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: { 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 , matrixMult#1(::(@l, @ls), @m2) -> ::(computeLine(@l, @m2, nil()), matrixMult(@ls, @m2)) , matrixMult#1(nil(), @m2) -> nil() , lineMult(@n, @l1, @l2) -> lineMult#1(@l1, @l2, @n) , +(@x, @y) -> #add(@x, @y) , 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) , *(@x, @y) -> #mult(@x, @y) , lineMult#1(::(@x, @xs), @l2, @n) -> lineMult#2(@l2, @n, @x, @xs) , lineMult#1(nil(), @l2, @n) -> nil() , computeLine#2(::(@l, @ls), @acc, @x, @xs) -> computeLine(@xs, @ls, lineMult(@x, @l, @acc)) , computeLine#2(nil(), @acc, @x, @xs) -> nil() } Weak Trs: { #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@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)) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We add the following dependency tuples: 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#1^#(nil(), @acc, @m) -> c_3() , computeLine#2^#(::(@l, @ls), @acc, @x, @xs) -> c_14(computeLine^#(@xs, @ls, lineMult(@x, @l, @acc)), lineMult^#(@x, @l, @acc)) , computeLine#2^#(nil(), @acc, @x, @xs) -> c_15() , matrixMult#1^#(::(@l, @ls), @m2) -> c_4(computeLine^#(@l, @m2, nil()), matrixMult^#(@ls, @m2)) , matrixMult#1^#(nil(), @m2) -> c_5() , matrixMult^#(@m1, @m2) -> c_10(matrixMult#1^#(@m1, @m2)) , lineMult^#(@n, @l1, @l2) -> c_6(lineMult#1^#(@l1, @l2, @n)) , lineMult#1^#(::(@x, @xs), @l2, @n) -> c_12(lineMult#2^#(@l2, @n, @x, @xs)) , lineMult#1^#(nil(), @l2, @n) -> c_13() , +^#(@x, @y) -> c_7(#add^#(@x, @y)) , lineMult#2^#(::(@y, @ys), @n, @x, @xs) -> c_8(+^#(*(@x, @n), @y), *^#(@x, @n), lineMult^#(@n, @xs, @ys)) , lineMult#2^#(nil(), @n, @x, @xs) -> c_9(*^#(@x, @n), lineMult^#(@n, @xs, nil())) , *^#(@x, @y) -> c_11(#mult^#(@x, @y)) } Weak DPs: { #add^#(#pos(#s(#0())), @y) -> c_31(#succ^#(@y)) , #add^#(#pos(#s(#s(@x))), @y) -> c_32(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , #add^#(#0(), @y) -> c_33() , #add^#(#neg(#s(#0())), @y) -> c_34(#pred^#(@y)) , #add^#(#neg(#s(#s(@x))), @y) -> c_35(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , #mult^#(#pos(@x), #pos(@y)) -> c_18(#natmult^#(@x, @y)) , #mult^#(#pos(@x), #0()) -> c_19() , #mult^#(#pos(@x), #neg(@y)) -> c_20(#natmult^#(@x, @y)) , #mult^#(#0(), #pos(@y)) -> c_21() , #mult^#(#0(), #0()) -> c_22() , #mult^#(#0(), #neg(@y)) -> c_23() , #mult^#(#neg(@x), #pos(@y)) -> c_24(#natmult^#(@x, @y)) , #mult^#(#neg(@x), #0()) -> c_25() , #mult^#(#neg(@x), #neg(@y)) -> c_26(#natmult^#(@x, @y)) , #natmult^#(#0(), @y) -> c_16() , #natmult^#(#s(@x), @y) -> c_17(#add^#(#pos(@y), #natmult(@x, @y)), #natmult^#(@x, @y)) , #succ^#(#pos(#s(@x))) -> c_27() , #succ^#(#0()) -> c_28() , #succ^#(#neg(#s(#0()))) -> c_29() , #succ^#(#neg(#s(#s(@x)))) -> c_30() , #pred^#(#pos(#s(#0()))) -> c_36() , #pred^#(#pos(#s(#s(@x)))) -> c_37() , #pred^#(#0()) -> c_38() , #pred^#(#neg(#s(@x))) -> c_39() } 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: { 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#1^#(nil(), @acc, @m) -> c_3() , computeLine#2^#(::(@l, @ls), @acc, @x, @xs) -> c_14(computeLine^#(@xs, @ls, lineMult(@x, @l, @acc)), lineMult^#(@x, @l, @acc)) , computeLine#2^#(nil(), @acc, @x, @xs) -> c_15() , matrixMult#1^#(::(@l, @ls), @m2) -> c_4(computeLine^#(@l, @m2, nil()), matrixMult^#(@ls, @m2)) , matrixMult#1^#(nil(), @m2) -> c_5() , matrixMult^#(@m1, @m2) -> c_10(matrixMult#1^#(@m1, @m2)) , lineMult^#(@n, @l1, @l2) -> c_6(lineMult#1^#(@l1, @l2, @n)) , lineMult#1^#(::(@x, @xs), @l2, @n) -> c_12(lineMult#2^#(@l2, @n, @x, @xs)) , lineMult#1^#(nil(), @l2, @n) -> c_13() , +^#(@x, @y) -> c_7(#add^#(@x, @y)) , lineMult#2^#(::(@y, @ys), @n, @x, @xs) -> c_8(+^#(*(@x, @n), @y), *^#(@x, @n), lineMult^#(@n, @xs, @ys)) , lineMult#2^#(nil(), @n, @x, @xs) -> c_9(*^#(@x, @n), lineMult^#(@n, @xs, nil())) , *^#(@x, @y) -> c_11(#mult^#(@x, @y)) } Weak DPs: { #add^#(#pos(#s(#0())), @y) -> c_31(#succ^#(@y)) , #add^#(#pos(#s(#s(@x))), @y) -> c_32(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , #add^#(#0(), @y) -> c_33() , #add^#(#neg(#s(#0())), @y) -> c_34(#pred^#(@y)) , #add^#(#neg(#s(#s(@x))), @y) -> c_35(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , #mult^#(#pos(@x), #pos(@y)) -> c_18(#natmult^#(@x, @y)) , #mult^#(#pos(@x), #0()) -> c_19() , #mult^#(#pos(@x), #neg(@y)) -> c_20(#natmult^#(@x, @y)) , #mult^#(#0(), #pos(@y)) -> c_21() , #mult^#(#0(), #0()) -> c_22() , #mult^#(#0(), #neg(@y)) -> c_23() , #mult^#(#neg(@x), #pos(@y)) -> c_24(#natmult^#(@x, @y)) , #mult^#(#neg(@x), #0()) -> c_25() , #mult^#(#neg(@x), #neg(@y)) -> c_26(#natmult^#(@x, @y)) , #natmult^#(#0(), @y) -> c_16() , #natmult^#(#s(@x), @y) -> c_17(#add^#(#pos(@y), #natmult(@x, @y)), #natmult^#(@x, @y)) , #succ^#(#pos(#s(@x))) -> c_27() , #succ^#(#0()) -> c_28() , #succ^#(#neg(#s(#0()))) -> c_29() , #succ^#(#neg(#s(#s(@x)))) -> c_30() , #pred^#(#pos(#s(#0()))) -> c_36() , #pred^#(#pos(#s(#s(@x)))) -> c_37() , #pred^#(#0()) -> c_38() , #pred^#(#neg(#s(@x))) -> c_39() } Weak