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/-
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.Data.Set.Lattice
import Mathlib.Data.Set.Pairwise.Basic
/-!
# Relations holding pairwise
In this file we prove many facts about `Pairwise` and the se... | obtain ⟨d, hd, hb⟩ := hb
obtain hcd | hcd := eq_or_ne (g c) (g d)
· exact hg d hd (hcd ▸ ha) hb hab
· exact (hs hc hd <| ne_of_apply_ne _ hcd).mono
(le_iSup₂ (f := fun i _ => f i) a ha)
(le_iSup₂ (f := fun i _ => f i) b hb)
/-- If the suprema of columns are pairwise disjoint and suprema of rows as ... | Mathlib/Data/Set/Pairwise/Lattice.lean | 72 | 84 |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.Composition
import Mathlib.Data.Matrix.ConjTranspose
/-!... | This is the block form of `Matrix.diag`, and the left-inverse of `Matrix.blockDiagonal'`. -/
def blockDiag' (M : Matrix (Σ i, m' i) (Σ i, n' i) α) (k : o) : Matrix (m' k) (n' k) α :=
of fun i j => M ⟨k, i⟩ ⟨k, j⟩
-- TODO: set as an equation lemma for `blockDiag'`, see https://github.com/leanprover-community/mathlib4... | Mathlib/Data/Matrix/Block.lean | 714 | 720 |
/-
Copyright (c) 2021 Christopher Hoskin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christopher Hoskin, Yaël Dillies
-/
import Mathlib.Algebra.Order.Group.Unbundled.Abs
import Mathlib.Algebra.Notation
/-!
# Positive & negative parts
Mathematical structures posse... | Mathlib/Algebra/Order/Group/PosPart.lean | 272 | 273 | |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Ralf Stephan, Neil Strickland, Ruben Van de Velde
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Positive.Ring
import Mathlib.... | Mathlib/Data/PNat/Basic.lean | 416 | 420 | |
/-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
import Mathlib.RingTheory.Derivation.ToSquareZero
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.IsTensorProduct
import Mathlib.... | · rw [zero_mul, map_zero, map_zero, zero_smul, smul_zero, add_zero]
swap
· intro x₁ y₁ h₁ h₂
rw [add_mul, map_add, map_add, map_add, add_smul, smul_add, h₁, h₂, add_add_add_comm]
intro x₁ x₂
refine TensorProduct.induction_on y ?_ ?_ ?_
· rw [mul_zero, map_zero, map_zero, zero_smul, smul_zero, add_zero]
... | Mathlib/RingTheory/Kaehler/Basic.lean | 82 | 103 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
/-!
# (Generalized) Boolean algebras
A Boolean algebra is a bounded distributive lattice with a complement ope... | sdiff_eq a b := by
refine hf ((map_sdiff _ _).trans (sdiff_eq.trans ?_))
| Mathlib/Order/BooleanAlgebra.lean | 781 | 782 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Kim Morrison
-/
import Mathlib.CategoryTheory.Functor.Currying
import Mathlib.CategoryTheory.Subobject.FactorThru
import Mathlib.CategoryTheory.Subobject.WellPowered
import... |
theorem inf_def {B : C} (m m' : Subobject B) : m ⊓ m' = (inf.obj m).obj m' :=
| Mathlib/CategoryTheory/Subobject/Lattice.lean | 426 | 427 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Rat
import Mathlib.Algebra.Ring.Int.Parity
import Mathlib.Data.PNat.Defs... | have := congr_arg Int.natAbs e
simp only [Int.natAbs_mul, Int.natAbs_natCast] at this; simp [this]
theorem den_dvd (a b : ℤ) : ((a /. b).den : ℤ) ∣ b := by
by_cases b0 : b = 0; · simp [b0]
rcases e : a /. b with ⟨n, d, h, c⟩
rw [mk'_eq_divInt, divInt_eq_iff b0 (ne_of_gt (Int.natCast_pos.2 (Nat.pos_of_ne_zero... | Mathlib/Data/Rat/Lemmas.lean | 24 | 30 |
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.CartanSubalgebra
import Mathlib.Algebra.Lie.Weights.Basic
/-!
# Weights and roots of Lie modules and Lie algebras with respect to Cartan subalge... | let xy : rootSpace H 0 ⊗[R] rootSpace H 0 := ⟨x, hx⟩ ⊗ₜ ⟨y, hy⟩
suffices (rootSpaceProduct R L H 0 0 0 (add_zero 0) xy : L) ∈ rootSpace H 0 by
rwa [rootSpaceProduct_tmul, Subtype.coe_mk, Subtype.coe_mk] at this
exact (rootSpaceProduct R L H 0 0 0 (add_zero 0) xy).property }
| Mathlib/Algebra/Lie/Weights/Cartan.lean | 154 | 157 |
/-
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... | @[reassoc]
theorem coprod.map_codiag {X Y : C} (f : X ⟶ Y) [HasBinaryCoproduct X X] [HasBinaryCoproduct Y Y] :
coprod.map f f ≫ codiag Y = codiag X ≫ f := by simp
| Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | 822 | 824 |
/-
Copyright (c) 2024 Sophie Morel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sophie Morel
-/
import Mathlib.Analysis.NormedSpace.PiTensorProduct.ProjectiveSeminorm
import Mathlib.LinearAlgebra.Isomorphisms
/-!
