MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: *(x,0()) -> 0() *(x,s(y)) -> +(*(x,y),x) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) 1() -> s(0()) fac(0()) -> 1() fac(0()) -> s(0()) fac(s(x)) -> *(s(x),fac(x)) floop(0(),y) -> y floop(s(x),y) -> floop(x,*(s(x),y)) - Signature: {*/2,+/2,1/0,fac/1,floop/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {*,+,1,fac,floop} and constructors {0,s} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs *#(x,0()) -> c_1() *#(x,s(y)) -> c_2(+#(*(x,y),x),*#(x,y)) +#(x,0()) -> c_3() +#(x,s(y)) -> c_4(+#(x,y)) 1#() -> c_5() fac#(0()) -> c_6(1#()) fac#(0()) -> c_7() fac#(s(x)) -> c_8(*#(s(x),fac(x)),fac#(x)) floop#(0(),y) -> c_9() floop#(s(x),y) -> c_10(floop#(x,*(s(x),y)),*#(s(x),y)) Weak DPs and mark the set of starting terms. * Step 2: UsableRules MAYBE + Considered Problem: - Strict DPs: *#(x,0()) -> c_1() *#(x,s(y)) -> c_2(+#(*(x,y),x),*#(x,y)) +#(x,0()) -> c_3() +#(x,s(y)) -> c_4(+#(x,y)) 1#() -> c_5() fac#(0()) -> c_6(1#()) fac#(0()) -> c_7() fac#(s(x)) -> c_8(*#(s(x),fac(x)),fac#(x)) floop#(0(),y) -> c_9() floop#(s(x),y) -> c_10(floop#(x,*(s(x),y)),*#(s(x),y)) - Weak TRS: *(x,0()) -> 0() *(x,s(y)) -> +(*(x,y),x) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) 1() -> s(0()) fac(0()) -> 1() fac(0()) -> s(0()) fac(s(x)) -> *(s(x),fac(x)) floop(0(),y) -> y floop(s(x),y) -> floop(x,*(s(x),y)) - Signature: {*/2,+/2,1/0,fac/1,floop/2,*#/2,+#/2,1#/0,fac#/1,floop#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1 ,c_7/0,c_8/2,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,1#,fac#,floop#} and constructors {0,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: *(x,0()) -> 0() *(x,s(y)) -> +(*(x,y),x) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) 1() -> s(0()) fac(0()) -> 1() fac(0()) -> s(0()) fac(s(x)) -> *(s(x),fac(x)) *#(x,0()) -> c_1() *#(x,s(y)) -> c_2(+#(*(x,y),x),*#(x,y)) +#(x,0()) -> c_3() +#(x,s(y)) -> c_4(+#(x,y)) 1#() -> c_5() fac#(0()) -> c_6(1#()) fac#(0()) -> c_7() fac#(s(x)) -> c_8(*#(s(x),fac(x)),fac#(x)) floop#(0(),y) -> c_9() floop#(s(x),y) -> c_10(floop#(x,*(s(x),y)),*#(s(x),y)) * Step 3: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: *#(x,0()) -> c_1() *#(x,s(y)) -> c_2(+#(*(x,y),x),*#(x,y)) +#(x,0()) -> c_3() +#(x,s(y)) -> c_4(+#(x,y)) 1#() -> c_5() fac#(0()) -> c_6(1#()) fac#(0()) -> c_7() fac#(s(x)) -> c_8(*#(s(x),fac(x)),fac#(x)) floop#(0(),y) -> c_9() floop#(s(x),y) -> c_10(floop#(x,*(s(x),y)),*#(s(x),y)) - Weak TRS: *(x,0()) -> 0() *(x,s(y)) -> +(*(x,y),x) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) 1() -> s(0()) fac(0()) -> 1() fac(0()) -> s(0()) fac(s(x)) -> *(s(x),fac(x)) - Signature: {*/2,+/2,1/0,fac/1,floop/2,*#/2,+#/2,1#/0,fac#/1,floop#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1 ,c_7/0,c_8/2,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,1#,fac#,floop#} and constructors {0,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,3,5,7,9} by application of Pre({1,3,5,7,9}) = {2,4,6,8,10}. Here rules are labelled as follows: 1: *#(x,0()) -> c_1() 2: *#(x,s(y)) -> c_2(+#(*(x,y),x),*#(x,y)) 3: +#(x,0()) -> c_3() 4: +#(x,s(y)) -> c_4(+#(x,y)) 5: 1#() -> c_5() 6: fac#(0()) -> c_6(1#()) 7: fac#(0()) -> c_7() 8: fac#(s(x)) -> c_8(*#(s(x),fac(x)),fac#(x)) 9: floop#(0(),y) -> c_9() 10: floop#(s(x),y) -> c_10(floop#(x,*(s(x),y)),*#(s(x),y)) * Step 4: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: *#(x,s(y)) -> c_2(+#(*(x,y),x),*#(x,y)) +#(x,s(y)) -> c_4(+#(x,y)) fac#(0()) -> c_6(1#()) fac#(s(x)) -> c_8(*#(s(x),fac(x)),fac#(x)) floop#(s(x),y) -> c_10(floop#(x,*(s(x),y)),*#(s(x),y)) - Weak DPs: *#(x,0()) -> c_1() +#(x,0()) -> c_3() 1#() -> c_5() fac#(0()) -> c_7() floop#(0(),y) -> c_9() - Weak TRS: *(x,0()) -> 0() *(x,s(y)) -> +(*(x,y),x) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) 1() -> s(0()) fac(0()) -> 