WORST_CASE(?,O(n^3)) * Step 1: DependencyPairs WORST_CASE(?,O(n^3)) + Considered Problem: - Strict TRS: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) ifMinus(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) -> ifMinus(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: {ifMinus/3,le/2,minus/2,quot/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {ifMinus,le,minus,quot} and constructors {0,false,s,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y)) ifMinus#(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)) minus#(0(),Y) -> c_6() minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y)) quot#(0(),s(Y)) -> c_8() quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y)) Weak DPs and mark the set of starting terms. * Step 2: UsableRules WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y)) ifMinus#(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)) minus#(0(),Y) -> c_6() minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y)) quot#(0(),s(Y)) -> c_8() quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y)) - Weak TRS: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) ifMinus(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) -> ifMinus(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: {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,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} - Obligation: innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s ,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) ifMinus(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) -> ifMinus(le(s(X),Y),s(X),Y) ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y)) ifMinus#(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)) minus#(0(),Y) -> c_6() minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y)) quot#(0(),s(Y)) -> c_8() quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y)) * Step 3: PredecessorEstimation WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y)) ifMinus#(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)) minus#(0(),Y) -> c_6() minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y)) quot#(0(),s(Y)) -> c_8() quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y)) - Weak TRS: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) ifMinus(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) -> ifMinus(le(s(X),Y),s(X),Y) - Signature: {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,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} - Obligation: innermost runtime complexity wrt. defined symbols {ifMinus#,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 {2,3,4,6,8} by application of Pre({2,3,4,6,8}) = {1,5,7,9}. Here rules are labelled as follows: 1: ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y)) 2: ifMinus#(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: minus#(0(),Y) -> c_6() 7: minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y)) 8: quot#(0(),s(Y)) -> c_8() 9: quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y)) * Step 4: RemoveWeakSuffixes WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y)) le#(s(X),s(Y)) -> c_5(le#(X,Y)) minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y)) quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y)) - Weak DPs: ifMinus#(true(),s(X),Y) -> c_2() le#(0(),Y) -> c_3() le#(s(X),0()) -> c_4() minus#(0(),Y) -> c_6() quot#(0(),s(Y)) -> c_8() - Weak TRS: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) ifMinus(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) -> ifMinus(le(s(X),Y),s(X),Y) - Signature: {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,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} - Obligation: innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s ,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y)) -->_1 minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y)):3 -->_1 minus#(0(),Y) -> c_6():8 2:S:le#(s(X),s(Y)) -> c_5(le#(X,Y)) -->_1 le#(s(X),0()) -> c_4():7 -->_1 le#(0(),Y) -> c_3():6 -->_1 le#(s(X),s(Y)) -> c_5(le#(X,Y)):2 3:S:minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y)) -->_2 le#(s(X),0()) -> c_4():7 -->_1 ifMinus#(true(),s(X),Y) -> c_2():5 -->_2 le#(s(X),s(Y)) -> c_5(le#(X,Y)):2 -->_1 ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y)):1 4:S:quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y)) -->_1 quot#(0(),s(Y)) -> c_8():9 -->_2 minus#(0(),Y) -> c_6():8 -->_1 quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y)):4 -->_2 minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y)):3 5:W:ifMinus#(true(),s(X),Y) -> c_2() 6:W:le#(0(),Y) -> c_3() 7:W:le#(s(X),0()) -> c_4() 8:W:minus#(0(),Y) -> c_6() 9:W:quot#(0(),s(Y)) -> c_8() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 9: quot#(0(),s(Y)) -> c_8() 8: minus#(0(),Y) -> c_6() 6: le#(0(),Y) -> c_3() 5: ifMinus#(true(),s(X),Y) -> c_2() 7: le#(s(X),0()) -> c_4() * Step 5: Decompose WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y)) le#(s(X),s(Y)) -> c_5(le#(X,Y)) minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y)) quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y)) - Weak