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) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.Group.Multiset.Defs
import Mathlib.A... | {x : M} (h : x ∈ closure s) :
p x h := by
simp_rw [closure_eq_mrange] at h
| Mathlib/Algebra/Group/Submonoid/Membership.lean | 241 | 243 |
/-
Copyright (c) 2020 Jean Lo, Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yury Kudryashov
-/
import Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set
import Mathlib.Algebra.Ring.Action.Pointwise.Set
import Mathlib.Topology.Bornology.Basic... | end GroupWithZero
section AddGroup
variable {M E : Type*} [Monoid M] [AddGroup E] [DistribMulAction M E] [Bornology M]
| Mathlib/Topology/Bornology/Absorbs.lean | 193 | 197 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Antoine Chambert-Loir
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.RingTheory.Ideal.Maps
/-!
# More operations on subalgebras
In this file we relate the ... | {ι : Type*} (ι' : Finset ι) (s : ι → S) (l : ι → S)
(e : ∑ i ∈ ι', l i * s i = 1) (hs : ∀ i, s i ∈ S') (hl : ∀ i, l i ∈ S') (x : S)
(H : ∀ i, ∃ n : ℕ, (s i ^ n : S) • x ∈ S') : x ∈ S' := by
suffices x ∈ Subalgebra.toSubmodule (Algebra.ofId S' S).range by
obtain ⟨x, rfl⟩ := this
exact x.2
choose ... | Mathlib/Algebra/Algebra/Subalgebra/Operations.lean | 40 | 68 |
/-
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.ShortComplex.QuasiIso
import Mathlib.CategoryTheory.Limits.Preserves.Finite
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels
/-!
... | Iso.hom_inv_id, comp_id, Iso.hom_inv_id_assoc]
@[reassoc]
lemma mapOpcyclesIso_hom_naturality [S₁.HasRightHomology] [S₂.HasRightHomology]
[F.PreservesRightHomologyOf S₁] [F.PreservesRightHomologyOf S₂] :
opcyclesMap (F.mapShortComplex.map φ) ≫ (S₂.mapOpcyclesIso F).hom =
(S₁.mapOpcyclesIso F).hom ≫ F... | Mathlib/Algebra/Homology/ShortComplex/PreservesHomology.lean | 528 | 536 |
/-
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
-/
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Analysis.NormedSpac... |
theorem log_nonpos_iff (hx : 0 ≤ x) : log x ≤ 0 ↔ x ≤ 1 := by
rcases hx.eq_or_lt with (rfl | hx)
· simp [le_refl, zero_le_one]
rw [← not_lt, log_pos_iff hx.le, not_lt]
| Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 194 | 198 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kim Morrison
-/
import Mathlib.Algebra.BigOperators.Finsupp.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Preimage
import Mathlib.Algebra.Module.Defs
import Ma... | @[simp]
theorem graph_zero : graph (0 : α →₀ M) = ∅ := by simp [graph]
@[simp]
theorem graph_eq_empty {f : α →₀ M} : f.graph = ∅ ↔ f = 0 :=
(graph_injective α M).eq_iff' graph_zero
| Mathlib/Data/Finsupp/Basic.lean | 101 | 106 |
/-
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.Field.IsField
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.Algebra.Polynomial.Monic
im... | end Ring
section CommRing
variable [CommRing R] {p p₁ p₂ q : R[X]}
| Mathlib/Algebra/Polynomial/Div.lean | 570 | 575 |
/-
Copyright (c) 2022 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Data.ENNReal.Lemmas
import Mathlib.Topology.MetricSpace.Thickening
import Mathlib.Topology.ContinuousMap.Bounded.Basic
/-!
# Thickened indicators
This fi... | unfold thickenedIndicatorAux
apply le_antisymm _ bot_le
have key := tsub_le_tsub
| Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | 84 | 86 |
/-
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.Set.Prod
import Mathlib.Logic.Equiv.Fin.Basic
import Mathlib.ModelTheory... | Mathlib/ModelTheory/Syntax.lean | 1,155 | 1,170 | |
/-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Dagur Asgeirsson
-/
import Mathlib.Topology.Category.CompHaus.Basic
import Mathlib.Topology.Category.CompHausLike.Limits
/-!
# Explicit limits and colimits
This file applies ... | Mathlib/Topology/Category/CompHaus/Limits.lean | 136 | 139 | |
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Yury Kudryashov
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Data.Nat.ModEq
/-!
# Pigeonhole principles
Given pigeons ... | there is nonnegative, then for at least one of these `n` pigeonholes, the total weight of the
pigeons in this pigeonhole is less than or equal to `b`. -/
theorem exists_sum_fiber_le_of_sum_fiber_nonneg_of_sum_le_nsmul
(hf : ∀ y ∉ t, (0 : M) ≤ ∑ x ∈ s with f x = y, w x) (ht : t.Nonempty)
(hb : ∑ x ∈ s, w x ≤ #t ... | Mathlib/Combinatorics/Pigeonhole.lean | 183 | 190 |
/-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Set.Defs
import Mathlib.Logic.Basic
import Mathlib.Logic.ExistsUnique
import Mathlib.Logic.Nonempty
import Mathlib.Logic.Nontrivia... | Surjective (fun g : β → γ ↦ g ∘ f) ↔ Injective f := by
classical
refine ⟨not_imp_not.mp fun not_inj surj ↦ not_subsingleton γ ⟨fun c c' ↦ ?_⟩,
(·.surjective_comp_right)⟩
| Mathlib/Logic/Function/Basic.lean | 736 | 739 |
/-
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.Calculus.InverseFunctionTheorem.FDeriv
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathl... | Mathlib/Analysis/Calculus/Implicit.lean | 464 | 467 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Floor.Defs
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Algebra.Order.Floor.Semiring
deprecated_module (sinc... | Mathlib/Algebra/Order/Floor.lean | 1,224 | 1,226 | |
/-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
/-!
