Commit 9df1dae0 authored by Ralf Jung's avatar Ralf Jung

appendix: base logic is now up-to-date

parent 0621aa23
......@@ -1002,9 +1002,6 @@ Proof.
by rewrite cmra_core_l cmra_core_idemp.
Qed.
Lemma always_later P : P P.
Proof. by unseal. Qed.
(* Always derived *)
Hint Resolve always_mono always_elim.
Global Instance always_mono' : Proper (() ==> ()) (@uPred_always M).
......@@ -1069,32 +1066,6 @@ Proof. intros; rewrite -always_and_sep_l'; auto. Qed.
Lemma always_entails_r' P Q : (P Q) P P Q.
Proof. intros; rewrite -always_and_sep_r'; auto. Qed.
(* Conditional always *)
Global Instance always_if_ne n p : Proper (dist n ==> dist n) (@uPred_always_if M p).
Proof. solve_proper. Qed.
Global Instance always_if_proper p : Proper (() ==> ()) (@uPred_always_if M p).
Proof. solve_proper. Qed.
Global Instance always_if_mono p : Proper (() ==> ()) (@uPred_always_if M p).
Proof. solve_proper. Qed.
Lemma always_if_elim p P : ?p P P.
Proof. destruct p; simpl; auto using always_elim. Qed.
Lemma always_elim_if p P : P ?p P.
Proof. destruct p; simpl; auto using always_elim. Qed.
Lemma always_if_pure p φ : ?p φ φ.
Proof. destruct p; simpl; auto using always_pure. Qed.
Lemma always_if_and p P Q : ?p (P Q) ?p P ?p Q.
Proof. destruct p; simpl; auto using always_and. Qed.
Lemma always_if_or p P Q : ?p (P Q) ?p P ?p Q.
Proof. destruct p; simpl; auto using always_or. Qed.
Lemma always_if_exist {A} p (Ψ : A uPred M) : (?p a, Ψ a) a, ?p Ψ a.
Proof. destruct p; simpl; auto using always_exist. Qed.
Lemma always_if_sep p P Q : ?p (P Q) ?p P ?p Q.
Proof. destruct p; simpl; auto using always_sep. Qed.
Lemma always_if_later p P : ?p P ?p P.
Proof. destruct p; simpl; auto using always_later. Qed.
(* Later *)
Lemma later_mono P Q : (P Q) P Q.
Proof.
......@@ -1128,6 +1099,10 @@ Proof.
eauto using uPred_closed, uPred_mono, cmra_included_includedN.
Qed.
Lemma always_later P : P P.
Proof. by unseal. Qed.
(* Later derived *)
Lemma later_proper P Q : (P Q) P Q.
Proof. by intros ->. Qed.
......@@ -1169,6 +1144,34 @@ Proof. apply wand_intro_r; rewrite -later_sep; eauto using wand_elim_l. Qed.
Lemma later_iff P Q : (P Q) P Q.
Proof. by rewrite /uPred_iff later_and !later_impl. Qed.
(* Conditional always *)
Global Instance always_if_ne n p : Proper (dist n ==> dist n) (@uPred_always_if M p).
Proof. solve_proper. Qed.
Global Instance always_if_proper p : Proper (() ==> ()) (@uPred_always_if M p).
Proof. solve_proper. Qed.
Global Instance always_if_mono p : Proper (() ==> ()) (@uPred_always_if M p).
Proof. solve_proper. Qed.
Lemma always_if_elim p P : ?p P P.
Proof. destruct p; simpl; auto using always_elim. Qed.
Lemma always_elim_if p P : P ?p P.
Proof. destruct p; simpl; auto using always_elim. Qed.
Lemma always_if_pure p φ : ?p φ φ.
Proof. destruct p; simpl; auto using always_pure. Qed.
Lemma always_if_and p P Q : ?p (P Q) ?p P ?p Q.
Proof. destruct p; simpl; auto using always_and. Qed.
Lemma always_if_or p P Q : ?p (P Q) ?p P ?p Q.
Proof. destruct p; simpl; auto using always_or. Qed.
Lemma always_if_exist {A} p (Ψ : A uPred M) : (?p a, Ψ a) a, ?p Ψ a.
Proof. destruct p; simpl; auto using always_exist. Qed.
Lemma always_if_sep p P Q : ?p (P Q) ?p P ?p Q.
Proof. destruct p; simpl; auto using always_sep. Qed.
Lemma always_if_later p P : ?p P ?p P.
Proof. destruct p; simpl; auto using always_later. Qed.
