Context stringlengths 227 76.5k | target stringlengths 0 11.6k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 16 3.69k |
|---|---|---|---|---|
/-
Copyright (c) 2020 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Utensil Song
-/
import Mathlib.Algebra.RingQuot
import Mathlib.LinearAlgebra.TensorAlgebra.Basic
import Mathlib.LinearAlgebra.QuadraticForm.Isometry
import Mathlib.LinearAlge... | let s : Subalgebra R (CliffordAlgebra Q) :=
{ carrier := C
mul_mem' := @mul
add_mem' := @add
algebraMap_mem' := algebraMap }
let of : { f : M →ₗ[R] s // ∀ m, f m * f m = _root_.algebraMap _ _ (Q m) } :=
| Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean | 191 | 196 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Functor.Category
import Mathlib.CategoryTheory.Iso
/-!
# Natural isomorphisms
For the most ... | @[simp]
theorem cancel_natIso_hom_right_assoc {W X X' : D} {Y : C} (f : W ⟶ X) (g : X ⟶ F.obj Y)
(f' : W ⟶ X') (g' : X' ⟶ F.obj Y) :
f ≫ g ≫ α.hom.app Y = f' ≫ g' ≫ α.hom.app Y ↔ f ≫ g = f' ≫ g' := by
| Mathlib/CategoryTheory/NatIso.lean | 153 | 156 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Topology.Algebra.Constructions
import Mathlib.Topology.Bases
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Topology.UniformSpac... |
theorem CauchySeq.comp_injective [SemilatticeSup β] [NoMaxOrder β] [Nonempty β] {u : ℕ → α}
(hu : CauchySeq u) {f : β → ℕ} (hf : Injective f) : CauchySeq (u ∘ f) :=
| Mathlib/Topology/UniformSpace/Cauchy.lean | 202 | 204 |
/-
Copyright (c) 2024 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Filtered.Connected
import Mathlib.CategoryTheory.Limits.Types.Filtered
import Mathlib.CategoryTheory.Limits.Sifted
/-!
# Final functors w... | simpa using IsFiltered.coeq_condition _ _
theorem isCofiltered_costructuredArrow_of_isCofiltered_of_exists [IsCofilteredOrEmpty C]
(h₁ : ∀ d, ∃ c, Nonempty (F.obj c ⟶ d)) (h₂ : ∀ {d : D} {c : C} (s s' : F.obj c ⟶ d),
∃ (c' : C) (t : c' ⟶ c), F.map t ≫ s = F.map t ≫ s') (d : D) :
IsCofiltered (Costruc... | Mathlib/CategoryTheory/Filtered/Final.lean | 74 | 87 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Group.Defs
/-!
# Invertible elements
This file defines a typeclass `Invertible a` for elements `a` with a two-sided
multiplicative inverse.
The in... | simp [mul_assoc]
example {G} [Group G] (a b : G) : a * b * b⁻¹ = a := mul_inv_cancel_right a b
| Mathlib/Algebra/Group/Invertible/Defs.lean | 133 | 134 |
/-
Copyright (c) 2020 Kexing Ying and Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.Gro... | rw [finprod_mem_finset_product']
simp
@[to_additive]
theorem finprod_mem_finset_product₃ {γ : Type*} (s : Finset (α × β × γ)) (f : α × β × γ → M) :
| Mathlib/Algebra/BigOperators/Finprod.lean | 1,096 | 1,100 |
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Limits.IsLimit
import Mathlib.CategoryTheory.Category.ULift
import Mathlib.CategoryTheory.Ess... | uniq s m w := by
dsimp
rw [← P.uniq s.op m.op]
· rfl
· dsimp
| Mathlib/CategoryTheory/Limits/HasLimits.lean | 1,136 | 1,140 |
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Analysis.Complex.Convex
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.Analysis.Calculus.Deriv.Shift
/-!
# Estimates for the complex ... | /-- Give a bound on `‖(1 + t * z)⁻¹‖` for `0 ≤ t ≤ 1` and `‖z‖ < 1`. -/
lemma norm_one_add_mul_inv_le {t : ℝ} (ht : t ∈ Set.Icc 0 1) {z : ℂ} (hz : ‖z‖ < 1) :
‖(1 + t * z)⁻¹‖ ≤ (1 - ‖z‖)⁻¹ := by
rw [Set.mem_Icc] at ht
rw [norm_inv]
refine inv_anti₀ (by linarith) ?_
calc 1 - ‖z‖
_ ≤ 1 - t * ‖z‖ := by
... | Mathlib/Analysis/SpecialFunctions/Complex/LogBounds.lean | 108 | 122 |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
/-!
# Recursive computation rules for the Clifford algebra
This file provides API for a special case `CliffordAlg... |
/-- This lemma demonstrates the origin of the `foldl` name. -/
theorem foldl_prod_map_ι (l : List M) (f : M →ₗ[R] N →ₗ[R] N) (hf) (n : N) :
| Mathlib/LinearAlgebra/CliffordAlgebra/Fold.lean | 115 | 117 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Complex.Circle
import Mathlib.Analysis.SpecialFunctions.Complex.Log
/-!
# Maps on the unit circle
In this file we prove some basic lemmas ... |
/-! ### Map from `AddCircle` to `Circle` -/
theorem scaled_exp_map_periodic : Function.Periodic (fun x => Circle.exp (2 * π / T * x)) T := by
| Mathlib/Analysis/SpecialFunctions/Complex/Circle.lean | 123 | 126 |
/-
Copyright (c) 2021 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin
-/
import Mathlib.Algebra.Lie.OfAssociative
/-!
