diff --git a/draft/draft.tex b/draft/draft.tex index 4c8a3ea..35627f8 100644 --- a/draft/draft.tex +++ b/draft/draft.tex @@ -2216,7 +2216,7 @@ Now, we prove that the given order on the maybe functor is a liftable one. \item $Fg(k)=\bot$: It entails that $k=\bot$. We either have $h=\bot$ or $h\in Y$. If $h=\bot$ then we take $k'=k$, and we are done. If $h\in Y$, then by the surjectivity of $g$, there exists $k'$ such that $g(k')=h$ that entails $Fg(k')=h$ as well.\qed \end{itemize} \end{proof} -So, proven by Dubut, for every symmetric AM-simulation relation over a coalgebra $(X,\alpha)$ of the maybe functor, we have a witness $\sigma\c R\to R+1$ such that $\alpha\comp p_1=Fp_1\comp\sigma$. +So, proven by Dubut, for every AM-simulation relation over a coalgebra $(X,\alpha)$ of a functor with a liftable order, we have a witness $\sigma\c R\to R+1$ such that $\alpha\comp p_1=Fp_1\comp\sigma$. %\begin{lemma}\label{lem:maybe-func-set} % Assuming that $R$ is a symmetric AM-simulation over an $F$-coalgebra $(X,\alpha)$ that $FX=X+1$, then for every $(x_1,x_2)\in R$ % \begin{gather*}