Plain-language statement
pr/exp of an indicator is just prob
Exact Lean statement
lemma pr_eq_prob (f : Prob α) (y : α) : f.pr (λ x ↦ x = y) = f.prob y
Formal artifact
Lean source
lemma pr_eq_prob (f : Prob α) (y : α) : f.pr (λ x ↦ x = y) = f.prob y := by simp only [pr, exp]; by_cases m : y ∈ f.supp · rw [Finsupp.sum, Finset.sum_eq_single y]; simp only [ite_true, mul_one, prob] · intro z _ zy; simp only [zy, if_false, mul_zero] · intro n; simp only [m] at n simp only [Finsupp.mem_support_iff, ne_eq, not_not] at n; simp only [n, ite_true, mul_one] · rw [Finsupp.sum, Finset.sum_eq_zero]; rw [mem_iff, not_not] at m; exact m.symm intro z n; by_cases zy : z = y · simp only [supp, Finsupp.mem_support_iff, not_not, zy] at m n; simp only [n] at m · simp only [zy, if_false, mul_zero]- Project
- debate
- License
- Apache-2.0
- Commit
- de3a6e500ae1
- Source
- Prob/Arith.lean:253-262
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