From 188a7f624a21ec1dae11ad4caee3d9a28af9ebe3 Mon Sep 17 00:00:00 2001
From: Marco Realacci <marco@marcorealacci.me>
Date: Mon, 31 Mar 2025 10:25:22 +0200
Subject: [PATCH] vault backup: 2025-03-31 10:25:22

---
 .obsidian/workspace.json                  | 26 +++--------------------
 Concurrent Systems/notes/9 - Consensus.md |  7 +++++-
 2 files changed, 9 insertions(+), 24 deletions(-)

diff --git a/.obsidian/workspace.json b/.obsidian/workspace.json
index 681a3d7..11dc2fd 100644
--- a/.obsidian/workspace.json
+++ b/.obsidian/workspace.json
@@ -6,7 +6,6 @@
       {
         "id": "e20232f4bbe0efa1",
         "type": "tabs",
-        "dimension": 43.69627507163324,
         "children": [
           {
             "id": "aa7ed62f1bcd712a",
@@ -14,31 +13,12 @@
             "state": {
               "type": "markdown",
               "state": {
-                "file": "Concurrent Systems/notes/9 - Consensus.md",
+                "file": "Concurrent Systems/notes/2 - Fast mutex by Lamport.md",
                 "mode": "source",
                 "source": false
               },
               "icon": "lucide-file",
-              "title": "9 - Consensus"
-            }
-          }
-        ]
-      },
-      {
-        "id": "a1beaf9777f9958a",
-        "type": "tabs",
-        "dimension": 56.30372492836676,
-        "children": [
-          {
-            "id": "3aaacd443fbaa72c",
-            "type": "leaf",
-            "state": {
-              "type": "pdf",
-              "state": {
-                "file": "Concurrent Systems/slides/class 9.pdf"
-              },
-              "icon": "lucide-file-text",
-              "title": "class 9"
+              "title": "2 - Fast mutex by Lamport"
             }
           }
         ]
@@ -213,8 +193,8 @@
   },
   "active": "aa7ed62f1bcd712a",
   "lastOpenFiles": [
-    "Concurrent Systems/slides/class 9.pdf",
     "Concurrent Systems/notes/9 - Consensus.md",
+    "Concurrent Systems/slides/class 9.pdf",
     "Concurrent Systems/notes/8 - Enhancing Liveness Properties.md",
     "Concurrent Systems/notes/3b - Aravind's algorithm and improvements.md",
     "Concurrent Systems/notes/4 - Semaphores.md",
diff --git a/Concurrent Systems/notes/9 - Consensus.md b/Concurrent Systems/notes/9 - Consensus.md
index cfe3097..2f633ca 100644
--- a/Concurrent Systems/notes/9 - Consensus.md	
+++ b/Concurrent Systems/notes/9 - Consensus.md	
@@ -228,4 +228,9 @@ bmv_propose(i,v) :=
 - Wait freedom: trivial
 - Integrity: trivial
 - Agreement: by agreement of the h binary consensus objects
-- Validity: for all k, we prove that, if dec is the decided value, then there exists a j such that $p_j$ is participating
\ No newline at end of file
+- Validity: for all k, we prove that, if dec is the decided value, then there exists a j such that $p_j$ is participating (`PROP[j] ≠ ⊥`) and `dec[1...k] = PROP[j][1...k]`
+	- by construction, P contains the k-th bits of the proposals whose first (k-1) bits coincide with the first k-1 bits decides so far:
+		- for every  b ∈ P, there exists a j such that `PROP[j] ≠ ⊥`,
+		- `PROP[j][1..k-1] = dec[1..k-1]` and `PROP[j][k] = b`
+	-  whatever b ∈ P is selected in the k-th binary consensus, there exists a j such that `PROP[j] ≠ ⊥` and `PROP[j][1..k] = dec[1..k]`
+	- by taking `k = h`, we can conclude.
\ No newline at end of file