WORST_CASE(?,O(n^2)) * Step 1: DependencyPairs WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: -(x,0()) -> x -(0(),s(y)) -> 0() -(s(x),s(y)) -> -(x,y) div(x,0()) -> 0() div(0(),y) -> 0() div(s(x),s(y)) -> if(lt(x,y),0(),s(div(-(x,y),s(y)))) if(false(),x,y) -> y if(true(),x,y) -> x lt(x,0()) -> false() lt(0(),s(y)) -> true() lt(s(x),s(y)) -> lt(x,y) - Signature: {-/2,div/2,if/3,lt/2} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {-,div,if,lt} and constructors {0,false,s,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs -#(x,0()) -> c_1() -#(0(),s(y)) -> c_2() -#(s(x),s(y)) -> c_3(-#(x,y)) div#(x,0()) -> c_4() div#(0(),y) -> c_5() div#(s(x),s(y)) -> c_6(if#(lt(x,y),0(),s(div(-(x,y),s(y)))),lt#(x,y),div#(-(x,y),s(y)),-#(x,y)) if#(false(),x,y) -> c_7() if#(true(),x,y) -> c_8() lt#(x,0()) -> c_9() lt#(0(),s(y)) -> c_10() lt#(s(x),s(y)) -> c_11(lt#(x,y)) Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: -#(x,0()) -> c_1() -#(0(),s(y)) -> c_2() -#(s(x),s(y)) -> c_3(-#(x,y)) div#(x,0()) -> c_4() div#(0(),y) -> c_5() div#(s(x),s(y)) -> c_6(if#(lt(x,y),0(),s(div(-(x,y),s(y)))),lt#(x,y),div#(-(x,y),s(y)),-#(x,y)) if#(false(),x,y) -> c_7() if#(true(),x,y) -> c_8() lt#(x,0()) -> c_9() lt#(0(),s(y)) -> c_10() lt#(s(x),s(y)) -> c_11(lt#(x,y)) - Weak TRS: -(x,0()) -> x -(0(),s(y)) -> 0() -(s(x),s(y)) -> -(x,y) div(x,0()) -> 0() div(0(),y) -> 0() div(s(x),s(y)) -> if(lt(x,y),0(),s(div(-(x,y),s(y)))) if(false(),x,y) -> y if(true(),x,y) -> x lt(x,0()) -> false() lt(0(),s(y)) -> true() lt(s(x),s(y)) -> lt(x,y) - Signature: {-/2,div/2,if/3,lt/2,-#/2,div#/2,if#/3,lt#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/4 ,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1} - Obligation: innermost runtime complexity wrt. defined symbols {-#,div#,if#,lt#} and constructors {0,false,s,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,4,5,7,8,9,10} by application of Pre({1,2,4,5,7,8,9,10}) = {3,6,11}. Here rules are labelled as follows: 1: -#(x,0()) -> c_1() 2: -#(0(),s(y)) -> c_2() 3: -#(s(x),s(y)) -> c_3(-#(x,y)) 4: div#(x,0()) -> c_4() 5: div#(0(),y) -> c_5() 6: div#(s(x),s(y)) -> c_6(if#(lt(x,y),0(),s(div(-(x,y),s(y)))),lt#(x,y),div#(-(x,y),s(y)),-#(x,y)) 7: if#(false(),x,y) -> c_7() 8: if#(true(),x,y) -> c_8() 9: lt#(x,0()) -> c_9() 10: lt#(0(),s(y)) -> c_10() 11: lt#(s(x),s(y)) -> c_11(lt#(x,y)) * Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: -#(s(x),s(y)) -> c_3(-#(x,y)) div#(s(x),s(y)) -> c_6(if#(lt(x,y),0(),s(div(-(x,y),s(y)))),lt#(x,y),div#(-(x,y),s(y)),-#(x,y)) lt#(s(x),s(y)) -> c_11(lt#(x,y)) - Weak DPs: -#(x,0()) -> c_1() -#(0(),s(y)) -> c_2() div#(x,0()) -> c_4() div#(0(),y) -> c_5() if#(false(),x,y) -> c_7() if#(true(),x,y) -> c_8() lt#(x,0()) -> c_9() lt#(0(),s(y)) -> c_10() - Weak TRS: -(x,0()) -> x -(0(),s(y)) -> 0() -(s(x),s(y)) -> -(x,y) div(x,0()) -> 0() div(0(),y) -> 