WORST_CASE(?,O(n^2)) * Step 1: DependencyPairs WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: checkF(queue(cons(x,xs),r)) -> queue(cons(x,xs),r) checkF(queue(nil(),r)) -> queue(rev(r),nil()) empty() -> queue(nil(),nil()) enq(0()) -> empty() enq(s(n)) -> snoc(enq(n),n) head(queue(cons(x,f),r)) -> x head(queue(nil(),r)) -> errorHead() rev(xs) -> rev'(xs,nil()) rev'(cons(x,xs),ys) -> rev'(xs,cons(x,ys)) rev'(nil(),ys) -> ys snoc(queue(f,r),x) -> checkF(queue(f,cons(x,r))) tail(queue(cons(x,f),r)) -> checkF(queue(f,r)) tail(queue(nil(),r)) -> errorTail() - Signature: {checkF/1,empty/0,enq/1,head/1,rev/1,rev'/2,snoc/2,tail/1} / {0/0,cons/2,errorHead/0,errorTail/0,nil/0 ,queue/2,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {checkF,empty,enq,head,rev,rev',snoc ,tail} and constructors {0,cons,errorHead,errorTail,nil,queue,s} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs checkF#(queue(cons(x,xs),r)) -> c_1() checkF#(queue(nil(),r)) -> c_2(rev#(r)) empty#() -> c_3() enq#(0()) -> c_4(empty#()) enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)) head#(queue(cons(x,f),r)) -> c_6() head#(queue(nil(),r)) -> c_7() rev#(xs) -> c_8(rev'#(xs,nil())) rev'#(cons(x,xs),ys) -> c_9(rev'#(xs,cons(x,ys))) rev'#(nil(),ys) -> c_10() snoc#(queue(f,r),x) -> c_11(checkF#(queue(f,cons(x,r)))) tail#(queue(cons(x,f),r)) -> c_12(checkF#(queue(f,r))) tail#(queue(nil(),r)) -> c_13() Weak DPs and mark the set of starting terms. * Step 2: UsableRules WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: checkF#(queue(cons(x,xs),r)) -> c_1() checkF#(queue(nil(),r)) -> c_2(rev#(r)) empty#() -> c_3() enq#(0()) -> c_4(empty#()) enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)) head#(queue(cons(x,f),r)) -> c_6() head#(queue(nil(),r)) -> c_7() rev#(xs) -> c_8(rev'#(xs,nil())) rev'#(cons(x,xs),ys) -> c_9(rev'#(xs,cons(x,ys))) rev'#(nil(),ys) -> c_10() snoc#(queue(f,r),x) -> c_11(checkF#(queue(f,cons(x,r)))) tail#(queue(cons(x,f),r)) -> c_12(checkF#(queue(f,r))) tail#(queue(nil(),r)) -> c_13() - Weak TRS: checkF(queue(cons(x,xs),r)) -> queue(cons(x,xs),r) checkF(queue(nil(),r)) -> queue(rev(r),nil()) empty() -> queue(nil(),nil()) enq(0()) -> empty() enq(s(n)) -> snoc(enq(n),n) head(queue(cons(x,f),r)) -> x head(queue(nil(),r)) -> errorHead() rev(xs) -> rev'(xs,nil()) rev'(cons(x,xs),ys) -> rev'(xs,cons(x,ys)) rev'(nil(),ys) -> ys snoc(queue(f,r),x) -> checkF(queue(f,cons(x,r))) tail(queue(cons(x,f),r)) -> checkF(queue(f,r)) tail(queue(nil(),r)) -> errorTail() - Signature: {checkF/1,empty/0,enq/1,head/1,rev/1,rev'/2,snoc/2,tail/1,checkF#/1,empty#/0,enq#/1,head#/1,rev#/1,rev'#/2 ,snoc#/2,tail#/1} / {0/0,cons/2,errorHead/0,errorTail/0,nil/0,queue/2,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2 ,c_6/0,c_7/0,c_8/1,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {checkF#,empty#,enq#,head#,rev#,rev'#,snoc# ,tail#} and constructors {0,cons,errorHead,errorTail,nil,queue,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: checkF(queue(cons(x,xs),r)) -> queue(cons(x,xs),r) checkF(queue(nil(),r)) -> queue(rev(r),nil()) empty() -> queue(nil(),nil()) enq(0()) -> empty() enq(s(n)) -> snoc(enq(n),n) rev(xs) -> rev'(xs,nil()) rev'(cons(x,xs),ys) -> rev'(xs,cons(x,ys)) rev'(nil(),ys) -> ys snoc(queue(f,r),x) -> checkF(queue(f,cons(x,r))) checkF#(queue(cons(x,xs),r)) -> c_1() checkF#(queue(nil(),r)) -> c_2(rev#(r)) empty#() -> c_3() enq#(0()) -> c_4(empty#()) enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)) head#(queue(cons(x,f),r)) -> c_6() head#(queue(nil(),r)) -> c_7() rev#(xs) -> c_8(rev'#(xs,nil())) rev'#(cons(x,xs),ys) -> c_9(rev'#(xs,cons(x,ys))) rev'#(nil(),ys) -> c_10() snoc#(queue(f,r),x) -> c_11(checkF#(queue(f,cons(x,r)))) tail#(queue(cons(x,f),r)) -> c_12(checkF#(queue(f,r))) tail#(queue(nil(),r)) -> c_13() * Step 3: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: checkF#(queue(cons(x,xs),r)) -> c_1() checkF#(queue(nil(),r)) -> c_2(rev#(r)) empty#() -> c_3() enq#(0()) -> c_4(empty#()) enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)) head#(queue(cons(x,f),r)) -> c_6() head#(queue(nil(),r)) -> c_7() rev#(xs) -> c_8(rev'#(xs,nil())) rev'#(cons(x,xs),ys) -> c_9(rev'#(xs,cons(x,ys))) rev'#(nil(),ys) -> c_10() snoc#(queue(f,r),x) -> c_11(checkF#(queue(f,cons(x,r)))) tail#(queue(cons(x,f),r)) -> c_12(checkF#(queue(f,r))) tail#(queue(nil(),r)) -> c_13() - Weak TRS: checkF(queue(cons(x,xs),r)) -> queue(cons(x,xs),r) checkF(queue(nil(),r)) -> queue(rev(r),nil()) empty() -> queue(nil(),nil()) enq(0()) -> empty() enq(s(n)) -> snoc(enq(n),n) rev(xs) -> rev'(xs,nil()) rev'(cons(x,xs),ys) -> rev'(xs,cons(x,ys)) rev'(nil(),ys) -> ys snoc(queue(f,r),x) -> checkF(queue(f,cons(x,r))) - Signature: {checkF/1,empty/0,enq/1,head/1,rev/1,rev'/2,snoc/2,tail/1,checkF#/1,empty#/0,enq#/1,head#/1,rev#/1,rev'#/2 ,snoc#/2,tail#/1} / {0/0,cons/2,errorHead/0,errorTail/0,nil/0,queue/2,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2 ,c_6/0,c_7/0,c_8/1,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {checkF#,empty#,enq#,head#,rev#,rev'#,snoc# ,tail#} and constructors {0,cons,errorHead,errorTail,nil,queue,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,3,6,7,10,13} by application of Pre({1,3,6,7,10,13}) = {4,8,9,11,12}. Here rules are labelled as follows: 1: checkF#(queue(cons(x,xs),r)) -> c_1() 2: checkF#(queue(nil(),r)) -> c_2(rev#(r)) 3: empty#() -> c_3() 4: enq#(0()) -> c_4(empty#()) 5: enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)) 6: head#(queue(cons(x,f),r)) -> c_6() 7: head#(queue(nil(),r)) -> c_7() 8: rev#(xs) -> c_8(rev'#(xs,nil())) 9: rev'#(cons(x,xs),ys) -> c_9(rev'#(xs,cons(x,ys))) 10: rev'#(nil(),ys) -> c_10() 11: snoc#(queue(f,r),x) -> c_11(checkF#(queue(f,cons(x,r)))) 12: tail#(queue(cons(x,f),r)) -> c_12(checkF#(queue(f,r))) 13: tail#(queue(nil(),r)) -> c_13() * Step 4: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: checkF#(queue(nil(),r)) -> c_2(rev#(r)) enq#(0()) -> c_4(empty#()) enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)) rev#(xs) -> c_8(rev'#(xs,nil())) rev'#(cons(x,xs),ys) -> c_9(rev'#(xs,cons(x,ys))) snoc#(queue(f,r),x) -> c_11(checkF#(queue(f,cons(x,r)))) tail#(queue(cons(x,f),r)) -> c_12(checkF#(queue(f,r))) - Weak DPs: checkF#(queue(cons(x,xs),r)) -> c_1() empty#() -> c_3() head#(queue(cons(x,f),r)) -> c_6() head#(queue(nil(),r)) -> c_7() rev'#(nil(),ys) -> c_10() tail#(queue(nil(),r)) -> c_13() - Weak TRS: checkF(queue(cons(x,xs),r)) -> queue(cons(x,xs),r) checkF(queue(nil(),r)) -> queue(rev(r),nil()) empty() -> queue(nil(),nil()) enq(0()) -> empty() enq(s(n)) -> snoc(enq(n),n) rev(xs) -> rev'(xs,nil()) rev'(cons(x,xs),ys) -> rev'(xs,cons(x,ys)) rev'(nil(),ys) -> ys snoc(queue(f,r),x) -> checkF(queue(f,cons(x,r))) - Signature: {checkF/1,empty/0,enq/1,head/1,rev/1,rev'/2,snoc/2,tail/1,checkF#/1,empty#/0,enq#/1,head#/1,rev#/1,rev'#/2 ,snoc#/2,tail#/1} / {0/0,cons/2,errorHead/0,errorTail/0,nil/0,queue/2,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2 ,c_6/0,c_7/0,c_8/1,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {checkF#,empty#,enq#,head#,rev#,rev'#,snoc# ,tail#} and constructors {0,cons,errorHead,errorTail,nil,queue,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2} by application of Pre({2}) = {3}. Here rules are labelled as follows: 1: checkF#(queue(nil(),r)) -> c_2(rev#(r)) 2: enq#(0()) -> c_4(empty#()) 3: enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)) 4: rev#(xs) -> c_8(rev'#(xs,nil())) 5: rev'#(cons(x,xs),ys) -> c_9(rev'#(xs,cons(x,ys))) 6: snoc#(queue(f,r),x) -> c_11(checkF#(queue(f,cons(x,r)))) 7: tail#(queue(cons(x,f),r)) -> c_12(checkF#(queue(f,r))) 8: checkF#(queue(cons(x,xs),r)) -> c_1() 9: empty#() -> c_3() 10: head#(queue(cons(x,f),r)) -> c_6() 11: head#(queue(nil(),r)) -> c_7() 12: rev'#(nil(),ys) -> c_10() 13: tail#(queue(nil(),r)) -> c_13() * Step 5: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: checkF#(queue(nil(),r)) -> c_2(rev#(r)) enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)) rev#(xs) -> c_8(rev'#(xs,nil())) rev'#(cons(x,xs),ys) -> c_9(rev'#(xs,cons(x,ys))) snoc#(queue(f,r),x) -> c_11(checkF#(queue(f,cons(x,r)))) tail#(queue(cons(x,f),r)) -> c_12(checkF#(queue(f,r))) - Weak DPs: checkF#(queue(cons(x,xs),r)) -> c_1() empty#() -> c_3() enq#(0()) -> c_4(empty#()) head#(queue(cons(x,f),r)) -> c_6() head#(queue(nil(),r)) -> c_7() rev'#(nil(),ys) -> c_10() tail#(queue(nil(),r)) -> c_13() - Weak TRS: checkF(queue(cons(x,xs),r)) -> queue(cons(x,xs),r) checkF(queue(nil(),r)) -> queue(rev(r),nil()) empty() -> queue(nil(),nil()) enq(0()) -> empty() enq(s(n)) -> snoc(enq(n),n) rev(xs) -> rev'(xs,nil()) rev'(cons(x,xs),ys) -> rev'(xs,cons(x,ys)) rev'(nil(),ys) -> ys snoc(queue(f,r),x) -> checkF(queue(f,cons(x,r))) - Signature: {checkF/1,empty/0,enq/1,head/1,rev/1,rev'/2,snoc/2,tail/1,checkF#/1,empty#/0,enq#/1,head#/1,rev#/1,rev'#/2 ,snoc#/2,tail#/1} / {0/0,cons/2,errorHead/0,errorTail/0,nil/0,queue/2,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2 ,c_6/0,c_7/0,c_8/1,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {checkF#,empty#,enq#,head#,rev#,rev'#,snoc# ,tail#} and constructors {0,cons,errorHead,errorTail,nil,queue,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:checkF#(queue(nil(),r)) -> c_2(rev#(r)) -->_1 rev#(xs) -> c_8(rev'#(xs,nil())):3 2:S:enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)) -->_2 enq#(0()) -> c_4(empty#()):9 -->_1 snoc#(queue(f,r),x) -> c_11(checkF#(queue(f,cons(x,r)))):5 -->_2 enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)):2 3:S:rev#(xs) -> c_8(rev'#(xs,nil())) -->_1 rev'#(cons(x,xs),ys) -> c_9(rev'#(xs,cons(x,ys))):4 -->_1 rev'#(nil(),ys) -> c_10():12 4:S:rev'#(cons(x,xs),ys) -> c_9(rev'#(xs,cons(x,ys))) -->_1 rev'#(nil(),ys) -> c_10():12 -->_1 rev'#(cons(x,xs),ys) -> c_9(rev'#(xs,cons(x,ys))):4 5:S:snoc#(queue(f,r),x) -> c_11(checkF#(queue(f,cons(x,r)))) -->_1 checkF#(queue(cons(x,xs),r)) -> c_1():7 -->_1 checkF#(queue(nil(),r)) -> c_2(rev#(r)):1 6:S:tail#(queue(cons(x,f),r)) -> c_12(checkF#(queue(f,r))) -->_1 checkF#(queue(cons(x,xs),r)) -> c_1():7 -->_1 checkF#(queue(nil(),r)) -> c_2(rev#(r)):1 7:W:checkF#(queue(cons(x,xs),r)) -> c_1() 8:W:empty#() -> c_3() 9:W:enq#(0()) -> c_4(empty#()) -->_1 empty#() -> c_3():8 10:W:head#(queue(cons(x,f),r)) -> c_6() 11:W:head#(queue(nil(),r)) -> c_7() 12:W:rev'#(nil(),ys) -> c_10() 13:W:tail#(queue(nil(),r)) -> c_13() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 13: tail#(queue(nil(),r)) -> c_13() 11: head#(queue(nil(),r)) -> c_7() 10: head#(queue(cons(x,f),r)) -> c_6() 7: checkF#(queue(cons(x,xs),r)) -> c_1() 9: enq#(0()) -> c_4(empty#()) 8: empty#() -> c_3() 12: rev'#(nil(),ys) -> c_10() * Step 6: RemoveHeads WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: checkF#(queue(nil(),r)) -> c_2(rev#(r)) enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)) rev#(xs) -> c_8(rev'#(xs,nil())) rev'#(cons(x,xs),ys) -> c_9(rev'#(xs,cons(x,ys))) snoc#(queue(f,r),x) -> c_11(checkF#(queue(f,cons(x,r)))) tail#(queue(cons(x,f),r)) -> c_12(checkF#(queue(f,r))) - Weak TRS: checkF(queue(cons(x,xs),r)) -> queue(cons(x,xs),r) checkF(queue(nil(),r)) -> queue(rev(r),nil()) empty() -> queue(nil(),nil()) enq(0()) -> empty() enq(s(n)) -> snoc(enq(n),n) rev(xs) -> rev'(xs,nil()) rev'(cons(x,xs),ys) -> rev'(xs,cons(x,ys)) rev'(nil(),ys) -> ys snoc(queue(f,r),x) -> checkF(queue(f,cons(x,r))) - Signature: {checkF/1,empty/0,enq/1,head/1,rev/1,rev'/2,snoc/2,tail/1,checkF#/1,empty#/0,enq#/1,head#/1,rev#/1,rev'#/2 ,snoc#/2,tail#/1} / {0/0,cons/2,errorHead/0,errorTail/0,nil/0,queue/2,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2 ,c_6/0,c_7/0,c_8/1,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {checkF#,empty#,enq#,head#,rev#,rev'#,snoc# ,tail#} and constructors {0,cons,errorHead,errorTail,nil,queue,s} + Applied Processor: RemoveHeads + Details: Consider the dependency graph 1:S:checkF#(queue(nil(),r)) -> c_2(rev#(r)) -->_1 rev#(xs) -> c_8(rev'#(xs,nil())):3 2:S:enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)) -->_1 snoc#(queue(f,r),x) -> c_11(checkF#(queue(f,cons(x,r)))):5 -->_2 enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)):2 3:S:rev#(xs) -> c_8(rev'#(xs,nil())) -->_1 rev'#(cons(x,xs),ys) -> c_9(rev'#(xs,cons(x,ys))):4 4:S:rev'#(cons(x,xs),ys) -> c_9(rev'#(xs,cons(x,ys))) -->_1 rev'#(cons(x,xs),ys) -> c_9(rev'#(xs,cons(x,ys))):4 5:S:snoc#(queue(f,r),x) -> c_11(checkF#(queue(f,cons(x,r)))) -->_1 checkF#(queue(nil(),r)) -> c_2(rev#(r)):1 6:S:tail#(queue(cons(x,f),r)) -> c_12(checkF#(queue(f,r))) -->_1 checkF#(queue(nil(),r)) -> c_2(rev#(r)):1 Following roots of the dependency graph are removed, as the considered set of starting terms is closed under reduction with respect to these rules (modulo compound contexts). [(6,tail#(queue(cons(x,f),r)) -> c_12(checkF#(queue(f,r))))] * Step 7: Decompose WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: checkF#(queue(nil(),r)) -> c_2(rev#(r)) enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)) rev#(xs) -> c_8(rev'#(xs,nil())) rev'#(cons(x,xs),ys) -> c_9(rev'#(xs,cons(x,ys))) snoc#(queue(f,r),x) -> c_11(checkF#(queue(f,cons(x,r)))) - Weak TRS: checkF(queue(cons(x,xs),r)) -> queue(cons(x,xs),r) checkF(queue(nil(),r)) -> queue(rev(r),nil()) empty() -> queue(nil(),nil()) enq(0()) -> empty() enq(s(n)) -> snoc(enq(n),n) rev(xs) -> rev'(xs,nil()) rev'(cons(x,xs),ys) -> rev'(xs,cons(x,ys)) rev'(nil(),ys) -> ys snoc(queue(f,r),x) -> checkF(queue(f,cons(x,r))) - Signature: {checkF/1,empty/0,enq/1,head/1,rev/1,rev'/2,snoc/2,tail/1,checkF#/1,empty#/0,enq#/1,head#/1,rev#/1,rev'#/2 ,snoc#/2,tail#/1} / {0/0,cons/2,errorHead/0,errorTail/0,nil/0,queue/2,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2 ,c_6/0,c_7/0,c_8/1,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {checkF#,empty#,enq#,head#,rev#,rev'#,snoc# ,tail#} and constructors {0,cons,errorHead,errorTail,nil,queue,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: checkF#(queue(nil(),r)) -> c_2(rev#(r)) rev#(xs) -> c_8(rev'#(xs,nil())) rev'#(cons(x,xs),ys) -> c_9(rev'#(xs,cons(x,ys))) - Weak DPs: enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)) snoc#(queue(f,r),x) -> c_11(checkF#(queue(f,cons(x,r)))) - Weak TRS: checkF(queue(cons(x,xs),r)) -> queue(cons(x,xs),r) checkF(queue(nil(),r)) -> queue(rev(r),nil()) empty() -> queue(nil(),nil()) enq(0()) -> empty() enq(s(n)) -> snoc(enq(n),n) rev(xs) -> rev'(xs,nil()) rev'(cons(x,xs),ys) -> rev'(xs,cons(x,ys)) rev'(nil(),ys) -> ys snoc(queue(f,r),x) -> checkF(queue(f,cons(x,r))) - Signature: {checkF/1,empty/0,enq/1,head/1,rev/1,rev'/2,snoc/2,tail/1,checkF#/1,empty#/0,enq#/1,head#/1,rev#/1,rev'#/2 ,snoc#/2,tail#/1} / {0/0,cons/2,errorHead/0,errorTail/0,nil/0,queue/2,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2 ,c_6/0,c_7/0,c_8/1,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {checkF#,empty#,enq#,head#,rev#,rev'#,snoc# ,tail#} and constructors {0,cons,errorHead,errorTail,nil,queue,s} Problem (S) - Strict DPs: enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)) snoc#(queue(f,r),x) -> c_11(checkF#(queue(f,cons(x,r)))) - Weak DPs: checkF#(queue(nil(),r)) -> c_2(rev#(r)) rev#(xs) -> c_8(rev'#(xs,nil())) rev'#(cons(x,xs),ys) -> c_9(rev'#(xs,cons(x,ys))) - Weak TRS: checkF(queue(cons(x,xs),r)) -> queue(cons(x,xs),r) checkF(queue(nil(),r)) -> queue(rev(r),nil()) empty() -> queue(nil(),nil()) enq(0()) -> empty() enq(s(n)) -> snoc(enq(n),n) rev(xs) -> rev'(xs,nil()) rev'(cons(x,xs),ys) -> rev'(xs,cons(x,ys)) rev'(nil(),ys) -> ys snoc(queue(f,r),x) -> checkF(queue(f,cons(x,r))) - Signature: {checkF/1,empty/0,enq/1,head/1,rev/1,rev'/2,snoc/2,tail/1,checkF#/1,empty#/0,enq#/1,head#/1,rev#/1,rev'#/2 ,snoc#/2,tail#/1} / {0/0,cons/2,errorHead/0,errorTail/0,nil/0,queue/2,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2 ,c_6/0,c_7/0,c_8/1,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {checkF#,empty#,enq#,head#,rev#,rev'#,snoc# ,tail#} and constructors {0,cons,errorHead,errorTail,nil,queue,s} ** Step 7.