# Jordan rings
Let `A` be a non-unital, non-associative ring. Then `A` is said to be a (commutative, linear) Jo... | Mathlib/Algebra/Jordan/Basic.lean | 240 | 246 | |
/-
Copyright (c) 2024 Lean FRO. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Data.List.InsertIdx
/-!
This is a stub file for importing `Mathlib.Data.List.InsertNth`,
which has been renamed to `Mathlib.Data.List.InsertIdx`.
This file c... | Mathlib/Data/List/InsertNth.lean | 59 | 67 | |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Morphisms.Constructors
import Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.Equaliz... | QuasiSeparatedSpace X ↔ QuasiSeparated (terminal.from X) :=
(quasiSeparated_over_affine_iff _).symm
| Mathlib/AlgebraicGeometry/Morphisms/QuasiSeparated.lean | 134 | 136 |
/-
Copyright (c) 2024 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.Tactic.Group
import Mathlib.Topology.UrysohnsLemma
/-!
# Everywhere positive sets in measure spaces
... |
/-- The everywhere positive subset of a set is obtained by removing an open set. -/
lemma exists_isOpen_everywherePosSubset_eq_diff (μ : Measure α) (s : Set α) :
∃ u, IsOpen u ∧ μ.everywherePosSubset s = s \ u := by
refine ⟨{x | ∃ n ∈ 𝓝[s] x, μ n = 0}, ?_, by ext x; simp [everywherePosSubset, zero_lt_iff]⟩
rw... | Mathlib/MeasureTheory/Measure/EverywherePos.lean | 59 | 73 |
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: María Inés de Frutos-Fernández
-/
import Mathlib.Order.Filter.Cofinite
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.UniqueFactorizationDomai... | by_contra h_nmem
rw [mem_union, mem_setOf_eq, mem_setOf_eq] at h_nmem
push_neg at h_nmem
rw [← Associates.count_ne_zero_iff_dvd ha_ne_zero hv_irred, not_not,
← Associates.count_ne_zero_iff_dvd hJ_ne_zero hv_irred, not_not] at h_nmem
rw [mem_setOf_eq, h_nmem.1, h_nmem.2, sub_self] at hv
exa... | Mathlib/RingTheory/DedekindDomain/Factorization.lean | 559 | 565 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Function.L1Space.Integrable
import Mathlib.MeasureTheory.Function.LpSpace.Indicator
/-! # Functions integrable on a set an... | exact ⟨s, sl, hs.neg⟩
protected theorem IntegrableAtFilter.sub {f g : α → E}
(hf : IntegrableAtFilter f l μ) (hg : IntegrableAtFilter g l μ) :
IntegrableAtFilter (f - g) l μ := by
rw [sub_eq_add_neg]
exact hf.add hg.neg
| Mathlib/MeasureTheory/Integral/IntegrableOn.lean | 414 | 420 |
/-
Copyright (c) 2023 Adrian Wüthrich. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adrian Wüthrich
-/
import Mathlib.Combinatorics.SimpleGraph.AdjMatrix
import Mathlib.LinearAlgebra.Matrix.PosDef
/-!
# Laplacian Matrix
This module defines the Laplacian matrix of a... | /-- Let $L$ be the graph Laplacian and let $x \in \mathbb{R}$, then
$$x^{\top} L x = \sum_{i \sim j} (x_{i}-x_{j})^{2}$$,
where $\sim$ denotes the adjacency relation -/
theorem lapMatrix_toLinearMap₂' [Field R] [CharZero R] (x : V → R) :
toLinearMap₂' R (G.lapMatrix R) x x =
(∑ i : V, ∑ j : V, if G.Adj i j then... | Mathlib/Combinatorics/SimpleGraph/LapMatrix.lean | 76 | 88 |
/-
Copyright (c) 2021 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, David Kurniadi Angdinata
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.CubicDiscriminant
import Mathlib.RingTheory.Nilpotent.Defs
import Mathlib.Tactic.Fiel... | Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean | 537 | 539 | |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym
/-!
# Exponentially tilted measures
The exponential tilting of a measure `μ` on `α` by a function `f : α... | lemma tilted_apply_eq_ofReal_integral [SFinite μ] (f : α → ℝ) (s : Set α) :
μ.tilted f s = ENNReal.ofReal (∫ a in s, exp (f a) / ∫ x, exp (f x) ∂μ ∂μ) := by
by_cases hf : Integrable (fun x ↦ exp (f x)) μ
· rw [tilted_apply _ _, ← ofReal_integral_eq_lintegral_ofReal]
· exact hf.integrableOn.div_const _
·... | Mathlib/MeasureTheory/Measure/Tilted.lean | 114 | 120 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Convert
import Mathlib.Tactic.SplitIfs
import Mathlib.Tactic.Tauto
/-!
