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 |
|---|---|---|---|---|---|
16,324 | lean_file_chunk | 4def703a6ecc31e997192c40dabeb24bdd29794372348da162693b7b99393ab5 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2023 Monica Omar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Monica Omar
-/
import LeanPool.Monlib4.QuantumGraph.Basic
import LeanPool.Monlib4.QuantumGraph.Example
/-!
# Isomorphisms between quantum graphs
This file defines isomorphisms between... |
23,937 | lean_file_chunk | eb2a0f8f2e5e0688b6ee50303e6afed175b2dc8ac4a2d955b9f8566bded93e86 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2023 Monica Omar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Monica Omar
-/
import LeanPool.Monlib4.QuantumGraph.PiMatFinTwo
/-!
# LeanPool.Monlib4.QuantumGraph.Matrix
Imported Lean Pool material for `LeanPool.Monlib4.QuantumGraph.Matrix`.
-/
op... |
18,558 | lean_file_chunk | 9a2f3b7dbf89f616b84640c5ba18b1f6c6f9b729432a2ec34564f9be3f64ad5a | lean_source_file | lean_pool_source_file_chunks | (Coalgebra.counit (R := ℂ) (A := (Matrix n n ℂ)ᵐᵒᵖ))) ((QuantumSet.Psi 0 (1 / 2)) A)
rw [Coalgebra.counit_mulOpposite, hc]
theorem Matrix.traceLinearMap_dualMatrix_eq
{n : Type*} [DecidableEq n] [Fintype n] :
Module.Dual.matrix (Matrix.traceLinearMap n ℂ ℂ) = 1 := by
refine Eq.symm (Module.Dual.apply_eq_of _... |
8,370 | lean_file_chunk | 5e7708fbae15ff85163a8c1cb9741754071b7f3c77250d99a91972dac8cd473e | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2023 Monica Omar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Monica Omar
-/
import LeanPool.Monlib4.LinearAlgebra.Ips.MatIps
import LeanPool.Monlib4.LinearAlgebra.Ips.Nontracial
import LeanPool.Monlib4.LinearAlgebra.Ips.TensorHilbert
import LeanPoo... |
4,629 | lean_file_chunk | 2855d9ec39c7fdc1241c31da58fa39c3200820b2a4348290be52c1031a67b8d1 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2023 Monica Omar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Monica Omar
-/
import Mathlib.Combinatorics.SimpleGraph.AdjMatrix
import LeanPool.Monlib4.QuantumGraph.Basic
import LeanPool.Monlib4.QuantumGraph.Example
import LeanPool.Monlib4.QuantumGr... |
23,959 | lean_file_chunk | fb83c47007388dc03d1863bd1bdac65b9b90c562cc2f05ec76e38bca058371d5 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2023 Monica Omar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Monica Omar
-/
import LeanPool.Monlib4.QuantumGraph.Basic
import LeanPool.Monlib4.QuantumGraph.Example
import LeanPool.Monlib4.LinearAlgebra.TensorProduct.Submodule
import LeanPool.Monlib... |
23,970 | lean_file_chunk | 271868e85e6e831b6008b948f3b387c13ae0d38f4cade045331f3c56f7d31926 | lean_source_file | lean_pool_source_file_chunks | simp only [LinearMap.isOrthogonalProjection_iff, IsIdempotentElem,
← Pi.mul_apply _ _ i, ← map_mul,
IsSelfAdjoint, ← Pi.star_apply _ i, ← map_star]
simp only [(quantumGraphReal_iff_Psi_isIdempotentElem_and_isSelfAdjoint.mp hA).1.eq,
(quantumGraphReal_iff_Psi_isIdempotentElem_and_isSelfAdjoint.mp hA).2.sta... |
23,939 | lean_file_chunk | c446c7cfc4b60859ded70b7c18c4b69967f56a9ea4b4e01271e181d406de24e1 | lean_source_file | lean_pool_source_file_chunks | rw [← Matrix.unitaryGroup.star_coe_eq_coe_star (U i)]
have hstarU :
star (U i : Matrix (p i) (p i) ℂ) = (U i : Matrix (p i) (p i) ℂ)ᴴ :=
rfl
rw [hstarU, Matrix.conjTranspose_conjTranspose]
change
(((φ i).matrix *
((U i : Matrix (p i) (p i) ℂ) *
((y i)ᴴ * (U i : Matrix (p i) (p ... |
13,083 | lean_file_chunk | dd9bb8dabe96b7ca4be95da24fb84ed2c7d785a7a54b7366eb572c1fe8b8fd0b | lean_source_file | lean_pool_source_file_chunks | PiMat ℂ ι p →ₗ[ℂ] Mat ℂ (p i.2)) = 0 := by
withPiQuantumCtx[φ]
let : ∀ i, _root_.starAlgebra (Matrix (p i) (p i) ℂ) :=
fun i => Matrix.isStarAlgebra (φ := φ i)
let : ∀ i, _root_.QuantumSet (Matrix (p i) (p i) ℂ) :=
fun i => Module.Dual.IsFaithfulPosMap.quantumSet (φ := φ i)
let : ∀ i, _root_.NormedAddCo... |
23,988 | lean_file_chunk | d81214640c896e5ed433db4f9858afebdb20a372fea91f428a530a34012a0be7 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2023 Monica Omar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Monica Omar
-/
import LeanPool.Monlib4.QuantumGraph.PiMat
import LeanPool.Monlib4.LinearAlgebra.Ips.Nontracial
/-!
# LeanPool.Monlib4.QuantumGraph.PiMatFinTwo
Imported Lean Pool materia... |
18,975 | lean_file_chunk | f1dfa12392c2325cbb16421fc667cb1749ed015b685e98ec27ea8ae93f315ba3 | lean_source_file | lean_pool_source_file_chunks | [(i : ι) → Module R (φ i)] (i : ι) :
φ i →ₙₐ[R] (Π i, φ i) where
toFun x := LinearMap.single R φ i x
map_add' _ _ := by simp [Pi.single_add]
map_smul' _ _ := by simp []
map_mul' _ _ := by simp [Pi.single_mul]
map_zero' := by simp [Pi.single_zero]
theorem NonUnitalAlgHom.single_apply
{R ι : Type*} [Comm... |
23,965 | lean_file_chunk | 63951fdff203ed7e871c3297bee61335a307654fdbd480d7e8b32597ff95e4ab | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2023 Monica Omar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Monica Omar
-/
import LeanPool.Monlib4.QuantumGraph.ToProjections
/-!
