WORST_CASE(Omega(n^1),O(n^2)) * Step 1: Sum WORST_CASE(Omega(n^1),O(n^2)) + 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: 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^2)) + 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: UsableRules WORST_CASE(?,O(n^2)) + 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: 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(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)) ** Step 1.b:3: PredecessorEstimation WORST_CASE(?,O(n^2)) + 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: 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:4: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + 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: 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:5: SimplifyRHS WORST_CASE(?,O(n^2)) + 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: 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:6: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + 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: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))) 2: minus#(x,s(y)) -> c_4(minus#(x,y)) The strictly oriented rules are moved into the weak component. *** Step 1.b:6.a:1: NaturalPI WORST_CASE(?,O(n^2)) + 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: NaturalPI {shape = Mixed 2, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(mixed(2)): The following argument positions are considered usable: uargs(c_2) = {1,2}, uargs(c_4) = {1}, uargs(c_7) = {1,2} Following symbols are considered usable: {minus,pred,quot,log#,minus#,pred#,quot#} TcT has computed the following interpretation: p(0) = 0 p(log) = x1^2 p(minus) = x1 p(pred) = x1 p(quot) = x1 p(s) = 1 + x1 p(log#) = 1 + x1^2 p(minus#) = x2 p(pred#) = x1^2 p(quot#) = 1 + x1*x2 p(c_1) = 1 p(c_2) = x1 + x2 p(c_3) = 1 p(c_4) = x1 p(c_5) = 0 p(c_6) = 1 p(c_7) = 1 + x1 + x2 Following rules are strictly oriented: log#(s(s(x))) = 5 + 4*x + x^2 > 3 + 4*x + x^2 = c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))) minus#(x,s(y)) = 1 + y > y = c_4(minus#(x,y)) Following rules are (at-least) weakly oriented: quot#(s(x),s(y)) = 2 + x + x*y + y >= 2 + x + x*y + y = c_7(quot#(minus(x,y),s(y)),minus#(x,y)) minus(x,0()) = x >= x = x minus(x,s(y)) = x >= x = pred(minus(x,y)) pred(s(x)) = 1 + x >= x = x quot(0(),s(y)) = 0 >= 0 = 0() quot(s(x),s(y)) = 1 + x >= 1 + x = s(quot(minus(x,y),s(y))) *** Step 1.b:6.a:2: Assumption WORST_CASE(?,O(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))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))) minus#(x,s(y)) -> c_4(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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 1.b:6.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + 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))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))) minus#(x,s(y)) -> c_4(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: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)) -->_2 minus#(x,s(y)) -> c_4(minus#(x,y)):3 -->_1 quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)):1 2:W: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())))):2 -->_2 quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y)),minus#(x,y)):1 3:W:minus#(x,s(y)) -> c_4(minus#(x,y)) -->_1 minus#(x,s(y)) -> c_4(minus#(x,y)):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: minus#(x,s(y)) -> c_4(minus#(x,y)) *** Step 1.b:6.b:2: SimplifyRHS WORST_CASE(?,O(n^2)) + 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))) -> 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: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))) -> 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())))):2 -->_2 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:3: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y))) - Weak 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_7/1} - Obligation: innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} and constructors {0,s} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y))) The strictly oriented rules are moved into the weak component. **** Step 1.b:6.b:3.a:1: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y))) - Weak 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_7/1} - Obligation: innermost runtime complexity wrt. defined symbols {log#,minus#,pred#,quot#} and constructors {0,s} + Applied Processor: NaturalMI {miDimension = 3, miDegree = 2, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Just first alternative for predecessorEstimation on any intersect of rules of CDG leaf and strict-rules} + Details: We apply a matrix interpretation of kind constructor based matrix interpretation (containing no more than 2 non-zero interpretation-entries in the diagonal of the component-wise maxima): The following argument positions are considered usable: uargs(c_2) = {1,2}, uargs(c_7) = {1} Following symbols are considered usable: {minus,pred,quot,log#,minus#,pred#,quot#} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(log) = [0] [0] [0] p(minus) = [1 0 0] [0] [0 1 0] x1 + [0] [0 1 1] [0] p(pred) = [1 0 0] [0] [0 1 0] x1 + [0] [0 1 0] [0] p(quot) = [1 0 0] [1 0 0] [0] [0 1 0] x1 + [0 0 0] x2 + [0] [0 0 0] [0 0 0] [0] p(s) = [1 1 0] [0] [0 1 1] x1 + [1] [0 0 0] [0] p(log#) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 0] [0] p(minus#) = [0] [0] [0] p(pred#) = [0] [0] [0] p(quot#) = [0 1 0] [0] [0 0 1] x1 + [0] [0 0 0] [0] p(c_1) = [0] [0] [0] p(c_2) = [1 0 0] [1 1 0] [0] [0 0 0] x1 + [0 0 0] x2 + [0] [0 0 0] [0 0 0] [0] p(c_3) = [0] [0] [0] p(c_4) = [0] [0] [0] p(c_5) = [0] [0] [0] p(c_6) = [0] [0] [0] p(c_7) = [1 0 0] [0] [0 0 0] x1 + [0] [0 0 0] [0] Following rules are strictly oriented: quot#(s(x),s(y)) = [0 1 1] [1] [0 0 0] x + [0] [0 0 0] [0] > [0 1 0] [0] [0 0 0] x + [0] [0 0 0] [0] = c_7(quot#(minus(x,y),s(y))) Following rules are (at-least) weakly oriented: log#(s(s(x))) = [1 2 1] [1] [0 0 0] x + [1] [0 0 0] [0] >= [1 2 1] [1] [0 0 0] x + [0] [0 0 0] [0] = c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))) minus(x,0()) = [1 0 0] [0] [0 1 0] x + [0] [0 1 1] [0] >= [1 0 0] [0] [0 1 0] x + [0] [0 0 1] [0] = x minus(x,s(y)) = [1 0 0] [0] [0 1 0] x + [0] [0 1 1] [0] >= [1 0 0] [0] [0 1 0] x + [0] [0 1 0] [0] = pred(minus(x,y)) pred(s(x)) = [1 1 0] [0] [0 1 1] x + [1] [0 1 1] [1] >= [1 0 0] [0] [0 1 0] x + [0] [0 0 1] [0] = x quot(0(),s(y)) = [1 1 0] [0] [0 0 0] y + [0] [0 0 0] [0] >= [0] [0] [0] = 0() quot(s(x),s(y)) = [1 1 0] [1 1 0] [0] [0 1 1] x + [0 0 0] y + [1] [0 0 0] [0 0 0] [0] >= [1 1 0] [1 1 0] [0] [0 1 0] x + [0 0 0] y + [1] [0 0 0] [0 0 0] [0] = s(quot(minus(x,y),s(y))) **** Step 1.b:6.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),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: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 1.b:6.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),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: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W: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))):2 -->_1 log#(s(s(x))) -> c_2(log#(s(quot(x,s(s(0()))))),quot#(x,s(s(0())))):1 2:W:quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y))) -->_1 quot#(s(x),s(y)) -> c_7(quot#(minus(x,y),s(y))):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: 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))) **** Step 1.b:6.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - 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). WORST_CASE(Omega(n^1),O(n^2))