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) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.RingTheory.LocalProperties.Reduced
/-!
# Basic properties of schemes
We provide some basic properties of sche... | obtain ⟨y, e⟩ := X.affineCover.covers x
convert h₂ (X.affineCover.map (X.affineCover.f x)) y _
· rw [e]
apply h₁
theorem eq_zero_of_basicOpen_eq_bot {X : Scheme} [hX : IsReduced X] {U : X.Opens}
(s : Γ(X, U)) (hs : X.basicOpen s = ⊥) : s = 0 := by
apply TopCat.Presheaf.section_ext X.sheaf U
intro x hx
| Mathlib/AlgebraicGeometry/Properties.lean | 143 | 151 |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Logic.IsEmpty
import Mathlib.Tactic.Inhabit
/-!
# Types with a unique term
In this file we define a typeclass `Unique`,
which expresses that a type h... | theorem Pi.default_apply {β : α → Sort v} [∀ a, Inhabited (β a)] (a : α) :
@default (∀ a, β a) _ a = default :=
rfl
| Mathlib/Logic/Unique.lean | 161 | 164 |
/-
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.GroupWithZero.Hom
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.Algebra.Ring.Defs
import Mathlib.Data.Nat.Lattice
/-!
# Definition... | nilpotencyClass x = 1 ↔ x = 0 :=
⟨eq_zero_of_nilpotencyClass_eq_one, fun hx ↦ hx ▸ nilpotencyClass_zero⟩
end MonoidWithZero
| Mathlib/RingTheory/Nilpotent/Defs.lean | 158 | 161 |
/-
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... | filter_upwards [eventually_ge_atTop 1] with _ hx
exact log_nonneg hx
| inr hp => -- p ≠ 0
refine GrowsPolynomially.of_isTheta (growsPolynomially_rpow (p-1))
(isTheta_deriv_rpow_p_mul_one_sub_smoothingFn hp) ?_
filter_upwards [eventually_gt_atTop 0] with _ _
positivity
lemma growsPolynomiall... | Mathlib/Computability/AkraBazzi/AkraBazzi.lean | 908 | 928 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Primrec
import Mathlib.Data.Nat.PSub
import Mathlib.Data.PFun
/-!
# The partial recursive functions
The partial recursive functions are... | theorem rfindOpt_dom {α} {f : ℕ → Option α} : (rfindOpt f).Dom ↔ ∃ n a, a ∈ f n :=
⟨fun h => (rfindOpt_spec ⟨h, rfl⟩).imp fun _ h => ⟨_, h⟩, fun h => by
have h' : ∃ n, (f n).isSome := h.imp fun n => Option.isSome_iff_exists.2
have s := Nat.find_spec h'
have fd : (rfind fun n => (f n).isSome).Dom :=
... | Mathlib/Computability/Partrec.lean | 133 | 141 |
/-
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.Measure.Comap
import Mathlib.MeasureTheory.Measure.QuasiMeasurePreserving
/-!
# Restricting a measure to a subset or a s... | alias ⟨_, ext_of_iUnion_eq_univ⟩ := ext_iff_of_iUnion_eq_univ
/-- Two measures are equal if they have equal restrictions on a spanning collection of sets
(formulated using `biUnion`). -/
theorem ext_iff_of_biUnion_eq_univ {S : Set ι} {s : ι → Set α} (hc : S.Countable)
| Mathlib/MeasureTheory/Measure/Restrict.lean | 404 | 408 |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.SmoothSeries
import Mathlib.Analysis.Calculus.BumpFunction.InnerProduct
import Mathlib.Analysis.Convolution
import Mathlib.Anal... | have C : D ^ finrank ℝ E ≠ 0 := by
norm_cast
exact pow_ne_zero _ Dpos.ne'
| Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.lean | 306 | 308 |
/-
Copyright (c) 2020 Zhangir Azerbayev. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Zhangir Azerbayev
-/
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.LinearAlgebra.LinearIndependent.Defs
import Mathlib.LinearAlgebra.Multilinear.Basis
/-!
# Alte... | rfl
theorem coeFn_smul (c : S) (f : M [⋀^ι]→ₗ[R] N) : ⇑(c • f) = c • ⇑f :=
rfl
| Mathlib/LinearAlgebra/Alternating/Basic.lean | 222 | 226 |
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Integral.Prod
import Mathlib.Meas... | end NoMeasurability
section Measurability
variable [MeasurableSpace G] {μ ν : Measure G}
/-- The convolution of `f` and `g` exists at `x` when the function `t ↦ L (f t) (g (x - t))` is
| Mathlib/Analysis/Convolution.lean | 150 | 155 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Primrec
import Mathlib.Data.Nat.PSub
import Mathlib.Data.PFun
/-!
# The partial recursive functions
The partial recursive functions are... |
theorem vector_ofFn {n} {f : Fin n → α → σ} (hf : ∀ i, Computable (f i)) :
Computable fun a => List.Vector.ofFn fun i => f i a :=
(Partrec.vector_mOfFn hf).of_eq fun a => by simp
| Mathlib/Computability/Partrec.lean | 664 | 667 |
/-
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.MeasureTheory.Function.LocallyIntegrable
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Integral.Prod
import Mathlib.Me... | congr! 4 with K _KU K_comp
exact measure_isMulLeftInvariant_eq_smul_of_ne_top μ' μ K_comp.measure_lt_top.ne
K_comp.measure_lt_top.ne
ext s _hs
rw [s.measure_eq_iInf_isOpen, s.measure_eq_iInf_isOpen]
congr! 4 with U _sU U_open
exact A U U_open
/-- **Uniqueness of left-invariant measures**:
Two Haa... | Mathlib/MeasureTheory/Measure/Haar/Unique.lean | 867 | 884 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.NAry
import Mathlib.Data.Finset.Slice
import Mathlib.Data.Set.Sups
/-!
