| import OAI.Combinatorics.Permanent.Model |
|
|
| namespace OAI |
|
|
| noncomputable section |
| namespace FourRow |
| open scoped BigOperators |
|
|
| /- Quantitative exponent-two stability estimates. -/ |
| def rowSq (a : Site → ℝ) : ℝ := ∑ j, a j ^ 2 |
|
|
| def rowDist (a : Site → ℝ) : ℝ := ∑ j, (a j - 1) ^ 2 |
|
|
| def rowDot (a b : Site → ℝ) : ℝ := ∑ j, a j * b j |
|
|
| def pairEntries (a b : Site → ℝ) : Fin 6 → ℝ := |
| ![a 0 * b 1 + a 1 * b 0, |
| a 0 * b 2 + a 2 * b 0, |
| a 0 * b 3 + a 3 * b 0, |
| a 1 * b 2 + a 2 * b 1, |
| a 1 * b 3 + a 3 * b 1, |
| a 2 * b 3 + a 3 * b 2] |
|
|
| def pairSq (a b : Site → ℝ) : ℝ := ∑ j, pairEntries a b j ^ 2 |
|
|
| theorem rowSq_nonneg (a : Site → ℝ) : 0 ≤ rowSq a := |
| Finset.sum_nonneg fun _ _ => sq_nonneg _ |
|
|
| theorem rowDist_nonneg (a : Site → ℝ) : 0 ≤ rowDist a := |
| Finset.sum_nonneg fun _ _ => sq_nonneg _ |
|
|
| theorem pairSq_nonneg (a b : Site → ℝ) : 0 ≤ pairSq a b := |
| Finset.sum_nonneg fun _ _ => sq_nonneg _ |
|
|
| theorem pairSq_identity (a b : Site → ℝ) : |
| pairSq a b = rowSq a * rowSq b + rowDot a b ^ 2 - |
| 2 * ∑ j, (a j * b j) ^ 2 := by |
| simp only [pairSq, pairEntries, rowSq, rowDot, Fin.sum_univ_six, |
| Fin.sum_univ_four, Matrix.cons_val_zero, Matrix.cons_val_one, |
| Matrix.cons_val] |
| ring |
|
|
| theorem pairSq_comm (a b : Site → ℝ) : pairSq a b = pairSq b a := by |
| simp only [pairSq_identity, rowDot, mul_comm] |
|
|
| theorem rowDot_sq_le (a b : Site → ℝ) : |
| rowDot a b ^ 2 ≤ rowSq a * rowSq b := |
| Finset.sum_mul_sq_le_sq_mul_sq Finset.univ a b |
|
|
| theorem rowDot_sq_le_four (a b : Site → ℝ) : |
| rowDot a b ^ 2 ≤ 4 * ∑ j, (a j * b j) ^ 2 := by |
| have h := Finset.sum_mul_sq_le_sq_mul_sq Finset.univ |
| (fun j : Site => (1 : ℝ)) (fun j => a j * b j) |
| simpa [rowDot] using h |
|
|
| theorem pair_stability_one (a b : Site → ℝ) |
| (ha : ∀ j, 0 ≤ a j) (hb : ∀ j, 0 ≤ b j) |
| (haq : rowSq a = 4) (hbq : rowSq b = 4) : |
| rowDist a ≤ 9 * (24 - pairSq a b) := by |
| let t := rowDot a b |
| let s := ∑ j, (a j * b j) ^ 2 |
| let d := ∑ j, (a j - b j) ^ 2 |
| let e := ∑ j, (a j * b j - 1) ^ 2 |
| let D := 24 - pairSq a b |
| have ht0 : 0 ≤ t := Finset.sum_nonneg fun j _ => mul_nonneg (ha j) (hb j) |
| have ht2 : t ^ 2 ≤ 16 := by |
| have h := rowDot_sq_le a b |
| rw [haq, hbq] at h |
| norm_num at h |
| exact h |
| have ht4 : t ≤ 4 := by nlinarith |
| have htprod : 0 ≤ t * (4 - t) := mul_nonneg ht0 (by linarith) |
| have hts : t ^ 2 ≤ 4 * s := rowDot_sq_le_four a b |
| have hD : D = 8 - t ^ 2 + 2 * s := by |
| dsimp [D, t, s] |
