WORST_CASE(?,O(n^4)) * Step 1: DependencyPairs WORST_CASE(?,O(n^4)) + Considered Problem: - Strict TRS: add(0(),y) -> y add(s(x),y) -> s(add(x,y)) dyade(cons(x,xs),ls) -> cons(mult(x,ls),dyade(xs,ls)) dyade(nil(),ls) -> nil() mult(n,cons(x,xs)) -> cons(times(n,x),mult(n,xs)) mult(n,nil()) -> nil() times(0(),y) -> 0() times(s(x),y) -> add(y,times(x,y)) - Signature: {add/2,dyade/2,mult/2,times/2} / {0/0,cons/2,nil/0,s/1} - Obligation: innermost runtime complexity wrt. defined symbols {add,dyade,mult,times} and constructors {0,cons,nil,s} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs add#(0(),y) -> c_1() add#(s(x),y) -> c_2(add#(x,y)) dyade#(cons(x,xs),ls) -> c_3(mult#(x,ls),dyade#(xs,ls)) dyade#(nil(),ls) -> c_4() mult#(n,cons(x,xs)) -> c_5(times#(n,x),mult#(n,xs)) mult#(n,nil()) -> c_6() times#(0(),y) -> c_7() times#(s(x),y) -> c_8(add#(y,times(x,y)),times#(x,y)) Weak DPs and mark the set of starting terms. * Step 2: PredecessorEstimation WORST_CASE(?,O(n^4)) + Considered Problem: - Strict DPs: add#(0(),y) -> c_1() add#(s(x),y) -> c_2(add#(x,y)) dyade#(cons(x,xs),ls) -> c_3(mult#(x,ls),dyade#(xs,ls)) dyade#(nil(),ls) -> c_4() mult#(n,cons(x,xs)) -> c_5(times#(n,x),mult#(n,xs)) mult#(n,nil()) -> c_6() times#(0(),y) -> c_7() times#(s(x),y) -> c_8(add#(y,times(x,y)),times#(x,y)) - Weak TRS: add(0(),y) -> y add(s(x),y) -> s(add(x,y)) dyade(cons(x,xs),ls) -> cons(mult(x,ls),dyade(xs,ls)) dyade(nil(),ls) -> nil() mult(n,cons(x,xs)) -> cons(times(n,x),mult(n,xs)) mult(n,nil()) -> nil() times(0(),y) -> 0() times(s(x),y) -> add(y,times(x,y)) - Signature: {add/2,dyade/2,mult/2,times/2,add#/2,dyade#/2,mult#/2,times#/2} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/2,c_6/0,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dyade#,mult#,times#} and constructors {0,cons,nil,s} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,4,6,7} by application of Pre({1,4,6,7}) = {2,3,5,8}. Here rules are labelled as follows: 1: add#(0(),y) -> c_1() 2: add#(s(x),y) -> c_2(add#(x,y)) 3: dyade#(cons(x,xs),ls) -> c_3(mult#(x,ls),dyade#(xs,ls)) 4: dyade#(nil(),ls) -> c_4() 5: mult#(n,cons(x,xs)) -> c_5(times#(n,x),mult#(n,xs)) 6: mult#(n,nil()) -> c_6() 7: times#(0(),y) -> c_7() 8: times#(s(x),y) -> c_8(add#(y,times(x,y)),times#(x,y)) * Step 3: RemoveWeakSuffixes WORST_CASE(?,O(n^4)) + Considered Problem: - Strict DPs: add#(s(x),y) -> c_2(add#(x,y)) dyade#(cons(x,xs),ls) -> c_3(mult#(x,ls),dyade#(xs,ls)) mult#(n,cons(x,xs)) -> c_5(times#(n,x),mult#(n,xs)) times#(s(x),y) -> c_8(add#(y,times(x,y)),times#(x,y)) - Weak DPs: add#(0(),y) -> c_1() dyade#(nil(),ls) -> c_4() mult#(n,nil()) -> c_6() times#(0(),y) -> c_7() - Weak TRS: add(0(),y) -> y add(s(x),y) -> s(add(x,y)) dyade(cons(x,xs),ls) -> cons(mult(x,ls),dyade(xs,ls)) dyade(nil(),ls) -> nil() mult(n,cons(x,xs)) -> cons(times(n,x),mult(n,xs)) mult(n,nil()) -> nil() times(0(),y) -> 0() times(s(x),y) -> add(y,times(x,y)) - Signature: {add/2,dyade/2,mult/2,times/2,add#/2,dyade#/2,mult#/2,times#/2} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/2,c_6/0,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dyade#,mult#,times#} and constructors {0,cons,nil,s} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:add#(s(x),y) -> c_2(add#(x,y)) -->_1 add#(0(),y) -> c_1():5 -->_1 add#(s(x),y) -> c_2(add#(x,y)):1 2:S:dyade#(cons(x,xs),ls) -> c_3(mult#(x,ls),dyade#(xs,ls)) -->_1 mult#(n,cons(x,xs)) -> c_5(times#(n,x),mult#(n,xs)):3 -->_1 mult#(n,nil()) -> c_6():7 -->_2 dyade#(nil(),ls) -> c_4():6 -->_2 dyade#(cons(x,xs),ls) -> c_3(mult#(x,ls),dyade#(xs,ls)):2 3:S:mult#(n,cons(x,xs)) -> c_5(times#(n,x),mult#(n,xs)) -->_1 times#(s(x),y) -> c_8(add#(y,times(x,y)),times#(x,y)):4 -->_1 times#(0(),y) -> c_7():8 -->_2 mult#(n,nil()) -> c_6():7 -->_2 mult#(n,cons(x,xs)) -> c_5(times#(n,x),mult#(n,xs)):3 4:S:times#(s(x),y) -> c_8(add#(y,times(x,y)),times#(x,y)) -->_2 times#(0(),y) -> c_7():8 -->_1 add#(0(),y) -> c_1():5 -->_2 times#(s(x),y) -> c_8(add#(y,times(x,y)),times#(x,y)):4 -->_1 add#(s(x),y) -> c_2(add#(x,y)):1 5:W:add#(0(),y) -> c_1() 6:W:dyade#(nil(),ls) -> c_4() 7:W:mult#(n,nil()) -> c_6() 8:W:times#(0(),y) -> c_7() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 6: dyade#(nil(),ls) -> c_4() 7: mult#(n,nil()) -> c_6() 8: times#(0(),y) -> c_7() 5: add#(0(),y) -> c_1() * Step 4: UsableRules WORST_CASE(?,O(n^4)) + Considered Problem: - Strict DPs: add#(s(x),y) -> c_2(add#(x,y)) dyade#(cons(x,xs),ls) -> c_3(mult#(x,ls),dyade#(xs,ls)) mult#(n,cons(x,xs)) -> c_5(times#(n,x),mult#(n,xs)) times#(s(x),y) -> c_8(add#(y,times(x,y)),times#(x,y)) - Weak TRS: add(0(),y) -> y add(s(x),y) -> s(add(x,y)) dyade(cons(x,xs),ls) -> cons(mult(x,ls),dyade(xs,ls)) dyade(nil(),ls) -> nil() mult(n,cons(x,xs)) -> cons(times(n,x),mult(n,xs)) mult(n,nil()) -> nil() times(0(),y) -> 0() times(s(x),y) -> add(y,times(x,y)) - Signature: {add/2,dyade/2,mult/2,times/2,add#/2,dyade#/2,mult#/2,times#/2} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/2,c_6/0,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dyade#,mult#,times#} and constructors {0,cons,nil,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: add(0(),y) -> y add(s(x),y) -> s(add(x,y)) times(0(),y) -> 0() times(s(x),y) -> add(y,times(x,y)) add#(s(x),y) -> c_2(add#(x,y)) dyade#(cons(x,xs),ls) -> c_3(mult#(x,ls),dyade#(xs,ls)) mult#(n,cons(x,xs)) -> c_5(times#(n,x),mult#(n,xs)) times#(s(x),y) -> c_8(add#(y,times(x,y)),times#(x,y)) * Step 5: DecomposeDG WORST_CASE(?,O(n^4)) + Considered Problem: - Strict DPs: add#(s(x),y) -> c_2(add#(x,y)) dyade#(cons(x,xs),ls) -> c_3(mult#(x,ls),dyade#(xs,ls)) mult#(n,cons(x,xs)) -> c_5(times#(n,x),mult#(n,xs)) times#(s(x),y) -> c_8(add#(y,times(x,y)),times#(x,y)) - Weak TRS: add(0(),y) -> y add(s(x),y) -> s(add(x,y)) times(0(),y) -> 0() times(s(x),y) -> add(y,times(x,y)) - Signature: {add/2,dyade/2,mult/2,times/2,add#/2,dyade#/2,mult#/2,times#/2} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/2,c_6/0,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dyade#,mult#,times#} and constructors {0,cons,nil,s} + 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 dyade#(cons(x,xs),ls) -> c_3(mult#(x,ls),dyade#(xs,ls)) and a lower component add#(s(x),y) -> c_2(add#(x,y)) mult#(n,cons(x,xs)) -> c_5(times#(n,x),mult#(n,xs)) times#(s(x),y) -> c_8(add#(y,times(x,y)),times#(x,y)) Further, following extension rules are added to the