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import OAI.NumberTheory.PiExponent.Jets.NormalBasisRigidity
namespace OAI
noncomputable section
namespace PiExponent.NormalBasisRigidity
open Module
variable {K : Type*} [Field K] (m : ℕ) (y : K) (hy : y ≠ 0)
def frameEquiv : (Fin (m + 1) → K) ≃ₗ[K] (Fin (m + 1) → K) where
toFun c := Fin.cases (y * c 0) (fun i => c i.succ + c 0)
invFun t := Fin.cases (t 0 / y) (fun i => t i.succ - t 0 / y)
left_inv c := by
funext i
refine Fin.cases ?_ (fun j => ?_) i
· simp [hy]
· simp [hy]
right_inv t := by
funext i
refine Fin.cases ?_ (fun j => ?_) i
· change y * (t 0 / y) = t 0
field_simp
· simp
map_add' c d := by
funext i
refine Fin.cases ?_ (fun j => ?_) i
· simp [mul_add]
· simp only [Pi.add_apply, Fin.cases_succ]
ring
map_smul' a c := by
funext i
refine Fin.cases ?_ (fun j => ?_) i
· simp only [Pi.smul_apply, Fin.cases_zero, smul_eq_mul, RingHom.id_apply]
ring
· simp [mul_add]
def frameBasis : Basis (Fin (m + 1)) K (Fin (m + 1) → K) :=
(Pi.basisFun K _).map (frameEquiv m y hy)
@[simp] lemma frameBasis_repr_zero (t : Fin (m + 1) → K) :
(frameBasis m y hy).repr t 0 = t 0 / y := by
change (Pi.basisFun K _).repr ((frameEquiv m y hy).symm t) 0 = _
rw [Pi.basisFun_repr]
rfl
@[simp] lemma frameBasis_repr_succ (t : Fin (m + 1) → K) (i : Fin m) :
(frameBasis m y hy).repr t i.succ = t i.succ - t 0 / y := by
change (Pi.basisFun K _).repr ((frameEquiv m y hy).symm t) i.succ = _
rw [Pi.basisFun_repr]
rfl
@[simp] lemma frameBasis_succ (i : Fin m) :
frameBasis m y hy i.succ = Pi.basisFun K _ i.succ := by
ext j
refine Fin.cases ?_ (fun k => ?_) j
· simp [frameBasis, frameEquiv, Pi.basisFun_apply]
· simp [frameBasis, frameEquiv, Pi.basisFun_apply, Pi.single_apply]
@[simp] lemma frameBasis_zero_zero : frameBasis m y hy 0 0 = y := by
simp [frameBasis, frameEquiv, Pi.basisFun_apply]
@[simp] lemma frameBasis_zero_succ (i : Fin m) : frameBasis m y hy 0 i.succ = 1 := by
simp [frameBasis, frameEquiv, Pi.basisFun_apply]
variable {V : Type*} [AddCommGroup V] [Module K V]
lemma mandatory_logarithmic_differential (q : (Fin (m + 1) → K) →ₗ[K] V)
(hq : Submodule.span K (Set.range (fun j => q (frameBasis m y hy j))) = ⊤)
(i : Fin m)
(hi : ∀ A, IsNormalBasis (K := K) (fun j => q (frameBasis m y hy j)) A → i.succ ∈ A)
(t : Fin (m + 1) → K) (ht : q t = 0) : t i.succ = t 0 / y := by
have h := mandatory_coordinate_eq_zero (frameBasis m y hy) q hq i.succ hi t ht
rw [frameBasis_repr_succ] at h
exact sub_eq_zero.mp h
lemma span_mapped_basis_eq_top {ι U : Type*} [Fintype ι]
[AddCommGroup U] [Module K U] (b : Basis ι K U) (q : U →ₗ[K] V)
(hq : Function.Surjective q) :
Submodule.span K (Set.range (fun j => q (b j))) = ⊤ := by
