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ae00908 | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 | import OAI.Algebra.FormulaHitting.Series
import OAI.Algebra.FormulaHitting.RationalTransfer
import OAI.Algebra.FormulaHitting.Generator
import OAI.Algebra.FormulaHitting.Evaluation
import OAI.Algebra.FormulaHitting.FormulaSeries
import Mathlib.Data.Rat.Cast.Lemmas
namespace OAI
noncomputable section
namespace NCHitting
open Matrix
variable {F : Type*} [Field F] [CharZero F]
omit [CharZero F] in
theorem jet_eq_zero_iff (d : ℕ) (a : PowerSeries F) :
jet d a = 0 ↔ ∀ r < d, PowerSeries.coeff r a = 0 := by
constructor
· intro h r hr
exact congrFun h ⟨r, hr⟩
· intro h
ext r
exact h r.val r.isLt
omit [CharZero F] in
theorem jet_ne_zero_of_order_lt (d : ℕ) (a : PowerSeries F)
(horder : a.order < (d : ℕ∞)) : jet d a ≠ 0 := by
intro h
exact (not_le_of_gt horder)
(PowerSeries.nat_le_order a d ((jet_eq_zero_iff d a).mp h))
def integrationMatrix (d i : ℕ) : Matrix (Fin d) (Fin d) F :=
fun r q => (matrixEntry i r.val q.val : F)
theorem integrationMatrix_apply (d i : ℕ) (r q : Fin d) :
integrationMatrix (F := F) d i r q =
if q.val < r.val then
(-1 : F) ^ (r.val - q.val - 1) /
((r.val : F) * (i : F) ^ (r.val - q.val))
else 0 := by
by_cases h : q.val < r.val <;> simp [integrationMatrix, matrixEntry, h]
theorem jet_J (d i : ℕ) (a : PowerSeries F) :
jet d (J i a) = integrationMatrix d i *ᵥ jet d a := by
ext r
rw [jet_apply, coeff_J]
simp only [Matrix.mulVec, dotProduct, integrationMatrix_apply, jet_apply]
let term := fun q =>
(if q < r.val then (-1 : F) ^ (r.val - q - 1) /
((r.val : F) * (i : F) ^ (r.val - q)) else 0) * PowerSeries.coeff q a
change _ = ∑ q : Fin d, term q.val
rw [Fin.sum_univ_eq_sum_range term d]
calc
_ = ∑ q ∈ Finset.range r.val, term q := by
apply Finset.sum_congr rfl
intro q hq
simp only [term, ite_eq_left (Finset.mem_range.mp hq)]
_ = ∑ q ∈ Finset.range d, term q := by
apply Finset.sum_subset (Finset.range_mono (Nat.le_of_lt r.isLt))
intro q hqd hqr
have h : ¬q < r.val := by simpa using hqr
simp [term, h]
theorem jet_wordSeries {n : ℕ} (d : ℕ) (w : Word n) :
jet d (wordSeries (F := F) w) =
(w.map (fun i => integrationMatrix (F := F) d (i.val + 1))).prod *ᵥ
jet d (1 : PowerSeries F) := by
induction w with
| nil => simp
| cons i w ih =>
rw [wordSeries_cons, jet_J, ih]
simp only [List.map_cons, List.prod_cons, Matrix.mulVec_mulVec]
omit [CharZero F] in
theorem generator_cast_eq_integrationMatrix {n s : ℕ} (hns : 2 ≤ n * s)
(i : Fin n) :
(fun r q => (generator n s i r q : F)) =
integrationMatrix (F := F) (dimension n s) (i.val + 1) := by
ext r q
simp only [integrationMatrix, generator_entry hns]
theorem Formula.jet_wordSum_eq_eval_mulVec {n : ℕ} (f : Formula F n) (d : ℕ) :
jet d (f.polynomial.sum fun w a => a • wordSeries (F := F) w) =
f.eval (algebraMap F (Matrix (Fin d) (Fin d) F))
(fun i => integrationMatrix (F := F) d (i.val + 1)) *ᵥ
jet d (1 : PowerSeries F) := by
rw [Formula.eval_eq_sum]
simp only [Finsupp.sum, map_sum, map_smul, Matrix.sum_mulVec,
← Algebra.smul_def, Matrix.smul_mulVec, jet_wordSeries]
theorem Formula.eval_ne_zero_of_jet_wordSum_ne_zero {n : ℕ}
(f : Formula F n) (d : ℕ)
(hjet : jet d (f.polynomial.sum fun w a => a • wordSeries (F := F) w) ≠ 0) :
f.eval (algebraMap F (Matrix (Fin d) (Fin d) F))
(fun i => integrationMatrix (F := F) d (i.val + 1)) ≠ 0 := by
intro h
apply hjet
rw [f.jet_wordSum_eq_eval_mulVec d, h, Matrix.zero_mulVec]
theorem Formula.jet_series_eq_eval_mulVec {n : ℕ} (f : Formula F n) (d : ℕ) :
jet d f.series =
f.eval (algebraMap F (Matrix (Fin d) (Fin d) F))
(fun i => integrationMatrix (F := F) d (i.val + 1)) *ᵥ
jet d (1 : PowerSeries F) := by
rw [f.series_eq_smul_sum, f.jet_wordSum_eq_eval_mulVec]
theorem Formula.eval_ne_zero_of_jet_series_ne_zero {n : ℕ} (f : Formula F n) (d : ℕ)
(hjet : jet d f.series ≠ 0) :
f.eval (algebraMap F (Matrix (Fin d) (Fin d) F))
(fun i => integrationMatrix (F := F) d (i.val + 1)) ≠ 0 := by
apply f.eval_ne_zero_of_jet_wordSum_ne_zero d
rwa [← f.series_eq_smul_sum]
theorem Formula.eval_generator_ne_zero_of_jet_series_ne_zero {n s : ℕ}
(f : Formula F n) (hns : 2 ≤ n * s)
(hjet : jet (dimension n s) f.series ≠ 0) :
f.eval (Matrix.scalar (Fin (dimension n s)))
(fun i r q => (generator n s i r q : F)) ≠ 0 := by
have hT : (fun (i : Fin n) r q => (generator n s i r q : F)) =
(fun i => integrationMatrix (F := F) (dimension n s) (i.val + 1)) := by
funext i
exact generator_cast_eq_integrationMatrix hns i
rw [hT]
exact f.eval_ne_zero_of_jet_series_ne_zero (dimension n s) hjet
end NCHitting
end
end OAI
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