openAI-math-dataset / lean /OAI /Combinatorics /BalancedRyser /Probability /FiniteKernelPotential.lean
| import Mathlib.Algebra.Order.BigOperators.Group.Finset | |
| import Mathlib.Tactic | |
| namespace OAI | |
| open scoped BigOperators | |
| namespace Balanced.FiniteKernelPotential | |
| noncomputable section | |
| variable {X : Type*} [Fintype X] | |
| def advance (K : X → X → ℝ) (w : X → ℝ) (y : X) : ℝ := | |
| ∑ x, w x * K x y | |
| def weights (K : ℕ → X → X → ℝ) (w₀ : X → ℝ) : ℕ → X → ℝ | |
| | 0 => w₀ | |
| | t + 1 => advance (K t) (weights K w₀ t) | |
| @[simp] theorem weights_zero (K : ℕ → X → X → ℝ) (w₀ : X → ℝ) : | |
| weights K w₀ 0 = w₀ := rfl | |
| @[simp] theorem weights_succ (K : ℕ → X → X → ℝ) (w₀ : X → ℝ) (t : ℕ) : | |
| weights K w₀ (t + 1) = advance (K t) (weights K w₀ t) := rfl | |
| theorem advance_nonneg (K : X → X → ℝ) (w : X → ℝ) | |
| (hK : ∀ x y, 0 ≤ K x y) (hw : ∀ x, 0 ≤ w x) (y : X) : | |
| 0 ≤ advance K w y := by | |
| apply Finset.sum_nonneg | |
| intro x hx | |
| exact mul_nonneg (hw x) (hK x y) | |
| theorem weights_nonneg (K : ℕ → X → X → ℝ) (w₀ : X → ℝ) | |
| (hK : ∀ t x y, 0 ≤ K t x y) (hw₀ : ∀ x, 0 ≤ w₀ x) | |
| (t : ℕ) (x : X) : 0 ≤ weights K w₀ t x := by | |
| induction t generalizing x with | |
| | zero => exact hw₀ x | |
| | succ t ih => | |
| exact advance_nonneg (K t) (weights K w₀ t) (hK t) | |
| ih x | |
| theorem advance_pairing (K : X → X → ℝ) (w Φ : X → ℝ) : | |
| ∑ y, advance K w y * Φ y = ∑ x, w x * ∑ y, K x y * Φ y := by | |
| simp only [advance, Finset.sum_mul, Finset.mul_sum, mul_assoc] | |
| exact Finset.sum_comm | |
| theorem advance_potential_le (K : X → X → ℝ) (w Φ Ψ : X → ℝ) | |
| (hw : ∀ x, 0 ≤ w x) | |
| (hstep : ∀ x, ∑ y, K x y * Ψ y ≤ Φ x) : | |
| ∑ y, advance K w y * Ψ y ≤ ∑ x, w x * Φ x := by | |
| rw [advance_pairing] | |
| apply Finset.sum_le_sum | |
| intro x hx | |
| exact mul_le_mul_of_nonneg_left (hstep x) (hw x) | |
| theorem expected_potential_le (K : ℕ → X → X → ℝ) (w₀ : X → ℝ) | |
| (Φ : ℕ → X → ℝ) | |
| (hK : ∀ t x y, 0 ≤ K t x y) (hw₀ : ∀ x, 0 ≤ w₀ x) | |
| (hstep : ∀ t x, ∑ y, K t x y * Φ (t + 1) y ≤ Φ t x) | |
| (T : ℕ) : | |
| ∑ y, weights K w₀ T y * Φ T y ≤ ∑ x, w₀ x * Φ 0 x := by | |
| induction T with | |
| | zero => exact le_rfl | |
| | succ T ih => | |
| exact (advance_potential_le (K T) (weights K w₀ T) (Φ T) (Φ (T + 1)) | |
| (weights_nonneg K w₀ hK hw₀ T) (hstep T)).trans ih | |
| theorem payoff_le (K : ℕ → X → X → ℝ) (w₀ : X → ℝ) | |
| (Φ : ℕ → X → ℝ) (payoff : X → ℝ) | |
| (hK : ∀ t x y, 0 ≤ K t x y) (hw₀ : ∀ x, 0 ≤ w₀ x) | |
| (hstep : ∀ t x, ∑ y, K t x y * Φ (t + 1) y ≤ Φ t x) | |
| (T : ℕ) (hpayoff : ∀ x, payoff x ≤ Φ T x) : | |
| ∑ y, weights K w₀ T y * payoff y ≤ ∑ x, w₀ x * Φ 0 x := by | |
| calc | |
