The full dataset viewer is not available (click to read why). Only showing a preview of the rows.
The dataset generation failed because of a cast error
Error code: DatasetGenerationCastError
Exception: DatasetGenerationCastError
Message: An error occurred while generating the dataset
All the data files must have the same columns, but at some point there are 1 new columns ({'split'}) and 2 missing columns ({'jsonFile', 'module'}).
This happened while the json dataset builder was generating data using
hf://datasets/l3lab/miniCTX/minictx/mathlib.jsonl (at revision 0bbb78f73b2cd61f8e0d5a7e108e4865763746c3)
Please either edit the data files to have matching columns, or separate them into different configurations (see docs at https://hf.co/docs/hub/datasets-manual-configuration#multiple-configurations)
Traceback: Traceback (most recent call last):
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1869, in _prepare_split_single
writer.write_table(table)
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/arrow_writer.py", line 580, in write_table
pa_table = table_cast(pa_table, self._schema)
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/table.py", line 2292, in table_cast
return cast_table_to_schema(table, schema)
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/table.py", line 2240, in cast_table_to_schema
raise CastError(
datasets.table.CastError: Couldn't cast
srcContext: string
theoremStatement: string
theoremName: string
fileCreated: struct<commit: string, date: timestamp[s]>
child 0, commit: string
child 1, date: timestamp[s]
theoremCreated: struct<commit: string, date: timestamp[s]>
child 0, commit: string
child 1, date: timestamp[s]
file: string
positionMetadata: struct<lineInFile: int64, tokenPositionInFile: int64, theoremPositionInFile: int64>
child 0, lineInFile: int64
child 1, tokenPositionInFile: int64
child 2, theoremPositionInFile: int64
dependencyMetadata: struct<inFilePremises: bool, repositoryPremises: bool>
child 0, inFilePremises: bool
child 1, repositoryPremises: bool
proofMetadata: struct<hasProof: bool, proof: string, proofType: string, proofLengthLines: int64, proofLengthTokens: int64>
child 0, hasProof: bool
child 1, proof: string
child 2, proofType: string
child 3, proofLengthLines: int64
child 4, proofLengthTokens: int64
split: string
to
{'srcContext': Value(dtype='string', id=None), 'theoremStatement': Value(dtype='string', id=None), 'theoremName': Value(dtype='string', id=None), 'fileCreated': {'commit': Value(dtype='string', id=None), 'date': Value(dtype='timestamp[s]', id=None)}, 'theoremCreated': {'commit': Value(dtype='string', id=None), 'date': Value(dtype='timestamp[s]', id=None)}, 'file': Value(dtype='string', id=None), 'module': Value(dtype='string', id=None), 'jsonFile': Value(dtype='string', id=None), 'positionMetadata': {'lineInFile': Value(dtype='int64', id=None), 'tokenPositionInFile': Value(dtype='int64', id=None), 'theoremPositionInFile': Value(dtype='int64', id=None)}, 'dependencyMetadata': {'inFilePremises': Value(dtype='bool', id=None), 'numInFilePremises': Value(dtype='int64', id=None), 'repositoryPremises': Value(dtype='bool', id=None), 'numRepositoryPremises': Value(dtype='int64', id=None), 'numPremises': Value(dtype='int64', id=None), 'importedModules': Sequence(feature=Value(dtype='string', id=None), length=-1, id=None)}, 'proofMetadata': {'hasProof': Value(dtype='bool', id=None), 'proof': Value(dtype='string', id=None), 'proofType': Value(dtype='string', id=None), 'proofLengthLines': Value(dtype='int64', id=None), 'proofLengthTokens': Value(dtype='int64', id=None)}}
because column names don't match
During handling of the above exception, another exception occurred:
Traceback (most recent call last):
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1392, in compute_config_parquet_and_info_response
parquet_operations = convert_to_parquet(builder)
File "/src/services/worker/src/worker/job_runners/config/parquet_and_info.py", line 1041, in convert_to_parquet
builder.download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 924, in download_and_prepare
self._download_and_prepare(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 999, in _download_and_prepare
self._prepare_split(split_generator, **prepare_split_kwargs)
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1740, in _prepare_split
for job_id, done, content in self._prepare_split_single(
File "/src/services/worker/.venv/lib/python3.9/site-packages/datasets/builder.py", line 1871, in _prepare_split_single
raise DatasetGenerationCastError.from_cast_error(
datasets.exceptions.DatasetGenerationCastError: An error occurred while generating the dataset
All the data files must have the same columns, but at some point there are 1 new columns ({'split'}) and 2 missing columns ({'jsonFile', 'module'}).
This happened while the json dataset builder was generating data using
hf://datasets/l3lab/miniCTX/minictx/mathlib.jsonl (at revision 0bbb78f73b2cd61f8e0d5a7e108e4865763746c3)
Please either edit the data files to have matching columns, or separate them into different configurations (see docs at https://hf.co/docs/hub/datasets-manual-configuration#multiple-configurations)Need help to make the dataset viewer work? Make sure to review how to configure the dataset viewer, and open a discussion for direct support.
srcContext
string | theoremStatement
string | theoremName
string | fileCreated
dict | theoremCreated
dict | file
string | module
string | jsonFile
string | positionMetadata
dict | dependencyMetadata
dict | proofMetadata
dict |
|---|---|---|---|---|---|---|---|---|---|---|
/-
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.Analysis.Complex.Basic
/-!
# The Clifford Algebra
This file defines the Gamma matrices.
-/
/-! TODO: Prove algebra generated by gamma matrices is isomorphic to Clifford algebra. -/
/-! TODO: Define relations between the gamma matrices. -/
namespace spaceTime
open Complex
noncomputable section diracRepresentation
/-- The γ⁰ gamma matrix in the Dirac representation. -/
def γ0 : Matrix (Fin 4) (Fin 4) ℂ :=
![![1, 0, 0, 0], ![0, 1, 0, 0], ![0, 0, -1, 0], ![0, 0, 0, -1]]
/-- The γ¹ gamma matrix in the Dirac representation. -/
def γ1 : Matrix (Fin 4) (Fin 4) ℂ :=
![![0, 0, 0, 1], ![0, 0, 1, 0], ![0, -1, 0, 0], ![-1, 0, 0, 0]]
/-- The γ² gamma matrix in the Dirac representation. -/
def γ2 : Matrix (Fin 4) (Fin 4) ℂ :=
![![0, 0, 0, - I], ![0, 0, I, 0], ![0, I, 0, 0], ![-I, 0, 0, 0]]
/-- The γ³ gamma matrix in the Dirac representation. -/
def γ3 : Matrix (Fin 4) (Fin 4) ℂ :=
![![0, 0, 1, 0], ![0, 0, 0, -1], ![-1, 0, 0, 0], ![0, 1, 0, 0]]
/-- The γ⁵ gamma matrix in the Dirac representation. -/
def γ5 : Matrix (Fin 4) (Fin 4) ℂ := I • (γ0 * γ1 * γ2 * γ3)
/-- The γ gamma matrices in the Dirac representation. -/
@[simp]
def γ : Fin 4 → Matrix (Fin 4) (Fin 4) ℂ := ![γ0, γ1, γ2, γ3]
namespace γ
open spaceTime
variable (μ ν : Fin 4)
/-- The subset of `Matrix (Fin 4) (Fin 4) ℂ` formed by the gamma matrices in the Dirac
representation. -/
@[simp]
def γSet : Set (Matrix (Fin 4) (Fin 4) ℂ) := {γ i | i : Fin 4}
| lemma γ_in_γSet (μ : Fin 4) : γ μ ∈ γSet | spaceTime.γ.γ_in_γSet | {
"commit": "7020263",
"date": "2024-05-14T00:00:00"
} | {
"commit": "b9c2086",
"date": "2024-05-10T00:00:00"
} | hep_lean/HepLean/SpaceTime/CliffordAlgebra.lean | HepLean.SpaceTime.CliffordAlgebra | HepLean.SpaceTime.CliffordAlgebra.jsonl | {
"lineInFile": 54,
"tokenPositionInFile": 1613,
"theoremPositionInFile": 7
} | {
"inFilePremises": true,
"numInFilePremises": 6,
"repositoryPremises": true,
"numRepositoryPremises": 6,
"numPremises": 23,
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"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Opposites",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Init.Order.LinearOrder",
"Mathlib.Order.Notation",
"Mathlib.Order.Basic",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Tactic.FBinop",
"Mathlib.Data.SProd",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.BoundedOrder",
"Mathlib.Order.Disjoint",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Data.Option.NAry",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.WithBot",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Data.Int.Basic",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Control.Functor",
"Mathlib.Data.List.Defs",
"Mathlib.Init.Data.List.Basic",
"Mathlib.Data.List.GetD",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Rat.Init",
"Mathlib.Data.Rat.Defs",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Data.Rat.Order",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.Algebra.Module.Pi",
"Mathlib.Algebra.Module.Prod",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Pi",
"Mathlib.Algebra.Order.Module.OrderedSMul",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Algebra.BigOperators.List.Defs",
"Mathlib.Init.Data.List.Instances",
"Mathlib.Init.Data.List.Lemmas",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Algebra.Parity",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Algebra.GeomSum",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Basic",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Archimedean",
"Mathlib.Order.Iterate",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
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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.Analysis.Complex.Basic
/-!
# The Clifford Algebra
This file defines the Gamma matrices.