0() div(s(x),s(y)) -> if(lt(x,y),0(),s(div(-(x,y),s(y)))) if(false(),x,y) -> y if(true(),x,y) -> x lt(x,0()) -> false() lt(0(),s(y)) -> true() lt(s(x),s(y)) -> lt(x,y) - Signature: {-/2,div/2,if/3,lt/2,-#/2,div#/2,if#/3,lt#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/4 ,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1} - Obligation: innermost runtime complexity wrt. defined symbols {-#,div#,if#,lt#} and constructors {0,false,s,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:-#(s(x),s(y)) -> c_3(-#(x,y)) -->_1 -#(0(),s(y)) -> c_2():5 -->_1 -#(x,0()) -> c_1():4 -->_1 -#(s(x),s(y)) -> c_3(-#(x,y)):1 2:S:div#(s(x),s(y)) -> c_6(if#(lt(x,y),0(),s(div(-(x,y),s(y)))),lt#(x,y),div#(-(x,y),s(y)),-#(x,y)) -->_2 lt#(s(x),s(y)) -> c_11(lt#(x,y)):3 -->_2 lt#(0(),s(y)) -> c_10():11 -->_2 lt#(x,0()) -> c_9():10 -->_1 if#(true(),x,y) -> c_8():9 -->_1 if#(false(),x,y) -> c_7():8 -->_3 div#(0(),y) -> c_5():7 -->_4 -#(0(),s(y)) -> c_2():5 -->_4 -#(x,0()) -> c_1():4 -->_3 div#(s(x),s(y)) -> c_6(if#(lt(x,y),0(),s(div(-(x,y),s(y)))),lt#(x,y),div#(-(x,y),s(y)),-#(x,y)):2 -->_4 -#(s(x),s(y)) -> c_3(-#(x,y)):1 3:S:lt#(s(x),s(y)) -> c_11(lt#(x,y)) -->_1 lt#(0(),s(y)) -> c_10():11 -->_1 lt#(x,0()) -> c_9():10 -->_1 lt#(s(x),s(y)) -> c_11(lt#(x,y)):3 4:W:-#(x,0()) -> c_1() 5:W:-#(0(),s(y)) -> c_2() 6:W:div#(x,0()) -> c_4() 7:W:div#(0(),y) -> c_5() 8:W:if#(false(),x,y) -> c_7() 9:W:if#(true(),x,y) -> c_8() 10:W:lt#(x,0()) -> c_9() 11:W:lt#(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. 6: div#(x,0()) -> c_4() 7: div#(0(),y) -> c_5() 8: if#(false(),x,y) -> c_7() 9: if#(true(),x,y) -> c_8() 10: lt#(x,0()) -> c_9() 11: lt#(0(),s(y)) -> c_10() 4: -#(x,0()) -> c_1() 5: -#(0(),s(y)) -> c_2() * Step 4: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: -#(s(x),s(y)) -> c_3(-#(x,y)) div#(s(x),s(y)) -> c_6(if#(lt(x,y),0(),s(div(-(x,y),s(y)))),lt#(x,y),div#(-(x,y),s(y)),-#(x,y)) lt#(s(x),s(y)) -> c_11(lt#(x,y)) - Weak TRS: -(x,0()) -> x -(0(),s(y)) -> 0() -(s(x),s(y)) -> -(x,y) div(x,0()) -> 0() div(0(),y) -> 0() div(s(x),s(y)) -> if(lt(x,y),0(),s(div(-(x,y),s(y)))) if(false(),x,y) -> y if(true(),x,y) -> x lt(x,0()) -> false() lt(0(),s(y)) -> true() lt(s(x),s(y)) -> lt(x,y) - Signature: {-/2,div/2,if/3,lt/2,-#/2,div#/2,if#/3,lt#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/4 ,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1} - Obligation: innermost runtime complexity wrt. defined symbols {-#,div#,if#,lt#} and constructors {0,false,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:-#(s(x),s(y)) -> c_3(-#(x,y)) -->_1 -#(s(x),s(y)) -> c_3(-#(x,y)):1 2:S:div#(s(x),s(y)) -> c_6(if#(lt(x,y),0(),s(div(-(x,y),s(y)))),lt#(x,y),div#(-(x,y),s(y)),-#(x,y)) -->_2 lt#(s(x),s(y)) -> c_11(lt#(x,y)):3 -->_3 div#(s(x),s(y)) -> c_6(if#(lt(x,y),0(),s(div(-(x,y),s(y)))),lt#(x,y),div#(-(x,y),s(y)),-#(x,y)):2 -->_4 -#(s(x),s(y)) -> c_3(-#(x,y)):1 