WORST_CASE(Omega(n^1),O(n^2)) * Step 1: Sum WORST_CASE(Omega(n^1),O(n^2)) + Considered Problem: - Strict TRS: gcd(0(),Y) -> 0() gcd(s(X),0()) -> s(X) gcd(s(X),s(Y)) -> if(le(Y,X),s(X),s(Y)) if(false(),s(X),s(Y)) -> gcd(minus(Y,X),s(X)) if(true(),s(X),s(Y)) -> gcd(minus(X,Y),s(Y)) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) minus(X,0()) -> X minus(X,s(Y)) -> pred(minus(X,Y)) pred(s(X)) -> X - Signature: {gcd/2,if/3,le/2,minus/2,pred/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd,if,le,minus,pred} and constructors {0,false,s,true} + Applied Processor: Sum {left = someStrategy, right = someStrategy} + Details: () ** Step 1.a:1: DecreasingLoops WORST_CASE(Omega(n^1),?) + Considered Problem: - Strict TRS: gcd(0(),Y) -> 0() gcd(s(X),0()) -> s(X) gcd(s(X),s(Y)) -> if(le(Y,X),s(X),s(Y)) if(false(),s(X),s(Y)) -> gcd(minus(Y,X),s(X)) if(true(),s(X),s(Y)) -> gcd(minus(X,Y),s(Y)) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) minus(X,0()) -> X minus(X,s(Y)) -> pred(minus(X,Y)) pred(s(X)) -> X - Signature: {gcd/2,if/3,le/2,minus/2,pred/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd,if,le,minus,pred} and constructors {0,false,s,true} + Applied Processor: DecreasingLoops {bound = AnyLoop, narrow = 10} + Details: The system has following decreasing Loops: le(x,y){x -> s(x),y -> s(y)} = le(s(x),s(y)) ->^+ le(x,y) = C[le(x,y) = le(x,y){}] ** Step 1.b:1: DependencyPairs WORST_CASE(?,O(n^2)) + Considered Problem: - Strict TRS: gcd(0(),Y) -> 0() gcd(s(X),0()) -> s(X) gcd(s(X),s(Y)) -> if(le(Y,X),s(X),s(Y)) if(false(),s(X),s(Y)) -> gcd(minus(Y,X),s(X)) if(true(),s(X),s(Y)) -> gcd(minus(X,Y),s(Y)) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) minus(X,0()) -> X minus(X,s(Y)) -> pred(minus(X,Y)) pred(s(X)) -> X - Signature: {gcd/2,if/3,le/2,minus/2,pred/1} / {0/0,false/0,s/1,true/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd,if,le,minus,pred} and constructors {0,false,s,true} + Applied Processor: DependencyPairs {dpKind_ = DT} + Details: We add the following dependency tuples: Strict DPs gcd#(0(),Y) -> c_1() gcd#(s(X),0()) -> c_2() gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X)) if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X)) if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y)) le#(0(),Y) -> c_6() le#(s(X),0()) -> c_7() le#(s(X),s(Y)) -> c_8(le#(X,Y)) minus#(X,0()) -> c_9() minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y)) pred#(s(X)) -> c_11() Weak DPs and mark the set of starting terms. ** Step 1.b:2: PredecessorEstimation WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: gcd#(0(),Y) -> c_1() gcd#(s(X),0()) -> c_2() gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X)) if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X)) if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y)) le#(0(),Y) -> c_6() le#(s(X),0()) -> c_7() le#(s(X),s(Y)) -> c_8(le#(X,Y)) minus#(X,0()) -> c_9() minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y)) pred#(s(X)) -> c_11() - Weak TRS: gcd(0(),Y) -> 0() gcd(s(X),0()) -> s(X) gcd(s(X),s(Y)) -> if(le(Y,X),s(X),s(Y)) if(false(),s(X),s(Y)) -> gcd(minus(Y,X),s(X)) if(true(),s(X),s(Y)) -> gcd(minus(X,Y),s(Y)) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) minus(X,0()) -> X