YES(O(1),O(n^1)) 3.61/1.37 YES(O(1),O(n^1)) 3.61/1.37 3.61/1.37 We are left with following problem, upon which TcT provides the 3.61/1.37 certificate YES(O(1),O(n^1)). 3.61/1.37 3.61/1.37 Strict Trs: 3.61/1.37 { terms(N) -> cons(recip(sqr(N)), n__terms(n__s(N))) 3.61/1.37 , terms(X) -> n__terms(X) 3.61/1.37 , sqr(X) -> n__sqr(X) 3.61/1.37 , sqr(0()) -> 0() 3.61/1.37 , sqr(s(X)) -> s(n__add(n__sqr(activate(X)), n__dbl(activate(X)))) 3.61/1.37 , s(X) -> n__s(X) 3.61/1.37 , activate(X) -> X 3.61/1.37 , activate(n__terms(X)) -> terms(activate(X)) 3.61/1.37 , activate(n__s(X)) -> s(X) 3.61/1.37 , activate(n__add(X1, X2)) -> add(activate(X1), activate(X2)) 3.61/1.37 , activate(n__sqr(X)) -> sqr(activate(X)) 3.61/1.37 , activate(n__dbl(X)) -> dbl(activate(X)) 3.61/1.37 , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) 3.61/1.37 , dbl(X) -> n__dbl(X) 3.61/1.37 , dbl(0()) -> 0() 3.61/1.37 , dbl(s(X)) -> s(n__s(n__dbl(activate(X)))) 3.61/1.37 , add(X1, X2) -> n__add(X1, X2) 3.61/1.37 , add(0(), X) -> X 3.61/1.37 , add(s(X), Y) -> s(n__add(activate(X), Y)) 3.61/1.37 , first(X1, X2) -> n__first(X1, X2) 3.61/1.37 , first(0(), X) -> nil() 3.61/1.37 , first(s(X), cons(Y, Z)) -> 3.61/1.37 cons(Y, n__first(activate(X), activate(Z))) } 3.61/1.37 Obligation: 3.61/1.37 innermost runtime complexity 3.61/1.37 Answer: 3.61/1.37 YES(O(1),O(n^1)) 3.61/1.37 3.61/1.37 Arguments of following rules are not normal-forms: 3.61/1.37 3.61/1.37 { sqr(s(X)) -> s(n__add(n__sqr(activate(X)), n__dbl(activate(X)))) 3.61/1.37 , dbl(s(X)) -> s(n__s(n__dbl(activate(X)))) 3.61/1.37 , add(s(X), Y) -> s(n__add(activate(X), Y)) 3.61/1.37 , first(s(X), cons(Y, Z)) -> 3.61/1.37 cons(Y, n__first(activate(X), activate(Z))) } 3.61/1.37 3.61/1.37 All above mentioned rules can be savely removed. 3.61/1.37 3.61/1.37 We are left with following problem, upon which TcT provides the 3.61/1.37 certificate YES(O(1),O(n^1)). 3.61/1.37 3.61/1.37 Strict Trs: 3.61/1.37 { terms(N) -> cons(recip(sqr(N)), n__terms(n__s(N))) 3.61/1.37 , terms(X) -> n__terms(X) 3.61/1.37 , sqr(X) -> n__sqr(X) 3.61/1.37 , sqr(0()) -> 0() 3.61/1.37 , s(X) -> n__s(X) 3.61/1.37 , activate(X) -> X 3.61/1.37 , activate(n__terms(X)) -> terms(activate(X)) 3.61/1.37 , activate(n__s(X)) -> s(X) 3.61/1.37 , activate(n__add(X1, X2)) -> add(activate(X1), activate(X2)) 3.61/1.37 , activate(n__sqr(X)) -> sqr(activate(X)) 3.61/1.37 , activate(n__dbl(X)) -> dbl(activate(X)) 3.61/1.37 , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) 3.61/1.37 , dbl(X) -> n__dbl(X) 3.61/1.37 , dbl(0()) -> 0() 3.61/1.37 , add(X1, X2) -> n__add(X1, X2) 3.61/1.37 , add(0(), X) -> X 3.61/1.37 , first(X1, X2) -> n__first(X1, X2) 3.61/1.37 , first(0(), X) -> nil() } 3.61/1.37 Obligation: 3.61/1.37 innermost runtime complexity 3.61/1.37 Answer: 3.61/1.37 YES(O(1),O(n^1)) 3.61/1.37 3.61/1.37 We add the following weak dependency pairs: 3.61/1.37 3.61/1.37 Strict DPs: 3.61/1.37 { terms^#(N) -> c_1(sqr^#(N)) 3.61/1.37 , terms^#(X) -> c_2() 3.61/1.37 , sqr^#(X) -> c_3() 3.61/1.37 , sqr^#(0()) -> c_4() 3.61/1.37 , s^#(X) -> c_5() 3.61/1.37 , activate^#(X) -> c_6() 3.61/1.37 , activate^#(n__terms(X)) -> c_7(terms^#(activate(X))) 3.61/1.37 , activate^#(n__s(X)) -> c_8(s^#(X)) 3.61/1.37 , activate^#(n__add(X1, X2)) -> 3.61/1.37 c_9(add^#(activate(X1), activate(X2))) 3.61/1.37 , activate^#(n__sqr(X)) -> c_10(sqr^#(activate(X))) 3.61/1.37 , activate^#(n__dbl(X)) -> c_11(dbl^#(activate(X))) 3.61/1.37 , activate^#(n__first(X1, X2)) -> 3.61/1.37 c_12(first^#(activate(X1), activate(X2))) 3.61/1.37 , add^#(X1, X2) -> c_15() 3.61/1.37 , add^#(0(), X) -> c_16() 3.61/1.37 , dbl^#(X) -> c_13() 3.61/1.37 , dbl^#(0()) -> c_14() 3.61/1.37 , first^#(X1, X2) -> c_17() 3.61/1.37 , first^#(0(), X) -> c_18() } 3.61/1.37 3.61/1.37 and mark the set of starting terms. 