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 |
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/-
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.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.Normed.Module.Convex
/-!
# Sides of affine subspaces
This ... | (s.map ↑f).WSameSide (f x) (f y) ↔ s.WSameSide x y :=
(show Function.Injective f.toAffineMap from f.injective).wSameSide_map_iff
@[simp]
| Mathlib/Analysis/Convex/Side.lean | 83 | 86 |
/-
Copyright (c) 2024 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.MvPolynomial.Monad
import Mathlib.LinearAlgebra.Charpoly.ToMatrix
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.Li... |
@[simp]
lemma toMvPolynomial_zero : (0 : M₁ →ₗ[R] M₂).toMvPolynomial b₁ b₂ = 0 := by
| Mathlib/Algebra/Module/LinearMap/Polynomial.lean | 184 | 186 |
/-
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 | 794 | 797 | |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Adam Topaz, Johan Commelin, Jakob von Raumer
-/
import Mathlib.Algebra.Homology.ImageToKernel
import Mathlib.Algebra.Homology.ShortComplex.Exact
import Mathlib.CategoryTh... | rw [ShortComplex.exact_iff_kernel_ι_comp_cokernel_π_zero] at hS ⊢
dsimp at hS
apply F.zero_of_map_zero
obtain ⟨k, hk⟩ :=
kernel.lift' (F.map S.g) (F.map (kernel.ι S.g))
| Mathlib/CategoryTheory/Abelian/Exact.lean | 198 | 202 |
/-
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.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Re... |
theorem inv_mul (t : TransvectionStruct n R) : t.inv.toMatrix * t.toMatrix = 1 := by
rcases t with ⟨_, _, t_hij⟩
simp [toMatrix, transvection_mul_transvection_same, t_hij]
| Mathlib/LinearAlgebra/Matrix/Transvection.lean | 184 | 188 |
/-
Copyright (c) 2024 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Principal
/-!
# Ordinal arithmetic with cardinals
This file co... | theorem principal_opow_ord {c : Cardinal} (hc : ℵ₀ ≤ c) : Principal (· ^ ·) c.ord := by
apply (isInitial_ord c).principal_opow
| Mathlib/SetTheory/Cardinal/Ordinal.lean | 151 | 152 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Jeremy Avigad, Simon Hudon
-/
import Batteries.WF
import Mathlib.Data.Part
import Mathlib.Data.Rel
import Mathlib.Tactic.GeneralizeProofs
/-!
# Partial functions
This... | Mathlib/Data/PFun.lean | 647 | 652 | |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nathaniel Thomas, Jeremy Avigad, Johannes Hölzl, Mario Carneiro, Anne Baanen,
Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Algebra.Modul... |
@[simp, norm_cast]
theorem id'_coe {σ : R →+* R} [RingHomId σ] : ((id' : M →ₛₗ[σ] M) : M → M) = _root_.id :=
rfl
end
| Mathlib/Algebra/Module/LinearMap/Defs.lean | 283 | 288 |
/-
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
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Ta... | rw [Algebra.algebraMap_eq_smul_one, smul_mul_smul_comm, e0_mul_e0, smul_neg]
rw [sub_eq_add_neg, sub_eq_add_neg, h1, h2, h3]
_ = ι (Q' Q) (m₁, r₁) * ι (Q' Q) (m₂, r₂) := by
rw [ι_eq_v_add_smul_e0, ι_eq_v_add_smul_e0, mul_add, add_mul, add_mul, add_assoc]
... | Mathlib/LinearAlgebra/CliffordAlgebra/EvenEquiv.lean | 185 | 215 |
/-
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
-/
import Mathlib.Data.ENNReal.Operations
/-!
# Results about division in extended non-negative reals
This file establishes basic properties related t... |
@[simp]
| Mathlib/Data/ENNReal/Inv.lean | 482 | 483 |
/-
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.Data.ENat.Lattice
import Mathlib.Order.OrderIsoNat
import Mathlib.Tactic.TFAE
/-!
# Maximal length of chains
This file contains lemmas to work with the ma... | singleton_mem_subchain_iff, Set.Nonempty]
@[simp]
theorem chainHeight_eq_zero_iff : s.chainHeight = 0 ↔ s = ∅ := by
rw [← not_iff_not, ← Ne, ← ENat.one_le_iff_ne_zero, one_le_chainHeight_iff,
| Mathlib/Order/Height.lean | 127 | 131 |
/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.ModelTheory.Substructures
/-!
# Elementary Maps Between First-Order Structures
## Main Definitions
- A `FirstOrder... | theorem theory_model_iff (f : M ↪ₑ[L] N) (T : L.Theory) : M ⊨ T ↔ N ⊨ T := by
simp only [Theory.model_iff, f.map_sentence]
| Mathlib/ModelTheory/ElementaryMaps.lean | 98 | 100 |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq
/-!
