YES(O(1),O(n^3)) 934.71/297.04 YES(O(1),O(n^3)) 934.71/297.04 934.71/297.04 We are left with following problem, upon which TcT provides the 934.71/297.04 certificate YES(O(1),O(n^3)). 934.71/297.04 934.71/297.04 Strict Trs: 934.71/297.04 { a__nats() -> a__adx(a__zeros()) 934.71/297.04 , a__nats() -> nats() 934.71/297.04 , a__adx(X) -> adx(X) 934.71/297.04 , a__adx(cons(X, Y)) -> a__incr(cons(X, adx(Y))) 934.71/297.04 , a__zeros() -> cons(0(), zeros()) 934.71/297.04 , a__zeros() -> zeros() 934.71/297.04 , a__incr(X) -> incr(X) 934.71/297.04 , a__incr(cons(X, Y)) -> cons(s(X), incr(Y)) 934.71/297.04 , a__hd(X) -> hd(X) 934.71/297.04 , a__hd(cons(X, Y)) -> mark(X) 934.71/297.04 , mark(cons(X1, X2)) -> cons(X1, X2) 934.71/297.04 , mark(0()) -> 0() 934.71/297.04 , mark(zeros()) -> a__zeros() 934.71/297.04 , mark(s(X)) -> s(X) 934.71/297.04 , mark(incr(X)) -> a__incr(mark(X)) 934.71/297.04 , mark(adx(X)) -> a__adx(mark(X)) 934.71/297.04 , mark(nats()) -> a__nats() 934.71/297.04 , mark(hd(X)) -> a__hd(mark(X)) 934.71/297.04 , mark(tl(X)) -> a__tl(mark(X)) 934.71/297.04 , a__tl(X) -> tl(X) 934.71/297.04 , a__tl(cons(X, Y)) -> mark(Y) } 934.71/297.04 Obligation: 934.71/297.04 innermost runtime complexity 934.71/297.04 Answer: 934.71/297.04 YES(O(1),O(n^3)) 934.71/297.04 934.71/297.04 The weightgap principle applies (using the following nonconstant 934.71/297.04 growth matrix-interpretation) 934.71/297.04 934.71/297.04 The following argument positions are usable: 934.71/297.04 Uargs(a__adx) = {1}, Uargs(a__incr) = {1}, Uargs(a__hd) = {1}, 934.71/297.04 Uargs(a__tl) = {1} 934.71/297.04 934.71/297.04 TcT has computed the following matrix interpretation satisfying 934.71/297.04 not(EDA) and not(IDA(1)). 934.71/297.04 934.71/297.04 [a__nats] = [3] 934.71/297.04 934.71/297.04 [a__adx](x1) = [1] x1 + [7] 934.71/297.04 934.71/297.04 [a__zeros] = [7] 934.71/297.04 934.71/297.04 [cons](x1, x2) = [1] x1 + [1] x2 + [1] 934.71/297.04 934.71/297.04 [0] = [7] 934.71/297.04 934.71/297.04 [zeros] = [7] 934.71/297.04 934.71/297.04 [a__incr](x1) = [1] x1 + [7] 934.71/297.04 934.71/297.04 [s](x1) = [1] x1 + [7] 934.71/297.04 934.71/297.04 [incr](x1) = [1] x1 + [7] 934.71/297.04 934.71/297.04 [adx](x1) = [1] x1 + [5] 934.71/297.04 934.71/297.04 [a__hd](x1) = [1] x1 + [5] 934.71/297.04 934.71/297.04 [mark](x1) = [1] x1 + [7] 934.71/297.04 934.71/297.04 [a__tl](x1) = [1] x1 + [7] 934.71/297.04 934.71/297.04 [nats] = [7] 934.71/297.04 934.71/297.04 [hd](x1) = [1] x1 + [7] 934.71/297.04 934.71/297.04 [tl](x1) = [1] x1 + [7] 934.71/297.04 934.71/297.04 The order satisfies the following ordering constraints: 934.71/297.04 934.71/297.04 [a__nats()] = [3] 934.71/297.04 ? [14] 934.71/297.04 = [a__adx(a__zeros())] 934.71/297.04 934.71/297.04 [a__nats()] = [3] 934.71/297.04 ? [7] 934.71/297.04 = [nats()] 934.71/297.04 934.71/297.04 [a__adx(X)] = [1] X + [7] 934.71/297.04 > [1] X + [5] 934.71/297.04 = [adx(X)] 934.71/297.04 934.71/297.04 [a__adx(cons(X, Y))] = [1] X + [1] Y + [8] 934.71/297.04 ? [1] X + [1] Y + [13] 934.71/297.04 = [a__incr(cons(X, adx(Y)))] 934.71/297.04 934.71/297.04 [a__zeros()] = [7] 934.71/297.04 ? [15] 934.71/297.04 = [cons(0(), zeros())] 934.71/297.04 934.71/297.04 [a__zeros()] = [7] 934.71/297.04 >= [7] 934.71/297.04 = [zeros()] 934.71/297.04 934.71/297.04 [a__incr(X)] = [1] X + [7] 934.71/297.04 >= [1] X + [7] 934.71/297.04 = [incr(X)] 934.71/297.04 934.71/297.04 [a__incr(cons(X, Y))] = [1] X + [1] Y + [8] 934.71/297.04 ? [1] X + [1] Y + [15] 934.71/297.04 = [cons(s(X), incr(Y))] 934.71/297.04 934.71/297.04 [a__hd(X)] = [1] X + [5] 934.71/297.04 ? [1] X + [7] 934.71/297.04 = [hd(X)] 934.71/297.04 934.71/297.04 [a__hd(cons(X, Y))] = [1] X + [1] Y + [6] 934.71/297.04 ? [1] X + [7] 934.71/297.04 = [mark(X)] 934.71/297.04 934.71/297.04 [mark(cons(X1, X2))] = [1] X1 + [1] X2 + [8] 934.71/297.04 > [1] X1 + [1] X2 + [1] 934.71/297.04 = [cons(X1, X2)] 934.71/297.04 934.71/297.04 [mark(0())] = [14] 934.71/297.04 > [7] 934.71/297.04 = [0()] 934.71/297.04 934.71/297.04 [mark(zeros())] = [14] 934.71/297.04 > [7] 934.71/297.04 = [a__zeros()] 934.71/297.04 934.71/297.04 [mark(s(X))] = [1] X + [14] 934.71/297.04 > [1] X + [7] 934.71/297.04 = [s(X)] 934.71/297.04 934.71/297.04 [mark(incr(X))] = [1] X + [14] 934.71/297.04 >= [1] X + [14] 934.71/297.04 = [a__incr(mark(X))] 934.71/297.04 934.71/297.04 [mark(adx(X))] = [1] X + [12] 934.71/297.04 ? [1] X + [14] 934.71/297.04 = [a__adx(mark(X))] 934.71/297.04 934.71/297.04 [mark(nats())] = [14] 934.71/297.04 > [3] 934.71/297.04 = [a__nats()] 934.71/297.04 934.71/297.04 [mark(hd(X))] = [1] X + [14] 934.71/297.04 > [1] X + [12] 934.71/297.04 = [a__hd(mark(X))] 934.71/297.04 934.71/297.04 [mark(tl(X))] = [1] X + [14] 934.71/297.04 >= [1] X + [14] 934.71/297.04 = [a__tl(mark(X))] 934.71/297.04 934.71/297.04 [a__tl(X)] = [1] X + [7] 934.71/297.04 >= [1] X + [7] 934.71/297.04 = [tl(X)] 934.71/297.04 934.71/297.04 [a__tl(cons(X, Y))] = [1] X + [1] Y + [8] 934.71/297.04 > [1] Y + [7] 934.71/297.04 = [mark(Y)] 934.71/297.04 934.71/297.04 934.71/297.04 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 934.71/297.04 934.71/297.04 We are left with following problem, upon which TcT provides the 934.71/297.04 certificate YES(O(1),O(n^3)). 934.71/297.04 934.71/297.04 Strict Trs: 934.71/297.04 { a__nats() -> a__adx(a__zeros()) 934.71/297.04 , a__nats() -> nats() 934.71/297.04 , a__adx(cons(X, Y)) -> a__incr(cons(X, adx(Y))) 934.71/297.04 , a__zeros() -> cons(0(), zeros()) 934.71/297.04 , a__zeros() -> zeros() 934.71/297.04 , a__incr(X) -> incr(X) 934.71/297.04 , a__incr(cons(X, Y)) -> cons(s(X), incr(Y)) 934.71/297.04 , a__hd(X) -> hd(X) 934.71/297.04 , a__hd(cons(X, Y)) -> mark(X) 934.71/297.04 , mark(incr(X)) -> a__incr(mark(X)) 934.71/297.04 , mark(adx(X)) -> a__adx(mark(X)) 934.71/297.04 , mark(tl(X)) -> a__tl(mark(X)) 934.71/297.04 , a__tl(X) -> tl(X) } 934.71/297.04 Weak Trs: 934.71/297.04 { a__adx(X) -> adx(X) 934.71/297.04 , mark(cons(X1, X2)) -> cons(X1, X2) 934.71/297.04 , mark(0()) -> 0() 934.71/297.04 , mark(zeros()) -> a__zeros() 934.71/297.04 , mark(s(X)) -> s(X) 934.71/297.04 , mark(nats()) -> a__nats() 934.71/297.04 , mark(hd(X)) -> a__hd(mark(X)) 934.71/297.04 , a__tl(cons(X, Y)) -> mark(Y) } 934.71/297.04 Obligation: 934.71/297.04 innermost runtime complexity 934.71/297.04 Answer: 934.71/297.04 YES(O(1),O(n^3)) 934.71/297.04 934.71/297.04 The weightgap principle applies (using the following nonconstant 934.71/297.04 growth matrix-interpretation) 934.71/297.04 934.71/297.04 The following argument positions are usable: 934.71/297.04 Uargs(a__adx) = {1}, Uargs(a__incr) = {1}, Uargs(a__hd) = {1}, 934.71/297.04 Uargs(a__tl) = {1} 934.71/297.04 934.71/297.04 TcT has computed the following matrix interpretation satisfying 934.71/297.04 not(EDA) and not(IDA(1)). 