MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { #equal(@x, @y) -> #eq(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , and(@x, @y) -> #and(@x, @y) , insert(@x, @l) -> insert#1(@l, @x) , insert#1(::(@y, @ys), @x) -> insert#2(leq(@x, @y), @x, @y, @ys) , insert#1(nil(), @x) -> ::(@x, nil()) , leq(@l1, @l2) -> leq#1(@l1, @l2) , insert#2(#false(), @x, @y, @ys) -> ::(@y, insert(@x, @ys)) , insert#2(#true(), @x, @y, @ys) -> ::(@x, ::(@y, @ys)) , isortlist(@l) -> isortlist#1(@l) , isortlist#1(::(@x, @xs)) -> insert(@x, isortlist(@xs)) , isortlist#1(nil()) -> nil() , leq#1(::(@x, @xs), @l2) -> leq#2(@l2, @x, @xs) , leq#1(nil(), @l2) -> #true() , leq#2(::(@y, @ys), @x, @xs) -> or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))) , leq#2(nil(), @x, @xs) -> #false() , or(@x, @y) -> #or(@x, @y) } Weak Trs: { #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(nil(), nil()) -> #true() , #eq(#0(), #0()) -> #true() , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #s(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#pos(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #s(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#pos(@x), #0()) -> #GT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , #cklt(#EQ()) -> #false() , #cklt(#GT()) -> #false() , #cklt(#LT()) -> #true() , #and(#false(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#true(), #false()) -> #false() , #and(#true(), #true()) -> #true() , #or(#false(), #false()) -> #false() , #or(#false(), #true()) -> #true() , #or(#true(), #false()) -> #true() , #or(#true(), #true()) -> #true() } Obligation: innermost runtime complexity Answer: MAYBE We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. Trs: { isortlist#1(nil()) -> nil() } The induced complexity on above rules (modulo remaining rules) is YES(?,O(n^1)) . These rules are moved into the corresponding weak component(s). Sub-proof: ---------- The following argument positions are usable: Uargs(#cklt) = {1}, Uargs(and) = {1, 2}, Uargs(#and) = {1, 2}, Uargs(insert) = {2}, Uargs(::) = {2}, Uargs(insert#2) = {1}, Uargs(or) = {1, 2} TcT has computed following constructor-based matrix interpretation satisfying not(EDA). [#equal](x1, x2) = [0] [#eq](x1, x2) = [0] [#less](x1, x2) = [0] [#compare](x1, x2) = [0] [#cklt](x1) = [2] x1 + [0] [and](x1, x2) = [1] x1 + [2] x2 + [0] [#and](x1, x2) = [1] x1 + [2] x2 + [0] [insert](x1, x2) = [1] x2 + [0] [insert#1](x1, x2) = [1] x1 + [0] [::](x1, x2) = [1] x2 + [0] [leq](x1, x2) = [0] [insert#2](x1, x2, x3, x4) = [2] x1 + [1] x4 + [0] [nil] = [1] [#false] = [0] [#true] = [0] [isortlist](x1) = [2] x1 + [1] [isortlist#1](x1) = [2] x1 + [1] [leq#1](x1, x2) = [0] [leq#2](x1, x2, x3) = [0] [or](x1, x2) = [2] x1 + [1] x2 + [0] [#or](x1, x2) = [0] [#EQ] = [0] [#GT] = [0] [#LT] = [0] [#0] = [0] [#neg](x1) = [1] x1 + [0] [#pos](x1) = [1] x1 + [0] [#s](x1) = [1] x1 + [0] This order satisfies following ordering constraints [#equal(@x, @y)] = [0] >= [0] = [#eq(@x, @y)] [#eq(::(@x_1, @x_2), ::(@y_1, @y_2))] = [0] >= [0] = [#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))] [#eq(::(@x_1, @x_2), nil())] = [0] >= [0] = [#false()] [#eq(nil(), ::(@y_1, @y_2))] = [0] >= [0] = [#false()] [#eq(nil(), nil())] = [0] >= [0] = [#true()] [#eq(#0(), #0())] = [0] >= [0] = [#true()] [#eq(#0(), #neg(@y))] = [0] >= [0] = [#false()] [#eq(#0(), #pos(@y))] = [0] >= [0] = [#false()] [#eq(#0(), #s(@y))] = [0] >= [0] = [#false()] [#eq(#neg(@x), #0())] = [0] >= [0] = [#false()] [#eq(#neg(@x), #neg(@y))] = [0] >= [0] = [#eq(@x, @y)] [#eq(#neg(@x), #pos(@y))] = [0] >= [0] = [#false()] [#eq(#pos(@x), #0())] = [0] >= [0] = [#false()] [#eq(#pos(@x), #neg(@y))] = [0] >= [0] = [#false()] [#eq(#pos(@x), #pos(@y))] = [0] >= [0] = [#eq(@x, @y)] [#eq(#s(@x), #0())] = [0] >= [0] = [#false()] [#eq(#s(@x), #s(@y))] = [0] >= [0] = [#eq(@x, @y)] [#less(@x, @y)] = [0] >= [0] = [#cklt(#compare(@x, @y))] [#compare(#0(), #0())] = [0] >= [0] = [#EQ()] [#compare(#0(), #neg(@y))] = [0] >= [0] = [#GT()] [#compare(#0(), #pos(@y))] = [0] >= [0] = [#LT()] [#compare(#0(), #s(@y))] = [0] >= [0] = [#LT()] [#compare(#neg(@x), #0())] = [0] >= [0] = [#LT()] [#compare(#neg(@x), #neg(@y))] = [0] >= [0] = [#compare(@y, @x)] [#compare(#neg(@x), #pos(@y))] = [0] >= [0] = [#LT()] [#compare(#pos(@x), #0())] = [0] >= [0] = [#GT()] [#compare(#pos(@x), #neg(@y))] = [0] >= [0] = [#GT()] [#compare(#pos(@x), #pos(@y))] = [0] >= [0] = [#compare(@x, @y)] [#compare(#s(@x), #0())] = [0] >= [0] = [#GT()] [#compare(#s(@x), #s(@y))] = [0] >= [0] = [#compare(@x, @y)] [#cklt(#EQ())] = [0] >= [0] = [#false()] [#cklt(#GT())] = [0] >= [0] = [#false()] [#cklt(#LT())] = [0] >= [0] = [#true()] [and(@x, @y)] = [1] @x + [2] @y + [0] >= [1] @x + [2] @y + [0] = [#and(@x, @y)] [#and(#false(), #false())] = [0] >= [0] = [#false()] [#and(#false(), #true())] = [0] >= [0] = [#false()] [#and(#true(), #false())] = [0] >= [0] = [#false()] [#and(#true(), #true())] = [0] >= [0] = [#true()] [insert(@x, @l)] = [1] @l + [0] >= [1] @l + [0] = [insert#1(@l, @x)] [insert#1(::(@y, @ys), @x)] = [1] @ys + [0] >= [1] @ys + [0] = [insert#2(leq(@x, @y), @x, @y, @ys)] [insert#1(nil(), @x)] = [1] >= [1] = [::(@x, nil())] [leq(@l1, @l2)] = [0] >= [0] = [leq#1(@l1, @l2)] [insert#2(#false(), @x, @y, @ys)] = [1] @ys + [0] >= [1] @ys + [0] = [::(@y, insert(@x, @ys))] [insert#2(#true(), @x, @y, @ys)] = [1] @ys + [0] >= [1] @ys + [0] = [::(@x, ::(@y, @ys))] [isortlist(@l)] = [2] @l + [1] >= [2] @l + [1] = [isortlist#1(@l)] [isortlist#1(::(@x, @xs))] = [2] @xs + [1] >= [2] @xs + [1] = [insert(@x, isortlist(@xs))] [isortlist#1(nil())] = [3] > [1] = [nil()] [leq#1(::(@x, @xs), @l2)] = [0] >= [0] = [leq#2(@l2, @x, @xs)] [leq#1(nil(), @l2)] = [0] >= [0] = [#true()] [leq#2(::(@y, @ys), @x, @xs)] = [0] >= [0] = [or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys)))] [leq#2(nil(), @x, @xs)] = [0] >= [0] = [#false()] [or(@x, @y)] = [2] @x + [1] @y + [0] >= [0] = [#or(@x, @y)] [#or(#false(), #false())] = [0] >= [0] = [#false()] [#or(#false(), #true())] = [0] >= [0] = [#true()] [#or(#true(), #false())] = [0] >= [0] = [#true()] [#or(#true(), #true())] = [0] >= [0] = [#true()] We return to the main proof. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { #equal(@x, @y) -> #eq(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , and(@x, @y) -> #and(@x, @y) , insert(@x, @l) -> insert#1(@l, @x) , insert#1(::(@y, @ys), @x) -> insert#2(leq(@x, @y), @x, @y, @ys) , insert#1(nil(), @x) -> ::(@x, nil()) , leq(@l1, @l2) -> leq#1(@l1, @l2) , insert#2(#false(), @x, @y, @ys) -> ::(@y, insert(@x, @ys)) , insert#2(#true(), @x, @y, @ys) -> ::(@x, ::(@y, @ys)) , isortlist(@l) -> isortlist#1(@l) , isortlist#1(::(@x, @xs)) -> insert(@x, isortlist(@xs)) , leq#1(::(@x, @xs), @l2) -> leq#2(@l2, @x, @xs) , leq#1(nil(), @l2) -> #true() , leq#2(::(@y, @ys), @x, @xs) -> or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))) , leq#2(nil(), @x, @xs) -> #false() , or(@x, @y) -> #or(@x, @y) } Weak Trs: { #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(nil(), nil()) -> #true() , #eq(#0(), #0()) -> #true() , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #s(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#pos(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #s(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#pos(@x), #0()) -> #GT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , #cklt(#EQ()) -> #false() , #cklt(#GT()) -> #false() , #cklt(#LT()) -> #true() , #and(#false(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#true(), #false()) -> #false() , #and(#true(), #true()) -> #true() , isortlist#1(nil()) -> nil() , #or(#false(), #false()) -> #false() , #or(#false(), #true()) -> #true() , #or(#true(), #false()) -> #true() , #or(#true(), #true()) -> #true() } Obligation: innermost runtime complexity Answer: MAYBE We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. Trs: { isortlist(@l) -> isortlist#1(@l) } The induced complexity on above rules (modulo remaining rules) is YES(?,O(n^1)) . These rules are moved into the corresponding weak component(s). Sub-proof: ---------- The following argument positions are usable: Uargs(#cklt) = {1}, Uargs(and) = {1, 2}, Uargs(#and) = {1, 2}, Uargs(insert) = {2}, Uargs(::) = {2}, Uargs(insert#2) = {1}, Uargs(or) = {1, 2} TcT has computed following constructor-based matrix interpretation satisfying not(EDA). [#equal](x1, x2) = [0] [#eq](x1, x2) = [0] [#less](x1, x2) = [0] [#compare](x1, x2) = [0] [#cklt](x1) = [2] x1 + [0] [and](x1, x2) = [1] x1 + [1] x2 + [0] [#and](x1, x2) = [1] x1 + [1] x2 + [0] [insert](x1, x2) = [1] x2 + [1] [insert#1](x1, x2) = [1] x1 + [1] [::](x1, x2) = [1] x2 + [1] [leq](x1, x2) = [0] [insert#2](x1, x2, x3, x4) = [2] x1 + [1] x4 + [2] [nil] = [2] [#false] = [0] [#true] = [0] [isortlist](x1) = [2] x1 + [2] [isortlist#1](x1) = [2] x1 + [1] [leq#1](x1, x2) = [0] [leq#2](x1, x2, x3) = [0] [or](x1, x2) = [1] x1 + [1] x2 + [0] [#or](x1, x2) = [0] [#EQ] = [0] [#GT] = [0] [#LT] = [0] [#0] = [0] [#neg](x1) = [1] x1 + [0] [#pos](x1) = [1] x1 + [0] [#s](x1) = [1] x1 + [0] This order satisfies following ordering constraints [#equal(@x, @y)] = [0] >= [0] = [#eq(@x, @y)] [#eq(::(@x_1, @x_2), ::(@y_1, @y_2))] = [0] >= [0] = [#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))] [#eq(::(@x_1, @x_2), nil())] = [0] >= [0] = [#false()] [#eq(nil(), ::(@y_1, @y_2))] = [0] >= [0] = [#false()] [#eq(nil(), nil())] = [0] >= [0] = [#true()] [#eq(#0(), #0())] = [0] >= [0] = [#true()] [#eq(#0(), #neg(@y))] = [0] >= [0] = [#false()] [#eq(#0(), #pos(@y))] = [0] >= [0] = [#false()] [#eq(#0(), #s(@y))] = [0] >= [0] = [#false()] [#eq(#neg(@x), #0())] = [0] >= [0] = [#false()] [#eq(#neg(@x), #neg(@y))] = [0] >= [0] = [#eq(@x, @y)] [#eq(#neg(@x), #pos(@y))] = [0] >= [0] = [#false()] [#eq(#pos(@x), #0())] = [0] >= [0] = [#false()] [#eq(#pos(@x), #neg(@y))] = [0] >= [0] = [#false()] [#eq(#pos(@x), #pos(@y))] = [0] >= [0] = [#eq(@x, @y)] [#eq(#s(@x), #0())] = [0] >= [0] = [#false()] [#eq(#s(@x), #s(@y))] = [0] >= [0] = [#eq(@x, @y)] [#less(@x, @y)] = [0] >= [0] = [#cklt(#compare(@x, @y))] [#compare(#0(), #0())] = [0] >= [0] = [#EQ()] [#compare(#0(), #neg(@y))] = [0] >= [0] = [#GT()] [#compare(#0(), #pos(@y))] = [0] >= [0] = [#LT()] [#compare(#0(), #s(@y))] = [0] >= [0] = [#LT()] [#compare(#neg(@x), #0())] = [0] >= [0] = [#LT()] [#compare(#neg(@x), #neg(@y))] = [0] >= [0] = [#compare(@y, @x)] [#compare(#neg(@x), #pos(@y))] = [0] >= [0] = [#LT()] [#compare(#pos(@x), #0())] = [0] >= [0] = [#GT()] [#compare(#pos(@x), #neg(@y))] = [0] >= [0] = [#GT()] [#compare(#pos(@x), #pos(@y))] = [0] >= [0] = [#compare(@x, @y)] [#compare(#s(@x), #0())] = [0] >= [0] = [#GT()] [#compare(#s(@x), #s(@y))] = [0] >= [0] = [#compare(@x, @y)] [#cklt(#EQ())] = [0] >= [0] = [#false()] [#cklt(#GT())] = [0] >= [0] = [#false()] [#cklt(#LT())] = [0] >= [0] = [#true()] [and(@x, @y)] = [1] @x + [1] @y + [0] >= [1] @x + [1] @y + [0] = [#and(@x, @y)] [#and(#false(), #false())] = [0] >= [0] = [#false()] [#and(#false(), #true())] = [0] >= [0] = [#false()] [#and(#true(), #false())] = [0] >= [0] = [#false()] [#and(#true(), #true())] = [0] >= [0] = [#true()] [insert(@x, @l)] = [1] @l + [1] >= [1] @l + [1] = [insert#1(@l, @x)] [insert#1(::(@y, @ys), @x)] = [1] @ys + [2] >= [1] @ys + [2] = [insert#2(leq(@x, @y), @x, @y, @ys)] [insert#1(nil(), @x)] = [3] >= [3] = [::(@x, nil())] [leq(@l1, @l2)] = [0] >= [0] = [leq#1(@l1, @l2)] [insert#2(#false(), @x, @y, @ys)] = [1] @ys + [2] >= [1] @ys + [2] = [::(@y, insert(@x, @ys))] [insert#2(#true(), @x, @y, @ys)] = [1] @ys + [2] >= [1] @ys + [2] = [::(@x, ::(@y, @ys))] [isortlist(@l)] = [2] @l + [2] > [2] @l + [1] = [isortlist#1(@l)] [isortlist#1(::(@x, @xs))] = [2] @xs + [3] >= [2] @xs + [3] = [insert(@x, isortlist(@xs))] [isortlist#1(nil())] = [5] > [2] = [nil()] [leq#1(::(@x, @xs), @l2)] = [0] >= [0] = [leq#2(@l2, @x, @xs)] [leq#1(nil(), @l2)] = [0] >= [0] = [#true()] [leq#2(::(@y, @ys), @x, @xs)] = [0] >= [0] = [or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys)))] [leq#2(nil(), @x, @xs)] = [0] >= [0] = [#false()] [or(@x, @y)] = [1] @x + [1] @y + [0] >= [0] = [#or(@x, @y)] [#or(#false(), #false())] = [0] >= [0] = [#false()] [#or(#false(), #true())] = [0] >= [0] = [#true()] [#or(#true(), #false())] = [0] >= [0] = [#true()] [#or(#true(), #true())] = [0] >= [0] = [#true()] We return to the main proof. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { #equal(@x, @y) -> #eq(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , and(@x, @y) -> #and(@x, @y) , insert(@x, @l) -> insert#1(@l, @x) , insert#1(::(@y, @ys), @x) -> insert#2(leq(@x, @y), @x, @y, @ys) , insert#1(nil(), @x) -> ::(@x, nil()) , leq(@l1, @l2) -> leq#1(@l1, @l2) , insert#2(#false(), @x, @y, @ys) -> ::(@y, insert(@x, @ys)) , insert#2(#true(), @x, @y, @ys) -> ::(@x, ::(@y, @ys)) , isortlist#1(::(@x, @xs)) -> insert(@x, isortlist(@xs)) , leq#1(::(@x, @xs), @l2) -> leq#2(@l2, @x, @xs) , leq#1(nil(), @l2) -> #true() , leq#2(::(@y, @ys), @x, @xs) -> or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))) , leq#2(nil(), @x, @xs) -> #false() , or(@x, @y) -> #or(@x, @y) } Weak Trs: { #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(nil(), nil()) -> #true() , #eq(#0(), #0()) -> #true() , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #s(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#pos(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #s(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#pos(@x), #0()) -> #GT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , #cklt(#EQ()) -> #false() , #cklt(#GT()) -> #false() , #cklt(#LT()) -> #true() , #and(#false(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#true(), #false()) -> #false() , #and(#true(), #true()) -> #true() , isortlist(@l) -> isortlist#1(@l) , isortlist#1(nil()) -> nil() , #or(#false(), #false()) -> #false() , #or(#false(), #true()) -> #true() , #or(#true(), #false()) -> #true() , #or(#true(), #true()) -> #true() } Obligation: innermost runtime complexity Answer: MAYBE We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. Trs: { isortlist#1(::(@x, @xs)) -> insert(@x, isortlist(@xs)) } The induced complexity on above rules (modulo remaining rules) is YES(?,O(n^1)) . These rules are moved into the corresponding weak component(s). Sub-proof: ---------- The following argument positions are usable: Uargs(#cklt) = {1}, Uargs(and) = {1, 2}, Uargs(#and) = {1, 2}, Uargs(insert) = {2}, Uargs(::) = {2}, Uargs(insert#2) = {1}, Uargs(or) = {1, 2} TcT has computed following constructor-based matrix interpretation satisfying not(EDA). [#equal](x1, x2) = [0] [#eq](x1, x2) = [0] [#less](x1, x2) = [0] [#compare](x1, x2) = [0] [#cklt](x1) = [2] x1 + [0] [and](x1, x2) = [2] x1 + [1] x2 + [0] [#and](x1, x2) = [1] x1 + [1] x2 + [0] [insert](x1, x2) = [1] x2 + [1] [insert#1](x1, x2) = [1] x1 + [1] [::](x1, x2) = [1] x2 + [1] [leq](x1, x2) = [0] [insert#2](x1, x2, x3, x4) = [1] x1 + [1] x4 + [2] [nil] = [0] [#false] = [0] [#true] = [0] [isortlist](x1) = [2] x1 + [1] [isortlist#1](x1) = [2] x1 + [1] [leq#1](x1, x2) = [0] [leq#2](x1, x2, x3) = [0] [or](x1, x2) = [2] x1 + [2] x2 + [0] [#or](x1, x2) = [0] [#EQ] = [0] [#GT] = [0] [#LT] = [0] [#0] = [0] [#neg](x1) = [1] x1 + [0] [#pos](x1) = [1] x1 + [0] [#s](x1) = [1] x1 + [0] This order satisfies following ordering constraints [#equal(@x, @y)] = [0] >= [0] = [#eq(@x, @y)] [#eq(::(@x_1, @x_2), ::(@y_1, @y_2))] = [0] >= [0] = [#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))] [#eq(::(@x_1, @x_2), nil())] = [0] >= [0] = [#false()] [#eq(nil(), ::(@y_1, @y_2))] = [0] >= [0] = [#false()] [#eq(nil(), nil())] = [0] >= [0] = [#true()] [#eq(#0(), #0())] = [0] >= [0] = [#true()] [#eq(#0(), #neg(@y))] = [0] >= [0] = [#false()] [#eq(#0(), #pos(@y))] = [0] >= [0] = [#false()] [#eq(#0(), #s(@y))] = [0] >= [0] = [#false()] [#eq(#neg(@x), #0())] = [0] >= [0] = [#false()] [#eq(#neg(@x), #neg(@y))] = [0] >= [0] = [#eq(@x, @y)] [#eq(#neg(@x), #pos(@y))] = [0] >= [0] = [#false()] [#eq(#pos(@x), #0())] = [0] >= [0] = [#false()] [#eq(#pos(@x), #neg(@y))] = [0] >= [0] = [#false()] [#eq(#pos(@x), #pos(@y))] = [0] >= [0] = [#eq(@x, @y)] [#eq(#s(@x), #0())] = [0] >= [0] = [#false()] [#eq(#s(@x), #s(@y))] = [0] >= [0] = [#eq(@x, @y)] [#less(@x, @y)] = [0] >= [0] = [#cklt(#compare(@x, @y))] [#compare(#0(), #0())] = [0] >= [0] = [#EQ()] [#compare(#0(), #neg(@y))] = [0] >= [0] = [#GT()] [#compare(#0(), #pos(@y))] = [0] >= [0] = [#LT()] [#compare(#0(), #s(@y))] = [0] >= [0] = [#LT()] [#compare(#neg(@x), #0())] = [0] >= [0] = [#LT()] [#compare(#neg(@x), #neg(@y))] = [0] >= [0] = [#compare(@y, @x)] [#compare(#neg(@x), #pos(@y))] = [0] >= [0] = [#LT()] [#compare(#pos(@x), #0())] = [0] >= [0] = [#GT()] [#compare(#pos(@x), #neg(@y))] = [0] >= [0] = [#GT()] [#compare(#pos(@x), #pos(@y))] = [0] >= [0] = [#compare(@x, @y)] [#compare(#s(@x), #0())] = [0] >= [0] = [#GT()] [#compare(#s(@x), #s(@y))] = [0] >= [0] = [#compare(@x, @y)] [#cklt(#EQ())] = [0] >= [0] = [#false()] [#cklt(#GT())] = [0] >= [0] = [#false()] [#cklt(#LT())] = [0] >= [0] = [#true()] [and(@x, @y)] = [2] @x + [1] @y + [0] >= [1] @x + [1] @y + [0] = [#and(@x, @y)] [#and(#false(), #false())] = [0] >= [0] = [#false()] [#and(#false(), #true())] = [0] >= [0] = [#false()] [#and(#true(), #false())] = [0] >= [0] = [#false()] [#and(#true(), #true())] = [0] >= [0] = [#true()] [insert(@x, @l)] = [1] @l + [1] >= [1] @l + [1] = [insert#1(@l, @x)] [insert#1(::(@y, @ys), @x)] = [1] @ys + [2] >= [1] @ys + [2] = [insert#2(leq(@x, @y), @x, @y, @ys)] [insert#1(nil(), @x)] = [1] >= [1] = [::(@x, nil())] [leq(@l1, @l2)] = [0] >= [0] = [leq#1(@l1, @l2)] [insert#2(#false(), @x, @y, @ys)] = [1] @ys + [2] >= [1] @ys + [2] = [::(@y, insert(@x, @ys))] [insert#2(#true(), @x, @y, @ys)] = [1] @ys + [2] >= [1] @ys + [2] = [::(@x, ::(@y, @ys))] [isortlist(@l)] = [2] @l + [1] >= [2] @l + [1] = [isortlist#1(@l)] [isortlist#1(::(@x, @xs))] = [2] @xs + [3] > [2] @xs + [2] = [insert(@x, isortlist(@xs))] [isortlist#1(nil())] = [1] > [0] = [nil()] [leq#1(::(@x, @xs), @l2)] = [0] >= [0] = [leq#2(@l2, @x, @xs)] [leq#1(nil(), @l2)] = [0] >= [0] = [#true()] [leq#2(::(@y, @ys), @x, @xs)] = [0] >= [0] = [or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys)))] [leq#2(nil(), @x, @xs)] = [0] >= [0] = [#false()] [or(@x, @y)] = [2] @x + [2] @y + [0] >= [0] = [#or(@x, @y)] [#or(#false(), #false())] = [0] >= [0] = [#false()] [#or(#false(), #true())] = [0] >= [0] = [#true()] [#or(#true(), #false())] = [0] >= [0] = [#true()] [#or(#true(), #true())] = [0] >= [0] = [#true()] We return to the main proof. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { #equal(@x, @y) -> #eq(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , and(@x, @y) -> #and(@x, @y) , insert(@x, @l) -> insert#1(@l, @x) , insert#1(::(@y, @ys), @x) -> insert#2(leq(@x, @y), @x, @y, @ys) , insert#1(nil(), @x) -> ::(@x, nil()) , leq(@l1, @l2) -> leq#1(@l1, @l2) , insert#2(#false(), @x, @y, @ys) -> ::(@y, insert(@x, @ys)) , insert#2(#true(), @x, @y, @ys) -> ::(@x, ::(@y, @ys)) , leq#1(::(@x, @xs), @l2) -> leq#2(@l2, @x, @xs) , leq#1(nil(), @l2) -> #true() , leq#2(::(@y, @ys), @x, @xs) -> or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))) , leq#2(nil(), @x, @xs) -> #false() , or(@x, @y) -> #or(@x, @y) } Weak Trs: { #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(nil(), nil()) -> #true() , #eq(#0(), #0()) -> #true() , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #s(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#pos(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #s(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#pos(@x), #0()) -> #GT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , #cklt(#EQ()) -> #false() , #cklt(#GT()) -> #false() , #cklt(#LT()) -> #true() , #and(#false(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#true(), #false()) -> #false() , #and(#true(), #true()) -> #true() , isortlist(@l) -> isortlist#1(@l) , isortlist#1(::(@x, @xs)) -> insert(@x, isortlist(@xs)) , isortlist#1(nil()) -> nil() , #or(#false(), #false()) -> #false() , #or(#false(), #true()) -> #true() , #or(#true(), #false()) -> #true() , #or(#true(), #true()) -> #true() } Obligation: innermost runtime complexity Answer: MAYBE We use the processor 'matrix interpretation of dimension 2' to orient following rules strictly. Trs: { insert#1(nil(), @x) -> ::(@x, nil()) } The induced complexity on above rules (modulo remaining rules) is YES(?,O(n^1)) . These rules are moved into the corresponding weak component(s). Sub-proof: ---------- The following argument positions are usable: Uargs(#cklt) = {1}, Uargs(and) = {1, 2}, Uargs(#and) = {1, 2}, Uargs(insert) = {2}, Uargs(::) = {2}, Uargs(insert#2) = {1}, Uargs(or) = {1, 2} TcT has computed following constructor-based matrix interpretation satisfying not(EDA) and not(IDA(1)). [#equal](x1, x2) = [0 0] x1 + [0] [2 2] [0] [#eq](x1, x2) = [0 0] x1 + [0] [2 0] [0] [#less](x1, x2) = [0 0] x2 + [0] [1 1] [1] [#compare](x1, x2) = [0] [0] [#cklt](x1) = [2 0] x1 + [0] [0 0] [1] [and](x1, x2) = [1 0] x1 + [1 0] x2 + [0] [0 0] [1 0] [2] [#and](x1, x2) = [1 0] x1 + [1 0] x2 + [0] [0 0] [0 0] [2] [insert](x1, x2) = [1 1] x2 + [1] [0 1] [0] [insert#1](x1, x2) = [1 1] x1 + [1] [0 1] [0] [::](x1, x2) = [1 0] x2 + [1] [0 1] [0] [leq](x1, x2) = [0] [1] [insert#2](x1, x2, x3, x4) = [2 2] x1 + [1 1] x4 + [0] [0 0] [0 1] [0] [nil] = [2] [1] [#false] = [0] [1] [#true] = [0] [1] [isortlist](x1) = [2 0] x1 + [2] [0 0] [1] [isortlist#1](x1) = [2 0] x1 + [2] [0 0] [1] [leq#1](x1, x2) = [0] [1] [leq#2](x1, x2, x3) = [0] [1] [or](x1, x2) = [1 0] x1 + [1 0] x2 + [0] [0 0] [0 0] [1] [#or](x1, x2) = [0] [1] [#EQ] = [0] [0] [#GT] = [0] [0] [#LT] = [0] [0] [#0] = [2] [0] [#neg](x1) = [1 0] x1 + [2] [0 1] [0] [#pos](x1) = [1 1] x1 + [2] [0 0] [0] [#s](x1) = [1 0] x1 + [2] [0 1] [0] This order satisfies following ordering constraints [#equal(@x, @y)] = [0 0] @x + [0] [2 2] [0] >= [0 0] @x + [0] [2 0] [0] = [#eq(@x, @y)] [#eq(::(@x_1, @x_2), ::(@y_1, @y_2))] = [0 0] @x_2 + [0] [2 0] [2] >= [0] [2] = [#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))] [#eq(::(@x_1, @x_2), nil())] = [0 0] @x_2 + [0] [2 0] [2] >= [0] [1] = [#false()] [#eq(nil(), ::(@y_1, @y_2))] = [0] [4] >= [0] [1] = [#false()] [#eq(nil(), nil())] = [0] [4] >= [0] [1] = [#true()] [#eq(#0(), #0())] = [0] [4] >= [0] [1] = [#true()] [#eq(#0(), #neg(@y))] = [0] [4] >= [0] [1] = [#false()] [#eq(#0(), #pos(@y))] = [0] [4] >= [0] [1] = [#false()] [#eq(#0(), #s(@y))] = [0] [4] >= [0] [1] = [#false()] [#eq(#neg(@x), #0())] = [0 0] @x + [0] [2 0] [4] >= [0] [1] = [#false()] [#eq(#neg(@x), #neg(@y))] = [0 0] @x + [0] [2 0] [4] >= [0 0] @x + [0] [2 0] [0] = [#eq(@x, @y)] [#eq(#neg(@x), #pos(@y))] = [0 0] @x + [0] [2 0] [4] >= [0] [1] = [#false()] [#eq(#pos(@x), #0())] = [0 0] @x + [0] [2 2] [4] >= [0] [1] = [#false()] [#eq(#pos(@x), #neg(@y))] = [0 0] @x + [0] [2 2] [4] >= [0] [1] = [#false()] [#eq(#pos(@x), #pos(@y))] = [0 0] @x + [0] [2 2] [4] >= [0 0] @x + [0] [2 0] [0] = [#eq(@x, @y)] [#eq(#s(@x), #0())] = [0 0] @x + [0] [2 0] [4] >= [0] [1] = [#false()] [#eq(#s(@x), #s(@y))] = [0 0] @x + [0] [2 0] [4] >= [0 0] @x + [0] [2 0] [0] = [#eq(@x, @y)] [#less(@x, @y)] = [0 0] @y + [0] [1 1] [1] >= [0] [1] = [#cklt(#compare(@x, @y))] [#compare(#0(), #0())] = [0] [0] >= [0] [0] = [#EQ()] [#compare(#0(), #neg(@y))] = [0] [0] >= [0] [0] = [#GT()] [#compare(#0(), #pos(@y))] = [0] [0] >= [0] [0] = [#LT()] [#compare(#0(), #s(@y))] = [0] [0] >= [0] [0] = [#LT()] [#compare(#neg(@x), #0())] = [0] [0] >= [0] [0] = [#LT()] [#compare(#neg(@x), #neg(@y))] = [0] [0] >= [0] [0] = [#compare(@y, @x)] [#compare(#neg(@x), #pos(@y))] = [0] [0] >= [0] [0] = [#LT()] [#compare(#pos(@x), #0())] = [0] [0] >= [0] [0] = [#GT()] [#compare(#pos(@x), #neg(@y))] = [0] [0] >= [0] [0] = [#GT()] [#compare(#pos(@x), #pos(@y))] = [0] [0] >= [0] [0] = [#compare(@x, @y)] [#compare(#s(@x), #0())] = [0] [0] >= [0] [0] = [#GT()] [#compare(#s(@x), #s(@y))] = [0] [0] >= [0] [0] = [#compare(@x, @y)] [#cklt(#EQ())] = [0] [1] >= [0] [1] = [#false()] [#cklt(#GT())] = [0] [1] >= [0] [1] = [#false()] [#cklt(#LT())] = [0] [1] >= [0] [1] = [#true()] [and(@x, @y)] = [1 0] @x + [1 0] @y + [0] [0 0] [1 0] [2] >= [1 0] @x + [1 0] @y + [0] [0 0] [0 0] [2] = [#and(@x, @y)] [#and(#false(), #false())] = [0] [2] >= [0] [1] = [#false()] [#and(#false(), #true())] = [0] [2] >= [0] [1] = [#false()] [#and(#true(), #false())] = [0] [2] >= [0] [1] = [#false()] [#and(#true(), #true())] = [0] [2] >= [0] [1] = [#true()] [insert(@x, @l)] = [1 1] @l + [1] [0 1] [0] >= [1 1] @l + [1] [0 1] [0] = [insert#1(@l, @x)] [insert#1(::(@y, @ys), @x)] = [1 1] @ys + [2] [0 1] [0] >= [1 1] @ys + [2] [0 1] [0] = [insert#2(leq(@x, @y), @x, @y, @ys)] [insert#1(nil(), @x)] = [4] [1] > [3] [1] = [::(@x, nil())] [leq(@l1, @l2)] = [0] [1] >= [0] [1] = [leq#1(@l1, @l2)] [insert#2(#false(), @x, @y, @ys)] = [1 1] @ys + [2] [0 1] [0] >= [1 1] @ys + [2] [0 1] [0] = [::(@y, insert(@x, @ys))] [insert#2(#true(), @x, @y, @ys)] = [1 1] @ys + [2] [0 1] [0] >= [1 0] @ys + [2] [0 1] [0] = [::(@x, ::(@y, @ys))] [isortlist(@l)] = [2 0] @l + [2] [0 0] [1] >= [2 0] @l + [2] [0 0] [1] = [isortlist#1(@l)] [isortlist#1(::(@x, @xs))] = [2 0] @xs + [4] [0 0] [1] >= [2 0] @xs + [4] [0 0] [1] = [insert(@x, isortlist(@xs))] [isortlist#1(nil())] = [6] [1] > [2] [1] = [nil()] [leq#1(::(@x, @xs), @l2)] = [0] [1] >= [0] [1] = [leq#2(@l2, @x, @xs)] [leq#1(nil(), @l2)] = [0] [1] >= [0] [1] = [#true()] [leq#2(::(@y, @ys), @x, @xs)] = [0] [1] >= [0] [1] = [or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys)))] [leq#2(nil(), @x, @xs)] = [0] [1] >= [0] [1] = [#false()] [or(@x, @y)] = [1 0] @x + [1 0] @y + [0] [0 0] [0 0] [1] >= [0] [1] = [#or(@x, @y)] [#or(#false(), #false())] = [0] [1] >= [0] [1] = [#false()] [#or(#false(), #true())] = [0] [1] >= [0] [1] = [#true()] [#or(#true(), #false())] = [0] [1] >= [0] [1] = [#true()] [#or(#true(), #true())] = [0] [1] >= [0] [1] = [#true()] We return to the main proof. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { #equal(@x, @y) -> #eq(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , and(@x, @y) -> #and(@x, @y) , insert(@x, @l) -> insert#1(@l, @x) , insert#1(::(@y, @ys), @x) -> insert#2(leq(@x, @y), @x, @y, @ys) , leq(@l1, @l2) -> leq#1(@l1, @l2) , insert#2(#false(), @x, @y, @ys) -> ::(@y, insert(@x, @ys)) , insert#2(#true(), @x, @y, @ys) -> ::(@x, ::(@y, @ys)) , leq#1(::(@x, @xs), @l2) -> leq#2(@l2, @x, @xs) , leq#1(nil(), @l2) -> #true() , leq#2(::(@y, @ys), @x, @xs) -> or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))) , leq#2(nil(), @x, @xs) -> #false() , or(@x, @y) -> #or(@x, @y) } Weak Trs: { #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(nil(), nil()) -> #true() , #eq(#0(), #0()) -> #true() , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #s(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#pos(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #s(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#pos(@x), #0()) -> #GT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , #cklt(#EQ()) -> #false() , #cklt(#GT()) -> #false() , #cklt(#LT()) -> #true() , #and(#false(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#true(), #false()) -> #false() , #and(#true(), #true()) -> #true() , insert#1(nil(), @x) -> ::(@x, nil()) , isortlist(@l) -> isortlist#1(@l) , isortlist#1(::(@x, @xs)) -> insert(@x, isortlist(@xs)) , isortlist#1(nil()) -> nil() , #or(#false(), #false()) -> #false() , #or(#false(), #true()) -> #true() , #or(#true(), #false()) -> #true() , #or(#true(), #true()) -> #true() } Obligation: innermost runtime complexity Answer: MAYBE We use the processor 'matrix interpretation of dimension 2' to orient following rules strictly. Trs: { insert#2(#true(), @x, @y, @ys) -> ::(@x, ::(@y, @ys)) } The induced complexity on above rules (modulo remaining rules) is YES(?,O(n^1)) . These rules are moved into the corresponding weak component(s). Sub-proof: ---------- The following argument positions are usable: Uargs(#cklt) = {1}, Uargs(and) = {1, 2}, Uargs(#and) = {1, 2}, Uargs(insert) = {2}, Uargs(::) = {2}, Uargs(insert#2) = {1}, Uargs(or) = {1, 2} TcT has computed following constructor-based matrix interpretation satisfying not(EDA) and not(IDA(1)). [#equal](x1, x2) = [0 0] x1 + [0 0] x2 + [0] [2 1] [2 0] [1] [#eq](x1, x2) = [0 0] x2 + [0] [1 0] [1] [#less](x1, x2) = [0 0] x1 + [0 0] x2 + [0] [0 2] [2 2] [0] [#compare](x1, x2) = [0 0] x1 + [0 0] x2 + [0] [0 1] [0 1] [0] [#cklt](x1) = [2 0] x1 + [0] [1 2] [0] [and](x1, x2) = [1 0] x1 + [1 0] x2 + [0] [2 0] [0 1] [0] [#and](x1, x2) = [1 0] x1 + [1 0] x2 + [0] [1 0] [0 1] [0] [insert](x1, x2) = [1 2] x1 + [1 1] x2 + [1] [0 0] [0 0] [1] [insert#1](x1, x2) = [1 1] x1 + [1 2] x2 + [1] [0 0] [0 0] [1] [::](x1, x2) = [1 1] x1 + [1 1] x2 + [0] [0 0] [0 0] [1] [leq](x1, x2) = [0] [1] [insert#2](x1, x2, x3, x4) = [1 2] x1 + [1 2] x2 + [1 1] x3 + [1 1] x4 + [0] [0 0] [0 0] [0 0] [0 0] [1] [nil] = [0] [0] [#false] = [0] [1] [#true] = [0] [1] [isortlist](x1) = [2 1] x1 + [2] [0 1] [0] [isortlist#1](x1) = [2 1] x1 + [2] [0 1] [0] [leq#1](x1, x2) = [0] [1] [leq#2](x1, x2, x3) = [0] [1] [or](x1, x2) = [1 0] x1 + [2 0] x2 + [0] [0 0] [0 0] [1] [#or](x1, x2) = [0] [1] [#EQ] = [0] [2] [#GT] = [0] [1] [#LT] = [0] [1] [#0] = [0] [1] [#neg](x1) = [1 0] x1 + [0] [0 1] [2] [#pos](x1) = [1 0] x1 + [0] [0 1] [2] [#s](x1) = [1 0] x1 + [0] [0 1] [0] This order satisfies following ordering constraints [#equal(@x, @y)] = [0 0] @x + [0 0] @y + [0] [2 1] [2 0] [1] >= [0 0] @y + [0] [1 0] [1] = [#eq(@x, @y)] [#eq(::(@x_1, @x_2), ::(@y_1, @y_2))] = [0 0] @y_1 + [0 0] @y_2 + [0] [1 1] [1 1] [1] >= [0 0] @y_2 + [0] [1 0] [1] = [#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))] [#eq(::(@x_1, @x_2), nil())] = [0] [1] >= [0] [1] = [#false()] [#eq(nil(), ::(@y_1, @y_2))] = [0 0] @y_1 + [0 0] @y_2 + [0] [1 1] [1 