MAYBE 292.89/161.47 MAYBE 292.89/161.47 292.89/161.47 We are left with following problem, upon which TcT provides the 292.89/161.47 certificate MAYBE. 292.89/161.47 292.89/161.47 Strict Trs: 292.89/161.47 { terms(N) -> cons(recip(sqr(N))) 292.89/161.47 , sqr(0()) -> 0() 292.89/161.47 , sqr(s(X)) -> s(add(sqr(X), dbl(X))) 292.89/161.47 , add(0(), X) -> X 292.89/161.47 , add(s(X), Y) -> s(add(X, Y)) 292.89/161.47 , dbl(0()) -> 0() 292.89/161.47 , dbl(s(X)) -> s(s(dbl(X))) 292.89/161.47 , first(0(), X) -> nil() 292.89/161.47 , first(s(X), cons(Y)) -> cons(Y) } 292.89/161.47 Obligation: 292.89/161.47 runtime complexity 292.89/161.47 Answer: 292.89/161.47 MAYBE 292.89/161.47 292.89/161.47 None of the processors succeeded. 292.89/161.47 292.89/161.47 Details of failed attempt(s): 292.89/161.47 ----------------------------- 292.89/161.47 1) 'Best' failed due to the following reason: 292.89/161.47 292.89/161.47 None of the processors succeeded. 292.89/161.47 292.89/161.47 Details of failed attempt(s): 292.89/161.47 ----------------------------- 292.89/161.47 1) 'With Problem ... (timeout of 148 seconds) (timeout of 297 292.89/161.47 seconds)' failed due to the following reason: 292.89/161.47 292.89/161.47 The weightgap principle applies (using the following nonconstant 292.89/161.47 growth matrix-interpretation) 292.89/161.47 292.89/161.47 The following argument positions are usable: 292.89/161.47 Uargs(cons) = {1}, Uargs(recip) = {1}, Uargs(s) = {1}, 292.89/161.47 Uargs(add) = {1, 2} 292.89/161.47 292.89/161.47 TcT has computed the following matrix interpretation satisfying 292.89/161.47 not(EDA) and not(IDA(1)). 292.89/161.47 292.89/161.47 [terms](x1) = [1] x1 + [7] 292.89/161.47 292.89/161.47 [cons](x1) = [1] x1 + [0] 292.89/161.47 292.89/161.47 [recip](x1) = [1] x1 + [1] 292.89/161.47 292.89/161.47 [sqr](x1) = [0] 292.89/161.47 292.89/161.47 [0] = [0] 292.89/161.47 292.89/161.47 [s](x1) = [1] x1 + [0] 292.89/161.47 292.89/161.47 [add](x1, x2) = [1] x1 + [1] x2 + [0] 292.89/161.47 292.89/161.47 [dbl](x1) = [0] 292.89/161.47 292.89/161.47 [first](x1, x2) = [1] x2 + [0] 292.89/161.47 292.89/161.47 [nil] = [7] 292.89/161.47 292.89/161.47 The order satisfies the following ordering constraints: 292.89/161.47 292.89/161.47 [terms(N)] = [1] N + [7] 292.89/161.47 > [1] 292.89/161.47 = [cons(recip(sqr(N)))] 292.89/161.47 292.89/161.47 [sqr(0())] = [0] 292.89/161.47 >= [0] 292.89/161.47 = [0()] 292.89/161.47 292.89/161.47 [sqr(s(X))] = [0] 292.89/161.47 >= [0] 292.89/161.47 = [s(add(sqr(X), dbl(X)))] 292.89/161.47 292.89/161.47 [add(0(), X)] = [1] X + [0] 292.89/161.47 >= [1] X + [0] 292.89/161.47 = [X] 292.89/161.47 292.89/161.47 [add(s(X), Y)] = [1] X + [1] Y + [0] 292.89/161.47 >= [1] X + [1] Y + [0] 292.89/161.47 = [s(add(X, Y))] 292.89/161.47 292.89/161.47 [dbl(0())] = [0] 292.89/161.47 >= [0] 292.89/161.47 = [0()] 292.89/161.47 292.89/161.47 [dbl(s(X))] = [0] 292.89/161.47 >= [0] 292.89/161.47 = [s(s(dbl(X)))] 292.89/161.47 292.89/161.47 [first(0(), X)] = [1] X + [0] 292.89/161.47 ? [7] 292.89/161.47 = [nil()] 292.89/161.47 292.89/161.47 [first(s(X), cons(Y))] = [1] Y + [0] 292.89/161.47 >= [1] Y + [0] 292.89/161.47 = [cons(Y)] 292.89/161.47 292.89/161.47 292.89/161.47 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 292.89/161.47 292.89/161.47 We are left with following problem, upon which TcT provides the 292.89/161.47 certificate MAYBE. 