chars int64 40 24k | content_type stringclasses 9
values | id stringlengths 13 64 | source stringclasses 5
values | subset stringclasses 31
values | text stringlengths 40 24k |
|---|---|---|---|---|---|
23,949 | lean_file_chunk | 5440e032df0b884f871f32dc62bdceca3797edeff08cb2be7b4c4c5e090db4de | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import LeanPool.OSforGFF.Spacetime.Euclidean
import Mathlib.Analysis.Complex.Exponent... |
20,430 | lean_file_chunk | 0ffd4cae58342806993879e13f9be2ffb076dbfaac293ec46f3152c605ef2235 | lean_source_file | lean_pool_source_file_chunks | exact Integrable.mono' h_bound_int hf_meas h_norm_bound
-- Part 2: Integrability on (1/m, ∞)
· -- At infinity: use besselK1_asymptotic for exponential decay
-- For r > 1/m, we have mr > 1, so K₁(mr) ≤ (sinh 1 + 2) exp(-mr)
set C := sinh 1 + 2 with hC_def
have hC_pos : 0 < C := by
simp only [hC_def... |
23,943 | lean_file_chunk | 91f99063ad8510adb89bdb7c4d453029b6f05c09f3e5242b728b87acdb8ee552 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.Fourier.Inve... |
11,710 | lean_file_chunk | f67b99a822a90d2c7a3cf6969784f75b4ae957f629ed61129a44a3a85f4ca833 | lean_source_file | lean_pool_source_file_chunks | /-- The exponential decay function is integrable over ℝ. -/
lemma integrable_expDecayFun (μ : ℝ) (hμ : 0 < μ) : Integrable (expDecayFun μ) volume := by
unfold expDecayFun
exact Integrable.ofReal (integrable_exponential_decay μ hμ)
/-- Mathlib's Fourier transform of expDecayFun equals the scaled Lorentzian.
FT_... |
10,223 | lean_file_chunk | 719f60d36500e01338dbabc7a3aad21c8125c28fc4dddf15a920d96414b5eaa7 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.LinearAlgebra.Matrix.PosDef
import Ma... |
23,990 | lean_file_chunk | 59f028d54d86fe149f762e4bdcc310c581a96be182f857b37d7904ba50230acb | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import Mathlib.Analysis.Distribution.SchwartzSpace.Deriv
import Mathlib.MeasureTheory... |
23,986 | lean_file_chunk | adeb1c4fed5d30965f9784c2c6c344d75f7916393d8f4096c42df9d8834adfee | lean_source_file | lean_pool_source_file_chunks | -- The map preserves measure (translation invariance of Lebesgue measure)
have he_preserves : MeasurePreserving e (volume.prod volume) (volume.prod volume) := by
-- Use measurePreserving_sub_prod: (x, y) ↦ (x - y, y) preserves measure
have := measurePreserving_sub_prod (G := EuclideanSpace ℝ (Fin d)) vo... |
651 | lean_file_chunk | e74d7e9dc0e0a0f0d65e21fa2c2c2b459bc433124c9c0d873a85677478d4001e | lean_source_file | lean_pool_source_file_chunks | _ < ‖a‖ := by linarith [norm_nonneg a]
linarith
change Integrable F (volume.prod volume)
rw [← MeasureTheory.integrableOn_iff_integrable_of_support_subset h_supp_F]
exact h_cont_F.integrableOn_compact hK_compact
rw [h_assoc]
-- Use Tendsto.congr' with the eventually equa... |
12,526 | lean_file_chunk | 074163054c977b643f5e41c6338b5496150ea3a843c1e8ccc24b6bf3c01f0d24 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import LeanPool.OSforGFF.General.PositiveDefinite
import Mathlib.Analysis.InnerProduc... |
23,965 | lean_file_chunk | 582edba6e95dd1564227dc2ee340e8fcc20bde866a4948f587e8e9f11fa3dbb8 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import LeanPool.OSforGFF.General.SchurProduct
import Mathlib.Analysis.SpecialFunction... |
685 | lean_file_chunk | 584a92fb2730097e2790180e034ed8def9402fe5b857acbf2f8c8c4fae203b70 | lean_source_file | lean_pool_source_file_chunks | -- Convert continuousAt to the right-sided limit
rw [← h_limit_simplify]
exact tendsto_nhdsWithin_of_tendsto_nhds h_cont_at_zero.tendsto
-- Apply ge_of_tendsto: if f(ε) ≥ 0 eventually and f → f(0), then f(0) ≥ 0
have h_final : 0 ≤ x ⬝ᵥ (entrywiseExp R).mulVec x :=
ge_of_tendsto h_tendsto (by
... |
23,967 | lean_file_chunk | 50d515e80c0ae2e2e7dc555cede38b0af2b9e5b539bab83cc9629dad703ee94f | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import Mathlib.MeasureTheory.Integral.Prod
import Mathlib.MeasureTheory.Integral.Boch... |
23,994 | lean_file_chunk | fb46d13f20de3acb55fe358c18d783b74c4636e2e53e425296a9a173497371ef | lean_source_file | lean_pool_source_file_chunks | rw [← integrableOn_univ, ← Iic_union_Ioi (a := (0:ℝ)),
show Iic (0:ℝ) = Iio 0 ∪ {0} from Eq.symm Iio_union_right]
exact (h_iio.union (integrableOn_singleton (hx := by simp))).union h_ioi
-- g ≥ 0 everywhere
have h_nonneg : ∀ t : ℝ, 0 ≤ g t := fun t => by positivity