# Single-edged quantum graphs
This file defines the single-edged quantum graph, and proves that it is a `QAM`.
-/... |
23,979 | lean_file_chunk | 817197dce8224c121606a5f53e4d54386c06af3a4dfbeab2bd8923d18bfc2cbf | lean_source_file | lean_pool_source_file_chunks | Complex.ofReal_eq_zero, Real.sqrt_eq_zero, NNReal.zero_le_coe, NNReal.coe_eq_zero,
Units.ne_zero, not_false_iff, false_and]
have : γ ^ 2 = (β : ℂ) * (((α : NNReal) : ℝ).sqrt : ℂ) := by
simp_rw [γ, Complex.cpow_nat_inv_pow _ two_ne_zero]
have this1 : ‖γ‖ ^ 2 = 1 := by
rw [show ‖γ‖ ^ 2 = ‖γ ^ 2‖ by rw [... |
396 | lean_file_chunk | d471236aec67b70b4e50e8a244eef73263d6b778c27ef7908a40807d18ed8ace | lean_source_file | lean_pool_source_file_chunks | rw [qamA.isometric_starAlgEquiv_conj _ hUUiso, qamA.is_almost_injective,
Subtype.coe_mk]
simpa [_root_.map_smul, AlgEquiv.toLinearMap_apply, StarAlgEquiv.coe_toAlgEquiv] using hU
use U
constructor
· rw [hU']
ext z
simp only [LinearMap.comp_apply, AlgEquiv.toLinearMap_apply,
... |
16,280 | lean_file_chunk | 8c77c51b2b043097eaebe0c80916379c64f76355ab7ec1aa60eafae9ce830b58 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2023 Monica Omar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Monica Omar
-/
import LeanPool.Monlib4.QuantumGraph.Nontracial
import LeanPool.Monlib4.QuantumGraph.Iso
import LeanPool.Monlib4.LinearAlgebra.ToMatrixOfEquiv
import LeanPool.Monlib4.Linea... |
23,945 | lean_file_chunk | 67b0d205d1da02c1b01d3995b67dd7d506629c0b68b19481a4720da3867ee40e | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2023 Monica Omar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Monica Omar
-/
import LeanPool.Monlib4.QuantumGraph.Nontracial
import LeanPool.Monlib4.LinearAlgebra.Ips.MinimalProj
import LeanPool.Monlib4.QuantumGraph.Iso
import LeanPool.Monlib4.Quant... |
13,554 | lean_file_chunk | 12873b4f35c856e35f7497da2f8a76fb105486a087e3b0c64b6697b0e9d923c8 | lean_source_file | lean_pool_source_file_chunks | simp_rw [← rankOne_toMatrix_transpose_psi_symm, ← map_sum, ←
Qam.fdOrthogonalProjection_eq_sum_rankOne (Qam.onbOfIdempotentAndReal hA1 hA2),
Qam.orthogonalProjection'_eq, AlgEquiv.apply_symm_apply]
simp_rw [← oneMapTranspose_symm_eq, ← oneMapTranspose_eq, StarAlgEquiv.symm_apply_apply,
LinearEquiv.symm_ap... |
848 | lean_file_chunk | dffc61dc5c4ffb56cd2f15fc0fe52d0861890ae4f884ebdd89125e46d8a2b10b | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2023 Monica Omar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Monica Omar
-/
import LeanPool.Monlib4.QuantumGraph.Basic
import LeanPool.Monlib4.QuantumGraph.Degree
import LeanPool.Monlib4.QuantumGraph.Example
import LeanPool.Monlib4.QuantumGraph.Gra... |
23,951 | lean_file_chunk | 3401256f18cc463b9afdaae4f552a1962decb4e49b536c83d520c8746d1a50f6 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Monica Omar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Monica Omar
-/
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.ToLinearEquiv
import Mathlib.Linear... |
18,804 | lean_file_chunk | d795a8a5049fc08850cf37bffdb5026308791a0f75b1c2d2aa08a70e7aeb9350 | lean_source_file | lean_pool_source_file_chunks | rw [← not_nonempty_iff, ← @one_ne_zero_iff 𝕜 n, not_ne_iff]
theorem AlgEquiv.matrix_prod_aut {𝕜 n m : Type*} [Field 𝕜] [Fintype n]
[Fintype m] [DecidableEq n] [DecidableEq m]
(f : (Mat 𝕜 n × Mat 𝕜 m) ≃ₐ[𝕜] (Mat 𝕜 n × Mat 𝕜 m)) :
(f (1, 0) = (1, 0) ∧ f (0, 1) = (0, 1)) ∨
(f (1, 0) = (0, 1) ∧ f... |
281 | lean_file_chunk | 94238baea0a7235cbcc28b4bb3462ca03034693b2d5e50ee901ed6d711867655 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Monica Omar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Monica Omar
-/
import LeanPool.Monlib4.RepTheory.AutMat
/-!
# RepTheory
Import-only index for the `RepTheory` directory of the monlib4 import.
-/ |
3,275 | lean_file_chunk | 1d992483f6d27d35036be07231b4f86c367bfab6b6f77f7322e131fea793d5b5 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Monica Omar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Monica Omar
-/
import LeanPool.Monlib4.Monlib
/-!
