YES(O(1),O(n^2)) 570.12/150.33 YES(O(1),O(n^2)) 570.12/150.33 570.12/150.33 We are left with following problem, upon which TcT provides the 570.12/150.33 certificate YES(O(1),O(n^2)). 570.12/150.33 570.12/150.33 Strict Trs: 570.12/150.33 { #equal(@x, @y) -> #eq(@x, @y) 570.12/150.33 , leq#2(nil(), @x, @xs) -> #false() 570.12/150.33 , leq#2(::(@y, @ys), @x, @xs) -> 570.12/150.33 or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))) 570.12/150.33 , or(@x, @y) -> #or(@x, @y) 570.12/150.33 , and(@x, @y) -> #and(@x, @y) 570.12/150.33 , isortlist(@l) -> isortlist#1(@l) 570.12/150.33 , leq#1(nil(), @l2) -> #true() 570.12/150.33 , leq#1(::(@x, @xs), @l2) -> leq#2(@l2, @x, @xs) 570.12/150.33 , insert(@x, @l) -> insert#1(@l, @x) 570.12/150.33 , insert#2(#true(), @x, @y, @ys) -> ::(@x, ::(@y, @ys)) 570.12/150.33 , insert#2(#false(), @x, @y, @ys) -> ::(@y, insert(@x, @ys)) 570.12/150.33 , leq(@l1, @l2) -> leq#1(@l1, @l2) 570.12/150.33 , isortlist#1(nil()) -> nil() 570.12/150.33 , isortlist#1(::(@x, @xs)) -> insert(@x, isortlist(@xs)) 570.12/150.33 , insert#1(nil(), @x) -> ::(@x, nil()) 570.12/150.33 , insert#1(::(@y, @ys), @x) -> insert#2(leq(@x, @y), @x, @y, @ys) 570.12/150.33 , #less(@x, @y) -> #cklt(#compare(@x, @y)) } 570.12/150.33 Weak Trs: 570.12/150.33 { #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 570.12/150.33 , #eq(#pos(@x), #0()) -> #false() 570.12/150.33 , #eq(#pos(@x), #neg(@y)) -> #false() 570.12/150.33 , #eq(nil(), nil()) -> #true() 570.12/150.33 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 570.12/150.33 , #eq(::(@x_1, @x_2), nil()) -> #false() 570.12/150.33 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 570.12/150.33 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 570.12/150.33 , #eq(#0(), #pos(@y)) -> #false() 570.12/150.33 , #eq(#0(), #0()) -> #true() 570.12/150.33 , #eq(#0(), #neg(@y)) -> #false() 570.12/150.33 , #eq(#0(), #s(@y)) -> #false() 570.12/150.33 , #eq(#neg(@x), #pos(@y)) -> #false() 570.12/150.33 , #eq(#neg(@x), #0()) -> #false() 570.12/150.33 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 570.12/150.33 , #eq(#s(@x), #0()) -> #false() 570.12/150.33 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 570.12/150.33 , #cklt(#EQ()) -> #false() 570.12/150.33 , #cklt(#LT()) -> #true() 570.12/150.33 , #cklt(#GT()) -> #false() 570.12/150.33 , #and(#true(), #true()) -> #true() 570.12/150.33 , #and(#true(), #false()) -> #false() 570.12/150.33 , #and(#false(), #true()) -> #false() 570.12/150.33 , #and(#false(), #false()) -> #false() 570.12/150.33 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 570.12/150.33 , #compare(#pos(@x), #0()) -> #GT() 570.12/150.33 , #compare(#pos(@x), #neg(@y)) -> #GT() 570.12/150.33 , #compare(#0(), #pos(@y)) -> #LT() 570.12/150.33 , #compare(#0(), #0()) -> #EQ() 570.12/150.33 , #compare(#0(), #neg(@y)) -> #GT() 570.12/150.33 , #compare(#0(), #s(@y)) -> #LT() 570.12/150.33 , #compare(#neg(@x), #pos(@y)) -> #LT() 570.12/150.33 , #compare(#neg(@x), #0()) -> #LT() 570.12/150.33 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 570.12/150.33 , #compare(#s(@x), #0()) -> #GT() 570.12/150.33 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 570.12/150.33 , #or(#true(), #true()) -> #true() 570.12/150.33 , #or(#true(), #false()) -> #true() 570.12/150.33 , #or(#false(), #true()) -> #true() 570.12/150.33 , #or(#false(), #false()) -> #false() } 570.12/150.33 Obligation: 570.12/150.33 innermost runtime complexity 570.12/150.33 Answer: 570.12/150.33 YES(O(1),O(n^2)) 570.12/150.33 570.12/150.33 We add the following dependency tuples: 570.12/150.33 570.12/150.33 Strict DPs: 570.12/150.33 { #equal^#(@x, @y) -> c_1(#eq^#(@x, @y)) 570.12/150.33 , leq#2^#(nil(), @x, @xs) -> c_2() 570.12/150.33 , leq#2^#(::(@y, @ys), @x, @xs) -> 570.12/150.33 c_3(or^#(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))), 570.12/150.33 #less^#(@x, @y), 570.12/150.33 and^#(#equal(@x, @y), leq(@xs, @ys)), 570.12/150.33 #equal^#(@x, @y), 570.12/150.33 leq^#(@xs, @ys)) 570.12/150.33 , or^#(@x, @y) -> c_4(#or^#(@x, @y)) 570.12/150.33 , #less^#(@x, @y) -> 570.12/150.33 c_17(#cklt^#(#compare(@x, @y)), #compare^#(@x, @y)) 570.12/150.33 , and^#(@x, @y) -> c_5(#and^#(@x, @y)) 570.12/150.33 , leq^#(@l1, @l2) -> c_12(leq#1^#(@l1, @l2)) 570.12/150.33 , isortlist^#(@l) -> c_6(isortlist#1^#(@l)) 570.12/150.33 , isortlist#1^#(nil()) -> c_13() 570.12/150.33 , isortlist#1^#(::(@x, @xs)) -> 570.12/150.33 c_14(insert^#(@x, isortlist(@xs)), isortlist^#(@xs)) 570.12/150.33 , leq#1^#(nil(), @l2) -> c_7() 570.12/150.33 , leq#1^#(::(@x, @xs), @l2) -> c_8(leq#2^#(@l2, @x, @xs)) 570.12/150.33 , insert^#(@x, @l) -> c_9(insert#1^#(@l, @x)) 570.12/150.33 , insert#1^#(nil(), @x) -> c_15() 570.12/150.33 , insert#1^#(::(@y, @ys), @x) -> 570.12/150.33 c_16(insert#2^#(leq(@x, @y), @x, @y, @ys), leq^#(@x, @y)) 570.12/150.33 , insert#2^#(#true(), @x, @y, @ys) -> c_10() 570.12/150.33 , insert#2^#(#false(), @x, @y, @ys) -> c_11(insert^#(@x, @ys)) } 570.12/150.33 Weak DPs: 570.12/150.33 { #eq^#(#pos(@x), #pos(@y)) -> c_18(#eq^#(@x, @y)) 570.12/150.33 , #eq^#(#pos(@x), #0()) -> c_19() 570.12/150.33 , #eq^#(#pos(@x), #neg(@y)) -> c_20() 570.12/150.33 , #eq^#(nil(), nil()) -> c_21() 570.12/150.33 , #eq^#(nil(), ::(@y_1, @y_2)) -> c_22() 570.12/150.33 , #eq^#(::(@x_1, @x_2), nil()) -> c_23() 570.12/150.33 , #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 570.12/150.33 c_24(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), 570.12/150.33 #eq^#(@x_1, @y_1), 570.12/150.33 #eq^#(@x_2, @y_2)) 570.12/150.33 , #eq^#(#0(), #pos(@y)) -> c_25() 570.12/150.33 , #eq^#(#0(), #0()) -> c_26() 570.12/150.33 , #eq^#(#0(), #neg(@y)) -> c_27() 570.12/150.33 , #eq^#(#0(), #s(@y)) -> c_28() 570.12/150.33 , #eq^#(#neg(@x), #pos(@y)) -> c_29() 570.12/150.33 , #eq^#(#neg(@x), #0()) -> c_30() 570.12/150.33 , #eq^#(#neg(@x), #neg(@y)) -> c_31(#eq^#(@x, @y)) 570.12/150.33 , #eq^#(#s(@x), #0()) -> c_32() 570.12/150.33 , #eq^#(#s(@x), #s(@y)) -> c_33(#eq^#(@x, @y)) 570.12/150.33 , #or^#(#true(), #true()) -> c_53() 570.12/150.33 , #or^#(#true(), #false()) -> c_54() 570.12/150.33 , #or^#(#false(), #true()) -> c_55() 570.12/150.33 , #or^#(#false(), #false()) -> c_56() 570.12/150.33 , #and^#(#true(), #true()) -> c_37() 570.12/150.33 , #and^#(#true(), #false()) -> c_38() 570.12/150.33 , #and^#(#false(), #true()) -> c_39() 570.12/150.33 , #and^#(#false(), #false()) -> c_40() 570.12/150.33 , #cklt^#(#EQ()) -> c_34() 570.12/150.33 , #cklt^#(#LT()) -> c_35() 570.12/150.33 , #cklt^#(#GT()) -> c_36() 570.12/150.33 , #compare^#(#pos(@x), #pos(@y)) -> c_41(#compare^#(@x, @y)) 570.12/150.33 , #compare^#(#pos(@x), #0()) -> c_42() 570.12/150.33 , #compare^#(#pos(@x), #neg(@y)) -> c_43() 570.12/150.33 , #compare^#(#0(), #pos(@y)) -> c_44() 570.12/150.33 , #compare^#(#0(), #0()) -> c_45() 570.12/150.33 , #compare^#(#0(), #neg(@y)) -> c_46() 570.12/150.33 , #compare^#(#0(), #s(@y)) -> c_47() 570.12/150.33 , #compare^#(#neg(@x), #pos(@y)) -> c_48() 570.12/150.33 , #compare^#(#neg(@x), #0()) -> c_49() 570.12/150.33 , #compare^#(#neg(@x), #neg(@y)) -> c_50(#compare^#(@y, @x)) 570.12/150.33 , #compare^#(#s(@x), #0()) -> c_51() 570.12/150.33 , #compare^#(#s(@x), #s(@y)) -> c_52(#compare^#(@x, @y)) } 570.12/150.33 570.12/150.33 and mark the set of starting terms. 570.12/150.33 570.12/150.33 We are left with following problem, upon which TcT provides the 570.12/150.33 certificate YES(O(1),O(n^2)). 570.12/150.33 570.12/150.33 Strict DPs: 570.12/150.33 { #equal^#(@x, @y) -> c_1(#eq^#(@x, @y)) 570.12/150.33 , leq#2^#(nil(), @x, @xs) -> c_2() 570.12/150.33 , leq#2^#(::(@y, @ys), @x, @xs) -> 570.12/150.33 c_3(or^#(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))), 570.12/150.33 #less^#(@x, @y), 570.12/150.33 and^#(#equal(@x, @y), leq(@xs, @ys)), 570.12/150.33 #equal^#(@x, @y), 570.12/150.33 leq^#(@xs, @ys)) 570.12/150.33 , or^#(@x, @y) -> c_4(#or^#(@x, @y)) 570.12/150.33 , #less^#(@x, @y) -> 570.12/150.33 c_17(#cklt^#(#compare(@x, @y)), #compare^#(@x, @y)) 570.12/150.33 , and^#(@x, @y) -> c_5(#and^#(@x, @y)) 570.12/150.33 , leq^#(@l1, @l2) -> c_12(leq#1^#(@l1, @l2)) 570.12/150.33 , isortlist^#(@l) -> c_6(isortlist#1^#(@l)) 570.12/150.33 , isortlist#1^#(nil()) -> c_13() 570.12/150.33 , isortlist#1^#(::(@x, @xs)) -> 570.12/150.33 c_14(insert^#(@x, isortlist(@xs)), isortlist^#(@xs)) 570.12/150.33 , leq#1^#(nil(), @l2) -> c_7() 570.12/150.33 , leq#1^#(::(@x, @xs), @l2) -> c_8(leq#2^#(@l2, @x, @xs)) 570.12/150.33 , insert^#(@x, @l) -> c_9(insert#1^#(@l, @x)) 570.12/150.33 , insert#1^#(nil(), @x) -> c_15() 570.12/150.33 , insert#1^#(::(@y, @ys), @x) -> 570.12/150.33 c_16(insert#2^#(leq(@x, @y), @x, @y, @ys), leq^#(@x, @y)) 570.12/150.33 , insert#2^#(#true(), @x, @y, @ys) -> c_10() 570.12/150.33 , insert#2^#(#false(), @x, @y, @ys) -> c_11(insert^#(@x, @ys)) } 570.12/150.33 Weak DPs: 570.12/150.33 { #eq^#(#pos(@x), #pos(@y)) -> c_18(#eq^#(@x, @y)) 570.12/150.33 , #eq^#(#pos(@x), #0()) -> c_19() 570.12/150.33 , #eq^#(#pos(@x), #neg(@y)) -> c_20() 570.12/150.33 , #eq^#(nil(), nil()) -> c_21() 570.12/150.33 , #eq^#(nil(), ::(@y_1, @y_2)) -> c_22() 570.12/150.33 , #eq^#(::(@x_1, @x_2), nil()) -> c_23() 570.12/150.33 , #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 570.12/150.33 c_24(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), 570.12/150.33 #eq^#(@x_1, @y_1), 570.12/150.33 #eq^#(@x_2, @y_2)) 570.12/150.33 , #eq^#(#0(), #pos(@y)) -> c_25() 570.12/150.33 , #eq^#(#0(), #0()) -> c_26() 570.12/150.33 , #eq^#(#0(), #neg(@y)) -> c_27() 570.12/150.33 , #eq^#(#0(), #s(@y)) -> c_28() 570.12/150.33 , #eq^#(#neg(@x), #pos(@y)) -> c_29() 570.12/150.33 , #eq^#(#neg(@x), #0()) -> c_30() 570.12/150.33 , #eq^#(#neg(@x), #neg(@y)) -> c_31(#eq^#(@x, @y)) 570.12/150.33 , #eq^#(#s(@x), #0()) -> c_32() 570.12/150.33 , #eq^#(#s(@x), #s(@y)) -> c_33(#eq^#(@x, @y)) 570.12/150.33 , #or^#(#true(), #true()) -> c_53() 570.12/150.33 , #or^#(#true(), #false()) -> c_54() 570.12/150.33 , #or^#(#false(), #true()) -> c_55() 570.12/150.33 , #or^#(#false(), #false()) -> c_56() 570.12/150.33 , #and^#(#true(), #true()) -> c_37() 570.12/150.33 , #and^#(#true(), #false()) -> c_38() 570.12/150.33 , #and^#(#false(), #true()) -> c_39() 570.12/150.33 , #and^#(#false(), #false()) -> c_40() 570.12/150.33 , #cklt^#(#EQ()) -> c_34() 570.12/150.33 , #cklt^#(#LT()) -> c_35() 570.12/150.33 , #cklt^#(#GT()) -> c_36() 570.12/150.33 , #compare^#(#pos(@x), #pos(@y)) -> c_41(#compare^#(@x, @y)) 570.12/150.33 , #compare^#(#pos(@x), #0()) -> c_42() 570.12/150.33 , #compare^#(#pos(@x), #neg(@y)) -> c_43() 570.12/150.33 , #compare^#(#0(), #pos(@y)) -> c_44() 570.12/150.33 , #compare^#(#0(), #0()) -> c_45() 570.12/150.33 , #compare^#(#0(), #neg(@y)) -> c_46() 570.12/150.33 , #compare^#(#0(), #s(@y)) -> c_47() 