3.61/1.37 3.61/1.37 We are left with following problem, upon which TcT provides the 3.61/1.37 certificate YES(O(1),O(n^1)). 3.61/1.37 3.61/1.37 Strict DPs: 3.61/1.37 { terms^#(N) -> c_1(sqr^#(N)) 3.61/1.37 , terms^#(X) -> c_2() 3.61/1.37 , sqr^#(X) -> c_3() 3.61/1.37 , sqr^#(0()) -> c_4() 3.61/1.37 , s^#(X) -> c_5() 3.61/1.37 , activate^#(X) -> c_6() 3.61/1.37 , activate^#(n__terms(X)) -> c_7(terms^#(activate(X))) 3.61/1.37 , activate^#(n__s(X)) -> c_8(s^#(X)) 3.61/1.37 , activate^#(n__add(X1, X2)) -> 3.61/1.37 c_9(add^#(activate(X1), activate(X2))) 3.61/1.37 , activate^#(n__sqr(X)) -> c_10(sqr^#(activate(X))) 3.61/1.37 , activate^#(n__dbl(X)) -> c_11(dbl^#(activate(X))) 3.61/1.37 , activate^#(n__first(X1, X2)) -> 3.61/1.37 c_12(first^#(activate(X1), activate(X2))) 3.61/1.37 , add^#(X1, X2) -> c_15() 3.61/1.37 , add^#(0(), X) -> c_16() 3.61/1.37 , dbl^#(X) -> c_13() 3.61/1.37 , dbl^#(0()) -> c_14() 3.61/1.37 , first^#(X1, X2) -> c_17() 3.61/1.37 , first^#(0(), X) -> c_18() } 3.61/1.37 Strict Trs: 3.61/1.37 { terms(N) -> cons(recip(sqr(N)), n__terms(n__s(N))) 3.61/1.37 , terms(X) -> n__terms(X) 3.61/1.37 , sqr(X) -> n__sqr(X) 3.61/1.37 , sqr(0()) -> 0() 3.61/1.37 , s(X) -> n__s(X) 3.61/1.37 , activate(X) -> X 3.61/1.37 , activate(n__terms(X)) -> terms(activate(X)) 3.61/1.37 , activate(n__s(X)) -> s(X) 3.61/1.37 , activate(n__add(X1, X2)) -> add(activate(X1), activate(X2)) 3.61/1.37 , activate(n__sqr(X)) -> sqr(activate(X)) 3.61/1.37 , activate(n__dbl(X)) -> dbl(activate(X)) 3.61/1.37 , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) 3.61/1.37 , dbl(X) -> n__dbl(X) 3.61/1.37 , dbl(0()) -> 0() 3.61/1.37 , add(X1, X2) -> n__add(X1, X2) 3.61/1.37 , add(0(), X) -> X 3.61/1.37 , first(X1, X2) -> n__first(X1, X2) 3.61/1.37 , first(0(), X) -> nil() } 3.61/1.37 Obligation: 3.61/1.37 innermost runtime complexity 3.61/1.37 Answer: 3.61/1.37 YES(O(1),O(n^1)) 3.61/1.37 3.61/1.37 The weightgap principle applies (using the following constant 3.61/1.37 growth matrix-interpretation) 3.61/1.37 3.61/1.37 The following argument positions are usable: 3.61/1.37 Uargs(terms) = {1}, Uargs(cons) = {1}, Uargs(recip) = {1}, 3.61/1.37 Uargs(sqr) = {1}, Uargs(dbl) = {1}, Uargs(add) = {1, 2}, 3.61/1.37 Uargs(first) = {1, 2}, Uargs(terms^#) = {1}, Uargs(c_1) = {1}, 3.61/1.37 Uargs(sqr^#) = {1}, Uargs(c_7) = {1}, Uargs(c_8) = {1}, 3.61/1.37 Uargs(c_9) = {1}, Uargs(add^#) = {1, 2}, Uargs(c_10) = {1}, 3.61/1.37 Uargs(c_11) = {1}, Uargs(dbl^#) = {1}, Uargs(c_12) = {1}, 3.61/1.37 Uargs(first^#) = {1, 2} 3.61/1.37 3.61/1.37 TcT has computed the following constructor-restricted matrix 3.61/1.37 interpretation. 3.61/1.37 3.61/1.37 [terms](x1) = [1 0] x1 + [2] 3.61/1.37 [0 1] [2] 3.61/1.37 3.61/1.37 [cons](x1, x2) = [1 0] x1 + [0] 3.61/1.37 [0 0] [1] 3.61/1.37 3.61/1.37 [recip](x1) = [1 0] x1 + [0] 3.61/1.37 [0 0] [2] 3.61/1.37 3.61/1.37 [sqr](x1) = [1 0] x1 + [1] 3.61/1.37 [0 1] [2] 3.61/1.37 3.61/1.37 [n__terms](x1) = [1 0] x1 + [0] 3.61/1.37 [0 1] [2] 3.61/1.37 3.61/1.37 [n__s](x1) = [1 0] x1 + [0] 3.61/1.37 [0 0] [2] 3.61/1.37 3.61/1.37 [0] = [0] 3.61/1.37 [0] 3.61/1.37 3.61/1.37 [s](x1) = [1 0] x1 + [1] 3.61/1.37 [0 0] [2] 3.61/1.37 3.61/1.37 [n__add](x1, x2) = [1 0] x1 + [1 0] x2 + [0] 3.61/1.37 [0 1] [0 1] [2] 3.61/1.37 3.61/1.37 [n__sqr](x1) = [1 0] x1 + [0] 3.61/1.37 [0 1] [2] 3.61/1.37 3.61/1.37 [activate](x1) = [1 2] x1 + [1] 3.61/1.37 [0 2] [0] 3.61/1.37 3.61/1.37 [n__dbl](x1) = [1 0] x1 + [0] 3.61/1.37 [0 1] [2] 3.61/1.37 3.61/1.37 [dbl](x1) = [1 0] x1 + [2] 3.61/1.37 [0 1] [2] 3.61/1.37 3.61/1.37 [add](x1, x2) = [1 0] x1 + [1 0] x2 + [2] 3.61/1.37 [0 1] [0 1] [2] 3.61/1.37 3.61/1.37 [first](x1, x2) = [1 0] x1 + [1 0] x2 + [1] 3.61/1.37 [0 1] [0 1] [2] 3.61/1.37 3.61/1.37 [nil] = [0] 3.61/1.37 [1] 3.61/1.37 3.61/1.37 [n__first](x1, x2) = [1 0] x1 + [1 0] x2 + [0] 3.61/1.37 [0 1] [0 1] [2] 3.61/1.37 3.61/1.37 [terms^#](x1) = [1 0] x1 + [1] 3.61/1.37 [0 0] [2] 3.61/1.37 3.61/1.37 [c_1](x1) = [1 0] x1 + [1] 3.61/1.37 [0 1] [1] 3.61/1.37 3.61/1.37 [sqr^#](x1) = [1 0] x1 + [1] 3.61/1.37 [0 0] [2] 3.61/1.37 3.61/1.37 [c_2] = [1] 3.61/1.37 [1] 3.61/1.37 3.61/1.37 [c_3] = [1] 3.61/1.37 [2] 3.61/1.37 3.61/1.37 [c_4] = [1] 3.61/1.37 [1] 3.61/1.37 3.61/1.37 [s^#](x1) = [1 0] x1 + [1] 3.61/1.37 [2 1] [1] 3.61/1.37 3.61/1.37 [c_5] = [1] 3.61/1.37 [1] 3.61/1.37 3.61/1.37 [activate^#](x1) = [1 2] x1 + [0] 3.61/1.37 [2 2] [1] 3.61/1.37 3.61/1.37 [c_6] = [2] 3.61/1.37 [1] 3.61/1.37 3.61/1.37 [c_7](x1) = [1 0] x1 + [2] 3.61/1.37 [0 1] [2] 3.61/1.37 3.61/1.37 [c_8](x1) = [1 0] x1 + [2] 3.61/1.37 [0 1] [2] 3.61/1.37 3.61/1.37 [c_9](x1) = [1 0] x1 + [2] 3.61/1.37 [0 1] [2] 3.61/1.37 3.61/1.37 [add^#](x1, x2) = [1 0] x1 + [1 0] x2 + [2] 3.61/1.37 [0 0] [0 0] [2] 3.61/1.37 3.61/1.37 [c_10](x1) = [1 0] x1 + [2] 3.61/1.37 [0 1] [2] 3.61/1.37 3.61/1.37 [c_11](x1) = [1 0] x1 + [2] 3.61/1.37 [0 1] [2] 3.61/1.37 3.61/1.37 [dbl^#](x1) = [1 0] x1 + [1] 3.61/1.37 [0 0] [2] 3.61/1.37 3.61/1.37 [c_12](x1) = [1 0] x1 + [2] 3.61/1.37 [0 1] [2] 3.61/1.37 3.61/1.37 [first^#](x1, x2) = [1 0] x1 + [1 0] x2 + [2] 3.61/1.37 [0 0] [0 0] [2] 3.61/1.37 3.61/1.37 [c_13] = [1] 3.61/1.37 [1] 3.61/1.37 3.61/1.37 [c_14] = [1] 3.61/1.37 [1] 3.61/1.37 3.61/1.37 [c_15] = [1] 3.61/1.37 [1] 3.61/1.37 3.61/1.37 [c_16] = [1] 3.61/1.37 [1] 3.61/1.37 3.61/1.37 [c_17] = [1] 3.61/1.37 [1] 3.61/1.37 3.61/1.37 [c_18] = [1] 3.61/1.37 [1] 3.61/1.37 3.61/1.37 The order satisfies the following ordering constraints: 3.61/1.37 3.61/1.37 [terms(N)] = [1 0] N + [2] 3.61/1.37 [0 1] [2] 3.61/1.37 > [1 0] N + [1] 3.61/1.37 [0 0] [1] 3.61/1.37 = [cons(recip(sqr(N)), n__terms(n__s(N)))] 3.61/1.37 3.61/1.37 [terms(X)] = [1 0] X + [2] 3.61/1.37 [0 1] [2] 3.61/1.37 > [1 0] X + [0] 3.61/1.37 [0 1] [2] 3.61/1.37 = [n__terms(X)] 3.61/1.37 3.61/1.37 [sqr(X)] = [1 0] X + [1] 3.61/1.37 [0 1] [2] 3.61/1.37 > [1 0] X + [0] 3.61/1.37 [0 1] [2] 3.61/1.37 = [n__sqr(X)] 3.61/1.37 3.61/1.37 [sqr(0())] = [1] 3.61/1.37 [2] 3.61/1.37 > [0] 3.61/1.37 [0] 3.61/1.37 = [0()] 3.61/1.37 3.61/1.37 [s(X)] = [1 0] X + [1] 3.61/1.37 [0 0] [2] 3.61/1.37 > [1 0] X + [0] 3.61/1.37 [0 0] [2] 3.61/1.37 = [n__s(X)] 3.61/1.37 3.61/1.37 [activate(X)] = [1 2] X + [1] 3.61/1.37 [0 2] [0] 3.61/1.37 > [1 0] X + [0] 3.61/1.37 [0 1] [0] 3.61/1.37 = [X] 3.61/1.37 3.61/1.37 [activate(n__terms(X))] = [1 2] X + [5] 3.61/1.37 [0 2] [4] 3.61/1.37 > [1 2] X + [3] 3.61/1.37 [0 2] [2] 3.61/1.37 = [terms(activate(X))] 3.61/1.37 3.61/1.37 [activate(n__s(X))] = [1 0] X + [5] 3.61/1.37 [0 0] [4] 3.61/1.37 > [1 0] X + [1] 3.61/1.37 [0 0] [2] 3.61/1.37 = [s(X)] 3.61/1.37 3.61/1.37 [activate(n__add(X1, X2))] = [1 2] X1 + [1 2] X2 + [5] 3.61/1.37 [0 2] [0 2] [4] 3.61/1.37 > [1 2] X1 + [1 2] X2 + [4] 