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) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Gluing
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.CategoryTheory.Limits.Sh... |
theorem t_id (i : 𝒰.J) : t 𝒰 f g i i = 𝟙 _ := by
apply pullback.hom_ext <;> rw [Category.id_comp]
· apply pullback.hom_ext
| Mathlib/AlgebraicGeometry/Pullbacks.lean | 78 | 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.Algebra.Group.Units.Equiv
import Mathlib.Algebra.Order.Group.End
import Mathlib.Logic.Function.Conjugate
import Mathlib.Order.Bounds.OrderIso
import ... | intro y
change IsLUB (e ⁻¹' { x | f x ≤ y }) (e.symm (g y))
rw [e.isLUB_preimage, e.apply_symm_apply]
| Mathlib/Order/SemiconjSup.lean | 68 | 70 |
/-
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, Heather Macbeth
-/
import Mathlib.Topology.FiberBundle.Trivialization
import Mathlib.Topology.Order.LeftRightNhds
/-!
# Fiber bundles
Mathemat... | rw [e.coe_fst]
· exact ⟨hz, a.mem_base_pretrivializationAt z.proj⟩
· rw [e.mem_source]
exact a.mem_base_pretrivializationAt z.proj
end FiberPrebundle
| Mathlib/Topology/FiberBundle/Basic.lean | 854 | 863 |
/-
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 ... | Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 1,289 | 1,291 | |
/-
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.Algebra.Subalgebra.Tower
import Mathlib.Data.Finite.Sum
import Mathlib.Data.Matrix.Block
import Mathl... | /-- A variant of `Matrix.mulVecLin_submatrix` that keeps around `LinearEquiv`s. -/
theorem Matrix.mulVecLin_reindex [Fintype n] [Fintype l] (e₁ : k ≃ m) (e₂ : l ≃ n)
(M : Matrix k l R) :
| Mathlib/LinearAlgebra/Matrix/ToLin.lean | 235 | 237 |
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ord... | div_lt_div₀ hac hbd c0 d0
| Mathlib/Algebra/Order/Field/Basic.lean | 58 | 58 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.ContMDiff.Defs
/-!
## Basic properties of `C^n` functions between manifolds
In this file, we show that stan... | `f` is `C^n` within `s` at `x`. -/
theorem ContMDiffAt.comp_contMDiffWithinAt {g : M' → M''} (x : M)
(hg : ContMDiffAt I' I'' n g (f x)) (hf : ContMDiffWithinAt I I' n f s x) :
ContMDiffWithinAt I I'' n (g ∘ f) s x :=
| Mathlib/Geometry/Manifold/ContMDiff/Basic.lean | 119 | 122 |
/-
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, Sander Dahmen, Kim Morrison
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.LinearAlgebra.LinearIndependent.Basic
import Mathlib.Data.Set.Card
/... | theorem Submodule.rank_le (s : Submodule R M) : Module.rank R s ≤ Module.rank R M := by
rw [← rank_top R M]
exact rank_mono le_top
theorem LinearMap.lift_rank_le_of_surjective (f : M →ₗ[R] M') (h : Surjective f) :
lift.{v} (Module.rank R M') ≤ lift.{v'} (Module.rank R M) := by
rw [← rank_range_of_surjective ... | Mathlib/LinearAlgebra/Dimension/Basic.lean | 358 | 371 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Frédéric Dupuis
-/
import Mathlib.Algebra.Star.SelfAdjoint
import Mathlib.Algebra.Module.Basic
import Mathlib.Algebra.Module.Equiv.Defs
import Mathlib.Algebra.Module.LinearMa... | (skewAdjointPart R).comp (selfAdjoint.submodule R A).subtype = 0 :=
LinearMap.ext fun x ↦ x.2.skewAdjointPart_apply R
| Mathlib/Algebra/Star/Module.lean | 179 | 181 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Eric Wieser
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Action
import Mathlib.Algebra.GroupWithZero.Invertible
import Mathlib.LinearAlgebra.Prod
/-!
# Trivial Square-Ze... | theorem map_comp_inlAlgHom (f : M →ₗ[R'] N) :
(map f).comp (inlAlgHom R' R' M) = inlAlgHom R' R' N :=
| Mathlib/Algebra/TrivSqZeroExt.lean | 1,070 | 1,071 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryTheor... | (f : X ⟶ Y) : (0 : C ⥤ D).map f = 0 :=
| Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | 316 | 316 |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Data.NNReal.Basic
import Mathlib.Topology.Algebra.Support
import Mathlib.To... | variable {E : Type*} [TopologicalSpace E] [ContinuousENorm E]
@[continuity, fun_prop]
| Mathlib/Analysis/Normed/Group/Basic.lean | 897 | 899 |
/-
Copyright (c) 2022 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Analysis.Complex.Polynomial.UnitTrinomial
import Mathlib.RingTheory.Polynomial.GaussLemma
import Mathlib.Tactic.LinearCombination
/-!
# Irreducibili... | have hn : 1 < n := Nat.one_lt_iff_ne_zero_and_ne_one.mpr ⟨hn0, hn1⟩
have h := (IsPrimitive.Int.irreducible_iff_irreducible_map_cast ?_).mp
(X_pow_sub_X_sub_one_irreducible hn1)
· rwa [Polynomial.map_sub, Polynomial.map_sub, Polynomial.map_pow, Polynomial.map_one,
Polynomial.map_X] at h
· exact hp.symm... | Mathlib/RingTheory/Polynomial/Selmer.lean | 71 | 82 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Batteries.Data.List.Perm
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.TakeWhile
import Mathlib.Order.Fin.Basic
/-!
