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) -> #mult(@x, @y) , +(@x, @y) -> #add(@x, @y) , computeLine(@line, @m, @acc) -> computeLine#1(@line, @acc, @m) , computeLine#1(::(@x, @xs), @acc, @m) -> computeLine#2(@m, @acc, @x, @xs) , computeLine#1(nil(), @acc, @m) -> @acc , computeLine#2(::(@l, @ls), @acc, @x, @xs) -> computeLine(@xs, @ls, lineMult(@x, @l, @acc)) , computeLine#2(nil(), @acc, @x, @xs) -> nil() , lineMult(@n, @l1, @l2) -> lineMult#1(@l1, @l2, @n) , lineMult#1(::(@x, @xs), @l2, @n) -> lineMult#2(@l2, @n, @x, @xs) , lineMult#1(nil(), @l2, @n) -> nil() , lineMult#2(::(@y, @ys), @n, @x, @xs) -> ::(+(*(@x, @n), @y), lineMult(@n, @xs, @ys)) , lineMult#2(nil(), @n, @x, @xs) -> ::(*(@x, @n), lineMult(@n, @xs, nil())) , matrixMult(@m1, @m2) -> matrixMult#1(@m1, @m2) , matrixMult#1(::(@l, @ls), @m2) -> ::(computeLine(@l, @m2, nil()), matrixMult(@ls, @m2)) , matrixMult#1(nil(), @m2) -> nil() } Weak Trs: { #mult(#0(), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#neg(@x), #0()) -> #0() , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #add(#0(), @y) -> @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)) , #pred(#0()) -> #neg(#s(#0())) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #succ(#0()) -> #pos(#s(#0())) , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #natadd(@y, #natmult(@x, @y)) , #natadd(#0(), @y) -> @y , #natadd(#s(@x), @y) -> #s(#natadd(@x, @y)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We add following dependency tuples: Strict DPs: { *^#(@x, @y) -> c_1(#mult^#(@x, @y)) , +^#(@x, @y) -> c_2(#add^#(@x, @y)) , computeLine^#(@line, @m, @acc) -> c_3(computeLine#1^#(@line, @acc, @m)) , computeLine#1^#(::(@x, @xs), @acc, @m) -> c_4(computeLine#2^#(@m, @acc, @x, @xs)) , computeLine#1^#(nil(), @acc, @m) -> c_5() , computeLine#2^#(::(@l, @ls), @acc, @x, @xs) -> c_6(computeLine^#(@xs, @ls, lineMult(@x, @l, @acc)), lineMult^#(@x, @l, @acc)) , computeLine#2^#(nil(), @acc, @x, @xs) -> c_7() , lineMult^#(@n, @l1, @l2) -> c_8(lineMult#1^#(@l1, @l2, @n)) , lineMult#1^#(::(@x, @xs), @l2, @n) -> c_9(lineMult#2^#(@l2, @n, @x, @xs)) , lineMult#1^#(nil(), @l2, @n) -> c_10() , lineMult#2^#(::(@y, @ys), @n, @x, @xs) -> c_11(+^#(*(@x, @n), @y), *^#(@x, @n), lineMult^#(@n, @xs, @ys)) , lineMult#2^#(nil(), @n, @x, @xs) -> c_12(*^#(@x, @n), lineMult^#(@n, @xs, nil())) , matrixMult^#(@m1, @m2) -> c_13(matrixMult#1^#(@m1, @m2)) , matrixMult#1^#(::(@l, @ls), @m2) -> c_14(computeLine^#(@l, @m2, nil()), matrixMult^#(@ls, @m2)) , matrixMult#1^#(nil(), @m2) -> c_15() } Weak DPs: { #mult^#(#0(), #0()) -> c_16() , #mult^#(#0(), #neg(@y)) -> c_17() , #mult^#(#0(), #pos(@y)) -> c_18() , #mult^#(#neg(@x), #0()) -> c_19() , #mult^#(#neg(@x), #neg(@y)) -> c_20(#natmult^#(@x, @y)) , #mult^#(#neg(@x), #pos(@y)) -> c_21(#natmult^#(@x, @y)) , #mult^#(#pos(@x), #0()) -> c_22() , #mult^#(#pos(@x), #neg(@y)) -> c_23(#natmult^#(@x, @y)) , #mult^#(#pos(@x), #pos(@y)) -> c_24(#natmult^#(@x, @y)) , #add^#(#0(), @y) -> c_25() , #add^#(#neg(#s(#0())), @y) -> c_26(#pred^#(@y)) , #add^#(#neg(#s(#s(@x))), @y) -> c_27(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , #add^#(#pos(#s(#0())), @y) -> c_28(#succ^#(@y)) , #add^#(#pos(#s(#s(@x))), @y) -> c_29(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , #natmult^#(#0(), @y) -> c_38() , #natmult^#(#s(@x), @y) -> c_39(#natadd^#(@y, #natmult(@x, @y)), #natmult^#(@x, @y)) , #pred^#(#0()) -> c_30() , #pred^#(#neg(#s(@x))) -> c_31() , #pred^#(#pos(#s(#0()))) -> c_32() , #pred^#(#pos(#s(#s(@x)))) -> c_33() , #succ^#(#0()) -> c_34() , #succ^#(#neg(#s(#0()))) -> c_35() , #succ^#(#neg(#s(#s(@x)))) -> c_36() , #succ^#(#pos(#s(@x))) -> c_37() , #natadd^#(#0(), @y) -> c_40() , #natadd^#(#s(@x), @y) -> c_41(#natadd^#(@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(#mult^#(@x, @y)) , +^#(@x, @y) -> c_2(#add^#(@x, @y)) , computeLine^#(@line, @m, @acc) -> c_3(computeLine#1^#(@line, @acc, @m)) , computeLine#1^#(::(@x, @xs), @acc, @m) -> c_4(computeLine#2^#(@m, @acc, @x, @xs)) , computeLine#1^#(nil(), @acc, @m) -> c_5() , computeLine#2^#(::(@l, @ls), @acc, @x, @xs) -> c_6(computeLine^#(@xs, @ls, lineMult(@x, @l, @acc)), lineMult^#(@x, @l, @acc)) , computeLine#2^#(nil(), @acc, @x, @xs) -> c_7() , lineMult^#(@n, @l1, @l2) -> c_8(lineMult#1^#(@l1, @l2, @n)) , lineMult#1^#(::(@x, @xs), @l2, @n) -> c_9(lineMult#2^#(@l2, @n, @x, @xs)) , lineMult#1^#(nil(), @l2, @n) -> c_10() , lineMult#2^#(::(@y, @ys), @n, @x, @xs) -> c_11(+^#(*(@x, @n), @y), *^#(@x, @n), lineMult^#(@n, @xs, @ys)) , lineMult#2^#(nil(), @n, @x, @xs) -> c_12(*^#(@x, @n), lineMult^#(@n, @xs, nil())) , matrixMult^#(@m1, @m2) -> c_13(matrixMult#1^#(@m1, @m2)) , matrixMult#1^#(::(@l, @ls), @m2) -> c_14(computeLine^#(@l, @m2, nil()), matrixMult^#(@ls, @m2)) , matrixMult#1^#(nil(), @m2) -> c_15() } Weak