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2f887db | 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 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 | import OAI.Computability.PerfectCompleteness.Algebra.LowerCrossRank
import OAI.Computability.PerfectCompleteness.Foundations.DualQuotientRowsLemmas
import OAI.Computability.PerfectCompleteness.Sampling.RestrictionRowsLaw
namespace OAI
section
namespace PerfectCompleteness.RestrictionCrossLaw
noncomputable section
open UniqueGamesTheorem.Integration.BinaryLinear (F2)
open UniqueGamesTheorem.Fourier.MatrixRestrictions
open UniqueGamesTheorem.Foundations.Games
open RestrictionRowsLaw DualQuotientRows
variable {E C I : Type*} [AddCommGroup E] [Module F2 E]
[AddCommGroup C] [Module F2 C] [Fintype I]
(W : Submodule F2 E) (C' : Submodule F2 C) (b : Module.Basis I F2 C')
def rowPairEquiv (m : Nat) :
((Fin m ⊕ Fin m) → Module.Dual F2 (E ⧸ W)) ≃ₗ[F2]
((Fin m → W.dualAnnihilator) × (Fin m → W.dualAnnihilator)) :=
(LinearEquiv.piCongrRight
(fun _ : Fin m ⊕ Fin m => quotientRowEquiv W)).trans
(LinearEquiv.sumArrowLequivProdArrow
(Fin m) (Fin m) F2 W.dualAnnihilator)
variable {m : Nat} (sel : (Fin m ⊕ Fin m) → I)
def selectedPair : Parameter W C' →ₗ[F2]
((Fin m → W.dualAnnihilator) × (Fin m → W.dualAnnihilator)) :=
(rowPairEquiv W m).toLinearMap.comp (selectedRowsLinear W C' b sel)
omit [Fintype I] in
@[simp] theorem selectedPair_fst (A : Parameter W C') (i : Fin m) :
(selectedPair W C' b sel A).1 i =
quotientRowEquiv W (rowsLinear W C' b A (sel (Sum.inl i))) := rfl
omit [Fintype I] in
@[simp] theorem selectedPair_snd (A : Parameter W C') (j : Fin m) :
(selectedPair W C' b sel A).2 j =
quotientRowEquiv W (rowsLinear W C' b A (sel (Sum.inr j))) := rfl
theorem selectedPair_surjective (hsel : Function.Injective sel) :
Function.Surjective (selectedPair W C' b sel) :=
(rowPairEquiv W m).surjective.comp (selectedRowsLinear_surjective W C' b sel hsel)
private theorem uniform_product {A B : Type*} [Fintype A] [Fintype B]
[Nonempty A] [Nonempty B] :
(FiniteDistribution.uniform A).product (FiniteDistribution.uniform B) =
FiniteDistribution.uniform (A × B) := by
apply FiniteDistribution.eq_of_weight_eq
intro p
change (1 / (Fintype.card A : ℝ)) * (1 / (Fintype.card B : ℝ)) =
1 / (Fintype.card (A × B) : ℝ)
simp only [Fintype.card_prod, Nat.cast_mul, one_div, mul_inv]
theorem selectedPair_law [Fintype (Parameter W C')] [Fintype W.dualAnnihilator]
(hsel : Function.Injective sel) :
(FiniteDistribution.uniform (Parameter W C')).pushforward (selectedPair W C' b sel) =
(FiniteDistribution.uniform (Fin m → W.dualAnnihilator)).product
(FiniteDistribution.uniform (Fin m → W.dualAnnihilator)) := by
rw [uniform_product]
exact UniformLinearImage.uniform_pushforward_linearMap
(selectedPair W C' b sel) (selectedPair_surjective W C' b sel hsel)
omit [Fintype I] in
theorem coordinate_translate_left (coordinates : I → Module.Dual F2 C)
(hcoordinates : ∀ i, (coordinates i).comp C'.subtype = b.coord i)
(X : E →ₗ[F2] C) (A : Parameter W C') (i : Fin m) :
(coordinates (sel (Sum.inl i))).comp (translate W C' X A) =
(coordinates (sel (Sum.inl i))).comp X +
((selectedPair W C' b sel A).1 i : Module.Dual F2 E) := by
simpa only [affineRow, selectedPair_fst, quotientRowEquiv_val] using
coordinate_translate W C' b coordinates hcoordinates X A (sel (Sum.inl i))
omit [Fintype I] in
theorem coordinate_translate_right (coordinates : I → Module.Dual F2 C)
(hcoordinates : ∀ i, (coordinates i).comp C'.subtype = b.coord i)
(X : E →ₗ[F2] C) (A : Parameter W C') (j : Fin m) :
(coordinates (sel (Sum.inr j))).comp (translate W C' X A) =
(coordinates (sel (Sum.inr j))).comp X +
((selectedPair W C' b sel A).2 j : Module.Dual F2 E) := by
simpa only [affineRow, selectedPair_snd, quotientRowEquiv_val] using
coordinate_translate W C' b coordinates hcoordinates X A (sel (Sum.inr j))
theorem crossMatrix_law [Fintype (Parameter W C')] [Fintype W.dualAnnihilator]
(hsel : Function.Injective sel)
(coordinates : I → Module.Dual F2 C)
(hcoordinates : ∀ i, (coordinates i).comp C'.subtype = b.coord i)
(X : E →ₗ[F2] C)
(F : Module.Dual F2 E →ₗ[F2] Module.Dual F2 (Module.Dual F2 E)) :
(FiniteDistribution.uniform (Parameter W C')).pushforward
(Γ := Matrix (Fin m) (Fin m) F2)
(fun A i j =>
F ((coordinates (sel (Sum.inl i))).comp (translate W C' X A))
((coordinates (sel (Sum.inr j))).comp (translate W C' X A))) =
((FiniteDistribution.uniform (Fin m → W.dualAnnihilator)).product
(FiniteDistribution.uniform (Fin m → W.dualAnnihilator))).pushforward
(Γ := Matrix (Fin m) (Fin m) F2)
(fun p => LowerCrossRank.crossMatrix F W.dualAnnihilator
(fun i => (coordinates (sel (Sum.inl i))).comp X)
(fun j => (coordinates (sel (Sum.inr j))).comp X) p.1 p.2) := by
have hcross :
(fun (A : Parameter W C') =>
(fun i j => F ((coordinates (sel (Sum.inl i))).comp (translate W C' X A))
((coordinates (sel (Sum.inr j))).comp (translate W C' X A)) :
Matrix (Fin m) (Fin m) F2)) =
(fun A => LowerCrossRank.crossMatrix F W.dualAnnihilator
(fun i => (coordinates (sel (Sum.inl i))).comp X)
(fun j => (coordinates (sel (Sum.inr j))).comp X)
(selectedPair W C' b sel A).1 (selectedPair W C' b sel A).2) := by
funext A i j
rw [coordinate_translate_left W C' b sel coordinates hcoordinates X A i,
coordinate_translate_right W C' b sel coordinates hcoordinates X A j]
rfl
rw [hcross, ← FiniteDistribution.pushforward_comp
(FiniteDistribution.uniform (Parameter W C')) (selectedPair W C' b sel)
(fun p => LowerCrossRank.crossMatrix F W.dualAnnihilator
(fun i => (coordinates (sel (Sum.inl i))).comp X)
(fun j => (coordinates (sel (Sum.inr j))).comp X) p.1 p.2),
selectedPair_law W C' b sel hsel]
end
end PerfectCompleteness.RestrictionCrossLaw
end
end OAI
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