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) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov, Hunter Monroe
-/
import Mathlib.Combinatorics.SimpleGraph.Init
import Mathlib.Data.Finite.Prod
import... | Mathlib/Combinatorics/SimpleGraph/Basic.lean | 878 | 881 | |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | Mathlib/Data/List/Basic.lean | 3,443 | 3,452 | |
/-
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.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Functor.EpiMono
import Mathlib.CategoryTheory.HomCongr
/-!
# Reflective functors
Ba... |
variable (j : C ⥤ D)
/-- The coreflector `D ⥤ C` when `L : C ⥤ D` is coreflective. -/
def coreflector [Coreflective j] : D ⥤ C := Coreflective.R (L := j)
/-- The adjunction `j ⊣ coreflector j` when `j` is coreflective. -/
def coreflectorAdjunction [Coreflective j] : j ⊣ coreflector j := Coreflective.adj
| Mathlib/CategoryTheory/Adjunction/Reflective.lean | 180 | 187 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Analysis.InnerProductSpace.Continuous
import Mathlib.Analysis.Normed.Module.Dual
import Mathlib.MeasureTheory.Function.AEEqOfLIntegral
import Mathlib.Measu... | ⟨u, u_sep, hu⟩
refine ae_eq_zero_of_forall_dual_of_isSeparable ℝ u_sep (fun c => ?_) hu
refine ae_eq_zero_restrict_of_forall_setIntegral_eq_zero_real ?_ ?_ ht hμt
· intro s hs hμs
exact ContinuousLinearMap.integrable_comp c (hf_int_finite s hs hμs)
· intro s hs hμs
rw [ContinuousLinearMap.integral_c... | Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean | 229 | 241 |
/-
Copyright (c) 2024 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Small.Module
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.LinearAlgebra.TensorProduc... | have hσ₂ : l ∘ b ∘ σ = Subtype.val := by ext i; exact congr_arg Subtype.val (hσ i)
refine ⟨(Set.finite_range (l ∘ b)).toFinset,
by simpa [Set.range_comp, LinearMap.range_eq_top], ?_⟩
let f : M →ₗ[R] (Set.finite_range (l ∘ b)).toFinset →₀ R :=
| Mathlib/Algebra/Module/FinitePresentation.lean | 131 | 134 |
/-
Copyright (c) 2024 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.MvPolynomial.Monad
import Mathlib.LinearAlgebra.Charpoly.ToMatrix
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.Li... | open Algebra.TensorProduct in
lemma toMvPolynomial_baseChange (f : M₁ →ₗ[R] M₂) (i : ι₂) (A : Type*) [CommRing A] [Algebra R A] :
(f.baseChange A).toMvPolynomial (basis A b₁) (basis A b₂) i =
MvPolynomial.map (algebraMap R A) (f.toMvPolynomial b₁ b₂ i) := by
| Mathlib/Algebra/Module/LinearMap/Polynomial.lean | 166 | 169 |
/-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Polynomial.Reverse
/-!
# "Mirror" of a univariate polynomial
In this file we define `Po... | rw [fgh, mirror_mul_of_domain, mirror_mul_of_domain, mirror_mirror, mul_assoc, mul_comm h,
mul_comm g.mirror, mul_assoc, ← mul_assoc]
have g_dvd_f : g ∣ f := by
rw [fgh]
exact dvd_mul_right g h
have h_dvd_f : h ∣ f := by
rw [fgh]
exact dvd_mul_left h g
| Mathlib/Algebra/Polynomial/Mirror.lean | 199 | 206 |
/-
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 | 981 | 987 | |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Order.Interval.Finset.Na... | Mathlib/Topology/EMetricSpace/Basic.lean | 959 | 960 | |
/-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.LinearAlgebra.Dimension.LinearMap
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.LinearAlgebra.Matrix.ToLin
/-!
# Finite... | rw [Matrix.vecMulVec_eq (Fin 1), Matrix.toLin'_mul]
refine le_trans (LinearMap.rank_comp_le_left _ _) ?_
refine (LinearMap.rank_le_domain _).trans_eq ?_
rw [rank_fun', Fintype.card_ofSubsingleton, Nat.cast_one]
| Mathlib/LinearAlgebra/FreeModule/Finite/Matrix.lean | 113 | 119 |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.EffectiveEpi.Preserves
import Mathlib.CategoryTheory.Limits.Final.ParallelPair
import Mathlib... | (c : PullbackCone π π) (hc : IsLimit c) :
(parallelPair (C := (Sieve.ofArrows (fun (_ : Unit) => X) (fun _ => π)).arrows.categoryᵒᵖ)
(Y := op ((Presieve.categoryMk _ (c.fst ≫ π) ⟨_, c.fst, π, ofArrows.mk (), rfl⟩)))
(X := op ((Presieve.categoryMk _ π (Sieve.ofArrows_mk _ _ Unit.unit))))
(Quiver.Hom.... | Mathlib/CategoryTheory/Sites/Coherent/RegularSheaves.lean | 160 | 178 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.MeasureTheory.Group.MeasurableEquiv
import Mathlib... | (hA : MeasurableSet A) (hA' : μ A ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
∃ U, U ⊇ A ∧ IsOpen U ∧ μ U < ∞ ∧ μ (U \ A) < ε := by
rcases A.exists_isOpen_lt_add hA' hε with ⟨U, hAU, hUo, hU⟩
use U, hAU, hUo, hU.trans_le le_top
exact measure_diff_lt_of_lt_add hA.nullMeasurableSet hAU hA' hU
| Mathlib/MeasureTheory/Measure/Regular.lean | 361 | 366 |
/-
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.Data.Set.Image
/-!