292.89/161.47 292.89/161.47 Strict Trs: 292.89/161.47 { sqr(0()) -> 0() 292.89/161.47 , sqr(s(X)) -> s(add(sqr(X), dbl(X))) 292.89/161.47 , add(0(), X) -> X 292.89/161.47 , add(s(X), Y) -> s(add(X, Y)) 292.89/161.47 , dbl(0()) -> 0() 292.89/161.47 , dbl(s(X)) -> s(s(dbl(X))) 292.89/161.47 , first(0(), X) -> nil() 292.89/161.47 , first(s(X), cons(Y)) -> cons(Y) } 292.89/161.47 Weak Trs: { terms(N) -> cons(recip(sqr(N))) } 292.89/161.47 Obligation: 292.89/161.47 runtime complexity 292.89/161.47 Answer: 292.89/161.47 MAYBE 292.89/161.47 292.89/161.47 The weightgap principle applies (using the following nonconstant 292.89/161.47 growth matrix-interpretation) 292.89/161.47 292.89/161.47 The following argument positions are usable: 292.89/161.47 Uargs(cons) = {1}, Uargs(recip) = {1}, Uargs(s) = {1}, 292.89/161.47 Uargs(add) = {1, 2} 292.89/161.47 292.89/161.47 TcT has computed the following matrix interpretation satisfying 292.89/161.47 not(EDA) and not(IDA(1)). 292.89/161.47 292.89/161.47 [terms](x1) = [1] x1 + [7] 292.89/161.47 292.89/161.47 [cons](x1) = [1] x1 + [0] 292.89/161.47 292.89/161.47 [recip](x1) = [1] x1 + [5] 292.89/161.47 292.89/161.47 [sqr](x1) = [0] 292.89/161.47 292.89/161.47 [0] = [1] 292.89/161.47 292.89/161.47 [s](x1) = [1] x1 + [0] 292.89/161.47 292.89/161.47 [add](x1, x2) = [1] x1 + [1] x2 + [0] 292.89/161.47 292.89/161.47 [dbl](x1) = [0] 292.89/161.47 292.89/161.47 [first](x1, x2) = [1] x2 + [0] 292.89/161.47 292.89/161.47 [nil] = [7] 292.89/161.47 292.89/161.47 The order satisfies the following ordering constraints: 292.89/161.47 292.89/161.47 [terms(N)] = [1] N + [7] 292.89/161.47 > [5] 292.89/161.47 = [cons(recip(sqr(N)))] 292.89/161.47 292.89/161.47 [sqr(0())] = [0] 292.89/161.47 ? [1] 292.89/161.47 = [0()] 292.89/161.47 292.89/161.47 [sqr(s(X))] = [0] 292.89/161.47 >= [0] 292.89/161.47 = [s(add(sqr(X), dbl(X)))] 292.89/161.47 292.89/161.47 [add(0(), X)] = [1] X + [1] 292.89/161.47 > [1] X + [0] 292.89/161.47 = [X] 292.89/161.47 292.89/161.47 [add(s(X), Y)] = [1] X + [1] Y + [0] 292.89/161.47 >= [1] X + [1] Y + [0] 292.89/161.47 = [s(add(X, Y))] 292.89/161.47 292.89/161.47 [dbl(0())] = [0] 292.89/161.47 ? [1] 292.89/161.47 = [0()] 292.89/161.47 292.89/161.47 [dbl(s(X))] = [0] 292.89/161.47 >= [0] 292.89/161.47 = [s(s(dbl(X)))] 292.89/161.47 292.89/161.47 [first(0(), X)] = [1] X + [0] 292.89/161.47 ? [7] 292.89/161.47 = [nil()] 292.89/161.47 292.89/161.47 [first(s(X), cons(Y))] = [1] Y + [0] 292.89/161.47 >= [1] Y + [0] 292.89/161.47 = [cons(Y)] 292.89/161.47 292.89/161.47 292.89/161.47 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 292.89/161.47 292.89/161.47 We are left with following problem, upon which TcT provides the 292.89/161.47 certificate MAYBE. 292.89/161.47 292.89/161.47 Strict Trs: 292.89/161.47 { sqr(0()) -> 0() 292.89/161.47 , sqr(s(X)) -> s(add(sqr(X), dbl(X))) 292.89/161.47 , add(s(X), Y) -> s(add(X, Y)) 292.89/161.47 , dbl(0()) -> 0() 292.89/161.47 , dbl(s(X)) -> s(s(dbl(X))) 292.89/161.47 , first(0(), X) -> nil() 292.89/161.47 , first(s(X), cons(Y)) -> cons(Y) } 292.89/161.47 Weak Trs: 292.89/161.47 { terms(N) -> cons(recip(sqr(N))) 292.89/161.47 , add(0(), X) -> X } 292.89/161.47 Obligation: 292.89/161.47 runtime complexity 292.89/161.47 Answer: 292.89/161.47 MAYBE 292.89/161.47 292.89/161.47 The weightgap principle applies (using the following nonconstant 292.89/161.47 growth matrix-interpretation) 292.89/161.47 292.89/161.47 The following argument positions are usable: 292.89/161.47 Uargs(cons) = {1}, Uargs(recip) = {1}, Uargs(s) = {1}, 292.89/161.47 Uargs(add) = {1, 2} 292.89/161.47 292.89/161.47 TcT has computed the following matrix interpretation satisfying 292.89/161.47 not(EDA) and not(IDA(1)). 