WORST_CASE(?,O(n^3)) * Step 1: DependencyPairs WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: choose(x,cons(v,w),y,0()) -> cons(x,cons(v,w)) choose(x,cons(v,w),0(),s(z)) -> cons(v,insert(x,w)) choose(x,cons(v,w),s(y),s(z)) -> choose(x,cons(v,w),y,z) insert(x,cons(v,w)) -> choose(x,cons(v,w),x,v) insert(x,nil()) -> cons(x,nil()) sort(cons(x,y)) -> insert(x,sort(y)) sort(nil()) -> nil() - Signature: {choose/4,insert/2,sort/1} / {0/0,cons/2,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {choose,insert,sort} and constructors {0,cons,nil,s} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs choose#(x,cons(v,w),y,0()) -> c_1() choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w)) choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z)) insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v)) insert#(x,nil()) -> c_5() sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y)) sort#(nil()) -> c_7() Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: choose#(x,cons(v,w),y,0()) -> c_1() choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w)) choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z)) insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v)) insert#(x,nil()) -> c_5() sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y)) sort#(nil()) -> c_7() - Weak TRS: choose(x,cons(v,w),y,0()) -> cons(x,cons(v,w)) choose(x,cons(v,w),0(),s(z)) -> cons(v,insert(x,w)) choose(x,cons(v,w),s(y),s(z)) -> choose(x,cons(v,w),y,z) insert(x,cons(v,w)) -> choose(x,cons(v,w),x,v) insert(x,nil()) -> cons(x,nil()) sort(cons(x,y)) -> insert(x,sort(y)) sort(nil()) -> nil() - Signature: {choose/4,insert/2,sort/1,choose#/4,insert#/2,sort#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0 ,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {choose#,insert#,sort#} and constructors {0,cons,nil,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,5,7} by application of Pre({1,5,7}) = {2,3,4,6}. Here rules are labelled as follows: 1: choose#(x,cons(v,w),y,0()) -> c_1() 2: choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w)) 3: choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z)) 4: insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v)) 5: insert#(x,nil()) -> c_5() 6: sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y)) 7: sort#(nil()) -> c_7() * Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w)) choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z)) insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v)) sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y)) - Weak DPs: choose#(x,cons(v,w),y,0()) -> c_1() insert#(x,nil()) -> c_5() sort#(nil()) -> c_7() - Weak TRS: choose(x,cons(v,w),y,0()) -> cons(x,cons(v,w)) choose(x,cons(v,w),0(),s(z)) -> cons(v,insert(x,w)) choose(x,cons(v,w),s(y),s(z)) -> choose(x,cons(v,w),y,z) insert(x,cons(v,w)) -> choose(x,cons(v,w),x,v) insert(x,nil()) -> cons(x,nil()) sort(cons(x,y)) -> insert(x,sort(y)) sort(nil()) -> nil() - Signature: {choose/4,insert/2,sort/1,choose#/4,insert#/2,sort#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0 ,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {choose#,insert#,sort#} and constructors {0,cons,nil,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w)) -->_1 insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v)):3 -->_1 insert#(x,nil()) -> c_5():6 2:S:choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z)) -->_1 choose#(x,cons(v,w),y,0()) -> c_1():5 -->_1 choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z)):2 -->_1 choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w)):1 3:S:insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v)) -->_1 choose#(x,cons(v,w),y,0()) -> c_1():5 -->_1 choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z)):2 -->_1 choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w)):1 4:S:sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y)) -->_2 sort#(nil()) -> c_7():7 -->_1 insert#(x,nil()) -> c_5():6 -->_2 sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y)):4 -->_1 insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v)):3 5:W:choose#(x,cons(v,w),y,0()) -> c_1() 6:W:insert#(x,nil()) -> c_5() 7:W:sort#(nil()) -> c_7() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 7: sort#(nil()) -> c_7() 6: insert#(x,nil()) -> c_5() 5: choose#(x,cons(v,w),y,0()) -> c_1() * Step 4: DecomposeDG WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w)) choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z)) insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v)) sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y)) - Weak TRS: choose(x,cons(v,w),y,0()) -> cons(x,cons(v,w)) choose(x,cons(v,w),0(),s(z)) -> cons(v,insert(x,w)) choose(x,cons(v,w),s(y),s(z)) -> choose(x,cons(v,w),y,z) insert(x,cons(v,w)) -> choose(x,cons(v,w),x,v) insert(x,nil()) -> cons(x,nil()) sort(cons(x,y)) -> insert(x,sort(y)) sort(nil()) -> nil() - Signature: {choose/4,insert/2,sort/1,choose#/4,insert#/2,sort#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0 ,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {choose#,insert#,sort#} and constructors {0,cons,nil,s} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y)) and a lower component choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w)) choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z)) insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v)) Further, following extension rules are added to the lower component. sort#(cons(x,y)) -> insert#(x,sort(y)) sort#(cons(x,y)) -> sort#(y) ** Step 4.a:1: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y)) - Weak TRS: choose(x,cons(v,w),y,0()) -> cons(x,cons(v,w)) choose(x,cons(v,w),0(),s(z)) -> cons(v,insert(x,w)) choose(x,cons(v,w),s(y),s(z)) -> choose(x,cons(v,w),y,z) insert(x,cons(v,w)) -> choose(x,cons(v,w),x,v) insert(x,nil()) -> cons(x,nil()) sort(cons(x,y)) -> insert(x,sort(y)) sort(nil()) -> nil() - Signature: {choose/4,insert/2,sort/1,choose#/4,insert#/2,sort#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0 ,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {choose#,insert#,sort#} and constructors {0,cons,nil,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y)) -->_2 sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: sort#(cons(x,y)) -> c_6(sort#(y)) ** Step 4.a:2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sort#(cons(x,y)) -> c_6(sort#(y)) - Weak TRS: choose(x,cons(v,w),y,0()) -> cons(x,cons(v,w)) choose(x,cons(v,w),0(),s(z)) -> cons(v,insert(x,w)) choose(x,cons(v,w),s(y),s(z)) -> choose(x,cons(v,w),y,z) insert(x,cons(v,w)) -> choose(x,cons(v,w),x,v) insert(x,nil()) -> cons(x,nil()) sort(cons(x,y)) -> insert(x,sort(y)) sort(nil()) -> nil() - Signature: {choose/4,insert/2,sort/1,choose#/4,insert#/2,sort#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0 ,c_6/1,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {choose#,insert#,sort#} and constructors {0,cons,nil,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: sort#(cons(x,y)) -> c_6(sort#(y)) ** Step 4.a:3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sort#(cons(x,y)) -> c_6(sort#(y)) - Signature: {choose/4,insert/2,sort/1,choose#/4,insert#/2,sort#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0 ,c_6/1,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {choose#,insert#,sort#} and constructors {0,cons,nil,s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(c_6) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(choose) = [0] p(cons) = [1] x1 + [1] x2 + [9] p(insert) = [0] p(nil) = [0] p(s) = [1] x1 + [0] p(sort) = [0] p(choose#) = [0] p(insert#) = [0] p(sort#) = [1] x1 + [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [1] x1 + [0] p(c_7) = [0] Following rules are strictly oriented: sort#(cons(x,y)) = [1] x + [1] y + [9] > [1] y + [0] = c_6(sort#(y)) Following rules are (at-least) weakly oriented: Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 4.a:4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: sort#(cons(x,y)) -> c_6(sort#(y)) - Signature: {choose/4,insert/2,sort/1,choose#/4,insert#/2,sort#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0 ,c_6/1,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {choose#,insert#,sort#} and constructors {0,cons,nil,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 4.b:1: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w)) choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z)) insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v)) - Weak DPs: sort#(cons(x,y)) -> insert#(x,sort(y)) sort#(cons(x,y)) -> sort#(y) - Weak TRS: choose(x,cons(v,w),y,0()) -> cons(x,cons(v,w)) choose(x,cons(v,w),0(),s(z)) -> cons(v,insert(x,w)) choose(x,cons(v,w),s(y),s(z)) -> choose(x,cons(v,w),y,z) insert(x,cons(v,w)) -> choose(x,cons(v,w),x,v) insert(x,nil()) -> cons(x,nil()) sort(cons(x,y)) -> insert(x,sort(y)) sort(nil()) -> nil() - Signature: {choose/4,insert/2,sort/1,choose#/4,insert#/2,sort#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0 ,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {choose#,insert#,sort#} and constructors {0,cons,nil,s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant 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(insert) = {2}, uargs(insert#) = {2}, uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(choose) = [1] x2 + [1] p(cons) = [1] x2 + [1] p(insert) = [1] x2 + [1] p(nil) = [0] p(s) = [1] x1 + [0] p(sort) = [1] x1 + [0] p(choose#) = [1] x2 + [3] p(insert#) = [1] x2 + [4] p(sort#) = [1] x1 + [4] p(c_1) = [0] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] Following rules are strictly oriented: insert#(x,cons(v,w)) = [1] w + [5] > [1] w + [4] = c_4(choose#(x,cons(v,w),x,v)) Following rules are (at-least) weakly oriented: choose#(x,cons(v,w),0(),s(z)) = [1] w + [4] >= [1] w + [4] = c_2(insert#(x,w)) choose#(x,cons(v,w),s(y),s(z)) = [1] w + [4] >= [1] w + [4] = c_3(choose#(x,cons(v,w),y,z)) sort#(cons(x,y)) = [1] y + [5] >= [1] y + [4] = insert#(x,sort(y)) sort#(cons(x,y)) = [1] y + [5] >= [1] y + [4] = sort#(y) choose(x,cons(v,w),y,0()) = [1] w + [2] >= [1] w + [2] = cons(x,cons(v,w)) choose(x,cons(v,w),0(),s(z)) = [1] w + [2] >= [1] w + [2] = cons(v,insert(x,w)) choose(x,cons(v,w),s(y),s(z)) = [1] w + [2] >= [1] w + [2] = choose(x,cons(v,w),y,z) insert(x,cons(v,w)) = [1] w + [2] >= [1] w + [2] = choose(x,cons(v,w),x,v) insert(x,nil()) = [1] >= [1] = cons(x,nil()) sort(cons(x,y)) = [1] y + [1] >= [1] y + [1] = insert(x,sort(y)) sort(nil()) = [0] >= [0] = nil() Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 4.b:2: WeightGap WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w)) choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z)) - Weak DPs: insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v)) sort#(cons(x,y)) -> insert#(x,sort(y)) sort#(cons(x,y)) -> sort#(y) - Weak TRS: choose(x,cons(v,w),y,0()) -> cons(x,cons(v,w)) choose(x,cons(v,w),0(),s(z)) -> cons(v,insert(x,w)) choose(x,cons(v,w),s(y),s(z)) -> choose(x,cons(v,w),y,z) insert(x,cons(v,w)) -> choose(x,cons(v,w),x,v) insert(x,nil()) -> cons(x,nil()) sort(cons(x,y)) -> insert(x,sort(y)) sort(nil()) -> nil() - Signature: {choose/4,insert/2,sort/1,choose#/4,insert#/2,sort#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0 ,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {choose#,insert#,sort#} and constructors {0,cons,nil,s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant 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(insert) = {2}, uargs(insert#) = {2}, uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [1] p(choose) = [1] x2 + [4] p(cons) = [1] x2 + [4] p(insert) = [1] x2 + [4] p(nil) = [0] p(s) = [4] p(sort) = [1] x1 + [4] p(choose#) = [1] x2 + [6] p(insert#) = [1] x2 + [6] p(sort#) = [1] x1 + [7] p(c_1) = [1] p(c_2) = [1] x1 + [3] p(c_3) = [1] x1 + [1] p(c_4) = [1] x1 + [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [1] Following rules are strictly oriented: choose#(x,cons(v,w),0(),s(z)) = [1] w + [10] > [1] w + [9] = c_2(insert#(x,w)) Following rules are (at-least) weakly oriented: choose#(x,cons(v,w),s(y),s(z)) = [1] w + [10] >= [1] w + [11] = c_3(choose#(x,cons(v,w),y,z)) insert#(x,cons(v,w)) = [1] w + [10] >= [1] w + [10] = c_4(choose#(x,cons(v,w),x,v)) sort#(cons(x,y)) = [1] y + [11] >= [1] y + [10] = insert#(x,sort(y)) sort#(cons(x,y)) = [1] y + [11] >= [1] y + [7] = sort#(y) choose(x,cons(v,w),y,0()) = [1] w + [8] >= [1] w + [8] = cons(x,cons(v,w)) choose(x,cons(v,w),0(),s(z)) = [1] w + [8] >= [1] w + [8] = cons(v,insert(x,w)) choose(x,cons(v,w),s(y),s(z)) = [1] w + [8] >= [1] w + [8] = choose(x,cons(v,w),y,z) insert(x,cons(v,w)) = [1] w + [8] >= [1] w + [8] = choose(x,cons(v,w),x,v) insert(x,nil()) = [4] >= [4] = cons(x,nil()) sort(cons(x,y)) = [1] y + [8] >= [1] y + [8] = insert(x,sort(y)) sort(nil()) = [4] >= [0] = nil() Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 4.b:3: Ara WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z)) - Weak DPs: choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w)) insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v)) sort#(cons(x,y)) -> insert#(x,sort(y)) sort#(cons(x,y)) -> sort#(y) - Weak TRS: choose(x,cons(v,w),y,0()) -> cons(x,cons(v,w)) choose(x,cons(v,w),0(),s(z)) -> cons(v,insert(x,w)) choose(x,cons(v,w),s(y),s(z)) -> choose(x,cons(v,w),y,z) insert(x,cons(v,w)) -> choose(x,cons(v,w),x,v) insert(x,nil()) -> cons(x,nil()) sort(cons(x,y)) -> insert(x,sort(y)) sort(nil()) -> nil() - Signature: {choose/4,insert/2,sort/1,choose#/4,insert#/2,sort#/1} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/0 ,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {choose#,insert#,sort#} and constructors {0,cons,nil,s} + Applied Processor: Ara {araHeuristics = NoHeuristics, minDegree = 2, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1} + Details: Signatures used: ---------------- 0 :: [] -(0)-> "A"(0, 0) choose :: ["A"(15, 0) x "A"(15, 0) x "A"(0, 0) x "A"(0, 0)] -(15)-> "A"(15, 0) cons :: ["A"(1, 0) x "A"(15, 0)] -(1)-> "A"(1, 14) cons :: ["A"(15, 0) x "A"(15, 0)] -(15)-> "A"(15, 0) insert :: ["A"(15, 0) x "A"(15, 0)] -(15)-> "A"(15, 0) nil :: [] -(0)-> "A"(15, 0) nil :: [] -(0)-> "A"(15, 12) nil :: [] -(0)-> "A"(15, 6) s :: ["A"(0, 0)] -(0)-> "A"(0, 0) s :: ["A"(14, 0)] -(14)-> "A"(14, 0) sort :: ["A"(15, 0)] -(0)-> "A"(15, 0) choose# :: ["A"(4, 0) x "A"(1, 14) x "A"(0, 0) x "A"(14, 0)] -(1)-> "A"(1, 0) insert# :: ["A"(4, 0) x "A"(15, 0)] -(2)-> "A"(14, 1) sort# :: ["A"(15, 0)] -(10)-> "A"(2, 1) c_2 :: ["A"(14, 0)] -(0)-> "A"(4, 10) c_3 :: ["A"(0, 0)] -(0)-> "A"(3, 10) c_4 :: ["A"(1, 0)] -(0)-> "A"(14, 1) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "0_A" :: [] -(0)-> "A"(1, 0) "0_A" :: [] -(0)-> "A"(0, 1) "c_2_A" :: ["A"(0)] -(0)-> "A"(1, 0) "c_2_A" :: ["A"(0)] -(0)-> "A"(0, 1) "c_3_A" :: ["A"(0)] -(0)-> "A"(1, 0) "c_3_A" :: ["A"(0)] -(0)-> "A"(0, 1) "c_4_A" :: ["A"(0)] -(0)-> "A"(1, 0) "c_4_A" :: ["A"(0)] -(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, 0)] -(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"(1, 0)] -(0)-> "A"(0, 1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z)) 2. Weak: WORST_CASE(?,O(n^3))