570.12/150.33 , #compare^#(#neg(@x), #pos(@y)) -> c_48() 570.12/150.34 , #compare^#(#neg(@x), #0()) -> c_49() 570.12/150.34 , #compare^#(#neg(@x), #neg(@y)) -> c_50(#compare^#(@y, @x)) 570.12/150.34 , #compare^#(#s(@x), #0()) -> c_51() 570.12/150.34 , #compare^#(#s(@x), #s(@y)) -> c_52(#compare^#(@x, @y)) } 570.12/150.34 Weak Trs: 570.12/150.34 { #equal(@x, @y) -> #eq(@x, @y) 570.12/150.34 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 570.12/150.34 , #eq(#pos(@x), #0()) -> #false() 570.12/150.34 , #eq(#pos(@x), #neg(@y)) -> #false() 570.12/150.34 , #eq(nil(), nil()) -> #true() 570.12/150.34 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 570.12/150.34 , #eq(::(@x_1, @x_2), nil()) -> #false() 570.12/150.34 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 570.12/150.34 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 570.12/150.34 , #eq(#0(), #pos(@y)) -> #false() 570.12/150.34 , #eq(#0(), #0()) -> #true() 570.12/150.34 , #eq(#0(), #neg(@y)) -> #false() 570.12/150.34 , #eq(#0(), #s(@y)) -> #false() 570.12/150.34 , #eq(#neg(@x), #pos(@y)) -> #false() 570.12/150.34 , #eq(#neg(@x), #0()) -> #false() 570.12/150.34 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 570.12/150.34 , #eq(#s(@x), #0()) -> #false() 570.12/150.34 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 570.12/150.34 , leq#2(nil(), @x, @xs) -> #false() 570.12/150.34 , leq#2(::(@y, @ys), @x, @xs) -> 570.12/150.34 or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))) 570.12/150.34 , or(@x, @y) -> #or(@x, @y) 570.12/150.34 , and(@x, @y) -> #and(@x, @y) 570.12/150.34 , isortlist(@l) -> isortlist#1(@l) 570.12/150.34 , leq#1(nil(), @l2) -> #true() 570.12/150.34 , leq#1(::(@x, @xs), @l2) -> leq#2(@l2, @x, @xs) 570.12/150.34 , #cklt(#EQ()) -> #false() 570.12/150.34 , #cklt(#LT()) -> #true() 570.12/150.34 , #cklt(#GT()) -> #false() 570.12/150.34 , insert(@x, @l) -> insert#1(@l, @x) 570.12/150.34 , insert#2(#true(), @x, @y, @ys) -> ::(@x, ::(@y, @ys)) 570.12/150.34 , insert#2(#false(), @x, @y, @ys) -> ::(@y, insert(@x, @ys)) 570.12/150.34 , #and(#true(), #true()) -> #true() 570.12/150.34 , #and(#true(), #false()) -> #false() 570.12/150.34 , #and(#false(), #true()) -> #false() 570.12/150.34 , #and(#false(), #false()) -> #false() 570.12/150.34 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 570.12/150.34 , #compare(#pos(@x), #0()) -> #GT() 570.12/150.34 , #compare(#pos(@x), #neg(@y)) -> #GT() 570.12/150.34 , #compare(#0(), #pos(@y)) -> #LT() 570.12/150.34 , #compare(#0(), #0()) -> #EQ() 570.12/150.34 , #compare(#0(), #neg(@y)) -> #GT() 570.12/150.34 , #compare(#0(), #s(@y)) -> #LT() 570.12/150.34 , #compare(#neg(@x), #pos(@y)) -> #LT() 570.12/150.34 , #compare(#neg(@x), #0()) -> #LT() 570.12/150.34 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 570.12/150.34 , #compare(#s(@x), #0()) -> #GT() 570.12/150.34 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 570.12/150.34 , leq(@l1, @l2) -> leq#1(@l1, @l2) 570.12/150.34 , isortlist#1(nil()) -> nil() 570.12/150.34 , isortlist#1(::(@x, @xs)) -> insert(@x, isortlist(@xs)) 570.12/150.34 , #or(#true(), #true()) -> #true() 570.12/150.34 , #or(#true(), #false()) -> #true() 570.12/150.34 , #or(#false(), #true()) -> #true() 570.12/150.34 , #or(#false(), #false()) -> #false() 570.12/150.34 , insert#1(nil(), @x) -> ::(@x, nil()) 570.12/150.34 , insert#1(::(@y, @ys), @x) -> insert#2(leq(@x, @y), @x, @y, @ys) 570.12/150.34 , #less(@x, @y) -> #cklt(#compare(@x, @y)) } 570.12/150.34 Obligation: 570.12/150.34 innermost runtime complexity 570.12/150.34 Answer: 570.12/150.34 YES(O(1),O(n^2)) 570.12/150.34 570.12/150.34 We estimate the number of application of {1,2,4,5,6,9,11,14,16} by 570.12/150.34 applications of Pre({1,2,4,5,6,9,11,14,16}) = {3,7,8,12,13,15}. 570.12/150.34 Here rules are labeled as follows: 570.12/150.34 570.12/150.34 DPs: 570.12/150.34 { 1: #equal^#(@x, @y) -> c_1(#eq^#(@x, @y)) 570.12/150.34 , 2: leq#2^#(nil(), @x, @xs) -> c_2() 570.12/150.34 , 3: leq#2^#(::(@y, @ys), @x, @xs) -> 570.12/150.34 c_3(or^#(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))), 570.12/150.34 #less^#(@x, @y), 570.12/150.34 and^#(#equal(@x, @y), leq(@xs, @ys)), 570.12/150.34 #equal^#(@x, @y), 570.12/150.34 leq^#(@xs, @ys)) 570.12/150.34 , 4: or^#(@x, @y) -> c_4(#or^#(@x, @y)) 570.12/150.34 , 5: #less^#(@x, @y) -> 570.12/150.34 c_17(#cklt^#(#compare(@x, @y)), #compare^#(@x, @y)) 570.12/150.34 , 6: and^#(@x, @y) -> c_5(#and^#(@x, @y)) 570.12/150.34 , 7: leq^#(@l1, @l2) -> c_12(leq#1^#(@l1, @l2)) 570.12/150.34 , 8: isortlist^#(@l) -> c_6(isortlist#1^#(@l)) 570.12/150.34 , 9: isortlist#1^#(nil()) -> c_13() 570.12/150.34 , 10: isortlist#1^#(::(@x, @xs)) -> 570.12/150.34 c_14(insert^#(@x, isortlist(@xs)), isortlist^#(@xs)) 570.12/150.34 , 11: leq#1^#(nil(), @l2) -> c_7() 570.12/150.34 , 12: leq#1^#(::(@x, @xs), @l2) -> c_8(leq#2^#(@l2, @x, @xs)) 570.12/150.34 , 13: insert^#(@x, @l) -> c_9(insert#1^#(@l, @x)) 570.12/150.34 , 14: insert#1^#(nil(), @x) -> c_15() 570.12/150.34 , 15: insert#1^#(::(@y, @ys), @x) -> 570.12/150.34 c_16(insert#2^#(leq(@x, @y), @x, @y, @ys), leq^#(@x, @y)) 570.12/150.34 , 16: insert#2^#(#true(), @x, @y, @ys) -> c_10() 570.12/150.34 , 17: insert#2^#(#false(), @x, @y, @ys) -> c_11(insert^#(@x, @ys)) 570.12/150.34 , 18: #eq^#(#pos(@x), #pos(@y)) -> c_18(#eq^#(@x, @y)) 570.12/150.34 , 19: #eq^#(#pos(@x), #0()) -> c_19() 570.12/150.34 , 20: #eq^#(#pos(@x), #neg(@y)) -> c_20() 570.12/150.34 , 21: #eq^#(nil(), nil()) -> c_21() 570.12/150.34 , 22: #eq^#(nil(), ::(@y_1, @y_2)) -> c_22() 570.12/150.34 , 23: #eq^#(::(@x_1, @x_2), nil()) -> c_23() 570.12/150.34 , 24: #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 570.12/150.34 c_24(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), 570.12/150.34 #eq^#(@x_1, @y_1), 570.12/150.34 #eq^#(@x_2, @y_2)) 570.12/150.34 , 25: #eq^#(#0(), #pos(@y)) -> c_25() 570.12/150.34 , 26: #eq^#(#0(), #0()) -> c_26() 570.12/150.34 , 27: #eq^#(#0(), #neg(@y)) -> c_27() 570.12/150.34 , 28: #eq^#(#0(), #s(@y)) -> c_28() 570.12/150.34 , 29: #eq^#(#neg(@x), #pos(@y)) -> c_29() 570.12/150.34 , 30: #eq^#(#neg(@x), #0()) -> c_30() 570.12/150.34 , 31: #eq^#(#neg(@x), #neg(@y)) -> c_31(#eq^#(@x, @y)) 570.12/150.34 , 32: #eq^#(#s(@x), #0()) -> c_32() 570.12/150.34 , 33: #eq^#(#s(@x), #s(@y)) -> c_33(#eq^#(@x, @y)) 570.12/150.34 , 34: #or^#(#true(), #true()) -> c_53() 570.12/150.34 , 35: #or^#(#true(), #false()) -> c_54() 570.12/150.34 , 36: #or^#(#false(), #true()) -> c_55() 570.12/150.34 , 37: #or^#(#false(), #false()) -> c_56() 570.12/150.34 , 38: #and^#(#true(), #true()) -> c_37() 570.12/150.34 , 39: #and^#(#true(), #false()) -> c_38() 570.12/150.34 , 40: #and^#(#false(), #true()) -> c_39() 570.12/150.34 , 41: #and^#(#false(), #false()) -> c_40() 570.12/150.34 , 42: #cklt^#(#EQ()) -> c_34() 570.12/150.34 , 43: #cklt^#(#LT()) -> c_35() 570.12/150.34 , 44: #cklt^#(#GT()) -> c_36() 570.12/150.34 , 45: #compare^#(#pos(@x), #pos(@y)) -> c_41(#compare^#(@x, @y)) 570.12/150.34 , 46: #compare^#(#pos(@x), #0()) -> c_42() 570.12/150.34 , 47: #compare^#(#pos(@x), #neg(@y)) -> c_43() 570.12/150.34 , 48: #compare^#(#0(), #pos(@y)) -> c_44() 570.12/150.34 , 49: #compare^#(#0(), #0()) -> c_45() 570.12/150.34 , 50: #compare^#(#0(), #neg(@y)) -> c_46() 570.12/150.34 , 51: #compare^#(#0(), #s(@y)) -> c_47() 570.12/150.34 , 52: #compare^#(#neg(@x), #pos(@y)) -> c_48() 570.12/150.34 , 53: #compare^#(#neg(@x), #0()) -> c_49() 570.12/150.34 , 54: #compare^#(#neg(@x), #neg(@y)) -> c_50(#compare^#(@y, @x)) 570.12/150.34 , 55: #compare^#(#s(@x), #0()) -> c_51() 570.12/150.34 , 56: #compare^#(#s(@x), #s(@y)) -> c_52(#compare^#(@x, @y)) } 570.12/150.34 570.12/150.34 We are left with following problem, upon which TcT provides the 570.12/150.34 certificate YES(O(1),O(n^2)). 570.12/150.34 570.12/150.34 Strict DPs: 570.12/150.34 { leq#2^#(::(@y, @ys), @x, @xs) -> 570.12/150.34 c_3(or^#(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))), 570.12/150.34 #less^#(@x, @y), 570.12/150.34 and^#(#equal(@x, @y), leq(@xs, @ys)), 570.12/150.34 #equal^#(@x, @y), 570.12/150.34 leq^#(@xs, @ys)) 570.12/150.34 , leq^#(@l1, @l2) -> c_12(leq#1^#(@l1, @l2)) 570.12/150.34 , isortlist^#(@l) -> c_6(isortlist#1^#(@l)) 570.12/150.34 , isortlist#1^#(::(@x, @xs)) -> 570.12/150.34 c_14(insert^#(@x, isortlist(@xs)), isortlist^#(@xs)) 570.12/150.34 , leq#1^#(::(@x, @xs), @l2) -> c_8(leq#2^#(@l2, @x, @xs)) 570.12/150.34 , insert^#(@x, @l) -> c_9(insert#1^#(@l, @x)) 570.12/150.34 , insert#1^#(::(@y, @ys), @x) -> 570.12/150.34 c_16(insert#2^#(leq(@x, @y), @x, @y, @ys), leq^#(@x, @y)) 570.12/150.34 , insert#2^#(#false(), @x, @y, @ys) -> c_11(insert^#(@x, @ys)) } 570.12/150.34 Weak DPs: 570.12/150.34 { #equal^#(@x, @y) -> c_1(#eq^#(@x, @y)) 570.12/150.34 , #eq^#(#pos(@x), #pos(@y)) -> c_18(#eq^#(@x, @y)) 570.12/150.34 , #eq^#(#pos(@x), #0()) -> c_19() 570.12/150.34 , #eq^#(#pos(@x), #neg(@y)) -> c_20() 570.12/150.34 , #eq^#(nil(), nil()) -> c_21() 570.12/150.34 , #eq^#(nil(), ::(@y_1, @y_2)) -> c_22() 570.12/150.34 , #eq^#(::(@x_1, @x_2), nil()) -> c_23() 570.12/150.34 , #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 570.12/150.34 c_24(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), 570.12/150.34 #eq^#(@x_1, @y_1), 570.12/150.34 #eq^#(@x_2, @y_2)) 570.12/150.34 , #eq^#(#0(), #pos(@y)) -> c_25() 570.12/150.34 , #eq^#(#0(), #0()) -> c_26() 570.12/150.34 , #eq^#(#0(), #neg(@y)) -> c_27() 570.12/150.34 , #eq^#(#0(), #s(@y)) -> c_28() 570.12/150.34 , #eq^#(#neg(@x), #pos(@y)) -> c_29() 570.12/150.34 , #eq^#(#neg(@x), #0()) -> c_30() 570.12/150.34 , #eq^#(#neg(@x), #neg(@y)) -> c_31(#eq^#(@x, @y)) 570.12/150.34 , #eq^#(#s(@x), #0()) -> c_32() 570.12/150.34 , #eq^#(#s(@x), #s(@y)) -> c_33(#eq^#(@x, @y)) 570.12/150.34 , leq#2^#(nil(), @x, @xs) -> c_2() 570.12/150.34 , or^#(@x, @y) -> c_4(#or^#(@x, @y)) 570.12/150.34 , #less^#(@x, @y) -> 570.12/150.34 c_17(#cklt^#(#compare(@x, @y)), #compare^#(@x, @y)) 570.12/150.34 , and^#(@x, @y) -> c_5(#and^#(@x, @y)) 570.12/150.34 , #or^#(#true(), #true()) -> c_53() 570.12/150.34 , #or^#(#true(), #false()) -> c_54() 570.12/150.34 , #or^#(#false(), #true()) -> c_55() 570.12/150.34 , #or^#(#false(), #false()) -> c_56() 570.12/150.34 , #and^#(#true(), #true()) -> c_37() 570.12/150.34 , #and^#(#true(), #false()) -> c_38() 570.12/150.34 , #and^#(#false(), #true()) -> c_39() 570.12/150.34 , #and^#(#false(), #false()) -> c_40() 570.12/150.34 , isortlist#1^#(nil()) -> c_13() 570.12/150.34 , leq#1^#(nil(), @l2) -> c_7() 570.12/150.34 , insert#1^#(nil(), @x) -> c_15() 570.12/150.34 , insert#2^#(#true(), @x, @y, @ys) -> c_10() 570.12/150.34 , #cklt^#(#EQ()) -> c_34() 570.12/150.34 , #cklt^#(#LT()) -> c_35() 570.12/150.34 , #cklt^#(#GT()) -> c_36() 570.12/150.34 , #compare^#(#pos(@x), #pos(@y)) -> c_41(#compare^#(@x, @y)) 570.12/150.34 , #compare^#(#pos(@x), #0()) -> c_42() 570.12/150.34 , #compare^#(#pos(@x), #neg(@y)) -> c_43() 570.12/150.34 , #compare^#(#0(), #pos(@y)) -> c_44() 570.12/150.34 , #compare^#(#0(), #0()) -> c_45() 570.12/150.34 , #compare^#(#0(), #neg(@y)) -> c_46() 570.12/150.34 , #compare^#(#0(), #s(@y)) -> c_47() 570.12/150.34 , #compare^#(#neg(@x), #pos(@y)) -> c_48() 570.12/150.34 , #compare^#(#neg(@x), #0()) -> c_49() 570.12/150.34 , #compare^#(#neg(@x), #neg(@y)) -> c_50(#compare^#(@y, @x)) 570.12/150.34 , #compare^#(#s(@x), #0()) -> c_51() 570.12/150.34 , #compare^#(#s(@x), #s(@y)) -> c_52(#compare^#(@x, @y)) } 570.12/150.34 Weak Trs: 570.12/150.34 { #equal(@x, @y) -> #eq(@x, @y) 570.12/150.34 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 570.12/150.34 , #eq(#pos(@x), #0()) -> #false() 570.12/150.34 , #eq(#pos(@x), #neg(@y)) -> #false() 570.12/150.34 , #eq(nil(), nil()) -> #true() 570.12/150.34 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 570.12/150.34 , #eq(::(@x_1, @x_2), nil()) -> #false() 570.12/150.34 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 570.12/150.34 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 570.12/150.34 , #eq(#0(), #pos(@y)) -> #false() 570.12/150.34 , #eq(#0(), #0()) -> #true() 570.12/150.34 , #eq(#0(), #neg(@y)) -> #false() 570.12/150.34 , #eq(#0(), #s(@y)) -> #false() 570.12/150.34 , #eq(#neg(@x), #pos(@y)) -> #false() 570.12/150.34 , #eq(#neg(@x), #0()) -> #false() 570.12/150.34 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 570.12/150.34 , #eq(#s(@x), #0()) -> #false() 570.12/150.34 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 570.12/150.34 , leq#2(nil(), @x, @xs) -> #false() 570.12/150.34 , leq#2(::(@y, @ys), @x, @xs) -> 570.12/150.34 or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))) 570.12/150.34 , or(@x, @y) -> #or(@x, @y) 570.12/150.34 , and(@x, @y) -> #and(@x, @y) 570.12/150.34 , isortlist(@l) -> isortlist#1(@l) 570.12/150.34 , leq#1(nil(), @l2) -> #true() 570.12/150.34 , leq#1(::(@x, @xs), @l2) -> leq#2(@l2, @x, @xs) 570.12/150.34 , #cklt(#EQ()) -> #false() 570.12/150.34 , #cklt(#LT()) -> #true() 570.12/150.34 , #cklt(#GT()) -> #false() 570.12/150.34 , insert(@x, @l) -> insert#1(@l, @x) 570.12/150.34 , insert#2(#true(), @x, @y, @ys) -> ::(@x, ::(@y, @ys)) 570.12/150.34 , insert#2(#false(), @x, @y, @ys) -> ::(@y, insert(@x, @ys)) 570.12/150.34 , #and(#true(), #true()) -> #true() 570.12/150.34 , #and(#true(), #false()) -> #false() 570.12/150.34 , #and(#false(), #true()) -> #false() 570.12/150.34 , #and(#false(), #false()) -> #false() 570.12/150.34 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 570.12/150.34 , #compare(#pos(@x), #0()) -> #GT() 570.12/150.34 , #compare(#pos(@x), #neg(@y)) -> #GT() 570.12/150.34 , #compare(#0(), #pos(@y)) -> #LT() 570.12/150.34 , #compare(#0(), #0()) -> #EQ() 570.12/150.34 , #compare(#0(), #neg(@y)) -> #GT() 570.12/150.34 , #compare(#0(), #s(@y)) -> #LT() 570.12/150.34 , #compare(#neg(@x), #pos(@y)) -> #LT() 570.12/150.34 , #compare(#neg(@x), #0()) -> #LT() 570.12/150.34 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 570.12/150.34 , #compare(#s(@x), #0()) -> #GT() 570.12/150.34 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 570.12/150.34 , leq(@l1, @l2) -> leq#1(@l1, @l2) 570.12/150.34 , isortlist#1(nil()) -> nil() 570.12/150.34 , isortlist#1(::(@x, @xs)) -> insert(@x, isortlist(@xs)) 570.12/150.34 , #or(#true(), #true()) -> #true() 570.12/150.34 , #or(#true(), #false()) -> #true() 570.12/150.34 , #or(#false(), #true()) -> #true() 570.12/150.34 , #or(#false(), #false()) -> #false() 570.12/150.34 , insert#1(nil(), @x) -> ::(@x, nil()) 570.12/150.34 , insert#1(::(@y, @ys), @x) -> insert#2(leq(@x, @y), @x, @y, @ys) 570.12/150.34 , #less(@x, @y) -> #cklt(#compare(@x, @y)) } 570.12/150.34 Obligation: 570.12/150.34 innermost runtime complexity 570.12/150.34 Answer: 570.12/150.34 YES(O(1),O(n^2)) 570.12/150.34 570.12/150.34 The following weak DPs constitute a sub-graph of the DG that is 570.12/150.34 closed under successors. The DPs are removed. 570.12/150.34 570.12/150.34 { #equal^#(@x, @y) -> c_1(#eq^#(@x, @y)) 570.12/150.34 , #eq^#(#pos(@x), #pos(@y)) -> c_18(#eq^#(@x, @y)) 570.12/150.34 , #eq^#(#pos(@x), #0()) -> c_19() 570.12/150.34 , #eq^#(#pos(@x), #neg(@y)) -> c_20() 570.12/150.34 , #eq^#(nil(), nil()) -> c_21() 570.12/150.34 , #eq^#(nil(), ::(@y_1, @y_2)) -> c_22() 570.12/150.34 , #eq^#(::(@x_1, @x_2), nil()) -> c_23() 570.12/150.34 , #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 570.12/150.34 c_24(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), 570.12/150.34 #eq^#(@x_1, @y_1), 570.12/150.34 #eq^#(@x_2, @y_2)) 570.12/150.34 , #eq^#(#0(), #pos(@y)) -> c_25() 570.12/150.34 , #eq^#(#0(), #0()) -> c_26() 570.12/150.34 , #eq^#(#0(), #neg(@y)) -> c_27() 570.12/150.34 , #eq^#(#0(), #s(@y)) -> c_28() 570.12/150.34 , #eq^#(#neg(@x), #pos(@y)) -> c_29() 570.12/150.34 , #eq^#(#neg(@x), #0()) -> c_30() 570.12/150.34 , #eq^#(#neg(@x), #neg(@y)) -> c_31(#eq^#(@x, @y)) 570.12/150.34 , #eq^#(#s(@x), #0()) -> c_32() 570.12/150.34 , #eq^#(#s(@x), #s(@y)) -> c_33(#eq^#(@x, @y)) 570.12/150.34 , leq#2^#(nil(), @x, @xs) -> c_2() 570.12/150.34 , or^#(@x, @y) -> c_4(#or^#(@x, @y)) 570.12/150.34 , #less^#(@x, @y) -> 570.12/150.34 c_17(#cklt^#(#compare(@x, @y)), #compare^#(@x, @y)) 570.12/150.34 , and^#(@x, @y) -> c_5(#and^#(@x, @y)) 570.12/150.34 , #or^#(#true(), #true()) -> c_53() 570.12/150.34 , #or^#(#true(), #false()) -> c_54() 570.12/150.34 , #or^#(#false(), #true()) -> c_55() 570.12/150.34 , #or^#(#false(), #false()) -> c_56() 570.12/150.34 , #and^#(#true(), #true()) -> c_37() 570.12/150.34 , #and^#(#true(), #false()) -> c_38() 570.12/150.34 , #and^#(#false(), #true()) -> c_39() 570.12/150.34 , #and^#(#false(), #false()) -> c_40() 570.12/150.34 , isortlist#1^#(nil()) -> c_13() 570.12/150.34 , leq#1^#(nil(), @l2) -> c_7() 570.12/150.34 , insert#1^#(nil(), @x) -> c_15() 570.12/150.34 , insert#2^#(#true(), @x, @y, @ys) -> c_10() 570.12/150.34 , #cklt^#(#EQ()) -> c_34() 570.12/150.34 , #cklt^#(#LT()) -> c_35() 570.12/150.34 , #cklt^#(#GT()) -> c_36() 570.12/150.34 , #compare^#(#pos(@x), #pos(@y)) -> c_41(#compare^#(@x, @y)) 570.12/150.34 , #compare^#(#pos(@x), #0()) -> c_42() 570.12/150.34 , #compare^#(#pos(@x), #neg(@y)) -> c_43() 570.12/150.34 , #compare^#(#0(), #pos(@y)) -> c_44() 570.12/150.34 , #compare^#(#0(), #0()) -> c_45() 570.12/150.34 , #compare^#(#0(), #neg(@y)) -> c_46() 570.12/150.34 , #compare^#(#0(), #s(@y)) -> c_47() 570.12/150.34 , #compare^#(#neg(@x), #pos(@y)) -> c_48() 570.12/150.34 , #compare^#(#neg(@x), #0()) -> c_49() 570.12/150.34 , #compare^#(#neg(@x), #neg(@y)) -> c_50(#compare^#(@y, @x)) 570.12/150.34 , #compare^#(#s(@x), #0()) -> c_51() 570.12/150.34 , #compare^#(#s(@x), #s(@y)) -> c_52(#compare^#(@x, @y)) } 570.12/150.34 570.12/150.34 We are left with following problem, upon which TcT provides the 570.12/150.34 certificate YES(O(1),O(n^2)). 570.12/150.34 570.12/150.34 Strict DPs: 570.12/150.34 { leq#2^#(::(@y, @ys), @x, @xs) -> 570.12/150.34 c_3(or^#(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))), 570.12/150.34 #less^#(@x, @y), 570.12/150.34 and^#(#equal(@x, @y), leq(@xs, @ys)), 570.12/150.34 #equal^#(@x, @y), 570.12/150.34 leq^#(@xs, @ys)) 570.12/150.34 , leq^#(@l1, @l2) -> c_12(leq#1^#(@l1, @l2)) 570.12/150.34 , isortlist^#(@l) -> c_6(isortlist#1^#(@l)) 570.12/150.34 , isortlist#1^#(::(@x, @xs)) -> 570.12/150.34 c_14(insert^#(@x, isortlist(@xs)), isortlist^#(@xs)) 570.12/150.34 , leq#1^#(::(@x, @xs), @l2) -> c_8(leq#2^#(@l2, @x, @xs)) 570.12/150.34 , insert^#(@x, @l) -> c_9(insert#1^#(@l, @x)) 570.12/150.34 , insert#1^#(::(@y, @ys), @x) -> 570.12/150.34 c_16(insert#2^#(leq(@x, @y), @x, @y, @ys), leq^#(@x, @y)) 570.12/150.34 , insert#2^#(#false(), @x, @y, @ys) -> c_11(insert^#(@x, @ys)) } 570.12/150.34 Weak Trs: 570.12/150.34 { #equal(@x, @y) -> #eq(@x, @y) 570.12/150.34 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 570.12/150.34 , #eq(#pos(@x), #0()) -> #false() 570.12/150.34 , #eq(#pos(@x), #neg(@y)) -> #false() 570.12/150.34 , #eq(nil(), nil()) -> #true() 570.12/150.34 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 570.12/150.34 , #eq(::(@x_1, @x_2), nil()) -> #false() 570.12/150.34 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 570.12/150.34 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 570.12/150.34 , #eq(#0(), #pos(@y)) -> #false() 570.12/150.34 , #eq(#0(), #0()) -> #true() 570.12/150.34 , #eq(#0(), #neg(@y)) -> #false() 570.12/150.34 , #eq(#0(), #s(@y)) -> #false() 570.12/150.34 , #eq(#neg(@x), #pos(@y)) -> #false() 570.12/150.34 , #eq(#neg(@x), #0()) -> #false() 570.12/150.34 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 570.12/150.34 , #eq(#s(@x), #0()) -> #false() 570.12/150.34 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 570.12/150.34 , leq#2(nil(), @x, @xs) -> #false() 570.12/150.34 , leq#2(::(@y, @ys), @x, @xs) -> 570.12/150.34 or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))) 570.12/150.34 , or(@x, @y) -> #or(@x, @y) 570.12/150.34 , and(@x, @y) -> #and(@x, @y) 570.12/150.34 , isortlist(@l) -> isortlist#1(@l) 570.12/150.34 , leq#1(nil(), @l2) -> #true() 570.12/150.34 , leq#1(::(@x, @xs), @l2) -> leq#2(@l2, @x, @xs) 570.12/150.34 , #cklt(#EQ()) -> #false() 570.12/150.34 , #cklt(#LT()) -> #true() 570.12/150.34 , #cklt(#GT()) -> #false() 570.12/150.34 , insert(@x, @l) -> insert#1(@l, @x) 570.12/150.34 , insert#2(#true(), @x, @y, @ys) -> ::(@x, ::(@y, @ys)) 570.12/150.34 , insert#2(#false(), @x, @y, @ys) -> ::(@y, insert(@x, @ys)) 570.12/150.34 , #and(#true(), #true()) -> #true() 570.12/150.34 , #and(#true(), #false()) -> #false() 570.12/150.34 , #and(#false(), #true()) -> #false() 570.12/150.34 , #and(#false(), #false()) -> #false() 570.12/150.34 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 570.12/150.34 , #compare(#pos(@x), #0()) -> #GT() 570.12/150.34 , #compare(#pos(@x), #neg(@y)) -> #GT() 570.12/150.34 , #compare(#0(), #pos(@y)) -> #LT() 570.12/150.34 , #compare(#0(), #0()) -> #EQ() 570.12/150.34 , #compare(#0(), #neg(@y)) -> #GT() 570.12/150.34 , #compare(#0(), #s(@y)) -> #LT() 570.12/150.34 , #compare(#neg(@x), #pos(@y)) -> #LT() 570.12/150.34 , #compare(#neg(@x), #0()) -> #LT() 570.12/150.34 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 570.12/150.34 , #compare(#s(@x), #0()) -> #GT() 570.12/150.34 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 570.12/150.34 , leq(@l1, @l2) -> leq#1(@l1, @l2) 570.12/150.34 , isortlist#1(nil()) -> nil() 570.12/150.34 , isortlist#1(::(@x, @xs)) -> insert(@x, isortlist(@xs)) 570.12/150.35 , #or(#true(), #true()) -> #true() 570.12/150.35 , #or(#true(), #false()) -> #true() 570.12/150.35 , #or(#false(), #true()) -> #true() 570.12/150.35 , #or(#false(), #false()) -> #false() 570.12/150.35 , insert#1(nil(), @x) -> ::(@x, nil()) 570.12/150.35 , insert#1(::(@y, @ys), @x) -> insert#2(leq(@x, @y), @x, @y, @ys) 570.12/150.35 , #less(@x, @y) -> #cklt(#compare(@x, @y)) } 570.12/150.35 Obligation: 570.12/150.35 innermost runtime complexity 570.12/150.35 Answer: 570.12/150.35 YES(O(1),O(n^2)) 570.12/150.35 570.12/150.35 Due to missing edges in the dependency-graph, the right-hand sides 570.12/150.35 of following rules could be simplified: 570.12/150.35 570.12/150.35 { leq#2^#(::(@y, @ys), @x, @xs) -> 570.12/150.35 c_3(or^#(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))), 570.12/150.35 #less^#(@x, @y), 570.12/150.35 and^#(#equal(@x, @y), leq(@xs, @ys)), 570.12/150.35 #equal^#(@x, @y), 570.12/150.35 leq^#(@xs, @ys)) } 570.12/150.35 570.12/150.35 We are left with following problem, upon which TcT provides the 570.12/150.35 certificate YES(O(1),O(n^2)). 