# So... | induction b generalizing a with
| zero => simp
| succ n ih =>
rw [List.range'_succ]
refine List.sorted_cons.mpr ⟨fun b hb ↦ ?_, @ih (a + s)⟩
exact lt_of_lt_of_le (Nat.lt_add_of_pos_right (Nat.zero_lt_of_ne_zero hs))
(List.left_le_of_mem_range' hb)
| Mathlib/Data/List/Sort.lean | 157 | 163 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Morphisms.Constructors
import Mathlib.RingTheory.LocalProperties.Basic
import Mathlib.RingTheory.RingHom.Locally
/-!
# Properties of morp... | simp
theorem Spec_iff {R S : CommRingCat.{u}} {φ : R ⟶ S} :
P (Spec.map φ) ↔ Q φ.hom := by
have H := (isLocal_ringHomProperty P).respectsIso
rw [iff_of_isAffine (P := P), ← H.cancel_right_isIso _ (Scheme.ΓSpecIso _).hom,
← CommRingCat.hom_comp, Scheme.ΓSpecIso_naturality, CommRingCat.hom_comp, H.cancel_l... | Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean | 346 | 370 |
/-
Copyright (c) 2021 Vladimir Goryachev. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Vladimir Goryachev, Kyle Miller, Kim Morrison, Eric Rodriguez
-/
import Mathlib.Data.List.GetD
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.SuccPred
import M... | noncomputable def giCountNth (hp : (setOf p).Infinite) : GaloisInsertion (count p) (nth p) :=
GaloisInsertion.monotoneIntro (nth_monotone hp) (count_monotone p) (le_nth_count hp)
(count_nth_of_infinite hp)
theorem gc_count_nth (hp : (setOf p).Infinite) : GaloisConnection (count p) (nth p) :=
(giCountNth hp).gc... | Mathlib/Data/Nat/Nth.lean | 421 | 428 |
/-
Copyright (c) 2021 Alex Kontorovich and Heather Macbeth and Marc Masdeu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, Heather Macbeth, Marc Masdeu
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.GeneralLinearG... |
theorem re_T_zpow_smul (n : ℤ) : (T ^ n • z).re = z.re + n := by
rw [← coe_re, coe_T_zpow_smul_eq, add_re, intCast_re, coe_re]
theorem im_T_zpow_smul (n : ℤ) : (T ^ n • z).im = z.im := by
rw [← coe_im, coe_T_zpow_smul_eq, add_im, intCast_im, add_zero, coe_im]
theorem re_T_smul : (T • z).re = z.re + 1 := by simpa... | Mathlib/NumberTheory/Modular.lean | 306 | 318 |
/-
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.OuterMeasure.Operations
import Mathlib.Analysis.SpecificLimits.Basic
/-!
# Outer measures from functions
Given an arbit... | @[simp]
theorem boundedBy_zero : boundedBy (0 : Set α → ℝ≥0∞) = 0 := by
rw [← coe_bot, eq_bot_iff]
| Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean | 291 | 293 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.Convex.StrictConvexBetween
import Mathlib.Analysis.InnerProductSpace.Convex
import Mathlib.Analysis.Normed.Affine.Convex
import Mathlib.Geometry.E... | simp [h.dist_left_right]
lemma IsDiameter.left_eq_right_iff (h : s.IsDiameter p₁ p₂) : p₁ = p₂ ↔ s.radius = 0 := by
rw [← dist_eq_zero, h.dist_left_right]
simp
lemma IsDiameter.left_ne_right_iff_radius_ne_zero (h : s.IsDiameter p₁ p₂) :
| Mathlib/Geometry/Euclidean/Sphere/Basic.lean | 279 | 285 |
/-
Copyright (c) 2021 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Yaël Dillies, Anthony DeRossi
-/
import Mathlib.Computability.NFA
import Mathlib.Data.List.ReduceOption
/-!
# Epsilon Nondeterministic Finite Automata
This file contains th... | M.evalFrom S (x ++ [a]) = M.stepSet (M.evalFrom S x) a := by
rw [evalFrom, List.foldl_append, List.foldl_cons, List.foldl_nil]
| Mathlib/Computability/EpsilonNFA.lean | 110 | 112 |
/-
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.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calcul... |
theorem LineDifferentiableAt.congr_of_eventuallyEq
(h : LineDifferentiableAt 𝕜 f x v) (hL : f₁ =ᶠ[𝓝 x] f) :
LineDifferentiableAt 𝕜 f₁ x v := by
apply DifferentiableAt.congr_of_eventuallyEq h
| Mathlib/Analysis/Calculus/LineDeriv/Basic.lean | 368 | 372 |
/-
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.Vector.Defs
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Scan
import Mathlib.Control.Applicative
import M... | conv_rhs => erw [← List.get_ofFn f ⟨i, by simp⟩]
simp only [get_eq_get_toList]
congr <;> simp [Fin.heq_ext_iff]
@[simp]
theorem ofFn_get (v : Vector α n) : ofFn (get v) = v := by
| Mathlib/Data/Vector/Basic.lean | 163 | 168 |
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | (T) The measures μsₙ converge weakly to the measure μ.
-/
variable {Ω : Type*} [MeasurableSpace Ω] [TopologicalSpace Ω] [OpensMeasurableSpace Ω]
lemma lintegral_le_liminf_lintegral_of_forall_isOpen_measure_le_liminf_measure
{μ : Measure Ω} {μs : ℕ → Measure Ω} {f : Ω → ℝ} (f_cont : Continuous f) (f_nn : 0 ≤ f)... | Mathlib/MeasureTheory/Measure/Portmanteau.lean | 462 | 474 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Topology.Continuous
import Mathlib.Topology.Defs.Induced
/-!
