YES(O(1),O(n^2)) 849.64/297.08 YES(O(1),O(n^2)) 849.64/297.08 849.64/297.08 We are left with following problem, upon which TcT provides the 849.64/297.08 certificate YES(O(1),O(n^2)). 849.64/297.08 849.64/297.08 Strict Trs: 849.64/297.08 { a__terms(N) -> cons(recip(a__sqr(mark(N))), terms(s(N))) 849.64/297.08 , a__terms(X) -> terms(X) 849.64/297.08 , a__sqr(X) -> sqr(X) 849.64/297.08 , a__sqr(s(X)) -> s(add(sqr(X), dbl(X))) 849.64/297.08 , a__sqr(0()) -> 0() 849.64/297.08 , mark(cons(X1, X2)) -> cons(mark(X1), X2) 849.64/297.08 , mark(recip(X)) -> recip(mark(X)) 849.64/297.08 , mark(terms(X)) -> a__terms(mark(X)) 849.64/297.08 , mark(s(X)) -> s(X) 849.64/297.08 , mark(0()) -> 0() 849.64/297.08 , mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) 849.64/297.08 , mark(sqr(X)) -> a__sqr(mark(X)) 849.64/297.08 , mark(dbl(X)) -> a__dbl(mark(X)) 849.64/297.08 , mark(nil()) -> nil() 849.64/297.08 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 849.64/297.08 , a__dbl(X) -> dbl(X) 849.64/297.08 , a__dbl(s(X)) -> s(s(dbl(X))) 849.64/297.08 , a__dbl(0()) -> 0() 849.64/297.08 , a__add(X1, X2) -> add(X1, X2) 849.64/297.08 , a__add(s(X), Y) -> s(add(X, Y)) 849.64/297.08 , a__add(0(), X) -> mark(X) 849.64/297.08 , a__first(X1, X2) -> first(X1, X2) 849.64/297.08 , a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z)) 849.64/297.08 , a__first(0(), X) -> nil() } 849.64/297.08 Obligation: 849.64/297.08 innermost runtime complexity 849.64/297.08 Answer: 849.64/297.08 YES(O(1),O(n^2)) 849.64/297.08 849.64/297.08 The weightgap principle applies (using the following nonconstant 849.64/297.08 growth matrix-interpretation) 849.64/297.08 849.64/297.08 The following argument positions are usable: 849.64/297.08 Uargs(a__terms) = {1}, Uargs(cons) = {1}, Uargs(recip) = {1}, 849.64/297.08 Uargs(a__sqr) = {1}, Uargs(a__dbl) = {1}, Uargs(a__add) = {1, 2}, 849.64/297.08 Uargs(a__first) = {1, 2} 849.64/297.08 849.64/297.08 TcT has computed the following matrix interpretation satisfying 849.64/297.08 not(EDA) and not(IDA(1)). 849.64/297.08 849.64/297.08 [a__terms](x1) = [1] x1 + [1] 849.64/297.08 849.64/297.08 [cons](x1, x2) = [1] x1 + [0] 849.64/297.08 849.64/297.08 [recip](x1) = [1] x1 + [0] 849.64/297.08 849.64/297.08 [a__sqr](x1) = [1] x1 + [0] 849.64/297.08 849.64/297.08 [mark](x1) = [0] 849.64/297.08 849.64/297.08 [terms](x1) = [1] x1 + [7] 849.64/297.08 849.64/297.08 [s](x1) = [0] 849.64/297.08 849.64/297.08 [0] = [0] 849.64/297.08 849.64/297.08 [add](x1, x2) = [5] 849.64/297.08 849.64/297.08 [sqr](x1) = [1] x1 + [5] 849.64/297.08 849.64/297.08 [dbl](x1) = [1] x1 + [0] 849.64/297.08 849.64/297.08 [a__dbl](x1) = [1] x1 + [0] 849.64/297.08 849.64/297.08 [a__add](x1, x2) = [1] x1 + [1] x2 + [0] 849.64/297.08 849.64/297.08 [a__first](x1, x2) = [1] x1 + [1] x2 + [0] 849.64/297.08 849.64/297.08 [nil] = [7] 849.64/297.08 849.64/297.08 [first](x1, x2) = [1] x1 + [1] x2 + [7] 849.64/297.08 849.64/297.08 The order satisfies the following ordering constraints: 849.64/297.08 849.64/297.08 [a__terms(N)] = [1] N + [1] 849.64/297.08 > [0] 849.64/297.08 = [cons(recip(a__sqr(mark(N))), terms(s(N)))] 849.64/297.08 849.64/297.08 [a__terms(X)] = [1] X + [1] 849.64/297.08 ? [1] X + [7] 849.64/297.08 = [terms(X)] 849.64/297.08 849.64/297.08 [a__sqr(X)] = [1] X + [0] 849.64/297.08 ? [1] X + [5] 849.64/297.08 = [sqr(X)] 849.64/297.08 849.64/297.08 [a__sqr(s(X))] = [0] 849.64/297.08 >= [0] 849.64/297.08 = [s(add(sqr(X), dbl(X)))] 849.64/297.08 849.64/297.08 [a__sqr(0())] = [0] 849.64/297.08 >= [0] 849.64/297.08 = [0()] 849.64/297.08 849.64/297.08 [mark(cons(X1, X2))] = [0] 849.64/297.08 >= [0] 849.64/297.08 = [cons(mark(X1), X2)] 849.64/297.08 849.64/297.08 [mark(recip(X))] = [0] 849.64/297.08 >= [0] 849.64/297.08 = [recip(mark(X))] 849.64/297.08 849.64/297.08 [mark(terms(X))] = [0] 849.64/297.08 ? [1] 849.64/297.08 = [a__terms(mark(X))] 849.64/297.08 849.64/297.08 [mark(s(X))] = [0] 849.64/297.08 >= [0] 849.64/297.08 = [s(X)] 849.64/297.08 849.64/297.08 [mark(0())] = [0] 849.64/297.08 >= [0] 849.64/297.08 = [0()] 849.64/297.08 849.64/297.08 [mark(add(X1, X2))] = [0] 849.64/297.08 >= [0] 849.64/297.08 = [a__add(mark(X1), mark(X2))] 849.64/297.08 849.64/297.08 [mark(sqr(X))] = [0] 849.64/297.08 >= [0] 849.64/297.08 = [a__sqr(mark(X))] 849.64/297.08 849.64/297.08 [mark(dbl(X))] = [0] 849.64/297.08 >= [0] 849.64/297.08 = [a__dbl(mark(X))] 849.64/297.08 849.64/297.08 [mark(nil())] = [0] 849.64/297.08 ? [7] 849.64/297.08 = [nil()] 849.64/297.08 849.64/297.08 [mark(first(X1, X2))] = [0] 849.64/297.08 >= [0] 849.64/297.08 = [a__first(mark(X1), mark(X2))] 849.64/297.08 849.64/297.08 [a__dbl(X)] = [1] X + [0] 849.64/297.08 >= [1] X + [0] 849.64/297.08 = [dbl(X)] 849.64/297.08 849.64/297.08 [a__dbl(s(X))] = [0] 849.64/297.08 >= [0] 849.64/297.08 = [s(s(dbl(X)))] 849.64/297.08 849.64/297.08 [a__dbl(0())] = [0] 849.64/297.08 >= [0] 849.64/297.08 = [0()] 849.64/297.08 849.64/297.08 [a__add(X1, X2)] = [1] X1 + [1] X2 + [0] 849.64/297.08 ? [5] 849.64/297.08 = [add(X1, X2)] 849.64/297.08 849.64/297.08 [a__add(s(X), Y)] = [1] Y + [0] 849.64/297.08 >= [0] 849.64/297.08 = [s(add(X, Y))] 849.64/297.08 849.64/297.08 [a__add(0(), X)] = [1] X + [0] 849.64/297.08 >= [0] 849.64/297.08 = [mark(X)] 849.64/297.08 849.64/297.08 [a__first(X1, X2)] = [1] X1 + [1] X2 + [0] 849.64/297.08 ? [1] X1 + [1] X2 + [7] 849.64/297.08 = [first(X1, X2)] 849.64/297.08 849.64/297.08 [a__first(s(X), cons(Y, Z))] = [1] Y + [0] 849.64/297.08 >= [0] 849.64/297.08 = [cons(mark(Y), first(X, Z))] 849.64/297.08 849.64/297.08 [a__first(0(), X)] = [1] X + [0] 849.64/297.08 ? [7] 849.64/297.08 = [nil()] 849.64/297.08 849.64/297.08 849.64/297.08 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 849.64/297.08 849.64/297.08 We are left with following problem, upon which TcT provides the 849.64/297.08 certificate YES(O(1),O(n^2)). 849.64/297.08 849.64/297.08 Strict Trs: 849.64/297.08 { a__terms(X) -> terms(X) 849.64/297.08 , a__sqr(X) -> sqr(X) 849.64/297.08 , a__sqr(s(X)) -> s(add(sqr(X), dbl(X))) 849.64/297.08 , a__sqr(0()) -> 0() 849.64/297.08 , mark(cons(X1, X2)) -> cons(mark(X1), X2) 849.64/297.08 , mark(recip(X)) -> recip(mark(X)) 849.64/297.08 , mark(terms(X)) -> a__terms(mark(X)) 849.64/297.08 , mark(s(X)) -> s(X) 849.64/297.08 , mark(0()) -> 0() 849.64/297.08 , mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) 849.64/297.08 , mark(sqr(X)) -> a__sqr(mark(X)) 849.64/297.08 , mark(dbl(X)) -> a__dbl(mark(X)) 849.64/297.08 , mark(nil()) -> nil() 849.64/297.08 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 849.64/297.08 , a__dbl(X) -> dbl(X) 849.64/297.08 , a__dbl(s(X)) -> s(s(dbl(X))) 849.64/297.08 , a__dbl(0()) -> 0() 849.64/297.08 , a__add(X1, X2) -> add(X1, X2) 849.64/297.08 , a__add(s(X), Y) -> s(add(X, Y)) 849.64/297.08 , a__add(0(), X) -> mark(X) 849.64/297.08 , a__first(X1, X2) -> first(X1, X2) 849.64/297.08 , a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z)) 849.64/297.08 , a__first(0(), X) -> nil() } 849.64/297.08 Weak Trs: 849.64/297.08 { a__terms(N) -> cons(recip(a__sqr(mark(N))), terms(s(N))) } 849.64/297.08 Obligation: 849.64/297.08 innermost runtime complexity 849.64/297.08 Answer: 849.64/297.08 YES(O(1),O(n^2)) 849.64/297.08 849.64/297.08 The weightgap principle applies (using the following nonconstant 849.64/297.08 growth matrix-interpretation) 849.64/297.08 849.64/297.08 The following argument positions are usable: 849.64/297.08 Uargs(a__terms) = {1}, Uargs(cons) = {1}, Uargs(recip) = {1}, 849.64/297.08 Uargs(a__sqr) = {1}, Uargs(a__dbl) = {1}, Uargs(a__add) = {1, 2}, 849.64/297.08 Uargs(a__first) = {1, 2} 849.64/297.08 849.64/297.08 TcT has computed the following matrix interpretation satisfying 849.64/297.08 not(EDA) and not(IDA(1)). 849.64/297.08 849.64/297.08 [a__terms](x1) = [1] x1 + [0] 849.64/297.08 849.64/297.08 [cons](x1, x2) = [1] x1 + [0] 849.64/297.08 849.64/297.08 [recip](x1) = [1] x1 + [0] 849.64/297.08 849.64/297.08 [a__sqr](x1) = [1] x1 + [0] 849.64/297.08 849.64/297.08 [mark](x1) = [0] 849.64/297.08 849.64/297.08 [terms](x1) = [1] x1 + [7] 849.64/297.08 849.64/297.08 [s](x1) = [0] 849.64/297.08 849.64/297.08 [0] = [0] 849.64/297.08 849.64/297.08 [add](x1, x2) = [5] 849.64/297.08 849.64/297.08 [sqr](x1) = [1] x1 + [5] 849.64/297.08 849.64/297.08 [dbl](x1) = [1] x1 + [0] 849.64/297.08 849.64/297.08 [a__dbl](x1) = [1] x1 + [0] 849.64/297.08 849.64/297.08 [a__add](x1, x2) = [1] x1 + [1] x2 + [0] 849.64/297.08 849.64/297.08 [a__first](x1, x2) = [1] x1 + [1] x2 + [1] 849.64/297.08 849.64/297.08 [nil] = [6] 849.64/297.08 849.64/297.08 [first](x1, x2) = [1] x1 + [1] x2 + [0] 849.64/297.08 849.64/297.08 The order satisfies the following ordering constraints: 849.64/297.08 849.64/297.08 [a__terms(N)] = [1] N + [0] 849.64/297.08 >= [0] 849.64/297.08 = [cons(recip(a__sqr(mark(N))), terms(s(N)))] 849.64/297.08 849.64/297.08 [a__terms(X)] = [1] X + [0] 849.64/297.08 ? [1] X + [7] 849.64/297.08 = [terms(X)] 849.64/297.08 849.64/297.08 [a__sqr(X)] = [1] X + [0] 849.64/297.08 ? [1] X + [5] 849.64/297.08 = [sqr(X)] 849.64/297.08 849.64/297.08 [a__sqr(s(X))] = [0] 849.64/297.08 >= [0] 849.64/297.08 = [s(add(sqr(X), dbl(X)))] 849.64/297.08 849.64/297.08 [a__sqr(0())] = [0] 849.64/297.08 >= [0] 849.64/297.08 = [0()] 849.64/297.08 849.64/297.08 [mark(cons(X1, X2))] = [0] 849.64/297.08 >= [0] 849.64/297.08 = [cons(mark(X1), X2)] 849.64/297.08 849.64/297.08 [mark(recip(X))] = [0] 849.64/297.08 >= [0] 849.64/297.08 = [recip(mark(X))] 849.64/297.08 849.64/297.08 [mark(terms(X))] = [0] 849.64/297.08 >= [0] 849.64/297.08 = [a__terms(mark(X))] 849.64/297.08 849.64/297.08 [mark(s(X))] = [0] 849.64/297.08 >= [0] 849.64/297.08 = [s(X)] 849.64/297.08 849.64/297.09 [mark(0())] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [0()] 849.64/297.09 849.64/297.09 [mark(add(X1, X2))] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [a__add(mark(X1), mark(X2))] 849.64/297.09 849.64/297.09 [mark(sqr(X))] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [a__sqr(mark(X))] 849.64/297.09 849.64/297.09 [mark(dbl(X))] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [a__dbl(mark(X))] 849.64/297.09 849.64/297.09 [mark(nil())] = [0] 849.64/297.09 ? [6] 849.64/297.09 = [nil()] 849.64/297.09 849.64/297.09 [mark(first(X1, X2))] = [0] 849.64/297.09 ? [1] 849.64/297.09 = [a__first(mark(X1), mark(X2))] 849.64/297.09 849.64/297.09 [a__dbl(X)] = [1] X + [0] 849.64/297.09 >= [1] X + [0] 849.64/297.09 = [dbl(X)] 849.64/297.09 849.64/297.09 [a__dbl(s(X))] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [s(s(dbl(X)))] 849.64/297.09 849.64/297.09 [a__dbl(0())] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [0()] 849.64/297.09 849.64/297.09 [a__add(X1, X2)] = [1] X1 + [1] X2 + [0] 849.64/297.09 ? [5] 849.64/297.09 = [add(X1, X2)] 849.64/297.09 849.64/297.09 [a__add(s(X), Y)] = [1] Y + [0] 849.64/297.09 >= [0] 849.64/297.09 = [s(add(X, Y))] 849.64/297.09 849.64/297.09 [a__add(0(), X)] = [1] X + [0] 849.64/297.09 >= [0] 849.64/297.09 = [mark(X)] 849.64/297.09 849.64/297.09 [a__first(X1, X2)] = [1] X1 + [1] X2 + [1] 849.64/297.09 > [1] X1 + [1] X2 + [0] 849.64/297.09 = [first(X1, X2)] 849.64/297.09 849.64/297.09 [a__first(s(X), cons(Y, Z))] = [1] Y + [1] 849.64/297.09 > [0] 849.64/297.09 = [cons(mark(Y), first(X, Z))] 849.64/297.09 849.64/297.09 [a__first(0(), X)] = [1] X + [1] 849.64/297.09 ? [6] 849.64/297.09 = [nil()] 849.64/297.09 849.64/297.09 849.64/297.09 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 849.64/297.09 849.64/297.09 We are left with following problem, upon which TcT provides the 849.64/297.09 certificate YES(O(1),O(n^2)). 