MAYBE * Step 1: DependencyPairs MAYBE + Considered Problem: - Strict TRS: *(0(),y) -> 0() *(s(x),y) -> +(*(x,y),y) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) fact(0()) -> s(0()) fact(s(x)) -> *(s(x),fact(p(s(x)))) p(s(x)) -> x - Signature: {*/2,+/2,fact/1,p/1} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {*,+,fact,p} and constructors {0,s} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs *#(0(),y) -> c_1() *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y)) +#(x,0()) -> c_3() +#(x,s(y)) -> c_4(+#(x,y)) fact#(0()) -> c_5() fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x))),p#(s(x))) p#(s(x)) -> c_7() Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation MAYBE + Considered Problem: - Strict DPs: *#(0(),y) -> c_1() *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y)) +#(x,0()) -> c_3() +#(x,s(y)) -> c_4(+#(x,y)) fact#(0()) -> c_5() fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x))),p#(s(x))) p#(s(x)) -> c_7() - Weak TRS: *(0(),y) -> 0() *(s(x),y) -> +(*(x,y),y) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) fact(0()) -> s(0()) fact(s(x)) -> *(s(x),fact(p(s(x)))) p(s(x)) -> x - Signature: {*/2,+/2,fact/1,p/1,*#/2,+#/2,fact#/1,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/3,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,fact#,p#} 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} by application of Pre({1,3,5,7}) = {2,4,6}. Here rules are labelled as follows: 1: *#(0(),y) -> c_1() 2: *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y)) 3: +#(x,0()) -> c_3() 4: +#(x,s(y)) -> c_4(+#(x,y)) 5: fact#(0()) -> c_5() 6: fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x))),p#(s(x))) 7: p#(s(x)) -> c_7() * Step 3: RemoveWeakSuffixes MAYBE + Considered Problem: - Strict DPs: *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y)) +#(x,s(y)) -> c_4(+#(x,y)) fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x))),p#(s(x))) - Weak DPs: *#(0(),y) -> c_1() +#(x,0()) -> c_3() fact#(0()) -> c_5() p#(s(x)) -> c_7() - Weak TRS: *(0(),y) -> 0() *(s(x),y) -> +(*(x,y),y) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) fact(0()) -> s(0()) fact(s(x)) -> *(s(x),fact(p(s(x)))) p(s(x)) -> x - Signature: {*/2,+/2,fact/1,p/1,*#/2,+#/2,fact#/1,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/3,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,fact#,p#} and constructors {0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:*#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y)) -->_1 +#(x,s(y)) -> c_4(+#(x,y)):2 -->_1 +#(x,0()) -> c_3():5 -->_2 *#(0(),y) -> c_1():4 -->_2 *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y)):1 2:S:+#(x,s(y)) -> c_4(+#(x,y)) -->_1 +#(x,0()) -> c_3():5 -->_1 +#(x,s(y)) -> c_4(+#(x,y)):2 3:S:fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x))),p#(s(x))) -->_3 p#(s(x)) -> c_7():7 -->_2 fact#(0()) -> c_5():6 -->_2 fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x))),p#(s(x))):3 -->_1 *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y)):1 4:W:*#(0(),y) -> c_1() 5:W:+#(x,0()) -> c_3() 6:W:fact#(0()) -> c_5() 7:W:p#(s(x)) -> c_7() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 6: fact#(0()) -> c_5() 7: p#(s(x)) -> c_7() 4: *#(0(),y) -> c_1() 5: +#(x,0()) -> c_3() * Step 4: SimplifyRHS MAYBE + Considered Problem: - Strict DPs: *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y)) +#(x,s(y)) -> c_4(+#(x,y)) fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x))),p#(s(x))) - Weak TRS: *(0(),y) -> 