# Relation between mono/epi and pullback/pushout squares
In this file, monomorphisms and epimorphisms are char... | lemma epi_iff_isPushout : Epi f ↔ IsPushout f f (𝟙 Y) (𝟙 Y) := by
constructor
· intro
exact IsPushout.of_isColimit (PushoutCocone.isColimitMkIdId f)
· intro hf
exact (epi_iff_inl_eq_inr hf.isColimit).2 rfl
| Mathlib/CategoryTheory/Limits/EpiMono.lean | 113 | 118 |
/-
Copyright (c) 2021 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.Convex.Extreme
import Mathlib.Analysis.Convex.Function
import Mathlib.Topology.Algebra.Module.LinearMap
import Mathlib... | Mathlib/Analysis/Convex/Exposed.lean | 243 | 250 | |
/-
Copyright (c) 2020 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Batteries.Tactic.Lint.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Order.ZeroLEOne... | lemma natCast_nonneg [IsOrderedRing α] (n : ℕ) : (0 : α) ≤ n := Nat.cast_nonneg n
-- used alongside `mul_neg` and `mul_nonpos`, so has the same argument pattern for uniformity
@[nolint unusedArguments]
| Mathlib/Tactic/Linarith/Lemmas.lean | 65 | 68 |
/-
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... | Mathlib/Data/Nat/Factorization/Basic.lean | 932 | 937 | |
/-
Copyright (c) 2024 Mitchell Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Lee
-/
import Mathlib.Topology.Algebra.GroupCompletion
import Mathlib.Topology.Algebra.InfiniteSum.Group
/-!
# Infinite sums in the completion of a topological group
-/
open U... | theorem summable_iff_cauchySeq_finset_and_tsum_mem (f : β → α) :
Summable f ↔ CauchySeq (fun s : Finset β ↦ ∑ b ∈ s, f b) ∧
∑' i, toCompl (f i) ∈ Set.range toCompl := by
classical
constructor
· rintro ⟨a, ha⟩
exact ⟨ha.cauchySeq, ((summable_iff_summable_compl_and_tsum_mem f).mp ⟨a, ha⟩).2⟩
· rintr... | Mathlib/Topology/Algebra/InfiniteSum/GroupCompletion.lean | 32 | 45 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.MeasureTheory.Group.MeasurableEquiv
import Mathlib... | theorem _root_.Set.exists_isOpen_lt_add [OuterRegular μ] (A : Set α) (hA : μ A ≠ ∞) {ε : ℝ≥0∞}
(hε : ε ≠ 0) : ∃ U, U ⊇ A ∧ IsOpen U ∧ μ U < μ A + ε :=
A.exists_isOpen_lt_of_lt _ (ENNReal.lt_add_right hA hε)
theorem _root_.Set.exists_isOpen_le_add (A : Set α) (μ : Measure α) [OuterRegular μ] {ε : ℝ≥0∞}
| Mathlib/MeasureTheory/Measure/Regular.lean | 349 | 353 |
/-
Copyright (c) 2024 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.RingTheory.MvPolynomial.Homogeneous
/-!
# The universal c... | (univ R n).map (eval₂Hom f M) = charpoly (Matrix.of M.curry) := by
rw [univ, ← charpoly_map, coe_eval₂Hom, ← mvPolynomialX_map_eval₂ f (Matrix.of M.curry)]
simp only [of_apply, Function.curry_apply, Prod.mk.eta]
lemma univ_map_map :
| Mathlib/LinearAlgebra/Matrix/Charpoly/Univ.lean | 54 | 58 |
/-
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... | open Structure
/-- Used for defining `FirstOrder.Language.Theory.ModelType.instInhabited`. -/
def Inhabited.trivialStructure {α : Type*} [Inhabited α] : L.Structure α :=
⟨default, default⟩
| Mathlib/ModelTheory/Basic.lean | 169 | 173 |
/-
Copyright (c) 2021 Vladimir Goryachev. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Vladimir Goryachev, Kyle Miller, Kim Morrison, Eric Rodriguez
-/
import Mathlib.Data.List.GetD
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.SuccPred
import M... | nth_mem n (lt_card_toFinset_of_nth_ne_zero h)
theorem nth_eq_zero_mono (h₀ : ¬p 0) {a b : ℕ} (hab : a ≤ b) (ha : nth p a = 0) : nth p b = 0 := by
simp only [nth_eq_zero, h₀, false_and, false_or] at ha ⊢
exact ha.imp fun hf hle => hle.trans hab
lemma nth_ne_zero_anti (h₀ : ¬p 0) {a b : ℕ} (hab : a ≤ b) (hb : nth... | Mathlib/Data/Nat/Nth.lean | 247 | 255 |
/-
Copyright (c) 2020 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Logic.IsEmpty
import Mathlib.Order.Basic
import Mathlib.Tactic.MkIffOfInductiveProp
import Batteries.WF
/-!
# Unbundled... |
lemma ssubset_asymm [IsAsymm α (· ⊂ ·)] {a b : α} : a ⊂ b → ¬b ⊂ a := asymm
| Mathlib/Order/RelClasses.lean | 552 | 553 |
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.Algebra.Algebra.ZMod
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
/-!
# Cyclotomic polynomials and `expand`.
We gather results relating cy... | expand R p (cyclotomic n R) = cyclotomic (n * p) R * cyclotomic n R := by
rcases Nat.eq_zero_or_pos n with (rfl | hnpos)
· simp
haveI := NeZero.of_pos hnpos
suffices expand ℤ p (cyclotomic n ℤ) = cyclotomic (n * p) ℤ * cyclotomic n ℤ by
rw [← map_cyclotomic_int, ← map_expand, this, Polynomial.map_mul, m... | Mathlib/RingTheory/Polynomial/Cyclotomic/Expand.lean | 36 | 72 |
/-
Copyright (c) 2024 Theodore Hwa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kim Morrison, Violeta Hernández Palacios, Junyan Xu, Theodore Hwa
-/
import Mathlib.Logic.Hydra
import Mathlib.SetTheory.Surreal.Basic
/-!
