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) 2019 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Monad.Basic
import Mathlib.Control.Monad.Writer
import Mathlib.Control.Lawful
import Batteries.Tactic.Congr
import Batteries.Lean.Except
import Batte... | instance {ε} [MonadCont m] [LawfulMonadCont m] : LawfulMonadCont (ExceptT ε m) where
callCC_bind_right := by
intros; simp only [callCC, ExceptT.callCC, ExceptT.run_bind, callCC_bind_right]; ext
| Mathlib/Control/Monad/Cont.lean | 128 | 130 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.Defs
impor... |
/-- **Cauchy–Schwarz inequality**. -/
theorem inner_mul_inner_self_le (x y : E) : ‖⟪x, y⟫‖ * ‖⟪y, x⟫‖ ≤ re ⟪x, x⟫ * re ⟪y, y⟫ :=
letI cd : PreInnerProductSpace.Core 𝕜 E := PreInnerProductSpace.toCore
| Mathlib/Analysis/InnerProductSpace/Basic.lean | 254 | 257 |
/-
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... | ⇑(toOrderIso f).symm = (f⁻¹ : CircleDeg1Liftˣ) :=
rfl
| Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 212 | 213 |
/-
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.SetTheory.Cardinal.Finite
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Topology.UniformSpace.Cauchy
import Mathlib.Topology.Algebra... | · exact tprod_eq_one_of_not_multipliable ht ▸ mem_of_mem_nhds ho
· obtain ⟨s, hs⟩ := vanish _ he
exact ⟨s, fun t hts ↦ (t.tprod_subtype f).symm ▸ hs _ hts⟩
variable [CompleteSpace α]
@[to_additive]
theorem multipliable_iff_vanishing :
Multipliable f ↔
∀ e ∈ 𝓝 (1 : α), ∃ s : Finset β, ∀ t, Disjoint ... | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | 262 | 277 |
/-
Copyright (c) 2023 Ziyu Wang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ziyu Wang, Chenyi Li, Sébastien Gouëzel, Penghao Yu, Zhipeng Cao
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.Calc... |
theorem gradient_eq_zero_of_not_differentiableAt (h : ¬DifferentiableAt 𝕜 f x) : ∇ f x = 0 := by
| Mathlib/Analysis/Calculus/Gradient/Basic.lean | 110 | 111 |
/-
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.NAry
import Mathlib.Order.SupClosed
import Mathlib.Order.UpperLower.Closure
/-!
# Set family operations
This file defines a few binary operation... | Mathlib/Data/Set/Sups.lean | 416 | 425 | |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.UniformSpace.CompactConvergence
import Mathlib.Topology.UniformSpace.Equicontinuity
import Mathlib.Topology.UniformSpace.Equiv
/-!
# Asco... | (F_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K) :
(UniformOnFun.uniformSpace X α 𝔖).comap F =
(Pi.uniformSpace _).comap ((⋃₀ 𝔖).restrict ∘ F) := by
-- Recall that the uniform structure on `X →ᵤ[𝔖] α` is the one induced by all the maps
-- `K.restrict : (X →ᵤ[𝔖] α) → (K →ᵤ α)` for `K ∈ 𝔖`. Its pullback a... | Mathlib/Topology/UniformSpace/Ascoli.lean | 212 | 240 |
/-
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.Normed.Group.Uniform
import Mathlib.Analysis.SpecificLimits.Basic
/-!
# Completeness of normed groups
This file includes a completeness c... | /-- In a complete normed additive group, every absolutely convergent series converges in the
space. -/
lemma NormedAddCommGroup.summable_imp_tendsto_of_complete [CompleteSpace E] (u : ℕ → E)
(hu : Summable (‖u ·‖)) : ∃ a, Tendsto (fun n => ∑ i ∈ range n, u i) atTop (𝓝 a) := by
refine cauchySeq_tendsto_of_complet... | Mathlib/Analysis/Normed/Group/Completeness.lean | 75 | 80 |
/-
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... |
theorem intCast_surjective : Function.Surjective ((↑) : ℤ → ZMod n) :=
intCast_rightInverse.surjective
| Mathlib/Data/ZMod/Basic.lean | 209 | 212 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Polynomial.Basi... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 621 | 622 | |
/-
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.GroupTheory.OrderOfElement
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Data.Nat.Factorization.B... | -/
@[to_additive "If there exists an injective, addition-preserving map from `G` to `H`,
then the exponent of `G` divides the exponent of `H`."]
theorem exponent_dvd_of_monoidHom (e : G →* H) (e_inj : Function.Injective e) :
Monoid.exponent G ∣ Monoid.exponent H :=
exponent_dvd_of_forall_pow_eq_one fun g => e_inj... | Mathlib/GroupTheory/Exponent.lean | 342 | 362 |
/-
Copyright (c) 2024 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Group.Action.Pi
import Mathlib.Algebra.Group.End
import Mathlib.Algebra.Module.NatInt
import Mathlib.Algebra.Order.Archimedean.Basic
/-!
