YES Problem: active(and(true(),X)) -> mark(X) active(and(false(),Y)) -> mark(false()) active(if(true(),X,Y)) -> mark(X) active(if(false(),X,Y)) -> mark(Y) active(add(0(),X)) -> mark(X) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(first(s(X),cons(Y,Z))) -> mark(cons(Y,first(X,Z))) active(from(X)) -> mark(cons(X,from(s(X)))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(true()) -> active(true()) mark(false()) -> active(false()) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(0()) -> active(0()) mark(s(X)) -> active(s(X)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(from(X)) -> active(from(X)) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) if(active(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) from(mark(X)) -> from(X) from(active(X)) -> from(X) Proof: Matrix Interpretation Processor: dim=3 interpretation: [0] [false] = [1] [1], [1 1 0] [1] [s](x0) = [0 0 0]x0 + [0] [0 0 0] [0], [1 0 0] [1 0 0] [1] [and](x0, x1) = [0 1 0]x0 + [0 1 0]x1 + [1] [0 0 0] [0 1 0] [1], [1 1 0] [mark](x0) = [0 1 0]x0 [0 0 0] , [1 0 0] [1 1 0] [1 0 0] [if](x0, x1, x2) = [0 1 0]x0 + [0 1 0]x1 + [0 1 0]x2 [0 0 0] [0 0 0] [0 1 0] , [0] [nil] = [0] [0], [0] [true] = [1] [0], [1 0 0] [1 0 0] [0] [add](x0, x1) = [0 1 0]x0 + [0 1 0]x1 + [1] [0 0 0] [0 1 0] [1], [1 1 0] [0] [from](x0) = [1 1 0]x0 + [1] [1 1 0] [1], [1 1 0] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [1 0 0] [first](x0, x1) = [0 1 0]x0 + [0 1 0]x1 [0 0 0] [0 0 0] , [0] [0] = [0] [0], [1 0 1] [active](x0) = [0 1 0]x0 [0 0 0] orientation: [1 1 0] [2] [1 1 0] active(and(true(),X)) = [0 1 0]X + [2] >= [0 1 0]X = mark(X) [0 0 0] [0] [0 0 0] [1 1 0] [2] [1] active(and(false(),Y)) = [0 1 0]Y + [2] >= [1] = mark(false()) [0 0 0] [0] [0] [1 1 0] [1 1 0] [0] [1 1 0] active(if(true(),X,Y)) = [0 1 0]X + [0 1 0]Y + [1] >= [0 1 0]X = mark(X) [0 0 0] [0 0 0] [0] [0 0 0] [1 1 0] [1 1 0] [0] [1 1 0] active(if(false(),X,Y)) = [0 1 0]X + [0 1 0]Y + [1] >= [0 1 0]Y = mark(Y) [0 0 0] [0 0 0] [0] [0 0 0] [1 1 0] [1] [1 1 0] active(add(0(),X)) = [0 1 0]X + [1] >= [0 1 0]X = mark(X) [0 0 0] [0] [0 0 0] [1 1 0] [1 1 0] [2] [1 1 0] [1 1 0] [2] active(add(s(X),Y)) = [0 0 0]X + [0 1 0]Y + [1] >= [0 0 0]X + [0 0 0]Y + [0] = mark(s(add(X,Y))) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [0] active(first(0(),X)) = [0 1 0]X >= [0] = mark(nil()) [0 0 0] [0] [1 1 0] [1 1 0] [1 0 0] [1] [1 0 0] [1 1 0] [1 0 0] active(first(s(X),cons(Y,Z))) = [0 0 0]X + [0 0 0]Y + [0 0 0]Z + [0] >= [0 0 0]X + [0 0 0]Y + [0 0 0]Z = mark(cons(Y,first(X,Z))) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [2 2 0] [1] [2 2 0] [1] active(from(X)) = [1 1 0]X + [1] >= [0 0 0]X + [0] = mark(cons(X,from(s(X)))) [0 0 0] [0] [0 0 0] [0] [1 1 0] [1 1 0] [2] [1 1 0] [1 1 0] [2] mark(and(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = active(and(mark(X1),X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1] [0] mark(true()) = [1] >= [1] = active(true()) [0] [0] [1] [1] mark(false()) = [1] >= [1] = active(false()) [0] [0] [1 1 0] [1 2 0] [1 1 0] [1 1 0] [1 1 0] [1 1 0] mark(if(X1,X2,X3)) = [0 1 0]X1 + [0 1 0]X2 + [0 1 0]X3 >= [0 1 0]X1 + [0 1 0]X2 + [0 1 0]X3 = active(if(mark(X1),X2,X3)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] [1] [1 1 0] [1 1 0] [1] mark(add(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = active(add(mark(X1),X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [0] [0] mark(0()) = [0] >= [0] = active(0()) [0] [0] [1 1 0] [1] [1 1 0] [1] mark(s(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = active(s(X)) [0 0 0] [0] [0 0 0] [0] [1 1 0] [1 1 0] [1 1 0] [1 1 0] mark(first(X1,X2)) = [0 1 0]X1 + [0 1 0]X2 >= [0 1 0]X1 + [0 1 0]X2 = active(first(mark(X1),mark(X2))) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] mark(nil()) = [0] >= [0] = active(nil()) [0] [0] [1 1 0] [1 0 0] [1 1 0] [1 0 0] mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = active(cons(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [2 2 0] [1] [2 2 0] [1] mark(from(X)) = [1 1 0]X + [1] >= [1 1 0]X + [1] = active(from(X)) [0 0 0] [0] [0 0 0] [0] [1 1 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] and(mark(X1),X2) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = and(X1,X2) [0 0 0] [0 1 0] [1] [0 0 0] [0 1 0] [1] [1 0 0] [1 1 0] [1] [1 0 0] [1 0 0] [1] and(X1,mark(X2)) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = and(X1,X2) [0 0 0] [0 1 0] [1] [0 0 0] [0 1 0] [1] [1 0 1] [1 0 0] [1] [1 0 0] [1 0 0] [1] and(active(X1),X2) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = and(X1,X2) [0 0 0] [0 1 0] [1] [0 0 0] [0 1 0] [1] [1 0 0] [1 0 1] [1] [1 0 0] [1 0 0] [1] and(X1,active(X2)) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = and(X1,X2) [0 0 0] [0 1 0] [1] [0 0 0] [0 1 0] [1] [1 1 0] [1 1 0] [1 0 0] [1 0 0] [1 1 0] [1 0 0] if(mark(X1),X2,X3) = [0 1 0]X1 + [0 1 0]X2 + [0 1 0]X3 >= [0 1 0]X1 + [0 1 0]X2 + [0 1 0]X3 = if(X1,X2,X3) [0 0 0] [0 0 0] [0 1 0] [0 0 0] [0 0 0] [0 1 0] [1 0 0] [1 2 0] [1 0 0] [1 0 0] [1 1 0] [1 0 0] if(X1,mark(X2),X3) = [0 1 0]X1 + [0 1 0]X2 + [0 1 0]X3 >= [0 1 0]X1 + [0 1 0]X2 + [0 1 0]X3 = if(X1,X2,X3) [0 0 0] [0 0 0] [0 1 0] [0 0 0] [0 0 0] [0 1 0] [1 0 0] [1 1 0] [1 1 0] [1 0 0] [1 1 0] [1 0 0] if(X1,X2,mark(X3)) = [0 1 0]X1 + [0 1 0]X2 + [0 1 0]X3 >= [0 1 0]X1 + [0 1 0]X2 + [0 1 0]X3 = if(X1,X2,X3) [0 0 0] [0 0 0] [0 1 0] [0 0 0] [0 0 0] [0 1 0] [1 0 1] [1 1 0] [1 0 0] [1 0 0] [1 1 0] [1 0 0] if(active(X1),X2,X3) = [0 