WORST_CASE(?,O(n^2)) * Step 1: DependencyPairs WORST_CASE(?,O(n^2)) + 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^2)) + 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^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)) 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: Decompose 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)) 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: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - 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)) -> 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} Problem (S) - Strict DPs: sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y)) - Weak 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 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} ** Step 4.a:1: PredecessorEstimationCP 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)) -> 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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w)) The strictly oriented rules are moved into the weak component. *** Step 4.a:1.a:1: NaturalPI 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)) -> 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: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_6) = {1,2} Following symbols are considered usable: {choose,insert,sort,choose#,insert#,sort#} TcT has computed the following interpretation: p(0) = 0 p(choose) = 2 + x2 p(cons) = 1 + x2 p(insert) = 2 + x2 p(nil) = 0 p(s) = 0 p(sort) = 2*x1 p(choose#) = 3*x2 p(insert#) = 3*x2 p(sort#) = 3*x1^2 p(c_1) = 2 p(c_2) = x1 p(c_3) = x1 p(c_4) = x1 p(c_5) = 0 p(c_6) = x1 + x2 p(c_7) = 0 Following rules are strictly oriented: choose#(x,cons(v,w),0(),s(z)) = 3 + 3*w > 3*w = c_2(insert#(x,w)) Following rules are (at-least) weakly oriented: choose#(x,cons(v,w),s(y),s(z)) = 3 + 3*w >= 3 + 3*w = c_3(choose#(x,cons(v,w),y,z)) insert#(x,cons(v,w)) = 3 + 3*w >= 3 + 3*w = c_4(choose#(x,cons(v,w),x,v)) sort#(cons(x,y)) = 3 + 6*y + 3*y^2 >= 6*y + 3*y^2 = c_6(insert#(x,sort(y)),sort#(y)) choose(x,cons(v,w),y,0()) = 3 + w >= 2 + w = cons(x,cons(v,w)) choose(x,cons(v,w),0(),s(z)) = 3 + w >= 3 + w = cons(v,insert(x,w)) choose(x,cons(v,w),s(y),s(z)) = 3 + w >= 3 + w = choose(x,cons(v,w),y,z) insert(x,cons(v,w)) = 3 + w >= 3 + w = choose(x,cons(v,w),x,v) insert(x,nil()) = 2 >= 1 = cons(x,nil()) sort(cons(x,y)) = 2 + 2*y >= 2 + 2*y = insert(x,sort(y)) sort(nil()) = 0 >= 0 = nil() *** Step 4.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: 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: choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w)) 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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 4.a:1.b:1: PredecessorEstimationCP 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)) insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v)) - Weak DPs: choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w)) 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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 2: insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v)) The strictly oriented rules are moved into the weak component. **** Step 4.a:1.b:1.a:1: NaturalPI 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)) insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v)) - Weak DPs: choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w)) 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: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_6) = {1,2} Following symbols are considered usable: {choose,insert,sort,choose#,insert#,sort#} TcT has computed the following interpretation: p(0) = 0 p(choose) = 1 + x2 p(cons) = 1 + x2 p(insert) = 1 + x2 p(nil) = 0 p(s) = x1 p(sort) = x1 p(choose#) = 2*x2 p(insert#) = 1 + 2*x2 p(sort#) = x1^2 p(c_1) = 0 p(c_2) = x1 p(c_3) = x1 p(c_4) = x1 p(c_5) = 0 p(c_6) = x1 + x2 p(c_7) = 0 Following rules are strictly oriented: insert#(x,cons(v,w)) = 3 + 2*w > 2 + 2*w = c_4(choose#(x,cons(v,w),x,v)) Following rules are (at-least) weakly oriented: choose#(x,cons(v,w),0(),s(z)) = 2 + 2*w >= 1 + 2*w = c_2(insert#(x,w)) choose#(x,cons(v,w),s(y),s(z)) = 2 + 2*w >= 2 + 2*w = c_3(choose#(x,cons(v,w),y,z)) sort#(cons(x,y)) = 1 + 2*y + y^2 >= 1 + 2*y + y^2 = c_6(insert#(x,sort(y)),sort#(y)) choose(x,cons(v,w),y,0()) = 2 + w >= 2 + w = cons(x,cons(v,w)) choose(x,cons(v,w),0(),s(z)) = 2 + w >= 2 + w = cons(v,insert(x,w)) choose(x,cons(v,w),s(y),s(z)) = 2 + w >= 2 + w = choose(x,cons(v,w),y,z) insert(x,cons(v,w)) = 2 + w >= 2 + w = choose(x,cons(v,w),x,v) insert(x,nil()) = 1 >= 1 = cons(x,nil()) sort(cons(x,y)) = 1 + y >= 1 + y = insert(x,sort(y)) sort(nil()) = 0 >= 0 = nil() **** Step 4.a:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + 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)) -> 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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 4.a:1.b:1.b:1: PredecessorEstimationCP 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)) -> 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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z)) The strictly oriented rules are moved into the weak component. ***** Step 4.a:1.b:1.b:1.a:1: NaturalPI 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)) -> 