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 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... | Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | 1,307 | 1,311 | |
/-
Copyright (c) 2021 Ashvni Narayanan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ashvni Narayanan, David Loeffler
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Numbe... | apply sum_congr rfl
rintro x hx
rw [mem_range_succ_iff] at hx
rw [choose_symm hx, tsub_tsub_cancel_of_le hx]
/-
### examples
| Mathlib/NumberTheory/BernoulliPolynomials.lean | 57 | 63 |
/-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.DeleteEdges
import Mathlib.Data.Fintype.Powerset
/-!
# Subgraphs of a simple graph
A subgraph of ... | simp
· use w
simp
| Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 909 | 911 |
/-
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.Calculus.TangentCone
import Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics
import Mathlib.Analysis.As... |
@[simp, fun_prop]
theorem differentiable_id : Differentiable 𝕜 (id : E → E) := fun _ => differentiableAt_id
| Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 1,029 | 1,031 |
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Order.Archimedean.IndicatorCard
import Mathlib.Probability.Martingale.Centering
import Mathlib.Probability.Martingale.Convergence
import Mathlib.Prob... | rw [Pi.one_apply, norm_one]
theorem integrable_process (μ : Measure Ω) [IsFiniteMeasure μ] (hs : ∀ n, MeasurableSet[ℱ n] (s n))
(n : ℕ) : Integrable (process s n) μ :=
integrable_finset_sum' _ fun _ _ =>
IntegrableOn.integrable_indicator (integrable_const 1) <| ℱ.le _ _ <| hs _
| Mathlib/Probability/Martingale/BorelCantelli.lean | 295 | 300 |
/-
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 Mathlib.Data.Set.Operations
import Mathlib.Order.Basic
import Mathlib.Order.BooleanAlgebra
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.B... | Mathlib/Data/Set/Basic.lean | 2,100 | 2,102 | |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.Algebra.Ring.NegOnePow
import Mathlib.Algebra.Category.Grp.Preadditive
import Mathlib.Tactic.Linarith
import Mathlib.Cat... | rfl
lemma δ_shape (hnm : ¬ n + 1 = m) (z : Cochain F G n) : δ n m z = 0 := by
ext p q hpq
dsimp only [δ]
rw [Cochain.mk_v, Cochain.zero_v, F.shape, G.shape, comp_zero, zero_add, zero_comp, smul_zero]
all_goals
| Mathlib/Algebra/Homology/HomotopyCategory/HomComplex.lean | 420 | 426 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.P... | Mathlib/RingTheory/Polynomial/Basic.lean | 382 | 382 | |
/-
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, Amelia Livingston, Yury Kudryashov,
Neil Strickland, Aaron Anderson
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Tactic.Commo... | ⟨exists_eq_mul_left_of_dvd, by
| Mathlib/Algebra/Divisibility/Basic.lean | 151 | 151 |
/-
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.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Order.Sub.WithTop
import Mathlib.Data.NNReal.Defs
import Mathlib.Order.Interval.Set.... |
/-- The set of numbers in `ℝ≥0∞` that are not equal to `∞` is equivalent to `ℝ≥0`. -/
def neTopEquivNNReal : { a | a ≠ ∞ } ≃ ℝ≥0 where
| Mathlib/Data/ENNReal/Basic.lean | 438 | 440 |
/-
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.Basic
/-!
# Maps between real and extended non-negative real numbers
This file focuses on the functions `ENNReal.toReal... |
@[simp]
theorem toNNReal_le_toNNReal (ha : a ≠ ∞) (hb : b ≠ ∞) : a.toNNReal ≤ b.toNNReal ↔ a ≤ b :=
⟨fun h => by rwa [← coe_toNNReal ha, ← coe_toNNReal hb, coe_le_coe], toNNReal_mono hb⟩
| Mathlib/Data/ENNReal/Real.lean | 87 | 90 |
/-
Copyright (c) 2019 Jan-David Salchow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo
-/
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
/-!
# Results about operator n... | variable [NormedAddCommGroup E] [NormedSpace 𝕜 E]
variable (𝕜) (R : Type*)
section
variable [NonUnitalNormedRing R] [NormedSpace 𝕜 R] [IsScalarTower 𝕜 R R]
| Mathlib/Analysis/NormedSpace/OperatorNorm/Mul.lean | 226 | 231 |
/-
Copyright (c) 2022 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.NumberTheory.Cyclotomic.Discriminant
import Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral
import Mathlib.RingTheory.Ideal.Norm.AbsNorm
import M... | (cyclotomic.irreducible_rat (by simp only [PNat.pow_coe, gt_iff_lt, PNat.pos, pow_pos]))
hs htwo]
/-- The norm, relative to `ℤ`, of `ζ ^ 2 ^ k - 1` in a `2 ^ (k + 1)`-th cyclotomic extension of `ℚ`
| Mathlib/NumberTheory/Cyclotomic/Rat.lean | 368 | 371 |
/-
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.Order.BooleanAlgebra
import Mathlib.Tactic.Common
/-!
# Co-Heyting boundary
The boundary of an element of a co-Heyting algebra is the intersection of its... | sup_le_sup (inf_le_inf_left _ <| hnot_anti le_sup_left)
(inf_le_inf_left _ <| hnot_anti le_sup_right)
| Mathlib/Order/Heyting/Boundary.lean | 80 | 82 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Joey van Langen, Casper Putz
-/
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Data.Nat.Find
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Order.Lattice
/-!
