MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: f(0(),0(),0(),0(),0()) -> 0() f(0(),0(),0(),0(),s(x5)) -> f(x5,x5,x5,x5,x5) f(0(),0(),0(),s(x4),x5) -> f(x4,x4,x4,x4,x5) f(0(),0(),s(x3),x4,x5) -> f(x3,x3,x3,x4,x5) f(0(),s(x2),x3,x4,x5) -> f(x2,x2,x3,x4,x5) f(s(x1),x2,x3,x4,x5) -> f(x1,x2,x3,x4,x5) - Signature: {f/5} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {f} and constructors {0,s} + Applied Processor: DependencyPairs {dpKind_ = WIDP} + Details: We add the following weak innermost dependency pairs: Strict DPs f#(0(),0(),0(),0(),0()) -> c_1() f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5)) f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5)) f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5)) f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5)) f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5)) Weak DPs and mark the set of starting terms. * Step 2: UsableRules MAYBE + Considered Problem: - Strict DPs: f#(0(),0(),0(),0(),0()) -> c_1() f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5)) f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5)) f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5)) f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5)) f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5)) - Strict TRS: f(0(),0(),0(),0(),0()) -> 0() f(0(),0(),0(),0(),s(x5)) -> f(x5,x5,x5,x5,x5) f(0(),0(),0(),s(x4),x5) -> f(x4,x4,x4,x4,x5) f(0(),0(),s(x3),x4,x5) -> f(x3,x3,x3,x4,x5) f(0(),s(x2),x3,x4,x5) -> f(x2,x2,x3,x4,x5) f(s(x1),x2,x3,x4,x5) -> f(x1,x2,x3,x4,x5) - Signature: {f/5,f#/5} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#} and constructors {0,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: f#(0(),0(),0(),0(),0()) -> c_1() f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5)) f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5)) f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5)) f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5)) f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5)) * Step 3: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: f#(0(),0(),0(),0(),0()) -> c_1() f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5)) f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5)) f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5)) f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5)) f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5)) - Signature: {f/5,f#/5} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#} and constructors {0,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1} by application of Pre({1}) = {2,3,4,5,6}. Here rules are labelled as follows: 1: f#(0(),0(),0(),0(),0()) -> c_1() 2: f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5)) 3: f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5)) 4: f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5)) 5: f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5)) 6: f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5)) * Step 4: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5)) f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5)) f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5)) f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5)) f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5)) - Weak DPs: f#(0(),0(),0(),0(),0()) -> c_1() - Signature: {f/5,f#/5} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#} and constructors {0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5)) -->_1 f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5)):5 -->_1 f#(0(),0(),0(),0(),0()) -> c_1():6 2:S:f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5)) -->_1 f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5)):5 -->_1 f#(0(),0(),0(),0(),0()) -> c_1():6 -->_1 f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5)):1 3:S:f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5)) -->_1 f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5)):5 -->_1 f#(0(),0(),0(),0(),0()) -> c_1():6 -->_1 f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5)):2 -->_1 f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5)):1 4:S:f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5)) -->_1 f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5)):5 -->_1 f#(0(),0(),0(),0(),0()) -> c_1():6 -->_1 f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5)):3 -->_1 f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5)):2 -->_1 f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5)):1 5:S:f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5)) -->_1 f#(0(),0(),0(),0(),0()) -> c_1():6 -->_1 f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5)):5 -->_1 f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5)):4 -->_1 f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5)):3 -->_1 f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5)):2 -->_1 f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5)):1 6:W:f#(0(),0(),0(),0(),0()) -> c_1() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 6: f#(0(),0(),0(),0(),0()) -> c_1() * Step 5: PredecessorEstimationCP MAYBE + Considered Problem: - Strict DPs: f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5)) f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5)) f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5)) f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5)) f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5)) - Signature: {f/5,f#/5} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#} and constructors {0,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: f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5)) The strictly oriented rules are moved into the weak component. ** Step 5.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5)) f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5)) f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5)) f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5)) f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5)) - Signature: {f/5,f#/5} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#} and constructors {0,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_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1} Following symbols are considered usable: {f#} TcT has computed the following interpretation: p(0) = [0] p(f) = [1] x1 + [1] x2 + [1] x4 + [1] x5 + [8] p(s) = [1] x1 + [4] p(f#) = [2] x5 + [0] p(c_1) = [1] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [0] p(c_6) = [1] x1 + [0] Following rules are strictly oriented: f#(0(),0(),0(),0(),s(x5)) = [2] x5 + [8] > [2] x5 + [0] = c_2(f#(x5,x5,x5,x5,x5)) Following rules are (at-least) weakly oriented: f#(0(),0(),0(),s(x4),x5) = [2] x5 + [0] >= [2] x5 + [0] = c_3(f#(x4,x4,x4,x4,x5)) f#(0(),0(),s(x3),x4,x5) = [2] x5 + [0] >= [2] x5 + [0] = c_4(f#(x3,x3,x3,x4,x5)) f#(0(),s(x2),x3,x4,x5) = [2] x5 + [0] >= [2] x5 + [0] = c_5(f#(x2,x2,x3,x4,x5)) f#(s(x1),x2,x3,x4,x5) = [2] x5 + [0] >= [2] x5 + [0] = c_6(f#(x1,x2,x3,x4,x5)) ** Step 5.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5)) f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5)) f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5)) f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5)) - Weak DPs: f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5)) - Signature: {f/5,f#/5} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#} and constructors {0,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ** Step 5.b:1: PredecessorEstimationCP MAYBE + Considered Problem: - Strict DPs: f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5)) f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5)) f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5)) f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5)) - Weak DPs: f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5)) - Signature: {f/5,f#/5} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#} and constructors {0,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: f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5)) The strictly oriented rules are moved into the weak component. *** Step 5.b:1.a:1: NaturalPI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5)) f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5)) f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5)) f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5)) - Weak DPs: f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5)) - Signature: {f/5,f#/5} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#} and constructors {0,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_5) = {1}, uargs(c_6) = {1} Following