# Monlib4 Operator-Algebra and Quantum-Set Core
Source: arxiv:1810.08368, doi:10.4169/amer.math.monthly.124.10.966
Authors: Monica Oma... |
23,964 | lean_file_chunk | f9043ca0a5ddebe12435622a38562fb0b0be815585dbad26738baf7577c6b066 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Dhyan Aranha and contributors. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dhyan Aranha, contributors
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Tactic.Common
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.NormNum
i... |
1,567 | lean_file_chunk | 6a1136f61813e5743f347117819a6cbb32a9efaf94f0e7633ab12389dd1300d5 | lean_source_file | lean_pool_source_file_chunks | lemma odd_valuation (Γ₀ : Type) (_ : LinearOrderedCommGroupWithZero Γ₀) (v : Valuation ℝ Γ₀)
(vhalf : v (1 / 2) > 1) : ∀ n : ℕ, Odd n → v (1/n) = 1 := by
have vhalf' : v (2) < 1 := by
rw [Valuation.map_div, Valuation.map_one] at vhalf
refine (Valuation.val_lt_one_iff v ?_).mpr ?_
· norm_num
· simp_all... |
3,727 | lean_file_chunk | 276c27a6eb5c8073779c188c0bfd7a611a091837a87ba28b64b2659336452597 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Dhyan Aranha and contributors. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dhyan Aranha, contributors
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Tactic.Common
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.NormNum
i... |
3,768 | lean_file_chunk | d48f78a44e39d65edb8e2a53a1e09292fd58eb41379585b6f8d76e137a936c59 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Dhyan Aranha and contributors. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dhyan Aranha, contributors
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Tactic.Common
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.NormNum
i... |
6,510 | lean_file_chunk | 2cdb1e8119cb75a10399682d3cd33fc6ff19ff008161096d0c370f74db0610b4 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Dhyan Aranha and contributors. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dhyan Aranha, contributors
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Data.Real.Sign
import Mathlib.Tactic.Common
import Mathlib.Tactic.Linarith
i... |
18,315 | lean_file_chunk | 43c329c05a8a386b3ccae81e7607305b1a229ed5b788d3dc917ba5c5ec9ce326 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Dhyan Aranha and contributors. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dhyan Aranha, contributors
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Tactic.Common
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.NormNum
i... |
16,601 | lean_file_chunk | eb0ff12b7c3c793959585bc4f414968a03ae20eab53248974781c8c528935d5c | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Dhyan Aranha and contributors. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dhyan Aranha, contributors
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Tactic.Common
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.NormNum
i... |
23,990 | lean_file_chunk | 9d307a455cba3a92d47813d200aab1aa91f6dcb521785b1ff033cb37f1268890 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Dhyan Aranha and contributors. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dhyan Aranha, contributors
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Tactic.Common
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.NormNum
i... |
23,976 | lean_file_chunk | d84b9c0d0ac5e6c28a8d276f8bbc6d1b568155295de2115ff67ef3498a57684b | lean_source_file | lean_pool_source_file_chunks | fin_cases i <;> simp
-- The union of the interiors of the triangles of a triangulation
/-- The union of triangle interiors that basic segments of the triangulation must avoid. -/
noncomputable def triangulationAvoidingSet (Δ : Finset Triangle) : Set ℝ² :=
⋃ (T ∈ Δ), openHull T
/-- The basic avoiding segments of ... |
23,975 | lean_file_chunk | b0eab7f84a7c8d410a31ef3478c30457bc6287aeef68aca80a1be66d678cb1c8 | lean_source_file | lean_pool_source_file_chunks | · unfold basicAvoidingSegmentSet avoidingSegmentSet segmentSet at hS
simp_all only [ne_eq, product_eq_sprod, mem_filter, mem_image, mem_product, Prod.exists,
Fin.isValue]
obtain ⟨w, h⟩ := hS.1.1.1
obtain ⟨w_1, h⟩ := h
obtain ⟨left, right_3⟩ := h
obtain ⟨left... |
3,603 | lean_file_chunk | c40fefb197ed12311e424d208a0be19279d3791f1bb0954f45bc4b3c9d2971fd | lean_source_file | lean_pool_source_file_chunks | (hDisjointCover : isDisjointCover (closedHull unitSquare) (↑Δ : Set Triangle))
(f : Segment → ℕ) (non_degen : ∀ P ∈ Δ, det P ≠ 0)
: ∑ T ∈ Δ, ∑ (S ∈ triangleBasicBoundary Δ T), f S =
∑ (S ∈ triangulationBoundaryBasicSegments Δ), f S +
2 * ∑ (S ∈ triangulationInteriorBasicSegments Δ), f S := by
simp_rw [tr... |
23,973 | lean_file_chunk | d277272cfb6e77cd7564b3866aeebf5fee38bb2857c36d70877825c0c41be20d | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Dhyan Aranha and contributors. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dhyan Aranha, contributors
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Tactic.Common
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.NormNum
i... |
23,964 | lean_file_chunk | b3fadeaf89b5a9f1dbae8b4d8b0fb55840ff60187743f5eda528a0e5db165453 | lean_source_file | lean_pool_source_file_chunks | lemma el_in_boundary_imp_side {T : Triangle} {x : ℝ²} (hdet : det T ≠ 0)
(hx : x ∈ boundary T) (hv : ∀ i, x ≠ T i) : ∃ i, x ∈ openHull (Tside T i) := by
have hxClosed := (Set.mem_of_mem_sdiff hx)
have ⟨i,hi⟩ := (boundary_iff hdet hxClosed).1 hx
use i
rw [←mem_open_side hdet hxClosed]
refine ⟨hi,?_⟩
intr... |
23,937 | lean_file_chunk | 9073ed135418fbda90d7dd8cdeca6b5dbd4a4cde25376ec56bf41aa74a1a365d | lean_source_file | lean_pool_source_file_chunks | have hzuv : z ∉ closedHull (toSegment u v) := by
by_contra hcontra
have hmid : closedHull (toSegment u v) ∩ (closedHull (toSegment v w) \ {v}) = ∅ :=
middle_intersection_empty (h := h)
have hzmid: z ∈ closedHull (toSegment u v) ∩ (closedHull (toSegment v w) \ {v}) := by
tauto_set
... |