### Surreal multiplication
In... | lemma P24_neg_left : P24 x₁ x₂ y ↔ P24 (-x₂) (-x₁) y := by rw [P24, P24, P2_neg_left, P4_neg_left]
| Mathlib/SetTheory/Surreal/Multiplication.lean | 127 | 127 |
/-
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.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.C... | Mathlib/Analysis/BoundedVariation.lean | 700 | 703 | |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Johan Commelin
-/
import Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic
import Mathlib.CategoryTheory.Monoidal.Types.Basic
import Mathlib.LinearAlgebra.DirectSum.Finsupp... | /-- The canonical isomorphism `(free R).obj X ⊗ (free R).obj Y ≅ (free R).obj (X ⊗ Y)`
for two types `X` and `Y`.
(This should not be used directly: it is is part of the implementation of the
monoidal structure on the functor `free R`.) -/
def μIso (X Y : Type u) :
(free R).obj X ⊗ (free R).obj Y ≅ (free R).obj (X ... | Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean | 132 | 149 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Mul
import ... | Mathlib/Analysis/Calculus/MeanValue.lean | 1,060 | 1,080 | |
/-
Copyright (c) 2021 Shing Tak Lam. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shing Tak Lam
-/
import Mathlib.Topology.Order.ProjIcc
import Mathlib.Topology.ContinuousMap.Ordered
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.UnitInterval
/-!
# Homo... | HomotopyRel g₀ g₁ S where
toHomotopy := Homotopy.cast F.toHomotopy h₀ h₁
| Mathlib/Topology/Homotopy/Basic.lean | 600 | 601 |
/-
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.Algebra.Group.Support
import Mathlib.Algebra.Order.Monoid.Unbundled.WithTop
import Mathlib.Order.WellFoundedSet
/-!
# Hahn Series
If `Γ` is ordered an... |
theorem single_ne_zero (h : r ≠ 0) : single a r ≠ 0 := fun con =>
| Mathlib/RingTheory/HahnSeries/Basic.lean | 210 | 211 |
/-
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, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.I... | -- see Note [lower instance priority]
| Mathlib/Algebra/Ring/Defs.lean | 352 | 352 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Aurélien Saue, Anne Baanen
-/
import Mathlib.Tactic.NormNum.Inv
import Mathlib.Tactic.NormNum.Pow
import Mathlib.Util.AtomM
/-!
# `ring` tactic
A tactic for solving e... | (va : ExBase sα a) (vb : ExProd sℕ b) :
ExProd sα q($a ^ $b * (nat_lit 1).rawCast) := .mul va vb (.const 1 none)
| Mathlib/Tactic/Ring/Basic.lean | 280 | 281 |
/-
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... | Subtype.fintype _
instance fintypeEdgeSetBot : Fintype (⊥ : SimpleGraph V).edgeSet := by
| Mathlib/Combinatorics/SimpleGraph/Basic.lean | 532 | 534 |
/-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.IntegralClosure.IntegrallyClosed
import Mathlib.RingTheory.Trace.Basic
import Mathlib.RingTheory.Norm.Basic
/-!
# Discriminant of a famil... | (-1) ^ (n * (n - 1) / 2) *
∏ i : Fin pb.dim, ∏ j ∈ Ioi i, (e j pb.gen - e i pb.gen) * (e i pb.gen - e j pb.gen) := by
rw [discr_powerBasis_eq_prod' _ _ _ e]
simp_rw [fun i j => neg_eq_neg_one_mul ((e j pb.gen - e i pb.gen) * (e i pb.gen - e j pb.gen)),
prod_mul_distrib]
congr
| Mathlib/RingTheory/Discriminant.lean | 171 | 176 |
/-
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.Analytic.Constructions
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Calculus.FDe... |
end DivisionRingInverse
| Mathlib/Analysis/Calculus/FDeriv/Mul.lean | 937 | 939 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Family
/-! # Ordinal exponential
In this file we define the power function and the lo... | (mul_le_mul_right' (opow_le_opow_left c ab.le) a).trans_lt
(mul_lt_mul_of_pos_left ab (opow_pos c ((Ordinal.zero_le a).trans_lt ab)))
theorem opow_add (a b c : Ordinal) : a ^ (b + c) = a ^ b * a ^ c := by
rcases eq_or_ne a 0 with (rfl | a0)
· rcases eq_or_ne c 0 with (rfl | c0)
· simp
have : b + ... | Mathlib/SetTheory/Ordinal/Exponential.lean | 187 | 208 |
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.NoZeroSMulDivisors.Basic
import Mathlib.SetTheory.Cardinal.Basic
/-!
# Cardinality of a module
This file proves that the cardinality of... | theorem mk_le_of_module (R : Type u) (E : Type v)
[AddCommGroup E] [Ring R] [Module R E] [Nontrivial E] [NoZeroSMulDivisors R E] :
Cardinal.lift.{v} (#R) ≤ Cardinal.lift.{u} (#E) := by
obtain ⟨x, hx⟩ : ∃ (x : E), x ≠ 0 := exists_ne 0
have : Injective (fun k ↦ k • x) := smul_left_injective R hx
exact lift_... | Mathlib/Algebra/Module/Card.lean | 24 | 29 |
/-
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... | @[simp]
theorem zero_toOuterMeasure {_m : MeasurableSpace α} : (0 : Measure α).toOuterMeasure = 0 :=
rfl
@[simp, norm_cast]
theorem coe_zero {_m : MeasurableSpace α} : ⇑(0 : Measure α) = 0 :=
rfl
| Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 748 | 755 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 2,292 | 2,294 | |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Simon Hudon
-/
import Mathlib.Data.Fin.Fin2
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Common
/-!