# Injective seminorm on the tensor of a finite famil... | apply ContinuousLinearMap.coe_injective
ext
simp only [ContinuousLinearMap.coe_comp, mapL_coe, LinearMap.compMultilinearMap_apply,
LinearMap.coe_comp, ContinuousLinearMap.coe_coe, Function.comp_apply, map_tprod,
liftIsometry_apply_apply, lift.tprod, ContinuousMultilinearMap.coe_coe,
ContinuousMultilin... | Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean | 365 | 372 |
/-
Copyright (c) 2022 Pierre-Alexandre Bazin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pierre-Alexandre Bazin
-/
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.ZMod
import Mathlib.GroupTheory.Torsion
import Mathlib.LinearAlgebra.Isomorphism... | rw [← mem_torsionBySet_iff, h]
trivial⟩
/-- An `a`-torsion module is a module whose `a`-torsion submodule is the full space. -/
theorem isTorsionBy_iff_torsionBy_eq_top : IsTorsionBy R M a ↔ torsionBy R M a = ⊤ := by
| Mathlib/Algebra/Module/Torsion.lean | 328 | 332 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Eval.Degree
import Mathlib.Algebra.Prime.Lemmas
/-!
# Theory of degrees of polynomials
S... | rw [coeff_add, natDegree_le_iff_coeff_eq_zero.mp qn _ hm, add_zero]
theorem natDegree_add_le_iff_right {n : ℕ} (p q : R[X]) (pn : p.natDegree ≤ n) :
(p + q).natDegree ≤ n ↔ q.natDegree ≤ n := by
| Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 84 | 87 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | Mathlib/Order/Filter/Basic.lean | 1,212 | 1,214 | |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Anatole Dedecker
-/
import Mathlib.Analysis.LocallyConvex.BalancedCoreHull
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.LinearAlgebra.FreeM... |
theorem isClosedEmbedding_smul_left [T2Space E] {c : E} (hc : c ≠ 0) :
IsClosedEmbedding fun x : 𝕜 => x • c :=
LinearMap.isClosedEmbedding_of_injective (LinearMap.ker_toSpanSingleton 𝕜 E hc)
| Mathlib/Topology/Algebra/Module/FiniteDimension.lean | 529 | 532 |
/-
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 | 546 | 554 | |
/-
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.Algebra.Order.Ring.Nat
import Mathlib.Logic.Encodable.Pi
import Mathlib.Logic.Function.Iterate
/-!
# The primitive recursive functions
The primitive ... | simpa using comp' (prec hg hh) (hf.cons idv)
theorem pred : @Primrec' 1 fun v => v.head.pred :=
(prec' head (const 0) head).of_eq fun v => by simp; cases v.head <;> rfl
theorem add : @Primrec' 2 fun v => v.head + v.tail.head :=
(prec head (succ.comp₁ _ (tail head))).of_eq fun v => by
simp; induction v.head ... | Mathlib/Computability/Primrec.lean | 1,321 | 1,328 |
/-
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 | 671 | 672 | |
/-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.SetTheory.Cardinal.Finite
/-!
# Cardinality of finite types
The cardinality of a finite type `α` is given by `Nat.card α`. This function has
the "junk val... | Mathlib/Data/Finite/Card.lean | 223 | 227 | |
/-
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
-/
import Mathlib.Data.Set.Piecewise
import Mathlib.Order.Filter.Tendsto
import Mathlib.Order.Filter.Bases.Finite
/-!
# (Co)product of a family of f... | have : ∀ i, (t i).Nonempty := fun i => nonempty_of_mem (htf i)
choose g hg using this
have : update g i x ∈ I'.pi t := fun j _ => by
rcases eq_or_ne j i with (rfl | hne) <;> simp [*]
simpa using hts this i hi
@[simp]
theorem pi_mem_pi_iff [∀ i, NeBot (f i)] {I : Set ι} (hI : I.Finite) :
I.pi s ∈ pi f ↔... | Mathlib/Order/Filter/Pi.lean | 80 | 88 |
/-
Copyright (c) 2024 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Order.Hom.Ring
import Mathlib.Data.ENat.Basic
import Mathlib.SetTheory.Cardinal.Basic
/-!
# Conversion between `Cardinal` and `ℕ∞`
In this ... | simp_rw [toENat_ofENat, enat_gc _]
| Mathlib/SetTheory/Cardinal/ENat.lean | 275 | 276 |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Localization.LocalizerMorphism
import Mathlib.CategoryTheory.HomCongr
/-!
# Bijections between morphisms in two localized categories
Given two ... | @[simp]
lemma homEquiv_refl (f : L₁.obj X ⟶ L₁.obj Y) :
homEquiv W L₁ L₁ f = f := by
apply LocalizerMorphism.id_homMap
| Mathlib/CategoryTheory/Localization/HomEquiv.lean | 137 | 140 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Sum.Basic
import Mathlib.Logic.Equiv.Option
import Mathlib.Logic.Equiv.Sum
import Mathlib.Logic.Function.Conjugate
impor... | Mathlib/Logic/Equiv/Basic.lean | 1,903 | 1,907 | |
/-
Copyright (c) 2021 Alena Gusakov, Bhavik Mehta, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alena Gusakov, Bhavik Mehta, Kyle Miller
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Fintype.Powerset
import Mathlib.Data.Set.Finite.Basic
/-!
#... | (ha : ∀ s : Finset ι, s.Nonempty → s ≠ univ → #s < #(s.biUnion t))
(s' : Finset { x' : ι | x' ≠ x }) : #s' ≤ #(s'.biUnion fun x' => (t x').erase a) := by
haveI := Classical.decEq ι
specialize ha (s'.image fun z => z.1)
rw [image_nonempty, Finset.card_image_of_injective s' Subtype.coe_injective] at ha
by... | Mathlib/Combinatorics/Hall/Finite.lean | 50 | 70 |
/-
Copyright (c) 2024 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Composition.MapComap
import Mathlib.Probability.Martingale.Convergence
import Mathlib.Probability.Process.PartitionFiltration
/-!