# More basic logic properties
A few more logic le... | Mathlib/Logic/Lemmas.lean | 76 | 78 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
/-! # P... |
theorem inv_rpow (hx : 0 ≤ x) (y : ℝ) : x⁻¹ ^ y = (x ^ y)⁻¹ := by
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 462 | 463 |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Rémy Degenne
-/
import Mathlib.Order.ConditionallyCompleteLattice.Indexed
import Mathlib.Order.Filter.IsBounded
import Mathlib.Order.Hom.CompleteL... | Mathlib/Order/LiminfLimsup.lean | 1,232 | 1,233 | |
/-
Copyright (c) 2021 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Ines Wright, Joachim Breitner
-/
import Mathlib.GroupTheory.Solvable
import Mathlib.GroupTheory.Sylow
import Mathlib.Algebra.Group.Subgroup.Order
import Mathlib.GroupTheo... | classical
constructor
· intro h
exact Nat.find_le h
· intro h
rw [eq_top_iff, ← upperCentralSeries_nilpotencyClass]
exact upperCentralSeries_mono _ h
| Mathlib/GroupTheory/Nilpotent.lean | 372 | 379 |
/-
Copyright (c) 2021 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, David Kurniadi Angdinata
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.CubicDiscriminant
import Mathlib.RingTheory.Nilpotent.Defs
import Mathlib.Tactic.Fiel... | Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean | 607 | 610 | |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomotopyCategory.HomComplex
import Mathlib.Algebra.Homology.HomotopyCategory.Shift
/-! Shifting cochains
Let `C` be a preadditive category. Gi... | change shiftAddHom K L n a (-γ) = _
apply map_neg
@[simp]
lemma rightUnshift_add {n' a : ℤ} (γ₁ γ₂ : Cochain K (L⟦a⟧) n') (n : ℤ) (hn : n' + a = n) :
| Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean | 258 | 262 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Order.Preorder.Chain
import Mathlib.Tactic.Linter.DeprecatedModule
deprecated_module (since := "2025-04-13")
| Mathlib/Order/Chain.lean | 107 | 110 | |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Data.NNReal.Basic
import Mathlib.Topology.Algebra.Support
import Mathlib.To... | /-- The norm of a normed group as a group norm. -/
@[to_additive "The norm of a normed group as an additive group norm."]
def normGroupNorm : GroupNorm E :=
| Mathlib/Analysis/Normed/Group/Basic.lean | 1,294 | 1,296 |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calc... | Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | 523 | 530 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Data.Nat.Choose.Sum
impo... | @[simp]
theorem intCast_coeff_zero {i : ℤ} {R : Type*} [Ring R] : (i : R[X]).coeff 0 = i := by
| Mathlib/Algebra/Polynomial/Coeff.lean | 350 | 351 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive
impo... | cases h' _ rfl h
| Mathlib/Data/Option/Basic.lean | 171 | 172 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker, Andrew Yang, Yuyang Zhao
-/
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.RingTheory.Polynomial.ScaleRoots
/-!
# The... | theorem integralNormalization_C {x : R} (hx : x ≠ 0) : integralNormalization (C x) = 1 := by
simp [integralNormalization, sum_def, support_C hx, degree_C hx]
| Mathlib/RingTheory/Polynomial/IntegralNormalization.lean | 44 | 45 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Order.Group.Finset
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Eva... | Irreducible p ↔ ∀ q, Monic q → natDegree q ∈ Finset.Ioc 0 (natDegree p / 2) → ¬ q ∣ p := by
have : p * C (leadingCoeff p)⁻¹ ≠ 1 := by
contrapose! hpu
exact isUnit_of_mul_eq_one _ _ hpu
rw [← irreducible_mul_leadingCoeff_inv,
(monic_mul_leadingCoeff_inv hp0).irreducible_iff_lt_natDegree_lt this,
... | Mathlib/Algebra/Polynomial/FieldDivision.lean | 643 | 652 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Grade
import Mathlib.Data.Finset.Powerset
import Mathlib.Order.Interval.Finset.Basic
/-!
# Intervals of finsets as finsets
This file provides... | /-- A function `f` from `Finset α` is strictly antitone if and only if `f (cons a s ha) < f s` for
all `s` and `a ∉ s`. -/
lemma strictAnti_iff_forall_cons_lt : StrictAnti f ↔ ∀ s ⦃a⦄ ha, f (cons a s ha) < f s :=
strictMono_iff_forall_lt_cons (β := βᵒᵈ)
| Mathlib/Data/Finset/Interval.lean | 139 | 142 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.Additive
import Mathlib.Algebra.Homology.ShortComplex.Exact
import Mathlib.Algebra.Homology.ShortComplex.Preadditive
import Mathlib.Tactic.Linar... | ShortComplex.opcyclesMapIso (K.isoSc' i j k hi hk)
@[reassoc (attr := simp)]
lemma pOpcycles_opcyclesIsoSc'_inv :
(K.sc' i j k).pOpcycles ≫ (K.opcyclesIsoSc' i j k hi hk).inv = K.pOpcycles j := by
| Mathlib/Algebra/Homology/ShortComplex/HomologicalComplex.lean | 798 | 802 |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Joseph Myers
-/
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.Normed.Group.AddTorsor
/-!
# Perpendicular bisector of a segment
We def... | refine ⟨fun h ↦ ?_, fun h ↦ h ▸ perpBisector_self _⟩
rw [← left_mem_perpBisector, h]
| Mathlib/Geometry/Euclidean/PerpBisector.lean | 109 | 110 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Data.Sign
import Mathlib.Topology.Order.Basic
/-!
# Topology on `SignType`
This file gives `SignType` the discrete topology, and proves continuity result... |
end LinearOrder
| Mathlib/Topology/Instances/Sign.lean | 50 | 53 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Algebra.Group.Action.Defs
import Mathlib.Algebra.Group.End
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.Common
/-!
