YES(O(1),O(n^6)) 876.13/288.95 YES(O(1),O(n^6)) 876.13/288.95 876.13/288.95 We are left with following problem, upon which TcT provides the 876.13/288.95 certificate YES(O(1),O(n^6)). 876.13/288.95 876.13/288.95 Strict Trs: 876.13/288.95 { #equal(@x, @y) -> #eq(@x, @y) 876.13/288.95 , splitAndSort(@l) -> sortAll(split(@l)) 876.13/288.95 , sortAll#2(tuple#2(@vals, @key), @xs) -> 876.13/288.95 ::(tuple#2(quicksort(@vals), @key), sortAll(@xs)) 876.13/288.95 , quicksort#2(tuple#2(@xs, @ys), @z) -> 876.13/288.95 append(quicksort(@xs), ::(@z, quicksort(@ys))) 876.13/288.95 , split(@l) -> split#1(@l) 876.13/288.95 , append(@l, @ys) -> append#1(@l, @ys) 876.13/288.95 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.13/288.95 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.13/288.95 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.13/288.95 , sortAll(@l) -> sortAll#1(@l) 876.13/288.95 , insert#2(nil(), @keyX, @valX, @x) -> 876.13/288.95 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.13/288.95 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.13/288.95 insert#3(@l1, @keyX, @ls, @valX, @x) 876.13/288.95 , sortAll#1(nil()) -> nil() 876.13/288.95 , sortAll#1(::(@x, @xs)) -> sortAll#2(@x, @xs) 876.13/288.95 , split#1(nil()) -> nil() 876.13/288.95 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.13/288.95 , #greater(@x, @y) -> #ckgt(#compare(@x, @y)) 876.13/288.95 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.13/288.95 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.13/288.95 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.13/288.95 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.13/288.95 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.13/288.95 insert#2(@l, @keyX, @valX, @x) 876.13/288.95 , quicksort#1(nil()) -> nil() 876.13/288.95 , quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) 876.13/288.95 , append#1(nil(), @ys) -> @ys 876.13/288.95 , append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) 876.13/288.95 , splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) 876.13/288.95 , quicksort(@l) -> quicksort#1(@l) 876.13/288.95 , splitqs#3(#true(), @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) 876.13/288.95 , splitqs#3(#false(), @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) 876.13/288.95 , splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> 876.13/288.95 splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) 876.13/288.95 , splitqs#1(nil(), @pivot) -> tuple#2(nil(), nil()) 876.13/288.95 , splitqs#1(::(@x, @xs), @pivot) -> 876.13/288.95 splitqs#2(splitqs(@pivot, @xs), @pivot, @x) } 876.13/288.95 Weak Trs: 876.13/288.95 { #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.13/288.95 , #eq(#pos(@x), #0()) -> #false() 876.13/288.95 , #eq(#pos(@x), #neg(@y)) -> #false() 876.13/288.95 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.13/288.95 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.13/288.95 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.13/288.95 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.13/288.95 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.13/288.95 , #eq(nil(), nil()) -> #true() 876.13/288.95 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.13/288.95 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.13/288.95 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.13/288.95 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.13/288.95 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.13/288.95 , #eq(#0(), #pos(@y)) -> #false() 876.13/288.95 , #eq(#0(), #0()) -> #true() 876.13/288.95 , #eq(#0(), #neg(@y)) -> #false() 876.13/288.95 , #eq(#0(), #s(@y)) -> #false() 876.13/288.95 , #eq(#neg(@x), #pos(@y)) -> #false() 876.13/288.95 , #eq(#neg(@x), #0()) -> #false() 876.13/288.95 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.13/288.95 , #eq(#s(@x), #0()) -> #false() 876.13/288.95 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.13/288.95 , #ckgt(#EQ()) -> #false() 876.13/288.95 , #ckgt(#LT()) -> #false() 876.13/288.95 , #ckgt(#GT()) -> #true() 876.13/288.95 , #and(#true(), #true()) -> #true() 876.13/288.95 , #and(#true(), #false()) -> #false() 876.13/288.95 , #and(#false(), #true()) -> #false() 876.13/288.95 , #and(#false(), #false()) -> #false() 876.13/288.95 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 876.13/288.95 , #compare(#pos(@x), #0()) -> #GT() 876.13/288.95 , #compare(#pos(@x), #neg(@y)) -> #GT() 876.13/288.95 , #compare(#0(), #pos(@y)) -> #LT() 876.13/288.95 , #compare(#0(), #0()) -> #EQ() 876.13/288.95 , #compare(#0(), #neg(@y)) -> #GT() 876.13/288.95 , #compare(#0(), #s(@y)) -> #LT() 876.13/288.95 , #compare(#neg(@x), #pos(@y)) -> #LT() 876.13/288.95 , #compare(#neg(@x), #0()) -> #LT() 876.13/288.95 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 876.13/288.95 , #compare(#s(@x), #0()) -> #GT() 876.13/288.95 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) } 876.13/288.95 Obligation: 876.13/288.95 innermost runtime complexity 876.13/288.95 Answer: 876.13/288.95 YES(O(1),O(n^6)) 876.13/288.95 876.13/288.95 We add the following dependency tuples: 876.13/288.95 876.13/288.95 Strict DPs: 876.13/288.95 { #equal^#(@x, @y) -> c_1(#eq^#(@x, @y)) 876.13/288.95 , splitAndSort^#(@l) -> c_2(sortAll^#(split(@l)), split^#(@l)) 876.13/288.95 , sortAll^#(@l) -> c_9(sortAll#1^#(@l)) 876.13/288.95 , split^#(@l) -> c_5(split#1^#(@l)) 876.13/288.95 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> 876.13/288.95 c_3(quicksort^#(@vals), sortAll^#(@xs)) 876.13/288.95 , quicksort^#(@l) -> c_25(quicksort#1^#(@l)) 876.13/288.95 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> 876.13/288.95 c_4(append^#(quicksort(@xs), ::(@z, quicksort(@ys))), 876.13/288.95 quicksort^#(@xs), 876.13/288.95 quicksort^#(@ys)) 876.13/288.95 , append^#(@l, @ys) -> c_6(append#1^#(@l, @ys)) 876.13/288.95 , split#1^#(nil()) -> c_14() 876.13/288.95 , split#1^#(::(@x, @xs)) -> 876.13/288.95 c_15(insert^#(@x, split(@xs)), split^#(@xs)) 876.13/288.95 , append#1^#(nil(), @ys) -> c_22() 876.13/288.95 , append#1^#(::(@x, @xs), @ys) -> c_23(append^#(@xs, @ys)) 876.13/288.95 , insert#3^#(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.13/288.95 c_7(insert#4^#(#equal(@key1, @keyX), 876.13/288.95 @key1, 876.13/288.95 @ls, 876.13/288.95 @valX, 876.13/288.95 @vals1, 876.13/288.95 @x), 876.13/288.95 #equal^#(@key1, @keyX)) 876.13/288.95 , insert#4^#(#true(), @key1, @ls, @valX, @vals1, @x) -> c_17() 876.13/288.95 , insert#4^#(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.13/288.95 c_18(insert^#(@x, @ls)) 876.13/288.95 , insert^#(@x, @l) -> c_8(insert#1^#(@x, @l, @x)) 876.13/288.95 , insert#1^#(tuple#2(@valX, @keyX), @l, @x) -> 876.13/288.95 c_19(insert#2^#(@l, @keyX, @valX, @x)) 876.13/288.95 , sortAll#1^#(nil()) -> c_12() 876.13/288.95 , sortAll#1^#(::(@x, @xs)) -> c_13(sortAll#2^#(@x, @xs)) 876.13/288.95 , insert#2^#(nil(), @keyX, @valX, @x) -> c_10() 876.13/288.95 , insert#2^#(::(@l1, @ls), @keyX, @valX, @x) -> 876.13/288.95 c_11(insert#3^#(@l1, @keyX, @ls, @valX, @x)) 876.13/288.95 , #greater^#(@x, @y) -> 876.13/288.95 c_16(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y)) 876.13/288.95 , quicksort#1^#(nil()) -> c_20() 876.13/288.95 , quicksort#1^#(::(@z, @zs)) -> 876.13/288.95 c_21(quicksort#2^#(splitqs(@z, @zs), @z), splitqs^#(@z, @zs)) 876.13/288.95 , splitqs^#(@pivot, @l) -> c_24(splitqs#1^#(@l, @pivot)) 876.13/288.95 , splitqs#1^#(nil(), @pivot) -> c_29() 876.13/288.95 , splitqs#1^#(::(@x, @xs), @pivot) -> 876.13/288.95 c_30(splitqs#2^#(splitqs(@pivot, @xs), @pivot, @x), 876.13/288.95 splitqs^#(@pivot, @xs)) 876.13/288.95 , splitqs#3^#(#true(), @ls, @rs, @x) -> c_26() 876.13/288.95 , splitqs#3^#(#false(), @ls, @rs, @x) -> c_27() 876.13/288.95 , splitqs#2^#(tuple#2(@ls, @rs), @pivot, @x) -> 876.13/288.95 c_28(splitqs#3^#(#greater(@x, @pivot), @ls, @rs, @x), 876.13/288.95 #greater^#(@x, @pivot)) } 876.13/288.95 Weak DPs: 876.13/288.95 { #eq^#(#pos(@x), #pos(@y)) -> c_31(#eq^#(@x, @y)) 876.13/288.95 , #eq^#(#pos(@x), #0()) -> c_32() 876.13/288.95 , #eq^#(#pos(@x), #neg(@y)) -> c_33() 876.13/288.95 , #eq^#(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.13/288.95 c_34(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), 876.13/288.95 #eq^#(@x_1, @y_1), 876.13/288.95 #eq^#(@x_2, @y_2)) 876.13/288.95 , #eq^#(tuple#2(@x_1, @x_2), nil()) -> c_35() 876.13/288.95 , #eq^#(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> c_36() 876.13/288.95 , #eq^#(nil(), tuple#2(@y_1, @y_2)) -> c_37() 876.13/288.95 , #eq^#(nil(), nil()) -> c_38() 876.13/288.95 , #eq^#(nil(), ::(@y_1, @y_2)) -> c_39() 876.13/288.95 , #eq^#(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> c_40() 876.13/288.95 , #eq^#(::(@x_1, @x_2), nil()) -> c_41() 876.13/288.95 , #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.13/288.95 c_42(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), 876.13/288.95 #eq^#(@x_1, @y_1), 876.13/288.95 #eq^#(@x_2, @y_2)) 876.13/288.95 , #eq^#(#0(), #pos(@y)) -> c_43() 876.13/288.95 , #eq^#(#0(), #0()) -> c_44() 876.13/288.95 , #eq^#(#0(), #neg(@y)) -> c_45() 876.13/288.95 , #eq^#(#0(), #s(@y)) -> c_46() 876.13/288.95 , #eq^#(#neg(@x), #pos(@y)) -> c_47() 876.13/288.95 , #eq^#(#neg(@x), #0()) -> c_48() 876.13/288.95 , #eq^#(#neg(@x), #neg(@y)) -> c_49(#eq^#(@x, @y)) 876.13/288.95 , #eq^#(#s(@x), #0()) -> c_50() 876.13/288.95 , #eq^#(#s(@x), #s(@y)) -> c_51(#eq^#(@x, @y)) 876.13/288.95 , #ckgt^#(#EQ()) -> c_52() 876.13/288.95 , #ckgt^#(#LT()) -> c_53() 876.13/288.95 , #ckgt^#(#GT()) -> c_54() 876.13/288.95 , #compare^#(#pos(@x), #pos(@y)) -> c_59(#compare^#(@x, @y)) 876.13/288.95 , #compare^#(#pos(@x), #0()) -> c_60() 876.13/288.95 , #compare^#(#pos(@x), #neg(@y)) -> c_61() 876.13/288.95 , #compare^#(#0(), #pos(@y)) -> c_62() 876.13/288.95 , #compare^#(#0(), #0()) -> c_63() 876.13/288.95 , #compare^#(#0(), #neg(@y)) -> c_64() 876.13/288.95 , #compare^#(#0(), #s(@y)) -> c_65() 876.13/288.95 , #compare^#(#neg(@x), #pos(@y)) -> c_66() 876.13/288.95 , #compare^#(#neg(@x), #0()) -> c_67() 876.13/288.95 , #compare^#(#neg(@x), #neg(@y)) -> c_68(#compare^#(@y, @x)) 876.13/288.95 , #compare^#(#s(@x), #0()) -> c_69() 876.13/288.95 , #compare^#(#s(@x), #s(@y)) -> c_70(#compare^#(@x, @y)) 876.13/288.95 , #and^#(#true(), #true()) -> c_55() 876.13/288.95 , #and^#(#true(), #false()) -> c_56() 876.13/288.95 , #and^#(#false(), #true()) -> c_57() 876.13/288.95 , #and^#(#false(), #false()) -> c_58() } 876.13/288.95 876.13/288.95 and mark the set of starting terms. 876.13/288.95 876.13/288.95 We are left with following problem, upon which TcT provides the 876.13/288.95 certificate YES(O(1),O(n^6)). 876.13/288.95 876.13/288.95 Strict DPs: 876.13/288.95 { #equal^#(@x, @y) -> c_1(#eq^#(@x, @y)) 876.13/288.95 , splitAndSort^#(@l) -> c_2(sortAll^#(split(@l)), split^#(@l)) 876.13/288.95 , sortAll^#(@l) -> c_9(sortAll#1^#(@l)) 876.13/288.95 , split^#(@l) -> c_5(split#1^#(@l)) 876.13/288.95 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> 876.13/288.95 c_3(quicksort^#(@vals), sortAll^#(@xs)) 876.13/288.95 , quicksort^#(@l) -> c_25(quicksort#1^#(@l)) 876.13/288.95 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> 876.13/288.95 c_4(append^#(quicksort(@xs), ::(@z, quicksort(@ys))), 876.13/288.95 quicksort^#(@xs), 876.13/288.95 quicksort^#(@ys)) 876.13/288.95 , append^#(@l, @ys) -> c_6(append#1^#(@l, @ys)) 876.13/288.95 , split#1^#(nil()) -> c_14() 876.13/288.95 , split#1^#(::(@x, @xs)) -> 876.13/288.95 c_15(insert^#(@x, split(@xs)), split^#(@xs)) 876.13/288.95 , append#1^#(nil(), @ys) -> c_22() 876.13/288.95 , append#1^#(::(@x, @xs), @ys) -> c_23(append^#(@xs, @ys)) 876.13/288.95 , insert#3^#(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.13/288.95 c_7(insert#4^#(#equal(@key1, @keyX), 876.13/288.95 @key1, 876.13/288.95 @ls, 876.13/288.95 @valX, 876.13/288.95 @vals1, 876.13/288.95 @x), 876.13/288.95 #equal^#(@key1, @keyX)) 876.13/288.95 , insert#4^#(#true(), @key1, @ls, @valX, @vals1, @x) -> c_17() 876.13/288.95 , insert#4^#(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.13/288.95 c_18(insert^#(@x, @ls)) 876.13/288.95 , insert^#(@x, @l) -> c_8(insert#1^#(@x, @l, @x)) 876.13/288.95 , insert#1^#(tuple#2(@valX, @keyX), @l, @x) -> 876.13/288.95 c_19(insert#2^#(@l, @keyX, @valX, @x)) 876.13/288.95 , sortAll#1^#(nil()) -> c_12() 876.13/288.95 , sortAll#1^#(::(@x, @xs)) -> c_13(sortAll#2^#(@x, @xs)) 876.13/288.95 , insert#2^#(nil(), @keyX, @valX, @x) -> c_10() 876.13/288.95 , insert#2^#(::(@l1, @ls), @keyX, @valX, @x) -> 876.13/288.95 c_11(insert#3^#(@l1, @keyX, @ls, @valX, @x)) 876.13/288.95 , #greater^#(@x, @y) -> 876.13/288.95 c_16(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y)) 876.13/288.95 , quicksort#1^#(nil()) -> c_20() 876.13/288.95 , quicksort#1^#(::(@z, @zs)) -> 876.13/288.95 c_21(quicksort#2^#(splitqs(@z, @zs), @z), splitqs^#(@z, @zs)) 876.13/288.95 , splitqs^#(@pivot, @l) -> c_24(splitqs#1^#(@l, @pivot)) 876.13/288.95 , splitqs#1^#(nil(), @pivot) -> c_29() 876.13/288.95 , splitqs#1^#(::(@x, @xs), @pivot) -> 876.13/288.95 c_30(splitqs#2^#(splitqs(@pivot, @xs), @pivot, @x), 876.13/288.95 splitqs^#(@pivot, @xs)) 876.13/288.95 , splitqs#3^#(#true(), @ls, @rs, @x) -> c_26() 876.13/288.95 , splitqs#3^#(#false(), @ls, @rs, @x) -> c_27() 876.13/288.95 , splitqs#2^#(tuple#2(@ls, @rs), @pivot, @x) -> 876.13/288.95 c_28(splitqs#3^#(#greater(@x, @pivot), @ls, @rs, @x), 876.13/288.95 #greater^#(@x, @pivot)) } 876.13/288.95 Weak DPs: 876.13/288.95 { #eq^#(#pos(@x), #pos(@y)) -> c_31(#eq^#(@x, @y)) 876.13/288.95 , #eq^#(#pos(@x), #0()) -> c_32() 876.13/288.95 , #eq^#(#pos(@x), #neg(@y)) -> c_33() 876.13/288.95 , #eq^#(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.13/288.95 c_34(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), 876.13/288.95 #eq^#(@x_1, @y_1), 876.13/288.95 #eq^#(@x_2, @y_2)) 876.13/288.95 , #eq^#(tuple#2(@x_1, @x_2), nil()) -> c_35() 876.13/288.95 , #eq^#(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> c_36() 876.13/288.95 , #eq^#(nil(), tuple#2(@y_1, @y_2)) -> c_37() 876.13/288.95 , #eq^#(nil(), nil()) -> c_38() 876.13/288.95 , #eq^#(nil(), ::(@y_1, @y_2)) -> c_39() 876.13/288.95 , #eq^#(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> c_40() 876.13/288.95 , #eq^#(::(@x_1, @x_2), nil()) -> c_41() 876.13/288.95 , #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.13/288.95 c_42(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), 876.13/288.95 #eq^#(@x_1, @y_1), 876.13/288.95 #eq^#(@x_2, @y_2)) 876.13/288.95 , #eq^#(#0(), #pos(@y)) -> c_43() 876.13/288.95 , #eq^#(#0(), #0()) -> c_44() 876.13/288.95 , #eq^#(#0(), #neg(@y)) -> c_45() 876.13/288.95 , #eq^#(#0(), #s(@y)) -> c_46() 876.13/288.95 , #eq^#(#neg(@x), #pos(@y)) -> c_47() 876.13/288.95 , #eq^#(#neg(@x), #0()) -> c_48() 876.13/288.95 , #eq^#(#neg(@x), #neg(@y)) -> c_49(#eq^#(@x, @y)) 876.13/288.95 , #eq^#(#s(@x), #0()) -> c_50() 876.13/288.95 , #eq^#(#s(@x), #s(@y)) -> c_51(#eq^#(@x, @y)) 876.13/288.95 , #ckgt^#(#EQ()) -> c_52() 876.13/288.95 , #ckgt^#(#LT()) -> c_53() 876.13/288.95 , #ckgt^#(#GT()) -> c_54() 876.13/288.95 , #compare^#(#pos(@x), #pos(@y)) -> c_59(#compare^#(@x, @y)) 876.13/288.95 , #compare^#(#pos(@x), #0()) -> c_60() 876.13/288.95 , #compare^#(#pos(@x), #neg(@y)) -> c_61() 876.13/288.95 , #compare^#(#0(), #pos(@y)) -> c_62() 876.13/288.95 , #compare^#(#0(), #0()) -> c_63() 876.13/288.95 , #compare^#(#0(), #neg(@y)) -> c_64() 876.13/288.95 , #compare^#(#0(), #s(@y)) -> c_65() 876.13/288.95 , #compare^#(#neg(@x), #pos(@y)) -> c_66() 876.13/288.95 , #compare^#(#neg(@x), #0()) -> c_67() 876.13/288.95 , #compare^#(#neg(@x), #neg(@y)) -> c_68(#compare^#(@y, @x)) 876.13/288.95 , #compare^#(#s(@x), #0()) -> c_69() 876.13/288.95 , #compare^#(#s(@x), #s(@y)) -> c_70(#compare^#(@x, @y)) 876.13/288.95 , #and^#(#true(), #true()) -> c_55() 876.13/288.95 , #and^#(#true(), #false()) -> c_56() 876.13/288.95 , #and^#(#false(), #true()) -> c_57() 876.13/288.95 , #and^#(#false(), #false()) -> c_58() } 876.13/288.95 Weak Trs: 876.13/288.95 { #equal(@x, @y) -> #eq(@x, @y) 876.13/288.95 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.13/288.95 , #eq(#pos(@x), #0()) -> #false() 876.13/288.95 , #eq(#pos(@x), #neg(@y)) -> #false() 876.13/288.95 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.13/288.95 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.13/288.95 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.13/288.95 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.13/288.95 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.13/288.95 , #eq(nil(), nil()) -> #true() 876.13/288.95 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.13/288.95 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.13/288.95 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.13/288.95 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.13/288.95 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.13/288.95 , #eq(#0(), #pos(@y)) -> #false() 876.13/288.95 , #eq(#0(), #0()) -> #true() 876.13/288.95 , #eq(#0(), #neg(@y)) -> #false() 876.13/288.95 , #eq(#0(), #s(@y)) -> #false() 876.13/288.95 , #eq(#neg(@x), #pos(@y)) -> #false() 876.13/288.95 , #eq(#neg(@x), #0()) -> #false() 876.13/288.95 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.13/288.95 , #eq(#s(@x), #0()) -> #false() 876.13/288.95 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.13/288.95 , splitAndSort(@l) -> sortAll(split(@l)) 876.13/288.95 , sortAll#2(tuple#2(@vals, @key), @xs) -> 876.13/288.95 ::(tuple#2(quicksort(@vals), @key), sortAll(@xs)) 876.13/288.95 , quicksort#2(tuple#2(@xs, @ys), @z) -> 876.13/288.95 append(quicksort(@xs), ::(@z, quicksort(@ys))) 876.13/288.95 , split(@l) -> split#1(@l) 876.13/288.95 , append(@l, @ys) -> append#1(@l, @ys) 876.13/288.95 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.13/288.95 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.13/288.95 , #ckgt(#EQ()) -> #false() 876.13/288.95 , #ckgt(#LT()) -> #false() 876.13/288.95 , #ckgt(#GT()) -> #true() 876.13/288.95 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.13/288.95 , sortAll(@l) -> sortAll#1(@l) 876.13/288.95 , insert#2(nil(), @keyX, @valX, @x) -> 876.13/288.95 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.13/288.95 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.13/288.95 insert#3(@l1, @keyX, @ls, @valX, @x) 876.13/288.95 , #and(#true(), #true()) -> #true() 876.13/288.95 , #and(#true(), #false()) -> #false() 876.13/288.95 , #and(#false(), #true()) -> #false() 876.13/288.95 , #and(#false(), #false()) -> #false() 876.13/288.95 , sortAll#1(nil()) -> nil() 876.13/288.95 , sortAll#1(::(@x, @xs)) -> sortAll#2(@x, @xs) 876.13/288.95 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 876.13/288.95 , #compare(#pos(@x), #0()) -> #GT() 876.13/288.95 , #compare(#pos(@x), #neg(@y)) -> #GT() 876.13/288.95 , #compare(#0(), #pos(@y)) -> #LT() 876.13/288.95 , #compare(#0(), #0()) -> #EQ() 876.13/288.95 , #compare(#0(), #neg(@y)) -> #GT() 876.13/288.95 , #compare(#0(), #s(@y)) -> #LT() 876.13/288.95 , #compare(#neg(@x), #pos(@y)) -> #LT() 876.13/288.95 , #compare(#neg(@x), #0()) -> #LT() 876.13/288.95 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 876.13/288.95 , #compare(#s(@x), #0()) -> #GT() 876.13/288.95 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 876.13/288.95 , split#1(nil()) -> nil() 876.13/288.95 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.13/288.95 , #greater(@x, @y) -> #ckgt(#compare(@x, @y)) 876.13/288.95 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.13/288.95 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.13/288.95 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.13/288.95 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.13/288.95 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.13/288.95 insert#2(@l, @keyX, @valX, @x) 876.13/288.95 , quicksort#1(nil()) -> nil() 876.13/288.95 , quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) 876.13/288.95 , append#1(nil(), @ys) -> @ys 876.13/288.95 , append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) 876.13/288.95 , splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) 876.13/288.95 , quicksort(@l) -> quicksort#1(@l) 876.13/288.95 , splitqs#3(#true(), @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) 876.13/288.95 , splitqs#3(#false(), @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) 876.13/288.95 , splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> 876.13/288.95 splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) 876.13/288.95 , splitqs#1(nil(), @pivot) -> tuple#2(nil(), nil()) 876.13/288.95 , splitqs#1(::(@x, @xs), @pivot) -> 876.13/288.95 splitqs#2(splitqs(@pivot, @xs), @pivot, @x) } 876.13/288.95 Obligation: 876.13/288.95 innermost runtime complexity 876.13/288.95 Answer: 876.13/288.95 YES(O(1),O(n^6)) 876.13/288.95 876.13/288.95 We estimate the number of application of 876.13/288.95 {1,9,11,14,18,20,22,23,26,28,29} by applications of 876.13/288.95 Pre({1,9,11,14,18,20,22,23,26,28,29}) = {3,4,6,8,13,17,25,30}. Here 876.13/288.95 rules are labeled as follows: 876.13/288.95 876.13/288.95 DPs: 876.13/288.95 { 1: #equal^#(@x, @y) -> c_1(#eq^#(@x, @y)) 876.13/288.96 , 2: splitAndSort^#(@l) -> c_2(sortAll^#(split(@l)), split^#(@l)) 876.13/288.96 , 3: sortAll^#(@l) -> c_9(sortAll#1^#(@l)) 876.13/288.96 , 4: split^#(@l) -> c_5(split#1^#(@l)) 876.13/288.96 , 5: sortAll#2^#(tuple#2(@vals, @key), @xs) -> 876.13/288.96 c_3(quicksort^#(@vals), sortAll^#(@xs)) 876.13/288.96 , 6: quicksort^#(@l) -> c_25(quicksort#1^#(@l)) 876.13/288.96 , 7: quicksort#2^#(tuple#2(@xs, @ys), @z) -> 876.13/288.96 c_4(append^#(quicksort(@xs), ::(@z, quicksort(@ys))), 876.13/288.96 quicksort^#(@xs), 876.13/288.96 quicksort^#(@ys)) 876.13/288.96 , 8: append^#(@l, @ys) -> c_6(append#1^#(@l, @ys)) 876.13/288.96 , 9: split#1^#(nil()) -> c_14() 876.13/288.96 , 10: split#1^#(::(@x, @xs)) -> 876.13/288.96 c_15(insert^#(@x, split(@xs)), split^#(@xs)) 876.13/288.96 , 11: append#1^#(nil(), @ys) -> c_22() 876.13/288.96 , 12: append#1^#(::(@x, @xs), @ys) -> c_23(append^#(@xs, @ys)) 876.13/288.96 , 13: insert#3^#(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.13/288.96 c_7(insert#4^#(#equal(@key1, @keyX), 876.13/288.96 @key1, 876.13/288.96 @ls, 876.13/288.96 @valX, 876.13/288.96 @vals1, 876.13/288.96 @x), 876.13/288.96 #equal^#(@key1, @keyX)) 876.13/288.96 , 14: insert#4^#(#true(), @key1, @ls, @valX, @vals1, @x) -> c_17() 876.13/288.96 , 15: insert#4^#(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.13/288.96 c_18(insert^#(@x, @ls)) 876.13/288.96 , 16: insert^#(@x, @l) -> c_8(insert#1^#(@x, @l, @x)) 876.13/288.96 , 17: insert#1^#(tuple#2(@valX, @keyX), @l, @x) -> 876.13/288.96 c_19(insert#2^#(@l, @keyX, @valX, @x)) 876.13/288.96 , 18: sortAll#1^#(nil()) -> c_12() 876.13/288.96 , 19: sortAll#1^#(::(@x, @xs)) -> c_13(sortAll#2^#(@x, @xs)) 876.13/288.96 , 20: insert#2^#(nil(), @keyX, @valX, @x) -> c_10() 876.13/288.96 , 21: insert#2^#(::(@l1, @ls), @keyX, @valX, @x) -> 876.13/288.96 c_11(insert#3^#(@l1, @keyX, @ls, @valX, @x)) 876.13/288.96 , 22: #greater^#(@x, @y) -> 876.13/288.96 c_16(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y)) 876.13/288.96 , 23: quicksort#1^#(nil()) -> c_20() 876.13/288.96 , 24: quicksort#1^#(::(@z, @zs)) -> 876.13/288.96 c_21(quicksort#2^#(splitqs(@z, @zs), @z), splitqs^#(@z, @zs)) 876.13/288.96 , 25: splitqs^#(@pivot, @l) -> c_24(splitqs#1^#(@l, @pivot)) 876.13/288.96 , 26: splitqs#1^#(nil(), @pivot) -> c_29() 876.13/288.96 , 27: splitqs#1^#(::(@x, @xs), @pivot) -> 876.13/288.96 c_30(splitqs#2^#(splitqs(@pivot, @xs), @pivot, @x), 876.13/288.96 splitqs^#(@pivot, @xs)) 876.13/288.96 , 28: splitqs#3^#(#true(), @ls, @rs, @x) -> c_26() 876.13/288.96 , 29: splitqs#3^#(#false(), @ls, @rs, @x) -> c_27() 876.13/288.96 , 30: splitqs#2^#(tuple#2(@ls, @rs), @pivot, @x) -> 876.13/288.96 c_28(splitqs#3^#(#greater(@x, @pivot), @ls, @rs, @x), 876.13/288.96 #greater^#(@x, @pivot)) 876.13/288.96 , 31: #eq^#(#pos(@x), #pos(@y)) -> c_31(#eq^#(@x, @y)) 876.13/288.96 , 32: #eq^#(#pos(@x), #0()) -> c_32() 876.13/288.96 , 33: #eq^#(#pos(@x), #neg(@y)) -> c_33() 876.13/288.96 , 34: #eq^#(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.13/288.96 c_34(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), 876.13/288.96 #eq^#(@x_1, @y_1), 876.13/288.96 #eq^#(@x_2, @y_2)) 876.13/288.96 , 35: #eq^#(tuple#2(@x_1, @x_2), nil()) -> c_35() 876.13/288.96 , 36: #eq^#(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> c_36() 876.13/288.96 , 37: #eq^#(nil(), tuple#2(@y_1, @y_2)) -> c_37() 876.13/288.96 , 38: #eq^#(nil(), nil()) -> c_38() 876.13/288.96 , 39: #eq^#(nil(), ::(@y_1, @y_2)) -> c_39() 876.13/288.96 , 40: #eq^#(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> c_40() 876.13/288.96 , 41: #eq^#(::(@x_1, @x_2), nil()) -> c_41() 876.13/288.96 , 42: #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.13/288.96 c_42(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), 876.13/288.96 #eq^#(@x_1, @y_1), 876.13/288.96 #eq^#(@x_2, @y_2)) 876.13/288.96 , 43: #eq^#(#0(), #pos(@y)) -> c_43() 876.13/288.96 , 44: #eq^#(#0(), #0()) -> c_44() 876.13/288.96 , 45: #eq^#(#0(), #neg(@y)) -> c_45() 876.13/288.96 , 46: #eq^#(#0(), #s(@y)) -> c_46() 876.13/288.96 , 47: #eq^#(#neg(@x), #pos(@y)) -> c_47() 876.13/288.96 , 48: #eq^#(#neg(@x), #0()) -> c_48() 876.13/288.96 , 49: #eq^#(#neg(@x), #neg(@y)) -> c_49(#eq^#(@x, @y)) 876.13/288.96 , 50: #eq^#(#s(@x), #0()) -> c_50() 876.13/288.96 , 51: #eq^#(#s(@x), #s(@y)) -> c_51(#eq^#(@x, @y)) 876.13/288.96 , 52: #ckgt^#(#EQ()) -> c_52() 876.13/288.96 , 53: #ckgt^#(#LT()) -> c_53() 876.13/288.96 , 54: #ckgt^#(#GT()) -> c_54() 876.13/288.96 , 55: #compare^#(#pos(@x), #pos(@y)) -> c_59(#compare^#(@x, @y)) 876.13/288.96 , 56: #compare^#(#pos(@x), #0()) -> c_60() 876.13/288.96 , 57: #compare^#(#pos(@x), #neg(@y)) -> c_61() 876.13/288.96 , 58: #compare^#(#0(), #pos(@y)) -> c_62() 876.13/288.96 , 59: #compare^#(#0(), #0()) -> c_63() 876.13/288.96 , 60: #compare^#(#0(), #neg(@y)) -> c_64() 876.13/288.96 , 61: #compare^#(#0(), #s(@y)) -> c_65() 876.13/288.96 , 62: #compare^#(#neg(@x), #pos(@y)) -> c_66() 876.13/288.96 , 63: #compare^#(#neg(@x), #0()) -> c_67() 876.13/288.96 , 64: #compare^#(#neg(@x), #neg(@y)) -> c_68(#compare^#(@y, @x)) 876.13/288.96 , 65: #compare^#(#s(@x), #0()) -> c_69() 876.13/288.96 , 66: #compare^#(#s(@x), #s(@y)) -> c_70(#compare^#(@x, @y)) 876.13/288.96 , 67: #and^#(#true(), #true()) -> c_55() 876.13/288.96 , 68: #and^#(#true(), #false()) -> c_56() 876.13/288.96 , 69: #and^#(#false(), #true()) -> c_57() 876.13/288.96 , 70: #and^#(#false(), #false()) -> c_58() } 876.13/288.96 876.13/288.96 We are left with following problem, upon which TcT provides the 876.13/288.96 certificate YES(O(1),O(n^6)). 876.13/288.96 876.13/288.96 Strict DPs: 876.13/288.96 { splitAndSort^#(@l) -> c_2(sortAll^#(split(@l)), split^#(@l)) 876.13/288.96 , sortAll^#(@l) -> c_9(sortAll#1^#(@l)) 876.13/288.96 , split^#(@l) -> c_5(split#1^#(@l)) 876.13/288.96 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> 876.13/288.96 c_3(quicksort^#(@vals), sortAll^#(@xs)) 876.13/288.96 , quicksort^#(@l) -> c_25(quicksort#1^#(@l)) 876.13/288.96 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> 876.13/288.96 c_4(append^#(quicksort(@xs), ::(@z, quicksort(@ys))), 876.13/288.96 quicksort^#(@xs), 876.13/288.96 quicksort^#(@ys)) 876.13/288.96 , append^#(@l, @ys) -> c_6(append#1^#(@l, @ys)) 876.13/288.96 , split#1^#(::(@x, @xs)) -> 876.13/288.96 c_15(insert^#(@x, split(@xs)), split^#(@xs)) 876.13/288.96 , append#1^#(::(@x, @xs), @ys) -> c_23(append^#(@xs, @ys)) 876.13/288.96 , insert#3^#(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.13/288.96 c_7(insert#4^#(#equal(@key1, @keyX), 876.13/288.96 @key1, 876.13/288.96 @ls, 876.13/288.96 @valX, 876.13/288.96 @vals1, 876.13/288.96 @x), 876.13/288.96 #equal^#(@key1, @keyX)) 876.13/288.96 , insert#4^#(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.13/288.96 c_18(insert^#(@x, @ls)) 876.13/288.96 , insert^#(@x, @l) -> c_8(insert#1^#(@x, @l, @x)) 876.13/288.96 , insert#1^#(tuple#2(@valX, @keyX), @l, @x) -> 876.13/288.96 c_19(insert#2^#(@l, @keyX, @valX, @x)) 876.13/288.96 , sortAll#1^#(::(@x, @xs)) -> c_13(sortAll#2^#(@x, @xs)) 876.13/288.96 , insert#2^#(::(@l1, @ls), @keyX, @valX, @x) -> 876.13/288.96 c_11(insert#3^#(@l1, @keyX, @ls, @valX, @x)) 876.13/288.96 , quicksort#1^#(::(@z, @zs)) -> 876.13/288.96 c_21(quicksort#2^#(splitqs(@z, @zs), @z), splitqs^#(@z, @zs)) 876.13/288.96 , splitqs^#(@pivot, @l) -> c_24(splitqs#1^#(@l, @pivot)) 876.13/288.96 , splitqs#1^#(::(@x, @xs), @pivot) -> 876.13/288.96 c_30(splitqs#2^#(splitqs(@pivot, @xs), @pivot, @x), 876.13/288.96 splitqs^#(@pivot, @xs)) 876.13/288.96 , splitqs#2^#(tuple#2(@ls, @rs), @pivot, @x) -> 876.13/288.96 c_28(splitqs#3^#(#greater(@x, @pivot), @ls, @rs, @x), 876.13/288.96 #greater^#(@x, @pivot)) } 876.13/288.96 Weak DPs: 876.13/288.96 { #equal^#(@x, @y) -> c_1(#eq^#(@x, @y)) 876.13/288.96 , #eq^#(#pos(@x), #pos(@y)) -> c_31(#eq^#(@x, @y)) 876.13/288.96 , #eq^#(#pos(@x), #0()) -> c_32() 876.13/288.96 , #eq^#(#pos(@x), #neg(@y)) -> c_33() 876.13/288.96 , #eq^#(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.13/288.96 c_34(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), 876.13/288.96 #eq^#(@x_1, @y_1), 876.13/288.96 #eq^#(@x_2, @y_2)) 876.13/288.96 , #eq^#(tuple#2(@x_1, @x_2), nil()) -> c_35() 876.13/288.96 , #eq^#(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> c_36() 876.13/288.96 , #eq^#(nil(), tuple#2(@y_1, @y_2)) -> c_37() 876.13/288.96 , #eq^#(nil(), nil()) -> c_38() 876.13/288.96 , #eq^#(nil(), ::(@y_1, @y_2)) -> c_39() 876.13/288.96 , #eq^#(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> c_40() 876.13/288.96 , #eq^#(::(@x_1, @x_2), nil()) -> c_41() 876.13/288.96 , #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.13/288.96 c_42(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), 876.13/288.96 #eq^#(@x_1, @y_1), 876.13/288.96 #eq^#(@x_2, @y_2)) 876.13/288.96 , #eq^#(#0(), #pos(@y)) -> c_43() 876.13/288.96 , #eq^#(#0(), #0()) -> c_44() 876.13/288.96 , #eq^#(#0(), #neg(@y)) -> c_45() 876.13/288.96 , #eq^#(#0(), #s(@y)) -> c_46() 876.13/288.96 , #eq^#(#neg(@x), #pos(@y)) -> c_47() 876.13/288.96 , #eq^#(#neg(@x), #0()) -> c_48() 876.13/288.96 , #eq^#(#neg(@x), #neg(@y)) -> c_49(#eq^#(@x, @y)) 876.13/288.96 , #eq^#(#s(@x), #0()) -> c_50() 876.13/288.96 , #eq^#(#s(@x), #s(@y)) -> c_51(#eq^#(@x, @y)) 876.13/288.96 , split#1^#(nil()) -> c_14() 876.13/288.96 , append#1^#(nil(), @ys) -> c_22() 876.13/288.96 , insert#4^#(#true(), @key1, @ls, @valX, @vals1, @x) -> c_17() 876.13/288.96 , sortAll#1^#(nil()) -> c_12() 876.13/288.96 , insert#2^#(nil(), @keyX, @valX, @x) -> c_10() 876.13/288.96 , #greater^#(@x, @y) -> 876.13/288.96 c_16(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y)) 876.13/288.96 , #ckgt^#(#EQ()) -> c_52() 876.13/288.96 , #ckgt^#(#LT()) -> c_53() 876.13/288.96 , #ckgt^#(#GT()) -> c_54() 876.13/288.96 , #compare^#(#pos(@x), #pos(@y)) -> c_59(#compare^#(@x, @y)) 876.13/288.96 , #compare^#(#pos(@x), #0()) -> c_60() 876.13/288.96 , #compare^#(#pos(@x), #neg(@y)) -> c_61() 876.13/288.96 , #compare^#(#0(), #pos(@y)) -> c_62() 876.13/288.96 , #compare^#(#0(), #0()) -> c_63() 876.13/288.96 , #compare^#(#0(), #neg(@y)) -> c_64() 876.13/288.96 , #compare^#(#0(), #s(@y)) -> c_65() 876.13/288.96 , #compare^#(#neg(@x), #pos(@y)) -> c_66() 876.13/288.96 , #compare^#(#neg(@x), #0()) -> c_67() 876.13/288.96 , #compare^#(#neg(@x), #neg(@y)) -> c_68(#compare^#(@y, @x)) 876.13/288.96 , #compare^#(#s(@x), #0()) -> c_69() 876.13/288.96 , #compare^#(#s(@x), #s(@y)) -> c_70(#compare^#(@x, @y)) 876.13/288.96 , quicksort#1^#(nil()) -> c_20() 876.13/288.96 , splitqs#1^#(nil(), @pivot) -> c_29() 876.13/288.96 , splitqs#3^#(#true(), @ls, @rs, @x) -> c_26() 876.13/288.96 , splitqs#3^#(#false(), @ls, @rs, @x) -> c_27() 876.13/288.96 , #and^#(#true(), #true()) -> c_55() 876.13/288.96 , #and^#(#true(), #false()) -> c_56() 876.13/288.96 , #and^#(#false(), #true()) -> c_57() 876.13/288.96 , #and^#(#false(), #false()) -> c_58() } 876.13/288.96 Weak Trs: 876.13/288.96 { #equal(@x, @y) -> #eq(@x, @y) 876.13/288.96 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.13/288.96 , #eq(#pos(@x), #0()) -> #false() 876.13/288.96 , #eq(#pos(@x), #neg(@y)) -> #false() 876.13/288.96 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.13/288.96 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.13/288.96 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.13/288.96 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.13/288.96 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.13/288.96 , #eq(nil(), nil()) -> #true() 876.13/288.96 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.13/288.96 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.13/288.96 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.13/288.96 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.13/288.96 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.13/288.96 , #eq(#0(), #pos(@y)) -> #false() 876.13/288.96 , #eq(#0(), #0()) -> #true() 876.13/288.96 , #eq(#0(), #neg(@y)) -> #false() 876.13/288.96 , #eq(#0(), #s(@y)) -> #false() 876.13/288.96 , #eq(#neg(@x), #pos(@y)) -> #false() 876.13/288.96 , #eq(#neg(@x), #0()) -> #false() 876.13/288.96 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.13/288.96 , #eq(#s(@x), #0()) -> #false() 876.13/288.96 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.13/288.96 , splitAndSort(@l) -> sortAll(split(@l)) 876.13/288.96 , sortAll#2(tuple#2(@vals, @key), @xs) -> 876.13/288.96 ::(tuple#2(quicksort(@vals), @key), sortAll(@xs)) 876.13/288.96 , quicksort#2(tuple#2(@xs, @ys), @z) -> 876.13/288.96 append(quicksort(@xs), ::(@z, quicksort(@ys))) 876.13/288.96 , split(@l) -> split#1(@l) 876.13/288.96 , append(@l, @ys) -> append#1(@l, @ys) 876.13/288.96 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.13/288.96 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.13/288.96 , #ckgt(#EQ()) -> #false() 876.13/288.96 , #ckgt(#LT()) -> #false() 876.13/288.96 , #ckgt(#GT()) -> #true() 876.13/288.96 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.13/288.96 , sortAll(@l) -> sortAll#1(@l) 876.13/288.96 , insert#2(nil(), @keyX, @valX, @x) -> 876.13/288.96 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.13/288.96 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.13/288.96 insert#3(@l1, @keyX, @ls, @valX, @x) 876.13/288.96 , #and(#true(), #true()) -> #true() 876.13/288.96 , #and(#true(), #false()) -> #false() 876.13/288.96 , #and(#false(), #true()) -> #false() 876.13/288.96 , #and(#false(), #false()) -> #false() 876.13/288.96 , sortAll#1(nil()) -> nil() 876.13/288.96 , sortAll#1(::(@x, @xs)) -> sortAll#2(@x, @xs) 876.13/288.96 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 876.13/288.96 , #compare(#pos(@x), #0()) -> #GT() 876.13/288.96 , #compare(#pos(@x), #neg(@y)) -> #GT() 876.13/288.96 , #compare(#0(), #pos(@y)) -> #LT() 876.13/288.96 , #compare(#0(), #0()) -> #EQ() 876.13/288.96 , #compare(#0(), #neg(@y)) -> #GT() 876.13/288.96 , #compare(#0(), #s(@y)) -> #LT() 876.13/288.96 , #compare(#neg(@x), #pos(@y)) -> #LT() 876.13/288.96 , #compare(#neg(@x), #0()) -> #LT() 876.13/288.96 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 876.13/288.96 , #compare(#s(@x), #0()) -> #GT() 876.13/288.96 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 876.13/288.96 , split#1(nil()) -> nil() 876.13/288.96 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.13/288.96 , #greater(@x, @y) -> #ckgt(#compare(@x, @y)) 876.13/288.96 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.13/288.96 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.13/288.96 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.13/288.96 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.13/288.96 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.13/288.96 insert#2(@l, @keyX, @valX, @x) 876.13/288.96 , quicksort#1(nil()) -> nil() 876.13/288.96 , quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) 876.13/288.96 , append#1(nil(), @ys) -> @ys 876.13/288.96 , append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) 876.13/288.96 , splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) 876.13/288.96 , quicksort(@l) -> quicksort#1(@l) 876.13/288.96 , splitqs#3(#true(), @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) 876.13/288.96 , splitqs#3(#false(), @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) 876.13/288.96 , splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> 876.13/288.96 splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) 876.13/288.96 , splitqs#1(nil(), @pivot) -> tuple#2(nil(), nil()) 876.13/288.96 , splitqs#1(::(@x, @xs), @pivot) -> 876.13/288.96 splitqs#2(splitqs(@pivot, @xs), @pivot, @x) } 876.13/288.96 Obligation: 876.13/288.96 innermost runtime complexity 876.13/288.96 Answer: 876.13/288.96 YES(O(1),O(n^6)) 876.13/288.96 876.13/288.96 We estimate the number of application of {19} by applications of 876.13/288.96 Pre({19}) = {18}. Here rules are labeled as follows: 876.13/288.96 876.13/288.96 DPs: 876.13/288.96 { 1: splitAndSort^#(@l) -> c_2(sortAll^#(split(@l)), split^#(@l)) 876.13/288.96 , 2: sortAll^#(@l) -> c_9(sortAll#1^#(@l)) 876.13/288.96 , 3: split^#(@l) -> c_5(split#1^#(@l)) 876.13/288.96 , 4: sortAll#2^#(tuple#2(@vals, @key), @xs) -> 876.13/288.96 c_3(quicksort^#(@vals), sortAll^#(@xs)) 876.13/288.96 , 5: quicksort^#(@l) -> c_25(quicksort#1^#(@l)) 876.13/288.96 , 6: quicksort#2^#(tuple#2(@xs, @ys), @z) -> 876.13/288.96 c_4(append^#(quicksort(@xs), ::(@z, quicksort(@ys))), 876.13/288.96 quicksort^#(@xs), 876.13/288.96 quicksort^#(@ys)) 876.13/288.96 , 7: append^#(@l, @ys) -> c_6(append#1^#(@l, @ys)) 876.13/288.96 , 8: split#1^#(::(@x, @xs)) -> 876.13/288.96 c_15(insert^#(@x, split(@xs)), split^#(@xs)) 876.13/288.96 , 9: append#1^#(::(@x, @xs), @ys) -> c_23(append^#(@xs, @ys)) 876.13/288.96 , 10: insert#3^#(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.13/288.96 c_7(insert#4^#(#equal(@key1, @keyX), 876.13/288.96 @key1, 876.13/288.96 @ls, 876.13/288.96 @valX, 876.13/288.96 @vals1, 876.13/288.96 @x), 876.13/288.96 #equal^#(@key1, @keyX)) 876.13/288.96 , 11: insert#4^#(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.13/288.96 c_18(insert^#(@x, @ls)) 876.13/288.96 , 12: insert^#(@x, @l) -> c_8(insert#1^#(@x, @l, @x)) 876.13/288.96 , 13: insert#1^#(tuple#2(@valX, @keyX), @l, @x) -> 876.13/288.96 c_19(insert#2^#(@l, @keyX, @valX, @x)) 876.13/288.96 , 14: sortAll#1^#(::(@x, @xs)) -> c_13(sortAll#2^#(@x, @xs)) 876.13/288.96 , 15: insert#2^#(::(@l1, @ls), @keyX, @valX, @x) -> 876.13/288.96 c_11(insert#3^#(@l1, @keyX, @ls, @valX, @x)) 876.13/288.96 , 16: quicksort#1^#(::(@z, @zs)) -> 876.13/288.96 c_21(quicksort#2^#(splitqs(@z, @zs), @z), splitqs^#(@z, @zs)) 876.13/288.96 , 17: splitqs^#(@pivot, @l) -> c_24(splitqs#1^#(@l, @pivot)) 876.13/288.96 , 18: splitqs#1^#(::(@x, @xs), @pivot) -> 876.13/288.96 c_30(splitqs#2^#(splitqs(@pivot, @xs), @pivot, @x), 876.13/288.96 splitqs^#(@pivot, @xs)) 876.13/288.96 , 19: splitqs#2^#(tuple#2(@ls, @rs), @pivot, @x) -> 876.13/288.96 c_28(splitqs#3^#(#greater(@x, @pivot), @ls, @rs, @x), 876.13/288.96 #greater^#(@x, @pivot)) 876.13/288.96 , 20: #equal^#(@x, @y) -> c_1(#eq^#(@x, @y)) 876.13/288.96 , 21: #eq^#(#pos(@x), #pos(@y)) -> c_31(#eq^#(@x, @y)) 876.13/288.96 , 22: #eq^#(#pos(@x), #0()) -> c_32() 876.13/288.96 , 23: #eq^#(#pos(@x), #neg(@y)) -> c_33() 876.13/288.96 , 24: #eq^#(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.13/288.96 c_34(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), 876.13/288.96 #eq^#(@x_1, @y_1), 876.13/288.96 #eq^#(@x_2, @y_2)) 876.13/288.96 , 25: #eq^#(tuple#2(@x_1, @x_2), nil()) -> c_35() 876.13/288.96 , 26: #eq^#(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> c_36() 876.13/288.96 , 27: #eq^#(nil(), tuple#2(@y_1, @y_2)) -> c_37() 876.13/288.96 , 28: #eq^#(nil(), nil()) -> c_38() 876.13/288.96 , 29: #eq^#(nil(), ::(@y_1, @y_2)) -> c_39() 876.13/288.96 , 30: #eq^#(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> c_40() 876.13/288.96 , 31: #eq^#(::(@x_1, @x_2), nil()) -> c_41() 876.13/288.96 , 32: #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.13/288.96 c_42(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), 876.13/288.96 #eq^#(@x_1, @y_1), 876.13/288.96 #eq^#(@x_2, @y_2)) 876.13/288.96 , 33: #eq^#(#0(), #pos(@y)) -> c_43() 876.13/288.96 , 34: #eq^#(#0(), #0()) -> c_44() 876.13/288.96 , 35: #eq^#(#0(), #neg(@y)) -> c_45() 876.13/288.96 , 36: #eq^#(#0(), #s(@y)) -> c_46() 876.13/288.96 , 37: #eq^#(#neg(@x), #pos(@y)) -> c_47() 876.13/288.96 , 38: #eq^#(#neg(@x), #0()) -> c_48() 876.13/288.96 , 39: #eq^#(#neg(@x), #neg(@y)) -> c_49(#eq^#(@x, @y)) 876.13/288.96 , 40: #eq^#(#s(@x), #0()) -> c_50() 876.13/288.96 , 41: #eq^#(#s(@x), #s(@y)) -> c_51(#eq^#(@x, @y)) 876.13/288.96 , 42: split#1^#(nil()) -> c_14() 876.13/288.96 , 43: append#1^#(nil(), @ys) -> c_22() 876.13/288.96 , 44: insert#4^#(#true(), @key1, @ls, @valX, @vals1, @x) -> c_17() 876.13/288.96 , 45: sortAll#1^#(nil()) -> c_12() 876.13/288.96 , 46: insert#2^#(nil(), @keyX, @valX, @x) -> c_10() 876.13/288.97 , 47: #greater^#(@x, @y) -> 876.13/288.97 c_16(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y)) 876.13/288.97 , 48: #ckgt^#(#EQ()) -> c_52() 876.13/288.97 , 49: #ckgt^#(#LT()) -> c_53() 876.13/288.97 , 50: #ckgt^#(#GT()) -> c_54() 876.13/288.97 , 51: #compare^#(#pos(@x), #pos(@y)) -> c_59(#compare^#(@x, @y)) 876.13/288.97 , 52: #compare^#(#pos(@x), #0()) -> c_60() 876.13/288.97 , 53: #compare^#(#pos(@x), #neg(@y)) -> c_61() 876.13/288.97 , 54: #compare^#(#0(), #pos(@y)) -> c_62() 876.13/288.97 , 55: #compare^#(#0(), #0()) -> c_63() 876.13/288.97 , 56: #compare^#(#0(), #neg(@y)) -> c_64() 876.13/288.97 , 57: #compare^#(#0(), #s(@y)) -> c_65() 876.13/288.97 , 58: #compare^#(#neg(@x), #pos(@y)) -> c_66() 876.13/288.97 , 59: #compare^#(#neg(@x), #0()) -> c_67() 876.13/288.97 , 60: #compare^#(#neg(@x), #neg(@y)) -> c_68(#compare^#(@y, @x)) 876.13/288.97 , 61: #compare^#(#s(@x), #0()) -> c_69() 876.13/288.97 , 62: #compare^#(#s(@x), #s(@y)) -> c_70(#compare^#(@x, @y)) 876.13/288.97 , 63: quicksort#1^#(nil()) -> c_20() 876.13/288.97 , 64: splitqs#1^#(nil(), @pivot) -> c_29() 876.13/288.97 , 65: splitqs#3^#(#true(), @ls, @rs, @x) -> c_26() 876.13/288.97 , 66: splitqs#3^#(#false(), @ls, @rs, @x) -> c_27() 876.13/288.97 , 67: #and^#(#true(), #true()) -> c_55() 876.13/288.97 , 68: #and^#(#true(), #false()) -> c_56() 876.13/288.97 , 69: #and^#(#false(), #true()) -> c_57() 876.13/288.97 , 70: #and^#(#false(), #false()) -> c_58() } 876.13/288.97 876.13/288.97 We are left with following problem, upon which TcT provides the 876.13/288.97 certificate YES(O(1),O(n^6)). 876.13/288.97 876.13/288.97 Strict DPs: 876.13/288.97 { splitAndSort^#(@l) -> c_2(sortAll^#(split(@l)), split^#(@l)) 876.13/288.97 , sortAll^#(@l) -> c_9(sortAll#1^#(@l)) 876.13/288.97 , split^#(@l) -> c_5(split#1^#(@l)) 876.13/288.97 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> 876.13/288.97 c_3(quicksort^#(@vals), sortAll^#(@xs)) 876.13/288.97 , quicksort^#(@l) -> c_25(quicksort#1^#(@l)) 876.13/288.97 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> 876.13/288.97 c_4(append^#(quicksort(@xs), ::(@z, quicksort(@ys))), 876.13/288.97 quicksort^#(@xs), 876.13/288.97 quicksort^#(@ys)) 876.13/288.97 , append^#(@l, @ys) -> c_6(append#1^#(@l, @ys)) 876.13/288.97 , split#1^#(::(@x, @xs)) -> 876.13/288.97 c_15(insert^#(@x, split(@xs)), split^#(@xs)) 876.13/288.97 , append#1^#(::(@x, @xs), @ys) -> c_23(append^#(@xs, @ys)) 876.13/288.97 , insert#3^#(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.13/288.97 c_7(insert#4^#(#equal(@key1, @keyX), 876.13/288.97 @key1, 876.13/288.97 @ls, 876.13/288.97 @valX, 876.13/288.97 @vals1, 876.13/288.97 @x), 876.13/288.97 #equal^#(@key1, @keyX)) 876.13/288.97 , insert#4^#(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.13/288.97 c_18(insert^#(@x, @ls)) 876.13/288.97 , insert^#(@x, @l) -> c_8(insert#1^#(@x, @l, @x)) 876.13/288.97 , insert#1^#(tuple#2(@valX, @keyX), @l, @x) -> 876.13/288.97 c_19(insert#2^#(@l, @keyX, @valX, @x)) 876.13/288.97 , sortAll#1^#(::(@x, @xs)) -> c_13(sortAll#2^#(@x, @xs)) 876.13/288.97 , insert#2^#(::(@l1, @ls), @keyX, @valX, @x) -> 876.13/288.97 c_11(insert#3^#(@l1, @keyX, @ls, @valX, @x)) 876.13/288.97 , quicksort#1^#(::(@z, @zs)) -> 876.13/288.97 c_21(quicksort#2^#(splitqs(@z, @zs), @z), splitqs^#(@z, @zs)) 876.13/288.97 , splitqs^#(@pivot, @l) -> c_24(splitqs#1^#(@l, @pivot)) 876.13/288.97 , splitqs#1^#(::(@x, @xs), @pivot) -> 876.13/288.97 c_30(splitqs#2^#(splitqs(@pivot, @xs), @pivot, @x), 876.13/288.97 splitqs^#(@pivot, @xs)) } 876.13/288.97 Weak DPs: 876.13/288.97 { #equal^#(@x, @y) -> c_1(#eq^#(@x, @y)) 876.13/288.97 , #eq^#(#pos(@x), #pos(@y)) -> c_31(#eq^#(@x, @y)) 876.13/288.97 , #eq^#(#pos(@x), #0()) -> c_32() 876.13/288.97 , #eq^#(#pos(@x), #neg(@y)) -> c_33() 876.13/288.97 , #eq^#(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.13/288.97 c_34(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), 876.13/288.97 #eq^#(@x_1, @y_1), 876.13/288.97 #eq^#(@x_2, @y_2)) 876.13/288.97 , #eq^#(tuple#2(@x_1, @x_2), nil()) -> c_35() 876.13/288.97 , #eq^#(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> c_36() 876.13/288.97 , #eq^#(nil(), tuple#2(@y_1, @y_2)) -> c_37() 876.13/288.97 , #eq^#(nil(), nil()) -> c_38() 876.13/288.97 , #eq^#(nil(), ::(@y_1, @y_2)) -> c_39() 876.13/288.97 , #eq^#(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> c_40() 876.13/288.97 , #eq^#(::(@x_1, @x_2), nil()) -> c_41() 876.13/288.97 , #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.13/288.97 c_42(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), 876.13/288.97 #eq^#(@x_1, @y_1), 876.13/288.97 #eq^#(@x_2, @y_2)) 876.13/288.97 , #eq^#(#0(), #pos(@y)) -> c_43() 876.13/288.97 , #eq^#(#0(), #0()) -> c_44() 876.13/288.97 , #eq^#(#0(), #neg(@y)) -> c_45() 876.13/288.97 , #eq^#(#0(), #s(@y)) -> c_46() 876.13/288.97 , #eq^#(#neg(@x), #pos(@y)) -> c_47() 876.13/288.97 , #eq^#(#neg(@x), #0()) -> c_48() 876.13/288.97 , #eq^#(#neg(@x), #neg(@y)) -> c_49(#eq^#(@x, @y)) 876.13/288.97 , #eq^#(#s(@x), #0()) -> c_50() 876.13/288.97 , #eq^#(#s(@x), #s(@y)) -> c_51(#eq^#(@x, @y)) 876.13/288.97 , split#1^#(nil()) -> c_14() 876.13/288.97 , append#1^#(nil(), @ys) -> c_22() 876.13/288.97 , insert#4^#(#true(), @key1, @ls, @valX, @vals1, @x) -> c_17() 876.13/288.97 , sortAll#1^#(nil()) -> c_12() 876.13/288.97 , insert#2^#(nil(), @keyX, @valX, @x) -> c_10() 876.13/288.97 , #greater^#(@x, @y) -> 876.13/288.97 c_16(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y)) 876.13/288.97 , #ckgt^#(#EQ()) -> c_52() 876.13/288.97 , #ckgt^#(#LT()) -> c_53() 876.13/288.97 , #ckgt^#(#GT()) -> c_54() 876.13/288.97 , #compare^#(#pos(@x), #pos(@y)) -> c_59(#compare^#(@x, @y)) 876.13/288.97 , #compare^#(#pos(@x), #0()) -> c_60() 876.13/288.97 , #compare^#(#pos(@x), #neg(@y)) -> c_61() 876.13/288.97 , #compare^#(#0(), #pos(@y)) -> c_62() 876.13/288.97 , #compare^#(#0(), #0()) -> c_63() 876.13/288.97 , #compare^#(#0(), #neg(@y)) -> c_64() 876.13/288.97 , #compare^#(#0(), #s(@y)) -> c_65() 876.13/288.97 , #compare^#(#neg(@x), #pos(@y)) -> c_66() 876.13/288.97 , #compare^#(#neg(@x), #0()) -> c_67() 876.13/288.97 , #compare^#(#neg(@x), #neg(@y)) -> c_68(#compare^#(@y, @x)) 876.13/288.97 , #compare^#(#s(@x), #0()) -> c_69() 876.13/288.97 , #compare^#(#s(@x), #s(@y)) -> c_70(#compare^#(@x, @y)) 876.13/288.97 , quicksort#1^#(nil()) -> c_20() 876.13/288.97 , splitqs#1^#(nil(), @pivot) -> c_29() 876.13/288.97 , splitqs#3^#(#true(), @ls, @rs, @x) -> c_26() 876.13/288.97 , splitqs#3^#(#false(), @ls, @rs, @x) -> c_27() 876.13/288.97 , splitqs#2^#(tuple#2(@ls, @rs), @pivot, @x) -> 876.13/288.97 c_28(splitqs#3^#(#greater(@x, @pivot), @ls, @rs, @x), 876.13/288.97 #greater^#(@x, @pivot)) 876.13/288.97 , #and^#(#true(), #true()) -> c_55() 876.13/288.97 , #and^#(#true(), #false()) -> c_56() 876.13/288.97 , #and^#(#false(), #true()) -> c_57() 876.13/288.97 , #and^#(#false(), #false()) -> c_58() } 876.13/288.97 Weak Trs: 876.13/288.97 { #equal(@x, @y) -> #eq(@x, @y) 876.13/288.97 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.13/288.97 , #eq(#pos(@x), #0()) -> #false() 876.13/288.97 , #eq(#pos(@x), #neg(@y)) -> #false() 876.13/288.97 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.13/288.97 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.13/288.97 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.13/288.97 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.13/288.97 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.13/288.97 , #eq(nil(), nil()) -> #true() 876.13/288.97 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.13/288.97 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.13/288.97 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.13/288.97 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.13/288.97 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.13/288.97 , #eq(#0(), #pos(@y)) -> #false() 876.13/288.97 , #eq(#0(), #0()) -> #true() 876.13/288.97 , #eq(#0(), #neg(@y)) -> #false() 876.13/288.97 , #eq(#0(), #s(@y)) -> #false() 876.13/288.97 , #eq(#neg(@x), #pos(@y)) -> #false() 876.13/288.97 , #eq(#neg(@x), #0()) -> #false() 876.13/288.97 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.13/288.97 , #eq(#s(@x), #0()) -> #false() 876.13/288.97 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.13/288.97 , splitAndSort(@l) -> sortAll(split(@l)) 876.13/288.97 , sortAll#2(tuple#2(@vals, @key), @xs) -> 876.13/288.97 ::(tuple#2(quicksort(@vals), @key), sortAll(@xs)) 876.13/288.97 , quicksort#2(tuple#2(@xs, @ys), @z) -> 876.13/288.97 append(quicksort(@xs), ::(@z, quicksort(@ys))) 876.13/288.97 , split(@l) -> split#1(@l) 876.13/288.97 , append(@l, @ys) -> append#1(@l, @ys) 876.13/288.97 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.13/288.97 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.13/288.97 , #ckgt(#EQ()) -> #false() 876.13/288.97 , #ckgt(#LT()) -> #false() 876.13/288.97 , #ckgt(#GT()) -> #true() 876.13/288.97 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.13/288.97 , sortAll(@l) -> sortAll#1(@l) 876.13/288.97 , insert#2(nil(), @keyX, @valX, @x) -> 876.13/288.97 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.13/288.97 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.13/288.97 insert#3(@l1, @keyX, @ls, @valX, @x) 876.13/288.97 , #and(#true(), #true()) -> #true() 876.13/288.97 , #and(#true(), #false()) -> #false() 876.13/288.97 , #and(#false(), #true()) -> #false() 876.13/288.97 , #and(#false(), #false()) -> #false() 876.13/288.97 , sortAll#1(nil()) -> nil() 876.13/288.97 , sortAll#1(::(@x, @xs)) -> sortAll#2(@x, @xs) 876.13/288.97 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 876.13/288.97 , #compare(#pos(@x), #0()) -> #GT() 876.13/288.97 , #compare(#pos(@x), #neg(@y)) -> #GT() 876.13/288.97 , #compare(#0(), #pos(@y)) -> #LT() 876.13/288.97 , #compare(#0(), #0()) -> #EQ() 876.13/288.97 , #compare(#0(), #neg(@y)) -> #GT() 876.13/288.97 , #compare(#0(), #s(@y)) -> #LT() 876.13/288.97 , #compare(#neg(@x), #pos(@y)) -> #LT() 876.13/288.97 , #compare(#neg(@x), #0()) -> #LT() 876.13/288.97 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 876.13/288.97 , #compare(#s(@x), #0()) -> #GT() 876.13/288.97 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 876.13/288.97 , split#1(nil()) -> nil() 876.13/288.97 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.13/288.97 , #greater(@x, @y) -> #ckgt(#compare(@x, @y)) 876.13/288.97 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.13/288.97 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.13/288.97 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.13/288.97 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.13/288.97 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.13/288.97 insert#2(@l, @keyX, @valX, @x) 876.13/288.97 , quicksort#1(nil()) -> nil() 876.13/288.97 , quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) 876.13/288.97 , append#1(nil(), @ys) -> @ys 876.13/288.97 , append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) 876.13/288.97 , splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) 876.13/288.97 , quicksort(@l) -> quicksort#1(@l) 876.13/288.97 , splitqs#3(#true(), @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) 876.13/288.97 , splitqs#3(#false(), @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) 876.13/288.97 , splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> 876.13/288.97 splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) 876.13/288.97 , splitqs#1(nil(), @pivot) -> tuple#2(nil(), nil()) 876.13/288.97 , splitqs#1(::(@x, @xs), @pivot) -> 876.13/288.97 splitqs#2(splitqs(@pivot, @xs), @pivot, @x) } 876.13/288.97 Obligation: 876.13/288.97 innermost runtime complexity 876.13/288.97 Answer: 876.13/288.97 YES(O(1),O(n^6)) 876.13/288.97 876.13/288.97 The following weak DPs constitute a sub-graph of the DG that is 876.13/288.97 closed under successors. The DPs are removed. 