YES(O(1),O(n^3)) 203.78/60.04 YES(O(1),O(n^3)) 203.78/60.04 203.78/60.04 We are left with following problem, upon which TcT provides the 203.78/60.04 certificate YES(O(1),O(n^3)). 203.78/60.04 203.78/60.04 Strict Trs: 203.78/60.04 { a__nats() -> a__adx(a__zeros()) 203.78/60.04 , a__nats() -> nats() 203.78/60.04 , a__adx(X) -> adx(X) 203.78/60.04 , a__adx(cons(X, Y)) -> a__incr(cons(X, adx(Y))) 203.78/60.04 , a__zeros() -> cons(0(), zeros()) 203.78/60.04 , a__zeros() -> zeros() 203.78/60.04 , a__incr(X) -> incr(X) 203.78/60.04 , a__incr(cons(X, Y)) -> cons(s(X), incr(Y)) 203.78/60.04 , a__hd(X) -> hd(X) 203.78/60.04 , a__hd(cons(X, Y)) -> mark(X) 203.78/60.04 , mark(cons(X1, X2)) -> cons(X1, X2) 203.78/60.04 , mark(0()) -> 0() 203.78/60.04 , mark(zeros()) -> a__zeros() 203.78/60.04 , mark(s(X)) -> s(X) 203.78/60.04 , mark(incr(X)) -> a__incr(mark(X)) 203.78/60.04 , mark(adx(X)) -> a__adx(mark(X)) 203.78/60.04 , mark(nats()) -> a__nats() 203.78/60.04 , mark(hd(X)) -> a__hd(mark(X)) 203.78/60.04 , mark(tl(X)) -> a__tl(mark(X)) 203.78/60.04 , a__tl(X) -> tl(X) 203.78/60.04 , a__tl(cons(X, Y)) -> mark(Y) } 203.78/60.04 Obligation: 203.78/60.04 derivational complexity 203.78/60.04 Answer: 203.78/60.04 YES(O(1),O(n^3)) 203.78/60.04 203.78/60.04 We use the processor 'matrix interpretation of dimension 3' to 203.78/60.04 orient following rules strictly. 203.78/60.04 203.78/60.04 Trs: 203.78/60.04 { a__nats() -> a__adx(a__zeros()) 203.78/60.04 , a__nats() -> nats() 203.78/60.04 , a__adx(X) -> adx(X) 203.78/60.04 , a__adx(cons(X, Y)) -> a__incr(cons(X, adx(Y))) 203.78/60.04 , a__zeros() -> cons(0(), zeros()) 203.78/60.04 , a__zeros() -> zeros() 203.78/60.04 , a__incr(X) -> incr(X) 203.78/60.04 , a__incr(cons(X, Y)) -> cons(s(X), incr(Y)) 203.78/60.04 , a__hd(X) -> hd(X) 203.78/60.04 , a__hd(cons(X, Y)) -> mark(X) 203.78/60.04 , mark(cons(X1, X2)) -> cons(X1, X2) 203.78/60.04 , mark(0()) -> 0() 203.78/60.04 , mark(zeros()) -> a__zeros() 203.78/60.04 , mark(s(X)) -> s(X) 203.78/60.04 , mark(incr(X)) -> a__incr(mark(X)) 203.78/60.04 , mark(adx(X)) -> a__adx(mark(X)) 203.78/60.04 , mark(nats()) -> a__nats() 203.78/60.04 , mark(hd(X)) -> a__hd(mark(X)) 203.78/60.04 , mark(tl(X)) -> a__tl(mark(X)) 203.78/60.04 , a__tl(X) -> tl(X) 203.78/60.04 , a__tl(cons(X, Y)) -> mark(Y) } 203.78/60.04 203.78/60.04 The induced complexity on above rules (modulo remaining rules) is 203.78/60.04 YES(?,O(n^3)) . These rules are removed from the problem. Note that 203.78/60.04 none of the weakly oriented rules is size-increasing. The overall 203.78/60.04 complexity is obtained by composition . 