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: checkF#(queue(nil(),r)) -> c_2(rev#(r)) rev#(xs) -> c_8(rev'#(xs,nil())) rev'#(cons(x,xs),ys) -> c_9(rev'#(xs,cons(x,ys))) - Weak DPs: enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)) snoc#(queue(f,r),x) -> c_11(checkF#(queue(f,cons(x,r)))) - Weak TRS: checkF(queue(cons(x,xs),r)) -> queue(cons(x,xs),r) checkF(queue(nil(),r)) -> queue(rev(r),nil()) empty() -> queue(nil(),nil()) enq(0()) -> empty() enq(s(n)) -> snoc(enq(n),n) rev(xs) -> rev'(xs,nil()) rev'(cons(x,xs),ys) -> rev'(xs,cons(x,ys)) rev'(nil(),ys) -> ys snoc(queue(f,r),x) -> checkF(queue(f,cons(x,r))) - Signature: {checkF/1,empty/0,enq/1,head/1,rev/1,rev'/2,snoc/2,tail/1,checkF#/1,empty#/0,enq#/1,head#/1,rev#/1,rev'#/2 ,snoc#/2,tail#/1} / {0/0,cons/2,errorHead/0,errorTail/0,nil/0,queue/2,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2 ,c_6/0,c_7/0,c_8/1,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {checkF#,empty#,enq#,head#,rev#,rev'#,snoc# ,tail#} and constructors {0,cons,errorHead,errorTail,nil,queue,s} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 4: rev'#(cons(x,xs),ys) -> c_9(rev'#(xs,cons(x,ys))) The strictly oriented rules are moved into the weak component. *** Step 7.a:1.a:1: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: checkF#(queue(nil(),r)) -> c_2(rev#(r)) rev#(xs) -> c_8(rev'#(xs,nil())) rev'#(cons(x,xs),ys) -> c_9(rev'#(xs,cons(x,ys))) - Weak DPs: enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)) snoc#(queue(f,r),x) -> c_11(checkF#(queue(f,cons(x,r)))) - Weak TRS: checkF(queue(cons(x,xs),r)) -> queue(cons(x,xs),r) checkF(queue(nil(),r)) -> queue(rev(r),nil()) empty() -> queue(nil(),nil()) enq(0()) -> empty() enq(s(n)) -> snoc(enq(n),n) rev(xs) -> rev'(xs,nil()) rev'(cons(x,xs),ys) -> rev'(xs,cons(x,ys)) rev'(nil(),ys) -> ys snoc(queue(f,r),x) -> checkF(queue(f,cons(x,r))) - Signature: {checkF/1,empty/0,enq/1,head/1,rev/1,rev'/2,snoc/2,tail/1,checkF#/1,empty#/0,enq#/1,head#/1,rev#/1,rev'#/2 ,snoc#/2,tail#/1} / {0/0,cons/2,errorHead/0,errorTail/0,nil/0,queue/2,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2 ,c_6/0,c_7/0,c_8/1,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {checkF#,empty#,enq#,head#,rev#,rev'#,snoc# ,tail#} and constructors {0,cons,errorHead,errorTail,nil,queue,s} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 2, 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 (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_5) = {1,2}, uargs(c_8) = {1}, uargs(c_9) = {1}, uargs(c_11) = {1} Following symbols are considered usable: {checkF,empty,enq,snoc,checkF#,empty#,enq#,head#,rev#,rev'#,snoc#,tail#} TcT has computed the following interpretation: p(0) = [1] [0] [0] p(checkF) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(cons) = [0 0 1] [0] [0 1 0] x2 + [1] [0 0 0] [0] p(empty) = [0] [1] [0] p(enq) = [1 0 0] [1] [0 1 0] x1 + [1] [0 0 0] [0] p(errorHead) = [0] [0] [0] p(errorTail) = [0] [0] [0] p(head) = [0] [0] [0] p(nil) = [0] [0] [0] p(queue) = [1 0 0] [0] [0 1 1] x2 + [1] [0 0 0] [0] p(rev) = [0] [0] [0] p(rev') = [1 0 0] [0 1 0] [1] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 0] [0 0 0] [0] p(s) = [1 1 0] [1] [0 1 1] x1 + [1] [0 0 0] [1] p(snoc) = [0 1 0] [0 0 0] [0] [0 1 0] x1 + [0 0 1] x2 + [1] [0 0 0] [0 0 0] [0] p(tail) = [0] [0] [0] p(checkF#) = [0 1 0] [0] [1 0 0] x1 + [1] [0 0 0] [0] p(empty#) = [0] [0] [0] p(enq#) = [1 1 1] [0] [1 1 0] x1 + [1] [1 1 0] [0] p(head#) = [0] [0] [0] p(rev#) = [0 1 0] [1] [1 1 1] x1 + [1] [0 1 1] [1] p(rev'#) = [0 1 0] [1 0 1] [0] [0 0 1] x1 + [1 1 0] x2 + [0] [0 0 0] [0 1 1] [1] p(snoc#) = [0 1 0] [0 0 0] [1] [0 1 0] x1 + [1 0 1] x2 + [0] [0 0 0] [0 1 0] [1] p(tail#) = [0] [0] [0] p(c_1) = [0] [0] [0] p(c_2) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(c_3) = [0] [0] [0] p(c_4) = [0] [0] [0] p(c_5) = [1 0 0] [1 0 0] [1] [1 0 1] x1 + [0 0 0] x2 + [0] [0 1 1] [0 0 0] [0] p(c_6) = [0] [0] [0] p(c_7) = [0] [0] [0] p(c_8) = [1 0 1] [0] [0 0 0] x1 + [0] [0 1 0] [0] p(c_9) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [1] p(c_10) = [0] [0] [0] p(c_11) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [1] p(c_12) = [0] [0] [0] p(c_13) = [0] [0] [0] Following rules are strictly oriented: rev'#(cons(x,xs),ys) = [0 1 0] [1 0 1] [1] [0 0 0] xs + [1 1 0] ys + [0] [0 0 0] [0 1 1] [1] > [0 1 0] [0 0 1] [0] [0 0 0] xs + [0 0 0] ys + [0] [0 0 0] [0 0 0] [1] = c_9(rev'#(xs,cons(x,ys))) Following rules are (at-least) weakly oriented: checkF#(queue(nil(),r)) = [0 1 1] [1] [1 0 0] r + [1] [0 0 0] [0] >= [0 1 0] [1] [0 0 0] r + [1] [0 0 0] [0] = c_2(rev#(r)) enq#(s(n)) = [1 2 1] [3] [1 2 1] n + [3] [1 2 1] [2] >= [1 2 1] [3] [0 2 0] n + [3] [1 2 1] [2] = c_5(snoc#(enq(n),n),enq#(n)) rev#(xs) = [0 1 0] [1] [1 1 1] xs + [1] [0 1 1] [1] >= [0 1 0] [1] [0 0 0] xs + [0] [0 0 1] [0] = c_8(rev'#(xs,nil())) snoc#(queue(f,r),x) = [0 1 1] [0 0 0] [2] [0 1 1] r + [1 0 1] x + [1] [0 0 0] [0 1 0] [1] >= [0 1 0] [2] [0 0 1] r + [1] [0 0 0] [1] = c_11(checkF#(queue(f,cons(x,r)))) checkF(queue(cons(x,xs),r)) = [1 0 0] [0] [0 1 1] r + [1] [0 0 0] [0] >= [1 0 0] [0] [0 1 1] r + [1] [0 0 0] [0] = queue(cons(x,xs),r) checkF(queue(nil(),r)) = [1 0 0] [0] [0 1 1] r + [1] [0 0 0] [0] >= [0] [1] [0] = queue(rev(r),nil()) empty() = [0] [1] [0] >= [0] [1] [0] = queue(nil(),nil()) enq(0()) = [2] [1] [0] >= [0] [1] [0] = empty() enq(s(n)) = [1 1 0] [2] [0 1 1] n + [2] [0 0 0] [0] >= [0 1 0] [1] [0 1 1] n + [2] [0 0 0] [0] = snoc(enq(n),n) snoc(queue(f,r),x) = [0 1 1] [0 0 0] [1] [0 1 1] r + [0 0 1] x + [2] [0 0 0] [0 0 0] [0] >= [0 0 1] [0] [0 1 0] r + [2] [0 0 0] [0] = checkF(queue(f,cons(x,r))) *** Step 7.