TRS: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) ifMinus(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) -> ifMinus(le(s(X),Y),s(X),Y) - Signature: {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,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} - Obligation: innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s ,true} + Applied Processor: Decompose {onSelection = all cycle independent sub-graph, withBound = RelativeAdd} + Details: We analyse the complexity of following sub-problems (R) and (S). Problem (S) is obtained from the input problem by shifting strict rules from (R) into the weak component. Problem (R) - Strict DPs: ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y)) le#(s(X),s(Y)) -> c_5(le#(X,Y)) minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y)) - Weak DPs: quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y)) - Weak TRS: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) ifMinus(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) -> ifMinus(le(s(X),Y),s(X),Y) - Signature: {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,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} - Obligation: innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s ,true} Problem (S) - Strict DPs: quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y)) - Weak DPs: ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y)) le#(s(X),s(Y)) -> c_5(le#(X,Y)) minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y)) - Weak TRS: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) ifMinus(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) -> ifMinus(le(s(X),Y),s(X),Y) - Signature: {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,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} - Obligation: innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s ,true} ** Step 5.a:1: DecomposeDG WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y)) le#(s(X),s(Y)) -> c_5(le#(X,Y)) minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y)) - Weak DPs: quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y)) - Weak TRS: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) ifMinus(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) -> ifMinus(le(s(X),Y),s(X),Y) - Signature: {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,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} - Obligation: innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s ,true} + Applied Processor: DecomposeDG {onSelection = all below first cut in WDG, onUpper = Just someStrategy, onLower = Nothing} + Details: We decompose the input problem according to the dependency graph into the upper component quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y)) and a lower component ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y)) le#(s(X),s(Y)) -> c_5(le#(X,Y)) minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y)) Further, following extension rules are added to the lower component. quot#(s(X),s(Y)) -> minus#(X,Y) quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y)) *** Step 5.a:1.a:1: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y)) - Weak TRS: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) ifMinus(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) -> ifMinus(le(s(X),Y),s(X),Y) - Signature: {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,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} - Obligation: innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s ,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y)) The strictly oriented rules are moved into the weak component. **** Step 5.a:1.a:1.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y)) - Weak TRS: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) ifMinus(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) -> ifMinus(le(s(X),Y),s(X),Y) - Signature: {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,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} - Obligation: innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s ,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, 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: The following argument positions are considered usable: uargs(c_9) = {1} Following symbols are considered usable: {ifMinus,minus,ifMinus#,le#,minus#,quot#} TcT has computed the following interpretation: p(0) = [0] p(false) = [0] p(ifMinus) = [1] x2 + [3] p(le) = [0] p(minus) = [1] x1 + [3] p(quot) = [1] x1 + [1] x2 + [1] p(s) = [1] x1 + [13] p(true) = [0] p(ifMinus#) = [1] x1 + [2] x2 + [1] p(le#) = [1] x1 + [2] x2 + [1] p(minus#) = [1] p(quot#) = [1] x1 + [1] p(c_1) = [1] p(c_2) = [0] p(c_3) = [4] p(c_4) = [2] p(c_5) = [2] x1 + [8] p(c_6) = [4] p(c_7) = [4] x1 + [8] x2 + [0] p(c_8) = [2] p(c_9) = [1] x1 + [7] x2 + [2] Following rules are strictly oriented: quot#(s(X),s(Y)) = [1] X + [14] > [1] X + [13] = c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y)) Following rules are (at-least) weakly oriented: ifMinus(false(),s(X),Y) = [1] X + [16] >= [1] X + [16] = s(minus(X,Y)) ifMinus(true(),s(X),Y) = [1] X + [16] >= [0] = 0() minus(0(),Y) = [3] >= [0] = 