3.61/1.37 [0 2] [0 2] [2] 3.61/1.37 = [add(activate(X1), activate(X2))] 3.61/1.37 3.61/1.37 [activate(n__sqr(X))] = [1 2] X + [5] 3.61/1.37 [0 2] [4] 3.61/1.37 > [1 2] X + [2] 3.61/1.37 [0 2] [2] 3.61/1.37 = [sqr(activate(X))] 3.61/1.37 3.61/1.37 [activate(n__dbl(X))] = [1 2] X + [5] 3.61/1.37 [0 2] [4] 3.61/1.38 > [1 2] X + [3] 3.61/1.38 [0 2] [2] 3.61/1.38 = [dbl(activate(X))] 3.61/1.38 3.61/1.38 [activate(n__first(X1, X2))] = [1 2] X1 + [1 2] X2 + [5] 3.61/1.38 [0 2] [0 2] [4] 3.61/1.38 > [1 2] X1 + [1 2] X2 + [3] 3.61/1.38 [0 2] [0 2] [2] 3.61/1.38 = [first(activate(X1), activate(X2))] 3.61/1.38 3.61/1.38 [dbl(X)] = [1 0] X + [2] 3.61/1.38 [0 1] [2] 3.61/1.38 > [1 0] X + [0] 3.61/1.38 [0 1] [2] 3.61/1.38 = [n__dbl(X)] 3.61/1.38 3.61/1.38 [dbl(0())] = [2] 3.61/1.38 [2] 3.61/1.38 > [0] 3.61/1.38 [0] 3.61/1.38 = [0()] 3.61/1.38 3.61/1.38 [add(X1, X2)] = [1 0] X1 + [1 0] X2 + [2] 3.61/1.38 [0 1] [0 1] [2] 3.61/1.38 > [1 0] X1 + [1 0] X2 + [0] 3.61/1.38 [0 1] [0 1] [2] 3.61/1.38 = [n__add(X1, X2)] 3.61/1.38 3.61/1.38 [add(0(), X)] = [1 0] X + [2] 3.61/1.38 [0 1] [2] 3.61/1.38 > [1 0] X + [0] 3.61/1.38 [0 1] [0] 3.61/1.38 = [X] 3.61/1.38 3.61/1.38 [first(X1, X2)] = [1 0] X1 + [1 0] X2 + [1] 3.61/1.38 [0 1] [0 1] [2] 3.61/1.38 > [1 0] X1 + [1 0] X2 + [0] 3.61/1.38 [0 1] [0 1] [2] 3.61/1.38 = [n__first(X1, X2)] 3.61/1.38 3.61/1.38 [first(0(), X)] = [1 0] X + [1] 3.61/1.38 [0 1] [2] 3.61/1.38 > [0] 3.61/1.38 [1] 3.61/1.38 = [nil()] 3.61/1.38 3.61/1.38 [terms^#(N)] = [1 0] N + [1] 3.61/1.38 [0 0] [2] 3.61/1.38 ? [1 0] N + [2] 3.61/1.38 [0 0] [3] 3.61/1.38 = [c_1(sqr^#(N))] 3.61/1.38 3.61/1.38 [terms^#(X)] = [1 0] X + [1] 3.61/1.38 [0 0] [2] 3.61/1.38 >= [1] 3.61/1.38 [1] 3.61/1.38 = [c_2()] 3.61/1.38 3.61/1.38 [sqr^#(X)] = [1 0] X + [1] 3.61/1.38 [0 0] [2] 3.61/1.38 >= [1] 3.61/1.38 [2] 3.61/1.38 = [c_3()] 3.61/1.38 3.61/1.38 [sqr^#(0())] = [1] 3.61/1.38 [2] 3.61/1.38 >= [1] 3.61/1.38 [1] 3.61/1.38 = [c_4()] 3.61/1.38 3.61/1.38 [s^#(X)] = [1 0] X + [1] 3.61/1.38 [2 1] [1] 3.61/1.38 >= [1] 3.61/1.38 [1] 3.61/1.38 = [c_5()] 3.61/1.38 3.61/1.38 [activate^#(X)] = [1 2] X + [0] 3.61/1.38 [2 2] [1] 3.61/1.38 ? [2] 3.61/1.38 [1] 3.61/1.38 = [c_6()] 3.61/1.38 3.61/1.38 [activate^#(n__terms(X))] = [1 2] X + [4] 3.61/1.38 [2 2] [5] 3.61/1.38 >= [1 2] X + [4] 3.61/1.38 [0 0] [4] 3.61/1.38 = [c_7(terms^#(activate(X)))] 3.61/1.38 3.61/1.38 [activate^#(n__s(X))] = [1 0] X + [4] 3.61/1.38 [2 0] [5] 3.61/1.38 ? [1 0] X + [3] 3.61/1.38 [2 1] [3] 3.61/1.38 = [c_8(s^#(X))] 3.61/1.38 3.61/1.38 [activate^#(n__add(X1, X2))] = [1 2] X1 + [1 2] X2 + [4] 3.61/1.38 [2 2] [2 2] [5] 3.61/1.38 ? [1 2] X1 + [1 2] X2 + [6] 3.61/1.38 [0 0] [0 0] [4] 3.61/1.38 = [c_9(add^#(activate(X1), activate(X2)))] 3.61/1.38 3.61/1.38 [activate^#(n__sqr(X))] = [1 2] X + [4] 3.61/1.38 [2 2] [5] 3.61/1.38 >= [1 2] X + [4] 3.61/1.38 [0 0] [4] 3.61/1.38 = [c_10(sqr^#(activate(X)))] 3.61/1.38 3.61/1.38 [activate^#(n__dbl(X))] = [1 2] X + [4] 3.61/1.38 [2 2] [5] 3.61/1.38 >= [1 2] X + [4] 3.61/1.38 [0 0] [4] 3.61/1.38 = [c_11(dbl^#(activate(X)))] 3.61/1.38 3.61/1.38 [activate^#(n__first(X1, X2))] = [1 2] X1 + [1 2] X2 + [4] 3.61/1.38 [2 2] [2 2] [5] 3.61/1.38 ? [1 2] X1 + [1 2] X2 + [6] 3.61/1.38 [0 0] [0 0] [4] 3.61/1.38 = [c_12(first^#(activate(X1), activate(X2)))] 3.61/1.38 3.61/1.38 [add^#(X1, X2)] = [1 0] X1 + [1 0] X2 + [2] 3.61/1.38 [0 0] [0 0] [2] 3.61/1.38 > [1] 3.61/1.38 [1] 3.61/1.38 = [c_15()] 3.61/1.38 3.61/1.38 [add^#(0(), X)] = [1 0] X + [2] 3.61/1.38 [0 0] [2] 