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) 2023 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Richard M. Hill
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Module.AEval
import Mathlib... |
lemma mkDerivation_one_eq_derivative (f : R[X]) : mkDerivation R (1 : R[X]) f = derivative f := by
rw [mkDerivation_one_eq_derivative']
| Mathlib/Algebra/Polynomial/Derivation.lean | 73 | 75 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Oliver Nash
-/
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Union
/-!
# Finsets in product types
This file defines finset constru... |
end Diag
| Mathlib/Data/Finset/Prod.lean | 388 | 389 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.MeasureTheory.Function.SimpleFuncDense
/-!
# Strongly measurable and finitely strongly meas... | have : ∀ n, ∃ c, ∀ x, fs n x = c := fun n => SimpleFunc.simpleFunc_bot (fs n)
let cs n := (this n).choose
have h_cs_eq : ∀ n, ⇑(fs n) = fun _ => cs n := fun n => funext (this n).choose_spec
conv at h_fs_tendsto => enter [x, 1, n]; rw [h_cs_eq]
have h_tendsto : Tendsto cs atTop (𝓝 (f hα.some)) := h_... | Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | 238 | 242 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.Interval.Set.Defs
/-!
# Intervals
In any pr... | rw [← Ici_inter_Iic, ← Iic_inter_Ici, inter_inter_inter_comm, Iic_inter_Ici]
simp [hab, hbc]
| Mathlib/Order/Interval/Set/Basic.lean | 835 | 836 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Polynomial.Basi... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 1,337 | 1,341 | |
/-
Copyright (c) 2023 Koundinya Vajjha. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Koundinya Vajjha, Thomas Browning
-/
import Mathlib.NumberTheory.Harmonic.Defs
import Mathlib.NumberTheory.Padics.PadicNumbers
import Mathlib.Tactic.Positivity
/-!
The nth Harmonic... | /-- The 2-adic norm of the n-th harmonic number is 2 raised to the logarithm of n in base 2. -/
lemma padicNorm_two_harmonic {n : ℕ} (hn : n ≠ 0) :
‖(harmonic n : ℚ_[2])‖ = 2 ^ (Nat.log 2 n) := by
rw [padicNormE.eq_padicNorm, padicNorm.eq_zpow_of_nonzero (harmonic_pos hn).ne',
padicValRat_two_harmonic, neg_ne... | Mathlib/NumberTheory/Harmonic/Int.lean | 42 | 46 |
/-
Copyright (c) 2020 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Algebra.Algebra.Spectrum.Basic
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathl... | theorem hasEigenvalue_iff_mem_spectrum [FiniteDimensional K V] {f : End K V} {μ : K} :
f.HasEigenvalue μ ↔ μ ∈ spectrum K f :=
hasUnifEigenvalue_iff_mem_spectrum
alias ⟨_, HasEigenvalue.of_mem_spectrum⟩ := hasEigenvalue_iff_mem_spectrum
| Mathlib/LinearAlgebra/Eigenspace/Basic.lean | 461 | 465 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Opposite
import Mathlib.Topology.Algebra.Group.Qu... | Mathlib/Topology/Algebra/Module/Basic.lean | 1,512 | 1,515 | |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.AlgebraicGeometry.Scheme
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.CommSq
import Mathlib.Categor... | lemma isIso_app (V : Y.Opens) (hV : V ≤ f.opensRange) : IsIso (f.app V) := by
rw [show V = f ''ᵁ f ⁻¹ᵁ V from Opens.ext (Set.image_preimage_eq_of_subset hV).symm]
infer_instance
/-- The isomorphism `Γ(Y, f(U)) ≅ Γ(X, U)` induced by an open immersion `f : X ⟶ Y`. -/
def appIso (U) : Γ(Y, f ''ᵁ U) ≅ Γ(X, U) :=
(as... | Mathlib/AlgebraicGeometry/OpenImmersion.lean | 155 | 170 |
/-
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.... | simp [KaehlerDifferential.derivationQuotKerTotal_apply]
theorem KaehlerDifferential.kerTotal_eq :
LinearMap.ker (Finsupp.linearCombination S (KaehlerDifferential.D R S)) =
KaehlerDifferential.kerTotal R S := by
apply le_antisymm
· conv_rhs => rw [← (KaehlerDifferential.kerTotal R S).ker_mkQ]
rw [← ... | Mathlib/RingTheory/Kaehler/Basic.lean | 566 | 574 |
