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: 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) 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: 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 3: 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) 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: 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 4: UsableRules 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) 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)) 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)) * Step 5: 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)) 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 = Nothing, 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: 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.a:2: WeightGap 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: 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(ifMinus) = {1}, uargs(s) = {1}, uargs(quot#) = {1}, uargs(c_9) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(false) = [0] p(ifMinus) = [1] x1 + [1] x2 + [0] p(le) = [0] p(minus) = [1] x1 + [0] p(quot) = [1] x1 + [1] x2 + [2] p(s) = [1] x1 + [4] p(true) = [0] p(ifMinus#) = [1] x1 + [2] p(le#) = [1] x2 + [1] p(minus#) = [8] x1 + [1] x2 + [0] p(quot#) = [1] x1 + [11] p(c_1) = [1] x1 + [2] p(c_2) = [2] p(c_3) = [2] p(c_4) = [1] p(c_5) = [8] p(c_6) = [1] p(c_7) = [8] x1 + [8] x2 + [2] p(c_8) = [1] p(c_9) = [1] x1 + [1] Following rules are strictly oriented: quot#(s(X),s(Y)) = [1] X + [15] > [1] X + [12] = c_9(quot#(minus(X,Y),s(Y))) Following rules are (at-least) weakly oriented: ifMinus(false(),s(X),Y) = [1] X + [4] >= [1] X + [4] = s(minus(X,Y)) ifMinus(true(),s(X),Y) = [1] X + [4] >= [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 + [4] >= [1] X + [4] = ifMinus(le(s(X),Y),s(X),Y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 5.a:3: EmptyProcessor 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: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 5.b:1: DecomposeDG 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: 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 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)) 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. ifMinus#(false(),s(X),Y) -> minus#(X,Y) minus#(s(X),Y) -> ifMinus#(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 5.b:1.a:1: 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 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),le#(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 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.b:1.a:2: WeightGap 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: 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(ifMinus) = {1}, uargs(s) = {1}, uargs(ifMinus#) = {1}, uargs(quot#) = {1}, uargs(c_1) = {1}, uargs(c_7) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(false) = [0] p(ifMinus) = [1] x1 + [0] p(le) = [0] p(minus) = [0] p(quot) = [4] x1 + [2] x2 + [0] p(s) = [1] x1 + [0] p(true) = [0] p(ifMinus#) = [1] x1 + [1] p(le#) = [1] x1 + [1] p(minus#) = [0] p(quot#) = [1] x1 + [4] x2 + [0] p(c_1) = [1] x1 + [0] p(c_2) = [0] p(c_3) = [1] p(c_4) = [1] p(c_5) = [1] x1 + [0] p(c_6) = [2] p(c_7) = [1] x1 + [3] p(c_8) = [1] p(c_9) = [4] x1 + [2] Following rules are strictly oriented: ifMinus#(false(),s(X),Y) = [1] > [0] = c_1(minus#(X,Y)) Following rules are (at-least) weakly oriented: minus#(s(X),Y) = [0] >= [4] = c_7(ifMinus#(le(s(X),Y),s(X),Y)) quot#(s(X),s(Y)) = [1] X + [4] Y + [0] >= [0] = minus#(X,Y) quot#(s(X),s(Y)) = [1] X + [4] Y + [0] >= [4] Y + [0] = quot#(minus(X,Y),s(Y)) ifMinus(false(),s(X),Y) = [0] >= [0] = s(minus(X,Y)) ifMinus(true(),s(X),Y) = [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) = [0] >= [0] = ifMinus(le(s(X),Y),s(X),Y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 5.b:1.a:3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: minus#(s(X),Y) -> c_7(ifMinus#(le(s(X),Y),s(X),Y)) - Weak DPs: ifMinus#(false(),s(X),Y) -> c_1(minus#(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: 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(ifMinus) = {1}, uargs(s) = {1}, uargs(ifMinus#) = {1}, uargs(quot#) = {1}, uargs(c_1) = {1}, uargs(c_7) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [2] p(false) = [0] p(ifMinus) = [1] x1 + [1] x2 + [0] p(le) = [0] p(minus) = [1] x1 + [0] p(quot) = [1] x2 + [2] p(s) = [1] x1 + [2] p(true) = [0] p(ifMinus#) = [1] x1 + [1] x2 + [4] p(le#) = [1] x1 + [1] x2 + [1] p(minus#) = [1] x1 + [5] p(quot#) = [1] x1 + [3] p(c_1) = [1] x1 + [1] p(c_2) = [0] p(c_3) = [1] p(c_4) = [0] p(c_5) = [2] x1 + [4] p(c_6) = [0] p(c_7) = [1] x1 + [0] p(c_8) = [4] p(c_9) = [4] x2 + [4] Following rules are strictly oriented: minus#(s(X),Y) = [1] X + [7] > [1] X + [6] = c_7(ifMinus#(le(s(X),Y),s(X),Y)) Following rules are (at-least) weakly oriented: ifMinus#(false(),s(X),Y) = [1] X + [6] >= [1] X + [6] = c_1(minus#(X,Y)) quot#(s(X),s(Y)) = [1] X + [5] >= [1] X + [5] = minus#(X,Y) quot#(s(X),s(Y)) = [1] X + [5] >= [1] X + [3] = quot#(minus(X,Y),s(Y)) ifMinus(false(),s(X),Y) = [1] X + [2] >= [1] X + [2] = s(minus(X,Y)) ifMinus(true(),s(X),Y) = [1] X + [2] >= [2] = 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) = [2] >= [2] = 0() minus(s(X),Y) = [1] X + [2] >= [1] X + [2] = ifMinus(le(s(X),Y),s(X),Y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 5.b:1.a:4: EmptyProcessor 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: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 5.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: ifMinus#(false(),s(X),Y) -> minus#(X,Y) minus#(s(X),Y) -> ifMinus#(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: 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: 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(ifMinus) = {1}, uargs(s) = {1}, uargs(ifMinus#) = {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(ifMinus) = [1] x1 + [1] x2 + [0] p(le) = [0] p(minus) = [1] x1 + [0] p(quot) = [1] x1 + [1] x2 + [2] p(s) = [1] x1 + [2] p(true) = [0] p(ifMinus#) = [1] x1 + [1] x2 + [1] p(le#) = [1] x1 + [2] p(minus#) = [1] x1 + [2] p(quot#) = [1] x1 + [0] p(c_1) = [4] p(c_2) = [4] p(c_3) = [1] p(c_4) = [0] p(c_5) = [1] x1 + [0] p(c_6) = [0] p(c_7) = [4] x1 + [4] p(c_8) = [0] p(c_9) = [2] x2 + [1] Following rules are strictly oriented: le#(s(X),s(Y)) = [1] X + [4] > [1] X + [2] = c_5(le#(X,Y)) Following rules are (at-least) weakly oriented: ifMinus#(false(),s(X),Y) = [1] X + [3] >= [1] X + [2] = minus#(X,Y) minus#(s(X),Y) = [1] X + [4] >= [1] X + [3] = ifMinus#(le(s(X),Y),s(X),Y) minus#(s(X),Y) = [1] X + [4] >= [1] X + [4] = le#(s(X),Y) quot#(s(X),s(Y)) = [1] X + [2] >= [1] X + [2] = minus#(X,Y) quot#(s(X),s(Y)) = [1] X + [2] >= [1] X + [0] = quot#(minus(X,Y),s(Y)) ifMinus(false(),s(X),Y) = [1] X + [2] >= [1] X + [2] = s(minus(X,Y)) ifMinus(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] = ifMinus(le(s(X),Y),s(X),Y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 5.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: ifMinus#(false(),s(X),Y) -> minus#(X,Y) le#(s(X),s(Y)) -> c_5(le#(X,Y)) minus#(s(X),Y) -> ifMinus#(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: 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). WORST_CASE(?,O(n^3))