# M... | @[scoped simp]
theorem map_add_int' [AddGroupWithOne G] [AddGroup H] [AddConstMapClass F G H 1 b]
(f : F) (x : G) (n : ℤ) : f (x + n) = f x + n • b := by
| Mathlib/Algebra/AddConstMap/Basic.lean | 197 | 199 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Sum.Basic
import Mathlib.Logic.Equiv.Option
import Mathlib.Logic.Equiv.Sum
import Mathlib.Logic.Function.Conjugate
impor... | /-- A function is invariant to a swap if it is equal at both elements -/
theorem apply_swap_eq_self {v : α → β} {i j : α} (hv : v i = v j) (k : α) :
v (swap i j k) = v k := by
by_cases hi : k = i
| Mathlib/Logic/Equiv/Basic.lean | 653 | 656 |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | Mathlib/Data/List/Basic.lean | 2,345 | 2,348 | |
/-
Copyright (c) 2023 Ashvni Narayanan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ashvni Narayanan, Moritz Firsching, Michael Stoll
-/
import Mathlib.Algebra.Group.EvenFunction
import Mathlib.Data.ZMod.Units
import Mathlib.NumberTheory.MulChar.Basic
/-!
# Dirichl... |
lemma even_or_odd [NoZeroDivisors S] : ψ.Even ∨ ψ.Odd := by
suffices ψ (-1) ^ 2 = 1 by convert sq_eq_one_iff.mp this
| Mathlib/NumberTheory/DirichletCharacter/Basic.lean | 298 | 300 |
/-
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.Convex.Between
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathli... | variable [NormedAddCommGroup E] [NormedSpace ℝ E] [MeasurableSpace P]
variable [MetricSpace P] [NormedAddTorsor E P] [BorelSpace P]
/-- Mapping a set of reals along a line segment scales the measure by the length of a segment.
This is an auxiliary result used to prove `hausdorffMeasure_affineSegment`. -/
theorem haus... | Mathlib/MeasureTheory/Measure/Hausdorff.lean | 1,066 | 1,094 |
/-
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... | @[deprecated (since := "2024-12-20")]
alias isAddCyclic_iff_exists_ofOrder_eq_natCard := isAddCyclic_iff_exists_addOrderOf_eq_natCard
@[deprecated (since := "2024-12-20")]
alias IsCyclic.iff_exists_ofOrder_eq_natCard_of_Fintype :=
isCyclic_iff_exists_orderOf_eq_natCard
| Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | 137 | 142 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.Defs
impor... | local notation "⟪" x ", " y "⟫" => @inner 𝕜 _ _ x y
| Mathlib/Analysis/InnerProductSpace/Basic.lean | 275 | 275 |
/-
Copyright (c) 2022 Jake Levinson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jake Levinson
-/
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Finset.Prod
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.UpperLower.Basic
/-!
# Young diagrams
A You... |
This section defines `μ.row` and `μ.rowLen`, with the following API:
1. `(i, j) ∈ μ ↔ j < μ.rowLen i`
2. `μ.row i = {i} ×ˢ (Finset.range (μ.rowLen i))`
3. `μ.rowLen i = (μ.row i).card`
| Mathlib/Combinatorics/Young/YoungDiagram.lean | 237 | 241 |
/-
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.Algebra.Group.Action.Pointwise.Finset
import Mathlib.Algebra.Ring.Nat
/-!
# e-transforms
e-transforms are a family of transformations of pairs of finite ... |
variable {e x}
@[to_additive]
theorem mulDysonETransform.smul_finset_snd_subset_fst :
e • (mulDysonETransform e x).2 ⊆ (mulDysonETransform e x).1 := by
dsimp
| Mathlib/Combinatorics/Additive/ETransform.lean | 75 | 81 |
/-
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.Basic
/-!
# Semiconjugate and commuting maps
We define the following predicates:
* `Function.Semiconj`: `f : α → β` semiconjugates ... | Mathlib/Logic/Function/Conjugate.lean | 204 | 206 | |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Order.Field.Power
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.RingTheory.Polynomial.Bernstein
import Mathlib.Topology.ContinuousMap... | rwa [dist_comm] at m
end bernsteinApproximation
open bernsteinApproximation
| Mathlib/Analysis/SpecialFunctions/Bernstein.lean | 193 | 197 |
/-
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.Fold
import Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
import Mathlib... | | add _ _ hx hy => rw [map_add, hx, hy]
| ι_mul _ _ hx => rw [changeForm_ι_mul, hx, LinearMap.zero_apply, map_zero, LinearMap.zero_apply,
| Mathlib/LinearAlgebra/CliffordAlgebra/Contraction.lean | 299 | 300 |
/-
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.Analysis.Normed.Lp.lpSpace
import Mathlib.Topology.Sets.Compacts
/-!