# Characteris... | let ⟨e, hmul⟩ := hdvd
| Mathlib/Algebra/CharP/Defs.lean | 206 | 206 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Commute.Defs
import Mathlib.Algebra.Opposites
import Mathlib.Tactic.Spread
/-!
# Definitions of group actions
This file de... | Mathlib/Algebra/Group/Action/Defs.lean | 661 | 664 | |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.InnerProductSpace.Sy... |
/-- If `K₁` is complete and contained in `K₂`, `K₁` and `K₁ᗮ ⊓ K₂` span `K₂`. -/
theorem sup_orthogonal_inf_of_completeSpace {K₁ K₂ : Submodule 𝕜 E} (h : K₁ ≤ K₂)
| Mathlib/Analysis/InnerProductSpace/Projection.lean | 738 | 740 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.Group.Action.Units
import Mathlib.Algebra.Group.Nat.Units
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra... | @[simp]
lemma Semifield.isCoprime_iff {R : Type*} [Semifield R] {m n : R} :
IsCoprime m n ↔ m ≠ 0 ∨ n ≠ 0 := by
obtain rfl | hn := eq_or_ne n 0
· simp [isCoprime_zero_right]
suffices IsCoprime m n by simpa [hn]
refine ⟨0, n⁻¹, ?_⟩
simp [inv_mul_cancel₀ hn]
namespace IsRelPrime
| Mathlib/RingTheory/Coprime/Basic.lean | 421 | 430 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Floris van Doorn, Sébastien Gouëzel, Alex J. Best
-/
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algeb... | @[simp] lemma sum_zipWith_distrib_left [NonUnitalNonAssocSemiring R] (f : ι → κ → R) (a : R) :
∀ (l₁ : List ι) (l₂ : List κ),
(zipWith (fun i j ↦ a * f i j) l₁ l₂).sum = a * (zipWith f l₁ l₂).sum
| [], _ => by simp
| _, [] => by simp
| Mathlib/Algebra/BigOperators/Ring/List.lean | 92 | 96 |
/-
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... | and_congr (Injective.of_comp_iff hf.injective _) (Surjective.of_comp_iff' hf _)
/-- **Cantor's diagonal argument** implies that there are no surjective functions from `α`
to `Set α`. -/
| Mathlib/Logic/Function/Basic.lean | 239 | 242 |
/-
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.Analysis.Calculus.LineDeriv.Measurable
import Mathlib.Analysis.Normed.Module.FiniteDimension
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar... | · apply Continuous.integrable_of_hasCompactSupport
· exact (hf.continuous.comp (continuous_add_right _)).mul hg.continuous
· exact h'g.mul_left
· exact (hf.continuous.mul hg.continuous).integrable_of_hasCompactSupport h'g.mul_left
/-- The line derivative of a Lipschitz function is almost everywhere linear ... | Mathlib/Analysis/Calculus/Rademacher.lean | 199 | 233 |
/-
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.Geometry.RingedSpace.OpenImmersion
import Mathlib.AlgebraicGeometry.Scheme
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq
import Mathlib.Categor... |
/-- The open sets of an open subscheme corresponds to the open sets containing in the image. -/
@[simps]
def IsOpenImmersion.opensEquiv {X Y : Scheme.{u}} (f : X ⟶ Y) [IsOpenImmersion f] :
| Mathlib/AlgebraicGeometry/OpenImmersion.lean | 219 | 222 |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.DoldKan.GammaCompN
import Mathlib.AlgebraicTopology.DoldKan.NReflectsIso
/-! The unit isomorphism of the Dold-Kan equivalence
In order to con... | (Γ₂N₂ToKaroubiIso.app X).inv ≫
Γ₂N₂.natTrans.app ((toKaroubi (SimplicialObject C)).obj X) := by
rw [Γ₂N₂.natTrans_app_f_app]
dsimp only [Karoubi.decompId_i_toKaroubi, Karoubi.decompId_p_toKaroubi, Functor.comp_map,
| Mathlib/AlgebraicTopology/DoldKan/NCompGamma.lean | 185 | 188 |
/-
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.Logic.Function.Defs
import Mathlib.Logic.Function.Iterate
import Aesop
import Mathlib.Tactic.Inhabit
/-!
# Extra facts about `Prod`
This file proves ... | let ⟨y, hy⟩ := hg p.2
⟨(x, y), Prod.ext hx hy⟩
| Mathlib/Data/Prod/Basic.lean | 224 | 225 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Alex Keizer
-/
import Mathlib.Algebra.Group.Nat.Even
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Ring.Nat
import Mathlib.Data.List.GetD
import Mathlib.Data.Nat.B... | rintro ⟨⟩
split_ifs <;> rfl
lemma bitwise_zero : bitwise f 0 0 = 0 := by
simp only [bitwise_zero_right, ite_self]
| Mathlib/Data/Nat/Bitwise.lean | 55 | 60 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Size
import Batteries.Data.Int
/-!
# Bitwise operations on integers
Possi... | Mathlib/Data/Int/Bitwise.lean | 448 | 449 | |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Module.BigOperators
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Squarefree
imp... |
/-- `Ω n` is the number of prime factors of `n`. -/
def cardFactors : ArithmeticFunction ℕ :=
| Mathlib/NumberTheory/ArithmeticFunction.lean | 868 | 870 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Partrec
import Mathlib.Data.Option.Basic
/-!