# Set family operations
This file defines a few binary operations on `... | Mathlib/Data/Finset/Sups.lean | 747 | 748 | |
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Algebra.BigOperators.Expect
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Convex.Jensen
import M... | theorem Lp_add_le_tsum {f g : ι → ℝ≥0} {p : ℝ} (hp : 1 ≤ p) (hf : Summable fun i => f i ^ p)
(hg : Summable fun i => g i ^ p) :
(Summable fun i => (f i + g i) ^ p) ∧
(∑' i, (f i + g i) ^ p) ^ (1 / p) ≤
(∑' i, f i ^ p) ^ (1 / p) + (∑' i, g i ^ p) ^ (1 / p) := by
have pos : 0 < p := lt_of_lt_of_le... | Mathlib/Analysis/MeanInequalities.lean | 646 | 654 |
/-
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
/-!
# Teichmüller lifts
This file defines `WittVector.teichmuller`, a monoid hom `R →* 𝕎 R`, which embeds `r : R` as the... | rw [RingHom.map_mul]
ext1 n
simp only [Pi.mul_apply, ghostMap_apply, ghostComponent_teichmullerFun, mul_pow]
private theorem teichmuller_mul_aux₂ {R : Type*} (x y : MvPolynomial R ℤ) :
teichmullerFun p (x * y) = teichmullerFun p x * teichmullerFun p y := by
| Mathlib/RingTheory/WittVector/Teichmuller.lean | 76 | 81 |
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Eric Wieser
-/
import Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal
/-!
This file contains a proof that the radical of any homogeneous ideal is a homogeneous ideal
... | (homogeneous_mem_or_mem :
∀ {x y : A}, IsHomogeneousElem 𝒜 x → IsHomogeneousElem 𝒜 y → x * y ∈ I → x ∈ I ∨ y ∈ I) :
Ideal.IsPrime I :=
⟨I_ne_top, by
intro x y hxy
by_contra! rid
obtain ⟨rid₁, rid₂⟩ := rid
classical
/-
The idea of the proof is the following :
since... | Mathlib/RingTheory/GradedAlgebra/Radical.lean | 47 | 136 |
/-
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 | 1,966 | 1,974 | |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.RingTheory.PowerSeries.Basic
/-!
# Formal power se... | rw [mem_range] at h
congr
rw [coeff_trunc, if_pos h]
@[simp] theorem trunc_X (n) : trunc (n + 2) X = (Polynomial.X : R[X]) := by
ext d
rw [coeff_trunc, coeff_X]
split_ifs with h₁ h₂
· rw [h₂, coeff_X_one]
· rw [coeff_X_of_ne_one h₂]
· rw [coeff_X_of_ne_one]
intro hd
apply h₁
| Mathlib/RingTheory/PowerSeries/Trunc.lean | 108 | 120 |
/-
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.Interval.Finset.Basic
/-!
# Intervals as multisets
This file defines intervals as multisets.
## Main declarations
In a `LocallyFiniteOrder`,
* `M... | @[simp]
theorem Ioo_eq_zero_of_le (h : b ≤ a) : Ioo a b = 0 :=
| Mathlib/Order/Interval/Multiset.lean | 153 | 154 |
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.Group.Action.Pi
import Mathlib.Data.Finset.Prod
import Mathlib.Data.SetLike.Basic
import Mathlib.Data.Sym.Basic
import Mathlib.Data.Sym.Sym2.Init
/-... | lift f s(a₁, a₂) = (f : α → α → β) a₁ a₂ :=
rfl
@[simp]
| Mathlib/Data/Sym/Sym2.lean | 192 | 195 |
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.GroupWithZero.Subgroup
import Mathlib.Data.Finite.Card
import Mathlib.Data.Finite.Pr... |
@[to_additive]
theorem relindex_dvd_card : H.relindex K ∣ Nat.card K :=
(H.subgroupOf K).index_dvd_card
| Mathlib/GroupTheory/Index.lean | 282 | 285 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Edward Ayers
-/
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.HasPullback
import Mathlib.Data.Set.BooleanAlgebra
/-!
# Theory of sieves
- For an object `X` of a ca... | constructor
· intro hle X f hf
apply hle
refine ⟨X, f, 𝟙 _, hf, ?_⟩
rw [id_comp]
· rintro hle Y f ⟨X, g, h, hg, rfl⟩
apply Sieve.downward_closed S
exact hle g hg
| Mathlib/CategoryTheory/Sites/Sieves.lean | 700 | 707 |
/-
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.Data.Set.SymmDiff
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Irreducible
/-!
# Connected subsets ... | (K : ∀ i, i ∈ t → ∀ j, j ∈ t → ReflTransGen (fun i j => (s i ∩ s j).Nonempty ∧ i ∈ t) i j) :
IsPreconnected (⋃ n ∈ t, s n) := by
let R := fun i j : ι => (s i ∩ s j).Nonempty ∧ i ∈ t
have P : ∀ i, i ∈ t → ∀ j, j ∈ t → ReflTransGen R i j →
∃ p, p ⊆ t ∧ i ∈ p ∧ j ∈ p ∧ IsPreconnected (⋃ j ∈ p, s j) := fu... | Mathlib/Topology/Connected/Basic.lean | 148 | 153 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau
-/
import Mathlib.Data.List.Forall2
/-!
