YES Problem: a__and(true(),X) -> mark(X) a__and(false(),Y) -> false() a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(true()) -> true() mark(false()) -> false() mark(0()) -> 0() mark(s(X)) -> s(X) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(X1,X2) a__and(X1,X2) -> and(X1,X2) a__if(X1,X2,X3) -> if(X1,X2,X3) a__add(X1,X2) -> add(X1,X2) a__first(X1,X2) -> first(X1,X2) a__from(X) -> from(X) Proof: DP Processor: DPs: a__and#(true(),X) -> mark#(X) a__if#(true(),X,Y) -> mark#(X) a__if#(false(),X,Y) -> mark#(Y) a__add#(0(),X) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(if(X1,X2,X3)) -> mark#(X1) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(add(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> a__add#(mark(X1),X2) mark#(first(X1,X2)) -> mark#(X2) mark#(first(X1,X2)) -> mark#(X1) mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) mark#(from(X)) -> a__from#(X) TRS: a__and(true(),X) -> mark(X) a__and(false(),Y) -> false() a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(true()) -> true() mark(false()) -> false() mark(0()) -> 0() mark(s(X)) -> s(X) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(X1,X2) a__and(X1,X2) -> and(X1,X2) a__if(X1,X2,X3) -> if(X1,X2,X3) a__add(X1,X2) -> add(X1,X2) a__first(X1,X2) -> first(X1,X2) a__from(X) -> from(X) TDG Processor: DPs: a__and#(true(),X) -> mark#(X) a__if#(true(),X,Y) -> mark#(X) a__if#(false(),X,Y) -> mark#(Y) a__add#(0(),X) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(if(X1,X2,X3)) -> mark#(X1) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(add(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> a__add#(mark(X1),X2) mark#(first(X1,X2)) -> mark#(X2) mark#(first(X1,X2)) -> mark#(X1) mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) mark#(from(X)) -> a__from#(X) TRS: a__and(true(),X) -> mark(X) a__and(false(),Y) -> false() a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(true()) -> true() mark(false()) -> false() mark(0()) -> 0() mark(s(X)) -> s(X) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(X1,X2) a__and(X1,X2) -> and(X1,X2) a__if(X1,X2,X3) -> if(X1,X2,X3) a__add(X1,X2) -> add(X1,X2) a__first(X1,X2) -> first(X1,X2) a__from(X) -> from(X) graph: a__add#(0(),X) -> mark#(X) -> mark#(from(X)) -> a__from#(X) a__add#(0(),X) -> mark#(X) -> mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) a__add#(0(),X) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X1) a__add#(0(),X) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X2) a__add#(0(),X) -> mark#(X) -> mark#(add(X1,X2)) -> a__add#(mark(X1),X2) a__add#(0(),X) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X1) a__add#(0(),X) -> mark#(X) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__add#(0(),X) -> mark#(X) -> mark#(if(X1,X2,X3)) -> mark#(X1) a__add#(0(),X) -> mark#(X) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__add#(0(),X) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) a__if#(false(),X,Y) -> mark#(Y) -> mark#(from(X)) -> a__from#(X) a__if#(false(),X,Y) -> mark#(Y) -> mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) a__if#(false(),X,Y) -> mark#(Y) -> mark#(first(X1,X2)) -> mark#(X1) a__if#(false(),X,Y) -> mark#(Y) -> mark#(first(X1,X2)) -> mark#(X2) a__if#(false(),X,Y) -> mark#(Y) -> mark#(add(X1,X2)) -> a__add#(mark(X1),X2) a__if#(false(),X,Y) -> mark#(Y) -> mark#(add