YES Problem: active(eq(0(),0())) -> mark(true()) active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(X,Y)) -> mark(false()) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(take(0(),X)) -> mark(nil()) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(length(nil())) -> mark(0()) active(length(cons(X,L))) -> mark(s(length(L))) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(0()) -> active(0()) mark(true()) -> active(true()) mark(s(X)) -> active(s(X)) mark(false()) -> active(false()) mark(inf(X)) -> active(inf(mark(X))) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(length(X)) -> active(length(mark(X))) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) eq(active(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) inf(mark(X)) -> inf(X) inf(active(X)) -> inf(X) 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) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) length(mark(X)) -> length(X) length(active(X)) -> length(X) Proof: Matrix Interpretation Processor: dim=3 interpretation: [1 1 0] [mark](x0) = [0 0 0]x0 [0 1 1] , [1 0 0] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 1 1] , [1 0 0] [1 0 1] [eq](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 1] [0 0 0] , [0] [true] = [0] [1], [1 0 0] [s](x0) = [0 0 0]x0 [0 1 1] , [0] [nil] = [1] [0], [0] [0] = [0] [1], [1 1 0] [inf](x0) = [1 1 0]x0 [0 0 0] , [1 1 0] [1] [length](x0) = [0 1 1]x0 + [0] [0 1 1] [0], [1 1 0] [1 1 0] [take](x0, x1) = [0 1 1]x0 + [0 1 1]x1 [0 0 0] [0 0 0] , [0] [false] = [0] [0], [1 1 0] [active](x0) = [0 0 0]x0 [0 1 1] orientation: [1] [0] active(eq(0(),0())) = [0] >= [0] = mark(true()) [1] [1] [1 0 0] [1 1 1] [1 0 0] [1 0 1] active(eq(s(X),s(Y))) = [0 0 0]X + [0 0 0]Y >= [0 0 0]X + [0 0 0]Y = mark(eq(X,Y)) [0 1 1] [0 0 0] [0 0 1] [0 0 0] [1 0 0] [1 0 1] [0] active(eq(X,Y)) = [0 0 0]X + [0 0 0]Y >= [0] = mark(false()) [0 0 1] [0 0 0] [0] [2 2 0] [2 0 0] active(inf(X)) = [0 0 0]X >= [0 0 0]X = mark(cons(X,inf(s(X)))) [1 1 0] [1 0 0] [1 2 1] [1] [1] active(take(0(),X)) = [0 0 0]X + [0] >= [0] = mark(nil()) [0 1 1] [1] [1] [1 1 1] [1 1 1] [1 0 0] [1 1 0] [1 1 0] [1 0 0] active(take(s(X),cons(Y,L))) = [0 0 0]L + [0 0 0]X + [0 0 0]Y >= [0 0 0]L + [0 0 0]X + [0 0 0]Y = mark(cons(Y,take(X,L))) [0 1 1] [0 1 1] [0 0 0] [0 1 1] [0 1 1] [0 0 0] [3] [0] active(length(nil())) = [0] >= [0] = mark(0()) [2] [1] [1 1 1] [1 0 0] [1] [1 1 0] [1] active(length(cons(X,L))) = [0 0 0]L + [0 0 0]X + [0] >= [0 0 0]L + [0] = mark(s(length(L))) [0 2 2] [0 0 0] [0] [0 2 2] [0] [1 0 0] [1 0 1] [1 0 0] [1 0 1] mark(eq(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = active(eq(X1,X2)) [0 0 1] [0 0 0] [0 0 1] [0 0 0] [0] [0] mark(0()) = [0] >= [0] = active(0()) [1] [1] [0] [0] mark(true()) = [0] >= [0] = active(true()) [1] [1] [1 0 0] [1 0 0] mark(s(X)) = [0 0 0]X >= [0 0 0]X = active(s(X)) [0 1 1] [0 1 1] [0] [0] mark(false()) = [0] >= [0] = active(false()) [0] [0] [2 2 0] [2 2 0] mark(inf(X)) = [0 0 0]X >= [0 0 0]X = active(inf(mark(X))) [1 1 0] [1 1 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 1 1] [0 0 0] [0 1 1] [1 2 1] [1 2 1] [1 2 1] [1 2 1] mark(take(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = active(take(mark(X1),mark(X2))) [0 1 1] [0 1 1] [0 1 1] [0 1 1] [1] [1] mark(nil()) = [0] >= [0] = active(nil()) [1] [1] [1 2 1] [1] [1 2 1] [1] mark(length(X)) = [0 0 0]X + [0] >= [0 0 0]X + [0] = active(length(mark(X))) [0 2 2] [0] [0 2 2] [0] [1 1 0] [1 