# Termination of a hydra game
This file deals with the following version... | Fibration (GameAdd (CutExpand r) (CutExpand r)) (CutExpand r) fun s ↦ s.1 + s.2 := by
rintro ⟨s₁, s₂⟩ s ⟨t, a, hr, he⟩; dsimp at he ⊢
classical
obtain ⟨ha, rfl⟩ := add_singleton_eq_iff.1 he
rw [add_assoc, mem_add] at ha
obtain h | h := ha
· refine ⟨(s₁.erase a + t, s₂), GameAdd.fst ⟨t, a, hr, ?_⟩, ?_⟩
... | Mathlib/Logic/Hydra.lean | 114 | 126 |
/-
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.Constructions
import Mathlib.Tactic.FunProp
/-!
# Measurable embeddings and equivalences
A measurable e... | congr
exact (MeasurableEquiv.ofInvolutive _ compl_involutive h).measurableEmbedding.comap_eq
@[simp] theorem MeasurableSpace.comap_not (p : α → Prop) :
MeasurableSpace.comap (fun a ↦ ¬p a) inferInstance = MeasurableSpace.comap p inferInstance :=
MeasurableSpace.comap_compl (fun _ _ ↦ measurableSet_top) _
| Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean | 753 | 759 |
/-
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.Calculus.FDeriv.Prod
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv
import Mathlib.GroupTheory.MonoidLocalization.Basic
impo... | cases nonempty_fintype ι
rw [Fintype.linearIndependent_iff]; push_neg
rcases hextr.exists_multipliers_of_hasStrictFDerivAt hf' hφ' with ⟨Λ, Λ₀, hΛ, hΛf⟩
refine ⟨Option.elim' Λ₀ Λ, ?_, ?_⟩
· simpa [add_comm] using hΛf
· simpa only [funext_iff, not_and_or, or_comm, Option.exists, Prod.mk_eq_zero, Ne,
no... | Mathlib/Analysis/Calculus/LagrangeMultipliers.lean | 132 | 142 |
/-
Copyright (c) 2021 Chris Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Bailey
-/
import Mathlib.Data.Nat.Notation
import Mathlib.Data.String.Defs
import Mathlib.Tactic.Basic
import Batteries.Tactic.Alias
/-!
# Miscellaneous lemmas about strings
-/
n... | | ⟨s⟩ => by simp only [leftpad, String.length, List.length_leftpad]
| Mathlib/Data/String/Lemmas.lean | 30 | 31 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.InnerProductSpace.Sy... | /-- The orthogonal projection onto `Kᗮ` of an element of `K` is zero. -/
theorem orthogonalProjection_mem_subspace_orthogonal_precomplement_eq_zero
[Kᗮ.HasOrthogonalProjection] {v : E} (hv : v ∈ K) : Kᗮ.orthogonalProjection v = 0 :=
orthogonalProjection_mem_subspace_orthogonalComplement_eq_zero (K.le_orthogonal_o... | Mathlib/Analysis/InnerProductSpace/Projection.lean | 849 | 853 |
/-
Copyright (c) 2022 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.BigOperators.Group.List.Basic
import Mathlib.Algebra.Group.Embedding
import Mathlib.Algebra.Group.Nat.Defs
import Mathlib.Data.Finsupp.Single... | cases lt_or_le n l₁.length with
| inl h =>
rw [getD_append _ _ _ _ h, Finsupp.embDomain_notin_range, add_zero]
| Mathlib/Data/List/ToFinsupp.lean | 93 | 95 |
/-
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.Computability.Partrec
import Mathlib.Data.Option.Basic
/-!
# Gödel Numbering for Partial Recursive Functions.
This file defines `Nat.Partrec.Code`, a... | | pair pf pg hf hg =>
| Mathlib/Computability/PartrecCode.lean | 531 | 531 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Init
import Mathlib.Data.Int.Init
import Mathlib.... | Mathlib/Algebra/Group/Basic.lean | 1,326 | 1,326 | |
/-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.Galois.Basic
/-!
# Separably Closed Field
In this file we define the typeclass for separably closed fields and separable closures,
and prove some of thei... | ∃ z, z ^ n = x := by
have hn' : 0 < n := Nat.pos_of_ne_zero fun h => by
rw [h, Nat.cast_zero] at hn
exact hn.out rfl
have : degree (X ^ n - C x) ≠ 0 := by
rw [degree_X_pow_sub_C hn' x]
| Mathlib/FieldTheory/IsSepClosed.lean | 131 | 136 |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.MeasureTheory.Integral.Bochner.FundThmCalculus
import Mathlib.MeasureT... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 835 | 837 | |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Batteries.Data.List.Perm
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.TakeWhile
import Mathlib.Order.Fin.Basic
/-!