849.64/297.09 849.64/297.09 Strict Trs: 849.64/297.09 { a__terms(X) -> terms(X) 849.64/297.09 , a__sqr(X) -> sqr(X) 849.64/297.09 , a__sqr(s(X)) -> s(add(sqr(X), dbl(X))) 849.64/297.09 , a__sqr(0()) -> 0() 849.64/297.09 , mark(cons(X1, X2)) -> cons(mark(X1), X2) 849.64/297.09 , mark(recip(X)) -> recip(mark(X)) 849.64/297.09 , mark(terms(X)) -> a__terms(mark(X)) 849.64/297.09 , mark(s(X)) -> s(X) 849.64/297.09 , mark(0()) -> 0() 849.64/297.09 , mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) 849.64/297.09 , mark(sqr(X)) -> a__sqr(mark(X)) 849.64/297.09 , mark(dbl(X)) -> a__dbl(mark(X)) 849.64/297.09 , mark(nil()) -> nil() 849.64/297.09 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 849.64/297.09 , a__dbl(X) -> dbl(X) 849.64/297.09 , a__dbl(s(X)) -> s(s(dbl(X))) 849.64/297.09 , a__dbl(0()) -> 0() 849.64/297.09 , a__add(X1, X2) -> add(X1, X2) 849.64/297.09 , a__add(s(X), Y) -> s(add(X, Y)) 849.64/297.09 , a__add(0(), X) -> mark(X) 849.64/297.09 , a__first(0(), X) -> nil() } 849.64/297.09 Weak Trs: 849.64/297.09 { a__terms(N) -> cons(recip(a__sqr(mark(N))), terms(s(N))) 849.64/297.09 , a__first(X1, X2) -> first(X1, X2) 849.64/297.09 , a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z)) } 849.64/297.09 Obligation: 849.64/297.09 innermost runtime complexity 849.64/297.09 Answer: 849.64/297.09 YES(O(1),O(n^2)) 849.64/297.09 849.64/297.09 The weightgap principle applies (using the following nonconstant 849.64/297.09 growth matrix-interpretation) 849.64/297.09 849.64/297.09 The following argument positions are usable: 849.64/297.09 Uargs(a__terms) = {1}, Uargs(cons) = {1}, Uargs(recip) = {1}, 849.64/297.09 Uargs(a__sqr) = {1}, Uargs(a__dbl) = {1}, Uargs(a__add) = {1, 2}, 849.64/297.09 Uargs(a__first) = {1, 2} 849.64/297.09 849.64/297.09 TcT has computed the following matrix interpretation satisfying 849.64/297.09 not(EDA) and not(IDA(1)). 849.64/297.09 849.64/297.09 [a__terms](x1) = [1] x1 + [4] 849.64/297.09 849.64/297.09 [cons](x1, x2) = [1] x1 + [0] 849.64/297.09 849.64/297.09 [recip](x1) = [1] x1 + [0] 849.64/297.09 849.64/297.09 [a__sqr](x1) = [1] x1 + [0] 849.64/297.09 849.64/297.09 [mark](x1) = [0] 849.64/297.09 849.64/297.09 [terms](x1) = [1] x1 + [3] 849.64/297.09 849.64/297.09 [s](x1) = [0] 849.64/297.09 849.64/297.09 [0] = [0] 849.64/297.09 849.64/297.09 [add](x1, x2) = [5] 849.64/297.09 849.64/297.09 [sqr](x1) = [1] x1 + [5] 849.64/297.09 849.64/297.09 [dbl](x1) = [1] x1 + [0] 849.64/297.09 849.64/297.09 [a__dbl](x1) = [1] x1 + [0] 849.64/297.09 849.64/297.09 [a__add](x1, x2) = [1] x1 + [1] x2 + [0] 849.64/297.09 849.64/297.09 [a__first](x1, x2) = [1] x1 + [1] x2 + [0] 849.64/297.09 849.64/297.09 [nil] = [7] 849.64/297.09 849.64/297.09 [first](x1, x2) = [1] x1 + [1] x2 + [0] 849.64/297.09 849.64/297.09 The order satisfies the following ordering constraints: 849.64/297.09 849.64/297.09 [a__terms(N)] = [1] N + [4] 849.64/297.09 > [0] 849.64/297.09 = [cons(recip(a__sqr(mark(N))), terms(s(N)))] 849.64/297.09 849.64/297.09 [a__terms(X)] = [1] X + [4] 849.64/297.09 > [1] X + [3] 849.64/297.09 = [terms(X)] 849.64/297.09 849.64/297.09 [a__sqr(X)] = [1] X + [0] 849.64/297.09 ? [1] X + [5] 849.64/297.09 = [sqr(X)] 849.64/297.09 849.64/297.09 [a__sqr(s(X))] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [s(add(sqr(X), dbl(X)))] 849.64/297.09 849.64/297.09 [a__sqr(0())] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [0()] 849.64/297.09 849.64/297.09 [mark(cons(X1, X2))] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [cons(mark(X1), X2)] 849.64/297.09 849.64/297.09 [mark(recip(X))] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [recip(mark(X))] 849.64/297.09 849.64/297.09 [mark(terms(X))] = [0] 849.64/297.09 ? [4] 849.64/297.09 = [a__terms(mark(X))] 849.64/297.09 849.64/297.09 [mark(s(X))] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [s(X)] 849.64/297.09 849.64/297.09 [mark(0())] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [0()] 849.64/297.09 849.64/297.09 [mark(add(X1, X2))] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [a__add(mark(X1), mark(X2))] 849.64/297.09 849.64/297.09 [mark(sqr(X))] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [a__sqr(mark(X))] 849.64/297.09 849.64/297.09 [mark(dbl(X))] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [a__dbl(mark(X))] 849.64/297.09 849.64/297.09 [mark(nil())] = [0] 849.64/297.09 ? [7] 849.64/297.09 = [nil()] 849.64/297.09 849.64/297.09 [mark(first(X1, X2))] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [a__first(mark(X1), mark(X2))] 849.64/297.09 849.64/297.09 [a__dbl(X)] = [1] X + [0] 849.64/297.09 >= [1] X + [0] 849.64/297.09 = [dbl(X)] 849.64/297.09 849.64/297.09 [a__dbl(s(X))] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [s(s(dbl(X)))] 849.64/297.09 849.64/297.09 [a__dbl(0())] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [0()] 849.64/297.09 849.64/297.09 [a__add(X1, X2)] = [1] X1 + [1] X2 + [0] 849.64/297.09 ? [5] 849.64/297.09 = [add(X1, X2)] 849.64/297.09 849.64/297.09 [a__add(s(X), Y)] = [1] Y + [0] 849.64/297.09 >= [0] 849.64/297.09 = [s(add(X, Y))] 849.64/297.09 849.64/297.09 [a__add(0(), X)] = [1] X + [0] 849.64/297.09 >= [0] 849.64/297.09 = [mark(X)] 849.64/297.09 849.64/297.09 [a__first(X1, X2)] = [1] X1 + [1] X2 + [0] 849.64/297.09 >= [1] X1 + [1] X2 + [0] 849.64/297.09 = [first(X1, X2)] 849.64/297.09 849.64/297.09 [a__first(s(X), cons(Y, Z))] = [1] Y + [0] 849.64/297.09 >= [0] 849.64/297.09 = [cons(mark(Y), first(X, Z))] 849.64/297.09 849.64/297.09 [a__first(0(), X)] = [1] X + [0] 849.64/297.09 ? [7] 849.64/297.09 = [nil()] 849.64/297.09 849.64/297.09 849.64/297.09 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 849.64/297.09 849.64/297.09 We are left with following problem, upon which TcT provides the 849.64/297.09 certificate YES(O(1),O(n^2)). 849.64/297.09 849.64/297.09 Strict Trs: 849.64/297.09 { a__sqr(X) -> sqr(X) 849.64/297.09 , a__sqr(s(X)) -> s(add(sqr(X), dbl(X))) 849.64/297.09 , a__sqr(0()) -> 0() 849.64/297.09 , mark(cons(X1, X2)) -> cons(mark(X1), X2) 849.64/297.09 , mark(recip(X)) -> recip(mark(X)) 849.64/297.09 , mark(terms(X)) -> a__terms(mark(X)) 849.64/297.09 , mark(s(X)) -> s(X) 849.64/297.09 , mark(0()) -> 0() 849.64/297.09 , mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) 849.64/297.09 , mark(sqr(X)) -> a__sqr(mark(X)) 849.64/297.09 , mark(dbl(X)) -> a__dbl(mark(X)) 849.64/297.09 , mark(nil()) -> nil() 849.64/297.09 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 849.64/297.09 , a__dbl(X) -> dbl(X) 849.64/297.09 , a__dbl(s(X)) -> s(s(dbl(X))) 849.64/297.09 , a__dbl(0()) -> 0() 849.64/297.09 , a__add(X1, X2) -> add(X1, X2) 849.64/297.09 , a__add(s(X), Y) -> s(add(X, Y)) 849.64/297.09 , a__add(0(), X) -> mark(X) 849.64/297.09 , a__first(0(), X) -> nil() } 849.64/297.09 Weak Trs: 849.64/297.09 { a__terms(N) -> cons(recip(a__sqr(mark(N))), terms(s(N))) 849.64/297.09 , a__terms(X) -> terms(X) 849.64/297.09 , a__first(X1, X2) -> first(X1, X2) 849.64/297.09 , a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z)) } 849.64/297.09 Obligation: 849.64/297.09 innermost runtime complexity 849.64/297.09 Answer: 849.64/297.09 YES(O(1),O(n^2)) 849.64/297.09 849.64/297.09 The weightgap principle applies (using the following nonconstant 849.64/297.09 growth matrix-interpretation) 849.64/297.09 849.64/297.09 The following argument positions are usable: 849.64/297.09 Uargs(a__terms) = {1}, Uargs(cons) = {1}, Uargs(recip) = {1}, 849.64/297.09 Uargs(a__sqr) = {1}, Uargs(a__dbl) = {1}, Uargs(a__add) = {1, 2}, 849.64/297.09 Uargs(a__first) = {1, 2} 849.64/297.09 849.64/297.09 TcT has computed the following matrix interpretation satisfying 849.64/297.09 not(EDA) and not(IDA(1)). 