minus(X,s(Y)) -> pred(minus(X,Y)) pred(s(X)) -> X - Signature: {gcd/2,if/3,le/2,minus/2,pred/1,gcd#/2,if#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0 ,c_3/2,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if#,le#,minus#,pred#} and constructors {0,false,s ,true} + Applied Processor: PredecessorEstimation {onSelection = all simple predecessor estimation selector} + Details: We estimate the number of application of {1,2,6,7,9,11} by application of Pre({1,2,6,7,9,11}) = {3,4,5,8,10}. Here rules are labelled as follows: 1: gcd#(0(),Y) -> c_1() 2: gcd#(s(X),0()) -> c_2() 3: gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X)) 4: if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X)) 5: if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y)) 6: le#(0(),Y) -> c_6() 7: le#(s(X),0()) -> c_7() 8: le#(s(X),s(Y)) -> c_8(le#(X,Y)) 9: minus#(X,0()) -> c_9() 10: minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y)) 11: pred#(s(X)) -> c_11() ** Step 1.b:3: RemoveWeakSuffixes WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X)) if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X)) if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y)) le#(s(X),s(Y)) -> c_8(le#(X,Y)) minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y)) - Weak DPs: gcd#(0(),Y) -> c_1() gcd#(s(X),0()) -> c_2() le#(0(),Y) -> c_6() le#(s(X),0()) -> c_7() minus#(X,0()) -> c_9() pred#(s(X)) -> c_11() - Weak TRS: gcd(0(),Y) -> 0() gcd(s(X),0()) -> s(X) gcd(s(X),s(Y)) -> if(le(Y,X),s(X),s(Y)) if(false(),s(X),s(Y)) -> gcd(minus(Y,X),s(X)) if(true(),s(X),s(Y)) -> gcd(minus(X,Y),s(Y)) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) minus(X,0()) -> X minus(X,s(Y)) -> pred(minus(X,Y)) pred(s(X)) -> X - Signature: {gcd/2,if/3,le/2,minus/2,pred/1,gcd#/2,if#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0 ,c_3/2,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if#,le#,minus#,pred#} and constructors {0,false,s ,true} + Applied Processor: RemoveWeakSuffixes + Details: Consider the dependency graph 1:S:gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X)) -->_2 le#(s(X),s(Y)) -> c_8(le#(X,Y)):4 -->_1 if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y)):3 -->_1 if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X)):2 -->_2 le#(s(X),0()) -> c_7():9 -->_2 le#(0(),Y) -> c_6():8 2:S:if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X)) -->_2 minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y)):5 -->_2 minus#(X,0()) -> c_9():10 -->_1 gcd#(0(),Y) -> c_1():6 -->_1 gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X)):1 3:S:if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y)) -->_2 minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y)):5 -->_2 minus#(X,0()) -> c_9():10 -->_1 gcd#(0(),Y) -> c_1():6 -->_1 gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X)):1 4:S:le#(s(X),s(Y)) -> c_8(le#(X,Y)) -->_1 le#(s(X),0()) -> c_7():9 -->_1 le#(0(),Y) -> c_6():8 -->_1 le#(s(X),s(Y)) -> c_8(le#(X,Y)):4 5:S:minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y)) -->_1 pred#(s(X)) -> c_11():11 -->_2 minus#(X,0()) -> c_9():10 -->_2 minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y)):5 6:W:gcd#(0(),Y) -> c_1() 7:W:gcd#(s(X),0()) -> c_2() 8:W:le#(0(),Y) -> c_6() 9:W:le#(s(X),0()) -> c_7() 10:W:minus#(X,0()) -> c_9() 11:W:pred#(s(X)) -> c_11() The following weak DPs constitute a sub-graph of the DG that is closed under successors. The DPs are removed. 