### Recursion on the natural numbers and `Set.range`
-/
namespace Nat
section Set
open Set
theorem zero_union_range_succ : {... | cases n <;> simp
@[simp]
| Mathlib/Data/Nat/Set.lean | 21 | 23 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Judith Ludwig, Christian Merten
-/
import Mathlib.Algebra.GeomSum
import Mathlib.LinearAlgebra.SModEq
import Mathlib.RingTheory.Jacobson.Ideal
import Mathlib.RingTheory.Ideal.Quo... | ⟨fun _ _ => Subsingleton.elim _ _⟩
variable {I M}
| Mathlib/RingTheory/AdicCompletion/Basic.lean | 108 | 111 |
/-
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... |
variable (R R' : Type*) {E F : Type*} [Ring R] [Invertible (2 : R)] [AddCommGroup E] [Module R E]
[Ring R'] [Invertible (2 : R')] [AddCommGroup F] [Module R' F]
| Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | 220 | 222 |
/-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Set.Defs
import Mathlib.Logic.Basic
import Mathlib.Logic.ExistsUnique
import Mathlib.Logic.Nonempty
import Mathlib.Logic.Nontrivia... | (hf : Injective f) (hf' : Injective f')
(im_disj : ∀ {x x' : α} {hx : p x} {hx' : ¬ p x'}, f ⟨x, hx⟩ ≠ f' ⟨x', hx'⟩) :
Function.Injective (fun x ↦ if h : p x then f ⟨x, h⟩ else f' ⟨x, h⟩) := fun x₁ x₂ h => by
dsimp only at h
by_cases h₁ : p x₁ <;> by_cases h₂ : p x₂
| Mathlib/Logic/Function/Basic.lean | 128 | 132 |
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Algebra.Module.Basic
/-!
# Group and ring filter bases
A `GroupFilterBasis` is a `FilterBasis` on a group with some properties relating
the ... | · intro x₀
rw [basis.tendsto_iff basis]
intro U U_in
exact conj x₀ U_in
| Mathlib/Topology/Algebra/FilterBasis.lean | 205 | 208 |
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Bhavik Mehta
-/
import Mathlib.Analysis.Calculus.Deriv.Support
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.MeasureTheory.Function.Jacobian
imp... | have F'int : IntegrableOn (F ∘ f') (Iic a) := by apply F.integrable_comp f'int
have A : Tendsto (F ∘ f) atBot (𝓝 (limUnder atBot (F ∘ f))) := by
apply tendsto_limUnder_of_hasDerivAt_of_integrableOn_Iic Fderiv F'int
have B : limUnder atBot (F ∘ f) = F 0 := by
| Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | 898 | 901 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.NoncommPi... | ((Nat.modEq_zero_iff_dvd.mpr h).symm.trans (card_sylow_modEq_one p G))))
variable {p} {G}
namespace Sylow
/-- Sylow subgroups are isomorphic -/
nonrec def equivSMul (P : Sylow p G) (g : G) : P ≃* (g • P : Sylow p G) :=
equivSMul (MulAut.conj g) P.toSubgroup
/-- Sylow subgroups are isomorphic -/
noncomputa... | Mathlib/GroupTheory/Sylow.lean | 325 | 339 |
/-
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 | 1,686 | 1,696 | |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Multiset.Bind
/-!
# The cartesian product of multisets
## Main definitions
* `Multiset.pi`: Cartesian product of multisets indexed by a multise... | intro f _
exact Pi.cons_swap eq)
@[simp]
theorem pi_zero (t : ∀ a, Multiset (β a)) : pi 0 t = {Pi.empty β} :=
rfl
@[simp]
theorem pi_cons (m : Multiset α) (t : ∀ a, Multiset (β a)) (a : α) :
pi (a ::ₘ m) t = (t a).bind fun b => (pi m t).map <| Pi.cons m a b :=
recOn_cons a m
theorem card_pi (... | Mathlib/Data/Multiset/Pi.lean | 120 | 136 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.P... |
/-- The push-forward of an ideal `I` of `R` to `MvPolynomial σ R` via inclusion
is exactly the set of polynomials whose coefficients are in `I` -/
theorem mem_map_C_iff {I : Ideal R} {f : MvPolynomial σ R} :
f ∈ (Ideal.map (C : R →+* MvPolynomial σ R) I : Ideal (MvPolynomial σ R)) ↔
∀ m : σ →₀ ℕ, f.coeff m ... | Mathlib/RingTheory/Polynomial/Basic.lean | 1,085 | 1,093 |
/-
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
-/
import Mathlib.Algebra.Order.SuccPred
import Mathlib.Data.Sum.Order
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.PPWithUniv
/-!
