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/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import Mathlib.Algebra.Lie.Classical
/-!
# The Minkowski Metric
This file introduces the Minkowski metric on sp... | lemma matrix_eq_id_iff : Λ = 1 ↔ ∀ w v, ⟪v, Λ *ᵥ w⟫ₘ = ⟪v, w⟫ₘ | minkowskiMetric.matrix_eq_id_iff | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | hep_lean/HepLean/SpaceTime/MinkowskiMetric.lean | HepLean.SpaceTime.MinkowskiMetric | HepLean.SpaceTime.MinkowskiMetric.jsonl | {
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"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
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"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n rw [matrix_eq_iff_eq_forall]\n simp",
"proofType": "tactic",
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/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzGroup.Basic
/-!
# The Proper Lorentz Group
The proper Lorentz group is the subgroup of the Lorentz group with determinant `1`... | lemma det_of_joined {Λ Λ' : LorentzGroup d} (h : Joined Λ Λ') : Λ.1.det = Λ'.1.det | LorentzGroup.det_of_joined | {
"commit": "89e940a0299f69637521d7b2b002748aaab92ff2",
"date": "2024-05-17T00:00:00"
} | {
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"date": "2024-05-17T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Proper.lean | HepLean.SpaceTime.LorentzGroup.Proper | HepLean.SpaceTime.LorentzGroup.Proper.jsonl | {
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"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
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"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":=\n det_on_connected_component $ pathComponent_subset_component _ h",
"proofType": "term",
"proofLengthLines": 1,
"proofLengthTokens": 68
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import Mathlib.Algebra.Lie.Classical
/-!
# The Minkowski Metric
This file introduces the Minkowski metric on sp... | @[simp]
lemma dual_id : @dual d 1 = 1 | minkowskiMetric.dual_id | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | hep_lean/HepLean/SpaceTime/MinkowskiMetric.lean | HepLean.SpaceTime.MinkowskiMetric | HepLean.SpaceTime.MinkowskiMetric.jsonl | {
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"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n simpa only [dual, transpose_one, mul_one] using minkowskiMatrix.sq",
"proofType": "tactic",
"proofLengthLines": 1,
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} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzGroup.Basic
import HepLean.Mathematics.SO3.Basic
import Mathlib.Topology.Constructions
/-!
# Rotations
This file describes th... | lemma SO3ToMatrix_in_LorentzGroup (A : SO(3)) : SO3ToMatrix A ∈ LorentzGroup 3 | LorentzGroup.SO3ToMatrix_in_LorentzGroup | {
"commit": "026ed8b85e7a3c83656d9de99ef81bf41ac20e9e",
"date": "2024-05-21T00:00:00"
} | {
"commit": "c64d926e7c2009e6d90d58343fc9d9ad4eb069cf",
"date": "2024-07-02T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Rotations.lean | HepLean.SpaceTime.LorentzGroup.Rotations | HepLean.SpaceTime.LorentzGroup.Rotations.jsonl | {
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"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n rw [LorentzGroup.mem_iff_dual_mul_self]\n simp only [minkowskiMetric.dual, minkowskiMatrix.as_block, SO3ToMatrix,\n Matrix.fromBlocks_transpose, Matrix.transpose_one, Matrix.transpose_zero,\n Matrix.fromBlocks_multiply, mul_one, Matrix.mul_zero, add_zero, Matrix.zero_mu... |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import Mathlib.Data.Complex.Exponential
import Mathlib.Geometry.Manifold.SmoothManifoldWithCorners
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Space ... | lemma stdBasis_3 : stdBasis 3 = ![0, 0, 0, 1] | SpaceTime.stdBasis_3 | {
"commit": "7020263053f9d96398aaa860dc7b31460214b0b0",
"date": "2024-05-14T00:00:00"
} | {
"commit": "db0f6de59c636e635219f2013bf9cad19a2d09ca",
"date": "2024-05-13T00:00:00"
} | hep_lean/HepLean/SpaceTime/Basic.lean | HepLean.SpaceTime.Basic | HepLean.SpaceTime.Basic.jsonl | {
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"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n funext i\n fin_cases i <;> simp [stdBasis_apply]",
"proofType": "tactic",
"proofLengthLines": 2,
"proofLengthTokens": 56
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.NormOne
import HepLean.SpaceTime.LorentzGroup.Proper
/-!
# The Orthochronous Lorentz Group
We define the give a series... | lemma orthchroMapReal_on_not_IsOrthochronous {Λ : LorentzGroup d} (h : ¬ IsOrthochronous Λ) :
orthchroMapReal Λ = - 1 | LorentzGroup.orthchroMapReal_on_not_IsOrthochronous | {
"commit": "89e940a0299f69637521d7b2b002748aaab92ff2",
"date": "2024-05-17T00:00:00"
} | {
"commit": "7ebd2af7a5878a0a87dc35be735a66a0e0e4764a",
"date": "2024-05-17T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Orthochronous.lean | HepLean.SpaceTime.LorentzGroup.Orthochronous | HepLean.SpaceTime.LorentzGroup.Orthochronous.jsonl | {
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"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n rw [not_orthochronous_iff_le_neg_one] at h\n change stepFunction (timeComp _)= - 1\n rw [stepFunction, if_pos h]",
"proofType": "tactic",
"proofLengthLines": 3,
"proofLengthTokens": 120
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.LorentzAction
import HepLean.SpaceTime.LorentzVector.Covariant
import HepLean.Tensors.Basic
/-!
# Contractions of Lore... | lemma asTenProd_diag :
@asTenProd d = ∑ μ, η μ μ • (LorentzVector.stdBasis μ ⊗ₜ[ℝ] LorentzVector.stdBasis μ) | minkowskiMatrix.asTenProd_diag | {
"commit": "99f4e858393555a74df554cd7fa7fc8f290b860a",
"date": "2024-07-29T00:00:00"
} | {
"commit": "7b0b979d51cceb4c96a445d2f78e43d69d510b60",
"date": "2024-07-31T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzVector/Contraction.lean | HepLean.SpaceTime.LorentzVector.Contraction | HepLean.SpaceTime.LorentzVector.Contraction.jsonl | {
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"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n simp only [asTenProd]\n refine Finset.sum_congr rfl (fun μ _ => ?_)\n rw [Finset.sum_eq_single μ]\n · intro ν _ hμν\n rw [minkowskiMatrix.off_diag_zero hμν.symm]\n exact TensorProduct.zero_smul (e μ ⊗ₜ[ℝ] e ν)\n · intro a\n rename_i j\n exact False.elim (a j)",... |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.LorentzAction
import HepLean.SpaceTime.LorentzVector.Covariant
import HepLean.Tensors.Basic
/-!
# Contractions of Lore... | lemma unitDown_rid (x : CovariantLorentzVector d) :
TensorStructure.contrLeftAux contrDownUp (x ⊗ₜ[ℝ] unitDown) = x | LorentzVector.unitDown_rid | {
"commit": "99f4e858393555a74df554cd7fa7fc8f290b860a",
"date": "2024-07-29T00:00:00"
} | {
"commit": "a97cb623798a037cb7f9ff27508a67bec560f6c7",
"date": "2024-07-30T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzVector/Contraction.lean | HepLean.SpaceTime.LorentzVector.Contraction | HepLean.SpaceTime.LorentzVector.Contraction.jsonl | {
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"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n simp only [unitDown, LinearEquiv.refl_toLinearMap]\n rw [tmul_sum]\n simp only [TensorStructure.contrLeftAux, contrDownUp, LinearEquiv.refl_toLinearMap,\n Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,\n Finset.sum_singleton, map_add, Linea... |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzGroup.Basic
/-!
# The Proper Lorentz Group
The proper Lorentz group is the subgroup of the Lorentz group with determinant `1`... | lemma id_IsProper : @IsProper d 1 | LorentzGroup.id_IsProper | {
"commit": "89e940a0299f69637521d7b2b002748aaab92ff2",
"date": "2024-05-17T00:00:00"
} | {
"commit": "8024aa92a17078b8ce8608d256f65d2d82aecb41",
"date": "2024-05-20T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Proper.lean | HepLean.SpaceTime.LorentzGroup.Proper | HepLean.SpaceTime.LorentzGroup.Proper.jsonl | {
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"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n simp [IsProper]",
"proofType": "tactic",
"proofLengthLines": 1,
"proofLengthTokens": 23
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import Mathlib.Algebra.Lie.Classical
/-!
