YES Problem 1: (VAR v_NonEmpty:S L:S N:S V:S V1:S V2:S X:S X1:S X2:S) (RULES a__U11(tt,L:S) -> s(a__length(mark(L:S))) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__and(tt,X:S) -> mark(X:S) a__and(X1:S,X2:S) -> and(X1:S,X2:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__isNatList(V1:S) a__isNat(s(V1:S)) -> a__isNat(V1:S) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__isNatList(V:S) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) a__isNatList(nil) -> tt a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(zeros) -> a__zeros ) (STRATEGY INNERMOST) Problem 1: Dependency Pairs Processor: -> Pairs: A__U11(tt,L:S) -> A__LENGTH(mark(L:S)) A__U11(tt,L:S) -> MARK(L:S) A__AND(tt,X:S) -> MARK(X:S) A__ISNAT(length(V1:S)) -> A__ISNATLIST(V1:S) A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) A__ISNATILIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatIList(V2:S)) A__ISNATILIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) A__ISNATILIST(V:S) -> A__ISNATLIST(V:S) A__ISNATLIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatList(V2:S)) A__ISNATLIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) A__LENGTH(cons(N:S,L:S)) -> A__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) A__LENGTH(cons(N:S,L:S)) -> A__AND(a__isNatList(L:S),isNat(N:S)) A__LENGTH(cons(N:S,L:S)) -> A__ISNATLIST(L:S) MARK(U11(X1:S,X2:S)) -> A__U11(mark(X1:S),X2:S) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> A__ISNAT(X:S) MARK(isNatIList(X:S)) -> A__ISNATILIST(X:S) MARK(isNatList(X:S)) -> A__ISNATLIST(X:S) MARK(length(X:S)) -> A__LENGTH(mark(X:S)) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> A__ZEROS -> Rules: a__U11(tt,L:S) -> s(a__length(mark(L:S))) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__and(tt,X:S) -> mark(X:S) a__and(X1:S,X2:S) -> and(X1:S,X2:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__isNatList(V1:S) a__isNat(s(V1:S)) -> a__isNat(V1:S) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__isNatList(V:S) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) a__isNatList(nil) -> tt a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(zeros) -> a__zeros Problem 1: SCC Processor: -> Pairs: A__U11(tt,L:S) -> A__LENGTH(mark(L:S)) A__U11(tt,L:S) -> MARK(L:S) A__AND(tt,X:S) -> MARK(X:S) A__ISNAT(length(V1:S)) -> A__ISNATLIST(V1:S) A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) A__ISNATILIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatIList(V2:S)) A__ISNATILIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) A__ISNATILIST(V:S) -> A__ISNATLIST(V:S) A__ISNATLIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatList(V2:S)) A__ISNATLIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) A__LENGTH(cons(N:S,L:S)) -> A__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) A__LENGTH(cons(N:S,L:S)) -> A__AND(a__isNatList(L:S),isNat(N:S)) A__LENGTH(cons(N:S,L:S)) -> A__ISNATLIST(L:S) MARK(U11(X1:S,X2:S)) -> A__U11(mark(X1:S),X2:S) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> A__ISNAT(X:S) MARK(isNatIList(X:S)) -> A__ISNATILIST(X:S) MARK(isNatList(X:S)) -> A__ISNATLIST(X:S) MARK(length(X:S)) -> A__LENGTH(mark(X:S)) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) MARK(zeros) -> A__ZEROS -> Rules: a__U11(tt,L:S) -> s(a__length(mark(L:S))) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__and(tt,X:S) -> mark(X:S) a__and(X1:S,X2:S) -> and(X1:S,X2:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__isNatList(V1:S) a__isNat(s(V1:S)) -> a__isNat(V1:S) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__isNatList(V:S) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) a__isNatList(nil) -> tt a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(zeros) -> a__zeros ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__U11(tt,L:S) -> A__LENGTH(mark(L:S)) A__U11(tt,L:S) -> MARK(L:S) A__AND(tt,X:S) -> MARK(X:S) A__ISNAT(length(V1:S)) -> A__ISNATLIST(V1:S) A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) A__ISNATILIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatIList(V2:S)) A__ISNATILIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) A__ISNATILIST(V:S) -> A__ISNATLIST(V:S) A__ISNATLIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatList(V2:S)) A__ISNATLIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) A__LENGTH(cons(N:S,L:S)) -> A__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) A__LENGTH(cons(N:S,L:S)) -> A__AND(a__isNatList(L:S),isNat(N:S)) A__LENGTH(cons(N:S,L:S)) -> A__ISNATLIST(L:S) MARK(U11(X1:S,X2:S)) -> A__U11(mark(X1:S),X2:S) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> A__ISNAT(X:S) MARK(isNatIList(X:S)) -> A__ISNATILIST(X:S) MARK(isNatList(X:S)) -> A__ISNATLIST(X:S) MARK(length(X:S)) -> A__LENGTH(mark(X:S)) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: a__U11(tt,L:S) -> s(a__length(mark(L:S))) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__and(tt,X:S) -> mark(X:S) a__and(X1:S,X2:S) -> and(X1:S,X2:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__isNatList(V1:S) a__isNat(s(V1:S)) -> a__isNat(V1:S) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__isNatList(V:S) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) a__isNatList(nil) -> tt a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(zeros) -> a__zeros Problem 1: Reduction Pairs Processor: -> Pairs: A__U11(tt,L:S) -> A__LENGTH(mark(L:S)) A__U11(tt,L:S) -> MARK(L:S) A__AND(tt,X:S) -> MARK(X:S) A__ISNAT(length(V1:S)) -> A__ISNATLIST(V1:S) A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) A__ISNATILIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatIList(V2:S)) A__ISNATILIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) A__ISNATILIST(V:S) -> A__ISNATLIST(V:S) A__ISNATLIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatList(V2:S)) A__ISNATLIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) A__LENGTH(cons(N:S,L:S)) -> A__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) A__LENGTH(cons(N:S,L:S)) -> A__AND(a__isNatList(L:S),isNat(N:S)) A__LENGTH(cons(N:S,L:S)) -> A__ISNATLIST(L:S) MARK(U11(X1:S,X2:S)) -> A__U11(mark(X1:S),X2:S) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> A__ISNAT(X:S) MARK(isNatIList(X:S)) -> A__ISNATILIST(X:S) MARK(isNatList(X:S)) -> A__ISNATLIST(X:S) MARK(length(X:S)) -> A__LENGTH(mark(X:S)) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) -> Rules: a__U11(tt,L:S) -> s(a__length(mark(L:S))) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__and(tt,X:S) -> mark(X:S) a__and(X1:S,X2:S) -> and(X1:S,X2:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__isNatList(V1:S) a__isNat(s(V1:S)) -> a__isNat(V1:S) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__isNatList(V:S) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) a__isNatList(nil) -> tt a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(zeros) -> a__zeros -> Usable rules: a__U11(tt,L:S) -> s(a__length(mark(L:S))) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__and(tt,X:S) -> mark(X:S) a__and(X1:S,X2:S) -> and(X1:S,X2:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__isNatList(V1:S) a__isNat(s(V1:S)) -> a__isNat(V1:S) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__isNatList(V:S) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) a__isNatList(nil) -> tt a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(zeros) -> a__zeros ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__U11](X1,X2) = X1 + X2 + 2 [a__and](X1,X2) = X1 + X2 [a__isNat](X) = X + 2 [a__isNatIList](X) = 2.X + 2 [a__isNatList](X) = X + 1 [a__length](X) = X + 2 [a__zeros] = 2 [mark](X) = X + 2 [0] = 0 [U11](X1,X2) = X1 + X2 + 2 [and](X1,X2) = X1 + X2 [cons](X1,X2) = X1 + 2.X2 + 2 [fSNonEmpty] = 0 [isNat](X) = X [isNatIList](X) = 2.X + 2 [isNatList](X) = X + 1 [length](X) = X + 2 [nil] = 1 [s](X) = X [tt] = 2 [zeros] = 0 [A__U11](X1,X2) = 2.X1 + 2.X2 + 1 [A__AND](X1,X2) = 2.X2 + 1 [A__ISNAT](X) = X + 1 [A__ISNATILIST](X) = 2.X + 2 [A__ISNATLIST](X) = X + 2 [A__LENGTH](X) = 2.X [A__ZEROS] = 0 [MARK](X) = 2.X + 1 Problem 1: SCC Processor: -> Pairs: A__U11(tt,L:S) -> MARK(L:S) A__AND(tt,X:S) -> MARK(X:S) A__ISNAT(length(V1:S)) -> A__ISNATLIST(V1:S) A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) A__ISNATILIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatIList(V2:S)) A__ISNATILIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) A__ISNATILIST(V:S) -> A__ISNATLIST(V:S) A__ISNATLIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatList(V2:S)) A__ISNATLIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) A__LENGTH(cons(N:S,L:S)) -> A__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) A__LENGTH(cons(N:S,L:S)) -> A__AND(a__isNatList(L:S),isNat(N:S)) A__LENGTH(cons(N:S,L:S)) -> A__ISNATLIST(L:S) MARK(U11(X1:S,X2:S)) -> A__U11(mark(X1:S),X2:S) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> A__ISNAT(X:S) MARK(isNatIList(X:S)) -> A__ISNATILIST(X:S) MARK(isNatList(X:S)) -> A__ISNATLIST(X:S) MARK(length(X:S)) -> A__LENGTH(mark(X:S)) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) -> Rules: a__U11(tt,L:S) -> s(a__length(mark(L:S))) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__and(tt,X:S) -> mark(X:S) a__and(X1:S,X2:S) -> and(X1:S,X2:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__isNatList(V1:S) a__isNat(s(V1:S)) -> a__isNat(V1:S) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__isNatList(V:S) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) a__isNatList(nil) -> tt a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(zeros) -> a__zeros ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__U11(tt,L:S) -> MARK(L:S) A__AND(tt,X:S) -> MARK(X:S) A__ISNAT(length(V1:S)) -> A__ISNATLIST(V1:S) A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) A__ISNATILIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatIList(V2:S)) A__ISNATILIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) A__ISNATILIST(V:S) -> A__ISNATLIST(V:S) A__ISNATLIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatList(V2:S)) A__ISNATLIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) A__LENGTH(cons(N:S,L:S)) -> A__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) A__LENGTH(cons(N:S,L:S)) -> A__AND(a__isNatList(L:S),isNat(N:S)) A__LENGTH(cons(N:S,L:S)) -> A__ISNATLIST(L:S) MARK(U11(X1:S,X2:S)) -> A__U11(mark(X1:S),X2:S) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> A__ISNAT(X:S) MARK(isNatIList(X:S)) -> A__ISNATILIST(X:S) MARK(isNatList(X:S)) -> A__ISNATLIST(X:S) MARK(length(X:S)) -> A__LENGTH(mark(X:S)) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: a__U11(tt,L:S) -> s(a__length(mark(L:S))) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__and(tt,X:S) -> mark(X:S) a__and(X1:S,X2:S) -> and(X1:S,X2:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__isNatList(V1:S) a__isNat(s(V1:S)) -> a__isNat(V1:S) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__isNatList(V:S) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) a__isNatList(nil) -> tt a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(zeros) -> a__zeros Problem 1: Reduction Pairs Processor: -> Pairs: A__U11(tt,L:S) -> MARK(L:S) A__AND(tt,X:S) -> MARK(X:S) A__ISNAT(length(V1:S)) -> A__ISNATLIST(V1:S) A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) A__ISNATILIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatIList(V2:S)) A__ISNATILIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) A__ISNATILIST(V:S) -> A__ISNATLIST(V:S) A__ISNATLIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatList(V2:S)) A__ISNATLIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) A__LENGTH(cons(N:S,L:S)) -> A__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) A__LENGTH(cons(N:S,L:S)) -> A__AND(a__isNatList(L:S),isNat(N:S)) A__LENGTH(cons(N:S,L:S)) -> A__ISNATLIST(L:S) MARK(U11(X1:S,X2:S)) -> A__U11(mark(X1:S),X2:S) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> A__ISNAT(X:S) MARK(isNatIList(X:S)) -> A__ISNATILIST(X:S) MARK(isNatList(X:S)) -> A__ISNATLIST(X:S) MARK(length(X:S)) -> A__LENGTH(mark(X:S)) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) -> Rules: a__U11(tt,L:S) -> s(a__length(mark(L:S))) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__and(tt,X:S) -> mark(X:S) a__and(X1:S,X2:S) -> and(X1:S,X2:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__isNatList(V1:S) a__isNat(s(V1:S)) -> a__isNat(V1:S) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__isNatList(V:S) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) a__isNatList(nil) -> tt a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(zeros) -> a__zeros -> Usable rules: a__U11(tt,L:S) -> s(a__length(mark(L:S))) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__and(tt,X:S) -> mark(X:S) a__and(X1:S,X2:S) -> and(X1:S,X2:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__isNatList(V1:S) a__isNat(s(V1:S)) -> a__isNat(V1:S) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__isNatList(V:S) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) a__isNatList(nil) -> tt a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(zeros) -> a__zeros ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__U11](X1,X2) = X1 + 2.X2 + 2 [a__and](X1,X2) = X1 + 2.X2 [a__isNat](X) = 2 [a__isNatIList](X) = 2.X + 2 [a__isNatList](X) = 2 [a__length](X) = X + 2 [a__zeros] = 2 [mark](X) = 2.X + 2 [0] = 0 [U11](X1,X2) = X1 + X2 + 1 [and](X1,X2) = X1 + X2 [cons](X1,X2) = 2.X1 + 2.X2 + 2 [fSNonEmpty] = 0 [isNat](X) = 0 [isNatIList](X) = 2.X + 2 [isNatList](X) = 0 [length](X) = X + 2 [nil] = 0 [s](X) = X [tt] = 2 [zeros] = 0 [A__U11](X1,X2) = X1 + 2.X2 [A__AND](X1,X2) = 2.X2 + 1 [A__ISNAT](X) = 1 [A__ISNATILIST](X) = 2.X + 2 [A__ISNATLIST](X) = 1 [A__LENGTH](X) = X [A__ZEROS] = 0 [MARK](X) = 2.X + 1 Problem 1: SCC Processor: -> Pairs: A__AND(tt,X:S) -> MARK(X:S) A__ISNAT(length(V1:S)) -> A__ISNATLIST(V1:S) A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) A__ISNATILIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatIList(V2:S)) A__ISNATILIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) A__ISNATILIST(V:S) -> A__ISNATLIST(V:S) A__ISNATLIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatList(V2:S)) A__ISNATLIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) A__LENGTH(cons(N:S,L:S)) -> A__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) A__LENGTH(cons(N:S,L:S)) -> A__AND(a__isNatList(L:S),isNat(N:S)) A__LENGTH(cons(N:S,L:S)) -> A__ISNATLIST(L:S) MARK(U11(X1:S,X2:S)) -> A__U11(mark(X1:S),X2:S) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> A__ISNAT(X:S) MARK(isNatIList(X:S)) -> A__ISNATILIST(X:S) MARK(isNatList(X:S)) -> A__ISNATLIST(X:S) MARK(length(X:S)) -> A__LENGTH(mark(X:S)) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) -> Rules: a__U11(tt,L:S) -> s(a__length(mark(L:S))) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__and(tt,X:S) -> mark(X:S) a__and(X1:S,X2:S) -> and(X1:S,X2:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__isNatList(V1:S) a__isNat(s(V1:S)) -> a__isNat(V1:S) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__isNatList(V:S) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) a__isNatList(nil) -> tt a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(zeros) -> a__zeros ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__AND(tt,X:S) -> MARK(X:S) A__ISNAT(length(V1:S)) -> A__ISNATLIST(V1:S) A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) A__ISNATILIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatIList(V2:S)) A__ISNATILIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) A__ISNATILIST(V:S) -> A__ISNATLIST(V:S) A__ISNATLIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatList(V2:S)) A__ISNATLIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) A__LENGTH(cons(N:S,L:S)) -> A__AND(a__isNatList(L:S),isNat(N:S)) A__LENGTH(cons(N:S,L:S)) -> A__ISNATLIST(L:S) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> A__ISNAT(X:S) MARK(isNatIList(X:S)) -> A__ISNATILIST(X:S) MARK(isNatList(X:S)) -> A__ISNATLIST(X:S) MARK(length(X:S)) -> A__LENGTH(mark(X:S)) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: a__U11(tt,L:S) -> s(a__length(mark(L:S))) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__and(tt,X:S) -> mark(X:S) a__and(X1:S,X2:S) -> and(X1:S,X2:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__isNatList(V1:S) a__isNat(s(V1:S)) -> a__isNat(V1:S) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__isNatList(V:S) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) a__isNatList(nil) -> tt a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(zeros) -> a__zeros Problem 1: Reduction Pairs Processor: -> Pairs: A__AND(tt,X:S) -> MARK(X:S) A__ISNAT(length(V1:S)) -> A__ISNATLIST(V1:S) A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) A__ISNATILIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatIList(V2:S)) A__ISNATILIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) A__ISNATILIST(V:S) -> A__ISNATLIST(V:S) A__ISNATLIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatList(V2:S)) A__ISNATLIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) A__LENGTH(cons(N:S,L:S)) -> A__AND(a__isNatList(L:S),isNat(N:S)) A__LENGTH(cons(N:S,L:S)) -> A__ISNATLIST(L:S) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> A__ISNAT(X:S) MARK(isNatIList(X:S)) -> A__ISNATILIST(X:S) MARK(isNatList(X:S)) -> A__ISNATLIST(X:S) MARK(length(X:S)) -> A__LENGTH(mark(X:S)) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) -> Rules: a__U11(tt,L:S) -> s(a__length(mark(L:S))) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__and(tt,X:S) -> mark(X:S) a__and(X1:S,X2:S) -> and(X1:S,X2:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__isNatList(V1:S) a__isNat(s(V1:S)) -> a__isNat(V1:S) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__isNatList(V:S) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) a__isNatList(nil) -> tt a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(zeros) -> a__zeros -> Usable rules: a__U11(tt,L:S) -> s(a__length(mark(L:S))) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__and(tt,X:S) -> mark(X:S) a__and(X1:S,X2:S) -> and(X1:S,X2:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__isNatList(V1:S) a__isNat(s(V1:S)) -> a__isNat(V1:S) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__isNatList(V:S) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) a__isNatList(nil) -> tt a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(zeros) -> a__zeros ->Interpretation type: Linear ->Coefficients: Natural Numbers ->Dimension: 1 ->Bound: 2 ->Interpretation: [a__U11](X1,X2) = X1 + X2 + 2 [a__and](X1,X2) = X1 + X2 + 1 [a__isNat](X) = X + 2 [a__isNatIList](X) = 2.X + 2 [a__isNatList](X) = X + 1 [a__length](X) = X + 2 [a__zeros] = 2 [mark](X) = X + 2 [0] = 0 [U11](X1,X2) = X1 + X2 + 2 [and](X1,X2) = X1 + X2 + 1 [cons](X1,X2) = X1 + 2.X2 + 2 [fSNonEmpty] = 0 [isNat](X) = X [isNatIList](X) = 2.X + 1 [isNatList](X) = X [length](X) = X + 2 [nil] = 1 [s](X) = X [tt] = 2 [zeros] = 0 [A__U11](X1,X2) = 0 [A__AND](X1,X2) = X1 + 2.X2 + 2 [A__ISNAT](X) = 2.X [A__ISNATILIST](X) = 2.X + 2 [A__ISNATLIST](X) = 2.X + 1 [A__LENGTH](X) = 2.X + 1 [A__ZEROS] = 0 [MARK](X) = 2.X + 2 Problem 1: SCC Processor: -> Pairs: A__ISNAT(length(V1:S)) -> A__ISNATLIST(V1:S) A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) A__ISNATILIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatIList(V2:S)) A__ISNATILIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) A__ISNATILIST(V:S) -> A__ISNATLIST(V:S) A__ISNATLIST(cons(V1:S,V2:S)) -> A__AND(a__isNat(V1:S),isNatList(V2:S)) A__ISNATLIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) A__LENGTH(cons(N:S,L:S)) -> A__AND(a__isNatList(L:S),isNat(N:S)) A__LENGTH(cons(N:S,L:S)) -> A__ISNATLIST(L:S) MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> A__AND(mark(X1:S),X2:S) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(isNat(X:S)) -> A__ISNAT(X:S) MARK(isNatIList(X:S)) -> A__ISNATILIST(X:S) MARK(isNatList(X:S)) -> A__ISNATLIST(X:S) MARK(length(X:S)) -> A__LENGTH(mark(X:S)) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) -> Rules: a__U11(tt,L:S) -> s(a__length(mark(L:S))) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__and(tt,X:S) -> mark(X:S) a__and(X1:S,X2:S) -> and(X1:S,X2:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__isNatList(V1:S) a__isNat(s(V1:S)) -> a__isNat(V1:S) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__isNatList(V:S) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) a__isNatList(nil) -> tt a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(zeros) -> a__zeros ->Strongly Connected Components: ->->Cycle: ->->-> Pairs: A__ISNAT(length(V1:S)) -> A__ISNATLIST(V1:S) A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) A__ISNATLIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) ->->-> Rules: a__U11(tt,L:S) -> s(a__length(mark(L:S))) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__and(tt,X:S) -> mark(X:S) a__and(X1:S,X2:S) -> and(X1:S,X2:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__isNatList(V1:S) a__isNat(s(V1:S)) -> a__isNat(V1:S) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__isNatList(V:S) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) a__isNatList(nil) -> tt