/export/starexec/sandbox/solver/bin/starexec_run_tct_rc /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1),O(n^2)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^2)) + Considered Problem: - Strict TRS: ++(.(x,y),z) -> .(x,++(y,z)) ++(nil(),y) -> y car(.(x,y)) -> x cdr(.(x,y)) -> y null(.(x,y)) -> false() null(nil()) -> true() rev(.(x,y)) -> ++(rev(y),.(x,nil())) rev(nil()) -> nil() - Signature: {++/2,car/1,cdr/1,null/1,rev/1} / {./2,false/0,nil/0,true/0} - Obligation: runtime complexity wrt. defined symbols {++,car,cdr,null,rev} and constructors {.,false,nil,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: ++(.(x,y),z) -> .(x,++(y,z)) ++(nil(),y) -> y car(.(x,y)) -> x cdr(.(x,y)) -> y null(.(x,y)) -> false() null(nil()) -> true() rev(.(x,y)) -> ++(rev(y),.(x,nil())) rev(nil()) -> nil() - Signature: {++/2,car/1,cdr/1,null/1,rev/1} / {./2,false/0,nil/0,true/0} - Obligation: runtime complexity wrt. defined symbols {++,car,cdr,null,rev} and constructors {.,false,nil,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: ++(.(x,y),z) -> .(x,++(y,z)) ++(nil(),y) -> y car(.(x,y)) -> x cdr(.(x,y)) -> y null(.(x,y)) -> false() null(nil()) -> true() rev(.(x,y)) -> ++(rev(y),.(x,nil())) rev(nil()) -> nil() - Signature: {++/2,car/1,cdr/1,null/1,rev/1} / {./2,false/0,nil/0,true/0} - Obligation: runtime complexity wrt. defined symbols {++,car,cdr,null,rev} and constructors {.,false,nil,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: ++(y,z){y -> .(x,y)} = ++(.(x,y),z) ->^+ .(x,++(y,z)) = C[++(y,z) = ++(y,z){}] ** Step 1.b:1: ToInnermost. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: ++(.(x,y),z) -> .(x,++(y,z)) ++(nil(),y) -> y car(.(x,y)) -> x cdr(.(x,y)) -> y null(.(x,y)) -> false() null(nil()) -> true() rev(.(x,y)) -> ++(rev(y),.(x,nil())) rev(nil()) -> nil() - Signature: {++/2,car/1,cdr/1,null/1,rev/1} / {./2,false/0,nil/0,true/0} - Obligation: runtime complexity wrt. defined symbols {++,car,cdr,null,rev} and constructors {.,false,nil,true} + Applied Processor: ToInnermost + Details: switch to innermost, as the system is overlay and right linear and does not contain weak rules ** Step 1.b:2: DependencyPairs. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: ++(.(x,y),z) -> .(x,++(y,z)) ++(nil(),y) -> y car(.(x,y)) -> x cdr(.(x,y)) -> y null(.(x,y)) -> false() null(nil()) -> true() rev(.(x,y)) -> ++(rev(y),.(x,nil())) rev(nil()) -> nil() - Signature: {++/2,car/1,cdr/1,null/1,rev/1} / {./2,false/0,nil/0,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {++,car,cdr,null,rev} and constructors {.,false,nil,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs ++#(.(x,y),z) -> c_1(++#(y,z)) ++#(nil(),y) -> c_2() car#(.(x,y)) -> c_3() cdr#(.(x,y)) -> c_4() null#(.(x,y)) -> c_5() null#(nil()) -> c_6() rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y)) rev#(nil()) -> c_8() Weak DPs and mark the set of starting terms. ** Step 1.b:3: UsableRules. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: ++#(.(x,y),z) -> c_1(++#(y,z)) ++#(nil(),y) -> c_2() car#(.