1() fac(0()) -> s(0()) fac(s(x)) -> *(s(x),fac(x)) - Signature: {*/2,+/2,1/0,fac/1,floop/2,*#/2,+#/2,1#/0,fac#/1,floop#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1 ,c_7/0,c_8/2,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,1#,fac#,floop#} and constructors {0,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {3} by application of Pre({3}) = {4}. Here rules are labelled as follows: 1: *#(x,s(y)) -> c_2(+#(*(x,y),x),*#(x,y)) 2: +#(x,s(y)) -> c_4(+#(x,y)) 3: fac#(0()) -> c_6(1#()) 4: fac#(s(x)) -> c_8(*#(s(x),fac(x)),fac#(x)) 5: floop#(s(x),y) -> c_10(floop#(x,*(s(x),y)),*#(s(x),y)) 6: *#(x,0()) -> c_1() 7: +#(x,0()) -> c_3() 8: 1#() -> c_5() 9: fac#(0()) -> c_7() 10: floop#(0(),y) -> c_9() * Step 5: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: *#(x,s(y)) -> c_2(+#(*(x,y),x),*#(x,y)) +#(x,s(y)) -> c_4(+#(x,y)) fac#(s(x)) -> c_8(*#(s(x),fac(x)),fac#(x)) floop#(s(x),y) -> c_10(floop#(x,*(s(x),y)),*#(s(x),y)) - Weak DPs: *#(x,0()) -> c_1() +#(x,0()) -> c_3() 1#() -> c_5() fac#(0()) -> c_6(1#()) fac#(0()) -> c_7() floop#(0(),y) -> c_9() - Weak TRS: *(x,0()) -> 0() *(x,s(y)) -> +(*(x,y),x) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) 1() -> s(0()) fac(0()) -> 1() fac(0()) -> s(0()) fac(s(x)) -> *(s(x),fac(x)) - Signature: {*/2,+/2,1/0,fac/1,floop/2,*#/2,+#/2,1#/0,fac#/1,floop#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1 ,c_7/0,c_8/2,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,1#,fac#,floop#} and constructors {0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:*#(x,s(y)) -> c_2(+#(*(x,y),x),*#(x,y)) -->_1 +#(x,s(y)) -> c_4(+#(x,y)):2 -->_1 +#(x,0()) -> c_3():6 -->_2 *#(x,0()) -> c_1():5 -->_2 *#(x,s(y)) -> c_2(+#(*(x,y),x),*#(x,y)):1 2:S:+#(x,s(y)) -> c_4(+#(x,y)) -->_1 +#(x,0()) -> c_3():6 -->_1 +#(x,s(y)) -> c_4(+#(x,y)):2 3:S:fac#(s(x)) -> c_8(*#(s(x),fac(x)),fac#(x)) -->_2 fac#(0()) -> c_6(1#()):8 -->_2 fac#(0()) -> c_7():9 -->_1 *#(x,0()) -> c_1():5 -->_2 fac#(s(x)) -> c_8(*#(s(x),fac(x)),fac#(x)):3 -->_1 *#(x,s(y)) -> c_2(+#(*(x,y),x),*#(x,y)):1 4:S:floop#(s(x),y) -> c_10(floop#(x,*(s(x),y)),*#(s(x),y)) -->_1 floop#(0(),y) -> c_9():10 -->_2 *#(x,0()) -> c_1():5 -->_1 floop#(s(x),y) -> c_10(floop#(x,*(s(x),y)),*#(s(x),y)):4 -->_2 *#(x,s(y)) -> c_2(+#(*(x,y),x),*#(x,y)):1 5:W:*#(x,0()) -> c_1() 6:W:+#(x,0()) -> c_3() 7:W:1#() -> c_5() 8:W:fac#(0()) -> c_6(1#()) -->_1 1#() -> c_5():7 9:W:fac#(0()) -> c_7() 10:W:floop#(0(),y) -> c_9() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 10: floop#(0(),y) -> c_9() 9: fac#(0()) -> c_7() 8: fac#(0()) -> c_6(1#()) 7: 1#() -> c_5() 5: *#(x,0()) -> c_1() 6: +#(x,0()) -> c_3() * Step 6: Decompose MAYBE + Considered Problem: - Strict DPs: *#(x,s(y)) -> c_2(+#(*(x,y),x),*#(x,y)) +#(x,s(y)) -> c_4(+#(x,y)) fac#(s(x)) -> c_8(*#(s(x),fac(x)),fac#(x)) floop#(s(x),y) -> c_10(floop#(x,*(s(x),y)),*#(s(x),y)) - Weak TRS: *(x,0()) -> 0() *(x,s(y)) -> +(*(x,y),x) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) 1() -> s(0()) fac(0()) -> 1() fac(0()) -> s(0()) fac(s(x)) -> *(s(x),fac(x)) - Signature: {*/2,+/2,1/0,fac/1,floop/2,*#/2,+#/2,1#/0,fac#/1,floop#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1 ,c_7/0,c_8/2,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,1#,fac#,floop#} and constructors {0,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: *#(x,s(y)) -> c_2(+#(*(x,y),x),*#(x,y)) +#(x,s(y)) -> c_4(+#(x,y)) - Weak DPs: fac#(s(x)) -> c_8(*#(s(x),fac(x)),fac#(x)) floop#(s(x),y) -> c_10(floop#(x,*(s(x),y)),*#(s(x),y)) - Weak TRS: *(x,0()) -> 0() *(x,s(y)) -> +(*(x,y),x) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) 1() -> s(0()) fac(0()) -> 1() fac(0()) -> s(0()) fac(s(x)) -> *(s(x),fac(x)) - Signature: {*/2,+/2,1/0,fac/1,floop/2,*#/2,+#/2,1#/0,fac#/1,floop#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1 ,c_7/0,c_8/2,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,1#,fac#,floop#} and constructors {0,s} Problem (S) - Strict DPs: fac#(s(x)) -> c_8(*#(s(x),fac(x)),fac#(x)) floop#(s(x),y) -> c_10(floop#(x,*(s(x),y)),*#(s(x),y)) - Weak DPs: *#(x,s(y)) -> c_2(+#(*(x,y),x),*#(x,y)) +#(x,s(y)) -> c_4(+#(x,y)) - Weak TRS: *(x,0()) -> 0() *(x,s(y)) -> +(*(x,y),x) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) 1() -> s(0()) fac(0()) -> 1() fac(0()) -> s(0()) fac(s(x)) -> *(s(x),fac(x)) - Signature: {*/2,+/2,1/0,fac/1,floop/2,*#/2,+#/2,1#/0,fac#/1,floop#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1 ,c_7/0,c_8/2,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,1#,fac#,floop#} and constructors {0,s} ** Step 6.a:1: DecomposeDG MAYBE + Considered Problem: - Strict DPs: *#(x,s(y)) -> c_2(+#(*(x,y),x),*#(x,y)) +#(x,s(y)) -> c_4(+#(x,y)) - Weak DPs: fac#(s(x)) -> c_8(*#(s(x),fac(x)),fac#(x)) floop#(s(x),y) -> c_10(floop#(x,*(s(x),y)),*#(s(x),y)) - Weak TRS: *(x,0()) -> 0() *(x,s(y)) -> +(*(x,y),x) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) 1() -> s(0()) fac(0()) -> 1() fac(0()) -> s(0()) fac(s(x)) -> *(s(x),fac(x)) - Signature: {*/2,+/2,1/0,fac/1,floop/2,*#/2,+#/2,1#/0,fac#/1,floop#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1 ,c_7/0,c_8/2,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,1#,fac#,floop#} and constructors {0,s} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component fac#(s(x)) -> c_8(*#(s(x),fac(x)),fac#(x)) floop#(s(x),y) -> c_10(floop#(x,*(s(x),y)),*#(s(x),y)) and a lower component *#(x,s(y)) -> c_2(+#(*(x,y),x),*#(x,y)) +#(x,s(y)) -> c_4(+#(x,y)) Further, following extension rules are added to the lower component. fac#(s(x)) -> *#(s(x),fac(x)) fac#(s(x)) -> fac#(x) floop#(s(x),y) -> *#(s(x),y) floop#(s(x),y) -> floop#(x,*(s(x),y)) *** Step 6.a:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: fac#(s(x)) -> c_8(*#(s(x),fac(x)),fac#(x)) floop#(s(x),y) -> c_10(floop#(x,*(s(x),y)),*#(s(x),y)) - Weak TRS: *(x,0()) -> 0() *(x,s(y)) -> +(*(x,y),x) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) 1() -> s(0()) fac(0()) -> 1() fac(0()) -> s(0()) fac(s(x)) -> *(s(x),fac(x)) - Signature: {*/2,+/2,1/0,fac/1,floop/2,*#/2,+#/2,1#/0,fac#/1,floop#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1 ,c_7/0,c_8/2,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,1#,fac#,floop#} 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: 2: floop#(s(x),y) -> c_10(floop#(x,*(s(x),y)),*#(s(x),y)) The strictly oriented rules are moved into the weak component. **** Step 6.a:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: fac#(s(x)) -> c_8(*#(s(x),fac(x)),fac#(x)) floop#(s(x),y) -> c_10(floop#(x,*(s(x),y)),*#(s(x),y)) - Weak TRS: *(x,0()) -> 0() *(x,s(y)) -> +(*(x,y),x) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) 1() -> s(0()) fac(0()) -> 1() fac(0()) -> s(0()) fac(s(x)) -> *(s(x),fac(x)) - Signature: {*/2,+/2,1/0,fac/1,floop/2,*#/2,+#/2,1#/0,fac#/1,floop#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1 ,c_7/0,c_8/2,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,1#,fac#,floop#} 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_8) = {1,2}, uargs(c_10) = {1} Following symbols are considered usable: {*#,+#,1#,fac#,floop#} TcT has computed the following interpretation: p(*) = [2] x1 + [3] x2 + [2] p(+) = [1] x2 + [4] p(0) = [3] p(1) = [2] p(fac) = [0] p(floop) = [1] x1 + [8] x2 + [8] p(s) = [1] x1 + [8] p(*#) = [0] p(+#) = [1] p(1#) = [8] p(fac#) = [0] p(floop#) = [2] x1 + [0] p(c_1) = [1] p(c_2) = [0] p(c_3) = [1] p(c_4) = [0] p(c_5) = [4] p(c_6) = [2] x1 + [1] p(c_7) = [2] p(c_8) = [2] x1 + [4] x2 + [0] p(c_9) = [1] p(c_10) = [1] x1 + [1] x2 + [15] Following rules are strictly oriented: floop#(s(x),y) = [2] x + [16] > [2] x + [15] = c_10(floop#(x,*(s(x),y)),*#(s(x),y)) Following rules are (at-least) weakly oriented: fac#(s(x)) = [0] >= [0] = c_8(*#(s(x),fac(x)),fac#(x)) **** Step 6.a:1.