934.71/297.04 934.71/297.04 [a__nats] = [3] 934.71/297.04 934.71/297.04 [a__adx](x1) = [1] x1 + [4] 934.71/297.04 934.71/297.04 [a__zeros] = [0] 934.71/297.04 934.71/297.04 [cons](x1, x2) = [1] x1 + [1] x2 + [4] 934.71/297.04 934.71/297.04 [0] = [0] 934.71/297.04 934.71/297.04 [zeros] = [0] 934.71/297.04 934.71/297.04 [a__incr](x1) = [1] x1 + [4] 934.71/297.04 934.71/297.04 [s](x1) = [1] x1 + [4] 934.71/297.04 934.71/297.04 [incr](x1) = [1] x1 + [4] 934.71/297.04 934.71/297.04 [adx](x1) = [1] x1 + [0] 934.71/297.04 934.71/297.04 [a__hd](x1) = [1] x1 + [4] 934.71/297.04 934.71/297.04 [mark](x1) = [1] x1 + [5] 934.71/297.04 934.71/297.04 [a__tl](x1) = [1] x1 + [4] 934.71/297.04 934.71/297.04 [nats] = [0] 934.71/297.04 934.71/297.04 [hd](x1) = [1] x1 + [7] 934.71/297.04 934.71/297.04 [tl](x1) = [1] x1 + [0] 934.71/297.04 934.71/297.04 The order satisfies the following ordering constraints: 934.71/297.04 934.71/297.04 [a__nats()] = [3] 934.71/297.04 ? [4] 934.71/297.04 = [a__adx(a__zeros())] 934.71/297.04 934.71/297.04 [a__nats()] = [3] 934.71/297.04 > [0] 934.71/297.04 = [nats()] 934.71/297.04 934.71/297.04 [a__adx(X)] = [1] X + [4] 934.71/297.04 > [1] X + [0] 934.71/297.04 = [adx(X)] 934.71/297.04 934.71/297.04 [a__adx(cons(X, Y))] = [1] X + [1] Y + [8] 934.71/297.04 >= [1] X + [1] Y + [8] 934.71/297.04 = [a__incr(cons(X, adx(Y)))] 934.71/297.04 934.71/297.04 [a__zeros()] = [0] 934.71/297.04 ? [4] 934.71/297.04 = [cons(0(), zeros())] 934.71/297.04 934.71/297.04 [a__zeros()] = [0] 934.71/297.04 >= [0] 934.71/297.04 = [zeros()] 934.71/297.04 934.71/297.04 [a__incr(X)] = [1] X + [4] 934.71/297.04 >= [1] X + [4] 934.71/297.04 = [incr(X)] 934.71/297.04 934.71/297.04 [a__incr(cons(X, Y))] = [1] X + [1] Y + [8] 934.71/297.04 ? [1] X + [1] Y + [12] 934.71/297.04 = [cons(s(X), incr(Y))] 934.71/297.04 934.71/297.04 [a__hd(X)] = [1] X + [4] 934.71/297.04 ? [1] X + [7] 934.71/297.04 = [hd(X)] 934.71/297.04 934.71/297.04 [a__hd(cons(X, Y))] = [1] X + [1] Y + [8] 934.71/297.04 > [1] X + [5] 934.71/297.04 = [mark(X)] 934.71/297.04 934.71/297.04 [mark(cons(X1, X2))] = [1] X1 + [1] X2 + [9] 934.71/297.04 > [1] X1 + [1] X2 + [4] 934.71/297.04 = [cons(X1, X2)] 934.71/297.04 934.71/297.04 [mark(0())] = [5] 934.71/297.04 > [0] 934.71/297.04 = [0()] 934.71/297.04 934.71/297.04 [mark(zeros())] = [5] 934.71/297.04 > [0] 934.71/297.04 = [a__zeros()] 934.71/297.04 934.71/297.04 [mark(s(X))] = [1] X + [9] 934.71/297.04 > [1] X + [4] 934.71/297.04 = [s(X)] 934.71/297.04 934.71/297.04 [mark(incr(X))] = [1] X + [9] 934.71/297.04 >= [1] X + [9] 934.71/297.04 = [a__incr(mark(X))] 934.71/297.04 934.71/297.04 [mark(adx(X))] = [1] X + [5] 934.71/297.04 ? [1] X + [9] 934.71/297.04 = [a__adx(mark(X))] 934.71/297.04 934.71/297.04 [mark(nats())] = [5] 934.71/297.04 > [3] 934.71/297.04 = [a__nats()] 934.71/297.04 934.71/297.04 [mark(hd(X))] = [1] X + [12] 934.71/297.04 > [1] X + [9] 934.71/297.04 = [a__hd(mark(X))] 934.71/297.04 934.71/297.04 [mark(tl(X))] = [1] X + [5] 934.71/297.04 ? [1] X + [9] 934.71/297.04 = [a__tl(mark(X))] 934.71/297.04 934.71/297.04 [a__tl(X)] = [1] X + [4] 934.71/297.04 > [1] X + [0] 934.71/297.04 = [tl(X)] 934.71/297.04 934.71/297.04 [a__tl(cons(X, Y))] = [1] X + [1] Y + [8] 934.71/297.04 > [1] Y + [5] 934.71/297.04 = [mark(Y)] 934.71/297.04 934.71/297.04 934.71/297.04 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 934.71/297.04 934.71/297.04 We are left with following problem, upon which TcT provides the 934.71/297.04 certificate YES(O(1),O(n^3)). 934.71/297.04 934.71/297.04 Strict Trs: 934.71/297.04 { a__nats() -> a__adx(a__zeros()) 934.71/297.04 , a__adx(cons(X, Y)) -> a__incr(cons(X, adx(Y))) 934.71/297.04 , a__zeros() -> cons(0(), zeros()) 934.71/297.04 , a__zeros() -> zeros() 934.71/297.04 , a__incr(X) -> incr(X) 934.71/297.04 , a__incr(cons(X, Y)) -> cons(s(X), incr(Y)) 934.71/297.04 , a__hd(X) -> hd(X) 934.71/297.04 , mark(incr(X)) -> a__incr(mark(X)) 934.71/297.04 , mark(adx(X)) -> a__adx(mark(X)) 934.71/297.04 , mark(tl(X)) -> a__tl(mark(X)) } 934.71/297.04 Weak Trs: 934.71/297.04 { a__nats() -> nats() 934.71/297.04 , a__adx(X) -> adx(X) 934.71/297.04 , a__hd(cons(X, Y)) -> mark(X) 934.71/297.04 , mark(cons(X1, X2)) -> cons(X1, X2) 934.71/297.04 , mark(0()) -> 0() 934.71/297.04 , mark(zeros()) -> a__zeros() 934.71/297.04 , mark(s(X)) -> s(X) 934.71/297.04 , mark(nats()) -> a__nats() 934.71/297.04 , mark(hd(X)) -> a__hd(mark(X)) 934.71/297.04 , a__tl(X) -> tl(X) 934.71/297.04 , a__tl(cons(X, Y)) -> mark(Y) } 934.71/297.04 Obligation: 934.71/297.04 innermost runtime complexity 934.71/297.04 Answer: 934.71/297.04 YES(O(1),O(n^3)) 934.71/297.04 934.71/297.04 The weightgap principle applies (using the following nonconstant 934.71/297.04 growth matrix-interpretation) 934.71/297.04 934.71/297.04 The following argument positions are usable: 934.71/297.04 Uargs(a__adx) = {1}, Uargs(a__incr) = {1}, Uargs(a__hd) = {1}, 934.71/297.04 Uargs(a__tl) = {1} 934.71/297.04 934.71/297.04 TcT has computed the following matrix interpretation satisfying 934.71/297.04 not(EDA) and not(IDA(1)). 934.71/297.04 934.71/297.04 [a__nats] = [3] 934.71/297.04 934.71/297.04 [a__adx](x1) = [1] x1 + [4] 934.71/297.04 934.71/297.04 [a__zeros] = [0] 934.71/297.04 934.71/297.04 [cons](x1, x2) = [1] x1 + [1] x2 + [0] 934.71/297.04 934.71/297.04 [0] = [0] 934.71/297.04 934.71/297.04 [zeros] = [0] 934.71/297.04 934.71/297.04 [a__incr](x1) = [1] x1 + [0] 934.71/297.04 934.71/297.04 [s](x1) = [1] x1 + [0] 934.71/297.04 934.71/297.04 [incr](x1) = [1] x1 + [0] 934.71/297.04 934.71/297.05 [adx](x1) = [1] x1 + [0] 934.71/297.05 934.71/297.05 [a__hd](x1) = [1] x1 + [4] 934.71/297.05 934.71/297.05 [mark](x1) = [1] x1 + [1] 934.71/297.05 934.71/297.05 [a__tl](x1) = [1] x1 + [4] 934.71/297.05 934.71/297.05 [nats] = [3] 934.71/297.05 934.71/297.05 [hd](x1) = [1] x1 + [7] 934.71/297.05 934.71/297.05 [tl](x1) = [1] x1 + [0] 934.71/297.05 934.71/297.05 The order satisfies the following ordering constraints: 934.71/297.05 934.71/297.05 [a__nats()] = [3] 934.71/297.05 ? [4] 934.71/297.05 = [a__adx(a__zeros())] 934.71/297.05 934.71/297.05 [a__nats()] = [3] 934.71/297.05 >= [3] 934.71/297.05 = [nats()] 934.71/297.05 934.71/297.05 [a__adx(X)] = [1] X + [4] 934.71/297.05 > [1] X + [0] 934.71/297.05 = [adx(X)] 934.71/297.05 934.71/297.05 [a__adx(cons(X, Y))] = [1] X + [1] Y + [4] 934.71/297.05 > [1] X + [1] Y + [0] 934.71/297.05 = [a__incr(cons(X, adx(Y)))] 934.71/297.05 934.71/297.05 [a__zeros()] = [0] 934.71/297.05 >= [0] 934.71/297.05 = [cons(0(), zeros())] 934.71/297.05 934.71/297.05 [a__zeros()] = [0] 934.71/297.05 >= [0] 934.71/297.05 = [zeros()] 934.71/297.05 934.71/297.05 [a__incr(X)] = [1] X + [0] 934.71/297.05 >= [1] X + [0] 934.71/297.05 = [incr(X)] 934.71/297.05 934.71/297.05 [a__incr(cons(X, Y))] = [1] X + [1] Y + [0] 934.71/297.05 >= [1] X + [1] Y + [0] 934.71/297.05 = [cons(s(X), incr(Y))] 934.71/297.05 934.71/297.05 [a__hd(X)] = [1] X + [4] 934.71/297.05 ? [1] X + [7] 934.71/297.05 = [hd(X)] 934.71/297.05 934.71/297.05 [a__hd(cons(X, Y))] = [1] X + [1] Y + [4] 934.71/297.05 > [1] X + [1] 934.71/297.05 = [mark(X)] 934.71/297.05 934.71/297.05 [mark(cons(X1, X2))] = [1] X1 + [1] X2 + [1] 934.71/297.05 > [1] X1 + [1] X2 + [0] 934.71/297.05 = [cons(X1, X2)] 934.71/297.05 934.71/297.05 [mark(0())] = [1] 934.71/297.05 > [0] 934.71/297.05 = [0()] 934.71/297.05 934.71/297.05 [mark(zeros())] = [1] 934.71/297.05 > [0] 934.71/297.05 = [a__zeros()] 934.71/297.05 934.71/297.05 [mark(s(X))] = [1] X + [1] 934.71/297.05 > [1] X + [0] 934.71/297.05 = [s(X)] 934.71/297.05 934.71/297.05 [mark(incr(X))] = [1] X + [1] 934.71/297.05 >= [1] X + [1] 934.71/297.05 = [a__incr(mark(X))] 934.71/297.05 934.71/297.05 [mark(adx(X))] = [1] X + [1] 934.71/297.05 ? [1] X + [5] 934.71/297.05 = [a__adx(mark(X))] 934.71/297.05 934.71/297.05 [mark(nats())] = [4] 934.71/297.05 > [3] 934.71/297.05 = [a__nats()] 934.71/297.05 934.71/297.05 [mark(hd(X))] = [1] X + [8] 934.71/297.05 > [1] X + [5] 934.71/297.05 = [a__hd(mark(X))] 934.71/297.05 934.71/297.05 [mark(tl(X))] = [1] X + [1] 934.71/297.05 ? [1] X + [5] 934.71/297.05 = [a__tl(mark(X))] 934.71/297.05 934.71/297.05 [a__tl(X)] = [1] X + [4] 934.71/297.05 > [1] X + [0] 934.71/297.05 = [tl(X)] 934.71/297.05 934.71/297.05 [a__tl(cons(X, Y))] = [1] X + [1] Y + [4] 934.71/297.05 > [1] Y + [1] 934.71/297.05 = [mark(Y)] 934.71/297.05 934.71/297.05 934.71/297.05 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 934.71/297.05 934.71/297.05 We are left with following problem, upon which TcT provides the 934.71/297.05 certificate YES(O(1),O(n^3)). 934.71/297.05 934.71/297.05 Strict Trs: 934.71/297.05 { a__nats() -> a__adx(a__zeros()) 934.71/297.05 , a__zeros() -> cons(0(), zeros()) 934.71/297.05 , a__zeros() -> zeros() 934.71/297.05 , a__incr(X) -> incr(X) 934.71/297.05 , a__incr(cons(X, Y)) -> cons(s(X), incr(Y)) 934.71/297.05 , a__hd(X) -> hd(X) 934.71/297.05 , mark(incr(X)) -> a__incr(mark(X)) 934.71/297.05 , mark(adx(X)) -> a__adx(mark(X)) 934.71/297.05 , mark(tl(X)) -> a__tl(mark(X)) } 934.71/297.05 Weak Trs: 934.71/297.05 { a__nats() -> nats() 934.71/297.05 , a__adx(X) -> adx(X) 934.71/297.05 , a__adx(cons(X, Y)) -> a__incr(cons(X, adx(Y))) 934.71/297.05 , a__hd(cons(X, Y)) -> mark(X) 934.71/297.05 , mark(cons(X1, X2)) -> cons(X1, X2) 934.71/297.05 , mark(0()) -> 0() 934.71/297.05 , mark(zeros()) -> a__zeros() 934.71/297.05 , mark(s(X)) -> s(X) 934.71/297.05 , mark(nats()) -> a__nats() 934.71/297.05 , mark(hd(X)) -> a__hd(mark(X)) 934.71/297.05 , a__tl(X) -> tl(X) 934.71/297.05 , a__tl(cons(X, Y)) -> mark(Y) } 934.71/297.05 Obligation: 934.71/297.05 innermost runtime complexity 934.71/297.05 Answer: 934.71/297.05 YES(O(1),O(n^3)) 934.71/297.05 934.71/297.05 The weightgap principle applies (using the following nonconstant 934.71/297.05 growth matrix-interpretation) 934.71/297.05 934.71/297.05 The following argument positions are usable: 934.71/297.05 Uargs(a__adx) = {1}, Uargs(a__incr) = {1}, Uargs(a__hd) = {1}, 934.71/297.05 Uargs(a__tl) = {1} 934.71/297.05 934.71/297.05 TcT has computed the following matrix interpretation satisfying 934.71/297.05 not(EDA) and not(IDA(1)). 934.71/297.05 934.71/297.05 [a__nats] = [3] 934.71/297.05 934.71/297.05 [a__adx](x1) = [1] x1 + [4] 934.71/297.05 934.71/297.05 [a__zeros] = [0] 934.71/297.05 934.71/297.05 [cons](x1, x2) = [1] x1 + [1] x2 + [4] 934.71/297.05 934.71/297.05 [0] = [0] 934.71/297.05 934.71/297.05 [zeros] = [0] 934.71/297.05 934.71/297.05 [a__incr](x1) = [1] x1 + [4] 934.71/297.05 934.71/297.05 [s](x1) = [1] x1 + [0] 934.71/297.05 934.71/297.05 [incr](x1) = [1] x1 + [0] 934.71/297.05 934.71/297.05 [adx](x1) = [1] x1 + [0] 934.71/297.05 934.71/297.05 [a__hd](x1) = [1] x1 + [0] 934.71/297.05 934.71/297.05 [mark](x1) = [1] x1 + [1] 934.71/297.05 934.71/297.05 [a__tl](x1) = [1] x1 + [0] 934.71/297.05 934.71/297.05 [nats] = [3] 934.71/297.05 934.71/297.05 [hd](x1) = [1] x1 + [0] 934.71/297.05 934.71/297.05 [tl](x1) = [1] x1 + [0] 934.71/297.05 934.71/297.05 The order satisfies the following ordering constraints: 934.71/297.05 934.71/297.05 [a__nats()] = [3] 934.71/297.05 ? [4] 934.71/297.05 = [a__adx(a__zeros())] 934.71/297.05 934.71/297.05 [a__nats()] = [3] 934.71/297.05 >= [3] 934.71/297.05 = [nats()] 934.71/297.05 934.71/297.05 [a__adx(X)] = [1] X + [4] 934.71/297.05 > [1] X + [0] 934.71/297.05 = [adx(X)] 934.71/297.05 934.71/297.05 [a__adx(cons(X, Y))] = [1] X + [1] Y + [8] 934.71/297.05 >= [1] X + [1] Y + [8] 934.71/297.05 = [a__incr(cons(X, adx(Y)))] 934.71/297.05 934.71/297.05 [a__zeros()] = [0] 934.71/297.05 ? [4] 934.71/297.05 = [cons(0(), zeros())] 934.71/297.05 934.71/297.05 [a__zeros()] = [0] 934.71/297.05 >= [0] 934.71/297.05 = [zeros()] 934.71/297.05 934.71/297.05 [a__incr(X)] = [1] X + [4] 934.71/297.05 > [1] X + [0] 934.71/297.05 = [incr(X)] 934.71/297.05 934.71/297.05 [a__incr(cons(X, Y))] = [1] X + [1] Y + [8] 934.71/297.05 > [1] X + [1] Y + [4] 934.71/297.05 = [cons(s(X), incr(Y))] 934.71/297.05 934.71/297.05 [a__hd(X)] = [1] X + [0] 934.71/297.05 >= [1] X + [0] 934.71/297.05 = [hd(X)] 934.71/297.05 934.71/297.05 [a__hd(cons(X, Y))] = [1] X + [1] Y + [4] 934.71/297.05 > [1] X + [1] 934.71/297.05 = [mark(X)] 934.71/297.05 934.71/297.05 [mark(cons(X1, X2))] = [1] X1 + [1] X2 + [5] 934.71/297.05 > [1] X1 + [1] X2 + [4] 934.71/297.05 = [cons(X1, X2)] 934.71/297.05 934.71/297.05 [mark(0())] = [1] 934.71/297.05 > [0] 934.71/297.05 = [0()] 934.71/297.05 934.71/297.05 [mark(zeros())] = [1] 934.71/297.05 > [0] 934.71/297.05 = [a__zeros()] 934.71/297.05 934.71/297.05 [mark(s(X))] = [1] X + [1] 934.71/297.05 > [1] X + [0] 934.71/297.05 = [s(X)] 934.71/297.05 934.71/297.05 [mark(incr(X))] = [1] X + [1] 934.71/297.05 ? [1] X + [5] 934.71/297.05 = [a__incr(mark(X))] 934.71/297.05 934.71/297.05 [mark(adx(X))] = [1] X + [1] 934.71/297.05 ? [1] X + [5] 934.71/297.05 = [a__adx(mark(X))] 934.71/297.05 934.71/297.05 [mark(nats())] = [4] 934.71/297.05 > [3] 934.71/297.05 = [a__nats()] 934.71/297.05 934.71/297.05 [mark(hd(X))] = [1] X + [1] 934.71/297.05 >= [1] X + [1] 934.71/297.05 = [a__hd(mark(X))] 934.71/297.05 934.71/297.05 [mark(tl(X))] = [1] X + [1] 934.71/297.05 >= [1] X + [1] 934.71/297.05 = [a__tl(mark(X))] 934.71/297.05 934.71/297.05 [a__tl(X)] = [1] X + [0] 934.71/297.05 >= [1] X + [0] 934.71/297.05 = [tl(X)] 934.71/297.05 934.71/297.05 [a__tl(cons(X, Y))] = [1] X + [1] Y + [4] 934.71/297.05 > [1] Y + [1] 934.71/297.05 = [mark(Y)] 934.71/297.05 934.71/297.05 934.71/297.05 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 934.71/297.05 934.71/297.05 We are left with following problem, upon which TcT provides the 934.71/297.05 certificate YES(O(1),O(n^3)). 