# Ker... | · exact fun _ _ _ ↦ (integrable_densityProcess hκν _ _ hs).integrableOn
· intro x hx _
rw [setIntegral_densityProcess hκν i a hs hx,
setIntegral_densityProcess_of_le hκν hij a hs hx]
· exact StronglyMeasurable.aestronglyMeasurable
(stronglyMeasurable_countableFiltration_densityProcess κ ν i a hs)
... | Mathlib/Probability/Kernel/Disintegration/Density.lean | 287 | 298 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathli... | _ ⊆ Set.range f := Set.image_subset_range f ↑s
/-- If the only elements outside `s` are those left fixed by `σ`, then mapping by `σ` has no effect.
-/
theorem map_perm {σ : Equiv.Perm α} (hs : { a | σ a ≠ a } ⊆ s) : s.map (σ : α ↪ α) = s :=
coe_injective <| (coe_map _ _).trans <| Set.image_perm hs
| Mathlib/Data/Finset/Image.lean | 101 | 106 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Joël Riou
-/
import Mathlib.CategoryTheory.Sites.Subsheaf
import Mathlib.CategoryTheory.Sites.CompatibleSheafification
import Mathlib.CategoryTheory.Sites.LocallyInjective
/-... | IsLocallySurjective ((sheafCompose J (forget A)).map φ) :=
(Presheaf.isLocallySurjective_iff_whisker_forget J φ.val).1 inferInstance
attribute [local instance] Types.instFunLike Types.instConcreteCategory in
theorem isLocallySurjective_iff_isIso {F G : Sheaf J (Type w)} (f : F ⟶ G) :
IsLocallySurjective f ↔ ... | Mathlib/CategoryTheory/Sites/LocallySurjective.lean | 356 | 361 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Data.Stream.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Data.List.Defs
import Mathlib.Data.Nat.Basic
import Mathlib.Tactic.Common... | fun ainl : a ∈ l => mem_cons_of_mem b (mem_append_stream_left s ainl)
@[simp]
theorem take_zero (s : Stream' α) : take 0 s = [] :=
rfl
| Mathlib/Data/Stream/Init.lean | 501 | 505 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Julian Kuelshammer
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.Pointwise.Set.Finite
import Mathlib.Alge... | simp_rw [← isPeriodicPt_mul_iff_pow_eq_one, ← minimalPeriod_eq_minimalPeriod_iff, orderOf]
/-- An injective homomorphism of monoids preserves orders of elements. -/
@[to_additive "An injective homomorphism of additive monoids preserves orders of elements."]
theorem orderOf_injective {H : Type*} [Monoid H] (f : G →* ... | Mathlib/GroupTheory/OrderOfElement.lean | 314 | 318 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Patrick Stevens
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.BigOperators.Ring.Finset
import ... | induction n with
| zero => simp
| succ n hn => rw [Nat.sum_antidiagonal_succ, hn, Nat.choose_succ_succ (d + (n + 1)), ← add_assoc]
end Finset
| Mathlib/Data/Nat/Choose/Sum.lean | 234 | 239 |
/-
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 ... | (hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) {t : R} (ht : 0 < t) : s.SSameSide x (t • (x -ᵥ p₁) +ᵥ p₂) :=
(sSameSide_smul_vsub_vadd_left hx hp₁ hp₂ ht).symm
theorem sSameSide_lineMap_left {s : AffineSubspace R P} {x y : P} (hx : x ∈ s) (hy : y ∉ s) {t : R}
(ht : 0 < t) : s.SSameSide (lineMap x y t) y :=
sSameSide_smul_v... | Mathlib/Analysis/Convex/Side.lean | 622 | 632 |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.ObjectProperty.ClosedUnderIsomorphisms
import Mathlib.CategoryTheory.MorphismProperty.Composition
import Mathlib.CategoryTheory.Localization.Adjun... | constructor
· intro hf
obtain ⟨g, hg⟩ := (hf _ hX).2 (𝟙 X)
exact ⟨g, hg, (hf _ hY).1 (by simp only [reassoc_of% hg, comp_id])⟩
· apply W_of_isIso
instance : (W P).RespectsIso where
| Mathlib/CategoryTheory/Localization/Bousfield.lean | 98 | 104 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jujian Zhang
-/
import Mathlib.Algebra.GroupWithZero.Subgroup
import Mathlib.Algebra.Order.Group.Action
import Mathlib.LinearAlgebra.Finsupp.Supported
import Mathlib.LinearAl... | zero_mem' := ⟨Submodule.zero_mem _, zero⟩
add_mem' := fun ⟨mem, h⟩ ⟨mem', h'⟩ ↦ ⟨_, add mem mem' h h'⟩
smul_mem' := fun r _ ⟨mem, h⟩ ↦ ⟨_, smul₁ r mem h⟩ }
| Mathlib/Algebra/Module/Submodule/Pointwise.lean | 404 | 406 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.ConditionalProbability
import Mathlib.Probability.Kernel.Basic
import Mathlib.Probability.Kernel.Composition.MeasureComp
import Mathlib.Tactic.... | simp [hωψ]
theorem IndepFun.neg_right {_mβ : MeasurableSpace β} {_mβ' : MeasurableSpace β'} [Neg β']
[MeasurableNeg β'] (hfg : IndepFun f g κ μ) :
IndepFun f (-g) κ μ := hfg.comp measurable_id measurable_neg
theorem IndepFun.neg_left {_mβ : MeasurableSpace β} {_mβ' : MeasurableSpace β'} [Neg β]
| Mathlib/Probability/Independence/Kernel.lean | 1,009 | 1,015 |
/-
Copyright (c) 2023 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.InvariantForm
import Mathlib.Algebra.Lie.Weights.Cartan
import Mathlib.Algebra.Lie.Weights.Linear
impo... | apply LinearMap.trace_comp_eq_zero_of_commute_of_trace_restrict_eq_zero
· exact IsTriangularizable.maxGenEigenspace_eq_top (1 ⊗ₜ[R] x)
· exact fun μ ↦ trace_toEnd_eq_zero_of_mem_lcs A (A ⊗[R] L)
(genWeightSpaceOf (A ⊗[R] M) μ ((1:A) ⊗ₜ[R] x)) (le_refl 1) hz
· exact commute_toEnd_of_mem_center_right (A ⊗[R... | Mathlib/Algebra/Lie/TraceForm.lean | 279 | 284 |
/-
Copyright (c) 2022 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.MeasureTheory.Integral.ExpDecay
/-!