1 0]X1 + [0 1 0]X2 + [0 1 0]X3 >= [0 1 0]X1 + [0 1 0]X2 + [0 1 0]X3 = if(X1,X2,X3) [0 0 0] [0 0 0] [0 1 0] [0 0 0] [0 0 0] [0 1 0] [1 0 0] [1 1 1] [1 0 0] [1 0 0] [1 1 0] [1 0 0] if(X1,active(X2),X3) = [0 1 0]X1 + [0 1 0]X2 + [0 1 0]X3 >= [0 1 0]X1 + [0 1 0]X2 + [0 1 0]X3 = if(X1,X2,X3) [0 0 0] [0 0 0] [0 1 0] [0 0 0] [0 0 0] [0 1 0] [1 0 0] [1 1 0] [1 0 1] [1 0 0] [1 1 0] [1 0 0] if(X1,X2,active(X3)) = [0 1 0]X1 + [0 1 0]X2 + [0 1 0]X3 >= [0 1 0]X1 + [0 1 0]X2 + [0 1 0]X3 = if(X1,X2,X3) [0 0 0] [0 0 0] [0 1 0] [0 0 0] [0 0 0] [0 1 0] [1 1 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] add(mark(X1),X2) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = add(X1,X2) [0 0 0] [0 1 0] [1] [0 0 0] [0 1 0] [1] [1 0 0] [1 1 0] [0] [1 0 0] [1 0 0] [0] add(X1,mark(X2)) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = add(X1,X2) [0 0 0] [0 1 0] [1] [0 0 0] [0 1 0] [1] [1 0 1] [1 0 0] [0] [1 0 0] [1 0 0] [0] add(active(X1),X2) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = add(X1,X2) [0 0 0] [0 1 0] [1] [0 0 0] [0 1 0] [1] [1 0 0] [1 0 1] [0] [1 0 0] [1 0 0] [0] add(X1,active(X2)) = [0 1 0]X1 + [0 1 0]X2 + [1] >= [0 1 0]X1 + [0 1 0]X2 + [1] = add(X1,X2) [0 0 0] [0 1 0] [1] [0 0 0] [0 1 0] [1] [1 2 0] [1] [1 1 0] [1] s(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = s(X) [0 0 0] [0] [0 0 0] [0] [1 1 1] [1] [1 1 0] [1] s(active(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = s(X) [0 0 0] [0] [0 0 0] [0] [1 1 0] [1 0 0] [1 0 0] [1 0 0] first(mark(X1),X2) = [0 1 0]X1 + [0 1 0]X2 >= [0 1 0]X1 + [0 1 0]X2 = first(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 1 0] [1 0 0] [1 0 0] first(X1,mark(X2)) = [0 1 0]X1 + [0 1 0]X2 >= [0 1 0]X1 + [0 1 0]X2 = first(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 0] [1 0 0] [1 0 0] first(active(X1),X2) = [0 1 0]X1 + [0 1 0]X2 >= [0 1 0]X1 + [0 1 0]X2 = first(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 1] [1 0 0] [1 0 0] first(X1,active(X2)) = [0 1 0]X1 + [0 1 0]X2 >= [0 1 0]X1 + [0 1 0]X2 = first(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 2 0] [1 0 0] [1 1 0] [1 0 0] cons(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] [1 1 0] [1 0 0] cons(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 1] [1 0 0] [1 1 0] [1 0 0] cons(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 0 1] [1 1 0] [1 0 0] cons(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 2 0] [0] [1 1 0] [0] from(mark(X)) = [1 2 0]X + [1] >= [1 1 0]X + [1] = from(X) [1 2 0] [1] [1 1 0] [1] [1 1 1] [0] [1 1 0] [0] from(active(X)) = [1 1 1]X + [1] >= [1 1 0]X + [1] = from(X) [1 1 1] [1] [1 1 0] [1] problem: active(if(true(),X,Y)) -> mark(X) active(if(false(),X,Y)) -> mark(Y) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(from(X)) -> mark(cons(X,from(s(X)))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(false()) -> active(false()) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(0()) -> active(0()) mark(s(X)) -> active(s(X)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(from(X)) -> active(from(X)) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) if(active(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) from(mark(X)) -> from(X) from(active(X)) -> from(X) Matrix Interpretation Processor: dim=3 interpretation: [0] [false] = [0] [0], [1 1 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 1 0] [1 0 0] [1] [and](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1] [0 0 0] [0 0 0] [0], [1 1 0] [mark](x0) = [0 0 0]x0 [0 0 0] , [1 1 0] [1 1 0] [1 0 0] [if](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [1 1 0]x2 [1 0 0] [0 0 0] [0 0 0] , [0] [nil] = [1] [0], [0] [true] = [1] [0], [1 1 0] [1 0 0] [1] [add](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [0 0 0] [0], [1 0 0] [0] [from](x0) = [1 1 0]x0 + [1] [0 0 0] [0], [1 0 0] [1 0 0] [0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1] [0 0 0] [0 0 0] [1], [1 1 0] [1 1 0] [0] [first](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1] [0 0 0] [0 0 0] [0], [0] [0] = [0] [0], [1 1 0] [active](x0) = [0 0 0]x0 [0 0 0] orientation: [1 1 0] [2 1 0] [1] [1 1 0] active(if(true(),X,Y)) = [0 0 0]X + [0 0 0]Y + [0] >= [0 0 0]X = mark(X) [0 0 0] [0 0 0] [0] [0 0 0] [1 1 0] [2 1 0] [1 1 0] active(if(false(),X,Y)) = [0 0 0]X + [0 0 0]Y >= [0 0 0]Y = mark(Y) [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 0 0] [1] [1 1 0] [1 0 0] [1] active(add(s(X),Y)) = [0 0 0]X + [0 0 0]Y + [0] >= [0 0 0]X + [0 0 0]Y + [0] = mark(s(add(X,Y))) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 1 0] [1] [1] active(first(0(),X)) = [0 0 0]X + [0] >= [0] = mark(nil()) [0 0 0] [0] [0] [2 1 0] [1] [2 1 0] [1] active(from(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = mark(cons(X,from(s(X)))) [0 0 0] [0] [0 0 0] [0] [1 1 0] [1 0 0] [2] [1 1 0] [1 0 0] [2] mark(and(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = active(and(mark(X1),X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [0] [0] mark(false()) = [0] >= [0] = active(false()) [0] [0] [1 1 0] [1 1 0] [2 1 0] [1 1 0] [1 1 0] [2 1 0] mark(if(X1,X2,X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = active(if(mark(X1),X2,X3)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 0 0] [1] [1 1 0] [1 0 0] [1] mark(add(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = active(add(mark(X1),X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [0] [0] mark(0()) = [0] >= [0] = active(0()) [0] [0] [1 1 0] [1 1 