934.71/297.05 934.71/297.05 Strict Trs: 934.71/297.05 { a__nats() -> a__adx(a__zeros()) 934.71/297.05 , a__zeros() -> cons(0(), zeros()) 934.71/297.05 , a__zeros() -> zeros() 934.71/297.05 , a__hd(X) -> hd(X) 934.71/297.05 , mark(incr(X)) -> a__incr(mark(X)) 934.71/297.05 , mark(adx(X)) -> a__adx(mark(X)) 934.71/297.05 , mark(tl(X)) -> a__tl(mark(X)) } 934.71/297.05 Weak Trs: 934.71/297.05 { a__nats() -> nats() 934.71/297.05 , a__adx(X) -> adx(X) 934.71/297.05 , a__adx(cons(X, Y)) -> a__incr(cons(X, adx(Y))) 934.71/297.05 , a__incr(X) -> incr(X) 934.71/297.05 , a__incr(cons(X, Y)) -> cons(s(X), incr(Y)) 934.71/297.05 , a__hd(cons(X, Y)) -> mark(X) 934.71/297.05 , mark(cons(X1, X2)) -> cons(X1, X2) 934.71/297.05 , mark(0()) -> 0() 934.71/297.05 , mark(zeros()) -> a__zeros() 934.71/297.05 , mark(s(X)) -> s(X) 934.71/297.05 , mark(nats()) -> a__nats() 934.71/297.05 , mark(hd(X)) -> a__hd(mark(X)) 934.71/297.05 , a__tl(X) -> tl(X) 934.71/297.05 , a__tl(cons(X, Y)) -> mark(Y) } 934.71/297.05 Obligation: 934.71/297.05 innermost runtime complexity 934.71/297.05 Answer: 934.71/297.05 YES(O(1),O(n^3)) 934.71/297.05 934.71/297.05 The weightgap principle applies (using the following nonconstant 934.71/297.05 growth matrix-interpretation) 934.71/297.05 934.71/297.05 The following argument positions are usable: 934.71/297.05 Uargs(a__adx) = {1}, Uargs(a__incr) = {1}, Uargs(a__hd) = {1}, 934.71/297.05 Uargs(a__tl) = {1} 934.71/297.05 934.71/297.05 TcT has computed the following matrix interpretation satisfying 934.71/297.05 not(EDA) and not(IDA(1)). 934.71/297.05 934.71/297.05 [a__nats] = [5] 934.71/297.05 934.71/297.05 [a__adx](x1) = [1] x1 + [4] 934.71/297.05 934.71/297.05 [a__zeros] = [0] 934.71/297.05 934.71/297.05 [cons](x1, x2) = [1] x1 + [1] x2 + [4] 934.71/297.05 934.71/297.05 [0] = [0] 934.71/297.05 934.71/297.05 [zeros] = [0] 934.71/297.05 934.71/297.05 [a__incr](x1) = [1] x1 + [4] 934.71/297.05 934.71/297.05 [s](x1) = [1] x1 + [0] 934.71/297.05 934.71/297.05 [incr](x1) = [1] x1 + [0] 934.71/297.05 934.71/297.05 [adx](x1) = [1] x1 + [0] 934.71/297.05 934.71/297.05 [a__hd](x1) = [1] x1 + [0] 934.71/297.05 934.71/297.05 [mark](x1) = [1] x1 + [1] 934.71/297.05 934.71/297.05 [a__tl](x1) = [1] x1 + [4] 934.71/297.05 934.71/297.05 [nats] = [4] 934.71/297.05 934.71/297.05 [hd](x1) = [1] x1 + [7] 934.71/297.05 934.71/297.05 [tl](x1) = [1] x1 + [0] 934.71/297.05 934.71/297.05 The order satisfies the following ordering constraints: 934.71/297.05 934.71/297.05 [a__nats()] = [5] 934.71/297.05 > [4] 934.71/297.05 = [a__adx(a__zeros())] 934.71/297.05 934.71/297.05 [a__nats()] = [5] 934.71/297.05 > [4] 934.71/297.05 = [nats()] 934.71/297.05 934.71/297.05 [a__adx(X)] = [1] X + [4] 934.71/297.05 > [1] X + [0] 934.71/297.05 = [adx(X)] 934.71/297.05 934.71/297.05 [a__adx(cons(X, Y))] = [1] X + [1] Y + [8] 934.71/297.05 >= [1] X + [1] Y + [8] 934.71/297.05 = [a__incr(cons(X, adx(Y)))] 934.71/297.05 934.71/297.05 [a__zeros()] = [0] 934.71/297.05 ? [4] 934.71/297.05 = [cons(0(), zeros())] 934.71/297.05 934.71/297.05 [a__zeros()] = [0] 934.71/297.05 >= [0] 934.71/297.05 = [zeros()] 934.71/297.05 934.71/297.05 [a__incr(X)] = [1] X + [4] 934.71/297.05 > [1] X + [0] 934.71/297.05 = [incr(X)] 934.71/297.05 934.71/297.05 [a__incr(cons(X, Y))] = [1] X + [1] Y + [8] 934.71/297.05 > [1] X + [1] Y + [4] 934.71/297.05 = [cons(s(X), incr(Y))] 934.71/297.05 934.71/297.05 [a__hd(X)] = [1] X + [0] 934.71/297.05 ? [1] X + [7] 934.71/297.05 = [hd(X)] 934.71/297.05 934.71/297.05 [a__hd(cons(X, Y))] = [1] X + [1] Y + [4] 934.71/297.05 > [1] X + [1] 934.71/297.05 = [mark(X)] 934.71/297.05 934.71/297.05 [mark(cons(X1, X2))] = [1] X1 + [1] X2 + [5] 934.71/297.05 > [1] X1 + [1] X2 + [4] 934.71/297.05 = [cons(X1, X2)] 934.71/297.05 934.71/297.05 [mark(0())] = [1] 934.71/297.05 > [0] 934.71/297.05 = [0()] 934.71/297.05 934.71/297.05 [mark(zeros())] = [1] 934.71/297.05 > [0] 934.71/297.05 = [a__zeros()] 934.71/297.05 934.71/297.05 [mark(s(X))] = [1] X + [1] 934.71/297.05 > [1] X + [0] 934.71/297.05 = [s(X)] 934.71/297.05 934.71/297.05 [mark(incr(X))] = [1] X + [1] 934.71/297.05 ? [1] X + [5] 934.71/297.05 = [a__incr(mark(X))] 934.71/297.05 934.71/297.05 [mark(adx(X))] = [1] X + [1] 934.71/297.05 ? [1] X + [5] 934.71/297.05 = [a__adx(mark(X))] 934.71/297.05 934.71/297.05 [mark(nats())] = [5] 934.71/297.05 >= [5] 934.71/297.05 = [a__nats()] 934.71/297.05 934.71/297.05 [mark(hd(X))] = [1] X + [8] 934.71/297.05 > [1] X + [1] 934.71/297.05 = [a__hd(mark(X))] 934.71/297.05 934.71/297.05 [mark(tl(X))] = [1] X + [1] 934.71/297.05 ? [1] X + [5] 934.71/297.05 = [a__tl(mark(X))] 934.71/297.05 934.71/297.05 [a__tl(X)] = [1] X + [4] 934.71/297.05 > [1] X + [0] 934.71/297.05 = [tl(X)] 934.71/297.05 934.71/297.05 [a__tl(cons(X, Y))] = [1] X + [1] Y + [8] 934.71/297.05 > [1] Y + [1] 934.71/297.05 = [mark(Y)] 934.71/297.05 934.71/297.05 934.71/297.05 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 934.71/297.05 934.71/297.05 We are left with following problem, upon which TcT provides the 934.71/297.05 certificate YES(O(1),O(n^3)). 934.71/297.05 934.71/297.05 Strict Trs: 934.71/297.05 { a__zeros() -> cons(0(), zeros()) 934.71/297.05 , a__zeros() -> zeros() 934.71/297.05 , a__hd(X) -> hd(X) 934.71/297.05 , mark(incr(X)) -> a__incr(mark(X)) 934.71/297.05 , mark(adx(X)) -> a__adx(mark(X)) 934.71/297.05 , mark(tl(X)) -> a__tl(mark(X)) } 934.71/297.05 Weak Trs: 934.71/297.05 { a__nats() -> a__adx(a__zeros()) 934.71/297.05 , a__nats() -> nats() 934.71/297.05 , a__adx(X) -> adx(X) 934.71/297.05 , a__adx(cons(X, Y)) -> a__incr(cons(X, adx(Y))) 934.71/297.05 , a__incr(X) -> incr(X) 934.71/297.05 , a__incr(cons(X, Y)) -> cons(s(X), incr(Y)) 934.71/297.05 , a__hd(cons(X, Y)) -> mark(X) 934.71/297.05 , mark(cons(X1, X2)) -> cons(X1, X2) 934.71/297.05 , mark(0()) -> 0() 934.71/297.05 , mark(zeros()) -> a__zeros() 934.71/297.05 , mark(s(X)) -> s(X) 934.71/297.05 , mark(nats()) -> a__nats() 934.71/297.05 , mark(hd(X)) -> a__hd(mark(X)) 934.71/297.05 , a__tl(X) -> tl(X) 934.71/297.05 , a__tl(cons(X, Y)) -> mark(Y) } 934.71/297.05 Obligation: 934.71/297.05 innermost runtime complexity 934.71/297.05 Answer: 934.71/297.05 YES(O(1),O(n^3)) 934.71/297.05 934.71/297.05 The weightgap principle applies (using the following nonconstant 934.71/297.05 growth matrix-interpretation) 934.71/297.05 934.71/297.05 The following argument positions are usable: 934.71/297.05 Uargs(a__adx) = {1}, Uargs(a__incr) = {1}, Uargs(a__hd) = {1}, 934.71/297.05 Uargs(a__tl) = {1} 934.71/297.05 934.71/297.05 TcT has computed the following matrix interpretation satisfying 934.71/297.05 not(EDA) and not(IDA(1)). 