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e320e82 | 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 | import OAI.Combinatorics.ExpectationThreshold.TupleCoefficients
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Finset.Lattice.Fold
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.BigOperators.GroupWithZero.Finset
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.NormNum
namespace OAI
/-!
# A finite geometric bound for nonzero tuples
Every finite family of tuples lies in a finite coordinate box. Factoring
the sum over this box reduces the estimate to ordinary finite geometric
sums. Adding the zero tuple contributes exactly one, so removing it gives
the selector's budget of strictly less than `1 / 20`.
-/
namespace TalagrandThreshold
open scoped BigOperators
private theorem finite_geom_sum_mul_one_sub (c : ℝ) (n : ℕ) :
(∑ k ∈ Finset.range n, c ^ k) * (1 - c) = 1 - c ^ n := by
induction n with
| zero => simp
| succ n ih =>
rw [Finset.sum_range_succ, pow_succ]
nlinarith [ih]
private theorem finite_geom_sum_le_inverse {c : ℝ}
(hc0 : 0 ≤ c) (hc1 : c < 1) (n : ℕ) :
(∑ k ∈ Finset.range n, c ^ k) ≤ (1 - c)⁻¹ := by
rw [← one_div]
apply (le_div_iff₀ (sub_pos.mpr hc1)).mpr
rw [finite_geom_sum_mul_one_sub]
exact sub_le_self 1 (pow_nonneg hc0 n)
/-- A finite tuple family is bounded by the product of geometric bounds. -/
theorem finite_tuple_sum_le_inverse_product {ι : Type*} [Fintype ι]
(T : Finset (ι → ℕ)) (c : ι → ℝ)
(hc0 : ∀ i, 0 ≤ c i) (hc1 : ∀ i, c i < 1) :
(∑ t ∈ T, ∏ i, (c i) ^ (t i)) ≤ ∏ i, (1 - c i)⁻¹ := by
classical
let m : ι → ℕ := fun i => T.sup (fun t => t i) + 1
let B : Finset (ι → ℕ) := Fintype.piFinset (fun i => Finset.range (m i))
have hTB : T ⊆ B := by
intro t ht
apply Fintype.mem_piFinset.mpr
intro i
apply Finset.mem_range.mpr
exact Nat.lt_succ_of_le (Finset.le_sup (f := fun u => u i) ht)
calc
(∑ t ∈ T, ∏ i, (c i) ^ (t i)) ≤
∑ t ∈ B, ∏ i, (c i) ^ (t i) :=
Finset.sum_le_sum_of_subset_of_nonneg hTB
(fun t _ _ => Finset.prod_nonneg fun i _ => pow_nonneg (hc0 i) (t i))
_ = ∏ i, ∑ k ∈ Finset.range (m i), (c i) ^ k :=
(Finset.prod_univ_sum (fun i => Finset.range (m i))
(fun i k => (c i) ^ k)).symm
_ ≤ ∏ i, (1 - c i)⁻¹ := by
apply Finset.prod_le_prod₀
· intro i _
exact Finset.sum_nonneg fun k _ => pow_nonneg (hc0 i) k
· intro i _
exact finite_geom_sum_le_inverse (hc0 i) (hc1 i) (m i)
/-- The sum of coefficients controls every finite tuple family. -/
theorem finite_tuple_sum_le_inverse_one_sub_sum {ι : Type*} [Fintype ι]
(T : Finset (ι → ℕ)) (c : ι → ℝ)
(hc0 : ∀ i, 0 ≤ c i) (hc1 : ∀ i, c i < 1)
(hsum : (∑ i, c i) < 1) :
(∑ t ∈ T, ∏ i, (c i) ^ (t i)) ≤ (1 - ∑ i, c i)⁻¹ := by
calc
(∑ t ∈ T, ∏ i, (c i) ^ (t i)) ≤ ∏ i, (1 - c i)⁻¹ :=
finite_tuple_sum_le_inverse_product T c hc0 hc1
_ = (∏ i, (1 - c i))⁻¹ := Finset.prod_inv_distrib _
_ ≤ (1 - ∑ i, c i)⁻¹ :=
inv_anti₀ (sub_pos.mpr hsum)
(one_sub_sum_le_prod_one_sub c hc0 (fun i => (hc1 i).le))
/-- The actual selector coefficients give a strict budget for every finite
family of nonzero tuples, without an infinite-series argument. -/
theorem tuple_geometric_sum_lt_one_twentieth {s : ℕ}
(T : Finset (Fin s → ℕ)) (hT : ∀ t ∈ T, t ≠ 0) :
(∑ t ∈ T, ∏ i : Fin s, ((3 : ℝ) / 64 ^ (i.val + 1)) ^ (t i)) < 1 / 20 := by
classical
let c : Fin s → ℝ := fun i => tupleCoefficient i.val
have hc0 : ∀ i, 0 ≤ c i := fun i => tupleCoefficient_nonneg i.val
have hc1 : ∀ i, c i < 1 := fun i => tupleCoefficient_lt_one i.val
have hrho : (∑ i, c i) < (1 / 21 : ℝ) := sum_tupleCoefficient_lt s
have hsum : (∑ i, c i) < 1 := hrho.trans (by norm_num)
have hz : (0 : Fin s → ℕ) ∉ T := fun hz => hT 0 hz rfl
have hall := finite_tuple_sum_le_inverse_one_sub_sum (insert 0 T) c hc0 hc1 hsum
have htotal : 1 + (∑ t ∈ T, ∏ i, (c i) ^ (t i)) ≤ (1 - ∑ i, c i)⁻¹ := by
simpa only [Finset.sum_insert hz, Pi.zero_apply, pow_zero, Finset.prod_const_one]
using hall
have hden : (20 / 21 : ℝ) < 1 - ∑ i, c i := by linarith
have hinv : (1 - ∑ i, c i)⁻¹ < (21 / 20 : ℝ) := by
have h := (inv_lt_inv₀ (sub_pos.mpr hsum)
(by norm_num : (0 : ℝ) < 20 / 21)).mpr hden
norm_num at h
exact h
change (∑ t ∈ T, ∏ i, (c i) ^ (t i)) < 1 / 20
linarith
theorem tuple_geometric_sum_le_one_twentieth {s : ℕ}
(T : Finset (Fin s → ℕ)) (hT : ∀ t ∈ T, t ≠ 0) :
(∑ t ∈ T, ∏ i : Fin s, ((3 : ℝ) / 64 ^ (i.val + 1)) ^ (t i)) ≤ 1 / 20 :=
(tuple_geometric_sum_lt_one_twentieth T hT).le
/-- A positive total degree is the form supplied by a nonempty changed set. -/
theorem tuple_geometric_sum_lt_one_twentieth_of_pos_total {s : ℕ}
(T : Finset (Fin s → ℕ)) (hT : ∀ t ∈ T, 0 < ∑ i, t i) :
(∑ t ∈ T, ∏ i : Fin s, ((3 : ℝ) / 64 ^ (i.val + 1)) ^ (t i)) < 1 / 20 := by
apply tuple_geometric_sum_lt_one_twentieth T
intro t ht hz
have hpos := hT t ht
simp [hz] at hpos
end TalagrandThreshold
end OAI
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