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ddc84c5 | 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 | import Mathlib.Analysis.Normed.Algebra.Exponential
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Analysis.Normed.Ring.InfiniteSum
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Ring
namespace OAI
namespace Generator
/-! The homogeneous expansion of an ordered exponential product,
with factorial weights and multiplication order retained. -/
noncomputable section
open scoped BigOperators
section Algebra
variable {R : Type*} [Ring R] [Algebra ℝ R]
def exponentialDegree : List R → ℕ → R
| [], d => if d = 0 then 1 else 0
| a :: as, d => ∑ k ∈ Finset.range (d + 1),
((k.factorial : ℝ)⁻¹ • a ^ k) * exponentialDegree as (d - k)
@[simp] theorem exponentialDegree_zero (xs : List R) : exponentialDegree xs 0 = 1 := by
induction xs with
| nil => simp [exponentialDegree]
| cons a as ih => simp [exponentialDegree, ih]
@[simp] theorem exponentialDegree_one (xs : List R) : exponentialDegree xs 1 = xs.sum := by
induction xs with
| nil => simp [exponentialDegree]
| cons a as ih =>
simp [exponentialDegree, Finset.sum_range_succ, ih, add_comm]
/-- Transport the finite original-degree polynomial through an algebra map. -/
theorem exponentialDegree_map {S : Type*} [Ring S] [Algebra ℝ S]
(φ : R →ₐ[ℝ] S) (xs : List R) (d : ℕ) :
φ (exponentialDegree xs d) = exponentialDegree (xs.map φ) d := by
induction xs generalizing d with
| nil => by_cases hd : d = 0 <;> simp [exponentialDegree, hd]
| cons a as ih => simp [exponentialDegree, ih]
end Algebra
variable {R : Type*} [NormedRing R] [NormedAlgebra ℝ R]
/-- Absolute convergence for fixed row count. This is not the uniform bound. -/
theorem exponentialDegree_summable_norm (xs : List R) :
Summable (fun d => ‖exponentialDegree xs d‖) := by
induction xs with
| nil =>
have he : (fun d => ‖exponentialDegree ([] : List R) d‖) =
(fun d : ℕ => if d = 0 then ‖(1 : R)‖ else 0) := by
funext d
by_cases hd : d = 0 <;> simp [exponentialDegree, hd]
rw [he]
exact (hasSum_ite_eq (0 : ℕ) ‖(1 : R)‖).summable
| cons a as ih =>
exact summable_norm_sum_mul_range_of_summable_norm
(NormedSpace.norm_expSeries_summable' (𝕂 := ℝ) a) ih
variable [CompleteSpace R]
/-- The series sums to the ordered product, without requiring commuting rows. -/
theorem exponentialDegree_hasSum (xs : List R) :
HasSum (exponentialDegree xs) ((xs.map NormedSpace.exp).prod) := by
induction xs with
| nil => simpa [exponentialDegree] using (hasSum_ite_eq (0 : ℕ) (1 : R))
| cons a as ih =>
have h := hasSum_sum_range_mul_of_summable_norm
(NormedSpace.norm_expSeries_summable' (𝕂 := ℝ) a)
(exponentialDegree_summable_norm as)
rw [(NormedSpace.exp_series_hasSum_exp' (𝕂 := ℝ) a).tsum_eq, ih.tsum_eq] at h
exact h
end
/-! Summation of homogeneous-degree remainder bounds, given the expansion and its coefficient estimates. -/
variable {E : Type*} [NormedAddCommGroup E]
theorem geometric_second_order_remainder {q : ℝ} (hq0 : 0 ≤ q) (hq1 : q < 1)
(f : ℕ → E) {S : E} (hs : HasSum f S)
(hb : ∀ n : ℕ, ‖f (n + 2)‖ ≤ q ^ (n + 2)) :
‖S - f 0 - f 1‖ ≤ q ^ 2 / (1 - q) := by
have hg : HasSum (fun n : ℕ => q ^ (n + 2)) (q ^ 2 / (1 - q)) := by
simpa only [pow_add, div_eq_mul_inv, mul_comm] using
(hasSum_geometric_of_lt_one hq0 hq1).mul_right (q ^ 2)
have ht : HasSum (fun n : ℕ => f (n + 2)) (S - f 0 - f 1) := by
simpa [Finset.sum_range_succ, sub_add_eq_sub_sub] using (hasSum_nat_add_iff' 2).mpr hs
exact ht.norm_le_of_bounded hg hb
/-- The source's 4α²/(1−2α) expression, with precisely 0 < α < 1/2. -/
theorem perturbation_remainder_from_degree_bounds {α : ℝ} (hα : 0 < α) (hhalf : α < 1 / 2)
(f : ℕ → E) {S : E} (hs : HasSum f S)
(hb : ∀ n : ℕ, ‖f (n + 2)‖ ≤ (2 * α) ^ (n + 2)) :
‖S - f 0 - f 1‖ ≤ 4 * α ^ 2 / (1 - 2 * α) := by
have h := geometric_second_order_remainder (by linarith : 0 ≤ 2 * α)
(by linarith : 2 * α < 1) f hs hb
convert h using 1
ring
end Generator
end OAI
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