# The Kuratowski embedding
Any separable metric space can be embedded isom... | have C : dist b (x n) - dist a (x n) = embeddingOfSubset x b n - embeddingOfSubset x a n := by
simp only [embeddingOfSubset_coe, sub_sub_sub_cancel_right]
have :=
calc
dist a b ≤ dist a (x n) + dist (x n) b := dist_triangle _ _ _
_ = 2 * dist a (x n) + (dist b (x n) - dist a (x n)) := by simp [d... | Mathlib/Topology/MetricSpace/Kuratowski.lean | 61 | 87 |
/-
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.Comma.Over.Basic
import Mathlib.CategoryTheory.Discrete.Basic
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryThe... |
/-- The product comparison morphism from `F(A ⨯ -)` to `FA ⨯ F-`, whose components are given by
`prodComparison`.
| Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | 1,093 | 1,095 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | Mathlib/Order/Filter/Basic.lean | 1,496 | 1,497 | |
/-
Copyright (c) 2020 Johan Commelin, Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.MvPolynomial.Variables
/-!
# Monad operations on `MvPolynomial`
... |
theorem aeval_bind₁ [Algebra R S] (f : τ → S) (g : σ → MvPolynomial τ R) (φ : MvPolynomial σ R) :
aeval f (bind₁ g φ) = aeval (fun i => aeval f (g i)) φ :=
eval₂Hom_bind₁ _ _ _ _
| Mathlib/Algebra/MvPolynomial/Monad.lean | 250 | 254 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
/-!
# Degrees of polynomials
This file establ... | (h : f.totalDegree < ∑ i ∈ d.support, d i) : coeff d f = 0 := by
classical
rw [totalDegree, Finset.sup_lt_iff] at h
· specialize h d
rw [mem_support_iff] at h
refine not_not.mp (mt h ?_)
| Mathlib/Algebra/MvPolynomial/Degrees.lean | 496 | 501 |
/-
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... | ωSup (iterateChain f x h) ≤ a := by
apply ωSup_le
intro n
| Mathlib/Order/OmegaCompletePartialOrder.lean | 816 | 818 |
/-
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... | rwa [Fin.ext_iff, Fin.val_zero, ← Ne, ← Nat.one_le_iff_ne_zero] at hk
· exact hk
theorem natCast_eq_iff (p : ℕ) (n : ℕ) (z : ZMod p) [NeZero p] :
| Mathlib/Data/ZMod/Basic.lean | 546 | 549 |
/-
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
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Order.Lattice
/-!
# Ordered Subtraction
This file proves l... | protected theorem le_tsub_of_add_le_left (ha : AddLECancellable a) (h : a + b ≤ c) : b ≤ c - a :=
ha <| h.trans le_add_tsub
| Mathlib/Algebra/Order/Sub/Defs.lean | 182 | 184 |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.EffectiveEpi.Preserves
import Mathlib.CategoryTheory.Limits.Final.ParallelPair
import Mathlib... | ∀ (X B : C) (π : X ⟶ B) [EffectiveEpi π] [HasPullback π π],
IsIso (equalizer.lift (P.map π.op) (equalizerCondition_w' P π)) := by
constructor
· intro hP X B π _ _
have h := hP.bijective_mapToEqualizer_pullback _ X B π
rw [← isIso_iff_bijective, mapToEqualizer_eq_comp] at h
exact IsIso.of_isIso... | Mathlib/CategoryTheory/Sites/Coherent/RegularSheaves.lean | 136 | 150 |
/-
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.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.C... | Mathlib/FieldTheory/RatFunc/Basic.lean | 1,101 | 1,107 | |
/-
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... | refine (μ.finiteSpanningSetsInOpen'.mono' fun U hU => ?_).outerRegular
have : Fact (μ U < ∞) := ⟨hU.2⟩
exact ⟨hU.1, inferInstance⟩
| Mathlib/MeasureTheory/Measure/Regular.lean | 961 | 963 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.Order.IsLUB
/-!
# Order topology on a densely ordered set
-/
open Set Filter TopologicalSpace Topology Func... | refine ⟨fun h => le_of_not_lt fun hab => ?_, by simp_all⟩
have := h.closure_eq
rw [closure_Ico hab.ne, Icc_eq_Ico_same_iff] at this
| Mathlib/Topology/Order/DenselyOrdered.lean | 390 | 392 |
/-
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 Mathlib.Algebra.Notation.Defs
import Mathlib.Data.Int.Notation
import Mathlib.Data.Nat.BinaryRec
import Mathlib.L... | /-- An additive monoid in which addition is right-cancellative.
Main examples are `ℕ` and groups. This is the right typeclass for many sum lemmas, as having a zero
| Mathlib/Algebra/Group/Defs.lean | 678 | 679 |
/-
Copyright (c) 2017 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Mario Carneiro
-/
import Mathlib.Algebra.Ring.CharZero
import Mathlib.Algebra.Star.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Order.Interval.Set.UnorderedInterva... | · intro hx
obtain ⟨x₁, hx₁, hx₁'⟩ := hx
| Mathlib/Data/Complex/Basic.lean | 831 | 832 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.FractionalIdeal.Basic
import Mathlib.RingTheory.IntegralClosure.IsIntegral.Basi... | @[simp]
theorem map_symm_map (I : FractionalIdeal S P') (g : P ≃ₐ[R] P') :
(I.map (g.symm : P' →ₐ[R] P)).map (g : P →ₐ[R] P') = I := by
| Mathlib/RingTheory/FractionalIdeal/Operations.lean | 117 | 119 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Sigma.Basic
import Mathlib.Algebra.Order.Ring.Nat
/-!