1] [1] >= [0] [1] = [#false()] [#eq(nil(), nil())] = [0] [1] >= [0] [1] = [#true()] [#eq(#0(), #0())] = [0] [1] >= [0] [1] = [#true()] [#eq(#0(), #neg(@y))] = [0 0] @y + [0] [1 0] [1] >= [0] [1] = [#false()] [#eq(#0(), #pos(@y))] = [0 0] @y + [0] [1 0] [1] >= [0] [1] = [#false()] [#eq(#0(), #s(@y))] = [0 0] @y + [0] [1 0] [1] >= [0] [1] = [#false()] [#eq(#neg(@x), #0())] = [0] [1] >= [0] [1] = [#false()] [#eq(#neg(@x), #neg(@y))] = [0 0] @y + [0] [1 0] [1] >= [0 0] @y + [0] [1 0] [1] = [#eq(@x, @y)] [#eq(#neg(@x), #pos(@y))] = [0 0] @y + [0] [1 0] [1] >= [0] [1] = [#false()] [#eq(#pos(@x), #0())] = [0] [1] >= [0] [1] = [#false()] [#eq(#pos(@x), #neg(@y))] = [0 0] @y + [0] [1 0] [1] >= [0] [1] = [#false()] [#eq(#pos(@x), #pos(@y))] = [0 0] @y + [0] [1 0] [1] >= [0 0] @y + [0] [1 0] [1] = [#eq(@x, @y)] [#eq(#s(@x), #0())] = [0] [1] >= [0] [1] = [#false()] [#eq(#s(@x), #s(@y))] = [0 0] @y + [0] [1 0] [1] >= [0 0] @y + [0] [1 0] [1] = [#eq(@x, @y)] [#less(@x, @y)] = [0 0] @x + [0 0] @y + [0] [0 2] [2 2] [0] >= [0 0] @x + [0 0] @y + [0] [0 2] [0 2] [0] = [#cklt(#compare(@x, @y))] [#compare(#0(), #0())] = [0] [2] >= [0] [2] = [#EQ()] [#compare(#0(), #neg(@y))] = [0 0] @y + [0] [0 1] [3] >= [0] [1] = [#GT()] [#compare(#0(), #pos(@y))] = [0 0] @y + [0] [0 1] [3] >= [0] [1] = [#LT()] [#compare(#0(), #s(@y))] = [0 0] @y + [0] [0 1] [1] >= [0] [1] = [#LT()] [#compare(#neg(@x), #0())] = [0 0] @x + [0] [0 1] [3] >= [0] [1] = [#LT()] [#compare(#neg(@x), #neg(@y))] = [0 0] @x + [0 0] @y + [0] [0 1] [0 1] [4] >= [0 0] @x + [0 0] @y + [0] [0 1] [0 1] [0] = [#compare(@y, @x)] [#compare(#neg(@x), #pos(@y))] = [0 0] @x + [0 0] @y + [0] [0 1] [0 1] [4] >= [0] [1] = [#LT()] [#compare(#pos(@x), #0())] = [0 0] @x + [0] [0 1] [3] >= [0] [1] = [#GT()] [#compare(#pos(@x), #neg(@y))] = [0 0] @x + [0 0] @y + [0] [0 1] [0 1] [4] >= [0] [1] = [#GT()] [#compare(#pos(@x), #pos(@y))] = [0 0] @x + [0 0] @y + [0] [0 1] [0 1] [4] >= [0 0] @x + [0 0] @y + [0] [0 1] [0 1] [0] = [#compare(@x, @y)] [#compare(#s(@x), #0())] = [0 0] @x + [0] [0 1] [1] >= [0] [1] = [#GT()] [#compare(#s(@x), #s(@y))] = [0 0] @x + [0 0] @y + [0] [0 1] [0 1] [0] >= [0 0] @x + [0 0] @y + [0] [0 1] [0 1] [0] = [#compare(@x, @y)] [#cklt(#EQ())] = [0] [4] >= [0] [1] = [#false()] [#cklt(#GT())] = [0] [2] >= [0] [1] = [#false()] [#cklt(#LT())] = [0] [2] >= [0] [1] = [#true()] [and(@x, @y)] = [1 0] @x + [1 0] @y + [0] [2 0] [0 1] [0] >= [1 0] @x + [1 0] @y + [0] [1 0] [0 1] [0] = [#and(@x, @y)] [#and(#false(), #false())] = [0] [1] >= [0] [1] = [#false()] [#and(#false(), #true())] = [0] [1] >= [0] [1] = [#false()] [#and(#true(), #false())] = [0] [1] >= [0] [1] = [#false()] [#and(#true(), #true())] = [0] [1] >= [0] [1] = [#true()] [insert(@x, @l)] = [1 1] @l + [1 2] @x + [1] [0 0] [0 0] [1] >= [1 1] @l + [1 2] @x + [1] [0 0] [0 0] [1] = [insert#1(@l, @x)] [insert#1(::(@y, @ys), @x)] = [1 2] @x + [1 1] @y + [1 1] @ys + [2] [0 0] [0 0] [0 0] [1] >= [1 2] @x + [1 1] @y + [1 1] @ys + [2] [0 0] [0 0] [0 0] [1] = [insert#2(leq(@x, @y), @x, @y, @ys)] [insert#1(nil(), @x)] = [1 2] @x + [1] [0 0] [1] > [1 1] @x + [0] [0 0] [1] = [::(@x, nil())] [leq(@l1, @l2)] = [0] [1] >= [0] [1] = [leq#1(@l1, @l2)] [insert#2(#false(), @x, @y, @ys)] = [1 2] @x + [1 1] @y + [1 1] @ys + [2] [0 0] [0 0] [0 0] [1] >= [1 2] @x + [1 1] @y + [1 1] @ys + [2] [0 0] [0 0] [0 0] [1] = [::(@y, insert(@x, @ys))] [insert#2(#true(), @x, @y, @ys)] = [1 2] @x + [1 1] @y + [1 1] @ys + [2] [0 0] [0 0] [0 0] [1] > [1 1] @x + [1 1] @y + [1 1] @ys + [1] [0 0] [0 0] [0 0] [1] = [::(@x, ::(@y, @ys))] [isortlist(@l)] = [2 1] @l + [2] [0 1] [0] >= [2 1] @l + [2] [0 1] [0] = [isortlist#1(@l)] [isortlist#1(::(@x, @xs))] = [2 2] @x + [2 2] @xs + [3] [0 0] [0 0] [1] >= [1 2] @x + [2 2] @xs + [3] [0 0] [0 0] [1] = [insert(@x, isortlist(@xs))] [isortlist#1(nil())] = [2] [0] > [0] [0] = [nil()] [leq#1(::(@x, @xs), @l2)] = [0] [1] >= [0] [1] = [leq#2(@l2, @x, @xs)] [leq#1(nil(), @l2)] = [0] [1] >= [0] [1] = [#true()] [leq#2(::(@y, @ys), @x, @xs)] = [0] [1] >= [0] [1] = [or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys)))] [leq#2(nil(), @x, @xs)] = [0] [1] >= [0] [1] = [#false()] [or(@x, @y)] = [1 0] @x + [2 0] @y + [0] [0 0] [0 0] [1] >= [0] [1] = [#or(@x, @y)] [#or(#false(), #false())] = [0] [1] >= [0] [1] = [#false()] [#or(#false(), #true())] = [0] [1] >= [0] [1] = [#true()] [#or(#true(), #false())] = [0] [1] >= [0] [1] = [#true()] [#or(#true(), #true())] = [0] [1] >= [0] [1] = [#true()] We return to the main proof. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { #equal(@x, @y) -> #eq(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , and(@x, @y) -> #and(@x, @y) , insert(@x, @l) -> insert#1(@l, @x) , insert#1(::(@y, @ys), @x) -> insert#2(leq(@x, @y), @x, @y, @ys) , leq(@l1, @l2) -> leq#1(@l1, @l2) , insert#2(#false(), @x, @y, @ys) -> ::(@y, insert(@x, @ys)) , leq#1(::(@x, @xs), @l2) -> leq#2(@l2, @x, @xs) , leq#1(nil(), @l2) -> #true() , leq#2(::(@y, @ys), @x, @xs) -> or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))) , leq#2(nil(), @x, @xs) -> #false() , or(@x, @y) -> #or(@x, @y) } Weak Trs: { #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(nil(), nil()) -> #true() , #eq(#0(), #0()) -> #true() , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #s(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#pos(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #s(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#pos(@x), #0()) -> #GT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , #cklt(#EQ()) -> #false() , #cklt(#GT()) -> #false() , #cklt(#LT()) -> #true() , #and(#false(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#true(), #false()) -> #false() , #and(#true(), #true()) -> #true() , insert#1(nil(), @x) -> ::(@x, nil()) , insert#2(#true(), @x, @y, @ys) -> ::(@x, ::(@y, @ys)) , isortlist(@l) -> isortlist#1(@l) , isortlist#1(::(@x, @xs)) -> insert(@x, isortlist(@xs)) , isortlist#1(nil()) -> nil() , #or(#false(), #false()) -> #false() , #or(#false(), #true()) -> #true() , #or(#true(), #false()) -> #true() , #or(#true(), #true()) -> #true() } Obligation: innermost runtime complexity Answer: MAYBE We use the processor 'matrix interpretation of dimension 4' to orient following rules strictly. Trs: { insert#1(::(@y, @ys), @x) -> insert#2(leq(@x, @y), @x, @y, @ys) } The induced complexity on above rules (modulo remaining rules) is YES(?,O(n^3)) . These rules are moved into the corresponding weak component(s). Sub-proof: ---------- The following argument positions are usable: Uargs(#cklt) = {1}, Uargs(and) = {1, 2}, Uargs(#and) = {1, 2}, Uargs(insert) = {2}, Uargs(::) = {2}, Uargs(insert#2) = {1}, Uargs(or) = {1, 2} TcT has computed following constructor-based matrix interpretation satisfying not(EDA) and not(IDA(3)). [0 0 0 0] [0 0 0 0] [0] [#equal](x1, x2) = [0 0 0 0] x1 + [1 0 0 0] x2 + [0] [1 0 0 1] [1 1 0 0] [1] [0 1 0 0] [0 0 0 0] [1] [0] [#eq](x1, x2) = [0] [1] [1] [0 0 0 0] [0 0 0 0] [0] [#less](x1, x2) = [0 0 0 1] x1 + [0 0 0 0] x2 + [0] [1 1 1 0] [1 0 1 0] [1] [0 1 1 0] [0 0 1 0] [0] [0 0 0 0] [0 0 0 0] [0] [#compare](x1, x2) = [0 0 0 0] x1 + [0 0 0 0] x2 + [0] [0 0 0 0] [0 0 0 0] [0] [0 1 1 0] [0 0 1 0] [0] [1 0 0 0] [0] [#cklt](x1) = [0 0 0 0] x1 + [0] [0 0 0 1] [0] [0 0 0 1] [0] [1 0 0 0] [1 0 0 0] [0] [and](x1, x2) = [1 0 0 1] x1 + [0 0 0 0] x2 + [0] [0 1 1 1] [0 0 0 0] [0] [0 1 1 0] [0 0 0 0] [0] [1 0 0 0] [1 0 0 0] [0] [#and](x1, x2) = [0 0 0 0] x1 + [0 0 0 0] x2 + [0] [0 1 0 1] [0 0 0 0] [0] [0 0 1 0] [0 0 0 0] [0] [1 0 0 1] [1] [insert](x1, x2) = [0 1 0 0] x2 + [1] [0 0 1 1] [0] [0 0 0 1] [1] [1 0 0 1] [1] [insert#1](x1, x2) = [0 1 0 0] x1 + [1] [0 0 1 1] [0] [0 0 0 1] [1] [1 0 0 0] [1] [::](x1, x2) = [0 0 0 1] x2 + [0] [0 1 1 0] [1] [0 0 0 1] [1] [0] [leq](x1, x2) = [0] [1] [1] [1 1 1 1] [1 0 0 1] [0] [insert#2](x1, x2, x3, x4) = [0 1 1 0] x1 + [0 0 0 1] x4 + [0] [0 0 1 1] [0 1 1 1] [0] [0 0 1 1] [0 0 0 1] [0] [1] [nil] = [0] [0] [1] [0] [#false] = [0] [1] [1] [0] [#true] = [0] [1] [1] [1 1 1 0] [0] [isortlist](x1) = [0 1 1 1] x1 + [0] [0 1 1 0] [0] [0 0 0 1] [1] [1 1 1 0] [0] [isortlist#1](x1) = [0 1 1 0] x1 + [0] [0 1 1 0] [0] [0 0 0 1] [1] [0] [leq#1](x1, x2) = [0] [1] [1] [0] [leq#2](x1, x2, x3) = [0] [1] [1] [1 0 0 0] [1 0 0 0] [0] [or](x1, x2) = [0 0 0 0] x1 + [0 0 0 0] x2 + [0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [1] [0] [#or](x1, x2) = [0] [1] [1] [0] [#EQ] = [0] [0] [1] [0] [#GT] = [0] [0] [1] [0] [#LT] = [0] [0] [1] [1] [#0] = [1] [0] [1] [1 0 0 0] [1] [#neg](x1) = [0 0 0 0] x1 + [1] [0 1 1 1] [0] [0 0 0 0] [1] [1 0 0 0] [1] [#pos](x1) = [0 0 0 0] x1 + [1] [0 1 1 1] [1] [0 0 0 0] [1] [1 0 0 0] [1] [#s](x1) = [0 0 0 0] x1 + [0] [0 1 1 1] [1] [0 0 0 0] [1] This order satisfies following ordering constraints [#equal(@x, @y)] = [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] @x + [1 0 0 0] @y + [0] [1 0 0 1] [1 1 0 0] [1] [0 1 0 0] [0 0 0 0] [1] >= [0] [0] [1] [1] = [#eq(@x, @y)] [#eq(::(@x_1, @x_2), ::(@y_1, @y_2))] = [0] [0] [1] [1] >= [0] [0] [1] [1] = [#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))] [#eq(::(@x_1, @x_2), nil())] = [0] [0] [1] [1] >= [0] [0] [1] [1] = [#false()] [#eq(nil(), ::(@y_1, @y_2))] = [0] [0] [1] [1] >= [0] [0] [1] [1] = [#false()] [#eq(nil(), nil())] = [0] [0] [1] [1] >= [0] [0] [1] [1] = [#true()] [#eq(#0(), #0())] = [0] [0] [1] [1] >= [0] [0] [1] [1] = [#true()] [#eq(#0(), #neg(@y))] = [0] [0] [1] [1] >= [0] [0] [1] [1] = [#false()] [#eq(#0(), #pos(@y))] = [0] [0] [1] [1] >= [0] [0] [1] [1] = [#false()] [#eq(#0(), #s(@y))] = [0] [0] [1] [1] >= [0] [0] [1] [1] = [#false()] [#eq(#neg(@x), #0())] = [0] [0] [1] [1] >= [0] [0] [1] [1] = [#false()] [#eq(#neg(@x), #neg(@y))] = [0] [0] [1] [1] >= [0] [0] [1] [1] = [#eq(@x, @y)] [#eq(#neg(@x), #pos(@y))] = [0] [0] [1] [1] >= [0] [0] [1] [1] = [#false()] [#eq(#pos(@x), #0())] = [0] [0] [1] [1] >= [0] [0] [1] [1] = [#false()] [#eq(#pos(@x), #neg(@y))] = [0] [0] [1] [1] >= [0] [0] [1] [1] = [#false()] [#eq(#pos(@x), #pos(@y))] = [0] [0] [1] [1] >= [0] [0] [1] [1] = [#eq(@x, @y)] [#eq(#s(@x), #0())] = [0] [0] [1] [1] >= [0] [0] [1] [1] = [#false()] [#eq(#s(@x), #s(@y))] = [0] [0] [1] [1] >= [0] [0] [1] [1] = [#eq(@x, @y)] [#less(@x, @y)] = [0 