/-
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.Group.Subgroup.Finite
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.Perm.Basic
import Mathlib.LinearAlgebra.Alternating.... | calc
∑ τ ∈ _, sign τ • domDomCongr τ (a.domCoprod b) =
∑ τ ∈ {τ | τ⁻¹ * σ ∈ f.range}, sign τ • domDomCongr τ (a.domCoprod b) := by
simp [QuotientGroup.leftRel_apply, f]
_ = ∑ τ ∈ {τ | τ⁻¹ ∈ f.range}, sign (σ * τ) • domDomCongr (σ * τ) (a.domCoprod b) := by
conv_lhs => rw [← Finset.map_univ... | Mathlib/LinearAlgebra/Alternating/DomCoprod.lean | 212 | 222 |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.NumberTheory.Padics.PadicNorm
import Mathlib.Analysis.Normed.Field.Lemmas
import Mathlib.Tactic.Peel
import... |
open PadicSeq Padic
variable {p : ℕ} [Fact p.Prime] (f : CauSeq _ (@padicNormE p _))
theorem rat_dense' (q : ℚ_[p]) {ε : ℚ} (hε : 0 < ε) : ∃ r : ℚ, padicNormE (q - r : ℚ_[p]) < ε :=
Quotient.inductionOn q fun q' ↦
have : ∃ N, ∀ m ≥ N, ∀ n ≥ N, padicNorm p (q' m - q' n) < ε := cauchy₂ _ hε
let ⟨N, hN⟩ := th... | Mathlib/NumberTheory/Padics/PadicNumbers.lean | 599 | 616 |
/-
Copyright (c) 2023 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Computability.AkraBazzi.GrowsPolynomially
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
/-!
# Divid... | lemma strictAntiOn_smoothingFn : StrictAntiOn ε (Set.Ioi 1) := by
show StrictAntiOn (fun x => 1 / log x) (Set.Ioi 1)
simp_rw [one_div]
| Mathlib/Computability/AkraBazzi/AkraBazzi.lean | 450 | 452 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Algebra.Group.Fin.Tuple
import Mathlib.Data.Matrix.RowCol
import Mathlib.Data.Fin.VecNotation
import Mathlib.Tactic.FinCases
import Mathlib.Alge... |
end Vec2AndVec3
| Mathlib/Data/Matrix/Notation.lean | 523 | 525 |
/-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.DeleteEdges
import Mathlib.Data.Fintype.Powerset
/-!
# Subgraphs of a simple graph
A subgraph of ... | simp +contextual
| Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 1,107 | 1,108 |
/-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Data.DFinsupp.Lex
import Mathlib.Order.Antisymmetrization
import Mathlib.Order.GameAdd
import Mathlib.SetTheory.Cardinal.Order
import Mathlib.Tactic.AdaptationNo... | (fun {i j} h => ((@IsTrichotomous.trichotomous ι r _ i j).resolve_left h.1).resolve_left h.2) hr
| Mathlib/Data/DFinsupp/WellFounded.lean | 161 | 162 |
/-
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 | 1,748 | 1,748 | |
/-
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.Homology.ComplexShape
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Algebra.Group.Nat.Defs
import Mathlib.Tactic.ByContra
/-! # Embeddings of comp... | complex shapes `e : Embedding c c'` is stable by `c'.prev`. -/
class IsTruncLE : Prop extends e.IsRelIff where
mem_prev {i' : ι'} {j : ι} (h : c'.Rel i' (e.f j)) :
∃ i, e.f i = i'
| Mathlib/Algebra/Homology/Embedding/Basic.lean | 122 | 126 |
/-
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... | theorem det_updateRow_sum_aux (M : Matrix n n R) {j : n} (s : Finset n) (hj : j ∉ s) (c : n → R)
(a : R) :
(M.updateRow j (a • M j + ∑ k ∈ s, (c k) • M k)).det = a • M.det := by
induction s using Finset.induction_on with
| Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean | 416 | 419 |
/-
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 | 710 | 712 | |
/-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann, Kyle Miller, Mario Carneiro
-/
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Data.Nat.BinaryRec
import Mathlib.Logic.F... | Mathlib/Data/Nat/Fib/Basic.lean | 292 | 298 | |
/-
Copyright (c) 2021 Jakob Scholbach. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob Scholbach, Joël Riou
-/
import Mathlib.CategoryTheory.CommSq
import Mathlib.CategoryTheory.Retract
/-!