0 1] [1 0 0] [1 0 1] eq(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = eq(X1,X2) [0 1 1] [0 0 0] [0 0 1] [0 0 0] [1 0 0] [1 2 1] [1 0 0] [1 0 1] eq(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = eq(X1,X2) [0 0 1] [0 0 0] [0 0 1] [0 0 0] [1 1 0] [1 0 1] [1 0 0] [1 0 1] eq(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = eq(X1,X2) [0 1 1] [0 0 0] [0 0 1] [0 0 0] [1 0 0] [1 2 1] [1 0 0] [1 0 1] eq(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = eq(X1,X2) [0 0 1] [0 0 0] [0 0 1] [0 0 0] [1 1 0] [1 0 0] s(mark(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 1 1] [0 1 1] [1 1 0] [1 0 0] s(active(X)) = [0 0 0]X >= [0 0 0]X = s(X) [0 1 1] [0 1 1] [1 1 0] [1 1 0] inf(mark(X)) = [1 1 0]X >= [1 1 0]X = inf(X) [0 0 0] [0 0 0] [1 1 0] [1 1 0] inf(active(X)) = [1 1 0]X >= [1 1 0]X = inf(X) [0 0 0] [0 0 0] [1 1 0] [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 1 1] [0 0 0] [0 1 1] [1 0 0] [1 1 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 1 1] [0 0 0] [0 1 1] [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 1 1] [0 0 0] [0 1 1] [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 1 1] [0 0 0] [0 1 1] [1 1 0] [1 1 0] [1 1 0] [1 1 0] take(mark(X1),X2) = [0 1 1]X1 + [0 1 1]X2 >= [0 1 1]X1 + [0 1 1]X2 = take(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] [1 1 0] [1 1 0] take(X1,mark(X2)) = [0 1 1]X1 + [0 1 1]X2 >= [0 1 1]X1 + [0 1 1]X2 = take(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] [1 1 0] [1 1 0] take(active(X1),X2) = [0 1 1]X1 + [0 1 1]X2 >= [0 1 1]X1 + [0 1 1]X2 = take(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1 1 0] [1 1 0] [1 1 0] take(X1,active(X2)) = [0 1 1]X1 + [0 1 1]X2 >= [0 1 1]X1 + [0 1 1]X2 = take(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 0] [1] [1 1 0] [1] length(mark(X)) = [0 1 1]X + [0] >= [0 1 1]X + [0] = length(X) [0 1 1] [0] [0 1 1] [0] [1 1 0] [1] [1 1 0] [1] length(active(X)) = [0 1 1]X + [0] >= [0 1 1]X + [0] = length(X) [0 1 1] [0] [0 1 1] [0] problem: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(eq(X,Y)) -> mark(false()) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(take(0(),X)) -> mark(nil()) active(take(s(X),cons(Y,L))) -> mark(cons(Y,take(X,L))) active(length(cons(X,L))) -> mark(s(length(L))) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(0()) -> active(0()) mark(true()) -> active(true()) mark(s(X)) -> active(s(X)) mark(false()) -> active(false()) mark(inf(X)) -> active(inf(mark(X))) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(length(X)) -> active(length(mark(X))) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) eq(active(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) inf(mark(X)) -> inf(X) inf(active(X)) -> inf(X) 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) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) length(mark(X)) -> length(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [mark](x0) = [0 1 1]x0 [0 0 0] , [1 0 0] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 1 1]x1 [0 0 0] [0 0 0] , [1 1 0] [1 0 0] [1] [eq](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 1 0] [0 0 0] [0], [0] [true] = [0] [0], [1 0 0] [s](x0) = [0 1 1]x0 [0 0 0] , [1] [nil] = [0] [0], [0] [0] = [0] [0], [1 0 1] [inf](x0) = [0 0 0]x0 [1 0 1] , [1 0 1] [length](x0) = [0 0 0]x0 [0 1 1] , [1 0 1] [1 0 1] [0] [take](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [0] [0 1 1] [0 1 1] [1], [0] [false] = [0] [0], [1 