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: fac#(s(x)) -> c_8(*#(s(x),fac(x)),fac#(x)) - Weak DPs: floop#(s(x),y) -> c_10(floop#(x,*(s(x),y)),*#(s(x),y)) - Weak TRS: *(x,0()) -> 0() *(x,s(y)) -> +(*(x,y),x) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) 1() -> s(0()) fac(0()) -> 1() fac(0()) -> s(0()) fac(s(x)) -> *(s(x),fac(x)) - Signature: {*/2,+/2,1/0,fac/1,floop/2,*#/2,+#/2,1#/0,fac#/1,floop#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1 ,c_7/0,c_8/2,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,1#,fac#,floop#} and constructors {0,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 6.a:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: fac#(s(x)) -> c_8(*#(s(x),fac(x)),fac#(x)) - Weak DPs: floop#(s(x),y) -> c_10(floop#(x,*(s(x),y)),*#(s(x),y)) - Weak TRS: *(x,0()) -> 0() *(x,s(y)) -> +(*(x,y),x) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) 1() -> s(0()) fac(0()) -> 1() fac(0()) -> s(0()) fac(s(x)) -> *(s(x),fac(x)) - Signature: {*/2,+/2,1/0,fac/1,floop/2,*#/2,+#/2,1#/0,fac#/1,floop#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1 ,c_7/0,c_8/2,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,1#,fac#,floop#} and constructors {0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:fac#(s(x)) -> c_8(*#(s(x),fac(x)),fac#(x)) -->_2 fac#(s(x)) -> c_8(*#(s(x),fac(x)),fac#(x)):1 2:W:floop#(s(x),y) -> c_10(floop#(x,*(s(x),y)),*#(s(x),y)) -->_1 floop#(s(x),y) -> c_10(floop#(x,*(s(x),y)),*#(s(x),y)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: floop#(s(x),y) -> c_10(floop#(x,*(s(x),y)),*#(s(x),y)) **** Step 6.a:1.a:1.b:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: fac#(s(x)) -> c_8(*#(s(x),fac(x)),fac#(x)) - Weak TRS: *(x,0()) -> 0() *(x,s(y)) -> +(*(x,y),x) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) 1() -> s(0()) fac(0()) -> 1() fac(0()) -> s(0()) fac(s(x)) -> *(s(x),fac(x)) - Signature: {*/2,+/2,1/0,fac/1,floop/2,*#/2,+#/2,1#/0,fac#/1,floop#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1 ,c_7/0,c_8/2,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,1#,fac#,floop#} and constructors {0,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:fac#(s(x)) -> c_8(*#(s(x),fac(x)),fac#(x)) -->_2 fac#(s(x)) -> c_8(*#(s(x),fac(x)),fac#(x)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: fac#(s(x)) -> c_8(fac#(x)) **** Step 6.a:1.a:1.b:3: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: fac#(s(x)) -> c_8(fac#(x)) - Weak TRS: *(x,0()) -> 0() *(x,s(y)) -> +(*(x,y),x) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) 1() -> s(0()) fac(0()) -> 1() fac(0()) -> s(0()) fac(s(x)) -> *(s(x),fac(x)) - Signature: {*/2,+/2,1/0,fac/1,floop/2,*#/2,+#/2,1#/0,fac#/1,floop#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1 ,c_7/0,c_8/1,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,1#,fac#,floop#} and constructors {0,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: fac#(s(x)) -> c_8(fac#(x)) **** Step 6.a:1.a:1.b:4: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: fac#(s(x)) -> c_8(fac#(x)) - Signature: {*/2,+/2,1/0,fac/1,floop/2,*#/2,+#/2,1#/0,fac#/1,floop#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1 ,c_7/0,c_8/1,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,1#,fac#,floop#} 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: fac#(s(x)) -> c_8(fac#(x)) The strictly oriented rules are moved into the weak component. ***** Step 6.a:1.a:1.b:4.