From 4bd8e1019a174922ccab5ebe68ad38be873ece2b Mon Sep 17 00:00:00 2001 From: oskar Date: Tue, 24 Mar 2026 21:24:26 +0100 Subject: [PATCH] device-56.home 2026-3-24:21:24:26 --- .obsidian/plugins/breadcrumbs/data.json | 2 +- .../plugins/obsidian-day-planner/data.json | 2 +- fonction partielle.md | 7 ++++++ schéma mu.md | 25 ++++++++++++++++--- 4 files changed, 30 insertions(+), 6 deletions(-) diff --git a/.obsidian/plugins/breadcrumbs/data.json b/.obsidian/plugins/breadcrumbs/data.json index d0db734c..5c80a004 100644 --- a/.obsidian/plugins/breadcrumbs/data.json +++ b/.obsidian/plugins/breadcrumbs/data.json @@ -651,7 +651,7 @@ "alias": false }, "lock_view": false, - "lock_path": "fonction totale.md" + "lock_path": "schéma mu.md" } }, "codeblocks": { diff --git a/.obsidian/plugins/obsidian-day-planner/data.json b/.obsidian/plugins/obsidian-day-planner/data.json index a20577dc..c6bfff24 100644 --- a/.obsidian/plugins/obsidian-day-planner/data.json +++ b/.obsidian/plugins/obsidian-day-planner/data.json @@ -128,7 +128,7 @@ "url": "https://p41-caldav.icloud.com/published/2/MTAxMTY2MzAzOTMxMDExNnNhSybinMIYEV7aX8u2ivwMf6mQTWIf-UHg5jE9lHwnhgwPCFGf3CeArlRQ0PoddDWYY4uAj2DMucc4RL02mc8", "color": "#40d040" }, - "text": "BEGIN:VCALENDAR\r\nVERSION:2.0\r\nPRODID:-//caldav.icloud.com//CALDAVJ 2610B722//EN\r\nX-WR-CALNAME:oscar EDT\r\nX-APPLE-CALENDAR-COLOR:#63DA38\r\nBEGIN:VEVENT\r\nCREATED:20260119T143640Z\r\nDTEND;TZID=Europe/Paris:20260121T123000\r\nDTSTAMP:20260128T094455Z\r\nDTSTART;TZID=Europe/Paris:20260121T103000\r\nLAST-MODIFIED:20260128T094433Z\r\nRRULE:FREQ=WEEKLY\r\nSEQUENCE:1\r\nSUMMARY:Syntax\\, semantics\\, discourse 2\r\nUID:036A467A-A175-433D-963D-639666B404D2\r\nURL;VALUE=URI:\r\nX-APPLE-CREATOR-IDENTITY:com.apple.calendar\r\nX-APPLE-CREATOR-TEAM-IDENTITY:0000000000\r\nEND:VEVENT\r\nBEGIN:VEVENT\r\nCREATED:20260215T172750Z\r\nDTEND;VALUE=DATE:20260224\r\nDTSTAMP:20260215T172751Z\r\nDTSTART;VALUE=DATE:20260218\r\nLAST-MODIFIED:20260215T172750Z\r\nSEQUENCE:0\r\nSUMMARY:Clara en vacances loin de Paris :(\r\nUID:72E3256D-9936-4E62-A0AC-63C901AE2DCF\r\nX-APPLE-CREATOR-IDENTITY:com.apple.calendar\r\nX-APPLE-CREATOR-TEAM-IDENTITY:0000000000\r\nEND:VEVENT\r\nBEGIN:VEVENT\r\nCREATED:20260111T214137Z\r\nDTEND;TZID=Europe/Paris:20260112T120000\r\nDTSTAMP:20260321T143104Z\r\nDTSTART;TZID=Europe/Paris:20260112T090000\r\nLAST-MODIFIED:20260111T214147Z\r\nLOCATION:137 ODG\r\nRRULE:FREQ=WEEKLY\r\nSEQUENCE:1\r\nSUMMARY:Mathématiques pour non-spécialistes\r\nUID:75E46102-DD64-440A-882D-B5690AAB97C5\r\nX-APPLE-CREATOR-IDENTITY:com.apple.calendar\r\nX-APPLE-CREATOR-TEAM-IDENTITY:0000000000\r\nEND:VEVENT\r\nBEGIN:VEVENT\r\nCREATED:20260321T143056Z\r\nDTEND;TZID=Europe/Paris:20260330T120000\r\nDTSTAMP:20260321T143104Z\r\nDTSTART;TZID=Europe/Paris:20260330T090000\r\nLAST-MODIFIED:20260321T143103Z\r\nLOCATION:137 