0 1] [active](x0) = [0 1 1]x0 [0 0 0] orientation: [1 2 2] [1 0 0] [1] [1 2 0] [1 0 0] [1] active(eq(s(X),s(Y))) = [0 1 1]X + [0 0 0]Y + [0] >= [0 1 0]X + [0 0 0]Y + [0] = mark(eq(X,Y)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 2 0] [1 0 0] [1] [0] active(eq(X,Y)) = [0 1 0]X + [0 0 0]Y + [0] >= [0] = mark(false()) [0 0 0] [0 0 0] [0] [0] [2 0 2] [2 0 0] active(inf(X)) = [1 0 1]X >= [1 0 0]X = mark(cons(X,inf(s(X)))) [0 0 0] [0 0 0] [1 1 2] [1] [1] active(take(0(),X)) = [0 1 1]X + [1] >= [0] = mark(nil()) [0 0 0] [0] [0] [1 1 1] [1 1 1] [1 0 0] [1] [1 0 1] [1 0 1] [1 0 0] [0] active(take(s(X),cons(Y,L))) = [0 1 1]L + [0 1 1]X + [0 0 0]Y + [1] >= [0 1 1]L + [0 1 1]X + [0 0 0]Y + [1] = mark(cons(Y,take(X,L))) [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 0] [1 0 1] active(length(cons(X,L))) = [0 1 1]L + [0 0 0]X >= [0 1 1]L = mark(s(length(L))) [0 0 0] [0 0 0] [0 0 0] [1 2 0] [1 0 0] [1] [1 2 0] [1 0 0] [1] mark(eq(X1,X2)) = [0 1 0]X1 + [0 0 0]X2 + [0] >= [0 1 0]X1 + [0 0 0]X2 + [0] = active(eq(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] [0] [0] mark(true()) = [0] >= [0] = active(true()) [0] [0] [1 0 0] [1 0 0] mark(s(X)) = [0 1 1]X >= [0 1 1]X = active(s(X)) [0 0 0] [0 0 0] [0] [0] mark(false()) = [0] >= [0] = active(false()) [0] [0] [2 0 2] [2 0 2] mark(inf(X)) = [1 0 1]X >= [1 0 1]X = active(inf(mark(X))) [0 0 0] [0 0 0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] mark(cons(X1,X2)) = [0 0 0]X1 + [0 1 1]X2 >= [0 0 0]X1 + [0 1 1]X2 = active(cons(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 2] [1 1 2] [1] [1 1 2] [1 1 2] [1] mark(take(X1,X2)) = [0 1 1]X1 + [0 1 1]X2 + [1] >= [0 1 1]X1 + [0 1 1]X2 + [1] = active(take(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 1 2] [1 1 2] mark(length(X)) = [0 1 1]X >= [0 1 1]X = active(length(mark(X))) [0 0 0] [0 0 0] [1 1 2] [1 0 0] [1] [1 1 0] [1 0 0] [1] eq(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = eq(X1,X2) [0 1 1] [0 0 0] [0] [0 1 0] [0 0 0] [0] [1 1 0] [1 0 1] [1] [1 1 0] [1 0 0] [1] eq(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = eq(X1,X2) [0 1 0] [0 0 0] [0] [0 1 0] [0 0 0] [0] [1 1 2] [1 0 0] [1] [1 1 0] [1 0 0] [1] eq(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = eq(X1,X2) [0 1 1] [0 0 0] [0] [0 1 0] [0 0 0] [0] [1 1 0] [1 0 1] [1] [1 1 0] [1 0 0] [1] eq(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = eq(X1,X2) [0 1 0] [0 0 0] [0] [0 1 0] [0 0 0] [0] [1 0 1] [1 0 0] s(mark(X)) = [0 1 1]X >= [0 1 1]X = s(X) [0 0 0] [0 0 0] [1 0 1] [1 0 0] s(active(X)) = [0 1 1]X >= [0 1 1]X = s(X) [0 0 0] [0 0 0] [1 0 1] [1 0 1] inf(mark(X)) = [0 0 0]X >= [0 0 0]X = inf(X) [1 0 1] [1 0 1] [1 0 1] [1 0 1] inf(active(X)) = [0 0 0]X >= [0 0 0]X = inf(X) [1 0 1] [1 0 1] [1 0 1] [1 0 0] [1 0 0] [1 0 0] cons(mark(X1),X2) = [0 0 0]X1 + [0 1 1]X2 >= [0 0 0]X1 + [0 1 1]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 1 1]X2 >= [0 0 0]X1 + [0 1 1]X2 = cons(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] cons(active(X1),X2) = [0 0 0]X1 + [0 1 1]X2 >= [0 0 0]X1 + [0 1 1]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,active(X2)) = [0 0 0]X1 + [0 1 1]X2 >= [0 0 0]X1 + [0 1 1]X2 = cons(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] [0] [1 0 1] [1 0 1] [0] take(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = take(X1,X2) [0 1 1] [0 1 1] [1] [0 1 1] [0 1 1] [1] [1 0 1] [1 0 1] [0] [1 0 1] [1 0 