lower component. dyade#(cons(x,xs),ls) -> dyade#(xs,ls) dyade#(cons(x,xs),ls) -> mult#(x,ls) ** Step 5.a:1: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: dyade#(cons(x,xs),ls) -> c_3(mult#(x,ls),dyade#(xs,ls)) - Weak TRS: add(0(),y) -> y add(s(x),y) -> s(add(x,y)) times(0(),y) -> 0() times(s(x),y) -> add(y,times(x,y)) - Signature: {add/2,dyade/2,mult/2,times/2,add#/2,dyade#/2,mult#/2,times#/2} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/2,c_6/0,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dyade#,mult#,times#} and constructors {0,cons,nil,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:dyade#(cons(x,xs),ls) -> c_3(mult#(x,ls),dyade#(xs,ls)) -->_2 dyade#(cons(x,xs),ls) -> c_3(mult#(x,ls),dyade#(xs,ls)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: dyade#(cons(x,xs),ls) -> c_3(dyade#(xs,ls)) ** Step 5.a:2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: dyade#(cons(x,xs),ls) -> c_3(dyade#(xs,ls)) - Weak TRS: add(0(),y) -> y add(s(x),y) -> s(add(x,y)) times(0(),y) -> 0() times(s(x),y) -> add(y,times(x,y)) - Signature: {add/2,dyade/2,mult/2,times/2,add#/2,dyade#/2,mult#/2,times#/2} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/1 ,c_4/0,c_5/2,c_6/0,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dyade#,mult#,times#} and constructors {0,cons,nil,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: dyade#(cons(x,xs),ls) -> c_3(dyade#(xs,ls)) ** Step 5.a:3: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: dyade#(cons(x,xs),ls) -> c_3(dyade#(xs,ls)) - Signature: {add/2,dyade/2,mult/2,times/2,add#/2,dyade#/2,mult#/2,times#/2} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/1 ,c_4/0,c_5/2,c_6/0,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dyade#,mult#,times#} and constructors {0,cons,nil,s} + 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_3) = {1} Following symbols are considered usable: {add#,dyade#,mult#,times#} TcT has computed the following interpretation: p(0) = [0] p(add) = [0] p(cons) = [1] x1 + [1] x2 + [2] p(dyade) = [0] p(mult) = [0] p(nil) = [0] p(s) = [1] x1 + [0] p(times) = [0] p(add#) = [0] p(dyade#) = [8] x1 + [0] p(mult#) = [2] x2 + [0] p(times#) = [2] p(c_1) = [8] p(c_2) = [1] x1 + [0] p(c_3) = [1] x1 + [6] p(c_4) = [0] p(c_5) = [4] p(c_6) = [1] p(c_7) = [0] p(c_8) = [1] x1 + [0] Following rules are strictly oriented: dyade#(cons(x,xs),ls) = [8] x + [8] xs + [16] > [8] xs + [6] = c_3(dyade#(xs,ls)) Following rules are (at-least) weakly oriented: ** Step 5.a:4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: dyade#(cons(x,xs),ls) -> c_3(dyade#(xs,ls)) - Signature: {add/2,dyade/2,mult/2,times/2,add#/2,dyade#/2,mult#/2,times#/2} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/1 ,c_4/0,c_5/2,c_6/0,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dyade#,mult#,times#} and constructors {0,cons,nil,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). ** Step 5.b:1: DecomposeDG WORST_CASE(?,O(n^3)) + Considered Problem: - Strict DPs: add#(s(x),y) -> c_2(add#(x,y)) mult#(n,cons(x,xs)) -> c_5(times#(n,x),mult#(n,xs)) times#(s(x),y) -> c_8(add#(y,times(x,y)),times#(x,y)) - Weak DPs: dyade#(cons(x,xs),ls) -> dyade#(xs,ls) dyade#(cons(x,xs),ls) -> mult#(x,ls) - Weak