(x,y)) -> c_3() cdr#(.(x,y)) -> c_4() null#(.(x,y)) -> c_5() null#(nil()) -> c_6() rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y)) rev#(nil()) -> c_8() - Weak TRS: ++(.(x,y),z) -> .(x,++(y,z)) ++(nil(),y) -> y car(.(x,y)) -> x cdr(.(x,y)) -> y null(.(x,y)) -> false() null(nil()) -> true() rev(.(x,y)) -> ++(rev(y),.(x,nil())) rev(nil()) -> nil() - Signature: {++/2,car/1,cdr/1,null/1,rev/1,++#/2,car#/1,cdr#/1,null#/1,rev#/1} / {./2,false/0,nil/0,true/0,c_1/1,c_2/0 ,c_3/0,c_4/0,c_5/0,c_6/0,c_7/2,c_8/0} - Obligation: innermost runtime complexity wrt. defined symbols {++#,car#,cdr#,null#,rev#} and constructors {.,false,nil ,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: ++(.(x,y),z) -> .(x,++(y,z)) ++(nil(),y) -> y rev(.(x,y)) -> ++(rev(y),.(x,nil())) rev(nil()) -> nil() ++#(.(x,y),z) -> c_1(++#(y,z)) ++#(nil(),y) -> c_2() car#(.(x,y)) -> c_3() cdr#(.(x,y)) -> c_4() null#(.(x,y)) -> c_5() null#(nil()) -> c_6() rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y)) rev#(nil()) -> c_8() ** Step 1.b:4: PredecessorEstimation. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: ++#(.(x,y),z) -> c_1(++#(y,z)) ++#(nil(),y) -> c_2() car#(.(x,y)) -> c_3() cdr#(.(x,y)) -> c_4() null#(.(x,y)) -> c_5() null#(nil()) -> c_6() rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y)) rev#(nil()) -> c_8() - Weak TRS: ++(.(x,y),z) -> .(x,++(y,z)) ++(nil(),y) -> y rev(.(x,y)) -> ++(rev(y),.(x,nil())) rev(nil()) -> nil() - Signature: {++/2,car/1,cdr/1,null/1,rev/1,++#/2,car#/1,cdr#/1,null#/1,rev#/1} / {./2,false/0,nil/0,true/0,c_1/1,c_2/0 ,c_3/0,c_4/0,c_5/0,c_6/0,c_7/2,c_8/0} - Obligation: innermost runtime complexity wrt. defined symbols {++#,car#,cdr#,null#,rev#} and constructors {.,false,nil ,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2,3,4,5,6,8} by application of Pre({2,3,4,5,6,8}) = {1,7}. Here rules are labelled as follows: 1: ++#(.(x,y),z) -> c_1(++#(y,z)) 2: ++#(nil(),y) -> c_2() 3: car#(.(x,y)) -> c_3() 4: cdr#(.(x,y)) -> c_4() 5: null#(.(x,y)) -> c_5() 6: null#(nil()) -> c_6() 7: rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y)) 8: rev#(nil()) -> c_8() ** Step 1.b:5: RemoveWeakSuffixes. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: ++#(.(x,y),z) -> c_1(++#(y,z)) rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y)) - Weak DPs: ++#(nil(),y) -> c_2() car#(.(x,y)) -> c_3() cdr#(.(x,y)) -> c_4() null#(.(x,y)) -> c_5() null#(nil()) -> c_6() rev#(nil()) -> c_8() - Weak TRS: ++(.(x,y),z) -> .(x,++(y,z)) ++(nil(),y) -> y rev(.(x,y)) -> ++(rev(y),.(x,nil())) rev(nil()) -> nil() - Signature: {++/2,car/1,cdr/1,null/1,rev/1,++#/2,car#/1,cdr#/1,null#/1,rev#/1} / {./2,false/0,nil/0,true/0,c_1/1,c_2/0 ,c_3/0,c_4/0,c_5/0,c_6/0,c_7/2,c_8/0} - Obligation: innermost runtime complexity wrt. defined symbols {++#,car#,cdr#,null#,rev#} and constructors {.,false,nil ,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:++#(.(x,y),z) -> c_1(++#(y,z)) -->_1 ++#(nil(),y) -> c_2():3 -->_1 ++#(.