{
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  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The first compilation pass is named _lowering_, and three main tasks are carried out: _indexification_, _substitution_, and _domain-alignment_. In this notebook, we will explore the Indexification step.\n",
    "\n",
    "\n",
    "# Indexification\n",
    "\n",
    "The _indexification_ consists of converting Functions into Indexeds, that is actual array accesses. An `Indexed` always keeps a reference to its originating `Function`. For instance, all accesses to `u` such as `u[t, x + 1]` and `u[t + 1, x − 2]` would store a pointer to the same, user-defined `Function` _u(t, x)_. This metadata is exploited throughout the various compilation passes. \n",
    "\n",
    "To see this, let's create a `TimeFunction` named `u`."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "u(time, x)\n"
     ]
    }
   ],
   "source": [
    "from devito import Grid, TimeFunction\n",
    "u = TimeFunction(name='u', grid=Grid((3,)), save=1)\n",
    "print(u)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that `u` has one temporal dimension and one spatial dimension, that we will call `time` and `x` here, respectively."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(time, x)"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "time, x = u.dimensions\n",
    "time, x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Functions (of type `Function`, `TimeFunction` and `SparseFunction`) can be indexified in two equivalent ways, either via the .indexify() method, as illustrated below, or via explicit indexing, as typically done in user code."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "u[time, x]\n"
     ]
    }
   ],
   "source": [
    "u_i = u.indexify() # For more details about the method `indexify`, see `devito/symbolics/manipulation.py`\n",
    "print(u_i)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, dimensions are interpreted as indices."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u_i.indices == u.dimensions"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The explicit indexing mode is when you write"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = u[time,x+1]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and you get exactly (representation shown below)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "u[time, x + 1]\n"
     ]
    }
   ],
   "source": [
    "print(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "ie, this is totally equivalent to indexify. Another example, you can write"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "b = u[time+1,x-2]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "and you get\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "u[time + 1, x - 2]\n"
     ]
    }
   ],
   "source": [
    "print(b)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Clearly, we now have that"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.indices == b.indices"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Thus"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a == b"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "However, we can verify that these accesses still point to the very same `TimeFunction` object."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "a.function is b.function "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note also that there is no `data` associated with `Indexed` objects."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "devito.types.basic.Indexed"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(u_i)\n",
    "# u_i.data   # This should return an error!"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The `data` representing the grid is accessed through the `Function` that the indices point to."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Data([[0., 0., 0.]], dtype=float32)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "u_i.function.data"
   ]
  }
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