Foundation Primitive Recognition Calculus Delta Probability Prob Disjoint Or

Adding the parts

Probability begins with counting. On a finite space of N+1 equally likely alternatives, an event is just a subset: a rule that picks out some of the alternatives. The probability of that event is the count of picked alternatives divided by N+1, a rational number between 0 and 1. The impossible event, which picks nothing, has probability 0; the certain event, which picks everything, has probability 1.

The classical addition rule follows directly. If two events E and F are disjoint, meaning no alternative satisfies both, then the count of alternatives satisfying E or F equals the count for E plus the count for F. Dividing by N+1 gives the same additivity for probabilities: prob(E or F) = prob(E) + prob(F). This is the statement of prob_disjoint_or, a theorem in the framework's machine-checked library of formal theorems.

In Recognition Science, this is the native probability layer. The framework models a recognition event, a discrete record of which alternatives are distinguished, as a finite subset. Probability here is not an axiom or a limit of frequencies; it is defined as rational counting over finite distinction alternatives. The library proves the whole package in one headline theorem: impossible event zero, certain event one, every event in [0,1], monotonicity under inclusion, and the disjoint additivity rule.

The rule is deliberately narrow. It holds for finite spaces only, and only for events that cannot overlap. It says nothing about infinite probability spaces, about conditional probability, or about events that share alternatives. Those are separate constructions. What prob_disjoint_or establishes is the simplest structural fact of probability: when the parts do not overlap, the whole is their sum.

/-- Disjoint finite events have additive probability. -/
theorem prob_disjoint_or {N : ℕ} {E F : Event N}
    (hdisj : ∀ i, ¬ (E i ∧ F i)) :
    prob (fun i => E i ∨ F i) = prob E + prob F := by
  unfold prob
  rw [count_disjoint_or hdisj]
  rw [Nat.cast_add]
  ring

/-- Uniform finite probability, as a rational counting ratio. -/
noncomputable def prob {N : ℕ} (E : Event N) : ℚ :=
  (count E : ℚ) / (N + 1 : ℚ)

/-- **Delta-native probability headline.** Probability at the native finite layer
is rational counting over finite distinction alternatives: impossible event zero,
certain event one, and every event has probability in `[0,1]`. -/
theorem delta_probability_headline (N : ℕ) :
    prob (N := N) (fun _ => False) = 0
      ∧ prob (N := N) (fun _ => True) = 1
      ∧ (∀ E : Event N, 0 ≤ prob E ∧ prob E ≤ 1)
      ∧ (∀ E F : Event N, (∀ i, E i → F i) → prob E ≤ prob F)
      ∧ (∀ E F : Event N, (∀ i, ¬ (E i ∧ F i)) →
          prob (fun i => E i ∨ F i) = prob E + prob F) :=
  ⟨prob_empty N, prob_univ N, fun E => ⟨prob_nonneg E, prob_le_one E⟩,
    fun _ _ h => prob_mono h, fun _ _ h => prob_disjoint_or h⟩

What this page does not claim

The rule does not extend to events that overlap, where the sum would double-count shared alternatives. The declaration does not define probability for infinite spaces or continuous distributions. It does not introduce conditional probability or independence.

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/Foundation/PrimitiveRecognitionCalculus/DeltaProbability.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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