(X1,X2)) -> mark#(X1) a__if#(false(),X,Y) -> mark#(Y) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__if#(false(),X,Y) -> mark#(Y) -> mark#(if(X1,X2,X3)) -> mark#(X1) a__if#(false(),X,Y) -> mark#(Y) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__if#(false(),X,Y) -> mark#(Y) -> mark#(and(X1,X2)) -> mark#(X1) a__if#(true(),X,Y) -> mark#(X) -> mark#(from(X)) -> a__from#(X) a__if#(true(),X,Y) -> mark#(X) -> mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) a__if#(true(),X,Y) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X1) a__if#(true(),X,Y) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X2) a__if#(true(),X,Y) -> mark#(X) -> mark#(add(X1,X2)) -> a__add#(mark(X1),X2) a__if#(true(),X,Y) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X1) a__if#(true(),X,Y) -> mark#(X) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__if#(true(),X,Y) -> mark#(X) -> mark#(if(X1,X2,X3)) -> mark#(X1) a__if#(true(),X,Y) -> mark#(X) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__if#(true(),X,Y) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) -> a__if#(false(),X,Y) -> mark#(Y) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) -> a__if#(true(),X,Y) -> mark#(X) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(from(X)) -> a__from#(X) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(first(X1,X2)) -> mark#(X1) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(first(X1,X2)) -> mark#(X2) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(add(X1,X2)) -> a__add#(mark(X1),X2) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(add(X1,X2)) -> mark#(X1) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(if(X1,X2,X3)) -> mark#(X1) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(if(X1,X2,X3)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> a__from#(X) mark#(and(X1,X2)) -> mark#(X1) -> mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) mark#(and(X1,X2)) -> mark#(X1) -> mark#(first(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(first(X1,X2)) -> mark#(X2) mark#(and(X1,X2)) -> mark#(X1) -> mark#(add(X1,X2)) -> a__add#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) -> mark#(add(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(and(X1,X2)) -> mark#(X1) -> mark#(if(X1,X2,X3)) -> mark#(X1) mark#(and(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(and(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) -> a__and#(true(),X) -> mark#(X) mark#(first(X1,X2)) -> mark#(X2) -> mark#(from(X)) -> a__from#(X) mark#(first(X1,X2)) -> mark#(X2) -> mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) mark#(first(X1,X2)) -> mark#(X2) -> mark#(first(X1,X2)) -> mark#(X1) mark#(first(X1,X2)) -> mark#(X2) -> mark#(first(X1,X2)) -> mark#(X2) mark#(first(X1,X2)) -> mark#(X2) -> mark#(add(X1,X2)) -> a__add#(mark(X1),X2) mark#(first(X1,X2)) -> mark#(X2) -> mark#(add(X1,X2)) -> mark#(X1) mark#(first(X1,X2)) -> mark#(X2) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(first(X1,X2)) -> mark#(X2) -> mark#(if(X1,X2,X3)) -> mark#(X1) mark#(first(X1,X2)) -> mark#(X2) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(first(X1,X2)) -> mark#(X2) -> mark#(and(X1,X2)) -> mark#(X1) mark#(first(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> a__from#(X) mark#(first(X1,X2)) -> mark#(X1) -> mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