-- C = ∫_ℝ g(t) dt > 0 (g > 0 on ope... |
2,053 | lean_file_chunk | a43ccaa33485221201b0840121a645313ef4ff8c1925ef3efffe4f1df232f159 | lean_source_file | lean_pool_source_file_chunks | have h_bound : ∀ x : Ω × (ℝ × ℝ),
(↑‖(A x.2.1 x.1 - c) * starRingEnd ℂ (A x.2.2 x.1 - c)‖₊ : ℝ≥0∞) ≤
↑‖A x.2.1 x.1 - c‖₊ ^ 2 + ↑‖A x.2.2 x.1 - c‖₊ ^ 2 := by
intro x
rw [nnnorm_mul, coe_mul, show ‖starRingEnd ℂ (A x.2.2 x.1 - c)‖₊ = ‖A x.2.2 x.1 - c‖₊
from nnnorm_star _]
exact ennreal_mul_l... |
23,937 | lean_file_chunk | f43d250a85ea5466faefc424953cbd4f4f0bf7bab2050762dc09cb54c6646973 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
import Mathlib.Me... |
3,169 | lean_file_chunk | 0e99b223e5d8e5479f2be483ba05858168213b5287140d4cb4502ca8e0cf603b | lean_source_file | lean_pool_source_file_chunks | refine setIntegral_congr_fun measurableSet_Ioi fun t ht => ?_
rw [show -a/t^2 - b*t^2 = -(a/t^2 + b*t^2) by ring, complete_square a b ha hb t ht, neg_add,
exp_add]
ring_nf
/-- Second substitution: u = √b · t, so √a/t - √b·t = √(ab)/u - u -/
lemma laplace_integral_subst_scale (a b : ℝ) (ha : 0 < a) (hb : 0 < b)... |
1,983 | lean_file_chunk | c2eb0fa33aea42ed76240dce79fbd1d9a9f563fb2787be6a4c7d63512ec4106b | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import Mathlib.Data.Complex.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.B... |
23,988 | lean_file_chunk | 4a59997982ca4d8d95a891923a3a88ffb3b310dc49c9f5396f8f7b10d92579bd | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import Mathlib.Analysis.Distribution.SchwartzSpace.Deriv
import Mathlib.MeasureTheor... |
23,988 | lean_file_chunk | cb88d47bfef9ecefdc22f3039be3d183a73fc61d30111b7f828f0a5cea2f1781 | lean_source_file | lean_pool_source_file_chunks | -- The integral bound follows from:
-- 1. ‖f x‖ ≤ C_f / (1 + ‖x‖)^N (Schwartz decay)
-- 2. On support of K_sing(x-y), Peetre gives: 1/(1+‖x‖)^N ≤ (1+R₀)^N/(1+‖y‖)^N
-- 3. Change of variables: ∫ |K_sing(x-y)| dx = ∫ |K_sing(z)| dz = I_Ksing
calc ‖∫ x, f x * (kernelSingular K R₀ (x - y) : ℂ)‖
≤ ∫ x, ‖f x * ... |
2,302 | lean_file_chunk | 158ed3f92fffad772a825def12a9236195d9df9a80362cc307ea939aa03a3f50 | lean_source_file | lean_pool_source_file_chunks | · exact le_of_lt hC_conv_pos
· intro a
have h_one_plus_pos : 0 < 1 + ‖a‖ := by positivity
have h_base_ge_one : 1 ≤ 1 + ‖a‖ := by linarith [norm_nonneg a]
-- The bound from convolutionPolynomialDecay gives us:
-- ‖∫ H(y) g_flip(a-y) dy‖ ≤ C_conv / (1 + ‖a‖)^N ≤ C_conv * (1 + ‖a‖)^(-α)
have h_conv_t... |
4,918 | lean_file_chunk | 642f3211364fc339b876c6a0e4f269ca7553a1f660e25c78fbb2d23d5ce00601 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import Mathlib.LinearAlgebra.Matrix.Hadamard
import Mathlib.LinearAlgebra.Matrix.Kron... |
23,982 | lean_file_chunk | 73829c0df62a9520e4638adac923fd8e2c7fdd73c8e120a8b3667f306f9722be | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import Mathlib.Analysis.Distribution.SchwartzSpace.Deriv
import Mathlib.MeasureTheor... |
23,959 | lean_file_chunk | 8c17e44bf8784f5942838a5a1f55287a5b6993f2b861bfff742a120b243ba982 | lean_source_file | lean_pool_source_file_chunks | · -- ae bound: ‖f(x) K_tail(x-y) g(y-a)‖ ≤ M_tail * ‖f(x) * g(y-a)‖
filter_upwards with ⟨x, y⟩
simp only [Function.uncurry]
have hKt_bnd : ‖(K_tail (x - y) : ℂ)‖ ≤ M_tail := by
rw [Complex.norm_real, Real.norm_eq_abs]
exact hM_tail (x - y)
calc ‖f x * (K_tail (x - y) : ℂ) * g (y ... |
5,704 | lean_file_chunk | 01419cd5de42f2083e6c85779b152b03c3d2e4d01f8c523df0be23ec1f6b5748 | lean_source_file | lean_pool_source_file_chunks | -- kernelTail K R₀ x = K x * indicator is measurable
have hind : Measurable ((closedBall (0 : E) R₀)ᶜ.indicator (fun _ => (1 : ℝ))) :=
measurable_const.indicator isClosed_closedBall.measurableSet.compl
have h1 : Measurable (kernelTail K R₀) := hK_meas.mul hind
have h2 : Measurable (fun... |
922 | lean_file_chunk | 4bf5e80bde8a58531158e45235a18a7df6e1943d20fcd0ee407c69644a385157 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import LeanPool.OSforGFF.General.BesselFunction
import LeanPool.OSforGFF.General.Four... |
673 | lean_file_chunk | 286f3a342063eed01119441712b4e4ee065e2b568d26351af599ee26a8e9e6a0 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Rémy Degenne, Peter Pfaffelhuber. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Peter Pfaffelhuber
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.MeasureTheory.MeasurableSpace.Defs
/-!
