# Source code for ufl_legacy.algorithms.formtransformations

    # -*- coding: utf-8 -*-
    """This module defines utilities for transforming
    complete Forms into new related Forms."""

    # Copyright (C) 2008-2016 Martin Sandve Alnæs
    #
    # This file is part of UFL (https://www.fenicsproject.org)
    #
    # SPDX-License-Identifier:    LGPL-3.0-or-later
    #
    # Modified by Anders Logg, 2008-2009.
    # Modified by Garth N. Wells, 2010.
    # Modified by Marie E. Rognes, 2010.

    from ufl_legacy.log import error, warning, debug

    # All classes:
    from ufl_legacy.core.expr import ufl_err_str
    from ufl_legacy.argument import Argument
    from ufl_legacy.coefficient import Coefficient
    from ufl_legacy.constantvalue import Zero
    from ufl_legacy.algebra import Conj

    # Other algorithms:
    from ufl_legacy.algorithms.map_integrands import map_integrands
    from ufl_legacy.algorithms.transformer import Transformer
    from ufl_legacy.algorithms.replace import replace

    # FIXME: Don't use this below, it makes partextracter more expensive than necessary
    def _expr_has_terminal_types(expr, ufl_types):
        input = [expr]
        while input:
            e = input.pop()
            ops = e.ufl_operands
            if ops:
                input.extend(ops)
            elif isinstance(e, ufl_types):
                return True
        return False

    def zero_expr(e):
        return Zero(e.ufl_shape, e.ufl_free_indices, e.ufl_index_dimensions)

    class PartExtracter(Transformer):
        """
        PartExtracter extracts those parts of a form that contain the
        given argument(s).
        """

        def __init__(self, arguments):
            Transformer.__init__(self)
            self._want = set(arguments)

        def expr(self, x):
            """The default is a nonlinear operator not accepting any
            Arguments among its children."""
            if _expr_has_terminal_types(x, Argument):
                error("Found Argument in %s, this is an invalid expression." % ufl_err_str(x))
            return (x, set())

        # Terminals that are not Variables or Arguments behave as default
        # expr-s.
        terminal = expr

        def variable(self, x):
            "Return relevant parts of this variable."

            # Extract parts/provides from this variable's expression
            expression, label = x.ufl_operands
            part, provides = self.visit(expression)

            # If the extracted part is zero or we provide more than we
            # want, return zero
            if isinstance(part, Zero) or (provides - self._want):
                return (zero_expr(x), set())

            # Reuse varible if possible (or reconstruct from part)
            x = self.reuse_if_possible(x, part, label)

            return (x, provides)

        def argument(self, x):
            "Return itself unless itself provides too much."

            # An argument provides itself
            provides = {x}

            # If we provide more than we want, return zero
            if provides - self._want:
                return (zero_expr(x), set())

            return (x, provides)

        def sum(self, x):
            """
            Return the terms that might eventually yield the correct
            parts(!)

            The logic required for sums is a bit elaborate:

            A sum may contain terms providing different arguments. We
            should return (a sum of) a suitable subset of these
            terms. Those should all provide the same arguments.

            For each term in a sum, there are 2 simple possibilities:

            1a) The relevant part of the term is zero -> skip.
            1b) The term provides more arguments than we want -> skip

            2) If all terms fall into the above category, we can just
            return zero.

            Any remaining terms may provide exactly the arguments we want,
            or fewer. This is where things start getting interesting.

            3) Bottom-line: if there are terms with providing different
            arguments -- provide terms that contain the most arguments. If
            there are terms providing different sets of same size -> throw
            error (e.g. Argument(-1) + Argument(-2))
            """

            parts_that_provide = {}

            # 1. Skip terms that provide too much
            original_terms = x.ufl_operands
            assert len(original_terms) == 2
            for term in original_terms:

                # Visit this term in the sum
                part, term_provides = self.visit(term)

                # If this part is zero or it provides more than we want,
                # skip it
                if isinstance(part, Zero) or (term_provides - self._want):
                    continue

                # Add the contributions from this part to temporary list
                term_provides = frozenset(term_provides)
                if term_provides in parts_that_provide:
                    parts_that_provide[term_provides] += [part]
                else:
                    parts_that_provide[term_provides] = [part]

            # 2. If there are no remaining terms, return zero
            if not parts_that_provide:
                return (zero_expr(x), set())

            # 3. Return the terms that provide the biggest set
            most_provided = frozenset()
            for (provideds, parts) in parts_that_provide.items():  # TODO: Just sort instead?