# So... | theorem sorted_le_listMap (hf : StrictAnti f) :
(l.map f).Sorted (· ≤ ·) ↔ l.Sorted (· ≥ ·) :=
hf.dual_right.sorted_ge_listMap
theorem sorted_ge_listMap (hf : StrictAnti f) :
(l.map f).Sorted (· ≥ ·) ↔ l.Sorted (· ≤ ·) :=
hf.dual_right.sorted_le_listMap
theorem sorted_lt_listMap (hf : StrictAnti f) :
... | Mathlib/Data/List/Sort.lean | 342 | 356 |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 2,315 | 2,317 | |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.GroupWithZero.Indicator
import Mathlib.Topology.Piecewise
import Mathlib.Topology.Instances.ENNReal.Lemmas
/-!
# Semicontinuous maps
A ... | theorem lowerSemicontinuousWithinAt_tsum {f : ι → α → ℝ≥0∞}
(h : ∀ i, LowerSemicontinuousWithinAt (f i) s x) :
LowerSemicontinuousWithinAt (fun x' => ∑' i, f i x') s x := by
simp_rw [ENNReal.tsum_eq_iSup_sum]
refine lowerSemicontinuousWithinAt_iSup fun b => ?_
exact lowerSemicontinuousWithinAt_sum fun i _... | Mathlib/Topology/Semicontinuous.lean | 643 | 651 |
/-
Copyright (c) 2020 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn
-/
import Mathlib.Data.Finset.Lattice.Fold
import Mathlib.Logic.Encodable.Basic
import Mathlib.Order.Atoms
import Mathlib.Order.Cofinal
import Mathlib.Order.UpperLower.Principal... | namespace Cofinal
variable [Preorder P]
instance : Inhabited (Cofinal P) :=
⟨_, .univ⟩
| Mathlib/Order/Ideal.lean | 477 | 482 |
/-
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.Group.Nat.Defs
import Mathlib.CategoryTheory.Category.Preorder
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Functor.Const
import M... | @[simp]
lemma homMk₅_app_five :
(homMk₅ app₀ app₁ app₂ app₃ app₄ app₅ w₀ w₁ w₂ w₃ w₄).app ⟨5, by valid⟩ = app₅ := rfl
| Mathlib/CategoryTheory/ComposableArrows.lean | 780 | 782 |
/-
Copyright (c) 2023 Jake Levinson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jake Levinson
-/
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Positivity.Core
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
/-!
# D... |
theorem doubleFactorial_eq_prod_even : ∀ n : ℕ, (2 * n)‼ = ∏ i ∈ Finset.range n, 2 * (i + 1)
| 0 => rfl
| n + 1 => by
rw [Finset.prod_range_succ, ← doubleFactorial_eq_prod_even _, mul_comm (2 * n)‼,
(by ring : 2 * (n + 1) = 2 * n + 2)]
| Mathlib/Data/Nat/Factorial/DoubleFactorial.lean | 63 | 68 |
/-
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.Homotopy
import Mathlib.Algebra.Ring.NegOnePow
import Mathlib.Algebra.Category.Grp.Preadditive
import Mathlib.Tactic.Linarith
import Mathlib.Cat... | simp only [comp_v _ _ h p _ q rfl (by omega), neg_v, comp_neg]
@[simp]
protected lemma comp_smul {n₁ n₂ n₁₂ : ℤ} (z₁ : Cochain F G n₁) (k : R) (z₂ : Cochain G K n₂)
(h : n₁ + n₂ = n₁₂ ) : z₁.comp (k • z₂) h = k • (z₁.comp z₂ h) := by
| Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean | 362 | 366 |
/-
Copyright (c) 2023 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.Data.Complex.FiniteDimensional
import Mathlib.MeasureTheory.Constructions.HaarToSphere
import Mathlib.MeasureTheory.Integral.Gamma
import Mathlib.Measure... | variable (ι : Type*) [Fintype ι] {p : ℝ}
theorem MeasureTheory.volume_sum_rpow_lt_one (hp : 1 ≤ p) :
volume {x : ι → ℝ | ∑ i, |x i| ^ p < 1} =
.ofReal ((2 * Gamma (1 / p + 1)) ^ card ι / Gamma (card ι / p + 1)) := by
have h₁ : 0 < p := by linarith
have h₂ : ∀ x : ι → ℝ, 0 ≤ ∑ i, |x i| ^ p := by
refin... | Mathlib/MeasureTheory/Measure/Lebesgue/VolumeOfBalls.lean | 166 | 196 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Devon Tuma, Oliver Nash
-/
import Mathlib.Algebra.Group.Action.Opposite
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.GroupWithZero.Associated
import M... | mul_right_cancel₀ (nonZeroDivisors.coe_ne_zero _) (by
simpa only [Subtype.ext_iff, Submonoid.coe_mul] using h)
| Mathlib/Algebra/GroupWithZero/NonZeroDivisors.lean | 157 | 159 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Peter Pfaffelhuber
-/
import Mathlib.Data.Nat.Lattice
import Mathlib.Data.Set.Accumulate
import Mathlib.Data.Set.Pairwise.Lattice
import Mathlib.MeasureTheory.PiSystem
/-!... | /-- In a semiring of sets `C`, for all set `s ∈ C` and finite set of sets `I ⊆ C`,
`disjointOfDiffUnion` is a finite set of sets in `C` such that
`s \ ⋃₀ I = ⋃₀ (hC.disjointOfDiffUnion hs I hI)`.