849.64/297.09 849.64/297.09 [a__terms](x1) = [1] x1 + [4] 849.64/297.09 849.64/297.09 [cons](x1, x2) = [1] x1 + [0] 849.64/297.09 849.64/297.09 [recip](x1) = [1] x1 + [0] 849.64/297.09 849.64/297.09 [a__sqr](x1) = [1] x1 + [1] 849.64/297.09 849.64/297.09 [mark](x1) = [0] 849.64/297.09 849.64/297.09 [terms](x1) = [2] 849.64/297.09 849.64/297.09 [s](x1) = [0] 849.64/297.09 849.64/297.09 [0] = [0] 849.64/297.09 849.64/297.09 [add](x1, x2) = [5] 849.64/297.09 849.64/297.09 [sqr](x1) = [1] x1 + [0] 849.64/297.09 849.64/297.09 [dbl](x1) = [1] x1 + [0] 849.64/297.09 849.64/297.09 [a__dbl](x1) = [1] x1 + [0] 849.64/297.09 849.64/297.09 [a__add](x1, x2) = [1] x1 + [1] x2 + [0] 849.64/297.09 849.64/297.09 [a__first](x1, x2) = [1] x1 + [1] x2 + [0] 849.64/297.09 849.64/297.09 [nil] = [7] 849.64/297.09 849.64/297.09 [first](x1, x2) = [1] x1 + [1] x2 + [0] 849.64/297.09 849.64/297.09 The order satisfies the following ordering constraints: 849.64/297.09 849.64/297.09 [a__terms(N)] = [1] N + [4] 849.64/297.09 > [1] 849.64/297.09 = [cons(recip(a__sqr(mark(N))), terms(s(N)))] 849.64/297.09 849.64/297.09 [a__terms(X)] = [1] X + [4] 849.64/297.09 > [2] 849.64/297.09 = [terms(X)] 849.64/297.09 849.64/297.09 [a__sqr(X)] = [1] X + [1] 849.64/297.09 > [1] X + [0] 849.64/297.09 = [sqr(X)] 849.64/297.09 849.64/297.09 [a__sqr(s(X))] = [1] 849.64/297.09 > [0] 849.64/297.09 = [s(add(sqr(X), dbl(X)))] 849.64/297.09 849.64/297.09 [a__sqr(0())] = [1] 849.64/297.09 > [0] 849.64/297.09 = [0()] 849.64/297.09 849.64/297.09 [mark(cons(X1, X2))] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [cons(mark(X1), X2)] 849.64/297.09 849.64/297.09 [mark(recip(X))] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [recip(mark(X))] 849.64/297.09 849.64/297.09 [mark(terms(X))] = [0] 849.64/297.09 ? [4] 849.64/297.09 = [a__terms(mark(X))] 849.64/297.09 849.64/297.09 [mark(s(X))] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [s(X)] 849.64/297.09 849.64/297.09 [mark(0())] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [0()] 849.64/297.09 849.64/297.09 [mark(add(X1, X2))] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [a__add(mark(X1), mark(X2))] 849.64/297.09 849.64/297.09 [mark(sqr(X))] = [0] 849.64/297.09 ? [1] 849.64/297.09 = [a__sqr(mark(X))] 849.64/297.09 849.64/297.09 [mark(dbl(X))] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [a__dbl(mark(X))] 849.64/297.09 849.64/297.09 [mark(nil())] = [0] 849.64/297.09 ? [7] 849.64/297.09 = [nil()] 849.64/297.09 849.64/297.09 [mark(first(X1, X2))] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [a__first(mark(X1), mark(X2))] 849.64/297.09 849.64/297.09 [a__dbl(X)] = [1] X + [0] 849.64/297.09 >= [1] X + [0] 849.64/297.09 = [dbl(X)] 849.64/297.09 849.64/297.09 [a__dbl(s(X))] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [s(s(dbl(X)))] 849.64/297.09 849.64/297.09 [a__dbl(0())] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [0()] 849.64/297.09 849.64/297.09 [a__add(X1, X2)] = [1] X1 + [1] X2 + [0] 849.64/297.09 ? [5] 849.64/297.09 = [add(X1, X2)] 849.64/297.09 849.64/297.09 [a__add(s(X), Y)] = [1] Y + [0] 849.64/297.09 >= [0] 849.64/297.09 = [s(add(X, Y))] 849.64/297.09 849.64/297.09 [a__add(0(), X)] = [1] X + [0] 849.64/297.09 >= [0] 849.64/297.09 = [mark(X)] 849.64/297.09 849.64/297.09 [a__first(X1, X2)] = [1] X1 + [1] X2 + [0] 849.64/297.09 >= [1] X1 + [1] X2 + [0] 849.64/297.09 = [first(X1, X2)] 849.64/297.09 849.64/297.09 [a__first(s(X), cons(Y, Z))] = [1] Y + [0] 849.64/297.09 >= [0] 849.64/297.09 = [cons(mark(Y), first(X, Z))] 849.64/297.09 849.64/297.09 [a__first(0(), X)] = [1] X + [0] 849.64/297.09 ? [7] 849.64/297.09 = [nil()] 849.64/297.09 849.64/297.09 849.64/297.09 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 849.64/297.09 849.64/297.09 We are left with following problem, upon which TcT provides the 849.64/297.09 certificate YES(O(1),O(n^2)). 849.64/297.09 849.64/297.09 Strict Trs: 849.64/297.09 { mark(cons(X1, X2)) -> cons(mark(X1), X2) 849.64/297.09 , mark(recip(X)) -> recip(mark(X)) 849.64/297.09 , mark(terms(X)) -> a__terms(mark(X)) 849.64/297.09 , mark(s(X)) -> s(X) 849.64/297.09 , mark(0()) -> 0() 849.64/297.09 , mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) 849.64/297.09 , mark(sqr(X)) -> a__sqr(mark(X)) 849.64/297.09 , mark(dbl(X)) -> a__dbl(mark(X)) 849.64/297.09 , mark(nil()) -> nil() 849.64/297.09 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 849.64/297.09 , a__dbl(X) -> dbl(X) 849.64/297.09 , a__dbl(s(X)) -> s(s(dbl(X))) 849.64/297.09 , a__dbl(0()) -> 0() 849.64/297.09 , a__add(X1, X2) -> add(X1, X2) 849.64/297.09 , a__add(s(X), Y) -> s(add(X, Y)) 849.64/297.09 , a__add(0(), X) -> mark(X) 849.64/297.09 , a__first(0(), X) -> nil() } 849.64/297.09 Weak Trs: 849.64/297.09 { a__terms(N) -> cons(recip(a__sqr(mark(N))), terms(s(N))) 849.64/297.09 , a__terms(X) -> terms(X) 849.64/297.09 , a__sqr(X) -> sqr(X) 849.64/297.09 , a__sqr(s(X)) -> s(add(sqr(X), dbl(X))) 849.64/297.09 , a__sqr(0()) -> 0() 849.64/297.09 , a__first(X1, X2) -> first(X1, X2) 849.64/297.09 , a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z)) } 849.64/297.09 Obligation: 849.64/297.09 innermost runtime complexity 849.64/297.09 Answer: 849.64/297.09 YES(O(1),O(n^2)) 849.64/297.09 849.64/297.09 The weightgap principle applies (using the following nonconstant 849.64/297.09 growth matrix-interpretation) 849.64/297.09 849.64/297.09 The following argument positions are usable: 849.64/297.09 Uargs(a__terms) = {1}, Uargs(cons) = {1}, Uargs(recip) = {1}, 849.64/297.09 Uargs(a__sqr) = {1}, Uargs(a__dbl) = {1}, Uargs(a__add) = {1, 2}, 849.64/297.09 Uargs(a__first) = {1, 2} 849.64/297.09 849.64/297.09 TcT has computed the following matrix interpretation satisfying 849.64/297.09 not(EDA) and not(IDA(1)). 849.64/297.09 849.64/297.09 [a__terms](x1) = [1] x1 + [0] 849.64/297.09 849.64/297.09 [cons](x1, x2) = [1] x1 + [0] 849.64/297.09 849.64/297.09 [recip](x1) = [1] x1 + [0] 849.64/297.09 849.64/297.09 [a__sqr](x1) = [1] x1 + [0] 849.64/297.09 849.64/297.09 [mark](x1) = [0] 849.64/297.09 849.64/297.09 [terms](x1) = [0] 849.64/297.09 849.64/297.09 [s](x1) = [0] 849.64/297.09 849.64/297.09 [0] = [4] 849.64/297.09 849.64/297.09 [add](x1, x2) = [5] 849.64/297.09 849.64/297.09 [sqr](x1) = [1] x1 + [0] 849.64/297.09 849.64/297.09 [dbl](x1) = [1] x1 + [0] 849.64/297.09 849.64/297.09 [a__dbl](x1) = [1] x1 + [0] 849.64/297.09 849.64/297.09 [a__add](x1, x2) = [1] x1 + [1] x2 + [0] 849.64/297.09 849.64/297.09 [a__first](x1, x2) = [1] x1 + [1] x2 + [0] 849.64/297.09 849.64/297.09 [nil] = [3] 849.64/297.09 849.64/297.09 [first](x1, x2) = [1] x1 + [1] x2 + [0] 849.64/297.09 849.64/297.09 The order satisfies the following ordering constraints: 849.64/297.09 849.64/297.09 [a__terms(N)] = [1] N + [0] 849.64/297.09 >= [0] 849.64/297.09 = [cons(recip(a__sqr(mark(N))), terms(s(N)))] 849.64/297.09 849.64/297.09 [a__terms(X)] = [1] X + [0] 849.64/297.09 >= [0] 849.64/297.09 = [terms(X)] 849.64/297.09 849.64/297.09 [a__sqr(X)] = [1] X + [0] 849.64/297.09 >= [1] X + [0] 849.64/297.09 = [sqr(X)] 849.64/297.09 849.64/297.09 [a__sqr(s(X))] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [s(add(sqr(X), dbl(X)))] 849.64/297.09 849.64/297.09 [a__sqr(0())] = [4] 849.64/297.09 >= [4] 849.64/297.09 = [0()] 849.64/297.09 849.64/297.09 [mark(cons(X1, X2))] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [cons(mark(X1), X2)] 849.64/297.09 849.64/297.09 [mark(recip(X))] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [recip(mark(X))] 849.64/297.09 849.64/297.09 [mark(terms(X))] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [a__terms(mark(X))] 849.64/297.09 849.64/297.09 [mark(s(X))] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [s(X)] 849.64/297.09 849.64/297.09 [mark(0())] = [0] 849.64/297.09 ? [4] 849.64/297.09 = [0()] 849.64/297.09 849.64/297.09 [mark(add(X1, X2))] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [a__add(mark(X1), mark(X2))] 849.64/297.09 849.64/297.09 [mark(sqr(X))] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [a__sqr(mark(X))] 849.64/297.09 849.64/297.09 [mark(dbl(X))] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [a__dbl(mark(X))] 849.64/297.09 849.64/297.09 [mark(nil())] = [0] 849.64/297.09 ? [3] 849.64/297.09 = [nil()] 849.64/297.09 849.64/297.09 [mark(first(X1, X2))] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [a__first(mark(X1), mark(X2))] 849.64/297.09 849.64/297.09 [a__dbl(X)] = [1] X + [0] 849.64/297.09 >= [1] X + [0] 849.64/297.09 = [dbl(X)] 849.64/297.09 849.64/297.09 [a__dbl(s(X))] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [s(s(dbl(X)))] 849.64/297.09 849.64/297.09 [a__dbl(0())] = [4] 849.64/297.09 >= [4] 849.64/297.09 = [0()] 849.64/297.09 849.64/297.09 [a__add(X1, X2)] = [1] X1 + [1] X2 + [0] 849.64/297.09 ? [5] 849.64/297.09 = [add(X1, X2)] 849.64/297.09 849.64/297.09 [a__add(s(X), Y)] = [1] Y + [0] 849.64/297.09 >= [0] 849.64/297.09 = [s(add(X, Y))] 849.64/297.09 849.64/297.09 [a__add(0(), X)] = [1] X + [4] 849.64/297.09 > [0] 849.64/297.09 = [mark(X)] 849.64/297.09 849.64/297.09 [a__first(X1, X2)] = [1] X1 + [1] X2 + [0] 849.64/297.09 >= [1] X1 + [1] X2 + [0] 849.64/297.09 = [first(X1, X2)] 849.64/297.09 849.64/297.09 [a__first(s(X), cons(Y, Z))] = [1] Y + [0] 849.64/297.09 >= [0] 849.64/297.09 = [cons(mark(Y), first(X, Z))] 849.64/297.09 849.64/297.09 [a__first(0(), X)] = [1] X + [4] 849.64/297.09 > [3] 849.64/297.09 = [nil()] 849.64/297.09 849.64/297.09 849.64/297.09 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 849.64/297.09 849.64/297.09 We are left with following problem, upon which TcT provides the 849.64/297.09 certificate YES(O(1),O(n^2)). 849.64/297.09 849.64/297.09 Strict Trs: 849.64/297.09 { mark(cons(X1, X2)) -> cons(mark(X1), X2) 849.64/297.09 , mark(recip(X)) -> recip(mark(X)) 849.64/297.09 , mark(terms(X)) -> a__terms(mark(X)) 849.64/297.09 , mark(s(X)) -> s(X) 849.64/297.09 , mark(0()) -> 0() 849.64/297.09 , mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) 849.64/297.09 , mark(sqr(X)) -> a__sqr(mark(X)) 849.64/297.09 , mark(dbl(X)) -> a__dbl(mark(X)) 849.64/297.09 , mark(nil()) -> nil() 849.64/297.09 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 849.64/297.09 , a__dbl(X) -> dbl(X) 849.64/297.09 , a__dbl(s(X)) -> s(s(dbl(X))) 849.64/297.09 , a__dbl(0()) -> 0() 849.64/297.09 , a__add(X1, X2) -> add(X1, X2) 849.64/297.09 , a__add(s(X), Y) -> s(add(X, Y)) } 849.64/297.09 Weak Trs: 849.64/297.09 { a__terms(N) -> cons(recip(a__sqr(mark(N))), terms(s(N))) 849.64/297.09 , a__terms(X) -> terms(X) 849.64/297.09 , a__sqr(X) -> sqr(X) 849.64/297.09 , a__sqr(s(X)) -> s(add(sqr(X), dbl(X))) 849.64/297.09 , a__sqr(0()) -> 0() 849.64/297.09 , a__add(0(), X) -> mark(X) 849.64/297.09 , a__first(X1, X2) -> first(X1, X2) 849.64/297.09 , a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z)) 849.64/297.09 , a__first(0(), X) -> nil() } 849.64/297.09 Obligation: 849.64/297.09 innermost runtime complexity 849.64/297.09 Answer: 849.64/297.09 YES(O(1),O(n^2)) 849.64/297.09 849.64/297.09 The weightgap principle applies (using the following nonconstant 849.64/297.09 growth matrix-interpretation) 849.64/297.09 849.64/297.09 The following argument positions are usable: 849.64/297.09 Uargs(a__terms) = {1}, Uargs(cons) = {1}, Uargs(recip) = {1}, 849.64/297.09 Uargs(a__sqr) = {1}, Uargs(a__dbl) = {1}, Uargs(a__add) = {1, 2}, 849.64/297.09 Uargs(a__first) = {1, 2} 849.64/297.09 849.64/297.09 TcT has computed the following matrix interpretation satisfying 849.64/297.09 not(EDA) and not(IDA(1)). 