YES(O(1),O(n^2)) 439.18/148.14 YES(O(1),O(n^2)) 439.18/148.14 439.18/148.14 We are left with following problem, upon which TcT provides the 439.18/148.14 certificate YES(O(1),O(n^2)). 439.18/148.14 439.18/148.14 Strict Trs: 439.18/148.14 { lgth(Nil()) -> Nil() 439.18/148.14 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 439.18/148.14 , gcd(Nil(), Nil()) -> Nil() 439.18/148.14 , gcd(Nil(), Cons(x, xs)) -> Nil() 439.18/148.14 , gcd(Cons(x, xs), Nil()) -> Nil() 439.18/148.14 , gcd(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.14 gcd[Ite][False][Ite](eqList(Cons(x', xs'), Cons(x, xs)), 439.18/148.14 Cons(x', xs'), 439.18/148.14 Cons(x, xs)) 439.18/148.14 , @(Nil(), ys) -> ys 439.18/148.14 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 439.18/148.14 , monus(x, y) -> 439.18/148.14 monus[Ite](eqList(lgth(y), Cons(Nil(), Nil())), x, y) 439.18/148.14 , eqList(Nil(), Nil()) -> True() 439.18/148.14 , eqList(Nil(), Cons(y, ys)) -> False() 439.18/148.14 , eqList(Cons(x, xs), Nil()) -> False() 439.18/148.14 , eqList(Cons(x, xs), Cons(y, ys)) -> 439.18/148.14 and(eqList(x, y), eqList(xs, ys)) 439.18/148.14 , gt0(Nil(), y) -> False() 439.18/148.14 , gt0(Cons(x, xs), Nil()) -> True() 439.18/148.14 , gt0(Cons(x', xs'), Cons(x, xs)) -> gt0(xs', xs) 439.18/148.14 , goal(x, y) -> gcd(x, y) } 439.18/148.14 Weak Trs: 439.18/148.14 { monus[Ite](True(), Cons(x, xs), y) -> xs 439.18/148.14 , monus[Ite](False(), Cons(x', xs'), Cons(x, xs)) -> monus(xs', xs) 439.18/148.14 , and(True(), True()) -> True() 439.18/148.14 , and(True(), False()) -> False() 439.18/148.14 , and(False(), True()) -> False() 439.18/148.14 , and(False(), False()) -> False() 439.18/148.14 , gcd[Ite][False][Ite](True(), x, y) -> x 439.18/148.14 , gcd[Ite][False][Ite](False(), x, y) -> 439.18/148.14 gcd[Ite][False][Ite][False][Ite](gt0(x, y), x, y) } 439.18/148.14 Obligation: 439.18/148.14 innermost runtime complexity 439.18/148.14 Answer: 439.18/148.14 YES(O(1),O(n^2)) 439.18/148.14 439.18/148.14 We add the following dependency tuples: 439.18/148.14 439.18/148.14 Strict DPs: 439.18/148.14 { lgth^#(Nil()) -> c_1() 439.18/148.14 , lgth^#(Cons(x, xs)) -> 439.18/148.14 c_2(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 439.18/148.14 , @^#(Nil(), ys) -> c_7() 439.18/148.14 , @^#(Cons(x, xs), ys) -> c_8(@^#(xs, ys)) 439.18/148.14 , gcd^#(Nil(), Nil()) -> c_3() 439.18/148.14 , gcd^#(Nil(), Cons(x, xs)) -> c_4() 439.18/148.14 , gcd^#(Cons(x, xs), Nil()) -> c_5() 439.18/148.14 , gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.14 c_6(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 439.18/148.14 Cons(x', xs'), 439.18/148.14 Cons(x, xs)), 439.18/148.14 eqList^#(Cons(x', xs'), Cons(x, xs))) 439.18/148.14 , eqList^#(Nil(), Nil()) -> c_10() 439.18/148.14 , eqList^#(Nil(), Cons(y, ys)) -> c_11() 439.18/148.14 , eqList^#(Cons(x, xs), Nil()) -> c_12() 439.18/148.14 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 439.18/148.14 c_13(and^#(eqList(x, y), eqList(xs, ys)), 439.18/148.14 eqList^#(x, y), 439.18/148.14 eqList^#(xs, ys)) 439.18/148.14 , monus^#(x, y) -> 439.18/148.14 c_9(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.14 eqList^#(lgth(y), Cons(Nil(), Nil())), 439.18/148.14 lgth^#(y)) 439.18/148.14 , gt0^#(Nil(), y) -> c_14() 439.18/148.14 , gt0^#(Cons(x, xs), Nil()) -> c_15() 439.18/148.14 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_16(gt0^#(xs', xs)) 439.18/148.14 , goal^#(x, y) -> c_17(gcd^#(x, y)) } 439.18/148.14 Weak DPs: 439.18/148.14 { gcd[Ite][False][Ite]^#(True(), x, y) -> c_24() 439.18/148.14 , gcd[Ite][False][Ite]^#(False(), x, y) -> c_25(gt0^#(x, y)) 439.18/148.14 , monus[Ite]^#(True(), Cons(x, xs), y) -> c_18() 439.18/148.14 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 439.18/148.14 c_19(monus^#(xs', xs)) 439.18/148.14 , and^#(True(), True()) -> c_20() 439.18/148.14 , and^#(True(), False()) -> c_21() 439.18/148.14 , and^#(False(), True()) -> c_22() 439.18/148.14 , and^#(False(), False()) -> c_23() } 439.18/148.14 439.18/148.14 and mark the set of starting terms. 439.18/148.14 439.18/148.14 We are left with following problem, upon which TcT provides the 439.18/148.14 certificate YES(O(1),O(n^2)). 439.18/148.14 439.18/148.14 Strict DPs: 439.18/148.14 { lgth^#(Nil()) -> c_1() 439.18/148.14 , lgth^#(Cons(x, xs)) -> 439.18/148.14 c_2(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 439.18/148.14 , @^#(Nil(), ys) -> c_7() 439.18/148.14 , @^#(Cons(x, xs), ys) -> c_8(@^#(xs, ys)) 439.18/148.14 , gcd^#(Nil(), Nil()) -> c_3() 439.18/148.14 , gcd^#(Nil(), Cons(x, xs)) -> c_4() 439.18/148.14 , gcd^#(Cons(x, xs), Nil()) -> c_5() 439.18/148.14 , gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.14 c_6(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 439.18/148.14 Cons(x', xs'), 439.18/148.14 Cons(x, xs)), 439.18/148.14 eqList^#(Cons(x', xs'), Cons(x, xs))) 439.18/148.14 , eqList^#(Nil(), Nil()) -> c_10() 439.18/148.14 , eqList^#(Nil(), Cons(y, ys)) -> c_11() 439.18/148.14 , eqList^#(Cons(x, xs), Nil()) -> c_12() 439.18/148.14 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 439.18/148.14 c_13(and^#(eqList(x, y), eqList(xs, ys)), 439.18/148.14 eqList^#(x, y), 439.18/148.14 eqList^#(xs, ys)) 439.18/148.14 , monus^#(x, y) -> 439.18/148.14 c_9(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.14 eqList^#(lgth(y), Cons(Nil(), Nil())), 439.18/148.14 lgth^#(y)) 439.18/148.14 , gt0^#(Nil(), y) -> c_14() 439.18/148.14 , gt0^#(Cons(x, xs), Nil()) -> c_15() 439.18/148.14 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_16(gt0^#(xs', xs)) 439.18/148.14 , goal^#(x, y) -> c_17(gcd^#(x, y)) } 439.18/148.14 Weak DPs: 439.18/148.14 { gcd[Ite][False][Ite]^#(True(), x, y) -> c_24() 439.18/148.14 , gcd[Ite][False][Ite]^#(False(), x, y) -> c_25(gt0^#(x, y)) 439.18/148.14 , monus[Ite]^#(True(), Cons(x, xs), y) -> c_18() 439.18/148.14 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 439.18/148.14 c_19(monus^#(xs', xs)) 439.18/148.14 , and^#(True(), True()) -> c_20() 439.18/148.14 , and^#(True(), False()) -> c_21() 439.18/148.14 , and^#(False(), True()) -> c_22() 439.18/148.14 , and^#(False(), False()) -> c_23() } 439.18/148.14 Weak Trs: 439.18/148.14 { lgth(Nil()) -> Nil() 439.18/148.14 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 439.18/148.14 , monus[Ite](True(), Cons(x, xs), y) -> xs 439.18/148.14 , monus[Ite](False(), Cons(x', xs'), Cons(x, xs)) -> monus(xs', xs) 439.18/148.14 , gcd(Nil(), Nil()) -> Nil() 439.18/148.14 , gcd(Nil(), Cons(x, xs)) -> Nil() 439.18/148.14 , gcd(Cons(x, xs), Nil()) -> Nil() 439.18/148.14 , gcd(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.14 gcd[Ite][False][Ite](eqList(Cons(x', xs'), Cons(x, xs)), 439.18/148.14 Cons(x', xs'), 439.18/148.14 Cons(x, xs)) 439.18/148.14 , @(Nil(), ys) -> ys 439.18/148.14 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 439.18/148.14 , monus(x, y) -> 439.18/148.14 monus[Ite](eqList(lgth(y), Cons(Nil(), Nil())), x, y) 439.18/148.14 , and(True(), True()) -> True() 439.18/148.14 , and(True(), False()) -> False() 439.18/148.14 , and(False(), True()) -> False() 439.18/148.14 , and(False(), False()) -> False() 439.18/148.14 , eqList(Nil(), Nil()) -> True() 439.18/148.14 , eqList(Nil(), Cons(y, ys)) -> False() 439.18/148.14 , eqList(Cons(x, xs), Nil()) -> False() 439.18/148.14 , eqList(Cons(x, xs), Cons(y, ys)) -> 439.18/148.14 and(eqList(x, y), eqList(xs, ys)) 439.18/148.14 , gt0(Nil(), y) -> False() 439.18/148.14 , gt0(Cons(x, xs), Nil()) -> True() 439.18/148.14 , gt0(Cons(x', xs'), Cons(x, xs)) -> gt0(xs', xs) 439.18/148.14 , gcd[Ite][False][Ite](True(), x, y) -> x 439.18/148.14 , gcd[Ite][False][Ite](False(), x, y) -> 439.18/148.14 gcd[Ite][False][Ite][False][Ite](gt0(x, y), x, y) 439.18/148.14 , goal(x, y) -> gcd(x, y) } 439.18/148.14 Obligation: 439.18/148.14 innermost runtime complexity 439.18/148.14 Answer: 439.18/148.14 YES(O(1),O(n^2)) 439.18/148.14 439.18/148.14 We estimate the number of application of {1,3,5,6,7,9,10,11} by 439.18/148.14 applications of Pre({1,3,5,6,7,9,10,11}) = {2,4,12,13,17}. Here 439.18/148.14 rules are labeled as follows: 439.18/148.14 439.18/148.14 DPs: 439.18/148.14 { 1: lgth^#(Nil()) -> c_1() 439.18/148.14 , 2: lgth^#(Cons(x, xs)) -> 439.18/148.14 c_2(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 439.18/148.14 , 3: @^#(Nil(), ys) -> c_7() 439.18/148.14 , 4: @^#(Cons(x, xs), ys) -> c_8(@^#(xs, ys)) 439.18/148.14 , 5: gcd^#(Nil(), Nil()) -> c_3() 439.18/148.14 , 6: gcd^#(Nil(), Cons(x, xs)) -> c_4() 439.18/148.14 , 7: gcd^#(Cons(x, xs), Nil()) -> c_5() 439.18/148.14 , 8: gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.14 c_6(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 439.18/148.14 Cons(x', xs'), 439.18/148.14 Cons(x, xs)), 439.18/148.14 eqList^#(Cons(x', xs'), Cons(x, xs))) 439.18/148.14 , 9: eqList^#(Nil(), Nil()) -> c_10() 439.18/148.14 , 10: eqList^#(Nil(), Cons(y, ys)) -> c_11() 439.18/148.14 , 11: eqList^#(Cons(x, xs), Nil()) -> c_12() 439.18/148.14 , 12: eqList^#(Cons(x, xs), Cons(y, ys)) -> 439.18/148.14 c_13(and^#(eqList(x, y), eqList(xs, ys)), 439.18/148.14 eqList^#(x, y), 439.18/148.14 eqList^#(xs, ys)) 439.18/148.14 , 13: monus^#(x, y) -> 439.18/148.14 c_9(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.14 eqList^#(lgth(y), Cons(Nil(), Nil())), 439.18/148.14 lgth^#(y)) 439.18/148.14 , 14: gt0^#(Nil(), y) -> c_14() 439.18/148.14 , 15: gt0^#(Cons(x, xs), Nil()) -> c_15() 439.18/148.14 , 16: gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_16(gt0^#(xs', xs)) 439.18/148.14 , 17: goal^#(x, y) -> c_17(gcd^#(x, y)) 439.18/148.14 , 18: gcd[Ite][False][Ite]^#(True(), x, y) -> c_24() 439.18/148.14 , 19: gcd[Ite][False][Ite]^#(False(), x, y) -> c_25(gt0^#(x, y)) 439.18/148.14 , 20: monus[Ite]^#(True(), Cons(x, xs), y) -> c_18() 439.18/148.14 , 21: monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 439.18/148.14 c_19(monus^#(xs', xs)) 439.18/148.14 , 22: and^#(True(), True()) -> c_20() 439.18/148.14 , 23: and^#(True(), False()) -> c_21() 439.18/148.14 , 24: and^#(False(), True()) -> c_22() 439.18/148.14 , 25: and^#(False(), False()) -> c_23() } 439.18/148.14 439.18/148.14 We are left with following problem, upon which TcT provides the 439.18/148.14 certificate YES(O(1),O(n^2)). 439.18/148.14 439.18/148.14 Strict DPs: 439.18/148.14 { lgth^#(Cons(x, xs)) -> 439.18/148.14 c_2(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 439.18/148.14 , @^#(Cons(x, xs), ys) -> c_8(@^#(xs, ys)) 439.18/148.14 , gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.14 c_6(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 439.18/148.14 Cons(x', xs'), 439.18/148.14 Cons(x, xs)), 439.18/148.14 eqList^#(Cons(x', xs'), Cons(x, xs))) 439.18/148.14 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 439.18/148.14 c_13(and^#(eqList(x, y), eqList(xs, ys)), 439.18/148.14 eqList^#(x, y), 439.18/148.14 eqList^#(xs, ys)) 439.18/148.14 , monus^#(x, y) -> 439.18/148.14 c_9(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.14 eqList^#(lgth(y), Cons(Nil(), Nil())), 439.18/148.14 lgth^#(y)) 439.18/148.14 , gt0^#(Nil(), y) -> c_14() 439.18/148.14 , gt0^#(Cons(x, xs), Nil()) -> c_15() 439.18/148.14 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_16(gt0^#(xs', xs)) 439.18/148.14 , goal^#(x, y) -> c_17(gcd^#(x, y)) } 439.18/148.14 Weak DPs: 439.18/148.14 { lgth^#(Nil()) -> c_1() 439.18/148.14 , @^#(Nil(), ys) -> c_7() 439.18/148.14 , gcd^#(Nil(), Nil()) -> c_3() 439.18/148.14 , gcd^#(Nil(), Cons(x, xs)) -> c_4() 439.18/148.14 , gcd^#(Cons(x, xs), Nil()) -> c_5() 439.18/148.14 , gcd[Ite][False][Ite]^#(True(), x, y) -> c_24() 439.18/148.14 , gcd[Ite][False][Ite]^#(False(), x, y) -> c_25(gt0^#(x, y)) 439.18/148.14 , eqList^#(Nil(), Nil()) -> c_10() 439.18/148.14 , eqList^#(Nil(), Cons(y, ys)) -> c_11() 439.18/148.14 , eqList^#(Cons(x, xs), Nil()) -> c_12() 439.18/148.14 , monus[Ite]^#(True(), Cons(x, xs), y) -> c_18() 439.18/148.14 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 439.18/148.14 c_19(monus^#(xs', xs)) 439.18/148.14 , and^#(True(), True()) -> c_20() 439.18/148.14 , and^#(True(), False()) -> c_21() 439.18/148.14 , and^#(False(), True()) -> c_22() 439.18/148.14 , and^#(False(), False()) -> c_23() } 439.18/148.14 Weak Trs: 439.18/148.14 { lgth(Nil()) -> Nil() 439.18/148.14 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 439.18/148.14 , monus[Ite](True(), Cons(x, xs), y) -> xs 439.18/148.14 , monus[Ite](False(), Cons(x', xs'), Cons(x, xs)) -> monus(xs', xs) 439.18/148.14 , gcd(Nil(), Nil()) -> Nil() 439.18/148.14 , gcd(Nil(), Cons(x, xs)) -> Nil() 439.18/148.14 , gcd(Cons(x, xs), Nil()) -> Nil() 439.18/148.14 , gcd(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.14 gcd[Ite][False][Ite](eqList(Cons(x', xs'), Cons(x, xs)), 439.18/148.14 Cons(x', xs'), 439.18/148.14 Cons(x, xs)) 439.18/148.14 , @(Nil(), ys) -> ys 439.18/148.14 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 439.18/148.14 , monus(x, y) -> 439.18/148.14 monus[Ite](eqList(lgth(y), Cons(Nil(), Nil())), x, y) 439.18/148.14 , and(True(), True()) -> True() 439.18/148.14 , and(True(), False()) -> False() 439.18/148.14 , and(False(), True()) -> False() 439.18/148.14 , and(False(), False()) -> False() 439.18/148.14 , eqList(Nil(), Nil()) -> True() 439.18/148.14 , eqList(Nil(), Cons(y, ys)) -> False() 439.18/148.14 , eqList(Cons(x, xs), Nil()) -> False() 439.18/148.14 , eqList(Cons(x, xs), Cons(y, ys)) -> 439.18/148.14 and(eqList(x, y), eqList(xs, ys)) 439.18/148.14 , gt0(Nil(), y) -> False() 439.18/148.14 , gt0(Cons(x, xs), Nil()) -> True() 439.18/148.14 , gt0(Cons(x', xs'), Cons(x, xs)) -> gt0(xs', xs) 439.18/148.14 , gcd[Ite][False][Ite](True(), x, y) -> x 439.18/148.14 , gcd[Ite][False][Ite](False(), x, y) -> 439.18/148.14 gcd[Ite][False][Ite][False][Ite](gt0(x, y), x, y) 439.18/148.14 , goal(x, y) -> gcd(x, y) } 439.18/148.14 Obligation: 439.18/148.14 innermost runtime complexity 439.18/148.14 Answer: 439.18/148.14 YES(O(1),O(n^2)) 439.18/148.14 439.18/148.14 The following weak DPs constitute a sub-graph of the DG that is 439.18/148.14 closed under successors. The DPs are removed. 439.18/148.14 439.18/148.14 { lgth^#(Nil()) -> c_1() 439.18/148.14 , @^#(Nil(), ys) -> c_7() 439.18/148.14 , gcd^#(Nil(), Nil()) -> c_3() 439.18/148.14 , gcd^#(Nil(), Cons(x, xs)) -> c_4() 439.18/148.14 , gcd^#(Cons(x, xs), Nil()) -> c_5() 439.18/148.14 , gcd[Ite][False][Ite]^#(True(), x, y) -> c_24() 439.18/148.14 , eqList^#(Nil(), Nil()) -> c_10() 439.18/148.14 , eqList^#(Nil(), Cons(y, ys)) -> c_11() 439.18/148.14 , eqList^#(Cons(x, xs), Nil()) -> c_12() 439.18/148.14 , monus[Ite]^#(True(), Cons(x, xs), y) -> c_18() 439.18/148.14 , and^#(True(), True()) -> c_20() 439.18/148.14 , and^#(True(), False()) -> c_21() 439.18/148.14 , and^#(False(), True()) -> c_22() 439.18/148.14 , and^#(False(), False()) -> c_23() } 439.18/148.14 439.18/148.14 We are left with following problem, upon which TcT provides the 439.18/148.14 certificate YES(O(1),O(n^2)). 439.18/148.14 439.18/148.14 Strict DPs: 439.18/148.14 { lgth^#(Cons(x, xs)) -> 439.18/148.14 c_2(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 439.18/148.14 , @^#(Cons(x, xs), ys) -> c_8(@^#(xs, ys)) 439.18/148.14 , gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.14 c_6(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 439.18/148.14 Cons(x', xs'), 439.18/148.14 Cons(x, xs)), 439.18/148.14 eqList^#(Cons(x', xs'), Cons(x, xs))) 439.18/148.14 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 439.18/148.14 c_13(and^#(eqList(x, y), eqList(xs, ys)), 439.18/148.14 eqList^#(x, y), 439.18/148.14 eqList^#(xs, ys)) 439.18/148.14 , monus^#(x, y) -> 439.18/148.14 c_9(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.14 eqList^#(lgth(y), Cons(Nil(), Nil())), 439.18/148.14 lgth^#(y)) 439.18/148.14 , gt0^#(Nil(), y) -> c_14() 439.18/148.14 , gt0^#(Cons(x, xs), Nil()) -> c_15() 439.18/148.14 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_16(gt0^#(xs', xs)) 439.18/148.14 , goal^#(x, y) -> c_17(gcd^#(x, y)) } 439.18/148.14 Weak DPs: 439.18/148.14 { gcd[Ite][False][Ite]^#(False(), x, y) -> c_25(gt0^#(x, y)) 439.18/148.14 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 439.18/148.14 c_19(monus^#(xs', xs)) } 439.18/148.14 Weak Trs: 439.18/148.14 { lgth(Nil()) -> Nil() 439.18/148.14 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 439.18/148.14 , monus[Ite](True(), Cons(x, xs), y) -> xs 439.18/148.14 , monus[Ite](False(), Cons(x', xs'), Cons(x, xs)) -> monus(xs', xs) 439.18/148.14 , gcd(Nil(), Nil()) -> Nil() 439.18/148.14 , gcd(Nil(), Cons(x, xs)) -> Nil() 439.18/148.14 , gcd(Cons(x, xs), Nil()) -> Nil() 439.18/148.14 , gcd(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.14 gcd[Ite][False][Ite](eqList(Cons(x', xs'), Cons(x, xs)), 439.18/148.14 Cons(x', xs'), 439.18/148.14 Cons(x, xs)) 439.18/148.14 , @(Nil(), ys) -> ys 439.18/148.14 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 439.18/148.14 , monus(x, y) -> 439.18/148.14 monus[Ite](eqList(lgth(y), Cons(Nil(), Nil())), x, y) 439.18/148.14 , and(True(), True()) -> True() 439.18/148.14 , and(True(), False()) -> False() 439.18/148.14 , and(False(), True()) -> False() 439.18/148.14 , and(False(), False()) -> False() 439.18/148.14 , eqList(Nil(), Nil()) -> True() 439.18/148.14 , eqList(Nil(), Cons(y, ys)) -> False() 439.18/148.14 , eqList(Cons(x, xs), Nil()) -> False() 439.18/148.14 , eqList(Cons(x, xs), Cons(y, ys)) -> 439.18/148.14 and(eqList(x, y), eqList(xs, ys)) 439.18/148.14 , gt0(Nil(), y) -> False() 439.18/148.14 , gt0(Cons(x, xs), Nil()) -> True() 439.18/148.14 , gt0(Cons(x', xs'), Cons(x, xs)) -> gt0(xs', xs) 439.18/148.14 , gcd[Ite][False][Ite](True(), x, y) -> x 439.18/148.14 , gcd[Ite][False][Ite](False(), x, y) -> 439.18/148.14 gcd[Ite][False][Ite][False][Ite](gt0(x, y), x, y) 439.18/148.14 , goal(x, y) -> gcd(x, y) } 439.18/148.14 Obligation: 439.18/148.14 innermost runtime complexity 439.18/148.14 Answer: 439.18/148.14 YES(O(1),O(n^2)) 439.18/148.14 439.18/148.14 Due to missing edges in the dependency-graph, the right-hand sides 439.18/148.14 of following rules could be simplified: 439.18/148.14 439.18/148.14 { eqList^#(Cons(x, xs), Cons(y, ys)) -> 439.18/148.14 c_13(and^#(eqList(x, y), eqList(xs, ys)), 439.18/148.14 eqList^#(x, y), 439.18/148.14 eqList^#(xs, ys)) } 439.18/148.14 439.18/148.14 We are left with following problem, upon which TcT provides the 439.18/148.14 certificate YES(O(1),O(n^2)). 439.18/148.14 439.18/148.14 Strict DPs: 439.18/148.14 { lgth^#(Cons(x, xs)) -> 439.18/148.14 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 439.18/148.14 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 439.18/148.14 , gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.14 c_3(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 439.18/148.14 Cons(x', xs'), 439.18/148.14 Cons(x, xs)), 439.18/148.14 eqList^#(Cons(x', xs'), Cons(x, xs))) 439.18/148.14 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 439.18/148.14 c_4(eqList^#(x, y), eqList^#(xs, ys)) 439.18/148.14 , monus^#(x, y) -> 439.18/148.14 c_5(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.14 eqList^#(lgth(y), Cons(Nil(), Nil())), 439.18/148.14 lgth^#(y)) 439.18/148.14 , gt0^#(Nil(), y) -> c_6() 439.18/148.14 , gt0^#(Cons(x, xs), Nil()) -> c_7() 439.18/148.14 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_8(gt0^#(xs', xs)) 439.18/148.14 , goal^#(x, y) -> c_9(gcd^#(x, y)) } 439.18/148.14 Weak DPs: 439.18/148.14 { gcd[Ite][False][Ite]^#(False(), x, y) -> c_10(gt0^#(x, y)) 439.18/148.14 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 439.18/148.14 c_11(monus^#(xs', xs)) } 439.18/148.14 Weak Trs: 439.18/148.14 { lgth(Nil()) -> Nil() 439.18/148.14 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 439.18/148.14 , monus[Ite](True(), Cons(x, xs), y) -> xs 439.18/148.14 , monus[Ite](False(), Cons(x', xs'), Cons(x, xs)) -> monus(xs', xs) 439.18/148.14 , gcd(Nil(), Nil()) -> Nil() 439.18/148.14 , gcd(Nil(), Cons(x, xs)) -> Nil() 439.18/148.14 , gcd(Cons(x, xs), Nil()) -> Nil() 439.18/148.14 , gcd(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.14 gcd[Ite][False][Ite](eqList(Cons(x', xs'), Cons(x, xs)), 439.18/148.14 Cons(x', xs'), 439.18/148.14 Cons(x, xs)) 439.18/148.14 , @(Nil(), ys) -> ys 439.18/148.14 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 439.18/148.14 , monus(x, y) -> 439.18/148.14 monus[Ite](eqList(lgth(y), Cons(Nil(), Nil())), x, y) 439.18/148.14 , and(True(), True()) -> True() 439.18/148.14 , and(True(), False()) -> False() 439.18/148.14 , and(False(), True()) -> False() 439.18/148.14 , and(False(), False()) -> False() 439.18/148.14 , eqList(Nil(), Nil()) -> True() 439.18/148.14 , eqList(Nil(), Cons(y, ys)) -> False() 439.18/148.14 , eqList(Cons(x, xs), Nil()) -> False() 439.18/148.15 , eqList(Cons(x, xs), Cons(y, ys)) -> 439.18/148.15 and(eqList(x, y), eqList(xs, ys)) 439.18/148.15 , gt0(Nil(), y) -> False() 439.18/148.15 , gt0(Cons(x, xs), Nil()) -> True() 439.18/148.15 , gt0(Cons(x', xs'), Cons(x, xs)) -> gt0(xs', xs) 439.18/148.15 , gcd[Ite][False][Ite](True(), x, y) -> x 439.18/148.15 , gcd[Ite][False][Ite](False(), x, y) -> 439.18/148.15 gcd[Ite][False][Ite][False][Ite](gt0(x, y), x, y) 439.18/148.15 , goal(x, y) -> gcd(x, y) } 439.18/148.15 Obligation: 439.18/148.15 innermost runtime complexity 439.18/148.15 Answer: 439.18/148.15 YES(O(1),O(n^2)) 439.18/148.15 439.18/148.15 We replace rewrite rules by usable rules: 439.18/148.15 439.18/148.15 Weak Usable Rules: 439.18/148.15 { lgth(Nil()) -> Nil() 439.18/148.15 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 439.18/148.15 , @(Nil(), ys) -> ys 439.18/148.15 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 439.18/148.15 , and(True(), True()) -> True() 439.18/148.15 , and(True(), False()) -> False() 439.18/148.15 , and(False(), True()) -> False() 439.18/148.15 , and(False(), False()) -> False() 439.18/148.15 , eqList(Nil(), Nil()) -> True() 439.18/148.15 , eqList(Nil(), Cons(y, ys)) -> False() 439.18/148.15 , eqList(Cons(x, xs), Nil()) -> False() 439.18/148.15 , eqList(Cons(x, xs), Cons(y, ys)) -> 439.18/148.15 and(eqList(x, y), eqList(xs, ys)) } 439.18/148.15 439.18/148.15 We are left with following problem, upon which TcT provides the 439.18/148.15 certificate YES(O(1),O(n^2)). 439.18/148.15 439.18/148.15 Strict DPs: 439.18/148.15 { lgth^#(Cons(x, xs)) -> 439.18/148.15 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 439.18/148.15 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 439.18/148.15 , gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.15 c_3(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 439.18/148.15 Cons(x', xs'), 439.18/148.15 Cons(x, xs)), 439.18/148.15 eqList^#(Cons(x', xs'), Cons(x, xs))) 439.18/148.15 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 439.18/148.15 c_4(eqList^#(x, y), eqList^#(xs, ys)) 439.18/148.15 , monus^#(x, y) -> 439.18/148.15 c_5(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.15 eqList^#(lgth(y), Cons(Nil(), Nil())), 439.18/148.15 lgth^#(y)) 439.18/148.15 , gt0^#(Nil(), y) -> c_6() 439.18/148.15 , gt0^#(Cons(x, xs), Nil()) -> c_7() 439.18/148.15 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_8(gt0^#(xs', xs)) 439.18/148.15 , goal^#(x, y) -> c_9(gcd^#(x, y)) } 439.18/148.15 Weak DPs: 439.18/148.15 { gcd[Ite][False][Ite]^#(False(), x, y) -> c_10(gt0^#(x, y)) 439.18/148.15 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 439.18/148.15 c_11(monus^#(xs', xs)) } 439.18/148.15 Weak Trs: 439.18/148.15 { lgth(Nil()) -> Nil() 439.18/148.15 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 439.18/148.15 , @(Nil(), ys) -> ys 439.18/148.15 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 439.18/148.15 , and(True(), True()) -> True() 439.18/148.15 , and(True(), False()) -> False() 439.18/148.15 , and(False(), True()) -> False() 439.18/148.15 , and(False(), False()) -> False() 439.18/148.15 , eqList(Nil(), Nil()) -> True() 439.18/148.15 , eqList(Nil(), Cons(y, ys)) -> False() 439.18/148.15 , eqList(Cons(x, xs), Nil()) -> False() 439.18/148.15 , eqList(Cons(x, xs), Cons(y, ys)) -> 439.18/148.15 and(eqList(x, y), eqList(xs, ys)) } 439.18/148.15 Obligation: 439.18/148.15 innermost runtime complexity 439.18/148.15 Answer: 439.18/148.15 YES(O(1),O(n^2)) 439.18/148.15 