876.13/288.97 876.13/288.97 { #equal^#(@x, @y) -> c_1(#eq^#(@x, @y)) 876.13/288.97 , #eq^#(#pos(@x), #pos(@y)) -> c_31(#eq^#(@x, @y)) 876.13/288.97 , #eq^#(#pos(@x), #0()) -> c_32() 876.13/288.97 , #eq^#(#pos(@x), #neg(@y)) -> c_33() 876.13/288.97 , #eq^#(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.13/288.97 c_34(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), 876.13/288.97 #eq^#(@x_1, @y_1), 876.13/288.97 #eq^#(@x_2, @y_2)) 876.13/288.97 , #eq^#(tuple#2(@x_1, @x_2), nil()) -> c_35() 876.13/288.97 , #eq^#(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> c_36() 876.13/288.97 , #eq^#(nil(), tuple#2(@y_1, @y_2)) -> c_37() 876.13/288.97 , #eq^#(nil(), nil()) -> c_38() 876.13/288.97 , #eq^#(nil(), ::(@y_1, @y_2)) -> c_39() 876.13/288.97 , #eq^#(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> c_40() 876.13/288.97 , #eq^#(::(@x_1, @x_2), nil()) -> c_41() 876.13/288.97 , #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.13/288.97 c_42(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), 876.13/288.97 #eq^#(@x_1, @y_1), 876.13/288.97 #eq^#(@x_2, @y_2)) 876.13/288.97 , #eq^#(#0(), #pos(@y)) -> c_43() 876.13/288.97 , #eq^#(#0(), #0()) -> c_44() 876.13/288.97 , #eq^#(#0(), #neg(@y)) -> c_45() 876.13/288.97 , #eq^#(#0(), #s(@y)) -> c_46() 876.13/288.97 , #eq^#(#neg(@x), #pos(@y)) -> c_47() 876.13/288.97 , #eq^#(#neg(@x), #0()) -> c_48() 876.13/288.97 , #eq^#(#neg(@x), #neg(@y)) -> c_49(#eq^#(@x, @y)) 876.13/288.97 , #eq^#(#s(@x), #0()) -> c_50() 876.13/288.97 , #eq^#(#s(@x), #s(@y)) -> c_51(#eq^#(@x, @y)) 876.13/288.97 , split#1^#(nil()) -> c_14() 876.13/288.97 , append#1^#(nil(), @ys) -> c_22() 876.13/288.97 , insert#4^#(#true(), @key1, @ls, @valX, @vals1, @x) -> c_17() 876.13/288.97 , sortAll#1^#(nil()) -> c_12() 876.13/288.97 , insert#2^#(nil(), @keyX, @valX, @x) -> c_10() 876.13/288.97 , #greater^#(@x, @y) -> 876.13/288.97 c_16(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y)) 876.13/288.97 , #ckgt^#(#EQ()) -> c_52() 876.13/288.97 , #ckgt^#(#LT()) -> c_53() 876.13/288.97 , #ckgt^#(#GT()) -> c_54() 876.13/288.97 , #compare^#(#pos(@x), #pos(@y)) -> c_59(#compare^#(@x, @y)) 876.13/288.97 , #compare^#(#pos(@x), #0()) -> c_60() 876.13/288.97 , #compare^#(#pos(@x), #neg(@y)) -> c_61() 876.13/288.97 , #compare^#(#0(), #pos(@y)) -> c_62() 876.13/288.97 , #compare^#(#0(), #0()) -> c_63() 876.13/288.97 , #compare^#(#0(), #neg(@y)) -> c_64() 876.13/288.97 , #compare^#(#0(), #s(@y)) -> c_65() 876.13/288.97 , #compare^#(#neg(@x), #pos(@y)) -> c_66() 876.13/288.97 , #compare^#(#neg(@x), #0()) -> c_67() 876.13/288.97 , #compare^#(#neg(@x), #neg(@y)) -> c_68(#compare^#(@y, @x)) 876.13/288.97 , #compare^#(#s(@x), #0()) -> c_69() 876.13/288.97 , #compare^#(#s(@x), #s(@y)) -> c_70(#compare^#(@x, @y)) 876.13/288.97 , quicksort#1^#(nil()) -> c_20() 876.13/288.97 , splitqs#1^#(nil(), @pivot) -> c_29() 876.13/288.97 , splitqs#3^#(#true(), @ls, @rs, @x) -> c_26() 876.13/288.97 , splitqs#3^#(#false(), @ls, @rs, @x) -> c_27() 876.13/288.97 , splitqs#2^#(tuple#2(@ls, @rs), @pivot, @x) -> 876.13/288.97 c_28(splitqs#3^#(#greater(@x, @pivot), @ls, @rs, @x), 876.13/288.97 #greater^#(@x, @pivot)) 876.13/288.97 , #and^#(#true(), #true()) -> c_55() 876.13/288.97 , #and^#(#true(), #false()) -> c_56() 876.13/288.97 , #and^#(#false(), #true()) -> c_57() 876.13/288.97 , #and^#(#false(), #false()) -> c_58() } 876.13/288.97 876.13/288.97 We are left with following problem, upon which TcT provides the 876.13/288.97 certificate YES(O(1),O(n^6)). 876.13/288.97 876.13/288.97 Strict DPs: 876.13/288.97 { splitAndSort^#(@l) -> c_2(sortAll^#(split(@l)), split^#(@l)) 876.13/288.97 , sortAll^#(@l) -> c_9(sortAll#1^#(@l)) 876.13/288.97 , split^#(@l) -> c_5(split#1^#(@l)) 876.13/288.97 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> 876.13/288.97 c_3(quicksort^#(@vals), sortAll^#(@xs)) 876.13/288.97 , quicksort^#(@l) -> c_25(quicksort#1^#(@l)) 876.13/288.97 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> 876.13/288.97 c_4(append^#(quicksort(@xs), ::(@z, quicksort(@ys))), 876.13/288.97 quicksort^#(@xs), 876.13/288.97 quicksort^#(@ys)) 876.13/288.97 , append^#(@l, @ys) -> c_6(append#1^#(@l, @ys)) 876.13/288.97 , split#1^#(::(@x, @xs)) -> 876.13/288.97 c_15(insert^#(@x, split(@xs)), split^#(@xs)) 876.13/288.97 , append#1^#(::(@x, @xs), @ys) -> c_23(append^#(@xs, @ys)) 876.13/288.97 , insert#3^#(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.13/288.97 c_7(insert#4^#(#equal(@key1, @keyX), 876.13/288.97 @key1, 876.13/288.97 @ls, 876.13/288.97 @valX, 876.13/288.97 @vals1, 876.13/288.97 @x), 876.13/288.97 #equal^#(@key1, @keyX)) 876.13/288.97 , insert#4^#(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.13/288.97 c_18(insert^#(@x, @ls)) 876.13/288.97 , insert^#(@x, @l) -> c_8(insert#1^#(@x, @l, @x)) 876.13/288.97 , insert#1^#(tuple#2(@valX, @keyX), @l, @x) -> 876.13/288.97 c_19(insert#2^#(@l, @keyX, @valX, @x)) 876.13/288.97 , sortAll#1^#(::(@x, @xs)) -> c_13(sortAll#2^#(@x, @xs)) 876.13/288.97 , insert#2^#(::(@l1, @ls), @keyX, @valX, @x) -> 876.13/288.97 c_11(insert#3^#(@l1, @keyX, @ls, @valX, @x)) 876.13/288.97 , quicksort#1^#(::(@z, @zs)) -> 876.13/288.97 c_21(quicksort#2^#(splitqs(@z, @zs), @z), splitqs^#(@z, @zs)) 876.13/288.97 , splitqs^#(@pivot, @l) -> c_24(splitqs#1^#(@l, @pivot)) 876.13/288.97 , splitqs#1^#(::(@x, @xs), @pivot) -> 876.13/288.97 c_30(splitqs#2^#(splitqs(@pivot, @xs), @pivot, @x), 876.13/288.97 splitqs^#(@pivot, @xs)) } 876.13/288.97 Weak Trs: 876.13/288.97 { #equal(@x, @y) -> #eq(@x, @y) 876.13/288.97 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.13/288.97 , #eq(#pos(@x), #0()) -> #false() 876.13/288.97 , #eq(#pos(@x), #neg(@y)) -> #false() 876.13/288.97 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.13/288.97 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.13/288.97 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.13/288.97 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.13/288.97 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.13/288.97 , #eq(nil(), nil()) -> #true() 876.13/288.97 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.13/288.97 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.13/288.97 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.13/288.97 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.13/288.97 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.13/288.97 , #eq(#0(), #pos(@y)) -> #false() 876.13/288.97 , #eq(#0(), #0()) -> #true() 876.13/288.97 , #eq(#0(), #neg(@y)) -> #false() 876.13/288.97 , #eq(#0(), #s(@y)) -> #false() 876.13/288.97 , #eq(#neg(@x), #pos(@y)) -> #false() 876.13/288.97 , #eq(#neg(@x), #0()) -> #false() 876.13/288.97 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.13/288.97 , #eq(#s(@x), #0()) -> #false() 876.13/288.97 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.13/288.97 , splitAndSort(@l) -> sortAll(split(@l)) 876.13/288.97 , sortAll#2(tuple#2(@vals, @key), @xs) -> 876.13/288.97 ::(tuple#2(quicksort(@vals), @key), sortAll(@xs)) 876.13/288.97 , quicksort#2(tuple#2(@xs, @ys), @z) -> 876.13/288.97 append(quicksort(@xs), ::(@z, quicksort(@ys))) 876.13/288.97 , split(@l) -> split#1(@l) 876.13/288.97 , append(@l, @ys) -> append#1(@l, @ys) 876.13/288.97 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.13/288.97 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.13/288.97 , #ckgt(#EQ()) -> #false() 876.13/288.97 , #ckgt(#LT()) -> #false() 876.13/288.97 , #ckgt(#GT()) -> #true() 876.13/288.97 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.13/288.97 , sortAll(@l) -> sortAll#1(@l) 876.13/288.97 , insert#2(nil(), @keyX, @valX, @x) -> 876.13/288.97 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.13/288.97 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.13/288.97 insert#3(@l1, @keyX, @ls, @valX, @x) 876.13/288.97 , #and(#true(), #true()) -> #true() 876.13/288.97 , #and(#true(), #false()) -> #false() 876.13/288.97 , #and(#false(), #true()) -> #false() 876.13/288.97 , #and(#false(), #false()) -> #false() 876.13/288.97 , sortAll#1(nil()) -> nil() 876.13/288.97 , sortAll#1(::(@x, @xs)) -> sortAll#2(@x, @xs) 876.13/288.97 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 876.13/288.97 , #compare(#pos(@x), #0()) -> #GT() 876.13/288.97 , #compare(#pos(@x), #neg(@y)) -> #GT() 876.13/288.97 , #compare(#0(), #pos(@y)) -> #LT() 876.13/288.97 , #compare(#0(), #0()) -> #EQ() 876.13/288.97 , #compare(#0(), #neg(@y)) -> #GT() 876.13/288.97 , #compare(#0(), #s(@y)) -> #LT() 876.13/288.97 , #compare(#neg(@x), #pos(@y)) -> #LT() 876.13/288.97 , #compare(#neg(@x), #0()) -> #LT() 876.13/288.97 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 876.13/288.97 , #compare(#s(@x), #0()) -> #GT() 876.13/288.97 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 876.13/288.97 , split#1(nil()) -> nil() 876.13/288.97 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.13/288.97 , #greater(@x, @y) -> #ckgt(#compare(@x, @y)) 876.13/288.97 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.13/288.97 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.13/288.97 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.13/288.98 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.13/288.98 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.13/288.98 insert#2(@l, @keyX, @valX, @x) 876.13/288.98 , quicksort#1(nil()) -> nil() 876.13/288.98 , quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) 876.13/288.98 , append#1(nil(), @ys) -> @ys 876.13/288.98 , append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) 876.13/288.98 , splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) 876.13/288.98 , quicksort(@l) -> quicksort#1(@l) 876.13/288.98 , splitqs#3(#true(), @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) 876.13/288.98 , splitqs#3(#false(), @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) 876.13/288.98 , splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> 876.13/288.98 splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) 876.13/288.98 , splitqs#1(nil(), @pivot) -> tuple#2(nil(), nil()) 876.13/288.98 , splitqs#1(::(@x, @xs), @pivot) -> 876.13/288.98 splitqs#2(splitqs(@pivot, @xs), @pivot, @x) } 876.13/288.98 Obligation: 876.13/288.98 innermost runtime complexity 876.13/288.98 Answer: 876.13/288.98 YES(O(1),O(n^6)) 876.13/288.98 876.13/288.98 Due to missing edges in the dependency-graph, the right-hand sides 876.13/288.98 of following rules could be simplified: 876.13/288.98 876.13/288.98 { insert#3^#(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.13/288.98 c_7(insert#4^#(#equal(@key1, @keyX), 876.13/288.98 @key1, 876.13/288.98 @ls, 876.13/288.98 @valX, 876.13/288.98 @vals1, 876.13/288.98 @x), 876.13/288.98 #equal^#(@key1, @keyX)) 876.13/288.98 , splitqs#1^#(::(@x, @xs), @pivot) -> 876.13/288.98 c_30(splitqs#2^#(splitqs(@pivot, @xs), @pivot, @x), 876.13/288.98 splitqs^#(@pivot, @xs)) } 876.13/288.98 876.13/288.98 We are left with following problem, upon which TcT provides the 876.13/288.98 certificate YES(O(1),O(n^6)). 876.13/288.98 876.13/288.98 Strict DPs: 876.13/288.98 { splitAndSort^#(@l) -> c_1(sortAll^#(split(@l)), split^#(@l)) 876.13/288.98 , sortAll^#(@l) -> c_2(sortAll#1^#(@l)) 876.13/288.98 , split^#(@l) -> c_3(split#1^#(@l)) 876.13/288.98 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> 876.13/288.98 c_4(quicksort^#(@vals), sortAll^#(@xs)) 876.13/288.98 , quicksort^#(@l) -> c_5(quicksort#1^#(@l)) 876.13/288.98 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> 876.13/288.98 c_6(append^#(quicksort(@xs), ::(@z, quicksort(@ys))), 876.13/288.98 quicksort^#(@xs), 876.13/288.98 quicksort^#(@ys)) 876.13/288.98 , append^#(@l, @ys) -> c_7(append#1^#(@l, @ys)) 876.13/288.98 , split#1^#(::(@x, @xs)) -> 876.13/288.98 c_8(insert^#(@x, split(@xs)), split^#(@xs)) 876.13/288.98 , append#1^#(::(@x, @xs), @ys) -> c_9(append^#(@xs, @ys)) 876.13/288.98 , insert#3^#(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.13/288.98 c_10(insert#4^#(#equal(@key1, @keyX), 876.13/288.98 @key1, 876.13/288.98 @ls, 876.13/288.98 @valX, 876.13/288.98 @vals1, 876.13/288.98 @x)) 876.13/288.98 , insert#4^#(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.13/288.98 c_11(insert^#(@x, @ls)) 876.13/288.98 , insert^#(@x, @l) -> c_12(insert#1^#(@x, @l, @x)) 876.13/288.98 , insert#1^#(tuple#2(@valX, @keyX), @l, @x) -> 876.13/288.98 c_13(insert#2^#(@l, @keyX, @valX, @x)) 876.13/288.98 , sortAll#1^#(::(@x, @xs)) -> c_14(sortAll#2^#(@x, @xs)) 876.13/288.98 , insert#2^#(::(@l1, @ls), @keyX, @valX, @x) -> 876.13/288.98 c_15(insert#3^#(@l1, @keyX, @ls, @valX, @x)) 876.13/288.98 , quicksort#1^#(::(@z, @zs)) -> 876.13/288.98 c_16(quicksort#2^#(splitqs(@z, @zs), @z), splitqs^#(@z, @zs)) 876.13/288.98 , splitqs^#(@pivot, @l) -> c_17(splitqs#1^#(@l, @pivot)) 876.13/288.98 , splitqs#1^#(::(@x, @xs), @pivot) -> 876.13/288.98 c_18(splitqs^#(@pivot, @xs)) } 876.13/288.98 Weak Trs: 876.13/288.98 { #equal(@x, @y) -> #eq(@x, @y) 876.13/288.98 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.13/288.98 , #eq(#pos(@x), #0()) -> #false() 876.13/288.98 , #eq(#pos(@x), #neg(@y)) -> #false() 876.13/288.98 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.13/288.98 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.13/288.98 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.13/288.98 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.13/288.98 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.13/288.98 , #eq(nil(), nil()) -> #true() 876.13/288.98 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.13/288.98 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.13/288.98 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.13/288.98 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.13/288.98 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.13/288.98 , #eq(#0(), #pos(@y)) -> #false() 876.13/288.98 , #eq(#0(), #0()) -> #true() 876.13/288.98 , #eq(#0(), #neg(@y)) -> #false() 876.13/288.98 , #eq(#0(), #s(@y)) -> #false() 876.13/288.98 , #eq(#neg(@x), #pos(@y)) -> #false() 876.13/288.98 , #eq(#neg(@x), #0()) -> #false() 876.13/288.98 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.13/288.98 , #eq(#s(@x), #0()) -> #false() 876.13/288.98 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.13/288.98 , splitAndSort(@l) -> sortAll(split(@l)) 876.13/288.98 , sortAll#2(tuple#2(@vals, @key), @xs) -> 876.13/288.98 ::(tuple#2(quicksort(@vals), @key), sortAll(@xs)) 876.13/288.98 , quicksort#2(tuple#2(@xs, @ys), @z) -> 876.13/288.98 append(quicksort(@xs), ::(@z, quicksort(@ys))) 876.13/288.98 , split(@l) -> split#1(@l) 876.13/288.98 , append(@l, @ys) -> append#1(@l, @ys) 876.13/288.98 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.13/288.98 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.13/288.98 , #ckgt(#EQ()) -> #false() 876.13/288.98 , #ckgt(#LT()) -> #false() 876.13/288.98 , #ckgt(#GT()) -> #true() 876.13/288.98 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.13/288.98 , sortAll(@l) -> sortAll#1(@l) 876.13/288.98 , insert#2(nil(), @keyX, @valX, @x) -> 876.13/288.98 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.13/288.98 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.13/288.98 insert#3(@l1, @keyX, @ls, @valX, @x) 876.13/288.98 , #and(#true(), #true()) -> #true() 876.13/288.98 , #and(#true(), #false()) -> #false() 876.13/288.98 , #and(#false(), #true()) -> #false() 876.13/288.98 , #and(#false(), #false()) -> #false() 876.13/288.98 , sortAll#1(nil()) -> nil() 876.13/288.98 , sortAll#1(::(@x, @xs)) -> sortAll#2(@x, @xs) 876.13/288.98 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 876.13/288.98 , #compare(#pos(@x), #0()) -> #GT() 876.13/288.98 , #compare(#pos(@x), #neg(@y)) -> #GT() 876.13/288.98 , #compare(#0(), #pos(@y)) -> #LT() 876.13/288.98 , #compare(#0(), #0()) -> #EQ() 876.13/288.98 , #compare(#0(), #neg(@y)) -> #GT() 876.13/288.98 , #compare(#0(), #s(@y)) -> #LT() 876.13/288.98 , #compare(#neg(@x), #pos(@y)) -> #LT() 876.13/288.98 , #compare(#neg(@x), #0()) -> #LT() 876.13/288.98 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 876.13/288.98 , #compare(#s(@x), #0()) -> #GT() 876.13/288.98 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 876.13/288.98 , split#1(nil()) -> nil() 876.13/288.98 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.13/288.98 , #greater(@x, @y) -> #ckgt(#compare(@x, @y)) 876.13/288.98 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.13/288.98 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.13/288.98 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.13/288.98 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.13/288.98 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.13/288.98 insert#2(@l, @keyX, @valX, @x) 876.13/288.98 , quicksort#1(nil()) -> nil() 876.13/288.98 , quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) 876.13/288.98 , append#1(nil(), @ys) -> @ys 876.13/288.98 , append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) 876.13/288.98 , splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) 876.13/288.98 , quicksort(@l) -> quicksort#1(@l) 876.13/288.98 , splitqs#3(#true(), @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) 876.13/288.98 , splitqs#3(#false(), @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) 876.13/288.98 , splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> 876.13/288.98 splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) 876.13/288.98 , splitqs#1(nil(), @pivot) -> tuple#2(nil(), nil()) 876.13/288.98 , splitqs#1(::(@x, @xs), @pivot) -> 876.13/288.98 splitqs#2(splitqs(@pivot, @xs), @pivot, @x) } 876.13/288.98 Obligation: 876.13/288.98 innermost runtime complexity 876.13/288.98 Answer: 876.13/288.98 YES(O(1),O(n^6)) 876.13/288.98 876.13/288.98 We replace rewrite rules by usable rules: 876.13/288.98 876.13/288.98 Weak Usable Rules: 876.13/288.98 { #equal(@x, @y) -> #eq(@x, @y) 876.13/288.98 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.13/288.98 , #eq(#pos(@x), #0()) -> #false() 876.13/288.98 , #eq(#pos(@x), #neg(@y)) -> #false() 876.13/288.98 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.13/288.98 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.13/288.98 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.13/288.98 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.13/288.98 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.13/288.98 , #eq(nil(), nil()) -> #true() 876.13/288.98 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.13/288.98 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.13/288.98 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.13/288.98 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.13/288.98 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.13/288.98 , #eq(#0(), #pos(@y)) -> #false() 876.13/288.98 , #eq(#0(), #0()) -> #true() 876.13/288.98 , #eq(#0(), #neg(@y)) -> #false() 876.13/288.98 , #eq(#0(), #s(@y)) -> #false() 876.13/288.98 , #eq(#neg(@x), #pos(@y)) -> #false() 876.13/288.98 , #eq(#neg(@x), #0()) -> #false() 876.13/288.98 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.13/288.98 , #eq(#s(@x), #0()) -> #false() 876.13/288.98 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.13/288.98 , quicksort#2(tuple#2(@xs, @ys), @z) -> 876.13/288.98 append(quicksort(@xs), ::(@z, quicksort(@ys))) 876.13/288.98 , split(@l) -> split#1(@l) 876.13/288.98 , append(@l, @ys) -> append#1(@l, @ys) 876.13/288.98 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.13/288.98 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.13/288.98 , #ckgt(#EQ()) -> #false() 876.13/288.98 , #ckgt(#LT()) -> #false() 876.13/288.98 , #ckgt(#GT()) -> #true() 876.13/288.98 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.13/288.98 , insert#2(nil(), @keyX, @valX, @x) -> 876.13/288.98 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.13/288.98 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.13/288.98 insert#3(@l1, @keyX, @ls, @valX, @x) 876.13/288.98 , #and(#true(), #true()) -> #true() 876.13/288.98 , #and(#true(), #false()) -> #false() 876.13/288.98 , #and(#false(), #true()) -> #false() 876.13/288.98 , #and(#false(), #false()) -> #false() 876.13/288.98 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 876.13/288.98 , #compare(#pos(@x), #0()) -> #GT() 876.13/288.98 , #compare(#pos(@x), #neg(@y)) -> #GT() 876.13/288.98 , #compare(#0(), #pos(@y)) -> #LT() 876.13/288.98 , #compare(#0(), #0()) -> #EQ() 876.13/288.98 , #compare(#0(), #neg(@y)) -> #GT() 876.13/288.98 , #compare(#0(), #s(@y)) -> #LT() 876.13/288.98 , #compare(#neg(@x), #pos(@y)) -> #LT() 876.13/288.98 , #compare(#neg(@x), #0()) -> #LT() 876.13/288.98 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 876.13/288.98 , #compare(#s(@x), #0()) -> #GT() 876.13/288.98 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 876.13/288.98 , split#1(nil()) -> nil() 876.13/288.98 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.13/288.98 , #greater(@x, @y) -> #ckgt(#compare(@x, @y)) 876.13/288.98 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.13/288.98 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.13/288.98 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.13/288.98 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.13/288.98 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.13/288.98 insert#2(@l, @keyX, @valX, @x) 876.13/288.98 , quicksort#1(nil()) -> nil() 876.13/288.98 , quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) 876.13/288.98 , append#1(nil(), @ys) -> @ys 876.13/288.98 , append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) 876.13/288.98 , splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) 876.13/288.98 , quicksort(@l) -> quicksort#1(@l) 876.13/288.98 , splitqs#3(#true(), @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) 876.13/288.98 , splitqs#3(#false(), @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) 876.13/288.98 , splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> 876.13/288.98 splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) 876.13/288.98 , splitqs#1(nil(), @pivot) -> tuple#2(nil(), nil()) 876.13/288.98 , splitqs#1(::(@x, @xs), @pivot) -> 876.13/288.98 splitqs#2(splitqs(@pivot, @xs), @pivot, @x) } 876.13/288.98 876.13/288.98 We are left with following problem, upon which TcT provides the 876.13/288.98 certificate YES(O(1),O(n^6)). 876.13/288.98 876.13/288.98 Strict DPs: 876.13/288.98 { splitAndSort^#(@l) -> c_1(sortAll^#(split(@l)), split^#(@l)) 876.13/288.98 , sortAll^#(@l) -> c_2(sortAll#1^#(@l)) 876.13/288.98 , split^#(@l) -> c_3(split#1^#(@l)) 876.13/288.98 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> 876.13/288.98 c_4(quicksort^#(@vals), sortAll^#(@xs)) 876.13/288.98 , quicksort^#(@l) -> c_5(quicksort#1^#(@l)) 876.13/288.98 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> 876.13/288.98 c_6(append^#(quicksort(@xs), ::(@z, quicksort(@ys))), 876.13/288.98 quicksort^#(@xs), 876.13/288.98 quicksort^#(@ys)) 876.13/288.98 , append^#(@l, @ys) -> c_7(append#1^#(@l, @ys)) 876.13/288.98 , split#1^#(::(@x, @xs)) -> 876.13/288.98 c_8(insert^#(@x, split(@xs)), split^#(@xs)) 876.13/288.98 , append#1^#(::(@x, @xs), @ys) -> c_9(append^#(@xs, @ys)) 876.13/288.98 , insert#3^#(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.13/288.98 c_10(insert#4^#(#equal(@key1, @keyX), 876.13/288.98 @key1, 876.13/288.98 @ls, 876.13/288.98 @valX, 876.13/288.98 @vals1, 876.13/288.98 @x)) 876.13/288.98 , insert#4^#(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.13/288.98 c_11(insert^#(@x, @ls)) 876.13/288.98 , insert^#(@x, @l) -> c_12(insert#1^#(@x, @l, @x)) 876.13/288.98 , insert#1^#(tuple#2(@valX, @keyX), @l, @x) -> 876.13/288.98 c_13(insert#2^#(@l, @keyX, @valX, @x)) 876.13/288.98 , sortAll#1^#(::(@x, @xs)) -> c_14(sortAll#2^#(@x, @xs)) 876.13/288.98 , insert#2^#(::(@l1, @ls), @keyX, @valX, @x) -> 876.13/288.98 c_15(insert#3^#(@l1, @keyX, @ls, @valX, @x)) 876.13/288.98 , quicksort#1^#(::(@z, @zs)) -> 876.13/288.98 c_16(quicksort#2^#(splitqs(@z, @zs), @z), splitqs^#(@z, @zs)) 876.13/288.98 , splitqs^#(@pivot, @l) -> c_17(splitqs#1^#(@l, @pivot)) 876.13/288.98 , splitqs#1^#(::(@x, @xs), @pivot) -> 876.13/288.98 c_18(splitqs^#(@pivot, @xs)) } 876.13/288.98 Weak Trs: 876.13/288.98 { #equal(@x, @y) -> #eq(@x, @y) 876.13/288.98 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.13/288.98 , #eq(#pos(@x), #0()) -> #false() 876.13/288.98 , #eq(#pos(@x), #neg(@y)) -> #false() 876.13/288.98 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.13/288.98 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.13/288.98 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.13/288.98 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.13/288.98 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.13/288.98 , #eq(nil(), nil()) -> #true() 876.13/288.98 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.13/288.98 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.13/288.98 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.13/288.98 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.13/288.98 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.13/288.98 , #eq(#0(), #pos(@y)) -> #false() 876.13/288.98 , #eq(#0(), #0()) -> #true() 876.13/288.98 , #eq(#0(), #neg(@y)) -> #false() 876.13/288.98 , #eq(#0(), #s(@y)) -> #false() 876.13/288.98 , #eq(#neg(@x), #pos(@y)) -> #false() 876.13/288.98 , #eq(#neg(@x), #0()) -> #false() 876.13/288.98 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.13/288.98 , #eq(#s(@x), #0()) -> #false() 876.13/288.98 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.13/288.98 , quicksort#2(tuple#2(@xs, @ys), @z) -> 876.13/288.98 append(quicksort(@xs), ::(@z, quicksort(@ys))) 876.13/288.98 , split(@l) -> split#1(@l) 876.13/288.98 , append(@l, @ys) -> append#1(@l, @ys) 876.13/288.98 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.13/288.98 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.13/288.98 , #ckgt(#EQ()) -> #false() 876.13/288.98 , #ckgt(#LT()) -> #false() 876.13/288.98 , #ckgt(#GT()) -> #true() 876.13/288.98 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.13/288.98 , insert#2(nil(), @keyX, @valX, @x) -> 876.13/288.98 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.13/288.98 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.13/288.98 insert#3(@l1, @keyX, @ls, @valX, @x) 876.13/288.98 , #and(#true(), #true()) -> #true() 876.13/288.98 , #and(#true(), #false()) -> #false() 876.13/288.98 , #and(#false(), #true()) -> #false() 876.13/288.98 , #and(#false(), #false()) -> #false() 876.13/288.98 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 876.13/288.98 , #compare(#pos(@x), #0()) -> #GT() 876.13/288.98 , #compare(#pos(@x), #neg(@y)) -> #GT() 876.13/288.98 , #compare(#0(), #pos(@y)) -> #LT() 876.13/288.98 , #compare(#0(), #0()) -> #EQ() 876.13/288.98 , #compare(#0(), #neg(@y)) -> #GT() 876.13/288.98 , #compare(#0(), #s(@y)) -> #LT() 876.13/288.98 , #compare(#neg(@x), #pos(@y)) -> #LT() 876.13/288.98 , #compare(#neg(@x), #0()) -> #LT() 876.13/288.98 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 876.13/288.98 , #compare(#s(@x), #0()) -> #GT() 876.13/288.98 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 876.13/288.98 , split#1(nil()) -> nil() 876.13/288.98 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.13/288.98 , #greater(@x, @y) -> #ckgt(#compare(@x, @y)) 876.13/288.98 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.13/288.98 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.13/288.98 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.13/288.98 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.13/288.98 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.13/288.98 insert#2(@l, @keyX, @valX, @x) 876.13/288.98 , quicksort#1(nil()) -> nil() 876.13/288.98 , quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) 876.13/288.98 , append#1(nil(), @ys) -> @ys 876.13/288.98 , append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) 876.13/288.98 , splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) 876.13/288.98 , quicksort(@l) -> quicksort#1(@l) 876.13/288.98 , splitqs#3(#true(), @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) 876.13/288.98 , splitqs#3(#false(), @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) 876.13/288.98 , splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> 876.13/288.98 splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) 876.13/288.98 , splitqs#1(nil(), @pivot) -> tuple#2(nil(), nil()) 876.13/288.98 , splitqs#1(::(@x, @xs), @pivot) -> 876.13/288.98 splitqs#2(splitqs(@pivot, @xs), @pivot, @x) } 876.13/288.98 Obligation: 876.13/288.98 innermost runtime complexity 876.13/288.98 Answer: 876.13/288.98 YES(O(1),O(n^6)) 876.13/288.98 876.13/288.98 We analyse the complexity of following sub-problems (R) and (S). 876.13/288.98 Problem (S) is obtained from the input problem by shifting strict 876.13/288.98 rules from (R) into the weak component: 876.13/288.98 876.13/288.98 Problem (R): 876.13/288.98 ------------ 876.13/288.98 Strict DPs: 876.13/288.98 { splitAndSort^#(@l) -> c_1(sortAll^#(split(@l)), split^#(@l)) 876.13/288.98 , split^#(@l) -> c_3(split#1^#(@l)) 876.13/288.98 , split#1^#(::(@x, @xs)) -> 876.13/288.98 c_8(insert^#(@x, split(@xs)), split^#(@xs)) 876.13/288.98 , insert#3^#(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.13/288.98 c_10(insert#4^#(#equal(@key1, @keyX), 876.13/288.98 @key1, 876.13/288.98 @ls, 876.13/288.98 @valX, 876.13/288.98 @vals1, 876.13/288.98 @x)) 876.13/288.98 , insert#4^#(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.13/288.98 c_11(insert^#(@x, @ls)) 876.13/288.98 , insert^#(@x, @l) -> c_12(insert#1^#(@x, @l, @x)) 876.13/288.98 , insert#1^#(tuple#2(@valX, @keyX), @l, @x) -> 876.13/288.98 c_13(insert#2^#(@l, @keyX, @valX, @x)) 876.13/288.98 , insert#2^#(::(@l1, @ls), @keyX, @valX, @x) -> 876.13/288.98 c_15(insert#3^#(@l1, @keyX, @ls, @valX, @x)) } 876.13/288.98 Weak DPs: 876.13/288.98 { sortAll^#(@l) -> c_2(sortAll#1^#(@l)) 876.13/288.98 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> 876.13/288.98 c_4(quicksort^#(@vals), sortAll^#(@xs)) 876.13/288.98 , quicksort^#(@l) -> c_5(quicksort#1^#(@l)) 876.13/288.98 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> 876.13/288.98 c_6(append^#(quicksort(@xs), ::(@z, quicksort(@ys))), 876.13/288.98 quicksort^#(@xs), 876.13/288.98 quicksort^#(@ys)) 876.13/288.98 , append^#(@l, @ys) -> c_7(append#1^#(@l, @ys)) 876.13/288.98 , append#1^#(::(@x, @xs), @ys) -> c_9(append^#(@xs, @ys)) 876.13/288.98 , sortAll#1^#(::(@x, @xs)) -> c_14(sortAll#2^#(@x, @xs)) 876.13/288.98 , quicksort#1^#(::(@z, @zs)) -> 876.13/288.98 c_16(quicksort#2^#(splitqs(@z, @zs), @z), splitqs^#(@z, @zs)) 876.13/288.98 , splitqs^#(@pivot, @l) -> c_17(splitqs#1^#(@l, @pivot)) 876.13/288.98 , splitqs#1^#(::(@x, @xs), @pivot) -> 876.13/288.98 c_18(splitqs^#(@pivot, @xs)) } 876.13/288.98 Weak Trs: 876.13/288.98 { #equal(@x, @y) -> #eq(@x, @y) 876.13/288.98 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.13/288.98 , #eq(#pos(@x), #0()) -> #false() 876.13/288.98 , #eq(#pos(@x), #neg(@y)) -> #false() 876.13/288.98 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.13/288.98 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.13/288.98 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.13/288.98 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.13/288.98 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.13/288.98 , #eq(nil(), nil()) -> #true() 876.13/288.98 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.13/288.98 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.13/288.98 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.13/288.98 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.13/288.98 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.13/288.98 , #eq(#0(), #pos(@y)) -> #false() 876.13/288.98 , #eq(#0(), #0()) -> #true() 876.13/288.98 , #eq(#0(), #neg(@y)) -> #false() 876.13/288.98 , #eq(#0(), #s(@y)) -> #false() 876.13/288.98 , #eq(#neg(@x), #pos(@y)) -> #false() 876.13/288.98 , #eq(#neg(@x), #0()) -> #false() 876.13/288.98 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.13/288.98 , #eq(#s(@x), #0()) -> #false() 876.13/288.98 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.13/288.98 , quicksort#2(tuple#2(@xs, @ys), @z) -> 876.13/288.98 append(quicksort(@xs), ::(@z, quicksort(@ys))) 876.13/288.98 , split(@l) -> split#1(@l) 876.13/288.98 , append(@l, @ys) -> append#1(@l, @ys) 876.13/288.98 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.13/288.98 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.13/288.98 , #ckgt(#EQ()) -> #false() 876.13/288.98 , #ckgt(#LT()) -> #false() 876.13/288.98 , #ckgt(#GT()) -> #true() 876.13/288.98 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.13/288.98 , insert#2(nil(), @keyX, @valX, @x) -> 876.13/288.98 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.13/288.98 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.13/288.98 insert#3(@l1, @keyX, @ls, @valX, @x) 876.13/288.98 , #and(#true(), #true()) -> #true() 876.13/288.98 , #and(#true(), #false()) -> #false() 876.13/288.98 , #and(#false(), #true()) -> #false() 876.13/288.98 , #and(#false(), #false()) -> #false() 876.13/288.98 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 876.13/288.98 , #compare(#pos(@x), #0()) -> #GT() 876.13/288.98 , #compare(#pos(@x), #neg(@y)) -> #GT() 876.13/288.98 , #compare(#0(), #pos(@y)) -> #LT() 876.13/288.98 , #compare(#0(), #0()) -> #EQ() 876.13/288.98 , #compare(#0(), #neg(@y)) -> #GT() 876.13/288.98 , #compare(#0(), #s(@y)) -> #LT() 876.13/288.98 , #compare(#neg(@x), #pos(@y)) -> #LT() 876.13/288.98 , #compare(#neg(@x), #0()) -> #LT() 876.13/288.98 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 876.13/288.98 , #compare(#s(@x), #0()) -> #GT() 876.13/288.98 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 876.13/288.98 , split#1(nil()) -> nil() 876.13/288.98 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.13/288.98 , #greater(@x, @y) -> #ckgt(#compare(@x, @y)) 876.13/288.98 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.13/288.98 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.13/288.98 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.13/288.98 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.13/288.98 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.13/288.98 insert#2(@l, @keyX, @valX, @x) 876.13/288.98 , quicksort#1(nil()) -> nil() 876.13/288.98 , quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) 876.13/288.98 , append#1(nil(), @ys) -> @ys 876.13/288.98 , append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) 876.13/288.98 , splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) 876.13/288.98 , quicksort(@l) -> quicksort#1(@l) 876.13/288.98 , splitqs#3(#true(), @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) 876.13/288.98 , splitqs#3(#false(), @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) 876.13/288.98 , splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> 876.13/288.98 splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) 876.13/288.98 , splitqs#1(nil(), @pivot) -> tuple#2(nil(), nil()) 876.13/288.98 , splitqs#1(::(@x, @xs), @pivot) -> 876.13/288.98 splitqs#2(splitqs(@pivot, @xs), @pivot, @x) } 876.13/288.98 StartTerms: basic terms 876.13/288.98 Strategy: innermost 876.13/288.98 876.13/288.98 Problem (S): 876.13/288.98 ------------ 876.13/288.98 Strict DPs: 876.13/288.98 { sortAll^#(@l) -> c_2(sortAll#1^#(@l)) 876.13/288.98 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> 876.13/288.98 c_4(quicksort^#(@vals), sortAll^#(@xs)) 876.13/288.98 , quicksort^#(@l) -> c_5(quicksort#1^#(@l)) 876.13/288.98 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> 876.13/288.98 c_6(append^#(quicksort(@xs), ::(@z, quicksort(@ys))), 876.13/288.98 quicksort^#(@xs), 876.13/288.98 quicksort^#(@ys)) 876.13/288.98 , append^#(@l, @ys) -> c_7(append#1^#(@l, @ys)) 876.13/288.98 , append#1^#(::(@x, @xs), @ys) -> c_9(append^#(@xs, @ys)) 876.13/288.98 , sortAll#1^#(::(@x, @xs)) -> c_14(sortAll#2^#(@x, @xs)) 876.13/288.98 , quicksort#1^#(::(@z, @zs)) -> 876.13/288.98 c_16(quicksort#2^#(splitqs(@z, @zs), @z), splitqs^#(@z, @zs)) 876.13/288.98 , splitqs^#(@pivot, @l) -> c_17(splitqs#1^#(@l, @pivot)) 876.13/288.98 , splitqs#1^#(::(@x, @xs), @pivot) -> 876.13/288.98 c_18(splitqs^#(@pivot, @xs)) } 876.13/288.98 Weak DPs: 876.13/288.98 { splitAndSort^#(@l) -> c_1(sortAll^#(split(@l)), split^#(@l)) 876.13/288.98 , split^#(@l) -> c_3(split#1^#(@l)) 876.13/288.98 , split#1^#(::(@x, @xs)) -> 876.13/288.98 c_8(insert^#(@x, split(@xs)), split^#(@xs)) 876.13/288.98 , insert#3^#(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.13/288.98 c_10(insert#4^#(#equal(@key1, @keyX), 876.13/288.98 @key1, 876.13/288.98 @ls, 876.13/288.98 @valX, 876.13/288.98 @vals1, 876.13/288.98 @x)) 876.13/288.98 , insert#4^#(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.13/288.98 c_11(insert^#(@x, @ls)) 876.13/288.98 , insert^#(@x, @l) -> c_12(insert#1^#(@x, @l, @x)) 876.13/288.98 , insert#1^#(tuple#2(@valX, @keyX), @l, @x) -> 876.13/288.98 c_13(insert#2^#(@l, @keyX, @valX, @x)) 876.13/288.98 , insert#2^#(::(@l1, @ls), @keyX, @valX, @x) -> 876.13/288.98 c_15(insert#3^#(@l1, @keyX, @ls, @valX, @x)) } 876.13/288.98 Weak Trs: 876.13/288.98 { #equal(@x, @y) -> #eq(@x, @y) 876.13/288.98 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.13/288.98 , #eq(#pos(@x), #0()) -> #false() 876.13/288.98 , #eq(#pos(@x), #neg(@y)) -> #false() 876.13/288.98 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.13/288.98 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.13/288.98 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.13/288.98 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.13/288.98 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.13/288.98 , #eq(nil(), nil()) -> #true() 876.13/288.98 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.13/288.98 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.13/288.98 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.13/288.98 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.13/288.98 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.13/288.98 , #eq(#0(), #pos(@y)) -> #false() 876.13/288.98 , #eq(#0(), #0()) -> #true() 876.13/288.98 , #eq(#0(), #neg(@y)) -> #false() 876.13/288.98 , #eq(#0(), #s(@y)) -> #false() 876.13/288.98 , #eq(#neg(@x), #pos(@y)) -> #false() 876.13/288.98 , #eq(#neg(@x), #0()) -> #false() 876.13/288.98 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.13/288.98 , #eq(#s(@x), #0()) -> #false() 876.13/288.98 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.13/288.98 , quicksort#2(tuple#2(@xs, @ys), @z) -> 876.13/288.98 append(quicksort(@xs), ::(@z, quicksort(@ys))) 876.13/288.98 , split(@l) -> split#1(@l) 876.13/288.98 , append(@l, @ys) -> append#1(@l, @ys) 876.13/288.98 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.13/288.98 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.13/288.98 , #ckgt(#EQ()) -> #false() 876.13/288.98 , #ckgt(#LT()) -> #false() 876.13/288.98 , #ckgt(#GT()) -> #true() 876.13/288.98 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.13/288.98 , insert#2(nil(), @keyX, @valX, @x) -> 876.13/288.98 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.13/288.98 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.13/288.98 insert#3(@l1, @keyX, @ls, @valX, @x) 876.13/288.98 , #and(#true(), #true()) -> #true() 876.13/288.98 , #and(#true(), #false()) -> #false() 876.13/288.98 , #and(#false(), #true()) -> #false() 876.13/288.98 , #and(#false(), #false()) -> #false() 876.13/288.98 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 876.13/288.98 , #compare(#pos(@x), #0()) -> #GT() 876.13/288.98 , #compare(#pos(@x), #neg(@y)) -> #GT() 876.13/288.98 , #compare(#0(), #pos(@y)) -> #LT() 876.13/288.98 , #compare(#0(), #0()) -> #EQ() 876.13/288.98 , #compare(#0(), #neg(@y)) -> #GT() 876.13/288.98 , #compare(#0(), #s(@y)) -> #LT() 876.13/288.98 , #compare(#neg(@x), #pos(@y)) -> #LT() 876.13/288.98 , #compare(#neg(@x), #0()) -> #LT() 876.13/288.98 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 876.13/288.99 , #compare(#s(@x), #0()) -> #GT() 876.13/288.99 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 876.13/288.99 , split#1(nil()) -> nil() 876.13/288.99 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.13/288.99 , #greater(@x, @y) -> #ckgt(#compare(@x, @y)) 876.13/288.99 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.13/288.99 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.13/288.99 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.13/288.99 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.13/288.99 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.13/288.99 insert#2(@l, @keyX, @valX, @x) 876.13/288.99 , quicksort#1(nil()) -> nil() 876.13/288.99 , quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) 876.13/288.99 , append#1(nil(), @ys) -> @ys 876.13/288.99 , append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) 876.13/288.99 , splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) 876.13/288.99 , quicksort(@l) -> quicksort#1(@l) 876.13/288.99 , splitqs#3(#true(), @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) 876.13/288.99 , splitqs#3(#false(), @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) 876.13/288.99 , splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> 876.13/288.99 splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) 876.13/288.99 , splitqs#1(nil(), @pivot) -> tuple#2(nil(), nil()) 876.13/288.99 , splitqs#1(::(@x, @xs), @pivot) -> 876.13/288.99 splitqs#2(splitqs(@pivot, @xs), @pivot, @x) } 876.13/288.99 StartTerms: basic terms 876.13/288.99 Strategy: innermost 876.13/288.99 876.13/288.99 Overall, the transformation results in the following sub-problem(s): 876.13/288.99 876.13/288.99 Generated new problems: 876.13/288.99 ----------------------- 876.13/288.99 R) Strict DPs: 876.13/288.99 { splitAndSort^#(@l) -> c_1(sortAll^#(split(@l)), split^#(@l)) 876.13/288.99 , split^#(@l) -> c_3(split#1^#(@l)) 876.13/288.99 , split#1^#(::(@x, @xs)) -> 876.13/288.99 c_8(insert^#(@x, split(@xs)), split^#(@xs)) 876.13/288.99 , insert#3^#(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.13/288.99 c_10(insert#4^#(#equal(@key1, @keyX), 876.13/288.99 @key1, 876.13/288.99 @ls, 876.13/288.99 @valX, 876.13/288.99 @vals1, 876.13/288.99 @x)) 876.13/288.99 , insert#4^#(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.13/288.99 c_11(insert^#(@x, @ls)) 876.13/288.99 , insert^#(@x, @l) -> c_12(insert#1^#(@x, @l, @x)) 876.13/288.99 , insert#1^#(tuple#2(@valX, @keyX), @l, @x) -> 876.13/288.99 c_13(insert#2^#(@l, @keyX, @valX, @x)) 876.13/288.99 , insert#2^#(::(@l1, @ls), @keyX, @valX, @x) -> 876.13/288.99 c_15(insert#3^#(@l1, @keyX, @ls, @valX, @x)) } 876.13/288.99 Weak DPs: 876.13/288.99 { sortAll^#(@l) -> c_2(sortAll#1^#(@l)) 876.13/288.99 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> 876.13/288.99 c_4(quicksort^#(@vals), sortAll^#(@xs)) 876.13/288.99 , quicksort^#(@l) -> c_5(quicksort#1^#(@l)) 876.13/288.99 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> 876.13/288.99 c_6(append^#(quicksort(@xs), ::(@z, quicksort(@ys))), 876.13/288.99 quicksort^#(@xs), 876.13/288.99 quicksort^#(@ys)) 876.13/288.99 , append^#(@l, @ys) -> c_7(append#1^#(@l, @ys)) 876.13/288.99 , append#1^#(::(@x, @xs), @ys) -> c_9(append^#(@xs, @ys)) 876.13/288.99 , sortAll#1^#(::(@x, @xs)) -> c_14(sortAll#2^#(@x, @xs)) 876.13/288.99 , quicksort#1^#(::(@z, @zs)) -> 876.13/288.99 c_16(quicksort#2^#(splitqs(@z, @zs), @z), splitqs^#(@z, @zs)) 876.13/288.99 , splitqs^#(@pivot, @l) -> c_17(splitqs#1^#(@l, @pivot)) 876.13/288.99 , splitqs#1^#(::(@x, @xs), @pivot) -> 876.13/288.99 c_18(splitqs^#(@pivot, @xs)) } 876.13/288.99 Weak Trs: 876.13/288.99 { #equal(@x, @y) -> #eq(@x, @y) 876.13/288.99 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.13/288.99 , #eq(#pos(@x), #0()) -> #false() 876.13/288.99 , #eq(#pos(@x), #neg(@y)) -> #false() 876.13/288.99 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.13/288.99 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.13/288.99 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.13/288.99 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.13/288.99 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.13/288.99 , #eq(nil(), nil()) -> #true() 876.13/288.99 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.13/288.99 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.13/288.99 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.13/288.99 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.13/288.99 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.13/288.99 , #eq(#0(), #pos(@y)) -> #false() 876.13/288.99 , #eq(#0(), #0()) -> #true() 876.13/288.99 , #eq(#0(), #neg(@y)) -> #false() 876.13/288.99 , #eq(#0(), #s(@y)) -> #false() 876.13/288.99 , #eq(#neg(@x), #pos(@y)) -> #false() 876.13/288.99 , #eq(#neg(@x), #0()) -> #false() 876.13/288.99 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.13/288.99 , #eq(#s(@x), #0()) -> #false() 876.13/288.99 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.13/288.99 , quicksort#2(tuple#2(@xs, @ys), @z) -> 876.13/288.99 append(quicksort(@xs), ::(@z, quicksort(@ys))) 876.13/288.99 , split(@l) -> split#1(@l) 876.13/288.99 , append(@l, @ys) -> append#1(@l, @ys) 876.13/288.99 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.13/288.99 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.13/288.99 , #ckgt(#EQ()) -> #false() 876.13/288.99 , #ckgt(#LT()) -> #false() 876.13/288.99 , #ckgt(#GT()) -> #true() 876.13/288.99 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.13/288.99 , insert#2(nil(), @keyX, @valX, @x) -> 876.13/288.99 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.13/288.99 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.13/288.99 insert#3(@l1, @keyX, @ls, @valX, @x) 876.13/288.99 , #and(#true(), #true()) -> #true() 876.13/288.99 , #and(#true(), #false()) -> #false() 876.13/288.99 , #and(#false(), #true()) -> #false() 876.13/288.99 , #and(#false(), #false()) -> #false() 876.13/288.99 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 876.13/288.99 , #compare(#pos(@x), #0()) -> #GT() 876.13/288.99 , #compare(#pos(@x), #neg(@y)) -> #GT() 876.13/288.99 , #compare(#0(), #pos(@y)) -> #LT() 876.13/288.99 , #compare(#0(), #0()) -> #EQ() 876.13/288.99 , #compare(#0(), #neg(@y)) -> #GT() 876.13/288.99 , #compare(#0(), #s(@y)) -> #LT() 876.13/288.99 , #compare(#neg(@x), #pos(@y)) -> #LT() 876.13/288.99 , #compare(#neg(@x), #0()) -> #LT() 876.13/288.99 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 876.13/288.99 , #compare(#s(@x), #0()) -> #GT() 876.13/288.99 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 876.13/288.99 , split#1(nil()) -> nil() 876.13/288.99 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.13/288.99 , #greater(@x, @y) -> #ckgt(#compare(@x, @y)) 876.13/288.99 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.13/288.99 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.13/288.99 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.13/288.99 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.13/288.99 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.13/288.99 insert#2(@l, @keyX, @valX, @x) 876.13/288.99 , quicksort#1(nil()) -> nil() 876.13/288.99 , quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) 876.13/288.99 , append#1(nil(), @ys) -> @ys 876.13/288.99 , append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) 876.13/288.99 , splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) 876.13/288.99 , quicksort(@l) -> quicksort#1(@l) 876.13/288.99 , splitqs#3(#true(), @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) 876.13/288.99 , splitqs#3(#false(), @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) 876.13/288.99 , splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> 876.13/288.99 splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) 876.13/288.99 , splitqs#1(nil(), @pivot) -> tuple#2(nil(), nil()) 876.13/288.99 , splitqs#1(::(@x, @xs), @pivot) -> 876.13/288.99 splitqs#2(splitqs(@pivot, @xs), @pivot, @x) } 876.13/288.99 StartTerms: basic terms 876.13/288.99 Strategy: innermost 876.13/288.99 876.13/288.99 This problem was proven YES(O(1),O(n^2)). 