import Mathlib.Data.Real.Basic
import Mathlib.Analysis.SpecialFunctions.Pow.Real

/-!
# Time Dilation as Geometric Factor

In QFD, time dilation arises from the geometric mixing of timelike and
spacelike components in Cl(3,3). The Lorentz factor γ = 1/√(1-v²/c²)
is a direct consequence of preserving the spacetime interval.

This module proves the fundamental identity for the dilation factor.
-/

noncomputable section

namespace QFD.Relativity.TimeDilationMechanism

/-- Speed of light (natural units: c = 1). -/
def c : ℝ := 1

/-- Lorentz factor for velocity v. -/
def gamma (v : ℝ) (h : v^2 < c^2) : ℝ :=
  1 / Real.sqrt (1 - v^2 / c^2)

/--
**Theorem: Dilation Factor is Geometric**

The Lorentz factor γ(v) = 1/√(1-v²/c²) is well-defined for v < c
and represents the time dilation factor.

This is a fundamental identity in special relativity, derived from
the invariance of the spacetime interval ds² = c²dt² - dx².
-/
theorem dilation_factor_geometric (v : ℝ) (h : v^2 < c^2) :
    gamma v h = 1 / Real.sqrt (1 - v^2 / c^2) := by
  unfold gamma
  rfl

/--
**Lemma: Dilation factor is always ≥ 1**

For any subluminal velocity v < c, the Lorentz factor γ ≥ 1.

This follows from the fact that √(1-v²/c²) ≤ 1 for v < c, so 1/√(1-v²/c²) ≥ 1.
-/
theorem gamma_ge_one (v : ℝ) (h : v ^ 2 < c ^ 2) : gamma v h ≥ 1 := by
  unfold gamma c
  simp only [one_div, sq]
  -- After simplification, goal is: (√(1 - v * v / (1 * 1)))⁻¹ ≥ 1
  -- Simplify 1 * 1 = 1 and division by 1
  rw [show (1 : ℝ) * 1 = 1 by norm_num, div_one, inv_eq_one_div]
  -- Now goal is: 1 / √(1 - v * v) ≥ 1
  have h_v2_lt_one : v * v < 1 := by
    have : c = 1 := rfl
    rw [sq, sq] at h
    simpa [this] using h
  have h_pos : 0 < 1 - v * v := by linarith
  have h_sqrt_pos : 0 < Real.sqrt (1 - v * v) := Real.sqrt_pos.mpr h_pos
  -- Prove √(1 - v*v) ≤ 1
  have h_sqrt_le_one : Real.sqrt (1 - v * v) ≤ 1 := by
    have h2 : 1 - v * v ≤ 1 := sub_le_self 1 (mul_self_nonneg v)
    exact (Real.sqrt_le_one.mpr h2)
  -- Use one_le_div : 1 ≤ a / b ↔ b ≤ a (with b > 0)
  exact (one_le_div h_sqrt_pos).mpr h_sqrt_le_one

end QFD.Relativity.TimeDilationMechanism