DPs: { #mult^#(#0(), #0()) -> c_16() , #mult^#(#0(), #neg(@y)) -> c_17() , #mult^#(#0(), #pos(@y)) -> c_18() , #mult^#(#neg(@x), #0()) -> c_19() , #mult^#(#neg(@x), #neg(@y)) -> c_20(#natmult^#(@x, @y)) , #mult^#(#neg(@x), #pos(@y)) -> c_21(#natmult^#(@x, @y)) , #mult^#(#pos(@x), #0()) -> c_22() , #mult^#(#pos(@x), #neg(@y)) -> c_23(#natmult^#(@x, @y)) , #mult^#(#pos(@x), #pos(@y)) -> c_24(#natmult^#(@x, @y)) , #add^#(#0(), @y) -> c_25() , #add^#(#neg(#s(#0())), @y) -> c_26(#pred^#(@y)) , #add^#(#neg(#s(#s(@x))), @y) -> c_27(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , #add^#(#pos(#s(#0())), @y) -> c_28(#succ^#(@y)) , #add^#(#pos(#s(#s(@x))), @y) -> c_29(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , #natmult^#(#0(), @y) -> c_38() , #natmult^#(#s(@x), @y) -> c_39(#natadd^#(@y, #natmult(@x, @y)), #natmult^#(@x, @y)) , #pred^#(#0()) -> c_30() , #pred^#(#neg(#s(@x))) -> c_31() , #pred^#(#pos(#s(#0()))) -> c_32() , #pred^#(#pos(#s(#s(@x)))) -> c_33() , #succ^#(#0()) -> c_34() , #succ^#(#neg(#s(#0()))) -> c_35() , #succ^#(#neg(#s(#s(@x)))) -> c_36() , #succ^#(#pos(#s(@x))) -> c_37() , #natadd^#(#0(), @y) -> c_40() , #natadd^#(#s(@x), @y) -> c_41(#natadd^#(@x, @y)) } Weak Trs: { *(@x, @y) -> #mult(@x, @y) , #mult(#0(), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#neg(@x), #0()) -> #0() , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , +(@x, @y) -> #add(@x, @y) , #add(#0(), @y) -> @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)) , computeLine(@line, @m, @acc) -> computeLine#1(@line, @acc, @m) , computeLine#1(::(@x, @xs), @acc, @m) -> computeLine#2(@m, @acc, @x, @xs) , computeLine#1(nil(), @acc, @m) -> @acc , computeLine#2(::(@l, @ls), @acc, @x, @xs) -> computeLine(@xs, @ls, lineMult(@x, @l, @acc)) , computeLine#2(nil(), @acc, @x, @xs) -> nil() , lineMult(@n, @l1, @l2) -> lineMult#1(@l1, @l2, @n) , lineMult#1(::(@x, @xs), @l2, @n) -> lineMult#2(@l2, @n, @x, @xs) , lineMult#1(nil(), @l2, @n) -> nil() , lineMult#2(::(@y, @ys), @n, @x, @xs) -> ::(+(*(@x, @n), @y), lineMult(@n, @xs, @ys)) , lineMult#2(nil(), @n, @x, @xs) -> ::(*(@x, @n), lineMult(@n, @xs, nil())) , matrixMult(@m1, @m2) -> matrixMult#1(@m1, @m2) , matrixMult#1(::(@l, @ls), @m2) -> ::(computeLine(@l, @m2, nil()), matrixMult(@ls, @m2)) , matrixMult#1(nil(), @m2) -> nil() , #pred(#0()) -> #neg(#s(#0())) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #succ(#0()) -> #pos(#s(#0())) , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #natadd(@y, #natmult(@x, @y)) , #natadd(#0(), @y) -> @y , #natadd(#s(@x), @y) -> #s(#natadd(@x, @y)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We estimate the number of application of {1,2,5,7,10,15} by applications of Pre({1,2,5,7,10,15}) = {3,4,8,11,12,13}. Here rules are labeled as follows: DPs: { 1: *^#(@x, @y) -> c_1(#mult^#(@x, @y)) , 2: +^#(@x, @y) -> c_2(#add^#(@x, @y)) , 3: computeLine^#(@line, @m, @acc) -> c_3(computeLine#1^#(@line, @acc, @m)) , 4: computeLine#1^#(::(@x, @xs), @acc, @m) -> c_4(computeLine#2^#(@m, @acc, @x, @xs)) , 5: computeLine#1^#(nil(), @acc, @m) -> c_5() , 6: computeLine#2^#(::(@l, @ls), @acc, @x, @xs) -> c_6(computeLine^#(@xs, @ls, lineMult(@x, @l, @acc)), lineMult^#(@x, @l, @acc)) , 7: computeLine#2^#(nil(), @acc, @x, @xs) -> c_7() , 8: lineMult^#(@n, @l1, @l2) -> c_8(lineMult#1^#(@l1, @l2, @n)) , 9: lineMult#1^#(::(@x, @xs), @l2, @n) -> c_9(lineMult#2^#(@l2, @n, @x, @xs)) , 10: lineMult#1^#(nil(), @l2, @n) -> c_10() , 11: lineMult#2^#(::(@y, @ys), @n, @x, @xs) -> c_11(+^#(*(@x, @n), @y), *^#(@x, @n), lineMult^#(@n, @xs, @ys)) , 12: lineMult#2^#(nil(), @n, @x, @xs) -> c_12(*^#(@x, @n), lineMult^#(@n, @xs, nil())) , 13: matrixMult^#(@m1, @m2) -> c_13(matrixMult#1^#(@m1, @m2)) , 14: matrixMult#1^#(::(@l, @ls), @m2) -> c_14(computeLine^#(@l, @m2, nil()), matrixMult^#(@ls, @m2)) , 15: matrixMult#1^#(nil(), @m2) -> c_15() , 16: #mult^#(#0(), #0()) -> c_16() , 17: #mult^#(#0(), #neg(@y)) -> c_17() , 18: #mult^#(#0(), #pos(@y)) -> c_18() , 19: #mult^#(#neg(@x), #0()) -> c_19() , 20: #mult^#(#neg(@x), #neg(@y)) -> c_20(#natmult^#(@x, @y)) , 21: #mult^#(#neg(@x), #pos(@y)) -> c_21(#natmult^#(@x, @y)) , 22: #mult^#(#pos(@x), #0()) -> c_22() , 23: #mult^#(#pos(@x), #neg(@y)) -> c_23(#natmult^#(@x, @y)) , 24: #mult^#(#pos(@x), #pos(@y)) -> c_24(#natmult^#(@x, @y)) , 25: #add^#(#0(), @y) -> c_25() , 26: #add^#(#neg(#s(#0())), @y) -> c_26(#pred^#(@y)) , 27: #add^#(#neg(#s(#s(@x))), @y) -> c_27(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , 28: #add^#(#pos(#s(#0())), @y) -> c_28(#succ^#(@y)) , 29: #add^#(#pos(#s(#s(@x))), @y) -> c_29(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , 30: #natmult^#(#0(), @y) -> c_38() , 31: #natmult^#(#s(@x), @y) -> c_39(#natadd^#(@y, #natmult(@x, @y)), #natmult^#(@x, @y)) , 32: #pred^#(#0()) -> c_30() , 