876.13/288.99 876.13/288.99 S) Strict DPs: 876.13/288.99 { sortAll^#(@l) -> c_2(sortAll#1^#(@l)) 876.13/288.99 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> 876.13/288.99 c_4(quicksort^#(@vals), sortAll^#(@xs)) 876.13/288.99 , quicksort^#(@l) -> c_5(quicksort#1^#(@l)) 876.13/288.99 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> 876.13/288.99 c_6(append^#(quicksort(@xs), ::(@z, quicksort(@ys))), 876.13/288.99 quicksort^#(@xs), 876.13/288.99 quicksort^#(@ys)) 876.13/288.99 , append^#(@l, @ys) -> c_7(append#1^#(@l, @ys)) 876.13/288.99 , append#1^#(::(@x, @xs), @ys) -> c_9(append^#(@xs, @ys)) 876.13/288.99 , sortAll#1^#(::(@x, @xs)) -> c_14(sortAll#2^#(@x, @xs)) 876.13/288.99 , quicksort#1^#(::(@z, @zs)) -> 876.13/288.99 c_16(quicksort#2^#(splitqs(@z, @zs), @z), splitqs^#(@z, @zs)) 876.13/288.99 , splitqs^#(@pivot, @l) -> c_17(splitqs#1^#(@l, @pivot)) 876.13/288.99 , splitqs#1^#(::(@x, @xs), @pivot) -> 876.13/288.99 c_18(splitqs^#(@pivot, @xs)) } 876.13/288.99 Weak DPs: 876.13/288.99 { splitAndSort^#(@l) -> c_1(sortAll^#(split(@l)), split^#(@l)) 876.13/288.99 , split^#(@l) -> c_3(split#1^#(@l)) 876.13/288.99 , split#1^#(::(@x, @xs)) -> 876.13/288.99 c_8(insert^#(@x, split(@xs)), split^#(@xs)) 876.13/288.99 , insert#3^#(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.13/288.99 c_10(insert#4^#(#equal(@key1, @keyX), 876.13/288.99 @key1, 876.13/288.99 @ls, 876.13/288.99 @valX, 876.13/288.99 @vals1, 876.13/288.99 @x)) 876.13/288.99 , insert#4^#(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.13/288.99 c_11(insert^#(@x, @ls)) 876.13/288.99 , insert^#(@x, @l) -> c_12(insert#1^#(@x, @l, @x)) 876.13/288.99 , insert#1^#(tuple#2(@valX, @keyX), @l, @x) -> 876.13/288.99 c_13(insert#2^#(@l, @keyX, @valX, @x)) 876.13/288.99 , insert#2^#(::(@l1, @ls), @keyX, @valX, @x) -> 876.13/288.99 c_15(insert#3^#(@l1, @keyX, @ls, @valX, @x)) } 876.13/288.99 Weak Trs: 876.13/288.99 { #equal(@x, @y) -> #eq(@x, @y) 876.13/288.99 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.13/288.99 , #eq(#pos(@x), #0()) -> #false() 876.13/288.99 , #eq(#pos(@x), #neg(@y)) -> #false() 876.13/288.99 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.13/288.99 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.13/288.99 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.13/288.99 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.13/288.99 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.13/288.99 , #eq(nil(), nil()) -> #true() 876.13/288.99 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.13/288.99 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.13/288.99 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.13/288.99 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.13/288.99 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.13/288.99 , #eq(#0(), #pos(@y)) -> #false() 876.13/288.99 , #eq(#0(), #0()) -> #true() 876.13/288.99 , #eq(#0(), #neg(@y)) -> #false() 876.13/288.99 , #eq(#0(), #s(@y)) -> #false() 876.13/288.99 , #eq(#neg(@x), #pos(@y)) -> #false() 876.13/288.99 , #eq(#neg(@x), #0()) -> #false() 876.13/288.99 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.13/288.99 , #eq(#s(@x), #0()) -> #false() 876.13/288.99 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.13/288.99 , quicksort#2(tuple#2(@xs, @ys), @z) -> 876.13/288.99 append(quicksort(@xs), ::(@z, quicksort(@ys))) 876.13/288.99 , split(@l) -> split#1(@l) 876.13/288.99 , append(@l, @ys) -> append#1(@l, @ys) 876.13/288.99 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.13/288.99 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.13/288.99 , #ckgt(#EQ()) -> #false() 876.13/288.99 , #ckgt(#LT()) -> #false() 876.13/288.99 , #ckgt(#GT()) -> #true() 876.13/288.99 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.13/288.99 , insert#2(nil(), @keyX, @valX, @x) -> 876.13/288.99 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.13/288.99 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.13/288.99 insert#3(@l1, @keyX, @ls, @valX, @x) 876.13/288.99 , #and(#true(), #true()) -> #true() 876.13/288.99 , #and(#true(), #false()) -> #false() 876.13/288.99 , #and(#false(), #true()) -> #false() 876.13/288.99 , #and(#false(), #false()) -> #false() 876.13/288.99 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 876.13/288.99 , #compare(#pos(@x), #0()) -> #GT() 876.13/288.99 , #compare(#pos(@x), #neg(@y)) -> #GT() 876.13/288.99 , #compare(#0(), #pos(@y)) -> #LT() 876.13/288.99 , #compare(#0(), #0()) -> #EQ() 876.13/288.99 , #compare(#0(), #neg(@y)) -> #GT() 876.13/288.99 , #compare(#0(), #s(@y)) -> #LT() 876.13/288.99 , #compare(#neg(@x), #pos(@y)) -> #LT() 876.13/288.99 , #compare(#neg(@x), #0()) -> #LT() 876.13/288.99 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 876.13/288.99 , #compare(#s(@x), #0()) -> #GT() 876.13/288.99 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 876.13/288.99 , split#1(nil()) -> nil() 876.13/288.99 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.13/288.99 , #greater(@x, @y) -> #ckgt(#compare(@x, @y)) 876.13/288.99 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.13/288.99 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.13/288.99 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.13/288.99 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.13/288.99 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.13/288.99 insert#2(@l, @keyX, @valX, @x) 876.13/288.99 , quicksort#1(nil()) -> nil() 876.13/288.99 , quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) 876.13/288.99 , append#1(nil(), @ys) -> @ys 876.13/288.99 , append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) 876.13/288.99 , splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) 876.13/288.99 , quicksort(@l) -> quicksort#1(@l) 876.13/288.99 , splitqs#3(#true(), @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) 876.13/288.99 , splitqs#3(#false(), @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) 876.13/288.99 , splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> 876.13/288.99 splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) 876.13/288.99 , splitqs#1(nil(), @pivot) -> tuple#2(nil(), nil()) 876.13/288.99 , splitqs#1(::(@x, @xs), @pivot) -> 876.13/288.99 splitqs#2(splitqs(@pivot, @xs), @pivot, @x) } 876.13/288.99 StartTerms: basic terms 876.13/288.99 Strategy: innermost 876.13/288.99 876.13/288.99 This problem was proven YES(O(1),O(n^6)). 876.13/288.99 876.13/288.99 876.13/288.99 Proofs for generated problems: 876.13/288.99 ------------------------------ 876.13/288.99 R) We are left with following problem, upon which TcT provides the 876.13/288.99 certificate YES(O(1),O(n^2)). 876.13/288.99 876.13/288.99 Strict DPs: 876.13/288.99 { splitAndSort^#(@l) -> c_1(sortAll^#(split(@l)), split^#(@l)) 876.13/288.99 , split^#(@l) -> c_3(split#1^#(@l)) 876.13/288.99 , split#1^#(::(@x, @xs)) -> 876.13/288.99 c_8(insert^#(@x, split(@xs)), split^#(@xs)) 876.13/288.99 , insert#3^#(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.13/288.99 c_10(insert#4^#(#equal(@key1, @keyX), 876.13/288.99 @key1, 876.13/288.99 @ls, 876.13/288.99 @valX, 876.13/288.99 @vals1, 876.13/288.99 @x)) 876.13/288.99 , insert#4^#(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.13/288.99 c_11(insert^#(@x, @ls)) 876.13/288.99 , insert^#(@x, @l) -> c_12(insert#1^#(@x, @l, @x)) 876.13/288.99 , insert#1^#(tuple#2(@valX, @keyX), @l, @x) -> 876.13/288.99 c_13(insert#2^#(@l, @keyX, @valX, @x)) 876.13/288.99 , insert#2^#(::(@l1, @ls), @keyX, @valX, @x) -> 876.13/288.99 c_15(insert#3^#(@l1, @keyX, @ls, @valX, @x)) } 876.13/288.99 Weak DPs: 876.13/288.99 { sortAll^#(@l) -> c_2(sortAll#1^#(@l)) 876.13/288.99 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> 876.13/288.99 c_4(quicksort^#(@vals), sortAll^#(@xs)) 876.13/288.99 , quicksort^#(@l) -> c_5(quicksort#1^#(@l)) 876.13/288.99 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> 876.13/288.99 c_6(append^#(quicksort(@xs), ::(@z, quicksort(@ys))), 876.13/288.99 quicksort^#(@xs), 876.13/288.99 quicksort^#(@ys)) 876.13/288.99 , append^#(@l, @ys) -> c_7(append#1^#(@l, @ys)) 876.13/288.99 , append#1^#(::(@x, @xs), @ys) -> c_9(append^#(@xs, @ys)) 876.13/288.99 , sortAll#1^#(::(@x, @xs)) -> c_14(sortAll#2^#(@x, @xs)) 876.13/288.99 , quicksort#1^#(::(@z, @zs)) -> 876.13/288.99 c_16(quicksort#2^#(splitqs(@z, @zs), @z), splitqs^#(@z, @zs)) 876.13/288.99 , splitqs^#(@pivot, @l) -> c_17(splitqs#1^#(@l, @pivot)) 876.13/288.99 , splitqs#1^#(::(@x, @xs), @pivot) -> 876.13/288.99 c_18(splitqs^#(@pivot, @xs)) } 876.13/288.99 Weak Trs: 876.13/288.99 { #equal(@x, @y) -> #eq(@x, @y) 876.13/288.99 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.13/288.99 , #eq(#pos(@x), #0()) -> #false() 876.13/288.99 , #eq(#pos(@x), #neg(@y)) -> #false() 876.13/288.99 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.13/288.99 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.13/288.99 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.13/288.99 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.13/288.99 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.13/288.99 , #eq(nil(), nil()) -> #true() 876.13/288.99 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.13/288.99 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.13/288.99 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.13/288.99 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.13/288.99 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.13/288.99 , #eq(#0(), #pos(@y)) -> #false() 876.13/288.99 , #eq(#0(), #0()) -> #true() 876.13/288.99 , #eq(#0(), #neg(@y)) -> #false() 876.13/288.99 , #eq(#0(), #s(@y)) -> #false() 876.13/288.99 , #eq(#neg(@x), #pos(@y)) -> #false() 876.13/288.99 , #eq(#neg(@x), #0()) -> #false() 876.13/288.99 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.13/288.99 , #eq(#s(@x), #0()) -> #false() 876.13/288.99 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.13/288.99 , quicksort#2(tuple#2(@xs, @ys), @z) -> 876.13/288.99 append(quicksort(@xs), ::(@z, quicksort(@ys))) 876.13/288.99 , split(@l) -> split#1(@l) 876.13/288.99 , append(@l, @ys) -> append#1(@l, @ys) 876.13/288.99 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.13/288.99 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.13/288.99 , #ckgt(#EQ()) -> #false() 876.13/288.99 , #ckgt(#LT()) -> #false() 876.13/288.99 , #ckgt(#GT()) -> #true() 876.13/288.99 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.13/288.99 , insert#2(nil(), @keyX, @valX, @x) -> 876.13/288.99 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.13/288.99 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.13/288.99 insert#3(@l1, @keyX, @ls, @valX, @x) 876.13/288.99 , #and(#true(), #true()) -> #true() 876.13/288.99 , #and(#true(), #false()) -> #false() 876.13/288.99 , #and(#false(), #true()) -> #false() 876.13/288.99 , #and(#false(), #false()) -> #false() 876.13/288.99 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 876.13/288.99 , #compare(#pos(@x), #0()) -> #GT() 876.13/288.99 , #compare(#pos(@x), #neg(@y)) -> #GT() 876.13/288.99 , #compare(#0(), #pos(@y)) -> #LT() 876.13/288.99 , #compare(#0(), #0()) -> #EQ() 876.13/288.99 , #compare(#0(), #neg(@y)) -> #GT() 876.13/288.99 , #compare(#0(), #s(@y)) -> #LT() 876.13/288.99 , #compare(#neg(@x), #pos(@y)) -> #LT() 876.13/288.99 , #compare(#neg(@x), #0()) -> #LT() 876.13/288.99 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 876.13/288.99 , #compare(#s(@x), #0()) -> #GT() 876.13/288.99 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 876.13/288.99 , split#1(nil()) -> nil() 876.13/288.99 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.13/289.00 , #greater(@x, @y) -> #ckgt(#compare(@x, @y)) 876.13/289.00 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.13/289.00 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.13/289.00 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.13/289.00 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.13/289.00 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.13/289.00 insert#2(@l, @keyX, @valX, @x) 876.13/289.00 , quicksort#1(nil()) -> nil() 876.13/289.00 , quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) 876.13/289.00 , append#1(nil(), @ys) -> @ys 876.13/289.00 , append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) 876.13/289.00 , splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) 876.13/289.00 , quicksort(@l) -> quicksort#1(@l) 876.13/289.00 , splitqs#3(#true(), @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) 876.13/289.00 , splitqs#3(#false(), @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) 876.13/289.00 , splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> 876.13/289.00 splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) 876.13/289.00 , splitqs#1(nil(), @pivot) -> tuple#2(nil(), nil()) 876.13/289.00 , splitqs#1(::(@x, @xs), @pivot) -> 876.13/289.00 splitqs#2(splitqs(@pivot, @xs), @pivot, @x) } 876.13/289.00 Obligation: 876.13/289.00 innermost runtime complexity 876.13/289.00 Answer: 876.13/289.00 YES(O(1),O(n^2)) 876.13/289.00 876.13/289.00 The following weak DPs constitute a sub-graph of the DG that is 876.13/289.00 closed under successors. The DPs are removed. 876.13/289.00 876.13/289.00 { sortAll^#(@l) -> c_2(sortAll#1^#(@l)) 876.13/289.00 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> 876.13/289.00 c_4(quicksort^#(@vals), sortAll^#(@xs)) 876.13/289.00 , quicksort^#(@l) -> c_5(quicksort#1^#(@l)) 876.13/289.00 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> 876.13/289.00 c_6(append^#(quicksort(@xs), ::(@z, quicksort(@ys))), 876.13/289.00 quicksort^#(@xs), 876.13/289.00 quicksort^#(@ys)) 876.13/289.00 , append^#(@l, @ys) -> c_7(append#1^#(@l, @ys)) 876.13/289.00 , append#1^#(::(@x, @xs), @ys) -> c_9(append^#(@xs, @ys)) 876.13/289.00 , sortAll#1^#(::(@x, @xs)) -> c_14(sortAll#2^#(@x, @xs)) 876.13/289.00 , quicksort#1^#(::(@z, @zs)) -> 876.13/289.00 c_16(quicksort#2^#(splitqs(@z, @zs), @z), splitqs^#(@z, @zs)) 876.13/289.00 , splitqs^#(@pivot, @l) -> c_17(splitqs#1^#(@l, @pivot)) 876.13/289.00 , splitqs#1^#(::(@x, @xs), @pivot) -> 876.13/289.00 c_18(splitqs^#(@pivot, @xs)) } 876.13/289.00 876.13/289.00 We are left with following problem, upon which TcT provides the 876.13/289.00 certificate YES(O(1),O(n^2)). 876.13/289.00 876.13/289.00 Strict DPs: 876.13/289.00 { splitAndSort^#(@l) -> c_1(sortAll^#(split(@l)), split^#(@l)) 876.13/289.00 , split^#(@l) -> c_3(split#1^#(@l)) 876.13/289.00 , split#1^#(::(@x, @xs)) -> 876.13/289.00 c_8(insert^#(@x, split(@xs)), split^#(@xs)) 876.13/289.00 , insert#3^#(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.13/289.00 c_10(insert#4^#(#equal(@key1, @keyX), 876.13/289.00 @key1, 876.13/289.00 @ls, 876.13/289.00 @valX, 876.13/289.00 @vals1, 876.13/289.00 @x)) 876.13/289.00 , insert#4^#(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.13/289.00 c_11(insert^#(@x, @ls)) 876.13/289.00 , insert^#(@x, @l) -> c_12(insert#1^#(@x, @l, @x)) 876.13/289.00 , insert#1^#(tuple#2(@valX, @keyX), @l, @x) -> 876.13/289.00 c_13(insert#2^#(@l, @keyX, @valX, @x)) 876.13/289.00 , insert#2^#(::(@l1, @ls), @keyX, @valX, @x) -> 876.13/289.00 c_15(insert#3^#(@l1, @keyX, @ls, @valX, @x)) } 876.13/289.00 Weak Trs: 876.13/289.00 { #equal(@x, @y) -> #eq(@x, @y) 876.13/289.00 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.13/289.00 , #eq(#pos(@x), #0()) -> #false() 876.13/289.00 , #eq(#pos(@x), #neg(@y)) -> #false() 876.13/289.00 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.13/289.00 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.13/289.00 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.13/289.00 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.13/289.00 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.13/289.00 , #eq(nil(), nil()) -> #true() 876.13/289.00 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.13/289.00 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.13/289.00 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.13/289.00 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.13/289.00 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.13/289.00 , #eq(#0(), #pos(@y)) -> #false() 876.13/289.00 , #eq(#0(), #0()) -> #true() 876.13/289.00 , #eq(#0(), #neg(@y)) -> #false() 876.13/289.00 , #eq(#0(), #s(@y)) -> #false() 876.13/289.00 , #eq(#neg(@x), #pos(@y)) -> #false() 876.13/289.00 , #eq(#neg(@x), #0()) -> #false() 876.13/289.00 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.13/289.00 , #eq(#s(@x), #0()) -> #false() 876.13/289.00 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.13/289.00 , quicksort#2(tuple#2(@xs, @ys), @z) -> 876.13/289.00 append(quicksort(@xs), ::(@z, quicksort(@ys))) 876.13/289.00 , split(@l) -> split#1(@l) 876.13/289.00 , append(@l, @ys) -> append#1(@l, @ys) 876.13/289.00 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.13/289.00 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.13/289.00 , #ckgt(#EQ()) -> #false() 876.13/289.00 , #ckgt(#LT()) -> #false() 876.13/289.00 , #ckgt(#GT()) -> #true() 876.13/289.00 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.13/289.00 , insert#2(nil(), @keyX, @valX, @x) -> 876.13/289.00 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.13/289.00 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.13/289.00 insert#3(@l1, @keyX, @ls, @valX, @x) 876.13/289.00 , #and(#true(), #true()) -> #true() 876.13/289.00 , #and(#true(), #false()) -> #false() 876.13/289.00 , #and(#false(), #true()) -> #false() 876.13/289.00 , #and(#false(), #false()) -> #false() 876.13/289.00 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 876.13/289.00 , #compare(#pos(@x), #0()) -> #GT() 876.13/289.00 , #compare(#pos(@x), #neg(@y)) -> #GT() 876.13/289.00 , #compare(#0(), #pos(@y)) -> #LT() 876.13/289.00 , #compare(#0(), #0()) -> #EQ() 876.13/289.00 , #compare(#0(), #neg(@y)) -> #GT() 876.46/289.00 , #compare(#0(), #s(@y)) -> #LT() 876.46/289.00 , #compare(#neg(@x), #pos(@y)) -> #LT() 876.46/289.00 , #compare(#neg(@x), #0()) -> #LT() 876.46/289.00 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 876.46/289.00 , #compare(#s(@x), #0()) -> #GT() 876.46/289.00 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 876.46/289.00 , split#1(nil()) -> nil() 876.46/289.00 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.46/289.00 , #greater(@x, @y) -> #ckgt(#compare(@x, @y)) 876.46/289.00 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.00 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.46/289.00 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.00 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.46/289.00 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.46/289.00 insert#2(@l, @keyX, @valX, @x) 876.46/289.00 , quicksort#1(nil()) -> nil() 876.46/289.00 , quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) 876.46/289.00 , append#1(nil(), @ys) -> @ys 876.46/289.00 , append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) 876.46/289.00 , splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) 876.46/289.00 , quicksort(@l) -> quicksort#1(@l) 876.46/289.00 , splitqs#3(#true(), @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) 876.46/289.00 , splitqs#3(#false(), @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) 876.46/289.00 , splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> 876.46/289.00 splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) 876.46/289.00 , splitqs#1(nil(), @pivot) -> tuple#2(nil(), nil()) 876.46/289.00 , splitqs#1(::(@x, @xs), @pivot) -> 876.46/289.00 splitqs#2(splitqs(@pivot, @xs), @pivot, @x) } 876.46/289.00 Obligation: 876.46/289.00 innermost runtime complexity 876.46/289.00 Answer: 876.46/289.00 YES(O(1),O(n^2)) 876.46/289.00 876.46/289.00 Due to missing edges in the dependency-graph, the right-hand sides 876.46/289.00 of following rules could be simplified: 876.46/289.00 876.46/289.00 { splitAndSort^#(@l) -> c_1(sortAll^#(split(@l)), split^#(@l)) } 876.46/289.00 876.46/289.00 We are left with following problem, upon which TcT provides the 876.46/289.00 certificate YES(O(1),O(n^2)). 876.46/289.00 876.46/289.00 Strict DPs: 876.46/289.00 { splitAndSort^#(@l) -> c_1(split^#(@l)) 876.46/289.00 , split^#(@l) -> c_2(split#1^#(@l)) 876.46/289.00 , split#1^#(::(@x, @xs)) -> 876.46/289.00 c_3(insert^#(@x, split(@xs)), split^#(@xs)) 876.46/289.00 , insert#3^#(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.46/289.00 c_4(insert#4^#(#equal(@key1, @keyX), 876.46/289.00 @key1, 876.46/289.00 @ls, 876.46/289.00 @valX, 876.46/289.00 @vals1, 876.46/289.00 @x)) 876.46/289.00 , insert#4^#(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.00 c_5(insert^#(@x, @ls)) 876.46/289.00 , insert^#(@x, @l) -> c_6(insert#1^#(@x, @l, @x)) 876.46/289.00 , insert#1^#(tuple#2(@valX, @keyX), @l, @x) -> 876.46/289.00 c_7(insert#2^#(@l, @keyX, @valX, @x)) 876.46/289.00 , insert#2^#(::(@l1, @ls), @keyX, @valX, @x) -> 876.46/289.00 c_8(insert#3^#(@l1, @keyX, @ls, @valX, @x)) } 876.46/289.00 Weak Trs: 876.46/289.00 { #equal(@x, @y) -> #eq(@x, @y) 876.46/289.00 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.46/289.00 , #eq(#pos(@x), #0()) -> #false() 876.46/289.00 , #eq(#pos(@x), #neg(@y)) -> #false() 876.46/289.00 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.46/289.00 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.00 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.46/289.00 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.46/289.00 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.00 , #eq(nil(), nil()) -> #true() 876.46/289.00 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.46/289.00 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.00 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.46/289.00 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.46/289.00 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.00 , #eq(#0(), #pos(@y)) -> #false() 876.46/289.00 , #eq(#0(), #0()) -> #true() 876.46/289.00 , #eq(#0(), #neg(@y)) -> #false() 876.46/289.00 , #eq(#0(), #s(@y)) -> #false() 876.46/289.00 , #eq(#neg(@x), #pos(@y)) -> #false() 876.46/289.00 , #eq(#neg(@x), #0()) -> #false() 876.46/289.00 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.46/289.00 , #eq(#s(@x), #0()) -> #false() 876.46/289.00 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.46/289.00 , quicksort#2(tuple#2(@xs, @ys), @z) -> 876.46/289.00 append(quicksort(@xs), ::(@z, quicksort(@ys))) 876.46/289.00 , split(@l) -> split#1(@l) 876.46/289.00 , append(@l, @ys) -> append#1(@l, @ys) 876.46/289.00 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.46/289.00 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.46/289.00 , #ckgt(#EQ()) -> #false() 876.46/289.00 , #ckgt(#LT()) -> #false() 876.46/289.00 , #ckgt(#GT()) -> #true() 876.46/289.00 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.46/289.00 , insert#2(nil(), @keyX, @valX, @x) -> 876.46/289.00 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.46/289.00 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.46/289.00 insert#3(@l1, @keyX, @ls, @valX, @x) 876.46/289.00 , #and(#true(), #true()) -> #true() 876.46/289.00 , #and(#true(), #false()) -> #false() 876.46/289.00 , #and(#false(), #true()) -> #false() 876.46/289.00 , #and(#false(), #false()) -> #false() 876.46/289.00 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 876.46/289.00 , #compare(#pos(@x), #0()) -> #GT() 876.46/289.00 , #compare(#pos(@x), #neg(@y)) -> #GT() 876.46/289.00 , #compare(#0(), #pos(@y)) -> #LT() 876.46/289.00 , #compare(#0(), #0()) -> #EQ() 876.46/289.00 , #compare(#0(), #neg(@y)) -> #GT() 876.46/289.00 , #compare(#0(), #s(@y)) -> #LT() 876.46/289.00 , #compare(#neg(@x), #pos(@y)) -> #LT() 876.46/289.00 , #compare(#neg(@x), #0()) -> #LT() 876.46/289.00 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 876.46/289.00 , #compare(#s(@x), #0()) -> #GT() 876.46/289.00 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 876.46/289.00 , split#1(nil()) -> nil() 876.46/289.00 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.46/289.00 , #greater(@x, @y) -> #ckgt(#compare(@x, @y)) 876.46/289.00 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.00 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.46/289.00 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.00 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.46/289.00 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.46/289.00 insert#2(@l, @keyX, @valX, @x) 876.46/289.00 , quicksort#1(nil()) -> nil() 876.46/289.00 , quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) 876.46/289.00 , append#1(nil(), @ys) -> @ys 876.46/289.00 , append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) 876.46/289.00 , splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) 876.46/289.00 , quicksort(@l) -> quicksort#1(@l) 876.46/289.00 , splitqs#3(#true(), @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) 876.46/289.00 , splitqs#3(#false(), @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) 876.46/289.00 , splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> 876.46/289.00 splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) 876.46/289.00 , splitqs#1(nil(), @pivot) -> tuple#2(nil(), nil()) 876.46/289.00 , splitqs#1(::(@x, @xs), @pivot) -> 876.46/289.00 splitqs#2(splitqs(@pivot, @xs), @pivot, @x) } 876.46/289.00 Obligation: 876.46/289.00 innermost runtime complexity 876.46/289.00 Answer: 876.46/289.00 YES(O(1),O(n^2)) 876.46/289.00 876.46/289.00 We replace rewrite rules by usable rules: 876.46/289.00 876.46/289.00 Weak Usable Rules: 876.46/289.00 { #equal(@x, @y) -> #eq(@x, @y) 876.46/289.00 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.46/289.00 , #eq(#pos(@x), #0()) -> #false() 876.46/289.00 , #eq(#pos(@x), #neg(@y)) -> #false() 876.46/289.00 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.46/289.00 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.00 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.46/289.00 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.46/289.00 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.00 , #eq(nil(), nil()) -> #true() 876.46/289.00 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.46/289.00 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.00 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.46/289.00 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.46/289.00 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.00 , #eq(#0(), #pos(@y)) -> #false() 876.46/289.00 , #eq(#0(), #0()) -> #true() 876.46/289.00 , #eq(#0(), #neg(@y)) -> #false() 876.46/289.00 , #eq(#0(), #s(@y)) -> #false() 876.46/289.00 , #eq(#neg(@x), #pos(@y)) -> #false() 876.46/289.00 , #eq(#neg(@x), #0()) -> #false() 876.46/289.00 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.46/289.00 , #eq(#s(@x), #0()) -> #false() 876.46/289.00 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.46/289.00 , split(@l) -> split#1(@l) 876.46/289.00 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.46/289.00 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.46/289.00 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.46/289.00 , insert#2(nil(), @keyX, @valX, @x) -> 876.46/289.00 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.46/289.00 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.46/289.00 insert#3(@l1, @keyX, @ls, @valX, @x) 876.46/289.00 , #and(#true(), #true()) -> #true() 876.46/289.00 , #and(#true(), #false()) -> #false() 876.46/289.00 , #and(#false(), #true()) -> #false() 876.46/289.00 , #and(#false(), #false()) -> #false() 876.46/289.00 , split#1(nil()) -> nil() 876.46/289.00 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.46/289.00 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.00 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.46/289.00 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.00 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.46/289.00 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.46/289.00 insert#2(@l, @keyX, @valX, @x) } 876.46/289.00 876.46/289.00 We are left with following problem, upon which TcT provides the 876.46/289.00 certificate YES(O(1),O(n^2)). 876.46/289.00 876.46/289.00 Strict DPs: 876.46/289.00 { splitAndSort^#(@l) -> c_1(split^#(@l)) 876.46/289.00 , split^#(@l) -> c_2(split#1^#(@l)) 876.46/289.00 , split#1^#(::(@x, @xs)) -> 876.46/289.00 c_3(insert^#(@x, split(@xs)), split^#(@xs)) 876.46/289.00 , insert#3^#(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.46/289.00 c_4(insert#4^#(#equal(@key1, @keyX), 876.46/289.00 @key1, 876.46/289.00 @ls, 876.46/289.00 @valX, 876.46/289.00 @vals1, 876.46/289.00 @x)) 876.46/289.00 , insert#4^#(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.00 c_5(insert^#(@x, @ls)) 876.46/289.00 , insert^#(@x, @l) -> c_6(insert#1^#(@x, @l, @x)) 876.46/289.00 , insert#1^#(tuple#2(@valX, @keyX), @l, @x) -> 876.46/289.00 c_7(insert#2^#(@l, @keyX, @valX, @x)) 876.46/289.00 , insert#2^#(::(@l1, @ls), @keyX, @valX, @x) -> 876.46/289.00 c_8(insert#3^#(@l1, @keyX, @ls, @valX, @x)) } 876.46/289.00 Weak Trs: 876.46/289.00 { #equal(@x, @y) -> #eq(@x, @y) 876.46/289.00 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.46/289.00 , #eq(#pos(@x), #0()) -> #false() 876.46/289.00 , #eq(#pos(@x), #neg(@y)) -> #false() 876.46/289.00 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.46/289.00 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.00 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.46/289.00 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.46/289.00 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.00 , #eq(nil(), nil()) -> #true() 876.46/289.00 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.46/289.00 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.00 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.46/289.00 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.46/289.00 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.00 , #eq(#0(), #pos(@y)) -> #false() 876.46/289.00 , #eq(#0(), #0()) -> #true() 876.46/289.00 , #eq(#0(), #neg(@y)) -> #false() 876.46/289.00 , #eq(#0(), #s(@y)) -> #false() 876.46/289.00 , #eq(#neg(@x), #pos(@y)) -> #false() 876.46/289.00 , #eq(#neg(@x), #0()) -> #false() 876.46/289.00 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.46/289.00 , #eq(#s(@x), #0()) -> #false() 876.46/289.00 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.46/289.00 , split(@l) -> split#1(@l) 876.46/289.00 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.46/289.00 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.46/289.00 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.46/289.00 , insert#2(nil(), @keyX, @valX, @x) -> 876.46/289.00 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.46/289.00 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.46/289.00 insert#3(@l1, @keyX, @ls, @valX, @x) 876.46/289.00 , #and(#true(), #true()) -> #true() 876.46/289.00 , #and(#true(), #false()) -> #false() 876.46/289.00 , #and(#false(), #true()) -> #false() 876.46/289.00 , #and(#false(), #false()) -> #false() 876.46/289.00 , split#1(nil()) -> nil() 876.46/289.00 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.46/289.00 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.00 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.46/289.00 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.00 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.46/289.00 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.46/289.00 insert#2(@l, @keyX, @valX, @x) } 876.46/289.00 Obligation: 876.46/289.00 innermost runtime complexity 876.46/289.00 Answer: 876.46/289.00 YES(O(1),O(n^2)) 876.46/289.00 876.46/289.00 We decompose the input problem according to the dependency graph 876.46/289.00 into the upper component 876.46/289.00 876.46/289.00 { splitAndSort^#(@l) -> c_1(split^#(@l)) 876.46/289.00 , split^#(@l) -> c_2(split#1^#(@l)) 876.46/289.00 , split#1^#(::(@x, @xs)) -> 876.46/289.00 c_3(insert^#(@x, split(@xs)), split^#(@xs)) } 876.46/289.00 876.46/289.00 and lower component 876.46/289.00 876.46/289.00 { insert#3^#(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.46/289.00 c_4(insert#4^#(#equal(@key1, @keyX), 876.46/289.00 @key1, 876.46/289.00 @ls, 876.46/289.00 @valX, 876.46/289.00 @vals1, 876.46/289.00 @x)) 876.46/289.00 , insert#4^#(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.00 c_5(insert^#(@x, @ls)) 876.46/289.00 , insert^#(@x, @l) -> c_6(insert#1^#(@x, @l, @x)) 876.46/289.00 , insert#1^#(tuple#2(@valX, @keyX), @l, @x) -> 876.46/289.00 c_7(insert#2^#(@l, @keyX, @valX, @x)) 876.46/289.00 , insert#2^#(::(@l1, @ls), @keyX, @valX, @x) -> 876.46/289.00 c_8(insert#3^#(@l1, @keyX, @ls, @valX, @x)) } 876.46/289.00 876.46/289.00 Further, following extension rules are added to the lower 876.46/289.00 component. 876.46/289.00 876.46/289.00 { splitAndSort^#(@l) -> split^#(@l) 876.46/289.00 , split^#(@l) -> split#1^#(@l) 876.46/289.00 , split#1^#(::(@x, @xs)) -> split^#(@xs) 876.46/289.00 , split#1^#(::(@x, @xs)) -> insert^#(@x, split(@xs)) } 876.46/289.00 876.46/289.00 TcT solves the upper component with certificate YES(O(1),O(n^1)). 876.46/289.00 876.46/289.00 Sub-proof: 876.46/289.00 ---------- 876.46/289.00 We are left with following problem, upon which TcT provides the 876.46/289.00 certificate YES(O(1),O(n^1)). 876.46/289.00 876.46/289.00 Strict DPs: 876.46/289.00 { splitAndSort^#(@l) -> c_1(split^#(@l)) 876.46/289.00 , split^#(@l) -> c_2(split#1^#(@l)) 876.46/289.00 , split#1^#(::(@x, @xs)) -> 876.46/289.00 c_3(insert^#(@x, split(@xs)), split^#(@xs)) } 876.46/289.00 Weak Trs: 876.46/289.00 { #equal(@x, @y) -> #eq(@x, @y) 876.46/289.00 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.46/289.00 , #eq(#pos(@x), #0()) -> #false() 876.46/289.00 , #eq(#pos(@x), #neg(@y)) -> #false() 876.46/289.00 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.46/289.00 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.00 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.46/289.00 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.46/289.00 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.00 , #eq(nil(), nil()) -> #true() 876.46/289.00 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.46/289.00 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.00 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.46/289.00 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.46/289.00 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.00 , #eq(#0(), #pos(@y)) -> #false() 876.46/289.00 , #eq(#0(), #0()) -> #true() 876.46/289.00 , #eq(#0(), #neg(@y)) -> #false() 876.46/289.00 , #eq(#0(), #s(@y)) -> #false() 876.46/289.00 , #eq(#neg(@x), #pos(@y)) -> #false() 876.46/289.00 , #eq(#neg(@x), #0()) -> #false() 876.46/289.00 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.46/289.00 , #eq(#s(@x), #0()) -> #false() 876.46/289.00 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.46/289.00 , split(@l) -> split#1(@l) 876.46/289.00 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.46/289.00 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.46/289.00 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.46/289.00 , insert#2(nil(), @keyX, @valX, @x) -> 876.46/289.00 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.46/289.00 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.46/289.00 insert#3(@l1, @keyX, @ls, @valX, @x) 876.46/289.00 , #and(#true(), #true()) -> #true() 876.46/289.00 , #and(#true(), #false()) -> #false() 876.46/289.00 , #and(#false(), #true()) -> #false() 876.46/289.00 , #and(#false(), #false()) -> #false() 876.46/289.00 , split#1(nil()) -> nil() 876.46/289.00 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.46/289.00 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.00 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.46/289.00 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.00 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.46/289.00 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.46/289.00 insert#2(@l, @keyX, @valX, @x) } 876.46/289.00 Obligation: 876.46/289.00 innermost runtime complexity 876.46/289.00 Answer: 876.46/289.00 YES(O(1),O(n^1)) 876.46/289.00 876.46/289.00 We use the processor 'matrix interpretation of dimension 1' to 876.46/289.00 orient following rules strictly. 876.46/289.00 876.46/289.00 DPs: 876.46/289.00 { 1: splitAndSort^#(@l) -> c_1(split^#(@l)) 876.46/289.00 , 3: split#1^#(::(@x, @xs)) -> 876.46/289.00 c_3(insert^#(@x, split(@xs)), split^#(@xs)) } 876.46/289.00 876.46/289.00 Sub-proof: 876.46/289.00 ---------- 876.46/289.00 The following argument positions are usable: 876.46/289.00 Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_3) = {1, 2} 876.46/289.00 876.46/289.00 TcT has computed the following constructor-based matrix 876.46/289.00 interpretation satisfying not(EDA). 876.46/289.00 876.46/289.00 [#equal](x1, x2) = [0] 876.46/289.00 876.46/289.00 [#eq](x1, x2) = [0] 876.46/289.00 876.46/289.00 [split](x1) = [0] 876.46/289.00 876.46/289.00 [#true] = [0] 876.46/289.00 876.46/289.00 [insert#3](x1, x2, x3, x4, x5) = [7] x2 + [4] x3 + [4] x4 + [5] x5 + [0] 876.46/289.00 876.46/289.00 [insert](x1, x2) = [0] 876.46/289.00 876.46/289.00 [#pos](x1) = [1] x1 + [0] 876.46/289.00 876.46/289.00 [insert#2](x1, x2, x3, x4) = [2] x2 + [4] x3 + [5] x4 + [0] 876.46/289.00 876.46/289.00 [#and](x1, x2) = [0] 876.46/289.00 876.46/289.00 [tuple#2](x1, x2) = [0] 876.46/289.00 876.46/289.00 [nil] = [0] 876.46/289.00 876.46/289.00 [split#1](x1) = [0] 876.46/289.00 876.46/289.00 [insert#4](x1, x2, x3, x4, x5, x6) = [1] x2 + [4] x3 + [4] x4 + [4] x5 + [5] x6 + [0] 876.46/289.00 876.46/289.00 [insert#1](x1, x2, x3) = [4] x2 + [1] x3 + [0] 876.46/289.00 876.46/289.00 [#false] = [0] 876.46/289.00 876.46/289.00 [::](x1, x2) = [1] x2 + [1] 876.46/289.00 876.46/289.00 [#0] = [0] 876.46/289.00 876.46/289.00 [#neg](x1) = [1] x1 + [0] 876.46/289.00 876.46/289.00 [#s](x1) = [1] x1 + [0] 876.46/289.00 876.46/289.00 [splitAndSort^#](x1) = [7] x1 + [7] 876.46/289.00 876.46/289.00 [split^#](x1) = [1] x1 + [0] 876.46/289.00 876.46/289.00 [split#1^#](x1) = [1] x1 + [0] 876.46/289.00 876.46/289.00 [insert^#](x1, x2) = [0] 876.46/289.00 876.46/289.00 [c_1](x1) = [7] x1 + [3] 876.46/289.00 876.46/289.00 [c_2](x1) = [1] x1 + [0] 876.46/289.00 876.46/289.00 [c_3](x1, x2) = [4] x1 + [1] x2 + [0] 876.46/289.00 876.46/289.00 The order satisfies the following ordering constraints: 876.46/289.00 876.46/289.00 [#equal(@x, @y)] = [0] 876.46/289.00 >= [0] 876.46/289.00 = [#eq(@x, @y)] 876.46/289.00 876.46/289.00 [#eq(#pos(@x), #pos(@y))] = [0] 876.46/289.00 >= [0] 876.46/289.00 = [#eq(@x, @y)] 876.46/289.00 876.46/289.00 [#eq(#pos(@x), #0())] = [0] 876.46/289.00 >= [0] 876.46/289.00 = [#false()] 876.46/289.00 876.46/289.00 [#eq(#pos(@x), #neg(@y))] = [0] 876.46/289.00 >= [0] 876.46/289.00 = [#false()] 876.46/289.00 876.46/289.00 [#eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2))] = [0] 876.46/289.00 >= [0] 876.46/289.00 = [#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))] 876.46/289.00 876.46/289.00 [#eq(tuple#2(@x_1, @x_2), nil())] = [0] 876.46/289.00 >= [0] 876.46/289.00 = [#false()] 876.46/289.00 876.46/289.00 [#eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2))] = [0] 876.46/289.00 >= [0] 876.46/289.00 = [#false()] 876.46/289.00 876.46/289.00 [#eq(nil(), tuple#2(@y_1, @y_2))] = [0] 876.46/289.00 >= [0] 876.46/289.00 = [#false()] 876.46/289.00 876.46/289.00 [#eq(nil(), nil())] = [0] 876.46/289.00 >= [0] 876.46/289.00 = [#true()] 876.46/289.00 876.46/289.00 [#eq(nil(), ::(@y_1, @y_2))] = [0] 876.46/289.00 >= [0] 876.46/289.00 = [#false()] 876.46/289.00 876.46/289.00 [#eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2))] = [0] 876.46/289.00 >= [0] 876.46/289.00 = [#false()] 876.46/289.00 876.46/289.00 [#eq(::(@x_1, @x_2), nil())] = [0] 876.46/289.00 >= [0] 876.46/289.00 = [#false()] 876.46/289.00 876.46/289.00 [#eq(::(@x_1, @x_2), ::(@y_1, @y_2))] = [0] 876.46/289.00 >= [0] 876.46/289.00 = [#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))] 876.46/289.00 876.46/289.00 [#eq(#0(), #pos(@y))] = [0] 876.46/289.00 >= [0] 876.46/289.00 = [#false()] 876.46/289.00 876.46/289.00 [#eq(#0(), #0())] = [0] 876.46/289.00 >= [0] 876.46/289.00 = [#true()] 876.46/289.00 876.46/289.00 [#eq(#0(), #neg(@y))] = [0] 876.46/289.00 >= [0] 876.46/289.00 = [#false()] 876.46/289.00 876.46/289.00 [#eq(#0(), #s(@y))] = [0] 876.46/289.00 >= [0] 876.46/289.00 = [#false()] 876.46/289.00 876.46/289.00 [#eq(#neg(@x), #pos(@y))] = [0] 876.46/289.00 >= [0] 876.46/289.00 = [#false()] 876.46/289.00 876.46/289.00 [#eq(#neg(@x), #0())] = [0] 876.46/289.00 >= [0] 876.46/289.00 = [#false()] 876.46/289.00 876.46/289.00 [#eq(#neg(@x), #neg(@y))] = [0] 876.46/289.00 >= [0] 876.46/289.00 = [#eq(@x, @y)] 876.46/289.00 876.46/289.00 [#eq(#s(@x), #0())] = [0] 876.46/289.00 >= [0] 876.46/289.00 = [#false()] 876.46/289.00 876.46/289.00 [#eq(#s(@x), #s(@y))] = [0] 876.46/289.00 >= [0] 876.46/289.00 = [#eq(@x, @y)] 876.46/289.00 876.46/289.00 [split(@l)] = [0] 876.46/289.00 >= [0] 876.46/289.00 = [split#1(@l)] 876.46/289.00 876.46/289.00 [insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x)] = [5] @x + [4] @valX + [7] @keyX + [4] @ls + [0] 876.46/289.00 ? [5] @x + [4] @valX + [4] @ls + [4] @vals1 + [1] @key1 + [0] 876.46/289.00 = [insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x)] 876.46/289.00 876.46/289.00 [insert(@x, @l)] = [0] 876.46/289.00 ? [1] @x + [4] @l + [0] 876.46/289.00 = [insert#1(@x, @l, @x)] 876.46/289.00 876.46/289.00 [insert#2(nil(), @keyX, @valX, @x)] = [5] @x + [4] @valX + [2] @keyX + [0] 876.46/289.00 ? [1] 876.46/289.00 = [::(tuple#2(::(@valX, nil()), @keyX), nil())] 876.46/289.00 876.46/289.00 [insert#2(::(@l1, @ls), @keyX, @valX, @x)] = [5] @x + [4] @valX + [2] @keyX + [0] 876.46/289.00 ? [5] @x + [4] @valX + [7] @keyX + [4] @ls + [0] 876.46/289.00 = [insert#3(@l1, @keyX, @ls, @valX, @x)] 876.46/289.00 876.46/289.00 [#and(#true(), #true())] = [0] 876.46/289.00 >= [0] 876.46/289.01 = [#true()] 876.46/289.01 876.46/289.01 [#and(#true(), #false())] = [0] 876.46/289.01 >= [0] 876.46/289.01 = [#false()] 876.46/289.01 876.46/289.01 [#and(#false(), #true())] = [0] 876.46/289.01 >= [0] 876.46/289.01 = [#false()] 876.46/289.01 876.46/289.01 [#and(#false(), #false())] = [0] 876.46/289.01 >= [0] 876.46/289.01 = [#false()] 876.46/289.01 876.46/289.01 [split#1(nil())] = [0] 876.46/289.01 >= [0] 876.46/289.01 = [nil()] 876.46/289.01 876.46/289.01 [split#1(::(@x, @xs))] = [0] 876.46/289.01 >= [0] 876.46/289.01 = [insert(@x, split(@xs))] 876.46/289.01 876.46/289.01 [insert#4(#true(), @key1, @ls, @valX, @vals1, @x)] = [5] @x + [4] @valX + [4] @ls + [4] @vals1 + [1] @key1 + [0] 876.46/289.01 ? [1] @ls + [1] 876.46/289.01 = [::(tuple#2(::(@valX, @vals1), @key1), @ls)] 876.46/289.01 876.46/289.01 [insert#4(#false(), @key1, @ls, @valX, @vals1, @x)] = [5] @x + [4] @valX + [4] @ls + [4] @vals1 + [1] @key1 + [0] 876.46/289.01 ? [1] 876.46/289.01 = [::(tuple#2(@vals1, @key1), insert(@x, @ls))] 876.46/289.01 876.46/289.01 [insert#1(tuple#2(@valX, @keyX), @l, @x)] = [1] @x + [4] @l + [0] 876.46/289.01 ? [5] @x + [4] @valX + [2] @keyX + [0] 876.46/289.01 = [insert#2(@l, @keyX, @valX, @x)] 876.46/289.01 876.46/289.01 [splitAndSort^#(@l)] = [7] @l + [7] 876.46/289.01 > [7] @l + [3] 876.46/289.01 = [c_1(split^#(@l))] 876.46/289.01 876.46/289.01 [split^#(@l)] = [1] @l + [0] 876.46/289.01 >= [1] @l + [0] 876.46/289.01 = [c_2(split#1^#(@l))] 876.46/289.01 876.46/289.01 [split#1^#(::(@x, @xs))] = [1] @xs + [1] 876.46/289.01 > [1] @xs + [0] 876.46/289.01 = [c_3(insert^#(@x, split(@xs)), split^#(@xs))] 876.46/289.01 876.46/289.01 876.46/289.01 We return to the main proof. Consider the set of all dependency 876.46/289.01 pairs 876.46/289.01 876.46/289.01 : 876.46/289.01 { 1: splitAndSort^#(@l) -> c_1(split^#(@l)) 876.46/289.01 , 2: split^#(@l) -> c_2(split#1^#(@l)) 876.46/289.01 , 3: split#1^#(::(@x, @xs)) -> 876.46/289.01 c_3(insert^#(@x, split(@xs)), split^#(@xs)) } 876.46/289.01 876.46/289.01 Processor 'matrix interpretation of dimension 1' induces the 876.46/289.01 complexity certificate YES(?,O(n^1)) on application of dependency 876.46/289.01 pairs {1,3}. These cover all (indirect) predecessors of dependency 876.46/289.01 pairs {1,2,3}, their number of application is equally bounded. The 876.46/289.01 dependency pairs are shifted into the weak component. 876.46/289.01 876.46/289.01 We are left with following problem, upon which TcT provides the 876.46/289.01 certificate YES(O(1),O(1)). 