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/-
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.Module.Basic
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
/-!
# Midpoint of a segment
## Main definitions
* `midpoint R x y`: midp... | theorem midpoint_sub_add (x y : V) : midpoint R (x - y) (x + y) = x := by
rw [sub_eq_add_neg, ← vadd_eq_add, ← vadd_eq_add, ← midpoint_vadd_midpoint]; simp
@[simp]
| Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | 210 | 213 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.Order.IsLUB
/-!
# Order topology on a densely ordered set
-/
open Set Filter TopologicalSpace Topology Func... | simp [frontier, ha]
theorem frontier_Ici [NoMinOrder α] {a : α} : frontier (Ici a) = {a} :=
frontier_Ici' nonempty_Iio
@[simp]
theorem frontier_Iic' {a : α} (ha : (Ioi a).Nonempty) : frontier (Iic a) = {a} := by
simp [frontier, ha]
| Mathlib/Topology/Order/DenselyOrdered.lean | 136 | 143 |
/-
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... | Mathlib/Algebra/Order/Field/Basic.lean | 76 | 76 | |
/-
Copyright (c) 2020 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Devon Tuma
-/
import Mathlib.Probability.ProbabilityMassFunction.Basic
/-!
# Monad Operations for Probability Mass Functions
This file constructs two operations on `PMF` ... | refine _root_.trans ENNReal.tsum_comm (tsum_congr fun b => tsum_congr fun a => ?_)
split_ifs with h1 h2 h2 <;> ring
section Measure
variable (s : Set β)
@[simp]
theorem toOuterMeasure_bindOnSupport_apply :
(p.bindOnSupport f).toOuterMeasure s =
∑' a, p a * if h : p a = 0 then 0 else (f a h).toOuterMeas... | Mathlib/Probability/ProbabilityMassFunction/Monad.lean | 276 | 292 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Devon Tuma
-/
import Mathlib.Probability.ProbabilityMassFunction.Monad
import Mathlib.Control.ULiftable
/-!
# Specific Constructions of Probability Mass Functions
Thi... | (id_map := id_map)
| Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | 136 | 136 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Finset.Card
import Mathlib.Data.Fintype.Basic
/-!
# Cardinalities of finite types
This file defines the cardinality `Fintype.card α` as the numb... | have h₁ : image f univ = univ :=
eq_of_subset_of_card_le (subset_univ _)
((card_image_of_injective univ hinj).symm ▸ le_rfl)
| Mathlib/Data/Fintype/Card.lean | 305 | 307 |
/-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck
-/
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
/-!
# Congruence subgroups
This defines congruence subgroups of `SL(2, ℤ)` such as `Γ(N)`, `Γ₀(N)` and `Γ₁(N)... | Int.cast_zero, and_self, and_true, true_and]
refine Iff.mpr (ZMod.intCast_zmod_eq_zero_iff_dvd M (Int.natAbs N)) ?_
simp only [Int.natCast_natAbs, abs_dvd, hNM]
| Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean | 73 | 75 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.FixedPoint
/-!
# Cofinality
This file co... | nfpFamily_lt_ord_lift hc (by rwa [(#ι).lift_id]) hf
theorem nfp_lt_ord {f : Ordinal → Ordinal} {c} (hc : ℵ₀ < cof c) (hf : ∀ i < c, f i < c) {a} :
| Mathlib/SetTheory/Cardinal/Cofinality.lean | 303 | 305 |
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Antoine Labelle, Rémi Bottinelli
-/
import Mathlib.Combinatorics.Quiver.Path
import Mathlib.Combinatorics.Quiver.Push
/-!
## Symmetric quivers and arrow reversal
This file co... | /-- Given a quiver `V'` with reversible arrows, a prefunctor to `V'` can be lifted to one from
`Symmetrify V` to `V'` -/
def lift [HasReverse V'] (φ : Prefunctor V V') :
Prefunctor (Symmetrify V) V' where
obj := φ.obj
map
| Mathlib/Combinatorics/Quiver/Symmetric.lean | 158 | 163 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | if_pos hx
| Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 273 | 274 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
/-!
# Intervals as finsets
This file provides basic results about all the `Finset.Ixx... | theorem Ico_diff_Ioo_self (h : a < b) : Ico a b \ Ioo a b = {a} := by simp [← coe_inj, h]
| Mathlib/Order/Interval/Finset/Basic.lean | 604 | 604 |
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Data.Int.Order.Units
import Mathlib.Data.ZMod.IntUnitsPower
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.LinearAlgebra.DirectSum.TensorProduc... | (-1 : ℤˣ)^(j * i) • lof R _ ℬ𝒜 (j, i) (b ⊗ₜ a) := by
rw [gradedCommAux]
dsimp
simp [mul_comm i j]
@[simp]
| Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean | 85 | 90 |
/-
Copyright (c) 2023 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.Combinatorics.SimpleGraph.Triangle.Basic
/-!
# Construct a tripartite graph from its triangles
This file contains the constru... | @[simp] lemma in₂₀_iff : (graph t).Adj (in₂ c) (in₀ a) ↔ ∃ b, (a, b, c) ∈ t :=
⟨by rintro ⟨⟩; exact ⟨_, ‹_›⟩, fun ⟨_, h⟩ ↦ in₂₀ h⟩
| Mathlib/Combinatorics/SimpleGraph/Triangle/Tripartite.lean | 78 | 79 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Complex.AbsMax
import Mathlib.Analysis.Complex.RemovableSingularity
/-!