# The Minkowski Metric
This file introduces the Minkowski metric on sp... | @[simp]
lemma dual_eta : @dual d η = η | minkowskiMetric.dual_eta | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | hep_lean/HepLean/SpaceTime/MinkowskiMetric.lean | HepLean.SpaceTime.MinkowskiMetric | HepLean.SpaceTime.MinkowskiMetric.jsonl | {
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"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n simp only [dual, eq_transpose]\n noncomm_ring [minkowskiMatrix.sq]",
"proofType": "tactic",
"proofLengthLines": 2,
"proofLengthTokens": 74
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.NormOne
import HepLean.SpaceTime.LorentzGroup.Proper
/-!
# The Orthochronous Lorentz Group
We define the give a series... | lemma mul_othchron_of_not_othchron_not_othchron {Λ Λ' : LorentzGroup d} (h : ¬ IsOrthochronous Λ)
(h' : ¬ IsOrthochronous Λ') : IsOrthochronous (Λ * Λ') | LorentzGroup.mul_othchron_of_not_othchron_not_othchron | {
"commit": "89e940a0299f69637521d7b2b002748aaab92ff2",
"date": "2024-05-17T00:00:00"
} | {
"commit": "7ebd2af7a5878a0a87dc35be735a66a0e0e4764a",
"date": "2024-05-17T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Orthochronous.lean | HepLean.SpaceTime.LorentzGroup.Orthochronous | HepLean.SpaceTime.LorentzGroup.Orthochronous.jsonl | {
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"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n rw [IsOrthochronous_iff_transpose] at h\n rw [IsOrthochronous_iff_futurePointing] at h h'\n rw [IsOrthochronous, timeComp_mul]\n exact NormOneLorentzVector.FuturePointing.metric_reflect_not_mem_not_mem h h'",
"proofType": "tactic",
"proofLengthLines": 4,
"proofLengthT... |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.LorentzAction
import HepLean.SpaceTime.LorentzVector.Covariant
import HepLean.Tensors.Basic
/-!
# Contractions of Lore... | lemma unitUp_rid (x : LorentzVector d) :
TensorStructure.contrLeftAux contrUpDown (x ⊗ₜ[ℝ] unitUp) = x | LorentzVector.unitUp_rid | {
"commit": "99f4e858393555a74df554cd7fa7fc8f290b860a",
"date": "2024-07-29T00:00:00"
} | {
"commit": "a97cb623798a037cb7f9ff27508a67bec560f6c7",
"date": "2024-07-30T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzVector/Contraction.lean | HepLean.SpaceTime.LorentzVector.Contraction | HepLean.SpaceTime.LorentzVector.Contraction.jsonl | {
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"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n simp only [unitUp, LinearEquiv.refl_toLinearMap]\n rw [tmul_sum]\n simp only [TensorStructure.contrLeftAux, LinearEquiv.refl_toLinearMap, Fintype.sum_sum_type,\n Finset.univ_unique, Fin.default_eq_zero, Fin.isValue, Finset.sum_singleton, map_add,\n LinearMap.coe_comp, ... |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzGroup.Basic
/-!
# The Proper Lorentz Group
The proper Lorentz group is the subgroup of the Lorentz group with determinant `1`... | lemma detContinuous_eq_iff_det_eq (Λ Λ' : LorentzGroup d) :
detContinuous Λ = detContinuous Λ' ↔ Λ.1.det = Λ'.1.det | LorentzGroup.detContinuous_eq_iff_det_eq | {
"commit": "89e940a0299f69637521d7b2b002748aaab92ff2",
"date": "2024-05-17T00:00:00"
} | {
"commit": "89e940a0299f69637521d7b2b002748aaab92ff2",
"date": "2024-05-17T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Proper.lean | HepLean.SpaceTime.LorentzGroup.Proper | HepLean.SpaceTime.LorentzGroup.Proper.jsonl | {
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"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n cases' det_eq_one_or_neg_one Λ with h1 h1\n · rw [h1, (detContinuous_eq_one Λ).mpr h1]\n cases' det_eq_one_or_neg_one Λ' with h2 h2\n · rw [h2, (detContinuous_eq_one Λ').mpr h2]\n simp only [toMul_zero]\n · rw [h2, (detContinuous_eq_zero Λ').mpr h2]\n erw [... |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import HepLean.SpaceTime.LorentzGroup.Basic
import Mathlib.RepresentationTheory.Basic
/-!
# Covariant Lorentz ve... | @[simp]
lemma decomp_stdBasis' (v : CovariantLorentzVector d) :
v (Sum.inl 0) • stdBasis (Sum.inl 0)
+ ∑ a₂ : Fin d, v (Sum.inr a₂) • stdBasis (Sum.inr a₂) = v | CovariantLorentzVector.decomp_stdBasis' | {
"commit": "99f4e858393555a74df554cd7fa7fc8f290b860a",
"date": "2024-07-29T00:00:00"
} | {
"commit": "99f4e858393555a74df554cd7fa7fc8f290b860a",
"date": "2024-07-29T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzVector/Covariant.lean | HepLean.SpaceTime.LorentzVector.Covariant | HepLean.SpaceTime.LorentzVector.Covariant.jsonl | {
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"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n trans ∑ i, v i • stdBasis i\n · simp only [Fin.isValue, Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero,\n Finset.sum_singleton]\n · exact decomp_stdBasis v",
"proofType": "tactic",
"proofLengthLines": 4,
"proofLengthTokens": 180
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.NormOne
import HepLean.SpaceTime.LorentzGroup.Proper
/-!
# The Orthochronous Lorentz Group
We define the give a series... | lemma IsOrthochronous_iff_transpose :
IsOrthochronous Λ ↔ IsOrthochronous (transpose Λ) | LorentzGroup.IsOrthochronous_iff_transpose | {
"commit": "89e940a0299f69637521d7b2b002748aaab92ff2",
"date": "2024-05-17T00:00:00"
} | {
"commit": "7ebd2af7a5878a0a87dc35be735a66a0e0e4764a",
"date": "2024-05-17T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Orthochronous.lean | HepLean.SpaceTime.LorentzGroup.Orthochronous | HepLean.SpaceTime.LorentzGroup.Orthochronous.jsonl | {
"lineInFile": 38,
"tokenPositionInFile": 1061,
"theoremPositionInFile": 2
} | {
"inFilePremises": true,
"numInFilePremises": 1,
"repositoryPremises": true,
"numRepositoryPremises": 3,
"numPremises": 13,
"importedModules": [
"Init.Prelude",
"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by rfl",
"proofType": "tactic",
"proofLengthLines": 0,
"proofLengthTokens": 9
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.LorentzAction
import HepLean.SpaceTime.LorentzVector.Covariant
import HepLean.Tensors.Basic
/-!
# Contractions of Lore... | @[simp]
lemma contrDownUp_invariant_lorentzAction : @contrDownUp d ((CovariantLorentzVector.rep g) x ⊗ₜ[ℝ]
(LorentzVector.rep g) y) = contrDownUp (x ⊗ₜ[ℝ] y) | LorentzVector.contrDownUp_invariant_lorentzAction | {
"commit": "99f4e858393555a74df554cd7fa7fc8f290b860a",
"date": "2024-07-29T00:00:00"
} | {
"commit": "a65fb06605cbfe5b3907ae8a3963ce7cd664a975",
"date": "2024-07-30T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzVector/Contraction.lean | HepLean.SpaceTime.LorentzVector.Contraction | HepLean.SpaceTime.LorentzVector.Contraction.jsonl | {
"lineInFile": 160,
"tokenPositionInFile": 6033,
"theoremPositionInFile": 11
} | {
"inFilePremises": true,
"numInFilePremises": 2,
"repositoryPremises": true,
"numRepositoryPremises": 14,
"numPremises": 102,
"importedModules": [
"Init.Prelude",
"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n rw [contrDownUp_tmul_eq_dotProduct, contrDownUp_tmul_eq_dotProduct]\n rw [dotProduct_comm, dotProduct_comm x y]\n simp only [rep_apply, CovariantLorentzVector.rep_apply]\n rw [Matrix.dotProduct_mulVec, vecMul_transpose, mulVec_mulVec]\n simp only [LorentzGroup.subtype_inv_... |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzGroup.Proper
import Mathlib.Topology.Constructions
import HepLean.SpaceTime.LorentzVector.NormOne
/-!
