MTtranslateService/Lib/site-packages/sympy/integrals/rubi/parsetools/parse.py

"""
Parser for FullForm[Downvalues[]] of Mathematica rules.

This parser is customised to parse the output in MatchPy rules format. Multiple
`Constraints` are divided into individual `Constraints` because it helps the
MatchPy's `ManyToOneReplacer` to backtrack earlier and improve the speed.

Parsed output is formatted into readable format by using `sympify` and print the
expression using `sstr`. This replaces `And`, `Mul`, 'Pow' by their respective
symbols.

Mathematica
===========

To get the full form from Wolfram Mathematica, type:
```
ShowSteps = False
Import["RubiLoader.m"]
Export["output.txt", ToString@FullForm@DownValues@Int]
```

The file ``output.txt`` will then contain the rules in parseable format.

References
==========
[1] http://reference.wolfram.com/language/ref/FullForm.html
[2] http://reference.wolfram.com/language/ref/DownValues.html
[3] https://gist.github.com/Upabjojr/bc07c49262944f9c1eb0
"""
import re
import os
import inspect
from sympy.core.function import Function
from sympy.core.symbol import Symbol
from sympy.core.sympify import sympify
from sympy.sets.sets import Set
from sympy.printing import StrPrinter
from sympy.utilities.misc import debug


class RubiStrPrinter(StrPrinter):
    def _print_Not(self, expr):
        return "Not(%s)" % self._print(expr.args[0])


def rubi_printer(expr, **settings):
    return RubiStrPrinter(settings).doprint(expr)


replacements = dict(  # Mathematica equivalent functions in SymPy
        Times="Mul",
        Plus="Add",
        Power="Pow",
        Log='log',
        Exp='exp',
        Sqrt='sqrt',
        Cos='cos',
        Sin='sin',
        Tan='tan',
        Cot='1/tan',
        cot='1/tan',
        Sec='1/cos',
        sec='1/cos',
        Csc='1/sin',
        csc='1/sin',
        ArcSin='asin',
        ArcCos='acos',
        # ArcTan='atan',
        ArcCot='acot',
        ArcSec='asec',
        ArcCsc='acsc',
        Sinh='sinh',
        Cosh='cosh',
        Tanh='tanh',
        Coth='1/tanh',
        coth='1/tanh',
        Sech='1/cosh',
        sech='1/cosh',
        Csch='1/sinh',
        csch='1/sinh',
        ArcSinh='asinh',
        ArcCosh='acosh',
        ArcTanh='atanh',
        ArcCoth='acoth',
        ArcSech='asech',
        ArcCsch='acsch',
        Expand='expand',
        Im='im',
        Re='re',
        Flatten='flatten',
        Polylog='polylog',
        Cancel='cancel',
        #Gamma='gamma',
        TrigExpand='expand_trig',
        Sign='sign',
        Simplify='simplify',
        Defer='UnevaluatedExpr',
        Identity = 'S',
        Sum = 'Sum_doit',
        Module = 'With',
        Block = 'With',
        Null = 'None'
)

temporary_variable_replacement = {  # Temporarily rename because it can raise errors while sympifying
    'gcd' : "_gcd",
    'jn' : "_jn",
}

permanent_variable_replacement = {  # Permamenely rename these variables
    r"\[ImaginaryI]" : 'ImaginaryI',
    "$UseGamma": '_UseGamma',
}

