| import OAI.Combinatorics.ContingencyTables.Counting.CompanionPhysicalNormalizer |
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| namespace OAI |
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| /-! |
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| namespace ContingencyTables.CompanionFarMass |
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| open scoped BigOperators Classical |
| open CompanionFactorialApplication CompanionPhysicalNormalizer TreeMarginalSignature |
| open BalancedCellFibres ForestPenaltyApplication |
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| variable {I J : Type*} [Fintype I] [Fintype J] |
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| noncomputable def actualWeight (A : Instance I J) (alpha : ℝ) (a : Profiles (cellWidth A)) : ℝ := |
| physicalWeight A.T A.b (ForestLiteralCount.lower A.T A.d A.b) |
| (ForestLiteralCount.width A.T A.d A.b) A.r A.c alpha (BalancedCellFibres.profile (cellWidth A) a) |
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| noncomputable def actualPenalty (A : Instance I J) (a : Profiles (cellWidth A)) : ℝ := |
| penalty A.T A.b (ForestLiteralCount.lower A.T A.d A.b) |
| (ForestLiteralCount.width A.T A.d A.b) A.r A.c (BalancedCellFibres.profile (cellWidth A) a) |
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| lemma weight_at_large_penalty (A : Instance I J) (alpha : ℝ) (K : ℕ) |
| (a : Profiles (cellWidth A)) (ha : 4*(K:ℝ) ≤ alpha*actualPenalty A a) : |
| actualWeight A alpha a ≤ (2:ℝ)^(-(4*K:ℕ):ℝ) := by |
| unfold actualWeight |
| rw [exact_eq] |
| split_ifs |
| · apply Real.rpow_le_rpow_of_exponent_le (by norm_num) |
| change -alpha*actualPenalty A a ≤ -((4*K:ℕ):ℝ) |
| push_cast |
| linarith |
| · exact (Real.rpow_pos_of_pos (by norm_num : (0:ℝ) < 2) _).le |
|
|
| /-- The true expanded physical box has at most 2^K summands, so the |
| entire terminal region whose penalty is at least 4K contributes at most |
| 2^(-3K). Neither a table-count approximation nor an oracle is assumed. -/ |
| theorem far_mass_le (A : Instance I J) (alpha : ℝ) (B K : ℕ) |
| (hW : 20*A.d ≤ A.W) (hB : (∏ a : I×J,(A.b a.1 a.2+1)) ≤ 2^B) (hBK : 2*B ≤ K) : |
| (∑ a : Profiles (cellWidth A),if 4*(K:ℝ) ≤ alpha*actualPenalty A a |
| then actualWeight A alpha a else 0) ≤ (2:ℝ)^(-(3*K:ℕ):ℝ) := by |
| have hcard : Fintype.card (Profiles (cellWidth A)) ≤ 2^K := |
| ForestBoxCardinality.cardinality_le_two_pow A hW B K hB hBK |
| have hcardR : (Fintype.card (Profiles (cellWidth A)):ℝ) ≤ (2:ℝ)^K := by exact_mod_cast hcard |
| have hp : 0 ≤ (2:ℝ)^(-(4*K:ℕ):ℝ) := (Real.rpow_pos_of_pos (by norm_num) _).le |
| calc |
| _ ≤ ∑ _a : Profiles (cellWidth A),(2:ℝ)^(-(4*K:ℕ):ℝ) := by |
| apply Finset.sum_le_sum |
| intro a _ |
| split_ifs with ha |
| · exact weight_at_large_penalty A alpha K a ha |
| · exact hp |
| _ = (Fintype.card (Profiles (cellWidth A)):ℝ)*(2:ℝ)^(-(4*K:ℕ):ℝ) := by simp |
| _ ≤ (2:ℝ)^K*(2:ℝ)^(-(4*K:ℕ):ℝ) := mul_le_mul_of_nonneg_right hcardR hp |
| _ = _ := by |
| rw [← Real.rpow_natCast,← Real.rpow_add (by norm_num : (0:ℝ) < 2)] |
| congr 1 |
| push_cast |
| ring |
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|
| end ContingencyTables.CompanionFarMass |
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| end OAI |
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