/export/starexec/sandbox/solver/bin/starexec_run_tct_rci /export/starexec/sandbox/benchmark/theBenchmark.xml /export/starexec/sandbox/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1),?) * Step 1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: fac(s(x)) -> times(fac(p(s(x))),s(x)) p(s(0())) -> 0() p(s(s(x))) -> s(p(s(x))) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) times(x,0()) -> 0() times(0(),y) -> 0() times(s(x),y) -> plus(times(x,y),y) - Signature: {fac/1,p/1,plus/2,times/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {fac,p,plus,times} 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: fac(s(x)) -> times(fac(p(s(x))),s(x)) p(s(0())) -> 0() p(s(s(x))) -> s(p(s(x))) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) times(x,0()) -> 0() times(0(),y) -> 0() times(s(x),y) -> plus(times(x,y),y) - Signature: {fac/1,p/1,plus/2,times/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {fac,p,plus,times} 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: fac(s(x)) -> times(fac(p(s(x))),s(x)) p(s(0())) -> 0() p(s(s(x))) -> s(p(s(x))) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) times(x,0()) -> 0() times(0(),y) -> 0() times(s(x),y) -> plus(times(x,y),y) - Signature: {fac/1,p/1,plus/2,times/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {fac,p,plus,times} and constructors {0,s} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: p(s(x)){x -> s(x)} = p(s(s(x))) ->^+ s(p(s(x))) = C[p(s(x)) = p(s(x)){}] ** Step 1.b:1: DependencyPairs. MAYBE + Considered Problem: - Strict TRS: fac(s(x)) -> times(fac(p(s(x))),s(x)) p(s(0())) -> 0() p(s(s(x))) -> s(p(s(x))) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) times(x,0()) -> 0() times(0(),y) -> 0() times(s(x),y) -> plus(times(x,y),y) - Signature: {fac/1,p/1,plus/2,times/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {fac,p,plus,times} and constructors {0,s} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs fac#(s(x)) -> c_1(times#(fac(p(s(x))),s(x)),fac#(p(s(x))),p#(s(x))) p#(s(0())) -> c_2() p#(s(s(x))) -> c_3(p#(s(x))) plus#(x,0()) -> c_4() plus#(x,s(y)) -> c_5(plus#(x,y)) times#(x,0()) -> c_6() times#(0(),y) -> c_7() times#(s(x),y) -> c_8(plus#(times(x,y),y),times#(x,y)) Weak DPs and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation. MAYBE + Considered Problem: - Strict DPs: fac#(s(x)) -> c_1(times#(fac(p(s(x))),s(x)),fac#(p(s(x))),p#(s(x))) p#(s(0())) -> c_2() p#(s(s(x))) -> c_3(p#(s(x))) plus#(x,0()) -> c_4() plus#(x,s(y)) -> c_5(plus#(x,y)) times#(x,0()) -> c_6() times#(0(),y) -> c_7() times#(s(x),y) -> c_8(plus#(times(x,y),y),times#(x,y)) - Weak TRS: fac(s(x)) -> times(fac(p(s(x))),s(x)) p(s(0())) -> 0() p(s(s(x))) -> s(p(s(x))) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) times(x,0()) -> 0() times(0(),y) -> 0() times(s(x),y) -> plus(times(x,y),y) - Signature: {fac/1,p/1,plus/2,times/2,fac#/1,p#/1,plus#/2,times#/2} / {0/0,s/1,c_1/3,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/0 ,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {fac#,p#,plus#,times#} and constructors {0,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {2,4,6,7} by