570.12/150.35 570.12/150.35 Strict DPs: 570.12/150.35 { leq#2^#(::(@y, @ys), @x, @xs) -> c_1(leq^#(@xs, @ys)) 570.12/150.35 , leq^#(@l1, @l2) -> c_2(leq#1^#(@l1, @l2)) 570.12/150.35 , isortlist^#(@l) -> c_3(isortlist#1^#(@l)) 570.12/150.35 , isortlist#1^#(::(@x, @xs)) -> 570.12/150.35 c_4(insert^#(@x, isortlist(@xs)), isortlist^#(@xs)) 570.12/150.35 , leq#1^#(::(@x, @xs), @l2) -> c_5(leq#2^#(@l2, @x, @xs)) 570.12/150.35 , insert^#(@x, @l) -> c_6(insert#1^#(@l, @x)) 570.12/150.35 , insert#1^#(::(@y, @ys), @x) -> 570.12/150.35 c_7(insert#2^#(leq(@x, @y), @x, @y, @ys), leq^#(@x, @y)) 570.12/150.35 , insert#2^#(#false(), @x, @y, @ys) -> c_8(insert^#(@x, @ys)) } 570.12/150.35 Weak Trs: 570.12/150.35 { #equal(@x, @y) -> #eq(@x, @y) 570.12/150.35 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 570.12/150.35 , #eq(#pos(@x), #0()) -> #false() 570.12/150.35 , #eq(#pos(@x), #neg(@y)) -> #false() 570.12/150.35 , #eq(nil(), nil()) -> #true() 570.12/150.35 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 570.12/150.35 , #eq(::(@x_1, @x_2), nil()) -> #false() 570.12/150.35 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 570.12/150.35 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 570.12/150.35 , #eq(#0(), #pos(@y)) -> #false() 570.12/150.35 , #eq(#0(), #0()) -> #true() 570.12/150.35 , #eq(#0(), #neg(@y)) -> #false() 570.12/150.35 , #eq(#0(), #s(@y)) -> #false() 570.12/150.35 , #eq(#neg(@x), #pos(@y)) -> #false() 570.12/150.35 , #eq(#neg(@x), #0()) -> #false() 570.12/150.35 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 570.12/150.35 , #eq(#s(@x), #0()) -> #false() 570.12/150.35 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 570.12/150.35 , leq#2(nil(), @x, @xs) -> #false() 570.12/150.35 , leq#2(::(@y, @ys), @x, @xs) -> 570.12/150.35 or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))) 570.12/150.35 , or(@x, @y) -> #or(@x, @y) 570.12/150.35 , and(@x, @y) -> #and(@x, @y) 570.12/150.35 , isortlist(@l) -> isortlist#1(@l) 570.12/150.35 , leq#1(nil(), @l2) -> #true() 570.12/150.35 , leq#1(::(@x, @xs), @l2) -> leq#2(@l2, @x, @xs) 570.12/150.35 , #cklt(#EQ()) -> #false() 570.12/150.35 , #cklt(#LT()) -> #true() 570.12/150.35 , #cklt(#GT()) -> #false() 570.12/150.35 , insert(@x, @l) -> insert#1(@l, @x) 570.12/150.35 , insert#2(#true(), @x, @y, @ys) -> ::(@x, ::(@y, @ys)) 570.12/150.35 , insert#2(#false(), @x, @y, @ys) -> ::(@y, insert(@x, @ys)) 570.12/150.35 , #and(#true(), #true()) -> #true() 570.12/150.35 , #and(#true(), #false()) -> #false() 570.12/150.35 , #and(#false(), #true()) -> #false() 570.12/150.35 , #and(#false(), #false()) -> #false() 570.12/150.35 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 570.12/150.35 , #compare(#pos(@x), #0()) -> #GT() 570.12/150.35 , #compare(#pos(@x), #neg(@y)) -> #GT() 570.12/150.35 , #compare(#0(), #pos(@y)) -> #LT() 570.12/150.35 , #compare(#0(), #0()) -> #EQ() 570.12/150.35 , #compare(#0(), #neg(@y)) -> #GT() 570.12/150.35 , #compare(#0(), #s(@y)) -> #LT() 570.12/150.35 , #compare(#neg(@x), #pos(@y)) -> #LT() 570.12/150.35 , #compare(#neg(@x), #0()) -> #LT() 570.12/150.35 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 570.12/150.35 , #compare(#s(@x), #0()) -> #GT() 570.12/150.35 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 570.12/150.35 , leq(@l1, @l2) -> leq#1(@l1, @l2) 570.12/150.35 , isortlist#1(nil()) -> nil() 570.12/150.35 , isortlist#1(::(@x, @xs)) -> insert(@x, isortlist(@xs)) 570.12/150.35 , #or(#true(), #true()) -> #true() 570.12/150.35 , #or(#true(), #false()) -> #true() 570.12/150.35 , #or(#false(), #true()) -> #true() 570.12/150.35 , #or(#false(), #false()) -> #false() 570.12/150.35 , insert#1(nil(), @x) -> ::(@x, nil()) 570.12/150.35 , insert#1(::(@y, @ys), @x) -> insert#2(leq(@x, @y), @x, @y, @ys) 570.12/150.35 , #less(@x, @y) -> #cklt(#compare(@x, @y)) } 570.12/150.35 Obligation: 570.12/150.35 innermost runtime complexity 570.12/150.35 Answer: 570.12/150.35 YES(O(1),O(n^2)) 570.12/150.35 570.12/150.35 We decompose the input problem according to the dependency graph 570.12/150.35 into the upper component 570.12/150.35 570.12/150.35 { isortlist^#(@l) -> c_3(isortlist#1^#(@l)) 570.12/150.35 , isortlist#1^#(::(@x, @xs)) -> 570.12/150.35 c_4(insert^#(@x, isortlist(@xs)), isortlist^#(@xs)) } 570.12/150.35 570.12/150.35 and lower component 570.12/150.35 570.12/150.35 { leq#2^#(::(@y, @ys), @x, @xs) -> c_1(leq^#(@xs, @ys)) 570.12/150.35 , leq^#(@l1, @l2) -> c_2(leq#1^#(@l1, @l2)) 570.12/150.35 , leq#1^#(::(@x, @xs), @l2) -> c_5(leq#2^#(@l2, @x, @xs)) 570.12/150.35 , insert^#(@x, @l) -> c_6(insert#1^#(@l, @x)) 570.12/150.35 , insert#1^#(::(@y, @ys), @x) -> 570.12/150.35 c_7(insert#2^#(leq(@x, @y), @x, @y, @ys), leq^#(@x, @y)) 570.12/150.35 , insert#2^#(#false(), @x, @y, @ys) -> c_8(insert^#(@x, @ys)) } 570.12/150.35 570.12/150.35 Further, following extension rules are added to the lower 570.12/150.35 component. 570.12/150.35 570.12/150.35 { isortlist^#(@l) -> isortlist#1^#(@l) 570.12/150.35 , isortlist#1^#(::(@x, @xs)) -> isortlist^#(@xs) 570.12/150.35 , isortlist#1^#(::(@x, @xs)) -> insert^#(@x, isortlist(@xs)) } 570.12/150.35 570.12/150.35 TcT solves the upper component with certificate YES(O(1),O(n^1)). 570.12/150.35 570.12/150.35 Sub-proof: 570.12/150.35 ---------- 570.12/150.35 We are left with following problem, upon which TcT provides the 570.12/150.35 certificate YES(O(1),O(n^1)). 570.12/150.35 570.12/150.35 Strict DPs: 570.12/150.35 { isortlist^#(@l) -> c_3(isortlist#1^#(@l)) 570.12/150.35 , isortlist#1^#(::(@x, @xs)) -> 570.12/150.35 c_4(insert^#(@x, isortlist(@xs)), isortlist^#(@xs)) } 570.12/150.35 Weak Trs: 570.12/150.35 { #equal(@x, @y) -> #eq(@x, @y) 570.12/150.35 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 570.12/150.35 , #eq(#pos(@x), #0()) -> #false() 570.12/150.35 , #eq(#pos(@x), #neg(@y)) -> #false() 570.12/150.35 , #eq(nil(), nil()) -> #true() 570.12/150.35 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 570.12/150.35 , #eq(::(@x_1, @x_2), nil()) -> #false() 570.12/150.35 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 570.12/150.35 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 570.12/150.35 , #eq(#0(), #pos(@y)) -> #false() 570.12/150.35 , #eq(#0(), #0()) -> #true() 570.12/150.35 , #eq(#0(), #neg(@y)) -> #false() 570.12/150.35 , #eq(#0(), #s(@y)) -> #false() 570.12/150.35 , #eq(#neg(@x), #pos(@y)) -> #false() 570.12/150.35 , #eq(#neg(@x), #0()) -> #false() 570.12/150.35 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 570.12/150.35 , #eq(#s(@x), #0()) -> #false() 570.12/150.35 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 570.12/150.35 , leq#2(nil(), @x, @xs) -> #false() 570.12/150.35 , leq#2(::(@y, @ys), @x, @xs) -> 570.12/150.35 or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))) 570.12/150.35 , or(@x, @y) -> #or(@x, @y) 570.12/150.35 , and(@x, @y) -> #and(@x, @y) 570.12/150.35 , isortlist(@l) -> isortlist#1(@l) 570.12/150.35 , leq#1(nil(), @l2) -> #true() 570.12/150.35 , leq#1(::(@x, @xs), @l2) -> leq#2(@l2, @x, @xs) 570.12/150.35 , #cklt(#EQ()) -> #false() 570.12/150.35 , #cklt(#LT()) -> #true() 570.12/150.35 , #cklt(#GT()) -> #false() 570.12/150.35 , insert(@x, @l) -> insert#1(@l, @x) 570.12/150.35 , insert#2(#true(), @x, @y, @ys) -> ::(@x, ::(@y, @ys)) 570.12/150.35 , insert#2(#false(), @x, @y, @ys) -> ::(@y, insert(@x, @ys)) 570.12/150.35 , #and(#true(), #true()) -> #true() 570.12/150.35 , #and(#true(), #false()) -> #false() 570.12/150.35 , #and(#false(), #true()) -> #false() 570.12/150.35 , #and(#false(), #false()) -> #false() 570.12/150.35 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 570.12/150.35 , #compare(#pos(@x), #0()) -> #GT() 570.12/150.35 , #compare(#pos(@x), #neg(@y)) -> #GT() 570.12/150.35 , #compare(#0(), #pos(@y)) -> #LT() 570.12/150.35 , #compare(#0(), #0()) -> #EQ() 570.12/150.35 , #compare(#0(), #neg(@y)) -> #GT() 570.12/150.35 , #compare(#0(), #s(@y)) -> #LT() 570.12/150.35 , #compare(#neg(@x), #pos(@y)) -> #LT() 570.12/150.35 , #compare(#neg(@x), #0()) -> #LT() 570.12/150.35 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 570.12/150.35 , #compare(#s(@x), #0()) -> #GT() 570.12/150.35 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 570.12/150.35 , leq(@l1, @l2) -> leq#1(@l1, @l2) 570.12/150.35 , isortlist#1(nil()) -> nil() 570.12/150.35 , isortlist#1(::(@x, @xs)) -> insert(@x, isortlist(@xs)) 570.12/150.35 , #or(#true(), #true()) -> #true() 570.12/150.35 , #or(#true(), #false()) -> #true() 570.12/150.35 , #or(#false(), #true()) -> #true() 570.12/150.35 , #or(#false(), #false()) -> #false() 570.12/150.35 , insert#1(nil(), @x) -> ::(@x, nil()) 570.12/150.35 , insert#1(::(@y, @ys), @x) -> insert#2(leq(@x, @y), @x, @y, @ys) 570.12/150.35 , #less(@x, @y) -> #cklt(#compare(@x, @y)) } 570.12/150.35 Obligation: 570.12/150.35 innermost runtime complexity 570.12/150.35 Answer: 570.12/150.35 YES(O(1),O(n^1)) 570.12/150.35 570.12/150.35 We use the processor 'matrix interpretation of dimension 1' to 570.12/150.35 orient following rules strictly. 570.12/150.35 570.12/150.35 DPs: 570.12/150.35 { 2: isortlist#1^#(::(@x, @xs)) -> 570.12/150.35 c_4(insert^#(@x, isortlist(@xs)), isortlist^#(@xs)) } 570.12/150.35 570.12/150.35 Sub-proof: 570.12/150.35 ---------- 570.12/150.35 The following argument positions are usable: 570.12/150.35 Uargs(c_3) = {1}, Uargs(c_4) = {1, 2} 570.12/150.35 570.12/150.35 TcT has computed the following constructor-based matrix 570.12/150.35 interpretation satisfying not(EDA). 