#... | Mathlib/GroupTheory/Perm/Basic.lean | 295 | 299 | |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Antoine Chambert-Loir
-/
import Mathlib.Algebra.Ring.CharZero
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.IndexNormal
import Mathlib.GroupTheory.Perm.F... | decide
theorem IsThreeCycle.mem_alternatingGroup {f : Perm α} (h : IsThreeCycle f) :
| Mathlib/GroupTheory/SpecificGroups/Alternating.lean | 89 | 91 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Measure.Decomposition.RadonNikodym
/-!
# Exponentially tilted measures
The exponential tilting of a measure `μ` on `α` by a function `f : α... | @[simp]
lemma tilted_of_not_aemeasurable (hf : ¬ AEMeasurable f μ) : μ.tilted f = 0 := by
refine tilted_of_not_integrable ?_
suffices ¬ AEMeasurable (fun x ↦ exp (f x)) μ by exact fun h ↦ this h.1.aemeasurable
exact fun h ↦ hf (aemeasurable_of_aemeasurable_exp h)
| Mathlib/MeasureTheory/Measure/Tilted.lean | 46 | 50 |
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
/-! # Jacobi's theta function
This file define... | intro n
simpa only [Int.cast_add, Int.cast_one] using norm_exp_mul_sq_le hτ (n + 1)
have s : HasSum (fun n : ℕ =>
rexp (-π * τ.im) ^ (n + 1)) (rexp (-π * τ.im) / (1 - rexp (-π * τ.im))) := by
simp_rw [pow_succ', div_eq_mul_inv, hasSum_mul_left_iff (Real.exp_ne_zero _)]
exact hasSum_geometric_of_... | Mathlib/NumberTheory/ModularForms/JacobiTheta/OneVariable.lean | 96 | 116 |
/-
Copyright (c) 2022 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Tactic.IntervalCases
/-!
# Cubics and discriminants
This file defines cubic polynomials ... | /-- The equivalence between cubic polynomials and polynomials of degree at most three. -/
@[simps]
def equiv : Cubic R ≃ { p : R[X] // p.degree ≤ 3 } where
| Mathlib/Algebra/CubicDiscriminant.lean | 241 | 243 |
/-
Copyright (c) 2017 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Data.PFunctor.Univariate.Basic
/-!
# M-types
M types are potentially infinite tree-like structures. They are defined
as the greatest fixpoint of a polynomi... | /-- Bisimulation is the standard proof technique for equality between
infinite tree-like structures -/
structure IsBisimulation : Prop where
/-- The head of the trees are equal -/
head : ∀ {a a'} {f f'}, M.mk ⟨a, f⟩ ~ M.mk ⟨a', f'⟩ → a = a'
| Mathlib/Data/PFunctor/Univariate/M.lean | 532 | 536 |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou
-/
import Mathlib.Data.Set.Order
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.Interval.Set.Image
import Mathlib.Order.Interval.Set.LinearOrder
import Mathlib.Tac... | alias dual_uIoo := uIoo_toDual
@[simp]
| Mathlib/Order/Interval/Set/UnorderedInterval.lean | 331 | 333 |
/-
Copyright (c) 2021 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, David Kurniadi Angdinata
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.CubicDiscriminant
import Mathlib.RingTheory.Nilpotent.Defs
import Mathlib.Tactic.Fiel... | rw [j, W.c₄_of_char_two, ← pow_mul]
/-- A variant of `WeierstrassCurve.j_eq_zero_iff_of_char_two` without assuming a reduced ring. -/
lemma j_eq_zero_iff_of_char_two' : W.j = 0 ↔ W.a₁ ^ 12 = 0 := by
| Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean | 396 | 399 |
/-
Copyright (c) 2023 Koundinya Vajjha. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Koundinya Vajjha, Thomas Browning
-/
import Mathlib.Data.Rat.Defs
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
/-!
This file defines the harmonic numbers.
* `Mathilb/Num... | Mathlib/NumberTheory/Harmonic/Defs.lean | 31 | 34 | |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stephen Morgan, Kim Morrison
-/
import Mathlib.CategoryTheory.Products.Basic
/-!
# Lemmas about functors out of product categories.
-/
open CategoryTheory
namespace CategoryTheory.Bi... | rw [← Functor.map_comp, prod_comp, Category.comp_id]
@[simp]
theorem map_comp_id (F : C × D ⥤ E) (X Y Z : C) (W : D) (f : X ⟶ Y) (g : Y ⟶ Z) :
| Mathlib/CategoryTheory/Products/Bifunctor.lean | 31 | 34 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot
-/
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.OrdConnected
/-!
# Projection of a line onto a closed interval
Given a linearly... | Mathlib/Order/Interval/Set/ProjIcc.lean | 116 | 116 | |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.CharP.Reduced
import Mathlib.RingTheory.IntegralDomain
-- TODO: remove Mathlib.Algebra.CharP.Reduced and move the last two lemmas to Lemmas
/-... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 398 | 401 | |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Finite.Defs
import Mathlib.Data.Finset.BooleanAlgebra
import Mathlib.Data.Finset.Image
import Mathlib.Data.Fintype.Defs
import Mathlib.Data.Fintyp... | Mathlib/Data/Fintype/Basic.lean | 828 | 830 | |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Logic.Encodable.Pi
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Topology.MetricSpace.Closeds
import Mathlib.Topology.MetricSpace.Comple... | have : IsCompact s := (isCompact_range ha.continuous).union (isCompact_range hb.continuous)
letI : MetricSpace (Subtype s) := by infer_instance
| Mathlib/Topology/MetricSpace/GromovHausdorff.lean | 190 | 191 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathli... | Mathlib/Analysis/Complex/Basic.lean | 149 | 149 | |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Kevin Buzzard, Kim Morrison, Johan Commelin, Chris Hughes,
Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.Notation.Pi
imp... | | Int.negSucc n => by rw [zpow_negSucc, hf, map_pow, ← zpow_negSucc]
@[to_additive (attr := simp)]
| Mathlib/Algebra/Group/Hom/Defs.lean | 468 | 470 |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Topology.Bases
import Mathlib.Topology.Separation.Regular
/-!
