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theorem proved term proof

different_rungs_different_barriers

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formal statement (Lean)

 160theorem different_rungs_different_barriers (r₁ r₂ : ℕ) (h : r₁ ≠ r₂) :
 161    activationBarrier (phi ^ r₁) ≠ activationBarrier (phi ^ r₂) := by

proof body

Term-mode proof.

 162  unfold activationBarrier J_one J
 163  simp only [sub_zero]
 164  intro heq
 165  have h1 : (0 : ℝ) < phi ^ r₁ := pow_pos phi_pos r₁
 166  have h2 : (0 : ℝ) < phi ^ r₂ := pow_pos phi_pos r₂
 167  -- If ½(φ^r₁ + φ^{-r₁}) = ½(φ^r₂ + φ^{-r₂}) then φ^r₁ = φ^r₂
 168  -- (J is strictly convex and symmetric about 1, so injective on (1,∞))
 169  have := pow_left_injective₀ (ne_of_gt phi_pos) h
 170  simp only [heq] at this
 171  exact h (this rfl)
 172
 173/-- A single-rung enzyme is an imperfect catalyst for a different-rung reaction. -/

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