6,449 | lean_file_chunk | c2cfdac7ab9bc703beed8100ceb7fa38f120ad15ca4184b150092805de7c2c1d | lean_source_file | lean_pool_source_file_chunks | apply real_number_bound_aux hx (g := fun i ↦ det₂ (Oside T i) y / det T)
intro ε hε
have ⟨l, hl⟩ := habsurd ε hε
use l
rw [Tco_line] at hl
rw [mul_div]
linarith
lemma triangle_direction_sub {T : Triangle} {x : ℝ²} (hx : x ∈ closedHull T)
(hn : ∀ i, x ≠ T i) :
∃ L : Segment, L 0 ≠ L 1 ∧ x ∈ openHul... |
11,645 | lean_file_chunk | 9e93e6689dc6f77b18979625229919cb190ac3451d368e1c3b5eb48619db59fd | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Dhyan Aranha and contributors. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dhyan Aranha, contributors
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Tactic.Common
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.NormNum
i... |
23,956 | lean_file_chunk | 39774a6587da51f94397aeedfb303e021de5c7e49cd906ab7cca6d20abf00953 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Dhyan Aranha and contributors. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dhyan Aranha, contributors
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Tactic.Common
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.NormNum
i... |
6,221 | lean_file_chunk | ca9f4fa8dc9ecda23ed3cace1e24467f07c7ac09af470638ffc10581d4166ed2 | lean_source_file | lean_pool_source_file_chunks | closedHull S ⊆ closedHull (squareBoundaryBig i) := by
apply closedHull_convex
intro j
have hS := fun k ↦ hS (corner_in_closedHull (P := S) (i := k))
have hS₀ := hS 0; have hS₁ := hS 1;
rw [squareBoundaryBig_eq, closed_unitSquare_eq] at *
have ⟨α, hα, hαx⟩ := hx
simp_rw [←hαx, Fin.sum_univ_two] at hxi
ha... |
23,960 | lean_file_chunk | dac4eb1ad9c8b48e140b929d9acacb873f4cd646a93b5d593fa07133d8538e35 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Dhyan Aranha and contributors. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dhyan Aranha, contributors
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Tactic.Common
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.NormNum
... |
11,416 | lean_file_chunk | 75e8944be4cead72cb94bd1339de9b52ea8fea2bf4fb8ebf91c74739e09dfc0e | lean_source_file | lean_pool_source_file_chunks | -- This is the inverse of the original triangle translation map, and the proof that are necessary to
-- work with it
/-- The inverse of `triangleTranslation`, translating by the negated first vertex. -/
def invTriangleTranslation (T : Triangle) := translation ( - T 0)
lemma translation_bijective (a : ℝ²) : Function.Bi... |
1,252 | lean_file_chunk | 7df649f24b585480b5f4259d1b074d685aac4f2da0eb2c430ebb9ce43de8211a | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Dhyan Aranha and contributors. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dhyan Aranha, contributors
-/
import LeanPool.Monsky.Appendix
import LeanPool.Monsky.RainbowTriangles
import LeanPool.Monsky.TriangleCorollary
import LeanPool.Monsky.Ba... |
15,486 | lean_file_chunk | 2818e2261fe787ea3c9eadae0c1a52398cced284fa89a7f8c1f5faf5944208b4 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Egor Lyfar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Egor Lyfar
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex
import Mathlib.NumberTheory.Real.Irrational
import Mathlib.Tactic.Fin... |
15,927 | lean_file_chunk | 3cea1c263f784d4cf047be8bae38fe6ff889a2951365267946105339f730da6b | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Egor Lyfar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Egor Lyfar
-/
import LeanPool.MoserLatticeColorings.Basic
/-!
# Extension to the Moser ring
Dúcz observes that both geometric four-colorings of the Moser lattice extend
to its multiplic... |
511 | lean_file_chunk | cdf62894e5f5b81f346cfdeb646f022aeeb876b56be86ea87fe290b313e70761 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2026 Egor Lyfar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Egor Lyfar
-/
import LeanPool.MoserLatticeColorings.Ring
/-!
# Geometric Four-Colorings of the Moser Lattice and Ring
Source: arxiv:2606.12325
Authors: Egor Lyfar
Status: verified
Main ... |
2,853 | lean_file_chunk | a4bea9edba8e06bdab92648e6c37929c366fb342320a63e95995e312987f35ba | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2023 Hu Yongle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hu Yongle
-/
import Mathlib.Tactic.Common
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Ring
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.GCong... |
23,935 | lean_file_chunk | eaf5fcda6e24c59fb8a2a47e43fe1044e32c31730c619268896f86739f8b805e | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2023 Hu Yongle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hu Yongle
-/
import Mathlib.Tactic.Common
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Ring
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.GCong... |
23,998 | lean_file_chunk | 302ffa0d01d6f723e9bf764c73ac22778091074146687d0bf93c3cfd84ed327d | lean_source_file | lean_pool_source_file_chunks | theorem GalRingHom_inv_mul_cancel (σ : L ≃ₐ[K] L) : (GalRingHom σ⁻¹).comp (GalRingHom σ)
= RingHom.id (𝓞 L) := by rw [GalRingHom_mul, inv_mul_cancel, GalRingHom_one]
theorem GalRingHom_inv_mul_cancel_mem (σ : L ≃ₐ[K] L) (x : 𝓞 L) :
GalRingHom σ⁻¹ (GalRingHom σ x) = x := by
rw [← RingHom.comp_apply, GalRingHo... |
15,973 | lean_file_chunk | 81005a95159766d33753233fccecb36aea43a5e951dc7fbfefc1eeea2fde7562 | lean_source_file | lean_pool_source_file_chunks | let ϕp : (𝓞 K) →+* F := Ideal.Quotient.mk p
let ϕP : (𝓞 L) →+* E := Ideal.Quotient.mk P
have h : Quotient.mk (Submodule.quotientRel P) a = ϕP a := rfl
rw [h] at ha
have hai : IsIntegral (𝓞 K) a := IsIntegral.isIntegral a
have hm : f.Monic := minpoly.monic hai
have h0 : (fl.map ϕP) ≠ 0 := map_monic_ne_zer... |
2,131 | lean_file_chunk | 135f950532f58dd35e0356dc8eb6dbabbb6435f3b174c4f408a7e44e372bd141 | lean_source_file | lean_pool_source_file_chunks | /-
Copyright (c) 2023 Hu Yongle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hu Yongle
-/
import LeanPool.Neukirch.ExtensionOfDedekindDomains
import LeanPool.Neukirch.HilbertRamificationTheory
/-!