# A computable model of ZFA without infinity
In this file we define finite hereditary list... | | ⟨_, l⟩ => a ∈ l
| Mathlib/SetTheory/Lists.lean | 249 | 250 |
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Joachim Breitner
-/
import Mathlib.Algebra.Group.Action.End
import Mathlib.Algebra.Group.Action.Pointwise.Set.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mat... |
/-- One can replace the first letter in a non-empty reduced word by an element of the same
group -/
def replaceHead : ∀ {i j : ι} (x : M i) (_hnotone : x ≠ 1) (_w : NeWord M i j), NeWord M i j
| _, _, x, h, singleton _ _ => singleton x h
| _, _, x, h, append w₁ hne w₂ => append (replaceHead x h w₁) hne w₂
@[simp]... | Mathlib/GroupTheory/CoprodI.lean | 740 | 758 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Grade
import Mathlib.Data.Finset.Powerset
import Mathlib.Order.Interval.Finset.Basic
/-!
# Intervals of finsets as finsets
This file provides... | exact ⟨le_sup_left, union_subset h <| hv.trans sdiff_subset⟩
theorem Ico_eq_image_ssubsets (h : s ⊆ t) : Ico s t = (t \ s).ssubsets.image (s ∪ ·) := by
ext u
simp_rw [mem_Ico, mem_image, mem_ssubsets]
constructor
· rintro ⟨hs, ht⟩
exact ⟨u \ s, sdiff_lt_sdiff_right ht hs, sup_sdiff_cancel_right hs⟩
| Mathlib/Data/Finset/Interval.lean | 80 | 87 |
/-
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... | le_antisymm (hn _ (orderOf_pos a))
(calc
orderOf a = @Fintype.card (zpowers a) (id _) := Fintype.card_zpowers.symm
_ ≤
@Fintype.card (({b : α | b ^ orderOf a = 1} : Finset _) : Set α)
(Fintype.ofFinset _ fun _ => Iff.rfl) :=
(@Fintype.card_le_of_injective (zpowers a)
... | Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | 438 | 462 |
/-
Copyright (c) 2023 Matthew Robert Ballard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Matthew Robert Ballard
-/
import Mathlib.Algebra.Divisibility.Units
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.Common
/-!
# The maximal power of one natural n... | have : 0 < p := lt_trans (b := 1) (by simp) hp
dsimp [maxPowDiv]
rw [maxPowDiv.go, if_pos, mul_div_right _ this]
· apply go_succ
· refine ⟨hp, ?_, by simp⟩
apply Nat.mul_pos this hn
theorem base_pow_mul {p n exp : ℕ} (hp : 1 < p) (hn : 0 < n) :
| Mathlib/Data/Nat/MaxPowDiv.lean | 68 | 75 |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Analysis.InnerProductSpace.Convex
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Combinatorics.Additive.AP.Three... | map_add' a b := by simp_rw [Pi.add_apply, add_mul, sum_add_distrib]
theorem map_zero (d : ℕ) (a : Fin 0 → ℕ) : map d a = 0 := by simp [map]
| Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | 132 | 135 |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.RingTheory.Localizati... | | succ n hn => convert QT _ _; simpa using hn
section Support
theorem support_C_mul_T (a : R) (n : ℤ) : Finsupp.support (C a * T n) ⊆ {n} := by
rw [← single_eq_C_mul_T]
exact support_single_subset
| Mathlib/Algebra/Polynomial/Laurent.lean | 373 | 380 |
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
import Mathlib.Comp... | Mathlib/Computability/TMToPartrec.lean | 1,912 | 1,913 | |
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Antoine Labelle, Rémi Bottinelli
-/
import Mathlib.Combinatorics.Quiver.Path
import Mathlib.Combinatorics.Quiver.Push
/-!
## Symmetric quivers and arrow reversal
This file co... | · rintro h
simpa using congr_arg Quiver.reverse h
| Mathlib/Combinatorics/Quiver/Symmetric.lean | 61 | 62 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Sites.Sieves
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono
/-!
# The sheaf condition for a presieve
We define what it means fo... | Mathlib/CategoryTheory/Sites/IsSheafFor.lean | 792 | 795 | |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Instances
import Mathlib.Algebra.Category.Ring.Limits
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq
import Mathlib.Catego... | commutes' := DFunLike.congr_fun <| congrArg Hom.hom
((s.ι.naturality Limits.WalkingSpan.Hom.snd).trans
(s.ι.naturality Limits.WalkingSpan.Hom.fst).symm) }
letI : Algebra R (pushoutCocone R A B).pt := show Algebra R (A ⊗[R] B) by infer_instance
-- The factor map is a ⊗ b ↦ f(a) * g(... | Mathlib/Algebra/Category/Ring/Constructions.lean | 76 | 81 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yury Kudryashov
-/
import Mathlib.MeasureTheory.OuterMeasure.Basic
/-!
