WORST_CASE(?,O(n^2)) * Step 1: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: cond_insert_ord_x_ys_1(False(),x3,x2,x1) -> Cons(x2,insert#3(x3,x1)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) insert#3(x2,Nil()) -> Cons(x2,Nil()) insert#3(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) main(x1) -> sort#2(x1) sort#2(Cons(x4,x2)) -> insert#3(x4,sort#2(x2)) sort#2(Nil()) -> Nil() - Signature: {cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1,insert#3,leq#2,main ,sort#2} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(Cons) = {2}, uargs(cond_insert_ord_x_ys_1) = {1}, uargs(insert#3) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(Cons) = [1] x1 + [1] x2 + [0] p(False) = [1] p(Nil) = [12] p(S) = [0] p(True) = [0] p(cond_insert_ord_x_ys_1) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [0] p(insert#3) = [1] x1 + [1] x2 + [0] p(leq#2) = [0] p(main) = [2] x1 + [0] p(sort#2) = [2] x1 + [0] Following rules are strictly oriented: cond_insert_ord_x_ys_1(False(),x3,x2,x1) = [1] x1 + [1] x2 + [1] x3 + [1] > [1] x1 + [1] x2 + [1] x3 + [0] = Cons(x2,insert#3(x3,x1)) sort#2(Nil()) = [24] > [12] = Nil() Following rules are (at-least) weakly oriented: cond_insert_ord_x_ys_1(True(),x3,x2,x1) = [1] x1 + [1] x2 + [1] x3 + [0] >= [1] x1 + [1] x2 + [1] x3 + [0] = Cons(x3,Cons(x2,x1)) insert#3(x2,Nil()) = [1] x2 + [12] >= [1] x2 + [12] = Cons(x2,Nil()) insert#3(x6,Cons(x4,x2)) = [1] x2 + [1] x4 + [1] x6 + [0] >= [1] x2 + [1] x4 + [1] x6 + [0] = cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) = [0] >= [0] = True() leq#2(S(x12),0()) = [0] >= [1] = False() leq#2(S(x4),S(x2)) = [0] >= [0] = leq#2(x4,x2) main(x1) = [2] x1 + [0] >= [2] x1 + [0] = sort#2(x1) sort#2(Cons(x4,x2)) = [2] x2 + [2] x4 + [0] >= [2] x2 + [1] x4 + [0] = insert#3(x4,sort#2(x2)) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 2: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) insert#3(x2,Nil()) -> Cons(x2,Nil()) insert#3(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) main(x1) -> sort#2(x1) sort#2(Cons(x4,x2)) -> insert#3(x4,sort#2(x2)) - Weak TRS: cond_insert_ord_x_ys_1(False(),x3,x2,x1) -> Cons(x2,insert#3(x3,x1)) sort#2(Nil()) -> Nil() - Signature: {cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1,insert#3,leq#2,main ,sort#2} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(Cons) = {2}, uargs(cond_insert_ord_x_ys_1) = {1}, uargs(insert#3) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(Cons) = [1] x1 + [1] x2 + [8] p(False) = [8] p(Nil) = [10] p(S) = [1] x1 + [0] p(True) = [0] p(cond_insert_ord_x_ys_1) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [0] p(insert#3) = [1] x1 + [1] x2 + [0] p(leq#2) = [0] p(main) = [1] x1 + [0] p(sort#2) = [1] x1 + [5] Following rules are strictly oriented: insert#3(x6,Cons(x4,x2)) = [1] x2 + [1] x4 + [1] x6 + [8] > [1] x2 + [1] x4 + [1] x6 + [0] = cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) sort#2(Cons(x4,x2)) = [1] x2 + [1] x4 + [13] > [1] x2 + [1] x4 + [5] = insert#3(x4,sort#2(x2)) Following rules are (at-least) weakly oriented: cond_insert_ord_x_ys_1(False(),x3,x2,x1) = [1] x1 + [1] x2 + [1] x3 + [8] >= [1] x1 + [1] x2 + [1] x3 + [8] = Cons(x2,insert#3(x3,x1)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) = [1] x1 + [1] x2 + [1] x3 + [0] >= [1] x1 + [1] x2 + [1] x3 + [16] = Cons(x3,Cons(x2,x1)) insert#3(x2,Nil()) = [1] x2 + [10] >= [1] x2 + [18] = Cons(x2,Nil()) leq#2(0(),x8) = [0] >= [0] = True() leq#2(S(x12),0()) = [0] >= [8] = False() leq#2(S(x4),S(x2)) = [0] >= [0] = leq#2(x4,x2) main(x1) = [1] x1 + [0] >= [1] x1 + [5] = sort#2(x1) sort#2(Nil()) = [15] >= [10] = Nil() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 3: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) insert#3(x2,Nil()) -> Cons(x2,Nil()) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) main(x1) -> sort#2(x1) - Weak TRS: cond_insert_ord_x_ys_1(False(),x3,x2,x1) -> Cons(x2,insert#3(x3,x1)) insert#3(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) sort#2(Cons(x4,x2)) -> insert#3(x4,sort#2(x2)) sort#2(Nil()) -> Nil() - Signature: {cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1,insert#3,leq#2,main ,sort#2} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(Cons) = {2}, uargs(cond_insert_ord_x_ys_1) = {1}, uargs(insert#3) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(Cons) = [1] x1 + [1] x2 + [1] p(False) = [9] p(Nil) = [0] p(S) = [1] x1 + [0] p(True) = [0] p(cond_insert_ord_x_ys_1) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [0] p(insert#3) = [1] x1 + [1] x2 + [8] p(leq#2) = [9] p(main) = [8] x1 + [2] p(sort#2) = [8] x1 + [9] Following rules are strictly oriented: insert#3(x2,Nil()) = [1] x2 + [8] > [1] x2 + [1] = Cons(x2,Nil()) leq#2(0(),x8) = [9] > [0] = True() Following rules are (at-least) weakly oriented: cond_insert_ord_x_ys_1(False(),x3,x2,x1) = [1] x1 + [1] x2 + [1] x3 + [9] >= [1] x1 + [1] x2 + [1] x3 + [9] = Cons(x2,insert#3(x3,x1)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) = [1] x1 + [1] x2 + [1] x3 + [0] >= [1] x1 + [1] x2 + [1] x3 + [2] = Cons(x3,Cons(x2,x1)) insert#3(x6,Cons(x4,x2)) = [1] x2 + [1] x4 + [1] x6 + [9] >= [1] x2 + [1] x4 + [1] x6 + [9] = cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(S(x12),0()) = [9] >= [9] = False() leq#2(S(x4),S(x2)) = [9] >= [9] = leq#2(x4,x2) main(x1) = [8] x1 + [2] >= [8] x1 + [9] = sort#2(x1) sort#2(Cons(x4,x2)) = [8] x2 + [8] x4 + [17] >= [8] x2 + [1] x4 + [17] = insert#3(x4,sort#2(x2)) sort#2(Nil()) = [9] >= [0] = Nil() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 4: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) main(x1) -> sort#2(x1) - Weak TRS: cond_insert_ord_x_ys_1(False(),x3,x2,x1) -> Cons(x2,insert#3(x3,x1)) insert#3(x2,Nil()) -> Cons(x2,Nil()) insert#3(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) -> True() sort#2(Cons(x4,x2)) -> insert#3(x4,sort#2(x2)) sort#2(Nil()) -> Nil() - Signature: {cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1,insert#3,leq#2,main ,sort#2} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(Cons) = {2}, uargs(cond_insert_ord_x_ys_1) = {1}, uargs(insert#3) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(Cons) = [1] x1 + [1] x2 + [0] p(False) = [0] p(Nil) = [0] p(S) = [0] p(True) = [0] p(cond_insert_ord_x_ys_1) = [1] x1 + [1] x2 + [1] x3 + [1] x4 + [0] p(insert#3) = [1] x1 + [1] x2 + [0] p(leq#2) = [0] p(main) = [1] x1 + [1] p(sort#2) = [1] x1 + [0] Following rules are strictly oriented: main(x1) = [1] x1 + [1] > [1] x1 + [0] = sort#2(x1) Following rules are (at-least) weakly oriented: cond_insert_ord_x_ys_1(False(),x3,x2,x1) = [1] x1 + [1] x2 + [1] x3 + [0] >= [1] x1 + [1] x2 + [1] x3 + [0] = Cons(x2,insert#3(x3,x1)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) = [1] x1 + [1] x2 + [1] x3 + [0] >= [1] x1 + [1] x2 + [1] x3 + [0] = Cons(x3,Cons(x2,x1)) insert#3(x2,Nil()) = [1] x2 + [0] >= [1] x2 + [0] = Cons(x2,Nil()) insert#3(x6,Cons(x4,x2)) = [1] x2 + [1] x4 + [1] x6 + [0] >= [1] x2 + [1] x4 + [1] x6 + [0] = cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) = [0] >= [0] = True() leq#2(S(x12),0()) = [0] >= [0] = False() leq#2(S(x4),S(x2)) = [0] >= [0] = leq#2(x4,x2) sort#2(Cons(x4,x2)) = [1] x2 + [1] x4 + [0] >= [1] x2 + [1] x4 + [0] = insert#3(x4,sort#2(x2)) sort#2(Nil()) = [0] >= [0] = Nil() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) - Weak TRS: cond_insert_ord_x_ys_1(False(),x3,x2,x1) -> Cons(x2,insert#3(x3,x1)) insert#3(x2,Nil()) -> Cons(x2,Nil()) insert#3(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) -> True() main(x1) -> sort#2(x1) sort#2(Cons(x4,x2)) -> insert#3(x4,sort#2(x2)) sort#2(Nil()) -> Nil() - Signature: {cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1,insert#3,leq#2,main ,sort#2} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following nonconstant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(Cons) = {2}, uargs(cond_insert_ord_x_ys_1) = {1}, uargs(insert#3) = {2} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(Cons) = [1] x2 + [2] p(False) = [2] p(Nil) = [1] p(S) = [0] p(True) = [2] p(cond_insert_ord_x_ys_1) = [1] x1 + [1] x4 + [4] p(insert#3) = [1] x2 + [4] p(leq#2) = [2] p(main) = [4] x1 + [5] p(sort#2) = [4] x1 + [5] Following rules are strictly oriented: cond_insert_ord_x_ys_1(True(),x3,x2,x1) = [1] x1 + [6] > [1] x1 + [4] = Cons(x3,Cons(x2,x1)) Following rules are (at-least) weakly oriented: cond_insert_ord_x_ys_1(False(),x3,x2,x1) = [1] x1 + [6] >= [1] x1 + [6] = Cons(x2,insert#3(x3,x1)) insert#3(x2,Nil()) = [5] >= [3] = Cons(x2,Nil()) insert#3(x6,Cons(x4,x2)) = [1] x2 + [6] >= [1] x2 + [6] = cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) = [2] >= [2] = True() leq#2(S(x12),0()) = [2] >= [2] = False() leq#2(S(x4),S(x2)) = [2] >= [2] = leq#2(x4,x2) main(x1) = [4] x1 + [5] >= [4] x1 + [5] = sort#2(x1) sort#2(Cons(x4,x2)) = [4] x2 + [13] >= [4] x2 + [9] = insert#3(x4,sort#2(x2)) sort#2(Nil()) = [9] >= [1] = Nil() Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 6: Ara WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) - Weak TRS: cond_insert_ord_x_ys_1(False(),x3,x2,x1) -> Cons(x2,insert#3(x3,x1)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) insert#3(x2,Nil()) -> Cons(x2,Nil()) insert#3(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) -> True() main(x1) -> sort#2(x1) sort#2(Cons(x4,x2)) -> insert#3(x4,sort#2(x2)) sort#2(Nil()) -> Nil() - Signature: {cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1,insert#3,leq#2,main ,sort#2} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: Ara {heuristics_ = NoHeuristics, minDegree = 2, maxDegree = 2, araTimeout = 15, araFindStrictRules = Just 1} + Details: Signatures used: ---------------- 0 :: [] -(0)-> A(0, 0) Cons :: [A(0, 0) x A(11, 5)] -(6)-> A(6, 5) Cons :: [A(0, 0) x A(15, 6)] -(9)-> A(9, 6) Cons :: [A(0, 0) x A(5, 5)] -(0)-> A(0, 5) Cons :: [A(0, 0) x A(10, 5)] -(5)-> A(5, 5) Cons :: [A(0, 0) x A(13, 13)] -(0)-> A(0, 13) False :: [] -(0)-> A(1, 9) False :: [] -(0)-> A(15, 15) Nil :: [] -(0)-> A(6, 5) Nil :: [] -(0)-> A(9, 6) Nil :: [] -(0)-> A(15, 15) Nil :: [] -(0)-> A(7, 11) S :: [A(0, 0)] -(0)-> A(0, 0) True :: [] -(0)-> A(1, 9) True :: [] -(0)-> A(15, 15) cond_insert_ord_x_ys_1 :: [A(1, 9) x A(0, 0) x A(0, 0) x A(11, 5)] -(5)-> A(0, 5) insert#3 :: [A(0, 0) x A(6, 5)] -(0)-> A(0, 5) leq#2 :: [A(0, 0) x A(0, 0)] -(1)-> A(8, 12) main :: [A(15, 8)] -(9)-> A(0, 2) sort#2 :: [A(9, 6)] -(7)-> A(0, 5) Cost-free Signatures used: -------------------------- 0 :: [] -(0)-> A_cf(0, 0) Cons :: [A_cf(0, 0) x A_cf(5, 0)] -(5)-> A_cf(5, 0) Cons :: [A_cf(0, 0) x A_cf(8, 3)] -(5)-> A_cf(5, 3) Cons :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) Cons :: [A_cf(0, 0) x A_cf(12, 12)] -(0)-> A_cf(0, 12) Cons :: [A_cf(0, 0) x A_cf(6, 0)] -(6)-> A_cf(6, 0) Cons :: [A_cf(0, 0) x A_cf(7, 1)] -(6)-> A_cf(6, 1) False :: [] -(0)-> A_cf(0, 0) False :: [] -(0)-> A_cf(11, 9) False :: [] -(0)-> A_cf(1, 1) False :: [] -(0)-> A_cf(3, 3) False :: [] -(0)-> A_cf(10, 8) Nil :: [] -(0)-> A_cf(5, 0) Nil :: [] -(0)-> A_cf(15, 11) Nil :: [] -(0)-> A_cf(0, 0) Nil :: [] -(0)-> A_cf(15, 15) Nil :: [] -(0)-> A_cf(6, 0) Nil :: [] -(0)-> A_cf(11, 11) Nil :: [] -(0)-> A_cf(10, 2) S :: [A_cf(0, 0)] -(0)-> A_cf(0, 0) True :: [] -(0)-> A_cf(0, 0) True :: [] -(0)-> A_cf(11, 11) True :: [] -(0)-> A_cf(1, 1) True :: [] -(0)-> A_cf(3, 3) True :: [] -(0)-> A_cf(10, 10) cond_insert_ord_x_ys_1 :: [A_cf(0, 0) x A_cf(0, 0) x A_cf(0, 0) x A_cf(5, 0)] -(10)-> A_cf(5, 0) cond_insert_ord_x_ys_1 :: [A_cf(1, 1) x A_cf(0, 0) x A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) cond_insert_ord_x_ys_1 :: [A_cf(0, 0) x A_cf(0, 0) x A_cf(0, 0) x A_cf(6, 0)] -(12)-> A_cf(6, 0) insert#3 :: [A_cf(0, 0) x A_cf(5, 0)] -(5)-> A_cf(5, 0) insert#3 :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) insert#3 :: [A_cf(0, 0) x A_cf(6, 0)] -(6)-> A_cf(6, 0) leq#2 :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) leq#2 :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(8, 8) leq#2 :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(3, 3) sort#2 :: [A_cf(6, 0)] -(0)-> A_cf(6, 0) Base Constructor Signatures used: --------------------------------- 0_A :: [] -(0)-> A(1, 0) 0_A :: [] -(0)-> A(0, 1) Cons_A :: [A(0, 0) x A(1, 0)] -(1)-> A(1, 0) Cons_A :: [A(0, 0) x A(1, 1)] -(0)-> A(0, 1) False_A :: [] -(0)-> A(1, 0) False_A :: [] -(0)-> A(0, 1) Nil_A :: [] -(0)-> A(1, 0) Nil_A :: [] -(0)-> A(0, 1) S_A :: [A(1, 0)] -(0)-> A(1, 0) S_A :: [A(0, 0)] -(0)-> A(0, 1) True_A :: [] -(0)-> A(1, 0) True_A :: [] -(0)-> A(0, 1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: leq#2(S(x12),0()) -> False() 2. Weak: leq#2(S(x4),S(x2)) -> leq#2(x4,x2) * Step 7: Ara WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: leq#2(S(x4),S(x2)) -> leq#2(x4,x2) - Weak TRS: cond_insert_ord_x_ys_1(False(),x3,x2,x1) -> Cons(x2,insert#3(x3,x1)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) insert#3(x2,Nil()) -> Cons(x2,Nil()) insert#3(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() main(x1) -> sort#2(x1) sort#2(Cons(x4,x2)) -> insert#3(x4,sort#2(x2)) sort#2(Nil()) -> Nil() - Signature: {cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1,insert#3,leq#2,main ,sort#2} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: Ara {heuristics_ = NoHeuristics, minDegree = 2, maxDegree = 2, araTimeout = 15, araFindStrictRules = Just 1} + Details: Signatures used: ---------------- 0 :: [] -(0)-> A(0, 0) 0 :: [] -(0)-> A(1, 0) Cons :: [A(1, 0) x A(1, 0)] -(0)-> A(1, 0) Cons :: [A(3, 0) x A(15, 12)] -(0)-> A(3, 12) Cons :: [A(0, 0) x A(0, 0)] -(0)-> A(0, 0) Cons :: [A(2, 0) x A(3, 1)] -(0)-> A(2, 1) False :: [] -(0)-> A(0, 0) False :: [] -(0)-> A(13, 13) Nil :: [] -(0)-> A(1, 0) Nil :: [] -(0)-> A(3, 12) Nil :: [] -(0)-> A(7, 15) Nil :: [] -(0)-> A(7, 7) S :: [A(0, 0)] -(0)-> A(0, 0) S :: [A(1, 0)] -(1)-> A(1, 0) True :: [] -(0)-> A(0, 0) True :: [] -(0)-> A(13, 13) cond_insert_ord_x_ys_1 :: [A(0, 0) x A(3, 0) x A(0, 0) x A(1, 0)] -(0)-> A(0, 0) insert#3 :: [A(3, 0) x A(1, 0)] -(0)-> A(0, 0) leq#2 :: [A(0, 0) x A(1, 0)] -(0)-> A(7, 6) main :: [A(13, 14)] -(14)-> A(0, 0) sort#2 :: [A(3, 12)] -(8)-> A(0, 0) Cost-free Signatures used: -------------------------- 0 :: [] -(0)-> A_cf(0, 0) Cons :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) Cons :: [A_cf(0, 0) x A_cf(6, 6)] -(0)-> A_cf(0, 6) Cons :: [A_cf(12, 0) x A_cf(12, 0)] -(0)-> A_cf(12, 0) Cons :: [A_cf(1, 0) x A_cf(1, 0)] -(0)-> A_cf(1, 0) False :: [] -(0)-> A_cf(0, 0) False :: [] -(0)-> A_cf(7, 6) False :: [] -(0)-> A_cf(15, 15) False :: [] -(0)-> A_cf(10, 10) Nil :: [] -(0)-> A_cf(0, 0) Nil :: [] -(0)-> A_cf(11, 13) Nil :: [] -(0)-> A_cf(12, 0) Nil :: [] -(0)-> A_cf(1, 0) Nil :: [] -(0)-> A_cf(15, 11) Nil :: [] -(0)-> A_cf(3, 2) S :: [A_cf(0, 0)] -(0)-> A_cf(0, 0) True :: [] -(0)-> A_cf(0, 0) True :: [] -(0)-> A_cf(7, 6) True :: [] -(0)-> A_cf(15, 15) True :: [] -(0)-> A_cf(10, 10) cond_insert_ord_x_ys_1 :: [A_cf(0, 0) x A_cf(0, 0) x A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) cond_insert_ord_x_ys_1 :: [A_cf(7, 6) x A_cf(0, 0) x A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) cond_insert_ord_x_ys_1 :: [A_cf(0, 0) x A_cf(12, 0) x A_cf(1, 0) x A_cf(1, 0)] -(0)-> A_cf(1, 0) insert#3 :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) insert#3 :: [A_cf(12, 0) x A_cf(1, 0)] -(0)-> A_cf(1, 0) leq#2 :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) leq#2 :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(15, 15) leq#2 :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(8, 8) sort#2 :: [A_cf(12, 0)] -(0)-> A_cf(1, 0) Base Constructor Signatures used: --------------------------------- 0_A :: [] -(0)-> A(1, 0) 0_A :: [] -(0)-> A(0, 1) Cons_A :: [A(1, 0) x A(1, 0)] -(0)-> A(1, 0) Cons_A :: [A(0, 0) x A(1, 1)] -(0)-> A(0, 1) False_A :: [] -(0)-> A(1, 0) False_A :: [] -(0)-> A(0, 1) Nil_A :: [] -(0)-> A(1, 0) Nil_A :: [] -(0)-> A(0, 1) S_A :: [A(1, 0)] -(1)-> A(1, 0) S_A :: [A(0, 0)] -(0)-> A(0, 1) True_A :: [] -(0)-> A(1, 0) True_A :: [] -(0)-> A(0, 1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: leq#2(S(x4),S(x2)) -> leq#2(x4,x2) 2. Weak: * Step 8: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: cond_insert_ord_x_ys_1(False(),x3,x2,x1) -> Cons(x2,insert#3(x3,x1)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) insert#3(x2,Nil()) -> Cons(x2,Nil()) insert#3(x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() leq#2(S(x4),S(x2)) -> leq#2(x4,x2) main(x1) -> sort#2(x1) sort#2(Cons(x4,x2)) -> insert#3(x4,sort#2(x2)) sort#2(Nil()) -> Nil() - Signature: {cond_insert_ord_x_ys_1/4,insert#3/2,leq#2/2,main/1,sort#2/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1,insert#3,leq#2,main ,sort#2} and constructors {0,Cons,False,Nil,S,True} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))