Trs: { #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@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 , matrixMult#1(::(@l, @ls), @m2) -> ::(computeLine(@l, @m2, nil()), matrixMult(@ls, @m2)) , matrixMult#1(nil(), @m2) -> nil() , lineMult(@n, @l1, @l2) -> lineMult#1(@l1, @l2, @n) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , +(@x, @y) -> #add(@x, @y) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , 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())) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@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)) , matrixMult(@m1, @m2) -> matrixMult#1(@m1, @m2) , *(@x, @y) -> #mult(@x, @y) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , lineMult#1(::(@x, @xs), @l2, @n) -> lineMult#2(@l2, @n, @x, @xs) , lineMult#1(nil(), @l2, @n) -> nil() , computeLine#2(::(@l, @ls), @acc, @x, @xs) -> computeLine(@xs, @ls, lineMult(@x, @l, @acc)) , computeLine#2(nil(), @acc, @x, @xs) -> nil() } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We estimate the number of application of {3,5,7,11,12,15} by applications of Pre({3,5,7,11,12,15}) = {1,2,8,9,13,14}. Here rules are labeled as follows: 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#1^#(nil(), @acc, @m) -> c_3() , 4: computeLine#2^#(::(@l, @ls), @acc, @x, @xs) -> c_14(computeLine^#(@xs, @ls, lineMult(@x, @l, @acc)), lineMult^#(@x, @l, @acc)) , 5: computeLine#2^#(nil(), @acc, @x, @xs) -> c_15() , 6: matrixMult#1^#(::(@l, @ls), @m2) -> c_4(computeLine^#(@l, @m2, nil()), matrixMult^#(@ls, @m2)) , 7: matrixMult#1^#(nil(), @m2) -> c_5() , 8: matrixMult^#(@m1, @m2) -> c_10(matrixMult#1^#(@m1, @m2)) , 9: lineMult^#(@n, @l1, @l2) -> c_6(lineMult#1^#(@l1, @l2, @n)) , 10: lineMult#1^#(::(@x, @xs), @l2, @n) -> c_12(lineMult#2^#(@l2, @n, @x, @xs)) , 11: lineMult#1^#(nil(), @l2, @n) -> c_13() , 12: +^#(@x, @y) -> c_7(#add^#(@x, @y)) , 13: lineMult#2^#(::(@y, @ys), @n, @x, @xs) -> c_8(+^#(*(@x, @n), @y), *^#(@x, @n), lineMult^#(@n, @xs, @ys)) , 14: lineMult#2^#(nil(), @n, @x, @xs) -> c_9(*^#(@x, @n), lineMult^#(@n, @xs, nil())) , 15: *^#(@x, @y) -> c_11(#mult^#(@x, @y)) , 16: #add^#(#pos(#s(#0())), @y) -> c_31(#succ^#(@y)) , 17: #add^#(#pos(#s(#s(@x))), @y) -> c_32(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , 18: #add^#(#0(), @y) -> c_33() , 19: #add^#(#neg(#s(#0())), @y) -> c_34(#pred^#(@y)) , 20: #add^#(#neg(#s(#s(@x))), @y) -> c_35(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , 21: #mult^#(#pos(@x), #pos(@y)) -> c_18(#natmult^#(@x, @y)) , 22: #mult^#(#pos(@x), #0()) -> c_19() , 23: #mult^#(#pos(@x), #neg(@y)) -> c_20(#natmult^#(@x, @y)) , 24: #mult^#(#0(), #pos(@y)) -> c_21() , 25: #mult^#(#0(), #0()) -> c_22() , 26: #mult^#(#0(), #neg(@y)) -> c_23() , 27: #mult^#(#neg(@x), #pos(@y)) -> c_24(#natmult^#(@x, @y)) , 28: #mult^#(#neg(@x), #0()) -> c_25() , 29: #mult^#(#neg(@x), #neg(@y)) -> c_26(#natmult^#(@x, @y)) , 30: #natmult^#(#0(), @y) -> c_16() , 31: #natmult^#(#s(@x), @y) -> c_17(#add^#(#pos(@y), #natmult(@x, @y)), #natmult^#(@x, @y)) , 32: #succ^#(#pos(#s(@x))) -> c_27() , 33: #succ^#(#0()) -> c_28() , 34: #succ^#(#neg(#s(#0()))) -> c_29() , 35: #succ^#(#neg(#s(#s(@x)))) -> c_30() , 36: #pred^#(#pos(#s(#0()))) -> c_36() , 37: #pred^#(#pos(#s(#s(@x)))) -> c_37() , 38: #pred^#(#0()) -> c_38() , 39: #pred^#(#neg(#s(@x))) -> c_39() } 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_14(computeLine^#(@xs, @ls, lineMult(@x, @l, @acc)), lineMult^#(@x, @l, @acc)) , matrixMult#1^#(::(@l, @ls), @m2) -> c_4(computeLine^#(@l, @m2, nil()), matrixMult^#(@ls, @m2)) , matrixMult^#(@m1, @m2) -> c_10(matrixMult#1^#(@m1, @m2)) , lineMult^#(@n, @l1, @l2) -> c_6(lineMult#1^#(@l1, @l2, @n)) , lineMult#1^#(::(@x, @xs), @l2, @n) -> c_12(lineMult#2^#(@l2, @n, @x, @xs)) , lineMult#2^#(::(@y, @ys), @n, @x, @xs) -> c_8(+^#(*(@x, @n), @y), *^#(@x, @n), lineMult^#(@n, @xs, @ys)) , lineMult#2^#(nil(), @n, @x, @xs) -> c_9(*^#(@x, @n), lineMult^#(@n, @xs, nil())) } Weak