# ... | the elements of `α`. -/
@[simps! symm_apply_coe]
def enum (r : α → α → Prop) [IsWellOrder α r] : (· < · : Iio (type r) → Iio (type r) → Prop) ≃r r :=
(typein r).subrelIso
| Mathlib/SetTheory/Ordinal/Basic.lean | 445 | 448 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Comma.Over.Basic
import Mathlib.CategoryTheory.Discrete.Basic
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryThe... | theorem coprod.map_desc {S T U V W : C} [HasBinaryCoproduct U W] [HasBinaryCoproduct T V]
(f : U ⟶ S) (g : W ⟶ S) (h : T ⟶ U) (k : V ⟶ W) :
coprod.map h k ≫ coprod.desc f g = coprod.desc (h ≫ f) (k ≫ g) := by
| Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | 765 | 767 |
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.RingTheory.Ro... | obtain ⟨hd, hdn⟩ := Nat.mem_properDivisors.mp h
have h0n : 0 < n := pos_of_gt hdn
have h0d : 0 < d := Nat.pos_of_dvd_of_pos hd h0n
rw [← prod_cyclotomic_eq_X_pow_sub_one h0d, ← prod_cyclotomic_eq_X_pow_sub_one h0n, mul_comm,
Finset.prod_sdiff (Nat.divisors_subset_of_dvd h0n.ne' hd)]
theorem X_pow_sub_one_m... | Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | 387 | 393 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.Module.End
import Mathlib.Algebra.Ring.Prod
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.GroupAc... | Mathlib/Data/ZMod/Basic.lean | 1,324 | 1,337 | |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
/-!
# Biproducts and binary biproducts
... | biproduct.hom_ext' _ _ fun j => by simp [← Iso.eq_comp_inv]
/-- If a category has biproducts of a shape `J`, its `colim` and `lim` functor on diagrams over `J`
are isomorphic. -/
| Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean | 530 | 533 |
/-
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
/-!
#... | haveI := (nonempty_prod.1 h.nonempty).2
exact ⟨h.preconnected.ofBoxProdRight⟩
@[simp]
| Mathlib/Combinatorics/SimpleGraph/Prod.lean | 198 | 201 |
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Riccardo Brasca
-/
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.RingTheory.Ideal.Quotient.Operations
import Mathl... | and only if `m ∈ S`."]
lemma norm_mk_eq_zero [hS : IsClosed (S : Set M)] : ‖(m : M ⧸ S)‖ = 0 ↔ m ∈ S := by
rw [norm_mk_eq_zero_iff_mem_closure, hS.closure_eq, SetLike.mem_coe]
| Mathlib/Analysis/Normed/Group/Quotient.lean | 180 | 183 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Sander Dahmen, Kim Morrison, Chris Hughes, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.LinearAlgebra.Basis.Prod
impo... | [∀ i : ι, Module R (M i)] [∀ i : ι, Module.Free R (M i)] [∀ i : ι, Module.Finite R (M i)] :
finrank R (∀ i, M i) = ∑ i, finrank R (M i) := by
letI := nontrivial_of_invariantBasisNumber R
| Mathlib/LinearAlgebra/Dimension/Constructions.lean | 306 | 308 |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.LSeries.AbstractFuncEq
import Mathlib.NumberTheory.ModularForms.JacobiTheta.Bounds
import Mathlib.Analysis.SpecialFunctions.Gamma.Deligne
... |
end FEPair
/-!
## Differentiability and residues
-/
section FEPair
| Mathlib/NumberTheory/LSeries/HurwitzZetaEven.lean | 390 | 398 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Init
import Mathlib.Data.Int.Init
import Mathlib.... | inv_involutive.injective
@[to_additive (attr := simp)]
| Mathlib/Algebra/Group/Basic.lean | 323 | 325 |
/-
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 | 863 | 866 | |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Subalgebra.Tower
import Mathlib.Data.Finite.Sum
import Mathlib.Data.Matrix.Block
import Mathl... | theorem LinearMap.toMatrixAlgEquiv_apply (f : M₁ →ₗ[R] M₁) (i j : n) :
LinearMap.toMatrixAlgEquiv v₁ f i j = v₁.repr (f (v₁ j)) i := by
simp [LinearMap.toMatrixAlgEquiv, LinearMap.toMatrix_apply]
| Mathlib/LinearAlgebra/Matrix/ToLin.lean | 701 | 703 |
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Bryan Gin-ge Chen, Patrick Massot, Wen Yang, Johan Commelin
-/
import Mathlib.Data.Set.Finite.Range
import Mathlib.Order.Partition.Finpartition
/-!
# Equivalenc... |
theorem finite_classes_ker {α β : Type*} [Finite β] (f : α → β) : (Setoid.ker f).classes.Finite :=
(Set.finite_range _).subset <| classes_ker_subset_fiber_set f
theorem card_classes_ker_le {α β : Type*} [Fintype β] (f : α → β)
| Mathlib/Data/Setoid/Partition.lean | 67 | 71 |
/-
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`... | /-- If a topological space `M` admits an atlas with locally path-connected charts,
then `M` itself is locally path-connected. -/
theorem ChartedSpace.locPathConnectedSpace [LocPathConnectedSpace H] : LocPathConnectedSpace M := by
refine ⟨fun x ↦ ⟨fun s ↦ ⟨fun hs ↦ ?_, fun ⟨u, hu⟩ ↦ Filter.mem_of_superset hu.1.1 hu.2⟩... | Mathlib/Geometry/Manifold/ChartedSpace.lean | 679 | 686 |
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Data.Setoid.Basic
import Mathlib.GroupTheory.Congruence.Hom
/-!
# Congruence relations
This f... | Mathlib/GroupTheory/Congruence/Basic.lean | 648 | 650 | |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Rémy Degenne
-/
import Mathlib.Order.ConditionallyCompleteLattice.Indexed
import Mathlib.Order.Filter.IsBounded
import Mathlib.Order.Hom.CompleteL... | ∀ᶠ b : β in atTop, x + ε < u b :=
eventually_lt_of_lt_liminf (lt_of_lt_of_le (add_lt_of_neg_right x hε) hu) hu_bdd
| Mathlib/Order/LiminfLimsup.lean | 789 | 791 |
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Basis
/-!