# Tuples of types, and their categorica... | Mathlib/Data/TypeVec.lean | 699 | 702 | |
/-
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.Equiv.PartialEquiv
import Mathlib.Topology.Homeomorph.Lemmas
import Mathlib.Topology.Sets.Opens
/-!
# Partial homeomorphisms
This file de... | (e.open_target.eventually_mem hx).mono e.right_inv'
theorem eventually_right_inverse' {x} (hx : x ∈ e.source) :
∀ᶠ y in 𝓝 (e x), e (e.symm y) = y :=
e.eventually_right_inverse (e.map_source hx)
| Mathlib/Topology/PartialHomeomorph.lean | 234 | 238 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Shing Tak Lam, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Int.ModEq
import Mathlib.Da... | · simp
· rcases b with - | b
· induction' n with n ih
· rfl
· rw [Nat.zero_add] at ih ⊢
simp only [ih, add_comm 1, ofDigits_one_cons, Nat.cast_id, digits_one_succ]
· induction n using Nat.strongRecOn with | ind n h => ?_
cases n
· rw [digits_zero]
rfl
· simp o... | Mathlib/Data/Nat/Digits.lean | 240 | 264 |
/-
Copyright (c) 2023 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Algebra.Order.ToIntervalMod
import Mathli... | lemma eventually_atTop_le {b : ℝ} (hb : b ∈ Set.Ioo 0 1) (hf : GrowsPolynomially f) :
∃ c > 0, ∀ᶠ x in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ≤ c * f x := by
obtain ⟨c₁, _, c₂, hc₂, h⟩ := hf b hb
refine ⟨c₂, hc₂, ?_⟩
filter_upwards [h]
exact fun _ H u hu => (H u hu).2
| Mathlib/Computability/AkraBazzi/GrowsPolynomially.lean | 61 | 66 |
/-
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... | theorem smeval_monomial (n : ℕ) :
(monomial n r).smeval x = r • x ^ n := by
simp only [smeval_eq_sum, smul_pow, zero_smul, sum_monomial_index]
| Mathlib/Algebra/Polynomial/Smeval.lean | 61 | 63 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Nat.Totient
import Mathlib.Data.ZMod.Aut
import Mathlib.Data.ZMod.QuotientGroup
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.Sub... | ∃! n : ZMod (Fintype.card α), x = a ^ n.val := by
obtain ⟨n, rfl⟩ := ha x
refine ⟨n, (?_ : a ^ n = _), fun y (hy : a ^ n = _) ↦ ?_⟩
· rw [← zpow_natCast, zpow_eq_zpow_iff_modEq, orderOf_eq_card_of_forall_mem_zpowers ha,
Int.modEq_comm, Int.modEq_iff_add_fac, Nat.card_eq_fintype_card, ← ZMod.intCast_eq_i... | Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | 396 | 404 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 820 | 820 | |
/-
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.RingTheory.FinitePresentation
import Mathlib.RingTheory.FiniteStability
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.Ideal.Quotient.Ni... |
variable {R : Type u} [CommRing R]
variable {A B : Type u} [CommRing A] [Algebra R A] [CommRing B] [Algebra R B]
| Mathlib/RingTheory/Smooth/Basic.lean | 344 | 347 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Family
/-! # Ordinal exponential
In this file we define the power function and the lo... | theorem opow_le_iff_le_log' {b x c : Ordinal} (hb : 1 < b) (hc : c ≠ 0) :
b ^ c ≤ x ↔ c ≤ log b x := by
obtain rfl | hx := eq_or_ne x 0
· rw [log_zero_right, Ordinal.le_zero, Ordinal.le_zero, opow_eq_zero]
simp [hc, (zero_lt_one.trans hb).ne']
· exact opow_le_iff_le_log hb hx
theorem le_log_of_opow_le {b... | Mathlib/SetTheory/Ordinal/Exponential.lean | 347 | 354 |
/-
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.AffineScheme
import Mathlib.AlgebraicGeometry.Pullbacks
import Mathlib.AlgebraicGeometry.Limits
import Mathlib.CategoryTheory.MorphismPrope... |
lemma of_openCover (H : ∀ i, P (𝒰.pullbackHom f i)) : P f := by
apply of_iSup_eq_top (fun i ↦ (𝒰.map i).opensRange) 𝒰.iSup_opensRange
exact fun i ↦ (P.arrow_mk_iso_iff (morphismRestrictOpensRange f _)).mpr (H i)
theorem iff_of_openCover (𝒰 : Y.OpenCover) :
P f ↔ ∀ i, P (𝒰.pullbackHom f i) :=
⟨fun H _ ↦... | Mathlib/AlgebraicGeometry/Morphisms/Basic.lean | 169 | 206 |
/-
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... | polar (s • f) x y = s • polar f x y := by simp only [polar, Pi.smul_apply, smul_sub]
| Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 111 | 112 |
/-
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.Data.Finset.NAry
import Mathlib.Data.Finset.Slice
import Mathlib.Data.Set.Sups
/-!
# Set family operations
This file defines a few binary operations on `... | variable (s s₁ s₂ t t₁ t₂ u v : Finset α)
| Mathlib/Data/Finset/Sups.lean | 193 | 193 |
/-
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.Algebra.Order.Group.Indicator
import Mathlib.Analysis.Normed.Affine.AddTorsor
import Mathlib.Analysis.NormedSpace.FunctionSeries
import Mathlib.Analy... | exact Real.dist_le_of_mem_Icc_01 (c.lim_mem_Icc _) (c.lim_mem_Icc _)
| succ n ihn =>
by_cases hxl : x ∈ c.left.U
· filter_upwards [IsOpen.mem_nhds c.left.open_U hxl, ihn c.left] with _ hyl hyd
rw [pow_succ', c.lim_eq_midpoint, c.lim_eq_midpoint,
c.right.lim_of_mem_C _ (c.left_U_subset_right_... | Mathlib/Topology/UrysohnsLemma.lean | 278 | 312 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
/-!