# Dense embeddings
This file defines three properties of f... | rw [di.nhds_eq_comap a, ((nhds_basis_opens _).comap _).mem_iff] at hs
rcases hs with ⟨U, ⟨haU, hUo⟩, sub : i ⁻¹' U ⊆ s⟩
refine mem_of_superset (hUo.mem_nhds haU) ?_
calc
U ⊆ closure (i '' (i ⁻¹' U)) := di.dense.subset_closure_image_preimage_of_isOpen hUo
_ ⊆ closure (i '' s) := closure_mono (image_subse... | Mathlib/Topology/DenseEmbedding.lean | 65 | 72 |
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.FieldSimp
/-!
# Jensen's inequality... |
/-- **Maximum principle** for convex functions on a segment. If a function `f` is convex on the
segment `[x, y]`, then the eventual maximum of `f` on `[x, y]` is at `x` or `y`. -/
lemma ConvexOn.le_max_of_mem_segment (hf : ConvexOn 𝕜 s f) (hx : x ∈ s) (hy : y ∈ s)
(hz : z ∈ [x -[𝕜] y]) : f z ≤ max (f x) (f y) :=... | Mathlib/Analysis/Convex/Jensen.lean | 296 | 303 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fin.VecNotation
import Mathlib.Logic.Small.Basic
import Mathlib.SetTheory.ZFC.PSet
/-!
# A model of ZFC
In this file, we model Zermelo-Fraenkel ... | Mathlib/SetTheory/ZFC/Basic.lean | 912 | 918 | |
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.Complex.Circle
import Mathlib.Analysis.NormedSpace.BallAction
import Mathlib.Analysis.SpecialFunc... | simp
convert congrArg LinearMap.range (this.comp 0 U.symm.toContinuousLinearEquiv.hasFDerivAt).fderiv
symm
convert
(U.symm : EuclideanSpace ℝ (Fin n) ≃ₗᵢ[ℝ] (ℝ ∙ (↑(-v) : E))ᗮ).range_comp
(ℝ ∙ (↑(-v) : E))ᗮ.subtype using 1
| Mathlib/Geometry/Manifold/Instances/Sphere.lean | 490 | 495 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.ULift
import Mathlib.Data.ZMod.Defs
import Mathlib.SetTheory.Cardinal.ToNat
import Mathlib.SetTheory.Cardinal.ENat
/-!
# Finite Cardinality Funct... |
lemma card_eq_card_toFinset (s : Set α) [Fintype s] : Nat.card s = s.toFinset.card := by
| Mathlib/SetTheory/Cardinal/Finite.lean | 52 | 53 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
/-!
# Grönwall's inequality
The main technical result of this file is the Grönwall-like inequality
`norm_le_gron... | exact continuous_const.add ((continuous_id.mul continuous_const).mul continuous_const)
/-! ### Inequality and corollaries -/
| Mathlib/Analysis/ODE/Gronwall.lean | 86 | 89 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Yaël Dillies
-/
import Mathlib.Data.Set.BooleanAlgebra
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.Hom.Basic
/-!
# Closure operators between preorders
We defin... | c = ofPred c c.IsClosed c.le_closure c.isClosed_closure fun _ _ ↦ closure_min := by
ext
| Mathlib/Order/Closure.lean | 204 | 205 |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Measure.Content
import Mathlib.MeasureTheory.Group.Prod
import Mathlib.Topology.Algebra.Group.Compact
/-!
# Haar measure
In this fi... | K₀.interior_nonempty.mono (interior_mono subset_closure)
have := measure_eq_div_smul μ (haarMeasure K₀)
| Mathlib/MeasureTheory/Measure/Haar/Basic.lean | 665 | 666 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Algebra.Group.TypeTags.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Piecewise
import Mathlib.Or... |
variable [Finite ι] [∀ i, DiscreteTopology (π i)]
| Mathlib/Topology/Constructions.lean | 985 | 986 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.SymmDiff
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Irreducible
/-!
# Connected subsets ... | (the quotient and subspace topologies of the image agree) whose fibers are preconnected. -/
theorem IsPreconnected.preimage_of_isOpenMap [TopologicalSpace β] {f : α → β} {s : Set β}
(hs : IsPreconnected s) (hinj : Function.Injective f) (hf : IsOpenMap f) (hsf : s ⊆ range f) :
IsPreconnected (f ⁻¹' s) := fun u ... | Mathlib/Topology/Connected/Basic.lean | 338 | 358 |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Analysis.InnerProductSpace.Convex
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Combinatorics.Additive.AP.Three... | theorem sum_sq_le_of_mem_box (hx : x ∈ box n d) : ∑ i : Fin n, x i ^ 2 ≤ n * (d - 1) ^ 2 := by
rw [mem_box] at hx
have : ∀ i, x i ^ 2 ≤ (d - 1) ^ 2 := fun i =>
Nat.pow_le_pow_left (Nat.le_sub_one_of_lt (hx i)) _
exact (sum_le_card_nsmul univ _ _ fun i _ => this i).trans (by rw [card_fin, smul_eq_mul])
theore... | Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | 193 | 202 |
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Fin.VecNotation
import Mathlib.SetTheory.Cardinal.Basic
/-!