0] mark(s(X)) = [0 0 0]X >= [0 0 0]X = active(s(X)) [0 0 0] [0 0 0] [1 1 0] [1 1 0] [1] [1 1 0] [1 1 0] [1] mark(first(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = active(first(mark(X1),mark(X2))) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1] [1] mark(nil()) = [0] >= [0] = active(nil()) [0] [0] [1 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = active(cons(X1,X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [2 1 0] [1] [2 1 0] [1] mark(from(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = active(from(X)) [0 0 0] [0] [0 0 0] [0] [1 1 0] [1 0 0] [1] [1 1 0] [1 0 0] [1] and(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = and(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 1 0] [1 1 0] [1] [1 1 0] [1 0 0] [1] and(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = and(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 1 0] [1 0 0] [1] [1 1 0] [1 0 0] [1] and(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = and(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 1 0] [1 1 0] [1] [1 1 0] [1 0 0] [1] and(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = and(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 1 0] [1 1 0] [1 0 0] [1 1 0] [1 1 0] [1 0 0] if(mark(X1),X2,X3) = [0 0 0]X1 + [0 0 0]X2 + [1 1 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [1 1 0]X3 = if(X1,X2,X3) [1 1 0] [0 0 0] [0 0 0] [1 0 0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] [1 0 0] [1 1 0] [1 1 0] [1 0 0] if(X1,mark(X2),X3) = [0 0 0]X1 + [0 0 0]X2 + [1 1 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [1 1 0]X3 = if(X1,X2,X3) [1 0 0] [0 0 0] [0 0 0] [1 0 0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] [1 1 0] [1 1 0] [1 1 0] [1 0 0] if(X1,X2,mark(X3)) = [0 0 0]X1 + [0 0 0]X2 + [1 1 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [1 1 0]X3 = if(X1,X2,X3) [1 0 0] [0 0 0] [0 0 0] [1 0 0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] [1 0 0] [1 1 0] [1 1 0] [1 0 0] if(active(X1),X2,X3) = [0 0 0]X1 + [0 0 0]X2 + [1 1 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [1 1 0]X3 = if(X1,X2,X3) [1 1 0] [0 0 0] [0 0 0] [1 0 0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] [1 0 0] [1 1 0] [1 1 0] [1 0 0] if(X1,active(X2),X3) = [0 0 0]X1 + [0 0 0]X2 + [1 1 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [1 1 0]X3 = if(X1,X2,X3) [1 0 0] [0 0 0] [0 0 0] [1 0 0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] [1 1 0] [1 1 0] [1 1 0] [1 0 0] if(X1,X2,active(X3)) = [0 0 0]X1 + [0 0 0]X2 + [1 1 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [1 1 0]X3 = if(X1,X2,X3) [1 0 0] [0 0 0] [0 0 0] [1 0 0] [0 0 0] [0 0 0] [1 1 0] [1 0 0] [1] [1 1 0] [1 0 0] [1] add(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = add(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 1 0] [1 1 0] [1] [1 1 0] [1 0 0] [1] add(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = add(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 1 0] [1 0 0] [1] [1 1 0] [1 0 0] [1] add(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = add(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 1 0] [1 1 0] [1] [1 1 0] [1 0 0] [1] add(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = add(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 1 0] [1 1 0] s(mark(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 0 0] [0 0 0] [1 1 0] [1 1 0] s(active(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 0 0] [0 0 0] [1 1 0] [1 1 0] [0] [1 1 0] [1 1 0] [0] first(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = first(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 1 0] [1 1 0] [0] [1 1 0] [1 1 0] [0] first(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = first(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 1 0] [1 1 0] [0] [1 1 0] [1 1 0] [0] first(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = first(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 1 0] [1 1 0] [0] [1 1 0] [1 1 0] [0] first(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = first(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 1 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] cons(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = cons(X1,X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [1 1 0] [0] [1 0 0] [1 0 0] [0] cons(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = cons(X1,X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 1 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] cons(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = cons(X1,X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [1 1 0] [0] [1 0 0] [1 0 0] [0] cons(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = cons(X1,X2) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 1 0] [0] [1 0 0] [0] from(mark(X)) = [1 1 0]X + [1] >= [1 1 0]X + [1] = from(X) [0 0 0] [0] [0 0 0] [0] [1 1 0] [0] [1 0 0] [0] from(active(X)) = [1 1 0]X + [1] >= [1 1 0]X + [1] = from(X) [0 0 0] [0] [0 0 0] [0] problem: active(if(false(),X,Y)) -> mark(Y) active(add(s(X),Y)) -> mark(s(add(X,Y))) active(first(0(),X)) -> mark(nil()) active(from(X)) -> mark(cons(X,from(s(X)))) mark(and(X1,X2)) -> active(and(mark(X1),X2)) mark(false()) -> active(false()) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(0()) -> active(0()) mark(s(X)) -> active(s(X)) mark(first(X1,X2)) -> active(first(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(from(X)) -> active(from(X)) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) if(active(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) from(mark(X)) -> from(X) from(active(X)) -> from(X) Matrix