7: gcd#(s(X),0()) -> c_2() 6: gcd#(0(),Y) -> c_1() 10: minus#(X,0()) -> c_9() 11: pred#(s(X)) -> c_11() 8: le#(0(),Y) -> c_6() 9: le#(s(X),0()) -> c_7() ** Step 1.b:4: SimplifyRHS WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X)) if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X)) if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y)) le#(s(X),s(Y)) -> c_8(le#(X,Y)) minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y)) - Weak TRS: gcd(0(),Y) -> 0() gcd(s(X),0()) -> s(X) gcd(s(X),s(Y)) -> if(le(Y,X),s(X),s(Y)) if(false(),s(X),s(Y)) -> gcd(minus(Y,X),s(X)) if(true(),s(X),s(Y)) -> gcd(minus(X,Y),s(Y)) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) minus(X,0()) -> X minus(X,s(Y)) -> pred(minus(X,Y)) pred(s(X)) -> X - Signature: {gcd/2,if/3,le/2,minus/2,pred/1,gcd#/2,if#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0 ,c_3/2,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/2,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if#,le#,minus#,pred#} and constructors {0,false,s ,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X)) -->_2 le#(s(X),s(Y)) -> c_8(le#(X,Y)):4 -->_1 if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y)):3 -->_1 if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X)):2 2:S:if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X)) -->_2 minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y)):5 -->_1 gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X)):1 3:S:if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y)) -->_2 minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y)):5 -->_1 gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X)):1 4:S:le#(s(X),s(Y)) -> c_8(le#(X,Y)) -->_1 le#(s(X),s(Y)) -> c_8(le#(X,Y)):4 5:S:minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y)) -->_2 minus#(X,s(Y)) -> c_10(pred#(minus(X,Y)),minus#(X,Y)):5 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: minus#(X,s(Y)) -> c_10(minus#(X,Y)) ** Step 1.b:5: UsableRules WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X)) if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X)) if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y)) le#(s(X),s(Y)) -> c_8(le#(X,Y)) minus#(X,s(Y)) -> c_10(minus#(X,Y)) - Weak TRS: gcd(0(),Y) -> 0() gcd(s(X),0()) -> s(X) gcd(s(X),s(Y)) -> if(le(Y,X),s(X),s(Y)) if(false(),s(X),s(Y)) -> gcd(minus(Y,X),s(X)) if(true(),s(X),s(Y)) -> gcd(minus(X,Y),s(Y)) le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) minus(X,0()) -> X minus(X,s(Y)) -> pred(minus(X,Y)) pred(s(X)) -> X - Signature: {gcd/2,if/3,le/2,minus/2,pred/1,gcd#/2,if#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0 ,c_3/2,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if#,le#,minus#,pred#} and constructors {0,false,s ,true} + Applied Processor: UsableRules + Details: We replace rewrite rules by usable rules: le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) minus(X,0()) -> X minus(X,s(Y)) -> pred(minus(X,Y)) pred(s(X)) -> X gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X)) if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X)) if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y)) le#(s(X),s(Y)) -> c_8(le#(X,Y)) minus#(X,s(Y)) -> c_10(minus#(X,Y)) ** Step 1.b:6: DecomposeDG WORST_CASE(?,O(n^2)) + Considered Problem: - Strict DPs: gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X)) if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X)) if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y)) le#(s(X),s(Y)) -> c_8(le#(X,Y)) minus#(X,s(Y)) -> c_10(minus#(X,Y)) - Weak TRS: le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) minus(X,0()) -> X minus(X,s(Y)) -> pred(minus(X,Y)) pred(s(X)) -> X - Signature: {gcd/2,if/3,le/2,minus/2,pred/1,gcd#/2,if#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0 ,c_3/2,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if#,le#,minus#,pred#} 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 gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X)) if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X)) if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y)) and a lower component le#(s(X),s(Y)) -> c_8(le#(X,Y)) minus#(X,s(Y)) -> c_10(minus#(X,Y)) Further, following extension rules are added to the lower component. gcd#(s(X),s(Y)) -> if#(le(Y,X),s(X),s(Y)) gcd#(s(X),s(Y)) -> le#(Y,X) if#(false(),s(X),s(Y)) -> gcd#(minus(Y,X),s(X)) if#(false(),s(X),s(Y)) -> minus#(Y,X) if#(true(),s(X),s(Y)) -> gcd#(minus(X,Y),s(Y)) if#(true(),s(X),s(Y)) -> minus#(X,Y) *** Step 1.b:6.a:1: SimplifyRHS WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X)) if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X)) if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y)) - Weak TRS: le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) minus(X,0()) -> X minus(X,s(Y)) -> pred(minus(X,Y)) pred(s(X)) -> X - Signature: {gcd/2,if/3,le/2,minus/2,pred/1,gcd#/2,if#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0 ,c_3/2,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if#,le#,minus#,pred#} and constructors {0,false,s ,true} + Applied Processor: SimplifyRHS + Details: Consider the dependency graph 1:S:gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X)) -->_1 if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y)):3 -->_1 if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X)):2 2:S:if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X)),minus#(Y,X)) -->_1 gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X)):1 3:S:if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y)),minus#(X,Y)) -->_1 gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y)),le#(Y,X)):1 Due to missing edges in the depndency graph, the right-hand sides of following rules could be simplified: gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y))) if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X))) if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y))) *** Step 1.b:6.a:2: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y))) if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X))) if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y))) - Weak TRS: le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) minus(X,0()) -> X