apply top_unique
intro x _
obtain ⟨t, rfl⟩ := hq x
rw [← b.sum_repr t, map_sum]
apply Submodule.sum_mem
intro i _
rw [map_smul]
exact Submodule.smul_mem _ _ (Submodule.subset_span (Set.mem_range_self i))
theorem normal_basis_differential_dichotomy
(q : (Fin (m + 1) → K) →ₗ[K] V) (hq : Function.Surjective q) (hq0 : q ≠ 0)
(hexclude : ∀ A B : Finset (Fin (m + 1)),
IsNormalBasis (K := K) (fun j => q (Pi.basisFun K _ j)) A →
IsNormalBasis (K := K) (fun j => q (frameBasis m y hy j)) B →
∀ i, i ≠ 0 → i ∈ A → i ∉ B →
(∀ j, i < j → (j ∈ A ↔ j ∈ B)) → False) :
(∀ t, q t = 0 → t 0 = 0) ∨
∃ i : Fin m, ∀ t, q t = 0 → t i.succ = t 0 / y := by
classical
let b := Pi.basisFun K (Fin (m + 1))
let S := Finset.univ.filter (fun i : Fin (m + 1) => i ≠ 0 ∧ q (b i) ≠ 0)
by_cases hS : S.Nonempty
· let i := S.max' hS
have hiS : i ∈ S := Finset.max'_mem _ _
have hi := (Finset.mem_filter.mp hiS).2
have hlarge : ∀ j, i < j → q (b j) = 0 := by
intro j hj
by_contra hjq
have hj0 : j ≠ 0 := by
intro hz
subst j
exact (not_lt_of_ge (Fin.zero_le _)) hj
have hjS : j ∈ S := Finset.mem_filter.mpr ⟨Finset.mem_univ _, hj0, hjq⟩
exact (not_lt_of_ge (Finset.le_max' _ _ hjS)) hj
have hvu : ∀ j : Fin (m + 1), j ≠ 0 → q (frameBasis m y hy j) = q (b j) := by
intro j
refine Fin.cases ?_ (fun k _ => ?_) j
· intro hz
exact (hz rfl).elim
· rw [frameBasis_succ]
have hmand : ∀ B, IsNormalBasis (K := K) (fun j => q (frameBasis m y hy j)) B → i ∈ B :=
largest_nontangent_mandatory (fun j => q (b j))
(fun j => q (frameBasis m y hy j)) (span_mapped_basis_eq_top b q hq)
hvu hexclude i hi.1 hi.2 hlarge
rcases Fin.eq_zero_or_eq_succ i with hz | ⟨j, hj⟩
· exact (hi.1 hz).elim
· right
refine ⟨j, ?_⟩
intro t ht
apply mandatory_logarithmic_differential m y hy q
(span_mapped_basis_eq_top (frameBasis m y hy) q hq) j ?_ t ht
intro B hB
rw [← hj]
exact hmand B hB
· have hpos : ∀ j : Fin m, q (b j.succ) = 0 := by
intro j
by_contra hj
exact hS ⟨j.succ, Finset.mem_filter.mpr ⟨Finset.mem_univ _, Fin.succ_ne_zero _, hj⟩⟩
have he0 : q (b 0) ≠ 0 := by
intro he
apply hq0
apply b.ext
intro j
refine Fin.cases ?_ (fun k => ?_) j
· exact he
· exact hpos k
have hqform (t : Fin (m + 1) → K) : q t = t 0 • q (b 0) := by
calc
q t = q (∑ j, b.repr t j • b j) := congrArg q (b.sum_repr t).symm
_ = ∑ j, b.repr t j • q (b j) := by simp only [map_sum, map_smul]
_ = t 0 • q (b 0) := by
rw [Fin.sum_univ_succ]
simp only [hpos, smul_zero, Finset.sum_const_zero, add_zero]
simp only [b, Pi.basisFun_repr]
left
intro t ht
rw [hqform] at ht
exact (smul_eq_zero.mp ht).resolve_right he0
end PiExponent.NormalBasisRigidity
end
end OAI
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