/export/starexec/sandbox2/solver/bin/starexec_run_tct_rci /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1),?) * Step 1: Sum. WORST_CASE(Omega(n^1),?) + 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: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + 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: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: DecreasingLoops. WORST_CASE(Omega(n^1),?) + 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: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: *(x,y){y -> s(y)} = *(x,s(y)) ->^+ +(*(x,y),x) = C[*(x,y) = *(x,y){}] ** Step 1.b: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 1.b:2: 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)) 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: 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 1.b:3: 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)) 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: 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 1.b:4: 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)) 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: 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 1.b:5: UsableRules. 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)) 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,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)) ** Step 1.b:6: DecomposeDG. 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: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, 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 1.b:6.a:1: 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 1.b:6.a:2: 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 1.b:6.a:3: NaturalMI. 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: NaturalMI {miDimension = 1, miDegree = 1, 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_8) = {1}, uargs(c_10) = {1} Following symbols are considered usable: {*#,+#,1#,fac#,floop#} TcT has computed the following interpretation: p(*) = [1] p(+) = [1] x1 + [8] x2 + [0] p(0) = [2] p(1) = [2] p(fac) = [2] x1 + [2] p(floop) = [1] x1 + [1] p(s) = [1] x1 + [2] p(*#) = [1] x1 + [1] p(+#) = [8] p(1#) = [2] p(fac#) = [2] x1 + [1] p(floop#) = [0] p(c_1) = [0] p(c_2) = [1] x1 + [1] x2 + [1] p(c_3) = [0] p(c_4) = [0] p(c_5) = [8] p(c_6) = [2] x1 + [0] p(c_7) = [0] p(c_8) = [1] x1 + [1] p(c_9) = [1] p(c_10) = [8] x1 + [0] Following rules are strictly oriented: fac#(s(x)) = [2] x + [5] > [2] x + [2] = c_8(fac#(x)) Following rules are (at-least) weakly oriented: floop#(s(x),y) = [0] >= [0] = c_10(floop#(x,*(s(x),y))) *** Step 1.b:6.a:4: NaturalMI. 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: NaturalMI {miDimension = 1, miDegree = 1, 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_8) = {1}, uargs(c_10) = {1} Following symbols are considered usable: {*#,+#,1#,fac#,floop#} TcT has computed the following interpretation: p(*) = [1] x1 + [15] p(+) = [0] p(0) = [0] p(1) = [0] p(fac) = [0] p(floop) = [0] p(s) = [1] x1 + [2] p(*#) = [0] p(+#) = [0] p(1#) = [0] p(fac#) = [0] p(floop#) = [8] x1 + [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [1] x1 + [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [1] x1 + [0] p(c_9) = [0] p(c_10) = [1] x1 + [12] Following rules are strictly oriented: floop#(s(x),y) = [8] x + [16] > [8] x + [12] = c_10(floop#(x,*(s(x),y))) Following rules are (at-least) weakly oriented: fac#(s(x)) = [0] >= [0] = c_8(fac#(x)) *** Step 1.b:6.a:5: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak 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: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 1.b:6.b: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)) -> *#(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: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Nothing, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component *#(x,s(y)) -> c_2(+#(*(x,y),x),*#(x,y)) 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)) and a lower component +#(x,s(y)) -> c_4(+#(x,y)) Further, following extension rules are added to the lower component. *#(x,s(y)) -> *#(x,y) *#(x,s(y)) -> +#(*(x,y),x) 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 1.b:6.b:1.a:1: SimplifyRHS. MAYBE + Considered Problem: - Strict DPs: *#(x,s(y)) -> c_2(+#(*(x,y),x),*#(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: SimplifyRHS + Details: Consider the dependency graph 1:S:*#(x,s(y)) -> c_2(+#(*(x,y),x),*#(x,y)) -->_2 *#(x,s(y)) -> c_2(+#(*(x,y),x),*#(x,y)):1 2:W:fac#(s(x)) -> *#(s(x),fac(x)) -->_1 *#(x,s(y)) -> c_2(+#(*(x,y),x),*#(x,y)):1 3:W:fac#(s(x)) -> fac#(x) -->_1 fac#(s(x)) -> fac#(x):3 -->_1 fac#(s(x)) -> *#(s(x),fac(x)):2 4:W:floop#(s(x),y) -> *#(s(x),y) -->_1 *#(x,s(y)) -> c_2(+#(*(x,y),x),*#(x,y)):1 5:W:floop#(s(x),y) -> floop#(x,*(s(x),y)) -->_1 floop#(s(x),y) -> floop#(x,*(s(x),y)):5 -->_1 floop#(s(x),y) -> *#(s(x),y):4 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: *#(x,s(y)) -> c_2(*#(x,y)) **** Step 1.b:6.b:1.a:2: Failure MAYBE + Considered Problem: - Strict DPs: *#(x,s(y)) -> c_2(*#(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_2/1,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 1.b:6.b:1.b:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: +#(x,s(y)) -> c_4(+#(x,y)) - Weak DPs: *#(x,s(y)) -> *#(x,y) *#(x,s(y)) -> +#(*(x,y),x) 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: NaturalMI {miDimension = 1, miDegree = 1, 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_4) = {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) = [0] p(floop) = [0] p(s) = [1] x1 + [1] p(*#) = [8] x1 + [8] p(+#) = [8] x2 + [8] p(1#) = [0] p(fac#) = [8] x1 + [9] p(floop#) = [15] x1 + [1] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [1] x1 + [6] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] Following rules are strictly oriented: +#(x,s(y)) = [8] y + [16] > [8] y + [14] = c_4(+#(x,y)) Following rules are (at-least) weakly oriented: *#(x,s(y)) = [8] x + [8] >= [8] x + [8] = *#(x,y) *#(x,s(y)) = [8] x + [8] >= [8] x + [8] = +#(*(x,y),x) fac#(s(x)) = [8] x + [17] >= [8] x + [16] = *#(s(x),fac(x)) fac#(s(x)) = [8] x + [17] >= [8] x + [9] = fac#(x) floop#(s(x),y) = [15] x + [16] >= [8] x + [16] = *#(s(x),y) floop#(s(x),y) = [15] x + [16] >= [15] x + [1] = floop#(x,*(s(x),y)) **** Step 1.b:6.b:1.b:2: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: *#(x,s(y)) -> *#(x,y) *#(x,s(y)) -> +#(*(x,y),x) +#(x,s(y)) -> c_4(+#(x,y)) 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 already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),?)