| _ ≤ ∑ y, weights K w₀ T y * Φ T y := by | |
| apply Finset.sum_le_sum | |
| intro y hy | |
| exact mul_le_mul_of_nonneg_left (hpayoff y) (weights_nonneg K w₀ hK hw₀ T y) | |
| _ ≤ _ := expected_potential_le K w₀ Φ hK hw₀ hstep T | |
| def pointMass [DecidableEq X] (x₀ : X) (x : X) : ℝ := if x = x₀ then 1 else 0 | |
| omit [Fintype X] in | |
| theorem pointMass_nonneg [DecidableEq X] (x₀ x : X) : | |
| 0 ≤ pointMass x₀ x := by unfold pointMass; split_ifs <;> norm_num | |
| @[simp] theorem pointMass_pairing [DecidableEq X] (x₀ : X) (f : X → ℝ) : | |
| ∑ x, pointMass x₀ x * f x = f x₀ := by simp [pointMass] | |
| theorem pointMass_potential_le [DecidableEq X] | |
| (K : ℕ → X → X → ℝ) (x₀ : X) (Φ : X → ℝ) | |
| (hK : ∀ t x y, 0 ≤ K t x y) | |
| (hstep : ∀ t x, ∑ y, K t x y * Φ y ≤ Φ x) (T : ℕ) : | |
| ∑ y, weights K (pointMass x₀) T y * Φ y ≤ Φ x₀ := by | |
| simpa using expected_potential_le K (pointMass x₀) (fun _ => Φ) hK | |
| (pointMass_nonneg x₀) hstep T | |
| theorem pointMass_payoff_le [DecidableEq X] | |
| (K : ℕ → X → X → ℝ) (x₀ : X) (Φ payoff : X → ℝ) | |
| (hK : ∀ t x y, 0 ≤ K t x y) | |
| (hstep : ∀ t x, ∑ y, K t x y * Φ y ≤ Φ x) | |
| (T : ℕ) (hpayoff : ∀ x, payoff x ≤ Φ x) : | |
| ∑ y, weights K (pointMass x₀) T y * payoff y ≤ Φ x₀ := by | |
| simpa using payoff_le K (pointMass x₀) (fun _ => Φ) payoff hK | |
| (pointMass_nonneg x₀) hstep T hpayoff | |
| theorem weights_eq_of_recurrence (K : ℕ → X → X → ℝ) (w : ℕ → X → ℝ) | |
| (hrec : ∀ t y, w (t + 1) y = ∑ x, w t x * K t x y) (T : ℕ) : | |
| weights K (w 0) T = w T := by | |
| induction T with | |
| | zero => rfl | |
| | succ T ih => | |
| funext y | |
| simp only [weights_succ, advance, ih] | |
| exact (hrec T y).symm | |
| theorem evolution_payoff_le (K : ℕ → X → X → ℝ) (w : ℕ → X → ℝ) | |
| (Φ : ℕ → X → ℝ) (payoff : X → ℝ) | |
| (hrec : ∀ t y, w (t + 1) y = ∑ x, w t x * K t x y) | |
| (hK : ∀ t x y, 0 ≤ K t x y) (hw₀ : ∀ x, 0 ≤ w 0 x) | |
| (hstep : ∀ t x, ∑ y, K t x y * Φ (t + 1) y ≤ Φ t x) | |
| (T : ℕ) (hpayoff : ∀ x, payoff x ≤ Φ T x) : | |
| ∑ y, w T y * payoff y ≤ ∑ x, w 0 x * Φ 0 x := by | |
| rw [← weights_eq_of_recurrence K w hrec T] | |
| exact payoff_le K (w 0) Φ payoff hK hw₀ hstep T hpayoff | |
| theorem pointMass_event_le [DecidableEq X] | |
| (K : ℕ → X → X → ℝ) (x₀ : X) (Φ : X → ℝ) | |
| (hK : ∀ t x y, 0 ≤ K t x y) | |
| (hstep : ∀ t x, ∑ y, K t x y * Φ y ≤ Φ x) | |
| (T : ℕ) (E : X → Prop) [DecidablePred E] | |
| (hE : ∀ x, (if E x then (1 : ℝ) else 0) ≤ Φ x) : | |
| (∑ y, if E y then weights K (pointMass x₀) T y else 0) ≤ Φ x₀ := by | |
| simpa only [mul_ite, mul_one, mul_zero] using | |
| pointMass_payoff_le K x₀ Φ (fun x => if E x then 1 else 0) hK hstep T hE | |
| end | |
| end Balanced.FiniteKernelPotential | |
| end OAI | |