849.64/297.09 849.64/297.09 [a__terms](x1) = [1] x1 + [0] 849.64/297.09 849.64/297.09 [cons](x1, x2) = [1] x1 + [0] 849.64/297.09 849.64/297.09 [recip](x1) = [1] x1 + [0] 849.64/297.09 849.64/297.09 [a__sqr](x1) = [1] x1 + [0] 849.64/297.09 849.64/297.09 [mark](x1) = [0] 849.64/297.09 849.64/297.09 [terms](x1) = [1] x1 + [0] 849.64/297.09 849.64/297.09 [s](x1) = [4] 849.64/297.09 849.64/297.09 [0] = [0] 849.64/297.09 849.64/297.09 [add](x1, x2) = [1] x1 + [1] x2 + [7] 849.64/297.09 849.64/297.09 [sqr](x1) = [1] x1 + [0] 849.64/297.09 849.64/297.09 [dbl](x1) = [1] x1 + [7] 849.64/297.09 849.64/297.09 [a__dbl](x1) = [1] x1 + [0] 849.64/297.09 849.64/297.09 [a__add](x1, x2) = [1] x1 + [1] x2 + [1] 849.64/297.09 849.64/297.09 [a__first](x1, x2) = [1] x1 + [1] x2 + [4] 849.64/297.09 849.64/297.09 [nil] = [3] 849.64/297.09 849.64/297.09 [first](x1, x2) = [1] x1 + [1] x2 + [3] 849.64/297.09 849.64/297.09 The order satisfies the following ordering constraints: 849.64/297.09 849.64/297.09 [a__terms(N)] = [1] N + [0] 849.64/297.09 >= [0] 849.64/297.09 = [cons(recip(a__sqr(mark(N))), terms(s(N)))] 849.64/297.09 849.64/297.09 [a__terms(X)] = [1] X + [0] 849.64/297.09 >= [1] X + [0] 849.64/297.09 = [terms(X)] 849.64/297.09 849.64/297.09 [a__sqr(X)] = [1] X + [0] 849.64/297.09 >= [1] X + [0] 849.64/297.09 = [sqr(X)] 849.64/297.09 849.64/297.09 [a__sqr(s(X))] = [4] 849.64/297.09 >= [4] 849.64/297.09 = [s(add(sqr(X), dbl(X)))] 849.64/297.09 849.64/297.09 [a__sqr(0())] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [0()] 849.64/297.09 849.64/297.09 [mark(cons(X1, X2))] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [cons(mark(X1), X2)] 849.64/297.09 849.64/297.09 [mark(recip(X))] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [recip(mark(X))] 849.64/297.09 849.64/297.09 [mark(terms(X))] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [a__terms(mark(X))] 849.64/297.09 849.64/297.09 [mark(s(X))] = [0] 849.64/297.09 ? [4] 849.64/297.09 = [s(X)] 849.64/297.09 849.64/297.09 [mark(0())] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [0()] 849.64/297.09 849.64/297.09 [mark(add(X1, X2))] = [0] 849.64/297.09 ? [1] 849.64/297.09 = [a__add(mark(X1), mark(X2))] 849.64/297.09 849.64/297.09 [mark(sqr(X))] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [a__sqr(mark(X))] 849.64/297.09 849.64/297.09 [mark(dbl(X))] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [a__dbl(mark(X))] 849.64/297.09 849.64/297.09 [mark(nil())] = [0] 849.64/297.09 ? [3] 849.64/297.09 = [nil()] 849.64/297.09 849.64/297.09 [mark(first(X1, X2))] = [0] 849.64/297.09 ? [4] 849.64/297.09 = [a__first(mark(X1), mark(X2))] 849.64/297.09 849.64/297.09 [a__dbl(X)] = [1] X + [0] 849.64/297.09 ? [1] X + [7] 849.64/297.09 = [dbl(X)] 849.64/297.09 849.64/297.09 [a__dbl(s(X))] = [4] 849.64/297.09 >= [4] 849.64/297.09 = [s(s(dbl(X)))] 849.64/297.09 849.64/297.09 [a__dbl(0())] = [0] 849.64/297.09 >= [0] 849.64/297.09 = [0()] 849.64/297.09 849.64/297.09 [a__add(X1, X2)] = [1] X1 + [1] X2 + [1] 849.64/297.09 ? [1] X1 + [1] X2 + [7] 849.64/297.09 = [add(X1, X2)] 849.64/297.09 849.64/297.09 [a__add(s(X), Y)] = [1] Y + [5] 849.64/297.09 > [4] 849.64/297.09 = [s(add(X, Y))] 849.64/297.09 849.64/297.09 [a__add(0(), X)] = [1] X + [1] 849.64/297.09 > [0] 849.64/297.09 = [mark(X)] 849.64/297.09 849.64/297.09 [a__first(X1, X2)] = [1] X1 + [1] X2 + [4] 849.64/297.09 > [1] X1 + [1] X2 + [3] 849.64/297.09 = [first(X1, X2)] 849.64/297.09 849.64/297.09 [a__first(s(X), cons(Y, Z))] = [1] Y + [8] 849.64/297.09 > [0] 849.64/297.09 = [cons(mark(Y), first(X, Z))] 849.64/297.09 849.64/297.09 [a__first(0(), X)] = [1] X + [4] 849.64/297.09 > [3] 849.64/297.09 = [nil()] 849.64/297.09 849.64/297.09 849.64/297.09 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 849.64/297.09 849.64/297.09 We are left with following problem, upon which TcT provides the 849.64/297.09 certificate YES(O(1),O(n^2)). 849.64/297.09 849.64/297.09 Strict Trs: 849.64/297.09 { mark(cons(X1, X2)) -> cons(mark(X1), X2) 849.64/297.09 , mark(recip(X)) -> recip(mark(X)) 849.64/297.09 , mark(terms(X)) -> a__terms(mark(X)) 849.64/297.09 , mark(s(X)) -> s(X) 849.64/297.09 , mark(0()) -> 0() 849.64/297.09 , mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) 849.64/297.09 , mark(sqr(X)) -> a__sqr(mark(X)) 849.64/297.09 , mark(dbl(X)) -> a__dbl(mark(X)) 849.64/297.09 , mark(nil()) -> nil() 849.64/297.09 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 849.64/297.09 , a__dbl(X) -> dbl(X) 849.64/297.09 , a__dbl(s(X)) -> s(s(dbl(X))) 849.64/297.09 , a__dbl(0()) -> 0() 849.64/297.09 , a__add(X1, X2) -> add(X1, X2) } 849.64/297.09 Weak Trs: 849.64/297.09 { a__terms(N) -> cons(recip(a__sqr(mark(N))), terms(s(N))) 849.64/297.09 , a__terms(X) -> terms(X) 849.64/297.09 , a__sqr(X) -> sqr(X) 849.64/297.09 , a__sqr(s(X)) -> s(add(sqr(X), dbl(X))) 849.64/297.09 , a__sqr(0()) -> 0() 849.64/297.09 , a__add(s(X), Y) -> s(add(X, Y)) 849.64/297.09 , a__add(0(), X) -> mark(X) 849.64/297.09 , a__first(X1, X2) -> first(X1, X2) 849.64/297.09 , a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z)) 849.64/297.09 , a__first(0(), X) -> nil() } 849.64/297.09 Obligation: 849.64/297.09 innermost runtime complexity 849.64/297.09 Answer: 849.64/297.09 YES(O(1),O(n^2)) 849.64/297.09 849.64/297.09 The weightgap principle applies (using the following nonconstant 849.64/297.09 growth matrix-interpretation) 849.64/297.09 849.64/297.09 The following argument positions are usable: 849.64/297.09 Uargs(a__terms) = {1}, Uargs(cons) = {1}, Uargs(recip) = {1}, 849.64/297.09 Uargs(a__sqr) = {1}, Uargs(a__dbl) = {1}, Uargs(a__add) = {1, 2}, 849.64/297.09 Uargs(a__first) = {1, 2} 849.64/297.09 849.64/297.09 TcT has computed the following matrix interpretation satisfying 849.64/297.09 not(EDA) and not(IDA(1)). 849.64/297.09 849.64/297.09 [a__terms](x1) = [1] x1 + [0] 849.64/297.09 849.64/297.09 [cons](x1, x2) = [1] x1 + [0] 849.64/297.09 849.64/297.09 [recip](x1) = [1] x1 + [0] 849.64/297.09 849.64/297.09 [a__sqr](x1) = [1] x1 + [0] 849.64/297.09 849.64/297.09 [mark](x1) = [0] 849.64/297.09 849.64/297.10 [terms](x1) = [1] x1 + [0] 849.64/297.10 849.64/297.10 [s](x1) = [4] 849.64/297.10 849.64/297.10 [0] = [0] 849.64/297.10 849.64/297.10 [add](x1, x2) = [1] x1 + [1] x2 + [7] 849.64/297.10 849.64/297.10 [sqr](x1) = [1] x1 + [0] 849.64/297.10 849.64/297.10 [dbl](x1) = [1] x1 + [7] 849.64/297.10 849.64/297.10 [a__dbl](x1) = [1] x1 + [4] 849.64/297.10 849.64/297.10 [a__add](x1, x2) = [1] x1 + [1] x2 + [0] 849.64/297.10 849.64/297.10 [a__first](x1, x2) = [1] x1 + [1] x2 + [4] 849.64/297.10 849.64/297.10 [nil] = [3] 849.64/297.10 849.64/297.10 [first](x1, x2) = [1] x1 + [1] x2 + [3] 849.64/297.10 849.64/297.10 The order satisfies the following ordering constraints: 849.64/297.10 849.64/297.10 [a__terms(N)] = [1] N + [0] 849.64/297.10 >= [0] 849.64/297.10 = [cons(recip(a__sqr(mark(N))), terms(s(N)))] 849.64/297.10 849.64/297.10 [a__terms(X)] = [1] X + [0] 849.64/297.10 >= [1] X + [0] 849.64/297.10 = [terms(X)] 849.64/297.10 849.64/297.10 [a__sqr(X)] = [1] X + [0] 849.64/297.10 >= [1] X + [0] 849.64/297.10 = [sqr(X)] 849.64/297.10 849.64/297.10 [a__sqr(s(X))] = [4] 849.64/297.10 >= [4] 849.64/297.10 = [s(add(sqr(X), dbl(X)))] 849.64/297.10 849.64/297.10 [a__sqr(0())] = [0] 849.64/297.10 >= [0] 849.64/297.10 = [0()] 849.64/297.10 849.64/297.10 [mark(cons(X1, X2))] = [0] 849.64/297.10 >= [0] 849.64/297.10 = [cons(mark(X1), X2)] 849.64/297.10 849.64/297.10 [mark(recip(X))] = [0] 849.64/297.10 >= [0] 849.64/297.10 = [recip(mark(X))] 849.64/297.10 849.64/297.10 [mark(terms(X))] = [0] 849.64/297.10 >= [0] 849.64/297.10 = [a__terms(mark(X))] 849.64/297.10 849.64/297.10 [mark(s(X))] = [0] 849.64/297.10 ? [4] 849.64/297.10 = [s(X)] 849.64/297.10 849.64/297.10 [mark(0())] = [0] 849.64/297.10 >= [0] 849.64/297.10 = [0()] 849.64/297.10 849.64/297.10 [mark(add(X1, X2))] = [0] 849.64/297.10 >= [0] 849.64/297.10 = [a__add(mark(X1), mark(X2))] 849.64/297.10 849.64/297.10 [mark(sqr(X))] = [0] 849.64/297.10 >= [0] 849.64/297.10 = [a__sqr(mark(X))] 849.64/297.10 849.64/297.10 [mark(dbl(X))] = [0] 849.64/297.10 ? [4] 849.64/297.10 = [a__dbl(mark(X))] 849.64/297.10 849.64/297.10 [mark(nil())] = [0] 849.64/297.10 ? [3] 849.64/297.10 = [nil()] 849.64/297.10 849.64/297.10 [mark(first(X1, X2))] = [0] 849.64/297.10 ? [4] 849.64/297.10 = [a__first(mark(X1), mark(X2))] 849.64/297.10 849.64/297.10 [a__dbl(X)] = [1] X + [4] 849.64/297.10 ? [1] X + [7] 849.64/297.10 = [dbl(X)] 849.64/297.10 849.64/297.10 [a__dbl(s(X))] = [8] 849.64/297.10 > [4] 849.64/297.10 = [s(s(dbl(X)))] 849.64/297.10 849.64/297.10 [a__dbl(0())] = [4] 849.64/297.10 > [0] 849.64/297.10 = [0()] 849.64/297.10 849.64/297.10 [a__add(X1, X2)] = [1] X1 + [1] X2 + [0] 849.64/297.10 ? [1] X1 + [1] X2 + [7] 849.64/297.10 = [add(X1, X2)] 849.64/297.10 849.64/297.10 [a__add(s(X), Y)] = [1] Y + [4] 849.64/297.10 >= [4] 849.64/297.10 = [s(add(X, Y))] 849.64/297.10 849.64/297.10 [a__add(0(), X)] = [1] X + [0] 849.64/297.10 >= [0] 849.64/297.10 = [mark(X)] 849.64/297.10 849.64/297.10 [a__first(X1, X2)] = [1] X1 + [1] X2 + [4] 849.64/297.10 > [1] X1 + [1] X2 + [3] 849.64/297.10 = [first(X1, X2)] 849.64/297.10 849.64/297.10 [a__first(s(X), cons(Y, Z))] = [1] Y + [8] 849.64/297.10 > [0] 849.64/297.10 = [cons(mark(Y), first(X, Z))] 849.86/297.10 849.86/297.10 [a__first(0(), X)] = [1] X + [4] 849.86/297.10 > [3] 849.86/297.10 = [nil()] 849.86/297.10 849.86/297.10 849.86/297.10 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 849.86/297.10 849.86/297.10 We are left with following problem, upon which TcT provides the 849.86/297.10 certificate YES(O(1),O(n^2)). 849.86/297.10 849.86/297.10 Strict Trs: 849.86/297.10 { mark(cons(X1, X2)) -> cons(mark(X1), X2) 849.86/297.10 , mark(recip(X)) -> recip(mark(X)) 849.86/297.10 , mark(terms(X)) -> a__terms(mark(X)) 849.86/297.10 , mark(s(X)) -> s(X) 849.86/297.10 , mark(0()) -> 0() 849.86/297.10 , mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) 849.86/297.10 , mark(sqr(X)) -> a__sqr(mark(X)) 849.86/297.10 , mark(dbl(X)) -> a__dbl(mark(X)) 849.86/297.10 , mark(nil()) -> nil() 849.86/297.10 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 849.86/297.10 , a__dbl(X) -> dbl(X) 849.86/297.10 , a__add(X1, X2) -> add(X1, X2) } 849.86/297.10 Weak Trs: 849.86/297.10 { a__terms(N) -> cons(recip(a__sqr(mark(N))), terms(s(N))) 849.86/297.10 , a__terms(X) -> terms(X) 849.86/297.10 , a__sqr(X) -> sqr(X) 849.86/297.10 , a__sqr(s(X)) -> s(add(sqr(X), dbl(X))) 849.86/297.10 , a__sqr(0()) -> 0() 849.86/297.10 , a__dbl(s(X)) -> s(s(dbl(X))) 849.86/297.10 , a__dbl(0()) -> 0() 849.86/297.10 , a__add(s(X), Y) -> s(add(X, Y)) 849.86/297.10 , a__add(0(), X) -> mark(X) 849.86/297.10 , a__first(X1, X2) -> first(X1, X2) 849.86/297.10 , a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z)) 849.86/297.10 , a__first(0(), X) -> nil() } 849.86/297.10 Obligation: 849.86/297.10 innermost runtime complexity 849.86/297.10 Answer: 849.86/297.10 YES(O(1),O(n^2)) 849.86/297.10 849.86/297.10 The weightgap principle applies (using the following nonconstant 849.86/297.10 growth matrix-interpretation) 849.86/297.10 849.86/297.10 The following argument positions are usable: 849.86/297.10 Uargs(a__terms) = {1}, Uargs(cons) = {1}, Uargs(recip) = {1}, 849.86/297.10 Uargs(a__sqr) = {1}, Uargs(a__dbl) = {1}, Uargs(a__add) = {1, 2}, 849.86/297.10 Uargs(a__first) = {1, 2} 849.86/297.10 849.86/297.10 TcT has computed the following matrix interpretation satisfying 849.86/297.10 not(EDA) and not(IDA(1)). 