(* True now *)
Global Instance except_last_ne n : Proper (dist n ==> dist n) (@uPred_except_last M).
Proof. solve_proper. Qed.
......@@ -1226,10 +1229,9 @@ Proof.
by rewrite (assoc op _ z1) -(comm op z1) (assoc op z1)
-(assoc op _ a2) (comm op z1) -Hy1 -Hy2.
Qed.
Lemma always_ownM (a : M) : Persistent a uPred_ownM a uPred_ownM a.
Lemma always_ownM_core (a : M) : uPred_ownM a uPred_ownM (core a).
Proof.
split=> n x /=; split; [by apply always_elim|unseal; intros Hx]; simpl.
rewrite -(persistent_core a). by apply cmra_core_monoN.
split=> n x /=; unseal; intros Hx. simpl. by apply cmra_core_monoN.
Qed.
Lemma ownM_empty : True uPred_ownM .
Proof. unseal; split=> n x ??; by exists x; rewrite left_id. Qed.
......@@ -1250,18 +1252,27 @@ Lemma cmra_valid_intro {A : cmraT} (a : A) : ✓ a → True ⊢ ✓ a.
Proof. unseal=> ?; split=> n x ? _ /=; by apply cmra_valid_validN. Qed.
Lemma cmra_valid_elim {A : cmraT} (a : A) : ¬ {0} a a False.
Proof. unseal=> Ha; split=> n x ??; apply Ha, cmra_validN_le with n; auto. Qed.
Lemma always_cmra_valid {A : cmraT} (a : A) : a a.
Lemma always_cmra_valid_1 {A : cmraT} (a : A) : a a.
Proof. by unseal. Qed.
Lemma cmra_valid_weaken {A : cmraT} (a b : A) : (a b) a.
Proof. unseal; split=> n x _; apply cmra_validN_op_l. Qed.
(* Own and valid derived *)
Lemma always_ownM (a : M) : Persistent a uPred_ownM a uPred_ownM a.
Proof.
intros; apply (anti_symm _); first by apply:always_elim.
by rewrite {1}always_ownM_core persistent_core.
Qed.
Lemma ownM_invalid (a : M) : ¬ {0} a uPred_ownM a False.
Proof. by intros; rewrite ownM_valid cmra_valid_elim. Qed.
Global Instance ownM_mono : Proper (flip () ==> ()) (@uPred_ownM M).
Proof. intros a b [b' ->]. rewrite ownM_op. eauto. Qed.
Lemma ownM_empty' : uPred_ownM True.
Proof. apply (anti_symm _); auto using ownM_empty. Qed.
Lemma always_cmra_valid {A : cmraT} (a : A) : a a.
intros; apply (anti_symm _); first by apply:always_elim.
apply:always_cmra_valid_1.
Qed.
(* Viewshifts *)
Lemma rvs_intro P : P =r=> P.
......
......@@ -299,78 +299,80 @@ Furthermore, we have the usual $\eta$ and $\beta$ laws for projections, $\lambda
{\prop \proves \propB \wand \propC}
\end{mathpar}
\paragraph{Laws for ghosts and physical resources.}
\paragraph{Laws for the always modality.}
\begin{mathpar}
\begin{array}{rMcMl}
\ownGGhost{\melt} * \ownGGhost{\meltB} &\provesIff& \ownGGhost{\melt \mtimes \meltB} \\
\ownGGhost{\melt} &\proves& \mval(\melt) \\
\TRUE &\proves& \ownGGhost{\munit}
\end{array}
\infer[$\always$-mono]
{\prop \proves \propB}
{\always{\prop} \proves \always{\propB}}
\and
\begin{array}[c]{rMcMl}
\always{\prop} &\proves& \prop \\
\always{(\prop \land \propB)} &\proves& \always{(\prop * \propB)} \\
\always{\prop} \land \propB &\proves& \always{\prop} * \propB
\end{array}
\and
\begin{array}{c}
\ownPhys{\state} * \ownPhys{\state'} \proves \FALSE
\begin{array}[c]{rMcMl}
\always{\prop} &\proves& \always\always\prop \\
\All x. \always{\prop} &\proves& \always{\All x. \prop} \\
\always{\Exists x. \prop} &\proves& \Exists x. \always{\prop}
\end{array}
\end{mathpar}
\paragraph{Laws for the later modality.}
\begin{mathpar}
\infer[$\later$-mono]
{\pfctx \proves \prop}
{\pfctx \proves \later{\prop}}
{\prop \proves \propB}
{\later\prop \proves \later{\propB}}
\and
\infer[L{\"o}b]
{}
{(\later\prop\Ra\prop) \proves \prop}