3.61/1.38 > [1] 3.61/1.38 [1] 3.61/1.38 = [c_16()] 3.61/1.38 3.61/1.38 [dbl^#(X)] = [1 0] X + [1] 3.61/1.38 [0 0] [2] 3.61/1.38 >= [1] 3.61/1.38 [1] 3.61/1.38 = [c_13()] 3.61/1.38 3.61/1.38 [dbl^#(0())] = [1] 3.61/1.38 [2] 3.61/1.38 >= [1] 3.61/1.38 [1] 3.61/1.38 = [c_14()] 3.61/1.38 3.61/1.38 [first^#(X1, X2)] = [1 0] X1 + [1 0] X2 + [2] 3.61/1.38 [0 0] [0 0] [2] 3.61/1.38 > [1] 3.61/1.38 [1] 3.61/1.38 = [c_17()] 3.61/1.38 3.61/1.38 [first^#(0(), X)] = [1 0] X + [2] 3.61/1.38 [0 0] [2] 3.61/1.38 > [1] 3.61/1.38 [1] 3.61/1.38 = [c_18()] 3.61/1.38 3.61/1.38 3.61/1.38 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 3.61/1.38 3.61/1.38 We are left with following problem, upon which TcT provides the 3.61/1.38 certificate YES(O(1),O(1)). 3.61/1.38 3.61/1.38 Strict DPs: 3.61/1.38 { terms^#(N) -> c_1(sqr^#(N)) 3.61/1.38 , terms^#(X) -> c_2() 3.61/1.38 , sqr^#(X) -> c_3() 3.61/1.38 , sqr^#(0()) -> c_4() 3.61/1.38 , s^#(X) -> c_5() 3.61/1.38 , activate^#(X) -> c_6() 3.61/1.38 , activate^#(n__terms(X)) -> c_7(terms^#(activate(X))) 3.61/1.38 , activate^#(n__s(X)) -> c_8(s^#(X)) 3.61/1.38 , activate^#(n__add(X1, X2)) -> 3.61/1.38 c_9(add^#(activate(X1), activate(X2))) 3.61/1.38 , activate^#(n__sqr(X)) -> c_10(sqr^#(activate(X))) 3.61/1.38 , activate^#(n__dbl(X)) -> c_11(dbl^#(activate(X))) 3.61/1.38 , activate^#(n__first(X1, X2)) -> 3.61/1.38 c_12(first^#(activate(X1), activate(X2))) 3.61/1.38 , dbl^#(X) -> c_13() 3.61/1.38 , dbl^#(0()) -> c_14() } 3.61/1.38 Weak DPs: 3.61/1.38 { add^#(X1, X2) -> c_15() 3.61/1.38 , add^#(0(), X) -> c_16() 3.61/1.38 , first^#(X1, X2) -> c_17() 3.61/1.38 , first^#(0(), X) -> c_18() } 3.61/1.38 Weak Trs: 3.61/1.38 { terms(N) -> cons(recip(sqr(N)), n__terms(n__s(N))) 3.61/1.38 , terms(X) -> n__terms(X) 3.61/1.38 , sqr(X) -> n__sqr(X) 3.61/1.38 , sqr(0()) -> 0() 3.61/1.38 , s(X) -> n__s(X) 3.61/1.38 , activate(X) -> X 3.61/1.38 , activate(n__terms(X)) -> terms(activate(X)) 3.61/1.38 , activate(n__s(X)) -> s(X) 3.61/1.38 , activate(n__add(X1, X2)) -> add(activate(X1), activate(X2)) 3.61/1.38 , activate(n__sqr(X)) -> sqr(activate(X)) 3.61/1.38 , activate(n__dbl(X)) -> dbl(activate(X)) 3.61/1.38 , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) 3.61/1.38 , dbl(X) -> n__dbl(X) 3.61/1.38 , dbl(0()) -> 0() 3.61/1.38 , add(X1, X2) -> n__add(X1, X2) 3.61/1.38 , add(0(), X) -> X 3.61/1.38 , first(X1, X2) -> n__first(X1, X2) 3.61/1.38 , first(0(), X) -> nil() } 3.61/1.38 Obligation: 3.61/1.38 innermost runtime complexity 3.61/1.38 Answer: 3.61/1.38 YES(O(1),O(1)) 3.61/1.38 3.61/1.38 We estimate the number of application of {2,3,4,5,6,9,12,13,14} by 3.61/1.38 applications of Pre({2,3,4,5,6,9,12,13,14}) = {1,7,8,10,11}. Here 3.61/1.38 rules are labeled as follows: 3.61/1.38 3.61/1.38 DPs: 3.61/1.38 { 1: terms^#(N) -> c_1(sqr^#(N)) 3.61/1.38 , 2: terms^#(X) -> c_2() 3.61/1.38 , 3: sqr^#(X) -> c_3() 3.61/1.38 , 4: sqr^#(0()) -> c_4() 3.61/1.38 , 5: s^#(X) -> c_5() 3.61/1.38 , 6: activate^#(X) -> c_6() 3.61/1.38 , 7: activate^#(n__terms(X)) -> c_7(terms^#(activate(X))) 3.61/1.38 , 8: activate^#(n__s(X)) -> c_8(s^#(X)) 3.61/1.38 , 9: activate^#(n__add(X1, X2)) -> 3.61/1.38 c_9(add^#(activate(X1), activate(X2))) 3.61/1.38 , 10: activate^#(n__sqr(X)) -> c_10(sqr^#(activate(X))) 3.61/1.38 , 11: activate^#(n__dbl(X)) -> c_11(dbl^#(activate(X))) 3.61/1.38 , 12: activate^#(n__first(X1, X2)) -> 3.61/1.38 c_12(first^#(activate(X1), activate(X2))) 3.61/1.38 , 13: dbl^#(X) -> c_13() 3.61/1.38 , 14: dbl^#(0()) -> c_14() 3.61/1.38 , 15: add^#(X1, X2) -> c_15() 3.61/1.38 , 16: add^#(0(), X) -> c_16() 3.61/1.38 , 17: first^#(X1, X2) -> c_17() 3.61/1.38 , 18: first^#(0(), X) -> c_18() } 3.61/1.38 3.61/1.38 We are left with following problem, upon which TcT provides the 3.61/1.38 certificate YES(O(1),O(1)). 