openAI-math-dataset / lean /OAI /Computability /VertexCover /Reduction /PrivateProjectionBatchLower.lean
| import Mathlib | |
| import OAI.Computability.VertexCover.Analysis.RestrictionSum | |
| namespace OAI | |
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| namespace VertexCover.Restriction | |
| theorem choose_step (d h : ℕ) (hd : 0 < d) (hh : 0 < h) : | |
| (d:ℝ)*((d-1).choose (h-1):ℝ) = (h:ℝ)*(d.choose h:ℝ) := by | |
| have he := Nat.add_one_mul_choose_eq (d-1) (h-1) | |
| rw [Nat.sub_add_cancel hd, Nat.sub_add_cancel hh] at he | |
| exact_mod_cast he.trans (Nat.mul_comm _ _) | |
| theorem batch_pair_scale (d h : ℕ) (hd : 4 ≤ d) (hh : 3 ≤ h) : | |
| (d:ℝ)*((d-2).choose (h-2):ℝ) ≤ 2*(h:ℝ)*((d-1).choose (h-1):ℝ) := by | |
| have he := choose_step (d-1) (h-1) (by omega) (by omega) | |
| have hed : d-1-1 = d-2 := by omega | |
| have heh : h-1-1 = h-2 := by omega | |
| rw [hed, heh, Nat.cast_sub (by omega : 1 ≤ d), Nat.cast_sub (by omega : 1 ≤ h)] at he | |
| norm_num only [Nat.cast_one] at he | |
| have hdR : (4:ℝ) ≤ d := by exact_mod_cast hd | |
| have hC1 : (0:ℝ) ≤ (d-1).choose (h-1) := Nat.cast_nonneg _ | |
| have hC2 : (0:ℝ) ≤ (d-2).choose (h-2) := Nat.cast_nonneg _ | |
| nlinarith [mul_nonneg (by linarith : (0:ℝ) ≤ d-2) hC2] | |
| theorem batch_triple_scale (d h : ℕ) (hd : 4 ≤ d) (hh : 3 ≤ h) : | |
| (d:ℝ)^2*((d-3).choose (h-3):ℝ) ≤ 4*(h:ℝ)^2*((d-1).choose (h-1):ℝ) := by | |
| have he := choose_step (d-2) (h-2) (by omega) (by omega) | |
| have hed : d-2-1 = d-3 := by omega | |
| have heh : h-2-1 = h-3 := by omega | |
| rw [hed, heh, Nat.cast_sub (by omega : 2 ≤ d), Nat.cast_sub (by omega : 2 ≤ h)] at he | |
| norm_num only [Nat.cast_ofNat] at he | |
| have hdR : (4:ℝ) ≤ d := by exact_mod_cast hd | |
| have hhR : (0:ℝ) ≤ h := Nat.cast_nonneg _ | |
| have hC2 : (0:ℝ) ≤ (d-2).choose (h-2) := Nat.cast_nonneg _ | |
| have hC3 : (0:ℝ) ≤ (d-3).choose (h-3) := Nat.cast_nonneg _ | |
| have hl : (d:ℝ)*((d-3).choose (h-3):ℝ) ≤ 2*(h:ℝ)*((d-2).choose (h-2):ℝ) := by | |
| nlinarith [mul_nonneg (by linarith : (0:ℝ) ≤ d-4) hC3] | |
| have hl' := mul_le_mul_of_nonneg_left hl (Nat.cast_nonneg d : (0:ℝ) ≤ d) | |
| have hp := mul_le_mul_of_nonneg_left (batch_pair_scale d h hd hh) (by positivity : (0:ℝ) ≤ 2*h) | |
| nlinarith | |
| end VertexCover.Restriction | |
| namespace VertexCover.LabelCover | |
| open VertexCover.Restriction | |
| open EnergyForm.Projection | |
| open scoped BigOperators | |
| theorem privateProjection_batch_lower (Φ : LabelCover) {d h : ℕ} | |
| (hd : 4 ≤ d) (hh : 3 ≤ h) (ν : ℝ) | |
| (hscale : 16*(h:ℝ)+32*(h:ℝ)^2 ≤ ν*d/2) | |
| (A : Finset (Φ.Coordinate d → ℝ)) (hA : A.Nonempty) | |
| (henergy : ν*d ≤ (Φ.restrictionForm d).energy (Φ.restrictionFunction A hA)) : | |
| (d.choose h:ℝ)*ν*h/2 ≤ | |
| ∑ J ∈ batches d h, ∑ j ∈ J, | |
| (Φ.restrictionForm d).energy (Φ.privateProjection J j (Φ.restrictionFunction A hA)) := by | |
| have hb := Φ.privateProjection_batch_bound hh A hA | |
| have hC1 : (0:ℝ) ≤ (d-1).choose (h-1) := Nat.cast_nonneg _ | |
| have hl := mul_le_mul_of_nonneg_left henergy hC1 | |
| have hs := mul_le_mul_of_nonneg_left hscale hC1 | |
| have hp := batch_pair_scale d h hd hh | |
| have ht := batch_triple_scale d h hd hh | |
| have he := choose_step d h (by omega) (by omega) | |
| have he' : ((d-1).choose (h-1):ℝ)*ν*d = (d.choose h:ℝ)*ν*h := by | |
| calc | |
| _ = ν*((d:ℝ)*((d-1).choose (h-1):ℝ)) := by ring | |
| _ = _ := by rw [he]; ring | |
| nlinarith | |
| end VertexCover.LabelCover | |
| end | |
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| end | |
| end OAI | |