570.12/150.35 570.12/150.35 [#equal](x1, x2) = [0] 570.12/150.35 570.12/150.35 [#eq](x1, x2) = [0] 570.12/150.35 570.12/150.35 [leq#2](x1, x2, x3) = [7] x2 + [7] x3 + [0] 570.12/150.35 570.12/150.35 [or](x1, x2) = [0] 570.12/150.35 570.12/150.35 [#true] = [0] 570.12/150.35 570.12/150.35 [and](x1, x2) = [0] 570.12/150.35 570.12/150.35 [isortlist](x1) = [0] 570.12/150.35 570.12/150.35 [leq#1](x1, x2) = [4] x2 + [0] 570.12/150.35 570.12/150.35 [#cklt](x1) = [0] 570.12/150.35 570.12/150.35 [insert](x1, x2) = [0] 570.12/150.35 570.12/150.35 [#pos](x1) = [1] x1 + [0] 570.12/150.35 570.12/150.35 [#EQ] = [0] 570.12/150.35 570.12/150.35 [insert#2](x1, x2, x3, x4) = [1] x2 + [1] x3 + [4] x4 + [0] 570.12/150.35 570.12/150.35 [#and](x1, x2) = [0] 570.12/150.35 570.12/150.35 [#compare](x1, x2) = [0] 570.12/150.35 570.12/150.35 [nil] = [0] 570.12/150.35 570.12/150.35 [leq](x1, x2) = [0] 570.12/150.35 570.12/150.35 [#false] = [0] 570.12/150.35 570.12/150.35 [::](x1, x2) = [1] x2 + [2] 570.12/150.35 570.12/150.35 [isortlist#1](x1) = [0] 570.12/150.35 570.12/150.35 [#LT] = [0] 570.12/150.35 570.12/150.35 [#or](x1, x2) = [0] 570.12/150.35 570.12/150.35 [insert#1](x1, x2) = [7] x2 + [0] 570.12/150.35 570.12/150.35 [#0] = [0] 570.12/150.35 570.12/150.35 [#neg](x1) = [1] x1 + [0] 570.12/150.35 570.12/150.35 [#less](x1, x2) = [0] 570.12/150.35 570.12/150.35 [#s](x1) = [1] x1 + [0] 570.12/150.35 570.12/150.35 [#GT] = [0] 570.12/150.35 570.12/150.35 [isortlist^#](x1) = [1] x1 + [0] 570.12/150.35 570.12/150.35 [isortlist#1^#](x1) = [1] x1 + [0] 570.12/150.35 570.12/150.35 [insert^#](x1, x2) = [0] 570.12/150.35 570.12/150.35 [c_3](x1) = [1] x1 + [0] 570.12/150.35 570.12/150.35 [c_4](x1, x2) = [2] x1 + [1] x2 + [1] 570.12/150.35 570.12/150.35 The order satisfies the following ordering constraints: 570.12/150.35 570.12/150.35 [#equal(@x, @y)] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#eq(@x, @y)] 570.12/150.35 570.12/150.35 [#eq(#pos(@x), #pos(@y))] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#eq(@x, @y)] 570.12/150.35 570.12/150.35 [#eq(#pos(@x), #0())] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#false()] 570.12/150.35 570.12/150.35 [#eq(#pos(@x), #neg(@y))] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#false()] 570.12/150.35 570.12/150.35 [#eq(nil(), nil())] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#true()] 570.12/150.35 570.12/150.35 [#eq(nil(), ::(@y_1, @y_2))] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#false()] 570.12/150.35 570.12/150.35 [#eq(::(@x_1, @x_2), nil())] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#false()] 570.12/150.35 570.12/150.35 [#eq(::(@x_1, @x_2), ::(@y_1, @y_2))] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))] 570.12/150.35 570.12/150.35 [#eq(#0(), #pos(@y))] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#false()] 570.12/150.35 570.12/150.35 [#eq(#0(), #0())] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#true()] 570.12/150.35 570.12/150.35 [#eq(#0(), #neg(@y))] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#false()] 570.12/150.35 570.12/150.35 [#eq(#0(), #s(@y))] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#false()] 570.12/150.35 570.12/150.35 [#eq(#neg(@x), #pos(@y))] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#false()] 570.12/150.35 570.12/150.35 [#eq(#neg(@x), #0())] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#false()] 570.12/150.35 570.12/150.35 [#eq(#neg(@x), #neg(@y))] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#eq(@x, @y)] 570.12/150.35 570.12/150.35 [#eq(#s(@x), #0())] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#false()] 570.12/150.35 570.12/150.35 [#eq(#s(@x), #s(@y))] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#eq(@x, @y)] 570.12/150.35 570.12/150.35 [leq#2(nil(), @x, @xs)] = [7] @x + [7] @xs + [0] 570.12/150.35 >= [0] 570.12/150.35 = [#false()] 570.12/150.35 570.12/150.35 [leq#2(::(@y, @ys), @x, @xs)] = [7] @x + [7] @xs + [0] 570.12/150.35 >= [0] 570.12/150.35 = [or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys)))] 570.12/150.35 570.12/150.35 [or(@x, @y)] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#or(@x, @y)] 570.12/150.35 570.12/150.35 [and(@x, @y)] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#and(@x, @y)] 570.12/150.35 570.12/150.35 [isortlist(@l)] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [isortlist#1(@l)] 570.12/150.35 570.12/150.35 [leq#1(nil(), @l2)] = [4] @l2 + [0] 570.12/150.35 >= [0] 570.12/150.35 = [#true()] 570.12/150.35 570.12/150.35 [leq#1(::(@x, @xs), @l2)] = [4] @l2 + [0] 570.12/150.35 ? [7] @x + [7] @xs + [0] 570.12/150.35 = [leq#2(@l2, @x, @xs)] 570.12/150.35 570.12/150.35 [#cklt(#EQ())] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#false()] 570.12/150.35 570.12/150.35 [#cklt(#LT())] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#true()] 570.12/150.35 570.12/150.35 [#cklt(#GT())] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#false()] 570.12/150.35 570.12/150.35 [insert(@x, @l)] = [0] 570.12/150.35 ? [7] @x + [0] 570.12/150.35 = [insert#1(@l, @x)] 570.12/150.35 570.12/150.35 [insert#2(#true(), @x, @y, @ys)] = [1] @x + [1] @y + [4] @ys + [0] 570.12/150.35 ? [1] @ys + [4] 570.12/150.35 = [::(@x, ::(@y, @ys))] 570.12/150.35 570.12/150.35 [insert#2(#false(), @x, @y, @ys)] = [1] @x + [1] @y + [4] @ys + [0] 570.12/150.35 ? [2] 570.12/150.35 = [::(@y, insert(@x, @ys))] 570.12/150.35 570.12/150.35 [#and(#true(), #true())] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#true()] 570.12/150.35 570.12/150.35 [#and(#true(), #false())] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#false()] 570.12/150.35 570.12/150.35 [#and(#false(), #true())] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#false()] 570.12/150.35 570.12/150.35 [#and(#false(), #false())] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#false()] 570.12/150.35 570.12/150.35 [#compare(#pos(@x), #pos(@y))] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#compare(@x, @y)] 570.12/150.35 570.12/150.35 [#compare(#pos(@x), #0())] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#GT()] 570.12/150.35 570.12/150.35 [#compare(#pos(@x), #neg(@y))] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#GT()] 570.12/150.35 570.12/150.35 [#compare(#0(), #pos(@y))] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#LT()] 570.12/150.35 570.12/150.35 [#compare(#0(), #0())] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#EQ()] 570.12/150.35 570.12/150.35 [#compare(#0(), #neg(@y))] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#GT()] 570.12/150.35 570.12/150.35 [#compare(#0(), #s(@y))] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#LT()] 570.12/150.35 570.12/150.35 [#compare(#neg(@x), #pos(@y))] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#LT()] 570.12/150.35 570.12/150.35 [#compare(#neg(@x), #0())] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#LT()] 570.12/150.35 570.12/150.35 [#compare(#neg(@x), #neg(@y))] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#compare(@y, @x)] 570.12/150.35 570.12/150.35 [#compare(#s(@x), #0())] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#GT()] 570.12/150.35 570.12/150.35 [#compare(#s(@x), #s(@y))] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#compare(@x, @y)] 570.12/150.35 570.12/150.35 [leq(@l1, @l2)] = [0] 570.12/150.35 ? [4] @l2 + [0] 570.12/150.35 = [leq#1(@l1, @l2)] 570.12/150.35 570.12/150.35 [isortlist#1(nil())] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [nil()] 570.12/150.35 570.12/150.35 [isortlist#1(::(@x, @xs))] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [insert(@x, isortlist(@xs))] 570.12/150.35 570.12/150.35 [#or(#true(), #true())] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#true()] 570.12/150.35 570.12/150.35 [#or(#true(), #false())] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#true()] 570.12/150.35 570.12/150.35 [#or(#false(), #true())] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#true()] 570.12/150.35 570.12/150.35 [#or(#false(), #false())] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#false()] 570.12/150.35 570.12/150.35 [insert#1(nil(), @x)] = [7] @x + [0] 570.12/150.35 ? [2] 570.12/150.35 = [::(@x, nil())] 570.12/150.35 570.12/150.35 [insert#1(::(@y, @ys), @x)] = [7] @x + [0] 570.12/150.35 ? [1] @x + [1] @y + [4] @ys + [0] 570.12/150.35 = [insert#2(leq(@x, @y), @x, @y, @ys)] 570.12/150.35 570.12/150.35 [#less(@x, @y)] = [0] 570.12/150.35 >= [0] 570.12/150.35 = [#cklt(#compare(@x, @y))] 570.12/150.35 570.12/150.35 [isortlist^#(@l)] = [1] @l + [0] 570.12/150.35 >= [1] @l + [0] 570.12/150.35 = [c_3(isortlist#1^#(@l))] 570.12/150.35 570.12/150.35 [isortlist#1^#(::(@x, @xs))] = [1] @xs + [2] 570.12/150.35 > [1] @xs + [1] 570.12/150.35 = [c_4(insert^#(@x, isortlist(@xs)), isortlist^#(@xs))] 570.12/150.35 570.12/150.35 570.12/150.35 We return to the main proof. Consider the set of all dependency 570.12/150.35 pairs 570.12/150.35 570.12/150.35 : 570.12/150.35 { 1: isortlist^#(@l) -> c_3(isortlist#1^#(@l)) 570.12/150.35 , 2: isortlist#1^#(::(@x, @xs)) -> 570.12/150.35 c_4(insert^#(@x, isortlist(@xs)), isortlist^#(@xs)) } 570.12/150.35 570.12/150.35 Processor 'matrix interpretation of dimension 1' induces the 570.12/150.35 complexity certificate YES(?,O(n^1)) on application of dependency 570.12/150.35 pairs {2}. These cover all (indirect) predecessors of dependency 570.12/150.35 pairs {1,2}, their number of application is equally bounded. The 570.12/150.35 dependency pairs are shifted into the weak component. 570.12/150.35 570.12/150.35 We are left with following problem, upon which TcT provides the 570.12/150.35 certificate YES(O(1),O(1)). 