(x,y),z) -> c_1(++#(y,z)):1 2:S:rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y)) -->_2 rev#(nil()) -> c_8():8 -->_1 ++#(nil(),y) -> c_2():3 -->_2 rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y)):2 -->_1 ++#(.(x,y),z) -> c_1(++#(y,z)):1 3:W:++#(nil(),y) -> c_2() 4:W:car#(.(x,y)) -> c_3() 5:W:cdr#(.(x,y)) -> c_4() 6:W:null#(.(x,y)) -> c_5() 7:W:null#(nil()) -> c_6() 8:W:rev#(nil()) -> c_8() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 7: null#(nil()) -> c_6() 6: null#(.(x,y)) -> c_5() 5: cdr#(.(x,y)) -> c_4() 4: car#(.(x,y)) -> c_3() 8: rev#(nil()) -> c_8() 3: ++#(nil(),y) -> c_2() ** Step 1.b:6: Decompose. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: ++#(.(x,y),z) -> c_1(++#(y,z)) rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y)) - Weak TRS: ++(.(x,y),z) -> .(x,++(y,z)) ++(nil(),y) -> y rev(.(x,y)) -> ++(rev(y),.(x,nil())) rev(nil()) -> nil() - Signature: {++/2,car/1,cdr/1,null/1,rev/1,++#/2,car#/1,cdr#/1,null#/1,rev#/1} / {./2,false/0,nil/0,true/0,c_1/1,c_2/0 ,c_3/0,c_4/0,c_5/0,c_6/0,c_7/2,c_8/0} - Obligation: innermost runtime complexity wrt. defined symbols {++#,car#,cdr#,null#,rev#} and constructors {.,false,nil ,true} + 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: ++#(.(x,y),z) -> c_1(++#(y,z)) - Weak DPs: rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y)) - Weak TRS: ++(.(x,y),z) -> .(x,++(y,z)) ++(nil(),y) -> y rev(.(x,y)) -> ++(rev(y),.(x,nil())) rev(nil()) -> nil() - Signature: {++/2,car/1,cdr/1,null/1,rev/1,++#/2,car#/1,cdr#/1,null#/1,rev#/1} / {./2,false/0,nil/0,true/0,c_1/1,c_2/0 ,c_3/0,c_4/0,c_5/0,c_6/0,c_7/2,c_8/0} - Obligation: innermost runtime complexity wrt. defined symbols {++#,car#,cdr#,null#,rev#} and constructors {.,false,nil ,true} Problem (S) - Strict DPs: rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y)) - Weak DPs: ++#(.(x,y),z) -> c_1(++#(y,z)) - Weak TRS: ++(.(x,y),z) -> .(x,++(y,z)) ++(nil(),y) -> y rev(.(x,y)) -> ++(rev(y),.(x,nil())) rev(nil()) -> nil() - Signature: {++/2,car/1,cdr/1,null/1,rev/1,++#/2,car#/1,cdr#/1,null#/1,rev#/1} / {./2,false/0,nil/0,true/0,c_1/1,c_2/0 ,c_3/0,c_4/0,c_5/0,c_6/0,c_7/2,c_8/0} - Obligation: innermost runtime complexity wrt. defined symbols {++#,car#,cdr#,null#,rev#} and constructors {.,false,nil ,true} *** Step 1.b:6.a:1: PredecessorEstimationCP. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: ++#(.(x,y),z) -> c_1(++#(y,z)) - Weak DPs: rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y)) - Weak TRS: ++(.(x,y),z) -> .(x,++(y,z)) ++(nil(),y) -> y rev(.(x,y)) -> ++(rev(y),.(x,nil())) rev(nil()) -> nil() - Signature: {++/2,car/1,cdr/1,null/1,rev/1,++#/2,car#/1,cdr#/1,null#/1,rev#/1} / {./2,false/0,nil/0,true/0,c_1/1,c_2/0 ,c_3/0,c_4/0,c_5/0,c_6/0,c_7/2,c_8/0} - Obligation: innermost runtime complexity wrt. defined symbols {++#,car#,cdr#,null#,rev#} and constructors {.,false,nil ,true} + 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: ++#(.(x,y),z) -> c_1(++#(y,z)) The strictly oriented rules are moved into the weak component. **** Step 1.b:6.a:1.a:1: NaturalPI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: ++#(.