-/
/-! TODO: Prove algebra generated by gamma matrices is isomorphic to Clifford algebra. -/
/-! TODO: Define relations between the gamma matrices. -/
namespace spaceTime
open Complex
noncomputable section diracRepresentation
/-- The γ⁰ gamma matrix in the Dirac representation. -/
def γ0 : Matrix (Fin 4) (Fin 4) ℂ :=
![![1, 0, 0, 0], ![0, 1, 0, 0], ![0, 0, -1, 0], ![0, 0, 0, -1]]
/-- The γ¹ gamma matrix in the Dirac representation. -/
def γ1 : Matrix (Fin 4) (Fin 4) ℂ :=
![![0, 0, 0, 1], ![0, 0, 1, 0], ![0, -1, 0, 0], ![-1, 0, 0, 0]]
/-- The γ² gamma matrix in the Dirac representation. -/
def γ2 : Matrix (Fin 4) (Fin 4) ℂ :=
![![0, 0, 0, - I], ![0, 0, I, 0], ![0, I, 0, 0], ![-I, 0, 0, 0]]
/-- The γ³ gamma matrix in the Dirac representation. -/
def γ3 : Matrix (Fin 4) (Fin 4) ℂ :=
![![0, 0, 1, 0], ![0, 0, 0, -1], ![-1, 0, 0, 0], ![0, 1, 0, 0]]
/-- The γ⁵ gamma matrix in the Dirac representation. -/
def γ5 : Matrix (Fin 4) (Fin 4) ℂ := I • (γ0 * γ1 * γ2 * γ3)
/-- The γ gamma matrices in the Dirac representation. -/
@[simp]
def γ : Fin 4 → Matrix (Fin 4) (Fin 4) ℂ := ![γ0, γ1, γ2, γ3]
namespace γ
open spaceTime
variable (μ ν : Fin 4)
/-- The subset of `Matrix (Fin 4) (Fin 4) ℂ` formed by the gamma matrices in the Dirac
representation. -/
@[simp]
def γSet : Set (Matrix (Fin 4) (Fin 4) ℂ) := {γ i | i : Fin 4}
lemma γ_in_γSet (μ : Fin 4) : γ μ ∈ γSet := by
simp [γSet]
/-- The algebra generated by the gamma matrices in the Dirac representation. -/
def diracAlgebra : Subalgebra ℝ (Matrix (Fin 4) (Fin 4) ℂ) :=
Algebra.adjoin ℝ γSet
| lemma γSet_subset_diracAlgebra : γSet ⊆ diracAlgebra | spaceTime.γ.γSet_subset_diracAlgebra | {
"commit": "7020263",
"date": "2024-05-14T00:00:00"
} | {
"commit": "b9c2086",
"date": "2024-05-10T00:00:00"
} | hep_lean/HepLean/SpaceTime/CliffordAlgebra.lean | HepLean.SpaceTime.CliffordAlgebra | HepLean.SpaceTime.CliffordAlgebra.jsonl | {
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"Lean.Elab.Deriving.Ord",
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"Lean.Elab.InheritDoc",
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"Lean.Elab.CheckTactic",
"Lean.Elab.MatchExpr",
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"Lean",
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"Std.Data.MLList.IO",
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"Std.Data.RBMap.Basic",
"Std.Data.RBMap.WF",
"Std.Data.RBMap.Alter",
"Std.Data.RBMap.Lemmas",
"Std.Data.RBMap",
"Std.Data.Range.Lemmas",
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"Std.Data.Rat.Lemmas",
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"Std.Data.String.Lemmas",
"Std.Data.String",
"Std.Data.Sum.Basic",
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"Std.Data.Sum",
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"Std.Data.UnionFind.Basic",
"Std.Data.UnionFind.Lemmas",
"Std.Data.UnionFind",
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"Std.Lean.AttributeExtra",
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"Std.Lean.IO.Process",
"Std.Lean.Json",
"Std.Lean.Meta.AssertHypotheses",
"Std.Lean.Meta.Clear",
"Std.Lean.Meta.Expr",
"Std.Lean.PersistentHashMap",
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"Std.Lean.MonadBacktrack",
"Std.Lean.Meta.SavedState",
"Std.Lean.Meta.Simp",
"Std.Lean.Meta.UnusedNames",
"Std.Lean.NameMap",
"Std.Lean.PersistentHashSet",
"Std.Lean.SMap",
"Std.Lean.Syntax",
"Std.Lean.Util.EnvSearch",
"Std.Lean.Util.Path",
"Std.Tactic.Unreachable",
"Std.Linter.UnreachableTactic",
"Std.Linter.UnnecessarySeqFocus",
"Std.Linter",
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"Std.Tactic.Classical",
"Std.Tactic.Congr",
"Std.Tactic.Exact",
"Std.Tactic.FalseOrByContra",
"Std.Tactic.Instances",
"Std.Tactic.NoMatch",
"Std.Tactic.PermuteGoals",
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"Std.Util.CheckTactic",
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"Std.Util.Pickle",
"Std.WF",
"Std",
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"Mathlib.Tactic.ExtendDoc",
"Mathlib.Tactic.TypeStar",
"Mathlib.Util.AssertExists",
"Mathlib.Algebra.Group.Defs",
"Mathlib.Mathport.Attributes",
"Mathlib.Tactic.ProjectionNotation",
"Mathlib.Init.Logic",
"Mathlib.Tactic.Cases",
"Mathlib.Algebra.Group.Semiconj.Defs",
"Mathlib.Util.CompileInductive",
"Mathlib.Init.Data.Nat.Basic",
"Mathlib.Init.Algebra.Classes",
"Mathlib.Init.Data.Ordering.Basic",
"Mathlib.Tactic.Core",
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"Mathlib.Init.Order.Defs",
"Mathlib.Init.Data.Nat.Lemmas",
"Mathlib.Tactic.Basic",
"Mathlib.Tactic.Attr.Register",
"Mathlib.Init.Function",
"Mathlib.Logic.Nonempty",
"Mathlib.Init.Set",
"Mathlib.Logic.Basic",
"Mathlib.Logic.Function.Basic",
"Mathlib.Logic.Nontrivial.Defs",
"Mathlib.Tactic.GCongr.ForwardAttr",
"Mathlib.Tactic.GCongr.Core",
"Mathlib.Tactic.Conv",
"Mathlib.Tactic.PushNeg",
"Mathlib.Tactic.Use",
"Mathlib.Data.Nat.Defs",
"Mathlib.Algebra.Group.Commute.Defs",
"Aesop.Check",
"Aesop.Nanos",
"Aesop.Util.UnionFind",
"Aesop.Util.UnorderedArraySet",
"Aesop.Util.Basic",
"Aesop.Rule.Name",
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"Aesop.RulePattern",
"Aesop.Index.Basic",
"Aesop.Options.Public",
"Aesop.Options.Internal",
"Aesop.Options",
"Aesop.Percent",
"Aesop.Util.Tactic",
"Aesop.Util.EqualUpToIds",
"Aesop.Script",
"Aesop.RuleTac.Basic",
"Aesop.Rule.Basic",
"Aesop.Index",
"Aesop.Rule",
"Aesop.RuleSet.Member",
"Aesop.RuleSet.Name",
"Aesop.RuleSet.Filter",
"Aesop.RuleSet",
"Aesop.Frontend.Extension.Init",
"Aesop.Frontend.Extension",
"Aesop.ElabM",
"Aesop.Frontend.Basic",
"Aesop.RuleTac.ElabRuleTerm",
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"Aesop.RuleTac.Cases",
"Aesop.Builder.Cases",
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"Aesop.Builder.NormSimp",
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"Aesop.Builder",
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"Aesop.RuleTac.Apply",
"Aesop.RuleTac.Forward",
"Aesop.RuleTac.Preprocess",
"Aesop.RuleTac.Tactic",
"Aesop.RuleTac",
"Aesop.Search.Expansion.Basic",
"Aesop.Search.Expansion.Simp",
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"Aesop.Tree.UnsafeQueue",
"Aesop.Tree.Data",
"Aesop.Tree.Traversal",
"Aesop.Tree.RunMetaM",
"Aesop.Tree.TreeM",
"Aesop.Tree.AddRapp",
"Aesop.Tree.State",
"Aesop.Tree.Check",
"Lean.Replay",
"Aesop.Tree.Tracing",
"Aesop.Tree.ExtractProof",
"Aesop.Tree.ExtractScript",
"Aesop.Tree.Free",
"Aesop.Tree",
"Aesop.Search.Queue.Class",
"Aesop.Stats.Basic",
"Aesop.Search.SearchM",
"Aesop.Search.RuleSelection",
"Aesop.Search.Expansion.Norm",
"Aesop.Search.Expansion",
"Aesop.Exception",
"Aesop.Search.ExpandSafePrefix",
"Aesop.Search.Queue",
"Aesop.Search.Main",
"Aesop.BuiltinRules.Assumption",
"Aesop.BuiltinRules.ApplyHyps",
"Aesop.BuiltinRules.DestructProducts",
"Aesop.BuiltinRules.Ext",
"Aesop.BuiltinRules.Intros",
"Aesop.BuiltinRules.Split",
"Aesop.BuiltinRules.Subst",
"Aesop.Stats.Extension",
"Aesop.Stats.Report",
"Aesop.Frontend.Command",
"Aesop.Frontend.Tactic",
"Aesop.Frontend",
"Aesop.BuiltinRules",
"Aesop.Main",
"Aesop",
"Mathlib.Tactic.SimpRw",
"Mathlib.Algebra.Group.Basic",
"Mathlib.Tactic.Inhabit",
"Mathlib.Data.Prod.Basic",
"Mathlib.Lean.Name",
"Mathlib.Tactic.MkIffOfInductiveProp",
"Mathlib.Data.Sum.Basic",
"Mathlib.Logic.IsEmpty",
"Mathlib.Logic.Unique",
"Mathlib.Tactic.Spread",
"Mathlib.Algebra.Group.Pi.Basic",
"Mathlib.Data.FunLike.Basic",
"Mathlib.Algebra.Group.Hom.Defs",
"Mathlib.Algebra.Group.Hom.Basic",
"Mathlib.Data.FunLike.Embedding",
"Mathlib.Data.FunLike.Equiv",
