import Mathlib import OAI.MathematicalPhysics.PEPSFilters.LocalOperators namespace OAI /-! Dimension-free row-column and scalar modular distortion bounds. -/ noncomputable section open scoped BigOperators ComplexOrder open scoped BigOperators ComplexOrder Matrix.Norms.L2Operator open scoped BigOperators open scoped Topology open Filter open scoped MatrixOrder open scoped BigOperators Matrix.Norms.L2Operator open PolynomialPEPS.PinnedEntropy open scoped BigOperators Matrix.Norms.L2Operator namespace PolynomialPEPS.Subvolume.MatrixRows open Matrix variable {ι κ : Type*} [Fintype ι] [Fintype κ] [DecidableEq ι] [DecidableEq κ] omit [DecidableEq ι] in lemma column_l2 (A : Matrix ι κ ℂ) (j : κ) : Real.sqrt (∑ i, ‖A i j‖^2) ≤ ‖A‖ := by have h := Matrix.l2_opNorm_mulVec A (PiLp.single 2 j (1 : ℂ)) have hx : ‖(PiLp.single 2 j (1 : ℂ) : EuclideanSpace ℂ κ)‖ = 1 := by simp rw [hx,mul_one] at h have he : (EuclideanSpace.equiv ι ℂ).symm (A *ᵥ (PiLp.single 2 j (1 : ℂ))) = WithLp.toLp 2 (fun i => A i j) := by ext i change (A *ᵥ Pi.single j (1 : ℂ)) i = A i j simp only [Matrix.mulVec_single,MulOpposite.op_one,one_smul] rfl rw [he,EuclideanSpace.norm_eq] at h exact h lemma row_l2 (A : Matrix ι κ ℂ) (i : ι) : Real.sqrt (∑ j, ‖A i j‖^2) ≤ ‖A‖ := by simpa [Matrix.conjTranspose_apply, Matrix.l2_opNorm_conjTranspose] using column_l2 A.conjTranspose i lemma row_product (A B : Matrix ι κ ℂ) (i : ι) : ∑ j, ‖A i j‖ * ‖B i j‖ ≤ ‖A‖ * ‖B‖ := by have h := Real.sum_mul_le_sqrt_mul_sqrt Finset.univ (fun j => ‖A i j‖) (fun j => ‖B i j‖) exact h.trans (mul_le_mul (row_l2 A i) (row_l2 B i) (Real.sqrt_nonneg _) (norm_nonneg _)) lemma column_product (A B : Matrix ι κ ℂ) (j : κ) : ∑ i, ‖A i j‖ * ‖B i j‖ ≤ ‖A‖ * ‖B‖ := by simpa [Matrix.conjTranspose_apply, Matrix.l2_opNorm_conjTranspose] using row_product A.conjTranspose B.conjTranspose j theorem weighted_sum (A B : Matrix ι ι ℂ) (p : ι → ℝ) (hp : ∀ i, 0 ≤ p i) : ∑ i, ∑ j, max (p i) (p j) * (‖A i j‖ * ‖B i j‖) ≤ 2 * (∑ i, p i) * (‖A‖ * ‖B‖) := by calc _ ≤ ∑ i, ∑ j, (p i+p j) * (‖A i j‖ * ‖B i j‖) := by gcongr with i j exact max_le (le_add_of_nonneg_right (hp j)) (le_add_of_nonneg_left (hp i)) _ = (∑ i, p i * (∑ j, ‖A i j‖ * ‖B i j‖)) + (∑ j, p j * (∑ i, ‖A i j‖ * ‖B i j‖)) := by simp_rw [add_mul,Finset.sum_add_distrib,← Finset.mul_sum] rw [Finset.sum_comm] simp_rw [Finset.mul_sum] _ ≤ (∑ i, p i * (‖A‖*‖B‖)) + (∑ j, p j * (‖A‖*‖B‖)) := by exact add_le_add (Finset.sum_le_sum fun i _ => mul_le_mul_of_nonneg_left (row_product A B i) (hp i)) (Finset.sum_le_sum fun j _ => mul_le_mul_of_nonneg_left (column_product A B j) (hp j)) _ = _ := by rw [← Finset.sum_mul]; ring theorem coefficient_sum (A B : Matrix ι ι ℂ) (p : ι → ℝ) (hp : ∀ i, 0 ≤ p i) (d : ι → ι → ℂ) (C : ℝ) (hC : 0 ≤ C) (hd : ∀ i j, ‖d i j‖ ≤ C * max (p i) (p j)) : ‖∑ i, ∑ j, d i j * A i j * B i j‖ ≤ 2*C*(∑ i, p