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lemma proved tactic proof

rateAction_pos

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formal statement (Lean)

  41lemma rateAction_pos (rot : TwoBranchRotation) : 0 < rateAction rot := by

proof body

Tactic-mode proof.

  42  unfold rateAction
  43  apply neg_pos.mpr
  44  have ⟨h1, h2⟩ := rot.θ_s_bounds
  45  have hsin_pos : 0 < Real.sin rot.θ_s :=
  46    sin_pos_of_pos_of_lt_pi h1 (by linarith : rot.θ_s < π)
  47  -- sin θ < 1 for 0 < θ < π/2
  48  have hsin_lt_one : Real.sin rot.θ_s < 1 := by
  49    have hx1 : -(π / 2) ≤ rot.θ_s := by linarith
  50    have hlt : rot.θ_s < π / 2 := h2
  51    have : Real.sin rot.θ_s < Real.sin (π / 2) :=
  52      sin_lt_sin_of_lt_of_le_pi_div_two hx1 le_rfl hlt
  53    simpa [Real.sin_pi_div_two] using this
  54  exact Real.log_neg hsin_pos hsin_lt_one
  55
  56/-- Born weight from rate action: exp(-2A) = sin²(θ_s) -/

depends on (7)

Lean names referenced from this declaration's body.