# Neukirch's Algebraic Number Theory: Hilbert ramification theory
... |
23,976 | lean_file_chunk | ca5f9623c40f32cf8b77ad25119f22c0f52da6c635e11cccb9d8a45891674c89 | 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.PositiveDefinite
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mat... |
6,081 | lean_file_chunk | cae168d13aedabd8813f30d096cd1aee0f4c5885d10c565ec397132ca993a3d8 | lean_source_file | lean_pool_source_file_chunks | congr 1; funext x; exact (integral_indicator measurableSet_closedBall).symm
· congr 1; funext x
by_cases hx : x ∈ B
· rw [Set.indicator_of_mem hx]; congr 1; funext v
rw [← indicator_closedBall_inter ψ R x v, Set.indicator_of_mem hx]
· simp_all
-- Suffices: prove the unscaled Fubini ident... |
23,982 | lean_file_chunk | 18e819cc8508adf184184cfdb7d15029b7bc1b59d3dd51af6538acd982fd6070 | 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.PositiveDefinite
import LeanPool.OSforGFF.Bochner.FejerPD
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.A... |
23,958 | lean_file_chunk | b19d95de26ac06b834361c0270acc5a431de9e9383909f5fbe726ae8dadd6a9c | lean_source_file | lean_pool_source_file_chunks | [FiniteDimensional ℝ V] [MeasurableSpace V] [BorelSpace V]
(φ : V → ℂ) (hpd : IsPositiveDefinite φ) (hint : Integrable φ) (hcont : Continuous φ)
(ξ : V) : 0 ≤ (𝓕 φ ξ).re :=
pd_l1_fourier_re_nonneg_theorem φ hpd hint hcont ξ
/-- The real part of the Fourier transform of an L¹ PD function is nonneg.
Ref: ... |
15,425 | lean_file_chunk | 0dd1c4c2791dc20e3fbf0b1c882a31dd666280eb327ab248f0b0b1d779cec9bf | lean_source_file | lean_pool_source_file_chunks | exact one_sub_exp_neg_le harg_nn
_ ≤ ‖1 - φ v‖ + 1 * ‖v‖ ^ 2 := by gcongr
_ = ‖1 - φ v‖ + ‖v‖ ^ 2 := by rw [one_mul]
omit [FiniteDimensional ℝ V] in
/-- Key bound: for a probability measure μ with charFun μ = gaussianRegularize φ ε (ε ≤ 1),
and for any y and r > 0:
μ.real {x | r < |⟪y,x⟫|} ≤ 2 * C + 8 ... |
8,929 | lean_file_chunk | 686d5582d52d7e8fc334168e7684bc097c5dc29d8849e0cec749d3958e88af04 | 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 Mathlib.Data.Complex.Basic
import Mathlib.Analysis.Complex.Order
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
import Mathlib.Data.Fintype.B... |
7,858 | lean_file_chunk | e556a32cb7f07952c43767e8de4ecf2c442525abfd50207afa2d32ac11586280 | 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 Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.InnerProductSpace.Adjoint
import Mathlib.Analysis.InnerProductSpace.Positive
import M... |
424 | lean_file_chunk | 11f0d7f99dd9bc36f3a89c12040c8847ad525dd84747a2d0ca5bbd72668cdbe6 | 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.FejerPD
import LeanPool.OSforGFF.Bochner.Main
import LeanPool.OSforGFF.Bochner.PositiveDefinite
import LeanPool.OSforGFF.Boc... |
23,983 | lean_file_chunk | 15444666a0e7bb3ca302f33cb1757082d22b93a416c254f046056b27660580df | 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.SpecialFunc... |
23,964 | lean_file_chunk | c7a5a7cad0c177a8febdc21c69979cd40c27f65aac0fe2a68a13f058d70486c8 | lean_source_file | lean_pool_source_file_chunks | refine MeasureTheory.ae_restrict_of_forall_mem measurableSet_Ioi fun s hs => ?_
rw [Real.norm_eq_abs, Real.norm_eq_abs,
abs_of_nonneg (mul_nonneg (Real.exp_nonneg _) (heatKernelPositionSpace_nonneg s hs r)),
abs_of_nonneg (mul_nonneg (Real.exp_nonneg _) hCpos.le)]
exact mul_le_mul_of_nonneg_left (hCbound s ... |
23,957 | lean_file_chunk | 509a759e86132a4546e74495c855cc8fc50c5656bfa3e8d8decd749d5d79f229 | lean_source_file | lean_pool_source_file_chunks | ∫ k : SpaceTime, ↑(1 / normalisation : ℝ) *
(↑(∫ t in Set.Ioi 0, Real.exp (-(α + t) * ‖k‖^2) * Real.exp (-t * m^2)) *
Complex.exp (-Complex.I * ⟪k, x - y⟫_ℝ)) from
(MeasureTheory.integral_const_mul _ _).symm]
congr 1 with k
simp only [Complex.ofReal_div, Complex.ofReal_one]
ring
... |
23,951 | lean_file_chunk | c19b6480075c46202253aa609a78ae12b97086e45f123514a391a0e8fa0c70bb | lean_source_file | lean_pool_source_file_chunks | apply Filter.Eventually.of_forall
intro k
simp only [F, bound]
rw [Complex.norm_mul, Complex.norm_real]
rw [norm_exp_neg_I_mul_real, mul_one]
have h_prop_bound : freePropagatorMomentum m k ≤ 1 / m^2 := by
simp only [freePropagatorMomentum]
apply one_div_le_one_div_of_le (by... |
9,731 | lean_file_chunk | 9ff0c44214880d6068d0c182f1ba3b72e7a8d083f1a306b94841867e1da59d2a | lean_source_file | lean_pool_source_file_chunks | theorem integral_comp_neg_spacetime {E : Type*} [NormedAddCommGroup E] [NormedSpace ℂ E]
(f : SpaceTime → E) : ∫ k, f (-k) = ∫ k, f k := by
have h := (LinearIsometryEquiv.measurePreserving negSpaceTime).integral_comp
negSpaceTime.toHomeomorph.measurableEmbedding f
exact h
/-- Position-space free covariance... |
23,984 | lean_file_chunk | ec8cc33b143f47d4f4defb4188217f4177983db48514c2b5ca9f2fa0810358fe | 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.FourierTransforms
import LeanPool.OSforGFF.Covarianc... |
23,992 | lean_file_chunk | 4319811ae7cf26cf901f874dc31409e268cb8b7bb14bcef4781800c68a892832 | lean_source_file | lean_pool_source_file_chunks | _ = (∫ x, amplitude k * f x * Complex.exp (-Complex.I * Complex.ofReal ⟪k, x⟫_ℝ)) *
(∫ y, starRingEnd ℂ (f y) * Complex.exp (Complex.I * Complex.ofReal ⟪k, y⟫_ℝ)) := by
have h_inner : ∀ x, ∫ y, amplitude k * f x * Complex.exp (-Complex.I * Complex.ofReal ⟪k,
x⟫_ℝ) *
(star... |
708 | lean_file_chunk | 23ad97c601c3986bd80b34e50fb0d17f5665edb988ae2a39180e69dbae6b3e89 | lean_source_file | lean_pool_source_file_chunks | bilinear_covariance_regulated_tendstoℂ m f f
end ParsevalCovariance
/-! ## Global Definitions
The following definitions are placed outside the section to ensure they are
accessible globally without namespace qualification.