minus(X,s(Y)) -> pred(minus(X,Y)) pred(s(X)) -> X - Signature: {gcd/2,if/3,le/2,minus/2,pred/1,gcd#/2,if#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0 ,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if#,le#,minus#,pred#} 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(pred) = {1}, uargs(gcd#) = {1}, uargs(if#) = {1}, uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(false) = [5] p(gcd) = [0] p(if) = [4] x1 + [1] x3 + [1] p(le) = [5] p(minus) = [1] x1 + [0] p(pred) = [1] x1 + [0] p(s) = [1] x1 + [3] p(true) = [0] p(gcd#) = [1] x1 + [1] x2 + [4] p(if#) = [1] x1 + [1] x2 + [1] x3 + [0] p(le#) = [2] x1 + [1] p(minus#) = [2] x1 + [0] p(pred#) = [4] x1 + [0] p(c_1) = [1] p(c_2) = [0] p(c_3) = [1] x1 + [1] p(c_4) = [1] x1 + [0] p(c_5) = [1] x1 + [0] p(c_6) = [4] p(c_7) = [4] p(c_8) = [4] x1 + [1] p(c_9) = [0] p(c_10) = [1] p(c_11) = [2] Following rules are strictly oriented: if#(false(),s(X),s(Y)) = [1] X + [1] Y + [11] > [1] X + [1] Y + [7] = c_4(gcd#(minus(Y,X),s(X))) Following rules are (at-least) weakly oriented: gcd#(s(X),s(Y)) = [1] X + [1] Y + [10] >= [1] X + [1] Y + [12] = c_3(if#(le(Y,X),s(X),s(Y))) if#(true(),s(X),s(Y)) = [1] X + [1] Y + [6] >= [1] X + [1] Y + [7] = c_5(gcd#(minus(X,Y),s(Y))) le(0(),Y) = [5] >= [0] = true() le(s(X),0()) = [5] >= [5] = false() le(s(X),s(Y)) = [5] >= [5] = le(X,Y) minus(X,0()) = [1] X + [0] >= [1] X + [0] = X minus(X,s(Y)) = [1] X + [0] >= [1] X + [0] = pred(minus(X,Y)) pred(s(X)) = [1] X + [3] >= [1] X + [0] = X Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 1.b:6.a:3: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y))) if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y))) - Weak DPs: if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X))) - Weak TRS: le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) minus(X,0()) -> X minus(X,s(Y)) -> pred(minus(X,Y)) pred(s(X)) -> X - Signature: {gcd/2,if/3,le/2,minus/2,pred/1,gcd#/2,if#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0 ,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if#,le#,minus#,pred#} and constructors {0,false,s ,true} + Applied Processor: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(c_3) = {1}, uargs(c_4) = {1}, uargs(c_5) = {1} Following symbols are considered usable: {minus,pred,gcd#,if#,le#,minus#,pred#} TcT has computed the following interpretation: p(0) = 2 p(false) = 5 p(gcd) = 1 + 2*x1 p(if) = 1 + 4*x2 p(le) = 4*x2 p(minus) = x1 p(pred) = x1 p(s) = 3 + x1 p(true) = 4 p(gcd#) = 4 + x1 + x2 p(if#) = 2 + x2 + x3 p(le#) = 2*x1 p(minus#) = 2 + 4*x1 p(pred#) = 0 p(c_1) = 0 p(c_2) = 1 p(c_3) = x1 p(c_4) = x1 p(c_5) = x1 p(c_6) = 1 p(c_7) = 0 p(c_8) = x1 p(c_9) = 0 p(c_10) = 1 p(c_11) = 1 Following rules are strictly oriented: gcd#(s(X),s(Y)) = 10 + X + Y > 8 + X + Y = c_3(if#(le(Y,X),s(X),s(Y))) if#(true(),s(X),s(Y)) = 8 + X + Y > 7 + X + Y = c_5(gcd#(minus(X,Y),s(Y))) Following rules are (at-least) weakly oriented: if#(false(),s(X),s(Y)) = 8 + X + Y >= 7 + X + Y = c_4(gcd#(minus(Y,X),s(X))) minus(X,0()) = X >= X = X minus(X,s(Y)) = X >= X = pred(minus(X,Y)) pred(s(X)) = 3 + X >= X = X *** Step 1.b:6.a:4: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: gcd#(s(X),s(Y)) -> c_3(if#(le(Y,X),s(X),s(Y))) if#(false(),s(X),s(Y)) -> c_4(gcd#(minus(Y,X),s(X))) if#(true(),s(X),s(Y)) -> c_5(gcd#(minus(X,Y),s(Y))) - Weak TRS: le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) minus(X,0()) -> X minus(X,s(Y)) -> pred(minus(X,Y)) pred(s(X)) -> X - Signature: {gcd/2,if/3,le/2,minus/2,pred/1,gcd#/2,if#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0 ,c_3/1,c_4/1,c_5/1,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if#,le#,minus#,pred#} and constructors {0,false,s ,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). *** Step 1.b:6.b:1: WeightGap WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: le#(s(X),s(Y)) -> c_8(le#(X,Y)) minus#(X,s(Y)) -> c_10(minus#(X,Y)) - Weak DPs: gcd#(s(X),s(Y)) -> if#(le(Y,X),s(X),s(Y)) gcd#(s(X),s(Y)) -> le#(Y,X) if#(false(),s(X),s(Y)) -> gcd#(minus(Y,X),s(X)) if#(false(),s(X),s(Y)) -> minus#(Y,X) if#(true(),s(X),s(Y)) -> gcd#(minus(X,Y),s(Y)) if#(true(),s(X),s(Y)) -> minus#(X,Y) - Weak TRS: le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) minus(X,0()) -> X minus(X,s(Y)) -> pred(minus(X,Y)) pred(s(X)) -> X - Signature: {gcd/2,if/3,le/2,minus/2,pred/1,gcd#/2,if#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0 ,c_3/2,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if#,le#,minus#,pred#} 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(pred) = {1}, uargs(gcd#) = {1}, uargs(if#) = {1}, uargs(c_8) = {1}, uargs(c_10) = {1} Following symbols are considered usable: all TcT has computed the following interpretation: p(0) = [0] p(false) = [0] p(gcd) = [0] p(if) = [0] p(le) = [1] p(minus) = [1] x1 + [0] p(pred) = [1] x1 + [0] p(s) = [1] x1 + [1] p(true) = [0] p(gcd#) = [1] x1 + [1] x2 + [5] p(if#) = [1] x1 + [1] x2 + [1] x3 + [4] p(le#) = [1] x1 + [1] x2 + [7] p(minus#) = [1] x1 + [1] x2 + [2] p(pred#) = [1] x1 + [1] p(c_1) = [4] p(c_2) = [0] p(c_3) = [2] x1 + [0] p(c_4) = [1] x1 + [1] x2 + [1] p(c_5) = [1] x2 + [1] p(c_6) = [0] p(c_7) = [2] p(c_8) = [1] x1 + [1] p(c_9) = [1] p(c_10) = [1] x1 + [5] p(c_11) = [1] Following rules are strictly oriented: le#(s(X),s(Y)) = [1] X + [1] Y + [9] > [1] X + [1] Y + [8] = c_8(le#(X,Y)) Following rules are (at-least) weakly oriented: gcd#(s(X),s(Y)) = [1] X + [1] Y + [7] >= [1] X + [1] Y + [7] = if#(le(Y,X),s(X),s(Y)) gcd#(s(X),s(Y)) = [1] X + [1] Y + [7] >= [1] X + [1] Y + [7] = le#(Y,X) if#(false(),s(X),s(Y)) = [1] X + [1] Y + [6] >= [1] X + [1] Y + [6] = gcd#(minus(Y,X),s(X)) if#(false(),s(X),s(Y)) = [1] X + [1] Y + [6] >= [1] X + [1] Y + [2] = minus#(Y,X) if#(true(),s(X),s(Y)) = [1] X + [1] Y + [6] >= [1] X + [1] Y + [6] = gcd#(minus(X,Y),s(Y)) if#(true(),s(X),s(Y)) = [1] X + [1] Y + [6] >= [1] X + [1] Y + [2] = minus#(X,Y) minus#(X,s(Y)) = [1] X + [1] Y + [3] >= [1] X + [1] Y + [7] = c_10(minus#(X,Y)) le(0(),Y) = [1] >= [0] = true() le(s(X),0()) = [1] >= [0] = false() le(s(X),s(Y)) = [1] >= [1] = le(X,Y) minus(X,0()) = [1] X + [0] >= [1] X + [0] = X minus(X,s(Y)) = [1] X + [0] >= [1] X + [0] = pred(minus(X,Y)) pred(s(X)) = [1] X + [1] >= [1] X + [0] = X Further, it can be verified that all rules not oriented are covered by the weightgap condition. *** Step 1.b:6.b:2: NaturalPI WORST_CASE(?,O(n^1)) + Considered Problem: - Strict DPs: minus#(X,s(Y)) -> c_10(minus#(X,Y)) - Weak DPs: gcd#(s(X),s(Y)) -> if#(le(Y,X),s(X),s(Y)) gcd#(s(X),s(Y)) -> le#(Y,X) if#(false(),s(X),s(Y)) -> gcd#(minus(Y,X),s(X)) if#(false(),s(X),s(Y)) -> minus#(Y,X) if#(true(),s(X),s(Y)) -> gcd#(minus(X,Y),s(Y)) if#(true(),s(X),s(Y)) -> minus#(X,Y) le#(s(X),s(Y)) -> c_8(le#(X,Y)) - Weak TRS: le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) minus(X,0()) -> X minus(X,s(Y)) -> pred(minus(X,Y)) pred(s(X)) -> X - Signature: {gcd/2,if/3,le/2,minus/2,pred/1,gcd#/2,if#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0 ,c_3/2,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if#,le#,minus#,pred#} and constructors {0,false,s ,true} + Applied Processor: NaturalPI {shape = Linear, restrict = Restrict, uargs = UArgs, urules = URules, selector = Just any strict-rules} + Details: We apply a polynomial interpretation of kind constructor-based(linear): The following argument positions are considered usable: uargs(c_8) = {1}, uargs(c_10) = {1} Following symbols are considered usable: {minus,pred,gcd#,if#,le#,minus#,pred#} TcT has computed the following interpretation: p(0) = 2 p(false) = 7 p(gcd) = 4 p(if) = 2 + x2 p(le) = 2 + x1 + 5*x2 p(minus) = 1 + x1 p(pred) = x1 p(s) = 1 + x1 p(true) = 0 p(gcd#) = 1 + 4*x1 + 4*x2 p(if#) = 1 + 4*x2 + 4*x3 p(le#) = x2 p(minus#) = 2*x1 + 2*x2 p(pred#) = 4 + 2*x1 p(c_1) = 0 p(c_2) = 2 p(c_3) = 1 + x2 p(c_4) = 1 + x1 + x2 p(c_5) = 1 + x2 p(c_6) = 0 p(c_7) = 4 p(c_8) = x1 p(c_9) = 1 p(c_10) = x1 p(c_11) = 2 Following rules are strictly oriented: minus#(X,s(Y)) = 2 + 2*X + 2*Y > 2*X + 2*Y = c_10(minus#(X,Y)) Following rules are (at-least) weakly oriented: gcd#(s(X),s(Y)) = 9 + 4*X + 4*Y >= 9 + 4*X + 4*Y = if#(le(Y,X),s(X),s(Y)) gcd#(s(X),s(Y)) = 9 + 4*X + 4*Y >= X = le#(Y,X) if#(false(),s(X),s(Y)) = 9 + 4*X + 4*Y >= 9 + 4*X + 4*Y = gcd#(minus(Y,X),s(X)) if#(false(),s(X),s(Y)) = 9 + 4*X + 4*Y >= 2*X + 2*Y = minus#(Y,X) if#(true(),s(X),s(Y)) = 9 + 4*X + 4*Y >= 9 + 4*X + 4*Y = gcd#(minus(X,Y),s(Y)) if#(true(),s(X),s(Y)) = 9 + 4*X + 4*Y >= 2*X + 2*Y = minus#(X,Y) le#(s(X),s(Y)) = 1 + Y >= Y = c_8(le#(X,Y)) minus(X,0()) = 1 + X >= X = X minus(X,s(Y)) = 1 + X >= 1 + X = pred(minus(X,Y)) pred(s(X)) = 1 + X >= X = X *** Step 1.b:6.b:3: EmptyProcessor WORST_CASE(?,O(1)) + Considered Problem: - Weak DPs: gcd#(s(X),s(Y)) -> if#(le(Y,X),s(X),s(Y)) gcd#(s(X),s(Y)) -> le#(Y,X) if#(false(),s(X),s(Y)) -> gcd#(minus(Y,X),s(X)) if#(false(),s(X),s(Y)) -> minus#(Y,X) if#(true(),s(X),s(Y)) -> gcd#(minus(X,Y),s(Y)) if#(true(),s(X),s(Y)) -> minus#(X,Y) le#(s(X),s(Y)) -> c_8(le#(X,Y)) minus#(X,s(Y)) -> c_10(minus#(X,Y)) - Weak TRS: le(0(),Y) -> true() le(s(X),0()) -> false() le(s(X),s(Y)) -> le(X,Y) minus(X,0()) -> X minus(X,s(Y)) -> pred(minus(X,Y)) pred(s(X)) -> X - Signature: {gcd/2,if/3,le/2,minus/2,pred/1,gcd#/2,if#/3,le#/2,minus#/2,pred#/1} / {0/0,false/0,s/1,true/0,c_1/0,c_2/0 ,c_3/2,c_4/2,c_5/2,c_6/0,c_7/0,c_8/1,c_9/0,c_10/1,c_11/0} - Obligation: innermost runtime complexity wrt. defined symbols {gcd#,if#,le#,minus#,pred#} and constructors {0,false,s ,true} + Applied Processor: EmptyProcessor + Details: The problem is already closed. The intended complexity is O(1). WORST_CASE(Omega(n^1),O(n^2))