THIS FILE IS NOT USED FOR TH... |
12,004 | lean_file_chunk | 97deb568483b646a96924246f58ec7abfb3d0d2bcd96bd5ce20ea3aa145748a5 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Rémy Degenne, Peter Pfaffelhuber. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Peter Pfaffelhuber
-/
import Mathlib.MeasureTheory.Constructions.ClosedCompactCylinders
import Mathlib.Topology.Compactness.CompactSystem
import Mathli... |
8,351 | lean_file_chunk | 7fa42267f66a01475d6b6bdfbd77dbfe52bd945edd26f54b2df6092feb08a1c1 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Rémy Degenne, Peter Pfaffelhuber. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Peter Pfaffelhuber
-/
import LeanPool.OSforGFF.KolmogorovExtension4.RegularContent
import Mathlib.MeasureTheory.Constructions.ProjectiveFamilyContent
i... |
3,826 | lean_file_chunk | 71b4382b8b4ddbf5303ee795f67533e7c3ff5bdf58aaa24d37bf59ddf2610e32 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Rémy Degenne, Peter Pfaffelhuber. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Peter Pfaffelhuber
-/
import LeanPool.OSforGFF.KolmogorovExtension4.CompactSystem
import Mathlib.MeasureTheory.Measure.AddContent
/-!
# Regular Conten... |
1,835 | lean_file_chunk | 17a308b52ec015fcbdaf357e97112de88ae2e168de24cc7d5f74cd2ee7a82770 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Rémy Degenne, Peter Pfaffelhuber. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Peter Pfaffelhuber
-/
import Mathlib.MeasureTheory.SetSemiring
/-! # Semirings of sets
A semi-ring of sets `C` is a family of sets containing `∅`, st... |
606 | lean_file_chunk | 0e18a43e186e810732acdda504efcbfeac050a3329e1d5a76004d32efee6998a | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Rémy Degenne, Peter Pfaffelhuber. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Peter Pfaffelhuber
-/
import LeanPool.OSforGFF.KolmogorovExtension4.AuxLemmas
import LeanPool.OSforGFF.KolmogorovExtension4.CompactSystem
import LeanPo... |
20,131 | lean_file_chunk | 92b97c773afbab5d4d9d1f14c54ad0ba0a155ccb73490f56d8600ad376255a89 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import Mathlib.Probability.Distributions.Gaussian.Fernique
import LeanPool.OSforGFF.... |
10,497 | lean_file_chunk | 61e4d8ff275eaa03d0966b0a2d22023d9a02631b696d3dbf7c292fdb9adcfff1 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Analytic.Construction... |
23,703 | lean_file_chunk | 876187dabe82d53868f946938c53621bea47f71b7b6a1024c1135e751e7a57af | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import LeanPool.OSforGFF.OS.OS0Analyticity
import LeanPool.OSforGFF.Schwinger.Gaussi... |
10,543 | lean_file_chunk | 44f509a51d95f5644c2caaf3e47bc8a36b28390caf9724b6215157b87bf3dcbd | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import Mathlib.Data.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.Basic
im... |
8,418 | lean_file_chunk | 53cb7512adb1bd08d8a73891339427896587c1fd4dd4cb0457b56c3d4b5b45fb | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import Mathlib.Data.Complex.Basic
import Mathlib.Analysis.LocallyConvex.Basic
import... |
4,381 | lean_file_chunk | dc5aa30c3e0efb06fd5798e80faeadef30bed0c2336e89241007393f5ae5591f | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback
-/
import LeanPool.OSforGFF.Minlos.NuclearSpace
import LeanPool.OSforGFF.Minlos.PietschBridge
import LeanPool.OSforGFF.Minlos.Main
i... |
598 | lean_file_chunk | 3dc0f68c7606fed545d61866278f05d1b33279fa655a4d6dd47321772e3809b8 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import LeanPool.OSforGFF.Measure.Construct
import LeanPool.OSforGFF.Measure.GaussianF... |
3,942 | lean_file_chunk | 4805b949321d3e57b3bd093bc0820b3220668c1974692e58ae4d85579f685abc | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas
-/
import LeanPool.OSforGFF.Bochner.Main
/-!