# Gödel Numbering for Partial Recursive Functions.
This file defines `Nat.Partrec.Code`, a... | rcases hf with ⟨cf, rfl⟩; rcases hg with ⟨cg, rfl⟩
exact ⟨prec cf cg, rfl⟩
| rfind pf hf =>
rcases hf with ⟨cf, rfl⟩
refine ⟨comp (rfind' cf) (pair Code.id zero), ?_⟩
| Mathlib/Computability/PartrecCode.lean | 538 | 542 |
/-
Copyright (c) 2020 Heather Macbeth, Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Patrick Massot
-/
import Mathlib.Algebra.Group.Subgroup.Order
import Mathlib.Algebra.Order.Archimedean.Basic
/-!
# Archimedean groups
This file prov... | /-- If a nontrivial subgroup of a linear ordered commutative group is disjoint
with the interval `Set.Ioo 1 a` for some `1 < a`, then the set of elements greater than 1 of this
group admits the least element. -/
@[to_additive "If a nontrivial additive subgroup of a linear ordered additive commutative group is
disjoint ... | Mathlib/GroupTheory/Archimedean.lean | 60 | 87 |
/-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.SeparableDegree
import Mathlib.FieldTheory.IsSepClosed
/-!
# Separable closure
This file contains basics about the (relative) separable closure of a fie... | (separableClosure F K).comap i = separableClosure F E := by
ext x
exact map_mem_separableClosure_iff i
| Mathlib/FieldTheory/SeparableClosure.lean | 100 | 103 |
/-
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.Module.Basic
import Mathlib.Algebra.Module.LinearMap.Defs
import Mathlib.RingTheory.HahnSeries.Basic
import Mathlib.Algebra.BigOperators.Group.... | exact HahnSeries.min_le_min_add hx hy hxy
theorem min_order_le_order_add {Γ} [Zero Γ] [LinearOrder Γ] {x y : HahnSeries Γ R}
| Mathlib/RingTheory/HahnSeries/Addition.lean | 167 | 169 |
/-
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,
Kim Morrison
-/
import Mathlib.Data.List.Basic
/-!
# Lattice structure of lists
This files pro... | · rw [← ab] at hb
rw [count_eq_zero.2 hb, Nat.min_zero, Nat.min_zero]
· rw [count_cons_of_ne (Ne.symm ab)]
| Mathlib/Data/List/Lattice.lean | 211 | 214 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.Matrix.RowCol
/-!
# Dot product of two vectors
This file conta... | simpa only [conjTranspose_conjTranspose] using conjTranspose_mul_self_mul_eq_zero Aᴴ _
lemma mul_self_mul_conjTranspose_eq_zero {p} (A : Matrix m n R) (B : Matrix p m R) :
| Mathlib/LinearAlgebra/Matrix/DotProduct.lean | 164 | 166 |
/-
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.LinearAlgebra.Dimension.LinearMap
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.LinearAlgebra.Matrix.ToLin
/-!
# Finite... |
variable [Module R S] [SMulCommClass R S S]
| Mathlib/LinearAlgebra/FreeModule/Finite/Matrix.lean | 59 | 61 |
/-
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... | Mathlib/Combinatorics/SimpleGraph/Basic.lean | 863 | 866 | |
/-
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 | 488 | 491 | |
/-
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, Yaël Dillies, Yuyang Zhao
-/
import Mathlib.Algebra.Order.Ring.Unbundled.Basic
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Alge... | Mathlib/Algebra/Order/Ring/Defs.lean | 552 | 559 | |
/-
Copyright (c) 2023 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.FieldTheory.PrimitiveElement
import Mathlib.FieldTheory.IsAlgClosed.Basic
/-!
# Results about `minpoly... | · rw [← natDegree_minpolyDiv_succ hx]; rfl
· rw [minpolyDiv_eq_zero hx, minpoly.eq_zero hx]; rfl
section PowerBasis
variable {K}
lemma sum_smul_minpolyDiv_eq_X_pow (E) [Field E] [Algebra K E] [IsAlgClosed E]
[FiniteDimensional K L] [Algebra.IsSeparable K L]
{x : L} (hxL : Algebra.adjoin K {x} = ⊤) {r : ... | Mathlib/FieldTheory/Minpoly/MinpolyDiv.lean | 185 | 212 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin
-/
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Logic.Equiv.Functor
import Mathlib.RingTheory.FreeRing
/-... | conv_lhs => reduce
rfl
/-- If α has size at most 1 then the natural map from the free ring on `α` to the
free commutative ring on `α` is an isomorphism of rings. -/
def subsingletonEquivFreeCommRing [Subsingleton α] : FreeRing α ≃+* FreeCommRing α :=
RingEquiv.ofBijective (coeRingHom _) (by
have ... | Mathlib/RingTheory/FreeCommRing.lean | 377 | 390 |
/-
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 | 1,030 | 1,036 | |
/-
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.Geometry.Euclidean.Sphere.Basic
/-!