876.46/289.01 876.46/289.01 Weak DPs: 876.46/289.01 { splitAndSort^#(@l) -> c_1(split^#(@l)) 876.46/289.01 , split^#(@l) -> c_2(split#1^#(@l)) 876.46/289.01 , split#1^#(::(@x, @xs)) -> 876.46/289.01 c_3(insert^#(@x, split(@xs)), split^#(@xs)) } 876.46/289.01 Weak Trs: 876.46/289.01 { #equal(@x, @y) -> #eq(@x, @y) 876.46/289.01 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.46/289.01 , #eq(#pos(@x), #0()) -> #false() 876.46/289.01 , #eq(#pos(@x), #neg(@y)) -> #false() 876.46/289.01 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.46/289.01 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.01 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.46/289.01 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.46/289.01 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.01 , #eq(nil(), nil()) -> #true() 876.46/289.01 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.46/289.01 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.01 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.46/289.01 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.46/289.01 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.01 , #eq(#0(), #pos(@y)) -> #false() 876.46/289.01 , #eq(#0(), #0()) -> #true() 876.46/289.01 , #eq(#0(), #neg(@y)) -> #false() 876.46/289.01 , #eq(#0(), #s(@y)) -> #false() 876.46/289.01 , #eq(#neg(@x), #pos(@y)) -> #false() 876.46/289.01 , #eq(#neg(@x), #0()) -> #false() 876.46/289.01 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.46/289.01 , #eq(#s(@x), #0()) -> #false() 876.46/289.01 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.46/289.01 , split(@l) -> split#1(@l) 876.46/289.01 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.46/289.01 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.46/289.01 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.46/289.01 , insert#2(nil(), @keyX, @valX, @x) -> 876.46/289.01 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.46/289.01 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.46/289.01 insert#3(@l1, @keyX, @ls, @valX, @x) 876.46/289.01 , #and(#true(), #true()) -> #true() 876.46/289.01 , #and(#true(), #false()) -> #false() 876.46/289.01 , #and(#false(), #true()) -> #false() 876.46/289.01 , #and(#false(), #false()) -> #false() 876.46/289.01 , split#1(nil()) -> nil() 876.46/289.01 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.46/289.01 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.01 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.46/289.01 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.01 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.46/289.01 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.46/289.01 insert#2(@l, @keyX, @valX, @x) } 876.46/289.01 Obligation: 876.46/289.01 innermost runtime complexity 876.46/289.01 Answer: 876.46/289.01 YES(O(1),O(1)) 876.46/289.01 876.46/289.01 The following weak DPs constitute a sub-graph of the DG that is 876.46/289.01 closed under successors. The DPs are removed. 876.46/289.01 876.46/289.01 { splitAndSort^#(@l) -> c_1(split^#(@l)) 876.46/289.01 , split^#(@l) -> c_2(split#1^#(@l)) 876.46/289.01 , split#1^#(::(@x, @xs)) -> 876.46/289.01 c_3(insert^#(@x, split(@xs)), split^#(@xs)) } 876.46/289.01 876.46/289.01 We are left with following problem, upon which TcT provides the 876.46/289.01 certificate YES(O(1),O(1)). 876.46/289.01 876.46/289.01 Weak Trs: 876.46/289.01 { #equal(@x, @y) -> #eq(@x, @y) 876.46/289.01 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.46/289.01 , #eq(#pos(@x), #0()) -> #false() 876.46/289.01 , #eq(#pos(@x), #neg(@y)) -> #false() 876.46/289.01 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.46/289.01 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.01 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.46/289.01 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.46/289.01 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.01 , #eq(nil(), nil()) -> #true() 876.46/289.01 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.46/289.01 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.01 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.46/289.01 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.46/289.01 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.01 , #eq(#0(), #pos(@y)) -> #false() 876.46/289.01 , #eq(#0(), #0()) -> #true() 876.46/289.01 , #eq(#0(), #neg(@y)) -> #false() 876.46/289.01 , #eq(#0(), #s(@y)) -> #false() 876.46/289.01 , #eq(#neg(@x), #pos(@y)) -> #false() 876.46/289.01 , #eq(#neg(@x), #0()) -> #false() 876.46/289.01 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.46/289.01 , #eq(#s(@x), #0()) -> #false() 876.46/289.01 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.46/289.01 , split(@l) -> split#1(@l) 876.46/289.01 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.46/289.01 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.46/289.01 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.46/289.01 , insert#2(nil(), @keyX, @valX, @x) -> 876.46/289.01 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.46/289.01 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.46/289.01 insert#3(@l1, @keyX, @ls, @valX, @x) 876.46/289.01 , #and(#true(), #true()) -> #true() 876.46/289.01 , #and(#true(), #false()) -> #false() 876.46/289.01 , #and(#false(), #true()) -> #false() 876.46/289.01 , #and(#false(), #false()) -> #false() 876.46/289.01 , split#1(nil()) -> nil() 876.46/289.01 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.46/289.01 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.01 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.46/289.01 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.01 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.46/289.01 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.46/289.01 insert#2(@l, @keyX, @valX, @x) } 876.46/289.01 Obligation: 876.46/289.01 innermost runtime complexity 876.46/289.01 Answer: 876.46/289.01 YES(O(1),O(1)) 876.46/289.01 876.46/289.01 No rule is usable, rules are removed from the input problem. 876.46/289.01 876.46/289.01 We are left with following problem, upon which TcT provides the 876.46/289.01 certificate YES(O(1),O(1)). 876.46/289.01 876.46/289.01 Rules: Empty 876.46/289.01 Obligation: 876.46/289.01 innermost runtime complexity 876.46/289.01 Answer: 876.46/289.01 YES(O(1),O(1)) 876.46/289.01 876.46/289.01 Empty rules are trivially bounded 876.46/289.01 876.46/289.01 We return to the main proof. 876.46/289.01 876.46/289.01 We are left with following problem, upon which TcT provides the 876.46/289.01 certificate YES(O(1),O(n^1)). 876.46/289.01 876.46/289.01 Strict DPs: 876.46/289.01 { insert#3^#(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.46/289.01 c_4(insert#4^#(#equal(@key1, @keyX), 876.46/289.01 @key1, 876.46/289.01 @ls, 876.46/289.01 @valX, 876.46/289.01 @vals1, 876.46/289.01 @x)) 876.46/289.01 , insert#4^#(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.01 c_5(insert^#(@x, @ls)) 876.46/289.01 , insert^#(@x, @l) -> c_6(insert#1^#(@x, @l, @x)) 876.46/289.01 , insert#1^#(tuple#2(@valX, @keyX), @l, @x) -> 876.46/289.01 c_7(insert#2^#(@l, @keyX, @valX, @x)) 876.46/289.01 , insert#2^#(::(@l1, @ls), @keyX, @valX, @x) -> 876.46/289.01 c_8(insert#3^#(@l1, @keyX, @ls, @valX, @x)) } 876.46/289.01 Weak DPs: 876.46/289.01 { splitAndSort^#(@l) -> split^#(@l) 876.46/289.01 , split^#(@l) -> split#1^#(@l) 876.46/289.01 , split#1^#(::(@x, @xs)) -> split^#(@xs) 876.46/289.01 , split#1^#(::(@x, @xs)) -> insert^#(@x, split(@xs)) } 876.46/289.01 Weak Trs: 876.46/289.01 { #equal(@x, @y) -> #eq(@x, @y) 876.46/289.01 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.46/289.01 , #eq(#pos(@x), #0()) -> #false() 876.46/289.01 , #eq(#pos(@x), #neg(@y)) -> #false() 876.46/289.01 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.46/289.01 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.01 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.46/289.01 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.46/289.01 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.01 , #eq(nil(), nil()) -> #true() 876.46/289.01 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.46/289.01 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.01 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.46/289.01 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.46/289.01 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.01 , #eq(#0(), #pos(@y)) -> #false() 876.46/289.01 , #eq(#0(), #0()) -> #true() 876.46/289.01 , #eq(#0(), #neg(@y)) -> #false() 876.46/289.01 , #eq(#0(), #s(@y)) -> #false() 876.46/289.01 , #eq(#neg(@x), #pos(@y)) -> #false() 876.46/289.01 , #eq(#neg(@x), #0()) -> #false() 876.46/289.01 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.46/289.01 , #eq(#s(@x), #0()) -> #false() 876.46/289.01 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.46/289.01 , split(@l) -> split#1(@l) 876.46/289.01 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.46/289.01 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.46/289.01 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.46/289.01 , insert#2(nil(), @keyX, @valX, @x) -> 876.46/289.01 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.46/289.01 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.46/289.01 insert#3(@l1, @keyX, @ls, @valX, @x) 876.46/289.01 , #and(#true(), #true()) -> #true() 876.46/289.01 , #and(#true(), #false()) -> #false() 876.46/289.01 , #and(#false(), #true()) -> #false() 876.46/289.01 , #and(#false(), #false()) -> #false() 876.46/289.01 , split#1(nil()) -> nil() 876.46/289.01 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.46/289.01 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.01 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.46/289.01 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.01 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.46/289.01 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.46/289.01 insert#2(@l, @keyX, @valX, @x) } 876.46/289.01 Obligation: 876.46/289.01 innermost runtime complexity 876.46/289.01 Answer: 876.46/289.01 YES(O(1),O(n^1)) 876.46/289.01 876.46/289.01 We use the processor 'matrix interpretation of dimension 1' to 876.46/289.01 orient following rules strictly. 876.46/289.01 876.46/289.01 DPs: 876.46/289.01 { 2: insert#4^#(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.01 c_5(insert^#(@x, @ls)) 876.46/289.01 , 5: insert#2^#(::(@l1, @ls), @keyX, @valX, @x) -> 876.46/289.01 c_8(insert#3^#(@l1, @keyX, @ls, @valX, @x)) 876.46/289.01 , 7: split^#(@l) -> split#1^#(@l) 876.46/289.01 , 8: split#1^#(::(@x, @xs)) -> split^#(@xs) 876.46/289.01 , 9: split#1^#(::(@x, @xs)) -> insert^#(@x, split(@xs)) } 876.46/289.01 Trs: 876.46/289.01 { split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.46/289.01 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.01 ::(tuple#2(::(@valX, @vals1), @key1), @ls) } 876.46/289.01 876.46/289.01 Sub-proof: 876.46/289.01 ---------- 876.46/289.01 The following argument positions are usable: 876.46/289.01 Uargs(c_4) = {1}, Uargs(c_5) = {1}, Uargs(c_6) = {1}, 876.46/289.01 Uargs(c_7) = {1}, Uargs(c_8) = {1} 876.46/289.01 876.46/289.01 TcT has computed the following constructor-based matrix 876.46/289.01 interpretation satisfying not(EDA). 876.46/289.01 876.46/289.01 [#equal](x1, x2) = [0] 876.46/289.01 876.46/289.01 [#eq](x1, x2) = [0] 876.46/289.01 876.46/289.01 [split](x1) = [2] x1 + [0] 876.46/289.01 876.46/289.01 [#true] = [0] 876.46/289.01 876.46/289.01 [insert#3](x1, x2, x3, x4, x5) = [1] x1 + [1] x3 + [5] 876.46/289.01 876.46/289.01 [insert](x1, x2) = [1] x2 + [3] 876.46/289.01 876.46/289.01 [#pos](x1) = [1] x1 + [0] 876.46/289.01 876.46/289.01 [insert#2](x1, x2, x3, x4) = [1] x1 + [3] 876.46/289.01 876.46/289.01 [#and](x1, x2) = [0] 876.46/289.01 876.46/289.01 [tuple#2](x1, x2) = [1] 876.46/289.01 876.46/289.01 [nil] = [0] 876.46/289.01 876.46/289.01 [split#1](x1) = [2] x1 + [0] 876.46/289.01 876.46/289.01 [insert#4](x1, x2, x3, x4, x5, x6) = [1] x3 + [6] 876.46/289.01 876.46/289.01 [insert#1](x1, x2, x3) = [1] x2 + [3] 876.46/289.01 876.46/289.01 [#false] = [0] 876.46/289.01 876.46/289.01 [::](x1, x2) = [1] x1 + [1] x2 + [2] 876.46/289.01 876.46/289.01 [#0] = [0] 876.46/289.01 876.46/289.01 [#neg](x1) = [1] x1 + [0] 876.46/289.01 876.46/289.01 [#s](x1) = [1] x1 + [0] 876.46/289.01 876.46/289.01 [splitAndSort^#](x1) = [7] x1 + [7] 876.46/289.01 876.46/289.01 [split^#](x1) = [4] x1 + [7] 876.46/289.01 876.46/289.01 [split#1^#](x1) = [4] x1 + [1] 876.46/289.01 876.46/289.01 [insert#3^#](x1, x2, x3, x4, x5) = [1] x1 + [2] x3 + [3] 876.46/289.01 876.46/289.01 [insert#4^#](x1, x2, x3, x4, x5, x6) = [2] x3 + [4] 876.46/289.01 876.46/289.01 [insert^#](x1, x2) = [2] x2 + [0] 876.46/289.01 876.46/289.01 [insert#1^#](x1, x2, x3) = [2] x2 + [0] 876.46/289.01 876.46/289.01 [insert#2^#](x1, x2, x3, x4) = [2] x1 + [0] 876.46/289.01 876.46/289.01 [c_4](x1) = [1] x1 + [0] 876.46/289.01 876.46/289.01 [c_5](x1) = [1] x1 + [1] 876.46/289.01 876.46/289.01 [c_6](x1) = [1] x1 + [0] 876.46/289.01 876.46/289.01 [c_7](x1) = [1] x1 + [0] 876.46/289.01 876.46/289.01 [c_8](x1) = [1] x1 + [0] 876.46/289.01 876.46/289.01 The order satisfies the following ordering constraints: 876.46/289.01 876.46/289.01 [#equal(@x, @y)] = [0] 876.46/289.01 >= [0] 876.46/289.01 = [#eq(@x, @y)] 876.46/289.01 876.46/289.01 [#eq(#pos(@x), #pos(@y))] = [0] 876.46/289.01 >= [0] 876.46/289.01 = [#eq(@x, @y)] 876.46/289.01 876.46/289.01 [#eq(#pos(@x), #0())] = [0] 876.46/289.01 >= [0] 876.46/289.01 = [#false()] 876.46/289.01 876.46/289.01 [#eq(#pos(@x), #neg(@y))] = [0] 876.46/289.01 >= [0] 876.46/289.01 = [#false()] 876.46/289.01 876.46/289.01 [#eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2))] = [0] 876.46/289.01 >= [0] 876.46/289.01 = [#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))] 876.46/289.01 876.46/289.01 [#eq(tuple#2(@x_1, @x_2), nil())] = [0] 876.46/289.01 >= [0] 876.46/289.01 = [#false()] 876.46/289.01 876.46/289.01 [#eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2))] = [0] 876.46/289.01 >= [0] 876.46/289.01 = [#false()] 876.46/289.01 876.46/289.01 [#eq(nil(), tuple#2(@y_1, @y_2))] = [0] 876.46/289.01 >= [0] 876.46/289.01 = [#false()] 876.46/289.01 876.46/289.01 [#eq(nil(), nil())] = [0] 876.46/289.01 >= [0] 876.46/289.01 = [#true()] 876.46/289.01 876.46/289.01 [#eq(nil(), ::(@y_1, @y_2))] = [0] 876.46/289.01 >= [0] 876.46/289.01 = [#false()] 876.46/289.01 876.46/289.01 [#eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2))] = [0] 876.46/289.01 >= [0] 876.46/289.01 = [#false()] 876.46/289.01 876.46/289.01 [#eq(::(@x_1, @x_2), nil())] = [0] 876.46/289.01 >= [0] 876.46/289.01 = [#false()] 876.46/289.01 876.46/289.01 [#eq(::(@x_1, @x_2), ::(@y_1, @y_2))] = [0] 876.46/289.01 >= [0] 876.46/289.01 = [#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))] 876.46/289.01 876.46/289.01 [#eq(#0(), #pos(@y))] = [0] 876.46/289.01 >= [0] 876.46/289.01 = [#false()] 876.46/289.01 876.46/289.01 [#eq(#0(), #0())] = [0] 876.46/289.01 >= [0] 876.46/289.01 = [#true()] 876.46/289.01 876.46/289.01 [#eq(#0(), #neg(@y))] = [0] 876.46/289.01 >= [0] 876.46/289.01 = [#false()] 876.46/289.01 876.46/289.01 [#eq(#0(), #s(@y))] = [0] 876.46/289.01 >= [0] 876.46/289.01 = [#false()] 876.46/289.01 876.46/289.01 [#eq(#neg(@x), #pos(@y))] = [0] 876.46/289.01 >= [0] 876.46/289.01 = [#false()] 876.46/289.01 876.46/289.01 [#eq(#neg(@x), #0())] = [0] 876.46/289.01 >= [0] 876.46/289.01 = [#false()] 876.46/289.01 876.46/289.01 [#eq(#neg(@x), #neg(@y))] = [0] 876.46/289.01 >= [0] 876.46/289.01 = [#eq(@x, @y)] 876.46/289.01 876.46/289.01 [#eq(#s(@x), #0())] = [0] 876.46/289.01 >= [0] 876.46/289.01 = [#false()] 876.46/289.01 876.46/289.01 [#eq(#s(@x), #s(@y))] = [0] 876.46/289.01 >= [0] 876.46/289.01 = [#eq(@x, @y)] 876.46/289.01 876.46/289.01 [split(@l)] = [2] @l + [0] 876.46/289.01 >= [2] @l + [0] 876.46/289.01 = [split#1(@l)] 876.46/289.01 876.46/289.01 [insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x)] = [1] @ls + [6] 876.46/289.01 >= [1] @ls + [6] 876.46/289.01 = [insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x)] 876.46/289.01 876.46/289.01 [insert(@x, @l)] = [1] @l + [3] 876.46/289.01 >= [1] @l + [3] 876.46/289.01 = [insert#1(@x, @l, @x)] 876.46/289.01 876.46/289.01 [insert#2(nil(), @keyX, @valX, @x)] = [3] 876.46/289.01 >= [3] 876.46/289.01 = [::(tuple#2(::(@valX, nil()), @keyX), nil())] 876.46/289.01 876.46/289.01 [insert#2(::(@l1, @ls), @keyX, @valX, @x)] = [1] @l1 + [1] @ls + [5] 876.46/289.01 >= [1] @l1 + [1] @ls + [5] 876.46/289.01 = [insert#3(@l1, @keyX, @ls, @valX, @x)] 876.46/289.01 876.46/289.01 [#and(#true(), #true())] = [0] 876.46/289.01 >= [0] 876.46/289.01 = [#true()] 876.46/289.01 876.46/289.01 [#and(#true(), #false())] = [0] 876.46/289.01 >= [0] 876.46/289.01 = [#false()] 876.46/289.01 876.46/289.01 [#and(#false(), #true())] = [0] 876.46/289.01 >= [0] 876.46/289.01 = [#false()] 876.46/289.01 876.46/289.01 [#and(#false(), #false())] = [0] 876.46/289.01 >= [0] 876.46/289.01 = [#false()] 876.46/289.01 876.46/289.01 [split#1(nil())] = [0] 876.46/289.01 >= [0] 876.46/289.01 = [nil()] 876.46/289.01 876.46/289.01 [split#1(::(@x, @xs))] = [2] @x + [2] @xs + [4] 876.46/289.01 > [2] @xs + [3] 876.46/289.01 = [insert(@x, split(@xs))] 876.46/289.01 876.46/289.01 [insert#4(#true(), @key1, @ls, @valX, @vals1, @x)] = [1] @ls + [6] 876.46/289.01 > [1] @ls + [3] 876.46/289.01 = [::(tuple#2(::(@valX, @vals1), @key1), @ls)] 876.46/289.01 876.46/289.01 [insert#4(#false(), @key1, @ls, @valX, @vals1, @x)] = [1] @ls + [6] 876.46/289.01 >= [1] @ls + [6] 876.46/289.01 = [::(tuple#2(@vals1, @key1), insert(@x, @ls))] 876.46/289.01 876.46/289.01 [insert#1(tuple#2(@valX, @keyX), @l, @x)] = [1] @l + [3] 876.46/289.01 >= [1] @l + [3] 876.46/289.01 = [insert#2(@l, @keyX, @valX, @x)] 876.46/289.01 876.46/289.01 [splitAndSort^#(@l)] = [7] @l + [7] 876.46/289.01 >= [4] @l + [7] 876.46/289.01 = [split^#(@l)] 876.46/289.02 876.46/289.02 [split^#(@l)] = [4] @l + [7] 876.46/289.02 > [4] @l + [1] 876.46/289.02 = [split#1^#(@l)] 876.46/289.02 876.46/289.02 [split#1^#(::(@x, @xs))] = [4] @x + [4] @xs + [9] 876.46/289.02 > [4] @xs + [7] 876.46/289.02 = [split^#(@xs)] 876.46/289.02 876.46/289.02 [split#1^#(::(@x, @xs))] = [4] @x + [4] @xs + [9] 876.46/289.02 > [4] @xs + [0] 876.46/289.02 = [insert^#(@x, split(@xs))] 876.46/289.02 876.46/289.02 [insert#3^#(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x)] = [2] @ls + [4] 876.46/289.02 >= [2] @ls + [4] 876.46/289.02 = [c_4(insert#4^#(#equal(@key1, @keyX), 876.46/289.02 @key1, 876.46/289.02 @ls, 876.46/289.02 @valX, 876.46/289.02 @vals1, 876.46/289.02 @x))] 876.46/289.02 876.46/289.02 [insert#4^#(#false(), @key1, @ls, @valX, @vals1, @x)] = [2] @ls + [4] 876.46/289.02 > [2] @ls + [1] 876.46/289.02 = [c_5(insert^#(@x, @ls))] 876.46/289.02 876.46/289.02 [insert^#(@x, @l)] = [2] @l + [0] 876.46/289.02 >= [2] @l + [0] 876.46/289.02 = [c_6(insert#1^#(@x, @l, @x))] 876.46/289.02 876.46/289.02 [insert#1^#(tuple#2(@valX, @keyX), @l, @x)] = [2] @l + [0] 876.46/289.02 >= [2] @l + [0] 876.46/289.02 = [c_7(insert#2^#(@l, @keyX, @valX, @x))] 876.46/289.02 876.46/289.02 [insert#2^#(::(@l1, @ls), @keyX, @valX, @x)] = [2] @l1 + [2] @ls + [4] 876.46/289.02 > [1] @l1 + [2] @ls + [3] 876.46/289.02 = [c_8(insert#3^#(@l1, @keyX, @ls, @valX, @x))] 876.46/289.02 876.46/289.02 876.46/289.02 We return to the main proof. Consider the set of all dependency 876.46/289.02 pairs 876.46/289.02 876.46/289.02 : 876.46/289.02 { 1: insert#3^#(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.46/289.02 c_4(insert#4^#(#equal(@key1, @keyX), 876.46/289.02 @key1, 876.46/289.02 @ls, 876.46/289.02 @valX, 876.46/289.02 @vals1, 876.46/289.02 @x)) 876.46/289.02 , 2: insert#4^#(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.02 c_5(insert^#(@x, @ls)) 876.46/289.02 , 3: insert^#(@x, @l) -> c_6(insert#1^#(@x, @l, @x)) 876.46/289.02 , 4: insert#1^#(tuple#2(@valX, @keyX), @l, @x) -> 876.46/289.02 c_7(insert#2^#(@l, @keyX, @valX, @x)) 876.46/289.02 , 5: insert#2^#(::(@l1, @ls), @keyX, @valX, @x) -> 876.46/289.02 c_8(insert#3^#(@l1, @keyX, @ls, @valX, @x)) 876.46/289.02 , 6: splitAndSort^#(@l) -> split^#(@l) 876.46/289.02 , 7: split^#(@l) -> split#1^#(@l) 876.46/289.02 , 8: split#1^#(::(@x, @xs)) -> split^#(@xs) 876.46/289.02 , 9: split#1^#(::(@x, @xs)) -> insert^#(@x, split(@xs)) } 876.46/289.02 876.46/289.02 Processor 'matrix interpretation of dimension 1' induces the 876.46/289.02 complexity certificate YES(?,O(n^1)) on application of dependency 876.46/289.02 pairs {2,5,7,8,9}. These cover all (indirect) predecessors of 876.46/289.02 dependency pairs {1,2,3,4,5,6,7,8,9}, their number of application 876.46/289.02 is equally bounded. The dependency pairs are shifted into the weak 876.46/289.02 component. 876.46/289.02 876.46/289.02 We are left with following problem, upon which TcT provides the 876.46/289.02 certificate YES(O(1),O(1)). 876.46/289.02 876.46/289.02 Weak DPs: 876.46/289.02 { splitAndSort^#(@l) -> split^#(@l) 876.46/289.02 , split^#(@l) -> split#1^#(@l) 876.46/289.02 , split#1^#(::(@x, @xs)) -> split^#(@xs) 876.46/289.02 , split#1^#(::(@x, @xs)) -> insert^#(@x, split(@xs)) 876.46/289.02 , insert#3^#(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.46/289.02 c_4(insert#4^#(#equal(@key1, @keyX), 876.46/289.02 @key1, 876.46/289.02 @ls, 876.46/289.02 @valX, 876.46/289.02 @vals1, 876.46/289.02 @x)) 876.46/289.02 , insert#4^#(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.02 c_5(insert^#(@x, @ls)) 876.46/289.02 , insert^#(@x, @l) -> c_6(insert#1^#(@x, @l, @x)) 876.46/289.02 , insert#1^#(tuple#2(@valX, @keyX), @l, @x) -> 876.46/289.02 c_7(insert#2^#(@l, @keyX, @valX, @x)) 876.46/289.02 , insert#2^#(::(@l1, @ls), @keyX, @valX, @x) -> 876.46/289.02 c_8(insert#3^#(@l1, @keyX, @ls, @valX, @x)) } 876.46/289.02 Weak Trs: 876.46/289.02 { #equal(@x, @y) -> #eq(@x, @y) 876.46/289.02 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.46/289.02 , #eq(#pos(@x), #0()) -> #false() 876.46/289.02 , #eq(#pos(@x), #neg(@y)) -> #false() 876.46/289.02 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.46/289.02 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.02 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.46/289.02 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.46/289.02 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.02 , #eq(nil(), nil()) -> #true() 876.46/289.02 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.46/289.02 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.02 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.46/289.02 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.46/289.02 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.02 , #eq(#0(), #pos(@y)) -> #false() 876.46/289.02 , #eq(#0(), #0()) -> #true() 876.46/289.02 , #eq(#0(), #neg(@y)) -> #false() 876.46/289.02 , #eq(#0(), #s(@y)) -> #false() 876.46/289.02 , #eq(#neg(@x), #pos(@y)) -> #false() 876.46/289.02 , #eq(#neg(@x), #0()) -> #false() 876.46/289.02 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.46/289.02 , #eq(#s(@x), #0()) -> #false() 876.46/289.02 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.46/289.02 , split(@l) -> split#1(@l) 876.46/289.02 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.46/289.02 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.46/289.02 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.46/289.02 , insert#2(nil(), @keyX, @valX, @x) -> 876.46/289.02 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.46/289.02 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.46/289.02 insert#3(@l1, @keyX, @ls, @valX, @x) 876.46/289.02 , #and(#true(), #true()) -> #true() 876.46/289.02 , #and(#true(), #false()) -> #false() 876.46/289.02 , #and(#false(), #true()) -> #false() 876.46/289.02 , #and(#false(), #false()) -> #false() 876.46/289.02 , split#1(nil()) -> nil() 876.46/289.02 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.46/289.02 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.02 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.46/289.02 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.02 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.46/289.02 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.46/289.02 insert#2(@l, @keyX, @valX, @x) } 876.46/289.02 Obligation: 876.46/289.02 innermost runtime complexity 876.46/289.02 Answer: 876.46/289.02 YES(O(1),O(1)) 876.46/289.02 876.46/289.02 The following weak DPs constitute a sub-graph of the DG that is 876.46/289.02 closed under successors. The DPs are removed. 876.46/289.02 876.46/289.02 { splitAndSort^#(@l) -> split^#(@l) 876.46/289.02 , split^#(@l) -> split#1^#(@l) 876.46/289.02 , split#1^#(::(@x, @xs)) -> split^#(@xs) 876.46/289.02 , split#1^#(::(@x, @xs)) -> insert^#(@x, split(@xs)) 876.46/289.02 , insert#3^#(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.46/289.02 c_4(insert#4^#(#equal(@key1, @keyX), 876.46/289.02 @key1, 876.46/289.02 @ls, 876.46/289.02 @valX, 876.46/289.02 @vals1, 876.46/289.02 @x)) 876.46/289.02 , insert#4^#(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.02 c_5(insert^#(@x, @ls)) 876.46/289.02 , insert^#(@x, @l) -> c_6(insert#1^#(@x, @l, @x)) 876.46/289.02 , insert#1^#(tuple#2(@valX, @keyX), @l, @x) -> 876.46/289.02 c_7(insert#2^#(@l, @keyX, @valX, @x)) 876.46/289.02 , insert#2^#(::(@l1, @ls), @keyX, @valX, @x) -> 876.46/289.02 c_8(insert#3^#(@l1, @keyX, @ls, @valX, @x)) } 876.46/289.02 876.46/289.02 We are left with following problem, upon which TcT provides the 876.46/289.02 certificate YES(O(1),O(1)). 876.46/289.02 876.46/289.02 Weak Trs: 876.46/289.02 { #equal(@x, @y) -> #eq(@x, @y) 876.46/289.02 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.46/289.02 , #eq(#pos(@x), #0()) -> #false() 876.46/289.02 , #eq(#pos(@x), #neg(@y)) -> #false() 876.46/289.02 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.46/289.02 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.02 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.46/289.02 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.46/289.02 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.02 , #eq(nil(), nil()) -> #true() 876.46/289.02 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.46/289.02 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.02 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.46/289.02 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.46/289.02 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.02 , #eq(#0(), #pos(@y)) -> #false() 876.46/289.02 , #eq(#0(), #0()) -> #true() 876.46/289.02 , #eq(#0(), #neg(@y)) -> #false() 876.46/289.02 , #eq(#0(), #s(@y)) -> #false() 876.46/289.02 , #eq(#neg(@x), #pos(@y)) -> #false() 876.46/289.02 , #eq(#neg(@x), #0()) -> #false() 876.46/289.02 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.46/289.02 , #eq(#s(@x), #0()) -> #false() 876.46/289.02 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.46/289.02 , split(@l) -> split#1(@l) 876.46/289.02 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.46/289.02 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.46/289.02 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.46/289.02 , insert#2(nil(), @keyX, @valX, @x) -> 876.46/289.02 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.46/289.02 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.46/289.02 insert#3(@l1, @keyX, @ls, @valX, @x) 876.46/289.02 , #and(#true(), #true()) -> #true() 876.46/289.02 , #and(#true(), #false()) -> #false() 876.46/289.02 , #and(#false(), #true()) -> #false() 876.46/289.02 , #and(#false(), #false()) -> #false() 876.46/289.02 , split#1(nil()) -> nil() 876.46/289.02 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.46/289.02 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.02 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.46/289.02 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.02 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.46/289.02 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.46/289.02 insert#2(@l, @keyX, @valX, @x) } 876.46/289.02 Obligation: 876.46/289.02 innermost runtime complexity 876.46/289.02 Answer: 876.46/289.02 YES(O(1),O(1)) 876.46/289.02 876.46/289.02 No rule is usable, rules are removed from the input problem. 876.46/289.02 876.46/289.02 We are left with following problem, upon which TcT provides the 876.46/289.02 certificate YES(O(1),O(1)). 876.46/289.02 876.46/289.02 Rules: Empty 876.46/289.02 Obligation: 876.46/289.02 innermost runtime complexity 876.46/289.02 Answer: 876.46/289.02 YES(O(1),O(1)) 876.46/289.02 876.46/289.02 Empty rules are trivially bounded 876.46/289.02 876.46/289.02 S) We are left with following problem, upon which TcT provides the 876.46/289.02 certificate YES(O(1),O(n^6)). 876.46/289.02 876.46/289.02 Strict DPs: 876.46/289.02 { sortAll^#(@l) -> c_2(sortAll#1^#(@l)) 876.46/289.02 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> 876.46/289.02 c_4(quicksort^#(@vals), sortAll^#(@xs)) 876.46/289.02 , quicksort^#(@l) -> c_5(quicksort#1^#(@l)) 876.46/289.02 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> 876.46/289.02 c_6(append^#(quicksort(@xs), ::(@z, quicksort(@ys))), 876.46/289.02 quicksort^#(@xs), 876.46/289.02 quicksort^#(@ys)) 876.46/289.02 , append^#(@l, @ys) -> c_7(append#1^#(@l, @ys)) 876.46/289.02 , append#1^#(::(@x, @xs), @ys) -> c_9(append^#(@xs, @ys)) 876.46/289.02 , sortAll#1^#(::(@x, @xs)) -> c_14(sortAll#2^#(@x, @xs)) 876.46/289.02 , quicksort#1^#(::(@z, @zs)) -> 876.46/289.02 c_16(quicksort#2^#(splitqs(@z, @zs), @z), splitqs^#(@z, @zs)) 876.46/289.02 , splitqs^#(@pivot, @l) -> c_17(splitqs#1^#(@l, @pivot)) 876.46/289.02 , splitqs#1^#(::(@x, @xs), @pivot) -> 876.46/289.02 c_18(splitqs^#(@pivot, @xs)) } 876.46/289.02 Weak DPs: 876.46/289.02 { splitAndSort^#(@l) -> c_1(sortAll^#(split(@l)), split^#(@l)) 876.46/289.02 , split^#(@l) -> c_3(split#1^#(@l)) 876.46/289.02 , split#1^#(::(@x, @xs)) -> 876.46/289.02 c_8(insert^#(@x, split(@xs)), split^#(@xs)) 876.46/289.02 , insert#3^#(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.46/289.02 c_10(insert#4^#(#equal(@key1, @keyX), 876.46/289.02 @key1, 876.46/289.02 @ls, 876.46/289.02 @valX, 876.46/289.02 @vals1, 876.46/289.02 @x)) 876.46/289.02 , insert#4^#(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.02 c_11(insert^#(@x, @ls)) 876.46/289.02 , insert^#(@x, @l) -> c_12(insert#1^#(@x, @l, @x)) 876.46/289.02 , insert#1^#(tuple#2(@valX, @keyX), @l, @x) -> 876.46/289.02 c_13(insert#2^#(@l, @keyX, @valX, @x)) 876.46/289.02 , insert#2^#(::(@l1, @ls), @keyX, @valX, @x) -> 876.46/289.02 c_15(insert#3^#(@l1, @keyX, @ls, @valX, @x)) } 876.46/289.02 Weak Trs: 876.46/289.02 { #equal(@x, @y) -> #eq(@x, @y) 876.46/289.02 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.46/289.02 , #eq(#pos(@x), #0()) -> #false() 876.46/289.02 , #eq(#pos(@x), #neg(@y)) -> #false() 876.46/289.02 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.46/289.02 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.02 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.46/289.02 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.46/289.02 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.02 , #eq(nil(), nil()) -> #true() 876.46/289.02 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.46/289.02 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.02 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.46/289.02 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.46/289.02 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.02 , #eq(#0(), #pos(@y)) -> #false() 876.46/289.02 , #eq(#0(), #0()) -> #true() 876.46/289.02 , #eq(#0(), #neg(@y)) -> #false() 876.46/289.02 , #eq(#0(), #s(@y)) -> #false() 876.46/289.02 , #eq(#neg(@x), #pos(@y)) -> #false() 876.46/289.02 , #eq(#neg(@x), #0()) -> #false() 876.46/289.02 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.46/289.02 , #eq(#s(@x), #0()) -> #false() 876.46/289.02 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.46/289.02 , quicksort#2(tuple#2(@xs, @ys), @z) -> 876.46/289.02 append(quicksort(@xs), ::(@z, quicksort(@ys))) 876.46/289.02 , split(@l) -> split#1(@l) 876.46/289.02 , append(@l, @ys) -> append#1(@l, @ys) 876.46/289.02 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.46/289.02 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.46/289.02 , #ckgt(#EQ()) -> #false() 876.46/289.02 , #ckgt(#LT()) -> #false() 876.46/289.02 , #ckgt(#GT()) -> #true() 876.46/289.02 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.46/289.02 , insert#2(nil(), @keyX, @valX, @x) -> 876.46/289.02 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.46/289.02 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.46/289.02 insert#3(@l1, @keyX, @ls, @valX, @x) 876.46/289.02 , #and(#true(), #true()) -> #true() 876.46/289.02 , #and(#true(), #false()) -> #false() 876.46/289.02 , #and(#false(), #true()) -> #false() 876.46/289.02 , #and(#false(), #false()) -> #false() 876.46/289.02 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 876.46/289.02 , #compare(#pos(@x), #0()) -> #GT() 876.46/289.02 , #compare(#pos(@x), #neg(@y)) -> #GT() 876.46/289.02 , #compare(#0(), #pos(@y)) -> #LT() 876.46/289.02 , #compare(#0(), #0()) -> #EQ() 876.46/289.02 , #compare(#0(), #neg(@y)) -> #GT() 876.46/289.02 , #compare(#0(), #s(@y)) -> #LT() 876.46/289.02 , #compare(#neg(@x), #pos(@y)) -> #LT() 876.46/289.02 , #compare(#neg(@x), #0()) -> #LT() 876.46/289.02 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 876.46/289.02 , #compare(#s(@x), #0()) -> #GT() 876.46/289.02 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 876.46/289.02 , split#1(nil()) -> nil() 876.46/289.02 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.46/289.02 , #greater(@x, @y) -> #ckgt(#compare(@x, @y)) 876.46/289.02 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.02 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.46/289.02 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.02 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.46/289.02 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.46/289.02 insert#2(@l, @keyX, @valX, @x) 876.46/289.02 , quicksort#1(nil()) -> nil() 876.46/289.02 , quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) 876.46/289.02 , append#1(nil(), @ys) -> @ys 876.46/289.02 , append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) 876.46/289.02 , splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) 876.46/289.02 , quicksort(@l) -> quicksort#1(@l) 876.46/289.02 , splitqs#3(#true(), @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) 876.46/289.02 , splitqs#3(#false(), @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) 876.46/289.02 , splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> 876.46/289.02 splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) 876.46/289.02 , splitqs#1(nil(), @pivot) -> tuple#2(nil(), nil()) 876.46/289.02 , splitqs#1(::(@x, @xs), @pivot) -> 876.46/289.02 splitqs#2(splitqs(@pivot, @xs), @pivot, @x) } 876.46/289.02 Obligation: 876.46/289.02 innermost runtime complexity 876.46/289.02 Answer: 876.46/289.02 YES(O(1),O(n^6)) 876.46/289.02 876.46/289.02 The following weak DPs constitute a sub-graph of the DG that is 876.46/289.02 closed under successors. The DPs are removed. 876.46/289.02 876.46/289.02 { split^#(@l) -> c_3(split#1^#(@l)) 876.46/289.02 , split#1^#(::(@x, @xs)) -> 876.46/289.02 c_8(insert^#(@x, split(@xs)), split^#(@xs)) 876.46/289.02 , insert#3^#(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.46/289.02 c_10(insert#4^#(#equal(@key1, @keyX), 876.46/289.02 @key1, 876.46/289.02 @ls, 876.46/289.02 @valX, 876.46/289.02 @vals1, 876.46/289.02 @x)) 876.46/289.02 , insert#4^#(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.02 c_11(insert^#(@x, @ls)) 876.46/289.02 , insert^#(@x, @l) -> c_12(insert#1^#(@x, @l, @x)) 876.46/289.02 , insert#1^#(tuple#2(@valX, @keyX), @l, @x) -> 876.46/289.02 c_13(insert#2^#(@l, @keyX, @valX, @x)) 876.46/289.02 , insert#2^#(::(@l1, @ls), @keyX, @valX, @x) -> 876.46/289.02 c_15(insert#3^#(@l1, @keyX, @ls, @valX, @x)) } 876.46/289.02 876.46/289.02 We are left with following problem, upon which TcT provides the 876.46/289.02 certificate YES(O(1),O(n^6)). 876.46/289.02 876.46/289.02 Strict DPs: 876.46/289.02 { sortAll^#(@l) -> c_2(sortAll#1^#(@l)) 876.46/289.02 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> 876.46/289.02 c_4(quicksort^#(@vals), sortAll^#(@xs)) 876.46/289.02 , quicksort^#(@l) -> c_5(quicksort#1^#(@l)) 876.46/289.02 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> 876.46/289.02 c_6(append^#(quicksort(@xs), ::(@z, quicksort(@ys))), 876.46/289.02 quicksort^#(@xs), 876.46/289.02 quicksort^#(@ys)) 876.46/289.02 , append^#(@l, @ys) -> c_7(append#1^#(@l, @ys)) 876.46/289.02 , append#1^#(::(@x, @xs), @ys) -> c_9(append^#(@xs, @ys)) 876.46/289.02 , sortAll#1^#(::(@x, @xs)) -> c_14(sortAll#2^#(@x, @xs)) 876.46/289.02 , quicksort#1^#(::(@z, @zs)) -> 876.46/289.02 c_16(quicksort#2^#(splitqs(@z, @zs), @z), splitqs^#(@z, @zs)) 876.46/289.02 , splitqs^#(@pivot, @l) -> c_17(splitqs#1^#(@l, @pivot)) 876.46/289.02 , splitqs#1^#(::(@x, @xs), @pivot) -> 876.46/289.02 c_18(splitqs^#(@pivot, @xs)) } 876.46/289.02 Weak DPs: 876.46/289.02 { splitAndSort^#(@l) -> c_1(sortAll^#(split(@l)), split^#(@l)) } 876.46/289.02 Weak Trs: 876.46/289.02 { #equal(@x, @y) -> #eq(@x, @y) 876.46/289.02 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.46/289.02 , #eq(#pos(@x), #0()) -> #false() 876.46/289.02 , #eq(#pos(@x), #neg(@y)) -> #false() 876.46/289.02 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.46/289.02 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.02 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.46/289.02 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.46/289.02 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.02 , #eq(nil(), nil()) -> #true() 876.46/289.02 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.46/289.02 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.02 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.46/289.02 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.46/289.02 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.02 , #eq(#0(), #pos(@y)) -> #false() 876.46/289.02 , #eq(#0(), #0()) -> #true() 876.46/289.02 , #eq(#0(), #neg(@y)) -> #false() 876.46/289.02 , #eq(#0(), #s(@y)) -> #false() 876.46/289.02 , #eq(#neg(@x), #pos(@y)) -> #false() 876.46/289.02 , #eq(#neg(@x), #0()) -> #false() 876.46/289.02 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.46/289.02 , #eq(#s(@x), #0()) -> #false() 876.46/289.02 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.46/289.02 , quicksort#2(tuple#2(@xs, @ys), @z) -> 876.46/289.02 append(quicksort(@xs), ::(@z, quicksort(@ys))) 876.46/289.02 , split(@l) -> split#1(@l) 876.46/289.02 , append(@l, @ys) -> append#1(@l, @ys) 876.46/289.02 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.46/289.02 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.46/289.02 , #ckgt(#EQ()) -> #false() 876.46/289.02 , #ckgt(#LT()) -> #false() 876.46/289.02 , #ckgt(#GT()) -> #true() 876.46/289.02 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.46/289.02 , insert#2(nil(), @keyX, @valX, @x) -> 876.46/289.02 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.46/289.02 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.46/289.02 insert#3(@l1, @keyX, @ls, @valX, @x) 876.46/289.02 , #and(#true(), #true()) -> #true() 876.46/289.02 , #and(#true(), #false()) -> #false() 876.46/289.02 , #and(#false(), #true()) -> #false() 876.46/289.02 , #and(#false(), #false()) -> #false() 876.46/289.02 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 876.46/289.02 , #compare(#pos(@x), #0()) -> #GT() 876.46/289.02 , #compare(#pos(@x), #neg(@y)) -> #GT() 876.46/289.02 , #compare(#0(), #pos(@y)) -> #LT() 876.46/289.02 , #compare(#0(), #0()) -> #EQ() 876.46/289.02 , #compare(#0(), #neg(@y)) -> #GT() 876.46/289.02 , #compare(#0(), #s(@y)) -> #LT() 876.46/289.02 , #compare(#neg(@x), #pos(@y)) -> #LT() 876.46/289.02 , #compare(#neg(@x), #0()) -> #LT() 876.46/289.02 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 876.46/289.02 , #compare(#s(@x), #0()) -> #GT() 876.46/289.02 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 876.46/289.02 , split#1(nil()) -> nil() 876.46/289.02 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.46/289.02 , #greater(@x, @y) -> #ckgt(#compare(@x, @y)) 876.46/289.02 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.02 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.46/289.02 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.02 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.46/289.02 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.46/289.02 insert#2(@l, @keyX, @valX, @x) 876.46/289.02 , quicksort#1(nil()) -> nil() 876.46/289.02 , quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) 876.46/289.02 , append#1(nil(), @ys) -> @ys 876.46/289.02 , append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) 876.46/289.02 , splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) 876.46/289.02 , quicksort(@l) -> quicksort#1(@l) 876.46/289.02 , splitqs#3(#true(), @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) 876.46/289.02 , splitqs#3(#false(), @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) 876.46/289.02 , splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> 876.46/289.02 splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) 876.46/289.02 , splitqs#1(nil(), @pivot) -> tuple#2(nil(), nil()) 876.46/289.02 , splitqs#1(::(@x, @xs), @pivot) -> 876.46/289.02 splitqs#2(splitqs(@pivot, @xs), @pivot, @x) } 876.46/289.02 Obligation: 876.46/289.02 innermost runtime complexity 876.46/289.02 Answer: 876.46/289.02 YES(O(1),O(n^6)) 876.46/289.02 876.46/289.02 Due to missing edges in the dependency-graph, the right-hand sides 876.46/289.02 of following rules could be simplified: 876.46/289.02 876.46/289.02 { splitAndSort^#(@l) -> c_1(sortAll^#(split(@l)), split^#(@l)) } 876.46/289.02 876.46/289.02 We are left with following problem, upon which TcT provides the 876.46/289.02 certificate YES(O(1),O(n^6)). 876.46/289.02 876.46/289.02 Strict DPs: 876.46/289.02 { sortAll^#(@l) -> c_1(sortAll#1^#(@l)) 876.46/289.02 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> 876.46/289.02 c_2(quicksort^#(@vals), sortAll^#(@xs)) 876.46/289.02 , quicksort^#(@l) -> c_3(quicksort#1^#(@l)) 876.46/289.02 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> 876.46/289.02 c_4(append^#(quicksort(@xs), ::(@z, quicksort(@ys))), 876.46/289.02 quicksort^#(@xs), 876.46/289.03 quicksort^#(@ys)) 876.46/289.03 , append^#(@l, @ys) -> c_5(append#1^#(@l, @ys)) 876.46/289.03 , append#1^#(::(@x, @xs), @ys) -> c_6(append^#(@xs, @ys)) 876.46/289.03 , sortAll#1^#(::(@x, @xs)) -> c_7(sortAll#2^#(@x, @xs)) 876.46/289.03 , quicksort#1^#(::(@z, @zs)) -> 876.46/289.03 c_8(quicksort#2^#(splitqs(@z, @zs), @z), splitqs^#(@z, @zs)) 876.46/289.03 , splitqs^#(@pivot, @l) -> c_9(splitqs#1^#(@l, @pivot)) 876.46/289.03 , splitqs#1^#(::(@x, @xs), @pivot) -> 876.46/289.03 c_10(splitqs^#(@pivot, @xs)) } 876.46/289.03 Weak DPs: { splitAndSort^#(@l) -> c_11(sortAll^#(split(@l))) } 876.46/289.03 Weak Trs: 876.46/289.03 { #equal(@x, @y) -> #eq(@x, @y) 876.46/289.03 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.46/289.03 , #eq(#pos(@x), #0()) -> #false() 876.46/289.03 , #eq(#pos(@x), #neg(@y)) -> #false() 876.46/289.03 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.46/289.03 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.03 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.46/289.03 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.46/289.03 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.03 , #eq(nil(), nil()) -> #true() 876.46/289.03 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.46/289.03 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.03 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.46/289.03 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.46/289.03 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.03 , #eq(#0(), #pos(@y)) -> #false() 876.46/289.03 , #eq(#0(), #0()) -> #true() 876.46/289.03 , #eq(#0(), #neg(@y)) -> #false() 876.46/289.03 , #eq(#0(), #s(@y)) -> #false() 876.46/289.03 , #eq(#neg(@x), #pos(@y)) -> #false() 876.46/289.03 , #eq(#neg(@x), #0()) -> #false() 876.46/289.03 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.46/289.03 , #eq(#s(@x), #0()) -> #false() 876.46/289.03 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.46/289.03 , quicksort#2(tuple#2(@xs, @ys), @z) -> 876.46/289.03 append(quicksort(@xs), ::(@z, quicksort(@ys))) 876.46/289.03 , split(@l) -> split#1(@l) 876.46/289.03 , append(@l, @ys) -> append#1(@l, @ys) 876.46/289.03 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.46/289.03 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.46/289.03 , #ckgt(#EQ()) -> #false() 876.46/289.03 , #ckgt(#LT()) -> #false() 876.46/289.03 , #ckgt(#GT()) -> #true() 876.46/289.03 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.46/289.03 , insert#2(nil(), @keyX, @valX, @x) -> 876.46/289.03 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.46/289.03 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.46/289.03 insert#3(@l1, @keyX, @ls, @valX, @x) 876.46/289.03 , #and(#true(), #true()) -> #true() 876.46/289.03 , #and(#true(), #false()) -> #false() 876.46/289.03 , #and(#false(), #true()) -> #false() 876.46/289.03 , #and(#false(), #false()) -> #false() 876.46/289.03 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 876.46/289.03 , #compare(#pos(@x), #0()) -> #GT() 876.46/289.03 , #compare(#pos(@x), #neg(@y)) -> #GT() 876.46/289.03 , #compare(#0(), #pos(@y)) -> #LT() 876.46/289.03 , #compare(#0(), #0()) -> #EQ() 876.46/289.03 , #compare(#0(), #neg(@y)) -> #GT() 876.46/289.03 , #compare(#0(), #s(@y)) -> #LT() 876.46/289.03 , #compare(#neg(@x), #pos(@y)) -> #LT() 876.46/289.03 , #compare(#neg(@x), #0()) -> #LT() 876.46/289.03 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 876.46/289.03 , #compare(#s(@x), #0()) -> #GT() 876.46/289.03 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 876.46/289.03 , split#1(nil()) -> nil() 876.46/289.03 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.46/289.03 , #greater(@x, @y) -> #ckgt(#compare(@x, @y)) 876.46/289.03 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.03 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.46/289.03 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.03 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.46/289.03 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.46/289.03 insert#2(@l, @keyX, @valX, @x) 876.46/289.03 , quicksort#1(nil()) -> nil() 876.46/289.03 , quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) 876.46/289.03 , append#1(nil(), @ys) -> @ys 876.46/289.03 , append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) 876.46/289.03 , splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) 876.46/289.03 , quicksort(@l) -> quicksort#1(@l) 876.46/289.03 , splitqs#3(#true(), @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) 876.46/289.03 , splitqs#3(#false(), @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) 876.46/289.03 , splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> 876.46/289.03 splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) 876.46/289.03 , splitqs#1(nil(), @pivot) -> tuple#2(nil(), nil()) 876.46/289.03 , splitqs#1(::(@x, @xs), @pivot) -> 876.46/289.03 splitqs#2(splitqs(@pivot, @xs), @pivot, @x) } 876.46/289.03 Obligation: 876.46/289.03 innermost runtime complexity 876.46/289.03 Answer: 876.46/289.03 YES(O(1),O(n^6)) 876.46/289.03 876.46/289.03 We decompose the input problem according to the dependency graph 876.46/289.03 into the upper component 876.46/289.03 876.46/289.03 { splitAndSort^#(@l) -> c_11(sortAll^#(split(@l))) 876.46/289.03 , sortAll^#(@l) -> c_1(sortAll#1^#(@l)) 876.46/289.03 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> 876.46/289.03 c_2(quicksort^#(@vals), sortAll^#(@xs)) 876.46/289.03 , sortAll#1^#(::(@x, @xs)) -> c_7(sortAll#2^#(@x, @xs)) } 876.46/289.03 876.46/289.03 and lower component 876.46/289.03 876.46/289.03 { quicksort^#(@l) -> c_3(quicksort#1^#(@l)) 876.46/289.03 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> 876.46/289.03 c_4(append^#(quicksort(@xs), ::(@z, quicksort(@ys))), 876.46/289.03 quicksort^#(@xs), 876.46/289.03 quicksort^#(@ys)) 876.46/289.03 , append^#(@l, @ys) -> c_5(append#1^#(@l, @ys)) 876.46/289.03 , append#1^#(::(@x, @xs), @ys) -> c_6(append^#(@xs, @ys)) 876.46/289.03 , quicksort#1^#(::(@z, @zs)) -> 876.46/289.03 c_8(quicksort#2^#(splitqs(@z, @zs), @z), splitqs^#(@z, @zs)) 876.46/289.03 , splitqs^#(@pivot, @l) -> c_9(splitqs#1^#(@l, @pivot)) 876.46/289.03 , splitqs#1^#(::(@x, @xs), @pivot) -> 876.46/289.03 c_10(splitqs^#(@pivot, @xs)) } 876.46/289.03 876.46/289.03 Further, following extension rules are added to the lower 876.46/289.03 component. 876.46/289.03 876.46/289.03 { splitAndSort^#(@l) -> sortAll^#(split(@l)) 876.46/289.03 , sortAll^#(@l) -> sortAll#1^#(@l) 876.46/289.03 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> sortAll^#(@xs) 876.46/289.03 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> quicksort^#(@vals) 876.46/289.03 , sortAll#1^#(::(@x, @xs)) -> sortAll#2^#(@x, @xs) } 876.46/289.03 876.46/289.03 TcT solves the upper component with certificate YES(O(1),O(n^1)). 876.46/289.03 876.46/289.03 Sub-proof: 876.46/289.03 ---------- 876.46/289.03 We are left with following problem, upon which TcT provides the 876.46/289.03 certificate YES(O(1),O(n^1)). 