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: fac#(s(x)) -> c_8(fac#(x)) - Signature: {*/2,+/2,1/0,fac/1,floop/2,*#/2,+#/2,1#/0,fac#/1,floop#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1 ,c_7/0,c_8/1,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,1#,fac#,floop#} 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_8) = {1} Following symbols are considered usable: {*#,+#,1#,fac#,floop#} TcT has computed the following interpretation: p(*) = [1] x2 + [4] p(+) = [2] x1 + [1] p(0) = [1] p(1) = [0] p(fac) = [1] p(floop) = [1] x2 + [0] p(s) = [1] x1 + [8] p(*#) = [1] x1 + [1] p(+#) = [2] p(1#) = [2] p(fac#) = [1] x1 + [0] p(floop#) = [1] x1 + [1] x2 + [2] p(c_1) = [0] p(c_2) = [1] x1 + [1] x2 + [2] p(c_3) = [1] p(c_4) = [4] x1 + [1] p(c_5) = [2] p(c_6) = [0] p(c_7) = [0] p(c_8) = [1] x1 + [0] p(c_9) = [1] p(c_10) = [8] x1 + [1] Following rules are strictly oriented: fac#(s(x)) = [1] x + [8] > [1] x + [0] = c_8(fac#(x)) Following rules are (at-least) weakly oriented: ***** Step 6.a:1.a:1.b:4.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: fac#(s(x)) -> c_8(fac#(x)) - Signature: {*/2,+/2,1/0,fac/1,floop/2,*#/2,+#/2,1#/0,fac#/1,floop#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1 ,c_7/0,c_8/1,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,1#,fac#,floop#} and constructors {0,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 6.a:1.a:1.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: fac#(s(x)) -> c_8(fac#(x)) - Signature: {*/2,+/2,1/0,fac/1,floop/2,*#/2,+#/2,1#/0,fac#/1,floop#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1 ,c_7/0,c_8/1,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,1#,fac#,floop#} and constructors {0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:fac#(s(x)) -> c_8(fac#(x)) -->_1 fac#(s(x)) -> c_8(fac#(x)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: fac#(s(x)) -> c_8(fac#(x)) ***** Step 6.a:1.a:1.b:4.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Signature: {*/2,+/2,1/0,fac/1,floop/2,*#/2,+#/2,1#/0,fac#/1,floop#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1 ,c_7/0,c_8/1,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,1#,fac#,floop#} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 6.a:1.b:1: Failure MAYBE + Considered Problem: - Strict DPs: *#(x,s(y)) -> c_2(+#(*(x,y),x),*#(x,y)) +#(x,s(y)) -> c_4(+#(x,y)) - Weak DPs: fac#(s(x)) -> *#(s(x),fac(x)) fac#(s(x)) -> fac#(x) floop#(s(x),y) -> *#(s(x),y) floop#(s(x),y) -> floop#(x,*(s(x),y)) - Weak TRS: *(x,0()) -> 0() *(x,s(y)) -> +(*(x,y),x) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) 1() -> s(0()) fac(0()) -> 1() fac(0()) -> s(0()) fac(s(x)) -> *(s(x),fac(x)) - Signature: {*/2,+/2,1/0,fac/1,floop/2,*#/2,+#/2,1#/0,fac#/1,floop#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1 ,c_7/0,c_8/2,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,1#,fac#,floop#} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is still open. ** Step 6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: fac#(s(x)) -> c_8(*#(s(x),fac(x)),fac#(x)) floop#(s(x),y) -> c_10(floop#(x,*(s(x),y)),*#(s(x),y)) - Weak DPs: *#(x,s(y)) -> c_2(+#(*(x,y),x),*#(x,y)) +#(x,s(y)) -> c_4(+#(x,y)) - Weak TRS: *(x,0()) -> 0() *(x,s(y)) -> +(*(x,y),x) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) 1() -> s(0()) fac(0()) -> 1() fac(0()) -> s(0()) fac(s(x)) -> *(s(x),fac(x)) - Signature: {*/2,+/2,1/0,fac/1,floop/2,*#/2,+#/2,1#/0,fac#/1,floop#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1 ,c_7/0,c_8/2,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,1#,fac#,floop#} and constructors {0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:fac#(s(x)) -> c_8(*#(s(x),fac(x)),fac#(x)) -->_1 *#(x,s(y)) -> c_2(+#(*(x,y),x),*#(x,y)):3 -->_2 fac#(s(x)) -> c_8(*#(s(x),fac(x)),fac#(x)):1 2:S:floop#(s(x),y) -> c_10(floop#(x,*(s(x),y)),*#(s(x),y)) -->_2 *#(x,s(y)) -> c_2(+#(*(x,y),x),*#(x,y)):3 -->_1 floop#(s(x),y) -> c_10(floop#(x,*(s(x),y)),*#(s(x),y)):2 3:W:*#(x,s(y)) -> c_2(+#(*(x,y),x),*#(x,y)) -->_1 +#(x,s(y)) -> c_4(+#(x,y)):4 -->_2 *#(x,s(y)) -> c_2(+#(*(x,y),x),*#(x,y)):3 4:W:+#(x,s(y)) -> c_4(+#(x,y)) -->_1 +#(x,s(y)) -> c_4(+#(x,y)):4 