934.71/297.05 934.71/297.05 [a__nats] = [3] 934.71/297.05 934.71/297.05 [a__adx](x1) = [1] x1 + [0] 934.71/297.05 934.71/297.05 [a__zeros] = [1] 934.71/297.05 934.71/297.05 [cons](x1, x2) = [1] x1 + [1] x2 + [0] 934.71/297.05 934.71/297.05 [0] = [0] 934.71/297.05 934.71/297.05 [zeros] = [0] 934.71/297.05 934.71/297.05 [a__incr](x1) = [1] x1 + [0] 934.71/297.05 934.71/297.05 [s](x1) = [1] x1 + [0] 934.71/297.05 934.71/297.05 [incr](x1) = [1] x1 + [0] 934.71/297.05 934.71/297.05 [adx](x1) = [1] x1 + [0] 934.71/297.05 934.71/297.05 [a__hd](x1) = [1] x1 + [4] 934.71/297.05 934.71/297.05 [mark](x1) = [1] x1 + [1] 934.71/297.05 934.71/297.05 [a__tl](x1) = [1] x1 + [4] 934.71/297.05 934.71/297.05 [nats] = [3] 934.71/297.05 934.71/297.05 [hd](x1) = [1] x1 + [7] 934.71/297.05 934.71/297.05 [tl](x1) = [1] x1 + [0] 934.71/297.05 934.71/297.05 The order satisfies the following ordering constraints: 934.71/297.05 934.71/297.05 [a__nats()] = [3] 934.71/297.05 > [1] 934.71/297.05 = [a__adx(a__zeros())] 934.71/297.05 934.71/297.05 [a__nats()] = [3] 934.71/297.05 >= [3] 934.71/297.05 = [nats()] 934.71/297.05 934.71/297.05 [a__adx(X)] = [1] X + [0] 934.71/297.05 >= [1] X + [0] 934.71/297.05 = [adx(X)] 934.71/297.05 934.71/297.05 [a__adx(cons(X, Y))] = [1] X + [1] Y + [0] 934.71/297.05 >= [1] X + [1] Y + [0] 934.71/297.05 = [a__incr(cons(X, adx(Y)))] 934.71/297.05 934.71/297.05 [a__zeros()] = [1] 934.71/297.05 > [0] 934.71/297.05 = [cons(0(), zeros())] 934.71/297.05 934.71/297.05 [a__zeros()] = [1] 934.71/297.05 > [0] 934.71/297.05 = [zeros()] 934.71/297.05 934.71/297.05 [a__incr(X)] = [1] X + [0] 934.71/297.05 >= [1] X + [0] 934.71/297.05 = [incr(X)] 934.71/297.05 934.71/297.05 [a__incr(cons(X, Y))] = [1] X + [1] Y + [0] 934.71/297.05 >= [1] X + [1] Y + [0] 934.71/297.05 = [cons(s(X), incr(Y))] 934.71/297.05 934.71/297.05 [a__hd(X)] = [1] X + [4] 934.71/297.05 ? [1] X + [7] 934.71/297.05 = [hd(X)] 934.71/297.05 934.71/297.05 [a__hd(cons(X, Y))] = [1] X + [1] Y + [4] 934.71/297.05 > [1] X + [1] 934.71/297.05 = [mark(X)] 934.71/297.05 934.71/297.05 [mark(cons(X1, X2))] = [1] X1 + [1] X2 + [1] 934.71/297.05 > [1] X1 + [1] X2 + [0] 934.71/297.05 = [cons(X1, X2)] 934.71/297.05 934.71/297.05 [mark(0())] = [1] 934.71/297.05 > [0] 934.71/297.05 = [0()] 934.71/297.05 934.71/297.05 [mark(zeros())] = [1] 934.71/297.05 >= [1] 934.71/297.05 = [a__zeros()] 934.71/297.05 934.71/297.05 [mark(s(X))] = [1] X + [1] 934.71/297.05 > [1] X + [0] 934.71/297.05 = [s(X)] 934.71/297.05 934.71/297.05 [mark(incr(X))] = [1] X + [1] 934.71/297.05 >= [1] X + [1] 934.71/297.05 = [a__incr(mark(X))] 934.71/297.05 934.71/297.05 [mark(adx(X))] = [1] X + [1] 934.71/297.05 >= [1] X + [1] 934.71/297.05 = [a__adx(mark(X))] 934.71/297.05 934.71/297.05 [mark(nats())] = [4] 934.71/297.05 > [3] 934.71/297.05 = [a__nats()] 934.71/297.05 934.71/297.05 [mark(hd(X))] = [1] X + [8] 934.71/297.05 > [1] X + [5] 934.71/297.05 = [a__hd(mark(X))] 934.71/297.05 934.71/297.05 [mark(tl(X))] = [1] X + [1] 934.71/297.05 ? [1] X + [5] 934.71/297.05 = [a__tl(mark(X))] 934.71/297.05 934.71/297.05 [a__tl(X)] = [1] X + [4] 934.71/297.05 > [1] X + [0] 934.71/297.05 = [tl(X)] 934.71/297.05 934.71/297.05 [a__tl(cons(X, Y))] = [1] X + [1] Y + [4] 934.71/297.05 > [1] Y + [1] 934.71/297.05 = [mark(Y)] 934.71/297.05 934.71/297.05 934.71/297.05 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 934.71/297.05 934.71/297.05 We are left with following problem, upon which TcT provides the 934.71/297.05 certificate YES(O(1),O(n^3)). 934.71/297.05 934.71/297.05 Strict Trs: 934.71/297.05 { a__hd(X) -> hd(X) 934.71/297.05 , mark(incr(X)) -> a__incr(mark(X)) 934.71/297.05 , mark(adx(X)) -> a__adx(mark(X)) 934.71/297.05 , mark(tl(X)) -> a__tl(mark(X)) } 934.71/297.05 Weak Trs: 934.71/297.05 { a__nats() -> a__adx(a__zeros()) 934.71/297.05 , a__nats() -> nats() 934.71/297.05 , a__adx(X) -> adx(X) 934.71/297.05 , a__adx(cons(X, Y)) -> a__incr(cons(X, adx(Y))) 934.71/297.05 , a__zeros() -> cons(0(), zeros()) 934.71/297.05 , a__zeros() -> zeros() 934.71/297.05 , a__incr(X) -> incr(X) 934.71/297.05 , a__incr(cons(X, Y)) -> cons(s(X), incr(Y)) 934.71/297.05 , a__hd(cons(X, Y)) -> mark(X) 934.71/297.05 , mark(cons(X1, X2)) -> cons(X1, X2) 934.71/297.05 , mark(0()) -> 0() 934.71/297.05 , mark(zeros()) -> a__zeros() 934.71/297.05 , mark(s(X)) -> s(X) 934.71/297.05 , mark(nats()) -> a__nats() 934.71/297.05 , mark(hd(X)) -> a__hd(mark(X)) 934.71/297.05 , a__tl(X) -> tl(X) 934.71/297.05 , a__tl(cons(X, Y)) -> mark(Y) } 934.71/297.05 Obligation: 934.71/297.05 innermost runtime complexity 934.71/297.05 Answer: 934.71/297.05 YES(O(1),O(n^3)) 934.71/297.05 934.71/297.05 We use the processor 'matrix interpretation of dimension 2' to 934.71/297.05 orient following rules strictly. 934.71/297.05 934.71/297.05 Trs: 934.71/297.05 { a__hd(X) -> hd(X) 934.71/297.05 , mark(tl(X)) -> a__tl(mark(X)) } 934.71/297.05 934.71/297.05 The induced complexity on above rules (modulo remaining rules) is 934.71/297.05 YES(?,O(n^1)) . These rules are moved into the corresponding weak 934.71/297.05 component(s). 934.71/297.05 934.71/297.05 Sub-proof: 934.71/297.05 ---------- 934.71/297.05 The following argument positions are usable: 934.71/297.05 Uargs(a__adx) = {1}, Uargs(a__incr) = {1}, Uargs(a__hd) = {1}, 934.71/297.05 Uargs(a__tl) = {1} 934.71/297.05 934.71/297.05 TcT has computed the following constructor-based matrix 934.71/297.05 interpretation satisfying not(EDA) and not(IDA(1)). 