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cdec422 | 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 | 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
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