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: *#(x,s(y)) -> c_2(+#(*(x,y),x),*#(x,y)) 4: +#(x,s(y)) -> c_4(+#(x,y)) ** Step 6.b:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: fac#(s(x)) -> c_8(*#(s(x),fac(x)),fac#(x)) floop#(s(x),y) -> c_10(floop#(x,*(s(x),y)),*#(s(x),y)) - Weak TRS: *(x,0()) -> 0() *(x,s(y)) -> +(*(x,y),x) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) 1() -> s(0()) fac(0()) -> 1() fac(0()) -> s(0()) fac(s(x)) -> *(s(x),fac(x)) - Signature: {*/2,+/2,1/0,fac/1,floop/2,*#/2,+#/2,1#/0,fac#/1,floop#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1 ,c_7/0,c_8/2,c_9/0,c_10/2} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,1#,fac#,floop#} and constructors {0,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:fac#(s(x)) -> c_8(*#(s(x),fac(x)),fac#(x)) -->_2 fac#(s(x)) -> c_8(*#(s(x),fac(x)),fac#(x)):1 2:S:floop#(s(x),y) -> c_10(floop#(x,*(s(x),y)),*#(s(x),y)) -->_1 floop#(s(x),y) -> c_10(floop#(x,*(s(x),y)),*#(s(x),y)):2 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: fac#(s(x)) -> c_8(fac#(x)) floop#(s(x),y) -> c_10(floop#(x,*(s(x),y))) ** Step 6.b:3: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: fac#(s(x)) -> c_8(fac#(x)) floop#(s(x),y) -> c_10(floop#(x,*(s(x),y))) - Weak TRS: *(x,0()) -> 0() *(x,s(y)) -> +(*(x,y),x) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) 1() -> s(0()) fac(0()) -> 1() fac(0()) -> s(0()) fac(s(x)) -> *(s(x),fac(x)) - Signature: {*/2,+/2,1/0,fac/1,floop/2,*#/2,+#/2,1#/0,fac#/1,floop#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1 ,c_7/0,c_8/1,c_9/0,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,1#,fac#,floop#} and constructors {0,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: *(x,0()) -> 0() *(x,s(y)) -> +(*(x,y),x) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) fac#(s(x)) -> c_8(fac#(x)) floop#(s(x),y) -> c_10(floop#(x,*(s(x),y))) ** Step 6.b:4: Decompose WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: fac#(s(x)) -> c_8(fac#(x)) floop#(s(x),y) -> c_10(floop#(x,*(s(x),y))) - Weak TRS: *(x,0()) -> 0() *(x,s(y)) -> +(*(x,y),x) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) - Signature: {*/2,+/2,1/0,fac/1,floop/2,*#/2,+#/2,1#/0,fac#/1,floop#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1 ,c_7/0,c_8/1,c_9/0,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,1#,fac#,floop#} and constructors {0,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: fac#(s(x)) -> c_8(fac#(x)) - Weak DPs: floop#(s(x),y) -> c_10(floop#(x,*(s(x),y))) - Weak TRS: *(x,0()) -> 0() *(x,s(y)) -> +(*(x,y),x) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) - Signature: {*/2,+/2,1/0,fac/1,floop/2,*#/2,+#/2,1#/0,fac#/1,floop#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1 ,c_7/0,c_8/1,c_9/0,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,1#,fac#,floop#} and constructors {0,s} Problem (S) - Strict DPs: floop#(s(x),y) -> c_10(floop#(x,*(s(x),y))) - Weak DPs: fac#(s(x)) -> c_8(fac#(x)) - Weak TRS: *(x,0()) -> 0() *(x,s(y)) -> +(*(x,y),x) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) - Signature: {*/2,+/2,1/0,fac/1,floop/2,*#/2,+#/2,1#/0,fac#/1,floop#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1 ,c_7/0,c_8/1,c_9/0,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,1#,fac#,floop#} and constructors {0,s} *** Step 6.b:4.a:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: fac#(s(x)) -> c_8(fac#(x)) - Weak DPs: floop#(s(x),y) -> c_10(floop#(x,*(s(x),y))) - Weak TRS: *(x,0()) -> 0() *(x,s(y)) -> +(*(x,y),x) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) - Signature: {*/2,+/2,1/0,fac/1,floop/2,*#/2,+#/2,1#/0,fac#/1,floop#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1 ,c_7/0,c_8/1,c_9/0,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,1#,fac#,floop#} and constructors {0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:fac#(s(x)) -> c_8(fac#(x)) -->_1 