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: checkF#(queue(nil(),r)) -> c_2(rev#(r)) rev#(xs) -> c_8(rev'#(xs,nil())) - Weak DPs: enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)) rev'#(cons(x,xs),ys) -> c_9(rev'#(xs,cons(x,ys))) snoc#(queue(f,r),x) -> c_11(checkF#(queue(f,cons(x,r)))) - Weak TRS: checkF(queue(cons(x,xs),r)) -> queue(cons(x,xs),r) checkF(queue(nil(),r)) -> queue(rev(r),nil()) empty() -> queue(nil(),nil()) enq(0()) -> empty() enq(s(n)) -> snoc(enq(n),n) rev(xs) -> rev'(xs,nil()) rev'(cons(x,xs),ys) -> rev'(xs,cons(x,ys)) rev'(nil(),ys) -> ys snoc(queue(f,r),x) -> checkF(queue(f,cons(x,r))) - Signature: {checkF/1,empty/0,enq/1,head/1,rev/1,rev'/2,snoc/2,tail/1,checkF#/1,empty#/0,enq#/1,head#/1,rev#/1,rev'#/2 ,snoc#/2,tail#/1} / {0/0,cons/2,errorHead/0,errorTail/0,nil/0,queue/2,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2 ,c_6/0,c_7/0,c_8/1,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {checkF#,empty#,enq#,head#,rev#,rev'#,snoc# ,tail#} and constructors {0,cons,errorHead,errorTail,nil,queue,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 7.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: checkF#(queue(nil(),r)) -> c_2(rev#(r)) rev#(xs) -> c_8(rev'#(xs,nil())) - Weak DPs: enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)) rev'#(cons(x,xs),ys) -> c_9(rev'#(xs,cons(x,ys))) snoc#(queue(f,r),x) -> c_11(checkF#(queue(f,cons(x,r)))) - Weak TRS: checkF(queue(cons(x,xs),r)) -> queue(cons(x,xs),r) checkF(queue(nil(),r)) -> queue(rev(r),nil()) empty() -> queue(nil(),nil()) enq(0()) -> empty() enq(s(n)) -> snoc(enq(n),n) rev(xs) -> rev'(xs,nil()) rev'(cons(x,xs),ys) -> rev'(xs,cons(x,ys)) rev'(nil(),ys) -> ys snoc(queue(f,r),x) -> checkF(queue(f,cons(x,r))) - Signature: {checkF/1,empty/0,enq/1,head/1,rev/1,rev'/2,snoc/2,tail/1,checkF#/1,empty#/0,enq#/1,head#/1,rev#/1,rev'#/2 ,snoc#/2,tail#/1} / {0/0,cons/2,errorHead/0,errorTail/0,nil/0,queue/2,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2 ,c_6/0,c_7/0,c_8/1,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {checkF#,empty#,enq#,head#,rev#,rev'#,snoc# ,tail#} and constructors {0,cons,errorHead,errorTail,nil,queue,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:checkF#(queue(nil(),r)) -> c_2(rev#(r)) -->_1 rev#(xs) -> c_8(rev'#(xs,nil())):2 2:S:rev#(xs) -> c_8(rev'#(xs,nil())) -->_1 rev'#(cons(x,xs),ys) -> c_9(rev'#(xs,cons(x,ys))):4 3:W:enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)) -->_1 snoc#(queue(f,r),x) -> c_11(checkF#(queue(f,cons(x,r)))):5 -->_2 enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)):3 4:W:rev'#(cons(x,xs),ys) -> c_9(rev'#(xs,cons(x,ys))) -->_1 rev'#(cons(x,xs),ys) -> c_9(rev'#(xs,cons(x,ys))):4 5:W:snoc#(queue(f,r),x) -> c_11(checkF#(queue(f,cons(x,r)))) -->_1 checkF#(queue(nil(),r)) -> c_2(rev#(r)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: rev'#(cons(x,xs),ys) -> c_9(rev'#(xs,cons(x,ys))) *** Step 7.a:1.b:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: checkF#(queue(nil(),r)) -> c_2(rev#(r)) rev#(xs) -> c_8(rev'#(xs,nil())) - Weak DPs: enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)) snoc#(queue(f,r),x) -> c_11(checkF#(queue(f,cons(x,r)))) - Weak TRS: checkF(queue(cons(x,xs),r)) -> queue(cons(x,xs),r) checkF(queue(nil(),r)) -> queue(rev(r),nil()) empty() -> queue(nil(),nil()) enq(0()) -> empty() enq(s(n)) -> snoc(enq(n),n) rev(xs) -> rev'(xs,nil()) rev'(cons(x,xs),ys) -> rev'(xs,cons(x,ys)) rev'(nil(),ys) -> ys snoc(queue(f,r),x) -> checkF(queue(f,cons(x,r))) - Signature: {checkF/1,empty/0,enq/1,head/1,rev/1,rev'/2,snoc/2,tail/1,checkF#/1,empty#/0,enq#/1,head#/1,rev#/1,rev'#/2 ,snoc#/2,tail#/1} / {0/0,cons/2,errorHead/0,errorTail/0,nil/0,queue/2,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2 ,c_6/0,c_7/0,c_8/1,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {checkF#,empty#,enq#,head#,rev#,rev'#,snoc# ,tail#} and constructors {0,cons,errorHead,errorTail,nil,queue,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:checkF#(queue(nil(),r)) -> c_2(rev#(r)) -->_1 rev#(xs) -> c_8(rev'#(xs,nil())):2 2:S:rev#(xs) -> c_8(rev'#(xs,nil())) 3:W:enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)) -->_1 snoc#(queue(f,r),x) -> c_11(checkF#(queue(f,cons(x,r)))):5 -->_2 enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)):3 5:W:snoc#(queue(f,r),x) -> c_11(checkF#(queue(f,cons(x,r)))) -->_1 checkF#(queue(nil(),r)) -> c_2(rev#(r)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: rev#(xs) -> c_8() *** Step 7.a:1.