# zip & unzip
This file provides results about `List.zipWith`, `List.zip` and `List.unzip` (definitions are in
core ... | @[deprecated (since := "2025-02-14")] alias get?_zipWith' := getElem?_zipWith'
@[deprecated (since := "2025-02-14")] alias get?_zipWith_eq_some := getElem?_zipWith_eq_some
@[deprecated (since := "2025-02-14")] alias get?_zip_eq_some := getElem?_zip_eq_some
| Mathlib/Data/List/Zip.lean | 120 | 122 |
/-
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... | Mathlib/Data/Complex/Exponential.lean | 1,381 | 1,386 | |
/-
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.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
/-!
# Special values of Hurwitz and Riemann zeta functions
This file gives t... |
-- private because it is superseded by `hurwitzZeta_neg_nat` below
private lemma hurwitzZeta_neg_two_mul_nat (hk : k ≠ 0) (hx : x ∈ Icc (0 : ℝ) 1) :
hurwitzZeta x (-(2 * k)) = -1 / (2 * k + 1) *
((Polynomial.bernoulli (2 * k + 1)).map (algebraMap ℚ ℂ)).eval (x : ℂ) := by
suffices hurwitzZetaEven x (-(2 * k... | Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 172 | 179 |
/-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
import Mathlib.SetTheory.Nimber.Basic
/-!
# Nim and the Sprague-Grundy theore... | theorem nim_equiv_iff_eq {o₁ o₂ : Ordinal} : (nim o₁ ≈ nim o₂) ↔ o₁ = o₂ := by
rw [Impartial.equiv_iff_add_equiv_zero, nim_add_equiv_zero_iff]
| Mathlib/SetTheory/Game/Nim.lean | 216 | 218 |
/-
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
-/
import Mathlib.MeasureTheory.Function.L1Space.Integrable
import Mathlib.MeasureTheory.Function.LpSpace.Indicator
/-! # Functions integrable on a set an... |
theorem IntegrableOn.mono_set_ae (h : IntegrableOn f t μ) (hst : s ≤ᵐ[μ] t) : IntegrableOn f s μ :=
| Mathlib/MeasureTheory/Integral/IntegrableOn.lean | 112 | 113 |
/-
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.MeasureTheory.Measure.AEMeasurable
import Mathlib.Order.Filter.EventuallyConst
/-!
# Measure preserving maps
We say that `f : α → β` is a measure p... | MeasurePreserving (e ∘ g) μa μc ↔ MeasurePreserving g μa μb := by
refine ⟨fun hg => ?_, fun hg => h.comp hg⟩
convert (MeasurePreserving.symm e h).comp hg
| Mathlib/Dynamics/Ergodic/MeasurePreserving.lean | 102 | 104 |
/-
Copyright (c) 2022 George Peter Banyard, Yaël Dillies, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: George Peter Banyard, Yaël Dillies, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Path
import Mathlib.Combinatorics.SimpleGraph.Metric
/-!
#... |
instance boxProdFintypeNeighborSet (x : α × β)
[Fintype (G.neighborSet x.1)] [Fintype (H.neighborSet x.2)] :
Fintype ((G □ H).neighborSet x) :=
| Mathlib/Combinatorics/SimpleGraph/Prod.lean | 204 | 207 |
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Int.DivMod
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.... | @[simps!]
def succAboveEmb (p : Fin (n + 1)) : Fin n ↪ Fin (n + 1) := ⟨p.succAbove, succAbove_right_injective⟩
| Mathlib/Data/Fin/Basic.lean | 1,018 | 1,019 |
/-
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... | · rfl
· simp only [coeff_zero]
| Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean | 432 | 434 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.WSeq.Basic
import Mathlib.Data.WSeq.Defs
import Mathlib.Data.WSeq.Productive
import Mathlib.Data.WSeq.Relation
deprecated_module (since :=... | Mathlib/Data/Seq/WSeq.lean | 504 | 518 | |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
/-!
# Neighborhoods and continuity relative to a subset
This file develops API on the relative versions
* `nhdsWithin` ... | (hf x hx).prodMap (hg y hy)
@[deprecated (since := "2025-03-10")]
alias ContinuousOn.prod_map := ContinuousOn.prodMap
theorem continuousWithinAt_prod_of_discrete_left [DiscreteTopology α]
{f : α × β → γ} {s : Set (α × β)} {x : α × β} :
| Mathlib/Topology/ContinuousOn.lean | 1,111 | 1,117 |
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.... | · simp only [alls, ih, Realize]
exact ⟨fun h xs => Fin.snoc_init_self xs ▸ h _ _, fun h xs x => h (Fin.snoc xs x)⟩
| Mathlib/ModelTheory/Semantics.lean | 745 | 747 |
/-
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, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | Mathlib/SetTheory/Cardinal/Basic.lean | 2,287 | 2,291 | |
/-
Copyright (c) 2024 Newell Jensen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Newell Jensen, Mitchell Lee, Óscar Álvarez
-/
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Algebra.Ring.Int.Parity
import Mathlib.GroupTheory.Coxeter.Matrix
import Mat... | · rintro _ ⟨i, rfl⟩ y _
exact mul_simple_left y i
· exact this
/-- If `p : W → Prop` holds for the identity and it is preserved under multiplying on the right
by a simple reflection, then it holds for all elements of `W`. -/
theorem simple_induction_right {p : W → Prop} (w : W) (one : p 1)
(mul_simple_righ... | Mathlib/GroupTheory/Coxeter/Basic.lean | 266 | 275 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau
-/
import Mathlib.Data.List.Forall2
/-!