0() minus(s(X),Y) = [1] X + [16] >= [1] X + [16] = ifMinus(le(s(X),Y),s(X),Y) **** Step 5.a:1.a:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y)) - Weak TRS: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) ifMinus(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) -> ifMinus(le(s(X),Y),s(X),Y) - Signature: {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,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} - Obligation: innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s ,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 5.a:1.a:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y)) - Weak TRS: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) ifMinus(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) -> ifMinus(le(s(X),Y),s(X),Y) - Signature: {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,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} - Obligation: innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s ,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y)) -->_1 quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y)):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y)) **** Step 5.a:1.a:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) ifMinus(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) -> ifMinus(le(s(X),Y),s(X),Y) - Signature: {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,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} - Obligation: innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s ,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 5.a:1.b:1: PredecessorEstimationCP WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y)) le#(s(X),s(Y)) -> c_5(le#(X,Y)) minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y)) - Weak DPs: quot#(s(X),s(Y)) -> minus#(X,Y) quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y)) - Weak TRS: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) ifMinus(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) -> ifMinus(le(s(X),Y),s(X),Y) - Signature: {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,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} - Obligation: innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s ,true} + 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: 2: le#(s(X),s(Y)) -> c_5(le#(X,Y)) The strictly oriented rules are moved into the weak component. **** Step 5.a:1.b:1.a:1: NaturalMI WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y)) le#(s(X),s(Y)) -> c_5(le#(X,Y)) minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y)) - Weak DPs: quot#(s(X),s(Y)) -> minus#(X,Y) quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y)) - Weak TRS: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) ifMinus(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) -> ifMinus(le(s(X),Y),s(X),Y) - Signature: {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,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} - Obligation: innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s ,true} + 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_1) = {1}, uargs(c_5) = {1}, uargs(c_7) = {1,2} Following symbols are considered usable: {ifMinus,le,minus,ifMinus#,le#,minus#,quot#} TcT has computed the following interpretation: p(0) = [0] [0] [0] p(false) = [0] [0] [1] p(ifMinus) = [0 0 0] [0 1 0] [0] [0 0 1] x1 + [1 0 0] x2 + [0] [0 0 0] [0 0 1] [0] p(le) = [0 0 0] [0] [0 1 1] x1 + [0] [0 0 0] [1] p(minus) = [0 1 0] [0] [0 1 0] x1 + [0] [0 0 1] [0] p(quot) = [0] [0] [0] p(s) = [0 1 1] [0] [0 1 1] x1 + [1] [0 0 1] [1] p(true) = [0] [0] [0] p(ifMinus#) = [1 0 0] [0 0 0] [1] [1 0 1] x2 + [0 1 1] x3 + [0] [1 0 0] [0 1 1] [1] p(le#) = [0 0 1] [0 0 0] [1] [1 0 0] x1 + [0 0 1] x2 + [1] [0 1 1] [0 1 0] [0] p(minus#) = [0 1 1] [0 0 0] [1] [0 1 0] x1 + [0 1 1] x2 + [1] [0 0 0] [0 0 0] [0] p(quot#) = [0 1 0] [0 0 0] [0] [1 0 1] x1 + [1 0 1] x2 + [0] [0 1 1] [0 0 0] [0] p(c_1) = [1 0 0] [0] [0 1 0] x1 + [0] [0 0 0] [0] p(c_2) = [0] [0] [0] p(c_3) = [0] [0] [0] p(c_4) = [0] [0] [0] p(c_5) = [1 0 0] [0] [0 0 0] x1 + [1] [0 0 1] [0] p(c_6) = [0] [0] [0] p(c_7) = [1 0 0] [1 0 0] [0] [0 0 1] x1 + [0 0 0] x2 + [1] [0 0 0] [0 0 0] [0] p(c_8) = [0] [0] [0] p(c_9) = [0] [0] [0] Following rules are strictly oriented: le#(s(X),s(Y)) = [0 0 1] [0 0 0] [2] [0 1 1] X + [0 0 1] Y + [2] [0 1 2] [0 1 1] [3] > [0 0 1] [0 0 0] [1] [0 0 0] X + [0 0 0] Y + [1] [0 1 1] [0 1 0] [0] = c_5(le#(X,Y)) Following rules are (at-least) weakly oriented: ifMinus#(false(),s(X),Y) = [0 1 1] [0 0 0] [1] [0 1 2] X + [0 1 1] Y + [1] [0 1 1] [0 1 1] [1] >= [0 1 1] [0 0 0] [1] [0 1 0] X + [0 1 1] Y + [1] [0 0 0] [0 0 0] [0] = c_1(minus#(X,Y)) minus#(s(X),Y) = [0 1 2] [0 0 0] [3] [0 1 1] X + [0 1 1] Y + [2] [0 0 0] [0 0 0] [0] >= [0 1 2] [0 0 0] [3] [0 1 1] X + [0 1 1] Y + [2] [0 0 0] [0 0 0] [0] = c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y)) quot#(s(X),s(Y)) = [0 1 1] [0 0 0] [1] [0 1 2] X + [0 1 2] Y + [2] [0 1 2] [0 0 0] [2] >= [0 1 1] [0 0 0] [1] [0 1 0] X + [0 1 1] Y + [1] [0 0 0] [0 0 0] [0] = minus#(X,Y) quot#(s(X),s(Y)) = [0 1 1] [0 0 0] [1] [0 1 2] X + [0 1 2] Y + [2] [0 1 2] [0 0 0] [2] >= [0 1 0] [0 0 0] [0] [0 1 1] X + [0 1 2] Y + [1] [0 1 1] [0 0 0] [0] = quot#(minus(X,Y),s(Y)) ifMinus(false(),s(X),Y) = [0 1 1] [1] [0 1 1] X + [1] [0 0 1] [1] >= [0 1 1] [0] [0 1 1] X + [1] [0 0 1] [1] = s(minus(X,Y)) ifMinus(true(),s(X),Y) = [0 1 1] [1] [0 1 1] X + [0] [0 0 1] [1] >= [0] [0] [0] = 0() le(0(),Y) = [0] [0] [1] >= [0] [0] [0] = true() le(s(X),0()) = [0 0 0] [0] [0 1 2] X + [2] [0 0 0] [1] >= [0] [0] [1] = false() le(s(X),s(Y)) = [0 0 0] [0] [0 1 2] X + [2] [0 0 0] [1] >= [0 0 0] [0] [0 1 1] X + [0] [0 0 0] [1] = le(X,Y) minus(0(),Y) = [0] [0] [0] >= [0] [0] [0] = 0() minus(s(X),Y) = [0 1 1] [1] [0 1 1] X + [1] [0 0 1] [1] >= [0 1 1] [1] [0 1 1] X + [1] [0 0 1] [1] = ifMinus(le(s(X),Y),s(X),Y) **** Step 5.a:1.b:1.