Interpretation Processor: dim=3 interpretation: [1] [false] = [1] [1], [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1 0 0] [1] [and](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 1] [0 0 0] [1], [1 0 1] [mark](x0) = [0 0 0]x0 [0 0 1] , [1 0 1] [1 0 0] [1 0 0] [if](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 1]x2 [0 0 1] [0 0 0] [0 0 1] , [0] [nil] = [0] [0], [1 0 0] [1 0 0] [add](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 1] [0 0 0] , [1 0 0] [from](x0) = [1 0 1]x0 [1 0 1] , [1 0 0] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [1 0 0] [1] [first](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 1] [0 0 1] [1], [1] [0] = [0] [1], [1 1 0] [active](x0) = [0 0 0]x0 [0 0 1] orientation: [1 0 0] [1 0 1] [2] [1 0 1] active(if(false(),X,Y)) = [0 0 0]X + [0 0 0]Y + [0] >= [0 0 0]Y = mark(Y) [0 0 0] [0 0 1] [1] [0 0 1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] active(add(s(X),Y)) = [0 0 0]X + [0 0 0]Y >= [0 0 0]X + [0 0 0]Y = mark(s(add(X,Y))) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [2] [0] active(first(0(),X)) = [0 0 0]X + [0] >= [0] = mark(nil()) [0 0 1] [2] [0] [2 0 1] [2 0 0] active(from(X)) = [0 0 0]X >= [0 0 0]X = mark(cons(X,from(s(X)))) [1 0 1] [0 0 0] [1 0 1] [1 0 0] [2] [1 0 1] [1 0 0] [1] mark(and(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = active(and(mark(X1),X2)) [0 0 1] [0 0 0] [1] [0 0 1] [0 0 0] [1] [2] [2] mark(false()) = [0] >= [0] = active(false()) [1] [1] [1 0 2] [1 0 0] [1 0 1] [1 0 2] [1 0 0] [1 0 1] mark(if(X1,X2,X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = active(if(mark(X1),X2,X3)) [0 0 1] [0 0 0] [0 0 1] [0 0 1] [0 0 0] [0 0 1] [1 0 1] [1 0 0] [1 0 1] [1 0 0] mark(add(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = active(add(mark(X1),X2)) [0 0 1] [0 0 0] [0 0 1] [0 0 0] [2] [1] mark(0()) = [0] >= [0] = active(0()) [1] [1] [1 0 0] [1 0 0] mark(s(X)) = [0 0 0]X >= [0 0 0]X = active(s(X)) [0 0 0] [0 0 0] [1 0 1] [1 0 1] [2] [1 0 1] [1 0 1] [1] mark(first(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = active(first(mark(X1),mark(X2))) [0 0 1] [0 0 1] [1] [0 0 1] [0 0 1] [1] [0] [0] mark(nil()) = [0] >= [0] = active(nil()) [0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = active(cons(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [2 0 1] [2 0 1] mark(from(X)) = [0 0 0]X >= [0 0 0]X = active(from(X)) [1 0 1] [1 0 1] [1 0 1] [1 0 0] [1] [1 0 0] [1 0 0] [1] and(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = and(X1,X2) [0 0 1] [0 0 0] [1] [0 0 1] [0 0 0] [1] [1 0 0] [1 0 1] [1] [1 0 0] [1 0 0] [1] and(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = and(X1,X2) [0 0 1] [0 0 0] [1] [0 0 1] [0 0 0] [1] [1 1 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] and(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = and(X1,X2) [0 0 1] [0 0 0] [1] [0 0 1] [0 0 0] [1] [1 0 0] [1 1 0] [1] [1 0 0] [1 0 0] [1] and(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = and(X1,X2) [0 0 1] [0 0 0] [1] [0 0 1] [0 0 0] [1] [1 0 2] [1 0 0] [1 0 0] [1 0 1] [1 0 0] [1 0 0] if(mark(X1),X2,X3) = [0 0 0]X1 + [0 0 0]X2 + [0 0 1]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 1]X3 = if(X1,X2,X3) [0 0 1] [0 0 0] [0 0 1] [0 0 1] [0 0 0] [0 0 1] [1 0 1] [1 0 1] [1 0 0] [1 0 1] [1 0 0] [1 0 0] if(X1,mark(X2),X3) = [0 0 0]X1 + [0 0 0]X2 + [0 0 1]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 1]X3 = if(X1,X2,X3) [0 0 1] [0 0 0] [0 0 1] [0 0 1] [0 0 0] [0 0 1] [1 0 1] [1 0 0] [1 0 1] [1 0 1] [1 0 0] [1 0 0] if(X1,X2,mark(X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 1]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 1]X3 = if(X1,X2,X3) [0 0 1] [0 0 0] [0 0 1] [0 0 1] [0 0 0] [0 0 1] [1 1 1] [1 0 0] [1 0 0] [1 0 1] [1 0 0] [1 0 0] if(active(X1),X2,X3) = [0 0 0]X1 + [0 0 0]X2 + [0 0 1]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 1]X3 = if(X1,X2,X3) [0 0 1] [0 0 0] [0 0 1] [0 0 1] [0 0 0] [0 0 1] [1 0 1] [1 1 0] [1 0 0] [1 0 1] [1 0 0] [1 0 0] if(X1,active(X2),X3) = [0 0 0]X1 + [0 0 0]X2 + [0 0 1]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 1]X3 = if(X1,X2,X3) [0 0 1] [0 0 0] [0 0 1] [0 0 1] [0 0 0] [0 0 1] [1 0 1] [1 0 0] [1 1 0] [1 0 1] [1 0 0] [1 0 0] if(X1,X2,active(X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 1]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 1]X3 = if(X1,X2,X3) [0 0 1] [0 0 0] [0 0 1] [0 0 1] [0 0 0] [0 0 1] [1 0 1] [1 0 0] [1 0 0] [1 0 0] add(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = add(X1,X2) [0 0 1] [0 0 0] [0 0 1] [0 0 0] [1 0 0] [1 0 1] [1 0 0] [1 0 0] add(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = add(X1,X2) [0 0 1] [0 0 0] [0 0 1] [0 0 0] [1 1 0] [1 0 0] [1 0 0] [1 0 0] add(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = add(X1,X2) [0 0 1] [0 0 0] [0 0 1] [0 0 0] [1 0 0] [1 1 0] [1 0 0] [1 0 0] add(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = add(X1,X2) [0 0 1] [0 0 0] [0 0 1] [0 0 0] [1 0 1] [1 0 0] s(mark(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 0 0] [0 0 0] [1 1 0] [1 0 0] s(active(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 0 0] [0 0 0] [1 0 1] [1 0 0] [1] [1 0 0] [1 0 0] [1] first(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = first(X1,X2) [0 0 1] [0 0 1] [1] [0 0 1] [0 0 1] [1] [1 