Learning.sum_rewardByCount_eq_sumRewards🔗

Learning.sum_rewardByCount_eq_sumRewards.{u_1, u_2, u_3} {𝓐 : Type u_1}
  {𝓨 : Type u_2} {Ω : Type u_3} [DecidableEq 𝓐] [AddCommGroup 𝓨]
  {A : ℕ → Ω → 𝓐} {R : ℕ → Ω → 𝓨} (a : 𝓐) (t : ℕ)
  (ω : Ω × (ℕ → 𝓐 → 𝓨)) :
  ∑ m ∈ Finset.Icc 1 (pullCount A a t (Prod.fst ω)),
      rewardByCount A R a m ω =
    sumRewards A R a t (Prod.fst ω)
Learning.sum_rewardByCount_eq_sumRewards.{u_1,
    u_2, u_3}
  {𝓐 : Type u_1} {𝓨 : Type u_2}
  {Ω : Type u_3} [DecidableEq 𝓐]
  [AddCommGroup 𝓨] {A : ℕ → Ω → 𝓐}
  {R : ℕ → Ω → 𝓨} (a : 𝓐) (t : ℕ)
  (ω : Ω × (ℕ → 𝓐 → 𝓨)) :
  ∑
      m ∈
        Finset.Icc 1
          (pullCount A a t (Prod.fst ω)),
      rewardByCount A R a m ω =
    sumRewards A R a t (Prod.fst ω)

lemma sum_rewardByCount_eq_sumRewards {R : ℕ → Ω → 𝓨} (a : 𝓐) (t : ℕ) (ω : Ω × (ℕ → 𝓐 → 𝓨)) :
    ∑ m ∈ Icc 1 (pullCount A a t ω.1), rewardByCount A R a m ω = sumRewards A R a t ω.1

by
  induction t with
  | zero => simp [pullCount, sumRewards]
  | succ t ht =>
    by_cases hta : A t ω.1 = a
    · rw [← hta] at ht ⊢
      rw [pullCount_action_eq_pullCount_add_one, sum_Icc_succ_top (Nat.le_add_left 1 _), ht]
      unfold sumRewards
      rw [sum_range_succ, ite_eq_left rfl, rewardByCount_pullCount_add_one_eq_reward]
    · unfold sumRewards
      rwa [pullCount_eq_pullCount_of_action_ne hta, sum_range_succ, ite_eq_right hta, add_zero]

Meaning last changed in v4.34.0-rc2-14-gf86702d (2026-08-25), the 3th recorded change.

Self-contained, with its dependencies inlined and proofs replaced by sorry: download the raw file · open it in the Lean web editor.

Dependency graph

Audit surface: 4 project declarations, 45 external constants

✓ Proved: no sorry anywhere in its closure

This is the tool's own reading of one build's recorded axioms, and it is not robust against an author who wants it to pass. Checking meant to be relied on should go through Comparator, which replays the proof through the kernel from an export against an explicit list of permitted axioms.