File size: 1,891 Bytes
67aee1d | 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 | import OAI.Analysis.GroupDeterminants.LampDenominators
namespace OAI
universe uH
noncomputable section
open scoped BigOperators Pointwise
namespace GroupRingDeterminant
variable {H : Type uH} [Group H]
lemma zmodTwo_eq_one_of_ne_zero (z : ZMod 2) (hz : z ≠ 0) : z = 1 := by
exact (by decide : ∀ z : ZMod 2, z ≠ 0 → z = 1) z hz
lemma binary_support_expansion (a : MonoidAlgebra (ZMod 2) H) (ha : a.coeff 1 = 1) :
∃ S : Finset H, 1 ∉ S ∧ a = 1 + ∑ s ∈ S, MonoidAlgebra.single s 1 := by
classical
have hm : 1 ∈ a.coeff.support := Finsupp.mem_support_iff.mpr (ha ▸ one_ne_zero)
refine ⟨a.coeff.support.erase 1, Finset.notMem_erase _ _, ?_⟩
calc
a = ∑ s ∈ a.coeff.support, MonoidAlgebra.single s (a.coeff s) :=
(MonoidAlgebra.sum_coeff_single a).symm
_ = MonoidAlgebra.single 1 (a.coeff 1) +
∑ s ∈ a.coeff.support.erase 1, MonoidAlgebra.single s (a.coeff s) :=
(Finset.add_sum_erase _ _ hm).symm
_ = _ := by
rw [ha, ← MonoidAlgebra.one_def]
congr 1
apply Finset.sum_congr rfl
intro s hs
rw [zmodTwo_eq_one_of_ne_zero (a.coeff s)
(Finsupp.mem_support_iff.mp (Finset.mem_of_mem_erase hs))]
lemma binary_coefficient_normalize (a : MonoidAlgebra (ZMod 2) H) (ha : a ≠ 0) :
∃ h : H, ∃ S : Finset H, 1 ∉ S ∧
MonoidAlgebra.single h⁻¹ 1 * a = 1 + ∑ s ∈ S, MonoidAlgebra.single s 1 := by
classical
obtain ⟨h, hh⟩ : ∃ h, a.coeff h ≠ 0 := by
by_contra! hn
apply ha
ext h
exact hn h
have he : (MonoidAlgebra.single h⁻¹ 1 * a).coeff 1 = 1 := by
rw [MonoidAlgebra.coeff_single_mul_eq_mul_coeff h (by simp [inv_mul_eq_one, eq_comm])]
exact one_mul _ |>.trans (zmodTwo_eq_one_of_ne_zero _ hh)
obtain ⟨S, hS, he⟩ := binary_support_expansion _ he
exact ⟨h, S, hS, he⟩
end GroupRingDeterminant
end
end OAI
|