From 10c9f6b716c5a3273d8645b2e7b5758b8cdf60ac Mon Sep 17 00:00:00 2001
From: Louis <me@louisgallet.fr>
Date: Thu, 14 Sep 2023 16:20:59 +0200
Subject: [PATCH] vault backup: 2023-09-14 16:20:59

---
 .obsidian/workspace.json                      | 15 +++++-----
 .../Chapter 4 - A bit of imperative.md        |  2 ++
 Algo/Séminaire/Chapter 5 - Recursivity.md     | 29 ++++++++++++++++++-
 3 files changed, 37 insertions(+), 9 deletions(-)

diff --git a/.obsidian/workspace.json b/.obsidian/workspace.json
index c06623d..d032a0e 100644
--- a/.obsidian/workspace.json
+++ b/.obsidian/workspace.json
@@ -13,7 +13,7 @@
             "state": {
               "type": "markdown",
               "state": {
-                "file": "Algo/Séminaire/Chapter 4 - A bit of imperative.md",
+                "file": "Algo/Séminaire/Chapter 5 - Recursivity.md",
                 "mode": "source",
                 "source": false
               }
@@ -85,7 +85,7 @@
             "state": {
               "type": "backlink",
               "state": {
-                "file": "Algo/Séminaire/Chapter 4 - A bit of imperative.md",
+                "file": "Algo/Séminaire/Chapter 5 - Recursivity.md",
                 "collapseAll": false,
                 "extraContext": false,
                 "sortOrder": "alphabetical",
@@ -102,7 +102,7 @@
             "state": {
               "type": "outgoing-link",
               "state": {
-                "file": "Algo/Séminaire/Chapter 4 - A bit of imperative.md",
+                "file": "Algo/Séminaire/Chapter 5 - Recursivity.md",
                 "linksCollapsed": false,
                 "unlinkedCollapsed": true
               }
@@ -125,7 +125,7 @@
             "state": {
               "type": "outline",
               "state": {
-                "file": "Algo/Séminaire/Chapter 4 - A bit of imperative.md"
+                "file": "Algo/Séminaire/Chapter 5 - Recursivity.md"
               }
             }
           },
@@ -158,11 +158,11 @@
   },
   "active": "9e7be1904dfd4451",
   "lastOpenFiles": [
-    "Algo/Séminaire/assets/unitaire-meme.png",
-    "Algo/Séminaire/Chapter 5 - Recursivity.md",
-    "Algo/Séminaire/Exercices seminaire.md",
     "Algo/Séminaire/Chapter 4 - A bit of imperative.md",
+    "Algo/Séminaire/Chapter 5 - Recursivity.md",
     "Algo/Séminaire/Chapter 3 - Case analysis.md",
+    "Algo/Séminaire/assets/unitaire-meme.png",
+    "Algo/Séminaire/Exercices seminaire.md",
     "Algo/Séminaire/Introduction.md",
     "Algo/Séminaire/Chapter 2 - Functions.md",
     "Algo/Séminaire/assets/fact function response.png",
@@ -185,7 +185,6 @@
     "Mathématiques/Séminaire/Logics",
     "Mathématiques/Séminaire/Logics/Pasted image 20230904101721.png",
     "Mathématiques/Séminaire/Logics/Pasted image 20230904101453.png",
-    "Mathématiques/Séminaire/Logics/Pasted image 20230904100934.png",
     "Mathématiques/Séminaire",
     "2023-09-01.md",
     "Algo",
diff --git a/Algo/Séminaire/Chapter 4 - A bit of imperative.md b/Algo/Séminaire/Chapter 4 - A bit of imperative.md
index 0e5b46e..21d1740 100644
--- a/Algo/Séminaire/Chapter 4 - A bit of imperative.md	
+++ b/Algo/Séminaire/Chapter 4 - A bit of imperative.md	
@@ -1,3 +1,5 @@
+
+![](https://gitea.louisgallet.fr/lgallet/epicours/raw/branch/main/Algo/S%C3%A9minaire/assets/unitaire-meme.png)
 ## 0.1. Print
 If we want to print something in CAML, we have to use this structure
 
diff --git a/Algo/Séminaire/Chapter 5 - Recursivity.md b/Algo/Séminaire/Chapter 5 - Recursivity.md
index 3a57428..0affc11 100644
--- a/Algo/Séminaire/Chapter 5 - Recursivity.md	
+++ b/Algo/Séminaire/Chapter 5 - Recursivity.md	
@@ -1,4 +1,5 @@
-![Alt](https://gitea.louisgallet.fr/lgallet/epicours/raw/branch/main/Algo/S%C3%A9minaire/assets/recursivite-meme.png "recursivity problem")
+
+<center><img src="https://gitea.louisgallet.fr/lgallet/epicours/raw/branch/main/Algo/S%C3%A9minaire/assets/recursivite-meme.png " width=512 height=auto /> </center>
 
 ### 5.1. Simple functions
 To create a recursive function we have to use this structure
@@ -64,3 +65,29 @@ A[ctd 3] --> B[ctd 2] --> C[ctd 1] --> D[ctd 0] --> E[ctd -1] --> F[ ] --> E -->
 > ctd is countdown
 
 ## 5.3. Several recursive calls
+A function that have several recursive calls is defind like this
+$$
+Fn = {1 -> \text{if } n\eqslantless{1} \brace{Fn+1+Fn+2} }
+$$
+**Fibonacci functions**
+```Ocaml
+let rec fibonacci n =
+	if n = 0 then
+		0
+	else if n = 1 then
+		1
+	else
+		fibonacci(n-1) + fibonacci(n-2);;
+
+let add n = 
+	if n = 0 then
+		false
+	else
+		even (n-1);;
+
+let even n =
+	if n = 0 then
+		true
+	else
+		add (n-1);;
+```
\ No newline at end of file