# Finite-Dimensional Marginals for Minlos' Theorem
For a continuous positive-definite normalized functional Φ on ... |
11,165 | lean_file_chunk | 9bfce9c54a1ea70106535ffa711e93df3ce78a282802e1be62d537b30ce40db8 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas
-/
import LeanPool.OSforGFF.Minlos.MeasurableModification
/-!
# Minlos' Theorem
Minlos' theorem: on a nuclear locally convex space E, a continuous positive-de... |
23,947 | lean_file_chunk | 334ceb77759b06b4e95bf9cd1114f8d2cd1f9f3b066aa885709474f24c05b1f8 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas
-/
import LeanPool.OSforGFF.Minlos.ProjectiveFamily
import LeanPool.OSforGFF.Minlos.MinlosConcentration
import Mathlib.Topology.Bases
import Mathlib.Topology.Ex... |
23,962 | lean_file_chunk | 4114a21671f346c9a20d2e6381ce2878a65d814dbc691eb446e34a2c7d157491 | lean_source_file | lean_pool_source_file_chunks | noncomputable def extensionCLM [SeparableSpace E] [IsHilbertNuclear E] [Nonempty E]
(d : ℕ → E) (hd : DenseRange d)
(p : ℕ → Seminorm ℝ E) (hp_top : WithSeminorms (fun n => p n))
(ω : E → ℝ) (hω : ω ∈ goodPaths d p) :
WeakDual ℝ E :=
⟨{ toFun := extensionFun d hd p ω hω
map_add' := extensionFun_m... |
6,248 | lean_file_chunk | 5ae54d3946189eda75a6591bdaf1361000521026ca509b23a3b7cc281a8ec4a0 | lean_source_file | lean_pool_source_file_chunks | (measurable_pi_apply (d (φ k))).sub (measurable_pi_apply f)
apply Measurable.aestronglyMeasurable
exact ((hY_meas.const_mul t).complex_ofReal.const_mul I).cexp
have h_bound : ∀ k, ∀ᵐ ω ∂ν, ‖exp (I * ↑(t * Y k ω))‖ ≤ (1 : ℝ) := by
intro k
filter_upwards with ω
simp [Complex.norm_exp, Co... |
23,974 | lean_file_chunk | fe88c4e67a82149cb31614aa37ff7ebccdc1d948fcc543d28630beddea358cda | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas
-/
import LeanPool.OSforGFF.Minlos.SazonovTightness
import LeanPool.OSforGFF.Minlos.NuclearSpace
import LeanPool.OSforGFF.Minlos.PietschBridge
import Mathlib.Da... |
23,948 | lean_file_chunk | 6907341bdc9c9da8e75fa55d3051f016edb2c0d2c0e06a239707764a198ae66f | lean_source_file | lean_pool_source_file_chunks | have hu_proj_orth : p.innerProd u (∑ j : Fin k, c j • e j) = 0 := by
rw [gs_innerProd_sum_right p hp]
simp_rw [gs_innerProd_smul_right p hp]; simp [hu_orth]
-- Pythagoras: u + proj = v
have hv_eq : u + ∑ j : Fin k, c j • e j = v := sub_add_cancel v _
have h_pyth := gs_pythagoras p hp u (∑ j : Fi... |
23,972 | lean_file_chunk | e950c85d237925b35fb7bc55c93b694de9e5f220fe6151fbf8238eab365be5e4 | lean_source_file | lean_pool_source_file_chunks | ν {ω : E → ℝ | ∃ c : ℕ →₀ ℚ, c.support ⊆ Finset.range N ∧
ω (c.sum (fun i a => (a : ℝ) • d i)) ≠
∑ j : Fin k, α_map c j * ω (e j)} ≤
ENNReal.ofReal ε_q := by
-- Define kernel elements z_c for each c
let z (c : ℕ →₀ ℚ) : E := c.sum (fun i a => (a : ℝ) • d i) - ∑ j, α_map c j • e j
-- Step 1: Li... |
15,317 | lean_file_chunk | 924958281eeff660d304c98864452a2af541d527cd8277d06cbfe004c601dc9f | lean_source_file | lean_pool_source_file_chunks | set T : (E → ℝ) → ℝ := fun ω => ∑ i : Fin (k + 1), ω (e i) ^ 2
have hA_ne_top : ν {ω | R ^ 2 < T ω} ≠ ⊤ :=
ne_top_of_le_ne_top (measure_ne_top ν Set.univ) (measure_mono (Set.subset_univ _))
rw [← ENNReal.ofReal_toReal hA_ne_top]
apply ENNReal.ofReal_le_ofReal
have h_T_meas : Measurable T :=
Finset.measu... |
4,011 | lean_file_chunk | 6a02de9d79b2692e9253daf451d716120c4945b74c3197ab38a7cd06111095ed | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback
-/
import Mathlib.Analysis.LocallyConvex.WithSeminorms
import Mathlib.Topology.Algebra.Module.Spaces.WeakDual
import Mathlib.Measure... |
23,948 | lean_file_chunk | 5b9b638cc71ecf72ef7c67e0a13b6298b8896b7fe9fc0f58cfd1939e972c3edb | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas
-/
import LeanPool.OSforGFF.Minlos.NuclearSpace
import Mathlib.Topology.Instances.RealVectorSpace
/-!