# Second intersection of a sphere and a line
This file defines and proves basic results about the second intersection... | intro he
rw [eq_comm, Sphere.secondInter_eq_self_iff, ← neg_neg (p' -ᵥ p), inner_neg_left,
neg_vsub_eq_vsub_rev, neg_eq_zero, eq_comm] at he
exact ((inner_pos_or_eq_of_dist_le_radius hp hp').resolve_right (Ne.symm h)).ne he
/-- If the vector passed to `secondInter` is given by a subtraction involving the poi... | Mathlib/Geometry/Euclidean/Sphere/SecondInter.lean | 149 | 154 |
/-
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... |
end CommSemiringNoZeroDivisors
| Mathlib/Algebra/Polynomial/Derivative.lean | 658 | 660 |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.Order.Antidiag.Finsupp
import Mathlib.Data.Finsupp.Weight
import Mathlib.Tactic.Linarith
import Mathlib.LinearAlgebra.Pi
import Mat... | rfl
| Mathlib/RingTheory/MvPowerSeries/Basic.lean | 205 | 206 |
/-
Copyright (c) 2020 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.Algebra.Lie.Abelian
import Mathlib.LinearAlgebra.Matrix.Trace
import Mathlib.Algebra.Lie.... | simpa [A, B, stdBasisMatrix, Matrix.mul_apply, hij] using congr_fun (congr_fun c' i) i
end SpecialLinear
namespace Symplectic
/-- The symplectic Lie algebra: skew-adjoint matrices with respect to the canonical skew-symmetric
bilinear form. -/
def sp [Fintype l] : LieSubalgebra R (Matrix (l ⊕ l) (l ⊕ l) R) :=
| Mathlib/Algebra/Lie/Classical.lean | 122 | 130 |
/-
Copyright (c) 2024 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata, Fabrizio Barroero, Laura Capuano, Nirvana Coppola,
María Inés de Frutos-Fernández, Sam van Gool, Silvain Rideau-Kikuchi, Amos Turchet,
Frances... | ext1 z
rw [← Rat.num_div_den z, map_div₀, map_div₀, h, eq_on_nat_iff_eq_on_int.mp h]
/-- The equivalence class of an absolute value on the rationals is determined by its values on
the natural numbers. -/
lemma equiv_on_nat_iff_equiv : (∃ c : ℝ, 0 < c ∧ ∀ n : ℕ , f n ^ c = g n) ↔ f ≈ g := by
refine ⟨fun ⟨c, hc, h... | Mathlib/NumberTheory/Ostrowski.lean | 87 | 101 |
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Eric Wieser, Jeremy Avigad, Johan Commelin
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.Linear... | (fromBlocksZero₁₂InvertibleEquiv _ _ _).nonempty_congr]
/-- An expression for the inverse of an upper block-triangular matrix, when either both elements of
diagonal are invertible, or both are not. -/
| Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 184 | 187 |
/-
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.Projection
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique
import Mathlib.MeasureTheory.Function.L2Space
/-... | alias integrable_condexpL2_indicator := integrable_condExpL2_indicator
end CondexpL2Indicator
| Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean | 407 | 409 |
/-
Copyright (c) 2014 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Jeremy Avigad
-/
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Algebra.Order.Ring.Canonical
/-!
# Distance function on ℕ
This file defines a simple dista... | Mathlib/Data/Nat/Dist.lean | 112 | 113 | |
/-
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... | have E : EqOn f g s := fun x hx => by
refine le_antisymm (le_ciInf fun y => hf.le_add_mul hx y.2) ?_
| Mathlib/Topology/MetricSpace/Lipschitz.lean | 360 | 361 |
/-
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.Group.Nat.Even
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Data.Nat.Cast.Commute
import Mathlib.Data.Set.Operations
import Mathlib.Logic.Fu... |
end Nat
| Mathlib/Algebra/Ring/Parity.lean | 300 | 301 |
/-
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.CategoryTheory.Sites.Sheaf
/-! Objects which cover the terminal object
In this file, given a site `(C, J)`, we introduce the notion of a family
of objects `Y :... | lemma coversTop_iff_of_isTerminal (X : C) (hX : IsTerminal X)
{I : Type*} (Y : I → C) :
J.CoversTop Y ↔ Sieve.ofObjects Y X ∈ J X := by
constructor
· tauto
· intro h W
apply J.superset_covering _ (J.pullback_stable (hX.from W) h)
rintro T a ⟨i, ⟨b⟩⟩
exact ⟨i, ⟨b⟩⟩
| Mathlib/CategoryTheory/Sites/CoversTop.lean | 40 | 48 |
/-
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... | variable {x y z} in
theorem dvd_gcd (hx : z ∣ x) (hy : z ∣ y) : z ∣ gcd x y := by
| Mathlib/RingTheory/PrincipalIdealDomain.lean | 189 | 190 |
/-
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.CategoryTheory.Limits.HasLimits
import Mathlib.CategoryTheory.Products.Basic
import Mathlib.CategoryTheory.Functor.Currying
import Mathlib.CategoryTheory.P... |
variable (F)
variable [HasColimitsOfShape K C]
/-- Given a functor `F : J ⥤ K ⥤ C`, with all needed colimits,
we can construct a diagram consisting of the colimit cocone over each functor `F.obj j`,
and the universal cocone morphisms between these.
| Mathlib/CategoryTheory/Limits/Fubini.lean | 423 | 429 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Field.IsField
import Mathlib.Algebra.GroupWithZero.N... | Mathlib/RingTheory/Localization/Basic.lean | 1,312 | 1,320 | |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Kim Morrison, Floris van Doorn
-/
import Mathlib.Tactic.CategoryTheory.Reassoc
/-!