`disjointOfDiff` is a special case of `disjointOfDiffUnion` where `I` is a
singleton. -/
| Mathlib/MeasureTheory/SetSemiring.lean | 214 | 218 |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | rfl
@[simp]
theorem mapMatrix_symm (f : α ≃+* β) : f.mapMatrix.symm = (f.symm.mapMatrix : Matrix m m β ≃+* _) :=
| Mathlib/Data/Matrix/Basic.lean | 536 | 539 |
/-
Copyright (c) 2024 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.GroupTheory.Perm.Cycle.Type
/-!
# Fixed-point-free automorphisms
This file defines fixed-point-free automorphisms and proves some basic properties.... |
section Involutive
theorem coe_eq_inv_of_involutive (hφ : FixedPointFree φ) (h2 : Function.Involutive φ) :
| Mathlib/GroupTheory/FixedPointFree.lean | 57 | 60 |
/-
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.Data.Fintype.Card
import Mathlib.Algebra.Order.BigOperators.Group.Multiset
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.Multiset.OrderedM... | See also `Finset.single_le_sum`."]
lemma _root_.CanonicallyOrderedCommMonoid.single_le_prod {i : ι} (hi : i ∈ s) :
f i ≤ ∏ j ∈ s, f j :=
single_le_prod' (fun _ _ ↦ one_le _) hi
| Mathlib/Algebra/Order/BigOperators/Group/Finset.lean | 365 | 368 |
/-
Copyright (c) 2024 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Disintegration.Density
import Mathlib.Probability.Kernel.WithDensity
/-!
# Radon-Nikodym derivative and Lebesgue decomposition for kern... | noncomputable
irreducible_def singularPart (κ η : Kernel α γ) [IsSFiniteKernel κ] [IsSFiniteKernel η] :
Kernel α γ :=
withDensity (κ + η) (fun a x ↦ Real.toNNReal (rnDerivAux κ (κ + η) a x)
- Real.toNNReal (1 - rnDerivAux κ (κ + η) a x) * rnDeriv κ η a x)
| Mathlib/Probability/Kernel/RadonNikodym.lean | 257 | 262 |
/-
Copyright (c) 2024 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn,
Mario Carneiro
-/
import Mathlib.Data.List.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Util... | rw [getD_eq_default _ _ h', getD_eq_default]
rwa [Nat.le_sub_iff_add_le' h, ← length_append]
theorem getD_eq_getD_getElem? (n : ℕ) : l.getD n d = l[n]?.getD d := by
| Mathlib/Data/List/GetD.lean | 83 | 86 |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
/-!
# Topological study of spaces `Π (n : ℕ), E n`
When `E n` are topological spaces, the space `Π (n : ... | exists_pos_mul_lt (half_pos εpos) _
apply @mem_iInf_of_iInter _ _ _ _ _ K.finite_toSet fun i =>
{ p : (∀ i : ι, F i) × ∀ i : ι, F i | dist (p.fst i) (p.snd i) < δ }
| Mathlib/Topology/MetricSpace/PiNat.lean | 846 | 848 |
/-
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.Basic
import Mathlib.Algebra.Module.End
import Mathlib.Algebra.Ring.Prod
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.GroupAc... | map_mul' := cast_mul h
@[simp]
theorem castHom_apply {h : m ∣ n} (i : ZMod n) : castHom h R i = cast i :=
rfl
@[simp]
theorem cast_sub (h : m ∣ n) (a b : ZMod n) : (cast (a - b : ZMod n) : R) = cast a - cast b :=
(castHom h R).map_sub a b
@[simp]
theorem cast_neg (h : m ∣ n) (a : ZMod n) : (cast (-a : ZMod n) ... | Mathlib/Data/ZMod/Basic.lean | 312 | 324 |
/-
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, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | Mathlib/SetTheory/Cardinal/Basic.lean | 1,005 | 1,008 | |
/-
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.Algebra.Group.Action.Pi
import Mathlib.Algebra.Order.AbsoluteValue.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.Mi... | a₁ ⊔ b₁ ≈ a₂ ⊔ b₂ := by
intro ε ε0
| Mathlib/Algebra/Order/CauSeq/Basic.lean | 750 | 751 |
/-
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.Logic.Function.Conjugate
/-!