(x,y),z) -> c_1(++#(y,z)) - Weak DPs: rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y)) - Weak TRS: ++(.(x,y),z) -> .(x,++(y,z)) ++(nil(),y) -> y rev(.(x,y)) -> ++(rev(y),.(x,nil())) rev(nil()) -> nil() - Signature: {++/2,car/1,cdr/1,null/1,rev/1,++#/2,car#/1,cdr#/1,null#/1,rev#/1} / {./2,false/0,nil/0,true/0,c_1/1,c_2/0 ,c_3/0,c_4/0,c_5/0,c_6/0,c_7/2,c_8/0} - Obligation: innermost runtime complexity wrt. defined symbols {++#,car#,cdr#,null#,rev#} and constructors {.,false,nil ,true} + 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_1) = {1}, uargs(c_7) = {1,2} Following symbols are considered usable: {++,rev,++#,car#,cdr#,null#,rev#} TcT has computed the following interpretation: p(++) = 4 + x1 + x2 p(.) = 2 + x2 p(car) = 1 p(cdr) = 1 + 2*x1 p(false) = 0 p(nil) = 0 p(null) = 2 p(rev) = 3*x1 p(true) = 2 p(++#) = 2 + 4*x1 + x2^2 p(car#) = 4 + x1^2 p(cdr#) = 1 + 2*x1 p(null#) = 1 p(rev#) = 3*x1^2 p(c_1) = 1 + x1 p(c_2) = 1 p(c_3) = 1 p(c_4) = 4 p(c_5) = 0 p(c_6) = 2 p(c_7) = x1 + x2 p(c_8) = 0 Following rules are strictly oriented: ++#(.(x,y),z) = 10 + 4*y + z^2 > 3 + 4*y + z^2 = c_1(++#(y,z)) Following rules are (at-least) weakly oriented: rev#(.(x,y)) = 12 + 12*y + 3*y^2 >= 6 + 12*y + 3*y^2 = c_7(++#(rev(y),.(x,nil())),rev#(y)) ++(.(x,y),z) = 6 + y + z >= 6 + y + z = .(x,++(y,z)) ++(nil(),y) = 4 + y >= y = y rev(.(x,y)) = 6 + 3*y >= 6 + 3*y = ++(rev(y),.(x,nil())) rev(nil()) = 0 >= 0 = nil() **** Step 1.b:6.a:1.a:2: Assumption. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: ++#(.(x,y),z) -> c_1(++#(y,z)) rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y)) - Weak TRS: ++(.(x,y),z) -> .(x,++(y,z)) ++(nil(),y) -> y rev(.(x,y)) -> ++(rev(y),.(x,nil())) rev(nil()) -> nil() - Signature: {++/2,car/1,cdr/1,null/1,rev/1,++#/2,car#/1,cdr#/1,null#/1,rev#/1} / {./2,false/0,nil/0,true/0,c_1/1,c_2/0 ,c_3/0,c_4/0,c_5/0,c_6/0,c_7/2,c_8/0} - Obligation: innermost runtime complexity wrt. defined symbols {++#,car#,cdr#,null#,rev#} and constructors {.,false,nil ,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}} + Details: () **** Step 1.b:6.a:1.b:1: RemoveWeakSuffixes. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: ++#(.(x,y),z) -> c_1(++#(y,z)) rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y)) - Weak TRS: ++(.(x,y),z) -> .(x,++(y,z)) ++(nil(),y) -> y rev(.(x,y)) -> ++(rev(y),.(x,nil())) rev(nil()) -> nil() - Signature: {++/2,car/1,cdr/1,null/1,rev/1,++#/2,car#/1,cdr#/1,null#/1,rev#/1} / {./2,false/0,nil/0,true/0,c_1/1,c_2/0 ,c_3/0,c_4/0,c_5/0,c_6/0,c_7/2,c_8/0} - Obligation: innermost runtime complexity wrt. defined symbols {++#,car#,cdr#,null#,rev#} and constructors {.,false,nil ,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:++#(.(x,y),z) -> c_1(++#(y,z)) -->_1 ++#(.(x,y),z) -> c_1(++#(y,z)):1 2:W:rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y)) -->_2 rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y)):2 -->_1 ++#(.