symbols are considered usable: {f#} TcT has computed the following interpretation: p(0) = 0 p(f) = 4 + 2*x1*x3 + x2 + x2*x4 + 2*x2*x5 + 2*x2^2 + x3*x5 + 2*x3^2 + x4*x5 + x5 p(s) = 2 + x1 p(f#) = 6*x4 + 3*x5^2 p(c_1) = 1 p(c_2) = x1 p(c_3) = 4 + x1 p(c_4) = x1 p(c_5) = x1 p(c_6) = x1 Following rules are strictly oriented: f#(0(),0(),0(),s(x4),x5) = 12 + 6*x4 + 3*x5^2 > 4 + 6*x4 + 3*x5^2 = c_3(f#(x4,x4,x4,x4,x5)) Following rules are (at-least) weakly oriented: f#(0(),0(),0(),0(),s(x5)) = 12 + 12*x5 + 3*x5^2 >= 6*x5 + 3*x5^2 = c_2(f#(x5,x5,x5,x5,x5)) f#(0(),0(),s(x3),x4,x5) = 6*x4 + 3*x5^2 >= 6*x4 + 3*x5^2 = c_4(f#(x3,x3,x3,x4,x5)) f#(0(),s(x2),x3,x4,x5) = 6*x4 + 3*x5^2 >= 6*x4 + 3*x5^2 = c_5(f#(x2,x2,x3,x4,x5)) f#(s(x1),x2,x3,x4,x5) = 6*x4 + 3*x5^2 >= 6*x4 + 3*x5^2 = c_6(f#(x1,x2,x3,x4,x5)) *** Step 5.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5)) f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5)) f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5)) - Weak DPs: f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5)) f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5)) - Signature: {f/5,f#/5} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#} and constructors {0,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 5.b:1.b:1: NaturalMI MAYBE + Considered Problem: - Strict DPs: f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5)) f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5)) f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5)) - Weak DPs: f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5)) f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5)) - Signature: {f/5,f#/5} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#} and constructors {0,s} + Applied Processor: NaturalMI {miDimension = 4, miDegree = 4, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any 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_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1} Following symbols are considered usable: {f#} TcT has computed the following interpretation: p(0) = [0] [0] [0] [0] p(f) = [0] [0] [0] [0] p(s) = [1 1 1 1] [0] [0 0 0 0] x1 + [0] [0 0 1 1] [0] [0 0 0 1] [1] p(f#) = [0 0 0 0] [0 0 0 0] [0 0 0 1] [0 0 1 1] [1 0 1 0] [0] [0 0 0 0] x1 + [0 1 0 1] x2 + [0 0 1 1] x3 + [0 0 0 0] x4 + [0 0 0 0] x5 + [0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 1 0 0] [0] [0 1 0 0] [1 1 0 0] [0 0 1 1] [1 0 0 0] [0 0 0 0] [0] p(c_1) = [0] [0] [0] [0] p(c_2) = [1 0 1 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] p(c_3) = [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] p(c_4) = [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [0] p(c_5) = [1 0 0 0] [0] [0 0 0 0] x1 + [1] [0 0 1 0] [0] [0 1 0 0] [0] p(c_6) = [1 0 0 0] [0] [0 0 0 0] x1 + [0] [0 0 1 0] [0] [0 0 0 0] [0] Following rules are strictly oriented: f#(0(),0(),s(x3),x4,x5) = [0 0 0 1] [0 0 1 1] [1 0 1 0] [1] [0 0 1 2] x3 + [0 0 0 0] x4 + [0 0 0 0] x5 + [1] [0 0 0 0] [0 0 0 0] [0 1 0 0] [0] [0 0 1 2] [1 0 0 0] [0 0 0 0] [1] > [0 0 0 1] [0 0 1 1] [1 0 1 0] [0] [0 0 0 0] x3 + [0 0 0 0] x4 + [0 0 0 0] x5 + [0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] = c_4(f#(x3,x3,x3,x4,x5)) Following rules are (at-least) weakly oriented: f#(0(),0(),0(),0(),s(x5)) = [1 1 2 2] [0] [0 0 0 0] x5 + [0] [0 0 0 0] [0] [0 0 0 0] [0] >= [1 1 2 2] [0] [0 0 0 0] x5 + [0] [0 0 0 0] [0] [0 0 0 0] [0] = c_2(f#(x5,x5,x5,x5,x5)) f#(0(),0(),0(),s(x4),x5) = [0 0 1 2] [1 0 1 0] [1] [0 0 0 0] x4 + [0 0 0 0] x5 + [0] [0 0 0 0] [0 1 0 0] [0] [1 1 1 1] [0 0 0 0] [0] >= [0 0 1 2] [1 0 1 0] [0] [0 0 0 0] x4 + [0 0 0 0] x5 + [0] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0] = c_3(f#(x4,x4,x4,x4,x5)) f#(0(),s(x2),x3,x4,x5) = [0 0 0 0] [0 0 0 1] [0 0 1 1] [1 0 1 0] [0] [0 0 0 1] x2 + [0 0 1 1] x3 + [0 0 0 0] x4 + [0 0 0 0] x5 + [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 1 0 0] [0] [1 1 1 1] [0 0 1 1] [1 0 0 0] [0 0 0 0] [0] >= [0 0 0 0] [0 0 0 1] [0 0 1 1] [1 0 1 0] [0] [0 0 0 0] x2 + [0 0 0 0] x3 + [0 0 0 0] x4 + [0 0 0 0] x5 + [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 1 0 0] [0] [0 1 0 1] [0 0 1 1] [0 0 0 0] [0 0 0 0] [0] = c_5(f#(x2,x2,x3,x4,x5)) f#(s(x1),x2,x3,x4,x5) = [0 0 0 0] [0 0 0 1] [0 0 1 1] [1 0 1 0] [0] [0 1 0 1] x2 + [0 0 1 1] x3 + [0 0 0 