# Iterations of a function
In this file we prove simple properties of `Nat.iterate f n` a.k.a. `f^[n]`:
* `iterate_ze... | fun f ↦ iterate_comm f m n
| Mathlib/Logic/Function/Iterate.lean | 200 | 201 |
/-
Copyright (c) 2023 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.InvariantForm
import Mathlib.Algebra.Lie.Weights.Cartan
import Mathlib.Algebra.Lie.Weights.Linear
impo... | lemma restrict_killingForm :
(killingForm R L).restrict I = LieModule.traceForm R I L :=
rfl
variable [Module.Free R L] [Module.Finite R L] [IsDomain R] [IsPrincipalIdealRing R]
lemma killingForm_eq :
| Mathlib/Algebra/Lie/TraceForm.lean | 376 | 382 |
/-
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.Data.Set.Constructions
import Mathlib.Order.Filter.AtTopBot.CountablyGenerated
import Mathlib.Topology.Constructions
import Mathlib.Top... | /-- A set is dense iff it has non-trivial intersection with all basis sets. -/
theorem IsTopologicalBasis.dense_iff {b : Set (Set α)} (hb : IsTopologicalBasis b) {s : Set α} :
Dense s ↔ ∀ o ∈ b, Set.Nonempty o → (o ∩ s).Nonempty := by
| Mathlib/Topology/Bases.lean | 208 | 210 |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Xavier Roblot
-/
import Mathlib.Algebra.Algebra.Hom.Rat
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.RingTh... | 0 < nrRealPlaces K := by
refine Nat.pos_of_ne_zero ?_
by_contra hc
refine (Nat.not_odd_iff_even.mpr ?_) h
rw [← card_add_two_mul_card_eq_rank, hc, zero_add]
exact even_two_mul (nrComplexPlaces K)
| Mathlib/NumberTheory/NumberField/Embeddings.lean | 667 | 672 |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.Bornology.Constructions
import Mathlib.Topology.MetricSpace.Pseudo.Def... | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | 436 | 440 | |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Group.PUnit
import Mathlib.Algebra.Group.Subgroup.Ker
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.GroupTheory.Congruence... | ← range_comp, Sum.range_eq]; rfl
@[to_additive (attr := simp)] theorem mrange_inl_sup_mrange_inr :
MonoidHom.mrange (inl : M →* M ∗ N) ⊔ MonoidHom.mrange (inr : N →* M ∗ N) = ⊤ := by
| Mathlib/GroupTheory/Coprod/Basic.lean | 253 | 256 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.InitTail
/-!
# Truncated Witt vectors
The ring of truncated Witt vectors (of length `n`) is a quotient of the... | invFun g :=
⟨fun k => (truncate k).comp g, by
intro _ _ h
simp only [← RingHom.comp_assoc, truncate_comp_wittVector_truncate]⟩
left_inv := by rintro ⟨f, hf⟩; simp only [truncate_comp_lift]
| Mathlib/RingTheory/WittVector/Truncated.lean | 467 | 471 |
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.Asymptotics.Theta
/-!
# Asymptotic equivalence
In this file, we define the relation `IsEquivalent l u v`, which means that `u-v` is litt... | theorem IsLittleO.add_isEquivalent (hu : u =o[l] w) (hv : v ~[l] w) : u + v ~[l] w :=
add_comm v u ▸ hv.add_isLittleO hu
theorem IsLittleO.isEquivalent (huv : (u - v) =o[l] v) : u ~[l] v := huv
theorem IsEquivalent.neg (huv : u ~[l] v) : (fun x ↦ -u x) ~[l] fun x ↦ -v x := by
rw [IsEquivalent]
convert huv.isLit... | Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean | 157 | 164 |
/-
Copyright (c) 2023 Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémi Bottinelli
-/
import Mathlib.Data.Set.Function
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.EMetricSpace.BoundedVariation
/-!
# Constant speed
This file defines... | rw [eVariationOn.subsingleton f (fun y hy z hz => hs hy.1 hz.1 : (s ∩ Icc x x).Subsingleton)]
simp only [sub_self, mul_zero, ENNReal.ofReal_zero]
theorem hasConstantSpeedOnWith_iff_ordered :
HasConstantSpeedOnWith f s l ↔ ∀ ⦃x⦄ (_ : x ∈ s) ⦃y⦄ (_ : y ∈ s),
| Mathlib/Analysis/ConstantSpeed.lean | 64 | 68 |
/-
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.Algebra.Order.Field.Basic
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Orde... |
theorem interedges_biUnion (s : Finset ι) (t : Finset κ) (f : ι → Finset α) (g : κ → Finset β) :
interedges r (s.biUnion f) (t.biUnion g) =
(s ×ˢ t).biUnion fun ab ↦ interedges r (f ab.1) (g ab.2) := by
| Mathlib/Combinatorics/SimpleGraph/Density.lean | 109 | 112 |
/-
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... | set_option linter.deprecated false in
@[deprecated attachFin_uIcc (since := "2025-04-06")]
| Mathlib/Order/Interval/Finset/Fin.lean | 109 | 110 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Gabin Kolly
-/
import Mathlib.Data.Fintype.Order
import Mathlib.Order.Closure
import Mathlib.ModelTheory.Semantics
import Mathlib.ModelTheory.Encoding
/-!
# First-Orde... | variable (L)
| Mathlib/ModelTheory/Substructures.lean | 222 | 223 |
/-
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.Order.Interval.Finset.Basic
/-!
# Intervals as multisets
This file defines intervals as multisets.
## Main declarations
In a `LocallyFiniteOrder`,
* `M... | rw [Ico, ← Finset.filter_val, Finset.Ico_filter_lt_of_le_right hcb]
| Mathlib/Order/Interval/Multiset.lean | 202 | 202 |
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subgroup.Ker
/-!
# Basic results on subgroups
We prove basic results... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 1,673 | 1,675 | |
/-
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.Algebra.Order.Group.Unbundled.Int
import Mathlib.Algebra.Order.Nonneg.Basic
import Mathlib.Algebra.Order.Ring.Unbundled.Rat
imp... | Mathlib/Data/NNRat/Defs.lean | 446 | 447 | |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Init
import Mathlib.Data.Int.Init
import Mathlib.... | Mathlib/Algebra/Group/Basic.lean | 1,331 | 1,332 | |
/-
Copyright (c) 2022 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer, Kevin Klinge, Andrew Yang
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Math... | simp
map_add' := fun x y => by
| Mathlib/RingTheory/OreLocalization/Ring.lean | 143 | 144 |
/-
Copyright (c) 2023 Scott Carnahan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Carnahan
-/
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Eval.SMul
/-!
