File size: 2,605 Bytes
a72760f | 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 | import OAI.NumberTheory.PiExponent.Analysis.FormalLogTruncation
import OAI.NumberTheory.PiExponent.Jets.BranchContact
namespace OAI
noncomputable section
namespace PiExponent.FormalBranchContact
open FormalLogTruncation
theorem logTail_order_ge (T : ℕ) : (T : ℕ∞) ≤ (logTail T).order :=
PowerSeries.nat_le_order (logTail T) T (coeff_logTail_eq_zero T)
theorem composed_logTail_order_ge (T : ℕ) (t : PowerSeries ℂ)
(ht : PowerSeries.constantCoeff t = 0) :
(T : ℕ∞) * t.order ≤ PowerSeries.order (PowerSeries.subst t (logTail T)) := by
calc
(T : ℕ∞) * t.order = MvPowerSeries.order t * T := by
rw [PowerSeries.order_eq_order, mul_comm]
_ ≤ MvPowerSeries.order t * (logTail T).order :=
mul_le_mul_right (logTail_order_ge T) _
_ ≤ MvPowerSeries.order (PowerSeries.subst t (logTail T)) :=
PowerSeries.le_order_subst t (PowerSeries.HasSubst.of_constantCoeff_zero' ht) _
_ = PowerSeries.order (PowerSeries.subst t (logTail T)) := PowerSeries.order_eq_order.symm
theorem composed_logTail_strict_contact {m : ℕ}
(v : Fin (m+1) → ℚ) (hv : ∀ i, 0 < v i)
(a : Fin (m+1) → PowerSeries ℂ) (hne : ∃ i, a i ≠ 0)
(ha : ∀ i, PowerSeries.constantCoeff (a i) = 0)
(T : Fin m → ℕ) (hT : ∀ i, v i.succ < (T i : ℚ) * v 0)
(i : Fin m) (k : ℕ)
(hk : PowerSeries.order (PowerSeries.subst (a 0) (logTail (T i))) = (k : ℕ∞)) :
BranchContact.contact v a hne * v i.succ < (k : ℚ) := by
have hTi : 0 < T i := by
by_contra h
have hz : T i = 0 := by omega
have hh := hT i
rw [hz, Nat.cast_zero, zero_mul] at hh
exact (not_lt_of_ge (hv i.succ).le) hh
have hb := composed_logTail_order_ge (T i) (a 0) (ha 0)
rw [hk] at hb
have hfinite : (a 0).order ≠ ⊤ := by
intro htop
have hTi0 : (T i : ℕ∞) ≠ 0 := by exact_mod_cast hTi.ne'
simp only [htop, ENat.mul_top hTi0] at hb
simp at hb
obtain ⟨n, hn⟩ := ENat.ne_top_iff_exists.mp hfinite
rw [← hn] at hb
have hnat : T i * n ≤ k := by exact_mod_cast hb
have hcontact := BranchContact.contact_bound v hv a hne 0 n hn.symm
have hμ := BranchContact.contact_pos v hv a hne ha
calc
BranchContact.contact v a hne * v i.succ <
BranchContact.contact v a hne * ((T i : ℚ) * v 0) :=
mul_lt_mul_of_pos_left (hT i) hμ
_ = (T i : ℚ) * (BranchContact.contact v a hne * v 0) := by ring
_ ≤ (T i : ℚ) * (n : ℚ) := mul_le_mul_of_nonneg_left hcontact (by positivity)
_ ≤ (k : ℚ) := by exact_mod_cast hnat
end PiExponent.FormalBranchContact
end
end OAI
|