/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: log(s(0())) -> 0() log(s(s(x))) -> s(log(s(quot(x,s(s(0())))))) minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {log/1,minus/2,pred/1,quot/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {log,minus,pred,quot} 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: log(s(0())) -> 0() log(s(s(x))) -> s(log(s(quot(x,s(s(0())))))) minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {log/1,minus/2,pred/1,quot/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {log,minus,pred,quot} and constructors {0,s} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () *** Step 1.a:1.a:1: Ara. MAYBE + Considered Problem: - Strict TRS: log(s(0())) -> 0() log(s(s(x))) -> s(log(s(quot(x,s(s(0())))))) minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {log/1,minus/2,pred/1,quot/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {log,minus,pred,quot} 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 "log") :: ["A"(0)] -(1)-> "A"(0) F (TrsFun "main") :: ["A"(1) x "A"(0)] -(1)-> "A"(0) F (TrsFun "minus") :: ["A"(1) x "A"(0)] -(1)-> "A"(1) F (TrsFun "pred") :: ["A"(1)] -(0)-> "A"(1) F (TrsFun "quot") :: ["A"(1) x "A"(0)] -(1)-> "A"(0) F (TrsFun "s") :: ["A"(0)] -(0)-> "A"(0) F (TrsFun "s") :: ["A"(1)] -(1)-> "A"(1) Cost-free Signatures used: -------------------------- Base Constructor Signatures used: --------------------------------- Following Still Strict Rules were Typed as: ------------------------------------------- 1. Strict: log(s(0())) -> 0() log(s(s(x))) -> s(log(s(quot(x,s(s(0())))))) minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) main(x1,x2) -> quot(x1,x2) 2. Weak: *** Step 1.a:1.b:1: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: log(s(0())) -> 0() log(s(s(x))) -> s(log(s(quot(x,s(s(0())))))) minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {log/1,minus/2,pred/1,quot/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {log,minus,pred,quot} and constructors {0,s} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: minus(x,y){y -> s(y)} = minus(x,s(y)) ->^+ pred(minus(x,y)) = C[minus(x,y) = minus(x,y){}] ** Step 1.b:1: DependencyPairs. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: log(s(0())) -> 0() log(s(s(x))) -> s(log(s(quot(x,s(s(0())))))) minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {log/1,minus/2,pred/1,quot/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {log,minus,pred,quot} and constructors {0,s} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs log#(s(0())) -> c_1() log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))) minus#(x,0()) -> c_3() minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y)) pred#(s(x)) -> c_5() quot#(0(),s(y)) -> c_6() quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(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: log#(s(0())) -> c_1() log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))) minus#(x,0()) -> c_3() minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y)) pred#(s(x)) -> c_5() quot#(0(),s(y)) -> c_6() quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)) - Weak TRS: log(s(0())) -> 0() log(s(s(x))) -> s(log(s(quot(x,s(s(0())))))) minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/0,c_6/0 ,c_7/2} - Obligation: innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} and constructors {0,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,3,5,6} by application of Pre({1,3,5,6}) = {2,4,7}. Here rules are labelled as follows: 1: log#(s(0())) -> c_1() 2: log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))) 3: minus#(x,0()) -> c_3() 4: minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y)) 5: pred#(s(x)) -> c_5() 6: quot#(0(),s(y)) -> c_6() 7: quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)) ** Step 1.b:3: RemoveWeakSuffixes. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))) minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y)) quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)) - Weak DPs: log#(s(0())) -> c_1() minus#(x,0()) -> c_3() pred#(s(x)) -> c_5() quot#(0(),s(y)) -> c_6() - Weak TRS: log(s(0())) -> 0() log(s(s(x))) -> s(log(s(quot(x,s(s(0())))))) minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/0,c_6/0 ,c_7/2} - Obligation: innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} and constructors {0,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))) -->_2 quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)):3 -->_2 quot#(0(),s(y)) -> c_6():7 -->_1 log#(s(0())) -> c_1():4 -->_1 log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))):1 2:S:minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y)) -->_1 pred#(s(x)) -> c_5():6 -->_2 minus#(x,0()) -> c_3():5 -->_2 minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y)):2 3:S:quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)) -->_1 quot#(0(),s(y)) -> c_6():7 -->_2 minus#(x,0()) -> c_3():5 -->_1 quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)):3 -->_2 minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y)):2 4:W:log#(s(0())) -> c_1() 5:W:minus#(x,0()) -> c_3() 6:W:pred#(s(x)) -> c_5() 7:W:quot#(0(),s(y)) -> c_6() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: log#(s(0())) -> c_1() 6: pred#(s(x)) -> c_5() 5: minus#(x,0()) -> c_3() 7: quot#(0(),s(y)) -> c_6() ** Step 1.b:4: SimplifyRHS. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))) minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y)) quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)) - Weak TRS: log(s(0())) -> 0() log(s(s(x))) -> s(log(s(quot(x,s(s(0())))))) minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/2,c_5/0,c_6/0 ,c_7/2} - Obligation: innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} and constructors {0,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))) -->_2 quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)):3 -->_1 log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))):1 2:S:minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y)) -->_2 minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y)):2 3:S:quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)) -->_1 quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)):3 -->_2 minus#(x,s(y)) -> c_4(pred#(minus(x,y)),minus#(x,y)):2 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: minus#(x,s(y)) -> c_4(minus#(x,y)) ** Step 1.b:5: UsableRules. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))) minus#(x,s(y)) -> c_4(minus#(x,y)) quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)) - Weak TRS: log(s(0())) -> 0() log(s(s(x))) -> s(log(s(quot(x,s(s(0())))))) minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/0 ,c_7/2} - Obligation: innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} and constructors {0,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))) minus#(x,s(y)) -> c_4(minus#(x,y)) quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)) ** Step 1.b:6: DecomposeDG. WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))) minus#(x,s(y)) -> c_4(minus#(x,y)) quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)) - Weak TRS: minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/0 ,c_7/2} - Obligation: innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} 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 log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))) and a lower component minus#(x,s(y)) -> c_4(minus#(x,y)) quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)) Further, following extension rules are added to the lower component. log#(s(s(x))) -> log#(s(quot(x,s(s(0()))))) log#(s(s(x))) -> quot#(x,s(s(0()))) *** Step 1.b:6.a:1: SimplifyRHS. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))) - Weak TRS: minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/0 ,c_7/2} - Obligation: innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} and constructors {0,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))) -->_1 log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0())))))) *** Step 1.b:6.a:2: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0())))))) - Weak TRS: minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0 ,c_7/2} - Obligation: innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} and constructors {0,s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(pred) = {1}, uargs(quot) = {1}, uargs(s) = {1}, uargs(log#) = {1}, uargs(c_2) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [2] p(log) = [1] x1 + [0] p(minus) = [1] x1 + [0] p(pred) = [1] x1 + [0] p(quot) = [1] x1 + [0] p(s) = [1] x1 + [1] p(log#) = [1] x1 + [6] p(minus#) = [1] x2 + [1] p(pred#) = [1] x1 + [1] p(quot#) = [1] x1 + [4] x2 + [1] p(c_1) = [1] p(c_2) = [1] x1 + [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [1] p(c_6) = [2] p(c_7) = [2] x2 + [1] Following rules are strictly oriented: log#(s(s(x))) = [1] x + [8] > [1] x + [7] = c_2(log#(s(quot(x,s(s(0())))))) Following rules are (at-least) weakly oriented: minus(x,0()) = [1] x + [0] >= [1] x + [0] = x minus(x,s(y)) = [1] x + [0] >= [1] x + [0] = pred(minus(x,y)) pred(s(x)) = [1] x + [1] >= [1] x + [0] = x quot(0(),s(y)) = [2] >= [2] = 0() quot(s(x),s(y)) = [1] x + [1] >= [1] x + [1] = s(quot(minus(x,y),s(y))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 1.b:6.a:3: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0())))))) - Weak TRS: minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/1,c_3/0,c_4/1,c_5/0,c_6/0 ,c_7/2} - Obligation: innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} 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. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: minus#(x,s(y)) -> c_4(minus#(x,y)) quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)) - Weak DPs: log#(s(s(x))) -> log#(s(quot(x,s(s(0()))))) log#(s(s(x))) -> quot#(x,s(s(0()))) - Weak TRS: minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/0 ,c_7/2} - Obligation: innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} 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 log#(s(s(x))) -> log#(s(quot(x,s(s(0()))))) log#(s(s(x))) -> quot#(x,s(s(0()))) quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)) and a lower component minus#(x,s(y)) -> c_4(minus#(x,y)) Further, following extension rules are added to the lower component. log#(s(s(x))) -> log#(s(quot(x,s(s(0()))))) log#(s(s(x))) -> quot#(x,s(s(0()))) quot#(s(x),s(y)) -> minus#(x,y) quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)) **** Step 1.b:6.b:1.a:1: SimplifyRHS. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)) - Weak DPs: log#(s(s(x))) -> log#(s(quot(x,s(s(0()))))) log#(s(s(x))) -> quot#(x,s(s(0()))) - Weak TRS: minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/0 ,c_7/2} - Obligation: innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} and constructors {0,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)) -->_1 quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)):1 2:W:log#(s(s(x))) -> log#(s(quot(x,s(s(0()))))) -->_1 log#(s(s(x))) -> quot#(x,s(s(0()))):3 -->_1 log#(s(s(x))) -> log#(s(quot(x,s(s(0()))))):2 3:W:log#(s(s(x))) -> quot#(x,s(s(0()))) -->_1 quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y))) **** Step 1.b:6.b:1.a:2: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y))) - Weak DPs: log#(s(s(x))) -> log#(s(quot(x,s(s(0()))))) log#(s(s(x))) -> quot#(x,s(s(0()))) - Weak TRS: minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/0 ,c_7/1} - Obligation: innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} and constructors {0,s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(pred) = {1}, uargs(quot) = {1}, uargs(s) = {1}, uargs(log#) = {1}, uargs(quot#) = {1}, uargs(c_7) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [7] p(log) = [1] x1 + [0] p(minus) = [1] x1 + [0] p(pred) = [1] x1 + [0] p(quot) = [1] x1 + [1] p(s) = [1] x1 + [4] p(log#) = [1] x1 + [4] p(minus#) = [1] x1 + [0] p(pred#) = [1] x1 + [0] p(quot#) = [1] x1 + [2] p(c_1) = [4] p(c_2) = [2] x1 + [2] x2 + [0] p(c_3) = [0] p(c_4) = [1] p(c_5) = [0] p(c_6) = [2] p(c_7) = [1] x1 + [2] Following rules are strictly oriented: quot#(s(x),s(y)) = [1] x + [6] > [1] x + [4] = c_7(quot#(minus(x,y),s(y))) Following rules are (at-least) weakly oriented: log#(s(s(x))) = [1] x + [12] >= [1] x + [9] = log#(s(quot(x,s(s(0()))))) log#(s(s(x))) = [1] x + [12] >= [1] x + [2] = quot#(x,s(s(0()))) minus(x,0()) = [1] x + [0] >= [1] x + [0] = x minus(x,s(y)) = [1] x + [0] >= [1] x + [0] = pred(minus(x,y)) pred(s(x)) = [1] x + [4] >= [1] x + [0] = x quot(0(),s(y)) = [8] >= [7] = 0() quot(s(x),s(y)) = [1] x + [5] >= [1] x + [5] = s(quot(minus(x,y),s(y))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. **** Step 1.b:6.b:1.a:3: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: log#(s(s(x))) -> log#(s(quot(x,s(s(0()))))) log#(s(s(x))) -> quot#(x,s(s(0()))) quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y))) - Weak TRS: minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/0 ,c_7/1} - Obligation: innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} 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.b:1: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: minus#(x,s(y)) -> c_4(minus#(x,y)) - Weak DPs: log#(s(s(x))) -> log#(s(quot(x,s(s(0()))))) log#(s(s(x))) -> quot#(x,s(s(0()))) quot#(s(x),s(y)) -> minus#(x,y) quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)) - Weak TRS: minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/0 ,c_7/2} - Obligation: innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} and constructors {0,s} + Applied Processor: WeightGap {wgDimension = 1, wgDegree = 1, wgKind = Algebraic, wgUArgs = UArgs, wgOn = WgOnAny} + Details: The weightgap principle applies using the following constant growth matrix-interpretation: We apply a matrix interpretation of kind constructor based matrix interpretation: The following argument positions are considered usable: uargs(pred) = {1}, uargs(quot) = {1}, uargs(s) = {1}, uargs(log#) = {1}, uargs(quot#) = {1}, uargs(c_4) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [2] p(log) = [4] x1 + [1] p(minus) = [1] x1 + [0] p(pred) = [1] x1 + [0] p(quot) = [1] x1 + [0] p(s) = [1] x1 + [1] p(log#) = [1] x1 + [3] p(minus#) = [1] x2 + [2] p(pred#) = [4] x1 + [2] p(quot#) = [1] x1 + [1] x2 + [1] p(c_1) = [4] p(c_2) = [2] x2 + [0] p(c_3) = [0] p(c_4) = [1] x1 + [0] p(c_5) = [1] p(c_6) = [4] p(c_7) = [1] x1 + [1] x2 + [0] Following rules are strictly oriented: minus#(x,s(y)) = [1] y + [3] > [1] y + [2] = c_4(minus#(x,y)) Following rules are (at-least) weakly oriented: log#(s(s(x))) = [1] x + [5] >= [1] x + [4] = log#(s(quot(x,s(s(0()))))) log#(s(s(x))) = [1] x + [5] >= [1] x + [5] = quot#(x,s(s(0()))) quot#(s(x),s(y)) = [1] x + [1] y + [3] >= [1] y + [2] = minus#(x,y) quot#(s(x),s(y)) = [1] x + [1] y + [3] >= [1] x + [1] y + [2] = quot#(minus(x,y),s(y)) minus(x,0()) = [1] x + [0] >= [1] x + [0] = x minus(x,s(y)) = [1] x + [0] >= [1] x + [0] = pred(minus(x,y)) pred(s(x)) = [1] x + [1] >= [1] x + [0] = x quot(0(),s(y)) = [2] >= [2] = 0() quot(s(x),s(y)) = [1] x + [1] >= [1] x + [1] = s(quot(minus(x,y),s(y))) Further, it can be verified that all rules not oriented are covered by the weightgap condition. **** Step 1.b:6.b:1.b:2: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: log#(s(s(x))) -> log#(s(quot(x,s(s(0()))))) log#(s(s(x))) -> quot#(x,s(s(0()))) minus#(x,s(y)) -> c_4(minus#(x,y)) quot#(s(x),s(y)) -> minus#(x,y) quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)) - Weak TRS: minus(x,0()) -> x minus(x,s(y)) -> pred(minus(x,y)) pred(s(x)) -> x quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {log/1,minus/2,pred/1,quot/2,log#/1,minus#/2,pred#/1,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/0 ,c_7/2} - Obligation: innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} 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))