# Boosts
This file defi... | /--
This lemma states that for a given four-velocity `u`, the general boost
transformation `genBoost u u` is equal to the identity linear map `LinearMap.id`.
-/
lemma self (u : FuturePointing d) : genBoost u u = LinearMap.id | LorentzGroup.genBoost.self | {
"commit": "7ebd2af7a5878a0a87dc35be735a66a0e0e4764a",
"date": "2024-05-17T00:00:00"
} | {
"commit": "28cc0ff7121fba90f032469b1685b796e604635b",
"date": "2024-04-17T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Boosts.lean | HepLean.SpaceTime.LorentzGroup.Boosts | HepLean.SpaceTime.LorentzGroup.Boosts.jsonl | {
"lineInFile": 69,
"tokenPositionInFile": 2202,
"theoremPositionInFile": 1
} | {
"inFilePremises": true,
"numInFilePremises": 3,
"repositoryPremises": true,
"numRepositoryPremises": 10,
"numPremises": 210,
"importedModules": [
"Init.Prelude",
"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n ext x\n simp only [genBoost, LinearMap.add_apply, LinearMap.id_coe, id_eq]\n rw [add_assoc, add_right_eq_self, add_eq_zero_iff_eq_neg, genBoostAux₁, genBoostAux₂]\n simp only [LinearMap.coe_mk, AddHom.coe_mk, map_add, smul_add, neg_smul, neg_add_rev, neg_neg]\n rw [← add_s... |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.NormOne
import HepLean.SpaceTime.LorentzGroup.Proper
/-!
# The Orthochronous Lorentz Group
We define the give a series... | lemma orthchroMap_not_IsOrthochronous {Λ : LorentzGroup d} (h : ¬ IsOrthochronous Λ) :
orthchroMap Λ = Additive.toMul (1 : ZMod 2) | LorentzGroup.orthchroMap_not_IsOrthochronous | {
"commit": "89e940a0299f69637521d7b2b002748aaab92ff2",
"date": "2024-05-17T00:00:00"
} | {
"commit": "7ebd2af7a5878a0a87dc35be735a66a0e0e4764a",
"date": "2024-05-17T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Orthochronous.lean | HepLean.SpaceTime.LorentzGroup.Orthochronous | HepLean.SpaceTime.LorentzGroup.Orthochronous.jsonl | {
"lineInFile": 121,
"tokenPositionInFile": 4526,
"theoremPositionInFile": 15
} | {
"inFilePremises": true,
"numInFilePremises": 5,
"repositoryPremises": true,
"numRepositoryPremises": 9,
"numPremises": 186,
"importedModules": [
"Init.Prelude",
"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n simp only [orthchroMap, ContinuousMap.comp_apply, ContinuousMap.coe_mk,\n orthchroMapReal_on_not_IsOrthochronous h, coeForℤ₂_apply, Subtype.mk.injEq, Nat.reduceAdd]\n rw [if_neg]\n · rfl\n · linarith",
"proofType": "tactic",
"proofLengthLines": 5,
"proofLengthToken... |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import Mathlib.Algebra.Lie.Classical
/-!
# The Minkowski Metric
This file introduces the Minkowski metric on sp... | @[simp]
lemma dual_mul : dual (Λ * Λ') = dual Λ' * dual Λ | minkowskiMetric.dual_mul | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | hep_lean/HepLean/SpaceTime/MinkowskiMetric.lean | HepLean.SpaceTime.MinkowskiMetric | HepLean.SpaceTime.MinkowskiMetric.jsonl | {
"lineInFile": 241,
"tokenPositionInFile": 7520,
"theoremPositionInFile": 23
} | {
"inFilePremises": true,
"numInFilePremises": 3,
"repositoryPremises": true,
"numRepositoryPremises": 3,
"numPremises": 57,
"importedModules": [
"Init.Prelude",
"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n simp only [dual, transpose_mul]\n trans η * Λ'ᵀ * (η * η) * Λᵀ * η\n · noncomm_ring [minkowskiMatrix.sq]\n · noncomm_ring",
"proofType": "tactic",
"proofLengthLines": 4,
"proofLengthTokens": 129
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.MinkowskiMetric
import Mathlib.Algebra.Lie.Classical
/-!
# The Lorentz Algebra
We define
- Define `lorentzAlgebra` via `LieAlgebra.... | lemma mem_iff {A : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ} :
A ∈ lorentzAlgebra ↔ Aᵀ * η = - η * A | SpaceTime.lorentzAlgebra.mem_iff | {
"commit": "a52d8ea452f8b6c9f0d5833190b01e18d19cc4b4",
"date": "2024-05-24T00:00:00"
} | {
"commit": "8fd0b63edbfd2814d1fba639a9838632881f48e4",
"date": "2024-05-09T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzAlgebra/Basic.lean | HepLean.SpaceTime.LorentzAlgebra.Basic | HepLean.SpaceTime.LorentzAlgebra.Basic.jsonl | {
"lineInFile": 41,
"tokenPositionInFile": 1330,
"theoremPositionInFile": 3
} | {
"inFilePremises": true,
"numInFilePremises": 3,
"repositoryPremises": true,
"numRepositoryPremises": 4,
"numPremises": 47,
"importedModules": [
"Init.Prelude",
"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":=\n Iff.intro (fun h => transpose_eta ⟨A, h⟩) (fun h => mem_of_transpose_eta_eq_eta_mul_self h)",
"proofType": "term",
"proofLengthLines": 1,
"proofLengthTokens": 96
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import Mathlib.Algebra.Lie.Classical
/-!
# The Minkowski Metric
This file introduces the Minkowski metric on sp... | @[simp]
lemma sq : @minkowskiMatrix d * minkowskiMatrix = 1 | minkowskiMatrix.sq | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | hep_lean/HepLean/SpaceTime/MinkowskiMetric.lean | HepLean.SpaceTime.MinkowskiMetric | HepLean.SpaceTime.MinkowskiMetric.jsonl | {
"lineInFile": 37,
"tokenPositionInFile": 907,
"theoremPositionInFile": 2
} | {
"inFilePremises": true,
"numInFilePremises": 1,
"repositoryPremises": true,
"numRepositoryPremises": 1,
"numPremises": 92,
"importedModules": [
"Init.Prelude",
"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_mul_diagonal]\n ext1 i j\n rcases i with i | i <;> rcases j with j | j\n · simp only [diagonal, of_apply, Sum.inl.injEq, Sum.elim_inl, mul_one]\n split\n · rename_i h\n subst h\n ... |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.NormOne
import HepLean.SpaceTime.LorentzGroup.Proper
/-!
# The Orthochronous Lorentz Group
We define the give a series... | lemma orthchroMapReal_on_IsOrthochronous {Λ : LorentzGroup d} (h : IsOrthochronous Λ) :
orthchroMapReal Λ = 1 | LorentzGroup.orthchroMapReal_on_IsOrthochronous | {
"commit": "89e940a0299f69637521d7b2b002748aaab92ff2",
"date": "2024-05-17T00:00:00"
} | {
"commit": "7ebd2af7a5878a0a87dc35be735a66a0e0e4764a",
"date": "2024-05-17T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Orthochronous.lean | HepLean.SpaceTime.LorentzGroup.Orthochronous | HepLean.SpaceTime.LorentzGroup.Orthochronous.jsonl | {
"lineInFile": 90,
"tokenPositionInFile": 3197,
"theoremPositionInFile": 10
} | {
"inFilePremises": true,
"numInFilePremises": 4,
"repositoryPremises": true,
"numRepositoryPremises": 7,
"numPremises": 174,
"importedModules": [
"Init.Prelude",
"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n rw [IsOrthochronous_iff_ge_one, timeComp] at h\n change stepFunction (Λ.1 _ _) = 1\n rw [stepFunction, if_pos h, if_neg]\n linarith",
"proofType": "tactic",
"proofLengthLines": 4,
"proofLengthTokens": 139
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.LorentzAction
import HepLean.SpaceTime.LorentzVector.Covariant
import HepLean.Tensors.Basic
/-!
