/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),O(n^3)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^3)) + Considered Problem: - Strict TRS: plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) plus(0(),x) -> x plus(s(x),y) -> s(plus(x,y)) times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) - Signature: {plus/2,times/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {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: plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) plus(0(),x) -> x plus(s(x),y) -> s(plus(x,y)) times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) - Signature: {plus/2,times/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {plus,times} and constructors {0,s} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () *** Step 1.a:1.a:1: Ara. MAYBE + Considered Problem: - Strict TRS: plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) plus(0(),x) -> x plus(s(x),y) -> s(plus(x,y)) times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) - Signature: {plus/2,times/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {plus,times} and constructors {0,s} + Applied Processor: Ara {minDegree = 1, maxDegree = 3, araTimeout = 15, araRuleShifting = Just 1, isBestCase = True, mkCompletelyDefined = False, verboseOutput = False} + Details: Signatures used: ---------------- F (TrsFun "0") :: [] -(0)-> "A"(0) F (TrsFun "0") :: [] -(0)-> "A"(1) F (TrsFun "main") :: ["A"(0) x "A"(1)] -(1)-> "A"(0) F (TrsFun "plus") :: ["A"(0) x "A"(0)] -(1)-> "A"(0) F (TrsFun "s") :: ["A"(0)] -(0)-> "A"(0) F (TrsFun "s") :: ["A"(1)] -(1)-> "A"(1) F (TrsFun "times") :: ["A"(0) x "A"(1)] -(1)-> "A"(0) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) plus(0(),x) -> x plus(s(x),y) -> s(plus(x,y)) times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) main(x1,x2) -> times(x1,x2) 2. Weak: *** Step 1.a:1.b:1: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) plus(0(),x) -> x plus(s(x),y) -> s(plus(x,y)) times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) - Signature: {plus/2,times/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {plus,times} and constructors {0,s} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: plus(x,y){y -> s(y)} = plus(x,s(y)) ->^+ s(plus(x,y)) = C[plus(x,y) = plus(x,y){}] ** Step 1.b:1: DependencyPairs. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) plus(0(),x) -> x plus(s(x),y) -> s(plus(x,y)) times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) - Signature: {plus/2,times/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {plus,times} and constructors {0,s} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs plus#(x,0()) -> c_1() plus#(x,s(y)) -> c_2(plus#(x,y)) plus#(0(),x) -> c_3() plus#(s(x),y) -> c_4(plus#(x,y)) times#(x,0()) -> c_5() times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y)) Weak DPs and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: plus#(x,0()) -> c_1() plus#(x,s(y)) -> c_2(plus#(x,y)) plus#(0(),x) -> c_3() plus#(s(x),y) -> c_4(plus#(x,y)) times#(x,0()) -> c_5() times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y)) - Weak TRS: plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) plus(0(),x) -> x plus(s(x),y) -> s(plus(x,y)) times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) - Signature: {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2} - Obligation: innermost runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,3,5} by application of Pre({1,3,5}) = {2,4,6}. Here rules are labelled as follows: 1: plus#(x,0()) -> c_1() 2: plus#(x,s(y)) -> c_2(plus#(x,y)) 3: plus#(0(),x) -> c_3() 4: plus#(s(x),y) -> c_4(plus#(x,y)) 5: times#(x,0()) -> c_5() 6: times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y)) ** Step 1.b:3: RemoveWeakSuffixes. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: plus#(x,s(y)) -> c_2(plus#(x,y)) plus#(s(x),y) -> c_4(plus#(x,y)) times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y)) - Weak DPs: plus#(x,0()) -> c_1() plus#(0(),x) -> c_3() times#(x,0()) -> c_5() - Weak TRS: plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) plus(0(),x) -> x plus(s(x),y) -> s(plus(x,y)) times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) - Signature: {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2} - Obligation: innermost runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:plus#(x,s(y)) -> c_2(plus#(x,y)) -->_1 plus#(s(x),y) -> c_4(plus#(x,y)):2 -->_1 plus#(0(),x) -> c_3():5 -->_1 plus#(x,0()) -> c_1():4 -->_1 plus#(x,s(y)) -> c_2(plus#(x,y)):1 2:S:plus#(s(x),y) -> c_4(plus#(x,y)) -->_1 plus#(0(),x) -> c_3():5 -->_1 plus#(x,0()) -> c_1():4 -->_1 plus#(s(x),y) -> c_4(plus#(x,y)):2 -->_1 plus#(x,s(y)) -> c_2(plus#(x,y)):1 3:S:times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y)) -->_2 times#(x,0()) -> c_5():6 -->_1 plus#(0(),x) -> c_3():5 -->_1 plus#(x,0()) -> c_1():4 -->_2 times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y)):3 -->_1 plus#(s(x),y) -> c_4(plus#(x,y)):2 -->_1 plus#(x,s(y)) -> c_2(plus#(x,y)):1 4:W:plus#(x,0()) -> c_1() 5:W:plus#(0(),x) -> c_3() 6:W:times#(x,0()) -> c_5() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 6: times#(x,0()) -> c_5() 4: plus#(x,0()) -> c_1() 5: plus#(0(),x) -> c_3() ** Step 1.b:4: Decompose. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: plus#(x,s(y)) -> c_2(plus#(x,y)) plus#(s(x),y) -> c_4(plus#(x,y)) times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y)) - Weak TRS: plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) plus(0(),x) -> x plus(s(x),y) -> s(plus(x,y)) times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) - Signature: {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2} - Obligation: innermost runtime complexity wrt. defined symbols {plus#,times#} 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: plus#(x,s(y)) -> c_2(plus#(x,y)) plus#(s(x),y) -> c_4(plus#(x,y)) - Weak DPs: times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y)) - Weak TRS: plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) plus(0(),x) -> x plus(s(x),y) -> s(plus(x,y)) times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) - Signature: {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2} - Obligation: innermost runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s} Problem (S) - Strict DPs: times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y)) - Weak DPs: plus#(x,s(y)) -> c_2(plus#(x,y)) plus#(s(x),y) -> c_4(plus#(x,y)) - Weak TRS: plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) plus(0(),x) -> x plus(s(x),y) -> s(plus(x,y)) times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) - Signature: {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2} - Obligation: innermost runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s} *** Step 1.b:4.a:1: PredecessorEstimationCP. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: plus#(x,s(y)) -> c_2(plus#(x,y)) plus#(s(x),y) -> c_4(plus#(x,y)) - Weak DPs: times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y)) - Weak TRS: plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) plus(0(),x) -> x plus(s(x),y) -> s(plus(x,y)) times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) - Signature: {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2} - Obligation: innermost runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: plus#(x,s(y)) -> c_2(plus#(x,y)) The strictly oriented rules are moved into the weak component. **** Step 1.b:4.a:1.a:1: NaturalPI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: plus#(x,s(y)) -> c_2(plus#(x,y)) plus#(s(x),y) -> c_4(plus#(x,y)) - Weak DPs: times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y)) - Weak TRS: plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) plus(0(),x) -> x plus(s(x),y) -> s(plus(x,y)) times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) - Signature: {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2} - Obligation: innermost runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_4) = {1}, uargs(c_6) = {1,2} Following symbols are considered usable: {plus#,times#} TcT has computed the following interpretation: p(0) = 0 p(plus) = 6 + 2*x1 + 2*x1*x2 + x1^2 + 2*x2^2 p(s) = 2 + x1 p(times) = 1 + 2*x1 + 2*x1*x2 p(plus#) = 5 + x2 p(times#) = 2 + x1 + 4*x1*x2 + 3*x2 p(c_1) = 4 p(c_2) = x1 p(c_3) = 0 p(c_4) = x1 p(c_5) = 0 p(c_6) = 1 + x1 + x2 Following rules are strictly oriented: plus#(x,s(y)) = 7 + y > 5 + y = c_2(plus#(x,y)) Following rules are (at-least) weakly oriented: plus#(s(x),y) = 5 + y >= 5 + y = c_4(plus#(x,y)) times#(x,s(y)) = 8 + 9*x + 4*x*y + 3*y >= 8 + 2*x + 4*x*y + 3*y = c_6(plus#(times(x,y),x),times#(x,y)) **** Step 1.b:4.a:1.a:2: Assumption. WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: plus#(s(x),y) -> c_4(plus#(x,y)) - Weak DPs: plus#(x,s(y)) -> c_2(plus#(x,y)) times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y)) - Weak TRS: plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) plus(0(),x) -> x plus(s(x),y) -> s(plus(x,y)) times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) - Signature: {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2} - Obligation: innermost runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}} + Details: () **** Step 1.b:4.a:1.b:1: DecomposeDG. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: plus#(s(x),y) -> c_4(plus#(x,y)) - Weak DPs: plus#(x,s(y)) -> c_2(plus#(x,y)) times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y)) - Weak TRS: plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) plus(0(),x) -> x plus(s(x),y) -> s(plus(x,y)) times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) - Signature: {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2} - Obligation: innermost runtime complexity wrt. defined symbols {plus#,times#} 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 times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y)) and a lower component plus#(x,s(y)) -> c_2(plus#(x,y)) plus#(s(x),y) -> c_4(plus#(x,y)) Further, following extension rules are added to the lower component. times#(x,s(y)) -> plus#(times(x,y),x) times#(x,s(y)) -> times#(x,y) ***** Step 1.b:4.a:1.b:1.a:1: PredecessorEstimationCP. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y)) - Weak TRS: plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) plus(0(),x) -> x plus(s(x),y) -> s(plus(x,y)) times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) - Signature: {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2} - Obligation: innermost runtime complexity wrt. defined symbols {plus#,times#} 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: times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y)) The strictly oriented rules are moved into the weak component. ****** Step 1.b:4.a:1.b:1.a:1.a:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y)) - Weak TRS: plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) plus(0(),x) -> x plus(s(x),y) -> s(plus(x,y)) times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) - Signature: {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2} - Obligation: innermost runtime complexity wrt. defined symbols {plus#,times#} 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_6) = {1,2} Following symbols are considered usable: {plus#,times#} TcT has computed the following interpretation: p(0) = [0] p(plus) = [0] p(s) = [1] x1 + [2] p(times) = [0] p(plus#) = [0] p(times#) = [10] x2 + [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [1] x1 + [1] x2 + [0] Following rules are strictly oriented: times#(x,s(y)) = [10] y + [20] > [10] y + [0] = c_6(plus#(times(x,y),x),times#(x,y)) Following rules are (at-least) weakly oriented: ****** Step 1.b:4.a:1.b:1.a:1.a:2: Assumption. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y)) - Weak TRS: plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) plus(0(),x) -> x plus(s(x),y) -> s(plus(x,y)) times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) - Signature: {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2} - Obligation: innermost runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}} + Details: () ****** Step 1.b:4.a:1.b:1.a:1.b:1: RemoveWeakSuffixes. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y)) - Weak TRS: plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) plus(0(),x) -> x plus(s(x),y) -> s(plus(x,y)) times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) - Signature: {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2} - Obligation: innermost runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y)) -->_2 times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y)) ****** Step 1.b:4.a:1.b:1.a:1.b:2: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) plus(0(),x) -> x plus(s(x),y) -> s(plus(x,y)) times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) - Signature: {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2} - Obligation: innermost runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ***** Step 1.b:4.a:1.b:1.b:1: PredecessorEstimationCP. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: plus#(s(x),y) -> c_4(plus#(x,y)) - Weak DPs: plus#(x,s(y)) -> c_2(plus#(x,y)) times#(x,s(y)) -> plus#(times(x,y),x) times#(x,s(y)) -> times#(x,y) - Weak TRS: plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) plus(0(),x) -> x plus(s(x),y) -> s(plus(x,y)) times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) - Signature: {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2} - Obligation: innermost runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: plus#(s(x),y) -> c_4(plus#(x,y)) The strictly oriented rules are moved into the weak component. ****** Step 1.b:4.a:1.b:1.b:1.a:1: NaturalPI. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: plus#(s(x),y) -> c_4(plus#(x,y)) - Weak DPs: plus#(x,s(y)) -> c_2(plus#(x,y)) times#(x,s(y)) -> plus#(times(x,y),x) times#(x,s(y)) -> times#(x,y) - Weak TRS: plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) plus(0(),x) -> x plus(s(x),y) -> s(plus(x,y)) times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) - Signature: {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2} - Obligation: innermost runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s} + Applied Processor: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_2) = {1}, uargs(c_4) = {1} Following symbols are considered usable: {plus,times,plus#,times#} TcT has computed the following interpretation: p(0) = 0 p(plus) = x1 + 3*x2 p(s) = 3 + x1 p(times) = x1*x2 p(plus#) = 4*x1 + x2 p(times#) = 2 + 2*x1 + 4*x1*x2 + 6*x1^2 + x2 p(c_1) = 4 p(c_2) = 2 + x1 p(c_3) = 1 p(c_4) = 5 + x1 p(c_5) = 1 p(c_6) = 1 Following rules are strictly oriented: plus#(s(x),y) = 12 + 4*x + y > 5 + 4*x + y = c_4(plus#(x,y)) Following rules are (at-least) weakly oriented: plus#(x,s(y)) = 3 + 4*x + y >= 2 + 4*x + y = c_2(plus#(x,y)) times#(x,s(y)) = 5 + 14*x + 4*x*y + 6*x^2 + y >= x + 4*x*y = plus#(times(x,y),x) times#(x,s(y)) = 5 + 14*x + 4*x*y + 6*x^2 + y >= 2 + 2*x + 4*x*y + 6*x^2 + y = times#(x,y) plus(x,0()) = x >= x = x plus(x,s(y)) = 9 + x + 3*y >= 3 + x + 3*y = s(plus(x,y)) plus(0(),x) = 3*x >= x = x plus(s(x),y) = 3 + x + 3*y >= 3 + x + 3*y = s(plus(x,y)) times(x,0()) = 0 >= 0 = 0() times(x,s(y)) = 3*x + x*y >= 3*x + x*y = plus(times(x,y),x) ****** Step 1.b:4.a:1.b:1.b:1.a:2: Assumption. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: plus#(x,s(y)) -> c_2(plus#(x,y)) plus#(s(x),y) -> c_4(plus#(x,y)) times#(x,s(y)) -> plus#(times(x,y),x) times#(x,s(y)) -> times#(x,y) - Weak TRS: plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) plus(0(),x) -> x plus(s(x),y) -> s(plus(x,y)) times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) - Signature: {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2} - Obligation: innermost runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}} + Details: () ****** Step 1.b:4.a:1.b:1.b:1.b:1: RemoveWeakSuffixes. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: plus#(x,s(y)) -> c_2(plus#(x,y)) plus#(s(x),y) -> c_4(plus#(x,y)) times#(x,s(y)) -> plus#(times(x,y),x) times#(x,s(y)) -> times#(x,y) - Weak TRS: plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) plus(0(),x) -> x plus(s(x),y) -> s(plus(x,y)) times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) - Signature: {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2} - Obligation: innermost runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:plus#(x,s(y)) -> c_2(plus#(x,y)) -->_1 plus#(s(x),y) -> c_4(plus#(x,y)):2 -->_1 plus#(x,s(y)) -> c_2(plus#(x,y)):1 2:W:plus#(s(x),y) -> c_4(plus#(x,y)) -->_1 plus#(s(x),y) -> c_4(plus#(x,y)):2 -->_1 plus#(x,s(y)) -> c_2(plus#(x,y)):1 3:W:times#(x,s(y)) -> plus#(times(x,y),x) -->_1 plus#(s(x),y) -> c_4(plus#(x,y)):2 -->_1 plus#(x,s(y)) -> c_2(plus#(x,y)):1 4:W:times#(x,s(y)) -> times#(x,y) -->_1 times#(x,s(y)) -> times#(x,y):4 -->_1 times#(x,s(y)) -> plus#(times(x,y),x):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: times#(x,s(y)) -> times#(x,y) 3: times#(x,s(y)) -> plus#(times(x,y),x) 1: plus#(x,s(y)) -> c_2(plus#(x,y)) 2: plus#(s(x),y) -> c_4(plus#(x,y)) ****** Step 1.b:4.a:1.b:1.b:1.b:2: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) plus(0(),x) -> x plus(s(x),y) -> s(plus(x,y)) times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) - Signature: {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2} - Obligation: innermost runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 1.b:4.b:1: RemoveWeakSuffixes. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y)) - Weak DPs: plus#(x,s(y)) -> c_2(plus#(x,y)) plus#(s(x),y) -> c_4(plus#(x,y)) - Weak TRS: plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) plus(0(),x) -> x plus(s(x),y) -> s(plus(x,y)) times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) - Signature: {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2} - Obligation: innermost runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y)) -->_1 plus#(s(x),y) -> c_4(plus#(x,y)):3 -->_1 plus#(x,s(y)) -> c_2(plus#(x,y)):2 -->_2 times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y)):1 2:W:plus#(x,s(y)) -> c_2(plus#(x,y)) -->_1 plus#(s(x),y) -> c_4(plus#(x,y)):3 -->_1 plus#(x,s(y)) -> c_2(plus#(x,y)):2 3:W:plus#(s(x),y) -> c_4(plus#(x,y)) -->_1 plus#(s(x),y) -> c_4(plus#(x,y)):3 -->_1 plus#(x,s(y)) -> c_2(plus#(x,y)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: plus#(s(x),y) -> c_4(plus#(x,y)) 2: plus#(x,s(y)) -> c_2(plus#(x,y)) *** Step 1.b:4.b:2: SimplifyRHS. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y)) - Weak TRS: plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) plus(0(),x) -> x plus(s(x),y) -> s(plus(x,y)) times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) - Signature: {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/2} - Obligation: innermost runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y)) -->_2 times#(x,s(y)) -> c_6(plus#(times(x,y),x),times#(x,y)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: times#(x,s(y)) -> c_6(times#(x,y)) *** Step 1.b:4.b:3: UsableRules. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: times#(x,s(y)) -> c_6(times#(x,y)) - Weak TRS: plus(x,0()) -> x plus(x,s(y)) -> s(plus(x,y)) plus(0(),x) -> x plus(s(x),y) -> s(plus(x,y)) times(x,0()) -> 0() times(x,s(y)) -> plus(times(x,y),x) - Signature: {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1} - Obligation: innermost runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: times#(x,s(y)) -> c_6(times#(x,y)) *** Step 1.b:4.b:4: PredecessorEstimationCP. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: times#(x,s(y)) -> c_6(times#(x,y)) - Signature: {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1} - Obligation: innermost runtime complexity wrt. defined symbols {plus#,times#} 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: times#(x,s(y)) -> c_6(times#(x,y)) The strictly oriented rules are moved into the weak component. **** Step 1.b:4.b:4.a:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: times#(x,s(y)) -> c_6(times#(x,y)) - Signature: {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1} - Obligation: innermost runtime complexity wrt. defined symbols {plus#,times#} 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_6) = {1} Following symbols are considered usable: {plus#,times#} TcT has computed the following interpretation: p(0) = [1] p(plus) = [1] x2 + [1] p(s) = [1] x1 + [8] p(times) = [1] x2 + [1] p(plus#) = [1] p(times#) = [1] x2 + [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] p(c_4) = [1] p(c_5) = [4] p(c_6) = [1] x1 + [4] Following rules are strictly oriented: times#(x,s(y)) = [1] y + [8] > [1] y + [4] = c_6(times#(x,y)) Following rules are (at-least) weakly oriented: **** Step 1.b:4.b:4.a:2: Assumption. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: times#(x,s(y)) -> c_6(times#(x,y)) - Signature: {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1} - Obligation: innermost runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown, timeBCUB = Unknown, timeBCLB = Unknown}} + Details: () **** Step 1.b:4.b:4.b:1: RemoveWeakSuffixes. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: times#(x,s(y)) -> c_6(times#(x,y)) - Signature: {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1} - Obligation: innermost runtime complexity wrt. defined symbols {plus#,times#} and constructors {0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:times#(x,s(y)) -> c_6(times#(x,y)) -->_1 times#(x,s(y)) -> c_6(times#(x,y)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: times#(x,s(y)) -> c_6(times#(x,y)) **** Step 1.b:4.b:4.b:2: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Signature: {plus/2,times/2,plus#/2,times#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/1} - Obligation: innermost runtime complexity wrt. defined symbols {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),O(n^3))