\and
\infer[$\later$-$\exists$]
{\text{$\type$ is inhabited}}
{\later{\Exists x:\type.\prop} \proves \Exists x:\type. \later\prop}
\\\\
\begin{array}[c]{rMcMl}
\later{(\prop \wedge \propB)} &\provesIff& \later{\prop} \wedge \later{\propB} \\
\later{(\prop \vee \propB)} &\provesIff& \later{\prop} \vee \later{\propB} \\
\All x. \later\prop &\proves& \later{\All x.\prop} \\
\later\Exists x. \prop &\proves& \later\FALSE \lor {\Exists x.\later\prop} \\
\later\prop &\proves& \later\FALSE \lor (\later\FALSE \Ra \prop) \\
\end{array}
\and
\begin{array}[c]{rMcMl}
\later{\All x.\prop} &\provesIff& \All x. \later\prop \\
\Exists x. \later\prop &\proves& \later{\Exists x.\prop} \\
\later{(\prop * \propB)} &\provesIff& \later\prop * \later\propB
\later{(\prop * \propB)} &\provesIff& \later\prop * \later\propB \\
\always{\later\prop} &\provesIff& \later\always{\prop} \\
\end{array}
\end{mathpar}
A type $\type$ being \emph{inhabited} means that $ \proves \wtt{\term}{\type}$ is derivable for some $\term$.
\paragraph{Laws for the always modality.}
\paragraph{Laws for ghosts and validity.}
\begin{mathpar}
\infer[$\always$I]
{\always{\pfctx} \proves \prop}
{\always{\pfctx} \proves \always{\prop}}
\and
\infer[$\always$E]{}
{\always{\prop} \proves \prop}
\and
\begin{array}[c]{rMcMl}
\always{(\prop \land \propB)} &\proves& \always{(\prop * \propB)} \\
\always{\prop} \land \propB &\proves& \always{\prop} * \propB \\
\always{\later\prop} &\provesIff& \later\always{\prop} \\
\end{array}
\and
\begin{array}[c]{rMcMl}
\always{(\prop \land \propB)} &\provesIff& \always{\prop} \land \always{\propB} \\
\always{(\prop \lor \propB)} &\provesIff& \always{\prop} \lor \always{\propB} \\
\always{\All x. \prop} &\provesIff& \All x. \always{\prop} \\
\always{\Exists x. \prop} &\provesIff& \Exists x. \always{\prop} \\
\begin{array}{rMcMl}
\ownGGhost{\melt} * \ownGGhost{\meltB} &\provesIff& \ownGGhost{\melt \mtimes \meltB} \\
\ownGGhost\melt &\proves& \always{\ownGGhost{\mcore\melt}} \\
\TRUE &\proves& \ownGGhost{\munit} \\
\later\ownGGhost\melt &\proves& \Exists\meltB. \ownGGhost\meltB \land \later(\melt = \meltB)
\end{array}
\and
{ \term =_\type \term' \proves \always \term =_\type \term'}
\infer[valid-intro]
{\melt \in \mval}
{\TRUE \vdash \mval(\melt)}
\and
{ \ownGGhost{\mcore\melt} \proves \always \ownGGhost{\mcore\melt}}
\infer[valid-elim]
{\melt \notin \mval_0}
{\mval(\melt) \proves \FALSE}
\and
{ \mval(\melt) \proves \always \mval(\melt)}
\begin{array}{rMcMl}
\ownGGhost{\melt} &\proves& \mval(\melt) \\
\mval(\melt \mtimes \meltB) &\proves& \mval(\melt) \\
\mval(\melt) &\proves& \always\mval(\melt)
\end{array}
\end{mathpar}
\paragraph{Laws for the update modality.}
\begin{mathpar}
\infer[upd-mono]
{\prop \proves \propB}
{\upd\prop \proves \upd\propB}
\infer[upd-intro]
{}{\prop \proves \upd \prop}
......@@ -388,7 +390,11 @@ A type $\type$ being \emph{inhabited} means that $ \proves \wtt{\term}{\type}$ i
\subsection{Soundness}
The soundness statement of the logic
The soundness statement of the logic reads as follows: For any $n$, we have
\begin{align*}
\lnot(\TRUE \vdash (\upd\later)^n \FALSE)
\end{align*}
where $(\upd\later)^n$ is short for $\upd\later$ being nested $n$ times.
%%% Local Variables:
......
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