439.18/148.15 Consider the dependency graph 439.18/148.15 439.18/148.15 1: lgth^#(Cons(x, xs)) -> 439.18/148.15 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 439.18/148.15 -->_1 @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) :2 439.18/148.15 -->_2 lgth^#(Cons(x, xs)) -> 439.18/148.15 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) :1 439.18/148.15 439.18/148.15 2: @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 439.18/148.15 -->_1 @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) :2 439.18/148.15 439.18/148.15 3: gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.15 c_3(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 439.18/148.15 Cons(x', xs'), 439.18/148.15 Cons(x, xs)), 439.18/148.15 eqList^#(Cons(x', xs'), Cons(x, xs))) 439.18/148.15 -->_1 gcd[Ite][False][Ite]^#(False(), x, y) -> 439.18/148.15 c_10(gt0^#(x, y)) :10 439.18/148.15 -->_2 eqList^#(Cons(x, xs), Cons(y, ys)) -> 439.18/148.15 c_4(eqList^#(x, y), eqList^#(xs, ys)) :4 439.18/148.15 439.18/148.15 4: eqList^#(Cons(x, xs), Cons(y, ys)) -> 439.18/148.15 c_4(eqList^#(x, y), eqList^#(xs, ys)) 439.18/148.15 -->_2 eqList^#(Cons(x, xs), Cons(y, ys)) -> 439.18/148.15 c_4(eqList^#(x, y), eqList^#(xs, ys)) :4 439.18/148.15 -->_1 eqList^#(Cons(x, xs), Cons(y, ys)) -> 439.18/148.15 c_4(eqList^#(x, y), eqList^#(xs, ys)) :4 439.18/148.15 439.18/148.15 5: monus^#(x, y) -> 439.18/148.15 c_5(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.15 eqList^#(lgth(y), Cons(Nil(), Nil())), 439.18/148.15 lgth^#(y)) 439.18/148.15 -->_1 monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 439.18/148.15 c_11(monus^#(xs', xs)) :11 439.18/148.15 -->_2 eqList^#(Cons(x, xs), Cons(y, ys)) -> 439.18/148.15 c_4(eqList^#(x, y), eqList^#(xs, ys)) :4 439.18/148.15 -->_3 lgth^#(Cons(x, xs)) -> 439.18/148.15 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) :1 439.18/148.15 439.18/148.15 6: gt0^#(Nil(), y) -> c_6() 439.18/148.15 439.18/148.15 7: gt0^#(Cons(x, xs), Nil()) -> c_7() 439.18/148.15 439.18/148.15 8: gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_8(gt0^#(xs', xs)) 439.18/148.15 -->_1 gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_8(gt0^#(xs', xs)) :8 439.18/148.15 -->_1 gt0^#(Cons(x, xs), Nil()) -> c_7() :7 439.18/148.15 -->_1 gt0^#(Nil(), y) -> c_6() :6 439.18/148.15 439.18/148.15 9: goal^#(x, y) -> c_9(gcd^#(x, y)) 439.18/148.15 -->_1 gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.15 c_3(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 439.18/148.15 Cons(x', xs'), 439.18/148.15 Cons(x, xs)), 439.18/148.15 eqList^#(Cons(x', xs'), Cons(x, xs))) :3 439.18/148.15 439.18/148.15 10: gcd[Ite][False][Ite]^#(False(), x, y) -> c_10(gt0^#(x, y)) 439.18/148.15 -->_1 gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_8(gt0^#(xs', xs)) :8 439.18/148.15 -->_1 gt0^#(Cons(x, xs), Nil()) -> c_7() :7 439.18/148.15 -->_1 gt0^#(Nil(), y) -> c_6() :6 439.18/148.15 439.18/148.15 11: monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 439.18/148.15 c_11(monus^#(xs', xs)) 439.18/148.15 -->_1 monus^#(x, y) -> 439.18/148.15 c_5(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.15 eqList^#(lgth(y), Cons(Nil(), Nil())), 439.18/148.15 lgth^#(y)) :5 439.18/148.15 439.18/148.15 439.18/148.15 Following roots of the dependency graph are removed, as the 439.18/148.15 considered set of starting terms is closed under reduction with 439.18/148.15 respect to these rules (modulo compound contexts). 439.18/148.15 439.18/148.15 { goal^#(x, y) -> c_9(gcd^#(x, y)) } 439.18/148.15 439.18/148.15 439.18/148.15 We are left with following problem, upon which TcT provides the 439.18/148.15 certificate YES(O(1),O(n^2)). 439.18/148.15 439.18/148.15 Strict DPs: 439.18/148.15 { lgth^#(Cons(x, xs)) -> 439.18/148.15 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 439.18/148.15 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 439.18/148.15 , gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.15 c_3(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 439.18/148.15 Cons(x', xs'), 439.18/148.15 Cons(x, xs)), 439.18/148.15 eqList^#(Cons(x', xs'), Cons(x, xs))) 439.18/148.15 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 439.18/148.15 c_4(eqList^#(x, y), eqList^#(xs, ys)) 439.18/148.15 , monus^#(x, y) -> 439.18/148.15 c_5(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.15 eqList^#(lgth(y), Cons(Nil(), Nil())), 439.18/148.15 lgth^#(y)) 439.18/148.15 , gt0^#(Nil(), y) -> c_6() 439.18/148.15 , gt0^#(Cons(x, xs), Nil()) -> c_7() 439.18/148.15 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_8(gt0^#(xs', xs)) } 439.18/148.15 Weak DPs: 439.18/148.15 { gcd[Ite][False][Ite]^#(False(), x, y) -> c_10(gt0^#(x, y)) 439.18/148.15 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 439.18/148.15 c_11(monus^#(xs', xs)) } 439.18/148.15 Weak Trs: 439.18/148.15 { lgth(Nil()) -> Nil() 439.18/148.15 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 439.18/148.15 , @(Nil(), ys) -> ys 439.18/148.15 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 439.18/148.15 , and(True(), True()) -> True() 439.18/148.15 , and(True(), False()) -> False() 439.18/148.15 , and(False(), True()) -> False() 439.18/148.15 , and(False(), False()) -> False() 439.18/148.15 , eqList(Nil(), Nil()) -> True() 439.18/148.15 , eqList(Nil(), Cons(y, ys)) -> False() 439.18/148.15 , eqList(Cons(x, xs), Nil()) -> False() 439.18/148.15 , eqList(Cons(x, xs), Cons(y, ys)) -> 439.18/148.15 and(eqList(x, y), eqList(xs, ys)) } 439.18/148.15 Obligation: 439.18/148.15 innermost runtime complexity 439.18/148.15 Answer: 439.18/148.15 YES(O(1),O(n^2)) 439.18/148.15 439.18/148.15 We analyse the complexity of following sub-problems (R) and (S). 439.18/148.15 Problem (S) is obtained from the input problem by shifting strict 439.18/148.15 rules from (R) into the weak component: 439.18/148.15 439.18/148.15 Problem (R): 439.18/148.15 ------------ 439.18/148.15 Strict DPs: 439.18/148.15 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.15 c_3(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 439.18/148.15 Cons(x', xs'), 439.18/148.15 Cons(x, xs)), 439.18/148.15 eqList^#(Cons(x', xs'), Cons(x, xs))) 439.18/148.15 , gt0^#(Nil(), y) -> c_6() 439.18/148.15 , gt0^#(Cons(x, xs), Nil()) -> c_7() 439.18/148.15 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_8(gt0^#(xs', xs)) } 439.18/148.15 Weak DPs: 439.18/148.15 { lgth^#(Cons(x, xs)) -> 439.18/148.15 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 439.18/148.15 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 439.18/148.15 , gcd[Ite][False][Ite]^#(False(), x, y) -> c_10(gt0^#(x, y)) 439.18/148.15 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 439.18/148.15 c_4(eqList^#(x, y), eqList^#(xs, ys)) 439.18/148.15 , monus^#(x, y) -> 439.18/148.15 c_5(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.15 eqList^#(lgth(y), Cons(Nil(), Nil())), 439.18/148.15 lgth^#(y)) 439.18/148.15 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 439.18/148.15 c_11(monus^#(xs', xs)) } 439.18/148.15 Weak Trs: 439.18/148.15 { lgth(Nil()) -> Nil() 439.18/148.15 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 439.18/148.15 , @(Nil(), ys) -> ys 439.18/148.15 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 439.18/148.15 , and(True(), True()) -> True() 439.18/148.15 , and(True(), False()) -> False() 439.18/148.15 , and(False(), True()) -> False() 439.18/148.15 , and(False(), False()) -> False() 439.18/148.15 , eqList(Nil(), Nil()) -> True() 439.18/148.15 , eqList(Nil(), Cons(y, ys)) -> False() 439.18/148.15 , eqList(Cons(x, xs), Nil()) -> False() 439.18/148.15 , eqList(Cons(x, xs), Cons(y, ys)) -> 439.18/148.15 and(eqList(x, y), eqList(xs, ys)) } 439.18/148.15 StartTerms: basic terms 439.18/148.15 Strategy: innermost 439.18/148.15 439.18/148.15 Problem (S): 439.18/148.15 ------------ 439.18/148.15 Strict DPs: 439.18/148.15 { lgth^#(Cons(x, xs)) -> 439.18/148.15 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 439.18/148.15 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 439.18/148.15 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 439.18/148.15 c_4(eqList^#(x, y), eqList^#(xs, ys)) 439.18/148.15 , monus^#(x, y) -> 439.18/148.15 c_5(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.15 eqList^#(lgth(y), Cons(Nil(), Nil())), 439.18/148.15 lgth^#(y)) } 439.18/148.15 Weak DPs: 439.18/148.15 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.15 c_3(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 439.18/148.15 Cons(x', xs'), 439.18/148.15 Cons(x, xs)), 439.18/148.15 eqList^#(Cons(x', xs'), Cons(x, xs))) 439.18/148.15 , gcd[Ite][False][Ite]^#(False(), x, y) -> c_10(gt0^#(x, y)) 439.18/148.15 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 439.18/148.15 c_11(monus^#(xs', xs)) 439.18/148.15 , gt0^#(Nil(), y) -> c_6() 439.18/148.15 , gt0^#(Cons(x, xs), Nil()) -> c_7() 439.18/148.15 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_8(gt0^#(xs', xs)) } 439.18/148.15 Weak Trs: 439.18/148.15 { lgth(Nil()) -> Nil() 439.18/148.15 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 439.18/148.15 , @(Nil(), ys) -> ys 439.18/148.15 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 439.18/148.15 , and(True(), True()) -> True() 439.18/148.15 , and(True(), False()) -> False() 439.18/148.15 , and(False(), True()) -> False() 439.18/148.15 , and(False(), False()) -> False() 439.18/148.15 , eqList(Nil(), Nil()) -> True() 439.18/148.15 , eqList(Nil(), Cons(y, ys)) -> False() 439.18/148.15 , eqList(Cons(x, xs), Nil()) -> False() 439.18/148.15 , eqList(Cons(x, xs), Cons(y, ys)) -> 439.18/148.15 and(eqList(x, y), eqList(xs, ys)) } 439.18/148.15 StartTerms: basic terms 439.18/148.15 Strategy: innermost 439.18/148.15 439.18/148.15 Overall, the transformation results in the following sub-problem(s): 439.18/148.15 439.18/148.15 Generated new problems: 439.18/148.15 ----------------------- 439.18/148.15 R) Strict DPs: 439.18/148.15 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.15 c_3(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 439.18/148.15 Cons(x', xs'), 439.18/148.15 Cons(x, xs)), 439.18/148.15 eqList^#(Cons(x', xs'), Cons(x, xs))) 439.18/148.15 , gt0^#(Nil(), y) -> c_6() 439.18/148.15 , gt0^#(Cons(x, xs), Nil()) -> c_7() 439.18/148.15 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_8(gt0^#(xs', xs)) } 439.18/148.15 Weak DPs: 439.18/148.15 { lgth^#(Cons(x, xs)) -> 439.18/148.15 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 439.18/148.15 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 439.18/148.15 , gcd[Ite][False][Ite]^#(False(), x, y) -> c_10(gt0^#(x, y)) 439.18/148.15 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 439.18/148.15 c_4(eqList^#(x, y), eqList^#(xs, ys)) 439.18/148.15 , monus^#(x, y) -> 439.18/148.15 c_5(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.15 eqList^#(lgth(y), Cons(Nil(), Nil())), 439.18/148.15 lgth^#(y)) 439.18/148.15 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 439.18/148.15 c_11(monus^#(xs', xs)) } 439.18/148.15 Weak Trs: 439.18/148.15 { lgth(Nil()) -> Nil() 439.18/148.15 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 439.18/148.15 , @(Nil(), ys) -> ys 439.18/148.15 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 439.18/148.15 , and(True(), True()) -> True() 439.18/148.15 , and(True(), False()) -> False() 439.18/148.15 , and(False(), True()) -> False() 439.18/148.15 , and(False(), False()) -> False() 439.18/148.15 , eqList(Nil(), Nil()) -> True() 439.18/148.15 , eqList(Nil(), Cons(y, ys)) -> False() 439.18/148.15 , eqList(Cons(x, xs), Nil()) -> False() 439.18/148.15 , eqList(Cons(x, xs), Cons(y, ys)) -> 439.18/148.15 and(eqList(x, y), eqList(xs, ys)) } 439.18/148.15 StartTerms: basic terms 439.18/148.15 Strategy: innermost 439.18/148.15 439.18/148.15 This problem was proven YES(O(1),O(n^1)). 