# Lemmas about `min` and `max` in an ordered monoid.
-/
op... | obtain h | h := le_total a b <;> simp [h, mul_comm]
@[to_additive]
| Mathlib/Algebra/Order/Monoid/Unbundled/MinMax.lean | 25 | 27 |
/-
Copyright (c) 2022 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Yury Kudryashov, Kevin H. Wilson, Heather Macbeth
-/
import Mathlib.Order.Filter.Tendsto
/-!
# Product and coproduct filters
In this file we define `F... | Tendsto f (x ×ˢ y) z ↔ ∀ W ∈ z, ∃ U ∈ x, ∃ V ∈ y, ∀ x y, x ∈ U → y ∈ V → f (x, y) ∈ W := by
simp only [tendsto_def, mem_prod_iff, prod_sub_preimage_iff, exists_prop]
| Mathlib/Order/Filter/Prod.lean | 416 | 418 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Piecewise
import Mathlib.Algebra.Group.Ext
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Ma... | simp [biproduct.ι_π_ne _ h, biproduct.ι_π_ne _ this]
inv_hom_id := by
classical
cases nonempty_fintype β
ext g g'
| Mathlib/CategoryTheory/Preadditive/Biproducts.lean | 283 | 287 |
/-
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.Limits
import Mathlib.Topology.Category.CompHausLike.EffectiveEpi
/-!
# Effective epimorphisms in `CompHaus`
Thi... | refine ⟨Sigma.ι X t x, ?_⟩
change (Sigma.ι X t ≫ Sigma.desc π) x = _
simpa using h
tfae_have 2 → 3
| e => by
rw [epi_iff_surjective] at e
let i : ∐ X ≅ finiteCoproduct X :=
(colimit.isColimit _).coconePointUniqueUpToIso (finiteCoproduct.isColimit _)
intro b
obtain ⟨t, rfl⟩ := e b
... | Mathlib/Topology/Category/CompHaus/EffectiveEpi.lean | 72 | 85 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Analysis.InnerProductSpace.Continuous
import Mathlib.Analysis.Normed.Module.Dual
import Mathlib.MeasureTheory.Function.AEEqOfLIntegral
import Mathlib.Measu... | Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean | 740 | 744 | |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yuyang Zhao
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Polynomial.AlgebraMap
/-!
# Algebra towers for polynomial
This file proves some basic results about t... |
end Polynomial
| Mathlib/RingTheory/Polynomial/Tower.lean | 68 | 70 |
/-
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.Algebra.Order.UpperLower
import Mathlib.Topology.Algebra.Group.Pointwise
/-!
# Topological facts about upper/lower/order-connected sets
The topological c... | rw [← h.upperClosure_inter_lowerClosure, interior_inter]
exact
(upperClosure s).upper.interior.ordConnected.inter (lowerClosure s).lower.interior.ordConnected
| Mathlib/Topology/Algebra/Order/UpperLower.lean | 100 | 102 |
/-
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 | 1,417 | 1,423 | |
/-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri
-/
import Mathlib.Geometry.Manifold.ContMDiffMap
import Mathlib.Geometry.Manifold.MFDeriv.Basic
/-!
# `C^n` monoid
A `C^n` monoid is a monoid that is also a `C^n` m... | Mathlib/Geometry/Manifold/Algebra/Monoid.lean | 536 | 538 | |
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Heather Macbeth, Johan Commelin
-/
import Mathlib.RingTheory.WittVector.Domain
import Mathlib.RingTheory.WittVector.MulCoeff
import Mathlib.RingTheory.DiscreteValuati... | Mathlib/RingTheory/WittVector/DiscreteValuationRing.lean | 160 | 164 | |
/-
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.Data.Set.Image
import Mathlib.Data.Set.BooleanAlgebra
/-!
# Sets in sigma types
This file defines `Set.sigma`, the indexed sum of sets.
-/
namespace Set... | simp [h.symm]
theorem image_sigmaMk_preimage_sigmaMap_subset {β : ι' → Type*} (f : ι → ι')
(g : ∀ i, α i → β (f i)) (i : ι) (s : Set (β (f i))) :
| Mathlib/Data/Set/Sigma.lean | 31 | 34 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Algebra.Group.Fin.Tuple
import Mathlib.Data.Matrix.RowCol
import Mathlib.Data.Fin.VecNotation
import Mathlib.Tactic.FinCases
import Mathlib.Alge... | ext fun r s => (congr_fun (updateRow_ne (Fin.succAbove_ne i r) : _ = A _) (f s) :)
/-- Updating a column then removing it is the same as removing it. -/
@[simp]
| Mathlib/Data/Matrix/Notation.lean | 396 | 399 |
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.Interval.Set.OrderEmbedding
import Mathlib.Order.SetNotation
/-!