(x,y),z) -> c_1(++#(y,z)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y)) 1: ++#(.(x,y),z) -> c_1(++#(y,z)) **** Step 1.b:6.a:1.b:2: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: ++(.(x,y),z) -> .(x,++(y,z)) ++(nil(),y) -> y rev(.(x,y)) -> ++(rev(y),.(x,nil())) rev(nil()) -> nil() - Signature: {++/2,car/1,cdr/1,null/1,rev/1,++#/2,car#/1,cdr#/1,null#/1,rev#/1} / {./2,false/0,nil/0,true/0,c_1/1,c_2/0 ,c_3/0,c_4/0,c_5/0,c_6/0,c_7/2,c_8/0} - Obligation: innermost runtime complexity wrt. defined symbols {++#,car#,cdr#,null#,rev#} and constructors {.,false,nil ,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 1.b:6.b:1: RemoveWeakSuffixes. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y)) - Weak DPs: ++#(.(x,y),z) -> c_1(++#(y,z)) - Weak TRS: ++(.(x,y),z) -> .(x,++(y,z)) ++(nil(),y) -> y rev(.(x,y)) -> ++(rev(y),.(x,nil())) rev(nil()) -> nil() - Signature: {++/2,car/1,cdr/1,null/1,rev/1,++#/2,car#/1,cdr#/1,null#/1,rev#/1} / {./2,false/0,nil/0,true/0,c_1/1,c_2/0 ,c_3/0,c_4/0,c_5/0,c_6/0,c_7/2,c_8/0} - Obligation: innermost runtime complexity wrt. defined symbols {++#,car#,cdr#,null#,rev#} and constructors {.,false,nil ,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y)) -->_1 ++#(.(x,y),z) -> c_1(++#(y,z)):2 -->_2 rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y)):1 2:W:++#(.(x,y),z) -> c_1(++#(y,z)) -->_1 ++#(.(x,y),z) -> c_1(++#(y,z)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: ++#(.(x,y),z) -> c_1(++#(y,z)) *** Step 1.b:6.b:2: SimplifyRHS. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y)) - Weak TRS: ++(.(x,y),z) -> .(x,++(y,z)) ++(nil(),y) -> y rev(.(x,y)) -> ++(rev(y),.(x,nil())) rev(nil()) -> nil() - Signature: {++/2,car/1,cdr/1,null/1,rev/1,++#/2,car#/1,cdr#/1,null#/1,rev#/1} / {./2,false/0,nil/0,true/0,c_1/1,c_2/0 ,c_3/0,c_4/0,c_5/0,c_6/0,c_7/2,c_8/0} - Obligation: innermost runtime complexity wrt. defined symbols {++#,car#,cdr#,null#,rev#} and constructors {.,false,nil ,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y)) -->_2 rev#(.(x,y)) -> c_7(++#(rev(y),.(x,nil())),rev#(y)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: rev#(.(x,y)) -> c_7(rev#(y)) *** Step 1.b:6.b:3: UsableRules. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: rev#(.(x,y)) -> c_7(rev#(y)) - Weak TRS: ++(.(x,y),z) -> .(x,++(y,z)) ++(nil(),y) -> y rev(.(x,y)) -> ++(rev(y),.(x,nil())) rev(nil()) -> nil() - Signature: {++/2,car/1,cdr/1,null/1,rev/1,++#/2,car#/1,cdr#/1,null#/1,rev#/1} / {./2,false/0,nil/0,true/0,c_1/1,c_2/0 ,c_3/0,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0} - Obligation: innermost runtime complexity wrt. defined symbols {++#,car#,cdr#,null#,rev#} and constructors {.,false,nil ,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: rev#(.(x,y)) -> c_7(rev#(y)) *** Step 1.b:6.b:4: PredecessorEstimationCP. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: rev#(.