0] x4 + [0 0 0 0] x5 + [0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 1 0 0] [0] [1 1 0 0] [0 0 1 1] [1 0 0 0] [0 0 0 0] [0] >= [0 0 0 1] [0 0 1 1] [1 0 1 0] [0] [0 0 0 0] x3 + [0 0 0 0] x4 + [0 0 0 0] x5 + [0] [0 0 0 0] [0 0 0 0] [0 1 0 0] [0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] = c_6(f#(x1,x2,x3,x4,x5)) *** Step 5.b:1.b:2: NaturalMI MAYBE + Considered Problem: - Strict DPs: f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5)) f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5)) - Weak DPs: f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5)) f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5)) f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5)) - Signature: {f/5,f#/5} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#} and constructors {0,s} + Applied Processor: NaturalMI {miDimension = 4, miDegree = 4, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just any 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_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1}, uargs(c_6) = {1} Following symbols are considered usable: {f#} TcT has computed the following interpretation: p(0) = [0] [0] [1] [0] p(f) = [0] [0] [0] [0] p(s) = [1 1 1 1] [0] [0 1 1 1] x1 + [0] [0 0 1 1] [1] [0 0 0 1] [1] p(f#) = [0 0 0 0] [0 0 0 1] [0 0 1 0] [0 1 0 0] [1 0 1 1] [0] [0 1 0 0] x1 + [0 1 0 0] x2 + [0 1 0 0] x3 + [0 0 1 1] x4 + [0 0 0 0] x5 + [0] [1 1 0 0] [0 0 1 0] [0 0 0 1] [0 0 0 0] [0 0 1 0] [1] [0 0 0 0] [0 1 1 1] [0 1 0 0] [0 0 0 0] [0 0 0 0] [0] p(c_1) = [0] [0] [0] [0] p(c_2) = [1 0 0 0] [1] [0 0 0 0] x1 + [1] [0 0 0 0] [1] [0 0 0 0] [0] p(c_3) = [1 0 0 0] [0] [0 0 0 0] x1 + [1] [0 0 0 0] [1] [0 0 0 0] [1] p(c_4) = [1 0 0 0] [1] [0 0 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [1] p(c_5) = [1 0 0 0] [0] [0 0 0 1] x1 + [0] [0 0 0 0] [0] [0 0 0 0] [1] p(c_6) = [1 0 0 0] [0] [0 1 0 0] x1 + [0] [0 0 0 0] [1] [0 0 0 0] [0] Following rules are strictly oriented: f#(0(),s(x2),x3,x4,x5) = [0 0 0 1] [0 0 1 0] [0 1 0 0] [1 0 1 1] [1] [0 1 1 1] x2 + [0 1 0 0] x3 + [0 0 1 1] x4 + [0 0 0 0] x5 + [0] [0 0 1 1] [0 0 0 1] [0 0 0 0] [0 0 1 0] [2] [0 1 2 3] [0 1 0 0] [0 0 0 0] [0 0 0 0] [2] > [0 0 0 1] [0 0 1 0] [0 1 0 0] [1 0 1 1] [0] [0 1 1 1] x2 + [0 1 0 0] x3 + [0 0 0 0] x4 + [0 0 0 0] x5 + [0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] = c_5(f#(x2,x2,x3,x4,x5)) Following rules are (at-least) weakly oriented: f#(0(),0(),0(),0(),s(x5)) = [1 1 2 3] [3] [0 0 0 0] x5 + [1] [0 0 1 1] [3] [0 0 0 0] [1] >= [1 1 2 2] [1] [0 0 0 0] x5 + [1] [0 0 0 0] [1] [0 0 0 0] [0] = c_2(f#(x5,x5,x5,x5,x5)) f#(0(),0(),0(),s(x4),x5) = [0 1 1 1] [1 0 1 1] [1] [0 0 1 2] x4 + [0 0 0 0] x5 + [2] [0 0 0 0] [0 0 1 0] [2] [0 0 0 0] [0 0 0 0] [1] >= [0 1 1 1] [1 0 1 1] [0] [0 0 0 0] x4 + [0 0 0 0] x5 + [1] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [1] = c_3(f#(x4,x4,x4,x4,x5)) f#(0(),0(),s(x3),x4,x5) = [0 0 1 1] [0 1 0 0] [1 0 1 1] [1] [0 1 1 1] x3 + [0 0 1 1] x4 + [0 0 0 0] x5 + [0] [0 0 0 1] [0 0 0 0] [0 0 1 0] [3] [0 1 1 1] [0 0 0 0] [0 0 0 0] [1] >= [0 0 1 1] [0 1 0 0] [1 0 1 1] [1] [0 0 0 0] x3 + [0 0 0 0] x4 + [0 0 0 0] x5 + [0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] = c_4(f#(x3,x3,x3,x4,x5)) f#(s(x1),x2,x3,x4,x5) = [0 0 0 0] [0 0 0 1] [0 0 1 0] [0 1 0 0] [1 0 1 1] [0] [0 1 1 1] x1 + [0 1 0 0] x2 + [0 1 0 0] x3 + [0 0 1 1] x4 + [0 0 0 0] x5 + [0] [1 2 2 2] [0 0 1 0] [0 0 0 1] [0 0 0 0] [0 0 1 0] [1] [0 0 0 0] [0 1 1 1] [0 1 0 0] [0 0 0 0] [0 0 0 0] [0] >= [0 0 0 0] [0 0 0 1] [0 0 1 0] [0 1 0 0] [1 0 1 1] [0] [0 1 0 0] x1 + [0 1 0 0] x2 + [0 1 0 0] x3 + [0 0 1 1] x4 + [0 0 0 0] x5 + [0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [1] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0 0 0 0] [0] = c_6(f#(x1,x2,x3,x4,x5)) *** Step 5.b:1.b:3: Failure MAYBE + Considered Problem: - Strict DPs: f#(s(x1),x2,x3,x4,x5) -> c_6(f#(x1,x2,x3,x4,x5)) - Weak DPs: f#(0(),0(),0(),0(),s(x5)) -> c_2(f#(x5,x5,x5,x5,x5)) f#(0(),0(),0(),s(x4),x5) -> c_3(f#(x4,x4,x4,x4,x5)) f#(0(),0(),s(x3),x4,x5) -> c_4(f#(x3,x3,x3,x4,x5)) f#(0(),s(x2),x3,x4,x5) -> c_5(f#(x2,x2,x3,x4,x5)) - Signature: {f/5,f#/5} / {0/0,s/1,c_1/0,c_2/1,c_3/1,c_4/1,c_5/1,c_6/1} - Obligation: innermost runtime complexity wrt. defined symbols {f#} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is still open. MAYBE