439.18/148.15 439.18/148.15 S) Strict DPs: 439.18/148.15 { lgth^#(Cons(x, xs)) -> 439.18/148.15 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 439.18/148.15 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 439.18/148.15 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 439.18/148.15 c_4(eqList^#(x, y), eqList^#(xs, ys)) 439.18/148.15 , monus^#(x, y) -> 439.18/148.15 c_5(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.15 eqList^#(lgth(y), Cons(Nil(), Nil())), 439.18/148.15 lgth^#(y)) } 439.18/148.15 Weak DPs: 439.18/148.15 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.15 c_3(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 439.18/148.15 Cons(x', xs'), 439.18/148.15 Cons(x, xs)), 439.18/148.15 eqList^#(Cons(x', xs'), Cons(x, xs))) 439.18/148.15 , gcd[Ite][False][Ite]^#(False(), x, y) -> c_10(gt0^#(x, y)) 439.18/148.15 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 439.18/148.15 c_11(monus^#(xs', xs)) 439.18/148.15 , gt0^#(Nil(), y) -> c_6() 439.18/148.15 , gt0^#(Cons(x, xs), Nil()) -> c_7() 439.18/148.15 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_8(gt0^#(xs', xs)) } 439.18/148.15 Weak Trs: 439.18/148.15 { lgth(Nil()) -> Nil() 439.18/148.15 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 439.18/148.15 , @(Nil(), ys) -> ys 439.18/148.15 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 439.18/148.15 , and(True(), True()) -> True() 439.18/148.15 , and(True(), False()) -> False() 439.18/148.15 , and(False(), True()) -> False() 439.18/148.15 , and(False(), False()) -> False() 439.18/148.15 , eqList(Nil(), Nil()) -> True() 439.18/148.15 , eqList(Nil(), Cons(y, ys)) -> False() 439.18/148.15 , eqList(Cons(x, xs), Nil()) -> False() 439.18/148.15 , eqList(Cons(x, xs), Cons(y, ys)) -> 439.18/148.15 and(eqList(x, y), eqList(xs, ys)) } 439.18/148.15 StartTerms: basic terms 439.18/148.15 Strategy: innermost 439.18/148.15 439.18/148.15 This problem was proven YES(O(1),O(n^2)). 439.18/148.15 439.18/148.15 439.18/148.15 Proofs for generated problems: 439.18/148.15 ------------------------------ 439.18/148.15 R) We are left with following problem, upon which TcT provides the 439.18/148.15 certificate YES(O(1),O(n^1)). 439.18/148.15 439.18/148.15 Strict DPs: 439.18/148.15 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.15 c_3(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 439.18/148.15 Cons(x', xs'), 439.18/148.15 Cons(x, xs)), 439.18/148.15 eqList^#(Cons(x', xs'), Cons(x, xs))) 439.18/148.15 , gt0^#(Nil(), y) -> c_6() 439.18/148.15 , gt0^#(Cons(x, xs), Nil()) -> c_7() 439.18/148.15 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_8(gt0^#(xs', xs)) } 439.18/148.15 Weak DPs: 439.18/148.15 { lgth^#(Cons(x, xs)) -> 439.18/148.15 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 439.18/148.15 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 439.18/148.15 , gcd[Ite][False][Ite]^#(False(), x, y) -> c_10(gt0^#(x, y)) 439.18/148.15 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 439.18/148.15 c_4(eqList^#(x, y), eqList^#(xs, ys)) 439.18/148.15 , monus^#(x, y) -> 439.18/148.15 c_5(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.15 eqList^#(lgth(y), Cons(Nil(), Nil())), 439.18/148.15 lgth^#(y)) 439.18/148.15 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 439.18/148.15 c_11(monus^#(xs', xs)) } 439.18/148.15 Weak Trs: 439.18/148.15 { lgth(Nil()) -> Nil() 439.18/148.15 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 439.18/148.15 , @(Nil(), ys) -> ys 439.18/148.15 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 439.18/148.15 , and(True(), True()) -> True() 439.18/148.15 , and(True(), False()) -> False() 439.18/148.15 , and(False(), True()) -> False() 439.18/148.15 , and(False(), False()) -> False() 439.18/148.15 , eqList(Nil(), Nil()) -> True() 439.18/148.15 , eqList(Nil(), Cons(y, ys)) -> False() 439.18/148.15 , eqList(Cons(x, xs), Nil()) -> False() 439.18/148.15 , eqList(Cons(x, xs), Cons(y, ys)) -> 439.18/148.15 and(eqList(x, y), eqList(xs, ys)) } 439.18/148.15 Obligation: 439.18/148.15 innermost runtime complexity 439.18/148.15 Answer: 439.18/148.15 YES(O(1),O(n^1)) 439.18/148.15 439.18/148.15 We estimate the number of application of {1} by applications of 439.18/148.15 Pre({1}) = {}. Here rules are labeled as follows: 439.18/148.15 439.18/148.15 DPs: 439.18/148.15 { 1: gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.15 c_3(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 439.18/148.15 Cons(x', xs'), 439.18/148.15 Cons(x, xs)), 439.18/148.15 eqList^#(Cons(x', xs'), Cons(x, xs))) 439.18/148.15 , 2: gt0^#(Nil(), y) -> c_6() 439.18/148.15 , 3: gt0^#(Cons(x, xs), Nil()) -> c_7() 439.18/148.15 , 4: gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_8(gt0^#(xs', xs)) 439.18/148.15 , 5: lgth^#(Cons(x, xs)) -> 439.18/148.15 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 439.18/148.16 , 6: @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 439.18/148.16 , 7: gcd[Ite][False][Ite]^#(False(), x, y) -> c_10(gt0^#(x, y)) 439.18/148.16 , 8: eqList^#(Cons(x, xs), Cons(y, ys)) -> 439.18/148.16 c_4(eqList^#(x, y), eqList^#(xs, ys)) 439.18/148.16 , 9: monus^#(x, y) -> 439.18/148.16 c_5(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.16 eqList^#(lgth(y), Cons(Nil(), Nil())), 439.18/148.16 lgth^#(y)) 439.18/148.16 , 10: monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 439.18/148.16 c_11(monus^#(xs', xs)) } 439.18/148.16 439.18/148.16 We are left with following problem, upon which TcT provides the 439.18/148.16 certificate YES(O(1),O(n^1)). 439.18/148.16 439.18/148.16 Strict DPs: 439.18/148.16 { gt0^#(Nil(), y) -> c_6() 439.18/148.16 , gt0^#(Cons(x, xs), Nil()) -> c_7() 439.18/148.16 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_8(gt0^#(xs', xs)) } 439.18/148.16 Weak DPs: 439.18/148.16 { lgth^#(Cons(x, xs)) -> 439.18/148.16 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 439.18/148.16 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 439.18/148.16 , gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.16 c_3(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 439.18/148.16 Cons(x', xs'), 439.18/148.16 Cons(x, xs)), 439.18/148.16 eqList^#(Cons(x', xs'), Cons(x, xs))) 439.18/148.16 , gcd[Ite][False][Ite]^#(False(), x, y) -> c_10(gt0^#(x, y)) 439.18/148.16 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 439.18/148.16 c_4(eqList^#(x, y), eqList^#(xs, ys)) 439.18/148.16 , monus^#(x, y) -> 439.18/148.16 c_5(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.16 eqList^#(lgth(y), Cons(Nil(), Nil())), 439.18/148.16 lgth^#(y)) 439.18/148.16 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 439.18/148.16 c_11(monus^#(xs', xs)) } 439.18/148.16 Weak Trs: 439.18/148.16 { lgth(Nil()) -> Nil() 439.18/148.16 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 439.18/148.16 , @(Nil(), ys) -> ys 439.18/148.16 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 439.18/148.16 , and(True(), True()) -> True() 439.18/148.16 , and(True(), False()) -> False() 439.18/148.16 , and(False(), True()) -> False() 439.18/148.16 , and(False(), False()) -> False() 439.18/148.16 , eqList(Nil(), Nil()) -> True() 439.18/148.16 , eqList(Nil(), Cons(y, ys)) -> False() 439.18/148.16 , eqList(Cons(x, xs), Nil()) -> False() 439.18/148.16 , eqList(Cons(x, xs), Cons(y, ys)) -> 439.18/148.16 and(eqList(x, y), eqList(xs, ys)) } 439.18/148.16 Obligation: 439.18/148.16 innermost runtime complexity 439.18/148.16 Answer: 439.18/148.16 YES(O(1),O(n^1)) 439.18/148.16 439.18/148.16 The following weak DPs constitute a sub-graph of the DG that is 439.18/148.16 closed under successors. The DPs are removed. 439.18/148.16 439.18/148.16 { lgth^#(Cons(x, xs)) -> 439.18/148.16 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 439.18/148.16 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 439.18/148.16 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 439.18/148.16 c_4(eqList^#(x, y), eqList^#(xs, ys)) 439.18/148.16 , monus^#(x, y) -> 439.18/148.16 c_5(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.16 eqList^#(lgth(y), Cons(Nil(), Nil())), 439.18/148.16 lgth^#(y)) 439.18/148.16 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 439.18/148.16 c_11(monus^#(xs', xs)) } 439.18/148.16 439.18/148.16 We are left with following problem, upon which TcT provides the 439.18/148.16 certificate YES(O(1),O(n^1)). 439.18/148.16 439.18/148.16 Strict DPs: 439.18/148.16 { gt0^#(Nil(), y) -> c_6() 439.18/148.16 , gt0^#(Cons(x, xs), Nil()) -> c_7() 439.18/148.16 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_8(gt0^#(xs', xs)) } 439.18/148.16 Weak DPs: 439.18/148.16 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.16 c_3(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 439.18/148.16 Cons(x', xs'), 439.18/148.16 Cons(x, xs)), 439.18/148.16 eqList^#(Cons(x', xs'), Cons(x, xs))) 439.18/148.16 , gcd[Ite][False][Ite]^#(False(), x, y) -> c_10(gt0^#(x, y)) } 439.18/148.16 Weak Trs: 439.18/148.16 { lgth(Nil()) -> Nil() 439.18/148.16 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 439.18/148.16 , @(Nil(), ys) -> ys 439.18/148.16 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 439.18/148.16 , and(True(), True()) -> True() 439.18/148.16 , and(True(), False()) -> False() 439.18/148.16 , and(False(), True()) -> False() 439.18/148.16 , and(False(), False()) -> False() 439.18/148.16 , eqList(Nil(), Nil()) -> True() 439.18/148.16 , eqList(Nil(), Cons(y, ys)) -> False() 439.18/148.16 , eqList(Cons(x, xs), Nil()) -> False() 439.18/148.16 , eqList(Cons(x, xs), Cons(y, ys)) -> 439.18/148.16 and(eqList(x, y), eqList(xs, ys)) } 439.18/148.16 Obligation: 439.18/148.16 innermost runtime complexity 439.18/148.16 Answer: 439.18/148.16 YES(O(1),O(n^1)) 439.18/148.16 439.18/148.16 Due to missing edges in the dependency-graph, the right-hand sides 439.18/148.16 of following rules could be simplified: 439.18/148.16 439.18/148.16 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.16 c_3(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 439.18/148.16 Cons(x', xs'), 439.18/148.16 Cons(x, xs)), 439.18/148.16 eqList^#(Cons(x', xs'), Cons(x, xs))) } 439.18/148.16 439.18/148.16 We are left with following problem, upon which TcT provides the 439.18/148.16 certificate YES(O(1),O(n^1)). 439.18/148.16 439.18/148.16 Strict DPs: 439.18/148.16 { gt0^#(Nil(), y) -> c_1() 439.18/148.16 , gt0^#(Cons(x, xs), Nil()) -> c_2() 439.18/148.16 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_3(gt0^#(xs', xs)) } 439.18/148.16 Weak DPs: 439.18/148.16 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.16 c_4(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 439.18/148.16 Cons(x', xs'), 439.18/148.16 Cons(x, xs))) 439.18/148.16 , gcd[Ite][False][Ite]^#(False(), x, y) -> c_5(gt0^#(x, y)) } 439.18/148.16 Weak Trs: 439.18/148.16 { lgth(Nil()) -> Nil() 439.18/148.16 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 439.18/148.16 , @(Nil(), ys) -> ys 439.18/148.16 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 439.18/148.16 , and(True(), True()) -> True() 439.18/148.16 , and(True(), False()) -> False() 439.18/148.16 , and(False(), True()) -> False() 439.18/148.16 , and(False(), False()) -> False() 439.18/148.16 , eqList(Nil(), Nil()) -> True() 439.18/148.16 , eqList(Nil(), Cons(y, ys)) -> False() 439.18/148.16 , eqList(Cons(x, xs), Nil()) -> False() 439.18/148.16 , eqList(Cons(x, xs), Cons(y, ys)) -> 439.18/148.16 and(eqList(x, y), eqList(xs, ys)) } 439.18/148.16 Obligation: 439.18/148.16 innermost runtime complexity 439.18/148.16 Answer: 439.18/148.16 YES(O(1),O(n^1)) 439.18/148.16 439.18/148.16 We replace rewrite rules by usable rules: 439.18/148.16 439.18/148.16 Weak Usable Rules: 439.18/148.16 { and(True(), True()) -> True() 439.18/148.16 , and(True(), False()) -> False() 439.18/148.16 , and(False(), True()) -> False() 439.18/148.16 , and(False(), False()) -> False() 439.18/148.16 , eqList(Nil(), Nil()) -> True() 439.18/148.16 , eqList(Nil(), Cons(y, ys)) -> False() 439.18/148.16 , eqList(Cons(x, xs), Nil()) -> False() 439.18/148.16 , eqList(Cons(x, xs), Cons(y, ys)) -> 439.18/148.16 and(eqList(x, y), eqList(xs, ys)) } 439.18/148.16 439.18/148.16 We are left with following problem, upon which TcT provides the 439.18/148.16 certificate YES(O(1),O(n^1)). 