i)*(‖A‖*‖B‖) := by calc _ ≤ ∑ i, ∑ j, ‖d i j * A i j * B i j‖ := (norm_sum_le _ _).trans (Finset.sum_le_sum fun i _ => norm_sum_le _ _) _ ≤ ∑ i, ∑ j, C*max (p i) (p j) * (‖A i j‖*‖B i j‖) := by simp only [norm_mul] exact Finset.sum_le_sum fun i _ => Finset.sum_le_sum fun j _ => by simpa only [mul_assoc] using mul_le_mul_of_nonneg_right (hd i j) (mul_nonneg (norm_nonneg _) (norm_nonneg _)) _ = C * (∑ i, ∑ j, max (p i) (p j) * (‖A i j‖*‖B i j‖)) := by simp_rw [Finset.mul_sum] congr 1 ext i congr 1 ext j ring _ ≤ C * (2*(∑ i, p i)*(‖A‖*‖B‖)) := mul_le_mul_of_nonneg_left (weighted_sum A B p hp) hC _ = _ := by ring end PolynomialPEPS.Subvolume.MatrixRows namespace PolynomialPEPS.Subvolume.ModularScalar lemma even_exp_remainder (x : ℝ) (hx : 0 ≤ x) : Real.exp x + Real.exp (-x) - 2 ≤ x^2 * Real.exp x := by let y := Real.exp (x/2) let z := Real.exp (-x/2) have hy : 0 < y := Real.exp_pos _ have hz : 0 < z := Real.exp_pos _ have hyz : y*z=1 := by dsimp [y,z] rw [← Real.exp_add, ← add_div, add_neg_cancel, zero_div, Real.exp_zero] have hysq : y^2 = Real.exp x := by dsimp [y] rw [sq,← Real.exp_add] congr 1 ring have hzsq : z^2 = Real.exp (-x) := by dsimp [z] rw [sq,← Real.exp_add] congr 1 ring have hd0 : 0 ≤ y-z := sub_nonneg.mpr (Real.exp_le_exp.mpr (by linarith)) have he := Real.add_one_le_exp (-x) have hmul : y*Real.exp (-x) = z := by dsimp [y,z] rw [← Real.exp_add] congr 1 ring have hd : y-z ≤ x*y := by have ht := mul_le_mul_of_nonneg_left he hy.le rw [hmul] at ht nlinarith have hs := sq_le_sq₀ hd0 (mul_nonneg hx hy.le) |>.mpr hd rw [mul_pow,hysq] at hs nlinarith [sq_nonneg (y-z)] lemma polynomial_exp_decay (x : ℝ) (hx : 0 ≤ x) : x^2 * Real.exp (-x/4) ≤ 32 := by have hp := Real.pow_div_factorial_le_exp (x/4) (show 0 ≤ x/4 by positivity) 2 norm_num at hp have he : Real.exp (x/4)*Real.exp (-x/4) = 1 := by rw [← Real.exp_add, ← add_div, add_neg_cancel, zero_div, Real.exp_zero] have hh := mul_le_mul_of_nonneg_right hp (Real.exp_pos (-x/4)).le nlinarith theorem quadratic_log_ratio (x a : ℝ) (hx : 0 ≤ x) (ha : |a| ≤ 1/4) : Real.exp (-x/2) * (Real.exp (a*x) + Real.exp (-a*x) - 2) ≤ 32*a^2 := by have heven : Real.exp (a*x) + Real.exp (-a*x) = Real.exp (|a| * x) + Real.exp (-(|a| * x)) := by rcases le_total 0 a with h|h · simp [abs_of_nonneg h] · simp [abs_of_nonpos h,add_comm] rw [heven] have ht := mul_le_mul_of_nonneg_left (even_exp_remainder (|a| * x) (by positivity)) (Real.exp_pos (-x/2)).le have hcomb : Real.exp (-x/2) * ((|a| * x)^2 * Real.exp (|a| * x)) = a^2 * (x^2 * Real.exp ((|a|-1/2)*x)) := by rw [mul_pow,sq_abs] calc _ = a^2*x^2*(Real.exp (-x/2)*Real.exp (|a| * x)) := by ring _ = _ := by rw [← Real.exp_add, show -x/2 + |a| * x = (|a|-1/2)*x by ring] ring rw [hcomb] at ht have he : Real.exp ((|a|-1/2)*x) ≤ Real.exp (-x/4) := Real.exp_le_exp.mpr (by nlinarith) calc _ ≤ a^2 * (x^2 * Real.exp ((|a|-1/2)*x)) := ht _ ≤ a^2 * (x^2 * Real.exp (-x/4)) := by gcongr _ ≤ a^2 * 32 := mul_le_mul_of_nonneg_left (polynomial_exp_decay x hx) (sq_nonneg a) _ = 32*a^2 := by ring end PolynomialPEPS.Subvolume.ModularScalar namespace PolynomialPEPS.Subvolume.ModularScalar lemma positive_weight_bound (s t a : ℝ) (hs : 0 < s) (ht : 0 < t) (ha : |a| ≤ 1/4) : Real.sqrt (s*t) * ((s/t)^a+(t/s)^a-2) ≤ 32*a^2*max s t := by suffices hordered : ∀ s t : ℝ, 0 < s → 0 < t → t ≤ s → Real.sqrt (s*t) * ((s/t)^a+(t/s)^a-2) ≤ 32*a^2*s by rcases le_total t s with h|h · simpa only [max_eq_left h] using hordered s t hs ht h · simpa only [max_eq_right h,mul_comm s t,add_comm ((s/t)^a) ((t/s)^a)] using hordered t s ht hs h intro s t hs ht hts let x := Real.log s - Real.log t have hx : 0 ≤ x := sub_nonneg.mpr (Real.log_le_log ht hts) have hst : (s/t)^a = Real.exp (a*x) := by rw [Real.rpow_def_of_pos (div_pos hs ht),Real.log_div hs.ne' ht.ne'] congr 1 dsimp [x] ring have hts' : (t/s)^a = Real.exp (-a*x) := by rw [Real.rpow_def_of_pos (div_pos ht hs),Real.log_div ht.ne' hs.ne'] congr 1 dsimp [x] ring have hroot : Real.sqrt (s*t) = s*Real.exp (-x/2) := by rw [Real.sqrt_eq_rpow,Real.rpow_def_of_pos (mul_pos hs ht),Real.log_mul hs.ne' ht.ne'] nth_rw 2 [← Real.exp_log hs] rw [← Real.exp_add] congr 1 dsimp [x] ring rw [hst,hts',hroot,mul_assoc] calc _ ≤ s*(32*a^2) := mul_le_mul_of_nonneg_left (quadratic_log_ratio x a hx ha) hs.le _ = _ := by ring def modularCoefficient (s t a : ℝ) : ℝ := if 0 < s ∧ 0 < t then Real.sqrt (s*t) * ((s/t)^a+(t/s)^a-2)/2 else 0 lemma modularCoefficient_nonneg (s t a : ℝ) : 0 ≤ modularCoefficient s t a := by unfold modularCoefficient split_ifs with h · have h₁ := Real.add_one_le_exp (Real.log (s/t)*a) have h₂ := Real.add_one_le_exp (Real.log (t/s)*a) have hlog : Real.log (t/s) = -Real.log (s/t) := by rw [Real.log_div h.2.ne' h.1.ne',Real.log_div h.1.ne' h.2.ne'] ring have hexp : 0 ≤ (s/t)^a+(t/s)^a-2 := by rw [Real.rpow_def_of_pos (div_pos h.1 h.2),Real.rpow_def_of_pos (div_pos h.2 h.1)] rw [hlog] at h₂ ⊢ nlinarith positivity · rfl theorem modularCoefficient_bound (s t a : ℝ) (hs : 0 ≤ s) (ht : 0 ≤ t) (ha : |a| ≤ 1/4) : |modularCoefficient s t a| ≤ 16*a^2*max s t := by rw [abs_of_nonneg (modularCoefficient_nonneg s t a)] unfold modularCoefficient split_ifs with h · have H := positive_weight_bound s t a h.1 h.2 ha linarith · exact mul_nonneg (mul_nonneg (by norm_num) (sq_nonneg a)) (by simpa using max_le_max hs ht) end PolynomialPEPS.Subvolume.ModularScalar namespace PolynomialPEPS.Subvolume.ModularScalar lemma modularCoefficient_formula (s t a : ℝ) (hs : 0 ≤ s) (ht : 0 ≤ t) : Real.sqrt s * Real.sqrt t * (((s^a)*(t^a)⁻¹ + (s^a)⁻¹*(t^a))/2 - 1) = modularCoefficient s t a := by by_cases hs0 : s=0 · simp [hs0,modularCoefficient] by_cases ht0 : t=0 · simp [ht0,modularCoefficient] have hs' : 0