0 0] [1 0 1] [1] [1 0 0] [1 0 0] [1] first(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = first(X1,X2) [0 0 1] [0 0 1] [1] [0 0 1] [0 0 1] [1] [1 1 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] first(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = first(X1,X2) [0 0 1] [0 0 1] [1] [0 0 1] [0 0 1] [1] [1 0 0] [1 1 0] [1] [1 0 0] [1 0 0] [1] first(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = first(X1,X2) [0 0 1] [0 0 1] [1] [0 0 1] [0 0 1] [1] [1 0 1] [1 0 0] [1 0 0] [1 0 0] cons(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 1] [1 0 0] [1 0 0] cons(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 0 0] [1 0 0] [1 0 0] cons(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 1 0] [1 0 0] [1 0 0] cons(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 0] from(mark(X)) = [1 0 2]X >= [1 0 1]X = from(X) [1 0 2] [1 0 1] [1 1 0] [1 0 0] from(active(X)) = [1 1 1]X >= [1 0 1]X = from(X) [1 1 1] [1 0 1] problem: active(add(s(X),Y)) -> mark(s(add(X,Y))) active(from(X)) -> mark(cons(X,from(s(X)))) mark(false()) -> active(false()) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(s(X)) -> active(s(X)) mark(nil()) -> active(nil()) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(from(X)) -> active(from(X)) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(X1,mark(X2),X3) -> if(X1,X2,X3) if(X1,X2,mark(X3)) -> if(X1,X2,X3) if(active(X1),X2,X3) -> if(X1,X2,X3) if(X1,active(X2),X3) -> if(X1,X2,X3) if(X1,X2,active(X3)) -> if(X1,X2,X3) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) cons(mark(X1),X2) -> cons(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) from(mark(X)) -> from(X) from(active(X)) -> from(X) Matrix Interpretation Processor: dim=3 interpretation: [0] [false] = [1] [0], [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1 0 0] [0] [and](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 0 0] [1 1 1] [1], [1 1 0] [0] [mark](x0) = [0 1 0]x0 + [0] [0 0 1] [1], [1 1 0] [1 0 1] [1 1 1] [1] [if](x0, x1, x2) = [0 1 0]x0 + [1 0 0]x1 + [0 0 0]x2 + [0] [0 0 0] [0 0 0] [0 0 0] [0], [0] [nil] = [0] [0], [1 0 0] [1 1 0] [add](x0, x1) = [0 1 0]x0 + [1 0 1]x1 [0 0 0] [0 0 0] , [1 0 1] [1] [from](x0) = [1 1 1]x0 + [1] [1 1 0] [1], [1 1 1] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [1 0 0] [1] [first](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [1 0 0] [0 0 0] [0], [1 0 1] [0] [active](x0) = [0 1 0]x0 + [0] [0 0 0] [1] orientation: [1 0 0] [1 1 0] [0] [1 0 0] [1 1 0] [0] active(add(s(X),Y)) = [0 0 0]X + [1 0 1]Y + [0] >= [0 0 0]X + [0 0 0]Y + [0] = mark(s(add(X,Y))) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [2 1 1] [2] [2 1 1] [1] active(from(X)) = [1 1 1]X + [1] >= [0 0 0]X + [0] = mark(cons(X,from(s(X)))) [0 0 0] [1] [0 0 0] [1] [1] [0] mark(false()) = [1] >= [1] = active(false()) [1] [1] [1 2 0] [2 0 1] [1 1 1] [1] [1 2 0] [1 0 1] [1 1 1] [1] mark(if(X1,X2,X3)) = [0 1 0]X1 + [1 0 0]X2 + [0 0 0]X3 + [0] >= [0 1 0]X1 + [1 0 0]X2 + [0 0 0]X3 + [0] = active(if(mark(X1),X2,X3)) [0 0 0] [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [0 0 0] [1] [1 1 0] [2 1 1] [0] [1 1 0] [1 1 0] [0] mark(add(X1,X2)) = [0 1 0]X1 + [1 0 1]X2 + [0] >= [0 1 0]X1 + [1 0 1]X2 + [0] = active(add(mark(X1),X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [1 0 0] [0] [1 0 0] [0] mark(s(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = active(s(X)) [0 0 0] [1] [0 0 0] [1] [0] [0] mark(nil()) = [0] >= [0] = active(nil()) [1] [1] [1 1 1] [1 0 0] [0] [1 1 1] [1 0 0] [0] mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = active(cons(X1,X2)) [0 0 0] [0 0 0] [1] [0 0 0] [0 0 0] [1] [2 1 2] [2] [2 1 1] [2] mark(from(X)) = [1 1 1]X + [1] >= [1 1 1]X + [1] = active(from(X)) [1 1 0] [2] [0 0 0] [1] [1 1 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] and(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = and(X1,X2) [0 0 0] [1 1 1] [1] [0 0 0] [1 1 1] [1] [1 0 0] [1 1 0] [0] [1 0 0] [1 0 0] [0] and(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = and(X1,X2) [0 0 0] [1 2 1] [2] [0 0 0] [1 1 1] [1] [1 0 1] [1 0 0] [0] [1 0 0] [1 0 0] [0] and(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = and(X1,X2) [0 0 0] [1 1 1] [1] [0 0 0] [1 1 1] [1] [1 0 0] [1 0 1] [0] [1 0 0] [1 0 0] [0] and(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = and(X1,X2) [0 0 0] [1 1 1] [2] [0 0 0] [1 1 1] [1] [1 2 0] [1 0 1] [1 1 1] [1] [1 1 0] [1 0 1] [1 1 1] [1] if(mark(X1),X2,X3) = [0 1 0]X1 + [1 0 0]X2 + [0 0 0]X3 + [0] >= [0 1 0]X1 + [1 0 0]X2 + [0 0 0]X3 + [0] = if(X1,X2,X3) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [1 1 0] [1 1 1] [1 1 1] [2] [1 1 0] [1 0 1] [1 1 1] [1] if(X1,mark(X2),X3) = [0 1 0]X1 + [1 1 0]X2 + [0 0 0]X3 + [0] >= [0 1 0]X1 + [1 0 0]X2 + [0 0 0]X3 + [0] = if(X1,X2,X3) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [1 1 0] [1 0 1] [1 2 1] [2] [1 1 0] [1 0 1] [1 1 1] [1] if(X1,X2,mark(X3)) = [0 1 0]X1 + [1 0 0]X2 + [0 0 0]X3 + [0] >= [0 1 0]X1 + [1 0 0]X2 + [0 0 0]X3 + [0] = if(X1,X2,X3) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [1 1 1] [1 0 1] [1 1 1] [1] [1 1 0] [1 0 1] [1 1 1] [1] if(active(X1),X2,X3) = [0 1 0]X1 + [1 0 0]X2 + [0 0 0]X3 + [0] >= [0 1 0]X1 + [1 0 0]X2 + [0 0 0]X3 + [0] = if(X1,X2,X3) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [1 1 0] [1 0 1] [1 1 1] [2] [1 1 0] [1 0 1] [1 1 1] [1] if(X1,active(X2),X3) = [0 1 0]X1 + [1 0 1]X2 + [0 0 0]X3 + [0] >= [0 1 0]X1 + [1 0 0]X2 + [0 0 0]X3 + [0] = if(X1,X2,X3) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [1 1 0] [1 0 1] [1 1 1] [2] [1 1 0] [1 0 1] [1 1 1] [1] if(X1,X2,active(X3)) = [0 1 0]X1 + [1 0 0]X2 + [0 0 0]X3 + [0] >= [0 1 0]X1 + [1 0 0]X2 + [0 0 0]X3 + [0] = if(X1,X2,X3) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [1 1 0] [1 1 0] [1 0 0] [1 1 0] add(mark(X1),X2) = [0 1 0]X1 + [1 0 1]X2 >= [0 1 0]X1 + [1 0 1]X2 = add(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 2 0] [0] [1 0 0] [1 1 0] add(X1,mark(X2)) = [0 1 0]X1 + [1 1 1]X2 + [1] >= [0 1 0]X1 + [1 0 1]X2 = add(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [1 0 1] [1 1 0] [1 0 0] [1 1 0] add(active(X1),X2) = [0 1 0]X1 + [1 0 1]X2 >= [0 1 0]X1 + [1 0 1]X2 = add(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 1 1] [0] [1 0 0] [1 1 0] add(X1,active(X2)) = [0 1 0]X1 + [1 0 1]X2 + [1] >= [0 1 0]X1 + [1 0 1]X2 = add(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [1 1 0] [1 0 0] s(mark(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 0 0] [0 0 0] [1 0 1] [1 0 0] s(active(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 0 0] [0 0 0] [1 1 0] [1 0 0] [1] [1 0 0] [1 0 0] [1] first(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = first(X1,X2) [1 1 0] [0 0 0] [0] [1 0 0] [0 0 0] [0] [1 0 0] [1 1 0] [1] [1 0 0] [1 0 0] [1] first(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = first(X1,X2) [1 0 0] [0 0 0] [0] [1 0 0] [0 0 0] [0] [1 0 1] [1 0 0] [1] [1 0 0] [1 0 0] [1] first(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = first(X1,X2) [1 0 1] [0 0 0] [0] [1 0 0] [0 0 0] [0] [1 0 0] [1 0 1] [1] [1 0 0] [1 0 0] [1] first(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = first(X1,X2) [1 0 0] [0 0 0] [0] [1 0 0] [0 0 0] [0] [1 2 1] [1 0 0] [1] [1 1 1] [1 0 0] cons(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [1 1 1] [1 1 0] [1 1 1] [1 0 0] cons(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 1] [1 0 0] [1] [1 1 1] [1 0 0] cons(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [1 1 1] [1 0 1] [1 1 1] [1 0 0] cons(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 1] [2] [1 0 1] [1] from(mark(X)) = [1 2 1]X + [2] >= [1 1 1]X + [1] = from(X) [1 2 0] [1] [1 1 0] [1] [1 0 1] [2] [1 0 1] [1] from(active(X)) = [1 1 1]X + [2] >= [1 1 1]X + [1] = from(X) [1 1 1] [1] [1 1 0] [1] problem: active(add(s(X),Y)) -> mark(s(add(X,Y))) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(s(X)) -> active(s(X)) mark(nil()) -> active(nil()) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(from(X)) -> active(from(X)) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(active(X1),X2,X3) -> if(X1,X2,X3) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) Matrix Interpretation Processor: dim=3 interpretation: [1 1 1] [s](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1 0 0] [and](x0, x1) = [1 0 0]x0 + [1 0 0]x1 [0 0 0] [1 0 0] , [1 0 0] [0] [mark](x0) = [0 0 0]x0 + [1] [0 1 1] [0], [1 0 0] [1 0 0] [1 0 0] [if](x0, x1, x2) = [0 0 0]x0 + [0 0 0]x1 + [0 0 0]x2 [0 0 0] [0 0 0] [0 0 0] , [1] [nil] = [0] [0], [1 0 0] [1 0 0] [add](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 1 0] [from](x0) = [0 0 0]x0 [0 0 0] , [1 0 0] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [1 0 0] [1 0 0] [0] [first](x0, x1) = [1 1 1]x0 + [0 0 0]x1 + [0] [0 0 0] [1 0 0] [1], [1 1 0] [0] [active](x0) = [0 0 1]x0 + [1] [0 0 0] [0] orientation: [1 1 1] [1 0 0] [0] [1 0 0] [1 0 0] [0] active(add(s(X),Y)) = [0 0 0]X + [0 0 0]Y + [1] >= [0 0 0]X + [0 0 0]Y + [1] = mark(s(add(X,Y))) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [0] mark(if(X1,X2,X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] = active(if(mark(X1),X2,X3)) [0 0 0] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] mark(add(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = active(add(mark(X1),X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 1 1] [0] [1 1 1] [0] mark(s(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = active(s(X)) [0 0 0] [0] [0 0 0] [0] [1] [1] mark(nil()) = [1] >= [1] = active(nil()) [0] [0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = active(cons(X1,X2)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 1 0] [0] [1 1 0] [0] mark(from(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = active(from(X)) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] and(mark(X1),X2) = [1 0 0]X1 + [1 0 0]X2 >= [1 0 0]X1 + [1 0 0]X2 = and(X1,X2) [0 0 0] [1 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] and(X1,mark(X2)) = [1 0 0]X1 + [1 0 0]X2 >= [1 0 0]X1 + [1 0 0]X2 = and(X1,X2) [0 0 0] [1 0 0] [0 0 0] [1 0 0] [1 1 0] [1 0 0] [1 0 0] [1 0 0] and(active(X1),X2) = [1 1 0]X1 + [1 0 0]X2 >= [1 0 0]X1 + [1 0 0]X2 = and(X1,X2) [0 0 0] [1 0 0] [0 0 0] [1 0 0] [1 0 0] [1 1 0] [1 0 0] [1 0 0] and(X1,active(X2)) = [1 0 0]X1 + [1 1 0]X2 >= [1 0 0]X1 + [1 0 0]X2 = and(X1,X2) [0 0 0] [1 1 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] if(mark(X1),X2,X3) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = if(X1,X2,X3) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] if(active(X1),X2,X3) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 = if(X1,X2,X3) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] add(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = add(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] add(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = add(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 0 0] [1 0 