# The “almost everywhere” filter of co-null sets.
If `μ` is an outer measure or a measure on `α... | theorem ae_eq_set_symmDiff {s' t' : Set α} (h : s =ᵐ[μ] t) (h' : s' =ᵐ[μ] t') :
s ∆ s' =ᵐ[μ] t ∆ t' :=
| Mathlib/MeasureTheory/OuterMeasure/AE.lean | 193 | 194 |
/-
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... | funext fun i ↦ hf i <| congrFun h _
| Mathlib/Logic/Function/Basic.lean | 109 | 110 |
/-
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.Primrec
import Mathlib.Data.Nat.PSub
import Mathlib.Data.PFun
/-!
# The partial recursive functions
The partial recursive functions are... | simp only [decode_prod_val, decode_nat, Option.map_some', PFun.coe_val, bind_eq_bind,
bind_some, Part.map_bind, map_some]
cases decode (α := α) n.unpair.1 <;> simp
cases decode (α := β) n.unpair.2 <;> simp,
fun hf => by
have :
Partrec fun a : α × ℕ =>
(encode (dec... | Mathlib/Computability/Partrec.lean | 565 | 573 |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.Algebra.Module.ZLattice.Basic
import Mathlib.Analysis.InnerProductSpace.ProdL2
import Mathlib.MeasureTheory.Measure.Haar.Unique
import Mathlib.NumberTheo... | simp_rw [Pi.nnnorm_def, apply_at]
theorem norm_le_iff [NumberField K] (x : K) (r : ℝ) :
| Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 72 | 74 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.CharP.Reduced
import Mathlib.RingTheory.IntegralDomain
-- TODO: remove Mathlib.Algebra.CharP.Reduced and move the last two lemmas to Lemmas
/-... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 416 | 429 | |
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Analysis.InnerProductSpace.Convex
import Mathlib.Algebra.Module.LinearMap.Rat
import Mathlib.Tactic.Module
/... | Mathlib/Analysis/InnerProductSpace/OfNorm.lean | 278 | 285 | |
/-
Copyright (c) 2020 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Algebra.Algebra.Spectrum.Basic
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathl... | lemma HasUnifEigenvalue.mem_spectrum {f : End R M} {μ : R} (hμ : HasUnifEigenvalue f μ 1) :
μ ∈ spectrum R f := by
refine spectrum.mem_iff.mpr fun h_unit ↦ ?_
set f' := LinearMap.GeneralLinearGroup.toLinearEquiv h_unit.unit
| Mathlib/LinearAlgebra/Eigenspace/Basic.lean | 209 | 212 |
/-
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.Algebra.Field.NegOnePow
import Mathlib.Algebra.Field.Periodic
import Mathlib.Algebra.Qua... | rw [← mul_self_eq_one_iff, ← sin_sq_add_cos_sq, sq, sq, ← sub_eq_iff_eq_add, sub_self]
exact ⟨fun h => by rw [h, mul_zero], eq_zero_of_mul_self_eq_zero ∘ Eq.symm⟩
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 988 | 990 |
/-
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.Probability.Independence.Basic
import Mathlib.Probability.Independence.Conditional
/-!
# Kolmogorov's 0-1 law
Let `s : ι → MeasurableSpace Ω` be an indep... |
/-- **Kolmogorov's 0-1 law**, kernel version: any event in the tail σ-algebra of an independent
sequence of sub-σ-algebras has probability 0 or 1 almost surely. -/
theorem Kernel.measure_zero_or_one_of_measurableSet_limsup_atBot (h_le : ∀ n, s n ≤ m0)
(h_indep : iIndep s κ μα) {t : Set Ω} (ht_tail : MeasurableSet[... | Mathlib/Probability/Independence/ZeroOne.lean | 329 | 333 |
/-
Copyright (c) 2024 Raghuram Sundararajan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Raghuram Sundararajan
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
/-!
# Extensionality lemmas for rings and similar structures
In this file we prove e... | Mathlib/Algebra/Ring/Ext.lean | 508 | 512 | |
/-
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... |
/-- This lemma works in the case in which `a ≠ 0`. The version where
`ZMod n` is not infinite, i.e. `n ≠ 0`, is `addOrderOf_coe`. -/
@[simp]
theorem addOrderOf_coe' {a : ℕ} (n : ℕ) (a0 : a ≠ 0) : addOrderOf (a : ZMod n) = n / n.gcd a := by
| Mathlib/Data/ZMod/Basic.lean | 122 | 126 |
/-
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.Algebra.Homology.ShortComplex.Exact
import Mathlib.CategoryTheory.Shift.ShiftSequence
import Mathlib.CategoryTheory.Triangulated.Functor
import Mathlib.CategoryT... | F.additive_of_preserves_binary_products
lemma isHomological_of_localization (L : C ⥤ D)
[L.CommShift ℤ] [L.IsTriangulated] [L.mapArrow.EssSurj] (F : D ⥤ A)
(G : C ⥤ A) (e : L ⋙ F ≅ G) [G.IsHomological] :
F.IsHomological := by
have : F.PreservesZeroMorphisms := preservesZeroMorphisms_of_map_zero_object
... | Mathlib/CategoryTheory/Triangulated/HomologicalFunctor.lean | 144 | 154 |
/-
Copyright (c) 2024 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Robin Carlier
-/
import Mathlib.CategoryTheory.Limits.FullSubcategory
import Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts
import Mathlib.CategoryT... | Construct a morphism to the product given its two components.
