YES(O(1),O(n^2)) 435.96/148.04 YES(O(1),O(n^2)) 435.96/148.04 435.96/148.04 We are left with following problem, upon which TcT provides the 435.96/148.04 certificate YES(O(1),O(n^2)). 435.96/148.04 435.96/148.04 Strict Trs: 435.96/148.04 { lgth(Nil()) -> Nil() 435.96/148.04 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 435.96/148.04 , gcd(Nil(), Nil()) -> Nil() 435.96/148.04 , gcd(Nil(), Cons(x, xs)) -> Nil() 435.96/148.04 , gcd(Cons(x, xs), Nil()) -> Nil() 435.96/148.04 , gcd(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.04 gcd[Ite][False][Ite](eqList(Cons(x', xs'), Cons(x, xs)), 435.96/148.04 Cons(x', xs'), 435.96/148.04 Cons(x, xs)) 435.96/148.04 , @(Nil(), ys) -> ys 435.96/148.04 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 435.96/148.04 , monus(x, y) -> 435.96/148.04 monus[Ite](eqList(lgth(y), Cons(Nil(), Nil())), x, y) 435.96/148.04 , eqList(Nil(), Nil()) -> True() 435.96/148.04 , eqList(Nil(), Cons(y, ys)) -> False() 435.96/148.04 , eqList(Cons(x, xs), Nil()) -> False() 435.96/148.04 , eqList(Cons(x, xs), Cons(y, ys)) -> 435.96/148.04 and(eqList(x, y), eqList(xs, ys)) 435.96/148.04 , gt0(Nil(), y) -> False() 435.96/148.04 , gt0(Cons(x, xs), Nil()) -> True() 435.96/148.04 , gt0(Cons(x', xs'), Cons(x, xs)) -> gt0(xs', xs) 435.96/148.04 , goal(x, y) -> gcd(x, y) } 435.96/148.04 Weak Trs: 435.96/148.04 { monus[Ite](True(), Cons(x, xs), y) -> xs 435.96/148.04 , monus[Ite](False(), Cons(x', xs'), Cons(x, xs)) -> monus(xs', xs) 435.96/148.04 , and(True(), True()) -> True() 435.96/148.04 , and(True(), False()) -> False() 435.96/148.04 , and(False(), True()) -> False() 435.96/148.04 , and(False(), False()) -> False() 435.96/148.04 , gcd[Ite][False][Ite](True(), x, y) -> x 435.96/148.04 , gcd[Ite][False][Ite](False(), x, y) -> 435.96/148.04 gcd[Ite][False][Ite][False][Ite](gt0(x, y), x, y) } 435.96/148.04 Obligation: 435.96/148.04 innermost runtime complexity 435.96/148.04 Answer: 435.96/148.04 YES(O(1),O(n^2)) 435.96/148.04 435.96/148.04 We add the following dependency tuples: 435.96/148.04 435.96/148.04 Strict DPs: 435.96/148.04 { lgth^#(Nil()) -> c_1() 435.96/148.04 , lgth^#(Cons(x, xs)) -> 435.96/148.04 c_2(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 435.96/148.04 , @^#(Nil(), ys) -> c_7() 435.96/148.04 , @^#(Cons(x, xs), ys) -> c_8(@^#(xs, ys)) 435.96/148.04 , gcd^#(Nil(), Nil()) -> c_3() 435.96/148.04 , gcd^#(Nil(), Cons(x, xs)) -> c_4() 435.96/148.04 , gcd^#(Cons(x, xs), Nil()) -> c_5() 435.96/148.04 , gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.04 c_6(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 435.96/148.04 Cons(x', xs'), 435.96/148.04 Cons(x, xs)), 435.96/148.04 eqList^#(Cons(x', xs'), Cons(x, xs))) 435.96/148.04 , eqList^#(Nil(), Nil()) -> c_10() 435.96/148.04 , eqList^#(Nil(), Cons(y, ys)) -> c_11() 435.96/148.04 , eqList^#(Cons(x, xs), Nil()) -> c_12() 435.96/148.04 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 435.96/148.04 c_13(and^#(eqList(x, y), eqList(xs, ys)), 435.96/148.04 eqList^#(x, y), 435.96/148.04 eqList^#(xs, ys)) 435.96/148.04 , monus^#(x, y) -> 435.96/148.04 c_9(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.04 eqList^#(lgth(y), Cons(Nil(), Nil())), 435.96/148.04 lgth^#(y)) 435.96/148.04 , gt0^#(Nil(), y) -> c_14() 435.96/148.04 , gt0^#(Cons(x, xs), Nil()) -> c_15() 435.96/148.04 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_16(gt0^#(xs', xs)) 435.96/148.04 , goal^#(x, y) -> c_17(gcd^#(x, y)) } 435.96/148.04 Weak DPs: 435.96/148.04 { gcd[Ite][False][Ite]^#(True(), x, y) -> c_24() 435.96/148.04 , gcd[Ite][False][Ite]^#(False(), x, y) -> c_25(gt0^#(x, y)) 435.96/148.04 , monus[Ite]^#(True(), Cons(x, xs), y) -> c_18() 435.96/148.04 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 435.96/148.04 c_19(monus^#(xs', xs)) 435.96/148.04 , and^#(True(), True()) -> c_20() 435.96/148.04 , and^#(True(), False()) -> c_21() 435.96/148.04 , and^#(False(), True()) -> c_22() 435.96/148.04 , and^#(False(), False()) -> c_23() } 435.96/148.04 435.96/148.04 and mark the set of starting terms. 435.96/148.04 435.96/148.04 We are left with following problem, upon which TcT provides the 435.96/148.04 certificate YES(O(1),O(n^2)). 435.96/148.04 435.96/148.04 Strict DPs: 435.96/148.04 { lgth^#(Nil()) -> c_1() 435.96/148.04 , lgth^#(Cons(x, xs)) -> 435.96/148.04 c_2(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 435.96/148.04 , @^#(Nil(), ys) -> c_7() 435.96/148.04 , @^#(Cons(x, xs), ys) -> c_8(@^#(xs, ys)) 435.96/148.04 , gcd^#(Nil(), Nil()) -> c_3() 435.96/148.04 , gcd^#(Nil(), Cons(x, xs)) -> c_4() 435.96/148.04 , gcd^#(Cons(x, xs), Nil()) -> c_5() 435.96/148.04 , gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.04 c_6(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 435.96/148.04 Cons(x', xs'), 435.96/148.04 Cons(x, xs)), 435.96/148.04 eqList^#(Cons(x', xs'), Cons(x, xs))) 435.96/148.04 , eqList^#(Nil(), Nil()) -> c_10() 435.96/148.04 , eqList^#(Nil(), Cons(y, ys)) -> c_11() 435.96/148.04 , eqList^#(Cons(x, xs), Nil()) -> c_12() 435.96/148.04 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 435.96/148.04 c_13(and^#(eqList(x, y), eqList(xs, ys)), 435.96/148.04 eqList^#(x, y), 435.96/148.04 eqList^#(xs, ys)) 435.96/148.04 , monus^#(x, y) -> 435.96/148.04 c_9(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.04 eqList^#(lgth(y), Cons(Nil(), Nil())), 435.96/148.04 lgth^#(y)) 435.96/148.04 , gt0^#(Nil(), y) -> c_14() 435.96/148.04 , gt0^#(Cons(x, xs), Nil()) -> c_15() 435.96/148.04 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_16(gt0^#(xs', xs)) 435.96/148.04 , goal^#(x, y) -> c_17(gcd^#(x, y)) } 435.96/148.04 Weak DPs: 435.96/148.04 { gcd[Ite][False][Ite]^#(True(), x, y) -> c_24() 435.96/148.04 , gcd[Ite][False][Ite]^#(False(), x, y) -> c_25(gt0^#(x, y)) 435.96/148.04 , monus[Ite]^#(True(), Cons(x, xs), y) -> c_18() 435.96/148.04 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 435.96/148.04 c_19(monus^#(xs', xs)) 435.96/148.04 , and^#(True(), True()) -> c_20() 435.96/148.04 , and^#(True(), False()) -> c_21() 435.96/148.04 , and^#(False(), True()) -> c_22() 435.96/148.04 , and^#(False(), False()) -> c_23() } 435.96/148.04 Weak Trs: 435.96/148.04 { lgth(Nil()) -> Nil() 435.96/148.04 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 435.96/148.04 , monus[Ite](True(), Cons(x, xs), y) -> xs 435.96/148.04 , monus[Ite](False(), Cons(x', xs'), Cons(x, xs)) -> monus(xs', xs) 435.96/148.04 , gcd(Nil(), Nil()) -> Nil() 435.96/148.04 , gcd(Nil(), Cons(x, xs)) -> Nil() 435.96/148.04 , gcd(Cons(x, xs), Nil()) -> Nil() 435.96/148.04 , gcd(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.04 gcd[Ite][False][Ite](eqList(Cons(x', xs'), Cons(x, xs)), 435.96/148.04 Cons(x', xs'), 435.96/148.04 Cons(x, xs)) 435.96/148.04 , @(Nil(), ys) -> ys 435.96/148.04 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 435.96/148.04 , monus(x, y) -> 435.96/148.04 monus[Ite](eqList(lgth(y), Cons(Nil(), Nil())), x, y) 435.96/148.04 , and(True(), True()) -> True() 435.96/148.04 , and(True(), False()) -> False() 435.96/148.04 , and(False(), True()) -> False() 435.96/148.04 , and(False(), False()) -> False() 435.96/148.04 , eqList(Nil(), Nil()) -> True() 435.96/148.04 , eqList(Nil(), Cons(y, ys)) -> False() 435.96/148.04 , eqList(Cons(x, xs), Nil()) -> False() 435.96/148.04 , eqList(Cons(x, xs), Cons(y, ys)) -> 435.96/148.04 and(eqList(x, y), eqList(xs, ys)) 435.96/148.04 , gt0(Nil(), y) -> False() 435.96/148.04 , gt0(Cons(x, xs), Nil()) -> True() 435.96/148.04 , gt0(Cons(x', xs'), Cons(x, xs)) -> gt0(xs', xs) 435.96/148.04 , gcd[Ite][False][Ite](True(), x, y) -> x 435.96/148.04 , gcd[Ite][False][Ite](False(), x, y) -> 435.96/148.04 gcd[Ite][False][Ite][False][Ite](gt0(x, y), x, y) 435.96/148.04 , goal(x, y) -> gcd(x, y) } 435.96/148.04 Obligation: 435.96/148.04 innermost runtime complexity 435.96/148.04 Answer: 435.96/148.04 YES(O(1),O(n^2)) 435.96/148.04 435.96/148.04 We estimate the number of application of {1,3,5,6,7,9,10,11} by 435.96/148.04 applications of Pre({1,3,5,6,7,9,10,11}) = {2,4,12,13,17}. Here 435.96/148.04 rules are labeled as follows: 435.96/148.04 435.96/148.04 DPs: 435.96/148.04 { 1: lgth^#(Nil()) -> c_1() 435.96/148.04 , 2: lgth^#(Cons(x, xs)) -> 435.96/148.04 c_2(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 435.96/148.04 , 3: @^#(Nil(), ys) -> c_7() 435.96/148.04 , 4: @^#(Cons(x, xs), ys) -> c_8(@^#(xs, ys)) 435.96/148.04 , 5: gcd^#(Nil(), Nil()) -> c_3() 435.96/148.04 , 6: gcd^#(Nil(), Cons(x, xs)) -> c_4() 435.96/148.04 , 7: gcd^#(Cons(x, xs), Nil()) -> c_5() 435.96/148.04 , 8: gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.04 c_6(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 435.96/148.04 Cons(x', xs'), 435.96/148.04 Cons(x, xs)), 435.96/148.04 eqList^#(Cons(x', xs'), Cons(x, xs))) 435.96/148.04 , 9: eqList^#(Nil(), Nil()) -> c_10() 435.96/148.04 , 10: eqList^#(Nil(), Cons(y, ys)) -> c_11() 435.96/148.04 , 11: eqList^#(Cons(x, xs), Nil()) -> c_12() 435.96/148.04 , 12: eqList^#(Cons(x, xs), Cons(y, ys)) -> 435.96/148.04 c_13(and^#(eqList(x, y), eqList(xs, ys)), 435.96/148.04 eqList^#(x, y), 435.96/148.04 eqList^#(xs, ys)) 435.96/148.04 , 13: monus^#(x, y) -> 435.96/148.04 c_9(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.04 eqList^#(lgth(y), Cons(Nil(), Nil())), 435.96/148.04 lgth^#(y)) 435.96/148.04 , 14: gt0^#(Nil(), y) -> c_14() 435.96/148.04 , 15: gt0^#(Cons(x, xs), Nil()) -> c_15() 435.96/148.04 , 16: gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_16(gt0^#(xs', xs)) 435.96/148.04 , 17: goal^#(x, y) -> c_17(gcd^#(x, y)) 435.96/148.04 , 18: gcd[Ite][False][Ite]^#(True(), x, y) -> c_24() 435.96/148.04 , 19: gcd[Ite][False][Ite]^#(False(), x, y) -> c_25(gt0^#(x, y)) 435.96/148.04 , 20: monus[Ite]^#(True(), Cons(x, xs), y) -> c_18() 435.96/148.04 , 21: monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 435.96/148.04 c_19(monus^#(xs', xs)) 435.96/148.04 , 22: and^#(True(), True()) -> c_20() 435.96/148.04 , 23: and^#(True(), False()) -> c_21() 435.96/148.04 , 24: and^#(False(), True()) -> c_22() 435.96/148.04 , 25: and^#(False(), False()) -> c_23() } 435.96/148.04 435.96/148.04 We are left with following problem, upon which TcT provides the 435.96/148.04 certificate YES(O(1),O(n^2)). 435.96/148.04 435.96/148.04 Strict DPs: 435.96/148.04 { lgth^#(Cons(x, xs)) -> 435.96/148.05 c_2(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 435.96/148.05 , @^#(Cons(x, xs), ys) -> c_8(@^#(xs, ys)) 435.96/148.05 , gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.05 c_6(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 435.96/148.05 Cons(x', xs'), 435.96/148.05 Cons(x, xs)), 435.96/148.05 eqList^#(Cons(x', xs'), Cons(x, xs))) 435.96/148.05 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 435.96/148.05 c_13(and^#(eqList(x, y), eqList(xs, ys)), 435.96/148.05 eqList^#(x, y), 435.96/148.05 eqList^#(xs, ys)) 435.96/148.05 , monus^#(x, y) -> 435.96/148.05 c_9(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.05 eqList^#(lgth(y), Cons(Nil(), Nil())), 435.96/148.05 lgth^#(y)) 435.96/148.05 , gt0^#(Nil(), y) -> c_14() 435.96/148.05 , gt0^#(Cons(x, xs), Nil()) -> c_15() 435.96/148.05 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_16(gt0^#(xs', xs)) 435.96/148.05 , goal^#(x, y) -> c_17(gcd^#(x, y)) } 435.96/148.05 Weak DPs: 435.96/148.05 { lgth^#(Nil()) -> c_1() 435.96/148.05 , @^#(Nil(), ys) -> c_7() 435.96/148.05 , gcd^#(Nil(), Nil()) -> c_3() 435.96/148.05 , gcd^#(Nil(), Cons(x, xs)) -> c_4() 435.96/148.05 , gcd^#(Cons(x, xs), Nil()) -> c_5() 435.96/148.05 , gcd[Ite][False][Ite]^#(True(), x, y) -> c_24() 435.96/148.05 , gcd[Ite][False][Ite]^#(False(), x, y) -> c_25(gt0^#(x, y)) 435.96/148.05 , eqList^#(Nil(), Nil()) -> c_10() 435.96/148.05 , eqList^#(Nil(), Cons(y, ys)) -> c_11() 435.96/148.05 , eqList^#(Cons(x, xs), Nil()) -> c_12() 435.96/148.05 , monus[Ite]^#(True(), Cons(x, xs), y) -> c_18() 435.96/148.05 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 435.96/148.05 c_19(monus^#(xs', xs)) 435.96/148.05 , and^#(True(), True()) -> c_20() 435.96/148.05 , and^#(True(), False()) -> c_21() 435.96/148.05 , and^#(False(), True()) -> c_22() 435.96/148.05 , and^#(False(), False()) -> c_23() } 435.96/148.05 Weak Trs: 435.96/148.05 { lgth(Nil()) -> Nil() 435.96/148.05 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 435.96/148.05 , monus[Ite](True(), Cons(x, xs), y) -> xs 435.96/148.05 , monus[Ite](False(), Cons(x', xs'), Cons(x, xs)) -> monus(xs', xs) 435.96/148.05 , gcd(Nil(), Nil()) -> Nil() 435.96/148.05 , gcd(Nil(), Cons(x, xs)) -> Nil() 435.96/148.05 , gcd(Cons(x, xs), Nil()) -> Nil() 435.96/148.05 , gcd(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.05 gcd[Ite][False][Ite](eqList(Cons(x', xs'), Cons(x, xs)), 435.96/148.05 Cons(x', xs'), 435.96/148.05 Cons(x, xs)) 435.96/148.05 , @(Nil(), ys) -> ys 435.96/148.05 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 435.96/148.05 , monus(x, y) -> 435.96/148.05 monus[Ite](eqList(lgth(y), Cons(Nil(), Nil())), x, y) 435.96/148.05 , and(True(), True()) -> True() 435.96/148.05 , and(True(), False()) -> False() 435.96/148.05 , and(False(), True()) -> False() 435.96/148.05 , and(False(), False()) -> False() 435.96/148.05 , eqList(Nil(), Nil()) -> True() 435.96/148.05 , eqList(Nil(), Cons(y, ys)) -> False() 435.96/148.05 , eqList(Cons(x, xs), Nil()) -> False() 435.96/148.05 , eqList(Cons(x, xs), Cons(y, ys)) -> 435.96/148.05 and(eqList(x, y), eqList(xs, ys)) 435.96/148.05 , gt0(Nil(), y) -> False() 435.96/148.05 , gt0(Cons(x, xs), Nil()) -> True() 435.96/148.05 , gt0(Cons(x', xs'), Cons(x, xs)) -> gt0(xs', xs) 435.96/148.05 , gcd[Ite][False][Ite](True(), x, y) -> x 435.96/148.05 , gcd[Ite][False][Ite](False(), x, y) -> 435.96/148.05 gcd[Ite][False][Ite][False][Ite](gt0(x, y), x, y) 435.96/148.05 , goal(x, y) -> gcd(x, y) } 435.96/148.05 Obligation: 435.96/148.05 innermost runtime complexity 435.96/148.05 Answer: 435.96/148.05 YES(O(1),O(n^2)) 435.96/148.05 435.96/148.05 The following weak DPs constitute a sub-graph of the DG that is 435.96/148.05 closed under successors. The DPs are removed. 435.96/148.05 435.96/148.05 { lgth^#(Nil()) -> c_1() 435.96/148.05 , @^#(Nil(), ys) -> c_7() 435.96/148.05 , gcd^#(Nil(), Nil()) -> c_3() 435.96/148.05 , gcd^#(Nil(), Cons(x, xs)) -> c_4() 435.96/148.05 , gcd^#(Cons(x, xs), Nil()) -> c_5() 435.96/148.05 , gcd[Ite][False][Ite]^#(True(), x, y) -> c_24() 435.96/148.05 , eqList^#(Nil(), Nil()) -> c_10() 435.96/148.05 , eqList^#(Nil(), Cons(y, ys)) -> c_11() 435.96/148.05 , eqList^#(Cons(x, xs), Nil()) -> c_12() 435.96/148.05 , monus[Ite]^#(True(), Cons(x, xs), y) -> c_18() 435.96/148.05 , and^#(True(), True()) -> c_20() 435.96/148.05 , and^#(True(), False()) -> c_21() 435.96/148.05 , and^#(False(), True()) -> c_22() 435.96/148.05 , and^#(False(), False()) -> c_23() } 435.96/148.05 435.96/148.05 We are left with following problem, upon which TcT provides the 435.96/148.05 certificate YES(O(1),O(n^2)). 435.96/148.05 435.96/148.05 Strict DPs: 435.96/148.05 { lgth^#(Cons(x, xs)) -> 435.96/148.05 c_2(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 435.96/148.05 , @^#(Cons(x, xs), ys) -> c_8(@^#(xs, ys)) 435.96/148.05 , gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.05 c_6(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 435.96/148.05 Cons(x', xs'), 435.96/148.05 Cons(x, xs)), 435.96/148.05 eqList^#(Cons(x', xs'), Cons(x, xs))) 435.96/148.05 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 435.96/148.05 c_13(and^#(eqList(x, y), eqList(xs, ys)), 435.96/148.05 eqList^#(x, y), 435.96/148.05 eqList^#(xs, ys)) 435.96/148.05 , monus^#(x, y) -> 435.96/148.05 c_9(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.05 eqList^#(lgth(y), Cons(Nil(), Nil())), 435.96/148.05 lgth^#(y)) 435.96/148.05 , gt0^#(Nil(), y) -> c_14() 435.96/148.05 , gt0^#(Cons(x, xs), Nil()) -> c_15() 435.96/148.05 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_16(gt0^#(xs', xs)) 435.96/148.05 , goal^#(x, y) -> c_17(gcd^#(x, y)) } 435.96/148.05 Weak DPs: 435.96/148.05 { gcd[Ite][False][Ite]^#(False(), x, y) -> c_25(gt0^#(x, y)) 435.96/148.05 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 435.96/148.05 c_19(monus^#(xs', xs)) } 435.96/148.05 Weak Trs: 435.96/148.05 { lgth(Nil()) -> Nil() 435.96/148.05 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 435.96/148.05 , monus[Ite](True(), Cons(x, xs), y) -> xs 435.96/148.05 , monus[Ite](False(), Cons(x', xs'), Cons(x, xs)) -> monus(xs', xs) 435.96/148.05 , gcd(Nil(), Nil()) -> Nil() 435.96/148.05 , gcd(Nil(), Cons(x, xs)) -> Nil() 435.96/148.05 , gcd(Cons(x, xs), Nil()) -> Nil() 435.96/148.05 , gcd(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.05 gcd[Ite][False][Ite](eqList(Cons(x', xs'), Cons(x, xs)), 435.96/148.05 Cons(x', xs'), 435.96/148.05 Cons(x, xs)) 435.96/148.05 , @(Nil(), ys) -> ys 435.96/148.05 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 435.96/148.05 , monus(x, y) -> 435.96/148.05 monus[Ite](eqList(lgth(y), Cons(Nil(), Nil())), x, y) 435.96/148.05 , and(True(), True()) -> True() 435.96/148.05 , and(True(), False()) -> False() 435.96/148.05 , and(False(), True()) -> False() 435.96/148.05 , and(False(), False()) -> False() 435.96/148.05 , eqList(Nil(), Nil()) -> True() 435.96/148.05 , eqList(Nil(), Cons(y, ys)) -> False() 435.96/148.05 , eqList(Cons(x, xs), Nil()) -> False() 435.96/148.05 , eqList(Cons(x, xs), Cons(y, ys)) -> 435.96/148.05 and(eqList(x, y), eqList(xs, ys)) 435.96/148.05 , gt0(Nil(), y) -> False() 435.96/148.05 , gt0(Cons(x, xs), Nil()) -> True() 435.96/148.05 , gt0(Cons(x', xs'), Cons(x, xs)) -> gt0(xs', xs) 435.96/148.05 , gcd[Ite][False][Ite](True(), x, y) -> x 435.96/148.05 , gcd[Ite][False][Ite](False(), x, y) -> 435.96/148.05 gcd[Ite][False][Ite][False][Ite](gt0(x, y), x, y) 435.96/148.05 , goal(x, y) -> gcd(x, y) } 435.96/148.05 Obligation: 435.96/148.05 innermost runtime complexity 435.96/148.05 Answer: 435.96/148.05 YES(O(1),O(n^2)) 435.96/148.05 435.96/148.05 Due to missing edges in the dependency-graph, the right-hand sides 435.96/148.05 of following rules could be simplified: 435.96/148.05 435.96/148.05 { eqList^#(Cons(x, xs), Cons(y, ys)) -> 435.96/148.05 c_13(and^#(eqList(x, y), eqList(xs, ys)), 435.96/148.05 eqList^#(x, y), 435.96/148.05 eqList^#(xs, ys)) } 435.96/148.05 435.96/148.05 We are left with following problem, upon which TcT provides the 435.96/148.05 certificate YES(O(1),O(n^2)). 435.96/148.05 435.96/148.05 Strict DPs: 435.96/148.05 { lgth^#(Cons(x, xs)) -> 435.96/148.05 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 435.96/148.05 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 435.96/148.05 , gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.05 c_3(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 435.96/148.05 Cons(x', xs'), 435.96/148.05 Cons(x, xs)), 435.96/148.05 eqList^#(Cons(x', xs'), Cons(x, xs))) 435.96/148.05 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 435.96/148.05 c_4(eqList^#(x, y), eqList^#(xs, ys)) 435.96/148.05 , monus^#(x, y) -> 435.96/148.05 c_5(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.05 eqList^#(lgth(y), Cons(Nil(), Nil())), 435.96/148.05 lgth^#(y)) 435.96/148.05 , gt0^#(Nil(), y) -> c_6() 435.96/148.05 , gt0^#(Cons(x, xs), Nil()) -> c_7() 435.96/148.05 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_8(gt0^#(xs', xs)) 435.96/148.05 , goal^#(x, y) -> c_9(gcd^#(x, y)) } 435.96/148.05 Weak DPs: 435.96/148.05 { gcd[Ite][False][Ite]^#(False(), x, y) -> c_10(gt0^#(x, y)) 435.96/148.05 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 435.96/148.05 c_11(monus^#(xs', xs)) } 435.96/148.05 Weak Trs: 435.96/148.05 { lgth(Nil()) -> Nil() 435.96/148.05 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 435.96/148.05 , monus[Ite](True(), Cons(x, xs), y) -> xs 435.96/148.05 , monus[Ite](False(), Cons(x', xs'), Cons(x, xs)) -> monus(xs', xs) 435.96/148.05 , gcd(Nil(), Nil()) -> Nil() 435.96/148.05 , gcd(Nil(), Cons(x, xs)) -> Nil() 435.96/148.05 , gcd(Cons(x, xs), Nil()) -> Nil() 435.96/148.05 , gcd(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.05 gcd[Ite][False][Ite](eqList(Cons(x', xs'), Cons(x, xs)), 435.96/148.05 Cons(x', xs'), 435.96/148.05 Cons(x, xs)) 435.96/148.05 , @(Nil(), ys) -> ys 435.96/148.05 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 435.96/148.05 , monus(x, y) -> 435.96/148.05 monus[Ite](eqList(lgth(y), Cons(Nil(), Nil())), x, y) 435.96/148.05 , and(True(), True()) -> True() 435.96/148.05 , and(True(), False()) -> False() 435.96/148.05 , and(False(), True()) -> False() 435.96/148.05 , and(False(), False()) -> False() 435.96/148.05 , eqList(Nil(), Nil()) -> True() 435.96/148.05 , eqList(Nil(), Cons(y, ys)) -> False() 435.96/148.05 , eqList(Cons(x, xs), Nil()) -> False() 435.96/148.05 , eqList(Cons(x, xs), Cons(y, ys)) -> 435.96/148.05 and(eqList(x, y), eqList(xs, ys)) 435.96/148.05 , gt0(Nil(), y) -> False() 435.96/148.05 , gt0(Cons(x, xs), Nil()) -> True() 435.96/148.05 , gt0(Cons(x', xs'), Cons(x, xs)) -> gt0(xs', xs) 435.96/148.05 , gcd[Ite][False][Ite](True(), x, y) -> x 435.96/148.05 , gcd[Ite][False][Ite](False(), x, y) -> 435.96/148.05 gcd[Ite][False][Ite][False][Ite](gt0(x, y), x, y) 435.96/148.05 , goal(x, y) -> gcd(x, y) } 435.96/148.05 Obligation: 435.96/148.05 innermost runtime complexity 435.96/148.05 Answer: 435.96/148.05 YES(O(1),O(n^2)) 435.96/148.05 435.96/148.05 We replace rewrite rules by usable rules: 435.96/148.05 435.96/148.05 Weak Usable Rules: 435.96/148.05 { lgth(Nil()) -> Nil() 435.96/148.05 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 435.96/148.05 , @(Nil(), ys) -> ys 435.96/148.05 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 435.96/148.05 , and(True(), True()) -> True() 435.96/148.05 , and(True(), False()) -> False() 435.96/148.05 , and(False(), True()) -> False() 435.96/148.05 , and(False(), False()) -> False() 435.96/148.05 , eqList(Nil(), Nil()) -> True() 435.96/148.05 , eqList(Nil(), Cons(y, ys)) -> False() 435.96/148.05 , eqList(Cons(x, xs), Nil()) -> False() 435.96/148.05 , eqList(Cons(x, xs), Cons(y, ys)) -> 435.96/148.05 and(eqList(x, y), eqList(xs, ys)) } 435.96/148.05 435.96/148.05 We are left with following problem, upon which TcT provides the 435.96/148.05 certificate YES(O(1),O(n^2)). 435.96/148.05 435.96/148.05 Strict DPs: 435.96/148.05 { lgth^#(Cons(x, xs)) -> 435.96/148.05 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 435.96/148.05 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 435.96/148.05 , gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.05 c_3(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 435.96/148.05 Cons(x', xs'), 435.96/148.05 Cons(x, xs)), 435.96/148.05 eqList^#(Cons(x', xs'), Cons(x, xs))) 435.96/148.05 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 435.96/148.05 c_4(eqList^#(x, y), eqList^#(xs, ys)) 435.96/148.05 , monus^#(x, y) -> 435.96/148.05 c_5(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.05 eqList^#(lgth(y), Cons(Nil(), Nil())), 435.96/148.05 lgth^#(y)) 435.96/148.05 , gt0^#(Nil(), y) -> c_6() 435.96/148.05 , gt0^#(Cons(x, xs), Nil()) -> c_7() 435.96/148.05 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_8(gt0^#(xs', xs)) 435.96/148.05 , goal^#(x, y) -> c_9(gcd^#(x, y)) } 435.96/148.05 Weak DPs: 435.96/148.05 { gcd[Ite][False][Ite]^#(False(), x, y) -> c_10(gt0^#(x, y)) 435.96/148.05 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 435.96/148.05 c_11(monus^#(xs', xs)) } 435.96/148.05 Weak Trs: 435.96/148.05 { lgth(Nil()) -> Nil() 435.96/148.05 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 435.96/148.05 , @(Nil(), ys) -> ys 435.96/148.05 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 435.96/148.05 , and(True(), True()) -> True() 435.96/148.05 , and(True(), False()) -> False() 435.96/148.05 , and(False(), True()) -> False() 435.96/148.05 , and(False(), False()) -> False() 435.96/148.05 , eqList(Nil(), Nil()) -> True() 435.96/148.05 , eqList(Nil(), Cons(y, ys)) -> False() 435.96/148.05 , eqList(Cons(x, xs), Nil()) -> False() 435.96/148.05 , eqList(Cons(x, xs), Cons(y, ys)) -> 435.96/148.05 and(eqList(x, y), eqList(xs, ys)) } 435.96/148.05 Obligation: 435.96/148.05 innermost runtime complexity 435.96/148.05 Answer: 435.96/148.05 YES(O(1),O(n^2)) 