fac#(s(x)) -> c_8(fac#(x)):1 2:W:floop#(s(x),y) -> c_10(floop#(x,*(s(x),y))) -->_1 floop#(s(x),y) -> c_10(floop#(x,*(s(x),y))):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: floop#(s(x),y) -> c_10(floop#(x,*(s(x),y))) *** Step 6.b:4.a:2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: fac#(s(x)) -> c_8(fac#(x)) - Weak TRS: *(x,0()) -> 0() *(x,s(y)) -> +(*(x,y),x) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) - Signature: {*/2,+/2,1/0,fac/1,floop/2,*#/2,+#/2,1#/0,fac#/1,floop#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1 ,c_7/0,c_8/1,c_9/0,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,1#,fac#,floop#} and constructors {0,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: fac#(s(x)) -> c_8(fac#(x)) *** Step 6.b:4.a:3: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: fac#(s(x)) -> c_8(fac#(x)) - Signature: {*/2,+/2,1/0,fac/1,floop/2,*#/2,+#/2,1#/0,fac#/1,floop#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1 ,c_7/0,c_8/1,c_9/0,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,1#,fac#,floop#} 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: fac#(s(x)) -> c_8(fac#(x)) The strictly oriented rules are moved into the weak component. **** Step 6.b:4.a:3.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: fac#(s(x)) -> c_8(fac#(x)) - Signature: {*/2,+/2,1/0,fac/1,floop/2,*#/2,+#/2,1#/0,fac#/1,floop#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1 ,c_7/0,c_8/1,c_9/0,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,1#,fac#,floop#} 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_8) = {1} Following symbols are considered usable: {*#,+#,1#,fac#,floop#} TcT has computed the following interpretation: p(*) = [0] p(+) = [0] p(0) = [0] p(1) = [0] p(fac) = [1] x1 + [1] p(floop) = [0] p(s) = [1] x1 + [3] p(*#) = [8] x1 + [2] p(+#) = [1] x1 + [8] p(1#) = [0] p(fac#) = [7] x1 + [0] p(floop#) = [2] x1 + [8] p(c_1) = [0] p(c_2) = [1] x1 + [1] x2 + [2] p(c_3) = [0] p(c_4) = [8] x1 + [1] p(c_5) = [2] p(c_6) = [0] p(c_7) = [0] p(c_8) = [1] x1 + [13] p(c_9) = [4] p(c_10) = [2] x1 + [2] Following rules are strictly oriented: fac#(s(x)) = [7] x + [21] > [7] x + [13] = c_8(fac#(x)) Following rules are (at-least) weakly oriented: **** Step 6.b:4.a:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: fac#(s(x)) -> c_8(fac#(x)) - Signature: {*/2,+/2,1/0,fac/1,floop/2,*#/2,+#/2,1#/0,fac#/1,floop#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1 ,c_7/0,c_8/1,c_9/0,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,1#,fac#,floop#} and constructors {0,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 6.b:4.a:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: fac#(s(x)) -> c_8(fac#(x)) - Signature: {*/2,+/2,1/0,fac/1,floop/2,*#/2,+#/2,1#/0,fac#/1,floop#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1 ,c_7/0,c_8/1,c_9/0,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,1#,fac#,floop#} and constructors {0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:fac#(s(x)) -> c_8(fac#(x)) -->_1 fac#(s(x)) -> c_8(fac#(x)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: fac#(s(x)) -> c_8(fac#(x)) **** Step 6.b:4.a:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Signature: {*/2,+/2,1/0,fac/1,floop/2,*#/2,+#/2,1#/0,fac#/1,floop#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1 ,c_7/0,c_8/1,c_9/0,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,1#,fac#,floop#} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 6.