# Isomorphisms
This file defines isomorphisms between objects of a categ... | Mathlib/CategoryTheory/Iso.lean | 657 | 658 | |
/-
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.Group.Nat.Even
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Data.Nat.Cast.Commute
import Mathlib.Data.Set.Operations
import Mathlib.Logic.Fu... | lemma neg_one_pow_congr (h : Even m ↔ Even n) : (-1 : R) ^ m = (-1) ^ n := by
simp [h, neg_one_pow_eq_ite]
| Mathlib/Algebra/Ring/Parity.lean | 351 | 352 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Yury Kudryashov
-/
import Mathlib.Tactic.ApplyFun
import Mathlib.Topology.Separation.Regular
import Mathlib.Topology.UniformSpace.Basic
/-!
# Hausdorff... | theorem comap_map_mk_uniformity : comap (Prod.map mk mk) (map (Prod.map mk mk) (𝓤 α)) = 𝓤 α := by
refine le_antisymm ?_ le_comap_map
refine ((((𝓤 α).basis_sets.map _).comap _).le_basis_iff uniformity_hasBasis_open).2 fun U hU ↦ ?_
refine ⟨U, hU.1, fun (x₁, x₂) ⟨(y₁, y₂), hyU, hxy⟩ ↦ ?_⟩
simp only [Prod.map, ... | Mathlib/Topology/UniformSpace/Separation.lean | 202 | 216 |
/-
Copyright (c) 2021 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Analysis.Convex.Cone.Basic
import Mathlib.Analysis.InnerProductSpace.Projection
/-!
# Convex cones in inner product spaces
We define `Set.inn... | Mathlib/Analysis/Convex/Cone/InnerDual.lean | 205 | 215 | |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.ContDiff.RCLike
import Mathlib.MeasureTheory.Measure.Hausdorff
/-!
# Hausdorff dimension
The Hausdorff dimension of a set `X` in ... | dimH (⋃ i, s i) = ⨆ i, dimH (s i) := by
borelize X
refine le_antisymm (dimH_le fun d hd => ?_) (iSup_le fun i => dimH_mono <| subset_iUnion _ _)
| Mathlib/Topology/MetricSpace/HausdorffDimension.lean | 168 | 170 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Eric Wieser
-/
import Mathlib.Algebra.DirectSum.Internal
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolyn... | lemma HomogeneousSubmodule.gradedMonoid :
SetLike.GradedMonoid (homogeneousSubmodule σ R) :=
WeightedHomogeneousSubmodule.gradedMonoid
/-- The decomposition of `MvPolynomial σ R` into homogeneous submodules. -/
| Mathlib/RingTheory/MvPolynomial/Homogeneous.lean | 513 | 517 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.MeasurableSpace.MeasurablyGenerated
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.Order.Interval.Set... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 1,945 | 1,945 | |
/-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann, Kyle Miller, Mario Carneiro
-/
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Data.Nat.BinaryRec
import Mathlib.Logic.F... | Mathlib/Data/Nat/Fib/Basic.lean | 261 | 261 | |
/-
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.CategoryTheory.Category.ULift
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Skeletal
import Mathlib.Logic.UnivLE
import Mathlib.Logic... | instance essentiallySmall_fullSubcategory_mem (s : Set C) [Small.{w} s] [LocallySmall.{w} C] :
EssentiallySmall.{w} (ObjectProperty.FullSubcategory (· ∈ s)) :=
suffices Small.{w} (ObjectProperty.FullSubcategory (· ∈ s)) from
| Mathlib/CategoryTheory/EssentiallySmall.lean | 231 | 233 |
/-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.GCDMonoid.Multiset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Algebra.Group.TypeTags.Finite
import Mathlib.Combinatorics.Enumerative... |
attribute [to_additive existing] exists_prime_orderOf_dvd_card
-- TODO: Make the `Finite` version of this theorem the default
/-- For every prime `p` dividing the order of a finite group `G` there exists an element of order
`p` in `G`. This is known as Cauchy's theorem. -/
@[to_additive]
theorem _root_.exists_prime_o... | Mathlib/GroupTheory/Perm/Cycle/Type.lean | 518 | 529 |
/-
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.PreservesHomology
import Mathlib.Algebra.Homology.ShortComplex.Abelian
import Mathlib.Algebra.Homology.ShortComplex.QuasiIso
import... | IsZero S.X₂ :=
(ShortComplex.HomologyData.ofZeros S hf hg).exact_iff.1 ex
end
section Preadditive
variable [Preadditive C] [Preadditive D] (S : ShortComplex C)
| Mathlib/Algebra/Homology/ShortComplex/Exact.lean | 257 | 264 |
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yury Kudryashov, Sébastien Gouëzel, Chris Hughes
-/
import Mathlib.Data.Fin.Rev
import Mathlib.Data.Nat.Find
/-!
