# ============================================================
# File: _solve_dae_differential_algebraic_equations.py
# Topic: Differential–Algebraic Equations (DAEs)
#
# This is a "template solver" (explanatory/derivation style),
# consistent with your other advanced buckets.
#
# Supported "kinds":
#   1) semi_explicit_index1_backward_euler_newton_template
#   2) fully_implicit_bdf_template
#   3) consistent_initial_conditions_template
#   4) index_discussion_and_flagging_template
#
# Output:
#   { "problem_s": "...", "soln_s": "...", "answer_s": "..." }
#
# Notes:
#   - Not a full numeric DAE integrator; focuses on canonical workflow.
#   - Uses robust string assembly (list join) to avoid syntax pitfalls.
# ============================================================

from __future__ import annotations

from typing import Optional, Dict, Any, Literal

import sympy as sp


DAEKind = Literal[
    "semi_explicit_index1_backward_euler_newton_template",
    "fully_implicit_bdf_template",
    "consistent_initial_conditions_template",
    "index_discussion_and_flagging_template",
]


def solve_dae_differential_algebraic_equations(
    *,
    kind: DAEKind,
    # symbols
    t_symbol: str = "t",
    # display / placeholder strings (kept as strings on purpose)
    f_str: str = r"f(t,y,z)",
    g_str: str = r"g(t,y,z)",
    F_str: str = r"F(t,y,\dot y)",
    # numeric method knobs (for display)
    method_name: str = "Backward Euler (BDF1)",
    bdf_order: int = 1,
    # misc
    locals_map: Optional[Dict[str, Any]] = None,
    simplify: bool = True,
) -> dict:
    """
    Return an explanation template for DAEs.

    Parameters are mainly for display/text generation.
    We intentionally do not sympify f_str/g_str/F_str since they can be non-parseable.
    """

    _ = simplify  # reserved for future expansion (kept for signature compatibility)

    t = sp.Symbol(t_symbol, real=True)

    base_locals: Dict[str, Any] = {
        t_symbol: t,
        "pi": sp.pi,
        "E": sp.E,
        "exp": sp.exp,
        "log": sp.log,
        "ln": sp.log,
        "sqrt": sp.sqrt,
        "sin": sp.sin,
        "cos": sp.cos,
        "Abs": sp.Abs,
        "Heaviside": sp.Heaviside,
        "DiracDelta": sp.DiracDelta,
    }
    if locals_map:
        base_locals.update(locals_map)

    def card(title: str, body_aligned: str, note: str = "") -> str:
        note_html = ("<div class='note'>" + note + "</div>") if note else ""
        return (
            "<div class='wrap'><div class='card'>"
            "<div class='badge'>" + title + "<br/></div>"
            "\\(\\displaystyle \\begin{aligned}"
            + body_aligned
            + "\\end{aligned}\\)"
            + note_html
            + "</div></div>"
        )

    # -----------------------------
    # 1) Semi-explicit index-1 template
    # -----------------------------
    if kind == "semi_explicit_index1_backward_euler_newton_template":
        # Build "evaluated at n+1" display safely (string-only)
        f_n1 = f_str.replace("t", "t_{n+1}").replace("y", "y_{n+1}").replace("z", "z_{n+1}")
        g_n1 = g_str.replace("t", "t_{n+1}").replace("y", "y_{n+1}").replace("z", "z_{n+1}")

        body_parts = [
            r"& \displaystyle \text{Semi-explicit DAE (index-1 typical):} \\",
            r"& \displaystyle \begin{cases}",
            r"y' = " + f_str + r" \\",
            r"0 = " + g_str,
            r"\end{cases} \\[10pt]",
            r"& \displaystyle \textbf{Idea: implicit step + enforce constraints each step.} \\[6pt]",
            r"& \displaystyle \text{Using " + method_name + r" with step }h: \\",
            r"& \displaystyle y_{n+1}=y_n + h\,(" + f_n1 + r") \\",
            r"& \displaystyle 0=(" + g_n1 + r") \\[10pt]",
            r"& \displaystyle \text{Define the coupled residual in unknowns }(y_{n+1},z_{n+1}): \\",
            r"& \displaystyle \mathbf{F}(y_{n+1},z_{n+1})="
            r"\begin{pmatrix}"
            r"y_{n+1}-y_n-h\,(" + f_n1 + r") \\"
            r"(" + g_n1 + r")"
            r"\end{pmatrix}=\mathbf{0} \\[10pt]",
            r"& \displaystyle \textbf{Newton solve (Type \#14 engine):} \\",
            r"& \displaystyle J\,\Delta = -\mathbf{F},\quad (y,z)\leftarrow (y,z)+\Delta \\",
            r"& \displaystyle \text{where } J=\frac{\partial \mathbf{F}}{\partial (y,z)}. \\",
        ]

