Cost Ndim Dalembert Jcost N Submult

The submultiplicative bound

In mathematics, a d'Alembert equation is a functional equation that characterizes solutions by their behavior under a product operation. For a function f on positive numbers, the classical form is f(x+y) + f(x−y) = 2f(x)f(y), whose continuous solutions are the cosine and hyperbolic cosine families. The Recognition Science framework generalizes this to vectors, where the product is taken componentwise. Its cost function JcostN assigns a nonnegative real number to each vector of positive entries.

The framework's machine-checked library of formal theorems proves that JcostN satisfies a multidimensional d'Alembert identity. For any two vectors x and y with positive entries, the cost of their componentwise product plus the cost of their componentwise quotient equals twice the cost of x plus twice the cost of y plus twice the product of the two costs. Written in symbols: JcostN(x∘y) + JcostN(x⊘y) = 2JcostN(x) + 2JcostN(y) + 2JcostN(x)JcostN(y).

Because the quotient term is always nonnegative, the identity immediately yields an inequality: the cost of the product is at most twice the cost of x plus twice the cost of y plus twice the product of the costs. This is the lemma JcostN_submult. It is a direct consequence of the identity, requiring only the nonnegativity of the cost function. The proof is a single arithmetic step after applying the identity and the nonnegativity fact.

In Recognition Science, this inequality is a structural constraint on how recognition costs compose. It says that combining two vectors never costs more than a specific function of their individual costs. The bound is sharp in the sense that it follows from an exact identity, not a loose estimate. The lemma holds for any dimension n and any parameter α, with no additional assumptions beyond positivity of the vector entries.

The lemma does not claim that the bound is tight, nor that equality holds for any particular vectors. It does not assert that the quotient term vanishes, nor that the cost function is subadditive in the usual sense. It is purely an upper bound derived from the exact identity. The framework's library proves the identity and the inequality as formal theorems, with no unproved assumptions beyond the standard axioms of the underlying logic.

theorem JcostN_dAlembert {n : ℕ} (α x y : Vec n)
    (hx : ∀ i, 0 < x i) (hy : ∀ i, 0 < y i) :
    JcostN α (hadamardMul x y) + JcostN α (hadamardDiv x y)
      = 2 * JcostN α x + 2 * JcostN α y + 2 * JcostN α x * JcostN α y := by
  let u : ℝ := Real.exp (dot α (logVec x))
  let v : ℝ := Real.exp (dot α (logVec y))
  have hu : 0 < u := by
    dsimp [u]
    exact Real.exp_pos _
  have hv : 0 < v := by
    dsimp [v]
    exact Real.exp_pos _
  have hmul :
      JcostN α (hadamardMul x y) = Jcost (u * v) := by
    calc
      JcostN α (hadamardMul x y)
          = Jcost (Real.exp (dot α (logVec (hadamardMul x y)))) := by
              simp [JcostN, JlogN]
      _ = Jcost (Real.exp (dot α (logVec x) + dot α (logVec y))) := by
            rw [dot_log_hadamardMul α x y hx hy]
      _ = Jcost (u * v) := by
            simp [u, v, Real.exp_add]
  have hdiv :
      JcostN α (hadamardDiv x y) = Jcost (u / v) := by
    calc
      JcostN α (hadamardDiv x y)
          = Jcost (Real.exp (dot α (logVec (hadamardDiv x y)))) := by
              simp [JcostN, JlogN]
      _ = Jcost (Real.exp (dot α (logVec x) - dot α (logVec y))) := by
            rw [dot_log_hadamardDiv α x y hx hy]
      _ = Jcost (u / v) := by
            simp [u, v, Real.exp_sub]
  have hbase := dalembert_identity (x := u) (y := v) hu hv
  calc
    JcostN α (hadamardMul x y) + JcostN α (hadamardDiv x y)
        = Jcost (u * v) + Jcost (u / v) := by rw [hmul, hdiv]
    _ = 2 * Jcost u + 2 * Jcost v + 2 * Jcost u * Jcost v := hbase
    _ = 2 * JcostN α x + 2 * JcostN α y + 2 * JcostN α x * JcostN α y := by
          simp [u, v, JcostN, JlogN]

lemma JcostN_submult {n : ℕ} (α x y : Vec n)
    (hx : ∀ i, 0 < x i) (hy : ∀ i, 0 < y i) :
    JcostN α (hadamardMul x y)
      ≤ 2 * JcostN α x + 2 * JcostN α y + 2 * JcostN α x * JcostN α y := by
  have h := JcostN_dAlembert α x y hx hy
  have hnonneg : 0 ≤ JcostN α (hadamardDiv x y) := JcostN_nonneg α (hadamardDiv x y)
  linarith

What this page does not claim

The lemma does not claim that the bound is tight or that equality holds for any vectors. It does not claim that the quotient term vanishes or that the cost is subadditive. It does not assert anything about the value of the parameter α or the dimension n beyond the stated theorem.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Cost/Ndim/DAlembert.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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