DPs: { computeLine#1^#(nil(), @acc, @m) -> c_3() , computeLine#2^#(nil(), @acc, @x, @xs) -> c_15() , matrixMult#1^#(nil(), @m2) -> c_5() , lineMult#1^#(nil(), @l2, @n) -> c_13() , +^#(@x, @y) -> c_7(#add^#(@x, @y)) , #add^#(#pos(#s(#0())), @y) -> c_31(#succ^#(@y)) , #add^#(#pos(#s(#s(@x))), @y) -> c_32(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , #add^#(#0(), @y) -> c_33() , #add^#(#neg(#s(#0())), @y) -> c_34(#pred^#(@y)) , #add^#(#neg(#s(#s(@x))), @y) -> c_35(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , *^#(@x, @y) -> c_11(#mult^#(@x, @y)) , #mult^#(#pos(@x), #pos(@y)) -> c_18(#natmult^#(@x, @y)) , #mult^#(#pos(@x), #0()) -> c_19() , #mult^#(#pos(@x), #neg(@y)) -> c_20(#natmult^#(@x, @y)) , #mult^#(#0(), #pos(@y)) -> c_21() , #mult^#(#0(), #0()) -> c_22() , #mult^#(#0(), #neg(@y)) -> c_23() , #mult^#(#neg(@x), #pos(@y)) -> c_24(#natmult^#(@x, @y)) , #mult^#(#neg(@x), #0()) -> c_25() , #mult^#(#neg(@x), #neg(@y)) -> c_26(#natmult^#(@x, @y)) , #natmult^#(#0(), @y) -> c_16() , #natmult^#(#s(@x), @y) -> c_17(#add^#(#pos(@y), #natmult(@x, @y)), #natmult^#(@x, @y)) , #succ^#(#pos(#s(@x))) -> c_27() , #succ^#(#0()) -> c_28() , #succ^#(#neg(#s(#0()))) -> c_29() , #succ^#(#neg(#s(#s(@x)))) -> c_30() , #pred^#(#pos(#s(#0()))) -> c_36() , #pred^#(#pos(#s(#s(@x)))) -> c_37() , #pred^#(#0()) -> c_38() , #pred^#(#neg(#s(@x))) -> c_39() } Weak Trs: { #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@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 , matrixMult#1(::(@l, @ls), @m2) -> ::(computeLine(@l, @m2, nil()), matrixMult(@ls, @m2)) , matrixMult#1(nil(), @m2) -> nil() , lineMult(@n, @l1, @l2) -> lineMult#1(@l1, @l2, @n) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , +(@x, @y) -> #add(@x, @y) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , 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())) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@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)) , matrixMult(@m1, @m2) -> matrixMult#1(@m1, @m2) , *(@x, @y) -> #mult(@x, @y) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , lineMult#1(::(@x, @xs), @l2, @n) -> lineMult#2(@l2, @n, @x, @xs) , lineMult#1(nil(), @l2, @n) -> nil() , computeLine#2(::(@l, @ls), @acc, @x, @xs) -> computeLine(@xs, @ls, lineMult(@x, @l, @acc)) , computeLine#2(nil(), @acc, @x, @xs) -> nil() } 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. { computeLine#1^#(nil(), @acc, @m) -> c_3() , computeLine#2^#(nil(), @acc, @x, @xs) -> c_15() , matrixMult#1^#(nil(), @m2) -> c_5() , lineMult#1^#(nil(), @l2, @n) -> c_13() , +^#(@x, @y) -> c_7(#add^#(@x, @y)) , #add^#(#pos(#s(#0())), @y) -> c_31(#succ^#(@y)) , #add^#(#pos(#s(#s(@x))), @y) -> c_32(#succ^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , #add^#(#0(), @y) -> c_33() , #add^#(#neg(#s(#0())), @y) -> c_34(#pred^#(@y)) , #add^#(#neg(#s(#s(@x))), @y) -> c_35(#pred^#(#add(#pos(#s(@x)), @y)), #add^#(#pos(#s(@x)), @y)) , *^#(@x, @y) -> c_11(#mult^#(@x, @y)) , #mult^#(#pos(@x), #pos(@y)) -> c_18(#natmult^#(@x, @y)) , #mult^#(#pos(@x), #0()) -> c_19() , #mult^#(#pos(@x), #neg(@y)) -> c_20(#natmult^#(@x, @y)) , #mult^#(#0(), #pos(@y)) -> c_21() , #mult^#(#0(), #0()) -> c_22() , #mult^#(#0(), #neg(@y)) -> c_23() , #mult^#(#neg(@x), #pos(@y)) -> c_24(#natmult^#(@x, @y)) , #mult^#(#neg(@x), #0()) -> c_25() , #mult^#(#neg(@x), #neg(@y)) -> c_26(#natmult^#(@x, @y)) , #natmult^#(#0(), @y) -> c_16() , #natmult^#(#s(@x), @y) -> c_17(#add^#(#pos(@y), #natmult(@x, @y)), #natmult^#(@x, @y)) , #succ^#(#pos(#s(@x))) -> c_27() , #succ^#(#0()) -> c_28() , #succ^#(#neg(#s(#0()))) -> c_29() , #succ^#(#neg(#s(#s(@x)))) -> c_30() , #pred^#(#pos(#s(#0()))) -> c_36() , #pred^#(#pos(#s(#s(@x)))) -> c_37() , #pred^#(#0()) -> c_38() , #pred^#(#neg(#s(@x))) -> c_39() } 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_14(computeLine^#(@xs, @ls, lineMult(@x, @l, @acc)), lineMult^#(@x, @l, @acc)) , matrixMult#1^#(::(@l, @ls), @m2) -> c_4(computeLine^#(@l, @m2, nil()), matrixMult^#(@ls, @m2)) , matrixMult^#(@m1, @m2) -> c_10(matrixMult#1^#(@m1, @m2)) , lineMult^#(@n, @l1, @l2) -> c_6(lineMult#1^#(@l1, @l2, @n)) , lineMult#1^#(::(@x, @xs), @l2, @n) -> c_12(lineMult#2^#(@l2, @n, @x, @xs)) , lineMult#2^#(::(@y, @ys), @n, @x, @xs) -> c_8(+^#(*(@x, @n), @y), *^#(@x, @n), lineMult^#(@n, @xs, @ys)) , lineMult#2^#(nil(), @n, @x, @xs) -> c_9(*^#(@x, @n), lineMult^#(@n, @xs, nil())) } Weak