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openAI-math-dataset / lean /OAI /Computability /VertexCover /Reduction /PrivateProjectionGroupLower.lean
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Add lean source and binary files (part 64)
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import Mathlib
import OAI.Computability.VertexCover.Reduction.PrivatePairProjectionRight
namespace OAI
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namespace VertexCover.LabelCover
open VertexCover.Restriction
open EnergyForm.Projection
open scoped BigOperators
noncomputable def otherWeightProjection (Φ : LabelCover) {d : ℕ}
(J : Finset (Fin d)) (j : Fin d) : (Φ.restrictionForm d).Projection :=
product Φ.weightProjection Φ.weightProjection_commute (J.erase j).toList
theorem privateSeed_weight_commute (Φ : LabelCover) {d : ℕ}
(J : Finset (Fin d)) (j k : Fin d) (f : Φ.RestrictionSpace d) :
Φ.privateSeedProjection J j (Φ.weightProjection k f) =
Φ.weightProjection k (Φ.privateSeedProjection J j f) :=
Φ.condition_weightProjection_commute (Φ.privateSeedKey J j) k f
theorem privateSeed_otherWeight_commute (Φ : LabelCover) {d : ℕ}
(J : Finset (Fin d)) (j : Fin d) (f : Φ.RestrictionSpace d) :
Φ.privateSeedProjection J j (Φ.otherWeightProjection J j f) =
Φ.otherWeightProjection J j (Φ.privateSeedProjection J j f) :=
(productMap_commute Φ.weightProjection (Φ.privateSeedProjection J j)
(fun k x => (Φ.privateSeed_weight_commute J j k x).symm) (J.erase j).toList f).symm
noncomputable def privateAverageProjection (Φ : LabelCover) {d : ℕ}
(J : Finset (Fin d)) (j : Fin d) : (Φ.restrictionForm d).Projection :=
(Φ.privateSeedProjection J j).comp (Φ.otherWeightProjection J j)
(Φ.privateSeed_otherWeight_commute J j)
theorem privateAverage_weight_commute (Φ : LabelCover) {d : ℕ}
(J : Finset (Fin d)) (j k : Fin d) (f : Φ.RestrictionSpace d) :
Φ.privateAverageProjection J j (Φ.weightProjection k f) =
Φ.weightProjection k (Φ.privateAverageProjection J j f) := by
change Φ.privateSeedProjection J j
(productMap Φ.weightProjection (J.erase j).toList (Φ.weightProjection k f)) = _
rw [productMap_commute Φ.weightProjection (Φ.weightProjection k)
(fun l => Φ.weightProjection_commute l k), Φ.privateSeed_weight_commute]
rfl
noncomputable def privateProjection (Φ : LabelCover) {d : ℕ}
(J : Finset (Fin d)) (j : Fin d) : (Φ.restrictionForm d).Projection :=
(Φ.weightProjection j).residual.comp (Φ.privateAverageProjection J j)
(residual_commute _ _ (fun f => (Φ.privateAverage_weight_commute J j j f).symm))
theorem privateProjection_weight_commute (Φ : LabelCover) {d : ℕ}
(J : Finset (Fin d)) (j k : Fin d) (f : Φ.RestrictionSpace d) :
Φ.privateProjection J j (Φ.weightProjection k f) =
Φ.weightProjection k (Φ.privateProjection J j f) := by
change (Φ.weightProjection j).residual
(Φ.privateAverageProjection J j (Φ.weightProjection k f)) = _
rw [Φ.privateAverage_weight_commute, residual_commute _ _ (Φ.weightProjection_commute j k)]
rfl
theorem privateProjection_group_le (Φ : LabelCover) {d : ℕ}
(J : Finset (Fin d)) (j : Fin d) (f : Φ.RestrictionSpace d) :
(Φ.restrictionForm d).energy (Φ.privateProjection J j (Φ.weightGroup j f)) ≤
(Φ.restrictionForm d).energy (Φ.privateProjection J j f) :=
finDifference_retained_le Φ.weightProjection Φ.weightProjection_commute
(Φ.privateProjection J j) (fun k x => (Φ.privateProjection_weight_commute J j k x).symm) j f
theorem privateProjection_group_eq (Φ : LabelCover) {d : ℕ}
(J : Finset (Fin d)) (j : Fin d) (f : Φ.RestrictionSpace d) :
Φ.privateProjection J j (Φ.weightGroup j f) =
Φ.privateAverageProjection J j (Φ.weightGroup j f) := by
change (Φ.weightProjection j).residual
(Φ.privateAverageProjection J j (Φ.weightGroup j f)) = _
rw [residual_commute _ _ (fun x => (Φ.privateAverage_weight_commute J j j x).symm),
Φ.weightGroup_centered]
theorem privateAverage_loss_le (Φ : LabelCover) {d : ℕ}
(J : Finset (Fin d)) (j : Fin d) (f : Φ.RestrictionSpace d) :
(Φ.restrictionForm d).energy ((Φ.privateAverageProjection J j).residual f) ≤
(∑ k ∈ J.erase j, (Φ.restrictionForm d).energy ((Φ.weightProjection k).residual f)) +
∑ e : PositionPair d, (Φ.restrictionForm d).energy ((Φ.privatePairProjection J j e).residual f) := by
have hc := comp_loss_le (Φ.privateSeedProjection J j) (Φ.otherWeightProjection J j)
(Φ.privateSeed_otherWeight_commute J j) f
have hw := product_loss_finset Φ.weightProjection Φ.weightProjection_commute (J.erase j) f
have hs := residual_le_of_nested (Φ.privateSeedProjection J j)
(product (Φ.privatePairProjection J j) (Φ.privatePairProjection_commute J j) Finset.univ.toList)
(Φ.privateSeedProjection_product J j) f
have hsp := product_loss_finset (Φ.privatePairProjection J j)
(Φ.privatePairProjection_commute J j) Finset.univ f
change (Φ.restrictionForm d).energy ((Φ.privateAverageProjection J j).residual f) ≤ _ at hc
change (Φ.restrictionForm d).energy ((Φ.otherWeightProjection J j).residual f) ≤ _ at hw
linarith
theorem privateProjection_group_lower (Φ : LabelCover) {d : ℕ}
(J : Finset (Fin d)) (j : Fin d) (f : Φ.RestrictionSpace d) :
(Φ.restrictionForm d).energy (Φ.weightGroup j f) ≤
(Φ.restrictionForm d).energy (Φ.privateProjection J j f) +
(∑ k ∈ J.erase j, (Φ.restrictionForm d).energy
((Φ.weightProjection k).residual (Φ.weightGroup j f))) +
∑ e : PositionPair d, (Φ.restrictionForm d).energy
((Φ.privatePairProjection J j e).residual (Φ.weightGroup j f)) := by
have he := (Φ.privateAverageProjection J j).pythagoras (Φ.weightGroup j f)
have hl := Φ.privateAverage_loss_le J j (Φ.weightGroup j f)
have hr := Φ.privateProjection_group_le J j f
rw [Φ.privateProjection_group_eq] at hr
linarith
end VertexCover.LabelCover
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end OAI