0() *(s(x),y) -> +(*(x,y),y) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) fact(0()) -> s(0()) fact(s(x)) -> *(s(x),fact(p(s(x)))) p(s(x)) -> x - Signature: {*/2,+/2,fact/1,p/1,*#/2,+#/2,fact#/1,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/3,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,fact#,p#} and constructors {0,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:*#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y)) -->_1 +#(x,s(y)) -> c_4(+#(x,y)):2 -->_2 *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y)):1 2:S:+#(x,s(y)) -> c_4(+#(x,y)) -->_1 +#(x,s(y)) -> c_4(+#(x,y)):2 3:S:fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x))),p#(s(x))) -->_2 fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x))),p#(s(x))):3 -->_1 *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x)))) * Step 5: Decompose MAYBE + Considered Problem: - Strict DPs: *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y)) +#(x,s(y)) -> c_4(+#(x,y)) fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x)))) - Weak TRS: *(0(),y) -> 0() *(s(x),y) -> +(*(x,y),y) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) fact(0()) -> s(0()) fact(s(x)) -> *(s(x),fact(p(s(x)))) p(s(x)) -> x - Signature: {*/2,+/2,fact/1,p/1,*#/2,+#/2,fact#/1,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,fact#,p#} 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: *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y)) +#(x,s(y)) -> c_4(+#(x,y)) - Weak DPs: fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x)))) - Weak TRS: *(0(),y) -> 0() *(s(x),y) -> +(*(x,y),y) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) fact(0()) -> s(0()) fact(s(x)) -> *(s(x),fact(p(s(x)))) p(s(x)) -> x - Signature: {*/2,+/2,fact/1,p/1,*#/2,+#/2,fact#/1,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,fact#,p#} and constructors {0,s} Problem (S) - Strict DPs: fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x)))) - Weak DPs: *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y)) +#(x,s(y)) -> c_4(+#(x,y)) - Weak TRS: *(0(),y) -> 0() *(s(x),y) -> +(*(x,y),y) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) fact(0()) -> s(0()) fact(s(x)) -> *(s(x),fact(p(s(x)))) p(s(x)) -> x - Signature: {*/2,+/2,fact/1,p/1,*#/2,+#/2,fact#/1,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,fact#,p#} and constructors {0,s} ** Step 5.a:1: DecomposeDG MAYBE + Considered Problem: - Strict DPs: *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y)) +#(x,s(y)) -> c_4(+#(x,y)) - Weak DPs: fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x)))) - Weak TRS: *(0(),y) -> 0() *(s(x),y) -> +(*(x,y),y) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) fact(0()) -> s(0()) fact(s(x)) -> *(s(x),fact(p(s(x)))) p(s(x)) -> x - Signature: {*/2,+/2,fact/1,p/1,*#/2,+#/2,fact#/1,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,fact#,p#} 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 fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x)))) and a lower component *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y)) +#(x,s(y)) -> c_4(+#(x,y)) Further, following extension rules are added to the lower component. fact#(s(x)) -> *#(s(x),fact(p(s(x)))) fact#(s(x)) -> fact#(p(s(x))) *** Step 5.a:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x)))) - Weak TRS: *(0(),y) -> 0() *(s(x),y) -> +(*(x,y),y) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) fact(0()) -> s(0()) fact(s(x)) -> *(s(x),fact(p(s(x)))) p(s(x)) -> x - Signature: {*/2,+/2,fact/1,p/1,*#/2,+#/2,fact#/1,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,fact#,p#} and