0] [1 0 0] add(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = add(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 1 0] [1 0 0] [1 0 0] add(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = add(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 1] [1] [1 1 1] s(mark(X)) = [0 0 0]X + [0] >= [0 0 0]X = s(X) [0 0 0] [0] [0 0 0] [1 1 1] [1] [1 1 1] s(active(X)) = [0 0 0]X + [0] >= [0 0 0]X = s(X) [0 0 0] [0] [0 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] first(mark(X1),X2) = [1 1 1]X1 + [0 0 0]X2 + [1] >= [1 1 1]X1 + [0 0 0]X2 + [0] = first(X1,X2) [0 0 0] [1 0 0] [1] [0 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] first(X1,mark(X2)) = [1 1 1]X1 + [0 0 0]X2 + [0] >= [1 1 1]X1 + [0 0 0]X2 + [0] = first(X1,X2) [0 0 0] [1 0 0] [1] [0 0 0] [1 0 0] [1] [1 1 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] first(active(X1),X2) = [1 1 1]X1 + [0 0 0]X2 + [1] >= [1 1 1]X1 + [0 0 0]X2 + [0] = first(X1,X2) [0 0 0] [1 0 0] [1] [0 0 0] [1 0 0] [1] [1 0 0] [1 1 0] [0] [1 0 0] [1 0 0] [0] first(X1,active(X2)) = [1 1 1]X1 + [0 0 0]X2 + [0] >= [1 1 1]X1 + [0 0 0]X2 + [0] = first(X1,X2) [0 0 0] [1 1 0] [1] [0 0 0] [1 0 0] [1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] cons(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 1 0] [1 0 0] [1 0 0] cons(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = cons(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] problem: active(add(s(X),Y)) -> mark(s(add(X,Y))) mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(s(X)) -> active(s(X)) mark(nil()) -> active(nil()) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(from(X)) -> active(from(X)) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(active(X1),X2,X3) -> if(X1,X2,X3) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) Matrix Interpretation Processor: dim=3 interpretation: [1 0 0] [0] [s](x0) = [0 0 0]x0 + [1] [0 1 1] [0], [1 0 0] [1 0 0] [and](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [1 0 1] , [1 0 0] [mark](x0) = [0 0 0]x0 [0 1 1] , [1 0 0] [1 0 0] [1 0 1] [0] [if](x0, x1, x2) = [0 1 1]x0 + [0 1 0]x1 + [1 0 0]x2 + [1] [0 0 0] [0 0 0] [0 1 0] [0], [0] [nil] = [0] [0], [1 1 1] [1 0 0] [add](x0, x1) = [0 1 1]x0 + [0 0 0]x1 [0 0 0] [0 1 1] , [1 0 1] [1] [from](x0) = [0 0 1]x0 + [1] [0 1 0] [0], [1 0 0] [1 0 1] [cons](x0, x1) = [0 0 0]x0 + [0 0 1]x1 [1 1 0] [1 0 0] , [1 0 0] [1 1 1] [first](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [1 0 0] , [1 0 0] [active](x0) = [0 0 0]x0 [0 1 1] orientation: [1 1 1] [1 0 0] [1] [1 1 1] [1 0 0] [0] active(add(s(X),Y)) = [0 0 0]X + [0 0 0]Y + [0] >= [0 0 0]X + [0 0 0]Y + [0] = mark(s(add(X,Y))) [0 1 1] [0 1 1] [1] [0 1 1] [0 1 1] [1] [1 0 0] [1 0 0] [1 0 1] [0] [1 0 0] [1 0 0] [1 0 1] [0] mark(if(X1,X2,X3)) = [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [0] = active(if(mark(X1),X2,X3)) [0 1 1] [0 1 0] [1 1 0] [1] [0 1 1] [0 1 0] [1 1 0] [1] [1 1 1] [1 0 0] [1 1 1] [1 0 0] mark(add(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = active(add(mark(X1),X2)) [0 1 1] [0 1 1] [0 1 1] [0 1 1] [1 0 0] [0] [1 0 0] [0] mark(s(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = active(s(X)) [0 1 1] [1] [0 1 1] [1] [0] [0] mark(nil()) = [0] >= [0] = active(nil()) [0] [0] [1 0 0] [1 0 1] [1 0 0] [1 0 1] mark(cons(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = active(cons(X1,X2)) [1 1 0] [1 0 1] [1 1 0] [1 0 1] [1 0 1] [1] [1 0 1] [1] mark(from(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = active(from(X)) [0 1 1] [1] [0 1 1] [1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] and(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = and(X1,X2) [0 0 0] [1 0 1] [0 0 0] [1 0 1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] and(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = and(X1,X2) [0 0 0] [1 1 1] [0 0 0] [1 0 1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] and(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = and(X1,X2) [0 0 0] [1 0 1] [0 0 0] [1 0 1] [1 0 0] [1 0 0] [1 0 0] [1 0 0] and(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = and(X1,X2) [0 0 0] [1 1 1] [0 0 0] [1 0 1] [1 0 0] [1 0 0] [1 0 1] [0] [1 0 0] [1 0 0] [1 0 1] [0] if(mark(X1),X2,X3) = [0 1 1]X1 + [0 1 0]X2 + [1 0 0]X3 + [1] >= [0 1 1]X1 + [0 1 0]X2 + [1 0 0]X3 + [1] = if(X1,X2,X3) [0 0 0] [0 0 0] [0 1 0] [0] [0 0 0] [0 0 0] [0 1 0] [0] [1 0 0] [1 0 0] [1 0 1] [0] [1 0 0] [1 0 0] [1 0 1] [0] if(active(X1),X2,X3) = [0 1 1]X1 + [0 1 0]X2 + [1 0 0]X3 + [1] >= [0 1 1]X1 + [0 1 0]X2 + [1 0 0]X3 + [1] = if(X1,X2,X3) [0 0 0] [0 0 0] [0 1 0] [0] [0 0 0] [0 0 0] [0 1 0] [0] [1 1 1] [1 0 0] [1 1 1] [1 0 0] add(mark(X1),X2) = [0 1 1]X1 + [0 0 0]X2 >= [0 1 1]X1 + [0 0 0]X2 = add(X1,X2) [0 0 0] [0 1 1] [0 0 0] [0 1 1] [1 1 1] [1 0 0] [1 1 1] [1 0 0] add(X1,mark(X2)) = [0 1 1]X1 + [0 0 0]X2 >= [0 1 1]X1 + [0 0 0]X2 = add(X1,X2) [0 0 0] [0 1 1] [0 0 0] [0 1 1] [1 1 1] [1 0 0] [1 1 1] [1 0 0] add(active(X1),X2) = [0 1 1]X1 + [0 0 0]X2 >= [0 1 1]X1 + [0 0 0]X2 = add(X1,X2) [0 0 0] [0 1 1] [0 0 0] [0 1 1] [1 1 1] [1 0 0] [1 1 1] [1 0 0] add(X1,active(X2)) = [0 1 1]X1 + [0 0 0]X2 >= [0 1 1]X1 + [0 0 0]X2 = add(X1,X2) [0 0 0] [0 1 1] [0 0 0] [0 1 1] [1 0 0] [1 1 1] [1 0 0] [1 1 1] first(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = first(X1,X2) [0 0 0] [1 0 0] [0 0 0] [1 0 0] [1 0 0] [1 1 1] [1 0 0] [1 1 1] first(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = first(X1,X2) [0 0 0] [1 0 0] [0 0 0] [1 