# Ordering on topologies and (co)induced topologies
Topologies on a fixe... | continuous_iff_coinduced_le.2 le_rfl
theorem continuous_coinduced_dom {g : β → γ} {t₁ : TopologicalSpace α} {t₂ : TopologicalSpace γ} :
Continuous[coinduced f t₁, t₂] g ↔ Continuous[t₁, t₂] (g ∘ f) := by
| Mathlib/Topology/Order.lean | 638 | 641 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.Cover
import Mathlib.Order.Iterate
/-!
# Successor and predecessor
This file defines succes... | apply (le_sInf fun b => succ_le_of_lt).antisymm
obtain rfl | ha := eq_or_ne a ⊤
| Mathlib/Order/SuccPred/Basic.lean | 542 | 543 |
/-
Copyright (c) 2021 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno
-/
import Mathlib.CategoryTheory.NatIso
/-!
# Bicategories
In this file we define typeclass for bicategories.
A bicategory `B` consists of
* objects `a : B`,
* 1-morphisms ... | theorem leftUnitor_inv_whiskerRight (f : a ⟶ b) (g : b ⟶ c) :
(λ_ f).inv ▷ g = (λ_ (f ≫ g)).inv ≫ (α_ (𝟙 a) f g).inv :=
| Mathlib/CategoryTheory/Bicategory/Basic.lean | 369 | 370 |
/-
Copyright (c) 2023 Scott Carnahan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Carnahan
-/
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Eval.SMul
/-!
# Scalar-multiple polynomial ev... | theorem smeval_eq_sum : p.smeval x = p.sum (smul_pow x) := by rw [smeval_def]
| Mathlib/Algebra/Polynomial/Smeval.lean | 54 | 54 |
/-
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, Mario Carneiro
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds
/-!
# Pi
This file contains lemmas which establish bounds on `Real.pi`.
Notably, t... |
theorem pi_lt_d6 : π < 3.141593 := by
-- bound[3141593*^-6, Iters -> 11, Rounding -> .5, Precision -> 17]
| Mathlib/Data/Real/Pi/Bounds.lean | 183 | 185 |
/-
Copyright (c) 2023 Kyle Miller, Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Rémi Bottinelli
-/
import Mathlib.Combinatorics.SimpleGraph.Path
import Mathlib.Data.Set.Card
/-!
# Connectivity of subgraphs and induced graphs
## Main de... | | cons h p => G.subgraphOfAdj h ⊔ p.toSubgraph
theorem toSubgraph_cons_nil_eq_subgraphOfAdj (h : G.Adj u v) :
(cons h nil).toSubgraph = G.subgraphOfAdj h := by simp
theorem mem_verts_toSubgraph (p : G.Walk u v) : w ∈ p.toSubgraph.verts ↔ w ∈ p.support := by
induction p with
| nil => simp
| cons h p' ih =>... | Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean | 134 | 146 |
/-
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... | rw [← image_union]
simp [image_univ_of_surjective H]
| Mathlib/Data/Set/Image.lean | 361 | 363 |
/-
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.... | exact h_inv _ ih
| Mathlib/Algebra/Group/Basic.lean | 915 | 915 |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Xavier Roblot
-/
import Mathlib.Algebra.Algebra.Hom.Rat
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.RingTh... | ¬ IsUnramified k w ↔ IsComplex w ∧ IsReal (w.comap (algebraMap k K)) := by
rw [IsUnramified, mult, mult, ← not_isReal_iff_isComplex]
split_ifs with h₁ h₂ h₂ <;>
simp only [not_true_eq_false, false_iff, and_self, forall_true_left, IsEmpty.forall_iff,
not_and, OfNat.one_ne_ofNat, not_false_eq_true, true... | Mathlib/NumberTheory/NumberField/Embeddings.lean | 839 | 845 |
/-
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.Function.LpSeminorm.Trim
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Function.StronglyMeasura... | haveI : Fact (m ≤ m0) := ⟨hm⟩
change CompleteSpace (lpMeasSubgroup F m p μ)
infer_instance
@[deprecated (since := "2025-04-09")]
alias isComplete_aeStronglyMeasurable' := isComplete_aestronglyMeasurable
theorem isClosed_aestronglyMeasurable [Fact (1 ≤ p)] [CompleteSpace F] (hm : m ≤ m0) :
IsClosed {f : Lp F... | Mathlib/MeasureTheory/Function/ConditionalExpectation/AEMeasurable.lean | 433 | 442 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.UniformSpace.Cauchy
/-!
# Uniform convergence
A sequence of functions `Fₙ` (with values in a metric space) converges uniformly on a se... | Mathlib/Topology/UniformSpace/UniformConvergence.lean | 615 | 619 | |
/-
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.CharP.Lemmas
import Mathlib.GroupTheory.OrderOfElement
/-!
# Lemmas about rings of characteristic two
This file contains results about `CharP R 2`,... | split_ifs with h
· rw [neg_one_eq_one_iff.2 h, orderOf_one]
apply orderOf_eq_prime
· simp
| Mathlib/Algebra/CharP/Two.lean | 127 | 130 |
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Algebra.Order.Pi
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Data.Real.Pointwise
/... | Real.norm_of_nonneg (apply_nonneg p _)]
theorem ball_zero_eq_preimage_ball {r : ℝ} : p.ball 0 r = p ⁻¹' Metric.ball 0 r := by
rw [ball_zero_eq, preimage_metric_ball]
theorem closedBall_zero_eq_preimage_closedBall {r : ℝ} :
p.closedBall 0 r = p ⁻¹' Metric.closedBall 0 r := by
rw [closedBall_zero_eq, preima... | Mathlib/Analysis/Seminorm.lean | 756 | 763 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov, David Loeffler
-/
import Mathlib.Analysis.Convex.Slope
import Mathlib.Analysis.Calculus.Deriv.MeanValue
/-!