435.96/148.05 435.96/148.05 Consider the dependency graph 435.96/148.05 435.96/148.05 1: lgth^#(Cons(x, xs)) -> 435.96/148.05 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 435.96/148.05 -->_1 @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) :2 435.96/148.05 -->_2 lgth^#(Cons(x, xs)) -> 435.96/148.05 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) :1 435.96/148.05 435.96/148.05 2: @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 435.96/148.05 -->_1 @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) :2 435.96/148.05 435.96/148.05 3: gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.05 c_3(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 435.96/148.05 Cons(x', xs'), 435.96/148.05 Cons(x, xs)), 435.96/148.05 eqList^#(Cons(x', xs'), Cons(x, xs))) 435.96/148.05 -->_1 gcd[Ite][False][Ite]^#(False(), x, y) -> 435.96/148.05 c_10(gt0^#(x, y)) :10 435.96/148.05 -->_2 eqList^#(Cons(x, xs), Cons(y, ys)) -> 435.96/148.05 c_4(eqList^#(x, y), eqList^#(xs, ys)) :4 435.96/148.05 435.96/148.05 4: eqList^#(Cons(x, xs), Cons(y, ys)) -> 435.96/148.05 c_4(eqList^#(x, y), eqList^#(xs, ys)) 435.96/148.05 -->_2 eqList^#(Cons(x, xs), Cons(y, ys)) -> 435.96/148.05 c_4(eqList^#(x, y), eqList^#(xs, ys)) :4 435.96/148.05 -->_1 eqList^#(Cons(x, xs), Cons(y, ys)) -> 435.96/148.05 c_4(eqList^#(x, y), eqList^#(xs, ys)) :4 435.96/148.05 435.96/148.05 5: monus^#(x, y) -> 435.96/148.05 c_5(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.05 eqList^#(lgth(y), Cons(Nil(), Nil())), 435.96/148.05 lgth^#(y)) 435.96/148.05 -->_1 monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 435.96/148.05 c_11(monus^#(xs', xs)) :11 435.96/148.05 -->_2 eqList^#(Cons(x, xs), Cons(y, ys)) -> 435.96/148.05 c_4(eqList^#(x, y), eqList^#(xs, ys)) :4 435.96/148.05 -->_3 lgth^#(Cons(x, xs)) -> 435.96/148.05 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) :1 435.96/148.05 435.96/148.05 6: gt0^#(Nil(), y) -> c_6() 435.96/148.05 435.96/148.05 7: gt0^#(Cons(x, xs), Nil()) -> c_7() 435.96/148.05 435.96/148.05 8: gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_8(gt0^#(xs', xs)) 435.96/148.05 -->_1 gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_8(gt0^#(xs', xs)) :8 435.96/148.05 -->_1 gt0^#(Cons(x, xs), Nil()) -> c_7() :7 435.96/148.05 -->_1 gt0^#(Nil(), y) -> c_6() :6 435.96/148.05 435.96/148.05 9: goal^#(x, y) -> c_9(gcd^#(x, y)) 435.96/148.05 -->_1 gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.05 c_3(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 435.96/148.05 Cons(x', xs'), 435.96/148.05 Cons(x, xs)), 435.96/148.05 eqList^#(Cons(x', xs'), Cons(x, xs))) :3 435.96/148.05 435.96/148.05 10: gcd[Ite][False][Ite]^#(False(), x, y) -> c_10(gt0^#(x, y)) 435.96/148.05 -->_1 gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_8(gt0^#(xs', xs)) :8 435.96/148.05 -->_1 gt0^#(Cons(x, xs), Nil()) -> c_7() :7 435.96/148.05 -->_1 gt0^#(Nil(), y) -> c_6() :6 435.96/148.05 435.96/148.05 11: monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 435.96/148.05 c_11(monus^#(xs', xs)) 435.96/148.05 -->_1 monus^#(x, y) -> 435.96/148.05 c_5(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.05 eqList^#(lgth(y), Cons(Nil(), Nil())), 435.96/148.05 lgth^#(y)) :5 435.96/148.05 435.96/148.05 435.96/148.05 Following roots of the dependency graph are removed, as the 435.96/148.05 considered set of starting terms is closed under reduction with 435.96/148.05 respect to these rules (modulo compound contexts). 435.96/148.05 435.96/148.05 { goal^#(x, y) -> c_9(gcd^#(x, y)) } 435.96/148.05 435.96/148.05 435.96/148.05 We are left with following problem, upon which TcT provides the 435.96/148.05 certificate YES(O(1),O(n^2)). 435.96/148.05 435.96/148.05 Strict DPs: 435.96/148.05 { lgth^#(Cons(x, xs)) -> 435.96/148.05 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 435.96/148.05 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 435.96/148.05 , gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.05 c_3(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 435.96/148.05 Cons(x', xs'), 435.96/148.05 Cons(x, xs)), 435.96/148.05 eqList^#(Cons(x', xs'), Cons(x, xs))) 435.96/148.05 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 435.96/148.05 c_4(eqList^#(x, y), eqList^#(xs, ys)) 435.96/148.05 , monus^#(x, y) -> 435.96/148.05 c_5(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.05 eqList^#(lgth(y), Cons(Nil(), Nil())), 435.96/148.05 lgth^#(y)) 435.96/148.05 , gt0^#(Nil(), y) -> c_6() 435.96/148.05 , gt0^#(Cons(x, xs), Nil()) -> c_7() 435.96/148.05 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_8(gt0^#(xs', xs)) } 435.96/148.05 Weak DPs: 435.96/148.05 { gcd[Ite][False][Ite]^#(False(), x, y) -> c_10(gt0^#(x, y)) 435.96/148.05 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 435.96/148.05 c_11(monus^#(xs', xs)) } 435.96/148.05 Weak Trs: 435.96/148.05 { lgth(Nil()) -> Nil() 435.96/148.05 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 435.96/148.05 , @(Nil(), ys) -> ys 435.96/148.05 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 435.96/148.05 , and(True(), True()) -> True() 435.96/148.05 , and(True(), False()) -> False() 435.96/148.05 , and(False(), True()) -> False() 435.96/148.05 , and(False(), False()) -> False() 435.96/148.05 , eqList(Nil(), Nil()) -> True() 435.96/148.05 , eqList(Nil(), Cons(y, ys)) -> False() 435.96/148.05 , eqList(Cons(x, xs), Nil()) -> False() 435.96/148.05 , eqList(Cons(x, xs), Cons(y, ys)) -> 435.96/148.05 and(eqList(x, y), eqList(xs, ys)) } 435.96/148.05 Obligation: 435.96/148.05 innermost runtime complexity 435.96/148.05 Answer: 435.96/148.05 YES(O(1),O(n^2)) 435.96/148.05 435.96/148.05 We analyse the complexity of following sub-problems (R) and (S). 435.96/148.05 Problem (S) is obtained from the input problem by shifting strict 435.96/148.05 rules from (R) into the weak component: 435.96/148.05 435.96/148.05 Problem (R): 435.96/148.05 ------------ 435.96/148.05 Strict DPs: 435.96/148.05 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.05 c_3(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 435.96/148.05 Cons(x', xs'), 435.96/148.05 Cons(x, xs)), 435.96/148.05 eqList^#(Cons(x', xs'), Cons(x, xs))) 435.96/148.05 , gt0^#(Nil(), y) -> c_6() 435.96/148.05 , gt0^#(Cons(x, xs), Nil()) -> c_7() 435.96/148.05 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_8(gt0^#(xs', xs)) } 435.96/148.05 Weak DPs: 435.96/148.05 { lgth^#(Cons(x, xs)) -> 435.96/148.05 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 435.96/148.05 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 435.96/148.05 , gcd[Ite][False][Ite]^#(False(), x, y) -> c_10(gt0^#(x, y)) 435.96/148.05 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 435.96/148.05 c_4(eqList^#(x, y), eqList^#(xs, ys)) 435.96/148.05 , monus^#(x, y) -> 435.96/148.05 c_5(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.05 eqList^#(lgth(y), Cons(Nil(), Nil())), 435.96/148.05 lgth^#(y)) 435.96/148.05 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 435.96/148.05 c_11(monus^#(xs', xs)) } 435.96/148.05 Weak Trs: 435.96/148.05 { lgth(Nil()) -> Nil() 435.96/148.05 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 435.96/148.05 , @(Nil(), ys) -> ys 435.96/148.05 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 435.96/148.05 , and(True(), True()) -> True() 435.96/148.05 , and(True(), False()) -> False() 435.96/148.05 , and(False(), True()) -> False() 435.96/148.05 , and(False(), False()) -> False() 435.96/148.05 , eqList(Nil(), Nil()) -> True() 435.96/148.05 , eqList(Nil(), Cons(y, ys)) -> False() 435.96/148.05 , eqList(Cons(x, xs), Nil()) -> False() 435.96/148.05 , eqList(Cons(x, xs), Cons(y, ys)) -> 435.96/148.05 and(eqList(x, y), eqList(xs, ys)) } 435.96/148.05 StartTerms: basic terms 435.96/148.05 Strategy: innermost 435.96/148.05 435.96/148.05 Problem (S): 435.96/148.05 ------------ 435.96/148.05 Strict DPs: 435.96/148.05 { lgth^#(Cons(x, xs)) -> 435.96/148.05 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 435.96/148.05 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 435.96/148.05 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 435.96/148.05 c_4(eqList^#(x, y), eqList^#(xs, ys)) 435.96/148.05 , monus^#(x, y) -> 435.96/148.05 c_5(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.05 eqList^#(lgth(y), Cons(Nil(), Nil())), 435.96/148.05 lgth^#(y)) } 435.96/148.05 Weak DPs: 435.96/148.05 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.05 c_3(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 435.96/148.05 Cons(x', xs'), 435.96/148.05 Cons(x, xs)), 435.96/148.05 eqList^#(Cons(x', xs'), Cons(x, xs))) 435.96/148.05 , gcd[Ite][False][Ite]^#(False(), x, y) -> c_10(gt0^#(x, y)) 435.96/148.05 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 435.96/148.05 c_11(monus^#(xs', xs)) 435.96/148.05 , gt0^#(Nil(), y) -> c_6() 435.96/148.05 , gt0^#(Cons(x, xs), Nil()) -> c_7() 435.96/148.05 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_8(gt0^#(xs', xs)) } 435.96/148.05 Weak Trs: 435.96/148.05 { lgth(Nil()) -> Nil() 435.96/148.05 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 435.96/148.05 , @(Nil(), ys) -> ys 435.96/148.05 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 435.96/148.05 , and(True(), True()) -> True() 435.96/148.05 , and(True(), False()) -> False() 435.96/148.05 , and(False(), True()) -> False() 435.96/148.05 , and(False(), False()) -> False() 435.96/148.05 , eqList(Nil(), Nil()) -> True() 435.96/148.05 , eqList(Nil(), Cons(y, ys)) -> False() 435.96/148.05 , eqList(Cons(x, xs), Nil()) -> False() 435.96/148.05 , eqList(Cons(x, xs), Cons(y, ys)) -> 435.96/148.05 and(eqList(x, y), eqList(xs, ys)) } 435.96/148.05 StartTerms: basic terms 435.96/148.05 Strategy: innermost 435.96/148.05 435.96/148.05 Overall, the transformation results in the following sub-problem(s): 435.96/148.05 435.96/148.05 Generated new problems: 435.96/148.05 ----------------------- 435.96/148.05 R) Strict DPs: 435.96/148.05 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.05 c_3(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 435.96/148.05 Cons(x', xs'), 435.96/148.05 Cons(x, xs)), 435.96/148.05 eqList^#(Cons(x', xs'), Cons(x, xs))) 435.96/148.05 , gt0^#(Nil(), y) -> c_6() 435.96/148.05 , gt0^#(Cons(x, xs), Nil()) -> c_7() 435.96/148.05 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_8(gt0^#(xs', xs)) } 435.96/148.05 Weak DPs: 435.96/148.05 { lgth^#(Cons(x, xs)) -> 435.96/148.05 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 435.96/148.05 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 435.96/148.05 , gcd[Ite][False][Ite]^#(False(), x, y) -> c_10(gt0^#(x, y)) 435.96/148.05 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 435.96/148.05 c_4(eqList^#(x, y), eqList^#(xs, ys)) 435.96/148.05 , monus^#(x, y) -> 435.96/148.05 c_5(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.05 eqList^#(lgth(y), Cons(Nil(), Nil())), 435.96/148.05 lgth^#(y)) 435.96/148.05 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 435.96/148.05 c_11(monus^#(xs', xs)) } 435.96/148.05 Weak Trs: 435.96/148.05 { lgth(Nil()) -> Nil() 435.96/148.05 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 435.96/148.05 , @(Nil(), ys) -> ys 435.96/148.05 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 435.96/148.05 , and(True(), True()) -> True() 435.96/148.05 , and(True(), False()) -> False() 435.96/148.05 , and(False(), True()) -> False() 435.96/148.05 , and(False(), False()) -> False() 435.96/148.05 , eqList(Nil(), Nil()) -> True() 435.96/148.05 , eqList(Nil(), Cons(y, ys)) -> False() 435.96/148.05 , eqList(Cons(x, xs), Nil()) -> False() 435.96/148.05 , eqList(Cons(x, xs), Cons(y, ys)) -> 435.96/148.05 and(eqList(x, y), eqList(xs, ys)) } 435.96/148.05 StartTerms: basic terms 435.96/148.05 Strategy: innermost 435.96/148.05 435.96/148.05 This problem was proven YES(O(1),O(n^1)). 