# These functions have different return type in different cases. So better to use a try and except in the constraints, when any of these appear
f_diff_return_type = ['BinomialParts', 'BinomialDegree', 'TrinomialParts', 'GeneralizedBinomialParts', 'GeneralizedTrinomialParts', 'PseudoBinomialParts', 'PerfectPowerTest',
    'SquareFreeFactorTest', 'SubstForFractionalPowerOfQuotientOfLinears', 'FractionalPowerOfQuotientOfLinears', 'InverseFunctionOfQuotientOfLinears',
    'FractionalPowerOfSquareQ', 'FunctionOfLinear', 'FunctionOfInverseLinear', 'FunctionOfTrig', 'FindTrigFactor', 'FunctionOfLog',
    'PowerVariableExpn', 'FunctionOfSquareRootOfQuadratic', 'SubstForFractionalPowerOfLinear', 'FractionalPowerOfLinear', 'InverseFunctionOfLinear',
    'Divides', 'DerivativeDivides', 'TrigSquare', 'SplitProduct', 'SubstForFractionalPowerOfQuotientOfLinears', 'InverseFunctionOfQuotientOfLinears',
    'FunctionOfHyperbolic', 'SplitSum']


def contains_diff_return_type(a):
    """
    This function returns whether an expression contains functions which have different return types in
    diiferent cases.
    """
    if isinstance(a, list):
        for i in a:
            if contains_diff_return_type(i):
                return True
    elif type(a) == Function('With') or type(a) == Function('Module'):
        for i in f_diff_return_type:
            if a.has(Function(i)):
                return True
    else:
        if a in f_diff_return_type:
            return True
    return False


def parse_full_form(wmexpr):
    """
    Parses FullForm[Downvalues[]] generated by Mathematica
    """
    out = []
    stack = [out]
    generator = re.finditer(r'[\[\],]', wmexpr)
    last_pos = 0
    for match in generator:
        if match is None:
            break
        position = match.start()
        last_expr = wmexpr[last_pos:position].replace(',', '').replace(']', '').replace('[', '').strip()

        if match.group() == ',':
            if last_expr != '':
                stack[-1].append(last_expr)
        elif match.group() == ']':
            if last_expr != '':
                stack[-1].append(last_expr)
            stack.pop()
        elif match.group() == '[':
            stack[-1].append([last_expr])
            stack.append(stack[-1][-1])
        last_pos = match.end()
    return out[0]


def get_default_values(parsed, default_values={}):
    """
    Returns Optional variables and their values in the pattern
    """
    if not isinstance(parsed, list):
        return default_values

    if parsed[0] == "Times":  # find Default arguments for "Times"
        for i in parsed[1:]:
            if i[0] == "Optional":
                default_values[(i[1][1])] = 1

    if parsed[0] == "Plus":  # find Default arguments for "Plus"
        for i in parsed[1:]:
            if i[0] == "Optional":
                default_values[(i[1][1])] = 0

    if parsed[0] == "Power":  # find Default arguments for "Power"
        for i in parsed[1:]:
            if i[0] == "Optional":
                default_values[(i[1][1])] = 1

    if len(parsed) == 1:
        return default_values

    for i in parsed:
        default_values = get_default_values(i, default_values)

    return default_values


def add_wildcards(string, optional={}):
    """
    Replaces `Pattern(variable)` by `variable` in `string`.
    Returns the free symbols present in the string.
    """
    symbols = []  # stores symbols present in the expression

    p = r'(Optional\(Pattern\((\w+), Blank\)\))'
    matches = re.findall(p, string)
    for i in matches:
        string = string.replace(i[0], "WC('{}', S({}))".format(i[1], optional[i[1]]))
        symbols.append(i[1])

    p = r'(Pattern\((\w+), Blank\))'
    matches = re.findall(p, string)
    for i in matches:
        string = string.replace(i[0], i[1] + '_')
        symbols.append(i[1])

    p = r'(Pattern\((\w+), Blank\(Symbol\)\))'
    matches = re.findall(p, string)
    for i in matches:
        string = string.replace(i[0], i[1] + '_')
        symbols.append(i[1])

    return string, symbols


def seperate_freeq(s, variables=[], x=None):
    """
    Returns list of symbols in FreeQ.
    """
    if s[0] == 'FreeQ':
        if len(s[1]) == 1:
            variables = [s[1]]
        else:
            variables = s[1][1:]
        x = s[2]
    else:
        for i in s[1:]:
            variables, x = seperate_freeq(i, variables, x)
            return variables, x
    return variables, x