# Contractions of Lore... | @[simp]
lemma contrLeft_asCoTenProd (x : LorentzVector d) :
contrLeftAux contrUpDown (x ⊗ₜ[ℝ] asCoTenProd) =
∑ μ, ((η μ μ * x μ) • CovariantLorentzVector.stdBasis μ) | minkowskiMatrix.contrLeft_asCoTenProd | {
"commit": "99f4e858393555a74df554cd7fa7fc8f290b860a",
"date": "2024-07-29T00:00:00"
} | {
"commit": "7b0b979d51cceb4c96a445d2f78e43d69d510b60",
"date": "2024-07-31T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzVector/Contraction.lean | HepLean.SpaceTime.LorentzVector.Contraction | HepLean.SpaceTime.LorentzVector.Contraction.jsonl | {
"lineInFile": 222,
"tokenPositionInFile": 8460,
"theoremPositionInFile": 17
} | {
"inFilePremises": true,
"numInFilePremises": 4,
"repositoryPremises": true,
"numRepositoryPremises": 15,
"numPremises": 131,
"importedModules": [
"Init.Prelude",
"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n simp only [asCoTenProd_diag]\n rw [tmul_sum]\n rw [map_sum]\n refine Finset.sum_congr rfl (fun μ _ => ?_)\n simp only [contrLeftAux, LinearEquiv.refl_toLinearMap, tmul_smul, map_smul,\n LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, assoc_symm_tmul, map_t... |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import Mathlib.Algebra.Lie.Classical
/-!
# The Minkowski Metric
This file introduces the Minkowski metric on sp... | lemma time_sq_eq_metric_add_space : v.time ^ 2 = ⟪v, v⟫ₘ + ‖v.space‖ ^ 2 | minkowskiMetric.time_sq_eq_metric_add_space | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | hep_lean/HepLean/SpaceTime/MinkowskiMetric.lean | HepLean.SpaceTime.MinkowskiMetric | HepLean.SpaceTime.MinkowskiMetric.jsonl | {
"lineInFile": 163,
"tokenPositionInFile": 5079,
"theoremPositionInFile": 14
} | {
"inFilePremises": true,
"numInFilePremises": 2,
"repositoryPremises": true,
"numRepositoryPremises": 7,
"numPremises": 66,
"importedModules": [
"Init.Prelude",
"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n rw [self_eq_time_minus_norm]\n exact Eq.symm (sub_add_cancel (v.time ^ 2) (‖v.space‖ ^ 2))",
"proofType": "tactic",
"proofLengthLines": 2,
"proofLengthTokens": 98
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import HepLean.SpaceTime.LorentzGroup.Basic
import Mathlib.RepresentationTheory.Basic
/-!
# Lorentz group action... | lemma rep_apply (g : LorentzGroup d) : rep g v = g *ᵥ v | LorentzVector.rep_apply | {
"commit": "99f4e858393555a74df554cd7fa7fc8f290b860a",
"date": "2024-07-29T00:00:00"
} | {
"commit": "a65fb06605cbfe5b3907ae8a3963ce7cd664a975",
"date": "2024-07-30T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzVector/LorentzAction.lean | HepLean.SpaceTime.LorentzVector.LorentzAction | HepLean.SpaceTime.LorentzVector.LorentzAction.jsonl | {
"lineInFile": 30,
"tokenPositionInFile": 800,
"theoremPositionInFile": 0
} | {
"inFilePremises": true,
"numInFilePremises": 1,
"repositoryPremises": true,
"numRepositoryPremises": 6,
"numPremises": 45,
"importedModules": [
"Init.Prelude",
"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= rfl",
"proofType": "term",
"proofLengthLines": 0,
"proofLengthTokens": 6
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzGroup.Proper
import Mathlib.Topology.Constructions
import HepLean.SpaceTime.LorentzVector.NormOne
/-!
# Boosts
This file defi... | lemma toMatrix_mulVec (u v : FuturePointing d) (x : LorentzVector d) :
(toMatrix u v).mulVec x = genBoost u v x | LorentzGroup.genBoost.toMatrix_mulVec | {
"commit": "7ebd2af7a5878a0a87dc35be735a66a0e0e4764a",
"date": "2024-05-17T00:00:00"
} | {
"commit": "fdf5fda1e73e043ba40a3836ed1f00ddcc259373",
"date": "2024-05-17T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Boosts.lean | HepLean.SpaceTime.LorentzGroup.Boosts | HepLean.SpaceTime.LorentzGroup.Boosts.jsonl | {
"lineInFile": 87,
"tokenPositionInFile": 2977,
"theoremPositionInFile": 3
} | {
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"numInFilePremises": 2,
"repositoryPremises": true,
"numRepositoryPremises": 8,
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"importedModules": [
"Init.Prelude",
"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":=\n LinearMap.toMatrix_mulVec_repr LorentzVector.stdBasis LorentzVector.stdBasis (genBoost u v) x",
"proofType": "term",
"proofLengthLines": 1,
"proofLengthTokens": 98
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.NormOne
import HepLean.SpaceTime.LorentzGroup.Proper
/-!
# The Orthochronous Lorentz Group
We define the give a series... | lemma mul_not_othchron_of_othchron_not_othchron {Λ Λ' : LorentzGroup d} (h : IsOrthochronous Λ)
(h' : ¬ IsOrthochronous Λ') : ¬ IsOrthochronous (Λ * Λ') | LorentzGroup.mul_not_othchron_of_othchron_not_othchron | {
"commit": "89e940a0299f69637521d7b2b002748aaab92ff2",
"date": "2024-05-17T00:00:00"
} | {
"commit": "7ebd2af7a5878a0a87dc35be735a66a0e0e4764a",
"date": "2024-05-17T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Orthochronous.lean | HepLean.SpaceTime.LorentzGroup.Orthochronous | HepLean.SpaceTime.LorentzGroup.Orthochronous.jsonl | {
"lineInFile": 143,
"tokenPositionInFile": 5598,
"theoremPositionInFile": 18
} | {
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"numInFilePremises": 4,
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"importedModules": [
"Init.Prelude",
"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n rw [not_orthochronous_iff_le_zero, timeComp_mul]\n rw [IsOrthochronous_iff_transpose] at h\n rw [IsOrthochronous_iff_futurePointing] at h h'\n exact NormOneLorentzVector.FuturePointing.metric_reflect_mem_not_mem h h'",
"proofType": "tactic",
"proofLengthLines": 4,
"pr... |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import Mathlib.Algebra.Lie.Classical
/-!
# The Minkowski Metric
This file introduces the Minkowski metric on sp... | lemma ge_abs_inner_product : v.time * w.time - ‖⟪v.space, w.space⟫_ℝ‖ ≤ ⟪v, w⟫ₘ | minkowskiMetric.ge_abs_inner_product | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | {
"commit": "fdf5fda1e73e043ba40a3836ed1f00ddcc259373",
"date": "2024-05-17T00:00:00"
} | hep_lean/HepLean/SpaceTime/MinkowskiMetric.lean | HepLean.SpaceTime.MinkowskiMetric | HepLean.SpaceTime.MinkowskiMetric.jsonl | {
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"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n rw [eq_time_minus_inner_prod, sub_le_sub_iff_left]\n exact Real.le_norm_self ⟪v.space, w.space⟫_ℝ",
"proofType": "tactic",
"proofLengthLines": 2,
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} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import Mathlib.Data.Complex.Exponential
import Mathlib.Geometry.Manifold.SmoothManifoldWithCorners
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Space ... | @[simp]
lemma smul_apply (x : SpaceTime) (a : ℝ) (i : Fin 4) : (a • x) i = a * x i | SpaceTime.smul_apply | {
"commit": "7020263053f9d96398aaa860dc7b31460214b0b0",
"date": "2024-05-14T00:00:00"
} | {
"commit": "5a5540ba782a4f6fd72fe122514f3c9c82c2b90a",
"date": "2024-04-26T00:00:00"
} | hep_lean/HepLean/SpaceTime/Basic.lean | HepLean.SpaceTime.Basic | HepLean.SpaceTime.Basic.jsonl | {
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"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= rfl",
"proofType": "term",
"proofLengthLines": 0,
"proofLengthTokens": 6
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzGroup.Basic
/-!