570.12/150.35 570.12/150.35 Weak DPs: 570.12/150.35 { isortlist^#(@l) -> c_3(isortlist#1^#(@l)) 570.12/150.35 , isortlist#1^#(::(@x, @xs)) -> 570.12/150.35 c_4(insert^#(@x, isortlist(@xs)), isortlist^#(@xs)) } 570.12/150.35 Weak Trs: 570.12/150.35 { #equal(@x, @y) -> #eq(@x, @y) 570.12/150.35 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 570.12/150.35 , #eq(#pos(@x), #0()) -> #false() 570.12/150.35 , #eq(#pos(@x), #neg(@y)) -> #false() 570.12/150.35 , #eq(nil(), nil()) -> #true() 570.12/150.35 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 570.12/150.35 , #eq(::(@x_1, @x_2), nil()) -> #false() 570.12/150.35 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 570.12/150.35 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 570.12/150.35 , #eq(#0(), #pos(@y)) -> #false() 570.12/150.35 , #eq(#0(), #0()) -> #true() 570.12/150.35 , #eq(#0(), #neg(@y)) -> #false() 570.12/150.35 , #eq(#0(), #s(@y)) -> #false() 570.12/150.35 , #eq(#neg(@x), #pos(@y)) -> #false() 570.12/150.35 , #eq(#neg(@x), #0()) -> #false() 570.12/150.35 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 570.12/150.35 , #eq(#s(@x), #0()) -> #false() 570.12/150.35 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 570.12/150.35 , leq#2(nil(), @x, @xs) -> #false() 570.12/150.35 , leq#2(::(@y, @ys), @x, @xs) -> 570.12/150.35 or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))) 570.12/150.35 , or(@x, @y) -> #or(@x, @y) 570.12/150.35 , and(@x, @y) -> #and(@x, @y) 570.12/150.35 , isortlist(@l) -> isortlist#1(@l) 570.12/150.35 , leq#1(nil(), @l2) -> #true() 570.12/150.35 , leq#1(::(@x, @xs), @l2) -> leq#2(@l2, @x, @xs) 570.12/150.35 , #cklt(#EQ()) -> #false() 570.12/150.35 , #cklt(#LT()) -> #true() 570.12/150.35 , #cklt(#GT()) -> #false() 570.12/150.35 , insert(@x, @l) -> insert#1(@l, @x) 570.12/150.35 , insert#2(#true(), @x, @y, @ys) -> ::(@x, ::(@y, @ys)) 570.12/150.35 , insert#2(#false(), @x, @y, @ys) -> ::(@y, insert(@x, @ys)) 570.12/150.35 , #and(#true(), #true()) -> #true() 570.12/150.35 , #and(#true(), #false()) -> #false() 570.12/150.35 , #and(#false(), #true()) -> #false() 570.12/150.35 , #and(#false(), #false()) -> #false() 570.12/150.35 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 570.12/150.35 , #compare(#pos(@x), #0()) -> #GT() 570.12/150.35 , #compare(#pos(@x), #neg(@y)) -> #GT() 570.12/150.35 , #compare(#0(), #pos(@y)) -> #LT() 570.12/150.35 , #compare(#0(), #0()) -> #EQ() 570.12/150.35 , #compare(#0(), #neg(@y)) -> #GT() 570.12/150.35 , #compare(#0(), #s(@y)) -> #LT() 570.12/150.35 , #compare(#neg(@x), #pos(@y)) -> #LT() 570.12/150.35 , #compare(#neg(@x), #0()) -> #LT() 570.12/150.35 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 570.12/150.35 , #compare(#s(@x), #0()) -> #GT() 570.12/150.35 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 570.12/150.35 , leq(@l1, @l2) -> leq#1(@l1, @l2) 570.12/150.35 , isortlist#1(nil()) -> nil() 570.12/150.35 , isortlist#1(::(@x, @xs)) -> insert(@x, isortlist(@xs)) 570.12/150.35 , #or(#true(), #true()) -> #true() 570.12/150.35 , #or(#true(), #false()) -> #true() 570.12/150.35 , #or(#false(), #true()) -> #true() 570.12/150.35 , #or(#false(), #false()) -> #false() 570.12/150.35 , insert#1(nil(), @x) -> ::(@x, nil()) 570.12/150.35 , insert#1(::(@y, @ys), @x) -> insert#2(leq(@x, @y), @x, @y, @ys) 570.12/150.35 , #less(@x, @y) -> #cklt(#compare(@x, @y)) } 570.12/150.35 Obligation: 570.12/150.35 innermost runtime complexity 570.12/150.35 Answer: 570.12/150.35 YES(O(1),O(1)) 570.12/150.35 570.12/150.35 The following weak DPs constitute a sub-graph of the DG that is 570.12/150.35 closed under successors. The DPs are removed. 570.12/150.35 570.12/150.35 { isortlist^#(@l) -> c_3(isortlist#1^#(@l)) 570.12/150.35 , isortlist#1^#(::(@x, @xs)) -> 570.12/150.35 c_4(insert^#(@x, isortlist(@xs)), isortlist^#(@xs)) } 570.12/150.35 570.12/150.35 We are left with following problem, upon which TcT provides the 570.12/150.35 certificate YES(O(1),O(1)). 570.12/150.35 570.12/150.35 Weak Trs: 570.12/150.35 { #equal(@x, @y) -> #eq(@x, @y) 570.12/150.35 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 570.12/150.35 , #eq(#pos(@x), #0()) -> #false() 570.12/150.35 , #eq(#pos(@x), #neg(@y)) -> #false() 570.12/150.35 , #eq(nil(), nil()) -> #true() 570.12/150.35 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 570.12/150.35 , #eq(::(@x_1, @x_2), nil()) -> #false() 570.12/150.35 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 570.12/150.35 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 570.12/150.35 , #eq(#0(), #pos(@y)) -> #false() 570.12/150.35 , #eq(#0(), #0()) -> #true() 570.12/150.35 , #eq(#0(), #neg(@y)) -> #false() 570.12/150.35 , #eq(#0(), #s(@y)) -> #false() 570.12/150.36 , #eq(#neg(@x), #pos(@y)) -> #false() 570.12/150.36 , #eq(#neg(@x), #0()) -> #false() 570.12/150.36 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 570.12/150.36 , #eq(#s(@x), #0()) -> #false() 570.12/150.36 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 570.12/150.36 , leq#2(nil(), @x, @xs) -> #false() 570.12/150.36 , leq#2(::(@y, @ys), @x, @xs) -> 570.12/150.36 or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))) 570.12/150.36 , or(@x, @y) -> #or(@x, @y) 570.12/150.36 , and(@x, @y) -> #and(@x, @y) 570.12/150.36 , isortlist(@l) -> isortlist#1(@l) 570.12/150.36 , leq#1(nil(), @l2) -> #true() 570.12/150.36 , leq#1(::(@x, @xs), @l2) -> leq#2(@l2, @x, @xs) 570.12/150.36 , #cklt(#EQ()) -> #false() 570.12/150.36 , #cklt(#LT()) -> #true() 570.12/150.36 , #cklt(#GT()) -> #false() 570.12/150.36 , insert(@x, @l) -> insert#1(@l, @x) 570.12/150.36 , insert#2(#true(), @x, @y, @ys) -> ::(@x, ::(@y, @ys)) 570.12/150.36 , insert#2(#false(), @x, @y, @ys) -> ::(@y, insert(@x, @ys)) 570.12/150.36 , #and(#true(), #true()) -> #true() 570.12/150.36 , #and(#true(), #false()) -> #false() 570.12/150.36 , #and(#false(), #true()) -> #false() 570.12/150.36 , #and(#false(), #false()) -> #false() 570.12/150.36 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 570.12/150.36 , #compare(#pos(@x), #0()) -> #GT() 570.12/150.36 , #compare(#pos(@x), #neg(@y)) -> #GT() 570.12/150.36 , #compare(#0(), #pos(@y)) -> #LT() 570.12/150.36 , #compare(#0(), #0()) -> #EQ() 570.12/150.36 , #compare(#0(), #neg(@y)) -> #GT() 570.12/150.36 , #compare(#0(), #s(@y)) -> #LT() 570.12/150.36 , #compare(#neg(@x), #pos(@y)) -> #LT() 570.12/150.36 , #compare(#neg(@x), #0()) -> #LT() 570.12/150.36 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 570.12/150.36 , #compare(#s(@x), #0()) -> #GT() 570.12/150.36 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 570.12/150.36 , leq(@l1, @l2) -> leq#1(@l1, @l2) 570.12/150.36 , isortlist#1(nil()) -> nil() 570.12/150.36 , isortlist#1(::(@x, @xs)) -> insert(@x, isortlist(@xs)) 570.12/150.36 , #or(#true(), #true()) -> #true() 570.12/150.36 , #or(#true(), #false()) -> #true() 570.12/150.36 , #or(#false(), #true()) -> #true() 570.12/150.36 , #or(#false(), #false()) -> #false() 570.12/150.36 , insert#1(nil(), @x) -> ::(@x, nil()) 570.12/150.36 , insert#1(::(@y, @ys), @x) -> insert#2(leq(@x, @y), @x, @y, @ys) 570.12/150.36 , #less(@x, @y) -> #cklt(#compare(@x, @y)) } 570.12/150.36 Obligation: 570.12/150.36 innermost runtime complexity 570.12/150.36 Answer: 570.12/150.36 YES(O(1),O(1)) 570.12/150.36 570.12/150.36 No rule is usable, rules are removed from the input problem. 570.12/150.36 570.12/150.36 We are left with following problem, upon which TcT provides the 570.12/150.36 certificate YES(O(1),O(1)). 570.12/150.36 570.12/150.36 Rules: Empty 570.12/150.36 Obligation: 570.12/150.36 innermost runtime complexity 570.12/150.36 Answer: 570.12/150.36 YES(O(1),O(1)) 570.12/150.36 570.12/150.36 Empty rules are trivially bounded 570.12/150.36 570.12/150.36 We return to the main proof. 570.12/150.36 570.12/150.36 We are left with following problem, upon which TcT provides the 570.12/150.36 certificate YES(O(1),O(n^1)). 570.12/150.36 570.12/150.36 Strict DPs: 570.12/150.36 { leq#2^#(::(@y, @ys), @x, @xs) -> c_1(leq^#(@xs, @ys)) 570.12/150.36 , leq^#(@l1, @l2) -> c_2(leq#1^#(@l1, @l2)) 570.12/150.36 , leq#1^#(::(@x, @xs), @l2) -> c_5(leq#2^#(@l2, @x, @xs)) 570.12/150.36 , insert^#(@x, @l) -> c_6(insert#1^#(@l, @x)) 570.12/150.36 , insert#1^#(::(@y, @ys), @x) -> 570.12/150.36 c_7(insert#2^#(leq(@x, @y), @x, @y, @ys), leq^#(@x, @y)) 570.12/150.36 , insert#2^#(#false(), @x, @y, @ys) -> c_8(insert^#(@x, @ys)) } 570.12/150.36 Weak DPs: 570.12/150.36 { isortlist^#(@l) -> isortlist#1^#(@l) 570.12/150.36 , isortlist#1^#(::(@x, @xs)) -> isortlist^#(@xs) 570.12/150.36 , isortlist#1^#(::(@x, @xs)) -> insert^#(@x, isortlist(@xs)) } 570.12/150.36 Weak Trs: 570.12/150.36 { #equal(@x, @y) -> #eq(@x, @y) 570.12/150.36 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 570.12/150.36 , #eq(#pos(@x), #0()) -> #false() 570.12/150.36 , #eq(#pos(@x), #neg(@y)) -> #false() 570.12/150.36 , #eq(nil(), nil()) -> #true() 570.12/150.36 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 570.12/150.36 , #eq(::(@x_1, @x_2), nil()) -> #false() 570.12/150.36 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 570.12/150.36 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 570.12/150.36 , #eq(#0(), #pos(@y)) -> #false() 570.12/150.36 , #eq(#0(), #0()) -> #true() 570.12/150.36 , #eq(#0(), #neg(@y)) -> #false() 570.12/150.36 , #eq(#0(), #s(@y)) -> #false() 570.12/150.36 , #eq(#neg(@x), #pos(@y)) -> #false() 570.12/150.36 , #eq(#neg(@x), #0()) -> #false() 570.12/150.36 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 570.12/150.36 , #eq(#s(@x), #0()) -> #false() 570.12/150.36 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 570.12/150.36 , leq#2(nil(), @x, @xs) -> #false() 570.12/150.36 , leq#2(::(@y, @ys), @x, @xs) -> 570.12/150.36 or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))) 570.12/150.36 , or(@x, @y) -> #or(@x, @y) 570.12/150.36 , and(@x, @y) -> #and(@x, @y) 570.12/150.36 , isortlist(@l) -> isortlist#1(@l) 570.12/150.36 , leq#1(nil(), @l2) -> #true() 570.12/150.36 , leq#1(::(@x, @xs), @l2) -> leq#2(@l2, @x, @xs) 570.12/150.36 , #cklt(#EQ()) -> #false() 570.12/150.36 , #cklt(#LT()) -> #true() 570.12/150.36 , #cklt(#GT()) -> #false() 570.12/150.36 , insert(@x, @l) -> insert#1(@l, @x) 570.12/150.36 , insert#2(#true(), @x, @y, @ys) -> ::(@x, ::(@y, @ys)) 570.12/150.36 , insert#2(#false(), @x, @y, @ys) -> ::(@y, insert(@x, @ys)) 570.12/150.36 , #and(#true(), #true()) -> #true() 570.12/150.36 , #and(#true(), #false()) -> #false() 570.12/150.36 , #and(#false(), #true()) -> #false() 570.12/150.36 , #and(#false(), #false()) -> #false() 570.12/150.36 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 570.12/150.36 , #compare(#pos(@x), #0()) -> #GT() 570.12/150.36 , #compare(#pos(@x), #neg(@y)) -> #GT() 570.12/150.36 , #compare(#0(), #pos(@y)) -> #LT() 570.12/150.36 , #compare(#0(), #0()) -> #EQ() 570.12/150.36 , #compare(#0(), #neg(@y)) -> #GT() 570.12/150.36 , #compare(#0(), #s(@y)) -> #LT() 570.12/150.36 , #compare(#neg(@x), #pos(@y)) -> #LT() 570.12/150.36 , #compare(#neg(@x), #0()) -> #LT() 570.12/150.36 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 570.12/150.36 , #compare(#s(@x), #0()) -> #GT() 570.12/150.36 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 570.12/150.36 , leq(@l1, @l2) -> leq#1(@l1, @l2) 570.12/150.36 , isortlist#1(nil()) -> nil() 570.12/150.36 , isortlist#1(::(@x, @xs)) -> insert(@x, isortlist(@xs)) 570.12/150.36 , #or(#true(), #true()) -> #true() 570.12/150.36 , #or(#true(), #false()) -> #true() 570.12/150.36 , #or(#false(), #true()) -> #true() 570.12/150.36 , #or(#false(), #false()) -> #false() 570.12/150.36 , insert#1(nil(), @x) -> ::(@x, nil()) 570.12/150.36 , insert#1(::(@y, @ys), @x) -> insert#2(leq(@x, @y), @x, @y, @ys) 570.12/150.36 , #less(@x, @y) -> #cklt(#compare(@x, @y)) } 570.12/150.36 Obligation: 570.12/150.36 innermost runtime complexity 570.12/150.36 Answer: 570.12/150.36 YES(O(1),O(n^1)) 570.12/150.36 570.12/150.36 We use the processor 'matrix interpretation of dimension 1' to 570.12/150.36 orient following rules strictly. 