1] [0] take(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = take(X1,X2) [0 1 1] [0 1 1] [1] [0 1 1] [0 1 1] [1] [1 0 1] [1 0 1] [0] [1 0 1] [1 0 1] [0] take(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = take(X1,X2) [0 1 1] [0 1 1] [1] [0 1 1] [0 1 1] [1] [1 0 1] [1 0 1] [0] [1 0 1] [1 0 1] [0] take(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 + [0] >= [0 0 0]X1 + [0 0 0]X2 + [0] = take(X1,X2) [0 1 1] [0 1 1] [1] [0 1 1] [0 1 1] [1] [1 0 1] [1 0 1] length(mark(X)) = [0 0 0]X >= [0 0 0]X = length(X) [0 1 1] [0 1 1] [1 0 1] [1 0 1] length(active(X)) = [0 0 0]X >= [0 0 0]X = length(X) [0 1 1] [0 1 1] problem: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(take(0(),X)) -> mark(nil()) active(length(cons(X,L))) -> mark(s(length(L))) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(0()) -> active(0()) mark(true()) -> active(true()) mark(s(X)) -> active(s(X)) mark(false()) -> active(false()) mark(inf(X)) -> active(inf(mark(X))) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(length(X)) -> active(length(mark(X))) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) eq(active(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) inf(mark(X)) -> inf(X) inf(active(X)) -> inf(X) 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) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) length(mark(X)) -> length(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [mark](x0) = [0 1 1]x0 [0 0 0] , [1 0 0] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [0 1 1]x1 [0 0 0] [0 0 0] , [1 0 0] [1 1 0] [eq](x0, x1) = [0 0 0]x0 + [1 1 0]x1 [0 0 0] [1 0 1] , [0] [true] = [0] [0], [1 0 0] [s](x0) = [0 1 1]x0 [0 0 0] , [0] [nil] = [0] [0], [0] [0] = [1] [0], [1 0 1] [0] [inf](x0) = [0 0 0]x0 + [1] [1 0 1] [0], [1 0 1] [length](x0) = [0 0 0]x0 [0 1 1] , [1 0 1] [1 0 1] [1] [take](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1] [0 1 1] [0 1 1] [1], [0] [false] = [1] [0], [1 0 1] [active](x0) = [0 1 1]x0 [0 0 0] orientation: [1 0 0] [2 1 1] [1 0 0] [2 1 1] active(eq(s(X),s(Y))) = [0 0 0]X + [2 1 1]Y >= [0 0 0]X + [2 1 1]Y = mark(eq(X,Y)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [2 0 2] [0] [2 0 0] [0] active(inf(X)) = [1 0 1]X + [1] >= [1 0 0]X + [1] = mark(cons(X,inf(s(X)))) [0 0 0] [0] [0 0 0] [0] [1 1 2] [3] [0] active(take(0(),X)) = [0 1 1]X + [3] >= [0] = mark(nil()) [0 0 0] [0] [0] [1 1 1] [1 0 0] [1 0 1] active(length(cons(X,L))) = [0 1 1]L + [0 0 0]X >= [0 1 1]L = mark(s(length(L))) [0 0 0] [0 0 0] [0 0 0] [1 0 0] [2 1 1] [1 0 0] [2 1 1] mark(eq(X1,X2)) = [0 0 0]X1 + [2 1 1]X2 >= [0 0 0]X1 + [2 1 1]X2 = active(eq(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [0] [0] mark(0()) = [1] >= [1] = active(0()) [0] [0] [0] [0] mark(true()) = [0] >= [0] = active(true()) [0] [0] [1 0 0] [1 0 0] mark(s(X)) = [0 1 1]X >= [0 1 1]X = active(s(X)) [0 0 0] [0 0 0] [0] [0] mark(false()) = [1] >= [1] = active(false()) [0] [0] [2 0 2] [0] [2 0 2] [0] mark(inf(X)) = [1 0 1]X + [1] >= [1 0 1]X + [1] = active(inf(mark(X))) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] mark(cons(X1,X2)) = [0 0 0]X1 + [0 1 1]X2 >= [0 0 0]X1 + [0 1 1]X2 = active(cons(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 1 2] [1 1 2] [2] [1 1 2] [1 1 2] [2] mark(take(X1,X2)) = [0 1 1]X1 + [0 1 1]X2 + [2] >= [0 1 1]X1 + [0 1 1]X2 + [2] = active(take(mark(X1),mark(X2))) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [0] [0] mark(nil()) = [0] >= [0] = active(nil()) [0] [0] [1 1 2] [1 