# The Proper Lorentz Group
The proper Lorentz group is the subgroup of the Lorentz group with determinant `1`... | @[simp]
lemma _root_.toMul_eq_one {α : Type*} [One α] {x : Additive α} :
Additive.toMul x = 1 ↔ x = 0 | toMul_eq_one | {
"commit": "89e940a0299f69637521d7b2b002748aaab92ff2",
"date": "2024-05-17T00:00:00"
} | {
"commit": "17f09022db0e73dffc3a3dff364831cc391abdb4",
"date": "2024-09-04T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Proper.lean | HepLean.SpaceTime.LorentzGroup.Proper | HepLean.SpaceTime.LorentzGroup.Proper.jsonl | {
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"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":=\n Iff.rfl",
"proofType": "term",
"proofLengthLines": 1,
"proofLengthTokens": 12
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.LorentzAction
import HepLean.SpaceTime.LorentzVector.Covariant
import HepLean.Tensors.Basic
/-!
# Contractions of Lore... | lemma contrDownUp_tmul_eq_dotProduct {x : CovariantLorentzVector d} {y : LorentzVector d} :
contrDownUp (x ⊗ₜ[ℝ] y) = x ⬝ᵥ y | LorentzVector.contrDownUp_tmul_eq_dotProduct | {
"commit": "99f4e858393555a74df554cd7fa7fc8f290b860a",
"date": "2024-07-29T00:00:00"
} | {
"commit": "a65fb06605cbfe5b3907ae8a3963ce7cd664a975",
"date": "2024-07-30T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzVector/Contraction.lean | HepLean.SpaceTime.LorentzVector.Contraction | HepLean.SpaceTime.LorentzVector.Contraction.jsonl | {
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"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n rw [dotProduct_comm x y]\n rfl",
"proofType": "tactic",
"proofLengthLines": 2,
"proofLengthTokens": 38
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.LorentzAction
import HepLean.SpaceTime.LorentzVector.Covariant
import HepLean.Tensors.Basic
/-!
# Contractions of Lore... | lemma asTenProd_invariant (g : LorentzGroup d) :
TensorProduct.map (LorentzVector.rep g) (LorentzVector.rep g) asTenProd = asTenProd | minkowskiMatrix.asTenProd_invariant | {
"commit": "99f4e858393555a74df554cd7fa7fc8f290b860a",
"date": "2024-07-29T00:00:00"
} | {
"commit": "7b0b979d51cceb4c96a445d2f78e43d69d510b60",
"date": "2024-07-31T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzVector/Contraction.lean | HepLean.SpaceTime.LorentzVector.Contraction | HepLean.SpaceTime.LorentzVector.Contraction.jsonl | {
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"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n simp only [asTenProd, map_sum, map_smul, map_tmul, rep_apply_stdBasis,\n Matrix.transpose_apply]\n trans ∑ μ : Fin 1 ⊕ Fin d, ∑ ν : Fin 1 ⊕ Fin d, ∑ φ : Fin 1 ⊕ Fin d,\n η μ ν • (g.1 φ μ • e φ) ⊗ₜ[ℝ] ∑ ρ : Fin 1 ⊕ Fin d, g.1 ρ ν • e ρ\n · refine Finset.sum_congr rfl ... |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import Mathlib.Algebra.Lie.Classical
/-!
# The Minkowski Metric
This file introduces the Minkowski metric on sp... | lemma eq_time_minus_inner_prod : ⟪v, w⟫ₘ = v.time * w.time - ⟪v.space, w.space⟫_ℝ | minkowskiMetric.eq_time_minus_inner_prod | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | {
"commit": "c64d926e7c2009e6d90d58343fc9d9ad4eb069cf",
"date": "2024-07-02T00:00:00"
} | hep_lean/HepLean/SpaceTime/MinkowskiMetric.lean | HepLean.SpaceTime.MinkowskiMetric | HepLean.SpaceTime.MinkowskiMetric.jsonl | {
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"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n rw [as_sum, @PiLp.inner_apply]\n rfl",
"proofType": "tactic",
"proofLengthLines": 2,
"proofLengthTokens": 44
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import Mathlib.Algebra.Lie.Classical
/-!
# The Minkowski Metric
This file introduces the Minkowski metric on sp... | @[simp]
lemma det_dual : (dual Λ).det = Λ.det | minkowskiMetric.det_dual | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | hep_lean/HepLean/SpaceTime/MinkowskiMetric.lean | HepLean.SpaceTime.MinkowskiMetric | HepLean.SpaceTime.MinkowskiMetric.jsonl | {
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"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n simp only [dual, det_mul, minkowskiMatrix.det_eq_neg_one_pow_d, det_transpose]\n group\n norm_cast\n simp",
"proofType": "tactic",
"proofLengthLines": 4,
"proofLengthTokens": 113
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.MinkowskiMetric
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
/-!
# Lorentz vector as a self-adjoint matrix
There is a line... | /-- The matrix `x.toMatrix` for `x ∈ spaceTime` is self adjoint. -/
lemma toMatrix_isSelfAdjoint (x : LorentzVector 3) : IsSelfAdjoint (toMatrix x) | SpaceTime.toMatrix_isSelfAdjoint | {
"commit": "c64d926e7c2009e6d90d58343fc9d9ad4eb069cf",
"date": "2024-07-02T00:00:00"
} | {
"commit": "ea4327aff59fde778912eb081ed0847eb114af44",
"date": "2024-06-12T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzVector/AsSelfAdjointMatrix.lean | HepLean.SpaceTime.LorentzVector.AsSelfAdjointMatrix | HepLean.SpaceTime.LorentzVector.AsSelfAdjointMatrix.jsonl | {
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"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n rw [isSelfAdjoint_iff, star_eq_conjTranspose, ← Matrix.ext_iff]\n intro i j\n fin_cases i <;> fin_cases j <;>\n simp [toMatrix, conj_ofReal]\n rfl",
"proofType": "tactic",
"proofLengthLines": 5,
"proofLengthTokens": 156
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.NormOne
import HepLean.SpaceTime.LorentzGroup.Proper
/-!
# The Orthochronous Lorentz Group
We define the give a series... | lemma mul_not_othchron_of_not_othchron_othchron {Λ Λ' : LorentzGroup d} (h : ¬ IsOrthochronous Λ)
(h' : IsOrthochronous Λ') : ¬ IsOrthochronous (Λ * Λ') | LorentzGroup.mul_not_othchron_of_not_othchron_othchron | {
"commit": "89e940a0299f69637521d7b2b002748aaab92ff2",
"date": "2024-05-17T00:00:00"
} | {
"commit": "7ebd2af7a5878a0a87dc35be735a66a0e0e4764a",
"date": "2024-05-17T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Orthochronous.lean | HepLean.SpaceTime.LorentzGroup.Orthochronous | HepLean.SpaceTime.LorentzGroup.Orthochronous.jsonl | {
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"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n rw [not_orthochronous_iff_le_zero, timeComp_mul]\n rw [IsOrthochronous_iff_transpose] at h\n rw [IsOrthochronous_iff_futurePointing] at h h'\n exact NormOneLorentzVector.FuturePointing.metric_reflect_not_mem_mem h h'",
"proofType": "tactic",
"proofLengthLines": 4,
"pr... |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import HepLean.SpaceTime.LorentzGroup.Basic
import Mathlib.RepresentationTheory.Basic
/-!
# Lorentz group action... | lemma rep_apply_stdBasis (g : LorentzGroup d) (μ : Fin 1 ⊕ Fin d) :
rep g (stdBasis μ) = ∑ ν, g.1.transpose μ ν • stdBasis ν | LorentzVector.rep_apply_stdBasis | {
"commit": "99f4e858393555a74df554cd7fa7fc8f290b860a",
"date": "2024-07-29T00:00:00"
} | {
"commit": "7b0b979d51cceb4c96a445d2f78e43d69d510b60",
"date": "2024-07-31T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzVector/LorentzAction.lean | HepLean.SpaceTime.LorentzVector.LorentzAction | HepLean.SpaceTime.LorentzVector.LorentzAction.jsonl | {
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"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n simp only [rep_apply, Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,\n Finset.sum_singleton, decomp_stdBasis']\n funext ν\n simp only [stdBasis, Matrix.transpose_apply]\n erw [Pi.basisFun_apply, Matrix.mulVec_single_one]\n rfl",
"proofType... |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.NormOne
import HepLean.SpaceTime.LorentzGroup.Proper
/-!