# Schwarz lemma
In this file we prove several versions of the Schw... | have hR₁ : 0 < R₁ := nonempty_ball.1 ⟨z, hz⟩
have hR₂ : 0 < R₂ := nonempty_ball.1 ⟨f z, h_maps hz⟩
rcases eq_or_ne (dslope f c z) 0 with hc | hc
· rw [hc, norm_zero]; exact div_nonneg hR₂.le hR₁.le
rcases exists_dual_vector ℂ _ hc with ⟨g, hg, hgf⟩
have hg' : ‖g‖₊ = 1 := NNReal.eq hg
have hg₀ : ‖g‖₊ ≠ 0 :... | Mathlib/Analysis/Complex/Schwarz.lean | 92 | 108 |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.Fintype.EquivFin
import Mathlib.Data.Fintype.Inv
/-! # Equivalence between fintypes
This file contains some basic results on equivalences wher... | `{e' : α ≃ α // ∀ x : {x // p x}, e' x = e x}` is known. -/
noncomputable def toCompl {p q : α → Prop} (e : { x // p x } ≃ { x // q x }) :
{ x // ¬p x } ≃ { x // ¬q x } := by
| Mathlib/Logic/Equiv/Fintype.lean | 91 | 93 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Sites.Sieves
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono
/-!
# The sheaf condition for a presieve
We define what it means fo... | -/
noncomputable
def familyOfElements : FamilyOfElements P (ofArrows X π) :=
(exists_familyOfElements hx).choose
| Mathlib/CategoryTheory/Sites/IsSheafFor.lean | 711 | 715 |
/-
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.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive
impo... |
theorem forall_mem_map {f : α → β} {o : Option α} {p : β → Prop} :
(∀ y ∈ o.map f, p y) ↔ ∀ x ∈ o, p (f x) := by simp
| Mathlib/Data/Option/Basic.lean | 53 | 55 |
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Algebra.BigOperators.Group.Multiset.Basic
import Mathlib.Data.PNat.Prime
import Mathlib.Data.Nat.Factors
import Mathlib.Data.Multiset.OrderedMonoid
i... | factorMultiset m ≤ v ↔ m ∣ v.prod := by
let h := @factorMultiset_le_iff m v.prod
| Mathlib/Data/PNat/Factors.lean | 288 | 289 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Set.Piecewise
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Core
import Mathlib.Tactic.Attr.Core
/-!
# Partial equivalences
This f... |
@[simp, mfld_simps]
theorem pi_refl : (PartialEquiv.pi fun i ↦ PartialEquiv.refl (αi i)) = .refl (∀ i, αi i) := by
ext <;> simp
| Mathlib/Logic/Equiv/PartialEquiv.lean | 867 | 871 |
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.Interval.Set.OrderEmbedding
import Mathlib.Order.SetNotation
/-!
# Properties of unbundled ... | Mathlib/Order/UpperLower/Basic.lean | 825 | 827 | |
/-
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.Logic.Function.Defs
import Mathlib.Tactic.TypeStar
/-!
# Nontrivial types
A type is *nontrivial* if it contains at least two elements. This is ... |
-- `x` and `y` are explicit here, as they are often needed to guide typechecking of `h`.
theorem nontrivial_of_ne (x y : α) (h : x ≠ y) : Nontrivial α :=
⟨⟨x, y, h⟩⟩
theorem nontrivial_iff_exists_ne (x : α) : Nontrivial α ↔ ∃ y, y ≠ x :=
| Mathlib/Logic/Nontrivial/Defs.lean | 45 | 50 |
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.Dimension.Constructions
/-!
# Some results on free modules over rings satisfying strong rank condition
T... | · rintro ⟨⟨v₀, hv₀⟩, h⟩
use v₀, hv₀
intro v hv
obtain ⟨r, hr⟩ := h ⟨v, hv⟩
use r
rwa [Subtype.ext_iff, coe_smul] at hr
· rintro ⟨v₀, hv₀, h⟩
use ⟨v₀, hv₀⟩
rintro ⟨v, hv⟩
obtain ⟨r, hr⟩ := h v hv
use r
rwa [Subtype.ext_iff, coe_smul]
/-- A submodule has dimension `1` if and o... | Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean | 124 | 139 |
/-
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... | slice_lhs 2 3 => rw [coequalizer.π_desc]
slice_rhs 1 2 => rw [coequalizer.π_desc]
dsimp [AssociatorBimod.homAux]
refine (cancel_epi ((tensorRight _).map (coequalizer.π _ _))).1 ?_
dsimp
slice_lhs 1 2 => rw [← comp_whiskerRight, coequalizer.π_desc]
slice_rhs 1 3 => rw [π_tensor_id_preserves_coequalizer_inv... | Mathlib/CategoryTheory/Monoidal/Bimod.lean | 906 | 922 |
/-
Copyright (c) 2024 Sophie Morel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sophie Morel
-/
import Mathlib.Analysis.NormedSpace.PiTensorProduct.ProjectiveSeminorm
import Mathlib.LinearAlgebra.Isomorphisms
/-!
# Injective seminorm on the tensor of a finite famil... | simp only [mapL, map_smul, liftIsometry_apply_apply, lift.tprod,
ContinuousMultilinearMap.coe_coe, ContinuousMultilinearMap.compContinuousLinearMap_apply,
tprodL_toFun, map_tprod, ContinuousLinearMap.coe_coe]
| add _ _ hx hy => simp only [map_add, hx, hy]
/-- Given submodules `pᵢ ⊆ Eᵢ`, this is the natur... | Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean | 345 | 350 |
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.Dimension.Constructions
/-!