1 2] mark(length(X)) = [0 1 1]X >= [0 1 1]X = active(length(mark(X))) [0 0 0] [0 0 0] [1 0 1] [1 1 0] [1 0 0] [1 1 0] eq(mark(X1),X2) = [0 0 0]X1 + [1 1 0]X2 >= [0 0 0]X1 + [1 1 0]X2 = eq(X1,X2) [0 0 0] [1 0 1] [0 0 0] [1 0 1] [1 0 0] [1 1 2] [1 0 0] [1 1 0] eq(X1,mark(X2)) = [0 0 0]X1 + [1 1 2]X2 >= [0 0 0]X1 + [1 1 0]X2 = eq(X1,X2) [0 0 0] [1 0 1] [0 0 0] [1 0 1] [1 0 1] [1 1 0] [1 0 0] [1 1 0] eq(active(X1),X2) = [0 0 0]X1 + [1 1 0]X2 >= [0 0 0]X1 + [1 1 0]X2 = eq(X1,X2) [0 0 0] [1 0 1] [0 0 0] [1 0 1] [1 0 0] [1 1 2] [1 0 0] [1 1 0] eq(X1,active(X2)) = [0 0 0]X1 + [1 1 2]X2 >= [0 0 0]X1 + [1 1 0]X2 = eq(X1,X2) [0 0 0] [1 0 1] [0 0 0] [1 0 1] [1 0 1] [1 0 0] s(mark(X)) = [0 1 1]X >= [0 1 1]X = s(X) [0 0 0] [0 0 0] [1 0 1] [1 0 0] s(active(X)) = [0 1 1]X >= [0 1 1]X = s(X) [0 0 0] [0 0 0] [1 0 1] [0] [1 0 1] [0] inf(mark(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = inf(X) [1 0 1] [0] [1 0 1] [0] [1 0 1] [0] [1 0 1] [0] inf(active(X)) = [0 0 0]X + [1] >= [0 0 0]X + [1] = inf(X) [1 0 1] [0] [1 0 1] [0] [1 0 1] [1 0 0] [1 0 0] [1 0 0] cons(mark(X1),X2) = [0 0 0]X1 + [0 1 1]X2 >= [0 0 0]X1 + [0 1 1]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 1 1]X2 >= [0 0 0]X1 + [0 1 1]X2 = cons(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] cons(active(X1),X2) = [0 0 0]X1 + [0 1 1]X2 >= [0 0 0]X1 + [0 1 1]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,active(X2)) = [0 0 0]X1 + [0 1 1]X2 >= [0 0 0]X1 + [0 1 1]X2 = cons(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] [1] take(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = take(X1,X2) [0 1 1] [0 1 1] [1] [0 1 1] [0 1 1] [1] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] [1] take(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = take(X1,X2) [0 1 1] [0 1 1] [1] [0 1 1] [0 1 1] [1] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] [1] take(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = take(X1,X2) [0 1 1] [0 1 1] [1] [0 1 1] [0 1 1] [1] [1 0 1] [1 0 1] [1] [1 0 1] [1 0 1] [1] take(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = take(X1,X2) [0 1 1] [0 1 1] [1] [0 1 1] [0 1 1] [1] [1 0 1] [1 0 1] length(mark(X)) = [0 0 0]X >= [0 0 0]X = length(X) [0 1 1] [0 1 1] [1 0 1] [1 0 1] length(active(X)) = [0 0 0]X >= [0 0 0]X = length(X) [0 1 1] [0 1 1] problem: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(inf(X)) -> mark(cons(X,inf(s(X)))) active(length(cons(X,L))) -> mark(s(length(L))) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(0()) -> active(0()) mark(true()) -> active(true()) mark(s(X)) -> active(s(X)) mark(false()) -> active(false()) mark(inf(X)) -> active(inf(mark(X))) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(length(X)) -> active(length(mark(X))) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) eq(active(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) inf(mark(X)) -> inf(X) inf(active(X)) -> inf(X) 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) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) length(mark(X)) -> length(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dim=3 interpretation: [1 0 1] [mark](x0) = [0 1 1]x0 [0 0 0] , [1 0 0] [1 0 0] [cons](x0, x1) = [0 0 0]x0 + [1 0 1]x1 [0 0 0] [0 0 0] , [1 0 0] [1 0 0] [0] [eq](x0, x1) = [0 0 0]x0 + [0 0 0]x1 + [1] [0 0 0] [0 0 0] [0], [0] [true] = [0] [0], [1 0 0] [s](x0) = [0 0 0]x0 [0 0 0] , [0] [nil] = [0] [0], [0] [0] = [0] [0], [1 0 1] [0] [inf](x0) = [1 