292.89/161.47 292.89/161.47 [terms](x1) = [1] x1 + [7] 292.89/161.47 292.89/161.47 [cons](x1) = [1] x1 + [0] 292.89/161.47 292.89/161.47 [recip](x1) = [1] x1 + [7] 292.89/161.47 292.89/161.47 [sqr](x1) = [0] 292.89/161.47 292.89/161.47 [0] = [1] 292.89/161.47 292.89/161.47 [s](x1) = [1] x1 + [0] 292.89/161.47 292.89/161.47 [add](x1, x2) = [1] x1 + [1] x2 + [0] 292.89/161.47 292.89/161.47 [dbl](x1) = [0] 292.89/161.47 292.89/161.47 [first](x1, x2) = [1] x1 + [1] x2 + [0] 292.89/161.47 292.89/161.47 [nil] = [0] 292.89/161.47 292.89/161.47 The order satisfies the following ordering constraints: 292.89/161.47 292.89/161.47 [terms(N)] = [1] N + [7] 292.89/161.47 >= [7] 292.89/161.47 = [cons(recip(sqr(N)))] 292.89/161.47 292.89/161.47 [sqr(0())] = [0] 292.89/161.47 ? [1] 292.89/161.47 = [0()] 292.89/161.47 292.89/161.47 [sqr(s(X))] = [0] 292.89/161.47 >= [0] 292.89/161.47 = [s(add(sqr(X), dbl(X)))] 292.89/161.47 292.89/161.47 [add(0(), X)] = [1] X + [1] 292.89/161.47 > [1] X + [0] 292.89/161.47 = [X] 292.89/161.47 292.89/161.47 [add(s(X), Y)] = [1] X + [1] Y + [0] 292.89/161.47 >= [1] X + [1] Y + [0] 292.89/161.47 = [s(add(X, Y))] 292.89/161.47 292.89/161.47 [dbl(0())] = [0] 292.89/161.47 ? [1] 292.89/161.47 = [0()] 292.89/161.47 292.89/161.47 [dbl(s(X))] = [0] 292.89/161.47 >= [0] 292.89/161.47 = [s(s(dbl(X)))] 292.89/161.47 292.89/161.47 [first(0(), X)] = [1] X + [1] 292.89/161.47 > [0] 292.89/161.47 = [nil()] 292.89/161.47 292.89/161.47 [first(s(X), cons(Y))] = [1] X + [1] Y + [0] 292.89/161.47 >= [1] Y + [0] 292.89/161.47 = [cons(Y)] 292.89/161.47 292.89/161.47 292.89/161.47 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 292.89/161.47 292.89/161.47 We are left with following problem, upon which TcT provides the 292.89/161.47 certificate MAYBE. 292.89/161.47 292.89/161.47 Strict Trs: 292.89/161.47 { sqr(0()) -> 0() 292.89/161.47 , sqr(s(X)) -> s(add(sqr(X), dbl(X))) 292.89/161.47 , add(s(X), Y) -> s(add(X, Y)) 292.89/161.47 , dbl(0()) -> 0() 292.89/161.47 , dbl(s(X)) -> s(s(dbl(X))) 292.89/161.47 , first(s(X), cons(Y)) -> cons(Y) } 292.89/161.47 Weak Trs: 292.89/161.47 { terms(N) -> cons(recip(sqr(N))) 292.89/161.47 , add(0(), X) -> X 292.89/161.47 , first(0(), X) -> nil() } 292.89/161.47 Obligation: 292.89/161.47 runtime complexity 292.89/161.47 Answer: 292.89/161.47 MAYBE 292.89/161.47 292.89/161.47 The weightgap principle applies (using the following nonconstant 292.89/161.47 growth matrix-interpretation) 292.89/161.47 292.89/161.47 The following argument positions are usable: 292.89/161.47 Uargs(cons) = {1}, Uargs(recip) = {1}, Uargs(s) = {1}, 292.89/161.47 Uargs(add) = {1, 2} 292.89/161.47 292.89/161.47 TcT has computed the following matrix interpretation satisfying 292.89/161.47 not(EDA) and not(IDA(1)). 292.89/161.47 292.89/161.47 [terms](x1) = [1] x1 + [7] 292.89/161.47 292.89/161.47 [cons](x1) = [1] x1 + [3] 292.89/161.47 292.89/161.47 [recip](x1) = [1] x1 + [3] 292.89/161.47 292.89/161.47 [sqr](x1) = [1] x1 + [0] 292.89/161.47 292.89/161.47 [0] = [6] 292.89/161.47 292.89/161.47 [s](x1) = [1] x1 + [1] 292.89/161.47 292.89/161.47 [add](x1, x2) = [1] x1 + [1] x2 + [7] 292.89/161.47 292.89/161.47 [dbl](x1) = [7] 292.89/161.47 292.89/161.47 [first](x1, x2) = [1] x1 + [1] x2 + [3] 292.89/161.47 292.89/161.47 [nil] = [1] 292.89/161.47 292.89/161.47 The order satisfies the following ordering constraints: 292.89/161.47 292.89/161.47 [terms(N)] = [1] N + [7] 292.89/161.47 > [1] N + [6] 292.89/161.47 = [cons(recip(sqr(N)))] 292.89/161.47 292.89/161.47 [sqr(0())] = [6] 292.89/161.47 >= [6] 292.89/161.47 = [0()] 292.89/161.47 292.89/161.47 [sqr(s(X))] = [1] X + [1] 292.89/161.47 ? [1] X + [15] 292.89/161.47 = [s(add(sqr(X), dbl(X)))] 292.89/161.47 292.89/161.47 [add(0(), X)] = [1] X + [13] 292.89/161.47 > [1] X + [0] 292.89/161.47 = [X] 292.89/161.47 292.89/161.47 [add(s(X), Y)] = [1] X + [1] Y + [8] 292.89/161.47 >= [1] X + [1] Y + [8] 292.89/161.47 = [s(add(X, Y))] 292.89/161.47 292.89/161.47 [dbl(0())] = [7] 292.89/161.47 > [6] 292.89/161.47 = [0()] 292.89/161.47 292.89/161.47 [dbl(s(X))] = [7] 292.89/161.47 ? [9] 292.89/161.47 = [s(s(dbl(X)))] 292.89/161.47 292.89/161.47 [first(0(), X)] = [1] X + [9] 292.89/161.47 > [1] 292.89/161.47 = [nil()] 292.89/161.47 292.89/161.47 [first(s(X), cons(Y))] = [1] X + [1] Y + [7] 292.89/161.47 > [1] Y + [3] 292.89/161.47 = [cons(Y)] 292.89/161.47 292.89/161.47 292.89/161.47 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 292.89/161.47 292.89/161.47 We are left with following problem, upon which TcT provides the 292.89/161.47 certificate MAYBE. 