# The Gamma function
This file defines the `Γ` functio... | Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean | 558 | 560 | |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Adam Topaz, Johan Commelin, Jakob von Raumer
-/
import Mathlib.Algebra.Homology.ImageToKernel
import Mathlib.Algebra.Homology.ShortComplex.Exact
import Mathlib.CategoryTh... | PreservesHomology L := by
apply preservesHomology_of_map_exact
intro S hS
let φ : (ShortComplex.mk _ _ (Abelian.comp_coimage_π_eq_zero S.zero)).map L ⟶ S.map L :=
{ τ₁ := 𝟙 _
τ₂ := 𝟙 _
τ₃ := L.map (Abelian.factorThruCoimage S.g)
comm₂₃ := by
dsimp
rw [Category.id_comp, ... | Mathlib/CategoryTheory/Abelian/Exact.lean | 271 | 282 |
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca, Johan Commelin
-/
import Mathlib.Algebra.GCDMonoid.IntegrallyClosed
import Mathlib.FieldTheory.Finite.Basic
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
impo... | theorem isIntegral (hpos : 0 < n) : IsIntegral ℤ μ := by
use X ^ n - 1
constructor
· exact monic_X_pow_sub_C 1 (ne_of_lt hpos).symm
· simp only [((IsPrimitiveRoot.iff_def μ n).mp h).left, eval₂_one, eval₂_X_pow, eval₂_sub,
sub_self]
| Mathlib/RingTheory/RootsOfUnity/Minpoly.lean | 40 | 45 |
/-
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.Deriv.Basic
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Covering.Bes... | = ENNReal.ofReal (∫ x in s, |(f' x).det| * g (f x) ∂μ) := by
rw [Measure.comap_apply f hf.injective (fun t ht ↦ hf.measurableSet_image' ht) _ hs,
withDensity_apply _ (hf.measurableSet_image' hs),
← ofReal_integral_eq_lintegral_ofReal hg_int
((ae_restrict_iff' (hf.measurableSet_image' hs)).mpr hg),... | Mathlib/MeasureTheory/Function/Jacobian.lean | 1,222 | 1,227 |
/-
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... | rw [Nat.pow_succ, Nat.coprime_mul_iff_left]
exact ⟨And.right, fun hab => ⟨hab.pow_left _, hab⟩⟩
| Mathlib/Data/Nat/GCD/Basic.lean | 192 | 193 |
/-
Copyright (c) 2023 Ashvni Narayanan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ashvni Narayanan, Moritz Firsching, Michael Stoll
-/
import Mathlib.Algebra.Group.EvenFunction
import Mathlib.Data.ZMod.Units
import Mathlib.NumberTheory.MulChar.Basic
/-!
# Dirichl... |
/-- The character of level `d` through which `χ` factors is uniquely determined. -/
lemma existsUnique {d : ℕ} [NeZero n] (h : FactorsThrough χ d) :
∃! χ' : DirichletCharacter R d, χ = changeLevel h.dvd χ' := by
rcases h with ⟨hd, χ₂, rfl⟩
| Mathlib/NumberTheory/DirichletCharacter/Basic.lean | 120 | 124 |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Algebra.EuclideanDomain.Int
import Mathlib.Data.Nat.Prime.Int
import Mathlib.Data.ZMod.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
/-!
# Coprim... | theorem eq_zero_iff_gcd_ne_one {a : ℤ} {p : ℕ} [pp : Fact p.Prime] :
(a : ZMod p) = 0 ↔ a.gcd p ≠ 1 := by
rw [Ne, Int.gcd_comm, ← Int.isCoprime_iff_gcd_eq_one,
(Nat.prime_iff_prime_int.1 pp.1).coprime_iff_not_dvd, Classical.not_not,
intCast_zmod_eq_zero_iff_dvd]
| Mathlib/Data/ZMod/Coprime.lean | 24 | 28 |
/-
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, Yaël Dillies, Kin Yau James Wong
-/
import Mathlib.MeasureTheory.MeasurableSpace.Constructions
import Mathlib.MeasureTheory.PiSystem
import Mathlib.Topo... | Set (∀ i : measurableCylinders.finset ht, α i) :=
((mem_measurableCylinders t).mp ht).choose_spec.choose
theorem measurableCylinders.measurableSet (ht : t ∈ measurableCylinders α) :
| Mathlib/MeasureTheory/Constructions/Cylinders.lean | 294 | 297 |
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Rémy Degenne
-/
import Mathlib.Probability.Process.Adapted
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Stopping times, stopped processes and stopped va... | theorem stoppedProcess_eq_stoppedValue {u : ι → Ω → β} {τ : Ω → ι} :
stoppedProcess u τ = fun i => stoppedValue u fun ω => min i (τ ω) :=
rfl
theorem stoppedValue_stoppedProcess {u : ι → Ω → β} {τ σ : Ω → ι} :
stoppedValue (stoppedProcess u τ) σ = stoppedValue u fun ω => min (σ ω) (τ ω) :=
rfl
theorem sto... | Mathlib/Probability/Process/Stopping.lean | 703 | 724 |
/-
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.Data.Finset.BooleanAlgebra
import Mathlib.Data.Set.Piecewise
import Mathlib.Order.Interval.Set.Basic
/-!
