/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: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(s(x),s(y)),s(y))) - Signature: {le/2,minus/2,quot/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {le,minus,quot} and constructors {0,false,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: Sum. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(s(x),s(y)),s(y))) - Signature: {le/2,minus/2,quot/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {le,minus,quot} and constructors {0,false,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:2: DecreasingLoops. WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(s(x),s(y)),s(y))) - Signature: {le/2,minus/2,quot/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {le,minus,quot} and constructors {0,false,s,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: le(x,y){x -> s(x),y -> s(y)} = le(s(x),s(y)) ->^+ le(x,y) = C[le(x,y) = le(x,y){}] ** Step 1.b:1: DependencyPairs. MAYBE + Considered Problem: - Strict TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(s(x),s(y)),s(y))) - Signature: {le/2,minus/2,quot/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {le,minus,quot} and constructors {0,false,s,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs le#(0(),y) -> c_1() le#(s(x),0()) -> c_2() le#(s(x),s(y)) -> c_3(le#(x,y)) minus#(x,0()) -> c_4() minus#(s(x),s(y)) -> c_5(minus#(x,y)) quot#(0(),s(y)) -> c_6() quot#(s(x),s(y)) -> c_7(quot#(minus(s(x),s(y)),s(y)),minus#(s(x),s(y))) Weak DPs and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation. MAYBE + Considered Problem: - Strict DPs: le#(0(),y) -> c_1() le#(s(x),0()) -> c_2() le#(s(x),s(y)) -> c_3(le#(x,y)) minus#(x,0()) -> c_4() minus#(s(x),s(y)) -> c_5(minus#(x,y)) quot#(0(),s(y)) -> c_6() quot#(s(x),s(y)) -> c_7(quot#(minus(s(x),s(y)),s(y)),minus#(s(x),s(y))) - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(s(x),s(y)),s(y))) - Signature: {le/2,minus/2,quot/2,le#/2,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0 ,c_7/2} - Obligation: innermost runtime complexity wrt. defined symbols {le#,minus#,quot#} and constructors {0,false,s,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,4,6} by application of Pre({1,2,4,6}) = {3,5,7}. Here rules are labelled as follows: 1: le#(0(),y) -> c_1() 2: le#(s(x),0()) -> c_2() 3: le#(s(x),s(y)) -> c_3(le#(x,y)) 4: minus#(x,0()) -> c_4() 5: minus#(s(x),s(y)) -> c_5(minus#(x,y)) 6: quot#(0(),s(y)) -> c_6() 7: quot#(s(x),s(y)) -> c_7(quot#(minus(s(x),s(y)),s(y)),minus#(s(x),s(y))) ** Step 1.b:3: RemoveWeakSuffixes. MAYBE + Considered Problem: - Strict DPs: le#(s(x),s(y)) -> c_3(le#(x,y)) minus#(s(x),s(y)) -> c_5(minus#(x,y)) quot#(s(x),s(y)) -> c_7(quot#(minus(s(x),s(y)),s(y)),minus#(s(x),s(y))) - Weak DPs: le#(0(),y) -> c_1() le#(s(x),0()) -> c_2() minus#(x,0()) -> c_4() quot#(0(),s(y)) -> c_6() - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(s(x),s(y)),s(y))) - Signature: {le/2,minus/2,quot/2,le#/2,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0 ,c_7/2} - Obligation: innermost runtime complexity wrt. defined symbols {le#,minus#,quot#} and constructors {0,false,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:le#(s(x),s(y)) -> c_3(le#(x,y)) -->_1 le#(s(x),0()) -> c_2():5 -->_1 le#(0(),y) -> c_1():4 -->_1 le#(s(x),s(y)) -> c_3(le#(x,y)):1 2:S:minus#(s(x),s(y)) -> c_5(minus#(x,y)) -->_1 minus#(x,0()) -> c_4():6 -->_1 minus#(s(x),s(y)) -> c_5(minus#(x,y)):2 3:S:quot#(s(x),s(y)) -> c_7(quot#(minus(s(x),s(y)),s(y)),minus#(s(x),s(y))) -->_1 quot#(0(),s(y)) -> c_6():7 -->_1 quot#(s(x),s(y)) -> c_7(quot#(minus(s(x),s(y)),s(y)),minus#(s(x),s(y))):3 -->_2 minus#(s(x),s(y)) -> c_5(minus#(x,y)):2 4:W:le#(0(),y) -> c_1() 5:W:le#(s(x),0()) -> c_2() 6:W:minus#(x,0()) -> c_4() 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. 