-/
def lift {T X Y : C} (f : T ⟶ X) (g : T ⟶ Y) : T ⟶ X ⊗ Y :=
| Mathlib/CategoryTheory/ChosenFiniteProducts.lean | 102 | 104 |
/-
Copyright (c) 2024 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.NumberTheory.DirichletCharacter.Bounds
import Mathlib.NumberTheory.LSeries.Convolution
import Mathlib.NumberTheory.LSeries.Deriv
import Mathlib.NumberThe... | lemma LSeriesHasSum_zeta {s : ℂ} (hs : 1 < s.re) : LSeriesHasSum ↗ζ s (riemannZeta s) :=
LSeries_zeta_eq_riemannZeta hs ▸ (LSeriesSummable_zeta_iff.mpr hs).LSeriesHasSum
/-- The L-series of the arithmetic function `ζ` and of the Möbius function are inverses. -/
lemma LSeries_zeta_mul_Lseries_moebius {s : ℂ} (hs : 1 ... | Mathlib/NumberTheory/LSeries/Dirichlet.lean | 275 | 280 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.Option
import Mathlib.Logic.Equiv.Fin.Rotate
import Mathlib.Logic.Equiv.Fintype
/-!
# Permutatio... | (Finset.univ : Finset <| Fin n.succ × Perm (Fin n)).map
Equiv.Perm.decomposeFin.symm.toEmbedding :=
(Finset.univ_map_equiv_to_embedding _).symm
| Mathlib/GroupTheory/Perm/Fin.lean | 64 | 66 |
/-
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_opow_left {a b : Ordinal} (c : Ordinal) (ab : a ≤ b) : a ^ c ≤ b ^ c := by
by_cases a0 : a = 0
-- Porting note: `le_refl` is required.
· subst a
by_cases c0 : c = 0
· subst c
simp only [opow_zero, le_refl]
· simp only [zero_opow c0, Ordinal.zero_le]
· induction c using limitRec... | Mathlib/SetTheory/Ordinal/Exponential.lean | 147 | 162 |
/-
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.MeasureTheory.Measure.GiryMonad
import Mathlib.MeasureTheory.Measure.Stieltjes
import Mathlib.Analysis.Normed.Order.Lattice
import Mathlib.MeasureTheory.Fu... | lemma monotone_defaultRatCDF : Monotone defaultRatCDF := by
unfold defaultRatCDF
intro x y hxy
| Mathlib/Probability/Kernel/Disintegration/MeasurableStieltjes.lean | 163 | 165 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Finset.Prod
import Mathlib.Order.Hom.WithTopBot
import Mathlib.Order.Interval.Set.UnorderedInterval
/-!
# Locally... | Mathlib/Order/Interval/Finset/Defs.lean | 1,353 | 1,354 | |
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
import Mathlib.Comp... | Mathlib/Computability/TMToPartrec.lean | 1,850 | 1,853 | |
/-
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.Field.Defs
import Mathlib.Algebra.Group.Action.Pi
import Mathlib.Algebra.Group.Indicator
imp... | Mathlib/Algebra/Module/Basic.lean | 192 | 194 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | Mathlib/Order/Filter/Basic.lean | 2,420 | 2,421 | |
/-
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.Properties
/-!
# Function field of integral schemes
We define the function field of an irreducible scheme as the stalk of the generic poi... | -- Porting note: this was `ext1`
apply Subtype.ext
exact (genericPoint_eq_of_isOpenImmersion U.ι).symm
theorem functionField_isFractionRing_of_isAffineOpen [IsIntegral X] (U : X.Opens)
(hU : IsAffineOpen U) [Nonempty U] :
IsFractionRing Γ(X, U) X.functionField := by
haveI : IsAffine _ := hU
haveI : I... | Mathlib/AlgebraicGeometry/FunctionField.lean | 141 | 154 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stephen Morgan, Kim Morrison, Johannes Hölzl, Reid Barton
-/
import Mathlib.CategoryTheory.Category.Init
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Tactic.PPWithUniv
import... | Hom X Y := X.down ⟶ Y.down
id X := 𝟙 X.down
comp f g := f ≫ g
-- We verify that this previous instance can lift small categories to large categories.