0 1]x0 + [0] [1 0 1] [1], [1 0 1] [length](x0) = [0 0 0]x0 [0 1 1] , [1 0 1] [1 0 1] [take](x0, x1) = [0 0 0]x0 + [0 0 0]x1 [0 0 0] [0 0 0] , [0] [false] = [0] [0], [1 0 1] [active](x0) = [0 1 1]x0 [0 0 0] orientation: [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] active(eq(s(X),s(Y))) = [0 0 0]X + [0 0 0]Y + [1] >= [0 0 0]X + [0 0 0]Y + [1] = mark(eq(X,Y)) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [2 0 2] [1] [2 0 0] [0] active(inf(X)) = [2 0 2]X + [1] >= [2 0 0]X + [1] = mark(cons(X,inf(s(X)))) [0 0 0] [0] [0 0 0] [0] [2 0 1] [1 0 0] [1 0 1] active(length(cons(X,L))) = [1 0 1]L + [0 0 0]X >= [0 0 0]L = mark(s(length(L))) [0 0 0] [0 0 0] [0 0 0] [1 0 0] [1 0 0] [0] [1 0 0] [1 0 0] [0] mark(eq(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = active(eq(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] [0] [0] mark(true()) = [0] >= [0] = active(true()) [0] [0] [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] [0] [0] mark(false()) = [0] >= [0] = active(false()) [0] [0] [2 0 2] [1] [2 0 2] [1] mark(inf(X)) = [2 0 2]X + [1] >= [2 0 2]X + [1] = active(inf(mark(X))) [0 0 0] [0] [0 0 0] [0] [1 0 0] [1 0 0] [1 0 0] [1 0 0] mark(cons(X1,X2)) = [0 0 0]X1 + [1 0 1]X2 >= [0 0 0]X1 + [1 0 1]X2 = active(cons(X1,X2)) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] [1 0 1] [1 0 1] mark(take(X1,X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = active(take(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 2] [1 1 2] mark(length(X)) = [0 1 1]X >= [0 1 1]X = active(length(mark(X))) [0 0 0] [0 0 0] [1 0 1] [1 0 0] [0] [1 0 0] [1 0 0] [0] eq(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = eq(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [1 0 1] [0] [1 0 0] [1 0 0] [0] eq(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = eq(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 1] [1 0 0] [0] [1 0 0] [1 0 0] [0] eq(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = eq(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 0 0] [0] [1 0 0] [1 0 1] [0] [1 0 0] [1 0 0] [0] eq(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 + [1] >= [0 0 0]X1 + [0 0 0]X2 + [1] = eq(X1,X2) [0 0 0] [0 0 0] [0] [0 0 0] [0 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 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 0 1] [0] [1 0 1] [0] inf(mark(X)) = [1 0 1]X + [0] >= [1 0 1]X + [0] = inf(X) [1 0 1] [1] [1 0 1] [1] [1 0 1] [0] [1 0 1] [0] inf(active(X)) = [1 0 1]X + [0] >= [1 0 1]X + [0] = inf(X) [1 0 1] [1] [1 0 1] [1] [1 0 1] [1 0 0] [1 0 0] [1 0 0] cons(mark(X1),X2) = [0 0 0]X1 + [1 0 1]X2 >= [0 0 0]X1 + [1 0 1]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 + [1 0 1]X2 >= [0 0 0]X1 + [1 0 1]X2 = cons(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] cons(active(X1),X2) = [0 0 0]X1 + [1 0 1]X2 >= [0 0 0]X1 + [1 0 1]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,active(X2)) = [0 0 0]X1 + [1 0 1]X2 >= [0 0 0]X1 + [1 0 1]X2 = cons(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] [1 0 1] [1 0 1] take(mark(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = take(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] [1 0 1] [1 0 1] take(X1,mark(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = take(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] [1 0 1] [1 0 1] take(active(X1),X2) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = take(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] [1 