a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(zeros) -> a__zeros ->->Cycle: ->->-> Pairs: MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) ->->-> Rules: a__U11(tt,L:S) -> s(a__length(mark(L:S))) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__and(tt,X:S) -> mark(X:S) a__and(X1:S,X2:S) -> and(X1:S,X2:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__isNatList(V1:S) a__isNat(s(V1:S)) -> a__isNat(V1:S) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__isNatList(V:S) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) a__isNatList(nil) -> tt a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(zeros) -> a__zeros The problem is decomposed in 2 subproblems. Problem 1.1: Subterm Processor: -> Pairs: A__ISNAT(length(V1:S)) -> A__ISNATLIST(V1:S) A__ISNAT(s(V1:S)) -> A__ISNAT(V1:S) A__ISNATLIST(cons(V1:S,V2:S)) -> A__ISNAT(V1:S) -> Rules: a__U11(tt,L:S) -> s(a__length(mark(L:S))) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__and(tt,X:S) -> mark(X:S) a__and(X1:S,X2:S) -> and(X1:S,X2:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__isNatList(V1:S) a__isNat(s(V1:S)) -> a__isNat(V1:S) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__isNatList(V:S) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) a__isNatList(nil) -> tt a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(zeros) -> a__zeros ->Projection: pi(A__ISNAT) = 1 pi(A__ISNATLIST) = 1 Problem 1.1: SCC Processor: -> Pairs: Empty -> Rules: a__U11(tt,L:S) -> s(a__length(mark(L:S))) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__and(tt,X:S) -> mark(X:S) a__and(X1:S,X2:S) -> and(X1:S,X2:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__isNatList(V1:S) a__isNat(s(V1:S)) -> a__isNat(V1:S) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__isNatList(V:S) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) a__isNatList(nil) -> tt a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(zeros) -> a__zeros ->Strongly Connected Components: There is no strongly connected component The problem is finite. Problem 1.2: Subterm Processor: -> Pairs: MARK(U11(X1:S,X2:S)) -> MARK(X1:S) MARK(and(X1:S,X2:S)) -> MARK(X1:S) MARK(cons(X1:S,X2:S)) -> MARK(X1:S) MARK(length(X:S)) -> MARK(X:S) MARK(s(X:S)) -> MARK(X:S) -> Rules: a__U11(tt,L:S) -> s(a__length(mark(L:S))) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__and(tt,X:S) -> mark(X:S) a__and(X1:S,X2:S) -> and(X1:S,X2:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__isNatList(V1:S) a__isNat(s(V1:S)) -> a__isNat(V1:S) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__isNatList(V:S) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) a__isNatList(nil) -> tt a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(zeros) -> a__zeros ->Projection: pi(MARK) = 1 Problem 1.2: SCC Processor: -> Pairs: Empty -> Rules: a__U11(tt,L:S) -> s(a__length(mark(L:S))) a__U11(X1:S,X2:S) -> U11(X1:S,X2:S) a__and(tt,X:S) -> mark(X:S) a__and(X1:S,X2:S) -> and(X1:S,X2:S) a__isNat(0) -> tt a__isNat(length(V1:S)) -> a__isNatList(V1:S) a__isNat(s(V1:S)) -> a__isNat(V1:S) a__isNat(X:S) -> isNat(X:S) a__isNatIList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatIList(V2:S)) a__isNatIList(zeros) -> tt a__isNatIList(V:S) -> a__isNatList(V:S) a__isNatIList(X:S) -> isNatIList(X:S) a__isNatList(cons(V1:S,V2:S)) -> a__and(a__isNat(V1:S),isNatList(V2:S)) a__isNatList(nil) -> tt a__isNatList(X:S) -> isNatList(X:S) a__length(cons(N:S,L:S)) -> a__U11(a__and(a__isNatList(L:S),isNat(N:S)),L:S) a__length(nil) -> 0 a__length(X:S) -> length(X:S) a__zeros -> cons(0,zeros) a__zeros -> zeros mark(0) -> 0 mark(U11(X1:S,X2:S)) -> a__U11(mark(X1:S),X2:S) mark(and(X1:S,X2:S)) -> a__and(mark(X1:S),X2:S) mark(cons(X1:S,X2:S)) -> cons(mark(X1:S),X2:S) mark(isNat(X:S)) -> a__isNat(X:S) mark(isNatIList(X:S)) -> a__isNatIList(X:S) mark(isNatList(X:S)) -> a__isNatList(X:S) mark(length(X:S)) -> a__length(mark(X:S)) mark(nil) -> nil mark(s(X:S)) -> s(mark(X:S)) mark(tt) -> tt mark(zeros) -> a__zeros ->Strongly Connected Components: There is no strongly connected component The problem is finite.