435.96/148.05 435.96/148.05 S) Strict DPs: 435.96/148.05 { lgth^#(Cons(x, xs)) -> 435.96/148.06 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 435.96/148.06 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 435.96/148.06 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 435.96/148.06 c_4(eqList^#(x, y), eqList^#(xs, ys)) 435.96/148.06 , monus^#(x, y) -> 435.96/148.06 c_5(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.06 eqList^#(lgth(y), Cons(Nil(), Nil())), 435.96/148.06 lgth^#(y)) } 435.96/148.06 Weak DPs: 435.96/148.06 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.06 c_3(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 435.96/148.06 Cons(x', xs'), 435.96/148.06 Cons(x, xs)), 435.96/148.06 eqList^#(Cons(x', xs'), Cons(x, xs))) 435.96/148.06 , gcd[Ite][False][Ite]^#(False(), x, y) -> c_10(gt0^#(x, y)) 435.96/148.06 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 435.96/148.06 c_11(monus^#(xs', xs)) 435.96/148.06 , gt0^#(Nil(), y) -> c_6() 435.96/148.06 , gt0^#(Cons(x, xs), Nil()) -> c_7() 435.96/148.06 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_8(gt0^#(xs', xs)) } 435.96/148.06 Weak Trs: 435.96/148.06 { lgth(Nil()) -> Nil() 435.96/148.06 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 435.96/148.06 , @(Nil(), ys) -> ys 435.96/148.06 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 435.96/148.06 , and(True(), True()) -> True() 435.96/148.06 , and(True(), False()) -> False() 435.96/148.06 , and(False(), True()) -> False() 435.96/148.06 , and(False(), False()) -> False() 435.96/148.06 , eqList(Nil(), Nil()) -> True() 435.96/148.06 , eqList(Nil(), Cons(y, ys)) -> False() 435.96/148.06 , eqList(Cons(x, xs), Nil()) -> False() 435.96/148.06 , eqList(Cons(x, xs), Cons(y, ys)) -> 435.96/148.06 and(eqList(x, y), eqList(xs, ys)) } 435.96/148.06 StartTerms: basic terms 435.96/148.06 Strategy: innermost 435.96/148.06 435.96/148.06 This problem was proven YES(O(1),O(n^2)). 435.96/148.06 435.96/148.06 435.96/148.06 Proofs for generated problems: 435.96/148.06 ------------------------------ 435.96/148.06 R) We are left with following problem, upon which TcT provides the 435.96/148.06 certificate YES(O(1),O(n^1)). 435.96/148.06 435.96/148.06 Strict DPs: 435.96/148.06 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.06 c_3(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 435.96/148.06 Cons(x', xs'), 435.96/148.06 Cons(x, xs)), 435.96/148.06 eqList^#(Cons(x', xs'), Cons(x, xs))) 435.96/148.06 , gt0^#(Nil(), y) -> c_6() 435.96/148.06 , gt0^#(Cons(x, xs), Nil()) -> c_7() 435.96/148.06 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_8(gt0^#(xs', xs)) } 435.96/148.06 Weak DPs: 435.96/148.06 { lgth^#(Cons(x, xs)) -> 435.96/148.06 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 435.96/148.06 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 435.96/148.06 , gcd[Ite][False][Ite]^#(False(), x, y) -> c_10(gt0^#(x, y)) 435.96/148.06 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 435.96/148.06 c_4(eqList^#(x, y), eqList^#(xs, ys)) 435.96/148.06 , monus^#(x, y) -> 435.96/148.06 c_5(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.06 eqList^#(lgth(y), Cons(Nil(), Nil())), 435.96/148.06 lgth^#(y)) 435.96/148.06 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 435.96/148.06 c_11(monus^#(xs', xs)) } 435.96/148.06 Weak Trs: 435.96/148.06 { lgth(Nil()) -> Nil() 435.96/148.06 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 435.96/148.06 , @(Nil(), ys) -> ys 435.96/148.06 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 435.96/148.06 , and(True(), True()) -> True() 435.96/148.06 , and(True(), False()) -> False() 435.96/148.06 , and(False(), True()) -> False() 435.96/148.06 , and(False(), False()) -> False() 435.96/148.06 , eqList(Nil(), Nil()) -> True() 435.96/148.06 , eqList(Nil(), Cons(y, ys)) -> False() 435.96/148.06 , eqList(Cons(x, xs), Nil()) -> False() 435.96/148.06 , eqList(Cons(x, xs), Cons(y, ys)) -> 435.96/148.06 and(eqList(x, y), eqList(xs, ys)) } 435.96/148.06 Obligation: 435.96/148.06 innermost runtime complexity 435.96/148.06 Answer: 435.96/148.06 YES(O(1),O(n^1)) 435.96/148.06 435.96/148.06 We estimate the number of application of {1} by applications of 435.96/148.06 Pre({1}) = {}. Here rules are labeled as follows: 435.96/148.06 435.96/148.06 DPs: 435.96/148.06 { 1: gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.06 c_3(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 435.96/148.06 Cons(x', xs'), 435.96/148.06 Cons(x, xs)), 435.96/148.06 eqList^#(Cons(x', xs'), Cons(x, xs))) 435.96/148.06 , 2: gt0^#(Nil(), y) -> c_6() 435.96/148.06 , 3: gt0^#(Cons(x, xs), Nil()) -> c_7() 435.96/148.06 , 4: gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_8(gt0^#(xs', xs)) 435.96/148.06 , 5: lgth^#(Cons(x, xs)) -> 435.96/148.06 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 435.96/148.06 , 6: @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 435.96/148.06 , 7: gcd[Ite][False][Ite]^#(False(), x, y) -> c_10(gt0^#(x, y)) 435.96/148.06 , 8: eqList^#(Cons(x, xs), Cons(y, ys)) -> 435.96/148.06 c_4(eqList^#(x, y), eqList^#(xs, ys)) 435.96/148.06 , 9: monus^#(x, y) -> 435.96/148.06 c_5(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.06 eqList^#(lgth(y), Cons(Nil(), Nil())), 435.96/148.06 lgth^#(y)) 435.96/148.06 , 10: monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 435.96/148.06 c_11(monus^#(xs', xs)) } 435.96/148.06 435.96/148.06 We are left with following problem, upon which TcT provides the 435.96/148.06 certificate YES(O(1),O(n^1)). 435.96/148.06 435.96/148.06 Strict DPs: 435.96/148.06 { gt0^#(Nil(), y) -> c_6() 435.96/148.06 , gt0^#(Cons(x, xs), Nil()) -> c_7() 435.96/148.06 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_8(gt0^#(xs', xs)) } 435.96/148.06 Weak DPs: 435.96/148.06 { lgth^#(Cons(x, xs)) -> 435.96/148.06 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 435.96/148.06 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 435.96/148.06 , gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.06 c_3(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 435.96/148.06 Cons(x', xs'), 435.96/148.06 Cons(x, xs)), 435.96/148.06 eqList^#(Cons(x', xs'), Cons(x, xs))) 435.96/148.06 , gcd[Ite][False][Ite]^#(False(), x, y) -> c_10(gt0^#(x, y)) 435.96/148.06 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 435.96/148.06 c_4(eqList^#(x, y), eqList^#(xs, ys)) 435.96/148.06 , monus^#(x, y) -> 435.96/148.06 c_5(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.06 eqList^#(lgth(y), Cons(Nil(), Nil())), 435.96/148.06 lgth^#(y)) 435.96/148.06 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 435.96/148.06 c_11(monus^#(xs', xs)) } 435.96/148.06 Weak Trs: 435.96/148.06 { lgth(Nil()) -> Nil() 435.96/148.06 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 435.96/148.06 , @(Nil(), ys) -> ys 435.96/148.06 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 435.96/148.06 , and(True(), True()) -> True() 435.96/148.06 , and(True(), False()) -> False() 435.96/148.06 , and(False(), True()) -> False() 435.96/148.06 , and(False(), False()) -> False() 435.96/148.06 , eqList(Nil(), Nil()) -> True() 435.96/148.06 , eqList(Nil(), Cons(y, ys)) -> False() 435.96/148.06 , eqList(Cons(x, xs), Nil()) -> False() 435.96/148.06 , eqList(Cons(x, xs), Cons(y, ys)) -> 435.96/148.06 and(eqList(x, y), eqList(xs, ys)) } 435.96/148.06 Obligation: 435.96/148.06 innermost runtime complexity 435.96/148.06 Answer: 435.96/148.06 YES(O(1),O(n^1)) 435.96/148.06 435.96/148.06 The following weak DPs constitute a sub-graph of the DG that is 435.96/148.06 closed under successors. The DPs are removed. 435.96/148.06 435.96/148.06 { lgth^#(Cons(x, xs)) -> 435.96/148.06 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 435.96/148.06 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 435.96/148.06 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 435.96/148.06 c_4(eqList^#(x, y), eqList^#(xs, ys)) 435.96/148.06 , monus^#(x, y) -> 435.96/148.06 c_5(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.06 eqList^#(lgth(y), Cons(Nil(), Nil())), 435.96/148.06 lgth^#(y)) 435.96/148.06 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 435.96/148.06 c_11(monus^#(xs', xs)) } 435.96/148.06 435.96/148.06 We are left with following problem, upon which TcT provides the 435.96/148.06 certificate YES(O(1),O(n^1)). 435.96/148.06 435.96/148.06 Strict DPs: 435.96/148.06 { gt0^#(Nil(), y) -> c_6() 435.96/148.06 , gt0^#(Cons(x, xs), Nil()) -> c_7() 435.96/148.06 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_8(gt0^#(xs', xs)) } 435.96/148.06 Weak DPs: 435.96/148.06 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.06 c_3(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 435.96/148.06 Cons(x', xs'), 435.96/148.06 Cons(x, xs)), 435.96/148.06 eqList^#(Cons(x', xs'), Cons(x, xs))) 435.96/148.06 , gcd[Ite][False][Ite]^#(False(), x, y) -> c_10(gt0^#(x, y)) } 435.96/148.06 Weak Trs: 435.96/148.06 { lgth(Nil()) -> Nil() 435.96/148.06 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 435.96/148.06 , @(Nil(), ys) -> ys 435.96/148.06 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 435.96/148.06 , and(True(), True()) -> True() 435.96/148.06 , and(True(), False()) -> False() 435.96/148.06 , and(False(), True()) -> False() 435.96/148.06 , and(False(), False()) -> False() 435.96/148.06 , eqList(Nil(), Nil()) -> True() 435.96/148.06 , eqList(Nil(), Cons(y, ys)) -> False() 435.96/148.06 , eqList(Cons(x, xs), Nil()) -> False() 435.96/148.06 , eqList(Cons(x, xs), Cons(y, ys)) -> 435.96/148.06 and(eqList(x, y), eqList(xs, ys)) } 435.96/148.06 Obligation: 435.96/148.06 innermost runtime complexity 435.96/148.06 Answer: 435.96/148.06 YES(O(1),O(n^1)) 435.96/148.06 435.96/148.06 Due to missing edges in the dependency-graph, the right-hand sides 435.96/148.06 of following rules could be simplified: 435.96/148.06 435.96/148.06 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.06 c_3(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 435.96/148.06 Cons(x', xs'), 435.96/148.06 Cons(x, xs)), 435.96/148.06 eqList^#(Cons(x', xs'), Cons(x, xs))) } 435.96/148.06 435.96/148.06 We are left with following problem, upon which TcT provides the 435.96/148.06 certificate YES(O(1),O(n^1)). 435.96/148.06 435.96/148.06 Strict DPs: 435.96/148.06 { gt0^#(Nil(), y) -> c_1() 435.96/148.06 , gt0^#(Cons(x, xs), Nil()) -> c_2() 435.96/148.06 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_3(gt0^#(xs', xs)) } 435.96/148.06 Weak DPs: 435.96/148.06 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.06 c_4(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 435.96/148.06 Cons(x', xs'), 435.96/148.06 Cons(x, xs))) 435.96/148.06 , gcd[Ite][False][Ite]^#(False(), x, y) -> c_5(gt0^#(x, y)) } 435.96/148.06 Weak Trs: 435.96/148.06 { lgth(Nil()) -> Nil() 435.96/148.06 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 435.96/148.06 , @(Nil(), ys) -> ys 435.96/148.06 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 435.96/148.06 , and(True(), True()) -> True() 435.96/148.06 , and(True(), False()) -> False() 435.96/148.06 , and(False(), True()) -> False() 435.96/148.06 , and(False(), False()) -> False() 435.96/148.06 , eqList(Nil(), Nil()) -> True() 435.96/148.06 , eqList(Nil(), Cons(y, ys)) -> False() 435.96/148.06 , eqList(Cons(x, xs), Nil()) -> False() 435.96/148.06 , eqList(Cons(x, xs), Cons(y, ys)) -> 435.96/148.06 and(eqList(x, y), eqList(xs, ys)) } 435.96/148.06 Obligation: 435.96/148.06 innermost runtime complexity 435.96/148.06 Answer: 435.96/148.06 YES(O(1),O(n^1)) 435.96/148.06 435.96/148.06 We replace rewrite rules by usable rules: 435.96/148.06 435.96/148.06 Weak Usable Rules: 435.96/148.06 { and(True(), True()) -> True() 435.96/148.06 , and(True(), False()) -> False() 435.96/148.06 , and(False(), True()) -> False() 435.96/148.06 , and(False(), False()) -> False() 435.96/148.06 , eqList(Nil(), Nil()) -> True() 435.96/148.06 , eqList(Nil(), Cons(y, ys)) -> False() 435.96/148.06 , eqList(Cons(x, xs), Nil()) -> False() 435.96/148.06 , eqList(Cons(x, xs), Cons(y, ys)) -> 435.96/148.06 and(eqList(x, y), eqList(xs, ys)) } 435.96/148.06 435.96/148.06 We are left with following problem, upon which TcT provides the 435.96/148.06 certificate YES(O(1),O(n^1)). 