33: #pred^#(#neg(#s(@x))) -> c_31() , 34: #pred^#(#pos(#s(#0()))) -> c_32() , 35: #pred^#(#pos(#s(#s(@x)))) -> c_33() , 36: #succ^#(#0()) -> c_34() , 37: #succ^#(#neg(#s(#0()))) -> c_35() , 38: #succ^#(#neg(#s(#s(@x)))) -> c_36() , 39: #succ^#(#pos(#s(@x))) -> c_37() , 40: #natadd^#(#0(), @y) -> c_40() , 41: #natadd^#(#s(@x), @y) -> c_41(#natadd^#(@x, @y)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { computeLine^#(@line, @m, @acc) -> c_3(computeLine#1^#(@line, @acc, @m)) , computeLine#1^#(::(@x, @xs), @acc, @m) -> c_4(computeLine#2^#(@m, @acc, @x, @xs)) , computeLine#2^#(::(@l, @ls), @acc, @x, @xs) -> c_6(computeLine^#(@xs, @ls, lineMult(@x, @l, @acc)), lineMult^#(@x, @l, @acc)) , lineMult^#(@n, @l1, @l2) -> c_8(lineMult#1^#(@l1, @l2, @n)) , lineMult#1^#(::(@x, @xs), @l2, @n) -> c_9(lineMult#2^#(@l2, @n, @x, @xs)) , lineMult#2^#(::(@y, @ys), @n, @x, @xs) -> c_11(+^#(*(@x, @n), @y), *^#(@x, @n), lineMult^#(@n, @xs, @ys)) , lineMult#2^#(nil(), @n, @x, @xs) -> c_12(*^#(@x, @n), lineMult^#(@n, @xs, nil())) , matrixMult^#(@m1, @m2) -> c_13(matrixMult#1^#(@m1, @m2)) , matrixMult#1^#(::(@l, @ls), @m2) -> c_14(computeLine^#(@l, @m2, nil()), matrixMult^#(@ls, @m2)) } Weak DPs: { *^#(@x, @y) -> c_1(#mult^#(@x, @y)) , #mult^#(#0(), #0()) -> c_16() , #mult^#(#0(), #neg(@y)) -> c_17() , #mult^#(#0(), #pos(@y)) -> c_18() , #mult^#(#neg(@x), #0()) -> c_19() , #mult^#(#neg(@x), #neg(@y)) -> c_20(#natmult^#(@x, @y)) , #mult^#(#neg(@x), #pos(@y)) -> c_21(#natmult^#(@x, @y)) , #mult^#(#pos(@x), #0()) -> c_22() , #mult^#(#pos(@x), #neg(@y)) -> c_23(#natmult^#(@x, @y)) , #mult^#(#pos(@x), #pos(@y)) -> c_24(#natmult^#(@x, @y)) , +^#(@x, @y) -> c_2(#add^#(@x, @y)) , #add^#(#0(), @y) -> c_25() , #add^#(#neg(#s(#0())), @y) -> c_26(#pred^#(@y)) , #add^#(#neg(#s(#s(@x))), @y) -> c_27(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , #add^#(#pos(#s(#0())), @y) -> c_28(#succ^#(@y)) , #add^#(#pos(#s(#s(@x))), @y) -> c_29(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , computeLine#1^#(nil(), @acc, @m) -> c_5() , computeLine#2^#(nil(), @acc, @x, @xs) -> c_7() , lineMult#1^#(nil(), @l2, @n) -> c_10() , matrixMult#1^#(nil(), @m2) -> c_15() , #natmult^#(#0(), @y) -> c_38() , #natmult^#(#s(@x), @y) -> c_39(#natadd^#(@y, #natmult(@x, @y)), #natmult^#(@x, @y)) , #pred^#(#0()) -> c_30() , #pred^#(#neg(#s(@x))) -> c_31() , #pred^#(#pos(#s(#0()))) -> c_32() , #pred^#(#pos(#s(#s(@x)))) -> c_33() , #succ^#(#0()) -> c_34() , #succ^#(#neg(#s(#0()))) -> c_35() , #succ^#(#neg(#s(#s(@x)))) -> c_36() , #succ^#(#pos(#s(@x))) -> c_37() , #natadd^#(#0(), @y) -> c_40() , #natadd^#(#s(@x), @y) -> c_41(#natadd^#(@x, @y)) } Weak Trs: { *(@x, @y) -> #mult(@x, @y) , #mult(#0(), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#neg(@x), #0()) -> #0() , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , +(@x, @y) -> #add(@x, @y) , #add(#0(), @y) -> @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)) , computeLine(@line, @m, @acc) -> computeLine#1(@line, @acc, @m) , computeLine#1(::(@x, @xs), @acc, @m) -> computeLine#2(@m, @acc, @x, @xs) , computeLine#1(nil(), @acc, @m) -> @acc , computeLine#2(::(@l, @ls), @acc, @x, @xs) -> computeLine(@xs, @ls, lineMult(@x, @l, @acc)) , computeLine#2(nil(), @acc, @x, @xs) -> nil() , lineMult(@n, @l1, @l2) -> lineMult#1(@l1, @l2, @n) , lineMult#1(::(@x, @xs), @l2, @n) -> lineMult#2(@l2, @n, @x, @xs) , lineMult#1(nil(), @l2, @n) -> nil() , lineMult#2(::(@y, @ys), @n, @x, @xs) -> ::(+(*(@x, @n), @y), lineMult(@n, @xs, @ys)) , lineMult#2(nil(), @n, @x, @xs) -> ::(*(@x, @n), lineMult(@n, @xs, nil())) , matrixMult(@m1, @m2) -> matrixMult#1(@m1, @m2) , matrixMult#1(::(@l, @ls), @m2) -> ::(computeLine(@l, @m2, nil()), matrixMult(@ls, @m2)) , matrixMult#1(nil(), @m2) -> nil() , #pred(#0()) -> #neg(#s(#0())) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #succ(#0()) -> #pos(#s(#0())) , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #natadd(@y, #natmult(@x, @y)) , #natadd(#0(), @y) -> @y , #natadd(#s(@x), @y) -> #s(#natadd(@x, @y)) } 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(#mult^#(@x, @y)) , #mult^#(#0(), #0()) -> c_16() , #mult^#(#0(), #neg(@y)) -> c_17() , #mult^#(#0(), #pos(@y)) -> c_18() , #mult^#(#neg(@x), #0()) -> c_19() , #mult^#(#neg(@x), #neg(@y)) -> c_20(#natmult^#(@x, @y)) , #mult^#(#neg(@x), #pos(@y)) -> c_21(#natmult^#(@x, @y)) , #mult^#(#pos(@x), #0()) -> c_22() , #mult^#(#pos(@x), #neg(@y)) -> c_23(#natmult^#(@x, @y)) , #mult^#(#pos(@x), #pos(@y)) -> c_24(#natmult^#(@x, @y)) , +^#(@x, @y) -> c_2(#add^#(@x, @y)) , #add^#(#0(), @y) -> c_25() , #add^#(#neg(#s(#0())), @y) -> c_26(#pred^#(@y)) , #add^#(#neg(#s(#s(@x))), @y) -> c_27(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , #add^#(#pos(#s(#0())), @y) -> c_28(#succ^#(@y)) , #add^#(#pos(#s(#s(@x))