849.86/297.10 849.86/297.10 [a__terms](x1) = [1] x1 + [0] 849.86/297.10 849.86/297.10 [cons](x1, x2) = [1] x1 + [0] 849.86/297.10 849.86/297.10 [recip](x1) = [1] x1 + [0] 849.86/297.10 849.86/297.10 [a__sqr](x1) = [1] x1 + [0] 849.86/297.10 849.86/297.10 [mark](x1) = [1] x1 + [0] 849.86/297.10 849.86/297.10 [terms](x1) = [1] x1 + [0] 849.86/297.10 849.86/297.10 [s](x1) = [4] 849.86/297.10 849.86/297.10 [0] = [0] 849.86/297.10 849.86/297.10 [add](x1, x2) = [1] x1 + [1] x2 + [4] 849.86/297.10 849.86/297.10 [sqr](x1) = [1] x1 + [0] 849.86/297.10 849.86/297.10 [dbl](x1) = [1] x1 + [0] 849.86/297.10 849.86/297.10 [a__dbl](x1) = [1] x1 + [0] 849.86/297.10 849.86/297.10 [a__add](x1, x2) = [1] x1 + [1] x2 + [0] 849.86/297.10 849.86/297.10 [a__first](x1, x2) = [1] x1 + [1] x2 + [4] 849.86/297.10 849.86/297.10 [nil] = [3] 849.86/297.10 849.86/297.10 [first](x1, x2) = [1] x1 + [1] x2 + [0] 849.86/297.10 849.86/297.10 The order satisfies the following ordering constraints: 849.86/297.10 849.86/297.10 [a__terms(N)] = [1] N + [0] 849.86/297.10 >= [1] N + [0] 849.86/297.10 = [cons(recip(a__sqr(mark(N))), terms(s(N)))] 849.86/297.10 849.86/297.10 [a__terms(X)] = [1] X + [0] 849.86/297.10 >= [1] X + [0] 849.86/297.10 = [terms(X)] 849.86/297.10 849.86/297.10 [a__sqr(X)] = [1] X + [0] 849.86/297.10 >= [1] X + [0] 849.86/297.10 = [sqr(X)] 849.86/297.10 849.86/297.10 [a__sqr(s(X))] = [4] 849.86/297.10 >= [4] 849.86/297.10 = [s(add(sqr(X), dbl(X)))] 849.86/297.10 849.86/297.10 [a__sqr(0())] = [0] 849.86/297.10 >= [0] 849.86/297.10 = [0()] 849.86/297.10 849.86/297.10 [mark(cons(X1, X2))] = [1] X1 + [0] 849.86/297.10 >= [1] X1 + [0] 849.86/297.10 = [cons(mark(X1), X2)] 849.86/297.10 849.86/297.10 [mark(recip(X))] = [1] X + [0] 849.86/297.10 >= [1] X + [0] 849.86/297.10 = [recip(mark(X))] 849.86/297.10 849.86/297.10 [mark(terms(X))] = [1] X + [0] 849.86/297.10 >= [1] X + [0] 849.86/297.10 = [a__terms(mark(X))] 849.86/297.10 849.86/297.10 [mark(s(X))] = [4] 849.86/297.10 >= [4] 849.86/297.10 = [s(X)] 849.86/297.10 849.86/297.10 [mark(0())] = [0] 849.86/297.10 >= [0] 849.86/297.10 = [0()] 849.86/297.10 849.86/297.10 [mark(add(X1, X2))] = [1] X1 + [1] X2 + [4] 849.86/297.10 > [1] X1 + [1] X2 + [0] 849.86/297.10 = [a__add(mark(X1), mark(X2))] 849.86/297.10 849.86/297.10 [mark(sqr(X))] = [1] X + [0] 849.86/297.10 >= [1] X + [0] 849.86/297.10 = [a__sqr(mark(X))] 849.86/297.10 849.86/297.10 [mark(dbl(X))] = [1] X + [0] 849.86/297.10 >= [1] X + [0] 849.86/297.10 = [a__dbl(mark(X))] 849.86/297.10 849.86/297.10 [mark(nil())] = [3] 849.86/297.10 >= [3] 849.86/297.10 = [nil()] 849.86/297.10 849.86/297.10 [mark(first(X1, X2))] = [1] X1 + [1] X2 + [0] 849.86/297.10 ? [1] X1 + [1] X2 + [4] 849.86/297.10 = [a__first(mark(X1), mark(X2))] 849.86/297.10 849.86/297.10 [a__dbl(X)] = [1] X + [0] 849.86/297.10 >= [1] X + [0] 849.86/297.10 = [dbl(X)] 849.86/297.10 849.86/297.10 [a__dbl(s(X))] = [4] 849.86/297.10 >= [4] 849.86/297.10 = [s(s(dbl(X)))] 849.86/297.10 849.86/297.10 [a__dbl(0())] = [0] 849.86/297.10 >= [0] 849.86/297.10 = [0()] 849.86/297.10 849.86/297.10 [a__add(X1, X2)] = [1] X1 + [1] X2 + [0] 849.86/297.10 ? [1] X1 + [1] X2 + [4] 849.86/297.10 = [add(X1, X2)] 849.86/297.10 849.86/297.10 [a__add(s(X), Y)] = [1] Y + [4] 849.86/297.10 >= [4] 849.86/297.10 = [s(add(X, Y))] 849.86/297.10 849.86/297.10 [a__add(0(), X)] = [1] X + [0] 849.86/297.10 >= [1] X + [0] 849.86/297.10 = [mark(X)] 849.86/297.10 849.86/297.10 [a__first(X1, X2)] = [1] X1 + [1] X2 + [4] 849.86/297.10 > [1] X1 + [1] X2 + [0] 849.86/297.10 = [first(X1, X2)] 849.86/297.10 849.86/297.10 [a__first(s(X), cons(Y, Z))] = [1] Y + [8] 849.86/297.10 > [1] Y + [0] 849.86/297.10 = [cons(mark(Y), first(X, Z))] 849.86/297.10 849.86/297.10 [a__first(0(), X)] = [1] X + [4] 849.86/297.10 > [3] 849.86/297.10 = [nil()] 849.86/297.10 849.86/297.10 849.86/297.10 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 849.86/297.10 849.86/297.10 We are left with following problem, upon which TcT provides the 849.86/297.10 certificate YES(O(1),O(n^2)). 849.86/297.10 849.86/297.10 Strict Trs: 849.86/297.10 { mark(cons(X1, X2)) -> cons(mark(X1), X2) 849.86/297.10 , mark(recip(X)) -> recip(mark(X)) 849.86/297.10 , mark(terms(X)) -> a__terms(mark(X)) 849.86/297.10 , mark(s(X)) -> s(X) 849.86/297.10 , mark(0()) -> 0() 849.86/297.10 , mark(sqr(X)) -> a__sqr(mark(X)) 849.86/297.10 , mark(dbl(X)) -> a__dbl(mark(X)) 849.86/297.10 , mark(nil()) -> nil() 849.86/297.10 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 849.86/297.10 , a__dbl(X) -> dbl(X) 849.86/297.10 , a__add(X1, X2) -> add(X1, X2) } 849.86/297.10 Weak Trs: 849.86/297.10 { a__terms(N) -> cons(recip(a__sqr(mark(N))), terms(s(N))) 849.86/297.10 , a__terms(X) -> terms(X) 849.86/297.10 , a__sqr(X) -> sqr(X) 849.86/297.10 , a__sqr(s(X)) -> s(add(sqr(X), dbl(X))) 849.86/297.10 , a__sqr(0()) -> 0() 849.86/297.10 , mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) 849.86/297.10 , a__dbl(s(X)) -> s(s(dbl(X))) 849.86/297.10 , a__dbl(0()) -> 0() 849.86/297.10 , a__add(s(X), Y) -> s(add(X, Y)) 849.86/297.10 , a__add(0(), X) -> mark(X) 849.86/297.10 , a__first(X1, X2) -> first(X1, X2) 849.86/297.10 , a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z)) 849.86/297.10 , a__first(0(), X) -> nil() } 849.86/297.10 Obligation: 849.86/297.10 innermost runtime complexity 849.86/297.10 Answer: 849.86/297.10 YES(O(1),O(n^2)) 849.86/297.10 849.86/297.10 The weightgap principle applies (using the following nonconstant 849.86/297.10 growth matrix-interpretation) 849.86/297.10 849.86/297.10 The following argument positions are usable: 849.86/297.10 Uargs(a__terms) = {1}, Uargs(cons) = {1}, Uargs(recip) = {1}, 849.86/297.10 Uargs(a__sqr) = {1}, Uargs(a__dbl) = {1}, Uargs(a__add) = {1, 2}, 849.86/297.10 Uargs(a__first) = {1, 2} 849.86/297.10 849.86/297.10 TcT has computed the following matrix interpretation satisfying 849.86/297.10 not(EDA) and not(IDA(1)). 849.86/297.10 849.86/297.10 [a__terms](x1) = [1] x1 + [0] 849.86/297.10 849.86/297.10 [cons](x1, x2) = [1] x1 + [0] 849.86/297.10 849.86/297.10 [recip](x1) = [1] x1 + [0] 849.86/297.10 849.86/297.10 [a__sqr](x1) = [1] x1 + [0] 849.86/297.10 849.86/297.10 [mark](x1) = [0] 849.86/297.10 849.86/297.10 [terms](x1) = [1] x1 + [0] 849.86/297.10 849.86/297.10 [s](x1) = [4] 849.86/297.10 849.86/297.10 [0] = [0] 849.86/297.10 849.86/297.10 [add](x1, x2) = [1] x1 + [1] x2 + [7] 849.86/297.10 849.86/297.10 [sqr](x1) = [1] x1 + [0] 849.86/297.10 849.86/297.10 [dbl](x1) = [1] x1 + [3] 849.86/297.10 849.86/297.10 [a__dbl](x1) = [1] x1 + [4] 849.86/297.10 849.86/297.10 [a__add](x1, x2) = [1] x1 + [1] x2 + [0] 849.86/297.10 849.86/297.10 [a__first](x1, x2) = [1] x1 + [1] x2 + [4] 849.86/297.10 849.86/297.10 [nil] = [3] 849.86/297.10 849.86/297.10 [first](x1, x2) = [1] x1 + [1] x2 + [3] 849.86/297.10 849.86/297.10 The order satisfies the following ordering constraints: 849.86/297.10 849.86/297.10 [a__terms(N)] = [1] N + [0] 849.86/297.10 >= [0] 849.86/297.10 = [cons(recip(a__sqr(mark(N))), terms(s(N)))] 849.86/297.10 849.86/297.10 [a__terms(X)] = [1] X + [0] 849.86/297.10 >= [1] X + [0] 849.86/297.10 = [terms(X)] 849.86/297.10 849.86/297.10 [a__sqr(X)] = [1] X + [0] 849.86/297.10 >= [1] X + [0] 849.86/297.10 = [sqr(X)] 849.86/297.10 849.86/297.10 [a__sqr(s(X))] = [4] 849.86/297.10 >= [4] 849.86/297.10 = [s(add(sqr(X), dbl(X)))] 849.86/297.10 849.86/297.10 [a__sqr(0())] = [0] 849.86/297.10 >= [0] 849.86/297.10 = [0()] 849.86/297.10 849.86/297.10 [mark(cons(X1, X2))] = [0] 849.86/297.10 >= [0] 849.86/297.10 = [cons(mark(X1), X2)] 849.86/297.10 849.86/297.10 [mark(recip(X))] = [0] 849.86/297.10 >= [0] 849.86/297.10 = [recip(mark(X))] 849.86/297.10 849.86/297.10 [mark(terms(X))] = [0] 849.86/297.10 >= [0] 849.86/297.10 = [a__terms(mark(X))] 849.86/297.10 849.86/297.10 [mark(s(X))] = [0] 849.86/297.10 ? [4] 849.86/297.10 = [s(X)] 849.86/297.10 849.86/297.10 [mark(0())] = [0] 849.86/297.10 >= [0] 849.86/297.10 = [0()] 849.86/297.10 849.86/297.10 [mark(add(X1, X2))] = [0] 849.86/297.10 >= [0] 849.86/297.10 = [a__add(mark(X1), mark(X2))] 849.86/297.10 849.86/297.10 [mark(sqr(X))] = [0] 849.86/297.10 >= [0] 849.86/297.10 = [a__sqr(mark(X))] 849.86/297.10 849.86/297.10 [mark(dbl(X))] = [0] 849.86/297.10 ? [4] 849.86/297.10 = [a__dbl(mark(X))] 849.86/297.10 849.86/297.10 [mark(nil())] = [0] 849.86/297.10 ? [3] 849.86/297.10 = [nil()] 849.86/297.10 849.86/297.10 [mark(first(X1, X2))] = [0] 849.86/297.10 ? [4] 849.86/297.10 = [a__first(mark(X1), mark(X2))] 849.86/297.10 849.86/297.10 [a__dbl(X)] = [1] X + [4] 849.86/297.10 > [1] X + [3] 849.86/297.10 = [dbl(X)] 849.86/297.10 849.86/297.10 [a__dbl(s(X))] = [8] 849.86/297.10 > [4] 849.86/297.10 = [s(s(dbl(X)))] 849.86/297.10 849.86/297.10 [a__dbl(0())] = [4] 849.86/297.10 > [0] 849.86/297.10 = [0()] 849.86/297.10 849.86/297.10 [a__add(X1, X2)] = [1] X1 + [1] X2 + [0] 849.86/297.10 ? [1] X1 + [1] X2 + [7] 849.86/297.10 = [add(X1, X2)] 849.86/297.10 849.86/297.10 [a__add(s(X), Y)] = [1] Y + [4] 849.86/297.10 >= [4] 849.86/297.10 = [s(add(X, Y))] 849.86/297.10 849.86/297.10 [a__add(0(), X)] = [1] X + [0] 849.86/297.10 >= [0] 849.86/297.10 = [mark(X)] 849.86/297.10 849.86/297.10 [a__first(X1, X2)] = [1] X1 + [1] X2 + [4] 849.86/297.10 > [1] X1 + [1] X2 + [3] 849.86/297.10 = [first(X1, X2)] 849.86/297.10 849.86/297.10 [a__first(s(X), cons(Y, Z))] = [1] Y + [8] 849.86/297.10 > [0] 849.86/297.10 = [cons(mark(Y), first(X, Z))] 849.86/297.10 849.86/297.10 [a__first(0(), X)] = [1] X + [4] 849.86/297.10 > [3] 849.86/297.10 = [nil()] 849.86/297.10 849.86/297.10 849.86/297.10 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 849.86/297.10 849.86/297.10 We are left with following problem, upon which TcT provides the 849.86/297.10 certificate YES(O(1),O(n^2)). 