# Convex combinations
... | rw [Finset.convexHull_eq, Set.mem_setOf_eq]
/-- This is a version of `Finset.mem_convexHull` stated without `Finset.centerMass`. -/
lemma Finset.mem_convexHull' {s : Finset E} {x : E} :
x ∈ convexHull R (s : Set E) ↔
∃ w : E → R, (∀ y ∈ s, 0 ≤ w y) ∧ ∑ y ∈ s, w y = 1 ∧ ∑ y ∈ s, w y • y = x := by
rw [mem_... | Mathlib/Analysis/Convex/Combination.lean | 379 | 385 |
/-
Copyright (c) 2024 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.DirectSum.LinearMap
import Mathlib.Algebra.Lie.Weights.Cartan
import Mathlib.Data.Int.Interval
import Mathlib.LinearAlgebra.Trace
import Mathlib.Ring... | · rw [hα, smul_zero, zero_add]; exact β.genWeightSpace_ne_bot
classical
| Mathlib/Algebra/Lie/Weights/Chain.lean | 310 | 311 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalcul... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 1,236 | 1,238 | |
/-
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.... | {a n' d' : α} → IsRat a (.ofNat n) d → n = n' → d = d' → a = n' / d'
| _, _, _, ⟨_, rfl⟩, rfl, rfl => by simp [div_eq_mul_inv]
theorem IsRat.of_raw (α) [DivisionRing α] (n : ℤ) (d : ℕ)
| Mathlib/Tactic/NormNum/Result.lean | 232 | 235 |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.TangentCone
import Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics
import Mathlib.Analysis.As... |
@[simp low, fun_prop]
theorem differentiableAt_ofNat (n : ℕ) [OfNat F n] (x : E) :
DifferentiableAt 𝕜 (ofNat(n) : E → F) x := differentiableAt_const _
| Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 1,186 | 1,189 |
/-
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... | instance : ENorm ℝ≥0∞ where
enorm x := x
| Mathlib/Analysis/Normed/Group/Basic.lean | 871 | 872 |
/-
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.Group.Submonoid.Operations
import Mathlib.Algebra.MonoidAlgebra.Defs
import Mathlib.Algebra.Order.Mon... |
theorem support_binomial' (k m : ℕ) (x y : R) :
Polynomial.support (C x * X ^ k + C y * X ^ m) ⊆ {k, m} :=
| Mathlib/Algebra/Polynomial/Basic.lean | 777 | 779 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Matrix.RowCol
import Mathli... | Fintype.prod_equiv (swap i j) _ _ (by simp [apply_swap_eq_self hpij])
simp [this, sign_swap hij, -sign_swap', prod_mul_distrib])
(fun σ _ _ => (not_congr mul_swap_eq_iff).mpr hij) (fun _ _ => mem_univ _) fun σ _ =>
mul_swap_involutive i j σ
@[simp]
theorem det_mul (M N : Matrix n n R) : d... | Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean | 128 | 141 |
/-
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... | theorem Finset.card_compl_add_card [DecidableEq α] [Fintype α] (s : Finset α) :
#sᶜ + #s = Fintype.card α := by
| Mathlib/Data/Fintype/Card.lean | 139 | 140 |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Real.Cardinality
import Mathlib.Topology.TietzeExtension
/-!
# Not normal topological spaces
In this file we prove (see `IsClosed.not_normal_of... | [DiscreteTopology s] : #s < 𝔠 := by
-- Proof by contradiction: assume `𝔠 ≤ #s`
by_contra! h
-- Choose a countable dense set `t : Set X`
rcases exists_countable_dense X with ⟨t, htc, htd⟩
haveI := htc.to_subtype
-- To obtain a contradiction, we will prove `2 ^ 𝔠 ≤ 𝔠`.
refine (Cardinal.cantor 𝔠).no... | Mathlib/Topology/Separation/NotNormal.lean | 26 | 53 |
/-
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 not_wOppSide_bot (x y : P) : ¬(⊥ : AffineSubspace R P).WOppSide x y :=
fun ⟨_, h, _⟩ => h.elim
| Mathlib/Analysis/Convex/Side.lean | 194 | 195 |
/-
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... |
/-! ### Limit ordinals -/
| Mathlib/SetTheory/Ordinal/Arithmetic.lean | 191 | 192 |
/-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.Tactic.CategoryTheory.Monoidal.Basic
import Mathlib.CategoryTheory.Closed.Monoidal
import Mathlib.Tactic.ApplyFun
/-!
# Rigid (autonomous) monoida... | /-- If `Y Y'` have an exact pairing,
then the functor `tensor_right Y` is left adjoint to `tensor_right Y'`.
-/
def tensorRightAdjunction (Y Y' : C) [ExactPairing Y Y'] : tensorRight Y ⊣ tensorRight Y' :=
| Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean | 346 | 349 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Data.Nat.Choose.Sum
impo... | theorem support_smul [SMulZeroClass S R] (r : S) (p : R[X]) :
support (r • p) ⊆ support p := by
intro i hi
simp? [mem_support_iff] at hi ⊢ says simp only [mem_support_iff, coeff_smul, ne_eq] at hi ⊢
contrapose! hi
| Mathlib/Algebra/Polynomial/Coeff.lean | 50 | 54 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Order.Sub.WithTop
import Mathlib.Data.NNReal.Defs
import Mathlib.Order.Interval.Set.... |
@[simp, norm_cast] lemma toNNReal_coe (r : ℝ≥0) : (r : ℝ≥0∞).toNNReal = r := rfl
| Mathlib/Data/ENNReal/Basic.lean | 212 | 213 |
/-
Copyright (c) 2021 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Alena Gusakov, Yaël Dillies
-/
import Mathlib.Data.Finset.Grade
import Mathlib.Data.Finset.Sups
import Mathlib.Logic.Function.Iterate
/-!