application of Pre({2,4,6,7}) = {1,3,5,8}. Here rules are labelled as follows: 1: fac#(s(x)) -> c_1(times#(fac(p(s(x))),s(x)),fac#(p(s(x))),p#(s(x))) 2: p#(s(0())) -> c_2() 3: p#(s(s(x))) -> c_3(p#(s(x))) 4: plus#(x,0()) -> c_4() 5: plus#(x,s(y)) -> c_5(plus#(x,y)) 6: times#(x,0()) -> c_6() 7: times#(0(),y) -> c_7() 8: times#(s(x),y) -> c_8(plus#(times(x,y),y),times#(x,y)) ** Step 1.b:3: RemoveWeakSuffixes. MAYBE + Considered Problem: - Strict DPs: fac#(s(x)) -> c_1(times#(fac(p(s(x))),s(x)),fac#(p(s(x))),p#(s(x))) p#(s(s(x))) -> c_3(p#(s(x))) plus#(x,s(y)) -> c_5(plus#(x,y)) times#(s(x),y) -> c_8(plus#(times(x,y),y),times#(x,y)) - Weak DPs: p#(s(0())) -> c_2() plus#(x,0()) -> c_4() times#(x,0()) -> c_6() times#(0(),y) -> c_7() - Weak TRS: fac(s(x)) -> times(fac(p(s(x))),s(x)) p(s(0())) -> 0() p(s(s(x))) -> s(p(s(x))) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) times(x,0()) -> 0() times(0(),y) -> 0() times(s(x),y) -> plus(times(x,y),y) - Signature: {fac/1,p/1,plus/2,times/2,fac#/1,p#/1,plus#/2,times#/2} / {0/0,s/1,c_1/3,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/0 ,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {fac#,p#,plus#,times#} and constructors {0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:fac#(s(x)) -> c_1(times#(fac(p(s(x))),s(x)),fac#(p(s(x))),p#(s(x))) -->_1 times#(s(x),y) -> c_8(plus#(times(x,y),y),times#(x,y)):4 -->_3 p#(s(s(x))) -> c_3(p#(s(x))):2 -->_1 times#(0(),y) -> c_7():8 -->_3 p#(s(0())) -> c_2():5 -->_2 fac#(s(x)) -> c_1(times#(fac(p(s(x))),s(x)),fac#(p(s(x))),p#(s(x))):1 2:S:p#(s(s(x))) -> c_3(p#(s(x))) -->_1 p#(s(0())) -> c_2():5 -->_1 p#(s(s(x))) -> c_3(p#(s(x))):2 3:S:plus#(x,s(y)) -> c_5(plus#(x,y)) -->_1 plus#(x,0()) -> c_4():6 -->_1 plus#(x,s(y)) -> c_5(plus#(x,y)):3 4:S:times#(s(x),y) -> c_8(plus#(times(x,y),y),times#(x,y)) -->_2 times#(0(),y) -> c_7():8 -->_2 times#(x,0()) -> c_6():7 -->_1 plus#(x,0()) -> c_4():6 -->_2 times#(s(x),y) -> c_8(plus#(times(x,y),y),times#(x,y)):4 -->_1 plus#(x,s(y)) -> c_5(plus#(x,y)):3 5:W:p#(s(0())) -> c_2() 6:W:plus#(x,0()) -> c_4() 7:W:times#(x,0()) -> c_6() 8:W:times#(0(),y) -> c_7() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 5: p#(s(0())) -> c_2() 6: plus#(x,0()) -> c_4() 7: times#(x,0()) -> c_6() 8: times#(0(),y) -> c_7() ** Step 1.b:4: DecomposeDG. MAYBE + Considered Problem: - Strict DPs: fac#(s(x)) -> c_1(times#(fac(p(s(x))),s(x)),fac#(p(s(x))),p#(s(x))) p#(s(s(x))) -> c_3(p#(s(x))) plus#(x,s(y)) -> c_5(plus#(x,y)) times#(s(x),y) -> c_8(plus#(times(x,y),y),times#(x,y)) - Weak TRS: fac(s(x)) -> times(fac(p(s(x))),s(x)) p(s(0())) -> 0() p(s(s(x))) -> s(p(s(x))) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) times(x,0()) -> 0() times(0(),y) -> 0() times(s(x),y) -> plus(times(x,y),y) - Signature: {fac/1,p/1,plus/2,times/2,fac#/1,p#/1,plus#/2,times#/2} / {0/0,s/1,c_1/3,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/0 ,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {fac#,p#,plus#,times#} 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_1(times#(fac(p(s(x))),s(x)),fac#(p(s(x))),p#(s(x))) and a lower component p#(s(s(x))) -> c_3(p#(s(x))) plus#(x,s(y)) -> c_5(plus#(x,y)) times#(s(x),y) -> c_8(plus#(times(x,y),y),times#(x,y)) Further, following extension rules are added to the lower component. fac#(s(x)) -> fac#(p(s(x))) fac#(s(x)) -> p#(s(x)) fac#(s(x)) -> times#(fac(p(s(x))),s(x)) *** Step 1.b:4.a:1: SimplifyRHS. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: fac#(s(x)) -> c_1(times#(fac(p(s(x))),s(x)),fac#(p(s(x))),p#(s(x))) - Weak TRS: fac(s(x)) -> times(fac(p(s(x))),s(x)) p(s(0())) -> 0() p(s(s(x))) -> s(p(s(x))) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) times(x,0()) -> 0() times(0(),y) -> 0() times(s(x),y) -> plus(times(x,y),y) - Signature: {fac/1,p/1,plus/2,times/2,fac#/1,p#/1,plus#/2,times#/2} / {0/0,s/1,c_1/3,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/0 ,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {fac#,p#,plus#,times#} and constructors {0,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:fac#(s(x)) -> c_1(times#(fac(p(s(x))),s(x)),fac#(p(s(x))),p#(s(x))) -->_2 fac#(s(x)) -> c_1(times#(fac(p(s(x))),s(x)),fac#(p(s(x))),p#(s(x))):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: fac#(s(x)) -> c_1(fac#(p(s(x)))) *** Step 1.b:4.a:2: UsableRules. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: fac#(s(x)) -> c_1(fac#(p(s(x)))) - Weak TRS: fac(s(x)) -> times(fac(p(s(x))),s(x)) p(s(0())) -> 0() p(s(s(x))) -> s(p(s(x))) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) times(x,0()) -> 0() times(0(),y) -> 0() times(s(x),y) -> plus(times(x,y),y) - Signature: {fac/1,p/1,plus/2,times/2,fac#/1,p#/1,plus#/2,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/0 ,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {fac#,p#,plus#,times#} and constructors {0,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: p(s(0())) -> 0() p(s(s(x))) -> s(p(s(x))) fac#(s(x)) -> c_1(fac#(p(s(x)))) *** Step 1.b:4.a:3: Ara. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: fac#(s(x)) -> c_1(fac#(p(s(x)))) - Weak TRS: p(s(0())) -> 0() p(s(s(x))) -> s(p(s(x))) - Signature: {fac/1,p/1,plus/2,times/2,fac#/1,p#/1,plus#/2,times#/2} / {0/0,s/1,c_1/1,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/0 ,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {fac#,p#,plus#,times#} and constructors {0,s} + Applied Processor: Ara {minDegree = 1, maxDegree = 2, araTimeout = 8, araRuleShifting = Just 1, isBestCase = False, mkCompletelyDefined = False, verboseOutput = False} + Details: Signatures used: ---------------- F (TrsFun "0") :: [] -(0)-> "A"(13, 13) F (TrsFun "0") :: [] -(0)-> "A"(15, 15) F (TrsFun "p") :: ["A"(0, 13)] -(1)-> "A"(13, 13) F (TrsFun "s") :: ["A"(13, 13)] -(13)-> "A"(13, 13) F (TrsFun "s") :: ["A"(13, 13)] -(0)-> "A"(0, 13) F (DpFun "fac") :: ["A"(13, 13)] -(7)-> "A"(0, 14) F (ComFun 1) :: ["A"(0, 14)] -(0)-> "A"(14, 14) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- "F (ComFun 1)_A" :: ["A"(0, 0)] -(0)-> "A"(1, 0) "F (ComFun 1)_A" :: ["A"(0, 1)] -(0)-> "A"(0, 1) "F (TrsFun \"0\")_A" :: [] -(0)-> "A"(1, 0) "F (TrsFun \"0\")_A" :: [] -(0)-> "A"(0, 1) "F (TrsFun \"s\")_A" :: ["A"(0, 0)] -(1)-> "A"(1, 