# Some results on free modules over rings satisfying strong rank condition
T... | simpa using finrank_eq_card_basis b
/-- A module has dimension 1 iff there is some nonzero `v : V` so every vector is a multiple of `v`.
-/
theorem finrank_eq_one_iff' [Module.Free K V] :
finrank K V = 1 ↔ ∃ v ≠ 0, ∀ w : V, ∃ c : K, c • v = w := by
rw [← rank_eq_one_iff]
| Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean | 192 | 198 |
/-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Order.Basic
import Mathlib.Order.Monotone.Defs
/-!
# Covariants and contravariants
This file contains general lemma... |
@[to_additive]
theorem Group.mulLeftReflectLE_of_mulLeftMono [Group N] [LE N]
[MulLeftMono N] : MulLeftReflectLE N :=
inferInstance
| Mathlib/Algebra/Order/Monoid/Unbundled/Defs.lean | 292 | 297 |
/-
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 ... | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | 622 | 626 | |
/-
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.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
/-!
# Oriented angles in right-angled triangles.
T... | Real.Angle.cos (∡ p₃ p₁ p₂) = dist p₁ p₂ / dist p₁ p₃ := by
have hs : (∡ p₃ p₁ p₂).sign = 1 := by rw [← oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two]
rw [oangle_eq_angle_of_sign_eq_one hs, angle_comm, Real.Angle.cos_coe,
cos_angle_of_angle_eq_pi_div_two (angle_rev_eq_pi_div_two_of_oangle_eq_pi_div_... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 584 | 588 |
/-
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... | which we will take the martingale part. -/
noncomputable def process (s : ℕ → Set Ω) (n : ℕ) : Ω → ℝ :=
∑ k ∈ Finset.range n, (s (k + 1)).indicator 1
variable {s : ℕ → Set Ω}
| Mathlib/Probability/Martingale/BorelCantelli.lean | 263 | 267 |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.CategoryTheory.EffectiveEpi.Basic
/-!
# Composition of effective epimorphisms
This file provides `EffectiveEpi` instances for certain composition... | (i : B ⟶ B')
theorem effectiveEpiFamilyStructCompIso_aux
{W : C} (e : (a : α) → X a ⟶ W)
(h : ∀ {Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂),
g₁ ≫ π a₁ ≫ i = g₂ ≫ π a₂ ≫ i → g₁ ≫ e a₁ = g₂ ≫ e a₂)
{Z : C} (a₁ a₂ : α) (g₁ : Z ⟶ X a₁) (g₂ : Z ⟶ X a₂) (hg : g₁ ≫ π a₁ = g₂ ≫ π a₂) :
g₁ ≫ e a... | Mathlib/CategoryTheory/EffectiveEpi/Comp.lean | 104 | 112 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.FinMeasAdditive
/-!
# Extension of a linear function from indicators to L1
Give... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 1,468 | 1,480 | |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Piecewise
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Pi
import Mathlib.Algebra.O... | refine univ.four_functions_theorem h₁ h₂ h₃ h₄ ?_ ?_ ?_ <;> simp [h]
end Finset
section DistribLattice
variable [DistribLattice α] [CommSemiring β] [LinearOrder β] [IsStrictOrderedRing β]
[ExistsAddOfLE β] (f f₁ f₂ f₃ f₄ g μ : α → β)
/-- The **Four Functions Theorem**, aka **Ahlswede-Daykin Inequality**. -/
lemm... | Mathlib/Combinatorics/SetFamily/FourFunctions.lean | 267 | 300 |
/-
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.IsPrimePow
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Algebra.Ring.CharZero
im... | lemma nonempty_properDivisors : n.properDivisors.Nonempty ↔ 1 < n :=
⟨fun ⟨_m, hm⟩ ↦ one_lt_of_mem_properDivisors hm, fun hn ↦
⟨1, one_mem_properDivisors_iff_one_lt.2 hn⟩⟩
| Mathlib/NumberTheory/Divisors.lean | 292 | 295 |
/-
Copyright (c) 2021 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.LinearAlgebra.Ray
import Mathlib.LinearAlgebra.Determinant
/-!
# Orientations of modules
This file defines orientations of modules.
## Main definitions
... |
/-- The value of `Orientation.map` when the index type has cardinality equal to the finite
dimension, in terms of `f.det`. -/
| Mathlib/LinearAlgebra/Orientation.lean | 346 | 348 |
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, María Inés de Frutos-Fernández
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Algebra.DirectSum.Decomposition
import Mathlib.Algeb... | has weighted degree zero if and only if `∀ x : σ, m x = 0`. -/
theorem weightedDegree_eq_zero_iff [CanonicallyOrderedAdd M]
(hw : NonTorsionWeight w) {m : σ →₀ ℕ} :
weight w m = 0 ↔ ∀ x : σ, m x = 0 := by
simp only [weight, Finsupp.linearCombination, LinearMap.toAddMonoidHom_coe, coe_lsum,
LinearMap.coe... | Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean | 529 | 542 |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Regular.Basic
import Mathlib.GroupTheory.MonoidLocalization.Basic
import Mathlib.LinearAlgebra.Matrix.MvPolynomial
import Mathlib.LinearAlgebra.Matri... | prod_update_of_mem (Finset.mem_univ _), sdiff_singleton_eq_erase, one_mul]
· rw [diagonal_apply_ne _ hij]
| Mathlib/LinearAlgebra/Matrix/Adjugate.lean | 311 | 312 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.Defs
impor... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 1,516 | 1,520 | |
/-
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... | deriv (fun x => 1 + ε x) =ᶠ[atTop] deriv ε := by
filter_upwards [eventually_gt_atTop 1] with x hx; rw [deriv_add] <;> aesop
_ =ᶠ[atTop] fun x => -x⁻¹ / (log x ^ 2) := by
filter_upwards [eventually_gt_atTop 1] with x hx
simp [deriv_smoothingFn hx]
| Mathlib/Computability/AkraBazzi/AkraBazzi.lean | 415 | 419 |
/-
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.Order.Group.Unbundled.Int
import Mathlib.Algebra.Ring.Nat
import Mathlib.Data.Int.GCD
/-!