| rw [pairSq_identity, haq, hbq] |
| ring |
| have hd : d = 8 - 2 * t := by |
| have h : d = rowSq a + rowSq b - 2 * t := by |
| dsimp [d, rowSq, t, rowDot] |
| simp only [Fin.sum_univ_four] |
| ring |
| rw [h, haq, hbq] |
| ring |
| have he : e = s - 2 * t + 4 := by |
| dsimp [e, s, t, rowDot] |
| simp only [Fin.sum_univ_four] |
| ring |
| have hdD : d ≤ D := by nlinarith |
| have heD : 2 * e ≤ D := by nlinarith |
| have hpoint (j : Site) : (a j - 1) ^ 2 ≤ |
| 8 * (a j - b j) ^ 2 + 2 * (a j * b j - 1) ^ 2 := by |
| have haj : a j ^ 2 ≤ 4 := by |
| calc |
| a j ^ 2 ≤ rowSq a := |
| Finset.single_le_sum (fun i _ => sq_nonneg (a i)) (Finset.mem_univ j) |
| _ = 4 := haq |
| have h1 : (a j - 1) ^ 2 ≤ (a j ^ 2 - 1) ^ 2 := by |
| nlinarith [mul_nonneg (mul_nonneg (ha j) (show 0 ≤ a j + 2 by linarith [ha j])) |
| (sq_nonneg (a j - 1))] |
| have h2 : (a j ^ 2 - a j * b j) ^ 2 ≤ 4 * (a j - b j) ^ 2 := by |
| nlinarith [mul_nonneg (show 0 ≤ 4 - a j ^ 2 by linarith) |
| (sq_nonneg (a j - b j))] |
| nlinarith [sq_nonneg (a j ^ 2 - 2 * a j * b j + 1)] |
| have hs := Finset.sum_le_sum (fun j (_ : j ∈ (Finset.univ : Finset Site)) => hpoint j) |
| have hbound : rowDist a ≤ 8 * d + 2 * e := by |
| simpa only [rowDist, Finset.sum_add_distrib, ← Finset.mul_sum] using hs |
| linarith |
|
|
| theorem pair_stability (a b : Site → ℝ) |
| (ha : ∀ j, 0 ≤ a j) (hb : ∀ j, 0 ≤ b j) |
| (haq : rowSq a = 4) (hbq : rowSq b = 4) : |
| rowDist a + rowDist b ≤ 18 * (24 - pairSq a b) := by |
| have h1 := pair_stability_one a b ha hb haq hbq |
| have h2 := pair_stability_one b a hb ha hbq haq |
| rw [pairSq_comm b a] at h2 |
| linarith |
|
|
|
|
| theorem sum_perm_succ {n : ℕ} (f : Equiv.Perm (Fin (n + 1)) → ℝ) : |
| ∑ σ, f σ = ∑ i : Fin (n + 1), ∑ τ : Equiv.Perm (Fin n), |
| f (Equiv.Perm.decomposeFin.symm (i, τ)) := by |
| rw [← Equiv.sum_comp Equiv.Perm.decomposeFin.symm f] |
| exact Fintype.sum_prod_type _ |
|
|
| theorem decompose4_two (i : Fin 4) (e : Equiv.Perm (Fin 3)) : |
| Equiv.Perm.decomposeFin.symm (i, e) 2 = Equiv.swap 0 i (e 1).succ := |
| Equiv.Perm.decomposeFin_symm_apply_succ e i 1 |
|
|
| theorem decompose4_three (i : Fin 4) (e : Equiv.Perm (Fin 3)) : |
| Equiv.Perm.decomposeFin.symm (i, e) 3 = Equiv.swap 0 i (e 2).succ := |
| Equiv.Perm.decomposeFin_symm_apply_succ e i 2 |
|
|
| theorem decompose3_two (i : Fin 3) (e : Equiv.Perm (Fin 2)) : |
| Equiv.Perm.decomposeFin.symm (i, e) 2 = Equiv.swap 0 i (e 1).succ := |
| Equiv.Perm.decomposeFin_symm_apply_succ e i 1 |
|
|
| theorem permanent_pair_identity (f : Functions) : |
| (∑ π : Perm, ∏ i, f i (π i)) = |