# Convexity of functions and derivat... | derivWithin f (Iio x) x = sSup (slope f x '' {y | y ∈ S ∧ y < x}) :=
(hfc.hasDerivWithinAt_sSup_slope_of_mem_interior hxs).derivWithin (uniqueDiffWithinAt_Iio x)
lemma monotoneOn_rightDeriv (hfc : ConvexOn ℝ S f) :
MonotoneOn (fun x ↦ derivWithin f (Ioi x) x) (interior S) := by
intro x hxs y hys hxy
rcas... | Mathlib/Analysis/Convex/Deriv.lean | 483 | 493 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Group.End
import Mathlib.Data.Set.Function
/-!
# Fixed points of a self-map
In this file we define
* the predicate `IsFixedPt f x := f x =... | protected theorem perm_inv (h : IsFixedPt e x) : IsFixedPt (⇑e⁻¹) x :=
h.equiv_symm
protected theorem perm_pow (h : IsFixedPt e x) (n : ℕ) : IsFixedPt (⇑(e ^ n)) x := h.iterate _
| Mathlib/Dynamics/FixedPoints/Basic.lean | 97 | 100 |
/-
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.MeasurableSpace.CountablyGenerated
import Mathlib.Probability.Process.Filtration
/-!
# Filtration built from the finite partitions of a coun... | lemma iSup_partitionFiltration_eq_generateFrom_range (ht : ∀ n, MeasurableSet (t n)) :
⨆ n, partitionFiltration ht n = generateFrom (Set.range t) := by
conv_rhs => rw [← generateFrom_iUnion_memPartition t, ← iSup_generateFrom]
rfl
| Mathlib/Probability/Process/PartitionFiltration.lean | 86 | 89 |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Set.Image
import Mathlib.Topology.Bases
import Mathlib.Topology.Inseparable
import Mathlib.Topology.Compactness.Exterior
/-!
# Alexandrov-discrete to... | simp_rw [sUnion_eq_biUnion, closure_iUnion]
| Mathlib/Topology/AlexandrovDiscrete.lean | 112 | 113 |
/-
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, Jeremy Avigad
-/
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Data.Set.Finite.Range
import Mathlib.Data.Set.Lattice
import Mathlib.Topology.Defs.... |
lemma IsOpen.isLocallyClosed (hs : IsOpen s) : IsLocallyClosed s :=
⟨_, _, hs, isClosed_univ, (inter_univ _).symm⟩
lemma IsClosed.isLocallyClosed (hs : IsClosed s) : IsLocallyClosed s :=
| Mathlib/Topology/Basic.lean | 129 | 133 |
/-
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.BigOperators.Ring.Finset
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
/-... | theorem IsCoprime.pow_iff (hm : 0 < m) (hn : 0 < n) : IsCoprime (x ^ m) (y ^ n) ↔ IsCoprime x y :=
(IsCoprime.pow_left_iff hm).trans <| IsCoprime.pow_right_iff hn
| Mathlib/RingTheory/Coprime/Lemmas.lean | 204 | 206 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Data.Finset.Lattice.Fold
import Mathlib.Data.Set.Sigma
import Mathlib.Order.CompleteLattice.Finset
/-!
# Finite sets in a ... |
theorem card_sigmaLift :
(sigmaLift f a b).card = dite (a.1 = b.1) (fun h => (f (h ▸ a.2) b.2).card) fun _ => 0 := by
simp_rw [sigmaLift]
split_ifs with h <;> simp [h]
| Mathlib/Data/Finset/Sigma.lean | 204 | 208 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark
-/
import Mathlib.Algebra.Polynomial.Monic
/-!
# Lemmas for the interaction between polynomials and `∑` and `∏`.
Recall that `∑` and `∏` are notation for ... |
theorem degree_list_sum_le_of_forall_degree_le (l : List S[X])
(n : WithBot ℕ) (hl : ∀ p ∈ l, degree p ≤ n) :
| Mathlib/Algebra/Polynomial/BigOperators.lean | 62 | 64 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
/-!
# Kernels and cokernels
In a category with zero morphisms, the kernel of a morphism `f : X... | { desc
fac := fun s j => by
| Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean | 581 | 582 |
/-
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.RingTheory.WittVector.Basic
import Mathlib.RingTheory.WittVector.IsPoly
/-!
## The Verschiebung operator
## References
* [Hazewinkel, *Witt Vectors*... | simp only [← aeval_verschiebungPoly]
exact eval₂Hom_congr (RingHom.ext_int _ _) rfl rfl
_ = _ := by rw [ghostComponent_verschiebung]; rfl
end
end WittVector
| Mathlib/RingTheory/WittVector/Verschiebung.lean | 179 | 196 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johan Commelin
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Combinatorics.Enumerative.Composition
/-!
# Composition of analytic functions
In this fi... | ext k
by_cases h : k = c.index j
· rw [h]
let r : Fin (c.blocksFun (c.index j)) → Fin n := c.embedding (c.index j)
simp only [Function.update_self]
change p (c.blocksFun (c.index j)) (Function.update v j z ∘ r) = _
let j' := c.invEmbedding j
suffices B : Function.update v j z ∘ r = Function.up... | Mathlib/Analysis/Analytic/Composition.lean | 140 | 162 |
/-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.Normed.Lp.lpSpace
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Hilbert sum of ... |
theorem _root_.Orthonormal.linearIsometryEquiv_symm_apply_single_one [DecidableEq ι] (h i) :
(hv.isHilbertSum h).linearIsometryEquiv.symm (lp.single 2 i 1) = v i := by
rw [IsHilbertSum.linearIsometryEquiv_symm_apply_single, LinearIsometry.toSpanSingleton_apply,
one_smul]
| Mathlib/Analysis/InnerProductSpace/l2Space.lean | 511 | 515 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.I... | Mathlib/Algebra/Ring/Defs.lean | 413 | 413 | |
/-
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.Nontrivial.Basic
import Mathlib.Order.TypeTags
import Mathlib.Data.Option.NAry
import Mathlib.Tactic.Contrapose
import Mathlib.Tactic.Lift
import... | Mathlib/Order/WithBot.lean | 1,179 | 1,181 | |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Oliver Nash
-/
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Union
/-!