# Congruences modulo a natural number
This file def... | @[simp] lemma add_modEq_left : n + a ≡ a [MOD n] := by rw [ModEq, add_mod_left]
@[simp] lemma add_modEq_right : a + n ≡ a [MOD n] := by rw [ModEq, add_mod_right]
| Mathlib/Data/Nat/ModEq.lean | 199 | 201 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yakov Pechersky
-/
import Mathlib.RingTheory.IsPrimary
import Mathlib.RingTheory.Ideal.Operations
/-!
# Primary ideals
A proper ideal `I` is primary iff `xy ∈ I` implies `x ∈ I... |
theorem isPrime_radical {I : Ideal R} (hi : I.IsPrimary) : IsPrime (radical I) :=
⟨mt radical_eq_top.1 hi.1,
fun {x y} ⟨m, hxy⟩ => by
rw [mul_pow] at hxy; rcases (isPrimary_iff.mp hi).2 hxy with h | h
· exact Or.inl ⟨m, h⟩
· exact Or.inr (mem_radical_of_pow_mem h)⟩
theorem isPrimary_inf {I J : Ideal ... | Mathlib/RingTheory/Ideal/IsPrimary.lean | 42 | 52 |
/-
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 =... | Mathlib/Dynamics/FixedPoints/Basic.lean | 185 | 187 | |
/-
Copyright (c) 2022 Antoine Labelle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle
-/
import Mathlib.RepresentationTheory.FDRep
import Mathlib.LinearAlgebra.Trace
import Mathlib.RepresentationTheory.Invariants
/-!
# Characters of representations
Th... | variable {G : Type u} [Group G] [IsAlgClosed k]
variable [Fintype G] [Invertible (Fintype.card G : k)]
| Mathlib/RepresentationTheory/Character.lean | 99 | 101 |
/-
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.Algebra.Algebra.Operations
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib... |
theorem bot_mul : ⊥ * I = ⊥ := by simp
@[simp]
theorem top_mul : ⊤ * I = I :=
Submodule.top_smul I
| Mathlib/RingTheory/Ideal/Operations.lean | 283 | 289 |
/-
Copyright (c) 2023 Claus Clausen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Claus Clausen, Patrick Massot
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.CDF
import Mathlib.Probability.Distributions.Gamma
/-! # Exponential distributions over ... | /-- A negative exponential function is integrable on intervals in `R≥0` -/
lemma exp_neg_integrableOn_Ioc {b x : ℝ} (hb : 0 < b) :
IntegrableOn (fun x ↦ rexp (-(b * x))) (Ioc 0 x) := by
simp only [neg_mul_eq_neg_mul]
exact (exp_neg_integrableOn_Ioi _ hb).mono_set Ioc_subset_Ioi_self
| Mathlib/Probability/Distributions/Exponential.lean | 121 | 125 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Topology.Connected.LocPathConnected
/-!
# Charted spaces
A smooth manifold is a topological space `M`... | { __ := e.singletonChartedSpace h
compatible := by
intro e' e'' he' he''
rw [e.singletonChartedSpace_mem_atlas_eq h e' he',
e.singletonChartedSpace_mem_atlas_eq h e'' he'']
refine G.mem_of_eqOnSource ?_ e.symm_trans_self
have hle : idRestrGroupoid ≤ G := (closedUnderRestriction_iff... | Mathlib/Geometry/Manifold/ChartedSpace.lean | 1,237 | 1,246 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.Order.Hom.Monoid
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Tac... | `Valuation R Γ₀ ≃ Valuation R Γ'₀`. -/
def congr (f : Γ₀ ≃*o Γ'₀) : Valuation R Γ₀ ≃ Valuation R Γ'₀ where
toFun := map f f.toOrderIso.monotone
invFun := map f.symm f.toOrderIso.symm.monotone
| Mathlib/RingTheory/Valuation/Basic.lean | 331 | 334 |
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Limits.IsLimit
import Mathlib.CategoryTheory.Category.ULift
import Mathlib.CategoryTheory.Ess... | · dsimp
| Mathlib/CategoryTheory/Limits/HasLimits.lean | 1,126 | 1,126 |
/-
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... | theorem fib_lt_fib_succ {n : ℕ} (hn : 2 ≤ n) : fib n < fib (n + 1) := by
rcases exists_add_of_le hn with ⟨n, rfl⟩
rw [← tsub_pos_iff_lt, add_comm 2, add_right_comm, fib_add_two, add_tsub_cancel_right, fib_pos]
exact succ_pos n
| Mathlib/Data/Nat/Fib/Basic.lean | 101 | 104 |
/-
Copyright (c) 2023 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Heather Macbeth
-/
import Mathlib.MeasureTheory.Constructions.Pi
/-!
# Marginals of multivariate functions
In this file, we define a convenient way to compute int... | (hst : Disjoint s t) : ∫⋯∫⁻_s ∪ t, f ∂μ = ∫⋯∫⁻_t, ∫⋯∫⁻_s, f ∂μ ∂μ := by
rw [Finset.union_comm, lmarginal_union μ f hf hst.symm]
variable {μ}
| Mathlib/MeasureTheory/Integral/Marginal.lean | 156 | 160 |
/-
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 | 1,243 | 1,244 | |
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
/-!
