YES(O(1),O(n^2)) We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict Trs: { -(@x, @y) -> #sub(@x, @y) , sub(@b1, @b2) -> sub#1(sub'(@b1, @b2, #abs(#0()))) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , div(@x, @y) -> #div(@x, @y) , bitToInt'#1(nil(), @n) -> #abs(#0()) , bitToInt'#1(::(@x, @xs), @n) -> +(*(@x, @n), bitToInt'(@xs, *(@n, #pos(#s(#s(#0())))))) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , mult3(@b1, @b2, @b3) -> mult(mult(@b1, @b2), @b2) , sub#1(tuple#2(@b, @_@1)) -> @b , leq(@b1, @b2) -> #less(compare(@b1, @b2), #pos(#s(#0()))) , #greater(@x, @y) -> #ckgt(#compare(@x, @y)) , bitToInt'(@b, @n) -> bitToInt'#1(@b, @n) , mult(@b1, @b2) -> mult#1(@b1, @b2) , bitToInt(@b) -> bitToInt'(@b, #abs(#pos(#s(#0())))) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , *(@x, @y) -> #mult(@x, @y) , sub'#5(#true(), @z, @zs) -> ::(#abs(#0()), @zs) , sub'#5(#false(), @z, @zs) -> ::(@z, @zs) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , sub'#3(tuple#2(@z, @r'), @xs, @ys) -> sub'#4(sub'(@xs, @ys, @r'), @z) , #equal(@x, @y) -> #eq(@x, @y) , sub'#2(nil(), @r, @x, @xs) -> tuple#2(nil(), @r) , sub'#2(::(@y, @ys), @r, @x, @xs) -> sub'#3(diff(@x, @y, @r), @xs, @ys) , compare#2(nil(), @x, @xs) -> #abs(#0()) , compare#2(::(@y, @ys), @x, @xs) -> compare#3(compare(@xs, @ys), @x, @y) , sub'#4(tuple#2(@zs, @s), @z) -> tuple#2(sub'#5(#equal(@s, #pos(#s(#0()))), @z, @zs), @s) , compare#5(#true(), @x, @y) -> -(#0(), #pos(#s(#0()))) , compare#5(#false(), @x, @y) -> compare#6(#greater(@x, @y)) , compare#3(@r, @x, @y) -> compare#4(#equal(@r, #0()), @r, @x, @y) , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , sub'#1(nil(), @b2, @r) -> tuple#2(nil(), @r) , sub'#1(::(@x, @xs), @b2, @r) -> sub'#2(@b2, @r, @x, @xs) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , sub'(@b1, @b2, @r) -> sub'#1(@b1, @b2, @r) , compare(@b1, @b2) -> compare#1(@b1, @b2) , compare#6(#true()) -> #abs(#pos(#s(#0()))) , compare#6(#false()) -> #abs(#0()) , compare#4(#true(), @r, @x, @y) -> compare#5(#less(@x, @y), @x, @y) , compare#4(#false(), @r, @x, @y) -> @r , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , compare#1(nil(), @b2) -> #abs(#0()) , compare#1(::(@x, @xs), @b2) -> compare#2(@b2, @x, @xs) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) } Weak Trs: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #ckgt(#EQ()) -> #false() , #ckgt(#LT()) -> #false() , #ckgt(#GT()) -> #true() , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We add the following dependency tuples: Strict DPs: { -^#(@x, @y) -> c_1(#sub^#(@x, @y)) , sub^#(@b1, @b2) -> c_2(sub#1^#(sub'(@b1, @b2, #abs(#0()))), sub'^#(@b1, @b2, #abs(#0())), #abs^#(#0())) , sub#1^#(tuple#2(@b, @_@1)) -> c_12() , sub'^#(@b1, @b2, @r) -> c_51(sub'#1^#(@b1, @b2, @r)) , #abs^#(#neg(@x)) -> c_62() , #abs^#(#pos(@x)) -> c_63() , #abs^#(#0()) -> c_64() , #abs^#(#s(@x)) -> c_65() , diff#1^#(#true()) -> c_3(#abs^#(#pos(#s(#0())))) , diff#1^#(#false()) -> c_4(#abs^#(#0())) , mult#2^#(@zs, @b2, @x) -> c_5(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs), #equal^#(@x, #pos(#s(#0())))) , mult#3^#(#true(), @b2, @zs) -> c_42(add^#(@b2, @zs)) , mult#3^#(#false(), @b2, @zs) -> c_43() , #equal^#(@x, @y) -> c_29(#eq^#(@x, @y)) , div^#(@x, @y) -> c_6(#div^#(@x, @y)) , bitToInt'#1^#(nil(), @n) -> c_7(#abs^#(#0())) , bitToInt'#1^#(::(@x, @xs), @n) -> c_8(+^#(*(@x, @n), bitToInt'(@xs, *(@n, #pos(#s(#s(#0())))))), *^#(@x, @n), bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0()))))), *^#(@n, #pos(#s(#s(#0()))))) , +^#(@x, @y) -> c_21(#add^#(@x, @y)) , *^#(@x, @y) -> c_24(#mult^#(@x, @y)) , bitToInt'^#(@b, @n) -> c_15(bitToInt'#1^#(@b, @n)) , sum^#(@x, @y, @r) -> c_9(sum#1^#(+(+(@x, @y), @r)), +^#(+(@x, @y), @r), +^#(@x, @y)) , sum#1^#(@s) -> c_20(sum#2^#(#equal(@s, #0()), @s), #equal^#(@s, #0())) , mod^#(@x, @y) -> c_10(-^#(@x, *(@x, div(@x, @y))), *^#(@x, div(@x, @y)), div^#(@x, @y)) , mult3^#(@b1, @b2, @b3) -> c_11(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , mult^#(@b1, @b2) -> c_16(mult#1^#(@b1, @b2)) , leq^#(@b1, @b2) -> c_13(#less^#(compare(@b1, @b2), #pos(#s(#0()))), compare^#(@b1, @b2)) , #less^#(@x, @y) -> c_27(#cklt^#(#compare(@x, @y)), #compare^#(@x, @y)) , compare^#(@b1, @b2) -> c_52(compare#1^#(@b1, @b2)) , #greater^#(@x, @y) -> c_14(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y)) , mult#1^#(nil(), @b2) -> c_49() , mult#1^#(::(@x, @xs), @b2) -> c_50(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), #abs^#(#0()), mult^#(@xs, @b2)) , bitToInt^#(@b) -> c_17(bitToInt'^#(@b, #abs(#pos(#s(#0())))), #abs^#(#pos(#s(#0())))) , sum#2^#(#true(), @s) -> c_18(#abs^#(#0()), #abs^#(#0())) , sum#2^#(#false(), @s) -> c_19(sum#3^#(#equal(@s, #pos(#s(#0()))), @s), #equal^#(@s, #pos(#s(#0())))) , sum#3^#(#true(), @s) -> c_57(#abs^#(#pos(#s(#0()))), #abs^#(#0())) , sum#3^#(#false(), @s) -> c_58(sum#4^#(#equal(@s, #pos(#s(#s(#0()))))), #equal^#(@s, #pos(#s(#s(#0()))))) , sum#4^#(#true()) -> c_22(#abs^#(#0()), #abs^#(#pos(#s(#0())))) , sum#4^#(#false()) -> c_23(#abs^#(#pos(#s(#0()))), #abs^#(#pos(#s(#0())))) , sub'#5^#(#true(), @z, @zs) -> c_25(#abs^#(#0())) , sub'#5^#(#false(), @z, @zs) -> c_26() , sub'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_28(sub'#4^#(sub'(@xs, @ys, @r'), @z), sub'^#(@xs, @ys, @r')) , sub'#4^#(tuple#2(@zs, @s), @z) -> c_34(sub'#5^#(#equal(@s, #pos(#s(#0()))), @z, @zs), #equal^#(@s, #pos(#s(#0())))) , sub'#2^#(nil(), @r, @x, @xs) -> c_30() , sub'#2^#(::(@y, @ys), @r, @x, @xs) -> c_31(sub'#3^#(diff(@x, @y, @r), @xs, @ys), diff^#(@x, @y, @r)) , diff^#(@x, @y, @r) -> c_48(mod^#(+(+(@x, @y), @r), #pos(#s(#s(#0())))), +^#(+(@x, @y), @r), +^#(@x, @y), diff#1^#(#less(-(-(@x, @y), @r), #0())), #less^#(-(-(@x, @y), @r), #0()), -^#(-(@x, @y), @r), -^#(@x, @y)) , compare#2^#(nil(), @x, @xs) -> c_32(#abs^#(#0())) , compare#2^#(::(@y, @ys), @x, @xs) -> c_33(compare#3^#(compare(@xs, @ys), @x, @y), compare^#(@xs, @ys)) , compare#3^#(@r, @x, @y) -> c_37(compare#4^#(#equal(@r, #0()), @r, @x, @y), #equal^#(@r, #0())) , compare#5^#(#true(), @x, @y) -> c_35(-^#(#0(), #pos(#s(#0())))) , compare#5^#(#false(), @x, @y) -> c_36(compare#6^#(#greater(@x, @y)), #greater^#(@x, @y)) , compare#6^#(#true()) -> c_53(#abs^#(#pos(#s(#0())))) , compare#6^#(#false()) -> c_54(#abs^#(#0())) , compare#4^#(#true(), @r, @x, @y) -> c_55(compare#5^#(#less(@x, @y), @x, @y), #less^#(@x, @y)) , compare#4^#(#false(), @r, @x, @y) -> c_56() , add^#(@b1, @b2) -> c_38(add'^#(@b1, @b2, #abs(#0())), #abs^#(#0())) , add'^#(@b1, @b2, @r) -> c_59(add'#1^#(@b1, @b2, @r)) , add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_39(add'^#(@xs, @ys, @r')) , sub'#1^#(nil(), @b2, @r) -> c_40() , sub'#1^#(::(@x, @xs), @b2, @r) -> c_41(sub'#2^#(@b2, @r, @x, @xs)) , add'#1^#(nil(), @b2, @r) -> c_44() , add'#1^#(::(@x, @xs), @b2, @r) -> c_45(add'#2^#(@b2, @r, @x, @xs)) , add'#2^#(nil(), @r, @x, @xs) -> c_46() , add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_47(add'#3^#(sum(@x, @y, @r), @xs, @ys), sum^#(@x, @y, @r)) , compare#1^#(nil(), @b2) -> c_60(#abs^#(#0())) , compare#1^#(::(@x, @xs), @b2) -> c_61(compare#2^#(@b2, @x, @xs)) } Weak DPs: { #sub^#(@x, #neg(@y)) -> c_120(#add^#(@x, #pos(@y))) , #sub^#(@x, #pos(@y)) -> c_121(#add^#(@x, #neg(@y))) , #sub^#(@x, #0()) -> c_122() , #div^#(#neg(@x), #neg(@y)) -> c_136(#natdiv^#(@x, @y)) , #div^#(#neg(@x), #pos(@y)) -> c_137(#natdiv^#(@x, @y)) , #div^#(#neg(@x), #0()) -> c_138() , #div^#(#pos(@x), #neg(@y)) -> c_139(#natdiv^#(@x, @y)) , #div^#(#pos(@x), #pos(@y)) -> c_140(#natdiv^#(@x, @y)) , #div^#(#pos(@x), #0()) -> c_141() , #div^#(#0(), #neg(@y)) -> c_142() , #div^#(#0(), #pos(@y)) -> c_143() , #div^#(#0(), #0()) -> c_144() , #ckgt^#(#EQ()) -> c_70() , #ckgt^#(#LT()) -> c_71() , #ckgt^#(#GT()) -> c_72() , #compare^#(#neg(@x), #neg(@y)) -> c_82(#compare^#(@y, @x)) , #compare^#(#neg(@x), #pos(@y)) -> c_83() , #compare^#(#neg(@x), #0()) -> c_84() , #compare^#(#pos(@x), #neg(@y)) -> c_85() , #compare^#(#pos(@x), #pos(@y)) -> c_86(#compare^#(@x, @y)) , #compare^#(#pos(@x), #0()) -> c_87() , #compare^#(#0(), #neg(@y)) -> c_88() , #compare^#(#0(), #pos(@y)) -> c_89() , #compare^#(#0(), #0()) -> c_90() , #compare^#(#0(), #s(@y)) -> c_91() , #compare^#(#s(@x), #0()) -> c_92() , #compare^#(#s(@x), #s(@y)) -> c_93(#compare^#(@x, @y)) , #add^#(#neg(#s(#0())), @y) -> c_73(#pred^#(@y)) , #add^#(#neg(#s(#s(@x))), @y) -> c_74(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , #add^#(#pos(#s(#0())), @y) -> c_75(#succ^#(@y)) , #add^#(#pos(#s(#s(@x))), @y) -> c_76(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , #add^#(#0(), @y) -> c_77() , #mult^#(#neg(@x), #neg(@y)) -> c_123(#natmult^#(@x, @y)) , #mult^#(#neg(@x), #pos(@y)) -> c_124(#natmult^#(@x, @y)) , #mult^#(#neg(@x), #0()) -> c_125() , #mult^#(#pos(@x), #neg(@y)) -> c_126(#natmult^#(@x, @y)) , #mult^#(#pos(@x), #pos(@y)) -> c_127(#natmult^#(@x, @y)) , #mult^#(#pos(@x), #0()) -> c_128() , #mult^#(#0(), #neg(@y)) -> c_129() , #mult^#(#0(), #pos(@y)) -> c_130() , #mult^#(#0(), #0()) -> c_131() , #cklt^#(#EQ()) -> c_117() , #cklt^#(#LT()) -> c_118() , #cklt^#(#GT()) -> c_119() , #eq^#(nil(), nil()) -> c_94() , #eq^#(nil(), tuple#2(@y_1, @y_2)) -> c_95() , #eq^#(nil(), ::(@y_1, @y_2)) -> c_96() , #eq^#(#neg(@x), #neg(@y)) -> c_97(#eq^#(@x, @y)) , #eq^#(#neg(@x), #pos(@y)) -> c_98() , #eq^#(#neg(@x), #0()) -> c_99() , #eq^#(#pos(@x), #neg(@y)) -> c_100() , #eq^#(#pos(@x), #pos(@y)) -> c_101(#eq^#(@x, @y)) , #eq^#(#pos(@x), #0()) -> c_102() , #eq^#(tuple#2(@x_1, @x_2), nil()) -> c_103() , #eq^#(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> c_104(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), #eq^#(@x_1, @y_1), #eq^#(@x_2, @y_2)) , #eq^#(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> c_105() , #eq^#(::(@x_1, @x_2), nil()) -> c_106() , #eq^#(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> c_107() , #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) -> c_108(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), #eq^#(@x_1, @y_1), #eq^#(@x_2, @y_2)) , #eq^#(#0(), #neg(@y)) -> c_109() , #eq^#(#0(), #pos(@y)) -> c_110() , #eq^#(#0(), #0()) -> c_111() , #eq^#(#0(), #s(@y)) -> c_112() , #eq^#(#s(@x), #0()) -> c_113() , #eq^#(#s(@x), #s(@y)) -> c_114(#eq^#(@x, @y)) , #natsub^#(@x, #0()) -> c_66() , #natsub^#(#s(@x), #s(@y)) -> c_67(#natsub^#(@x, @y)) , #natdiv^#(#0(), #0()) -> c_68() , #natdiv^#(#s(@x), #s(@y)) -> c_69(#natdiv^#(#natsub(@x, @y), #s(@y)), #natsub^#(@x, @y)) , #pred^#(#neg(#s(@x))) -> c_145() , #pred^#(#pos(#s(#0()))) -> c_146() , #pred^#(#pos(#s(#s(@x)))) -> c_147() , #pred^#(#0()) -> c_148() , #succ^#(#neg(#s(#0()))) -> c_132() , #succ^#(#neg(#s(#s(@x)))) -> c_133() , #succ^#(#pos(#s(@x))) -> c_134() , #succ^#(#0()) -> c_135() , #and^#(#true(), #true()) -> c_78() , #and^#(#true(), #false()) -> c_79() , #and^#(#false(), #true()) -> c_80() , #and^#(#false(), #false()) -> c_81() , #natmult^#(#0(), @y) -> c_115() , #natmult^#(#s(@x), @y) -> c_116(#add^#(#pos(@y), #natmult(@x, @y)), #natmult^#(@x, @y)) } and mark the set of starting terms. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { -^#(@x, @y) -> c_1(#sub^#(@x, @y)) , sub^#(@b1, @b2) -> c_2(sub#1^#(sub'(@b1, @b2, #abs(#0()))), sub'^#(@b1, @b2, #abs(#0())), #abs^#(#0())) , sub#1^#(tuple#2(@b, @_@1)) -> c_12() , sub'^#(@b1, @b2, @r) -> c_51(sub'#1^#(@b1, @b2, @r)) , #abs^#(#neg(@x)) -> c_62() , #abs^#(#pos(@x)) -> c_63() , #abs^#(#0()) -> c_64() , #abs^#(#s(@x)) -> c_65() , diff#1^#(#true()) -> c_3(#abs^#(#pos(#s(#0())))) , diff#1^#(#false()) -> c_4(#abs^#(#0())) , mult#2^#(@zs, @b2, @x) -> c_5(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs), #equal^#(@x, #pos(#s(#0())))) , mult#3^#(#true(), @b2, @zs) -> c_42(add^#(@b2, @zs)) , mult#3^#(#false(), @b2, @zs) -> c_43() , #equal^#(@x, @y) -> c_29(#eq^#(@x, @y)) , div^#(@x, @y) -> c_6(#div^#(@x, @y)) , bitToInt'#1^#(nil(), @n) -> c_7(#abs^#(#0())) , bitToInt'#1^#(::(@x, @xs), @n) -> c_8(+^#(*(@x, @n), bitToInt'(@xs, *(@n, #pos(#s(#s(#0())))))), *^#(@x, @n), bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0()))))), *^#(@n, #pos(#s(#s(#0()))))) , +^#(@x, @y) -> c_21(#add^#(@x, @y)) , *^#(@x, @y) -> c_24(#mult^#(@x, @y)) , bitToInt'^#(@b, @n) -> c_15(bitToInt'#1^#(@b, @n)) , sum^#(@x, @y, @r) -> c_9(sum#1^#(+(+(@x, @y), @r)), +^#(+(@x, @y), @r), +^#(@x, @y)) , sum#1^#(@s) -> c_20(sum#2^#(#equal(@s, #0()), @s), #equal^#(@s, #0())) , mod^#(@x, @y) -> c_10(-^#(@x, *(@x, div(@x, @y))), *^#(@x, div(@x, @y)), div^#(@x, @y)) , mult3^#(@b1, @b2, @b3) -> c_11(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , mult^#(@b1, @b2) -> c_16(mult#1^#(@b1, @b2)) , leq^#(@b1, @b2) -> c_13(#less^#(compare(@b1, @b2), #pos(#s(#0()))), compare^#(@b1, @b2)) , #less^#(@x, @y) -> c_27(#cklt^#(#compare(@x, @y)), #compare^#(@x, @y)) , compare^#(@b1, @b2) -> c_52(compare#1^#(@b1, @b2)) , #greater^#(@x, @y) -> c_14(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y)) , mult#1^#(nil(), @b2) -> c_49() , mult#1^#(::(@x, @xs), @b2) -> c_50(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), #abs^#(#0()), mult^#(@xs, @b2)) , bitToInt^#(@b) -> c_17(bitToInt'^#(@b, #abs(#pos(#s(#0())))), #abs^#(#pos(#s(#0())))) , sum#2^#(#true(), @s) -> c_18(#abs^#(#0()), #abs^#(#0())) , sum#2^#(#false(), @s) -> c_19(sum#3^#(#equal(@s, #pos(#s(#0()))), @s), #equal^#(@s, #pos(#s(#0())))) , sum#3^#(#true(), @s) -> c_57(#abs^#(#pos(#s(#0()))), #abs^#(#0())) , sum#3^#(#false(), @s) -> c_58(sum#4^#(#equal(@s, #pos(#s(#s(#0()))))), #equal^#(@s, #pos(#s(#s(#0()))))) , sum#4^#(#true()) -> c_22(#abs^#(#0()), #abs^#(#pos(#s(#0())))) , sum#4^#(#false()) -> c_23(#abs^#(#pos(#s(#0()))), #abs^#(#pos(#s(#0())))) , sub'#5^#(#true(), @z, @zs) -> c_25(#abs^#(#0())) , sub'#5^#(#false(), @z, @zs) -> c_26() , sub'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_28(sub'#4^#(sub'(@xs, @ys, @r'), @z), sub'^#(@xs, @ys, @r')) , sub'#4^#(tuple#2(@zs, @s), @z) -> c_34(sub'#5^#(#equal(@s, #pos(#s(#0()))), @z, @zs), #equal^#(@s, #pos(#s(#0())))) , sub'#2^#(nil(), @r, @x, @xs) -> c_30() , sub'#2^#(::(@y, @ys), @r, @x, @xs) -> c_31(sub'#3^#(diff(@x, @y, @r), @xs, @ys), diff^#(@x, @y, @r)) , diff^#(@x, @y, @r) -> c_48(mod^#(+(+(@x, @y), @r), #pos(#s(#s(#0())))), +^#(+(@x, @y), @r), +^#(@x, @y), diff#1^#(#less(-(-(@x, @y), @r), #0())), #less^#(-(-(@x, @y), @r), #0()), -^#(-(@x, @y), @r), -^#(@x, @y)) , compare#2^#(nil(), @x, @xs) -> c_32(#abs^#(#0())) , compare#2^#(::(@y, @ys), @x, @xs) -> c_33(compare#3^#(compare(@xs, @ys), @x, @y), compare^#(@xs, @ys)) , compare#3^#(@r, @x, @y) -> c_37(compare#4^#(#equal(@r, #0()), @r, @x, @y), #equal^#(@r, #0())) , compare#5^#(#true(), @x, @y) -> c_35(-^#(#0(), #pos(#s(#0())))) , compare#5^#(#false(), @x, @y) -> c_36(compare#6^#(#greater(@x, @y)), #greater^#(@x, @y)) , compare#6^#(#true()) -> c_53(#abs^#(#pos(#s(#0())))) , compare#6^#(#false()) -> c_54(#abs^#(#0())) , compare#4^#(#true(), @r, @x, @y) -> c_55(compare#5^#(#less(@x, @y), @x, @y), #less^#(@x, @y)) , compare#4^#(#false(), @r, @x, @y) -> c_56() , add^#(@b1, @b2) -> c_38(add'^#(@b1, @b2, #abs(#0())), #abs^#(#0())) , add'^#(@b1, @b2, @r) -> c_59(add'#1^#(@b1, @b2, @r)) , add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_39(add'^#(@xs, @ys, @r')) , sub'#1^#(nil(), @b2, @r) -> c_40() , sub'#1^#(::(@x, @xs), @b2, @r) -> c_41(sub'#2^#(@b2, @r, @x, @xs)) , add'#1^#(nil(), @b2, @r) -> c_44() , add'#1^#(::(@x, @xs), @b2, @r) -> c_45(add'#2^#(@b2, @r, @x, @xs)) , add'#2^#(nil(), @r, @x, @xs) -> c_46() , add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_47(add'#3^#(sum(@x, @y, @r), @xs, @ys), sum^#(@x, @y, @r)) , compare#1^#(nil(), @b2) -> c_60(#abs^#(#0())) , compare#1^#(::(@x, @xs), @b2) -> c_61(compare#2^#(@b2, @x, @xs)) } Weak DPs: { #sub^#(@x, #neg(@y)) -> c_120(#add^#(@x, #pos(@y))) , #sub^#(@x, #pos(@y)) -> c_121(#add^#(@x, #neg(@y))) , #sub^#(@x, #0()) -> c_122() , #div^#(#neg(@x), #neg(@y)) -> c_136(#natdiv^#(@x, @y)) , #div^#(#neg(@x), #pos(@y)) -> c_137(#natdiv^#(@x, @y)) , #div^#(#neg(@x), #0()) -> c_138() , #div^#(#pos(@x), #neg(@y)) -> c_139(#natdiv^#(@x, @y)) , #div^#(#pos(@x), #pos(@y)) -> c_140(#natdiv^#(@x, @y)) , #div^#(#pos(@x), #0()) -> c_141() , #div^#(#0(), #neg(@y)) -> c_142() , #div^#(#0(), #pos(@y)) -> c_143() , #div^#(#0(), #0()) -> c_144() , #ckgt^#(#EQ()) -> c_70() , #ckgt^#(#LT()) -> c_71() , #ckgt^#(#GT()) -> c_72() , #compare^#(#neg(@x), #neg(@y)) -> c_82(#compare^#(@y, @x)) , #compare^#(#neg(@x), #pos(@y)) -> c_83() , #compare^#(#neg(@x), #0()) -> c_84() , #compare^#(#pos(@x), #neg(@y)) -> c_85() , #compare^#(#pos(@x), #pos(@y)) -> c_86(#compare^#(@x, @y)) , #compare^#(#pos(@x), #0()) -> c_87() , #compare^#(#0(), #neg(@y)) -> c_88() , #compare^#(#0(), #pos(@y)) -> c_89() , #compare^#(#0(), #0()) -> c_90() , #compare^#(#0(), #s(@y)) -> c_91() , #compare^#(#s(@x), #0()) -> c_92() , #compare^#(#s(@x), #s(@y)) -> c_93(#compare^#(@x, @y)) , #add^#(#neg(#s(#0())), @y) -> c_73(#pred^#(@y)) , #add^#(#neg(#s(#s(@x))), @y) -> c_74(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , #add^#(#pos(#s(#0())), @y) -> c_75(#succ^#(@y)) , #add^#(#pos(#s(#s(@x))), @y) -> c_76(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , #add^#(#0(), @y) -> c_77() , #mult^#(#neg(@x), #neg(@y)) -> c_123(#natmult^#(@x, @y)) , #mult^#(#neg(@x), #pos(@y)) -> c_124(#natmult^#(@x, @y)) , #mult^#(#neg(@x), #0()) -> c_125() , #mult^#(#pos(@x), #neg(@y)) -> c_126(#natmult^#(@x, @y)) , #mult^#(#pos(@x), #pos(@y)) -> c_127(#natmult^#(@x, @y)) , #mult^#(#pos(@x), #0()) -> c_128() , #mult^#(#0(), #neg(@y)) -> c_129() , #mult^#(#0(), #pos(@y)) -> c_130() , #mult^#(#0(), #0()) -> c_131() , #cklt^#(#EQ()) -> c_117() , #cklt^#(#LT()) -> c_118() , #cklt^#(#GT()) -> c_119() , #eq^#(nil(), nil()) -> c_94() , #eq^#(nil(), tuple#2(@y_1, @y_2)) -> c_95() , #eq^#(nil(), ::(@y_1, @y_2)) -> c_96() , #eq^#(#neg(@x), #neg(@y)) -> c_97(#eq^#(@x, @y)) , #eq^#(#neg(@x), #pos(@y)) -> c_98() , #eq^#(#neg(@x), #0()) -> c_99() , #eq^#(#pos(@x), #neg(@y)) -> c_100() , #eq^#(#pos(@x), #pos(@y)) -> c_101(#eq^#(@x, @y)) , #eq^#(#pos(@x), #0()) -> c_102() , #eq^#(tuple#2(@x_1, @x_2), nil()) -> c_103() , #eq^#(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> c_104(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), #eq^#(@x_1, @y_1), #eq^#(@x_2, @y_2)) , #eq^#(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> c_105() , #eq^#(::(@x_1, @x_2), nil()) -> c_106() , #eq^#(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> c_107() , #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) -> c_108(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), #eq^#(@x_1, @y_1), #eq^#(@x_2, @y_2)) , #eq^#(#0(), #neg(@y)) -> c_109() , #eq^#(#0(), #pos(@y)) -> c_110() , #eq^#(#0(), #0()) -> c_111() , #eq^#(#0(), #s(@y)) -> c_112() , #eq^#(#s(@x), #0()) -> c_113() , #eq^#(#s(@x), #s(@y)) -> c_114(#eq^#(@x, @y)) , #natsub^#(@x, #0()) -> c_66() , #natsub^#(#s(@x), #s(@y)) -> c_67(#natsub^#(@x, @y)) , #natdiv^#(#0(), #0()) -> c_68() , #natdiv^#(#s(@x), #s(@y)) -> c_69(#natdiv^#(#natsub(@x, @y), #s(@y)), #natsub^#(@x, @y)) , #pred^#(#neg(#s(@x))) -> c_145() , #pred^#(#pos(#s(#0()))) -> c_146() , #pred^#(#pos(#s(#s(@x)))) -> c_147() , #pred^#(#0()) -> c_148() , #succ^#(#neg(#s(#0()))) -> c_132() , #succ^#(#neg(#s(#s(@x)))) -> c_133() , #succ^#(#pos(#s(@x))) -> c_134() , #succ^#(#0()) -> c_135() , #and^#(#true(), #true()) -> c_78() , #and^#(#true(), #false()) -> c_79() , #and^#(#false(), #true()) -> c_80() , #and^#(#false(), #false()) -> c_81() , #natmult^#(#0(), @y) -> c_115() , #natmult^#(#s(@x), @y) -> c_116(#add^#(#pos(@y), #natmult(@x, @y)), #natmult^#(@x, @y)) } Weak Trs: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , sub(@b1, @b2) -> sub#1(sub'(@b1, @b2, #abs(#0()))) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #ckgt(#EQ()) -> #false() , #ckgt(#LT()) -> #false() , #ckgt(#GT()) -> #true() , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , div(@x, @y) -> #div(@x, @y) , bitToInt'#1(nil(), @n) -> #abs(#0()) , bitToInt'#1(::(@x, @xs), @n) -> +(*(@x, @n), bitToInt'(@xs, *(@n, #pos(#s(#s(#0())))))) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , mult3(@b1, @b2, @b3) -> mult(mult(@b1, @b2), @b2) , sub#1(tuple#2(@b, @_@1)) -> @b , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , leq(@b1, @b2) -> #less(compare(@b1, @b2), #pos(#s(#0()))) , #greater(@x, @y) -> #ckgt(#compare(@x, @y)) , bitToInt'(@b, @n) -> bitToInt'#1(@b, @n) , mult(@b1, @b2) -> mult#1(@b1, @b2) , bitToInt(@b) -> bitToInt'(@b, #abs(#pos(#s(#0())))) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , *(@x, @y) -> #mult(@x, @y) , sub'#5(#true(), @z, @zs) -> ::(#abs(#0()), @zs) , sub'#5(#false(), @z, @zs) -> ::(@z, @zs) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , sub'#3(tuple#2(@z, @r'), @xs, @ys) -> sub'#4(sub'(@xs, @ys, @r'), @z) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , sub'#2(nil(), @r, @x, @xs) -> tuple#2(nil(), @r) , sub'#2(::(@y, @ys), @r, @x, @xs) -> sub'#3(diff(@x, @y, @r), @xs, @ys) , compare#2(nil(), @x, @xs) -> #abs(#0()) , compare#2(::(@y, @ys), @x, @xs) -> compare#3(compare(@xs, @ys), @x, @y) , sub'#4(tuple#2(@zs, @s), @z) -> tuple#2(sub'#5(#equal(@s, #pos(#s(#0()))), @z, @zs), @s) , compare#5(#true(), @x, @y) -> -(#0(), #pos(#s(#0()))) , compare#5(#false(), @x, @y) -> compare#6(#greater(@x, @y)) , compare#3(@r, @x, @y) -> compare#4(#equal(@r, #0()), @r, @x, @y) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , sub'#1(nil(), @b2, @r) -> tuple#2(nil(), @r) , sub'#1(::(@x, @xs), @b2, @r) -> sub'#2(@b2, @r, @x, @xs) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sub'(@b1, @b2, @r) -> sub'#1(@b1, @b2, @r) , compare(@b1, @b2) -> compare#1(@b1, @b2) , compare#6(#true()) -> #abs(#pos(#s(#0()))) , compare#6(#false()) -> #abs(#0()) , compare#4(#true(), @r, @x, @y) -> compare#5(#less(@x, @y), @x, @y) , compare#4(#false(), @r, @x, @y) -> @r , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , compare#1(nil(), @b2) -> #abs(#0()) , compare#1(::(@x, @xs), @b2) -> compare#2(@b2, @x, @xs) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We estimate the number of application of {1,3,5,6,7,8,13,14,15,18,19,27,29,30,40,43,54,58,60,62} by applications of Pre({1,3,5,6,7,8,13,14,15,18,19,27,29,30,40,43,54,58,60,62}) = {2,4,9,10,11,16,17,21,22,23,25,26,31,32,33,34,35,36,37,38,39,42,45,46,48,49,50,51,52,53,55,56,59,61,64}. Here rules are labeled as follows: DPs: { 1: -^#(@x, @y) -> c_1(#sub^#(@x, @y)) , 2: sub^#(@b1, @b2) -> c_2(sub#1^#(sub'(@b1, @b2, #abs(#0()))), sub'^#(@b1, @b2, #abs(#0())), #abs^#(#0())) , 3: sub#1^#(tuple#2(@b, @_@1)) -> c_12() , 4: sub'^#(@b1, @b2, @r) -> c_51(sub'#1^#(@b1, @b2, @r)) , 5: #abs^#(#neg(@x)) -> c_62() , 6: #abs^#(#pos(@x)) -> c_63() , 7: #abs^#(#0()) -> c_64() , 8: #abs^#(#s(@x)) -> c_65() , 9: diff#1^#(#true()) -> c_3(#abs^#(#pos(#s(#0())))) , 10: diff#1^#(#false()) -> c_4(#abs^#(#0())) , 11: mult#2^#(@zs, @b2, @x) -> c_5(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs), #equal^#(@x, #pos(#s(#0())))) , 12: mult#3^#(#true(), @b2, @zs) -> c_42(add^#(@b2, @zs)) , 13: mult#3^#(#false(), @b2, @zs) -> c_43() , 14: #equal^#(@x, @y) -> c_29(#eq^#(@x, @y)) , 15: div^#(@x, @y) -> c_6(#div^#(@x, @y)) , 16: bitToInt'#1^#(nil(), @n) -> c_7(#abs^#(#0())) , 17: bitToInt'#1^#(::(@x, @xs), @n) -> c_8(+^#(*(@x, @n), bitToInt'(@xs, *(@n, #pos(#s(#s(#0())))))), *^#(@x, @n), bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0()))))), *^#(@n, #pos(#s(#s(#0()))))) , 18: +^#(@x, @y) -> c_21(#add^#(@x, @y)) , 19: *^#(@x, @y) -> c_24(#mult^#(@x, @y)) , 20: bitToInt'^#(@b, @n) -> c_15(bitToInt'#1^#(@b, @n)) , 21: sum^#(@x, @y, @r) -> c_9(sum#1^#(+(+(@x, @y), @r)), +^#(+(@x, @y), @r), +^#(@x, @y)) , 22: sum#1^#(@s) -> c_20(sum#2^#(#equal(@s, #0()), @s), #equal^#(@s, #0())) , 23: mod^#(@x, @y) -> c_10(-^#(@x, *(@x, div(@x, @y))), *^#(@x, div(@x, @y)), div^#(@x, @y)) , 24: mult3^#(@b1, @b2, @b3) -> c_11(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , 25: mult^#(@b1, @b2) -> c_16(mult#1^#(@b1, @b2)) , 26: leq^#(@b1, @b2) -> c_13(#less^#(compare(@b1, @b2), #pos(#s(#0()))), compare^#(@b1, @b2)) , 27: #less^#(@x, @y) -> c_27(#cklt^#(#compare(@x, @y)), #compare^#(@x, @y)) , 28: compare^#(@b1, @b2) -> c_52(compare#1^#(@b1, @b2)) , 29: #greater^#(@x, @y) -> c_14(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y)) , 30: mult#1^#(nil(), @b2) -> c_49() , 31: mult#1^#(::(@x, @xs), @b2) -> c_50(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), #abs^#(#0()), mult^#(@xs, @b2)) , 32: bitToInt^#(@b) -> c_17(bitToInt'^#(@b, #abs(#pos(#s(#0())))), #abs^#(#pos(#s(#0())))) , 33: sum#2^#(#true(), @s) -> c_18(#abs^#(#0()), #abs^#(#0())) , 34: sum#2^#(#false(), @s) -> c_19(sum#3^#(#equal(@s, #pos(#s(#0()))), @s), #equal^#(@s, #pos(#s(#0())))) , 35: sum#3^#(#true(), @s) -> c_57(#abs^#(#pos(#s(#0()))), #abs^#(#0())) , 36: sum#3^#(#false(), @s) -> c_58(sum#4^#(#equal(@s, #pos(#s(#s(#0()))))), #equal^#(@s, #pos(#s(#s(#0()))))) , 37: sum#4^#(#true()) -> c_22(#abs^#(#0()), #abs^#(#pos(#s(#0())))) , 38: sum#4^#(#false()) -> c_23(#abs^#(#pos(#s(#0()))), #abs^#(#pos(#s(#0())))) , 39: sub'#5^#(#true(), @z, @zs) -> c_25(#abs^#(#0())) , 40: sub'#5^#(#false(), @z, @zs) -> c_26() , 41: sub'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_28(sub'#4^#(sub'(@xs, @ys, @r'), @z), sub'^#(@xs, @ys, @r')) , 42: sub'#4^#(tuple#2(@zs, @s), @z) -> c_34(sub'#5^#(#equal(@s, #pos(#s(#0()))), @z, @zs), #equal^#(@s, #pos(#s(#0())))) , 43: sub'#2^#(nil(), @r, @x, @xs) -> c_30() , 44: sub'#2^#(::(@y, @ys), @r, @x, @xs) -> c_31(sub'#3^#(diff(@x, @y, @r), @xs, @ys), diff^#(@x, @y, @r)) , 45: diff^#(@x, @y, @r) -> c_48(mod^#(+(+(@x, @y), @r), #pos(#s(#s(#0())))), +^#(+(@x, @y), @r), +^#(@x, @y), diff#1^#(#less(-(-(@x, @y), @r), #0())), #less^#(-(-(@x, @y), @r), #0()), -^#(-(@x, @y), @r), -^#(@x, @y)) , 46: compare#2^#(nil(), @x, @xs) -> c_32(#abs^#(#0())) , 47: compare#2^#(::(@y, @ys), @x, @xs) -> c_33(compare#3^#(compare(@xs, @ys), @x, @y), compare^#(@xs, @ys)) , 48: compare#3^#(@r, @x, @y) -> c_37(compare#4^#(#equal(@r, #0()), @r, @x, @y), #equal^#(@r, #0())) , 49: compare#5^#(#true(), @x, @y) -> c_35(-^#(#0(), #pos(#s(#0())))) , 50: compare#5^#(#false(), @x, @y) -> c_36(compare#6^#(#greater(@x, @y)), #greater^#(@x, @y)) , 51: compare#6^#(#true()) -> c_53(#abs^#(#pos(#s(#0())))) , 52: compare#6^#(#false()) -> c_54(#abs^#(#0())) , 53: compare#4^#(#true(), @r, @x, @y) -> c_55(compare#5^#(#less(@x, @y), @x, @y), #less^#(@x, @y)) , 54: compare#4^#(#false(), @r, @x, @y) -> c_56() , 55: add^#(@b1, @b2) -> c_38(add'^#(@b1, @b2, #abs(#0())), #abs^#(#0())) , 56: add'^#(@b1, @b2, @r) -> c_59(add'#1^#(@b1, @b2, @r)) , 57: add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_39(add'^#(@xs, @ys, @r')) , 58: sub'#1^#(nil(), @b2, @r) -> c_40() , 59: sub'#1^#(::(@x, @xs), @b2, @r) -> c_41(sub'#2^#(@b2, @r, @x, @xs)) , 60: add'#1^#(nil(), @b2, @r) -> c_44() , 61: add'#1^#(::(@x, @xs), @b2, @r) -> c_45(add'#2^#(@b2, @r, @x, @xs)) , 62: add'#2^#(nil(), @r, @x, @xs) -> c_46() , 63: add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_47(add'#3^#(sum(@x, @y, @r), @xs, @ys), sum^#(@x, @y, @r)) , 64: compare#1^#(nil(), @b2) -> c_60(#abs^#(#0())) , 65: compare#1^#(::(@x, @xs), @b2) -> c_61(compare#2^#(@b2, @x, @xs)) , 66: #sub^#(@x, #neg(@y)) -> c_120(#add^#(@x, #pos(@y))) , 67: #sub^#(@x, #pos(@y)) -> c_121(#add^#(@x, #neg(@y))) , 68: #sub^#(@x, #0()) -> c_122() , 69: #div^#(#neg(@x), #neg(@y)) -> c_136(#natdiv^#(@x, @y)) , 70: #div^#(#neg(@x), #pos(@y)) -> c_137(#natdiv^#(@x, @y)) , 71: #div^#(#neg(@x), #0()) -> c_138() , 72: #div^#(#pos(@x), #neg(@y)) -> c_139(#natdiv^#(@x, @y)) , 73: #div^#(#pos(@x), #pos(@y)) -> c_140(#natdiv^#(@x, @y)) , 74: #div^#(#pos(@x), #0()) -> c_141() , 75: #div^#(#0(), #neg(@y)) -> c_142() , 76: #div^#(#0(), #pos(@y)) -> c_143() , 77: #div^#(#0(), #0()) -> c_144() , 78: #ckgt^#(#EQ()) -> c_70() , 79: #ckgt^#(#LT()) -> c_71() , 80: #ckgt^#(#GT()) -> c_72() , 81: #compare^#(#neg(@x), #neg(@y)) -> c_82(#compare^#(@y, @x)) , 82: #compare^#(#neg(@x), #pos(@y)) -> c_83() , 83: #compare^#(#neg(@x), #0()) -> c_84() , 84: #compare^#(#pos(@x), #neg(@y)) -> c_85() , 85: #compare^#(#pos(@x), #pos(@y)) -> c_86(#compare^#(@x, @y)) , 86: #compare^#(#pos(@x), #0()) -> c_87() , 87: #compare^#(#0(), #neg(@y)) -> c_88() , 88: #compare^#(#0(), #pos(@y)) -> c_89() , 89: #compare^#(#0(), #0()) -> c_90() , 90: #compare^#(#0(), #s(@y)) -> c_91() , 91: #compare^#(#s(@x), #0()) -> c_92() , 92: #compare^#(#s(@x), #s(@y)) -> c_93(#compare^#(@x, @y)) , 93: #add^#(#neg(#s(#0())), @y) -> c_73(#pred^#(@y)) , 94: #add^#(#neg(#s(#s(@x))), @y) -> c_74(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , 95: #add^#(#pos(#s(#0())), @y) -> c_75(#succ^#(@y)) , 96: #add^#(#pos(#s(#s(@x))), @y) -> c_76(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , 97: #add^#(#0(), @y) -> c_77() , 98: #mult^#(#neg(@x), #neg(@y)) -> c_123(#natmult^#(@x, @y)) , 99: #mult^#(#neg(@x), #pos(@y)) -> c_124(#natmult^#(@x, @y)) , 100: #mult^#(#neg(@x), #0()) -> c_125() , 101: #mult^#(#pos(@x), #neg(@y)) -> c_126(#natmult^#(@x, @y)) , 102: #mult^#(#pos(@x), #pos(@y)) -> c_127(#natmult^#(@x, @y)) , 103: #mult^#(#pos(@x), #0()) -> c_128() , 104: #mult^#(#0(), #neg(@y)) -> c_129() , 105: #mult^#(#0(), #pos(@y)) -> c_130() , 106: #mult^#(#0(), #0()) -> c_131() , 107: #cklt^#(#EQ()) -> c_117() , 108: #cklt^#(#LT()) -> c_118() , 109: #cklt^#(#GT()) -> c_119() , 110: #eq^#(nil(), nil()) -> c_94() , 111: #eq^#(nil(), tuple#2(@y_1, @y_2)) -> c_95() , 112: #eq^#(nil(), ::(@y_1, @y_2)) -> c_96() , 113: #eq^#(#neg(@x), #neg(@y)) -> c_97(#eq^#(@x, @y)) , 114: #eq^#(#neg(@x), #pos(@y)) -> c_98() , 115: #eq^#(#neg(@x), #0()) -> c_99() , 116: #eq^#(#pos(@x), #neg(@y)) -> c_100() , 117: #eq^#(#pos(@x), #pos(@y)) -> c_101(#eq^#(@x, @y)) , 118: #eq^#(#pos(@x), #0()) -> c_102() , 119: #eq^#(tuple#2(@x_1, @x_2), nil()) -> c_103() , 120: #eq^#(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> c_104(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), #eq^#(@x_1, @y_1), #eq^#(@x_2, @y_2)) , 121: #eq^#(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> c_105() , 122: #eq^#(::(@x_1, @x_2), nil()) -> c_106() , 123: #eq^#(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> c_107() , 124: #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) -> c_108(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), #eq^#(@x_1, @y_1), #eq^#(@x_2, @y_2)) , 125: #eq^#(#0(), #neg(@y)) -> c_109() , 126: #eq^#(#0(), #pos(@y)) -> c_110() , 127: #eq^#(#0(), #0()) -> c_111() , 128: #eq^#(#0(), #s(@y)) -> c_112() , 129: #eq^#(#s(@x), #0()) -> c_113() , 130: #eq^#(#s(@x), #s(@y)) -> c_114(#eq^#(@x, @y)) , 131: #natsub^#(@x, #0()) -> c_66() , 132: #natsub^#(#s(@x), #s(@y)) -> c_67(#natsub^#(@x, @y)) , 133: #natdiv^#(#0(), #0()) -> c_68() , 134: #natdiv^#(#s(@x), #s(@y)) -> c_69(#natdiv^#(#natsub(@x, @y), #s(@y)), #natsub^#(@x, @y)) , 135: #pred^#(#neg(#s(@x))) -> c_145() , 136: #pred^#(#pos(#s(#0()))) -> c_146() , 137: #pred^#(#pos(#s(#s(@x)))) -> c_147() , 138: #pred^#(#0()) -> c_148() , 139: #succ^#(#neg(#s(#0()))) -> c_132() , 140: #succ^#(#neg(#s(#s(@x)))) -> c_133() , 141: #succ^#(#pos(#s(@x))) -> c_134() , 142: #succ^#(#0()) -> c_135() , 143: #and^#(#true(), #true()) -> c_78() , 144: #and^#(#true(), #false()) -> c_79() , 145: #and^#(#false(), #true()) -> c_80() , 146: #and^#(#false(), #false()) -> c_81() , 147: #natmult^#(#0(), @y) -> c_115() , 148: #natmult^#(#s(@x), @y) -> c_116(#add^#(#pos(@y), #natmult(@x, @y)), #natmult^#(@x, @y)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { sub^#(@b1, @b2) -> c_2(sub#1^#(sub'(@b1, @b2, #abs(#0()))), sub'^#(@b1, @b2, #abs(#0())), #abs^#(#0())) , sub'^#(@b1, @b2, @r) -> c_51(sub'#1^#(@b1, @b2, @r)) , diff#1^#(#true()) -> c_3(#abs^#(#pos(#s(#0())))) , diff#1^#(#false()) -> c_4(#abs^#(#0())) , mult#2^#(@zs, @b2, @x) -> c_5(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs), #equal^#(@x, #pos(#s(#0())))) , mult#3^#(#true(), @b2, @zs) -> c_42(add^#(@b2, @zs)) , bitToInt'#1^#(nil(), @n) -> c_7(#abs^#(#0())) , bitToInt'#1^#(::(@x, @xs), @n) -> c_8(+^#(*(@x, @n), bitToInt'(@xs, *(@n, #pos(#s(#s(#0())))))), *^#(@x, @n), bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0()))))), *^#(@n, #pos(#s(#s(#0()))))) , bitToInt'^#(@b, @n) -> c_15(bitToInt'#1^#(@b, @n)) , sum^#(@x, @y, @r) -> c_9(sum#1^#(+(+(@x, @y), @r)), +^#(+(@x, @y), @r), +^#(@x, @y)) , sum#1^#(@s) -> c_20(sum#2^#(#equal(@s, #0()), @s), #equal^#(@s, #0())) , mod^#(@x, @y) -> c_10(-^#(@x, *(@x, div(@x, @y))), *^#(@x, div(@x, @y)), div^#(@x, @y)) , mult3^#(@b1, @b2, @b3) -> c_11(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , mult^#(@b1, @b2) -> c_16(mult#1^#(@b1, @b2)) , leq^#(@b1, @b2) -> c_13(#less^#(compare(@b1, @b2), #pos(#s(#0()))), compare^#(@b1, @b2)) , compare^#(@b1, @b2) -> c_52(compare#1^#(@b1, @b2)) , mult#1^#(::(@x, @xs), @b2) -> c_50(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), #abs^#(#0()), mult^#(@xs, @b2)) , bitToInt^#(@b) -> c_17(bitToInt'^#(@b, #abs(#pos(#s(#0())))), #abs^#(#pos(#s(#0())))) , sum#2^#(#true(), @s) -> c_18(#abs^#(#0()), #abs^#(#0())) , sum#2^#(#false(), @s) -> c_19(sum#3^#(#equal(@s, #pos(#s(#0()))), @s), #equal^#(@s, #pos(#s(#0())))) , sum#3^#(#true(), @s) -> c_57(#abs^#(#pos(#s(#0()))), #abs^#(#0())) , sum#3^#(#false(), @s) -> c_58(sum#4^#(#equal(@s, #pos(#s(#s(#0()))))), #equal^#(@s, #pos(#s(#s(#0()))))) , sum#4^#(#true()) -> c_22(#abs^#(#0()), #abs^#(#pos(#s(#0())))) , sum#4^#(#false()) -> c_23(#abs^#(#pos(#s(#0()))), #abs^#(#pos(#s(#0())))) , sub'#5^#(#true(), @z, @zs) -> c_25(#abs^#(#0())) , sub'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_28(sub'#4^#(sub'(@xs, @ys, @r'), @z), sub'^#(@xs, @ys, @r')) , sub'#4^#(tuple#2(@zs, @s), @z) -> c_34(sub'#5^#(#equal(@s, #pos(#s(#0()))), @z, @zs), #equal^#(@s, #pos(#s(#0())))) , sub'#2^#(::(@y, @ys), @r, @x, @xs) -> c_31(sub'#3^#(diff(@x, @y, @r), @xs, @ys), diff^#(@x, @y, @r)) , diff^#(@x, @y, @r) -> c_48(mod^#(+(+(@x, @y), @r), #pos(#s(#s(#0())))), +^#(+(@x, @y), @r), +^#(@x, @y), diff#1^#(#less(-(-(@x, @y), @r), #0())), #less^#(-(-(@x, @y), @r), #0()), -^#(-(@x, @y), @r), -^#(@x, @y)) , compare#2^#(nil(), @x, @xs) -> c_32(#abs^#(#0())) , compare#2^#(::(@y, @ys), @x, @xs) -> c_33(compare#3^#(compare(@xs, @ys), @x, @y), compare^#(@xs, @ys)) , compare#3^#(@r, @x, @y) -> c_37(compare#4^#(#equal(@r, #0()), @r, @x, @y), #equal^#(@r, #0())) , compare#5^#(#true(), @x, @y) -> c_35(-^#(#0(), #pos(#s(#0())))) , compare#5^#(#false(), @x, @y) -> c_36(compare#6^#(#greater(@x, @y)), #greater^#(@x, @y)) , compare#6^#(#true()) -> c_53(#abs^#(#pos(#s(#0())))) , compare#6^#(#false()) -> c_54(#abs^#(#0())) , compare#4^#(#true(), @r, @x, @y) -> c_55(compare#5^#(#less(@x, @y), @x, @y), #less^#(@x, @y)) , add^#(@b1, @b2) -> c_38(add'^#(@b1, @b2, #abs(#0())), #abs^#(#0())) , add'^#(@b1, @b2, @r) -> c_59(add'#1^#(@b1, @b2, @r)) , add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_39(add'^#(@xs, @ys, @r')) , sub'#1^#(::(@x, @xs), @b2, @r) -> c_41(sub'#2^#(@b2, @r, @x, @xs)) , add'#1^#(::(@x, @xs), @b2, @r) -> c_45(add'#2^#(@b2, @r, @x, @xs)) , add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_47(add'#3^#(sum(@x, @y, @r), @xs, @ys), sum^#(@x, @y, @r)) , compare#1^#(nil(), @b2) -> c_60(#abs^#(#0())) , compare#1^#(::(@x, @xs), @b2) -> c_61(compare#2^#(@b2, @x, @xs)) } Weak DPs: { -^#(@x, @y) -> c_1(#sub^#(@x, @y)) , #sub^#(@x, #neg(@y)) -> c_120(#add^#(@x, #pos(@y))) , #sub^#(@x, #pos(@y)) -> c_121(#add^#(@x, #neg(@y))) , #sub^#(@x, #0()) -> c_122() , sub#1^#(tuple#2(@b, @_@1)) -> c_12() , #abs^#(#neg(@x)) -> c_62() , #abs^#(#pos(@x)) -> c_63() , #abs^#(#0()) -> c_64() , #abs^#(#s(@x)) -> c_65() , mult#3^#(#false(), @b2, @zs) -> c_43() , #equal^#(@x, @y) -> c_29(#eq^#(@x, @y)) , div^#(@x, @y) -> c_6(#div^#(@x, @y)) , #div^#(#neg(@x), #neg(@y)) -> c_136(#natdiv^#(@x, @y)) , #div^#(#neg(@x), #pos(@y)) -> c_137(#natdiv^#(@x, @y)) , #div^#(#neg(@x), #0()) -> c_138() , #div^#(#pos(@x), #neg(@y)) -> c_139(#natdiv^#(@x, @y)) , #div^#(#pos(@x), #pos(@y)) -> c_140(#natdiv^#(@x, @y)) , #div^#(#pos(@x), #0()) -> c_141() , #div^#(#0(), #neg(@y)) -> c_142() , #div^#(#0(), #pos(@y)) -> c_143() , #div^#(#0(), #0()) -> c_144() , +^#(@x, @y) -> c_21(#add^#(@x, @y)) , *^#(@x, @y) -> c_24(#mult^#(@x, @y)) , #less^#(@x, @y) -> c_27(#cklt^#(#compare(@x, @y)), #compare^#(@x, @y)) , #greater^#(@x, @y) -> c_14(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y)) , #ckgt^#(#EQ()) -> c_70() , #ckgt^#(#LT()) -> c_71() , #ckgt^#(#GT()) -> c_72() , #compare^#(#neg(@x), #neg(@y)) -> c_82(#compare^#(@y, @x)) , #compare^#(#neg(@x), #pos(@y)) -> c_83() , #compare^#(#neg(@x), #0()) -> c_84() , #compare^#(#pos(@x), #neg(@y)) -> c_85() , #compare^#(#pos(@x), #pos(@y)) -> c_86(#compare^#(@x, @y)) , #compare^#(#pos(@x), #0()) -> c_87() , #compare^#(#0(), #neg(@y)) -> c_88() , #compare^#(#0(), #pos(@y)) -> c_89() , #compare^#(#0(), #0()) -> c_90() , #compare^#(#0(), #s(@y)) -> c_91() , #compare^#(#s(@x), #0()) -> c_92() , #compare^#(#s(@x), #s(@y)) -> c_93(#compare^#(@x, @y)) , mult#1^#(nil(), @b2) -> c_49() , #add^#(#neg(#s(#0())), @y) -> c_73(#pred^#(@y)) , #add^#(#neg(#s(#s(@x))), @y) -> c_74(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , #add^#(#pos(#s(#0())), @y) -> c_75(#succ^#(@y)) , #add^#(#pos(#s(#s(@x))), @y) -> c_76(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , #add^#(#0(), @y) -> c_77() , #mult^#(#neg(@x), #neg(@y)) -> c_123(#natmult^#(@x, @y)) , #mult^#(#neg(@x), #pos(@y)) -> c_124(#natmult^#(@x, @y)) , #mult^#(#neg(@x), #0()) -> c_125() , #mult^#(#pos(@x), #neg(@y)) -> c_126(#natmult^#(@x, @y)) , #mult^#(#pos(@x), #pos(@y)) -> c_127(#natmult^#(@x, @y)) , #mult^#(#pos(@x), #0()) -> c_128() , #mult^#(#0(), #neg(@y)) -> c_129() , #mult^#(#0(), #pos(@y)) -> c_130() , #mult^#(#0(), #0()) -> c_131() , sub'#5^#(#false(), @z, @zs) -> c_26() , #cklt^#(#EQ()) -> c_117() , #cklt^#(#LT()) -> c_118() , #cklt^#(#GT()) -> c_119() , #eq^#(nil(), nil()) -> c_94() , #eq^#(nil(), tuple#2(@y_1, @y_2)) -> c_95() , #eq^#(nil(), ::(@y_1, @y_2)) -> c_96() , #eq^#(#neg(@x), #neg(@y)) -> c_97(#eq^#(@x, @y)) , #eq^#(#neg(@x), #pos(@y)) -> c_98() , #eq^#(#neg(@x), #0()) -> c_99() , #eq^#(#pos(@x), #neg(@y)) -> c_100() , #eq^#(#pos(@x), #pos(@y)) -> c_101(#eq^#(@x, @y)) , #eq^#(#pos(@x), #0()) -> c_102() , #eq^#(tuple#2(@x_1, @x_2), nil()) -> c_103() , #eq^#(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> c_104(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), #eq^#(@x_1, @y_1), #eq^#(@x_2, @y_2)) , #eq^#(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> c_105() , #eq^#(::(@x_1, @x_2), nil()) -> c_106() , #eq^#(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> c_107() , #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) -> c_108(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), #eq^#(@x_1, @y_1), #eq^#(@x_2, @y_2)) , #eq^#(#0(), #neg(@y)) -> c_109() , #eq^#(#0(), #pos(@y)) -> c_110() , #eq^#(#0(), #0()) -> c_111() , #eq^#(#0(), #s(@y)) -> c_112() , #eq^#(#s(@x), #0()) -> c_113() , #eq^#(#s(@x), #s(@y)) -> c_114(#eq^#(@x, @y)) , sub'#2^#(nil(), @r, @x, @xs) -> c_30() , compare#4^#(#false(), @r, @x, @y) -> c_56() , sub'#1^#(nil(), @b2, @r) -> c_40() , add'#1^#(nil(), @b2, @r) -> c_44() , add'#2^#(nil(), @r, @x, @xs) -> c_46() , #natsub^#(@x, #0()) -> c_66() , #natsub^#(#s(@x), #s(@y)) -> c_67(#natsub^#(@x, @y)) , #natdiv^#(#0(), #0()) -> c_68() , #natdiv^#(#s(@x), #s(@y)) -> c_69(#natdiv^#(#natsub(@x, @y), #s(@y)), #natsub^#(@x, @y)) , #pred^#(#neg(#s(@x))) -> c_145() , #pred^#(#pos(#s(#0()))) -> c_146() , #pred^#(#pos(#s(#s(@x)))) -> c_147() , #pred^#(#0()) -> c_148() , #succ^#(#neg(#s(#0()))) -> c_132() , #succ^#(#neg(#s(#s(@x)))) -> c_133() , #succ^#(#pos(#s(@x))) -> c_134() , #succ^#(#0()) -> c_135() , #and^#(#true(), #true()) -> c_78() , #and^#(#true(), #false()) -> c_79() , #and^#(#false(), #true()) -> c_80() , #and^#(#false(), #false()) -> c_81() , #natmult^#(#0(), @y) -> c_115() , #natmult^#(#s(@x), @y) -> c_116(#add^#(#pos(@y), #natmult(@x, @y)), #natmult^#(@x, @y)) } Weak Trs: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , sub(@b1, @b2) -> sub#1(sub'(@b1, @b2, #abs(#0()))) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #ckgt(#EQ()) -> #false() , #ckgt(#LT()) -> #false() , #ckgt(#GT()) -> #true() , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , div(@x, @y) -> #div(@x, @y) , bitToInt'#1(nil(), @n) -> #abs(#0()) , bitToInt'#1(::(@x, @xs), @n) -> +(*(@x, @n), bitToInt'(@xs, *(@n, #pos(#s(#s(#0())))))) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , mult3(@b1, @b2, @b3) -> mult(mult(@b1, @b2), @b2) , sub#1(tuple#2(@b, @_@1)) -> @b , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , leq(@b1, @b2) -> #less(compare(@b1, @b2), #pos(#s(#0()))) , #greater(@x, @y) -> #ckgt(#compare(@x, @y)) , bitToInt'(@b, @n) -> bitToInt'#1(@b, @n) , mult(@b1, @b2) -> mult#1(@b1, @b2) , bitToInt(@b) -> bitToInt'(@b, #abs(#pos(#s(#0())))) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , *(@x, @y) -> #mult(@x, @y) , sub'#5(#true(), @z, @zs) -> ::(#abs(#0()), @zs) , sub'#5(#false(), @z, @zs) -> ::(@z, @zs) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , sub'#3(tuple#2(@z, @r'), @xs, @ys) -> sub'#4(sub'(@xs, @ys, @r'), @z) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , sub'#2(nil(), @r, @x, @xs) -> tuple#2(nil(), @r) , sub'#2(::(@y, @ys), @r, @x, @xs) -> sub'#3(diff(@x, @y, @r), @xs, @ys) , compare#2(nil(), @x, @xs) -> #abs(#0()) , compare#2(::(@y, @ys), @x, @xs) -> compare#3(compare(@xs, @ys), @x, @y) , sub'#4(tuple#2(@zs, @s), @z) -> tuple#2(sub'#5(#equal(@s, #pos(#s(#0()))), @z, @zs), @s) , compare#5(#true(), @x, @y) -> -(#0(), #pos(#s(#0()))) , compare#5(#false(), @x, @y) -> compare#6(#greater(@x, @y)) , compare#3(@r, @x, @y) -> compare#4(#equal(@r, #0()), @r, @x, @y) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , sub'#1(nil(), @b2, @r) -> tuple#2(nil(), @r) , sub'#1(::(@x, @xs), @b2, @r) -> sub'#2(@b2, @r, @x, @xs) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sub'(@b1, @b2, @r) -> sub'#1(@b1, @b2, @r) , compare(@b1, @b2) -> compare#1(@b1, @b2) , compare#6(#true()) -> #abs(#pos(#s(#0()))) , compare#6(#false()) -> #abs(#0()) , compare#4(#true(), @r, @x, @y) -> compare#5(#less(@x, @y), @x, @y) , compare#4(#false(), @r, @x, @y) -> @r , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , compare#1(nil(), @b2) -> #abs(#0()) , compare#1(::(@x, @xs), @b2) -> compare#2(@b2, @x, @xs) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We estimate the number of application of {3,4,7,12,19,21,23,24,25,30,33,35,36,44} by applications of Pre({3,4,7,12,19,21,23,24,25,30,33,35,36,44}) = {9,11,16,20,22,27,29,34,37,45}. Here rules are labeled as follows: DPs: { 1: sub^#(@b1, @b2) -> c_2(sub#1^#(sub'(@b1, @b2, #abs(#0()))), sub'^#(@b1, @b2, #abs(#0())), #abs^#(#0())) , 2: sub'^#(@b1, @b2, @r) -> c_51(sub'#1^#(@b1, @b2, @r)) , 3: diff#1^#(#true()) -> c_3(#abs^#(#pos(#s(#0())))) , 4: diff#1^#(#false()) -> c_4(#abs^#(#0())) , 5: mult#2^#(@zs, @b2, @x) -> c_5(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs), #equal^#(@x, #pos(#s(#0())))) , 6: mult#3^#(#true(), @b2, @zs) -> c_42(add^#(@b2, @zs)) , 7: bitToInt'#1^#(nil(), @n) -> c_7(#abs^#(#0())) , 8: bitToInt'#1^#(::(@x, @xs), @n) -> c_8(+^#(*(@x, @n), bitToInt'(@xs, *(@n, #pos(#s(#s(#0())))))), *^#(@x, @n), bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0()))))), *^#(@n, #pos(#s(#s(#0()))))) , 9: bitToInt'^#(@b, @n) -> c_15(bitToInt'#1^#(@b, @n)) , 10: sum^#(@x, @y, @r) -> c_9(sum#1^#(+(+(@x, @y), @r)), +^#(+(@x, @y), @r), +^#(@x, @y)) , 11: sum#1^#(@s) -> c_20(sum#2^#(#equal(@s, #0()), @s), #equal^#(@s, #0())) , 12: mod^#(@x, @y) -> c_10(-^#(@x, *(@x, div(@x, @y))), *^#(@x, div(@x, @y)), div^#(@x, @y)) , 13: mult3^#(@b1, @b2, @b3) -> c_11(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , 14: mult^#(@b1, @b2) -> c_16(mult#1^#(@b1, @b2)) , 15: leq^#(@b1, @b2) -> c_13(#less^#(compare(@b1, @b2), #pos(#s(#0()))), compare^#(@b1, @b2)) , 16: compare^#(@b1, @b2) -> c_52(compare#1^#(@b1, @b2)) , 17: mult#1^#(::(@x, @xs), @b2) -> c_50(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), #abs^#(#0()), mult^#(@xs, @b2)) , 18: bitToInt^#(@b) -> c_17(bitToInt'^#(@b, #abs(#pos(#s(#0())))), #abs^#(#pos(#s(#0())))) , 19: sum#2^#(#true(), @s) -> c_18(#abs^#(#0()), #abs^#(#0())) , 20: sum#2^#(#false(), @s) -> c_19(sum#3^#(#equal(@s, #pos(#s(#0()))), @s), #equal^#(@s, #pos(#s(#0())))) , 21: sum#3^#(#true(), @s) -> c_57(#abs^#(#pos(#s(#0()))), #abs^#(#0())) , 22: sum#3^#(#false(), @s) -> c_58(sum#4^#(#equal(@s, #pos(#s(#s(#0()))))), #equal^#(@s, #pos(#s(#s(#0()))))) , 23: sum#4^#(#true()) -> c_22(#abs^#(#0()), #abs^#(#pos(#s(#0())))) , 24: sum#4^#(#false()) -> c_23(#abs^#(#pos(#s(#0()))), #abs^#(#pos(#s(#0())))) , 25: sub'#5^#(#true(), @z, @zs) -> c_25(#abs^#(#0())) , 26: sub'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_28(sub'#4^#(sub'(@xs, @ys, @r'), @z), sub'^#(@xs, @ys, @r')) , 27: sub'#4^#(tuple#2(@zs, @s), @z) -> c_34(sub'#5^#(#equal(@s, #pos(#s(#0()))), @z, @zs), #equal^#(@s, #pos(#s(#0())))) , 28: sub'#2^#(::(@y, @ys), @r, @x, @xs) -> c_31(sub'#3^#(diff(@x, @y, @r), @xs, @ys), diff^#(@x, @y, @r)) , 29: diff^#(@x, @y, @r) -> c_48(mod^#(+(+(@x, @y), @r), #pos(#s(#s(#0())))), +^#(+(@x, @y), @r), +^#(@x, @y), diff#1^#(#less(-(-(@x, @y), @r), #0())), #less^#(-(-(@x, @y), @r), #0()), -^#(-(@x, @y), @r), -^#(@x, @y)) , 30: compare#2^#(nil(), @x, @xs) -> c_32(#abs^#(#0())) , 31: compare#2^#(::(@y, @ys), @x, @xs) -> c_33(compare#3^#(compare(@xs, @ys), @x, @y), compare^#(@xs, @ys)) , 32: compare#3^#(@r, @x, @y) -> c_37(compare#4^#(#equal(@r, #0()), @r, @x, @y), #equal^#(@r, #0())) , 33: compare#5^#(#true(), @x, @y) -> c_35(-^#(#0(), #pos(#s(#0())))) , 34: compare#5^#(#false(), @x, @y) -> c_36(compare#6^#(#greater(@x, @y)), #greater^#(@x, @y)) , 35: compare#6^#(#true()) -> c_53(#abs^#(#pos(#s(#0())))) , 36: compare#6^#(#false()) -> c_54(#abs^#(#0())) , 37: compare#4^#(#true(), @r, @x, @y) -> c_55(compare#5^#(#less(@x, @y), @x, @y), #less^#(@x, @y)) , 38: add^#(@b1, @b2) -> c_38(add'^#(@b1, @b2, #abs(#0())), #abs^#(#0())) , 39: add'^#(@b1, @b2, @r) -> c_59(add'#1^#(@b1, @b2, @r)) , 40: add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_39(add'^#(@xs, @ys, @r')) , 41: sub'#1^#(::(@x, @xs), @b2, @r) -> c_41(sub'#2^#(@b2, @r, @x, @xs)) , 42: add'#1^#(::(@x, @xs), @b2, @r) -> c_45(add'#2^#(@b2, @r, @x, @xs)) , 43: add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_47(add'#3^#(sum(@x, @y, @r), @xs, @ys), sum^#(@x, @y, @r)) , 44: compare#1^#(nil(), @b2) -> c_60(#abs^#(#0())) , 45: compare#1^#(::(@x, @xs), @b2) -> c_61(compare#2^#(@b2, @x, @xs)) , 46: -^#(@x, @y) -> c_1(#sub^#(@x, @y)) , 47: #sub^#(@x, #neg(@y)) -> c_120(#add^#(@x, #pos(@y))) , 48: #sub^#(@x, #pos(@y)) -> c_121(#add^#(@x, #neg(@y))) , 49: #sub^#(@x, #0()) -> c_122() , 50: sub#1^#(tuple#2(@b, @_@1)) -> c_12() , 51: #abs^#(#neg(@x)) -> c_62() , 52: #abs^#(#pos(@x)) -> c_63() , 53: #abs^#(#0()) -> c_64() , 54: #abs^#(#s(@x)) -> c_65() , 55: mult#3^#(#false(), @b2, @zs) -> c_43() , 56: #equal^#(@x, @y) -> c_29(#eq^#(@x, @y)) , 57: div^#(@x, @y) -> c_6(#div^#(@x, @y)) , 58: #div^#(#neg(@x), #neg(@y)) -> c_136(#natdiv^#(@x, @y)) , 59: #div^#(#neg(@x), #pos(@y)) -> c_137(#natdiv^#(@x, @y)) , 60: #div^#(#neg(@x), #0()) -> c_138() , 61: #div^#(#pos(@x), #neg(@y)) -> c_139(#natdiv^#(@x, @y)) , 62: #div^#(#pos(@x), #pos(@y)) -> c_140(#natdiv^#(@x, @y)) , 63: #div^#(#pos(@x), #0()) -> c_141() , 64: #div^#(#0(), #neg(@y)) -> c_142() , 65: #div^#(#0(), #pos(@y)) -> c_143() , 66: #div^#(#0(), #0()) -> c_144() , 67: +^#(@x, @y) -> c_21(#add^#(@x, @y)) , 68: *^#(@x, @y) -> c_24(#mult^#(@x, @y)) , 69: #less^#(@x, @y) -> c_27(#cklt^#(#compare(@x, @y)), #compare^#(@x, @y)) , 70: #greater^#(@x, @y) -> c_14(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y)) , 71: #ckgt^#(#EQ()) -> c_70() , 72: #ckgt^#(#LT()) -> c_71() , 73: #ckgt^#(#GT()) -> c_72() , 74: #compare^#(#neg(@x), #neg(@y)) -> c_82(#compare^#(@y, @x)) , 75: #compare^#(#neg(@x), #pos(@y)) -> c_83() , 76: #compare^#(#neg(@x), #0()) -> c_84() , 77: #compare^#(#pos(@x), #neg(@y)) -> c_85() , 78: #compare^#(#pos(@x), #pos(@y)) -> c_86(#compare^#(@x, @y)) , 79: #compare^#(#pos(@x), #0()) -> c_87() , 80: #compare^#(#0(), #neg(@y)) -> c_88() , 81: #compare^#(#0(), #pos(@y)) -> c_89() , 82: #compare^#(#0(), #0()) -> c_90() , 83: #compare^#(#0(), #s(@y)) -> c_91() , 84: #compare^#(#s(@x), #0()) -> c_92() , 85: #compare^#(#s(@x), #s(@y)) -> c_93(#compare^#(@x, @y)) , 86: mult#1^#(nil(), @b2) -> c_49() , 87: #add^#(#neg(#s(#0())), @y) -> c_73(#pred^#(@y)) , 88: #add^#(#neg(#s(#s(@x))), @y) -> c_74(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , 89: #add^#(#pos(#s(#0())), @y) -> c_75(#succ^#(@y)) , 90: #add^#(#pos(#s(#s(@x))), @y) -> c_76(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , 91: #add^#(#0(), @y) -> c_77() , 92: #mult^#(#neg(@x), #neg(@y)) -> c_123(#natmult^#(@x, @y)) , 93: #mult^#(#neg(@x), #pos(@y)) -> c_124(#natmult^#(@x, @y)) , 94: #mult^#(#neg(@x), #0()) -> c_125() , 95: #mult^#(#pos(@x), #neg(@y)) -> c_126(#natmult^#(@x, @y)) , 96: #mult^#(#pos(@x), #pos(@y)) -> c_127(#natmult^#(@x, @y)) , 97: #mult^#(#pos(@x), #0()) -> c_128() , 98: #mult^#(#0(), #neg(@y)) -> c_129() , 99: #mult^#(#0(), #pos(@y)) -> c_130() , 100: #mult^#(#0(), #0()) -> c_131() , 101: sub'#5^#(#false(), @z, @zs) -> c_26() , 102: #cklt^#(#EQ()) -> c_117() , 103: #cklt^#(#LT()) -> c_118() , 104: #cklt^#(#GT()) -> c_119() , 105: #eq^#(nil(), nil()) -> c_94() , 106: #eq^#(nil(), tuple#2(@y_1, @y_2)) -> c_95() , 107: #eq^#(nil(), ::(@y_1, @y_2)) -> c_96() , 108: #eq^#(#neg(@x), #neg(@y)) -> c_97(#eq^#(@x, @y)) , 109: #eq^#(#neg(@x), #pos(@y)) -> c_98() , 110: #eq^#(#neg(@x), #0()) -> c_99() , 111: #eq^#(#pos(@x), #neg(@y)) -> c_100() , 112: #eq^#(#pos(@x), #pos(@y)) -> c_101(#eq^#(@x, @y)) , 113: #eq^#(#pos(@x), #0()) -> c_102() , 114: #eq^#(tuple#2(@x_1, @x_2), nil()) -> c_103() , 115: #eq^#(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> c_104(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), #eq^#(@x_1, @y_1), #eq^#(@x_2, @y_2)) , 116: #eq^#(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> c_105() , 117: #eq^#(::(@x_1, @x_2), nil()) -> c_106() , 118: #eq^#(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> c_107() , 119: #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) -> c_108(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), #eq^#(@x_1, @y_1), #eq^#(@x_2, @y_2)) , 120: #eq^#(#0(), #neg(@y)) -> c_109() , 121: #eq^#(#0(), #pos(@y)) -> c_110() , 122: #eq^#(#0(), #0()) -> c_111() , 123: #eq^#(#0(), #s(@y)) -> c_112() , 124: #eq^#(#s(@x), #0()) -> c_113() , 125: #eq^#(#s(@x), #s(@y)) -> c_114(#eq^#(@x, @y)) , 126: sub'#2^#(nil(), @r, @x, @xs) -> c_30() , 127: compare#4^#(#false(), @r, @x, @y) -> c_56() , 128: sub'#1^#(nil(), @b2, @r) -> c_40() , 129: add'#1^#(nil(), @b2, @r) -> c_44() , 130: add'#2^#(nil(), @r, @x, @xs) -> c_46() , 131: #natsub^#(@x, #0()) -> c_66() , 132: #natsub^#(#s(@x), #s(@y)) -> c_67(#natsub^#(@x, @y)) , 133: #natdiv^#(#0(), #0()) -> c_68() , 134: #natdiv^#(#s(@x), #s(@y)) -> c_69(#natdiv^#(#natsub(@x, @y), #s(@y)), #natsub^#(@x, @y)) , 135: #pred^#(#neg(#s(@x))) -> c_145() , 136: #pred^#(#pos(#s(#0()))) -> c_146() , 137: #pred^#(#pos(#s(#s(@x)))) -> c_147() , 138: #pred^#(#0()) -> c_148() , 139: #succ^#(#neg(#s(#0()))) -> c_132() , 140: #succ^#(#neg(#s(#s(@x)))) -> c_133() , 141: #succ^#(#pos(#s(@x))) -> c_134() , 142: #succ^#(#0()) -> c_135() , 143: #and^#(#true(), #true()) -> c_78() , 144: #and^#(#true(), #false()) -> c_79() , 145: #and^#(#false(), #true()) -> c_80() , 146: #and^#(#false(), #false()) -> c_81() , 147: #natmult^#(#0(), @y) -> c_115() , 148: #natmult^#(#s(@x), @y) -> c_116(#add^#(#pos(@y), #natmult(@x, @y)), #natmult^#(@x, @y)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { sub^#(@b1, @b2) -> c_2(sub#1^#(sub'(@b1, @b2, #abs(#0()))), sub'^#(@b1, @b2, #abs(#0())), #abs^#(#0())) , sub'^#(@b1, @b2, @r) -> c_51(sub'#1^#(@b1, @b2, @r)) , mult#2^#(@zs, @b2, @x) -> c_5(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs), #equal^#(@x, #pos(#s(#0())))) , mult#3^#(#true(), @b2, @zs) -> c_42(add^#(@b2, @zs)) , bitToInt'#1^#(::(@x, @xs), @n) -> c_8(+^#(*(@x, @n), bitToInt'(@xs, *(@n, #pos(#s(#s(#0())))))), *^#(@x, @n), bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0()))))), *^#(@n, #pos(#s(#s(#0()))))) , bitToInt'^#(@b, @n) -> c_15(bitToInt'#1^#(@b, @n)) , sum^#(@x, @y, @r) -> c_9(sum#1^#(+(+(@x, @y), @r)), +^#(+(@x, @y), @r), +^#(@x, @y)) , sum#1^#(@s) -> c_20(sum#2^#(#equal(@s, #0()), @s), #equal^#(@s, #0())) , mult3^#(@b1, @b2, @b3) -> c_11(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , mult^#(@b1, @b2) -> c_16(mult#1^#(@b1, @b2)) , leq^#(@b1, @b2) -> c_13(#less^#(compare(@b1, @b2), #pos(#s(#0()))), compare^#(@b1, @b2)) , compare^#(@b1, @b2) -> c_52(compare#1^#(@b1, @b2)) , mult#1^#(::(@x, @xs), @b2) -> c_50(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), #abs^#(#0()), mult^#(@xs, @b2)) , bitToInt^#(@b) -> c_17(bitToInt'^#(@b, #abs(#pos(#s(#0())))), #abs^#(#pos(#s(#0())))) , sum#2^#(#false(), @s) -> c_19(sum#3^#(#equal(@s, #pos(#s(#0()))), @s), #equal^#(@s, #pos(#s(#0())))) , sum#3^#(#false(), @s) -> c_58(sum#4^#(#equal(@s, #pos(#s(#s(#0()))))), #equal^#(@s, #pos(#s(#s(#0()))))) , sub'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_28(sub'#4^#(sub'(@xs, @ys, @r'), @z), sub'^#(@xs, @ys, @r')) , sub'#4^#(tuple#2(@zs, @s), @z) -> c_34(sub'#5^#(#equal(@s, #pos(#s(#0()))), @z, @zs), #equal^#(@s, #pos(#s(#0())))) , sub'#2^#(::(@y, @ys), @r, @x, @xs) -> c_31(sub'#3^#(diff(@x, @y, @r), @xs, @ys), diff^#(@x, @y, @r)) , diff^#(@x, @y, @r) -> c_48(mod^#(+(+(@x, @y), @r), #pos(#s(#s(#0())))), +^#(+(@x, @y), @r), +^#(@x, @y), diff#1^#(#less(-(-(@x, @y), @r), #0())), #less^#(-(-(@x, @y), @r), #0()), -^#(-(@x, @y), @r), -^#(@x, @y)) , compare#2^#(::(@y, @ys), @x, @xs) -> c_33(compare#3^#(compare(@xs, @ys), @x, @y), compare^#(@xs, @ys)) , compare#3^#(@r, @x, @y) -> c_37(compare#4^#(#equal(@r, #0()), @r, @x, @y), #equal^#(@r, #0())) , compare#5^#(#false(), @x, @y) -> c_36(compare#6^#(#greater(@x, @y)), #greater^#(@x, @y)) , compare#4^#(#true(), @r, @x, @y) -> c_55(compare#5^#(#less(@x, @y), @x, @y), #less^#(@x, @y)) , add^#(@b1, @b2) -> c_38(add'^#(@b1, @b2, #abs(#0())), #abs^#(#0())) , add'^#(@b1, @b2, @r) -> c_59(add'#1^#(@b1, @b2, @r)) , add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_39(add'^#(@xs, @ys, @r')) , sub'#1^#(::(@x, @xs), @b2, @r) -> c_41(sub'#2^#(@b2, @r, @x, @xs)) , add'#1^#(::(@x, @xs), @b2, @r) -> c_45(add'#2^#(@b2, @r, @x, @xs)) , add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_47(add'#3^#(sum(@x, @y, @r), @xs, @ys), sum^#(@x, @y, @r)) , compare#1^#(::(@x, @xs), @b2) -> c_61(compare#2^#(@b2, @x, @xs)) } Weak DPs: { -^#(@x, @y) -> c_1(#sub^#(@x, @y)) , #sub^#(@x, #neg(@y)) -> c_120(#add^#(@x, #pos(@y))) , #sub^#(@x, #pos(@y)) -> c_121(#add^#(@x, #neg(@y))) , #sub^#(@x, #0()) -> c_122() , sub#1^#(tuple#2(@b, @_@1)) -> c_12() , #abs^#(#neg(@x)) -> c_62() , #abs^#(#pos(@x)) -> c_63() , #abs^#(#0()) -> c_64() , #abs^#(#s(@x)) -> c_65() , diff#1^#(#true()) -> c_3(#abs^#(#pos(#s(#0())))) , diff#1^#(#false()) -> c_4(#abs^#(#0())) , mult#3^#(#false(), @b2, @zs) -> c_43() , #equal^#(@x, @y) -> c_29(#eq^#(@x, @y)) , div^#(@x, @y) -> c_6(#div^#(@x, @y)) , #div^#(#neg(@x), #neg(@y)) -> c_136(#natdiv^#(@x, @y)) , #div^#(#neg(@x), #pos(@y)) -> c_137(#natdiv^#(@x, @y)) , #div^#(#neg(@x), #0()) -> c_138() , #div^#(#pos(@x), #neg(@y)) -> c_139(#natdiv^#(@x, @y)) , #div^#(#pos(@x), #pos(@y)) -> c_140(#natdiv^#(@x, @y)) , #div^#(#pos(@x), #0()) -> c_141() , #div^#(#0(), #neg(@y)) -> c_142() , #div^#(#0(), #pos(@y)) -> c_143() , #div^#(#0(), #0()) -> c_144() , bitToInt'#1^#(nil(), @n) -> c_7(#abs^#(#0())) , +^#(@x, @y) -> c_21(#add^#(@x, @y)) , *^#(@x, @y) -> c_24(#mult^#(@x, @y)) , mod^#(@x, @y) -> c_10(-^#(@x, *(@x, div(@x, @y))), *^#(@x, div(@x, @y)), div^#(@x, @y)) , #less^#(@x, @y) -> c_27(#cklt^#(#compare(@x, @y)), #compare^#(@x, @y)) , #greater^#(@x, @y) -> c_14(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y)) , #ckgt^#(#EQ()) -> c_70() , #ckgt^#(#LT()) -> c_71() , #ckgt^#(#GT()) -> c_72() , #compare^#(#neg(@x), #neg(@y)) -> c_82(#compare^#(@y, @x)) , #compare^#(#neg(@x), #pos(@y)) -> c_83() , #compare^#(#neg(@x), #0()) -> c_84() , #compare^#(#pos(@x), #neg(@y)) -> c_85() , #compare^#(#pos(@x), #pos(@y)) -> c_86(#compare^#(@x, @y)) , #compare^#(#pos(@x), #0()) -> c_87() , #compare^#(#0(), #neg(@y)) -> c_88() , #compare^#(#0(), #pos(@y)) -> c_89() , #compare^#(#0(), #0()) -> c_90() , #compare^#(#0(), #s(@y)) -> c_91() , #compare^#(#s(@x), #0()) -> c_92() , #compare^#(#s(@x), #s(@y)) -> c_93(#compare^#(@x, @y)) , mult#1^#(nil(), @b2) -> c_49() , sum#2^#(#true(), @s) -> c_18(#abs^#(#0()), #abs^#(#0())) , sum#3^#(#true(), @s) -> c_57(#abs^#(#pos(#s(#0()))), #abs^#(#0())) , #add^#(#neg(#s(#0())), @y) -> c_73(#pred^#(@y)) , #add^#(#neg(#s(#s(@x))), @y) -> c_74(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , #add^#(#pos(#s(#0())), @y) -> c_75(#succ^#(@y)) , #add^#(#pos(#s(#s(@x))), @y) -> c_76(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , #add^#(#0(), @y) -> c_77() , sum#4^#(#true()) -> c_22(#abs^#(#0()), #abs^#(#pos(#s(#0())))) , sum#4^#(#false()) -> c_23(#abs^#(#pos(#s(#0()))), #abs^#(#pos(#s(#0())))) , #mult^#(#neg(@x), #neg(@y)) -> c_123(#natmult^#(@x, @y)) , #mult^#(#neg(@x), #pos(@y)) -> c_124(#natmult^#(@x, @y)) , #mult^#(#neg(@x), #0()) -> c_125() , #mult^#(#pos(@x), #neg(@y)) -> c_126(#natmult^#(@x, @y)) , #mult^#(#pos(@x), #pos(@y)) -> c_127(#natmult^#(@x, @y)) , #mult^#(#pos(@x), #0()) -> c_128() , #mult^#(#0(), #neg(@y)) -> c_129() , #mult^#(#0(), #pos(@y)) -> c_130() , #mult^#(#0(), #0()) -> c_131() , sub'#5^#(#true(), @z, @zs) -> c_25(#abs^#(#0())) , sub'#5^#(#false(), @z, @zs) -> c_26() , #cklt^#(#EQ()) -> c_117() , #cklt^#(#LT()) -> c_118() , #cklt^#(#GT()) -> c_119() , #eq^#(nil(), nil()) -> c_94() , #eq^#(nil(), tuple#2(@y_1, @y_2)) -> c_95() , #eq^#(nil(), ::(@y_1, @y_2)) -> c_96() , #eq^#(#neg(@x), #neg(@y)) -> c_97(#eq^#(@x, @y)) , #eq^#(#neg(@x), #pos(@y)) -> c_98() , #eq^#(#neg(@x), #0()) -> c_99() , #eq^#(#pos(@x), #neg(@y)) -> c_100() , #eq^#(#pos(@x), #pos(@y)) -> c_101(#eq^#(@x, @y)) , #eq^#(#pos(@x), #0()) -> c_102() , #eq^#(tuple#2(@x_1, @x_2), nil()) -> c_103() , #eq^#(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> c_104(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), #eq^#(@x_1, @y_1), #eq^#(@x_2, @y_2)) , #eq^#(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> c_105() , #eq^#(::(@x_1, @x_2), nil()) -> c_106() , #eq^#(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> c_107() , #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) -> c_108(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), #eq^#(@x_1, @y_1), #eq^#(@x_2, @y_2)) , #eq^#(#0(), #neg(@y)) -> c_109() , #eq^#(#0(), #pos(@y)) -> c_110() , #eq^#(#0(), #0()) -> c_111() , #eq^#(#0(), #s(@y)) -> c_112() , #eq^#(#s(@x), #0()) -> c_113() , #eq^#(#s(@x), #s(@y)) -> c_114(#eq^#(@x, @y)) , sub'#2^#(nil(), @r, @x, @xs) -> c_30() , compare#2^#(nil(), @x, @xs) -> c_32(#abs^#(#0())) , compare#5^#(#true(), @x, @y) -> c_35(-^#(#0(), #pos(#s(#0())))) , compare#6^#(#true()) -> c_53(#abs^#(#pos(#s(#0())))) , compare#6^#(#false()) -> c_54(#abs^#(#0())) , compare#4^#(#false(), @r, @x, @y) -> c_56() , sub'#1^#(nil(), @b2, @r) -> c_40() , add'#1^#(nil(), @b2, @r) -> c_44() , add'#2^#(nil(), @r, @x, @xs) -> c_46() , compare#1^#(nil(), @b2) -> c_60(#abs^#(#0())) , #natsub^#(@x, #0()) -> c_66() , #natsub^#(#s(@x), #s(@y)) -> c_67(#natsub^#(@x, @y)) , #natdiv^#(#0(), #0()) -> c_68() , #natdiv^#(#s(@x), #s(@y)) -> c_69(#natdiv^#(#natsub(@x, @y), #s(@y)), #natsub^#(@x, @y)) , #pred^#(#neg(#s(@x))) -> c_145() , #pred^#(#pos(#s(#0()))) -> c_146() , #pred^#(#pos(#s(#s(@x)))) -> c_147() , #pred^#(#0()) -> c_148() , #succ^#(#neg(#s(#0()))) -> c_132() , #succ^#(#neg(#s(#s(@x)))) -> c_133() , #succ^#(#pos(#s(@x))) -> c_134() , #succ^#(#0()) -> c_135() , #and^#(#true(), #true()) -> c_78() , #and^#(#true(), #false()) -> c_79() , #and^#(#false(), #true()) -> c_80() , #and^#(#false(), #false()) -> c_81() , #natmult^#(#0(), @y) -> c_115() , #natmult^#(#s(@x), @y) -> c_116(#add^#(#pos(@y), #natmult(@x, @y)), #natmult^#(@x, @y)) } Weak Trs: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , sub(@b1, @b2) -> sub#1(sub'(@b1, @b2, #abs(#0()))) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #ckgt(#EQ()) -> #false() , #ckgt(#LT()) -> #false() , #ckgt(#GT()) -> #true() , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , div(@x, @y) -> #div(@x, @y) , bitToInt'#1(nil(), @n) -> #abs(#0()) , bitToInt'#1(::(@x, @xs), @n) -> +(*(@x, @n), bitToInt'(@xs, *(@n, #pos(#s(#s(#0())))))) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , mult3(@b1, @b2, @b3) -> mult(mult(@b1, @b2), @b2) , sub#1(tuple#2(@b, @_@1)) -> @b , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , leq(@b1, @b2) -> #less(compare(@b1, @b2), #pos(#s(#0()))) , #greater(@x, @y) -> #ckgt(#compare(@x, @y)) , bitToInt'(@b, @n) -> bitToInt'#1(@b, @n) , mult(@b1, @b2) -> mult#1(@b1, @b2) , bitToInt(@b) -> bitToInt'(@b, #abs(#pos(#s(#0())))) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , *(@x, @y) -> #mult(@x, @y) , sub'#5(#true(), @z, @zs) -> ::(#abs(#0()), @zs) , sub'#5(#false(), @z, @zs) -> ::(@z, @zs) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , sub'#3(tuple#2(@z, @r'), @xs, @ys) -> sub'#4(sub'(@xs, @ys, @r'), @z) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , sub'#2(nil(), @r, @x, @xs) -> tuple#2(nil(), @r) , sub'#2(::(@y, @ys), @r, @x, @xs) -> sub'#3(diff(@x, @y, @r), @xs, @ys) , compare#2(nil(), @x, @xs) -> #abs(#0()) , compare#2(::(@y, @ys), @x, @xs) -> compare#3(compare(@xs, @ys), @x, @y) , sub'#4(tuple#2(@zs, @s), @z) -> tuple#2(sub'#5(#equal(@s, #pos(#s(#0()))), @z, @zs), @s) , compare#5(#true(), @x, @y) -> -(#0(), #pos(#s(#0()))) , compare#5(#false(), @x, @y) -> compare#6(#greater(@x, @y)) , compare#3(@r, @x, @y) -> compare#4(#equal(@r, #0()), @r, @x, @y) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , sub'#1(nil(), @b2, @r) -> tuple#2(nil(), @r) , sub'#1(::(@x, @xs), @b2, @r) -> sub'#2(@b2, @r, @x, @xs) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sub'(@b1, @b2, @r) -> sub'#1(@b1, @b2, @r) , compare(@b1, @b2) -> compare#1(@b1, @b2) , compare#6(#true()) -> #abs(#pos(#s(#0()))) , compare#6(#false()) -> #abs(#0()) , compare#4(#true(), @r, @x, @y) -> compare#5(#less(@x, @y), @x, @y) , compare#4(#false(), @r, @x, @y) -> @r , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , compare#1(nil(), @b2) -> #abs(#0()) , compare#1(::(@x, @xs), @b2) -> compare#2(@b2, @x, @xs) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We estimate the number of application of {16,18,20,23} by applications of Pre({16,18,20,23}) = {15,17,19,24}. Here rules are labeled as follows: DPs: { 1: sub^#(@b1, @b2) -> c_2(sub#1^#(sub'(@b1, @b2, #abs(#0()))), sub'^#(@b1, @b2, #abs(#0())), #abs^#(#0())) , 2: sub'^#(@b1, @b2, @r) -> c_51(sub'#1^#(@b1, @b2, @r)) , 3: mult#2^#(@zs, @b2, @x) -> c_5(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs), #equal^#(@x, #pos(#s(#0())))) , 4: mult#3^#(#true(), @b2, @zs) -> c_42(add^#(@b2, @zs)) , 5: bitToInt'#1^#(::(@x, @xs), @n) -> c_8(+^#(*(@x, @n), bitToInt'(@xs, *(@n, #pos(#s(#s(#0())))))), *^#(@x, @n), bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0()))))), *^#(@n, #pos(#s(#s(#0()))))) , 6: bitToInt'^#(@b, @n) -> c_15(bitToInt'#1^#(@b, @n)) , 7: sum^#(@x, @y, @r) -> c_9(sum#1^#(+(+(@x, @y), @r)), +^#(+(@x, @y), @r), +^#(@x, @y)) , 8: sum#1^#(@s) -> c_20(sum#2^#(#equal(@s, #0()), @s), #equal^#(@s, #0())) , 9: mult3^#(@b1, @b2, @b3) -> c_11(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , 10: mult^#(@b1, @b2) -> c_16(mult#1^#(@b1, @b2)) , 11: leq^#(@b1, @b2) -> c_13(#less^#(compare(@b1, @b2), #pos(#s(#0()))), compare^#(@b1, @b2)) , 12: compare^#(@b1, @b2) -> c_52(compare#1^#(@b1, @b2)) , 13: mult#1^#(::(@x, @xs), @b2) -> c_50(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), #abs^#(#0()), mult^#(@xs, @b2)) , 14: bitToInt^#(@b) -> c_17(bitToInt'^#(@b, #abs(#pos(#s(#0())))), #abs^#(#pos(#s(#0())))) , 15: sum#2^#(#false(), @s) -> c_19(sum#3^#(#equal(@s, #pos(#s(#0()))), @s), #equal^#(@s, #pos(#s(#0())))) , 16: sum#3^#(#false(), @s) -> c_58(sum#4^#(#equal(@s, #pos(#s(#s(#0()))))), #equal^#(@s, #pos(#s(#s(#0()))))) , 17: sub'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_28(sub'#4^#(sub'(@xs, @ys, @r'), @z), sub'^#(@xs, @ys, @r')) , 18: sub'#4^#(tuple#2(@zs, @s), @z) -> c_34(sub'#5^#(#equal(@s, #pos(#s(#0()))), @z, @zs), #equal^#(@s, #pos(#s(#0())))) , 19: sub'#2^#(::(@y, @ys), @r, @x, @xs) -> c_31(sub'#3^#(diff(@x, @y, @r), @xs, @ys), diff^#(@x, @y, @r)) , 20: diff^#(@x, @y, @r) -> c_48(mod^#(+(+(@x, @y), @r), #pos(#s(#s(#0())))), +^#(+(@x, @y), @r), +^#(@x, @y), diff#1^#(#less(-(-(@x, @y), @r), #0())), #less^#(-(-(@x, @y), @r), #0()), -^#(-(@x, @y), @r), -^#(@x, @y)) , 21: compare#2^#(::(@y, @ys), @x, @xs) -> c_33(compare#3^#(compare(@xs, @ys), @x, @y), compare^#(@xs, @ys)) , 22: compare#3^#(@r, @x, @y) -> c_37(compare#4^#(#equal(@r, #0()), @r, @x, @y), #equal^#(@r, #0())) , 23: compare#5^#(#false(), @x, @y) -> c_36(compare#6^#(#greater(@x, @y)), #greater^#(@x, @y)) , 24: compare#4^#(#true(), @r, @x, @y) -> c_55(compare#5^#(#less(@x, @y), @x, @y), #less^#(@x, @y)) , 25: add^#(@b1, @b2) -> c_38(add'^#(@b1, @b2, #abs(#0())), #abs^#(#0())) , 26: add'^#(@b1, @b2, @r) -> c_59(add'#1^#(@b1, @b2, @r)) , 27: add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_39(add'^#(@xs, @ys, @r')) , 28: sub'#1^#(::(@x, @xs), @b2, @r) -> c_41(sub'#2^#(@b2, @r, @x, @xs)) , 29: add'#1^#(::(@x, @xs), @b2, @r) -> c_45(add'#2^#(@b2, @r, @x, @xs)) , 30: add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_47(add'#3^#(sum(@x, @y, @r), @xs, @ys), sum^#(@x, @y, @r)) , 31: compare#1^#(::(@x, @xs), @b2) -> c_61(compare#2^#(@b2, @x, @xs)) , 32: -^#(@x, @y) -> c_1(#sub^#(@x, @y)) , 33: #sub^#(@x, #neg(@y)) -> c_120(#add^#(@x, #pos(@y))) , 34: #sub^#(@x, #pos(@y)) -> c_121(#add^#(@x, #neg(@y))) , 35: #sub^#(@x, #0()) -> c_122() , 36: sub#1^#(tuple#2(@b, @_@1)) -> c_12() , 37: #abs^#(#neg(@x)) -> c_62() , 38: #abs^#(#pos(@x)) -> c_63() , 39: #abs^#(#0()) -> c_64() , 40: #abs^#(#s(@x)) -> c_65() , 41: diff#1^#(#true()) -> c_3(#abs^#(#pos(#s(#0())))) , 42: diff#1^#(#false()) -> c_4(#abs^#(#0())) , 43: mult#3^#(#false(), @b2, @zs) -> c_43() , 44: #equal^#(@x, @y) -> c_29(#eq^#(@x, @y)) , 45: div^#(@x, @y) -> c_6(#div^#(@x, @y)) , 46: #div^#(#neg(@x), #neg(@y)) -> c_136(#natdiv^#(@x, @y)) , 47: #div^#(#neg(@x), #pos(@y)) -> c_137(#natdiv^#(@x, @y)) , 48: #div^#(#neg(@x), #0()) -> c_138() , 49: #div^#(#pos(@x), #neg(@y)) -> c_139(#natdiv^#(@x, @y)) , 50: #div^#(#pos(@x), #pos(@y)) -> c_140(#natdiv^#(@x, @y)) , 51: #div^#(#pos(@x), #0()) -> c_141() , 52: #div^#(#0(), #neg(@y)) -> c_142() , 53: #div^#(#0(), #pos(@y)) -> c_143() , 54: #div^#(#0(), #0()) -> c_144() , 55: bitToInt'#1^#(nil(), @n) -> c_7(#abs^#(#0())) , 56: +^#(@x, @y) -> c_21(#add^#(@x, @y)) , 57: *^#(@x, @y) -> c_24(#mult^#(@x, @y)) , 58: mod^#(@x, @y) -> c_10(-^#(@x, *(@x, div(@x, @y))), *^#(@x, div(@x, @y)), div^#(@x, @y)) , 59: #less^#(@x, @y) -> c_27(#cklt^#(#compare(@x, @y)), #compare^#(@x, @y)) , 60: #greater^#(@x, @y) -> c_14(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y)) , 61: #ckgt^#(#EQ()) -> c_70() , 62: #ckgt^#(#LT()) -> c_71() , 63: #ckgt^#(#GT()) -> c_72() , 64: #compare^#(#neg(@x), #neg(@y)) -> c_82(#compare^#(@y, @x)) , 65: #compare^#(#neg(@x), #pos(@y)) -> c_83() , 66: #compare^#(#neg(@x), #0()) -> c_84() , 67: #compare^#(#pos(@x), #neg(@y)) -> c_85() , 68: #compare^#(#pos(@x), #pos(@y)) -> c_86(#compare^#(@x, @y)) , 69: #compare^#(#pos(@x), #0()) -> c_87() , 70: #compare^#(#0(), #neg(@y)) -> c_88() , 71: #compare^#(#0(), #pos(@y)) -> c_89() , 72: #compare^#(#0(), #0()) -> c_90() , 73: #compare^#(#0(), #s(@y)) -> c_91() , 74: #compare^#(#s(@x), #0()) -> c_92() , 75: #compare^#(#s(@x), #s(@y)) -> c_93(#compare^#(@x, @y)) , 76: mult#1^#(nil(), @b2) -> c_49() , 77: sum#2^#(#true(), @s) -> c_18(#abs^#(#0()), #abs^#(#0())) , 78: sum#3^#(#true(), @s) -> c_57(#abs^#(#pos(#s(#0()))), #abs^#(#0())) , 79: #add^#(#neg(#s(#0())), @y) -> c_73(#pred^#(@y)) , 80: #add^#(#neg(#s(#s(@x))), @y) -> c_74(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , 81: #add^#(#pos(#s(#0())), @y) -> c_75(#succ^#(@y)) , 82: #add^#(#pos(#s(#s(@x))), @y) -> c_76(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , 83: #add^#(#0(), @y) -> c_77() , 84: sum#4^#(#true()) -> c_22(#abs^#(#0()), #abs^#(#pos(#s(#0())))) , 85: sum#4^#(#false()) -> c_23(#abs^#(#pos(#s(#0()))), #abs^#(#pos(#s(#0())))) , 86: #mult^#(#neg(@x), #neg(@y)) -> c_123(#natmult^#(@x, @y)) , 87: #mult^#(#neg(@x), #pos(@y)) -> c_124(#natmult^#(@x, @y)) , 88: #mult^#(#neg(@x), #0()) -> c_125() , 89: #mult^#(#pos(@x), #neg(@y)) -> c_126(#natmult^#(@x, @y)) , 90: #mult^#(#pos(@x), #pos(@y)) -> c_127(#natmult^#(@x, @y)) , 91: #mult^#(#pos(@x), #0()) -> c_128() , 92: #mult^#(#0(), #neg(@y)) -> c_129() , 93: #mult^#(#0(), #pos(@y)) -> c_130() , 94: #mult^#(#0(), #0()) -> c_131() , 95: sub'#5^#(#true(), @z, @zs) -> c_25(#abs^#(#0())) , 96: sub'#5^#(#false(), @z, @zs) -> c_26() , 97: #cklt^#(#EQ()) -> c_117() , 98: #cklt^#(#LT()) -> c_118() , 99: #cklt^#(#GT()) -> c_119() , 100: #eq^#(nil(), nil()) -> c_94() , 101: #eq^#(nil(), tuple#2(@y_1, @y_2)) -> c_95() , 102: #eq^#(nil(), ::(@y_1, @y_2)) -> c_96() , 103: #eq^#(#neg(@x), #neg(@y)) -> c_97(#eq^#(@x, @y)) , 104: #eq^#(#neg(@x), #pos(@y)) -> c_98() , 105: #eq^#(#neg(@x), #0()) -> c_99() , 106: #eq^#(#pos(@x), #neg(@y)) -> c_100() , 107: #eq^#(#pos(@x), #pos(@y)) -> c_101(#eq^#(@x, @y)) , 108: #eq^#(#pos(@x), #0()) -> c_102() , 109: #eq^#(tuple#2(@x_1, @x_2), nil()) -> c_103() , 110: #eq^#(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> c_104(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), #eq^#(@x_1, @y_1), #eq^#(@x_2, @y_2)) , 111: #eq^#(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> c_105() , 112: #eq^#(::(@x_1, @x_2), nil()) -> c_106() , 113: #eq^#(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> c_107() , 114: #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) -> c_108(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), #eq^#(@x_1, @y_1), #eq^#(@x_2, @y_2)) , 115: #eq^#(#0(), #neg(@y)) -> c_109() , 116: #eq^#(#0(), #pos(@y)) -> c_110() , 117: #eq^#(#0(), #0()) -> c_111() , 118: #eq^#(#0(), #s(@y)) -> c_112() , 119: #eq^#(#s(@x), #0()) -> c_113() , 120: #eq^#(#s(@x), #s(@y)) -> c_114(#eq^#(@x, @y)) , 121: sub'#2^#(nil(), @r, @x, @xs) -> c_30() , 122: compare#2^#(nil(), @x, @xs) -> c_32(#abs^#(#0())) , 123: compare#5^#(#true(), @x, @y) -> c_35(-^#(#0(), #pos(#s(#0())))) , 124: compare#6^#(#true()) -> c_53(#abs^#(#pos(#s(#0())))) , 125: compare#6^#(#false()) -> c_54(#abs^#(#0())) , 126: compare#4^#(#false(), @r, @x, @y) -> c_56() , 127: sub'#1^#(nil(), @b2, @r) -> c_40() , 128: add'#1^#(nil(), @b2, @r) -> c_44() , 129: add'#2^#(nil(), @r, @x, @xs) -> c_46() , 130: compare#1^#(nil(), @b2) -> c_60(#abs^#(#0())) , 131: #natsub^#(@x, #0()) -> c_66() , 132: #natsub^#(#s(@x), #s(@y)) -> c_67(#natsub^#(@x, @y)) , 133: #natdiv^#(#0(), #0()) -> c_68() , 134: #natdiv^#(#s(@x), #s(@y)) -> c_69(#natdiv^#(#natsub(@x, @y), #s(@y)), #natsub^#(@x, @y)) , 135: #pred^#(#neg(#s(@x))) -> c_145() , 136: #pred^#(#pos(#s(#0()))) -> c_146() , 137: #pred^#(#pos(#s(#s(@x)))) -> c_147() , 138: #pred^#(#0()) -> c_148() , 139: #succ^#(#neg(#s(#0()))) -> c_132() , 140: #succ^#(#neg(#s(#s(@x)))) -> c_133() , 141: #succ^#(#pos(#s(@x))) -> c_134() , 142: #succ^#(#0()) -> c_135() , 143: #and^#(#true(), #true()) -> c_78() , 144: #and^#(#true(), #false()) -> c_79() , 145: #and^#(#false(), #true()) -> c_80() , 146: #and^#(#false(), #false()) -> c_81() , 147: #natmult^#(#0(), @y) -> c_115() , 148: #natmult^#(#s(@x), @y) -> c_116(#add^#(#pos(@y), #natmult(@x, @y)), #natmult^#(@x, @y)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { sub^#(@b1, @b2) -> c_2(sub#1^#(sub'(@b1, @b2, #abs(#0()))), sub'^#(@b1, @b2, #abs(#0())), #abs^#(#0())) , sub'^#(@b1, @b2, @r) -> c_51(sub'#1^#(@b1, @b2, @r)) , mult#2^#(@zs, @b2, @x) -> c_5(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs), #equal^#(@x, #pos(#s(#0())))) , mult#3^#(#true(), @b2, @zs) -> c_42(add^#(@b2, @zs)) , bitToInt'#1^#(::(@x, @xs), @n) -> c_8(+^#(*(@x, @n), bitToInt'(@xs, *(@n, #pos(#s(#s(#0())))))), *^#(@x, @n), bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0()))))), *^#(@n, #pos(#s(#s(#0()))))) , bitToInt'^#(@b, @n) -> c_15(bitToInt'#1^#(@b, @n)) , sum^#(@x, @y, @r) -> c_9(sum#1^#(+(+(@x, @y), @r)), +^#(+(@x, @y), @r), +^#(@x, @y)) , sum#1^#(@s) -> c_20(sum#2^#(#equal(@s, #0()), @s), #equal^#(@s, #0())) , mult3^#(@b1, @b2, @b3) -> c_11(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , mult^#(@b1, @b2) -> c_16(mult#1^#(@b1, @b2)) , leq^#(@b1, @b2) -> c_13(#less^#(compare(@b1, @b2), #pos(#s(#0()))), compare^#(@b1, @b2)) , compare^#(@b1, @b2) -> c_52(compare#1^#(@b1, @b2)) , mult#1^#(::(@x, @xs), @b2) -> c_50(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), #abs^#(#0()), mult^#(@xs, @b2)) , bitToInt^#(@b) -> c_17(bitToInt'^#(@b, #abs(#pos(#s(#0())))), #abs^#(#pos(#s(#0())))) , sum#2^#(#false(), @s) -> c_19(sum#3^#(#equal(@s, #pos(#s(#0()))), @s), #equal^#(@s, #pos(#s(#0())))) , sub'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_28(sub'#4^#(sub'(@xs, @ys, @r'), @z), sub'^#(@xs, @ys, @r')) , sub'#2^#(::(@y, @ys), @r, @x, @xs) -> c_31(sub'#3^#(diff(@x, @y, @r), @xs, @ys), diff^#(@x, @y, @r)) , compare#2^#(::(@y, @ys), @x, @xs) -> c_33(compare#3^#(compare(@xs, @ys), @x, @y), compare^#(@xs, @ys)) , compare#3^#(@r, @x, @y) -> c_37(compare#4^#(#equal(@r, #0()), @r, @x, @y), #equal^#(@r, #0())) , compare#4^#(#true(), @r, @x, @y) -> c_55(compare#5^#(#less(@x, @y), @x, @y), #less^#(@x, @y)) , add^#(@b1, @b2) -> c_38(add'^#(@b1, @b2, #abs(#0())), #abs^#(#0())) , add'^#(@b1, @b2, @r) -> c_59(add'#1^#(@b1, @b2, @r)) , add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_39(add'^#(@xs, @ys, @r')) , sub'#1^#(::(@x, @xs), @b2, @r) -> c_41(sub'#2^#(@b2, @r, @x, @xs)) , add'#1^#(::(@x, @xs), @b2, @r) -> c_45(add'#2^#(@b2, @r, @x, @xs)) , add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_47(add'#3^#(sum(@x, @y, @r), @xs, @ys), sum^#(@x, @y, @r)) , compare#1^#(::(@x, @xs), @b2) -> c_61(compare#2^#(@b2, @x, @xs)) } Weak DPs: { -^#(@x, @y) -> c_1(#sub^#(@x, @y)) , #sub^#(@x, #neg(@y)) -> c_120(#add^#(@x, #pos(@y))) , #sub^#(@x, #pos(@y)) -> c_121(#add^#(@x, #neg(@y))) , #sub^#(@x, #0()) -> c_122() , sub#1^#(tuple#2(@b, @_@1)) -> c_12() , #abs^#(#neg(@x)) -> c_62() , #abs^#(#pos(@x)) -> c_63() , #abs^#(#0()) -> c_64() , #abs^#(#s(@x)) -> c_65() , diff#1^#(#true()) -> c_3(#abs^#(#pos(#s(#0())))) , diff#1^#(#false()) -> c_4(#abs^#(#0())) , mult#3^#(#false(), @b2, @zs) -> c_43() , #equal^#(@x, @y) -> c_29(#eq^#(@x, @y)) , div^#(@x, @y) -> c_6(#div^#(@x, @y)) , #div^#(#neg(@x), #neg(@y)) -> c_136(#natdiv^#(@x, @y)) , #div^#(#neg(@x), #pos(@y)) -> c_137(#natdiv^#(@x, @y)) , #div^#(#neg(@x), #0()) -> c_138() , #div^#(#pos(@x), #neg(@y)) -> c_139(#natdiv^#(@x, @y)) , #div^#(#pos(@x), #pos(@y)) -> c_140(#natdiv^#(@x, @y)) , #div^#(#pos(@x), #0()) -> c_141() , #div^#(#0(), #neg(@y)) -> c_142() , #div^#(#0(), #pos(@y)) -> c_143() , #div^#(#0(), #0()) -> c_144() , bitToInt'#1^#(nil(), @n) -> c_7(#abs^#(#0())) , +^#(@x, @y) -> c_21(#add^#(@x, @y)) , *^#(@x, @y) -> c_24(#mult^#(@x, @y)) , mod^#(@x, @y) -> c_10(-^#(@x, *(@x, div(@x, @y))), *^#(@x, div(@x, @y)), div^#(@x, @y)) , #less^#(@x, @y) -> c_27(#cklt^#(#compare(@x, @y)), #compare^#(@x, @y)) , #greater^#(@x, @y) -> c_14(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y)) , #ckgt^#(#EQ()) -> c_70() , #ckgt^#(#LT()) -> c_71() , #ckgt^#(#GT()) -> c_72() , #compare^#(#neg(@x), #neg(@y)) -> c_82(#compare^#(@y, @x)) , #compare^#(#neg(@x), #pos(@y)) -> c_83() , #compare^#(#neg(@x), #0()) -> c_84() , #compare^#(#pos(@x), #neg(@y)) -> c_85() , #compare^#(#pos(@x), #pos(@y)) -> c_86(#compare^#(@x, @y)) , #compare^#(#pos(@x), #0()) -> c_87() , #compare^#(#0(), #neg(@y)) -> c_88() , #compare^#(#0(), #pos(@y)) -> c_89() , #compare^#(#0(), #0()) -> c_90() , #compare^#(#0(), #s(@y)) -> c_91() , #compare^#(#s(@x), #0()) -> c_92() , #compare^#(#s(@x), #s(@y)) -> c_93(#compare^#(@x, @y)) , mult#1^#(nil(), @b2) -> c_49() , sum#2^#(#true(), @s) -> c_18(#abs^#(#0()), #abs^#(#0())) , sum#3^#(#true(), @s) -> c_57(#abs^#(#pos(#s(#0()))), #abs^#(#0())) , sum#3^#(#false(), @s) -> c_58(sum#4^#(#equal(@s, #pos(#s(#s(#0()))))), #equal^#(@s, #pos(#s(#s(#0()))))) , #add^#(#neg(#s(#0())), @y) -> c_73(#pred^#(@y)) , #add^#(#neg(#s(#s(@x))), @y) -> c_74(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , #add^#(#pos(#s(#0())), @y) -> c_75(#succ^#(@y)) , #add^#(#pos(#s(#s(@x))), @y) -> c_76(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , #add^#(#0(), @y) -> c_77() , sum#4^#(#true()) -> c_22(#abs^#(#0()), #abs^#(#pos(#s(#0())))) , sum#4^#(#false()) -> c_23(#abs^#(#pos(#s(#0()))), #abs^#(#pos(#s(#0())))) , #mult^#(#neg(@x), #neg(@y)) -> c_123(#natmult^#(@x, @y)) , #mult^#(#neg(@x), #pos(@y)) -> c_124(#natmult^#(@x, @y)) , #mult^#(#neg(@x), #0()) -> c_125() , #mult^#(#pos(@x), #neg(@y)) -> c_126(#natmult^#(@x, @y)) , #mult^#(#pos(@x), #pos(@y)) -> c_127(#natmult^#(@x, @y)) , #mult^#(#pos(@x), #0()) -> c_128() , #mult^#(#0(), #neg(@y)) -> c_129() , #mult^#(#0(), #pos(@y)) -> c_130() , #mult^#(#0(), #0()) -> c_131() , sub'#5^#(#true(), @z, @zs) -> c_25(#abs^#(#0())) , sub'#5^#(#false(), @z, @zs) -> c_26() , #cklt^#(#EQ()) -> c_117() , #cklt^#(#LT()) -> c_118() , #cklt^#(#GT()) -> c_119() , sub'#4^#(tuple#2(@zs, @s), @z) -> c_34(sub'#5^#(#equal(@s, #pos(#s(#0()))), @z, @zs), #equal^#(@s, #pos(#s(#0())))) , #eq^#(nil(), nil()) -> c_94() , #eq^#(nil(), tuple#2(@y_1, @y_2)) -> c_95() , #eq^#(nil(), ::(@y_1, @y_2)) -> c_96() , #eq^#(#neg(@x), #neg(@y)) -> c_97(#eq^#(@x, @y)) , #eq^#(#neg(@x), #pos(@y)) -> c_98() , #eq^#(#neg(@x), #0()) -> c_99() , #eq^#(#pos(@x), #neg(@y)) -> c_100() , #eq^#(#pos(@x), #pos(@y)) -> c_101(#eq^#(@x, @y)) , #eq^#(#pos(@x), #0()) -> c_102() , #eq^#(tuple#2(@x_1, @x_2), nil()) -> c_103() , #eq^#(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> c_104(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), #eq^#(@x_1, @y_1), #eq^#(@x_2, @y_2)) , #eq^#(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> c_105() , #eq^#(::(@x_1, @x_2), nil()) -> c_106() , #eq^#(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> c_107() , #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) -> c_108(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), #eq^#(@x_1, @y_1), #eq^#(@x_2, @y_2)) , #eq^#(#0(), #neg(@y)) -> c_109() , #eq^#(#0(), #pos(@y)) -> c_110() , #eq^#(#0(), #0()) -> c_111() , #eq^#(#0(), #s(@y)) -> c_112() , #eq^#(#s(@x), #0()) -> c_113() , #eq^#(#s(@x), #s(@y)) -> c_114(#eq^#(@x, @y)) , sub'#2^#(nil(), @r, @x, @xs) -> c_30() , diff^#(@x, @y, @r) -> c_48(mod^#(+(+(@x, @y), @r), #pos(#s(#s(#0())))), +^#(+(@x, @y), @r), +^#(@x, @y), diff#1^#(#less(-(-(@x, @y), @r), #0())), #less^#(-(-(@x, @y), @r), #0()), -^#(-(@x, @y), @r), -^#(@x, @y)) , compare#2^#(nil(), @x, @xs) -> c_32(#abs^#(#0())) , compare#5^#(#true(), @x, @y) -> c_35(-^#(#0(), #pos(#s(#0())))) , compare#5^#(#false(), @x, @y) -> c_36(compare#6^#(#greater(@x, @y)), #greater^#(@x, @y)) , compare#6^#(#true()) -> c_53(#abs^#(#pos(#s(#0())))) , compare#6^#(#false()) -> c_54(#abs^#(#0())) , compare#4^#(#false(), @r, @x, @y) -> c_56() , sub'#1^#(nil(), @b2, @r) -> c_40() , add'#1^#(nil(), @b2, @r) -> c_44() , add'#2^#(nil(), @r, @x, @xs) -> c_46() , compare#1^#(nil(), @b2) -> c_60(#abs^#(#0())) , #natsub^#(@x, #0()) -> c_66() , #natsub^#(#s(@x), #s(@y)) -> c_67(#natsub^#(@x, @y)) , #natdiv^#(#0(), #0()) -> c_68() , #natdiv^#(#s(@x), #s(@y)) -> c_69(#natdiv^#(#natsub(@x, @y), #s(@y)), #natsub^#(@x, @y)) , #pred^#(#neg(#s(@x))) -> c_145() , #pred^#(#pos(#s(#0()))) -> c_146() , #pred^#(#pos(#s(#s(@x)))) -> c_147() , #pred^#(#0()) -> c_148() , #succ^#(#neg(#s(#0()))) -> c_132() , #succ^#(#neg(#s(#s(@x)))) -> c_133() , #succ^#(#pos(#s(@x))) -> c_134() , #succ^#(#0()) -> c_135() , #and^#(#true(), #true()) -> c_78() , #and^#(#true(), #false()) -> c_79() , #and^#(#false(), #true()) -> c_80() , #and^#(#false(), #false()) -> c_81() , #natmult^#(#0(), @y) -> c_115() , #natmult^#(#s(@x), @y) -> c_116(#add^#(#pos(@y), #natmult(@x, @y)), #natmult^#(@x, @y)) } Weak Trs: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , sub(@b1, @b2) -> sub#1(sub'(@b1, @b2, #abs(#0()))) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #ckgt(#EQ()) -> #false() , #ckgt(#LT()) -> #false() , #ckgt(#GT()) -> #true() , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , div(@x, @y) -> #div(@x, @y) , bitToInt'#1(nil(), @n) -> #abs(#0()) , bitToInt'#1(::(@x, @xs), @n) -> +(*(@x, @n), bitToInt'(@xs, *(@n, #pos(#s(#s(#0())))))) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , mult3(@b1, @b2, @b3) -> mult(mult(@b1, @b2), @b2) , sub#1(tuple#2(@b, @_@1)) -> @b , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , leq(@b1, @b2) -> #less(compare(@b1, @b2), #pos(#s(#0()))) , #greater(@x, @y) -> #ckgt(#compare(@x, @y)) , bitToInt'(@b, @n) -> bitToInt'#1(@b, @n) , mult(@b1, @b2) -> mult#1(@b1, @b2) , bitToInt(@b) -> bitToInt'(@b, #abs(#pos(#s(#0())))) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , *(@x, @y) -> #mult(@x, @y) , sub'#5(#true(), @z, @zs) -> ::(#abs(#0()), @zs) , sub'#5(#false(), @z, @zs) -> ::(@z, @zs) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , sub'#3(tuple#2(@z, @r'), @xs, @ys) -> sub'#4(sub'(@xs, @ys, @r'), @z) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , sub'#2(nil(), @r, @x, @xs) -> tuple#2(nil(), @r) , sub'#2(::(@y, @ys), @r, @x, @xs) -> sub'#3(diff(@x, @y, @r), @xs, @ys) , compare#2(nil(), @x, @xs) -> #abs(#0()) , compare#2(::(@y, @ys), @x, @xs) -> compare#3(compare(@xs, @ys), @x, @y) , sub'#4(tuple#2(@zs, @s), @z) -> tuple#2(sub'#5(#equal(@s, #pos(#s(#0()))), @z, @zs), @s) , compare#5(#true(), @x, @y) -> -(#0(), #pos(#s(#0()))) , compare#5(#false(), @x, @y) -> compare#6(#greater(@x, @y)) , compare#3(@r, @x, @y) -> compare#4(#equal(@r, #0()), @r, @x, @y) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , sub'#1(nil(), @b2, @r) -> tuple#2(nil(), @r) , sub'#1(::(@x, @xs), @b2, @r) -> sub'#2(@b2, @r, @x, @xs) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sub'(@b1, @b2, @r) -> sub'#1(@b1, @b2, @r) , compare(@b1, @b2) -> compare#1(@b1, @b2) , compare#6(#true()) -> #abs(#pos(#s(#0()))) , compare#6(#false()) -> #abs(#0()) , compare#4(#true(), @r, @x, @y) -> compare#5(#less(@x, @y), @x, @y) , compare#4(#false(), @r, @x, @y) -> @r , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , compare#1(nil(), @b2) -> #abs(#0()) , compare#1(::(@x, @xs), @b2) -> compare#2(@b2, @x, @xs) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We estimate the number of application of {15,20} by applications of Pre({15,20}) = {8,19}. Here rules are labeled as follows: DPs: { 1: sub^#(@b1, @b2) -> c_2(sub#1^#(sub'(@b1, @b2, #abs(#0()))), sub'^#(@b1, @b2, #abs(#0())), #abs^#(#0())) , 2: sub'^#(@b1, @b2, @r) -> c_51(sub'#1^#(@b1, @b2, @r)) , 3: mult#2^#(@zs, @b2, @x) -> c_5(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs), #equal^#(@x, #pos(#s(#0())))) , 4: mult#3^#(#true(), @b2, @zs) -> c_42(add^#(@b2, @zs)) , 5: bitToInt'#1^#(::(@x, @xs), @n) -> c_8(+^#(*(@x, @n), bitToInt'(@xs, *(@n, #pos(#s(#s(#0())))))), *^#(@x, @n), bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0()))))), *^#(@n, #pos(#s(#s(#0()))))) , 6: bitToInt'^#(@b, @n) -> c_15(bitToInt'#1^#(@b, @n)) , 7: sum^#(@x, @y, @r) -> c_9(sum#1^#(+(+(@x, @y), @r)), +^#(+(@x, @y), @r), +^#(@x, @y)) , 8: sum#1^#(@s) -> c_20(sum#2^#(#equal(@s, #0()), @s), #equal^#(@s, #0())) , 9: mult3^#(@b1, @b2, @b3) -> c_11(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , 10: mult^#(@b1, @b2) -> c_16(mult#1^#(@b1, @b2)) , 11: leq^#(@b1, @b2) -> c_13(#less^#(compare(@b1, @b2), #pos(#s(#0()))), compare^#(@b1, @b2)) , 12: compare^#(@b1, @b2) -> c_52(compare#1^#(@b1, @b2)) , 13: mult#1^#(::(@x, @xs), @b2) -> c_50(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), #abs^#(#0()), mult^#(@xs, @b2)) , 14: bitToInt^#(@b) -> c_17(bitToInt'^#(@b, #abs(#pos(#s(#0())))), #abs^#(#pos(#s(#0())))) , 15: sum#2^#(#false(), @s) -> c_19(sum#3^#(#equal(@s, #pos(#s(#0()))), @s), #equal^#(@s, #pos(#s(#0())))) , 16: sub'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_28(sub'#4^#(sub'(@xs, @ys, @r'), @z), sub'^#(@xs, @ys, @r')) , 17: sub'#2^#(::(@y, @ys), @r, @x, @xs) -> c_31(sub'#3^#(diff(@x, @y, @r), @xs, @ys), diff^#(@x, @y, @r)) , 18: compare#2^#(::(@y, @ys), @x, @xs) -> c_33(compare#3^#(compare(@xs, @ys), @x, @y), compare^#(@xs, @ys)) , 19: compare#3^#(@r, @x, @y) -> c_37(compare#4^#(#equal(@r, #0()), @r, @x, @y), #equal^#(@r, #0())) , 20: compare#4^#(#true(), @r, @x, @y) -> c_55(compare#5^#(#less(@x, @y), @x, @y), #less^#(@x, @y)) , 21: add^#(@b1, @b2) -> c_38(add'^#(@b1, @b2, #abs(#0())), #abs^#(#0())) , 22: add'^#(@b1, @b2, @r) -> c_59(add'#1^#(@b1, @b2, @r)) , 23: add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_39(add'^#(@xs, @ys, @r')) , 24: sub'#1^#(::(@x, @xs), @b2, @r) -> c_41(sub'#2^#(@b2, @r, @x, @xs)) , 25: add'#1^#(::(@x, @xs), @b2, @r) -> c_45(add'#2^#(@b2, @r, @x, @xs)) , 26: add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_47(add'#3^#(sum(@x, @y, @r), @xs, @ys), sum^#(@x, @y, @r)) , 27: compare#1^#(::(@x, @xs), @b2) -> c_61(compare#2^#(@b2, @x, @xs)) , 28: -^#(@x, @y) -> c_1(#sub^#(@x, @y)) , 29: #sub^#(@x, #neg(@y)) -> c_120(#add^#(@x, #pos(@y))) , 30: #sub^#(@x, #pos(@y)) -> c_121(#add^#(@x, #neg(@y))) , 31: #sub^#(@x, #0()) -> c_122() , 32: sub#1^#(tuple#2(@b, @_@1)) -> c_12() , 33: #abs^#(#neg(@x)) -> c_62() , 34: #abs^#(#pos(@x)) -> c_63() , 35: #abs^#(#0()) -> c_64() , 36: #abs^#(#s(@x)) -> c_65() , 37: diff#1^#(#true()) -> c_3(#abs^#(#pos(#s(#0())))) , 38: diff#1^#(#false()) -> c_4(#abs^#(#0())) , 39: mult#3^#(#false(), @b2, @zs) -> c_43() , 40: #equal^#(@x, @y) -> c_29(#eq^#(@x, @y)) , 41: div^#(@x, @y) -> c_6(#div^#(@x, @y)) , 42: #div^#(#neg(@x), #neg(@y)) -> c_136(#natdiv^#(@x, @y)) , 43: #div^#(#neg(@x), #pos(@y)) -> c_137(#natdiv^#(@x, @y)) , 44: #div^#(#neg(@x), #0()) -> c_138() , 45: #div^#(#pos(@x), #neg(@y)) -> c_139(#natdiv^#(@x, @y)) , 46: #div^#(#pos(@x), #pos(@y)) -> c_140(#natdiv^#(@x, @y)) , 47: #div^#(#pos(@x), #0()) -> c_141() , 48: #div^#(#0(), #neg(@y)) -> c_142() , 49: #div^#(#0(), #pos(@y)) -> c_143() , 50: #div^#(#0(), #0()) -> c_144() , 51: bitToInt'#1^#(nil(), @n) -> c_7(#abs^#(#0())) , 52: +^#(@x, @y) -> c_21(#add^#(@x, @y)) , 53: *^#(@x, @y) -> c_24(#mult^#(@x, @y)) , 54: mod^#(@x, @y) -> c_10(-^#(@x, *(@x, div(@x, @y))), *^#(@x, div(@x, @y)), div^#(@x, @y)) , 55: #less^#(@x, @y) -> c_27(#cklt^#(#compare(@x, @y)), #compare^#(@x, @y)) , 56: #greater^#(@x, @y) -> c_14(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y)) , 57: #ckgt^#(#EQ()) -> c_70() , 58: #ckgt^#(#LT()) -> c_71() , 59: #ckgt^#(#GT()) -> c_72() , 60: #compare^#(#neg(@x), #neg(@y)) -> c_82(#compare^#(@y, @x)) , 61: #compare^#(#neg(@x), #pos(@y)) -> c_83() , 62: #compare^#(#neg(@x), #0()) -> c_84() , 63: #compare^#(#pos(@x), #neg(@y)) -> c_85() , 64: #compare^#(#pos(@x), #pos(@y)) -> c_86(#compare^#(@x, @y)) , 65: #compare^#(#pos(@x), #0()) -> c_87() , 66: #compare^#(#0(), #neg(@y)) -> c_88() , 67: #compare^#(#0(), #pos(@y)) -> c_89() , 68: #compare^#(#0(), #0()) -> c_90() , 69: #compare^#(#0(), #s(@y)) -> c_91() , 70: #compare^#(#s(@x), #0()) -> c_92() , 71: #compare^#(#s(@x), #s(@y)) -> c_93(#compare^#(@x, @y)) , 72: mult#1^#(nil(), @b2) -> c_49() , 73: sum#2^#(#true(), @s) -> c_18(#abs^#(#0()), #abs^#(#0())) , 74: sum#3^#(#true(), @s) -> c_57(#abs^#(#pos(#s(#0()))), #abs^#(#0())) , 75: sum#3^#(#false(), @s) -> c_58(sum#4^#(#equal(@s, #pos(#s(#s(#0()))))), #equal^#(@s, #pos(#s(#s(#0()))))) , 76: #add^#(#neg(#s(#0())), @y) -> c_73(#pred^#(@y)) , 77: #add^#(#neg(#s(#s(@x))), @y) -> c_74(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , 78: #add^#(#pos(#s(#0())), @y) -> c_75(#succ^#(@y)) , 79: #add^#(#pos(#s(#s(@x))), @y) -> c_76(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , 80: #add^#(#0(), @y) -> c_77() , 81: sum#4^#(#true()) -> c_22(#abs^#(#0()), #abs^#(#pos(#s(#0())))) , 82: sum#4^#(#false()) -> c_23(#abs^#(#pos(#s(#0()))), #abs^#(#pos(#s(#0())))) , 83: #mult^#(#neg(@x), #neg(@y)) -> c_123(#natmult^#(@x, @y)) , 84: #mult^#(#neg(@x), #pos(@y)) -> c_124(#natmult^#(@x, @y)) , 85: #mult^#(#neg(@x), #0()) -> c_125() , 86: #mult^#(#pos(@x), #neg(@y)) -> c_126(#natmult^#(@x, @y)) , 87: #mult^#(#pos(@x), #pos(@y)) -> c_127(#natmult^#(@x, @y)) , 88: #mult^#(#pos(@x), #0()) -> c_128() , 89: #mult^#(#0(), #neg(@y)) -> c_129() , 90: #mult^#(#0(), #pos(@y)) -> c_130() , 91: #mult^#(#0(), #0()) -> c_131() , 92: sub'#5^#(#true(), @z, @zs) -> c_25(#abs^#(#0())) , 93: sub'#5^#(#false(), @z, @zs) -> c_26() , 94: #cklt^#(#EQ()) -> c_117() , 95: #cklt^#(#LT()) -> c_118() , 96: #cklt^#(#GT()) -> c_119() , 97: sub'#4^#(tuple#2(@zs, @s), @z) -> c_34(sub'#5^#(#equal(@s, #pos(#s(#0()))), @z, @zs), #equal^#(@s, #pos(#s(#0())))) , 98: #eq^#(nil(), nil()) -> c_94() , 99: #eq^#(nil(), tuple#2(@y_1, @y_2)) -> c_95() , 100: #eq^#(nil(), ::(@y_1, @y_2)) -> c_96() , 101: #eq^#(#neg(@x), #neg(@y)) -> c_97(#eq^#(@x, @y)) , 102: #eq^#(#neg(@x), #pos(@y)) -> c_98() , 103: #eq^#(#neg(@x), #0()) -> c_99() , 104: #eq^#(#pos(@x), #neg(@y)) -> c_100() , 105: #eq^#(#pos(@x), #pos(@y)) -> c_101(#eq^#(@x, @y)) , 106: #eq^#(#pos(@x), #0()) -> c_102() , 107: #eq^#(tuple#2(@x_1, @x_2), nil()) -> c_103() , 108: #eq^#(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> c_104(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), #eq^#(@x_1, @y_1), #eq^#(@x_2, @y_2)) , 109: #eq^#(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> c_105() , 110: #eq^#(::(@x_1, @x_2), nil()) -> c_106() , 111: #eq^#(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> c_107() , 112: #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) -> c_108(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), #eq^#(@x_1, @y_1), #eq^#(@x_2, @y_2)) , 113: #eq^#(#0(), #neg(@y)) -> c_109() , 114: #eq^#(#0(), #pos(@y)) -> c_110() , 115: #eq^#(#0(), #0()) -> c_111() , 116: #eq^#(#0(), #s(@y)) -> c_112() , 117: #eq^#(#s(@x), #0()) -> c_113() , 118: #eq^#(#s(@x), #s(@y)) -> c_114(#eq^#(@x, @y)) , 119: sub'#2^#(nil(), @r, @x, @xs) -> c_30() , 120: diff^#(@x, @y, @r) -> c_48(mod^#(+(+(@x, @y), @r), #pos(#s(#s(#0())))), +^#(+(@x, @y), @r), +^#(@x, @y), diff#1^#(#less(-(-(@x, @y), @r), #0())), #less^#(-(-(@x, @y), @r), #0()), -^#(-(@x, @y), @r), -^#(@x, @y)) , 121: compare#2^#(nil(), @x, @xs) -> c_32(#abs^#(#0())) , 122: compare#5^#(#true(), @x, @y) -> c_35(-^#(#0(), #pos(#s(#0())))) , 123: compare#5^#(#false(), @x, @y) -> c_36(compare#6^#(#greater(@x, @y)), #greater^#(@x, @y)) , 124: compare#6^#(#true()) -> c_53(#abs^#(#pos(#s(#0())))) , 125: compare#6^#(#false()) -> c_54(#abs^#(#0())) , 126: compare#4^#(#false(), @r, @x, @y) -> c_56() , 127: sub'#1^#(nil(), @b2, @r) -> c_40() , 128: add'#1^#(nil(), @b2, @r) -> c_44() , 129: add'#2^#(nil(), @r, @x, @xs) -> c_46() , 130: compare#1^#(nil(), @b2) -> c_60(#abs^#(#0())) , 131: #natsub^#(@x, #0()) -> c_66() , 132: #natsub^#(#s(@x), #s(@y)) -> c_67(#natsub^#(@x, @y)) , 133: #natdiv^#(#0(), #0()) -> c_68() , 134: #natdiv^#(#s(@x), #s(@y)) -> c_69(#natdiv^#(#natsub(@x, @y), #s(@y)), #natsub^#(@x, @y)) , 135: #pred^#(#neg(#s(@x))) -> c_145() , 136: #pred^#(#pos(#s(#0()))) -> c_146() , 137: #pred^#(#pos(#s(#s(@x)))) -> c_147() , 138: #pred^#(#0()) -> c_148() , 139: #succ^#(#neg(#s(#0()))) -> c_132() , 140: #succ^#(#neg(#s(#s(@x)))) -> c_133() , 141: #succ^#(#pos(#s(@x))) -> c_134() , 142: #succ^#(#0()) -> c_135() , 143: #and^#(#true(), #true()) -> c_78() , 144: #and^#(#true(), #false()) -> c_79() , 145: #and^#(#false(), #true()) -> c_80() , 146: #and^#(#false(), #false()) -> c_81() , 147: #natmult^#(#0(), @y) -> c_115() , 148: #natmult^#(#s(@x), @y) -> c_116(#add^#(#pos(@y), #natmult(@x, @y)), #natmult^#(@x, @y)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { sub^#(@b1, @b2) -> c_2(sub#1^#(sub'(@b1, @b2, #abs(#0()))), sub'^#(@b1, @b2, #abs(#0())), #abs^#(#0())) , sub'^#(@b1, @b2, @r) -> c_51(sub'#1^#(@b1, @b2, @r)) , mult#2^#(@zs, @b2, @x) -> c_5(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs), #equal^#(@x, #pos(#s(#0())))) , mult#3^#(#true(), @b2, @zs) -> c_42(add^#(@b2, @zs)) , bitToInt'#1^#(::(@x, @xs), @n) -> c_8(+^#(*(@x, @n), bitToInt'(@xs, *(@n, #pos(#s(#s(#0())))))), *^#(@x, @n), bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0()))))), *^#(@n, #pos(#s(#s(#0()))))) , bitToInt'^#(@b, @n) -> c_15(bitToInt'#1^#(@b, @n)) , sum^#(@x, @y, @r) -> c_9(sum#1^#(+(+(@x, @y), @r)), +^#(+(@x, @y), @r), +^#(@x, @y)) , sum#1^#(@s) -> c_20(sum#2^#(#equal(@s, #0()), @s), #equal^#(@s, #0())) , mult3^#(@b1, @b2, @b3) -> c_11(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , mult^#(@b1, @b2) -> c_16(mult#1^#(@b1, @b2)) , leq^#(@b1, @b2) -> c_13(#less^#(compare(@b1, @b2), #pos(#s(#0()))), compare^#(@b1, @b2)) , compare^#(@b1, @b2) -> c_52(compare#1^#(@b1, @b2)) , mult#1^#(::(@x, @xs), @b2) -> c_50(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), #abs^#(#0()), mult^#(@xs, @b2)) , bitToInt^#(@b) -> c_17(bitToInt'^#(@b, #abs(#pos(#s(#0())))), #abs^#(#pos(#s(#0())))) , sub'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_28(sub'#4^#(sub'(@xs, @ys, @r'), @z), sub'^#(@xs, @ys, @r')) , sub'#2^#(::(@y, @ys), @r, @x, @xs) -> c_31(sub'#3^#(diff(@x, @y, @r), @xs, @ys), diff^#(@x, @y, @r)) , compare#2^#(::(@y, @ys), @x, @xs) -> c_33(compare#3^#(compare(@xs, @ys), @x, @y), compare^#(@xs, @ys)) , compare#3^#(@r, @x, @y) -> c_37(compare#4^#(#equal(@r, #0()), @r, @x, @y), #equal^#(@r, #0())) , add^#(@b1, @b2) -> c_38(add'^#(@b1, @b2, #abs(#0())), #abs^#(#0())) , add'^#(@b1, @b2, @r) -> c_59(add'#1^#(@b1, @b2, @r)) , add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_39(add'^#(@xs, @ys, @r')) , sub'#1^#(::(@x, @xs), @b2, @r) -> c_41(sub'#2^#(@b2, @r, @x, @xs)) , add'#1^#(::(@x, @xs), @b2, @r) -> c_45(add'#2^#(@b2, @r, @x, @xs)) , add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_47(add'#3^#(sum(@x, @y, @r), @xs, @ys), sum^#(@x, @y, @r)) , compare#1^#(::(@x, @xs), @b2) -> c_61(compare#2^#(@b2, @x, @xs)) } Weak DPs: { -^#(@x, @y) -> c_1(#sub^#(@x, @y)) , #sub^#(@x, #neg(@y)) -> c_120(#add^#(@x, #pos(@y))) , #sub^#(@x, #pos(@y)) -> c_121(#add^#(@x, #neg(@y))) , #sub^#(@x, #0()) -> c_122() , sub#1^#(tuple#2(@b, @_@1)) -> c_12() , #abs^#(#neg(@x)) -> c_62() , #abs^#(#pos(@x)) -> c_63() , #abs^#(#0()) -> c_64() , #abs^#(#s(@x)) -> c_65() , diff#1^#(#true()) -> c_3(#abs^#(#pos(#s(#0())))) , diff#1^#(#false()) -> c_4(#abs^#(#0())) , mult#3^#(#false(), @b2, @zs) -> c_43() , #equal^#(@x, @y) -> c_29(#eq^#(@x, @y)) , div^#(@x, @y) -> c_6(#div^#(@x, @y)) , #div^#(#neg(@x), #neg(@y)) -> c_136(#natdiv^#(@x, @y)) , #div^#(#neg(@x), #pos(@y)) -> c_137(#natdiv^#(@x, @y)) , #div^#(#neg(@x), #0()) -> c_138() , #div^#(#pos(@x), #neg(@y)) -> c_139(#natdiv^#(@x, @y)) , #div^#(#pos(@x), #pos(@y)) -> c_140(#natdiv^#(@x, @y)) , #div^#(#pos(@x), #0()) -> c_141() , #div^#(#0(), #neg(@y)) -> c_142() , #div^#(#0(), #pos(@y)) -> c_143() , #div^#(#0(), #0()) -> c_144() , bitToInt'#1^#(nil(), @n) -> c_7(#abs^#(#0())) , +^#(@x, @y) -> c_21(#add^#(@x, @y)) , *^#(@x, @y) -> c_24(#mult^#(@x, @y)) , mod^#(@x, @y) -> c_10(-^#(@x, *(@x, div(@x, @y))), *^#(@x, div(@x, @y)), div^#(@x, @y)) , #less^#(@x, @y) -> c_27(#cklt^#(#compare(@x, @y)), #compare^#(@x, @y)) , #greater^#(@x, @y) -> c_14(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y)) , #ckgt^#(#EQ()) -> c_70() , #ckgt^#(#LT()) -> c_71() , #ckgt^#(#GT()) -> c_72() , #compare^#(#neg(@x), #neg(@y)) -> c_82(#compare^#(@y, @x)) , #compare^#(#neg(@x), #pos(@y)) -> c_83() , #compare^#(#neg(@x), #0()) -> c_84() , #compare^#(#pos(@x), #neg(@y)) -> c_85() , #compare^#(#pos(@x), #pos(@y)) -> c_86(#compare^#(@x, @y)) , #compare^#(#pos(@x), #0()) -> c_87() , #compare^#(#0(), #neg(@y)) -> c_88() , #compare^#(#0(), #pos(@y)) -> c_89() , #compare^#(#0(), #0()) -> c_90() , #compare^#(#0(), #s(@y)) -> c_91() , #compare^#(#s(@x), #0()) -> c_92() , #compare^#(#s(@x), #s(@y)) -> c_93(#compare^#(@x, @y)) , mult#1^#(nil(), @b2) -> c_49() , sum#2^#(#true(), @s) -> c_18(#abs^#(#0()), #abs^#(#0())) , sum#2^#(#false(), @s) -> c_19(sum#3^#(#equal(@s, #pos(#s(#0()))), @s), #equal^#(@s, #pos(#s(#0())))) , sum#3^#(#true(), @s) -> c_57(#abs^#(#pos(#s(#0()))), #abs^#(#0())) , sum#3^#(#false(), @s) -> c_58(sum#4^#(#equal(@s, #pos(#s(#s(#0()))))), #equal^#(@s, #pos(#s(#s(#0()))))) , #add^#(#neg(#s(#0())), @y) -> c_73(#pred^#(@y)) , #add^#(#neg(#s(#s(@x))), @y) -> c_74(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , #add^#(#pos(#s(#0())), @y) -> c_75(#succ^#(@y)) , #add^#(#pos(#s(#s(@x))), @y) -> c_76(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , #add^#(#0(), @y) -> c_77() , sum#4^#(#true()) -> c_22(#abs^#(#0()), #abs^#(#pos(#s(#0())))) , sum#4^#(#false()) -> c_23(#abs^#(#pos(#s(#0()))), #abs^#(#pos(#s(#0())))) , #mult^#(#neg(@x), #neg(@y)) -> c_123(#natmult^#(@x, @y)) , #mult^#(#neg(@x), #pos(@y)) -> c_124(#natmult^#(@x, @y)) , #mult^#(#neg(@x), #0()) -> c_125() , #mult^#(#pos(@x), #neg(@y)) -> c_126(#natmult^#(@x, @y)) , #mult^#(#pos(@x), #pos(@y)) -> c_127(#natmult^#(@x, @y)) , #mult^#(#pos(@x), #0()) -> c_128() , #mult^#(#0(), #neg(@y)) -> c_129() , #mult^#(#0(), #pos(@y)) -> c_130() , #mult^#(#0(), #0()) -> c_131() , sub'#5^#(#true(), @z, @zs) -> c_25(#abs^#(#0())) , sub'#5^#(#false(), @z, @zs) -> c_26() , #cklt^#(#EQ()) -> c_117() , #cklt^#(#LT()) -> c_118() , #cklt^#(#GT()) -> c_119() , sub'#4^#(tuple#2(@zs, @s), @z) -> c_34(sub'#5^#(#equal(@s, #pos(#s(#0()))), @z, @zs), #equal^#(@s, #pos(#s(#0())))) , #eq^#(nil(), nil()) -> c_94() , #eq^#(nil(), tuple#2(@y_1, @y_2)) -> c_95() , #eq^#(nil(), ::(@y_1, @y_2)) -> c_96() , #eq^#(#neg(@x), #neg(@y)) -> c_97(#eq^#(@x, @y)) , #eq^#(#neg(@x), #pos(@y)) -> c_98() , #eq^#(#neg(@x), #0()) -> c_99() , #eq^#(#pos(@x), #neg(@y)) -> c_100() , #eq^#(#pos(@x), #pos(@y)) -> c_101(#eq^#(@x, @y)) , #eq^#(#pos(@x), #0()) -> c_102() , #eq^#(tuple#2(@x_1, @x_2), nil()) -> c_103() , #eq^#(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> c_104(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), #eq^#(@x_1, @y_1), #eq^#(@x_2, @y_2)) , #eq^#(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> c_105() , #eq^#(::(@x_1, @x_2), nil()) -> c_106() , #eq^#(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> c_107() , #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) -> c_108(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), #eq^#(@x_1, @y_1), #eq^#(@x_2, @y_2)) , #eq^#(#0(), #neg(@y)) -> c_109() , #eq^#(#0(), #pos(@y)) -> c_110() , #eq^#(#0(), #0()) -> c_111() , #eq^#(#0(), #s(@y)) -> c_112() , #eq^#(#s(@x), #0()) -> c_113() , #eq^#(#s(@x), #s(@y)) -> c_114(#eq^#(@x, @y)) , sub'#2^#(nil(), @r, @x, @xs) -> c_30() , diff^#(@x, @y, @r) -> c_48(mod^#(+(+(@x, @y), @r), #pos(#s(#s(#0())))), +^#(+(@x, @y), @r), +^#(@x, @y), diff#1^#(#less(-(-(@x, @y), @r), #0())), #less^#(-(-(@x, @y), @r), #0()), -^#(-(@x, @y), @r), -^#(@x, @y)) , compare#2^#(nil(), @x, @xs) -> c_32(#abs^#(#0())) , compare#5^#(#true(), @x, @y) -> c_35(-^#(#0(), #pos(#s(#0())))) , compare#5^#(#false(), @x, @y) -> c_36(compare#6^#(#greater(@x, @y)), #greater^#(@x, @y)) , compare#6^#(#true()) -> c_53(#abs^#(#pos(#s(#0())))) , compare#6^#(#false()) -> c_54(#abs^#(#0())) , compare#4^#(#true(), @r, @x, @y) -> c_55(compare#5^#(#less(@x, @y), @x, @y), #less^#(@x, @y)) , compare#4^#(#false(), @r, @x, @y) -> c_56() , sub'#1^#(nil(), @b2, @r) -> c_40() , add'#1^#(nil(), @b2, @r) -> c_44() , add'#2^#(nil(), @r, @x, @xs) -> c_46() , compare#1^#(nil(), @b2) -> c_60(#abs^#(#0())) , #natsub^#(@x, #0()) -> c_66() , #natsub^#(#s(@x), #s(@y)) -> c_67(#natsub^#(@x, @y)) , #natdiv^#(#0(), #0()) -> c_68() , #natdiv^#(#s(@x), #s(@y)) -> c_69(#natdiv^#(#natsub(@x, @y), #s(@y)), #natsub^#(@x, @y)) , #pred^#(#neg(#s(@x))) -> c_145() , #pred^#(#pos(#s(#0()))) -> c_146() , #pred^#(#pos(#s(#s(@x)))) -> c_147() , #pred^#(#0()) -> c_148() , #succ^#(#neg(#s(#0()))) -> c_132() , #succ^#(#neg(#s(#s(@x)))) -> c_133() , #succ^#(#pos(#s(@x))) -> c_134() , #succ^#(#0()) -> c_135() , #and^#(#true(), #true()) -> c_78() , #and^#(#true(), #false()) -> c_79() , #and^#(#false(), #true()) -> c_80() , #and^#(#false(), #false()) -> c_81() , #natmult^#(#0(), @y) -> c_115() , #natmult^#(#s(@x), @y) -> c_116(#add^#(#pos(@y), #natmult(@x, @y)), #natmult^#(@x, @y)) } Weak Trs: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , sub(@b1, @b2) -> sub#1(sub'(@b1, @b2, #abs(#0()))) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #ckgt(#EQ()) -> #false() , #ckgt(#LT()) -> #false() , #ckgt(#GT()) -> #true() , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , div(@x, @y) -> #div(@x, @y) , bitToInt'#1(nil(), @n) -> #abs(#0()) , bitToInt'#1(::(@x, @xs), @n) -> +(*(@x, @n), bitToInt'(@xs, *(@n, #pos(#s(#s(#0())))))) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , mult3(@b1, @b2, @b3) -> mult(mult(@b1, @b2), @b2) , sub#1(tuple#2(@b, @_@1)) -> @b , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , leq(@b1, @b2) -> #less(compare(@b1, @b2), #pos(#s(#0()))) , #greater(@x, @y) -> #ckgt(#compare(@x, @y)) , bitToInt'(@b, @n) -> bitToInt'#1(@b, @n) , mult(@b1, @b2) -> mult#1(@b1, @b2) , bitToInt(@b) -> bitToInt'(@b, #abs(#pos(#s(#0())))) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , *(@x, @y) -> #mult(@x, @y) , sub'#5(#true(), @z, @zs) -> ::(#abs(#0()), @zs) , sub'#5(#false(), @z, @zs) -> ::(@z, @zs) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , sub'#3(tuple#2(@z, @r'), @xs, @ys) -> sub'#4(sub'(@xs, @ys, @r'), @z) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , sub'#2(nil(), @r, @x, @xs) -> tuple#2(nil(), @r) , sub'#2(::(@y, @ys), @r, @x, @xs) -> sub'#3(diff(@x, @y, @r), @xs, @ys) , compare#2(nil(), @x, @xs) -> #abs(#0()) , compare#2(::(@y, @ys), @x, @xs) -> compare#3(compare(@xs, @ys), @x, @y) , sub'#4(tuple#2(@zs, @s), @z) -> tuple#2(sub'#5(#equal(@s, #pos(#s(#0()))), @z, @zs), @s) , compare#5(#true(), @x, @y) -> -(#0(), #pos(#s(#0()))) , compare#5(#false(), @x, @y) -> compare#6(#greater(@x, @y)) , compare#3(@r, @x, @y) -> compare#4(#equal(@r, #0()), @r, @x, @y) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , sub'#1(nil(), @b2, @r) -> tuple#2(nil(), @r) , sub'#1(::(@x, @xs), @b2, @r) -> sub'#2(@b2, @r, @x, @xs) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sub'(@b1, @b2, @r) -> sub'#1(@b1, @b2, @r) , compare(@b1, @b2) -> compare#1(@b1, @b2) , compare#6(#true()) -> #abs(#pos(#s(#0()))) , compare#6(#false()) -> #abs(#0()) , compare#4(#true(), @r, @x, @y) -> compare#5(#less(@x, @y), @x, @y) , compare#4(#false(), @r, @x, @y) -> @r , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , compare#1(nil(), @b2) -> #abs(#0()) , compare#1(::(@x, @xs), @b2) -> compare#2(@b2, @x, @xs) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We estimate the number of application of {8,18} by applications of Pre({8,18}) = {7,17}. Here rules are labeled as follows: DPs: { 1: sub^#(@b1, @b2) -> c_2(sub#1^#(sub'(@b1, @b2, #abs(#0()))), sub'^#(@b1, @b2, #abs(#0())), #abs^#(#0())) , 2: sub'^#(@b1, @b2, @r) -> c_51(sub'#1^#(@b1, @b2, @r)) , 3: mult#2^#(@zs, @b2, @x) -> c_5(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs), #equal^#(@x, #pos(#s(#0())))) , 4: mult#3^#(#true(), @b2, @zs) -> c_42(add^#(@b2, @zs)) , 5: bitToInt'#1^#(::(@x, @xs), @n) -> c_8(+^#(*(@x, @n), bitToInt'(@xs, *(@n, #pos(#s(#s(#0())))))), *^#(@x, @n), bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0()))))), *^#(@n, #pos(#s(#s(#0()))))) , 6: bitToInt'^#(@b, @n) -> c_15(bitToInt'#1^#(@b, @n)) , 7: sum^#(@x, @y, @r) -> c_9(sum#1^#(+(+(@x, @y), @r)), +^#(+(@x, @y), @r), +^#(@x, @y)) , 8: sum#1^#(@s) -> c_20(sum#2^#(#equal(@s, #0()), @s), #equal^#(@s, #0())) , 9: mult3^#(@b1, @b2, @b3) -> c_11(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , 10: mult^#(@b1, @b2) -> c_16(mult#1^#(@b1, @b2)) , 11: leq^#(@b1, @b2) -> c_13(#less^#(compare(@b1, @b2), #pos(#s(#0()))), compare^#(@b1, @b2)) , 12: compare^#(@b1, @b2) -> c_52(compare#1^#(@b1, @b2)) , 13: mult#1^#(::(@x, @xs), @b2) -> c_50(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), #abs^#(#0()), mult^#(@xs, @b2)) , 14: bitToInt^#(@b) -> c_17(bitToInt'^#(@b, #abs(#pos(#s(#0())))), #abs^#(#pos(#s(#0())))) , 15: sub'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_28(sub'#4^#(sub'(@xs, @ys, @r'), @z), sub'^#(@xs, @ys, @r')) , 16: sub'#2^#(::(@y, @ys), @r, @x, @xs) -> c_31(sub'#3^#(diff(@x, @y, @r), @xs, @ys), diff^#(@x, @y, @r)) , 17: compare#2^#(::(@y, @ys), @x, @xs) -> c_33(compare#3^#(compare(@xs, @ys), @x, @y), compare^#(@xs, @ys)) , 18: compare#3^#(@r, @x, @y) -> c_37(compare#4^#(#equal(@r, #0()), @r, @x, @y), #equal^#(@r, #0())) , 19: add^#(@b1, @b2) -> c_38(add'^#(@b1, @b2, #abs(#0())), #abs^#(#0())) , 20: add'^#(@b1, @b2, @r) -> c_59(add'#1^#(@b1, @b2, @r)) , 21: add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_39(add'^#(@xs, @ys, @r')) , 22: sub'#1^#(::(@x, @xs), @b2, @r) -> c_41(sub'#2^#(@b2, @r, @x, @xs)) , 23: add'#1^#(::(@x, @xs), @b2, @r) -> c_45(add'#2^#(@b2, @r, @x, @xs)) , 24: add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_47(add'#3^#(sum(@x, @y, @r), @xs, @ys), sum^#(@x, @y, @r)) , 25: compare#1^#(::(@x, @xs), @b2) -> c_61(compare#2^#(@b2, @x, @xs)) , 26: -^#(@x, @y) -> c_1(#sub^#(@x, @y)) , 27: #sub^#(@x, #neg(@y)) -> c_120(#add^#(@x, #pos(@y))) , 28: #sub^#(@x, #pos(@y)) -> c_121(#add^#(@x, #neg(@y))) , 29: #sub^#(@x, #0()) -> c_122() , 30: sub#1^#(tuple#2(@b, @_@1)) -> c_12() , 31: #abs^#(#neg(@x)) -> c_62() , 32: #abs^#(#pos(@x)) -> c_63() , 33: #abs^#(#0()) -> c_64() , 34: #abs^#(#s(@x)) -> c_65() , 35: diff#1^#(#true()) -> c_3(#abs^#(#pos(#s(#0())))) , 36: diff#1^#(#false()) -> c_4(#abs^#(#0())) , 37: mult#3^#(#false(), @b2, @zs) -> c_43() , 38: #equal^#(@x, @y) -> c_29(#eq^#(@x, @y)) , 39: div^#(@x, @y) -> c_6(#div^#(@x, @y)) , 40: #div^#(#neg(@x), #neg(@y)) -> c_136(#natdiv^#(@x, @y)) , 41: #div^#(#neg(@x), #pos(@y)) -> c_137(#natdiv^#(@x, @y)) , 42: #div^#(#neg(@x), #0()) -> c_138() , 43: #div^#(#pos(@x), #neg(@y)) -> c_139(#natdiv^#(@x, @y)) , 44: #div^#(#pos(@x), #pos(@y)) -> c_140(#natdiv^#(@x, @y)) , 45: #div^#(#pos(@x), #0()) -> c_141() , 46: #div^#(#0(), #neg(@y)) -> c_142() , 47: #div^#(#0(), #pos(@y)) -> c_143() , 48: #div^#(#0(), #0()) -> c_144() , 49: bitToInt'#1^#(nil(), @n) -> c_7(#abs^#(#0())) , 50: +^#(@x, @y) -> c_21(#add^#(@x, @y)) , 51: *^#(@x, @y) -> c_24(#mult^#(@x, @y)) , 52: mod^#(@x, @y) -> c_10(-^#(@x, *(@x, div(@x, @y))), *^#(@x, div(@x, @y)), div^#(@x, @y)) , 53: #less^#(@x, @y) -> c_27(#cklt^#(#compare(@x, @y)), #compare^#(@x, @y)) , 54: #greater^#(@x, @y) -> c_14(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y)) , 55: #ckgt^#(#EQ()) -> c_70() , 56: #ckgt^#(#LT()) -> c_71() , 57: #ckgt^#(#GT()) -> c_72() , 58: #compare^#(#neg(@x), #neg(@y)) -> c_82(#compare^#(@y, @x)) , 59: #compare^#(#neg(@x), #pos(@y)) -> c_83() , 60: #compare^#(#neg(@x), #0()) -> c_84() , 61: #compare^#(#pos(@x), #neg(@y)) -> c_85() , 62: #compare^#(#pos(@x), #pos(@y)) -> c_86(#compare^#(@x, @y)) , 63: #compare^#(#pos(@x), #0()) -> c_87() , 64: #compare^#(#0(), #neg(@y)) -> c_88() , 65: #compare^#(#0(), #pos(@y)) -> c_89() , 66: #compare^#(#0(), #0()) -> c_90() , 67: #compare^#(#0(), #s(@y)) -> c_91() , 68: #compare^#(#s(@x), #0()) -> c_92() , 69: #compare^#(#s(@x), #s(@y)) -> c_93(#compare^#(@x, @y)) , 70: mult#1^#(nil(), @b2) -> c_49() , 71: sum#2^#(#true(), @s) -> c_18(#abs^#(#0()), #abs^#(#0())) , 72: sum#2^#(#false(), @s) -> c_19(sum#3^#(#equal(@s, #pos(#s(#0()))), @s), #equal^#(@s, #pos(#s(#0())))) , 73: sum#3^#(#true(), @s) -> c_57(#abs^#(#pos(#s(#0()))), #abs^#(#0())) , 74: sum#3^#(#false(), @s) -> c_58(sum#4^#(#equal(@s, #pos(#s(#s(#0()))))), #equal^#(@s, #pos(#s(#s(#0()))))) , 75: #add^#(#neg(#s(#0())), @y) -> c_73(#pred^#(@y)) , 76: #add^#(#neg(#s(#s(@x))), @y) -> c_74(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , 77: #add^#(#pos(#s(#0())), @y) -> c_75(#succ^#(@y)) , 78: #add^#(#pos(#s(#s(@x))), @y) -> c_76(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , 79: #add^#(#0(), @y) -> c_77() , 80: sum#4^#(#true()) -> c_22(#abs^#(#0()), #abs^#(#pos(#s(#0())))) , 81: sum#4^#(#false()) -> c_23(#abs^#(#pos(#s(#0()))), #abs^#(#pos(#s(#0())))) , 82: #mult^#(#neg(@x), #neg(@y)) -> c_123(#natmult^#(@x, @y)) , 83: #mult^#(#neg(@x), #pos(@y)) -> c_124(#natmult^#(@x, @y)) , 84: #mult^#(#neg(@x), #0()) -> c_125() , 85: #mult^#(#pos(@x), #neg(@y)) -> c_126(#natmult^#(@x, @y)) , 86: #mult^#(#pos(@x), #pos(@y)) -> c_127(#natmult^#(@x, @y)) , 87: #mult^#(#pos(@x), #0()) -> c_128() , 88: #mult^#(#0(), #neg(@y)) -> c_129() , 89: #mult^#(#0(), #pos(@y)) -> c_130() , 90: #mult^#(#0(), #0()) -> c_131() , 91: sub'#5^#(#true(), @z, @zs) -> c_25(#abs^#(#0())) , 92: sub'#5^#(#false(), @z, @zs) -> c_26() , 93: #cklt^#(#EQ()) -> c_117() , 94: #cklt^#(#LT()) -> c_118() , 95: #cklt^#(#GT()) -> c_119() , 96: sub'#4^#(tuple#2(@zs, @s), @z) -> c_34(sub'#5^#(#equal(@s, #pos(#s(#0()))), @z, @zs), #equal^#(@s, #pos(#s(#0())))) , 97: #eq^#(nil(), nil()) -> c_94() , 98: #eq^#(nil(), tuple#2(@y_1, @y_2)) -> c_95() , 99: #eq^#(nil(), ::(@y_1, @y_2)) -> c_96() , 100: #eq^#(#neg(@x), #neg(@y)) -> c_97(#eq^#(@x, @y)) , 101: #eq^#(#neg(@x), #pos(@y)) -> c_98() , 102: #eq^#(#neg(@x), #0()) -> c_99() , 103: #eq^#(#pos(@x), #neg(@y)) -> c_100() , 104: #eq^#(#pos(@x), #pos(@y)) -> c_101(#eq^#(@x, @y)) , 105: #eq^#(#pos(@x), #0()) -> c_102() , 106: #eq^#(tuple#2(@x_1, @x_2), nil()) -> c_103() , 107: #eq^#(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> c_104(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), #eq^#(@x_1, @y_1), #eq^#(@x_2, @y_2)) , 108: #eq^#(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> c_105() , 109: #eq^#(::(@x_1, @x_2), nil()) -> c_106() , 110: #eq^#(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> c_107() , 111: #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) -> c_108(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), #eq^#(@x_1, @y_1), #eq^#(@x_2, @y_2)) , 112: #eq^#(#0(), #neg(@y)) -> c_109() , 113: #eq^#(#0(), #pos(@y)) -> c_110() , 114: #eq^#(#0(), #0()) -> c_111() , 115: #eq^#(#0(), #s(@y)) -> c_112() , 116: #eq^#(#s(@x), #0()) -> c_113() , 117: #eq^#(#s(@x), #s(@y)) -> c_114(#eq^#(@x, @y)) , 118: sub'#2^#(nil(), @r, @x, @xs) -> c_30() , 119: diff^#(@x, @y, @r) -> c_48(mod^#(+(+(@x, @y), @r), #pos(#s(#s(#0())))), +^#(+(@x, @y), @r), +^#(@x, @y), diff#1^#(#less(-(-(@x, @y), @r), #0())), #less^#(-(-(@x, @y), @r), #0()), -^#(-(@x, @y), @r), -^#(@x, @y)) , 120: compare#2^#(nil(), @x, @xs) -> c_32(#abs^#(#0())) , 121: compare#5^#(#true(), @x, @y) -> c_35(-^#(#0(), #pos(#s(#0())))) , 122: compare#5^#(#false(), @x, @y) -> c_36(compare#6^#(#greater(@x, @y)), #greater^#(@x, @y)) , 123: compare#6^#(#true()) -> c_53(#abs^#(#pos(#s(#0())))) , 124: compare#6^#(#false()) -> c_54(#abs^#(#0())) , 125: compare#4^#(#true(), @r, @x, @y) -> c_55(compare#5^#(#less(@x, @y), @x, @y), #less^#(@x, @y)) , 126: compare#4^#(#false(), @r, @x, @y) -> c_56() , 127: sub'#1^#(nil(), @b2, @r) -> c_40() , 128: add'#1^#(nil(), @b2, @r) -> c_44() , 129: add'#2^#(nil(), @r, @x, @xs) -> c_46() , 130: compare#1^#(nil(), @b2) -> c_60(#abs^#(#0())) , 131: #natsub^#(@x, #0()) -> c_66() , 132: #natsub^#(#s(@x), #s(@y)) -> c_67(#natsub^#(@x, @y)) , 133: #natdiv^#(#0(), #0()) -> c_68() , 134: #natdiv^#(#s(@x), #s(@y)) -> c_69(#natdiv^#(#natsub(@x, @y), #s(@y)), #natsub^#(@x, @y)) , 135: #pred^#(#neg(#s(@x))) -> c_145() , 136: #pred^#(#pos(#s(#0()))) -> c_146() , 137: #pred^#(#pos(#s(#s(@x)))) -> c_147() , 138: #pred^#(#0()) -> c_148() , 139: #succ^#(#neg(#s(#0()))) -> c_132() , 140: #succ^#(#neg(#s(#s(@x)))) -> c_133() , 141: #succ^#(#pos(#s(@x))) -> c_134() , 142: #succ^#(#0()) -> c_135() , 143: #and^#(#true(), #true()) -> c_78() , 144: #and^#(#true(), #false()) -> c_79() , 145: #and^#(#false(), #true()) -> c_80() , 146: #and^#(#false(), #false()) -> c_81() , 147: #natmult^#(#0(), @y) -> c_115() , 148: #natmult^#(#s(@x), @y) -> c_116(#add^#(#pos(@y), #natmult(@x, @y)), #natmult^#(@x, @y)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { sub^#(@b1, @b2) -> c_2(sub#1^#(sub'(@b1, @b2, #abs(#0()))), sub'^#(@b1, @b2, #abs(#0())), #abs^#(#0())) , sub'^#(@b1, @b2, @r) -> c_51(sub'#1^#(@b1, @b2, @r)) , mult#2^#(@zs, @b2, @x) -> c_5(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs), #equal^#(@x, #pos(#s(#0())))) , mult#3^#(#true(), @b2, @zs) -> c_42(add^#(@b2, @zs)) , bitToInt'#1^#(::(@x, @xs), @n) -> c_8(+^#(*(@x, @n), bitToInt'(@xs, *(@n, #pos(#s(#s(#0())))))), *^#(@x, @n), bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0()))))), *^#(@n, #pos(#s(#s(#0()))))) , bitToInt'^#(@b, @n) -> c_15(bitToInt'#1^#(@b, @n)) , sum^#(@x, @y, @r) -> c_9(sum#1^#(+(+(@x, @y), @r)), +^#(+(@x, @y), @r), +^#(@x, @y)) , mult3^#(@b1, @b2, @b3) -> c_11(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , mult^#(@b1, @b2) -> c_16(mult#1^#(@b1, @b2)) , leq^#(@b1, @b2) -> c_13(#less^#(compare(@b1, @b2), #pos(#s(#0()))), compare^#(@b1, @b2)) , compare^#(@b1, @b2) -> c_52(compare#1^#(@b1, @b2)) , mult#1^#(::(@x, @xs), @b2) -> c_50(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), #abs^#(#0()), mult^#(@xs, @b2)) , bitToInt^#(@b) -> c_17(bitToInt'^#(@b, #abs(#pos(#s(#0())))), #abs^#(#pos(#s(#0())))) , sub'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_28(sub'#4^#(sub'(@xs, @ys, @r'), @z), sub'^#(@xs, @ys, @r')) , sub'#2^#(::(@y, @ys), @r, @x, @xs) -> c_31(sub'#3^#(diff(@x, @y, @r), @xs, @ys), diff^#(@x, @y, @r)) , compare#2^#(::(@y, @ys), @x, @xs) -> c_33(compare#3^#(compare(@xs, @ys), @x, @y), compare^#(@xs, @ys)) , add^#(@b1, @b2) -> c_38(add'^#(@b1, @b2, #abs(#0())), #abs^#(#0())) , add'^#(@b1, @b2, @r) -> c_59(add'#1^#(@b1, @b2, @r)) , add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_39(add'^#(@xs, @ys, @r')) , sub'#1^#(::(@x, @xs), @b2, @r) -> c_41(sub'#2^#(@b2, @r, @x, @xs)) , add'#1^#(::(@x, @xs), @b2, @r) -> c_45(add'#2^#(@b2, @r, @x, @xs)) , add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_47(add'#3^#(sum(@x, @y, @r), @xs, @ys), sum^#(@x, @y, @r)) , compare#1^#(::(@x, @xs), @b2) -> c_61(compare#2^#(@b2, @x, @xs)) } Weak DPs: { -^#(@x, @y) -> c_1(#sub^#(@x, @y)) , #sub^#(@x, #neg(@y)) -> c_120(#add^#(@x, #pos(@y))) , #sub^#(@x, #pos(@y)) -> c_121(#add^#(@x, #neg(@y))) , #sub^#(@x, #0()) -> c_122() , sub#1^#(tuple#2(@b, @_@1)) -> c_12() , #abs^#(#neg(@x)) -> c_62() , #abs^#(#pos(@x)) -> c_63() , #abs^#(#0()) -> c_64() , #abs^#(#s(@x)) -> c_65() , diff#1^#(#true()) -> c_3(#abs^#(#pos(#s(#0())))) , diff#1^#(#false()) -> c_4(#abs^#(#0())) , mult#3^#(#false(), @b2, @zs) -> c_43() , #equal^#(@x, @y) -> c_29(#eq^#(@x, @y)) , div^#(@x, @y) -> c_6(#div^#(@x, @y)) , #div^#(#neg(@x), #neg(@y)) -> c_136(#natdiv^#(@x, @y)) , #div^#(#neg(@x), #pos(@y)) -> c_137(#natdiv^#(@x, @y)) , #div^#(#neg(@x), #0()) -> c_138() , #div^#(#pos(@x), #neg(@y)) -> c_139(#natdiv^#(@x, @y)) , #div^#(#pos(@x), #pos(@y)) -> c_140(#natdiv^#(@x, @y)) , #div^#(#pos(@x), #0()) -> c_141() , #div^#(#0(), #neg(@y)) -> c_142() , #div^#(#0(), #pos(@y)) -> c_143() , #div^#(#0(), #0()) -> c_144() , bitToInt'#1^#(nil(), @n) -> c_7(#abs^#(#0())) , +^#(@x, @y) -> c_21(#add^#(@x, @y)) , *^#(@x, @y) -> c_24(#mult^#(@x, @y)) , sum#1^#(@s) -> c_20(sum#2^#(#equal(@s, #0()), @s), #equal^#(@s, #0())) , mod^#(@x, @y) -> c_10(-^#(@x, *(@x, div(@x, @y))), *^#(@x, div(@x, @y)), div^#(@x, @y)) , #less^#(@x, @y) -> c_27(#cklt^#(#compare(@x, @y)), #compare^#(@x, @y)) , #greater^#(@x, @y) -> c_14(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y)) , #ckgt^#(#EQ()) -> c_70() , #ckgt^#(#LT()) -> c_71() , #ckgt^#(#GT()) -> c_72() , #compare^#(#neg(@x), #neg(@y)) -> c_82(#compare^#(@y, @x)) , #compare^#(#neg(@x), #pos(@y)) -> c_83() , #compare^#(#neg(@x), #0()) -> c_84() , #compare^#(#pos(@x), #neg(@y)) -> c_85() , #compare^#(#pos(@x), #pos(@y)) -> c_86(#compare^#(@x, @y)) , #compare^#(#pos(@x), #0()) -> c_87() , #compare^#(#0(), #neg(@y)) -> c_88() , #compare^#(#0(), #pos(@y)) -> c_89() , #compare^#(#0(), #0()) -> c_90() , #compare^#(#0(), #s(@y)) -> c_91() , #compare^#(#s(@x), #0()) -> c_92() , #compare^#(#s(@x), #s(@y)) -> c_93(#compare^#(@x, @y)) , mult#1^#(nil(), @b2) -> c_49() , sum#2^#(#true(), @s) -> c_18(#abs^#(#0()), #abs^#(#0())) , sum#2^#(#false(), @s) -> c_19(sum#3^#(#equal(@s, #pos(#s(#0()))), @s), #equal^#(@s, #pos(#s(#0())))) , sum#3^#(#true(), @s) -> c_57(#abs^#(#pos(#s(#0()))), #abs^#(#0())) , sum#3^#(#false(), @s) -> c_58(sum#4^#(#equal(@s, #pos(#s(#s(#0()))))), #equal^#(@s, #pos(#s(#s(#0()))))) , #add^#(#neg(#s(#0())), @y) -> c_73(#pred^#(@y)) , #add^#(#neg(#s(#s(@x))), @y) -> c_74(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , #add^#(#pos(#s(#0())), @y) -> c_75(#succ^#(@y)) , #add^#(#pos(#s(#s(@x))), @y) -> c_76(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , #add^#(#0(), @y) -> c_77() , sum#4^#(#true()) -> c_22(#abs^#(#0()), #abs^#(#pos(#s(#0())))) , sum#4^#(#false()) -> c_23(#abs^#(#pos(#s(#0()))), #abs^#(#pos(#s(#0())))) , #mult^#(#neg(@x), #neg(@y)) -> c_123(#natmult^#(@x, @y)) , #mult^#(#neg(@x), #pos(@y)) -> c_124(#natmult^#(@x, @y)) , #mult^#(#neg(@x), #0()) -> c_125() , #mult^#(#pos(@x), #neg(@y)) -> c_126(#natmult^#(@x, @y)) , #mult^#(#pos(@x), #pos(@y)) -> c_127(#natmult^#(@x, @y)) , #mult^#(#pos(@x), #0()) -> c_128() , #mult^#(#0(), #neg(@y)) -> c_129() , #mult^#(#0(), #pos(@y)) -> c_130() , #mult^#(#0(), #0()) -> c_131() , sub'#5^#(#true(), @z, @zs) -> c_25(#abs^#(#0())) , sub'#5^#(#false(), @z, @zs) -> c_26() , #cklt^#(#EQ()) -> c_117() , #cklt^#(#LT()) -> c_118() , #cklt^#(#GT()) -> c_119() , sub'#4^#(tuple#2(@zs, @s), @z) -> c_34(sub'#5^#(#equal(@s, #pos(#s(#0()))), @z, @zs), #equal^#(@s, #pos(#s(#0())))) , #eq^#(nil(), nil()) -> c_94() , #eq^#(nil(), tuple#2(@y_1, @y_2)) -> c_95() , #eq^#(nil(), ::(@y_1, @y_2)) -> c_96() , #eq^#(#neg(@x), #neg(@y)) -> c_97(#eq^#(@x, @y)) , #eq^#(#neg(@x), #pos(@y)) -> c_98() , #eq^#(#neg(@x), #0()) -> c_99() , #eq^#(#pos(@x), #neg(@y)) -> c_100() , #eq^#(#pos(@x), #pos(@y)) -> c_101(#eq^#(@x, @y)) , #eq^#(#pos(@x), #0()) -> c_102() , #eq^#(tuple#2(@x_1, @x_2), nil()) -> c_103() , #eq^#(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> c_104(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), #eq^#(@x_1, @y_1), #eq^#(@x_2, @y_2)) , #eq^#(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> c_105() , #eq^#(::(@x_1, @x_2), nil()) -> c_106() , #eq^#(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> c_107() , #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) -> c_108(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), #eq^#(@x_1, @y_1), #eq^#(@x_2, @y_2)) , #eq^#(#0(), #neg(@y)) -> c_109() , #eq^#(#0(), #pos(@y)) -> c_110() , #eq^#(#0(), #0()) -> c_111() , #eq^#(#0(), #s(@y)) -> c_112() , #eq^#(#s(@x), #0()) -> c_113() , #eq^#(#s(@x), #s(@y)) -> c_114(#eq^#(@x, @y)) , sub'#2^#(nil(), @r, @x, @xs) -> c_30() , diff^#(@x, @y, @r) -> c_48(mod^#(+(+(@x, @y), @r), #pos(#s(#s(#0())))), +^#(+(@x, @y), @r), +^#(@x, @y), diff#1^#(#less(-(-(@x, @y), @r), #0())), #less^#(-(-(@x, @y), @r), #0()), -^#(-(@x, @y), @r), -^#(@x, @y)) , compare#2^#(nil(), @x, @xs) -> c_32(#abs^#(#0())) , compare#3^#(@r, @x, @y) -> c_37(compare#4^#(#equal(@r, #0()), @r, @x, @y), #equal^#(@r, #0())) , compare#5^#(#true(), @x, @y) -> c_35(-^#(#0(), #pos(#s(#0())))) , compare#5^#(#false(), @x, @y) -> c_36(compare#6^#(#greater(@x, @y)), #greater^#(@x, @y)) , compare#6^#(#true()) -> c_53(#abs^#(#pos(#s(#0())))) , compare#6^#(#false()) -> c_54(#abs^#(#0())) , compare#4^#(#true(), @r, @x, @y) -> c_55(compare#5^#(#less(@x, @y), @x, @y), #less^#(@x, @y)) , compare#4^#(#false(), @r, @x, @y) -> c_56() , sub'#1^#(nil(), @b2, @r) -> c_40() , add'#1^#(nil(), @b2, @r) -> c_44() , add'#2^#(nil(), @r, @x, @xs) -> c_46() , compare#1^#(nil(), @b2) -> c_60(#abs^#(#0())) , #natsub^#(@x, #0()) -> c_66() , #natsub^#(#s(@x), #s(@y)) -> c_67(#natsub^#(@x, @y)) , #natdiv^#(#0(), #0()) -> c_68() , #natdiv^#(#s(@x), #s(@y)) -> c_69(#natdiv^#(#natsub(@x, @y), #s(@y)), #natsub^#(@x, @y)) , #pred^#(#neg(#s(@x))) -> c_145() , #pred^#(#pos(#s(#0()))) -> c_146() , #pred^#(#pos(#s(#s(@x)))) -> c_147() , #pred^#(#0()) -> c_148() , #succ^#(#neg(#s(#0()))) -> c_132() , #succ^#(#neg(#s(#s(@x)))) -> c_133() , #succ^#(#pos(#s(@x))) -> c_134() , #succ^#(#0()) -> c_135() , #and^#(#true(), #true()) -> c_78() , #and^#(#true(), #false()) -> c_79() , #and^#(#false(), #true()) -> c_80() , #and^#(#false(), #false()) -> c_81() , #natmult^#(#0(), @y) -> c_115() , #natmult^#(#s(@x), @y) -> c_116(#add^#(#pos(@y), #natmult(@x, @y)), #natmult^#(@x, @y)) } Weak Trs: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , sub(@b1, @b2) -> sub#1(sub'(@b1, @b2, #abs(#0()))) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #ckgt(#EQ()) -> #false() , #ckgt(#LT()) -> #false() , #ckgt(#GT()) -> #true() , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , div(@x, @y) -> #div(@x, @y) , bitToInt'#1(nil(), @n) -> #abs(#0()) , bitToInt'#1(::(@x, @xs), @n) -> +(*(@x, @n), bitToInt'(@xs, *(@n, #pos(#s(#s(#0())))))) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , mult3(@b1, @b2, @b3) -> mult(mult(@b1, @b2), @b2) , sub#1(tuple#2(@b, @_@1)) -> @b , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , leq(@b1, @b2) -> #less(compare(@b1, @b2), #pos(#s(#0()))) , #greater(@x, @y) -> #ckgt(#compare(@x, @y)) , bitToInt'(@b, @n) -> bitToInt'#1(@b, @n) , mult(@b1, @b2) -> mult#1(@b1, @b2) , bitToInt(@b) -> bitToInt'(@b, #abs(#pos(#s(#0())))) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , *(@x, @y) -> #mult(@x, @y) , sub'#5(#true(), @z, @zs) -> ::(#abs(#0()), @zs) , sub'#5(#false(), @z, @zs) -> ::(@z, @zs) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , sub'#3(tuple#2(@z, @r'), @xs, @ys) -> sub'#4(sub'(@xs, @ys, @r'), @z) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , sub'#2(nil(), @r, @x, @xs) -> tuple#2(nil(), @r) , sub'#2(::(@y, @ys), @r, @x, @xs) -> sub'#3(diff(@x, @y, @r), @xs, @ys) , compare#2(nil(), @x, @xs) -> #abs(#0()) , compare#2(::(@y, @ys), @x, @xs) -> compare#3(compare(@xs, @ys), @x, @y) , sub'#4(tuple#2(@zs, @s), @z) -> tuple#2(sub'#5(#equal(@s, #pos(#s(#0()))), @z, @zs), @s) , compare#5(#true(), @x, @y) -> -(#0(), #pos(#s(#0()))) , compare#5(#false(), @x, @y) -> compare#6(#greater(@x, @y)) , compare#3(@r, @x, @y) -> compare#4(#equal(@r, #0()), @r, @x, @y) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , sub'#1(nil(), @b2, @r) -> tuple#2(nil(), @r) , sub'#1(::(@x, @xs), @b2, @r) -> sub'#2(@b2, @r, @x, @xs) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sub'(@b1, @b2, @r) -> sub'#1(@b1, @b2, @r) , compare(@b1, @b2) -> compare#1(@b1, @b2) , compare#6(#true()) -> #abs(#pos(#s(#0()))) , compare#6(#false()) -> #abs(#0()) , compare#4(#true(), @r, @x, @y) -> compare#5(#less(@x, @y), @x, @y) , compare#4(#false(), @r, @x, @y) -> @r , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , compare#1(nil(), @b2) -> #abs(#0()) , compare#1(::(@x, @xs), @b2) -> compare#2(@b2, @x, @xs) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We estimate the number of application of {7} by applications of Pre({7}) = {22}. Here rules are labeled as follows: DPs: { 1: sub^#(@b1, @b2) -> c_2(sub#1^#(sub'(@b1, @b2, #abs(#0()))), sub'^#(@b1, @b2, #abs(#0())), #abs^#(#0())) , 2: sub'^#(@b1, @b2, @r) -> c_51(sub'#1^#(@b1, @b2, @r)) , 3: mult#2^#(@zs, @b2, @x) -> c_5(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs), #equal^#(@x, #pos(#s(#0())))) , 4: mult#3^#(#true(), @b2, @zs) -> c_42(add^#(@b2, @zs)) , 5: bitToInt'#1^#(::(@x, @xs), @n) -> c_8(+^#(*(@x, @n), bitToInt'(@xs, *(@n, #pos(#s(#s(#0())))))), *^#(@x, @n), bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0()))))), *^#(@n, #pos(#s(#s(#0()))))) , 6: bitToInt'^#(@b, @n) -> c_15(bitToInt'#1^#(@b, @n)) , 7: sum^#(@x, @y, @r) -> c_9(sum#1^#(+(+(@x, @y), @r)), +^#(+(@x, @y), @r), +^#(@x, @y)) , 8: mult3^#(@b1, @b2, @b3) -> c_11(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , 9: mult^#(@b1, @b2) -> c_16(mult#1^#(@b1, @b2)) , 10: leq^#(@b1, @b2) -> c_13(#less^#(compare(@b1, @b2), #pos(#s(#0()))), compare^#(@b1, @b2)) , 11: compare^#(@b1, @b2) -> c_52(compare#1^#(@b1, @b2)) , 12: mult#1^#(::(@x, @xs), @b2) -> c_50(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), #abs^#(#0()), mult^#(@xs, @b2)) , 13: bitToInt^#(@b) -> c_17(bitToInt'^#(@b, #abs(#pos(#s(#0())))), #abs^#(#pos(#s(#0())))) , 14: sub'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_28(sub'#4^#(sub'(@xs, @ys, @r'), @z), sub'^#(@xs, @ys, @r')) , 15: sub'#2^#(::(@y, @ys), @r, @x, @xs) -> c_31(sub'#3^#(diff(@x, @y, @r), @xs, @ys), diff^#(@x, @y, @r)) , 16: compare#2^#(::(@y, @ys), @x, @xs) -> c_33(compare#3^#(compare(@xs, @ys), @x, @y), compare^#(@xs, @ys)) , 17: add^#(@b1, @b2) -> c_38(add'^#(@b1, @b2, #abs(#0())), #abs^#(#0())) , 18: add'^#(@b1, @b2, @r) -> c_59(add'#1^#(@b1, @b2, @r)) , 19: add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_39(add'^#(@xs, @ys, @r')) , 20: sub'#1^#(::(@x, @xs), @b2, @r) -> c_41(sub'#2^#(@b2, @r, @x, @xs)) , 21: add'#1^#(::(@x, @xs), @b2, @r) -> c_45(add'#2^#(@b2, @r, @x, @xs)) , 22: add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_47(add'#3^#(sum(@x, @y, @r), @xs, @ys), sum^#(@x, @y, @r)) , 23: compare#1^#(::(@x, @xs), @b2) -> c_61(compare#2^#(@b2, @x, @xs)) , 24: -^#(@x, @y) -> c_1(#sub^#(@x, @y)) , 25: #sub^#(@x, #neg(@y)) -> c_120(#add^#(@x, #pos(@y))) , 26: #sub^#(@x, #pos(@y)) -> c_121(#add^#(@x, #neg(@y))) , 27: #sub^#(@x, #0()) -> c_122() , 28: sub#1^#(tuple#2(@b, @_@1)) -> c_12() , 29: #abs^#(#neg(@x)) -> c_62() , 30: #abs^#(#pos(@x)) -> c_63() , 31: #abs^#(#0()) -> c_64() , 32: #abs^#(#s(@x)) -> c_65() , 33: diff#1^#(#true()) -> c_3(#abs^#(#pos(#s(#0())))) , 34: diff#1^#(#false()) -> c_4(#abs^#(#0())) , 35: mult#3^#(#false(), @b2, @zs) -> c_43() , 36: #equal^#(@x, @y) -> c_29(#eq^#(@x, @y)) , 37: div^#(@x, @y) -> c_6(#div^#(@x, @y)) , 38: #div^#(#neg(@x), #neg(@y)) -> c_136(#natdiv^#(@x, @y)) , 39: #div^#(#neg(@x), #pos(@y)) -> c_137(#natdiv^#(@x, @y)) , 40: #div^#(#neg(@x), #0()) -> c_138() , 41: #div^#(#pos(@x), #neg(@y)) -> c_139(#natdiv^#(@x, @y)) , 42: #div^#(#pos(@x), #pos(@y)) -> c_140(#natdiv^#(@x, @y)) , 43: #div^#(#pos(@x), #0()) -> c_141() , 44: #div^#(#0(), #neg(@y)) -> c_142() , 45: #div^#(#0(), #pos(@y)) -> c_143() , 46: #div^#(#0(), #0()) -> c_144() , 47: bitToInt'#1^#(nil(), @n) -> c_7(#abs^#(#0())) , 48: +^#(@x, @y) -> c_21(#add^#(@x, @y)) , 49: *^#(@x, @y) -> c_24(#mult^#(@x, @y)) , 50: sum#1^#(@s) -> c_20(sum#2^#(#equal(@s, #0()), @s), #equal^#(@s, #0())) , 51: mod^#(@x, @y) -> c_10(-^#(@x, *(@x, div(@x, @y))), *^#(@x, div(@x, @y)), div^#(@x, @y)) , 52: #less^#(@x, @y) -> c_27(#cklt^#(#compare(@x, @y)), #compare^#(@x, @y)) , 53: #greater^#(@x, @y) -> c_14(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y)) , 54: #ckgt^#(#EQ()) -> c_70() , 55: #ckgt^#(#LT()) -> c_71() , 56: #ckgt^#(#GT()) -> c_72() , 57: #compare^#(#neg(@x), #neg(@y)) -> c_82(#compare^#(@y, @x)) , 58: #compare^#(#neg(@x), #pos(@y)) -> c_83() , 59: #compare^#(#neg(@x), #0()) -> c_84() , 60: #compare^#(#pos(@x), #neg(@y)) -> c_85() , 61: #compare^#(#pos(@x), #pos(@y)) -> c_86(#compare^#(@x, @y)) , 62: #compare^#(#pos(@x), #0()) -> c_87() , 63: #compare^#(#0(), #neg(@y)) -> c_88() , 64: #compare^#(#0(), #pos(@y)) -> c_89() , 65: #compare^#(#0(), #0()) -> c_90() , 66: #compare^#(#0(), #s(@y)) -> c_91() , 67: #compare^#(#s(@x), #0()) -> c_92() , 68: #compare^#(#s(@x), #s(@y)) -> c_93(#compare^#(@x, @y)) , 69: mult#1^#(nil(), @b2) -> c_49() , 70: sum#2^#(#true(), @s) -> c_18(#abs^#(#0()), #abs^#(#0())) , 71: sum#2^#(#false(), @s) -> c_19(sum#3^#(#equal(@s, #pos(#s(#0()))), @s), #equal^#(@s, #pos(#s(#0())))) , 72: sum#3^#(#true(), @s) -> c_57(#abs^#(#pos(#s(#0()))), #abs^#(#0())) , 73: sum#3^#(#false(), @s) -> c_58(sum#4^#(#equal(@s, #pos(#s(#s(#0()))))), #equal^#(@s, #pos(#s(#s(#0()))))) , 74: #add^#(#neg(#s(#0())), @y) -> c_73(#pred^#(@y)) , 75: #add^#(#neg(#s(#s(@x))), @y) -> c_74(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , 76: #add^#(#pos(#s(#0())), @y) -> c_75(#succ^#(@y)) , 77: #add^#(#pos(#s(#s(@x))), @y) -> c_76(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , 78: #add^#(#0(), @y) -> c_77() , 79: sum#4^#(#true()) -> c_22(#abs^#(#0()), #abs^#(#pos(#s(#0())))) , 80: sum#4^#(#false()) -> c_23(#abs^#(#pos(#s(#0()))), #abs^#(#pos(#s(#0())))) , 81: #mult^#(#neg(@x), #neg(@y)) -> c_123(#natmult^#(@x, @y)) , 82: #mult^#(#neg(@x), #pos(@y)) -> c_124(#natmult^#(@x, @y)) , 83: #mult^#(#neg(@x), #0()) -> c_125() , 84: #mult^#(#pos(@x), #neg(@y)) -> c_126(#natmult^#(@x, @y)) , 85: #mult^#(#pos(@x), #pos(@y)) -> c_127(#natmult^#(@x, @y)) , 86: #mult^#(#pos(@x), #0()) -> c_128() , 87: #mult^#(#0(), #neg(@y)) -> c_129() , 88: #mult^#(#0(), #pos(@y)) -> c_130() , 89: #mult^#(#0(), #0()) -> c_131() , 90: sub'#5^#(#true(), @z, @zs) -> c_25(#abs^#(#0())) , 91: sub'#5^#(#false(), @z, @zs) -> c_26() , 92: #cklt^#(#EQ()) -> c_117() , 93: #cklt^#(#LT()) -> c_118() , 94: #cklt^#(#GT()) -> c_119() , 95: sub'#4^#(tuple#2(@zs, @s), @z) -> c_34(sub'#5^#(#equal(@s, #pos(#s(#0()))), @z, @zs), #equal^#(@s, #pos(#s(#0())))) , 96: #eq^#(nil(), nil()) -> c_94() , 97: #eq^#(nil(), tuple#2(@y_1, @y_2)) -> c_95() , 98: #eq^#(nil(), ::(@y_1, @y_2)) -> c_96() , 99: #eq^#(#neg(@x), #neg(@y)) -> c_97(#eq^#(@x, @y)) , 100: #eq^#(#neg(@x), #pos(@y)) -> c_98() , 101: #eq^#(#neg(@x), #0()) -> c_99() , 102: #eq^#(#pos(@x), #neg(@y)) -> c_100() , 103: #eq^#(#pos(@x), #pos(@y)) -> c_101(#eq^#(@x, @y)) , 104: #eq^#(#pos(@x), #0()) -> c_102() , 105: #eq^#(tuple#2(@x_1, @x_2), nil()) -> c_103() , 106: #eq^#(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> c_104(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), #eq^#(@x_1, @y_1), #eq^#(@x_2, @y_2)) , 107: #eq^#(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> c_105() , 108: #eq^#(::(@x_1, @x_2), nil()) -> c_106() , 109: #eq^#(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> c_107() , 110: #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) -> c_108(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), #eq^#(@x_1, @y_1), #eq^#(@x_2, @y_2)) , 111: #eq^#(#0(), #neg(@y)) -> c_109() , 112: #eq^#(#0(), #pos(@y)) -> c_110() , 113: #eq^#(#0(), #0()) -> c_111() , 114: #eq^#(#0(), #s(@y)) -> c_112() , 115: #eq^#(#s(@x), #0()) -> c_113() , 116: #eq^#(#s(@x), #s(@y)) -> c_114(#eq^#(@x, @y)) , 117: sub'#2^#(nil(), @r, @x, @xs) -> c_30() , 118: diff^#(@x, @y, @r) -> c_48(mod^#(+(+(@x, @y), @r), #pos(#s(#s(#0())))), +^#(+(@x, @y), @r), +^#(@x, @y), diff#1^#(#less(-(-(@x, @y), @r), #0())), #less^#(-(-(@x, @y), @r), #0()), -^#(-(@x, @y), @r), -^#(@x, @y)) , 119: compare#2^#(nil(), @x, @xs) -> c_32(#abs^#(#0())) , 120: compare#3^#(@r, @x, @y) -> c_37(compare#4^#(#equal(@r, #0()), @r, @x, @y), #equal^#(@r, #0())) , 121: compare#5^#(#true(), @x, @y) -> c_35(-^#(#0(), #pos(#s(#0())))) , 122: compare#5^#(#false(), @x, @y) -> c_36(compare#6^#(#greater(@x, @y)), #greater^#(@x, @y)) , 123: compare#6^#(#true()) -> c_53(#abs^#(#pos(#s(#0())))) , 124: compare#6^#(#false()) -> c_54(#abs^#(#0())) , 125: compare#4^#(#true(), @r, @x, @y) -> c_55(compare#5^#(#less(@x, @y), @x, @y), #less^#(@x, @y)) , 126: compare#4^#(#false(), @r, @x, @y) -> c_56() , 127: sub'#1^#(nil(), @b2, @r) -> c_40() , 128: add'#1^#(nil(), @b2, @r) -> c_44() , 129: add'#2^#(nil(), @r, @x, @xs) -> c_46() , 130: compare#1^#(nil(), @b2) -> c_60(#abs^#(#0())) , 131: #natsub^#(@x, #0()) -> c_66() , 132: #natsub^#(#s(@x), #s(@y)) -> c_67(#natsub^#(@x, @y)) , 133: #natdiv^#(#0(), #0()) -> c_68() , 134: #natdiv^#(#s(@x), #s(@y)) -> c_69(#natdiv^#(#natsub(@x, @y), #s(@y)), #natsub^#(@x, @y)) , 135: #pred^#(#neg(#s(@x))) -> c_145() , 136: #pred^#(#pos(#s(#0()))) -> c_146() , 137: #pred^#(#pos(#s(#s(@x)))) -> c_147() , 138: #pred^#(#0()) -> c_148() , 139: #succ^#(#neg(#s(#0()))) -> c_132() , 140: #succ^#(#neg(#s(#s(@x)))) -> c_133() , 141: #succ^#(#pos(#s(@x))) -> c_134() , 142: #succ^#(#0()) -> c_135() , 143: #and^#(#true(), #true()) -> c_78() , 144: #and^#(#true(), #false()) -> c_79() , 145: #and^#(#false(), #true()) -> c_80() , 146: #and^#(#false(), #false()) -> c_81() , 147: #natmult^#(#0(), @y) -> c_115() , 148: #natmult^#(#s(@x), @y) -> c_116(#add^#(#pos(@y), #natmult(@x, @y)), #natmult^#(@x, @y)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { sub^#(@b1, @b2) -> c_2(sub#1^#(sub'(@b1, @b2, #abs(#0()))), sub'^#(@b1, @b2, #abs(#0())), #abs^#(#0())) , sub'^#(@b1, @b2, @r) -> c_51(sub'#1^#(@b1, @b2, @r)) , mult#2^#(@zs, @b2, @x) -> c_5(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs), #equal^#(@x, #pos(#s(#0())))) , mult#3^#(#true(), @b2, @zs) -> c_42(add^#(@b2, @zs)) , bitToInt'#1^#(::(@x, @xs), @n) -> c_8(+^#(*(@x, @n), bitToInt'(@xs, *(@n, #pos(#s(#s(#0())))))), *^#(@x, @n), bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0()))))), *^#(@n, #pos(#s(#s(#0()))))) , bitToInt'^#(@b, @n) -> c_15(bitToInt'#1^#(@b, @n)) , mult3^#(@b1, @b2, @b3) -> c_11(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , mult^#(@b1, @b2) -> c_16(mult#1^#(@b1, @b2)) , leq^#(@b1, @b2) -> c_13(#less^#(compare(@b1, @b2), #pos(#s(#0()))), compare^#(@b1, @b2)) , compare^#(@b1, @b2) -> c_52(compare#1^#(@b1, @b2)) , mult#1^#(::(@x, @xs), @b2) -> c_50(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), #abs^#(#0()), mult^#(@xs, @b2)) , bitToInt^#(@b) -> c_17(bitToInt'^#(@b, #abs(#pos(#s(#0())))), #abs^#(#pos(#s(#0())))) , sub'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_28(sub'#4^#(sub'(@xs, @ys, @r'), @z), sub'^#(@xs, @ys, @r')) , sub'#2^#(::(@y, @ys), @r, @x, @xs) -> c_31(sub'#3^#(diff(@x, @y, @r), @xs, @ys), diff^#(@x, @y, @r)) , compare#2^#(::(@y, @ys), @x, @xs) -> c_33(compare#3^#(compare(@xs, @ys), @x, @y), compare^#(@xs, @ys)) , add^#(@b1, @b2) -> c_38(add'^#(@b1, @b2, #abs(#0())), #abs^#(#0())) , add'^#(@b1, @b2, @r) -> c_59(add'#1^#(@b1, @b2, @r)) , add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_39(add'^#(@xs, @ys, @r')) , sub'#1^#(::(@x, @xs), @b2, @r) -> c_41(sub'#2^#(@b2, @r, @x, @xs)) , add'#1^#(::(@x, @xs), @b2, @r) -> c_45(add'#2^#(@b2, @r, @x, @xs)) , add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_47(add'#3^#(sum(@x, @y, @r), @xs, @ys), sum^#(@x, @y, @r)) , compare#1^#(::(@x, @xs), @b2) -> c_61(compare#2^#(@b2, @x, @xs)) } Weak DPs: { -^#(@x, @y) -> c_1(#sub^#(@x, @y)) , #sub^#(@x, #neg(@y)) -> c_120(#add^#(@x, #pos(@y))) , #sub^#(@x, #pos(@y)) -> c_121(#add^#(@x, #neg(@y))) , #sub^#(@x, #0()) -> c_122() , sub#1^#(tuple#2(@b, @_@1)) -> c_12() , #abs^#(#neg(@x)) -> c_62() , #abs^#(#pos(@x)) -> c_63() , #abs^#(#0()) -> c_64() , #abs^#(#s(@x)) -> c_65() , diff#1^#(#true()) -> c_3(#abs^#(#pos(#s(#0())))) , diff#1^#(#false()) -> c_4(#abs^#(#0())) , mult#3^#(#false(), @b2, @zs) -> c_43() , #equal^#(@x, @y) -> c_29(#eq^#(@x, @y)) , div^#(@x, @y) -> c_6(#div^#(@x, @y)) , #div^#(#neg(@x), #neg(@y)) -> c_136(#natdiv^#(@x, @y)) , #div^#(#neg(@x), #pos(@y)) -> c_137(#natdiv^#(@x, @y)) , #div^#(#neg(@x), #0()) -> c_138() , #div^#(#pos(@x), #neg(@y)) -> c_139(#natdiv^#(@x, @y)) , #div^#(#pos(@x), #pos(@y)) -> c_140(#natdiv^#(@x, @y)) , #div^#(#pos(@x), #0()) -> c_141() , #div^#(#0(), #neg(@y)) -> c_142() , #div^#(#0(), #pos(@y)) -> c_143() , #div^#(#0(), #0()) -> c_144() , bitToInt'#1^#(nil(), @n) -> c_7(#abs^#(#0())) , +^#(@x, @y) -> c_21(#add^#(@x, @y)) , *^#(@x, @y) -> c_24(#mult^#(@x, @y)) , sum^#(@x, @y, @r) -> c_9(sum#1^#(+(+(@x, @y), @r)), +^#(+(@x, @y), @r), +^#(@x, @y)) , sum#1^#(@s) -> c_20(sum#2^#(#equal(@s, #0()), @s), #equal^#(@s, #0())) , mod^#(@x, @y) -> c_10(-^#(@x, *(@x, div(@x, @y))), *^#(@x, div(@x, @y)), div^#(@x, @y)) , #less^#(@x, @y) -> c_27(#cklt^#(#compare(@x, @y)), #compare^#(@x, @y)) , #greater^#(@x, @y) -> c_14(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y)) , #ckgt^#(#EQ()) -> c_70() , #ckgt^#(#LT()) -> c_71() , #ckgt^#(#GT()) -> c_72() , #compare^#(#neg(@x), #neg(@y)) -> c_82(#compare^#(@y, @x)) , #compare^#(#neg(@x), #pos(@y)) -> c_83() , #compare^#(#neg(@x), #0()) -> c_84() , #compare^#(#pos(@x), #neg(@y)) -> c_85() , #compare^#(#pos(@x), #pos(@y)) -> c_86(#compare^#(@x, @y)) , #compare^#(#pos(@x), #0()) -> c_87() , #compare^#(#0(), #neg(@y)) -> c_88() , #compare^#(#0(), #pos(@y)) -> c_89() , #compare^#(#0(), #0()) -> c_90() , #compare^#(#0(), #s(@y)) -> c_91() , #compare^#(#s(@x), #0()) -> c_92() , #compare^#(#s(@x), #s(@y)) -> c_93(#compare^#(@x, @y)) , mult#1^#(nil(), @b2) -> c_49() , sum#2^#(#true(), @s) -> c_18(#abs^#(#0()), #abs^#(#0())) , sum#2^#(#false(), @s) -> c_19(sum#3^#(#equal(@s, #pos(#s(#0()))), @s), #equal^#(@s, #pos(#s(#0())))) , sum#3^#(#true(), @s) -> c_57(#abs^#(#pos(#s(#0()))), #abs^#(#0())) , sum#3^#(#false(), @s) -> c_58(sum#4^#(#equal(@s, #pos(#s(#s(#0()))))), #equal^#(@s, #pos(#s(#s(#0()))))) , #add^#(#neg(#s(#0())), @y) -> c_73(#pred^#(@y)) , #add^#(#neg(#s(#s(@x))), @y) -> c_74(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , #add^#(#pos(#s(#0())), @y) -> c_75(#succ^#(@y)) , #add^#(#pos(#s(#s(@x))), @y) -> c_76(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , #add^#(#0(), @y) -> c_77() , sum#4^#(#true()) -> c_22(#abs^#(#0()), #abs^#(#pos(#s(#0())))) , sum#4^#(#false()) -> c_23(#abs^#(#pos(#s(#0()))), #abs^#(#pos(#s(#0())))) , #mult^#(#neg(@x), #neg(@y)) -> c_123(#natmult^#(@x, @y)) , #mult^#(#neg(@x), #pos(@y)) -> c_124(#natmult^#(@x, @y)) , #mult^#(#neg(@x), #0()) -> c_125() , #mult^#(#pos(@x), #neg(@y)) -> c_126(#natmult^#(@x, @y)) , #mult^#(#pos(@x), #pos(@y)) -> c_127(#natmult^#(@x, @y)) , #mult^#(#pos(@x), #0()) -> c_128() , #mult^#(#0(), #neg(@y)) -> c_129() , #mult^#(#0(), #pos(@y)) -> c_130() , #mult^#(#0(), #0()) -> c_131() , sub'#5^#(#true(), @z, @zs) -> c_25(#abs^#(#0())) , sub'#5^#(#false(), @z, @zs) -> c_26() , #cklt^#(#EQ()) -> c_117() , #cklt^#(#LT()) -> c_118() , #cklt^#(#GT()) -> c_119() , sub'#4^#(tuple#2(@zs, @s), @z) -> c_34(sub'#5^#(#equal(@s, #pos(#s(#0()))), @z, @zs), #equal^#(@s, #pos(#s(#0())))) , #eq^#(nil(), nil()) -> c_94() , #eq^#(nil(), tuple#2(@y_1, @y_2)) -> c_95() , #eq^#(nil(), ::(@y_1, @y_2)) -> c_96() , #eq^#(#neg(@x), #neg(@y)) -> c_97(#eq^#(@x, @y)) , #eq^#(#neg(@x), #pos(@y)) -> c_98() , #eq^#(#neg(@x), #0()) -> c_99() , #eq^#(#pos(@x), #neg(@y)) -> c_100() , #eq^#(#pos(@x), #pos(@y)) -> c_101(#eq^#(@x, @y)) , #eq^#(#pos(@x), #0()) -> c_102() , #eq^#(tuple#2(@x_1, @x_2), nil()) -> c_103() , #eq^#(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> c_104(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), #eq^#(@x_1, @y_1), #eq^#(@x_2, @y_2)) , #eq^#(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> c_105() , #eq^#(::(@x_1, @x_2), nil()) -> c_106() , #eq^#(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> c_107() , #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) -> c_108(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), #eq^#(@x_1, @y_1), #eq^#(@x_2, @y_2)) , #eq^#(#0(), #neg(@y)) -> c_109() , #eq^#(#0(), #pos(@y)) -> c_110() , #eq^#(#0(), #0()) -> c_111() , #eq^#(#0(), #s(@y)) -> c_112() , #eq^#(#s(@x), #0()) -> c_113() , #eq^#(#s(@x), #s(@y)) -> c_114(#eq^#(@x, @y)) , sub'#2^#(nil(), @r, @x, @xs) -> c_30() , diff^#(@x, @y, @r) -> c_48(mod^#(+(+(@x, @y), @r), #pos(#s(#s(#0())))), +^#(+(@x, @y), @r), +^#(@x, @y), diff#1^#(#less(-(-(@x, @y), @r), #0())), #less^#(-(-(@x, @y), @r), #0()), -^#(-(@x, @y), @r), -^#(@x, @y)) , compare#2^#(nil(), @x, @xs) -> c_32(#abs^#(#0())) , compare#3^#(@r, @x, @y) -> c_37(compare#4^#(#equal(@r, #0()), @r, @x, @y), #equal^#(@r, #0())) , compare#5^#(#true(), @x, @y) -> c_35(-^#(#0(), #pos(#s(#0())))) , compare#5^#(#false(), @x, @y) -> c_36(compare#6^#(#greater(@x, @y)), #greater^#(@x, @y)) , compare#6^#(#true()) -> c_53(#abs^#(#pos(#s(#0())))) , compare#6^#(#false()) -> c_54(#abs^#(#0())) , compare#4^#(#true(), @r, @x, @y) -> c_55(compare#5^#(#less(@x, @y), @x, @y), #less^#(@x, @y)) , compare#4^#(#false(), @r, @x, @y) -> c_56() , sub'#1^#(nil(), @b2, @r) -> c_40() , add'#1^#(nil(), @b2, @r) -> c_44() , add'#2^#(nil(), @r, @x, @xs) -> c_46() , compare#1^#(nil(), @b2) -> c_60(#abs^#(#0())) , #natsub^#(@x, #0()) -> c_66() , #natsub^#(#s(@x), #s(@y)) -> c_67(#natsub^#(@x, @y)) , #natdiv^#(#0(), #0()) -> c_68() , #natdiv^#(#s(@x), #s(@y)) -> c_69(#natdiv^#(#natsub(@x, @y), #s(@y)), #natsub^#(@x, @y)) , #pred^#(#neg(#s(@x))) -> c_145() , #pred^#(#pos(#s(#0()))) -> c_146() , #pred^#(#pos(#s(#s(@x)))) -> c_147() , #pred^#(#0()) -> c_148() , #succ^#(#neg(#s(#0()))) -> c_132() , #succ^#(#neg(#s(#s(@x)))) -> c_133() , #succ^#(#pos(#s(@x))) -> c_134() , #succ^#(#0()) -> c_135() , #and^#(#true(), #true()) -> c_78() , #and^#(#true(), #false()) -> c_79() , #and^#(#false(), #true()) -> c_80() , #and^#(#false(), #false()) -> c_81() , #natmult^#(#0(), @y) -> c_115() , #natmult^#(#s(@x), @y) -> c_116(#add^#(#pos(@y), #natmult(@x, @y)), #natmult^#(@x, @y)) } Weak Trs: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , sub(@b1, @b2) -> sub#1(sub'(@b1, @b2, #abs(#0()))) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #ckgt(#EQ()) -> #false() , #ckgt(#LT()) -> #false() , #ckgt(#GT()) -> #true() , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , div(@x, @y) -> #div(@x, @y) , bitToInt'#1(nil(), @n) -> #abs(#0()) , bitToInt'#1(::(@x, @xs), @n) -> +(*(@x, @n), bitToInt'(@xs, *(@n, #pos(#s(#s(#0())))))) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , mult3(@b1, @b2, @b3) -> mult(mult(@b1, @b2), @b2) , sub#1(tuple#2(@b, @_@1)) -> @b , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , leq(@b1, @b2) -> #less(compare(@b1, @b2), #pos(#s(#0()))) , #greater(@x, @y) -> #ckgt(#compare(@x, @y)) , bitToInt'(@b, @n) -> bitToInt'#1(@b, @n) , mult(@b1, @b2) -> mult#1(@b1, @b2) , bitToInt(@b) -> bitToInt'(@b, #abs(#pos(#s(#0())))) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , *(@x, @y) -> #mult(@x, @y) , sub'#5(#true(), @z, @zs) -> ::(#abs(#0()), @zs) , sub'#5(#false(), @z, @zs) -> ::(@z, @zs) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , sub'#3(tuple#2(@z, @r'), @xs, @ys) -> sub'#4(sub'(@xs, @ys, @r'), @z) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , sub'#2(nil(), @r, @x, @xs) -> tuple#2(nil(), @r) , sub'#2(::(@y, @ys), @r, @x, @xs) -> sub'#3(diff(@x, @y, @r), @xs, @ys) , compare#2(nil(), @x, @xs) -> #abs(#0()) , compare#2(::(@y, @ys), @x, @xs) -> compare#3(compare(@xs, @ys), @x, @y) , sub'#4(tuple#2(@zs, @s), @z) -> tuple#2(sub'#5(#equal(@s, #pos(#s(#0()))), @z, @zs), @s) , compare#5(#true(), @x, @y) -> -(#0(), #pos(#s(#0()))) , compare#5(#false(), @x, @y) -> compare#6(#greater(@x, @y)) , compare#3(@r, @x, @y) -> compare#4(#equal(@r, #0()), @r, @x, @y) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , sub'#1(nil(), @b2, @r) -> tuple#2(nil(), @r) , sub'#1(::(@x, @xs), @b2, @r) -> sub'#2(@b2, @r, @x, @xs) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sub'(@b1, @b2, @r) -> sub'#1(@b1, @b2, @r) , compare(@b1, @b2) -> compare#1(@b1, @b2) , compare#6(#true()) -> #abs(#pos(#s(#0()))) , compare#6(#false()) -> #abs(#0()) , compare#4(#true(), @r, @x, @y) -> compare#5(#less(@x, @y), @x, @y) , compare#4(#false(), @r, @x, @y) -> @r , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , compare#1(nil(), @b2) -> #abs(#0()) , compare#1(::(@x, @xs), @b2) -> compare#2(@b2, @x, @xs) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { -^#(@x, @y) -> c_1(#sub^#(@x, @y)) , #sub^#(@x, #neg(@y)) -> c_120(#add^#(@x, #pos(@y))) , #sub^#(@x, #pos(@y)) -> c_121(#add^#(@x, #neg(@y))) , #sub^#(@x, #0()) -> c_122() , sub#1^#(tuple#2(@b, @_@1)) -> c_12() , #abs^#(#neg(@x)) -> c_62() , #abs^#(#pos(@x)) -> c_63() , #abs^#(#0()) -> c_64() , #abs^#(#s(@x)) -> c_65() , diff#1^#(#true()) -> c_3(#abs^#(#pos(#s(#0())))) , diff#1^#(#false()) -> c_4(#abs^#(#0())) , mult#3^#(#false(), @b2, @zs) -> c_43() , #equal^#(@x, @y) -> c_29(#eq^#(@x, @y)) , div^#(@x, @y) -> c_6(#div^#(@x, @y)) , #div^#(#neg(@x), #neg(@y)) -> c_136(#natdiv^#(@x, @y)) , #div^#(#neg(@x), #pos(@y)) -> c_137(#natdiv^#(@x, @y)) , #div^#(#neg(@x), #0()) -> c_138() , #div^#(#pos(@x), #neg(@y)) -> c_139(#natdiv^#(@x, @y)) , #div^#(#pos(@x), #pos(@y)) -> c_140(#natdiv^#(@x, @y)) , #div^#(#pos(@x), #0()) -> c_141() , #div^#(#0(), #neg(@y)) -> c_142() , #div^#(#0(), #pos(@y)) -> c_143() , #div^#(#0(), #0()) -> c_144() , bitToInt'#1^#(nil(), @n) -> c_7(#abs^#(#0())) , +^#(@x, @y) -> c_21(#add^#(@x, @y)) , *^#(@x, @y) -> c_24(#mult^#(@x, @y)) , sum^#(@x, @y, @r) -> c_9(sum#1^#(+(+(@x, @y), @r)), +^#(+(@x, @y), @r), +^#(@x, @y)) , sum#1^#(@s) -> c_20(sum#2^#(#equal(@s, #0()), @s), #equal^#(@s, #0())) , mod^#(@x, @y) -> c_10(-^#(@x, *(@x, div(@x, @y))), *^#(@x, div(@x, @y)), div^#(@x, @y)) , #less^#(@x, @y) -> c_27(#cklt^#(#compare(@x, @y)), #compare^#(@x, @y)) , #greater^#(@x, @y) -> c_14(#ckgt^#(#compare(@x, @y)), #compare^#(@x, @y)) , #ckgt^#(#EQ()) -> c_70() , #ckgt^#(#LT()) -> c_71() , #ckgt^#(#GT()) -> c_72() , #compare^#(#neg(@x), #neg(@y)) -> c_82(#compare^#(@y, @x)) , #compare^#(#neg(@x), #pos(@y)) -> c_83() , #compare^#(#neg(@x), #0()) -> c_84() , #compare^#(#pos(@x), #neg(@y)) -> c_85() , #compare^#(#pos(@x), #pos(@y)) -> c_86(#compare^#(@x, @y)) , #compare^#(#pos(@x), #0()) -> c_87() , #compare^#(#0(), #neg(@y)) -> c_88() , #compare^#(#0(), #pos(@y)) -> c_89() , #compare^#(#0(), #0()) -> c_90() , #compare^#(#0(), #s(@y)) -> c_91() , #compare^#(#s(@x), #0()) -> c_92() , #compare^#(#s(@x), #s(@y)) -> c_93(#compare^#(@x, @y)) , mult#1^#(nil(), @b2) -> c_49() , sum#2^#(#true(), @s) -> c_18(#abs^#(#0()), #abs^#(#0())) , sum#2^#(#false(), @s) -> c_19(sum#3^#(#equal(@s, #pos(#s(#0()))), @s), #equal^#(@s, #pos(#s(#0())))) , sum#3^#(#true(), @s) -> c_57(#abs^#(#pos(#s(#0()))), #abs^#(#0())) , sum#3^#(#false(), @s) -> c_58(sum#4^#(#equal(@s, #pos(#s(#s(#0()))))), #equal^#(@s, #pos(#s(#s(#0()))))) , #add^#(#neg(#s(#0())), @y) -> c_73(#pred^#(@y)) , #add^#(#neg(#s(#s(@x))), @y) -> c_74(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , #add^#(#pos(#s(#0())), @y) -> c_75(#succ^#(@y)) , #add^#(#pos(#s(#s(@x))), @y) -> c_76(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , #add^#(#0(), @y) -> c_77() , sum#4^#(#true()) -> c_22(#abs^#(#0()), #abs^#(#pos(#s(#0())))) , sum#4^#(#false()) -> c_23(#abs^#(#pos(#s(#0()))), #abs^#(#pos(#s(#0())))) , #mult^#(#neg(@x), #neg(@y)) -> c_123(#natmult^#(@x, @y)) , #mult^#(#neg(@x), #pos(@y)) -> c_124(#natmult^#(@x, @y)) , #mult^#(#neg(@x), #0()) -> c_125() , #mult^#(#pos(@x), #neg(@y)) -> c_126(#natmult^#(@x, @y)) , #mult^#(#pos(@x), #pos(@y)) -> c_127(#natmult^#(@x, @y)) , #mult^#(#pos(@x), #0()) -> c_128() , #mult^#(#0(), #neg(@y)) -> c_129() , #mult^#(#0(), #pos(@y)) -> c_130() , #mult^#(#0(), #0()) -> c_131() , sub'#5^#(#true(), @z, @zs) -> c_25(#abs^#(#0())) , sub'#5^#(#false(), @z, @zs) -> c_26() , #cklt^#(#EQ()) -> c_117() , #cklt^#(#LT()) -> c_118() , #cklt^#(#GT()) -> c_119() , sub'#4^#(tuple#2(@zs, @s), @z) -> c_34(sub'#5^#(#equal(@s, #pos(#s(#0()))), @z, @zs), #equal^#(@s, #pos(#s(#0())))) , #eq^#(nil(), nil()) -> c_94() , #eq^#(nil(), tuple#2(@y_1, @y_2)) -> c_95() , #eq^#(nil(), ::(@y_1, @y_2)) -> c_96() , #eq^#(#neg(@x), #neg(@y)) -> c_97(#eq^#(@x, @y)) , #eq^#(#neg(@x), #pos(@y)) -> c_98() , #eq^#(#neg(@x), #0()) -> c_99() , #eq^#(#pos(@x), #neg(@y)) -> c_100() , #eq^#(#pos(@x), #pos(@y)) -> c_101(#eq^#(@x, @y)) , #eq^#(#pos(@x), #0()) -> c_102() , #eq^#(tuple#2(@x_1, @x_2), nil()) -> c_103() , #eq^#(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> c_104(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), #eq^#(@x_1, @y_1), #eq^#(@x_2, @y_2)) , #eq^#(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> c_105() , #eq^#(::(@x_1, @x_2), nil()) -> c_106() , #eq^#(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> c_107() , #eq^#(::(@x_1, @x_2), ::(@y_1, @y_2)) -> c_108(#and^#(#eq(@x_1, @y_1), #eq(@x_2, @y_2)), #eq^#(@x_1, @y_1), #eq^#(@x_2, @y_2)) , #eq^#(#0(), #neg(@y)) -> c_109() , #eq^#(#0(), #pos(@y)) -> c_110() , #eq^#(#0(), #0()) -> c_111() , #eq^#(#0(), #s(@y)) -> c_112() , #eq^#(#s(@x), #0()) -> c_113() , #eq^#(#s(@x), #s(@y)) -> c_114(#eq^#(@x, @y)) , sub'#2^#(nil(), @r, @x, @xs) -> c_30() , diff^#(@x, @y, @r) -> c_48(mod^#(+(+(@x, @y), @r), #pos(#s(#s(#0())))), +^#(+(@x, @y), @r), +^#(@x, @y), diff#1^#(#less(-(-(@x, @y), @r), #0())), #less^#(-(-(@x, @y), @r), #0()), -^#(-(@x, @y), @r), -^#(@x, @y)) , compare#2^#(nil(), @x, @xs) -> c_32(#abs^#(#0())) , compare#3^#(@r, @x, @y) -> c_37(compare#4^#(#equal(@r, #0()), @r, @x, @y), #equal^#(@r, #0())) , compare#5^#(#true(), @x, @y) -> c_35(-^#(#0(), #pos(#s(#0())))) , compare#5^#(#false(), @x, @y) -> c_36(compare#6^#(#greater(@x, @y)), #greater^#(@x, @y)) , compare#6^#(#true()) -> c_53(#abs^#(#pos(#s(#0())))) , compare#6^#(#false()) -> c_54(#abs^#(#0())) , compare#4^#(#true(), @r, @x, @y) -> c_55(compare#5^#(#less(@x, @y), @x, @y), #less^#(@x, @y)) , compare#4^#(#false(), @r, @x, @y) -> c_56() , sub'#1^#(nil(), @b2, @r) -> c_40() , add'#1^#(nil(), @b2, @r) -> c_44() , add'#2^#(nil(), @r, @x, @xs) -> c_46() , compare#1^#(nil(), @b2) -> c_60(#abs^#(#0())) , #natsub^#(@x, #0()) -> c_66() , #natsub^#(#s(@x), #s(@y)) -> c_67(#natsub^#(@x, @y)) , #natdiv^#(#0(), #0()) -> c_68() , #natdiv^#(#s(@x), #s(@y)) -> c_69(#natdiv^#(#natsub(@x, @y), #s(@y)), #natsub^#(@x, @y)) , #pred^#(#neg(#s(@x))) -> c_145() , #pred^#(#pos(#s(#0()))) -> c_146() , #pred^#(#pos(#s(#s(@x)))) -> c_147() , #pred^#(#0()) -> c_148() , #succ^#(#neg(#s(#0()))) -> c_132() , #succ^#(#neg(#s(#s(@x)))) -> c_133() , #succ^#(#pos(#s(@x))) -> c_134() , #succ^#(#0()) -> c_135() , #and^#(#true(), #true()) -> c_78() , #and^#(#true(), #false()) -> c_79() , #and^#(#false(), #true()) -> c_80() , #and^#(#false(), #false()) -> c_81() , #natmult^#(#0(), @y) -> c_115() , #natmult^#(#s(@x), @y) -> c_116(#add^#(#pos(@y), #natmult(@x, @y)), #natmult^#(@x, @y)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { sub^#(@b1, @b2) -> c_2(sub#1^#(sub'(@b1, @b2, #abs(#0()))), sub'^#(@b1, @b2, #abs(#0())), #abs^#(#0())) , sub'^#(@b1, @b2, @r) -> c_51(sub'#1^#(@b1, @b2, @r)) , mult#2^#(@zs, @b2, @x) -> c_5(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs), #equal^#(@x, #pos(#s(#0())))) , mult#3^#(#true(), @b2, @zs) -> c_42(add^#(@b2, @zs)) , bitToInt'#1^#(::(@x, @xs), @n) -> c_8(+^#(*(@x, @n), bitToInt'(@xs, *(@n, #pos(#s(#s(#0())))))), *^#(@x, @n), bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0()))))), *^#(@n, #pos(#s(#s(#0()))))) , bitToInt'^#(@b, @n) -> c_15(bitToInt'#1^#(@b, @n)) , mult3^#(@b1, @b2, @b3) -> c_11(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , mult^#(@b1, @b2) -> c_16(mult#1^#(@b1, @b2)) , leq^#(@b1, @b2) -> c_13(#less^#(compare(@b1, @b2), #pos(#s(#0()))), compare^#(@b1, @b2)) , compare^#(@b1, @b2) -> c_52(compare#1^#(@b1, @b2)) , mult#1^#(::(@x, @xs), @b2) -> c_50(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), #abs^#(#0()), mult^#(@xs, @b2)) , bitToInt^#(@b) -> c_17(bitToInt'^#(@b, #abs(#pos(#s(#0())))), #abs^#(#pos(#s(#0())))) , sub'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_28(sub'#4^#(sub'(@xs, @ys, @r'), @z), sub'^#(@xs, @ys, @r')) , sub'#2^#(::(@y, @ys), @r, @x, @xs) -> c_31(sub'#3^#(diff(@x, @y, @r), @xs, @ys), diff^#(@x, @y, @r)) , compare#2^#(::(@y, @ys), @x, @xs) -> c_33(compare#3^#(compare(@xs, @ys), @x, @y), compare^#(@xs, @ys)) , add^#(@b1, @b2) -> c_38(add'^#(@b1, @b2, #abs(#0())), #abs^#(#0())) , add'^#(@b1, @b2, @r) -> c_59(add'#1^#(@b1, @b2, @r)) , add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_39(add'^#(@xs, @ys, @r')) , sub'#1^#(::(@x, @xs), @b2, @r) -> c_41(sub'#2^#(@b2, @r, @x, @xs)) , add'#1^#(::(@x, @xs), @b2, @r) -> c_45(add'#2^#(@b2, @r, @x, @xs)) , add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_47(add'#3^#(sum(@x, @y, @r), @xs, @ys), sum^#(@x, @y, @r)) , compare#1^#(::(@x, @xs), @b2) -> c_61(compare#2^#(@b2, @x, @xs)) } Weak Trs: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , sub(@b1, @b2) -> sub#1(sub'(@b1, @b2, #abs(#0()))) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #ckgt(#EQ()) -> #false() , #ckgt(#LT()) -> #false() , #ckgt(#GT()) -> #true() , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , div(@x, @y) -> #div(@x, @y) , bitToInt'#1(nil(), @n) -> #abs(#0()) , bitToInt'#1(::(@x, @xs), @n) -> +(*(@x, @n), bitToInt'(@xs, *(@n, #pos(#s(#s(#0())))))) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , mult3(@b1, @b2, @b3) -> mult(mult(@b1, @b2), @b2) , sub#1(tuple#2(@b, @_@1)) -> @b , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , leq(@b1, @b2) -> #less(compare(@b1, @b2), #pos(#s(#0()))) , #greater(@x, @y) -> #ckgt(#compare(@x, @y)) , bitToInt'(@b, @n) -> bitToInt'#1(@b, @n) , mult(@b1, @b2) -> mult#1(@b1, @b2) , bitToInt(@b) -> bitToInt'(@b, #abs(#pos(#s(#0())))) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , *(@x, @y) -> #mult(@x, @y) , sub'#5(#true(), @z, @zs) -> ::(#abs(#0()), @zs) , sub'#5(#false(), @z, @zs) -> ::(@z, @zs) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , sub'#3(tuple#2(@z, @r'), @xs, @ys) -> sub'#4(sub'(@xs, @ys, @r'), @z) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , sub'#2(nil(), @r, @x, @xs) -> tuple#2(nil(), @r) , sub'#2(::(@y, @ys), @r, @x, @xs) -> sub'#3(diff(@x, @y, @r), @xs, @ys) , compare#2(nil(), @x, @xs) -> #abs(#0()) , compare#2(::(@y, @ys), @x, @xs) -> compare#3(compare(@xs, @ys), @x, @y) , sub'#4(tuple#2(@zs, @s), @z) -> tuple#2(sub'#5(#equal(@s, #pos(#s(#0()))), @z, @zs), @s) , compare#5(#true(), @x, @y) -> -(#0(), #pos(#s(#0()))) , compare#5(#false(), @x, @y) -> compare#6(#greater(@x, @y)) , compare#3(@r, @x, @y) -> compare#4(#equal(@r, #0()), @r, @x, @y) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , sub'#1(nil(), @b2, @r) -> tuple#2(nil(), @r) , sub'#1(::(@x, @xs), @b2, @r) -> sub'#2(@b2, @r, @x, @xs) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sub'(@b1, @b2, @r) -> sub'#1(@b1, @b2, @r) , compare(@b1, @b2) -> compare#1(@b1, @b2) , compare#6(#true()) -> #abs(#pos(#s(#0()))) , compare#6(#false()) -> #abs(#0()) , compare#4(#true(), @r, @x, @y) -> compare#5(#less(@x, @y), @x, @y) , compare#4(#false(), @r, @x, @y) -> @r , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , compare#1(nil(), @b2) -> #abs(#0()) , compare#1(::(@x, @xs), @b2) -> compare#2(@b2, @x, @xs) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) Due to missing edges in the dependency-graph, the right-hand sides of following rules could be simplified: { sub^#(@b1, @b2) -> c_2(sub#1^#(sub'(@b1, @b2, #abs(#0()))), sub'^#(@b1, @b2, #abs(#0())), #abs^#(#0())) , mult#2^#(@zs, @b2, @x) -> c_5(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs), #equal^#(@x, #pos(#s(#0())))) , bitToInt'#1^#(::(@x, @xs), @n) -> c_8(+^#(*(@x, @n), bitToInt'(@xs, *(@n, #pos(#s(#s(#0())))))), *^#(@x, @n), bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0()))))), *^#(@n, #pos(#s(#s(#0()))))) , leq^#(@b1, @b2) -> c_13(#less^#(compare(@b1, @b2), #pos(#s(#0()))), compare^#(@b1, @b2)) , mult#1^#(::(@x, @xs), @b2) -> c_50(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), #abs^#(#0()), mult^#(@xs, @b2)) , bitToInt^#(@b) -> c_17(bitToInt'^#(@b, #abs(#pos(#s(#0())))), #abs^#(#pos(#s(#0())))) , sub'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_28(sub'#4^#(sub'(@xs, @ys, @r'), @z), sub'^#(@xs, @ys, @r')) , sub'#2^#(::(@y, @ys), @r, @x, @xs) -> c_31(sub'#3^#(diff(@x, @y, @r), @xs, @ys), diff^#(@x, @y, @r)) , compare#2^#(::(@y, @ys), @x, @xs) -> c_33(compare#3^#(compare(@xs, @ys), @x, @y), compare^#(@xs, @ys)) , add^#(@b1, @b2) -> c_38(add'^#(@b1, @b2, #abs(#0())), #abs^#(#0())) , add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_47(add'#3^#(sum(@x, @y, @r), @xs, @ys), sum^#(@x, @y, @r)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { sub^#(@b1, @b2) -> c_1(sub'^#(@b1, @b2, #abs(#0()))) , sub'^#(@b1, @b2, @r) -> c_2(sub'#1^#(@b1, @b2, @r)) , mult#2^#(@zs, @b2, @x) -> c_3(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs)) , mult#3^#(#true(), @b2, @zs) -> c_4(add^#(@b2, @zs)) , bitToInt'#1^#(::(@x, @xs), @n) -> c_5(bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0())))))) , bitToInt'^#(@b, @n) -> c_6(bitToInt'#1^#(@b, @n)) , mult3^#(@b1, @b2, @b3) -> c_7(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , mult^#(@b1, @b2) -> c_8(mult#1^#(@b1, @b2)) , leq^#(@b1, @b2) -> c_9(compare^#(@b1, @b2)) , compare^#(@b1, @b2) -> c_10(compare#1^#(@b1, @b2)) , mult#1^#(::(@x, @xs), @b2) -> c_11(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), mult^#(@xs, @b2)) , bitToInt^#(@b) -> c_12(bitToInt'^#(@b, #abs(#pos(#s(#0()))))) , sub'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_13(sub'^#(@xs, @ys, @r')) , sub'#2^#(::(@y, @ys), @r, @x, @xs) -> c_14(sub'#3^#(diff(@x, @y, @r), @xs, @ys)) , compare#2^#(::(@y, @ys), @x, @xs) -> c_15(compare^#(@xs, @ys)) , add^#(@b1, @b2) -> c_16(add'^#(@b1, @b2, #abs(#0()))) , add'^#(@b1, @b2, @r) -> c_17(add'#1^#(@b1, @b2, @r)) , add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_18(add'^#(@xs, @ys, @r')) , sub'#1^#(::(@x, @xs), @b2, @r) -> c_19(sub'#2^#(@b2, @r, @x, @xs)) , add'#1^#(::(@x, @xs), @b2, @r) -> c_20(add'#2^#(@b2, @r, @x, @xs)) , add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_21(add'#3^#(sum(@x, @y, @r), @xs, @ys)) , compare#1^#(::(@x, @xs), @b2) -> c_22(compare#2^#(@b2, @x, @xs)) } Weak Trs: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , sub(@b1, @b2) -> sub#1(sub'(@b1, @b2, #abs(#0()))) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #ckgt(#EQ()) -> #false() , #ckgt(#LT()) -> #false() , #ckgt(#GT()) -> #true() , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , div(@x, @y) -> #div(@x, @y) , bitToInt'#1(nil(), @n) -> #abs(#0()) , bitToInt'#1(::(@x, @xs), @n) -> +(*(@x, @n), bitToInt'(@xs, *(@n, #pos(#s(#s(#0())))))) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , mult3(@b1, @b2, @b3) -> mult(mult(@b1, @b2), @b2) , sub#1(tuple#2(@b, @_@1)) -> @b , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , leq(@b1, @b2) -> #less(compare(@b1, @b2), #pos(#s(#0()))) , #greater(@x, @y) -> #ckgt(#compare(@x, @y)) , bitToInt'(@b, @n) -> bitToInt'#1(@b, @n) , mult(@b1, @b2) -> mult#1(@b1, @b2) , bitToInt(@b) -> bitToInt'(@b, #abs(#pos(#s(#0())))) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , *(@x, @y) -> #mult(@x, @y) , sub'#5(#true(), @z, @zs) -> ::(#abs(#0()), @zs) , sub'#5(#false(), @z, @zs) -> ::(@z, @zs) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , sub'#3(tuple#2(@z, @r'), @xs, @ys) -> sub'#4(sub'(@xs, @ys, @r'), @z) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , sub'#2(nil(), @r, @x, @xs) -> tuple#2(nil(), @r) , sub'#2(::(@y, @ys), @r, @x, @xs) -> sub'#3(diff(@x, @y, @r), @xs, @ys) , compare#2(nil(), @x, @xs) -> #abs(#0()) , compare#2(::(@y, @ys), @x, @xs) -> compare#3(compare(@xs, @ys), @x, @y) , sub'#4(tuple#2(@zs, @s), @z) -> tuple#2(sub'#5(#equal(@s, #pos(#s(#0()))), @z, @zs), @s) , compare#5(#true(), @x, @y) -> -(#0(), #pos(#s(#0()))) , compare#5(#false(), @x, @y) -> compare#6(#greater(@x, @y)) , compare#3(@r, @x, @y) -> compare#4(#equal(@r, #0()), @r, @x, @y) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , sub'#1(nil(), @b2, @r) -> tuple#2(nil(), @r) , sub'#1(::(@x, @xs), @b2, @r) -> sub'#2(@b2, @r, @x, @xs) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sub'(@b1, @b2, @r) -> sub'#1(@b1, @b2, @r) , compare(@b1, @b2) -> compare#1(@b1, @b2) , compare#6(#true()) -> #abs(#pos(#s(#0()))) , compare#6(#false()) -> #abs(#0()) , compare#4(#true(), @r, @x, @y) -> compare#5(#less(@x, @y), @x, @y) , compare#4(#false(), @r, @x, @y) -> @r , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , compare#1(nil(), @b2) -> #abs(#0()) , compare#1(::(@x, @xs), @b2) -> compare#2(@b2, @x, @xs) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We replace rewrite rules by usable rules: Weak Usable Rules: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , div(@x, @y) -> #div(@x, @y) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , mult(@b1, @b2) -> mult#1(@b1, @b2) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , *(@x, @y) -> #mult(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { sub^#(@b1, @b2) -> c_1(sub'^#(@b1, @b2, #abs(#0()))) , sub'^#(@b1, @b2, @r) -> c_2(sub'#1^#(@b1, @b2, @r)) , mult#2^#(@zs, @b2, @x) -> c_3(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs)) , mult#3^#(#true(), @b2, @zs) -> c_4(add^#(@b2, @zs)) , bitToInt'#1^#(::(@x, @xs), @n) -> c_5(bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0())))))) , bitToInt'^#(@b, @n) -> c_6(bitToInt'#1^#(@b, @n)) , mult3^#(@b1, @b2, @b3) -> c_7(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , mult^#(@b1, @b2) -> c_8(mult#1^#(@b1, @b2)) , leq^#(@b1, @b2) -> c_9(compare^#(@b1, @b2)) , compare^#(@b1, @b2) -> c_10(compare#1^#(@b1, @b2)) , mult#1^#(::(@x, @xs), @b2) -> c_11(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), mult^#(@xs, @b2)) , bitToInt^#(@b) -> c_12(bitToInt'^#(@b, #abs(#pos(#s(#0()))))) , sub'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_13(sub'^#(@xs, @ys, @r')) , sub'#2^#(::(@y, @ys), @r, @x, @xs) -> c_14(sub'#3^#(diff(@x, @y, @r), @xs, @ys)) , compare#2^#(::(@y, @ys), @x, @xs) -> c_15(compare^#(@xs, @ys)) , add^#(@b1, @b2) -> c_16(add'^#(@b1, @b2, #abs(#0()))) , add'^#(@b1, @b2, @r) -> c_17(add'#1^#(@b1, @b2, @r)) , add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_18(add'^#(@xs, @ys, @r')) , sub'#1^#(::(@x, @xs), @b2, @r) -> c_19(sub'#2^#(@b2, @r, @x, @xs)) , add'#1^#(::(@x, @xs), @b2, @r) -> c_20(add'#2^#(@b2, @r, @x, @xs)) , add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_21(add'#3^#(sum(@x, @y, @r), @xs, @ys)) , compare#1^#(::(@x, @xs), @b2) -> c_22(compare#2^#(@b2, @x, @xs)) } Weak Trs: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , div(@x, @y) -> #div(@x, @y) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , mult(@b1, @b2) -> mult#1(@b1, @b2) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , *(@x, @y) -> #mult(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) Consider the dependency graph 1: sub^#(@b1, @b2) -> c_1(sub'^#(@b1, @b2, #abs(#0()))) -->_1 sub'^#(@b1, @b2, @r) -> c_2(sub'#1^#(@b1, @b2, @r)) :2 2: sub'^#(@b1, @b2, @r) -> c_2(sub'#1^#(@b1, @b2, @r)) -->_1 sub'#1^#(::(@x, @xs), @b2, @r) -> c_19(sub'#2^#(@b2, @r, @x, @xs)) :19 3: mult#2^#(@zs, @b2, @x) -> c_3(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs)) -->_1 mult#3^#(#true(), @b2, @zs) -> c_4(add^#(@b2, @zs)) :4 4: mult#3^#(#true(), @b2, @zs) -> c_4(add^#(@b2, @zs)) -->_1 add^#(@b1, @b2) -> c_16(add'^#(@b1, @b2, #abs(#0()))) :16 5: bitToInt'#1^#(::(@x, @xs), @n) -> c_5(bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0())))))) -->_1 bitToInt'^#(@b, @n) -> c_6(bitToInt'#1^#(@b, @n)) :6 6: bitToInt'^#(@b, @n) -> c_6(bitToInt'#1^#(@b, @n)) -->_1 bitToInt'#1^#(::(@x, @xs), @n) -> c_5(bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0())))))) :5 7: mult3^#(@b1, @b2, @b3) -> c_7(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) -->_2 mult^#(@b1, @b2) -> c_8(mult#1^#(@b1, @b2)) :8 -->_1 mult^#(@b1, @b2) -> c_8(mult#1^#(@b1, @b2)) :8 8: mult^#(@b1, @b2) -> c_8(mult#1^#(@b1, @b2)) -->_1 mult#1^#(::(@x, @xs), @b2) -> c_11(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), mult^#(@xs, @b2)) :11 9: leq^#(@b1, @b2) -> c_9(compare^#(@b1, @b2)) -->_1 compare^#(@b1, @b2) -> c_10(compare#1^#(@b1, @b2)) :10 10: compare^#(@b1, @b2) -> c_10(compare#1^#(@b1, @b2)) -->_1 compare#1^#(::(@x, @xs), @b2) -> c_22(compare#2^#(@b2, @x, @xs)) :22 11: mult#1^#(::(@x, @xs), @b2) -> c_11(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), mult^#(@xs, @b2)) -->_2 mult^#(@b1, @b2) -> c_8(mult#1^#(@b1, @b2)) :8 -->_1 mult#2^#(@zs, @b2, @x) -> c_3(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs)) :3 12: bitToInt^#(@b) -> c_12(bitToInt'^#(@b, #abs(#pos(#s(#0()))))) -->_1 bitToInt'^#(@b, @n) -> c_6(bitToInt'#1^#(@b, @n)) :6 13: sub'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_13(sub'^#(@xs, @ys, @r')) -->_1 sub'^#(@b1, @b2, @r) -> c_2(sub'#1^#(@b1, @b2, @r)) :2 14: sub'#2^#(::(@y, @ys), @r, @x, @xs) -> c_14(sub'#3^#(diff(@x, @y, @r), @xs, @ys)) -->_1 sub'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_13(sub'^#(@xs, @ys, @r')) :13 15: compare#2^#(::(@y, @ys), @x, @xs) -> c_15(compare^#(@xs, @ys)) -->_1 compare^#(@b1, @b2) -> c_10(compare#1^#(@b1, @b2)) :10 16: add^#(@b1, @b2) -> c_16(add'^#(@b1, @b2, #abs(#0()))) -->_1 add'^#(@b1, @b2, @r) -> c_17(add'#1^#(@b1, @b2, @r)) :17 17: add'^#(@b1, @b2, @r) -> c_17(add'#1^#(@b1, @b2, @r)) -->_1 add'#1^#(::(@x, @xs), @b2, @r) -> c_20(add'#2^#(@b2, @r, @x, @xs)) :20 18: add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_18(add'^#(@xs, @ys, @r')) -->_1 add'^#(@b1, @b2, @r) -> c_17(add'#1^#(@b1, @b2, @r)) :17 19: sub'#1^#(::(@x, @xs), @b2, @r) -> c_19(sub'#2^#(@b2, @r, @x, @xs)) -->_1 sub'#2^#(::(@y, @ys), @r, @x, @xs) -> c_14(sub'#3^#(diff(@x, @y, @r), @xs, @ys)) :14 20: add'#1^#(::(@x, @xs), @b2, @r) -> c_20(add'#2^#(@b2, @r, @x, @xs)) -->_1 add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_21(add'#3^#(sum(@x, @y, @r), @xs, @ys)) :21 21: add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_21(add'#3^#(sum(@x, @y, @r), @xs, @ys)) -->_1 add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_18(add'^#(@xs, @ys, @r')) :18 22: compare#1^#(::(@x, @xs), @b2) -> c_22(compare#2^#(@b2, @x, @xs)) -->_1 compare#2^#(::(@y, @ys), @x, @xs) -> c_15(compare^#(@xs, @ys)) :15 Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). { leq^#(@b1, @b2) -> c_9(compare^#(@b1, @b2)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { sub^#(@b1, @b2) -> c_1(sub'^#(@b1, @b2, #abs(#0()))) , sub'^#(@b1, @b2, @r) -> c_2(sub'#1^#(@b1, @b2, @r)) , mult#2^#(@zs, @b2, @x) -> c_3(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs)) , mult#3^#(#true(), @b2, @zs) -> c_4(add^#(@b2, @zs)) , bitToInt'#1^#(::(@x, @xs), @n) -> c_5(bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0())))))) , bitToInt'^#(@b, @n) -> c_6(bitToInt'#1^#(@b, @n)) , mult3^#(@b1, @b2, @b3) -> c_7(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , mult^#(@b1, @b2) -> c_8(mult#1^#(@b1, @b2)) , compare^#(@b1, @b2) -> c_10(compare#1^#(@b1, @b2)) , mult#1^#(::(@x, @xs), @b2) -> c_11(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), mult^#(@xs, @b2)) , bitToInt^#(@b) -> c_12(bitToInt'^#(@b, #abs(#pos(#s(#0()))))) , sub'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_13(sub'^#(@xs, @ys, @r')) , sub'#2^#(::(@y, @ys), @r, @x, @xs) -> c_14(sub'#3^#(diff(@x, @y, @r), @xs, @ys)) , compare#2^#(::(@y, @ys), @x, @xs) -> c_15(compare^#(@xs, @ys)) , add^#(@b1, @b2) -> c_16(add'^#(@b1, @b2, #abs(#0()))) , add'^#(@b1, @b2, @r) -> c_17(add'#1^#(@b1, @b2, @r)) , add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_18(add'^#(@xs, @ys, @r')) , sub'#1^#(::(@x, @xs), @b2, @r) -> c_19(sub'#2^#(@b2, @r, @x, @xs)) , add'#1^#(::(@x, @xs), @b2, @r) -> c_20(add'#2^#(@b2, @r, @x, @xs)) , add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_21(add'#3^#(sum(@x, @y, @r), @xs, @ys)) , compare#1^#(::(@x, @xs), @b2) -> c_22(compare#2^#(@b2, @x, @xs)) } Weak Trs: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , div(@x, @y) -> #div(@x, @y) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , mult(@b1, @b2) -> mult#1(@b1, @b2) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , *(@x, @y) -> #mult(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component: Problem (R): ------------ Strict DPs: { sub^#(@b1, @b2) -> c_1(sub'^#(@b1, @b2, #abs(#0()))) , sub'^#(@b1, @b2, @r) -> c_2(sub'#1^#(@b1, @b2, @r)) , sub'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_13(sub'^#(@xs, @ys, @r')) , sub'#2^#(::(@y, @ys), @r, @x, @xs) -> c_14(sub'#3^#(diff(@x, @y, @r), @xs, @ys)) , sub'#1^#(::(@x, @xs), @b2, @r) -> c_19(sub'#2^#(@b2, @r, @x, @xs)) } Weak DPs: { mult#2^#(@zs, @b2, @x) -> c_3(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs)) , mult#3^#(#true(), @b2, @zs) -> c_4(add^#(@b2, @zs)) , bitToInt'#1^#(::(@x, @xs), @n) -> c_5(bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0())))))) , bitToInt'^#(@b, @n) -> c_6(bitToInt'#1^#(@b, @n)) , mult3^#(@b1, @b2, @b3) -> c_7(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , mult^#(@b1, @b2) -> c_8(mult#1^#(@b1, @b2)) , compare^#(@b1, @b2) -> c_10(compare#1^#(@b1, @b2)) , mult#1^#(::(@x, @xs), @b2) -> c_11(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), mult^#(@xs, @b2)) , bitToInt^#(@b) -> c_12(bitToInt'^#(@b, #abs(#pos(#s(#0()))))) , compare#2^#(::(@y, @ys), @x, @xs) -> c_15(compare^#(@xs, @ys)) , add^#(@b1, @b2) -> c_16(add'^#(@b1, @b2, #abs(#0()))) , add'^#(@b1, @b2, @r) -> c_17(add'#1^#(@b1, @b2, @r)) , add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_18(add'^#(@xs, @ys, @r')) , add'#1^#(::(@x, @xs), @b2, @r) -> c_20(add'#2^#(@b2, @r, @x, @xs)) , add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_21(add'#3^#(sum(@x, @y, @r), @xs, @ys)) , compare#1^#(::(@x, @xs), @b2) -> c_22(compare#2^#(@b2, @x, @xs)) } Weak Trs: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , div(@x, @y) -> #div(@x, @y) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , mult(@b1, @b2) -> mult#1(@b1, @b2) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , *(@x, @y) -> #mult(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } StartTerms: basic terms Defined Symbols: -^# #sub^# sub^# sub#1^# sub'^# #abs^# diff#1^# mult#2^# mult#3^# #equal^# div^# #div^# bitToInt'#1^# +^# *^# bitToInt'^# sum^# sum#1^# mod^# mult3^# mult^# leq^# #less^# compare^# #greater^# #ckgt^# #compare^# mult#1^# bitToInt^# sum#2^# sum#3^# #add^# sum#4^# #mult^# sub'#5^# #cklt^# sub'#3^# sub'#4^# #eq^# sub'#2^# diff^# compare#2^# compare#3^# compare#5^# compare#6^# compare#4^# add^# add'^# add'#3^# sub'#1^# add'#1^# add'#2^# compare#1^# #natsub^# #natdiv^# #pred^# #succ^# #and^# #natmult^# Constructors: #EQ nil #neg #divByZero #true #pos tuple#2 #false :: #LT #0 #s #GT Strategy: innermost Problem (S): ------------ Strict DPs: { mult#2^#(@zs, @b2, @x) -> c_3(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs)) , mult#3^#(#true(), @b2, @zs) -> c_4(add^#(@b2, @zs)) , bitToInt'#1^#(::(@x, @xs), @n) -> c_5(bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0())))))) , bitToInt'^#(@b, @n) -> c_6(bitToInt'#1^#(@b, @n)) , mult3^#(@b1, @b2, @b3) -> c_7(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , mult^#(@b1, @b2) -> c_8(mult#1^#(@b1, @b2)) , compare^#(@b1, @b2) -> c_10(compare#1^#(@b1, @b2)) , mult#1^#(::(@x, @xs), @b2) -> c_11(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), mult^#(@xs, @b2)) , bitToInt^#(@b) -> c_12(bitToInt'^#(@b, #abs(#pos(#s(#0()))))) , compare#2^#(::(@y, @ys), @x, @xs) -> c_15(compare^#(@xs, @ys)) , add^#(@b1, @b2) -> c_16(add'^#(@b1, @b2, #abs(#0()))) , add'^#(@b1, @b2, @r) -> c_17(add'#1^#(@b1, @b2, @r)) , add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_18(add'^#(@xs, @ys, @r')) , add'#1^#(::(@x, @xs), @b2, @r) -> c_20(add'#2^#(@b2, @r, @x, @xs)) , add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_21(add'#3^#(sum(@x, @y, @r), @xs, @ys)) , compare#1^#(::(@x, @xs), @b2) -> c_22(compare#2^#(@b2, @x, @xs)) } Weak DPs: { sub^#(@b1, @b2) -> c_1(sub'^#(@b1, @b2, #abs(#0()))) , sub'^#(@b1, @b2, @r) -> c_2(sub'#1^#(@b1, @b2, @r)) , sub'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_13(sub'^#(@xs, @ys, @r')) , sub'#2^#(::(@y, @ys), @r, @x, @xs) -> c_14(sub'#3^#(diff(@x, @y, @r), @xs, @ys)) , sub'#1^#(::(@x, @xs), @b2, @r) -> c_19(sub'#2^#(@b2, @r, @x, @xs)) } Weak Trs: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , div(@x, @y) -> #div(@x, @y) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , mult(@b1, @b2) -> mult#1(@b1, @b2) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , *(@x, @y) -> #mult(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } StartTerms: basic terms Defined Symbols: -^# #sub^# sub^# sub#1^# sub'^# #abs^# diff#1^# mult#2^# mult#3^# #equal^# div^# #div^# bitToInt'#1^# +^# *^# bitToInt'^# sum^# sum#1^# mod^# mult3^# mult^# leq^# #less^# compare^# #greater^# #ckgt^# #compare^# mult#1^# bitToInt^# sum#2^# sum#3^# #add^# sum#4^# #mult^# sub'#5^# #cklt^# sub'#3^# sub'#4^# #eq^# sub'#2^# diff^# compare#2^# compare#3^# compare#5^# compare#6^# compare#4^# add^# add'^# add'#3^# sub'#1^# add'#1^# add'#2^# compare#1^# #natsub^# #natdiv^# #pred^# #succ^# #and^# #natmult^# Constructors: #EQ nil #neg #divByZero #true #pos tuple#2 #false :: #LT #0 #s #GT Strategy: innermost Overall, the transformation results in the following sub-problem(s): Generated new problems: ----------------------- R) Strict DPs: { sub^#(@b1, @b2) -> c_1(sub'^#(@b1, @b2, #abs(#0()))) , sub'^#(@b1, @b2, @r) -> c_2(sub'#1^#(@b1, @b2, @r)) , sub'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_13(sub'^#(@xs, @ys, @r')) , sub'#2^#(::(@y, @ys), @r, @x, @xs) -> c_14(sub'#3^#(diff(@x, @y, @r), @xs, @ys)) , sub'#1^#(::(@x, @xs), @b2, @r) -> c_19(sub'#2^#(@b2, @r, @x, @xs)) } Weak DPs: { mult#2^#(@zs, @b2, @x) -> c_3(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs)) , mult#3^#(#true(), @b2, @zs) -> c_4(add^#(@b2, @zs)) , bitToInt'#1^#(::(@x, @xs), @n) -> c_5(bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0())))))) , bitToInt'^#(@b, @n) -> c_6(bitToInt'#1^#(@b, @n)) , mult3^#(@b1, @b2, @b3) -> c_7(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , mult^#(@b1, @b2) -> c_8(mult#1^#(@b1, @b2)) , compare^#(@b1, @b2) -> c_10(compare#1^#(@b1, @b2)) , mult#1^#(::(@x, @xs), @b2) -> c_11(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), mult^#(@xs, @b2)) , bitToInt^#(@b) -> c_12(bitToInt'^#(@b, #abs(#pos(#s(#0()))))) , compare#2^#(::(@y, @ys), @x, @xs) -> c_15(compare^#(@xs, @ys)) , add^#(@b1, @b2) -> c_16(add'^#(@b1, @b2, #abs(#0()))) , add'^#(@b1, @b2, @r) -> c_17(add'#1^#(@b1, @b2, @r)) , add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_18(add'^#(@xs, @ys, @r')) , add'#1^#(::(@x, @xs), @b2, @r) -> c_20(add'#2^#(@b2, @r, @x, @xs)) , add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_21(add'#3^#(sum(@x, @y, @r), @xs, @ys)) , compare#1^#(::(@x, @xs), @b2) -> c_22(compare#2^#(@b2, @x, @xs)) } Weak Trs: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , div(@x, @y) -> #div(@x, @y) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , mult(@b1, @b2) -> mult#1(@b1, @b2) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , *(@x, @y) -> #mult(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } StartTerms: basic terms Defined Symbols: -^# #sub^# sub^# sub#1^# sub'^# #abs^# diff#1^# mult#2^# mult#3^# #equal^# div^# #div^# bitToInt'#1^# +^# *^# bitToInt'^# sum^# sum#1^# mod^# mult3^# mult^# leq^# #less^# compare^# #greater^# #ckgt^# #compare^# mult#1^# bitToInt^# sum#2^# sum#3^# #add^# sum#4^# #mult^# sub'#5^# #cklt^# sub'#3^# sub'#4^# #eq^# sub'#2^# diff^# compare#2^# compare#3^# compare#5^# compare#6^# compare#4^# add^# add'^# add'#3^# sub'#1^# add'#1^# add'#2^# compare#1^# #natsub^# #natdiv^# #pred^# #succ^# #and^# #natmult^# Constructors: #EQ nil #neg #divByZero #true #pos tuple#2 #false :: #LT #0 #s #GT Strategy: innermost This problem was proven YES(O(1),O(n^1)). S) Strict DPs: { mult#2^#(@zs, @b2, @x) -> c_3(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs)) , mult#3^#(#true(), @b2, @zs) -> c_4(add^#(@b2, @zs)) , bitToInt'#1^#(::(@x, @xs), @n) -> c_5(bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0())))))) , bitToInt'^#(@b, @n) -> c_6(bitToInt'#1^#(@b, @n)) , mult3^#(@b1, @b2, @b3) -> c_7(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , mult^#(@b1, @b2) -> c_8(mult#1^#(@b1, @b2)) , compare^#(@b1, @b2) -> c_10(compare#1^#(@b1, @b2)) , mult#1^#(::(@x, @xs), @b2) -> c_11(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), mult^#(@xs, @b2)) , bitToInt^#(@b) -> c_12(bitToInt'^#(@b, #abs(#pos(#s(#0()))))) , compare#2^#(::(@y, @ys), @x, @xs) -> c_15(compare^#(@xs, @ys)) , add^#(@b1, @b2) -> c_16(add'^#(@b1, @b2, #abs(#0()))) , add'^#(@b1, @b2, @r) -> c_17(add'#1^#(@b1, @b2, @r)) , add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_18(add'^#(@xs, @ys, @r')) , add'#1^#(::(@x, @xs), @b2, @r) -> c_20(add'#2^#(@b2, @r, @x, @xs)) , add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_21(add'#3^#(sum(@x, @y, @r), @xs, @ys)) , compare#1^#(::(@x, @xs), @b2) -> c_22(compare#2^#(@b2, @x, @xs)) } Weak DPs: { sub^#(@b1, @b2) -> c_1(sub'^#(@b1, @b2, #abs(#0()))) , sub'^#(@b1, @b2, @r) -> c_2(sub'#1^#(@b1, @b2, @r)) , sub'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_13(sub'^#(@xs, @ys, @r')) , sub'#2^#(::(@y, @ys), @r, @x, @xs) -> c_14(sub'#3^#(diff(@x, @y, @r), @xs, @ys)) , sub'#1^#(::(@x, @xs), @b2, @r) -> c_19(sub'#2^#(@b2, @r, @x, @xs)) } Weak Trs: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , div(@x, @y) -> #div(@x, @y) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , mult(@b1, @b2) -> mult#1(@b1, @b2) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , *(@x, @y) -> #mult(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } StartTerms: basic terms Defined Symbols: -^# #sub^# sub^# sub#1^# sub'^# #abs^# diff#1^# mult#2^# mult#3^# #equal^# div^# #div^# bitToInt'#1^# +^# *^# bitToInt'^# sum^# sum#1^# mod^# mult3^# mult^# leq^# #less^# compare^# #greater^# #ckgt^# #compare^# mult#1^# bitToInt^# sum#2^# sum#3^# #add^# sum#4^# #mult^# sub'#5^# #cklt^# sub'#3^# sub'#4^# #eq^# sub'#2^# diff^# compare#2^# compare#3^# compare#5^# compare#6^# compare#4^# add^# add'^# add'#3^# sub'#1^# add'#1^# add'#2^# compare#1^# #natsub^# #natdiv^# #pred^# #succ^# #and^# #natmult^# Constructors: #EQ nil #neg #divByZero #true #pos tuple#2 #false :: #LT #0 #s #GT Strategy: innermost This problem was proven YES(O(1),O(n^2)). Proofs for generated problems: ------------------------------ R) We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { sub^#(@b1, @b2) -> c_1(sub'^#(@b1, @b2, #abs(#0()))) , sub'^#(@b1, @b2, @r) -> c_2(sub'#1^#(@b1, @b2, @r)) , sub'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_13(sub'^#(@xs, @ys, @r')) , sub'#2^#(::(@y, @ys), @r, @x, @xs) -> c_14(sub'#3^#(diff(@x, @y, @r), @xs, @ys)) , sub'#1^#(::(@x, @xs), @b2, @r) -> c_19(sub'#2^#(@b2, @r, @x, @xs)) } Weak DPs: { mult#2^#(@zs, @b2, @x) -> c_3(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs)) , mult#3^#(#true(), @b2, @zs) -> c_4(add^#(@b2, @zs)) , bitToInt'#1^#(::(@x, @xs), @n) -> c_5(bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0())))))) , bitToInt'^#(@b, @n) -> c_6(bitToInt'#1^#(@b, @n)) , mult3^#(@b1, @b2, @b3) -> c_7(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , mult^#(@b1, @b2) -> c_8(mult#1^#(@b1, @b2)) , compare^#(@b1, @b2) -> c_10(compare#1^#(@b1, @b2)) , mult#1^#(::(@x, @xs), @b2) -> c_11(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), mult^#(@xs, @b2)) , bitToInt^#(@b) -> c_12(bitToInt'^#(@b, #abs(#pos(#s(#0()))))) , compare#2^#(::(@y, @ys), @x, @xs) -> c_15(compare^#(@xs, @ys)) , add^#(@b1, @b2) -> c_16(add'^#(@b1, @b2, #abs(#0()))) , add'^#(@b1, @b2, @r) -> c_17(add'#1^#(@b1, @b2, @r)) , add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_18(add'^#(@xs, @ys, @r')) , add'#1^#(::(@x, @xs), @b2, @r) -> c_20(add'#2^#(@b2, @r, @x, @xs)) , add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_21(add'#3^#(sum(@x, @y, @r), @xs, @ys)) , compare#1^#(::(@x, @xs), @b2) -> c_22(compare#2^#(@b2, @x, @xs)) } Weak Trs: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , div(@x, @y) -> #div(@x, @y) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , mult(@b1, @b2) -> mult#1(@b1, @b2) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , *(@x, @y) -> #mult(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { mult#2^#(@zs, @b2, @x) -> c_3(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs)) , mult#3^#(#true(), @b2, @zs) -> c_4(add^#(@b2, @zs)) , bitToInt'#1^#(::(@x, @xs), @n) -> c_5(bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0())))))) , bitToInt'^#(@b, @n) -> c_6(bitToInt'#1^#(@b, @n)) , mult3^#(@b1, @b2, @b3) -> c_7(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , mult^#(@b1, @b2) -> c_8(mult#1^#(@b1, @b2)) , compare^#(@b1, @b2) -> c_10(compare#1^#(@b1, @b2)) , mult#1^#(::(@x, @xs), @b2) -> c_11(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), mult^#(@xs, @b2)) , bitToInt^#(@b) -> c_12(bitToInt'^#(@b, #abs(#pos(#s(#0()))))) , compare#2^#(::(@y, @ys), @x, @xs) -> c_15(compare^#(@xs, @ys)) , add^#(@b1, @b2) -> c_16(add'^#(@b1, @b2, #abs(#0()))) , add'^#(@b1, @b2, @r) -> c_17(add'#1^#(@b1, @b2, @r)) , add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_18(add'^#(@xs, @ys, @r')) , add'#1^#(::(@x, @xs), @b2, @r) -> c_20(add'#2^#(@b2, @r, @x, @xs)) , add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_21(add'#3^#(sum(@x, @y, @r), @xs, @ys)) , compare#1^#(::(@x, @xs), @b2) -> c_22(compare#2^#(@b2, @x, @xs)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { sub^#(@b1, @b2) -> c_1(sub'^#(@b1, @b2, #abs(#0()))) , sub'^#(@b1, @b2, @r) -> c_2(sub'#1^#(@b1, @b2, @r)) , sub'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_13(sub'^#(@xs, @ys, @r')) , sub'#2^#(::(@y, @ys), @r, @x, @xs) -> c_14(sub'#3^#(diff(@x, @y, @r), @xs, @ys)) , sub'#1^#(::(@x, @xs), @b2, @r) -> c_19(sub'#2^#(@b2, @r, @x, @xs)) } Weak Trs: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , div(@x, @y) -> #div(@x, @y) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , mult(@b1, @b2) -> mult#1(@b1, @b2) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , *(@x, @y) -> #mult(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We replace rewrite rules by usable rules: Weak Usable Rules: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , div(@x, @y) -> #div(@x, @y) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , +(@x, @y) -> #add(@x, @y) , *(@x, @y) -> #mult(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { sub^#(@b1, @b2) -> c_1(sub'^#(@b1, @b2, #abs(#0()))) , sub'^#(@b1, @b2, @r) -> c_2(sub'#1^#(@b1, @b2, @r)) , sub'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_13(sub'^#(@xs, @ys, @r')) , sub'#2^#(::(@y, @ys), @r, @x, @xs) -> c_14(sub'#3^#(diff(@x, @y, @r), @xs, @ys)) , sub'#1^#(::(@x, @xs), @b2, @r) -> c_19(sub'#2^#(@b2, @r, @x, @xs)) } Weak Trs: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , div(@x, @y) -> #div(@x, @y) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , +(@x, @y) -> #add(@x, @y) , *(@x, @y) -> #mult(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 5: sub'#1^#(::(@x, @xs), @b2, @r) -> c_19(sub'#2^#(@b2, @r, @x, @xs)) } Trs: { diff#1(#false()) -> #abs(#0()) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() } Sub-proof: ---------- The following argument positions are usable: Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_13) = {1}, Uargs(c_14) = {1}, Uargs(c_19) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [#natsub](x1, x2) = [1] x1 + [0] [-](x1, x2) = [1] x1 + [0] [sub](x1, x2) = [7] x1 + [7] x2 + [0] [diff#1](x1) = [4] x1 + [0] [#natdiv](x1, x2) = [0] [#ckgt](x1) = [7] x1 + [0] [#add](x1, x2) = [4] x2 + [0] [mult#2](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [div](x1, x2) = [4] x1 + [0] [bitToInt'#1](x1, x2) = [7] x1 + [7] x2 + [0] [sum](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [#EQ] = [0] [mod](x1, x2) = [4] x1 + [7] [#and](x1, x2) = [7] x1 + [7] x2 + [0] [mult3](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [sub#1](x1) = [7] x1 + [0] [#compare](x1, x2) = [4] [nil] = [0] [leq](x1, x2) = [7] x1 + [7] x2 + [0] [#greater](x1, x2) = [7] x1 + [7] x2 + [0] [bitToInt'](x1, x2) = [7] x1 + [7] x2 + [0] [mult](x1, x2) = [7] x1 + [7] x2 + [0] [bitToInt](x1) = [7] x1 + [0] [sum#2](x1, x2) = [7] x1 + [7] x2 + [0] [sum#1](x1) = [7] x1 + [0] [+](x1, x2) = [4] x2 + [0] [sum#4](x1) = [7] x1 + [0] [*](x1, x2) = [4] x1 + [4] x2 + [4] [#neg](x1) = [0] [sub'#5](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [#less](x1, x2) = [4] [sub'#3](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [#equal](x1, x2) = [7] x1 + [7] x2 + [0] [#eq](x1, x2) = [7] x1 + [7] x2 + [0] [#natmult](x1, x2) = [0] [#divByZero] = [0] [sub'#2](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0] [compare#2](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [#true] = [0] [sub'#4](x1, x2) = [7] x1 + [7] x2 + [0] [compare#5](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [compare#3](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [#cklt](x1) = [4] [add](x1, x2) = [7] x1 + [7] x2 + [0] [#sub](x1, x2) = [1] x1 + [0] [#pos](x1) = [0] [add'#3](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [sub'#1](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [mult#3](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [add'#1](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [add'#2](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0] [tuple#2](x1, x2) = [0] [diff](x1, x2, x3) = [4] x1 + [4] x2 + [0] [#false] = [2] [mult#1](x1, x2) = [7] x1 + [7] x2 + [0] [::](x1, x2) = [1] x1 + [1] x2 + [4] [#LT] = [0] [#mult](x1, x2) = [4] [#succ](x1) = [0] [sub'](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [compare](x1, x2) = [7] x1 + [7] x2 + [0] [compare#6](x1) = [7] x1 + [0] [compare#4](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0] [#0] = [0] [sum#3](x1, x2) = [7] x1 + [7] x2 + [0] [#div](x1, x2) = [0] [add'](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [compare#1](x1, x2) = [7] x1 + [7] x2 + [0] [#abs](x1) = [0] [#pred](x1) = [0] [#s](x1) = [1] x1 + [0] [#GT] = [0] [-^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_1](x1) = [7] x1 + [0] [#sub^#](x1, x2) = [7] x1 + [7] x2 + [0] [sub^#](x1, x2) = [7] x1 + [7] x2 + [7] [c_2](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [sub#1^#](x1) = [7] x1 + [0] [sub'^#](x1, x2, x3) = [1] x1 + [0] [#abs^#](x1) = [7] x1 + [0] [diff#1^#](x1) = [7] x1 + [0] [c_3](x1) = [7] x1 + [0] [c_4](x1) = [7] x1 + [0] [mult#2^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_5](x1, x2) = [7] x1 + [7] x2 + [0] [mult#3^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [#equal^#](x1, x2) = [7] x1 + [7] x2 + [0] [div^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_6](x1) = [7] x1 + [0] [#div^#](x1, x2) = [7] x1 + [7] x2 + [0] [bitToInt'#1^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_7](x1) = [7] x1 + [0] [c_8](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0] [+^#](x1, x2) = [7] x1 + [7] x2 + [0] [*^#](x1, x2) = [7] x1 + [7] x2 + [0] [bitToInt'^#](x1, x2) = [7] x1 + [7] x2 + [0] [sum^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_9](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [sum#1^#](x1) = [7] x1 + [0] [mod^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_10](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [mult3^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_11](x1, x2) = [7] x1 + [7] x2 + [0] [mult^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_12] = [0] [leq^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_13](x1, x2) = [7] x1 + [7] x2 + [0] [#less^#](x1, x2) = [7] x1 + [7] x2 + [0] [compare^#](x1, x2) = [7] x1 + [7] x2 + [0] [#greater^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_14](x1, x2) = [7] x1 + [7] x2 + [0] [#ckgt^#](x1) = [7] x1 + [0] [#compare^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_15](x1) = [7] x1 + [0] [c_16](x1) = [7] x1 + [0] [mult#1^#](x1, x2) = [7] x1 + [7] x2 + [0] [bitToInt^#](x1) = [7] x1 + [0] [c_17](x1, x2) = [7] x1 + [7] x2 + [0] [sum#2^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_18](x1, x2) = [7] x1 + [7] x2 + [0] [c_19](x1, x2) = [7] x1 + [7] x2 + [0] [sum#3^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_20](x1, x2) = [7] x1 + [7] x2 + [0] [c_21](x1) = [7] x1 + [0] [#add^#](x1, x2) = [7] x1 + [7] x2 + [0] [sum#4^#](x1) = [7] x1 + [0] [c_22](x1, x2) = [7] x1 + [7] x2 + [0] [c_23](x1, x2) = [7] x1 + [7] x2 + [0] [c_24](x1) = [7] x1 + [0] [#mult^#](x1, x2) = [7] x1 + [7] x2 + [0] [sub'#5^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_25](x1) = [7] x1 + [0] [c_26] = [0] [c_27](x1, x2) = [7] x1 + [7] x2 + [0] [#cklt^#](x1) = [7] x1 + [0] [sub'#3^#](x1, x2, x3) = [1] x2 + [0] [c_28](x1, x2) = [7] x1 + [7] x2 + [0] [sub'#4^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_29](x1) = [7] x1 + [0] [#eq^#](x1, x2) = [7] x1 + [7] x2 + [0] [sub'#2^#](x1, x2, x3, x4) = [1] x4 + [0] [c_30] = [0] [c_31](x1, x2) = [7] x1 + [7] x2 + [0] [diff^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [compare#2^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_32](x1) = [7] x1 + [0] [c_33](x1, x2) = [7] x1 + [7] x2 + [0] [compare#3^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_34](x1, x2) = [7] x1 + [7] x2 + [0] [compare#5^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_35](x1) = [7] x1 + [0] [c_36](x1, x2) = [7] x1 + [7] x2 + [0] [compare#6^#](x1) = [7] x1 + [0] [c_37](x1, x2) = [7] x1 + [7] x2 + [0] [compare#4^#](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0] [add^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_38](x1, x2) = [7] x1 + [7] x2 + [0] [add'^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [add'#3^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_39](x1) = [7] x1 + [0] [sub'#1^#](x1, x2, x3) = [1] x1 + [0] [c_40] = [0] [c_41](x1) = [7] x1 + [0] [c_42](x1) = [7] x1 + [0] [c_43] = [0] [add'#1^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_44] = [0] [c_45](x1) = [7] x1 + [0] [add'#2^#](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0] [c_46] = [0] [c_47](x1, x2) = [7] x1 + [7] x2 + [0] [c_48](x1, x2, x3, x4, x5, x6, x7) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [7] x5 + [7] x6 + [7] x7 + [0] [c_49] = [0] [c_50](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_51](x1) = [7] x1 + [0] [c_52](x1) = [7] x1 + [0] [compare#1^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_53](x1) = [7] x1 + [0] [c_54](x1) = [7] x1 + [0] [c_55](x1, x2) = [7] x1 + [7] x2 + [0] [c_56] = [0] [c_57](x1, x2) = [7] x1 + [7] x2 + [0] [c_58](x1, x2) = [7] x1 + [7] x2 + [0] [c_59](x1) = [7] x1 + [0] [c_60](x1) = [7] x1 + [0] [c_61](x1) = [7] x1 + [0] [c_62] = [0] [c_63] = [0] [c_64] = [0] [c_65] = [0] [#natsub^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_66] = [0] [c_67](x1) = [7] x1 + [0] [#natdiv^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_68] = [0] [c_69](x1, x2) = [7] x1 + [7] x2 + [0] [c_70] = [0] [c_71] = [0] [c_72] = [0] [c_73](x1) = [7] x1 + [0] [#pred^#](x1) = [7] x1 + [0] [c_74](x1, x2) = [7] x1 + [7] x2 + [0] [c_75](x1) = [7] x1 + [0] [#succ^#](x1) = [7] x1 + [0] [c_76](x1, x2) = [7] x1 + [7] x2 + [0] [c_77] = [0] [#and^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_78] = [0] [c_79] = [0] [c_80] = [0] [c_81] = [0] [c_82](x1) = [7] x1 + [0] [c_83] = [0] [c_84] = [0] [c_85] = [0] [c_86](x1) = [7] x1 + [0] [c_87] = [0] [c_88] = [0] [c_89] = [0] [c_90] = [0] [c_91] = [0] [c_92] = [0] [c_93](x1) = [7] x1 + [0] [c_94] = [0] [c_95] = [0] [c_96] = [0] [c_97](x1) = [7] x1 + [0] [c_98] = [0] [c_99] = [0] [c_100] = [0] [c_101](x1) = [7] x1 + [0] [c_102] = [0] [c_103] = [0] [c_104](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_105] = [0] [c_106] = [0] [c_107] = [0] [c_108](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_109] = [0] [c_110] = [0] [c_111] = [0] [c_112] = [0] [c_113] = [0] [c_114](x1) = [7] x1 + [0] [#natmult^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_115] = [0] [c_116](x1, x2) = [7] x1 + [7] x2 + [0] [c_117] = [0] [c_118] = [0] [c_119] = [0] [c_120](x1) = [7] x1 + [0] [c_121](x1) = [7] x1 + [0] [c_122] = [0] [c_123](x1) = [7] x1 + [0] [c_124](x1) = [7] x1 + [0] [c_125] = [0] [c_126](x1) = [7] x1 + [0] [c_127](x1) = [7] x1 + [0] [c_128] = [0] [c_129] = [0] [c_130] = [0] [c_131] = [0] [c_132] = [0] [c_133] = [0] [c_134] = [0] [c_135] = [0] [c_136](x1) = [7] x1 + [0] [c_137](x1) = [7] x1 + [0] [c_138] = [0] [c_139](x1) = [7] x1 + [0] [c_140](x1) = [7] x1 + [0] [c_141] = [0] [c_142] = [0] [c_143] = [0] [c_144] = [0] [c_145] = [0] [c_146] = [0] [c_147] = [0] [c_148] = [0] [c] = [0] [c_1](x1) = [4] x1 + [7] [c_2](x1) = [1] x1 + [0] [c_3](x1) = [7] x1 + [0] [c_4](x1) = [7] x1 + [0] [c_5](x1) = [7] x1 + [0] [c_6](x1) = [7] x1 + [0] [c_7](x1, x2) = [7] x1 + [7] x2 + [0] [c_8](x1) = [7] x1 + [0] [c_9](x1) = [7] x1 + [0] [c_10](x1) = [7] x1 + [0] [c_11](x1, x2) = [7] x1 + [7] x2 + [0] [c_12](x1) = [7] x1 + [0] [c_13](x1) = [1] x1 + [0] [c_14](x1) = [1] x1 + [0] [c_15](x1) = [7] x1 + [0] [c_16](x1) = [7] x1 + [0] [c_17](x1) = [7] x1 + [0] [c_18](x1) = [7] x1 + [0] [c_19](x1) = [1] x1 + [1] [c_20](x1) = [7] x1 + [0] [c_21](x1) = [7] x1 + [0] [c_22](x1) = [7] x1 + [0] The following symbols are considered usable {#natsub, -, diff#1, #natdiv, #add, div, mod, #compare, +, *, #less, #natmult, #cklt, #sub, diff, #mult, #succ, #div, #abs, #pred, sub^#, sub'^#, sub'#3^#, sub'#2^#, sub'#1^#} The order satisfies the following ordering constraints: [#natsub(@x, #0())] = [1] @x + [0] >= [1] @x + [0] = [@x] [#natsub(#s(@x), #s(@y))] = [1] @x + [0] >= [1] @x + [0] = [#natsub(@x, @y)] [-(@x, @y)] = [1] @x + [0] >= [1] @x + [0] = [#sub(@x, @y)] [diff#1(#true())] = [0] >= [0] = [#abs(#pos(#s(#0())))] [diff#1(#false())] = [8] > [0] = [#abs(#0())] [#natdiv(#0(), #0())] = [0] >= [0] = [#divByZero()] [#natdiv(#s(@x), #s(@y))] = [0] >= [0] = [#s(#natdiv(#natsub(@x, @y), #s(@y)))] [#add(#neg(#s(#0())), @y)] = [4] @y + [0] >= [0] = [#pred(@y)] [#add(#neg(#s(#s(@x))), @y)] = [4] @y + [0] >= [0] = [#pred(#add(#pos(#s(@x)), @y))] [#add(#pos(#s(#0())), @y)] = [4] @y + [0] >= [0] = [#succ(@y)] [#add(#pos(#s(#s(@x))), @y)] = [4] @y + [0] >= [0] = [#succ(#add(#pos(#s(@x)), @y))] [#add(#0(), @y)] = [4] @y + [0] >= [1] @y + [0] = [@y] [div(@x, @y)] = [4] @x + [0] >= [0] = [#div(@x, @y)] [mod(@x, @y)] = [4] @x + [7] > [1] @x + [0] = [-(@x, *(@x, div(@x, @y)))] [#compare(#neg(@x), #neg(@y))] = [4] >= [4] = [#compare(@y, @x)] [#compare(#neg(@x), #pos(@y))] = [4] > [0] = [#LT()] [#compare(#neg(@x), #0())] = [4] > [0] = [#LT()] [#compare(#pos(@x), #neg(@y))] = [4] > [0] = [#GT()] [#compare(#pos(@x), #pos(@y))] = [4] >= [4] = [#compare(@x, @y)] [#compare(#pos(@x), #0())] = [4] > [0] = [#GT()] [#compare(#0(), #neg(@y))] = [4] > [0] = [#GT()] [#compare(#0(), #pos(@y))] = [4] > [0] = [#LT()] [#compare(#0(), #0())] = [4] > [0] = [#EQ()] [#compare(#0(), #s(@y))] = [4] > [0] = [#LT()] [#compare(#s(@x), #0())] = [4] > [0] = [#GT()] [#compare(#s(@x), #s(@y))] = [4] >= [4] = [#compare(@x, @y)] [+(@x, @y)] = [4] @y + [0] >= [4] @y + [0] = [#add(@x, @y)] [*(@x, @y)] = [4] @x + [4] @y + [4] >= [4] = [#mult(@x, @y)] [#less(@x, @y)] = [4] >= [4] = [#cklt(#compare(@x, @y))] [#natmult(#0(), @y)] = [0] >= [0] = [#0()] [#natmult(#s(@x), @y)] = [0] >= [0] = [#add(#pos(@y), #natmult(@x, @y))] [#cklt(#EQ())] = [4] > [2] = [#false()] [#cklt(#LT())] = [4] > [0] = [#true()] [#cklt(#GT())] = [4] > [2] = [#false()] [#sub(@x, #neg(@y))] = [1] @x + [0] >= [0] = [#add(@x, #pos(@y))] [#sub(@x, #pos(@y))] = [1] @x + [0] >= [0] = [#add(@x, #neg(@y))] [#sub(@x, #0())] = [1] @x + [0] >= [1] @x + [0] = [@x] [diff(@x, @y, @r)] = [4] @x + [4] @y + [0] >= [0] = [tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0())))] [#mult(#neg(@x), #neg(@y))] = [4] > [0] = [#pos(#natmult(@x, @y))] [#mult(#neg(@x), #pos(@y))] = [4] > [0] = [#neg(#natmult(@x, @y))] [#mult(#neg(@x), #0())] = [4] > [0] = [#0()] [#mult(#pos(@x), #neg(@y))] = [4] > [0] = [#neg(#natmult(@x, @y))] [#mult(#pos(@x), #pos(@y))] = [4] > [0] = [#pos(#natmult(@x, @y))] [#mult(#pos(@x), #0())] = [4] > [0] = [#0()] [#mult(#0(), #neg(@y))] = [4] > [0] = [#0()] [#mult(#0(), #pos(@y))] = [4] > [0] = [#0()] [#mult(#0(), #0())] = [4] > [0] = [#0()] [#succ(#neg(#s(#0())))] = [0] >= [0] = [#0()] [#succ(#neg(#s(#s(@x))))] = [0] >= [0] = [#neg(#s(@x))] [#succ(#pos(#s(@x)))] = [0] >= [0] = [#pos(#s(#s(@x)))] [#succ(#0())] = [0] >= [0] = [#pos(#s(#0()))] [#div(#neg(@x), #neg(@y))] = [0] >= [0] = [#pos(#natdiv(@x, @y))] [#div(#neg(@x), #pos(@y))] = [0] >= [0] = [#neg(#natdiv(@x, @y))] [#div(#neg(@x), #0())] = [0] >= [0] = [#divByZero()] [#div(#pos(@x), #neg(@y))] = [0] >= [0] = [#neg(#natdiv(@x, @y))] [#div(#pos(@x), #pos(@y))] = [0] >= [0] = [#pos(#natdiv(@x, @y))] [#div(#pos(@x), #0())] = [0] >= [0] = [#divByZero()] [#div(#0(), #neg(@y))] = [0] >= [0] = [#0()] [#div(#0(), #pos(@y))] = [0] >= [0] = [#0()] [#div(#0(), #0())] = [0] >= [0] = [#divByZero()] [#abs(#neg(@x))] = [0] >= [0] = [#pos(@x)] [#abs(#pos(@x))] = [0] >= [0] = [#pos(@x)] [#abs(#0())] = [0] >= [0] = [#0()] [#abs(#s(@x))] = [0] >= [0] = [#pos(#s(@x))] [#pred(#neg(#s(@x)))] = [0] >= [0] = [#neg(#s(#s(@x)))] [#pred(#pos(#s(#0())))] = [0] >= [0] = [#0()] [#pred(#pos(#s(#s(@x))))] = [0] >= [0] = [#pos(#s(@x))] [#pred(#0())] = [0] >= [0] = [#neg(#s(#0()))] [sub^#(@b1, @b2)] = [7] @b1 + [7] @b2 + [7] >= [4] @b1 + [7] = [c_1(sub'^#(@b1, @b2, #abs(#0())))] [sub'^#(@b1, @b2, @r)] = [1] @b1 + [0] >= [1] @b1 + [0] = [c_2(sub'#1^#(@b1, @b2, @r))] [sub'#3^#(tuple#2(@z, @r'), @xs, @ys)] = [1] @xs + [0] >= [1] @xs + [0] = [c_13(sub'^#(@xs, @ys, @r'))] [sub'#2^#(::(@y, @ys), @r, @x, @xs)] = [1] @xs + [0] >= [1] @xs + [0] = [c_14(sub'#3^#(diff(@x, @y, @r), @xs, @ys))] [sub'#1^#(::(@x, @xs), @b2, @r)] = [1] @x + [1] @xs + [4] > [1] @xs + [1] = [c_19(sub'#2^#(@b2, @r, @x, @xs))] We return to the main proof. Consider the set of all dependency pairs : { 1: sub^#(@b1, @b2) -> c_1(sub'^#(@b1, @b2, #abs(#0()))) , 2: sub'^#(@b1, @b2, @r) -> c_2(sub'#1^#(@b1, @b2, @r)) , 3: sub'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_13(sub'^#(@xs, @ys, @r')) , 4: sub'#2^#(::(@y, @ys), @r, @x, @xs) -> c_14(sub'#3^#(diff(@x, @y, @r), @xs, @ys)) , 5: sub'#1^#(::(@x, @xs), @b2, @r) -> c_19(sub'#2^#(@b2, @r, @x, @xs)) } Processor 'matrix interpretation of dimension 1' induces the complexity certificate YES(?,O(n^1)) on application of dependency pairs {5}. These cover all (indirect) predecessors of dependency pairs {1,2,3,4,5}, their number of application is equally bounded. The dependency pairs are shifted into the weak component. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak DPs: { sub^#(@b1, @b2) -> c_1(sub'^#(@b1, @b2, #abs(#0()))) , sub'^#(@b1, @b2, @r) -> c_2(sub'#1^#(@b1, @b2, @r)) , sub'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_13(sub'^#(@xs, @ys, @r')) , sub'#2^#(::(@y, @ys), @r, @x, @xs) -> c_14(sub'#3^#(diff(@x, @y, @r), @xs, @ys)) , sub'#1^#(::(@x, @xs), @b2, @r) -> c_19(sub'#2^#(@b2, @r, @x, @xs)) } Weak Trs: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , div(@x, @y) -> #div(@x, @y) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , +(@x, @y) -> #add(@x, @y) , *(@x, @y) -> #mult(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { sub^#(@b1, @b2) -> c_1(sub'^#(@b1, @b2, #abs(#0()))) , sub'^#(@b1, @b2, @r) -> c_2(sub'#1^#(@b1, @b2, @r)) , sub'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_13(sub'^#(@xs, @ys, @r')) , sub'#2^#(::(@y, @ys), @r, @x, @xs) -> c_14(sub'#3^#(diff(@x, @y, @r), @xs, @ys)) , sub'#1^#(::(@x, @xs), @b2, @r) -> c_19(sub'#2^#(@b2, @r, @x, @xs)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak Trs: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , div(@x, @y) -> #div(@x, @y) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , +(@x, @y) -> #add(@x, @y) , *(@x, @y) -> #mult(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) No rule is usable, rules are removed from the input problem. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Rules: Empty Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) Empty rules are trivially bounded S) We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { mult#2^#(@zs, @b2, @x) -> c_3(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs)) , mult#3^#(#true(), @b2, @zs) -> c_4(add^#(@b2, @zs)) , bitToInt'#1^#(::(@x, @xs), @n) -> c_5(bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0())))))) , bitToInt'^#(@b, @n) -> c_6(bitToInt'#1^#(@b, @n)) , mult3^#(@b1, @b2, @b3) -> c_7(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , mult^#(@b1, @b2) -> c_8(mult#1^#(@b1, @b2)) , compare^#(@b1, @b2) -> c_10(compare#1^#(@b1, @b2)) , mult#1^#(::(@x, @xs), @b2) -> c_11(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), mult^#(@xs, @b2)) , bitToInt^#(@b) -> c_12(bitToInt'^#(@b, #abs(#pos(#s(#0()))))) , compare#2^#(::(@y, @ys), @x, @xs) -> c_15(compare^#(@xs, @ys)) , add^#(@b1, @b2) -> c_16(add'^#(@b1, @b2, #abs(#0()))) , add'^#(@b1, @b2, @r) -> c_17(add'#1^#(@b1, @b2, @r)) , add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_18(add'^#(@xs, @ys, @r')) , add'#1^#(::(@x, @xs), @b2, @r) -> c_20(add'#2^#(@b2, @r, @x, @xs)) , add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_21(add'#3^#(sum(@x, @y, @r), @xs, @ys)) , compare#1^#(::(@x, @xs), @b2) -> c_22(compare#2^#(@b2, @x, @xs)) } Weak DPs: { sub^#(@b1, @b2) -> c_1(sub'^#(@b1, @b2, #abs(#0()))) , sub'^#(@b1, @b2, @r) -> c_2(sub'#1^#(@b1, @b2, @r)) , sub'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_13(sub'^#(@xs, @ys, @r')) , sub'#2^#(::(@y, @ys), @r, @x, @xs) -> c_14(sub'#3^#(diff(@x, @y, @r), @xs, @ys)) , sub'#1^#(::(@x, @xs), @b2, @r) -> c_19(sub'#2^#(@b2, @r, @x, @xs)) } Weak Trs: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , div(@x, @y) -> #div(@x, @y) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , mult(@b1, @b2) -> mult#1(@b1, @b2) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , *(@x, @y) -> #mult(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { sub^#(@b1, @b2) -> c_1(sub'^#(@b1, @b2, #abs(#0()))) , sub'^#(@b1, @b2, @r) -> c_2(sub'#1^#(@b1, @b2, @r)) , sub'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_13(sub'^#(@xs, @ys, @r')) , sub'#2^#(::(@y, @ys), @r, @x, @xs) -> c_14(sub'#3^#(diff(@x, @y, @r), @xs, @ys)) , sub'#1^#(::(@x, @xs), @b2, @r) -> c_19(sub'#2^#(@b2, @r, @x, @xs)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { mult#2^#(@zs, @b2, @x) -> c_3(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs)) , mult#3^#(#true(), @b2, @zs) -> c_4(add^#(@b2, @zs)) , bitToInt'#1^#(::(@x, @xs), @n) -> c_5(bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0())))))) , bitToInt'^#(@b, @n) -> c_6(bitToInt'#1^#(@b, @n)) , mult3^#(@b1, @b2, @b3) -> c_7(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , mult^#(@b1, @b2) -> c_8(mult#1^#(@b1, @b2)) , compare^#(@b1, @b2) -> c_10(compare#1^#(@b1, @b2)) , mult#1^#(::(@x, @xs), @b2) -> c_11(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), mult^#(@xs, @b2)) , bitToInt^#(@b) -> c_12(bitToInt'^#(@b, #abs(#pos(#s(#0()))))) , compare#2^#(::(@y, @ys), @x, @xs) -> c_15(compare^#(@xs, @ys)) , add^#(@b1, @b2) -> c_16(add'^#(@b1, @b2, #abs(#0()))) , add'^#(@b1, @b2, @r) -> c_17(add'#1^#(@b1, @b2, @r)) , add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_18(add'^#(@xs, @ys, @r')) , add'#1^#(::(@x, @xs), @b2, @r) -> c_20(add'#2^#(@b2, @r, @x, @xs)) , add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_21(add'#3^#(sum(@x, @y, @r), @xs, @ys)) , compare#1^#(::(@x, @xs), @b2) -> c_22(compare#2^#(@b2, @x, @xs)) } Weak Trs: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , div(@x, @y) -> #div(@x, @y) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , mult(@b1, @b2) -> mult#1(@b1, @b2) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , *(@x, @y) -> #mult(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component: Problem (R): ------------ Strict DPs: { mult#2^#(@zs, @b2, @x) -> c_3(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs)) , mult#3^#(#true(), @b2, @zs) -> c_4(add^#(@b2, @zs)) , mult3^#(@b1, @b2, @b3) -> c_7(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , mult^#(@b1, @b2) -> c_8(mult#1^#(@b1, @b2)) , mult#1^#(::(@x, @xs), @b2) -> c_11(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), mult^#(@xs, @b2)) , add^#(@b1, @b2) -> c_16(add'^#(@b1, @b2, #abs(#0()))) , add'^#(@b1, @b2, @r) -> c_17(add'#1^#(@b1, @b2, @r)) , add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_18(add'^#(@xs, @ys, @r')) , add'#1^#(::(@x, @xs), @b2, @r) -> c_20(add'#2^#(@b2, @r, @x, @xs)) , add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_21(add'#3^#(sum(@x, @y, @r), @xs, @ys)) } Weak DPs: { bitToInt'#1^#(::(@x, @xs), @n) -> c_5(bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0())))))) , bitToInt'^#(@b, @n) -> c_6(bitToInt'#1^#(@b, @n)) , compare^#(@b1, @b2) -> c_10(compare#1^#(@b1, @b2)) , bitToInt^#(@b) -> c_12(bitToInt'^#(@b, #abs(#pos(#s(#0()))))) , compare#2^#(::(@y, @ys), @x, @xs) -> c_15(compare^#(@xs, @ys)) , compare#1^#(::(@x, @xs), @b2) -> c_22(compare#2^#(@b2, @x, @xs)) } Weak Trs: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , div(@x, @y) -> #div(@x, @y) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , mult(@b1, @b2) -> mult#1(@b1, @b2) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , *(@x, @y) -> #mult(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } StartTerms: basic terms Defined Symbols: -^# #sub^# sub^# sub#1^# sub'^# #abs^# diff#1^# mult#2^# mult#3^# #equal^# div^# #div^# bitToInt'#1^# +^# *^# bitToInt'^# sum^# sum#1^# mod^# mult3^# mult^# leq^# #less^# compare^# #greater^# #ckgt^# #compare^# mult#1^# bitToInt^# sum#2^# sum#3^# #add^# sum#4^# #mult^# sub'#5^# #cklt^# sub'#3^# sub'#4^# #eq^# sub'#2^# diff^# compare#2^# compare#3^# compare#5^# compare#6^# compare#4^# add^# add'^# add'#3^# sub'#1^# add'#1^# add'#2^# compare#1^# #natsub^# #natdiv^# #pred^# #succ^# #and^# #natmult^# Constructors: #EQ nil #neg #divByZero #true #pos tuple#2 #false :: #LT #0 #s #GT Strategy: innermost Problem (S): ------------ Strict DPs: { bitToInt'#1^#(::(@x, @xs), @n) -> c_5(bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0())))))) , bitToInt'^#(@b, @n) -> c_6(bitToInt'#1^#(@b, @n)) , compare^#(@b1, @b2) -> c_10(compare#1^#(@b1, @b2)) , bitToInt^#(@b) -> c_12(bitToInt'^#(@b, #abs(#pos(#s(#0()))))) , compare#2^#(::(@y, @ys), @x, @xs) -> c_15(compare^#(@xs, @ys)) , compare#1^#(::(@x, @xs), @b2) -> c_22(compare#2^#(@b2, @x, @xs)) } Weak DPs: { mult#2^#(@zs, @b2, @x) -> c_3(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs)) , mult#3^#(#true(), @b2, @zs) -> c_4(add^#(@b2, @zs)) , mult3^#(@b1, @b2, @b3) -> c_7(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , mult^#(@b1, @b2) -> c_8(mult#1^#(@b1, @b2)) , mult#1^#(::(@x, @xs), @b2) -> c_11(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), mult^#(@xs, @b2)) , add^#(@b1, @b2) -> c_16(add'^#(@b1, @b2, #abs(#0()))) , add'^#(@b1, @b2, @r) -> c_17(add'#1^#(@b1, @b2, @r)) , add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_18(add'^#(@xs, @ys, @r')) , add'#1^#(::(@x, @xs), @b2, @r) -> c_20(add'#2^#(@b2, @r, @x, @xs)) , add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_21(add'#3^#(sum(@x, @y, @r), @xs, @ys)) } Weak Trs: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , div(@x, @y) -> #div(@x, @y) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , mult(@b1, @b2) -> mult#1(@b1, @b2) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , *(@x, @y) -> #mult(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } StartTerms: basic terms Defined Symbols: -^# #sub^# sub^# sub#1^# sub'^# #abs^# diff#1^# mult#2^# mult#3^# #equal^# div^# #div^# bitToInt'#1^# +^# *^# bitToInt'^# sum^# sum#1^# mod^# mult3^# mult^# leq^# #less^# compare^# #greater^# #ckgt^# #compare^# mult#1^# bitToInt^# sum#2^# sum#3^# #add^# sum#4^# #mult^# sub'#5^# #cklt^# sub'#3^# sub'#4^# #eq^# sub'#2^# diff^# compare#2^# compare#3^# compare#5^# compare#6^# compare#4^# add^# add'^# add'#3^# sub'#1^# add'#1^# add'#2^# compare#1^# #natsub^# #natdiv^# #pred^# #succ^# #and^# #natmult^# Constructors: #EQ nil #neg #divByZero #true #pos tuple#2 #false :: #LT #0 #s #GT Strategy: innermost Overall, the transformation results in the following sub-problem(s): Generated new problems: ----------------------- R) Strict DPs: { mult#2^#(@zs, @b2, @x) -> c_3(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs)) , mult#3^#(#true(), @b2, @zs) -> c_4(add^#(@b2, @zs)) , mult3^#(@b1, @b2, @b3) -> c_7(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , mult^#(@b1, @b2) -> c_8(mult#1^#(@b1, @b2)) , mult#1^#(::(@x, @xs), @b2) -> c_11(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), mult^#(@xs, @b2)) , add^#(@b1, @b2) -> c_16(add'^#(@b1, @b2, #abs(#0()))) , add'^#(@b1, @b2, @r) -> c_17(add'#1^#(@b1, @b2, @r)) , add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_18(add'^#(@xs, @ys, @r')) , add'#1^#(::(@x, @xs), @b2, @r) -> c_20(add'#2^#(@b2, @r, @x, @xs)) , add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_21(add'#3^#(sum(@x, @y, @r), @xs, @ys)) } Weak DPs: { bitToInt'#1^#(::(@x, @xs), @n) -> c_5(bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0())))))) , bitToInt'^#(@b, @n) -> c_6(bitToInt'#1^#(@b, @n)) , compare^#(@b1, @b2) -> c_10(compare#1^#(@b1, @b2)) , bitToInt^#(@b) -> c_12(bitToInt'^#(@b, #abs(#pos(#s(#0()))))) , compare#2^#(::(@y, @ys), @x, @xs) -> c_15(compare^#(@xs, @ys)) , compare#1^#(::(@x, @xs), @b2) -> c_22(compare#2^#(@b2, @x, @xs)) } Weak Trs: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , div(@x, @y) -> #div(@x, @y) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , mult(@b1, @b2) -> mult#1(@b1, @b2) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , *(@x, @y) -> #mult(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } StartTerms: basic terms Defined Symbols: -^# #sub^# sub^# sub#1^# sub'^# #abs^# diff#1^# mult#2^# mult#3^# #equal^# div^# #div^# bitToInt'#1^# +^# *^# bitToInt'^# sum^# sum#1^# mod^# mult3^# mult^# leq^# #less^# compare^# #greater^# #ckgt^# #compare^# mult#1^# bitToInt^# sum#2^# sum#3^# #add^# sum#4^# #mult^# sub'#5^# #cklt^# sub'#3^# sub'#4^# #eq^# sub'#2^# diff^# compare#2^# compare#3^# compare#5^# compare#6^# compare#4^# add^# add'^# add'#3^# sub'#1^# add'#1^# add'#2^# compare#1^# #natsub^# #natdiv^# #pred^# #succ^# #and^# #natmult^# Constructors: #EQ nil #neg #divByZero #true #pos tuple#2 #false :: #LT #0 #s #GT Strategy: innermost This problem was proven YES(O(1),O(n^2)). S) Strict DPs: { bitToInt'#1^#(::(@x, @xs), @n) -> c_5(bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0())))))) , bitToInt'^#(@b, @n) -> c_6(bitToInt'#1^#(@b, @n)) , compare^#(@b1, @b2) -> c_10(compare#1^#(@b1, @b2)) , bitToInt^#(@b) -> c_12(bitToInt'^#(@b, #abs(#pos(#s(#0()))))) , compare#2^#(::(@y, @ys), @x, @xs) -> c_15(compare^#(@xs, @ys)) , compare#1^#(::(@x, @xs), @b2) -> c_22(compare#2^#(@b2, @x, @xs)) } Weak DPs: { mult#2^#(@zs, @b2, @x) -> c_3(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs)) , mult#3^#(#true(), @b2, @zs) -> c_4(add^#(@b2, @zs)) , mult3^#(@b1, @b2, @b3) -> c_7(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , mult^#(@b1, @b2) -> c_8(mult#1^#(@b1, @b2)) , mult#1^#(::(@x, @xs), @b2) -> c_11(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), mult^#(@xs, @b2)) , add^#(@b1, @b2) -> c_16(add'^#(@b1, @b2, #abs(#0()))) , add'^#(@b1, @b2, @r) -> c_17(add'#1^#(@b1, @b2, @r)) , add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_18(add'^#(@xs, @ys, @r')) , add'#1^#(::(@x, @xs), @b2, @r) -> c_20(add'#2^#(@b2, @r, @x, @xs)) , add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_21(add'#3^#(sum(@x, @y, @r), @xs, @ys)) } Weak Trs: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , div(@x, @y) -> #div(@x, @y) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , mult(@b1, @b2) -> mult#1(@b1, @b2) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , *(@x, @y) -> #mult(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } StartTerms: basic terms Defined Symbols: -^# #sub^# sub^# sub#1^# sub'^# #abs^# diff#1^# mult#2^# mult#3^# #equal^# div^# #div^# bitToInt'#1^# +^# *^# bitToInt'^# sum^# sum#1^# mod^# mult3^# mult^# leq^# #less^# compare^# #greater^# #ckgt^# #compare^# mult#1^# bitToInt^# sum#2^# sum#3^# #add^# sum#4^# #mult^# sub'#5^# #cklt^# sub'#3^# sub'#4^# #eq^# sub'#2^# diff^# compare#2^# compare#3^# compare#5^# compare#6^# compare#4^# add^# add'^# add'#3^# sub'#1^# add'#1^# add'#2^# compare#1^# #natsub^# #natdiv^# #pred^# #succ^# #and^# #natmult^# Constructors: #EQ nil #neg #divByZero #true #pos tuple#2 #false :: #LT #0 #s #GT Strategy: innermost This problem was proven YES(O(1),O(n^1)). Proofs for generated problems: ------------------------------ R) We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { mult#2^#(@zs, @b2, @x) -> c_3(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs)) , mult#3^#(#true(), @b2, @zs) -> c_4(add^#(@b2, @zs)) , mult3^#(@b1, @b2, @b3) -> c_7(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , mult^#(@b1, @b2) -> c_8(mult#1^#(@b1, @b2)) , mult#1^#(::(@x, @xs), @b2) -> c_11(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), mult^#(@xs, @b2)) , add^#(@b1, @b2) -> c_16(add'^#(@b1, @b2, #abs(#0()))) , add'^#(@b1, @b2, @r) -> c_17(add'#1^#(@b1, @b2, @r)) , add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_18(add'^#(@xs, @ys, @r')) , add'#1^#(::(@x, @xs), @b2, @r) -> c_20(add'#2^#(@b2, @r, @x, @xs)) , add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_21(add'#3^#(sum(@x, @y, @r), @xs, @ys)) } Weak DPs: { bitToInt'#1^#(::(@x, @xs), @n) -> c_5(bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0())))))) , bitToInt'^#(@b, @n) -> c_6(bitToInt'#1^#(@b, @n)) , compare^#(@b1, @b2) -> c_10(compare#1^#(@b1, @b2)) , bitToInt^#(@b) -> c_12(bitToInt'^#(@b, #abs(#pos(#s(#0()))))) , compare#2^#(::(@y, @ys), @x, @xs) -> c_15(compare^#(@xs, @ys)) , compare#1^#(::(@x, @xs), @b2) -> c_22(compare#2^#(@b2, @x, @xs)) } Weak Trs: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , div(@x, @y) -> #div(@x, @y) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , mult(@b1, @b2) -> mult#1(@b1, @b2) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , *(@x, @y) -> #mult(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { bitToInt'#1^#(::(@x, @xs), @n) -> c_5(bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0())))))) , bitToInt'^#(@b, @n) -> c_6(bitToInt'#1^#(@b, @n)) , compare^#(@b1, @b2) -> c_10(compare#1^#(@b1, @b2)) , bitToInt^#(@b) -> c_12(bitToInt'^#(@b, #abs(#pos(#s(#0()))))) , compare#2^#(::(@y, @ys), @x, @xs) -> c_15(compare^#(@xs, @ys)) , compare#1^#(::(@x, @xs), @b2) -> c_22(compare#2^#(@b2, @x, @xs)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { mult#2^#(@zs, @b2, @x) -> c_3(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs)) , mult#3^#(#true(), @b2, @zs) -> c_4(add^#(@b2, @zs)) , mult3^#(@b1, @b2, @b3) -> c_7(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , mult^#(@b1, @b2) -> c_8(mult#1^#(@b1, @b2)) , mult#1^#(::(@x, @xs), @b2) -> c_11(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), mult^#(@xs, @b2)) , add^#(@b1, @b2) -> c_16(add'^#(@b1, @b2, #abs(#0()))) , add'^#(@b1, @b2, @r) -> c_17(add'#1^#(@b1, @b2, @r)) , add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_18(add'^#(@xs, @ys, @r')) , add'#1^#(::(@x, @xs), @b2, @r) -> c_20(add'#2^#(@b2, @r, @x, @xs)) , add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_21(add'#3^#(sum(@x, @y, @r), @xs, @ys)) } Weak Trs: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , div(@x, @y) -> #div(@x, @y) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , mult(@b1, @b2) -> mult#1(@b1, @b2) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , *(@x, @y) -> #mult(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We replace rewrite rules by usable rules: Weak Usable Rules: { #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , mult(@b1, @b2) -> mult#1(@b1, @b2) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { mult#2^#(@zs, @b2, @x) -> c_3(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs)) , mult#3^#(#true(), @b2, @zs) -> c_4(add^#(@b2, @zs)) , mult3^#(@b1, @b2, @b3) -> c_7(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , mult^#(@b1, @b2) -> c_8(mult#1^#(@b1, @b2)) , mult#1^#(::(@x, @xs), @b2) -> c_11(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), mult^#(@xs, @b2)) , add^#(@b1, @b2) -> c_16(add'^#(@b1, @b2, #abs(#0()))) , add'^#(@b1, @b2, @r) -> c_17(add'#1^#(@b1, @b2, @r)) , add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_18(add'^#(@xs, @ys, @r')) , add'#1^#(::(@x, @xs), @b2, @r) -> c_20(add'#2^#(@b2, @r, @x, @xs)) , add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_21(add'#3^#(sum(@x, @y, @r), @xs, @ys)) } Weak Trs: { #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , mult(@b1, @b2) -> mult#1(@b1, @b2) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We decompose the input problem according to the dependency graph into the upper component { mult3^#(@b1, @b2, @b3) -> c_7(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , mult^#(@b1, @b2) -> c_8(mult#1^#(@b1, @b2)) , mult#1^#(::(@x, @xs), @b2) -> c_11(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), mult^#(@xs, @b2)) } and lower component { mult#2^#(@zs, @b2, @x) -> c_3(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs)) , mult#3^#(#true(), @b2, @zs) -> c_4(add^#(@b2, @zs)) , add^#(@b1, @b2) -> c_16(add'^#(@b1, @b2, #abs(#0()))) , add'^#(@b1, @b2, @r) -> c_17(add'#1^#(@b1, @b2, @r)) , add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_18(add'^#(@xs, @ys, @r')) , add'#1^#(::(@x, @xs), @b2, @r) -> c_20(add'#2^#(@b2, @r, @x, @xs)) , add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_21(add'#3^#(sum(@x, @y, @r), @xs, @ys)) } Further, following extension rules are added to the lower component. { mult3^#(@b1, @b2, @b3) -> mult^#(@b1, @b2) , mult3^#(@b1, @b2, @b3) -> mult^#(mult(@b1, @b2), @b2) , mult^#(@b1, @b2) -> mult#1^#(@b1, @b2) , mult#1^#(::(@x, @xs), @b2) -> mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , mult#1^#(::(@x, @xs), @b2) -> mult^#(@xs, @b2) } TcT solves the upper component with certificate YES(O(1),O(n^1)). Sub-proof: ---------- We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { mult3^#(@b1, @b2, @b3) -> c_7(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , mult^#(@b1, @b2) -> c_8(mult#1^#(@b1, @b2)) , mult#1^#(::(@x, @xs), @b2) -> c_11(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), mult^#(@xs, @b2)) } Weak Trs: { #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , mult(@b1, @b2) -> mult#1(@b1, @b2) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 3: mult#1^#(::(@x, @xs), @b2) -> c_11(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), mult^#(@xs, @b2)) } Trs: { sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #abs(#s(@x)) -> #pos(#s(@x)) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_7) = {1, 2}, Uargs(c_8) = {1}, Uargs(c_11) = {1, 2} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [#natsub](x1, x2) = [7] x1 + [7] x2 + [0] [-](x1, x2) = [7] x1 + [7] x2 + [0] [sub](x1, x2) = [7] x1 + [7] x2 + [0] [diff#1](x1) = [7] x1 + [0] [#natdiv](x1, x2) = [7] x1 + [7] x2 + [0] [#ckgt](x1) = [7] x1 + [0] [#add](x1, x2) = [4] x2 + [0] [mult#2](x1, x2, x3) = [1] x1 + [0] [div](x1, x2) = [7] x1 + [7] x2 + [0] [bitToInt'#1](x1, x2) = [7] x1 + [7] x2 + [0] [sum](x1, x2, x3) = [4] x1 + [4] x3 + [4] [#EQ] = [0] [mod](x1, x2) = [7] x1 + [7] x2 + [0] [#and](x1, x2) = [0] [mult3](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [sub#1](x1) = [7] x1 + [0] [#compare](x1, x2) = [7] x1 + [7] x2 + [0] [nil] = [0] [leq](x1, x2) = [7] x1 + [7] x2 + [0] [#greater](x1, x2) = [7] x1 + [7] x2 + [0] [bitToInt'](x1, x2) = [7] x1 + [7] x2 + [0] [mult](x1, x2) = [4] x1 + [0] [bitToInt](x1) = [7] x1 + [0] [sum#2](x1, x2) = [0] [sum#1](x1) = [0] [+](x1, x2) = [4] x2 + [0] [sum#4](x1) = [0] [*](x1, x2) = [7] x1 + [7] x2 + [0] [#neg](x1) = [0] [sub'#5](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [#less](x1, x2) = [7] x1 + [7] x2 + [0] [sub'#3](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [#equal](x1, x2) = [4] x2 + [0] [#eq](x1, x2) = [0] [#natmult](x1, x2) = [7] x1 + [7] x2 + [0] [#divByZero] = [0] [sub'#2](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0] [compare#2](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [#true] = [0] [sub'#4](x1, x2) = [7] x1 + [7] x2 + [0] [compare#5](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [compare#3](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [#cklt](x1) = [7] x1 + [0] [add](x1, x2) = [1] x2 + [0] [#sub](x1, x2) = [7] x1 + [7] x2 + [0] [#pos](x1) = [0] [add'#3](x1, x2, x3) = [1] x3 + [1] [sub'#1](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [mult#3](x1, x2, x3) = [1] x3 + [0] [add'#1](x1, x2, x3) = [1] x2 + [0] [add'#2](x1, x2, x3, x4) = [1] x1 + [0] [tuple#2](x1, x2) = [0] [diff](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [#false] = [0] [mult#1](x1, x2) = [4] x1 + [0] [::](x1, x2) = [1] x2 + [1] [#LT] = [0] [#mult](x1, x2) = [7] x1 + [7] x2 + [0] [#succ](x1) = [0] [sub'](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [compare](x1, x2) = [7] x1 + [7] x2 + [0] [compare#6](x1) = [7] x1 + [0] [compare#4](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0] [#0] = [0] [sum#3](x1, x2) = [0] [#div](x1, x2) = [7] x1 + [7] x2 + [0] [add'](x1, x2, x3) = [1] x2 + [0] [compare#1](x1, x2) = [7] x1 + [7] x2 + [0] [#abs](x1) = [4] x1 + [0] [#pred](x1) = [0] [#s](x1) = [2] [#GT] = [0] [-^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_1](x1) = [7] x1 + [0] [#sub^#](x1, x2) = [7] x1 + [7] x2 + [0] [sub^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_2](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [sub#1^#](x1) = [7] x1 + [0] [sub'^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [#abs^#](x1) = [7] x1 + [0] [diff#1^#](x1) = [7] x1 + [0] [c_3](x1) = [7] x1 + [0] [c_4](x1) = [7] x1 + [0] [mult#2^#](x1, x2, x3) = [0] [c_5](x1, x2) = [7] x1 + [7] x2 + [0] [mult#3^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [#equal^#](x1, x2) = [7] x1 + [7] x2 + [0] [div^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_6](x1) = [7] x1 + [0] [#div^#](x1, x2) = [7] x1 + [7] x2 + [0] [bitToInt'#1^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_7](x1) = [7] x1 + [0] [c_8](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0] [+^#](x1, x2) = [7] x1 + [7] x2 + [0] [*^#](x1, x2) = [7] x1 + [7] x2 + [0] [bitToInt'^#](x1, x2) = [7] x1 + [7] x2 + [0] [sum^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_9](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [sum#1^#](x1) = [7] x1 + [0] [mod^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_10](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [mult3^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [7] [c_11](x1, x2) = [7] x1 + [7] x2 + [0] [mult^#](x1, x2) = [1] x1 + [0] [c_12] = [0] [leq^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_13](x1, x2) = [7] x1 + [7] x2 + [0] [#less^#](x1, x2) = [7] x1 + [7] x2 + [0] [compare^#](x1, x2) = [7] x1 + [7] x2 + [0] [#greater^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_14](x1, x2) = [7] x1 + [7] x2 + [0] [#ckgt^#](x1) = [7] x1 + [0] [#compare^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_15](x1) = [7] x1 + [0] [c_16](x1) = [7] x1 + [0] [mult#1^#](x1, x2) = [1] x1 + [0] [bitToInt^#](x1) = [7] x1 + [0] [c_17](x1, x2) = [7] x1 + [7] x2 + [0] [sum#2^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_18](x1, x2) = [7] x1 + [7] x2 + [0] [c_19](x1, x2) = [7] x1 + [7] x2 + [0] [sum#3^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_20](x1, x2) = [7] x1 + [7] x2 + [0] [c_21](x1) = [7] x1 + [0] [#add^#](x1, x2) = [7] x1 + [7] x2 + [0] [sum#4^#](x1) = [7] x1 + [0] [c_22](x1, x2) = [7] x1 + [7] x2 + [0] [c_23](x1, x2) = [7] x1 + [7] x2 + [0] [c_24](x1) = [7] x1 + [0] [#mult^#](x1, x2) = [7] x1 + [7] x2 + [0] [sub'#5^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_25](x1) = [7] x1 + [0] [c_26] = [0] [c_27](x1, x2) = [7] x1 + [7] x2 + [0] [#cklt^#](x1) = [7] x1 + [0] [sub'#3^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_28](x1, x2) = [7] x1 + [7] x2 + [0] [sub'#4^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_29](x1) = [7] x1 + [0] [#eq^#](x1, x2) = [7] x1 + [7] x2 + [0] [sub'#2^#](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0] [c_30] = [0] [c_31](x1, x2) = [7] x1 + [7] x2 + [0] [diff^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [compare#2^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_32](x1) = [7] x1 + [0] [c_33](x1, x2) = [7] x1 + [7] x2 + [0] [compare#3^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_34](x1, x2) = [7] x1 + [7] x2 + [0] [compare#5^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_35](x1) = [7] x1 + [0] [c_36](x1, x2) = [7] x1 + [7] x2 + [0] [compare#6^#](x1) = [7] x1 + [0] [c_37](x1, x2) = [7] x1 + [7] x2 + [0] [compare#4^#](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0] [add^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_38](x1, x2) = [7] x1 + [7] x2 + [0] [add'^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [add'#3^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_39](x1) = [7] x1 + [0] [sub'#1^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_40] = [0] [c_41](x1) = [7] x1 + [0] [c_42](x1) = [7] x1 + [0] [c_43] = [0] [add'#1^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_44] = [0] [c_45](x1) = [7] x1 + [0] [add'#2^#](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0] [c_46] = [0] [c_47](x1, x2) = [7] x1 + [7] x2 + [0] [c_48](x1, x2, x3, x4, x5, x6, x7) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [7] x5 + [7] x6 + [7] x7 + [0] [c_49] = [0] [c_50](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_51](x1) = [7] x1 + [0] [c_52](x1) = [7] x1 + [0] [compare#1^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_53](x1) = [7] x1 + [0] [c_54](x1) = [7] x1 + [0] [c_55](x1, x2) = [7] x1 + [7] x2 + [0] [c_56] = [0] [c_57](x1, x2) = [7] x1 + [7] x2 + [0] [c_58](x1, x2) = [7] x1 + [7] x2 + [0] [c_59](x1) = [7] x1 + [0] [c_60](x1) = [7] x1 + [0] [c_61](x1) = [7] x1 + [0] [c_62] = [0] [c_63] = [0] [c_64] = [0] [c_65] = [0] [#natsub^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_66] = [0] [c_67](x1) = [7] x1 + [0] [#natdiv^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_68] = [0] [c_69](x1, x2) = [7] x1 + [7] x2 + [0] [c_70] = [0] [c_71] = [0] [c_72] = [0] [c_73](x1) = [7] x1 + [0] [#pred^#](x1) = [7] x1 + [0] [c_74](x1, x2) = [7] x1 + [7] x2 + [0] [c_75](x1) = [7] x1 + [0] [#succ^#](x1) = [7] x1 + [0] [c_76](x1, x2) = [7] x1 + [7] x2 + [0] [c_77] = [0] [#and^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_78] = [0] [c_79] = [0] [c_80] = [0] [c_81] = [0] [c_82](x1) = [7] x1 + [0] [c_83] = [0] [c_84] = [0] [c_85] = [0] [c_86](x1) = [7] x1 + [0] [c_87] = [0] [c_88] = [0] [c_89] = [0] [c_90] = [0] [c_91] = [0] [c_92] = [0] [c_93](x1) = [7] x1 + [0] [c_94] = [0] [c_95] = [0] [c_96] = [0] [c_97](x1) = [7] x1 + [0] [c_98] = [0] [c_99] = [0] [c_100] = [0] [c_101](x1) = [7] x1 + [0] [c_102] = [0] [c_103] = [0] [c_104](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_105] = [0] [c_106] = [0] [c_107] = [0] [c_108](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_109] = [0] [c_110] = [0] [c_111] = [0] [c_112] = [0] [c_113] = [0] [c_114](x1) = [7] x1 + [0] [#natmult^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_115] = [0] [c_116](x1, x2) = [7] x1 + [7] x2 + [0] [c_117] = [0] [c_118] = [0] [c_119] = [0] [c_120](x1) = [7] x1 + [0] [c_121](x1) = [7] x1 + [0] [c_122] = [0] [c_123](x1) = [7] x1 + [0] [c_124](x1) = [7] x1 + [0] [c_125] = [0] [c_126](x1) = [7] x1 + [0] [c_127](x1) = [7] x1 + [0] [c_128] = [0] [c_129] = [0] [c_130] = [0] [c_131] = [0] [c_132] = [0] [c_133] = [0] [c_134] = [0] [c_135] = [0] [c_136](x1) = [7] x1 + [0] [c_137](x1) = [7] x1 + [0] [c_138] = [0] [c_139](x1) = [7] x1 + [0] [c_140](x1) = [7] x1 + [0] [c_141] = [0] [c_142] = [0] [c_143] = [0] [c_144] = [0] [c_145] = [0] [c_146] = [0] [c_147] = [0] [c_148] = [0] [c] = [0] [c_1](x1) = [7] x1 + [0] [c_2](x1) = [7] x1 + [0] [c_3](x1) = [7] x1 + [0] [c_4](x1) = [7] x1 + [0] [c_5](x1) = [7] x1 + [0] [c_6](x1) = [7] x1 + [0] [c_7](x1, x2) = [1] x1 + [1] x2 + [7] [c_8](x1) = [1] x1 + [0] [c_9](x1) = [7] x1 + [0] [c_10](x1) = [7] x1 + [0] [c_11](x1, x2) = [2] x1 + [1] x2 + [0] [c_12](x1) = [7] x1 + [0] [c_13](x1) = [7] x1 + [0] [c_14](x1) = [7] x1 + [0] [c_15](x1) = [7] x1 + [0] [c_16](x1) = [7] x1 + [0] [c_17](x1) = [7] x1 + [0] [c_18](x1) = [7] x1 + [0] [c_19](x1) = [7] x1 + [0] [c_20](x1) = [7] x1 + [0] [c_21](x1) = [7] x1 + [0] [c_22](x1) = [7] x1 + [0] The following symbols are considered usable {#add, mult#2, sum, #and, mult, sum#2, sum#1, +, sum#4, #equal, #eq, add, add'#3, mult#3, add'#1, add'#2, mult#1, #succ, sum#3, add', #abs, #pred, mult3^#, mult^#, mult#1^#} The order satisfies the following ordering constraints: [#add(#neg(#s(#0())), @y)] = [4] @y + [0] >= [0] = [#pred(@y)] [#add(#neg(#s(#s(@x))), @y)] = [4] @y + [0] >= [0] = [#pred(#add(#pos(#s(@x)), @y))] [#add(#pos(#s(#0())), @y)] = [4] @y + [0] >= [0] = [#succ(@y)] [#add(#pos(#s(#s(@x))), @y)] = [4] @y + [0] >= [0] = [#succ(#add(#pos(#s(@x)), @y))] [#add(#0(), @y)] = [4] @y + [0] >= [1] @y + [0] = [@y] [mult#2(@zs, @b2, @x)] = [1] @zs + [0] >= [1] @zs + [0] = [mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs)] [sum(@x, @y, @r)] = [4] @x + [4] @r + [4] > [0] = [sum#1(+(+(@x, @y), @r))] [#and(#true(), #true())] = [0] >= [0] = [#true()] [#and(#true(), #false())] = [0] >= [0] = [#false()] [#and(#false(), #true())] = [0] >= [0] = [#false()] [#and(#false(), #false())] = [0] >= [0] = [#false()] [mult(@b1, @b2)] = [4] @b1 + [0] >= [4] @b1 + [0] = [mult#1(@b1, @b2)] [sum#2(#true(), @s)] = [0] >= [0] = [tuple#2(#abs(#0()), #abs(#0()))] [sum#2(#false(), @s)] = [0] >= [0] = [sum#3(#equal(@s, #pos(#s(#0()))), @s)] [sum#1(@s)] = [0] >= [0] = [sum#2(#equal(@s, #0()), @s)] [+(@x, @y)] = [4] @y + [0] >= [4] @y + [0] = [#add(@x, @y)] [sum#4(#true())] = [0] >= [0] = [tuple#2(#abs(#0()), #abs(#pos(#s(#0()))))] [sum#4(#false())] = [0] >= [0] = [tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0()))))] [#equal(@x, @y)] = [4] @y + [0] >= [0] = [#eq(@x, @y)] [#eq(nil(), nil())] = [0] >= [0] = [#true()] [#eq(nil(), tuple#2(@y_1, @y_2))] = [0] >= [0] = [#false()] [#eq(nil(), ::(@y_1, @y_2))] = [0] >= [0] = [#false()] [#eq(#neg(@x), #neg(@y))] = [0] >= [0] = [#eq(@x, @y)] [#eq(#neg(@x), #pos(@y))] = [0] >= [0] = [#false()] [#eq(#neg(@x), #0())] = [0] >= [0] = [#false()] [#eq(#pos(@x), #neg(@y))] = [0] >= [0] = [#false()] [#eq(#pos(@x), #pos(@y))] = [0] >= [0] = [#eq(@x, @y)] [#eq(#pos(@x), #0())] = [0] >= [0] = [#false()] [#eq(tuple#2(@x_1, @x_2), nil())] = [0] >= [0] = [#false()] [#eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2))] = [0] >= [0] = [#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))] [#eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2))] = [0] >= [0] = [#false()] [#eq(::(@x_1, @x_2), nil())] = [0] >= [0] = [#false()] [#eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2))] = [0] >= [0] = [#false()] [#eq(::(@x_1, @x_2), ::(@y_1, @y_2))] = [0] >= [0] = [#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))] [#eq(#0(), #neg(@y))] = [0] >= [0] = [#false()] [#eq(#0(), #pos(@y))] = [0] >= [0] = [#false()] [#eq(#0(), #0())] = [0] >= [0] = [#true()] [#eq(#0(), #s(@y))] = [0] >= [0] = [#false()] [#eq(#s(@x), #0())] = [0] >= [0] = [#false()] [#eq(#s(@x), #s(@y))] = [0] >= [0] = [#eq(@x, @y)] [add(@b1, @b2)] = [1] @b2 + [0] >= [1] @b2 + [0] = [add'(@b1, @b2, #abs(#0()))] [add'#3(tuple#2(@z, @r'), @xs, @ys)] = [1] @ys + [1] >= [1] @ys + [1] = [::(@z, add'(@xs, @ys, @r'))] [mult#3(#true(), @b2, @zs)] = [1] @zs + [0] >= [1] @zs + [0] = [add(@b2, @zs)] [mult#3(#false(), @b2, @zs)] = [1] @zs + [0] >= [1] @zs + [0] = [@zs] [add'#1(nil(), @b2, @r)] = [1] @b2 + [0] >= [0] = [nil()] [add'#1(::(@x, @xs), @b2, @r)] = [1] @b2 + [0] >= [1] @b2 + [0] = [add'#2(@b2, @r, @x, @xs)] [add'#2(nil(), @r, @x, @xs)] = [0] >= [0] = [nil()] [add'#2(::(@y, @ys), @r, @x, @xs)] = [1] @ys + [1] >= [1] @ys + [1] = [add'#3(sum(@x, @y, @r), @xs, @ys)] [mult#1(nil(), @b2)] = [0] >= [0] = [nil()] [mult#1(::(@x, @xs), @b2)] = [4] @xs + [4] > [4] @xs + [1] = [mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x)] [#succ(#neg(#s(#0())))] = [0] >= [0] = [#0()] [#succ(#neg(#s(#s(@x))))] = [0] >= [0] = [#neg(#s(@x))] [#succ(#pos(#s(@x)))] = [0] >= [0] = [#pos(#s(#s(@x)))] [#succ(#0())] = [0] >= [0] = [#pos(#s(#0()))] [sum#3(#true(), @s)] = [0] >= [0] = [tuple#2(#abs(#pos(#s(#0()))), #abs(#0()))] [sum#3(#false(), @s)] = [0] >= [0] = [sum#4(#equal(@s, #pos(#s(#s(#0())))))] [add'(@b1, @b2, @r)] = [1] @b2 + [0] >= [1] @b2 + [0] = [add'#1(@b1, @b2, @r)] [#abs(#neg(@x))] = [0] >= [0] = [#pos(@x)] [#abs(#pos(@x))] = [0] >= [0] = [#pos(@x)] [#abs(#0())] = [0] >= [0] = [#0()] [#abs(#s(@x))] = [8] > [0] = [#pos(#s(@x))] [#pred(#neg(#s(@x)))] = [0] >= [0] = [#neg(#s(#s(@x)))] [#pred(#pos(#s(#0())))] = [0] >= [0] = [#0()] [#pred(#pos(#s(#s(@x))))] = [0] >= [0] = [#pos(#s(@x))] [#pred(#0())] = [0] >= [0] = [#neg(#s(#0()))] [mult3^#(@b1, @b2, @b3)] = [7] @b1 + [7] @b2 + [7] @b3 + [7] >= [5] @b1 + [7] = [c_7(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2))] [mult^#(@b1, @b2)] = [1] @b1 + [0] >= [1] @b1 + [0] = [c_8(mult#1^#(@b1, @b2))] [mult#1^#(::(@x, @xs), @b2)] = [1] @xs + [1] > [1] @xs + [0] = [c_11(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), mult^#(@xs, @b2))] We return to the main proof. Consider the set of all dependency pairs : { 1: mult3^#(@b1, @b2, @b3) -> c_7(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , 2: mult^#(@b1, @b2) -> c_8(mult#1^#(@b1, @b2)) , 3: mult#1^#(::(@x, @xs), @b2) -> c_11(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), mult^#(@xs, @b2)) } Processor 'matrix interpretation of dimension 1' induces the complexity certificate YES(?,O(n^1)) on application of dependency pairs {3}. These cover all (indirect) predecessors of dependency pairs {1,2,3}, their number of application is equally bounded. The dependency pairs are shifted into the weak component. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak DPs: { mult3^#(@b1, @b2, @b3) -> c_7(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , mult^#(@b1, @b2) -> c_8(mult#1^#(@b1, @b2)) , mult#1^#(::(@x, @xs), @b2) -> c_11(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), mult^#(@xs, @b2)) } Weak Trs: { #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , mult(@b1, @b2) -> mult#1(@b1, @b2) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { mult3^#(@b1, @b2, @b3) -> c_7(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , mult^#(@b1, @b2) -> c_8(mult#1^#(@b1, @b2)) , mult#1^#(::(@x, @xs), @b2) -> c_11(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), mult^#(@xs, @b2)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak Trs: { #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , mult(@b1, @b2) -> mult#1(@b1, @b2) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) No rule is usable, rules are removed from the input problem. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Rules: Empty Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) Empty rules are trivially bounded We return to the main proof. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { mult#2^#(@zs, @b2, @x) -> c_3(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs)) , mult#3^#(#true(), @b2, @zs) -> c_4(add^#(@b2, @zs)) , add^#(@b1, @b2) -> c_16(add'^#(@b1, @b2, #abs(#0()))) , add'^#(@b1, @b2, @r) -> c_17(add'#1^#(@b1, @b2, @r)) , add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_18(add'^#(@xs, @ys, @r')) , add'#1^#(::(@x, @xs), @b2, @r) -> c_20(add'#2^#(@b2, @r, @x, @xs)) , add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_21(add'#3^#(sum(@x, @y, @r), @xs, @ys)) } Weak DPs: { mult3^#(@b1, @b2, @b3) -> mult^#(@b1, @b2) , mult3^#(@b1, @b2, @b3) -> mult^#(mult(@b1, @b2), @b2) , mult^#(@b1, @b2) -> mult#1^#(@b1, @b2) , mult#1^#(::(@x, @xs), @b2) -> mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , mult#1^#(::(@x, @xs), @b2) -> mult^#(@xs, @b2) } Weak Trs: { #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , mult(@b1, @b2) -> mult#1(@b1, @b2) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 5: add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_18(add'^#(@xs, @ys, @r')) , 7: add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_21(add'#3^#(sum(@x, @y, @r), @xs, @ys)) , 8: mult3^#(@b1, @b2, @b3) -> mult^#(@b1, @b2) , 9: mult3^#(@b1, @b2, @b3) -> mult^#(mult(@b1, @b2), @b2) , 12: mult#1^#(::(@x, @xs), @b2) -> mult^#(@xs, @b2) } Trs: { sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_3) = {1}, Uargs(c_4) = {1}, Uargs(c_16) = {1}, Uargs(c_17) = {1}, Uargs(c_18) = {1}, Uargs(c_20) = {1}, Uargs(c_21) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [#natsub](x1, x2) = [7] x1 + [7] x2 + [0] [-](x1, x2) = [7] x1 + [7] x2 + [0] [sub](x1, x2) = [7] x1 + [7] x2 + [0] [diff#1](x1) = [7] x1 + [0] [#natdiv](x1, x2) = [7] x1 + [7] x2 + [0] [#ckgt](x1) = [7] x1 + [0] [#add](x1, x2) = [4] x2 + [0] [mult#2](x1, x2, x3) = [1] x1 + [0] [div](x1, x2) = [7] x1 + [7] x2 + [0] [bitToInt'#1](x1, x2) = [7] x1 + [7] x2 + [0] [sum](x1, x2, x3) = [1] [#EQ] = [0] [mod](x1, x2) = [7] x1 + [7] x2 + [0] [#and](x1, x2) = [0] [mult3](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [sub#1](x1) = [7] x1 + [0] [#compare](x1, x2) = [7] x1 + [7] x2 + [0] [nil] = [0] [leq](x1, x2) = [7] x1 + [7] x2 + [0] [#greater](x1, x2) = [7] x1 + [7] x2 + [0] [bitToInt'](x1, x2) = [7] x1 + [7] x2 + [0] [mult](x1, x2) = [1] x1 + [0] [bitToInt](x1) = [7] x1 + [0] [sum#2](x1, x2) = [0] [sum#1](x1) = [0] [+](x1, x2) = [4] x2 + [0] [sum#4](x1) = [0] [*](x1, x2) = [7] x1 + [7] x2 + [0] [#neg](x1) = [0] [sub'#5](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [#less](x1, x2) = [7] x1 + [7] x2 + [0] [sub'#3](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [#equal](x1, x2) = [0] [#eq](x1, x2) = [0] [#natmult](x1, x2) = [7] x1 + [7] x2 + [0] [#divByZero] = [0] [sub'#2](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0] [compare#2](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [#true] = [0] [sub'#4](x1, x2) = [7] x1 + [7] x2 + [0] [compare#5](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [compare#3](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [#cklt](x1) = [7] x1 + [0] [add](x1, x2) = [1] x2 + [0] [#sub](x1, x2) = [7] x1 + [7] x2 + [0] [#pos](x1) = [0] [add'#3](x1, x2, x3) = [1] x3 + [4] [sub'#1](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [mult#3](x1, x2, x3) = [1] x3 + [0] [add'#1](x1, x2, x3) = [1] x2 + [0] [add'#2](x1, x2, x3, x4) = [1] x1 + [0] [tuple#2](x1, x2) = [0] [diff](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [#false] = [0] [mult#1](x1, x2) = [1] x1 + [0] [::](x1, x2) = [1] x2 + [4] [#LT] = [0] [#mult](x1, x2) = [7] x1 + [7] x2 + [0] [#succ](x1) = [0] [sub'](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [compare](x1, x2) = [7] x1 + [7] x2 + [0] [compare#6](x1) = [7] x1 + [0] [compare#4](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0] [#0] = [0] [sum#3](x1, x2) = [0] [#div](x1, x2) = [7] x1 + [7] x2 + [0] [add'](x1, x2, x3) = [1] x2 + [0] [compare#1](x1, x2) = [7] x1 + [7] x2 + [0] [#abs](x1) = [0] [#pred](x1) = [0] [#s](x1) = [0] [#GT] = [0] [-^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_1](x1) = [7] x1 + [0] [#sub^#](x1, x2) = [7] x1 + [7] x2 + [0] [sub^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_2](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [sub#1^#](x1) = [7] x1 + [0] [sub'^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [#abs^#](x1) = [7] x1 + [0] [diff#1^#](x1) = [7] x1 + [0] [c_3](x1) = [7] x1 + [0] [c_4](x1) = [7] x1 + [0] [mult#2^#](x1, x2, x3) = [2] x1 + [0] [c_5](x1, x2) = [7] x1 + [7] x2 + [0] [mult#3^#](x1, x2, x3) = [2] x3 + [0] [#equal^#](x1, x2) = [7] x1 + [7] x2 + [0] [div^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_6](x1) = [7] x1 + [0] [#div^#](x1, x2) = [7] x1 + [7] x2 + [0] [bitToInt'#1^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_7](x1) = [7] x1 + [0] [c_8](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0] [+^#](x1, x2) = [7] x1 + [7] x2 + [0] [*^#](x1, x2) = [7] x1 + [7] x2 + [0] [bitToInt'^#](x1, x2) = [7] x1 + [7] x2 + [0] [sum^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_9](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [sum#1^#](x1) = [7] x1 + [0] [mod^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_10](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [mult3^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [7] [c_11](x1, x2) = [7] x1 + [7] x2 + [0] [mult^#](x1, x2) = [2] x1 + [7] x2 + [0] [c_12] = [0] [leq^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_13](x1, x2) = [7] x1 + [7] x2 + [0] [#less^#](x1, x2) = [7] x1 + [7] x2 + [0] [compare^#](x1, x2) = [7] x1 + [7] x2 + [0] [#greater^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_14](x1, x2) = [7] x1 + [7] x2 + [0] [#ckgt^#](x1) = [7] x1 + [0] [#compare^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_15](x1) = [7] x1 + [0] [c_16](x1) = [7] x1 + [0] [mult#1^#](x1, x2) = [2] x1 + [7] x2 + [0] [bitToInt^#](x1) = [7] x1 + [0] [c_17](x1, x2) = [7] x1 + [7] x2 + [0] [sum#2^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_18](x1, x2) = [7] x1 + [7] x2 + [0] [c_19](x1, x2) = [7] x1 + [7] x2 + [0] [sum#3^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_20](x1, x2) = [7] x1 + [7] x2 + [0] [c_21](x1) = [7] x1 + [0] [#add^#](x1, x2) = [7] x1 + [7] x2 + [0] [sum#4^#](x1) = [7] x1 + [0] [c_22](x1, x2) = [7] x1 + [7] x2 + [0] [c_23](x1, x2) = [7] x1 + [7] x2 + [0] [c_24](x1) = [7] x1 + [0] [#mult^#](x1, x2) = [7] x1 + [7] x2 + [0] [sub'#5^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_25](x1) = [7] x1 + [0] [c_26] = [0] [c_27](x1, x2) = [7] x1 + [7] x2 + [0] [#cklt^#](x1) = [7] x1 + [0] [sub'#3^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_28](x1, x2) = [7] x1 + [7] x2 + [0] [sub'#4^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_29](x1) = [7] x1 + [0] [#eq^#](x1, x2) = [7] x1 + [7] x2 + [0] [sub'#2^#](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0] [c_30] = [0] [c_31](x1, x2) = [7] x1 + [7] x2 + [0] [diff^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [compare#2^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_32](x1) = [7] x1 + [0] [c_33](x1, x2) = [7] x1 + [7] x2 + [0] [compare#3^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_34](x1, x2) = [7] x1 + [7] x2 + [0] [compare#5^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_35](x1) = [7] x1 + [0] [c_36](x1, x2) = [7] x1 + [7] x2 + [0] [compare#6^#](x1) = [7] x1 + [0] [c_37](x1, x2) = [7] x1 + [7] x2 + [0] [compare#4^#](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0] [add^#](x1, x2) = [2] x2 + [0] [c_38](x1, x2) = [7] x1 + [7] x2 + [0] [add'^#](x1, x2, x3) = [2] x2 + [0] [add'#3^#](x1, x2, x3) = [3] x1 + [2] x3 + [4] [c_39](x1) = [7] x1 + [0] [sub'#1^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_40] = [0] [c_41](x1) = [7] x1 + [0] [c_42](x1) = [7] x1 + [0] [c_43] = [0] [add'#1^#](x1, x2, x3) = [2] x2 + [0] [c_44] = [0] [c_45](x1) = [7] x1 + [0] [add'#2^#](x1, x2, x3, x4) = [2] x1 + [0] [c_46] = [0] [c_47](x1, x2) = [7] x1 + [7] x2 + [0] [c_48](x1, x2, x3, x4, x5, x6, x7) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [7] x5 + [7] x6 + [7] x7 + [0] [c_49] = [0] [c_50](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_51](x1) = [7] x1 + [0] [c_52](x1) = [7] x1 + [0] [compare#1^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_53](x1) = [7] x1 + [0] [c_54](x1) = [7] x1 + [0] [c_55](x1, x2) = [7] x1 + [7] x2 + [0] [c_56] = [0] [c_57](x1, x2) = [7] x1 + [7] x2 + [0] [c_58](x1, x2) = [7] x1 + [7] x2 + [0] [c_59](x1) = [7] x1 + [0] [c_60](x1) = [7] x1 + [0] [c_61](x1) = [7] x1 + [0] [c_62] = [0] [c_63] = [0] [c_64] = [0] [c_65] = [0] [#natsub^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_66] = [0] [c_67](x1) = [7] x1 + [0] [#natdiv^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_68] = [0] [c_69](x1, x2) = [7] x1 + [7] x2 + [0] [c_70] = [0] [c_71] = [0] [c_72] = [0] [c_73](x1) = [7] x1 + [0] [#pred^#](x1) = [7] x1 + [0] [c_74](x1, x2) = [7] x1 + [7] x2 + [0] [c_75](x1) = [7] x1 + [0] [#succ^#](x1) = [7] x1 + [0] [c_76](x1, x2) = [7] x1 + [7] x2 + [0] [c_77] = [0] [#and^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_78] = [0] [c_79] = [0] [c_80] = [0] [c_81] = [0] [c_82](x1) = [7] x1 + [0] [c_83] = [0] [c_84] = [0] [c_85] = [0] [c_86](x1) = [7] x1 + [0] [c_87] = [0] [c_88] = [0] [c_89] = [0] [c_90] = [0] [c_91] = [0] [c_92] = [0] [c_93](x1) = [7] x1 + [0] [c_94] = [0] [c_95] = [0] [c_96] = [0] [c_97](x1) = [7] x1 + [0] [c_98] = [0] [c_99] = [0] [c_100] = [0] [c_101](x1) = [7] x1 + [0] [c_102] = [0] [c_103] = [0] [c_104](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_105] = [0] [c_106] = [0] [c_107] = [0] [c_108](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_109] = [0] [c_110] = [0] [c_111] = [0] [c_112] = [0] [c_113] = [0] [c_114](x1) = [7] x1 + [0] [#natmult^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_115] = [0] [c_116](x1, x2) = [7] x1 + [7] x2 + [0] [c_117] = [0] [c_118] = [0] [c_119] = [0] [c_120](x1) = [7] x1 + [0] [c_121](x1) = [7] x1 + [0] [c_122] = [0] [c_123](x1) = [7] x1 + [0] [c_124](x1) = [7] x1 + [0] [c_125] = [0] [c_126](x1) = [7] x1 + [0] [c_127](x1) = [7] x1 + [0] [c_128] = [0] [c_129] = [0] [c_130] = [0] [c_131] = [0] [c_132] = [0] [c_133] = [0] [c_134] = [0] [c_135] = [0] [c_136](x1) = [7] x1 + [0] [c_137](x1) = [7] x1 + [0] [c_138] = [0] [c_139](x1) = [7] x1 + [0] [c_140](x1) = [7] x1 + [0] [c_141] = [0] [c_142] = [0] [c_143] = [0] [c_144] = [0] [c_145] = [0] [c_146] = [0] [c_147] = [0] [c_148] = [0] [c] = [0] [c_1](x1) = [7] x1 + [0] [c_2](x1) = [7] x1 + [0] [c_3](x1) = [1] x1 + [0] [c_4](x1) = [1] x1 + [0] [c_5](x1) = [7] x1 + [0] [c_6](x1) = [7] x1 + [0] [c_7](x1, x2) = [7] x1 + [7] x2 + [0] [c_8](x1) = [7] x1 + [0] [c_9](x1) = [7] x1 + [0] [c_10](x1) = [7] x1 + [0] [c_11](x1, x2) = [7] x1 + [7] x2 + [0] [c_12](x1) = [7] x1 + [0] [c_13](x1) = [7] x1 + [0] [c_14](x1) = [7] x1 + [0] [c_15](x1) = [7] x1 + [0] [c_16](x1) = [1] x1 + [0] [c_17](x1) = [1] x1 + [0] [c_18](x1) = [1] x1 + [3] [c_19](x1) = [7] x1 + [0] [c_20](x1) = [1] x1 + [0] [c_21](x1) = [1] x1 + [0] [c_22](x1) = [7] x1 + [0] The following symbols are considered usable {#add, mult#2, sum, #and, mult, sum#2, sum#1, +, sum#4, #equal, #eq, add, add'#3, mult#3, add'#1, add'#2, mult#1, #succ, sum#3, add', #abs, #pred, mult#2^#, mult#3^#, mult3^#, mult^#, mult#1^#, add^#, add'^#, add'#3^#, add'#1^#, add'#2^#} The order satisfies the following ordering constraints: [#add(#neg(#s(#0())), @y)] = [4] @y + [0] >= [0] = [#pred(@y)] [#add(#neg(#s(#s(@x))), @y)] = [4] @y + [0] >= [0] = [#pred(#add(#pos(#s(@x)), @y))] [#add(#pos(#s(#0())), @y)] = [4] @y + [0] >= [0] = [#succ(@y)] [#add(#pos(#s(#s(@x))), @y)] = [4] @y + [0] >= [0] = [#succ(#add(#pos(#s(@x)), @y))] [#add(#0(), @y)] = [4] @y + [0] >= [1] @y + [0] = [@y] [mult#2(@zs, @b2, @x)] = [1] @zs + [0] >= [1] @zs + [0] = [mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs)] [sum(@x, @y, @r)] = [1] > [0] = [sum#1(+(+(@x, @y), @r))] [#and(#true(), #true())] = [0] >= [0] = [#true()] [#and(#true(), #false())] = [0] >= [0] = [#false()] [#and(#false(), #true())] = [0] >= [0] = [#false()] [#and(#false(), #false())] = [0] >= [0] = [#false()] [mult(@b1, @b2)] = [1] @b1 + [0] >= [1] @b1 + [0] = [mult#1(@b1, @b2)] [sum#2(#true(), @s)] = [0] >= [0] = [tuple#2(#abs(#0()), #abs(#0()))] [sum#2(#false(), @s)] = [0] >= [0] = [sum#3(#equal(@s, #pos(#s(#0()))), @s)] [sum#1(@s)] = [0] >= [0] = [sum#2(#equal(@s, #0()), @s)] [+(@x, @y)] = [4] @y + [0] >= [4] @y + [0] = [#add(@x, @y)] [sum#4(#true())] = [0] >= [0] = [tuple#2(#abs(#0()), #abs(#pos(#s(#0()))))] [sum#4(#false())] = [0] >= [0] = [tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0()))))] [#equal(@x, @y)] = [0] >= [0] = [#eq(@x, @y)] [#eq(nil(), nil())] = [0] >= [0] = [#true()] [#eq(nil(), tuple#2(@y_1, @y_2))] = [0] >= [0] = [#false()] [#eq(nil(), ::(@y_1, @y_2))] = [0] >= [0] = [#false()] [#eq(#neg(@x), #neg(@y))] = [0] >= [0] = [#eq(@x, @y)] [#eq(#neg(@x), #pos(@y))] = [0] >= [0] = [#false()] [#eq(#neg(@x), #0())] = [0] >= [0] = [#false()] [#eq(#pos(@x), #neg(@y))] = [0] >= [0] = [#false()] [#eq(#pos(@x), #pos(@y))] = [0] >= [0] = [#eq(@x, @y)] [#eq(#pos(@x), #0())] = [0] >= [0] = [#false()] [#eq(tuple#2(@x_1, @x_2), nil())] = [0] >= [0] = [#false()] [#eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2))] = [0] >= [0] = [#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))] [#eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2))] = [0] >= [0] = [#false()] [#eq(::(@x_1, @x_2), nil())] = [0] >= [0] = [#false()] [#eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2))] = [0] >= [0] = [#false()] [#eq(::(@x_1, @x_2), ::(@y_1, @y_2))] = [0] >= [0] = [#and(#eq(@x_1, @y_1), #eq(@x_2, @y_2))] [#eq(#0(), #neg(@y))] = [0] >= [0] = [#false()] [#eq(#0(), #pos(@y))] = [0] >= [0] = [#false()] [#eq(#0(), #0())] = [0] >= [0] = [#true()] [#eq(#0(), #s(@y))] = [0] >= [0] = [#false()] [#eq(#s(@x), #0())] = [0] >= [0] = [#false()] [#eq(#s(@x), #s(@y))] = [0] >= [0] = [#eq(@x, @y)] [add(@b1, @b2)] = [1] @b2 + [0] >= [1] @b2 + [0] = [add'(@b1, @b2, #abs(#0()))] [add'#3(tuple#2(@z, @r'), @xs, @ys)] = [1] @ys + [4] >= [1] @ys + [4] = [::(@z, add'(@xs, @ys, @r'))] [mult#3(#true(), @b2, @zs)] = [1] @zs + [0] >= [1] @zs + [0] = [add(@b2, @zs)] [mult#3(#false(), @b2, @zs)] = [1] @zs + [0] >= [1] @zs + [0] = [@zs] [add'#1(nil(), @b2, @r)] = [1] @b2 + [0] >= [0] = [nil()] [add'#1(::(@x, @xs), @b2, @r)] = [1] @b2 + [0] >= [1] @b2 + [0] = [add'#2(@b2, @r, @x, @xs)] [add'#2(nil(), @r, @x, @xs)] = [0] >= [0] = [nil()] [add'#2(::(@y, @ys), @r, @x, @xs)] = [1] @ys + [4] >= [1] @ys + [4] = [add'#3(sum(@x, @y, @r), @xs, @ys)] [mult#1(nil(), @b2)] = [0] >= [0] = [nil()] [mult#1(::(@x, @xs), @b2)] = [1] @xs + [4] >= [1] @xs + [4] = [mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x)] [#succ(#neg(#s(#0())))] = [0] >= [0] = [#0()] [#succ(#neg(#s(#s(@x))))] = [0] >= [0] = [#neg(#s(@x))] [#succ(#pos(#s(@x)))] = [0] >= [0] = [#pos(#s(#s(@x)))] [#succ(#0())] = [0] >= [0] = [#pos(#s(#0()))] [sum#3(#true(), @s)] = [0] >= [0] = [tuple#2(#abs(#pos(#s(#0()))), #abs(#0()))] [sum#3(#false(), @s)] = [0] >= [0] = [sum#4(#equal(@s, #pos(#s(#s(#0())))))] [add'(@b1, @b2, @r)] = [1] @b2 + [0] >= [1] @b2 + [0] = [add'#1(@b1, @b2, @r)] [#abs(#neg(@x))] = [0] >= [0] = [#pos(@x)] [#abs(#pos(@x))] = [0] >= [0] = [#pos(@x)] [#abs(#0())] = [0] >= [0] = [#0()] [#abs(#s(@x))] = [0] >= [0] = [#pos(#s(@x))] [#pred(#neg(#s(@x)))] = [0] >= [0] = [#neg(#s(#s(@x)))] [#pred(#pos(#s(#0())))] = [0] >= [0] = [#0()] [#pred(#pos(#s(#s(@x))))] = [0] >= [0] = [#pos(#s(@x))] [#pred(#0())] = [0] >= [0] = [#neg(#s(#0()))] [mult#2^#(@zs, @b2, @x)] = [2] @zs + [0] >= [2] @zs + [0] = [c_3(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs))] [mult#3^#(#true(), @b2, @zs)] = [2] @zs + [0] >= [2] @zs + [0] = [c_4(add^#(@b2, @zs))] [mult3^#(@b1, @b2, @b3)] = [7] @b1 + [7] @b2 + [7] @b3 + [7] > [2] @b1 + [7] @b2 + [0] = [mult^#(@b1, @b2)] [mult3^#(@b1, @b2, @b3)] = [7] @b1 + [7] @b2 + [7] @b3 + [7] > [2] @b1 + [7] @b2 + [0] = [mult^#(mult(@b1, @b2), @b2)] [mult^#(@b1, @b2)] = [2] @b1 + [7] @b2 + [0] >= [2] @b1 + [7] @b2 + [0] = [mult#1^#(@b1, @b2)] [mult#1^#(::(@x, @xs), @b2)] = [7] @b2 + [2] @xs + [8] >= [2] @xs + [8] = [mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x)] [mult#1^#(::(@x, @xs), @b2)] = [7] @b2 + [2] @xs + [8] > [7] @b2 + [2] @xs + [0] = [mult^#(@xs, @b2)] [add^#(@b1, @b2)] = [2] @b2 + [0] >= [2] @b2 + [0] = [c_16(add'^#(@b1, @b2, #abs(#0())))] [add'^#(@b1, @b2, @r)] = [2] @b2 + [0] >= [2] @b2 + [0] = [c_17(add'#1^#(@b1, @b2, @r))] [add'#3^#(tuple#2(@z, @r'), @xs, @ys)] = [2] @ys + [4] > [2] @ys + [3] = [c_18(add'^#(@xs, @ys, @r'))] [add'#1^#(::(@x, @xs), @b2, @r)] = [2] @b2 + [0] >= [2] @b2 + [0] = [c_20(add'#2^#(@b2, @r, @x, @xs))] [add'#2^#(::(@y, @ys), @r, @x, @xs)] = [2] @ys + [8] > [2] @ys + [7] = [c_21(add'#3^#(sum(@x, @y, @r), @xs, @ys))] We return to the main proof. Consider the set of all dependency pairs : { 1: mult#2^#(@zs, @b2, @x) -> c_3(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs)) , 2: mult#3^#(#true(), @b2, @zs) -> c_4(add^#(@b2, @zs)) , 3: add^#(@b1, @b2) -> c_16(add'^#(@b1, @b2, #abs(#0()))) , 4: add'^#(@b1, @b2, @r) -> c_17(add'#1^#(@b1, @b2, @r)) , 5: add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_18(add'^#(@xs, @ys, @r')) , 6: add'#1^#(::(@x, @xs), @b2, @r) -> c_20(add'#2^#(@b2, @r, @x, @xs)) , 7: add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_21(add'#3^#(sum(@x, @y, @r), @xs, @ys)) , 8: mult3^#(@b1, @b2, @b3) -> mult^#(@b1, @b2) , 9: mult3^#(@b1, @b2, @b3) -> mult^#(mult(@b1, @b2), @b2) , 10: mult^#(@b1, @b2) -> mult#1^#(@b1, @b2) , 11: mult#1^#(::(@x, @xs), @b2) -> mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , 12: mult#1^#(::(@x, @xs), @b2) -> mult^#(@xs, @b2) } Processor 'matrix interpretation of dimension 1' induces the complexity certificate YES(?,O(n^1)) on application of dependency pairs {5,7,8,9,12}. These cover all (indirect) predecessors of dependency pairs {1,2,3,4,5,6,7,8,9,10,11,12}, their number of application is equally bounded. The dependency pairs are shifted into the weak component. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak DPs: { mult#2^#(@zs, @b2, @x) -> c_3(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs)) , mult#3^#(#true(), @b2, @zs) -> c_4(add^#(@b2, @zs)) , mult3^#(@b1, @b2, @b3) -> mult^#(@b1, @b2) , mult3^#(@b1, @b2, @b3) -> mult^#(mult(@b1, @b2), @b2) , mult^#(@b1, @b2) -> mult#1^#(@b1, @b2) , mult#1^#(::(@x, @xs), @b2) -> mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , mult#1^#(::(@x, @xs), @b2) -> mult^#(@xs, @b2) , add^#(@b1, @b2) -> c_16(add'^#(@b1, @b2, #abs(#0()))) , add'^#(@b1, @b2, @r) -> c_17(add'#1^#(@b1, @b2, @r)) , add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_18(add'^#(@xs, @ys, @r')) , add'#1^#(::(@x, @xs), @b2, @r) -> c_20(add'#2^#(@b2, @r, @x, @xs)) , add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_21(add'#3^#(sum(@x, @y, @r), @xs, @ys)) } Weak Trs: { #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , mult(@b1, @b2) -> mult#1(@b1, @b2) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { mult#2^#(@zs, @b2, @x) -> c_3(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs)) , mult#3^#(#true(), @b2, @zs) -> c_4(add^#(@b2, @zs)) , mult3^#(@b1, @b2, @b3) -> mult^#(@b1, @b2) , mult3^#(@b1, @b2, @b3) -> mult^#(mult(@b1, @b2), @b2) , mult^#(@b1, @b2) -> mult#1^#(@b1, @b2) , mult#1^#(::(@x, @xs), @b2) -> mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , mult#1^#(::(@x, @xs), @b2) -> mult^#(@xs, @b2) , add^#(@b1, @b2) -> c_16(add'^#(@b1, @b2, #abs(#0()))) , add'^#(@b1, @b2, @r) -> c_17(add'#1^#(@b1, @b2, @r)) , add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_18(add'^#(@xs, @ys, @r')) , add'#1^#(::(@x, @xs), @b2, @r) -> c_20(add'#2^#(@b2, @r, @x, @xs)) , add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_21(add'#3^#(sum(@x, @y, @r), @xs, @ys)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak Trs: { #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , mult(@b1, @b2) -> mult#1(@b1, @b2) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) No rule is usable, rules are removed from the input problem. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Rules: Empty Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) Empty rules are trivially bounded S) We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { bitToInt'#1^#(::(@x, @xs), @n) -> c_5(bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0())))))) , bitToInt'^#(@b, @n) -> c_6(bitToInt'#1^#(@b, @n)) , compare^#(@b1, @b2) -> c_10(compare#1^#(@b1, @b2)) , bitToInt^#(@b) -> c_12(bitToInt'^#(@b, #abs(#pos(#s(#0()))))) , compare#2^#(::(@y, @ys), @x, @xs) -> c_15(compare^#(@xs, @ys)) , compare#1^#(::(@x, @xs), @b2) -> c_22(compare#2^#(@b2, @x, @xs)) } Weak DPs: { mult#2^#(@zs, @b2, @x) -> c_3(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs)) , mult#3^#(#true(), @b2, @zs) -> c_4(add^#(@b2, @zs)) , mult3^#(@b1, @b2, @b3) -> c_7(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , mult^#(@b1, @b2) -> c_8(mult#1^#(@b1, @b2)) , mult#1^#(::(@x, @xs), @b2) -> c_11(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), mult^#(@xs, @b2)) , add^#(@b1, @b2) -> c_16(add'^#(@b1, @b2, #abs(#0()))) , add'^#(@b1, @b2, @r) -> c_17(add'#1^#(@b1, @b2, @r)) , add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_18(add'^#(@xs, @ys, @r')) , add'#1^#(::(@x, @xs), @b2, @r) -> c_20(add'#2^#(@b2, @r, @x, @xs)) , add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_21(add'#3^#(sum(@x, @y, @r), @xs, @ys)) } Weak Trs: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , div(@x, @y) -> #div(@x, @y) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , mult(@b1, @b2) -> mult#1(@b1, @b2) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , *(@x, @y) -> #mult(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { mult#2^#(@zs, @b2, @x) -> c_3(mult#3^#(#equal(@x, #pos(#s(#0()))), @b2, @zs)) , mult#3^#(#true(), @b2, @zs) -> c_4(add^#(@b2, @zs)) , mult3^#(@b1, @b2, @b3) -> c_7(mult^#(mult(@b1, @b2), @b2), mult^#(@b1, @b2)) , mult^#(@b1, @b2) -> c_8(mult#1^#(@b1, @b2)) , mult#1^#(::(@x, @xs), @b2) -> c_11(mult#2^#(::(#abs(#0()), mult(@xs, @b2)), @b2, @x), mult^#(@xs, @b2)) , add^#(@b1, @b2) -> c_16(add'^#(@b1, @b2, #abs(#0()))) , add'^#(@b1, @b2, @r) -> c_17(add'#1^#(@b1, @b2, @r)) , add'#3^#(tuple#2(@z, @r'), @xs, @ys) -> c_18(add'^#(@xs, @ys, @r')) , add'#1^#(::(@x, @xs), @b2, @r) -> c_20(add'#2^#(@b2, @r, @x, @xs)) , add'#2^#(::(@y, @ys), @r, @x, @xs) -> c_21(add'#3^#(sum(@x, @y, @r), @xs, @ys)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { bitToInt'#1^#(::(@x, @xs), @n) -> c_5(bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0())))))) , bitToInt'^#(@b, @n) -> c_6(bitToInt'#1^#(@b, @n)) , compare^#(@b1, @b2) -> c_10(compare#1^#(@b1, @b2)) , bitToInt^#(@b) -> c_12(bitToInt'^#(@b, #abs(#pos(#s(#0()))))) , compare#2^#(::(@y, @ys), @x, @xs) -> c_15(compare^#(@xs, @ys)) , compare#1^#(::(@x, @xs), @b2) -> c_22(compare#2^#(@b2, @x, @xs)) } Weak Trs: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , div(@x, @y) -> #div(@x, @y) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , mult(@b1, @b2) -> mult#1(@b1, @b2) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , *(@x, @y) -> #mult(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component: Problem (R): ------------ Strict DPs: { bitToInt'#1^#(::(@x, @xs), @n) -> c_5(bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0())))))) , bitToInt'^#(@b, @n) -> c_6(bitToInt'#1^#(@b, @n)) , bitToInt^#(@b) -> c_12(bitToInt'^#(@b, #abs(#pos(#s(#0()))))) } Weak DPs: { compare^#(@b1, @b2) -> c_10(compare#1^#(@b1, @b2)) , compare#2^#(::(@y, @ys), @x, @xs) -> c_15(compare^#(@xs, @ys)) , compare#1^#(::(@x, @xs), @b2) -> c_22(compare#2^#(@b2, @x, @xs)) } Weak Trs: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , div(@x, @y) -> #div(@x, @y) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , mult(@b1, @b2) -> mult#1(@b1, @b2) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , *(@x, @y) -> #mult(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } StartTerms: basic terms Defined Symbols: -^# #sub^# sub^# sub#1^# sub'^# #abs^# diff#1^# mult#2^# mult#3^# #equal^# div^# #div^# bitToInt'#1^# +^# *^# bitToInt'^# sum^# sum#1^# mod^# mult3^# mult^# leq^# #less^# compare^# #greater^# #ckgt^# #compare^# mult#1^# bitToInt^# sum#2^# sum#3^# #add^# sum#4^# #mult^# sub'#5^# #cklt^# sub'#3^# sub'#4^# #eq^# sub'#2^# diff^# compare#2^# compare#3^# compare#5^# compare#6^# compare#4^# add^# add'^# add'#3^# sub'#1^# add'#1^# add'#2^# compare#1^# #natsub^# #natdiv^# #pred^# #succ^# #and^# #natmult^# Constructors: #EQ nil #neg #divByZero #true #pos tuple#2 #false :: #LT #0 #s #GT Strategy: innermost Problem (S): ------------ Strict DPs: { compare^#(@b1, @b2) -> c_10(compare#1^#(@b1, @b2)) , compare#2^#(::(@y, @ys), @x, @xs) -> c_15(compare^#(@xs, @ys)) , compare#1^#(::(@x, @xs), @b2) -> c_22(compare#2^#(@b2, @x, @xs)) } Weak DPs: { bitToInt'#1^#(::(@x, @xs), @n) -> c_5(bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0())))))) , bitToInt'^#(@b, @n) -> c_6(bitToInt'#1^#(@b, @n)) , bitToInt^#(@b) -> c_12(bitToInt'^#(@b, #abs(#pos(#s(#0()))))) } Weak Trs: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , div(@x, @y) -> #div(@x, @y) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , mult(@b1, @b2) -> mult#1(@b1, @b2) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , *(@x, @y) -> #mult(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } StartTerms: basic terms Defined Symbols: -^# #sub^# sub^# sub#1^# sub'^# #abs^# diff#1^# mult#2^# mult#3^# #equal^# div^# #div^# bitToInt'#1^# +^# *^# bitToInt'^# sum^# sum#1^# mod^# mult3^# mult^# leq^# #less^# compare^# #greater^# #ckgt^# #compare^# mult#1^# bitToInt^# sum#2^# sum#3^# #add^# sum#4^# #mult^# sub'#5^# #cklt^# sub'#3^# sub'#4^# #eq^# sub'#2^# diff^# compare#2^# compare#3^# compare#5^# compare#6^# compare#4^# add^# add'^# add'#3^# sub'#1^# add'#1^# add'#2^# compare#1^# #natsub^# #natdiv^# #pred^# #succ^# #and^# #natmult^# Constructors: #EQ nil #neg #divByZero #true #pos tuple#2 #false :: #LT #0 #s #GT Strategy: innermost Overall, the transformation results in the following sub-problem(s): Generated new problems: ----------------------- R) Strict