RCL_holds

formal source

  95
  96/-- **THEOREM: J satisfies the RCL.**
  97    This is the foundational identity — everything else follows. -/
  98theorem RCL_holds : SatisfiesRCL J := by
  99  intro x y hx hy
 100  unfold J Jcost
 101  have hx0 : x ≠ 0 := ne_of_gt hx
 102  have hy0 : y ≠ 0 := ne_of_gt hy
 103  have hxy0 : x * y ≠ 0 := mul_ne_zero hx0 hy0
 104  have hxy_div0 : x / y ≠ 0 := div_ne_zero hx0 hy0
 105  field_simp [hx0, hy0, hxy0, hxy_div0]
 106  ring
 107
 108/-! ## §3. Cost Composition as Algebraic Operation -/
 109
 110/-- **Cost-composition**: The binary operation on costs induced by the RCL.
 111    Given two "cost levels" a = J(x) and b = J(y), the composed cost is:
 112    a ★ b = 2ab + 2a + 2b = 2(a+1)(b+1) − 2
 113
 114    This captures how costs combine under multiplication of ratios. -/
 115noncomputable def costCompose (a b : ℝ) : ℝ := 2 * a * b + 2 * a + 2 * b
 116
 117/-- Notation for cost composition -/
 118infixl:70 " ★ " => costCompose
 119
 120/-- **THEOREM: Cost composition is commutative.** -/
 121theorem costCompose_comm (a b : ℝ) : a ★ b = b ★ a := by
 122  unfold costCompose; ring
 123
 124/-- **THEOREM: Associator defect for raw RCL composition.**
 125    The unnormalized RCL form is not strictly associative; the defect is `2*(a-c)`. -/
 126theorem costCompose_assoc_defect (a b c : ℝ) :
 127    (a ★ b) ★ c = a ★ (b ★ c) + 2 * (a - c) := by
 128  unfold costCompose