3.61/1.38 3.61/1.38 Strict DPs: 3.61/1.38 { terms^#(N) -> c_1(sqr^#(N)) 3.61/1.38 , activate^#(n__terms(X)) -> c_7(terms^#(activate(X))) 3.61/1.38 , activate^#(n__s(X)) -> c_8(s^#(X)) 3.61/1.38 , activate^#(n__sqr(X)) -> c_10(sqr^#(activate(X))) 3.61/1.38 , activate^#(n__dbl(X)) -> c_11(dbl^#(activate(X))) } 3.61/1.38 Weak DPs: 3.61/1.38 { terms^#(X) -> c_2() 3.61/1.38 , sqr^#(X) -> c_3() 3.61/1.38 , sqr^#(0()) -> c_4() 3.61/1.38 , s^#(X) -> c_5() 3.61/1.38 , activate^#(X) -> c_6() 3.61/1.38 , activate^#(n__add(X1, X2)) -> 3.61/1.38 c_9(add^#(activate(X1), activate(X2))) 3.61/1.38 , activate^#(n__first(X1, X2)) -> 3.61/1.38 c_12(first^#(activate(X1), activate(X2))) 3.61/1.38 , add^#(X1, X2) -> c_15() 3.61/1.38 , add^#(0(), X) -> c_16() 3.61/1.38 , dbl^#(X) -> c_13() 3.61/1.38 , dbl^#(0()) -> c_14() 3.61/1.38 , first^#(X1, X2) -> c_17() 3.61/1.38 , first^#(0(), X) -> c_18() } 3.61/1.38 Weak Trs: 3.61/1.38 { terms(N) -> cons(recip(sqr(N)), n__terms(n__s(N))) 3.61/1.38 , terms(X) -> n__terms(X) 3.61/1.38 , sqr(X) -> n__sqr(X) 3.61/1.38 , sqr(0()) -> 0() 3.61/1.38 , s(X) -> n__s(X) 3.61/1.38 , activate(X) -> X 3.61/1.38 , activate(n__terms(X)) -> terms(activate(X)) 3.61/1.38 , activate(n__s(X)) -> s(X) 3.61/1.38 , activate(n__add(X1, X2)) -> add(activate(X1), activate(X2)) 3.61/1.38 , activate(n__sqr(X)) -> sqr(activate(X)) 3.61/1.38 , activate(n__dbl(X)) -> dbl(activate(X)) 3.61/1.38 , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) 3.61/1.38 , dbl(X) -> n__dbl(X) 3.61/1.38 , dbl(0()) -> 0() 3.61/1.38 , add(X1, X2) -> n__add(X1, X2) 3.61/1.38 , add(0(), X) -> X 3.61/1.38 , first(X1, X2) -> n__first(X1, X2) 3.61/1.38 , first(0(), X) -> nil() } 3.61/1.38 Obligation: 3.61/1.38 innermost runtime complexity 3.61/1.38 Answer: 3.61/1.38 YES(O(1),O(1)) 3.61/1.38 3.61/1.38 We estimate the number of application of {1,3,4,5} by applications 3.61/1.38 of Pre({1,3,4,5}) = {2}. Here rules are labeled as follows: 3.61/1.38 3.61/1.38 DPs: 3.61/1.38 { 1: terms^#(N) -> c_1(sqr^#(N)) 3.61/1.38 , 2: activate^#(n__terms(X)) -> c_7(terms^#(activate(X))) 3.61/1.38 , 3: activate^#(n__s(X)) -> c_8(s^#(X)) 3.61/1.38 , 4: activate^#(n__sqr(X)) -> c_10(sqr^#(activate(X))) 3.61/1.38 , 5: activate^#(n__dbl(X)) -> c_11(dbl^#(activate(X))) 3.61/1.38 , 6: terms^#(X) -> c_2() 3.61/1.38 , 7: sqr^#(X) -> c_3() 3.61/1.38 , 8: sqr^#(0()) -> c_4() 3.61/1.38 , 9: s^#(X) -> c_5() 3.61/1.38 , 10: activate^#(X) -> c_6() 3.61/1.38 , 11: activate^#(n__add(X1, X2)) -> 3.61/1.38 c_9(add^#(activate(X1), activate(X2))) 3.61/1.38 , 12: activate^#(n__first(X1, X2)) -> 3.61/1.38 c_12(first^#(activate(X1), activate(X2))) 3.61/1.38 , 13: add^#(X1, X2) -> c_15() 3.61/1.38 , 14: add^#(0(), X) -> c_16() 3.61/1.38 , 15: dbl^#(X) -> c_13() 3.61/1.38 , 16: dbl^#(0()) -> c_14() 3.61/1.38 , 17: first^#(X1, X2) -> c_17() 3.61/1.38 , 18: first^#(0(), X) -> c_18() } 3.61/1.38 3.61/1.38 We are left with following problem, upon which TcT provides the 3.61/1.38 certificate YES(O(1),O(1)). 