TRS: add(0(),y) -> y add(s(x),y) -> s(add(x,y)) times(0(),y) -> 0() times(s(x),y) -> add(y,times(x,y)) - Signature: {add/2,dyade/2,mult/2,times/2,add#/2,dyade#/2,mult#/2,times#/2} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/2,c_6/0,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dyade#,mult#,times#} and constructors {0,cons,nil,s} + 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 dyade#(cons(x,xs),ls) -> dyade#(xs,ls) dyade#(cons(x,xs),ls) -> mult#(x,ls) mult#(n,cons(x,xs)) -> c_5(times#(n,x),mult#(n,xs)) and a lower component add#(s(x),y) -> c_2(add#(x,y)) times#(s(x),y) -> c_8(add#(y,times(x,y)),times#(x,y)) Further, following extension rules are added to the lower component. dyade#(cons(x,xs),ls) -> dyade#(xs,ls) dyade#(cons(x,xs),ls) -> mult#(x,ls) mult#(n,cons(x,xs)) -> mult#(n,xs) mult#(n,cons(x,xs)) -> times#(n,x) *** Step 5.b:1.a:1: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mult#(n,cons(x,xs)) -> c_5(times#(n,x),mult#(n,xs)) - Weak DPs: dyade#(cons(x,xs),ls) -> dyade#(xs,ls) dyade#(cons(x,xs),ls) -> mult#(x,ls) - Weak TRS: add(0(),y) -> y add(s(x),y) -> s(add(x,y)) times(0(),y) -> 0() times(s(x),y) -> add(y,times(x,y)) - Signature: {add/2,dyade/2,mult/2,times/2,add#/2,dyade#/2,mult#/2,times#/2} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/2,c_6/0,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dyade#,mult#,times#} and constructors {0,cons,nil,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:mult#(n,cons(x,xs)) -> c_5(times#(n,x),mult#(n,xs)) -->_2 mult#(n,cons(x,xs)) -> c_5(times#(n,x),mult#(n,xs)):1 2:W:dyade#(cons(x,xs),ls) -> dyade#(xs,ls) -->_1 dyade#(cons(x,xs),ls) -> mult#(x,ls):3 -->_1 dyade#(cons(x,xs),ls) -> dyade#(xs,ls):2 3:W:dyade#(cons(x,xs),ls) -> mult#(x,ls) -->_1 mult#(n,cons(x,xs)) -> c_5(times#(n,x),mult#(n,xs)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: mult#(n,cons(x,xs)) -> c_5(mult#(n,xs)) *** Step 5.b:1.a:2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mult#(n,cons(x,xs)) -> c_5(mult#(n,xs)) - Weak DPs: dyade#(cons(x,xs),ls) -> dyade#(xs,ls) dyade#(cons(x,xs),ls) -> mult#(x,ls) - Weak TRS: add(0(),y) -> y add(s(x),y) -> s(add(x,y)) times(0(),y) -> 0() times(s(x),y) -> add(y,times(x,y)) - Signature: {add/2,dyade/2,mult/2,times/2,add#/2,dyade#/2,mult#/2,times#/2} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/1,c_6/0,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dyade#,mult#,times#} and constructors {0,cons,nil,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: dyade#(cons(x,xs),ls) -> dyade#(xs,ls) dyade#(cons(x,xs),ls) -> mult#(x,ls) mult#(n,cons(x,xs)) -> c_5(mult#(n,xs)) *** Step 5.b:1.a:3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: mult#(n,cons(x,xs)) -> c_5(mult#(n,xs)) - Weak DPs: dyade#(cons(x,xs),ls) -> dyade#(xs,ls) dyade#(cons(x,xs),ls) -> mult#(x,ls) - Signature: {add/2,dyade/2,mult/2,times/2,add#/2,dyade#/2,mult#/2,times#/2} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/1,c_6/0,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dyade#,mult#,times#} and constructors {0,cons,nil,s} + 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(c_5) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(add) = [0] p(cons) = [1] x2 + [5] p(dyade) = [0] p(mult) = [0] p(nil) = [0] p(s) = [1] x1 + [0] p(times) = [0] p(add#) = [0] p(dyade#) = [1] x1 + [2] x2 + [0] p(mult#) = [2] x2 + [5] p(times#) = [0] p(c_1) = [0] p(c_2) = [0] p(c_3) = [0] p(c_4) = [1] p(c_5) = [1] x1 + [9] p(c_6) = [0] p(c_7) = [0] p(c_8) = [0] Following rules are strictly oriented: mult#(n,cons(x,xs)) = [2] xs + [15] > [2] xs + [14] = c_5(mult#(n,xs)) Following rules are (at-least) weakly oriented: dyade#(cons(x,xs),ls) = [2] ls + [1] xs + [5] >= [2] ls + [1] xs + [0] = dyade#(xs,ls) dyade#(cons(x,xs),ls) = [2] ls + [1] xs + [5] >= [2] ls + [5] = mult#(x,ls) 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: dyade#(cons(x,xs),ls) -> dyade#(xs,ls) dyade#(cons(x,xs),ls) -> mult#(x,ls) mult#(n,cons(x,xs)) -> c_5(mult#(n,xs)) - Signature: {add/2,dyade/2,mult/2,times/2,add#/2,dyade#/2,mult#/2,times#/2} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/1,c_6/0,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dyade#,mult#,times#} and constructors {0,cons,nil,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 5.b:1.b:1: DecomposeDG WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: add#(s(x),y) -> c_2(add#(x,y)) times#(s(x),y) -> c_8(add#(y,times(x,y)),times#(x,y)) - Weak DPs: dyade#(cons(x,xs),ls) -> dyade#(xs,ls) dyade#(cons(x,xs),ls) -> mult#(x,ls) mult#(n,cons(x,xs)) -> mult#(n,xs) mult#(n,cons(x,xs)) -> times#(n,x) - Weak TRS: add(0(),y) -> y add(s(x),y) -> s(add(x,y)) times(0(),y) -> 0() times(s(x),y) -> add(y,times(x,y)) - Signature: {add/2,dyade/2,mult/2,times/2,add#/2,dyade#/2,mult#/2,times#/2} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/2,c_6/0,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dyade#,mult#,times#} and constructors {0,cons,nil,s} + 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 dyade#(cons(x,xs),ls) -> dyade#(xs,ls) dyade#(cons(x,xs),ls) -> mult#(x,ls) mult#(n,cons(x,xs)) -> mult#(n,xs) mult#(n,cons(x,xs)) -> times#(n,x) times#(s(x),y) -> c_8(add#(y,times(x,y)),times#(x,y)) and a lower component add#(s(x),y) -> c_2(add#(x,y)) Further, following extension rules are added to the lower component. dyade#(cons(x,xs),ls) -> dyade#(xs,ls) dyade#(cons(x,xs),ls) -> mult#(x,ls) mult#(n,cons(x,xs)) -> mult#(n,xs) mult#(n,cons(x,xs)) -> times#(n,x) times#(s(x),y) -> add#(y,times(x,y)) times#(s(x),y) -> times#(x,y) **** Step 5.b:1.b:1.a:1: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: times#(s(x),y) -> c_8(add#(y,times(x,y)),times#(x,y)) - Weak DPs: dyade#(cons(x,xs),ls) -> dyade#(xs,ls) dyade#(cons(x,xs),ls) -> mult#(x,ls) mult#(n,cons(x,xs)) -> mult#(n,xs) mult#(n,cons(x,xs)) -> times#(n,x) - Weak TRS: add(0(),y) -> y add(s(x),y) -> s(add(x,y)) times(0(),y) -> 0() times(s(x),y) -> add(y,times(x,y)) - Signature: {add/2,dyade/2,mult/2,times/2,add#/2,dyade#/2,mult#/2,times#/2} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/2,c_6/0,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dyade#,mult#,times#} and constructors {0,cons,nil,s} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:times#(s(x),y) -> c_8(add#(y,times(x,y)),times#(x,y)) -->_2 