# Lifting properties
This file defines the lifting property of two morph... | HasLiftingProperty i p ↔ HasLiftingProperty i' p := by
constructor <;> intro
exacts [of_arrow_iso_left e p, of_arrow_iso_left e.symm p]
theorem iff_of_arrow_iso_right {A B X Y X' Y' : C} (i : A ⟶ B) {p : X ⟶ Y} {p' : X' ⟶ Y'}
| Mathlib/CategoryTheory/LiftingProperties/Basic.lean | 121 | 125 |
/-
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.InnerProductSpace.Basic
import Mathlib.Analysis.Normed.Ring.InfiniteSum
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.NumberTheo... | f ⍟ g = f' ⍟ g' := by
simp [convolution, toArithmeticFunction_congr hf, toArithmeticFunction_congr hg]
/-- The product of two arithmetic functions defines the same function as the Dirichlet convolution
| Mathlib/NumberTheory/LSeries/Convolution.lean | 66 | 69 |
/-
Copyright (c) 2020 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.Algebra.GeomSum
import Mathlib.Data.Finset.Slice
import Mathlib.Data.Nat.BitIndices
import Mathlib.Order.SupClosed
import Math... | refine ⟨w, hwt, hws, fun a has hat ↦ ?_⟩
| Mathlib/Combinatorics/Colex.lean | 357 | 357 |
/-
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.MeasurableSpace.MeasurablyGenerated
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.Order.Interval.Set... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 2,184 | 2,188 | |
/-
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... | (F : Cᵒᵖ ⥤ Type*) : S.IsSheafFor F := by
obtain ⟨Y, f, rfl, hf⟩ := Presieve.regular.single_epi (R := S)
rw [isSheafFor_arrows_iff]
refine fun x hx ↦ ⟨F.map (Projective.factorThru (𝟙 _) f).op <| x (), fun _ ↦ ?_, fun y h ↦ ?_⟩
· simpa using (hx () () Y (𝟙 Y) (f ≫ (Projective.factorThru (𝟙 _) f)) (by simp)... | Mathlib/CategoryTheory/Sites/Coherent/RegularSheaves.lean | 224 | 231 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.SuccPred.Initial... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 2,035 | 2,036 | |
/-
Copyright (c) 2024 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker, Devon Tuma, Kexing Ying
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.Density
import Mathlib.Probability.ConditionalProbability
import Mathlib.Probabili... | (by
let ⟨a, ha⟩ := hs
| Mathlib/Probability/Distributions/Uniform.lean | 243 | 244 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Lattice.Prod
import Mathlib.Data.Finite.Prod
import Mathlib.Data.Set.Lattice.Image
/-!
# N-ary images of finsets
This file defines `Finset.im... | lemma le_inf_image₂ {g : γ → δ} {a : δ} :
a ≤ inf (image₂ f s t) g ↔ ∀ x ∈ s, ∀ y ∈ t, a ≤ g (f x y) :=
sup_image₂_le (δ := δᵒᵈ)
variable (s t)
| Mathlib/Data/Finset/NAry.lean | 566 | 570 |
/-
Copyright (c) 2023 Alex Keizer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Keizer
-/
import Mathlib.Data.Vector.Basic
import Mathlib.Data.Vector.Snoc
/-!