                # Throw error if size of sets are equal (and not zero)
                if len(provideds) == len(most_provided) and len(most_provided):
                    error("Don't know what to do with sums with different Arguments.")

                if provideds > most_provided:
                    most_provided = provideds

            terms = parts_that_provide[most_provided]
            if len(terms) == 2:
                x = self.reuse_if_possible(x, *terms)
            else:
                x, = terms

            return (x, most_provided)

        def product(self, x, *ops):
            """ Note: Product is a visit-children-first handler. ops are
            the visited factors."""

            provides = set()
            factors = []

            for factor, factor_provides in ops:

                # If any factor is zero, return
                if isinstance(factor, Zero):
                    return (zero_expr(x), set())

                # Add factor to factors and extend provides
                factors.append(factor)
                provides = provides | factor_provides

                # If we provide more than we want, return zero
                if provides - self._want:
                    return (zero_expr(x), provides)

            # Reuse product if possible (or reconstruct from factors)
            x = self.reuse_if_possible(x, *factors)

            return (x, provides)

        # inner, outer and dot all behave as product
        inner = product
        outer = product
        dot = product

        def division(self, x):
            "Return parts_of_numerator/denominator."

            # Get numerator and denominator
            numerator, denominator = x.ufl_operands

            # Check for Arguments in the denominator
            if _expr_has_terminal_types(denominator, Argument):
                error("Found Argument in denominator of %s , this is an invalid expression." % ufl_err_str(x))

            # Visit numerator
            numerator_parts, provides = self.visit(numerator)

            # If numerator is zero, return zero. (No need to check whether
            # it provides too much, already checked by visit.)
            if isinstance(numerator_parts, Zero):
                return (zero_expr(x), set())

            # Reuse x if possible, otherwise reconstruct from (parts of)
            # numerator and denominator
            x = self.reuse_if_possible(x, numerator_parts, denominator)

            return (x, provides)

        def linear_operator(self, x, arg):
            """A linear operator with a single operand accepting arity > 0,
            providing whatever Argument its operand does."""

            # linear_operator is a visit-children-first handler. Handled
            # arguments are in arg.
            part, provides = arg

            # Discard if part is zero. (No need to check whether we
            # provide too much, already checked by children.)
            if isinstance(part, Zero):
                return (zero_expr(x), set())

            x = self.reuse_if_possible(x, part)

            return (x, provides)

        # Positive and negative restrictions behave as linear operators
        positive_restricted = linear_operator
        negative_restricted = linear_operator

        # Cell and facet average are linear operators
        cell_avg = linear_operator
        facet_avg = linear_operator

        # Grad is a linear operator
        grad = linear_operator

        # Conj, Real, Imag are linear operators
        conj = linear_operator
        real = linear_operator
        imag = linear_operator

        def linear_indexed_type(self, x):
            """Return parts of expression belonging to this indexed
            expression."""

            expression, index = x.ufl_operands
            part, provides = self.visit(expression)

            # Return zero if extracted part is zero. (The expression
            # should already have checked if it provides too much.)
            if isinstance(part, Zero):
                return (zero_expr(x), set())

            # Reuse x if possible (or reconstruct by indexing part)
            x = self.reuse_if_possible(x, part, index)

            return (x, provides)

        # All of these indexed thingies behave as a linear_indexed_type
        indexed = linear_indexed_type
        index_sum = linear_indexed_type
        component_tensor = linear_indexed_type

        def list_tensor(self, x, *ops):
            # list_tensor is a visit-children-first handler. ops contains
            # the visited operands with their provides. (It follows that
            # none of the visited operands provide more than wanted.)

            # Extract the most arguments provided by any of the components
            most_provides = ops[0][1]
            for (component, provides) in ops:
                if (provides - most_provides):
                    most_provides = provides

            # Check that all components either provide the same arguments
            # or vanish. (This check is here b/c it is not obvious what to
            # return if the components provide different arguments, at
            # least with the current transformer design.)
            for (component, provides) in ops:
                if (provides != most_provides and not isinstance(component, Zero)):
                    error("PartExtracter does not know how to handle list_tensors with non-zero components providing fewer arguments")

            # Return components
            components = [op[0] for op in ops]
            x = self.reuse_if_possible(x, *components)

            return (x, most_provides)

    def compute_form_with_arity(form, arity, arguments=None):
        """Compute parts of form of given arity."""

        # Extract all arguments in form
        if arguments is None:
            arguments = form.arguments()

        parts = [arg.part() for arg in arguments]
        if set(parts) - {None}:
            error("compute_form_with_arity cannot handle parts.")

        if len(arguments) < arity:
            warning("Form has no parts with arity %d." % arity)
            return 0 * form

        # Assuming that the form is not a sum of terms
        # that depend on different arguments, e.g. (u+v)*dx
        # would result in just v*dx. But that doesn't make
        # any sense anyway.
        sub_arguments = set(arguments[:arity])
        pe = PartExtracter(sub_arguments)

        def _transform(e):
            e, provides = pe.visit(e)
            if provides == sub_arguments:
                return e
            return Zero()
        return map_integrands(_transform, form)

    def compute_form_arities(form):
        """Return set of arities of terms present in form."""