# Functions defined piecewise on a fins... | s.piecewise (t.piecewise f₁ f₂) g = s.piecewise f₁ g :=
s.piecewise_congr (fun _i hi => piecewise_eq_of_mem _ _ _ (h hi)) fun _ _ => rfl
| Mathlib/Data/Finset/Piecewise.lean | 79 | 81 |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.DualNumber
import Mathlib.Algebra.QuaternionBasis
import Mathlib.Data.Complex.Module
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import ... | (Algebra.ofId _ _, ι (R := R) _ 1),
ι_mul_ι (1 : R) 1,
fun _ => (Algebra.commutes _ _).symm⟩)
| Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean | 358 | 360 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
/-!
# Intervals as finsets
This file provides basic results about all the `Finset.Ixx... |
theorem mem_uIcc' : a ∈ [[b, c]] ↔ b ≤ a ∧ a ≤ c ∨ c ≤ a ∧ a ≤ b := by simp [uIcc_eq_union]
| Mathlib/Order/Interval/Finset/Basic.lean | 1,051 | 1,053 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Functor.Hom
import Mathlib.CategoryTheory.Products.Basic
import Mathlib.Data.ULift
import Mathlib.Logic.Function.ULift
/-!
# The Yoneda emb... | lemma yonedaEquiv_naturality {X Y : C} {F : Cᵒᵖ ⥤ Type v₁} (f : yoneda.obj X ⟶ F)
(g : Y ⟶ X) : F.map g.op (yonedaEquiv f) = yonedaEquiv (yoneda.map g ≫ f) := by
change (f.app (op X) ≫ F.map g.op) (𝟙 X) = f.app (op Y) (𝟙 Y ≫ g)
rw [← f.naturality]
dsimp
| Mathlib/CategoryTheory/Yoneda.lean | 468 | 472 |
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calcul... | theorem lineDerivWithin_univ : lineDerivWithin 𝕜 f univ x v = lineDeriv 𝕜 f x v := by
simp [lineDerivWithin, lineDeriv]
| Mathlib/Analysis/Calculus/LineDeriv/Basic.lean | 170 | 171 |
/-
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.Data.List.Lemmas
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Data.Li... | induction' ys with ys_hd _ ys_ih generalizing f f'
· simp
· simp only [map, permutationsAux2_snd_cons, cons_append, cons.injEq]
rw [ys_ih]
· refine ⟨?_, rfl⟩
simp only [← map_cons, ← map_append]; apply H
· intro a; apply H
/-- The `f` argument to `permutationsAux2` when `r = []` can be eliminat... | Mathlib/Data/List/Permutation.lean | 90 | 100 |
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.Degree.TrailingDegree
import Mathlib.Algebra.Polynomial.EraseLead
/-!
# Reverse of a univariate polynomial
The main definition is `r... | zero_mul]
end Eval₂
/-- The reverse of a polynomial f is the polynomial obtained by "reading f backwards".
Even though this is not the actual definition, `reverse f = f (1/X) * X ^ f.natDegree`. -/
noncomputable def reverse (f : R[X]) : R[X] :=
reflect f.natDegree f
theorem coeff_reverse (f : R[X]) (n : ℕ) :... | Mathlib/Algebra/Polynomial/Reverse.lean | 221 | 232 |
/-
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... | simpa only [Real.norm_eq_abs, abs_pow, abs_of_nonneg ha.1.le] using hC (pow_ne_zero n ha.1.ne')
tfae_have 6 → 5 := fun ⟨a, ha, C, H₀, H⟩ ↦ ⟨a, ha.2, C, Or.inl H₀, H⟩
tfae_have 5 → 3
| Mathlib/Analysis/SpecificLimits/Normed.lean | 90 | 92 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Algebra.Group.Action.Defs
import Mathlib.Algebra.Group.End
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.Common
/-!
#... | Mathlib/GroupTheory/Perm/Basic.lean | 626 | 627 | |
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.Degree.Support
import Mathlib.Data.ENat.Basic
/-!
# Trailing degree of univariate polynomials
## Main definitions
* `trailingDegree... | refine (h₂ (Prod.ext_iff.mpr ?_).symm).elim
exact (add_eq_add_iff_eq_and_eq hi hj).mp h₁.symm
| Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean | 314 | 316 |
/-
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.List.Sublists
import Mathlib.Data.List.Zip
import Mathlib.Data.Multiset.Bind
import Mathlib.Data.Multiset.Range
/-!
# The powerset of a multiset
... |
/-- Helper function for the powerset of a multiset. Given a list `l`, returns a list
| Mathlib/Data/Multiset/Powerset.lean | 35 | 36 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.ConditionalProbability
import Mathlib.Probability.Kernel.Basic
import Mathlib.Probability.Kernel.Composition.MeasureComp
import Mathlib.Tactic.... | exact fun n => (h_indep n).indepSets
-- now go from π-systems to σ-algebras
refine IndepSets.indep (iSup_le h_le) h_le' hp_supr_pi hp'_pi ?_ h_gen' h_pi_system_indep
exact (generateFrom_iUnion_measurableSet _).symm
theorem iIndepSet.indep_generateFrom_lt [Preorder ι] {s : ι → Set Ω}
(hsm : ∀ n, Measurabl... | Mathlib/Probability/Independence/Kernel.lean | 659 | 665 |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Data.Matrix.ConjTranspose
/-!
# Row and column matrices
This file provides results about row and column matrices.
## Main definitions
* `Mat... |
@[deprecated (since := "2025-03-20")] alias col_smul := replicateCol_smul
| Mathlib/Data/Matrix/RowCol.lean | 94 | 96 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Pi
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
/-!