# Finsets in product types
This file defines finset constru... | theorem inter_product [DecidableEq α] [DecidableEq β] : (s ∩ s') ×ˢ t = s ×ˢ t ∩ s' ×ˢ t := by
ext ⟨x, y⟩
simp only [← and_and_right, mem_inter, mem_product]
| Mathlib/Data/Finset/Prod.lean | 233 | 235 |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Homology.Linear
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Tactic.Abel
/-!
# Chain homotopies
We define chain... | theorem nullHomotopicMap'_f_of_not_rel_right {k₁ k₀ : ι} (r₁₀ : c.Rel k₁ k₀)
(hk₁ : ∀ l : ι, ¬c.Rel l k₁) (h : ∀ i j, c.Rel j i → (C.X i ⟶ D.X j)) :
(nullHomotopicMap' h).f k₁ = C.d k₁ k₀ ≫ h k₀ k₁ r₁₀ := by
simp only [nullHomotopicMap']
rw [nullHomotopicMap_f_of_not_rel_right r₁₀ hk₁]
| Mathlib/Algebra/Homology/Homotopy.lean | 386 | 390 |
/-
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.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Discriminant
import Mathlib.RingTheory.DedekindDomain.IntegralClosure
import Mathlib.NumberTheory.K... | rw [coeSubmodule, ← this]
have H : RingHom.comp (algebraMap (FractionRing A) (FractionRing B))
↑(FractionRing.algEquiv A K).symm.toRingEquiv =
RingHom.comp ↑(FractionRing.algEquiv B L).symm.toRingEquiv (algebraMap K L) := by
apply IsLocalization.ringHom_ext A⁰
ext
simp only [AlgEquiv.toRin... | Mathlib/RingTheory/DedekindDomain/Different.lean | 426 | 433 |
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
/-!
# Thickenings in pseudo-metric spaces
## Main definitions
* `Metric.thickening δ s`, the open thickening by ra... | ⟨h.1, disjoint_compl_right_iff_subset.1 <| h.2.mono_right <| self_subset_cthickening _⟩
theorem _root_.IsCompact.exists_isCompact_cthickening [LocallyCompactSpace α] (hs : IsCompact s) :
∃ δ, 0 < δ ∧ IsCompact (cthickening δ s) := by
rcases exists_compact_superset hs with ⟨K, K_compact, hK⟩
rcases hs.exist... | Mathlib/Topology/MetricSpace/Thickening.lean | 412 | 420 |
/-
Copyright (c) 2023 Felix Weilacher. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Felix Weilacher, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.MeasureTheory.MeasurableSpace.Embedding
import Mathlib.Data.Set.MemPartition
import Mathlib.Order.Filter.CountableSepa... | refine MeasurableSet.union ?_ ?_ <;>
· refine measurableSet_generateFrom ?_
rw [memPartition_succ]
exact ⟨s, hs, by simp⟩
lemma generateFrom_memPartition_le_succ (t : ℕ → Set α) (n : ℕ) :
generateFrom (memPartition t n) ≤ generateFrom (memPartition t (n + 1)) :=
generateFrom_le (fun _ hs ↦ meas... | Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean | 323 | 359 |
/-
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... | · exact hS.epi_f
lemma Exact.isZero_X₂ (hS : S.Exact) (hf : S.f = 0) (hg : S.g = 0) : IsZero S.X₂ := by
have := hS.mono_g hf
rw [IsZero.iff_id_eq_zero, ← cancel_mono S.g, hg, comp_zero, comp_zero]
| Mathlib/Algebra/Homology/ShortComplex/Exact.lean | 453 | 457 |
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
/-!
# One-dimensional derivatives
This ... | theorem Filter.EventuallyEq.hasDerivAt_iff (h : f₀ =ᶠ[𝓝 x] f₁) :
HasDerivAt f₀ f' x ↔ HasDerivAt f₁ f' x :=
⟨fun h' ↦ h'.congr_of_eventuallyEq h.symm, fun h' ↦ h'.congr_of_eventuallyEq h⟩
| Mathlib/Analysis/Calculus/Deriv/Basic.lean | 576 | 578 |
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.Module.Card
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.SetTheory.Cardinal.Continuum
import Mathlib.SetTheory.Cardinal.C... |
/-- In a topological vector space over a nontrivially normed field, any neighborhood of zero has
the same cardinality as the whole space.
See also `cardinal_eq_of_mem_nhds`. -/
lemma cardinal_eq_of_mem_nhds_zero
{E : Type*} (𝕜 : Type*) [NontriviallyNormedField 𝕜] [Zero E] [MulActionWithZero 𝕜 E]
[Topologic... | Mathlib/Topology/Algebra/Module/Cardinality.lean | 56 | 93 |
/-
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... | ring
/-- When the reference set is dense, the embedding map is an isometry on its image. -/
theorem embeddingOfSubset_isometry (H : DenseRange x) : Isometry (embeddingOfSubset x) := by
refine Isometry.of_dist_eq fun a b => ?_
refine (embeddingOfSubset_dist_le x a b).antisymm (le_of_forall_pos_le_add fun e epos =... | Mathlib/Topology/MetricSpace/Kuratowski.lean | 52 | 57 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.MeasureTheory.Integral.Bochner.L1
import Mathlib.Me... | Mathlib/MeasureTheory/Integral/Bochner.lean | 322 | 325 | |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
import Mathlib.LinearAlgebra.Orientation
/-!