0 0 0] [0 0 0 0] [0] [0 0 0 1] @x + [0 0 0 0] @y + [0] [1 1 1 0] [1 0 1 0] [1] [0 1 1 0] [0 0 1 0] [0] >= [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] @x + [0 0 0 0] @y + [0] [0 1 1 0] [0 0 1 0] [0] [0 1 1 0] [0 0 1 0] [0] = [#cklt(#compare(@x, @y))] [#compare(#0(), #0())] = [0] [0] [0] [1] >= [0] [0] [0] [1] = [#EQ()] [#compare(#0(), #neg(@y))] = [0 0 0 0] [0] [0 0 0 0] @y + [0] [0 0 0 0] [0] [0 1 1 1] [1] >= [0] [0] [0] [1] = [#GT()] [#compare(#0(), #pos(@y))] = [0 0 0 0] [0] [0 0 0 0] @y + [0] [0 0 0 0] [0] [0 1 1 1] [2] >= [0] [0] [0] [1] = [#LT()] [#compare(#0(), #s(@y))] = [0 0 0 0] [0] [0 0 0 0] @y + [0] [0 0 0 0] [0] [0 1 1 1] [2] >= [0] [0] [0] [1] = [#LT()] [#compare(#neg(@x), #0())] = [0 0 0 0] [0] [0 0 0 0] @x + [0] [0 0 0 0] [0] [0 1 1 1] [1] >= [0] [0] [0] [1] = [#LT()] [#compare(#neg(@x), #neg(@y))] = [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] @x + [0 0 0 0] @y + [0] [0 0 0 0] [0 0 0 0] [0] [0 1 1 1] [0 1 1 1] [1] >= [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] @x + [0 0 0 0] @y + [0] [0 0 0 0] [0 0 0 0] [0] [0 0 1 0] [0 1 1 0] [0] = [#compare(@y, @x)] [#compare(#neg(@x), #pos(@y))] = [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] @x + [0 0 0 0] @y + [0] [0 0 0 0] [0 0 0 0] [0] [0 1 1 1] [0 1 1 1] [2] >= [0] [0] [0] [1] = [#LT()] [#compare(#pos(@x), #0())] = [0 0 0 0] [0] [0 0 0 0] @x + [0] [0 0 0 0] [0] [0 1 1 1] [2] >= [0] [0] [0] [1] = [#GT()] [#compare(#pos(@x), #neg(@y))] = [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] @x + [0 0 0 0] @y + [0] [0 0 0 0] [0 0 0 0] [0] [0 1 1 1] [0 1 1 1] [2] >= [0] [0] [0] [1] = [#GT()] [#compare(#pos(@x), #pos(@y))] = [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] @x + [0 0 0 0] @y + [0] [0 0 0 0] [0 0 0 0] [0] [0 1 1 1] [0 1 1 1] [3] >= [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] @x + [0 0 0 0] @y + [0] [0 0 0 0] [0 0 0 0] [0] [0 1 1 0] [0 0 1 0] [0] = [#compare(@x, @y)] [#compare(#s(@x), #0())] = [0 0 0 0] [0] [0 0 0 0] @x + [0] [0 0 0 0] [0] [0 1 1 1] [1] >= [0] [0] [0] [1] = [#GT()] [#compare(#s(@x), #s(@y))] = [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] @x + [0 0 0 0] @y + [0] [0 0 0 0] [0 0 0 0] [0] [0 1 1 1] [0 1 1 1] [2] >= [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] @x + [0 0 0 0] @y + [0] [0 0 0 0] [0 0 0 0] [0] [0 1 1 0] [0 0 1 0] [0] = [#compare(@x, @y)] [#cklt(#EQ())] = [0] [0] [1] [1] >= [0] [0] [1] [1] = [#false()] [#cklt(#GT())] = [0] [0] [1] [1] >= [0] [0] [1] [1] = [#false()] [#cklt(#LT())] = [0] [0] [1] [1] >= [0] [0] [1] [1] = [#true()] [and(@x, @y)] = [1 0 0 0] [1 0 0 0] [0] [1 0 0 1] @x + [0 0 0 0] @y + [0] [0 1 1 1] [0 0 0 0] [0] [0 1 1 0] [0 0 0 0] [0] >= [1 0 0 0] [1 0 0 0] [0] [0 0 0 0] @x + [0 0 0 0] @y + [0] [0 1 0 1] [0 0 0 0] [0] [0 0 1 0] [0 0 0 0] [0] = [#and(@x, @y)] [#and(#false(), #false())] = [0] [0] [1] [1] >= [0] [0] [1] [1] = [#false()] [#and(#false(), #true())] = [0] [0] [1] [1] >= [0] [0] [1] [1] = [#false()] [#and(#true(), #false())] = [0] [0] [1] [1] >= [0] [0] [1] [1] = [#false()] [#and(#true(), #true())] = [0] [0] [1] [1] >= [0] [0] [1] [1] = [#true()] [insert(@x, @l)] = [1 0 0 1] [1] [0 1 0 0] @l + [1] [0 0 1 1] [0] [0 0 0 1] [1] >= [1 0 0 1] [1] [0 1 0 0] @l + [1] [0 0 1 1] [0] [0 0 0 1] [1] = [insert#1(@l, @x)] [insert#1(::(@y, @ys), @x)] = [1 0 0 1] [3] [0 0 0 1] @ys + [1] [0 1 1 1] [2] [0 0 0 1] [2] > [1 0 0 1] [2] [0 0 0 1] @ys + [1] [0 1 1 1] [2] [0 0 0 1] [2] = [insert#2(leq(@x, @y), @x, @y, @ys)] [insert#1(nil(), @x)] = [3] [1] [1] [2] > [2] [1] [1] [2] = [::(@x, nil())] [leq(@l1, @l2)] = [0] [0] [1] [1] >= [0] [0] [1] [1] = [leq#1(@l1, @l2)] [insert#2(#false(), @x, @y, @ys)] = [1 0 0 1] [2] [0 0 0 1] @ys + [1] [0 1 1 1] [2] [0 0 0 1] [2] >= [1 0 0 1] [2] [0 0 0 1] @ys + [1] [0 1 1 1] [2] [0 0 0 1] [2] = [::(@y, insert(@x, @ys))] [insert#2(#true(), @x, @y, @ys)] = [1 0 0 1] [2] [0 0 0 1] @ys + [1] [0 1 1 1] [2] [0 0 0 1] [2] >= [1 0 0 0] [2] [0 0 0 1] @ys + [1] [0 1 1 1] [2] [0 0 0 1] [2] = [::(@x, ::(@y, @ys))] [isortlist(@l)] = [1 1 1 0] [0] [0 1 1 1] @l + [0] [0 1 1 0] [0] [0 0 0 1] [1] >= [1 1 1 0] [0] [0 1 1 0] @l + [0] [0 1 1 0] [0] [0 0 0 1] [1] = [isortlist#1(@l)] [isortlist#1(::(@x, @xs))] = [1 1 1 1] [2] [0 1 1 1] @xs + [1] [0 1 1 1] [1] [0 0 0 1] [2] >= [1 1 1 1] [2] [0 1 1 1] @xs + [1] [0 1 1 1] [1] [0 0 0 1] [2] = [insert(@x, isortlist(@xs))] [isortlist#1(nil())] = [1] [0] [0] [2] >= [1] [0] [0] [1] = [nil()] [leq#1(::(@x, @xs), @l2)] = [0] [0] [1] [1] >= [0] [0] [1] [1] = [leq#2(@l2, @x, @xs)] [leq#1(nil(), @l2)] = [0] [0] [1] [1] >= [0] [0] [1] [1] = [#true()] [leq#2(::(@y, @ys), @x, @xs)] = [0] [0] [1] [1] >= [0] [0] [1] [1] = [or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys)))] [leq#2(nil(), @x, @xs)] = [0] [0] [1] [1] >= [0] [0] [1] [1] = [#false()] [or(@x, @y)] = [1 0 0 0] [1 0 0 0] [0] [0 0 0 0] @x + [0 0 0 0] @y + [0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [1] >= [0] [0] [1] [1] = [#or(@x, @y)] [#or(#false(), #false())] = [0] [0] [1] [1] >= [0] [0] [1] [1] = [#false()] [#or(#false(), #true())] = [0] [0] [1] [1] >= [0] [0] [1] [1] = [#true()] [#or(#true(), #false())] = [0] [0] [1] [1] >= [0] [0] [1] [1] = [#true()] [#or(#true(), #true())] = [0] [0] [1] [1] >= [0] [0] [1] [1] = [#true()] We return to the main proof. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { #equal(@x, @y) -> #eq(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , and(@x, @y) -> #and(@x, @y) , insert(@x, @l) -> insert#1(@l, @x) , leq(@l1, @l2) -> leq#1(@l1, @l2) , insert#2(#false(), @x, @y, @ys) -> ::(@y, insert(@x, @ys)) , leq#1(::(@x, @xs), @l2) -> leq#2(@l2, @x, @xs) , leq#1(nil(), @l2) -> #true() , leq#2(::(@y, @ys), @x, @xs) -> or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))) , leq#2(nil(), @x, @xs) -> #false() , or(@x, @y) -> #or(@x, @y) } Weak Trs: { #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(nil(), nil()) -> #true() , #eq(#0(), #0()) -> #true() , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #s(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#pos(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #s(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#pos(@x), #0()) -> #GT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , #cklt(#EQ()) -> #false() , #cklt(#GT()) -> #false() , #cklt(#LT()) -> #true() , #and(#false(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#true(), #false()) -> #false() , #and(#true(), #true()) -> #true() , insert#1(::(@y, @ys), @x) -> insert#2(leq(@x, @y), @x, @y, @ys) , insert#1(nil(), @x) -> ::(@x, nil()) , insert#2(#true(), @x, @y, @ys) -> ::(@x, ::(@y, @ys)) , isortlist(@l) -> isortlist#1(@l) , isortlist#1(::(@x, @xs)) -> insert(@x, isortlist(@xs)) , isortlist#1(nil()) -> nil() , #or(#false(), #false()) -> #false() , #or(#false(), #true()) -> #true() , #or(#true(), #false()) -> #true() , #or(#true(), #true()) -> #true() } Obligation: innermost runtime complexity Answer: MAYBE We use the processor 'matrix interpretation of dimension 4' to orient following rules strictly. Trs: { leq#1(nil(), @l2) -> #true() , leq#2(nil(), @x, @xs) -> #false() } The induced complexity on above rules (modulo remaining rules) is YES(?,O(n^3)) . These rules are moved into the corresponding weak component(s). Sub-proof: ---------- The following argument positions are usable: Uargs(#cklt) = {1}, Uargs(and) = {1, 2}, Uargs(#and) = {1, 2}, Uargs(insert) = {2}, Uargs(::) = {2}, Uargs(insert#2) = {1}, Uargs(or) = {1, 2} TcT has computed following constructor-based matrix interpretation satisfying not(EDA) and not(IDA(3)). [0 0 0 0] [0 0 0 0] [0] [#equal](x1, x2) = [0 0 0 0] x1 + [0 0 0 0] x2 + [1] [0 0 0 0] [1 1 1 1] [0] [1 1 1 1] [0 0 0 0] [0] [0] [#eq](x1, x2) = [1] [0] [0] [0 0 0 0] [0 0 0 0] [0] [#less](x1, x2) = [0 0 0 0] x1 + [0 0 0 0] x2 + [1] [0 0 1 1] [0 0 1 0] [1] [0 0 0 1] [0 0 0 1] [0] [0] [#compare](x1, x2) = [1] [0] [0] [1 0 0 0] [0] [#cklt](x1) = [0 0 0 0] x1 + [1] [0 1 1 0] [0] [0 0 0 0] [0] [1 0 0 0] [1 0 0 0] [0] [and](x1, x2) = [0 0 0 0] x1 + [0 1 0 0] x2 + [0] [1 0 1 0] [0 0 0 0] [0] [0 1 0 0] [0 0 0 1] [0] [1 0 0 0] [1 0 0 0] [0] [#and](x1, x2) = [0 0 0 0] x1 + [0 1 0 0] x2 + [0] [0 0 1 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0] [1 1 0 0] [1] [insert](x1, x2) = [0 1 0 0] x2 + [1] [0 0 1 1] [0] [0 1 0 0] [0] [1 1 0 0] [1] [insert#1](x1, x2) = [0 1 0 0] x1 + [1] [0 0 1 1] [0] [0 1 0 0] [0] [1 0 0 0] [0] [::](x1, x2) = [0 1 0 0] x2 + [1] [0 0 1 1] [0] [0 1 0 0] [0] [1] [leq](x1, x2) = [1] [1] [0] [1 1 0 1] [1 1 0 0] [0] [insert#2](x1, x2, x3, x4) = [0 1 0 1] x1 + [0 1 0 0] x4 + [1] [0 0 0 1] [0 1 1 1] [0] [0 1 0 1] [0 1 0 0] [0] [0] [nil] = [0] [0] [0] [0] [#false] = [1] [0] [0] [0] [#true] = [1] [0] [0] [0 1 1 1] [0] [isortlist](x1) = [0 1 0 0] x1 + [0] [0 1 1 1] [0] [0 1 0 0] [1] [0 1 1 1] [0] [isortlist#1](x1) = [0 1 0 0] x1 + [0] [0 1 1 1] [0] [0 1 0 0] [0] [1] [leq#1](x1, x2) = [1] [0] [0] [1] [leq#2](x1, x2, x3) = [1] [0] [0] [1 0 0 0] [1 0 0 0] [0] [or](x1, x2) = [0 0 0 0] x1 + [0 0 0 0] x2 + [1] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0] [0] [#or](x1, x2) = [1] [0] [0] [0] [#EQ] = [1] [0] [0] [0] [#GT] = [1] [0] [0] [0] [#LT] = [1] [0] [0] [1] [#0] = [1] [1] [1] [1 0 0 0] [1] [#neg](x1) = [0 0 0 0] x1 + [0] [0 1 1 0] [0] [0 1 0 0] [0] [1 0 0 0] [1] [#pos](x1) = [0 1 0 0] x1 + [0] [0 0 1 0] [0] [0 0 0 1] [0] [1 0 0 0] [1] [#s](x1) = [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 1 0 1] [0] This order satisfies following ordering constraints [#equal(@x, @y)] = [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] @x + [0 0 0 0] @y + [1] [0 0 0 0] [1 1 1 1] [0] [1 1 1 1] [0 0 0 