Remove axiom that emp is timeless.

As Aleš observed, in the ordered RA model it is not, unless the order on the
unit is timeless.

theories/base_logic/upred.v

@@ -461,7 +461,7 @@ Proof.
 Qed.
 
 Lemma uPred_sbi_mixin (M : ucmraT) : SBIMixin
-  uPred_entails uPred_emp uPred_pure uPred_or uPred_impl
+  uPred_entails uPred_pure uPred_or uPred_impl
   (@uPred_forall M) (@uPred_exist M) (@uPred_internal_eq M)
   uPred_sep uPred_persistently uPred_later.
 Proof.
@@ -483,8 +483,6 @@ Proof.
     intros A Φ. unseal; by split=> -[|n] x.
   - (* (▷ ∃ a, Φ a) ⊢ ▷ False ∨ (∃ a, ▷ Φ a) *)
     intros A Φ. unseal; split=> -[|[|n]] x /=; eauto.
-  - (* ▷ emp ⊢ ▷ False ∨ emp *)
-    rewrite /uPred_emp. unseal; split; by right.
   - (* ▷ (P ∗ Q) ⊢ ▷ P ∗ ▷ Q *)
     intros P Q. unseal; split=> -[|n] x ? /=.
     { by exists x, (core x); rewrite cmra_core_r. }

theories/bi/big_op.v

@@ -687,10 +687,10 @@ Section list.
     ▷^n ([∗ list] k↦x ∈ l, Φ k x) ⊣⊢ ([∗ list] k↦x ∈ l, ▷^n Φ k x).
   Proof. apply (big_opL_commute _). Qed.
 
-  Global Instance big_sepL_nil_timeless Φ :
+  Global Instance big_sepL_nil_timeless `{!Timeless (emp%I : PROP)} Φ :
     Timeless ([∗ list] k↦x ∈ [], Φ k x).
   Proof. simpl; apply _. Qed.
-  Global Instance big_sepL_timeless Φ l :
+  Global Instance big_sepL_timeless `{!Timeless (emp%I : PROP)} Φ l :
     (∀ k x, Timeless (Φ k x)) → Timeless ([∗ list] k↦x ∈ l, Φ k x).
   Proof. revert Φ. induction l as [|x l IH]=> Φ ? /=; apply _. Qed.
   Global Instance big_sepL_timeless_id `{!Timeless (emp%I : PROP)} Ps :
@@ -712,10 +712,10 @@ Section gmap.
     ▷^n ([∗ map] k↦x ∈ m, Φ k x) ⊣⊢ ([∗ map] k↦x ∈ m, ▷^n Φ k x).
   Proof. apply (big_opM_commute _). Qed.
 
-  Global Instance big_sepM_nil_timeless Φ :
+  Global Instance big_sepM_nil_timeless `{!Timeless (emp%I : PROP)} Φ :
     Timeless ([∗ map] k↦x ∈ ∅, Φ k x).
   Proof. rewrite /big_opM map_to_list_empty. apply _. Qed.
-  Global Instance big_sepM_timeless Φ m :
+  Global Instance big_sepM_timeless `{!Timeless (emp%I : PROP)} Φ m :
     (∀ k x, Timeless (Φ k x)) → Timeless ([∗ map] k↦x ∈ m, Φ k x).
   Proof. intros. apply big_sepL_timeless=> _ [??]; apply _. Qed.
 End gmap.
@@ -734,9 +734,10 @@ Section gset.
     ▷^n ([∗ set] y ∈ X, Φ y) ⊣⊢ ([∗ set] y ∈ X, ▷^n Φ y).
   Proof. apply (big_opS_commute _). Qed.
 
-  Global Instance big_sepS_nil_timeless Φ : Timeless ([∗ set] x ∈ ∅, Φ x).
+  Global Instance big_sepS_nil_timeless `{!Timeless (emp%I : PROP)} Φ :
+    Timeless ([∗ set] x ∈ ∅, Φ x).
   Proof. rewrite /big_opS elements_empty. apply _. Qed.
-  Global Instance big_sepS_timeless Φ X :
+  Global Instance big_sepS_timeless `{!Timeless (emp%I : PROP)} Φ X :
     (∀ x, Timeless (Φ x)) → Timeless ([∗ set] x ∈ X, Φ x).
   Proof. rewrite /big_opS. apply _. Qed.
 End gset.
@@ -755,9 +756,10 @@ Section gmultiset.
     ▷^n ([∗ mset] y ∈ X, Φ y) ⊣⊢ ([∗ mset] y ∈ X, ▷^n Φ y).
   Proof. apply (big_opMS_commute _). Qed.
 