# Shadows
This file defines shad... | constructor
· rintro ⟨u, rfl, hut, htu⟩
exact ⟨_, htu, sdiff_subset, by rw [sdiff_sdiff_eq_self hut]⟩
· rintro ⟨s, hs, hst, rfl⟩
exact ⟨_, rfl, sdiff_subset, by rwa [sdiff_sdiff_eq_self hst]⟩
/-- `t ∈ ∂⁺^k 𝒜` iff `t` is exactly `k` elements less than something in `𝒜`.
See also `Finset.mem_upShadow_ite... | Mathlib/Combinatorics/SetFamily/Shadow.lean | 247 | 258 |
/-
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... | ContinuousWithinAt (fun x => setToFun μ T hT (fs x)) s x₀ :=
tendsto_setToFun_filter_of_dominated_convergence hT bound ‹_› ‹_› ‹_› ‹_›
theorem continuousAt_setToFun_of_dominated (hT : DominatedFinMeasAdditive μ T C) {fs : X → α → E}
{x₀ : X} {bound : α → ℝ} (hfs_meas : ∀ᶠ x in 𝓝 x₀, AEStronglyMeasurable (fs... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 1,089 | 1,098 |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.GroupTheory.GroupAction.Pointwise
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.LocallyConvex.BalancedCoreHull
import Mathlib.Analysis.... | IsVonNBounded 𝕜 (s + t) ↔ IsVonNBounded 𝕜 s ∧ IsVonNBounded 𝕜 t :=
⟨fun h ↦ ⟨h.of_add_left ht, h.of_add_right hs⟩, and_imp.2 IsVonNBounded.add⟩
| Mathlib/Analysis/LocallyConvex/Bounded.lean | 303 | 304 |
/-
Copyright (c) 2019 Johannes Hölzl, Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Zhouhang Zhou
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable
import M... |
end compQuasiMeasurePreserving
section compMeasurePreserving
| Mathlib/MeasureTheory/Function/AEEqFun.lean | 228 | 231 |
/-
Copyright (c) 2024 Brendan Murphy. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Brendan Murphy
-/
import Mathlib.RingTheory.Regular.IsSMulRegular
import Mathlib.RingTheory.Artinian.Module
import Mathlib.RingTheory.Nakayama
import Mathlib.Algebra.Equiv.TransferInst... | Iff.trans (isWeaklyRegular_cons_iff M r []) (and_iff_left (.nil R _))
lemma isWeaklyRegular_append_iff (rs₁ rs₂ : List R) :
IsWeaklyRegular M (rs₁ ++ rs₂) ↔
IsWeaklyRegular M rs₁ ∧
IsWeaklyRegular (M ⧸ (Ideal.ofList rs₁ • ⊤ : Submodule R M)) rs₂ := by
induction rs₁ generalizing M with
| nil =>
... | Mathlib/RingTheory/Regular/RegularSequence.lean | 369 | 381 |
/-
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, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Data.Nat.Prime.Basic
import Ma... | · rw [List.subperm_iff]
rintro ⟨u, hu1, hu2⟩
rw [← Nat.prod_primeFactorsList hb, ← hu1.prod_eq, ← prod_replicate]
exact hu2.prod_dvd_prod
· rintro ⟨c, rfl⟩
rw [Ne, pow_succ', mul_assoc, mul_eq_zero, _root_.not_or] at hb
rw [pow_succ', mul_assoc, replicate_succ,
(Nat.perm_pr... | Mathlib/Data/Nat/Factors.lean | 245 | 257 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Sébastien Gouëzel, Yury Kudryashov, Dylan MacKenzie, Patrick Massot
-/
import Mathlib.Algebra.BigOperators.Module
import Mathlib.Algebra.Order.Field.Power
import M... | apply cauchySeq_sum_of_eventually_eq this
(NormedAddCommGroup.cauchy_series_of_le_geometric' hr₁ _)
· exact C
| Mathlib/Analysis/SpecificLimits/Normed.lean | 562 | 564 |
/-
Copyright (c) 2020 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Matrix
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.Tactic.NoncommRing
/-!
# Lie algebras of skew-adjoint endomorphisms o... | change Bᵀ * J = J * (-B) at hB
rw [Matrix.lie_transpose, LieRing.of_associative_ring_bracket,
| Mathlib/Algebra/Lie/SkewAdjoint.lean | 98 | 99 |
/-
Copyright (c) 2022 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky, Floris van Doorn
-/
import Mathlib.Data.Nat.Find
import Mathlib.Data.PNat.Basic
/-!
# Explicit least witnesses to existentials on positive natural numbers
Implement... | @[simp]
theorem find_lt_iff (n : ℕ+) : PNat.find h < n ↔ ∃ m < n, p m :=
⟨fun h2 => ⟨PNat.find h, h2, PNat.find_spec h⟩, fun ⟨_, hmn, hm⟩ =>
(PNat.find_min' h hm).trans_lt hmn⟩
@[simp]
| Mathlib/Data/PNat/Find.lean | 71 | 76 |
/-
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 not_modEq_iff_toIcoMod_eq_toIocMod : ¬a ≡ b [PMOD p] ↔ toIcoMod hp a b = toIocMod hp a b :=
(modEq_iff_toIcoMod_ne_toIocMod _).not_left
| Mathlib/Algebra/Order/ToIntervalMod.lean | 552 | 554 |
/-
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.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sort
/-!