constructors {0,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: 1: fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x)))) The strictly oriented rules are moved into the weak component. **** Step 5.a:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x)))) - Weak TRS: *(0(),y) -> 0() *(s(x),y) -> +(*(x,y),y) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) fact(0()) -> s(0()) fact(s(x)) -> *(s(x),fact(p(s(x)))) p(s(x)) -> x - Signature: {*/2,+/2,fact/1,p/1,*#/2,+#/2,fact#/1,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,fact#,p#} and constructors {0,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_6) = {1,2} Following symbols are considered usable: {p,*#,+#,fact#,p#} TcT has computed the following interpretation: p(*) = [0 0 0] [1 0 0] [0] [0 1 1] x1 + [0 0 0] x2 + [1] [0 0 0] [0 0 0] [0] p(+) = [0 0 0] [0 1 0] [0] [0 0 0] x1 + [0 0 0] x2 + [1] [0 1 0] [0 0 0] [0] p(0) = [0] [0] [0] p(fact) = [1 0 0] [0] [0 0 0] x1 + [1] [0 1 0] [0] p(p) = [1 0 0] [0] [1 0 1] x1 + [1] [0 1 0] [0] p(s) = [1 1 0] [1] [0 0 1] x1 + [0] [0 0 1] [1] p(*#) = [0] [0] [1] p(+#) = [0] [0] [0] p(fact#) = [0 0 1] [0] [1 0 1] x1 + [0] [0 1 1] [0] p(p#) = [0] [0] [0] p(c_1) = [0] [0] [0] p(c_2) = [0] [0] [0] p(c_3) = [0] [0] [0] p(c_4) = [0] [0] [0] p(c_5) = [0] [0] [0] p(c_6) = [1 0 0] [1 0 0] [0] [0 0 0] x1 + [1 0 0] x2 + [0] [0 0 1] [0 0 0] [0] p(c_7) = [0] [0] [0] Following rules are strictly oriented: fact#(s(x)) = [0 0 1] [1] [1 1 1] x + [2] [0 0 2] [1] > [0 0 1] [0] [0 0 1] x + [0] [0 0 0] [1] = c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x)))) Following rules are (at-least) weakly oriented: p(s(x)) = [1 1 0] [1] [1 1 1] x + [3] [0 0 1] [0] >= [1 0 0] [0] [0 1 0] x + [0] [0 0 1] [0] = x **** Step 5.a:1.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x)))) - Weak TRS: *(0(),y) -> 0() *(s(x),y) -> +(*(x,y),y) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) fact(0()) -> s(0()) fact(s(x)) -> *(s(x),fact(p(s(x)))) p(s(x)) -> x - Signature: {*/2,+/2,fact/1,p/1,*#/2,+#/2,fact#/1,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,fact#,p#} and constructors {0,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 5.a:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x)))) - Weak TRS: *(0(),y) -> 0() *(s(x),y) -> +(*(x,y),y) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) fact(0()) -> s(0()) fact(s(x)) -> *(s(x),fact(p(s(x)))) p(s(x)) -> x - Signature: {*/2,+/2,fact/1,p/1,*#/2,+#/2,fact#/1,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,fact#,p#} and constructors {0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x)))) -->_2 fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x)))):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x)))) **** Step 5.a:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: *(0(),y) -> 0() *(s(x),y) -> +(*(x,y),y) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) fact(0()) -> s(0()) fact(s(x)) -> *(s(x),fact(p(s(x)))) p(s(x)) -> x - Signature: {*/2,+/2,fact/1,p/1,*#/2,+#/2,fact#/1,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,fact#,p#} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 5.a:1.b:1: DecomposeDG MAYBE + Considered Problem: - Strict DPs: *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y)) +#(x,s(y)) -> c_4(+#(x,y)) - Weak DPs: fact#(s(x)) -> *#(s(x),fact(p(s(x)))) fact#(s(x)) -> fact#(p(s(x))) - Weak TRS: *(0(),y) -> 0() *(s(x),y) -> +(*(x,y),y) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) fact(0()) -> s(0()) fact(s(x)) -> *(s(x),fact(p(s(x)))) p(s(x)) -> x - Signature: {*/2,+/2,fact/1,p/1,*#/2,+#/2,fact#/1,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,fact#,p#} 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 *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y)) fact#(s(x)) -> *#(s(x),fact(p(s(x)))) fact#(s(x)) -> fact#(p(s(x))) and a lower component +#(x,s(y)) -> c_4(+#(x,y)) Further, following extension rules are added to the lower component. *#(s(x),y) -> *#(x,y) *#(s(x),y) -> +#(*(x,y),y) fact#(s(x)) -> *#(s(x),fact(p(s(x)))) fact#(s(x)) -> fact#(p(s(x))) **** Step 5.a:1.b:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y)) - Weak DPs: fact#(s(x)) -> *#(s(x),fact(p(s(x)))) fact#(s(x)) -> fact#(p(s(x))) - Weak TRS: *(0(),y) -> 0() *(s(x),y) -> +(*(x,y),y) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) fact(0()) -> s(0()) fact(s(x)) -> *(s(x),fact(p(s(x)))) p(s(x)) -> x - Signature: {*/2,+/2,fact/1,p/1,*#/2,+#/2,fact#/1,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,fact#,p#} 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: *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y)) The strictly oriented rules are moved into the weak component. ***** Step 5.a:1.b:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y)) - Weak DPs: fact#(s(x)) -> *#(s(x),fact(p(s(x)))) fact#(s(x)) -> fact#(p(s(x))) - Weak TRS: *(0(),y) -> 0() *(s(x),y) -> +(*(x,y),y) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) fact(0()) -> s(0()) fact(s(x)) -> *(s(x),fact(p(s(x)))) p(s(x)) -> x - Signature: {*/2,+/2,fact/1,p/1,*#/2,+#/2,fact#/1,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,fact#,p#} 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) = {1,2} Following symbols are considered usable: {p,*#,+#,fact#,p#} TcT has computed the following interpretation: p(*) = [2] x1 + [0] p(+) = [0] p(0) = [8] p(fact) = [1] x1 + [0] p(p) = [1] x1 + [0] p(s) = [1] x1 + [8] p(*#) = [2] x1 + [0] p(+#) = [1] p(fact#) = [2] x1 + [0] p(p#) = [4] x1 + [1] p(c_1) = [2] p(c_2) = [4] x1 + [1] x2 + [8] p(c_3) = [1] p(c_4) = [4] x1 + [1] p(c_5) = [4] p(c_6) = [1] x1 + [2] p(c_7) = [0] Following rules are strictly oriented: *#(s(x),y) = [2] x + [16] > [2] x + [12] = c_2(+#(*(x,y),y),*#(x,y)) Following rules are (at-least) weakly oriented: fact#(s(x)) = [2] x + [16] >= [2] x + [16] = *#(s(x),fact(p(s(x)))) fact#(s(x)) = [2] x + [16] >= [2] x + [16] = fact#(p(s(x))) p(s(x)) = [1] x + [8] >= [1] x + [0] = x ***** Step 5.a:1.b:1.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y)) fact#(s(x)) -> *#(s(x),fact(p(s(x)))) fact#(s(x)) -> fact#(p(s(x))) - Weak TRS: *(0(),y) -> 0() *(s(x),y) -> +(*(x,y),y) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) fact(0()) -> s(0()) fact(s(x)) -> *(s(x),fact(p(s(x)))) p(s(x)) -> x - Signature: {*/2,+/2,fact/1,p/1,*#/2,+#/2,fact#/1,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,fact#,p#} and constructors {0,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 5.a:1.b:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y)) fact#(s(x)) -> *#(s(x),fact(p(s(x)))) fact#(s(x)) -> fact#(p(s(x))) - Weak TRS: *(0(),y) -> 0() *(s(x),y) -> +(*(x,y),y) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) fact(0()) -> s(0()) fact(s(x)) -> *(s(x),fact(p(s(x)))) p(s(x)) -> x - Signature: {*/2,+/2,fact/1,p/1,*#/2,+#/2,fact#/1,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,fact#,p#} and constructors {0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:*#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y)) -->_2 *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y)):1 2:W:fact#(s(x)) -> *#(s(x),fact(p(s(x)))) -->_1 *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y)):1 3:W:fact#(s(x)) -> fact#(p(s(x))) -->_1 fact#(s(x)) -> fact#(p(s(x))):3 -->_1 fact#(s(x)) -> *#(s(x),fact(p(s(x)))):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: fact#(s(x)) -> fact#(p(s(x))) 2: fact#(s(x)) -> *#(s(x),fact(p(s(x)))) 1: *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y)) ***** Step 5.a:1.b:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: *(0(),y) -> 0() *(s(x),y) -> +(*(x,y),y) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) fact(0()) -> s(0()) fact(s(x)) -> *(s(x),fact(p(s(x)))) p(s(x)) -> x - Signature: {*/2,+/2,fact/1,p/1,*#/2,+#/2,fact#/1,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,fact#,p#} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). **** Step 5.a:1.b:1.b:1: Failure MAYBE + Considered Problem: - Strict DPs: +#(x,s(y)) -> c_4(+#(x,y)) - Weak DPs: *#(s(x),y) -> *#(x,y) *#(s(x),y) -> +#(*(x,y),y) fact#(s(x)) -> *#(s(x),fact(p(s(x)))) fact#(s(x)) -> fact#(p(s(x))) - Weak TRS: *(0(),y) -> 0() *(s(x),y) -> +(*(x,y),y) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) fact(0()) -> s(0()) fact(s(x)) -> *(s(x),fact(p(s(x)))) p(s(x)) -> x - Signature: {*/2,+/2,fact/1,p/1,*#/2,+#/2,fact#/1,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,fact#,p#} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is still open. ** Step 5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x)))) - Weak DPs: *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y)) +#(x,s(y)) -> c_4(+#(x,y)) - Weak TRS: *(0(),y) -> 0() *(s(x),y) -> +(*(x,y),y) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) fact(0()) -> s(0()) fact(s(x)) -> *(s(x),fact(p(s(x)))) p(s(x)) -> x - Signature: {*/2,+/2,fact/1,p/1,*#/2,+#/2,fact#/1,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,fact#,p#} and constructors {0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x)))) -->_1 *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y)):2 -->_2 fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x)))):1 2:W:*#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y)) -->_1 +#(x,s(y)) -> c_4(+#(x,y)):3 -->_2 *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y)):2 3:W:+#(x,s(y)) -> c_4(+#(x,y)) -->_1 +#(x,s(y)) -> c_4(+#(x,y)):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: *#(s(x),y) -> c_2(+#(*(x,y),y),*#(x,y)) 3: +#(x,s(y)) -> c_4(+#(x,y)) ** Step 5.b:2: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x)))) - Weak TRS: *(0(),y) -> 0() *(s(x),y) -> +(*(x,y),y) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) fact(0()) -> s(0()) fact(s(x)) -> *(s(x),fact(p(s(x)))) p(s(x)) -> x - Signature: {*/2,+/2,fact/1,p/1,*#/2,+#/2,fact#/1,p#/1} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/2,c_7/0} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,fact#,p#} and constructors {0,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x)))) -->_2 fact#(s(x)) -> c_6(*#(s(x),fact(p(s(x)))),fact#(p(s(x)))):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: fact#(s(x)) -> c_6(fact#(p(s(x)))) ** Step 5.b:3: UsableRules WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: fact#(s(x)) -> c_6(fact#(p(s(x)))) - Weak TRS: *(0(),y) -> 0() *(s(x),y) -> +(*(x,y),y) +(x,0()) -> x +(x,s(y)) -> s(+(x,y)) fact(0()) -> s(0()) fact(s(x)) -> *(s(x),fact(p(s(x)))) p(s(x)) -> x - Signature: {*/2,+/2,fact/1,p/1,*#/2,+#/2,fact#/1,p#/1} / {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} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,fact#,p#} and