# Bridge: Pietsch Nuclearity → Bochner IsHilbertNuclear
... |
14,821 | lean_file_chunk | 10832d4d1f191f2a80b3100bd8ffa2ca3b94fd13a017913597628fee8b301cbf | lean_source_file | lean_pool_source_file_chunks | Finset.sum_le_sum (fun j _ => hCS j)
_ = (∑' n, c n) * ∑ j, (∑' n, (f n (e j)) ^ 2 * c n) := by rw [← Finset.mul_sum]
_ = (∑' n, c n) * (∑' n, ∑ j, (f n (e j)) ^ 2 * c n) := by
-- Swap finite sum and tsum
congr 1
rw [← Summable.tsum_finsetSum (fun ... |
12,828 | lean_file_chunk | 0e50e97d95068ad632a5d33bd778c6f5ba1d2c002f6c3df591e1c135abd7653d | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas
-/
import LeanPool.OSforGFF.Minlos.FinDimMarginals
import LeanPool.OSforGFF.KolmogorovExtension4.KolmogorovExtension
/-!
# Bridge: Bochner Marginals → Projecti... |
23,983 | lean_file_chunk | 15189d5a056982d5bc4c0b2c8039d412b3153c2788a1acadf660500877eb37d4 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas
-/
import LeanPool.OSforGFF.Bochner.Sazonov
import LeanPool.OSforGFF.Bochner.PositiveDefinite
import Mathlib.MeasureTheory.Measure.CharacteristicFunction.Basic
... |
23,990 | lean_file_chunk | 1fa69536374f1f534bfd20b364615018ebf1587014e936934d1d7f47c5d0728b | lean_source_file | lean_pool_source_file_chunks | have htI : t * I = ∫ x : V, gaussDensity σ x * (c * @inner ℝ V _ w x) ^ 2 := by
rw [show t = c ^ 2 from hc.symm, ← integral_const_mul]; congr 1; ext x; ring
rw [htI]
calc ∫ x, gaussDensity σ x * (c * @inner ℝ V _ w x) ^ 2
≤ ∫ x, gaussDensity σ x * (2 * (Real.cosh (c * @inner ℝ V _ w x) - 1)) := by... |
4,814 | lean_file_chunk | a4baa6cdf4749f7099b085bc57708be0263f61f1ac9c81eb26e7768ef2d18104 | lean_source_file | lean_pool_source_file_chunks | intro i; rw [restrictOp_apply]
simp_all
simp_all
exact lhs_eq.trans ((h1.symm.trans h2).trans rhs_eq)
/-- The trace of the restricted operator is bounded by the trace of the
original on any Hilbert basis. Combines trace basis independence on
EuclideanSpace with the diagonal bound.
-/
lemma restrict... |
633 | lean_file_chunk | d63fa2190786cae52a9fdb078a37f41478fd0700e6ccec0c7734685365210dc4 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas
-/
import LeanPool.OSforGFF.Minlos.FinDimMarginals
import LeanPool.OSforGFF.Minlos.Main
import LeanPool.OSforGFF.Minlos.MeasurableModification
import LeanPool.O... |
9,718 | lean_file_chunk | 23f385f20b0be601d53a39e94f84ee8beb595def0127d2ddb283c819fe3b177b | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Norm... |
3,037 | lean_file_chunk | 8a47a4c68ea0cfbfd813b4a5765538969549daa2519ef2c2d517f329e5100845 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import LeanPool.OSforGFF.Measure.GaussianFreeField
import LeanPool.OSforGFF.OS.OS3Ref... |
16,223 | lean_file_chunk | 8730fbb540f363ba72bf976ecadc35bae43f03be5f5d9b7dc782db6eec361af0 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import LeanPool.OSforGFF.Measure.Construct
import LeanPool.OSforGFF.Covariance.RealFo... |
23,984 | lean_file_chunk | e53547a29797f4184741b1f6db63f2c9b6fabf6729430f3f2d690d91610bd250 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import LeanPool.OSforGFF.Spacetime.Basic
import LeanPool.OSforGFF.OS.Axioms
import L... |
7,268 | lean_file_chunk | 9101c3d9b90cb69cf195e580c860e52a4c109443cc061763adfe03d911033511 | lean_source_file | lean_pool_source_file_chunks | (((analyticOnNhd_id (𝕜 := ℂ)).pow 2).mul analyticOnNhd_const)
-- 2 * t * Q_ri is linear in t
have : AnalyticOnNhd ℂ (fun t : ℂ =>
(2 * (freeCovarianceFormR m f_re f_im : ℂ)) * t) Set.univ :=
AnalyticOnNhd.mul analyticOnNhd_const analyticOnNhd_id
have heq : (fun t : ℂ => 2 * t * (freeCovarianc... |
23,982 | lean_file_chunk | 1c69017e22ed935352134395a66f858e8c3ade7f25ed62ab7a9fee1f9274b675 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import LeanPool.OSforGFF.Spacetime.Basic
import LeanPool.OSforGFF.OS.Axioms
import L... |
2,561 | lean_file_chunk | c6a3d3ddb2876aabe6c0f24ff568ae4c0047c2df4680d76cf8090e58e690e50e | lean_source_file | lean_pool_source_file_chunks | `c = 1/(2 m^2)`. This uses `gff_generating_L2_bound` and
`gff_two_point_locally_integrable` established above.