This file establishes a set of normalization lemmas for `map`/`mapAccumr` operations on vectors
-/
... | (m.fst.fst, m.snd) := by
induction xs, ys using Vector.revInductionOn₂ generalizing s₁ s₂ <;> simp_all
@[simp]
theorem map₂_map_right (f₁ : α → γ → ζ) (f₂ : β → γ) :
map₂ f₁ xs (map f₂ ys) = map₂ (fun x y => f₁ x (f₂ y)) xs ys := by
induction xs, ys using Vector.revInductionOn₂ <;> simp_all
@[simp]
| Mathlib/Data/Vector/MapLemmas.lean | 92 | 100 |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.CategoryTheory.Functor.Flat
import Mathlib.CategoryTheory.Sites.Continuous
import Mathlib.Tactic.ApplyFun
/-!
# Cover-preserving functors between sites.
In ... | (_ : x.Compatible) {Y₁ Y₂} {X} (f₁ : X ⟶ G.obj Y₁) (f₂ : X ⟶ G.obj Y₂) {g₁ : Y₁ ⟶ Z}
{g₂ : Y₂ ⟶ Z} (hg₁ : T g₁) (hg₂ : T g₂) (_ : f₁ ≫ G.map g₁ = f₂ ≫ G.map g₂),
ℱ.val.map f₁.op (x g₁ hg₁) = ℱ.val.map f₂.op (x g₂ hg₂)
section
| Mathlib/CategoryTheory/Sites/CoverPreserving.lean | 77 | 81 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.NNRat.Order
import Mathlib.Topology.Algebra.Order.Archimedean
import Mathlib.Topology.Algebra.Ring.Real
import Mathlib.Topology.In... | isUniformEmbedding_bot_of_pairwise_le_dist zero_lt_one <| by simpa using Int.pairwise_one_le_dist
| Mathlib/Topology/Instances/Rat.lean | 65 | 66 |
/-
Copyright (c) 2021 Lu-Ming Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lu-Ming Zhang
-/
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.LinearAlgebra.Matrix.Orthogonal
import Mathlib.Data.Matrix.Kronecker
/-!
# Diagonal matrices
This file co... | · rw [diagonal_apply_eq, diag]
· rw [diagonal_apply_ne _ hij, h hij]
/-- `Matrix.IsDiag.diagonal_diag` as an iff. -/
theorem isDiag_iff_diagonal_diag [Zero α] [DecidableEq n] (A : Matrix n n α) :
A.IsDiag ↔ diagonal (diag A) = A :=
| Mathlib/LinearAlgebra/Matrix/IsDiag.lean | 46 | 51 |
/-
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 | 546 | 547 | |
/-
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 | 595 | 598 | |
/-
Copyright (c) 2021 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Adjoint of operators on Hilbert spaces
Given ... | simp [adjoint_inner_left, adjoint_inner_right]
/-- The Gram operator T†T is a positive operator. -/
theorem re_inner_adjoint_mul_self_nonneg (T : E →ₗ[𝕜] E) (x : E) :
0 ≤ re ⟪x, (LinearMap.adjoint T * T) x⟫ := by
| Mathlib/Analysis/InnerProductSpace/Adjoint.lean | 439 | 443 |
/-
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.Data.ENNReal.Operations
/-!
# Results about division in extended non-negative reals
This file establishes basic properties related t... | rwa [← ENNReal.div_lt_iff (Or.inl ha) (Or.inr hb)]⟩
theorem exists_nat_mul_gt (ha : a ≠ 0) (hb : b ≠ ∞) : ∃ n : ℕ, b < n * a :=
(exists_nat_pos_mul_gt ha hb).imp fun _ => And.right
theorem exists_nat_pos_inv_mul_lt (ha : a ≠ ∞) (hb : b ≠ 0) :
| Mathlib/Data/ENNReal/Inv.lean | 595 | 600 |
/-
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
... | theorem Finset.centroid_eq_centerMass (s : Finset ι) (hs : s.Nonempty) (p : ι → E) :
s.centroid R p = s.centerMass (s.centroidWeights R) p :=
affineCombination_eq_centerMass (s.sum_centroidWeights_eq_one_of_nonempty R hs)
theorem Finset.centroid_mem_convexHull (s : Finset E) (hs : s.Nonempty) :
| Mathlib/Analysis/Convex/Combination.lean | 260 | 264 |
/-
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.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Grou... | exact to_map_eq_zero_iff.mp h
variable (A K C)
/-- An element of a ring is algebraic over the ring `A` iff it is algebraic
over the field of fractions of `A`.
