tau0_planck_relation

formal source

 169/-- **Theorem**: τ₀ = t_P / √π
 170
 171This relation shows τ₀ is the Planck time divided by √π. -/
 172theorem tau0_planck_relation : tau0 = planck_time / sqrt Real.pi := by
 173  unfold tau0 planck_time
 174  have hc : c_codata ≠ 0 := c_codata_ne_zero
 175  have hpi : Real.pi ≠ 0 := ne_of_gt Real.pi_pos
 176  have hpi_pos : 0 < Real.pi := Real.pi_pos
 177  have hc_pos : 0 < c_codata := c_codata_pos
 178  have hinner_pos : 0 < hbar_codata * G_codata := mul_pos hbar_codata_pos G_codata_pos
 179  have hsqrt_pi_pos : 0 < sqrt Real.pi := sqrt_pos.mpr hpi_pos
 180  have hsqrt_pi_ne : sqrt Real.pi ≠ 0 := ne_of_gt hsqrt_pi_pos
 181  have hc3_pos : 0 < c_codata ^ 3 := pow_pos hc_pos 3
 182  have hc5_pos : 0 < c_codata ^ 5 := pow_pos hc_pos 5
 183  have hinner5_nonneg : 0 ≤ hbar_codata * G_codata / c_codata ^ 5 :=
 184    le_of_lt (div_pos hinner_pos hc5_pos)
 185  have hc3 : c_codata ^ 3 ≠ 0 := pow_ne_zero 3 hc
 186  have hc5 : c_codata ^ 5 ≠ 0 := pow_ne_zero 5 hc
 187  have hinner3_div_pos : 0 < hbar_codata * G_codata / (Real.pi * c_codata ^ 3) :=
 188    div_pos hinner_pos (mul_pos hpi_pos hc3_pos)
 189  have hinner3_div_nonneg : 0 ≤ hbar_codata * G_codata / (Real.pi * c_codata ^ 3) :=
 190    le_of_lt hinner3_div_pos
 191  -- Strategy: show both sides equal by direct calculation
 192  -- LHS = sqrt(ℏG/(πc³))/c
 193  -- RHS = sqrt(ℏG/c⁵)/sqrt(π)
 194  -- Show: LHS² = RHS² and both are positive
 195  have hlhs_pos : 0 < sqrt (hbar_codata * G_codata / (Real.pi * c_codata ^ 3)) / c_codata :=
 196    div_pos (sqrt_pos.mpr hinner3_div_pos) hc_pos
 197  have hrhs_pos : 0 < sqrt (hbar_codata * G_codata / c_codata ^ 5) / sqrt Real.pi :=
 198    div_pos (sqrt_pos.mpr (div_pos hinner_pos hc5_pos)) hsqrt_pi_pos
 199  have hlhs_sq : (sqrt (hbar_codata * G_codata / (Real.pi * c_codata ^ 3)) / c_codata) ^ 2 =
 200                 hbar_codata * G_codata / (Real.pi * c_codata ^ 5) := by
 201    rw [div_pow, sq_sqrt hinner3_div_nonneg]
 202    field_simp