435.96/148.06 435.96/148.06 Strict DPs: 435.96/148.06 { gt0^#(Nil(), y) -> c_1() 435.96/148.06 , gt0^#(Cons(x, xs), Nil()) -> c_2() 435.96/148.06 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_3(gt0^#(xs', xs)) } 435.96/148.06 Weak DPs: 435.96/148.06 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.06 c_4(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 435.96/148.06 Cons(x', xs'), 435.96/148.06 Cons(x, xs))) 435.96/148.06 , gcd[Ite][False][Ite]^#(False(), x, y) -> c_5(gt0^#(x, y)) } 435.96/148.06 Weak Trs: 435.96/148.06 { and(True(), True()) -> True() 435.96/148.06 , and(True(), False()) -> False() 435.96/148.06 , and(False(), True()) -> False() 435.96/148.06 , and(False(), False()) -> False() 435.96/148.06 , eqList(Nil(), Nil()) -> True() 435.96/148.06 , eqList(Nil(), Cons(y, ys)) -> False() 435.96/148.06 , eqList(Cons(x, xs), Nil()) -> False() 435.96/148.06 , eqList(Cons(x, xs), Cons(y, ys)) -> 435.96/148.06 and(eqList(x, y), eqList(xs, ys)) } 435.96/148.06 Obligation: 435.96/148.06 innermost runtime complexity 435.96/148.06 Answer: 435.96/148.06 YES(O(1),O(n^1)) 435.96/148.06 435.96/148.06 We use the processor 'Small Polynomial Path Order (PS,1-bounded)' 435.96/148.06 to orient following rules strictly. 435.96/148.06 435.96/148.06 DPs: 435.96/148.06 { 1: gt0^#(Nil(), y) -> c_1() 435.96/148.06 , 2: gt0^#(Cons(x, xs), Nil()) -> c_2() 435.96/148.06 , 3: gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_3(gt0^#(xs', xs)) 435.96/148.06 , 4: gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.06 c_4(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 435.96/148.06 Cons(x', xs'), 435.96/148.06 Cons(x, xs))) 435.96/148.06 , 5: gcd[Ite][False][Ite]^#(False(), x, y) -> c_5(gt0^#(x, y)) } 435.96/148.06 Trs: 435.96/148.06 { and(True(), True()) -> True() 435.96/148.06 , and(True(), False()) -> False() 435.96/148.06 , and(False(), True()) -> False() 435.96/148.06 , and(False(), False()) -> False() 435.96/148.06 , eqList(Nil(), Nil()) -> True() 435.96/148.06 , eqList(Nil(), Cons(y, ys)) -> False() 435.96/148.06 , eqList(Cons(x, xs), Nil()) -> False() } 435.96/148.06 435.96/148.06 Sub-proof: 435.96/148.06 ---------- 435.96/148.06 The input was oriented with the instance of 'Small Polynomial Path 435.96/148.06 Order (PS,1-bounded)' as induced by the safe mapping 435.96/148.06 435.96/148.06 safe(True) = {}, safe(Nil) = {}, safe(and) = {}, safe(eqList) = {}, 435.96/148.06 safe(Cons) = {1, 2}, safe(False) = {}, safe(gcd^#) = {}, 435.96/148.06 safe(gcd[Ite][False][Ite]^#) = {}, safe(gt0^#) = {2}, 435.96/148.06 safe(c_1) = {}, safe(c_2) = {}, safe(c_3) = {}, safe(c_4) = {}, 435.96/148.06 safe(c_5) = {} 435.96/148.06 435.96/148.06 and precedence 435.96/148.06 435.96/148.06 eqList > and, gcd^# > and, gcd^# > gcd[Ite][False][Ite]^#, 435.96/148.06 gcd^# > gt0^#, gcd[Ite][False][Ite]^# > and, 435.96/148.06 gcd[Ite][False][Ite]^# > gt0^# . 435.96/148.06 435.96/148.06 Following symbols are considered recursive: 435.96/148.06 435.96/148.06 {gt0^#} 435.96/148.06 435.96/148.06 The recursion depth is 1. 435.96/148.06 435.96/148.06 Further, following argument filtering is employed: 435.96/148.06 435.96/148.06 pi(True) = [], pi(Nil) = [], pi(and) = [1], pi(eqList) = [], 435.96/148.06 pi(Cons) = [2], pi(False) = [], pi(gcd^#) = [1], 435.96/148.06 pi(gcd[Ite][False][Ite]^#) = [2], pi(gt0^#) = [1], pi(c_1) = [], 435.96/148.06 pi(c_2) = [], pi(c_3) = [1], pi(c_4) = [1], pi(c_5) = [1] 435.96/148.06 435.96/148.06 Usable defined function symbols are a subset of: 435.96/148.06 435.96/148.06 {and, gcd^#, gcd[Ite][False][Ite]^#, gt0^#} 435.96/148.06 435.96/148.06 For your convenience, here are the satisfied ordering constraints: 435.96/148.06 435.96/148.06 pi(gcd^#(Cons(x', xs'), Cons(x, xs))) = gcd^#(Cons(; xs');) 435.96/148.06 > c_4(gcd[Ite][False][Ite]^#(Cons(; xs'););) 435.96/148.06 = pi(c_4(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 435.96/148.06 Cons(x', xs'), 435.96/148.06 Cons(x, xs)))) 435.96/148.06 435.96/148.06 pi(gcd[Ite][False][Ite]^#(False(), x, y)) = gcd[Ite][False][Ite]^#(x;) 435.96/148.06 > c_5(gt0^#(x;);) 435.96/148.06 = pi(c_5(gt0^#(x, y))) 435.96/148.06 435.96/148.06 pi(gt0^#(Nil(), y)) = gt0^#(Nil();) 435.96/148.06 > c_1() 435.96/148.06 = pi(c_1()) 435.96/148.06 435.96/148.06 pi(gt0^#(Cons(x, xs), Nil())) = gt0^#(Cons(; xs);) 435.96/148.06 > c_2() 435.96/148.06 = pi(c_2()) 435.96/148.06 435.96/148.06 pi(gt0^#(Cons(x', xs'), Cons(x, xs))) = gt0^#(Cons(; xs');) 435.96/148.06 > c_3(gt0^#(xs';);) 435.96/148.06 = pi(c_3(gt0^#(xs', xs))) 435.96/148.06 435.96/148.06 pi(and(True(), True())) = and(True();) 435.96/148.06 > True() 435.96/148.06 = pi(True()) 435.96/148.06 435.96/148.06 pi(and(True(), False())) = and(True();) 435.96/148.06 > False() 435.96/148.06 = pi(False()) 435.96/148.06 435.96/148.06 pi(and(False(), True())) = and(False();) 435.96/148.06 > False() 435.96/148.06 = pi(False()) 435.96/148.06 435.96/148.06 pi(and(False(), False())) = and(False();) 435.96/148.06 > False() 435.96/148.06 = pi(False()) 435.96/148.06 435.96/148.06 435.96/148.06 The strictly oriented rules are moved into the weak component. 435.96/148.06 435.96/148.06 We are left with following problem, upon which TcT provides the 435.96/148.06 certificate YES(O(1),O(1)). 435.96/148.06 435.96/148.06 Weak DPs: 435.96/148.06 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.06 c_4(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 435.96/148.06 Cons(x', xs'), 435.96/148.06 Cons(x, xs))) 435.96/148.06 , gcd[Ite][False][Ite]^#(False(), x, y) -> c_5(gt0^#(x, y)) 435.96/148.06 , gt0^#(Nil(), y) -> c_1() 435.96/148.06 , gt0^#(Cons(x, xs), Nil()) -> c_2() 435.96/148.06 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_3(gt0^#(xs', xs)) } 435.96/148.06 Weak Trs: 435.96/148.06 { and(True(), True()) -> True() 435.96/148.06 , and(True(), False()) -> False() 435.96/148.06 , and(False(), True()) -> False() 435.96/148.06 , and(False(), False()) -> False() 435.96/148.06 , eqList(Nil(), Nil()) -> True() 435.96/148.06 , eqList(Nil(), Cons(y, ys)) -> False() 435.96/148.06 , eqList(Cons(x, xs), Nil()) -> False() 435.96/148.06 , eqList(Cons(x, xs), Cons(y, ys)) -> 435.96/148.06 and(eqList(x, y), eqList(xs, ys)) } 435.96/148.06 Obligation: 435.96/148.06 innermost runtime complexity 435.96/148.06 Answer: 435.96/148.06 YES(O(1),O(1)) 435.96/148.06 435.96/148.06 The following weak DPs constitute a sub-graph of the DG that is 435.96/148.06 closed under successors. The DPs are removed. 435.96/148.06 435.96/148.06 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.06 c_4(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 435.96/148.06 Cons(x', xs'), 435.96/148.06 Cons(x, xs))) 435.96/148.06 , gcd[Ite][False][Ite]^#(False(), x, y) -> c_5(gt0^#(x, y)) 435.96/148.06 , gt0^#(Nil(), y) -> c_1() 435.96/148.06 , gt0^#(Cons(x, xs), Nil()) -> c_2() 435.96/148.06 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_3(gt0^#(xs', xs)) } 435.96/148.06 435.96/148.06 We are left with following problem, upon which TcT provides the 435.96/148.06 certificate YES(O(1),O(1)). 435.96/148.06 435.96/148.06 Weak Trs: 435.96/148.06 { and(True(), True()) -> True() 435.96/148.06 , and(True(), False()) -> False() 435.96/148.06 , and(False(), True()) -> False() 435.96/148.06 , and(False(), False()) -> False() 435.96/148.06 , eqList(Nil(), Nil()) -> True() 435.96/148.06 , eqList(Nil(), Cons(y, ys)) -> False() 435.96/148.06 , eqList(Cons(x, xs), Nil()) -> False() 435.96/148.06 , eqList(Cons(x, xs), Cons(y, ys)) -> 435.96/148.06 and(eqList(x, y), eqList(xs, ys)) } 435.96/148.06 Obligation: 435.96/148.06 innermost runtime complexity 435.96/148.06 Answer: 435.96/148.06 YES(O(1),O(1)) 435.96/148.06 435.96/148.06 No rule is usable, rules are removed from the input problem. 435.96/148.06 435.96/148.06 We are left with following problem, upon which TcT provides the 435.96/148.06 certificate YES(O(1),O(1)). 435.96/148.06 435.96/148.06 Rules: Empty 435.96/148.06 Obligation: 435.96/148.06 innermost runtime complexity 435.96/148.06 Answer: 435.96/148.06 YES(O(1),O(1)) 435.96/148.06 435.96/148.06 Empty rules are trivially bounded 435.96/148.06 435.96/148.06 S) We are left with following problem, upon which TcT provides the 435.96/148.07 certificate YES(O(1),O(n^2)). 435.96/148.07 435.96/148.07 Strict DPs: 435.96/148.07 { lgth^#(Cons(x, xs)) -> 435.96/148.07 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 435.96/148.07 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 435.96/148.07 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 435.96/148.07 c_4(eqList^#(x, y), eqList^#(xs, ys)) 435.96/148.07 , monus^#(x, y) -> 435.96/148.07 c_5(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.07 eqList^#(lgth(y), Cons(Nil(), Nil())), 435.96/148.07 lgth^#(y)) } 435.96/148.07 Weak DPs: 435.96/148.07 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.07 c_3(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 435.96/148.07 Cons(x', xs'), 435.96/148.07 Cons(x, xs)), 435.96/148.07 eqList^#(Cons(x', xs'), Cons(x, xs))) 435.96/148.07 , gcd[Ite][False][Ite]^#(False(), x, y) -> c_10(gt0^#(x, y)) 435.96/148.07 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 435.96/148.07 c_11(monus^#(xs', xs)) 435.96/148.07 , gt0^#(Nil(), y) -> c_6() 435.96/148.07 , gt0^#(Cons(x, xs), Nil()) -> c_7() 435.96/148.07 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_8(gt0^#(xs', xs)) } 435.96/148.07 Weak Trs: 435.96/148.07 { lgth(Nil()) -> Nil() 435.96/148.07 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 435.96/148.07 , @(Nil(), ys) -> ys 435.96/148.07 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 435.96/148.07 , and(True(), True()) -> True() 435.96/148.07 , and(True(), False()) -> False() 435.96/148.07 , and(False(), True()) -> False() 435.96/148.07 , and(False(), False()) -> False() 435.96/148.07 , eqList(Nil(), Nil()) -> True() 435.96/148.07 , eqList(Nil(), Cons(y, ys)) -> False() 435.96/148.07 , eqList(Cons(x, xs), Nil()) -> False() 435.96/148.07 , eqList(Cons(x, xs), Cons(y, ys)) -> 435.96/148.07 and(eqList(x, y), eqList(xs, ys)) } 435.96/148.07 Obligation: 435.96/148.07 innermost runtime complexity 435.96/148.07 Answer: 435.96/148.07 YES(O(1),O(n^2)) 435.96/148.07 435.96/148.07 The following weak DPs constitute a sub-graph of the DG that is 435.96/148.07 closed under successors. The DPs are removed. 435.96/148.07 435.96/148.07 { gcd[Ite][False][Ite]^#(False(), x, y) -> c_10(gt0^#(x, y)) 435.96/148.07 , gt0^#(Nil(), y) -> c_6() 435.96/148.07 , gt0^#(Cons(x, xs), Nil()) -> c_7() 435.96/148.07 , gt0^#(Cons(x', xs'), Cons(x, xs)) -> c_8(gt0^#(xs', xs)) } 435.96/148.07 435.96/148.07 We are left with following problem, upon which TcT provides the 435.96/148.07 certificate YES(O(1),O(n^2)). 435.96/148.07 435.96/148.07 Strict DPs: 435.96/148.07 { lgth^#(Cons(x, xs)) -> 435.96/148.07 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 435.96/148.07 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 435.96/148.07 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 435.96/148.07 c_4(eqList^#(x, y), eqList^#(xs, ys)) 435.96/148.07 , monus^#(x, y) -> 435.96/148.07 c_5(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.07 eqList^#(lgth(y), Cons(Nil(), Nil())), 435.96/148.07 lgth^#(y)) } 435.96/148.07 Weak DPs: 435.96/148.07 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.07 c_3(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 435.96/148.07 Cons(x', xs'), 435.96/148.07 Cons(x, xs)), 435.96/148.07 eqList^#(Cons(x', xs'), Cons(x, xs))) 435.96/148.07 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 435.96/148.07 c_11(monus^#(xs', xs)) } 435.96/148.07 Weak Trs: 435.96/148.07 { lgth(Nil()) -> Nil() 435.96/148.07 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 435.96/148.07 , @(Nil(), ys) -> ys 435.96/148.07 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 435.96/148.07 , and(True(), True()) -> True() 435.96/148.07 , and(True(), False()) -> False() 435.96/148.07 , and(False(), True()) -> False() 435.96/148.07 , and(False(), False()) -> False() 435.96/148.07 , eqList(Nil(), Nil()) -> True() 435.96/148.07 , eqList(Nil(), Cons(y, ys)) -> False() 435.96/148.07 , eqList(Cons(x, xs), Nil()) -> False() 435.96/148.07 , eqList(Cons(x, xs), Cons(y, ys)) -> 435.96/148.07 and(eqList(x, y), eqList(xs, ys)) } 435.96/148.07 Obligation: 435.96/148.07 innermost runtime complexity 435.96/148.07 Answer: 435.96/148.07 YES(O(1),O(n^2)) 435.96/148.07 435.96/148.07 Due to missing edges in the dependency-graph, the right-hand sides 435.96/148.07 of following rules could be simplified: 435.96/148.07 435.96/148.07 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.07 c_3(gcd[Ite][False][Ite]^#(eqList(Cons(x', xs'), Cons(x, xs)), 435.96/148.07 Cons(x', xs'), 435.96/148.07 Cons(x, xs)), 435.96/148.07 eqList^#(Cons(x', xs'), Cons(x, xs))) } 435.96/148.07 435.96/148.07 We are left with following problem, upon which TcT provides the 435.96/148.07 certificate YES(O(1),O(n^2)). 