# Simple functions
A function `f` from a measurable ... | lemma iSup_coe_eapprox (hf : Measurable f) : ⨆ n, ⇑(eapprox f n) = f := by
simpa [funext_iff] using iSup_eapprox_apply hf
| Mathlib/MeasureTheory/Function/SimpleFunc.lean | 796 | 798 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Manuel Candales
-/
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
import Mat... | have hf1 : (f x).1 ≠ 0 := by simp [f, hx12]
have hf2 : (f x).2 ≠ 0 := by simp [f, hx32]
exact (InnerProductGeometry.continuousAt_angle hf1 hf2).comp (by fun_prop)
@[simp]
theorem _root_.AffineIsometry.angle_map {V₂ P₂ : Type*} [NormedAddCommGroup V₂]
[InnerProductSpace ℝ V₂] [MetricSpace P₂] [NormedAddTorsor... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean | 50 | 57 |
/-
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 ... | def maxTrivHom (f : M →ₗ⁅R,L⁆ N) : maxTrivSubmodule R L M →ₗ⁅R,L⁆ maxTrivSubmodule R L N where
toFun m := ⟨f m, fun x =>
(LieModuleHom.map_lie _ _ _).symm.trans <|
(congr_arg f (m.property x)).trans (LieModuleHom.map_zero _)⟩
map_add' m n := by ext; simp
| Mathlib/Algebra/Lie/Abelian.lean | 160 | 164 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Group.Subgroup.Ker
import Mathlib.Algebra.BigOperators.Group.List.Basic
/-!
# Free groups
This file defines free groups over a type. Furthermore, it is... | Mathlib/GroupTheory/FreeGroup/Basic.lean | 1,295 | 1,296 | |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
import Mathlib.Data.Set.Finite.Powerset
/-!
# Noncomputable Set Cardinality
We define the cardinality of set `s` as a term `Set... | rw [encard, encard, ENat.card_congr (Equiv.Set.univ ↑s)]
| Mathlib/Data/Set/Card.lean | 66 | 67 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Group.Multiset.Basic
/-!
# Bind operation for multisets
This file defines a few basic operations on `Multiset`, notably the mona... | Multiset.induction_on s (by simp) (by simp +contextual)
| Mathlib/Data/Multiset/Bind.lean | 158 | 159 |
/-
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.Topology.Category.Profinite.Nobeling.Basic
import Mathlib.Topology.Category.Profinite.Nobeling.Induction
import Mathlib.Topology.Category.Profinite... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 1,060 | 1,072 | |
/-
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... | rw [fderiv]
end TVS
section
| Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 219 | 223 |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Lu-Ming Zhang
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.Data.Matrix.Kronecker
import Mathlib.LinearAlgebra.FiniteDimensional.Basic
import Mathlib.LinearAlgebra.... | Function.Injective (fun x : Matrix m l α => B * x) :=
fun _ _ g => by simpa only [← Matrix.mul_assoc, Matrix.one_mul, h] using congr_arg (A * ·) g
| Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 301 | 302 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Regular.Basic
import Mathlib.Algebra.Ring.Defs
/-!
# Lemmas about regular... |
/-- A ring with no zero divisors is a `CancelMonoidWithZero`.
Note this is not an instance as it forms a typeclass loop. -/
abbrev NoZeroDivisors.toCancelMonoidWithZero [Ring α] [NoZeroDivisors α] : CancelMonoidWithZero α :=
| Mathlib/Algebra/Ring/Regular.lean | 42 | 46 |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Lu-Ming Zhang
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.Data.Matrix.Kronecker
import Mathlib.LinearAlgebra.FiniteDimensional.Basic
import Mathlib.LinearAlgebra.... | (diagonalInvertibleEquivInvertible v).nonempty_congr]
| Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 556 | 557 |
/-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.GCDMonoid.Multiset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Algebra.Group.TypeTags.Finite
import Mathlib.Combinatorics.Enumerative... |
namespace IsThreeCycle
| Mathlib/GroupTheory/Perm/Cycle/Type.lean | 589 | 590 |
/-
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.Equiv.Basic
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Part
import Mathlib.Tactic.NormNum
/-!
# Natural numbers with infinity
The... | ⟨fun hz => ⟨hx₁ hz, hy₁ hz⟩, fun _ => sup_le (hx₂ _) (hy₂ _)⟩ }
instance orderBot : OrderBot PartENat where
bot := ⊥
bot_le _ := ⟨fun _ => trivial, fun _ => Nat.zero_le _⟩
instance orderTop : OrderTop PartENat where
top := ⊤
le_top _ := ⟨fun h => False.elim h, fun hy => False.elim hy⟩
instance : ZeroLE... | Mathlib/Data/Nat/PartENat.lean | 259 | 277 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.ConditionalProbability
import Mathlib.Probability.Kernel.Basic
import Mathlib.Probability.Kernel.Composition.MeasureComp
import Mathlib.Tactic.... | theorem indep_bot_left (m' : MeasurableSpace Ω) {_mΩ : MeasurableSpace Ω}
{κ : Kernel α Ω} {μ : Measure α} [IsZeroOrMarkovKernel κ] :
Indep ⊥ m' κ μ := (indep_bot_right m').symm
theorem indepSet_empty_right {_mΩ : MeasurableSpace Ω}
{κ : Kernel α Ω} {μ : Measure α} [IsZeroOrMarkovKernel κ] (s : Set Ω) :
... | Mathlib/Probability/Independence/Kernel.lean | 304 | 327 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... |
theorem arg_neg_eq_arg_add_pi_iff {x : ℂ} :
arg (-x) = arg x + π ↔ x.im < 0 ∨ x.im = 0 ∧ 0 < x.re := by
| Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 421 | 423 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Patrick Massot
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
/-!
# Sets in product and pi types
This file proves basic properties of prod... | rw [Set.ext_iff] at he₁ he₂
aesop (add simp [Prod.swap_eq_iff_eq_swap])
exact ⟨
| Mathlib/Data/Set/Prod.lean | 1,019 | 1,021 |
/-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Simon Hudon, Kenny Lau
-/
import Mathlib.Data.Multiset.Bind
import Mathlib.Control.Traversable.Lemmas
import Mathlib.Control.Traversable.Instances
/-!