0] [0] >= [0] [1] [0] [0] = [#eq(@x, @y)] [#eq(::(@x_1, @x_2), ::(@y_1, @y_2))] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))] [#eq(::(@x_1, @x_2), nil())] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#false()] [#eq(nil(), ::(@y_1, @y_2))] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#false()] [#eq(nil(), nil())] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#true()] [#eq(#0(), #0())] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#true()] [#eq(#0(), #neg(@y))] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#false()] [#eq(#0(), #pos(@y))] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#false()] [#eq(#0(), #s(@y))] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#false()] [#eq(#neg(@x), #0())] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#false()] [#eq(#neg(@x), #neg(@y))] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#eq(@x, @y)] [#eq(#neg(@x), #pos(@y))] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#false()] [#eq(#pos(@x), #0())] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#false()] [#eq(#pos(@x), #neg(@y))] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#false()] [#eq(#pos(@x), #pos(@y))] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#eq(@x, @y)] [#eq(#s(@x), #0())] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#false()] [#eq(#s(@x), #s(@y))] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#eq(@x, @y)] [#less(@x, @y)] = [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] @x + [0 0 0 0] @y + [1] [0 0 1 1] [0 0 1 0] [1] [0 0 0 1] [0 0 0 1] [0] >= [0] [1] [1] [0] = [#cklt(#compare(@x, @y))] [#compare(#0(), #0())] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#EQ()] [#compare(#0(), #neg(@y))] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#GT()] [#compare(#0(), #pos(@y))] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#LT()] [#compare(#0(), #s(@y))] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#LT()] [#compare(#neg(@x), #0())] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#LT()] [#compare(#neg(@x), #neg(@y))] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#compare(@y, @x)] [#compare(#neg(@x), #pos(@y))] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#LT()] [#compare(#pos(@x), #0())] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#GT()] [#compare(#pos(@x), #neg(@y))] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#GT()] [#compare(#pos(@x), #pos(@y))] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#compare(@x, @y)] [#compare(#s(@x), #0())] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#GT()] [#compare(#s(@x), #s(@y))] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#compare(@x, @y)] [#cklt(#EQ())] = [0] [1] [1] [0] >= [0] [1] [0] [0] = [#false()] [#cklt(#GT())] = [0] [1] [1] [0] >= [0] [1] [0] [0] = [#false()] [#cklt(#LT())] = [0] [1] [1] [0] >= [0] [1] [0] [0] = [#true()] [and(@x, @y)] = [1 0 0 0] [1 0 0 0] [0] [0 0 0 0] @x + [0 1 0 0] @y + [0] [1 0 1 0] [0 0 0 0] [0] [0 1 0 0] [0 0 0 1] [0] >= [1 0 0 0] [1 0 0 0] [0] [0 0 0 0] @x + [0 1 0 0] @y + [0] [0 0 1 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0] = [#and(@x, @y)] [#and(#false(), #false())] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#false()] [#and(#false(), #true())] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#false()] [#and(#true(), #false())] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#false()] [#and(#true(), #true())] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#true()] [insert(@x, @l)] = [1 1 0 0] [1] [0 1 0 0] @l + [1] [0 0 1 1] [0] [0 1 0 0] [0] >= [1 1 0 0] [1] [0 1 0 0] @l + [1] [0 0 1 1] [0] [0 1 0 0] [0] = [insert#1(@l, @x)] [insert#1(::(@y, @ys), @x)] = [1 1 0 0] [2] [0 1 0 0] @ys + [2] [0 1 1 1] [0] [0 1 0 0] [1] >= [1 1 0 0] [2] [0 1 0 0] @ys + [2] [0 1 1 1] [0] [0 1 0 0] [1] = [insert#2(leq(@x, @y), @x, @y, @ys)] [insert#1(nil(), @x)] = [1] [1] [0] [0] > [0] [1] [0] [0] = [::(@x, nil())] [leq(@l1, @l2)] = [1] [1] [1] [0] >= [1] [1] [0] [0] = [leq#1(@l1, @l2)] [insert#2(#false(), @x, @y, @ys)] = [1 1 0 0] [1] [0 1 0 0] @ys + [2] [0 1 1 1] [0] [0 1 0 0] [1] >= [1 1 0 0] [1] [0 1 0 0] @ys + [2] [0 1 1 1] [0] [0 1 0 0] [1] = [::(@y, insert(@x, @ys))] [insert#2(#true(), @x, @y, @ys)] = [1 1 0 0] [1] [0 1 0 0] @ys + [2] [0 1 1 1] [0] [0 1 0 0] [1] > [1 0 0 0] [0] [0 1 0 0] @ys + [2] [0 1 1 1] [0] [0 1 0 0] [1] = [::(@x, ::(@y, @ys))] [isortlist(@l)] = [0 1 1 1] [0] [0 1 0 0] @l + [0] [0 1 1 1] [0] [0 1 0 0] [1] >= [0 1 1 1] [0] [0 1 0 0] @l + [0] [0 1 1 1] [0] [0 1 0 0] [0] = [isortlist#1(@l)] [isortlist#1(::(@x, @xs))] = [0 2 1 1] [1] [0 1 0 0] @xs + [1] [0 2 1 1] [1] [0 1 0 0] [1] >= [0 2 1 1] [1] [0 1 0 0] @xs + [1] [0 2 1 1] [1] [0 1 0 0] [0] = [insert(@x, isortlist(@xs))] [isortlist#1(nil())] = [0] [0] [0] [0] >= [0] [0] [0] [0] = [nil()] [leq#1(::(@x, @xs), @l2)] = [1] [1] [0] [0] >= [1] [1] [0] [0] = [leq#2(@l2, @x, @xs)] [leq#1(nil(), @l2)] = [1] [1] [0] [0] > [0] [1] [0] [0] = [#true()] [leq#2(::(@y, @ys), @x, @xs)] = [1] [1] [0] [0] >= [1] [1] [0] [0] = [or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys)))] [leq#2(nil(), @x, @xs)] = [1] [1] [0] [0] > [0] [1] [0] [0] = [#false()] [or(@x, @y)] = [1 0 0 0] [1 0 0 0] [0] [0 0 0 0] @x + [0 0 0 0] @y + [1] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0] >= [0] [1] [0] [0] = [#or(@x, @y)] [#or(#false(), #false())] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#false()] [#or(#false(), #true())] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#true()] [#or(#true(), #false())] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#true()] [#or(#true(), #true())] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#true()] We return to the main proof. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { #equal(@x, @y) -> #eq(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , and(@x, @y) -> #and(@x, @y) , insert(@x, @l) -> insert#1(@l, @x) , leq(@l1, @l2) -> leq#1(@l1, @l2) , insert#2(#false(), @x, @y, @ys) -> ::(@y, insert(@x, @ys)) , leq#1(::(@x, @xs), @l2) -> leq#2(@l2, @x, @xs) , leq#2(::(@y, @ys), @x, @xs) -> or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))) , or(@x, @y) -> #or(@x, @y) } Weak Trs: { #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(nil(), nil()) -> #true() , #eq(#0(), #0()) -> #true() , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #s(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#pos(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #s(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#pos(@x), #0()) -> #GT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , #cklt(#EQ()) -> #false() , #cklt(#GT()) -> #false() , #cklt(#LT()) -> #true() , #and(#false(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#true(), #false()) -> #false() , #and(#true(), #true()) -> #true() , insert#1(::(@y, @ys), @x) -> insert#2(leq(@x, @y), @x, @y, @ys) , insert#1(nil(), @x) -> ::(@x, nil()) , insert#2(#true(), @x, @y, @ys) -> ::(@x, ::(@y, @ys)) , isortlist(@l) -> isortlist#1(@l) , isortlist#1(::(@x, @xs)) -> insert(@x, isortlist(@xs)) , isortlist#1(nil()) -> nil() , leq#1(nil(), @l2) -> #true() , leq#2(nil(), @x, @xs) -> #false() , #or(#false(), #false()) -> #false() , #or(#false(), #true()) -> #true() , #or(#true(), #false()) -> #true() , #or(#true(), #true()) -> #true() } Obligation: innermost runtime complexity Answer: MAYBE We use the processor 'matrix interpretation of dimension 4' to orient following rules strictly. Trs: { insert#2(#false(), @x, @y, @ys) -> ::(@y, insert(@x, @ys)) } The induced complexity on above rules (modulo remaining rules) is YES(?,O(n^3)) . These rules are moved into the corresponding weak component(s). Sub-proof: ---------- The following argument positions are usable: Uargs(#cklt) = {1}, Uargs(and) = {1, 2}, Uargs(#and) = {1, 2}, Uargs(insert) = {2}, Uargs(::) = {2}, Uargs(insert#2) = {1}, Uargs(or) = {1, 2} TcT has computed following constructor-based matrix interpretation satisfying not(EDA) and not(IDA(3)). [0 0 0 0] [0 0 0 0] [0] [#equal](x1, x2) = [1 0 1 0] x1 + [0 1 0 0] x2 + [1] [0 0 0 0] [0 0 0 0] [1] [0 1 1 0] [0 1 1 0] [1] [0] [#eq](x1, x2) = [0] [0] [1] [0] [#less](x1, x2) = [1] [0] [1] [0] [#compare](x1, x2) = [1] [0] [0] [1 0 0 0] [0] [#cklt](x1) = [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [1] [1 0 0 0] [1 0 0 0] [0] [and](x1, x2) = [0 1 1 0] x1 + [0 0 0 0] x2 + [0] [1 0 0 0] [0 0 0 0] [0] [1 1 1 1] [0 0 0 0] [0] [1 0 0 0] [1 0 0 0] [0] [#and](x1, x2) = [0 0 0 0] x1 + [0 0 0 0] x2 + [0] [0 0 0 0] [0 0 0 0] [0] [0 1 1 1] [0 0 0 0] [0] [1 0 1 0] [0] [insert](x1, x2) = [0 1 0 1] x2 + [0] [0 0 1 0] [1] [0 0 1 0] [0] [1 0 1 0] [0] [insert#1](x1, x2) = [0 1 0 1] x1 + [0] [0 0 1 0] [1] [0 0 1 0] [0] [1 0 0 0] [0] [::](x1, x2) = [0 1 0 1] x2 + [0] [0 0 1 0] [1] [0 0 1 0] [0] [0] [leq](x1, x2) = [1] [1] [1] [1 0 0 1] [1 0 1 0] [0] [insert#2](x1, x2, x3, x4) = [1 0 0 0] x1 + [0 1 1 1] x4 + [0] [0 0 0 1] [0 0 1 0] [1] [0 0 0 1] [0 0 1 0] [0] [0] [nil] = [1] [0] [0] [0] [#false] = [0] [0] [1] [0] [#true] = [0] [0] [1] [0 1 0 1] [0] [isortlist](x1) = [0 1 0 1] x1 + [0] [0 0 1 0] [0] [0 0 1 0] [0] [0 1 0 1] [0] [isortlist#1](x1) = [0 1 0 1] x1 + [0] [0 0 1 0] [0] [0 0 1 0] [0] [0] [leq#1](x1, x2) = [1] [1] [1] [0] [leq#2](x1, x2, x3) = [1] [0] [1] [1 0 0 0] [1 0 1 0] [0] [or](x1, x2) = [1 0 0 0] x1 + [0 0 0 0] x2 + [0] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [1] [0] [#or](x1, x2) = [0] [0] [1] [0] [#EQ] = [0] [0] [0] [0] [#GT] = [0] [0] [0] [0] [#LT] = [0] [0] [0] [0] [#0] = [0] [0] [0] [1 0 0 0] [0] [#neg](x1) = [0 0 0 0] x1 + [0] [0 0 1 0] [0] [0 0 1 0] [0] [1 0 0 0] [0] [#pos](x1) = [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 1 1] [0] [1 0 0 0] [0] [#s](x1) = [0 0 0 0] x1 + [0] [0 0 1 0] [0] [0 0 1 0] [0] This order satisfies following ordering constraints [#equal(@x, @y)] = [0 0 0 0] [0 0 0 0] [0] [1 0 1 0] @x + [0 1 0 0] @y + [1] [0 0 0 0] [0 0 0 0] [1] [0 1 1 0] [0 1 1 