) mark#(first(X1,X2)) -> mark#(X1) -> mark#(first(X1,X2)) -> mark#(X1) mark#(first(X1,X2)) -> mark#(X1) -> mark#(first(X1,X2)) -> mark#(X2) mark#(first(X1,X2)) -> mark#(X1) -> mark#(add(X1,X2)) -> a__add#(mark(X1),X2) mark#(first(X1,X2)) -> mark#(X1) -> mark#(add(X1,X2)) -> mark#(X1) mark#(first(X1,X2)) -> mark#(X1) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(first(X1,X2)) -> mark#(X1) -> mark#(if(X1,X2,X3)) -> mark#(X1) mark#(first(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(first(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> a__add#(mark(X1),X2) -> a__add#(0(),X) -> mark#(X) mark#(add(X1,X2)) -> mark#(X1) -> mark#(from(X)) -> a__from#(X) mark#(add(X1,X2)) -> mark#(X1) -> mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) mark#(add(X1,X2)) -> mark#(X1) -> mark#(first(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> mark#(X1) -> mark#(first(X1,X2)) -> mark#(X2) mark#(add(X1,X2)) -> mark#(X1) -> mark#(add(X1,X2)) -> a__add#(mark(X1),X2) mark#(add(X1,X2)) -> mark#(X1) -> mark#(add(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> mark#(X1) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) mark#(add(X1,X2)) -> mark#(X1) -> mark#(if(X1,X2,X3)) -> mark#(X1) mark#(add(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) mark#(add(X1,X2)) -> mark#(X1) -> mark#(and(X1,X2)) -> mark#(X1) a__and#(true(),X) -> mark#(X) -> mark#(from(X)) -> a__from#(X) a__and#(true(),X) -> mark#(X) -> mark#(first(X1,X2)) -> a__first#(mark(X1),mark(X2)) a__and#(true(),X) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X1) a__and#(true(),X) -> mark#(X) -> mark#(first(X1,X2)) -> mark#(X2) a__and#(true(),X) -> mark#(X) -> mark#(add(X1,X2)) -> a__add#(mark(X1),X2) a__and#(true(),X) -> mark#(X) -> mark#(add(X1,X2)) -> mark#(X1) a__and#(true(),X) -> mark#(X) -> mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__and#(true(),X) -> mark#(X) -> mark#(if(X1,X2,X3)) -> mark#(X1) a__and#(true(),X) -> mark#(X) -> mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__and#(true(),X) -> mark#(X) -> mark#(and(X1,X2)) -> mark#(X1) SCC Processor: #sccs: 1 #rules: 12 #arcs: 94/196 DPs: a__add#(0(),X) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__and#(true(),X) -> mark#(X) mark#(if(X1,X2,X3)) -> mark#(X1) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__if#(true(),X,Y) -> mark#(X) mark#(add(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> a__add#(mark(X1),X2) mark#(first(X1,X2)) -> mark#(X2) mark#(first(X1,X2)) -> mark#(X1) a__if#(false(),X,Y) -> mark#(Y) TRS: a__and(true(),X) -> mark(X) a__and(false(),Y) -> false() a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(true()) -> true() mark(false()) -> false() mark(0()) -> 0() mark(s(X)) -> s(X) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(X1,X2) a__and(X1,X2) -> and(X1,X2) a__if(X1,X2,X3) -> if(X1,X2,X3) a__add(X1,X2) -> add(X1,X2) a__first(X1,X2) -> first(X1,X2) a__from(X) -> from(X) Arctic Interpretation Processor: dimension: 1 interpretation: [a__add#](x0, x1) = x1 + 0, [a__if#](x0, x1, x2) = 4x1 + x2 + 0, [mark#](x0) = x0 + 0, [a__and#](x0, x1) = 4x1 + 0, [if](x0, x1, x2) = x0 + 4x1 + 1x2, [and](x0, x1) = x0 + 4x1, [from](x0) = x0 + 3, [a__from](x0) = x0 + 3, [first](x0, x1) = x0 + 4x1 + 4, [cons](x0, x1) = x1 + 1, [nil] = 3, [a__first](x0, x1) = x0 + 4x1 + 4, [add](x0, x1) = x0 + 1x1, [s](x0) = x0 + 2, [a__add](x0, x1) = x0 + 1x1, [0] = 0, [a__if](x0, x1, x2) = x0 + 4x1 + 1x2, [false] = 7, [mark](x0) = x0, [a__and](x0, x1) = x0 + 4x1, [true] = 5 orientation: a__add#(0(),X) = X + 0 >= X + 0 = mark#(X) mark#(and(X1,X2)) = X1 + 4X2 + 0 >= X1 + 0 = mark#(X1) mark#(and(X1,X2)) = X1 + 4X2 + 0 >= 4X2 + 0 = a__and#(mark(X1),X2) a__and#(true(),X) = 4X + 0 >= X + 0 = mark#(X) mark#(if(X1,X2,X3)) = X1 + 4X2 + 1X3 + 0 >= X1 + 0 = mark#(X1) mark#(if(X1,X2,X3)) = X1 + 4X2 + 1X3 + 0 >= 4X2 + X3 + 0 = a__if#(mark(X1),X2,X3) a__if#(true(),X,Y) = 4X + Y + 0 >= X + 0 = mark#(X) mark#(add(X1,X2)) = X1 + 1X2 + 0 >= X1 + 0 = mark#(X1) mark#(add(X1,X2)) = X1 + 1X2 + 0 >= X2 + 0 = a__add#(mark(X1),X2) mark#(first(X1,X2)) = X1 + 4X2 + 4 >= X2 + 0 = mark#(X2) mark#(first(X1,X2)) = X1 + 4X2 + 4 >= X1 + 0 = mark#(X1) a__if#(false(),X,Y) = 4X + Y + 0 >= Y + 0 = mark#(Y) a__and(true(),X) = 4X + 5 >= X = mark(X) a__and(false(),Y) = 4Y + 7 >= 7 = false() a__if(true(),X,Y) = 4X + 1Y + 5 >= X = mark(X) a__if(false(),X,Y) = 4X + 1Y + 7 >= Y = mark(Y) a__add(0(),X) = 1X + 0 >= X = mark(X) a__add(s(X),Y) = X + 1Y + 2 >= X + 1Y + 2 = s(add(X,Y)) a__first(0(),X) = 4X + 4 >= 3 = nil() a__first(s(X),cons(Y,Z)) = X + 4Z + 5 >= X + 4Z + 4 = cons(Y,first(X,Z)) a__from(X) = X + 3 >= X + 3 = cons(X,from(s(X))) mark(and(X1,X2)) = X1 + 4X2 >= X1 + 4X2 = a__and(mark(X1),X2) mark(if(X1,X2,X3)) = X1 + 4X2 + 1X3 >= X1 + 4X2 + 1X3 = a__if(mark(X1),X2,X3) mark(add(X1,X2)) = X1 + 1X2 >= X1 + 1X2 = a__add(mark(X1),X2) mark(first(X1,X2)) = X1 + 4X2 + 4 >= X1 + 4X2 + 4 = a__first(mark(X1),mark(X2)) mark(from(X)) = X + 3 >= X + 3 = a__from(X) mark(true()) = 5 >= 5 = true() mark(false()) = 7 >= 7 = false() mark(0()) = 0 >= 0 = 0() mark(s(X)) = X + 2 >= X + 2 = s(X) mark(nil()) = 3 >= 3 = nil() mark(cons(X1,X2)) = X2 + 1 >= X2 + 1 = cons(X1,X2) a__and(X1,X2) = X1 + 4X2 >= X1 + 4X2 = and(X1,X2) a__if(X1,X2,X3) = X1 + 4X2 + 1X3 >= X1 + 4X2 + 1X3 = if(X1,X2,X3) a__add(X1,X2) = X1 + 1X2 >= X1 + 1X2 = add(X1,X2) a__first(X1,X2) = X1 + 4X2 + 4 >= X1 + 4X2 + 4 = first(X1,X2) a__from(X) = X + 3 >= X + 3 = from(X) problem: DPs: a__add#(0(),X) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__and#(true(),X) -> mark#(X) mark#(if(X1,X2,X3)) -> mark#(X1) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__if#(true(),X,Y) -> mark#(X) mark#(add(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> a__add#(mark(X1),X2) mark#(first(X1,X2)) -> mark#(X1) a__if#(false(),X,Y) -> mark#(Y) TRS: a__and(true(),X) -> mark(X) a__and(false(),Y) -> false() a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(true()) -> true() mark(false()) -> false() mark(0()) -> 0() mark(s(X)) -> s(X) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(X1,X2) a__and(X1,X2) -> and(X1,X2) a__if(X1,X2,X3) -> if(X1,X2,X3) a__add(X1,X2) -> add(X1,X2) a__first(X1,X2) -> first(X1,X2) a__from(X) -> from(X) Arctic Interpretation Processor: dimension: 1 interpretation: [a__add#](x0, x1) = x0 + x1 + 3, [a__if#](x0, x1, x2) = x0 + x1 + x2 + 7, [mark#](x0) = x0, [a__and#](x0, x1) = x0 + x1 + 0, [if](x0, x1, x2) = x0 + x1 + x2 + 7, [and](x0, x1) = x0 + x1 + 0, [from](x0) = x0 + 0, [a__from](x0) = x0 + 0, [first](x0, x1) = 7x0 + x1 + 0, [cons](x0, x1) = x1, [nil] = 0, [a__first](x0, x1) = 7x0 + x1 + 0, [add](x0, x1) = x0 + x1 + 