# Basics on... | theorem funMap_eq_coe_constants {c : L.Constants} {x : Fin 0 → M} : funMap c x = c :=
| Mathlib/ModelTheory/Basic.lean | 235 | 235 |
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.MeasureTheory.Measure.Doubling
import Mathlib.MeasureTheory.Covering.Vitali
import Mathlib.MeasureTheory.Covering.Differentiation
/-!
# Uniformly locally do... |
/-- A version of **Lebesgue differentiation theorem** for a sequence of closed balls whose
centers are not required to be fixed. -/
theorem ae_tendsto_average [NormedSpace ℝ E] [CompleteSpace E]
{f : α → E} (hf : LocallyIntegrable f μ) (K : ℝ) : ∀ᵐ x ∂μ,
∀ {ι : Type*} {l : Filter ι} (w : ι → α) (δ : ι → ℝ) (... | Mathlib/MeasureTheory/Covering/DensityTheorem.lean | 156 | 161 |
/-
Copyright (c) 2023 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.Degree.Lemmas
/-!
# `compute_degree` and `monicity`: tactics for explicit polynomials
This file defines two related tactics: `comp... | end ring
| Mathlib/Tactic/ComputeDegree.lean | 187 | 188 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yury Kudryashov
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.NonUnitalSubalgebra
import Mathlib.RingTheory.SimpleRing.Basic
/-!
# Subalgebras over Comm... | Mathlib/Algebra/Algebra/Subalgebra/Basic.lean | 1,175 | 1,178 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPi... | rw [P.sylow_mem_fixedPoints_iff, ← inf_eq_left, hP.inf_normalizer_sylow, inf_eq_left]
/-- A generalization of **Sylow's second theorem**.
| Mathlib/GroupTheory/Sylow.lean | 275 | 277 |
/-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
import Mathlib.CategoryTheory.Galois.GaloisObjects
import Mathlib.CategoryTheory.Limits.Shapes.CombinedProducts
import Mathlib.Data.Finite.Sum
/-!
# Decompositio... | in the fiber of `selfProd F X`. Applying `connected_component_unique` yields the result. -/
private lemma subobj_selfProd_trans [Mono u] (b : F.obj A) : ∃ (f : A ≅ A), F.map f.hom b = a := by
apply connected_component_unique F b a u (selfProdPermIncl h b)
exact selfProdTermIncl_fib_eq h b
end GaloisRepAux
/-- The... | Mathlib/CategoryTheory/Galois/Decomposition.lean | 265 | 284 |
/-
Copyright (c) 2022 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Fi... | /-- Casting `SignType → ℤ → α` is the same as casting directly `SignType → α`. -/
@[simp, norm_cast]
lemma intCast_cast {α : Type*} [AddGroupWithOne α] (s : SignType) : ((s : ℤ) : α) = s :=
map_cast' _ Int.cast_one Int.cast_zero (@Int.cast_one α _ ▸ Int.cast_neg 1) _
| Mathlib/Data/Sign.lean | 252 | 256 |
/-
Copyright (c) 2024 Newell Jensen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Newell Jensen, Mitchell Lee, Óscar Álvarez
-/
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Algebra.Ring.Int.Parity
import Mathlib.GroupTheory.Coxeter.Matrix
import Mat... | | succ k h' =>
have hk : k < 2 * p := by omega
apply h' at hk
by_cases h_even : Even k
· simp only [h_even, ↓reduceIte] at hk
simp only [Nat.not_even_iff_odd.mpr (Even.add_one h_even), ↓reduceIte]
rw [← List.take_concat_get (by simp [h]; omega), alternatingWord_succ, ← hk]
| Mathlib/GroupTheory/Coxeter/Basic.lean | 462 | 468 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.ConditionalProbability
import Mathlib.Probability.Kernel.Basic
import Mathlib.Probability.Kernel.Composition.MeasureComp
import Mathlib.Tactic.... | end Indep
/-! ### Deducing `Indep` from `iIndep` -/
section FromiIndepToIndep
variable {_mα : MeasurableSpace α}
theorem iIndepSets.indepSets {s : ι → Set (Set Ω)} {_mΩ : MeasurableSpace Ω}
{κ : Kernel α Ω} {μ : Measure α} (h_indep : iIndepSets s κ μ) {i j : ι} (hij : i ≠ j) :
IndepSets (s i) (s j) κ μ := ... | Mathlib/Probability/Independence/Kernel.lean | 410 | 446 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.MeasurableSpace.MeasurablyGenerated
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.Order.Interval.Set... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 1,913 | 1,927 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Integral.Le... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 778 | 780 | |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
/-!
# Intervals as finsets
This file provides basic results about all the `Finset.Ixx... | rw [← coe_eq_empty, coe_Ioo, Set.Ioo_eq_empty_iff]
| Mathlib/Order/Interval/Finset/Basic.lean | 94 | 95 |
/-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
import Mathlib.CategoryTheory.Galois.GaloisObjects
import Mathlib.CategoryTheory.Limits.Shapes.CombinedProducts
import Mathlib.Data.Finite.Sum
/-!