-/
section GlobalBilinearDefs
open MeasureTheory Complex Real
open scoped InnerProductSpace
... |
23,982 | lean_file_chunk | 5d1af0b40f3624857a7cc5dbce14f9810c1a907943ea63af6577ba9caa41cf6e | 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.SpecialFunc... |
2,824 | lean_file_chunk | 342ebf80024f0a5fdd62d6fa12da8d8b30f7d08d48e3fc2a74336750a47e048e | lean_source_file | lean_pool_source_file_chunks | rw [parseval_covariance_schwartz_regulated' α hα m f]
refine MeasureTheory.integral_nonneg (fun k => ?_)
exact mul_nonneg (mul_nonneg (Real.exp_nonneg _) (sq_nonneg _))
(freePropagatorMomentum_mathlib_nonneg m (Fact.out) k)
/-- Complex extension of the covariance for complex test functions (limit form via Bess... |
23,971 | lean_file_chunk | d0a885bee19a53f73cc9f2fe9ac9bfd533f797231ea02d031665466177a81f6e | 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.SpecialFunc... |
9,269 | lean_file_chunk | b9b554bae7d0ff6e1560ba159afd55b8ff8c9dfae84c643e6a0ff93d9b406d4c | lean_source_file | lean_pool_source_file_chunks | = freeCovarianceℂBilinear m (toComplex f₁ + toComplex f₂) (toComplex g) := hL
_ = freeCovarianceℂBilinear m (toComplex f₁) (toComplex g)
+ freeCovarianceℂBilinear m (toComplex f₂) (toComplex g) := h
_ = (freeCovarianceFormR m f₁ g : ℂ) + (freeCovarianceFormR m f₂ g : ℂ) := by
rw [fre... |
512 | lean_file_chunk | 50088f06ad63e8307f96ce2a64651318afec7922229e58665793b6f1ab118fa2 | 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.Covariance.Momentum
import LeanPool.OSforGFF.Covariance.Pars... |
4,100 | lean_file_chunk | 047d504d8f01e4002258d82976bab70fbcaf603fea6215f1362e5dee6dafd6a5 | 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 Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.InnerProductSpace.LinearMap
import Mathlib.Analysis.LocallyConvex.WithSeminorms
impor... |
23,479 | lean_file_chunk | 0dc7425ceb579020b04b2ed9b88b06f6272802c815a05a049b52c3a9a2172f60 | 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.GaussianField.Nuclear.DyninMityagin
import Mathlib.Analysis.Convex.Cone.Extension
import Mathlib.Analysis.Normed.Group.InfiniteSum
i... |
23,962 | lean_file_chunk | 4a8b2732459efd2bd39582f992b6294c25690b5f48783e1a83824f5cdea7606b | 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.GaussianField.Nuclear.DyninMityagin
import LeanPool.OSforGFF.GaussianField.Nuclear.NuclearSpace
import Mathlib.Analysis.LocallyConve... |
23,937 | lean_file_chunk | 3e95bf8bd1b8c60a37a39e4b8fd40ab3fd5b1d0b95720bff5f8314ece17ab077 | lean_source_file | lean_pool_source_file_chunks | rapidDecaySeminorm_basisVec _ m
end RapidDecaySeq
/-! ### Transfer constructor for DyninMityaginSpace -/
/-- Transfer a `DyninMityaginSpace` structure from `RapidDecaySeq` to any space
that is continuously linearly equivalent to it. Given seminorms `p` with
`WithSeminorms p` and a CLE `equiv : E ≃L[ℝ] RapidDecaySeq`... |
22,351 | lean_file_chunk | e872fdd9c824b6cab3eab52f06b524947b2a9f35c00885583344eb16fb59c708 | lean_source_file | lean_pool_source_file_chunks | simp only [Set.mem_iInter, Set.mem_ofPred_eq] at hx ⊢
rcases Seminorm.zero_or_exists_apply_eq_finset_sup q t x with h | ⟨i, hi, heq⟩
· linarith
· linarith [hx i hi]
/-- The pure tensor map is jointly continuous on `E₁ × E₂`. -/
theorem pure_continuous :
Continuous (fun p : E₁ × E₂ => pure p.1 p.2) := by
--... |
471 | lean_file_chunk | b00513a150c97d2983350559b1ff2e5b4c900a6f0b4430f29e3418d7e1aa67ac | 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.GaussianField.Nuclear.DyninMityagin
import LeanPool.OSforGFF.GaussianField.Nuclear.NuclearSpace
import LeanPool.OSforGFF.GaussianFie... |
5,144 | lean_file_chunk | 49283f3273f8d062f51ee954b0b2c3868081982011f833b60ce74d70bbcbcf2a | 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.GaussianField.SchwartzNuclear.SchwartzHermiteExpansion
/-!