3, [s](x0) = x0, [a__add](x0, x1) = x0 + x1 + 3, [0] = 0, [a__if](x0, x1, x2) = x0 + x1 + x2 + 7, [false] = 0, [mark](x0) = x0, [a__and](x0, x1) = x0 + x1 + 0, [true] = 0 orientation: a__add#(0(),X) = X + 3 >= X = mark#(X) mark#(and(X1,X2)) = X1 + X2 + 0 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__and#(mark(X1),X2) a__and#(true(),X) = X + 0 >= X = mark#(X) mark#(if(X1,X2,X3)) = X1 + X2 + X3 + 7 >= X1 = mark#(X1) mark#(if(X1,X2,X3)) = X1 + X2 + X3 + 7 >= X1 + X2 + X3 + 7 = a__if#(mark(X1),X2,X3) a__if#(true(),X,Y) = X + Y + 7 >= X = mark#(X) mark#(add(X1,X2)) = X1 + X2 + 3 >= X1 = mark#(X1) mark#(add(X1,X2)) = X1 + X2 + 3 >= X1 + X2 + 3 = a__add#(mark(X1),X2) mark#(first(X1,X2)) = 7X1 + X2 + 0 >= X1 = mark#(X1) a__if#(false(),X,Y) = X + Y + 7 >= Y = mark#(Y) a__and(true(),X) = X + 0 >= X = mark(X) a__and(false(),Y) = Y + 0 >= 0 = false() a__if(true(),X,Y) = X + Y + 7 >= X = mark(X) a__if(false(),X,Y) = X + Y + 7 >= Y = mark(Y) a__add(0(),X) = X + 3 >= X = mark(X) a__add(s(X),Y) = X + Y + 3 >= X + Y + 3 = s(add(X,Y)) a__first(0(),X) = X + 7 >= 0 = nil() a__first(s(X),cons(Y,Z)) = 7X + Z + 0 >= 7X + Z + 0 = cons(Y,first(X,Z)) a__from(X) = X + 0 >= X + 0 = cons(X,from(s(X))) mark(and(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__and(mark(X1),X2) mark(if(X1,X2,X3)) = X1 + X2 + X3 + 7 >= X1 + X2 + X3 + 7 = a__if(mark(X1),X2,X3) mark(add(X1,X2)) = X1 + X2 + 3 >= X1 + X2 + 3 = a__add(mark(X1),X2) mark(first(X1,X2)) = 7X1 + X2 + 0 >= 7X1 + X2 + 0 = a__first(mark(X1),mark(X2)) mark(from(X)) = X + 0 >= X + 0 = a__from(X) mark(true()) = 0 >= 0 = true() mark(false()) = 0 >= 0 = false() mark(0()) = 0 >= 0 = 0() mark(s(X)) = X >= X = s(X) mark(nil()) = 0 >= 0 = nil() mark(cons(X1,X2)) = X2 >= X2 = cons(X1,X2) a__and(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = and(X1,X2) a__if(X1,X2,X3) = X1 + X2 + X3 + 7 >= X1 + X2 + X3 + 7 = if(X1,X2,X3) a__add(X1,X2) = X1 + X2 + 3 >= X1 + X2 + 3 = add(X1,X2) a__first(X1,X2) = 7X1 + X2 + 0 >= 7X1 + X2 + 0 = first(X1,X2) a__from(X) = X + 0 >= X + 0 = from(X) problem: DPs: a__add#(0(),X) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__and#(true(),X) -> mark#(X) mark#(if(X1,X2,X3)) -> mark#(X1) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__if#(true(),X,Y) -> mark#(X) mark#(add(X1,X2)) -> mark#(X1) mark#(add(X1,X2)) -> a__add#(mark(X1),X2) a__if#(false(),X,Y) -> mark#(Y) TRS: a__and(true(),X) -> mark(X) a__and(false(),Y) -> false() a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(true()) -> true() mark(false()) -> false() mark(0()) -> 0() mark(s(X)) -> s(X) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(X1,X2) a__and(X1,X2) -> and(X1,X2) a__if(X1,X2,X3) -> if(X1,X2,X3) a__add(X1,X2) -> add(X1,X2) a__first(X1,X2) -> first(X1,X2) a__from(X) -> from(X) Arctic Interpretation Processor: dimension: 1 interpretation: [a__add#](x0, x1) = x1 + 0, [a__if#](x0, x1, x2) = x0 + x1 + x2, [mark#](x0) = x0, [a__and#](x0, x1) = x1, [if](x0, x1, x2) = x0 + x1 + x2, [and](x0, x1) = x0 + x1, [from](x0) = x0, [a__from](x0) = x0, [first](x0, x1) = x0 + 1x1, [cons](x0, x1) = x0 + x1, [nil] = 1, [a__first](x0, x1) = x0 + 1x1, [add](x0, x1) = 5x0 + x1 + 0, [s](x0) = x0, [a__add](x0, x1) = 5x0 + x1 + 0, [0] = 1, [a__if](x0, x1, x2) = x0 + x1 + x2, [false] = 0, [mark](x0) = x0, [a__and](x0, x1) = x0 + x1, [true] = 0 orientation: a__add#(0(),X) = X + 0 >= X = mark#(X) mark#(and(X1,X2)) = X1 + X2 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 >= X2 = a__and#(mark(X1),X2) a__and#(true(),X) = X >= X = mark#(X) mark#(if(X1,X2,X3)) = X1 + X2 + X3 >= X1 = mark#(X1) mark#(if(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = a__if#(mark(X1),X2,X3) a__if#(true(),X,Y) = X + Y + 0 >= X = mark#(X) mark#(add(X1,X2)) = 5X1 + X2 + 0 >= X1 = mark#(X1) mark#(add(X1,X2)) = 5X1 + X2 + 0 >= X2 + 0 = a__add#(mark(X1),X2) a__if#(false(),X,Y) = X + Y + 0 >= Y = mark#(Y) a__and(true(),X) = X + 0 >= X = mark(X) a__and(false(),Y) = Y + 0 >= 0 = false() a__if(true(),X,Y) = X + Y + 0 >= X = mark(X) a__if(false(),X,Y) = X + Y + 0 >= Y = mark(Y) a__add(0(),X) = X + 6 >= X = mark(X) a__add(s(X),Y) = 5X + Y + 0 >= 5X + Y + 0 = s(add(X,Y)) a__first(0(),X) = 1X + 1 >= 1 = nil() a__first(s(X),cons(Y,Z)) = X + 1Y + 1Z >= X + Y + 1Z = cons(Y,first(X,Z)) a__from(X) = X >= X = cons(X,from(s(X))) mark(and(X1,X2)) = X1 + X2 >= X1 + X2 = a__and(mark(X1),X2) mark(if(X1,X2,X3)) = X1 + X2 + X3 >= X1 + X2 + X3 = a__if(mark(X1),X2,X3) mark(add(X1,X2)) = 5X1 + X2 + 0 >= 5X1 + X2 + 0 = a__add(mark(X1),X2) mark(first(X1,X2)) = X1 + 1X2 >= X1 + 1X2 = a__first(mark(X1),mark(X2)) mark(from(X)) = X >= X = a__from(X) mark(true()) = 0 >= 0 = true() mark(false()) = 0 >= 0 = false() mark(0()) = 1 >= 1 = 0() mark(s(X)) = X >= X = s(X) mark(nil()) = 1 >= 1 = nil() mark(cons(X1,X2)) = X1 + X2 >= X1 + X2 = cons(X1,X2) a__and(X1,X2) = X1 + X2 >= X1 + X2 = and(X1,X2) a__if(X1,X2,X3) = X1 + X2 + X3 >= X1 + X2 + X3 = if(X1,X2,X3) a__add(X1,X2) = 5X1 + X2 + 0 >= 5X1 + X2 + 0 = add(X1,X2) a__first(X1,X2) = X1 + 1X2 >= X1 + 1X2 = first(X1,X2) a__from(X) = X >= X = from(X) problem: DPs: a__add#(0(),X) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__and#(true(),X) -> mark#(X) mark#(if(X1,X2,X3)) -> mark#(X1) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__if#(true(),X,Y) -> mark#(X) mark#(add(X1,X2)) -> a__add#(mark(X1),X2) a__if#(false(),X,Y) -> mark#(Y) TRS: a__and(true(),X) -> mark(X) a__and(false(),Y) -> false() a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(true()) -> true() mark(false()) -> false() mark(0()) -> 0() mark(s(X)) -> s(X) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(X1,X2) a__and(X1,X2) -> and(X1,X2) a__if(X1,X2,X3) -> if(X1,X2,X3) a__add(X1,X2) -> add(X1,X2) a__first(X1,X2) -> first(X1,X2) a__from(X) -> from(X) Arctic Interpretation Processor: dimension: 1 interpretation: [a__add#](x0, x1) = x0 + 1x1 + 0, [a__if#](x0, x1, x2) = x0 + 1x1 + 1x2 + 0, [mark#](x0) = 1x0 + 0, [a__and#](x0, x1) = x0 + 1x1 + 0, [if](x0, x1, x2) = x0 + x1 + x2, [and](x0, x1) = x0 + 3x1, [from](x0) = x0, [a__from](x0) = x0, [first](x0, x1) = x0 + x1, [cons](x0, x1) = x0, [nil] = 0, [a__first](x0, x1) = x0 + x1 + 0, [add](x0, x1) = x0 + 1x1 + 0, [s](x0) = x0 + 0, [a__add](x0, x1) = x0 + 1x1 + 0, [0] = 0, [a__if](x0, x1, x2) = x0 + x1 + x2 + 0, [false] = 0, [mark](x0) = x0 + 0, [a__and](x0, x1) = x0 + 3x1 + 0, [true] = 0 orientation: a__add#(0(),X) = 1X + 0 >= 1X + 0 = mark#(X) mark#(and(X1,X2)) = 1X1 + 4X2 + 0 >= 1X1 + 0 = mark#(X1) mark#(and(X1,X2)) = 1X1 + 4X2 + 0 >= X1 + 1X2 + 0 = a__and#(mark(X1),X2) a__and#(true(),X) = 1X + 0 >= 1X + 0 = mark#(X) mark#(if(X1,X2,X3)) = 1X1 + 1X2 + 1X3 + 0 >= 1X1 + 0 = mark#(X1) mark#(if(X1,X2,X3)) = 1X1 + 1X2 + 