435.96/148.07 435.96/148.07 Strict DPs: 435.96/148.07 { lgth^#(Cons(x, xs)) -> 435.96/148.07 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 435.96/148.07 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 435.96/148.07 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 435.96/148.07 c_3(eqList^#(x, y), eqList^#(xs, ys)) 435.96/148.07 , monus^#(x, y) -> 435.96/148.07 c_4(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.07 eqList^#(lgth(y), Cons(Nil(), Nil())), 435.96/148.07 lgth^#(y)) } 435.96/148.07 Weak DPs: 435.96/148.07 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.07 c_5(eqList^#(Cons(x', xs'), Cons(x, xs))) 435.96/148.07 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 435.96/148.07 c_6(monus^#(xs', xs)) } 435.96/148.07 Weak Trs: 435.96/148.07 { lgth(Nil()) -> Nil() 435.96/148.07 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 435.96/148.07 , @(Nil(), ys) -> ys 435.96/148.07 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 435.96/148.07 , and(True(), True()) -> True() 435.96/148.07 , and(True(), False()) -> False() 435.96/148.07 , and(False(), True()) -> False() 435.96/148.07 , and(False(), False()) -> False() 435.96/148.07 , eqList(Nil(), Nil()) -> True() 435.96/148.07 , eqList(Nil(), Cons(y, ys)) -> False() 435.96/148.07 , eqList(Cons(x, xs), Nil()) -> False() 435.96/148.07 , eqList(Cons(x, xs), Cons(y, ys)) -> 435.96/148.07 and(eqList(x, y), eqList(xs, ys)) } 435.96/148.07 Obligation: 435.96/148.07 innermost runtime complexity 435.96/148.07 Answer: 435.96/148.07 YES(O(1),O(n^2)) 435.96/148.07 435.96/148.07 We analyse the complexity of following sub-problems (R) and (S). 435.96/148.07 Problem (S) is obtained from the input problem by shifting strict 435.96/148.07 rules from (R) into the weak component: 435.96/148.07 435.96/148.07 Problem (R): 435.96/148.07 ------------ 435.96/148.07 Strict DPs: 435.96/148.07 { eqList^#(Cons(x, xs), Cons(y, ys)) -> 435.96/148.07 c_3(eqList^#(x, y), eqList^#(xs, ys)) } 435.96/148.07 Weak DPs: 435.96/148.07 { lgth^#(Cons(x, xs)) -> 435.96/148.07 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 435.96/148.07 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 435.96/148.07 , gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.07 c_5(eqList^#(Cons(x', xs'), Cons(x, xs))) 435.96/148.07 , monus^#(x, y) -> 435.96/148.07 c_4(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.07 eqList^#(lgth(y), Cons(Nil(), Nil())), 435.96/148.07 lgth^#(y)) 435.96/148.07 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 435.96/148.07 c_6(monus^#(xs', xs)) } 435.96/148.07 Weak Trs: 435.96/148.07 { lgth(Nil()) -> Nil() 435.96/148.07 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 435.96/148.07 , @(Nil(), ys) -> ys 435.96/148.07 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 435.96/148.07 , and(True(), True()) -> True() 435.96/148.07 , and(True(), False()) -> False() 435.96/148.07 , and(False(), True()) -> False() 435.96/148.07 , and(False(), False()) -> False() 435.96/148.07 , eqList(Nil(), Nil()) -> True() 435.96/148.07 , eqList(Nil(), Cons(y, ys)) -> False() 435.96/148.07 , eqList(Cons(x, xs), Nil()) -> False() 435.96/148.07 , eqList(Cons(x, xs), Cons(y, ys)) -> 435.96/148.07 and(eqList(x, y), eqList(xs, ys)) } 435.96/148.07 StartTerms: basic terms 435.96/148.07 Strategy: innermost 435.96/148.07 435.96/148.07 Problem (S): 435.96/148.07 ------------ 435.96/148.07 Strict DPs: 435.96/148.07 { lgth^#(Cons(x, xs)) -> 435.96/148.07 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 435.96/148.07 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 435.96/148.07 , monus^#(x, y) -> 435.96/148.07 c_4(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.07 eqList^#(lgth(y), Cons(Nil(), Nil())), 435.96/148.07 lgth^#(y)) } 435.96/148.07 Weak DPs: 435.96/148.07 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.07 c_5(eqList^#(Cons(x', xs'), Cons(x, xs))) 435.96/148.07 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 435.96/148.07 c_3(eqList^#(x, y), eqList^#(xs, ys)) 435.96/148.07 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 435.96/148.07 c_6(monus^#(xs', xs)) } 435.96/148.07 Weak Trs: 435.96/148.07 { lgth(Nil()) -> Nil() 435.96/148.07 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 435.96/148.07 , @(Nil(), ys) -> ys 435.96/148.07 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 435.96/148.07 , and(True(), True()) -> True() 435.96/148.07 , and(True(), False()) -> False() 435.96/148.07 , and(False(), True()) -> False() 435.96/148.07 , and(False(), False()) -> False() 435.96/148.07 , eqList(Nil(), Nil()) -> True() 435.96/148.07 , eqList(Nil(), Cons(y, ys)) -> False() 435.96/148.07 , eqList(Cons(x, xs), Nil()) -> False() 435.96/148.07 , eqList(Cons(x, xs), Cons(y, ys)) -> 435.96/148.07 and(eqList(x, y), eqList(xs, ys)) } 435.96/148.07 StartTerms: basic terms 435.96/148.07 Strategy: innermost 435.96/148.07 435.96/148.07 Overall, the transformation results in the following sub-problem(s): 435.96/148.07 435.96/148.07 Generated new problems: 435.96/148.07 ----------------------- 435.96/148.07 R) Strict DPs: 435.96/148.07 { eqList^#(Cons(x, xs), Cons(y, ys)) -> 435.96/148.07 c_3(eqList^#(x, y), eqList^#(xs, ys)) } 435.96/148.07 Weak DPs: 435.96/148.07 { lgth^#(Cons(x, xs)) -> 435.96/148.07 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 435.96/148.07 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 435.96/148.07 , gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.07 c_5(eqList^#(Cons(x', xs'), Cons(x, xs))) 435.96/148.07 , monus^#(x, y) -> 435.96/148.07 c_4(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.07 eqList^#(lgth(y), Cons(Nil(), Nil())), 435.96/148.07 lgth^#(y)) 435.96/148.07 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 435.96/148.07 c_6(monus^#(xs', xs)) } 435.96/148.07 Weak Trs: 435.96/148.07 { lgth(Nil()) -> Nil() 435.96/148.07 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 435.96/148.07 , @(Nil(), ys) -> ys 435.96/148.07 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 435.96/148.07 , and(True(), True()) -> True() 435.96/148.07 , and(True(), False()) -> False() 435.96/148.07 , and(False(), True()) -> False() 435.96/148.07 , and(False(), False()) -> False() 435.96/148.07 , eqList(Nil(), Nil()) -> True() 435.96/148.07 , eqList(Nil(), Cons(y, ys)) -> False() 435.96/148.07 , eqList(Cons(x, xs), Nil()) -> False() 435.96/148.07 , eqList(Cons(x, xs), Cons(y, ys)) -> 435.96/148.07 and(eqList(x, y), eqList(xs, ys)) } 435.96/148.07 StartTerms: basic terms 435.96/148.07 Strategy: innermost 435.96/148.07 435.96/148.07 This problem was proven YES(O(1),O(n^1)). 435.96/148.07 435.96/148.07 S) Strict DPs: 435.96/148.07 { lgth^#(Cons(x, xs)) -> 435.96/148.07 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 435.96/148.07 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 435.96/148.07 , monus^#(x, y) -> 435.96/148.07 c_4(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.07 eqList^#(lgth(y), Cons(Nil(), Nil())), 435.96/148.07 lgth^#(y)) } 435.96/148.07 Weak DPs: 435.96/148.07 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.07 c_5(eqList^#(Cons(x', xs'), Cons(x, xs))) 435.96/148.07 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 435.96/148.07 c_3(eqList^#(x, y), eqList^#(xs, ys)) 435.96/148.07 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 435.96/148.07 c_6(monus^#(xs', xs)) } 435.96/148.07 Weak Trs: 435.96/148.07 { lgth(Nil()) -> Nil() 435.96/148.07 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 435.96/148.07 , @(Nil(), ys) -> ys 435.96/148.07 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 435.96/148.07 , and(True(), True()) -> True() 435.96/148.07 , and(True(), False()) -> False() 435.96/148.07 , and(False(), True()) -> False() 435.96/148.07 , and(False(), False()) -> False() 435.96/148.07 , eqList(Nil(), Nil()) -> True() 435.96/148.07 , eqList(Nil(), Cons(y, ys)) -> False() 435.96/148.07 , eqList(Cons(x, xs), Nil()) -> False() 435.96/148.07 , eqList(Cons(x, xs), Cons(y, ys)) -> 435.96/148.07 and(eqList(x, y), eqList(xs, ys)) } 435.96/148.07 StartTerms: basic terms 435.96/148.07 Strategy: innermost 435.96/148.07 435.96/148.07 This problem was proven YES(O(1),O(n^2)). 435.96/148.07 435.96/148.07 435.96/148.07 Proofs for generated problems: 435.96/148.07 ------------------------------ 435.96/148.07 R) We are left with following problem, upon which TcT provides the 435.96/148.07 certificate YES(O(1),O(n^1)). 435.96/148.07 435.96/148.07 Strict DPs: 435.96/148.07 { eqList^#(Cons(x, xs), Cons(y, ys)) -> 435.96/148.07 c_3(eqList^#(x, y), eqList^#(xs, ys)) } 435.96/148.07 Weak DPs: 435.96/148.07 { lgth^#(Cons(x, xs)) -> 435.96/148.07 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 435.96/148.07 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 435.96/148.07 , gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.07 c_5(eqList^#(Cons(x', xs'), Cons(x, xs))) 435.96/148.07 , monus^#(x, y) -> 435.96/148.07 c_4(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.07 eqList^#(lgth(y), Cons(Nil(), Nil())), 435.96/148.07 lgth^#(y)) 435.96/148.07 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 435.96/148.07 c_6(monus^#(xs', xs)) } 435.96/148.07 Weak Trs: 435.96/148.07 { lgth(Nil()) -> Nil() 435.96/148.07 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 435.96/148.07 , @(Nil(), ys) -> ys 435.96/148.07 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 435.96/148.07 , and(True(), True()) -> True() 435.96/148.07 , and(True(), False()) -> False() 435.96/148.07 , and(False(), True()) -> False() 435.96/148.07 , and(False(), False()) -> False() 435.96/148.07 , eqList(Nil(), Nil()) -> True() 435.96/148.07 , eqList(Nil(), Cons(y, ys)) -> False() 435.96/148.07 , eqList(Cons(x, xs), Nil()) -> False() 435.96/148.07 , eqList(Cons(x, xs), Cons(y, ys)) -> 435.96/148.07 and(eqList(x, y), eqList(xs, ys)) } 435.96/148.07 Obligation: 435.96/148.07 innermost runtime complexity 435.96/148.07 Answer: 435.96/148.07 YES(O(1),O(n^1)) 435.96/148.07 435.96/148.07 The following weak DPs constitute a sub-graph of the DG that is 435.96/148.07 closed under successors. The DPs are removed. 435.96/148.07 435.96/148.07 { lgth^#(Cons(x, xs)) -> 435.96/148.07 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 435.96/148.07 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) } 435.96/148.07 435.96/148.07 We are left with following problem, upon which TcT provides the 435.96/148.07 certificate YES(O(1),O(n^1)). 