From 5ea9eab2179b7880c5ac1393b9fa7ec3701509a8 Mon Sep 17 00:00:00 2001
From: Robbert Krebbers <mail@robbertkrebbers.nl>
Date: Fri, 3 Apr 2020 18:37:11 +0200
Subject: [PATCH] =?UTF-8?q?Add=20lemma=20`singleton=5Fincluded=20:=20{[=20?=
 =?UTF-8?q?i=20:=3D=20x=20]}=20=E2=89=BC=20({[=20i=20:=3D=20y=20]}=20?=
 =?UTF-8?q?=E2=86=94=20x=20=E2=89=A1=20y=20=E2=88=A8=20x=20=E2=89=BC=20y`.?=
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Rename existing asymmetric lemma `singleton_included` into `singleton_included_l`.
---
 theories/algebra/gmap.v             | 16 ++++++++++++----
 theories/base_logic/lib/gen_heap.v  |  2 +-
 theories/base_logic/lib/proph_map.v |  2 +-
 3 files changed, 14 insertions(+), 6 deletions(-)

diff --git a/theories/algebra/gmap.v b/theories/algebra/gmap.v
index 3a5cbe52a..51ae2084d 100644
--- a/theories/algebra/gmap.v
+++ b/theories/algebra/gmap.v
@@ -308,7 +308,7 @@ Global Instance gmap_singleton_core_id i (x : A) :
   CoreId x → CoreId {[ i := x ]}.
 Proof. intros. by apply core_id_total, core_singleton'. Qed.
 
-Lemma singleton_includedN n m i x :
+Lemma singleton_includedN_l n m i x :
   {[ i := x ]} ≼{n} m ↔ ∃ y, m !! i ≡{n}≡ Some y ∧ Some x ≼{n} Some y.
 Proof.
   split.
@@ -321,7 +321,7 @@ Proof.
     + by rewrite lookup_op lookup_singleton_ne// lookup_partial_alter_ne// left_id.
 Qed.
 (* We do not have [x ≼ y ↔ ∀ n, x ≼{n} y], so we cannot use the previous lemma *)
-Lemma singleton_included m i x :
+Lemma singleton_included_l m i x :
   {[ i := x ]} ≼ m ↔ ∃ y, m !! i ≡ Some y ∧ Some x ≼ Some y.
 Proof.
   split.
@@ -333,13 +333,21 @@ Proof.
     + by rewrite lookup_op lookup_singleton lookup_partial_alter Hi.
     + by rewrite lookup_op lookup_singleton_ne// lookup_partial_alter_ne// left_id.
 Qed.
-Lemma singleton_included_exclusive m i x :
+Lemma singleton_included_exclusive_l m i x :
   Exclusive x → ✓ m →
   {[ i := x ]} ≼ m ↔ m !! i ≡ Some x.
 Proof.
-  intros ? Hm. rewrite singleton_included. split; last by eauto.
+  intros ? Hm. rewrite singleton_included_l. split; last by eauto.
   intros (y&?&->%(Some_included_exclusive _)); eauto using lookup_valid_Some.
 Qed.
+Lemma singleton_included i x y :
+  {[ i := x ]} ≼ ({[ i := y ]} : gmap K A) ↔ x ≡ y ∨ x ≼ y.
+Proof.
+  rewrite singleton_included_l. split.
+  - intros (y'&Hi&?). rewrite lookup_insert in Hi.
+    apply Some_included. by rewrite Hi.
+  - intros ?. exists y. by rewrite lookup_insert Some_included.
+Qed.
 
 Global Instance singleton_cancelable i x :
   Cancelable (Some x) → Cancelable {[ i := x ]}.
diff --git a/theories/base_logic/lib/gen_heap.v b/theories/base_logic/lib/gen_heap.v
index 4a3065576..1ee1d569d 100644
--- a/theories/base_logic/lib/gen_heap.v
+++ b/theories/base_logic/lib/gen_heap.v
@@ -146,7 +146,7 @@ Section to_gen_heap.
   Lemma gen_heap_singleton_included σ l q v :
     {[l := (q, to_agree v)]} ≼ to_gen_heap σ → σ !! l = Some v.
   Proof.
-    rewrite singleton_included=> -[[q' av] []].
+    rewrite singleton_included_l=> -[[q' av] []].
     rewrite /to_gen_heap lookup_fmap fmap_Some_equiv => -[v' [Hl [/= -> ->]]].
     move=> /Some_pair_included_total_2 [_] /to_agree_included /leibniz_equiv_iff -> //.
   Qed.
diff --git a/theories/base_logic/lib/proph_map.v b/theories/base_logic/lib/proph_map.v
index 4b4c1c117..6793cd7a7 100644
--- a/theories/base_logic/lib/proph_map.v
+++ b/theories/base_logic/lib/proph_map.v
@@ -102,7 +102,7 @@ Section to_proph_map.
   Lemma proph_map_singleton_included R p vs :
     {[p := Excl vs]} ≼ to_proph_map R → R !! p = Some vs.
   Proof.
-    rewrite singleton_included_exclusive; last by apply to_proph_map_valid.
+    rewrite singleton_included_exclusive_l; last by apply to_proph_map_valid.
     by rewrite leibniz_equiv_iff /to_proph_map lookup_fmap fmap_Some=> -[v' [-> [->]]].
   Qed.
 End to_proph_map.
-- 
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