Trs: { #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@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 , matrixMult#1(::(@l, @ls), @m2) -> ::(computeLine(@l, @m2, nil()), matrixMult(@ls, @m2)) , matrixMult#1(nil(), @m2) -> nil() , lineMult(@n, @l1, @l2) -> lineMult#1(@l1, @l2, @n) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , +(@x, @y) -> #add(@x, @y) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , 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())) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@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)) , matrixMult(@m1, @m2) -> matrixMult#1(@m1, @m2) , *(@x, @y) -> #mult(@x, @y) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , lineMult#1(::(@x, @xs), @l2, @n) -> lineMult#2(@l2, @n, @x, @xs) , lineMult#1(nil(), @l2, @n) -> nil() , computeLine#2(::(@l, @ls), @acc, @x, @xs) -> computeLine(@xs, @ls, lineMult(@x, @l, @acc)) , computeLine#2(nil(), @acc, @x, @xs) -> nil() } 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_8(+^#(*(@x, @n), @y), *^#(@x, @n), lineMult^#(@n, @xs, @ys)) , lineMult#2^#(nil(), @n, @x, @xs) -> c_9(*^#(@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)) , matrixMult#1^#(::(@l, @ls), @m2) -> c_4(computeLine^#(@l, @m2, nil()), matrixMult^#(@ls, @m2)) , matrixMult^#(@m1, @m2) -> c_5(matrixMult#1^#(@m1, @m2)) , lineMult^#(@n, @l1, @l2) -> c_6(lineMult#1^#(@l1, @l2, @n)) , lineMult#1^#(::(@x, @xs), @l2, @n) -> c_7(lineMult#2^#(@l2, @n, @x, @xs)) , lineMult#2^#(::(@y, @ys), @n, @x, @xs) -> c_8(lineMult^#(@n, @xs, @ys)) , lineMult#2^#(nil(), @n, @x, @xs) -> c_9(lineMult^#(@n, @xs, nil())) } Weak Trs: { #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@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 , matrixMult#1(::(@l, @ls), @m2) -> ::(computeLine(@l, @m2, nil()), matrixMult(@ls, @m2)) , matrixMult#1(nil(), @m2) -> nil() , lineMult(@n, @l1, @l2) -> lineMult#1(@l1, @l2, @n) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , +(@x, @y) -> #add(@x, @y) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , 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())) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@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)) , matrixMult(@m1, @m2) -> matrixMult#1(@m1, @m2) , *(@x, @y) -> #mult(@x, @y) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , lineMult#1(::(@x, @xs), @l2, @n) -> lineMult#2(@l2, @n, @x, @xs) , lineMult#1(nil(), @l2, @n) -> nil() , computeLine#2(::(@l, @ls), @acc, @x, @xs) -> computeLine(@xs, @ls, lineMult(@x, @l, @acc)) , computeLine#2(nil(), @acc, @x, @xs) -> nil() } Obligation: innermost runtime complexity Answer: YES(O(1),O(n^2)) We replace rewrite rules by usable rules: Weak Usable Rules: { #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , lineMult(@n, @l1, @l2) -> lineMult#1(@l1, @l2, @n) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , +(@x, @y) -> #add(@x, @y) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , 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())) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@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)) , *(@x, @y) -> #mult(@x, @y) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , lineMult#1(::(@x, @xs), @l2, @n) -> lineMult#2(@l2, @n, @x, @xs) , lineMult#1(nil(), @l2, @n) -> 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)) , matrixMult#1^#(::(@l, @ls), @m2) -> c_4(computeLine^#(@l, @m2, nil()), matrixMult^#(@ls, @m2)) , matrixMult^#(@m1, @m2) -> c_5(matrixMult#1^#(@m1, @m2)) , lineMult^#(@n, @l1, @l2) -> c_6(lineMult#1^#(@l1, @l2, @n)) , lineMult#1^#(::(@x, @xs), @l2, @n) -> c_7(lineMult#2^#(@l2, @n, @x, @xs)) , lineMult#2^#(::(@y, @ys), @n, @x, @xs) -> c_8(lineMult^#(@n, @xs, @ys)) , lineMult#2^#(nil(), @n, @x, @xs) -> c_9(lineMult^#(@n, @xs, nil())) } Weak Trs: { #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , lineMult(@n, @l1, @l2) -> lineMult#1(@l1, @l2, @n) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , +(@x, @y) -> #add(@x, @y) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , 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())) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@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)) , *(@x, @y) -> #mult(@x, @y) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , lineMult#1(::(@x, @xs), @l2, @n) -> lineMult#2(@l2, @n, @x, @xs) , lineMult#1(nil(), @l2, @n) -> nil() } 