times#(s(x),y) -> c_8(add#(y,times(x,y)),times#(x,y)):1 2:W:dyade#(cons(x,xs),ls) -> dyade#(xs,ls) -->_1 dyade#(cons(x,xs),ls) -> mult#(x,ls):3 -->_1 dyade#(cons(x,xs),ls) -> dyade#(xs,ls):2 3:W:dyade#(cons(x,xs),ls) -> mult#(x,ls) -->_1 mult#(n,cons(x,xs)) -> times#(n,x):5 -->_1 mult#(n,cons(x,xs)) -> mult#(n,xs):4 4:W:mult#(n,cons(x,xs)) -> mult#(n,xs) -->_1 mult#(n,cons(x,xs)) -> times#(n,x):5 -->_1 mult#(n,cons(x,xs)) -> mult#(n,xs):4 5:W:mult#(n,cons(x,xs)) -> times#(n,x) -->_1 times#(s(x),y) -> c_8(add#(y,times(x,y)),times#(x,y)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: times#(s(x),y) -> c_8(times#(x,y)) **** Step 5.b:1.b:1.a:2: UsableRules WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: times#(s(x),y) -> c_8(times#(x,y)) - Weak DPs: dyade#(cons(x,xs),ls) -> dyade#(xs,ls) dyade#(cons(x,xs),ls) -> mult#(x,ls) mult#(n,cons(x,xs)) -> mult#(n,xs) mult#(n,cons(x,xs)) -> times#(n,x) - Weak TRS: add(0(),y) -> y add(s(x),y) -> s(add(x,y)) times(0(),y) -> 0() times(s(x),y) -> add(y,times(x,y)) - Signature: {add/2,dyade/2,mult/2,times/2,add#/2,dyade#/2,mult#/2,times#/2} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/2,c_6/0,c_7/0,c_8/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dyade#,mult#,times#} and constructors {0,cons,nil,s} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: dyade#(cons(x,xs),ls) -> dyade#(xs,ls) dyade#(cons(x,xs),ls) -> mult#(x,ls) mult#(n,cons(x,xs)) -> mult#(n,xs) mult#(n,cons(x,xs)) -> times#(n,x) times#(s(x),y) -> c_8(times#(x,y)) **** Step 5.b:1.b:1.a:3: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: times#(s(x),y) -> c_8(times#(x,y)) - Weak DPs: dyade#(cons(x,xs),ls) -> dyade#(xs,ls) dyade#(cons(x,xs),ls) -> mult#(x,ls) mult#(n,cons(x,xs)) -> mult#(n,xs) mult#(n,cons(x,xs)) -> times#(n,x) - Signature: {add/2,dyade/2,mult/2,times/2,add#/2,dyade#/2,mult#/2,times#/2} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/2,c_6/0,c_7/0,c_8/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dyade#,mult#,times#} and constructors {0,cons,nil,s} + 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(c_8) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(add) = [1] x1 + [0] p(cons) = [1] x1 + [1] x2 + [12] p(dyade) = [0] p(mult) = [0] p(nil) = [0] p(s) = [1] x1 + [2] p(times) = [1] p(add#) = [1] x1 + [1] x2 + [0] p(dyade#) = [2] x1 + [1] x2 + [0] p(mult#) = [2] x1 + [1] x2 + [0] p(times#) = [2] x1 + [11] p(c_1) = [1] p(c_2) = [0] p(c_3) = [8] x1 + [1] x2 + [0] p(c_4) = [1] p(c_5) = [1] x1 + [1] x2 + [1] p(c_6) = [0] p(c_7) = [0] p(c_8) = [1] x1 + [2] Following rules are strictly oriented: times#(s(x),y) = [2] x + [15] > [2] x + [13] = c_8(times#(x,y)) Following rules are (at-least) weakly oriented: dyade#(cons(x,xs),ls) = [1] ls + [2] x + [2] xs + [24] >= [1] ls + [2] xs + [0] = dyade#(xs,ls) dyade#(cons(x,xs),ls) = [1] ls + [2] x + [2] xs + [24] >= [1] ls + [2] x + [0] = mult#(x,ls) mult#(n,cons(x,xs)) = [2] n + [1] x + [1] xs + [12] >= [2] n + [1] xs + [0] = mult#(n,xs) mult#(n,cons(x,xs)) = [2] n + [1] x + [1] xs + [12] >= [2] n + [11] = times#(n,x) Further, it can be verified that all rules not oriented are covered by the weightgap condition. **** Step 