"Mathlib.Data.Bool.Basic",
"Mathlib.Data.Option.Defs",
"Mathlib.Data.Sigma.Basic",
"Mathlib.Data.Subtype",
"Mathlib.Init.Data.Sigma.Basic",
"Mathlib.Init.Data.Quot",
"Mathlib.Logic.Relator",
"Mathlib.Lean.Elab.Term",
"Mathlib.Lean.PrettyPrinter.Delaborator",
"Mathlib.Util.WithWeakNamespace",
"Mathlib.Tactic.ScopedNS",
"Mathlib.Mathport.Notation",
"Mathlib.Data.Quot",
"Mathlib.Tactic.Coe",
"Mathlib.Init.Data.Bool.Lemmas",
"Mathlib.Tactic.Substs",
"Mathlib.Logic.Equiv.Defs",
"Mathlib.Logic.Function.Conjugate",
"Mathlib.Tactic.Lift",
"Mathlib.Lean.Meta.CongrTheorems",
"Mathlib.Tactic.Relation.Rfl",
"Mathlib.Tactic.Congr!",
"Mathlib.Tactic.Convert",
"Mathlib.Tactic.Contrapose",
"Mathlib.Tactic.GeneralizeProofs",
"Mathlib.Logic.Equiv.Basic",
"Mathlib.Algebra.Group.Equiv.Basic",
"Mathlib.Data.Nat.Cast.Defs",
"Mathlib.Data.Int.Cast.Defs",
"Mathlib.Data.Int.Cast.Basic",
"Mathlib.Algebra.Group.InjSurj",
"Mathlib.Algebra.Group.Semiconj.Basic",
"Mathlib.Algebra.Group.Commute.Basic",
"Mathlib.Algebra.GroupWithZero.Defs",
"Mathlib.Data.Int.Defs",
"Qq.ForLean.ReduceEval",
"Qq.ForLean.ToExpr",
"Qq.Typ",
"Qq.Macro",
"Qq.Delab",
"Qq.MetaM",
"Qq.ForLean.Do",
"Qq.SortLocalDecls",
"Qq.Match",
"Qq.AssertInstancesCommute",
"Qq",
"ImportGraph.RequiredModules",
"ImportGraph.Imports",
"Mathlib.Tactic.ApplyCongr",
"Mathlib.Lean.Meta.Basic",
"Mathlib.Tactic.ApplyAt",
"Mathlib.Tactic.ApplyWith",
"Mathlib.Tactic.ByContra",
"Mathlib.Tactic.CasesM",
"Mathlib.Tactic.Check",
"Mathlib.Util.Tactic",
"Mathlib.Tactic.Choose",
"Mathlib.Tactic.Clear!",
"Mathlib.Tactic.ClearExcept",
"Mathlib.Tactic.Clear_",
"Mathlib.Tactic.TermCongr",
"Mathlib.Tactic.Congrm",
"Mathlib.Tactic.Constructor",
"Mathlib.Tactic.DefEqTransformations",
"Mathlib.Tactic.ToLevel",
"Mathlib.Tactic.DeriveToExpr",
"Mathlib.Tactic.Existsi",
"Mathlib.Tactic.ExtractGoal",
"Mathlib.Tactic.ExtractLets",
"Mathlib.Tactic.FailIfNoProgress",
"Mathlib.Tactic.Find",
"Mathlib.Tactic.GuardGoalNums",
"Mathlib.Tactic.GuardHypNums",
"Mathlib.Tactic.HelpCmd",
"Mathlib.Tactic.HigherOrder",
"Mathlib.Tactic.Hint",
"Mathlib.Tactic.InferParam",
"Mathlib.Tactic.IrreducibleDef",
"Mathlib.Tactic.Lint",
"Mathlib.Tactic.NthRewrite",
"Mathlib.Tactic.Observe",
"Mathlib.Tactic.Propose",
"Mathlib.Tactic.RSuffices",
"Mathlib.Tactic.Recover",
"Mathlib.Tactic.Rename",
"Mathlib.Tactic.RenameBVar",
"Mathlib.Init.Core",
"Mathlib.Init.Control.Combinators",
"Mathlib.Tactic.Attr.Core",
"Mathlib.Control.Basic",
"Mathlib.Data.MLList.Dedup",
"Mathlib.Lean.Meta.DiscrTree",
"Mathlib.Tactic.Rewrites",
"Mathlib.Tactic.Says",
"Mathlib.Tactic.Set",
"Mathlib.Tactic.SimpIntro",
"Mathlib.Tactic.SuccessIfFailWithMsg",
"Mathlib.Tactic.SudoSetOption",
"Mathlib.Tactic.SwapVar",
"Mathlib.Tactic.Tauto",
"Mathlib.Util.WhatsNew",
"Mathlib.Tactic.ToExpr",
"Mathlib.Tactic.Trace",
"Mathlib.Tactic.TypeCheck",
"Mathlib.Tactic.UnsetOption",
"Mathlib.Tactic.Variable",
"ProofWidgets.Compat",
"ProofWidgets.Component.Basic",
"ProofWidgets.Component.MakeEditLink",
"ProofWidgets.Data.Html",
"ProofWidgets.Cancellable",
"ProofWidgets.Component.OfRpcMethod",
"Mathlib.Tactic.Widget.SelectInsertParamsClass",
"Mathlib.Tactic.Widget.SelectPanelUtils",
"Mathlib.Tactic.Widget.Calc",
"Mathlib.Tactic.Widget.Congrm",
"Mathlib.Tactic.Widget.Conv",
"Mathlib.Tactic.WLOG",
"Mathlib.Util.CountHeartbeats",
"Mathlib.Tactic.Common",
"Mathlib.Algebra.GroupPower.Basic",
"Mathlib.Logic.Nontrivial.Basic",
"Mathlib.Tactic.Nontriviality.Core",
"Mathlib.Tactic.Nontriviality",
"Mathlib.Algebra.Group.Units",
"Mathlib.Algebra.Opposites",
"Mathlib.Algebra.Group.Opposite",
"Mathlib.Algebra.Group.Units.Hom",
"Mathlib.Algebra.NeZero",
"Mathlib.Algebra.GroupWithZero.Hom",
"Mathlib.Algebra.GroupWithZero.NeZero",
"Mathlib.Init.Order.LinearOrder",
"Mathlib.Order.Notation",
"Mathlib.Order.Basic",
"Mathlib.Order.Synonym",
"Mathlib.Algebra.Group.OrderSynonym",
"Mathlib.Algebra.GroupWithZero.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Basic",
"Mathlib.Algebra.Group.Prod",
"Mathlib.Logic.Function.Iterate",
"Mathlib.Init.Data.Int.Order",
"Mathlib.Order.Compare",
"Mathlib.Order.Max",
"Mathlib.Init.Propext",
"Mathlib.Logic.Relation",
"Mathlib.Order.RelClasses",
"Mathlib.Order.Monotone.Basic",
"Mathlib.Order.ULift",
"Mathlib.Order.Lattice",
"Mathlib.Order.MinMax",
"Mathlib.Tactic.FBinop",
"Mathlib.Data.SProd",
"Mathlib.Data.Set.Defs",
"Mathlib.Order.BoundedOrder",
"Mathlib.Order.Disjoint",
"Mathlib.Order.PropInstances",
"Mathlib.Order.Heyting.Basic",
"Mathlib.Order.BooleanAlgebra",
"Mathlib.Order.SymmDiff",
"Mathlib.Util.Delaborators",
"Mathlib.Data.Set.Basic",
"Mathlib.Data.Set.Intervals.Basic",
"Mathlib.Data.Option.NAry",
"Mathlib.Data.Option.Basic",
"Mathlib.Order.WithBot",
"Mathlib.Data.Set.Image",
"Mathlib.Data.Set.Prod",
"Mathlib.Data.Set.Function",
"Mathlib.Order.Directed",
"Mathlib.Data.Set.Intervals.Image",
"Mathlib.Data.Set.NAry",
"Mathlib.Order.Bounds.Basic",
"Mathlib.Data.Set.Intervals.UnorderedInterval",
"Mathlib.Control.EquivFunctor",
"Mathlib.Logic.Equiv.Option",
"Mathlib.Data.Prod.PProd",
"Mathlib.Logic.Embedding.Basic",
"Mathlib.Order.RelIso.Basic",
"Mathlib.Tactic.Monotonicity.Attr",
"Mathlib.Order.Hom.Basic",
"Mathlib.Data.Set.Intervals.OrderEmbedding",
"Mathlib.Logic.Pairwise",
"Mathlib.Data.Set.Pairwise.Basic",
"Mathlib.Logic.Equiv.Set",
"Mathlib.Order.Hom.Set",
"Mathlib.Order.Antichain",
"Mathlib.Order.SetNotation",
"Mathlib.Data.Set.Intervals.OrdConnected",
"Mathlib.Order.Antisymmetrization",
"Mathlib.Order.Cover",
"Mathlib.Algebra.Function.Support",
"Mathlib.Algebra.Ring.Defs",
"Mathlib.Algebra.Ring.Basic",
"Mathlib.Algebra.Group.Hom.Instances",
"Mathlib.Algebra.Group.Pi.Lemmas",
"Mathlib.Algebra.Function.Indicator",
"Mathlib.Algebra.Ring.Hom.Defs",
"Mathlib.Algebra.Ring.Opposite",
"Mathlib.Data.Finite.Defs",
"Mathlib.Algebra.Group.TypeTags",
"Mathlib.GroupTheory.GroupAction.Defs",
"Mathlib.GroupTheory.GroupAction.Opposite",
"Mathlib.GroupTheory.GroupAction.Prod",
"Mathlib.Algebra.SMulWithZero",
"Mathlib.Data.Int.Basic",
"Mathlib.Algebra.Divisibility.Basic",
"Mathlib.Data.Nat.Basic",
"Mathlib.Data.Nat.Cast.Basic",
"Mathlib.Algebra.CharZero.Defs",
"Mathlib.Algebra.CovariantAndContravariant",
"Mathlib.Algebra.Order.Monoid.Lemmas",
"Mathlib.Algebra.Order.Monoid.Defs",
"Mathlib.Algebra.Order.Monoid.OrderDual",
"Mathlib.Algebra.GroupPower.CovariantClass",
"Mathlib.Algebra.Group.Units.Equiv",
"Mathlib.Algebra.Order.Sub.Defs",
"Mathlib.Algebra.Order.Group.Defs",
"Mathlib.Algebra.Order.Group.OrderIso",
"Mathlib.Algebra.Order.Group.Lattice",
"Mathlib.Algebra.Order.Group.Abs",
"Mathlib.Data.Nat.Cast.NeZero",
"Mathlib.Algebra.Group.WithOne.Defs",
"Mathlib.Algebra.GroupWithZero.InjSurj",
"Mathlib.Algebra.GroupWithZero.Units.Equiv",
"Mathlib.Algebra.Order.Monoid.Units",
"Mathlib.Algebra.Order.Group.Units",
"Mathlib.Algebra.Order.Monoid.Basic",
"Mathlib.Algebra.Order.Monoid.Canonical.Defs",
"Mathlib.Algebra.Order.Monoid.TypeTags",
"Mathlib.Algebra.Order.ZeroLEOne",
"Mathlib.Algebra.Order.Monoid.WithZero",
"Mathlib.Algebra.Order.Monoid.MinMax",
"Mathlib.Algebra.Order.Monoid.NatCast",
"Mathlib.Algebra.Order.Ring.Lemmas",
"Mathlib.Algebra.Order.Ring.Defs",
"Mathlib.Algebra.Order.Sub.Canonical",
"Mathlib.Algebra.Order.Ring.Canonical",
"Mathlib.Data.Nat.Order.Basic",
"Mathlib.Data.Nat.Cast.Order",
"Mathlib.Algebra.Order.Group.InjSurj",
"Mathlib.Algebra.Ring.InjSurj",
"Mathlib.Algebra.Order.Ring.InjSurj",
"Mathlib.Order.WellFounded",
"Mathlib.Data.Bool.Set",
"Mathlib.Data.Nat.Set",
"Mathlib.Control.ULift",
"Mathlib.Data.ULift",
"Mathlib.Order.CompleteLattice",
"Mathlib.Order.CompleteBooleanAlgebra",