292.89/161.47 292.89/161.47 Strict Trs: 292.89/161.47 { sqr(0()) -> 0() 292.89/161.47 , sqr(s(X)) -> s(add(sqr(X), dbl(X))) 292.89/161.47 , add(s(X), Y) -> s(add(X, Y)) 292.89/161.47 , dbl(s(X)) -> s(s(dbl(X))) } 292.89/161.47 Weak Trs: 292.89/161.47 { terms(N) -> cons(recip(sqr(N))) 292.89/161.47 , add(0(), X) -> X 292.89/161.47 , dbl(0()) -> 0() 292.89/161.47 , first(0(), X) -> nil() 292.89/161.47 , first(s(X), cons(Y)) -> cons(Y) } 292.89/161.47 Obligation: 292.89/161.47 runtime complexity 292.89/161.47 Answer: 292.89/161.47 MAYBE 292.89/161.47 292.89/161.47 The weightgap principle applies (using the following nonconstant 292.89/161.47 growth matrix-interpretation) 292.89/161.47 292.89/161.47 The following argument positions are usable: 292.89/161.47 Uargs(cons) = {1}, Uargs(recip) = {1}, Uargs(s) = {1}, 292.89/161.47 Uargs(add) = {1, 2} 292.89/161.47 292.89/161.47 TcT has computed the following matrix interpretation satisfying 292.89/161.47 not(EDA) and not(IDA(1)). 292.89/161.47 292.89/161.47 [terms](x1) = [1] x1 + [7] 292.89/161.47 292.89/161.47 [cons](x1) = [1] x1 + [3] 292.89/161.47 292.89/161.47 [recip](x1) = [1] x1 + [3] 292.89/161.47 292.89/161.47 [sqr](x1) = [1] x1 + [1] 292.89/161.47 292.89/161.47 [0] = [0] 292.89/161.47 292.89/161.47 [s](x1) = [1] x1 + [0] 292.89/161.47 292.89/161.47 [add](x1, x2) = [1] x1 + [1] x2 + [0] 292.89/161.47 292.89/161.47 [dbl](x1) = [0] 292.89/161.47 292.89/161.47 [first](x1, x2) = [1] x1 + [1] x2 + [7] 292.89/161.47 292.89/161.47 [nil] = [7] 292.89/161.47 292.89/161.47 The order satisfies the following ordering constraints: 292.89/161.47 292.89/161.47 [terms(N)] = [1] N + [7] 292.89/161.47 >= [1] N + [7] 292.89/161.47 = [cons(recip(sqr(N)))] 292.89/161.47 292.89/161.47 [sqr(0())] = [1] 292.89/161.47 > [0] 292.89/161.47 = [0()] 292.89/161.47 292.89/161.47 [sqr(s(X))] = [1] X + [1] 292.89/161.47 >= [1] X + [1] 292.89/161.47 = [s(add(sqr(X), dbl(X)))] 292.89/161.47 292.89/161.47 [add(0(), X)] = [1] X + [0] 292.89/161.47 >= [1] X + [0] 292.89/161.47 = [X] 292.89/161.47 292.89/161.47 [add(s(X), Y)] = [1] X + [1] Y + [0] 292.89/161.47 >= [1] X + [1] Y + [0] 292.89/161.47 = [s(add(X, Y))] 292.89/161.47 292.89/161.47 [dbl(0())] = [0] 292.89/161.47 >= [0] 292.89/161.47 = [0()] 292.89/161.47 292.89/161.47 [dbl(s(X))] = [0] 292.89/161.47 >= [0] 292.89/161.47 = [s(s(dbl(X)))] 292.89/161.47 292.89/161.47 [first(0(), X)] = [1] X + [7] 292.89/161.47 >= [7] 292.89/161.47 = [nil()] 292.89/161.47 292.89/161.47 [first(s(X), cons(Y))] = [1] X + [1] Y + [10] 292.89/161.47 > [1] Y + [3] 292.89/161.47 = [cons(Y)] 292.89/161.47 292.89/161.47 292.89/161.47 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 292.89/161.47 292.89/161.47 We are left with following problem, upon which TcT provides the 292.89/161.47 certificate MAYBE. 292.89/161.47 292.89/161.47 Strict Trs: 292.89/161.47 { sqr(s(X)) -> s(add(sqr(X), dbl(X))) 292.89/161.47 , add(s(X), Y) -> s(add(X, Y)) 292.89/161.47 , dbl(s(X)) -> s(s(dbl(X))) } 292.89/161.47 Weak Trs: 292.89/161.47 { terms(N) -> cons(recip(sqr(N))) 292.89/161.47 , sqr(0()) -> 0() 292.89/161.47 , add(0(), X) -> X 292.89/161.47 , dbl(0()) -> 0() 292.89/161.47 , first(0(), X) -> nil() 292.89/161.47 , first(s(X), cons(Y)) -> cons(Y) } 292.89/161.47 Obligation: 292.89/161.47 runtime complexity 292.89/161.47 Answer: 292.89/161.47 MAYBE 292.89/161.47 292.89/161.47 None of the processors succeeded. 