5.b:1.b:1.a:4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: dyade#(cons(x,xs),ls) -> dyade#(xs,ls) dyade#(cons(x,xs),ls) -> mult#(x,ls) mult#(n,cons(x,xs)) -> mult#(n,xs) mult#(n,cons(x,xs)) -> times#(n,x) times#(s(x),y) -> c_8(times#(x,y)) - Signature: {add/2,dyade/2,mult/2,times/2,add#/2,dyade#/2,mult#/2,times#/2} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/2,c_6/0,c_7/0,c_8/1} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dyade#,mult#,times#} and constructors {0,cons,nil,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). **** Step 5.b:1.b:1.b:1: NaturalMI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: add#(s(x),y) -> c_2(add#(x,y)) - Weak DPs: dyade#(cons(x,xs),ls) -> dyade#(xs,ls) dyade#(cons(x,xs),ls) -> mult#(x,ls) mult#(n,cons(x,xs)) -> mult#(n,xs) mult#(n,cons(x,xs)) -> times#(n,x) times#(s(x),y) -> add#(y,times(x,y)) times#(s(x),y) -> times#(x,y) - Weak TRS: add(0(),y) -> y add(s(x),y) -> s(add(x,y)) times(0(),y) -> 0() times(s(x),y) -> add(y,times(x,y)) - Signature: {add/2,dyade/2,mult/2,times/2,add#/2,dyade#/2,mult#/2,times#/2} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/2,c_6/0,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dyade#,mult#,times#} and constructors {0,cons,nil,s} + 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_2) = {1} Following symbols are considered usable: {add#,dyade#,mult#,times#} TcT has computed the following interpretation: p(0) = [1] p(add) = [2] x1 + [4] p(cons) = [1] x1 + [1] x2 + [0] p(dyade) = [0] p(mult) = [1] x2 + [2] p(nil) = [0] p(s) = [1] x1 + [1] p(times) = [8] x1 + [0] p(add#) = [2] x1 + [2] p(dyade#) = [1] x1 + [8] x2 + [8] p(mult#) = [2] x2 + [8] p(times#) = [2] x2 + [4] p(c_1) = [1] p(c_2) = [1] x1 + [0] p(c_3) = [0] p(c_4) = [1] p(c_5) = [1] x2 + [4] p(c_6) = [0] p(c_7) = [1] p(c_8) = [2] x2 + [1] Following rules are strictly oriented: add#(s(x),y) = [2] x + [4] > [2] x + [2] = c_2(add#(x,y)) Following rules are (at-least) weakly oriented: dyade#(cons(x,xs),ls) = [8] ls + [1] x + [1] xs + [8] >= [8] ls + [1] xs + [8] = dyade#(xs,ls) dyade#(cons(x,xs),ls) = [8] ls + [1] x + [1] xs + [8] >= [2] ls + [8] = mult#(x,ls) mult#(n,cons(x,xs)) = [2] x + [2] xs + [8] >= [2] xs + [8] = mult#(n,xs) mult#(n,cons(x,xs)) = [2] x + [2] xs + [8] >= [2] x + [4] = times#(n,x) times#(s(x),y) = [2] y + [4] >= [2] y + [2] = add#(y,times(x,y)) times#(s(x),y) = [2] y + [4] >= [2] y + [4] = times#(x,y) **** Step 5.b:1.b:1.b:2: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: add#(s(x),y) -> c_2(add#(x,y)) dyade#(cons(x,xs),ls) -> dyade#(xs,ls) dyade#(cons(x,xs),ls) -> mult#(x,ls) mult#(n,cons(x,xs)) -> mult#(n,xs) mult#(n,cons(x,xs)) -> times#(n,x) times#(s(x),y) -> add#(y,times(x,y)) times#(s(x),y) -> times#(x,y) - Weak TRS: add(0(),y) -> y add(s(x),y) -> s(add(x,y)) times(0(),y) -> 0() times(s(x),y) -> add(y,times(x,y)) - Signature: {add/2,dyade/2,mult/2,times/2,add#/2,dyade#/2,mult#/2,times#/2} / {0/0,cons/2,nil/0,s/1,c_1/0,c_2/1,c_3/2 ,c_4/0,c_5/2,c_6/0,c_7/0,c_8/2} - Obligation: innermost runtime complexity wrt. defined symbols {add#,dyade#,mult#,times#} and constructors {0,cons,nil,s} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(?,O(n^4))