DPs: { bitToInt'#1^#(::(@x, @xs), @n) -> c_5(bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0())))))) , bitToInt'^#(@b, @n) -> c_6(bitToInt'#1^#(@b, @n)) , bitToInt^#(@b) -> c_12(bitToInt'^#(@b, #abs(#pos(#s(#0()))))) } Weak DPs: { compare^#(@b1, @b2) -> c_10(compare#1^#(@b1, @b2)) , compare#2^#(::(@y, @ys), @x, @xs) -> c_15(compare^#(@xs, @ys)) , compare#1^#(::(@x, @xs), @b2) -> c_22(compare#2^#(@b2, @x, @xs)) } Weak Trs: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , div(@x, @y) -> #div(@x, @y) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , mult(@b1, @b2) -> mult#1(@b1, @b2) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , *(@x, @y) -> #mult(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } StartTerms: basic terms Defined Symbols: -^# #sub^# sub^# sub#1^# sub'^# #abs^# diff#1^# mult#2^# mult#3^# #equal^# div^# #div^# bitToInt'#1^# +^# *^# bitToInt'^# sum^# sum#1^# mod^# mult3^# mult^# leq^# #less^# compare^# #greater^# #ckgt^# #compare^# mult#1^# bitToInt^# sum#2^# sum#3^# #add^# sum#4^# #mult^# sub'#5^# #cklt^# sub'#3^# sub'#4^# #eq^# sub'#2^# diff^# compare#2^# compare#3^# compare#5^# compare#6^# compare#4^# add^# add'^# add'#3^# sub'#1^# add'#1^# add'#2^# compare#1^# #natsub^# #natdiv^# #pred^# #succ^# #and^# #natmult^# Constructors: #EQ nil #neg #divByZero #true #pos tuple#2 #false :: #LT #0 #s #GT Strategy: innermost This problem was proven YES(O(1),O(n^1)). S) Strict DPs: { compare^#(@b1, @b2) -> c_10(compare#1^#(@b1, @b2)) , compare#2^#(::(@y, @ys), @x, @xs) -> c_15(compare^#(@xs, @ys)) , compare#1^#(::(@x, @xs), @b2) -> c_22(compare#2^#(@b2, @x, @xs)) } Weak DPs: { bitToInt'#1^#(::(@x, @xs), @n) -> c_5(bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0())))))) , bitToInt'^#(@b, @n) -> c_6(bitToInt'#1^#(@b, @n)) , bitToInt^#(@b) -> c_12(bitToInt'^#(@b, #abs(#pos(#s(#0()))))) } Weak Trs: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , div(@x, @y) -> #div(@x, @y) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , mult(@b1, @b2) -> mult#1(@b1, @b2) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , *(@x, @y) -> #mult(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } StartTerms: basic terms Defined Symbols: -^# #sub^# sub^# sub#1^# sub'^# #abs^# diff#1^# mult#2^# mult#3^# #equal^# div^# #div^# bitToInt'#1^# +^# *^# bitToInt'^# sum^# sum#1^# mod^# mult3^# mult^# leq^# #less^# compare^# #greater^# #ckgt^# #compare^# mult#1^# bitToInt^# sum#2^# sum#3^# #add^# sum#4^# #mult^# sub'#5^# #cklt^# sub'#3^# sub'#4^# #eq^# sub'#2^# diff^# compare#2^# compare#3^# compare#5^# compare#6^# compare#4^# add^# add'^# add'#3^# sub'#1^# add'#1^# add'#2^# compare#1^# #natsub^# #natdiv^# #pred^# #succ^# #and^# #natmult^# Constructors: #EQ nil #neg #divByZero #true #pos tuple#2 #false :: #LT #0 #s #GT Strategy: innermost This problem was proven YES(O(1),O(n^1)). Proofs for generated problems: ------------------------------ R) We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { bitToInt'#1^#(::(@x, @xs), @n) -> c_5(bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0())))))) , bitToInt'^#(@b, @n) -> c_6(bitToInt'#1^#(@b, @n)) , bitToInt^#(@b) -> c_12(bitToInt'^#(@b, #abs(#pos(#s(#0()))))) } Weak DPs: { compare^#(@b1, @b2) -> c_10(compare#1^#(@b1, @b2)) , compare#2^#(::(@y, @ys), @x, @xs) -> c_15(compare^#(@xs, @ys)) , compare#1^#(::(@x, @xs), @b2) -> c_22(compare#2^#(@b2, @x, @xs)) } Weak Trs: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , div(@x, @y) -> #div(@x, @y) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , mult(@b1, @b2) -> mult#1(@b1, @b2) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , *(@x, @y) -> #mult(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { compare^#(@b1, @b2) -> c_10(compare#1^#(@b1, @b2)) , compare#2^#(::(@y, @ys), @x, @xs) -> c_15(compare^#(@xs, @ys)) , compare#1^#(::(@x, @xs), @b2) -> c_22(compare#2^#(@b2, @x, @xs)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { bitToInt'#1^#(::(@x, @xs), @n) -> c_5(bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0())))))) , bitToInt'^#(@b, @n) -> c_6(bitToInt'#1^#(@b, @n)) , bitToInt^#(@b) -> c_12(bitToInt'^#(@b, #abs(#pos(#s(#0()))))) } Weak Trs: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , div(@x, @y) -> #div(@x, @y) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , mult(@b1, @b2) -> mult#1(@b1, @b2) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , *(@x, @y) -> #mult(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We replace rewrite rules by usable rules: Weak Usable Rules: { #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , *(@x, @y) -> #mult(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { bitToInt'#1^#(::(@x, @xs), @n) -> c_5(bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0())))))) , bitToInt'^#(@b, @n) -> c_6(bitToInt'#1^#(@b, @n)) , bitToInt^#(@b) -> c_12(bitToInt'^#(@b, #abs(#pos(#s(#0()))))) } Weak Trs: { #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , *(@x, @y) -> #mult(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 1: bitToInt'#1^#(::(@x, @xs), @n) -> c_5(bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0())))))) , 2: bitToInt'^#(@b, @n) -> c_6(bitToInt'#1^#(@b, @n)) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_5) = {1}, Uargs(c_6) = {1}, Uargs(c_12) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [#natsub](x1, x2) = [7] x1 + [7] x2 + [0] [-](x1, x2) = [7] x1 + [7] x2 + [0] [sub](x1, x2) = [7] x1 + [7] x2 + [0] [diff#1](x1) = [7] x1 + [0] [#natdiv](x1, x2) = [7] x1 + [7] x2 + [0] [#ckgt](x1) = [7] x1 + [0] [#add](x1, x2) = [4] x2 + [0] [mult#2](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [div](x1, x2) = [7] x1 + [7] x2 + [0] [bitToInt'#1](x1, x2) = [7] x1 + [7] x2 + [0] [sum](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [#EQ] = [0] [mod](x1, x2) = [7] x1 + [7] x2 + [0] [#and](x1, x2) = [7] x1 + [7] x2 + [0] [mult3](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [sub#1](x1) = [7] x1 + [0] [#compare](x1, x2) = [7] x1 + [7] x2 + [0] [nil] = [0] [leq](x1, x2) = [7] x1 + [7] x2 + [0] [#greater](x1, x2) = [7] x1 + [7] x2 + [0] [bitToInt'](x1, x2) = [7] x1 + [7] x2 + [0] [mult](x1, x2) = [7] x1 + [7] x2 + [0] [bitToInt](x1) = [7] x1 + [0] [sum#2](x1, x2) = [7] x1 + [7] x2 + [0] [sum#1](x1) = [7] x1 + [0] [+](x1, x2) = [7] x1 + [7] x2 + [0] [sum#4](x1) = [7] x1 + [0] [*](x1, x2) = [4] x1 + [4] x2 + [0] [#neg](x1) = [1] x1 + [0] [sub'#5](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [#less](x1, x2) = [7] x1 + [7] x2 + [0] [sub'#3](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [#equal](x1, x2) = [7] x1 + [7] x2 + [0] [#eq](x1, x2) = [7] x1 + [7] x2 + [0] [#natmult](x1, x2) = [0] [#divByZero] = [0] [sub'#2](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0] [compare#2](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [#true] = [0] [sub'#4](x1, x2) = [7] x1 + [7] x2 + [0] [compare#5](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [compare#3](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [#cklt](x1) = [7] x1 + [0] [add](x1, x2) = [7] x1 + [7] x2 + [0] [#sub](x1, x2) = [7] x1 + [7] x2 + [0] [#pos](x1) = [1] x1 + [0] [add'#3](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [sub'#1](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [mult#3](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [add'#1](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [add'#2](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0] [tuple#2](x1, x2) = [1] x1 + [1] x2 + [0] [diff](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [#false] = [0] [mult#1](x1, x2) = [7] x1 + [7] x2 + [0] [::](x1, x2) = [1] x1 + [1] x2 + [4] [#LT] = [0] [#mult](x1, x2) = [4] x1 + [4] x2 + [0] [#succ](x1) = [0] [sub'](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [compare](x1, x2) = [7] x1 + [7] x2 + [0] [compare#6](x1) = [7] x1 + [0] [compare#4](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0] [#0] = [0] [sum#3](x1, x2) = [7] x1 + [7] x2 + [0] [#div](x1, x2) = [7] x1 + [7] x2 + [0] [add'](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [compare#1](x1, x2) = [7] x1 + [7] x2 + [0] [#abs](x1) = [4] x1 + [0] [#pred](x1) = [0] [#s](x1) = [0] [#GT] = [0] [-^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_1](x1) = [7] x1 + [0] [#sub^#](x1, x2) = [7] x1 + [7] x2 + [0] [sub^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_2](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [sub#1^#](x1) = [7] x1 + [0] [sub'^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [#abs^#](x1) = [7] x1 + [0] [diff#1^#](x1) = [7] x1 + [0] [c_3](x1) = [7] x1 + [0] [c_4](x1) = [7] x1 + [0] [mult#2^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_5](x1, x2) = [7] x1 + [7] x2 + [0] [mult#3^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [#equal^#](x1, x2) = [7] x1 + [7] x2 + [0] [div^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_6](x1) = [7] x1 + [0] [#div^#](x1, x2) = [7] x1 + [7] x2 + [0] [bitToInt'#1^#](x1, x2) = [2] x1 + [0] [c_7](x1) = [7] x1 + [0] [c_8](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0] [+^#](x1, x2) = [7] x1 + [7] x2 + [0] [*^#](x1, x2) = [7] x1 + [7] x2 + [0] [bitToInt'^#](x1, x2) = [2] x1 + [4] [sum^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_9](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [sum#1^#](x1) = [7] x1 + [0] [mod^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_10](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [mult3^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_11](x1, x2) = [7] x1 + [7] x2 + [0] [mult^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_12] = [0] [leq^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_13](x1, x2) = [7] x1 + [7] x2 + [0] [#less^#](x1, x2) = [7] x1 + [7] x2 + [0] [compare^#](x1, x2) = [7] x1 + [7] x2 + [0] [#greater^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_14](x1, x2) = [7] x1 + [7] x2 + [0] [#ckgt^#](x1) = [7] x1 + [0] [#compare^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_15](x1) = [7] x1 + [0] [c_16](x1) = [7] x1 + [0] [mult#1^#](x1, x2) = [7] x1 + [7] x2 + [0] [bitToInt^#](x1) = [7] x1 + [7] [c_17](x1, x2) = [7] x1 + [7] x2 + [0] [sum#2^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_18](x1, x2) = [7] x1 + [7] x2 + [0] [c_19](x1, x2) = [7] x1 + [7] x2 + [0] [sum#3^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_20](x1, x2) = [7] x1 + [7] x2 + [0] [c_21](x1) = [7] x1 + [0] [#add^#](x1, x2) = [7] x1 + [7] x2 + [0] [sum#4^#](x1) = [7] x1 + [0] [c_22](x1, x2) = [7] x1 + [7] x2 + [0] [c_23](x1, x2) = [7] x1 + [7] x2 + [0] [c_24](x1) = [7] x1 + [0] [#mult^#](x1, x2) = [7] x1 + [7] x2 + [0] [sub'#5^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_25](x1) = [7] x1 + [0] [c_26] = [0] [c_27](x1, x2) = [7] x1 + [7] x2 + [0] [#cklt^#](x1) = [7] x1 + [0] [sub'#3^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_28](x1, x2) = [7] x1 + [7] x2 + [0] [sub'#4^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_29](x1) = [7] x1 + [0] [#eq^#](x1, x2) = [7] x1 + [7] x2 + [0] [sub'#2^#](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0] [c_30] = [0] [c_31](x1, x2) = [7] x1 + [7] x2 + [0] [diff^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [compare#2^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_32](x1) = [7] x1 + [0] [c_33](x1, x2) = [7] x1 + [7] x2 + [0] [compare#3^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_34](x1, x2) = [7] x1 + [7] x2 + [0] [compare#5^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_35](x1) = [7] x1 + [0] [c_36](x1, x2) = [7] x1 + [7] x2 + [0] [compare#6^#](x1) = [7] x1 + [0] [c_37](x1, x2) = [7] x1 + [7] x2 + [0] [compare#4^#](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0] [add^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_38](x1, x2) = [7] x1 + [7] x2 + [0] [add'^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [add'#3^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_39](x1) = [7] x1 + [0] [sub'#1^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_40] = [0] [c_41](x1) = [7] x1 + [0] [c_42](x1) = [7] x1 + [0] [c_43] = [0] [add'#1^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_44] = [0] [c_45](x1) = [7] x1 + [0] [add'#2^#](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0] [c_46] = [0] [c_47](x1, x2) = [7] x1 + [7] x2 + [0] [c_48](x1, x2, x3, x4, x5, x6, x7) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [7] x5 + [7] x6 + [7] x7 + [0] [c_49] = [0] [c_50](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_51](x1) = [7] x1 + [0] [c_52](x1) = [7] x1 + [0] [compare#1^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_53](x1) = [7] x1 + [0] [c_54](x1) = [7] x1 + [0] [c_55](x1, x2) = [7] x1 + [7] x2 + [0] [c_56] = [0] [c_57](x1, x2) = [7] x1 + [7] x2 + [0] [c_58](x1, x2) = [7] x1 + [7] x2 + [0] [c_59](x1) = [7] x1 + [0] [c_60](x1) = [7] x1 + [0] [c_61](x1) = [7] x1 + [0] [c_62] = [0] [c_63] = [0] [c_64] = [0] [c_65] = [0] [#natsub^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_66] = [0] [c_67](x1) = [7] x1 + [0] [#natdiv^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_68] = [0] [c_69](x1, x2) = [7] x1 + [7] x2 + [0] [c_70] = [0] [c_71] = [0] [c_72] = [0] [c_73](x1) = [7] x1 + [0] [#pred^#](x1) = [7] x1 + [0] [c_74](x1, x2) = [7] x1 + [7] x2 + [0] [c_75](x1) = [7] x1 + [0] [#succ^#](x1) = [7] x1 + [0] [c_76](x1, x2) = [7] x1 + [7] x2 + [0] [c_77] = [0] [#and^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_78] = [0] [c_79] = [0] [c_80] = [0] [c_81] = [0] [c_82](x1) = [7] x1 + [0] [c_83] = [0] [c_84] = [0] [c_85] = [0] [c_86](x1) = [7] x1 + [0] [c_87] = [0] [c_88] = [0] [c_89] = [0] [c_90] = [0] [c_91] = [0] [c_92] = [0] [c_93](x1) = [7] x1 + [0] [c_94] = [0] [c_95] = [0] [c_96] = [0] [c_97](x1) = [7] x1 + [0] [c_98] = [0] [c_99] = [0] [c_100] = [0] [c_101](x1) = [7] x1 + [0] [c_102] = [0] [c_103] = [0] [c_104](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_105] = [0] [c_106] = [0] [c_107] = [0] [c_108](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_109] = [0] [c_110] = [0] [c_111] = [0] [c_112] = [0] [c_113] = [0] [c_114](x1) = [7] x1 + [0] [#natmult^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_115] = [0] [c_116](x1, x2) = [7] x1 + [7] x2 + [0] [c_117] = [0] [c_118] = [0] [c_119] = [0] [c_120](x1) = [7] x1 + [0] [c_121](x1) = [7] x1 + [0] [c_122] = [0] [c_123](x1) = [7] x1 + [0] [c_124](x1) = [7] x1 + [0] [c_125] = [0] [c_126](x1) = [7] x1 + [0] [c_127](x1) = [7] x1 + [0] [c_128] = [0] [c_129] = [0] [c_130] = [0] [c_131] = [0] [c_132] = [0] [c_133] = [0] [c_134] = [0] [c_135] = [0] [c_136](x1) = [7] x1 + [0] [c_137](x1) = [7] x1 + [0] [c_138] = [0] [c_139](x1) = [7] x1 + [0] [c_140](x1) = [7] x1 + [0] [c_141] = [0] [c_142] = [0] [c_143] = [0] [c_144] = [0] [c_145] = [0] [c_146] = [0] [c_147] = [0] [c_148] = [0] [c] = [0] [c_1](x1) = [7] x1 + [0] [c_2](x1) = [7] x1 + [0] [c_3](x1) = [7] x1 + [0] [c_4](x1) = [7] x1 + [0] [c_5](x1) = [1] x1 + [3] [c_6](x1) = [1] x1 + [3] [c_7](x1, x2) = [7] x1 + [7] x2 + [0] [c_8](x1) = [7] x1 + [0] [c_9](x1) = [7] x1 + [0] [c_10](x1) = [7] x1 + [0] [c_11](x1, x2) = [7] x1 + [7] x2 + [0] [c_12](x1) = [1] x1 + [3] [c_13](x1) = [7] x1 + [0] [c_14](x1) = [7] x1 + [0] [c_15](x1) = [7] x1 + [0] [c_16](x1) = [7] x1 + [0] [c_17](x1) = [7] x1 + [0] [c_18](x1) = [7] x1 + [0] [c_19](x1) = [7] x1 + [0] [c_20](x1) = [7] x1 + [0] [c_21](x1) = [7] x1 + [0] [c_22](x1) = [7] x1 + [0] The following symbols are considered usable {#add, *, #natmult, #mult, #succ, #abs, #pred, bitToInt'#1^#, bitToInt'^#, bitToInt^#} The order satisfies the following ordering constraints: [#add(#neg(#s(#0())), @y)] = [4] @y + [0] >= [0] = [#pred(@y)] [#add(#neg(#s(#s(@x))), @y)] = [4] @y + [0] >= [0] = [#pred(#add(#pos(#s(@x)), @y))] [#add(#pos(#s(#0())), @y)] = [4] @y + [0] >= [0] = [#succ(@y)] [#add(#pos(#s(#s(@x))), @y)] = [4] @y + [0] >= [0] = [#succ(#add(#pos(#s(@x)), @y))] [#add(#0(), @y)] = [4] @y + [0] >= [1] @y + [0] = [@y] [*(@x, @y)] = [4] @x + [4] @y + [0] >= [4] @x + [4] @y + [0] = [#mult(@x, @y)] [#natmult(#0(), @y)] = [0] >= [0] = [#0()] [#natmult(#s(@x), @y)] = [0] >= [0] = [#add(#pos(@y), #natmult(@x, @y))] [#mult(#neg(@x), #neg(@y))] = [4] @x + [4] @y + [0] >= [0] = [#pos(#natmult(@x, @y))] [#mult(#neg(@x), #pos(@y))] = [4] @x + [4] @y + [0] >= [0] = [#neg(#natmult(@x, @y))] [#mult(#neg(@x), #0())] = [4] @x + [0] >= [0] = [#0()] [#mult(#pos(@x), #neg(@y))] = [4] @x + [4] @y + [0] >= [0] = [#neg(#natmult(@x, @y))] [#mult(#pos(@x), #pos(@y))] = [4] @x + [4] @y + [0] >= [0] = [#pos(#natmult(@x, @y))] [#mult(#pos(@x), #0())] = [4] @x + [0] >= [0] = [#0()] [#mult(#0(), #neg(@y))] = [4] @y + [0] >= [0] = [#0()] [#mult(#0(), #pos(@y))] = [4] @y + [0] >= [0] = [#0()] [#mult(#0(), #0())] = [0] >= [0] = [#0()] [#succ(#neg(#s(#0())))] = [0] >= [0] = [#0()] [#succ(#neg(#s(#s(@x))))] = [0] >= [0] = [#neg(#s(@x))] [#succ(#pos(#s(@x)))] = [0] >= [0] = [#pos(#s(#s(@x)))] [#succ(#0())] = [0] >= [0] = [#pos(#s(#0()))] [#abs(#neg(@x))] = [4] @x + [0] >= [1] @x + [0] = [#pos(@x)] [#abs(#pos(@x))] = [4] @x + [0] >= [1] @x + [0] = [#pos(@x)] [#abs(#0())] = [0] >= [0] = [#0()] [#abs(#s(@x))] = [0] >= [0] = [#pos(#s(@x))] [#pred(#neg(#s(@x)))] = [0] >= [0] = [#neg(#s(#s(@x)))] [#pred(#pos(#s(#0())))] = [0] >= [0] = [#0()] [#pred(#pos(#s(#s(@x))))] = [0] >= [0] = [#pos(#s(@x))] [#pred(#0())] = [0] >= [0] = [#neg(#s(#0()))] [bitToInt'#1^#(::(@x, @xs), @n)] = [2] @x + [2] @xs + [8] > [2] @xs + [7] = [c_5(bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0()))))))] [bitToInt'^#(@b, @n)] = [2] @b + [4] > [2] @b + [3] = [c_6(bitToInt'#1^#(@b, @n))] [bitToInt^#(@b)] = [7] @b + [7] >= [2] @b + [7] = [c_12(bitToInt'^#(@b, #abs(#pos(#s(#0())))))] We return to the main proof. Consider the set of all dependency pairs : { 1: bitToInt'#1^#(::(@x, @xs), @n) -> c_5(bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0())))))) , 2: bitToInt'^#(@b, @n) -> c_6(bitToInt'#1^#(@b, @n)) , 3: bitToInt^#(@b) -> c_12(bitToInt'^#(@b, #abs(#pos(#s(#0()))))) } Processor 'matrix interpretation of dimension 1' induces the complexity certificate YES(?,O(n^1)) on application of dependency pairs {1,2}. These cover all (indirect) predecessors of dependency pairs {1,2,3}, their number of application is equally bounded. The dependency pairs are shifted into the weak component. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak DPs: { bitToInt'#1^#(::(@x, @xs), @n) -> c_5(bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0())))))) , bitToInt'^#(@b, @n) -> c_6(bitToInt'#1^#(@b, @n)) , bitToInt^#(@b) -> c_12(bitToInt'^#(@b, #abs(#pos(#s(#0()))))) } Weak Trs: { #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , *(@x, @y) -> #mult(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { bitToInt'#1^#(::(@x, @xs), @n) -> c_5(bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0())))))) , bitToInt'^#(@b, @n) -> c_6(bitToInt'#1^#(@b, @n)) , bitToInt^#(@b) -> c_12(bitToInt'^#(@b, #abs(#pos(#s(#0()))))) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak Trs: { #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , *(@x, @y) -> #mult(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) No rule is usable, rules are removed from the input problem. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Rules: Empty Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) Empty rules are trivially bounded S) We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { compare^#(@b1, @b2) -> c_10(compare#1^#(@b1, @b2)) , compare#2^#(::(@y, @ys), @x, @xs) -> c_15(compare^#(@xs, @ys)) , compare#1^#(::(@x, @xs), @b2) -> c_22(compare#2^#(@b2, @x, @xs)) } Weak DPs: { bitToInt'#1^#(::(@x, @xs), @n) -> c_5(bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0())))))) , bitToInt'^#(@b, @n) -> c_6(bitToInt'#1^#(@b, @n)) , bitToInt^#(@b) -> c_12(bitToInt'^#(@b, #abs(#pos(#s(#0()))))) } Weak Trs: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , div(@x, @y) -> #div(@x, @y) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , mult(@b1, @b2) -> mult#1(@b1, @b2) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , *(@x, @y) -> #mult(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { bitToInt'#1^#(::(@x, @xs), @n) -> c_5(bitToInt'^#(@xs, *(@n, #pos(#s(#s(#0())))))) , bitToInt'^#(@b, @n) -> c_6(bitToInt'#1^#(@b, @n)) , bitToInt^#(@b) -> c_12(bitToInt'^#(@b, #abs(#pos(#s(#0()))))) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { compare^#(@b1, @b2) -> c_10(compare#1^#(@b1, @b2)) , compare#2^#(::(@y, @ys), @x, @xs) -> c_15(compare^#(@xs, @ys)) , compare#1^#(::(@x, @xs), @b2) -> c_22(compare#2^#(@b2, @x, @xs)) } Weak Trs: { #natsub(@x, #0()) -> @x , #natsub(#s(@x), #s(@y)) -> #natsub(@x, @y) , -(@x, @y) -> #sub(@x, @y) , diff#1(#true()) -> #abs(#pos(#s(#0()))) , diff#1(#false()) -> #abs(#0()) , #natdiv(#0(), #0()) -> #divByZero() , #natdiv(#s(@x), #s(@y)) -> #s(#natdiv(#natsub(@x, @y), #s(@y))) , #add(#neg(#s(#0())), @y) -> #pred(@y) , #add(#neg(#s(#s(@x))), @y) -> #pred(#add(#pos(#s(@x)), @y)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@x)), @y)) , #add(#0(), @y) -> @y , mult#2(@zs, @b2, @x) -> mult#3(#equal(@x, #pos(#s(#0()))), @b2, @zs) , div(@x, @y) -> #div(@x, @y) , sum(@x, @y, @r) -> sum#1(+(+(@x, @y), @r)) , mod(@x, @y) -> -(@x, *(@x, div(@x, @y))) , #and(#true(), #true()) -> #true() , #and(#true(), #false()) -> #false() , #and(#false(), #true()) -> #false() , #and(#false(), #false()) -> #false() , #compare(#neg(@x), #neg(@y)) -> #compare(@y, @x) , #compare(#neg(@x), #pos(@y)) -> #LT() , #compare(#neg(@x), #0()) -> #LT() , #compare(#pos(@x), #neg(@y)) -> #GT() , #compare(#pos(@x), #pos(@y)) -> #compare(@x, @y) , #compare(#pos(@x), #0()) -> #GT() , #compare(#0(), #neg(@y)) -> #GT() , #compare(#0(), #pos(@y)) -> #LT() , #compare(#0(), #0()) -> #EQ() , #compare(#0(), #s(@y)) -> #LT() , #compare(#s(@x), #0()) -> #GT() , #compare(#s(@x), #s(@y)) -> #compare(@x, @y) , mult(@b1, @b2) -> mult#1(@b1, @b2) , sum#2(#true(), @s) -> tuple#2(#abs(#0()), #abs(#0())) , sum#2(#false(), @s) -> sum#3(#equal(@s, #pos(#s(#0()))), @s) , sum#1(@s) -> sum#2(#equal(@s, #0()), @s) , +(@x, @y) -> #add(@x, @y) , sum#4(#true()) -> tuple#2(#abs(#0()), #abs(#pos(#s(#0())))) , sum#4(#false()) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#pos(#s(#0())))) , *(@x, @y) -> #mult(@x, @y) , #less(@x, @y) -> #cklt(#compare(@x, @y)) , #equal(@x, @y) -> #eq(@x, @y) , #eq(nil(), nil()) -> #true() , #eq(nil(), tuple#2(@y_1, @y_2)) -> #false() , #eq(nil(), ::(@y_1, @y_2)) -> #false() , #eq(#neg(@x), #neg(@y)) -> #eq(@x, @y) , #eq(#neg(@x), #pos(@y)) -> #false() , #eq(#neg(@x), #0()) -> #false() , #eq(#pos(@x), #neg(@y)) -> #false() , #eq(#pos(@x), #pos(@y)) -> #eq(@x, @y) , #eq(#pos(@x), #0()) -> #false() , #eq(tuple#2(@x_1, @x_2), nil()) -> #false() , #eq(tuple#2(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(tuple#2(@x_1, @x_2), ::(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), nil()) -> #false() , #eq(::(@x_1, @x_2), tuple#2(@y_1, @y_2)) -> #false() , #eq(::(@x_1, @x_2), ::(@y_1, @y_2)) -> #and(#eq(@x_1, @y_1), #eq(@x_2, @y_2)) , #eq(#0(), #neg(@y)) -> #false() , #eq(#0(), #pos(@y)) -> #false() , #eq(#0(), #0()) -> #true() , #eq(#0(), #s(@y)) -> #false() , #eq(#s(@x), #0()) -> #false() , #eq(#s(@x), #s(@y)) -> #eq(@x, @y) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , #cklt(#EQ()) -> #false() , #cklt(#LT()) -> #true() , #cklt(#GT()) -> #false() , add(@b1, @b2) -> add'(@b1, @b2, #abs(#0())) , #sub(@x, #neg(@y)) -> #add(@x, #pos(@y)) , #sub(@x, #pos(@y)) -> #add(@x, #neg(@y)) , #sub(@x, #0()) -> @x , add'#3(tuple#2(@z, @r'), @xs, @ys) -> ::(@z, add'(@xs, @ys, @r')) , mult#3(#true(), @b2, @zs) -> add(@b2, @zs) , mult#3(#false(), @b2, @zs) -> @zs , add'#1(nil(), @b2, @r) -> nil() , add'#1(::(@x, @xs), @b2, @r) -> add'#2(@b2, @r, @x, @xs) , add'#2(nil(), @r, @x, @xs) -> nil() , add'#2(::(@y, @ys), @r, @x, @xs) -> add'#3(sum(@x, @y, @r), @xs, @ys) , diff(@x, @y, @r) -> tuple#2(mod(+(+(@x, @y), @r), #pos(#s(#s(#0())))), diff#1(#less(-(-(@x, @y), @r), #0()))) , mult#1(nil(), @b2) -> nil() , mult#1(::(@x, @xs), @b2) -> mult#2(::(#abs(#0()), mult(@xs, @b2)), @b2, @x) , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , sum#3(#true(), @s) -> tuple#2(#abs(#pos(#s(#0()))), #abs(#0())) , sum#3(#false(), @s) -> sum#4(#equal(@s, #pos(#s(#s(#0()))))) , #div(#neg(@x), #neg(@y)) -> #pos(#natdiv(@x, @y)) , #div(#neg(@x), #pos(@y)) -> #neg(#natdiv(@x, @y)) , #div(#neg(@x), #0()) -> #divByZero() , #div(#pos(@x), #neg(@y)) -> #neg(#natdiv(@x, @y)) , #div(#pos(@x), #pos(@y)) -> #pos(#natdiv(@x, @y)) , #div(#pos(@x), #0()) -> #divByZero() , #div(#0(), #neg(@y)) -> #0() , #div(#0(), #pos(@y)) -> #0() , #div(#0(), #0()) -> #divByZero() , add'(@b1, @b2, @r) -> add'#1(@b1, @b2, @r) , #abs(#neg(@x)) -> #pos(@x) , #abs(#pos(@x)) -> #pos(@x) , #abs(#0()) -> #0() , #abs(#s(@x)) -> #pos(#s(@x)) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) No rule is usable, rules are removed from the input problem. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { compare^#(@b1, @b2) -> c_10(compare#1^#(@b1, @b2)) , compare#2^#(::(@y, @ys), @x, @xs) -> c_15(compare^#(@xs, @ys)) , compare#1^#(::(@x, @xs), @b2) -> c_22(compare#2^#(@b2, @x, @xs)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. DPs: { 2: compare#2^#(::(@y, @ys), @x, @xs) -> c_15(compare^#(@xs, @ys)) , 3: compare#1^#(::(@x, @xs), @b2) -> c_22(compare#2^#(@b2, @x, @xs)) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_10) = {1}, Uargs(c_15) = {1}, Uargs(c_22) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [#natsub](x1, x2) = [7] x1 + [7] x2 + [0] [-](x1, x2) = [7] x1 + [7] x2 + [0] [sub](x1, x2) = [7] x1 + [7] x2 + [0] [diff#1](x1) = [7] x1 + [0] [#natdiv](x1, x2) = [7] x1 + [7] x2 + [0] [#ckgt](x1) = [7] x1 + [0] [#add](x1, x2) = [7] x1 + [7] x2 + [0] [mult#2](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [div](x1, x2) = [7] x1 + [7] x2 + [0] [bitToInt'#1](x1, x2) = [7] x1 + [7] x2 + [0] [sum](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [#EQ] = [0] [mod](x1, x2) = [7] x1 + [7] x2 + [0] [#and](x1, x2) = [7] x1 + [7] x2 + [0] [mult3](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [sub#1](x1) = [7] x1 + [0] [#compare](x1, x2) = [7] x1 + [7] x2 + [0] [nil] = [0] [leq](x1, x2) = [7] x1 + [7] x2 + [0] [#greater](x1, x2) = [7] x1 + [7] x2 + [0] [bitToInt'](x1, x2) = [7] x1 + [7] x2 + [0] [mult](x1, x2) = [7] x1 + [7] x2 + [0] [bitToInt](x1) = [7] x1 + [0] [sum#2](x1, x2) = [7] x1 + [7] x2 + [0] [sum#1](x1) = [7] x1 + [0] [+](x1, x2) = [7] x1 + [7] x2 + [0] [sum#4](x1) = [7] x1 + [0] [*](x1, x2) = [7] x1 + [7] x2 + [0] [#neg](x1) = [1] x1 + [0] [sub'#5](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [#less](x1, x2) = [7] x1 + [7] x2 + [0] [sub'#3](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [#equal](x1, x2) = [7] x1 + [7] x2 + [0] [#eq](x1, x2) = [7] x1 + [7] x2 + [0] [#natmult](x1, x2) = [7] x1 + [7] x2 + [0] [#divByZero] = [0] [sub'#2](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0] [compare#2](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [#true] = [0] [sub'#4](x1, x2) = [7] x1 + [7] x2 + [0] [compare#5](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [compare#3](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [#cklt](x1) = [7] x1 + [0] [add](x1, x2) = [7] x1 + [7] x2 + [0] [#sub](x1, x2) = [7] x1 + [7] x2 + [0] [#pos](x1) = [1] x1 + [0] [add'#3](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [sub'#1](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [mult#3](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [add'#1](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [add'#2](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0] [tuple#2](x1, x2) = [1] x1 + [1] x2 + [0] [diff](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [#false] = [0] [mult#1](x1, x2) = [7] x1 + [7] x2 + [0] [::](x1, x2) = [1] x1 + [1] x2 + [2] [#LT] = [0] [#mult](x1, x2) = [7] x1 + [7] x2 + [0] [#succ](x1) = [7] x1 + [0] [sub'](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [compare](x1, x2) = [7] x1 + [7] x2 + [0] [compare#6](x1) = [7] x1 + [0] [compare#4](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0] [#0] = [0] [sum#3](x1, x2) = [7] x1 + [7] x2 + [0] [#div](x1, x2) = [7] x1 + [7] x2 + [0] [add'](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [compare#1](x1, x2) = [7] x1 + [7] x2 + [0] [#abs](x1) = [7] x1 + [0] [#pred](x1) = [7] x1 + [0] [#s](x1) = [1] x1 + [0] [#GT] = [0] [-^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_1](x1) = [7] x1 + [0] [#sub^#](x1, x2) = [7] x1 + [7] x2 + [0] [sub^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_2](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [sub#1^#](x1) = [7] x1 + [0] [sub'^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [#abs^#](x1) = [7] x1 + [0] [diff#1^#](x1) = [7] x1 + [0] [c_3](x1) = [7] x1 + [0] [c_4](x1) = [7] x1 + [0] [mult#2^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_5](x1, x2) = [7] x1 + [7] x2 + [0] [mult#3^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [#equal^#](x1, x2) = [7] x1 + [7] x2 + [0] [div^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_6](x1) = [7] x1 + [0] [#div^#](x1, x2) = [7] x1 + [7] x2 + [0] [bitToInt'#1^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_7](x1) = [7] x1 + [0] [c_8](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0] [+^#](x1, x2) = [7] x1 + [7] x2 + [0] [*^#](x1, x2) = [7] x1 + [7] x2 + [0] [bitToInt'^#](x1, x2) = [7] x1 + [7] x2 + [0] [sum^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_9](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [sum#1^#](x1) = [7] x1 + [0] [mod^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_10](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [mult3^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_11](x1, x2) = [7] x1 + [7] x2 + [0] [mult^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_12] = [0] [leq^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_13](x1, x2) = [7] x1 + [7] x2 + [0] [#less^#](x1, x2) = [7] x1 + [7] x2 + [0] [compare^#](x1, x2) = [2] x1 + [0] [#greater^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_14](x1, x2) = [7] x1 + [7] x2 + [0] [#ckgt^#](x1) = [7] x1 + [0] [#compare^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_15](x1) = [7] x1 + [0] [c_16](x1) = [7] x1 + [0] [mult#1^#](x1, x2) = [7] x1 + [7] x2 + [0] [bitToInt^#](x1) = [7] x1 + [0] [c_17](x1, x2) = [7] x1 + [7] x2 + [0] [sum#2^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_18](x1, x2) = [7] x1 + [7] x2 + [0] [c_19](x1, x2) = [7] x1 + [7] x2 + [0] [sum#3^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_20](x1, x2) = [7] x1 + [7] x2 + [0] [c_21](x1) = [7] x1 + [0] [#add^#](x1, x2) = [7] x1 + [7] x2 + [0] [sum#4^#](x1) = [7] x1 + [0] [c_22](x1, x2) = [7] x1 + [7] x2 + [0] [c_23](x1, x2) = [7] x1 + [7] x2 + [0] [c_24](x1) = [7] x1 + [0] [#mult^#](x1, x2) = [7] x1 + [7] x2 + [0] [sub'#5^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_25](x1) = [7] x1 + [0] [c_26] = [0] [c_27](x1, x2) = [7] x1 + [7] x2 + [0] [#cklt^#](x1) = [7] x1 + [0] [sub'#3^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_28](x1, x2) = [7] x1 + [7] x2 + [0] [sub'#4^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_29](x1) = [7] x1 + [0] [#eq^#](x1, x2) = [7] x1 + [7] x2 + [0] [sub'#2^#](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0] [c_30] = [0] [c_31](x1, x2) = [7] x1 + [7] x2 + [0] [diff^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [compare#2^#](x1, x2, x3) = [2] x2 + [2] x3 + [1] [c_32](x1) = [7] x1 + [0] [c_33](x1, x2) = [7] x1 + [7] x2 + [0] [compare#3^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_34](x1, x2) = [7] x1 + [7] x2 + [0] [compare#5^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_35](x1) = [7] x1 + [0] [c_36](x1, x2) = [7] x1 + [7] x2 + [0] [compare#6^#](x1) = [7] x1 + [0] [c_37](x1, x2) = [7] x1 + [7] x2 + [0] [compare#4^#](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0] [add^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_38](x1, x2) = [7] x1 + [7] x2 + [0] [add'^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [add'#3^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_39](x1) = [7] x1 + [0] [sub'#1^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_40] = [0] [c_41](x1) = [7] x1 + [0] [c_42](x1) = [7] x1 + [0] [c_43] = [0] [add'#1^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_44] = [0] [c_45](x1) = [7] x1 + [0] [add'#2^#](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0] [c_46] = [0] [c_47](x1, x2) = [7] x1 + [7] x2 + [0] [c_48](x1, x2, x3, x4, x5, x6, x7) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [7] x5 + [7] x6 + [7] x7 + [0] [c_49] = [0] [c_50](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_51](x1) = [7] x1 + [0] [c_52](x1) = [7] x1 + [0] [compare#1^#](x1, x2) = [2] x1 + [0] [c_53](x1) = [7] x1 + [0] [c_54](x1) = [7] x1 + [0] [c_55](x1, x2) = [7] x1 + [7] x2 + [0] [c_56] = [0] [c_57](x1, x2) = [7] x1 + [7] x2 + [0] [c_58](x1, x2) = [7] x1 + [7] x2 + [0] [c_59](x1) = [7] x1 + [0] [c_60](x1) = [7] x1 + [0] [c_61](x1) = [7] x1 + [0] [c_62] = [0] [c_63] = [0] [c_64] = [0] [c_65] = [0] [#natsub^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_66] = [0] [c_67](x1) = [7] x1 + [0] [#natdiv^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_68] = [0] [c_69](x1, x2) = [7] x1 + [7] x2 + [0] [c_70] = [0] [c_71] = [0] [c_72] = [0] [c_73](x1) = [7] x1 + [0] [#pred^#](x1) = [7] x1 + [0] [c_74](x1, x2) = [7] x1 + [7] x2 + [0] [c_75](x1) = [7] x1 + [0] [#succ^#](x1) = [7] x1 + [0] [c_76](x1, x2) = [7] x1 + [7] x2 + [0] [c_77] = [0] [#and^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_78] = [0] [c_79] = [0] [c_80] = [0] [c_81] = [0] [c_82](x1) = [7] x1 + [0] [c_83] = [0] [c_84] = [0] [c_85] = [0] [c_86](x1) = [7] x1 + [0] [c_87] = [0] [c_88] = [0] [c_89] = [0] [c_90] = [0] [c_91] = [0] [c_92] = [0] [c_93](x1) = [7] x1 + [0] [c_94] = [0] [c_95] = [0] [c_96] = [0] [c_97](x1) = [7] x1 + [0] [c_98] = [0] [c_99] = [0] [c_100] = [0] [c_101](x1) = [7] x1 + [0] [c_102] = [0] [c_103] = [0] [c_104](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_105] = [0] [c_106] = [0] [c_107] = [0] [c_108](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_109] = [0] [c_110] = [0] [c_111] = [0] [c_112] = [0] [c_113] = [0] [c_114](x1) = [7] x1 + [0] [#natmult^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_115] = [0] [c_116](x1, x2) = [7] x1 + [7] x2 + [0] [c_117] = [0] [c_118] = [0] [c_119] = [0] [c_120](x1) = [7] x1 + [0] [c_121](x1) = [7] x1 + [0] [c_122] = [0] [c_123](x1) = [7] x1 + [0] [c_124](x1) = [7] x1 + [0] [c_125] = [0] [c_126](x1) = [7] x1 + [0] [c_127](x1) = [7] x1 + [0] [c_128] = [0] [c_129] = [0] [c_130] = [0] [c_131] = [0] [c_132] = [0] [c_133] = [0] [c_134] = [0] [c_135] = [0] [c_136](x1) = [7] x1 + [0] [c_137](x1) = [7] x1 + [0] [c_138] = [0] [c_139](x1) = [7] x1 + [0] [c_140](x1) = [7] x1 + [0] [c_141] = [0] [c_142] = [0] [c_143] = [0] [c_144] = [0] [c_145] = [0] [c_146] = [0] [c_147] = [0] [c_148] = [0] [c] = [0] [c_1](x1) = [7] x1 + [0] [c_2](x1) = [7] x1 + [0] [c_3](x1) = [7] x1 + [0] [c_4](x1) = [7] x1 + [0] [c_5](x1) = [7] x1 + [0] [c_6](x1) = [7] x1 + [0] [c_7](x1, x2) = [7] x1 + [7] x2 + [0] [c_8](x1) = [7] x1 + [0] [c_9](x1) = [7] x1 + [0] [c_10](x1) = [1] x1 + [0] [c_11](x1, x2) = [7] x1 + [7] x2 + [0] [c_12](x1) = [7] x1 + [0] [c_13](x1) = [7] x1 + [0] [c_14](x1) = [7] x1 + [0] [c_15](x1) = [1] x1 + [0] [c_16](x1) = [7] x1 + [0] [c_17](x1) = [7] x1 + [0] [c_18](x1) = [7] x1 + [0] [c_19](x1) = [7] x1 + [0] [c_20](x1) = [7] x1 + [0] [c_21](x1) = [7] x1 + [0] [c_22](x1) = [1] x1 + [1] The following symbols are considered usable {compare^#, compare#2^#, compare#1^#} The order satisfies the following ordering constraints: [compare^#(@b1, @b2)] = [2] @b1 + [0] >= [2] @b1 + [0] = [c_10(compare#1^#(@b1, @b2))] [compare#2^#(::(@y, @ys), @x, @xs)] = [2] @x + [2] @xs + [1] > [2] @xs + [0] = [c_15(compare^#(@xs, @ys))] [compare#1^#(::(@x, @xs), @b2)] = [2] @x + [2] @xs + [4] > [2] @x + [2] @xs + [2] = [c_22(compare#2^#(@b2, @x, @xs))] We return to the main proof. Consider the set of all dependency pairs : { 1: compare^#(@b1, @b2) -> c_10(compare#1^#(@b1, @b2)) , 2: compare#2^#(::(@y, @ys), @x, @xs) -> c_15(compare^#(@xs, @ys)) , 3: compare#1^#(::(@x, @xs), @b2) -> c_22(compare#2^#(@b2, @x, @xs)) } Processor 'matrix interpretation of dimension 1' induces the complexity certificate YES(?,O(n^1)) on application of dependency pairs {2,3}. These cover all (indirect) predecessors of dependency pairs {1,2,3}, their number of application is equally bounded. The dependency pairs are shifted into the weak component. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak DPs: { compare^#(@b1, @b2) -> c_10(compare#1^#(@b1, @b2)) , compare#2^#(::(@y, @ys), @x, @xs) -> c_15(compare^#(@xs, @ys)) , compare#1^#(::(@x, @xs), @b2) -> c_22(compare#2^#(@b2, @x, @xs)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { compare^#(@b1, @b2) -> c_10(compare#1^#(@b1, @b2)) , compare#2^#(::(@y, @ys), @x, @xs) -> c_15(compare^#(@xs, @ys)) , compare#1^#(::(@x, @xs), @b2) -> c_22(compare#2^#(@b2, @x, @xs)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Rules: Empty Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) Empty rules are trivially bounded Hurray, we answered YES(O(1),O(n^2))