constructors {0,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: p(s(x)) -> x fact#(s(x)) -> c_6(fact#(p(s(x)))) ** Step 5.b:4: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: fact#(s(x)) -> c_6(fact#(p(s(x)))) - Weak TRS: p(s(x)) -> x - Signature: {*/2,+/2,fact/1,p/1,*#/2,+#/2,fact#/1,p#/1} / {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} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,fact#,p#} and constructors {0,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: 1: fact#(s(x)) -> c_6(fact#(p(s(x)))) The strictly oriented rules are moved into the weak component. *** Step 5.b:4.a:1: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: fact#(s(x)) -> c_6(fact#(p(s(x)))) - Weak TRS: p(s(x)) -> x - Signature: {*/2,+/2,fact/1,p/1,*#/2,+#/2,fact#/1,p#/1} / {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} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,fact#,p#} and constructors {0,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_6) = {1} Following symbols are considered usable: {p,*#,+#,fact#,p#} TcT has computed the following interpretation: p(*) = [0 0 0] [0 0 0] [1] [0 0 0] x1 + [0 0 0] x2 + [2] [1 1 1] [0 0 1] [0] p(+) = [0 2 4] [1 0 0] [1] [0 1 4] x1 + [2 1 2] x2 + [0] [1 1 1] [1 0 0] [4] p(0) = [0] [2] [1] p(fact) = [0 0 0] [1] [0 0 0] x1 + [1] [4 0 2] [0] p(p) = [4 5 1] [3] [4 2 4] x1 + [4] [0 1 0] [0] p(s) = [1 1 1] [0] [0 0 1] x1 + [0] [0 0 1] [1] p(*#) = [0 0 0] [2 0 2] [1] [2 1 4] x1 + [2 1 1] x2 + [2] [0 2 1] [1 0 1] [1] p(+#) = [0 2 0] [4 0 1] [1] [0 1 1] x1 + [1 4 1] x2 + [0] [0 1 4] [1 0 2] [2] p(fact#) = [0 0 4] [0] [0 0 0] x1 + [0] [0 0 0] [0] p(p#) = [0] [1] [0] p(c_1) = [0] [4] [1] p(c_2) = [0 0 2] [0 1 2] [1] [0 4 2] x1 + [1 1 0] x2 + [4] [1 0 2] [0 2 1] [1] p(c_3) = [0] [1] [4] p(c_4) = [0 0 0] [4] [0 1 1] x1 + [2] [1 0 1] [4] p(c_5) = [4] [0] [4] p(c_6) = [1 0 0] [3] [0 0 0] x1 + [0] [0 0 0] [0] p(c_7) = [0] [0] [0] Following rules are strictly oriented: fact#(s(x)) = [0 0 4] [4] [0 0 0] x + [0] [0 0 0] [0] > [0 0 4] [3] [0 0 0] x + [0] [0 0 0] [0] = c_6(fact#(p(s(x)))) Following rules are (at-least) weakly oriented: p(s(x)) = [4 4 10] [4] [4 4 10] x + [8] [0 0 1] [0] >= [1 0 0] [0] [0 1 0] x + [0] [0 0 1] [0] = x *** Step 5.b:4.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: fact#(s(x)) -> c_6(fact#(p(s(x)))) - Weak TRS: p(s(x)) -> x - Signature: {*/2,+/2,fact/1,p/1,*#/2,+#/2,fact#/1,p#/1} / {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} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,fact#,p#} and constructors {0,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 5.b:4.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: fact#(s(x)) -> c_6(fact#(p(s(x)))) - Weak TRS: p(s(x)) -> x - Signature: {*/2,+/2,fact/1,p/1,*#/2,+#/2,fact#/1,p#/1} / {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} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,fact#,p#} and constructors {0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:fact#(s(x)) -> c_6(fact#(p(s(x)))) -->_1 fact#(s(x)) -> c_6(fact#(p(s(x)))):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: fact#(s(x)) -> c_6(fact#(p(s(x)))) *** Step 5.b:4.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: p(s(x)) -> x - Signature: {*/2,+/2,fact/1,p/1,*#/2,+#/2,fact#/1,p#/1} / {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} - Obligation: innermost runtime complexity wrt. defined symbols {*#,+#,fact#,p#} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). MAYBE