        # Extract all arguments present in form
        arguments = form.arguments()

        parts = [arg.part() for arg in arguments]
        if set(parts) - {None}:
            error("compute_form_arities cannot handle parts.")

        arities = set()
        for arity in range(len(arguments) + 1):

            # Compute parts with arity "arity"
            parts = compute_form_with_arity(form, arity, arguments)

            # Register arity if "parts" does not vanish
            if parts and parts.integrals():
                arities.add(arity)

        return arities

    def compute_form_lhs(form):
        """Compute the left hand side of a form.

        Example:
        -------
            a = u*v*dx + f*v*dx
            a = lhs(a) -> u*v*dx

        """
        return compute_form_with_arity(form, 2)

    def compute_form_rhs(form):
        """Compute the right hand side of a form.

        Example:
        -------
            a = u*v*dx + f*v*dx
            L = rhs(a) -> -f*v*dx

        """
        return -compute_form_with_arity(form, 1)

    def compute_form_functional(form):
        """Compute the functional part of a form, that
        is the terms independent of Arguments.

        (Used for testing, not sure if it's useful for anything?)"""
        return compute_form_with_arity(form, 0)

    def compute_form_action(form, coefficient):
        """Compute the action of a form on a Coefficient.

        This works simply by replacing the last Argument
        with a Coefficient on the same function space (element).
        The form returned will thus have one Argument less
        and one additional Coefficient at the end if no
        Coefficient has been provided.
        """
        # TODO: Check whatever makes sense for coefficient

        # Extract all arguments
        arguments = form.arguments()

        parts = [arg.part() for arg in arguments]
        if set(parts) - {None}:
            error("compute_form_action cannot handle parts.")

        # Pick last argument (will be replaced)
        u = arguments[-1]

        fs = u.ufl_function_space()
        if coefficient is None:
            coefficient = Coefficient(fs)
        elif coefficient.ufl_function_space() != fs:
            debug("Computing action of form on a coefficient in a different function space.")
        return replace(form, {u: coefficient})

    def compute_energy_norm(form, coefficient):
        """Compute the a-norm of a Coefficient given a form a.

        This works simply by replacing the two Arguments
        with a Coefficient on the same function space (element).
        The Form returned will thus be a functional with no
        Arguments, and one additional Coefficient at the
        end if no coefficient has been provided.
        """
        arguments = form.arguments()

        parts = [arg.part() for arg in arguments]
        if set(parts) - {None}:
            error("compute_energy_norm cannot handle parts.")

        if len(arguments) != 2:
            error("Expecting bilinear form.")
        v, u = arguments
        U = u.ufl_function_space()
        V = v.ufl_function_space()
        if U != V:
            error("Expecting equal finite elements for test and trial functions, got '%s' and '%s'." % (U, V))
        if coefficient is None:
            coefficient = Coefficient(V)
        else:
            if coefficient.ufl_function_space() != U:
                error("Trying to compute action of form on a "
                      "coefficient in an incompatible element space.")
        return replace(form, {u: coefficient, v: coefficient})

    def compute_form_adjoint(form, reordered_arguments=None):
        """Compute the adjoint of a bilinear form.

        This works simply by swapping the number and part of the two arguments,
        but keeping their elements and places in the integrand expressions.
        """
        arguments = form.arguments()

        parts = [arg.part() for arg in arguments]
        if set(parts) - {None}:
            error("compute_form_adjoint cannot handle parts.")

        if len(arguments) != 2:
            error("Expecting bilinear form.")

        v, u = arguments
        if v.number() >= u.number():
            error("Mistaken assumption in code!")

        if reordered_arguments is None:
            reordered_u = Argument(u.ufl_function_space(), number=v.number(),
                                   part=v.part())
            reordered_v = Argument(v.ufl_function_space(), number=u.number(),
                                   part=u.part())
        else:
            reordered_u, reordered_v = reordered_arguments

        if reordered_u.number() >= reordered_v.number():
            error("Ordering of new arguments is the same as the old arguments!")

        if reordered_u.part() != v.part():
            error("Ordering of new arguments is the same as the old arguments!")
        if reordered_v.part() != u.part():
            error("Ordering of new arguments is the same as the old arguments!")

        if reordered_u.ufl_function_space() != u.ufl_function_space():
            error("Element mismatch between new and old arguments (trial functions).")
        if reordered_v.ufl_function_space() != v.ufl_function_space():
            error("Element mismatch between new and old arguments (test functions).")

        return map_integrands(Conj, replace(form, {v: reordered_v, u: reordered_u}))