| ∑ j : Fin 6, pairEntries (f 0) (f 1) j * |
| pairEntries (f 2) (f 3) (Fin.rev j) := by |
| change (∑ π : Equiv.Perm (Fin 4), ∏ i : Fin 4, f i (π i)) = _ |
| simp only [sum_perm_succ, Fin.sum_univ_succ, Fintype.sum_unique, |
| Fin.prod_univ_four, pairEntries] |
| simp [decompose4_two, decompose4_three, decompose3_two, |
| Equiv.swap_apply_def, Fin.rev] |
| ring |
|
|
|
|
| def uniformLaw : Law := fun _ => 1 / 24 |
|
|
| def totalDist (f : Functions) : ℝ := ∑ i, rowDist (f i) |
|
|
| theorem totalDist_nonneg (f : Functions) : 0 ≤ totalDist f := |
| Finset.sum_nonneg fun i _ => rowDist_nonneg (f i) |
|
|
| theorem pairSq_le (a b : Site → ℝ) |
| (ha : ∀ j, 0 ≤ a j) (hb : ∀ j, 0 ≤ b j) |
| (haq : rowSq a = 4) (hbq : rowSq b = 4) : pairSq a b ≤ 24 := by |
| have h := pair_stability_one a b ha hb haq hbq |
| have h0 := rowDist_nonneg a |
| linarith |
|
|
| theorem uniform_expectation_eq (f : Functions) : |
| permanentExpectation uniformLaw f = (∑ π : Perm, ∏ i, f i (π i)) / 24 := by |
| simp only [permanentExpectation, uniformLaw, ← Finset.mul_sum] |
| ring |
|
|
| theorem normalized_uniform_stability (f : Functions) |
| (hf : ∀ i j, 0 ≤ f i j) (hn : ∀ i, rowSq (f i) = 4) : |
| permanentExpectation uniformLaw f ≤ 1 - totalDist f / 1728 := by |
| let P := permanentExpectation uniformLaw f |
| let U := pairSq (f 0) (f 1) |
| let V := pairSq (f 2) (f 3) |
| have hU0 : 0 ≤ U := pairSq_nonneg _ _ |
| have hV0 : 0 ≤ V := pairSq_nonneg _ _ |
| have hU : U ≤ 24 := pairSq_le _ _ (hf 0) (hf 1) (hn 0) (hn 1) |
| have hV : V ≤ 24 := pairSq_le _ _ (hf 2) (hf 3) (hn 2) (hn 3) |
| have hcs := Finset.sum_mul_sq_le_sq_mul_sq Finset.univ |
| (pairEntries (f 0) (f 1)) (fun j => pairEntries (f 2) (f 3) (Fin.rev j)) |
| have hrev : (∑ j : Fin 6, pairEntries (f 2) (f 3) (Fin.rev j) ^ 2) = V := |
| Equiv.sum_comp Fin.revPerm (fun j => pairEntries (f 2) (f 3) j ^ 2) |
| rw [← permanent_pair_identity, hrev] at hcs |
| change (∑ π : Perm, ∏ i, f i (π i)) ^ 2 ≤ U * V at hcs |
| have hP2 : 576 * P ^ 2 ≤ U * V := by |
| dsimp [P] |
| rw [uniform_expectation_eq] |
| nlinarith [hcs] |
| have hUV : U * V ≤ 12 * (U + V) := by |
| nlinarith [mul_nonneg hU0 (show 0 ≤ 24 - V by linarith), |
| mul_nonneg hV0 (show 0 ≤ 24 - U by linarith)] |
| have hd : totalDist f ≤ 18 * (48 - U - V) := by |
| have h1 := pair_stability _ _ (hf 0) (hf 1) (hn 0) (hn 1) |
| have h2 := pair_stability _ _ (hf 2) (hf 3) (hn 2) (hn 3) |
| simp only [totalDist, Fin.sum_univ_four] |
| dsimp [U, V] |
| linarith |
| change P ≤ _ |
| nlinarith [sq_nonneg (P - 1)] |
|
|
| end FourRow |
| end |
|
|
| end OAI |
|
|