# F... | intro
simp only [quot_mk_to_coe, traverse, lift_coe, Function.comp_apply,
ApplicativeTransformation.preserves_map, LawfulTraversable.naturality]
end Multiset
| Mathlib/Data/Multiset/Functor.lean | 123 | 128 |
/-
Copyright (c) 2021 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis... | ((tendsto_id.inv₀ one_ne_zero).smul tendsto_const_nhds).mono_left inf_le_left
rw [inv_one, one_smul] at H₁
have H₂ : ∀ᶠ r in 𝓝[<] (1 : ℝ), x ∈ r • s ∧ 0 < r ∧ r < 1 := by
filter_upwards [H₁ (mem_interior_iff_mem_nhds.1 hx), Ioo_mem_nhdsLT one_pos] with r h₁ h₂
exact ⟨(mem_smul_set_iff_inv_smul_mem₀ h₂.... | Mathlib/Analysis/Convex/Gauge.lean | 347 | 354 |
/-
Copyright (c) 2023 Bhavik Mehta, Rishi Mehta, Linus Sommer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Rishi Mehta, Linus Sommer
-/
import Mathlib.Algebra.GroupWithZero.Nat
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Combinatorics.SimpleG... | /-- A hamiltonian path visits every vertex. -/
@[simp] lemma IsHamiltonian.mem_support (hp : p.IsHamiltonian) (c : α) : c ∈ p.support := by
simp only [← List.count_pos_iff, hp _, Nat.zero_lt_one]
| Mathlib/Combinatorics/SimpleGraph/Hamiltonian.lean | 38 | 40 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Wen Yang
-/
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.RingTheory.... | obtain ⟨r, _, hr⟩ := mem_center_iff.mp A.property
simpa [← hr, Submonoid.smul_def, Units.smul_def] using smul_one_eq_diagonal r
right_inv a := by
obtain ⟨⟨a, _⟩, ha⟩ := a
exact SetCoe.ext <| Units.eq_iff.mp <| by simp
map_mul' A B := by
| Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean | 271 | 276 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Aurélien Saue, Anne Baanen
-/
import Mathlib.Tactic.NormNum.Inv
import Mathlib.Tactic.NormNum.Pow
import Mathlib.Util.AtomM
/-!
# `ring` tactic
A tactic for solving e... | * `(a ^ b * c)⁻¹ = a⁻¹ ^ b * c⁻¹`
-/
| Mathlib/Tactic/Ring/Basic.lean | 986 | 987 |
/-
Copyright (c) 2023 Mohanad ahmed. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mohanad Ahmed
-/
import Mathlib.Data.Matrix.Block
import Mathlib.LinearAlgebra.Matrix.SemiringInverse
/-! # Block Matrices from Rows and Columns
This file provides the basic definiti... | ext i (j | j) <;> simp
@[deprecated (since := "2024-12-11")] alias fromColumns_toColumns := fromCols_toCols
| Mathlib/Data/Matrix/ColumnRowPartitioned.lean | 118 | 121 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Wen Yang
-/
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.RingTheory.... | Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean | 514 | 521 | |
/-
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.Defs
import Mathlib.Order.Directed
import Mathlib.Order.BoundedOrder.Monotone
import Mathlib.Order.Interval.Set.Basic
/-... | rw [IsGLB, IsGLB, h]
| Mathlib/Order/Bounds/Basic.lean | 126 | 127 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Johan Commelin
-/
import Mathlib.Algebra.Algebra.RestrictScalars
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Module.Rat
import Mathlib.GroupThe... | @[simp]
theorem map_comp_includeRight (f : A →ₐ[R] C) (g : B →ₐ[R] D) :
| Mathlib/RingTheory/TensorProduct/Basic.lean | 1,031 | 1,032 |
/-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Data.Nat.PrimeFin
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Order.Interval.Finset.Nat
import... | lemma factorizationLCMLeft_zero_right : factorizationLCMLeft a 0 = 1 := by
simp [factorizationLCMLeft]
@[simp]
lemma factorizationLCRight_zero_left : factorizationLCMRight 0 b = 1 := by
simp [factorizationLCMRight]
@[simp]
lemma factorizationLCMRight_zero_right : factorizationLCMRight a 0 = 1 := by
simp [factori... | Mathlib/Data/Nat/Factorization/Basic.lean | 409 | 419 |
/-
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... | end ExactPairing
attribute [reassoc (attr := simp)] ExactPairing.coevaluation_evaluation
| Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean | 123 | 125 |
/-
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... |
theorem liminf_finset_inf' [ConditionallyCompleteLinearOrder β] {f : Filter α}
{F : ι → α → β} {s : Finset ι} (hs : s.Nonempty)
| Mathlib/Order/LiminfLimsup.lean | 1,183 | 1,185 |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Category.ModuleCat.ChangeOfRings
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.RingTheory.Kaehler.Basic
/-!
# The differentials of a morphis... | @[simp]
lemma d_add (b b' : B) : D.d (b + b') = D.d b + D.d b' := by simp [d]
| Mathlib/Algebra/Category/ModuleCat/Differentials/Basic.lean | 69 | 70 |
/-
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.Polynomial.FieldDivision
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Data.List.Prime
import Mathlib.RingTheory.Polynomial.Tower
/-!
# S... | fun _ p ha0 hp ih hfi =>
splits_mul _
(splits_of_degree_eq_one _
((splits_of_splits_mul _ (mul_ne_zero hp.1 ha0) hfi).1.def.resolve_left hp.1 hp.irreducible
(by rw [map_id])))
(ih (splits_of_splits_mul _ (mul_ne_zero hp.1 ha0) hfi).2)
end UFD
theorem splits_iff_exists_multiset {f... | Mathlib/Algebra/Polynomial/Splits.lean | 481 | 491 |
/-
Copyright (c) 2021 Martin Zinkevich. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Martin Zinkevich, Vincent Beffara
-/
import Mathlib.MeasureTheory.Integral.Bochner.Set
import Mathlib.Probability.Independence.Basic
/-!