7: quot#(0(),s(y)) -> c_6() 6: minus#(x,0()) -> c_4() 4: le#(0(),y) -> c_1() 5: le#(s(x),0()) -> c_2() ** Step 1.b:4: UsableRules. MAYBE + Considered Problem: - Strict DPs: le#(s(x),s(y)) -> c_3(le#(x,y)) minus#(s(x),s(y)) -> c_5(minus#(x,y)) quot#(s(x),s(y)) -> c_7(quot#(minus(s(x),s(y)),s(y)),minus#(s(x),s(y))) - Weak TRS: le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(s(x),s(y)),s(y))) - Signature: {le/2,minus/2,quot/2,le#/2,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0 ,c_7/2} - Obligation: innermost runtime complexity wrt. defined symbols {le#,minus#,quot#} and constructors {0,false,s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) le#(s(x),s(y)) -> c_3(le#(x,y)) minus#(s(x),s(y)) -> c_5(minus#(x,y)) quot#(s(x),s(y)) -> c_7(quot#(minus(s(x),s(y)),s(y)),minus#(s(x),s(y))) ** Step 1.b:5: DecomposeDG. MAYBE + Considered Problem: - Strict DPs: le#(s(x),s(y)) -> c_3(le#(x,y)) minus#(s(x),s(y)) -> c_5(minus#(x,y)) quot#(s(x),s(y)) -> c_7(quot#(minus(s(x),s(y)),s(y)),minus#(s(x),s(y))) - Weak TRS: minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) - Signature: {le/2,minus/2,quot/2,le#/2,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0 ,c_7/2} - Obligation: innermost runtime complexity wrt. defined symbols {le#,minus#,quot#} and constructors {0,false,s,true} + 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 le#(s(x),s(y)) -> c_3(le#(x,y)) quot#(s(x),s(y)) -> c_7(quot#(minus(s(x),s(y)),s(y)),minus#(s(x),s(y))) and a lower component minus#(s(x),s(y)) -> c_5(minus#(x,y)) Further, following extension rules are added to the lower component. le#(s(x),s(y)) -> le#(x,y) quot#(s(x),s(y)) -> minus#(s(x),s(y)) quot#(s(x),s(y)) -> quot#(minus(s(x),s(y)),s(y)) *** Step 1.b:5.a:1: SimplifyRHS. MAYBE + Considered Problem: - Strict DPs: le#(s(x),s(y)) -> c_3(le#(x,y)) quot#(s(x),s(y)) -> c_7(quot#(minus(s(x),s(y)),s(y)),minus#(s(x),s(y))) - Weak TRS: minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) - Signature: {le/2,minus/2,quot/2,le#/2,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0 ,c_7/2} - Obligation: innermost runtime complexity wrt. defined symbols {le#,minus#,quot#} and constructors {0,false,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:le#(s(x),s(y)) -> c_3(le#(x,y)) -->_1 le#(s(x),s(y)) -> c_3(le#(x,y)):1 2:S:quot#(s(x),s(y)) -> c_7(quot#(minus(s(x),s(y)),s(y)),minus#(s(x),s(y))) -->_1 quot#(s(x),s(y)) -> c_7(quot#(minus(s(x),s(y)),s(y)),minus#(s(x),s(y))):2 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(s(x),s(y)),s(y))) *** Step 1.b:5.a:2: NaturalMI. MAYBE + Considered Problem: - Strict DPs: le#(s(x),s(y)) -> c_3(le#(x,y)) quot#(s(x),s(y)) -> c_7(quot#(minus(s(x),s(y)),s(y))) - Weak TRS: minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) - Signature: {le/2,minus/2,quot/2,le#/2,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0 ,c_7/1} - Obligation: innermost runtime complexity wrt. defined symbols {le#,minus#,quot#} and constructors {0,false,s,true} + 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_7) = {1} Following symbols are considered usable: {le#,minus#,quot#} TcT has computed the following interpretation: p(0) = [1] p(false) = [1] p(le) = [8] x1 + [2] x2 + [1] p(minus) = [1] x1 + [8] p(quot) = [8] x1 + [1] p(s) = [1] x1 + [8] p(true) = [0] p(le#) = [2] x1 + [0] p(minus#) = [0] p(quot#) = [0] p(c_1) = [1] p(c_2) = [2] p(c_3) = [1] x1 + [8] p(c_4) = [1] p(c_5) = [1] x1 + [1] p(c_6) = [8] p(c_7) = [1] x1 + [0] Following rules are strictly oriented: le#(s(x),s(y)) = [2] x + [16] > [2] x + [8] = c_3(le#(x,y)) Following rules are (at-least) weakly oriented: quot#(s(x),s(y)) = [0] >= [0] = c_7(quot#(minus(s(x),s(y)),s(y))) *** Step 1.b:5.a:3: Failure MAYBE + Considered Problem: - Strict DPs: quot#(s(x),s(y)) -> c_7(quot#(minus(s(x),s(y)),s(y))) - Weak DPs: le#(s(x),s(y)) -> c_3(le#(x,y)) - Weak TRS: minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) - Signature: {le/2,minus/2,quot/2,le#/2,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0 ,c_7/1} - Obligation: innermost runtime complexity wrt. defined symbols {le#,minus#,quot#} and constructors {0,false,s,true} + Applied Processor: EmptyProcessor + Details: The problem is still open. *** Step 1.b:5.b:1: RemoveWeakSuffixes. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: minus#(s(x),s(y)) -> c_5(minus#(x,y)) - Weak DPs: le#(s(x),s(y)) -> le#(x,y) quot#(s(x),s(y)) -> minus#(s(x),s(y)) quot#(s(x),s(y)) -> quot#(minus(s(x),s(y)),s(y)) - Weak TRS: minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) - Signature: {le/2,minus/2,quot/2,le#/2,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0 ,c_7/2} - Obligation: innermost runtime complexity wrt. defined symbols {le#,minus#,quot#} and constructors {0,false,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:minus#(s(x),s(y)) -> c_5(minus#(x,y)) -->_1 minus#(s(x),s(y)) -> c_5(minus#(x,y)):1 2:W:le#(s(x),s(y)) -> le#(x,y) -->_1 le#(s(x),s(y)) -> le#(x,y):2 3:W:quot#(s(x),s(y)) -> minus#(s(x),s(y)) -->_1 minus#(s(x),s(y)) -> c_5(minus#(x,y)):1 4:W:quot#(s(x),s(y)) -> quot#(minus(s(x),s(y)),s(y)) -->_1 quot#(s(x),s(y)) -> quot#(minus(s(x),s(y)),s(y)):4 -->_1 quot#(s(x),s(y)) -> minus#(s(x),s(y)):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 2: le#(s(x),s(y)) -> le#(x,y) *** Step 1.b:5.b:2: NaturalMI. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: minus#(s(x),s(y)) -> c_5(minus#(x,y)) - Weak DPs: quot#(s(x),s(y)) -> minus#(s(x),s(y)) quot#(s(x),s(y)) -> quot#(minus(s(x),s(y)),s(y)) - Weak TRS: minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) - Signature: {le/2,minus/2,quot/2,le#/2,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0 ,c_7/2} - Obligation: innermost runtime complexity wrt. defined symbols {le#,minus#,quot#} and constructors {0,false,s,true} + 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: {le#,minus#,quot#} TcT has computed the following interpretation: p(0) = [0] p(false) = [0] p(le) = [0] p(minus) = [1] x1 + [0] p(quot) = [8] x2 + [0] p(s) = [1] x1 + [10] p(true) = [1] p(le#) = [1] x1 + [0] p(minus#) = [1] x2 + [0] p(quot#) = [1] x2 + [2] p(c_1) = [4] p(c_2) = [2] p(c_3) = [1] x1 + [1] p(c_4) = [1] p(c_5) = [1] x1 + [2] p(c_6) = [0] p(c_7) = [1] x2 + [1] Following rules are strictly oriented: minus#(s(x),s(y)) = [1] y + [10] > [1] y + [2] = c_5(minus#(x,y)) Following rules are (at-least) weakly oriented: quot#(s(x),s(y)) = [1] y + [12] >= [1] y + [10] = minus#(s(x),s(y)) quot#(s(x),s(y)) = [1] y + [12] >= [1] y + [12] = quot#(minus(s(x),s(y)),s(y)) *** Step 1.b:5.b:3: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: minus#(s(x),s(y)) -> c_5(minus#(x,y)) quot#(s(x),s(y)) -> minus#(s(x),s(y)) quot#(s(x),s(y)) -> quot#(minus(s(x),s(y)),s(y)) - Weak TRS: minus(x,0()) -> x minus(s(x),s(y)) -> minus(x,y) - Signature: {le/2,minus/2,quot/2,le#/2,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/1,c_6/0 ,c_7/2} - Obligation: innermost runtime complexity wrt. defined symbols {le#,minus#,quot#} and constructors {0,false,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),?)