0) "F (TrsFun \"s\")_A" :: ["A"(1, 1)] -(0)-> "A"(0, 1) Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: fac#(s(x)) -> c_1(fac#(p(s(x)))) 2. Weak: *** Step 1.b:4.b:1: DecomposeDG. MAYBE + Considered Problem: - Strict DPs: p#(s(s(x))) -> c_3(p#(s(x))) plus#(x,s(y)) -> c_5(plus#(x,y)) times#(s(x),y) -> c_8(plus#(times(x,y),y),times#(x,y)) - Weak DPs: fac#(s(x)) -> fac#(p(s(x))) fac#(s(x)) -> p#(s(x)) fac#(s(x)) -> times#(fac(p(s(x))),s(x)) - Weak TRS: fac(s(x)) -> times(fac(p(s(x))),s(x)) p(s(0())) -> 0() p(s(s(x))) -> s(p(s(x))) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) times(x,0()) -> 0() times(0(),y) -> 0() times(s(x),y) -> plus(times(x,y),y) - Signature: {fac/1,p/1,plus/2,times/2,fac#/1,p#/1,plus#/2,times#/2} / {0/0,s/1,c_1/3,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/0 ,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {fac#,p#,plus#,times#} 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)) -> fac#(p(s(x))) fac#(s(x)) -> p#(s(x)) fac#(s(x)) -> times#(fac(p(s(x))),s(x)) p#(s(s(x))) -> c_3(p#(s(x))) times#(s(x),y) -> c_8(plus#(times(x,y),y),times#(x,y)) and a lower component plus#(x,s(y)) -> c_5(plus#(x,y)) Further, following extension rules are added to the lower component. fac#(s(x)) -> fac#(p(s(x))) fac#(s(x)) -> p#(s(x)) fac#(s(x)) -> times#(fac(p(s(x))),s(x)) p#(s(s(x))) -> p#(s(x)) times#(s(x),y) -> plus#(times(x,y),y) times#(s(x),y) -> times#(x,y) **** Step 1.b:4.b:1.a:1: SimplifyRHS. MAYBE + Considered Problem: - Strict DPs: p#(s(s(x))) -> c_3(p#(s(x))) times#(s(x),y) -> c_8(plus#(times(x,y),y),times#(x,y)) - Weak DPs: fac#(s(x)) -> fac#(p(s(x))) fac#(s(x)) -> p#(s(x)) fac#(s(x)) -> times#(fac(p(s(x))),s(x)) - Weak TRS: fac(s(x)) -> times(fac(p(s(x))),s(x)) p(s(0())) -> 0() p(s(s(x))) -> s(p(s(x))) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) times(x,0()) -> 0() times(0(),y) -> 0() times(s(x),y) -> plus(times(x,y),y) - Signature: {fac/1,p/1,plus/2,times/2,fac#/1,p#/1,plus#/2,times#/2} / {0/0,s/1,c_1/3,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/0 ,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {fac#,p#,plus#,times#} and constructors {0,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:p#(s(s(x))) -> c_3(p#(s(x))) -->_1 p#(s(s(x))) -> c_3(p#(s(x))):1 2:S:times#(s(x),y) -> c_8(plus#(times(x,y),y),times#(x,y)) -->_2 times#(s(x),y) -> c_8(plus#(times(x,y),y),times#(x,y)):2 3:W:fac#(s(x)) -> fac#(p(s(x))) -->_1 fac#(s(x)) -> times#(fac(p(s(x))),s(x)):5 -->_1 fac#(s(x)) -> p#(s(x)):4 -->_1 fac#(s(x)) -> fac#(p(s(x))):3 4:W:fac#(s(x)) -> p#(s(x)) -->_1 p#(s(s(x))) -> c_3(p#(s(x))):1 5:W:fac#(s(x)) -> times#(fac(p(s(x))),s(x)) -->_1 times#(s(x),y) -> c_8(plus#(times(x,y),y),times#(x,y)):2 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: times#(s(x),y) -> c_8(times#(x,y)) **** Step 1.b:4.b:1.a:2: NaturalMI. MAYBE + Considered Problem: - Strict DPs: p#(s(s(x))) -> c_3(p#(s(x))) times#(s(x),y) -> c_8(times#(x,y)) - Weak DPs: fac#(s(x)) -> fac#(p(s(x))) fac#(s(x)) -> p#(s(x)) fac#(s(x)) -> times#(fac(p(s(x))),s(x)) - Weak TRS: fac(s(x)) -> times(fac(p(s(x))),s(x)) p(s(0())) -> 