# Decompositio... | rw [hfi1, ← hfi2]
exact congr_fun (F.mapIso fi2).hom_inv_id y
· refine ⟨evaluation_injective_of_isConnected F A X a, ?_⟩
intro x
use u ≫ Pi.π _ x
exact (selfProdProj_fiber h1) x
| Mathlib/CategoryTheory/Galois/Decomposition.lean | 286 | 292 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomologicalComplexLimits
import Mathlib.Algebra.Homology.Additive
/-! Binary biproducts of homological complexes
In this file, it is shown tha... | @[reassoc (attr := simp)]
lemma biprod_inr_desc_f (α : K ⟶ M) (β : L ⟶ M) (i : ι) :
(biprod.inr : L ⟶ K ⊞ L).f i ≫ (biprod.desc α β).f i = β.f i := by
rw [← comp_f, biprod.inr_desc]
| Mathlib/Algebra/Homology/HomologicalComplexBiprod.lean | 93 | 96 |
/-
Copyright (c) 2021 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno
-/
import Mathlib.CategoryTheory.NatIso
/-!
# Bicategories
In this file we define typeclass for bicategories.
A bicategory `B` consists of
* objects `a : B`,
* 1-morphisms ... | theorem associator_naturality_left {f f' : a ⟶ b} (η : f ⟶ f') (g : b ⟶ c) (h : c ⟶ d) :
η ▷ g ▷ h ≫ (α_ f' g h).hom = (α_ f g h).hom ≫ η ▷ (g ≫ h) := by simp
@[reassoc]
| Mathlib/CategoryTheory/Bicategory/Basic.lean | 296 | 299 |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Finite.Prod
import Mathlib.Data.Matroid.Init
import Mathlib.Data.Set.Card
import Mathlib.Data.Set.Finite.Powerset
import Mathlib.Order.UpperLower.Clos... | M.subset_ground B hB
| Mathlib/Data/Matroid/Basic.lean | 391 | 392 |
/-
Copyright (c) 2023 Adrian Wüthrich. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adrian Wüthrich
-/
import Mathlib.Combinatorics.SimpleGraph.AdjMatrix
import Mathlib.LinearAlgebra.Matrix.PosDef
/-!
# Laplacian Matrix
This module defines the Laplacian matrix of a... | theorem isSymm_lapMatrix [AddGroupWithOne R] : (G.lapMatrix R).IsSymm :=
(isSymm_degMatrix _).sub (isSymm_adjMatrix _)
| Mathlib/Combinatorics/SimpleGraph/LapMatrix.lean | 53 | 55 |
/-
Copyright (c) 2024 Emilie Burgun. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Emilie Burgun
-/
import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Dynamics.PeriodicPts.Defs
import Mathlib.GroupTheory.G... | theorem fixedBy_subset_fixedBy_zpow (g : G) (j : ℤ) :
fixedBy α g ⊆ fixedBy α (g ^ j) := by
intro a a_in_fixedBy
rw [mem_fixedBy, zpow_smul_eq_iff_minimalPeriod_dvd,
minimalPeriod_eq_one_iff_fixedBy.mpr a_in_fixedBy, Int.natCast_one]
exact one_dvd j
| Mathlib/GroupTheory/GroupAction/FixedPoints.lean | 82 | 87 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPi... | · exact heq
· haveI := hnc _ hK
have hPK : P ≤ K := le_trans le_normalizer hNK
refine (hK.1 ?_).elim
rw [← sup_of_le_right hNK, P.normalizer_sup_eq_top' hPK])
theorem normal_of_normalizerCondition (hnc : NormalizerCondition G) {p : ℕ} [Fact p.Prime]
[Finite (Sylow p G)] (P : Syl... | Mathlib/GroupTheory/Sylow.lean | 763 | 770 |
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov, Hunter Monroe
-/
import Mathlib.Combinatorics.SimpleGraph.Init
import Mathlib.Data.Finite.Prod
import... | simp [incidenceSet]
theorem mem_incidence_iff_neighbor {v w : V} :
s(v, w) ∈ G.incidenceSet v ↔ w ∈ G.neighborSet v := by
simp only [mem_incidenceSet, mem_neighborSet]
| Mathlib/Combinatorics/SimpleGraph/Basic.lean | 684 | 688 |
/-
Copyright (c) 2022 Paul Reichert. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul Reichert
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
/-!
# Affine map restrictions
This file defines restrictions of affine maps.
## Main definitions
* The... | obtain ⟨x, hx⟩ := id Ene
exact ⟨⟨φ x, AffineSubspace.mem_map.mpr ⟨x, hx, rfl⟩⟩⟩
attribute [local instance] AffineSubspace.nonempty_map AffineSubspace.toAddTorsor
| Mathlib/LinearAlgebra/AffineSpace/Restrict.lean | 33 | 36 |
/-
Copyright (c) 2024 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.DirectSum.LinearMap
import Mathlib.Algebra.Lie.Weights.Cartan
import Mathlib.Data.Int.Interval
import Mathlib.LinearAlgebra.Trace
import Mathlib.Ring... | section
variable (hα : α ≠ 0)
include hα
lemma chainTopCoeff_add_one :
letI := Classical.propDecidable
chainTopCoeff α β + 1 =
| Mathlib/Algebra/Lie/Weights/Chain.lean | 271 | 277 |
/-
Copyright (c) 2021 Manuel Candales. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Manuel Candales, Benjamin Davidson
-/
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
import Mathlib.Geometry.Euclidean.Sphere.Basic
/-!