# Some results on the rank... |
theorem rank_sup_eq_rank_left_mul_rank_of_free :
Module.rank R ↥(A ⊔ B) = Module.rank R A * Module.rank A (Algebra.adjoin A (B : Set S)) := by
rcases subsingleton_or_nontrivial R with _ | _
· haveI := Module.subsingleton R S; simp
nontriviality S using rank_subsingleton'
letI : Algebra A (Algebra.adjoin A ... | Mathlib/Algebra/Algebra/Subalgebra/Rank.lean | 30 | 41 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.Multilinear.Curry
/-!
# Formal multilinear series
In this file we define `FormalMultilinearSeries 𝕜 E F` to be a family o... | rfl
/-- Convenience congruence lemma stating in a dependent setting that, if the arguments to a formal
multilinear series are equal, then the values are also equal. -/
| Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean | 116 | 119 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Group.Action.Opposite
import Mathlib.Algebra.Group.Action.Units
import Mathlib.Algebra.Group.Invertible.Defs
import Mathlib.Algebra.GroupWithZero.U... | instance [NonUnitalSemiring R] [StarRing R] : StarRing Rᵐᵒᵖ where
star_add x y := unop_injective (star_add x.unop y.unop)
instance {M : Type*} [Star R] [Star M] [SMul R M] [StarModule R M] :
StarModule R Mᵐᵒᵖ where
star_smul r x := unop_injective (star_smul r x.unop)
| Mathlib/Algebra/Star/Basic.lean | 550 | 555 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Eric Wieser
-/
import Mathlib.Algebra.Quaternion
import Mathlib.Analysis.InnerProductSpace.Continuous
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.To... | Mathlib/Analysis/Quaternion.lean | 251 | 254 | |
/-
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... | @[to_additive (attr := simp) abs_norm]
theorem abs_norm' (z : E) : |‖z‖| = ‖z‖ := abs_of_nonneg <| norm_nonneg' _
| Mathlib/Analysis/Normed/Group/Basic.lean | 470 | 472 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.AdaptationNote
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
/-!
# Slop... | simp only [Pi.neg_apply, ← neg_div, neg_sub', neg_neg] at this
exact this
/-- If `f : 𝕜 → 𝕜` is strictly convex, then for any three points `x < y < z` the slope of the
secant line of `f` on `[x, y]` is strictly less than the slope of the secant line of `f` on
| Mathlib/Analysis/Convex/Slope.lean | 50 | 54 |
/-
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.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
/-!
# Oriented angles in right-angled triangles.
T... | have hs : (o.oangle x (x + y)).sign = 1 := by
rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two]
rw [o.oangle_eq_angle_of_sign_eq_one hs,
InnerProductGeometry.angle_add_eq_arccos_of_inner_eq_zero
(o.inner_eq_zero_of_oangle_eq_pi_div_two h)]
/-- An angle in a right-angled triangle expressed... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 36 | 42 |
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
/-!
# Symmetric difference and bi-implication
This file defines... | ⟨bihimp_comm⟩
| Mathlib/Order/SymmDiff.lean | 209 | 210 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Finite.Defs
import Mathlib.Data.Finset.BooleanAlgebra
import Mathlib.Data.Finset.Image
import Mathlib.Data.Fintype.Defs
import Mathlib.Data.Fintyp... | Mathlib/Data/Fintype/Basic.lean | 621 | 622 | |
/-
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... | apply separating_of_generateFrom (range (natGeneratingSequence _))
rintro - ⟨n, rfl⟩
rw [← decide_eq_decide]
exact congr_fun hxy n
/-- If a measurable space is countably generated and separates points, it is measure equivalent
to some subset of the Cantor space `ℕ → Bool` (equipped with the product sigma algeb... | Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean | 272 | 284 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kevin Buzzard
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.PowerSeries.WellKnown
/-!
# Bernoull... | mul_zero, and_false, sub_self, ← RingHom.map_mul, ← map_sum]
cases n with | zero => simp | succ n =>
rw [if_neg n.succ_succ_ne_one]
have hfact : ∀ m, (m ! : ℚ) ≠ 0 := fun m => mod_cast factorial_ne_zero m
have hite2 : ite (n.succ = 0) 1 0 = (0 : ℚ) := if_neg n.succ_ne_zero
simp only [CharP.cast_eq_zero, z... | Mathlib/NumberTheory/Bernoulli.lean | 248 | 268 |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Chris Hughes, Daniel Weber
-/
import Batteries.Data.Nat.Gcd
import Mathlib.Algebra.GroupWithZero.Associated
import Mathlib.Algebra.Ring.Divisibility.Basic
import Math... | multiplicity a b.natAbs = multiplicity (a : ℤ) b :=
multiplicity_eq_of_emultiplicity_eq (Int.emultiplicity_natAbs a b)
| Mathlib/RingTheory/Multiplicity.lean | 657 | 658 |
/-
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... | rfl
lemma some_ne_zero {x y : R} (h : W'.Nonsingular x y) : Point.some h ≠ 0 := by
rintro (_ | _)
/-- The negation of a nonsingular point on a Weierstrass curve in affine coordinates.