# Lists with no duplicates
`List.Nodup` is defined in `Data/List/Basic`. In this file we prove various properties of... | · apply (d.1 _ h₁ h₃).elim
· apply ih d.2 h₁ h₂ h₃
theorem nodup_map_iff_inj_on {f : α → β} {l : List α} (d : Nodup l) :
Nodup (map f l) ↔ ∀ x ∈ l, ∀ y ∈ l, f x = f y → x = y :=
| Mathlib/Data/List/Nodup.lean | 212 | 216 |
/-
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... | /-- A set of points, whose `vectorSpan` is finite-dimensional, is coplanar if and only if their
`vectorSpan` has dimension at most `2`. -/
theorem coplanar_iff_finrank_le_two {s : Set P} [FiniteDimensional k (vectorSpan k s)] :
Coplanar k s ↔ finrank k (vectorSpan k s) ≤ 2 := by
have h : Coplanar k s ↔ Module.ran... | Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | 627 | 632 |
/-
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... | not_bddAbove_iff.1 hs.not_bddAbove
| Mathlib/Order/Interval/Finset/Basic.lean | 920 | 921 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.AffineMap
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Mul
import ... | Mathlib/Analysis/Calculus/MeanValue.lean | 1,041 | 1,048 | |
/-
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.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
/-!
# One-dimensional iterated derivatives
We define the `n`-th de... | reformulated in terms of the one-dimensional derivative on sets with unique derivatives. -/
theorem contDiffOn_nat_iff_continuousOn_differentiableOn_deriv {n : ℕ} (hs : UniqueDiffOn 𝕜 s) :
ContDiffOn 𝕜 n f s ↔ (∀ m : ℕ, m ≤ n → ContinuousOn (iteratedDerivWithin m f s) s) ∧
∀ m : ℕ, m < n → DifferentiableOn ... | Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean | 167 | 170 |
/-
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.Data.Nat.Gcd
import Mathlib.Algebra.Group.Nat.Units
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.GroupWi... | Mathlib/Data/Nat/GCD/Basic.lean | 265 | 270 | |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Violeta Hernández Palacios, Grayson Burton, Floris van Doorn
-/
import Mathlib.Order.Antisymmetrization
import Mathlib.Order.Hom.WithTopBot
import Mathlib.Order.Interval.Se... | end WithTop
| Mathlib/Order/Cover.lean | 588 | 589 |
/-
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.Data.Set.Countable
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Tactic.FunProp.Attr
import Mathlib.Tactic.Mea... | set_option quotPrecheck false in
/-- Notation for `Measurable` with respect to a non-standard σ-algebra in the domain. -/
scoped notation "Measurable[" m "]" => @Measurable _ _ m _
| Mathlib/MeasureTheory/MeasurableSpace/Defs.lean | 476 | 478 |
/-
Copyright (c) 2021 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.Spectrum.Basic
import Mathlib.FieldTheory.IsAlgClosed.Basic
/-!
# Spectrum mapping theorem
This file develops proves the spectral mappi... | have hprod := eq_prod_roots_of_splits_id (IsAlgClosed.splits (C k - p))
have h_ne : C k - p ≠ 0 := ne_zero_of_degree_gt <| by
rwa [degree_sub_eq_right_of_degree_lt (lt_of_le_of_lt degree_C_le hdeg)]
have lead_ne := leadingCoeff_ne_zero.mpr h_ne
have lead_unit := (Units.map ↑ₐ.toMonoidHom (Units.mk0 _ lead_n... | Mathlib/FieldTheory/IsAlgClosed/Spectrum.lean | 97 | 115 |
/-
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... | theorem image_univ_of_surjective {ι : Type*} {f : ι → β} (H : Surjective f) : f '' univ = univ :=
eq_univ_of_forall <| by simpa [image]
| Mathlib/Data/Set/Image.lean | 273 | 275 |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.GroupWithZero.Basic
/-!
# Indicator functions and support of a function in groups with z... | simp only [indicator]
split_ifs
· rfl
· rw [mul_zero]
lemma indicator_mul_const (s : Set ι) (f : ι → M₀) (a : M₀) (i : ι) :
| Mathlib/Algebra/GroupWithZero/Indicator.lean | 38 | 43 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Amelia Livingston, Yury Kudryashov,
Neil Strickland, Aaron Anderson
-/
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathli... | /-- Given two elements `a`, `b` of a commutative `CancelMonoidWithZero` and a nonzero
element `c`, `a*c` divides `b*c` iff `a` divides `b`. -/
theorem mul_dvd_mul_iff_right [CancelCommMonoidWithZero α] {a b c : α} (hc : c ≠ 0) :
| Mathlib/Algebra/GroupWithZero/Divisibility.lean | 49 | 51 |
/-
Copyright (c) 2022 Sam van Gool and Jake Levinson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sam van Gool, Jake Levinson
-/
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.Stalks
import Mathlib.CategoryTheory.Limits.Preserves.Filtered
i... | -- Let g ∈ Γₛₜ 𝒢 x be a germ.
intro x g
-- Represent it on an open set U ⊆ X as ⟨t, U⟩.
obtain ⟨U, hxU, t, rfl⟩ := 𝒢.germ_exist x g
-- By local surjectivity, pass to a smaller open set V
-- on which there exists s ∈ Γ_ ℱ V mapping to t |_ V.
rcases hT.imageSieve_mem t x hxU with ⟨V, ι, ⟨s,... | Mathlib/Topology/Sheaves/LocallySurjective.lean | 78 | 118 |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.FieldTheory.Minpoly.Basic
import Mathlib.RingT... | exact Or.imp (dvd A x) (dvd A x) this
/-- If `L/K` is a field extension and an element `y` of `K` is a root of the minimal polynomial
| Mathlib/FieldTheory/Minpoly/Field.lean | 270 | 272 |
/-
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.Fin.Tuple.Basic
/-!