), @y) -> c_29(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , computeLine#1^#(nil(), @acc, @m) -> c_5() , computeLine#2^#(nil(), @acc, @x, @xs) -> c_7() , lineMult#1^#(nil(), @l2, @n) -> c_10() , matrixMult#1^#(nil(), @m2) -> c_15() , #natmult^#(#0(), @y) -> c_38() , #natmult^#(#s(@x), @y) -> c_39(#natadd^#(@y, #natmult(@x, @y)), #natmult^#(@x, @y)) , #pred^#(#0()) -> c_30() , #pred^#(#neg(#s(@x))) -> c_31() , #pred^#(#pos(#s(#0()))) -> c_32() , #pred^#(#pos(#s(#s(@x)))) -> c_33() , #succ^#(#0()) -> c_34() , #succ^#(#neg(#s(#0()))) -> c_35() , #succ^#(#neg(#s(#s(@x)))) -> c_36() , #succ^#(#pos(#s(@x))) -> c_37() , #natadd^#(#0(), @y) -> c_40() , #natadd^#(#s(@x), @y) -> c_41(#natadd^#(@x, @y)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { computeLine^#(@line, @m, @acc) -> c_3(computeLine#1^#(@line, @acc, @m)) , computeLine#1^#(::(@x, @xs), @acc, @m) -> c_4(computeLine#2^#(@m, @acc, @x, @xs)) , computeLine#2^#(::(@l, @ls), @acc, @x, @xs) -> c_6(computeLine^#(@xs, @ls, lineMult(@x, @l, @acc)), lineMult^#(@x, @l, @acc)) , lineMult^#(@n, @l1, @l2) -> c_8(lineMult#1^#(@l1, @l2, @n)) , lineMult#1^#(::(@x, @xs), @l2, @n) -> c_9(lineMult#2^#(@l2, @n, @x, @xs)) , lineMult#2^#(::(@y, @ys), @n, @x, @xs) -> c_11(+^#(*(@x, @n), @y), *^#(@x, @n), lineMult^#(@n, @xs, @ys)) , lineMult#2^#(nil(), @n, @x, @xs) -> c_12(*^#(@x, @n), lineMult^#(@n, @xs, nil())) , matrixMult^#(@m1, @m2) -> c_13(matrixMult#1^#(@m1, @m2)) , matrixMult#1^#(::(@l, @ls), @m2) -> c_14(computeLine^#(@l, @m2, nil()), matrixMult^#(@ls, @m2)) } Weak Trs: { *(@x, @y) -> #mult(@x, @y) , #mult(#0(), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#neg(@x), #0()) -> #0() , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , +(@x, @y) -> #add(@x, @y) , #add(#0(), @y) -> @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)) , computeLine(@line, @m, @acc) -> computeLine#1(@line, @acc, @m) , computeLine#1(::(@x, @xs), @acc, @m) -> computeLine#2(@m, @acc, @x, @xs) , computeLine#1(nil(), @acc, @m) -> @acc , computeLine#2(::(@l, @ls), @acc, @x, @xs) -> computeLine(@xs, @ls, lineMult(@x, @l, @acc)) , computeLine#2(nil(), @acc, @x, @xs) -> nil() , lineMult(@n, @l1, @l2) -> lineMult#1(@l1, @l2, @n) , lineMult#1(::(@x, @xs), @l2, @n) -> lineMult#2(@l2, @n, @x, @xs) , lineMult#1(nil(), @l2, @n) -> nil() , lineMult#2(::(@y, @ys), @n, @x, @xs) -> ::(+(*(@x, @n), @y), lineMult(@n, @xs, @ys)) , lineMult#2(nil(), @n, @x, @xs) -> ::(*(@x, @n), lineMult(@n, @xs, nil())) , matrixMult(@m1, @m2) -> matrixMult#1(@m1, @m2) , matrixMult#1(::(@l, @ls), @m2) -> ::(computeLine(@l, @m2, nil()), matrixMult(@ls, @m2)) , matrixMult#1(nil(), @m2) -> nil() , #pred(#0()) -> #neg(#s(#0())) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #succ(#0()) -> #pos(#s(#0())) , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #natadd(@y, #natmult(@x, @y)) , #natadd(#0(), @y) -> @y , #natadd(#s(@x), @y) -> #s(#natadd(@x, @y)) } 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: { lineMult#2^#(::(@y, @ys), @n, @x, @xs) -> c_11(+^#(*(@x, @n), @y), *^#(@x, @n), lineMult^#(@n, @xs, @ys)) , lineMult#2^#(nil(), @n, @x, @xs) -> c_12(*^#(@x, @n), lineMult^#(@n, @xs, nil())) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { computeLine^#(@line, @m, @acc) -> c_1(computeLine#1^#(@line, @acc, @m)) , computeLine#1^#(::(@x, @xs), @acc, @m) -> c_2(computeLine#2^#(@m, @acc, @x, @xs)) , computeLine#2^#(::(@l, @ls), @acc, @x, @xs) -> c_3(computeLine^#(@xs, @ls, lineMult(@x, @l, @acc)), lineMult^#(@x, @l, @acc)) , lineMult^#(@n, @l1, @l2) -> c_4(lineMult#1^#(@l1, @l2, @n)) , lineMult#1^#(::(@x, @xs), @l2, @n) -> c_5(lineMult#2^#(@l2, @n, @x, @xs)) , lineMult#2^#(::(@y, @ys), @n, @x, @xs) -> c_6(lineMult^#(@n, @xs, @ys)) , lineMult#2^#(nil(), @n, @x, @xs) -> c_7(lineMult^#(@n, @xs, nil())) , matrixMult^#(@m1, @m2) -> c_8(matrixMult#1^#(@m1, @m2)) , matrixMult#1^#(::(@l, @ls), @m2) -> c_9(computeLine^#(@l, @m2, nil()), matrixMult^#(@ls, @m2)) } Weak Trs: { *(@x, @y) -> #mult(@x, @y) , #mult(#0(), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#neg(@x), #0()) -> #0() , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , +(@x, @y) -> #add(@x, @y) , #add(#0(), @y) -> @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)) , computeLine(@line, @m, @acc) -> computeLine#1(@line, @acc, @m) , computeLine#1(::(@x, @xs), @acc, @m) -> computeLine#2(@m, @acc, @x, @xs) , computeLine#1(nil(), @acc, @m) -> @acc , computeLine#2(::(@l, @ls), @acc, @x, @xs) -> computeLine(@xs, @ls, lineMult(@x, @l, @acc)) , computeLine#2(nil(), @acc, @x, @xs) -> nil() , lineMult(@n, @l1, @l2) -> lineMult#1(@l1, @l2, @n) , lineMult#1(::(@x, @xs), @l2, @n) -> lineMult#2(@l2, @n, @x, @xs) , lineMult#1(nil(), @l2, @n) -> nil() , lineMult#2(::(@y, @ys), @n, @x, @xs) -> ::(+(*(@x, @n), @y), lineMult(@n, @xs, @ys)) , lineMult#2(nil(), @n, @x, @xs) -> ::(*(@x, @n), lineMult(@n, @xs, nil())) , matrixMult(@m1, @m2) -> matrixMult#1(@m1, @m2) , matrixMult#1(::(@l, @ls), @m2) -> ::(computeLine(@l, @m2, nil()), matrixMult(@ls, @m2)) , matrixMult#1(nil(), @m2) -> nil() , #pred(#0()) -> #neg(#s(#0())) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #succ(#0()) -> #pos(#s(#0())) , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #natadd(@y, #natmult(@x, @y)) , #natadd(#0(), @y) -> @y , #natadd(#s(@x), @y) -> #s(#natadd(@x, @y)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We replace rewrite rules by usable rules: Weak Usable Rules: { *(@x, @y) -> #mult(@x, @y) , #mult(#0(), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#neg(@x), #0()) -> #0() , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , +(@x, @y) -> #add(@x, @y) , #add(#0(), @y) -> @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)) , lineMult(@n, @l1, @l2) -> lineMult#1(@l1, @l2, @n) , lineMult#1(::(@x, @xs), @l2, @n) -> lineMult#2(@l2, @n, @x, @xs) , lineMult#1(nil(), @l2, @n) -> nil() , lineMult#2(::(@y, @ys), @n, @x, @xs) -> ::(+(*(@x, @n), @y), lineMult(@n, @xs, @ys)) , lineMult#2(nil(), @n, @x, @xs) -> ::(*(@x, @n), lineMult(@n, @xs, nil())) , #pred(#0()) -> #neg(#s(#0())) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #succ(#0()) -> #pos(#s(#0())) , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #natadd(@y, #natmult(@x, @y)) , #natadd(#0(), @y) -> @y , #natadd(#s(@x), @y) -> #s(#natadd(@x, @y)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^2)). Strict DPs: { computeLine^#(@line, @m, @acc) -> c_1(computeLine#1^#(@line, @acc, @m)) , computeLine#1^#(::(@x, @xs), @acc, @m) -> c_2(computeLine#2^#(@m, @acc, @x, @xs)) , computeLine#2^#(::(@l, @ls), @acc, @x, @xs) -> c_3(computeLine^#(@xs, @ls, lineMult(@x, @l, @acc)), lineMult^#(@x, @l, @acc)) , lineMult^#(@n, @l1, @l2) -> c_4(lineMult#1^#(@l1, @l2, @n)) , lineMult#1^#(::(@x, @xs), @l2, @n) -> c_5(lineMult#2^#(@l2, @n, @x, @xs)) , lineMult#2^#(::(@y, @ys), @n, @x, @xs) -> c_6(lineMult^#(@n, @xs, @ys)) , lineMult#2^#(nil(), @n, @x, @xs) -> c_7(lineMult^#(@n, @xs, nil())) , matrixMult^#(@m1, @m2) -> c_8(matrixMult#1^#(@m1, @m2)) , matrixMult#1^#(::(@l, @ls), @m2) -> c_9(computeLine^#(@l, @m2, nil()), matrixMult^#(@ls, @m2)) } Weak Trs: { *(@x, @y) -> #mult(@x, @y) , #mult(#0(), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#neg(@x), #0()) -> #0() , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , +(@x, @y) -> #add(@x, @y) , #add(#0(), @y) -> @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)) , lineMult(@n, @l1, @l2) -> lineMult#1(@l1, @l2, @n) , lineMult#1(::(@x, @xs), @l2, @n) -> lineMult#2(@l2, @n, @x, @xs) , lineMult#1(nil(), @l2, @n) -> nil() , lineMult#2(::(@y, @ys), @n, @x, @xs) -> ::(+(*(@x, @n), @y), lineMult(@n, @xs, @ys)) , lineMult#2(nil(), @n, @x, @xs) -> ::(*(@x, @n), lineMult(@n, @xs, nil())) , #pred(#0()) -> #neg(#s(#0())) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #succ(#0()) -> #pos(#s(#0())) , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #natadd(@y, #natmult(@x, @y)) , #natadd(#0(), @y) -> @y , #natadd(#s(@x), @y) -> #s(#natadd(@x, @y)) } 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 { matrixMult^#(@m1, @m2) -> c_8(matrixMult#1^#(@m1, @m2)) , matrixMult#1^#(::(@l, @ls), @m2) -> c_9(computeLine^#(@l, @m2, nil()), matrixMult^#(@ls, @m2)) } and lower component { computeLine^#(@line, @m, @acc) -> c_1(computeLine#1^#(@line, @acc, @m)) , computeLine#1^#(::(@x, @xs), @acc, @m) -> c_2(computeLine#2^#(@m, @acc, @x, @xs)) , computeLine#2^#(::(@l, @ls), @acc, @x, @xs) -> c_3(computeLine^#(@xs, @ls, lineMult(@x, @l, @acc)), lineMult^#(@x, @l, @acc)) , lineMult^#(@n, @l1, @l2) -> c_4(lineMult#1^#(@l1, @l2, @n)) , lineMult#1^#(::(@x, @xs), @l2, @n) -> c_5(lineMult#2^#(@l2, @n, @x, @xs)) , lineMult#2^#(::(@y, @ys), @n, @x, @xs) -> c_6(lineMult^#(@n, @xs, @ys)) , lineMult#2^#(nil(), @n, @x, @xs) -> c_7(lineMult^#(@n, @xs, nil())) } Further, following extension rules are added to the lower component. { matrixMult^#(@m1, @m2) -> matrixMult#1^#(@m1, @m2) , matrixMult#1^#(::(@l, @ls), @m2) -> computeLine^#(@l, @m2, nil()) , matrixMult#1^#(::(@l, @ls), @m2) -> matrixMult^#(@ls, @m2) } 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: { matrixMult^#(@m1, @m2) -> c_8(matrixMult#1^#(@m1, @m2)) , matrixMult#1^#(::(@l, @ls), @m2) -> c_9(computeLine^#(@l, @m2, nil()), matrixMult^#(@ls, @m2)) } Weak Trs: { *(@x, @y) -> #mult(@x, @y) , #mult(#0(), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#neg(@x), #0()) -> #0() , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , +(@x, @y) -> #add(@x, @y) , #add(#0(), @y) -> @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)) , lineMult(@n, @l1, @l2) -> lineMult#1(@l1, @l2, @n) , lineMult#1(::(@x, @xs), @l2, @n) -> lineMult#2(@l2, @n, @x, @xs) , lineMult#1(nil(), @l2, @n) -> nil() , lineMult#2(::(@y, @ys), @n, @x, @xs) -> ::(+(*(@x, @n), @y), lineMult(@n, @xs, @ys)) , lineMult#2(nil(), @n, @x, @xs) -> ::(*(@x, @n), lineMult(@n, @xs, nil())) , #pred(#0()) -> #neg(#s(#0())) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #succ(#0()) -> #pos(#s(#0())) , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #natadd(@y, #natmult(@x, @y)) , #natadd(#0(), @y) -> @y , #natadd(#s(@x), @y) -> #s(#natadd(@x, @y)) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^1)) Due to missing edges in the dependency-graph, the right-hand sides of following rules could be simplified: { matrixMult#1^#(::(@l, @ls), @m2) -> c_9(computeLine^#(@l, @m2, nil()), matrixMult^#(@ls, @m2)) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(n^1)). Strict DPs: { matrixMult^#(@m1, @m2) -> c_1(matrixMult#1^#(@m1, @m2)) , matrixMult#1^#(::(@l, @ls), @m2) -> c_2(matrixMult^#(@ls, @m2)) } Weak Trs: { *(@x, @y) -> #mult(@x, @y) , #mult(#0(), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#neg(@x), #0()) -> #0() , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , +(@x, @y) -> #add(@x, @y) , #add(#0(), @y) -> @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)) , lineMult(@n, @l1, @l2) -> lineMult#1(@l1, @l2, @n) , lineMult#1(::(@x, @xs), @l2, @n) -> lineMult#2(@l2, @n, @x, @xs) , lineMult#1(nil(), @l2, @n) -> nil() , lineMult#2(::(@y, @ys), @n, @x, @xs) -> ::(+(*(@x, @n), @y), lineMult(@n, @xs, @ys)) , lineMult#2(nil(), @n, @x, @xs) -> ::(*(@x, @n), lineMult(@n, @xs, nil())) , #pred(#0()) -> #neg(#s(#0())) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #succ(#0()) -> #pos(#s(#0())) , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #natadd(@y, #natmult(@x, @y)) , #natadd(#0(), @y) -> @y , #natadd(#s(@x), @y) -> #s(#natadd(@x, @y)) } 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: { matrixMult^#(@m1, @m2) -> c_1(matrixMult#1^#(@m1, @m2)) , matrixMult#1^#(::(@l, @ls), @m2) -> c_2(matrixMult^#(@ls, @m2)) } 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: matrixMult^#(@m1, @m2) -> c_1(matrixMult#1^#(@m1, @m2)) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_1) = {1}, Uargs(c_2) = {1} TcT has computed following constructor-based matrix interpretation satisfying not(EDA). [*](x1, x2) = [0] [#mult](x1, x2) = [0] [+](x1, x2) = [0] [#add](x1, x2) = [0] [::](x1, x2) = [1] x2 + [1] [nil] = [0] [lineMult](x1, x2, x3) = [0] [lineMult#1](x1, x2, x3) = [0] [lineMult#2](x1, x2, x3, x4) = [0] [#0] = [0] [#s](x1) = [0] [#neg](x1) = [0] [#pred](x1) = [0] [#pos](x1) = [0] [#succ](x1) = [0] [#natmult](x1, x2) = [0] [#natadd](x1, x2) = [0] [*^#](x1, x2) = [0] [#mult^#](x1, x2) = [0] [+^#](x1, x2) = [0] [#add^#](x1, x2) = [0] [computeLine^#](x1, x2, x3) = [0] [computeLine#1^#](x1, x2, x3) = [0] [computeLine#2^#](x1, x2, x3, x4) = [0] [lineMult^#](x1, x2, x3) = [0] [lineMult#1^#](x1, x2, x3) = [0] [lineMult#2^#](x1, x2, x3, x4) = [0] [matrixMult^#](x1, x2) = [1] x1 + [1] [matrixMult#1^#](x1, x2) = [1] x1 + [0] [#natmult^#](x1, x2) = [0] [#pred^#](x1) = [0] [#succ^#](x1) = [0] [#natadd^#](x1, x2) = [0] [c_8](x1) = [0] [c_9](x1, x2) = [0] [c] = [0] [c_1](x1) = [1] x1 + [0] [c_2](x1) = [1] x1 + [0] This order satisfies following ordering constraints [matrixMult^#(@m1, @m2)] = [1] @m1 + [1] > [1] @m1 + [0] = [c_1(matrixMult#1^#(@m1, @m2))] [matrixMult#1^#(::(@l, @ls), @m2)] = [1] @ls + [1] >= [1] @ls + [1] = [c_2(matrixMult^#(@ls, @m2))] Consider the set of all dependency pairs DPs: { 1: matrixMult^#(@m1, @m2) -> c_1(matrixMult#1^#(@m1, @m2)) , 2: matrixMult#1^#(::(@l, @ls), @m2) -> c_2(matrixMult^#(@ls, @m2)) } Processor 'matrix interpretation of dimension 1' induces the complexity certificate YES(?,O(n^1)) on application of dependency pairs {1}. These cover all (indirect) predecessors of dependency pairs {1,2}, their number of application is equally bounded. The dependency pairs are shifted into the corresponding weak component(s). We apply the transformation 'removetails' on the sub-problem: Weak DPs: { matrixMult^#(@m1, @m2) -> c_1(matrixMult#1^#(@m1, @m2)) , matrixMult#1^#(::(@l, @ls), @m2) -> c_2(matrixMult^#(@ls, @m2)) } StartTerms: basic terms Strategy: innermost The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { matrixMult^#(@m1, @m2) -> c_1(matrixMult#1^#(@m1, @m2)) , matrixMult#1^#(::(@l, @ls), @m2) -> c_2(matrixMult^#(@ls, @m2)) } 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: { computeLine^#(@line, @m, @acc) -> c_1(computeLine#1^#(@line, @acc, @m)) , computeLine#1^#(::(@x, @xs), @acc, @m) -> c_2(computeLine#2^#(@m, @acc, @x, @xs)) , computeLine#2^#(::(@l, @ls), @acc, @x, @xs) -> c_3(computeLine^#(@xs, @ls, lineMult(@x, @l, @acc)), lineMult^#(@x, @l, @acc)) , lineMult^#(@n, @l1, @l2) -> c_4(lineMult#1^#(@l1, @l2, @n)) , lineMult#1^#(::(@x, @xs), @l2, @n) -> c_5(lineMult#2^#(@l2, @n, @x, @xs)) , lineMult#2^#(::(@y, @ys), @n, @x, @xs) -> c_6(lineMult^#(@n, @xs, @ys)) , lineMult#2^#(nil(), @n, @x, @xs) -> c_7(lineMult^#(@n, @xs, nil())) } Weak