435.96/148.07 435.96/148.07 Strict DPs: 435.96/148.07 { eqList^#(Cons(x, xs), Cons(y, ys)) -> 435.96/148.07 c_3(eqList^#(x, y), eqList^#(xs, ys)) } 435.96/148.07 Weak DPs: 435.96/148.07 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.07 c_5(eqList^#(Cons(x', xs'), Cons(x, xs))) 435.96/148.07 , monus^#(x, y) -> 435.96/148.07 c_4(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.07 eqList^#(lgth(y), Cons(Nil(), Nil())), 435.96/148.07 lgth^#(y)) 435.96/148.07 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 435.96/148.07 c_6(monus^#(xs', xs)) } 435.96/148.07 Weak Trs: 435.96/148.07 { lgth(Nil()) -> Nil() 435.96/148.07 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 435.96/148.07 , @(Nil(), ys) -> ys 435.96/148.07 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 435.96/148.07 , and(True(), True()) -> True() 435.96/148.07 , and(True(), False()) -> False() 435.96/148.07 , and(False(), True()) -> False() 435.96/148.07 , and(False(), False()) -> False() 435.96/148.07 , eqList(Nil(), Nil()) -> True() 435.96/148.07 , eqList(Nil(), Cons(y, ys)) -> False() 435.96/148.07 , eqList(Cons(x, xs), Nil()) -> False() 435.96/148.07 , eqList(Cons(x, xs), Cons(y, ys)) -> 435.96/148.07 and(eqList(x, y), eqList(xs, ys)) } 435.96/148.07 Obligation: 435.96/148.07 innermost runtime complexity 435.96/148.07 Answer: 435.96/148.07 YES(O(1),O(n^1)) 435.96/148.07 435.96/148.07 Due to missing edges in the dependency-graph, the right-hand sides 435.96/148.07 of following rules could be simplified: 435.96/148.07 435.96/148.07 { monus^#(x, y) -> 435.96/148.07 c_4(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.07 eqList^#(lgth(y), Cons(Nil(), Nil())), 435.96/148.07 lgth^#(y)) } 435.96/148.07 435.96/148.07 We are left with following problem, upon which TcT provides the 435.96/148.07 certificate YES(O(1),O(n^1)). 435.96/148.07 435.96/148.07 Strict DPs: 435.96/148.07 { eqList^#(Cons(x, xs), Cons(y, ys)) -> 435.96/148.07 c_1(eqList^#(x, y), eqList^#(xs, ys)) } 435.96/148.07 Weak DPs: 435.96/148.07 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.07 c_2(eqList^#(Cons(x', xs'), Cons(x, xs))) 435.96/148.07 , monus^#(x, y) -> 435.96/148.07 c_3(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.07 eqList^#(lgth(y), Cons(Nil(), Nil()))) 435.96/148.07 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 435.96/148.07 c_4(monus^#(xs', xs)) } 435.96/148.07 Weak Trs: 435.96/148.07 { lgth(Nil()) -> Nil() 435.96/148.07 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 435.96/148.07 , @(Nil(), ys) -> ys 435.96/148.07 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 435.96/148.07 , and(True(), True()) -> True() 435.96/148.07 , and(True(), False()) -> False() 435.96/148.07 , and(False(), True()) -> False() 435.96/148.07 , and(False(), False()) -> False() 435.96/148.07 , eqList(Nil(), Nil()) -> True() 435.96/148.07 , eqList(Nil(), Cons(y, ys)) -> False() 435.96/148.07 , eqList(Cons(x, xs), Nil()) -> False() 435.96/148.07 , eqList(Cons(x, xs), Cons(y, ys)) -> 435.96/148.07 and(eqList(x, y), eqList(xs, ys)) } 435.96/148.07 Obligation: 435.96/148.07 innermost runtime complexity 435.96/148.07 Answer: 435.96/148.07 YES(O(1),O(n^1)) 435.96/148.07 435.96/148.07 We use the processor 'matrix interpretation of dimension 1' to 435.96/148.07 orient following rules strictly. 435.96/148.07 435.96/148.07 DPs: 435.96/148.07 { 1: eqList^#(Cons(x, xs), Cons(y, ys)) -> 435.96/148.07 c_1(eqList^#(x, y), eqList^#(xs, ys)) 435.96/148.07 , 2: gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.07 c_2(eqList^#(Cons(x', xs'), Cons(x, xs))) } 435.96/148.07 435.96/148.07 Sub-proof: 435.96/148.07 ---------- 435.96/148.07 The following argument positions are usable: 435.96/148.07 Uargs(c_1) = {1, 2}, Uargs(c_2) = {1}, Uargs(c_3) = {1, 2}, 435.96/148.07 Uargs(c_4) = {1} 435.96/148.07 435.96/148.07 TcT has computed the following constructor-based matrix 435.96/148.07 interpretation satisfying not(EDA). 435.96/148.07 435.96/148.07 [lgth](x1) = [0] 435.96/148.07 435.96/148.07 [True] = [0] 435.96/148.07 435.96/148.07 [Nil] = [0] 435.96/148.07 435.96/148.07 [@](x1, x2) = [0] 435.96/148.07 435.96/148.07 [and](x1, x2) = [0] 435.96/148.07 435.96/148.07 [eqList](x1, x2) = [0] 435.96/148.07 435.96/148.07 [Cons](x1, x2) = [1] x1 + [1] x2 + [1] 435.96/148.07 435.96/148.07 [False] = [0] 435.96/148.07 435.96/148.07 [gcd^#](x1, x2) = [4] x2 + [7] 435.96/148.07 435.96/148.07 [eqList^#](x1, x2) = [4] x2 + [0] 435.96/148.07 435.96/148.07 [monus^#](x1, x2) = [4] x1 + [4] 435.96/148.07 435.96/148.07 [monus[Ite]^#](x1, x2, x3) = [4] x2 + [0] 435.96/148.07 435.96/148.07 [c_1](x1, x2) = [1] x1 + [1] x2 + [1] 435.96/148.07 435.96/148.07 [c_2](x1) = [1] x1 + [3] 435.96/148.07 435.96/148.07 [c_3](x1, x2) = [1] x1 + [1] x2 + [0] 435.96/148.07 435.96/148.07 [c_4](x1) = [1] x1 + [0] 435.96/148.07 435.96/148.07 The order satisfies the following ordering constraints: 435.96/148.07 435.96/148.07 [lgth(Nil())] = [0] 435.96/148.07 >= [0] 435.96/148.07 = [Nil()] 435.96/148.07 435.96/148.07 [lgth(Cons(x, xs))] = [0] 435.96/148.07 >= [0] 435.96/148.07 = [@(Cons(Nil(), Nil()), lgth(xs))] 435.96/148.07 435.96/148.07 [@(Nil(), ys)] = [0] 435.96/148.07 ? [1] ys + [0] 435.96/148.07 = [ys] 435.96/148.07 435.96/148.07 [@(Cons(x, xs), ys)] = [0] 435.96/148.07 ? [1] x + [1] 435.96/148.07 = [Cons(x, @(xs, ys))] 435.96/148.07 435.96/148.07 [and(True(), True())] = [0] 435.96/148.07 >= [0] 435.96/148.07 = [True()] 435.96/148.07 435.96/148.07 [and(True(), False())] = [0] 435.96/148.07 >= [0] 435.96/148.07 = [False()] 435.96/148.07 435.96/148.07 [and(False(), True())] = [0] 435.96/148.07 >= [0] 435.96/148.07 = [False()] 435.96/148.07 435.96/148.07 [and(False(), False())] = [0] 435.96/148.07 >= [0] 435.96/148.07 = [False()] 435.96/148.07 435.96/148.07 [eqList(Nil(), Nil())] = [0] 435.96/148.07 >= [0] 435.96/148.07 = [True()] 435.96/148.07 435.96/148.07 [eqList(Nil(), Cons(y, ys))] = [0] 435.96/148.07 >= [0] 435.96/148.07 = [False()] 435.96/148.07 435.96/148.07 [eqList(Cons(x, xs), Nil())] = [0] 435.96/148.07 >= [0] 435.96/148.07 = [False()] 435.96/148.07 435.96/148.07 [eqList(Cons(x, xs), Cons(y, ys))] = [0] 435.96/148.07 >= [0] 435.96/148.07 = [and(eqList(x, y), eqList(xs, ys))] 435.96/148.07 435.96/148.07 [gcd^#(Cons(x', xs'), Cons(x, xs))] = [4] x + [4] xs + [11] 435.96/148.08 > [4] x + [4] xs + [7] 435.96/148.08 = [c_2(eqList^#(Cons(x', xs'), Cons(x, xs)))] 435.96/148.08 435.96/148.08 [eqList^#(Cons(x, xs), Cons(y, ys))] = [4] ys + [4] y + [4] 435.96/148.08 > [4] ys + [4] y + [1] 435.96/148.08 = [c_1(eqList^#(x, y), eqList^#(xs, ys))] 435.96/148.08 435.96/148.08 [monus^#(x, y)] = [4] x + [4] 435.96/148.08 >= [4] x + [4] 435.96/148.08 = [c_3(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.08 eqList^#(lgth(y), Cons(Nil(), Nil())))] 435.96/148.08 435.96/148.08 [monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs))] = [4] x' + [4] xs' + [4] 435.96/148.08 >= [4] xs' + [4] 435.96/148.08 = [c_4(monus^#(xs', xs))] 435.96/148.08 435.96/148.08 435.96/148.08 The strictly oriented rules are moved into the weak component. 435.96/148.08 435.96/148.08 We are left with following problem, upon which TcT provides the 435.96/148.08 certificate YES(O(1),O(1)). 435.96/148.08 435.96/148.08 Weak DPs: 435.96/148.08 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.08 c_2(eqList^#(Cons(x', xs'), Cons(x, xs))) 435.96/148.08 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 435.96/148.08 c_1(eqList^#(x, y), eqList^#(xs, ys)) 435.96/148.08 , monus^#(x, y) -> 435.96/148.08 c_3(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.08 eqList^#(lgth(y), Cons(Nil(), Nil()))) 435.96/148.08 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 435.96/148.08 c_4(monus^#(xs', xs)) } 435.96/148.08 Weak Trs: 435.96/148.08 { lgth(Nil()) -> Nil() 435.96/148.08 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 435.96/148.08 , @(Nil(), ys) -> ys 435.96/148.08 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 435.96/148.08 , and(True(), True()) -> True() 435.96/148.08 , and(True(), False()) -> False() 435.96/148.08 , and(False(), True()) -> False() 435.96/148.08 , and(False(), False()) -> False() 435.96/148.08 , eqList(Nil(), Nil()) -> True() 435.96/148.08 , eqList(Nil(), Cons(y, ys)) -> False() 435.96/148.08 , eqList(Cons(x, xs), Nil()) -> False() 435.96/148.08 , eqList(Cons(x, xs), Cons(y, ys)) -> 435.96/148.08 and(eqList(x, y), eqList(xs, ys)) } 435.96/148.08 Obligation: 435.96/148.08 innermost runtime complexity 435.96/148.08 Answer: 435.96/148.08 YES(O(1),O(1)) 435.96/148.08 435.96/148.08 The following weak DPs constitute a sub-graph of the DG that is 435.96/148.08 closed under successors. The DPs are removed. 435.96/148.08 435.96/148.08 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.08 c_2(eqList^#(Cons(x', xs'), Cons(x, xs))) 435.96/148.08 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 435.96/148.08 c_1(eqList^#(x, y), eqList^#(xs, ys)) 435.96/148.08 , monus^#(x, y) -> 435.96/148.08 c_3(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.08 eqList^#(lgth(y), Cons(Nil(), Nil()))) 435.96/148.08 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 435.96/148.08 c_4(monus^#(xs', xs)) } 435.96/148.08 435.96/148.08 We are left with following problem, upon which TcT provides the 435.96/148.08 certificate YES(O(1),O(1)). 435.96/148.08 435.96/148.08 Weak Trs: 435.96/148.08 { lgth(Nil()) -> Nil() 435.96/148.08 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 435.96/148.08 , @(Nil(), ys) -> ys 435.96/148.08 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 435.96/148.08 , and(True(), True()) -> True() 435.96/148.08 , and(True(), False()) -> False() 435.96/148.08 , and(False(), True()) -> False() 435.96/148.08 , and(False(), False()) -> False() 435.96/148.08 , eqList(Nil(), Nil()) -> True() 435.96/148.08 , eqList(Nil(), Cons(y, ys)) -> False() 435.96/148.08 , eqList(Cons(x, xs), Nil()) -> False() 435.96/148.08 , eqList(Cons(x, xs), Cons(y, ys)) -> 435.96/148.08 and(eqList(x, y), eqList(xs, ys)) } 435.96/148.08 Obligation: 435.96/148.08 innermost runtime complexity 435.96/148.08 Answer: 435.96/148.08 YES(O(1),O(1)) 435.96/148.08 435.96/148.08 No rule is usable, rules are removed from the input problem. 435.96/148.08 435.96/148.08 We are left with following problem, upon which TcT provides the 435.96/148.08 certificate YES(O(1),O(1)). 435.96/148.08 435.96/148.08 Rules: Empty 435.96/148.08 Obligation: 435.96/148.08 innermost runtime complexity 435.96/148.08 Answer: 435.96/148.08 YES(O(1),O(1)) 435.96/148.08 435.96/148.08 Empty rules are trivially bounded 435.96/148.08 435.96/148.08 S) We are left with following problem, upon which TcT provides the 435.96/148.08 certificate YES(O(1),O(n^2)). 435.96/148.08 435.96/148.08 Strict DPs: 435.96/148.08 { lgth^#(Cons(x, xs)) -> 435.96/148.08 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 435.96/148.08 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 435.96/148.08 , monus^#(x, y) -> 435.96/148.08 c_4(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.08 eqList^#(lgth(y), Cons(Nil(), Nil())), 435.96/148.08 lgth^#(y)) } 435.96/148.08 Weak DPs: 435.96/148.08 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.08 c_5(eqList^#(Cons(x', xs'), Cons(x, xs))) 435.96/148.08 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 435.96/148.08 c_3(eqList^#(x, y), eqList^#(xs, ys)) 435.96/148.08 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 435.96/148.08 c_6(monus^#(xs', xs)) } 435.96/148.08 Weak Trs: 435.96/148.08 { lgth(Nil()) -> Nil() 435.96/148.08 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 435.96/148.08 , @(Nil(), ys) -> ys 435.96/148.08 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 435.96/148.08 , and(True(), True()) -> True() 435.96/148.08 , and(True(), False()) -> False() 435.96/148.08 , and(False(), True()) -> False() 435.96/148.08 , and(False(), False()) -> False() 435.96/148.08 , eqList(Nil(), Nil()) -> True() 435.96/148.08 , eqList(Nil(), Cons(y, ys)) -> False() 435.96/148.08 , eqList(Cons(x, xs), Nil()) -> False() 435.96/148.08 , eqList(Cons(x, xs), Cons(y, ys)) -> 435.96/148.08 and(eqList(x, y), eqList(xs, ys)) } 435.96/148.08 Obligation: 435.96/148.08 innermost runtime complexity 435.96/148.08 Answer: 435.96/148.08 YES(O(1),O(n^2)) 435.96/148.08 435.96/148.08 The following weak DPs constitute a sub-graph of the DG that is 435.96/148.08 closed under successors. The DPs are removed. 435.96/148.08 435.96/148.08 { gcd^#(Cons(x', xs'), Cons(x, xs)) -> 435.96/148.08 c_5(eqList^#(Cons(x', xs'), Cons(x, xs))) 435.96/148.08 , eqList^#(Cons(x, xs), Cons(y, ys)) -> 435.96/148.08 c_3(eqList^#(x, y), eqList^#(xs, ys)) } 435.96/148.08 435.96/148.08 We are left with following problem, upon which TcT provides the 435.96/148.08 certificate YES(O(1),O(n^2)). 