/export/starexec/sandbox2/solver/bin/starexec_run_tct_rci /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files -------------------------------------------------------------------------------- WORST_CASE(Omega(n^1),O(n^4)) * Step 1: Sum. WORST_CASE(Omega(n^1),O(n^4)) + Considered Problem: - Strict TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) log(s(0())) -> 0() log(s(s(x))) -> s(log(s(quot(x,s(s(0())))))) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {if_minus/3,le/2,log/1,minus/2,quot/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus,le,log,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: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) log(s(0())) -> 0() log(s(s(x))) -> s(log(s(quot(x,s(s(0())))))) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {if_minus/3,le/2,log/1,minus/2,quot/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus,le,log,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: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) log(s(0())) -> 0() log(s(s(x))) -> s(log(s(quot(x,s(s(0())))))) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {if_minus/3,le/2,log/1,minus/2,quot/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus,le,log,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. WORST_CASE(?,O(n^4)) + Considered Problem: - Strict TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) log(s(0())) -> 0() log(s(s(x))) -> s(log(s(quot(x,s(s(0())))))) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {if_minus/3,le/2,log/1,minus/2,quot/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus,le,log,minus,quot} and constructors {0,false,s ,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs if_minus#(false(),s(x),y) -> c_1(minus#(x,y)) if_minus#(true(),s(x),y) -> c_2() le#(0(),y) -> c_3() le#(s(x),0()) -> c_4() le#(s(x),s(y)) -> c_5(le#(x,y)) log#(s(0())) -> c_6() log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))) minus#(0(),y) -> c_8() minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)) quot#(0(),s(y)) -> c_10() quot#(s(x),s(y)) -> c_11(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^4)) + Considered Problem: - Strict DPs: if_minus#(false(),s(x),y) -> c_1(minus#(x,y)) if_minus#(true(),s(x),y) -> c_2() le#(0(),y) -> c_3() le#(s(x),0()) -> c_4() le#(s(x),s(y)) -> c_5(le#(x,y)) log#(s(0())) -> c_6() log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))) minus#(0(),y) -> c_8() minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)) quot#(0(),s(y)) -> c_10() quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)) - Weak TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) log(s(0())) -> 0() log(s(s(x))) -> s(log(s(quot(x,s(s(0())))))) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0 ,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/2} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,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 {2,3,4,6,8,10} by application of Pre({2,3,4,6,8,10}) = {1,5,7,9,11}. Here rules are labelled as follows: 1: if_minus#(false(),s(x),y) -> c_1(minus#(x,y)) 2: if_minus#(true(),s(x),y) -> c_2() 3: le#(0(),y) -> c_3() 4: le#(s(x),0()) -> c_4() 5: le#(s(x),s(y)) -> c_5(le#(x,y)) 6: log#(s(0())) -> c_6() 7: log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))) 8: minus#(0(),y) -> c_8() 9: minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)) 10: quot#(0(),s(y)) -> c_10() 11: quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)) ** Step 1.b:3: RemoveWeakSuffixes. WORST_CASE(?,O(n^4)) + Considered Problem: - Strict DPs: if_minus#(false(),s(x),y) -> c_1(minus#(x,y)) le#(s(x),s(y)) -> c_5(le#(x,y)) log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))) minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)) quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)) - Weak DPs: if_minus#(true(),s(x),y) -> c_2() le#(0(),y) -> c_3() le#(s(x),0()) -> c_4() log#(s(0())) -> c_6() minus#(0(),y) -> c_8() quot#(0(),s(y)) -> c_10() - Weak TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) log(s(0())) -> 