0() p(s(s(x))) -> s(p(s(x))) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) times(x,0()) -> 0() times(0(),y) -> 0() times(s(x),y) -> plus(times(x,y),y) - Signature: {fac/1,p/1,plus/2,times/2,fac#/1,p#/1,plus#/2,times#/2} / {0/0,s/1,c_1/3,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/0 ,c_8/1} - Obligation: innermost runtime complexity wrt. defined symbols {fac#,p#,plus#,times#} 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_3) = {1}, uargs(c_8) = {1} Following symbols are considered usable: {p,fac#,p#,plus#,times#} TcT has computed the following interpretation: p(0) = [4] p(fac) = [1] x1 + [14] p(p) = [1] x1 + [0] p(plus) = [12] p(s) = [1] x1 + [4] p(times) = [2] p(fac#) = [5] x1 + [4] p(p#) = [1] x1 + [12] p(plus#) = [1] x1 + [2] x2 + [0] p(times#) = [2] x2 + [8] p(c_1) = [0] p(c_2) = [4] p(c_3) = [1] x1 + [0] p(c_4) = [1] p(c_5) = [8] x1 + [4] p(c_6) = [8] p(c_7) = [1] p(c_8) = [1] x1 + [0] Following rules are strictly oriented: p#(s(s(x))) = [1] x + [20] > [1] x + [16] = c_3(p#(s(x))) Following rules are (at-least) weakly oriented: fac#(s(x)) = [5] x + [24] >= [5] x + [24] = fac#(p(s(x))) fac#(s(x)) = [5] x + [24] >= [1] x + [16] = p#(s(x)) fac#(s(x)) = [5] x + [24] >= [2] x + [16] = times#(fac(p(s(x))),s(x)) times#(s(x),y) = [2] y + [8] >= [2] y + [8] = c_8(times#(x,y)) p(s(0())) = [8] >= [4] = 0() p(s(s(x))) = [1] x + [8] >= [1] x + [8] = s(p(s(x))) **** Step 1.b:4.b:1.a:3: Failure MAYBE + Considered Problem: - Strict DPs: times#(s(x),y) -> c_8(times#(x,y)) - Weak DPs: fac#(s(x)) -> fac#(p(s(x))) fac#(s(x)) -> p#(s(x)) fac#(s(x)) -> times#(fac(p(s(x))),s(x)) p#(s(s(x))) -> c_3(p#(s(x))) - Weak TRS: fac(s(x)) -> times(fac(p(s(x))),s(x)) p(s(0())) -> 0() p(s(s(x))) -> s(p(s(x))) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) times(x,0()) -> 0() times(0(),y) -> 0() times(s(x),y) -> plus(times(x,y),y) - Signature: {fac/1,p/1,plus/2,times/2,fac#/1,p#/1,plus#/2,times#/2} / {0/0,s/1,c_1/3,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/0 ,c_8/1} - Obligation: innermost runtime complexity wrt. defined symbols {fac#,p#,plus#,times#} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is still open. **** Step 1.b:4.b:1.b:1: RemoveWeakSuffixes. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: plus#(x,s(y)) -> c_5(plus#(x,y)) - Weak DPs: fac#(s(x)) -> fac#(p(s(x))) fac#(s(x)) -> p#(s(x)) fac#(s(x)) -> times#(fac(p(s(x))),s(x)) p#(s(s(x))) -> p#(s(x)) times#(s(x),y) -> plus#(times(x,y),y) times#(s(x),y) -> times#(x,y) - Weak TRS: fac(s(x)) -> times(fac(p(s(x))),s(x)) p(s(0())) -> 0() p(s(s(x))) -> s(p(s(x))) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) times(x,0()) -> 0() times(0(),y) -> 0() times(s(x),y) -> plus(times(x,y),y) - Signature: {fac/1,p/1,plus/2,times/2,fac#/1,p#/1,plus#/2,times#/2} / {0/0,s/1,c_1/3,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/0 ,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {fac#,p#,plus#,times#} and constructors {0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:plus#(x,s(y)) -> c_5(plus#(x,y)) -->_1 plus#(x,s(y)) -> c_5(plus#(x,y)):1 2:W:fac#(s(x)) -> fac#(p(s(x))) -->_1 