# Compositions
A compositio... | motive n c :=
reverse_reverse c ▸ c.reverse.recOnSingleAppend zero fun k n c ih ↦ by
convert append_single k n c.reverse ih using 1
· apply add_comm
· rw [reverse_append, reverse_single]
apply cast_heq
end Composition
| Mathlib/Combinatorics/Enumerative/Composition.lean | 668 | 676 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Order.Filter.AtTopBot.Finset
import Mathlib.Topology.Algebra.InfiniteSum.Group
import Mathlib.Topology.Algebra.Star
/-!
# Topological sums and functor... | @[to_additive]
lemma HasProd.sum {α β M : Type*} [CommMonoid M] [TopologicalSpace M] [ContinuousMul M]
{f : α ⊕ β → M} {a b : M}
(h₁ : HasProd (f ∘ Sum.inl) a) (h₂ : HasProd (f ∘ Sum.inr) b) : HasProd f (a * b) := by
have : Tendsto ((∏ b ∈ ·, f b) ∘ sumEquiv.symm) (atTop.map sumEquiv) (nhds (a * b)) := by
... | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | 84 | 101 |
/-
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 | 2,038 | 2,046 | |
/-
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... |
@[simp]
theorem restrict_add_restrict_compl (hs : MeasurableSet s) :
μ.restrict s + μ.restrict sᶜ = μ := by
| Mathlib/MeasureTheory/Measure/Restrict.lean | 255 | 258 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalcul... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 605 | 609 | |
/-
Copyright (c) 2023 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Sym.Sym2
/-! # Unordered tuples of elements of a list
Defines `List.sym` and the specialized `List.sym2` for comp... | simp only [mem_cons, not_or] at h
rw [List.sym2, map_cons, map_cons, disjoint_cons_left, disjoint_append_right,
disjoint_cons_right]
| Mathlib/Data/List/Sym.lean | 142 | 144 |
/-
Copyright (c) 2023 Yaël Dillies, Chenyi Li. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chenyi Li, Ziyu Wang, Yaël Dillies
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.InnerProductSpace.Basic
/-!
# Uniformly and strongly convex functions
I... |
lemma UniformConcaveOn.neg (hf : UniformConcaveOn s φ f) : UniformConvexOn s φ (-f) := by
refine ⟨hf.1, fun x hx y hy a b ha hb hab ↦ le_of_neg_le_neg ?_⟩
simpa [add_comm, -neg_le_neg_iff, ← le_sub_iff_add_le', sub_eq_add_neg, neg_add]
| Mathlib/Analysis/Convex/Strong.lean | 100 | 103 |
/-
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.Bundle
import Mathlib.Data.Set.Image
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Topology.O... | have : ∀ z ∈ e.baseSet ×ˢ (univ : Set F),
TotalSpace.mk z.1 (e.symm z.1 z.2) = e.toPartialHomeomorph.symm z := by
rintro x ⟨hx : x.1 ∈ e.baseSet, _⟩
rw [e.mk_symm hx]
refine ContinuousOn.congr ?_ this
rw [← e.target_eq]
exact e.toPartialHomeomorph.continuousOn_symm
| Mathlib/Topology/FiberBundle/Trivialization.lean | 564 | 571 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Ordmap.Invariants
/-!
# Verification of `Ordnode`
This file uses the invariants defined in `Mathlib.Data.Ordmap.Invariants` to construct `Ordset... | Mathlib/Data/Ordmap/Ordset.lean | 1,232 | 1,233 | |
/-
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.Analytic.Within
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries
/-!
# Higher d... | simpa [continuousWithinAt_univ] using h.continuousWithinAt
theorem ContDiffAt.analyticAt (h : ContDiffAt 𝕜 ω f x) : AnalyticAt 𝕜 f x := by
rw [← contDiffWithinAt_univ] at h
rw [← analyticWithinAt_univ]
exact h.analyticWithinAt
| Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 955 | 961 |
/-
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, Yaël Dillies
-/
import Mathlib.Algebra.Order.Group.DenselyOrdered
import Mathlib.Data.Real.Archimedean
import Mathlib.Order.LiminfLimsu... | intro x hx
by_contra!
have : (Set.Ioo c F.limsSup).Nonempty := ⟨x, ⟨hx, this⟩⟩
simp only [hc, Set.not_nonempty_empty] at this
apply liminf_le_of_frequently_le _ (bdd_above.isBoundedUnder f_decr)
exact B.mono fun x hx ↦ f_decr hx
push_neg at h'
| Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | 312 | 318 |
/-
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... | Mathlib/Data/Set/Function.lean | 1,953 | 1,955 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Floor.Defs
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Algebra.Order.Floor.Semiring
deprecated_module (sinc... | Mathlib/Algebra/Order/Floor.lean | 296 | 297 | |
/-
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... | by_cases h : ∀ j, r j i → f j < f i
· refine ⟨typein r' ⟨i, h⟩, typein_lt_type _ _, ?_⟩
rw [bfamilyOfFamily'_typein]
· push_neg at h
obtain ⟨hji, hij⟩ := wo.wf.min_mem _ h
refine ⟨typein r' ⟨_, fun k hkj => lt_of_lt_of_le ?_ hij⟩, typein_lt_type _ _, ?_⟩
· by_contra! H
exact ... | Mathlib/SetTheory/Cardinal/Cofinality.lean | 516 | 541 |
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Algebra.Order.Group.Pointwise.Interval
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Cardinal
import Mathlib.SetTheory.Cardi... | @[simp]
theorem cantorFunctionAux_false (h : f n = false) : cantorFunctionAux c f n = 0 := by
| Mathlib/Data/Real/Cardinality.lean | 64 | 65 |
/-
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... | subst hf
exact IsZero.of_epi_zero X Y
| Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | 210 | 212 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Subalgebra.Prod
import Mathlib.Algebra.Algebra.Subalgebra.Tower
import Mathlib.LinearAlgebra.Basis.Basic
import Mathlib.LinearAlgebra.Prod
/-!
