openAI-math-dataset / lean /OAI /LinearAlgebra /MatrixMultiplication /AuxiliarySeparation /Spectrum /Catalyst.lean
| import Mathlib.GroupTheory.MonoidLocalization.GrothendieckGroup | |
| import Mathlib.Tactic | |
| namespace OAI | |
| /-! | |
| # Catalysts from additive completion certificates | |
| Equality in the Grothendieck additive group means equality after adding a | |
| common element of the original semiring. This gives catalytic inequalities | |
| without assuming additive cancellation. | |
| -/ | |
| namespace MatrixMultiplication.AuxiliarySeparation | |
| open Algebra | |
| variable {S : Type*} [CommSemiring S] [Preorder S] | |
| /-- A negative-unit certificate in the additive completion gives a catalytic | |
| inequality in the original semiring, even when its addition is not cancellative. -/ | |
| theorem exists_catalyst_of_completion_eq | |
| (hadd : ∀ a b c d : S, a ≤ b → c ≤ d → a + c ≤ b + d) | |
| (m k : ℕ) (d X Y s : S) (hXY : X ≤ Y) | |
| (heq : -(m : ℤ) • GrothendieckAddGroup.of (1 : S) = | |
| GrothendieckAddGroup.of Y - GrothendieckAddGroup.of X + | |
| GrothendieckAddGroup.of (d * s) - k • GrothendieckAddGroup.of s) : | |
| ∃ D : S, D + (m : S) + d * s ≤ D + (k : S) * s := by | |
| let η : S →+ GrothendieckAddGroup S := GrothendieckAddGroup.of | |
| have hm : η (m : S) = m • η 1 := by | |
| rw [← nsmul_one m, map_nsmul] | |
| have hk : η ((k : S) * s) = k • η s := by | |
| rw [← nsmul_eq_mul, map_nsmul] | |
| have heq' : η ((m : S) + Y + d * s) = η (X + (k : S) * s) := by | |
| simp only [map_add, hm, hk] | |
| change -(m : ℤ) • η 1 = η Y - η X + η (d * s) - k • η s at heq | |
| simp only [neg_zsmul, natCast_zsmul] at heq | |
| calc | |
| m • η 1 + η Y + η (d * s) = | |
| m • η 1 + (η Y - η X + η (d * s) - k • η s) + η X + k • η s := by abel | |
| _ = m • η 1 + -(m • η 1) + η X + k • η s := by rw [← heq] | |
| _ = η X + k • η s := by abel | |
| obtain ⟨C, hC⟩ := (AddLocalization.addMonoidOf (⊤ : AddSubmonoid S)).exists_of_eq heq' | |
| refine ⟨(C : S) + Y, ?_⟩ | |
| calc | |
| ((C : S) + Y) + (m : S) + d * s = (C : S) + ((m : S) + Y + d * s) := by ac_rfl | |
| _ = (C : S) + (X + (k : S) * s) := hC | |
| _ ≤ (C : S) + (Y + (k : S) * s) := | |
| hadd _ _ _ _ le_rfl (hadd _ _ _ _ hXY le_rfl) | |
| _ = ((C : S) + Y) + (k : S) * s := by ac_rfl | |
| /-- A finite nonnegative integral combination of completion relations gives | |
| one catalytic inequality. The catalyst absorbs the aggregate of the ordered | |
| relations, and the resulting test element is the same combination of the | |
| original test elements. -/ | |
| theorem exists_catalyst_of_completion_sum_eq | |
| (hadd : ∀ a b c d : S, a ≤ b → c ≤ d → a + c ≤ b + d) | |
| {I : Type*} [Fintype I] (m k : ℕ) (d : S) | |
| (n : I → ℕ) (X Y Z : I → S) (hXY : ∀ i, X i ≤ Y i) | |
| (heq : -(m : ℤ) • GrothendieckAddGroup.of (1 : S) = | |
| ∑ i, n i • (GrothendieckAddGroup.of (Y i) - GrothendieckAddGroup.of (X i) + | |
| GrothendieckAddGroup.of (d * Z i) - k • GrothendieckAddGroup.of (Z i))) : | |
| ∃ D : S, D + (m : S) + d * (∑ i, n i • Z i) ≤ | |
| D + (k : S) * (∑ i, n i • Z i) := by | |
| classical | |
| have hnsmul (a b : S) (hab : a ≤ b) (l : ℕ) : l • a ≤ l • b := by | |
| induction l with | |
| | zero => simp | |
| | succ l ih => simpa only [succ_nsmul] using hadd _ _ _ _ ih hab | |
| have hsum (t : Finset I) : ∑ i ∈ t, n i • X i ≤ ∑ i ∈ t, n i • Y i := by | |
| induction t using Finset.induction_on with | |
| | empty => simp | |
| | @insert i t hi ih => | |
| simpa only [Finset.sum_insert hi] using | |
| hadd _ _ _ _ (hnsmul (X i) (Y i) (hXY i) (n i)) ih | |
| apply exists_catalyst_of_completion_eq hadd m k d | |
| (∑ i, n i • X i) (∑ i, n i • Y i) (∑ i, n i • Z i) (hsum Finset.univ) | |
| convert heq using 1 | |
| simp only [smul_sub, smul_add, Finset.sum_sub_distrib, Finset.sum_add_distrib, map_sum, map_nsmul, Finset.mul_sum, mul_smul_comm, ← smul_comm k] | |
| rw [Finset.smul_sum] | |
| end MatrixMultiplication.AuxiliarySeparation | |
| end OAI | |