| import OAI.Combinatorics.ContingencyTables.Transport.DyadicAcceptance |
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| namespace OAI |
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| /-! # Bounded bit allowance for every dense Metropolis decision -/ |
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| namespace ContingencyTables.GridBoundary |
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| open scoped BigOperators Classical |
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| variable {I J : Type*} [Fintype I] [Fintype J] [DecidableEq I] [DecidableEq J] |
| variable {q : ℕ} |
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| theorem denseExponent_upper (R : Option I → ℕ) (P : Option J → ℕ) |
| (d : ℕ) (hd : 14 ≤ d) (hR : ∀ i, d ^ 12 ≤ R i) (hP : ∀ j, d ^ 12 ≤ P j) |
| (hRmax : ∀ i, R (some i) ≤ R none) (hPmax : ∀ j, P (some j) ≤ P none) |
| (hdimR : Fintype.card (Option I) ≤ d) (hdimP : Fintype.card (Option J) ≤ d) |
| (hfree : Fintype.card (I × J) ≤ d) (htotal : ∑ i, R i = ∑ j, P j) |
| (e : I × J ≃ Fin q) (a : Grid (I := Fin q) (4 * d ^ 8)) : |
| denseExponent R P d e a ≤ 4 * d ^ 5 := by |
| have hcube (p : I × J) : 0 ≤ realCorner a (e p) ∧ realCorner a (e p) ≤ 4 * (d : ℝ) ^ 8 := by |
| have hlo : (0 : ℤ) ≤ corner a (e p) := (a (e p)).property.1 |
| have hhi : corner a (e p) ≤ (4 * d ^ 8 : ℕ) := by |
| have h : corner a (e p) ≤ (4 * d ^ 8 : ℕ) - 1 := (a (e p)).property.2 |
| omega |
| constructor |
| · change (0 : ℝ) ≤ (corner a (e p) : ℝ) |
| exact_mod_cast hlo |
| · change (corner a (e p) : ℝ) ≤ 4 * (d : ℝ) ^ 8 |
| exact_mod_cast hhi |
| have hp := (DenseAffine.densePenalty_cube_bound R P d hd hR hP hRmax hPmax |
| hdimR hdimP hfree htotal _ hcube).2 |
| have he : (denseExponent R P d e a : ℝ) = |
| (⌊(d : ℝ) ^ 4 * denseGridPenalty R P d e (realCorner a)⌋ : ℤ) := by |
| exact_mod_cast denseExponent_eq_floor R P d e a |
| have hf := Int.floor_le ((d : ℝ) ^ 4 * denseGridPenalty R P d e (realCorner a)) |
| have hmul := mul_le_mul_of_nonneg_left hp (show (0 : ℝ) ≤ (d : ℝ) ^ 4 by positivity) |
| change (d : ℝ) ^ 4 * denseGridPenalty R P d e (realCorner a) ≤ (d : ℝ) ^ 4 * (4 * d) at hmul |
| have hbound : (denseExponent R P d e a : ℝ) ≤ 4 * (d : ℝ) ^ 5 := by |
| rw [he] |
| nlinarith |
| exact_mod_cast hbound |
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| / |
| probability exactly, including a proposed move whose probability is one. -/ |
| theorem denseMetropolis_finite_coin (R : Option I → ℕ) (P : Option J → ℕ) |
| (d : ℕ) (hd : 14 ≤ d) (hR : ∀ i, d ^ 12 ≤ R i) (hP : ∀ j, d ^ 12 ≤ P j) |
| (hRmax : ∀ i, R (some i) ≤ R none) (hPmax : ∀ j, P (some j) ≤ P none) |
| (hdimR : Fintype.card (Option I) ≤ d) (hdimP : Fintype.card (Option J) ≤ d) |
| (hfree : Fintype.card (I × J) ≤ d) (htotal : ∑ i, R i = ∑ j, P j) |
| (e : I × J ≃ Fin q) (a b : Grid (I := Fin q) (4 * d ^ 8)) : |
| ((Finset.univ.filter (FairBits.acceptDyadic |
| (denseExponent R P d e b - denseExponent R P d e a) (8 * d ^ 5))).card : ℝ) / |
| (2 : ℝ) ^ (8 * d ^ 5) = |
| min 1 (denseBinWeight R P d e b / denseBinWeight R P d e a) := by |
| rw [FairBits.acceptDyadic_probability] |
| · exact (dense_acceptance_dyadic R P d e a b).symm |
| · exact (Nat.sub_le _ _).trans ((denseExponent_upper R P d hd hR hP hRmax hPmax |
| hdimR hdimP hfree htotal e b).trans (by omega)) |
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| end ContingencyTables.GridBoundary |
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| end OAI |
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