| 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 |
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|