3:S:lt#(s(x),s(y)) -> c_11(lt#(x,y)) -->_1 lt#(s(x),s(y)) -> c_11(lt#(x,y)):3 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: div#(s(x),s(y)) -> c_6(lt#(x,y),div#(-(x,y),s(y)),-#(x,y)) * Step 5: UsableRules WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: -#(s(x),s(y)) -> c_3(-#(x,y)) div#(s(x),s(y)) -> c_6(lt#(x,y),div#(-(x,y),s(y)),-#(x,y)) lt#(s(x),s(y)) -> c_11(lt#(x,y)) - Weak TRS: -(x,0()) -> x -(0(),s(y)) -> 0() -(s(x),s(y)) -> -(x,y) div(x,0()) -> 0() div(0(),y) -> 0() div(s(x),s(y)) -> if(lt(x,y),0(),s(div(-(x,y),s(y)))) if(false(),x,y) -> y if(true(),x,y) -> x lt(x,0()) -> false() lt(0(),s(y)) -> true() lt(s(x),s(y)) -> lt(x,y) - Signature: {-/2,div/2,if/3,lt/2,-#/2,div#/2,if#/3,lt#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/3 ,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1} - Obligation: innermost runtime complexity wrt. defined symbols {-#,div#,if#,lt#} and constructors {0,false,s,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: -(x,0()) -> x -(0(),s(y)) -> 0() -(s(x),s(y)) -> -(x,y) -#(s(x),s(y)) -> c_3(-#(x,y)) div#(s(x),s(y)) -> c_6(lt#(x,y),div#(-(x,y),s(y)),-#(x,y)) lt#(s(x),s(y)) -> c_11(lt#(x,y)) * Step 6: DecomposeDG WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: -#(s(x),s(y)) -> c_3(-#(x,y)) div#(s(x),s(y)) -> c_6(lt#(x,y),div#(-(x,y),s(y)),-#(x,y)) lt#(s(x),s(y)) -> c_11(lt#(x,y)) - Weak TRS: -(x,0()) -> x -(0(),s(y)) -> 0() -(s(x),s(y)) -> -(x,y) - Signature: {-/2,div/2,if/3,lt/2,-#/2,div#/2,if#/3,lt#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/3 ,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1} - Obligation: innermost runtime complexity wrt. defined symbols {-#,div#,if#,lt#} 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 div#(s(x),s(y)) -> c_6(lt#(x,y),div#(-(x,y),s(y)),-#(x,y)) and a lower component -#(s(x),s(y)) -> c_3(-#(x,y)) lt#(s(x),s(y)) -> c_11(lt#(x,y)) Further, following extension rules are added to the lower component. div#(s(x),s(y)) -> -#(x,y) div#(s(x),s(y)) -> div#(-(x,y),s(y)) div#(s(x),s(y)) -> lt#(x,y) ** Step 6.a:1: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: div#(s(x),s(y)) -> c_6(lt#(x,y),div#(-(x,y),s(y)),-#(x,y)) - Weak TRS: -(x,0()) -> x -(0(),s(y)) -> 0() -(s(x),s(y)) -> -(x,y) - Signature: {-/2,div/2,if/3,lt/2,-#/2,div#/2,if#/3,lt#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/3 ,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1} - Obligation: innermost runtime complexity wrt. defined symbols {-#,div#,if#,lt#} and constructors {0,false,s,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:div#(s(x),s(y)) -> c_6(lt#(x,y),div#(-(x,y),s(y)),-#(x,y)) -->_2 div#(s(x),s(y)) -> c_6(lt#(x,y),div#(-(x,y),s(y)),-#(x,y)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: div#(s(x),s(y)) -> c_6(div#(-(x,y),s(y))) ** Step 6.a:2: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: div#(s(x),s(y)) -> c_6(div#(-(x,y),s(y))) - Weak TRS: -(x,0()) -> x -(0(),s(y)) -> 0() -(s(x),s(y)) -> -(x,y) - Signature: {-/2,div/2,if/3,lt/2,-#/2,div#/2,if#/3,lt#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1 ,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1} - Obligation: innermost runtime complexity wrt. defined symbols {-#,div#,if#,lt#} 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_6) = {1} Following symbols are considered usable: {-,-#,div#,if#,lt#} TcT has computed the following interpretation: p(-) = [1] x1 + [0] p(0) = [2] p(div) = [2] x1 + [1] x2 + [4] p(false) = [8] p(if) = [4] p(lt) = [8] x2 + [1] p(s) = [1] x1 + [2] p(true) = [0] p(-#) = [8] x1 + [1] p(div#) = [8] x1 + [0] p(if#) = [1] x2 + [1] x3 + [0] p(lt#) = [1] x2 + [0] p(c_1) = [2] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] p(c_5) = [0] p(c_6) = [1] x1 + [14] p(c_7) = [0] p(c_8) = [1] p(c_9) = [2] p(c_10) = [2] p(c_11) = [2] x1 + [1] Following rules are strictly oriented: div#(s(x),s(y)) = [8] x + [16] > [8] x + [14] = c_6(div#(-(x,y),s(y))) Following rules are (at-least) weakly oriented: -(x,0()) = [1] x + [0] >= [1] x + [0] = x -(0(),s(y)) = [2] >= [2] = 0() -(s(x),s(y)) = [1] x + [2] >= [1] x + [0] = -(x,y) ** Step 6.a:3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: div#(s(x),s(y)) -> c_6(div#(-(x,y),s(y))) - Weak TRS: -(x,0()) -> x -(0(),s(y)) -> 0() -(s(x),s(y)) -> -(x,y) - Signature: {-/2,div/2,if/3,lt/2,-#/2,div#/2,if#/3,lt#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/1 ,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1} - Obligation: innermost runtime complexity wrt. defined symbols {-#,div#,if#,lt#} and constructors {0,false,s,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 6.b:1: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: -#(s(x),s(y)) -> c_3(-#(x,y)) lt#(s(x),s(y)) -> c_11(lt#(x,y)) - Weak DPs: div#(s(x),s(y)) -> -#(x,y) div#(s(x),s(y)) -> div#(-(x,y),s(y)) div#(s(x),s(y)) -> lt#(x,y) - Weak TRS: -(x,0()) -> x -(0(),s(y)) -> 0() -(s(x),s(y)) -> -(x,y) - Signature: {-/2,div/2,if/3,lt/2,-#/2,div#/2,if#/3,lt#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/3 ,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1} - Obligation: innermost runtime complexity wrt. defined symbols {-#,div#,if#,lt#} 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(div#) = {1}, uargs(c_3) = {1}, uargs(c_11) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(-) = [1] x1 + [2] p(0) = [5] p(div) = [0] p(false) = [0] p(if) = [0] p(lt) = [0] p(s) = [1] x1 + [3] p(true) = [0] p(-#) = [1] x2 + [6] p(div#) = [1] x1 + [1] x2 + [0] p(if#) = [0] p(lt#) = [1] x1 + [1] x2 + [3] p(c_1) = [0] p(c_2) = [0] p(c_3) = [1] x1 + [13] p(c_4) = [0] p(c_5) = [0] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [1] x1 + [5] Following rules are strictly oriented: lt#(s(x),s(y)) = [1] x + [1] y + [9] > [1] x + [1] y + [8] = c_11(lt#(x,y)) Following rules are (at-least) weakly oriented: -#(s(x),s(y)) = [1] y + [9] >= [1] y + [19] = c_3(-#(x,y)) div#(s(x),s(y)) = [1] x + [1] y + [6] >= [1] y + [6] = -#(x,y) div#(s(x),s(y)) = [1] x + [1] y + [6] >= [1] x + [1] y + [5] = div#(-(x,y),s(y)) div#(s(x),s(y)) = [1] x + [1] y + [6] >= [1] x + [1] y + [3] = lt#(x,y) -(x,0()) = [1] x + [2] >= [1] x + [0] = x -(0(),s(y)) = [7] >= [5] = 0() -(s(x),s(y)) = [1] x + [5] >= [1] x + [2] = -(x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 6.b:2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: -#(s(x),s(y)) -> c_3(-#(x,y)) - Weak DPs: div#(s(x),s(y)) -> -#(x,y) div#(s(x),s(y)) -> div#(-(x,y),s(y)) div#(s(x),s(y)) -> lt#(x,y) lt#(s(x),s(y)) -> c_11(lt#(x,y)) - Weak TRS: -(x,0()) -> x -(0(),s(y)) -> 0() -(s(x),s(y)) -> -(x,y) - Signature: {-/2,div/2,if/3,lt/2,-#/2,div#/2,if#/3,lt#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/3 ,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1} - Obligation: innermost runtime complexity wrt. defined symbols {-#,div#,if#,lt#} 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(div#) = {1}, uargs(c_3) = {1}, uargs(c_11) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(-) = [1] x1 + [3] p(0) = [0] p(div) = [2] x1 + [0] p(false) = [1] p(if) = [2] p(lt) = [1] x1 + [1] x2 + [1] p(s) = [1] x1 + [4] p(true) = [0] p(-#) = [2] x2 + [1] p(div#) = [1] x1 + [2] x2 + [1] p(if#) = [2] x2 + [1] x3 + [2] p(lt#) = [4] p(c_1) = [1] p(c_2) = [0] p(c_3) = [1] x1 + [0] p(c_4) = [1] p(c_5) = [2] p(c_6) = [1] x2 + [0] p(c_7) = [2] p(c_8) = [0] p(c_9) = [0] p(c_10) = [0] p(c_11) = [1] x1 + [0] Following rules are strictly oriented: -#(s(x),s(y)) = [2] y + [9] > [2] y + [1] = c_3(-#(x,y)) Following rules are (at-least) weakly oriented: div#(s(x),s(y)) = [1] x + [2] y + [13] >= [2] y + [1] = -#(x,y) div#(s(x),s(y)) = [1] x + [2] y + [13] >= [1] x + [2] y + [12] = div#(-(x,y),s(y)) div#(s(x),s(y)) = [1] x + [2] y + [13] >= [4] = lt#(x,y) lt#(s(x),s(y)) = [4] >= [4] = c_11(lt#(x,y)) -(x,0()) = [1] x + [3] >= [1] x + [0] = x -(0(),s(y)) = [3] >= [0] = 0() -(s(x),s(y)) = [1] x + [7] >= [1] x + [3] = -(x,y) Further, it can be verified that all rules not oriented are covered by the weightgap condition. ** Step 6.b:3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: -#(s(x),s(y)) -> c_3(-#(x,y)) div#(s(x),s(y)) -> -#(x,y) div#(s(x),s(y)) -> div#(-(x,y),s(y)) div#(s(x),s(y)) -> lt#(x,y) lt#(s(x),s(y)) -> c_11(lt#(x,y)) - Weak TRS: -(x,0()) -> x -(0(),s(y)) -> 0() -(s(x),s(y)) -> -(x,y) - Signature: {-/2,div/2,if/3,lt/2,-#/2,div#/2,if#/3,lt#/2} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0,c_3/1,c_4/0,c_5/0,c_6/3 ,c_7/0,c_8/0,c_9/0,c_10/0,c_11/1} - Obligation: innermost runtime complexity wrt. defined symbols {-#,div#,if#,lt#} 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^2))