-- SpacetimeDynamics.lean
-- Formalizing the emergence of Time and Gravity from QLF ZFA Latency
-- Also provides Form (Pauli basis) and EventSynthesisField for downstream modules.

import Mathlib.LinearAlgebra.Matrix.Hermitian
import Mathlib.Data.Complex.Basic
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic

open Matrix

/-! # Form: Pauli-basis representation of logical forms -/

structure Form where
  t : ℝ
  x : ℝ
  y : ℝ
  z : ℝ

noncomputable def Form.toMatrix (f : Form) : Matrix (Fin 2) (Fin 2) ℂ :=
  !![↑f.t + ↑f.z, ↑f.x - Complex.I * ↑f.y;
     ↑f.x + Complex.I * ↑f.y, ↑f.t - ↑f.z]

noncomputable def Form.fromMatrix (M : Matrix (Fin 2) (Fin 2) ℂ) : Form :=
  { t := ((M 0 0 + M 1 1) / 2).re
    x := ((M 0 1 + M 1 0) / 2).re
    y := ((Complex.I * (M 0 1 - M 1 0)) / 2).re
    z := ((M 0 0 - M 1 1) / 2).re }

theorem Form.toMatrix_adjoint (f : Form) :
    f.toMatrix.conjTranspose = f.toMatrix := by
  apply Matrix.ext; intro i j
  fin_cases i <;> fin_cases j <;>
  simp only [Matrix.conjTranspose_apply, Form.toMatrix,
    Matrix.cons_val_zero, Matrix.cons_val_one,
    Matrix.head_cons, Matrix.head_fin_const] <;>
  apply Complex.ext <;>
  simp [Complex.add_re, Complex.add_im, Complex.sub_re, Complex.sub_im,
        Complex.mul_re, Complex.mul_im, Complex.neg_re, Complex.neg_im,
        Complex.ofReal_re, Complex.ofReal_im, Complex.I_re, Complex.I_im,
        Complex.conj_re, Complex.conj_im, Complex.star_def]

theorem Form.equal_and_opposite_self (f : Form) :
    (f.toMatrix + f.toMatrix).IsHermitian := by
  unfold Matrix.IsHermitian
  rw [Matrix.conjTranspose_add, Form.toMatrix_adjoint]

/-! # Event-Synthesis Scalar Field φ -/

structure EventSynthesisField where
  phi : ℝ
  dphi_dt : ℝ
  V_phi : ℝ

noncomputable def EventSynthesisField.stressEnergy (φ : EventSynthesisField) : ℝ × ℝ :=
  (0.5 * φ.dphi_dt ^ 2 + φ.V_phi, 0.5 * φ.dphi_dt ^ 2 - φ.V_phi)

noncomputable def EventSynthesisField.Λ_eff (φ : EventSynthesisField) : ℝ :=
  8 * Real.pi * φ.stressEnergy.1

noncomputable def EventSynthesisField.w (φ : EventSynthesisField) : ℝ :=
  if φ.stressEnergy.1 ≠ 0 then φ.stressEnergy.2 / φ.stressEnergy.1 else -1

/-! # EventContext and TimeLatency -/

namespace QLF

/-- Represents a localized context within the QuCalc generative tree. -/
structure EventContext where
  w_zfa : Nat
  h_realizable : w_zfa > 0

/-- TimeLatency represents the delay (1 / w_zfa). Larger denominator = smaller latency. -/
structure TimeLatency where
  denominator : Nat

-- Ordering: t1 < t2 iff t1 has larger denominator (slower = more latency)
instance : LT TimeLatency where
  lt t1 t2 := t1.denominator > t2.denominator

/-- The fundamental time delay required to synthesize the logic bit. -/
def time_latency (e : EventContext) : TimeLatency :=
  ⟨e.w_zfa⟩

/-- Time Dilation Theorem: a more constrained context has higher time latency. -/
theorem time_dilation_in_constrained_space (e1 e2 : EventContext)
    (h_constraint : e1.w_zfa < e2.w_zfa) :
    time_latency e1 > time_latency e2 := h_constraint

/-- The Possibilist Gradient (Gravity). -/
def is_gravitationally_attractive (source target : EventContext) : Prop :=
  target.w_zfa < source.w_zfa

/-- Gravity implies time dilation. -/
theorem gravity_is_time_dilation (source target : EventContext)
    (h_gravity : is_gravitationally_attractive source target) :
    time_latency target > time_latency source := h_gravity

end QLF