292.89/161.47 292.89/161.47 Details of failed attempt(s): 292.89/161.47 ----------------------------- 292.89/161.47 1) 'empty' failed due to the following reason: 292.89/161.47 292.89/161.47 Empty strict component of the problem is NOT empty. 292.89/161.47 292.89/161.47 2) 'With Problem ...' failed due to the following reason: 292.89/161.47 292.89/161.47 None of the processors succeeded. 292.89/161.47 292.89/161.47 Details of failed attempt(s): 292.89/161.47 ----------------------------- 292.89/161.47 1) 'empty' failed due to the following reason: 292.89/161.47 292.89/161.47 Empty strict component of the problem is NOT empty. 292.89/161.47 292.89/161.47 2) 'Fastest' failed due to the following reason: 292.89/161.47 292.89/161.47 None of the processors succeeded. 292.89/161.47 292.89/161.47 Details of failed attempt(s): 292.89/161.47 ----------------------------- 292.89/161.47 1) 'With Problem ...' failed due to the following reason: 292.89/161.48 292.89/161.48 None of the processors succeeded. 292.89/161.48 292.89/161.48 Details of failed attempt(s): 292.89/161.48 ----------------------------- 292.89/161.48 1) 'empty' failed due to the following reason: 292.89/161.48 292.89/161.48 Empty strict component of the problem is NOT empty. 292.89/161.48 292.89/161.48 2) 'With Problem ...' failed due to the following reason: 292.89/161.48 292.89/161.48 The weightgap principle applies (using the following nonconstant 292.89/161.48 growth matrix-interpretation) 292.89/161.48 292.89/161.48 The following argument positions are usable: 292.89/161.48 Uargs(cons) = {1}, Uargs(recip) = {1}, Uargs(s) = {1}, 292.89/161.48 Uargs(add) = {1, 2} 292.89/161.48 292.89/161.48 TcT has computed the following matrix interpretation satisfying 292.89/161.48 not(EDA) and not(IDA(1)). 292.89/161.48 292.89/161.48 [terms](x1) = [0 1] x1 + [7] 292.89/161.48 [0 0] [7] 292.89/161.48 292.89/161.48 [cons](x1) = [1 0] x1 + [0] 292.89/161.48 [0 0] [0] 292.89/161.48 292.89/161.48 [recip](x1) = [1 0] x1 + [4] 292.89/161.48 [0 0] [4] 292.89/161.48 292.89/161.48 [sqr](x1) = [0 1] x1 + [0] 292.89/161.48 [0 0] [4] 292.89/161.48 292.89/161.48 [0] = [0] 292.89/161.48 [0] 292.89/161.48 292.89/161.48 [s](x1) = [1 0] x1 + [0] 292.89/161.48 [0 1] [1] 292.89/161.48 292.89/161.48 [add](x1, x2) = [1 0] x1 + [1 0] x2 + [0] 292.89/161.48 [0 0] [0 1] [0] 292.89/161.48 292.89/161.48 [dbl](x1) = [0] 292.89/161.48 [2] 292.89/161.48 292.89/161.48 [first](x1, x2) = [1 0] x2 + [4] 292.89/161.48 [0 0] [0] 292.89/161.48 292.89/161.48 [nil] = [4] 292.89/161.48 [0] 292.89/161.48 292.89/161.48 The order satisfies the following ordering constraints: 292.89/161.48 292.89/161.48 [terms(N)] = [0 1] N + [7] 292.89/161.48 [0 0] [7] 292.89/161.48 > [0 1] N + [4] 292.89/161.48 [0 0] [0] 292.89/161.48 = [cons(recip(sqr(N)))] 292.89/161.48 292.89/161.48 [sqr(0())] = [0] 292.89/161.48 [4] 292.89/161.48 >= [0] 292.89/161.48 [0] 292.89/161.48 = [0()] 292.89/161.48 292.89/161.48 [sqr(s(X))] = [0 1] X + [1] 292.89/161.48 [0 0] [4] 292.89/161.48 > [0 1] X + [0] 292.89/161.48 [0 0] [3] 292.89/161.48 = [s(add(sqr(X), dbl(X)))] 292.89/161.48 292.89/161.48 [add(0(), X)] = [1 0] X + [0] 292.89/161.48 [0 1] [0] 292.89/161.48 >= [1 0] X + [0] 292.89/161.48 [0 1] [0] 292.89/161.48 = [X] 292.89/161.48 292.89/161.48 [add(s(X), Y)] = [1 0] X + [1 0] Y + [0] 292.89/161.48 [0 0] [0 1] [0] 292.89/161.48 ? [1 0] X + [1 0] Y + [0] 292.89/161.48 [0 0] [0 1] [1] 292.89/161.48 = [s(add(X, Y))] 292.89/161.48 292.89/161.48 [dbl(0())] = [0] 292.89/161.48 [2] 292.89/161.48 >= [0] 292.89/161.48 [0] 292.89/161.48 = [0()] 292.89/161.48 292.89/161.48 [dbl(s(X))] = [0] 292.89/161.48 [2] 292.89/161.48 ? [0] 292.89/161.48 [4] 292.89/161.48 = [s(s(dbl(X)))] 292.89/161.48 292.89/161.48 [first(0(), X)] = [1 0] X + [4] 292.89/161.48 [0 0] [0] 292.89/161.48 >= [4] 292.89/161.48 [0] 292.89/161.48 = [nil()] 292.89/161.48 292.89/161.48 [first(s(X), cons(Y))] = [1 0] Y + [4] 292.89/161.48 [0 0] [0] 292.89/161.48 > [1 0] Y + [0] 292.89/161.48 [0 0] [0] 292.89/161.48 = [cons(Y)] 292.89/161.48 292.89/161.48 292.89/161.48 Further, it can be verified that all rules not oriented are covered by the weightgap condition. 