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: floop#(s(x),y) -> c_10(floop#(x,*(s(x),y))) - Weak DPs: fac#(s(x)) -> c_8(fac#(x)) - Weak TRS: *(x,0()) -> 0() *(x,s(y)) -> +(*(x,y),x) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) - Signature: {*/2,+/2,1/0,fac/1,floop/2,*#/2,+#/2,1#/0,fac#/1,floop#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1 ,c_7/0,c_8/1,c_9/0,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,1#,fac#,floop#} and constructors {0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:floop#(s(x),y) -> c_10(floop#(x,*(s(x),y))) -->_1 floop#(s(x),y) -> c_10(floop#(x,*(s(x),y))):1 2:W:fac#(s(x)) -> c_8(fac#(x)) -->_1 fac#(s(x)) -> c_8(fac#(x)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: fac#(s(x)) -> c_8(fac#(x)) *** Step 6.b:4.b:2: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: floop#(s(x),y) -> c_10(floop#(x,*(s(x),y))) - Weak TRS: *(x,0()) -> 0() *(x,s(y)) -> +(*(x,y),x) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) - Signature: {*/2,+/2,1/0,fac/1,floop/2,*#/2,+#/2,1#/0,fac#/1,floop#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1 ,c_7/0,c_8/1,c_9/0,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,1#,fac#,floop#} 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: floop#(s(x),y) -> c_10(floop#(x,*(s(x),y))) The strictly oriented rules are moved into the weak component. **** Step 6.b:4.b:2.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: floop#(s(x),y) -> c_10(floop#(x,*(s(x),y))) - Weak TRS: *(x,0()) -> 0() *(x,s(y)) -> +(*(x,y),x) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) - Signature: {*/2,+/2,1/0,fac/1,floop/2,*#/2,+#/2,1#/0,fac#/1,floop#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1 ,c_7/0,c_8/1,c_9/0,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,1#,fac#,floop#} 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_10) = {1} Following symbols are considered usable: {*#,+#,1#,fac#,floop#} TcT has computed the following interpretation: p(*) = [4] x1 + [8] p(+) = [1] x1 + [4] x2 + [0] p(0) = [0] p(1) = [1] p(fac) = [1] x1 + [2] p(floop) = [4] x1 + [0] p(s) = [1] x1 + [2] p(*#) = [0] p(+#) = [1] x1 + [1] x2 + [1] p(1#) = [1] p(fac#) = [8] p(floop#) = [8] x1 + [0] p(c_1) = [2] p(c_2) = [1] x2 + [1] p(c_3) = [0] p(c_4) = [2] p(c_5) = [0] p(c_6) = [0] p(c_7) = [2] p(c_8) = [1] p(c_9) = [1] p(c_10) = [1] x1 + [8] Following rules are strictly oriented: floop#(s(x),y) = [8] x + [16] > [8] x + [8] = c_10(floop#(x,*(s(x),y))) Following rules are (at-least) weakly oriented: **** Step 6.b:4.b:2.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: floop#(s(x),y) -> c_10(floop#(x,*(s(x),y))) - Weak TRS: *(x,0()) -> 0() *(x,s(y)) -> +(*(x,y),x) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) - Signature: {*/2,+/2,1/0,fac/1,floop/2,*#/2,+#/2,1#/0,fac#/1,floop#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1 ,c_7/0,c_8/1,c_9/0,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,1#,fac#,floop#} and constructors {0,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 6.b:4.b:2.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: floop#(s(x),y) -> c_10(floop#(x,*(s(x),y))) - Weak TRS: *(x,0()) -> 0() *(x,s(y)) -> +(*(x,y),x) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) - Signature: {*/2,+/2,1/0,fac/1,floop/2,*#/2,+#/2,1#/0,fac#/1,floop#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1 ,c_7/0,c_8/1,c_9/0,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,1#,fac#,floop#} and constructors {0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:floop#(s(x),y) -> c_10(floop#(x,*(s(x),y))) -->_1 floop#(s(x),y) -> c_10(floop#(x,*(s(x),y))):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: floop#(s(x),y) -> c_10(floop#(x,*(s(x),y))) **** Step 6.b:4.b:2.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: *(x,0()) -> 0() *(x,s(y)) -> +(*(x,y),x) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) - Signature: {*/2,+/2,1/0,fac/1,floop/2,*#/2,+#/2,1#/0,fac#/1,floop#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1 ,c_7/0,c_8/1,c_9/0,c_10/1} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,1#,fac#,floop#} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). MAYBE