# Operation on tuples
We interpret maps `∀ i : Fi... | simpa using insertNth_rev 0 a f i
theorem cons_comp_rev {α n} (a : α) (f : Fin n → α) :
| Mathlib/Data/Fin/Tuple/Basic.lean | 910 | 912 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.CharP.Two
import Mathlib.Data.Nat.Cast.Field
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Factorization.Induction
import Mat... | exact h b ⟨lt_of_mul_lt_mul_left ha (zero_le _), mul_comm _ _⟩)
_ = _ := by
have h1 : Function.Injective (· * p) := mul_left_injective₀ hp.ne_zero
have h2 : (range (p ^ n)).image (· * p) ⊆ range (p ^ (n + 1)) := fun a => by
simp only [mem_image, mem_range, exists_imp]
rintro ... | Mathlib/Data/Nat/Totient.lean | 187 | 213 |
/-
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.ConcreteCategory
import Mathlib.Algebra.Category.ModuleCat.Colimits
/-!
# Homology and exactness of short complexes of modules
In... | lemma ShortExact.moduleCat_exact_iff_function_exact :
S.Exact ↔ Function.Exact S.f S.g := by
rw [moduleCat_exact_iff_range_eq_ker, LinearMap.exact_iff]
| Mathlib/Algebra/Homology/ShortComplex/ModuleCat.lean | 74 | 76 |
/-
Copyright (c) 2021 Gabriel Moise. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Moise, Yaël Dillies, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Matrix.Mul
/-!
# Incidence matrix of a si... | rw [incMatrix_apply_mul_incMatrix_apply, Set.indicator_of_not_mem]
rw [G.incidenceSet_inter_incidenceSet_of_not_adj h hab]
exact Set.not_mem_empty e
| Mathlib/Combinatorics/SimpleGraph/IncMatrix.lean | 79 | 82 |
/-
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.Computability.AkraBazzi.GrowsPolynomially
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
/-!
# Divid... | simp_rw [div_eq_mul_inv]
_ =o[atTop] fun (n : ℕ) => (n : ℝ) * 1⁻¹ := by
refine IsBigO.mul_isLittleO (isBigO_refl _ _) ?_
refine IsLittleO.inv_rev ?main ?zero
| Mathlib/Computability/AkraBazzi/AkraBazzi.lean | 133 | 136 |
/-
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
-/
import Mathlib.Tactic.Attr.Register
import Mathlib.Tactic.Basic
import Batteries.Logic
import Batteries.Tactic.Trans
import Batteries.Util.LibraryNot... | Mathlib/Logic/Basic.lean | 1,212 | 1,213 | |
/-
Copyright (c) 2024 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul Lezeau, Calle Sönne
-/
import Mathlib.CategoryTheory.Functor.Category
import Mathlib.CategoryTheory.CommSq
/-!
# HomLift
Given a functor `p : 𝒳 ⥤ 𝒮`, this file provides API for... | subst_hom_lift p f φ; simp
| Mathlib/CategoryTheory/FiberedCategory/HomLift.lean | 88 | 89 |
/-
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.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
/-!
# Oriented an... | points is weakly between the other two. -/
theorem oangle_eq_zero_iff_wbtw {p₁ p₂ p₃ : P} :
| Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 465 | 466 |
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Dynamics.Ergodic.AddCircle
import Mathlib.MeasureTheory.Covering.LiminfLimsup
/-!
# Well-approximable numbers and Gallagher's ergodic theorem
Gallagher's e... |
local notation "𝕊" => AddCircle T
/-- **Gallagher's ergodic theorem** on Diophantine approximation. -/
theorem addWellApproximable_ae_empty_or_univ (δ : ℕ → ℝ) (hδ : Tendsto δ atTop (𝓝 0)) :
(∀ᵐ x, ¬addWellApproximable 𝕊 δ x) ∨ ∀ᵐ x, addWellApproximable 𝕊 δ x := by
/- Sketch of proof:
Let `E := addWell... | Mathlib/NumberTheory/WellApproximable.lean | 183 | 191 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Kim Morrison, Jakob von Raumer
-/
import Mathlib.Algebra.Category.ModuleCat.Basic
import Mathlib.LinearAlgebra.TensorProduct.Associator
import Mathlib.CategoryTheory.Monoi... | (associator_naturality := fun f g h ↦ MonoidalCategory.associator_naturality f g h)
(leftUnitor_naturality := fun f ↦ MonoidalCategory.leftUnitor_naturality f)
(rightUnitor_naturality := fun f ↦ rightUnitor_naturality f)
(pentagon := fun M N K L ↦ pentagon M N K L)
(triangle := fun M N ↦ triangle M N)
/-- Re... | Mathlib/Algebra/Category/ModuleCat/Monoidal/Basic.lean | 164 | 175 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.MeasurableSpace.Constructions
import Mathlib.Tactic.FunProp
/-!
# Measurable embeddings and equivalences
A measurable e... |
/-- Any two types with unique elements are measurably equivalent. -/
def ofUniqueOfUnique (α β : Type*) [MeasurableSpace α] [MeasurableSpace β] [Unique α] [Unique β] :
α ≃ᵐ β where
toEquiv := ofUnique α β
measurable_toFun := Subsingleton.measurable
measurable_invFun := Subsingleton.measurable
| Mathlib/MeasureTheory/MeasurableSpace/Embedding.lean | 357 | 363 |
/-
Copyright (c) 2022 María Inés de Frutos-Fernández, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: María Inés de Frutos-Fernández, Yaël Dillies
-/
import Mathlib.Data.NNReal.Defs
import Mathlib.Order.ConditionallyCompleteLattice.Group
import Mathlib.Tac... | instance (priority := 100) NonarchAddGroupSeminormClass.toAddGroupSeminormClass
[FunLike F E ℝ] [AddGroup E] [NonarchAddGroupSeminormClass F E] : AddGroupSeminormClass F E ℝ :=
{ ‹NonarchAddGroupSeminormClass F E› with
| Mathlib/Analysis/Normed/Group/Seminorm.lean | 148 | 150 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Simon Hudon
-/
import Mathlib.CategoryTheory.Monoidal.Category
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.PEmpty
/-!