# Properties of unbundled ... | Mathlib/Order/UpperLower/Basic.lean | 1,719 | 1,720 | |
/-
Copyright (c) 2021 Justus Springer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Justus Springer, Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Colimits
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.Algebra.Category.Ring.Limits
import ... | lemma exists_res_eq_zero_of_germ_eq_zero (U : Opens X) (f : X.presheaf.obj (op U)) (x : U)
(h : X.presheaf.germ U x.val x.property f = 0) :
∃ (V : Opens X) (i : V ⟶ U) (_ : x.1 ∈ V), X.presheaf.map i.op f = 0 := by
have h1 : X.presheaf.germ U x.val x.property f = X.presheaf.germ U x.val x.property 0 := by sim... | Mathlib/Geometry/RingedSpace/Basic.lean | 58 | 79 |
/-
Copyright (c) 2022 Niels Voss. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Niels Voss
-/
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.FieldTheory.Finite.Basic
import Mathlib.Order.Filter.Cofinite
/-!
# Fermat Pseudoprimes
In this file we define... | _ = (b ^ p + b) * (b ^ p - b) := by rw [Nat.sq_sub_sq]
_ = (b ^ p - b) * (b ^ p + b) := by rw [mul_comm]
_ = (b ^ (p - 1 + 1) - b) * (b ^ p + b) := by rw [Nat.sub_add_cancel hp]
_ = (b * b ^ (p - 1) - b) * (b ^ p + b) := by rw [pow_succ']
_ = (b * b ^ (p - 1) - b * 1) * (b ^ p + b) := by rw [mul_one... | Mathlib/NumberTheory/FermatPsp.lean | 159 | 173 |
/-
Copyright (c) 2018 Rohan Mitta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rohan Mitta, Kevin Buzzard, Alistair Tucker, Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Order.Interval.Set.ProjIcc
import Mathlib.Topology.Algebra.Order.Field
import Mathlib.Topolo... | rcases cauchySeq_iff_le_tendsto_0.1 hu with ⟨b, b_nonneg, hb, blim⟩
refine cauchySeq_iff_le_tendsto_0.2 ⟨fun n ↦ K * b n, ?_, ?_, ?_⟩
· exact fun n ↦ mul_nonneg (by positivity) (b_nonneg n)
| Mathlib/Topology/MetricSpace/Lipschitz.lean | 170 | 172 |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Lattice.Fold
/-!
# Irreducible and prime elements in an order
This file defines irreducible and prime elements in an order and shows that in ... | · exact ⟨{a}, by simp [ha]⟩
rw [not_supIrred] at ha
obtain ha | ⟨b, c, rfl, hb, hc⟩ := ha
· exact ⟨∅, by simp [ha.eq_bot]⟩
obtain ⟨s, rfl, hs⟩ := ih _ hb
| Mathlib/Order/Irreducible.lean | 119 | 123 |
/-
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, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalcul... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 753 | 755 | |
/-
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... | `MeasureTheory.Measure.ext_of_Ioc` is an extensionality lemma with weaker assumptions on `μ` and
`ν`. -/
theorem ext_of_Ioc_finite {α : Type*} [TopologicalSpace α] {m : MeasurableSpace α}
[SecondCountableTopology α] [LinearOrder α] [OrderTopology α] [BorelSpace α] (μ ν : Measure α)
[IsFiniteMeasure μ] (hμν : μ ... | Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean | 356 | 364 |
/-
Copyright (c) 2022 Antoine Labelle, Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle, Rémi Bottinelli
-/
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Combinatorics.Quiver.Path
/-!
# Rewriting arrows and paths along vertex... |
theorem Hom.cast_heq {u v u' v' : U} (hu : u = u') (hv : v = v') (e : u ⟶ v) :
HEq (e.cast hu hv) e := by
subst_vars
rfl
| Mathlib/Combinatorics/Quiver/Cast.lean | 50 | 54 |
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Kim Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Monoidal.Category
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory... | @[simp]
lemma comp_ε : ε (F ⋙ G) = ε G ≫ G.map (ε F) := rfl
@[simp]
| Mathlib/CategoryTheory/Monoidal/Functor.lean | 231 | 234 |
/-
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... |
/-- The union of an entourage of the diagonal in each set of a disjoint union is again an entourage
of the diagonal. -/
theorem union_mem_uniformity_sum {a : Set (α × α)} (ha : a ∈ 𝓤 α) {b : Set (β × β)} (hb : b ∈ 𝓤 β) :
Prod.map inl inl '' a ∪ Prod.map inr inr '' b ∈ 𝓤 (α ⊕ β) :=
union_mem_sup (image_mem_map... | Mathlib/Topology/UniformSpace/Basic.lean | 873 | 888 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.Shapes.IsTerminal
import Mathlib.CategoryTheory.Limits.HasLimits
/-!
# Initial and terminal objects in a category.
##... | Mathlib/CategoryTheory/Limits/Shapes/Terminal.lean | 397 | 397 | |
/-
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 | 2,116 | 2,117 | |
/-
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.Nat.Choose.Basic
import Mathlib.Data.Nat.Factorial.Cast
/-!
# Cast of binomial coefficients
This file allows calculating the binomial coefficient `a... | theorem cast_choose {a b : ℕ} (h : a ≤ b) : (b.choose a : K) = b ! / (a ! * (b - a)!) := by
have : ∀ {n : ℕ}, (n ! : K) ≠ 0 := Nat.cast_ne_zero.2 (factorial_ne_zero _)
rw [eq_div_iff_mul_eq (mul_ne_zero this this)]
rw_mod_cast [← mul_assoc, choose_mul_factorial_mul_factorial h]
| Mathlib/Data/Nat/Choose/Cast.lean | 25 | 28 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Morenikeji Neri
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.Algebra.EuclideanDomain.Field
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.RingTheor... |
@[deprecated (since := "2025-01-23")] alias dvd_or_coprime := dvd_or_isCoprime
/-- See also `Irreducible.isRelPrime_iff_not_dvd`. -/
theorem Irreducible.coprime_iff_not_dvd {p n : R} (hp : Irreducible p) :
| Mathlib/RingTheory/PrincipalIdealDomain.lean | 399 | 403 |
/-
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.Topology.EMetricSpace.Paracompact
import Mathlib.Topology.Instances.ENNReal.Lemmas
import Mathlib.Analysis.Convex.PartitionOfUnity
/-!