# Lists from functions
Theorems and lemmas for dealing with `List.ofFn`, which converts a function on `Fin n` to a list
of l... | theorem ofFn_const : ∀ (n : ℕ) (c : α), (ofFn fun _ : Fin n => c) = replicate n c
| 0, c => by rw [ofFn_zero, replicate_zero]
| Mathlib/Data/List/OfFn.lean | 124 | 125 |
/-
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.MeasureTheory.Measure.Haar.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Additive Haar measure constructed from a basis
Given a ba... | simp [(v.prod w).addHaar_eq_iff, Basis.prod_parallelepiped, Basis.addHaar_self]
end NormedSpace
end Fintype
| Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean | 280 | 284 |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Kim Morrison
-/
import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
/-!
# Filtered categories
A category is filtered if every finite diagram admits a cocone.
We give a... | IsFilteredOrEmpty.of_equivalence (opOpEquivalence _)
/-- If Cᵒᵖ is filtered, then C is cofiltered. -/
lemma isCofiltered_of_isFiltered_op [IsFiltered Cᵒᵖ] : IsCofiltered C :=
IsCofiltered.of_equivalence (opOpEquivalence _)
/-- If Cᵒᵖ is cofiltered, then C is filtered. -/
lemma isFiltered_of_isCofiltered_op [IsCof... | Mathlib/CategoryTheory/Filtered/Basic.lean | 860 | 876 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kim Morrison
-/
import Mathlib.Algebra.Category.Ring.Colimits
import Mathlib.Algebra.Category.Ring.Instances
import Mathlib.Algebra.Category.Ring.Limits
import Mathlib.Al... | Mathlib/AlgebraicGeometry/StructureSheaf.lean | 1,201 | 1,213 | |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Group.Defs
/-!
# Invertible elements
This file defines a typeclass `Invertible a` for elements `a` with a two-sided
multiplicative inverse.
The in... |
theorem invOf_eq_left_inv [Monoid α] {a b : α} [Invertible a] (hac : b * a = 1) : ⅟ a = b :=
| Mathlib/Algebra/Group/Invertible/Defs.lean | 141 | 142 |
/-
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... | /-- Typeclass for monotonicity of multiplication on the left,
namely `b₁ < b₂ → a * b₁ < a * b₂`.
You should usually not use this very granular typeclass directly, but rather a typeclass like
`OrderedCommGroup`. -/
abbrev MulLeftStrictMono [Mul M] [LT M] : Prop :=
CovariantClass M M (· * ·) (· < ·)
| Mathlib/Algebra/Order/Monoid/Unbundled/Defs.lean | 154 | 160 |
/-
Copyright (c) 2022 Sebastian Monnet. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sebastian Monnet
-/
import Mathlib.FieldTheory.Galois.Basic
import Mathlib.Topology.Algebra.FilterBasis
import Mathlib.Topology.Algebra.OpenSubgroup
import Mathlib.Tactic.ByContra
/... | /-- Given a tower of fields `L/E/K`, with `E/K` finite, the subgroup `Gal(L/E) ≤ L ≃ₐ[K] L` is
closed. -/
theorem IntermediateField.fixingSubgroup_isClosed {K L : Type*} [Field K] [Field L] [Algebra K L]
(E : IntermediateField K L) [FiniteDimensional K E] :
IsClosed (E.fixingSubgroup : Set (L ≃ₐ[K] L)) :=
O... | Mathlib/FieldTheory/KrullTopology.lean | 221 | 253 |
/-
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... | theorem Function.Embedding.toEquivRange_eq_ofInjective :
f.toEquivRange = Equiv.ofInjective f f.injective := by
| Mathlib/Logic/Equiv/Fintype.lean | 50 | 51 |
/-
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... | (selfAdjointPart R x : A) + skewAdjointPart R x = x := by
simp only [smul_sub, selfAdjointPart_apply_coe, smul_add, skewAdjointPart_apply_coe,
add_add_sub_cancel, invOf_two_smul_add_invOf_two_smul]
| Mathlib/Algebra/Star/Module.lean | 156 | 158 |
/-
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.OfAssociative
import Mathlib.Algebra.Lie.IdealOperations
/-!
# Trivial Lie modules and Abelian Lie algebras
The action of a Lie algebra `L` on ... | ⟨fun h z x y hz =>
hz.symm.trans
(((I : LieSubalgebra R L).coe_bracket x y).symm.trans
((coe_zero_iff_zero _ _).mpr (by apply h.trivial))),
| Mathlib/Algebra/Lie/Abelian.lean | 312 | 315 |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.C... | Mathlib/Analysis/BoundedVariation.lean | 177 | 181 | |
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Analysis.InnerProductSpace.Spectrum
import Mathlib.Data.Matrix.Rank
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.M... |
lemma eigenvectorUnitary_coe {𝕜 : Type*} [RCLike 𝕜] {n : Type*} [Fintype n]
{A : Matrix n n 𝕜} [DecidableEq n] (hA : Matrix.IsHermitian A) :
eigenvectorUnitary hA =
| Mathlib/LinearAlgebra/Matrix/Spectrum.lean | 70 | 73 |
/-
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.Order.Interval.Set.OrderEmbedding
import Mathlib.Order.Antichain
import Mathlib.Order.SetNotation
/-!