570.12/150.36 570.12/150.36 DPs: 570.12/150.36 { 1: leq#2^#(::(@y, @ys), @x, @xs) -> c_1(leq^#(@xs, @ys)) 570.12/150.36 , 5: insert#1^#(::(@y, @ys), @x) -> 570.12/150.36 c_7(insert#2^#(leq(@x, @y), @x, @y, @ys), leq^#(@x, @y)) 570.12/150.36 , 7: isortlist^#(@l) -> isortlist#1^#(@l) 570.12/150.36 , 8: isortlist#1^#(::(@x, @xs)) -> isortlist^#(@xs) 570.12/150.36 , 9: isortlist#1^#(::(@x, @xs)) -> insert^#(@x, isortlist(@xs)) } 570.12/150.36 570.12/150.36 Sub-proof: 570.12/150.36 ---------- 570.12/150.36 The following argument positions are usable: 570.12/150.36 Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_5) = {1}, 570.12/150.36 Uargs(c_6) = {1}, Uargs(c_7) = {1, 2}, Uargs(c_8) = {1} 570.12/150.36 570.12/150.36 TcT has computed the following constructor-based matrix 570.12/150.36 interpretation satisfying not(EDA). 570.12/150.36 570.12/150.36 [#equal](x1, x2) = [0] 570.12/150.36 570.12/150.36 [#eq](x1, x2) = [0] 570.12/150.36 570.12/150.36 [leq#2](x1, x2, x3) = [7] x2 + [7] x3 + [0] 570.12/150.36 570.12/150.36 [or](x1, x2) = [0] 570.12/150.36 570.12/150.36 [#true] = [0] 570.12/150.36 570.12/150.36 [and](x1, x2) = [0] 570.12/150.36 570.12/150.36 [isortlist](x1) = [1] x1 + [0] 570.12/150.36 570.12/150.36 [leq#1](x1, x2) = [4] x2 + [0] 570.12/150.36 570.12/150.36 [#cklt](x1) = [0] 570.12/150.36 570.12/150.36 [insert](x1, x2) = [1] x1 + [1] x2 + [1] 570.12/150.36 570.12/150.36 [#pos](x1) = [1] x1 + [0] 570.12/150.36 570.12/150.36 [#EQ] = [0] 570.12/150.36 570.12/150.36 [insert#2](x1, x2, x3, x4) = [1] x2 + [1] x3 + [1] x4 + [2] 570.12/150.36 570.12/150.36 [#and](x1, x2) = [0] 570.12/150.36 570.12/150.36 [#compare](x1, x2) = [0] 570.12/150.36 570.12/150.36 [nil] = [0] 570.12/150.36 570.12/150.36 [leq](x1, x2) = [0] 570.12/150.36 570.12/150.36 [#false] = [0] 570.12/150.36 570.12/150.36 [::](x1, x2) = [1] x1 + [1] x2 + [1] 570.12/150.36 570.12/150.36 [isortlist#1](x1) = [1] x1 + [0] 570.12/150.36 570.12/150.36 [#LT] = [0] 570.12/150.36 570.12/150.36 [#or](x1, x2) = [0] 570.12/150.36 570.12/150.36 [insert#1](x1, x2) = [1] x1 + [1] x2 + [1] 570.12/150.36 570.12/150.36 [#0] = [0] 570.12/150.36 570.12/150.36 [#neg](x1) = [1] x1 + [0] 570.12/150.36 570.12/150.36 [#less](x1, x2) = [0] 570.12/150.36 570.12/150.36 [#s](x1) = [1] x1 + [0] 570.12/150.36 570.12/150.36 [#GT] = [0] 570.12/150.36 570.12/150.36 [leq#2^#](x1, x2, x3) = [2] x1 + [0] 570.12/150.36 570.12/150.36 [leq^#](x1, x2) = [2] x2 + [0] 570.12/150.36 570.12/150.36 [isortlist^#](x1) = [6] x1 + [7] 570.12/150.36 570.12/150.36 [isortlist#1^#](x1) = [6] x1 + [3] 570.12/150.36 570.12/150.36 [leq#1^#](x1, x2) = [2] x2 + [0] 570.12/150.36 570.12/150.36 [insert^#](x1, x2) = [4] x2 + [0] 570.12/150.36 570.12/150.36 [insert#1^#](x1, x2) = [4] x1 + [0] 570.12/150.36 570.12/150.36 [insert#2^#](x1, x2, x3, x4) = [4] x4 + [0] 570.12/150.36 570.12/150.36 [c_1](x1) = [1] x1 + [1] 570.12/150.36 570.12/150.36 [c_2](x1) = [1] x1 + [0] 570.12/150.36 570.12/150.36 [c_5](x1) = [1] x1 + [0] 570.12/150.36 570.12/150.36 [c_6](x1) = [1] x1 + [0] 570.12/150.36 570.12/150.36 [c_7](x1, x2) = [1] x1 + [2] x2 + [3] 570.12/150.36 570.12/150.36 [c_8](x1) = [1] x1 + [0] 570.12/150.36 570.12/150.36 The order satisfies the following ordering constraints: 570.12/150.36 570.12/150.36 [#equal(@x, @y)] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#eq(@x, @y)] 570.12/150.36 570.12/150.36 [#eq(#pos(@x), #pos(@y))] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#eq(@x, @y)] 570.12/150.36 570.12/150.36 [#eq(#pos(@x), #0())] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#false()] 570.12/150.36 570.12/150.36 [#eq(#pos(@x), #neg(@y))] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#false()] 570.12/150.36 570.12/150.36 [#eq(nil(), nil())] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#true()] 570.12/150.36 570.12/150.36 [#eq(nil(), ::(@y_1, @y_2))] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#false()] 570.12/150.36 570.12/150.36 [#eq(::(@x_1, @x_2), nil())] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#false()] 570.12/150.36 570.12/150.36 [#eq(::(@x_1, @x_2), ::(@y_1, @y_2))] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))] 570.12/150.36 570.12/150.36 [#eq(#0(), #pos(@y))] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#false()] 570.12/150.36 570.12/150.36 [#eq(#0(), #0())] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#true()] 570.12/150.36 570.12/150.36 [#eq(#0(), #neg(@y))] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#false()] 570.12/150.36 570.12/150.36 [#eq(#0(), #s(@y))] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#false()] 570.12/150.36 570.12/150.36 [#eq(#neg(@x), #pos(@y))] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#false()] 570.12/150.36 570.12/150.36 [#eq(#neg(@x), #0())] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#false()] 570.12/150.36 570.12/150.36 [#eq(#neg(@x), #neg(@y))] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#eq(@x, @y)] 570.12/150.36 570.12/150.36 [#eq(#s(@x), #0())] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#false()] 570.12/150.36 570.12/150.36 [#eq(#s(@x), #s(@y))] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#eq(@x, @y)] 570.12/150.36 570.12/150.36 [leq#2(nil(), @x, @xs)] = [7] @x + [7] @xs + [0] 570.12/150.36 >= [0] 570.12/150.36 = [#false()] 570.12/150.36 570.12/150.36 [leq#2(::(@y, @ys), @x, @xs)] = [7] @x + [7] @xs + [0] 570.12/150.36 >= [0] 570.12/150.36 = [or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys)))] 570.12/150.36 570.12/150.36 [or(@x, @y)] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#or(@x, @y)] 570.12/150.36 570.12/150.36 [and(@x, @y)] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#and(@x, @y)] 570.12/150.36 570.12/150.36 [isortlist(@l)] = [1] @l + [0] 570.12/150.36 >= [1] @l + [0] 570.12/150.36 = [isortlist#1(@l)] 570.12/150.36 570.12/150.36 [leq#1(nil(), @l2)] = [4] @l2 + [0] 570.12/150.36 >= [0] 570.12/150.36 = [#true()] 570.12/150.36 570.12/150.36 [leq#1(::(@x, @xs), @l2)] = [4] @l2 + [0] 570.12/150.36 ? [7] @x + [7] @xs + [0] 570.12/150.36 = [leq#2(@l2, @x, @xs)] 570.12/150.36 570.12/150.36 [#cklt(#EQ())] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#false()] 570.12/150.36 570.12/150.36 [#cklt(#LT())] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#true()] 570.12/150.36 570.12/150.36 [#cklt(#GT())] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#false()] 570.12/150.36 570.12/150.36 [insert(@x, @l)] = [1] @x + [1] @l + [1] 570.12/150.36 >= [1] @x + [1] @l + [1] 570.12/150.36 = [insert#1(@l, @x)] 570.12/150.36 570.12/150.36 [insert#2(#true(), @x, @y, @ys)] = [1] @x + [1] @y + [1] @ys + [2] 570.12/150.36 >= [1] @x + [1] @y + [1] @ys + [2] 570.12/150.36 = [::(@x, ::(@y, @ys))] 570.12/150.36 570.12/150.36 [insert#2(#false(), @x, @y, @ys)] = [1] @x + [1] @y + [1] @ys + [2] 570.12/150.36 >= [1] @x + [1] @y + [1] @ys + [2] 570.12/150.36 = [::(@y, insert(@x, @ys))] 570.12/150.36 570.12/150.36 [#and(#true(), #true())] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#true()] 570.12/150.36 570.12/150.36 [#and(#true(), #false())] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#false()] 570.12/150.36 570.12/150.36 [#and(#false(), #true())] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#false()] 570.12/150.36 570.12/150.36 [#and(#false(), #false())] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#false()] 570.12/150.36 570.12/150.36 [#compare(#pos(@x), #pos(@y))] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#compare(@x, @y)] 570.12/150.36 570.12/150.36 [#compare(#pos(@x), #0())] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#GT()] 570.12/150.36 570.12/150.36 [#compare(#pos(@x), #neg(@y))] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#GT()] 570.12/150.36 570.12/150.36 [#compare(#0(), #pos(@y))] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#LT()] 570.12/150.36 570.12/150.36 [#compare(#0(), #0())] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#EQ()] 570.12/150.36 570.12/150.36 [#compare(#0(), #neg(@y))] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#GT()] 570.12/150.36 570.12/150.36 [#compare(#0(), #s(@y))] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#LT()] 570.12/150.36 570.12/150.36 [#compare(#neg(@x), #pos(@y))] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#LT()] 570.12/150.36 570.12/150.36 [#compare(#neg(@x), #0())] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#LT()] 570.12/150.36 570.12/150.36 [#compare(#neg(@x), #neg(@y))] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#compare(@y, @x)] 570.12/150.36 570.12/150.36 [#compare(#s(@x), #0())] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#GT()] 570.12/150.36 570.12/150.36 [#compare(#s(@x), #s(@y))] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#compare(@x, @y)] 570.12/150.36 570.12/150.36 [leq(@l1, @l2)] = [0] 570.12/150.36 ? [4] @l2 + [0] 570.12/150.36 = [leq#1(@l1, @l2)] 570.12/150.36 570.12/150.36 [isortlist#1(nil())] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [nil()] 570.12/150.36 570.12/150.36 [isortlist#1(::(@x, @xs))] = [1] @x + [1] @xs + [1] 570.12/150.36 >= [1] @x + [1] @xs + [1] 570.12/150.36 = [insert(@x, isortlist(@xs))] 570.12/150.36 570.12/150.36 [#or(#true(), #true())] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#true()] 570.12/150.36 570.12/150.36 [#or(#true(), #false())] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#true()] 570.12/150.36 570.12/150.36 [#or(#false(), #true())] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#true()] 570.12/150.36 570.12/150.36 [#or(#false(), #false())] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#false()] 570.12/150.36 570.12/150.36 [insert#1(nil(), @x)] = [1] @x + [1] 570.12/150.36 >= [1] @x + [1] 570.12/150.36 = [::(@x, nil())] 570.12/150.36 570.12/150.36 [insert#1(::(@y, @ys), @x)] = [1] @x + [1] @y + [1] @ys + [2] 570.12/150.36 >= [1] @x + [1] @y + [1] @ys + [2] 570.12/150.36 = [insert#2(leq(@x, @y), @x, @y, @ys)] 570.12/150.36 570.12/150.36 [#less(@x, @y)] = [0] 570.12/150.36 >= [0] 570.12/150.36 = [#cklt(#compare(@x, @y))] 570.12/150.36 570.12/150.36 [leq#2^#(::(@y, @ys), @x, @xs)] = [2] @y + [2] @ys + [2] 570.12/150.36 > [2] @ys + [1] 570.12/150.36 = [c_1(leq^#(@xs, @ys))] 570.12/150.36 570.12/150.36 [leq^#(@l1, @l2)] = [2] @l2 + [0] 570.12/150.36 >= [2] @l2 + [0] 570.12/150.36 = [c_2(leq#1^#(@l1, @l2))] 570.12/150.36 570.12/150.36 [isortlist^#(@l)] = [6] @l + [7] 570.12/150.36 > [6] @l + [3] 570.12/150.36 = [isortlist#1^#(@l)] 570.12/150.36 570.12/150.36 [isortlist#1^#(::(@x, @xs))] = [6] @x + [6] @xs + [9] 570.12/150.36 > [6] @xs + [7] 570.12/150.36 = [isortlist^#(@xs)] 570.12/150.36 570.12/150.36 [isortlist#1^#(::(@x, @xs))] = [6] @x + [6] @xs + [9] 570.12/150.36 > [4] @xs + [0] 570.12/150.36 = [insert^#(@x, isortlist(@xs))] 570.12/150.36 570.12/150.36 [leq#1^#(::(@x, @xs), @l2)] = [2] @l2 + [0] 570.12/150.36 >= [2] @l2 + [0] 570.12/150.36 = [c_5(leq#2^#(@l2, @x, @xs))] 570.12/150.36 570.12/150.36 [insert^#(@x, @l)] = [4] @l + [0] 570.12/150.36 >= [4] @l + [0] 570.12/150.36 = [c_6(insert#1^#(@l, @x))] 570.12/150.36 570.12/150.36 [insert#1^#(::(@y, @ys), @x)] = [4] @y + [4] @ys + [4] 570.12/150.36 > [4] @y + [4] @ys + [3] 570.12/150.36 = [c_7(insert#2^#(leq(@x, @y), @x, @y, @ys), leq^#(@x, @y))] 570.12/150.36 570.12/150.36 [insert#2^#(#false(), @x, @y, @ys)] = [4] @ys + [0] 570.12/150.36 >= [4] @ys + [0] 570.12/150.36 = [c_8(insert^#(@x, @ys))] 570.12/150.36 570.12/150.36 570.12/150.36 We return to the main proof. Consider the set of all dependency 570.12/150.36 pairs 570.12/150.36 570.12/150.36 : 570.12/150.36 { 1: leq#2^#(::(@y, @ys), @x, @xs) -> c_1(leq^#(@xs, @ys)) 570.12/150.36 , 2: leq^#(@l1, @l2) -> c_2(leq#1^#(@l1, @l2)) 570.12/150.36 , 3: leq#1^#(::(@x, @xs), @l2) -> c_5(leq#2^#(@l2, @x, @xs)) 570.12/150.36 , 4: insert^#(@x, @l) -> c_6(insert#1^#(@l, @x)) 570.12/150.36 , 5: insert#1^#(::(@y, @ys), @x) -> 570.12/150.36 c_7(insert#2^#(leq(@x, @y), @x, @y, @ys), leq^#(@x, @y)) 570.12/150.36 , 6: insert#2^#(#false(), @x, @y, @ys) -> c_8(insert^#(@x, @ys)) 570.12/150.36 , 7: isortlist^#(@l) -> isortlist#1^#(@l) 570.12/150.36 , 8: isortlist#1^#(::(@x, @xs)) -> isortlist^#(@xs) 570.12/150.36 , 9: isortlist#1^#(::(@x, @xs)) -> insert^#(@x, isortlist(@xs)) } 570.12/150.36 570.12/150.36 Processor 'matrix interpretation of dimension 1' induces the 570.12/150.36 complexity certificate YES(?,O(n^1)) on application of dependency 570.12/150.36 pairs {1,5,7,8,9}. These cover all (indirect) predecessors of 570.12/150.36 dependency pairs {1,2,3,4,5,6,7,8,9}, their number of application 570.12/150.36 is equally bounded. The dependency pairs are shifted into the weak 570.12/150.36 component. 570.12/150.36 570.12/150.36 We are left with following problem, upon which TcT provides the 570.12/150.36 certificate YES(O(1),O(1)). 570.12/150.36 570.12/150.36 Weak DPs: 570.12/150.36 { leq#2^#(::(@y, @ys), @x, @xs) -> c_1(leq^#(@xs, @ys)) 570.12/150.36 , leq^#(@l1, @l2) -> c_2(leq#1^#(@l1, @l2)) 570.12/150.36 , isortlist^#(@l) -> isortlist#1^#(@l) 570.12/150.36 , isortlist#1^#(::(@x, @xs)) -> isortlist^#(@xs) 570.12/150.36 , isortlist#1^#(::(@x, @xs)) -> insert^#(@x, isortlist(@xs)) 570.12/150.36 , leq#1^#(::(@x, @xs), @l2) -> c_5(leq#2^#(@l2, @x, @xs)) 570.12/150.36 , insert^#(@x, @l) -> c_6(insert#1^#(@l, @x)) 570.12/150.36 , insert#1^#(::(@y, @ys), @x) -> 570.12/150.36 c_7(insert#2^#(leq(@x, @y), @x, @y, @ys), leq^#(@x, @y)) 570.12/150.36 , insert#2^#(#false(), @x, @y, @ys) -> c_8(insert^#(@x, @ys)) } 570.12/150.36 Weak Trs: 570.12/150.36 { #equal(@x, @y) -> #eq(@x, @y) 570.12/150.36 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 570.12/150.36 , #eq(#pos(@x), #0()) -> #false() 570.12/150.36 , #eq(#pos(@x), #neg(@y)) -> #false() 570.12/150.36 , #eq(nil(), nil()) -> #true() 570.12/150.36 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 570.12/150.36 , #eq(::(@x_1, @x_2), nil()) -> #false() 570.12/150.36 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 570.12/150.36 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 570.12/150.36 , #eq(#0(), #pos(@y)) -> #false() 570.12/150.36 , #eq(#0(), #0()) -> #true() 570.12/150.36 , #eq(#0(), #neg(@y)) -> #false() 570.12/150.36 , #eq(#0(), #s(@y)) -> #false() 570.12/150.36 , #eq(#neg(@x), #pos(@y)) -> #false() 570.12/150.36 , #eq(#neg(@x), #0()) -> #false() 570.12/150.36 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 570.12/150.36 , #eq(#s(@x), #0()) -> #false() 570.12/150.36 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 570.12/150.36 , leq#2(nil(), @x, @xs) -> #false() 570.12/150.36 , leq#2(::(@y, @ys), @x, @xs) -> 570.12/150.36 or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))) 570.12/150.36 , or(@x, @y) -> #or(@x, @y) 570.12/150.36 , and(@x, @y) -> #and(@x, @y) 570.12/150.36 , isortlist(@l) -> isortlist#1(@l) 570.12/150.36 , leq#1(nil(), @l2) -> #true() 570.12/150.36 , leq#1(::(@x, @xs), @l2) -> leq#2(@l2, @x, @xs) 570.12/150.36 , #cklt(#EQ()) -> #false() 570.12/150.36 , #cklt(#LT()) -> #true() 570.12/150.36 , #cklt(#GT()) -> #false() 570.12/150.36 , insert(@x, @l) -> insert#1(@l, @x) 570.12/150.36 , insert#2(#true(), @x, @y, @ys) -> ::(@x, ::(@y, @ys)) 570.12/150.36 , insert#2(#false(), @x, @y, @ys) -> ::(@y, insert(@x, @ys)) 570.12/150.36 , #and(#true(), #true()) -> #true() 570.12/150.36 , #and(#true(), #false()) -> #false() 570.12/150.36 , #and(#false(), #true()) -> #false() 570.12/150.36 , #and(#false(), #false()) -> #false() 570.12/150.36 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 570.12/150.36 , #compare(#pos(@x), #0()) -> #GT() 570.12/150.36 , #compare(#pos(@x), #neg(@y)) -> #GT() 570.12/150.36 , #compare(#0(), #pos(@y)) -> #LT() 570.12/150.36 , #compare(#0(), #0()) -> #EQ() 570.12/150.36 , #compare(#0(), #neg(@y)) -> #GT() 570.12/150.36 , #compare(#0(), #s(@y)) -> #LT() 570.12/150.36 , #compare(#neg(@x), #pos(@y)) -> #LT() 570.12/150.36 , #compare(#neg(@x), #0()) -> #LT() 570.12/150.36 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 570.12/150.36 , #compare(#s(@x), #0()) -> #GT() 570.12/150.36 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 570.12/150.36 , leq(@l1, @l2) -> leq#1(@l1, @l2) 570.12/150.36 , isortlist#1(nil()) -> nil() 570.12/150.36 , isortlist#1(::(@x, @xs)) -> insert(@x, isortlist(@xs)) 570.12/150.36 , #or(#true(), #true()) -> #true() 570.12/150.36 , #or(#true(), #false()) -> #true() 570.12/150.36 , #or(#false(), #true()) -> #true() 570.12/150.36 , #or(#false(), #false()) -> #false() 570.12/150.36 , insert#1(nil(), @x) -> ::(@x, nil()) 570.12/150.36 , insert#1(::(@y, @ys), @x) -> insert#2(leq(@x, @y), @x, @y, @ys) 570.12/150.36 , #less(@x, @y) -> #cklt(#compare(@x, @y)) } 570.12/150.36 Obligation: 570.12/150.36 innermost runtime complexity 570.12/150.36 Answer: 570.12/150.36 YES(O(1),O(1)) 570.12/150.36 570.12/150.36 The following weak DPs constitute a sub-graph of the DG that is 570.12/150.36 closed under successors. The DPs are removed. 570.12/150.36 570.12/150.36 { leq#2^#(::(@y, @ys), @x, @xs) -> c_1(leq^#(@xs, @ys)) 570.12/150.36 , leq^#(@l1, @l2) -> c_2(leq#1^#(@l1, @l2)) 570.12/150.36 , isortlist^#(@l) -> isortlist#1^#(@l) 570.12/150.36 , isortlist#1^#(::(@x, @xs)) -> isortlist^#(@xs) 570.12/150.36 , isortlist#1^#(::(@x, @xs)) -> insert^#(@x, isortlist(@xs)) 570.12/150.36 , leq#1^#(::(@x, @xs), @l2) -> c_5(leq#2^#(@l2, @x, @xs)) 570.12/150.36 , insert^#(@x, @l) -> c_6(insert#1^#(@l, @x)) 570.12/150.36 , insert#1^#(::(@y, @ys), @x) -> 570.12/150.36 c_7(insert#2^#(leq(@x, @y), @x, @y, @ys), leq^#(@x, @y)) 570.12/150.36 , insert#2^#(#false(), @x, @y, @ys) -> c_8(insert^#(@x, @ys)) } 570.12/150.36 570.12/150.36 We are left with following problem, upon which TcT provides the 570.12/150.36 certificate YES(O(1),O(1)). 570.12/150.36 570.12/150.36 Weak Trs: 570.12/150.36 { #equal(@x, @y) -> #eq(@x, @y) 570.12/150.36 , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) 570.12/150.36 , #eq(#pos(@x), #0()) -> #false() 570.12/150.36 , #eq(#pos(@x), #neg(@y)) -> #false() 570.12/150.36 , #eq(nil(), nil()) -> #true() 570.12/150.36 , #eq(nil(), ::(@y_1, @y_2)) -> #false() 570.12/150.36 , #eq(::(@x_1, @x_2), nil()) -> #false() 570.12/150.36 , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> 570.12/150.36 #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) 570.12/150.36 , #eq(#0(), #pos(@y)) -> #false() 570.12/150.36 , #eq(#0(), #0()) -> #true() 570.12/150.36 , #eq(#0(), #neg(@y)) -> #false() 570.12/150.36 , #eq(#0(), #s(@y)) -> #false() 570.12/150.36 , #eq(#neg(@x), #pos(@y)) -> #false() 570.12/150.36 , #eq(#neg(@x), #0()) -> #false() 570.12/150.36 , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) 570.12/150.36 , #eq(#s(@x), #0()) -> #false() 570.12/150.36 , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) 570.12/150.36 , leq#2(nil(), @x, @xs) -> #false() 570.12/150.36 , leq#2(::(@y, @ys), @x, @xs) -> 570.12/150.36 or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))) 570.12/150.36 , or(@x, @y) -> #or(@x, @y) 570.12/150.36 , and(@x, @y) -> #and(@x, @y) 570.12/150.36 , isortlist(@l) -> isortlist#1(@l) 570.12/150.36 , leq#1(nil(), @l2) -> #true() 570.12/150.36 , leq#1(::(@x, @xs), @l2) -> leq#2(@l2, @x, @xs) 570.12/150.36 , #cklt(#EQ()) -> #false() 570.12/150.36 , #cklt(#LT()) -> #true() 570.12/150.36 , #cklt(#GT()) -> #false() 570.12/150.36 , insert(@x, @l) -> insert#1(@l, @x) 570.12/150.36 , insert#2(#true(), @x, @y, @ys) -> ::(@x, ::(@y, @ys)) 570.12/150.36 , insert#2(#false(), @x, @y, @ys) -> ::(@y, insert(@x, @ys)) 570.12/150.36 , #and(#true(), #true()) -> #true() 570.12/150.36 , #and(#true(), #false()) -> #false() 570.12/150.36 , #and(#false(), #true()) -> #false() 570.12/150.36 , #and(#false(), #false()) -> #false() 570.12/150.36 , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) 570.12/150.36 , #compare(#pos(@x), #0()) -> #GT() 570.12/150.36 , #compare(#pos(@x), #neg(@y)) -> #GT() 570.12/150.36 , #compare(#0(), #pos(@y)) -> #LT() 570.12/150.36 , #compare(#0(), #0()) -> #EQ() 570.12/150.36 , #compare(#0(), #neg(@y)) -> #GT() 570.12/150.36 , #compare(#0(), #s(@y)) -> #LT() 570.12/150.36 , #compare(#neg(@x), #pos(@y)) -> #LT() 570.12/150.36 , #compare(#neg(@x), #0()) -> #LT() 570.12/150.36 , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) 570.12/150.36 , #compare(#s(@x), #0()) -> #GT() 570.12/150.36 , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) 570.12/150.36 , leq(@l1, @l2) -> leq#1(@l1, @l2) 570.12/150.36 , isortlist#1(nil()) -> nil() 570.12/150.36 , isortlist#1(::(@x, @xs)) -> insert(@x, isortlist(@xs)) 570.12/150.36 , #or(#true(), #true()) -> #true() 570.12/150.36 , #or(#true(), #false()) -> #true() 570.12/150.36 , #or(#false(), #true()) -> #true() 570.12/150.36 , #or(#false(), #false()) -> #false() 570.12/150.36 , insert#1(nil(), @x) -> ::(@x, nil()) 570.12/150.36 , insert#1(::(@y, @ys), @x) -> insert#2(leq(@x, @y), @x, @y, @ys) 570.12/150.36 , #less(@x, @y) -> #cklt(#compare(@x, @y)) } 570.12/150.36 Obligation: 570.12/150.36 innermost runtime complexity 570.12/150.36 Answer: 570.12/150.36 YES(O(1),O(1)) 570.12/150.36 570.12/150.36 No rule is usable, rules are removed from the input problem. 570.12/150.36 570.12/150.36 We are left with following problem, upon which TcT provides the 570.12/150.36 certificate YES(O(1),O(1)). 570.12/150.36 570.12/150.36 Rules: Empty 570.12/150.36 Obligation: 570.12/150.36 innermost runtime complexity 570.12/150.36 Answer: 570.12/150.36 YES(O(1),O(1)) 570.12/150.36 570.12/150.36 Empty rules are trivially bounded 570.12/150.36 570.12/150.36 Hurray, we answered YES(O(1),O(n^2)) 570.94/151.02 EOF