WORST_CASE(?,O(n^2)) * Step 1: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) fold#3(insert_ord(x2),Nil()) -> Nil() fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2)) insert_ord#2(leq(),x2,Nil()) -> Cons(x2,Nil()) insert_ord#2(leq(),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(x3) -> fold#3(insert_ord(leq()),x3) - Signature: {cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0 ,insert_ord/1,leq/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1,fold#3,insert_ord#2,leq#2 ,main} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq} + 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_ord#2) = {3} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(Cons) = [1] x1 + [1] x2 + [0] p(False) = [2] p(Nil) = [3] p(S) = [0] p(True) = [0] p(cond_insert_ord_x_ys_1) = [1] x1 + [8] x2 + [1] x3 + [1] x4 + [0] p(fold#3) = [8] x2 + [0] p(insert_ord) = [1] x1 + [0] p(insert_ord#2) = [8] x2 + [1] x3 + [0] p(leq) = [0] p(leq#2) = [13] p(main) = [9] x1 + [0] Following rules are strictly oriented: cond_insert_ord_x_ys_1(False(),x0,x5,x2) = [8] x0 + [1] x2 + [1] x5 + [2] > [8] x0 + [1] x2 + [1] x5 + [0] = Cons(x5,insert_ord#2(leq(),x0,x2)) fold#3(insert_ord(x2),Nil()) = [24] > [3] = Nil() leq#2(0(),x8) = [13] > [0] = True() leq#2(S(x12),0()) = [13] > [2] = False() Following rules are (at-least) weakly oriented: cond_insert_ord_x_ys_1(True(),x3,x2,x1) = [1] x1 + [1] x2 + [8] x3 + [0] >= [1] x1 + [1] x2 + [1] x3 + [0] = Cons(x3,Cons(x2,x1)) fold#3(insert_ord(x6),Cons(x4,x2)) = [8] x2 + [8] x4 + [0] >= [8] x2 + [8] x4 + [0] = insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2)) insert_ord#2(leq(),x2,Nil()) = [8] x2 + [3] >= [1] x2 + [3] = Cons(x2,Nil()) insert_ord#2(leq(),x6,Cons(x4,x2)) = [1] x2 + [1] x4 + [8] x6 + [0] >= [1] x2 + [1] x4 + [8] x6 + [13] = cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(S(x4),S(x2)) = [13] >= [13] = leq#2(x4,x2) main(x3) = [9] x3 + [0] >= [8] x3 + [0] = fold#3(insert_ord(leq()),x3) 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)) fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2)) insert_ord#2(leq(),x2,Nil()) -> Cons(x2,Nil()) insert_ord#2(leq(),x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(S(x4),S(x2)) -> leq#2(x4,x2) main(x3) -> fold#3(insert_ord(leq()),x3) - Weak TRS: cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2)) fold#3(insert_ord(x2),Nil()) -> Nil() leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() - Signature: {cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0 ,insert_ord/1,leq/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1,fold#3,insert_ord#2,leq#2 ,main} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq} + 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_ord#2) = {3} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(Cons) = [1] x2 + [0] p(False) = [0] p(Nil) = [7] p(S) = [1] x1 + [0] p(True) = [0] p(cond_insert_ord_x_ys_1) = [1] x1 + [1] x4 + [5] p(fold#3) = [3] x1 + [7] p(insert_ord) = [1] x1 + [0] p(insert_ord#2) = [1] x3 + [2] p(leq) = [0] p(leq#2) = [0] p(main) = [0] Following rules are strictly oriented: cond_insert_ord_x_ys_1(True(),x3,x2,x1) = [1] x1 + [5] > [1] x1 + [0] = Cons(x3,Cons(x2,x1)) insert_ord#2(leq(),x2,Nil()) = [9] > [7] = Cons(x2,Nil()) Following rules are (at-least) weakly oriented: cond_insert_ord_x_ys_1(False(),x0,x5,x2) = [1] x2 + [5] >= [1] x2 + [2] = Cons(x5,insert_ord#2(leq(),x0,x2)) fold#3(insert_ord(x2),Nil()) = [3] x2 + [7] >= [7] = Nil() fold#3(insert_ord(x6),Cons(x4,x2)) = [3] x6 + [7] >= [3] x6 + [9] = insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2)) insert_ord#2(leq(),x6,Cons(x4,x2)) = [1] x2 + [2] >= [1] x2 + [5] = 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) main(x3) = [0] >= [7] = fold#3(insert_ord(leq()),x3) 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: fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2)) insert_ord#2(leq(),x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(S(x4),S(x2)) -> leq#2(x4,x2) main(x3) -> fold#3(insert_ord(leq()),x3) - Weak TRS: cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) fold#3(insert_ord(x2),Nil()) -> Nil() insert_ord#2(leq(),x2,Nil()) -> Cons(x2,Nil()) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() - Signature: {cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0 ,insert_ord/1,leq/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1,fold#3,insert_ord#2,leq#2 ,main} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq} + 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_ord#2) = {3} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(Cons) = [1] x2 + [0] p(False) = [2] p(Nil) = [0] p(S) = [1] x1 + [0] p(True) = [0] p(cond_insert_ord_x_ys_1) = [1] x1 + [1] x4 + [0] p(fold#3) = [3] x1 + [0] p(insert_ord) = [1] x1 + [0] p(insert_ord#2) = [1] x3 + [0] p(leq) = [0] p(leq#2) = [4] p(main) = [1] Following rules are strictly oriented: main(x3) = [1] > [0] = fold#3(insert_ord(leq()),x3) Following rules are (at-least) weakly oriented: cond_insert_ord_x_ys_1(False(),x0,x5,x2) = [1] x2 + [2] >= [1] x2 + [0] = Cons(x5,insert_ord#2(leq(),x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) = [1] x1 + [0] >= [1] x1 + [0] = Cons(x3,Cons(x2,x1)) fold#3(insert_ord(x2),Nil()) = [3] x2 + [0] >= [0] = Nil() fold#3(insert_ord(x6),Cons(x4,x2)) = [3] x6 + [0] >= [3] x6 + [0] = insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2)) insert_ord#2(leq(),x2,Nil()) = [0] >= [0] = Cons(x2,Nil()) insert_ord#2(leq(),x6,Cons(x4,x2)) = [1] x2 + [0] >= [1] x2 + [4] = cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) = [4] >= [0] = True() leq#2(S(x12),0()) = [4] >= [2] = False() leq#2(S(x4),S(x2)) = [4] >= [4] = leq#2(x4,x2) 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: fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2)) insert_ord#2(leq(),x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(S(x4),S(x2)) -> leq#2(x4,x2) - Weak TRS: cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) fold#3(insert_ord(x2),Nil()) -> Nil() insert_ord#2(leq(),x2,Nil()) -> Cons(x2,Nil()) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() main(x3) -> fold#3(insert_ord(leq()),x3) - Signature: {cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0 ,insert_ord/1,leq/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1,fold#3,insert_ord#2,leq#2 ,main} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq} + 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_ord#2) = {3} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [2] p(Cons) = [1] x1 + [1] x2 + [2] p(False) = [3] p(Nil) = [0] p(S) = [1] x1 + [0] p(True) = [2] p(cond_insert_ord_x_ys_1) = [1] x1 + [2] x2 + [1] x3 + [1] x4 + [2] p(fold#3) = [2] x2 + [0] p(insert_ord) = [2] p(insert_ord#2) = [2] x2 + [1] x3 + [2] p(leq) = [0] p(leq#2) = [5] p(main) = [4] x1 + [0] Following rules are strictly oriented: fold#3(insert_ord(x6),Cons(x4,x2)) = [2] x2 + [2] x4 + [4] > [2] x2 + [2] x4 + [2] = insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2)) Following rules are (at-least) weakly oriented: cond_insert_ord_x_ys_1(False(),x0,x5,x2) = [2] x0 + [1] x2 + [1] x5 + [5] >= [2] x0 + [1] x2 + [1] x5 + [4] = Cons(x5,insert_ord#2(leq(),x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) = [1] x1 + [1] x2 + [2] x3 + [4] >= [1] x1 + [1] x2 + [1] x3 + [4] = Cons(x3,Cons(x2,x1)) fold#3(insert_ord(x2),Nil()) = [0] >= [0] = Nil() insert_ord#2(leq(),x2,Nil()) = [2] x2 + [2] >= [1] x2 + [2] = Cons(x2,Nil()) insert_ord#2(leq(),x6,Cons(x4,x2)) = [1] x2 + [1] x4 + [2] x6 + [4] >= [1] x2 + [1] x4 + [2] x6 + [7] = cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(0(),x8) = [5] >= [2] = True() leq#2(S(x12),0()) = [5] >= [3] = False() leq#2(S(x4),S(x2)) = [5] >= [5] = leq#2(x4,x2) main(x3) = [4] x3 + [0] >= [2] x3 + [0] = fold#3(insert_ord(leq()),x3) Further, it can be verified that all rules not oriented are covered by the weightgap condition. * Step 5: Ara WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: insert_ord#2(leq(),x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) leq#2(S(x4),S(x2)) -> leq#2(x4,x2) - Weak TRS: cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) fold#3(insert_ord(x2),Nil()) -> Nil() fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2)) insert_ord#2(leq(),x2,Nil()) -> Cons(x2,Nil()) leq#2(0(),x8) -> True() leq#2(S(x12),0()) -> False() main(x3) -> fold#3(insert_ord(leq()),x3) - Signature: {cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0 ,insert_ord/1,leq/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1,fold#3,insert_ord#2,leq#2 ,main} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq} + 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(5, 0) x A(5, 0)] -(5)-> A(5, 0) Cons :: [A(10, 10) x A(10, 10)] -(10)-> A(0, 10) Cons :: [A(1, 0) x A(1, 0)] -(1)-> A(1, 0) Cons :: [A(2, 0) x A(2, 0)] -(2)-> A(2, 0) Cons :: [A(4, 0) x A(4, 0)] -(4)-> A(4, 0) False :: [] -(0)-> A(0, 0) False :: [] -(0)-> A(13, 13) Nil :: [] -(0)-> A(0, 10) Nil :: [] -(0)-> A(5, 0) Nil :: [] -(0)-> A(7, 7) Nil :: [] -(0)-> A(7, 13) 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(4, 8) x A(4, 0) x A(5, 0)] -(6)-> A(1, 0) fold#3 :: [A(9, 5) x A(0, 10)] -(0)-> A(1, 0) insert_ord :: [A(14, 0)] -(0)-> A(9, 5) insert_ord :: [A(15, 0)] -(0)-> A(9, 6) insert_ord#2 :: [A(0, 0) x A(4, 8) x A(5, 0)] -(1)-> A(1, 0) leq :: [] -(0)-> A(0, 0) leq :: [] -(0)-> A(5, 5) leq :: [] -(0)-> A(15, 13) leq#2 :: [A(0, 0) x A(1, 0)] -(0)-> A(7, 6) main :: [A(14, 14)] -(14)-> 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(8, 