/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())))))) min(X,0()) -> X min(s(X),s(Y)) -> min(X,Y) quot(0(),s(Y)) -> 0() quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y))) - Signature: {log/1,min/2,quot/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {log,min,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())))))) min(X,0()) -> X min(s(X),s(Y)) -> min(X,Y) quot(0(),s(Y)) -> 0() quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y))) - Signature: {log/1,min/2,quot/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {log,min,quot} 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: log(s(0())) -> 0() log(s(s(X))) -> s(log(s(quot(X,s(s(0())))))) min(X,0()) -> X min(s(X),s(Y)) -> min(X,Y) quot(0(),s(Y)) -> 0() quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y))) - Signature: {log/1,min/2,quot/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {log,min,quot} and constructors {0,s} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: min(x,y){x -> s(x),y -> s(y)} = min(s(x),s(y)) ->^+ min(x,y) = C[min(x,y) = min(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())))))) min(X,0()) -> X min(s(X),s(Y)) -> min(X,Y) quot(0(),s(Y)) -> 0() quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y))) - Signature: {log/1,min/2,quot/2} / {0/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {log,min,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())))) min#(X,0()) -> c_3() min#(s(X),s(Y)) -> c_4(min#(X,Y)) quot#(0(),s(Y)) -> c_5() quot#(s(X),s(Y)) -> c_6(quot#(min(X,Y),s(Y)),min#(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())))) min#(X,0()) -> c_3() min#(s(X),s(Y)) -> c_4(min#(X,Y)) quot#(0(),s(Y)) -> c_5() quot#(s(X),s(Y)) -> c_6(quot#(min(X,Y),s(Y)),min#(X,Y)) - Weak TRS: log(s(0())) -> 0() log(s(s(X))) -> s(log(s(quot(X,s(s(0())))))) min(X,0()) -> X min(s(X),s(Y)) -> min(X,Y) quot(0(),s(Y)) -> 0() quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y))) - Signature: {log/1,min/2,quot/2,log#/1,min#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/2} - Obligation: innermost runtime complexity wrt. defined symbols {log#,min#,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} by application of Pre({1,3,5}) = {2,4,6}. 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: min#(X,0()) -> c_3() 4: min#(s(X),s(Y)) -> c_4(min#(X,Y)) 5: quot#(0(),s(Y)) -> c_5() 6: quot#(s(X),s(Y)) -> c_6(quot#(min(X,Y),s(Y)),min#(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())))) min#(s(X),s(Y)) -> c_4(min#(X,Y)) quot#(s(X),s(Y)) -> c_6(quot#(min(X,Y),s(Y)),min#(X,Y)) - Weak DPs: log#(s(0())) -> c_1() min#(X,0()) -> c_3() quot#(0(),s(Y)) -> c_5() - Weak TRS: log(s(0())) -> 0() log(s(s(X))) -> s(log(s(quot(X,s(s(0())))))) min(X,0()) -> X min(s(X),s(Y)) -> min(X,Y) quot(0(),s(Y)) -> 0() quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y))) - Signature: {log/1,min/2,quot/2,log#/1,min#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/2} - Obligation: innermost runtime complexity wrt. defined symbols {log#,min#,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_6(quot#(min(X,Y),s(Y)),min#(X,Y)):3 -->_2 quot#(0(),s(Y)) -> c_5():6 -->_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:min#(s(X),s(Y)) -> c_4(min#(X,Y)) -->_1 min#(X,0()) -> c_3():5 -->_1 min#(s(X),s(Y)) -> c_4(min#(X,Y)):2 3:S:quot#(s(X),s(Y)) -> c_6(quot#(min(X,Y),s(Y)),min#(X,Y)) -->_1 quot#(0(),s(Y)) -> c_5():6 -->_2 min#(X,0()) -> c_3():5 -->_1 quot#(s(X),s(Y)) -> c_6(quot#(min(X,Y),s(Y)),min#(X,Y)):3 -->_2 min#(s(X),s(Y)) -> c_4(min#(X,Y)):2 4:W:log#(s(0())) -> c_1() 5:W:min#(X,0()) -> c_3() 6:W:quot#(0(),s(Y)) -> c_5() 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() 5: min#(X,0()) -> c_3() 6: quot#(0(),s(Y)) -> c_5() ** Step 1.b:4: 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())))) min#(s(X),s(Y)) -> c_4(min#(X,Y)) quot#(s(X),s(Y)) -> c_6(quot#(min(X,Y),s(Y)),min#(X,Y)) - Weak TRS: log(s(0())) -> 0() log(s(s(X))) -> s(log(s(quot(X,s(s(0())))))) min(X,0()) -> X min(s(X),s(Y)) -> min(X,Y) quot(0(),s(Y)) -> 0() quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y))) - Signature: {log/1,min/2,quot/2,log#/1,min#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/2} - Obligation: innermost runtime complexity wrt. defined symbols {log#,min#,quot#} and constructors {0,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: min(X,0()) -> X min(s(X),s(Y)) -> min(X,Y) quot(0(),s(Y)) -> 0() quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y))) log#(s(s(X))) -> c_2(log#(s(quot(X,s(s(0()))))),quot#(X,s(s(0())))) min#(s(X),s(Y)) -> c_4(min#(X,Y)) quot#(s(X),s(Y)) -> c_6(quot#(min(X,Y),s(Y)),min#(X,Y)) ** Step 1.b:5: 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())))) min#(s(X),s(Y)) -> c_4(min#(X,Y)) quot#(s(X),s(Y)) -> c_6(quot#(min(X,Y),s(Y)),min#(X,Y)) - Weak TRS: min(X,0()) -> X min(s(X),s(Y)) -> min(X,Y) quot(0(),s(Y)) -> 0() quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y))) - Signature: {log/1,min/2,quot/2,log#/1,min#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/2} - Obligation: innermost runtime complexity wrt. defined symbols {log#,min#,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 min#(s(X),s(Y)) -> c_4(min#(X,Y)) quot#(s(X),s(Y)) -> c_6(quot#(min(X,Y),s(Y)),min#(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:5.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: min(X,0()) -> X min(s(X),s(Y)) -> min(X,Y) quot(0(),s(Y)) -> 0() quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y))) - Signature: {log/1,min/2,quot/2,log#/1,min#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/2} - Obligation: innermost runtime complexity wrt. defined symbols {log#,min#,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:5.a:2: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: log#(s(s(X))) -> c_2(log#(s(quot(X,s(s(0())))))) - Weak TRS: min(X,0()) -> X min(s(X),s(Y)) -> min(X,Y) quot(0(),s(Y)) -> 0() quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y))) - Signature: {log/1,min/2,quot/2,log#/1,min#/2,quot#/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 {log#,min#,quot#} 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_2) = {1} Following symbols are considered usable: {min,quot,log#,min#,quot#} TcT has computed the following interpretation: p(0) = [12] p(log) = [1] x1 + [1] p(min) = [1] x1 + [0] p(quot) = [1] x1 + [1] p(s) = [1] x1 + [3] p(log#) = [4] x1 + [0] p(min#) = [1] x2 + [1] p(quot#) = [1] x1 + [2] p(c_1) = [0] p(c_2) = [1] x1 + [4] p(c_3) = [2] p(c_4) = [1] x1 + [0] p(c_5) = [4] p(c_6) = [1] x2 + [0] Following rules are strictly oriented: log#(s(s(X))) = [4] X + [24] > [4] X + [20] = c_2(log#(s(quot(X,s(s(0())))))) Following rules are (at-least) weakly oriented: min(X,0()) = [1] X + [0] >= [1] X + [0] = X min(s(X),s(Y)) = [1] X + [3] >= [1] X + [0] = min(X,Y) quot(0(),s(Y)) = [13] >= [12] = 0() quot(s(X),s(Y)) = [1] X + [4] >= [1] X + [4] = s(quot(min(X,Y),s(Y))) *** Step 1.b:5.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: min(X,0()) -> X min(s(X),s(Y)) -> min(X,Y) quot(0(),s(Y)) -> 0() quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y))) - Signature: {log/1,min/2,quot/2,log#/1,min#/2,quot#/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 {log#,min#,quot#} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 1.b:5.b:1: DecomposeDG. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: min#(s(X),s(Y)) -> c_4(min#(X,Y)) quot#(s(X),s(Y)) -> c_6(quot#(min(X,Y),s(Y)),min#(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: min(X,0()) -> X min(s(X),s(Y)) -> min(X,Y) quot(0(),s(Y)) -> 0() quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y))) - Signature: {log/1,min/2,quot/2,log#/1,min#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/2} - Obligation: innermost runtime complexity wrt. defined symbols {log#,min#,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_6(quot#(min(X,Y),s(Y)),min#(X,Y)) and a lower component min#(s(X),s(Y)) -> c_4(min#(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)) -> min#(X,Y) quot#(s(X),s(Y)) -> quot#(min(X,Y),s(Y)) **** Step 1.b:5.b:1.a:1: SimplifyRHS. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: quot#(s(X),s(Y)) -> c_6(quot#(min(X,Y),s(Y)),min#(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: min(X,0()) -> X min(s(X),s(Y)) -> min(X,Y) quot(0(),s(Y)) -> 0() quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y))) - Signature: {log/1,min/2,quot/2,log#/1,min#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/2} - Obligation: innermost runtime complexity wrt. defined symbols {log#,min#,quot#} and constructors {0,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:quot#(s(X),s(Y)) -> c_6(quot#(min(X,Y),s(Y)),min#(X,Y)) -->_1 quot#(s(X),s(Y)) -> c_6(quot#(min(X,Y),s(Y)),min#(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_6(quot#(min(X,Y),s(Y)),min#(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_6(quot#(min(X,Y),s(Y))) **** Step 1.b:5.b:1.a:2: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: quot#(s(X),s(Y)) -> c_6(quot#(min(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: min(X,0()) -> X min(s(X),s(Y)) -> min(X,Y) quot(0(),s(Y)) -> 0() quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y))) - Signature: {log/1,min/2,quot/2,log#/1,min#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1} - Obligation: innermost runtime complexity wrt. defined symbols {log#,min#,quot#} 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_6) = {1} Following symbols are considered usable: {min,quot,log#,min#,quot#} TcT has computed the following interpretation: p(0) = [12] p(log) = [1] x1 + [2] p(min) = [1] x1 + [0] p(quot) = [1] x1 + [0] p(s) = [1] x1 + [4] p(log#) = [2] x1 + [8] p(min#) = [1] x1 + [1] x2 + [0] p(quot#) = [2] x1 + [0] p(c_1) = [1] p(c_2) = [1] x1 + [1] p(c_3) = [1] p(c_4) = [2] x1 + [8] p(c_5) = [2] p(c_6) = [1] x1 + [4] Following rules are strictly oriented: quot#(s(X),s(Y)) = [2] X + [8] > [2] X + [4] = c_6(quot#(min(X,Y),s(Y))) Following rules are (at-least) weakly oriented: log#(s(s(X))) = [2] X + [24] >= [2] X + [16] = log#(s(quot(X,s(s(0()))))) log#(s(s(X))) = [2] X + [24] >= [2] X + [0] = quot#(X,s(s(0()))) min(X,0()) = [1] X + [0] >= [1] X + [0] = X min(s(X),s(Y)) = [1] X + [4] >= [1] X + [0] = min(X,Y) quot(0(),s(Y)) = [12] >= [12] = 0() quot(s(X),s(Y)) = [1] X + [4] >= [1] X + [4] = s(quot(min(X,Y),s(Y))) **** Step 1.b:5.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_6(quot#(min(X,Y),s(Y))) - Weak TRS: min(X,0()) -> X min(s(X),s(Y)) -> min(X,Y) quot(0(),s(Y)) -> 0() quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y))) - Signature: {log/1,min/2,quot/2,log#/1,min#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/1} - Obligation: innermost runtime complexity wrt. defined symbols {log#,min#,quot#} and constructors {0,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). **** Step 1.b:5.b:1.b:1: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: min#(s(X),s(Y)) -> c_4(min#(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)) -> min#(X,Y) quot#(s(X),s(Y)) -> quot#(min(X,Y),s(Y)) - Weak TRS: min(X,0()) -> X min(s(X),s(Y)) -> min(X,Y) quot(0(),s(Y)) -> 0() quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y))) - Signature: {log/1,min/2,quot/2,log#/1,min#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/2} - Obligation: innermost runtime complexity wrt. defined symbols {log#,min#,quot#} 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: {min,quot,log#,min#,quot#} TcT has computed the following interpretation: p(0) = [1] p(log) = [1] x1 + [0] p(min) = [1] x1 + [0] p(quot) = [1] x1 + [1] p(s) = [1] x1 + [1] p(log#) = [9] x1 + [8] p(min#) = [8] x2 + [1] p(quot#) = [8] x2 + [0] p(c_1) = [0] p(c_2) = [1] x1 + [2] x2 + [1] p(c_3) = [0] p(c_4) = [1] x1 + [5] p(c_5) = [1] p(c_6) = [2] x2 + [2] Following rules are strictly oriented: min#(s(X),s(Y)) = [8] Y + [9] > [8] Y + [6] = c_4(min#(X,Y)) Following rules are (at-least) weakly oriented: log#(s(s(X))) = [9] X + [26] >= [9] X + [26] = log#(s(quot(X,s(s(0()))))) log#(s(s(X))) = [9] X + [26] >= [24] = quot#(X,s(s(0()))) quot#(s(X),s(Y)) = [8] Y + [8] >= [8] Y + [1] = min#(X,Y) quot#(s(X),s(Y)) = [8] Y + [8] >= [8] Y + [8] = quot#(min(X,Y),s(Y)) min(X,0()) = [1] X + [0] >= [1] X + [0] = X min(s(X),s(Y)) = [1] X + [1] >= [1] X + [0] = min(X,Y) quot(0(),s(Y)) = [2] >= [1] = 0() quot(s(X),s(Y)) = [1] X + [2] >= [1] X + [2] = s(quot(min(X,Y),s(Y))) **** Step 1.b:5.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()))) min#(s(X),s(Y)) -> c_4(min#(X,Y)) quot#(s(X),s(Y)) -> min#(X,Y) quot#(s(X),s(Y)) -> quot#(min(X,Y),s(Y)) - Weak TRS: min(X,0()) -> X min(s(X),s(Y)) -> min(X,Y) quot(0(),s(Y)) -> 0() quot(s(X),s(Y)) -> s(quot(min(X,Y),s(Y))) - Signature: {log/1,min/2,quot/2,log#/1,min#/2,quot#/2} / {0/0,s/1,c_1/0,c_2/2,c_3/0,c_4/1,c_5/0,c_6/2} - Obligation: innermost runtime complexity wrt. defined symbols {log#,min#,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))