Add `wp_stuckness_mono`.

This is derived from `wp_forget_not_stuck` and a trivial preorder on
stuckness bits. (The two lemmas are redundant, but I have examples
where each seems more natural than the other.)

I did *not* bake `wp_stuckness_mono` into `strong_mono` for two
reasons. Mainly, I didn't see a nice way to combine the two proofs
(beyond `cut`). Less important, changing the type of `wp_strong_mono`
will break code.

theories/program_logic/hoare.v

@@ -57,6 +57,9 @@ Proof. solve_proper. Qed.
 Lemma ht_mono s E P P' Φ Φ' e :
   (P ⊢ P') → (∀ v, Φ' v ⊢ Φ v) → {{ P' }} e @ s; E {{ Φ' }} ⊢ {{ P }} e @ s; E {{ Φ }}.
 Proof. by intros; apply persistently_mono, wand_mono, wp_mono. Qed.
+Lemma ht_stuckness_mono s1 s2 E P Φ e :
+  (s1 ≤ s2)%stuckness → {{ P }} e @ s1; E {{ Φ }} ⊢ {{ P }} e @ s2; E {{ Φ }}.
+Proof. by intros; apply persistently_mono, wand_mono, wp_stuckness_mono. Qed.
 Global Instance ht_mono' s E :
   Proper (flip (⊢) ==> eq ==> pointwise_relation _ (⊢) ==> (⊢)) (ht s E).
 Proof. solve_proper. Qed.

theories/program_logic/language.v

@@ -44,6 +44,18 @@ Instance language_ctx_id Λ : LanguageCtx Λ id.
 Proof. constructor; naive_solver. Qed.
 
 Variant stuckness := not_stuck | maybe_stuck.
+Definition stuckness_le (s1 s2 : stuckness) : bool :=
+  match s1, s2 with
+  | maybe_stuck, not_stuck => false
+  | _, _ => true
+  end.
+Instance: @PreOrder stuckness stuckness_le.
+Proof.
+  split; first by case. move=>s1 s2 s3. by case: s1; case: s2; case: s3.
+Qed.
+Bind Scope stuckness_scope with stuckness.
+Delimit Scope stuckness_scope with stuckness.
+Infix "≤" := stuckness_le : stuckness_scope.
 
 Section language.
   Context {Λ : language}.

theories/program_logic/weakestpre.v

@@ -311,6 +311,9 @@ Proof.
   iIntros (HΦ) "H"; iApply (wp_strong_mono s E E); auto.
   iIntros "{$H}" (v) "?". by iApply HΦ.
 Qed.
+Lemma wp_stuckness_mono s1 s2 E e Φ :
+  (s1 ≤ s2)%stuckness → WP e @ s1; E {{ Φ }} ⊢ WP e @ s2; E {{ Φ }}.
+Proof. case: s1; case: s2 => // _. exact: wp_forget_not_stuck. Qed.
 Lemma wp_mask_mono s E1 E2 e Φ : E1 ⊆ E2 → WP e @ s; E1 {{ Φ }} ⊢ WP e @ s; E2 {{ Φ }}.
 Proof. iIntros (?) "H"; iApply (wp_strong_mono s E1 E2); auto. iFrame; eauto. Qed.
 Global Instance wp_mono' s E e :