# The Orthochronous Lorentz Group
We define the give a series... | lemma IsOrthochronous_iff_futurePointing :
IsOrthochronous Λ ↔ (toNormOneLorentzVector Λ) ∈ NormOneLorentzVector.FuturePointing d | LorentzGroup.IsOrthochronous_iff_futurePointing | {
"commit": "89e940a0299f69637521d7b2b002748aaab92ff2",
"date": "2024-05-17T00:00:00"
} | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Orthochronous.lean | HepLean.SpaceTime.LorentzGroup.Orthochronous | HepLean.SpaceTime.LorentzGroup.Orthochronous.jsonl | {
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"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n simp only [IsOrthochronous, timeComp_eq_toNormOneLorentzVector]\n rw [NormOneLorentzVector.FuturePointing.mem_iff_time_nonneg]",
"proofType": "tactic",
"proofLengthLines": 2,
"proofLengthTokens": 134
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.LorentzAction
import HepLean.SpaceTime.LorentzVector.Covariant
import HepLean.Tensors.Basic
/-!
# Contractions of Lore... | @[simp]
lemma asCoTenProd_contr_asTenProd :
(contrMidAux (contrDownUp) (asCoTenProd ⊗ₜ[ℝ] asTenProd))
= TensorProduct.comm ℝ _ _ (@unitDown d) | minkowskiMatrix.asCoTenProd_contr_asTenProd | {
"commit": "99f4e858393555a74df554cd7fa7fc8f290b860a",
"date": "2024-07-29T00:00:00"
} | {
"commit": "7b0b979d51cceb4c96a445d2f78e43d69d510b60",
"date": "2024-07-31T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzVector/Contraction.lean | HepLean.SpaceTime.LorentzVector.Contraction | HepLean.SpaceTime.LorentzVector.Contraction.jsonl | {
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"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n simp only [contrMidAux, LinearEquiv.refl_toLinearMap, asCoTenProd_diag,\n LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply]\n rw [sum_tmul, map_sum, map_sum, unitDown]\n simp only [Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,\n Finset.sum_singleton... |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import Mathlib.Algebra.Lie.Classical
/-!
# The Minkowski Metric
This file introduces the Minkowski metric on sp... | @[simp]
lemma dual_mulVec_right : ⟪x, (dual Λ) *ᵥ y⟫ₘ = ⟪Λ *ᵥ x, y⟫ₘ | minkowskiMetric.dual_mulVec_right | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | hep_lean/HepLean/SpaceTime/MinkowskiMetric.lean | HepLean.SpaceTime.MinkowskiMetric | HepLean.SpaceTime.MinkowskiMetric.jsonl | {
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"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n simp only [minkowskiMetric, LinearMap.coe_mk, AddHom.coe_mk, dual, minkowskiLinearForm_apply,\n mulVec_mulVec, ← mul_assoc, minkowskiMatrix.sq, one_mul, (vecMul_transpose Λ x).symm, ←\n dotProduct_mulVec]",
"proofType": "tactic",
"proofLengthLines": 3,
"proofLength... |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.MinkowskiMetric
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
/-!
# Lorentz vector as a self-adjoint matrix
There is a line... | lemma det_eq_ηLin (x : LorentzVector 3) : det (toSelfAdjointMatrix x).1 = ⟪x, x⟫ₘ | SpaceTime.det_eq_ηLin | {
"commit": "c64d926e7c2009e6d90d58343fc9d9ad4eb069cf",
"date": "2024-07-02T00:00:00"
} | {
"commit": "de89fd7ef0ab94b46123a4a24ff680e72d957dbf",
"date": "2024-06-13T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzVector/AsSelfAdjointMatrix.lean | HepLean.SpaceTime.LorentzVector.AsSelfAdjointMatrix | HepLean.SpaceTime.LorentzVector.AsSelfAdjointMatrix.jsonl | {
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"Init.Prelude",
"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n simp only [toSelfAdjointMatrix, LinearEquiv.coe_mk, toSelfAdjointMatrix'_coe, Fin.isValue,\n det_fin_two_of, eq_time_minus_inner_prod, LorentzVector.time, LorentzVector.space,\n PiLp.inner_apply, Function.comp_apply, RCLike.inner_apply, conj_trivial, Fin.sum_univ_three,\... |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.LorentzAction
import HepLean.SpaceTime.LorentzVector.Covariant
import HepLean.Tensors.Basic
/-!
# Contractions of Lore... | lemma asCoTenProd_diag :
@asCoTenProd d = ∑ μ, η μ μ • (CovariantLorentzVector.stdBasis μ ⊗ₜ[ℝ]
CovariantLorentzVector.stdBasis μ) | minkowskiMatrix.asCoTenProd_diag | {
"commit": "99f4e858393555a74df554cd7fa7fc8f290b860a",
"date": "2024-07-29T00:00:00"
} | {
"commit": "7b0b979d51cceb4c96a445d2f78e43d69d510b60",
"date": "2024-07-31T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzVector/Contraction.lean | HepLean.SpaceTime.LorentzVector.Contraction | HepLean.SpaceTime.LorentzVector.Contraction.jsonl | {
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"Init.Prelude",
"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n simp only [asCoTenProd]\n refine Finset.sum_congr rfl (fun μ _ => ?_)\n rw [Finset.sum_eq_single μ]\n · intro ν _ hμν\n rw [minkowskiMatrix.off_diag_zero hμν.symm]\n simp only [zero_smul]\n · intro a\n simp_all only",
"proofType": "tactic",
"proofLengthLines":... |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import Mathlib.Algebra.Lie.Classical
/-!
# The Minkowski Metric
This file introduces the Minkowski metric on sp... | @[simp]
lemma basis_left (μ : Fin 1 ⊕ Fin d) : ⟪e μ, v⟫ₘ = η μ μ * v μ | minkowskiMetric.basis_left | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | {
"commit": "c64d926e7c2009e6d90d58343fc9d9ad4eb069cf",
"date": "2024-07-02T00:00:00"
} | hep_lean/HepLean/SpaceTime/MinkowskiMetric.lean | HepLean.SpaceTime.MinkowskiMetric | HepLean.SpaceTime.MinkowskiMetric.jsonl | {
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"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n rw [as_sum]\n rcases μ with μ | μ\n · fin_cases μ\n simp [stdBasis_apply, minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]\n · simp [stdBasis_apply, minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]",
"proofType": "tactic",
"proofLengthLines": 5,
... |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import Mathlib.Data.Complex.Exponential
import Mathlib.Geometry.Manifold.SmoothManifoldWithCorners
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Space ... | lemma explicit (x : SpaceTime) : x = ![x 0, x 1, x 2, x 3] | SpaceTime.explicit | {
"commit": "7020263053f9d96398aaa860dc7b31460214b0b0",
"date": "2024-05-14T00:00:00"
} | {
"commit": "b9119dccf1cb5708c9237596fe29562ca085614a",
"date": "2024-04-18T00:00:00"
} | hep_lean/HepLean/SpaceTime/Basic.lean | HepLean.SpaceTime.Basic | HepLean.SpaceTime.Basic.jsonl | {
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"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n funext i\n fin_cases i <;> rfl",
"proofType": "tactic",
"proofLengthLines": 2,
"proofLengthTokens": 38
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import HepLean.SpaceTime.LorentzGroup.Basic
import Mathlib.RepresentationTheory.Basic
/-!
# Covariant Lorentz ve... | lemma rep_apply_stdBasis (g : LorentzGroup d) (μ : Fin 1 ⊕ Fin d) :
rep g (stdBasis μ) = ∑ ν, g⁻¹.1 μ ν • stdBasis ν | CovariantLorentzVector.rep_apply_stdBasis | {
"commit": "99f4e858393555a74df554cd7fa7fc8f290b860a",
"date": "2024-07-29T00:00:00"
} | {
"commit": "7b0b979d51cceb4c96a445d2f78e43d69d510b60",
"date": "2024-07-31T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzVector/Covariant.lean | HepLean.SpaceTime.LorentzVector.Covariant | HepLean.SpaceTime.LorentzVector.Covariant.jsonl | {
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"Init.Prelude",
"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n simp only [rep_apply, Fintype.sum_sum_type, Finset.univ_unique, Fin.default_eq_zero, Fin.isValue,\n Finset.sum_singleton, decomp_stdBasis']\n funext ν\n simp only [lorentzGroupIsGroup_inv]\n erw [Pi.basisFun_apply, Matrix.mulVec_single_one]\n rw [← LorentzGroup.coe_inv]... |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import Mathlib.Data.Complex.Exponential
import Mathlib.Geometry.Manifold.SmoothManifoldWithCorners
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Space ... | lemma stdBasis_not_eq {μ ν : Fin 4} (h : μ ≠ ν) : stdBasis μ ν = 0 | SpaceTime.stdBasis_not_eq | {
"commit": "7020263053f9d96398aaa860dc7b31460214b0b0",
"date": "2024-05-14T00:00:00"
} | {
"commit": "db0f6de59c636e635219f2013bf9cad19a2d09ca",
"date": "2024-05-13T00:00:00"
} | hep_lean/HepLean/SpaceTime/Basic.lean | HepLean.SpaceTime.Basic | HepLean.SpaceTime.Basic.jsonl | {
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"Init.Prelude",
"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n rw [stdBasis_apply]\n exact if_neg h",
"proofType": "tactic",
"proofLengthLines": 2,
"proofLengthTokens": 44
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.MinkowskiMetric
import Mathlib.Algebra.Lie.Classical
/-!