292.89/161.48 292.89/161.48 We are left with following problem, upon which TcT provides the 292.89/161.48 certificate MAYBE. 292.89/161.48 292.89/161.48 Strict Trs: 292.89/161.48 { add(s(X), Y) -> s(add(X, Y)) 292.89/161.48 , dbl(s(X)) -> s(s(dbl(X))) } 292.89/161.48 Weak Trs: 292.89/161.48 { terms(N) -> cons(recip(sqr(N))) 292.89/161.48 , sqr(0()) -> 0() 292.89/161.48 , sqr(s(X)) -> s(add(sqr(X), dbl(X))) 292.89/161.48 , add(0(), X) -> X 292.89/161.48 , dbl(0()) -> 0() 292.89/161.48 , first(0(), X) -> nil() 292.89/161.48 , first(s(X), cons(Y)) -> cons(Y) } 292.89/161.48 Obligation: 292.89/161.48 runtime complexity 292.89/161.48 Answer: 292.89/161.48 MAYBE 292.89/161.48 292.89/161.48 None of the processors succeeded. 292.89/161.48 292.89/161.48 Details of failed attempt(s): 292.89/161.48 ----------------------------- 292.89/161.48 1) 'empty' failed due to the following reason: 292.89/161.48 292.89/161.48 Empty strict component of the problem is NOT empty. 292.89/161.48 292.89/161.48 2) 'With Problem ...' failed due to the following reason: 292.89/161.48 292.89/161.48 None of the processors succeeded. 292.89/161.48 292.89/161.48 Details of failed attempt(s): 292.89/161.48 ----------------------------- 292.89/161.48 1) 'empty' failed due to the following reason: 292.89/161.48 292.89/161.48 Empty strict component of the problem is NOT empty. 292.89/161.48 292.89/161.48 2) 'With Problem ...' failed due to the following reason: 292.89/161.48 292.89/161.48 Empty strict component of the problem is NOT empty. 292.89/161.48 292.89/161.48 292.89/161.48 292.89/161.48 292.89/161.48 2) 'With Problem ...' failed due to the following reason: 292.89/161.48 292.89/161.48 We use the processor 'polynomial interpretation' to orient 292.89/161.48 following rules strictly. 292.89/161.48 292.89/161.48 Trs: { sqr(s(X)) -> s(add(sqr(X), dbl(X))) } 292.89/161.48 292.89/161.48 The induced complexity on above rules (modulo remaining rules) is 292.89/161.48 YES(?,O(n^2)) . These rules are moved into the corresponding weak 292.89/161.48 component(s). 292.89/161.48 292.89/161.48 Sub-proof: 292.89/161.48 ---------- 292.89/161.48 We consider the following typing: 292.89/161.48 292.89/161.48 terms :: b -> c 292.89/161.48 cons :: a -> c 292.89/161.48 recip :: b -> a 292.89/161.48 sqr :: b -> b 292.89/161.48 0 :: b 292.89/161.48 s :: b -> b 292.89/161.48 add :: (b,b) -> b 292.89/161.48 dbl :: b -> b 292.89/161.48 first :: (b,c) -> c 292.89/161.48 nil :: c 292.89/161.48 292.89/161.48 The following argument positions are considered usable: 292.89/161.48 292.89/161.48 Uargs(cons) = {1}, Uargs(recip) = {1}, Uargs(s) = {1}, 292.89/161.48 Uargs(add) = {1, 2} 292.89/161.48 292.89/161.48 TcT has computed the following constructor-restricted 292.89/161.48 typedpolynomial interpretation. 292.89/161.48 292.89/161.48 [terms](x1) = 3 + 3*x1 + 3*x1^2 292.89/161.48 292.89/161.48 [cons](x1) = x1 292.89/161.48 292.89/161.48 [recip](x1) = 3 + 3*x1 292.89/161.48 292.89/161.48 [sqr](x1) = x1^2 292.89/161.48 292.89/161.48 [0]() = 0 292.89/161.48 292.89/161.48 [s](x1) = 2 + x1 292.89/161.48 292.89/161.48 [add](x1, x2) = x1 + 2*x2 292.89/161.48 292.89/161.48 [dbl](x1) = 2*x1 292.89/161.48 292.89/161.48 [first](x1, x2) = 1 + 2*x1*x2 292.89/161.48 292.89/161.48 [nil]() = 0 292.89/161.48 292.89/161.48 292.89/161.48 This order satisfies the following ordering constraints. 