File size: 2,799 Bytes
216fb31 | 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 | import Mathlib
namespace OAI
noncomputable section
open scoped BigOperators
namespace ControlledBoundary.FormalDivisor
variable {Prime : Type*}
def positivePart (D : Prime →₀ ℚ) : Prime →₀ ℚ :=
D.mapRange (fun x => max x 0) (by simp)
def negativePart (D : Prime →₀ ℚ) : Prime →₀ ℚ :=
D.mapRange (fun x => max (-x) 0) (by simp)
lemma positivePart_nonneg (D : Prime →₀ ℚ) : 0 ≤ positivePart D :=
fun _ => le_max_right _ _
lemma negativePart_nonneg (D : Prime →₀ ℚ) : 0 ≤ negativePart D :=
fun _ => le_max_right _ _
lemma signed_decomposition (D : Prime →₀ ℚ) :
positivePart D - negativePart D = D := by
ext p
change max (D p) 0 - max (-D p) 0 = D p
rcases le_total 0 (D p) with h | h
· rw [max_eq_left h, max_eq_right (neg_nonpos.mpr h), sub_zero]
· rw [max_eq_right h, max_eq_left (neg_nonneg.mpr h)]
ring
lemma positivePart_coeff_lt_one (D : Prime →₀ ℚ) (h : ∀ p, D p < 1) :
∀ p, positivePart D p < 1 := by
intro p
exact max_lt (h p) zero_lt_one
lemma negativePart_support_subset (D : Prime →₀ ℚ) (S : Finset Prime)
(h : ∀ p, p ∉ S → 0 ≤ D p) : (negativePart D).support ⊆ S := by
intro p hp
by_contra hn
have hz : negativePart D p = 0 := max_eq_right (neg_nonpos.mpr (h p hn))
exact (Finsupp.mem_support_iff.mp hp) hz
lemma negativePart_signed_support [DecidableEq Prime] (G J K : Prime →₀ ℚ)
(s : ℚ) (hs : 0 ≤ s) (hG : 0 ≤ G) :
(negativePart (s • G - J - K)).support ⊆ J.support ∪ K.support := by
classical
apply negativePart_support_subset
intro p hp
have hJ : J p = 0 := Finsupp.notMem_support_iff.mp (fun h =>
hp (Finset.mem_union_left _ h))
have hK : K p = 0 := Finsupp.notMem_support_iff.mp (fun h =>
hp (Finset.mem_union_right _ h))
simpa only [Finsupp.sub_apply, Finsupp.smul_apply, smul_eq_mul, hJ, hK, sub_zero]
using mul_nonneg hs (hG p)
theorem normalize_correction {n : ℕ}
(M F E C : Prime →₀ ℚ) (P : Fin n → Prime →₀ ℚ) (a : Fin n → ℚ)
(t : ℚ) (ht : 0 < t) (hF : 0 ≤ F) (hE : 0 ≤ E)
(hSupp : F.support ⊆ C.support)
(hrep : E = t • M + F + ∑ j, a j • P j) :
∃ C₀ E₀ : Prime →₀ ℚ,
0 ≤ C₀ ∧ C₀.support ⊆ C.support ∧ 0 ≤ E₀ ∧
E₀ = M + C₀ + ∑ j, (t⁻¹ * a j) • P j := by
refine ⟨t⁻¹ • F, t⁻¹ • E, ?_, ?_, ?_, ?_⟩
· intro p
exact mul_nonneg (inv_nonneg.mpr ht.le) (hF p)
· exact (Finsupp.support_smul).trans hSupp
· intro p
exact mul_nonneg (inv_nonneg.mpr ht.le) (hE p)
· rw [hrep, smul_add, smul_add, smul_smul, inv_mul_cancel₀ (ne_of_gt ht),
one_smul, Finset.smul_sum]
simp only [smul_smul]
end ControlledBoundary.FormalDivisor
end
end OAI
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