1)) on application of dependency pairs {2}. These cover all (indirect) predecessors of dependency pairs {2,3,4,5}, their number of application is equally bounded. The dependency pairs are shifted into the corresponding weak component(s). We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { merge^#(@l1, @l2) -> c_1(merge#1^#(@l1, @l2)) } Weak DPs: { merge#1^#(::(@x, @xs), @l2) -> c_2(merge#2^#(@l2, @x, @xs)) , merge#2^#(::(@y, @ys), @x, @xs) -> c_3(merge#3^#(#less(@x, @y), @x, @xs, @y, @ys)) , merge#3^#(#false(), @x, @xs, @y, @ys) -> c_4(merge^#(::(@x, @xs), @ys)) , merge#3^#(#true(), @x, @xs, @y, @ys) -> c_5(merge^#(@xs, ::(@y, @ys))) , mergesort^#(@l) -> mergesort#1^#(@l) , mergesort#1^#(::(@x1, @xs)) -> mergesort#2^#(@xs, @x1) , mergesort#2^#(::(@x2, @xs'), @x1) -> mergesort#3^#(msplit(::(@x1, ::(@x2, @xs')))) , mergesort#3^#(tuple#2(@l1, @l2)) -> merge^#(mergesort(@l1), mergesort(@l2)) , mergesort#3^#(tuple#2(@l1, @l2)) -> mergesort^#(@l1) , mergesort#3^#(tuple#2(@l1, @l2)) -> mergesort^#(@l2) } Weak Trs: { #less(@x, @y) -> #cklt(#compare(@x, @y)) , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #s(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#pos(@x), #0()) -> #GT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , #cklt(#EQ()) -> #false() , #cklt(#GT()) -> #false() , #cklt(#LT()) -> #true() , merge(@l1, @l2) -> merge#1(@l1, @l2) , merge#1(::(@x, @xs), @l2) -> merge#2(@l2, @x, @xs) , merge#1(nil(), @l2) -> @l2 , merge#2(::(@y, @ys), @x, @xs) -> merge#3(#less(@x, @y), @x, @xs, @y, @ys) , merge#2(nil(), @x, @xs) -> ::(@x, @xs) , merge#3(#false(), @x, @xs, @y, @ys) -> ::(@y, merge(::(@x, @xs), @ys)) , merge#3(#true(), @x, @xs, @y, @ys) -> ::(@x, merge(@xs, ::(@y, @ys))) , mergesort(@l) -> mergesort#1(@l) , mergesort#1(::(@x1, @xs)) -> mergesort#2(@xs, @x1) , mergesort#1(nil()) -> nil() , mergesort#2(::(@x2, @xs'), @x1) -> mergesort#3(msplit(::(@x1, ::(@x2, @xs')))) , mergesort#2(nil(), @x1) -> ::(@x1, nil()) , msplit(@l) -> msplit#1(@l) , mergesort#3(tuple#2(@l1, @l2)) -> merge(mergesort(@l1), mergesort(@l2)) , msplit#1(::(@x1, @xs)) -> msplit#2(@xs, @x1) , msplit#1(nil()) -> tuple#2(nil(), nil()) , msplit#2(::(@x2, @xs'), @x1) -> msplit#3(msplit(@xs'), @x1, @x2) , msplit#2(nil(), @x1) -> tuple#2(::(@x1, nil()), nil()) , msplit#3(tuple#2(@l1, @l2), @x1, @x2) -> tuple#2(::(@x1, @l1), ::(@x2, @l2)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 2: merge#1^#(::(@x, @xs), @l2) -> c_2(merge#2^#(@l2, @x, @xs)) , 9: mergesort#3^#(tuple#2(@l1, @l2)) -> merge^#(mergesort(@l1), mergesort(@l2)) } Trs: { #cklt(#LT()) -> #true() } Sub-proof: ---------- The following argument positions are usable: Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_3) = {1}, Uargs(c_4) = {1}, Uargs(c_5) = {1} TcT