439.18/148.16 439.18/148.16 Strict DPs: 439.18/148.16 { gt0^#(Nil(), y) -> c_1() 439.18/148.16 , gt0^#(Cons(x, xs), Nil()) -> c_2() 439.18/148.16 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_3(gt0^#(xs', xs)) } 439.18/148.16 Weak DPs: 439.18/148.16 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.16 c_4(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 439.18/148.16 Cons(x', xs'), 439.18/148.16 Cons(x, xs))) 439.18/148.16 , gcd[Ite][False][Ite]^#(False(), x, y) -> c_5(gt0^#(x, y)) } 439.18/148.16 Weak Trs: 439.18/148.16 { and(True(), True()) -> True() 439.18/148.16 , and(True(), False()) -> False() 439.18/148.16 , and(False(), True()) -> False() 439.18/148.16 , and(False(), False()) -> False() 439.18/148.16 , eqList(Nil(), Nil()) -> True() 439.18/148.16 , eqList(Nil(), Cons(y, ys)) -> False() 439.18/148.16 , eqList(Cons(x, xs), Nil()) -> False() 439.18/148.16 , eqList(Cons(x, xs), Cons(y, ys)) -> 439.18/148.16 and(eqList(x, y), eqList(xs, ys)) } 439.18/148.16 Obligation: 439.18/148.16 innermost runtime complexity 439.18/148.16 Answer: 439.18/148.16 YES(O(1),O(n^1)) 439.18/148.16 439.18/148.16 We use the processor 'Small Polynomial Path Order (PS,1-bounded)' 439.18/148.16 to orient following rules strictly. 439.18/148.16 439.18/148.16 DPs: 439.18/148.16 { 1: gt0^#(Nil(), y) -> c_1() 439.18/148.16 , 2: gt0^#(Cons(x, xs), Nil()) -> c_2() 439.18/148.16 , 3: gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_3(gt0^#(xs', xs)) 439.18/148.16 , 4: gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.16 c_4(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 439.18/148.16 Cons(x', xs'), 439.18/148.16 Cons(x, xs))) 439.18/148.16 , 5: gcd[Ite][False][Ite]^#(False(), x, y) -> c_5(gt0^#(x, y)) } 439.18/148.16 Trs: 439.18/148.16 { and(True(), True()) -> True() 439.18/148.16 , and(True(), False()) -> False() 439.18/148.16 , and(False(), True()) -> False() 439.18/148.16 , and(False(), False()) -> False() 439.18/148.16 , eqList(Nil(), Nil()) -> True() 439.18/148.16 , eqList(Nil(), Cons(y, ys)) -> False() 439.18/148.16 , eqList(Cons(x, xs), Nil()) -> False() } 439.18/148.16 439.18/148.16 Sub-proof: 439.18/148.16 ---------- 439.18/148.16 The input was oriented with the instance of 'Small Polynomial Path 439.18/148.16 Order (PS,1-bounded)' as induced by the safe mapping 439.18/148.16 439.18/148.16 safe(True) = {}, safe(Nil) = {}, safe(and) = {}, safe(eqList) = {}, 439.18/148.16 safe(Cons) = {1, 2}, safe(False) = {}, safe(gcd^#) = {}, 439.18/148.16 safe(gcd[Ite][False][Ite]^#) = {}, safe(gt0^#) = {2}, 439.18/148.16 safe(c_1) = {}, safe(c_2) = {}, safe(c_3) = {}, safe(c_4) = {}, 439.18/148.16 safe(c_5) = {} 439.18/148.16 439.18/148.16 and precedence 439.18/148.16 439.18/148.16 eqList > and, gcd^# > and, gcd^# > gcd[Ite][False][Ite]^#, 439.18/148.16 gcd^# > gt0^#, gcd[Ite][False][Ite]^# > and, 439.18/148.16 gcd[Ite][False][Ite]^# > gt0^# . 439.18/148.16 439.18/148.16 Following symbols are considered recursive: 439.18/148.16 439.18/148.16 {gt0^#} 439.18/148.16 439.18/148.16 The recursion depth is 1. 439.18/148.16 439.18/148.16 Further, following argument filtering is employed: 439.18/148.16 439.18/148.16 pi(True) = [], pi(Nil) = [], pi(and) = [1], pi(eqList) = [], 439.18/148.16 pi(Cons) = [2], pi(False) = [], pi(gcd^#) = [1], 439.18/148.16 pi(gcd[Ite][False][Ite]^#) = [2], pi(gt0^#) = [1], pi(c_1) = [], 439.18/148.16 pi(c_2) = [], pi(c_3) = [1], pi(c_4) = [1], pi(c_5) = [1] 439.18/148.16 439.18/148.16 Usable defined function symbols are a subset of: 439.18/148.16 439.18/148.16 {and, gcd^#, gcd[Ite][False][Ite]^#, gt0^#} 439.18/148.16 439.18/148.16 For your convenience, here are the satisfied ordering constraints: 439.18/148.16 439.18/148.16 pi(gcd^#(Cons(x', xs'), Cons(x, xs))) = gcd^#(Cons(; xs');) 439.18/148.16 > c_4(gcd[Ite][False][Ite]^#(Cons(; xs'););) 439.18/148.16 = pi(c_4(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 439.18/148.16 Cons(x', xs'), 439.18/148.16 Cons(x, xs)))) 439.18/148.16 439.18/148.16 pi(gcd[Ite][False][Ite]^#(False(), x, y)) = gcd[Ite][False][Ite]^#(x;) 439.18/148.16 > c_5(gt0^#(x;);) 439.18/148.16 = pi(c_5(gt0^#(x, y))) 439.18/148.16 439.18/148.16 pi(gt0^#(Nil(), y)) = gt0^#(Nil();) 439.18/148.16 > c_1() 439.18/148.16 = pi(c_1()) 439.18/148.16 439.18/148.16 pi(gt0^#(Cons(x, xs), Nil())) = gt0^#(Cons(; xs);) 439.18/148.16 > c_2() 439.18/148.16 = pi(c_2()) 439.18/148.16 439.18/148.16 pi(gt0^#(Cons(x', xs'), Cons(x, xs))) = gt0^#(Cons(; xs');) 439.18/148.16 > c_3(gt0^#(xs';);) 439.18/148.16 = pi(c_3(gt0^#(xs', xs))) 439.18/148.16 439.18/148.16 pi(and(True(), True())) = and(True();) 439.18/148.16 > True() 439.18/148.16 = pi(True()) 439.18/148.16 439.18/148.16 pi(and(True(), False())) = and(True();) 439.18/148.16 > False() 439.18/148.16 = pi(False()) 439.18/148.16 439.18/148.16 pi(and(False(), True())) = and(False();) 439.18/148.16 > False() 439.18/148.16 = pi(False()) 439.18/148.16 439.18/148.16 pi(and(False(), False())) = and(False();) 439.18/148.16 > False() 439.18/148.16 = pi(False()) 439.18/148.16 439.18/148.16 439.18/148.16 The strictly oriented rules are moved into the weak component. 439.18/148.16 439.18/148.16 We are left with following problem, upon which TcT provides the 439.18/148.16 certificate YES(O(1),O(1)). 439.18/148.16 439.18/148.16 Weak DPs: 439.18/148.16 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.16 c_4(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 439.18/148.16 Cons(x', xs'), 439.18/148.16 Cons(x, xs))) 439.18/148.16 , gcd[Ite][False][Ite]^#(False(), x, y) -> c_5(gt0^#(x, y)) 439.18/148.16 , gt0^#(Nil(), y) -> c_1() 439.18/148.16 , gt0^#(Cons(x, xs), Nil()) -> c_2() 439.18/148.16 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_3(gt0^#(xs', xs)) } 439.18/148.16 Weak Trs: 439.18/148.16 { and(True(), True()) -> True() 439.18/148.16 , and(True(), False()) -> False() 439.18/148.16 , and(False(), True()) -> False() 439.18/148.16 , and(False(), False()) -> False() 439.18/148.16 , eqList(Nil(), Nil()) -> True() 439.18/148.16 , eqList(Nil(), Cons(y, ys)) -> False() 439.18/148.16 , eqList(Cons(x, xs), Nil()) -> False() 439.18/148.16 , eqList(Cons(x, xs), Cons(y, ys)) -> 439.18/148.16 and(eqList(x, y), eqList(xs, ys)) } 439.18/148.16 Obligation: 439.18/148.16 innermost runtime complexity 439.18/148.16 Answer: 439.18/148.16 YES(O(1),O(1)) 439.18/148.16 439.18/148.16 The following weak DPs constitute a sub-graph of the DG that is 439.18/148.16 closed under successors. The DPs are removed. 439.18/148.16 439.18/148.16 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.16 c_4(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 439.18/148.16 Cons(x', xs'), 439.18/148.16 Cons(x, xs))) 439.18/148.16 , gcd[Ite][False][Ite]^#(False(), x, y) -> c_5(gt0^#(x, y)) 439.18/148.16 , gt0^#(Nil(), y) -> c_1() 439.18/148.16 , gt0^#(Cons(x, xs), Nil()) -> c_2() 439.18/148.16 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_3(gt0^#(xs', xs)) } 439.18/148.16 439.18/148.16 We are left with following problem, upon which TcT provides the 439.18/148.16 certificate YES(O(1),O(1)). 439.18/148.16 439.18/148.16 Weak Trs: 439.18/148.16 { and(True(), True()) -> True() 439.18/148.16 , and(True(), False()) -> False() 439.18/148.16 , and(False(), True()) -> False() 439.18/148.16 , and(False(), False()) -> False() 439.18/148.16 , eqList(Nil(), Nil()) -> True() 439.18/148.16 , eqList(Nil(), Cons(y, ys)) -> False() 439.18/148.16 , eqList(Cons(x, xs), Nil()) -> False() 439.18/148.16 , eqList(Cons(x, xs), Cons(y, ys)) -> 439.18/148.16 and(eqList(x, y), eqList(xs, ys)) } 439.18/148.16 Obligation: 439.18/148.16 innermost runtime complexity 439.18/148.16 Answer: 439.18/148.16 YES(O(1),O(1)) 439.18/148.16 439.18/148.16 No rule is usable, rules are removed from the input problem. 439.18/148.16 439.18/148.16 We are left with following problem, upon which TcT provides the 439.18/148.16 certificate YES(O(1),O(1)). 439.18/148.16 439.18/148.16 Rules: Empty 439.18/148.16 Obligation: 439.18/148.16 innermost runtime complexity 439.18/148.16 Answer: 439.18/148.16 YES(O(1),O(1)) 439.18/148.16 439.18/148.16 Empty rules are trivially bounded 439.18/148.16 439.18/148.16 S) We are left with following problem, upon which TcT provides the 439.18/148.16 certificate YES(O(1),O(n^2)). 439.18/148.16 439.18/148.16 Strict DPs: 439.18/148.16 { lgth^#(Cons(x, xs)) -> 439.18/148.16 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 439.18/148.16 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 439.18/148.16 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 439.18/148.16 c_4(eqList^#(x, y), eqList^#(xs, ys)) 439.18/148.16 , monus^#(x, y) -> 439.18/148.16 c_5(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.16 eqList^#(lgth(y), Cons(Nil(), Nil())), 439.18/148.16 lgth^#(y)) } 439.18/148.16 Weak DPs: 439.18/148.16 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.16 c_3(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 439.18/148.16 Cons(x', xs'), 439.18/148.16 Cons(x, xs)), 439.18/148.16 eqList^#(Cons(x', xs'), Cons(x, xs))) 439.18/148.16 , gcd[Ite][False][Ite]^#(False(), x, y) -> c_10(gt0^#(x, y)) 439.18/148.16 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 439.18/148.16 c_11(monus^#(xs', xs)) 439.18/148.16 , gt0^#(Nil(), y) -> c_6() 439.18/148.16 , gt0^#(Cons(x, xs), Nil()) -> c_7() 439.18/148.16 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_8(gt0^#(xs', xs)) } 439.18/148.16 Weak Trs: 439.18/148.16 { lgth(Nil()) -> Nil() 439.18/148.16 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 439.18/148.16 , @(Nil(), ys) -> ys 439.18/148.16 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 439.18/148.16 , and(True(), True()) -> True() 439.18/148.16 , and(True(), False()) -> False() 439.18/148.16 , and(False(), True()) -> False() 439.18/148.16 , and(False(), False()) -> False() 439.18/148.16 , eqList(Nil(), Nil()) -> True() 439.18/148.16 , eqList(Nil(), Cons(y, ys)) -> False() 439.18/148.16 , eqList(Cons(x, xs), Nil()) -> False() 439.18/148.16 , eqList(Cons(x, xs), Cons(y, ys)) -> 439.18/148.16 and(eqList(x, y), eqList(xs, ys)) } 439.18/148.16 Obligation: 439.18/148.16 innermost runtime complexity 439.18/148.16 Answer: 439.18/148.16 YES(O(1),O(n^2)) 439.18/148.16 439.18/148.16 The following weak DPs constitute a sub-graph of the DG that is 439.18/148.16 closed under successors. The DPs are removed. 439.18/148.16 439.18/148.16 { gcd[Ite][False][Ite]^#(False(), x, y) -> c_10(gt0^#(x, y)) 439.18/148.16 , gt0^#(Nil(), y) -> c_6() 439.18/148.16 , gt0^#(Cons(x, xs), Nil()) -> c_7() 439.18/148.16 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_8(gt0^#(xs', xs)) } 439.18/148.16 439.18/148.16 We are left with following problem, upon which TcT provides the 439.18/148.16 certificate YES(O(1),O(n^2)). 439.18/148.16 439.18/148.16 Strict DPs: 439.18/148.16 { lgth^#(Cons(x, xs)) -> 439.18/148.16 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 439.18/148.16 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 439.18/148.16 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 439.18/148.16 c_4(eqList^#(x, y), eqList^#(xs, ys)) 439.18/148.16 , monus^#(x, y) -> 439.18/148.16 c_5(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.16 eqList^#(lgth(y), Cons(Nil(), Nil())), 439.18/148.16 lgth^#(y)) } 439.18/148.16 Weak DPs: 439.18/148.16 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.16 c_3(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 439.18/148.16 Cons(x', xs'), 439.18/148.16 Cons(x, xs)), 439.18/148.16 eqList^#(Cons(x', xs'), Cons(x, xs))) 439.18/148.16 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 439.18/148.16 c_11(monus^#(xs', xs)) } 439.18/148.16 Weak Trs: 439.18/148.16 { lgth(Nil()) -> Nil() 439.18/148.16 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 439.18/148.16 , @(Nil(), ys) -> ys 439.18/148.16 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 439.18/148.16 , and(True(), True()) -> True() 439.18/148.17 , and(True(), False()) -> False() 439.18/148.17 , and(False(), True()) -> False() 439.18/148.17 , and(False(), False()) -> False() 439.18/148.17 , eqList(Nil(), Nil()) -> True() 439.18/148.17 , eqList(Nil(), Cons(y, ys)) -> False() 439.18/148.17 , eqList(Cons(x, xs), Nil()) -> False() 439.18/148.17 , eqList(Cons(x, xs), Cons(y, ys)) -> 439.18/148.17 and(eqList(x, y), eqList(xs, ys)) } 439.18/148.17 Obligation: 439.18/148.17 innermost runtime complexity 439.18/148.17 Answer: 439.18/148.17 YES(O(1),O(n^2)) 439.18/148.17 439.18/148.17 Due to missing edges in the dependency-graph, the right-hand sides 439.18/148.17 of following rules could be simplified: 439.18/148.17 439.18/148.17 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.17 c_3(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 439.18/148.17 Cons(x', xs'), 439.18/148.17 Cons(x, xs)), 439.18/148.17 eqList^#(Cons(x', xs'), Cons(x, xs))) } 439.18/148.17 439.18/148.17 We are left with following problem, upon which TcT provides the 439.18/148.17 certificate YES(O(1),O(n^2)). 