876.46/289.03 876.46/289.03 Strict DPs: 876.46/289.03 { sortAll^#(@l) -> c_1(sortAll#1^#(@l)) 876.46/289.03 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> 876.46/289.03 c_2(quicksort^#(@vals), sortAll^#(@xs)) 876.46/289.03 , sortAll#1^#(::(@x, @xs)) -> c_7(sortAll#2^#(@x, @xs)) } 876.46/289.03 Weak DPs: { splitAndSort^#(@l) -> c_11(sortAll^#(split(@l))) } 876.46/289.03 Weak Trs: 876.46/289.03 { #equal(@x, @y) -> #eq(@x, @y) 876.46/289.03 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.46/289.03 , #eq(#pos(@x), #0()) -> #false() 876.46/289.03 , #eq(#pos(@x), #neg(@y)) -> #false() 876.46/289.03 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.46/289.03 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.03 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.46/289.03 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.46/289.03 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.03 , #eq(nil(), nil()) -> #true() 876.46/289.03 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.46/289.03 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.03 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.46/289.03 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.46/289.03 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.03 , #eq(#0(), #pos(@y)) -> #false() 876.46/289.03 , #eq(#0(), #0()) -> #true() 876.46/289.03 , #eq(#0(), #neg(@y)) -> #false() 876.46/289.03 , #eq(#0(), #s(@y)) -> #false() 876.46/289.03 , #eq(#neg(@x), #pos(@y)) -> #false() 876.46/289.03 , #eq(#neg(@x), #0()) -> #false() 876.46/289.03 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.46/289.03 , #eq(#s(@x), #0()) -> #false() 876.46/289.03 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.46/289.03 , quicksort#2(tuple#2(@xs, @ys), @z) -> 876.46/289.03 append(quicksort(@xs), ::(@z, quicksort(@ys))) 876.46/289.03 , split(@l) -> split#1(@l) 876.46/289.03 , append(@l, @ys) -> append#1(@l, @ys) 876.46/289.03 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.46/289.03 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.46/289.03 , #ckgt(#EQ()) -> #false() 876.46/289.03 , #ckgt(#LT()) -> #false() 876.46/289.03 , #ckgt(#GT()) -> #true() 876.46/289.03 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.46/289.03 , insert#2(nil(), @keyX, @valX, @x) -> 876.46/289.03 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.46/289.03 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.46/289.03 insert#3(@l1, @keyX, @ls, @valX, @x) 876.46/289.03 , #and(#true(), #true()) -> #true() 876.46/289.03 , #and(#true(), #false()) -> #false() 876.46/289.03 , #and(#false(), #true()) -> #false() 876.46/289.03 , #and(#false(), #false()) -> #false() 876.46/289.03 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 876.46/289.03 , #compare(#pos(@x), #0()) -> #GT() 876.46/289.03 , #compare(#pos(@x), #neg(@y)) -> #GT() 876.46/289.03 , #compare(#0(), #pos(@y)) -> #LT() 876.46/289.03 , #compare(#0(), #0()) -> #EQ() 876.46/289.03 , #compare(#0(), #neg(@y)) -> #GT() 876.46/289.03 , #compare(#0(), #s(@y)) -> #LT() 876.46/289.03 , #compare(#neg(@x), #pos(@y)) -> #LT() 876.46/289.03 , #compare(#neg(@x), #0()) -> #LT() 876.46/289.03 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 876.46/289.03 , #compare(#s(@x), #0()) -> #GT() 876.46/289.03 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 876.46/289.03 , split#1(nil()) -> nil() 876.46/289.03 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.46/289.03 , #greater(@x, @y) -> #ckgt(#compare(@x, @y)) 876.46/289.03 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.03 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.46/289.03 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.03 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.46/289.03 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.46/289.03 insert#2(@l, @keyX, @valX, @x) 876.46/289.03 , quicksort#1(nil()) -> nil() 876.46/289.03 , quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) 876.46/289.03 , append#1(nil(), @ys) -> @ys 876.46/289.03 , append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) 876.46/289.03 , splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) 876.46/289.03 , quicksort(@l) -> quicksort#1(@l) 876.46/289.03 , splitqs#3(#true(), @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) 876.46/289.03 , splitqs#3(#false(), @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) 876.46/289.03 , splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> 876.46/289.03 splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) 876.46/289.03 , splitqs#1(nil(), @pivot) -> tuple#2(nil(), nil()) 876.46/289.03 , splitqs#1(::(@x, @xs), @pivot) -> 876.46/289.03 splitqs#2(splitqs(@pivot, @xs), @pivot, @x) } 876.46/289.03 Obligation: 876.46/289.03 innermost runtime complexity 876.46/289.03 Answer: 876.46/289.03 YES(O(1),O(n^1)) 876.46/289.03 876.46/289.03 We replace rewrite rules by usable rules: 876.46/289.03 876.46/289.03 Weak Usable Rules: 876.46/289.03 { #equal(@x, @y) -> #eq(@x, @y) 876.46/289.03 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.46/289.03 , #eq(#pos(@x), #0()) -> #false() 876.46/289.03 , #eq(#pos(@x), #neg(@y)) -> #false() 876.46/289.03 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.46/289.03 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.03 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.46/289.03 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.46/289.03 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.03 , #eq(nil(), nil()) -> #true() 876.46/289.03 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.46/289.03 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.03 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.46/289.03 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.46/289.03 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.03 , #eq(#0(), #pos(@y)) -> #false() 876.46/289.03 , #eq(#0(), #0()) -> #true() 876.46/289.03 , #eq(#0(), #neg(@y)) -> #false() 876.46/289.03 , #eq(#0(), #s(@y)) -> #false() 876.46/289.03 , #eq(#neg(@x), #pos(@y)) -> #false() 876.46/289.03 , #eq(#neg(@x), #0()) -> #false() 876.46/289.03 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.46/289.03 , #eq(#s(@x), #0()) -> #false() 876.46/289.03 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.46/289.03 , split(@l) -> split#1(@l) 876.46/289.03 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.46/289.03 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.46/289.03 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.46/289.03 , insert#2(nil(), @keyX, @valX, @x) -> 876.46/289.03 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.46/289.03 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.46/289.03 insert#3(@l1, @keyX, @ls, @valX, @x) 876.46/289.03 , #and(#true(), #true()) -> #true() 876.46/289.03 , #and(#true(), #false()) -> #false() 876.46/289.03 , #and(#false(), #true()) -> #false() 876.46/289.03 , #and(#false(), #false()) -> #false() 876.46/289.03 , split#1(nil()) -> nil() 876.46/289.03 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.46/289.03 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.03 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.46/289.03 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.03 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.46/289.03 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.46/289.03 insert#2(@l, @keyX, @valX, @x) } 876.46/289.03 876.46/289.03 We are left with following problem, upon which TcT provides the 876.46/289.03 certificate YES(O(1),O(n^1)). 876.46/289.03 876.46/289.03 Strict DPs: 876.46/289.03 { sortAll^#(@l) -> c_1(sortAll#1^#(@l)) 876.46/289.03 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> 876.46/289.03 c_2(quicksort^#(@vals), sortAll^#(@xs)) 876.46/289.03 , sortAll#1^#(::(@x, @xs)) -> c_7(sortAll#2^#(@x, @xs)) } 876.46/289.03 Weak DPs: { splitAndSort^#(@l) -> c_11(sortAll^#(split(@l))) } 876.46/289.03 Weak Trs: 876.46/289.03 { #equal(@x, @y) -> #eq(@x, @y) 876.46/289.03 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.46/289.03 , #eq(#pos(@x), #0()) -> #false() 876.46/289.03 , #eq(#pos(@x), #neg(@y)) -> #false() 876.46/289.03 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.46/289.03 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.03 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.46/289.03 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.46/289.03 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.03 , #eq(nil(), nil()) -> #true() 876.46/289.03 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.46/289.03 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.03 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.46/289.03 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.46/289.03 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.03 , #eq(#0(), #pos(@y)) -> #false() 876.46/289.03 , #eq(#0(), #0()) -> #true() 876.46/289.03 , #eq(#0(), #neg(@y)) -> #false() 876.46/289.03 , #eq(#0(), #s(@y)) -> #false() 876.46/289.03 , #eq(#neg(@x), #pos(@y)) -> #false() 876.46/289.03 , #eq(#neg(@x), #0()) -> #false() 876.46/289.03 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.46/289.03 , #eq(#s(@x), #0()) -> #false() 876.46/289.03 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.46/289.03 , split(@l) -> split#1(@l) 876.46/289.03 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.46/289.03 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.46/289.03 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.46/289.03 , insert#2(nil(), @keyX, @valX, @x) -> 876.46/289.03 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.46/289.03 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.46/289.03 insert#3(@l1, @keyX, @ls, @valX, @x) 876.46/289.03 , #and(#true(), #true()) -> #true() 876.46/289.03 , #and(#true(), #false()) -> #false() 876.46/289.03 , #and(#false(), #true()) -> #false() 876.46/289.03 , #and(#false(), #false()) -> #false() 876.46/289.03 , split#1(nil()) -> nil() 876.46/289.03 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.46/289.03 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.03 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.46/289.03 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.03 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.46/289.03 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.46/289.03 insert#2(@l, @keyX, @valX, @x) } 876.46/289.03 Obligation: 876.46/289.03 innermost runtime complexity 876.46/289.03 Answer: 876.46/289.03 YES(O(1),O(n^1)) 876.46/289.03 876.46/289.03 We use the processor 'matrix interpretation of dimension 1' to 876.46/289.03 orient following rules strictly. 876.46/289.03 876.46/289.03 DPs: 876.46/289.03 { 3: sortAll#1^#(::(@x, @xs)) -> c_7(sortAll#2^#(@x, @xs)) 876.46/289.03 , 4: splitAndSort^#(@l) -> c_11(sortAll^#(split(@l))) } 876.46/289.03 Trs: 876.46/289.03 { #eq(nil(), nil()) -> #true() 876.46/289.03 , #eq(#0(), #0()) -> #true() 876.46/289.03 , insert#2(nil(), @keyX, @valX, @x) -> 876.46/289.03 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.46/289.03 , #and(#true(), #true()) -> #true() } 876.46/289.03 876.46/289.03 Sub-proof: 876.46/289.03 ---------- 876.46/289.03 The following argument positions are usable: 876.46/289.03 Uargs(c_1) = {1}, Uargs(c_2) = {2}, Uargs(c_7) = {1}, 876.46/289.03 Uargs(c_11) = {1} 876.46/289.03 876.46/289.03 TcT has computed the following constructor-based matrix 876.46/289.03 interpretation satisfying not(EDA). 876.46/289.03 876.46/289.03 [#equal](x1, x2) = [1] 876.46/289.03 876.46/289.03 [#eq](x1, x2) = [1] 876.46/289.03 876.46/289.03 [quicksort#2](x1, x2) = [7] x1 + [7] x2 + [0] 876.46/289.03 876.46/289.03 [split](x1) = [4] x1 + [0] 876.46/289.03 876.46/289.03 [#true] = [0] 876.46/289.03 876.46/289.03 [append](x1, x2) = [7] x1 + [7] x2 + [0] 876.46/289.03 876.46/289.03 [insert#3](x1, x2, x3, x4, x5) = [1] x3 + [5] 876.46/289.03 876.46/289.03 [#ckgt](x1) = [7] x1 + [0] 876.46/289.03 876.46/289.03 [insert](x1, x2) = [1] x2 + [4] 876.46/289.03 876.46/289.03 [#pos](x1) = [1] x1 + [0] 876.46/289.03 876.46/289.03 [#EQ] = [0] 876.46/289.03 876.46/289.03 [insert#2](x1, x2, x3, x4) = [1] x1 + [4] 876.46/289.03 876.46/289.03 [#and](x1, x2) = [1] 876.46/289.03 876.46/289.03 [#compare](x1, x2) = [7] x1 + [7] x2 + [0] 876.46/289.03 876.46/289.03 [tuple#2](x1, x2) = [0] 876.46/289.03 876.46/289.03 [nil] = [0] 876.46/289.03 876.46/289.03 [split#1](x1) = [4] x1 + [0] 876.46/289.03 876.46/289.03 [#greater](x1, x2) = [7] x1 + [7] x2 + [0] 876.46/289.03 876.46/289.03 [insert#4](x1, x2, x3, x4, x5, x6) = [4] x1 + [1] x3 + [1] 876.46/289.03 876.46/289.03 [insert#1](x1, x2, x3) = [1] x2 + [4] 876.46/289.03 876.46/289.03 [quicksort#1](x1) = [7] x1 + [0] 876.46/289.03 876.46/289.03 [append#1](x1, x2) = [7] x1 + [7] x2 + [0] 876.46/289.03 876.46/289.03 [splitqs](x1, x2) = [7] x1 + [7] x2 + [0] 876.46/289.03 876.46/289.03 [#false] = [1] 876.46/289.03 876.46/289.03 [quicksort](x1) = [7] x1 + [0] 876.46/289.03 876.46/289.03 [::](x1, x2) = [1] x2 + [1] 876.46/289.03 876.46/289.03 [#LT] = [0] 876.46/289.03 876.46/289.03 [splitqs#3](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0] 876.46/289.03 876.46/289.03 [splitqs#2](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] 876.46/289.03 876.46/289.03 [#0] = [0] 876.46/289.03 876.46/289.03 [#neg](x1) = [1] x1 + [0] 876.46/289.03 876.46/289.03 [#s](x1) = [1] x1 + [0] 876.46/289.03 876.46/289.03 [#GT] = [0] 876.46/289.03 876.46/289.03 [splitqs#1](x1, x2) = [7] x1 + [7] x2 + [0] 876.46/289.03 876.46/289.03 [splitAndSort^#](x1) = [7] x1 + [7] 876.46/289.03 876.46/289.03 [sortAll^#](x1) = [1] x1 + [0] 876.46/289.03 876.46/289.03 [sortAll#2^#](x1, x2) = [1] x2 + [0] 876.46/289.03 876.46/289.03 [quicksort^#](x1) = [0] 876.46/289.03 876.46/289.03 [sortAll#1^#](x1) = [1] x1 + [0] 876.46/289.03 876.46/289.03 [c_1](x1) = [1] x1 + [0] 876.46/289.03 876.46/289.03 [c_2](x1, x2) = [4] x1 + [1] x2 + [0] 876.46/289.03 876.46/289.03 [c_7](x1) = [1] x1 + [0] 876.46/289.03 876.46/289.03 [c_11](x1) = [1] x1 + [3] 876.46/289.03 876.46/289.03 The order satisfies the following ordering constraints: 876.46/289.03 876.46/289.03 [#equal(@x, @y)] = [1] 876.46/289.03 >= [1] 876.46/289.03 = [#eq(@x, @y)] 876.46/289.03 876.46/289.03 [#eq(#pos(@x), #pos(@y))] = [1] 876.46/289.03 >= [1] 876.46/289.03 = [#eq(@x, @y)] 876.46/289.03 876.46/289.03 [#eq(#pos(@x), #0())] = [1] 876.46/289.03 >= [1] 876.46/289.03 = [#false()] 876.46/289.03 876.46/289.03 [#eq(#pos(@x), #neg(@y))] = [1] 876.46/289.03 >= [1] 876.46/289.03 = [#false()] 876.46/289.03 876.46/289.03 [#eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2))] = [1] 876.46/289.03 >= [1] 876.46/289.03 = [#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))] 876.46/289.03 876.46/289.03 [#eq(tuple#2(@x_1, @x_2), nil())] = [1] 876.46/289.03 >= [1] 876.46/289.03 = [#false()] 876.46/289.03 876.46/289.03 [#eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2))] = [1] 876.46/289.03 >= [1] 876.46/289.03 = [#false()] 876.46/289.03 876.46/289.03 [#eq(nil(), tuple#2(@y_1, @y_2))] = [1] 876.46/289.03 >= [1] 876.46/289.03 = [#false()] 876.46/289.03 876.46/289.03 [#eq(nil(), nil())] = [1] 876.46/289.03 > [0] 876.46/289.03 = [#true()] 876.46/289.03 876.46/289.03 [#eq(nil(), ::(@y_1, @y_2))] = [1] 876.46/289.03 >= [1] 876.46/289.03 = [#false()] 876.46/289.03 876.46/289.03 [#eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2))] = [1] 876.46/289.03 >= [1] 876.46/289.03 = [#false()] 876.46/289.03 876.46/289.03 [#eq(::(@x_1, @x_2), nil())] = [1] 876.46/289.03 >= [1] 876.46/289.03 = [#false()] 876.46/289.03 876.46/289.03 [#eq(::(@x_1, @x_2), ::(@y_1, @y_2))] = [1] 876.46/289.03 >= [1] 876.46/289.03 = [#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))] 876.46/289.03 876.46/289.03 [#eq(#0(), #pos(@y))] = [1] 876.46/289.03 >= [1] 876.46/289.03 = [#false()] 876.46/289.03 876.46/289.03 [#eq(#0(), #0())] = [1] 876.46/289.03 > [0] 876.46/289.03 = [#true()] 876.46/289.03 876.46/289.03 [#eq(#0(), #neg(@y))] = [1] 876.46/289.03 >= [1] 876.46/289.03 = [#false()] 876.46/289.03 876.46/289.03 [#eq(#0(), #s(@y))] = [1] 876.46/289.03 >= [1] 876.46/289.03 = [#false()] 876.46/289.03 876.46/289.03 [#eq(#neg(@x), #pos(@y))] = [1] 876.46/289.03 >= [1] 876.46/289.03 = [#false()] 876.46/289.03 876.46/289.03 [#eq(#neg(@x), #0())] = [1] 876.46/289.03 >= [1] 876.46/289.03 = [#false()] 876.46/289.03 876.46/289.03 [#eq(#neg(@x), #neg(@y))] = [1] 876.46/289.03 >= [1] 876.46/289.03 = [#eq(@x, @y)] 876.46/289.03 876.46/289.03 [#eq(#s(@x), #0())] = [1] 876.46/289.03 >= [1] 876.46/289.03 = [#false()] 876.46/289.03 876.46/289.03 [#eq(#s(@x), #s(@y))] = [1] 876.46/289.03 >= [1] 876.46/289.03 = [#eq(@x, @y)] 876.46/289.03 876.46/289.03 [split(@l)] = [4] @l + [0] 876.46/289.03 >= [4] @l + [0] 876.46/289.03 = [split#1(@l)] 876.46/289.03 876.46/289.03 [insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x)] = [1] @ls + [5] 876.46/289.03 >= [1] @ls + [5] 876.46/289.03 = [insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x)] 876.46/289.03 876.46/289.03 [insert(@x, @l)] = [1] @l + [4] 876.46/289.03 >= [1] @l + [4] 876.46/289.03 = [insert#1(@x, @l, @x)] 876.46/289.03 876.46/289.03 [insert#2(nil(), @keyX, @valX, @x)] = [4] 876.46/289.03 > [1] 876.46/289.03 = [::(tuple#2(::(@valX, nil()), @keyX), nil())] 876.46/289.03 876.46/289.03 [insert#2(::(@l1, @ls), @keyX, @valX, @x)] = [1] @ls + [5] 876.46/289.03 >= [1] @ls + [5] 876.46/289.03 = [insert#3(@l1, @keyX, @ls, @valX, @x)] 876.46/289.03 876.46/289.03 [#and(#true(), #true())] = [1] 876.46/289.03 > [0] 876.46/289.03 = [#true()] 876.46/289.03 876.46/289.03 [#and(#true(), #false())] = [1] 876.46/289.03 >= [1] 876.46/289.03 = [#false()] 876.46/289.03 876.46/289.03 [#and(#false(), #true())] = [1] 876.46/289.03 >= [1] 876.46/289.03 = [#false()] 876.46/289.03 876.46/289.03 [#and(#false(), #false())] = [1] 876.46/289.03 >= [1] 876.46/289.03 = [#false()] 876.46/289.03 876.46/289.03 [split#1(nil())] = [0] 876.46/289.03 >= [0] 876.46/289.03 = [nil()] 876.46/289.03 876.46/289.03 [split#1(::(@x, @xs))] = [4] @xs + [4] 876.46/289.03 >= [4] @xs + [4] 876.46/289.03 = [insert(@x, split(@xs))] 876.46/289.04 876.46/289.04 [insert#4(#true(), @key1, @ls, @valX, @vals1, @x)] = [1] @ls + [1] 876.46/289.04 >= [1] @ls + [1] 876.46/289.04 = [::(tuple#2(::(@valX, @vals1), @key1), @ls)] 876.46/289.04 876.46/289.04 [insert#4(#false(), @key1, @ls, @valX, @vals1, @x)] = [1] @ls + [5] 876.46/289.04 >= [1] @ls + [5] 876.46/289.04 = [::(tuple#2(@vals1, @key1), insert(@x, @ls))] 876.46/289.04 876.46/289.04 [insert#1(tuple#2(@valX, @keyX), @l, @x)] = [1] @l + [4] 876.46/289.04 >= [1] @l + [4] 876.46/289.04 = [insert#2(@l, @keyX, @valX, @x)] 876.46/289.04 876.46/289.04 [splitAndSort^#(@l)] = [7] @l + [7] 876.46/289.04 > [4] @l + [3] 876.46/289.04 = [c_11(sortAll^#(split(@l)))] 876.46/289.04 876.46/289.04 [sortAll^#(@l)] = [1] @l + [0] 876.46/289.04 >= [1] @l + [0] 876.46/289.04 = [c_1(sortAll#1^#(@l))] 876.46/289.04 876.46/289.04 [sortAll#2^#(tuple#2(@vals, @key), @xs)] = [1] @xs + [0] 876.46/289.04 >= [1] @xs + [0] 876.46/289.04 = [c_2(quicksort^#(@vals), sortAll^#(@xs))] 876.46/289.04 876.46/289.04 [sortAll#1^#(::(@x, @xs))] = [1] @xs + [1] 876.46/289.04 > [1] @xs + [0] 876.46/289.04 = [c_7(sortAll#2^#(@x, @xs))] 876.46/289.04 876.46/289.04 876.46/289.04 We return to the main proof. Consider the set of all dependency 876.46/289.04 pairs 876.46/289.04 876.46/289.04 : 876.46/289.04 { 1: sortAll^#(@l) -> c_1(sortAll#1^#(@l)) 876.46/289.04 , 2: sortAll#2^#(tuple#2(@vals, @key), @xs) -> 876.46/289.04 c_2(quicksort^#(@vals), sortAll^#(@xs)) 876.46/289.04 , 3: sortAll#1^#(::(@x, @xs)) -> c_7(sortAll#2^#(@x, @xs)) 876.46/289.04 , 4: splitAndSort^#(@l) -> c_11(sortAll^#(split(@l))) } 876.46/289.04 876.46/289.04 Processor 'matrix interpretation of dimension 1' induces the 876.46/289.04 complexity certificate YES(?,O(n^1)) on application of dependency 876.46/289.04 pairs {3,4}. These cover all (indirect) predecessors of dependency 876.46/289.04 pairs {1,2,3,4}, their number of application is equally bounded. 876.46/289.04 The dependency pairs are shifted into the weak component. 876.46/289.04 876.46/289.04 We are left with following problem, upon which TcT provides the 876.46/289.04 certificate YES(O(1),O(1)). 876.46/289.04 876.46/289.04 Weak DPs: 876.46/289.04 { splitAndSort^#(@l) -> c_11(sortAll^#(split(@l))) 876.46/289.04 , sortAll^#(@l) -> c_1(sortAll#1^#(@l)) 876.46/289.04 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> 876.46/289.04 c_2(quicksort^#(@vals), sortAll^#(@xs)) 876.46/289.04 , sortAll#1^#(::(@x, @xs)) -> c_7(sortAll#2^#(@x, @xs)) } 876.46/289.04 Weak Trs: 876.46/289.04 { #equal(@x, @y) -> #eq(@x, @y) 876.46/289.04 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.46/289.04 , #eq(#pos(@x), #0()) -> #false() 876.46/289.04 , #eq(#pos(@x), #neg(@y)) -> #false() 876.46/289.04 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.46/289.04 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.04 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.46/289.04 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.46/289.04 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.04 , #eq(nil(), nil()) -> #true() 876.46/289.04 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.46/289.04 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.04 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.46/289.04 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.46/289.04 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.04 , #eq(#0(), #pos(@y)) -> #false() 876.46/289.04 , #eq(#0(), #0()) -> #true() 876.46/289.04 , #eq(#0(), #neg(@y)) -> #false() 876.46/289.04 , #eq(#0(), #s(@y)) -> #false() 876.46/289.04 , #eq(#neg(@x), #pos(@y)) -> #false() 876.46/289.04 , #eq(#neg(@x), #0()) -> #false() 876.46/289.04 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.46/289.04 , #eq(#s(@x), #0()) -> #false() 876.46/289.04 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.46/289.04 , split(@l) -> split#1(@l) 876.46/289.04 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.46/289.04 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.46/289.04 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.46/289.04 , insert#2(nil(), @keyX, @valX, @x) -> 876.46/289.04 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.46/289.04 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.46/289.04 insert#3(@l1, @keyX, @ls, @valX, @x) 876.46/289.04 , #and(#true(), #true()) -> #true() 876.46/289.04 , #and(#true(), #false()) -> #false() 876.46/289.04 , #and(#false(), #true()) -> #false() 876.46/289.04 , #and(#false(), #false()) -> #false() 876.46/289.04 , split#1(nil()) -> nil() 876.46/289.04 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.46/289.04 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.04 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.46/289.04 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.04 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.46/289.04 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.46/289.04 insert#2(@l, @keyX, @valX, @x) } 876.46/289.04 Obligation: 876.46/289.04 innermost runtime complexity 876.46/289.04 Answer: 876.46/289.04 YES(O(1),O(1)) 876.46/289.04 876.46/289.04 The following weak DPs constitute a sub-graph of the DG that is 876.46/289.04 closed under successors. The DPs are removed. 876.46/289.04 876.46/289.04 { splitAndSort^#(@l) -> c_11(sortAll^#(split(@l))) 876.46/289.04 , sortAll^#(@l) -> c_1(sortAll#1^#(@l)) 876.46/289.04 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> 876.46/289.04 c_2(quicksort^#(@vals), sortAll^#(@xs)) 876.46/289.04 , sortAll#1^#(::(@x, @xs)) -> c_7(sortAll#2^#(@x, @xs)) } 876.46/289.04 876.46/289.04 We are left with following problem, upon which TcT provides the 876.46/289.04 certificate YES(O(1),O(1)). 876.46/289.04 876.46/289.04 Weak Trs: 876.46/289.04 { #equal(@x, @y) -> #eq(@x, @y) 876.46/289.04 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.46/289.04 , #eq(#pos(@x), #0()) -> #false() 876.46/289.04 , #eq(#pos(@x), #neg(@y)) -> #false() 876.46/289.04 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.46/289.04 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.04 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.46/289.04 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.46/289.04 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.04 , #eq(nil(), nil()) -> #true() 876.46/289.04 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.46/289.04 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.04 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.46/289.04 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.46/289.04 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.04 , #eq(#0(), #pos(@y)) -> #false() 876.46/289.04 , #eq(#0(), #0()) -> #true() 876.46/289.04 , #eq(#0(), #neg(@y)) -> #false() 876.46/289.04 , #eq(#0(), #s(@y)) -> #false() 876.46/289.04 , #eq(#neg(@x), #pos(@y)) -> #false() 876.46/289.04 , #eq(#neg(@x), #0()) -> #false() 876.46/289.04 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.46/289.04 , #eq(#s(@x), #0()) -> #false() 876.46/289.04 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.46/289.04 , split(@l) -> split#1(@l) 876.46/289.04 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.46/289.04 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.46/289.04 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.46/289.04 , insert#2(nil(), @keyX, @valX, @x) -> 876.46/289.04 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.46/289.04 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.46/289.04 insert#3(@l1, @keyX, @ls, @valX, @x) 876.46/289.04 , #and(#true(), #true()) -> #true() 876.46/289.04 , #and(#true(), #false()) -> #false() 876.46/289.04 , #and(#false(), #true()) -> #false() 876.46/289.04 , #and(#false(), #false()) -> #false() 876.46/289.04 , split#1(nil()) -> nil() 876.46/289.04 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.46/289.04 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.04 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.46/289.04 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.04 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.46/289.04 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.46/289.04 insert#2(@l, @keyX, @valX, @x) } 876.46/289.04 Obligation: 876.46/289.04 innermost runtime complexity 876.46/289.04 Answer: 876.46/289.04 YES(O(1),O(1)) 876.46/289.04 876.46/289.04 No rule is usable, rules are removed from the input problem. 876.46/289.04 876.46/289.04 We are left with following problem, upon which TcT provides the 876.46/289.04 certificate YES(O(1),O(1)). 876.46/289.04 876.46/289.04 Rules: Empty 876.46/289.04 Obligation: 876.46/289.04 innermost runtime complexity 876.46/289.04 Answer: 876.46/289.04 YES(O(1),O(1)) 876.46/289.04 876.46/289.04 Empty rules are trivially bounded 876.46/289.04 876.46/289.04 We return to the main proof. 876.46/289.04 876.46/289.04 We are left with following problem, upon which TcT provides the 876.46/289.04 certificate YES(O(1),O(n^5)). 876.46/289.04 876.46/289.04 Strict DPs: 876.46/289.04 { quicksort^#(@l) -> c_3(quicksort#1^#(@l)) 876.46/289.04 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> 876.46/289.04 c_4(append^#(quicksort(@xs), ::(@z, quicksort(@ys))), 876.46/289.04 quicksort^#(@xs), 876.46/289.04 quicksort^#(@ys)) 876.46/289.04 , append^#(@l, @ys) -> c_5(append#1^#(@l, @ys)) 876.46/289.04 , append#1^#(::(@x, @xs), @ys) -> c_6(append^#(@xs, @ys)) 876.46/289.04 , quicksort#1^#(::(@z, @zs)) -> 876.46/289.04 c_8(quicksort#2^#(splitqs(@z, @zs), @z), splitqs^#(@z, @zs)) 876.46/289.04 , splitqs^#(@pivot, @l) -> c_9(splitqs#1^#(@l, @pivot)) 876.46/289.04 , splitqs#1^#(::(@x, @xs), @pivot) -> 876.46/289.04 c_10(splitqs^#(@pivot, @xs)) } 876.46/289.04 Weak DPs: 876.46/289.04 { splitAndSort^#(@l) -> sortAll^#(split(@l)) 876.46/289.04 , sortAll^#(@l) -> sortAll#1^#(@l) 876.46/289.04 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> sortAll^#(@xs) 876.46/289.04 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> quicksort^#(@vals) 876.46/289.04 , sortAll#1^#(::(@x, @xs)) -> sortAll#2^#(@x, @xs) } 876.46/289.04 Weak Trs: 876.46/289.04 { #equal(@x, @y) -> #eq(@x, @y) 876.46/289.04 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.46/289.04 , #eq(#pos(@x), #0()) -> #false() 876.46/289.04 , #eq(#pos(@x), #neg(@y)) -> #false() 876.46/289.04 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.46/289.04 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.04 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.46/289.04 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.46/289.04 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.04 , #eq(nil(), nil()) -> #true() 876.46/289.04 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.46/289.04 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.04 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.46/289.04 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.46/289.04 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.04 , #eq(#0(), #pos(@y)) -> #false() 876.46/289.04 , #eq(#0(), #0()) -> #true() 876.46/289.04 , #eq(#0(), #neg(@y)) -> #false() 876.46/289.04 , #eq(#0(), #s(@y)) -> #false() 876.46/289.04 , #eq(#neg(@x), #pos(@y)) -> #false() 876.46/289.04 , #eq(#neg(@x), #0()) -> #false() 876.46/289.04 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.46/289.04 , #eq(#s(@x), #0()) -> #false() 876.46/289.04 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.46/289.04 , quicksort#2(tuple#2(@xs, @ys), @z) -> 876.46/289.04 append(quicksort(@xs), ::(@z, quicksort(@ys))) 876.46/289.04 , split(@l) -> split#1(@l) 876.46/289.04 , append(@l, @ys) -> append#1(@l, @ys) 876.46/289.04 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.46/289.04 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.46/289.04 , #ckgt(#EQ()) -> #false() 876.46/289.04 , #ckgt(#LT()) -> #false() 876.46/289.04 , #ckgt(#GT()) -> #true() 876.46/289.04 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.46/289.04 , insert#2(nil(), @keyX, @valX, @x) -> 876.46/289.04 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.46/289.04 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.46/289.04 insert#3(@l1, @keyX, @ls, @valX, @x) 876.46/289.04 , #and(#true(), #true()) -> #true() 876.46/289.04 , #and(#true(), #false()) -> #false() 876.46/289.04 , #and(#false(), #true()) -> #false() 876.46/289.04 , #and(#false(), #false()) -> #false() 876.46/289.04 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 876.46/289.04 , #compare(#pos(@x), #0()) -> #GT() 876.46/289.04 , #compare(#pos(@x), #neg(@y)) -> #GT() 876.46/289.04 , #compare(#0(), #pos(@y)) -> #LT() 876.46/289.04 , #compare(#0(), #0()) -> #EQ() 876.46/289.04 , #compare(#0(), #neg(@y)) -> #GT() 876.46/289.04 , #compare(#0(), #s(@y)) -> #LT() 876.46/289.04 , #compare(#neg(@x), #pos(@y)) -> #LT() 876.46/289.04 , #compare(#neg(@x), #0()) -> #LT() 876.46/289.04 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 876.46/289.04 , #compare(#s(@x), #0()) -> #GT() 876.46/289.04 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 876.46/289.04 , split#1(nil()) -> nil() 876.46/289.04 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.46/289.04 , #greater(@x, @y) -> #ckgt(#compare(@x, @y)) 876.46/289.04 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.04 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.46/289.04 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.04 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.46/289.04 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.46/289.04 insert#2(@l, @keyX, @valX, @x) 876.46/289.04 , quicksort#1(nil()) -> nil() 876.46/289.04 , quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) 876.46/289.04 , append#1(nil(), @ys) -> @ys 876.46/289.04 , append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) 876.46/289.04 , splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) 876.46/289.04 , quicksort(@l) -> quicksort#1(@l) 876.46/289.04 , splitqs#3(#true(), @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) 876.46/289.04 , splitqs#3(#false(), @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) 876.46/289.04 , splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> 876.46/289.04 splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) 876.46/289.04 , splitqs#1(nil(), @pivot) -> tuple#2(nil(), nil()) 876.46/289.04 , splitqs#1(::(@x, @xs), @pivot) -> 876.46/289.04 splitqs#2(splitqs(@pivot, @xs), @pivot, @x) } 876.46/289.04 Obligation: 876.46/289.04 innermost runtime complexity 876.46/289.04 Answer: 876.46/289.04 YES(O(1),O(n^5)) 876.46/289.04 876.46/289.04 We decompose the input problem according to the dependency graph 876.46/289.04 into the upper component 876.46/289.04 876.46/289.04 { splitAndSort^#(@l) -> sortAll^#(split(@l)) 876.46/289.04 , sortAll^#(@l) -> sortAll#1^#(@l) 876.46/289.04 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> sortAll^#(@xs) 876.46/289.04 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> quicksort^#(@vals) 876.46/289.04 , quicksort^#(@l) -> c_3(quicksort#1^#(@l)) 876.46/289.04 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> 876.46/289.04 c_4(append^#(quicksort(@xs), ::(@z, quicksort(@ys))), 876.46/289.04 quicksort^#(@xs), 876.46/289.04 quicksort^#(@ys)) 876.46/289.04 , sortAll#1^#(::(@x, @xs)) -> sortAll#2^#(@x, @xs) 876.46/289.04 , quicksort#1^#(::(@z, @zs)) -> 876.46/289.04 c_8(quicksort#2^#(splitqs(@z, @zs), @z), splitqs^#(@z, @zs)) } 876.46/289.04 876.46/289.04 and lower component 876.46/289.04 876.46/289.04 { append^#(@l, @ys) -> c_5(append#1^#(@l, @ys)) 876.46/289.04 , append#1^#(::(@x, @xs), @ys) -> c_6(append^#(@xs, @ys)) 876.46/289.04 , splitqs^#(@pivot, @l) -> c_9(splitqs#1^#(@l, @pivot)) 876.46/289.04 , splitqs#1^#(::(@x, @xs), @pivot) -> 876.46/289.04 c_10(splitqs^#(@pivot, @xs)) } 876.46/289.04 876.46/289.04 Further, following extension rules are added to the lower 876.46/289.04 component. 876.46/289.04 876.46/289.04 { splitAndSort^#(@l) -> sortAll^#(split(@l)) 876.46/289.04 , sortAll^#(@l) -> sortAll#1^#(@l) 876.46/289.04 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> sortAll^#(@xs) 876.46/289.04 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> quicksort^#(@vals) 876.46/289.04 , quicksort^#(@l) -> quicksort#1^#(@l) 876.46/289.04 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> quicksort^#(@xs) 876.46/289.04 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> quicksort^#(@ys) 876.46/289.04 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> 876.46/289.04 append^#(quicksort(@xs), ::(@z, quicksort(@ys))) 876.46/289.04 , sortAll#1^#(::(@x, @xs)) -> sortAll#2^#(@x, @xs) 876.46/289.04 , quicksort#1^#(::(@z, @zs)) -> quicksort#2^#(splitqs(@z, @zs), @z) 876.46/289.04 , quicksort#1^#(::(@z, @zs)) -> splitqs^#(@z, @zs) } 876.46/289.04 876.46/289.04 TcT solves the upper component with certificate YES(O(1),O(n^1)). 876.46/289.04 876.46/289.04 Sub-proof: 876.46/289.04 ---------- 876.46/289.04 We are left with following problem, upon which TcT provides the 876.46/289.04 certificate YES(O(1),O(n^1)). 876.46/289.04 876.46/289.04 Strict DPs: 876.46/289.04 { quicksort^#(@l) -> c_3(quicksort#1^#(@l)) 876.46/289.04 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> 876.46/289.04 c_4(append^#(quicksort(@xs), ::(@z, quicksort(@ys))), 876.46/289.04 quicksort^#(@xs), 876.46/289.04 quicksort^#(@ys)) 876.46/289.04 , quicksort#1^#(::(@z, @zs)) -> 876.46/289.04 c_8(quicksort#2^#(splitqs(@z, @zs), @z), splitqs^#(@z, @zs)) } 876.46/289.04 Weak DPs: 876.46/289.04 { splitAndSort^#(@l) -> sortAll^#(split(@l)) 876.46/289.04 , sortAll^#(@l) -> sortAll#1^#(@l) 876.46/289.04 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> sortAll^#(@xs) 876.46/289.04 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> quicksort^#(@vals) 876.46/289.04 , sortAll#1^#(::(@x, @xs)) -> sortAll#2^#(@x, @xs) } 876.46/289.04 Weak Trs: 876.46/289.04 { #equal(@x, @y) -> #eq(@x, @y) 876.46/289.04 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.46/289.04 , #eq(#pos(@x), #0()) -> #false() 876.46/289.04 , #eq(#pos(@x), #neg(@y)) -> #false() 876.46/289.04 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.46/289.04 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.04 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.46/289.04 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.46/289.04 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.04 , #eq(nil(), nil()) -> #true() 876.46/289.04 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.46/289.04 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.04 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.46/289.04 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.46/289.04 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.04 , #eq(#0(), #pos(@y)) -> #false() 876.46/289.04 , #eq(#0(), #0()) -> #true() 876.46/289.04 , #eq(#0(), #neg(@y)) -> #false() 876.46/289.04 , #eq(#0(), #s(@y)) -> #false() 876.46/289.04 , #eq(#neg(@x), #pos(@y)) -> #false() 876.46/289.04 , #eq(#neg(@x), #0()) -> #false() 876.46/289.04 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.46/289.04 , #eq(#s(@x), #0()) -> #false() 876.46/289.04 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.46/289.04 , quicksort#2(tuple#2(@xs, @ys), @z) -> 876.46/289.04 append(quicksort(@xs), ::(@z, quicksort(@ys))) 876.46/289.04 , split(@l) -> split#1(@l) 876.46/289.04 , append(@l, @ys) -> append#1(@l, @ys) 876.46/289.04 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.46/289.04 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.46/289.04 , #ckgt(#EQ()) -> #false() 876.46/289.04 , #ckgt(#LT()) -> #false() 876.46/289.04 , #ckgt(#GT()) -> #true() 876.46/289.04 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.46/289.04 , insert#2(nil(), @keyX, @valX, @x) -> 876.46/289.04 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.46/289.04 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.46/289.04 insert#3(@l1, @keyX, @ls, @valX, @x) 876.46/289.04 , #and(#true(), #true()) -> #true() 876.46/289.04 , #and(#true(), #false()) -> #false() 876.46/289.04 , #and(#false(), #true()) -> #false() 876.46/289.04 , #and(#false(), #false()) -> #false() 876.46/289.04 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 876.46/289.04 , #compare(#pos(@x), #0()) -> #GT() 876.46/289.04 , #compare(#pos(@x), #neg(@y)) -> #GT() 876.46/289.04 , #compare(#0(), #pos(@y)) -> #LT() 876.46/289.04 , #compare(#0(), #0()) -> #EQ() 876.46/289.04 , #compare(#0(), #neg(@y)) -> #GT() 876.46/289.04 , #compare(#0(), #s(@y)) -> #LT() 876.46/289.04 , #compare(#neg(@x), #pos(@y)) -> #LT() 876.46/289.04 , #compare(#neg(@x), #0()) -> #LT() 876.46/289.04 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 876.46/289.04 , #compare(#s(@x), #0()) -> #GT() 876.46/289.04 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 876.46/289.04 , split#1(nil()) -> nil() 876.46/289.04 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.46/289.04 , #greater(@x, @y) -> #ckgt(#compare(@x, @y)) 876.46/289.04 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.04 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.46/289.04 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.04 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.46/289.04 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.46/289.04 insert#2(@l, @keyX, @valX, @x) 876.46/289.04 , quicksort#1(nil()) -> nil() 876.46/289.04 , quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) 876.46/289.04 , append#1(nil(), @ys) -> @ys 876.46/289.04 , append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) 876.46/289.04 , splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) 876.46/289.04 , quicksort(@l) -> quicksort#1(@l) 876.46/289.04 , splitqs#3(#true(), @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) 876.46/289.04 , splitqs#3(#false(), @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) 876.46/289.04 , splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> 876.46/289.04 splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) 876.46/289.04 , splitqs#1(nil(), @pivot) -> tuple#2(nil(), nil()) 876.46/289.04 , splitqs#1(::(@x, @xs), @pivot) -> 876.46/289.04 splitqs#2(splitqs(@pivot, @xs), @pivot, @x) } 876.46/289.04 Obligation: 876.46/289.04 innermost runtime complexity 876.46/289.04 Answer: 876.46/289.04 YES(O(1),O(n^1)) 876.46/289.04 876.46/289.04 We use the processor 'matrix interpretation of dimension 2' to 876.46/289.04 orient following rules strictly. 876.46/289.04 876.46/289.04 DPs: 876.46/289.04 { 3: quicksort#1^#(::(@z, @zs)) -> 876.46/289.04 c_8(quicksort#2^#(splitqs(@z, @zs), @z), splitqs^#(@z, @zs)) 876.46/289.04 , 4: splitAndSort^#(@l) -> sortAll^#(split(@l)) } 876.46/289.04 876.46/289.04 Sub-proof: 876.46/289.04 ---------- 876.46/289.04 The following argument positions are usable: 876.46/289.04 Uargs(c_3) = {1}, Uargs(c_4) = {1, 2, 3}, Uargs(c_8) = {1} 876.46/289.04 876.46/289.04 TcT has computed the following constructor-based matrix 876.46/289.04 interpretation satisfying not(EDA) and not(IDA(1)). 876.46/289.04 876.46/289.04 [#equal](x1, x2) = [0] 876.46/289.04 [0] 876.46/289.04 876.46/289.04 [#eq](x1, x2) = [0] 876.46/289.04 [0] 876.46/289.04 876.46/289.04 [quicksort#2](x1, x2) = [0] 876.46/289.04 [0] 876.46/289.04 876.46/289.04 [split](x1) = [0 1] x1 + [0] 876.46/289.04 [0 1] [0] 876.46/289.04 876.46/289.04 [#true] = [0] 876.46/289.04 [0] 876.46/289.04 876.46/289.04 [append](x1, x2) = [0] 876.46/289.04 [0] 876.46/289.04 876.46/289.04 [insert#3](x1, x2, x3, x4, x5) = [0 1] x1 + [1 0] x3 + [1] 876.46/289.04 [0 0] [0 1] [2] 876.46/289.04 876.46/289.04 [#ckgt](x1) = [0] 876.46/289.04 [0] 876.46/289.04 876.46/289.04 [insert](x1, x2) = [1 0] x2 + [1] 876.46/289.04 [0 1] [1] 876.46/289.04 876.46/289.04 [#pos](x1) = [1 0] x1 + [0] 876.46/289.04 [0 1] [0] 876.46/289.05 876.46/289.05 [#EQ] = [0] 876.46/289.05 [0] 876.46/289.05 876.46/289.05 [insert#2](x1, x2, x3, x4) = [1 0] x1 + [1] 876.46/289.05 [0 1] [1] 876.46/289.05 876.46/289.05 [#and](x1, x2) = [0] 876.46/289.05 [0] 876.46/289.05 876.46/289.05 [#compare](x1, x2) = [0] 876.46/289.05 [0] 876.46/289.05 876.46/289.05 [tuple#2](x1, x2) = [0 1] x1 + [0 1] x2 + [0] 876.46/289.05 [0 1] [0 0] [0] 876.46/289.05 876.46/289.05 [nil] = [0] 876.46/289.05 [0] 876.46/289.05 876.46/289.05 [split#1](x1) = [0 1] x1 + [0] 876.46/289.05 [0 1] [0] 876.46/289.05 876.46/289.05 [#greater](x1, x2) = [0] 876.46/289.05 [0] 876.46/289.05 876.46/289.05 [insert#4](x1, x2, x3, x4, x5, x6) = [1 0] x3 + [0 1] x5 + [1] 876.46/289.05 [0 1] [0 0] [2] 876.46/289.05 876.46/289.05 [insert#1](x1, x2, x3) = [1 0] x2 + [1] 876.46/289.05 [0 1] [1] 876.46/289.05 876.46/289.05 [quicksort#1](x1) = [0] 876.46/289.05 [0] 876.46/289.05 876.46/289.05 [append#1](x1, x2) = [6 0] x2 + [0] 876.46/289.05 [6 6] [0] 876.46/289.05 876.46/289.05 [splitqs](x1, x2) = [0 0] x1 + [0 1] x2 + [0] 876.46/289.05 [4 4] [4 4] [0] 876.46/289.05 876.46/289.05 [#false] = [0] 876.46/289.05 [0] 876.46/289.05 876.46/289.05 [quicksort](x1) = [0] 876.46/289.05 [0] 876.46/289.05 876.46/289.05 [::](x1, x2) = [0 1] x1 + [1 0] x2 + [0] 876.46/289.05 [0 0] [0 1] [1] 876.46/289.05 876.46/289.05 [#LT] = [0] 876.46/289.05 [0] 876.46/289.05 876.46/289.05 [splitqs#3](x1, x2, x3, x4) = [0 1] x2 + [0 1] x3 + [1] 876.46/289.05 [0 1] [0 0] [1] 876.46/289.05 876.46/289.05 [splitqs#2](x1, x2, x3) = [1 0] x1 + [0 0] x2 + [1] 876.46/289.05 [1 0] [0 4] [1] 876.46/289.05 876.46/289.05 [#0] = [0] 876.46/289.05 [0] 876.46/289.05 876.46/289.05 [#neg](x1) = [0] 876.46/289.05 [0] 876.46/289.05 876.46/289.05 [#s](x1) = [0] 876.46/289.05 [0] 876.46/289.05 876.46/289.05 [#GT] = [0] 876.46/289.05 [0] 876.46/289.05 876.46/289.05 [splitqs#1](x1, x2) = [0 1] x1 + [0 0] x2 + [0] 876.46/289.05 [0 2] [2 4] [0] 876.46/289.05 876.46/289.05 [splitAndSort^#](x1) = [7 7] x1 + [7] 876.46/289.05 [7 7] [7] 876.46/289.05 876.46/289.05 [sortAll^#](x1) = [4 0] x1 + [0] 876.46/289.05 [0 0] [0] 876.46/289.05 876.46/289.05 [sortAll#2^#](x1, x2) = [0 1] x1 + [4 0] x2 + [0] 876.46/289.05 [0 0] [0 0] [0] 876.46/289.05 876.46/289.05 [quicksort^#](x1) = [0 1] x1 + [0] 876.46/289.05 [0 0] [0] 876.46/289.05 876.46/289.05 [quicksort#2^#](x1, x2) = [1 0] x1 + [0 0] x2 + [0] 876.46/289.05 [0 0] [0 4] [0] 876.46/289.05 876.46/289.05 [append^#](x1, x2) = [0] 876.46/289.05 [0] 876.46/289.05 876.46/289.05 [sortAll#1^#](x1) = [4 0] x1 + [0] 876.46/289.05 [0 0] [0] 876.46/289.05 876.46/289.05 [quicksort#1^#](x1) = [0 1] x1 + [0] 876.46/289.05 [0 0] [0] 876.46/289.05 876.46/289.05 [splitqs^#](x1, x2) = [0] 876.46/289.05 [0] 876.46/289.05 876.46/289.05 [c_3](x1) = [1 0] x1 + [0] 876.46/289.05 [0 0] [0] 876.46/289.05 876.46/289.05 [c_4](x1, x2, x3) = [4 0] x1 + [1 0] x2 + [1 876.46/289.05 0] x3 + [0] 876.46/289.05 [0 0] [0 0] [0 876.46/289.05 0] [0] 876.46/289.05 876.46/289.05 [c_8](x1, x2) = [1 0] x1 + [0] 876.46/289.05 [0 0] [0] 876.46/289.05 876.46/289.05 The order satisfies the following ordering constraints: 876.46/289.05 876.46/289.05 [#equal(@x, @y)] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#eq(@x, @y)] 876.46/289.05 876.46/289.05 [#eq(#pos(@x), #pos(@y))] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#eq(@x, @y)] 876.46/289.05 876.46/289.05 [#eq(#pos(@x), #0())] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#false()] 876.46/289.05 876.46/289.05 [#eq(#pos(@x), #neg(@y))] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#false()] 876.46/289.05 876.46/289.05 [#eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2))] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))] 876.46/289.05 876.46/289.05 [#eq(tuple#2(@x_1, @x_2), nil())] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#false()] 876.46/289.05 876.46/289.05 [#eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2))] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#false()] 876.46/289.05 876.46/289.05 [#eq(nil(), tuple#2(@y_1, @y_2))] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#false()] 876.46/289.05 876.46/289.05 [#eq(nil(), nil())] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#true()] 876.46/289.05 876.46/289.05 [#eq(nil(), ::(@y_1, @y_2))] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#false()] 876.46/289.05 876.46/289.05 [#eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2))] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#false()] 876.46/289.05 876.46/289.05 [#eq(::(@x_1, @x_2), nil())] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#false()] 876.46/289.05 876.46/289.05 [#eq(::(@x_1, @x_2), ::(@y_1, @y_2))] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))] 876.46/289.05 876.46/289.05 [#eq(#0(), #pos(@y))] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#false()] 876.46/289.05 876.46/289.05 [#eq(#0(), #0())] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#true()] 876.46/289.05 876.46/289.05 [#eq(#0(), #neg(@y))] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#false()] 876.46/289.05 876.46/289.05 [#eq(#0(), #s(@y))] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#false()] 876.46/289.05 876.46/289.05 [#eq(#neg(@x), #pos(@y))] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#false()] 876.46/289.05 876.46/289.05 [#eq(#neg(@x), #0())] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#false()] 876.46/289.05 876.46/289.05 [#eq(#neg(@x), #neg(@y))] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#eq(@x, @y)] 876.46/289.05 876.46/289.05 [#eq(#s(@x), #0())] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#false()] 876.46/289.05 876.46/289.05 [#eq(#s(@x), #s(@y))] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#eq(@x, @y)] 876.46/289.05 876.46/289.05 [quicksort#2(tuple#2(@xs, @ys), @z)] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [append(quicksort(@xs), ::(@z, quicksort(@ys)))] 876.46/289.05 876.46/289.05 [split(@l)] = [0 1] @l + [0] 876.46/289.05 [0 1] [0] 876.46/289.05 >= [0 1] @l + [0] 876.46/289.05 [0 1] [0] 876.46/289.05 = [split#1(@l)] 876.46/289.05 876.46/289.05 [append(@l, @ys)] = [0] 876.46/289.05 [0] 876.46/289.05 ? [6 0] @ys + [0] 876.46/289.05 [6 6] [0] 876.46/289.05 = [append#1(@l, @ys)] 876.46/289.05 876.46/289.05 [insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x)] = [1 0] @ls + [0 1] @vals1 + [1] 876.46/289.05 [0 1] [0 0] [2] 876.46/289.05 >= [1 0] @ls + [0 1] @vals1 + [1] 876.46/289.05 [0 1] [0 0] [2] 876.46/289.05 = [insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x)] 876.46/289.05 876.46/289.05 [#ckgt(#EQ())] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#false()] 876.46/289.05 876.46/289.05 [#ckgt(#LT())] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#false()] 876.46/289.05 876.46/289.05 [#ckgt(#GT())] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#true()] 876.46/289.05 876.46/289.05 [insert(@x, @l)] = [1 0] @l + [1] 876.46/289.05 [0 1] [1] 876.46/289.05 >= [1 0] @l + [1] 876.46/289.05 [0 1] [1] 876.46/289.05 = [insert#1(@x, @l, @x)] 876.46/289.05 876.46/289.05 [insert#2(nil(), @keyX, @valX, @x)] = [1] 876.46/289.05 [1] 876.46/289.05 >= [1] 876.46/289.05 [1] 876.46/289.05 = [::(tuple#2(::(@valX, nil()), @keyX), nil())] 876.46/289.05 876.46/289.05 [insert#2(::(@l1, @ls), @keyX, @valX, @x)] = [0 1] @l1 + [1 0] @ls + [1] 876.46/289.05 [0 0] [0 1] [2] 876.46/289.05 >= [0 1] @l1 + [1 0] @ls + [1] 876.46/289.05 [0 0] [0 1] [2] 876.46/289.05 = [insert#3(@l1, @keyX, @ls, @valX, @x)] 876.46/289.05 876.46/289.05 [#and(#true(), #true())] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#true()] 876.46/289.05 876.46/289.05 [#and(#true(), #false())] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#false()] 876.46/289.05 876.46/289.05 [#and(#false(), #true())] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#false()] 876.46/289.05 876.46/289.05 [#and(#false(), #false())] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#false()] 876.46/289.05 876.46/289.05 [#compare(#pos(@x), #pos(@y))] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#compare(@x, @y)] 876.46/289.05 876.46/289.05 [#compare(#pos(@x), #0())] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#GT()] 