"Mathlib.Order.GaloisConnection",
"Mathlib.Data.Set.Lattice",
"Mathlib.Order.ConditionallyCompleteLattice.Basic",
"Mathlib.Order.LatticeIntervals",
"Mathlib.Order.CompleteLatticeIntervals",
"Mathlib.Algebra.Order.Nonneg.Ring",
"Mathlib.Control.Functor",
"Mathlib.Data.List.Defs",
"Mathlib.Init.Data.List.Basic",
"Mathlib.Data.List.GetD",
"Mathlib.Init.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Bits",
"Mathlib.Data.Nat.Bitwise",
"Mathlib.Data.Nat.Size",
"Mathlib.Init.Data.Int.Bitwise",
"Mathlib.Data.Int.Bitwise",
"Mathlib.Algebra.GroupPower.Hom",
"Mathlib.Algebra.Group.Semiconj.Units",
"Mathlib.Init.Classical",
"Mathlib.Algebra.GroupWithZero.Semiconj",
"Mathlib.Algebra.Group.Commute.Units",
"Mathlib.Algebra.GroupWithZero.Commute",
"Mathlib.Algebra.Divisibility.Units",
"Mathlib.Algebra.GroupWithZero.Divisibility",
"Mathlib.Algebra.Ring.Semiconj",
"Mathlib.Algebra.Ring.Units",
"Mathlib.Data.Bracket",
"Mathlib.Algebra.Ring.Commute",
"Mathlib.Algebra.Ring.Divisibility.Basic",
"Mathlib.Algebra.GroupPower.Ring",
"Mathlib.Algebra.GroupPower.Order",
"Mathlib.Algebra.Order.Ring.CharZero",
"Mathlib.Data.Int.Order.Basic",
"Mathlib.Algebra.Order.Ring.Abs",
"Mathlib.Data.Int.Order.Lemmas",
"Mathlib.Data.Int.Lemmas",
"Mathlib.Data.Rat.Init",
"Mathlib.Data.Rat.Defs",
"Mathlib.Algebra.Ring.Hom.Basic",
"Mathlib.Data.Nat.Cast.Commute",
"Mathlib.Data.Int.Cast.Lemmas",
"Mathlib.Data.Rat.Order",
"Mathlib.Data.NNRat.Defs",
"Mathlib.Algebra.GroupPower.IterateHom",
"Mathlib.GroupTheory.Perm.Basic",
"Mathlib.Algebra.Group.Aut",
"Mathlib.Algebra.Invertible.Defs",
"Mathlib.Algebra.Invertible.GroupWithZero",
"Mathlib.Algebra.Invertible.Basic",
"Mathlib.GroupTheory.GroupAction.Units",
"Mathlib.GroupTheory.GroupAction.Group",
"Mathlib.GroupTheory.GroupAction.Pi",
"Mathlib.Algebra.Field.Defs",
"Mathlib.Tactic.NormNum.Result",
"Mathlib.Util.Qq",
"Mathlib.Tactic.NormNum.Core",
"Mathlib.Tactic.HaveI",
"Mathlib.Tactic.NormNum.Basic",
"Mathlib.Algebra.GroupWithZero.Units.Lemmas",
"Mathlib.Algebra.Order.Field.Defs",
"Mathlib.Data.Rat.Field",
"Mathlib.Data.Int.Dvd.Basic",
"Mathlib.Data.Nat.Units",
"Mathlib.Data.Nat.Order.Lemmas",
"Mathlib.Algebra.Regular.Basic",
"Mathlib.Algebra.Ring.Regular",
"Mathlib.Data.Int.Div",
"Mathlib.Data.PNat.Defs",
"Mathlib.Data.Rat.Lemmas",
"Mathlib.Data.Rat.Cast.Defs",
"Mathlib.Tactic.NormNum.OfScientific",
"Mathlib.Data.Int.Cast.Field",
"Mathlib.Data.Int.CharZero",
"Mathlib.Data.Rat.Cast.CharZero",
"Mathlib.Algebra.Field.Basic",
"Mathlib.Tactic.NormNum.Inv",
"Mathlib.Tactic.NormNum.Eq",
"Mathlib.Algebra.Order.Monoid.WithTop",
"Mathlib.Algebra.Order.Invertible",
"Mathlib.Tactic.NormNum.Ineq",
"Mathlib.Tactic.NormNum.Pow",
"Mathlib.Tactic.NormNum.DivMod",
"Mathlib.Tactic.Positivity.Core",
"Mathlib.Data.Rat.Cast.Order",
"Mathlib.Tactic.NormNum",
"Mathlib.Util.AtomM",
"Mathlib.Tactic.Abel",
"Mathlib.Algebra.Module.Basic",
"Mathlib.Algebra.Regular.SMul",
"Mathlib.Algebra.Field.IsField",
"Mathlib.Algebra.Ring.Equiv",
"Mathlib.Algebra.Ring.CompTypeclasses",
"Mathlib.Algebra.Ring.Pi",
"Mathlib.Algebra.Module.Pi",
"Mathlib.Algebra.Module.Prod",
"Mathlib.Algebra.Ring.OrderSynonym",
"Mathlib.Algebra.Order.Module.Synonym",
"Mathlib.Algebra.Order.Module.Defs",
"Mathlib.Data.Prod.Lex",
"Mathlib.Algebra.Order.Monoid.Prod",
"Mathlib.Algebra.Order.Pi",
"Mathlib.Algebra.Order.Module.OrderedSMul",
"Mathlib.Order.Bounds.OrderIso",
"Mathlib.Algebra.Order.Field.Basic",
"Mathlib.Data.Nat.Cast.Field",
"Mathlib.Algebra.CharZero.Lemmas",
"Mathlib.Algebra.GroupWithZero.Power",
"Mathlib.Algebra.GroupWithZero.Bitwise",
"Mathlib.Algebra.Field.Opposite",
"Mathlib.Algebra.GroupRingAction.Basic",
"Mathlib.Algebra.Ring.Aut",
"Mathlib.Tactic.SetLike",
"Mathlib.Data.SetLike.Basic",
"Mathlib.Algebra.Star.Basic",
"Mathlib.Data.Nat.Factorial.Basic",
"Mathlib.Tactic.Positivity.Basic",
"Mathlib.Algebra.Order.Hom.Basic",
"Mathlib.Algebra.Order.AbsoluteValue",
"Mathlib.Algebra.Order.Group.MinMax",
"Mathlib.GroupTheory.GroupAction.Ring",
"Mathlib.Init.Align",
"Mathlib.Tactic.GCongr",
"Mathlib.Tactic.Ring.Basic",
"Mathlib.Tactic.Ring.RingNF",
"Mathlib.Algebra.Order.Positive.Ring",
"Mathlib.Data.PNat.Basic",
"Mathlib.Tactic.Ring.PNat",
"Mathlib.Tactic.Ring",
"Mathlib.Algebra.Order.CauSeq.Basic",
"Mathlib.Algebra.Order.CauSeq.Completion",
"Mathlib.Data.Real.Basic",
"Mathlib.Data.Complex.Basic",
"Mathlib.Algebra.BigOperators.List.Defs",
"Mathlib.Init.Data.List.Instances",
"Mathlib.Init.Data.List.Lemmas",
"Mathlib.Data.List.Basic",
"Mathlib.Data.List.Infix",
"Mathlib.Data.List.Forall2",
"Mathlib.Init.Data.Fin.Basic",
"Mathlib.Data.List.Nodup",
"Mathlib.Data.List.Dedup",
"Mathlib.Data.List.ProdSigma",
"Mathlib.Data.List.Lex",
"Mathlib.Data.List.Chain",
"Mathlib.Data.List.Zip",
"Mathlib.Data.List.Pairwise",
"Mathlib.Data.List.Range",
"Mathlib.Data.List.Rotate",
"Mathlib.Algebra.BigOperators.List.Basic",
"Mathlib.Data.Int.Units",
"Mathlib.Algebra.BigOperators.List.Lemmas",
"Mathlib.Data.Set.List",
"Mathlib.Data.List.Join",
"Mathlib.Data.List.Permutation",
"Mathlib.Data.List.InsertNth",
"Mathlib.Data.List.Lattice",
"Mathlib.Data.List.Count",
"Mathlib.Data.List.Perm",
"Mathlib.Init.Quot",
"Mathlib.Data.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Basic",
"Mathlib.Algebra.BigOperators.Multiset.Lemmas",
"Mathlib.Algebra.BigOperators.Multiset.Order",
"Mathlib.Algebra.Group.Embedding",
"Mathlib.Data.Fin.Basic",
"Mathlib.Data.Finset.Attr",
"Mathlib.Data.Multiset.Range",
"Mathlib.Data.Multiset.Nodup",
"Mathlib.Data.Multiset.Dedup",
"Mathlib.Data.Multiset.FinsetOps",
"Mathlib.Data.Finset.Basic",
"Mathlib.Data.Multiset.Bind",
"Mathlib.Data.Finset.Union",
"Mathlib.Data.Finset.Image",
"Mathlib.Data.List.MinMax",
"Mathlib.Data.Multiset.Fold",
"Mathlib.Data.Finset.Fold",
"Mathlib.Data.Finset.Card",
"Mathlib.Data.Finset.Option",
"Mathlib.Data.Multiset.Pi",
"Mathlib.Data.Finset.Pi",
"Mathlib.Data.Finset.Prod",
"Mathlib.Data.Multiset.Lattice",
"Mathlib.Order.Hom.Bounded",
"Mathlib.Order.Hom.Lattice",
"Mathlib.Data.Finset.Lattice",
"Mathlib.Data.Nat.Choose.Basic",
"Mathlib.Data.List.Sublists",
"Mathlib.Data.Multiset.Powerset",
"Mathlib.Data.Finset.Powerset",
"Mathlib.Data.Fin.OrderHom",
"Mathlib.Data.Fintype.Basic",
"Mathlib.Data.Finset.Piecewise",
"Mathlib.Data.Pi.Lex",
"Mathlib.Data.Fin.Tuple.Basic",
"Mathlib.Data.List.OfFn",
"Mathlib.Data.List.Sort",
"Mathlib.Data.List.Duplicate",
"Mathlib.Data.List.NodupEquivFin",
"Mathlib.Data.Fintype.Card",
"Mathlib.Data.Fintype.Powerset",
"Mathlib.Data.Fintype.Prod",
"Mathlib.Data.Set.Sigma",
"Mathlib.Data.Finset.Sigma",
"Mathlib.Data.Fintype.Sigma",
"Mathlib.Data.Multiset.Sum",
"Mathlib.Data.Finset.Sum",
"Mathlib.Logic.Embedding.Set",
"Mathlib.Data.Fintype.Sum",
"Mathlib.Data.Fintype.Pi",
"Mathlib.Data.Vector",
"Mathlib.Control.Applicative",
"Mathlib.Control.Traversable.Basic",
"Mathlib.Data.Vector.Basic",
"Mathlib.Data.Setoid.Basic",
"Mathlib.Tactic.ApplyFun",
"Mathlib.Data.Sym.Basic",
"Mathlib.Data.Fintype.Vector",
"Mathlib.Data.Finite.Basic",
"Mathlib.Lean.Expr.ExtraRecognizers",
"Mathlib.Data.Set.Functor",
"Mathlib.Data.Set.Finite",
"Mathlib.Data.Finset.Preimage",
"Mathlib.Algebra.BigOperators.Basic",
"Mathlib.Algebra.BigOperators.Ring",
"Mathlib.Algebra.BigOperators.Order",
"Mathlib.Order.LocallyFinite",
"Mathlib.Data.Set.Intervals.Monoid",
"Mathlib.Data.Finset.LocallyFinite.Basic",
"Mathlib.Data.Nat.Interval",
"Mathlib.Data.HashMap",
"Mathlib.Tactic.Linarith.Lemmas",
"Mathlib.Util.SynthesizeUsing",
"Mathlib.Tactic.Linarith.Datatypes",
"Mathlib.Tactic.Linarith.Elimination",
"Mathlib.Tactic.Linarith.Parsing",