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Strict DPs: ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y)) minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y)) - Weak DPs: le#(s(X),s(Y)) -> c_5(le#(X,Y)) quot#(s(X),s(Y)) -> minus#(X,Y) quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y)) - Weak TRS: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) ifMinus(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) -> ifMinus(le(s(X),Y),s(X),Y) - Signature: {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,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} - Obligation: innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s ,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () **** Step 5.a:1.b:1.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y)) minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y)) - Weak DPs: le#(s(X),s(Y)) -> c_5(le#(X,Y)) quot#(s(X),s(Y)) -> minus#(X,Y) quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y)) - Weak TRS: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) ifMinus(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) -> ifMinus(le(s(X),Y),s(X),Y) - Signature: {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,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} - Obligation: innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s ,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y)) -->_1 minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y)):2 2:S:minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y)) -->_2 le#(s(X),s(Y)) -> c_5(le#(X,Y)):3 -->_1 ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y)):1 3:W:le#(s(X),s(Y)) -> c_5(le#(X,Y)) -->_1 le#(s(X),s(Y)) -> c_5(le#(X,Y)):3 4:W:quot#(s(X),s(Y)) -> minus#(X,Y) -->_1 minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y)):2 5:W:quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y)) -->_1 quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y)):5 -->_1 quot#(s(X),s(Y)) -> minus#(X,Y):4 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 3: le#(s(X),s(Y)) -> c_5(le#(X,Y)) **** Step 5.a:1.b:1.b:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y)) minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y)) - Weak DPs: quot#(s(X),s(Y)) -> minus#(X,Y) quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y)) - Weak TRS: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) ifMinus(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) -> ifMinus(le(s(X),Y),s(X),Y) - Signature: {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,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} - Obligation: innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s ,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y)) -->_1 minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y)):2 2:S:minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y)) -->_1 ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y)):1 4:W:quot#(s(X),s(Y)) -> minus#(X,Y) -->_1 minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y)):2 5:W:quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y)) -->_1 quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y)):5 -->_1 quot#(s(X),s(Y)) -> minus#(X,Y):4 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y)) **** Step 5.a:1.b:1.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y)) minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y)) - Weak DPs: quot#(s(X),s(Y)) -> minus#(X,Y) quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y)) - Weak TRS: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) ifMinus(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) -> ifMinus(le(s(X),Y),s(X),Y) - Signature: {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,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} - Obligation: innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s ,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y)) 2: minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y)) The strictly oriented rules are moved into the weak component. ***** Step 5.a:1.b:1.