934.71/297.05 934.71/297.05 [a__nats] = [0] 934.71/297.05 [0] 934.71/297.05 934.71/297.05 [a__adx](x1) = [1 0] x1 + [0] 934.71/297.05 [0 1] [0] 934.71/297.05 934.71/297.05 [a__zeros] = [0] 934.71/297.05 [0] 934.71/297.05 934.71/297.05 [cons](x1, x2) = [0 1] x1 + [0 1] x2 + [0] 934.71/297.05 [0 1] [0 1] [0] 934.71/297.05 934.71/297.05 [0] = [0] 934.71/297.05 [0] 934.71/297.05 934.71/297.05 [zeros] = [0] 934.71/297.05 [0] 934.71/297.05 934.71/297.05 [a__incr](x1) = [1 0] x1 + [0] 934.71/297.05 [0 1] [0] 934.71/297.05 934.71/297.05 [s](x1) = [0] 934.71/297.05 [0] 934.71/297.05 934.71/297.05 [incr](x1) = [0 0] x1 + [0] 934.71/297.05 [0 1] [0] 934.71/297.05 934.71/297.05 [adx](x1) = [0 0] x1 + [0] 934.71/297.05 [0 1] [0] 934.71/297.05 934.71/297.05 [a__hd](x1) = [1 0] x1 + [1] 934.71/297.05 [0 1] [4] 934.71/297.05 934.71/297.05 [mark](x1) = [0 1] x1 + [0] 934.71/297.05 [0 1] [0] 934.71/297.05 934.71/297.05 [a__tl](x1) = [1 0] x1 + [0] 934.71/297.05 [0 1] [4] 934.71/297.05 934.71/297.05 [nats] = [0] 934.71/297.05 [0] 934.71/297.05 934.71/297.05 [hd](x1) = [0 0] x1 + [0] 934.71/297.05 [0 1] [4] 934.71/297.05 934.71/297.05 [tl](x1) = [0 0] x1 + [0] 934.71/297.05 [0 1] [4] 934.71/297.05 934.71/297.05 The order satisfies the following ordering constraints: 934.71/297.05 934.71/297.05 [a__nats()] = [0] 934.71/297.05 [0] 934.71/297.05 >= [0] 934.71/297.05 [0] 934.71/297.05 = [a__adx(a__zeros())] 934.71/297.05 934.71/297.05 [a__nats()] = [0] 934.71/297.05 [0] 934.71/297.05 >= [0] 934.71/297.05 [0] 934.71/297.05 = [nats()] 934.71/297.05 934.71/297.05 [a__adx(X)] = [1 0] X + [0] 934.71/297.05 [0 1] [0] 934.71/297.05 >= [0 0] X + [0] 934.71/297.05 [0 1] [0] 934.71/297.05 = [adx(X)] 934.71/297.05 934.71/297.05 [a__adx(cons(X, Y))] = [0 1] X + [0 1] Y + [0] 934.71/297.05 [0 1] [0 1] [0] 934.71/297.05 >= [0 1] X + [0 1] Y + [0] 934.71/297.05 [0 1] [0 1] [0] 934.71/297.05 = [a__incr(cons(X, adx(Y)))] 934.71/297.05 934.71/297.05 [a__zeros()] = [0] 934.71/297.05 [0] 934.71/297.05 >= [0] 934.71/297.05 [0] 934.71/297.05 = [cons(0(), zeros())] 934.71/297.05 934.71/297.05 [a__zeros()] = [0] 934.71/297.05 [0] 934.71/297.05 >= [0] 934.71/297.05 [0] 934.71/297.05 = [zeros()] 934.71/297.05 934.71/297.05 [a__incr(X)] = [1 0] X + [0] 934.71/297.05 [0 1] [0] 934.71/297.05 >= [0 0] X + [0] 934.71/297.05 [0 1] [0] 934.71/297.05 = [incr(X)] 934.71/297.05 934.71/297.05 [a__incr(cons(X, Y))] = [0 1] X + [0 1] Y + [0] 934.71/297.05 [0 1] [0 1] [0] 934.71/297.05 >= [0 1] Y + [0] 934.71/297.05 [0 1] [0] 934.71/297.05 = [cons(s(X), incr(Y))] 934.71/297.05 934.71/297.05 [a__hd(X)] = [1 0] X + [1] 934.71/297.05 [0 1] [4] 934.71/297.05 > [0 0] X + [0] 934.71/297.05 [0 1] [4] 934.71/297.05 = [hd(X)] 934.71/297.05 934.71/297.05 [a__hd(cons(X, Y))] = [0 1] X + [0 1] Y + [1] 934.71/297.05 [0 1] [0 1] [4] 934.71/297.05 > [0 1] X + [0] 934.71/297.05 [0 1] [0] 934.71/297.05 = [mark(X)] 934.71/297.05 934.71/297.05 [mark(cons(X1, X2))] = [0 1] X1 + [0 1] X2 + [0] 934.71/297.05 [0 1] [0 1] [0] 934.71/297.05 >= [0 1] X1 + [0 1] X2 + [0] 934.71/297.05 [0 1] [0 1] [0] 934.71/297.05 = [cons(X1, X2)] 934.71/297.05 934.71/297.05 [mark(0())] = [0] 934.71/297.05 [0] 934.71/297.05 >= [0] 934.71/297.05 [0] 934.71/297.05 = [0()] 934.71/297.05 934.71/297.05 [mark(zeros())] = [0] 934.71/297.05 [0] 934.71/297.05 >= [0] 934.71/297.05 [0] 934.71/297.05 = [a__zeros()] 934.71/297.05 934.71/297.05 [mark(s(X))] = [0] 934.71/297.05 [0] 934.71/297.05 >= [0] 934.71/297.05 [0] 934.71/297.05 = [s(X)] 934.71/297.05 934.71/297.05 [mark(incr(X))] = [0 1] X + [0] 934.71/297.05 [0 1] [0] 934.71/297.05 >= [0 1] X + [0] 934.71/297.05 [0 1] [0] 934.71/297.05 = [a__incr(mark(X))] 934.71/297.05 934.71/297.05 [mark(adx(X))] = [0 1] X + [0] 934.71/297.05 [0 1] [0] 934.71/297.05 >= [0 1] X + [0] 934.71/297.05 [0 1] [0] 934.71/297.05 = [a__adx(mark(X))] 934.71/297.05 934.71/297.05 [mark(nats())] = [0] 934.71/297.05 [0] 934.71/297.05 >= [0] 934.71/297.05 [0] 934.71/297.05 = [a__nats()] 934.71/297.05 934.71/297.05 [mark(hd(X))] = [0 1] X + [4] 934.71/297.05 [0 1] [4] 934.71/297.05 > [0 1] X + [1] 934.71/297.05 [0 1] [4] 934.71/297.05 = [a__hd(mark(X))] 934.71/297.05 934.71/297.05 [mark(tl(X))] = [0 1] X + [4] 934.71/297.05 [0 1] [4] 934.71/297.05 > [0 1] X + [0] 934.71/297.05 [0 1] [4] 934.71/297.05 = [a__tl(mark(X))] 934.71/297.05 934.71/297.05 [a__tl(X)] = [1 0] X + [0] 934.71/297.05 [0 1] [4] 934.71/297.05 >= [0 0] X + [0] 934.71/297.05 [0 1] [4] 934.71/297.05 = [tl(X)] 934.71/297.05 934.71/297.05 [a__tl(cons(X, Y))] = [0 1] X + [0 1] Y + [0] 934.71/297.05 [0 1] [0 1] [4] 934.71/297.05 >= [0 1] Y + [0] 934.71/297.05 [0 1] [0] 934.71/297.05 = [mark(Y)] 934.71/297.05 934.71/297.05 934.71/297.05 We return to the main proof. 934.71/297.05 934.71/297.05 We are left with following problem, upon which TcT provides the 934.71/297.05 certificate YES(O(1),O(n^3)). 934.71/297.05 934.71/297.05 Strict Trs: 934.71/297.05 { mark(incr(X)) -> a__incr(mark(X)) 934.71/297.05 , mark(adx(X)) -> a__adx(mark(X)) } 934.71/297.05 Weak Trs: 934.71/297.05 { a__nats() -> a__adx(a__zeros()) 934.71/297.05 , a__nats() -> nats() 934.71/297.05 , a__adx(X) -> adx(X) 934.71/297.05 , a__adx(cons(X, Y)) -> a__incr(cons(X, adx(Y))) 934.71/297.05 , a__zeros() -> cons(0(), zeros()) 934.71/297.05 , a__zeros() -> zeros() 934.71/297.05 , a__incr(X) -> incr(X) 934.71/297.05 , a__incr(cons(X, Y)) -> cons(s(X), incr(Y)) 934.71/297.05 , a__hd(X) -> hd(X) 934.71/297.05 , a__hd(cons(X, Y)) -> mark(X) 934.71/297.05 , mark(cons(X1, X2)) -> cons(X1, X2) 934.71/297.05 , mark(0()) -> 0() 934.71/297.05 , mark(zeros()) -> a__zeros() 934.71/297.05 , mark(s(X)) -> s(X) 934.71/297.05 , mark(nats()) -> a__nats() 934.71/297.05 , mark(hd(X)) -> a__hd(mark(X)) 934.71/297.05 , mark(tl(X)) -> a__tl(mark(X)) 934.71/297.05 , a__tl(X) -> tl(X) 934.71/297.05 , a__tl(cons(X, Y)) -> mark(Y) } 934.71/297.05 Obligation: 934.71/297.05 innermost runtime complexity 934.71/297.05 Answer: 934.71/297.05 YES(O(1),O(n^3)) 934.71/297.05 934.71/297.05 We use the processor 'matrix interpretation of dimension 2' to 934.71/297.05 orient following rules strictly. 934.71/297.05 934.71/297.05 Trs: { mark(adx(X)) -> a__adx(mark(X)) } 934.71/297.05 934.71/297.05 The induced complexity on above rules (modulo remaining rules) is 934.71/297.05 YES(?,O(n^2)) . These rules are moved into the corresponding weak 934.71/297.05 component(s). 934.71/297.05 934.71/297.05 Sub-proof: 934.71/297.05 ---------- 934.71/297.05 The following argument positions are usable: 934.71/297.05 Uargs(a__adx) = {1}, Uargs(a__incr) = {1}, Uargs(a__hd) = {1}, 934.71/297.05 Uargs(a__tl) = {1} 934.71/297.05 934.71/297.05 TcT has computed the following constructor-based matrix 934.71/297.05 interpretation satisfying not(EDA). 