0) x A_cf(8, 0)] -(8)-> A_cf(8, 0) Cons :: [A_cf(4, 0) x A_cf(4, 0)] -(4)-> A_cf(4, 0) False :: [] -(0)-> A_cf(7, 6) False :: [] -(0)-> A_cf(11, 15) False :: [] -(0)-> A_cf(0, 0) False :: [] -(0)-> A_cf(10, 10) Nil :: [] -(0)-> A_cf(0, 0) Nil :: [] -(0)-> A_cf(3, 11) Nil :: [] -(0)-> A_cf(15, 11) Nil :: [] -(0)-> A_cf(8, 0) Nil :: [] -(0)-> A_cf(7, 3) Nil :: [] -(0)-> A_cf(4, 0) S :: [A_cf(0, 0)] -(0)-> A_cf(0, 0) True :: [] -(0)-> A_cf(7, 6) True :: [] -(0)-> A_cf(11, 15) True :: [] -(0)-> A_cf(0, 0) True :: [] -(0)-> A_cf(10, 10) 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(0, 0) x A_cf(0, 0) x A_cf(0, 0)] -(2)-> A_cf(0, 0) cond_insert_ord_x_ys_1 :: [A_cf(0, 0) x A_cf(8, 0) x A_cf(4, 0) x A_cf(4, 0)] -(10)-> A_cf(4, 0) fold#3 :: [A_cf(0, 0) x A_cf(8, 0)] -(1)-> A_cf(4, 0) insert_ord :: [A_cf(0, 0)] -(0)-> A_cf(0, 0) insert_ord#2 :: [A_cf(4, 4) x A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) insert_ord#2 :: [A_cf(2, 2) x A_cf(0, 0) x A_cf(0, 0)] -(2)-> A_cf(0, 0) insert_ord#2 :: [A_cf(0, 0) x A_cf(8, 0) x A_cf(4, 0)] -(6)-> A_cf(4, 0) leq :: [] -(0)-> A_cf(4, 4) leq :: [] -(0)-> A_cf(11, 11) leq :: [] -(0)-> A_cf(2, 2) leq :: [] -(0)-> A_cf(0, 0) leq :: [] -(0)-> A_cf(3, 3) leq#2 :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(9, 15) 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) 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)] -(1)-> A(1, 0) Cons_A :: [A(1, 1) x A(1, 1)] -(1)-> 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) insert_ord_A :: [A(0)] -(0)-> A(1, 0) insert_ord_A :: [A(0)] -(0)-> A(0, 1) leq_A :: [] -(0)-> A(1, 0) leq_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: insert_ord#2(leq(),x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) * Step 6: Ara WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: insert_ord#2(leq(),x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) - Weak TRS: cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) fold#3(insert_ord(x2),Nil()) -> Nil() fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2)) insert_ord#2(leq(),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(x3) -> fold#3(insert_ord(leq()),x3) - Signature: {cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0 ,insert_ord/1,leq/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1,fold#3,insert_ord#2,leq#2 ,main} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq} + Applied Processor: Ara {heuristics_ = NoHeuristics, minDegree = 2, maxDegree = 2, araTimeout = 15, araFindStrictRules = Just 1} + Details: Signatures used: ---------------- 0 :: [] -(0)-> A(0, 0) Cons :: [A(8, 0) x A(8, 0)] -(8)-> A(8, 0) Cons :: [A(10, 0) x A(15, 5)] -(10)-> A(10, 5) Cons :: [A(7, 0) x A(7, 0)] -(7)-> A(7, 0) False :: [] -(0)-> A(10, 9) False :: [] -(0)-> A(15, 15) Nil :: [] -(0)-> A(10, 5) Nil :: [] -(0)-> A(8, 0) Nil :: [] -(0)-> A(13, 7) Nil :: [] -(0)-> A(15, 13) S :: [A(0, 0)] -(0)-> A(0, 0) True :: [] -(0)-> A(10, 9) True :: [] -(0)-> A(15, 15) cond_insert_ord_x_ys_1 :: [A(10, 9) x A(10, 0) x A(8, 0) x A(8, 0)] -(15)-> A(7, 0) fold#3 :: [A(1, 1) x A(10, 5)] -(6)-> A(6, 0) insert_ord :: [A(0, 0)] -(0)-> A(1, 1) insert_ord :: [A(0, 0)] -(0)-> A(10, 10) insert_ord :: [A(0, 0)] -(0)-> A(9, 9) insert_ord#2 :: [A(0, 0) x A(10, 0) x A(8, 0)] -(8)-> A(7, 0) leq :: [] -(0)-> A(0, 0) leq :: [] -(0)-> A(13, 7) leq :: [] -(0)-> A(7, 15) leq#2 :: [A(0, 0) x A(0, 0)] -(0)-> A(12, 12) main :: [A(14, 11)] -(14)-> 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(4, 4)] -(0)-> A_cf(0, 4) Cons :: [A_cf(5, 0) x A_cf(5, 0)] -(5)-> A_cf(5, 0) Cons :: [A_cf(2, 0) x A_cf(2, 0)] -(2)-> A_cf(2, 0) Cons :: [A_cf(2, 0) x A_cf(10, 8)] -(2)-> A_cf(2, 8) False :: [] -(0)-> A_cf(0, 1) False :: [] -(0)-> A_cf(3, 3) False :: [] -(0)-> A_cf(0, 0) False :: [] -(0)-> A_cf(10, 10) Nil :: [] -(0)-> A_cf(0, 0) Nil :: [] -(0)-> A_cf(15, 11) Nil :: [] -(0)-> A_cf(11, 11) Nil :: [] -(0)-> A_cf(5, 0) Nil :: [] -(0)-> A_cf(3, 3) Nil :: [] -(0)-> A_cf(2, 0) S :: [A_cf(0, 0)] -(0)-> A_cf(0, 0) True :: [] -(0)-> A_cf(0, 1) True :: [] -(0)-> A_cf(3, 3) True :: [] -(0)-> A_cf(0, 0) True :: [] -(0)-> A_cf(10, 10) cond_insert_ord_x_ys_1 :: [A_cf(0, 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(0, 0)] -(0)-> A_cf(0, 0) cond_insert_ord_x_ys_1 :: [A_cf(0, 0) x A_cf(4, 0) x A_cf(2, 0) x A_cf(2, 0)] -(4)-> A_cf(2, 0) fold#3 :: [A_cf(9, 2) x A_cf(5, 0)] -(1)-> A_cf(2, 0) insert_ord :: [A_cf(0, 0)] -(0)-> A_cf(9, 2) insert_ord :: [A_cf(0, 0)] -(0)-> A_cf(10, 4) insert_ord#2 :: [A_cf(4, 4) x A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) insert_ord#2 :: [A_cf(7, 1) x A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(0, 0) insert_ord#2 :: [A_cf(0, 0) x A_cf(4, 0) x A_cf(2, 0)] -(2)-> A_cf(2, 0) leq :: [] -(0)-> A_cf(4, 4) leq :: [] -(0)-> A_cf(11, 11) leq :: [] -(0)-> A_cf(7, 1) leq :: [] -(0)-> A_cf(15, 15) leq :: [] -(0)-> A_cf(0, 0) leq :: [] -(0)-> A_cf(3, 3) leq#2 :: [A_cf(0, 0) x A_cf(0, 0)] -(0)-> A_cf(3, 3) 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(0, 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)] -(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(0, 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) insert_ord_A :: [A(0)] -(0)-> A(1, 0) insert_ord_A :: [A(0)] -(0)-> A(0, 1) leq_A :: [] -(0)-> A(1, 0) leq_A :: [] -(0)-> A(0, 1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: insert_ord#2(leq(),x6,Cons(x4,x2)) -> cond_insert_ord_x_ys_1(leq#2(x6,x4),x6,x4,x2) 2. Weak: * Step 7: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: cond_insert_ord_x_ys_1(False(),x0,x5,x2) -> Cons(x5,insert_ord#2(leq(),x0,x2)) cond_insert_ord_x_ys_1(True(),x3,x2,x1) -> Cons(x3,Cons(x2,x1)) fold#3(insert_ord(x2),Nil()) -> Nil() fold#3(insert_ord(x6),Cons(x4,x2)) -> insert_ord#2(x6,x4,fold#3(insert_ord(x6),x2)) insert_ord#2(leq(),x2,Nil()) -> Cons(x2,Nil()) insert_ord#2(leq(),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(x3) -> fold#3(insert_ord(leq()),x3) - Signature: {cond_insert_ord_x_ys_1/4,fold#3/2,insert_ord#2/3,leq#2/2,main/1} / {0/0,Cons/2,False/0,Nil/0,S/1,True/0 ,insert_ord/1,leq/0} - Obligation: innermost runtime complexity wrt. defined symbols {cond_insert_ord_x_ys_1,fold#3,insert_ord#2,leq#2 ,main} and constructors {0,Cons,False,Nil,S,True,insert_ord,leq} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))