# Orientations of real inner product spaces.
Th... |
/-- The volume form on an oriented real inner product space can be evaluated as the determinant with
respect to any orthonormal basis of the space compatible with the orientation. -/
theorem volumeForm_robust (b : OrthonormalBasis (Fin n) ℝ E) (hb : b.toBasis.orientation = o) :
| Mathlib/Analysis/InnerProductSpace/Orientation.lean | 190 | 193 |
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.RingTheory.PowerBasis
/-!
# A predicate on adjoining root... | { AdjoinRoot.isAdjoinRoot f with Monic := hf }
| Mathlib/RingTheory/IsAdjoinRoot.lean | 288 | 289 |
/-
Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.RingTheory.Spect... | rw [inv_eq, inv_eq, le_div_iff_mul_le hI, le_div_iff_mul_le hJ, mul_comm]
lemma FractionalIdeal.inv_le_comm {I J : FractionalIdeal A⁰ K} (hI : I ≠ 0) (hJ : J ≠ 0) :
I⁻¹ ≤ J ↔ J⁻¹ ≤ I := by
simpa using le_inv_comm (A := A) (K := K) (inv_ne_zero hI) (inv_ne_zero hJ)
open FractionalIdeal
| Mathlib/RingTheory/DedekindDomain/Ideal.lean | 741 | 748 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yakov Pechersky
-/
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Infix
import Mathlib.Data.Quot
/-!
# List rotation
This file proves basic results about `List.r... | (a :: l : List α).rotate (n + 1) = (l ++ [a]).rotate n := by
rw [rotate_eq_rotate', rotate_eq_rotate', rotate'_cons_succ]
@[simp]
theorem mem_rotate : ∀ {l : List α} {a : α} {n : ℕ}, a ∈ l.rotate n ↔ a ∈ l
| [], _, n => by simp
| Mathlib/Data/List/Rotate.lean | 103 | 108 |
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Riccardo Brasca
-/
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.RingTheory.Ideal.Quotient.Operations
import Mathl... | ← IsometryEquiv.preimage_symm]
simp
/-- The norm of the projection is smaller or equal to the norm of the original element. -/
@[to_additive "The norm of the projection is smaller or equal to the norm of the original element."]
| Mathlib/Analysis/Normed/Group/Quotient.lean | 161 | 165 |
/-
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... | funext <| snoc_rev a f
theorem insertNth_binop (op : ∀ j, α j → α j → α j) (i : Fin (n + 1)) (x y : α i)
(p q : ∀ j, α (i.succAbove j)) :
| Mathlib/Data/Fin/Tuple/Basic.lean | 922 | 925 |
/-
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 add {β} [AddZeroClass β] {f g : α →ₛ β} (hf : f.FinMeasSupp μ)
(hg : g.FinMeasSupp μ) : (f + g).FinMeasSupp μ := by
| Mathlib/MeasureTheory/Function/SimpleFunc.lean | 1,098 | 1,100 |
/-
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, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Div
import Mathlib.RingTheory... | Mathlib/Algebra/Polynomial/RingDivision.lean | 509 | 513 | |
/-
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.PropInstances
import Mathlib.Order.GaloisConnection.Defs
/-!
# Heyting algebras
This file defines Heyting, co-Heyting and bi-Heyting algebras.
A H... |
@[simp]
| Mathlib/Order/Heyting/Basic.lean | 847 | 848 |
/-
Copyright (c) 2023 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.Analysis.SpecialFunctions.PolarCoord
import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
/-!
# Integrals involving the Gamma function
In this file, we... | theorem integral_rpow_mul_exp_neg_rpow {p q : ℝ} (hp : 0 < p) (hq : -1 < q) :
∫ x in Ioi (0 : ℝ), x ^ q * exp (- x ^ p) = (1 / p) * Gamma ((q + 1) / p) := by
calc
_ = ∫ (x : ℝ) in Ioi 0, (1 / p * x ^ (1 / p - 1)) • ((x ^ (1 / p)) ^ q * exp (-x)) := by
rw [← integral_comp_rpow_Ioi _ (one_div_ne_zero (ne_... | Mathlib/MeasureTheory/Integral/Gamma.lean | 21 | 37 |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Xavier Roblot
-/
import Mathlib.Algebra.Algebra.Hom.Rat
import Mathlib.Analysis.Complex.Polynomial.Basic
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.RingTh... | IsUnramified k w ↔ IsReal w ∨ IsComplex (w.comap (algebraMap k K)) := by
rw [← not_iff_not, not_isUnramified_iff, not_or,
not_isReal_iff_isComplex, not_isComplex_iff_isReal]
| Mathlib/NumberTheory/NumberField/Embeddings.lean | 847 | 850 |
/-
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 | 1,434 | 1,436 | |
/-
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.LinearAlgebra.Matrix.GeneralLinearGroup.Defs
import Mathlib.LinearAlgebra.Matrix.Nondegenerate
import Mathlib... | end Matrix
| Mathlib/LinearAlgebra/Matrix/ToLinearEquiv.lean | 231 | 241 |
/-
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, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.... | Pairwise (Disjoint on fun n : ℤ => Ico (n : α) (n + 1)) := by
simpa only [zero_add] using pairwise_disjoint_Ico_add_intCast (0 : α)
| Mathlib/Algebra/Order/Interval/Set/Group.lean | 226 | 228 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Moritz Doll
-/
import Mathlib.LinearAlgebra.Prod
/-!
