| import Mathlib.Algebra.MonoidAlgebra.Defs |
| import Mathlib.Algebra.Order.Ring.Basic |
| import Mathlib.Analysis.Complex.Basic |
| import Mathlib.Data.Finsupp.Order |
|
|
| namespace OAI |
|
|
| noncomputable section |
|
|
| namespace Generator |
|
|
| /-! Coefficientwise absolute-value product comparison, retaining all convolution terms contributing to the same group element. -/ |
| section |
| open scoped BigOperators |
| variable {G : Type*} [Monoid G] |
|
|
| def normCoefficients (f : MonoidAlgebra ℂ G) : MonoidAlgebra ℝ G := |
| .ofCoeff (f.coeff.mapRange (fun z => ‖z‖) norm_zero) |
|
|
| omit [Monoid G] in |
| @[simp] theorem normCoefficients_apply (f : MonoidAlgebra ℂ G) (g : G) : |
| (normCoefficients f).coeff g = ‖f.coeff g‖ := rfl |
|
|
| omit [Monoid G] in |
| @[simp] theorem normCoefficients_support (f : MonoidAlgebra ℂ G) : |
| (normCoefficients f).coeff.support = f.coeff.support := by |
| ext g |
| simp only [Finsupp.mem_support_iff, normCoefficients_apply, ne_eq, norm_eq_zero] |
|
|
| omit [Monoid G] in |
| theorem normCoefficients_nonnegative (f : MonoidAlgebra ℂ G) (g : G) : |
| 0 ≤ (normCoefficients f).coeff g := by |
| simpa only [normCoefficients_apply] using norm_nonneg (f.coeff g) |
|
|
| /-- Exact source inequality |fg|(w) ≤ (|f| |g|)(w), including cancellations. -/ |
| theorem normCoefficients_mul_le (f g : MonoidAlgebra ℂ G) (w : G) : |
| (normCoefficients (f * g)).coeff w ≤ |
| (normCoefficients f * normCoefficients g).coeff w := by |
| classical |
| rw [normCoefficients_apply, MonoidAlgebra.coeff_mul, MonoidAlgebra.coeff_mul] |
| simp only [Finsupp.sum, normCoefficients_support, normCoefficients_apply] |
| calc |
| _ ≤ ∑ a ∈ f.coeff.support, |
| ‖∑ b ∈ g.coeff.support, if a * b = w then f.coeff a * g.coeff b else 0‖ := |
| norm_sum_le _ _ |
| _ ≤ ∑ a ∈ f.coeff.support, ∑ b ∈ g.coeff.support, |
| ‖if a * b = w then f.coeff a * g.coeff b else 0‖ := |
| Finset.sum_le_sum fun _ _ => norm_sum_le _ _ |
| _ = _ := by |
| apply Finset.sum_congr rfl |
| intro a _ |
| apply Finset.sum_congr rfl |
| intro b _ |
| split_ifs <;> simp |
| |
| /-- The corresponding additive coefficient comparison. -/ |
| theorem normCoefficients_add_le (f g : MonoidAlgebra ℂ G) (w : G) : |
| (normCoefficients (f + g)).coeff w ≤ |
| (normCoefficients f + normCoefficients g).coeff w := |
| by simpa only [normCoefficients_apply, MonoidAlgebra.coeff_add, Finsupp.add_apply] using |
| norm_add_le (f.coeff w) (g.coeff w) |
|
|
| end |
|
|
| /-! The coefficientwise ordered ring of real finite group-ring sums. |
| The multiplication is actual collected convolution. -/ |
| open scoped BigOperators |
| namespace CoefficientOrder |
| variable {G : Type*} [Monoid G] |
|
|
| instance coefficientPartialOrder : PartialOrder (MonoidAlgebra ℝ G) := |
| PartialOrder.lift MonoidAlgebra.coeff MonoidAlgebra.coeff_injective |
|
|
| omit [Monoid G] in |
| @[simp] theorem le_iff (f g : MonoidAlgebra ℝ G) : f ≤ g ↔ ∀ w, f.coeff w ≤ g.coeff w := Iff.rfl |
|
|
| instance coefficientOrderedAdd : IsOrderedAddMonoid (MonoidAlgebra ℝ G) where |
| add_le_add_left _f _g h _a w := add_le_add (h w) le_rfl |
|
|
| instance coefficientZeroLEOne : ZeroLEOneClass (MonoidAlgebra ℝ G) where |
| zero_le_one := by |
| intro w |
| classical |
| simp only [MonoidAlgebra.one_def, MonoidAlgebra.coeff_single, Finsupp.single_apply] |
| split_ifs <;> norm_num |
| |
| theorem mul_nonnegative {f g : MonoidAlgebra ℝ G} (hf : 0 ≤ f) (hg : 0 ≤ g) : 0 ≤ f * g := by |
| classical |
| intro w |
| rw [MonoidAlgebra.coeff_mul] |
| apply Finset.sum_nonneg |
| intro a _ |
| apply Finset.sum_nonneg |
| intro b _ |
| dsimp only |
| split_ifs |
| · exact mul_nonneg (hf a) (hg b) |
| · exact le_refl _ |
|
|
| theorem mul_le_left {f g : MonoidAlgebra ℝ G} (h : f ≤ g) |
| (a : MonoidAlgebra ℝ G) (ha : 0 ≤ a) : a * f ≤ a * g := by |
| have hd : 0 ≤ g - f := fun w => sub_nonneg.mpr (h w) |
| have hm := mul_nonnegative ha hd |
| rw [mul_sub] at hm |
| exact fun w => sub_nonneg.mp (hm w) |
|
|
| theorem mul_le_right {f g : MonoidAlgebra ℝ G} (h : f ≤ g) |
| (a : MonoidAlgebra ℝ G) (ha : 0 ≤ a) : f * a ≤ g * a := by |
| have hd : 0 ≤ g - f := fun w => sub_nonneg.mpr (h w) |
| have hm := mul_nonnegative hd ha |
| rw [sub_mul] at hm |
| exact fun w => sub_nonneg.mp (hm w) |
|
|
| instance coefficientOrderedRing : IsOrderedRing (MonoidAlgebra ℝ G) where |
| mul_le_mul_of_nonneg_left := by |
| intro a ha b c h |
| exact mul_le_left h a ha |
| mul_le_mul_of_nonneg_right := by |
| intro a ha b c h |
| exact mul_le_right h a ha |
|
|
| end CoefficientOrder |
|
|
| end Generator |
|
|
| end |
|
|
| end OAI |
|
|