439.18/148.17 439.18/148.17 Strict DPs: 439.18/148.17 { lgth^#(Cons(x, xs)) -> 439.18/148.17 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 439.18/148.17 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 439.18/148.17 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 439.18/148.17 c_3(eqList^#(x, y), eqList^#(xs, ys)) 439.18/148.17 , monus^#(x, y) -> 439.18/148.17 c_4(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.17 eqList^#(lgth(y), Cons(Nil(), Nil())), 439.18/148.17 lgth^#(y)) } 439.18/148.17 Weak DPs: 439.18/148.17 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.17 c_5(eqList^#(Cons(x', xs'), Cons(x, xs))) 439.18/148.17 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 439.18/148.17 c_6(monus^#(xs', xs)) } 439.18/148.17 Weak Trs: 439.18/148.17 { lgth(Nil()) -> Nil() 439.18/148.17 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 439.18/148.17 , @(Nil(), ys) -> ys 439.18/148.17 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 439.18/148.17 , and(True(), True()) -> True() 439.18/148.17 , and(True(), False()) -> False() 439.18/148.17 , and(False(), True()) -> False() 439.18/148.17 , and(False(), False()) -> False() 439.18/148.17 , eqList(Nil(), Nil()) -> True() 439.18/148.17 , eqList(Nil(), Cons(y, ys)) -> False() 439.18/148.17 , eqList(Cons(x, xs), Nil()) -> False() 439.18/148.17 , eqList(Cons(x, xs), Cons(y, ys)) -> 439.18/148.17 and(eqList(x, y), eqList(xs, ys)) } 439.18/148.17 Obligation: 439.18/148.17 innermost runtime complexity 439.18/148.17 Answer: 439.18/148.17 YES(O(1),O(n^2)) 439.18/148.17 439.18/148.17 We analyse the complexity of following sub-problems (R) and (S). 439.18/148.17 Problem (S) is obtained from the input problem by shifting strict 439.18/148.17 rules from (R) into the weak component: 439.18/148.17 439.18/148.17 Problem (R): 439.18/148.17 ------------ 439.18/148.17 Strict DPs: 439.18/148.17 { eqList^#(Cons(x, xs), Cons(y, ys)) -> 439.18/148.17 c_3(eqList^#(x, y), eqList^#(xs, ys)) } 439.18/148.17 Weak DPs: 439.18/148.17 { lgth^#(Cons(x, xs)) -> 439.18/148.17 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 439.18/148.17 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 439.18/148.17 , gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.17 c_5(eqList^#(Cons(x', xs'), Cons(x, xs))) 439.18/148.17 , monus^#(x, y) -> 439.18/148.17 c_4(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.17 eqList^#(lgth(y), Cons(Nil(), Nil())), 439.18/148.17 lgth^#(y)) 439.18/148.17 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 439.18/148.17 c_6(monus^#(xs', xs)) } 439.18/148.17 Weak Trs: 439.18/148.17 { lgth(Nil()) -> Nil() 439.18/148.17 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 439.18/148.17 , @(Nil(), ys) -> ys 439.18/148.17 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 439.18/148.17 , and(True(), True()) -> True() 439.18/148.17 , and(True(), False()) -> False() 439.18/148.17 , and(False(), True()) -> False() 439.18/148.17 , and(False(), False()) -> False() 439.18/148.17 , eqList(Nil(), Nil()) -> True() 439.18/148.17 , eqList(Nil(), Cons(y, ys)) -> False() 439.18/148.17 , eqList(Cons(x, xs), Nil()) -> False() 439.18/148.17 , eqList(Cons(x, xs), Cons(y, ys)) -> 439.18/148.17 and(eqList(x, y), eqList(xs, ys)) } 439.18/148.17 StartTerms: basic terms 439.18/148.17 Strategy: innermost 439.18/148.17 439.18/148.17 Problem (S): 439.18/148.17 ------------ 439.18/148.17 Strict DPs: 439.18/148.17 { lgth^#(Cons(x, xs)) -> 439.18/148.17 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 439.18/148.17 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 439.18/148.17 , monus^#(x, y) -> 439.18/148.17 c_4(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.17 eqList^#(lgth(y), Cons(Nil(), Nil())), 439.18/148.17 lgth^#(y)) } 439.18/148.17 Weak DPs: 439.18/148.17 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.17 c_5(eqList^#(Cons(x', xs'), Cons(x, xs))) 439.18/148.17 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 439.18/148.17 c_3(eqList^#(x, y), eqList^#(xs, ys)) 439.18/148.17 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 439.18/148.17 c_6(monus^#(xs', xs)) } 439.18/148.17 Weak Trs: 439.18/148.17 { lgth(Nil()) -> Nil() 439.18/148.17 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 439.18/148.17 , @(Nil(), ys) -> ys 439.18/148.17 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 439.18/148.17 , and(True(), True()) -> True() 439.18/148.17 , and(True(), False()) -> False() 439.18/148.17 , and(False(), True()) -> False() 439.18/148.17 , and(False(), False()) -> False() 439.18/148.17 , eqList(Nil(), Nil()) -> True() 439.18/148.17 , eqList(Nil(), Cons(y, ys)) -> False() 439.18/148.17 , eqList(Cons(x, xs), Nil()) -> False() 439.18/148.17 , eqList(Cons(x, xs), Cons(y, ys)) -> 439.18/148.17 and(eqList(x, y), eqList(xs, ys)) } 439.18/148.17 StartTerms: basic terms 439.18/148.17 Strategy: innermost 439.18/148.17 439.18/148.17 Overall, the transformation results in the following sub-problem(s): 439.18/148.17 439.18/148.17 Generated new problems: 439.18/148.17 ----------------------- 439.18/148.17 R) Strict DPs: 439.18/148.17 { eqList^#(Cons(x, xs), Cons(y, ys)) -> 439.18/148.17 c_3(eqList^#(x, y), eqList^#(xs, ys)) } 439.18/148.17 Weak DPs: 439.18/148.17 { lgth^#(Cons(x, xs)) -> 439.18/148.17 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 439.18/148.17 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 439.18/148.17 , gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.17 c_5(eqList^#(Cons(x', xs'), Cons(x, xs))) 439.18/148.17 , monus^#(x, y) -> 439.18/148.17 c_4(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.17 eqList^#(lgth(y), Cons(Nil(), Nil())), 439.18/148.17 lgth^#(y)) 439.18/148.17 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 439.18/148.17 c_6(monus^#(xs', xs)) } 439.18/148.17 Weak Trs: 439.18/148.17 { lgth(Nil()) -> Nil() 439.18/148.17 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 439.18/148.17 , @(Nil(), ys) -> ys 439.18/148.17 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 439.18/148.17 , and(True(), True()) -> True() 439.18/148.17 , and(True(), False()) -> False() 439.18/148.17 , and(False(), True()) -> False() 439.18/148.17 , and(False(), False()) -> False() 439.18/148.17 , eqList(Nil(), Nil()) -> True() 439.18/148.17 , eqList(Nil(), Cons(y, ys)) -> False() 439.18/148.17 , eqList(Cons(x, xs), Nil()) -> False() 439.18/148.17 , eqList(Cons(x, xs), Cons(y, ys)) -> 439.18/148.17 and(eqList(x, y), eqList(xs, ys)) } 439.18/148.17 StartTerms: basic terms 439.18/148.17 Strategy: innermost 439.18/148.17 439.18/148.17 This problem was proven YES(O(1),O(n^1)). 439.18/148.17 439.18/148.17 S) Strict DPs: 439.18/148.17 { lgth^#(Cons(x, xs)) -> 439.18/148.17 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 439.18/148.17 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 439.18/148.17 , monus^#(x, y) -> 439.18/148.17 c_4(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.17 eqList^#(lgth(y), Cons(Nil(), Nil())), 439.18/148.17 lgth^#(y)) } 439.18/148.17 Weak DPs: 439.18/148.17 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.17 c_5(eqList^#(Cons(x', xs'), Cons(x, xs))) 439.18/148.17 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 439.18/148.17 c_3(eqList^#(x, y), eqList^#(xs, ys)) 439.18/148.17 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 439.18/148.17 c_6(monus^#(xs', xs)) } 439.18/148.17 Weak Trs: 439.18/148.17 { lgth(Nil()) -> Nil() 439.18/148.17 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 439.18/148.17 , @(Nil(), ys) -> ys 439.18/148.17 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 439.18/148.17 , and(True(), True()) -> True() 439.18/148.17 , and(True(), False()) -> False() 439.18/148.17 , and(False(), True()) -> False() 439.18/148.17 , and(False(), False()) -> False() 439.18/148.17 , eqList(Nil(), Nil()) -> True() 439.18/148.17 , eqList(Nil(), Cons(y, ys)) -> False() 439.18/148.17 , eqList(Cons(x, xs), Nil()) -> False() 439.18/148.17 , eqList(Cons(x, xs), Cons(y, ys)) -> 439.18/148.17 and(eqList(x, y), eqList(xs, ys)) } 439.18/148.17 StartTerms: basic terms 439.18/148.17 Strategy: innermost 439.18/148.17 439.18/148.17 This problem was proven YES(O(1),O(n^2)). 439.18/148.17 439.18/148.17 439.18/148.17 Proofs for generated problems: 439.18/148.17 ------------------------------ 439.18/148.17 R) We are left with following problem, upon which TcT provides the 439.18/148.17 certificate YES(O(1),O(n^1)). 439.18/148.17 439.18/148.17 Strict DPs: 439.18/148.17 { eqList^#(Cons(x, xs), Cons(y, ys)) -> 439.18/148.17 c_3(eqList^#(x, y), eqList^#(xs, ys)) } 439.18/148.17 Weak DPs: 439.18/148.17 { lgth^#(Cons(x, xs)) -> 439.18/148.17 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 439.18/148.17 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 439.18/148.17 , gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.17 c_5(eqList^#(Cons(x', xs'), Cons(x, xs))) 439.18/148.17 , monus^#(x, y) -> 439.18/148.17 c_4(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.17 eqList^#(lgth(y), Cons(Nil(), Nil())), 439.18/148.17 lgth^#(y)) 439.18/148.17 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 439.18/148.17 c_6(monus^#(xs', xs)) } 439.18/148.17 Weak Trs: 439.18/148.17 { lgth(Nil()) -> Nil() 439.18/148.17 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 439.18/148.17 , @(Nil(), ys) -> ys 439.18/148.17 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 439.18/148.17 , and(True(), True()) -> True() 439.18/148.17 , and(True(), False()) -> False() 439.18/148.17 , and(False(), True()) -> False() 439.18/148.17 , and(False(), False()) -> False() 439.18/148.17 , eqList(Nil(), Nil()) -> True() 439.18/148.17 , eqList(Nil(), Cons(y, ys)) -> False() 439.18/148.17 , eqList(Cons(x, xs), Nil()) -> False() 439.18/148.17 , eqList(Cons(x, xs), Cons(y, ys)) -> 439.18/148.17 and(eqList(x, y), eqList(xs, ys)) } 439.18/148.17 Obligation: 439.18/148.17 innermost runtime complexity 439.18/148.17 Answer: 439.18/148.17 YES(O(1),O(n^1)) 439.18/148.17 439.18/148.17 The following weak DPs constitute a sub-graph of the DG that is 439.18/148.17 closed under successors. The DPs are removed. 439.18/148.17 439.18/148.17 { lgth^#(Cons(x, xs)) -> 439.18/148.17 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 439.18/148.17 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) } 439.18/148.17 439.18/148.17 We are left with following problem, upon which TcT provides the 439.18/148.17 certificate YES(O(1),O(n^1)). 439.18/148.17 439.18/148.17 Strict DPs: 439.18/148.17 { eqList^#(Cons(x, xs), Cons(y, ys)) -> 439.18/148.17 c_3(eqList^#(x, y), eqList^#(xs, ys)) } 439.18/148.17 Weak DPs: 439.18/148.17 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.17 c_5(eqList^#(Cons(x', xs'), Cons(x, xs))) 439.18/148.17 , monus^#(x, y) -> 439.18/148.17 c_4(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.17 eqList^#(lgth(y), Cons(Nil(), Nil())), 439.18/148.17 lgth^#(y)) 439.18/148.17 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 439.18/148.17 c_6(monus^#(xs', xs)) } 439.18/148.17 Weak Trs: 439.18/148.17 { lgth(Nil()) -> Nil() 439.18/148.17 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 439.18/148.17 , @(Nil(), ys) -> ys 439.18/148.17 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 439.18/148.17 , and(True(), True()) -> True() 439.18/148.17 , and(True(), False()) -> False() 439.18/148.17 , and(False(), True()) -> False() 439.18/148.17 , and(False(), False()) -> False() 439.18/148.17 , eqList(Nil(), Nil()) -> True() 439.18/148.17 , eqList(Nil(), Cons(y, ys)) -> False() 439.18/148.17 , eqList(Cons(x, xs), Nil()) -> False() 439.18/148.17 , eqList(Cons(x, xs), Cons(y, ys)) -> 439.18/148.17 and(eqList(x, y), eqList(xs, ys)) } 439.18/148.17 Obligation: 439.18/148.17 innermost runtime complexity 439.18/148.17 Answer: 439.18/148.17 YES(O(1),O(n^1)) 439.18/148.17 439.18/148.17 Due to missing edges in the dependency-graph, the right-hand sides 439.18/148.17 of following rules could be simplified: 439.18/148.17 439.18/148.17 { monus^#(x, y) -> 439.18/148.17 c_4(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.17 eqList^#(lgth(y), Cons(Nil(), Nil())), 439.18/148.17 lgth^#(y)) } 439.18/148.17 439.18/148.17 We are left with following problem, upon which TcT provides the 439.18/148.17 certificate YES(O(1),O(n^1)). 