# The mo... | dsimp [tensorHom]
simp
theorem rightUnitor_naturality {X₁ X₂ : C} (f : X₁ ⟶ X₂) :
tensorHom ℬ f (𝟙 𝒯.cone.pt) ≫ (BinaryFan.rightUnitor 𝒯.isLimit (ℬ X₂ 𝒯.cone.pt).isLimit).hom =
| Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Basic.lean | 283 | 287 |
/-
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.Group.Basic
import Mathlib.Algebra.GroupWithZero.NeZero
import Mathlib.Logic.Unique
import Mathlib.Tactic.Conv
/-!
# Groups with an adjoined z... | section GroupWithZero
| Mathlib/Algebra/GroupWithZero/Basic.lean | 253 | 254 |
/-
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... | rwa [← IsRoot.def, isRoot_cyclotomic_iff] at h
· rw [← isRoot_cyclotomic_iff, IsRoot.def] at h
rw [cyclotomic_mul_prime_pow_eq R (NeZero.not_char_dvd R p m) hk, IsRoot.def, eval_pow,
h, zero_pow]
exact Nat.sub_ne_zero_of_lt <| pow_right_strictMono₀ hp.out.one_lt <| Nat.pred_lt hk.ne'
end CharP
end... | Mathlib/RingTheory/Polynomial/Cyclotomic/Expand.lean | 166 | 179 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.LinearAlgebra.SModEq
import Mathlib.RingTheory.Ideal.BigOperators
/-!
# Power basis
This file defines a structure ... | rcases subsingleton_or_nontrivial B
· exact ⟨0, by rw [SModEq, Subsingleton.eq_zero b, map_zero]⟩
refine ⟨pb.basis.repr b ⟨0, pb.dim_pos⟩, ?_⟩
have H := pb.basis.sum_repr b
| Mathlib/RingTheory/PowerBasis.lean | 133 | 136 |
/-
Copyright (c) 2022 Kyle Miller, Adam Topaz, Rémi Bottinelli, Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Adam Topaz, Rémi Bottinelli, Junyan Xu
-/
import Mathlib.Topology.Category.TopCat.Limits.Konig
/-!
# Cofiltered systems
This file de... | ext
rfl
map_comp f g := by
| Mathlib/CategoryTheory/CofilteredSystem.lean | 174 | 176 |
/-
Copyright (c) 2023 Mark Andrew Gerads. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mark Andrew Gerads, Junyan Xu, Eric Wieser
-/
import Mathlib.Tactic.Ring
/-!
# Hyperoperation sequence
This file defines the Hyperoperation sequence.
`hyperoperation 0 m k = k + ... | @[simp]
theorem hyperoperation_two : hyperoperation 2 = (· * ·) := by
ext m k
induction' k with bn bih
· rw [hyperoperation]
exact (Nat.mul_zero m).symm
| Mathlib/Data/Nat/Hyperoperation.lean | 60 | 65 |
/-
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.Algebra.GradedMonoid
import Mathlib.Algebra.DirectSum.Basic
/-!
# Additively-graded multiplicative structures on `⨁ i, A i`
This module provides a set of h... | ext ai ax bi bx ci cx : 6
dsimp only [coe_comp, Function.comp_apply, AddMonoidHom.compHom_apply_apply, flip_apply,
AddMonoidHom.flipHom_apply]
simp_rw [mulHom_of_of]
exact of_eq_of_gradedMonoid_eq (_root_.mul_assoc (GradedMonoid.mk ai ax) ⟨bi, bx⟩ ⟨ci, cx⟩)
| Mathlib/Algebra/DirectSum/Ring.lean | 238 | 243 |
/-
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.Polynomial.Splits
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.RingTheory.Ro... | theorem cyclotomic_prime (R : Type*) [Ring R] (p : ℕ) [hp : Fact p.Prime] :
cyclotomic p R = ∑ i ∈ Finset.range p, X ^ i := by
suffices cyclotomic p ℤ = ∑ i ∈ range p, X ^ i by
| Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | 355 | 357 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.QuasiIso
import Mathlib.CategoryTheory.Limits.Preserves.Finite
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels
/-!
... | ← rightHomologyMap'_comp, comp_id, id_comp]
@[reassoc]
lemma mapRightHomologyIso_inv_naturality [S₁.HasRightHomology] [S₂.HasRightHomology]
[F.PreservesRightHomologyOf S₁] [F.PreservesRightHomologyOf S₂] :
F.map (rightHomologyMap φ) ≫ (S₂.mapRightHomologyIso F).inv =
(S₁.mapRightHomologyIso F).inv ≫ ... | Mathlib/Algebra/Homology/ShortComplex/PreservesHomology.lean | 556 | 562 |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Algebra.Ring.Regular
import Mathlib.Algebra.Equiv.TransferInstance
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.BigOperators.Ring.Finset... |
end fromAddGrouptoDivisionMonoid
| Mathlib/Algebra/Group/AddChar.lean | 388 | 389 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Init
import Mathlib.Data.Int.Init
import Mathlib.... | Mathlib/Algebra/Group/Basic.lean | 564 | 564 | |
/-
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.Algebra.Group.TypeTags.Hom
import Mathlib.Algebra.Ring.Hom.Basic
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Algebra.Ring.Parity
/-!