# The Lorentz Algebra
We define
- Define `lorentzAlgebra` via `LieAlgebra.... | lemma mem_of_transpose_eta_eq_eta_mul_self {A : Matrix (Fin 1 ⊕ Fin 3) (Fin 1 ⊕ Fin 3) ℝ}
(h : Aᵀ * η = - η * A) : A ∈ lorentzAlgebra | SpaceTime.lorentzAlgebra.mem_of_transpose_eta_eq_eta_mul_self | {
"commit": "a52d8ea452f8b6c9f0d5833190b01e18d19cc4b4",
"date": "2024-05-24T00:00:00"
} | {
"commit": "a52d8ea452f8b6c9f0d5833190b01e18d19cc4b4",
"date": "2024-05-24T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzAlgebra/Basic.lean | HepLean.SpaceTime.LorentzAlgebra.Basic | HepLean.SpaceTime.LorentzAlgebra.Basic.jsonl | {
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"Init.Prelude",
"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n erw [mem_skewAdjointMatricesLieSubalgebra]\n simpa [LieAlgebra.Orthogonal.so', IsSkewAdjoint, IsAdjointPair] using h",
"proofType": "tactic",
"proofLengthLines": 2,
"proofLengthTokens": 124
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import Mathlib.Algebra.Lie.Classical
/-!
# The Minkowski Metric
This file introduces the Minkowski metric on sp... | lemma matrix_eq_iff_eq_forall' : (∀ v, Λ *ᵥ v= Λ' *ᵥ v) ↔ ∀ w v, ⟪v, Λ *ᵥ w⟫ₘ = ⟪v, Λ' *ᵥ w⟫ₘ | minkowskiMetric.matrix_eq_iff_eq_forall' | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | hep_lean/HepLean/SpaceTime/MinkowskiMetric.lean | HepLean.SpaceTime.MinkowskiMetric | HepLean.SpaceTime.MinkowskiMetric.jsonl | {
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"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n refine Iff.intro (fun h ↦ fun w v ↦ ?_) (fun h ↦ fun v ↦ ?_)\n · rw [h w]\n · simp only [matrix_apply_eq_iff_sub] at h\n refine sub_eq_zero.1 ?_\n have h1 := h v\n rw [nondegenerate] at h1\n simp only [sub_mulVec] at h1\n exact h1",
"proofType": "tactic",
... |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import Mathlib.Data.Complex.Exponential
import Mathlib.Geometry.Manifold.SmoothManifoldWithCorners
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Space ... | lemma stdBasis_apply (μ ν : Fin 4) : stdBasis μ ν = if μ = ν then 1 else 0 | SpaceTime.stdBasis_apply | {
"commit": "7020263053f9d96398aaa860dc7b31460214b0b0",
"date": "2024-05-14T00:00:00"
} | {
"commit": "db0f6de59c636e635219f2013bf9cad19a2d09ca",
"date": "2024-05-13T00:00:00"
} | hep_lean/HepLean/SpaceTime/Basic.lean | HepLean.SpaceTime.Basic | HepLean.SpaceTime.Basic.jsonl | {
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"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n erw [stdBasis, Pi.basisFun_apply, LinearMap.stdBasis_apply']",
"proofType": "tactic",
"proofLengthLines": 1,
"proofLengthTokens": 68
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.LorentzAction
import HepLean.SpaceTime.LorentzVector.Covariant
import HepLean.Tensors.Basic
/-!
# Contractions of Lore... | @[simp]
lemma contrUpDown_invariant_lorentzAction : @contrUpDown d ((LorentzVector.rep g) x ⊗ₜ[ℝ]
(CovariantLorentzVector.rep g) y) = contrUpDown (x ⊗ₜ[ℝ] y) | LorentzVector.contrUpDown_invariant_lorentzAction | {
"commit": "99f4e858393555a74df554cd7fa7fc8f290b860a",
"date": "2024-07-29T00:00:00"
} | {
"commit": "a65fb06605cbfe5b3907ae8a3963ce7cd664a975",
"date": "2024-07-30T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzVector/Contraction.lean | HepLean.SpaceTime.LorentzVector.Contraction | HepLean.SpaceTime.LorentzVector.Contraction.jsonl | {
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"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n rw [contrUpDown_tmul_eq_dotProduct, contrUpDown_tmul_eq_dotProduct]\n simp only [rep_apply, CovariantLorentzVector.rep_apply]\n rw [Matrix.dotProduct_mulVec, vecMul_transpose, mulVec_mulVec]\n simp only [LorentzGroup.subtype_inv_mul, one_mulVec]",
"proofType": "tactic",
... |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import Mathlib.Algebra.Lie.Classical
/-!
# The Minkowski Metric
This file introduces the Minkowski metric on sp... | /-- The Minkowski metric expressed as a sum. -/
lemma as_sum :
⟪v, w⟫ₘ = v.time * w.time - ∑ i, v.space i * w.space i | minkowskiMetric.as_sum | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | hep_lean/HepLean/SpaceTime/MinkowskiMetric.lean | HepLean.SpaceTime.MinkowskiMetric | HepLean.SpaceTime.MinkowskiMetric.jsonl | {
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"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n simp only [minkowskiMetric, LinearMap.coe_mk, AddHom.coe_mk, minkowskiLinearForm_apply,\n dotProduct, LieAlgebra.Orthogonal.indefiniteDiagonal, mulVec_diagonal, Fintype.sum_sum_type,\n Finset.univ_unique, Fin.default_eq_zero, Fin.isValue, Sum.elim_inl, one_mul,\n Fins... |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.NormOne
import HepLean.SpaceTime.LorentzGroup.Proper
/-!
# The Orthochronous Lorentz Group
We define the give a series... | lemma stepFunction_continuous : Continuous stepFunction | LorentzGroup.stepFunction_continuous | {
"commit": "89e940a0299f69637521d7b2b002748aaab92ff2",
"date": "2024-05-17T00:00:00"
} | {
"commit": "fdf5fda1e73e043ba40a3836ed1f00ddcc259373",
"date": "2024-05-17T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Orthochronous.lean | HepLean.SpaceTime.LorentzGroup.Orthochronous | HepLean.SpaceTime.LorentzGroup.Orthochronous.jsonl | {
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"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n apply Continuous.if ?_ continuous_const (Continuous.if ?_ continuous_const continuous_id)\n <;> intro a ha\n · rw [@Set.Iic_def, @frontier_Iic, @Set.mem_singleton_iff] at ha\n rw [ha]\n simp only [le_neg_self_iff, id_eq]\n have h1 : ¬ (1 : ℝ) ≤ 0 := by simp\n e... |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzGroup.Basic
/-!
# The Proper Lorentz Group
The proper Lorentz group is the subgroup of the Lorentz group with determinant `1`... | lemma detRep_continuous : Continuous (@detRep d) | LorentzGroup.detRep_continuous | {
"commit": "89e940a0299f69637521d7b2b002748aaab92ff2",
"date": "2024-05-17T00:00:00"
} | {
"commit": "89e940a0299f69637521d7b2b002748aaab92ff2",
"date": "2024-05-17T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Proper.lean | HepLean.SpaceTime.LorentzGroup.Proper | HepLean.SpaceTime.LorentzGroup.Proper.jsonl | {
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"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= detContinuous.2",
"proofType": "term",
"proofLengthLines": 0,
"proofLengthTokens": 18
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzGroup.Proper
import Mathlib.Topology.Constructions
import HepLean.SpaceTime.LorentzVector.NormOne
/-!