# Order-connected sets
We say that a set `s :... | @[simp]
theorem dual_ordConnected_iff {s : Set α} : OrdConnected (ofDual ⁻¹' s) ↔ OrdConnected s := by
simp_rw [ordConnected_def, toDual.surjective.forall, Icc_toDual, Subtype.forall']
exact forall_swap
| Mathlib/Order/Interval/Set/OrdConnected.lean | 227 | 230 |
/-
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... | rw [IsZero.iff_id_eq_zero]
apply hX.hom_ext
| Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | 614 | 616 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kim Morrison
-/
import Mathlib.Algebra.BigOperators.Finsupp.Basic
import Mathlib.Algebra.BigOperators.Group.Finset.Preimage
import Mathlib.Algebra.Module.Defs
import Ma... | toFun a := l (f a)
mem_support_toFun := by
intro a
simp only [Finset.mem_def.symm, Finset.mem_preimage]
exact l.mem_support_toFun (f a)
| Mathlib/Data/Finsupp/Basic.lean | 612 | 617 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Peter Pfaffelhuber
-/
import Mathlib.Data.Nat.Lattice
import Mathlib.Data.Set.Accumulate
import Mathlib.Data.Set.Pairwise.Lattice
import Mathlib.MeasureTheory.PiSystem
/-!... | refine PairwiseDisjoint.insert_of_not_mem h (hC.nmem_disjointOfDiff hs ht) fun u hu ↦ ?_
simp_rw [id]
refine Disjoint.mono_right ?_ (hC.disjoint_sUnion_disjointOfDiff hs ht)
simp only [Set.le_eq_subset]
exact subset_sUnion_of_mem hu
end disjointOfDiff
section disjointOfDiffUnion
variable {I : Finset (Set α... | Mathlib/MeasureTheory/SetSemiring.lean | 134 | 199 |
/-
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.Single
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
/-!
# The homology of single complexes
The main definition in this file ... | Mathlib/Algebra/Homology/SingleHomology.lean | 248 | 250 | |
/-
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.Nat.ModEq
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.NumberTheory.Zsqrtd.Basic
/-!
# Pell's equation and Matiyasevic's theorem
This file... |
theorem strictMono_x : StrictMono (xn a1)
| _, 0, h => absurd h <| Nat.not_lt_zero _
| m, n + 1, h => by
have : xn a1 m ≤ xn a1 n :=
Or.elim (lt_or_eq_of_le <| Nat.le_of_succ_le_succ h) (fun hl => le_of_lt <| strictMono_x hl)
fun e => by rw [e]
simp only [xn_succ, gt_iff_lt]
refine lt_of_... | Mathlib/NumberTheory/PellMatiyasevic.lean | 347 | 356 |
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Defs
/-!
# Basic kernels
This file contains basic results about kernels in general and definitions of some particular
kernels.
## Mai... | (t : Set β) : ∫⁻ b in t, f b ∂κ.restrict hs a = ∫⁻ b in t ∩ s, f b ∂κ a := by
rw [restrict_apply, Measure.restrict_restrict' hs]
| Mathlib/Probability/Kernel/Basic.lean | 264 | 267 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Tactic.Attr.Register
import Mathlib.Tactic.Basic
import Batteries.Logic
import Batteries.Tactic.Trans
import Batteries.Util.LibraryNot... | Mathlib/Logic/Basic.lean | 1,314 | 1,316 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Subgroup.Lattice
import Mathlib.Algebra.Group.Submonoid.BigOperators
import Mathlib.Data.Finset.Fin
import ... | congr
rw [← l.2.1]
simp_rw [← hl'₂]
rw [sign_prod_list_swap l.2.2, sign_prod_list_swap hl', List.length_map]
| Mathlib/GroupTheory/Perm/Sign.lean | 472 | 475 |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
import Mathlib.MeasureTheory.Integral.Bochner.FundThmCalculus
import Mathlib.MeasureT... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 1,562 | 1,568 | |
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Sébastien Gouëzel
-/
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
import Mathlib.MeasureTheory... | _ = μ s / μ t := by
rw [ENNReal.mul_inv_cancel, one_mul, div_eq_mul_inv]
· simp only [pow_pos (abs_pos.mpr hr), ENNReal.ofReal_eq_zero, not_le, Ne]
· simp only [ENNReal.ofReal_ne_top, Ne, not_false_iff]
instance (priority := 100) isUnifLocDoublingMeasureOfIsAddHaarMeasure :
IsUnifLocDoublingM... | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 544 | 560 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Order.Bounds.Image
import Mathlib.Order.Hom.Set
/-!
# Order isomorphisms and bounds.
-/
open Set
namespace OrderIso
variable {α β ... |
theorem isGLB_image' {s : Set α} {x : α} : IsGLB (f '' s) (f x) ↔ IsGLB s x :=
| Mathlib/Order/Bounds/OrderIso.lean | 39 | 40 |
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
/-! # Positive Definite Matrices
This file defi... | (hA : Matrix.PosSemidef A) (i : n) : 0 ≤ hA.1.eigenvalues i :=
(hA.re_dotProduct_nonneg _).trans_eq (hA.1.eigenvalues_eq _).symm
section sqrt
variable [DecidableEq n] {A : Matrix n n 𝕜} (hA : PosSemidef A)
/-- The positive semidefinite square root of a positive semidefinite matrix -/
noncomputable def sqrt : ... | Mathlib/LinearAlgebra/Matrix/PosDef.lean | 169 | 180 |
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov, Hunter Monroe
-/
import Mathlib.Combinatorics.SimpleGraph.Init
import Mathlib.Data.Finite.Prod
import... | G.incidenceSet a ∩ G.incidenceSet b = {s(a, b)} := by
refine (G.incidenceSet_inter_incidenceSet_subset <| h.ne).antisymm ?_
rintro _ (rfl : _ = s(a, b))
| Mathlib/Combinatorics/SimpleGraph/Basic.lean | 656 | 658 |
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Cast.Order.Basic
import Math... | | 0, pos _q => (zero_mul _).symm
| pos _p, 0 => rfl
| pos _p, pos _q => PosNum.mul_to_nat _ _
| Mathlib/Data/Num/Lemmas.lean | 266 | 269 |
/-
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.Algebra.Rat
import Mathlib.Data.Nat.Cast.Field
import Mathlib.RingTheory.PowerSeries.Basic
/-!