849.86/297.10 849.86/297.10 Strict Trs: 849.86/297.10 { mark(cons(X1, X2)) -> cons(mark(X1), X2) 849.86/297.10 , mark(recip(X)) -> recip(mark(X)) 849.86/297.10 , mark(terms(X)) -> a__terms(mark(X)) 849.86/297.10 , mark(s(X)) -> s(X) 849.86/297.10 , mark(0()) -> 0() 849.86/297.10 , mark(sqr(X)) -> a__sqr(mark(X)) 849.86/297.10 , mark(dbl(X)) -> a__dbl(mark(X)) 849.86/297.10 , mark(nil()) -> nil() 849.86/297.10 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 849.86/297.10 , a__add(X1, X2) -> add(X1, X2) } 849.86/297.10 Weak Trs: 849.86/297.10 { a__terms(N) -> cons(recip(a__sqr(mark(N))), terms(s(N))) 849.86/297.10 , a__terms(X) -> terms(X) 849.86/297.10 , a__sqr(X) -> sqr(X) 849.86/297.10 , a__sqr(s(X)) -> s(add(sqr(X), dbl(X))) 849.86/297.10 , a__sqr(0()) -> 0() 849.86/297.10 , mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) 849.86/297.10 , a__dbl(X) -> dbl(X) 849.86/297.10 , a__dbl(s(X)) -> s(s(dbl(X))) 849.86/297.10 , a__dbl(0()) -> 0() 849.86/297.10 , a__add(s(X), Y) -> s(add(X, Y)) 849.86/297.10 , a__add(0(), X) -> mark(X) 849.86/297.10 , a__first(X1, X2) -> first(X1, X2) 849.86/297.10 , a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z)) 849.86/297.10 , a__first(0(), X) -> nil() } 849.86/297.10 Obligation: 849.86/297.10 innermost runtime complexity 849.86/297.10 Answer: 849.86/297.10 YES(O(1),O(n^2)) 849.86/297.10 849.86/297.10 The weightgap principle applies (using the following nonconstant 849.86/297.10 growth matrix-interpretation) 849.86/297.10 849.86/297.10 The following argument positions are usable: 849.86/297.10 Uargs(a__terms) = {1}, Uargs(cons) = {1}, Uargs(recip) = {1}, 849.86/297.10 Uargs(a__sqr) = {1}, Uargs(a__dbl) = {1}, Uargs(a__add) = {1, 2}, 849.86/297.10 Uargs(a__first) = {1, 2} 849.86/297.10 849.86/297.10 TcT has computed the following matrix interpretation satisfying 849.86/297.10 not(EDA) and not(IDA(1)). 849.86/297.10 849.86/297.10 [a__terms](x1) = [1] x1 + [3] 849.86/297.10 849.86/297.10 [cons](x1, x2) = [1] x1 + [0] 849.86/297.10 849.86/297.10 [recip](x1) = [1] x1 + [0] 849.86/297.10 849.86/297.10 [a__sqr](x1) = [1] x1 + [0] 849.86/297.10 849.86/297.10 [mark](x1) = [1] x1 + [1] 849.86/297.10 849.86/297.10 [terms](x1) = [1] x1 + [0] 849.86/297.10 849.86/297.10 [s](x1) = [3] 849.86/297.10 849.86/297.10 [0] = [3] 849.86/297.10 849.86/297.10 [add](x1, x2) = [1] x1 + [1] x2 + [7] 849.86/297.10 849.86/297.10 [sqr](x1) = [1] x1 + [0] 849.86/297.10 849.86/297.10 [dbl](x1) = [1] x1 + [0] 849.86/297.10 849.86/297.10 [a__dbl](x1) = [1] x1 + [7] 849.86/297.10 849.86/297.10 [a__add](x1, x2) = [1] x1 + [1] x2 + [2] 849.86/297.10 849.86/297.10 [a__first](x1, x2) = [1] x1 + [1] x2 + [6] 849.86/297.10 849.86/297.10 [nil] = [2] 849.86/297.10 849.86/297.10 [first](x1, x2) = [1] x1 + [1] x2 + [0] 849.86/297.10 849.86/297.10 The order satisfies the following ordering constraints: 849.86/297.10 849.86/297.10 [a__terms(N)] = [1] N + [3] 849.86/297.10 > [1] N + [1] 849.86/297.10 = [cons(recip(a__sqr(mark(N))), terms(s(N)))] 849.86/297.10 849.86/297.10 [a__terms(X)] = [1] X + [3] 849.86/297.10 > [1] X + [0] 849.86/297.10 = [terms(X)] 849.86/297.10 849.86/297.10 [a__sqr(X)] = [1] X + [0] 849.86/297.10 >= [1] X + [0] 849.86/297.10 = [sqr(X)] 849.86/297.10 849.86/297.10 [a__sqr(s(X))] = [3] 849.86/297.10 >= [3] 849.86/297.10 = [s(add(sqr(X), dbl(X)))] 849.86/297.10 849.86/297.10 [a__sqr(0())] = [3] 849.86/297.10 >= [3] 849.86/297.10 = [0()] 849.86/297.10 849.86/297.10 [mark(cons(X1, X2))] = [1] X1 + [1] 849.86/297.10 >= [1] X1 + [1] 849.86/297.10 = [cons(mark(X1), X2)] 849.86/297.10 849.86/297.10 [mark(recip(X))] = [1] X + [1] 849.86/297.10 >= [1] X + [1] 849.86/297.10 = [recip(mark(X))] 849.86/297.10 849.86/297.10 [mark(terms(X))] = [1] X + [1] 849.86/297.10 ? [1] X + [4] 849.86/297.10 = [a__terms(mark(X))] 849.86/297.10 849.86/297.10 [mark(s(X))] = [4] 849.86/297.10 > [3] 849.86/297.10 = [s(X)] 849.86/297.10 849.86/297.10 [mark(0())] = [4] 849.86/297.10 > [3] 849.86/297.10 = [0()] 849.86/297.10 849.86/297.10 [mark(add(X1, X2))] = [1] X1 + [1] X2 + [8] 849.86/297.10 > [1] X1 + [1] X2 + [4] 849.86/297.10 = [a__add(mark(X1), mark(X2))] 849.86/297.10 849.86/297.10 [mark(sqr(X))] = [1] X + [1] 849.86/297.10 >= [1] X + [1] 849.86/297.10 = [a__sqr(mark(X))] 849.86/297.10 849.86/297.10 [mark(dbl(X))] = [1] X + [1] 849.86/297.10 ? [1] X + [8] 849.86/297.10 = [a__dbl(mark(X))] 849.86/297.10 849.86/297.10 [mark(nil())] = [3] 849.86/297.10 > [2] 849.86/297.10 = [nil()] 849.86/297.10 849.86/297.10 [mark(first(X1, X2))] = [1] X1 + [1] X2 + [1] 849.86/297.10 ? [1] X1 + [1] X2 + [8] 849.86/297.10 = [a__first(mark(X1), mark(X2))] 849.86/297.10 849.86/297.10 [a__dbl(X)] = [1] X + [7] 849.86/297.10 > [1] X + [0] 849.86/297.10 = [dbl(X)] 849.86/297.10 849.86/297.10 [a__dbl(s(X))] = [10] 849.86/297.10 > [3] 849.86/297.10 = [s(s(dbl(X)))] 849.86/297.10 849.86/297.10 [a__dbl(0())] = [10] 849.86/297.10 > [3] 849.86/297.10 = [0()] 849.86/297.10 849.86/297.10 [a__add(X1, X2)] = [1] X1 + [1] X2 + [2] 849.86/297.10 ? [1] X1 + [1] X2 + [7] 849.86/297.10 = [add(X1, X2)] 849.86/297.10 849.86/297.10 [a__add(s(X), Y)] = [1] Y + [5] 849.86/297.10 > [3] 849.86/297.10 = [s(add(X, Y))] 849.86/297.10 849.86/297.10 [a__add(0(), X)] = [1] X + [5] 849.86/297.10 > [1] X + [1] 849.86/297.10 = [mark(X)] 849.86/297.10 849.86/297.10 [a__first(X1, X2)] = [1] X1 + [1] X2 + [6] 849.86/297.10 > [1] X1 + [1] X2 + [0] 849.86/297.10 = [first(X1, X2)] 849.86/297.10 849.86/297.10 [a__first(s(X), cons(Y, Z))] = [1] Y + [9] 849.86/297.10 > [1] Y + [1] 849.86/297.10 = [cons(mark(Y), first(X, Z))] 849.86/297.10 849.86/297.10 [a__first(0(), X)] = [1] X + [9] 849.86/297.10 > [2] 849.86/297.10 = [nil()] 849.86/297.10 849.86/297.10 849.86/297.10 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 849.86/297.10 849.86/297.10 We are left with following problem, upon which TcT provides the 849.86/297.10 certificate YES(O(1),O(n^2)). 849.86/297.10 849.86/297.10 Strict Trs: 849.86/297.10 { mark(cons(X1, X2)) -> cons(mark(X1), X2) 849.86/297.10 , mark(recip(X)) -> recip(mark(X)) 849.86/297.10 , mark(terms(X)) -> a__terms(mark(X)) 849.86/297.10 , mark(sqr(X)) -> a__sqr(mark(X)) 849.86/297.10 , mark(dbl(X)) -> a__dbl(mark(X)) 849.86/297.10 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 849.86/297.10 , a__add(X1, X2) -> add(X1, X2) } 849.86/297.10 Weak Trs: 849.86/297.10 { a__terms(N) -> cons(recip(a__sqr(mark(N))), terms(s(N))) 849.86/297.10 , a__terms(X) -> terms(X) 849.86/297.10 , a__sqr(X) -> sqr(X) 849.86/297.10 , a__sqr(s(X)) -> s(add(sqr(X), dbl(X))) 849.86/297.10 , a__sqr(0()) -> 0() 849.86/297.10 , mark(s(X)) -> s(X) 849.86/297.10 , mark(0()) -> 0() 849.86/297.10 , mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) 849.86/297.10 , mark(nil()) -> nil() 849.86/297.10 , a__dbl(X) -> dbl(X) 849.86/297.10 , a__dbl(s(X)) -> s(s(dbl(X))) 849.86/297.10 , a__dbl(0()) -> 0() 849.86/297.10 , a__add(s(X), Y) -> s(add(X, Y)) 849.86/297.10 , a__add(0(), X) -> mark(X) 849.86/297.10 , a__first(X1, X2) -> first(X1, X2) 849.86/297.10 , a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z)) 849.86/297.10 , a__first(0(), X) -> nil() } 849.86/297.10 Obligation: 849.86/297.10 innermost runtime complexity 849.86/297.10 Answer: 849.86/297.10 YES(O(1),O(n^2)) 849.86/297.10 849.86/297.10 We use the processor 'matrix interpretation of dimension 2' to 849.86/297.10 orient following rules strictly. 849.86/297.10 849.86/297.10 Trs: 849.86/297.10 { mark(cons(X1, X2)) -> cons(mark(X1), X2) 849.86/297.10 , mark(terms(X)) -> a__terms(mark(X)) 849.86/297.10 , mark(dbl(X)) -> a__dbl(mark(X)) } 849.86/297.10 849.86/297.10 The induced complexity on above rules (modulo remaining rules) is 849.86/297.10 YES(?,O(n^2)) . These rules are moved into the corresponding weak 849.86/297.10 component(s). 849.86/297.10 849.86/297.10 Sub-proof: 849.86/297.10 ---------- 849.86/297.10 The following argument positions are usable: 849.86/297.10 Uargs(a__terms) = {1}, Uargs(cons) = {1}, Uargs(recip) = {1}, 849.86/297.10 Uargs(a__sqr) = {1}, Uargs(a__dbl) = {1}, Uargs(a__add) = {1, 2}, 849.86/297.10 Uargs(a__first) = {1, 2} 849.86/297.10 849.86/297.10 TcT has computed the following constructor-based matrix 849.86/297.10 interpretation satisfying not(EDA). 849.86/297.10 849.86/297.10 [a__terms](x1) = [1 4] x1 + [0] 849.86/297.10 [0 1] [2] 849.86/297.10 849.86/297.10 [cons](x1, x2) = [1 0] x1 + [0] 849.86/297.10 [0 1] [2] 849.86/297.10 849.86/297.10 [recip](x1) = [1 0] x1 + [0] 849.86/297.10 [0 1] [0] 849.86/297.10 849.86/297.10 [a__sqr](x1) = [1 0] x1 + [0] 849.86/297.10 [0 1] [0] 849.86/297.10 849.86/297.10 [mark](x1) = [1 4] x1 + [0] 849.86/297.10 [0 1] [0] 849.86/297.10 849.86/297.10 [terms](x1) = [1 4] x1 + [0] 849.86/297.10 [0 1] [2] 849.86/297.10 849.86/297.10 [s](x1) = [0] 849.86/297.10 [1] 849.86/297.10 849.86/297.10 [0] = [0] 849.86/297.10 [1] 849.86/297.10 849.86/297.10 [add](x1, x2) = [1 0] x1 + [1 4] x2 + [0] 849.86/297.10 [0 1] [0 1] [1] 849.86/297.10 849.86/297.10 [sqr](x1) = [1 0] x1 + [0] 849.86/297.10 [0 1] [0] 849.86/297.10 849.86/297.10 [dbl](x1) = [1 0] x1 + [0] 849.86/297.10 [0 1] [1] 849.86/297.10 849.86/297.10 [a__dbl](x1) = [1 0] x1 + [0] 849.86/297.10 [0 1] [1] 849.86/297.10 849.86/297.10 [a__add](x1, x2) = [1 0] x1 + [1 4] x2 + [0] 849.86/297.10 [0 1] [0 1] [1] 849.86/297.10 849.86/297.10 [a__first](x1, x2) = [1 0] x1 + [1 4] x2 + [0] 849.86/297.10 [0 1] [0 1] [0] 849.86/297.10 849.86/297.10 [nil] = [0] 849.86/297.10 [0] 849.86/297.10 849.86/297.10 [first](x1, x2) = [1 0] x1 + [1 4] x2 + [0] 849.86/297.10 [0 1] [0 1] [0] 849.86/297.10 849.86/297.10 The order satisfies the following ordering constraints: 849.86/297.10 849.86/297.10 [a__terms(N)] = [1 4] N + [0] 849.86/297.10 [0 1] [2] 849.86/297.10 >= [1 4] N + [0] 849.86/297.10 [0 1] [2] 849.86/297.10 = [cons(recip(a__sqr(mark(N))), terms(s(N)))] 849.86/297.10 849.86/297.10 [a__terms(X)] = [1 4] X + [0] 849.86/297.10 [0 1] [2] 849.86/297.10 >= [1 4] X + [0] 849.86/297.10 [0 1] [2] 849.86/297.10 = [terms(X)] 849.86/297.10 849.86/297.10 [a__sqr(X)] = [1 0] X + [0] 849.86/297.10 [0 1] [0] 849.86/297.10 >= [1 0] X + [0] 849.86/297.10 [0 1] [0] 849.86/297.10 = [sqr(X)] 849.86/297.10 849.86/297.10 [a__sqr(s(X))] = [0] 849.86/297.10 [1] 849.86/297.10 >= [0] 849.86/297.10 [1] 849.86/297.10 = [s(add(sqr(X), dbl(X)))] 