# Scalar-multiple polynomial ev... | rw [npow_add, smeval_mul, smeval_pow n, pow_one, npow_add, npow_one]
theorem smeval_comp : (p.comp q).smeval x = p.smeval (q.smeval x) := by
induction p using Polynomial.induction_on' with
| add r s hr hs => simp [add_comp, hr, hs, smeval_add]
| monomial n a => simp [smeval_monomial, smeval_C_mul, smeval_po... | Mathlib/Algebra/Polynomial/Smeval.lean | 277 | 282 |
/-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Order.Interval.Finset.Nat
import... | refine coprime_pow_primes _ _ (prime_of_mem_primeFactors hp) (prime_of_mem_primeFactors hq) ?_
contrapose! h'; rwa [← h']
variable {a b}
| Mathlib/Data/Nat/Factorization/Basic.lean | 445 | 448 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.FieldTheory.Finiteness
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
/-!
# Finite-dimensional subsp... | family's cardinality is one more than that of the finite-dimensional space. -/
theorem AffineIndependent.affineSpan_eq_top_iff_card_eq_finrank_add_one [FiniteDimensional k V]
[Fintype ι] {p : ι → P} (hi : AffineIndependent k p) :
affineSpan k (Set.range p) = ⊤ ↔ Fintype.card ι = finrank k V + 1 := by
construc... | Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | 313 | 320 |
/-
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.ShortComplex.Homology
/-!
# Quasi-isomorphisms of short complexes
This file introduces the typeclass `QuasiIso φ` for a morphism `φ : S₁ ⟶ S₂... | lemma quasiIso_of_epi_of_isIso_of_mono (φ : S₁ ⟶ S₂) [Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] :
QuasiIso φ := by
rw [((LeftHomologyMapData.ofEpiOfIsIsoOfMono φ) S₁.leftHomologyData).quasiIso_iff]
dsimp
infer_instance
| Mathlib/Algebra/Homology/ShortComplex/QuasiIso.lean | 138 | 142 |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Set.Lattice
import Mathlib.Order.ConditionallyCompleteLattice.Defs
/-!
# Theory of conditionally complete lattices
A conditionally complet... | WithTop.coe_sInf' (α := αᵒᵈ) hs h's
@[norm_cast]
| Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | 108 | 110 |
/-
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.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import Mathlib.Topology.Order.MonotoneContinuity
import M... | theorem map_add_int (x : ℝ) (m : ℤ) : f (x + m) = f x + m :=
f.commute_add_int m x
| Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 319 | 321 |
/-
Copyright (c) 2024 Mitchell Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Lee, Óscar Álvarez
-/
import Mathlib.GroupTheory.Coxeter.Length
import Mathlib.Data.List.GetD
import Mathlib.Tactic.Group
/-!
# Reflections, inversions, and inversion sequences... | theorem inv : t⁻¹ = t := by
| Mathlib/GroupTheory/Coxeter/Inversion.lean | 80 | 80 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov, Kexing Ying
-/
import Mathlib.Topology.Semicontinuous
import Mathlib.MeasureTheory.Function.AEMeasurableSequence
import Mathlib.MeasureTheory.Order.Lat... | lemma measurableSet_bddBelow_range {ι} [Countable ι] {f : ι → δ → α} (hf : ∀ i, Measurable (f i)) :
MeasurableSet {b | BddBelow (range (fun i ↦ f i b))} :=
measurableSet_bddAbove_range (α := αᵒᵈ) hf
end LinearOrder
section ConditionallyCompleteLattice
@[measurability, fun_prop]
theorem Measurable.iSup_Prop {α}... | Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean | 720 | 736 |
/-
Copyright (c) 2019 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.Data.EReal.Basic
deprecated_module (since := "2025-04-13")
| Mathlib/Data/Real/EReal.lean | 491 | 495 | |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Data.ENNReal.Real
import Mathlib.Tactic.Bound.Attribute
import Mathlib.Topology... | with_reducible_and_instances rfl
section Real
/-- Instantiate the reals as a pseudometric space. -/
| Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | 1,037 | 1,041 |
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Order.Filter.Cofinite
/-!
# Basic theory of bornology
We develop the basic theory of bornologies. Instead of axiomatizing bounded sets and defining
bor... | exact h isBounded_empty
| Mathlib/Topology/Bornology/Basic.lean | 143 | 144 |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Data.Fintype.Parity
import Mathlib.NumberTheory.LegendreSymbol.ZModChar
import Mathlib.FieldTheory.Finite.Basic
/-!
# Quadratic characters of finite fie... |
/-- The sum over the values of the quadratic character is zero when the characteristic is odd. -/
theorem quadraticChar_sum_zero (hF : ringChar F ≠ 2) : ∑ a : F, quadraticChar F a = 0 :=
sum_eq_zero_of_ne_one (quadraticChar_ne_one hF)
| Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean | 236 | 239 |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, David Loeffler, Heather Macbeth, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.Analysis.Calculus.ContDiff.CPolynomial
import... | @[simp 1100] lemma fourierPowSMulRight_apply {f : V → E} {v : V} {n : ℕ} {m : Fin n → W} :
fourierPowSMulRight L f v n m = (- (2 * π * I))^n • (∏ i, L v (m i)) • f v := by
simp [fourierPowSMulRight]
| Mathlib/Analysis/Fourier/FourierTransformDeriv.lean | 292 | 294 |
/-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
import Mathlib.RingTheory.Derivation.ToSquareZero
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.IsTensorProduct
import Mathlib.... |
variable {R}
| Mathlib/RingTheory/Kaehler/Basic.lean | 66 | 67 |
/-
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.Homology.Linear
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Tactic.Abel
/-!