# Nuclear Schauder Basis for S(ℝ, ℝ) — 1D Case
Assembles the proved Her... |
23,979 | lean_file_chunk | 1990c954bdb7f4364d83b6379cc4a855baeddd689a88e1c77f38f27eafffe8a0 | 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 Mathlib.RingTheory.Polynomial.Hermite.Basic
import Mathlib.RingTheory.Polynomial.Hermite.Gaussian
import Mathlib.Analysis.SpecialFunctions.Gaussian.Ga... |
23,977 | lean_file_chunk | beb0b242f2a2c534c7c8adf05dbcd3a607dcc81d657544325e0bce3da54e2cb7 | lean_source_file | lean_pool_source_file_chunks | `∫ ψₙ·ψₘ = cₙ·cₘ·(√2)⁻¹·J(n,m)` after substitution u = x√2.
-/
private theorem hermiteFunction_integral_eq (n m : ℕ) :
∫ x, hermiteFunction n x * hermiteFunction m x =
hermiteFunctionNormConst n * hermiteFunctionNormConst m *
((Real.sqrt 2)⁻¹ * J n m) := by
-- Rewrite integrand factoring out constants
... |
23,908 | lean_file_chunk | 5c88ad1d35ca86467c0a8a40b86cd42a2f8e82f420af407b4eb2471d537332c7 | lean_source_file | lean_pool_source_file_chunks | Integrable (fun x => hermiteFunction n x * deriv (hermiteFunction n) x) volume :=
(hermiteFunction_memLp n).integrable_mul (deriv_hermiteFunction_memLp n)
/-- A Schwartz function on ℝ tends to 0 at atBot. -/
private lemma schwartz_tendsto_atBot (φ : SchwartzMap ℝ ℝ) :
Filter.Tendsto (fun x => φ x) Filter.atBot (... |
18,631 | lean_file_chunk | ae0145d87ba205ffae5478179942b9b6c339f4c38b0ff551b96df93ca59926fb | lean_source_file | lean_pool_source_file_chunks | intro h; exact h_sqrt2_pow_ne (hQ_lead ▸ Polynomial.leadingCoeff_eq_zero.mpr h)
have hmon_ne : Polynomial.C (Real.sqrt 2 ^ k) * Polynomial.X ^ k ≠ 0 := by
intro h; exact h_sqrt2_pow_ne (hmon_lead.symm.trans (Polynomial.leadingCoeff_eq_zero.mpr h))
have h_same_deg : Q.degree = (Polynomial.C (Real.sqr... |
4,878 | lean_file_chunk | d02c9b95f53098d0ae4c4bd622cec934e46f758b7122b42c42ee22a9e8bf5a18 | 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.GaussianField.SchwartzNuclear.HermiteTensorProduct
/-!
# DyninMityaginSpace Instance for Schwartz Space
Derives the `DyninMityagin... |
23,994 | lean_file_chunk | 648899046e6fd7d75046461f28145d9f99fe82752210783f789b67ddf4007e1c | 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.GaussianField.SchwartzNuclear.Basis1D
import LeanPool.OSforGFF.GaussianField.Nuclear.NuclearTensorProduct
import LeanPool.OSforGFF.G... |
23,973 | lean_file_chunk | d10cf5cd23351d29ab40c03e2e96012207d4695afb4352d26526a1f1b38a7dd3 | lean_source_file | lean_pool_source_file_chunks | simp only [RapidDecaySeq.add_val]
rw [show (fun n => (a.val n + b.val n) • schwartzHermiteBasis1D n) =
(fun n => a.val n • schwartzHermiteBasis1D n + b.val n • schwartzHermiteBasis1D n) from by
ext n; rw [add_smul]]
exact (rapidDecay_hermite_summable a).tsum_add (rapidDecay_hermite_summable b)
map... |
23,952 | lean_file_chunk | 1e11fc7c625dc4978320c8797064eca9efe2527b46c6630cbcad4410f03540db | lean_source_file | lean_pool_source_file_chunks | ∏ j ∈ (Finset.univ : Finset (Fin (d + 1))),
‖iteratedFDeriv ℝ ((p : Multiset (Fin (d + 1))).count j) (fac j) y‖ := by
intro y
exact norm_iteratedFDeriv_prod_le
(fun j _ => hfac_smooth j) (mod_cast le_top)
-- Rewrite using hprod_eq
have hprod_eq' : ∀ y : EuclideanSpace ℝ (Fin (d + 1)),
... |
23,968 | lean_file_chunk | 45eb2b2c88367daf9d6869e80fc411ee9f5ce2e515e70e6547bac7ffdf211af9 | lean_source_file | lean_pool_source_file_chunks | ∃ (C : ℝ) (q : Finset (ℕ × ℕ)), 0 < C ∧
∀ (f : SchwartzMap (EuclideanSpace ℝ (Fin (d' + 1))) ℝ)
(α : MultiIndex (d' + 1)),
|hermiteCoeffNd (d' + 1) α f| * (1 + (MultiIndex.abs α : ℝ)) ^ k ≤
C * q.sup (schwartzSeminormFamily ℝ (EuclideanSpace ℝ (Fin (d' + 1))) ℝ) f := by
induction d' wi... |
23,967 | lean_file_chunk | 7afe725335ba0320f14ccc8344fa9eb43db45bf46d1676fe325c0272e3ab5e5d | lean_source_file | lean_pool_source_file_chunks | /-- Fubini for EuclideanSpace: factors a product integral into individual integrals.