def parse_freeq(l, x, cons_index, cons_dict, cons_import, symbols=None):
    """
    Converts FreeQ constraints into MatchPy constraint
    """
    res = []
    cons = ''
    for i in l:
        if isinstance(i, str):
            r = '        return FreeQ({}, {})'.format(i, x)
            # First it checks if a constraint is already present in `cons_dict`, If yes, use it else create a new one.
            if r not in cons_dict.values():
                cons_index += 1
                c = '\n    def cons_f{}({}, {}):\n'.format(cons_index, i, x)
                c += r
                c += '\n\n    cons{} = CustomConstraint({})\n'.format(cons_index, 'cons_f{}'.format(cons_index))
                cons_name = 'cons{}'.format(cons_index)
                cons_dict[cons_name] = r
            else:
                c = ''
                cons_name = next(key for key, value in sorted(cons_dict.items()) if value == r)

        elif isinstance(i, list):
            s = sorted(set(get_free_symbols(i, symbols)))
            s = ', '.join(s)
            r = '        return FreeQ({}, {})'.format(generate_sympy_from_parsed(i), x)
            if r not in cons_dict.values():
                cons_index += 1
                c = '\n    def cons_f{}({}):\n'.format(cons_index, s)
                c += r
                c += '\n\n    cons{} = CustomConstraint({})\n'.format(cons_index, 'cons_f{}'.format(cons_index))
                cons_name = 'cons{}'.format(cons_index)
                cons_dict[cons_name] = r
            else:
                c = ''
                cons_name = next(key for key, value in cons_dict.items() if value == r)

        if cons_name not in cons_import:
            cons_import.append(cons_name)

        res.append(cons_name)
        cons += c

    if res != []:
        return ', ' + ', '.join(res), cons, cons_index
    return '', cons, cons_index


def generate_sympy_from_parsed(parsed, wild=False, symbols=(), replace_Int=False):
    """
    Parses list into Python syntax.

    Parameters
    ==========
    wild : When set to True, the symbols are replaced as wild symbols.
    symbols : Symbols already present in the pattern.
    replace_Int: when set to True, `Int` is replaced by `Integral`(used to parse pattern).
    """
    out = ""

    if not isinstance(parsed, list):
        try:  # return S(number) if parsed is Number
            float(parsed)
            return "S({})".format(parsed)
        except:
            pass
        if parsed in symbols:
            if wild:
                return parsed + '_'
        return parsed

    if parsed[0] == 'Rational':
        return 'S({})/S({})'.format(generate_sympy_from_parsed(parsed[1], wild=wild, symbols=symbols, replace_Int=replace_Int), generate_sympy_from_parsed(parsed[2], wild=wild, symbols=symbols, replace_Int=replace_Int))

    if parsed[0] in replacements:
        out += replacements[parsed[0]]
    elif parsed[0] == 'Int' and replace_Int:
        out += 'Integral'
    else:
        out += parsed[0]

    if len(parsed) == 1:
        return out

    result = [generate_sympy_from_parsed(i, wild=wild, symbols=symbols, replace_Int=replace_Int) for i in parsed[1:]]
    if '' in result:
        result.remove('')

    out += "("
    out += ", ".join(result)
    out += ")"

    return out


def get_free_symbols(s, symbols, free_symbols=None):
    """
    Returns free_symbols present in `s`.
    """
    free_symbols = free_symbols or []
    if not isinstance(s, list):
        if s in symbols:
            free_symbols.append(s)
        return free_symbols

    for i in s:
        free_symbols = get_free_symbols(i, symbols, free_symbols)