# Partially defined linear maps
A `LinearPMap R E F` or `E →ₗ.[R] F` is a linear map from a submodule of `E` to... | (x : g.map (LinearMap.fst R E F)) :
g.toLinearPMap x = valFromGraph hg x.2 := by
| Mathlib/LinearAlgebra/LinearPMap.lean | 916 | 917 |
/-
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... | cases atTop_neBot_iff.mp h
let y := ((⨆ i, U i).topologicalClosure.orthogonalProjection x : E)
have proj_x : ∀ i, (U i).orthogonalProjection x = (U i).orthogonalProjection y := fun i =>
(orthogonalProjection_orthogonalProjection_of_le
((le_iSup U i).trans (iSup U).le_topologicalClosure) _).symm
suff... | Mathlib/Analysis/InnerProductSpace/Projection.lean | 872 | 877 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.Complex.Asymptotics
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Data.Complex.Trig... |
namespace Complex
| Mathlib/Analysis/SpecialFunctions/Exp.lean | 443 | 445 |
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Calculus.SmoothSeries
import Mathlib.Analysis.NormedSpace.OperatorNorm.Prod
import Mathlib.Analysis.SpecialFunctions.Gaussian.PoissonSummation... | refine tsum_congr (fun n ↦ ?_)
simp only [mul_add, Complex.exp_add, mul_one, mul_comm _ (n : ℂ), exp_int_mul_two_pi_mul_I,
mul_one]
lemma jacobiTheta₂'_add_left' (z τ : ℂ) :
jacobiTheta₂' (z + τ) τ =
cexp (-π * I * (τ + 2 * z)) * (jacobiTheta₂' z τ - 2 * π * I * jacobiTheta₂ z τ) := by
rcases le_or... | Mathlib/NumberTheory/ModularForms/JacobiTheta/TwoVariable.lean | 427 | 443 |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.Calculus.ContDiff.Bounds
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysi... | (hf : ∀ x, ‖f x‖ ≤ C₁) (h'f : ∀ x, ‖x‖ ^ (k + μ.integrablePower) * ‖f x‖ ≤ C₂)
(h''f : AEStronglyMeasurable f μ) :
Integrable (fun x ↦ ‖x‖ ^ k * ‖f x‖) μ := by
apply ((integrable_pow_neg_integrablePower μ).const_mul (2 ^ μ.integrablePower * (C₁ + C₂))).mono'
· exact AEStronglyMeasurable.mul (aestronglyM... | Mathlib/Analysis/Distribution/SchwartzSpace.lean | 643 | 647 |
/-
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 | 1,398 | 1,399 | |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.IntermediateField.Basic
imp... |
end IsScalarTower
section
variable [Field E] [Field E'] [Algebra F E] [Algebra F E']
(f : E →ₐ[F] E')
| Mathlib/FieldTheory/Separable.lean | 680 | 686 |
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Yury Kudryashov
-/
import Mathlib.Topology.Instances.NNReal.Lemmas
import Mathlib.Topology.Order.MonotoneContinuity
/-!
# Square root of a real numbe... | lemma sqrt_eq_iff_eq_sq : sqrt x = y ↔ x = y ^ 2 := sqrt.toEquiv.apply_eq_iff_eq_symm_apply
| Mathlib/Data/Real/Sqrt.lean | 59 | 59 |
/-
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.Nilpotent
import Mathlib.Algebra.Lie.Normalizer
/-!
# Cartan subalgebras
Cartan subalgebras are one of the most important concepts in Lie theor... | le_antisymm hk
(LieSubmodule.ucs_le_of_normalizer_eq_self H.normalizer_eq_self_of_isCartanSubalgebra k)
refine ⟨k, fun l hl => ?_⟩
rw [← Nat.sub_add_cancel hl, LieSubmodule.ucs_add, ← hk,
LieSubalgebra.ucs_eq_self_of_isCartanSubalgebra]
· rintro ⟨k, hk⟩
exact
{ nilpotent := by
... | Mathlib/Algebra/Lie/CartanSubalgebra.lean | 72 | 94 |
/-
Copyright (c) 2023 Yaël Dillies, Vladimir Ivanov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Vladimir Ivanov
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.BigOperators.... | rw [← sum_add_distrib, ← sum_add_distrib, sum_congr rfl fun s _ ↦ _]
simp_rw [div_add_div_same, ← Nat.cast_add, card_truncatedSup_union_add_card_truncatedSup_infs]
simp
lemma infSum_union_add_infSum_sups (𝒜 ℬ : Finset (Finset α)) :
infSum (𝒜 ∪ ℬ) + infSum (𝒜 ⊻ ℬ) = infSum 𝒜 + infSum ℬ := by
| Mathlib/Combinatorics/SetFamily/AhlswedeZhang.lean | 345 | 350 |
/-
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.RingTheory.Localization.Integer
import Mathlib.RingTheory.Localization.Submodule
/-!