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: checkF#(queue(nil(),r)) -> c_2(rev#(r)) rev#(xs) -> c_8() - Weak DPs: enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)) snoc#(queue(f,r),x) -> c_11(checkF#(queue(f,cons(x,r)))) - Weak TRS: checkF(queue(cons(x,xs),r)) -> queue(cons(x,xs),r) checkF(queue(nil(),r)) -> queue(rev(r),nil()) empty() -> queue(nil(),nil()) enq(0()) -> empty() enq(s(n)) -> snoc(enq(n),n) rev(xs) -> rev'(xs,nil()) rev'(cons(x,xs),ys) -> rev'(xs,cons(x,ys)) rev'(nil(),ys) -> ys snoc(queue(f,r),x) -> checkF(queue(f,cons(x,r))) - Signature: {checkF/1,empty/0,enq/1,head/1,rev/1,rev'/2,snoc/2,tail/1,checkF#/1,empty#/0,enq#/1,head#/1,rev#/1,rev'#/2 ,snoc#/2,tail#/1} / {0/0,cons/2,errorHead/0,errorTail/0,nil/0,queue/2,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2 ,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {checkF#,empty#,enq#,head#,rev#,rev'#,snoc# ,tail#} and constructors {0,cons,errorHead,errorTail,nil,queue,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: checkF#(queue(nil(),r)) -> c_2(rev#(r)) Consider the set of all dependency pairs 1: checkF#(queue(nil(),r)) -> c_2(rev#(r)) 2: rev#(xs) -> c_8() 3: enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)) 4: snoc#(queue(f,r),x) -> c_11(checkF#(queue(f,cons(x,r)))) Processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}induces the complexity certificateTIME (?,O(n^1)) SPACE(?,?)on application of the dependency pairs {1} These cover all (indirect) predecessors of dependency pairs {1,2} their number of applications is equally bounded. The dependency pairs are shifted into the weak component. **** Step 7.a:1.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: checkF#(queue(nil(),r)) -> c_2(rev#(r)) rev#(xs) -> c_8() - Weak DPs: enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)) snoc#(queue(f,r),x) -> c_11(checkF#(queue(f,cons(x,r)))) - Weak TRS: checkF(queue(cons(x,xs),r)) -> queue(cons(x,xs),r) checkF(queue(nil(),r)) -> queue(rev(r),nil()) empty() -> queue(nil(),nil()) enq(0()) -> empty() enq(s(n)) -> snoc(enq(n),n) rev(xs) -> rev'(xs,nil()) rev'(cons(x,xs),ys) -> rev'(xs,cons(x,ys)) rev'(nil(),ys) -> ys snoc(queue(f,r),x) -> checkF(queue(f,cons(x,r))) - Signature: {checkF/1,empty/0,enq/1,head/1,rev/1,rev'/2,snoc/2,tail/1,checkF#/1,empty#/0,enq#/1,head#/1,rev#/1,rev'#/2 ,snoc#/2,tail#/1} / {0/0,cons/2,errorHead/0,errorTail/0,nil/0,queue/2,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2 ,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {checkF#,empty#,enq#,head#,rev#,rev'#,snoc# ,tail#} and constructors {0,cons,errorHead,errorTail,nil,queue,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_2) = {1}, uargs(c_5) = {1,2}, uargs(c_11) = {1} Following symbols are considered usable: {checkF#,empty#,enq#,head#,rev#,rev'#,snoc#,tail#} TcT has computed the following interpretation: p(0) = [0] p(checkF) = [4] x1 + [4] p(cons) = [1] x2 + [3] p(empty) = [0] p(enq) = [0] p(errorHead) = [2] p(errorTail) = [4] p(head) = [0] p(nil) = [3] p(queue) = [1] x2 + [1] p(rev) = [1] x1 + [2] p(rev') = [1] x1 + [8] p(s) = [1] x1 + [2] p(snoc) = [4] x2 + [6] p(tail) = [2] x1 + [1] p(checkF#) = [1] p(empty#) = [2] p(enq#) = [8] x1 + [0] p(head#) = [0] p(rev#) = [0] p(rev'#) = [1] x1 + [8] x2 + [1] p(snoc#) = [2] p(tail#) = [1] x1 + [0] p(c_1) = [0] p(c_2) = [8] x1 + [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [8] x1 + [1] x2 + [0] p(c_6) = [1] p(c_7) = [0] p(c_8) = [0] p(c_9) = [2] x1 + [8] p(c_10) = [0] p(c_11) = [2] x1 + [0] p(c_12) = [2] x1 + [0] p(c_13) = [4] Following rules are strictly oriented: checkF#(queue(nil(),r)) = [1] > [0] = c_2(rev#(r)) Following rules are (at-least) weakly oriented: enq#(s(n)) = [8] n + [16] >= [8] n + [16] = c_5(snoc#(enq(n),n),enq#(n)) rev#(xs) = [0] >= [0] = c_8() snoc#(queue(f,r),x) = [2] >= [2] = c_11(checkF#(queue(f,cons(x,r)))) **** Step 7.a:1.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: rev#(xs) -> c_8() - Weak DPs: checkF#(queue(nil(),r)) -> c_2(rev#(r)) enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)) snoc#(queue(f,r),x) -> c_11(checkF#(queue(f,cons(x,r)))) - Weak TRS: checkF(queue(cons(x,xs),r)) -> queue(cons(x,xs),r) checkF(queue(nil(),r)) -> queue(rev(r),nil()) empty() -> queue(nil(),nil()) enq(0()) -> empty() enq(s(n)) -> snoc(enq(n),n) rev(xs) -> rev'(xs,nil()) rev'(cons(x,xs),ys) -> rev'(xs,cons(x,ys)) rev'(nil(),ys) -> ys snoc(queue(f,r),x) -> checkF(queue(f,cons(x,r))) - Signature: {checkF/1,empty/0,enq/1,head/1,rev/1,rev'/2,snoc/2,tail/1,checkF#/1,empty#/0,enq#/1,head#/1,rev#/1,rev'#/2 ,snoc#/2,tail#/1} / {0/0,cons/2,errorHead/0,errorTail/0,nil/0,queue/2,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2 ,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {checkF#,empty#,enq#,head#,rev#,rev'#,snoc# ,tail#} and constructors {0,cons,errorHead,errorTail,nil,queue,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 7.a:1.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: checkF#(queue(nil(),r)) -> c_2(rev#(r)) enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)) rev#(xs) -> c_8() snoc#(queue(f,r),x) -> c_11(checkF#(queue(f,cons(x,r)))) - Weak TRS: checkF(queue(cons(x,xs),r)) -> queue(cons(x,xs),r) checkF(queue(nil(),r)) -> queue(rev(r),nil()) empty() -> queue(nil(),nil()) enq(0()) -> empty() enq(s(n)) -> snoc(enq(n),n) rev(xs) -> rev'(xs,nil()) rev'(cons(x,xs),ys) -> rev'(xs,cons(x,ys)) rev'(nil(),ys) -> ys snoc(queue(f,r),x) -> checkF(queue(f,cons(x,r))) - Signature: {checkF/1,empty/0,enq/1,head/1,rev/1,rev'/2,snoc/2,tail/1,checkF#/1,empty#/0,enq#/1,head#/1,rev#/1,rev'#/2 ,snoc#/2,tail#/1} / {0/0,cons/2,errorHead/0,errorTail/0,nil/0,queue/2,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2 ,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {checkF#,empty#,enq#,head#,rev#,rev'#,snoc# ,tail#} and constructors {0,cons,errorHead,errorTail,nil,queue,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:checkF#(queue(nil(),r)) -> c_2(rev#(r)) -->_1 rev#(xs) -> c_8():3 2:W:enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)) -->_1 snoc#(queue(f,r),x) -> c_11(checkF#(queue(f,cons(x,r)))):4 -->_2 enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)):2 3:W:rev#(xs) -> c_8() 4:W:snoc#(queue(f,r),x) -> c_11(checkF#(queue(f,cons(x,r)))) -->_1 checkF#(queue(nil(),r)) -> c_2(rev#(r)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)) 4: snoc#(queue(f,r),x) -> c_11(checkF#(queue(f,cons(x,r)))) 1: checkF#(queue(nil(),r)) -> c_2(rev#(r)) 3: rev#(xs) -> c_8() **** Step 7.a:1.