(x,y)) -> c_7(rev#(y)) - Signature: {++/2,car/1,cdr/1,null/1,rev/1,++#/2,car#/1,cdr#/1,null#/1,rev#/1} / {./2,false/0,nil/0,true/0,c_1/1,c_2/0 ,c_3/0,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0} - Obligation: innermost runtime complexity wrt. defined symbols {++#,car#,cdr#,null#,rev#} and constructors {.,false,nil ,true} + 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: rev#(.(x,y)) -> c_7(rev#(y)) The strictly oriented rules are moved into the weak component. **** Step 1.b:6.b:4.a:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: rev#(.(x,y)) -> c_7(rev#(y)) - Signature: {++/2,car/1,cdr/1,null/1,rev/1,++#/2,car#/1,cdr#/1,null#/1,rev#/1} / {./2,false/0,nil/0,true/0,c_1/1,c_2/0 ,c_3/0,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0} - Obligation: innermost runtime complexity wrt. defined symbols {++#,car#,cdr#,null#,rev#} and constructors {.,false,nil ,true} + 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_7) = {1} Following symbols are considered usable: {++#,car#,cdr#,null#,rev#} TcT has computed the following interpretation: p(++) = [2] x1 + [1] x2 + [0] p(.) = [1] x2 + [4] p(car) = [2] x1 + [1] p(cdr) = [0] p(false) = [0] p(nil) = [1] p(null) = [0] p(rev) = [4] x1 + [1] p(true) = [0] p(++#) = [1] x1 + [1] p(car#) = [1] x1 + [1] p(cdr#) = [1] x1 + [2] p(null#) = [1] x1 + [4] p(rev#) = [4] x1 + [8] p(c_1) = [1] x1 + [0] p(c_2) = [1] p(c_3) = [1] p(c_4) = [1] p(c_5) = [0] p(c_6) = [0] p(c_7) = [1] x1 + [12] p(c_8) = [0] Following rules are strictly oriented: rev#(.(x,y)) = [4] y + [24] > [4] y + [20] = c_7(rev#(y)) Following rules are (at-least) weakly oriented: **** Step 1.b:6.b:4.a:2: Assumption. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: rev#(.(x,y)) -> c_7(rev#(y)) - Signature: {++/2,car/1,cdr/1,null/1,rev/1,++#/2,car#/1,cdr#/1,null#/1,rev#/1} / {./2,false/0,nil/0,true/0,c_1/1,c_2/0 ,c_3/0,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0} - Obligation: innermost runtime complexity wrt. defined symbols {++#,car#,cdr#,null#,rev#} and constructors {.,false,nil ,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}} + Details: () **** Step 1.b:6.b:4.b:1: RemoveWeakSuffixes. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: rev#(.(x,y)) -> c_7(rev#(y)) - Signature: {++/2,car/1,cdr/1,null/1,rev/1,++#/2,car#/1,cdr#/1,null#/1,rev#/1} / {./2,false/0,nil/0,true/0,c_1/1,c_2/0 ,c_3/0,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0} - Obligation: innermost runtime complexity wrt. defined symbols {++#,car#,cdr#,null#,rev#} and constructors {.,false,nil ,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:rev#(.(x,y)) -> c_7(rev#(y)) -->_1 rev#(.(x,y)) -> c_7(rev#(y)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: rev#(.(x,y)) -> c_7(rev#(y)) **** Step 1.b:6.b:4.b:2: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Signature: {++/2,car/1,cdr/1,null/1,rev/1,++#/2,car#/1,cdr#/1,null#/1,rev#/1} / {./2,false/0,nil/0,true/0,c_1/1,c_2/0 ,c_3/0,c_4/0,c_5/0,c_6/0,c_7/1,c_8/0} - Obligation: innermost runtime complexity wrt. defined symbols {++#,car#,cdr#,null#,rev#} and constructors {.,false,nil ,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^2))