# Integration in Probability Theory
Integra... | (h_ind : IndepSets {s | MeasurableSet[Mf] s} {T} μ) (h_meas_f : Measurable[Mf] f) :
(∫⁻ ω, f ω * T.indicator (fun _ => c) ω ∂μ) =
(∫⁻ ω, f ω ∂μ) * ∫⁻ ω, T.indicator (fun _ => c) ω ∂μ := by
revert f
have h_mul_indicator : ∀ g, Measurable g → Measurable fun a => g a * T.indicator (fun _ => c) a :=
f... | Mathlib/Probability/Integration.lean | 45 | 73 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Round
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Ring
/-... | rw [← neg_inj, ← floor_neg, ← floor_neg, ← Rat.cast_neg, Rat.floor_cast]
@[simp, norm_cast]
theorem round_cast (x : ℚ) : round (x : α) = round x := by
have : ((x + 1 / 2 : ℚ) : α) = x + 1 / 2 := by simp
rw [round_eq, round_eq, ← this, floor_cast]
| Mathlib/Data/Rat/Floor.lean | 96 | 101 |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Order.Floor.Semiring
import Mathlib.Data.Nat.Log
/-!
# Integer logarithms in a field with respect to a natural base
This file defines two `ℤ`-value... |
/-- `Int.clog b` and `zpow b` (almost) form a Galois connection. -/
theorem zpow_lt_iff_lt_clog {b : ℕ} (hb : 1 < b) {x : ℤ} {r : R} (hr : 0 < r) :
(b : R) ^ x < r ↔ x < clog b r :=
| Mathlib/Data/Int/Log.lean | 283 | 286 |
/-
Copyright (c) 2024 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Analysis.Complex.CauchyIntegral
/-!
# Convergence of Taylor series of holomorphic functions
We show that the Taylor series around some point `c : ℂ` of... | include hf in
/-- A function that is complex differentiable on the complex plane is given by evaluating
its Taylor series at any point `c`. -/
lemma hasSum_taylorSeries_of_entire :
HasSum (fun n : ℕ ↦ (n ! : ℂ)⁻¹ • (z - c) ^ n • iteratedDeriv n f c) (f z) :=
hasSum_taylorSeries_on_emetric_ball hf.differentiableOn... | Mathlib/Analysis/Complex/TaylorSeries.lean | 120 | 125 |
/-
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.List.MinMax
import Mathlib.Algebra.Tropical.Basic
import Mathlib.Order.ConditionallyCompleteLattice.Finset
import Mathlib.Algebra.BigOperators.G... |
theorem trop_sum [AddCommMonoid R] (s : Finset S) (f : S → R) :
trop (∑ i ∈ s, f i) = ∏ i ∈ s, trop (f i) := by
| Mathlib/Algebra/Tropical/BigOperators.lean | 46 | 48 |
/-
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... | (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : x ∈ s) : ContDiffWithinAt 𝕜 n f₁ s x :=
h.congr_of_eventuallyEq h₁ <| h₁.self_of_nhdsWithin hx
theorem Filter.EventuallyEq.congr_contDiffWithinAt_of_mem (h₁ : f₁ =ᶠ[𝓝[s] x] f) (hx : x ∈ s):
ContDiffWithinAt 𝕜 n f₁ s x ↔ ContDiffWithinAt 𝕜 n f s x :=
⟨fun H ↦ H.congr_of_eve... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 240 | 250 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kim Morrison, Ainsley Pahljina
-/
import Mathlib.RingTheory.Fintype
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Zify
/-!
# The Lucas-L... | theorem sZMod_eq_s (p' : ℕ) (i : ℕ) : sZMod (p' + 2) i = (s i : ZMod (2 ^ (p' + 2) - 1)) := by
| Mathlib/NumberTheory/LucasLehmer.lean | 145 | 145 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Field.IsField
import Mathlib.Algebra.GroupWithZero.N... | /-- If the square below commutes, the bottom map is uniquely specified:
```
R → S
↓ ↓
Rₘ → Sₘ
```
-/
| Mathlib/RingTheory/Localization/Basic.lean | 484 | 490 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
/-!
# Intervals as finsets
This file provides basic results about all the `Finset.Ixx... | theorem _root_.Set.Infinite.exists_lt (hs : s.Infinite) : ∀ a, ∃ b ∈ s, b < a :=
| Mathlib/Order/Interval/Finset/Basic.lean | 930 | 930 |
/-
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 | 799 | 801 | |
/-
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.Group.Indicator
import Mathlib.Algebra.Group.InjSurj
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Tactic.FastInstance
impo... | ⟨fun n v => v.mapRange (n • ·) (nsmul_zero _)⟩
instance instAddMonoid : AddMonoid (α →₀ M) :=
fast_instance% DFunLike.coe_injective.addMonoid _ coe_zero coe_add fun _ _ => rfl
| Mathlib/Data/Finsupp/Defs.lean | 569 | 572 |
/-
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... |
instance fintypeEdgeSetSdiff [DecidableEq V] [Fintype G₁.edgeSet] [Fintype G₂.edgeSet] :
| Mathlib/Combinatorics/SimpleGraph/Basic.lean | 547 | 548 |
/-
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.Adjunction.Unique
import Mathlib.CategoryTheory.Adjunction.Reflective
import Mathlib.CategoryTheory.Sites.Sheaf
import Mathlib.Categ... | rw [toSheafify] at h
rw [sheafifyLift]
let γ' : (presheafToSheaf J D).obj P ⟶ ⟨Q, hQ⟩ := ⟨γ⟩
change γ'.val = _
| Mathlib/CategoryTheory/Sites/Sheafification.lean | 184 | 187 |
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