        soln_s = card(
            "DAE (Index-1): Implicit step + Newton constraint solve",
            "".join(body_parts),
            note=(
                "For index-1 DAEs, the constraint typically determines z (possibly implicitly). "
                "You solve a coupled nonlinear system each step; do not explicitly invert J—solve JΔ=-F."
            ),
        )

        problem_s = (
            "Explain the standard numeric template to advance an index-1 semi-explicit DAE "
            "\\(y'=f(t,y,z),\\;0=g(t,y,z)\\) using an implicit step and Newton."
        )
        answer_s = (
            "\\(\\text{At each step, solve }"
            "\\mathbf{F}(y_{n+1},z_{n+1})=\\mathbf{0}\\text{ via Newton; "
            "this enforces the constraint and advances the differential variables.}\\)"
        )
        return {"problem_s": problem_s.strip(), "soln_s": soln_s.strip(), "answer_s": answer_s.strip()}

    # -----------------------------
    # 2) Fully-implicit DAE via BDF template
    # -----------------------------
    if kind == "fully_implicit_bdf_template":
        p = int(bdf_order) if int(bdf_order) >= 1 else 1

        if p == 1:
            yprime_approx = r"\dot y_{n+1}\approx \frac{y_{n+1}-y_n}{h}"
        elif p == 2:
            yprime_approx = r"\dot y_{n+1}\approx \frac{3y_{n+1}-4y_n+y_{n-1}}{2h}"
        else:
            yprime_approx = r"\dot y_{n+1}\approx \text{BDF" + str(p) + r" combination of past values}"

        body_parts = [
            r"& \displaystyle \text{Fully-implicit DAE: } " + F_str + r"=0 \\[8pt]",
            r"& \displaystyle \textbf{BDF idea: approximate }\dot y \text{ and solve an algebraic system each step.} \\[6pt]",
            r"& \displaystyle \text{Using " + (method_name if method_name else ("BDF" + str(p))) + r": \\",
            r"& \displaystyle " + yprime_approx + r" \\[8pt]",
            r"& \displaystyle \text{Then solve:} \\",
            r"& \displaystyle "
            + F_str.replace(r"\dot y", r"\left(" + yprime_approx + r"\right)")
            .replace("t", "t_{n+1}")
            .replace("y", "y_{n+1}")
            + r"=0 \\[10pt]",
            r"& \displaystyle \textbf{Newton solve: } F(y_{n+1})=0 \text{ (scalar) or } \mathbf{F}(\mathbf{y}_{n+1})=\mathbf{0} \text{ (vector).} \\",
        ]

        soln_s = card(
            "DAE (Fully implicit): BDF stepping + Newton",
            "".join(body_parts),
            note=(
                "Production DAE solvers (DASSL/IDA-style) use BDF families + Newton, "
                "with consistent initial conditions and step-size control."
            ),
        )

        problem_s = (
            "Explain the standard template to advance a fully implicit DAE "
            "\\(F(t,y,\\dot y)=0\\) using BDF-style time discretization."
        )
        answer_s = (
            "\\(\\text{Replace }\\dot y\\text{ by a BDF approximation and solve the resulting algebraic "
            "system for }y_{n+1}\\text{ via Newton at each step.}\\)"
        )
        return {"problem_s": problem_s.strip(), "soln_s": soln_s.strip(), "answer_s": answer_s.strip()}