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: checkF(queue(cons(x,xs),r)) -> queue(cons(x,xs),r) checkF(queue(nil(),r)) -> queue(rev(r),nil()) empty() -> queue(nil(),nil()) enq(0()) -> empty() enq(s(n)) -> snoc(enq(n),n) rev(xs) -> rev'(xs,nil()) rev'(cons(x,xs),ys) -> rev'(xs,cons(x,ys)) rev'(nil(),ys) -> ys snoc(queue(f,r),x) -> checkF(queue(f,cons(x,r))) - Signature: {checkF/1,empty/0,enq/1,head/1,rev/1,rev'/2,snoc/2,tail/1,checkF#/1,empty#/0,enq#/1,head#/1,rev#/1,rev'#/2 ,snoc#/2,tail#/1} / {0/0,cons/2,errorHead/0,errorTail/0,nil/0,queue/2,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2 ,c_6/0,c_7/0,c_8/0,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {checkF#,empty#,enq#,head#,rev#,rev'#,snoc# ,tail#} and constructors {0,cons,errorHead,errorTail,nil,queue,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 7.b:1: PredecessorEstimation WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)) snoc#(queue(f,r),x) -> c_11(checkF#(queue(f,cons(x,r)))) - Weak DPs: checkF#(queue(nil(),r)) -> c_2(rev#(r)) rev#(xs) -> c_8(rev'#(xs,nil())) rev'#(cons(x,xs),ys) -> c_9(rev'#(xs,cons(x,ys))) - Weak TRS: checkF(queue(cons(x,xs),r)) -> queue(cons(x,xs),r) checkF(queue(nil(),r)) -> queue(rev(r),nil()) empty() -> queue(nil(),nil()) enq(0()) -> empty() enq(s(n)) -> snoc(enq(n),n) rev(xs) -> rev'(xs,nil()) rev'(cons(x,xs),ys) -> rev'(xs,cons(x,ys)) rev'(nil(),ys) -> ys snoc(queue(f,r),x) -> checkF(queue(f,cons(x,r))) - Signature: {checkF/1,empty/0,enq/1,head/1,rev/1,rev'/2,snoc/2,tail/1,checkF#/1,empty#/0,enq#/1,head#/1,rev#/1,rev'#/2 ,snoc#/2,tail#/1} / {0/0,cons/2,errorHead/0,errorTail/0,nil/0,queue/2,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2 ,c_6/0,c_7/0,c_8/1,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {checkF#,empty#,enq#,head#,rev#,rev'#,snoc# ,tail#} and constructors {0,cons,errorHead,errorTail,nil,queue,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2} by application of Pre({2}) = {1}. Here rules are labelled as follows: 1: enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)) 2: snoc#(queue(f,r),x) -> c_11(checkF#(queue(f,cons(x,r)))) 3: checkF#(queue(nil(),r)) -> c_2(rev#(r)) 4: rev#(xs) -> c_8(rev'#(xs,nil())) 5: rev'#(cons(x,xs),ys) -> c_9(rev'#(xs,cons(x,ys))) ** Step 7.b:2: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)) - Weak DPs: checkF#(queue(nil(),r)) -> c_2(rev#(r)) rev#(xs) -> c_8(rev'#(xs,nil())) rev'#(cons(x,xs),ys) -> c_9(rev'#(xs,cons(x,ys))) snoc#(queue(f,r),x) -> c_11(checkF#(queue(f,cons(x,r)))) - Weak TRS: checkF(queue(cons(x,xs),r)) -> queue(cons(x,xs),r) checkF(queue(nil(),r)) -> queue(rev(r),nil()) empty() -> queue(nil(),nil()) enq(0()) -> empty() enq(s(n)) -> snoc(enq(n),n) rev(xs) -> rev'(xs,nil()) rev'(cons(x,xs),ys) -> rev'(xs,cons(x,ys)) rev'(nil(),ys) -> ys snoc(queue(f,r),x) -> checkF(queue(f,cons(x,r))) - Signature: {checkF/1,empty/0,enq/1,head/1,rev/1,rev'/2,snoc/2,tail/1,checkF#/1,empty#/0,enq#/1,head#/1,rev#/1,rev'#/2 ,snoc#/2,tail#/1} / {0/0,cons/2,errorHead/0,errorTail/0,nil/0,queue/2,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2 ,c_6/0,c_7/0,c_8/1,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {checkF#,empty#,enq#,head#,rev#,rev'#,snoc# ,tail#} and constructors {0,cons,errorHead,errorTail,nil,queue,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)) -->_1 snoc#(queue(f,r),x) -> c_11(checkF#(queue(f,cons(x,r)))):5 -->_2 enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)):1 2:W:checkF#(queue(nil(),r)) -> c_2(rev#(r)) -->_1 rev#(xs) -> c_8(rev'#(xs,nil())):3 3:W:rev#(xs) -> c_8(rev'#(xs,nil())) -->_1 rev'#(cons(x,xs),ys) -> c_9(rev'#(xs,cons(x,ys))):4 4:W:rev'#(cons(x,xs),ys) -> c_9(rev'#(xs,cons(x,ys))) -->_1 rev'#(cons(x,xs),ys) -> c_9(rev'#(xs,cons(x,ys))):4 5:W:snoc#(queue(f,r),x) -> c_11(checkF#(queue(f,cons(x,r)))) -->_1 checkF#(queue(nil(),r)) -> c_2(rev#(r)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: snoc#(queue(f,r),x) -> c_11(checkF#(queue(f,cons(x,r)))) 2: checkF#(queue(nil(),r)) -> c_2(rev#(r)) 3: rev#(xs) -> c_8(rev'#(xs,nil())) 4: rev'#(cons(x,xs),ys) -> c_9(rev'#(xs,cons(x,ys))) ** Step 7.b:3: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)) - Weak TRS: checkF(queue(cons(x,xs),r)) -> queue(cons(x,xs),r) checkF(queue(nil(),r)) -> queue(rev(r),nil()) empty() -> queue(nil(),nil()) enq(0()) -> empty() enq(s(n)) -> snoc(enq(n),n) rev(xs) -> rev'(xs,nil()) rev'(cons(x,xs),ys) -> rev'(xs,cons(x,ys)) rev'(nil(),ys) -> ys snoc(queue(f,r),x) -> checkF(queue(f,cons(x,r))) - Signature: {checkF/1,empty/0,enq/1,head/1,rev/1,rev'/2,snoc/2,tail/1,checkF#/1,empty#/0,enq#/1,head#/1,rev#/1,rev'#/2 ,snoc#/2,tail#/1} / {0/0,cons/2,errorHead/0,errorTail/0,nil/0,queue/2,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/2 ,c_6/0,c_7/0,c_8/1,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {checkF#,empty#,enq#,head#,rev#,rev'#,snoc# ,tail#} and constructors {0,cons,errorHead,errorTail,nil,queue,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)) -->_2 enq#(s(n)) -> c_5(snoc#(enq(n),n),enq#(n)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: enq#(s(n)) -> c_5(enq#(n)) ** Step 7.b:4: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: enq#(s(n)) -> c_5(enq#(n)) - Weak TRS: checkF(queue(cons(x,xs),r)) -> queue(cons(x,xs),r) checkF(queue(nil(),r)) -> queue(rev(r),nil()) empty() -> queue(nil(),nil()) enq(0()) -> empty() enq(s(n)) -> snoc(enq(n),n) rev(xs) -> rev'(xs,nil()) rev'(cons(x,xs),ys) -> rev'(xs,cons(x,ys)) rev'(nil(),ys) -> ys snoc(queue(f,r),x) -> checkF(queue(f,cons(x,r))) - Signature: {checkF/1,empty/0,enq/1,head/1,rev/1,rev'/2,snoc/2,tail/1,checkF#/1,empty#/0,enq#/1,head#/1,rev#/1,rev'#/2 ,snoc#/2,tail#/1} / {0/0,cons/2,errorHead/0,errorTail/0,nil/0,queue/2,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/1 ,c_6/0,c_7/0,c_8/1,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {checkF#,empty#,enq#,head#,rev#,rev'#,snoc# ,tail#} and constructors {0,cons,errorHead,errorTail,nil,queue,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: enq#(s(n)) -> c_5(enq#(n)) ** Step 7.b:5: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: enq#(s(n)) -> c_5(enq#(n)) - Signature: {checkF/1,empty/0,enq/1,head/1,rev/1,rev'/2,snoc/2,tail/1,checkF#/1,empty#/0,enq#/1,head#/1,rev#/1,rev'#/2 ,snoc#/2,tail#/1} / {0/0,cons/2,errorHead/0,errorTail/0,nil/0,queue/2,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/1 ,c_6/0,c_7/0,c_8/1,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {checkF#,empty#,enq#,head#,rev#,rev'#,snoc# ,tail#} and constructors {0,cons,errorHead,errorTail,nil,queue,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: enq#(s(n)) -> c_5(enq#(n)) The strictly oriented rules are moved into the weak component. *** Step 7.b:5.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: enq#(s(n)) -> c_5(enq#(n)) - Signature: {checkF/1,empty/0,enq/1,head/1,rev/1,rev'/2,snoc/2,tail/1,checkF#/1,empty#/0,enq#/1,head#/1,rev#/1,rev'#/2 ,snoc#/2,tail#/1} / {0/0,cons/2,errorHead/0,errorTail/0,nil/0,queue/2,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/1 ,c_6/0,c_7/0,c_8/1,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {checkF#,empty#,enq#,head#,rev#,rev'#,snoc# ,tail#} and constructors {0,cons,errorHead,errorTail,nil,queue,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_5) = {1} Following symbols are considered usable: {checkF#,empty#,enq#,head#,rev#,rev'#,snoc#,tail#} TcT has computed the following interpretation: p(0) = [1] p(checkF) = [1] x1 + [8] p(cons) = [1] x1 + [1] x2 + [0] p(empty) = [0] p(enq) = [0] p(errorHead) = [0] p(errorTail) = [0] p(head) = [0] p(nil) = [0] p(queue) = [1] x1 + [1] x2 + [0] p(rev) = [0] p(rev') = [0] p(s) = [1] x1 + [9] p(snoc) = [0] p(tail) = [0] p(checkF#) = [0] p(empty#) = [0] p(enq#) = [1] x1 + [7] p(head#) = [0] p(rev#) = [0] p(rev'#) = [1] p(snoc#) = [1] x2 + [1] p(tail#) = [1] x1 + [8] p(c_1) = [1] p(c_2) = [8] p(c_3) = [1] p(c_4) = [1] x1 + [1] p(c_5) = [1] x1 + [0] p(c_6) = [1] p(c_7) = [1] p(c_8) = [1] p(c_9) = [1] x1 + [2] p(c_10) = [2] p(c_11) = [1] x1 + [1] p(c_12) = [1] x1 + [0] p(c_13) = [0] Following rules are strictly oriented: enq#(s(n)) = [1] n + [16] > [1] n + [7] = c_5(enq#(n)) Following rules are (at-least) weakly oriented: *** Step 7.b:5.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: enq#(s(n)) -> c_5(enq#(n)) - Signature: {checkF/1,empty/0,enq/1,head/1,rev/1,rev'/2,snoc/2,tail/1,checkF#/1,empty#/0,enq#/1,head#/1,rev#/1,rev'#/2 ,snoc#/2,tail#/1} / {0/0,cons/2,errorHead/0,errorTail/0,nil/0,queue/2,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/1 ,c_6/0,c_7/0,c_8/1,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {checkF#,empty#,enq#,head#,rev#,rev'#,snoc# ,tail#} and constructors {0,cons,errorHead,errorTail,nil,queue,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 7.b:5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: enq#(s(n)) -> c_5(enq#(n)) - Signature: {checkF/1,empty/0,enq/1,head/1,rev/1,rev'/2,snoc/2,tail/1,checkF#/1,empty#/0,enq#/1,head#/1,rev#/1,rev'#/2 ,snoc#/2,tail#/1} / {0/0,cons/2,errorHead/0,errorTail/0,nil/0,queue/2,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/1 ,c_6/0,c_7/0,c_8/1,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {checkF#,empty#,enq#,head#,rev#,rev'#,snoc# ,tail#} and constructors {0,cons,errorHead,errorTail,nil,queue,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:enq#(s(n)) -> c_5(enq#(n)) -->_1 enq#(s(n)) -> c_5(enq#(n)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: enq#(s(n)) -> c_5(enq#(n)) *** Step 7.b:5.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Signature: {checkF/1,empty/0,enq/1,head/1,rev/1,rev'/2,snoc/2,tail/1,checkF#/1,empty#/0,enq#/1,head#/1,rev#/1,rev'#/2 ,snoc#/2,tail#/1} / {0/0,cons/2,errorHead/0,errorTail/0,nil/0,queue/2,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/1 ,c_6/0,c_7/0,c_8/1,c_9/1,c_10/0,c_11/1,c_12/1,c_13/0} - Obligation: innermost runtime complexity wrt. defined symbols {checkF#,empty#,enq#,head#,rev#,rev'#,snoc# ,tail#} and constructors {0,cons,errorHead,errorTail,nil,queue,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^2))