# ... | Mathlib/RingTheory/Adjoin/Basic.lean | 376 | 378 | |
/-
Copyright (c) 2021 Aaron Anderson, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Kevin Buzzard, Yaël Dillies, Eric Wieser
-/
import Mathlib.Data.Finset.Lattice.Union
import Mathlib.Data.Finset.Pairwise
import Mathlib.Data.Finset.Prod
i... |
@[deprecated (since := "2024-11-24")]
alias CompleteLattice.SetIndependent.pairwiseDisjoint := sSupIndep.pairwiseDisjoint
theorem sSupIndep_singleton (a : α) : sSupIndep ({a} : Set α) := fun i hi ↦ by
simp_all
@[deprecated (since := "2024-11-24")]
alias CompleteLattice.setIndependent_singleton := sSupIndep_singlet... | Mathlib/Order/SupIndep.lean | 264 | 273 |
/-
Copyright (c) 2023 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.MeasureTheory.Constructions.Pi
import Mathlib.MeasureTheory.Integral.Prod
/-!
# Integration with respect to a finite product of measures
On a finite pr... |
end Integrable
variable {𝕜 : Type*} [RCLike 𝕜]
/-- A version of **Fubini's theorem** in `n` variables, for a natural number `n`. -/
theorem integral_fin_nat_prod_eq_prod {n : ℕ} {E : Fin n → Type*}
[∀ i, MeasureSpace (E i)] [∀ i, SigmaFinite (volume : Measure (E i))]
(f : (i : Fin n) → E i → 𝕜) :
∫ x ... | Mathlib/MeasureTheory/Integral/Pi.lean | 65 | 84 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Sigma.Basic
import Mathlib.Algebra.Order.Ring.Nat
/-!
# A computable model of ZFA without infinity
In this file we define finite hereditary list... | · exact Subset.nil
· simp only [ofList, List.find?, List.mem_cons, forall_eq_or_imp] at *
| Mathlib/SetTheory/Lists.lean | 180 | 181 |
/-
Copyright (c) 2024 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Sanyam Gupta, Omar Haddad, David Lowry-Duda,
Lorenzo Luccioli, Pietro Monticone, Alexis Saurin, Florent Schaffhauser
-/
import Mathlib.NumberTheory.FLT.Basic
import... | have hm := S'.multiplicity_lambda_c_finite
suffices 2 ≤ multiplicity (hζ.toInteger - 1) S'.c by
obtain ⟨x, hx⟩ := pow_multiplicity_dvd (hζ.toInteger - 1) S'.c
refine ⟨λ ^ (multiplicity (hζ.toInteger - 1) S'.c - 2) * x, ?_⟩
rw [← mul_assoc, ← pow_add]
convert hx using 3
simp [this]
have := lamb... | Mathlib/NumberTheory/FLT/Three.lean | 279 | 307 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker
-/
import Mathlib.Algebra.Ring.Associated
import Mathlib.Algebra.Ring.Regular
/-!
# Monoids with normalization functions, `gcd`, and `lcm`
This file de... | apply (dvd_gcd_mul_of_dvd_mul h2).trans
rw [h1.mul_left_dvd]
have h4 : d₁ ^ k ≠ 0 := by
| Mathlib/Algebra/GCDMonoid/Basic.lean | 538 | 540 |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Kevin Buzzard, Kim Morrison, Johan Commelin, Chris Hughes,
Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.Notation.Pi
imp... | (n : ℤ) : f ∘ (g ^ n) = f ∘ g ^ n := by simp
end mul_one
| Mathlib/Algebra/Group/Hom/Defs.lean | 485 | 488 |
/-
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.Nat.Find
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.Common
/-!
# Coinductive formalization of unbounded computations.
This fil... | <;> apply recOn s (fun a => _) fun s' => _ <;> intro e h
· simp only [ret_bind] at h
| Mathlib/Data/Seq/Computation.lean | 689 | 690 |
/-
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 | 1,331 | 1,337 | |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Peter Nelson
-/
import Mathlib.Order.Antichain
/-!
# Minimality and Maximality
This file proves basic facts about minimality and maximality
of an element with respect to ... | Minimal (· ∈ f '' s) (f a) ↔ Minimal (· ∈ s) a := by
refine ⟨fun h ↦ ⟨ha, fun y hys ↦ ?_⟩, minimal_mem_image_monotone hf⟩
rw [← hf ha hys, ← hf hys ha]
exact h.le_of_le (mem_image_of_mem f hys)
| Mathlib/Order/Minimal.lean | 440 | 444 |
/-
Copyright (c) 2020 Jalex Stark. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jalex Stark, Kim Morrison, Eric Wieser, Oliver Nash, Wen Yang
-/
import Mathlib.Data.Matrix.Basic
/-!