0() log(s(s(x))) -> s(log(s(quot(x,s(s(0())))))) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0 ,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/2} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0 ,false,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:if_minus#(false(),s(x),y) -> c_1(minus#(x,y)) -->_1 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):4 -->_1 minus#(0(),y) -> c_8():10 2:S:le#(s(x),s(y)) -> c_5(le#(x,y)) -->_1 le#(s(x),0()) -> c_4():8 -->_1 le#(0(),y) -> c_3():7 -->_1 le#(s(x),s(y)) -> c_5(le#(x,y)):2 3:S:log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))) -->_2 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):5 -->_2 quot#(0(),s(y)) -> c_10():11 -->_1 log#(s(0())) -> c_6():9 -->_1 log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))):3 4:S:minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)) -->_2 le#(s(x),0()) -> c_4():8 -->_1 if_minus#(true(),s(x),y) -> c_2():6 -->_2 le#(s(x),s(y)) -> c_5(le#(x,y)):2 -->_1 if_minus#(false(),s(x),y) -> c_1(minus#(x,y)):1 5:S:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)) -->_1 quot#(0(),s(y)) -> c_10():11 -->_2 minus#(0(),y) -> c_8():10 -->_1 quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)):5 -->_2 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):4 6:W:if_minus#(true(),s(x),y) -> c_2() 7:W:le#(0(),y) -> c_3() 8:W:le#(s(x),0()) -> c_4() 9:W:log#(s(0())) -> c_6() 10:W:minus#(0(),y) -> c_8() 11:W:quot#(0(),s(y)) -> c_10() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 9: log#(s(0())) -> c_6() 11: quot#(0(),s(y)) -> c_10() 10: minus#(0(),y) -> c_8() 7: le#(0(),y) -> c_3() 6: if_minus#(true(),s(x),y) -> c_2() 8: le#(s(x),0()) -> c_4() ** Step 1.b:4: UsableRules. WORST_CASE(?,O(n^4)) + Considered Problem: - Strict DPs: if_minus#(false(),s(x),y) -> c_1(minus#(x,y)) le#(s(x),s(y)) -> c_5(le#(x,y)) log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))) minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)) quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)) - Weak TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) log(s(0())) -> 0() log(s(s(x))) -> s(log(s(quot(x,s(s(0())))))) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0 ,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/2} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0 ,false,s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) if_minus#(false(),s(x),y) -> c_1(minus#(x,y)) le#(s(x),s(y)) -> c_5(le#(x,y)) log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))) minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)) quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)) ** Step 1.b:5: DecomposeDG. WORST_CASE(?,O(n^4)) + Considered Problem: - Strict DPs: if_minus#(false(),s(x),y) -> c_1(minus#(x,y)) le#(s(x),s(y)) -> c_5(le#(x,y)) log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))) minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)) quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)) - Weak TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0 ,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/2} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,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 log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))) and a lower component if_minus#(false(),s(x),y) -> c_1(minus#(x,y)) le#(s(x),s(y)) -> c_5(le#(x,y)) minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)) quot#(s(x),s(y)) -> c_11(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:5.a:1: SimplifyRHS. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))) - Weak TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0 ,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/2} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0 ,false,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))) -->_1 log#(s(s(x))) -> c_7(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_7(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_7(log#(s(quot(x,s(s(0())))))) - Weak TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0 ,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/2} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,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_7) = {1} Following symbols are considered usable: {if_minus,minus,quot,if_minus#,le#,log#,minus#,quot#} TcT has computed the following interpretation: p(0) = [1] p(false) = [0] p(if_minus) = [1] x2 + [0] p(le) = [0] p(log) = [2] x1 + [1] p(minus) = [1] x1 + [0] p(quot) = [1] x1 + [1] p(s) = [1] x1 + [3] p(true) = [0] p(if_minus#) = [1] x3 + [2] p(le#) = [2] x1 + [1] x2 + [1] p(log#) = [4] x1 + [0] p(minus#) = [1] x1 + [1] x2 + [0] p(quot#) = [1] p(c_1) = [8] x1 + [0] p(c_2) = [2] p(c_3) = [0] p(c_4) = [8] p(c_5) = [2] x1 + [4] p(c_6) = [0] p(c_7) = [1] x1 + [4] p(c_8) = [8] p(c_9) = [1] x1 + [8] x2 + [0] p(c_10) = [1] p(c_11) = [1] x2 + [2] Following rules are strictly oriented: log#(s(s(x))) = [4] x + [24] > [4] x + [20] = c_7(log#(s(quot(x,s(s(0())))))) Following rules are (at-least) weakly oriented: if_minus(false(),s(x),y) = [1] x + [3] >= [1] x + [3] = s(minus(x,y)) if_minus(true(),s(x),y) = [1] x + [3] >= [1] = 0() minus(0(),y) = [1] >= [1] = 0() minus(s(x),y) = [1] x + [3] >= [1] x + [3] = if_minus(le(s(x),y),s(x),y) quot(0(),s(y)) = [2] >= [1] = 0() quot(s(x),s(y)) = [1] x + [4] >= [1] x + [4] = s(quot(minus(x,y),s(y))) *** Step 1.b:5.a:3: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: log#(s(s(x))) -> c_7(log#(s(quot(x,s(s(0())))))) - Weak TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0 ,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/1,c_8/0,c_9/2,c_10/0,c_11/2} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0 ,false,s,true} + 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^3)) + Considered Problem: - Strict DPs: if_minus#(false(),s(x),y) -> c_1(minus#(x,y)) le#(s(x),s(y)) -> c_5(le#(x,y)) minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)) quot#(s(x),s(y)) -> c_11(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: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0 ,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/2} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,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 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_11(quot#(minus(x,y),s(y)),minus#(x,y)) and a lower component if_minus#(false(),s(x),y) -> c_1(minus#(x,y)) le#(s(x),s(y)) -> c_5(le#(x,y)) minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(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:5.b:1.a:1: SimplifyRHS. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: quot#(s(x),s(y)) -> c_11(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: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0 ,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/2} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0 ,false,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:quot#(s(x),s(y)) -> c_11(quot#(minus(x,y),s(y)),minus#(x,y)) -->_1 quot#(s(x),s(y)) -> c_11(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_11(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_11(quot#(minus(x,y),s(y))) **** Step 1.b:5.b:1.a:2: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: quot#(s(x),s(y)) -> c_11(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: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0 ,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/1} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0 ,false,s,true} + 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(if_minus) = {1}, uargs(quot) = {1}, uargs(s) = {1}, uargs(log#) = {1}, uargs(quot#) = {1}, uargs(c_11) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(false) = [0] p(if_minus) = [1] x1 + [1] x2 + [0] p(le) = [0] p(log) = [1] x1 + [1] p(minus) = [1] x1 + [0] p(quot) = [1] x1 + [0] p(s) = [1] x1 + [1] p(true) = [0] p(if_minus#) = [0] p(le#) = [4] x2 + [1] p(log#) = [1] x1 + [7] p(minus#) = [1] x1 + [1] x2 + [1] p(quot#) = [1] x1 + [1] p(c_1) = [1] x1 + [1] p(c_2) = [2] p(c_3) = [1] p(c_4) = [0] p(c_5) = [2] x1 + [0] p(c_6) = [0] p(c_7) = [1] x2 + [1] p(c_8) = [0] p(c_9) = [1] x1 + [4] x2 + [0] p(c_10) = [0] p(c_11) = [1] x1 + [0] Following rules are strictly oriented: quot#(s(x),s(y)) = [1] x + [2] > [1] x + [1] = c_11(quot#(minus(x,y),s(y))) Following rules are (at-least) weakly oriented: log#(s(s(x))) = [1] x + [9] >= [1] x + [8] = log#(s(quot(x,s(s(0()))))) log#(s(s(x))) = [1] x + [9] >= [1] x + [1] = quot#(x,s(s(0()))) if_minus(false(),s(x),y) = [1] x + [1] >= [1] x + [1] = s(minus(x,y)) if_minus(true(),s(x),y) = [1] x + [1] >= [0] = 0() le(0(),y) = [0] >= [0] = true() le(s(x),0()) = [0] >= [0] = false() le(s(x),s(y)) = [0] >= [0] = le(x,y) minus(0(),y) = [0] >= [0] = 0() minus(s(x),y) = [1] x + [1] >= [1] x + [1] = if_minus(le(s(x),y),s(x),y) quot(0(),s(y)) = [0] >= [0] = 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: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_11(quot#(minus(x,y),s(y))) - Weak TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0 ,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/1} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0 ,false,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). **** Step 1.b:5.b:1.b:1: DecomposeDG. WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: if_minus#(false(),s(x),y) -> c_1(minus#(x,y)) le#(s(x),s(y)) -> c_5(le#(x,y)) minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(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: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0 ,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/2} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,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 if_minus#(false(),s(x),y) -> c_1(minus#(x,y)) log#(s(s(x))) -> log#(s(quot(x,s(s(0()))))) log#(s(s(x))) -> quot#(x,s(s(0()))) minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)) quot#(s(x),s(y)) -> minus#(x,y) quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)) and a lower component le#(s(x),s(y)) -> c_5(le#(x,y)) Further, following extension rules are added to the lower component. if_minus#(false(),s(x),y) -> minus#(x,y) log#(s(s(x))) -> log#(s(quot(x,s(s(0()))))) log#(s(s(x))) -> quot#(x,s(s(0()))) minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y) minus#(s(x),y) -> le#(s(x),y) quot#(s(x),s(y)) -> minus#(x,y) quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)) ***** Step 1.b:5.b:1.b:1.a:1: SimplifyRHS. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: if_minus#(false(),s(x),y) -> c_1(minus#(x,y)) minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(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: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0 ,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/2} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0 ,false,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:if_minus#(false(),s(x),y) -> c_1(minus#(x,y)) -->_1 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):2 2:S:minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)) -->_1 if_minus#(false(),s(x),y) -> c_1(minus#(x,y)):1 3:W:log#(s(s(x))) -> log#(s(quot(x,s(s(0()))))) -->_1 log#(s(s(x))) -> quot#(x,s(s(0()))):4 -->_1 log#(s(s(x))) -> log#(s(quot(x,s(s(0()))))):3 4:W:log#(s(s(x))) -> quot#(x,s(s(0()))) -->_1 quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)):6 -->_1 quot#(s(x),s(y)) -> minus#(x,y):5 5:W:quot#(s(x),s(y)) -> minus#(x,y) -->_1 minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y),le#(s(x),y)):2 6:W:quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)) -->_1 quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)):6 -->_1 quot#(s(x),s(y)) -> minus#(x,y):5 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y)) ***** Step 1.b:5.b:1.b:1.a:2: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: if_minus#(false(),s(x),y) -> c_1(minus#(x,y)) minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(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: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0 ,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/1,c_10/0,c_11/2} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0 ,false,s,true} + 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(if_minus) = {1}, uargs(quot) = {1}, uargs(s) = {1}, uargs(if_minus#) = {1}, uargs(log#) = {1}, uargs(quot#) = {1}, uargs(c_1) = {1}, uargs(c_9) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(false) = [0] p(if_minus) = [1] x1 + [1] x2 + [0] p(le) = [0] p(log) = [0] p(minus) = [1] x1 + [0] p(quot) = [1] x1 + [0] p(s) = [1] x1 + [0] p(true) = [0] p(if_minus#) = [1] x1 + [1] x2 + [0] p(le#) = [0] p(log#) = [1] x1 + [5] p(minus#) = [1] x1 + [5] p(quot#) = [1] x1 + [5] p(c_1) = [1] x1 + [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [1] x1 + [0] p(c_10) = [0] p(c_11) = [0] Following rules are strictly oriented: minus#(s(x),y) = [1] x + [5] > [1] x + [0] = c_9(if_minus#(le(s(x),y),s(x),y)) Following rules are (at-least) weakly oriented: if_minus#(false(),s(x),y) = [1] x + [0] >= [1] x + [5] = c_1(minus#(x,y)) log#(s(s(x))) = [1] x + [5] >= [1] x + [5] = 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 + [5] >= [1] x + [5] = minus#(x,y) quot#(s(x),s(y)) = [1] x + [5] >= [1] x + [5] = quot#(minus(x,y),s(y)) if_minus(false(),s(x),y) = [1] x + [0] >= [1] x + [0] = s(minus(x,y)) if_minus(true(),s(x),y) = [1] x + [0] >= [0] = 0() le(0(),y) = [0] >= [0] = true() le(s(x),0()) = [0] >= [0] = false() le(s(x),s(y)) = [0] >= [0] = le(x,y) minus(0(),y) = [0] >= [0] = 0() minus(s(x),y) = [1] x + [0] >= [1] x + [0] = if_minus(le(s(x),y),s(x),y) quot(0(),s(y)) = [0] >= [0] = 0() quot(s(x),s(y)) = [1] x + [0] >= [1] x + [0] = 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:5.b:1.b:1.a:3: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: if_minus#(false(),s(x),y) -> c_1(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()))) minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y)) quot#(s(x),s(y)) -> minus#(x,y) quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)) - Weak TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0 ,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/1,c_10/0,c_11/2} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0 ,false,s,true} + 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(if_minus) = {1}, uargs(quot) = {1}, uargs(s) = {1}, uargs(if_minus#) = {1}, uargs(log#) = {1}, uargs(quot#) = {1}, uargs(c_1) = {1}, uargs(c_9) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(false) = [0] p(if_minus) = [1] x1 + [1] x2 + [0] p(le) = [0] p(log) = [2] x1 + [1] p(minus) = [1] x1 + [0] p(quot) = [1] x1 + [0] p(s) = [1] x1 + [1] p(true) = [0] p(if_minus#) = [1] x1 + [1] x2 + [1] x3 + [0] p(le#) = [2] x1 + [2] x2 + [1] p(log#) = [1] x1 + [6] p(minus#) = [1] x1 + [1] x2 + [0] p(quot#) = [1] x1 + [4] x2 + [0] p(c_1) = [1] x1 + [0] p(c_2) = [1] p(c_3) = [0] p(c_4) = [1] p(c_5) = [1] x1 + [1] p(c_6) = [1] p(c_7) = [2] x2 + [1] p(c_8) = [4] p(c_9) = [1] x1 + [0] p(c_10) = [0] p(c_11) = [1] x1 + [1] x2 + [4] Following rules are strictly oriented: if_minus#(false(),s(x),y) = [1] x + [1] y + [1] > [1] x + [1] y + [0] = c_1(minus#(x,y)) Following rules are (at-least) weakly oriented: log#(s(s(x))) = [1] x + [8] >= [1] x + [7] = log#(s(quot(x,s(s(0()))))) log#(s(s(x))) = [1] x + [8] >= [1] x + [8] = quot#(x,s(s(0()))) minus#(s(x),y) = [1] x + [1] y + [1] >= [1] x + [1] y + [1] = c_9(if_minus#(le(s(x),y),s(x),y)) quot#(s(x),s(y)) = [1] x + [4] y + [5] >= [1] x + [1] y + [0] = minus#(x,y) quot#(s(x),s(y)) = [1] x + [4] y + [5] >= [1] x + [4] y + [4] = quot#(minus(x,y),s(y)) if_minus(false(),s(x),y) = [1] x + [1] >= [1] x + [1] = s(minus(x,y)) if_minus(true(),s(x),y) = [1] x + [1] >= [0] = 0() le(0(),y) = [0] >= [0] = true() le(s(x),0()) = [0] >= [0] = false() le(s(x),s(y)) = [0] >= [0] = le(x,y) minus(0(),y) = [0] >= [0] = 0() minus(s(x),y) = [1] x + [1] >= [1] x + [1] = if_minus(le(s(x),y),s(x),y) quot(0(),s(y)) = [0] >= [0] = 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:5.b:1.b:1.a:4: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: if_minus#(false(),s(x),y) -> c_1(minus#(x,y)) log#(s(s(x))) -> log#(s(quot(x,s(s(0()))))) log#(s(s(x))) -> quot#(x,s(s(0()))) minus#(s(x),y) -> c_9(if_minus#(le(s(x),y),s(x),y)) quot#(s(x),s(y)) -> minus#(x,y) quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)) - Weak TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0 ,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/1,c_10/0,c_11/2} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0 ,false,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ***** Step 1.b:5.b:1.b:1.b:1: WeightGap. WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: le#(s(x),s(y)) -> c_5(le#(x,y)) - Weak DPs: if_minus#(false(),s(x),y) -> minus#(x,y) log#(s(s(x))) -> log#(s(quot(x,s(s(0()))))) log#(s(s(x))) -> quot#(x,s(s(0()))) minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y) minus#(s(x),y) -> le#(s(x),y) quot#(s(x),s(y)) -> minus#(x,y) quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)) - Weak TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0 ,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/2} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,minus#,quot#} and constructors {0 ,false,s,true} + 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(if_minus) = {1}, uargs(quot) = {1}, uargs(s) = {1}, uargs(if_minus#) = {1}, uargs(log#) = {1}, uargs(quot#) = {1}, uargs(c_5) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(false) = [0] p(if_minus) = [1] x1 + [1] x2 + [0] p(le) = [0] p(log) = [1] x1 + [0] p(minus) = [1] x1 + [0] p(quot) = [1] x1 + [2] p(s) = [1] x1 + [2] p(true) = [0] p(if_minus#) = [1] x1 + [1] x2 + [6] p(le#) = [1] x1 + [0] p(log#) = [1] x1 + [6] p(minus#) = [1] x1 + [6] p(quot#) = [1] x1 + [1] x2 + [5] p(c_1) = [1] x1 + [0] p(c_2) = [1] p(c_3) = [0] p(c_4) = [0] p(c_5) = [1] x1 + [0] p(c_6) = [2] p(c_7) = [1] p(c_8) = [0] p(c_9) = [2] x1 + [2] p(c_10) = [4] p(c_11) = [1] x1 + [1] Following rules are strictly oriented: le#(s(x),s(y)) = [1] x + [2] > [1] x + [0] = c_5(le#(x,y)) Following rules are (at-least) weakly oriented: if_minus#(false(),s(x),y) = [1] x + [8] >= [1] x + [6] = minus#(x,y) log#(s(s(x))) = [1] x + [10] >= [1] x + [10] = log#(s(quot(x,s(s(0()))))) log#(s(s(x))) = [1] x + [10] >= [1] x + [9] = quot#(x,s(s(0()))) minus#(s(x),y) = [1] x + [8] >= [1] x + [8] = if_minus#(le(s(x),y),s(x),y) minus#(s(x),y) = [1] x + [8] >= [1] x + [2] = le#(s(x),y) quot#(s(x),s(y)) = [1] x + [1] y + [9] >= [1] x + [6] = minus#(x,y) quot#(s(x),s(y)) = [1] x + [1] y + [9] >= [1] x + [1] y + [7] = quot#(minus(x,y),s(y)) if_minus(false(),s(x),y) = [1] x + [2] >= [1] x + [2] = s(minus(x,y)) if_minus(true(),s(x),y) = [1] x + [2] >= [0] = 0() le(0(),y) = [0] >= [0] = true() le(s(x),0()) = [0] >= [0] = false() le(s(x),s(y)) = [0] >= [0] = le(x,y) minus(0(),y) = [0] >= [0] = 0() minus(s(x),y) = [1] x + [2] >= [1] x + [2] = if_minus(le(s(x),y),s(x),y) quot(0(),s(y)) = [2] >= [0] = 0() quot(s(x),s(y)) = [1] x + [4] >= [1] x + [4] = 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:5.b:1.b:1.b:2: EmptyProcessor. WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: if_minus#(false(),s(x),y) -> minus#(x,y) le#(s(x),s(y)) -> c_5(le#(x,y)) log#(s(s(x))) -> log#(s(quot(x,s(s(0()))))) log#(s(s(x))) -> quot#(x,s(s(0()))) minus#(s(x),y) -> if_minus#(le(s(x),y),s(x),y) minus#(s(x),y) -> le#(s(x),y) quot#(s(x),s(y)) -> minus#(x,y) quot#(s(x),s(y)) -> quot#(minus(x,y),s(y)) - Weak TRS: if_minus(false(),s(x),y) -> s(minus(x,y)) if_minus(true(),s(x),y) -> 0() le(0(),y) -> true() le(s(x),0()) -> false() le(s(x),s(y)) -> le(x,y) minus(0(),y) -> 0() minus(s(x),y) -> if_minus(le(s(x),y),s(x),y) quot(0(),s(y)) -> 0() quot(s(x),s(y)) -> s(quot(minus(x,y),s(y))) - Signature: {if_minus/3,le/2,log/1,minus/2,quot/2,if_minus#/3,le#/2,log#/1,minus#/2,quot#/2} / {0/0,false/0,s/1,true/0 ,c_1/1,c_2/0,c_3/0,c_4/0,c_5/1,c_6/0,c_7/2,c_8/0,c_9/2,c_10/0,c_11/2} - Obligation: innermost runtime complexity wrt. defined symbols {if_minus#,le#,log#,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),O(n^4))