1X3 + 0 >= X1 + 1X2 + 1X3 + 0 = a__if#(mark(X1),X2,X3) a__if#(true(),X,Y) = 1X + 1Y + 0 >= 1X + 0 = mark#(X) mark#(add(X1,X2)) = 1X1 + 2X2 + 1 >= X1 + 1X2 + 0 = a__add#(mark(X1),X2) a__if#(false(),X,Y) = 1X + 1Y + 0 >= 1Y + 0 = mark#(Y) a__and(true(),X) = 3X + 0 >= X + 0 = mark(X) a__and(false(),Y) = 3Y + 0 >= 0 = false() a__if(true(),X,Y) = X + Y + 0 >= X + 0 = mark(X) a__if(false(),X,Y) = X + Y + 0 >= Y + 0 = mark(Y) a__add(0(),X) = 1X + 0 >= X + 0 = mark(X) a__add(s(X),Y) = X + 1Y + 0 >= X + 1Y + 0 = s(add(X,Y)) a__first(0(),X) = X + 0 >= 0 = nil() a__first(s(X),cons(Y,Z)) = X + Y + 0 >= Y = cons(Y,first(X,Z)) a__from(X) = X >= X = cons(X,from(s(X))) mark(and(X1,X2)) = X1 + 3X2 + 0 >= X1 + 3X2 + 0 = a__and(mark(X1),X2) mark(if(X1,X2,X3)) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 + 0 = a__if(mark(X1),X2,X3) mark(add(X1,X2)) = X1 + 1X2 + 0 >= X1 + 1X2 + 0 = a__add(mark(X1),X2) mark(first(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__first(mark(X1),mark(X2)) mark(from(X)) = X + 0 >= X = a__from(X) mark(true()) = 0 >= 0 = true() mark(false()) = 0 >= 0 = false() mark(0()) = 0 >= 0 = 0() mark(s(X)) = X + 0 >= X + 0 = s(X) mark(nil()) = 0 >= 0 = nil() mark(cons(X1,X2)) = X1 + 0 >= X1 = cons(X1,X2) a__and(X1,X2) = X1 + 3X2 + 0 >= X1 + 3X2 = and(X1,X2) a__if(X1,X2,X3) = X1 + X2 + X3 + 0 >= X1 + X2 + X3 = if(X1,X2,X3) a__add(X1,X2) = X1 + 1X2 + 0 >= X1 + 1X2 + 0 = add(X1,X2) a__first(X1,X2) = X1 + X2 + 0 >= X1 + X2 = first(X1,X2) a__from(X) = X >= X = from(X) problem: DPs: a__add#(0(),X) -> mark#(X) mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__and#(true(),X) -> mark#(X) mark#(if(X1,X2,X3)) -> mark#(X1) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__if#(true(),X,Y) -> mark#(X) a__if#(false(),X,Y) -> mark#(Y) TRS: a__and(true(),X) -> mark(X) a__and(false(),Y) -> false() a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(true()) -> true() mark(false()) -> false() mark(0()) -> 0() mark(s(X)) -> s(X) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(X1,X2) a__and(X1,X2) -> and(X1,X2) a__if(X1,X2,X3) -> if(X1,X2,X3) a__add(X1,X2) -> add(X1,X2) a__first(X1,X2) -> first(X1,X2) a__from(X) -> from(X) SCC Processor: #sccs: 1 #rules: 7 #arcs: 76/64 DPs: mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__and#(true(),X) -> mark#(X) mark#(if(X1,X2,X3)) -> mark#(X1) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__if#(true(),X,Y) -> mark#(X) a__if#(false(),X,Y) -> mark#(Y) TRS: a__and(true(),X) -> mark(X) a__and(false(),Y) -> false() a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(true()) -> true() mark(false()) -> false() mark(0()) -> 0() mark(s(X)) -> s(X) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(X1,X2) a__and(X1,X2) -> and(X1,X2) a__if(X1,X2,X3) -> if(X1,X2,X3) a__add(X1,X2) -> add(X1,X2) a__first(X1,X2) -> first(X1,X2) a__from(X) -> from(X) Arctic Interpretation Processor: dimension: 1 interpretation: [a__if#](x0, x1, x2) = x0 + 7x1 + x2 + 2, [mark#](x0) = x0, [a__and#](x0, x1) = x1 + 0, [if](x0, x1, x2) = x0 + 7x1 + x2 + 2, [and](x0, x1) = x0 + x1 + 0, [from](x0) = x0 + 0, [a__from](x0) = x0 + 0, [first](x0, x1) = 6x0 + x1 + 0, [cons](x0, x1) = x0 + 0, [nil] = 0, [a__first](x0, x1) = 6x0 + x1 + 0, [add](x0, x1) = x0 + x1 + 4, [s](x0) = x0, [a__add](x0, x1) = x0 + x1 + 4, [0] = 0, [a__if](x0, x1, x2) = x0 + 7x1 + x2 + 