File size: 4,099 Bytes
216fb31 | 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 | 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
|