-  Global Instance big_sepMS_nil_timeless Φ : Timeless ([∗ mset] x ∈ ∅, Φ x).
+  Global Instance big_sepMS_nil_timeless `{!Timeless (emp%I : PROP)} Φ :
+    Timeless ([∗ mset] x ∈ ∅, Φ x).
   Proof. rewrite /big_opMS gmultiset_elements_empty. apply _. Qed.
-  Global Instance big_sepMS_timeless Φ X :
+  Global Instance big_sepMS_timeless `{!Timeless (emp%I : PROP)} Φ X :
     (∀ x, Timeless (Φ x)) → Timeless ([∗ mset] x ∈ X, Φ x).
   Proof. rewrite /big_opMS. apply _. Qed.
 End gmultiset.

theories/bi/derived.v

@@ -1554,13 +1554,14 @@ Proof. by rewrite {1}(except_0_intro Q) except_0_sep. Qed.
 Global Instance Timeless_proper : Proper ((≡) ==> iff) (@Timeless PROP).
 Proof. solve_proper. Qed.
 
-Global Instance emp_timeless : Timeless (emp : PROP)%I.
-Proof. apply later_emp_false. Qed.
 Global Instance pure_timeless φ : Timeless (⌜φ⌝ : PROP)%I.
 Proof.
   rewrite /Timeless /bi_except_0 pure_alt later_exist_false.
   apply or_elim, exist_elim; [auto|]=> Hφ. rewrite -(exist_intro Hφ). auto.
 Qed.
+Global Instance emp_timeless `{AffineBI PROP} : Timeless (emp : PROP)%I.
+Proof. rewrite -True_emp. apply _. Qed.
+
 Global Instance and_timeless P Q : Timeless P → Timeless Q → Timeless (P ∧ Q).
 Proof. intros; rewrite /Timeless except_0_and later_and; auto. Qed.
 Global Instance or_timeless P Q : Timeless P → Timeless Q → Timeless (P ∨ Q).

theories/bi/interface.v

@@ -127,7 +127,6 @@ Section bi_mixin.
     sbi_mixin_later_forall_2 {A} (Φ : A → PROP) : (∀ a, ▷ Φ a) ⊢ ▷ ∀ a, Φ a;
     sbi_mixin_later_exist_false {A} (Φ : A → PROP) :
       (▷ ∃ a, Φ a) ⊢ ▷ False ∨ (∃ a, ▷ Φ a);
-    sbi_mixin_later_emp_false : ▷ emp ⊢ ▷ False ∨ emp;
     sbi_mixin_later_sep_1 P Q : ▷ (P ∗ Q) ⊢ ▷ P ∗ ▷ Q;
     sbi_mixin_later_sep_2 P Q : ▷ P ∗ ▷ Q ⊢ ▷ (P ∗ Q);
     sbi_mixin_later_persistently_1 P : ▷ □ P ⊢ □ ▷ P;
@@ -213,7 +212,7 @@ Structure sbi := SBI {
   sbi_bi_mixin : BIMixin sbi_entails sbi_emp sbi_pure sbi_and sbi_or sbi_impl
                          sbi_forall sbi_exist sbi_internal_eq
                          sbi_sep sbi_wand sbi_persistently;
-  sbi_sbi_mixin : SBIMixin sbi_entails sbi_emp sbi_pure sbi_or sbi_impl
+  sbi_sbi_mixin : SBIMixin sbi_entails sbi_pure sbi_or sbi_impl
                            sbi_forall sbi_exist sbi_internal_eq
                            sbi_sep sbi_persistently bi_later;
 }.
@@ -447,8 +446,6 @@ Proof. eapply sbi_mixin_later_forall_2, sbi_sbi_mixin. Qed.
 Lemma later_exist_false {A} (Φ : A → PROP) :
   (▷ ∃ a, Φ a) ⊢ ▷ False ∨ (∃ a, ▷ Φ a).
 Proof. eapply sbi_mixin_later_exist_false, sbi_sbi_mixin. Qed.
-Lemma later_emp_false : ▷ (emp : PROP) ⊢ ▷ False ∨ emp.
-Proof. eapply sbi_mixin_later_emp_false, sbi_sbi_mixin. Qed.
 Lemma later_sep_1 P Q : ▷ (P ∗ Q) ⊢ ▷ P ∗ ▷ Q.
 Proof. eapply sbi_mixin_later_sep_1, sbi_sbi_mixin. Qed.
 Lemma later_sep_2 P Q : ▷ P ∗ ▷ Q ⊢ ▷ (P ∗ Q).