MAYBE We are left with following problem, upon which TcT provides the certificate MAYBE. 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: MAYBE We use the processor 'matrix interpretation of dimension 1' to orient following rules strictly. Trs: { matrixMult#1(nil(), @m2) -> nil() } The induced complexity on above rules (modulo remaining rules) is YES(?,O(n^1)) . These rules are moved into the corresponding weak component(s). Sub-proof: ---------- The following argument positions are usable: Uargs(+) = {1}, Uargs(computeLine) = {3}, Uargs(::) = {1, 2}, Uargs(#s) = {1}, Uargs(#neg) = {1}, Uargs(#pred) = {1}, Uargs(#pos) = {1}, Uargs(#succ) = {1}, Uargs(#natadd) = {2} TcT has computed following constructor-based matrix interpretation satisfying not(EDA). [*](x1, x2) = [0] [#mult](x1, x2) = [0] [+](x1, x2) = [2] x1 + [1] x2 + [0] [#add](x1, x2) = [1] x2 + [0] [computeLine](x1, x2, x3) = [2] x3 + [0] [computeLine#1](x1, x2, x3) = [2] x2 + [0] [::](x1, x2) = [1] x1 + [1] x2 + [0] [computeLine#2](x1, x2, x3, x4) = [2] x2 + [0] [nil] = [0] [lineMult](x1, x2, x3) = [1] x3 + [0] [lineMult#1](x1, x2, x3) = [1] x2 + [0] [lineMult#2](x1, x2, x3, x4) = [1] x1 + [0] [matrixMult](x1, x2) = [1] [matrixMult#1](x1, x2) = [1] [#0] = [0] [#s](x1) = [1] x1 + [0] [#neg](x1) = [1] x1 + [0] [#pred](x1) = [1] x1 + [0] [#pos](x1) = [1] x1 + [0] [#succ](x1) = [1] x1 + [0] [#natmult](x1, x2) = [0] [#natadd](x1, x2) = [2] x2 + [0] This order satisfies following ordering constraints [*(@x, @y)] = [0] >= [0] = [#mult(@x, @y)] [#mult(#0(), #0())] = [0] >= [0] = [#0()] [#mult(#0(), #neg(@y))] = [0] >= [0] = [#0()] [#mult(#0(), #pos(@y))] = [0] >= [0] = [#0()] [#mult(#neg(@x), #0())] = [0] >= [0] = [#0()] [#mult(#neg(@x), #neg(@y))] = [0] >= [0] = [#pos(#natmult(@x, @y))] [#mult(#neg(@x), #pos(@y))] = [0] >= [0] = [#neg(#natmult(@x, @y))] [#mult(#pos(@x), #0())] = [0] >= [0] = [#0()] [#mult(#pos(@x), #neg(@y))] = [0] >= [0] = [#neg(#natmult(@x, @y))] [#mult(#pos(@x), #pos(@y))] = [0] >= [0] = [#pos(#natmult(@x, @y))] [+(@x, @y)] = [2] @x + [1] @y + [0] >= [1] @y + [0] = [#add(@x, @y)] [#add(#0(), @y)] = [1] @y + [0] >= [1] @y + [0] = [@y] [#add(#neg(#s(#0())), @y)] = [1] @y + [0] >= [1] @y + [0] = [#pred(@y)] [#add(#neg(#s(#s(@x))), @y)] = [1] @y + [0] >= [1] @y + [0] = [#pred(#add(#pos(#s(@x)), @y))] [#add(#pos(#s(#0())), @y)] = [1] @y + [0] >= [1] @y + [0] = [#succ(@y)] [#add(#pos(#s(#s(@x))), @y)] = [1] @y + [0] >= [1] @y + [0] = [#succ(#add(#pos(#s(@x)), @y))] [computeLine(@line, @m, @acc)] = [2] @acc + [0] >= [2] @acc + [0] = [computeLine#1(@line, @acc, @m)] [computeLine#1(::(@x, @xs), @acc, @m)] = [2] @acc + [0] >= [2] @acc + [0] = [computeLine#2(@m, @acc, @x, @xs)] [computeLine#1(nil(), @acc, @m)] = [2] @acc + [0] >= [1] @acc + [0] = [@acc] [computeLine#2(::(@l, @ls), @acc, @x, @xs)] = [2] @acc + [0] >= [2] @acc + [0] = [computeLine(@xs, @ls, lineMult(@x, @l, @acc))] [computeLine#2(nil(), @acc, @x, @xs)] = [2] @acc + [0] >= [0] = [nil()] [lineMult(@n, @l1, @l2)] = [1] @l2 + [0] >= [1] @l2 + [0] = [lineMult#1(@l1, @l2, @n)] [lineMult#1(::(@x, @xs), @l2, @n)] = [1] @l2 + [0] >= [1] @l2 + [0] = [lineMult#2(@l2, @n, @x, @xs)] [lineMult#1(nil(), @l2, @n)] = [1] @l2 + [0] >= [0] = [nil()] [lineMult#2(::(@y, @ys), @n, @x, @xs)] = [1] @y + [1] @ys + [0] >= [1] @y + [1] @ys + [0] = [::(+(*(@x, @n), @y), lineMult(@n, @xs, @ys))] [lineMult#2(nil(), @n, @x, @xs)] = [0] >= [0] = [::(*(@x, @n), lineMult(@n, @xs, nil()))] [matrixMult(@m1, @m2)] = [1] >= [1] = [matrixMult#1(@m1, @m2)] [matrixMult#1(::(@l, @ls), @m2)] = [1] >= [1] = [::(computeLine(@l, @m2, nil()), matrixMult(@ls, @m2))] [matrixMult#1(nil(), @m2)] = [1] > [0] = [nil()] [#pred(#0())] = [0] >= [0] = [#neg(#s(#0()))] [#pred(#neg(#s(@x)))] = [1] @x + [0] >= [1] @x + [0] = [#neg(#s(#s(@x)))] [#pred(#pos(#s(#0())))] = [0] >= [0] = [#0()] [#pred(#pos(#s(#s(@x))))] = [1] @x + [0] >= [1] @x + [0] = [#pos(#s(@x))] [#succ(#0())] = [0] >= [0] = [#pos(#s(#0()))] [#succ(#neg(#s(#0())))] = [0] >= [0] = [#0()] [#succ(#neg(#s(#s(@x))))] = [1] @x + [0] >= [1] @x + [0] = [#neg(#s(@x))] [#succ(#pos(#s(@x)))] = [1] @x + [0] >= [1] @x + [0] = [#pos(#s(#s(@x)))] [#natmult(#0(), @y)] = [0] >= [0] = [#0()] [#natmult(#s(@x), @y)] = [0] >= [0] = [#natadd(@y, #natmult(@x, @y))] [#natadd(#0(), @y)] = [2] @y + [0] >= [1] @y + [0] = [@y] [#natadd(#s(@x), @y)] = [2] @y + [0] >= [2] @y + [0] = [#s(#natadd(@x, @y))] We return to the main proof. We are left with following problem, upon which TcT provides the certificate MAYBE. 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)) } 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)) , 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: MAYBE None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'Fastest' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'trivial' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'empty' failed due to the following reason: Empty strict component of the problem is NOT empty. 2) 'Sequentially' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'Polynomial Path Order (PS)' failed due to the following reason: The input cannot be shown compatible 2) 'Polynomial Path Order' failed due to the following reason: The input cannot be shown compatible 2) 'Fastest (timeout of 5 seconds)' failed due to the following reason: None of the processors succeeded. Details of failed attempt(s): ----------------------------- 1) 'Bounds with minimal-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. 2) 'Bounds with perSymbol-enrichment and initial automaton 'match'' failed due to the following reason: match-boundness of the problem could not be verified. Arrrr..