Note: Named `_revised` because the alternative OS0 proof in `GaussianFreeField.lean`
uses the same module; both are valid, and `OS.Master` uses this one.
-/
theorem gaussianFreeField_satisfies_OS1... |
6,345 | lean_file_chunk | c27c8a5603291806df2b9f17f997ec6d85f6aa8bf63d67fd1d98063028bb5818 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import LeanPool.OSforGFF.Measure.IsGaussian
import LeanPool.OSforGFF.Covariance.Posi... |
20,313 | lean_file_chunk | 2727abeaa9aebe0b465dec1fe4e03878844b88ca524f64eec21600bc7d6b3398 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import LeanPool.OSforGFF.Spacetime.Basic
import LeanPool.OSforGFF.Spacetime.Positive... |
23,993 | lean_file_chunk | eb2c782f2ac9c7d238e0f7991094f6a674c3a4262b1f62e08a496c22fd89a346 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import LeanPool.OSforGFF.OS.OS3MixedRepInfra
/-!
# OS3 — Mixed Representation via S... |
23,958 | lean_file_chunk | 71750bab16c468fafa02b118d2b0a860a373227468c6c1f6c977842b0aad5952 | lean_source_file | lean_pool_source_file_chunks | -- Step 3: Goal is ∫ x in S, ↑(f x) = ↑(result)
-- Use integral_complex_ofReal: ∫ x in S, ↑(f x) = ↑(∫ x in S, f x)
rw [integral_complex_ofReal]
-- Now goal is: ↑(∫ x in S, f x) = ↑(result), which follows from s_integral_eval
exact congrArg Complex.ofReal (s_integral_eval t ω hω)
/-- **Complex-valued s-integra... |
15,730 | lean_file_chunk | dc4a7ac0c550a4159caade869b8179f1f241a0e39a60bf5b7291caa15b761748 | lean_source_file | lean_pool_source_file_chunks | -- The formal proof requires showing:
-- (a) ∫_x ∫_y ∫_s F = ∫_{(x,y)} ∫_s F = ∫_{(s,x,y)} F (by integral_integral twice)
-- (b) ∫_s ∫_x ∫_y F = ∫_s ∫_{(x,y)} F = ∫_{(s,x,y)} F (by integral_integral twice)
-- Hence (a) = (b).
exact schwinger_fubini_core m f
/-- The kernel-level Schwinger representation holds... |
23,965 | lean_file_chunk | 09a025a0a2c5a7b27c5650219ef63c0c885c7091de26aefbefaf745e20627548 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import Mathlib.MeasureTheory.Integral.Gamma
import Mathlib.Analysis.SpecialFunctions... |
23,974 | lean_file_chunk | 7b6e963c31dac41314572fb6f50acf1a18c123eea128a51fe9b489bc2abb0d02 | lean_source_file | lean_pool_source_file_chunks | have h_maps : Set.MapsTo (fun r : ℝ => (s, r)) Set.univ (Set.Ioi 0 ×ˢ Set.univ) := by
intro r _; exact ⟨hs, Set.mem_univ _⟩
exact heatKernelPositionSpace_continuousOn.comp h_proj h_maps
exact (continuousOn_univ.mp h_cont_on)
exact (h_cont_r.comp h_norm).aestronglyMeasurable
have ... |
23,985 | lean_file_chunk | 4ea597e901836c5d15a26677ab3fde4a7fb46ea66445bb263f71e543dbcb8cae | lean_source_file | lean_pool_source_file_chunks | apply (EuclideanSpace.equiv (Fin (STDimension - 1)) ℝ).symm.continuous.comp
apply continuous_pi
intro i
have h : i.val + 1 < STDimension := by simp [STDimension]; omega
exact PiLp.continuous_apply 2 (fun _ : Fin STDimension => ℝ) (⟨i.val + 1, h⟩ : Fin STDimension)
/-- `spatialPart` is measurable. -/
lemma spat... |
23,969 | lean_file_chunk | 21a3f0829a7c30c6f56e8ae36497e476ddc181621a1a492913e6e7d19de74c71 | lean_source_file | lean_pool_source_file_chunks | _ = (1/6) * ∫ u in Set.Ioi 0, u^3 * Real.exp (-u^2 / (4 * s)) := by
conv_lhs => arg 2; ext u; rw [show Real.exp (-u^2 / (4 * s)) * (u^3 / 6) =
(1/6) * (u^3 * Real.exp (-u^2 / (4 * s))) by ring]
rw [MeasureTheory.integral_const_mul]
_ = (1/6) * (8 * s^2) := by rw [h_u_int]
-- Co... |
23,931 | lean_file_chunk | 1768148aa33cfa9ca12847c705ab6405d03a9e7288e1a558a70562c86eb018d8 | lean_source_file | lean_pool_source_file_chunks | exact (SchwartzMap.continuous f).comp h_sts_cont
-- Continuous implies strongly measurable
have h_joint_sm : MeasureTheory.StronglyMeasurable (fun p : ℝ × EuclideanSpace ℝ (Fin 3) =>
‖f (spacetimeOfTimeSpace p.1 p.2)‖) := h_joint_cont.stronglyMeasurable
-- Use StronglyMeasurable.integral_prod_right
... |
23,948 | lean_file_chunk | 85b2bffb4cd7db0e15bd71a35580e5f7821dc4da6a9b88f8db8e66dab6f59a6a | lean_source_file | lean_pool_source_file_chunks | -- (This is the atomic fact about Schwartz functions vanishing at t=0)