849.86/297.10 849.86/297.10 [a__sqr(0())] = [0] 849.86/297.10 [1] 849.86/297.10 >= [0] 849.86/297.10 [1] 849.86/297.10 = [0()] 849.86/297.10 849.86/297.10 [mark(cons(X1, X2))] = [1 4] X1 + [8] 849.86/297.10 [0 1] [2] 849.86/297.10 > [1 4] X1 + [0] 849.86/297.10 [0 1] [2] 849.86/297.10 = [cons(mark(X1), X2)] 849.86/297.10 849.86/297.10 [mark(recip(X))] = [1 4] X + [0] 849.86/297.10 [0 1] [0] 849.86/297.10 >= [1 4] X + [0] 849.86/297.10 [0 1] [0] 849.86/297.10 = [recip(mark(X))] 849.86/297.10 849.86/297.10 [mark(terms(X))] = [1 8] X + [8] 849.86/297.10 [0 1] [2] 849.86/297.10 > [1 8] X + [0] 849.86/297.10 [0 1] [2] 849.86/297.10 = [a__terms(mark(X))] 849.86/297.10 849.86/297.10 [mark(s(X))] = [4] 849.86/297.10 [1] 849.86/297.10 > [0] 849.86/297.10 [1] 849.86/297.10 = [s(X)] 849.86/297.10 849.86/297.11 [mark(0())] = [4] 849.86/297.11 [1] 849.86/297.11 > [0] 849.86/297.11 [1] 849.86/297.11 = [0()] 849.86/297.11 849.86/297.11 [mark(add(X1, X2))] = [1 4] X1 + [1 8] X2 + [4] 849.86/297.11 [0 1] [0 1] [1] 849.86/297.11 > [1 4] X1 + [1 8] X2 + [0] 849.86/297.11 [0 1] [0 1] [1] 849.86/297.11 = [a__add(mark(X1), mark(X2))] 849.86/297.11 849.86/297.11 [mark(sqr(X))] = [1 4] X + [0] 849.86/297.11 [0 1] [0] 849.86/297.11 >= [1 4] X + [0] 849.86/297.11 [0 1] [0] 849.86/297.11 = [a__sqr(mark(X))] 849.86/297.11 849.86/297.11 [mark(dbl(X))] = [1 4] X + [4] 849.86/297.11 [0 1] [1] 849.86/297.11 > [1 4] X + [0] 849.86/297.11 [0 1] [1] 849.86/297.11 = [a__dbl(mark(X))] 849.86/297.11 849.86/297.11 [mark(nil())] = [0] 849.86/297.11 [0] 849.86/297.11 >= [0] 849.86/297.11 [0] 849.86/297.11 = [nil()] 849.86/297.11 849.86/297.11 [mark(first(X1, X2))] = [1 4] X1 + [1 8] X2 + [0] 849.86/297.11 [0 1] [0 1] [0] 849.86/297.11 >= [1 4] X1 + [1 8] X2 + [0] 849.86/297.11 [0 1] [0 1] [0] 849.86/297.11 = [a__first(mark(X1), mark(X2))] 849.86/297.11 849.86/297.11 [a__dbl(X)] = [1 0] X + [0] 849.86/297.11 [0 1] [1] 849.86/297.11 >= [1 0] X + [0] 849.86/297.11 [0 1] [1] 849.86/297.11 = [dbl(X)] 849.86/297.11 849.86/297.11 [a__dbl(s(X))] = [0] 849.86/297.11 [2] 849.86/297.11 >= [0] 849.86/297.11 [1] 849.86/297.11 = [s(s(dbl(X)))] 849.86/297.11 849.86/297.11 [a__dbl(0())] = [0] 849.86/297.11 [2] 849.86/297.11 >= [0] 849.86/297.11 [1] 849.86/297.11 = [0()] 849.86/297.11 849.86/297.11 [a__add(X1, X2)] = [1 0] X1 + [1 4] X2 + [0] 849.86/297.11 [0 1] [0 1] [1] 849.86/297.11 >= [1 0] X1 + [1 4] X2 + [0] 849.86/297.11 [0 1] [0 1] [1] 849.86/297.11 = [add(X1, X2)] 849.86/297.11 849.86/297.11 [a__add(s(X), Y)] = [1 4] Y + [0] 849.86/297.11 [0 1] [2] 849.86/297.11 >= [0] 849.86/297.11 [1] 849.86/297.11 = [s(add(X, Y))] 849.86/297.11 849.86/297.11 [a__add(0(), X)] = [1 4] X + [0] 849.86/297.11 [0 1] [2] 849.86/297.11 >= [1 4] X + [0] 849.86/297.11 [0 1] [0] 849.86/297.11 = [mark(X)] 849.86/297.11 849.86/297.11 [a__first(X1, X2)] = [1 0] X1 + [1 4] X2 + [0] 849.86/297.11 [0 1] [0 1] [0] 849.86/297.11 >= [1 0] X1 + [1 4] X2 + [0] 849.86/297.11 [0 1] [0 1] [0] 849.86/297.11 = [first(X1, X2)] 849.86/297.11 849.86/297.11 [a__first(s(X), cons(Y, Z))] = [1 4] Y + [8] 849.86/297.11 [0 1] [3] 849.86/297.11 > [1 4] Y + [0] 849.86/297.11 [0 1] [2] 849.86/297.11 = [cons(mark(Y), first(X, Z))] 849.86/297.11 849.86/297.11 [a__first(0(), X)] = [1 4] X + [0] 849.86/297.11 [0 1] [1] 849.86/297.11 >= [0] 849.86/297.11 [0] 849.86/297.11 = [nil()] 849.86/297.11 849.86/297.11 849.86/297.11 We return to the main proof. 849.86/297.11 849.86/297.11 We are left with following problem, upon which TcT provides the 849.86/297.11 certificate YES(O(1),O(n^2)). 849.86/297.11 849.86/297.11 Strict Trs: 849.86/297.11 { mark(recip(X)) -> recip(mark(X)) 849.86/297.11 , mark(sqr(X)) -> a__sqr(mark(X)) 849.86/297.11 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 849.86/297.11 , a__add(X1, X2) -> add(X1, X2) } 849.86/297.11 Weak Trs: 849.86/297.11 { a__terms(N) -> cons(recip(a__sqr(mark(N))), terms(s(N))) 849.86/297.11 , a__terms(X) -> terms(X) 849.86/297.11 , a__sqr(X) -> sqr(X) 849.86/297.11 , a__sqr(s(X)) -> s(add(sqr(X), dbl(X))) 849.86/297.11 , a__sqr(0()) -> 0() 849.86/297.11 , mark(cons(X1, X2)) -> cons(mark(X1), X2) 849.86/297.11 , mark(terms(X)) -> a__terms(mark(X)) 849.86/297.11 , mark(s(X)) -> s(X) 849.86/297.11 , mark(0()) -> 0() 849.86/297.11 , mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) 849.86/297.11 , mark(dbl(X)) -> a__dbl(mark(X)) 849.86/297.11 , mark(nil()) -> nil() 849.86/297.11 , a__dbl(X) -> dbl(X) 849.86/297.11 , a__dbl(s(X)) -> s(s(dbl(X))) 849.86/297.11 , a__dbl(0()) -> 0() 849.86/297.11 , a__add(s(X), Y) -> s(add(X, Y)) 849.86/297.11 , a__add(0(), X) -> mark(X) 849.86/297.11 , a__first(X1, X2) -> first(X1, X2) 849.86/297.11 , a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z)) 849.86/297.11 , a__first(0(), X) -> nil() } 849.86/297.11 Obligation: 849.86/297.11 innermost runtime complexity 849.86/297.11 Answer: 849.86/297.11 YES(O(1),O(n^2)) 849.86/297.11 849.86/297.11 We use the processor 'matrix interpretation of dimension 2' to 849.86/297.11 orient following rules strictly. 849.86/297.11 849.86/297.11 Trs: 849.86/297.11 { mark(recip(X)) -> recip(mark(X)) 849.86/297.11 , mark(sqr(X)) -> a__sqr(mark(X)) } 849.86/297.11 849.86/297.11 The induced complexity on above rules (modulo remaining rules) is 849.86/297.11 YES(?,O(n^2)) . These rules are moved into the corresponding weak 849.86/297.11 component(s). 849.86/297.11 849.86/297.11 Sub-proof: 849.86/297.11 ---------- 849.86/297.11 The following argument positions are usable: 849.86/297.11 Uargs(a__terms) = {1}, Uargs(cons) = {1}, Uargs(recip) = {1}, 849.86/297.11 Uargs(a__sqr) = {1}, Uargs(a__dbl) = {1}, Uargs(a__add) = {1, 2}, 849.86/297.11 Uargs(a__first) = {1, 2} 849.86/297.11 849.86/297.11 TcT has computed the following constructor-based matrix 849.86/297.11 interpretation satisfying not(EDA). 849.86/297.11 849.86/297.11 [a__terms](x1) = [1 1] x1 + [7] 849.86/297.11 [0 1] [3] 849.86/297.11 849.86/297.11 [cons](x1, x2) = [1 0] x1 + [0] 849.86/297.11 [0 1] [0] 849.86/297.11 849.86/297.11 [recip](x1) = [1 0] x1 + [7] 849.86/297.11 [0 1] [1] 849.86/297.11 849.86/297.11 [a__sqr](x1) = [1 0] x1 + [0] 849.86/297.11 [0 1] [2] 849.86/297.11 849.86/297.11 [mark](x1) = [1 1] x1 + [0] 849.86/297.11 [0 1] [0] 849.86/297.11 849.86/297.11 [terms](x1) = [1 1] x1 + [5] 849.86/297.11 [0 1] [3] 849.86/297.11 849.86/297.11 [s](x1) = [0] 849.86/297.11 [4] 849.86/297.11 849.86/297.11 [0] = [0] 849.86/297.11 [0] 849.86/297.11 849.86/297.11 [add](x1, x2) = [1 0] x1 + [1 7] x2 + [0] 849.86/297.11 [0 1] [0 1] [0] 849.86/297.11 849.86/297.11 [sqr](x1) = [1 0] x1 + [0] 849.86/297.11 [0 1] [2] 849.86/297.11 849.86/297.11 [dbl](x1) = [1 2] x1 + [0] 849.86/297.11 [0 1] [0] 849.86/297.11 849.86/297.11 [a__dbl](x1) = [1 2] x1 + [0] 849.86/297.11 [0 1] [0] 849.86/297.11 849.86/297.11 [a__add](x1, x2) = [1 0] x1 + [1 7] x2 + [0] 849.86/297.11 [0 1] [0 1] [0] 849.86/297.11 849.86/297.11 [a__first](x1, x2) = [1 0] x1 + [1 1] x2 + [0] 849.86/297.11 [0 1] [0 1] [0] 849.86/297.11 849.86/297.11 [nil] = [0] 849.86/297.11 [0] 849.86/297.11 849.86/297.11 [first](x1, x2) = [1 0] x1 + [1 1] x2 + [0] 849.86/297.11 [0 1] [0 1] [0] 849.86/297.11 849.86/297.11 The order satisfies the following ordering constraints: 849.86/297.11 849.86/297.11 [a__terms(N)] = [1 1] N + [7] 849.86/297.11 [0 1] [3] 849.86/297.11 >= [1 1] N + [7] 849.86/297.11 [0 1] [3] 849.86/297.11 = [cons(recip(a__sqr(mark(N))), terms(s(N)))] 849.86/297.11 849.86/297.11 [a__terms(X)] = [1 1] X + [7] 849.86/297.11 [0 1] [3] 849.86/297.11 > [1 1] X + [5] 849.86/297.11 [0 1] [3] 849.86/297.11 = [terms(X)] 849.86/297.11 849.86/297.11 [a__sqr(X)] = [1 0] X + [0] 849.86/297.11 [0 1] [2] 849.86/297.11 >= [1 0] X + [0] 849.86/297.11 [0 1] [2] 849.86/297.11 = [sqr(X)] 849.86/297.11 849.86/297.11 [a__sqr(s(X))] = [0] 849.86/297.11 [6] 849.86/297.11 >= [0] 849.86/297.11 [4] 849.86/297.11 = [s(add(sqr(X), dbl(X)))] 849.86/297.11 849.86/297.11 [a__sqr(0())] = [0] 849.86/297.11 [2] 849.86/297.11 >= [0] 849.86/297.11 [0] 849.86/297.11 = [0()] 849.86/297.11 849.86/297.11 [mark(cons(X1, X2))] = [1 1] X1 + [0] 849.86/297.11 [0 1] [0] 849.86/297.11 >= [1 1] X1 + [0] 849.86/297.11 [0 1] [0] 849.86/297.11 = [cons(mark(X1), X2)] 849.86/297.11 849.86/297.11 [mark(recip(X))] = [1 1] X + [8] 849.86/297.11 [0 1] [1] 849.86/297.11 > [1 1] X + [7] 849.86/297.11 [0 1] [1] 849.86/297.11 = [recip(mark(X))] 849.86/297.11 849.86/297.11 [mark(terms(X))] = [1 2] X + [8] 849.86/297.11 [0 1] [3] 849.86/297.11 > [1 2] X + [7] 849.86/297.11 [0 1] [3] 849.86/297.11 = [a__terms(mark(X))] 849.86/297.11 849.86/297.11 [mark(s(X))] = [4] 849.86/297.11 [4] 849.86/297.11 > [0] 849.86/297.11 [4] 849.86/297.11 = [s(X)] 849.86/297.11 849.86/297.11 [mark(0())] = [0] 849.86/297.11 [0] 849.86/297.11 >= [0] 849.86/297.11 [0] 849.86/297.11 = [0()] 849.86/297.11 849.86/297.11 [mark(add(X1, X2))] = [1 1] X1 + [1 8] X2 + [0] 849.86/297.11 [0 1] [0 1] [0] 849.86/297.11 >= [1 1] X1 + [1 8] X2 + [0] 849.86/297.11 [0 1] [0 1] [0] 849.86/297.11 = [a__add(mark(X1), mark(X2))] 849.86/297.11 849.86/297.11 [mark(sqr(X))] = [1 1] X + [2] 849.86/297.11 [0 1] [2] 849.86/297.11 > [1 1] X + [0] 849.86/297.11 [0 1] [2] 849.86/297.11 = [a__sqr(mark(X))] 849.86/297.11 849.86/297.11 [mark(dbl(X))] = [1 3] X + [0] 849.86/297.11 [0 1] [0] 849.86/297.11 >= [1 3] X + [0] 849.86/297.11 [0 1] [0] 849.86/297.11 = [a__dbl(mark(X))] 849.86/297.11 849.86/297.11 [mark(nil())] = [0] 849.86/297.11 [0] 849.86/297.11 >= [0] 849.86/297.11 [0] 849.86/297.11 = [nil()] 849.86/297.11 849.86/297.11 [mark(first(X1, X2))] = [1 1] X1 + [1 2] X2 + [0] 849.86/297.11 [0 1] [0 1] [0] 849.86/297.11 >= [1 1] X1 + [1 2] X2 + [0] 849.86/297.11 [0 1] [0 1] [0] 849.86/297.11 = [a__first(mark(X1), mark(X2))] 849.86/297.11 849.86/297.11 [a__dbl(X)] = [1 2] X + [0] 849.86/297.11 [0 1] [0] 849.86/297.11 >= [1 2] X + [0] 849.86/297.11 [0 1] [0] 849.86/297.11 = [dbl(X)] 849.86/297.11 849.86/297.11 [a__dbl(s(X))] = [8] 849.86/297.11 [4] 849.86/297.11 > [0] 849.86/297.11 [4] 849.86/297.11 = [s(s(dbl(X)))] 849.86/297.11 849.86/297.11 [a__dbl(0())] = [0] 849.86/297.11 [0] 849.86/297.11 >= [0] 849.86/297.11 [0] 849.86/297.11 = [0()] 849.86/297.11 849.86/297.11 [a__add(X1, X2)] = [1 0] X1 + [1 7] X2 + [0] 849.86/297.11 [0 1] [0 1] [0] 849.86/297.11 >= [1 0] X1 + [1 7] X2 + [0] 849.86/297.11 [0 1] [0 1] [0] 849.86/297.11 = [add(X1, X2)] 849.86/297.11 849.86/297.11 [a__add(s(X), Y)] = [1 7] Y + [0] 849.86/297.11 [0 1] [4] 849.86/297.11 >= [0] 849.86/297.11 [4] 849.86/297.11 = [s(add(X, Y))] 849.86/297.11 849.86/297.11 [a__add(0(), X)] = [1 7] X + [0] 849.86/297.11 [0 1] [0] 849.86/297.11 >= [1 1] X + [0] 849.86/297.11 [0 1] [0] 849.86/297.11 = [mark(X)] 849.86/297.11 849.86/297.11 [a__first(X1, X2)] = [1 0] X1 + [1 1] X2 + [0] 849.86/297.11 [0 1] [0 1] [0] 849.86/297.11 >= [1 0] X1 + [1 1] X2 + [0] 849.86/297.11 [0 1] [0 1] [0] 849.86/297.11 = [first(X1, X2)] 849.86/297.11 849.86/297.11 [a__first(s(X), cons(Y, Z))] = [1 1] Y + [0] 849.86/297.11 [0 1] [4] 849.86/297.11 >= [1 1] Y + [0] 849.86/297.11 [0 1] [0] 849.86/297.11 = [cons(mark(Y), first(X, Z))] 849.86/297.11 849.86/297.11 [a__first(0(), X)] = [1 1] X + [0] 849.86/297.11 [0 1] [0] 849.86/297.11 >= [0] 849.86/297.11 [0] 849.86/297.11 = [nil()] 849.86/297.11 849.86/297.11 849.86/297.11 We return to the main proof. 