0 0] [1 0 0] [1 1 1] [1 0 0] [1 1 1] first(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = first(X1,X2) [0 0 0] [1 0 0] [0 0 0] [1 0 0] [1 0 0] [1 1 1] [1 0 0] [1 1 1] first(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = first(X1,X2) [0 0 0] [1 0 0] [0 0 0] [1 0 0] [1 0 0] [1 1 1] [1 0 0] [1 0 1] cons(X1,mark(X2)) = [0 0 0]X1 + [0 1 1]X2 >= [0 0 0]X1 + [0 0 1]X2 = cons(X1,X2) [1 1 0] [1 0 0] [1 1 0] [1 0 0] [1 0 0] [1 1 1] [1 0 0] [1 0 1] cons(X1,active(X2)) = [0 0 0]X1 + [0 1 1]X2 >= [0 0 0]X1 + [0 0 1]X2 = cons(X1,X2) [1 1 0] [1 0 0] [1 1 0] [1 0 0] problem: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(s(X)) -> active(s(X)) mark(nil()) -> active(nil()) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(from(X)) -> active(from(X)) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(active(X1),X2,X3) -> if(X1,X2,X3) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) Matrix Interpretation Processor: dim=1 interpretation: [s](x0) = 2x0 + 4, [and](x0, x1) = 2x0 + 4x1, [mark](x0) = 4x0, [if](x0, x1, x2) = 2x0 + 4x1 + 2x2, [nil] = 0, [add](x0, x1) = x0 + 4x1, [from](x0) = 2x0, [cons](x0, x1) = 2x0 + 4x1 + 2, [first](x0, x1) = x0 + 2x1 + 4, [active](x0) = x0 orientation: mark(if(X1,X2,X3)) = 8X1 + 16X2 + 8X3 >= 8X1 + 4X2 + 2X3 = active(if(mark(X1),X2,X3)) mark(add(X1,X2)) = 4X1 + 16X2 >= 4X1 + 4X2 = active(add(mark(X1),X2)) mark(s(X)) = 8X + 16 >= 2X + 4 = active(s(X)) mark(nil()) = 0 >= 0 = active(nil()) mark(cons(X1,X2)) = 8X1 + 16X2 + 8 >= 2X1 + 4X2 + 2 = active(cons(X1,X2)) mark(from(X)) = 8X >= 2X = active(from(X)) and(mark(X1),X2) = 8X1 + 4X2 >= 2X1 + 4X2 = and(X1,X2) and(X1,mark(X2)) = 2X1 + 16X2 >= 2X1 + 4X2 = and(X1,X2) and(active(X1),X2) = 2X1 + 4X2 >= 2X1 + 4X2 = and(X1,X2) and(X1,active(X2)) = 2X1 + 4X2 >= 2X1 + 4X2 = and(X1,X2) if(mark(X1),X2,X3) = 8X1 + 4X2 + 2X3 >= 2X1 + 4X2 + 2X3 = if(X1,X2,X3) if(active(X1),X2,X3) = 2X1 + 4X2 + 2X3 >= 2X1 + 4X2 + 2X3 = if(X1,X2,X3) add(mark(X1),X2) = 4X1 + 4X2 >= X1 + 4X2 = add(X1,X2) add(X1,mark(X2)) = X1 + 16X2 >= X1 + 4X2 = add(X1,X2) add(active(X1),X2) = X1 + 4X2 >= X1 + 4X2 = add(X1,X2) add(X1,active(X2)) = X1 + 4X2 >= X1 + 4X2 = add(X1,X2) first(mark(X1),X2) = 4X1 + 2X2 + 4 >= X1 + 2X2 + 4 = first(X1,X2) first(X1,mark(X2)) = X1 + 8X2 + 4 >= X1 + 2X2 + 4 = first(X1,X2) first(active(X1),X2) = X1 + 2X2 + 4 >= X1 + 2X2 + 4 = first(X1,X2) first(X1,active(X2)) = X1 + 2X2 + 4 >= X1 + 2X2 + 4 = first(X1,X2) cons(X1,mark(X2)) = 2X1 + 16X2 + 2 >= 2X1 + 4X2 + 2 = cons(X1,X2) cons(X1,active(X2)) = 2X1 + 4X2 + 2 >= 2X1 + 4X2 + 2 = cons(X1,X2) problem: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) mark(add(X1,X2)) -> active(add(mark(X1),X2)) mark(nil()) -> active(nil()) mark(from(X)) -> active(from(X)) and(mark(X1),X2) -> and(X1,X2) and(X1,mark(X2)) -> and(X1,X2) and(active(X1),X2) -> and(X1,X2) and(X1,active(X2)) -> and(X1,X2) if(mark(X1),X2,X3) -> if(X1,X2,X3) if(active(X1),X2,X3) -> if(X1,X2,X3) add(mark(X1),X2) -> add(X1,X2) add(X1,mark(X2)) -> add(X1,X2) add(active(X1),X2) -> add(X1,X2) add(X1,active(X2)) -> add(X1,X2) first(mark(X1),X2) -> first(X1,X2) first(X1,mark(X2)) -> first(X1,X2) first(active(X1),X2) -> first(X1,X2) first(X1,active(X2)) -> first(X1,X2) cons(X1,mark(X2)) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) Matrix Interpretation Processor: dim=1 interpretation: [and](x0, x1) = x0 + 6x1, [mark](x0) = 4x0 + 1, [if](x0, x1, x2) = 2x0 + x1 + 4x2 + 1, [nil] = 1, [add](x0, x1) = 4x0 + 2x1 + 4, [from](x0) = 4x0 + 1, [cons](x0, x1) = 4x0 + 4x1, [first](x0, x1) = x0 + 4x1, [active](x0) = x0 + 2 orientation: mark(if(X1,X2,X3)) = 8X1 + 4X2 + 16X3 + 5 >= 8X1 + X2 + 4X3 + 5 = active(if(mark(X1),X2,X3)) mark(add(X1,X2)) = 16X1 + 8X2 + 17 >= 16X1 + 2X2 + 10 = active(add(mark(X1),X2)) mark(nil()) = 5 >= 3 = active(nil()) mark(from(X)) = 16X + 5 >= 4X + 3 = active(from(X)) and(mark(X1),X2) = 4X1 + 6X2 + 1 >= X1 + 6X2 = and(X1,X2) and(X1,mark(X2)) = X1 + 24X2 + 6 >= X1 + 6X2 = and(X1,X2) and(active(X1),X2) = X1 + 6X2 + 2 >= X1 + 6X2 = and(X1,X2) and(X1,active(X2)) = X1 + 6X2 + 12 >= X1 + 6X2 = and(X1,X2) if(mark(X1),X2,X3) = 8X1 + X2 + 4X3 + 3 >= 2X1 + X2 + 4X3 + 1 = if(X1,X2,X3) if(active(X1),X2,X3) = 2X1 + X2 + 4X3 + 5 >= 2X1 + X2 + 4X3 + 1 = if(X1,X2,X3) add(mark(X1),X2) = 16X1 + 2X2 + 8 >= 4X1 + 2X2 + 4 = add(X1,X2) add(X1,mark(X2)) = 4X1 + 8X2 + 6 >= 4X1 + 2X2 + 4 = add(X1,X2) add(active(X1),X2) = 4X1 + 2X2 + 12 >= 4X1 + 2X2 + 4 = add(X1,X2) add(X1,active(X2)) = 4X1 + 2X2 + 8 >= 4X1 + 2X2 + 4 = add(X1,X2) first(mark(X1),X2) = 4X1 + 4X2 + 1 >= X1 + 4X2 = first(X1,X2) first(X1,mark(X2)) = X1 + 16X2 + 4 >= X1 + 4X2 = first(X1,X2) first(active(X1),X2) = X1 + 4X2 + 2 >= X1 + 4X2 = first(X1,X2) first(X1,active(X2)) = X1 + 4X2 + 8 >= X1 + 4X2 = first(X1,X2) cons(X1,mark(X2)) = 4X1 + 16X2 + 4 >= 4X1 + 4X2 = cons(X1,X2) cons(X1,active(X2)) = 4X1 + 4X2 + 8 >= 4X1 + 4X2 = cons(X1,X2) problem: mark(if(X1,X2,X3)) -> active(if(mark(X1),X2,X3)) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [mark](x0) = [0 0 1]x0 [0 1 0] , [1 0 1] [1 1 0] [1 1 1] [0] [if](x0, x1, x2) = [0 0 1]x0 + [0 0 0]x1 + [0 0 0]x2 + [0] [0 1 0] [0 0 0] [0 0 0] [1], [1 0 0] [active](x0) = [0 1 0]x0 [0 0 0] orientation: [1 1 1] [1 1 0] [1 1 1] [1] [1 1 1] [1 1 0] [1 1 1] mark(if(X1,X2,X3)) = [0 1 0]X1 + [0 0 0]X2 + [0 0 0]X3 + [1] >= [0 1 0]X1 + [0 0 0]X2 + [0 0 0]X3 = active(if(mark(X1),X2,X3)) [0 0 1] [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0 0 0] problem: Qed