# Lemmas abou... | exists_continuous_forall_mem_convex_of_local_const exists_forall_closedBall_subset_aux₂
(exists_forall_closedBall_subset_aux₁ hK hU hKU hfin)
/-- Let `X` be an extended metric space. Let `K : ι → Set X` be a locally finite family of closed
sets, let `U : ι → Set X` be a family of open sets such that `K i ⊆ U... | Mathlib/Topology/MetricSpace/PartitionOfUnity.lean | 87 | 93 |
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.Data.Fintype.Order
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.Me... | rw [enorm_indicator_eq_indicator_enorm]
exact s.indicator_le_self _ x
lemma eLpNormEssSup_indicator_le (s : Set α) (f : α → ε) :
eLpNormEssSup (s.indicator f) μ ≤ eLpNormEssSup f μ := by
refine essSup_mono_ae (Eventually.of_forall fun x => ?_)
simp_rw [enorm_indicator_eq_indicator_enorm]
exact Set.indica... | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 667 | 674 |
/-
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.Polynomial.Degree.CardPowDegree
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.NumberTheory.ClassNumber.AdmissibleAbsoluteValue
imp... | IsAdmissible (cardPowDegree : AbsoluteValue Fq[X] ℤ) :=
{ @cardPowDegree_isEuclidean Fq _
_ with
card := fun ε => Fintype.card Fq ^ ⌈-log ε / log (Fintype.card Fq)⌉₊
exists_partition' := fun n _ hε _ hb => exists_partition_polynomial n hε hb }
| Mathlib/NumberTheory/ClassNumber/AdmissibleCardPowDegree.lean | 248 | 253 |
/-
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.Finset.Card
import Mathlib.Data.Fintype.Basic
/-!
# Cardinalities of finite types
This file defines the cardinality `Fintype.card α` as the numb... |
alias ⟨_root_.Function.Injective.surjective_of_fintype,
_root_.Function.Surjective.injective_of_fintype⟩ :=
injective_iff_surjective_of_equiv
| Mathlib/Data/Fintype/Card.lean | 334 | 337 |
/-
Copyright (c) 2024 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Lie.Engel
import Mathlib.Algebra.Lie.Normalizer
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.D... | lemma self_mem_engel (x : L) : x ∈ engel R x := by
simp only [mem_engel_iff]
exact ⟨1, by simp⟩
| Mathlib/Algebra/Lie/EngelSubalgebra.lean | 66 | 68 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.IntermediateField.Basic
imp... | /-- If `IsUnit n` in a `CommRing R`, then `X ^ n - u` is separable for any unit `u`. -/
theorem separable_X_pow_sub_C_unit {n : ℕ} (u : Rˣ) (hn : IsUnit (n : R)) :
Separable (X ^ n - C (u : R)) := by
nontriviality R
rcases n.eq_zero_or_pos with (rfl | hpos)
· simp at hn
apply (separable_def' (X ^ n - C (u :... | Mathlib/FieldTheory/Separable.lean | 242 | 249 |
/-
Copyright (c) 2022 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Topology.Algebra.Module.WeakDual
import Mathlib.Algebra.Algebra.Spectrum.Basic
import Mathlib.Topology.ContinuousMap.Algebra
import Mathlib.Data.Set.... | theorem union_zero_isClosed [T2Space 𝕜] [ContinuousMul 𝕜] :
IsClosed (characterSpace 𝕜 A ∪ {0}) := by
simp only [union_zero, Set.setOf_forall]
exact
isClosed_iInter fun x =>
isClosed_iInter fun y =>
isClosed_eq (eval_continuous _) <| (eval_continuous _).mul (eval_continuous _)
| Mathlib/Topology/Algebra/Module/CharacterSpace.lean | 118 | 124 |
/-
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, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.FinMeasAdditive
/-!
# Extension of a linear function from indicators to L1
Give... | · exact fun n => (h _ (self_le_add_left _ _)).2
· filter_upwards [h_lim]
refine fun a h_lin => @Tendsto.comp _ _ _ (fun n => x (n + k)) (fun n => fs n a) _ _ _ h_lin ?_
rwa [tendsto_add_atTop_iff_nat]
variable {X : Type*} [TopologicalSpace X] [FirstCountableTopology X]
theorem continuousWithinAt_setToFun_... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 1,077 | 1,086 |
/-
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.Polynomial.Degree.Domain
import Mathlib.Algebra.Polynomial.Degree.Support
import Mathlib.Algebra.Poly... | Mathlib/Algebra/Polynomial/Derivative.lean | 675 | 677 | |
/-
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 (leftShiftAddEquiv K L n a n' hn).symm (-γ) = _
apply map_neg
@[simp]
lemma shift_neg (a : ℤ) :
| Mathlib/Algebra/Homology/HomotopyCategory/HomComplexShift.lean | 252 | 256 |
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Andrew Yang
-/
import Mathlib.AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf
import Mathlib.AlgebraicGeometry.GammaSpecAdjunction
import Mathlib.RingTheory.GradedAlgeb... | · rintro n ⟨a, ha⟩ i
simp_rw [proj_apply, smul_eq_mul, coe_decompose_mul_of_left_mem 𝒜 i ha]
let product : A⁰_ f :=
(HomogeneousLocalization.mk
⟨_, ⟨a ^ m, pow_mem_graded m ha⟩, ⟨_, ?_⟩, ⟨n, rfl⟩⟩ : A⁰_ f) *
| Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean | 392 | 396 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Devon Tuma
-/
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.RingTheory.Coprime.Basic
import Mathlib.Tactic.... | exact map_ne_zero_of_mem_nonZeroDivisors _ hf hs
| Mathlib/RingTheory/Polynomial/ScaleRoots.lean | 159 | 160 |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Support
/-!