849.86/297.11 849.86/297.11 We are left with following problem, upon which TcT provides the 849.86/297.11 certificate YES(O(1),O(n^2)). 849.86/297.11 849.86/297.11 Strict Trs: 849.86/297.11 { mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 849.86/297.11 , a__add(X1, X2) -> add(X1, X2) } 849.86/297.11 Weak Trs: 849.86/297.11 { a__terms(N) -> cons(recip(a__sqr(mark(N))), terms(s(N))) 849.86/297.11 , a__terms(X) -> terms(X) 849.86/297.11 , a__sqr(X) -> sqr(X) 849.86/297.11 , a__sqr(s(X)) -> s(add(sqr(X), dbl(X))) 849.86/297.11 , a__sqr(0()) -> 0() 849.86/297.11 , mark(cons(X1, X2)) -> cons(mark(X1), X2) 849.86/297.11 , mark(recip(X)) -> recip(mark(X)) 849.86/297.11 , mark(terms(X)) -> a__terms(mark(X)) 849.86/297.11 , mark(s(X)) -> s(X) 849.86/297.11 , mark(0()) -> 0() 849.86/297.11 , mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) 849.86/297.11 , mark(sqr(X)) -> a__sqr(mark(X)) 849.86/297.11 , mark(dbl(X)) -> a__dbl(mark(X)) 849.86/297.11 , mark(nil()) -> nil() 849.86/297.11 , a__dbl(X) -> dbl(X) 849.86/297.11 , a__dbl(s(X)) -> s(s(dbl(X))) 849.86/297.11 , a__dbl(0()) -> 0() 849.86/297.11 , a__add(s(X), Y) -> s(add(X, Y)) 849.86/297.11 , a__add(0(), X) -> mark(X) 849.86/297.11 , a__first(X1, X2) -> first(X1, X2) 849.86/297.11 , a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z)) 849.86/297.11 , a__first(0(), X) -> nil() } 849.86/297.11 Obligation: 849.86/297.11 innermost runtime complexity 849.86/297.11 Answer: 849.86/297.11 YES(O(1),O(n^2)) 849.86/297.11 849.86/297.11 We use the processor 'matrix interpretation of dimension 2' to 849.86/297.11 orient following rules strictly. 849.86/297.11 849.86/297.11 Trs: 849.86/297.11 { mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 849.86/297.11 , a__add(X1, X2) -> add(X1, X2) } 849.86/297.11 849.86/297.11 The induced complexity on above rules (modulo remaining rules) is 849.86/297.11 YES(?,O(n^2)) . These rules are moved into the corresponding weak 849.86/297.11 component(s). 849.86/297.11 849.86/297.11 Sub-proof: 849.86/297.11 ---------- 849.86/297.11 The following argument positions are usable: 849.86/297.11 Uargs(a__terms) = {1}, Uargs(cons) = {1}, Uargs(recip) = {1}, 849.86/297.11 Uargs(a__sqr) = {1}, Uargs(a__dbl) = {1}, Uargs(a__add) = {1, 2}, 849.86/297.11 Uargs(a__first) = {1, 2} 849.86/297.11 849.86/297.11 TcT has computed the following constructor-based matrix 849.86/297.11 interpretation satisfying not(EDA). 849.86/297.11 849.86/297.11 [a__terms](x1) = [1 2] x1 + [0] 849.86/297.11 [0 1] [4] 849.86/297.11 849.86/297.11 [cons](x1, x2) = [1 0] x1 + [0] 849.86/297.11 [0 1] [0] 849.86/297.11 849.86/297.11 [recip](x1) = [1 0] x1 + [0] 849.86/297.11 [0 1] [1] 849.86/297.11 849.86/297.11 [a__sqr](x1) = [1 0] x1 + [0] 849.86/297.11 [0 1] [3] 849.86/297.11 849.86/297.11 [mark](x1) = [1 2] x1 + [0] 849.86/297.11 [0 1] [0] 849.86/297.11 849.86/297.11 [terms](x1) = [1 2] x1 + [0] 849.86/297.11 [0 1] [4] 849.86/297.11 849.86/297.11 [s](x1) = [0] 849.86/297.11 [5] 849.86/297.11 849.86/297.11 [0] = [0] 849.86/297.11 [0] 849.86/297.11 849.86/297.11 [add](x1, x2) = [1 0] x1 + [1 2] x2 + [0] 849.86/297.11 [0 1] [0 1] [4] 849.86/297.11 849.86/297.11 [sqr](x1) = [1 0] x1 + [0] 849.86/297.11 [0 1] [3] 849.86/297.11 849.86/297.11 [dbl](x1) = [1 0] x1 + [0] 849.86/297.11 [0 1] [4] 849.86/297.11 849.86/297.11 [a__dbl](x1) = [1 0] x1 + [0] 849.86/297.11 [0 1] [4] 849.86/297.11 849.86/297.11 [a__add](x1, x2) = [1 0] x1 + [1 2] x2 + [4] 849.86/297.11 [0 1] [0 1] [4] 849.86/297.11 849.86/297.11 [a__first](x1, x2) = [1 0] x1 + [1 2] x2 + [0] 849.86/297.11 [0 1] [0 1] [4] 849.86/297.11 849.86/297.11 [nil] = [0] 849.86/297.11 [0] 849.86/297.11 849.86/297.11 [first](x1, x2) = [1 0] x1 + [1 2] x2 + [0] 849.86/297.11 [0 1] [0 1] [4] 849.86/297.11 849.86/297.11 The order satisfies the following ordering constraints: 849.86/297.11 849.86/297.11 [a__terms(N)] = [1 2] N + [0] 849.86/297.11 [0 1] [4] 849.86/297.11 >= [1 2] N + [0] 849.86/297.11 [0 1] [4] 849.86/297.11 = [cons(recip(a__sqr(mark(N))), terms(s(N)))] 849.86/297.11 849.86/297.11 [a__terms(X)] = [1 2] X + [0] 849.86/297.11 [0 1] [4] 849.86/297.11 >= [1 2] X + [0] 849.86/297.11 [0 1] [4] 849.86/297.11 = [terms(X)] 849.86/297.11 849.86/297.11 [a__sqr(X)] = [1 0] X + [0] 849.86/297.11 [0 1] [3] 849.86/297.11 >= [1 0] X + [0] 849.86/297.11 [0 1] [3] 849.86/297.11 = [sqr(X)] 849.86/297.11 849.86/297.11 [a__sqr(s(X))] = [0] 849.86/297.11 [8] 849.86/297.11 >= [0] 849.86/297.11 [5] 849.86/297.11 = [s(add(sqr(X), dbl(X)))] 849.86/297.11 849.86/297.11 [a__sqr(0())] = [0] 849.86/297.11 [3] 849.86/297.11 >= [0] 849.86/297.11 [0] 849.86/297.11 = [0()] 849.86/297.11 849.86/297.11 [mark(cons(X1, X2))] = [1 2] X1 + [0] 849.86/297.11 [0 1] [0] 849.86/297.11 >= [1 2] X1 + [0] 849.86/297.11 [0 1] [0] 849.86/297.11 = [cons(mark(X1), X2)] 849.86/297.11 849.86/297.11 [mark(recip(X))] = [1 2] X + [2] 849.86/297.11 [0 1] [1] 849.86/297.11 > [1 2] X + [0] 849.86/297.11 [0 1] [1] 849.86/297.11 = [recip(mark(X))] 849.86/297.11 849.86/297.11 [mark(terms(X))] = [1 4] X + [8] 849.86/297.11 [0 1] [4] 849.86/297.11 > [1 4] X + [0] 849.86/297.11 [0 1] [4] 849.86/297.11 = [a__terms(mark(X))] 849.86/297.11 849.86/297.11 [mark(s(X))] = [10] 849.86/297.11 [5] 849.86/297.11 > [0] 849.86/297.11 [5] 849.86/297.11 = [s(X)] 849.86/297.11 849.86/297.11 [mark(0())] = [0] 849.86/297.11 [0] 849.86/297.11 >= [0] 849.86/297.11 [0] 849.86/297.11 = [0()] 849.86/297.11 849.86/297.11 [mark(add(X1, X2))] = [1 2] X1 + [1 4] X2 + [8] 849.86/297.11 [0 1] [0 1] [4] 849.86/297.11 > [1 2] X1 + [1 4] X2 + [4] 849.86/297.11 [0 1] [0 1] [4] 849.86/297.11 = [a__add(mark(X1), mark(X2))] 849.86/297.11 849.86/297.11 [mark(sqr(X))] = [1 2] X + [6] 849.86/297.11 [0 1] [3] 849.86/297.11 > [1 2] X + [0] 849.86/297.11 [0 1] [3] 849.86/297.11 = [a__sqr(mark(X))] 849.86/297.11 849.86/297.11 [mark(dbl(X))] = [1 2] X + [8] 849.86/297.11 [0 1] [4] 849.86/297.11 > [1 2] X + [0] 849.86/297.11 [0 1] [4] 849.86/297.11 = [a__dbl(mark(X))] 849.86/297.11 849.86/297.11 [mark(nil())] = [0] 849.86/297.11 [0] 849.86/297.11 >= [0] 849.86/297.11 [0] 849.86/297.11 = [nil()] 849.86/297.11 849.86/297.11 [mark(first(X1, X2))] = [1 2] X1 + [1 4] X2 + [8] 849.86/297.11 [0 1] [0 1] [4] 849.86/297.11 > [1 2] X1 + [1 4] X2 + [0] 849.86/297.11 [0 1] [0 1] [4] 849.86/297.11 = [a__first(mark(X1), mark(X2))] 849.86/297.11 849.86/297.11 [a__dbl(X)] = [1 0] X + [0] 849.86/297.11 [0 1] [4] 849.86/297.11 >= [1 0] X + [0] 849.86/297.11 [0 1] [4] 849.86/297.11 = [dbl(X)] 849.86/297.11 849.86/297.11 [a__dbl(s(X))] = [0] 849.86/297.11 [9] 849.86/297.11 >= [0] 849.86/297.11 [5] 849.86/297.11 = [s(s(dbl(X)))] 849.86/297.11 849.86/297.11 [a__dbl(0())] = [0] 849.86/297.11 [4] 849.86/297.11 >= [0] 849.86/297.11 [0] 849.86/297.11 = [0()] 849.86/297.11 849.86/297.11 [a__add(X1, X2)] = [1 0] X1 + [1 2] X2 + [4] 849.86/297.11 [0 1] [0 1] [4] 849.86/297.11 > [1 0] X1 + [1 2] X2 + [0] 849.86/297.11 [0 1] [0 1] [4] 849.86/297.11 = [add(X1, X2)] 849.86/297.11 849.86/297.11 [a__add(s(X), Y)] = [1 2] Y + [4] 849.86/297.11 [0 1] [9] 849.86/297.11 > [0] 849.86/297.11 [5] 849.86/297.11 = [s(add(X, Y))] 849.86/297.11 849.86/297.11 [a__add(0(), X)] = [1 2] X + [4] 849.86/297.11 [0 1] [4] 849.86/297.11 > [1 2] X + [0] 849.86/297.11 [0 1] [0] 849.86/297.11 = [mark(X)] 849.86/297.11 849.86/297.11 [a__first(X1, X2)] = [1 0] X1 + [1 2] X2 + [0] 849.86/297.11 [0 1] [0 1] [4] 849.86/297.11 >= [1 0] X1 + [1 2] X2 + [0] 849.86/297.11 [0 1] [0 1] [4] 849.86/297.11 = [first(X1, X2)] 849.86/297.11 849.86/297.11 [a__first(s(X), cons(Y, Z))] = [1 2] Y + [0] 849.86/297.11 [0 1] [9] 849.86/297.11 >= [1 2] Y + [0] 849.86/297.11 [0 1] [0] 849.86/297.11 = [cons(mark(Y), first(X, Z))] 849.86/297.11 849.86/297.11 [a__first(0(), X)] = [1 2] X + [0] 849.86/297.11 [0 1] [4] 849.86/297.11 >= [0] 849.86/297.11 [0] 849.86/297.11 = [nil()] 849.86/297.11 849.86/297.11 849.86/297.11 We return to the main proof. 849.86/297.11 849.86/297.11 We are left with following problem, upon which TcT provides the 849.86/297.11 certificate YES(O(1),O(1)). 849.86/297.11 849.86/297.11 Weak Trs: 849.86/297.11 { a__terms(N) -> cons(recip(a__sqr(mark(N))), terms(s(N))) 849.86/297.11 , a__terms(X) -> terms(X) 849.86/297.11 , a__sqr(X) -> sqr(X) 849.86/297.11 , a__sqr(s(X)) -> s(add(sqr(X), dbl(X))) 849.86/297.11 , a__sqr(0()) -> 0() 849.86/297.11 , mark(cons(X1, X2)) -> cons(mark(X1), X2) 849.86/297.11 , mark(recip(X)) -> recip(mark(X)) 849.86/297.11 , mark(terms(X)) -> a__terms(mark(X)) 849.86/297.11 , mark(s(X)) -> s(X) 849.86/297.11 , mark(0()) -> 0() 849.86/297.11 , mark(add(X1, X2)) -> a__add(mark(X1), mark(X2)) 849.86/297.11 , mark(sqr(X)) -> a__sqr(mark(X)) 849.86/297.11 , mark(dbl(X)) -> a__dbl(mark(X)) 849.86/297.11 , mark(nil()) -> nil() 849.86/297.11 , mark(first(X1, X2)) -> a__first(mark(X1), mark(X2)) 849.86/297.11 , a__dbl(X) -> dbl(X) 849.86/297.11 , a__dbl(s(X)) -> s(s(dbl(X))) 849.86/297.11 , a__dbl(0()) -> 0() 849.86/297.11 , a__add(X1, X2) -> add(X1, X2) 849.86/297.11 , a__add(s(X), Y) -> s(add(X, Y)) 849.86/297.11 , a__add(0(), X) -> mark(X) 849.86/297.11 , a__first(X1, X2) -> first(X1, X2) 849.86/297.11 , a__first(s(X), cons(Y, Z)) -> cons(mark(Y), first(X, Z)) 849.86/297.11 , a__first(0(), X) -> nil() } 849.86/297.11 Obligation: 849.86/297.11 innermost runtime complexity 849.86/297.11 Answer: 849.86/297.11 YES(O(1),O(1)) 849.86/297.11 849.86/297.11 Empty rules are trivially bounded 849.86/297.11 849.86/297.11 Hurray, we answered YES(O(1),O(n^2)) 849.98/297.24 EOF