-/
theorem isAlgebraic_iff [Algebra A C] [Algebra K C] [IsScalarTower A K C] {x : C} :
IsAlgebraic A x ↔ IsAlgebraic K x := by
constructor <;> rintro... | Mathlib/RingTheory/Localization/Integral.lean | 129 | 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.Option.NAry
import Mathlib.Data.Seq.Computation
import Mathlib.Tactic.ApplyFun
import Mathlib.Data.List.Basic
/-!
# Possibly infinite lists
This... | statement of the where the sequence is not known to terminate see `length_le_iff'` -/
theorem length_le_iff {s : Seq α} {n : ℕ} {h : s.Terminates} :
s.length h ≤ n ↔ s.TerminatedAt n := by
| Mathlib/Data/Seq/Seq.lean | 755 | 757 |
/-
Copyright (c) 2020 Paul van Wamelen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul van Wamelen
-/
import Mathlib.Data.Int.NatPrime
import Mathlib.Data.ZMod.Basic
import Mathlib.RingTheory.Int.Basic
import Mathlib.Tactic.FieldSimp
/-!
# Pythagorean Triples
Th... | · intro h
obtain ⟨m, n, H⟩ := h.left.isPrimitiveClassified_of_coprime h.right
use m, n
rcases H with ⟨⟨rfl, rfl⟩ | ⟨rfl, rfl⟩, co, pp⟩
· refine ⟨Or.inl ⟨rfl, rfl⟩, ?_, co, pp⟩
have : z ^ 2 = (m ^ 2 + n ^ 2) ^ 2 := by
rw [sq, ← h.left.eq]
| Mathlib/NumberTheory/PythagoreanTriples.lean | 565 | 571 |
/-
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.Algebra.EuclideanDomain.Basic
import Mathlib.Algebra.EuclideanDomain.Field
import Mathlib.Algebra.Polynomial.Module.Basic
import Mathlib.Analysis.Calculus.Co... | | zero =>
simp only [taylor_within_zero_eval, Nat.factorial_zero, Nat.cast_one, inv_one, pow_zero,
mul_one, zero_add, one_smul]
simp only [iteratedDerivWithin_zero] at hf'
rw [iteratedDerivWithin_one]
exact hf'.hasDerivWithinAt.mono h
| succ k hk =>
simp_rw [Nat.add_succ, taylorWithinEval_... | Mathlib/Analysis/Calculus/Taylor.lean | 176 | 196 |
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Simon Hudon
-/
import Mathlib.Control.Functor.Multivariate
import Mathlib.Data.PFunctor.Univariate.Basic
/-!
# Multivariate polynomial functors.
Multivariate polynomial... | LiftR @r x y ↔ ∃ a f₀ f₁, x = ⟨a, f₀⟩ ∧ y = ⟨a, f₁⟩ ∧ ∀ i j, r (f₀ i j) (f₁ i j) := by
constructor
· rintro ⟨u, xeq, yeq⟩
rcases h : u with ⟨a, f⟩
use a, fun i j => (f i j).val.fst, fun i j => (f i j).val.snd
constructor
· rw [← xeq, h]
rfl
constructor
· rw [← yeq, h]
rfl
| Mathlib/Data/PFunctor/Multivariate/Basic.lean | 160 | 170 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Analysis.Convex.Hull
/-!