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#1^#(::(@l, @ls), @m2) -> c_4(computeLine^#(@l, @m2, nil()), matrixMult^#(@ls, @m2)) , matrixMult^#(@m1, @m2) -> c_5(matrixMult#1^#(@m1, @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_6(lineMult#1^#(@l1, @l2, @n)) , lineMult#1^#(::(@x, @xs), @l2, @n) -> c_7(lineMult#2^#(@l2, @n, @x, @xs)) , lineMult#2^#(::(@y, @ys), @n, @x, @xs) -> c_8(lineMult^#(@n, @xs, @ys)) , lineMult#2^#(nil(), @n, @x, @xs) -> c_9(lineMult^#(@n, @xs, nil())) } Further, following extension rules are added to the lower component. { matrixMult#1^#(::(@l, @ls), @m2) -> computeLine^#(@l, @m2, nil()) , matrixMult#1^#(::(@l, @ls), @m2) -> matrixMult^#(@ls, @m2) , matrixMult^#(@m1, @m2) -> matrixMult#1^#(@m1, @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#1^#(::(@l, @ls), @m2) -> c_4(computeLine^#(@l, @m2, nil()), matrixMult^#(@ls, @m2)) , matrixMult^#(@m1, @m2) -> c_5(matrixMult#1^#(@m1, @m2)) } Weak Trs: { #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , lineMult(@n, @l1, @l2) -> lineMult#1(@l1, @l2, @n) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , +(@x, @y) -> #add(@x, @y) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , 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())) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@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)) , *(@x, @y) -> #mult(@x, @y) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , lineMult#1(::(@x, @xs), @l2, @n) -> lineMult#2(@l2, @n, @x, @xs) , lineMult#1(nil(), @l2, @n) -> nil() } 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#1^#(::(@l, @ls), @m2) -> c_4(computeLine^#(@l, @m2, nil()), matrixMult^#(@ls, @m2)) , matrixMult^#(@m1, @m2) -> c_5(matrixMult#1^#(@m1, @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#1^#(::(@l, @ls), @m2) -> c_4(computeLine^#(@l, @m2, nil()), matrixMult^#(@ls, @m2)) , 2: matrixMult^#(@m1, @m2) -> c_5(matrixMult#1^#(@m1, @m2)) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_4) = {2}, Uargs(c_5) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [#natmult](x1, x2) = [7] x1 + [7] x2 + [0] [computeLine](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [computeLine#1](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [matrixMult#1](x1, x2) = [7] x1 + [7] x2 + [0] [lineMult](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [::](x1, x2) = [1] x1 + [1] x2 + [3] [#mult](x1, x2) = [7] x1 + [7] x2 + [0] [+](x1, x2) = [7] x1 + [7] x2 + [0] [#succ](x1) = [7] x1 + [0] [lineMult#2](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0] [#pos](x1) = [1] x1 + [0] [#add](x1, x2) = [7] x1 + [7] x2 + [0] [#0] = [0] [#neg](x1) = [1] x1 + [0] [matrixMult](x1, x2) = [7] x1 + [7] x2 + [0] [*](x1, x2) = [7] x1 + [7] x2 + [0] [#pred](x1) = [7] x1 + [0] [lineMult#1](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [#s](x1) = [1] x1 + [0] [nil] = [4] [computeLine#2](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0] [computeLine^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [7] [c_1](x1) = [7] x1 + [0] [computeLine#1^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_2](x1) = [7] x1 + [0] [computeLine#2^#](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0] [c_3] = [0] [matrixMult#1^#](x1, x2) = [3] x1 + [0] [c_4](x1, x2) = [7] x1 + [7] x2 + [0] [matrixMult^#](x1, x2) = [3] x1 + [4] [c_5] = [0] [lineMult^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [c_6](x1) = [7] x1 + [0] [lineMult#1^#](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [+^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_7](x1) = [7] x1 + [0] [#add^#](x1, x2) = [7] x1 + [7] x2 + [0] [lineMult#2^#](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0] [c_8](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [*^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_9](x1, x2) = [7] x1 + [7] x2 + [0] [c_10](x1) = [7] x1 + [0] [c_11](x1) = [7] x1 + [0] [#mult^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_12](x1) = [7] x1 + [0] [c_13] = [0] [c_14](x1, x2) = [7] x1 + [7] x2 + [0] [c_15] = [0] [#natmult^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_16] = [0] [c_17](x1, x2) = [7] x1 + [7] x2 + [0] [c_18](x1) = [7] x1 + [0] [c_19] = [0] [c_20](x1) = [7] x1 + [0] [c_21] = [0] [c_22] = [0] [c_23] = [0] [c_24](x1) = [7] x1 + [0] [c_25] = [0] [c_26](x1) = [7] x1 + [0] [#succ^#](x1) = [7] x1 + [0] [c_27] = [0] [c_28] = [0] [c_29] = [0] [c_30] = [0] [c_31](x1) = [7] x1 + [0] [c_32](x1, x2) = [7] x1 + [7] x2 + [0] [c_33] = [0] [c_34](x1) = [7] x1 + [0] [#pred^#](x1) = [7] x1 + [0] [c_35](x1, x2) = [7] x1 + [7] x2 + [0] [c_36] = [0] [c_37] = [0] [c_38] = [0] [c_39] = [0] [c] = [0] [c_1](x1) = [7] x1 + [0] [c_2](x1) = [7] x1 + [0] [c_3](x1, x2) = [7] x1 + [7] x2 + [0] [c_4](x1, x2) = [1] x2 + [4] [c_5](x1) = [1] x1 + [3] [c_6](x1) = [7] x1 + [0] [c_7](x1) = [7] x1 + [0] [c_8](x1) = [7] x1 + [0] [c_9](x1) = [7] x1 + [0] The following symbols are considered usable {matrixMult#1^#, matrixMult^#} The order satisfies the following ordering constraints: [matrixMult#1^#(::(@l, @ls), @m2)] = [3] @l + [3] @ls + [9] > [3] @ls + [8] = [c_4(computeLine^#(@l, @m2, nil()), matrixMult^#(@ls, @m2))] [matrixMult^#(@m1, @m2)] = [3] @m1 + [4] > [3] @m1 + [3] = [c_5(matrixMult#1^#(@m1, @m2))] The strictly oriented rules are moved into the weak component. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak DPs: { matrixMult#1^#(::(@l, @ls), @m2) -> c_4(computeLine^#(@l, @m2, nil()), matrixMult^#(@ls, @m2)) , matrixMult^#(@m1, @m2) -> c_5(matrixMult#1^#(@m1, @m2)) } 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. { matrixMult#1^#(::(@l, @ls), @m2) -> c_4(computeLine^#(@l, @m2, nil()), matrixMult^#(@ls, @m2)) , matrixMult^#(@m1, @m2) -> c_5(matrixMult#1^#(@m1, @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_6(lineMult#1^#(@l1, @l2, @n)) , lineMult#1^#(::(@x, @xs), @l2, @n) -> c_7(lineMult#2^#(@l2, @n, @x, @xs)) , lineMult#2^#(::(@y, @ys), @n, @x, @xs) -> c_8(lineMult^#(@n, @xs, @ys)) , lineMult#2^#(nil(), @n, @x, @xs) -> c_9(lineMult^#(@n, @xs, nil())) } Weak DPs: { matrixMult#1^#(::(@l, @ls), @m2) -> computeLine^#(@l, @m2, nil()) , matrixMult#1^#(::(@l, @ls), @m2) -> matrixMult^#(@ls, @m2) , matrixMult^#(@m1, @m2) -> matrixMult#1^#(@m1, @m2) } Weak Trs: { #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , lineMult(@n, @l1, @l2) -> lineMult#1(@l1, @l2, @n) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , +(@x, @y) -> #add(@x, @y) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , 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())) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@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)) , *(@x, @y) -> #mult(@x, @y) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , lineMult#1(::(@x, @xs), @l2, @n) -> lineMult#2(@l2, @n, @x, @xs) , lineMult#1(nil(), @l2, @n) -> nil() } 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: computeLine#1^#(::(@x, @xs), @acc, @m) -> c_2(computeLine#2^#(@m, @acc, @x, @xs)) , 5: lineMult#1^#(::(@x, @xs), @l2, @n) -> c_7(lineMult#2^#(@l2, @n, @x, @xs)) , 8: matrixMult#1^#(::(@l, @ls), @m2) -> computeLine^#(@l, @m2, nil()) , 9: matrixMult#1^#(::(@l, @ls), @m2) -> matrixMult^#(@ls, @m2) , 10: matrixMult^#(@m1, @m2) -> matrixMult#1^#(@m1, @m2) } Trs: { lineMult#2(::(@y, @ys), @n, @x, @xs) -> ::(+(*(@x, @n), @y), lineMult(@n, @xs, @ys)) } Sub-proof: ---------- The following argument positions are usable: Uargs(c_1) = {1}, Uargs(c_2) = {1}, Uargs(c_3) = {1, 2}, Uargs(c_6) = {1}, Uargs(c_7) = {1}, Uargs(c_8) = {1}, Uargs(c_9) = {1} TcT has computed the following constructor-based matrix interpretation satisfying not(EDA). [#natmult](x1, x2) = [0] [computeLine](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [computeLine#1](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [matrixMult#1](x1, x2) = [7] x1 + [7] x2 + [0] [lineMult](x1, x2, x3) = [1] x2 + [2] x3 + [0] [::](x1, x2) = [1] x1 + [1] x2 + [1] [#mult](x1, x2) = [0] [+](x1, x2) = [2] x2 + [0] [#succ](x1) = [0] [lineMult#2](x1, x2, x3, x4) = [2] x1 + [1] x4 + [1] [#pos](x1) = [0] [#add](x1, x2) = [2] x2 + [0] [#0] = [0] [#neg](x1) = [0] [matrixMult](x1, x2) = [7] x1 + [7] x2 + [0] [*](x1, x2) = [0] [#pred](x1) = [0] [lineMult#1](x1, x2, x3) = [1] x1 + [2] x2 + [0] [#s](x1) = [0] [nil] = [0] [computeLine#2](x1, x2, x3, x4) = [7] x1 + [7] x2 + [7] x3 + [7] x4 + [0] [computeLine^#](x1, x2, x3) = [4] x1 + [1] x2 + [0] [c_1](x1) = [7] x1 + [0] [computeLine#1^#](x1, x2, x3) = [4] x1 + [1] x3 + [0] [c_2](x1) = [7] x1 + [0] [computeLine#2^#](x1, x2, x3, x4) = [1] x1 + [4] x3 + [4] x4 + [0] [c_3] = [0] [matrixMult#1^#](x1, x2) = [4] x1 + [7] x2 + [1] [c_4](x1, x2) = [7] x1 + [7] x2 + [0] [matrixMult^#](x1, x2) = [4] x1 + [7] x2 + [3] [c_5] = [0] [lineMult^#](x1, x2, x3) = [1] x2 + [0] [c_6](x1) = [7] x1 + [0] [lineMult#1^#](x1, x2, x3) = [1] x1 + [0] [+^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_7](x1) = [7] x1 + [0] [#add^#](x1, x2) = [7] x1 + [7] x2 + [0] [lineMult#2^#](x1, x2, x3, x4) = [1] x3 + [1] x4 + [0] [c_8](x1, x2, x3) = [7] x1 + [7] x2 + [7] x3 + [0] [*^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_9](x1, x2) = [7] x1 + [7] x2 + [0] [c_10](x1) = [7] x1 + [0] [c_11](x1) = [7] x1 + [0] [#mult^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_12](x1) = [7] x1 + [0] [c_13] = [0] [c_14](x1, x2) = [7] x1 + [7] x2 + [0] [c_15] = [0] [#natmult^#](x1, x2) = [7] x1 + [7] x2 + [0] [c_16] = [0] [c_17](x1, x2) = [7] x1 + [7] x2 + [0] [c_18](x1) = [7] x1 + [0] [c_19] = [0] [c_20](x1) = [7] x1 + [0] [c_21] = [0] [c_22] = [0] [c_23] = [0] [c_24](x1) = [7] x1 + [0] [c_25] = [0] [c_26](x1) = [7] x1 + [0] [#succ^#](x1) = [7] x1 + [0] [c_27] = [0] [c_28] = [0] [c_29] = [0] [c_30] = [0] [c_31](x1) = [7] x1 + [0] [c_32](x1, x2) = [7] x1 + [7] x2 + [0] [c_33] = [0] [c_34](x1) = [7] x1 + [0] [#pred^#](x1) = [7] x1 + [0] [c_35](x1, x2) = [7] x1 + [7] x2 + [0] [c_36] = [0] [c_37] = [0] [c_38] = [0] [c_39] = [0] [c] = [0] [c_1](x1) = [1] x1 + [0] [c_2](x1) = [1] x1 + [1] [c_3](x1, x2) = [1] x1 + [1] x2 + [1] [c_4](x1, x2) = [7] x1 + [7] x2 + [0] [c_5](x1) = [7] x1 + [0] [c_6](x1) = [1] x1 + [0] [c_7](x1) = [1] x1 + [0] [c_8](x1) = [1] x1 + [0] [c_9](x1) = [1] x1 + [0] The following symbols are considered usable {#natmult, lineMult, #mult, +, #succ, lineMult#2, #add, *, #pred, lineMult#1, computeLine^#, computeLine#1^#, computeLine#2^#, matrixMult#1^#, matrixMult^#, lineMult^#, lineMult#1^#, lineMult#2^#} The order satisfies the following ordering constraints: [#natmult(#0(), @y)] = [0] >= [0] = [#0()] [#natmult(#s(@x), @y)] = [0] >= [0] = [#add(#pos(@y), #natmult(@x, @y))] [lineMult(@n, @l1, @l2)] = [1] @l1 + [2] @l2 + [0] >= [1] @l1 + [2] @l2 + [0] = [lineMult#1(@l1, @l2, @n)] [#mult(#pos(@x), #pos(@y))] = [0] >= [0] = [#pos(#natmult(@x, @y))] [#mult(#pos(@x), #0())] = [0] >= [0] = [#0()] [#mult(#pos(@x), #neg(@y))] = [0] >= [0] = [#neg(#natmult(@x, @y))] [#mult(#0(), #pos(@y))] = [0] >= [0] = [#0()] [#mult(#0(), #0())] = [0] >= [0] = [#0()] [#mult(#0(), #neg(@y))] = [0] >= [0] = [#0()] [#mult(#neg(@x), #pos(@y))] = [0] >= [0] = [#neg(#natmult(@x, @y))] [#mult(#neg(@x), #0())] = [0] >= [0] = [#0()] [#mult(#neg(@x), #neg(@y))] = [0] >= [0] = [#pos(#natmult(@x, @y))] [+(@x, @y)] = [2] @y + [0] >= [2] @y + [0] = [#add(@x, @y)] [#succ(#pos(#s(@x)))] = [0] >= [0] = [#pos(#s(#s(@x)))] [#succ(#0())] = [0] >= [0] = [#pos(#s(#0()))] [#succ(#neg(#s(#0())))] = [0] >= [0] = [#0()] [#succ(#neg(#s(#s(@x))))] = [0] >= [0] = [#neg(#s(@x))] [lineMult#2(::(@y, @ys), @n, @x, @xs)] = [2] @y + [1] @xs + [2] @ys + [3] > [2] @y + [1] @xs + [2] @ys + [1] = [::(+(*(@x, @n), @y), lineMult(@n, @xs, @ys))] [lineMult#2(nil(), @n, @x, @xs)] = [1] @xs + [1] >= [1] @xs + [1] = [::(*(@x, @n), lineMult(@n, @xs, nil()))] [#add(#pos(#s(#0())), @y)] = [2] @y + [0] >= [0] = [#succ(@y)] [#add(#pos(#s(#s(@x))), @y)] = [2] @y + [0] >= [0] = [#succ(#add(#pos(#s(@x)), @y))] [#add(#0(), @y)] = [2] @y + [0] >= [1] @y + [0] = [@y] [#add(#neg(#s(#0())), @y)] = [2] @y + [0] >= [0] = [#pred(@y)] [#add(#neg(#s(#s(@x))), @y)] = [2] @y + [0] >= [0] = [#pred(#add(#pos(#s(@x)), @y))] [*(@x, @y)] = [0] >= [0] = [#mult(@x, @y)] [#pred(#pos(#s(#0())))] = [0] >= [0] = [#0()] [#pred(#pos(#s(#s(@x))))] = [0] >= [0] = [#pos(#s(@x))] [#pred(#0())] = [0] >= [0] = [#neg(#s(#0()))] [#pred(#neg(#s(@x)))] = [0] >= [0] = [#neg(#s(#s(@x)))] [lineMult#1(::(@x, @xs), @l2, @n)] = [1] @x + [1] @xs + [2] @l2 + [1] >= [1] @xs + [2] @l2 + [1] = [lineMult#2(@l2, @n, @x, @xs)] [lineMult#1(nil(), @l2, @n)] = [2] @l2 + [0] >= [0] = [nil()] [computeLine^#(@line, @m, @acc)] = [4] @line + [1] @m + [0] >= [4] @line + [1] @m + [0] = [c_1(computeLine#1^#(@line, @acc, @m))] [computeLine#1^#(::(@x, @xs), @acc, @m)] = [4] @x + [1] @m + [4] @xs + [4] > [4] @x + [1] @m + [4] @xs + [1] = [c_2(computeLine#2^#(@m, @acc, @x, @xs))] [computeLine#2^#(::(@l, @ls), @acc, @x, @xs)] = [4] @x + [4] @xs + [1] @l + [1] @ls + [1] >= [4] @xs + [1] @l + [1] @ls + [1] = [c_3(computeLine^#(@xs, @ls, lineMult(@x, @l, @acc)), lineMult^#(@x, @l, @acc))] [matrixMult#1^#(::(@l, @ls), @m2)] = [4] @l + [4] @ls + [7] @m2 + [5] > [4] @l + [1] @m2 + [0] = [computeLine^#(@l, @m2, nil())] [matrixMult#1^#(::(@l, @ls), @m2)] = [4] @l + [4] @ls + [7] @m2 + [5] > [4] @ls + [7] @m2 + [3] = [matrixMult^#(@ls, @m2)] [matrixMult^#(@m1, @m2)] = [4] @m1 + [7] @m2 + [3] > [4] @m1 + [7] @m2 + [1] = [matrixMult#1^#(@m1, @m2)] [lineMult^#(@n, @l1, @l2)] = [1] @l1 + [0] >= [1] @l1 + [0] = [c_6(lineMult#1^#(@l1, @l2, @n))] [lineMult#1^#(::(@x, @xs), @l2, @n)] = [1] @x + [1] @xs + [1] > [1] @x + [1] @xs + [0] = [c_7(lineMult#2^#(@l2, @n, @x, @xs))] [lineMult#2^#(::(@y, @ys), @n, @x, @xs)] = [1] @x + [1] @xs + [0] >= [1] @xs + [0] = [c_8(lineMult^#(@n, @xs, @ys))] [lineMult#2^#(nil(), @n, @x, @xs)] = [1] @x + [1] @xs + [0] >= [1] @xs + [0] = [c_9(lineMult^#(@n, @xs, nil()))] We return to the main proof. Consider the set of all dependency pairs : { 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_6(lineMult#1^#(@l1, @l2, @n)) , 5: lineMult#1^#(::(@x, @xs), @l2, @n) -> c_7(lineMult#2^#(@l2, @n, @x, @xs)) , 6: lineMult#2^#(::(@y, @ys), @n, @x, @xs) -> c_8(lineMult^#(@n, @xs, @ys)) , 7: lineMult#2^#(nil(), @n, @x, @xs) -> c_9(lineMult^#(@n, @xs, nil())) , 8: matrixMult#1^#(::(@l, @ls), @m2) -> computeLine^#(@l, @m2, nil()) , 9: matrixMult#1^#(::(@l, @ls), @m2) -> matrixMult^#(@ls, @m2) , 10: matrixMult^#(@m1, @m2) -> matrixMult#1^#(@m1, @m2) } Processor 'matrix interpretation of dimension 1' induces the complexity certificate YES(?,O(n^1)) on application of dependency pairs {2,5,8,9,10}. 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 weak component. We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). 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)) , matrixMult#1^#(::(@l, @ls), @m2) -> computeLine^#(@l, @m2, nil()) , matrixMult#1^#(::(@l, @ls), @m2) -> matrixMult^#(@ls, @m2) , matrixMult^#(@m1, @m2) -> matrixMult#1^#(@m1, @m2) , lineMult^#(@n, @l1, @l2) -> c_6(lineMult#1^#(@l1, @l2, @n)) , lineMult#1^#(::(@x, @xs), @l2, @n) -> c_7(lineMult#2^#(@l2, @n, @x, @xs)) , lineMult#2^#(::(@y, @ys), @n, @x, @xs) -> c_8(lineMult^#(@n, @xs, @ys)) , lineMult#2^#(nil(), @n, @x, @xs) -> c_9(lineMult^#(@n, @xs, nil())) } Weak Trs: { #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , lineMult(@n, @l1, @l2) -> lineMult#1(@l1, @l2, @n) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , +(@x, @y) -> #add(@x, @y) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , 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())) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@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)) , *(@x, @y) -> #mult(@x, @y) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , lineMult#1(::(@x, @xs), @l2, @n) -> lineMult#2(@l2, @n, @x, @xs) , lineMult#1(nil(), @l2, @n) -> nil() } 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. { 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)) , matrixMult#1^#(::(@l, @ls), @m2) -> computeLine^#(@l, @m2, nil()) , matrixMult#1^#(::(@l, @ls), @m2) -> matrixMult^#(@ls, @m2) , matrixMult^#(@m1, @m2) -> matrixMult#1^#(@m1, @m2) , lineMult^#(@n, @l1, @l2) -> c_6(lineMult#1^#(@l1, @l2, @n)) , lineMult#1^#(::(@x, @xs), @l2, @n) -> c_7(lineMult#2^#(@l2, @n, @x, @xs)) , lineMult#2^#(::(@y, @ys), @n, @x, @xs) -> c_8(lineMult^#(@n, @xs, @ys)) , lineMult#2^#(nil(), @n, @x, @xs) -> c_9(lineMult^#(@n, @xs, nil())) } We are left with following problem, upon which TcT provides the certificate YES(O(1),O(1)). Weak Trs: { #natmult(#0(), @y) -> #0() , #natmult(#s(@x), @y) -> #add(#pos(@y), #natmult(@x, @y)) , lineMult(@n, @l1, @l2) -> lineMult#1(@l1, @l2, @n) , #mult(#pos(@x), #pos(@y)) -> #pos(#natmult(@x, @y)) , #mult(#pos(@x), #0()) -> #0() , #mult(#pos(@x), #neg(@y)) -> #neg(#natmult(@x, @y)) , #mult(#0(), #pos(@y)) -> #0() , #mult(#0(), #0()) -> #0() , #mult(#0(), #neg(@y)) -> #0() , #mult(#neg(@x), #pos(@y)) -> #neg(#natmult(@x, @y)) , #mult(#neg(@x), #0()) -> #0() , #mult(#neg(@x), #neg(@y)) -> #pos(#natmult(@x, @y)) , +(@x, @y) -> #add(@x, @y) , #succ(#pos(#s(@x))) -> #pos(#s(#s(@x))) , #succ(#0()) -> #pos(#s(#0())) , #succ(#neg(#s(#0()))) -> #0() , #succ(#neg(#s(#s(@x)))) -> #neg(#s(@x)) , 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())) , #add(#pos(#s(#0())), @y) -> #succ(@y) , #add(#pos(#s(#s(@x))), @y) -> #succ(#add(#pos(#s(@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)) , *(@x, @y) -> #mult(@x, @y) , #pred(#pos(#s(#0()))) -> #0() , #pred(#pos(#s(#s(@x)))) -> #pos(#s(@x)) , #pred(#0()) -> #neg(#s(#0())) , #pred(#neg(#s(@x))) -> #neg(#s(#s(@x))) , lineMult#1(::(@x, @xs), @l2, @n) -> lineMult#2(@l2, @n, @x, @xs) , lineMult#1(nil(), @l2, @n) -> nil() } 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 Hurray, we answered YES(O(1),O(n^2))