876.46/289.05 876.46/289.05 [#compare(#pos(@x), #neg(@y))] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#GT()] 876.46/289.05 876.46/289.05 [#compare(#0(), #pos(@y))] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#LT()] 876.46/289.05 876.46/289.05 [#compare(#0(), #0())] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#EQ()] 876.46/289.05 876.46/289.05 [#compare(#0(), #neg(@y))] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#GT()] 876.46/289.05 876.46/289.05 [#compare(#0(), #s(@y))] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#LT()] 876.46/289.05 876.46/289.05 [#compare(#neg(@x), #pos(@y))] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#LT()] 876.46/289.05 876.46/289.05 [#compare(#neg(@x), #0())] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#LT()] 876.46/289.05 876.46/289.05 [#compare(#neg(@x), #neg(@y))] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#compare(@y, @x)] 876.46/289.05 876.46/289.05 [#compare(#s(@x), #0())] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#GT()] 876.46/289.05 876.46/289.05 [#compare(#s(@x), #s(@y))] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#compare(@x, @y)] 876.46/289.05 876.46/289.05 [split#1(nil())] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [nil()] 876.46/289.05 876.46/289.05 [split#1(::(@x, @xs))] = [0 1] @xs + [1] 876.46/289.05 [0 1] [1] 876.46/289.05 >= [0 1] @xs + [1] 876.46/289.05 [0 1] [1] 876.46/289.05 = [insert(@x, split(@xs))] 876.46/289.05 876.46/289.05 [#greater(@x, @y)] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [#ckgt(#compare(@x, @y))] 876.46/289.05 876.46/289.05 [insert#4(#true(), @key1, @ls, @valX, @vals1, @x)] = [1 0] @ls + [0 1] @vals1 + [1] 876.46/289.05 [0 1] [0 0] [2] 876.46/289.05 >= [1 0] @ls + [0 1] @vals1 + [1] 876.46/289.05 [0 1] [0 0] [1] 876.46/289.05 = [::(tuple#2(::(@valX, @vals1), @key1), @ls)] 876.46/289.05 876.46/289.05 [insert#4(#false(), @key1, @ls, @valX, @vals1, @x)] = [1 0] @ls + [0 1] @vals1 + [1] 876.46/289.05 [0 1] [0 0] [2] 876.46/289.05 >= [1 0] @ls + [0 1] @vals1 + [1] 876.46/289.05 [0 1] [0 0] [2] 876.46/289.05 = [::(tuple#2(@vals1, @key1), insert(@x, @ls))] 876.46/289.05 876.46/289.05 [insert#1(tuple#2(@valX, @keyX), @l, @x)] = [1 0] @l + [1] 876.46/289.05 [0 1] [1] 876.46/289.05 >= [1 0] @l + [1] 876.46/289.05 [0 1] [1] 876.46/289.05 = [insert#2(@l, @keyX, @valX, @x)] 876.46/289.05 876.46/289.05 [quicksort#1(nil())] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [nil()] 876.46/289.05 876.46/289.05 [quicksort#1(::(@z, @zs))] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [quicksort#2(splitqs(@z, @zs), @z)] 876.46/289.05 876.46/289.05 [append#1(nil(), @ys)] = [6 0] @ys + [0] 876.46/289.05 [6 6] [0] 876.46/289.05 >= [1 0] @ys + [0] 876.46/289.05 [0 1] [0] 876.46/289.05 = [@ys] 876.46/289.05 876.46/289.05 [append#1(::(@x, @xs), @ys)] = [6 0] @ys + [0] 876.46/289.05 [6 6] [0] 876.46/289.05 ? [0 1] @x + [0] 876.46/289.05 [0 0] [1] 876.46/289.05 = [::(@x, append(@xs, @ys))] 876.46/289.05 876.46/289.05 [splitqs(@pivot, @l)] = [0 1] @l + [0 0] @pivot + [0] 876.46/289.05 [4 4] [4 4] [0] 876.46/289.05 >= [0 1] @l + [0 0] @pivot + [0] 876.46/289.05 [0 2] [2 4] [0] 876.46/289.05 = [splitqs#1(@l, @pivot)] 876.46/289.05 876.46/289.05 [quicksort(@l)] = [0] 876.46/289.05 [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [quicksort#1(@l)] 876.46/289.05 876.46/289.05 [splitqs#3(#true(), @ls, @rs, @x)] = [0 1] @ls + [0 1] @rs + [1] 876.46/289.05 [0 1] [0 0] [1] 876.46/289.05 >= [0 1] @ls + [0 1] @rs + [1] 876.46/289.05 [0 1] [0 0] [0] 876.46/289.05 = [tuple#2(@ls, ::(@x, @rs))] 876.46/289.05 876.46/289.05 [splitqs#3(#false(), @ls, @rs, @x)] = [0 1] @ls + [0 1] @rs + [1] 876.46/289.05 [0 1] [0 0] [1] 876.46/289.05 >= [0 1] @ls + [0 1] @rs + [1] 876.46/289.05 [0 1] [0 0] [1] 876.46/289.05 = [tuple#2(::(@x, @ls), @rs)] 876.46/289.05 876.46/289.05 [splitqs#2(tuple#2(@ls, @rs), @pivot, @x)] = [0 1] @ls + [0 0] @pivot + [0 1] @rs + [1] 876.46/289.05 [0 1] [0 4] [0 1] [1] 876.46/289.05 >= [0 1] @ls + [0 1] @rs + [1] 876.46/289.05 [0 1] [0 0] [1] 876.46/289.05 = [splitqs#3(#greater(@x, @pivot), @ls, @rs, @x)] 876.46/289.05 876.46/289.05 [splitqs#1(nil(), @pivot)] = [0 0] @pivot + [0] 876.46/289.05 [2 4] [0] 876.46/289.05 >= [0] 876.46/289.05 [0] 876.46/289.05 = [tuple#2(nil(), nil())] 876.46/289.05 876.46/289.05 [splitqs#1(::(@x, @xs), @pivot)] = [0 1] @xs + [0 0] @pivot + [1] 876.46/289.05 [0 2] [2 4] [2] 876.46/289.05 >= [0 1] @xs + [0 0] @pivot + [1] 876.46/289.05 [0 1] [0 4] [1] 876.46/289.05 = [splitqs#2(splitqs(@pivot, @xs), @pivot, @x)] 876.46/289.05 876.46/289.05 [splitAndSort^#(@l)] = [7 7] @l + [7] 876.46/289.05 [7 7] [7] 876.46/289.05 > [0 4] @l + [0] 876.46/289.05 [0 0] [0] 876.46/289.05 = [sortAll^#(split(@l))] 876.46/289.05 876.46/289.05 [sortAll^#(@l)] = [4 0] @l + [0] 876.46/289.05 [0 0] [0] 876.46/289.05 >= [4 0] @l + [0] 876.46/289.05 [0 0] [0] 876.46/289.05 = [sortAll#1^#(@l)] 876.46/289.05 876.46/289.05 [sortAll#2^#(tuple#2(@vals, @key), @xs)] = [4 0] @xs + [0 1] @vals + [0] 876.46/289.05 [0 0] [0 0] [0] 876.46/289.05 >= [4 0] @xs + [0] 876.46/289.05 [0 0] [0] 876.46/289.05 = [sortAll^#(@xs)] 876.46/289.05 876.46/289.05 [sortAll#2^#(tuple#2(@vals, @key), @xs)] = [4 0] @xs + [0 1] @vals + [0] 876.46/289.05 [0 0] [0 0] [0] 876.46/289.05 >= [0 1] @vals + [0] 876.46/289.05 [0 0] [0] 876.46/289.05 = [quicksort^#(@vals)] 876.46/289.05 876.46/289.05 [quicksort^#(@l)] = [0 1] @l + [0] 876.46/289.05 [0 0] [0] 876.46/289.05 >= [0 1] @l + [0] 876.46/289.05 [0 0] [0] 876.46/289.05 = [c_3(quicksort#1^#(@l))] 876.46/289.05 876.46/289.05 [quicksort#2^#(tuple#2(@xs, @ys), @z)] = [0 1] @ys + [0 1] @xs + [0 0] @z + [0] 876.46/289.05 [0 0] [0 0] [0 4] [0] 876.46/289.05 >= [0 1] @ys + [0 1] @xs + [0] 876.46/289.05 [0 0] [0 0] [0] 876.46/289.05 = [c_4(append^#(quicksort(@xs), ::(@z, quicksort(@ys))), 876.46/289.05 quicksort^#(@xs), 876.46/289.05 quicksort^#(@ys))] 876.46/289.05 876.46/289.05 [sortAll#1^#(::(@x, @xs))] = [0 4] @x + [4 0] @xs + [0] 876.46/289.05 [0 0] [0 0] [0] 876.46/289.05 >= [0 1] @x + [4 0] @xs + [0] 876.46/289.05 [0 0] [0 0] [0] 876.46/289.05 = [sortAll#2^#(@x, @xs)] 876.46/289.05 876.46/289.05 [quicksort#1^#(::(@z, @zs))] = [0 1] @zs + [1] 876.46/289.05 [0 0] [0] 876.46/289.05 > [0 1] @zs + [0] 876.46/289.05 [0 0] [0] 876.46/289.05 = [c_8(quicksort#2^#(splitqs(@z, @zs), @z), splitqs^#(@z, @zs))] 876.46/289.05 876.46/289.05 876.46/289.05 We return to the main proof. Consider the set of all dependency 876.46/289.05 pairs 876.46/289.05 876.46/289.05 : 876.46/289.05 { 1: quicksort^#(@l) -> c_3(quicksort#1^#(@l)) 876.46/289.05 , 2: quicksort#2^#(tuple#2(@xs, @ys), @z) -> 876.46/289.05 c_4(append^#(quicksort(@xs), ::(@z, quicksort(@ys))), 876.46/289.05 quicksort^#(@xs), 876.46/289.05 quicksort^#(@ys)) 876.46/289.05 , 3: quicksort#1^#(::(@z, @zs)) -> 876.46/289.05 c_8(quicksort#2^#(splitqs(@z, @zs), @z), splitqs^#(@z, @zs)) 876.46/289.05 , 4: splitAndSort^#(@l) -> sortAll^#(split(@l)) 876.46/289.05 , 5: sortAll^#(@l) -> sortAll#1^#(@l) 876.46/289.05 , 6: sortAll#2^#(tuple#2(@vals, @key), @xs) -> sortAll^#(@xs) 876.46/289.05 , 7: sortAll#2^#(tuple#2(@vals, @key), @xs) -> quicksort^#(@vals) 876.46/289.05 , 8: sortAll#1^#(::(@x, @xs)) -> sortAll#2^#(@x, @xs) } 876.46/289.05 876.46/289.05 Processor 'matrix interpretation of dimension 2' induces the 876.46/289.05 complexity certificate YES(?,O(n^1)) on application of dependency 876.46/289.05 pairs {3,4}. These cover all (indirect) predecessors of dependency 876.46/289.05 pairs {2,3,4}, their number of application is equally bounded. The 876.46/289.05 dependency pairs are shifted into the weak component. 876.46/289.05 876.46/289.05 We are left with following problem, upon which TcT provides the 876.46/289.05 certificate YES(O(1),O(n^1)). 876.46/289.05 876.46/289.05 Strict DPs: { quicksort^#(@l) -> c_3(quicksort#1^#(@l)) } 876.46/289.05 Weak DPs: 876.46/289.05 { splitAndSort^#(@l) -> sortAll^#(split(@l)) 876.46/289.05 , sortAll^#(@l) -> sortAll#1^#(@l) 876.46/289.05 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> sortAll^#(@xs) 876.46/289.05 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> quicksort^#(@vals) 876.46/289.05 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> 876.46/289.05 c_4(append^#(quicksort(@xs), ::(@z, quicksort(@ys))), 876.46/289.05 quicksort^#(@xs), 876.46/289.05 quicksort^#(@ys)) 876.46/289.05 , sortAll#1^#(::(@x, @xs)) -> sortAll#2^#(@x, @xs) 876.46/289.05 , quicksort#1^#(::(@z, @zs)) -> 876.46/289.05 c_8(quicksort#2^#(splitqs(@z, @zs), @z), splitqs^#(@z, @zs)) } 876.46/289.05 Weak Trs: 876.46/289.05 { #equal(@x, @y) -> #eq(@x, @y) 876.46/289.05 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.46/289.05 , #eq(#pos(@x), #0()) -> #false() 876.46/289.05 , #eq(#pos(@x), #neg(@y)) -> #false() 876.46/289.05 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.46/289.05 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.05 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.46/289.05 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.46/289.05 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.05 , #eq(nil(), nil()) -> #true() 876.46/289.05 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.46/289.05 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.05 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.46/289.05 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.46/289.05 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.05 , #eq(#0(), #pos(@y)) -> #false() 876.46/289.05 , #eq(#0(), #0()) -> #true() 876.46/289.05 , #eq(#0(), #neg(@y)) -> #false() 876.46/289.05 , #eq(#0(), #s(@y)) -> #false() 876.46/289.05 , #eq(#neg(@x), #pos(@y)) -> #false() 876.46/289.05 , #eq(#neg(@x), #0()) -> #false() 876.46/289.05 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.46/289.05 , #eq(#s(@x), #0()) -> #false() 876.46/289.05 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.46/289.05 , quicksort#2(tuple#2(@xs, @ys), @z) -> 876.46/289.05 append(quicksort(@xs), ::(@z, quicksort(@ys))) 876.46/289.05 , split(@l) -> split#1(@l) 876.46/289.05 , append(@l, @ys) -> append#1(@l, @ys) 876.46/289.05 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.46/289.05 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.46/289.05 , #ckgt(#EQ()) -> #false() 876.46/289.05 , #ckgt(#LT()) -> #false() 876.46/289.05 , #ckgt(#GT()) -> #true() 876.46/289.05 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.46/289.05 , insert#2(nil(), @keyX, @valX, @x) -> 876.46/289.05 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.46/289.05 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.46/289.05 insert#3(@l1, @keyX, @ls, @valX, @x) 876.46/289.05 , #and(#true(), #true()) -> #true() 876.46/289.05 , #and(#true(), #false()) -> #false() 876.46/289.05 , #and(#false(), #true()) -> #false() 876.46/289.05 , #and(#false(), #false()) -> #false() 876.46/289.05 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 876.46/289.05 , #compare(#pos(@x), #0()) -> #GT() 876.46/289.05 , #compare(#pos(@x), #neg(@y)) -> #GT() 876.46/289.05 , #compare(#0(), #pos(@y)) -> #LT() 876.46/289.05 , #compare(#0(), #0()) -> #EQ() 876.46/289.05 , #compare(#0(), #neg(@y)) -> #GT() 876.46/289.05 , #compare(#0(), #s(@y)) -> #LT() 876.46/289.05 , #compare(#neg(@x), #pos(@y)) -> #LT() 876.46/289.05 , #compare(#neg(@x), #0()) -> #LT() 876.46/289.05 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 876.46/289.05 , #compare(#s(@x), #0()) -> #GT() 876.46/289.05 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 876.46/289.05 , split#1(nil()) -> nil() 876.46/289.05 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.46/289.05 , #greater(@x, @y) -> #ckgt(#compare(@x, @y)) 876.46/289.05 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.05 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.46/289.05 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.05 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.46/289.05 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.46/289.05 insert#2(@l, @keyX, @valX, @x) 876.46/289.05 , quicksort#1(nil()) -> nil() 876.46/289.06 , quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) 876.46/289.06 , append#1(nil(), @ys) -> @ys 876.46/289.06 , append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) 876.46/289.06 , splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) 876.46/289.06 , quicksort(@l) -> quicksort#1(@l) 876.46/289.06 , splitqs#3(#true(), @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) 876.46/289.06 , splitqs#3(#false(), @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) 876.46/289.06 , splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> 876.46/289.06 splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) 876.46/289.06 , splitqs#1(nil(), @pivot) -> tuple#2(nil(), nil()) 876.46/289.06 , splitqs#1(::(@x, @xs), @pivot) -> 876.46/289.06 splitqs#2(splitqs(@pivot, @xs), @pivot, @x) } 876.46/289.06 Obligation: 876.46/289.06 innermost runtime complexity 876.46/289.06 Answer: 876.46/289.06 YES(O(1),O(n^1)) 876.46/289.06 876.46/289.06 We use the processor 'matrix interpretation of dimension 2' to 876.46/289.06 orient following rules strictly. 876.46/289.06 876.46/289.06 DPs: 876.46/289.06 { 2: splitAndSort^#(@l) -> sortAll^#(split(@l)) 876.46/289.06 , 5: sortAll#2^#(tuple#2(@vals, @key), @xs) -> quicksort^#(@vals) 876.46/289.06 , 8: quicksort#1^#(::(@z, @zs)) -> 876.46/289.06 c_8(quicksort#2^#(splitqs(@z, @zs), @z), splitqs^#(@z, @zs)) } 876.46/289.06 Trs: 876.46/289.06 { splitqs#3(#true(), @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) 876.46/289.06 , splitqs#1(::(@x, @xs), @pivot) -> 876.46/289.06 splitqs#2(splitqs(@pivot, @xs), @pivot, @x) } 876.46/289.06 876.46/289.06 Sub-proof: 876.46/289.06 ---------- 876.46/289.06 The following argument positions are usable: 876.46/289.06 Uargs(c_3) = {1}, Uargs(c_4) = {1, 2, 3}, Uargs(c_8) = {1} 876.46/289.06 876.46/289.06 TcT has computed the following constructor-based matrix 876.46/289.06 interpretation satisfying not(EDA) and not(IDA(1)). 876.46/289.06 876.46/289.06 [#equal](x1, x2) = [0] 876.46/289.06 [0] 876.46/289.06 876.46/289.06 [#eq](x1, x2) = [0] 876.46/289.06 [0] 876.46/289.06 876.46/289.06 [quicksort#2](x1, x2) = [0] 876.46/289.06 [0] 876.46/289.06 876.46/289.06 [split](x1) = [0 1] x1 + [0] 876.46/289.06 [0 3] [1] 876.46/289.06 876.46/289.06 [#true] = [0] 876.46/289.06 [0] 876.46/289.06 876.46/289.06 [append](x1, x2) = [0] 876.46/289.06 [0] 876.46/289.06 876.46/289.06 [insert#3](x1, x2, x3, x4, x5) = [1 0] x1 + [1 0] x3 + [1] 876.46/289.06 [0 0] [0 1] [3] 876.46/289.06 876.46/289.06 [#ckgt](x1) = [0] 876.46/289.06 [0] 876.46/289.06 876.46/289.06 [insert](x1, x2) = [1 0] x2 + [1] 876.46/289.06 [0 1] [2] 876.46/289.06 876.46/289.06 [#pos](x1) = [0] 876.46/289.06 [0] 876.46/289.06 876.46/289.06 [#EQ] = [0] 876.46/289.06 [0] 876.46/289.06 876.46/289.06 [insert#2](x1, x2, x3, x4) = [1 0] x1 + [1] 876.46/289.06 [0 1] [2] 876.46/289.06 876.46/289.06 [#and](x1, x2) = [0] 876.46/289.06 [0] 876.46/289.06 876.46/289.06 [#compare](x1, x2) = [0] 876.46/289.06 [0] 876.46/289.06 876.46/289.06 [tuple#2](x1, x2) = [0 1] x1 + [0 0] x2 + [0] 876.46/289.06 [0 1] [0 1] [0] 876.46/289.06 876.46/289.06 [nil] = [0] 876.46/289.06 [0] 876.46/289.06 876.46/289.06 [split#1](x1) = [0 1] x1 + [0] 876.46/289.06 [0 3] [1] 876.46/289.06 876.46/289.06 [#greater](x1, x2) = [0] 876.46/289.06 [0] 876.46/289.06 876.46/289.06 [insert#4](x1, x2, x3, x4, x5, x6) = [1 0] x3 + [0 1] x5 + [1] 876.46/289.06 [0 1] [0 0] [3] 876.46/289.06 876.46/289.06 [insert#1](x1, x2, x3) = [1 0] x2 + [1] 876.46/289.06 [0 1] [2] 876.46/289.06 876.46/289.06 [quicksort#1](x1) = [0] 876.46/289.06 [0] 876.46/289.06 876.46/289.06 [append#1](x1, x2) = [6 0] x2 + [0] 876.46/289.06 [6 6] [0] 876.46/289.06 876.46/289.06 [splitqs](x1, x2) = [0 4] x1 + [4 4] x2 + [0] 876.46/289.06 [0 0] [0 1] [0] 876.46/289.06 876.46/289.06 [#false] = [0] 876.46/289.06 [0] 876.46/289.06 876.46/289.06 [quicksort](x1) = [0] 876.46/289.06 [0] 876.46/289.06 876.46/289.06 [::](x1, x2) = [1 0] x1 + [1 0] x2 + [0] 876.46/289.06 [0 0] [0 1] [1] 876.46/289.06 876.46/289.06 [#LT] = [0] 876.46/289.06 [0] 876.46/289.06 876.46/289.06 [splitqs#3](x1, x2, x3, x4) = [0 2] x2 + [0 0] x3 + [1] 876.46/289.06 [0 1] [0 1] [1] 876.46/289.06 876.46/289.06 [splitqs#2](x1, x2, x3) = [0 2] x1 + [1] 876.46/289.06 [0 1] [1] 876.46/289.06 876.46/289.06 [#0] = [0] 876.46/289.06 [0] 876.46/289.06 876.46/289.06 [#neg](x1) = [0] 876.46/289.06 [0] 876.46/289.06 876.46/289.06 [#s](x1) = [0] 876.46/289.06 [0] 876.46/289.06 876.46/289.06 [#GT] = [0] 876.46/289.06 [0] 876.46/289.06 876.46/289.06 [splitqs#1](x1, x2) = [0 2] x1 + [0] 876.46/289.06 [0 1] [0] 876.46/289.06 876.46/289.06 [splitAndSort^#](x1) = [7 7] x1 + [7] 876.46/289.06 [7 7] [7] 876.46/289.06 876.46/289.06 [sortAll^#](x1) = [4 0] x1 + [1] 876.46/289.06 [0 0] [0] 876.46/289.06 876.46/289.06 [sortAll#2^#](x1, x2) = [2 0] x1 + [4 0] x2 + [1] 876.46/289.06 [0 0] [0 0] [0] 876.46/289.06 876.46/289.06 [quicksort^#](x1) = [0 2] x1 + [0] 876.46/289.06 [0 0] [0] 876.46/289.06 876.46/289.06 [quicksort#2^#](x1, x2) = [0 2] x1 + [0 0] x2 + [0] 876.46/289.06 [0 0] [4 0] [4] 876.46/289.06 876.46/289.06 [append^#](x1, x2) = [0] 876.46/289.06 [0] 876.46/289.06 876.46/289.06 [sortAll#1^#](x1) = [4 0] x1 + [1] 876.46/289.06 [0 0] [0] 876.46/289.06 876.46/289.06 [quicksort#1^#](x1) = [0 2] x1 + [0] 876.46/289.06 [0 0] [0] 876.46/289.06 876.46/289.06 [splitqs^#](x1, x2) = [0] 876.46/289.06 [0] 876.46/289.06 876.46/289.06 [c_3](x1) = [1 0] x1 + [0] 876.46/289.06 [0 0] [0] 876.46/289.06 876.46/289.06 [c_4](x1, x2, x3) = [4 0] x1 + [1 0] x2 + [1 876.46/289.06 0] x3 + [0] 876.46/289.06 [0 0] [0 0] [0 876.46/289.06 0] [3] 876.46/289.06 876.46/289.06 [c_8](x1, x2) = [1 0] x1 + [1] 876.46/289.06 [0 0] [0] 876.46/289.06 876.46/289.06 The order satisfies the following ordering constraints: 876.46/289.06 876.46/289.06 [#equal(@x, @y)] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#eq(@x, @y)] 876.46/289.06 876.46/289.06 [#eq(#pos(@x), #pos(@y))] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#eq(@x, @y)] 876.46/289.06 876.46/289.06 [#eq(#pos(@x), #0())] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#false()] 876.46/289.06 876.46/289.06 [#eq(#pos(@x), #neg(@y))] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#false()] 876.46/289.06 876.46/289.06 [#eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2))] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))] 876.46/289.06 876.46/289.06 [#eq(tuple#2(@x_1, @x_2), nil())] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#false()] 876.46/289.06 876.46/289.06 [#eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2))] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#false()] 876.46/289.06 876.46/289.06 [#eq(nil(), tuple#2(@y_1, @y_2))] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#false()] 876.46/289.06 876.46/289.06 [#eq(nil(), nil())] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#true()] 876.46/289.06 876.46/289.06 [#eq(nil(), ::(@y_1, @y_2))] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#false()] 876.46/289.06 876.46/289.06 [#eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2))] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#false()] 876.46/289.06 876.46/289.06 [#eq(::(@x_1, @x_2), nil())] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#false()] 876.46/289.06 876.46/289.06 [#eq(::(@x_1, @x_2), ::(@y_1, @y_2))] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))] 876.46/289.06 876.46/289.06 [#eq(#0(), #pos(@y))] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#false()] 876.46/289.06 876.46/289.06 [#eq(#0(), #0())] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#true()] 876.46/289.06 876.46/289.06 [#eq(#0(), #neg(@y))] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#false()] 876.46/289.06 876.46/289.06 [#eq(#0(), #s(@y))] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#false()] 876.46/289.06 876.46/289.06 [#eq(#neg(@x), #pos(@y))] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#false()] 876.46/289.06 876.46/289.06 [#eq(#neg(@x), #0())] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#false()] 876.46/289.06 876.46/289.06 [#eq(#neg(@x), #neg(@y))] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#eq(@x, @y)] 876.46/289.06 876.46/289.06 [#eq(#s(@x), #0())] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#false()] 876.46/289.06 876.46/289.06 [#eq(#s(@x), #s(@y))] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#eq(@x, @y)] 876.46/289.06 876.46/289.06 [quicksort#2(tuple#2(@xs, @ys), @z)] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [append(quicksort(@xs), ::(@z, quicksort(@ys)))] 876.46/289.06 876.46/289.06 [split(@l)] = [0 1] @l + [0] 876.46/289.06 [0 3] [1] 876.46/289.06 >= [0 1] @l + [0] 876.46/289.06 [0 3] [1] 876.46/289.06 = [split#1(@l)] 876.46/289.06 876.46/289.06 [append(@l, @ys)] = [0] 876.46/289.06 [0] 876.46/289.06 ? [6 0] @ys + [0] 876.46/289.06 [6 6] [0] 876.46/289.06 = [append#1(@l, @ys)] 876.46/289.06 876.46/289.06 [insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x)] = [1 0] @ls + [0 1] @vals1 + [1] 876.46/289.06 [0 1] [0 0] [3] 876.46/289.06 >= [1 0] @ls + [0 1] @vals1 + [1] 876.46/289.06 [0 1] [0 0] [3] 876.46/289.06 = [insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x)] 876.46/289.06 876.46/289.06 [#ckgt(#EQ())] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#false()] 876.46/289.06 876.46/289.06 [#ckgt(#LT())] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#false()] 876.46/289.06 876.46/289.06 [#ckgt(#GT())] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#true()] 876.46/289.06 876.46/289.06 [insert(@x, @l)] = [1 0] @l + [1] 876.46/289.06 [0 1] [2] 876.46/289.06 >= [1 0] @l + [1] 876.46/289.06 [0 1] [2] 876.46/289.06 = [insert#1(@x, @l, @x)] 876.46/289.06 876.46/289.06 [insert#2(nil(), @keyX, @valX, @x)] = [1] 876.46/289.06 [2] 876.46/289.06 >= [1] 876.46/289.06 [1] 876.46/289.06 = [::(tuple#2(::(@valX, nil()), @keyX), nil())] 876.46/289.06 876.46/289.06 [insert#2(::(@l1, @ls), @keyX, @valX, @x)] = [1 0] @l1 + [1 0] @ls + [1] 876.46/289.06 [0 0] [0 1] [3] 876.46/289.06 >= [1 0] @l1 + [1 0] @ls + [1] 876.46/289.06 [0 0] [0 1] [3] 876.46/289.06 = [insert#3(@l1, @keyX, @ls, @valX, @x)] 876.46/289.06 876.46/289.06 [#and(#true(), #true())] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#true()] 876.46/289.06 876.46/289.06 [#and(#true(), #false())] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#false()] 876.46/289.06 876.46/289.06 [#and(#false(), #true())] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#false()] 876.46/289.06 876.46/289.06 [#and(#false(), #false())] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#false()] 876.46/289.06 876.46/289.06 [#compare(#pos(@x), #pos(@y))] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#compare(@x, @y)] 876.46/289.06 876.46/289.06 [#compare(#pos(@x), #0())] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#GT()] 876.46/289.06 876.46/289.06 [#compare(#pos(@x), #neg(@y))] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#GT()] 876.46/289.06 876.46/289.06 [#compare(#0(), #pos(@y))] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#LT()] 876.46/289.06 876.46/289.06 [#compare(#0(), #0())] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#EQ()] 876.46/289.06 876.46/289.06 [#compare(#0(), #neg(@y))] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#GT()] 876.46/289.06 876.46/289.06 [#compare(#0(), #s(@y))] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#LT()] 876.46/289.06 876.46/289.06 [#compare(#neg(@x), #pos(@y))] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#LT()] 876.46/289.06 876.46/289.06 [#compare(#neg(@x), #0())] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#LT()] 876.46/289.06 876.46/289.06 [#compare(#neg(@x), #neg(@y))] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#compare(@y, @x)] 876.46/289.06 876.46/289.06 [#compare(#s(@x), #0())] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#GT()] 876.46/289.06 876.46/289.06 [#compare(#s(@x), #s(@y))] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#compare(@x, @y)] 876.46/289.06 876.46/289.06 [split#1(nil())] = [0] 876.46/289.06 [1] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [nil()] 876.46/289.06 876.46/289.06 [split#1(::(@x, @xs))] = [0 1] @xs + [1] 876.46/289.06 [0 3] [4] 876.46/289.06 >= [0 1] @xs + [1] 876.46/289.06 [0 3] [3] 876.46/289.06 = [insert(@x, split(@xs))] 876.46/289.06 876.46/289.06 [#greater(@x, @y)] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [#ckgt(#compare(@x, @y))] 876.46/289.06 876.46/289.06 [insert#4(#true(), @key1, @ls, @valX, @vals1, @x)] = [1 0] @ls + [0 1] @vals1 + [1] 876.46/289.06 [0 1] [0 0] [3] 876.46/289.06 >= [1 0] @ls + [0 1] @vals1 + [1] 876.46/289.06 [0 1] [0 0] [1] 876.46/289.06 = [::(tuple#2(::(@valX, @vals1), @key1), @ls)] 876.46/289.06 876.46/289.06 [insert#4(#false(), @key1, @ls, @valX, @vals1, @x)] = [1 0] @ls + [0 1] @vals1 + [1] 876.46/289.06 [0 1] [0 0] [3] 876.46/289.06 >= [1 0] @ls + [0 1] @vals1 + [1] 876.46/289.06 [0 1] [0 0] [3] 876.46/289.06 = [::(tuple#2(@vals1, @key1), insert(@x, @ls))] 876.46/289.06 876.46/289.06 [insert#1(tuple#2(@valX, @keyX), @l, @x)] = [1 0] @l + [1] 876.46/289.06 [0 1] [2] 876.46/289.06 >= [1 0] @l + [1] 876.46/289.06 [0 1] [2] 876.46/289.06 = [insert#2(@l, @keyX, @valX, @x)] 876.46/289.06 876.46/289.06 [quicksort#1(nil())] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [nil()] 876.46/289.06 876.46/289.06 [quicksort#1(::(@z, @zs))] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [quicksort#2(splitqs(@z, @zs), @z)] 876.46/289.06 876.46/289.06 [append#1(nil(), @ys)] = [6 0] @ys + [0] 876.46/289.06 [6 6] [0] 876.46/289.06 >= [1 0] @ys + [0] 876.46/289.06 [0 1] [0] 876.46/289.06 = [@ys] 876.46/289.06 876.46/289.06 [append#1(::(@x, @xs), @ys)] = [6 0] @ys + [0] 876.46/289.06 [6 6] [0] 876.46/289.06 ? [1 0] @x + [0] 876.46/289.06 [0 0] [1] 876.46/289.06 = [::(@x, append(@xs, @ys))] 876.46/289.06 876.46/289.06 [splitqs(@pivot, @l)] = [4 4] @l + [0 4] @pivot + [0] 876.46/289.06 [0 1] [0 0] [0] 876.46/289.06 >= [0 2] @l + [0] 876.46/289.06 [0 1] [0] 876.46/289.06 = [splitqs#1(@l, @pivot)] 876.46/289.06 876.46/289.06 [quicksort(@l)] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [quicksort#1(@l)] 876.46/289.06 876.46/289.06 [splitqs#3(#true(), @ls, @rs, @x)] = [0 2] @ls + [0 0] @rs + [1] 876.46/289.06 [0 1] [0 1] [1] 876.46/289.06 > [0 1] @ls + [0 0] @rs + [0] 876.46/289.06 [0 1] [0 1] [1] 876.46/289.06 = [tuple#2(@ls, ::(@x, @rs))] 876.46/289.06 876.46/289.06 [splitqs#3(#false(), @ls, @rs, @x)] = [0 2] @ls + [0 0] @rs + [1] 876.46/289.06 [0 1] [0 1] [1] 876.46/289.06 >= [0 1] @ls + [0 0] @rs + [1] 876.46/289.06 [0 1] [0 1] [1] 876.46/289.06 = [tuple#2(::(@x, @ls), @rs)] 876.46/289.06 876.46/289.06 [splitqs#2(tuple#2(@ls, @rs), @pivot, @x)] = [0 2] @ls + [0 2] @rs + [1] 876.46/289.06 [0 1] [0 1] [1] 876.46/289.06 >= [0 2] @ls + [0 0] @rs + [1] 876.46/289.06 [0 1] [0 1] [1] 876.46/289.06 = [splitqs#3(#greater(@x, @pivot), @ls, @rs, @x)] 876.46/289.06 876.46/289.06 [splitqs#1(nil(), @pivot)] = [0] 876.46/289.06 [0] 876.46/289.06 >= [0] 876.46/289.06 [0] 876.46/289.06 = [tuple#2(nil(), nil())] 876.46/289.06 876.46/289.06 [splitqs#1(::(@x, @xs), @pivot)] = [0 2] @xs + [2] 876.46/289.06 [0 1] [1] 876.46/289.06 > [0 2] @xs + [1] 876.46/289.06 [0 1] [1] 876.46/289.06 = [splitqs#2(splitqs(@pivot, @xs), @pivot, @x)] 876.46/289.06 876.46/289.06 [splitAndSort^#(@l)] = [7 7] @l + [7] 876.46/289.06 [7 7] [7] 876.46/289.06 > [0 4] @l + [1] 876.46/289.06 [0 0] [0] 876.46/289.06 = [sortAll^#(split(@l))] 876.46/289.06 876.46/289.06 [sortAll^#(@l)] = [4 0] @l + [1] 876.46/289.06 [0 0] [0] 876.46/289.06 >= [4 0] @l + [1] 876.46/289.06 [0 0] [0] 876.46/289.06 = [sortAll#1^#(@l)] 876.46/289.06 876.46/289.06 [sortAll#2^#(tuple#2(@vals, @key), @xs)] = [4 0] @xs + [0 2] @vals + [1] 876.46/289.06 [0 0] [0 0] [0] 876.46/289.06 >= [4 0] @xs + [1] 876.46/289.06 [0 0] [0] 876.46/289.06 = [sortAll^#(@xs)] 876.46/289.06 876.46/289.06 [sortAll#2^#(tuple#2(@vals, @key), @xs)] = [4 0] @xs + [0 2] @vals + [1] 876.46/289.06 [0 0] [0 0] [0] 876.46/289.06 > [0 2] @vals + [0] 876.46/289.06 [0 0] [0] 876.46/289.06 = [quicksort^#(@vals)] 876.46/289.06 876.46/289.06 [quicksort^#(@l)] = [0 2] @l + [0] 876.46/289.06 [0 0] [0] 876.46/289.06 >= [0 2] @l + [0] 876.46/289.06 [0 0] [0] 876.46/289.06 = [c_3(quicksort#1^#(@l))] 876.46/289.06 876.46/289.06 [quicksort#2^#(tuple#2(@xs, @ys), @z)] = [0 2] @ys + [0 2] @xs + [0 0] @z + [0] 876.46/289.06 [0 0] [0 0] [4 0] [4] 876.46/289.06 >= [0 2] @ys + [0 2] @xs + [0] 876.46/289.06 [0 0] [0 0] [3] 876.46/289.06 = [c_4(append^#(quicksort(@xs), ::(@z, quicksort(@ys))), 876.46/289.06 quicksort^#(@xs), 876.46/289.06 quicksort^#(@ys))] 876.46/289.06 876.46/289.06 [sortAll#1^#(::(@x, @xs))] = [4 0] @x + [4 0] @xs + [1] 876.46/289.06 [0 0] [0 0] [0] 876.46/289.06 >= [2 0] @x + [4 0] @xs + [1] 876.46/289.06 [0 0] [0 0] [0] 876.46/289.06 = [sortAll#2^#(@x, @xs)] 876.46/289.06 876.46/289.06 [quicksort#1^#(::(@z, @zs))] = [0 2] @zs + [2] 876.46/289.06 [0 0] [0] 876.46/289.06 > [0 2] @zs + [1] 876.46/289.06 [0 0] [0] 876.46/289.06 = [c_8(quicksort#2^#(splitqs(@z, @zs), @z), splitqs^#(@z, @zs))] 876.46/289.06 876.46/289.06 876.46/289.06 We return to the main proof. Consider the set of all dependency 876.46/289.06 pairs 876.46/289.06 876.46/289.06 : 876.46/289.06 { 1: quicksort^#(@l) -> c_3(quicksort#1^#(@l)) 876.46/289.06 , 2: splitAndSort^#(@l) -> sortAll^#(split(@l)) 876.46/289.06 , 3: sortAll^#(@l) -> sortAll#1^#(@l) 876.46/289.06 , 4: sortAll#2^#(tuple#2(@vals, @key), @xs) -> sortAll^#(@xs) 876.46/289.06 , 5: sortAll#2^#(tuple#2(@vals, @key), @xs) -> quicksort^#(@vals) 876.46/289.06 , 6: quicksort#2^#(tuple#2(@xs, @ys), @z) -> 876.46/289.06 c_4(append^#(quicksort(@xs), ::(@z, quicksort(@ys))), 876.46/289.06 quicksort^#(@xs), 876.46/289.06 quicksort^#(@ys)) 876.46/289.06 , 7: sortAll#1^#(::(@x, @xs)) -> sortAll#2^#(@x, @xs) 876.46/289.06 , 8: quicksort#1^#(::(@z, @zs)) -> 876.46/289.06 c_8(quicksort#2^#(splitqs(@z, @zs), @z), splitqs^#(@z, @zs)) } 876.46/289.06 876.46/289.06 Processor 'matrix interpretation of dimension 2' induces the 876.46/289.06 complexity certificate YES(?,O(n^1)) on application of dependency 876.46/289.06 pairs {2,5,8}. These cover all (indirect) predecessors of 876.46/289.07 dependency pairs {1,2,5,6,8}, their number of application is 876.46/289.07 equally bounded. The dependency pairs are shifted into the weak 876.46/289.07 component. 876.46/289.07 876.46/289.07 We are left with following problem, upon which TcT provides the 876.46/289.07 certificate YES(O(1),O(1)). 876.46/289.07 876.46/289.07 Weak DPs: 876.46/289.07 { splitAndSort^#(@l) -> sortAll^#(split(@l)) 876.46/289.07 , sortAll^#(@l) -> sortAll#1^#(@l) 876.46/289.07 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> sortAll^#(@xs) 876.46/289.07 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> quicksort^#(@vals) 876.46/289.07 , quicksort^#(@l) -> c_3(quicksort#1^#(@l)) 876.46/289.07 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> 876.46/289.07 c_4(append^#(quicksort(@xs), ::(@z, quicksort(@ys))), 876.46/289.07 quicksort^#(@xs), 876.46/289.07 quicksort^#(@ys)) 876.46/289.07 , sortAll#1^#(::(@x, @xs)) -> sortAll#2^#(@x, @xs) 876.46/289.07 , quicksort#1^#(::(@z, @zs)) -> 876.46/289.07 c_8(quicksort#2^#(splitqs(@z, @zs), @z), splitqs^#(@z, @zs)) } 876.46/289.07 Weak Trs: 876.46/289.07 { #equal(@x, @y) -> #eq(@x, @y) 876.46/289.07 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.46/289.07 , #eq(#pos(@x), #0()) -> #false() 876.46/289.07 , #eq(#pos(@x), #neg(@y)) -> #false() 876.46/289.07 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.46/289.07 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.07 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.46/289.07 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.46/289.07 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.07 , #eq(nil(), nil()) -> #true() 876.46/289.07 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.46/289.07 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.07 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.46/289.07 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.46/289.07 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.07 , #eq(#0(), #pos(@y)) -> #false() 876.46/289.07 , #eq(#0(), #0()) -> #true() 876.46/289.07 , #eq(#0(), #neg(@y)) -> #false() 876.46/289.07 , #eq(#0(), #s(@y)) -> #false() 876.46/289.07 , #eq(#neg(@x), #pos(@y)) -> #false() 876.46/289.07 , #eq(#neg(@x), #0()) -> #false() 876.46/289.07 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.46/289.07 , #eq(#s(@x), #0()) -> #false() 876.46/289.07 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.46/289.07 , quicksort#2(tuple#2(@xs, @ys), @z) -> 876.46/289.07 append(quicksort(@xs), ::(@z, quicksort(@ys))) 876.46/289.07 , split(@l) -> split#1(@l) 876.46/289.07 , append(@l, @ys) -> append#1(@l, @ys) 876.46/289.07 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.46/289.07 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.46/289.07 , #ckgt(#EQ()) -> #false() 876.46/289.07 , #ckgt(#LT()) -> #false() 876.46/289.07 , #ckgt(#GT()) -> #true() 876.46/289.07 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.46/289.07 , insert#2(nil(), @keyX, @valX, @x) -> 876.46/289.07 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.46/289.07 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.46/289.07 insert#3(@l1, @keyX, @ls, @valX, @x) 876.46/289.07 , #and(#true(), #true()) -> #true() 876.46/289.07 , #and(#true(), #false()) -> #false() 876.46/289.07 , #and(#false(), #true()) -> #false() 876.46/289.07 , #and(#false(), #false()) -> #false() 876.46/289.07 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 876.46/289.07 , #compare(#pos(@x), #0()) -> #GT() 876.46/289.07 , #compare(#pos(@x), #neg(@y)) -> #GT() 876.46/289.07 , #compare(#0(), #pos(@y)) -> #LT() 876.46/289.07 , #compare(#0(), #0()) -> #EQ() 876.46/289.07 , #compare(#0(), #neg(@y)) -> #GT() 876.46/289.07 , #compare(#0(), #s(@y)) -> #LT() 876.46/289.07 , #compare(#neg(@x), #pos(@y)) -> #LT() 876.46/289.07 , #compare(#neg(@x), #0()) -> #LT() 876.46/289.07 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 876.46/289.07 , #compare(#s(@x), #0()) -> #GT() 876.46/289.07 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 876.46/289.07 , split#1(nil()) -> nil() 876.46/289.07 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.46/289.07 , #greater(@x, @y) -> #ckgt(#compare(@x, @y)) 876.46/289.07 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.07 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.46/289.07 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.07 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.46/289.07 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.46/289.07 insert#2(@l, @keyX, @valX, @x) 876.46/289.07 , quicksort#1(nil()) -> nil() 876.46/289.07 , quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) 876.46/289.07 , append#1(nil(), @ys) -> @ys 876.46/289.07 , append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) 876.46/289.07 , splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) 876.46/289.07 , quicksort(@l) -> quicksort#1(@l) 876.46/289.07 , splitqs#3(#true(), @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) 876.46/289.07 , splitqs#3(#false(), @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) 876.46/289.07 , splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> 876.46/289.07 splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) 876.46/289.07 , splitqs#1(nil(), @pivot) -> tuple#2(nil(), nil()) 876.46/289.07 , splitqs#1(::(@x, @xs), @pivot) -> 876.46/289.07 splitqs#2(splitqs(@pivot, @xs), @pivot, @x) } 876.46/289.07 Obligation: 876.46/289.07 innermost runtime complexity 876.46/289.07 Answer: 876.46/289.07 YES(O(1),O(1)) 876.46/289.07 876.46/289.07 The following weak DPs constitute a sub-graph of the DG that is 876.46/289.07 closed under successors. The DPs are removed. 876.46/289.07 876.46/289.07 { splitAndSort^#(@l) -> sortAll^#(split(@l)) 876.46/289.07 , sortAll^#(@l) -> sortAll#1^#(@l) 876.46/289.07 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> sortAll^#(@xs) 876.46/289.07 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> quicksort^#(@vals) 876.46/289.07 , quicksort^#(@l) -> c_3(quicksort#1^#(@l)) 876.46/289.07 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> 876.46/289.07 c_4(append^#(quicksort(@xs), ::(@z, quicksort(@ys))), 876.46/289.07 quicksort^#(@xs), 876.46/289.07 quicksort^#(@ys)) 876.46/289.07 , sortAll#1^#(::(@x, @xs)) -> sortAll#2^#(@x, @xs) 876.46/289.07 , quicksort#1^#(::(@z, @zs)) -> 876.46/289.07 c_8(quicksort#2^#(splitqs(@z, @zs), @z), splitqs^#(@z, @zs)) } 876.46/289.07 876.46/289.07 We are left with following problem, upon which TcT provides the 876.46/289.07 certificate YES(O(1),O(1)). 876.46/289.07 876.46/289.07 Weak Trs: 876.46/289.07 { #equal(@x, @y) -> #eq(@x, @y) 876.46/289.07 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.46/289.07 , #eq(#pos(@x), #0()) -> #false() 876.46/289.07 , #eq(#pos(@x), #neg(@y)) -> #false() 876.46/289.07 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.46/289.07 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.07 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.46/289.07 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.46/289.07 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.07 , #eq(nil(), nil()) -> #true() 876.46/289.07 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.46/289.07 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.07 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.46/289.07 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.46/289.07 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.07 , #eq(#0(), #pos(@y)) -> #false() 876.46/289.07 , #eq(#0(), #0()) -> #true() 876.46/289.07 , #eq(#0(), #neg(@y)) -> #false() 876.46/289.07 , #eq(#0(), #s(@y)) -> #false() 876.46/289.07 , #eq(#neg(@x), #pos(@y)) -> #false() 876.46/289.07 , #eq(#neg(@x), #0()) -> #false() 876.46/289.07 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.46/289.07 , #eq(#s(@x), #0()) -> #false() 876.46/289.07 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.46/289.07 , quicksort#2(tuple#2(@xs, @ys), @z) -> 876.46/289.07 append(quicksort(@xs), ::(@z, quicksort(@ys))) 876.46/289.07 , split(@l) -> split#1(@l) 876.46/289.07 , append(@l, @ys) -> append#1(@l, @ys) 876.46/289.07 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.46/289.07 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.46/289.07 , #ckgt(#EQ()) -> #false() 876.46/289.07 , #ckgt(#LT()) -> #false() 876.46/289.07 , #ckgt(#GT()) -> #true() 876.46/289.07 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.46/289.07 , insert#2(nil(), @keyX, @valX, @x) -> 876.46/289.07 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.46/289.07 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.46/289.07 insert#3(@l1, @keyX, @ls, @valX, @x) 876.46/289.07 , #and(#true(), #true()) -> #true() 876.46/289.07 , #and(#true(), #false()) -> #false() 876.46/289.07 , #and(#false(), #true()) -> #false() 876.46/289.07 , #and(#false(), #false()) -> #false() 876.46/289.07 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 876.46/289.07 , #compare(#pos(@x), #0()) -> #GT() 876.46/289.07 , #compare(#pos(@x), #neg(@y)) -> #GT() 876.46/289.07 , #compare(#0(), #pos(@y)) -> #LT() 876.46/289.07 , #compare(#0(), #0()) -> #EQ() 876.46/289.07 , #compare(#0(), #neg(@y)) -> #GT() 876.46/289.07 , #compare(#0(), #s(@y)) -> #LT() 876.46/289.07 , #compare(#neg(@x), #pos(@y)) -> #LT() 876.46/289.07 , #compare(#neg(@x), #0()) -> #LT() 876.46/289.07 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 876.46/289.07 , #compare(#s(@x), #0()) -> #GT() 876.46/289.07 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 876.46/289.07 , split#1(nil()) -> nil() 876.46/289.07 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.46/289.07 , #greater(@x, @y) -> #ckgt(#compare(@x, @y)) 876.46/289.07 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.07 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.46/289.07 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.07 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.46/289.07 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.46/289.07 insert#2(@l, @keyX, @valX, @x) 876.46/289.07 , quicksort#1(nil()) -> nil() 876.46/289.07 , quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) 876.46/289.07 , append#1(nil(), @ys) -> @ys 876.46/289.07 , append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) 876.46/289.07 , splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) 876.46/289.07 , quicksort(@l) -> quicksort#1(@l) 876.46/289.07 , splitqs#3(#true(), @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) 876.46/289.07 , splitqs#3(#false(), @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) 876.46/289.07 , splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> 876.46/289.07 splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) 876.46/289.07 , splitqs#1(nil(), @pivot) -> tuple#2(nil(), nil()) 876.46/289.07 , splitqs#1(::(@x, @xs), @pivot) -> 876.46/289.07 splitqs#2(splitqs(@pivot, @xs), @pivot, @x) } 876.46/289.07 Obligation: 876.46/289.07 innermost runtime complexity 876.46/289.07 Answer: 876.46/289.07 YES(O(1),O(1)) 876.46/289.07 876.46/289.07 No rule is usable, rules are removed from the input problem. 876.46/289.07 876.46/289.07 We are left with following problem, upon which TcT provides the 876.46/289.07 certificate YES(O(1),O(1)). 876.46/289.07 876.46/289.07 Rules: Empty 876.46/289.07 Obligation: 876.46/289.07 innermost runtime complexity 876.46/289.07 Answer: 876.46/289.07 YES(O(1),O(1)) 876.46/289.07 876.46/289.07 Empty rules are trivially bounded 876.46/289.07 876.46/289.07 We return to the main proof. 876.46/289.07 876.46/289.07 We are left with following problem, upon which TcT provides the 876.46/289.07 certificate YES(O(1),O(n^4)). 876.46/289.07 876.46/289.07 Strict DPs: 876.46/289.07 { append^#(@l, @ys) -> c_5(append#1^#(@l, @ys)) 876.46/289.07 , append#1^#(::(@x, @xs), @ys) -> c_6(append^#(@xs, @ys)) 876.46/289.07 , splitqs^#(@pivot, @l) -> c_9(splitqs#1^#(@l, @pivot)) 876.46/289.07 , splitqs#1^#(::(@x, @xs), @pivot) -> 876.46/289.07 c_10(splitqs^#(@pivot, @xs)) } 876.46/289.07 Weak DPs: 876.46/289.07 { splitAndSort^#(@l) -> sortAll^#(split(@l)) 876.46/289.07 , sortAll^#(@l) -> sortAll#1^#(@l) 876.46/289.07 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> sortAll^#(@xs) 876.46/289.07 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> quicksort^#(@vals) 876.46/289.07 , quicksort^#(@l) -> quicksort#1^#(@l) 876.46/289.07 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> quicksort^#(@xs) 876.46/289.07 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> quicksort^#(@ys) 876.46/289.07 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> 876.46/289.07 append^#(quicksort(@xs), ::(@z, quicksort(@ys))) 876.46/289.07 , sortAll#1^#(::(@x, @xs)) -> sortAll#2^#(@x, @xs) 876.46/289.07 , quicksort#1^#(::(@z, @zs)) -> quicksort#2^#(splitqs(@z, @zs), @z) 876.46/289.07 , quicksort#1^#(::(@z, @zs)) -> splitqs^#(@z, @zs) } 876.46/289.07 Weak Trs: 876.46/289.07 { #equal(@x, @y) -> #eq(@x, @y) 876.46/289.07 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.46/289.07 , #eq(#pos(@x), #0()) -> #false() 876.46/289.07 , #eq(#pos(@x), #neg(@y)) -> #false() 876.46/289.07 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.46/289.07 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.07 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.46/289.07 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.46/289.07 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.07 , #eq(nil(), nil()) -> #true() 876.46/289.07 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.46/289.07 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.07 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.46/289.07 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.46/289.07 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.07 , #eq(#0(), #pos(@y)) -> #false() 876.46/289.07 , #eq(#0(), #0()) -> #true() 876.46/289.07 , #eq(#0(), #neg(@y)) -> #false() 876.46/289.07 , #eq(#0(), #s(@y)) -> #false() 876.46/289.07 , #eq(#neg(@x), #pos(@y)) -> #false() 876.46/289.07 , #eq(#neg(@x), #0()) -> #false() 876.46/289.07 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.46/289.07 , #eq(#s(@x), #0()) -> #false() 876.46/289.07 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.46/289.07 , quicksort#2(tuple#2(@xs, @ys), @z) -> 876.46/289.07 append(quicksort(@xs), ::(@z, quicksort(@ys))) 876.46/289.07 , split(@l) -> split#1(@l) 876.46/289.07 , append(@l, @ys) -> append#1(@l, @ys) 876.46/289.07 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.46/289.07 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.46/289.07 , #ckgt(#EQ()) -> #false() 876.46/289.07 , #ckgt(#LT()) -> #false() 876.46/289.07 , #ckgt(#GT()) -> #true() 876.46/289.07 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.46/289.07 , insert#2(nil(), @keyX, @valX, @x) -> 876.46/289.07 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.46/289.07 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.46/289.07 insert#3(@l1, @keyX, @ls, @valX, @x) 876.46/289.07 , #and(#true(), #true()) -> #true() 876.46/289.07 , #and(#true(), #false()) -> #false() 876.46/289.07 , #and(#false(), #true()) -> #false() 876.46/289.07 , #and(#false(), #false()) -> #false() 876.46/289.07 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 876.46/289.07 , #compare(#pos(@x), #0()) -> #GT() 876.46/289.07 , #compare(#pos(@x), #neg(@y)) -> #GT() 876.46/289.07 , #compare(#0(), #pos(@y)) -> #LT() 876.46/289.07 , #compare(#0(), #0()) -> #EQ() 876.46/289.07 , #compare(#0(), #neg(@y)) -> #GT() 876.46/289.07 , #compare(#0(), #s(@y)) -> #LT() 876.46/289.07 , #compare(#neg(@x), #pos(@y)) -> #LT() 876.46/289.07 , #compare(#neg(@x), #0()) -> #LT() 876.46/289.07 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 876.46/289.07 , #compare(#s(@x), #0()) -> #GT() 876.46/289.07 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 876.46/289.07 , split#1(nil()) -> nil() 876.46/289.07 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.46/289.07 , #greater(@x, @y) -> #ckgt(#compare(@x, @y)) 876.46/289.07 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.07 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.46/289.07 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.07 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.46/289.07 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.46/289.07 insert#2(@l, @keyX, @valX, @x) 876.46/289.07 , quicksort#1(nil()) -> nil() 876.46/289.07 , quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) 876.46/289.07 , append#1(nil(), @ys) -> @ys 876.46/289.07 , append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) 876.46/289.07 , splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) 876.46/289.07 , quicksort(@l) -> quicksort#1(@l) 876.46/289.07 , splitqs#3(#true(), @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) 876.46/289.07 , splitqs#3(#false(), @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) 876.46/289.07 , splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> 876.46/289.07 splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) 876.46/289.07 , splitqs#1(nil(), @pivot) -> tuple#2(nil(), nil()) 876.46/289.07 , splitqs#1(::(@x, @xs), @pivot) -> 876.46/289.07 splitqs#2(splitqs(@pivot, @xs), @pivot, @x) } 876.46/289.07 Obligation: 876.46/289.07 innermost runtime complexity 876.46/289.07 Answer: 876.46/289.07 YES(O(1),O(n^4)) 876.46/289.07 876.46/289.07 We use the processor 'matrix interpretation of dimension 2' to 876.46/289.07 orient following rules strictly. 876.46/289.07 876.46/289.07 DPs: 876.46/289.07 { 3: splitqs^#(@pivot, @l) -> c_9(splitqs#1^#(@l, @pivot)) 876.46/289.07 , 5: splitAndSort^#(@l) -> sortAll^#(split(@l)) 876.46/289.07 , 10: quicksort#2^#(tuple#2(@xs, @ys), @z) -> quicksort^#(@xs) 876.46/289.07 , 11: quicksort#2^#(tuple#2(@xs, @ys), @z) -> quicksort^#(@ys) 876.46/289.07 , 12: quicksort#2^#(tuple#2(@xs, @ys), @z) -> 876.46/289.07 append^#(quicksort(@xs), ::(@z, quicksort(@ys))) 876.46/289.07 , 15: quicksort#1^#(::(@z, @zs)) -> splitqs^#(@z, @zs) } 876.46/289.07 Trs: 876.46/289.07 { insert#2(nil(), @keyX, @valX, @x) -> 876.46/289.07 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.46/289.07 , split#1(nil()) -> nil() 876.46/289.07 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.46/289.07 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.07 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.46/289.07 , splitqs#3(#true(), @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) 876.46/289.07 , splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> 876.46/289.07 splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) } 876.46/289.07 876.46/289.07 Sub-proof: 876.46/289.07 ---------- 876.46/289.07 The following argument positions are usable: 876.46/289.07 Uargs(c_5) = {1}, Uargs(c_6) = {1}, Uargs(c_9) = {1}, 876.46/289.07 Uargs(c_10) = {1} 876.46/289.07 876.46/289.07 TcT has computed the following constructor-based matrix 876.46/289.07 interpretation satisfying not(EDA) and not(IDA(1)). 