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theorem proved term proof

zero_entropy_deterministic

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formal statement (Lean)

 111theorem zero_entropy_deterministic {n : ℕ} (d : ProbDist n) (i : Fin n)
 112    (hdet : d.probs i = 1) (hother : ∀ j ≠ i, d.probs j = 0) :
 113    shannonEntropy d = 0 := by

proof body

Term-mode proof.

 114  unfold shannonEntropy
 115  simp only [neg_eq_zero]
 116  apply Finset.sum_eq_zero
 117  intro j _
 118  by_cases heq : j = i
 119  · simp [heq, hdet, Real.log_one]
 120  · simp [hother j heq]
 121
 122/-! ## J-Cost Connection -/
 123
 124/-- The J-cost of a probability (relative to uniform).
 125    For p ∈ (0,1], this measures how "surprising" the probability is. -/

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