203.78/60.04 203.78/60.04 Sub-proof: 203.78/60.04 ---------- 203.78/60.04 TcT has computed the following matrix interpretation satisfying 203.78/60.04 not(EDA). 203.78/60.04 203.78/60.04 [7] 203.78/60.04 [a__nats] = [1] 203.78/60.04 [4] 203.78/60.04 203.78/60.04 [1 0 1] [2] 203.78/60.04 [a__adx](x1) = [0 1 0] x1 + [1] 203.78/60.04 [0 0 1] [1] 203.78/60.04 203.78/60.04 [1] 203.78/60.04 [a__zeros] = [0] 203.78/60.04 [3] 203.78/60.04 203.78/60.04 [1 0 0] [1 0 0] [0] 203.78/60.04 [cons](x1, x2) = [0 1 0] x1 + [0 1 0] x2 + [0] 203.78/60.04 [0 0 1] [0 0 1] [0] 203.78/60.04 203.78/60.04 [0] 203.78/60.04 [0] = [0] 203.78/60.04 [0] 203.78/60.04 203.78/60.04 [0] 203.78/60.04 [zeros] = [0] 203.78/60.04 [3] 203.78/60.04 203.78/60.04 [1 0 0] [1] 203.78/60.04 [a__incr](x1) = [0 1 0] x1 + [1] 203.78/60.04 [0 0 1] [0] 203.78/60.04 203.78/60.04 [1 0 0] [0] 203.78/60.04 [s](x1) = [0 0 0] x1 + [0] 203.78/60.04 [0 0 0] [0] 203.78/60.04 203.78/60.04 [1 0 0] [0] 203.78/60.04 [incr](x1) = [0 1 0] x1 + [1] 203.78/60.04 [0 0 1] [0] 203.78/60.04 203.78/60.04 [1 0 1] [0] 203.78/60.04 [adx](x1) = [0 1 0] x1 + [0] 203.78/60.04 [0 0 1] [1] 203.78/60.04 203.78/60.04 [1 2 4] [3] 203.78/60.04 [a__hd](x1) = [0 1 1] x1 + [4] 203.78/60.04 [0 0 1] [0] 203.78/60.04 203.78/60.04 [1 2 3] [2] 203.78/60.04 [mark](x1) = [0 1 1] x1 + [0] 203.78/60.04 [0 0 1] [0] 203.78/60.04 203.78/60.04 [1 2 3] [4] 203.78/60.04 [a__tl](x1) = [0 1 1] x1 + [4] 203.78/60.04 [0 0 1] [0] 203.78/60.04 203.78/60.04 [0] 203.78/60.04 [nats] = [0] 203.78/60.04 [4] 203.78/60.04 203.78/60.04 [1 2 4] [0] 203.78/60.04 [hd](x1) = [0 1 1] x1 + [4] 203.78/60.04 [0 0 1] [0] 203.78/60.04 203.78/60.04 [1 2 3] [0] 203.78/60.04 [tl](x1) = [0 1 1] x1 + [4] 203.78/60.04 [0 0 1] [0] 203.78/60.04 203.78/60.04 The order satisfies the following ordering constraints: 203.78/60.04 203.78/60.04 [a__nats()] = [7] 203.78/60.04 [1] 203.78/60.04 [4] 203.78/60.04 > [6] 203.78/60.04 [1] 203.78/60.04 [4] 203.78/60.04 = [a__adx(a__zeros())] 203.78/60.04 203.78/60.04 [a__nats()] = [7] 203.78/60.04 [1] 203.78/60.04 [4] 203.78/60.04 > [0] 203.78/60.04 [0] 203.78/60.04 [4] 203.78/60.04 = [nats()] 203.78/60.04 203.78/60.04 [a__adx(X)] = [1 0 1] [2] 203.78/60.04 [0 1 0] X + [1] 203.78/60.04 [0 0 1] [1] 203.78/60.04 > [1 0 1] [0] 203.78/60.04 [0 1 0] X + [0] 203.78/60.04 [0 0 1] [1] 203.78/60.04 = [adx(X)] 203.78/60.04 203.78/60.04 [a__adx(cons(X, Y))] = [1 0 1] [1 0 1] [2] 203.78/60.04 [0 1 0] X + [0 1 0] Y + [1] 203.78/60.04 [0 0 1] [0 0 1] [1] 203.78/60.04 > [1 0 0] [1 0 1] [1] 203.78/60.04 [0 1 0] X + [0 1 0] Y + [1] 203.78/60.04 [0 0 1] [0 0 1] [1] 203.78/60.04 = [a__incr(cons(X, adx(Y)))] 203.78/60.04 203.78/60.04 [a__zeros()] = [1] 203.78/60.04 [0] 203.78/60.04 [3] 203.78/60.04 > [0] 203.78/60.04 [0] 203.78/60.04 [3] 203.78/60.04 = [cons(0(), zeros())] 203.78/60.04 203.78/60.04 [a__zeros()] = [1] 203.78/60.04 [0] 203.78/60.04 [3] 203.78/60.04 > [0] 203.78/60.04 [0] 203.78/60.04 [3] 203.78/60.04 = [zeros()] 203.78/60.04 203.78/60.04 [a__incr(X)] = [1 0 0] [1] 203.78/60.04 [0 1 0] X + [1] 203.78/60.04 [0 0 1] [0] 203.78/60.04 > [1 0 0] [0] 203.78/60.04 [0 1 0] X + [1] 203.78/60.04 [0 0 1] [0] 203.78/60.04 = [incr(X)] 203.78/60.04 203.78/60.04 [a__incr(cons(X, Y))] = [1 0 0] [1 0 0] [1] 203.78/60.04 [0 1 0] X + [0 1 0] Y + [1] 203.78/60.04 [0 0 1] [0 0 1] [0] 203.78/60.04 > [1 0 0] [1 0 0] [0] 203.78/60.04 [0 0 0] X + [0 1 0] Y + [1] 203.78/60.04 [0 0 0] [0 0 1] [0] 203.78/60.04 = [cons(s(X), incr(Y))] 203.78/60.04 203.78/60.04 [a__hd(X)] = [1 2 4] [3] 203.78/60.04 [0 1 1] X + [4] 203.78/60.04 [0 0 1] [0] 203.78/60.04 > [1 2 4] [0] 203.78/60.04 [0 1 1] X + [4] 203.78/60.04 [0 0 1] [0] 203.78/60.04 = [hd(X)] 203.78/60.04 203.78/60.04 [a__hd(cons(X, Y))] = [1 2 4] [1 2 4] [3] 203.78/60.04 [0 1 1] X + [0 1 1] Y + [4] 203.78/60.04 [0 0 1] [0 0 1] [0] 203.78/60.04 > [1 2 3] [2] 203.78/60.04 [0 1 1] X + [0] 203.78/60.04 [0 0 1] [0] 203.78/60.04 = [mark(X)] 203.78/60.04 203.78/60.04 [mark(cons(X1, X2))] = [1 2 3] [1 2 3] [2] 203.78/60.04 [0 1 1] X1 + [0 1 1] X2 + [0] 203.78/60.04 [0 0 1] [0 0 1] [0] 203.78/60.04 > [1 0 0] [1 0 0] [0] 203.78/60.04 [0 1 0] X1 + [0 1 0] X2 + [0] 203.78/60.04 [0 0 1] [0 0 1] [0] 203.78/60.04 = [cons(X1, X2)] 203.78/60.04 203.78/60.04 [mark(0())] = [2] 203.78/60.04 [0] 203.78/60.04 [0] 203.78/60.04 > [0] 203.78/60.04 [0] 203.78/60.04 [0] 203.78/60.04 = [0()] 203.78/60.04 203.78/60.04 [mark(zeros())] = [11] 