439.18/148.17 439.18/148.17 Strict DPs: 439.18/148.17 { eqList^#(Cons(x, xs), Cons(y, ys)) -> 439.18/148.17 c_1(eqList^#(x, y), eqList^#(xs, ys)) } 439.18/148.17 Weak DPs: 439.18/148.17 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.17 c_2(eqList^#(Cons(x', xs'), Cons(x, xs))) 439.18/148.17 , monus^#(x, y) -> 439.18/148.17 c_3(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.17 eqList^#(lgth(y), Cons(Nil(), Nil()))) 439.18/148.17 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 439.18/148.17 c_4(monus^#(xs', xs)) } 439.18/148.17 Weak Trs: 439.18/148.17 { lgth(Nil()) -> Nil() 439.18/148.17 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 439.18/148.17 , @(Nil(), ys) -> ys 439.18/148.17 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 439.18/148.17 , and(True(), True()) -> True() 439.18/148.17 , and(True(), False()) -> False() 439.18/148.17 , and(False(), True()) -> False() 439.18/148.17 , and(False(), False()) -> False() 439.18/148.17 , eqList(Nil(), Nil()) -> True() 439.18/148.17 , eqList(Nil(), Cons(y, ys)) -> False() 439.18/148.17 , eqList(Cons(x, xs), Nil()) -> False() 439.18/148.17 , eqList(Cons(x, xs), Cons(y, ys)) -> 439.18/148.17 and(eqList(x, y), eqList(xs, ys)) } 439.18/148.17 Obligation: 439.18/148.17 innermost runtime complexity 439.18/148.17 Answer: 439.18/148.17 YES(O(1),O(n^1)) 439.18/148.17 439.18/148.17 We use the processor 'matrix interpretation of dimension 1' to 439.18/148.17 orient following rules strictly. 439.18/148.17 439.18/148.17 DPs: 439.18/148.17 { 1: eqList^#(Cons(x, xs), Cons(y, ys)) -> 439.18/148.17 c_1(eqList^#(x, y), eqList^#(xs, ys)) 439.18/148.17 , 2: gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.17 c_2(eqList^#(Cons(x', xs'), Cons(x, xs))) } 439.18/148.17 439.18/148.17 Sub-proof: 439.18/148.17 ---------- 439.18/148.17 The following argument positions are usable: 439.18/148.17 Uargs(c_1) = {1, 2}, Uargs(c_2) = {1}, Uargs(c_3) = {1, 2}, 439.18/148.17 Uargs(c_4) = {1} 439.18/148.17 439.18/148.17 TcT has computed the following constructor-based matrix 439.18/148.17 interpretation satisfying not(EDA). 439.18/148.17 439.18/148.17 [lgth](x1) = [0] 439.18/148.17 439.18/148.17 [True] = [0] 439.18/148.17 439.18/148.17 [Nil] = [0] 439.18/148.17 439.18/148.17 [@](x1, x2) = [0] 439.18/148.17 439.18/148.17 [and](x1, x2) = [0] 439.18/148.17 439.18/148.17 [eqList](x1, x2) = [0] 439.18/148.17 439.18/148.17 [Cons](x1, x2) = [1] x1 + [1] x2 + [1] 439.18/148.17 439.18/148.17 [False] = [0] 439.18/148.17 439.18/148.17 [gcd^#](x1, x2) = [4] x2 + [7] 439.18/148.17 439.18/148.17 [eqList^#](x1, x2) = [4] x2 + [0] 439.18/148.17 439.18/148.17 [monus^#](x1, x2) = [4] x1 + [4] 439.18/148.17 439.18/148.17 [monus[Ite]^#](x1, x2, x3) = [4] x2 + [0] 439.18/148.17 439.18/148.17 [c_1](x1, x2) = [1] x1 + [1] x2 + [1] 439.18/148.17 439.18/148.17 [c_2](x1) = [1] x1 + [3] 439.18/148.17 439.18/148.17 [c_3](x1, x2) = [1] x1 + [1] x2 + [0] 439.18/148.17 439.18/148.17 [c_4](x1) = [1] x1 + [0] 439.18/148.17 439.18/148.17 The order satisfies the following ordering constraints: 439.18/148.17 439.18/148.17 [lgth(Nil())] = [0] 439.18/148.17 >= [0] 439.18/148.17 = [Nil()] 439.18/148.17 439.18/148.17 [lgth(Cons(x, xs))] = [0] 439.18/148.17 >= [0] 439.18/148.17 = [@(Cons(Nil(), Nil()), lgth(xs))] 439.18/148.17 439.18/148.17 [@(Nil(), ys)] = [0] 439.18/148.17 ? [1] ys + [0] 439.18/148.17 = [ys] 439.18/148.17 439.18/148.17 [@(Cons(x, xs), ys)] = [0] 439.18/148.17 ? [1] x + [1] 439.18/148.17 = [Cons(x, @(xs, ys))] 439.18/148.17 439.18/148.17 [and(True(), True())] = [0] 439.18/148.17 >= [0] 439.18/148.17 = [True()] 439.18/148.17 439.18/148.17 [and(True(), False())] = [0] 439.18/148.17 >= [0] 439.18/148.17 = [False()] 439.18/148.17 439.18/148.17 [and(False(), True())] = [0] 439.18/148.17 >= [0] 439.18/148.17 = [False()] 439.18/148.17 439.18/148.17 [and(False(), False())] = [0] 439.18/148.17 >= [0] 439.18/148.17 = [False()] 439.18/148.17 439.18/148.17 [eqList(Nil(), Nil())] = [0] 439.18/148.17 >= [0] 439.18/148.17 = [True()] 439.18/148.17 439.18/148.17 [eqList(Nil(), Cons(y, ys))] = [0] 439.18/148.17 >= [0] 439.18/148.17 = [False()] 439.18/148.17 439.18/148.17 [eqList(Cons(x, xs), Nil())] = [0] 439.18/148.17 >= [0] 439.18/148.17 = [False()] 439.18/148.17 439.18/148.17 [eqList(Cons(x, xs), Cons(y, ys))] = [0] 439.18/148.17 >= [0] 439.18/148.17 = [and(eqList(x, y), eqList(xs, ys))] 439.18/148.17 439.18/148.17 [gcd^#(Cons(x', xs'), Cons(x, xs))] = [4] x + [4] xs + [11] 439.18/148.17 > [4] x + [4] xs + [7] 439.18/148.17 = [c_2(eqList^#(Cons(x', xs'), Cons(x, xs)))] 439.18/148.17 439.18/148.17 [eqList^#(Cons(x, xs), Cons(y, ys))] = [4] ys + [4] y + [4] 439.18/148.17 > [4] ys + [4] y + [1] 439.18/148.17 = [c_1(eqList^#(x, y), eqList^#(xs, ys))] 439.18/148.17 439.18/148.17 [monus^#(x, y)] = [4] x + [4] 439.18/148.18 >= [4] x + [4] 439.18/148.18 = [c_3(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.18 eqList^#(lgth(y), Cons(Nil(), Nil())))] 439.18/148.18 439.18/148.18 [monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs))] = [4] x' + [4] xs' + [4] 439.18/148.18 >= [4] xs' + [4] 439.18/148.18 = [c_4(monus^#(xs', xs))] 439.18/148.18 439.18/148.18 439.18/148.18 The strictly oriented rules are moved into the weak component. 439.18/148.18 439.18/148.18 We are left with following problem, upon which TcT provides the 439.18/148.18 certificate YES(O(1),O(1)). 439.18/148.18 439.18/148.18 Weak DPs: 439.18/148.18 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.18 c_2(eqList^#(Cons(x', xs'), Cons(x, xs))) 439.18/148.18 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 439.18/148.18 c_1(eqList^#(x, y), eqList^#(xs, ys)) 439.18/148.18 , monus^#(x, y) -> 439.18/148.18 c_3(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.18 eqList^#(lgth(y), Cons(Nil(), Nil()))) 439.18/148.18 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 439.18/148.18 c_4(monus^#(xs', xs)) } 439.18/148.18 Weak Trs: 439.18/148.18 { lgth(Nil()) -> Nil() 439.18/148.18 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 439.18/148.18 , @(Nil(), ys) -> ys 439.18/148.18 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 439.18/148.18 , and(True(), True()) -> True() 439.18/148.18 , and(True(), False()) -> False() 439.18/148.18 , and(False(), True()) -> False() 439.18/148.18 , and(False(), False()) -> False() 439.18/148.18 , eqList(Nil(), Nil()) -> True() 439.18/148.18 , eqList(Nil(), Cons(y, ys)) -> False() 439.18/148.18 , eqList(Cons(x, xs), Nil()) -> False() 439.18/148.18 , eqList(Cons(x, xs), Cons(y, ys)) -> 439.18/148.18 and(eqList(x, y), eqList(xs, ys)) } 439.18/148.18 Obligation: 439.18/148.18 innermost runtime complexity 439.18/148.18 Answer: 439.18/148.18 YES(O(1),O(1)) 439.18/148.18 439.18/148.18 The following weak DPs constitute a sub-graph of the DG that is 439.18/148.18 closed under successors. The DPs are removed. 439.18/148.18 439.18/148.18 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.18 c_2(eqList^#(Cons(x', xs'), Cons(x, xs))) 439.18/148.18 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 439.18/148.18 c_1(eqList^#(x, y), eqList^#(xs, ys)) 439.18/148.18 , monus^#(x, y) -> 439.18/148.18 c_3(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.18 eqList^#(lgth(y), Cons(Nil(), Nil()))) 439.18/148.18 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 439.18/148.18 c_4(monus^#(xs', xs)) } 439.18/148.18 439.18/148.18 We are left with following problem, upon which TcT provides the 439.18/148.18 certificate YES(O(1),O(1)). 439.18/148.18 439.18/148.18 Weak Trs: 439.18/148.18 { lgth(Nil()) -> Nil() 439.18/148.18 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 439.18/148.18 , @(Nil(), ys) -> ys 439.18/148.18 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 439.18/148.18 , and(True(), True()) -> True() 439.18/148.18 , and(True(), False()) -> False() 439.18/148.18 , and(False(), True()) -> False() 439.18/148.18 , and(False(), False()) -> False() 439.18/148.18 , eqList(Nil(), Nil()) -> True() 439.18/148.18 , eqList(Nil(), Cons(y, ys)) -> False() 439.18/148.18 , eqList(Cons(x, xs), Nil()) -> False() 439.18/148.18 , eqList(Cons(x, xs), Cons(y, ys)) -> 439.18/148.18 and(eqList(x, y), eqList(xs, ys)) } 439.18/148.18 Obligation: 439.18/148.18 innermost runtime complexity 439.18/148.18 Answer: 439.18/148.18 YES(O(1),O(1)) 439.18/148.18 439.18/148.18 No rule is usable, rules are removed from the input problem. 439.18/148.18 439.18/148.18 We are left with following problem, upon which TcT provides the 439.18/148.18 certificate YES(O(1),O(1)). 439.18/148.18 439.18/148.18 Rules: Empty 439.18/148.18 Obligation: 439.18/148.18 innermost runtime complexity 439.18/148.18 Answer: 439.18/148.18 YES(O(1),O(1)) 439.18/148.18 439.18/148.18 Empty rules are trivially bounded 439.18/148.18 439.18/148.18 S) We are left with following problem, upon which TcT provides the 439.18/148.18 certificate YES(O(1),O(n^2)). 439.18/148.18 439.18/148.18 Strict DPs: 439.18/148.18 { lgth^#(Cons(x, xs)) -> 439.18/148.18 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 439.18/148.18 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 439.18/148.18 , monus^#(x, y) -> 439.18/148.18 c_4(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.18 eqList^#(lgth(y), Cons(Nil(), Nil())), 439.18/148.18 lgth^#(y)) } 439.18/148.18 Weak DPs: 439.18/148.18 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.18 c_5(eqList^#(Cons(x', xs'), Cons(x, xs))) 439.18/148.18 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 439.18/148.18 c_3(eqList^#(x, y), eqList^#(xs, ys)) 439.18/148.18 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 439.18/148.18 c_6(monus^#(xs', xs)) } 439.18/148.18 Weak Trs: 439.18/148.18 { lgth(Nil()) -> Nil() 439.18/148.18 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 439.18/148.18 , @(Nil(), ys) -> ys 439.18/148.18 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 439.18/148.18 , and(True(), True()) -> True() 439.18/148.18 , and(True(), False()) -> False() 439.18/148.18 , and(False(), True()) -> False() 439.18/148.18 , and(False(), False()) -> False() 439.18/148.18 , eqList(Nil(), Nil()) -> True() 439.18/148.18 , eqList(Nil(), Cons(y, ys)) -> False() 439.18/148.18 , eqList(Cons(x, xs), Nil()) -> False() 439.18/148.18 , eqList(Cons(x, xs), Cons(y, ys)) -> 439.18/148.18 and(eqList(x, y), eqList(xs, ys)) } 439.18/148.18 Obligation: 439.18/148.18 innermost runtime complexity 439.18/148.18 Answer: 439.18/148.18 YES(O(1),O(n^2)) 439.18/148.18 439.18/148.18 The following weak DPs constitute a sub-graph of the DG that is 439.18/148.18 closed under successors. The DPs are removed. 439.18/148.18 439.18/148.18 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 439.18/148.18 c_5(eqList^#(Cons(x', xs'), Cons(x, xs))) 439.18/148.18 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 439.18/148.18 c_3(eqList^#(x, y), eqList^#(xs, ys)) } 439.18/148.18 439.18/148.18 We are left with following problem, upon which TcT provides the 439.18/148.18 certificate YES(O(1),O(n^2)). 