876.46/289.07 876.46/289.07 [#equal](x1, x2) = [0] 876.46/289.07 [0] 876.46/289.07 876.46/289.07 [#eq](x1, x2) = [0] 876.46/289.07 [0] 876.46/289.07 876.46/289.07 [quicksort#2](x1, x2) = [0] 876.46/289.07 [0] 876.46/289.07 876.46/289.07 [split](x1) = [7 0] x1 + [1] 876.46/289.07 [1 0] [0] 876.46/289.07 876.46/289.07 [#true] = [0] 876.46/289.07 [0] 876.46/289.07 876.46/289.07 [append](x1, x2) = [0] 876.46/289.07 [0] 876.46/289.07 876.46/289.07 [insert#3](x1, x2, x3, x4, x5) = [0 0] x1 + [1 0] x3 + [5] 876.46/289.07 [1 0] [0 1] [1] 876.46/289.07 876.46/289.07 [#ckgt](x1) = [0] 876.46/289.07 [0] 876.46/289.07 876.46/289.07 [insert](x1, x2) = [1 0] x2 + [4] 876.46/289.07 [0 1] [1] 876.46/289.07 876.46/289.07 [#pos](x1) = [0] 876.46/289.07 [0] 876.46/289.07 876.46/289.07 [#EQ] = [0] 876.46/289.07 [0] 876.46/289.07 876.46/289.07 [insert#2](x1, x2, x3, x4) = [1 0] x1 + [4] 876.46/289.07 [0 1] [1] 876.46/289.07 876.46/289.07 [#and](x1, x2) = [0] 876.46/289.07 [0] 876.46/289.07 876.46/289.07 [#compare](x1, x2) = [0] 876.46/289.07 [0] 876.46/289.07 876.46/289.07 [tuple#2](x1, x2) = [1 0] x1 + [0 0] x2 + [0] 876.46/289.07 [1 0] [1 0] [1] 876.46/289.07 876.46/289.07 [nil] = [0] 876.46/289.07 [0] 876.46/289.07 876.46/289.07 [split#1](x1) = [7 0] x1 + [1] 876.46/289.07 [1 0] [0] 876.46/289.07 876.46/289.07 [#greater](x1, x2) = [0] 876.46/289.07 [0] 876.46/289.07 876.46/289.07 [insert#4](x1, x2, x3, x4, x5, x6) = [1 0] x3 + [0 0] x5 + [5] 876.46/289.07 [0 1] [1 0] [1] 876.46/289.07 876.46/289.07 [insert#1](x1, x2, x3) = [1 0] x2 + [4] 876.46/289.07 [0 1] [1] 876.46/289.07 876.46/289.07 [quicksort#1](x1) = [0] 876.46/289.07 [0] 876.46/289.07 876.46/289.07 [append#1](x1, x2) = [6 0] x2 + [0] 876.46/289.07 [6 6] [0] 876.46/289.07 876.46/289.07 [splitqs](x1, x2) = [4 4] x2 + [0] 876.46/289.07 [1 0] [1] 876.46/289.07 876.46/289.07 [#false] = [0] 876.46/289.07 [0] 876.46/289.07 876.46/289.07 [quicksort](x1) = [0] 876.46/289.07 [0] 876.46/289.07 876.46/289.07 [::](x1, x2) = [0 0] x1 + [1 0] x2 + [1] 876.46/289.07 [1 0] [0 1] [0] 876.46/289.07 876.46/289.07 [#LT] = [0] 876.46/289.07 [0] 876.46/289.07 876.46/289.07 [splitqs#3](x1, x2, x3, x4) = [4 0] x2 + [4 0] x3 + [1] 876.46/289.07 [1 0] [1 0] [2] 876.46/289.07 876.46/289.07 [splitqs#2](x1, x2, x3) = [0 4] x1 + [0] 876.46/289.07 [0 1] [1] 876.46/289.07 876.46/289.07 [#0] = [0] 876.46/289.07 [0] 876.46/289.07 876.46/289.07 [#neg](x1) = [0] 876.46/289.07 [0] 876.46/289.07 876.46/289.07 [#s](x1) = [0] 876.46/289.07 [0] 876.46/289.07 876.46/289.07 [#GT] = [0] 876.46/289.07 [0] 876.46/289.07 876.46/289.07 [splitqs#1](x1, x2) = [4 0] x1 + [0] 876.46/289.07 [1 0] [1] 876.46/289.07 876.46/289.07 [splitAndSort^#](x1) = [7 7] x1 + [7] 876.46/289.07 [7 7] [7] 876.46/289.07 876.46/289.07 [sortAll^#](x1) = [0 4] x1 + [0] 876.46/289.07 [0 0] [0] 876.46/289.07 876.46/289.07 [sortAll#2^#](x1, x2) = [4 0] x1 + [0 4] x2 + [0] 876.46/289.07 [0 0] [0 0] [0] 876.46/289.07 876.46/289.07 [quicksort^#](x1) = [4 0] x1 + [0] 876.46/289.07 [0 0] [0] 876.46/289.07 876.46/289.07 [quicksort#2^#](x1, x2) = [0 4] x1 + [0] 876.46/289.07 [0 0] [0] 876.46/289.07 876.46/289.07 [append^#](x1, x2) = [0] 876.46/289.07 [0] 876.46/289.07 876.46/289.07 [append#1^#](x1, x2) = [0 0] x1 + [0 0] x2 + [0] 876.46/289.07 [2 0] [4 0] [6] 876.46/289.07 876.46/289.07 [sortAll#1^#](x1) = [0 4] x1 + [0] 876.46/289.07 [0 0] [0] 876.46/289.07 876.46/289.07 [quicksort#1^#](x1) = [4 0] x1 + [0] 876.46/289.07 [0 0] [0] 876.46/289.07 876.46/289.07 [splitqs^#](x1, x2) = [1 0] x2 + [1] 876.46/289.07 [0 0] [0] 876.46/289.07 876.46/289.07 [splitqs#1^#](x1, x2) = [1 0] x1 + [0] 876.46/289.07 [0 0] [4] 876.46/289.07 876.46/289.07 [c_5](x1) = [4 0] x1 + [0] 876.46/289.07 [0 0] [0] 876.46/289.07 876.46/289.07 [c_6](x1) = [4 0] x1 + [0] 876.46/289.07 [0 0] [3] 876.46/289.07 876.46/289.07 [c_9](x1) = [1 0] x1 + [0] 876.46/289.07 [0 0] [0] 876.46/289.07 876.46/289.07 [c_10](x1) = [1 0] x1 + [0] 876.46/289.07 [0 0] [3] 876.46/289.07 876.46/289.07 The order satisfies the following ordering constraints: 876.46/289.07 876.46/289.07 [#equal(@x, @y)] = [0] 876.46/289.07 [0] 876.46/289.07 >= [0] 876.46/289.07 [0] 876.46/289.07 = [#eq(@x, @y)] 876.46/289.07 876.46/289.07 [#eq(#pos(@x), #pos(@y))] = [0] 876.46/289.07 [0] 876.46/289.07 >= [0] 876.46/289.07 [0] 876.46/289.07 = [#eq(@x, @y)] 876.46/289.07 876.46/289.07 [#eq(#pos(@x), #0())] = [0] 876.46/289.07 [0] 876.46/289.07 >= [0] 876.46/289.07 [0] 876.46/289.07 = [#false()] 876.46/289.07 876.46/289.07 [#eq(#pos(@x), #neg(@y))] = [0] 876.46/289.07 [0] 876.46/289.07 >= [0] 876.46/289.07 [0] 876.46/289.07 = [#false()] 876.46/289.07 876.46/289.07 [#eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2))] = [0] 876.46/289.07 [0] 876.46/289.07 >= [0] 876.46/289.07 [0] 876.46/289.07 = [#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))] 876.46/289.07 876.46/289.07 [#eq(tuple#2(@x_1, @x_2), nil())] = [0] 876.46/289.07 [0] 876.46/289.07 >= [0] 876.46/289.07 [0] 876.46/289.07 = [#false()] 876.46/289.07 876.46/289.07 [#eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2))] = [0] 876.46/289.07 [0] 876.46/289.07 >= [0] 876.46/289.07 [0] 876.46/289.07 = [#false()] 876.46/289.07 876.46/289.07 [#eq(nil(), tuple#2(@y_1, @y_2))] = [0] 876.46/289.07 [0] 876.46/289.07 >= [0] 876.46/289.07 [0] 876.46/289.07 = [#false()] 876.46/289.07 876.46/289.07 [#eq(nil(), nil())] = [0] 876.46/289.07 [0] 876.46/289.07 >= [0] 876.46/289.07 [0] 876.46/289.07 = [#true()] 876.46/289.07 876.46/289.07 [#eq(nil(), ::(@y_1, @y_2))] = [0] 876.46/289.07 [0] 876.46/289.07 >= [0] 876.46/289.07 [0] 876.46/289.07 = [#false()] 876.46/289.07 876.46/289.07 [#eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2))] = [0] 876.46/289.07 [0] 876.46/289.07 >= [0] 876.46/289.07 [0] 876.46/289.07 = [#false()] 876.46/289.07 876.46/289.07 [#eq(::(@x_1, @x_2), nil())] = [0] 876.46/289.07 [0] 876.46/289.07 >= [0] 876.46/289.07 [0] 876.46/289.07 = [#false()] 876.46/289.07 876.46/289.07 [#eq(::(@x_1, @x_2), ::(@y_1, @y_2))] = [0] 876.46/289.07 [0] 876.46/289.07 >= [0] 876.46/289.07 [0] 876.46/289.07 = [#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))] 876.46/289.07 876.46/289.07 [#eq(#0(), #pos(@y))] = [0] 876.46/289.07 [0] 876.46/289.07 >= [0] 876.46/289.07 [0] 876.46/289.07 = [#false()] 876.46/289.07 876.46/289.07 [#eq(#0(), #0())] = [0] 876.46/289.07 [0] 876.46/289.07 >= [0] 876.46/289.07 [0] 876.46/289.07 = [#true()] 876.46/289.07 876.46/289.07 [#eq(#0(), #neg(@y))] = [0] 876.46/289.07 [0] 876.46/289.07 >= [0] 876.46/289.07 [0] 876.46/289.07 = [#false()] 876.46/289.07 876.46/289.07 [#eq(#0(), #s(@y))] = [0] 876.46/289.07 [0] 876.46/289.07 >= [0] 876.46/289.07 [0] 876.46/289.07 = [#false()] 876.46/289.07 876.46/289.07 [#eq(#neg(@x), #pos(@y))] = [0] 876.46/289.07 [0] 876.46/289.07 >= [0] 876.46/289.07 [0] 876.46/289.07 = [#false()] 876.46/289.07 876.46/289.07 [#eq(#neg(@x), #0())] = [0] 876.46/289.07 [0] 876.46/289.07 >= [0] 876.46/289.07 [0] 876.46/289.07 = [#false()] 876.46/289.07 876.46/289.07 [#eq(#neg(@x), #neg(@y))] = [0] 876.46/289.07 [0] 876.46/289.07 >= [0] 876.46/289.07 [0] 876.46/289.07 = [#eq(@x, @y)] 876.46/289.07 876.46/289.07 [#eq(#s(@x), #0())] = [0] 876.46/289.07 [0] 876.46/289.07 >= [0] 876.46/289.07 [0] 876.46/289.07 = [#false()] 876.46/289.07 876.46/289.07 [#eq(#s(@x), #s(@y))] = [0] 876.46/289.07 [0] 876.46/289.07 >= [0] 876.46/289.07 [0] 876.46/289.07 = [#eq(@x, @y)] 876.46/289.07 876.46/289.07 [quicksort#2(tuple#2(@xs, @ys), @z)] = [0] 876.46/289.07 [0] 876.46/289.07 >= [0] 876.46/289.07 [0] 876.46/289.07 = [append(quicksort(@xs), ::(@z, quicksort(@ys)))] 876.46/289.07 876.46/289.07 [split(@l)] = [7 0] @l + [1] 876.46/289.07 [1 0] [0] 876.46/289.07 >= [7 0] @l + [1] 876.46/289.07 [1 0] [0] 876.46/289.07 = [split#1(@l)] 876.46/289.07 876.46/289.07 [append(@l, @ys)] = [0] 876.46/289.07 [0] 876.46/289.07 ? [6 0] @ys + [0] 876.46/289.07 [6 6] [0] 876.46/289.07 = [append#1(@l, @ys)] 876.46/289.07 876.46/289.07 [insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x)] = [1 0] @ls + [0 0] @vals1 + [5] 876.46/289.07 [0 1] [1 0] [1] 876.46/289.07 >= [1 0] @ls + [0 0] @vals1 + [5] 876.46/289.07 [0 1] [1 0] [1] 876.46/289.07 = [insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x)] 876.46/289.07 876.46/289.08 [#ckgt(#EQ())] = [0] 876.46/289.08 [0] 876.46/289.08 >= [0] 876.46/289.08 [0] 876.46/289.08 = [#false()] 876.46/289.08 876.46/289.08 [#ckgt(#LT())] = [0] 876.46/289.08 [0] 876.46/289.08 >= [0] 876.46/289.08 [0] 876.46/289.08 = [#false()] 876.46/289.08 876.46/289.08 [#ckgt(#GT())] = [0] 876.46/289.08 [0] 876.46/289.08 >= [0] 876.46/289.08 [0] 876.46/289.08 = [#true()] 876.46/289.08 876.46/289.08 [insert(@x, @l)] = [1 0] @l + [4] 876.46/289.08 [0 1] [1] 876.46/289.08 >= [1 0] @l + [4] 876.46/289.08 [0 1] [1] 876.46/289.08 = [insert#1(@x, @l, @x)] 876.46/289.08 876.46/289.08 [insert#2(nil(), @keyX, @valX, @x)] = [4] 876.46/289.08 [1] 876.46/289.08 > [1] 876.46/289.08 [1] 876.46/289.08 = [::(tuple#2(::(@valX, nil()), @keyX), nil())] 876.46/289.08 876.46/289.08 [insert#2(::(@l1, @ls), @keyX, @valX, @x)] = [0 0] @l1 + [1 0] @ls + [5] 876.46/289.08 [1 0] [0 1] [1] 876.46/289.08 >= [0 0] @l1 + [1 0] @ls + [5] 876.46/289.08 [1 0] [0 1] [1] 876.46/289.08 = [insert#3(@l1, @keyX, @ls, @valX, @x)] 876.46/289.08 876.46/289.08 [#and(#true(), #true())] = [0] 876.46/289.08 [0] 876.46/289.08 >= [0] 876.46/289.08 [0] 876.46/289.08 = [#true()] 876.46/289.08 876.46/289.08 [#and(#true(), #false())] = [0] 876.46/289.08 [0] 876.46/289.08 >= [0] 876.46/289.08 [0] 876.46/289.08 = [#false()] 876.46/289.08 876.46/289.08 [#and(#false(), #true())] = [0] 876.46/289.08 [0] 876.46/289.08 >= [0] 876.46/289.08 [0] 876.46/289.08 = [#false()] 876.46/289.08 876.46/289.08 [#and(#false(), #false())] = [0] 876.46/289.08 [0] 876.46/289.08 >= [0] 876.46/289.08 [0] 876.46/289.08 = [#false()] 876.46/289.08 876.46/289.08 [#compare(#pos(@x), #pos(@y))] = [0] 876.46/289.08 [0] 876.46/289.08 >= [0] 876.46/289.08 [0] 876.46/289.08 = [#compare(@x, @y)] 876.46/289.08 876.46/289.08 [#compare(#pos(@x), #0())] = [0] 876.46/289.08 [0] 876.46/289.08 >= [0] 876.46/289.08 [0] 876.46/289.08 = [#GT()] 876.46/289.08 876.46/289.08 [#compare(#pos(@x), #neg(@y))] = [0] 876.46/289.08 [0] 876.46/289.08 >= [0] 876.46/289.08 [0] 876.46/289.08 = [#GT()] 876.46/289.08 876.46/289.08 [#compare(#0(), #pos(@y))] = [0] 876.46/289.08 [0] 876.46/289.08 >= [0] 876.46/289.08 [0] 876.46/289.08 = [#LT()] 876.46/289.08 876.46/289.08 [#compare(#0(), #0())] = [0] 876.46/289.08 [0] 876.46/289.08 >= [0] 876.46/289.08 [0] 876.46/289.08 = [#EQ()] 876.46/289.08 876.46/289.08 [#compare(#0(), #neg(@y))] = [0] 876.46/289.08 [0] 876.46/289.08 >= [0] 876.46/289.08 [0] 876.46/289.08 = [#GT()] 876.46/289.08 876.46/289.08 [#compare(#0(), #s(@y))] = [0] 876.46/289.08 [0] 876.46/289.08 >= [0] 876.46/289.08 [0] 876.46/289.08 = [#LT()] 876.46/289.08 876.46/289.08 [#compare(#neg(@x), #pos(@y))] = [0] 876.46/289.08 [0] 876.46/289.08 >= [0] 876.46/289.08 [0] 876.46/289.08 = [#LT()] 876.46/289.08 876.46/289.08 [#compare(#neg(@x), #0())] = [0] 876.46/289.08 [0] 876.46/289.08 >= [0] 876.46/289.08 [0] 876.46/289.08 = [#LT()] 876.46/289.08 876.46/289.08 [#compare(#neg(@x), #neg(@y))] = [0] 876.46/289.08 [0] 876.46/289.08 >= [0] 876.46/289.08 [0] 876.46/289.08 = [#compare(@y, @x)] 876.46/289.08 876.46/289.08 [#compare(#s(@x), #0())] = [0] 876.46/289.08 [0] 876.46/289.08 >= [0] 876.46/289.08 [0] 876.46/289.08 = [#GT()] 876.46/289.08 876.46/289.08 [#compare(#s(@x), #s(@y))] = [0] 876.46/289.08 [0] 876.46/289.08 >= [0] 876.46/289.08 [0] 876.46/289.08 = [#compare(@x, @y)] 876.46/289.08 876.46/289.08 [split#1(nil())] = [1] 876.46/289.08 [0] 876.46/289.08 > [0] 876.46/289.08 [0] 876.46/289.08 = [nil()] 876.46/289.08 876.46/289.08 [split#1(::(@x, @xs))] = [7 0] @xs + [8] 876.46/289.08 [1 0] [1] 876.46/289.08 > [7 0] @xs + [5] 876.46/289.08 [1 0] [1] 876.46/289.08 = [insert(@x, split(@xs))] 876.46/289.08 876.46/289.08 [#greater(@x, @y)] = [0] 876.46/289.08 [0] 876.46/289.08 >= [0] 876.46/289.08 [0] 876.46/289.08 = [#ckgt(#compare(@x, @y))] 876.46/289.08 876.46/289.08 [insert#4(#true(), @key1, @ls, @valX, @vals1, @x)] = [1 0] @ls + [0 0] @vals1 + [5] 876.46/289.08 [0 1] [1 0] [1] 876.46/289.08 > [1 0] @ls + [0 0] @vals1 + [1] 876.46/289.08 [0 1] [1 0] [1] 876.46/289.08 = [::(tuple#2(::(@valX, @vals1), @key1), @ls)] 876.46/289.08 876.46/289.08 [insert#4(#false(), @key1, @ls, @valX, @vals1, @x)] = [1 0] @ls + [0 0] @vals1 + [5] 876.46/289.08 [0 1] [1 0] [1] 876.46/289.08 >= [1 0] @ls + [0 0] @vals1 + [5] 876.46/289.08 [0 1] [1 0] [1] 876.46/289.08 = [::(tuple#2(@vals1, @key1), insert(@x, @ls))] 876.46/289.08 876.46/289.08 [insert#1(tuple#2(@valX, @keyX), @l, @x)] = [1 0] @l + [4] 876.46/289.08 [0 1] [1] 876.46/289.08 >= [1 0] @l + [4] 876.46/289.08 [0 1] [1] 876.46/289.08 = [insert#2(@l, @keyX, @valX, @x)] 876.46/289.08 876.46/289.08 [quicksort#1(nil())] = [0] 876.46/289.08 [0] 876.46/289.08 >= [0] 876.46/289.08 [0] 876.46/289.08 = [nil()] 876.46/289.08 876.46/289.08 [quicksort#1(::(@z, @zs))] = [0] 876.46/289.08 [0] 876.46/289.08 >= [0] 876.46/289.08 [0] 876.46/289.08 = [quicksort#2(splitqs(@z, @zs), @z)] 876.46/289.08 876.46/289.08 [append#1(nil(), @ys)] = [6 0] @ys + [0] 876.46/289.08 [6 6] [0] 876.46/289.08 >= [1 0] @ys + [0] 876.46/289.08 [0 1] [0] 876.46/289.08 = [@ys] 876.46/289.08 876.46/289.08 [append#1(::(@x, @xs), @ys)] = [6 0] @ys + [0] 876.46/289.08 [6 6] [0] 876.46/289.08 ? [0 0] @x + [1] 876.46/289.08 [1 0] [0] 876.46/289.08 = [::(@x, append(@xs, @ys))] 876.46/289.08 876.46/289.08 [splitqs(@pivot, @l)] = [4 4] @l + [0] 876.46/289.08 [1 0] [1] 876.46/289.08 >= [4 0] @l + [0] 876.46/289.08 [1 0] [1] 876.46/289.08 = [splitqs#1(@l, @pivot)] 876.46/289.08 876.46/289.08 [quicksort(@l)] = [0] 876.46/289.08 [0] 876.46/289.08 >= [0] 876.46/289.08 [0] 876.46/289.08 = [quicksort#1(@l)] 876.46/289.08 876.46/289.08 [splitqs#3(#true(), @ls, @rs, @x)] = [4 0] @ls + [4 0] @rs + [1] 876.46/289.08 [1 0] [1 0] [2] 876.46/289.08 > [1 0] @ls + [0 0] @rs + [0] 876.46/289.08 [1 0] [1 0] [2] 876.46/289.08 = [tuple#2(@ls, ::(@x, @rs))] 876.46/289.08 876.46/289.08 [splitqs#3(#false(), @ls, @rs, @x)] = [4 0] @ls + [4 0] @rs + [1] 876.46/289.08 [1 0] [1 0] [2] 876.46/289.08 >= [1 0] @ls + [0 0] @rs + [1] 876.46/289.08 [1 0] [1 0] [2] 876.46/289.08 = [tuple#2(::(@x, @ls), @rs)] 876.46/289.08 876.46/289.08 [splitqs#2(tuple#2(@ls, @rs), @pivot, @x)] = [4 0] @ls + [4 0] @rs + [4] 876.46/289.08 [1 0] [1 0] [2] 876.46/289.08 > [4 0] @ls + [4 0] @rs + [1] 876.46/289.08 [1 0] [1 0] [2] 876.46/289.08 = [splitqs#3(#greater(@x, @pivot), @ls, @rs, @x)] 876.46/289.08 876.46/289.08 [splitqs#1(nil(), @pivot)] = [0] 876.46/289.08 [1] 876.46/289.08 >= [0] 876.46/289.08 [1] 876.46/289.08 = [tuple#2(nil(), nil())] 876.46/289.08 876.46/289.08 [splitqs#1(::(@x, @xs), @pivot)] = [4 0] @xs + [4] 876.46/289.08 [1 0] [2] 876.46/289.08 >= [4 0] @xs + [4] 876.46/289.08 [1 0] [2] 876.46/289.08 = [splitqs#2(splitqs(@pivot, @xs), @pivot, @x)] 876.46/289.08 876.46/289.08 [splitAndSort^#(@l)] = [7 7] @l + [7] 876.46/289.08 [7 7] [7] 876.46/289.08 > [4 0] @l + [0] 876.46/289.08 [0 0] [0] 876.46/289.08 = [sortAll^#(split(@l))] 876.46/289.08 876.46/289.08 [sortAll^#(@l)] = [0 4] @l + [0] 876.46/289.08 [0 0] [0] 876.46/289.08 >= [0 4] @l + [0] 876.46/289.08 [0 0] [0] 876.46/289.08 = [sortAll#1^#(@l)] 876.46/289.08 876.46/289.08 [sortAll#2^#(tuple#2(@vals, @key), @xs)] = [0 4] @xs + [4 0] @vals + [0] 876.46/289.08 [0 0] [0 0] [0] 876.46/289.08 >= [0 4] @xs + [0] 876.46/289.08 [0 0] [0] 876.46/289.08 = [sortAll^#(@xs)] 876.46/289.08 876.46/289.08 [sortAll#2^#(tuple#2(@vals, @key), @xs)] = [0 4] @xs + [4 0] @vals + [0] 876.46/289.08 [0 0] [0 0] [0] 876.46/289.08 >= [4 0] @vals + [0] 876.46/289.08 [0 0] [0] 876.46/289.08 = [quicksort^#(@vals)] 876.46/289.08 876.46/289.08 [quicksort^#(@l)] = [4 0] @l + [0] 876.46/289.08 [0 0] [0] 876.46/289.08 >= [4 0] @l + [0] 876.46/289.08 [0 0] [0] 876.46/289.08 = [quicksort#1^#(@l)] 876.46/289.08 876.46/289.08 [quicksort#2^#(tuple#2(@xs, @ys), @z)] = [4 0] @ys + [4 0] @xs + [4] 876.46/289.08 [0 0] [0 0] [0] 876.46/289.08 > [4 0] @xs + [0] 876.46/289.08 [0 0] [0] 876.46/289.08 = [quicksort^#(@xs)] 876.46/289.08 876.46/289.08 [quicksort#2^#(tuple#2(@xs, @ys), @z)] = [4 0] @ys + [4 0] @xs + [4] 876.46/289.08 [0 0] [0 0] [0] 876.46/289.08 > [4 0] @ys + [0] 876.46/289.08 [0 0] [0] 876.46/289.08 = [quicksort^#(@ys)] 876.46/289.08 876.46/289.08 [quicksort#2^#(tuple#2(@xs, @ys), @z)] = [4 0] @ys + [4 0] @xs + [4] 876.46/289.08 [0 0] [0 0] [0] 876.46/289.08 > [0] 876.46/289.08 [0] 876.46/289.08 = [append^#(quicksort(@xs), ::(@z, quicksort(@ys)))] 876.46/289.08 876.46/289.08 [append^#(@l, @ys)] = [0] 876.46/289.08 [0] 876.46/289.08 >= [0] 876.46/289.08 [0] 876.46/289.08 = [c_5(append#1^#(@l, @ys))] 876.46/289.08 876.46/289.08 [append#1^#(::(@x, @xs), @ys)] = [0 0] @ys + [0 0] @xs + [0] 876.46/289.08 [4 0] [2 0] [8] 876.46/289.08 >= [0] 876.46/289.08 [3] 876.46/289.08 = [c_6(append^#(@xs, @ys))] 876.46/289.08 876.46/289.08 [sortAll#1^#(::(@x, @xs))] = [4 0] @x + [0 4] @xs + [0] 876.46/289.08 [0 0] [0 0] [0] 876.46/289.08 >= [4 0] @x + [0 4] @xs + [0] 876.46/289.08 [0 0] [0 0] [0] 876.46/289.08 = [sortAll#2^#(@x, @xs)] 876.46/289.08 876.46/289.08 [quicksort#1^#(::(@z, @zs))] = [4 0] @zs + [4] 876.46/289.08 [0 0] [0] 876.46/289.08 >= [4 0] @zs + [4] 876.46/289.08 [0 0] [0] 876.46/289.08 = [quicksort#2^#(splitqs(@z, @zs), @z)] 876.46/289.08 876.46/289.08 [quicksort#1^#(::(@z, @zs))] = [4 0] @zs + [4] 876.46/289.08 [0 0] [0] 876.46/289.08 > [1 0] @zs + [1] 876.46/289.08 [0 0] [0] 876.46/289.08 = [splitqs^#(@z, @zs)] 876.46/289.08 876.46/289.08 [splitqs^#(@pivot, @l)] = [1 0] @l + [1] 876.46/289.08 [0 0] [0] 876.46/289.08 > [1 0] @l + [0] 876.46/289.08 [0 0] [0] 876.46/289.08 = [c_9(splitqs#1^#(@l, @pivot))] 876.46/289.08 876.46/289.08 [splitqs#1^#(::(@x, @xs), @pivot)] = [1 0] @xs + [1] 876.46/289.08 [0 0] [4] 876.46/289.08 >= [1 0] @xs + [1] 876.46/289.08 [0 0] [3] 876.46/289.08 = [c_10(splitqs^#(@pivot, @xs))] 876.46/289.08 876.46/289.08 876.46/289.08 We return to the main proof. Consider the set of all dependency 876.46/289.08 pairs 876.46/289.08 876.46/289.08 : 876.46/289.08 { 1: append^#(@l, @ys) -> c_5(append#1^#(@l, @ys)) 876.46/289.08 , 2: append#1^#(::(@x, @xs), @ys) -> c_6(append^#(@xs, @ys)) 876.46/289.08 , 3: splitqs^#(@pivot, @l) -> c_9(splitqs#1^#(@l, @pivot)) 876.46/289.08 , 4: splitqs#1^#(::(@x, @xs), @pivot) -> 876.46/289.08 c_10(splitqs^#(@pivot, @xs)) 876.46/289.08 , 5: splitAndSort^#(@l) -> sortAll^#(split(@l)) 876.46/289.08 , 6: sortAll^#(@l) -> sortAll#1^#(@l) 876.46/289.08 , 7: sortAll#2^#(tuple#2(@vals, @key), @xs) -> sortAll^#(@xs) 876.46/289.08 , 8: sortAll#2^#(tuple#2(@vals, @key), @xs) -> quicksort^#(@vals) 876.46/289.08 , 9: quicksort^#(@l) -> quicksort#1^#(@l) 876.46/289.08 , 10: quicksort#2^#(tuple#2(@xs, @ys), @z) -> quicksort^#(@xs) 876.46/289.08 , 11: quicksort#2^#(tuple#2(@xs, @ys), @z) -> quicksort^#(@ys) 876.46/289.08 , 12: quicksort#2^#(tuple#2(@xs, @ys), @z) -> 876.46/289.08 append^#(quicksort(@xs), ::(@z, quicksort(@ys))) 876.46/289.08 , 13: sortAll#1^#(::(@x, @xs)) -> sortAll#2^#(@x, @xs) 876.46/289.08 , 14: quicksort#1^#(::(@z, @zs)) -> 876.46/289.08 quicksort#2^#(splitqs(@z, @zs), @z) 876.46/289.08 , 15: quicksort#1^#(::(@z, @zs)) -> splitqs^#(@z, @zs) } 876.46/289.08 876.46/289.08 Processor 'matrix interpretation of dimension 2' induces the 876.46/289.08 complexity certificate YES(?,O(n^1)) on application of dependency 876.46/289.08 pairs {3,5,10,11,12,15}. These cover all (indirect) predecessors of 876.46/289.08 dependency pairs {3,4,5,10,11,12,15}, their number of application 876.46/289.08 is equally bounded. The dependency pairs are shifted into the weak 876.46/289.08 component. 876.46/289.08 876.46/289.08 We are left with following problem, upon which TcT provides the 876.46/289.08 certificate YES(O(1),O(n^4)). 876.46/289.08 876.46/289.08 Strict DPs: 876.46/289.08 { append^#(@l, @ys) -> c_5(append#1^#(@l, @ys)) 876.46/289.08 , append#1^#(::(@x, @xs), @ys) -> c_6(append^#(@xs, @ys)) } 876.46/289.08 Weak DPs: 876.46/289.08 { splitAndSort^#(@l) -> sortAll^#(split(@l)) 876.46/289.08 , sortAll^#(@l) -> sortAll#1^#(@l) 876.46/289.08 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> sortAll^#(@xs) 876.46/289.08 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> quicksort^#(@vals) 876.46/289.08 , quicksort^#(@l) -> quicksort#1^#(@l) 876.46/289.08 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> quicksort^#(@xs) 876.46/289.08 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> quicksort^#(@ys) 876.46/289.08 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> 876.46/289.08 append^#(quicksort(@xs), ::(@z, quicksort(@ys))) 876.46/289.08 , sortAll#1^#(::(@x, @xs)) -> sortAll#2^#(@x, @xs) 876.46/289.08 , quicksort#1^#(::(@z, @zs)) -> quicksort#2^#(splitqs(@z, @zs), @z) 876.46/289.08 , quicksort#1^#(::(@z, @zs)) -> splitqs^#(@z, @zs) 876.46/289.08 , splitqs^#(@pivot, @l) -> c_9(splitqs#1^#(@l, @pivot)) 876.46/289.08 , splitqs#1^#(::(@x, @xs), @pivot) -> 876.46/289.08 c_10(splitqs^#(@pivot, @xs)) } 876.46/289.08 Weak Trs: 876.46/289.08 { #equal(@x, @y) -> #eq(@x, @y) 876.46/289.08 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.46/289.08 , #eq(#pos(@x), #0()) -> #false() 876.46/289.08 , #eq(#pos(@x), #neg(@y)) -> #false() 876.46/289.08 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.46/289.08 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.08 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.46/289.08 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.46/289.08 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.08 , #eq(nil(), nil()) -> #true() 876.46/289.08 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.46/289.08 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.08 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.46/289.08 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.46/289.08 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.08 , #eq(#0(), #pos(@y)) -> #false() 876.46/289.08 , #eq(#0(), #0()) -> #true() 876.46/289.08 , #eq(#0(), #neg(@y)) -> #false() 876.46/289.08 , #eq(#0(), #s(@y)) -> #false() 876.46/289.08 , #eq(#neg(@x), #pos(@y)) -> #false() 876.46/289.08 , #eq(#neg(@x), #0()) -> #false() 876.46/289.08 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.46/289.08 , #eq(#s(@x), #0()) -> #false() 876.46/289.08 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.46/289.08 , quicksort#2(tuple#2(@xs, @ys), @z) -> 876.46/289.08 append(quicksort(@xs), ::(@z, quicksort(@ys))) 876.46/289.08 , split(@l) -> split#1(@l) 876.46/289.08 , append(@l, @ys) -> append#1(@l, @ys) 876.46/289.08 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.46/289.08 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.46/289.08 , #ckgt(#EQ()) -> #false() 876.46/289.08 , #ckgt(#LT()) -> #false() 876.46/289.08 , #ckgt(#GT()) -> #true() 876.46/289.08 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.46/289.08 , insert#2(nil(), @keyX, @valX, @x) -> 876.46/289.08 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.46/289.08 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.46/289.08 insert#3(@l1, @keyX, @ls, @valX, @x) 876.46/289.08 , #and(#true(), #true()) -> #true() 876.46/289.08 , #and(#true(), #false()) -> #false() 876.46/289.08 , #and(#false(), #true()) -> #false() 876.46/289.08 , #and(#false(), #false()) -> #false() 876.46/289.08 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 876.46/289.08 , #compare(#pos(@x), #0()) -> #GT() 876.46/289.08 , #compare(#pos(@x), #neg(@y)) -> #GT() 876.46/289.08 , #compare(#0(), #pos(@y)) -> #LT() 876.46/289.08 , #compare(#0(), #0()) -> #EQ() 876.46/289.08 , #compare(#0(), #neg(@y)) -> #GT() 876.46/289.08 , #compare(#0(), #s(@y)) -> #LT() 876.46/289.08 , #compare(#neg(@x), #pos(@y)) -> #LT() 876.46/289.08 , #compare(#neg(@x), #0()) -> #LT() 876.46/289.08 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 876.46/289.08 , #compare(#s(@x), #0()) -> #GT() 876.46/289.08 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 876.46/289.08 , split#1(nil()) -> nil() 876.46/289.08 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.46/289.08 , #greater(@x, @y) -> #ckgt(#compare(@x, @y)) 876.46/289.08 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.08 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.46/289.08 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.08 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.46/289.08 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.46/289.08 insert#2(@l, @keyX, @valX, @x) 876.46/289.08 , quicksort#1(nil()) -> nil() 876.46/289.08 , quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) 876.46/289.08 , append#1(nil(), @ys) -> @ys 876.46/289.08 , append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) 876.46/289.08 , splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) 876.46/289.08 , quicksort(@l) -> quicksort#1(@l) 876.46/289.08 , splitqs#3(#true(), @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) 876.46/289.08 , splitqs#3(#false(), @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) 876.46/289.08 , splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> 876.46/289.08 splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) 876.46/289.08 , splitqs#1(nil(), @pivot) -> tuple#2(nil(), nil()) 876.46/289.08 , splitqs#1(::(@x, @xs), @pivot) -> 876.46/289.08 splitqs#2(splitqs(@pivot, @xs), @pivot, @x) } 876.46/289.08 Obligation: 876.46/289.08 innermost runtime complexity 876.46/289.08 Answer: 876.46/289.08 YES(O(1),O(n^4)) 876.46/289.08 876.46/289.08 The following weak DPs constitute a sub-graph of the DG that is 876.46/289.08 closed under successors. The DPs are removed. 876.46/289.08 876.46/289.08 { quicksort#1^#(::(@z, @zs)) -> splitqs^#(@z, @zs) 876.46/289.08 , splitqs^#(@pivot, @l) -> c_9(splitqs#1^#(@l, @pivot)) 876.46/289.08 , splitqs#1^#(::(@x, @xs), @pivot) -> 876.46/289.08 c_10(splitqs^#(@pivot, @xs)) } 876.46/289.08 876.46/289.08 We are left with following problem, upon which TcT provides the 876.46/289.08 certificate YES(O(1),O(n^4)). 876.46/289.08 876.46/289.08 Strict DPs: 876.46/289.08 { append^#(@l, @ys) -> c_5(append#1^#(@l, @ys)) 876.46/289.08 , append#1^#(::(@x, @xs), @ys) -> c_6(append^#(@xs, @ys)) } 876.46/289.08 Weak DPs: 876.46/289.08 { splitAndSort^#(@l) -> sortAll^#(split(@l)) 876.46/289.08 , sortAll^#(@l) -> sortAll#1^#(@l) 876.46/289.08 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> sortAll^#(@xs) 876.46/289.08 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> quicksort^#(@vals) 876.46/289.08 , quicksort^#(@l) -> quicksort#1^#(@l) 876.46/289.08 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> quicksort^#(@xs) 876.46/289.08 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> quicksort^#(@ys) 876.46/289.08 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> 876.46/289.08 append^#(quicksort(@xs), ::(@z, quicksort(@ys))) 876.46/289.08 , sortAll#1^#(::(@x, @xs)) -> sortAll#2^#(@x, @xs) 876.46/289.08 , quicksort#1^#(::(@z, @zs)) -> 876.46/289.08 quicksort#2^#(splitqs(@z, @zs), @z) } 876.46/289.08 Weak Trs: 876.46/289.08 { #equal(@x, @y) -> #eq(@x, @y) 876.46/289.08 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.46/289.08 , #eq(#pos(@x), #0()) -> #false() 876.46/289.08 , #eq(#pos(@x), #neg(@y)) -> #false() 876.46/289.08 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.46/289.08 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.08 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.46/289.08 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.46/289.09 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.09 , #eq(nil(), nil()) -> #true() 876.46/289.09 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.46/289.09 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.46/289.09 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.46/289.09 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.46/289.09 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.46/289.09 , #eq(#0(), #pos(@y)) -> #false() 876.46/289.09 , #eq(#0(), #0()) -> #true() 876.46/289.09 , #eq(#0(), #neg(@y)) -> #false() 876.46/289.09 , #eq(#0(), #s(@y)) -> #false() 876.46/289.09 , #eq(#neg(@x), #pos(@y)) -> #false() 876.46/289.09 , #eq(#neg(@x), #0()) -> #false() 876.46/289.09 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.46/289.09 , #eq(#s(@x), #0()) -> #false() 876.46/289.09 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.46/289.09 , quicksort#2(tuple#2(@xs, @ys), @z) -> 876.46/289.09 append(quicksort(@xs), ::(@z, quicksort(@ys))) 876.46/289.09 , split(@l) -> split#1(@l) 876.46/289.09 , append(@l, @ys) -> append#1(@l, @ys) 876.46/289.09 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.46/289.09 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.46/289.09 , #ckgt(#EQ()) -> #false() 876.46/289.09 , #ckgt(#LT()) -> #false() 876.46/289.09 , #ckgt(#GT()) -> #true() 876.46/289.09 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.46/289.09 , insert#2(nil(), @keyX, @valX, @x) -> 876.46/289.09 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.46/289.09 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.46/289.09 insert#3(@l1, @keyX, @ls, @valX, @x) 876.46/289.09 , #and(#true(), #true()) -> #true() 876.46/289.09 , #and(#true(), #false()) -> #false() 876.46/289.09 , #and(#false(), #true()) -> #false() 876.46/289.09 , #and(#false(), #false()) -> #false() 876.46/289.09 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 876.46/289.09 , #compare(#pos(@x), #0()) -> #GT() 876.46/289.09 , #compare(#pos(@x), #neg(@y)) -> #GT() 876.46/289.09 , #compare(#0(), #pos(@y)) -> #LT() 876.46/289.09 , #compare(#0(), #0()) -> #EQ() 876.46/289.09 , #compare(#0(), #neg(@y)) -> #GT() 876.46/289.09 , #compare(#0(), #s(@y)) -> #LT() 876.46/289.09 , #compare(#neg(@x), #pos(@y)) -> #LT() 876.46/289.09 , #compare(#neg(@x), #0()) -> #LT() 876.46/289.09 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 876.46/289.09 , #compare(#s(@x), #0()) -> #GT() 876.46/289.09 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 876.46/289.09 , split#1(nil()) -> nil() 876.46/289.09 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.46/289.09 , #greater(@x, @y) -> #ckgt(#compare(@x, @y)) 876.46/289.09 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.09 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.46/289.09 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.46/289.09 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.46/289.09 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.46/289.09 insert#2(@l, @keyX, @valX, @x) 876.46/289.09 , quicksort#1(nil()) -> nil() 876.46/289.09 , quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) 876.46/289.09 , append#1(nil(), @ys) -> @ys 876.46/289.09 , append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) 876.46/289.09 , splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) 876.46/289.09 , quicksort(@l) -> quicksort#1(@l) 876.46/289.09 , splitqs#3(#true(), @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) 876.46/289.09 , splitqs#3(#false(), @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) 876.46/289.09 , splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> 876.46/289.09 splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) 876.46/289.09 , splitqs#1(nil(), @pivot) -> tuple#2(nil(), nil()) 876.46/289.09 , splitqs#1(::(@x, @xs), @pivot) -> 876.46/289.09 splitqs#2(splitqs(@pivot, @xs), @pivot, @x) } 876.46/289.09 Obligation: 876.46/289.09 innermost runtime complexity 876.46/289.09 Answer: 876.46/289.09 YES(O(1),O(n^4)) 876.46/289.09 876.46/289.09 We use the processor 'matrix interpretation of dimension 4' to 876.46/289.09 orient following rules strictly. 876.46/289.09 876.46/289.09 DPs: { append#1^#(::(@x, @xs), @ys) -> c_6(append^#(@xs, @ys)) } 876.46/289.09 876.46/289.09 The induced complexity on above rules (modulo remaining rules) is 876.46/289.09 YES(?,O(n^4)) . These rules are moved into the corresponding weak 876.46/289.09 component(s). 876.46/289.09 876.46/289.09 Sub-proof: 876.46/289.09 ---------- 876.46/289.09 The following argument positions are usable: 876.46/289.09 Uargs(c_5) = {1}, Uargs(c_6) = {1} 876.46/289.09 876.46/289.09 TcT has computed the following constructor-based matrix 876.46/289.09 interpretation satisfying not(EDA). 876.46/289.09 876.46/289.09 [0] 876.46/289.09 [#equal](x1, x2) = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 876.46/289.09 [0] 876.46/289.09 [#eq](x1, x2) = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 876.46/289.09 [0 0 0 0] [0 0 0 876.46/289.09 0] [1] 876.46/289.09 [quicksort#2](x1, x2) = [0 0 0 0] x1 + [0 0 0 876.46/289.09 0] x2 + [0] 876.46/289.09 [1 0 0 0] [0 0 1 876.46/289.09 0] [0] 876.46/289.09 [0 0 0 1] [0 0 0 876.46/289.09 0] [1] 876.46/289.09 876.46/289.09 [0 0 0 0] [1] 876.46/289.09 [split](x1) = [0 0 0 0] x1 + [0] 876.46/289.09 [0 0 0 1] [0] 876.46/289.09 [0 0 0 1] [0] 876.46/289.09 876.46/289.09 [0] 876.46/289.09 [#true] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 876.46/289.09 [0 0 0 0] [1 0 0 876.46/289.09 0] [0] 876.46/289.09 [append](x1, x2) = [0 0 0 0] x1 + [0 1 0 876.46/289.09 0] x2 + [0] 876.46/289.09 [0 0 1 0] [0 0 1 876.46/289.09 0] [0] 876.46/289.09 [0 0 0 1] [0 0 0 876.46/289.09 1] [0] 876.46/289.09 876.46/289.09 [0 0 0 0] [1 0 0 876.46/289.09 0] [0] 876.46/289.09 [insert#3](x1, x2, x3, x4, x5) = [0 0 0 0] x1 + [0 0 0 876.46/289.09 0] x3 + [0] 876.46/289.09 [0 0 1 0] [0 0 1 876.46/289.09 0] [1] 876.46/289.09 [0 0 0 0] [1 0 0 876.46/289.09 1] [1] 876.46/289.09 876.46/289.09 [0] 876.46/289.09 [#ckgt](x1) = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 876.46/289.09 [1 0 0 0] [0] 876.46/289.09 [insert](x1, x2) = [0 0 0 0] x2 + [0] 876.46/289.09 [0 0 1 0] [1] 876.46/289.09 [1 0 0 1] [0] 876.46/289.09 876.46/289.09 [0] 876.46/289.09 [#pos](x1) = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 876.46/289.09 [0] 876.46/289.09 [#EQ] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 876.46/289.09 [1 0 0 0] [0] 876.46/289.09 [insert#2](x1, x2, x3, x4) = [0 0 0 0] x1 + [0] 876.46/289.09 [0 0 1 0] [1] 876.46/289.09 [1 0 0 1] [0] 876.46/289.09 876.46/289.09 [0] 876.46/289.09 [#and](x1, x2) = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 876.46/289.09 [0] 876.46/289.09 [#compare](x1, x2) = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 876.46/289.09 [0 0 1 0] [0 0 1 876.46/289.09 0] [0] 876.46/289.09 [tuple#2](x1, x2) = [0 0 0 1] x1 + [0 0 0 876.46/289.09 1] x2 + [0] 876.46/289.09 [0 0 0 1] [0 0 0 876.46/289.09 0] [0] 876.46/289.09 [0 0 0 1] [0 0 0 876.46/289.09 1] [0] 876.46/289.09 876.46/289.09 [1] 876.46/289.09 [nil] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 876.46/289.09 [0 0 0 0] [1] 876.46/289.09 [split#1](x1) = [0 0 0 0] x1 + [0] 876.46/289.09 [0 0 0 1] [0] 876.46/289.09 [0 0 0 1] [0] 876.46/289.09 876.46/289.09 [0] 876.46/289.09 [#greater](x1, x2) = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 876.46/289.09 [1 0 0 0] [0 0 0 876.46/289.09 0] [0] 876.46/289.09 [insert#4](x1, x2, x3, x4, x5, x6) = [0 0 0 0] x3 + [0 0 0 876.46/289.09 0] x5 + [0] 876.46/289.09 [0 0 1 0] [0 0 0 876.46/289.09 1] [1] 876.46/289.09 [1 0 0 1] [0 0 0 876.46/289.09 0] [1] 876.46/289.09 876.46/289.09 [1 0 0 0] [0] 876.46/289.09 [insert#1](x1, x2, x3) = [0 0 0 0] x2 + [0] 876.46/289.09 [0 0 1 0] [1] 876.46/289.09 [1 0 0 1] [0] 876.46/289.09 876.46/289.09 [0 0 0 0] [1] 876.46/289.09 [quicksort#1](x1) = [0 0 0 0] x1 + [0] 876.46/289.09 [0 0 1 0] [0] 876.46/289.09 [0 0 0 1] [0] 876.46/289.09 876.46/289.09 [0 0 0 0] [1 0 0 876.46/289.09 0] [0] 876.46/289.09 [append#1](x1, x2) = [0 0 0 0] x1 + [0 1 0 876.46/289.09 0] x2 + [0] 876.46/289.09 [0 0 1 0] [0 0 1 876.46/289.09 0] [0] 876.46/289.09 [0 0 0 1] [0 0 0 876.46/289.09 1] [0] 876.46/289.09 876.46/289.09 [0 0 1 0] [0] 876.46/289.09 [splitqs](x1, x2) = [0 0 0 1] x2 + [0] 876.46/289.09 [0 0 0 1] [0] 876.46/289.09 [0 0 0 1] [0] 876.46/289.09 876.46/289.09 [0] 876.46/289.09 [#false] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 876.46/289.09 [0 0 0 0] [1] 876.46/289.09 [quicksort](x1) = [0 0 0 0] x1 + [0] 876.46/289.09 [0 0 1 0] [0] 876.46/289.09 [0 0 0 1] [0] 876.46/289.09 876.46/289.09 [0 0 0 0] [1 0 0 876.46/289.09 0] [0] 876.46/289.09 [::](x1, x2) = [0 0 0 0] x1 + [0 0 0 876.46/289.09 0] x2 + [0] 876.46/289.09 [0 0 1 0] [0 0 1 876.46/289.09 0] [0] 876.46/289.09 [0 0 0 0] [0 0 0 876.46/289.09 1] [1] 876.46/289.09 876.46/289.09 [0] 876.46/289.09 [#LT] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 876.46/289.09 [0 0 1 0] [0 0 1 876.46/289.09 0] [0 0 1 0] [0] 876.46/289.09 [splitqs#3](x1, x2, x3, x4) = [0 0 0 1] x2 + [0 0 0 876.46/289.09 1] x3 + [0 0 0 0] x4 + [1] 876.46/289.09 [0 0 0 1] [0 0 0 876.46/289.09 1] [0 0 0 0] [1] 876.46/289.09 [0 0 0 1] [0 0 0 876.46/289.09 1] [0 0 0 0] [1] 876.46/289.09 876.46/289.09 [1 0 0 0] [0 0 1 876.46/289.09 0] [0] 876.46/289.09 [splitqs#2](x1, x2, x3) = [0 1 0 0] x1 + [0 0 0 876.46/289.09 0] x3 + [1] 876.46/289.09 [0 1 0 0] [0 0 0 876.46/289.09 0] [1] 876.46/289.09 [0 1 0 0] [0 0 0 876.46/289.09 0] [1] 876.46/289.09 876.46/289.09 [0] 876.46/289.09 [#0] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 876.46/289.09 [1 0 0 0] [0] 876.46/289.09 [#neg](x1) = [0 0 0 0] x1 + [0] 876.46/289.09 [0 0 1 0] [0] 876.46/289.09 [0 0 0 1] [0] 876.46/289.09 876.46/289.09 [0] 876.46/289.09 [#s](x1) = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 876.46/289.09 [0] 876.46/289.09 [#GT] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 876.46/289.09 [0 0 1 0] [0] 876.46/289.09 [splitqs#1](x1, x2) = [0 0 0 1] x1 + [0] 876.46/289.09 [0 0 0 1] [0] 876.46/289.09 [0 0 0 1] [0] 876.46/289.09 876.46/289.09 [1 0 1 1] [1] 876.46/289.09 [splitAndSort^#](x1) = [1 1 1 1] x1 + [1] 876.46/289.09 [1 1 1 1] [1] 876.46/289.09 [1 1 1 1] [1] 876.46/289.09 876.46/289.09 [0 0 1 0] [1] 876.46/289.09 [sortAll^#](x1) = [1 0 0 0] x1 + [0] 876.46/289.09 [1 0 0 0] [0] 876.46/289.09 [0 0 0 1] [1] 876.46/289.09 876.46/289.09 [0 0 1 0] [0 0 1 876.46/289.09 0] [1] 876.46/289.09 [sortAll#2^#](x1, x2) = [0 0 0 0] x1 + [1 0 0 876.46/289.09 0] x2 + [0] 876.46/289.09 [0 0 0 0] [1 0 0 876.46/289.09 0] [0] 876.46/289.09 [0 0 0 0] [0 0 0 876.46/289.09 1] [1] 876.46/289.09 876.46/289.09 [0 0 0 1] [0] 876.46/289.09 [quicksort^#](x1) = [0 0 0 0] x1 + [0] 876.46/289.09 [0 0 0 0] [0] 876.46/289.09 [0 0 0 0] [0] 876.46/289.09 876.46/289.09 [0 0 0 1] [0] 876.46/289.09 [quicksort#2^#](x1, x2) = [0 0 0 0] x1 + [0] 876.46/289.09 [0 0 0 0] [0] 876.46/289.09 [0 0 0 0] [0] 876.46/289.09 876.46/289.09 [0 0 0 1] [0] 876.46/289.09 [append^#](x1, x2) = [0 0 0 0] x1 + [0] 876.46/289.09 [0 0 0 0] [0] 876.46/289.09 [0 0 0 0] [0] 876.46/289.09 876.46/289.09 [0 0 0 1] [0] 876.46/289.09 [append#1^#](x1, x2) = [0 0 0 0] x1 + [0] 876.46/289.09 [0 0 0 0] [0] 876.46/289.09 [0 0 0 0] [0] 876.46/289.09 876.46/289.09 [0 0 1 0] [1] 876.46/289.09 [sortAll#1^#](x1) = [1 0 0 0] x1 + [0] 876.46/289.09 [1 0 0 0] [0] 876.46/289.09 [0 0 0 1] [1] 876.46/289.09 876.46/289.09 [0 0 0 1] [0] 876.46/289.09 [quicksort#1^#](x1) = [0 0 0 0] x1 + [0] 876.46/289.09 [0 0 0 0] [0] 876.46/289.09 [0 0 0 0] [0] 876.46/289.09 876.46/289.09 [1 0 0 0] [0] 876.46/289.09 [c_5](x1) = [0 0 0 0] x1 + [0] 876.46/289.09 [0 0 0 0] [0] 876.46/289.09 [0 0 0 0] [0] 876.46/289.09 876.46/289.09 [1 0 0 0] [0] 876.46/289.09 [c_6](x1) = [0 0 0 0] x1 + [0] 876.46/289.09 [0 0 0 0] [0] 876.46/289.09 [0 0 0 0] [0] 876.46/289.09 876.46/289.09 The order satisfies the following ordering constraints: 876.46/289.09 876.46/289.09 [#equal(@x, @y)] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#eq(@x, @y)] 876.46/289.09 876.46/289.09 [#eq(#pos(@x), #pos(@y))] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#eq(@x, @y)] 876.46/289.09 876.46/289.09 [#eq(#pos(@x), #0())] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#false()] 876.46/289.09 876.46/289.09 [#eq(#pos(@x), #neg(@y))] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#false()] 876.46/289.09 876.46/289.09 [#eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2))] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))] 876.46/289.09 876.46/289.09 [#eq(tuple#2(@x_1, @x_2), nil())] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#false()] 876.46/289.09 876.46/289.09 [#eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2))] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#false()] 876.46/289.09 876.46/289.09 [#eq(nil(), tuple#2(@y_1, @y_2))] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#false()] 876.46/289.09 876.46/289.09 [#eq(nil(), nil())] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#true()] 876.46/289.09 876.46/289.09 [#eq(nil(), ::(@y_1, @y_2))] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#false()] 876.46/289.09 876.46/289.09 [#eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2))] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#false()] 876.46/289.09 876.46/289.09 [#eq(::(@x_1, @x_2), nil())] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#false()] 876.46/289.09 876.46/289.09 [#eq(::(@x_1, @x_2), ::(@y_1, @y_2))] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))] 876.46/289.09 876.46/289.09 [#eq(#0(), #pos(@y))] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#false()] 876.46/289.09 876.46/289.09 [#eq(#0(), #0())] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#true()] 876.46/289.09 876.46/289.09 [#eq(#0(), #neg(@y))] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#false()] 876.46/289.09 876.46/289.09 [#eq(#0(), #s(@y))] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#false()] 876.46/289.09 876.46/289.09 [#eq(#neg(@x), #pos(@y))] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#false()] 876.46/289.09 876.46/289.09 [#eq(#neg(@x), #0())] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#false()] 876.46/289.09 876.46/289.09 [#eq(#neg(@x), #neg(@y))] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#eq(@x, @y)] 876.46/289.09 876.46/289.09 [#eq(#s(@x), #0())] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#false()] 876.46/289.09 876.46/289.09 [#eq(#s(@x), #s(@y))] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#eq(@x, @y)] 876.46/289.09 876.46/289.09 [quicksort#2(tuple#2(@xs, @ys), @z)] = [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] 876.46/289.09 [0 0 0 0] @ys + [0 0 0 0] @xs + [0 0 0 0] @z + [0] 876.46/289.09 [0 0 1 0] [0 0 1 0] [0 0 1 0] [0] 876.46/289.09 [0 0 0 1] [0 0 0 1] [0 0 0 0] [1] 876.46/289.09 >= [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] 876.46/289.09 [0 0 0 0] @ys + [0 0 0 0] @xs + [0 0 0 0] @z + [0] 876.46/289.09 [0 0 1 0] [0 0 1 0] [0 0 1 0] [0] 876.46/289.09 [0 0 0 1] [0 0 0 1] [0 0 0 0] [1] 876.46/289.09 = [append(quicksort(@xs), ::(@z, quicksort(@ys)))] 876.46/289.09 876.46/289.09 [split(@l)] = [0 0 0 0] [1] 876.46/289.09 [0 0 0 0] @l + [0] 876.46/289.09 [0 0 0 1] [0] 876.46/289.09 [0 0 0 1] [0] 876.46/289.09 >= [0 0 0 0] [1] 876.46/289.09 [0 0 0 0] @l + [0] 876.46/289.09 [0 0 0 1] [0] 876.46/289.09 [0 0 0 1] [0] 876.46/289.09 = [split#1(@l)] 876.46/289.09 876.46/289.09 [append(@l, @ys)] = [0 0 0 0] [1 0 0 0] [0] 876.46/289.09 [0 0 0 0] @l + [0 1 0 0] @ys + [0] 876.46/289.09 [0 0 1 0] [0 0 1 0] [0] 876.46/289.09 [0 0 0 1] [0 0 0 1] [0] 876.46/289.09 >= [0 0 0 0] [1 0 0 0] [0] 876.46/289.09 [0 0 0 0] @l + [0 1 0 0] @ys + [0] 876.46/289.09 [0 0 1 0] [0 0 1 0] [0] 876.46/289.09 [0 0 0 1] [0 0 0 1] [0] 876.46/289.09 = [append#1(@l, @ys)] 876.46/289.09 876.46/289.09 [insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x)] = [1 0 0 0] [0 0 0 0] [0] 876.46/289.09 [0 0 0 0] @ls + [0 0 0 0] @vals1 + [0] 876.46/289.09 [0 0 1 0] [0 0 0 1] [1] 876.46/289.09 [1 0 0 1] [0 0 0 0] [1] 876.46/289.09 >= [1 0 0 0] [0 0 0 0] [0] 876.46/289.09 [0 0 0 0] @ls + [0 0 0 0] @vals1 + [0] 876.46/289.09 [0 0 1 0] [0 0 0 1] [1] 876.46/289.09 [1 0 0 1] [0 0 0 0] [1] 876.46/289.09 = [insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x)] 876.46/289.09 876.46/289.09 [#ckgt(#EQ())] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#false()] 876.46/289.09 876.46/289.09 [#ckgt(#LT())] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#false()] 876.46/289.09 876.46/289.09 [#ckgt(#GT())] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#true()] 876.46/289.09 876.46/289.09 [insert(@x, @l)] = [1 0 0 0] [0] 876.46/289.09 [0 0 0 0] @l + [0] 876.46/289.09 [0 0 1 0] [1] 876.46/289.09 [1 0 0 1] [0] 876.46/289.09 >= [1 0 0 0] [0] 876.46/289.09 [0 0 0 0] @l + [0] 876.46/289.09 [0 0 1 0] [1] 876.46/289.09 [1 0 0 1] [0] 876.46/289.09 = [insert#1(@x, @l, @x)] 876.46/289.09 876.46/289.09 [insert#2(nil(), @keyX, @valX, @x)] = [1] 876.46/289.09 [0] 876.46/289.09 [1] 876.46/289.09 [1] 876.46/289.09 >= [1] 876.46/289.09 [0] 876.46/289.09 [1] 876.46/289.09 [1] 876.46/289.09 = [::(tuple#2(::(@valX, nil()), @keyX), nil())] 876.46/289.09 876.46/289.09 [insert#2(::(@l1, @ls), @keyX, @valX, @x)] = [0 0 0 0] [1 0 0 0] [0] 876.46/289.09 [0 0 0 0] @l1 + [0 0 0 0] @ls + [0] 876.46/289.09 [0 0 1 0] [0 0 1 0] [1] 876.46/289.09 [0 0 0 0] [1 0 0 1] [1] 876.46/289.09 >= [0 0 0 0] [1 0 0 0] [0] 876.46/289.09 [0 0 0 0] @l1 + [0 0 0 0] @ls + [0] 876.46/289.09 [0 0 1 0] [0 0 1 0] [1] 876.46/289.09 [0 0 0 0] [1 0 0 1] [1] 876.46/289.09 = [insert#3(@l1, @keyX, @ls, @valX, @x)] 876.46/289.09 876.46/289.09 [#and(#true(), #true())] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#true()] 876.46/289.09 876.46/289.09 [#and(#true(), #false())] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#false()] 876.46/289.09 876.46/289.09 [#and(#false(), #true())] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#false()] 876.46/289.09 876.46/289.09 [#and(#false(), #false())] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#false()] 876.46/289.09 876.46/289.09 [#compare(#pos(@x), #pos(@y))] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#compare(@x, @y)] 876.46/289.09 876.46/289.09 [#compare(#pos(@x), #0())] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#GT()] 876.46/289.09 876.46/289.09 [#compare(#pos(@x), #neg(@y))] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#GT()] 876.46/289.09 876.46/289.09 [#compare(#0(), #pos(@y))] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#LT()] 876.46/289.09 876.46/289.09 [#compare(#0(), #0())] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#EQ()] 876.46/289.09 876.46/289.09 [#compare(#0(), #neg(@y))] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#GT()] 876.46/289.09 876.46/289.09 [#compare(#0(), #s(@y))] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#LT()] 876.46/289.09 876.46/289.09 [#compare(#neg(@x), #pos(@y))] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#LT()] 876.46/289.09 876.46/289.09 [#compare(#neg(@x), #0())] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#LT()] 876.46/289.09 876.46/289.09 [#compare(#neg(@x), #neg(@y))] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#compare(@y, @x)] 876.46/289.09 876.46/289.09 [#compare(#s(@x), #0())] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#GT()] 876.46/289.09 876.46/289.09 [#compare(#s(@x), #s(@y))] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#compare(@x, @y)] 876.46/289.09 876.46/289.09 [split#1(nil())] = [1] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [1] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [nil()] 876.46/289.09 876.46/289.09 [split#1(::(@x, @xs))] = [0 0 0 0] [1] 876.46/289.09 [0 0 0 0] @xs + [0] 876.46/289.09 [0 0 0 1] [1] 876.46/289.09 [0 0 0 1] [1] 876.46/289.09 >= [0 0 0 0] [1] 876.46/289.09 [0 0 0 0] @xs + [0] 876.46/289.09 [0 0 0 1] [1] 876.46/289.09 [0 0 0 1] [1] 876.46/289.09 = [insert(@x, split(@xs))] 876.46/289.09 876.46/289.09 [#greater(@x, @y)] = [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 >= [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 [0] 876.46/289.09 = [#ckgt(#compare(@x, @y))] 876.46/289.09 876.46/289.09 [insert#4(#true(), @key1, @ls, @valX, @vals1, @x)] = [1 0 0 0] [0 0 0 0] [0] 876.46/289.09 [0 0 0 0] @ls + [0 0 0 0] @vals1 + [0] 876.46/289.09 [0 0 1 0] [0 0 0 1] [1] 876.46/289.09 [1 0 0 1] [0 0 0 0] [1] 876.46/289.09 >= [1 0 0 0] [0 0 0 0] [0] 876.46/289.09 [0 0 0 0] @ls + [0 0 0 0] @vals1 + [0] 876.46/289.09 [0 0 1 0] [0 0 0 1] [1] 876.46/289.09 [0 0 0 1] [0 0 0 0] [1] 876.46/289.09 = [::(tuple#2(::(@valX, @vals1), @key1), @ls)] 876.46/289.09 876.46/289.09 [insert#4(#false(), @key1, @ls, @valX, @vals1, @x)] = [1 0 0 0] [0 0 0 0] [0] 876.46/289.09 [0 0 0 0] @ls + [0 0 0 0] @vals1 + [0] 876.46/289.09 [0 0 1 0] [0 0 0 1] [1] 876.46/289.09 [1 0 0 1] [0 0 0 0] [1] 876.46/289.09 >= [1 0 0 0] [0 0 0 0] [0] 876.46/289.09 [0 0 0 0] @ls + [0 0 0 0] @vals1 + [0] 876.46/289.09 [0 0 1 0] [0 0 0 1] [1] 876.46/289.09 [1 0 0 1] [0 0 0 0] [1] 876.46/289.09 = [::(tuple#2(@vals1, @key1), insert(@x, @ls))] 876.46/289.09 876.46/289.09 [insert#1(tuple#2(@valX, @keyX), @l, @x)] = [1 0 0 0] [0] 876.46/289.09 [0 0 0 0] @l + [0] 876.46/289.09 [0 0 1 0] [1] 876.46/289.10 [1 0 0 1] [0] 876.46/289.10 >= [1 0 0 0] [0] 876.46/289.10 [0 0 0 0] @l + [0] 876.46/289.10 [0 0 1 0] [1] 876.46/289.10 [1 0 0 1] [0] 876.46/289.10 = [insert#2(@l, @keyX, @valX, @x)] 876.46/289.10 876.46/289.10 [quicksort#1(nil())] = [1] 876.46/289.10 [0] 876.46/289.10 [0] 876.46/289.10 [0] 876.46/289.10 >= [1] 876.46/289.10 [0] 876.46/289.10 [0] 876.46/289.10 [0] 876.46/289.10 = [nil()] 876.46/289.10 876.46/289.10 [quicksort#1(::(@z, @zs))] = [0 0 0 0] [0 0 0 0] [1] 876.46/289.10 [0 0 0 0] @z + [0 0 0 0] @zs + [0] 876.46/289.10 [0 0 1 0] [0 0 1 0] [0] 876.46/289.10 [0 0 0 0] [0 0 0 1] [1] 876.46/289.10 >= [0 0 0 0] [0 0 0 0] [1] 876.46/289.10 [0 0 0 0] @z + [0 0 0 0] @zs + [0] 876.46/289.10 [0 0 1 0] [0 0 1 0] [0] 876.46/289.10 [0 0 0 0] [0 0 0 1] [1] 876.46/289.10 = [quicksort#2(splitqs(@z, @zs), @z)] 876.46/289.10 876.46/289.10 [append#1(nil(), @ys)] = [1 0 0 0] [0] 876.46/289.10 [0 1 0 0] @ys + [0] 876.46/289.10 [0 0 1 0] [0] 876.46/289.10 [0 0 0 1] [0] 876.46/289.10 >= [1 0 0 0] [0] 876.46/289.10 [0 1 0 0] @ys + [0] 876.46/289.10 [0 0 1 0] [0] 876.46/289.10 [0 0 0 1] [0] 876.46/289.10 = [@ys] 876.46/289.10 876.46/289.10 [append#1(::(@x, @xs), @ys)] = [0 0 0 0] [1 0 0 0] [0 0 0 0] [0] 876.46/289.10 [0 0 0 0] @x + [0 1 0 0] @ys + [0 0 0 0] @xs + [0] 876.46/289.10 [0 0 1 0] [0 0 1 0] [0 0 1 0] [0] 876.46/289.10 [0 0 0 0] [0 0 0 1] [0 0 0 1] [1] 876.46/289.10 >= [0 0 0 0] [1 0 0 0] [0 0 0 0] [0] 876.46/289.10 [0 0 0 0] @x + [0 0 0 0] @ys + [0 0 0 0] @xs + [0] 876.46/289.10 [0 0 1 0] [0 0 1 0] [0 0 1 0] [0] 876.46/289.10 [0 0 0 0] [0 0 0 1] [0 0 0 1] [1] 876.46/289.10 = [::(@x, append(@xs, @ys))] 876.46/289.10 876.46/289.10 [splitqs(@pivot, @l)] = [0 0 1 0] [0] 876.46/289.10 [0 0 0 1] @l + [0] 876.46/289.10 [0 0 0 1] [0] 876.46/289.10 [0 0 0 1] [0] 876.46/289.10 >= [0 0 1 0] [0] 876.46/289.10 [0 0 0 1] @l + [0] 876.46/289.10 [0 0 0 1] [0] 876.46/289.10 [0 0 0 1] [0] 876.46/289.10 = [splitqs#1(@l, @pivot)] 876.46/289.10 876.46/289.10 [quicksort(@l)] = [0 0 0 0] [1] 876.46/289.10 [0 0 0 0] @l + [0] 876.46/289.10 [0 0 1 0] [0] 876.46/289.10 [0 0 0 1] [0] 876.69/289.10 >= [0 0 0 0] [1] 876.69/289.10 [0 0 0 0] @l + [0] 876.69/289.10 [0 0 1 0] [0] 876.69/289.10 [0 0 0 1] [0] 876.69/289.10 = [quicksort#1(@l)] 876.69/289.10 876.69/289.10 [splitqs#3(#true(), @ls, @rs, @x)] = [0 0 1 0] [0 0 1 0] [0 0 1 0] [0] 876.69/289.10 [0 0 0 0] @x + [0 0 0 1] @ls + [0 0 0 1] @rs + [1] 876.69/289.10 [0 0 0 0] [0 0 0 1] [0 0 0 1] [1] 876.69/289.10 [0 0 0 0] [0 0 0 1] [0 0 0 1] [1] 876.69/289.10 >= [0 0 1 0] [0 0 1 0] [0 0 1 0] [0] 876.69/289.10 [0 0 0 0] @x + [0 0 0 1] @ls + [0 0 0 1] @rs + [1] 876.69/289.10 [0 0 0 0] [0 0 0 1] [0 0 0 0] [0] 876.69/289.10 [0 0 0 0] [0 0 0 1] [0 0 0 1] [1] 876.69/289.10 = [tuple#2(@ls, ::(@x, @rs))] 876.69/289.10 876.69/289.10 [splitqs#3(#false(), @ls, @rs, @x)] = [0 0 1 0] [0 0 1 0] [0 0 1 0] [0] 876.69/289.10 [0 0 0 0] @x + [0 0 0 1] @ls + [0 0 0 1] @rs + [1] 876.69/289.10 [0 0 0 0] [0 0 0 1] [0 0 0 1] [1] 876.69/289.10 [0 0 0 0] [0 0 0 1] [0 0 0 1] [1] 876.69/289.10 >= [0 0 1 0] [0 0 1 0] [0 0 1 0] [0] 876.69/289.10 [0 0 0 0] @x + [0 0 0 1] @ls + [0 0 0 1] @rs + [1] 876.69/289.10 [0 0 0 0] [0 0 0 1] [0 0 0 0] [1] 876.69/289.10 [0 0 0 0] [0 0 0 1] [0 0 0 1] [1] 876.69/289.10 = [tuple#2(::(@x, @ls), @rs)] 876.69/289.10 876.69/289.10 [splitqs#2(tuple#2(@ls, @rs), @pivot, @x)] = [0 0 1 0] [0 0 1 0] [0 0 1 0] [0] 876.69/289.10 [0 0 0 0] @x + [0 0 0 1] @ls + [0 0 0 1] @rs + [1] 876.69/289.10 [0 0 0 0] [0 0 0 1] [0 0 0 1] [1] 876.69/289.10 [0 0 0 0] [0 0 0 1] [0 0 0 1] [1] 876.69/289.10 >= [0 0 1 0] [0 0 1 0] [0 0 1 0] [0] 876.69/289.10 [0 0 0 0] @x + [0 0 0 1] @ls + [0 0 0 1] @rs + [1] 876.69/289.10 [0 0 0 0] [0 0 0 1] [0 0 0 1] [1] 876.69/289.10 [0 0 0 0] [0 0 0 1] [0 0 0 1] [1] 876.69/289.10 = [splitqs#3(#greater(@x, @pivot), @ls, @rs, @x)] 876.69/289.10 876.69/289.10 [splitqs#1(nil(), @pivot)] = [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 >= [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 = [tuple#2(nil(), nil())] 876.69/289.10 876.69/289.10 [splitqs#1(::(@x, @xs), @pivot)] = [0 0 1 0] [0 0 1 0] [0] 876.69/289.10 [0 0 0 0] @x + [0 0 0 1] @xs + [1] 876.69/289.10 [0 0 0 0] [0 0 0 1] [1] 876.69/289.10 [0 0 0 0] [0 0 0 1] [1] 876.69/289.10 >= [0 0 1 0] [0 0 1 0] [0] 876.69/289.10 [0 0 0 0] @x + [0 0 0 1] @xs + [1] 876.69/289.10 [0 0 0 0] [0 0 0 1] [1] 876.69/289.10 [0 0 0 0] [0 0 0 1] [1] 876.69/289.10 = [splitqs#2(splitqs(@pivot, @xs), @pivot, @x)] 876.69/289.10 876.69/289.10 [splitAndSort^#(@l)] = [1 0 1 1] [1] 876.69/289.10 [1 1 1 1] @l + [1] 876.69/289.10 [1 1 1 1] [1] 876.69/289.10 [1 1 1 1] [1] 876.69/289.10 >= [0 0 0 1] [1] 876.69/289.10 [0 0 0 0] @l + [1] 876.69/289.10 [0 0 0 0] [1] 876.69/289.10 [0 0 0 1] [1] 876.69/289.10 = [sortAll^#(split(@l))] 876.69/289.10 876.69/289.10 [sortAll^#(@l)] = [0 0 1 0] [1] 876.69/289.10 [1 0 0 0] @l + [0] 876.69/289.10 [1 0 0 0] [0] 876.69/289.10 [0 0 0 1] [1] 876.69/289.10 >= [0 0 1 0] [1] 876.69/289.10 [1 0 0 0] @l + [0] 876.69/289.10 [1 0 0 0] [0] 876.69/289.10 [0 0 0 1] [1] 876.69/289.10 = [sortAll#1^#(@l)] 876.69/289.10 876.69/289.10 [sortAll#2^#(tuple#2(@vals, @key), @xs)] = [0 0 1 0] [0 0 0 1] [1] 876.69/289.10 [1 0 0 0] @xs + [0 0 0 0] @vals + [0] 876.69/289.10 [1 0 0 0] [0 0 0 0] [0] 876.69/289.10 [0 0 0 1] [0 0 0 0] [1] 876.69/289.10 >= [0 0 1 0] [1] 876.69/289.10 [1 0 0 0] @xs + [0] 876.69/289.10 [1 0 0 0] [0] 876.69/289.10 [0 0 0 1] [1] 876.69/289.10 = [sortAll^#(@xs)] 876.69/289.10 876.69/289.10 [sortAll#2^#(tuple#2(@vals, @key), @xs)] = [0 0 1 0] [0 0 0 1] [1] 876.69/289.10 [1 0 0 0] @xs + [0 0 0 0] @vals + [0] 876.69/289.10 [1 0 0 0] [0 0 0 0] [0] 876.69/289.10 [0 0 0 1] [0 0 0 0] [1] 876.69/289.10 > [0 0 0 1] [0] 876.69/289.10 [0 0 0 0] @vals + [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 = [quicksort^#(@vals)] 876.69/289.10 876.69/289.10 [quicksort^#(@l)] = [0 0 0 1] [0] 876.69/289.10 [0 0 0 0] @l + [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 >= [0 0 0 1] [0] 876.69/289.10 [0 0 0 0] @l + [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 = [quicksort#1^#(@l)] 876.69/289.10 876.69/289.10 [quicksort#2^#(tuple#2(@xs, @ys), @z)] = [0 0 0 1] [0 0 0 1] [0] 876.69/289.10 [0 0 0 0] @ys + [0 0 0 0] @xs + [0] 876.69/289.10 [0 0 0 0] [0 0 0 0] [0] 876.69/289.10 [0 0 0 0] [0 0 0 0] [0] 876.69/289.10 >= [0 0 0 1] [0] 876.69/289.10 [0 0 0 0] @xs + [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 = [quicksort^#(@xs)] 876.69/289.10 876.69/289.10 [quicksort#2^#(tuple#2(@xs, @ys), @z)] = [0 0 0 1] [0 0 0 1] [0] 876.69/289.10 [0 0 0 0] @ys + [0 0 0 0] @xs + [0] 876.69/289.10 [0 0 0 0] [0 0 0 0] [0] 876.69/289.10 [0 0 0 0] [0 0 0 0] [0] 876.69/289.10 >= [0 0 0 1] [0] 876.69/289.10 [0 0 0 0] @ys + [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 = [quicksort^#(@ys)] 876.69/289.10 876.69/289.10 [quicksort#2^#(tuple#2(@xs, @ys), @z)] = [0 0 0 1] [0 0 0 1] [0] 876.69/289.10 [0 0 0 0] @ys + [0 0 0 0] @xs + [0] 876.69/289.10 [0 0 0 0] [0 0 0 0] [0] 876.69/289.10 [0 0 0 0] [0 0 0 0] [0] 876.69/289.10 >= [0 0 0 1] [0] 876.69/289.10 [0 0 0 0] @xs + [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 = [append^#(quicksort(@xs), ::(@z, quicksort(@ys)))] 876.69/289.10 876.69/289.10 [append^#(@l, @ys)] = [0 0 0 1] [0] 876.69/289.10 [0 0 0 0] @l + [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 >= [0 0 0 1] [0] 876.69/289.10 [0 0 0 0] @l + [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 = [c_5(append#1^#(@l, @ys))] 876.69/289.10 876.69/289.10 [append#1^#(::(@x, @xs), @ys)] = [0 0 0 1] [1] 876.69/289.10 [0 0 0 0] @xs + [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 > [0 0 0 1] [0] 876.69/289.10 [0 0 0 0] @xs + [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 = [c_6(append^#(@xs, @ys))] 876.69/289.10 876.69/289.10 [sortAll#1^#(::(@x, @xs))] = [0 0 1 0] [0 0 1 0] [1] 876.69/289.10 [0 0 0 0] @x + [1 0 0 0] @xs + [0] 876.69/289.10 [0 0 0 0] [1 0 0 0] [0] 876.69/289.10 [0 0 0 0] [0 0 0 1] [2] 876.69/289.10 >= [0 0 1 0] [0 0 1 0] [1] 876.69/289.10 [0 0 0 0] @x + [1 0 0 0] @xs + [0] 876.69/289.10 [0 0 0 0] [1 0 0 0] [0] 876.69/289.10 [0 0 0 0] [0 0 0 1] [1] 876.69/289.10 = [sortAll#2^#(@x, @xs)] 876.69/289.10 876.69/289.10 [quicksort#1^#(::(@z, @zs))] = [0 0 0 1] [1] 876.69/289.10 [0 0 0 0] @zs + [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 > [0 0 0 1] [0] 876.69/289.10 [0 0 0 0] @zs + [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 = [quicksort#2^#(splitqs(@z, @zs), @z)] 876.69/289.10 876.69/289.10 876.69/289.10 We return to the main proof. 876.69/289.10 876.69/289.10 We are left with following problem, upon which TcT provides the 876.69/289.10 certificate YES(O(1),O(n^4)). 876.69/289.10 876.69/289.10 Strict DPs: { append^#(@l, @ys) -> c_5(append#1^#(@l, @ys)) } 876.69/289.10 Weak DPs: 876.69/289.10 { splitAndSort^#(@l) -> sortAll^#(split(@l)) 876.69/289.10 , sortAll^#(@l) -> sortAll#1^#(@l) 876.69/289.10 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> sortAll^#(@xs) 876.69/289.10 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> quicksort^#(@vals) 876.69/289.10 , quicksort^#(@l) -> quicksort#1^#(@l) 876.69/289.10 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> quicksort^#(@xs) 876.69/289.10 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> quicksort^#(@ys) 876.69/289.10 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> 876.69/289.10 append^#(quicksort(@xs), ::(@z, quicksort(@ys))) 876.69/289.10 , append#1^#(::(@x, @xs), @ys) -> c_6(append^#(@xs, @ys)) 876.69/289.10 , sortAll#1^#(::(@x, @xs)) -> sortAll#2^#(@x, @xs) 876.69/289.10 , quicksort#1^#(::(@z, @zs)) -> 876.69/289.10 quicksort#2^#(splitqs(@z, @zs), @z) } 876.69/289.10 Weak Trs: 876.69/289.10 { #equal(@x, @y) -> #eq(@x, @y) 876.69/289.10 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.69/289.10 , #eq(#pos(@x), #0()) -> #false() 876.69/289.10 , #eq(#pos(@x), #neg(@y)) -> #false() 876.69/289.10 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.69/289.10 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.69/289.10 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.69/289.10 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.69/289.10 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.69/289.10 , #eq(nil(), nil()) -> #true() 876.69/289.10 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.69/289.10 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.69/289.10 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.69/289.10 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.69/289.10 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.69/289.10 , #eq(#0(), #pos(@y)) -> #false() 876.69/289.10 , #eq(#0(), #0()) -> #true() 876.69/289.10 , #eq(#0(), #neg(@y)) -> #false() 876.69/289.10 , #eq(#0(), #s(@y)) -> #false() 876.69/289.10 , #eq(#neg(@x), #pos(@y)) -> #false() 876.69/289.10 , #eq(#neg(@x), #0()) -> #false() 876.69/289.10 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.69/289.10 , #eq(#s(@x), #0()) -> #false() 876.69/289.10 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.69/289.10 , quicksort#2(tuple#2(@xs, @ys), @z) -> 876.69/289.10 append(quicksort(@xs), ::(@z, quicksort(@ys))) 876.69/289.10 , split(@l) -> split#1(@l) 876.69/289.10 , append(@l, @ys) -> append#1(@l, @ys) 876.69/289.10 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.69/289.10 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.69/289.10 , #ckgt(#EQ()) -> #false() 876.69/289.10 , #ckgt(#LT()) -> #false() 876.69/289.10 , #ckgt(#GT()) -> #true() 876.69/289.10 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.69/289.10 , insert#2(nil(), @keyX, @valX, @x) -> 876.69/289.10 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.69/289.10 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.69/289.10 insert#3(@l1, @keyX, @ls, @valX, @x) 876.69/289.10 , #and(#true(), #true()) -> #true() 876.69/289.10 , #and(#true(), #false()) -> #false() 876.69/289.10 , #and(#false(), #true()) -> #false() 876.69/289.10 , #and(#false(), #false()) -> #false() 876.69/289.10 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 876.69/289.10 , #compare(#pos(@x), #0()) -> #GT() 876.69/289.10 , #compare(#pos(@x), #neg(@y)) -> #GT() 876.69/289.10 , #compare(#0(), #pos(@y)) -> #LT() 876.69/289.10 , #compare(#0(), #0()) -> #EQ() 876.69/289.10 , #compare(#0(), #neg(@y)) -> #GT() 876.69/289.10 , #compare(#0(), #s(@y)) -> #LT() 876.69/289.10 , #compare(#neg(@x), #pos(@y)) -> #LT() 876.69/289.10 , #compare(#neg(@x), #0()) -> #LT() 876.69/289.10 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 876.69/289.10 , #compare(#s(@x), #0()) -> #GT() 876.69/289.10 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 876.69/289.10 , split#1(nil()) -> nil() 876.69/289.10 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.69/289.10 , #greater(@x, @y) -> #ckgt(#compare(@x, @y)) 876.69/289.10 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.69/289.10 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.69/289.10 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.69/289.10 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.69/289.10 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.69/289.10 insert#2(@l, @keyX, @valX, @x) 876.69/289.10 , quicksort#1(nil()) -> nil() 876.69/289.10 , quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) 876.69/289.10 , append#1(nil(), @ys) -> @ys 876.69/289.10 , append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) 876.69/289.10 , splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) 876.69/289.10 , quicksort(@l) -> quicksort#1(@l) 876.69/289.10 , splitqs#3(#true(), @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) 876.69/289.10 , splitqs#3(#false(), @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) 876.69/289.10 , splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> 876.69/289.10 splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) 876.69/289.10 , splitqs#1(nil(), @pivot) -> tuple#2(nil(), nil()) 876.69/289.10 , splitqs#1(::(@x, @xs), @pivot) -> 876.69/289.10 splitqs#2(splitqs(@pivot, @xs), @pivot, @x) } 876.69/289.10 Obligation: 876.69/289.10 innermost runtime complexity 876.69/289.10 Answer: 876.69/289.10 YES(O(1),O(n^4)) 876.69/289.10 876.69/289.10 We use the processor 'matrix interpretation of dimension 4' to 876.69/289.10 orient following rules strictly. 876.69/289.10 876.69/289.10 DPs: { append^#(@l, @ys) -> c_5(append#1^#(@l, @ys)) } 876.69/289.10 876.69/289.10 The induced complexity on above rules (modulo remaining rules) is 876.69/289.10 YES(?,O(n^4)) . These rules are moved into the corresponding weak 876.69/289.10 component(s). 876.69/289.10 876.69/289.10 Sub-proof: 876.69/289.10 ---------- 876.69/289.10 The following argument positions are usable: 876.69/289.10 Uargs(c_5) = {1}, Uargs(c_6) = {1} 876.69/289.10 876.69/289.10 TcT has computed the following constructor-based matrix 876.69/289.10 interpretation satisfying not(EDA). 876.69/289.10 876.69/289.10 [0] 876.69/289.10 [#equal](x1, x2) = [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 876.69/289.10 [0] 876.69/289.10 [#eq](x1, x2) = [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 876.69/289.10 [0 0 0 1] [1 0 0 876.69/289.10 0] [0] 876.69/289.10 [quicksort#2](x1, x2) = [0 0 0 0] x1 + [0 0 0 876.69/289.10 0] x2 + [0] 876.69/289.10 [0 0 0 0] [0 0 0 876.69/289.10 0] [1] 876.69/289.10 [0 1 0 0] [0 0 0 876.69/289.10 0] [1] 876.69/289.10 876.69/289.10 [0 0 0 1] [0] 876.69/289.10 [split](x1) = [0 0 0 1] x1 + [0] 876.69/289.10 [0 0 0 0] [1] 876.69/289.10 [0 0 0 1] [0] 876.69/289.10 876.69/289.10 [0] 876.69/289.10 [#true] = [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 876.69/289.10 [1 0 0 0] [1 0 0 876.69/289.10 0] [0] 876.69/289.10 [append](x1, x2) = [0 0 0 0] x1 + [0 1 0 876.69/289.10 0] x2 + [0] 876.69/289.10 [0 0 0 0] [0 0 1 876.69/289.10 0] [0] 876.69/289.10 [0 0 0 1] [0 0 0 876.69/289.10 1] [0] 876.69/289.10 876.69/289.10 [1 0 0 0] [1 0 0 876.69/289.10 0] [1] 876.69/289.10 [insert#3](x1, x2, x3, x4, x5) = [1 0 0 0] x1 + [1 0 0 876.69/289.10 0] x3 + [0] 876.69/289.10 [0 0 0 0] [0 0 1 876.69/289.10 0] [0] 876.69/289.10 [0 0 0 0] [0 0 1 876.69/289.10 1] [1] 876.69/289.10 876.69/289.10 [0] 876.69/289.10 [#ckgt](x1) = [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 876.69/289.10 [1 0 0 0] [1] 876.69/289.10 [insert](x1, x2) = [1 0 0 0] x2 + [0] 876.69/289.10 [0 0 1 0] [0] 876.69/289.10 [0 0 1 1] [0] 876.69/289.10 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 [#pos](x1) = [0 0 0 0] x1 + [0] 876.69/289.10 [0 0 1 0] [0] 876.69/289.10 [0 0 0 1] [0] 876.69/289.10 876.69/289.10 [0] 876.69/289.10 [#EQ] = [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 876.69/289.10 [1 0 0 0] [1] 876.69/289.10 [insert#2](x1, x2, x3, x4) = [1 0 0 0] x1 + [0] 876.69/289.10 [0 0 1 0] [0] 876.69/289.10 [0 0 1 1] [0] 876.69/289.10 876.69/289.10 [0] 876.69/289.10 [#and](x1, x2) = [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 876.69/289.10 [0] 876.69/289.10 [#compare](x1, x2) = [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 876.69/289.10 [0 0 0 1] [0 0 0 876.69/289.10 0] [0] 876.69/289.10 [tuple#2](x1, x2) = [0 0 0 1] x1 + [0 0 0 876.69/289.10 1] x2 + [0] 876.69/289.10 [0 0 0 0] [0 0 0 876.69/289.10 0] [0] 876.69/289.10 [1 0 0 0] [1 0 0 876.69/289.10 0] [0] 876.69/289.10 876.69/289.10 [0] 876.69/289.10 [nil] = [0] 876.69/289.10 [1] 876.69/289.10 [0] 876.69/289.10 876.69/289.10 [0 0 0 1] [0] 876.69/289.10 [split#1](x1) = [0 0 0 1] x1 + [0] 876.69/289.10 [0 0 0 0] [1] 876.69/289.10 [0 0 0 1] [0] 876.69/289.10 876.69/289.10 [0] 876.69/289.10 [#greater](x1, x2) = [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 876.69/289.10 [1 0 0 0] [0 0 0 876.69/289.10 1] [1] 876.69/289.10 [insert#4](x1, x2, x3, x4, x5, x6) = [1 0 0 0] x3 + [0 0 0 876.69/289.10 1] x5 + [0] 876.69/289.10 [0 0 1 0] [0 0 0 876.69/289.10 0] [0] 876.69/289.10 [0 0 1 1] [0 0 0 876.69/289.10 0] [1] 876.69/289.10 876.69/289.10 [1 0 0 0] [1] 876.69/289.10 [insert#1](x1, x2, x3) = [1 0 0 0] x2 + [0] 876.69/289.10 [0 0 1 0] [0] 876.69/289.10 [0 0 1 1] [0] 876.69/289.10 876.69/289.10 [1 0 0 0] [0] 876.69/289.10 [quicksort#1](x1) = [0 0 0 0] x1 + [0] 876.69/289.10 [0 0 0 0] [1] 876.69/289.10 [0 0 0 1] [0] 876.69/289.10 876.69/289.10 [1 0 0 0] [1 0 0 876.69/289.10 0] [0] 876.69/289.10 [append#1](x1, x2) = [0 0 0 0] x1 + [0 1 0 876.69/289.10 0] x2 + [0] 876.69/289.10 [0 0 0 0] [0 0 1 876.69/289.10 0] [0] 876.69/289.10 [0 0 0 1] [0 0 0 876.69/289.10 1] [0] 876.69/289.10 876.69/289.10 [0 0 0 1] [0] 876.69/289.10 [splitqs](x1, x2) = [0 0 0 1] x2 + [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 [1 0 0 0] [0] 876.69/289.10 876.69/289.10 [0] 876.69/289.10 [#false] = [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 876.69/289.10 [1 0 0 0] [0] 876.69/289.10 [quicksort](x1) = [0 0 0 0] x1 + [0] 876.69/289.10 [0 0 0 0] [1] 876.69/289.10 [0 0 0 1] [0] 876.69/289.10 876.69/289.10 [1 0 0 0] [1 0 0 876.69/289.10 0] [0] 876.69/289.10 [::](x1, x2) = [0 0 0 0] x1 + [0 0 0 876.69/289.10 0] x2 + [0] 876.69/289.10 [0 0 0 0] [0 0 1 876.69/289.10 0] [0] 876.69/289.10 [0 0 0 0] [0 0 0 876.69/289.10 1] [1] 876.69/289.10 876.69/289.10 [0] 876.69/289.10 [#LT] = [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 876.69/289.10 [0 0 0 1] [0 0 0 876.69/289.10 0] [0 0 0 0] [1] 876.69/289.10 [splitqs#3](x1, x2, x3, x4) = [0 0 0 1] x2 + [0 0 0 876.69/289.10 1] x3 + [0 0 0 0] x4 + [1] 876.69/289.10 [0 0 0 0] [0 0 0 876.69/289.10 0] [0 0 0 0] [0] 876.69/289.10 [1 0 0 0] [1 0 0 876.69/289.10 0] [1 0 0 0] [0] 876.69/289.10 876.69/289.10 [1 0 0 0] [0 0 0 876.69/289.10 0] [1] 876.69/289.10 [splitqs#2](x1, x2, x3) = [0 1 0 0] x1 + [0 0 0 876.69/289.10 0] x3 + [1] 876.69/289.10 [0 0 0 0] [0 0 0 876.69/289.10 0] [0] 876.69/289.10 [0 0 0 1] [1 0 0 876.69/289.10 0] [0] 876.69/289.10 876.69/289.10 [0] 876.69/289.10 [#0] = [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 876.69/289.10 [0] 876.69/289.10 [#neg](x1) = [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 876.69/289.10 [0 0 0 1] [0] 876.69/289.10 [#s](x1) = [0 0 0 1] x1 + [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 [1 0 0 0] [0] 876.69/289.10 876.69/289.10 [0] 876.69/289.10 [#GT] = [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 876.69/289.10 [0 0 0 1] [0] 876.69/289.10 [splitqs#1](x1, x2) = [0 0 0 1] x1 + [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 [1 0 0 0] [0] 876.69/289.10 876.69/289.10 [1 1 0 1] [1] 876.69/289.10 [splitAndSort^#](x1) = [1 1 1 1] x1 + [1] 876.69/289.10 [1 1 1 1] [1] 876.69/289.10 [1 1 1 1] [1] 876.69/289.10 876.69/289.10 [1 0 0 0] [1] 876.69/289.10 [sortAll^#](x1) = [0 0 0 0] x1 + [1] 876.69/289.10 [0 0 0 0] [1] 876.69/289.10 [0 0 0 0] [1] 876.69/289.10 876.69/289.10 [1 0 0 0] [1 0 0 876.69/289.10 0] [1] 876.69/289.10 [sortAll#2^#](x1, x2) = [0 0 0 0] x1 + [0 0 0 876.69/289.10 0] x2 + [1] 876.69/289.10 [0 0 0 0] [0 0 0 876.69/289.10 0] [1] 876.69/289.10 [0 0 0 0] [0 0 0 876.69/289.10 0] [1] 876.69/289.10 876.69/289.10 [0 0 0 1] [1] 876.69/289.10 [quicksort^#](x1) = [0 0 0 0] x1 + [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 876.69/289.10 [0 1 0 0] [1] 876.69/289.10 [quicksort#2^#](x1, x2) = [0 0 0 0] x1 + [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 876.69/289.10 [0 0 0 1] [1] 876.69/289.10 [append^#](x1, x2) = [0 0 0 0] x1 + [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 876.69/289.10 [0 0 0 1] [0] 876.69/289.10 [append#1^#](x1, x2) = [0 0 0 0] x1 + [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 876.69/289.10 [1 0 0 0] [1] 876.69/289.10 [sortAll#1^#](x1) = [0 0 0 0] x1 + [1] 876.69/289.10 [0 0 0 0] [1] 876.69/289.10 [0 0 0 0] [1] 876.69/289.10 876.69/289.10 [0 0 0 1] [0] 876.69/289.10 [quicksort#1^#](x1) = [0 0 0 0] x1 + [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 876.69/289.10 [1 0 0 0] [0] 876.69/289.10 [c_5](x1) = [0 0 0 0] x1 + [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 876.69/289.10 [1 0 0 0] [0] 876.69/289.10 [c_6](x1) = [0 0 0 0] x1 + [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 [0 0 0 0] [0] 876.69/289.10 876.69/289.10 The order satisfies the following ordering constraints: 876.69/289.10 876.69/289.10 [#equal(@x, @y)] = [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 >= [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 = [#eq(@x, @y)] 876.69/289.10 876.69/289.10 [#eq(#pos(@x), #pos(@y))] = [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 >= [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 = [#eq(@x, @y)] 876.69/289.10 876.69/289.10 [#eq(#pos(@x), #0())] = [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 >= [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 = [#false()] 876.69/289.10 876.69/289.10 [#eq(#pos(@x), #neg(@y))] = [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 >= [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 = [#false()] 876.69/289.10 876.69/289.10 [#eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2))] = [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 >= [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 = [#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))] 876.69/289.10 876.69/289.10 [#eq(tuple#2(@x_1, @x_2), nil())] = [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 >= [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 = [#false()] 876.69/289.10 876.69/289.10 [#eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2))] = [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 >= [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 [0] 876.69/289.10 = [#false()] 876.69/289.10 876.69/289.10 [#eq(nil(), tuple#2(@y_1, @y_2))] = [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 >= [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 = [#false()] 876.69/289.11 876.69/289.11 [#eq(nil(), nil())] = [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 >= [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 = [#true()] 876.69/289.11 876.69/289.11 [#eq(nil(), ::(@y_1, @y_2))] = [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 >= [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 = [#false()] 876.69/289.11 876.69/289.11 [#eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2))] = [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 >= [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 = [#false()] 876.69/289.11 876.69/289.11 [#eq(::(@x_1, @x_2), nil())] = [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 >= [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 = [#false()] 876.69/289.11 876.69/289.11 [#eq(::(@x_1, @x_2), ::(@y_1, @y_2))] = [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 >= [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 = [#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))] 876.69/289.11 876.69/289.11 [#eq(#0(), #pos(@y))] = [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 >= [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 = [#false()] 876.69/289.11 876.69/289.11 [#eq(#0(), #0())] = [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 >= [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 = [#true()] 876.69/289.11 876.69/289.11 [#eq(#0(), #neg(@y))] = [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 >= [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 = [#false()] 876.69/289.11 876.69/289.11 [#eq(#0(), #s(@y))] = [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 >= [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 = [#false()] 876.69/289.11 876.69/289.11 [#eq(#neg(@x), #pos(@y))] = [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 >= [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 = [#false()] 876.69/289.11 876.69/289.11 [#eq(#neg(@x), #0())] = [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 >= [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 = [#false()] 876.69/289.11 876.69/289.11 [#eq(#neg(@x), #neg(@y))] = [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 >= [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 = [#eq(@x, @y)] 876.69/289.11 876.69/289.11 [#eq(#s(@x), #0())] = [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 >= [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 = [#false()] 876.69/289.11 876.69/289.11 [#eq(#s(@x), #s(@y))] = [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 >= [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 = [#eq(@x, @y)] 876.69/289.11 876.69/289.11 [quicksort#2(tuple#2(@xs, @ys), @z)] = [1 0 0 0] [1 0 0 0] [1 0 0 0] [0] 876.69/289.11 [0 0 0 0] @ys + [0 0 0 0] @xs + [0 0 0 0] @z + [0] 876.69/289.11 [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] 876.69/289.11 [0 0 0 1] [0 0 0 1] [0 0 0 0] [1] 876.69/289.11 >= [1 0 0 0] [1 0 0 0] [1 0 0 0] [0] 876.69/289.11 [0 0 0 0] @ys + [0 0 0 0] @xs + [0 0 0 0] @z + [0] 876.69/289.11 [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] 876.69/289.11 [0 0 0 1] [0 0 0 1] [0 0 0 0] [1] 876.69/289.11 = [append(quicksort(@xs), ::(@z, quicksort(@ys)))] 876.69/289.11 876.69/289.11 [split(@l)] = [0 0 0 1] [0] 876.69/289.11 [0 0 0 1] @l + [0] 876.69/289.11 [0 0 0 0] [1] 876.69/289.11 [0 0 0 1] [0] 876.69/289.11 >= [0 0 0 1] [0] 876.69/289.11 [0 0 0 1] @l + [0] 876.69/289.11 [0 0 0 0] [1] 876.69/289.11 [0 0 0 1] [0] 876.69/289.11 = [split#1(@l)] 876.69/289.11 876.69/289.11 [append(@l, @ys)] = [1 0 0 0] [1 0 0 0] [0] 876.69/289.11 [0 0 0 0] @l + [0 1 0 0] @ys + [0] 876.69/289.11 [0 0 0 0] [0 0 1 0] [0] 876.69/289.11 [0 0 0 1] [0 0 0 1] [0] 876.69/289.11 >= [1 0 0 0] [1 0 0 0] [0] 876.69/289.11 [0 0 0 0] @l + [0 1 0 0] @ys + [0] 876.69/289.11 [0 0 0 0] [0 0 1 0] [0] 876.69/289.11 [0 0 0 1] [0 0 0 1] [0] 876.69/289.11 = [append#1(@l, @ys)] 876.69/289.11 876.69/289.11 [insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x)] = [1 0 0 0] [0 0 0 1] [1] 876.69/289.11 [1 0 0 0] @ls + [0 0 0 1] @vals1 + [0] 876.69/289.11 [0 0 1 0] [0 0 0 0] [0] 876.69/289.11 [0 0 1 1] [0 0 0 0] [1] 876.69/289.11 >= [1 0 0 0] [0 0 0 1] [1] 876.69/289.11 [1 0 0 0] @ls + [0 0 0 1] @vals1 + [0] 876.69/289.11 [0 0 1 0] [0 0 0 0] [0] 876.69/289.11 [0 0 1 1] [0 0 0 0] [1] 876.69/289.11 = [insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x)] 876.69/289.11 876.69/289.11 [#ckgt(#EQ())] = [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 >= [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 = [#false()] 876.69/289.11 876.69/289.11 [#ckgt(#LT())] = [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 >= [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 = [#false()] 876.69/289.11 876.69/289.11 [#ckgt(#GT())] = [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 >= [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 = [#true()] 876.69/289.11 876.69/289.11 [insert(@x, @l)] = [1 0 0 0] [1] 876.69/289.11 [1 0 0 0] @l + [0] 876.69/289.11 [0 0 1 0] [0] 876.69/289.11 [0 0 1 1] [0] 876.69/289.11 >= [1 0 0 0] [1] 876.69/289.11 [1 0 0 0] @l + [0] 876.69/289.11 [0 0 1 0] [0] 876.69/289.11 [0 0 1 1] [0] 876.69/289.11 = [insert#1(@x, @l, @x)] 876.69/289.11 876.69/289.11 [insert#2(nil(), @keyX, @valX, @x)] = [1] 876.69/289.11 [0] 876.69/289.11 [1] 876.69/289.11 [1] 876.69/289.11 >= [1] 876.69/289.11 [0] 876.69/289.11 [1] 876.69/289.11 [1] 876.69/289.11 = [::(tuple#2(::(@valX, nil()), @keyX), nil())] 876.69/289.11 876.69/289.11 [insert#2(::(@l1, @ls), @keyX, @valX, @x)] = [1 0 0 0] [1 0 0 0] [1] 876.69/289.11 [1 0 0 0] @l1 + [1 0 0 0] @ls + [0] 876.69/289.11 [0 0 0 0] [0 0 1 0] [0] 876.69/289.11 [0 0 0 0] [0 0 1 1] [1] 876.69/289.11 >= [1 0 0 0] [1 0 0 0] [1] 876.69/289.11 [1 0 0 0] @l1 + [1 0 0 0] @ls + [0] 876.69/289.11 [0 0 0 0] [0 0 1 0] [0] 876.69/289.11 [0 0 0 0] [0 0 1 1] [1] 876.69/289.11 = [insert#3(@l1, @keyX, @ls, @valX, @x)] 876.69/289.11 876.69/289.11 [#and(#true(), #true())] = [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 >= [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 = [#true()] 876.69/289.11 876.69/289.11 [#and(#true(), #false())] = [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 >= [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 = [#false()] 876.69/289.11 876.69/289.11 [#and(#false(), #true())] = [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 >= [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 = [#false()] 876.69/289.11 876.69/289.11 [#and(#false(), #false())] = [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 >= [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 = [#false()] 876.69/289.11 876.69/289.11 [#compare(#pos(@x), #pos(@y))] = [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 >= [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 = [#compare(@x, @y)] 876.69/289.11 876.69/289.11 [#compare(#pos(@x), #0())] = [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 >= [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 = [#GT()] 876.69/289.11 876.69/289.11 [#compare(#pos(@x), #neg(@y))] = [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 >= [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 = [#GT()] 876.69/289.11 876.69/289.11 [#compare(#0(), #pos(@y))] = [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 >= [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 = [#LT()] 876.69/289.11 876.69/289.11 [#compare(#0(), #0())] = [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 >= [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 = [#EQ()] 876.69/289.11 876.69/289.11 [#compare(#0(), #neg(@y))] = [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 >= [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 = [#GT()] 876.69/289.11 876.69/289.11 [#compare(#0(), #s(@y))] = [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 >= [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 = [#LT()] 876.69/289.11 876.69/289.11 [#compare(#neg(@x), #pos(@y))] = [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 >= [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 = [#LT()] 876.69/289.11 876.69/289.11 [#compare(#neg(@x), #0())] = [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 >= [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 = [#LT()] 876.69/289.11 876.69/289.11 [#compare(#neg(@x), #neg(@y))] = [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 >= [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 = [#compare(@y, @x)] 876.69/289.11 876.69/289.11 [#compare(#s(@x), #0())] = [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 >= [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 = [#GT()] 876.69/289.11 876.69/289.11 [#compare(#s(@x), #s(@y))] = [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 >= [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 = [#compare(@x, @y)] 876.69/289.11 876.69/289.11 [split#1(nil())] = [0] 876.69/289.11 [0] 876.69/289.11 [1] 876.69/289.11 [0] 876.69/289.11 >= [0] 876.69/289.11 [0] 876.69/289.11 [1] 876.69/289.11 [0] 876.69/289.11 = [nil()] 876.69/289.11 876.69/289.11 [split#1(::(@x, @xs))] = [0 0 0 1] [1] 876.69/289.11 [0 0 0 1] @xs + [1] 876.69/289.11 [0 0 0 0] [1] 876.69/289.11 [0 0 0 1] [1] 876.69/289.11 >= [0 0 0 1] [1] 876.69/289.11 [0 0 0 1] @xs + [0] 876.69/289.11 [0 0 0 0] [1] 876.69/289.11 [0 0 0 1] [1] 876.69/289.11 = [insert(@x, split(@xs))] 876.69/289.11 876.69/289.11 [#greater(@x, @y)] = [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 >= [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 = [#ckgt(#compare(@x, @y))] 876.69/289.11 876.69/289.11 [insert#4(#true(), @key1, @ls, @valX, @vals1, @x)] = [1 0 0 0] [0 0 0 1] [1] 876.69/289.11 [1 0 0 0] @ls + [0 0 0 1] @vals1 + [0] 876.69/289.11 [0 0 1 0] [0 0 0 0] [0] 876.69/289.11 [0 0 1 1] [0 0 0 0] [1] 876.69/289.11 >= [1 0 0 0] [0 0 0 1] [1] 876.69/289.11 [0 0 0 0] @ls + [0 0 0 0] @vals1 + [0] 876.69/289.11 [0 0 1 0] [0 0 0 0] [0] 876.69/289.11 [0 0 0 1] [0 0 0 0] [1] 876.69/289.11 = [::(tuple#2(::(@valX, @vals1), @key1), @ls)] 876.69/289.11 876.69/289.11 [insert#4(#false(), @key1, @ls, @valX, @vals1, @x)] = [1 0 0 0] [0 0 0 1] [1] 876.69/289.11 [1 0 0 0] @ls + [0 0 0 1] @vals1 + [0] 876.69/289.11 [0 0 1 0] [0 0 0 0] [0] 876.69/289.11 [0 0 1 1] [0 0 0 0] [1] 876.69/289.11 >= [1 0 0 0] [0 0 0 1] [1] 876.69/289.11 [0 0 0 0] @ls + [0 0 0 0] @vals1 + [0] 876.69/289.11 [0 0 1 0] [0 0 0 0] [0] 876.69/289.11 [0 0 1 1] [0 0 0 0] [1] 876.69/289.11 = [::(tuple#2(@vals1, @key1), insert(@x, @ls))] 876.69/289.11 876.69/289.11 [insert#1(tuple#2(@valX, @keyX), @l, @x)] = [1 0 0 0] [1] 876.69/289.11 [1 0 0 0] @l + [0] 876.69/289.11 [0 0 1 0] [0] 876.69/289.11 [0 0 1 1] [0] 876.69/289.11 >= [1 0 0 0] [1] 876.69/289.11 [1 0 0 0] @l + [0] 876.69/289.11 [0 0 1 0] [0] 876.69/289.11 [0 0 1 1] [0] 876.69/289.11 = [insert#2(@l, @keyX, @valX, @x)] 876.69/289.11 876.69/289.11 [quicksort#1(nil())] = [0] 876.69/289.11 [0] 876.69/289.11 [1] 876.69/289.11 [0] 876.69/289.11 >= [0] 876.69/289.11 [0] 876.69/289.11 [1] 876.69/289.11 [0] 876.69/289.11 = [nil()] 876.69/289.11 876.69/289.11 [quicksort#1(::(@z, @zs))] = [1 0 0 0] [1 0 0 0] [0] 876.69/289.11 [0 0 0 0] @z + [0 0 0 0] @zs + [0] 876.69/289.11 [0 0 0 0] [0 0 0 0] [1] 876.69/289.11 [0 0 0 0] [0 0 0 1] [1] 876.69/289.11 >= [1 0 0 0] [1 0 0 0] [0] 876.69/289.11 [0 0 0 0] @z + [0 0 0 0] @zs + [0] 876.69/289.11 [0 0 0 0] [0 0 0 0] [1] 876.69/289.11 [0 0 0 0] [0 0 0 1] [1] 876.69/289.11 = [quicksort#2(splitqs(@z, @zs), @z)] 876.69/289.11 876.69/289.11 [append#1(nil(), @ys)] = [1 0 0 0] [0] 876.69/289.11 [0 1 0 0] @ys + [0] 876.69/289.11 [0 0 1 0] [0] 876.69/289.11 [0 0 0 1] [0] 876.69/289.11 >= [1 0 0 0] [0] 876.69/289.11 [0 1 0 0] @ys + [0] 876.69/289.11 [0 0 1 0] [0] 876.69/289.11 [0 0 0 1] [0] 876.69/289.11 = [@ys] 876.69/289.11 876.69/289.11 [append#1(::(@x, @xs), @ys)] = [1 0 0 0] [1 0 0 0] [1 0 0 0] [0] 876.69/289.11 [0 0 0 0] @x + [0 1 0 0] @ys + [0 0 0 0] @xs + [0] 876.69/289.11 [0 0 0 0] [0 0 1 0] [0 0 0 0] [0] 876.69/289.11 [0 0 0 0] [0 0 0 1] [0 0 0 1] [1] 876.69/289.11 >= [1 0 0 0] [1 0 0 0] [1 0 0 0] [0] 876.69/289.11 [0 0 0 0] @x + [0 0 0 0] @ys + [0 0 0 0] @xs + [0] 876.69/289.11 [0 0 0 0] [0 0 1 0] [0 0 0 0] [0] 876.69/289.11 [0 0 0 0] [0 0 0 1] [0 0 0 1] [1] 876.69/289.11 = [::(@x, append(@xs, @ys))] 876.69/289.11 876.69/289.11 [splitqs(@pivot, @l)] = [0 0 0 1] [0] 876.69/289.11 [0 0 0 1] @l + [0] 876.69/289.11 [0 0 0 0] [0] 876.69/289.11 [1 0 0 0] [0] 876.69/289.11 >= [0 0 0 1] [0] 876.69/289.11 [0 0 0 1] @l + [0] 876.69/289.11 [0 0 0 0] [0] 876.69/289.11 [1 0 0 0] [0] 876.69/289.11 = [splitqs#1(@l, @pivot)] 876.69/289.11 876.69/289.11 [quicksort(@l)] = [1 0 0 0] [0] 876.69/289.11 [0 0 0 0] @l + [0] 876.69/289.11 [0 0 0 0] [1] 876.69/289.11 [0 0 0 1] [0] 876.69/289.11 >= [1 0 0 0] [0] 876.69/289.11 [0 0 0 0] @l + [0] 876.69/289.11 [0 0 0 0] [1] 876.69/289.11 [0 0 0 1] [0] 876.69/289.11 = [quicksort#1(@l)] 876.69/289.11 876.69/289.11 [splitqs#3(#true(), @ls, @rs, @x)] = [0 0 0 0] [0 0 0 1] [0 0 0 0] [1] 876.69/289.11 [0 0 0 0] @x + [0 0 0 1] @ls + [0 0 0 1] @rs + [1] 876.69/289.11 [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] 876.69/289.11 [1 0 0 0] [1 0 0 0] [1 0 0 0] [0] 876.69/289.11 > [0 0 0 0] [0 0 0 1] [0 0 0 0] [0] 876.69/289.11 [0 0 0 0] @x + [0 0 0 1] @ls + [0 0 0 1] @rs + [1] 876.69/289.11 [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] 876.69/289.11 [1 0 0 0] [1 0 0 0] [1 0 0 0] [0] 876.69/289.11 = [tuple#2(@ls, ::(@x, @rs))] 876.69/289.11 876.69/289.11 [splitqs#3(#false(), @ls, @rs, @x)] = [0 0 0 0] [0 0 0 1] [0 0 0 0] [1] 876.69/289.11 [0 0 0 0] @x + [0 0 0 1] @ls + [0 0 0 1] @rs + [1] 876.69/289.11 [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] 876.69/289.11 [1 0 0 0] [1 0 0 0] [1 0 0 0] [0] 876.69/289.11 >= [0 0 0 0] [0 0 0 1] [0 0 0 0] [1] 876.69/289.11 [0 0 0 0] @x + [0 0 0 1] @ls + [0 0 0 1] @rs + [1] 876.69/289.11 [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] 876.69/289.11 [1 0 0 0] [1 0 0 0] [1 0 0 0] [0] 876.69/289.11 = [tuple#2(::(@x, @ls), @rs)] 876.69/289.11 876.69/289.11 [splitqs#2(tuple#2(@ls, @rs), @pivot, @x)] = [0 0 0 0] [0 0 0 1] [0 0 0 0] [1] 876.69/289.11 [0 0 0 0] @x + [0 0 0 1] @ls + [0 0 0 1] @rs + [1] 876.69/289.11 [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] 876.69/289.11 [1 0 0 0] [1 0 0 0] [1 0 0 0] [0] 876.69/289.11 >= [0 0 0 0] [0 0 0 1] [0 0 0 0] [1] 876.69/289.11 [0 0 0 0] @x + [0 0 0 1] @ls + [0 0 0 1] @rs + [1] 876.69/289.11 [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] 876.69/289.11 [1 0 0 0] [1 0 0 0] [1 0 0 0] [0] 876.69/289.11 = [splitqs#3(#greater(@x, @pivot), @ls, @rs, @x)] 876.69/289.11 876.69/289.11 [splitqs#1(nil(), @pivot)] = [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 >= [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 [0] 876.69/289.11 = [tuple#2(nil(), nil())] 876.69/289.11 876.69/289.11 [splitqs#1(::(@x, @xs), @pivot)] = [0 0 0 0] [0 0 0 1] [1] 876.69/289.11 [0 0 0 0] @x + [0 0 0 1] @xs + [1] 876.69/289.11 [0 0 0 0] [0 0 0 0] [0] 876.69/289.11 [1 0 0 0] [1 0 0 0] [0] 876.69/289.11 >= [0 0 0 0] [0 0 0 1] [1] 876.69/289.11 [0 0 0 0] @x + [0 0 0 1] @xs + [1] 876.69/289.11 [0 0 0 0] [0 0 0 0] [0] 876.69/289.11 [1 0 0 0] [1 0 0 0] [0] 876.69/289.11 = [splitqs#2(splitqs(@pivot, @xs), @pivot, @x)] 876.69/289.11 876.69/289.11 [splitAndSort^#(@l)] = [1 1 0 1] [1] 876.69/289.11 [1 1 1 1] @l + [1] 876.69/289.11 [1 1 1 1] [1] 876.69/289.11 [1 1 1 1] [1] 876.69/289.11 >= [0 0 0 1] [1] 876.69/289.11 [0 0 0 0] @l + [1] 876.69/289.11 [0 0 0 0] [1] 876.69/289.11 [0 0 0 0] [1] 876.69/289.11 = [sortAll^#(split(@l))] 876.69/289.11 876.69/289.11 [sortAll^#(@l)] = [1 0 0 0] [1] 876.69/289.11 [0 0 0 0] @l + [1] 876.69/289.11 [0 0 0 0] [1] 876.69/289.11 [0 0 0 0] [1] 876.69/289.11 >= [1 0 0 0] [1] 876.69/289.11 [0 0 0 0] @l + [1] 876.69/289.11 [0 0 0 0] [1] 876.69/289.11 [0 0 0 0] [1] 876.69/289.11 = [sortAll#1^#(@l)] 876.69/289.11 876.69/289.11 [sortAll#2^#(tuple#2(@vals, @key), @xs)] = [1 0 0 0] [0 0 0 1] [1] 876.69/289.11 [0 0 0 0] @xs + [0 0 0 0] @vals + [1] 876.69/289.11 [0 0 0 0] [0 0 0 0] [1] 876.69/289.11 [0 0 0 0] [0 0 0 0] [1] 876.69/289.11 >= [1 0 0 0] [1] 876.69/289.11 [0 0 0 0] @xs + [1] 876.69/289.11 [0 0 0 0] [1] 876.69/289.11 [0 0 0 0] [1] 876.69/289.11 = [sortAll^#(@xs)] 876.69/289.11 876.69/289.11 [sortAll#2^#(tuple#2(@vals, @key), @xs)] = [1 0 0 0] [0 0 0 1] [1] 876.69/289.11 [0 0 0 0] @xs + [0 0 0 0] @vals + [1] 876.69/289.11 [0 0 0 0] [0 0 0 0] [1] 876.69/289.11 [0 0 0 0] [0 0 0 0] [1] 876.69/289.11 >= [0 0 0 1] [1] 876.69/289.11 [0 0 0 0] @vals + [0] 876.69/289.11 [0 0 0 0] [0] 876.69/289.11 [0 0 0 0] [0] 876.69/289.11 = [quicksort^#(@vals)] 876.69/289.11 876.69/289.11 [quicksort^#(@l)] = [0 0 0 1] [1] 876.69/289.11 [0 0 0 0] @l + [0] 876.69/289.11 [0 0 0 0] [0] 876.69/289.11 [0 0 0 0] [0] 876.69/289.11 > [0 0 0 1] [0] 876.69/289.11 [0 0 0 0] @l + [0] 876.69/289.11 [0 0 0 0] [0] 876.69/289.11 [0 0 0 0] [0] 876.69/289.11 = [quicksort#1^#(@l)] 876.69/289.11 876.69/289.11 [quicksort#2^#(tuple#2(@xs, @ys), @z)] = [0 0 0 1] [0 0 0 1] [1] 876.69/289.11 [0 0 0 0] @ys + [0 0 0 0] @xs + [0] 876.69/289.11 [0 0 0 0] [0 0 0 0] [0] 876.69/289.11 [0 0 0 0] [0 0 0 0] [0] 876.69/289.11 >= [0 0 0 1] [1] 876.69/289.11 [0 0 0 0] @xs + [0] 876.69/289.11 [0 0 0 0] [0] 876.69/289.11 [0 0 0 0] [0] 876.69/289.11 = [quicksort^#(@xs)] 876.69/289.11 876.69/289.11 [quicksort#2^#(tuple#2(@xs, @ys), @z)] = [0 0 0 1] [0 0 0 1] [1] 876.69/289.11 [0 0 0 0] @ys + [0 0 0 0] @xs + [0] 876.69/289.11 [0 0 0 0] [0 0 0 0] [0] 876.69/289.11 [0 0 0 0] [0 0 0 0] [0] 876.69/289.11 >= [0 0 0 1] [1] 876.69/289.11 [0 0 0 0] @ys + [0] 876.69/289.11 [0 0 0 0] [0] 876.69/289.11 [0 0 0 0] [0] 876.69/289.11 = [quicksort^#(@ys)] 876.69/289.11 876.69/289.11 [quicksort#2^#(tuple#2(@xs, @ys), @z)] = [0 0 0 1] [0 0 0 1] [1] 876.69/289.11 [0 0 0 0] @ys + [0 0 0 0] @xs + [0] 876.69/289.11 [0 0 0 0] [0 0 0 0] [0] 876.69/289.11 [0 0 0 0] [0 0 0 0] [0] 876.69/289.11 >= [0 0 0 1] [1] 876.69/289.11 [0 0 0 0] @xs + [0] 876.69/289.11 [0 0 0 0] [0] 876.69/289.11 [0 0 0 0] [0] 876.69/289.11 = [append^#(quicksort(@xs), ::(@z, quicksort(@ys)))] 876.69/289.11 876.69/289.11 [append^#(@l, @ys)] = [0 0 0 1] [1] 876.69/289.11 [0 0 0 0] @l + [0] 876.69/289.11 [0 0 0 0] [0] 876.69/289.11 [0 0 0 0] [0] 876.69/289.11 > [0 0 0 1] [0] 876.69/289.11 [0 0 0 0] @l + [0] 876.69/289.11 [0 0 0 0] [0] 876.69/289.11 [0 0 0 0] [0] 876.69/289.11 = [c_5(append#1^#(@l, @ys))] 876.69/289.11 876.69/289.11 [append#1^#(::(@x, @xs), @ys)] = [0 0 0 1] [1] 876.69/289.11 [0 0 0 0] @xs + [0] 876.69/289.11 [0 0 0 0] [0] 876.69/289.11 [0 0 0 0] [0] 876.69/289.11 >= [0 0 0 1] [1] 876.69/289.11 [0 0 0 0] @xs + [0] 876.69/289.11 [0 0 0 0] [0] 876.69/289.11 [0 0 0 0] [0] 876.69/289.11 = [c_6(append^#(@xs, @ys))] 876.69/289.11 876.69/289.11 [sortAll#1^#(::(@x, @xs))] = [1 0 0 0] [1 0 0 0] [1] 876.69/289.11 [0 0 0 0] @x + [0 0 0 0] @xs + [1] 876.69/289.11 [0 0 0 0] [0 0 0 0] [1] 876.69/289.12 [0 0 0 0] [0 0 0 0] [1] 876.69/289.12 >= [1 0 0 0] [1 0 0 0] [1] 876.69/289.12 [0 0 0 0] @x + [0 0 0 0] @xs + [1] 876.69/289.12 [0 0 0 0] [0 0 0 0] [1] 876.69/289.12 [0 0 0 0] [0 0 0 0] [1] 876.69/289.12 = [sortAll#2^#(@x, @xs)] 876.69/289.12 876.69/289.12 [quicksort#1^#(::(@z, @zs))] = [0 0 0 1] [1] 876.69/289.12 [0 0 0 0] @zs + [0] 876.69/289.12 [0 0 0 0] [0] 876.69/289.12 [0 0 0 0] [0] 876.69/289.12 >= [0 0 0 1] [1] 876.69/289.12 [0 0 0 0] @zs + [0] 876.69/289.12 [0 0 0 0] [0] 876.69/289.12 [0 0 0 0] [0] 876.69/289.12 = [quicksort#2^#(splitqs(@z, @zs), @z)] 876.69/289.12 876.69/289.12 876.69/289.12 We return to the main proof. 876.69/289.12 876.69/289.12 We are left with following problem, upon which TcT provides the 876.69/289.12 certificate YES(O(1),O(1)). 876.69/289.12 876.69/289.12 Weak DPs: 876.69/289.12 { splitAndSort^#(@l) -> sortAll^#(split(@l)) 876.69/289.12 , sortAll^#(@l) -> sortAll#1^#(@l) 876.69/289.12 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> sortAll^#(@xs) 876.69/289.12 , sortAll#2^#(tuple#2(@vals, @key), @xs) -> quicksort^#(@vals) 876.69/289.12 , quicksort^#(@l) -> quicksort#1^#(@l) 876.69/289.12 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> quicksort^#(@xs) 876.69/289.12 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> quicksort^#(@ys) 876.69/289.12 , quicksort#2^#(tuple#2(@xs, @ys), @z) -> 876.69/289.12 append^#(quicksort(@xs), ::(@z, quicksort(@ys))) 876.69/289.12 , append^#(@l, @ys) -> c_5(append#1^#(@l, @ys)) 876.69/289.12 , append#1^#(::(@x, @xs), @ys) -> c_6(append^#(@xs, @ys)) 876.69/289.12 , sortAll#1^#(::(@x, @xs)) -> sortAll#2^#(@x, @xs) 876.69/289.12 , quicksort#1^#(::(@z, @zs)) -> 876.69/289.12 quicksort#2^#(splitqs(@z, @zs), @z) } 876.69/289.12 Weak Trs: 876.69/289.12 { #equal(@x, @y) -> #eq(@x, @y) 876.69/289.12 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 876.69/289.12 , #eq(#pos(@x), #0()) -> #false() 876.69/289.12 , #eq(#pos(@x), #neg(@y)) -> #false() 876.69/289.12 , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> 876.69/289.12 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.69/289.12 , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() 876.69/289.12 , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() 876.69/289.12 , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() 876.69/289.12 , #eq(nil(), nil()) -> #true() 876.69/289.12 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 876.69/289.12 , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() 876.69/289.12 , #eq(::(@x_1, @x_2), nil()) -> #false() 876.69/289.12 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 876.69/289.12 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 876.69/289.12 , #eq(#0(), #pos(@y)) -> #false() 876.69/289.12 , #eq(#0(), #0()) -> #true() 876.69/289.12 , #eq(#0(), #neg(@y)) -> #false() 876.69/289.12 , #eq(#0(), #s(@y)) -> #false() 876.69/289.12 , #eq(#neg(@x), #pos(@y)) -> #false() 876.69/289.12 , #eq(#neg(@x), #0()) -> #false() 876.69/289.12 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 876.69/289.12 , #eq(#s(@x), #0()) -> #false() 876.69/289.12 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 876.69/289.12 , quicksort#2(tuple#2(@xs, @ys), @z) -> 876.69/289.12 append(quicksort(@xs), ::(@z, quicksort(@ys))) 876.69/289.12 , split(@l) -> split#1(@l) 876.69/289.12 , append(@l, @ys) -> append#1(@l, @ys) 876.69/289.12 , insert#3(tuple#2(@vals1, @key1), @keyX, @ls, @valX, @x) -> 876.69/289.12 insert#4(#equal(@key1, @keyX), @key1, @ls, @valX, @vals1, @x) 876.69/289.12 , #ckgt(#EQ()) -> #false() 876.69/289.12 , #ckgt(#LT()) -> #false() 876.69/289.12 , #ckgt(#GT()) -> #true() 876.69/289.12 , insert(@x, @l) -> insert#1(@x, @l, @x) 876.69/289.12 , insert#2(nil(), @keyX, @valX, @x) -> 876.69/289.12 ::(tuple#2(::(@valX, nil()), @keyX), nil()) 876.69/289.12 , insert#2(::(@l1, @ls), @keyX, @valX, @x) -> 876.69/289.12 insert#3(@l1, @keyX, @ls, @valX, @x) 876.69/289.12 , #and(#true(), #true()) -> #true() 876.69/289.12 , #and(#true(), #false()) -> #false() 876.69/289.12 , #and(#false(), #true()) -> #false() 876.69/289.12 , #and(#false(), #false()) -> #false() 876.69/289.12 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 876.69/289.12 , #compare(#pos(@x), #0()) -> #GT() 876.69/289.12 , #compare(#pos(@x), #neg(@y)) -> #GT() 876.69/289.12 , #compare(#0(), #pos(@y)) -> #LT() 876.69/289.12 , #compare(#0(), #0()) -> #EQ() 876.69/289.12 , #compare(#0(), #neg(@y)) -> #GT() 876.69/289.12 , #compare(#0(), #s(@y)) -> #LT() 876.69/289.12 , #compare(#neg(@x), #pos(@y)) -> #LT() 876.69/289.12 , #compare(#neg(@x), #0()) -> #LT() 876.69/289.12 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 876.69/289.12 , #compare(#s(@x), #0()) -> #GT() 876.69/289.12 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 876.69/289.12 , split#1(nil()) -> nil() 876.69/289.12 , split#1(::(@x, @xs)) -> insert(@x, split(@xs)) 876.69/289.12 , #greater(@x, @y) -> #ckgt(#compare(@x, @y)) 876.69/289.12 , insert#4(#true(), @key1, @ls, @valX, @vals1, @x) -> 876.69/289.12 ::(tuple#2(::(@valX, @vals1), @key1), @ls) 876.69/289.12 , insert#4(#false(), @key1, @ls, @valX, @vals1, @x) -> 876.69/289.12 ::(tuple#2(@vals1, @key1), insert(@x, @ls)) 876.69/289.12 , insert#1(tuple#2(@valX, @keyX), @l, @x) -> 876.69/289.12 insert#2(@l, @keyX, @valX, @x) 876.69/289.12 , quicksort#1(nil()) -> nil() 876.69/289.12 , quicksort#1(::(@z, @zs)) -> quicksort#2(splitqs(@z, @zs), @z) 876.69/289.12 , append#1(nil(), @ys) -> @ys 876.69/289.12 , append#1(::(@x, @xs), @ys) -> ::(@x, append(@xs, @ys)) 876.69/289.12 , splitqs(@pivot, @l) -> splitqs#1(@l, @pivot) 876.69/289.12 , quicksort(@l) -> quicksort#1(@l) 876.69/289.12 , splitqs#3(#true(), @ls, @rs, @x) -> tuple#2(@ls, ::(@x, @rs)) 876.69/289.12 , splitqs#3(#false(), @ls, @rs, @x) -> tuple#2(::(@x, @ls), @rs) 876.69/289.12 , splitqs#2(tuple#2(@ls, @rs), @pivot, @x) -> 876.69/289.12 splitqs#3(#greater(@x, @pivot), @ls, @rs, @x) 876.69/289.12 , splitqs#1(nil(), @pivot) -> tuple#2(nil(), nil()) 876.69/289.12 , splitqs#1(::(@x, @xs), @pivot) -> 876.69/289.12 splitqs#2(splitqs(@pivot, @xs), @pivot, @x) } 876.69/289.12 Obligation: 876.69/289.12 innermost runtime complexity 876.69/289.12 Answer: 876.69/289.12 YES(O(1),O(1)) 876.69/289.12 876.69/289.12 Empty rules are trivially bounded 876.69/289.12 876.69/289.12 876.69/289.12 Hurray, we answered YES(O(1),O(n^6)) 876.69/289.12 EOF