Given a nonsingular point `P` in affine coordinates, use `-P` instead of `neg P`. -/
| Mathlib/AlgebraicGeometry/EllipticCurve/Affine.lean | 636 | 643 |
/-
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.Algebra.ModEq
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Algebra.Ring.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.Order.Cir... | theorem toIcoMod_sub_eq_sub (a b c : α) : toIcoMod hp a (b - c) = toIcoMod hp (a + c) b - c := by
simp_rw [toIcoMod, toIcoDiv_sub_eq_toIcoDiv_add, sub_right_comm]
| Mathlib/Algebra/Order/ToIntervalMod.lean | 442 | 443 |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.NumberTheory.Padics.PadicVal.Basic
/-!
# p-adic norm
This file defines the `p`-adic norm on `ℚ`.
The `p`-adic valuation on `ℚ` is the difference o... |
/-- The `p`-adic norm of `q` is `1` if `q` is prime and not equal to `p`. -/
| Mathlib/NumberTheory/Padics/PadicNorm.lean | 81 | 82 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Data.Set.Restrict
/-!
# Functions over sets
This file contains... | theorem preimage_invFun_of_mem [n : Nonempty α] {f : α → β} (hf : Injective f) {s : Set α}
(h : Classical.choice n ∈ s) : invFun f ⁻¹' s = f '' s ∪ (range f)ᶜ := by
| Mathlib/Data/Set/Function.lean | 1,042 | 1,043 |
/-
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.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Polynomial.Basic
/... | ring
theorem map_expand_pow_char (f : R[X]) (n : ℕ) :
map (frobenius R p ^ n) (expand R (p ^ n) f) = f ^ p ^ n := by
induction n with
| zero => simp [RingHom.one_def]
| Mathlib/Algebra/Polynomial/Expand.lean | 263 | 268 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Field.IsField
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.Algebra.Polynomial.Monic
im... | have : ¬degree q ≤ degree p := not_le_of_gt h
unfold divByMonic divModByMonicAux; dsimp; rw [dif_pos hq, if_neg (mt And.left this)]⟩
theorem degree_add_divByMonic (hq : Monic q) (h : degree q ≤ degree p) :
degree q + degree (p /ₘ q) = degree p := by
nontriviality R
have hdiv0 : p /ₘ q ≠ 0 := by rwa [Ne... | Mathlib/Algebra/Polynomial/Div.lean | 266 | 273 |
/-
Copyright (c) 2021 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson, Yaël Dillies
-/
import Mathlib.Data.Finset.Lattice.Fold
import Mathlib.Data.Finset.Order
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Data.Set.Finite.Range
import Ma... | Fintype.toCompleteAtomicBooleanAlgebra _
/-! ### Directed Orders -/
| Mathlib/Data/Fintype/Order.lean | 210 | 214 |
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll, Yaël Dillies
-/
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Comp
/-!
### Invariance of the derivative under translation
We show th... | variable {𝕜 F : Type*} [NontriviallyNormedField 𝕜] [NormedAddCommGroup F] [NormedSpace 𝕜 F]
{f : 𝕜 → F} {f' : F}
/-- Translation in the domain does not change the derivative. -/
lemma HasDerivAt.comp_const_add (a x : 𝕜) (hf : HasDerivAt f f' (a + x)) :
HasDerivAt (fun x ↦ f (a + x)) f' x := by
| Mathlib/Analysis/Calculus/Deriv/Shift.lean | 16 | 21 |
/-
Copyright (c) 2020 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Patrick Massot, Eric Wieser
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Sum.Basic
import Mathlib.Logic.Unique
import Mathlib.T... | Mathlib/Algebra/Group/Pi/Basic.lean | 395 | 397 | |
/-
Copyright (c) 2022 Pierre-Alexandre Bazin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pierre-Alexandre Bazin
-/
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.ZMod
import Mathlib.GroupTheory.Torsion
import Mathlib.LinearAlgebra.Isomorphism... |
end Submodule
namespace Module
variable [Ring R] [AddCommGroup M] [Module R M]
variable {I : Ideal R} {r : R}
/-- can't be an instance because `hM` can't be inferred -/
def IsTorsionBySet.hasSMul (hM : IsTorsionBySet R M I) : SMul (R ⧸ I) M where
smul b := QuotientAddGroup.lift I.toAddSubgroup (smulAddHom R M)
| Mathlib/Algebra/Module/Torsion.lean | 487 | 497 |
/-
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... | Mathlib/NumberTheory/ArithmeticFunction.lean | 1,349 | 1,358 | |
/-
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 ... | from its edge set, if present.
See also: `SimpleGraph.deleteEdges`. -/
def deleteEdges (G' : G.Subgraph) (s : Set (Sym2 V)) : G.Subgraph where
verts := G'.verts
Adj := G'.Adj \ Sym2.ToRel s
adj_sub h' := G'.adj_sub h'.1
edge_vert h' := G'.edge_vert h'.1
| Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 1,028 | 1,035 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Algebra.Basic
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Eval.Algebra
impo... | ascPochhammer S n * (ascPochhammer S m).comp (X + (n : S[X])) = ascPochhammer S (n + m) := by
induction' m with m ih
· simp
· rw [ascPochhammer_succ_right, Polynomial.mul_X_add_natCast_comp, ← mul_assoc, ih,
← add_assoc, ascPochhammer_succ_right, Nat.cast_add, add_assoc]
theorem ascPochhammer_nat_eq_as... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 143 | 152 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 1,108 | 1,109 | |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Logic.Equiv.PartialEquiv
import Mathlib.Topology.Homeomorph.Lemmas
import Mathlib.Topology.Sets.Opens
/-!