203.78/60.04 [3] 203.78/60.04 [3] 203.78/60.04 > [1] 203.78/60.04 [0] 203.78/60.04 [3] 203.78/60.04 = [a__zeros()] 203.78/60.04 203.78/60.04 [mark(s(X))] = [1 0 0] [2] 203.78/60.04 [0 0 0] X + [0] 203.78/60.04 [0 0 0] [0] 203.78/60.04 > [1 0 0] [0] 203.78/60.04 [0 0 0] X + [0] 203.78/60.04 [0 0 0] [0] 203.78/60.04 = [s(X)] 203.78/60.04 203.78/60.04 [mark(incr(X))] = [1 2 3] [4] 203.78/60.04 [0 1 1] X + [1] 203.78/60.04 [0 0 1] [0] 203.78/60.04 > [1 2 3] [3] 203.78/60.04 [0 1 1] X + [1] 203.78/60.04 [0 0 1] [0] 203.78/60.04 = [a__incr(mark(X))] 203.78/60.04 203.78/60.04 [mark(adx(X))] = [1 2 4] [5] 203.78/60.04 [0 1 1] X + [1] 203.78/60.04 [0 0 1] [1] 203.78/60.04 > [1 2 4] [4] 203.78/60.04 [0 1 1] X + [1] 203.78/60.04 [0 0 1] [1] 203.78/60.04 = [a__adx(mark(X))] 203.78/60.04 203.78/60.04 [mark(nats())] = [14] 203.78/60.04 [4] 203.78/60.04 [4] 203.78/60.04 > [7] 203.78/60.04 [1] 203.78/60.04 [4] 203.78/60.04 = [a__nats()] 203.78/60.04 203.78/60.04 [mark(hd(X))] = [1 4 9] [10] 203.78/60.04 [0 1 2] X + [4] 203.78/60.04 [0 0 1] [0] 203.78/60.04 > [1 4 9] [5] 203.78/60.04 [0 1 2] X + [4] 203.78/60.04 [0 0 1] [0] 203.78/60.04 = [a__hd(mark(X))] 203.78/60.04 203.78/60.04 [mark(tl(X))] = [1 4 8] [10] 203.78/60.04 [0 1 2] X + [4] 203.78/60.04 [0 0 1] [0] 203.78/60.04 > [1 4 8] [6] 203.78/60.04 [0 1 2] X + [4] 203.78/60.04 [0 0 1] [0] 203.78/60.04 = [a__tl(mark(X))] 203.78/60.04 203.78/60.04 [a__tl(X)] = [1 2 3] [4] 203.78/60.04 [0 1 1] X + [4] 203.78/60.04 [0 0 1] [0] 203.78/60.04 > [1 2 3] [0] 203.78/60.04 [0 1 1] X + [4] 203.78/60.04 [0 0 1] [0] 203.78/60.04 = [tl(X)] 203.78/60.04 203.78/60.04 [a__tl(cons(X, Y))] = [1 2 3] [1 2 3] [4] 203.78/60.04 [0 1 1] X + [0 1 1] Y + [4] 203.78/60.04 [0 0 1] [0 0 1] [0] 203.78/60.04 > [1 2 3] [2] 203.78/60.04 [0 1 1] Y + [0] 203.78/60.04 [0 0 1] [0] 203.78/60.04 = [mark(Y)] 203.78/60.04 203.78/60.04 203.78/60.04 We return to the main proof. 203.78/60.04 203.78/60.04 We are left with following problem, upon which TcT provides the 203.78/60.04 certificate YES(O(1),O(1)). 203.78/60.04 203.78/60.04 Rules: Empty 203.78/60.04 Obligation: 203.78/60.04 derivational complexity 203.78/60.04 Answer: 203.78/60.04 YES(O(1),O(1)) 203.78/60.04 203.78/60.04 Empty rules are trivially bounded 203.78/60.04 203.78/60.04 Hurray, we answered YES(O(1),O(n^3)) 203.78/60.05 EOF