    return free_symbols


def set_matchq_in_constraint(a, cons_index):
    """
    Takes care of the case, when a pattern matching has to be done inside a constraint.
    """
    lst = []
    res = ''
    if isinstance(a, list):
        if a[0] == 'MatchQ':
            s = a
            optional = get_default_values(s, {})
            r = generate_sympy_from_parsed(s, replace_Int=True)
            r, free_symbols = add_wildcards(r, optional=optional)
            free_symbols = sorted(set(free_symbols))  # remove common symbols
            r = sympify(r, locals={"Or": Function("Or"), "And": Function("And"), "Not":Function("Not")})
            pattern = r.args[1].args[0]
            cons = r.args[1].args[1]
            pattern = rubi_printer(pattern, sympy_integers=True)
            pattern = setWC(pattern)
            res = '        def _cons_f_{}({}):\n            return {}\n'.format(cons_index, ', '.join(free_symbols), cons)
            res += '        _cons_{} = CustomConstraint(_cons_f_{})\n'.format(cons_index, cons_index)
            res += '        pat = Pattern(UtilityOperator({}, x), _cons_{})\n'.format(pattern, cons_index)
            res += '        result_matchq = is_match(UtilityOperator({}, x), pat)'.format(r.args[0])
            return "result_matchq", res

        else:
            for i in a:
                if isinstance(i, list):
                    r = set_matchq_in_constraint(i, cons_index)
                    lst.append(r[0])
                    res = r[1]
                else:
                    lst.append(i)
    return lst, res


def _divide_constriant(s, symbols, cons_index, cons_dict, cons_import):
    # Creates a CustomConstraint of the form `CustomConstraint(lambda a, x: FreeQ(a, x))`
    lambda_symbols = sorted(set(get_free_symbols(s, symbols, [])))
    r = generate_sympy_from_parsed(s)
    r = sympify(r, locals={"Or": Function("Or"), "And": Function("And"), "Not":Function("Not")})
    if r.has(Function('MatchQ')):
        match_res = set_matchq_in_constraint(s, cons_index)
        res = match_res[1]
        res += '\n        return {}'.format(rubi_printer(sympify(generate_sympy_from_parsed(match_res[0]), locals={"Or": Function("Or"), "And": Function("And"), "Not":Function("Not")}), sympy_integers = True))

    elif contains_diff_return_type(s):
        res = '        try:\n            return {}\n        except (TypeError, AttributeError):\n            return False'.format(rubi_printer(r, sympy_integers=True))
    else:
        res = '        return {}'.format(rubi_printer(r, sympy_integers=True))

    # First it checks if a constraint is already present in `cons_dict`, If yes, use it else create a new one.
    if res not in cons_dict.values():
        cons_index += 1
        cons = '\n    def cons_f{}({}):\n'.format(cons_index, ', '.join(lambda_symbols))
        if 'x' in lambda_symbols:
            cons += '        if isinstance(x, (int, Integer, float, Float)):\n            return False\n'
        cons += res
        cons += '\n\n    cons{} = CustomConstraint({})\n'.format(cons_index, 'cons_f{}'.format(cons_index))
        cons_name = 'cons{}'.format(cons_index)
        cons_dict[cons_name] = res
    else:
        cons = ''
        cons_name = next(key for key, value in cons_dict.items() if value == res)

    if cons_name not in cons_import:
        cons_import.append(cons_name)
    return cons_name, cons, cons_index


def divide_constraint(s, symbols, cons_index, cons_dict, cons_import):
    """
    Divides multiple constraints into smaller constraints.

    Parameters
    ==========
    s : constraint as list
    symbols : all the symbols present in the expression
    """
    result =[]
    cons = ''
    if s[0] == 'And':
        for i in s[1:]:
            if i[0]!= 'FreeQ':
                a = _divide_constriant(i, symbols, cons_index, cons_dict, cons_import)
                result.append(a[0])
                cons += a[1]
                cons_index = a[2]
    else:
        a = _divide_constriant(s, symbols, cons_index, cons_dict, cons_import)
        result.append(a[0])
        cons += a[1]
        cons_index = a[2]

    r = ['']
    for i in result:
        if i != '':
            r.append(i)