# Partial homeomorphisms
This file de... | theorem isOpen_inter_preimage_symm {s : Set X} (hs : IsOpen s) : IsOpen (e.target ∩ e.symm ⁻¹' s) :=
e.symm.continuousOn.isOpen_inter_preimage e.open_target hs
| Mathlib/Topology/PartialHomeomorph.lean | 390 | 391 |
/-
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, Wen Yang
-/
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mat... | -- as the LHS simplifies already to `M.toSquareBlock id i ⟨i, ⋯⟩ ⟨i, ⋯⟩`
theorem det_toSquareBlock_id (M : Matrix m m R) (i : m) : (M.toSquareBlock id i).det = M i i :=
letI : Unique { a // id a = i } := ⟨⟨⟨i, rfl⟩⟩, fun j => Subtype.ext j.property⟩
(det_unique _).trans rfl
theorem det_toBlock (M : Matrix m m R) (... | Mathlib/LinearAlgebra/Matrix/Block.lean | 201 | 208 |
/-
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... | simp only [IsSeparable, minpoly.algEquiv_eq e x]
| Mathlib/FieldTheory/Separable.lean | 598 | 599 |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.Group.Support
import Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set
/-!
# Support of a function composed with a scalar action
We sh... | simp only [mem_smul_set_iff_inv_smul_mem, mem_support]
attribute [to_additive existing support_comp_inv_smul] mulSupport_comp_inv_smul
| Mathlib/Data/Set/Pointwise/Support.lean | 34 | 37 |
/-
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... | PreservesPullback.iso_hom_fst Scheme.forgetToTop g f, TopCat.coe_comp, Set.range_comp,
Set.range_eq_univ.mpr, ← @Set.preimage_univ _ _ (pullback.snd g.base f.base)]
-- Porting note (https://github.com/leanprover-community/mathlib4/issues/11224): was `rw`
· erw [TopCat.pullback_fst_image_snd_preimage]
rw... | Mathlib/AlgebraicGeometry/OpenImmersion.lean | 532 | 539 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Patrick Massot
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
/-!
# Sets in product and pi types
This file proves basic properties of prod... | image_subset (fun f : ∀ i, α i => f i) h
theorem univ_pi_subset_univ_pi_iff :
pi univ t₁ ⊆ pi univ t₂ ↔ (∀ i, t₁ i ⊆ t₂ i) ∨ ∃ i, t₁ i = ∅ := by simp [pi_subset_pi_iff]
theorem eval_preimage [DecidableEq ι] {s : Set (α i)} :
eval i ⁻¹' s = pi univ (update (fun _ => univ) i s) := by
ext x
simp [@forall... | Mathlib/Data/Set/Prod.lean | 848 | 865 |
/-
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... | obtain ⟨v, hv, -, f', hvf', hf'⟩ :=
(hf.of_le <| (add_le_add_right hkm 1).trans hmn).hasFDerivWithinAt_nhds (by simp)
(hg.of_le hkm) hgt
refine hf'.congr_of_eventuallyEq_insert ?_
filter_upwards [hv, ht]
exact fun y hy h2y => (hvf' y hy).fderivWithin h2y
match m with
| ω =>
obtain ... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 1,083 | 1,092 |
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Analysis.Complex.Convex
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.Analysis.Calculus.Deriv.Shift
/-!
# Estimates for the complex ... | lemma hasDerivAt_log_sub_logTaylor (n : ℕ) {z : ℂ} (hz : 1 + z ∈ slitPlane) :
HasDerivAt (fun z : ℂ ↦ log (1 + z) - logTaylor (n + 1) z) ((-z) ^ n * (1 + z)⁻¹) z := by
convert ((hasDerivAt_log hz).comp_const_add 1 z).sub (hasDerivAt_logTaylor n z) using 1
have hz' : -z ≠ 1 := by
intro H
rw [neg_eq_iff_e... | Mathlib/Analysis/SpecialFunctions/Complex/LogBounds.lean | 97 | 106 |
/-
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.Fintype.Lattice
import Mathlib.Data.Fintype.Sum
import Mathlib.Topology.Homeomorph.Lemmas
import Mathlib.Topology.MetricSpace.Antilipschitz
... | simp only [edist_pi_def, (hf _).edist_eq, Pi.map_apply]
/-- The composition of isometries is an isometry. -/
| Mathlib/Topology/MetricSpace/Isometry.lean | 97 | 99 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.SuccPred.Initial... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,505 | 1,510 | |
/-
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.MeasureTheory.Integral.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Integral.Le... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 2,104 | 2,120 | |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Neil Strickland
-/
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Data.PNat.Basic
/-!
# Primality and GCD on pnat
This file extends the theory of `ℕ+` with ... |
theorem Coprime.gcd_mul (k : ℕ+) {m n : ℕ+} (h : m.Coprime n) :
k.gcd (m * n) = k.gcd m * k.gcd n := by
| Mathlib/Data/PNat/Prime.lean | 251 | 253 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.ZeroCons
/-!