fac#(s(x)) -> times#(fac(p(s(x))),s(x)):4 -->_1 fac#(s(x)) -> p#(s(x)):3 -->_1 fac#(s(x)) -> fac#(p(s(x))):2 3:W:fac#(s(x)) -> p#(s(x)) -->_1 p#(s(s(x))) -> p#(s(x)):5 4:W:fac#(s(x)) -> times#(fac(p(s(x))),s(x)) -->_1 times#(s(x),y) -> times#(x,y):7 -->_1 times#(s(x),y) -> plus#(times(x,y),y):6 5:W:p#(s(s(x))) -> p#(s(x)) -->_1 p#(s(s(x))) -> p#(s(x)):5 6:W:times#(s(x),y) -> plus#(times(x,y),y) -->_1 plus#(x,s(y)) -> c_5(plus#(x,y)):1 7:W:times#(s(x),y) -> times#(x,y) -->_1 times#(s(x),y) -> times#(x,y):7 -->_1 times#(s(x),y) -> plus#(times(x,y),y):6 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: fac#(s(x)) -> p#(s(x)) 5: p#(s(s(x))) -> p#(s(x)) **** Step 1.b:4.b:1.b:2: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: plus#(x,s(y)) -> c_5(plus#(x,y)) - Weak DPs: fac#(s(x)) -> fac#(p(s(x))) fac#(s(x)) -> times#(fac(p(s(x))),s(x)) times#(s(x),y) -> plus#(times(x,y),y) times#(s(x),y) -> times#(x,y) - Weak TRS: fac(s(x)) -> times(fac(p(s(x))),s(x)) p(s(0())) -> 0() p(s(s(x))) -> s(p(s(x))) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) times(x,0()) -> 0() times(0(),y) -> 0() times(s(x),y) -> plus(times(x,y),y) - Signature: {fac/1,p/1,plus/2,times/2,fac#/1,p#/1,plus#/2,times#/2} / {0/0,s/1,c_1/3,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/0 ,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {fac#,p#,plus#,times#} 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_5) = {1} Following symbols are considered usable: {p,fac#,p#,plus#,times#} TcT has computed the following interpretation: p(0) = [0] p(fac) = [2] x1 + [4] p(p) = [1] x1 + [0] p(plus) = [8] p(s) = [1] x1 + [1] p(times) = [1] x1 + [1] x2 + [4] p(fac#) = [1] x1 + [8] p(p#) = [1] x1 + [0] p(plus#) = [1] x2 + [0] p(times#) = [1] x2 + [8] p(c_1) = [1] x2 + [4] x3 + [1] p(c_2) = [1] p(c_3) = [1] p(c_4) = [0] p(c_5) = [1] x1 + [0] p(c_6) = [0] p(c_7) = [8] p(c_8) = [4] Following rules are strictly oriented: plus#(x,s(y)) = [1] y + [1] > [1] y + [0] = c_5(plus#(x,y)) Following rules are (at-least) weakly oriented: fac#(s(x)) = [1] x + [9] >= [1] x + [9] = fac#(p(s(x))) fac#(s(x)) = [1] x + [9] >= [1] x + [9] = times#(fac(p(s(x))),s(x)) times#(s(x),y) = [1] y + [8] >= [1] y + [0] = plus#(times(x,y),y) times#(s(x),y) = [1] y + [8] >= [1] y + [8] = times#(x,y) p(s(0())) = [1] >= [0] = 0() p(s(s(x))) = [1] x + [2] >= [1] x + [2] = s(p(s(x))) **** Step 1.b:4.b:1.b:3: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: fac#(s(x)) -> fac#(p(s(x))) fac#(s(x)) -> times#(fac(p(s(x))),s(x)) plus#(x,s(y)) -> c_5(plus#(x,y)) times#(s(x),y) -> plus#(times(x,y),y) times#(s(x),y) -> times#(x,y) - Weak TRS: fac(s(x)) -> times(fac(p(s(x))),s(x)) p(s(0())) -> 0() p(s(s(x))) -> s(p(s(x))) plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) times(x,0()) -> 0() times(0(),y) -> 0() times(s(x),y) -> plus(times(x,y),y) - Signature: {fac/1,p/1,plus/2,times/2,fac#/1,p#/1,plus#/2,times#/2} / {0/0,s/1,c_1/3,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0,c_7/0 ,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {fac#,p#,plus#,times#} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),?)