    return ', '.join(r),cons, cons_index


def setWC(string):
    """
    Replaces `WC(a, b)` by `WC('a', S(b))`
    """
    p = r'(WC\((\w+), S\(([-+]?\d)\)\))'
    matches = re.findall(p, string)
    for i in matches:
        string = string.replace(i[0], "WC('{}', S({}))".format(i[1], i[2]))

    return string


def process_return_type(a1, L):
    """
    Functions like `Set`, `With` and `CompoundExpression` has to be taken special care.
    """
    a = sympify(a1[1])
    x = ''
    processed = False
    return_value = ''
    if type(a) == Function('With') or type(a) == Function('Module'):
        for i in a.args:
            for s in i.args:
                if isinstance(s, Set) and s not in L:
                    x += '\n        {} = {}'.format(s.args[0], rubi_printer(s.args[1], sympy_integers=True))

            if not type(i) in (Function('List'),  Function('CompoundExpression')) and not i.has(Function('CompoundExpression')):
                return_value = i
                processed = True

            elif type(i) == Function('CompoundExpression'):
                return_value = i.args[-1]
                processed = True

            elif type(i.args[0]) == Function('CompoundExpression'):
                C = i.args[0]
                return_value = '{}({}, {})'.format(i.func, C.args[-1], i.args[1])
                processed = True
    return x, return_value, processed


def extract_set(s, L):
    """
    this function extracts all `Set` functions
    """
    lst = []
    if isinstance(s, Set) and s not in L:
        lst.append(s)
    else:
        try:
            for i in s.args:
                lst += extract_set(i, L)
        except:  # when s has no attribute args (like `bool`)
            pass
    return lst


def replaceWith(s, symbols, index):
    """
    Replaces `With` and `Module by python functions`
    """
    return_type = None
    with_value = ''
    if type(s) == Function('With') or type(s) == Function('Module'):
        constraints = ' '
        result = '\n\n\ndef With{}({}):'.format(index, ', '.join(symbols))
        if type(s.args[0]) == Function('List'):  # get all local variables of With and Module
            L = list(s.args[0].args)
        else:
            L = [s.args[0]]
        lst = []
        for i in s.args[1:]:
            lst += extract_set(i, L)

        L += lst
        for i in L:  # define local variables
            if isinstance(i, Set):
                with_value += '\n    {} = {}'.format(i.args[0], rubi_printer(i.args[1], sympy_integers=True))

            elif isinstance(i, Symbol):
                with_value += "\n    {} = Symbol('{}')".format(i, i)
        #result += with_value
        if type(s.args[1]) == Function('CompoundExpression'):  # Expand CompoundExpression
            C = s.args[1]
            result += with_value
            if isinstance(C.args[0], Set):
                result += '\n    {} = {}'.format(C.args[0].args[0], C.args[0].args[1])
            result += '\n    return {}'.format(rubi_printer(C.args[1], sympy_integers=True))
            return result, constraints, return_type

        elif type(s.args[1]) == Function('Condition'):
            C = s.args[1]
            if len(C.args) == 2:
                if all(j in symbols for j in [str(i) for i in C.free_symbols]):
                    result += with_value
                    #constraints += 'CustomConstraint(lambda {}: {})'.format(', '.join([str(i) for i in C.free_symbols]), sstr(C.args[1], sympy_integers=True))
                    result += '\n    return {}'.format(rubi_printer(C.args[0], sympy_integers=True))
                else:
                    if 'x' in symbols:
                        result += '\n    if isinstance(x, (int, Integer, float, Float)):\n        return False'

                    if contains_diff_return_type(s):
                        n_with_value = with_value.replace('\n', '\n    ')
                        result += '\n    try:{}\n        res = {}'.format(n_with_value, rubi_printer(C.args[1], sympy_integers=True))
                        result += '\n    except (TypeError, AttributeError):\n        return False'
                        result += '\n    if res:'