# Boosts
This file defi... | lemma toMatrix_continuous (u : FuturePointing d) : Continuous (toMatrix u) | LorentzGroup.genBoost.toMatrix_continuous | {
"commit": "7ebd2af7a5878a0a87dc35be735a66a0e0e4764a",
"date": "2024-05-17T00:00:00"
} | {
"commit": "7ebd2af7a5878a0a87dc35be735a66a0e0e4764a",
"date": "2024-05-17T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Boosts.lean | HepLean.SpaceTime.LorentzGroup.Boosts | HepLean.SpaceTime.LorentzGroup.Boosts.jsonl | {
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"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n refine continuous_matrix ?_\n intro i j\n simp only [toMatrix_apply]\n refine (continuous_mul_left (η i i)).comp' ?_\n refine Continuous.sub ?_ ?_\n · refine .comp' (continuous_add_left (minkowskiMetric (e i) (e j))) ?_\n refine .comp' (continuous_mul_left (2 * ⟪e j, u... |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import Mathlib.Algebra.Lie.Classical
/-!
# The Minkowski Metric
This file introduces the Minkowski metric on sp... | @[simp]
lemma det_eq_neg_one_pow_d : (@minkowskiMatrix d).det = (- 1) ^ d | minkowskiMatrix.det_eq_neg_one_pow_d | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | hep_lean/HepLean/SpaceTime/MinkowskiMetric.lean | HepLean.SpaceTime.MinkowskiMetric | HepLean.SpaceTime.MinkowskiMetric.jsonl | {
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"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]",
"proofType": "tactic",
"proofLengthLines": 1,
"proofLengthTokens": 72
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import Mathlib.Algebra.Lie.Classical
/-!
# The Minkowski Metric
This file introduces the Minkowski metric on sp... | @[simp]
lemma eq_transpose : minkowskiMatrixᵀ = @minkowskiMatrix d | minkowskiMatrix.eq_transpose | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | hep_lean/HepLean/SpaceTime/MinkowskiMetric.lean | HepLean.SpaceTime.MinkowskiMetric | HepLean.SpaceTime.MinkowskiMetric.jsonl | {
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"Init.Prelude",
"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_transpose]",
"proofType": "tactic",
"proofLengthLines": 1,
"proofLengthTokens": 97
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import Mathlib.Data.Complex.Exponential
import Mathlib.Geometry.Manifold.SmoothManifoldWithCorners
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Space ... | lemma stdBasis_2 : stdBasis 2 = ![0, 0, 1, 0] | SpaceTime.stdBasis_2 | {
"commit": "7020263053f9d96398aaa860dc7b31460214b0b0",
"date": "2024-05-14T00:00:00"
} | {
"commit": "db0f6de59c636e635219f2013bf9cad19a2d09ca",
"date": "2024-05-13T00:00:00"
} | hep_lean/HepLean/SpaceTime/Basic.lean | HepLean.SpaceTime.Basic | HepLean.SpaceTime.Basic.jsonl | {
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} | {
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"importedModules": [
"Init.Prelude",
"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n funext i\n fin_cases i <;> simp [stdBasis_apply]",
"proofType": "tactic",
"proofLengthLines": 2,
"proofLengthTokens": 56
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import Mathlib.Algebra.Lie.Classical
/-!
# The Minkowski Metric
This file introduces the Minkowski metric on sp... | @[simp]
lemma η_apply_mul_η_apply_diag (μ : Fin 1 ⊕ Fin d) : η μ μ * η μ μ = 1 | minkowskiMatrix.η_apply_mul_η_apply_diag | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | {
"commit": "99f4e858393555a74df554cd7fa7fc8f290b860a",
"date": "2024-07-29T00:00:00"
} | hep_lean/HepLean/SpaceTime/MinkowskiMetric.lean | HepLean.SpaceTime.MinkowskiMetric | HepLean.SpaceTime.MinkowskiMetric.jsonl | {
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"tokenPositionInFile": 1924,
"theoremPositionInFile": 5
} | {
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"importedModules": [
"Init.Prelude",
"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n match μ with\n | Sum.inl _ => simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]\n | Sum.inr _ => simp [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal]",
"proofType": "tactic",
"proofLengthLines": 3,
"proofLengthTokens": 184
} |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.LorentzVector.Basic
import Mathlib.Algebra.Lie.Classical
/-!
# The Minkowski Metric
This file introduces the Minkowski metric on sp... | lemma on_basis (μ ν : Fin 1 ⊕ Fin d) : ⟪e μ, e ν⟫ₘ = η μ ν | minkowskiMetric.on_basis | {
"commit": "675b9a989a4d3f142c67d682ef68b22ea477e83a",
"date": "2024-07-01T00:00:00"
} | {
"commit": "c64d926e7c2009e6d90d58343fc9d9ad4eb069cf",
"date": "2024-07-02T00:00:00"
} | hep_lean/HepLean/SpaceTime/MinkowskiMetric.lean | HepLean.SpaceTime.MinkowskiMetric | HepLean.SpaceTime.MinkowskiMetric.jsonl | {
"lineInFile": 331,
"tokenPositionInFile": 10585,
"theoremPositionInFile": 37
} | {
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"importedModules": [
"Init.Prelude",
"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n rw [basis_left, stdBasis_apply]\n by_cases h : μ = ν\n · simp [h]\n · simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, diagonal_apply_eq,\n mul_ite, mul_one, mul_zero, ne_eq, h, not_false_eq_true, diagonal_apply_ne, ite_eq_right_iff]\n exact fun a... |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.MinkowskiMetric
import Mathlib.Algebra.Lie.Classical
/-!
# The Lorentz Algebra
We define
- Define `lorentzAlgebra` via `LieAlgebra.... | lemma diag_comp (Λ : lorentzAlgebra) (μ : Fin 1 ⊕ Fin 3) : Λ.1 μ μ = 0 | SpaceTime.lorentzAlgebra.diag_comp | {
"commit": "a52d8ea452f8b6c9f0d5833190b01e18d19cc4b4",
"date": "2024-05-24T00:00:00"
} | {
"commit": "28c9086f0df4cb6b566a6adbd391e5b6fcb0ee7f",
"date": "2024-06-12T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzAlgebra/Basic.lean | HepLean.SpaceTime.LorentzAlgebra.Basic | HepLean.SpaceTime.LorentzAlgebra.Basic.jsonl | {
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"tokenPositionInFile": 1979,
"theoremPositionInFile": 5
} | {
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"importedModules": [
"Init.Prelude",
"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n have h := congrArg (fun M ↦ M μ μ) $ mem_iff.mp Λ.2\n simp only [minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, mul_diagonal,\n transpose_apply, diagonal_neg, diagonal_mul, neg_mul] at h\n rcases μ with μ | μ\n · simp only [Sum.elim_inl, mul_one, one_mul] at ... |
/-
Copyright (c) 2024 Joseph Tooby-Smith. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Tooby-Smith
-/
import HepLean.SpaceTime.MinkowskiMetric
import Mathlib.Algebra.Lie.Classical
/-!
# The Lorentz Algebra
We define
- Define `lorentzAlgebra` via `LieAlgebra.... | lemma time_comps (Λ : lorentzAlgebra) (i : Fin 3) :
Λ.1 (Sum.inr i) (Sum.inl 0) = Λ.1 (Sum.inl 0) (Sum.inr i) | SpaceTime.lorentzAlgebra.time_comps | {
"commit": "a52d8ea452f8b6c9f0d5833190b01e18d19cc4b4",
"date": "2024-05-24T00:00:00"
} | {
"commit": "28c9086f0df4cb6b566a6adbd391e5b6fcb0ee7f",
"date": "2024-06-12T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzAlgebra/Basic.lean | HepLean.SpaceTime.LorentzAlgebra.Basic | HepLean.SpaceTime.LorentzAlgebra.Basic.jsonl | {
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"Init.Coe",
"Init.Notation",
"Init.Tactics",
"Init.SizeOf",
"Init.Core",
"Init.MetaTypes",
"Init.SimpLemmas",
... | {
"hasProof": true,
"proof": ":= by\n simpa only [Fin.isValue, minkowskiMatrix, LieAlgebra.Orthogonal.indefiniteDiagonal, mul_diagonal,\n transpose_apply, Sum.elim_inr, mul_neg, mul_one, diagonal_neg, diagonal_mul, Sum.elim_inl,\n neg_mul, one_mul, neg_inj] using congrArg (fun M ↦ M (Sum.inl 0) (Sum.inr i)) ... |
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