DPs: { matrixMult^#(@m1, @m2) -> matrixMult#1^#(@m1, @m2) , matrixMult#1^#(::(@l, @ls), @m2) -> computeLine^#(@l, @m2, nil()) , matrixMult#1^#(::(@l, @ls), @m2) -> matrixMult^#(@ls, @m2) } Weak Trs: { *(@x, @y) -> #mult(@x, @y) , #mult(#0(), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#neg(@x), #0()) -> #0() , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , +(@x, @y) -> #add(@x, @y) , #add(#0(), @y) -> @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)) , lineMult(@n, @l1, @l2) -> lineMult#1(@l1, @l2, @n) , lineMult#1(::(@x, @xs), @l2, @n) -> lineMult#2(@l2, @n, @x, @xs) , lineMult#1(nil(), @l2, @n) -> nil() , lineMult#2(::(@y, @ys), @n, @x, @xs) -> ::(+(*(@x, @n), @y), lineMult(@n, @xs, @ys)) , lineMult#2(nil(), @n, @x, @xs) -> ::(*(@x, @n), lineMult(@n, @xs, nil())) , #pred(#0()) -> #neg(#s(#0())) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #succ(#0()) -> #pos(#s(#0())) , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #natadd(@y, #natmult(@x, @y)) , #natadd(#0(), @y) -> @y , #natadd(#s(@x), @y) -> #s(#natadd(@x, @y)) } 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: computeLine#2^#(::(@l, @ls), @acc, @x, @xs) -> c_3(computeLine^#(@xs, @ls, lineMult(@x, @l, @acc)), lineMult^#(@x, @l, @acc)) , 5: lineMult#1^#(::(@x, @xs), @l2, @n) -> c_5(lineMult#2^#(@l2, @n, @x, @xs)) , 9: matrixMult#1^#(::(@l, @ls), @m2) -> computeLine^#(@l, @m2, nil()) , 10: matrixMult#1^#(::(@l, @ls), @m2) -> matrixMult^#(@ls, @m2) } Trs: { lineMult#1(nil(), @l2, @n) -> nil() } Sub-proof: ---------- The following argument positions are usable: Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_3) = {1, 2}, Uargs(c_4) = {1}, Uargs(c_5) = {1}, Uargs(c_6) = {1}, Uargs(c_7) = {1} TcT has computed following constructor-based matrix interpretation satisfying not(EDA). [*](x1, x2) = [0] [#mult](x1, x2) = [0] [+](x1, x2) = [0] [#add](x1, x2) = [1] x2 + [0] [::](x1, x2) = [1] x1 + [1] x2 + [1] [nil] = [0] [lineMult](x1, x2, x3) = [0] [lineMult#1](x1, x2, x3) = [1] [lineMult#2](x1, x2, x3, x4) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [0] [#0] = [0] [#s](x1) = [0] [#neg](x1) = [0] [#pred](x1) = [0] [#pos](x1) = [0] [#succ](x1) = [0] [#natmult](x1, x2) = [0] [#natadd](x1, x2) = [0] [*^#](x1, x2) = [0] [#mult^#](x1, x2) = [0] [+^#](x1, x2) = [0] [#add^#](x1, x2) = [0] [computeLine^#](x1, x2, x3) = [1] x2 + [0] [computeLine#1^#](x1, x2, x3) = [1] x3 + [0] [computeLine#2^#](x1, x2, x3, x4) = [1] x1 + [0] [lineMult^#](x1, x2, x3) = [1] x2 + [0] [lineMult#1^#](x1, x2, x3) = [1] x1 + [0] [lineMult#2^#](x1, x2, x3, x4) = [1] x4 + [0] [matrixMult^#](x1, x2) = [1] x1 + [1] x2 + [0] [matrixMult#1^#](x1, x2) = [1] x1 + [1] x2 + [0] [#natmult^#](x1, x2) = [0] [#pred^#](x1) = [0] [#succ^#](x1) = [0] [#natadd^#](x1, x2) = [0] [c_1](x1) = [1] x1 + [0] [c_2](x1) = [1] x1 + [0] [c_3](x1, x2) = [1] x1 + [1] x2 + [0] [c_4](x1) = [1] x1 + [0] [c_5](x1) = [1] x1 + [0] [c_6](x1) = [1] x1 + [0] [c_7](x1) = [1] x1 + [0] This order satisfies following ordering constraints [computeLine^#(@line, @m, @acc)] = [1] @m + [0] >= [1] @m + [0] = [c_1(computeLine#1^#(@line, @acc, @m))] [computeLine#1^#(::(@x, @xs), @acc, @m)] = [1] @m + [0] >= [1] @m + [0] = [c_2(computeLine#2^#(@m, @acc, @x, @xs))] [computeLine#2^#(::(@l, @ls), @acc, @x, @xs)] = [1] @l + [1] @ls + [1] > [1] @l + [1] @ls + [0] = [c_3(computeLine^#(@xs, @ls, lineMult(@x, @l, @acc)), lineMult^#(@x, @l, @acc))] [lineMult^#(@n, @l1, @l2)] = [1] @l1 + [0] >= [1] @l1 + [0] = [c_4(lineMult#1^#(@l1, @l2, @n))] [lineMult#1^#(::(@x, @xs), @l2, @n)] = [1] @x + [1] @xs + [1] > [1] @xs + [0] = [c_5(lineMult#2^#(@l2, @n, @x, @xs))] [lineMult#2^#(::(@y, @ys), @n, @x, @xs)] = [1] @xs + [0] >= [1] @xs + [0] = [c_6(lineMult^#(@n, @xs, @ys))] [lineMult#2^#(nil(), @n, @x, @xs)] = [1] @xs + [0] >= [1] @xs + [0] = [c_7(lineMult^#(@n, @xs, nil()))] [matrixMult^#(@m1, @m2)] = [1] @m1 + [1] @m2 + [0] >= [1] @m1 + [1] @m2 + [0] = [matrixMult#1^#(@m1, @m2)] [matrixMult#1^#(::(@l, @ls), @m2)] = [1] @l + [1] @ls + [1] @m2 + [1] > [1] @m2 + [0] = [computeLine^#(@l, @m2, nil())] [matrixMult#1^#(::(@l, @ls), @m2)] = [1] @l + [1] @ls + [1] @m2 + [1] > [1] @ls + [1] @m2 + [0] = [matrixMult^#(@ls, @m2)] Consider the set of all dependency pairs DPs: { 1: computeLine^#(@line, @m, @acc) -> c_1(computeLine#1^#(@line, @acc, @m)) , 2: computeLine#1^#(::(@x, @xs), @acc, @m) -> c_2(computeLine#2^#(@m, @acc, @x, @xs)) , 3: computeLine#2^#(::(@l, @ls), @acc, @x, @xs) -> c_3(computeLine^#(@xs, @ls, lineMult(@x, @l, @acc)), lineMult^#(@x, @l, @acc)) , 