"Mathlib.Tactic.Linarith.Verification",
"Mathlib.Tactic.Zify",
"Mathlib.Data.Num.Basic",
"Mathlib.Data.Tree",
"Mathlib.Tactic.CancelDenoms.Core",
"Mathlib.Tactic.Linarith.Preprocessing",
"Mathlib.Tactic.Linarith.Frontend",
"Mathlib.Tactic.Linarith",
"Mathlib.Algebra.BigOperators.Intervals",
"Mathlib.Algebra.Parity",
"Mathlib.Data.Nat.GCD.Basic",
"Mathlib.Data.Int.GCD",
"Mathlib.Data.Nat.ModEq",
"Mathlib.Data.Nat.Parity",
"Mathlib.Algebra.GeomSum",
"Mathlib.Algebra.Order.Ring.Pow",
"Mathlib.Algebra.Order.Field.Power",
"Mathlib.Data.Int.LeastGreatest",
"Mathlib.Data.Set.Intervals.Group",
"Mathlib.Tactic.Positivity",
"Mathlib.Algebra.Order.Floor",
"Mathlib.Algebra.EuclideanDomain.Defs",
"Mathlib.Algebra.EuclideanDomain.Instances",
"Mathlib.Util.DischargerAsTactic",
"Mathlib.Tactic.FieldSimp",
"Mathlib.Data.Rat.Floor",
"Mathlib.Algebra.Order.Archimedean",
"Mathlib.Algebra.Order.Group.Instances",
"Mathlib.Data.Set.Intervals.Disjoint",
"Mathlib.Data.Set.Intervals.OrderIso",
"Mathlib.Data.Nat.ForSqrt",
"Mathlib.Data.Nat.Sqrt",
"Mathlib.Data.Nat.Pairing",
"Mathlib.Logic.Equiv.Nat",
"Mathlib.Data.Fin.VecNotation",
"Mathlib.Logic.Equiv.Fin",
"Mathlib.Data.Countable.Defs",
"Mathlib.Data.Countable.Basic",
"Mathlib.Order.RelIso.Set",
"Mathlib.Data.Multiset.Sort",
"Mathlib.Data.Finset.Sort",
"Mathlib.Logic.Encodable.Basic",
"Mathlib.Logic.Denumerable",
"Mathlib.Logic.Equiv.List",
"Mathlib.Data.Set.Countable",
"Mathlib.Order.Filter.Basic",
"Mathlib.Order.Filter.Prod",
"Mathlib.Order.Filter.Ker",
"Mathlib.Order.Filter.Bases",
"Mathlib.Order.Filter.AtTopBot",
"Mathlib.Order.Filter.Archimedean",
"Mathlib.Order.Iterate",
"Mathlib.Order.Filter.Lift",
"Mathlib.Tactic.Continuity.Init",
"Mathlib.Tactic.Continuity",
"Mathlib.Tactic.FunProp.Decl",
"Mathlib.Tactic.FunProp.ToStd",
"Mathlib.Tactic.FunProp.Mor",
"Mathlib.Tactic.FunProp.FunctionData",
"Mathlib.Tactic.FunProp.Types",
"Mathlib.Tactic.FunProp.StateList",
"Mathlib.Tactic.FunProp.RefinedDiscrTree",
"Mathlib.Tactic.FunProp.Theorems",
"Mathlib.Tactic.FunProp.Attr",
"Mathlib.Tactic.FunProp.Core",
"Mathlib.Tactic.FunProp.Elab",
"Mathlib.Tactic.FunProp",
"Mathlib.Topology.Defs.Basic",
"Mathlib.Order.Filter.Pi",
"Mathlib.Order.Filter.Cofinite",
"Mathlib.Order.Chain",
"Mathlib.Order.Zorn",
"Mathlib.Order.ModularLattice",
"Mathlib.Order.Atoms",
"Mathlib.Order.ZornAtoms",
"Mathlib.Order.Filter.Ultrafilter",
"Mathlib.Topology.Defs.Filter",
"Mathlib.Topology.Basic",
"Mathlib.Topology.Defs.Induced",
"Mathlib.Topology.Order",
"Mathlib.Topology.Maps",
"Mathlib.Topology.NhdsSet",
"Mathlib.Topology.Constructions",
"Mathlib.Topology.ContinuousOn",
"Mathlib.Topology.Bases",
"Mathlib.Data.Set.Accumulate",
"Mathlib.Topology.Bornology.Basic",
"Mathlib.Order.Filter.SmallSets",
"Mathlib.Topology.LocallyFinite",
"Mathlib.Topology.Compactness.Compact",
"Mathlib.Topology.Compactness.LocallyCompact",
"Mathlib.Topology.Compactness.SigmaCompact",
"Mathlib.Order.SuccPred.Basic",
"Mathlib.Order.SuccPred.Relation",
"Mathlib.Data.Set.BoolIndicator",
"Mathlib.Topology.Clopen",
"Mathlib.Order.UpperLower.Basic",
"Mathlib.Order.Minimal",
"Mathlib.Topology.Irreducible",
"Mathlib.Topology.Connected.Basic",
"Mathlib.Topology.Connected.TotallyDisconnected",
"Mathlib.Data.List.TFAE",
"Mathlib.Tactic.TFAE",
"Mathlib.Topology.Inseparable",
"Mathlib.Topology.Separation",
"Mathlib.Topology.DenseEmbedding",
"Mathlib.Topology.Support",
"Mathlib.Topology.Connected.LocallyConnected",
"Mathlib.Topology.Homeomorph",
"Mathlib.Data.Set.Intervals.Pi",
"Mathlib.Data.Set.Pointwise.Basic",
"Mathlib.Order.Filter.Interval",
"Mathlib.Topology.Order.LeftRight",
"Mathlib.Topology.Order.OrderClosed",
"Mathlib.Topology.Order.Basic",
"Mathlib.Topology.Order.MonotoneContinuity",
"Mathlib.Algebra.Function.Finite",
"Mathlib.Algebra.BigOperators.Finprod",
"Mathlib.Topology.Algebra.InfiniteSum.Defs",
"Mathlib.Data.Fintype.Option",
"Mathlib.Algebra.BigOperators.Option",
"Mathlib.Data.Fintype.BigOperators",
"Mathlib.Data.Set.Pointwise.SMul",
"Mathlib.Order.Filter.NAry",
"Mathlib.Order.Filter.Pointwise",
"Mathlib.Algebra.AddTorsor",
"Mathlib.Topology.Algebra.Constructions",
"Mathlib.Algebra.Group.Conj",
"Mathlib.GroupTheory.Subsemigroup.Basic",
"Mathlib.GroupTheory.Subsemigroup.Operations",
"Mathlib.GroupTheory.Subsemigroup.Center",
"Mathlib.GroupTheory.Subsemigroup.Centralizer",
"Mathlib.GroupTheory.Submonoid.Basic",
"Mathlib.GroupTheory.Submonoid.Operations",
"Mathlib.GroupTheory.Submonoid.Center",
"Mathlib.GroupTheory.Submonoid.Centralizer",
"Mathlib.GroupTheory.Subgroup.Basic",
"Mathlib.GroupTheory.GroupAction.Basic",
"Mathlib.GroupTheory.GroupAction.Hom",
"Mathlib.GroupTheory.GroupAction.SubMulAction",
"Mathlib.Algebra.Group.ULift",
"Mathlib.Algebra.Ring.ULift",
"Mathlib.GroupTheory.GroupAction.DomAct.Basic",
"Mathlib.Algebra.Module.LinearMap.Basic",
"Mathlib.Algebra.Module.LinearMap.End",
"Mathlib.Algebra.Module.Equiv",
"Mathlib.Algebra.Module.ULift",
"Mathlib.Topology.Algebra.ConstMulAction",
"Mathlib.Topology.Algebra.MulAction",
"Mathlib.Algebra.BigOperators.Pi",
"Mathlib.Data.Set.UnionLift",
"Mathlib.Topology.ContinuousFunction.Basic",
"Mathlib.Topology.Algebra.Monoid",
"Mathlib.Topology.Algebra.InfiniteSum.Basic",
"Mathlib.Tactic.Monotonicity.Basic",
"Mathlib.Tactic.Monotonicity.Lemmas",
"Mathlib.Tactic.Monotonicity",
"Mathlib.Topology.UniformSpace.Basic",
"Mathlib.Topology.UniformSpace.Cauchy",
"Mathlib.Topology.UniformSpace.UniformConvergence",
"Mathlib.Topology.UniformSpace.Separation",
"Mathlib.Topology.UniformSpace.UniformEmbedding",
"Mathlib.Topology.UniformSpace.CompleteSeparated",
"Mathlib.Topology.UniformSpace.Pi",
"Mathlib.Topology.UniformSpace.Equiv",
"Mathlib.Topology.UniformSpace.UniformConvergenceTopology",
"Mathlib.Topology.UniformSpace.Equicontinuity",
"Mathlib.Topology.UniformSpace.Compact",
"Mathlib.GroupTheory.Subgroup.ZPowers",
"Mathlib.GroupTheory.GroupAction.ConjAct",
"Mathlib.Data.ZMod.Defs",
"Mathlib.Algebra.Order.Hom.Monoid",
"Mathlib.Algebra.Order.Hom.Ring",
"Mathlib.Data.Nat.SuccPred",
"Mathlib.Algebra.Order.Sub.WithTop",
"Mathlib.Algebra.Order.Ring.WithTop",
"Mathlib.Data.ENat.Basic",
"Mathlib.Data.Finsupp.Defs",
"Mathlib.Logic.Small.Defs",
"Mathlib.Logic.Small.Basic",
"Mathlib.Logic.Small.Set",
"Mathlib.Order.SuccPred.Limit",
"Mathlib.Order.SuccPred.CompleteLinearOrder",
"Mathlib.Dynamics.FixedPoints.Basic",
"Mathlib.Order.Hom.Order",
"Mathlib.Order.FixedPoints",
"Mathlib.SetTheory.Cardinal.SchroederBernstein",
"Mathlib.SetTheory.Cardinal.Basic",
"Mathlib.SetTheory.Cardinal.ENat",
"Mathlib.SetTheory.Cardinal.ToNat",
"Mathlib.Order.ConditionallyCompleteLattice.Finset",
"Mathlib.Data.Nat.Lattice",
"Mathlib.Data.ENat.Lattice",
"Mathlib.Data.Part",
"Mathlib.Data.Nat.PartENat",
"Mathlib.SetTheory.Cardinal.PartENat",
"Mathlib.SetTheory.Cardinal.Finite",
"Mathlib.Data.Fintype.Units",
"Mathlib.Algebra.Group.ConjFinite",
"Mathlib.Data.Fintype.List",
"Mathlib.Data.List.Cycle",
"Mathlib.Algebra.Associated",
"Mathlib.Data.Nat.Prime",
"Mathlib.Dynamics.PeriodicPts",
"Mathlib.Algebra.Group.Commutator",
"Mathlib.Algebra.FreeMonoid.Basic",
"Mathlib.Algebra.Group.Commute.Hom",
"Mathlib.Data.Finset.NoncommProd",
"Mathlib.GroupTheory.Submonoid.MulOpposite",
"Mathlib.GroupTheory.Submonoid.Membership",
"Mathlib.GroupTheory.Subgroup.Finite",
"Mathlib.Tactic.Group",
"Mathlib.GroupTheory.Commutator",
"Mathlib.Algebra.Quotient",
"Mathlib.GroupTheory.Subgroup.MulOpposite",
"Mathlib.GroupTheory.Subgroup.Actions",
"Mathlib.GroupTheory.Coset",
"Mathlib.GroupTheory.GroupAction.Quotient",
"Mathlib.GroupTheory.Congruence",