0] [1] >= [0] [0] [0] [1] = [#eq(@x, @y)] [#eq(::(@x_1, @x_2), ::(@y_1, @y_2))] = [0] [0] [0] [1] >= [0] [0] [0] [1] = [#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))] [#eq(::(@x_1, @x_2), nil())] = [0] [0] [0] [1] >= [0] [0] [0] [1] = [#false()] [#eq(nil(), ::(@y_1, @y_2))] = [0] [0] [0] [1] >= [0] [0] [0] [1] = [#false()] [#eq(nil(), nil())] = [0] [0] [0] [1] >= [0] [0] [0] [1] = [#true()] [#eq(#0(), #0())] = [0] [0] [0] [1] >= [0] [0] [0] [1] = [#true()] [#eq(#0(), #neg(@y))] = [0] [0] [0] [1] >= [0] [0] [0] [1] = [#false()] [#eq(#0(), #pos(@y))] = [0] [0] [0] [1] >= [0] [0] [0] [1] = [#false()] [#eq(#0(), #s(@y))] = [0] [0] [0] [1] >= [0] [0] [0] [1] = [#false()] [#eq(#neg(@x), #0())] = [0] [0] [0] [1] >= [0] [0] [0] [1] = [#false()] [#eq(#neg(@x), #neg(@y))] = [0] [0] [0] [1] >= [0] [0] [0] [1] = [#eq(@x, @y)] [#eq(#neg(@x), #pos(@y))] = [0] [0] [0] [1] >= [0] [0] [0] [1] = [#false()] [#eq(#pos(@x), #0())] = [0] [0] [0] [1] >= [0] [0] [0] [1] = [#false()] [#eq(#pos(@x), #neg(@y))] = [0] [0] [0] [1] >= [0] [0] [0] [1] = [#false()] [#eq(#pos(@x), #pos(@y))] = [0] [0] [0] [1] >= [0] [0] [0] [1] = [#eq(@x, @y)] [#eq(#s(@x), #0())] = [0] [0] [0] [1] >= [0] [0] [0] [1] = [#false()] [#eq(#s(@x), #s(@y))] = [0] [0] [0] [1] >= [0] [0] [0] [1] = [#eq(@x, @y)] [#less(@x, @y)] = [0] [1] [0] [1] >= [0] [0] [0] [1] = [#cklt(#compare(@x, @y))] [#compare(#0(), #0())] = [0] [1] [0] [0] >= [0] [0] [0] [0] = [#EQ()] [#compare(#0(), #neg(@y))] = [0] [1] [0] [0] >= [0] [0] [0] [0] = [#GT()] [#compare(#0(), #pos(@y))] = [0] [1] [0] [0] >= [0] [0] [0] [0] = [#LT()] [#compare(#0(), #s(@y))] = [0] [1] [0] [0] >= [0] [0] [0] [0] = [#LT()] [#compare(#neg(@x), #0())] = [0] [1] [0] [0] >= [0] [0] [0] [0] = [#LT()] [#compare(#neg(@x), #neg(@y))] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#compare(@y, @x)] [#compare(#neg(@x), #pos(@y))] = [0] [1] [0] [0] >= [0] [0] [0] [0] = [#LT()] [#compare(#pos(@x), #0())] = [0] [1] [0] [0] >= [0] [0] [0] [0] = [#GT()] [#compare(#pos(@x), #neg(@y))] = [0] [1] [0] [0] >= [0] [0] [0] [0] = [#GT()] [#compare(#pos(@x), #pos(@y))] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#compare(@x, @y)] [#compare(#s(@x), #0())] = [0] [1] [0] [0] >= [0] [0] [0] [0] = [#GT()] [#compare(#s(@x), #s(@y))] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#compare(@x, @y)] [#cklt(#EQ())] = [0] [0] [0] [1] >= [0] [0] [0] [1] = [#false()] [#cklt(#GT())] = [0] [0] [0] [1] >= [0] [0] [0] [1] = [#false()] [#cklt(#LT())] = [0] [0] [0] [1] >= [0] [0] [0] [1] = [#true()] [and(@x, @y)] = [1 0 0 0] [1 0 0 0] [0] [0 1 1 0] @x + [0 0 0 0] @y + [0] [1 0 0 0] [0 0 0 0] [0] [1 1 1 1] [0 0 0 0] [0] >= [1 0 0 0] [1 0 0 0] [0] [0 0 0 0] @x + [0 0 0 0] @y + [0] [0 0 0 0] [0 0 0 0] [0] [0 1 1 1] [0 0 0 0] [0] = [#and(@x, @y)] [#and(#false(), #false())] = [0] [0] [0] [1] >= [0] [0] [0] [1] = [#false()] [#and(#false(), #true())] = [0] [0] [0] [1] >= [0] [0] [0] [1] = [#false()] [#and(#true(), #false())] = [0] [0] [0] [1] >= [0] [0] [0] [1] = [#false()] [#and(#true(), #true())] = [0] [0] [0] [1] >= [0] [0] [0] [1] = [#true()] [insert(@x, @l)] = [1 0 1 0] [0] [0 1 0 1] @l + [0] [0 0 1 0] [1] [0 0 1 0] [0] >= [1 0 1 0] [0] [0 1 0 1] @l + [0] [0 0 1 0] [1] [0 0 1 0] [0] = [insert#1(@l, @x)] [insert#1(::(@y, @ys), @x)] = [1 0 1 0] [1] [0 1 1 1] @ys + [0] [0 0 1 0] [2] [0 0 1 0] [1] >= [1 0 1 0] [1] [0 1 1 1] @ys + [0] [0 0 1 0] [2] [0 0 1 0] [1] = [insert#2(leq(@x, @y), @x, @y, @ys)] [insert#1(nil(), @x)] = [0] [1] [1] [0] >= [0] [1] [1] [0] = [::(@x, nil())] [leq(@l1, @l2)] = [0] [1] [1] [1] >= [0] [1] [1] [1] = [leq#1(@l1, @l2)] [insert#2(#false(), @x, @y, @ys)] = [1 0 1 0] [1] [0 1 1 1] @ys + [0] [0 0 1 0] [2] [0 0 1 0] [1] > [1 0 1 0] [0] [0 1 1 1] @ys + [0] [0 0 1 0] [2] [0 0 1 0] [1] = [::(@y, insert(@x, @ys))] [insert#2(#true(), @x, @y, @ys)] = [1 0 1 0] [1] [0 1 1 1] @ys + [0] [0 0 1 0] [2] [0 0 1 0] [1] > [1 0 0 0] [0] [0 1 1 1] @ys + [0] [0 0 1 0] [2] [0 0 1 0] [1] = [::(@x, ::(@y, @ys))] [isortlist(@l)] = [0 1 0 1] [0] [0 1 0 1] @l + [0] [0 0 1 0] [0] [0 0 1 0] [0] >= [0 1 0 1] [0] [0 1 0 1] @l + [0] [0 0 1 0] [0] [0 0 1 0] [0] = [isortlist#1(@l)] [isortlist#1(::(@x, @xs))] = [0 1 1 1] [0] [0 1 1 1] @xs + [0] [0 0 1 0] [1] [0 0 1 0] [1] >= [0 1 1 1] [0] [0 1 1 1] @xs + [0] [0 0 1 0] [1] [0 0 1 0] [0] = [insert(@x, isortlist(@xs))] [isortlist#1(nil())] = [1] [1] [0] [0] > [0] [1] [0] [0] = [nil()] [leq#1(::(@x, @xs), @l2)] = [0] [1] [1] [1] >= [0] [1] [0] [1] = [leq#2(@l2, @x, @xs)] [leq#1(nil(), @l2)] = [0] [1] [1] [1] >= [0] [0] [0] [1] = [#true()] [leq#2(::(@y, @ys), @x, @xs)] = [0] [1] [0] [1] >= [0] [0] [0] [1] = [or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys)))] [leq#2(nil(), @x, @xs)] = [0] [1] [0] [1] >= [0] [0] [0] [1] = [#false()] [or(@x, @y)] = [1 0 0 0] [1 0 1 0] [0] [1 0 0 0] @x + [0 0 0 0] @y + [0] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [1] >= [0] [0] [0] [1] = [#or(@x, @y)] [#or(#false(), #false())] = [0] [0] [0] [1] >= [0] [0] [0] [1] = [#false()] [#or(#false(), #true())] = [0] [0] [0] [1] >= [0] [0] [0] [1] = [#true()] [#or(#true(), #false())] = [0] [0] [0] [1] >= [0] [0] [0] [1] = [#true()] [#or(#true(), #true())] = [0] [0] [0] [1] >= [0] [0] [0] [1] = [#true()] We return to the main proof. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { #equal(@x, @y) -> #eq(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , and(@x, @y) -> #and(@x, @y) , insert(@x, @l) -> insert#1(@l, @x) , leq(@l1, @l2) -> leq#1(@l1, @l2) , leq#1(::(@x, @xs), @l2) -> leq#2(@l2, @x, @xs) , leq#2(::(@y, @ys), @x, @xs) -> or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))) , or(@x, @y) -> #or(@x, @y) } Weak Trs: { #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(nil(), nil()) -> #true() , #eq(#0(), #0()) -> #true() , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #s(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#pos(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #s(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#pos(@x), #0()) -> #GT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , #cklt(#EQ()) -> #false() , #cklt(#GT()) -> #false() , #cklt(#LT()) -> #true() , #and(#false(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#true(), #false()) -> #false() , #and(#true(), #true()) -> #true() , insert#1(::(@y, @ys), @x) -> insert#2(leq(@x, @y), @x, @y, @ys) , insert#1(nil(), @x) -> ::(@x, nil()) , insert#2(#false(), @x, @y, @ys) -> ::(@y, insert(@x, @ys)) , insert#2(#true(), @x, @y, @ys) -> ::(@x, ::(@y, @ys)) , isortlist(@l) -> isortlist#1(@l) , isortlist#1(::(@x, @xs)) -> insert(@x, isortlist(@xs)) , isortlist#1(nil()) -> nil() , leq#1(nil(), @l2) -> #true() , leq#2(nil(), @x, @xs) -> #false() , #or(#false(), #false()) -> #false() , #or(#false(), #true()) -> #true() , #or(#true(), #false()) -> #true() , #or(#true(), #true()) -> #true() } Obligation: innermost runtime complexity Answer: MAYBE We use the processor 'matrix interpretation of dimension 4' to orient following rules strictly. Trs: { insert(@x, @l) -> insert#1(@l, @x) } The induced complexity on above rules (modulo remaining rules) is YES(?,O(n^3)) . These rules are moved into the corresponding weak component(s). Sub-proof: ---------- The following argument positions are usable: Uargs(#cklt) = {1}, Uargs(and) = {1, 2}, Uargs(#and) = {1, 2}, Uargs(insert) = {2}, Uargs(::) = {2}, Uargs(insert#2) = {1}, Uargs(or) = {1, 2} TcT has computed following constructor-based matrix interpretation satisfying not(EDA) and not(IDA(3)). [0 0 0 0] [0 0 0 0] [0] [#equal](x1, x2) = [0 0 0 0] x1 + [0 0 1 1] x2 + [1] [0 0 0 0] [0 0 0 0] [0] [1 1 1 0] [1 0 1 0] [0] [0 0 0 0] [0 0 0 0] [0] [#eq](x1, x2) = [0 0 0 0] x1 + [0 0 0 0] x2 + [1] [0 0 0 0] [0 0 0 0] [0] [1 0 1 0] [1 0 1 0] [0] [0 0 0 0] [0 0 0 0] [0] [#less](x1, x2) = [0 0 0 1] x1 + [0 1 0 0] x2 + [1] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0] [#compare](x1, x2) = [0 0 0 0] x1 + [0 0 0 0] x2 + [0] [1 0 0 0] [1 0 0 0] [0] [0 0 0 0] [0 0 0 0] [1] [1 0 0 0] [0] [#cklt](x1) = [0 1 0 1] x1 + [0] [0 0 0 0] [0] [0 1 0 1] [0] [1 0 0 0] [1 0 0 0] [0] [and](x1, x2) = [0 0 1 0] x1 + [0 0 0 0] x2 + [1] [0 0 0 0] [0 0 0 0] [0] [0 1 0 1] [0 0 0 0] [0] [1 0 0 0] [1 0 0 0] [0] [#and](x1, x2) = [0 0 0 0] x1 + [0 0 0 0] x2 + [1] [0 0 0 0] [0 0 0 0] [0] [0 1 0 0] [0 0 0 0] [0] [1 0 1 0] [1] [insert](x1, x2) = [0 0 1 0] x2 + [0] [0 0 1 0] [1] [0 1 0 1] [1] [1 0 1 0] [0] [insert#1](x1, x2) = [0 0 1 0] x1 + [0] [0 0 1 0] [1] [0 1 0 1] [1] [1 0 0 0] [0] [::](x1, x2) = [0 0 1 0] x2 + [0] [0 0 1 0] [1] [0 1 0 1] [1] [0] [leq](x1, x2) = [1] [0] [1] [1 0 1 1] [1 0 1 0] [0] [insert#2](x1, x2, x3, x4) = [1 0 0 0] x1 + [0 0 1 0] x4 + [1] [0 1 1 0] [0 0 1 0] [1] [1 1 0 0] [0 1 1 1] [1] [0] [nil] = [1] [0] [1] [0] [#false] = [1] [0] [1] [0] [#true] = [1] [0] [0] [0 1 0 1] [0] [isortlist](x1) = [0 0 1 0] x1 + [1] [0 0 1 0] [0] [0 1 1 1] [0] [0 1 0 1] [0] [isortlist#1](x1) = [0 0 1 0] x1 + [1] [0 0 1 0] [0] [0 1 1 1] [0] [0] [leq#1](x1, x2) = [1] [0] [1] [0] [leq#2](x1, x2, x3) = [1] [0] [1] [1 0 0 0] [1 0 0 0] [0] [or](x1, x2) = [0 0 0 0] x1 + [0 1 1 0] x2 + [0] [0 0 0 0] [0 0 1 0] [0] [0 0 0 0] [0 1 1 0] [0] [0 0 0 0] [0] [#or](x1, x2) = [0 1 0 0] x2 + [0] [0 0 0 0] [0] [0 1 0 0] [0] [0] [#EQ] = [0] [0] [1] [0] [#GT] = [0] [1] [1] [0] [#LT] = [0] [0] [1] [1] [#0] = [0] [0] [0] [1 0 1 0] [0] [#neg](x1) = [0 1 0 0] x1 + [0] [0 0 0 0] [0] [0 1 0 0] [0] [1 0 0 0] [1] [#pos](x1) = [0 0 1 0] x1 + [0] [0 0 1 0] [0] [0 0 0 1] [0] [1 0 1 0] [0] [#s](x1) = [0 1 0 0] x1 + [0] [0 0 0 0] [0] [0 0 1 1] [0] This order satisfies following ordering constraints [#equal(@x, @y)] = [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] @x + [0 0 1 1] @y + [1] [0 0 0 0] [0 0 0 0] [0] [1 1 1 0] [1 0 1 0] [0] >= [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] @x + [0 0 0 0] @y + [1] [0 0 0 0] [0 0 0 0] [0] [1 0 1 0] [1 0 1 0] [0] = [#eq(@x, @y)] [#eq(::(@x_1, @x_2), ::(@y_1, @y_2))] = [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] @x_2 + [0 0 0 0] @y_2 + [1] [0 0 0 0] [0 0 0 0] [0] [1 0 1 0] [1 0 1 