CH1 Programmation fonctionnelle et récursivité #1.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### CH1 Programmation fonctionnelle et récursivité  #1\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 1 Pourquoi des fonctions ? \n",
    "### Modularité et tests unitaires\n",
    "\n",
    "\n",
    "Conjecture de Golbach \n",
    "\n",
    "> \"Tout nombre entier pair supérieur à 3 peut s’écrire comme la somme de deux nombres premiers\"\n",
    "\n",
    "ex : 4 = 3+1  6 = 5+1 , 8 = 5+3 etc ..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import math"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Nous avons tout d'abord besoin d'un **Test de primalité** :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "votre nombre ?20\n",
      "20 = 2 x 10\n",
      "False\n"
     ]
    }
   ],
   "source": [
    "# a est-il un nombre premier ?\n",
    "a=int(input('votre nombre ?'))\n",
    "lim = int(math.sqrt(a))\n",
    "premier=True\n",
    "for i in range(2,lim+1):\n",
    "    if a%i==0:\n",
    "        premier=False\n",
    "        print(a,'=',i,'x',a//i)\n",
    "        break\n",
    "print(premier)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\n",
    "\n",
    "Nous allons réaliser un programme qui va vérifier la conjecture de Golbach\n",
    "```\n",
    "Entrer a\n",
    "verifié = Faux\n",
    "pour i allant de 1 à a :\n",
    "    si i est premier et a-i est premier \n",
    "    alors vérifié = vrai \n",
    "```\n",
    "il y a deux tests de primalité, alors nous allons créer une fonction réutilisable deux fois !\n",
    "Elle sera aussi réutilisable pour d'autres problèmes d'arithmétique utilisant la primalité.\n",
    "On appelle cela la ***modularité***.\n",
    "Les avantages de la modularité :\n",
    "\n",
    "- code utilisable plusieurs fois dans et à l'extérieur de notre programme\n",
    "- possibilité de faire des tests unitaires\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "def premier(a:int):\n",
    "    '''\n",
    "    :param  a(int): le nombte entier\n",
    "    :returns  bool, Vrai si a est premier\n",
    "    nécessite le module math\n",
    "    exemples :\n",
    "    >>> premier(3)\n",
    "    True\n",
    "    >>> premier(6)\n",
    "    False\n",
    "    >>> premier(1)\n",
    "    False\n",
    "    '''\n",
    "    if a==1 :\n",
    "        return False     # 1 n'est pas un nombre premier\n",
    "    lim = int(math.sqrt(a))\n",
    "    for i in range(2,lim+1):\n",
    "        if a%i==0:\n",
    "            return False        \n",
    "    return True"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Test unitaires : le code suivant va exécuter les tests saisis dans le docstring après le signe >>>.\n",
    "\n",
    "On parle de TDD Test Driven Dev (developpement piloté par les tests) : les tests sont définis avant de faire la fonction."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "import doctest"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Trying:\n",
      "    premier(3)\n",
      "Expecting:\n",
      "    True\n",
      "ok\n",
      "Trying:\n",
      "    premier(6)\n",
      "Expecting:\n",
      "    False\n",
      "ok\n",
      "Trying:\n",
      "    premier(1)\n",
      "Expecting:\n",
      "    False\n",
      "ok\n",
      "3 items had no tests:\n",
      "    __main__\n",
      "    __main__.goldbach\n",
      "    __main__.test\n",
      "1 items passed all tests:\n",
      "   3 tests in __main__.premier\n",
      "3 tests in 4 items.\n",
      "3 passed and 0 failed.\n",
      "Test passed.\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "TestResults(failed=0, attempted=3)"
      ]
     },
     "execution_count": 22,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "doctest.testmod(verbose = True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Vérification de la conjecture de Goldbach\n",
    "\n",
    "Au passage, on voit en début de fonction la commande assert qui vérifie la précondition"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "def goldbach(n:int):\n",
    "    '''\n",
    "    :param n (int): nombre entier pair > 3\n",
    "    :returns (tuple) : la somme trouvée, (n,0) sinon\n",
    "    '''\n",
    "    assert n%2==0 and n>3, \"le nombre n'est pas pair et > 3\"\n",
    "    for a in range(n):\n",
    "        if premier(a) and premier(n-a):\n",
    "            return a,n-a\n",
    "    return n,0  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(3, 997)\n"
     ]
    }
   ],
   "source": [
    "print(goldbach(1000))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "def test(n:int):\n",
    "    '''\n",
    "    param n (int) : limite du test\n",
    "    returns : None affichage des couples\n",
    "    '''\n",
    "    for i in range(4,n+1,2):\n",
    "        print(i,goldbach(i))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
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      "908 (31, 877)\n",
      "910 (3, 907)\n",
      "912 (5, 907)\n",
      "914 (3, 911)\n",
      "916 (5, 911)\n",
      "918 (7, 911)\n",
      "920 (13, 907)\n",
      "922 (3, 919)\n",
      "924 (5, 919)\n",
      "926 (7, 919)\n",
      "928 (17, 911)\n",
      "930 (11, 919)\n",
      "932 (3, 929)\n",
      "934 (5, 929)\n",
      "936 (7, 929)\n",
      "938 (19, 919)\n",
      "940 (3, 937)\n",
      "942 (5, 937)\n",
      "944 (3, 941)\n",
      "946 (5, 941)\n",
      "948 (7, 941)\n",
      "950 (3, 947)\n",
      "952 (5, 947)\n",
      "954 (7, 947)\n",
      "956 (3, 953)\n",
      "958 (5, 953)\n",
      "960 (7, 953)\n",
      "962 (43, 919)\n",
      "964 (11, 953)\n",
      "966 (13, 953)\n",
      "968 (31, 937)\n",
      "970 (3, 967)\n",
      "972 (5, 967)\n",
      "974 (3, 971)\n",
      "976 (5, 971)\n",
      "978 (7, 971)\n",
      "980 (3, 977)\n",
      "982 (5, 977)\n",
      "984 (7, 977)\n",
      "986 (3, 983)\n",
      "988 (5, 983)\n",
      "990 (7, 983)\n",
      "992 (73, 919)\n",
      "994 (3, 991)\n",
      "996 (5, 991)\n",
      "998 (7, 991)\n",
      "1000 (3, 997)\n"
     ]
    }
   ],
   "source": [
    "test(1000)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 2 . Le paradigme de la programmation fonctionnelle"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# exemple 1 \n",
    "a = 0\n",
    "def increment():\n",
    "    global a\n",
    "    a += 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# exemple 2 \n",
    "def increment(a):\n",
    "    return a + 1"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Les deux éléments fondamentaux (qui sont liés) de la programmation fonctionnelle :\n",
    "\n",
    "- pas d'utilisation de variables globales : le contexte est exclusivement fourni en paramètres de la fonction : elle n'a pas de comportement erratique en fonction du contexte global\n",
    "- pas d'effet de bord (affichages compris)\n",
    "\n",
    "Bien évidemment, l'exemple 2 est fidèle à ces exigences de la programmation fonctionnelle\n",
    "\n",
    "Les deux façons de faire des jeux de test :\n",
    "\n",
    "- avec le module doctest qui va tester les exemples donnés dans le docstring\n",
    "- ou plus simplement en créant au besoin un jeu de tests avec la commande assert :"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [],
   "source": [
    "assert premier(5)==True\n",
    "assert premier(1)==False\n",
    "assert premier(9)==False\n",
    "assert premier(17)==True"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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