"Mathlib.Init.Data.Sigma.Lex",
"Mathlib.Data.Sigma.Lex",
"Mathlib.Order.OrderIsoNat",
"Mathlib.Order.WellFoundedSet",
"Mathlib.GroupTheory.Submonoid.Pointwise",
"Mathlib.GroupTheory.Subgroup.Pointwise",
"Mathlib.GroupTheory.QuotientGroup",
"Mathlib.Topology.Algebra.Group.Basic",
"Mathlib.Topology.DiscreteSubset",
"Mathlib.Topology.Algebra.UniformGroup",
"Mathlib.Topology.Algebra.InfiniteSum.Group",
"Mathlib.Logic.Encodable.Lattice",
"Mathlib.Topology.Algebra.InfiniteSum.NatInt",
"Mathlib.Topology.Algebra.Order.Group",
"Mathlib.Data.Nat.Cast.Prod",
"Mathlib.Data.Int.Cast.Prod",
"Mathlib.Algebra.Order.Group.Prod",
"Mathlib.Algebra.Ring.Prod",
"Mathlib.Algebra.GroupRingAction.Subobjects",
"Mathlib.GroupTheory.Subsemigroup.Membership",
"Mathlib.RingTheory.NonUnitalSubsemiring.Basic",
"Mathlib.RingTheory.Subsemiring.Basic",
"Mathlib.RingTheory.Subring.Basic",
"Mathlib.Topology.Algebra.Ring.Basic",
"Mathlib.Topology.Algebra.GroupWithZero",
"Mathlib.Order.Filter.Extr",
"Mathlib.Topology.Order.LocalExtr",
"Mathlib.Algebra.Module.Submodule.Basic",
"Mathlib.Algebra.GCDMonoid.Basic",
"Mathlib.Algebra.PUnitInstances",
"Mathlib.Algebra.Module.Submodule.Lattice",
"Mathlib.Algebra.Module.Submodule.LinearMap",
"Mathlib.Algebra.Module.Submodule.Map",
"Mathlib.Algebra.Module.Submodule.Ker",
"Mathlib.Order.Hom.CompleteLattice",
"Mathlib.Algebra.Module.Submodule.RestrictScalars",
"Mathlib.Algebra.Algebra.Basic",
"Mathlib.FieldTheory.Subfield",
"Mathlib.Topology.Algebra.Field",
"Mathlib.Data.Set.Pointwise.Interval",
"Mathlib.Topology.Algebra.Order.Field",
"Mathlib.Topology.Order.MonotoneConvergence",
"Mathlib.Topology.Algebra.InfiniteSum.Order",
"Mathlib.Data.Int.Parity",
"Mathlib.Data.Int.Order.Units",
"Mathlib.Data.Int.ModEq",
"Mathlib.Data.Nat.Log",
"Mathlib.Data.List.Indexes",
"Mathlib.Data.List.Palindrome",
"Mathlib.Tactic.FinCases",
"Mathlib.Tactic.IntervalCases",
"Mathlib.Data.Nat.Digits",
"Mathlib.RingTheory.Multiplicity",
"Mathlib.Data.Nat.Multiplicity",
"Mathlib.Data.List.NatAntidiagonal",
"Mathlib.Data.Multiset.NatAntidiagonal",
"Mathlib.Data.Finset.Antidiagonal",
"Mathlib.Data.Finset.NatAntidiagonal",
"Mathlib.Algebra.BigOperators.NatAntidiagonal",
"Mathlib.Data.Nat.Choose.Sum",
"Mathlib.Data.Set.Intervals.Infinite",
"Mathlib.Data.Finite.Card",
"Mathlib.Data.Set.Pointwise.Finite",
"Mathlib.GroupTheory.Finiteness",
"Mathlib.GroupTheory.Index",
"Mathlib.GroupTheory.OrderOfElement",
"Mathlib.Algebra.CharP.Basic",
"Mathlib.Data.Finsupp.Indicator",
"Mathlib.Data.Fin.Interval",
"Mathlib.Data.Fintype.Fin",
"Mathlib.Data.List.FinRange",
"Mathlib.Algebra.BigOperators.Fin",
"Mathlib.Data.Finsupp.Fin",
"Mathlib.Algebra.BigOperators.Finsupp",
"Mathlib.Algebra.Algebra.Hom",
"Mathlib.Algebra.Algebra.NonUnitalHom",
"Mathlib.Algebra.Ring.Idempotents",
"Mathlib.Algebra.Module.Hom",
"Mathlib.LinearAlgebra.Basic",
"Mathlib.Order.Closure",
"Mathlib.Order.SupClosed",
"Mathlib.Data.Finset.Pairwise",
"Mathlib.Order.SupIndep",
"Mathlib.Data.Finset.Order",
"Mathlib.Data.Finite.Set",
"Mathlib.Order.CompactlyGenerated.Basic",
"Mathlib.Control.Monad.Basic",
"Mathlib.Order.OmegaCompletePartialOrder",
"Mathlib.LinearAlgebra.Span",
"Mathlib.LinearAlgebra.BilinearMap",
"Mathlib.Algebra.Module.Submodule.Bilinear",
"Mathlib.Tactic.SuppressCompilation",
"Mathlib.LinearAlgebra.TensorProduct.Basic",
"Mathlib.Algebra.Algebra.Bilinear",
"Mathlib.Algebra.Algebra.Equiv",
"Mathlib.Algebra.Module.Opposites",
"Mathlib.Algebra.Algebra.Opposite",
"Mathlib.Algebra.GroupWithZero.NonZeroDivisors",
"Mathlib.GroupTheory.GroupAction.BigOperators",
"Mathlib.Data.DFinsupp.Basic",
"Mathlib.Data.Rat.BigOperators",
"Mathlib.Data.Finsupp.Basic",
"Mathlib.Data.Finsupp.ToDFinsupp",
"Mathlib.Data.DFinsupp.Encodable",
"Mathlib.Data.Finsupp.Encodable",
"Mathlib.LinearAlgebra.Pi",
"Mathlib.LinearAlgebra.Finsupp",
"Mathlib.RingTheory.Ideal.Basic",
"Mathlib.Algebra.Module.BigOperators",
"Mathlib.Algebra.Order.Group.Action",
"Mathlib.Algebra.Module.Submodule.Pointwise",
"Mathlib.Algebra.Order.Kleene",
"Mathlib.Data.Finset.NAry",
"Mathlib.Data.Set.Pointwise.ListOfFn",
"Mathlib.Data.Finset.Pointwise",
"Mathlib.Data.Set.Pointwise.BigOperators",
"Mathlib.Data.Set.Semiring",
"Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise",
"Mathlib.Algebra.Algebra.Operations",
"Mathlib.Data.Fintype.Lattice",
"Mathlib.RingTheory.Coprime.Basic",
"Mathlib.RingTheory.Coprime.Lemmas",
"Mathlib.Algebra.Algebra.Prod",
"Mathlib.Order.PartialSups",
"Mathlib.LinearAlgebra.Prod",
"Mathlib.Tactic.LinearCombination",
"Mathlib.LinearAlgebra.LinearIndependent",
"Mathlib.Data.Finsupp.Order",
"Mathlib.Data.Finsupp.Multiset",
"Mathlib.Order.Bounded",
"Mathlib.Data.Sum.Order",
"Mathlib.Order.InitialSeg",
"Mathlib.SetTheory.Ordinal.Basic",
"Mathlib.SetTheory.Ordinal.Arithmetic",
"Mathlib.SetTheory.Ordinal.Exponential",
"Mathlib.SetTheory.Ordinal.FixedPoint",
"Mathlib.SetTheory.Ordinal.Principal",
"Mathlib.SetTheory.Cardinal.Ordinal",
"Mathlib.SetTheory.Cardinal.Cofinality",
"Mathlib.LinearAlgebra.Basis",
"Mathlib.LinearAlgebra.Quotient",
"Mathlib.RingTheory.Ideal.Operations",
"Mathlib.Data.ZMod.Basic",
"Mathlib.Data.ZMod.IntUnitsPower",
"Mathlib.Algebra.GroupPower.NegOnePow",
"Mathlib.Algebra.Periodic",
"Mathlib.Topology.UniformSpace.AbstractCompletion",
"Mathlib.Topology.UniformSpace.Completion",
"Mathlib.Topology.Algebra.UniformMulAction",
"Mathlib.Algebra.Star.Pi",
"Mathlib.Algebra.Star.Prod",
"Mathlib.Topology.Algebra.Star",
"Mathlib.Data.Int.Interval",
"Mathlib.Data.Int.SuccPred",
"Mathlib.Data.Int.ConditionallyCompleteOrder",
"Mathlib.Topology.Order.IntermediateValue",
"Mathlib.Topology.Algebra.Order.Compact",
"Mathlib.Algebra.BigOperators.WithTop",
"Mathlib.Algebra.Order.Field.Canonical.Defs",
"Mathlib.Algebra.Order.Field.Canonical.Basic",
"Mathlib.Algebra.Order.Field.InjSurj",
"Mathlib.Algebra.Order.Nonneg.Field",
"Mathlib.Algebra.Order.Nonneg.Floor",
"Mathlib.Algebra.Order.Module.Pointwise",
"Mathlib.Algebra.Bounds",
"Mathlib.Data.Real.Archimedean",
"Mathlib.Data.Real.Pointwise",
"Mathlib.Order.ConditionallyCompleteLattice.Group",
"Mathlib.Data.Real.NNReal",
"Mathlib.Data.Set.Intervals.WithBotTop",
"Mathlib.Data.ENNReal.Basic",
"Mathlib.Data.ENNReal.Operations",
"Mathlib.Data.ENNReal.Inv",
"Mathlib.Data.ENNReal.Real",
"Mathlib.Topology.EMetricSpace.Basic",
"Mathlib.Topology.Bornology.Constructions",
"Mathlib.Topology.MetricSpace.PseudoMetric",
"Mathlib.Topology.MetricSpace.ProperSpace",
"Mathlib.Topology.MetricSpace.Basic",
"Mathlib.Topology.Metrizable.Basic",
"Mathlib.Topology.Metrizable.Uniformity",
"Mathlib.Topology.Instances.Discrete",
"Mathlib.Topology.MetricSpace.Cauchy",
"Mathlib.Topology.MetricSpace.Bounded",
"Mathlib.Topology.Instances.Int",
"Mathlib.Topology.Instances.Real",
"Mathlib.Topology.Algebra.InfiniteSum.Real",
"Mathlib.Algebra.Order.Support",
"Mathlib.Order.LiminfLimsup",
"Mathlib.Order.Filter.CountableInter",
"Mathlib.Topology.Algebra.Order.LiminfLimsup",
"Mathlib.Topology.Algebra.InfiniteSum.Constructions",
"Mathlib.Topology.Algebra.InfiniteSum.Ring",
"Mathlib.Topology.Instances.NNReal",
"Mathlib.Topology.EMetricSpace.Lipschitz",
"Mathlib.Data.Set.Intervals.OrdConnectedComponent",
"Mathlib.Topology.Order.T5",
"Mathlib.Topology.Instances.ENNReal",
"Mathlib.Algebra.Algebra.Subalgebra.Basic",
"Mathlib.LinearAlgebra.Projection",
"Mathlib.Topology.Algebra.Module.Basic",
"Mathlib.Algebra.Algebra.Tower",
"Mathlib.Algebra.Algebra.Subalgebra.Tower",
"Mathlib.RingTheory.Adjoin.Basic",
"Mathlib.Topology.Algebra.Algebra",
"Mathlib.Analysis.SpecificLimits.Basic",