0 1] [1 0 1] take(X1,active(X2)) = [0 0 0]X1 + [0 0 0]X2 >= [0 0 0]X1 + [0 0 0]X2 = take(X1,X2) [0 0 0] [0 0 0] [0 0 0] [0 0 0] [1 0 1] [1 0 1] length(mark(X)) = [0 0 0]X >= [0 0 0]X = length(X) [0 1 1] [0 1 1] [1 0 1] [1 0 1] length(active(X)) = [0 0 0]X >= [0 0 0]X = length(X) [0 1 1] [0 1 1] problem: active(eq(s(X),s(Y))) -> mark(eq(X,Y)) active(length(cons(X,L))) -> mark(s(length(L))) mark(eq(X1,X2)) -> active(eq(X1,X2)) mark(0()) -> active(0()) mark(true()) -> active(true()) mark(s(X)) -> active(s(X)) mark(false()) -> active(false()) mark(inf(X)) -> active(inf(mark(X))) mark(cons(X1,X2)) -> active(cons(X1,X2)) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(nil()) -> active(nil()) mark(length(X)) -> active(length(mark(X))) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) eq(active(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) inf(mark(X)) -> inf(X) inf(active(X)) -> inf(X) 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) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) length(mark(X)) -> length(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dim=1 interpretation: [mark](x0) = 2x0, [cons](x0, x1) = 2x0 + 4x1 + 1, [eq](x0, x1) = 2x0 + 2x1 + 1, [true] = 0, [s](x0) = 2x0 + 2, [nil] = 4, [0] = 0, [inf](x0) = 2x0, [length](x0) = 4x0, [take](x0, x1) = x0 + x1, [false] = 0, [active](x0) = x0 orientation: active(eq(s(X),s(Y))) = 4X + 4Y + 9 >= 4X + 4Y + 2 = mark(eq(X,Y)) active(length(cons(X,L))) = 16L + 8X + 4 >= 16L + 4 = mark(s(length(L))) mark(eq(X1,X2)) = 4X1 + 4X2 + 2 >= 2X1 + 2X2 + 1 = active(eq(X1,X2)) mark(0()) = 0 >= 0 = active(0()) mark(true()) = 0 >= 0 = active(true()) mark(s(X)) = 4X + 4 >= 2X + 2 = active(s(X)) mark(false()) = 0 >= 0 = active(false()) mark(inf(X)) = 4X >= 4X = active(inf(mark(X))) mark(cons(X1,X2)) = 4X1 + 8X2 + 2 >= 2X1 + 4X2 + 1 = active(cons(X1,X2)) mark(take(X1,X2)) = 2X1 + 2X2 >= 2X1 + 2X2 = active(take(mark(X1),mark(X2))) mark(nil()) = 8 >= 4 = active(nil()) mark(length(X)) = 8X >= 8X = active(length(mark(X))) eq(mark(X1),X2) = 4X1 + 2X2 + 1 >= 2X1 + 2X2 + 1 = eq(X1,X2) eq(X1,mark(X2)) = 2X1 + 4X2 + 1 >= 2X1 + 2X2 + 1 = eq(X1,X2) eq(active(X1),X2) = 2X1 + 2X2 + 1 >= 2X1 + 2X2 + 1 = eq(X1,X2) eq(X1,active(X2)) = 2X1 + 2X2 + 1 >= 2X1 + 2X2 + 1 = eq(X1,X2) s(mark(X)) = 4X + 2 >= 2X + 2 = s(X) s(active(X)) = 2X + 2 >= 2X + 2 = s(X) inf(mark(X)) = 4X >= 2X = inf(X) inf(active(X)) = 2X >= 2X = inf(X) cons(mark(X1),X2) = 4X1 + 4X2 + 1 >= 2X1 + 4X2 + 1 = cons(X1,X2) cons(X1,mark(X2)) = 2X1 + 8X2 + 1 >= 2X1 + 4X2 + 1 = cons(X1,X2) cons(active(X1),X2) = 2X1 + 4X2 + 1 >= 2X1 + 4X2 + 1 = cons(X1,X2) cons(X1,active(X2)) = 2X1 + 4X2 + 1 >= 2X1 + 4X2 + 1 = cons(X1,X2) take(mark(X1),X2) = 2X1 + X2 >= X1 + X2 = take(X1,X2) take(X1,mark(X2)) = X1 + 2X2 >= X1 + X2 = take(X1,X2) take(active(X1),X2) = X1 + X2 >= X1 + X2 = take(X1,X2) take(X1,active(X2)) = X1 + X2 >= X1 + X2 = take(X1,X2) length(mark(X)) = 8X >= 4X = length(X) length(active(X)) = 4X >= 4X = length(X) problem: active(length(cons(X,L))) -> mark(s(length(L))) mark(0()) -> active(0()) mark(true()) -> active(true()) mark(false()) -> active(false()) mark(inf(X)) -> active(inf(mark(X))) mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) mark(length(X)) -> active(length(mark(X))) eq(mark(X1),X2) -> eq(X1,X2) eq(X1,mark(X2)) -> eq(X1,X2) eq(active(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) s(mark(X)) -> s(X) s(active(X)) -> s(X) inf(mark(X)) -> inf(X) inf(active(X)) -> inf(X) 