has computed following constructor-based matrix interpretation satisfying not(EDA). [#less](x1, x2) = [1] [#compare](x1, x2) = [0] [#cklt](x1) = [1] x1 + [1] [merge](x1, x2) = [1] x1 + [1] x2 + [0] [merge#1](x1, x2) = [1] x1 + [1] x2 + [0] [::](x1, x2) = [1] x2 + [1] [merge#2](x1, x2, x3) = [1] x1 + [1] x3 + [1] [nil] = [0] [merge#3](x1, x2, x3, x4, x5) = [1] x3 + [1] x5 + [2] [#false] = [1] [#true] = [0] [mergesort](x1) = [1] x1 + [0] [mergesort#1](x1) = [1] x1 + [0] [mergesort#2](x1, x2) = [1] x1 + [1] [msplit](x1) = [1] x1 + [0] [mergesort#3](x1) = [1] x1 + [0] [tuple#2](x1, x2) = [1] x1 + [1] x2 + [0] [msplit#1](x1) = [1] x1 + [0] [msplit#2](x1, x2) = [1] x1 + [1] [msplit#3](x1, x2, x3) = [1] x1 + [2] [#EQ] = [0] [#GT] = [0] [#LT] = [0] [#0] = [0] [#neg](x1) = [1] x1 + [0] [#pos](x1) = [1] x1 + [0] [#s](x1) = [1] x1 + [1] [#less^#](x1, x2) = [0] [#cklt^#](x1) = [0] [#compare^#](x1, x2) = [0] [merge^#](x1, x2) = [1] x1 + [1] x2 + [0] [merge#1^#](x1, x2) = [1] x1 + [1] x2 + [0] [merge#2^#](x1, x2, x3) = [1] x1 + [1] x3 + [0] [merge#3^#](x1, x2, x3, x4, x5) = [1] x3 + [1] x5 + [1] [mergesort^#](x1) = [1] x1 + [1] [mergesort#1^#](x1) = [1] x1 + [1] [mergesort#2^#](x1, x2) = [1] x1 + [2] [mergesort#3^#](x1) = [1] x1 + [1] [msplit^#](x1) = [0] [msplit#1^#](x1) = [0] [msplit#2^#](x1, x2) = [0] [msplit#3^#](x1, x2, x3) = [0] [c_1](x1) = [1] x1 + [0] [c_2](x1) = [1] x1 + [0] [c_3](x1) = [1] x1 + [0] [c_4](x1) = [1] x1 + [0] [c_5](x1) = [1] x1 + [0] [c_10](x1) = [0] [c_11](x1) = [0] [c_12](x1) = [0] This order satisfies following ordering constraints [#compare(#0(), #0())] = [0] >= [0] = [#EQ()] [#compare(#0(), #neg(@y))] = [0] >= [0] = [#GT()] [#compare(#0(), #pos(@y))] = [0] >= [0] = [#LT()] [#compare(#0(), #s(@y))] = [0] >= [0] = [#LT()] [#compare(#neg(@x), #0())] = [0] >= [0] = [#LT()] [#compare(#neg(@x), #neg(@y))] = [0] >= [0] = [#compare(@y, @x)] [#compare(#neg(@x), #pos(@y))] = [0] >= [0] = [#LT()] [#compare(#pos(@x), #0())] = [0] >= [0] = [#GT()] [#compare(#pos(@x), #neg(@y))] = [0] >= [0] = [#GT()] [#compare(#pos(@x), #pos(@y))] = [0] >= [0] = [#compare(@x, @y)] [#compare(#s(@x), #0())] = [0] >= [0] = [#GT()] [#compare(#s(@x), #s(@y))] = [0] >= [0] = [#compare(@x, @y)] [merge(@l1, @l2)] = [1] @l1 + [1] @l2 + [0] >= [1] @l1 + [1] @l2 + [0] = [merge#1(@l1, @l2)] [merge#1(::(@x, @xs), @l2)] = [1] @l2 + [1] @xs + [1] >= [1] @l2 + [1] @xs + [1] = [merge#2(@l2, @x, @xs)] [merge#1(nil(), @l2)] = [1] @l2 + [0] >= [1] @l2 + [0] = [@l2] [merge#2(::(@y, @ys), @x, @xs)] = [1] @xs + [1] @ys + [2] >= [1] @xs + [1] @ys + [2] = [merge#3(#less(@x, @y), @x, @xs, @y, @ys)] [merge#2(nil(), @x, @xs)] = [1] @xs + [1] >= [1] @xs + [1] = [::(@x, @xs)] [merge#3(#false(), @x, @xs, @y, @ys)] = [1] @xs + [1] @ys + [2] >= [1] @xs + [1] @ys + [2] = [::(@y, merge(::(@x, @xs), @ys))] [merge#3(#true(), @x, @xs, @y, @ys)] = [1] @xs + [1] @ys + [2] >= [1] @xs + [1] @ys + [2] = [::(@x, merge(@xs, ::(@y, @ys)))] [mergesort(@l)] = [1] @l + [0] >= [1] @l + [0] = [mergesort#1(@l)] [mergesort#1(::(@x1, @xs))] = [1] @xs + [1] >= [1] @xs + [1] = [mergesort#2(@xs, @x1)] [mergesort#1(nil())] = [0] >= [0] = [nil()] [mergesort#2(::(@x2, @xs'), @x1)] = [1] @xs' + [2] >= [1] @xs' + [2] = [mergesort#3(msplit(::(@x1, ::(@x2, @xs'))))] [mergesort#2(nil(), @x1)] = [1] >= [1] = [::(@x1, nil())] [msplit(@l)] = [1] @l + [0] >= [1] @l + [0] = [msplit#1(@l)] [mergesort#3(tuple#2(@l1, @l2))] = [1] @l1 + [1] @l2 + [0] >= [1] @l1 + [1] @l2 + [0] = [merge(mergesort(@l1), mergesort(@l2))] [msplit#1(::(@x1, @xs))] = [1] @xs + [1] >= [1] @xs + [1] = [msplit#2(@xs, @x1)] [msplit#1(nil())] = [0] >= [0] = [tuple#2(nil(), nil())] [msplit#2(::(@x2, @xs'), @x1)] = [1] @xs' + [2] >= [1] @xs' + [2] = [msplit#3(msplit(@xs'), @x1, @x2)] [msplit#2(nil(), @x1)] = [1] >= [1] = [tuple#2(::(@x1, nil()), nil())] [msplit#3(tuple#2(@l1, @l2), @x1, @x2)] = [1] @l1 + [1] @l2 + [2] >= [1] @l1 + [1] @l2 + [2] = [tuple#2(::(@x1, @l1), ::(@x2, @l2))] [merge^#(@l1, @l2)] = [1] @l1 + [1] @l2 + [0] >= [1] @l1 + [1] @l2 + [0] = [c_1(merge#1^#(@l1, @l2))] [merge#1^#(::(@x, @xs), @l2)] = [1] @l2 + [1] @xs + [1] > [1] @l2 + [1] @xs + [0] = [c_2(merge#2^#(@l2, @x, @xs))] [merge#2^#(::(@y, @ys), @x, @xs)] = [1] @xs + [1] @ys + [1] >= [1] @xs + [1] @ys + [1] = [c_3(merge#3^#(#less(@x, @y), @x, @xs, @y, @ys))] [merge#3^#(#false(), @x, @xs, @y, @ys)] = [1] @xs + [1] @ys + [1] >= [1] @xs + [1] @ys + [1] = [c_4(merge^#(::(@x, @xs), @ys))] [merge#3^#(#true(), @x, @xs, @y, @ys)] = [1] @xs + [1] @ys + [1] >= [1] @xs + [1] @ys + [1] = [c_5(merge^#(@xs, ::(@y, @ys)))] [mergesort^#(@l)] = [1] @l + [1] >= [1] @l + [1] = [mergesort#1^#(@l)] [mergesort#1^#(::(@x1, @xs))] = [1] @xs + [2] >= [1] @xs + [2] = [mergesort#2^#(@xs, @x1)] [mergesort#2^#(::(@x2, @xs'), @x1)] = [1] @xs' + [3] >= [1] @xs' + [3] = [mergesort#3^#(msplit(::(@x1, ::(@x2, @xs'))))] [mergesort#3^#(tuple#2(@l1, @l2))] = [1] @l1 + [1] @l2 + [1] > [1] @l1 + [1] @l2 + [0] = [merge^#(mergesort(@l1), mergesort(@l2))] [mergesort#3^#(tuple#2(@l1, @l2))] = [1] @l1 + [1] @l2 + [1] >= [1] @l1 + [1] = [mergesort^#(@l1)] [mergesort#3^#(tuple#2(@l1, @l2))] = [1] @l1 + [1] @l2 + [1] >= [1] @l2 + [1] = [mergesort^#(@l2)] Consider the set of all dependency pairs DPs: { 1: merge^#(@l1, @l2) -> c_1(merge#1^#(@l1, @l2)) , 2: merge#1^#(::(@x, @xs), @l2) -> c_2(merge#2^#(@l2, @x, @xs)) , 3: merge#2^#(::(@y, @ys), @x, @xs) -> c_3(merge#3^#(#less(@x, @y), @x, @xs, @y, @ys)) , 4: merge#3^#(#false(), @x, @xs, @y, @ys) -> c_4(merge^#(::(@x, @xs), @ys)) , 5: merge#3^#(#true(), @x, @xs, @y, @ys) -> c_5(merge^#(@xs, ::(@y, @ys))) , 6: mergesort^#(@l) -> mergesort#1^#(@l) , 7: mergesort#1^#(::(@x1, @xs)) -> mergesort#2^#(@xs, @x1) , 8: mergesort#2^#(::(@x2, @xs'), @x1) -> mergesort#3^#(msplit(::(@x1, ::(@x2, @xs')))) , 9: mergesort#3^#(tuple#2(@l1, @l2)) -> merge^#(mergesort(@l1), mergesort(@l2)) , 10: mergesort#3^#(tuple#2(@l1, @l2)) -> mergesort^#(@l1) , 11: mergesort#3^#(tuple#2(@l1, @l2)) -> mergesort^#(@l2) } Processor 'matrix interpretation of dimension 1' induces the complexity certificate YES(?,O(n^1)) on application of dependency pairs {2,9}. These cover all (indirect) predecessors of dependency pairs {1,2,3,4,5,9}, their number of application is equally bounded. The dependency pairs are shifted into the corresponding weak component(s). We apply the transformation 'removetails' on the sub-problem: Weak DPs: { merge^#(@l1, @l2) -> c_1(merge#1^#(@l1, @l2)) , merge#1^#(::(@x, @xs), @l2) -> c_2(merge#2^#(@l2, @x, @xs)) , merge#2^#(::(@y, @ys), @x, @xs) -> c_3(merge#3^#(#less(@x, @y), @x, @xs, @y, @ys)) , merge#3^#(#false(), @x, @xs, @y, @ys) -> c_4(merge^#(::(@x, @xs), @ys)) , merge#3^#(#true(), @x, @xs, @y, @ys) -> c_5(merge^#(@xs, ::(@y, @ys))) , mergesort^#(@l) -> mergesort#1^#(@l) , mergesort#1^#(::(@x1, @xs)) -> mergesort#2^#(@xs, @x1) , mergesort#2^#(::(@x2, @xs'), @x1) -> mergesort#3^#(msplit(::(@x1, ::(@x2, @xs')))) , mergesort#3^#(tuple#2(@l1, @l2)) -> merge^#(mergesort(@l1), mergesort(@l2)) , mergesort#3^#(tuple#2(@l1, @l2)) -> mergesort^#(@l1) , mergesort#3^#(tuple#2(@l1, @l2)) -> mergesort^#(@l2) } Weak Trs: { #less(@x, @y) -> #cklt(#compare(@x, @y)) , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #s(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#pos(@x), #0()) -> #GT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , #cklt(#EQ()) -> #false() , #cklt(#GT()) -> #false() , #cklt(#LT()) -> #true() , merge(@l1, @l2) -> merge#1(@l1, @l2) , merge#1(::(@x, @xs), @l2) -> merge#2(@l2, @x, @xs) , merge#1(nil(), @l2) -> @l2 , merge#2(::(@y, @ys), @x, @xs) -> merge#3(#less(@x, @y), @x, @xs, @y, @ys) , merge#2(nil(), @x, @xs) -> ::(@x, @xs) , merge#3(#false(), @x, @xs, @y, @ys) -> ::(@y, merge(::(@x, @xs), @ys)) , merge#3(#true(), @x, @xs, @y, @ys) -> ::(@x, merge(@xs, ::(@y, @ys))) , mergesort(@l) -> mergesort#1(@l) , mergesort#1(::(@x1, @xs)) -> mergesort#2(@xs, @x1) , mergesort#1(nil()) -> nil() , mergesort#2(::(@x2, @xs'), @x1) -> mergesort#3(msplit(::(@x1, ::(@x2, @xs')))) , mergesort#2(nil(), @x1) -> ::(@x1, nil()) , msplit(@l) -> msplit#1(@l) , mergesort#3(tuple#2(@l1, @l2)) -> merge(mergesort(@l1), mergesort(@l2)) , msplit#1(::(@x1, @xs)) -> msplit#2(@xs, @x1) , msplit#1(nil()) -> tuple#2(nil(), nil()) , msplit#2(::(@x2, @xs'), @x1) -> msplit#3(msplit(@xs'), @x1, @x2) , msplit#2(nil(), @x1) -> tuple#2(::(@x1, nil()), nil()) , msplit#3(tuple#2(@l1, @l2), @x1, @x2) -> tuple#2(::(@x1, @l1), ::(@x2, @l2)) } StartTerms: basic terms Strategy: innermost The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { merge^#(@l1, @l2) -> c_1(merge#1^#(@l1, @l2)) , merge#1^#(::(@x, @xs), @l2) -> c_2(merge#2^#(@l2, @x, @xs)) , merge#2^#(::(@y, @ys), @x, @xs) -> c_3(merge#3^#(#less(@x, @y), @x, @xs, @y, @ys)) , merge#3^#(#false(), @x, @xs, @y, @ys) -> c_4(merge^#(::(@x, @xs), @ys)) , merge#3^#(#true(), @x, @xs, @y, @ys) -> c_5(merge^#(@xs, ::(@y, @ys))) , mergesort^#(@l) -> mergesort#1^#(@l) , mergesort#1^#(::(@x1, @xs)) -> mergesort#2^#(@xs, @x1) , mergesort#2^#(::(@x2, @xs'), @x1) -> mergesort#3^#(msplit(::(@x1, ::(@x2, @xs')))) , mergesort#3^#(tuple#2(@l1, @l2)) -> merge^#(mergesort(@l1), mergesort(@l2)) , mergesort#3^#(tuple#2(@l1, @l2)) -> mergesort^#(@l1) , mergesort#3^#(tuple#2(@l1, @l2)) -> mergesort^#(@l2) } We apply the transformation 'usablerules' on the sub-problem: Weak Trs: { #less(@x, @y) -> #cklt(#compare(@x, @y)) , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #s(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#pos(@x), #0()) -> #GT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , #cklt(#EQ()) -> #false() , #cklt(#GT()) -> #false() , #cklt(#LT()) -> #true() , merge(@l1, @l2) -> merge#1(@l1, @l2) , merge#1(::(@x, @xs), @l2) -> merge#2(@l2, @x, @xs) , merge#1(nil(), @l2) -> @l2 , merge#2(::(@y, @ys), @x, @xs) -> merge#3(#less(@x, @y), @x, @xs, @y, @ys) , merge#2(nil(), @x, @xs) -> ::(@x, @xs) , merge#3(#false(), @x, @xs, @y, @ys) -> ::(@y, merge(::(@x, @xs), @ys)) , merge#3(#true(), @x, @xs, @y, @ys) -> ::(@x, merge(@xs, ::(@y, @ys))) , mergesort(@l) -> mergesort#1(@l) , mergesort#1(::(@x1, @xs)) -> mergesort#2(@xs, @x1) , mergesort#1(nil()) -> nil() , mergesort#2(::(@x2, @xs'), @x1) -> mergesort#3(msplit(::(@x1, ::(@x2, @xs')))) , mergesort#2(nil(), @x1) -> ::(@x1, nil()) , msplit(@l) -> msplit#1(@l) , mergesort#3(tuple#2(@l1, @l2)) -> merge(mergesort(@l1), mergesort(@l2)) , msplit#1(::(@x1, @xs)) -> msplit#2(@xs, @x1) , msplit#1(nil()) -> tuple#2(nil(), nil()) , msplit#2(::(@x2, @xs'), @x1) -> msplit#3(msplit(@xs'), @x1, @x2) , msplit#2(nil(), @x1) -> tuple#2(::(@x1, nil()), nil()) , msplit#3(tuple#2(@l1, @l2), @x1, @x2) -> tuple#2(::(@x1, @l1), ::(@x2, @l2)) } StartTerms: basic terms Strategy: innermost No rule is usable, rules are removed from the input problem. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Rules: Empty Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) Empty rules are trivially bounded Wall-time: 4.697835s CPU-time: 20.425s Wall-time: 28.713693s CPU-time: 93.799s Hurray, we answered YES(O(1),O(n^2))