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f0f05de | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 | 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
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