934.71/297.05 934.71/297.05 [a__nats] = [0] 934.71/297.05 [4] 934.71/297.05 934.71/297.05 [a__adx](x1) = [1 0] x1 + [0] 934.71/297.05 [0 1] [4] 934.71/297.05 934.71/297.05 [a__zeros] = [0] 934.71/297.05 [0] 934.71/297.05 934.71/297.05 [cons](x1, x2) = [1 1] x1 + [1 0] x2 + [0] 934.71/297.05 [0 1] [0 1] [0] 934.71/297.05 934.71/297.05 [0] = [0] 934.71/297.05 [0] 934.71/297.05 934.71/297.05 [zeros] = [0] 934.71/297.05 [0] 934.71/297.05 934.71/297.05 [a__incr](x1) = [1 0] x1 + [0] 934.71/297.05 [0 1] [0] 934.71/297.05 934.71/297.05 [s](x1) = [0] 934.71/297.05 [0] 934.71/297.05 934.71/297.05 [incr](x1) = [1 0] x1 + [0] 934.71/297.05 [0 1] [0] 934.71/297.05 934.71/297.05 [adx](x1) = [1 0] x1 + [0] 934.71/297.05 [0 1] [4] 934.71/297.05 934.71/297.05 [a__hd](x1) = [1 1] x1 + [0] 934.71/297.05 [0 1] [4] 934.71/297.05 934.71/297.05 [mark](x1) = [1 2] x1 + [0] 934.71/297.05 [0 1] [3] 934.71/297.06 934.71/297.06 [a__tl](x1) = [1 4] x1 + [0] 934.71/297.06 [0 1] [6] 934.71/297.06 934.71/297.06 [nats] = [0] 934.71/297.06 [1] 934.71/297.06 934.71/297.06 [hd](x1) = [1 1] x1 + [0] 934.71/297.06 [0 1] [4] 934.71/297.06 934.71/297.06 [tl](x1) = [1 4] x1 + [0] 934.71/297.06 [0 1] [6] 934.71/297.06 934.71/297.06 The order satisfies the following ordering constraints: 934.71/297.06 934.71/297.06 [a__nats()] = [0] 934.71/297.06 [4] 934.71/297.06 >= [0] 934.71/297.06 [4] 934.71/297.06 = [a__adx(a__zeros())] 934.71/297.06 934.71/297.06 [a__nats()] = [0] 934.71/297.06 [4] 934.71/297.06 >= [0] 934.71/297.06 [1] 934.71/297.06 = [nats()] 934.71/297.06 934.71/297.06 [a__adx(X)] = [1 0] X + [0] 934.71/297.06 [0 1] [4] 934.71/297.06 >= [1 0] X + [0] 934.71/297.06 [0 1] [4] 934.71/297.06 = [adx(X)] 934.71/297.06 934.71/297.06 [a__adx(cons(X, Y))] = [1 1] X + [1 0] Y + [0] 934.71/297.06 [0 1] [0 1] [4] 934.71/297.06 >= [1 1] X + [1 0] Y + [0] 934.71/297.06 [0 1] [0 1] [4] 934.71/297.06 = [a__incr(cons(X, adx(Y)))] 934.71/297.06 934.71/297.06 [a__zeros()] = [0] 934.71/297.06 [0] 934.71/297.06 >= [0] 934.71/297.06 [0] 934.71/297.06 = [cons(0(), zeros())] 934.71/297.06 934.71/297.06 [a__zeros()] = [0] 934.71/297.06 [0] 934.71/297.06 >= [0] 934.71/297.06 [0] 934.71/297.06 = [zeros()] 934.71/297.06 934.71/297.06 [a__incr(X)] = [1 0] X + [0] 934.71/297.06 [0 1] [0] 934.71/297.06 >= [1 0] X + [0] 934.71/297.06 [0 1] [0] 934.71/297.06 = [incr(X)] 934.71/297.06 934.71/297.06 [a__incr(cons(X, Y))] = [1 1] X + [1 0] Y + [0] 934.71/297.06 [0 1] [0 1] [0] 934.71/297.06 >= [1 0] Y + [0] 934.71/297.06 [0 1] [0] 934.71/297.06 = [cons(s(X), incr(Y))] 934.71/297.06 934.71/297.06 [a__hd(X)] = [1 1] X + [0] 934.71/297.06 [0 1] [4] 934.71/297.06 >= [1 1] X + [0] 934.71/297.06 [0 1] [4] 934.71/297.06 = [hd(X)] 934.71/297.06 934.71/297.06 [a__hd(cons(X, Y))] = [1 2] X + [1 1] Y + [0] 934.71/297.06 [0 1] [0 1] [4] 934.71/297.06 >= [1 2] X + [0] 934.71/297.06 [0 1] [3] 934.71/297.06 = [mark(X)] 934.71/297.06 934.71/297.06 [mark(cons(X1, X2))] = [1 3] X1 + [1 2] X2 + [0] 934.71/297.06 [0 1] [0 1] [3] 934.71/297.06 >= [1 1] X1 + [1 0] X2 + [0] 934.71/297.06 [0 1] [0 1] [0] 934.71/297.06 = [cons(X1, X2)] 934.71/297.06 934.71/297.06 [mark(0())] = [0] 934.71/297.06 [3] 934.71/297.06 >= [0] 934.71/297.06 [0] 934.71/297.06 = [0()] 934.71/297.06 934.71/297.06 [mark(zeros())] = [0] 934.71/297.06 [3] 934.71/297.06 >= [0] 934.71/297.06 [0] 934.71/297.06 = [a__zeros()] 934.71/297.06 934.71/297.06 [mark(s(X))] = [0] 934.71/297.06 [3] 934.71/297.06 >= [0] 934.71/297.06 [0] 934.71/297.06 = [s(X)] 934.71/297.06 934.71/297.06 [mark(incr(X))] = [1 2] X + [0] 934.71/297.06 [0 1] [3] 934.71/297.06 >= [1 2] X + [0] 934.71/297.06 [0 1] [3] 934.71/297.06 = [a__incr(mark(X))] 934.71/297.06 934.71/297.06 [mark(adx(X))] = [1 2] X + [8] 934.71/297.06 [0 1] [7] 934.71/297.06 > [1 2] X + [0] 934.71/297.06 [0 1] [7] 934.71/297.06 = [a__adx(mark(X))] 934.71/297.06 934.71/297.06 [mark(nats())] = [2] 934.71/297.06 [4] 934.71/297.06 > [0] 934.71/297.06 [4] 934.71/297.06 = [a__nats()] 934.71/297.06 934.71/297.06 [mark(hd(X))] = [1 3] X + [8] 934.71/297.06 [0 1] [7] 934.71/297.06 > [1 3] X + [3] 934.71/297.06 [0 1] [7] 934.71/297.06 = [a__hd(mark(X))] 934.71/297.06 934.71/297.06 [mark(tl(X))] = [1 6] X + [12] 934.71/297.06 [0 1] [9] 934.71/297.06 >= [1 6] X + [12] 934.71/297.06 [0 1] [9] 934.71/297.06 = [a__tl(mark(X))] 934.71/297.06 934.71/297.06 [a__tl(X)] = [1 4] X + [0] 934.71/297.06 [0 1] [6] 934.71/297.06 >= [1 4] X + [0] 934.71/297.06 [0 1] [6] 934.71/297.06 = [tl(X)] 934.71/297.06 934.71/297.06 [a__tl(cons(X, Y))] = [1 5] X + [1 4] Y + [0] 934.71/297.06 [0 1] [0 1] [6] 934.71/297.06 >= [1 2] Y + [0] 934.71/297.06 [0 1] [3] 934.71/297.06 = [mark(Y)] 934.71/297.06 934.71/297.06 934.71/297.06 We return to the main proof. 934.71/297.06 934.71/297.06 We are left with following problem, upon which TcT provides the 934.71/297.06 certificate YES(O(1),O(n^3)). 934.71/297.06 934.71/297.06 Strict Trs: { mark(incr(X)) -> a__incr(mark(X)) } 934.71/297.06 Weak Trs: 934.71/297.06 { a__nats() -> a__adx(a__zeros()) 934.71/297.06 , a__nats() -> nats() 934.71/297.06 , a__adx(X) -> adx(X) 934.71/297.06 , a__adx(cons(X, Y)) -> a__incr(cons(X, adx(Y))) 934.71/297.06 , a__zeros() -> cons(0(), zeros()) 934.71/297.06 , a__zeros() -> zeros() 934.71/297.06 , a__incr(X) -> incr(X) 934.71/297.06 , a__incr(cons(X, Y)) -> cons(s(X), incr(Y)) 934.71/297.06 , a__hd(X) -> hd(X) 934.71/297.06 , a__hd(cons(X, Y)) -> mark(X) 934.71/297.06 , mark(cons(X1, X2)) -> cons(X1, X2) 934.71/297.06 , mark(0()) -> 0() 934.71/297.06 , mark(zeros()) -> a__zeros() 934.71/297.06 , mark(s(X)) -> s(X) 934.71/297.06 , mark(adx(X)) -> a__adx(mark(X)) 934.71/297.06 , mark(nats()) -> a__nats() 934.71/297.06 , mark(hd(X)) -> a__hd(mark(X)) 934.71/297.06 , mark(tl(X)) -> a__tl(mark(X)) 934.71/297.06 , a__tl(X) -> tl(X) 934.71/297.06 , a__tl(cons(X, Y)) -> mark(Y) } 934.71/297.06 Obligation: 934.71/297.06 innermost runtime complexity 934.71/297.06 Answer: 934.71/297.06 YES(O(1),O(n^3)) 934.71/297.06 934.71/297.06 We use the processor 'matrix interpretation of dimension 3' to 934.71/297.06 orient following rules strictly. 934.71/297.06 934.71/297.06 Trs: { mark(incr(X)) -> a__incr(mark(X)) } 934.71/297.06 934.71/297.06 The induced complexity on above rules (modulo remaining rules) is 934.71/297.06 YES(?,O(n^3)) . These rules are moved into the corresponding weak 934.71/297.06 component(s). 934.71/297.06 934.71/297.06 Sub-proof: 934.71/297.06 ---------- 934.71/297.06 The following argument positions are usable: 934.71/297.06 Uargs(a__adx) = {1}, Uargs(a__incr) = {1}, Uargs(a__hd) = {1}, 934.71/297.06 Uargs(a__tl) = {1} 934.71/297.06 934.71/297.06 TcT has computed the following constructor-based matrix 934.71/297.06 interpretation satisfying not(EDA). 