import QFD.GA.BasisOperations
import Mathlib.LinearAlgebra.CliffordAlgebra.Grading

/-!
# Multivector Grade Projections

This module provides tools for working with grade-specific components of multivectors
in the Clifford algebra Cl(3,3):

* Grade 0: Scalars
* Grade 1: Vectors
* Grade 2: Bivectors
* Grade 3: Trivectors
* etc.

## Usage

These tools enable extraction and manipulation of specific grade components,
useful for separating electric/magnetic fields, extracting scalar parts, etc.

## Future Extensions

* Full `gradeProjection : ℕ → Cl33 → Cl33` using Mathlib's grading machinery
* Tactics for automatic grade separation
* Grade-specific simplification lemmas
-/

namespace QFD.GA.Grade

open QFD.GA
open CliffordAlgebra

/-! ## Grade Classification Predicates -/

/-- A multivector is a scalar if it's in the image of the canonical embedding ℝ → Cl33 -/
def isScalar (x : Cl33) : Prop :=
  ∃ (r : ℝ), x = algebraMap ℝ Cl33 r

/-- A multivector is a vector (grade 1) if it's a linear combination of basis vectors -/
def isVector (x : Cl33) : Prop :=
  ∃ f : Fin 6 → ℝ, x = ∑ i, f i • e i

/-- A multivector is a bivector (grade 2) if it's a linear combination of basis bivectors -/
def isBivector (x : Cl33) : Prop :=
  ∃ f : Fin 6 → Fin 6 → ℝ,
    x = ∑ i, ∑ j, if i < j then f i j • (e i * e j) else 0

/-! ## Helper Lemmas -/

/-- The zero element is a scalar -/
theorem zero_isScalar : isScalar (0 : Cl33) := by
  use 0
  simp [map_zero]

/-- Scalar multiples of scalars are scalars -/
theorem smul_scalar_isScalar {r : ℝ} {x : Cl33} (h : isScalar x) :
    isScalar (r • x) := by
  obtain ⟨rx, hx⟩ := h
  refine ⟨r * rx, ?_⟩
  simp [hx, isScalar, Algebra.smul_def, map_mul]

/-- Sum of scalars is a scalar -/
theorem add_scalar_isScalar {x y : Cl33} (hx : isScalar x) (hy : isScalar y) :
    isScalar (x + y) := by
  obtain ⟨rx, hx⟩ := hx
  obtain ⟨ry, hy⟩ := hy
  use rx + ry
  rw [hx, hy, map_add]

/-! ## Grade Extraction (Placeholder) -/

/-- Extract the scalar (grade 0) component of a multivector
Note: This is a placeholder. Full implementation requires Mathlib's grading structure -/
noncomputable def scalarPart (x : Cl33) : ℝ :=
  0

/-- Extract the vector (grade 1) component of a multivector
Note: This is a placeholder. Full implementation requires Mathlib's grading structure -/
noncomputable def vectorPart (x : Cl33) : Cl33 :=
  0

/-- Extract the bivector (grade 2) component of a multivector
Note: This is a placeholder. Full implementation requires Mathlib's grading structure -/
noncomputable def bivectorPart (x : Cl33) : Cl33 :=
  0

/-! ## Usage Examples

```lean
-- Check if an element is a scalar
example : isScalar (algebraMap ℝ Cl33 5) := by
  use 5; rfl

-- Check if a basis vector is a vector
example : isVector (e 0) := by
  use fun i => if i = 0 then 1 else 0
  simp [Finset.sum_ite_eq']
```
-/

end QFD.GA.Grade