                    else:
                        result+=with_value
                        result += '\n    if {}:'.format(rubi_printer(C.args[1], sympy_integers=True))
                    return_type = (with_value, rubi_printer(C.args[0], sympy_integers=True))
                    return_type1 = process_return_type(return_type, L)
                    if return_type1[2]:
                        return_type = (with_value+return_type1[0], rubi_printer(return_type1[1]))
                    result += '\n        return True'
                    result += '\n    return False'
            constraints = ', CustomConstraint(With{})'.format(index)
            return result, constraints, return_type

        elif type(s.args[1]) == Function('Module') or type(s.args[1]) == Function('With'):
            C = s.args[1]
            result += with_value
            return_type = (with_value, rubi_printer(C, sympy_integers=True))
            return_type1 = process_return_type(return_type, L)
            if return_type1[2]:
                return_type = (with_value+return_type1[0], rubi_printer(return_type1[1]))
            result += return_type1[0]
            result += '\n    return {}'.format(rubi_printer(return_type1[1]))
            return result, constraints, None

        elif s.args[1].has(Function("CompoundExpression")):
            C = s.args[1].args[0]
            result += with_value
            if isinstance(C.args[0], Set):
                result += '\n    {} = {}'.format(C.args[0].args[0], C.args[0].args[1])
            result += '\n    return {}({}, {})'.format(s.args[1].func, C.args[-1], s.args[1].args[1])
            return result, constraints, None

        result += with_value
        result += '\n    return {}'.format(rubi_printer(s.args[1], sympy_integers=True))
        return result, constraints, return_type
    else:
        return rubi_printer(s, sympy_integers=True), '', return_type


def downvalues_rules(r, header, cons_dict, cons_index, index):
    """
    Function which generates parsed rules by substituting all possible
    combinations of default values.
    """
    rules = '['
    parsed = '\n\n'
    repl_funcs = '\n\n'
    cons = ''
    cons_import = [] # it contains name of constraints that need to be imported for rules.
    for i in r:
        debug('parsing rule {}'.format(r.index(i) + 1))
        # Parse Pattern
        if i[1][1][0] == 'Condition':
            p = i[1][1][1].copy()
        else:
            p = i[1][1].copy()

        optional = get_default_values(p, {})
        pattern = generate_sympy_from_parsed(p.copy(), replace_Int=True)
        pattern, free_symbols = add_wildcards(pattern, optional=optional)
        free_symbols = sorted(set(free_symbols)) #remove common symbols
        # Parse Transformed Expression and Constraints
        if i[2][0] == 'Condition':  # parse rules without constraints separately
            constriant, constraint_def, cons_index = divide_constraint(i[2][2], free_symbols, cons_index, cons_dict, cons_import) # separate And constraints into individual constraints
            FreeQ_vars, FreeQ_x = seperate_freeq(i[2][2].copy()) # separate FreeQ into individual constraints
            transformed = generate_sympy_from_parsed(i[2][1].copy(), symbols=free_symbols)
        else:
            constriant = ''
            constraint_def = ''
            FreeQ_vars, FreeQ_x = [], []
            transformed = generate_sympy_from_parsed(i[2].copy(), symbols=free_symbols)
        FreeQ_constraint, free_cons_def, cons_index = parse_freeq(FreeQ_vars, FreeQ_x, cons_index, cons_dict, cons_import, free_symbols)
        pattern = sympify(pattern, locals={"Or": Function("Or"), "And": Function("And"), "Not":Function("Not") })
        pattern = rubi_printer(pattern, sympy_integers=True)

        pattern = setWC(pattern)
        transformed = sympify(transformed, locals={"Or": Function("Or"), "And": Function("And"), "Not":Function("Not") })
        constraint_def = constraint_def + free_cons_def
        cons += constraint_def
        index += 1