934.71/297.06 934.71/297.06 [0] 934.71/297.06 [a__nats] = [6] 934.71/297.06 [4] 934.71/297.06 934.71/297.06 [1 0 0] [0] 934.71/297.06 [a__adx](x1) = [0 1 0] x1 + [1] 934.71/297.06 [0 0 1] [4] 934.71/297.06 934.71/297.06 [0] 934.71/297.06 [a__zeros] = [4] 934.71/297.06 [0] 934.71/297.06 934.71/297.06 [1 1 0] [1 0 0] [0] 934.71/297.06 [cons](x1, x2) = [0 1 0] x1 + [0 1 0] x2 + [0] 934.71/297.06 [0 0 1] [0 0 1] [0] 934.71/297.06 934.71/297.06 [0] 934.71/297.06 [0] = [0] 934.71/297.06 [0] 934.71/297.06 934.71/297.06 [0] 934.71/297.06 [zeros] = [1] 934.71/297.06 [0] 934.71/297.06 934.71/297.06 [1 0 0] [0] 934.71/297.06 [a__incr](x1) = [0 1 0] x1 + [1] 934.71/297.06 [0 0 1] [0] 934.71/297.06 934.71/297.06 [0] 934.71/297.06 [s](x1) = [0] 934.71/297.06 [0] 934.71/297.06 934.71/297.06 [1 0 0] [0] 934.71/297.06 [incr](x1) = [0 1 0] x1 + [1] 934.71/297.06 [0 0 1] [0] 934.71/297.06 934.71/297.06 [1 0 0] [0] 934.71/297.06 [adx](x1) = [0 1 0] x1 + [0] 934.71/297.06 [0 0 1] [4] 934.71/297.06 934.71/297.06 [1 0 0] [0] 934.71/297.06 [a__hd](x1) = [0 1 2] x1 + [4] 934.71/297.06 [0 0 1] [1] 934.71/297.06 934.71/297.06 [1 1 0] [0] 934.71/297.06 [mark](x1) = [0 1 2] x1 + [3] 934.71/297.06 [0 0 1] [0] 934.71/297.06 934.71/297.06 [1 1 0] [1] 934.71/297.06 [a__tl](x1) = [0 1 2] x1 + [3] 934.71/297.06 [0 0 1] [0] 934.71/297.06 934.71/297.06 [0] 934.71/297.06 [nats] = [0] 934.71/297.06 [4] 934.71/297.06 934.71/297.06 [1 0 0] [0] 934.71/297.06 [hd](x1) = [0 1 2] x1 + [3] 934.71/297.06 [0 0 1] [1] 934.71/297.06 934.71/297.06 [1 1 0] [1] 934.71/297.06 [tl](x1) = [0 1 2] x1 + [3] 934.71/297.06 [0 0 1] [0] 934.71/297.06 934.71/297.06 The order satisfies the following ordering constraints: 934.71/297.06 934.71/297.06 [a__nats()] = [0] 934.71/297.06 [6] 934.71/297.06 [4] 934.71/297.06 >= [0] 934.71/297.06 [5] 934.71/297.06 [4] 934.71/297.06 = [a__adx(a__zeros())] 934.71/297.06 934.71/297.06 [a__nats()] = [0] 934.71/297.06 [6] 934.71/297.06 [4] 934.71/297.06 >= [0] 934.71/297.06 [0] 934.71/297.06 [4] 934.71/297.06 = [nats()] 934.71/297.06 934.71/297.06 [a__adx(X)] = [1 0 0] [0] 934.71/297.06 [0 1 0] X + [1] 934.71/297.06 [0 0 1] [4] 934.71/297.06 >= [1 0 0] [0] 934.71/297.06 [0 1 0] X + [0] 934.71/297.06 [0 0 1] [4] 934.71/297.06 = [adx(X)] 934.71/297.06 934.71/297.06 [a__adx(cons(X, Y))] = [1 1 0] [1 0 0] [0] 934.71/297.06 [0 1 0] X + [0 1 0] Y + [1] 934.71/297.06 [0 0 1] [0 0 1] [4] 934.71/297.06 >= [1 1 0] [1 0 0] [0] 934.71/297.06 [0 1 0] X + [0 1 0] Y + [1] 934.71/297.06 [0 0 1] [0 0 1] [4] 934.71/297.06 = [a__incr(cons(X, adx(Y)))] 934.71/297.06 934.71/297.06 [a__zeros()] = [0] 934.71/297.06 [4] 934.71/297.06 [0] 934.71/297.06 >= [0] 934.71/297.06 [1] 934.71/297.06 [0] 934.71/297.06 = [cons(0(), zeros())] 934.71/297.06 934.71/297.06 [a__zeros()] = [0] 934.71/297.06 [4] 934.71/297.06 [0] 934.71/297.06 >= [0] 934.71/297.06 [1] 934.71/297.06 [0] 934.71/297.06 = [zeros()] 934.71/297.06 934.71/297.06 [a__incr(X)] = [1 0 0] [0] 934.71/297.06 [0 1 0] X + [1] 934.71/297.06 [0 0 1] [0] 934.71/297.06 >= [1 0 0] [0] 934.71/297.06 [0 1 0] X + [1] 934.71/297.06 [0 0 1] [0] 934.71/297.06 = [incr(X)] 934.71/297.06 934.71/297.06 [a__incr(cons(X, Y))] = [1 1 0] [1 0 0] [0] 934.71/297.06 [0 1 0] X + [0 1 0] Y + [1] 934.71/297.06 [0 0 1] [0 0 1] [0] 934.71/297.06 >= [1 0 0] [0] 934.71/297.06 [0 1 0] Y + [1] 934.71/297.06 [0 0 1] [0] 934.71/297.06 = [cons(s(X), incr(Y))] 934.71/297.06 934.71/297.06 [a__hd(X)] = [1 0 0] [0] 934.71/297.06 [0 1 2] X + [4] 934.71/297.06 [0 0 1] [1] 934.71/297.06 >= [1 0 0] [0] 934.71/297.06 [0 1 2] X + [3] 934.71/297.06 [0 0 1] [1] 934.71/297.06 = [hd(X)] 934.71/297.06 934.71/297.06 [a__hd(cons(X, Y))] = [1 1 0] [1 0 0] [0] 934.71/297.06 [0 1 2] X + [0 1 2] Y + [4] 934.71/297.06 [0 0 1] [0 0 1] [1] 934.71/297.06 >= [1 1 0] [0] 934.71/297.06 [0 1 2] X + [3] 934.71/297.06 [0 0 1] [0] 934.71/297.06 = [mark(X)] 934.71/297.06 934.71/297.06 [mark(cons(X1, X2))] = [1 2 0] [1 1 0] [0] 934.71/297.06 [0 1 2] X1 + [0 1 2] X2 + [3] 934.71/297.06 [0 0 1] [0 0 1] [0] 934.71/297.06 >= [1 1 0] [1 0 0] [0] 934.71/297.06 [0 1 0] X1 + [0 1 0] X2 + [0] 934.71/297.06 [0 0 1] [0 0 1] [0] 934.71/297.06 = [cons(X1, X2)] 934.71/297.06 934.71/297.06 [mark(0())] = [0] 934.71/297.06 [3] 934.71/297.06 [0] 934.71/297.06 >= [0] 934.71/297.06 [0] 934.71/297.06 [0] 934.71/297.06 = [0()] 934.71/297.06 934.71/297.06 [mark(zeros())] = [1] 934.71/297.06 [4] 934.71/297.06 [0] 934.71/297.06 > [0] 934.71/297.06 [4] 934.71/297.06 [0] 934.71/297.06 = [a__zeros()] 934.71/297.06 934.71/297.06 [mark(s(X))] = [0] 934.71/297.06 [3] 934.71/297.06 [0] 934.71/297.06 >= [0] 934.71/297.06 [0] 934.71/297.06 [0] 934.71/297.06 = [s(X)] 934.71/297.06 934.71/297.06 [mark(incr(X))] = [1 1 0] [1] 934.71/297.06 [0 1 2] X + [4] 934.71/297.06 [0 0 1] [0] 934.71/297.06 > [1 1 0] [0] 934.71/297.06 [0 1 2] X + [4] 934.71/297.06 [0 0 1] [0] 934.71/297.06 = [a__incr(mark(X))] 934.71/297.06 934.71/297.06 [mark(adx(X))] = [1 1 0] [0] 934.71/297.06 [0 1 2] X + [11] 934.71/297.06 [0 0 1] [4] 934.71/297.06 >= [1 1 0] [0] 934.71/297.06 [0 1 2] X + [4] 934.71/297.06 [0 0 1] [4] 934.71/297.06 = [a__adx(mark(X))] 934.71/297.06 934.71/297.06 [mark(nats())] = [0] 934.71/297.06 [11] 934.71/297.06 [4] 934.71/297.06 >= [0] 934.71/297.06 [6] 934.71/297.06 [4] 934.71/297.06 = [a__nats()] 934.71/297.06 934.71/297.06 [mark(hd(X))] = [1 1 2] [3] 934.71/297.06 [0 1 4] X + [8] 934.71/297.06 [0 0 1] [1] 934.71/297.06 > [1 1 0] [0] 934.71/297.06 [0 1 4] X + [7] 934.71/297.06 [0 0 1] [1] 934.71/297.06 = [a__hd(mark(X))] 934.71/297.06 934.71/297.06 [mark(tl(X))] = [1 2 2] [4] 934.71/297.06 [0 1 4] X + [6] 934.71/297.06 [0 0 1] [0] 934.71/297.06 >= [1 2 2] [4] 934.71/297.06 [0 1 4] X + [6] 934.71/297.06 [0 0 1] [0] 934.71/297.06 = [a__tl(mark(X))] 934.71/297.06 934.71/297.06 [a__tl(X)] = [1 1 0] [1] 934.71/297.06 [0 1 2] X + [3] 934.71/297.06 [0 0 1] [0] 934.71/297.06 >= [1 1 0] [1] 934.71/297.06 [0 1 2] X + [3] 934.71/297.06 [0 0 1] [0] 934.71/297.06 = [tl(X)] 934.71/297.06 934.71/297.06 [a__tl(cons(X, Y))] = [1 2 0] [1 1 0] [1] 934.71/297.06 [0 1 2] X + [0 1 2] Y + [3] 934.71/297.06 [0 0 1] [0 0 1] [0] 934.71/297.06 > [1 1 0] [0] 934.71/297.06 [0 1 2] Y + [3] 934.71/297.06 [0 0 1] [0] 934.71/297.06 = [mark(Y)] 934.71/297.06 934.71/297.06 934.71/297.06 We return to the main proof. 934.71/297.06 934.71/297.06 We are left with following problem, upon which TcT provides the 934.71/297.06 certificate YES(O(1),O(1)). 934.71/297.06 934.71/297.06 Weak Trs: 934.71/297.06 { a__nats() -> a__adx(a__zeros()) 934.71/297.06 , a__nats() -> nats() 934.71/297.06 , a__adx(X) -> adx(X) 934.71/297.06 , a__adx(cons(X, Y)) -> a__incr(cons(X, adx(Y))) 934.71/297.06 , a__zeros() -> cons(0(), zeros()) 934.71/297.06 , a__zeros() -> zeros() 934.71/297.06 , a__incr(X) -> incr(X) 934.71/297.06 , a__incr(cons(X, Y)) -> cons(s(X), incr(Y)) 934.71/297.06 , a__hd(X) -> hd(X) 934.71/297.06 , a__hd(cons(X, Y)) -> mark(X) 934.71/297.06 , mark(cons(X1, X2)) -> cons(X1, X2) 934.71/297.06 , mark(0()) -> 0() 934.71/297.06 , mark(zeros()) -> a__zeros() 934.71/297.06 , mark(s(X)) -> s(X) 934.71/297.06 , mark(incr(X)) -> a__incr(mark(X)) 934.71/297.06 , mark(adx(X)) -> a__adx(mark(X)) 934.71/297.06 , mark(nats()) -> a__nats() 934.71/297.06 , mark(hd(X)) -> a__hd(mark(X)) 934.71/297.06 , mark(tl(X)) -> a__tl(mark(X)) 934.71/297.06 , a__tl(X) -> tl(X) 934.71/297.06 , a__tl(cons(X, Y)) -> mark(Y) } 934.71/297.06 Obligation: 934.71/297.06 innermost runtime complexity 934.71/297.06 Answer: 934.71/297.06 YES(O(1),O(1)) 934.71/297.06 934.71/297.06 Empty rules are trivially bounded 934.71/297.06 934.71/297.06 Hurray, we answered YES(O(1),O(n^3)) 934.98/297.23 EOF