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) take(mark(X1),X2) -> take(X1,X2) take(X1,mark(X2)) -> take(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) length(mark(X)) -> length(X) length(active(X)) -> length(X) Matrix Interpretation Processor: dim=1 interpretation: [mark](x0) = 2x0 + 1, [cons](x0, x1) = 4x0 + 4x1 + 6, [eq](x0, x1) = 4x0 + 4x1 + 4, [true] = 0, [s](x0) = 2x0 + 1, [0] = 6, [inf](x0) = x0 + 2, [length](x0) = 2x0 + 2, [take](x0, x1) = x0 + x1 + 1, [false] = 0, [active](x0) = x0 orientation: active(length(cons(X,L))) = 8L + 8X + 14 >= 8L + 11 = mark(s(length(L))) mark(0()) = 13 >= 6 = active(0()) mark(true()) = 1 >= 0 = active(true()) mark(false()) = 1 >= 0 = active(false()) mark(inf(X)) = 2X + 5 >= 2X + 3 = active(inf(mark(X))) mark(take(X1,X2)) = 2X1 + 2X2 + 3 >= 2X1 + 2X2 + 3 = active(take(mark(X1),mark(X2))) mark(length(X)) = 4X + 5 >= 4X + 4 = active(length(mark(X))) eq(mark(X1),X2) = 8X1 + 4X2 + 8 >= 4X1 + 4X2 + 4 = eq(X1,X2) eq(X1,mark(X2)) = 4X1 + 8X2 + 8 >= 4X1 + 4X2 + 4 = eq(X1,X2) eq(active(X1),X2) = 4X1 + 4X2 + 4 >= 4X1 + 4X2 + 4 = eq(X1,X2) eq(X1,active(X2)) = 4X1 + 4X2 + 4 >= 4X1 + 4X2 + 4 = eq(X1,X2) s(mark(X)) = 4X + 3 >= 2X + 1 = s(X) s(active(X)) = 2X + 1 >= 2X + 1 = s(X) inf(mark(X)) = 2X + 3 >= X + 2 = inf(X) inf(active(X)) = X + 2 >= X + 2 = inf(X) cons(mark(X1),X2) = 8X1 + 4X2 + 10 >= 4X1 + 4X2 + 6 = cons(X1,X2) cons(X1,mark(X2)) = 4X1 + 8X2 + 10 >= 4X1 + 4X2 + 6 = cons(X1,X2) cons(active(X1),X2) = 4X1 + 4X2 + 6 >= 4X1 + 4X2 + 6 = cons(X1,X2) cons(X1,active(X2)) = 4X1 + 4X2 + 6 >= 4X1 + 4X2 + 6 = cons(X1,X2) take(mark(X1),X2) = 2X1 + X2 + 2 >= X1 + X2 + 1 = take(X1,X2) take(X1,mark(X2)) = X1 + 2X2 + 2 >= X1 + X2 + 1 = take(X1,X2) take(active(X1),X2) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,X2) take(X1,active(X2)) = X1 + X2 + 1 >= X1 + X2 + 1 = take(X1,X2) length(mark(X)) = 4X + 4 >= 2X + 2 = length(X) length(active(X)) = 2X + 2 >= 2X + 2 = length(X) problem: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) eq(active(X1),X2) -> eq(X1,X2) eq(X1,active(X2)) -> eq(X1,X2) s(active(X)) -> s(X) inf(active(X)) -> inf(X) cons(active(X1),X2) -> cons(X1,X2) cons(X1,active(X2)) -> cons(X1,X2) take(active(X1),X2) -> take(X1,X2) take(X1,active(X2)) -> take(X1,X2) length(active(X)) -> length(X) Matrix Interpretation Processor: dim=1 interpretation: [mark](x0) = 4x0, [cons](x0, x1) = x0 + x1, [eq](x0, x1) = x0 + x1, [s](x0) = x0, [inf](x0) = x0, [length](x0) = x0, [take](x0, x1) = 2x0 + x1 + 1, [active](x0) = x0 + 3 orientation: mark(take(X1,X2)) = 8X1 + 4X2 + 4 >= 8X1 + 4X2 + 4 = active(take(mark(X1),mark(X2))) eq(active(X1),X2) = X1 + X2 + 3 >= X1 + X2 = eq(X1,X2) eq(X1,active(X2)) = X1 + X2 + 3 >= X1 + X2 = eq(X1,X2) s(active(X)) = X + 3 >= X = s(X) inf(active(X)) = X + 3 >= X = inf(X) cons(active(X1),X2) = X1 + X2 + 3 >= X1 + X2 = cons(X1,X2) cons(X1,active(X2)) = X1 + X2 + 3 >= X1 + X2 = cons(X1,X2) take(active(X1),X2) = 2X1 + X2 + 7 >= 2X1 + X2 + 1 = take(X1,X2) take(X1,active(X2)) = 2X1 + X2 + 4 >= 2X1 + X2 + 1 = take(X1,X2) length(active(X)) = X + 3 >= X = length(X) problem: mark(take(X1,X2)) -> active(take(mark(X1),mark(X2))) Matrix Interpretation Processor: dim=1 interpretation: [mark](x0) = 2x0, [take](x0, x1) = x0 + 2x1 + 2, [active](x0) = x0 + 1 orientation: mark(take(X1,X2)) = 2X1 + 4X2 + 4 >= 2X1 + 4X2 + 3 = active(take(mark(X1),mark(X2))) problem: Qed