        # below are certain if - else condition depending on various situation that may be encountered
        if type(transformed) == Function('With') or type(transformed) == Function('Module'): # define separate function when With appears
            transformed, With_constraints, return_type = replaceWith(transformed, free_symbols, index)
            if return_type is None:
                repl_funcs += '{}'.format(transformed)
                parsed += '\n    pattern' + str(index) + ' = Pattern(' + pattern + '' + FreeQ_constraint + '' + constriant + ')'
                parsed += '\n    ' + 'rule' + str(index) + ' = ReplacementRule(' + 'pattern' + rubi_printer(index, sympy_integers=True) + ', With{}'.format(index) + ')\n'
            else:
                repl_funcs += '{}'.format(transformed)
                parsed += '\n    pattern' + str(index) + ' = Pattern(' + pattern + '' + FreeQ_constraint + '' + constriant + With_constraints + ')'
                repl_funcs += '\n\n\ndef replacement{}({}):\n'.format(
                    index, ', '.join(free_symbols)
                ) + return_type[0] + '\n    return '.format(index) + return_type[1]
                parsed += '\n    ' + 'rule' + str(index) + ' = ReplacementRule(' + 'pattern' + rubi_printer(index, sympy_integers=True) + ', replacement{}'.format(index) + ')\n'

        else:
            transformed = rubi_printer(transformed, sympy_integers=True)
            parsed += '\n    pattern' + str(index) + ' = Pattern(' + pattern + '' + FreeQ_constraint + '' + constriant + ')'
            repl_funcs += '\n\n\ndef replacement{}({}):\n    return '.format(index, ', '.join(free_symbols), index) + transformed
            parsed += '\n    ' + 'rule' + str(index) + ' = ReplacementRule(' + 'pattern' + rubi_printer(index, sympy_integers=True) + ', replacement{}'.format(index) + ')\n'
        rules += 'rule{}, '.format(index)
    rules += ']'
    parsed += '    return ' + rules +'\n'

    header += '    from sympy.integrals.rubi.constraints import ' + ', '.join(word for word in cons_import)
    parsed = header + parsed + repl_funcs
    return parsed, cons_index, cons, index


def rubi_rule_parser(fullform, header=None, module_name='rubi_object'):
    """
    Parses rules in MatchPy format.

    Parameters
    ==========
    fullform : FullForm of the rule as string.
    header : Header imports for the file. Uses default imports if None.
    module_name : name of RUBI module

    References
    ==========
    [1] http://reference.wolfram.com/language/ref/FullForm.html
    [2] http://reference.wolfram.com/language/ref/DownValues.html
    [3] https://gist.github.com/Upabjojr/bc07c49262944f9c1eb0
    """

    if header is None:  # use default header values
        path_header = os.path.dirname(os.path.abspath(inspect.getfile(inspect.currentframe())))
        header = open(os.path.join(path_header, "header.py.txt")).read()
        header = header.format(module_name)

    cons_dict = {}  # dict keeps track of constraints that has been encountered, thus avoids repetition of constraints.
    cons_index = 0  # for index of a constraint
    index = 0  # indicates the number of a rule.
    cons = ''

    # Temporarily rename these variables because it
    # can raise errors while sympifying
    for i in temporary_variable_replacement:
        fullform = fullform.replace(i, temporary_variable_replacement[i])
    # Permanently rename these variables
    for i in permanent_variable_replacement:
        fullform = fullform.replace(i, permanent_variable_replacement[i])

    rules = []
    for i in parse_full_form(fullform): # separate all rules
        if i[0] == 'RuleDelayed':
            rules.append(i)
    parsed = downvalues_rules(rules, header, cons_dict, cons_index, index)
    result = parsed[0].strip() + '\n'
    cons += parsed[2]
    # Replace temporary variables by actual values
    for i in temporary_variable_replacement:
        cons = cons.replace(temporary_variable_replacement[i], i)
        result = result.replace(temporary_variable_replacement[i], i)
    cons = "\n".join(header.split("\n")[:-2]) + '\n' + cons
    return result, cons