{
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<span style=\"font-size:8pt\">*ENSAM-Bordeaux, Mathématiques et informatique. Date : le 11/10/19. Auteur : Éric Ducasse. Version : 1.0*</span>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div class=\"alert alert-block alert-danger\"> <span style=\"color:#800000\"> *On pourra faire exécuter ce notebook cellule par cellule $\\left(\\mathtt{Maj+Entrée}\\right)$ ou intégralement par $\\mathtt{\\;Kernel\\rightarrow Restart\\;\\&\\;Run\\;All}$.* </span> </div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy as sb\n",
    "sb.init_printing()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "print(f\"Version de sympy : {sb.__version__}\") "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Déclaration de symboles"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Noms de symboles"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = sb.symbols(\"a\") ; a"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "b,c,d = sb.symbols(\"b,c,d\") ; c"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a_0,a_1,a_2 = sb.symbols(\"a_0^*,a_1^*,a_2^*\") ; a_2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "a_3 = sb.symbols(\"a_3^[c]\") ; a_3"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Hypothèses sur les symboles"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### *Non nul*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "y = sb.symbols(\"y\") # Symbole sans hypothèse\n",
    "(sb.Eq(y,0),sb.Ne(y,0))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x  = sb.symbols(\"x\", zero=False)\n",
    "(sb.Eq(x,0),sb.Ne(x,0))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x.assumptions0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### *Réels*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = sb.symbols(\"x\", real=True)\n",
    "x.assumptions0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x  = sb.symbols(\"x\", nonzero=True) # Réel non nul\n",
    "x.assumptions0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = sb.symbols(\"x\", positive=True) # Strictement positif\n",
    "x.assumptions0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x = sb.symbols(\"x\", negative=True) # Strictement positif\n",
    "(x.equals(0),x.is_zero,x.is_real,x.is_nonzero,x.is_nonpositive)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### *Entiers*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "i,j,k = sb.symbols(\"i,j,k\", integer=True)\n",
    "(sb.sin(sb.pi*k),sb.cos(sb.pi*k))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "((i+j).is_integer,(i-j).is_integer,(i*j).is_integer)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "m = sb.symbols(\"m\", integer=True, positive=True)\n",
    "m.assumptions0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "(m > 0, m+1 > 0, 2*m > 0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### *Complexe*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "z = sb.symbols(\"z\", complex=True)\n",
    "z.assumptions0"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### *Imaginaire pur*"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "b = sb.symbols(\"b\", imaginary=True)\n",
    "b.conjugate()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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