# Convex join
This file defines the convex join of two sets. The convex join of `s` and `t` is the union of the
segments with on... | theorem convexJoin_mono (hs : s₁ ⊆ s₂) (ht : t₁ ⊆ t₂) : convexJoin 𝕜 s₁ t₁ ⊆ convexJoin 𝕜 s₂ t₂ :=
biUnion_mono hs fun _ _ => biUnion_subset_biUnion_left ht
| Mathlib/Analysis/Convex/Join.lean | 40 | 41 |
/-
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... |
section
| Mathlib/Data/Matrix/Basis.lean | 168 | 170 |
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Andrew Yang
-/
import Mathlib.AlgebraicGeometry.ProjectiveSpectrum.StructureSheaf
import Mathlib.AlgebraicGeometry.GammaSpecAdjunction
import Mathlib.RingTheory.GradedAlgeb... | include hm f_deg
variable {𝒜} in
lemma image_basicOpen_eq_basicOpen (a : A) (i : ℕ) :
toSpec 𝒜 f '' (Subtype.val ⁻¹' (pbo (decompose 𝒜 a i) : Set (ProjectiveSpectrum 𝒜))) =
| Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Scheme.lean | 526 | 530 |
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Alex Meiburg
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Degree.Monomial
/-!
# Erase the... | all_goals positivity
theorem nextCoeff_eq_zero_of_eraseLead_eq_zero (h : f.eraseLead = 0) : f.nextCoeff = 0 := by
by_contra h₂
exact leadingCoeff_ne_zero.mp (leadingCoeff_eraseLead_eq_nextCoeff h₂ ▸ h₂) h
end EraseLead
| Mathlib/Algebra/Polynomial/EraseLead.lean | 250 | 257 |
/-
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... | obtain rfl | x0 := eq_or_ne x 0
· by_cases n0 : n = 0 <;> simp [n0]
· induction n with
| zero => simp [x0]
| succ n ih => rw [pow_succ, arg_mul_coe_angle (pow_ne_zero n x0) x0, ih, succ_nsmul]
theorem arg_zpow_coe_angle (x : ℂ) (n : ℤ) :
| Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 496 | 502 |
/-
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.Fintype.Card
import Mathlib.Algebra.Order.BigOperators.Group.Multiset
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.Multiset.OrderedM... |
@[to_additive sum_eq_zero_iff_of_nonpos]
theorem prod_eq_one_iff_of_le_one' :
(∀ i ∈ s, f i ≤ 1) → ((∏ i ∈ s, f i) = 1 ↔ ∀ i ∈ s, f i = 1) :=
prod_eq_one_iff_of_one_le' (N := Nᵒᵈ)
@[to_additive single_le_sum]
| Mathlib/Algebra/Order/BigOperators/Group/Finset.lean | 155 | 161 |
/-
Copyright (c) 2021 Alex Kontorovich and Heather Macbeth and Marc Masdeu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, Heather Macbeth, Marc Masdeu
-/
import Mathlib.Analysis.Complex.Basic
import Mathlib.Data.Fintype.Parity
import Mathlib.LinearAl... | /-- Denominator of the formula for a fractional linear transformation -/
def denom (g : GL(2, ℝ)⁺) (z : ℍ) : ℂ := g 1 0 * z + g 1 1
| Mathlib/Analysis/Complex/UpperHalfPlane/Basic.lean | 189 | 190 |
/-
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.Closed.Cartesian
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Adjunction.FullyFaithful... |
theorem frobeniusMorphism_mate (h : L ⊣ F) (A : C) :
conjugateEquiv (h.comp (exp.adjunction A)) ((exp.adjunction (F.obj A)).comp h)
(frobeniusMorphism F h A).natTrans = (expComparison F A).natTrans := by
unfold expComparison frobeniusMorphism
have conjeq := iterated_mateEquiv_conjugateEquiv h h
(ex... | Mathlib/CategoryTheory/Closed/Functor.lean | 128 | 149 |
/-
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... |
theorem corec_id_id_eq_const (a : α) : corec id id a = const a := by
rw [corec_def, map_id, iterate_id]
theorem corec_id_f_eq_iterate (f : α → α) (a : α) : corec id f a = iterate f a :=
rfl
end Corec
| Mathlib/Data/Stream/Init.lean | 325 | 333 |
/-
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.Group.Action.Pi
import Mathlib.Algebra.Group.End
import Mathlib.Algebra.Module.NatInt
import Mathlib.Algebra.Order.Archimedean.Basic
/-!
