| import OAI.Algebra.Finitistic.Encoding.QuadraticPresentation |
| import Mathlib.LinearAlgebra.Finsupp.LinearCombination |
|
|
| namespace OAI |
| noncomputable section |
| namespace LittleFinitistic.QuadraticPresentation |
|
|
| variable (R : Type*) [CommRing R] (α : Type*) |
|
|
| abbrev Term := Unit ⊕ (α ⊕ (α × α)) |
|
|
| def term : Term α → FreeAlgebra R α |
| | .inl _ => 1 |
| | .inr (.inl a) => FreeAlgebra.ι R a |
| | .inr (.inr (a,b)) => FreeAlgebra.ι R a * FreeAlgebra.ι R b |
|
|
| /-- The degree bound is exactly the span of constants, generators, and ordered |
| products of two generators. The two factors are never commuted. -/ |
| theorem degreeLE_two_eq_span : |
| degreeLE R α 2 = Submodule.span R (Set.range (term R α)) := by |
| apply le_antisymm |
| · apply Submodule.span_le.mpr |
| rintro f ⟨w,hw,rfl⟩ |
| apply Submodule.subset_span |
| cases w with |
| | nil => exact ⟨.inl (), rfl⟩ |
| | cons a w => |
| cases w with |
| | nil => exact ⟨.inr (.inl a), by simp [term, word]⟩ |
| | cons b w => |
| cases w with |
| | nil => exact ⟨.inr (.inr (a,b)), by simp [term, word]⟩ |
| | cons c w => simp only [List.length_cons] at hw; omega |
| · apply Submodule.span_le.mpr |
| rintro f ⟨i,rfl⟩ |
| rcases i with u | (a | ⟨a,b⟩) |
| · exact one_mem_degreeLE 2 |
| · exact generator_mem_degreeLE a |
| · exact product_mem_degreeLE a b |
|
|
| /-- Finite coefficient form, suitable for the source's homogenization rule. -/ |
| def coefficientPolynomial [Fintype α] (c : R) (l : α → R) (q : α → α → R) : |
| FreeAlgebra R α := |
| algebraMap R (FreeAlgebra R α) c + |
| ∑ a, l a • FreeAlgebra.ι R a + |
| ∑ a, ∑ b, q a b • (FreeAlgebra.ι R a * FreeAlgebra.ι R b) |
|
|
| theorem exists_coefficients [Fintype α] (p : FreeAlgebra R α) |
| (h : p ∈ degreeLE R α 2) : |
| ∃ (c : R) (l : α → R) (q : α → α → R), coefficientPolynomial R α c l q = p := by |
| rw [degreeLE_two_eq_span] at h |
| obtain ⟨f,hf⟩ := (Submodule.mem_span_range_iff_exists_fun R).mp h |
| refine ⟨f (.inl ()), fun a => f (.inr (.inl a)), |
| fun a b => f (.inr (.inr (a,b))), ?_⟩ |
| simpa [coefficientPolynomial, Fintype.sum_sum_type, Fintype.sum_prod_type, |
| term, Algebra.smul_def, add_assoc] using hf |
|
|
| theorem coefficientPolynomial_degree [Fintype α] (c : R) (l : α → R) (q : α → α → R) : |
| coefficientPolynomial R α c l q ∈ degreeLE R α 2 := by |
| unfold coefficientPolynomial |
| apply Submodule.add_mem |
| · apply Submodule.add_mem |
| · simpa only [Algebra.smul_def, mul_one] using |
| (degreeLE R α 2).smul_mem c (one_mem_degreeLE (R := R) (α := α) 2) |
| · exact Submodule.sum_mem _ (fun a _ => |
| (degreeLE R α 2).smul_mem (l a) (generator_mem_degreeLE a)) |
| · exact Submodule.sum_mem _ (fun a _ => Submodule.sum_mem _ (fun b _ => |
| (degreeLE R α 2).smul_mem (q a b) (product_mem_degreeLE a b))) |
|
|
| /-- An exact characterization; the degree predicate hides no higher terms. -/ |
| theorem degreeLE_two_iff_coefficients [Fintype α] (p : FreeAlgebra R α) : |
| p ∈ degreeLE R α 2 ↔ |
| ∃ (c : R) (l : α → R) (q : α → α → R), coefficientPolynomial R α c l q = p := by |
| refine ⟨exists_coefficients R α p, ?_⟩ |
| rintro ⟨c,l,q,rfl⟩ |
| exact coefficientPolynomial_degree R α c l q |
|
|
| /-- Every new quadratic relation has explicit constant, linear, and ordered |
| quadratic coefficients, with a simultaneous choice for the whole finite family. -/ |
| theorem relation_coefficients [Finite α] {κ : Type*} (p : κ → FreeAlgebra R α) (N : ℕ) : |
| letI := Fintype.ofFinite (Letter α N) |
| ∃ (c : EquationIndex α N (κ := κ) → R) |
| (l : EquationIndex α N (κ := κ) → Letter α N → R) |
| (q : EquationIndex α N (κ := κ) → Letter α N → Letter α N → R), |
| ∀ i, coefficientPolynomial R (Letter α N) (c i) (l i) (q i) = relation R α p N i := by |
| let := Fintype.ofFinite (Letter α N) |
| have h := fun i => exists_coefficients R (Letter α N) (relation R α p N i) |
| (relation_degree R α p N i) |
| choose c l q h using h |
| exact ⟨c,l,q,h⟩ |
|
|
| end LittleFinitistic.QuadraticPresentation |
|
|
| end |
|
|
| end OAI |
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|