292.89/161.48 292.89/161.48 [terms(N)] = 3 + 3*N + 3*N^2 292.89/161.48 >= 3 + 3*N^2 292.89/161.48 = [cons(recip(sqr(N)))] 292.89/161.48 292.89/161.48 [sqr(0())] = 292.89/161.48 >= 292.89/161.48 = [0()] 292.89/161.48 292.89/161.48 [sqr(s(X))] = 4 + 4*X + X^2 292.89/161.48 > 2 + X^2 + 4*X 292.89/161.48 = [s(add(sqr(X), dbl(X)))] 292.89/161.48 292.89/161.48 [add(0(), X)] = 2*X 292.89/161.48 >= X 292.89/161.48 = [X] 292.89/161.48 292.89/161.48 [add(s(X), Y)] = 2 + X + 2*Y 292.89/161.48 >= 2 + X + 2*Y 292.89/161.48 = [s(add(X, Y))] 292.89/161.48 292.89/161.48 [dbl(0())] = 292.89/161.48 >= 292.89/161.48 = [0()] 292.89/161.48 292.89/161.48 [dbl(s(X))] = 4 + 2*X 292.89/161.48 >= 4 + 2*X 292.89/161.48 = [s(s(dbl(X)))] 292.89/161.48 292.89/161.48 [first(0(), X)] = 1 292.89/161.48 > 292.89/161.48 = [nil()] 292.89/161.48 292.89/161.48 [first(s(X), cons(Y))] = 1 + 4*Y + 2*X*Y 292.89/161.48 > Y 292.89/161.48 = [cons(Y)] 292.89/161.48 292.89/161.48 292.89/161.48 We return to the main proof. 292.89/161.48 292.89/161.48 We are left with following problem, upon which TcT provides the 292.89/161.48 certificate MAYBE. 292.89/161.48 292.89/161.48 Strict Trs: 292.89/161.48 { add(s(X), Y) -> s(add(X, Y)) 292.89/161.48 , dbl(s(X)) -> s(s(dbl(X))) } 292.89/161.48 Weak Trs: 292.89/161.48 { terms(N) -> cons(recip(sqr(N))) 292.89/161.48 , sqr(0()) -> 0() 292.89/161.48 , sqr(s(X)) -> s(add(sqr(X), dbl(X))) 292.89/161.48 , add(0(), X) -> X 292.89/161.48 , dbl(0()) -> 0() 292.89/161.48 , first(0(), X) -> nil() 292.89/161.48 , first(s(X), cons(Y)) -> cons(Y) } 292.89/161.48 Obligation: 292.89/161.48 runtime complexity 292.89/161.48 Answer: 292.89/161.48 MAYBE 292.89/161.48 292.89/161.48 None of the processors succeeded. 292.89/161.48 292.89/161.48 Details of failed attempt(s): 292.89/161.48 ----------------------------- 292.89/161.48 1) 'empty' failed due to the following reason: 292.89/161.48 292.89/161.48 Empty strict component of the problem is NOT empty. 292.89/161.48 292.89/161.48 2) 'With Problem ...' failed due to the following reason: 292.89/161.48 292.89/161.48 Empty strict component of the problem is NOT empty. 292.89/161.48 292.89/161.48 292.89/161.48 292.89/161.48 292.89/161.48 292.89/161.48 2) 'Fastest (timeout of 24 seconds) (timeout of 297 seconds)' 292.89/161.48 failed due to the following reason: 292.89/161.48 292.89/161.48 None of the processors succeeded. 292.89/161.48 292.89/161.48 Details of failed attempt(s): 292.89/161.48 ----------------------------- 292.89/161.48 1) 'Bounds with perSymbol-enrichment and initial automaton 'match'' 292.89/161.48 failed due to the following reason: 292.89/161.48 292.89/161.48 match-boundness of the problem could not be verified. 292.89/161.48 292.89/161.48 2) 'Bounds with minimal-enrichment and initial automaton 'match'' 292.89/161.48 failed due to the following reason: 292.89/161.48 292.89/161.48 match-boundness of the problem could not be verified. 292.89/161.48 292.89/161.48 292.89/161.48 3) 'Best' failed due to the following reason: 292.89/161.48 292.89/161.48 None of the processors succeeded. 292.89/161.48 292.89/161.48 Details of failed attempt(s): 292.89/161.48 ----------------------------- 292.89/161.48 1) 'bsearch-popstar (timeout of 297 seconds)' failed due to the 292.89/161.48 following reason: 292.89/161.48 292.89/161.48 The processor is inapplicable, reason: 292.89/161.48 Processor only applicable for innermost runtime complexity analysis 292.89/161.48 292.89/161.48 2) 'Polynomial Path Order (PS) (timeout of 297 seconds)' failed due 292.89/161.48 to the following reason: 292.89/161.48 292.89/161.48 The processor is inapplicable, reason: 292.89/161.48 Processor only applicable for innermost