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y)) minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y)) - Weak DPs: quot#(s(X),s(Y)) -> minus#(X,Y) quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y)) - Weak TRS: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) ifMinus(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) -> ifMinus(le(s(X),Y),s(X),Y) - Signature: {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,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} - Obligation: innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s ,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, 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: The following argument positions are considered usable: uargs(c_1) = {1}, uargs(c_7) = {1} Following symbols are considered usable: {ifMinus,minus,ifMinus#,le#,minus#,quot#} TcT has computed the following interpretation: p(0) = [11] p(false) = [0] p(ifMinus) = [1] x2 + [5] p(le) = [0] p(minus) = [1] x1 + [5] p(quot) = [8] x1 + [0] p(s) = [1] x1 + [8] p(true) = [2] p(ifMinus#) = [2] x2 + [0] p(le#) = [0] p(minus#) = [2] x1 + [11] p(quot#) = [3] x1 + [2] p(c_1) = [1] x1 + [1] p(c_2) = [1] p(c_3) = [8] p(c_4) = [8] p(c_5) = [2] x1 + [0] p(c_6) = [1] p(c_7) = [1] x1 + [0] p(c_8) = [2] p(c_9) = [1] x1 + [8] Following rules are strictly oriented: ifMinus#(false(),s(X),Y) = [2] X + [16] > [2] X + [12] = c_1(minus#(X,Y)) minus#(s(X),Y) = [2] X + [27] > [2] X + [16] = c_7(ifMinus#(le(s(X),Y),s(X),Y)) Following rules are (at-least) weakly oriented: quot#(s(X),s(Y)) = [3] X + [26] >= [2] X + [11] = minus#(X,Y) quot#(s(X),s(Y)) = [3] X + [26] >= [3] X + [17] = quot#(minus(X,Y),s(Y)) ifMinus(false(),s(X),Y) = [1] X + [13] >= [1] X + [13] = s(minus(X,Y)) ifMinus(true(),s(X),Y) = [1] X + [13] >= [11] = 0() minus(0(),Y) = [16] >= [11] = 0() minus(s(X),Y) = [1] X + [13] >= [1] X + [13] = ifMinus(le(s(X),Y),s(X),Y) ***** Step 5.a:1.b:1.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y)) minus#(s(X),Y) -> c_7(ifMinus#(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: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) ifMinus(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) -> ifMinus(le(s(X),Y),s(X),Y) - Signature: {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,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} - Obligation: innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s ,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () ***** Step 5.a:1.b:1.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y)) minus#(s(X),Y) -> c_7(ifMinus#(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: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) ifMinus(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) -> ifMinus(le(s(X),Y),s(X),Y) - Signature: {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,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} - Obligation: innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s ,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y)) -->_1 minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y)):2 2:W:minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y)) -->_1 ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y)):1 3:W:quot#(s(X),s(Y)) -> minus#(X,Y) -->_1 minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y)):2 4:W:quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y)) -->_1 quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y)):4 -->_1 quot#(s(X),s(Y)) -> minus#(X,Y):3 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: quot#(s(X),s(Y)) -> quot#(minus(X,Y),s(Y)) 3: quot#(s(X),s(Y)) -> minus#(X,Y) 1: ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y)) 2: minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y)) ***** Step 5.a:1.b:1.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) ifMinus(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) -> ifMinus(le(s(X),Y),s(X),Y) - Signature: {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,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} - Obligation: innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s ,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 5.b:1: RemoveWeakSuffixes WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y)) - Weak DPs: ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y)) le#(s(X),s(Y)) -> c_5(le#(X,Y)) minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y)) - Weak TRS: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) ifMinus(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) -> ifMinus(le(s(X),Y),s(X),Y) - Signature: {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,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} - Obligation: innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s ,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y)) -->_2 minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y)):4 -->_1 quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y)):1 2:W:ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y)) -->_1 minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y)):4 3:W:le#(s(X),s(Y)) -> c_5(le#(X,Y)) -->_1 le#(s(X),s(Y)) -> c_5(le#(X,Y)):3 4:W:minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y)) -->_2 le#(s(X),s(Y)) -> c_5(le#(X,Y)):3 -->_1 ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y)):2 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 4: minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y),le#(s(X),Y)) 2: ifMinus#(false(),s(X),Y) -> c_1(minus#(X,Y)) 3: le#(s(X),s(Y)) -> c_5(le#(X,Y)) ** Step 5.b:2: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y)) - Weak TRS: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) ifMinus(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) -> ifMinus(le(s(X),Y),s(X),Y) - Signature: {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,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} - Obligation: innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s ,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y)),minus#(X,Y)) -->_1 quot#(s(X),s(Y)) -> c_9(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_9(quot#(minus(X,Y),s(Y))) ** Step 5.b:3: PredecessorEstimationCP WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y))) - Weak TRS: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) ifMinus(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) -> ifMinus(le(s(X),Y),s(X),Y) - Signature: {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,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} - Obligation: innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s ,true} + Applied Processor: PredecessorEstimationCP {onSelectionCP = any intersect of rules of CDG leaf and strict-rules, withComplexityPair = NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing}} + Details: We first use the processor NaturalMI {miDimension = 1, miDegree = 1, miKind = Algebraic, uargs = UArgs, urules = URules, selector = Nothing} to orient following rules strictly: 1: quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y))) The strictly oriented rules are moved into the weak component. *** Step 5.b:3.a:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y))) - Weak TRS: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) ifMinus(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) -> ifMinus(le(s(X),Y),s(X),Y) - Signature: {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,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} - Obligation: innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s ,true} + Applied Processor: NaturalMI {miDimension = 1, miDegree = 1, 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: The following argument positions are considered usable: uargs(c_9) = {1} Following symbols are considered usable: {ifMinus,minus,ifMinus#,le#,minus#,quot#} TcT has computed the following interpretation: p(0) = [0] p(false) = [0] p(ifMinus) = [1] x2 + [11] p(le) = [0] p(minus) = [1] x1 + [11] p(quot) = [1] x1 + [0] p(s) = [1] x1 + [13] p(true) = [0] p(ifMinus#) = [8] x2 + [4] x3 + [2] p(le#) = [2] x2 + [2] p(minus#) = [1] x2 + [1] p(quot#) = [2] x1 + [0] p(c_1) = [1] x1 + [0] p(c_2) = [1] p(c_3) = [1] p(c_4) = [1] p(c_5) = [2] x1 + [0] p(c_6) = [2] p(c_7) = [1] x1 + [2] x2 + [8] p(c_8) = [2] p(c_9) = [1] x1 + [0] Following rules are strictly oriented: quot#(s(X),s(Y)) = [2] X + [26] > [2] X + [22] = c_9(quot#(minus(X,Y),s(Y))) Following rules are (at-least) weakly oriented: ifMinus(false(),s(X),Y) = [1] X + [24] >= [1] X + [24] = s(minus(X,Y)) ifMinus(true(),s(X),Y) = [1] X + [24] >= [0] = 0() minus(0(),Y) = [11] >= [0] = 0() minus(s(X),Y) = [1] X + [24] >= [1] X + [24] = ifMinus(le(s(X),Y),s(X),Y) *** Step 5.b:3.a:2: Assumption WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y))) - Weak TRS: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) ifMinus(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) -> ifMinus(le(s(X),Y),s(X),Y) - Signature: {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,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} - Obligation: innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s ,true} + Applied Processor: Assumption {assumed = Certificate {spaceUB = Unknown, spaceLB = Unknown, timeUB = Poly (Just 0), timeLB = Unknown}} + Details: () *** Step 5.b:3.b:1: RemoveWeakSuffixes WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y))) - Weak TRS: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) ifMinus(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) -> ifMinus(le(s(X),Y),s(X),Y) - Signature: {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,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} - Obligation: innermost runtime complexity wrt. defined symbols {ifMinus#,le#,minus#,quot#} and constructors {0,false,s ,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:W:quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y))) -->_1 quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y))):1 The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 1: quot#(s(X),s(Y)) -> c_9(quot#(minus(X,Y),s(Y))) *** Step 5.b:3.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak TRS: ifMinus(false(),s(X),Y) -> s(minus(X,Y)) ifMinus(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) -> ifMinus(le(s(X),Y),s(X),Y) - Signature: {ifMinus/3,le/2,minus/2,quot/2,ifMinus#/3,le#/2,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} - Obligation: innermost runtime complexity wrt. defined symbols {ifMinus#,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(?,O(n^3))