# Basic results on multisets
-/
-- No algebra should be required
assert_not_exists Monoid
universe v
open List S... | Mathlib/Data/Multiset/Basic.lean | 2,676 | 2,677 | |
/-
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.Data.Nat.Choose.Dvd
import Mathlib.RingTheory.IntegralClosure.IntegrallyClosed
import Mathlib.RingTheory.Norm.Basic
import Mathlib.RingTheory.Polynom... | rw [aeval_eq_sum_range, sum_range] at hz
conv_lhs at hz =>
congr
next => skip
ext i
rw [hf i (mem_range.2 (Fin.is_lt i)), ← smul_smul]
rw [← smul_sum] at hz
rw [← smul_right_injective _ hp hz]
exact
Subalgebra.sum_mem _ fun _ _ =>
Subalgebra.smul_mem _ (Subalgebra.pow_mem _ (subset_a... | Mathlib/RingTheory/Polynomial/Eisenstein/IsIntegral.lean | 215 | 230 |
/-
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... | rcases lt_or_le m (l.drop (n % l.length)).length with hm | hm
· rw [getElem?_append_left hm, getElem?_drop, ← add_mod_mod]
rw [length_drop, Nat.lt_sub_iff_add_lt] at hm
rw [mod_eq_of_lt hm, Nat.add_comm]
· have hlt : n % length l < length l := mod_lt _ (m.zero_le.trans_lt hml)
rw [getElem?_append_righ... | Mathlib/Data/List/Rotate.lean | 192 | 197 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Logic.Equiv.PartialEquiv
import Mathlib.Topology.Homeomorph.Lemmas
import Mathlib.Topology.Sets.Opens
/-!
# Partial homeomorphisms
This file de... | open_source := hf.isOpenMap _ e.open_source
open_target := e.open_target
continuousOn_toFun := by
| Mathlib/Topology/PartialHomeomorph.lean | 1,384 | 1,386 |
/-
Copyright (c) 2019 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Module.Submodule.Equiv
import Mathlib.Algebra.Module.Equiv.Basic
import Mathlib.Algebra.Module.Rat
im... | theorem self_trans_symm (e : M ≃ₗ⁅R,L⁆ N) : e.trans e.symm = refl :=
ext _ _ e.symm_apply_apply
@[simp]
theorem symm_trans_self (e : M ≃ₗ⁅R,L⁆ N) : e.symm.trans e = refl :=
ext _ _ e.apply_symm_apply
| Mathlib/Algebra/Lie/Basic.lean | 1,039 | 1,044 |
/-
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... |
variable {e : M → H} (h : IsOpenEmbedding e) {n : WithTop ℕ∞}
/-- If the `ChartedSpace` structure on a manifold `M` is given by an open embedding `e : M → H`,
then `e` is `C^n`. -/
| Mathlib/Geometry/Manifold/ContMDiff/Basic.lean | 344 | 348 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.FieldTheory.Finiteness
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
/-!
# Finite-dimensional subsp... | rw [← finrank_top, ← direction_top k V P] at hc
exact hi.affineSpan_eq_of_le_of_card_eq_finrank_add_one le_top hc
theorem Affine.Simplex.span_eq_top [FiniteDimensional k V] {n : ℕ} (T : Affine.Simplex k V n)
(hrank : finrank k V = n) : affineSpan k (Set.range T.points) = ⊤ := by
rw [AffineIndependent.aff... | Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | 325 | 337 |
/-
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.Basic
import Mathlib.RingTheory.Artinian.Module
/-!
# Lie subalgebras
This file defines Lie subalgebras of a Lie algebra and provides basic rel... | /-- An induction principle for span membership. If `p` holds for 0 and all elements of `s`, and is
preserved under addition, scalar multiplication and the Lie bracket, then `p` holds for all
elements of the Lie algebra spanned by `s`. -/
@[elab_as_elim]
theorem lieSpan_induction {p : (x : L) → x ∈ lieSpan R L s → Prop}... | Mathlib/Algebra/Lie/Subalgebra.lean | 677 | 681 |
/-
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... |
theorem subset_union_of_subset_right {s u : Set α} (h : s ⊆ u) (t : Set α) : s ⊆ t ∪ u :=
| Mathlib/Data/Set/Basic.lean | 693 | 694 |
/-
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.NumberTheory.ModularForms.JacobiTheta.TwoVariable
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
/-! # Jacobi's theta function
This file define... | simp_rw [UpperHalfPlane.modular_S_smul, jacobiTheta_eq_jacobiTheta₂, ← ofReal_zero]
norm_cast
simp_rw [jacobiTheta₂_functional_equation 0 τ, zero_pow two_ne_zero, mul_zero, zero_div,
Complex.exp_zero, mul_one, ← mul_assoc, mul_one_div, div_self h1, one_mul,
UpperHalfPlane.coe_mk, inv_neg, neg_div, one_div... | Mathlib/NumberTheory/ModularForms/JacobiTheta/OneVariable.lean | 45 | 54 |
/-
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.Algebra.Constructions
import Mathlib.Topology.Bases
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Topology.UniformSpac... | /-- A union of a finite family of sets is totally bounded
if and only if each set of the family is totally bounded. -/
lemma totallyBounded_sUnion {S : Set (Set α)} (hS : S.Finite) :
TotallyBounded (⋃₀ S) ↔ ∀ s ∈ S, TotallyBounded s := by
rw [sUnion_eq_biUnion, totallyBounded_biUnion hS]
/-- A finite set is tota... | Mathlib/Topology/UniformSpace/Cauchy.lean | 516 | 530 |
/-
Copyright (c) 2023 Luke Mantle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Mantle
-/
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Factorial.DoubleFactorial
/-!
# Hermite polynomials
This file defines `Polynomial.hermite n`, the `n`... | ∀ n k : ℕ,
hermite_explicit (n + 1) (k + 1) =
hermite_explicit (n + 1) k - (k + 2) * hermite_explicit n (k + 2) := by
intro n k
simp only [hermite_explicit]
-- Factor out (-1)'s.
rw [mul_comm (↑k + _ : ℤ), sub_eq_add_neg]
nth_rw 3 [neg_eq_neg_one_mul]
simp onl... | Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | 153 | 195 |
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