3.61/1.38 3.61/1.38 Strict DPs: 3.61/1.38 { activate^#(n__terms(X)) -> c_7(terms^#(activate(X))) } 3.61/1.38 Weak DPs: 3.61/1.38 { terms^#(N) -> c_1(sqr^#(N)) 3.61/1.38 , terms^#(X) -> c_2() 3.61/1.38 , sqr^#(X) -> c_3() 3.61/1.38 , sqr^#(0()) -> c_4() 3.61/1.38 , s^#(X) -> c_5() 3.61/1.38 , activate^#(X) -> c_6() 3.61/1.38 , activate^#(n__s(X)) -> c_8(s^#(X)) 3.61/1.38 , activate^#(n__add(X1, X2)) -> 3.61/1.38 c_9(add^#(activate(X1), activate(X2))) 3.61/1.38 , activate^#(n__sqr(X)) -> c_10(sqr^#(activate(X))) 3.61/1.38 , activate^#(n__dbl(X)) -> c_11(dbl^#(activate(X))) 3.61/1.38 , activate^#(n__first(X1, X2)) -> 3.61/1.38 c_12(first^#(activate(X1), activate(X2))) 3.61/1.38 , add^#(X1, X2) -> c_15() 3.61/1.38 , add^#(0(), X) -> c_16() 3.61/1.38 , dbl^#(X) -> c_13() 3.61/1.38 , dbl^#(0()) -> c_14() 3.61/1.38 , first^#(X1, X2) -> c_17() 3.61/1.38 , first^#(0(), X) -> c_18() } 3.61/1.38 Weak Trs: 3.61/1.38 { terms(N) -> cons(recip(sqr(N)), n__terms(n__s(N))) 3.61/1.38 , terms(X) -> n__terms(X) 3.61/1.38 , sqr(X) -> n__sqr(X) 3.61/1.38 , sqr(0()) -> 0() 3.61/1.38 , s(X) -> n__s(X) 3.61/1.38 , activate(X) -> X 3.61/1.38 , activate(n__terms(X)) -> terms(activate(X)) 3.61/1.38 , activate(n__s(X)) -> s(X) 3.61/1.38 , activate(n__add(X1, X2)) -> add(activate(X1), activate(X2)) 3.61/1.38 , activate(n__sqr(X)) -> sqr(activate(X)) 3.61/1.38 , activate(n__dbl(X)) -> dbl(activate(X)) 3.61/1.38 , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) 3.61/1.38 , dbl(X) -> n__dbl(X) 3.61/1.38 , dbl(0()) -> 0() 3.61/1.38 , add(X1, X2) -> n__add(X1, X2) 3.61/1.38 , add(0(), X) -> X 3.61/1.38 , first(X1, X2) -> n__first(X1, X2) 3.61/1.38 , first(0(), X) -> nil() } 3.61/1.38 Obligation: 3.61/1.38 innermost runtime complexity 3.61/1.38 Answer: 3.61/1.38 YES(O(1),O(1)) 3.61/1.38 3.61/1.38 We estimate the number of application of {1} by applications of 3.61/1.38 Pre({1}) = {}. Here rules are labeled as follows: 3.61/1.38 3.61/1.38 DPs: 3.61/1.38 { 1: activate^#(n__terms(X)) -> c_7(terms^#(activate(X))) 3.61/1.38 , 2: terms^#(N) -> c_1(sqr^#(N)) 3.61/1.38 , 3: terms^#(X) -> c_2() 3.61/1.38 , 4: sqr^#(X) -> c_3() 3.61/1.38 , 5: sqr^#(0()) -> c_4() 3.61/1.38 , 6: s^#(X) -> c_5() 3.61/1.38 , 7: activate^#(X) -> c_6() 3.61/1.38 , 8: activate^#(n__s(X)) -> c_8(s^#(X)) 3.61/1.38 , 9: activate^#(n__add(X1, X2)) -> 3.61/1.38 c_9(add^#(activate(X1), activate(X2))) 3.61/1.38 , 10: activate^#(n__sqr(X)) -> c_10(sqr^#(activate(X))) 3.61/1.38 , 11: activate^#(n__dbl(X)) -> c_11(dbl^#(activate(X))) 3.61/1.38 , 12: activate^#(n__first(X1, X2)) -> 3.61/1.38 c_12(first^#(activate(X1), activate(X2))) 3.61/1.38 , 13: add^#(X1, X2) -> c_15() 3.61/1.38 , 14: add^#(0(), X) -> c_16() 3.61/1.38 , 15: dbl^#(X) -> c_13() 3.61/1.38 , 16: dbl^#(0()) -> c_14() 3.61/1.38 , 17: first^#(X1, X2) -> c_17() 3.61/1.38 , 18: first^#(0(), X) -> c_18() } 3.61/1.38 3.61/1.38 We are left with following problem, upon which TcT provides the 3.61/1.38 certificate YES(O(1),O(1)). 3.61/1.38 3.61/1.38 Weak DPs: 3.61/1.38 { terms^#(N) -> c_1(sqr^#(N)) 3.61/1.38 , terms^#(X) -> c_2() 3.61/1.38 , sqr^#(X) -> c_3() 3.61/1.38 , sqr^#(0()) -> c_4() 3.61/1.38 , s^#(X) -> c_5() 3.61/1.38 , activate^#(X) -> c_6() 3.61/1.38 , activate^#(n__terms(X)) -> c_7(terms^#(activate(X))) 3.61/1.38 , activate^#(n__s(X)) -> c_8(s^#(X)) 3.61/1.38 , activate^#(n__add(X1, X2)) -> 3.61/1.38 c_9(add^#(activate(X1), activate(X2))) 3.61/1.38 , activate^#(n__sqr(X)) -> c_10(sqr^#(activate(X))) 3.61/1.38 , activate^#(n__dbl(X)) -> c_11(dbl^#(activate(X))) 3.61/1.38 , activate^#(n__first(X1, X2)) -> 3.61/1.38 c_12(first^#(activate(X1), activate(X2))) 3.61/1.38 , add^#(X1, X2) -> c_15() 3.61/1.38 , add^#(0(), X) -> c_16() 3.61/1.38 , dbl^#(X) -> c_13() 