"Mathlib.Data.Rat.Denumerable",
"Mathlib.SetTheory.Cardinal.Continuum",
"Mathlib.Data.Real.Cardinality",
"Mathlib.Data.Complex.Cardinality",
"Mathlib.RingTheory.AlgebraTower",
"Mathlib.Data.Finsupp.Fintype",
"Mathlib.Algebra.DirectSum.Basic",
"Mathlib.LinearAlgebra.DFinsupp",
"Mathlib.Algebra.DirectSum.Module",
"Mathlib.Algebra.DirectSum.Finsupp",
"Mathlib.LinearAlgebra.DirectSum.TensorProduct",
"Mathlib.LinearAlgebra.DirectSum.Finsupp",
"Mathlib.Algebra.Algebra.Pi",
"Mathlib.Algebra.BigOperators.RingEquiv",
"Mathlib.Algebra.Star.BigOperators",
"Mathlib.Init.Data.Subtype.Basic",
"Mathlib.Algebra.Star.SelfAdjoint",
"Mathlib.Algebra.Star.Module",
"Mathlib.Data.Matrix.Basic",
"Mathlib.Data.Matrix.Block",
"Mathlib.Data.Matrix.RowCol",
"Mathlib.LinearAlgebra.Matrix.Trace",
"Mathlib.Data.Matrix.Basis",
"Mathlib.LinearAlgebra.StdBasis",
"Mathlib.LinearAlgebra.FinsuppVectorSpace",
"Mathlib.LinearAlgebra.TensorProduct.Basis",
"Mathlib.LinearAlgebra.FreeModule.Basic",
"Mathlib.LinearAlgebra.LinearPMap",
"Mathlib.LinearAlgebra.Basis.VectorSpace",
"Mathlib.LinearAlgebra.Dimension.Basic",
"Mathlib.LinearAlgebra.Dimension.Finrank",
"Mathlib.RingTheory.Congruence",
"Mathlib.RingTheory.Ideal.Quotient",
"Mathlib.Algebra.BigOperators.Associated",
"Mathlib.Order.Filter.Subsingleton",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.Algebra.Algebra.RestrictScalars",
"Mathlib.RingTheory.Finiteness",
"Mathlib.Data.Matrix.Notation",
"Mathlib.LinearAlgebra.Matrix.ToLin",
"Mathlib.RingTheory.Nilpotent",
"Mathlib.RingTheory.Noetherian",
"Mathlib.RingTheory.UniqueFactorizationDomain",
"Mathlib.RingTheory.PrincipalIdealDomain",
"Mathlib.LinearAlgebra.InvariantBasisNumber",
"Mathlib.LinearAlgebra.Dimension.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Basic",
"Mathlib.LinearAlgebra.Dimension.Free",
"Mathlib.LinearAlgebra.Isomorphisms",
"Mathlib.Algebra.Group.Equiv.TypeTags",
"Mathlib.Algebra.Ring.Fin",
"Mathlib.RingTheory.Ideal.QuotientOperations",
"Mathlib.Algebra.EuclideanDomain.Basic",
"Mathlib.Data.List.Prime",
"Mathlib.Data.Nat.Factors",
"Mathlib.RingTheory.Int.Basic",
"Mathlib.Data.ZMod.Quotient",
"Mathlib.Data.Nat.GCD.BigOperators",
"Mathlib.GroupTheory.NoncommPiCoprod",
"Mathlib.Algebra.GCDMonoid.Multiset",
"Mathlib.Algebra.GCDMonoid.Finset",
"Mathlib.Data.Nat.PrimeFin",
"Mathlib.NumberTheory.Divisors",
"Mathlib.Data.Nat.MaxPowDiv",
"Mathlib.NumberTheory.Padics.PadicVal",
"Mathlib.Data.Nat.Factorization.Basic",
"Mathlib.Tactic.Peel",
"Mathlib.GroupTheory.Exponent",
"Mathlib.Combinatorics.Enumerative.Composition",
"Mathlib.Combinatorics.Enumerative.Partition",
"Mathlib.Data.Fintype.Perm",
"Mathlib.GroupTheory.Perm.Support",
"Mathlib.GroupTheory.Perm.List",
"Mathlib.Data.Finset.Fin",
"Mathlib.GroupTheory.Perm.Sign",
"Mathlib.Logic.Equiv.Fintype",
"Mathlib.GroupTheory.Perm.Finite",
"Mathlib.GroupTheory.Perm.Cycle.Basic",
"Mathlib.GroupTheory.Perm.Cycle.Factors",
"Mathlib.GroupTheory.Perm.Closure",
"Mathlib.Tactic.NormNum.GCD",
"Mathlib.GroupTheory.Perm.Cycle.Type",
"Mathlib.Init.Data.Prod",
"Mathlib.GroupTheory.MonoidLocalization",
"Mathlib.RingTheory.Localization.Basic",
"Mathlib.RingTheory.Localization.FractionRing",
"Mathlib.Algebra.MonoidAlgebra.Basic",
"Mathlib.Algebra.Group.UniqueProds",
"Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors",
"Mathlib.Algebra.FreeAlgebra",
"Mathlib.Algebra.CharP.Algebra",
"Mathlib.Algebra.CharP.ExpChar",
"Mathlib.Algebra.CharP.Two",
"Mathlib.Data.Nat.Count",
"Mathlib.Data.Nat.Periodic",
"Mathlib.Data.Nat.Totient",
"Mathlib.GroupTheory.Subgroup.Simple",
"Mathlib.GroupTheory.SpecificGroups.Cyclic",
"Mathlib.GroupTheory.PGroup",
"Mathlib.GroupTheory.Torsion",
"Mathlib.RingTheory.Coprime.Ideal",
"Mathlib.Algebra.Module.Torsion",
"Mathlib.LinearAlgebra.Dimension.Constructions",
"Mathlib.LinearAlgebra.Dimension.Finite",
"Mathlib.FieldTheory.Finiteness",
"Mathlib.Data.W.Basic",
"Mathlib.Data.W.Cardinal",
"Mathlib.SetTheory.Cardinal.Subfield",
"Mathlib.LinearAlgebra.Dimension.DivisionRing",
"Mathlib.LinearAlgebra.FiniteDimensional",
"Mathlib.LinearAlgebra.Dimension.LinearMap",
"Mathlib.Algebra.MonoidAlgebra.Support",
"Mathlib.Algebra.MonoidAlgebra.Degree",
"Mathlib.Algebra.Regular.Pow",
"Mathlib.Data.Multiset.Antidiagonal",
"Mathlib.Data.Finsupp.Antidiagonal",
"Mathlib.Data.MvPolynomial.Basic",
"Mathlib.Data.MvPolynomial.Rename",
"Mathlib.Data.MvPolynomial.Degrees",
"Mathlib.Data.MvPolynomial.Variables",
"Mathlib.Data.MvPolynomial.CommRing",
"Mathlib.Data.Nat.WithBot",
"Mathlib.Data.Polynomial.Basic",
"Mathlib.Data.Polynomial.Monomial",
"Mathlib.Data.Polynomial.Coeff",
"Mathlib.Data.Nat.Cast.WithTop",
"Mathlib.Data.Polynomial.Degree.Definitions",
"Mathlib.Data.Polynomial.Induction",
"Mathlib.Data.Polynomial.Eval",
"Mathlib.Data.Polynomial.AlgebraMap",
"Mathlib.Data.MvPolynomial.Equiv",
"Mathlib.Data.Polynomial.Derivative",
"Mathlib.Data.Polynomial.Degree.Lemmas",
"Mathlib.Algebra.MonoidAlgebra.Division",
"Mathlib.Data.Polynomial.EraseLead",
"Mathlib.Data.Polynomial.Inductions",
"Mathlib.Data.Polynomial.Degree.TrailingDegree",
"Mathlib.Data.Polynomial.Reverse",
"Mathlib.Data.Polynomial.Monic",
"Mathlib.Data.Polynomial.Div",
"Mathlib.Algebra.Polynomial.BigOperators",
"Mathlib.Data.Polynomial.RingDivision",
"Mathlib.RingTheory.EuclideanDomain",
"Mathlib.Data.Polynomial.FieldDivision",
"Mathlib.Tactic.ComputeDegree",
"Mathlib.Data.Polynomial.CancelLeads",
"Mathlib.RingTheory.Polynomial.Content",
"Mathlib.RingTheory.Polynomial.Basic",
"Mathlib.RingTheory.Polynomial.Quotient",
"Mathlib.RingTheory.JacobsonIdeal",
"Mathlib.Logic.Equiv.TransferInstance",
"Mathlib.RingTheory.Ideal.LocalRing",
"Mathlib.Data.Polynomial.Expand",
"Mathlib.Data.Polynomial.Laurent",
"Mathlib.Data.PEquiv",
"Mathlib.Data.Matrix.PEquiv",
"Mathlib.GroupTheory.Perm.Option",
"Mathlib.GroupTheory.Perm.Fin",
"Mathlib.Data.Fintype.Sort",
"Mathlib.LinearAlgebra.Multilinear.Basic",
"Mathlib.LinearAlgebra.Multilinear.Basis",
"Mathlib.LinearAlgebra.Alternating.Basic",
"Mathlib.LinearAlgebra.Matrix.Determinant",
"Mathlib.LinearAlgebra.Matrix.MvPolynomial",
"Mathlib.LinearAlgebra.Matrix.Polynomial",
"Mathlib.LinearAlgebra.Matrix.Adjugate",
"Mathlib.LinearAlgebra.TensorProduct.Tower",
"Mathlib.RingTheory.TensorProduct.Basic",
"Mathlib.RingTheory.MatrixAlgebra",
"Mathlib.Data.Matrix.DMatrix",
"Mathlib.RingTheory.PolynomialAlgebra",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Basic",
"Mathlib.LinearAlgebra.Matrix.Reindex",
"Mathlib.Data.Polynomial.Identities",
"Mathlib.RingTheory.Polynomial.Tower",
"Mathlib.RingTheory.Polynomial.Nilpotent",
"Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff",
"Mathlib.LinearAlgebra.Matrix.Charpoly.LinearMap",
"Mathlib.RingTheory.Adjoin.FG",
"Mathlib.RingTheory.Adjoin.Tower",
"Mathlib.Data.Polynomial.Module.Basic",
"Mathlib.RingTheory.FiniteType",
"Mathlib.RingTheory.Polynomial.ScaleRoots",
"Mathlib.RingTheory.IntegralClosure",
"Mathlib.FieldTheory.Minpoly.Basic",
"Mathlib.RingTheory.Polynomial.IntegralNormalization",
"Mathlib.RingTheory.Algebraic",
"Mathlib.FieldTheory.Minpoly.Field",
"Mathlib.LinearAlgebra.Charpoly.Basic",
"Mathlib.LinearAlgebra.FreeModule.StrongRankCondition",
"Mathlib.LinearAlgebra.FreeModule.Finite.Matrix",
"Mathlib.FieldTheory.Tower",
"Mathlib.Algebra.CharP.Invertible",
"Mathlib.Data.Complex.Module",
"Mathlib.Algebra.Star.Order",
"Mathlib.Data.Real.Sqrt",
"Mathlib.Data.Complex.Abs",
"Mathlib.Data.Complex.Order",
"Mathlib.Algebra.Order.CauSeq.BigOperators",
"Mathlib.Data.Complex.BigOperators",
"Mathlib.Data.Complex.Exponential",
"Mathlib.Analysis.Normed.Group.Seminorm",