# Definition of well-known power series
In t... | section Field
variable (A A' : Type*) [Ring A] [Ring A'] [Algebra ℚ A] [Algebra ℚ A']
| Mathlib/RingTheory/PowerSeries/WellKnown.lean | 174 | 176 |
/-
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.List
import Mathlib.Data.Fintype.OfMap
/-!
# Cycles of a list
Lists have an equivalence relation of whether they are rotational permut... | contradiction
by_cases h : x = y
· rw [h, nextOr_self_cons_cons, nextOr_self_cons_cons]
· rw [nextOr, nextOr, IH]
· simpa [h] using x_mem
· simpa using x_ne
theorem mem_of_nextOr_ne {xs : List α} {x d : α} (h : nextOr xs x d ≠ d) : x ∈ xs := by
induction' xs with y ys IH
· simp at h
rcases ys w... | Mathlib/Data/List/Cycle.lean | 62 | 73 |
/-
Copyright (c) 2021 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, David Kurniadi Angdinata
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.CubicDiscriminant
import Mathlib.RingTheory.Nilpotent.Defs
import Mathlib.Tactic.Fiel... | Mathlib/AlgebraicGeometry/EllipticCurve/Weierstrass.lean | 757 | 759 | |
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.GroupWithZero.Invertible
import Mathlib.Data.Sigma.Basic
import Mathlib.Algebra.Ring.Nat
import Mathlib.Data.... | | _, _, ⟨rfl⟩ => ⟨by simp⟩
| Mathlib/Tactic/NormNum/Result.lean | 168 | 169 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.GroupTheory.Perm.Basic
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.Lis... | end
end List
namespace Finset
| Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 860 | 865 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.Normed.Module.Convex
/-!
# Sides of affine subspaces
This ... | theorem wSameSide_self_iff {s : AffineSubspace R P} {x : P} :
s.WSameSide x x ↔ (s : Set P).Nonempty :=
⟨fun h => h.nonempty, fun ⟨p, hp⟩ => ⟨p, hp, p, hp, SameRay.rfl⟩⟩
theorem sSameSide_self_iff {s : AffineSubspace R P} {x : P} :
s.SSameSide x x ↔ (s : Set P).Nonempty ∧ x ∉ s :=
⟨fun ⟨h, hx, _⟩ => ⟨wSame... | Mathlib/Analysis/Convex/Side.lean | 201 | 208 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Alex Kontorovich, Heather Macbeth
-/
import Mathlib.MeasureTheory.Group.Action
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Integral.Lebe... | (hf : Integrable f ν) : ∫ x, f x ∂ν = ∑' g : G, ∫ x in g • s, f x ∂ν := by
rw [← MeasureTheory.integral_sum_measure, h.sum_restrict_of_ac hν]
rw [h.sum_restrict_of_ac hν]
exact hf
@[to_additive]
theorem integral_eq_tsum (h : IsFundamentalDomain G s μ) (f : α → E) (hf : Integrable f μ) :
| Mathlib/MeasureTheory/Group/FundamentalDomain.lean | 369 | 375 |
/-
Copyright (c) 2020 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Algebra.Algebra.Spectrum.Basic
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathl... | apply HasUnifEigenvalue.le (k := 1) (Order.one_le_iff_pos.mpr hm)
intro contra; apply hk
rw [genEigenspace_one, LinearMap.ker_eq_bot] at contra
rw [eq_bot_iff]
intro x hx
rw [mem_genEigenspace] at hx
rcases hx with ⟨l, -, hx⟩
rwa [LinearMap.ker_eq_bot.mpr] at hx
| Mathlib/LinearAlgebra/Eigenspace/Basic.lean | 306 | 313 |
/-
Copyright (c) 2020 Thomas Browning and Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.GroupTheory.Solvable
import Mathlib.FieldTheory.PolynomialGaloisGroup
import Mathlib.RingTheory.RootsOfUnity.Basic
/-... | simp only [Set.mem_insert_iff, Set.mem_singleton_iff] at hx
cases hx with rw [hx]
| inl hx => exact ⟨isIntegral α, hpq.1⟩
| inr hx => exact ⟨isIntegral β, hpq.2⟩
have key : minpoly F γ = minpoly F (f ⟨γ, hγ⟩) := by
refine minpoly.eq_of_irreducible_of_monic
(minpoly.irreducible (isInt... | Mathlib/FieldTheory/AbelRuffini.lean | 308 | 331 |
/-
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 | 2,548 | 2,550 | |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Logic.Relator
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
import Mathlib.Logic.Basic
import Mathl... |
theorem tail'_iff : TransGen r a c ↔ ∃ b, ReflTransGen r a b ∧ r b c := by
refine ⟨fun h ↦ ?_, fun ⟨b, hab, hbc⟩ ↦ tail' hab hbc⟩
cases h with
| single hac => exact ⟨_, by rfl, hac⟩
| tail hab hbc => exact ⟨_, hab.to_reflTransGen, hbc⟩
| Mathlib/Logic/Relation.lean | 427 | 433 |
/-
Copyright (c) 2024 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker, Devon Tuma, Kexing Ying
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.Density
import Mathlib.Probability.ConditionalProbability
import Mathlib.Probabili... | def uniformPDF (s : Set E) (x : E) (μ : Measure E := by volume_tac) : ℝ≥0∞ :=
s.indicator ((μ s)⁻¹ • (1 : E → ℝ≥0∞)) x
/-- Check that indeed any uniform random variable has the uniformPDF. -/
| Mathlib/Probability/Distributions/Uniform.lean | 187 | 190 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash, Yury Kudryashov
-/
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.LocallyFinite
import Mathlib.Topology.Maps.Proper.Basic
import Mathlib.Topology.UniformSpace.Un... |
theorem uniformContinuous_comp (g : C(β, δ)) (hg : UniformContinuous g) :
UniformContinuous (ContinuousMap.comp g : C(α, β) → C(α, δ)) :=
isUniformEmbedding_toUniformOnFunIsCompact.uniformContinuous_iff.mpr <|
UniformOnFun.postcomp_uniformContinuous hg |>.comp
isUniformEmbedding_toUniformOnFunIsCompact... | Mathlib/Topology/UniformSpace/CompactConvergence.lean | 256 | 261 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
/-!