3.61/1.38 , dbl^#(0()) -> c_14() 3.61/1.38 , first^#(X1, X2) -> c_17() 3.61/1.38 , first^#(0(), X) -> c_18() } 3.61/1.38 Weak Trs: 3.61/1.38 { terms(N) -> cons(recip(sqr(N)), n__terms(n__s(N))) 3.61/1.38 , terms(X) -> n__terms(X) 3.61/1.38 , sqr(X) -> n__sqr(X) 3.61/1.38 , sqr(0()) -> 0() 3.61/1.38 , s(X) -> n__s(X) 3.61/1.38 , activate(X) -> X 3.61/1.38 , activate(n__terms(X)) -> terms(activate(X)) 3.61/1.38 , activate(n__s(X)) -> s(X) 3.61/1.38 , activate(n__add(X1, X2)) -> add(activate(X1), activate(X2)) 3.61/1.38 , activate(n__sqr(X)) -> sqr(activate(X)) 3.61/1.38 , activate(n__dbl(X)) -> dbl(activate(X)) 3.61/1.38 , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) 3.61/1.38 , dbl(X) -> n__dbl(X) 3.61/1.38 , dbl(0()) -> 0() 3.61/1.38 , add(X1, X2) -> n__add(X1, X2) 3.61/1.38 , add(0(), X) -> X 3.61/1.38 , first(X1, X2) -> n__first(X1, X2) 3.61/1.38 , first(0(), X) -> nil() } 3.61/1.38 Obligation: 3.61/1.38 innermost runtime complexity 3.61/1.38 Answer: 3.61/1.38 YES(O(1),O(1)) 3.61/1.38 3.61/1.38 The following weak DPs constitute a sub-graph of the DG that is 3.61/1.38 closed under successors. The DPs are removed. 3.61/1.38 3.61/1.38 { terms^#(N) -> c_1(sqr^#(N)) 3.61/1.38 , terms^#(X) -> c_2() 3.61/1.38 , sqr^#(X) -> c_3() 3.61/1.38 , sqr^#(0()) -> c_4() 3.61/1.38 , s^#(X) -> c_5() 3.61/1.38 , activate^#(X) -> c_6() 3.61/1.38 , activate^#(n__terms(X)) -> c_7(terms^#(activate(X))) 3.61/1.38 , activate^#(n__s(X)) -> c_8(s^#(X)) 3.61/1.38 , activate^#(n__add(X1, X2)) -> 3.61/1.38 c_9(add^#(activate(X1), activate(X2))) 3.61/1.38 , activate^#(n__sqr(X)) -> c_10(sqr^#(activate(X))) 3.61/1.38 , activate^#(n__dbl(X)) -> c_11(dbl^#(activate(X))) 3.61/1.38 , activate^#(n__first(X1, X2)) -> 3.61/1.38 c_12(first^#(activate(X1), activate(X2))) 3.61/1.38 , add^#(X1, X2) -> c_15() 3.61/1.38 , add^#(0(), X) -> c_16() 3.61/1.38 , dbl^#(X) -> c_13() 3.61/1.38 , dbl^#(0()) -> c_14() 3.61/1.38 , first^#(X1, X2) -> c_17() 3.61/1.38 , first^#(0(), X) -> c_18() } 3.61/1.38 3.61/1.38 We are left with following problem, upon which TcT provides the 3.61/1.38 certificate YES(O(1),O(1)). 3.61/1.38 3.61/1.38 Weak Trs: 3.61/1.38 { terms(N) -> cons(recip(sqr(N)), n__terms(n__s(N))) 3.61/1.38 , terms(X) -> n__terms(X) 3.61/1.38 , sqr(X) -> n__sqr(X) 3.61/1.38 , sqr(0()) -> 0() 3.61/1.38 , s(X) -> n__s(X) 3.61/1.38 , activate(X) -> X 3.61/1.38 , activate(n__terms(X)) -> terms(activate(X)) 3.61/1.38 , activate(n__s(X)) -> s(X) 3.61/1.38 , activate(n__add(X1, X2)) -> add(activate(X1), activate(X2)) 3.61/1.38 , activate(n__sqr(X)) -> sqr(activate(X)) 3.61/1.38 , activate(n__dbl(X)) -> dbl(activate(X)) 3.61/1.38 , activate(n__first(X1, X2)) -> first(activate(X1), activate(X2)) 3.61/1.38 , dbl(X) -> n__dbl(X) 3.61/1.38 , dbl(0()) -> 0() 3.61/1.38 , add(X1, X2) -> n__add(X1, X2) 3.61/1.38 , add(0(), X) -> X 3.61/1.38 , first(X1, X2) -> n__first(X1, X2) 3.61/1.38 , first(0(), X) -> nil() } 3.61/1.38 Obligation: 3.61/1.38 innermost runtime complexity 3.61/1.38 Answer: 3.61/1.38 YES(O(1),O(1)) 3.61/1.38 3.61/1.38 No rule is usable, rules are removed from the input problem. 3.61/1.38 3.61/1.38 We are left with following problem, upon which TcT provides the 3.61/1.38 certificate YES(O(1),O(1)). 3.61/1.38 3.61/1.38 Rules: Empty 3.61/1.38 Obligation: 3.61/1.38 innermost runtime complexity 3.61/1.38 Answer: 3.61/1.38 YES(O(1),O(1)) 3.61/1.38 3.61/1.38 Empty rules are trivially bounded 3.61/1.38 3.61/1.38 Hurray, we answered YES(O(1),O(n^1)) 3.61/1.39 EOF