Merge branch 'ralf/misc-lemmas' into 'master'

add pair_included, auth_frag_core, singleton_core_total

See merge request iris/iris!421

theories/algebra/auth.v

@@ -286,6 +286,8 @@ Lemma auth_frag_op a b : ◯ (a ⋅ b) = ◯ a ⋅ ◯ b.
 Proof. done. Qed.
 Lemma auth_frag_mono a b : a ≼ b → ◯ a ≼ ◯ b.
 Proof. intros [c ->]. rewrite auth_frag_op. apply cmra_included_l. Qed.
+Lemma auth_frag_core a : core (◯ a) = ◯ (core a).
+Proof. done. Qed.
 
 Global Instance auth_frag_sep_homomorphism :
   MonoidHomomorphism op op (≡) (@auth_frag A).

theories/algebra/cmra.v

@@ -1221,6 +1221,9 @@ Section prod.
   Proof. done. Qed.
   Lemma pair_valid (a : A) (b : B) : ✓ (a, b) ↔ ✓ a ∧ ✓ b.
   Proof. done. Qed.
+  Lemma pair_included (a a' : A) (b b' : B) :
+    (a, b) ≼ (a', b') ↔ a ≼ a' ∧ b ≼ b'.
+  Proof. apply prod_included. Qed.
 
   Lemma pair_core `{!CmraTotal A, !CmraTotal B} (a : A) (b : B) :
     core (a, b) = (core a, core b).

theories/algebra/gmap.v

@@ -285,15 +285,18 @@ Proof.
 Qed.
 
 Lemma core_singleton (i : K) (x : A) cx :
-  pcore x = Some cx → core ({[ i := x ]} : gmap K A) = {[ i := cx ]}.
+  pcore x = Some cx → core {[ i := x ]} =@{gmap K A} {[ i := cx ]}.
 Proof. apply omap_singleton. Qed.
 Lemma core_singleton' (i : K) (x : A) cx :
-  pcore x ≡ Some cx → core ({[ i := x ]} : gmap K A) ≡ {[ i := cx ]}.
+  pcore x ≡ Some cx → core {[ i := x ]} ≡@{gmap K A} {[ i := cx ]}.
 Proof.
   intros (cx'&?&->)%equiv_Some_inv_r'. by rewrite (core_singleton _ _ cx').
 Qed.
+Lemma singleton_core_total `{!CmraTotal A} (i : K) (x : A) :
+  core {[ i := x ]} =@{gmap K A} {[ i := core x ]}.
+Proof. apply core_singleton. rewrite cmra_pcore_core //. Qed.
 Lemma op_singleton (i : K) (x y : A) :
-  {[ i := x ]} ⋅ {[ i := y ]} = ({[ i := x ⋅ y ]} : gmap K A).
+  {[ i := x ]} ⋅ {[ i := y ]} =@{gmap K A} {[ i := x ⋅ y ]}.
 Proof. by apply (merge_singleton _ _ _ x y). Qed.
 Global Instance is_op_singleton i a a1 a2 :
   IsOp a a1 a2 → IsOp' ({[ i := a ]} : gmap K A) {[ i := a1 ]} {[ i := a2 ]}.