# Fractional ideals
This file defines fractional... | section Semiring
instance : Add (FractionalIdeal S P) :=
⟨(· ⊔ ·)⟩
@[simp]
theorem sup_eq_add (I J : FractionalIdeal S P) : I ⊔ J = I + J :=
rfl
@[simp, norm_cast]
theorem coe_add (I J : FractionalIdeal S P) : (↑(I + J) : Submodule R P) = I + J :=
| Mathlib/RingTheory/FractionalIdeal/Basic.lean | 452 | 462 |
/-
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.Data.Set.Lattice
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.ModularLattice
import Mathlib.Order.SuccPred.Basic
import Mathlib.Order.WellFou... | simp_rw [exists_and_left, and_assoc, and_congr_right_iff, ← and_assoc, and_comm, exists_and_left,
SetLike.not_le_iff_exists, ne_comm, implies_true]
end SetLike
| Mathlib/Order/Atoms.lean | 215 | 218 |
/-
Copyright (c) 2021 Apurva Nakade. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Apurva Nakade
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.GroupTheory.MonoidLocalization.Away
impo... | Mathlib/SetTheory/Surreal/Dyadic.lean | 265 | 268 | |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.CharP.Frobenius
import Mathlib.Algebra.CharP.Pi
import Mathlib.Algebra.CharP.Quotient
import Mathlib.Algebra.CharP.Subring
import Mathlib.Analysis.Specia... | else 0
variable {K v O p}
open scoped Classical in
theorem coeff_nat_find_add_ne_zero {f : PreTilt O p} {h : ∃ n, coeff _ _ n f ≠ 0} (k : ℕ) :
coeff _ _ (Nat.find h + k) f ≠ 0 :=
coeff_add_ne_zero (Nat.find_spec h) k
theorem valAux_zero : valAux K v O p 0 = 0 :=
dif_neg fun ⟨_, hn⟩ => hn rfl
include hv
t... | Mathlib/RingTheory/Perfection.lean | 472 | 489 |
/-
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
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | _ ≤ ‖x‖ ^ 1 * ((Nat.succ 1 : ℝ) * ((Nat.factorial 1) * (1 : ℕ) : ℝ)⁻¹) :=
(exp_bound hx (by decide))
| Mathlib/Data/Complex/Exponential.lean | 436 | 437 |
/-
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... | homogeneousSubmodule σ R n = Finsupp.supported _ R { d | d.degree = n } := by
simp_rw [degree_eq_weight_one]
exact weightedHomogeneousSubmodule_eq_finsupp_supported R 1 n
| Mathlib/RingTheory/MvPolynomial/Homogeneous.lean | 99 | 102 |
/-
Copyright (c) 2022 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Finset.Pairwise
import Mathlib.Algebra.BigOperators.Group.Finset.Basic
/-!
# Sums of collections of Finsupp, ... | · simp only [foldr, Function.comp_apply, Finset.mem_union, Finsupp.mem_support_iff, ne_eq, IH,
find?, mem_cons, exists_eq_or_imp]
theorem Multiset.mem_sup_map_support_iff [Zero M] {s : Multiset (ι →₀ M)} {x : ι} :
x ∈ (s.map Finsupp.support).sup ↔ ∃ f ∈ s, x ∈ f.support :=
Quot.inductionOn s fun _ ↦ by
... | Mathlib/Data/Finsupp/BigOperators.lean | 60 | 66 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.RingTheory.Ideal.Operations
/-!
# Maps on modules and ideals
Main definitions include `Ideal.map`, `Ideal.comap`, `RingHom.ker`, `Module.annihilator`
and `Subm... | def ker : Ideal R :=
Ideal.comap f ⊥
| Mathlib/RingTheory/Ideal/Maps.lean | 655 | 656 |
/-
Copyright (c) 2023 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Bivariate
import Mathlib.AlgebraicGeometry.EllipticCurve.Weierstrass
import Mathlib.AlgebraicGeometry.EllipticCu... | [IsScalarTower R S K] (f : F →ₐ[S] K) (x₁ x₂ y₁ y₂ : F) :
(W'.baseChange K).toAffine.slope (f x₁) (f x₂) (f y₁) (f y₂) =
f ((W'.baseChange F).toAffine.slope x₁ x₂ y₁ y₂) := by
rw [← RingHom.coe_coe, ← map_slope, map_baseChange]
| Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean | 882 | 886 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.MFDeriv.Basic
/-!
### Relations between vector space derivative and manifold derivative
The manifold deriva... | mdifferentiableWithinAt_iff_differentiableWithinAt]
alias ⟨MDifferentiableOn.differentiableOn, DifferentiableOn.mdifferentiableOn⟩ :=
mdifferentiableOn_iff_differentiableOn
| Mathlib/Geometry/Manifold/MFDeriv/FDeriv.lean | 84 | 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... | finInsepDegree F E = finInsepDegree F K := by
simpa only [Cardinal.toNat_lift] using congr_arg Cardinal.toNat
| Mathlib/FieldTheory/SeparableClosure.lean | 313 | 314 |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.LSeries.AbstractFuncEq
import Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds
import Mathlib.NumberTheory.LSeries.MellinEqDirichlet
i... | Mathlib/NumberTheory/LSeries/HurwitzZetaOdd.lean | 566 | 572 | |
/-
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.Data.Finset.Lattice.Prod
import Mathlib.Data.Fintype.Powerset
import Mathlib.Data.Setoid.Basic
import Mathlib.Order.Atoms
impor... |
lemma exists_enumeration : ∃ f : s ≃ Σ t : P.parts, Fin #t.1,
∀ a b : s, P.part a = P.part b ↔ (f a).1 = (f b).1 := by
use P.equivSigmaParts.trans ((Equiv.refl _).sigmaCongr (fun t ↦ t.1.equivFin))
simp [equivSigmaParts, Equiv.sigmaCongr, Equiv.sigmaCongrLeft]
| Mathlib/Order/Partition/Finpartition.lean | 583 | 588 |
/-
Copyright (c) 2022 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Oleksandr Manzyuk
-/
import Mathlib.CategoryTheory.Bicategory.Basic
import Mathlib.CategoryTheory.Monoidal.Mon_
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equali... | Mathlib/CategoryTheory/Monoidal/Bimod.lean | 1,043 | 1,063 | |
/-
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 | 1,409 | 1,411 | |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Set.Lattice
import Mathlib.Order.ConditionallyCompleteLattice.Defs
/-!
# Theory of conditionally complete lattices
A conditionally complet... | exact Option.some_injective _
| Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | 91 | 92 |
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