# Power function... | by_cases hp_zero : p = 0
· simp [hp_zero, zero_lt_one]
· rw [← Ne] at hp_zero
have hp_pos := lt_of_le_of_ne hp_nonneg hp_zero.symm
rw [← zero_rpow_of_pos hp_pos]
| Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 808 | 812 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Yaël Dillies
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
/-!
# Integral average of a function
In this file we define `MeasureTheory.average... | ⨍ x, (f x, g x) ∂μ = (⨍ x, f x ∂μ, ⨍ x, g x ∂μ) :=
integral_pair hfi.to_average hgi.to_average
| Mathlib/MeasureTheory/Integral/Average.lean | 361 | 362 |
/-
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.Sites.Coherent.RegularSheaves
/-!
# Description of the covering sieves of the regular topology
This file characterises the coverin... | theorem mem_sieves_of_hasEffectiveEpi (S : Sieve X) :
(∃ (Y : C) (π : Y ⟶ X), EffectiveEpi π ∧ S.arrows π) → (S ∈ (regularTopology C) X) := by
rintro ⟨Y, π, h⟩
have h_le : Sieve.generate (Presieve.ofArrows (fun () ↦ Y) (fun _ ↦ π)) ≤ S := by
rw [Sieve.generate_le_iff (Presieve.ofArrows _ _) S]
apply Pre... | Mathlib/CategoryTheory/Sites/Coherent/RegularTopology.lean | 30 | 41 |
/-
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]
theorem diam_image (s : Set α) : Metric.diam (h '' s) = Metric.diam s :=
h.isometry.diam_image s
| Mathlib/Topology/MetricSpace/Isometry.lean | 562 | 564 |
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.Plus
import Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory
/-!
# Sheafification
We construct the sheafification of a presheaf ov... | mk x = mk y ↔ ∃ (W : J.Cover X) (h1 : W ⟶ S) (h2 : W ⟶ T), x.refine h1 = y.refine h2 := by
constructor
· intro h
obtain ⟨W, h1, h2, hh⟩ := Concrete.colimit_exists_of_rep_eq.{u} (C := D) _ _ _ h
use W.unop, h1.unop, h2.unop
ext I
apply_fun Multiequalizer.ι (W.unop.index P) I at hh
convert hh
... | Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean | 213 | 222 |
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.Maps
import Mathlib.Data.Finset.Max
import Mathlib.Data.Sy... | simp_rw [degree, neighborFinset_eq_filter, bot_adj, filter_False]
exact Finset.card_empty
theorem IsRegularOfDegree.top [DecidableEq V] :
(⊤ : SimpleGraph V).IsRegularOfDegree (Fintype.card V - 1) := by
| Mathlib/Combinatorics/SimpleGraph/Finite.lean | 279 | 283 |
/-
Copyright (c) 2018 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.NatIso
import Mathlib.Logic.Equiv.Defs
/-!
# Full and faithful functors
We define typeclasses `Full` and `Faithful`, decorating functors. ... |
section
| Mathlib/CategoryTheory/Functor/FullyFaithful.lean | 228 | 229 |
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Kim Morrison, Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Functor.Trifunctor
import Mathlib.CategoryTheo... | Mathlib/CategoryTheory/Monoidal/Category.lean | 1,019 | 1,022 | |
/-
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.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
/-!
# GCD and LCM operations on finsets
## Main definitions
- `Finset.gcd` - the greatest... | · intro a as
rw [Multiset.mem_map] at as
rcases as with ⟨b, ⟨bs, rfl⟩⟩
| Mathlib/Algebra/GCDMonoid/Finset.lean | 184 | 186 |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Kexing Ying
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.Integration
import Mathlib.MeasureTheory.Function.L2Space
/-!
# Variance of random ... | ext ω
rw [Pi.add_apply, sub_add_cancel]
theorem evariance_lt_top_iff_memLp [IsFiniteMeasure μ] (hX : AEStronglyMeasurable X μ) :
evariance X μ < ∞ ↔ MemLp X 2 μ where
mp := by contrapose!; rw [top_le_iff]; exact evariance_eq_top hX
mpr := evariance_lt_top
| Mathlib/Probability/Variance.lean | 103 | 110 |
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