Project documentation
The no-ratio version of Bayes' theorem holds unconditionally
Exact Lean statement
lemma bayes' (f : Prob α) (a b : α → Prop) : f.cond a b * f.pr b = f.cond b a * f.pr a
Formal artifact
Lean source
lemma bayes' (f : Prob α) (a b : α → Prop) : f.cond a b * f.pr b = f.cond b a * f.pr a := by simp only [cond] by_cases a0 : f.pr a = 0 · have ab0 : f.pr (λ x ↦ a x ∧ b x) = 0 := by apply le_antisymm _ pr_nonneg; rw [←a0]; apply pr_mono; intro x _; exact And.left simp only [a0, ab0, div_zero, mul_zero, zero_div, zero_mul] by_cases b0 : f.pr b = 0 · have ab0 : f.pr (λ x ↦ b x ∧ a x) = 0 := by apply le_antisymm _ pr_nonneg; rw [←b0]; apply pr_mono; intro x _; exact And.left simp only [b0, ab0, div_zero, mul_zero, zero_div, zero_mul] simp only [div_mul_cancel₀ _ a0, div_mul_cancel₀ _ b0, and_comm]- Project
- debate
- License
- Apache-2.0
- Commit
- de3a6e500ae1
- Source
- Prob/Cond.lean:163-173
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