-- Step 2: Define the dominating function using the constant from
-- F_norm_bound_via_linear_vanishing
-- Get the constant C_bound from the linear vanishing bound
obtain ⟨C_bound, hC_pos, h_F_bound⟩ := F_norm_bound_via_linear_vanishi... |
15,023 | lean_file_chunk | 6ab07f315785206609a954a90acb9771b09c5c9c7b41d872a954a222a65c6cfb | lean_source_file | lean_pool_source_file_chunks | -- The integrand is Schwartz × const × exp(real) × exp(real) × exp(pure imaginary)
-- Each factor is continuous, hence the whole product is continuous
have hf_cont : Continuous f := SchwartzMap.continuous f
have h1 : Continuous (fun (p : SpatialCoords × SpaceTime × SpaceTime) => (starRingEnd ℂ) (f
p.2... |
23,988 | lean_file_chunk | dfdfbd18b8965f5e4618658d141b3938a9fe85aa223c4d8d35ffc61b3c0a394a | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import LeanPool.OSforGFF.Spacetime.Basic
import LeanPool.OSforGFF.Spacetime.Positive... |
20,848 | lean_file_chunk | 67f8604f1fcc9f7731c86698559ed39d1aa33d08273be77898f0e61bea8072b2 | lean_source_file | lean_pool_source_file_chunks | have h_int : Integrable (fun p : SpaceTime × SpaceTime =>
starRingEnd ℂ (f p.1) * (freeCovariance m p.1 p.2 : ℂ) * starRingEnd ℂ (g p.2))
(MeasureTheory.volume.prod MeasureTheory.volume) := by
-- (star f)(x) = conj(f(Θx)), so conj(f(x)) = (star f)(Θ⁻¹ x) = (star f)(Θx) since Θ² = id
-- Actually, let... |
23,939 | lean_file_chunk | cc8078b5f980a4bc8612ff6b93a8a847bba98862e76fb36199fd875533989304 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import LeanPool.OSforGFF.Spacetime.Basic
import LeanPool.OSforGFF.Schwinger.Defs
imp... |
13,475 | lean_file_chunk | 475c704af27c5beaa21780432a00abcc3870e07ebde560e234705f5fc94b4933 | lean_source_file | lean_pool_source_file_chunks | covarianceClusteringReal (gaussianFreeFieldFree m) := by
intro f g ε hε
exact schwartz_cross_covariance_decay_real m f g ε hε
/-! ## OS4 Polynomial Clustering for GFF
The GFF satisfies polynomial clustering with any decay rate α > 0, because
the underlying covariance has exponential decay (from the mass gap).
Th... |
23,999 | lean_file_chunk | 73f862db12dcc1bd362bff55d7d14c82c0ea4a8d66a360605ef1d75c15cfa64f | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import Mathlib.Analysis.Distribution.SchwartzSpace.Deriv
import Mathlib.Analysis.Dis... |
23,964 | lean_file_chunk | 5a4715ceb2d3b90e57b1d02e1201d1fe717d707d88d6d59e170e7a3f5b4de263 | lean_source_file | lean_pool_source_file_chunks | apply Integrable.const_mul
have h_transl : ∀ x y, freeCovariance m x y = freeCovarianceKernel m (x - y) := by
intro x y; simp only [freeCovariance, freeCovarianceBessel, freeCovarianceKernel, zero_sub,
norm_neg]
have h_eq2 : (fun p : SpaceTime × SpaceTime => ‖(freeCovariance m p.1 p.2 : ℂ)‖ * ‖g p... |
16,885 | lean_file_chunk | ec610daf6fdf036806e10c2d0b9a235fcc5e405135fe8201be8bb55ecdd35cd3 | lean_source_file | lean_pool_source_file_chunks | have h_DecayFn_cont : Continuous (Function.uncurry DecayFn) := h_decay_cont
have h1 : ∫ s in Set.Icc (0 : ℝ) T, ∫ u in Set.Icc (0 : ℝ) T, ‖Cov s u‖ ≤
∫ s in Set.Icc (0 : ℝ) T, ∫ u in Set.Icc (0 : ℝ) T, c * (1 + |s - u|)^(-(3 : ℝ)) := by
apply MeasureTheory.setIntegral_mono_on
· -- Integr... |
12,022 | lean_file_chunk | 3894feec9fe063224e866402aa1ecf50599a68aabf4ea9d78dc7e67a7f04062b | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import Mathlib.Analysis.Distribution.SchwartzSpace.Deriv
import Mathlib.Analysis.Dis... |
852 | lean_file_chunk | 4641e49c8d76450ca80fc51804a0fccb097f5812251ff3134b60b58052a8af56 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import LeanPool.OSforGFF.OS.Axioms
import LeanPool.OSforGFF.OS.Master
import LeanPool... |
11,978 | lean_file_chunk | 2d6f0bdc88a3159b2242991889f7a3c0dd81d7d4419bc26fd811d60871e420a2 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import LeanPool.OSforGFF.Spacetime.Basic
import LeanPool.OSforGFF.Spacetime.ComplexT... |
13,640 | lean_file_chunk | 1d8bc5bd41138e993ef50937282a9a0a53bb600fac83d18da492a94cc70e4c43 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.MeasureTheory.Integral... |