# Power of a point (intersecting ch... | have h1 := vsub_sub_vsub_cancel_left a p m
have h2 := vsub_sub_vsub_cancel_left p q m
have h3 := vsub_sub_vsub_cancel_left a q m
have h : ∀ r, b -ᵥ r = m -ᵥ r + (m -ᵥ a) := fun r => by
rw [midpoint_vsub_left, ← right_vsub_midpoint, add_comm, vsub_add_vsub_cancel]
iterate 4 rw [dist_eq_norm_vsub V]
rw [←... | Mathlib/Geometry/Euclidean/Sphere/Power.lean | 84 | 96 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Analysis.Convex.Hull
/-!
# Convex join
This file defines the convex join of two sets. The convex join of `s` and `t` is the union of the
segments with on... |
protected theorem Convex.convexHull_union (hs : Convex 𝕜 s) (ht : Convex 𝕜 t) (hs₀ : s.Nonempty)
(ht₀ : t.Nonempty) : convexHull 𝕜 (s ∪ t) = convexJoin 𝕜 s t :=
| Mathlib/Analysis/Convex/Join.lean | 160 | 162 |
/-
Copyright (c) 2022 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.BernoulliPolynomials
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import... | · linarith
end Examples
| Mathlib/NumberTheory/ZetaValues.lean | 368 | 376 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.CharP.Two
import Mathlib.Data.Nat.Cast.Field
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Factorization.Induction
import Mat... | exact periodic_coprime a
| Mathlib/Data/Nat/Totient.lean | 74 | 74 |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Measure.Content
import Mathlib.MeasureTheory.Group.Prod
import Mathlib.Topology.Algebra.Group.Compact
/-!
# Haar measure
In this fi... | apply
isHaarMeasure_of_isCompact_nonempty_interior (haarMeasure K₀) K₀ K₀.isCompact
| Mathlib/MeasureTheory/Measure/Haar/Basic.lean | 568 | 569 |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis, Eric Wieser
-/
import Mathlib.LinearAlgebra.Multilinear.TensorProduct
import Mathlib.Tactic.AdaptationNote
import Mathlib.LinearAlgebra.Multilinear.Curry
/-!
# Tenso... |
@[simp]
theorem reindex_refl : reindex R s (Equiv.refl ι) = LinearEquiv.refl R _ := by
| Mathlib/LinearAlgebra/PiTensorProduct.lean | 753 | 755 |
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.BigOperators.Expect
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Canonical
import Mathlib.Algebra.O... | Mathlib/Data/NNReal/Basic.lean | 817 | 819 | |
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Rémy Degenne
-/
import Mathlib.Probability.Process.Adapted
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Stopping times, stopped processes and stopped va... | ext1 ω; simp only [Set.mem_setOf_eq, Set.mem_compl_iff, not_lt]
rw [this]
exact (hτ.measurableSet_lt i).compl
theorem IsStoppingTime.measurableSet_eq (hτ : IsStoppingTime f τ) (i : ι) :
MeasurableSet[f i] {ω | τ ω = i} := by
have : {ω | τ ω = i} = {ω | τ ω ≤ i} ∩ {ω | τ ω ≥ i} := by
ext1 ω; simp only... | Mathlib/Probability/Process/Stopping.lean | 191 | 199 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.GroupWithZero.Pointwise.Set.Basic
import Mathlib.Algebra.Ring.Pointwise.Set
import Mathlib.Topology.MetricSpace.Isometry
import Mathlib.Topol... |
@[to_additive]
instance Pi.isIsometricSMul'' {ι} {M : ι → Type*} [Fintype ι] [∀ i, Mul (M i)]
| Mathlib/Topology/MetricSpace/IsometricSMul.lean | 420 | 422 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Order.Field.Power
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.RingTheory.Polynomial.Bernstein
import Mathlib.Topology.ContinuousMap... | theorem δ_pos {f : C(I, ℝ)} {ε : ℝ} {h : 0 < ε} : 0 < δ f ε h :=
f.modulus_pos
| Mathlib/Analysis/SpecialFunctions/Bernstein.lean | 170 | 172 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Yury Kudryashov
-/
import Mathlib.Data.Finset.Fin
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Order.Interval.Set.Fin
/-!
# Finite intervals in `Fin n`
This fi... | theorem finsetImage_val_Ioi : (Ioi a).image val = Ioo (a : ℕ) n := by simp [← coe_inj]
@[simp]
| Mathlib/Order/Interval/Finset/Fin.lean | 161 | 163 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Simon Hudon
-/
import Mathlib.Data.PFunctor.Multivariate.Basic
/-!
# Multivariate quotients of polynomial functors.
Basic definition of multivariate QPF. QPFs form a co... | constructor
· intro h
| Mathlib/Data/QPF/Multivariate/Basic.lean | 176 | 177 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Batteries.Data.Rat.Lemmas
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Rat.Init
import Mathlib.Order.Basic
import Mathlib.Tactic.Commo... | Mathlib/Data/Rat/Defs.lean | 310 | 310 | |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.HomotopyCategory.HomComplex
import Mathlib.Algebra.Homology.HomotopyCofiber
/-! # The mapping cone of a morphism of cochain complexes
In this ... | exact ⟨h₁, h₂⟩
lemma id :
(fst φ).1.comp (inl φ) (add_neg_cancel 1) +
| Mathlib/Algebra/Homology/HomotopyCategory/MappingCone.lean | 222 | 225 |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.MeasureTheory.Group.GeometryOfNumbers
import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
import Mathlib.NumberTheory.NumberField.CanonicalEmbedd... | obtain ⟨a, ha, rfl⟩ := hx
exact ⟨a, ha, by simpa using h_nz, (convexBodyLT_mem K f).mp h_mem⟩
open scoped Classical in
| Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | 491 | 494 |
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