# Permutations from a list
A list `l : List α` ... | obtain ⟨⟨n, hn⟩, hx'⟩ := get_of_mem hx
have hl : (x :: y :: l).length = (x' :: y' :: l').length := by
rw [← dedup_eq_self.mpr hd, ← dedup_eq_self.mpr hd', ← card_toFinset, ← card_toFinset]
refine congr_arg Finset.card ?_
rw [← Finset.coe_inj, ← support_formPerm_of_nodup' _ hd (by simp), ←
| Mathlib/GroupTheory/Perm/List.lean | 275 | 279 |
/-
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... | ext y
rw [mem_closedBall, dist_comm, Real.dist_eq, abs_sub_le_iff, mem_Icc, ← sub_le_iff_le_add',
sub_le_comm]
| Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | 1,067 | 1,069 |
/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Set.Finite.Lemmas
import Mathlib.ModelTheory.Substructures
/-!
# Finitely Generated First-Order Structures
This file defines what it means for a... |
theorem fg_def {N : L.Substructure M} : N.FG ↔ ∃ S : Set M, S.Finite ∧ closure L S = N :=
⟨fun ⟨t, h⟩ => ⟨_, Finset.finite_toSet t, h⟩, by
rintro ⟨t', h, rfl⟩
rcases Finite.exists_finset_coe h with ⟨t, rfl⟩
| Mathlib/ModelTheory/FinitelyGenerated.lean | 45 | 49 |
/-
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.Calculus.MeanValue
/-!
# Extending differentiability to the boundary
We investigate how differentiable functions inside a set extend t... | · exact f_diff y hne
| Mathlib/Analysis/Calculus/FDeriv/Extend.lean | 212 | 217 |
/-
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.Algebra.Group.Embedding
import Mathlib.Order.Interval.Multiset
/-!
# Finite intervals of naturals
This file proves that `ℕ` is a `LocallyFiniteOrder` and... | rw [Ico_succ_right, ← Ico_insert_right h]
| Mathlib/Order/Interval/Finset/Nat.lean | 144 | 144 |
/-
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.... | (indepSet_iff_indepSets_singleton hs_meas ht_meas κ μ).trans indepSets_singleton_iff
theorem IndepSet.measure_inter_eq_mul {_m0 : MeasurableSpace Ω} (κ : Kernel α Ω) (μ : Measure α)
(h : IndepSet s t κ μ) : ∀ᵐ a ∂μ, κ a (s ∩ t) = κ a s * κ a t :=
Indep.indepSets h _ _ (by simp) (by simp)
theorem IndepSets.ind... | Mathlib/Probability/Independence/Kernel.lean | 823 | 829 |
/-
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.FieldTheory.Finite.Polynomial
import Mathlib.NumberTheory.Basic
import Mathlib.RingTheory.WittVector.WittPolynomial
/-!
# Witt struct... | C ((p : ℚ) ^ i) * wittStructureRat p (map (Int.castRingHom ℚ) Φ) i ^ p ^ (n - i) := by
rw [map_sum]
apply Finset.sum_congr rfl
intro i hi
rw [Finset.mem_range] at hi
simp only [IH i hi, RingHom.map_mul, RingHom.map_pow, map_C]
rfl
simp only [← sum_induction_steps, ← map_wittPolynomial ... | Mathlib/RingTheory/WittVector/StructurePolynomial.lean | 275 | 299 |
/-
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.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Covering.Besicovitch
import Mathlib.Tactic.AdaptationNote
import Mathlib.Algeb... | i.e., `s.card ≤ 5^dim`. -/
borelize E
let μ : Measure E := Measure.addHaar
let δ : ℝ := (1 : ℝ) / 2
let ρ : ℝ := (5 : ℝ) / 2
have ρpos : 0 < ρ := by norm_num
set A := ⋃ c ∈ s, ball (c : E) δ with hA
have D : Set.Pairwise (s : Set E) (Disjoint on fun c => ball (c : E) δ) := by
rintro c hc d hd hcd
... | Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean | 110 | 150 |
/-
Copyright (c) 2021 Julian Kuelshammer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Kuelshammer
-/
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.CharP.Lemmas
import Mathlib.Algebra.EuclideanDomain.Field
... | section Dickson
/-!
### A Lambda structure on `ℤ[X]`
Mathlib doesn't currently know what a Lambda ring is.
But once it does, we can endow `ℤ[X]` with a Lambda structure
| Mathlib/RingTheory/Polynomial/Dickson.lean | 105 | 112 |
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Algebra.Order.Pi
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Data.Real.Pointwise
/... | p.closedBall x r = ∅ := by
ext
rw [Seminorm.mem_closedBall, Set.mem_empty_iff_false, iff_false, not_le]
exact hr.trans_le (apply_nonneg _ _)
| Mathlib/Analysis/Seminorm.lean | 820 | 823 |
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Int.DivMod
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.... | Mathlib/Data/Fin/Basic.lean | 1,540 | 1,546 |
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