# Cast of... | /-- `coe : ℤ → α` as a `RingHom`. -/
| Mathlib/Data/Int/Cast/Lemmas.lean | 85 | 85 |
/-
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.Ordmap.Invariants
/-!
# Verification of `Ordnode`
This file uses the invariants defined in `Mathlib.Data.Ordmap.Invariants` to construct `Ordset... | rw [eraseMax, size_balanceL H.3.2.1 h.3 H.2.2.1 h.2 (Or.inr ⟨_, Or.inr e, H.3.1⟩)]
rw [size_node, e]; rfl
| Mathlib/Data/Ordmap/Ordset.lean | 385 | 387 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.InnerProductSpace.Sy... | theorem orthogonalProjection_orthogonalComplement_singleton_eq_zero (v : E) :
(𝕜 ∙ v)ᗮ.orthogonalProjection v = 0 :=
orthogonalProjection_mem_subspace_orthogonal_precomplement_eq_zero
| Mathlib/Analysis/InnerProductSpace/Projection.lean | 961 | 963 |
/-
Copyright (c) 2020 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.GroupTheory.Complement
/-!
# Semidirect product
This file defines semidirect products of groups, and the canonical maps in and out of the
semidirect prod... | Function.surjective_iff_hasRightInverse.2 ⟨inr, rightHom_inr⟩
| Mathlib/GroupTheory/SemidirectProduct.lean | 170 | 170 |
/-
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... | theorem eventually_const {f : Filter α} [t : NeBot f] {p : Prop} : (∀ᶠ _ in f, p) ↔ p := by
by_cases h : p <;> simp [h, t.ne]
| Mathlib/Order/Filter/Basic.lean | 600 | 601 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Positivity.Basic
/-!
# Algebraic order homomorphism classes
This file defines hom classes for common properties at the ... | theorem le_map_div_mul_map_div [Group α] [Mul β] [LE β] [SubmultiplicativeHomClass F α β]
(f : F) (a b c : α) : f (a / c) ≤ f (a / b) * f (b / c) := by
simpa only [div_mul_div_cancel] using map_mul_le_mul f (a / b) (b / c)
| Mathlib/Algebra/Order/Hom/Basic.lean | 131 | 133 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Floor.Defs
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Algebra.Order.Floor.Semiring
deprecated_module (sinc... | Mathlib/Algebra/Order/Floor.lean | 813 | 814 | |
/-
Copyright (c) 2021 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Group.Subgroup.Defs
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Star.Pi
import Mathlib.Algebra.Star.Rat
/-!
# Self-adjoint, sk... |
section Monoid
| Mathlib/Algebra/Star/SelfAdjoint.lean | 177 | 178 |
/-
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... | exact image_subset_iff.mpr st
· have : AnalyticOn 𝕜 f w := by
have : AnalyticOn 𝕜 (fun y ↦ (continuousMultilinearCurryFin0 𝕜 E F).symm (f y)) w :=
((h'p 0).mono wu).congr fun y hy ↦ (hp.zero_eq' (wu hy)).symm
have : AnalyticOn 𝕜 (fun y ↦ (continuousMultilinearCurryFin0 𝕜 E F)
... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 590 | 629 |
/-
Copyright (c) 2022 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Data.ENNReal.Lemmas
import Mathlib.Topology.MetricSpace.Thickening
import Mathlib.Topology.ContinuousMap.Bounded.Basic
/-!
# Thickened indicators
This fi... | This statement is for the unbundled `ℝ≥0∞`-valued functions `thickenedIndicatorAux δ E`, see
`thickenedIndicator_tendsto_indicator_closure` for the version for bundled `ℝ≥0`-valued
bounded continuous functions. -/
theorem thickenedIndicatorAux_tendsto_indicator_closure {δseq : ℕ → ℝ}
(δseq_lim : Tendsto δseq atTop ... | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | 110 | 115 |
/-
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.Group.Multiset
/-!
# Disjoint sum of multisets
This file defines the disjoint sum of two multisets as `Multiset (α ⊕ β)`. Beware not to con... | · simp only [inl.injEq, exists_eq_right]
rintro ⟨b, _, hb⟩
| Mathlib/Data/Multiset/Sum.lean | 50 | 51 |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | theorem sin_eq_iff_eq_or_add_eq_pi {θ ψ : Angle} : sin θ = sin ψ ↔ θ = ψ ∨ θ + ψ = π := by
induction ψ using Real.Angle.induction_on
exact sin_eq_real_sin_iff_eq_or_add_eq_pi
@[simp]
theorem sin_zero : sin (0 : Angle) = 0 := by rw [← coe_zero, sin_coe, Real.sin_zero]
theorem sin_coe_pi : sin (π : Angle) = 0 := by... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 292 | 302 |
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