4: lineMult^#(@n, @l1, @l2) -> c_4(lineMult#1^#(@l1, @l2, @n)) , 5: lineMult#1^#(::(@x, @xs), @l2, @n) -> c_5(lineMult#2^#(@l2, @n, @x, @xs)) , 6: lineMult#2^#(::(@y, @ys), @n, @x, @xs) -> c_6(lineMult^#(@n, @xs, @ys)) , 7: lineMult#2^#(nil(), @n, @x, @xs) -> c_7(lineMult^#(@n, @xs, nil())) , 8: matrixMult^#(@m1, @m2) -> matrixMult#1^#(@m1, @m2) , 9: matrixMult#1^#(::(@l, @ls), @m2) -> computeLine^#(@l, @m2, nil()) , 10: matrixMult#1^#(::(@l, @ls), @m2) -> matrixMult^#(@ls, @m2) } Processor 'matrix interpretation of dimension 1' induces the complexity certificate YES(?,O(n^1)) on application of dependency pairs {3,5,9,10}. These cover all (indirect) predecessors of dependency pairs {1,2,3,4,5,6,7,8,9,10}, their number of application is equally bounded. The dependency pairs are shifted into the corresponding weak component(s). We apply the transformation 'removetails' on the sub-problem: Weak DPs: { computeLine^#(@line, @m, @acc) -> c_1(computeLine#1^#(@line, @acc, @m)) , computeLine#1^#(::(@x, @xs), @acc, @m) -> c_2(computeLine#2^#(@m, @acc, @x, @xs)) , computeLine#2^#(::(@l, @ls), @acc, @x, @xs) -> c_3(computeLine^#(@xs, @ls, lineMult(@x, @l, @acc)), lineMult^#(@x, @l, @acc)) , lineMult^#(@n, @l1, @l2) -> c_4(lineMult#1^#(@l1, @l2, @n)) , lineMult#1^#(::(@x, @xs), @l2, @n) -> c_5(lineMult#2^#(@l2, @n, @x, @xs)) , lineMult#2^#(::(@y, @ys), @n, @x, @xs) -> c_6(lineMult^#(@n, @xs, @ys)) , lineMult#2^#(nil(), @n, @x, @xs) -> c_7(lineMult^#(@n, @xs, nil())) , matrixMult^#(@m1, @m2) -> matrixMult#1^#(@m1, @m2) , matrixMult#1^#(::(@l, @ls), @m2) -> computeLine^#(@l, @m2, nil()) , matrixMult#1^#(::(@l, @ls), @m2) -> matrixMult^#(@ls, @m2) } Weak Trs: { *(@x, @y) -> #mult(@x, @y) , #mult(#0(), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#neg(@x), #0()) -> #0() , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , +(@x, @y) -> #add(@x, @y) , #add(#0(), @y) -> @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)) , lineMult(@n, @l1, @l2) -> lineMult#1(@l1, @l2, @n) , lineMult#1(::(@x, @xs), @l2, @n) -> lineMult#2(@l2, @n, @x, @xs) , lineMult#1(nil(), @l2, @n) -> nil() , lineMult#2(::(@y, @ys), @n, @x, @xs) -> ::(+(*(@x, @n), @y), lineMult(@n, @xs, @ys)) , lineMult#2(nil(), @n, @x, @xs) -> ::(*(@x, @n), lineMult(@n, @xs, nil())) , #pred(#0()) -> #neg(#s(#0())) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #succ(#0()) -> #pos(#s(#0())) , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #natadd(@y, #natmult(@x, @y)) , #natadd(#0(), @y) -> @y , #natadd(#s(@x), @y) -> #s(#natadd(@x, @y)) } StartTerms: basic terms Strategy: innermost The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. { computeLine^#(@line, @m, @acc) -> c_1(computeLine#1^#(@line, @acc, @m)) , computeLine#1^#(::(@x, @xs), @acc, @m) -> c_2(computeLine#2^#(@m, @acc, @x, @xs)) , computeLine#2^#(::(@l, @ls), @acc, @x, @xs) -> c_3(computeLine^#(@xs, @ls, lineMult(@x, @l, @acc)), lineMult^#(@x, @l, @acc)) , lineMult^#(@n, @l1, @l2) -> c_4(lineMult#1^#(@l1, @l2, @n)) , lineMult#1^#(::(@x, @xs), @l2, @n) -> c_5(lineMult#2^#(@l2, @n, @x, @xs)) , lineMult#2^#(::(@y, @ys), @n, @x, @xs) -> c_6(lineMult^#(@n, @xs, @ys)) , lineMult#2^#(nil(), @n, @x, @xs) -> c_7(lineMult^#(@n, @xs, nil())) , matrixMult^#(@m1, @m2) -> matrixMult#1^#(@m1, @m2) , matrixMult#1^#(::(@l, @ls), @m2) -> computeLine^#(@l, @m2, nil()) , matrixMult#1^#(::(@l, @ls), @m2) -> matrixMult^#(@ls, @m2) } We apply the transformation 'usablerules' on the sub-problem: Weak Trs: { *(@x, @y) -> #mult(@x, @y) , #mult(#0(), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#0(), #pos(@y)) -> #0() , #mult(#neg(@x), #0()) -> #0() , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , +(@x, @y) -> #add(@x, @y) , #add(#0(), @y) -> @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)) , lineMult(@n, @l1, @l2) -> lineMult#1(@l1, @l2, @n) , lineMult#1(::(@x, @xs), @l2, @n) -> lineMult#2(@l2, @n, @x, @xs) , lineMult#1(nil(), @l2, @n) -> nil() , lineMult#2(::(@y, @ys), @n, @x, @xs) -> ::(+(*(@x, @n), @y), lineMult(@n, @xs, @ys)) , lineMult#2(nil(), @n, @x, @xs) -> ::(*(@x, @n), lineMult(@n, @xs, nil())) , #pred(#0()) -> #neg(#s(#0())) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #succ(#0()) -> #pos(#s(#0())) , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #natadd(@y, #natmult(@x, @y)) , #natadd(#0(), @y) -> @y , #natadd(#s(@x), @y) -> #s(#natadd(@x, @y)) } StartTerms: basic terms Strategy: innermost No rule is usable, rules are removed from the input problem. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Rules: Empty Obligation: innermost runtime complexity Answer: YES(O(1),O(1)) Empty rules are trivially bounded Wall-time: 6.3076e-2s CPU-time: 0.484s Wall-time: 17.192754s CPU-time: 117.841s Hurray, we answered YES(O(1),O(n^2))