ODG\r\nRECURRENCE-ID;TZID=Europe/Paris:20260330T090000\r\nSEQUENCE:1\r\nSUMMARY:ORAL Mathématiques pour non-spécialistes\r\nUID:75E46102-DD64-440A-882D-B5690AAB97C5\r\nX-APPLE-CREATOR-IDENTITY:com.apple.calendar\r\nX-APPLE-CREATOR-TEAM-IDENTITY:0000000000\r\nEND:VEVENT\r\nBEGIN:VEVENT\r\nCREATED:20260111T214305Z\r\nDTEND;TZID=Europe/Paris:20260112T163000\r\nDTSTAMP:20260111T224752Z\r\nDTSTART;TZID=Europe/Paris:20260112T133000\r\nLAST-MODIFIED:20260111T224751Z\r\nLOCATION:137 ODG\r\nRRULE:FREQ=WEEKLY\r\nSEQUENCE:2\r\nSUMMARY:Philosophie des mathématiques\r\nUID:E3D4F355-29EF-4058-985D-50A9451DDC3B\r\nX-APPLE-CREATOR-IDENTITY:com.apple.calendar\r\nX-APPLE-CREATOR-TEAM-IDENTITY:0000000000\r\nEND:VEVENT\r\nBEGIN:VEVENT\r\nCREATED:20260111T214443Z\r\nDTEND;TZID=Europe/Paris:20260114T163000\r\nDTSTAMP:20260111T230044Z\r\nDTSTART;TZID=Europe/Paris:20260114T133000\r\nLAST-MODIFIED:20260111T230044Z\r\nLOCATION:370 HAF\r\nRRULE:FREQ=WEEKLY\r\nSEQUENCE:1\r\nSUMMARY:Le savoir mathématique\r\nUID:ED1E92FE-047D-4967-B6F9-E72FE1AAE600\r\nURL;VALUE=URI:\r\nX-APPLE-CREATOR-IDENTITY:com.apple.calendar\r\nX-APPLE-CREATOR-TEAM-IDENTITY:0000000000\r\nEND:VEVENT\r\nBEGIN:VEVENT\r\nCREATED:20260215T143959Z\r\nDTEND;TZID=Europe/Paris:20260326T103000\r\nDTSTAMP:20260215T144000Z\r\nDTSTART;TZID=Europe/Paris:20260326T083000\r\nLAST-MODIFIED:20260215T143959Z\r\nSEQUENCE:0\r\nSUMMARY:partiel incomplétude \r\nUID:F78C5568-42F6-4285-BF8F-D29D356CE702\r\nURL;VALUE=URI:message:%3Cf994109f597b0c45f9dc8368fe08e65a52f8387355fab681\r\n 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+ "text": "BEGIN:VCALENDAR\r\nVERSION:2.0\r\nPRODID:-//caldav.icloud.com//CALDAVJ 2610B820//EN\r\nX-WR-CALNAME:oscar EDT\r\nX-APPLE-CALENDAR-COLOR:#63DA38\r\nBEGIN:VEVENT\r\nCREATED:20260119T143640Z\r\nDTEND;TZID=Europe/Paris:20260121T123000\r\nDTSTAMP:20260128T094455Z\r\nDTSTART;TZID=Europe/Paris:20260121T103000\r\nLAST-MODIFIED:20260128T094433Z\r\nRRULE:FREQ=WEEKLY\r\nSEQUENCE:1\r\nSUMMARY:Syntax\\, semantics\\, discourse 2\r\nUID:036A467A-A175-433D-963D-639666B404D2\r\nURL;VALUE=URI:\r\nX-APPLE-CREATOR-IDENTITY:com.apple.calendar\r\nX-APPLE-CREATOR-TEAM-IDENTITY:0000000000\r\nEND:VEVENT\r\nBEGIN:VEVENT\r\nCREATED:20260324T200649Z\r\nDTEND;TZID=Europe/Paris:20260404T230000\r\nDTSTAMP:20260324T200653Z\r\nDTSTART;TZID=Europe/Paris:20260404T190000\r\nLAST-MODIFIED:20260324T200652Z\r\nSEQUENCE:1\r\nSUMMARY:option lecture de spinoza\r\nUID:328073C4-A8F2-4D5B-9EFC-E48919453390\r\nX-APPLE-CREATOR-IDENTITY:com.apple.calendar\r\nX-APPLE-CREATOR-TEAM-IDENTITY:0000000000\r\nEND:VEVENT\r\nBEGIN:VEVENT\r\nCREATED:20260215T172750Z\r\nDTEND;VALUE=DATE:20260224\r\nDTSTAMP:20260215T172751Z\r\nDTSTART;VALUE=DATE:20260218\r\nLAST-MODIFIED:20260215T172750Z\r\nSEQUENCE:0\r\nSUMMARY:Clara en vacances loin de Paris :(\r\nUID:72E3256D-9936-4E62-A0AC-63C901AE2DCF\r\nX-APPLE-CREATOR-IDENTITY:com.apple.calendar\r\nX-APPLE-CREATOR-TEAM-IDENTITY:0000000000\r\nEND:VEVENT\r\nBEGIN:VEVENT\r\nCREATED:20260111T214137Z\r\nDTEND;TZID=Europe/Paris:20260112T120000\r\nDTSTAMP:20260321T143104Z\r\nDTSTART;TZID=Europe/Paris:20260112T090000\r\nLAST-MODIFIED:20260111T214147Z\r\nLOCATION:137 ODG\r\nRRULE:FREQ=WEEKLY\r\nSEQUENCE:1\r\nSUMMARY:Mathématiques pour non-spécialistes\r\nUID:75E46102-DD64-440A-882D-B5690AAB97C5\r\nX-APPLE-CREATOR-IDENTITY:com.apple.calendar\r\nX-APPLE-CREATOR-TEAM-IDENTITY:0000000000\r\nEND:VEVENT\r\nBEGIN:VEVENT\r\nCREATED:20260321T143056Z\r\nDTEND;TZID=Europe/Paris:20260330T120000\r\nDTSTAMP:20260321T143104Z\r\nDTSTART;TZID=Europe/Paris:20260330T090000\r\nLAST-MODIFIED:20260321T143103Z\r\nLOCATION:137 ODG\r\nRECURRENCE-ID;TZID=Europe/Paris:20260330T090000\r\nSEQUENCE:1\r\nSUMMARY:ORAL Mathématiques pour non-spécialistes\r\nUID:75E46102-DD64-440A-882D-B5690AAB97C5\r\nX-APPLE-CREATOR-IDENTITY:com.apple.calendar\r\nX-APPLE-CREATOR-TEAM-IDENTITY:0000000000\r\nEND:VEVENT\r\nBEGIN:VEVENT\r\nCREATED:20260111T214305Z\r\nDTEND;TZID=Europe/Paris:20260112T163000\r\nDTSTAMP:20260111T224752Z\r\nDTSTART;TZID=Europe/Paris:20260112T133000\r\nLAST-MODIFIED:20260111T224751Z\r\nLOCATION:137 ODG\r\nRRULE:FREQ=WEEKLY\r\nSEQUENCE:2\r\nSUMMARY:Philosophie des mathématiques\r\nUID:E3D4F355-29EF-4058-985D-50A9451DDC3B\r\nX-APPLE-CREATOR-IDENTITY:com.apple.calendar\r\nX-APPLE-CREATOR-TEAM-IDENTITY:0000000000\r\nEND:VEVENT\r\nBEGIN:VEVENT\r\nCREATED:20260111T214443Z\r\nDTEND;TZID=Europe/Paris:20260114T163000\r\nDTSTAMP:20260111T230044Z\r\nDTSTART;TZID=Europe/Paris:20260114T133000\r\nLAST-MODIFIED:20260111T230044Z\r\nLOCATION:370 HAF\r\nRRULE:FREQ=WEEKLY\r\nSEQUENCE:1\r\nSUMMARY:Le savoir mathématique\r\nUID:ED1E92FE-047D-4967-B6F9-E72FE1AAE600\r\nURL;VALUE=URI:\r\nX-APPLE-CREATOR-IDENTITY:com.apple.calendar\r\nX-APPLE-CREATOR-TEAM-IDENTITY:0000000000\r\nEND:VEVENT\r\nBEGIN:VEVENT\r\nCREATED:20260215T143959Z\r\nDTEND;TZID=Europe/Paris:20260326T103000\r\nDTSTAMP:20260215T144000Z\r\nDTSTART;TZID=Europe/Paris:20260326T083000\r\nLAST-MODIFIED:20260215T143959Z\r\nSEQUENCE:0\r\nSUMMARY:partiel incomplétude \r\nUID:F78C5568-42F6-4285-BF8F-D29D356CE702\r\nURL;VALUE=URI:message:%3Cf994109f597b0c45f9dc8368fe08e65a52f8387355fab681\r\n 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} ] } \ No newline at end of file diff --git a/fonction partielle.md b/fonction partielle.md index ea83c67a..60beb6ed 100644 --- a/fonction partielle.md +++ b/fonction partielle.md @@ -30,6 +30,13 @@ aliases: > - $h(x_1, x_2, \dots, x_{p})$ n'est pas définie si l'une des $f_{i}(x_1, x_2, \dots, x_{p})$ n'est pas défini ou si, toutes l'étant, $g(f_1(x_1, x_2, \dots, x_{p}), f_2(x_1, x_2, \dots, x_{p}), \dots, f_{n}(x_1, x_2, \dots, x_{p}))$ n'est pas définie > - Dans le cas contraire, $h(x_1, x_2, \dots, x_{p}) := g(f_1(x_1, x_2, \dots, x_{p}), f_2(x_1, x_2, \dots, x_{p}), \dots, f_{n}(x_1, x_2, \dots, x_{p}))$ +> [!definition] Récurrence +> Soient $g \in \mathscr{F}^{*}_{p}$ et $h \in \mathscr{F}^{*}_{p+2}$ +> Il existe une et une seule fonction $f \in \mathscr{F}^{*}_{p+1}$ vérifiant les conditions suivantes : +> - $\forall (\overline{x}) \in \mathbb{N}^{p},\quad f(\overline{x}, 0) = g(\overline{x})$ (donc, $f(\overline{x}, 0)$ est définie si et seulement si $g(\overline{x})$ l'est, et lui est égale dans ce cas) +> - $\forall (\overline{x}, y) \in \mathbb{N}^{p+1},\quad f(\overline{x}, y+1) = h (\overline{x}, y, f(\overline{x}, y)$ (même remarque que plus haut) +> - i on dira que $f$ est **définie par récurrence à partir de $g$ et $h$** + # Propriétés # Exemples diff --git a/schéma mu.md b/schéma mu.md index 7bc0c6c8..60307ab7 100644 --- a/schéma mu.md +++ b/schéma mu.md @@ -1,6 +1,6 @@ --- up: - - "[[fonction récursive]]" + - "[[fonction partielle]]" tags: - s/maths/logique - s/informatique @@ -10,11 +10,28 @@ aliases: --- > [!definition] [[schéma mu]] -> Soit $A \subseteq \mathbb{N}^{p+1}$ -> La fonction $F$ définie par le schéma $\mu$ sur $A$ est alors la fonction : -> $\begin{array}{rrcl} f :& \mathbb{N}^{p} &\to& \mathbb{N}\\ &(x_1, x_2, \dots, x_{p}) &\mapsto& \text{le plus petit } z \text{ tel que } (x_1, x_2, \dots, x_{p}, z) \in A \end{array}$ +> Soit $f \in \mathscr{F}^{*}_{p+1}$, alors la [[fonction partielle]] : +> $g(\overline{x}) = \mu y(f(\overline{x}, y) = 0)$ +> est définie de la façon suivante : +> - S'il existe au moins un entier $z$ tel que $f(\overline{x}, z)$ soit nul et que, pour tout $z' < z$, $f(\overline{x}, z')$ soit définie, alors $g(\overline{x})$ est le plus petit de ces entiers $z$ +> - Dans le as contraire, $g(\overline{x})$ n'est pas définie. +> --- +> Pour les ensembles: +> Si $A \subseteq \mathbb{N}^{p+1}$, alors : +> $\mu y((\overline{x}, y) \in A) = \mu y(1 \dot{-} \chi _{A}(\overline{x}, y) = 0)$ ^definition +> [!definition] [[schéma mu]] – Définition courte +> Soit $f \in \mathscr{F}^{*}_{p+1}$ +> Le schéma $\mu$ permet de définir $g(\overline{x}) = \mu y(f(\overline{x}, y) = 0)$ +> Cette fonction $g$ est la fonction qui donne le plus petit $z$ + +> [!idea] Intuition +> L'idée est de préserver la propriété de posséder un algorithme de calcul. +> Pour calculer $\mu y (f(\overline{x}, y)=0)$, on cherchera itérativement une valeur de $y$ en commençant par 0. +> Le problème est que, si $f(\overline{x}, y)$ n'est pas définie pour l'une de ces valeurs, notre algorithme ne pourrait pas la calculer. +> C'est pour cela que, si une des valeurs n'est pas dans le [[fonction partielle#^definition|domaine de définition]] de $f$, on considèrera que la recherche s'arrête ici. + # Propriétés # Exemples