    # -----------------------------
    # 3) Consistent initial conditions template
    # -----------------------------
    if kind == "consistent_initial_conditions_template":
        body_parts = [
            r"& \displaystyle \text{Why DAEs need consistent initial conditions:} \\",
            r"& \displaystyle \text{For semi-explicit: }\begin{cases}y'=f(t,y,z)\\ 0=g(t,y,z)\end{cases} \\[6pt]",
            r"& \displaystyle \textbf{Constraint must hold at }t_0: \quad g(t_0,y_0,z_0)=0. \\[8pt]",
            r"& \displaystyle \text{If }z_0\text{ is unknown, solve }g(t_0,y_0,z)=0\text{ for }z. \\[8pt]",
            r"& \displaystyle \textbf{Derivative consistency (often needed):} \\",
            r"& \displaystyle 0=\frac{d}{dt}g(t,y,z)=g_t + g_y\,y' + g_z\,z' \\",
            r"& \displaystyle \text{Substitute }y'=f(t,y,z)\text{ to relate }z'\text{ and ensure consistency.} \\[10pt]",
            r"& \displaystyle \text{Practical step: use Newton to solve constraint(s) for missing initial values.} \\",
        ]

        soln_s = card(
            "DAE: Consistent initial conditions",
            "".join(body_parts),
            note=(
                "For higher-index DAEs, you may need to differentiate constraints multiple times "
                "or use stabilization/projection methods."
            ),
        )

        problem_s = (
            "Explain what 'consistent initial conditions' means for DAEs and how to obtain them."
        )
        answer_s = (
            "\\(\\text{Choose }(y_0,z_0)\\text{ so that constraints hold }g(t_0,y_0,z_0)=0\\text{ "
            "(and possibly derivative constraints), typically by solving with Newton.}\\)"
        )
        return {"problem_s": problem_s.strip(), "soln_s": soln_s.strip(), "answer_s": answer_s.strip()}

    # -----------------------------
    # 4) Index discussion + flagging template
    # -----------------------------
    if kind == "index_discussion_and_flagging_template":
        body_parts = [
            r"& \displaystyle \text{DAE index (informal): how many times constraints must be differentiated to get an ODE.} \\[6pt]",
            r"& \displaystyle \textbf{Index-1: } \text{constraints can be solved (possibly implicitly) each step.} \\",
            r"& \displaystyle \textbf{Index-2/3: } \text{constraints require differentiation; naive stepping can drift/instability.} \\[8pt]",
            r"& \displaystyle \text{Typical flags for higher index:} \\",
            r"& \displaystyle \bullet\ \text{Constraint }g(t,y,z)=0\text{ cannot be solved for }z\text{ locally.} \\",
            r"& \displaystyle \bullet\ \text{Need }g_t+g_y y' + g_z z'=0\text{ (or more derivatives) to determine hidden variables.} \\",
            r"& \displaystyle \bullet\ \text{Numerically: Newton struggles or constraint residual grows ('drift').} \\[10pt]",
            r"& \displaystyle \text{Practical remedies (advanced): stabilization (Baumgarte), projection, reformulation, or specialized solvers.} \\",
        ]

        soln_s = card(
            "DAE: Index intuition + when to flag higher-index behavior",
            "".join(body_parts),
            note=(
                "For your app: treat index-1 as the primary supported template; "
                "for suspected higher-index systems, show a warning and route to "
                "stabilization/projection discussion if you implement Type #20 later."
            ),
        )

        problem_s = "Explain DAE index (index-1 vs higher) and how to detect/flag higher-index behavior."
        answer_s = (
            "\\(\\text{Index-1 DAEs are solvable by constraint solve each step; higher-index DAEs "
            "require differentiating constraints and need stabilization/special handling.}\\)"
        )
        return {"problem_s": problem_s.strip(), "soln_s": soln_s.strip(), "answer_s": answer_s.strip()}

    # Fallback (should not happen if kind is validated)
    return {
        "problem_s": "DAE template requested.",
        "soln_s": card("DAE", r"& \displaystyle \text{Unknown kind requested.}\\"),
        "answer_s": r"\(\text{Unknown DAE template kind.}\)",
    }