example (D : Type u) [SmallCategory D] : LargeCategory (ULift.{u + 1} D) := by infer_instance
| Mathlib/CategoryTheory/Category/Basic.lean | 356 | 362 |
/-
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_iff_isIso_leftHomologyMap' (φ : S₁ ⟶ S₂)
(h₁ : S₁.LeftHomologyData) (h₂ : S₂.LeftHomologyData) :
QuasiIso φ ↔ IsIso (leftHomologyMap' φ h₁ h₂) := by
have γ : LeftHomologyMapData φ h₁ h₂ := default
rw [γ.quasiIso_iff, γ.leftHomologyMap'_eq]
| Mathlib/Algebra/Homology/ShortComplex/QuasiIso.lean | 121 | 125 |
/-
Copyright (c) 2018 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.BigOperators.Expect
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Canonical
import Mathlib.Algebra.O... | Mathlib/Data/NNReal/Basic.lean | 1,133 | 1,135 | |
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Algebra.BigOperators.Expect
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Convex.Jensen
import M... | simpa [← rpow_mul, ha, hb, hpq.ne_zero, hpq.symm.ne_zero, _root_.div_eq_inv_mul] using
geom_mean_le_arith_mean2_weighted hpq.inv_nonneg hpq.symm.inv_nonneg
(rpow_nonneg ha p) (rpow_nonneg hb q) hpq.inv_add_inv_eq_one
/-- **Young's inequality**, a version for arbitrary real numbers. -/
theorem young_inequal... | Mathlib/Analysis/MeanInequalities.lean | 410 | 418 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Thomas Read, Andrew Yang, Dagur Asgeirsson, Joël Riou
-/
import Mathlib.CategoryTheory.Adjunction.Mates
/-!
# Uniqueness of adjoints
This file shows that adjoints are uni... | Mathlib/CategoryTheory/Adjunction/Unique.lean | 251 | 254 | |
/-
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 - 1) / (b - 1) * ((b ^ p + 1) / (b + 1)) = (b ^ (2 * p) - 1) / (b ^ 2 - 1) := by
have q₁ : b - 1 ∣ b ^ p - 1 := by simpa only [one_pow] using nat_sub_dvd_pow_sub_pow b 1 p
have q₂ : b + 1 ∣ b ^ p + 1 := by simpa only [one_pow] using hp.nat_add_dvd_pow_add_pow b 1
convert Nat.div_mul_div_comm q₁ q₂ usin... | Mathlib/NumberTheory/FermatPsp.lean | 142 | 148 |
/-
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.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.MeasureTheory.Measure.Count
import Mathlib.Order.Filter.ENNReal
import Mathlib.Probability.Unif... | rw [eventually_map] at h_le ⊢
rw [ae_map_iff hf (measurableSet_le hg measurable_const)]
exact h_le
theorem essSup_map_measure (hg : AEMeasurable g (Measure.map f μ)) (hf : AEMeasurable f μ) :
essSup g (Measure.map f μ) = essSup (g ∘ f) μ := by
rw [essSup_congr_ae hg.ae_eq_mk, essSup_map_measure_of_measurab... | Mathlib/MeasureTheory/Function/EssSup.lean | 245 | 251 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.WSeq.Basic
import Mathlib.Data.WSeq.Defs
import Mathlib.Data.WSeq.Productive
import Mathlib.Data.WSeq.Relation
deprecated_module (since :=... | Mathlib/Data/Seq/WSeq.lean | 909 | 920 | |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... |
/-! ### reverse -/
| Mathlib/Data/List/Basic.lean | 251 | 253 |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll, Thomas Zhu, Mario Carneiro
-/
import Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity
/-!
# The Jacobi Symbol
We define the Jacobi symbol and prove its main pro... | simp_rw [qrSign.mul_left x y a, Nat.cast_mul, mul_left, mul_mul_mul_comm] }
have rhs_apply : ∀ a b : ℕ, rhs a b = qrSign b a * J(b | a) := fun a b => rfl
| Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean | 406 | 407 |
/-
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 | 965 | 966 | |
/-
Copyright (c) 2019 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.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno
import Mathlib.Analysis.Calculus.FDeriv.Ad... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 2,011 | 2,029 | |
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll, Ralf Stephan
-/
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Data.Nat.Squarefree
/-!
# Smooth numbers
For `s : Finset ℕ` we define the set `Nat.factoredNum... |
lemma mem_primesBelow {k n : ℕ} :
| Mathlib/NumberTheory/SmoothNumbers.lean | 42 | 43 |
/-
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... | (∀ p : ℕ, ordCompl[p] a ∣ ordCompl[p] b) ↔ a ∣ b := by
refine ⟨fun h => ?_, fun hab p => ordCompl_dvd_ordCompl_of_dvd hab p⟩
| Mathlib/Data/Nat/Factorization/Basic.lean | 336 | 337 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Pi
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Simple functions
A function `f` from a measurable ... | protected theorem measurable [MeasurableSpace β] (f : α →ₛ β) : Measurable f := fun s _ =>
measurableSet_preimage f s
| Mathlib/MeasureTheory/Function/SimpleFunc.lean | 165 | 166 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Set.Function
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Says
/-!