435.96/148.08 435.96/148.08 Strict DPs: 435.96/148.08 { lgth^#(Cons(x, xs)) -> 435.96/148.08 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 435.96/148.08 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 435.96/148.08 , monus^#(x, y) -> 435.96/148.08 c_4(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.08 eqList^#(lgth(y), Cons(Nil(), Nil())), 435.96/148.08 lgth^#(y)) } 435.96/148.08 Weak DPs: 435.96/148.08 { monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 435.96/148.08 c_6(monus^#(xs', xs)) } 435.96/148.08 Weak Trs: 435.96/148.08 { lgth(Nil()) -> Nil() 435.96/148.08 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 435.96/148.08 , @(Nil(), ys) -> ys 435.96/148.08 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 435.96/148.08 , and(True(), True()) -> True() 435.96/148.08 , and(True(), False()) -> False() 435.96/148.08 , and(False(), True()) -> False() 435.96/148.08 , and(False(), False()) -> False() 435.96/148.08 , eqList(Nil(), Nil()) -> True() 435.96/148.08 , eqList(Nil(), Cons(y, ys)) -> False() 435.96/148.08 , eqList(Cons(x, xs), Nil()) -> False() 435.96/148.08 , eqList(Cons(x, xs), Cons(y, ys)) -> 435.96/148.08 and(eqList(x, y), eqList(xs, ys)) } 435.96/148.08 Obligation: 435.96/148.08 innermost runtime complexity 435.96/148.08 Answer: 435.96/148.08 YES(O(1),O(n^2)) 435.96/148.08 435.96/148.08 Due to missing edges in the dependency-graph, the right-hand sides 435.96/148.08 of following rules could be simplified: 435.96/148.08 435.96/148.08 { monus^#(x, y) -> 435.96/148.08 c_4(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.08 eqList^#(lgth(y), Cons(Nil(), Nil())), 435.96/148.08 lgth^#(y)) } 435.96/148.08 435.96/148.08 We are left with following problem, upon which TcT provides the 435.96/148.08 certificate YES(O(1),O(n^2)). 435.96/148.08 435.96/148.08 Strict DPs: 435.96/148.08 { lgth^#(Cons(x, xs)) -> 435.96/148.08 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 435.96/148.08 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 435.96/148.08 , monus^#(x, y) -> 435.96/148.08 c_3(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.08 lgth^#(y)) } 435.96/148.08 Weak DPs: 435.96/148.08 { monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 435.96/148.08 c_4(monus^#(xs', xs)) } 435.96/148.08 Weak Trs: 435.96/148.08 { lgth(Nil()) -> Nil() 435.96/148.08 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 435.96/148.08 , @(Nil(), ys) -> ys 435.96/148.08 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 435.96/148.08 , and(True(), True()) -> True() 435.96/148.08 , and(True(), False()) -> False() 435.96/148.08 , and(False(), True()) -> False() 435.96/148.08 , and(False(), False()) -> False() 435.96/148.08 , eqList(Nil(), Nil()) -> True() 435.96/148.08 , eqList(Nil(), Cons(y, ys)) -> False() 435.96/148.08 , eqList(Cons(x, xs), Nil()) -> False() 435.96/148.08 , eqList(Cons(x, xs), Cons(y, ys)) -> 435.96/148.08 and(eqList(x, y), eqList(xs, ys)) } 435.96/148.08 Obligation: 435.96/148.08 innermost runtime complexity 435.96/148.08 Answer: 435.96/148.08 YES(O(1),O(n^2)) 435.96/148.08 435.96/148.08 We use the processor 'matrix interpretation of dimension 2' to 435.96/148.08 orient following rules strictly. 435.96/148.08 435.96/148.08 DPs: 435.96/148.08 { 1: lgth^#(Cons(x, xs)) -> 435.96/148.08 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 435.96/148.08 , 2: @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 435.96/148.08 , 4: monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 435.96/148.08 c_4(monus^#(xs', xs)) } 435.96/148.08 435.96/148.08 Sub-proof: 435.96/148.08 ---------- 435.96/148.08 The following argument positions are usable: 435.96/148.08 Uargs(c_1) = {1, 2}, Uargs(c_2) = {1}, Uargs(c_3) = {1, 2}, 435.96/148.08 Uargs(c_4) = {1} 435.96/148.08 435.96/148.08 TcT has computed the following constructor-based matrix 435.96/148.08 interpretation satisfying not(EDA). 435.96/148.08 435.96/148.08 [lgth](x1) = [0] 435.96/148.08 [0] 435.96/148.08 435.96/148.08 [True] = [0] 435.96/148.08 [0] 435.96/148.08 435.96/148.08 [Nil] = [0] 435.96/148.08 [0] 435.96/148.08 435.96/148.08 [@](x1, x2) = [0] 435.96/148.08 [0] 435.96/148.08 435.96/148.08 [and](x1, x2) = [0] 435.96/148.08 [0] 435.96/148.08 435.96/148.08 [eqList](x1, x2) = [0] 435.96/148.08 [0] 435.96/148.08 435.96/148.08 [Cons](x1, x2) = [1 4] x2 + [1] 435.96/148.08 [0 1] [2] 435.96/148.08 435.96/148.08 [False] = [0] 435.96/148.08 [0] 435.96/148.08 435.96/148.08 [lgth^#](x1) = [0 2] x1 + [0] 435.96/148.08 [0 0] [0] 435.96/148.08 435.96/148.08 [@^#](x1, x2) = [1 0] x1 + [0] 435.96/148.08 [2 1] [0] 435.96/148.08 435.96/148.08 [monus^#](x1, x2) = [0 1] x1 + [1 4] x2 + [1] 435.96/148.08 [0 0] [4 4] [0] 435.96/148.08 435.96/148.08 [monus[Ite]^#](x1, x2, x3) = [0 1] x2 + [1 0] x3 + [1] 435.96/148.08 [0 0] [0 0] [0] 435.96/148.08 435.96/148.08 [c_1](x1, x2) = [1 0] x1 + [1 0] x2 + [1] 435.96/148.08 [0 0] [0 0] [0] 435.96/148.08 435.96/148.08 [c_2](x1) = [1 0] x1 + [0] 435.96/148.08 [0 0] [3] 435.96/148.08 435.96/148.08 [c_3](x1, x2) = [1 0] x1 + [2 0] x2 + [0] 435.96/148.08 [0 0] [0 0] [0] 435.96/148.08 435.96/148.08 [c_4](x1) = [1 0] x1 + [1] 435.96/148.08 [0 0] [0] 435.96/148.08 435.96/148.08 The order satisfies the following ordering constraints: 435.96/148.08 435.96/148.08 [lgth(Nil())] = [0] 435.96/148.08 [0] 435.96/148.08 >= [0] 435.96/148.08 [0] 435.96/148.08 = [Nil()] 435.96/148.08 435.96/148.08 [lgth(Cons(x, xs))] = [0] 435.96/148.08 [0] 435.96/148.08 >= [0] 435.96/148.08 [0] 435.96/148.08 = [@(Cons(Nil(), Nil()), lgth(xs))] 435.96/148.08 435.96/148.08 [@(Nil(), ys)] = [0] 435.96/148.08 [0] 435.96/148.08 ? [1 0] ys + [0] 435.96/148.08 [0 1] [0] 435.96/148.08 = [ys] 435.96/148.08 435.96/148.08 [@(Cons(x, xs), ys)] = [0] 435.96/148.08 [0] 435.96/148.08 ? [1] 435.96/148.08 [2] 435.96/148.08 = [Cons(x, @(xs, ys))] 435.96/148.08 435.96/148.08 [and(True(), True())] = [0] 435.96/148.08 [0] 435.96/148.08 >= [0] 435.96/148.08 [0] 435.96/148.08 = [True()] 435.96/148.08 435.96/148.08 [and(True(), False())] = [0] 435.96/148.08 [0] 435.96/148.08 >= [0] 435.96/148.08 [0] 435.96/148.08 = [False()] 435.96/148.08 435.96/148.08 [and(False(), True())] = [0] 435.96/148.08 [0] 435.96/148.08 >= [0] 435.96/148.08 [0] 435.96/148.08 = [False()] 435.96/148.08 435.96/148.08 [and(False(), False())] = [0] 435.96/148.08 [0] 435.96/148.08 >= [0] 435.96/148.08 [0] 435.96/148.08 = [False()] 435.96/148.08 435.96/148.08 [eqList(Nil(), Nil())] = [0] 435.96/148.08 [0] 435.96/148.08 >= [0] 435.96/148.08 [0] 435.96/148.08 = [True()] 435.96/148.08 435.96/148.08 [eqList(Nil(), Cons(y, ys))] = [0] 435.96/148.08 [0] 435.96/148.08 >= [0] 435.96/148.08 [0] 435.96/148.08 = [False()] 435.96/148.08 435.96/148.08 [eqList(Cons(x, xs), Nil())] = [0] 435.96/148.08 [0] 435.96/148.08 >= [0] 435.96/148.08 [0] 435.96/148.08 = [False()] 435.96/148.08 435.96/148.08 [eqList(Cons(x, xs), Cons(y, ys))] = [0] 435.96/148.08 [0] 435.96/148.08 >= [0] 435.96/148.08 [0] 435.96/148.08 = [and(eqList(x, y), eqList(xs, ys))] 435.96/148.08 435.96/148.08 [lgth^#(Cons(x, xs))] = [0 2] xs + [4] 435.96/148.08 [0 0] [0] 435.96/148.08 > [0 2] xs + [2] 435.96/148.08 [0 0] [0] 435.96/148.08 = [c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs))] 435.96/148.08 435.96/148.08 [@^#(Cons(x, xs), ys)] = [1 4] xs + [1] 435.96/148.08 [2 9] [4] 435.96/148.08 > [1 0] xs + [0] 435.96/148.08 [0 0] [3] 435.96/148.08 = [c_2(@^#(xs, ys))] 435.96/148.08 435.96/148.08 [monus^#(x, y)] = [0 1] x + [1 4] y + [1] 435.96/148.08 [0 0] [4 4] [0] 435.96/148.08 >= [0 1] x + [1 4] y + [1] 435.96/148.08 [0 0] [0 0] [0] 435.96/148.08 = [c_3(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.08 lgth^#(y))] 435.96/148.08 435.96/148.08 [monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs))] = [1 4] xs + [0 1] xs' + [4] 435.96/148.08 [0 0] [0 0] [0] 435.96/148.08 > [1 4] xs + [0 1] xs' + [2] 435.96/148.08 [0 0] [0 0] [0] 435.96/148.08 = [c_4(monus^#(xs', xs))] 435.96/148.08 435.96/148.08 435.96/148.08 We return to the main proof. Consider the set of all dependency 435.96/148.08 pairs 435.96/148.08 435.96/148.08 : 435.96/148.08 { 1: lgth^#(Cons(x, xs)) -> 435.96/148.08 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 435.96/148.08 , 2: @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 435.96/148.08 , 3: monus^#(x, y) -> 435.96/148.08 c_3(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.08 lgth^#(y)) 435.96/148.08 , 4: monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 435.96/148.08 c_4(monus^#(xs', xs)) } 435.96/148.08 435.96/148.08 Processor 'matrix interpretation of dimension 2' induces the 435.96/148.08 complexity certificate YES(?,O(n^2)) on application of dependency 435.96/148.08 pairs {1,2,4}. These cover all (indirect) predecessors of 435.96/148.08 dependency pairs {1,2,3,4}, their number of application is equally 435.96/148.08 bounded. The dependency pairs are shifted into the weak component. 435.96/148.08 435.96/148.08 We are left with following problem, upon which TcT provides the 435.96/148.08 certificate YES(O(1),O(1)). 435.96/148.08 435.96/148.08 Weak DPs: 435.96/148.08 { lgth^#(Cons(x, xs)) -> 435.96/148.08 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 435.96/148.08 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 435.96/148.08 , monus^#(x, y) -> 435.96/148.08 c_3(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.08 lgth^#(y)) 435.96/148.08 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 435.96/148.08 c_4(monus^#(xs', xs)) } 435.96/148.08 Weak Trs: 435.96/148.08 { lgth(Nil()) -> Nil() 435.96/148.08 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 435.96/148.08 , @(Nil(), ys) -> ys 435.96/148.08 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 435.96/148.08 , and(True(), True()) -> True() 435.96/148.08 , and(True(), False()) -> False() 435.96/148.08 , and(False(), True()) -> False() 435.96/148.08 , and(False(), False()) -> False() 435.96/148.08 , eqList(Nil(), Nil()) -> True() 435.96/148.08 , eqList(Nil(), Cons(y, ys)) -> False() 435.96/148.08 , eqList(Cons(x, xs), Nil()) -> False() 435.96/148.08 , eqList(Cons(x, xs), Cons(y, ys)) -> 435.96/148.08 and(eqList(x, y), eqList(xs, ys)) } 435.96/148.08 Obligation: 435.96/148.08 innermost runtime complexity 435.96/148.08 Answer: 435.96/148.08 YES(O(1),O(1)) 435.96/148.08 435.96/148.08 The following weak DPs constitute a sub-graph of the DG that is 435.96/148.08 closed under successors. The DPs are removed. 435.96/148.08 435.96/148.08 { lgth^#(Cons(x, xs)) -> 435.96/148.08 c_1(@^#(Cons(Nil(), Nil()), lgth(xs)), lgth^#(xs)) 435.96/148.08 , @^#(Cons(x, xs), ys) -> c_2(@^#(xs, ys)) 435.96/148.08 , monus^#(x, y) -> 435.96/148.08 c_3(monus[Ite]^#(eqList(lgth(y), Cons(Nil(), Nil())), x, y), 435.96/148.08 lgth^#(y)) 435.96/148.08 , monus[Ite]^#(False(), Cons(x', xs'), Cons(x, xs)) -> 435.96/148.08 c_4(monus^#(xs', xs)) } 435.96/148.08 435.96/148.08 We are left with following problem, upon which TcT provides the 435.96/148.08 certificate YES(O(1),O(1)). 435.96/148.08 435.96/148.08 Weak Trs: 435.96/148.08 { lgth(Nil()) -> Nil() 435.96/148.08 , lgth(Cons(x, xs)) -> @(Cons(Nil(), Nil()), lgth(xs)) 435.96/148.08 , @(Nil(), ys) -> ys 435.96/148.08 , @(Cons(x, xs), ys) -> Cons(x, @(xs, ys)) 435.96/148.08 , and(True(), True()) -> True() 435.96/148.08 , and(True(), False()) -> False() 435.96/148.08 , and(False(), True()) -> False() 435.96/148.08 , and(False(), False()) -> False() 435.96/148.08 , eqList(Nil(), Nil()) -> True() 435.96/148.08 , eqList(Nil(), Cons(y, ys)) -> False() 435.96/148.08 , eqList(Cons(x, xs), Nil()) -> False() 435.96/148.08 , eqList(Cons(x, xs), Cons(y, ys)) -> 435.96/148.08 and(eqList(x, y), eqList(xs, ys)) } 435.96/148.08 Obligation: 435.96/148.08 innermost runtime complexity 435.96/148.08 Answer: 435.96/148.08 YES(O(1),O(1)) 435.96/148.08 435.96/148.08 No rule is usable, rules are removed from the input problem. 435.96/148.08 435.96/148.08 We are left with following problem, upon which TcT provides the 435.96/148.08 certificate YES(O(1),O(1)). 435.96/148.08 435.96/148.08 Rules: Empty 435.96/148.08 Obligation: 435.96/148.08 innermost runtime complexity 435.96/148.08 Answer: 435.96/148.08 YES(O(1),O(1)) 435.96/148.08 435.96/148.08 Empty rules are trivially bounded 435.96/148.08 435.96/148.08 435.96/148.08 435.96/148.08 Hurray, we answered YES(O(1),O(n^2)) 436.40/148.33 EOF