439.18/148.18 439.18/148.18 Strict DPs: 439.18/148.18 { lgth^#(Cons(x, xs)) -> 439.18/148.18 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 439.18/148.18 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 439.18/148.18 , monus^#(x, y) -> 439.18/148.18 c_4(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.18 eqList^#(lgth(y), Cons(Nil(), Nil())), 439.18/148.18 lgth^#(y)) } 439.18/148.18 Weak DPs: 439.18/148.18 { monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 439.18/148.18 c_6(monus^#(xs', xs)) } 439.18/148.18 Weak Trs: 439.18/148.18 { lgth(Nil()) -> Nil() 439.18/148.18 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 439.18/148.18 , @(Nil(), ys) -> ys 439.18/148.18 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 439.18/148.18 , and(True(), True()) -> True() 439.18/148.18 , and(True(), False()) -> False() 439.18/148.18 , and(False(), True()) -> False() 439.18/148.18 , and(False(), False()) -> False() 439.18/148.18 , eqList(Nil(), Nil()) -> True() 439.18/148.18 , eqList(Nil(), Cons(y, ys)) -> False() 439.18/148.18 , eqList(Cons(x, xs), Nil()) -> False() 439.18/148.18 , eqList(Cons(x, xs), Cons(y, ys)) -> 439.18/148.18 and(eqList(x, y), eqList(xs, ys)) } 439.18/148.18 Obligation: 439.18/148.18 innermost runtime complexity 439.18/148.18 Answer: 439.18/148.18 YES(O(1),O(n^2)) 439.18/148.18 439.18/148.18 Due to missing edges in the dependency-graph, the right-hand sides 439.18/148.18 of following rules could be simplified: 439.18/148.18 439.18/148.18 { monus^#(x, y) -> 439.18/148.18 c_4(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.18 eqList^#(lgth(y), Cons(Nil(), Nil())), 439.18/148.18 lgth^#(y)) } 439.18/148.18 439.18/148.18 We are left with following problem, upon which TcT provides the 439.18/148.18 certificate YES(O(1),O(n^2)). 439.18/148.18 439.18/148.18 Strict DPs: 439.18/148.18 { lgth^#(Cons(x, xs)) -> 439.18/148.18 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 439.18/148.18 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 439.18/148.18 , monus^#(x, y) -> 439.18/148.18 c_3(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.18 lgth^#(y)) } 439.18/148.18 Weak DPs: 439.18/148.18 { monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 439.18/148.18 c_4(monus^#(xs', xs)) } 439.18/148.18 Weak Trs: 439.18/148.18 { lgth(Nil()) -> Nil() 439.18/148.18 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 439.18/148.18 , @(Nil(), ys) -> ys 439.18/148.18 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 439.18/148.18 , and(True(), True()) -> True() 439.18/148.18 , and(True(), False()) -> False() 439.18/148.18 , and(False(), True()) -> False() 439.18/148.18 , and(False(), False()) -> False() 439.18/148.18 , eqList(Nil(), Nil()) -> True() 439.18/148.18 , eqList(Nil(), Cons(y, ys)) -> False() 439.18/148.18 , eqList(Cons(x, xs), Nil()) -> False() 439.18/148.18 , eqList(Cons(x, xs), Cons(y, ys)) -> 439.18/148.18 and(eqList(x, y), eqList(xs, ys)) } 439.18/148.18 Obligation: 439.18/148.18 innermost runtime complexity 439.18/148.18 Answer: 439.18/148.18 YES(O(1),O(n^2)) 439.18/148.18 439.18/148.18 We use the processor 'matrix interpretation of dimension 2' to 439.18/148.18 orient following rules strictly. 439.18/148.18 439.18/148.18 DPs: 439.18/148.18 { 1: lgth^#(Cons(x, xs)) -> 439.18/148.18 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 439.18/148.18 , 2: @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 439.18/148.18 , 4: monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 439.18/148.18 c_4(monus^#(xs', xs)) } 439.18/148.18 439.18/148.18 Sub-proof: 439.18/148.18 ---------- 439.18/148.18 The following argument positions are usable: 439.18/148.18 Uargs(c_1) = {1, 2}, Uargs(c_2) = {1}, Uargs(c_3) = {1, 2}, 439.18/148.18 Uargs(c_4) = {1} 439.18/148.18 439.18/148.18 TcT has computed the following constructor-based matrix 439.18/148.18 interpretation satisfying not(EDA). 439.18/148.18 439.18/148.18 [lgth](x1) = [0] 439.18/148.18 [0] 439.18/148.18 439.18/148.18 [True] = [0] 439.18/148.18 [0] 439.18/148.18 439.18/148.18 [Nil] = [0] 439.18/148.18 [0] 439.18/148.18 439.18/148.18 [@](x1, x2) = [0] 439.18/148.18 [0] 439.18/148.18 439.18/148.18 [and](x1, x2) = [0] 439.18/148.18 [0] 439.18/148.18 439.18/148.18 [eqList](x1, x2) = [0] 439.18/148.18 [0] 439.18/148.18 439.18/148.18 [Cons](x1, x2) = [1 4] x2 + [1] 439.18/148.18 [0 1] [2] 439.18/148.18 439.18/148.18 [False] = [0] 439.18/148.18 [0] 439.18/148.18 439.18/148.18 [lgth^#](x1) = [0 2] x1 + [0] 439.18/148.18 [0 0] [0] 439.18/148.18 439.18/148.18 [@^#](x1, x2) = [1 0] x1 + [0] 439.18/148.18 [2 1] [0] 439.18/148.18 439.18/148.18 [monus^#](x1, x2) = [0 1] x1 + [1 4] x2 + [1] 439.18/148.18 [0 0] [4 4] [0] 439.18/148.18 439.18/148.18 [monus[Ite]^#](x1, x2, x3) = [0 1] x2 + [1 0] x3 + [1] 439.18/148.18 [0 0] [0 0] [0] 439.18/148.18 439.18/148.18 [c_1](x1, x2) = [1 0] x1 + [1 0] x2 + [1] 439.18/148.18 [0 0] [0 0] [0] 439.18/148.18 439.18/148.18 [c_2](x1) = [1 0] x1 + [0] 439.18/148.18 [0 0] [3] 439.18/148.18 439.18/148.18 [c_3](x1, x2) = [1 0] x1 + [2 0] x2 + [0] 439.18/148.18 [0 0] [0 0] [0] 439.18/148.18 439.18/148.18 [c_4](x1) = [1 0] x1 + [1] 439.18/148.18 [0 0] [0] 439.18/148.18 439.18/148.18 The order satisfies the following ordering constraints: 439.18/148.18 439.18/148.18 [lgth(Nil())] = [0] 439.18/148.18 [0] 439.18/148.18 >= [0] 439.18/148.18 [0] 439.18/148.18 = [Nil()] 439.18/148.18 439.18/148.18 [lgth(Cons(x, xs))] = [0] 439.18/148.18 [0] 439.18/148.18 >= [0] 439.18/148.18 [0] 439.18/148.18 = [@(Cons(Nil(), Nil()), lgth(xs))] 439.18/148.18 439.18/148.18 [@(Nil(), ys)] = [0] 439.18/148.18 [0] 439.18/148.18 ? [1 0] ys + [0] 439.18/148.18 [0 1] [0] 439.18/148.18 = [ys] 439.18/148.18 439.18/148.18 [@(Cons(x, xs), ys)] = [0] 439.18/148.18 [0] 439.18/148.18 ? [1] 439.18/148.18 [2] 439.18/148.18 = [Cons(x, @(xs, ys))] 439.18/148.18 439.18/148.18 [and(True(), True())] = [0] 439.18/148.18 [0] 439.18/148.18 >= [0] 439.18/148.18 [0] 439.18/148.18 = [True()] 439.18/148.18 439.18/148.18 [and(True(), False())] = [0] 439.18/148.18 [0] 439.18/148.18 >= [0] 439.18/148.18 [0] 439.18/148.18 = [False()] 439.18/148.18 439.18/148.18 [and(False(), True())] = [0] 439.18/148.18 [0] 439.18/148.18 >= [0] 439.18/148.18 [0] 439.18/148.18 = [False()] 439.18/148.18 439.18/148.18 [and(False(), False())] = [0] 439.18/148.18 [0] 439.18/148.18 >= [0] 439.18/148.18 [0] 439.18/148.18 = [False()] 439.18/148.18 439.18/148.18 [eqList(Nil(), Nil())] = [0] 439.18/148.18 [0] 439.18/148.18 >= [0] 439.18/148.18 [0] 439.18/148.18 = [True()] 439.18/148.18 439.18/148.18 [eqList(Nil(), Cons(y, ys))] = [0] 439.18/148.18 [0] 439.18/148.18 >= [0] 439.18/148.18 [0] 439.18/148.18 = [False()] 439.18/148.18 439.18/148.18 [eqList(Cons(x, xs), Nil())] = [0] 439.18/148.18 [0] 439.18/148.18 >= [0] 439.18/148.18 [0] 439.18/148.18 = [False()] 439.18/148.18 439.18/148.18 [eqList(Cons(x, xs), Cons(y, ys))] = [0] 439.18/148.18 [0] 439.18/148.18 >= [0] 439.18/148.18 [0] 439.18/148.18 = [and(eqList(x, y), eqList(xs, ys))] 439.18/148.18 439.18/148.18 [lgth^#(Cons(x, xs))] = [0 2] xs + [4] 439.18/148.18 [0 0] [0] 439.18/148.18 > [0 2] xs + [2] 439.18/148.18 [0 0] [0] 439.18/148.18 = [c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs))] 439.18/148.18 439.18/148.18 [@^#(Cons(x, xs), ys)] = [1 4] xs + [1] 439.18/148.18 [2 9] [4] 439.18/148.18 > [1 0] xs + [0] 439.18/148.18 [0 0] [3] 439.18/148.18 = [c_2(@^#(xs, ys))] 439.18/148.18 439.18/148.18 [monus^#(x, y)] = [0 1] x + [1 4] y + [1] 439.18/148.18 [0 0] [4 4] [0] 439.18/148.18 >= [0 1] x + [1 4] y + [1] 439.18/148.18 [0 0] [0 0] [0] 439.18/148.18 = [c_3(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.18 lgth^#(y))] 439.18/148.18 439.18/148.18 [monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs))] = [1 4] xs + [0 1] xs' + [4] 439.18/148.18 [0 0] [0 0] [0] 439.18/148.18 > [1 4] xs + [0 1] xs' + [2] 439.18/148.18 [0 0] [0 0] [0] 439.18/148.18 = [c_4(monus^#(xs', xs))] 439.18/148.18 439.18/148.18 439.18/148.18 We return to the main proof. Consider the set of all dependency 439.18/148.18 pairs 439.18/148.18 439.18/148.18 : 439.18/148.18 { 1: lgth^#(Cons(x, xs)) -> 439.18/148.18 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 439.18/148.18 , 2: @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 439.18/148.18 , 3: monus^#(x, y) -> 439.18/148.18 c_3(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.18 lgth^#(y)) 439.18/148.18 , 4: monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 439.18/148.18 c_4(monus^#(xs', xs)) } 439.18/148.18 439.18/148.18 Processor 'matrix interpretation of dimension 2' induces the 439.18/148.18 complexity certificate YES(?,O(n^2)) on application of dependency 439.18/148.18 pairs {1,2,4}. These cover all (indirect) predecessors of 439.18/148.18 dependency pairs {1,2,3,4}, their number of application is equally 439.18/148.18 bounded. The dependency pairs are shifted into the weak component. 439.18/148.18 439.18/148.18 We are left with following problem, upon which TcT provides the 439.18/148.18 certificate YES(O(1),O(1)). 439.18/148.18 439.18/148.18 Weak DPs: 439.18/148.18 { lgth^#(Cons(x, xs)) -> 439.18/148.18 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 439.18/148.18 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 439.18/148.18 , monus^#(x, y) -> 439.18/148.18 c_3(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.18 lgth^#(y)) 439.18/148.18 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 439.18/148.18 c_4(monus^#(xs', xs)) } 439.18/148.18 Weak Trs: 439.18/148.18 { lgth(Nil()) -> Nil() 439.18/148.18 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 439.18/148.18 , @(Nil(), ys) -> ys 439.18/148.18 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 439.18/148.18 , and(True(), True()) -> True() 439.18/148.18 , and(True(), False()) -> False() 439.18/148.18 , and(False(), True()) -> False() 439.18/148.18 , and(False(), False()) -> False() 439.18/148.18 , eqList(Nil(), Nil()) -> True() 439.18/148.18 , eqList(Nil(), Cons(y, ys)) -> False() 439.18/148.18 , eqList(Cons(x, xs), Nil()) -> False() 439.18/148.18 , eqList(Cons(x, xs), Cons(y, ys)) -> 439.18/148.18 and(eqList(x, y), eqList(xs, ys)) } 439.18/148.18 Obligation: 439.18/148.18 innermost runtime complexity 439.18/148.18 Answer: 439.18/148.18 YES(O(1),O(1)) 439.18/148.18 439.18/148.18 The following weak DPs constitute a sub-graph of the DG that is 439.18/148.18 closed under successors. The DPs are removed. 439.18/148.18 439.18/148.18 { lgth^#(Cons(x, xs)) -> 439.18/148.18 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 439.18/148.18 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 439.18/148.18 , monus^#(x, y) -> 439.18/148.18 c_3(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 439.18/148.18 lgth^#(y)) 439.18/148.18 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 439.18/148.18 c_4(monus^#(xs', xs)) } 439.18/148.18 439.18/148.18 We are left with following problem, upon which TcT provides the 439.18/148.18 certificate YES(O(1),O(1)). 439.18/148.18 439.18/148.18 Weak Trs: 439.18/148.18 { lgth(Nil()) -> Nil() 439.18/148.18 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 439.18/148.18 , @(Nil(), ys) -> ys 439.18/148.18 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 439.18/148.18 , and(True(), True()) -> True() 439.18/148.18 , and(True(), False()) -> False() 439.18/148.18 , and(False(), True()) -> False() 439.18/148.18 , and(False(), False()) -> False() 439.18/148.18 , eqList(Nil(), Nil()) -> True() 439.18/148.18 , eqList(Nil(), Cons(y, ys)) -> False() 439.18/148.18 , eqList(Cons(x, xs), Nil()) -> False() 439.18/148.18 , eqList(Cons(x, xs), Cons(y, ys)) -> 439.18/148.18 and(eqList(x, y), eqList(xs, ys)) } 439.18/148.18 Obligation: 439.18/148.18 innermost runtime complexity 439.18/148.18 Answer: 439.18/148.18 YES(O(1),O(1)) 439.18/148.18 439.18/148.18 No rule is usable, rules are removed from the input problem. 439.18/148.18 439.18/148.18 We are left with following problem, upon which TcT provides the 439.18/148.18 certificate YES(O(1),O(1)). 439.18/148.18 439.18/148.18 Rules: Empty 439.18/148.18 Obligation: 439.18/148.18 innermost runtime complexity 439.18/148.18 Answer: 439.18/148.18 YES(O(1),O(1)) 439.18/148.18 439.18/148.18 Empty rules are trivially bounded 439.18/148.18 439.18/148.18 439.18/148.18 439.18/148.18 Hurray, we answered YES(O(1),O(n^2)) 439.65/148.50 EOF