# Chain homotopies
We define chain... | To simplify the situation, we only construct homotopies of the form `Homotopy e 0`.
`Homotopy.equivSubZero` can provide the general case.
Notice however, that this construction does not have particularly good definitional properties:
we have to insert `eqToHom` in several places.
Hopefully this is okay in most applica... | Mathlib/Algebra/Homology/Homotopy.lean | 413 | 419 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Order.Filter.AtTopBot.Archimedean
import Mathlib.Order.Filter.AtTopBot.Finite
import Mathlib.Order.Filter.AtTopBot.Prod
import Mathlib.Topology.A... | · refine eventually_atTop.2
⟨1, fun n hn => hl.trans_le (h.lim_le_div hbdd (zero_lt_one.trans_le hn).ne')⟩
· obtain ⟨n, npos, hn⟩ : ∃ n : ℕ, 0 < n ∧ u n / n < L := by
rw [Subadditive.lim] at hL
rcases exists_lt_of_csInf_lt (by simp) hL with ⟨x, hx, xL⟩
rcases (mem_image _ _ _).1 hx with ⟨n, ... | Mathlib/Analysis/Subadditive.lean | 85 | 95 |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Data.ENNReal.Real
import Mathlib.Tactic.Bound.Attribute
import Mathlib.Topology... | compl_compl]
| Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | 592 | 593 |
/-
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, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | convert arg_mul_cos_add_sin_mul_I hr hi using 3
simp [toIocMod, cos_sub_int_mul_two_pi, sin_sub_int_mul_two_pi]
theorem arg_cos_add_sin_mul_I_eq_toIocMod (θ : ℝ) :
arg (cos θ + sin θ * I) = toIocMod Real.two_pi_pos (-π) θ := by
rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_l... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 453 | 465 |
/-
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.Order.Filter.SmallSets
import Mathlib.Topology.UniformSpace.Defs
import Mathlib.Topology.ContinuousOn
/-!
# Basic resu... |
open Sum
-- Obsolete auxiliary definitions and lemmas
| Mathlib/Topology/UniformSpace/Basic.lean | 851 | 854 |
/-
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.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.Ordering.Lemmas
import Mathlib.Data.PNat.Basic
import Mathlib.SetTheory.Ordinal.Principal
import Ma... |
theorem oadd_add (e n a o) : oadd e n a + o = addAux e n (a + o) :=
rfl
| Mathlib/SetTheory/Ordinal/Notation.lean | 382 | 384 |
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Reduced
import Mathlib.FieldTheory.KummerPolynomial
import Mathlib.FieldTheory.Separable
/-!
# Perfect fie... | /-- If `f` is a polynomial over an integral domain `R` of characteristic `p`, then there is
a map from the set of roots of `Polynomial.expand R p f` to the set of roots of `f`.
It's given by `x ↦ x ^ p`, see `rootsExpandToRoots_apply`. -/
noncomputable def rootsExpandToRoots : (expand R p f).roots.toFinset ↪ f.roots.to... | Mathlib/FieldTheory/Perfect.lean | 331 | 334 |
/-
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.Algebra.Order.Ring.Nat
import Mathlib.Logic.Encodable.Pi
import Mathlib.Logic.Function.Iterate
/-!
# The primitive recursive functions
The primitive ... |
theorem ulower_down : Primrec (ULower.down : α → ULower α) :=
letI : ∀ a, Decidable (a ∈ Set.range (encode : α → ℕ)) := decidableRangeEncode _
subtype_mk .encode
theorem ulower_up : Primrec (ULower.up : ULower α → α) :=
letI : ∀ a, Decidable (a ∈ Set.range (encode : α → ℕ)) := decidableRangeEncode _
option_ge... | Mathlib/Computability/Primrec.lean | 1,166 | 1,174 |
/-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.DeleteEdges
import Mathlib.Data.Fintype.Powerset
/-!
# Subgraphs of a simple graph
A subgraph of ... | gives induced subgraphs. See also `SimpleGraph.induce` for the `SimpleGraph` version, which,
unlike for subgraphs, results in a graph with a different vertex type. -/
/-- The induced subgraph of a subgraph. The expectation is that `s ⊆ G'.verts` for the usual
| Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 1,119 | 1,121 |
/-
Copyright (c) 2015 Nathaniel Thomas. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nathaniel Thomas, Jeremy Avigad, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.GroupWithZero.Action.Defs
import Mathlib.Algebra.Ring.Defs
/-!
# Modules over a ring
In th... | @[simp]
| Mathlib/Algebra/Module/Defs.lean | 104 | 104 |
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.Plus
import Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory
/-!