# Matrices with a single non-zero element.
This file provides `Matrix.stdBasisMatri... | (h : ∀ i j, f ∘ₗ stdBasisMatrixLinearMap R i j = g ∘ₗ stdBasisMatrixLinearMap R i j) :
f = g :=
LinearMap.toAddMonoidHom_injective <| ext_addMonoidHom fun i j =>
| Mathlib/Data/Matrix/Basis.lean | 160 | 162 |
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Algebra.Polynomial.Eval.Defs
import Mathlib.LinearAlgebra.Dimension.Constructions
/-!
# Linear recurrence
Informally, a "linear recurrence" is an... | (heq : ∀ n : Fin E.order, u n = init n) : ∀ n, u n = E.mkSol init n := by
intro n
rw [mkSol]
split_ifs with h'
| Mathlib/Algebra/LinearRecurrence.lean | 92 | 95 |
/-
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.Group.Nat.Defs
import Mathlib.CategoryTheory.Category.Preorder
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Functor.Const
import M... |
@[simp]
| Mathlib/CategoryTheory/ComposableArrows.lean | 324 | 325 |
/-
Copyright (c) 2024 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Lie.CartanSubalgebra
import Mathlib.Algebra.Lie.Rank
/-!
# Existence of Cartan subalgebras
In this file we prove existence of Cartan subalge... | Suppose that `engel K x` is minimal amongst the Engel subalgebras `engel K y` for `y ∈ U`.
Then `engel K x ≤ engel K y` for all `y ∈ U`.
Lemma 2 in [barnes1967]. -/
lemma engel_isBot_of_isMin (hLK : finrank K L ≤ #K) (U : LieSubalgebra K L)
(E : {engel K x | x ∈ U}) (hUle : U ≤ E) (hmin : IsMin E) :
IsBot E :=... | Mathlib/Algebra/Lie/CartanExists.lean | 124 | 345 |
/-
Copyright (c) 2023 Geoffrey Irving. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler, Geoffrey Irving, Stefan Kebekus
-/
import Mathlib.Analysis.Analytic.Composition
import Mathlib.Analysis.Analytic.Linear
import Mathlib.Analysis.NormedSpace.OperatorNor... | fun z hz ↦ (hf z hz).neg
theorem AnalyticOn.sub (hf : AnalyticOn 𝕜 f s) (hg : AnalyticOn 𝕜 g s) :
AnalyticOn 𝕜 (f - g) s :=
fun z hz => (hf z hz).sub (hg z hz)
theorem AnalyticOnNhd.sub (hf : AnalyticOnNhd 𝕜 f s) (hg : AnalyticOnNhd 𝕜 g s) :
AnalyticOnNhd 𝕜 (f - g) s :=
fun z hz => (hf z hz).sub (... | Mathlib/Analysis/Analytic/Constructions.lean | 238 | 251 |
/-
Copyright (c) 2021 Henry Swanson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Henry Swanson
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Combinatorics.Derangements.Basic
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Tactic.Ring
/-!... | ring
theorem card_derangements_fin_eq_numDerangements {n : ℕ} :
card (derangements (Fin n)) = numDerangements n := by
induction n using Nat.strongRecOn with | ind n hyp => _
rcases n with _ | _ | n
| Mathlib/Combinatorics/Derangements/Finite.lean | 87 | 92 |
/-
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... | end Complex
| Mathlib/Data/Complex/Exponential.lean | 694 | 694 |
/-
Copyright (c) 2021 Julian Kuelshammer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Kuelshammer
-/
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Data.Nat.Factorization.B... | @[to_additive]
protected alias ⟨_, ExponentExists.exponent_ne_zero⟩ := exponent_ne_zero
@[to_additive]
theorem exponent_pos : 0 < exponent G ↔ ExponentExists G :=
pos_iff_ne_zero.trans exponent_ne_zero
| Mathlib/GroupTheory/Exponent.lean | 106 | 111 |
/-
Copyright (c) 2023 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Joël Riou
-/
import Mathlib.Algebra.Category.ModuleCat.ChangeOfRings
import Mathlib.Algebra.Category.Ring.Basic
/-!
# Presheaves of modules over a presheaf of rings.
Give... | @[ext]
lemma hom_ext {f g : M₁ ⟶ M₂} (h : ∀ (X : Cᵒᵖ), f.app X = g.app X) :
f = g := Hom.ext (by ext1; apply h)
| Mathlib/Algebra/Category/ModuleCat/Presheaf.lean | 85 | 88 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2
import Mathlib.MeasureTheory.Measure.Real
/-! # Conditional expectation in L1
This file contains ... | have hS_meas : ∀ i, MeasurableSet[m] (S i) := measurableSet_spanningSets (μ.trim hm)
have hS_meas0 : ∀ i, MeasurableSet (S i) := fun i => hm _ (hS_meas i)
have hs_eq : s = ⋃ i, S i ∩ s := by
simp_rw [Set.inter_comm]
rw [← Set.inter_iUnion, iUnion_spanningSets (μ.trim hm), Set.inter_univ]
have hS_finite ... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean | 473 | 488 |
/-
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.Option.NAry
import Mathlib.Data.Seq.Computation
import Mathlib.Tactic.ApplyFun
import Mathlib.Data.List.Basic
/-!
# Possibly infinite lists
This... | cases s
· simp [join_cons_cons, join_cons_nil]
· simpa [join_cons_cons, join_cons_nil] using Or.inr ⟨_, _, S, rfl, rfl⟩
@[simp]
theorem join_append (S T : Seq (Seq1 α)) : join (append S T) = append (join S) (join T) := by
| Mathlib/Data/Seq/Seq.lean | 964 | 969 |
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