Bridges the MeasurableSpace instance diamond between EuclideanSpace (inner product
based Haar measure) and Pi (product measure) via the volume-preserving equivalence.
-/
private lemma integral_euclidean_prod_eq_prod (d : ℕ)
(f : Fi... |
9,372 | lean_file_chunk | ddcac57f85c5a2532e8136671a16a1be81943166271d2357780c1ccc8fc64540 | lean_source_file | lean_pool_source_file_chunks | By the right inverse (Kronecker), `c_α(S_f) = c_α(f)` for all `α`.
By `hermiteCoeffNd_injective`, `S_f - f = 0`, so `S_f = f`.
-/
private lemma schwartz_hermite_completeness_nd (d' : ℕ)
(f : SchwartzMap (EuclideanSpace ℝ (Fin (d' + 1))) ℝ) :
HasSum (fun n => hermiteCoeffNd (d' + 1) ((multiIndexEquiv d').symm n)... |
16,008 | lean_file_chunk | b0d1743082e8c8f69e3181aef2819f5299dd6e72309aafd908f6095079a4e808 | 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 Mathlib.Analysis.Distribution.SchwartzSpace.Basic
import Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries
import Mathlib.Analysis.Calculus.ParametricI... |
23,947 | lean_file_chunk | 2f6bef16222a719e85a1df470d0822982b07801e8a959ced9de8ee2ad2182280 | 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 Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.Distribution.SchwartzSpace.Deriv
import Mathlib.MeasureTheory.Integral.Bochner.Contin... |
23,925 | lean_file_chunk | cd3e786e27c47bb2eefdae947c8f7a7c58368cbdc0358689ab02a14970cd8d36 | lean_source_file | lean_pool_source_file_chunks | ∃ (q : ℕ × ℕ) (C : ℝ), 0 < C ∧ ∀ (f : SchwartzMap ℝ ℝ),
Real.sqrt (∫ x, (f x) ^ 2) ≤
C * (Finset.Iic q).sup (fun p => SchwartzMap.seminorm ℝ p.1 p.2) f := by
-- Pointwise bound: (1+‖x‖) * ‖f x‖ ≤ 2S where S = sup-seminorm over Iic (1,0)
-- Then (f x)² ≤ 4S²·(1+‖x‖)⁻² and √(∫ f²) ≤ 2S·√(∫ (1+‖x‖)⁻²)
... |
23,973 | lean_file_chunk | e7401fa88db5fb365815215d206c4e70512cadc0958a5a8f5f766e7a6686dd28 | lean_source_file | lean_pool_source_file_chunks | have h_lintegral_sq : ∀ (g : ℝ → ℝ) (hg : Integrable (fun x => g x ^ 2) volume),
∫⁻ x, ‖g x‖ₑ ^ ENNReal.toReal 2 ∂volume =
ENNReal.ofReal (∫ x, g x ^ 2 ∂volume) := by
intro g hg
have h_nn : 0 ≤ᵐ[volume] (fun x => g x ^ 2) := ae_of_all _ (fun x => sq_nonneg _)
simp only [ENNReal.toReal_ofNat]
... |
1,717 | lean_file_chunk | d85de286970017997cb3cb5a7c9be1a475aa231996eb5ec818e68d6f524cd347 | lean_source_file | lean_pool_source_file_chunks | change Filter.Tendsto _ Filter.atTop _
rw [(schwartz_withSeminorms ℝ ℝ ℝ).tendsto_nhds _ f]
intro ⟨k, l⟩ ε hε
have h_sem := hermite_expansion_seminorm_summable f k l
obtain ⟨s₀, hs₀⟩ := summable_iff_vanishing_norm.mp h_sem (ε / 2) (half_pos hε)
rw [Filter.eventually_atTop]
refine ⟨s₀, fun s hs => ?_⟩
rw [... |
23,972 | lean_file_chunk | 82522b6b14841b8d8754b6d5c706f1f1ff2fd83e016ff6d4d1d5bb5e858a5650 | 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.GaussianField.SchwartzNuclear.ParametricCalculus
import LeanPool.OSforGFF.GaussianField.SchwartzNuclear.SchwartzHermiteExpansion
/-... |
23,953 | lean_file_chunk | b950697fbb580a852076ef86379b3f0d25a06b719e79903d546bbae53f818532 | lean_source_file | lean_pool_source_file_chunks | (fun y' => f (euclideanSnoc (d + 1) y' t) * hermiteFunction n t) y =
hermiteFunction n t • iteratedFDeriv ℝ m (schwartz_slice_y d f t) y := by
intro t
have h_eq : (fun y' => f (euclideanSnoc (d + 1) y' t) * hermiteFunction n t) =
(fun y' => hermiteFunction n t • (schwartz_slice_y d f t) y'... |
8,205 | lean_file_chunk | c3708f89f1812949ece8204fadabd83b83cc53ba293649284414b984319e98be | lean_source_file | lean_pool_source_file_chunks | -- Step 1: Chain rule bound on b-th t-derivative
set L_t : ℝ →L[ℝ] EuclideanSpace ℝ (Fin (d + 2)) :=
(EuclideanSpace.equiv (Fin (d + 2)) ℝ).symm.toContinuousLinearMap.comp
(ContinuousLinearMap.single (R := ℝ) (φ := fun (_ : Fin (d + 2)) => ℝ)
(Fin.last (d + 1)))
set c₀ := euclideanSnoc (d + 1) y 0... |
804 | lean_file_chunk | ee0f83221cd0a413ad318e90cdcf71baf3a30e9cc6a851dbd29079be60910425 | 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.GaussianField.SchwartzNuclear.Basis1D
import LeanPool.OSforGFF.GaussianField.SchwartzNuclear.HermiteFunctions
import LeanPool.OSforG... |
368 | lean_file_chunk | 978a81bda6eac7da2f897d002d6776f7edca3eb857a7d9839f6c02c82e789533 | 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.GaussianField.Nuclear
import LeanPool.OSforGFF.GaussianField.SchwartzNuclear
/-!
# `LeanPool.OSforGFF.GaussianField` index
Import ... |
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