2, [false] = 0, [mark](x0) = x0, [a__and](x0, x1) = x0 + x1 + 0, [true] = 0 orientation: mark#(and(X1,X2)) = X1 + X2 + 0 >= X1 = mark#(X1) mark#(and(X1,X2)) = X1 + X2 + 0 >= X2 + 0 = a__and#(mark(X1),X2) a__and#(true(),X) = X + 0 >= X = mark#(X) mark#(if(X1,X2,X3)) = X1 + 7X2 + X3 + 2 >= X1 = mark#(X1) mark#(if(X1,X2,X3)) = X1 + 7X2 + X3 + 2 >= X1 + 7X2 + X3 + 2 = a__if#(mark(X1),X2,X3) a__if#(true(),X,Y) = 7X + Y + 2 >= X = mark#(X) a__if#(false(),X,Y) = 7X + Y + 2 >= Y = mark#(Y) a__and(true(),X) = X + 0 >= X = mark(X) a__and(false(),Y) = Y + 0 >= 0 = false() a__if(true(),X,Y) = 7X + Y + 2 >= X = mark(X) a__if(false(),X,Y) = 7X + Y + 2 >= Y = mark(Y) a__add(0(),X) = X + 4 >= X = mark(X) a__add(s(X),Y) = X + Y + 4 >= X + Y + 4 = s(add(X,Y)) a__first(0(),X) = X + 6 >= 0 = nil() a__first(s(X),cons(Y,Z)) = 6X + Y + 0 >= Y + 0 = cons(Y,first(X,Z)) a__from(X) = X + 0 >= X + 0 = cons(X,from(s(X))) mark(and(X1,X2)) = X1 + X2 + 0 >= X1 + X2 + 0 = a__and(mark(X1),X2) mark(if(X1,X2,X3)) = X1 + 7X2 + X3 + 2 >= X1 + 7X2 + X3 + 2 = a__if(mark(X1),X2,X3) mark(add(X1,X2)) = X1 + X2 + 4 >= X1 + X2 + 4 = a__add(mark(X1),X2) mark(first(X1,X2)) = 6X1 + X2 + 0 >= 6X1 + X2 + 0 = a__first(mark(X1),mark(X2)) mark(from(X)) = X + 0 >= X + 0 = a__from(X) mark(true()) = 0 >= 0 = true() mark(false()) = 0 >= 0 = false() mark(0()) = 0 >= 0 = 0() mark(s(X)) = X >= X = s(X) mark(nil()) = 0 >= 0 = nil() mark(cons(X1,X2)) = X1 + 0 >= X1 + 0 = cons(X1,X2) a__and(X1,X2) = X1 + X2 + 0 >= X1 + X2 + 0 = and(X1,X2) a__if(X1,X2,X3) = X1 + 7X2 + X3 + 2 >= X1 + 7X2 + X3 + 2 = if(X1,X2,X3) a__add(X1,X2) = X1 + X2 + 4 >= X1 + X2 + 4 = add(X1,X2) a__first(X1,X2) = 6X1 + X2 + 0 >= 6X1 + X2 + 0 = first(X1,X2) a__from(X) = X + 0 >= X + 0 = from(X) problem: DPs: mark#(and(X1,X2)) -> mark#(X1) mark#(and(X1,X2)) -> a__and#(mark(X1),X2) a__and#(true(),X) -> mark#(X) mark#(if(X1,X2,X3)) -> mark#(X1) mark#(if(X1,X2,X3)) -> a__if#(mark(X1),X2,X3) a__if#(false(),X,Y) -> mark#(Y) TRS: a__and(true(),X) -> mark(X) a__and(false(),Y) -> false() a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(true()) -> true() mark(false()) -> false() mark(0()) -> 0() mark(s(X)) -> s(X) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(X1,X2) a__and(X1,X2) -> and(X1,X2) a__if(X1,X2,X3) -> if(X1,X2,X3) a__add(X1,X2) -> add(X1,X2) a__first(X1,X2) -> first(X1,X2) a__from(X) -> from(X) Subterm Criterion Processor: simple projection: pi(a__and#) = 1 pi(mark#) = 0 pi(a__if#) = 2 problem: DPs: a__and#(true(),X) -> mark#(X) a__if#(false(),X,Y) -> mark#(Y) TRS: a__and(true(),X) -> mark(X) a__and(false(),Y) -> false() a__if(true(),X,Y) -> mark(X) a__if(false(),X,Y) -> mark(Y) a__add(0(),X) -> mark(X) a__add(s(X),Y) -> s(add(X,Y)) a__first(0(),X) -> nil() a__first(s(X),cons(Y,Z)) -> cons(Y,first(X,Z)) a__from(X) -> cons(X,from(s(X))) mark(and(X1,X2)) -> a__and(mark(X1),X2) mark(if(X1,X2,X3)) -> a__if(mark(X1),X2,X3) mark(add(X1,X2)) -> a__add(mark(X1),X2) mark(first(X1,X2)) -> a__first(mark(X1),mark(X2)) mark(from(X)) -> a__from(X) mark(true()) -> true() mark(false()) -> false() mark(0()) -> 0() mark(s(X)) -> s(X) mark(nil()) -> nil() mark(cons(X1,X2)) -> cons(X1,X2) a__and(X1,X2) -> and(X1,X2) a__if(X1,X2,X3) -> if(X1,X2,X3) a__add(X1,X2) -> add(X1,X2) a__first(X1,X2) -> first(X1,X2) a__from(X) -> from(X) SCC Processor: #sccs: 0 #rules: 0 #arcs: 23/4