"Mathlib.GroupTheory.Archimedean",
"Mathlib.Topology.Algebra.Order.Archimedean",
"Mathlib.Topology.Instances.Nat",
"Mathlib.Topology.Instances.Rat",
"Mathlib.Data.Set.Intervals.ProjIcc",
"Mathlib.Topology.Bornology.Hom",
"Mathlib.Topology.MetricSpace.Lipschitz",
"Mathlib.Topology.MetricSpace.Algebra",
"Mathlib.Topology.MetricSpace.Antilipschitz",
"Mathlib.Topology.MetricSpace.Isometry",
"Mathlib.Topology.MetricSpace.IsometricSMul",
"Mathlib.Topology.Defs.Sequences",
"Mathlib.Topology.Sequences",
"Mathlib.Analysis.Normed.Group.Basic",
"Mathlib.Analysis.Normed.Group.Hom",
"Mathlib.RingTheory.NonUnitalSubring.Basic",
"Mathlib.Algebra.Algebra.NonUnitalSubalgebra",
"Mathlib.Topology.MetricSpace.Dilation",
"Mathlib.Topology.MetricSpace.DilationEquiv",
"Mathlib.Analysis.Normed.Field.Basic",
"Mathlib.Analysis.Normed.MulAction",
"Mathlib.Analysis.NormedSpace.Basic",
"Mathlib.Analysis.NormedSpace.LinearIsometry",
"Mathlib.Algebra.Star.Pointwise",
"Mathlib.Algebra.Star.Center",
"Mathlib.Algebra.Star.StarAlgHom",
"Mathlib.Algebra.Star.Subalgebra",
"Mathlib.Algebra.Star.Unitary",
"Mathlib.Topology.Algebra.Module.Star",
"Mathlib.Analysis.NormedSpace.Star.Basic",
"Mathlib.Analysis.NormedSpace.ContinuousLinearMap",
"Mathlib.Analysis.RCLike.Basic",
"Mathlib.Topology.Algebra.InfiniteSum.Module",
"Mathlib.Topology.Instances.RealVectorSpace",
"Mathlib.Analysis.Complex.Basic"
]
} | {
"hasProof": true,
"proof": ":=\n Algebra.subset_adjoin",
"proofType": "term",
"proofLengthLines": 1,
"proofLengthTokens": 26
} |
"/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license (...TRUNCATED) | lemma γ_in_diracAlgebra (μ : Fin 4) : γ μ ∈ diracAlgebra | spaceTime.γ.γ_in_diracAlgebra | {
"commit": "7020263",
"date": "2024-05-14T00:00:00"
} | {
"commit": "b9c2086",
"date": "2024-05-10T00:00:00"
} | hep_lean/HepLean/SpaceTime/CliffordAlgebra.lean | HepLean.SpaceTime.CliffordAlgebra | HepLean.SpaceTime.CliffordAlgebra.jsonl | {
"lineInFile": 64,
"tokenPositionInFile": 1923,
"theoremPositionInFile": 10
} | {"inFilePremises":true,"numInFilePremises":4,"repositoryPremises":true,"numRepositoryPremises":4,"nu(...TRUNCATED) | {"hasProof":true,"proof":":=\n γSet_subset_diracAlgebra (γ_in_γSet μ)","proofType":"term","proo(...TRUNCATED) |
"/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license (...TRUNCATED) | "lemma mem_iff_norm : Λ ∈ LorentzGroup d ↔\n ∀ (x : LorentzVector d), ⟪Λ *ᵥ x, Λ *(...TRUNCATED) | LorentzGroup.mem_iff_norm | {
"commit": "89e940a",
"date": "2024-05-17T00:00:00"
} | {
"commit": "1133b88",
"date": "2024-07-12T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Basic.lean | HepLean.SpaceTime.LorentzGroup.Basic | HepLean.SpaceTime.LorentzGroup.Basic.jsonl | {
"lineInFile": 55,
"tokenPositionInFile": 1355,
"theoremPositionInFile": 2
} | {"inFilePremises":true,"numInFilePremises":1,"repositoryPremises":true,"numRepositoryPremises":7,"nu(...TRUNCATED) | {"hasProof":true,"proof":":= by\n refine Iff.intro (fun h x => h x x) (fun h x y => ?_)\n have hp (...TRUNCATED) |
"/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license (...TRUNCATED) | "lemma mem_iff_on_right : Λ ∈ LorentzGroup d ↔\n ∀ (x y : LorentzVector d), ⟪x, (dual Λ(...TRUNCATED) | LorentzGroup.mem_iff_on_right | {
"commit": "89e940a",
"date": "2024-05-17T00:00:00"
} | {
"commit": "675b9a9",
"date": "2024-07-01T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Basic.lean | HepLean.SpaceTime.LorentzGroup.Basic | HepLean.SpaceTime.LorentzGroup.Basic.jsonl | {
"lineInFile": 66,
"tokenPositionInFile": 1784,
"theoremPositionInFile": 3
} | {"inFilePremises":true,"numInFilePremises":1,"repositoryPremises":true,"numRepositoryPremises":7,"nu(...TRUNCATED) | {"hasProof":true,"proof":":= by\n refine Iff.intro (fun h x y ↦ ?_) (fun h x y ↦ ?_)\n · have(...TRUNCATED) |
"/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license (...TRUNCATED) | lemma mem_iff_dual_mul_self : Λ ∈ LorentzGroup d ↔ dual Λ * Λ = 1 | LorentzGroup.mem_iff_dual_mul_self | {
"commit": "89e940a",
"date": "2024-05-17T00:00:00"
} | {
"commit": "675b9a9",
"date": "2024-07-01T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Basic.lean | HepLean.SpaceTime.LorentzGroup.Basic | HepLean.SpaceTime.LorentzGroup.Basic.jsonl | {
"lineInFile": 75,
"tokenPositionInFile": 2099,
"theoremPositionInFile": 4
} | {"inFilePremises":true,"numInFilePremises":2,"repositoryPremises":true,"numRepositoryPremises":8,"nu(...TRUNCATED) | {"hasProof":true,"proof":":= by\n rw [mem_iff_on_right, matrix_eq_id_iff]\n exact forall_comm","pr(...TRUNCATED) |
"/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license (...TRUNCATED) | lemma mem_iff_self_mul_dual : Λ ∈ LorentzGroup d ↔ Λ * dual Λ = 1 | LorentzGroup.mem_iff_self_mul_dual | {
"commit": "89e940a",
"date": "2024-05-17T00:00:00"
} | {
"commit": "1133b88",
"date": "2024-07-12T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Basic.lean | HepLean.SpaceTime.LorentzGroup.Basic | HepLean.SpaceTime.LorentzGroup.Basic.jsonl | {
"lineInFile": 79,
"tokenPositionInFile": 2234,
"theoremPositionInFile": 5
} | {"inFilePremises":true,"numInFilePremises":2,"repositoryPremises":true,"numRepositoryPremises":3,"nu(...TRUNCATED) | {"hasProof":true,"proof":":= by\n rw [mem_iff_dual_mul_self]\n exact mul_eq_one_comm","proofType":(...TRUNCATED) |
"/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license (...TRUNCATED) | lemma mem_iff_transpose : Λ ∈ LorentzGroup d ↔ Λᵀ ∈ LorentzGroup d | LorentzGroup.mem_iff_transpose | {
"commit": "89e940a",
"date": "2024-05-17T00:00:00"
} | {
"commit": "675b9a9",
"date": "2024-07-01T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Basic.lean | HepLean.SpaceTime.LorentzGroup.Basic | HepLean.SpaceTime.LorentzGroup.Basic.jsonl | {
"lineInFile": 83,
"tokenPositionInFile": 2360,
"theoremPositionInFile": 6
} | {"inFilePremises":true,"numInFilePremises":3,"repositoryPremises":true,"numRepositoryPremises":6,"nu(...TRUNCATED) | {"hasProof":true,"proof":":= by\n refine Iff.intro (fun h ↦ ?_) (fun h ↦ ?_)\n · have h1 := c(...TRUNCATED) |
"/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license (...TRUNCATED) | "lemma mem_mul (hΛ : Λ ∈ LorentzGroup d) (hΛ' : Λ' ∈ LorentzGroup d) : Λ * Λ' ∈ LorentzG(...TRUNCATED) | LorentzGroup.mem_mul | {
"commit": "89e940a",
"date": "2024-05-17T00:00:00"
} | {
"commit": "675b9a9",
"date": "2024-07-01T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Basic.lean | HepLean.SpaceTime.LorentzGroup.Basic | HepLean.SpaceTime.LorentzGroup.Basic.jsonl | {
"lineInFile": 96,
"tokenPositionInFile": 3008,
"theoremPositionInFile": 7
} | {"inFilePremises":true,"numInFilePremises":2,"repositoryPremises":true,"numRepositoryPremises":4,"nu(...TRUNCATED) | {"hasProof":true,"proof":":= by\n rw [mem_iff_dual_mul_self, dual_mul]\n trans dual Λ' * (dual Λ(...TRUNCATED) |
"/-\nCopyright (c) 2024 Joseph Tooby-Smith. All rights reserved.\nReleased under Apache 2.0 license (...TRUNCATED) | lemma one_mem : 1 ∈ LorentzGroup d | LorentzGroup.one_mem | {
"commit": "89e940a",
"date": "2024-05-17T00:00:00"
} | {
"commit": "675b9a9",
"date": "2024-07-01T00:00:00"
} | hep_lean/HepLean/SpaceTime/LorentzGroup/Basic.lean | HepLean.SpaceTime.LorentzGroup.Basic | HepLean.SpaceTime.LorentzGroup.Basic.jsonl | {
"lineInFile": 103,
"tokenPositionInFile": 3282,
"theoremPositionInFile": 8
} | {"inFilePremises":true,"numInFilePremises":2,"repositoryPremises":true,"numRepositoryPremises":4,"nu(...TRUNCATED) | {"hasProof":true,"proof":":= by\n rw [mem_iff_dual_mul_self]\n simp","proofType":"tactic","proofLe(...TRUNCATED) |
End of preview.