runtime complexity analysis 292.89/161.48 292.89/161.48 292.89/161.48 292.89/161.48 2) 'Weak Dependency Pairs (timeout of 297 seconds)' failed due to 292.89/161.48 the following reason: 292.89/161.48 292.89/161.48 We add the following weak dependency pairs: 292.89/161.48 292.89/161.48 Strict DPs: 292.89/161.48 { terms^#(N) -> c_1(sqr^#(N)) 292.89/161.48 , sqr^#(0()) -> c_2() 292.89/161.48 , sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X))) 292.89/161.48 , add^#(0(), X) -> c_4(X) 292.89/161.48 , add^#(s(X), Y) -> c_5(add^#(X, Y)) 292.89/161.48 , dbl^#(0()) -> c_6() 292.89/161.48 , dbl^#(s(X)) -> c_7(dbl^#(X)) 292.89/161.48 , first^#(0(), X) -> c_8() 292.89/161.48 , first^#(s(X), cons(Y)) -> c_9(Y) } 292.89/161.48 292.89/161.48 and mark the set of starting terms. 292.89/161.48 292.89/161.48 We are left with following problem, upon which TcT provides the 292.89/161.48 certificate MAYBE. 292.89/161.48 292.89/161.48 Strict DPs: 292.89/161.48 { terms^#(N) -> c_1(sqr^#(N)) 292.89/161.48 , sqr^#(0()) -> c_2() 292.89/161.48 , sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X))) 292.89/161.48 , add^#(0(), X) -> c_4(X) 292.89/161.48 , add^#(s(X), Y) -> c_5(add^#(X, Y)) 292.89/161.48 , dbl^#(0()) -> c_6() 292.89/161.48 , dbl^#(s(X)) -> c_7(dbl^#(X)) 292.89/161.48 , first^#(0(), X) -> c_8() 292.89/161.48 , first^#(s(X), cons(Y)) -> c_9(Y) } 292.89/161.48 Strict Trs: 292.89/161.48 { terms(N) -> cons(recip(sqr(N))) 292.89/161.48 , sqr(0()) -> 0() 292.89/161.48 , sqr(s(X)) -> s(add(sqr(X), dbl(X))) 292.89/161.48 , add(0(), X) -> X 292.89/161.48 , add(s(X), Y) -> s(add(X, Y)) 292.89/161.48 , dbl(0()) -> 0() 292.89/161.48 , dbl(s(X)) -> s(s(dbl(X))) 292.89/161.48 , first(0(), X) -> nil() 292.89/161.48 , first(s(X), cons(Y)) -> cons(Y) } 292.89/161.48 Obligation: 292.89/161.48 runtime complexity 292.89/161.48 Answer: 292.89/161.48 MAYBE 292.89/161.48 292.89/161.48 We estimate the number of application of {2,6,8} by applications of 292.89/161.48 Pre({2,6,8}) = {1,4,7,9}. Here rules are labeled as follows: 292.89/161.48 292.89/161.48 DPs: 292.89/161.48 { 1: terms^#(N) -> c_1(sqr^#(N)) 292.89/161.48 , 2: sqr^#(0()) -> c_2() 292.89/161.48 , 3: sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X))) 292.89/161.48 , 4: add^#(0(), X) -> c_4(X) 292.89/161.48 , 5: add^#(s(X), Y) -> c_5(add^#(X, Y)) 292.89/161.48 , 6: dbl^#(0()) -> c_6() 292.89/161.48 , 7: dbl^#(s(X)) -> c_7(dbl^#(X)) 292.89/161.48 , 8: first^#(0(), X) -> c_8() 292.89/161.48 , 9: first^#(s(X), cons(Y)) -> c_9(Y) } 292.89/161.48 292.89/161.48 We are left with following problem, upon which TcT provides the 292.89/161.48 certificate MAYBE. 292.89/161.48 292.89/161.48 Strict DPs: 292.89/161.48 { terms^#(N) -> c_1(sqr^#(N)) 292.89/161.48 , sqr^#(s(X)) -> c_3(add^#(sqr(X), dbl(X))) 292.89/161.48 , add^#(0(), X) -> c_4(X) 292.89/161.48 , add^#(s(X), Y) -> c_5(add^#(X, Y)) 292.89/161.48 , dbl^#(s(X)) -> c_7(dbl^#(X)) 292.89/161.48 , first^#(s(X), cons(Y)) -> c_9(Y) } 292.89/161.48 Strict Trs: 292.89/161.48 { terms(N) -> cons(recip(sqr(N))) 292.89/161.48 , sqr(0()) -> 0() 292.89/161.48 , sqr(s(X)) -> s(add(sqr(X), dbl(X))) 292.89/161.48 , add(0(), X) -> X 292.89/161.48 , add(s(X), Y) -> s(add(X, Y)) 292.89/161.48 , dbl(0()) -> 0() 292.89/161.48 , dbl(s(X)) -> s(s(dbl(X))) 292.89/161.48 , first(0(), X) -> nil() 292.89/161.48 , first(s(X), cons(Y)) -> cons(Y) } 292.89/161.48 Weak DPs: 292.89/161.48 { sqr^#(0()) -> c_2() 292.89/161.48 , dbl^#(0()) -> c_6() 292.89/161.48 , first^#(0(), X) -> c_8() } 292.89/161.48 Obligation: 292.89/161.48 runtime complexity 292.89/161.48 Answer: 292.89/161.48 MAYBE 292.89/161.48 292.89/161.48 Empty strict component of the problem is NOT empty. 292.89/161.48 292.89/161.48 292.89/161.48 Arrrr.. 293.25/161.73 EOF