0] [2] >= [0] [1] [0] [1] = [#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))] [#eq(::(@x_1, @x_2), nil())] = [0 0 0 0] [0] [0 0 0 0] @x_2 + [1] [0 0 0 0] [0] [1 0 1 0] [1] >= [0] [1] [0] [1] = [#false()] [#eq(nil(), ::(@y_1, @y_2))] = [0 0 0 0] [0] [0 0 0 0] @y_2 + [1] [0 0 0 0] [0] [1 0 1 0] [1] >= [0] [1] [0] [1] = [#false()] [#eq(nil(), nil())] = [0] [1] [0] [0] >= [0] [1] [0] [0] = [#true()] [#eq(#0(), #0())] = [0] [1] [0] [2] >= [0] [1] [0] [0] = [#true()] [#eq(#0(), #neg(@y))] = [0 0 0 0] [0] [0 0 0 0] @y + [1] [0 0 0 0] [0] [1 0 1 0] [1] >= [0] [1] [0] [1] = [#false()] [#eq(#0(), #pos(@y))] = [0 0 0 0] [0] [0 0 0 0] @y + [1] [0 0 0 0] [0] [1 0 1 0] [2] >= [0] [1] [0] [1] = [#false()] [#eq(#0(), #s(@y))] = [0 0 0 0] [0] [0 0 0 0] @y + [1] [0 0 0 0] [0] [1 0 1 0] [1] >= [0] [1] [0] [1] = [#false()] [#eq(#neg(@x), #0())] = [0 0 0 0] [0] [0 0 0 0] @x + [1] [0 0 0 0] [0] [1 0 1 0] [1] >= [0] [1] [0] [1] = [#false()] [#eq(#neg(@x), #neg(@y))] = [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] @x + [0 0 0 0] @y + [1] [0 0 0 0] [0 0 0 0] [0] [1 0 1 0] [1 0 1 0] [0] >= [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] @x + [0 0 0 0] @y + [1] [0 0 0 0] [0 0 0 0] [0] [1 0 1 0] [1 0 1 0] [0] = [#eq(@x, @y)] [#eq(#neg(@x), #pos(@y))] = [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] @x + [0 0 0 0] @y + [1] [0 0 0 0] [0 0 0 0] [0] [1 0 1 0] [1 0 1 0] [1] >= [0] [1] [0] [1] = [#false()] [#eq(#pos(@x), #0())] = [0 0 0 0] [0] [0 0 0 0] @x + [1] [0 0 0 0] [0] [1 0 1 0] [2] >= [0] [1] [0] [1] = [#false()] [#eq(#pos(@x), #neg(@y))] = [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] @x + [0 0 0 0] @y + [1] [0 0 0 0] [0 0 0 0] [0] [1 0 1 0] [1 0 1 0] [1] >= [0] [1] [0] [1] = [#false()] [#eq(#pos(@x), #pos(@y))] = [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] @x + [0 0 0 0] @y + [1] [0 0 0 0] [0 0 0 0] [0] [1 0 1 0] [1 0 1 0] [2] >= [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] @x + [0 0 0 0] @y + [1] [0 0 0 0] [0 0 0 0] [0] [1 0 1 0] [1 0 1 0] [0] = [#eq(@x, @y)] [#eq(#s(@x), #0())] = [0 0 0 0] [0] [0 0 0 0] @x + [1] [0 0 0 0] [0] [1 0 1 0] [1] >= [0] [1] [0] [1] = [#false()] [#eq(#s(@x), #s(@y))] = [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] @x + [0 0 0 0] @y + [1] [0 0 0 0] [0 0 0 0] [0] [1 0 1 0] [1 0 1 0] [0] >= [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] @x + [0 0 0 0] @y + [1] [0 0 0 0] [0 0 0 0] [0] [1 0 1 0] [1 0 1 0] [0] = [#eq(@x, @y)] [#less(@x, @y)] = [0 0 0 0] [0 0 0 0] [0] [0 0 0 1] @x + [0 1 0 0] @y + [1] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [1] >= [0] [1] [0] [1] = [#cklt(#compare(@x, @y))] [#compare(#0(), #0())] = [0] [0] [2] [1] >= [0] [0] [0] [1] = [#EQ()] [#compare(#0(), #neg(@y))] = [0 0 0 0] [0] [0 0 0 0] @y + [0] [1 0 1 0] [1] [0 0 0 0] [1] >= [0] [0] [1] [1] = [#GT()] [#compare(#0(), #pos(@y))] = [0 0 0 0] [0] [0 0 0 0] @y + [0] [1 0 0 0] [2] [0 0 0 0] [1] >= [0] [0] [0] [1] = [#LT()] [#compare(#0(), #s(@y))] = [0 0 0 0] [0] [0 0 0 0] @y + [0] [1 0 1 0] [1] [0 0 0 0] [1] >= [0] [0] [0] [1] = [#LT()] [#compare(#neg(@x), #0())] = [0 0 0 0] [0] [0 0 0 0] @x + [0] [1 0 1 0] [1] [0 0 0 0] [1] >= [0] [0] [0] [1] = [#LT()] [#compare(#neg(@x), #neg(@y))] = [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] @x + [0 0 0 0] @y + [0] [1 0 1 0] [1 0 1 0] [0] [0 0 0 0] [0 0 0 0] [1] >= [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] @x + [0 0 0 0] @y + [0] [1 0 0 0] [1 0 0 0] [0] [0 0 0 0] [0 0 0 0] [1] = [#compare(@y, @x)] [#compare(#neg(@x), #pos(@y))] = [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] @x + [0 0 0 0] @y + [0] [1 0 1 0] [1 0 0 0] [1] [0 0 0 0] [0 0 0 0] [1] >= [0] [0] [0] [1] = [#LT()] [#compare(#pos(@x), #0())] = [0 0 0 0] [0] [0 0 0 0] @x + [0] [1 0 0 0] [2] [0 0 0 0] [1] >= [0] [0] [1] [1] = [#GT()] [#compare(#pos(@x), #neg(@y))] = [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] @x + [0 0 0 0] @y + [0] [1 0 0 0] [1 0 1 0] [1] [0 0 0 0] [0 0 0 0] [1] >= [0] [0] [1] [1] = [#GT()] [#compare(#pos(@x), #pos(@y))] = [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] @x + [0 0 0 0] @y + [0] [1 0 0 0] [1 0 0 0] [2] [0 0 0 0] [0 0 0 0] [1] >= [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] @x + [0 0 0 0] @y + [0] [1 0 0 0] [1 0 0 0] [0] [0 0 0 0] [0 0 0 0] [1] = [#compare(@x, @y)] [#compare(#s(@x), #0())] = [0 0 0 0] [0] [0 0 0 0] @x + [0] [1 0 1 0] [1] [0 0 0 0] [1] >= [0] [0] [1] [1] = [#GT()] [#compare(#s(@x), #s(@y))] = [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] @x + [0 0 0 0] @y + [0] [1 0 1 0] [1 0 1 0] [0] [0 0 0 0] [0 0 0 0] [1] >= [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] @x + [0 0 0 0] @y + [0] [1 0 0 0] [1 0 0 0] [0] [0 0 0 0] [0 0 0 0] [1] = [#compare(@x, @y)] [#cklt(#EQ())] = [0] [1] [0] [1] >= [0] [1] [0] [1] = [#false()] [#cklt(#GT())] = [0] [1] [0] [1] >= [0] [1] [0] [1] = [#false()] [#cklt(#LT())] = [0] [1] [0] [1] >= [0] [1] [0] [0] = [#true()] [and(@x, @y)] = [1 0 0 0] [1 0 0 0] [0] [0 0 1 0] @x + [0 0 0 0] @y + [1] [0 0 0 0] [0 0 0 0] [0] [0 1 0 1] [0 0 0 0] [0] >= [1 0 0 0] [1 0 0 0] [0] [0 0 0 0] @x + [0 0 0 0] @y + [1] [0 0 0 0] [0 0 0 0] [0] [0 1 0 0] [0 0 0 0] [0] = [#and(@x, @y)] [#and(#false(), #false())] = [0] [1] [0] [1] >= [0] [1] [0] [1] = [#false()] [#and(#false(), #true())] = [0] [1] [0] [1] >= [0] [1] [0] [1] = [#false()] [#and(#true(), #false())] = [0] [1] [0] [1] >= [0] [1] [0] [1] = [#false()] [#and(#true(), #true())] = [0] [1] [0] [1] >= [0] [1] [0] [0] = [#true()] [insert(@x, @l)] = [1 0 1 0] [1] [0 0 1 0] @l + [0] [0 0 1 0] [1] [0 1 0 1] [1] > [1 0 1 0] [0] [0 0 1 0] @l + [0] [0 0 1 0] [1] [0 1 0 1] [1] = [insert#1(@l, @x)] [insert#1(::(@y, @ys), @x)] = [1 0 1 0] [1] [0 0 1 0] @ys + [1] [0 0 1 0] [2] [0 1 1 1] [2] >= [1 0 1 0] [1] [0 0 1 0] @ys + [1] [0 0 1 0] [2] [0 1 1 1] [2] = [insert#2(leq(@x, @y), @x, @y, @ys)] [insert#1(nil(), @x)] = [0] [0] [1] [3] >= [0] [0] [1] [3] = [::(@x, nil())] [leq(@l1, @l2)] = [0] [1] [0] [1] >= [0] [1] [0] [1] = [leq#1(@l1, @l2)] [insert#2(#false(), @x, @y, @ys)] = [1 0 1 0] [1] [0 0 1 0] @ys + [1] [0 0 1 0] [2] [0 1 1 1] [2] >= [1 0 1 0] [1] [0 0 1 0] @ys + [1] [0 0 1 0] [2] [0 1 1 1] [2] = [::(@y, insert(@x, @ys))] [insert#2(#true(), @x, @y, @ys)] = [1 0 1 0] [0] [0 0 1 0] @ys + [1] [0 0 1 0] [2] [0 1 1 1] [2] >= [1 0 0 0] [0] [0 0 1 0] @ys + [1] [0 0 1 0] [2] [0 1 1 1] [2] = [::(@x, ::(@y, @ys))] [isortlist(@l)] = [0 1 0 1] [0] [0 0 1 0] @l + [1] [0 0 1 0] [0] [0 1 1 1] [0] >= [0 1 0 1] [0] [0 0 1 0] @l + [1] [0 0 1 0] [0] [0 1 1 1] [0] = [isortlist#1(@l)] [isortlist#1(::(@x, @xs))] = [0 1 1 1] [1] [0 0 1 0] @xs + [2] [0 0 1 0] [1] [0 1 2 1] [2] >= [0 1 1 1] [1] [0 0 1 0] @xs + [0] [0 0 1 0] [1] [0 1 2 1] [2] = [insert(@x, isortlist(@xs))] [isortlist#1(nil())] = [2] [1] [0] [2] > [0] [1] [0] [1] = [nil()] [leq#1(::(@x, @xs), @l2)] = [0] [1] [0] [1] >= [0] [1] [0] [1] = [leq#2(@l2, @x, @xs)] [leq#1(nil(), @l2)] = [0] [1] [0] [1] >= [0] [1] [0] [0] = [#true()] [leq#2(::(@y, @ys), @x, @xs)] = [0] [1] [0] [1] >= [0] [1] [0] [1] = [or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys)))] [leq#2(nil(), @x, @xs)] = [0] [1] [0] [1] >= [0] [1] [0] [1] = [#false()] [or(@x, @y)] = [1 0 0 0] [1 0 0 0] [0] [0 0 0 0] @x + [0 1 1 0] @y + [0] [0 0 0 0] [0 0 1 0] [0] [0 0 0 0] [0 1 1 0] [0] >= [0 0 0 0] [0] [0 1 0 0] @y + [0] [0 0 0 0] [0] [0 1 0 0] [0] = [#or(@x, @y)] [#or(#false(), #false())] = [0] [1] [0] [1] >= [0] [1] [0] [1] = [#false()] [#or(#false(), #true())] = [0] [1] [0] [1] >= [0] [1] [0] [0] = [#true()] [#or(#true(), #false())] = [0] [1] [0] [1] >= [0] [1] [0] [0] = [#true()] [#or(#true(), #true())] = [0] [1] [0] [1] >= [0] [1] [0] [0] = [#true()] We return to the main proof. We are left with following problem, upon which TcT provides the certificate MAYBE. Strict Trs: { #equal(@x, @y) -> #eq(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , and(@x, @y) -> #and(@x, @y) , leq(@l1, @l2) -> leq#1(@l1, @l2) , leq#1(::(@x, @xs), @l2) -> leq#2(@l2, @x, @xs) , leq#2(::(@y, @ys), @x, @xs) -> or(#less(@x, @y), and(#equal(@x, @y), leq(@xs, @ys))) , or(@x, @y) -> #or(@x, @y) } Weak Trs: { #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(nil(), nil()) -> #true() , #eq(#0(), #0()) -> #true() , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #s(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#pos(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #s(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#pos(@x), #0()) -> #GT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , #cklt(#EQ()) -> #false() , #cklt(#GT()) -> #false() , #cklt(#LT()) -> #true() , #and(#false(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#true(), #false()) -> #false() , #and(#true(), #true()) -> #true() , insert(@x, @l) -> insert#1(@l, @x) , insert#1(::(@y, @ys), @x) -> insert#2(leq(@x, @y), @x, @y, @ys) , insert#1(nil(), @x) -> ::(@x, nil()) , insert#2(#false(), @x, @y, @ys) -> ::(@y, insert(@x, @ys)) , insert#2(#true(), @x, @y, @ys) -> ::(@x, ::(@y, @ys)) , isortlist(@l) -> isortlist#1(@l) , isortlist#1(::(@x, @xs)) -> insert(@x, isortlist(@xs)) , isortlist#1(nil()) -> nil() , leq#1(nil(), @l2) -> #true() , leq#2(nil(), @x, @xs) -> #false() , #or(#false(), #false()) -> #false() , #or(#false(), #true()) -> #true() , #or(#true(), #false()) -> #true() , #or(#true(), #true()) -> #true() } Obligation: innermost runtime complexity Answer: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'Fastest' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'trivial' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'Sequentially' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'Polynomial Path Order (PS)' failed due to the following reason: The input cannot be shown compatible 2) 'Polynomial Path Order' failed due to the following reason: The input cannot be shown compatible 2) 'Fastest (timeout of 5 seconds)' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 2) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. Arrrr..