# Equivalences and sets
In this file we pr... | s ⊆ e.symm '' t ↔ e '' s ⊆ t :=
calc
s ⊆ e.symm '' t ↔ e.symm.symm '' s ⊆ t := by rw [e.symm.symm_image_subset]
_ ↔ e '' s ⊆ t := by rw [e.symm_symm]
| Mathlib/Logic/Equiv/Set.lean | 73 | 77 |
/-
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.Data.Matrix.Notation
import Mathlib.Data.Fin.Tuple.Reflection
/-!
# Lemmas for concrete matrices `Matrix (Fin m) (Fin n) α`
This file contains alternative ... | ```lean
example [NonUnitalNonAssocSemiring α] (a₁₁ a₁₂ a₂₁ a₂₂ b₁ b₂ : α) :
![b₁, b₂] ᵥ* !![a₁₁, a₁₂;
a₂₁, a₂₂] = ![b₁*a₁₁ + b₂*a₂₁, b₁*a₁₂ + b₂*a₂₂] :=
| Mathlib/Data/Matrix/Reflection.lean | 185 | 188 |
/-
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.Solvable
import Mathlib.Algebra.Lie.Quotient
import Mathlib.Algebra.Lie.Normalizer
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.... | rw [derivedSeries_def, derivedSeriesOfIdeal_succ, lowerCentralSeries_succ]
exact LieSubmodule.mono_lie h' h
/-- A Lie module is nilpotent if its lower central series reaches 0 (in a finite number of
steps). -/
@[mk_iff isNilpotent_iff_int]
class IsNilpotent : Prop where
| Mathlib/Algebra/Lie/Nilpotent.lean | 254 | 260 |
/-
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... | calc
0 = (0 : ZMod n).val := by rw [val_zero]
_ = (1 : ZMod n).val := congr_arg ZMod.val h
_ = 1 := val_one n
⟩⟩
instance nontrivial' : Nontrivial (ZMod 0) := by
delta ZMod; infer_instance
| Mathlib/Data/ZMod/Basic.lean | 676 | 684 |
/-
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... | x ∈ I.killingCompl ↔ ∀ y ∈ I, killingForm R L y x = 0 := by
rfl
| Mathlib/Algebra/Lie/TraceForm.lean | 368 | 370 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Order.Filter.Bases.Finit... | Mathlib/Topology/Algebra/Group/Basic.lean | 1,494 | 1,496 | |
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.DualNumber
import Mathlib.Analysis.Normed.Algebra.TrivSqZeroExt
/-!
# Results on `DualNumber R` related to the norm
These are just restatements of ... | end DualNumber
| Mathlib/Analysis/NormedSpace/DualNumber.lean | 38 | 39 |
/-
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.Sheaf
/-!
# The plus construction for presheaves.
This file contains the construction of `P⁺`, for a presheaf `P : Cᵒᵖ ⥤ D`
where `C` i... | J.diagramNatTrans (η ≫ γ) X = J.diagramNatTrans η X ≫ J.diagramNatTrans γ X := by
ext : 2
refine Multiequalizer.hom_ext _ _ _ (fun i => ?_)
simp
variable (D) in
| Mathlib/CategoryTheory/Sites/Plus.lean | 81 | 86 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Michael Stoll
-/
import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic
/-!
# Legendre symbol
This file contains results about Legendre symbols.
We define the Le... |
end Values
| Mathlib/NumberTheory/LegendreSymbol/Basic.lean | 294 | 296 |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | Mathlib/Data/List/Basic.lean | 1,882 | 1,885 | |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.ULift
import Mathlib.Data.ZMod.Defs
import Mathlib.SetTheory.Cardinal.ToNat
import Mathlib.SetTheory.Cardinal.ENat
/-!
# Finite Cardinality Funct... | card (α ⊕ β) = card α + card β := by
simp only [card, mk_sum, map_add, toENat_lift]
| Mathlib/SetTheory/Cardinal/Finite.lean | 285 | 287 |
/-
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
-/
import Batteries.Tactic.Congr
import Mathlib.Data.Option.Basic
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Set.Subsingleton
import Mathlib.Dat... | Mathlib/Data/Set/Image.lean | 1,394 | 1,395 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
/-!
# Limits and a... |
@[deprecated (since := "2025-03-02")]
alias tendsto_log_div_rpow_nhds_zero := tendsto_log_div_rpow_nhdsGT_zero
theorem tendsto_log_mul_rpow_nhdsGT_zero {r : ℝ} (hr : 0 < r) :
| Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | 375 | 379 |
/-
Copyright (c) 2019 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Action.Defs
import Mathlib.Algebra.Group.Units.Defs
import Mathlib.Algebra.BigOperators.Group.List.Basic
import Mathlib.Algebr... | -- This should be removed from the simp set and deprecated once those defeq abuses are cleaned up.
@[to_additive (attr := simp)]
| Mathlib/Algebra/FreeMonoid/Basic.lean | 117 | 118 |
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