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: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_6) = {1,2} Following symbols are considered usable: {choose,insert,sort,choose#,insert#,sort#} TcT has computed the following interpretation: p(0) = 2 p(choose) = 2 + 2*x1 + x2 p(cons) = 1 + x1 + x2 p(insert) = 2 + 2*x1 + x2 p(nil) = 0 p(s) = 1 + x1 p(sort) = 1 + 2*x1 p(choose#) = x1 + x1*x2 + x3 p(insert#) = 2*x1 + x1*x2 p(sort#) = 2*x1^2 p(c_1) = 0 p(c_2) = 1 + x1 p(c_3) = x1 p(c_4) = x1 p(c_5) = 0 p(c_6) = 1 + x1 + x2 p(c_7) = 2 Following rules are strictly oriented: choose#(x,cons(v,w),s(y),s(z)) = 1 + v*x + w*x + 2*x + y > v*x + w*x + 2*x + y = c_3(choose#(x,cons(v,w),y,z)) Following rules are (at-least) weakly oriented: choose#(x,cons(v,w),0(),s(z)) = 2 + v*x + w*x + 2*x >= 1 + w*x + 2*x = c_2(insert#(x,w)) insert#(x,cons(v,w)) = v*x + w*x + 3*x >= v*x + w*x + 3*x = c_4(choose#(x,cons(v,w),x,v)) sort#(cons(x,y)) = 2 + 4*x + 4*x*y + 2*x^2 + 4*y + 2*y^2 >= 1 + 3*x + 2*x*y + 2*y^2 = c_6(insert#(x,sort(y)),sort#(y)) choose(x,cons(v,w),y,0()) = 3 + v + w + 2*x >= 2 + v + w + x = cons(x,cons(v,w)) choose(x,cons(v,w),0(),s(z)) = 3 + v + w + 2*x >= 3 + v + w + 2*x = cons(v,insert(x,w)) choose(x,cons(v,w),s(y),s(z)) = 3 + v + w + 2*x >= 3 + v + w + 2*x = choose(x,cons(v,w),y,z) insert(x,cons(v,w)) = 3 + v + w + 2*x >= 3 + v + w + 2*x = choose(x,cons(v,w),x,v) insert(x,nil()) = 2 + 2*x >= 1 + x = cons(x,nil()) sort(cons(x,y)) = 3 + 2*x + 2*y >= 3 + 2*x + 2*y = insert(x,sort(y)) sort(nil()) = 1 >= 0 = nil() ***** Step 4.a:1.b:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak 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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 4.a:1.b:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak 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: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W: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 2:W:choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z)) -->_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:W:insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v)) -->_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:W: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)):4 -->_1 insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v)):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y)) 1: choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w)) 3: insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v)) 2: choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z)) ***** Step 4.a:1.b:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y)) - Weak 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 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:sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y)) -->_1 insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v)):4 -->_2 sort#(cons(x,y)) -> c_6(insert#(x,sort(y)),sort#(y)):1 2:W: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)):4 3:W:choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z)) -->_1 choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z)):3 -->_1 choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w)):2 4:W:insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v)) -->_1 choose#(x,cons(v,w),s(y),s(z)) -> c_3(choose#(x,cons(v,w),y,z)):3 -->_1 choose#(x,cons(v,w),0(),s(z)) -> c_2(insert#(x,w)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: insert#(x,cons(v,w)) -> c_4(choose#(x,cons(v,w),x,v)) 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)) ** Step 4.b:2: 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.b:3: 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.b:4: PredecessorEstimationCP 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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: sort#(cons(x,y)) -> c_6(sort#(y)) The strictly oriented rules are moved into the weak component. *** Step 4.b:4.a:1: NaturalMI 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: NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: 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: {choose#,insert#,sort#} TcT has computed the following interpretation: p(0) = [1] p(choose) = [2] x2 + [0] p(cons) = [1] x1 + [1] x2 + [8] p(insert) = [1] p(nil) = [1] p(s) = [4] p(sort) = [2] x1 + [1] p(choose#) = [1] x1 + [4] x3 + [1] p(insert#) = [1] x2 + [0] p(sort#) = [2] x1 + [1] p(c_1) = [1] p(c_2) = [1] p(c_3) = [1] x1 + [0] p(c_4) = [2] p(c_5) = [0] p(c_6) = [1] x1 + [9] p(c_7) = [0] Following rules are strictly oriented: sort#(cons(x,y)) = [2] x + [2] y + [17] > [2] y + [10] = c_6(sort#(y)) Following rules are (at-least) weakly oriented: *** Step 4.b:4.a:2: Assumption 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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 4.b:4.b:1: RemoveWeakSuffixes 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: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:sort#(cons(x,y)) -> c_6(sort#(y)) -->_1 sort#(cons(x,y)) -> c_6(sort#(y)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: sort#(cons(x,y)) -> c_6(sort#(y)) *** Step 4.b:4.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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). WORST_CASE(?,O(n^2))