# Sheafification
We construct the sheafification of a presheaf ov... | val I := t I.fromMiddle I.toMiddle
property := by
intro II
apply inj_of_sep P hsep
rw [← ConcreteCategory.comp_apply, ← ConcreteCategory.comp_apply, (J.toPlus P).naturality,
(J.toPlus P).naturality, ConcreteCategory.comp_apply, ConcreteCategory.comp_apply]
erw [toPlus_apply (T II.fst.fromMiddl... | Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean | 310 | 322 |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 901 | 908 | |
/-
Copyright (c) 2020 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Adam Topaz, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.FreeMonoid.UniqueProds
imp... | theorem hom_ext {f g : FreeAlgebra R X →ₐ[R] A}
(w : (f : FreeAlgebra R X → A) ∘ ι R = (g : FreeAlgebra R X → A) ∘ ι R) : f = g := by
rw [← lift_symm_apply, ← lift_symm_apply] at w
exact (lift R).symm.injective w
| Mathlib/Algebra/FreeAlgebra.lean | 439 | 442 |
/-
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.SetTheory.Cardinal.Finite
import Mathlib.Data.Set.Finite.Powerset
/-!
# Noncomputable Set Cardinality
We define the cardinality of set `s` as a term `Set... | have hdiff := WithTop.le_of_add_le_add_right (ht.subset hst).encard_lt_top.ne hts
rw [nonpos_iff_eq_zero, encard_eq_zero, diff_eq_empty] at hdiff
| Mathlib/Data/Set/Card.lean | 219 | 220 |
/-
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.Algebra.Star.SelfAdjoint
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
/-!
# Quate... | Mathlib/Algebra/Quaternion.lean | 1,556 | 1,557 | |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.GroupWithZero.Action.Units
import Mathlib.Algebra.Module.Basic
import Mathlib.Algebra.Order.AbsoluteValue.Basic
import Mathlib.Algebra.Ring.Int.Units... |
@[simp]
| Mathlib/Data/Int/AbsoluteValue.lean | 33 | 34 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.ZeroCons
/-!
# Basic results on multisets
-/
-- No algebra should be required
assert_not_exists Monoid
universe v
open List S... | Mathlib/Data/Multiset/Basic.lean | 1,403 | 1,410 | |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.ZeroCons
/-!
# Basic results on multisets
-/
-- No algebra should be required
assert_not_exists Monoid
universe v
open List S... | Mathlib/Data/Multiset/Basic.lean | 2,168 | 2,176 | |
/-
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.Independence.Kernel
import Mathlib.Probability.Kernel.Condexp
/-!
# Conditional Independence
We define conditional independence of sets/σ-alg... | iCondIndepSet m' hm' s μ ↔ iCondIndepSets m' hm' (fun i ↦ {s i}) μ :=
Kernel.iIndepSet_iff_iIndepSets_singleton hs
| Mathlib/Probability/Independence/Conditional.lean | 291 | 293 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.ChartedSpace
/-!
# Local properties invariant under a groupoid
We study properties of a triple `(g, s, x)` ... | LiftPropWithinAt P g (s ∩ t) x ↔ LiftPropWithinAt P g s x := by
rw [liftPropWithinAt_iff', liftPropWithinAt_iff', continuousWithinAt_inter' ht, hG.congr_set]
simp_rw [eventuallyEq_set, mem_preimage,
(chartAt _ x).eventually_nhds' (fun x ↦ x ∈ s ∩ t ↔ x ∈ s) (mem_chart_source _ x)]
exact (mem_nhdsWithin_if... | Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | 331 | 338 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.LinearAlgebra.Finsupp.Span
/-!
# Lie submodules of a Lie algebra
In this file we define Lie submodules, we construct ... |
instance subsingleton_of_bot : Subsingleton (LieSubmodule R L (⊥ : LieSubmodule R L M)) := by
apply subsingleton_of_bot_eq_top
| Mathlib/Algebra/Lie/Submodule.lean | 557 | 559 |
/-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Ira Fesefeldt
-/
import Mathlib.Control.Monad.Basic
import Mathlib.Dynamics.FixedPoints.Basic
import Mathlib.Order.CompleteLattice.Basic
import Mathlib.Order.Iterate
import M... | lemma ωScottContinuous.const {x : β} : ωScottContinuous (Function.const α x) := by
simp [ωScottContinuous, ScottContinuousOn, Set.range_nonempty]
end Continuity
| Mathlib/Order/OmegaCompletePartialOrder.lean | 294 | 298 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kexing Ying, Eric Wieser
-/
import Mathlib.Data.Finset.Sym
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.Linea... | end
section AssociatedHom
| Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 865 | 867 |
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä, Moritz Doll
-/
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.Topology.Algebra.Module.LinearMap
import Mathlib.Topology.Algebra.Module.WeakBilin
/-!
# Weak dual t... | Mathlib/Topology/Algebra/Module/WeakDual.lean | 347 | 348 | |
/-
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... | @[simp] lemma uIoc_of_ge (h : b ≤ a) : Ι a b = Ioc b a := by simp [uIoc, h]
lemma uIoc_eq_union : Ι a b = Ioc a b ∪ Ioc b a := by
cases le_total a b <;> simp [uIoc, *]
| Mathlib/Order/Interval/Set/UnorderedInterval.lean | 250 | 253 |
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Sébastien Gouëzel
-/
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
import Mathlib.MeasureTheory... | NullMeasurableSet (r • s) μ := by
obtain rfl | hs' := s.eq_empty_or_nonempty
· simp
obtain rfl | hr := eq_or_ne r 0
· simpa [zero_smul_set hs'] using nullMeasurableSet_singleton _
obtain ⟨t, ht, hst⟩ := hs
refine ⟨_, ht.const_smul_of_ne_zero hr, ?_⟩
rw [← measure_symmDiff_eq_zero_iff] at hst ⊢
rw [←... | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 386 | 399 |
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