| import OAI.Analysis.GroupDeterminants.LampDenominators |
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| namespace OAI |
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| universe uH |
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| noncomputable section |
| open scoped BigOperators Pointwise |
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| namespace GroupRingDeterminant |
| variable {H : Type uH} [Group H] |
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| 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 |
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| 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))] |
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| 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 |
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| end |
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| end OAI |
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