10,802 | lean_file_chunk | 573ed9297d5ae59f5a92ceeb36196e858976e0d118d83ff4a8cdb0fb4216b90f | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import LeanPool.OSforGFF.Spacetime.Basic
import LeanPool.OSforGFF.Schwinger.Defs
/-... |
465 | lean_file_chunk | b545626ec861b399264bbe1af81d60a121e134304a3188aad2584883fa3fba88 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import LeanPool.OSforGFF.Schwinger.Defs
import LeanPool.OSforGFF.Schwinger.GaussianMo... |
10,760 | lean_file_chunk | e586ca8edf5f807194ef6e1e39731c0f0c38d8de42ceff58923f11c98d9ff038 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
-- Import our functional analysis utilities
import LeanPool.OSforGFF.General.Functio... |
16,931 | lean_file_chunk | c40dde39e8ddf512214df6caacd4bd9689b06eae61ac9ceb3668d11c13609680 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import Mathlib.Data.Complex.Basic
import Mathlib.Analysis.Distribution.SchwartzSpace... |
6,622 | lean_file_chunk | 9960eb5002df13e941f7ed709891e327c09c0d2c7be6a8f4a3233e4c2dcabaa4 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import LeanPool.OSforGFF.Spacetime.Basic
import LeanPool.OSforGFF.Spacetime.ProdInte... |
7,603 | lean_file_chunk | 198eb182c902e3122232bf831785a3b61daf22b7374dd47b2fd5dd9e49fd4375 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Norm... |
13,772 | lean_file_chunk | 1a20685b9fdda0aebcf1b3c72b358eca0d142502453ae282d263b3be21688d9c | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import LeanPool.OSforGFF.Spacetime.Basic
/-!
# Euclidean Group E(4) and Its Actions... |
6,480 | lean_file_chunk | 3463e0cc499ad85f2b01104c86255985f3c10667db4ac03799163894fa274783 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import Mathlib.LinearAlgebra.Complex.FiniteDimensional
import LeanPool.OSforGFF.Space... |
23,977 | lean_file_chunk | fb6d99af3ab02b6437d0e8fdaec17aa6cfbd5e71e877cc9072ebd7363a96797b | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import Mathlib.Analysis.Distribution.SchwartzSpace.Deriv
import Mathlib.Analysis.Spec... |
5,685 | lean_file_chunk | a671b9e0506ebeafb1eb242da6d7dcd8194345dd360bfaf516b1c812a92d1b93 | lean_source_file | lean_pool_source_file_chunks | have h_eq : (fun s => spacetimeOfTimeSpace s x_sp) =
(fun s => spacetimeOfTimeSpace 0 x_sp + s • e₀) := funext h_path_eq
rw [h_eq]
-- Constant + linear is differentiable
exact (differentiable_const _).add (differentiable_id.smul_const e₀) |>.differentiableAt
-- Bound on the derivative: ‖de... |
23,955 | lean_file_chunk | dfdb9b02c6bd4140d6870f7eb4854322aa5828768ce770c70819b0d2b96bdbf5 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import LeanPool.OSforGFF.Spacetime.Basic
/-!
# Time Translation Operators
This fil... |
15,293 | lean_file_chunk | 41f37cd7ce5ec6c7db0b1f1eac61df68e7499287147a132569828590dadc3c5e | lean_source_file | lean_pool_source_file_chunks | -- MVT + derivative bound step using chain rule and currying
-- The key technical step is computing the derivative norm using fderiv_iteratedFDeriv
have h_mvt_bound : ‖g 1 - g 0‖ ≤ |h| * ⨆ t ∈ Set.Icc (0 : ℝ) 1,
‖iteratedFDeriv ℝ (n + 1) f (x + t • y)‖ :=
schwartz_timeTranslation_mvt_bound n f h x y hy
... |
15,479 | lean_file_chunk | 0aa5765f87d06a7bf395d29eaca027ab426d10018a9fc4b4eadfe6efd26b3027 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import LeanPool.OSforGFF.Spacetime.Basic
import LeanPool.OSforGFF.Spacetime.Decompos... |
776 | lean_file_chunk | 5b5f0559f64721f62bea2a6de4ecb42b1a3daead352492191da61aa3c924ced9 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import LeanPool.OSforGFF.Spacetime.Basic
import LeanPool.OSforGFF.Spacetime.ComplexTe... |
947 | lean_file_chunk | f1852f8956f71fac56ee8a1ad0fd8dbdb6a01d96b7b6bcfe0d7615cbcd452e5d | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael R. Douglas, Sarah Hoback, Anna Mei, Ron Nissim
-/
import LeanPool.OSforGFF.Bochner
import LeanPool.OSforGFF.Covariance
import LeanPool.... |
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