# M... | @[scoped simp]
theorem map_zsmul_const [AddGroup G] [AddGroup H] [AddConstMapClass F G H a b]
(f : F) (n : ℤ) : f (n • a) = f 0 + n • b := by
| Mathlib/Algebra/AddConstMap/Basic.lean | 192 | 194 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Antoine Chambert-Loir
-/
import Mathlib.Algebra.Ring.CharZero
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.IndexNormal
import Mathlib.GroupTheory.Perm.F... | mul_mem
(IsSwap.mul_mem_closure_three_cycles (hl a List.mem_cons_self)
(hl b (List.mem_cons_of_mem a List.mem_cons_self)))
(ih _ (fun g hg => hl g (List.mem_cons_of_mem _ (List.mem_cons_of_mem _ hg))) hn)
/-- A key lemma to prove $A_5$ is simple. Shows that any normal subgroup of an alt... | Mathlib/GroupTheory/SpecificGroups/Alternating.lean | 178 | 191 |
/-
Copyright (c) 2019 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Module.Submodule.Equiv
import Mathlib.Algebra.Module.Equiv.Basic
import Mathlib.Algebra.Module.Rat
im... | @[norm_cast, simp]
theorem toLinearMap_comp (f : N →ₗ⁅R,L⁆ P) (g : M →ₗ⁅R,L⁆ N) :
(f.comp g : M →ₗ[R] P) = (f : N →ₗ[R] P).comp (g : M →ₗ[R] N) :=
| Mathlib/Algebra/Lie/Basic.lean | 803 | 805 |
/-
Copyright (c) 2022 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Star.Module
import Mathlib.Algebra.Star.NonUn... | theorem iInf_toSubalgebra {ι : Sort*} (S : ι → StarSubalgebra R A) :
(⨅ i, S i).toSubalgebra = ⨅ i, (S i).toSubalgebra :=
SetLike.coe_injective <| by simp
| Mathlib/Algebra/Star/Subalgebra.lean | 689 | 691 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibili... |
end Algebra
end coeffsIn
| Mathlib/Algebra/MvPolynomial/Basic.lean | 984 | 986 |
/-
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... | | 0 => rfl
| 1 => rfl
lemma map_comp {i j k : Fin 2} (hij : i ≤ j) (hjk : j ≤ k) :
map f i k (hij.trans hjk) = map f i j hij ≫ map f j k hjk := by
obtain rfl | rfl : i = j ∨ j = k := by omega
· rw [map_id, id_comp]
· rw [map_id, comp_id]
end Mk₁
/-- Constructor for `ComposableArrows C 1`. -/
@[simp... | Mathlib/CategoryTheory/ComposableArrows.lean | 143 | 155 |
/-
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... | simpa only [one_zsmul] using toIocMod_sub_zsmul hp a b 1
@[simp]
| Mathlib/Algebra/Order/ToIntervalMod.lean | 436 | 438 |
/-
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... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | 575 | 579 | |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.MeasureTheory.Measure.AEMeasurable
import Mathlib.Order.Filter.EventuallyConst
/-!
# Measure preserving maps
We say that `f : α → β` is a measure p... | theorem measure_preimage {f : α → β} (hf : MeasurePreserving f μa μb) {s : Set β}
(hs : NullMeasurableSet s μb) : μa (f ⁻¹' s) = μb s := by
rw [← hf.map_eq] at hs ⊢
rw [map_apply₀ hf.1.aemeasurable hs]
| Mathlib/Dynamics/Ergodic/MeasurePreserving.lean | 122 | 126 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.UniformSpace.Cauchy
/-!
# Uniform convergence
A sequence of functions `Fₙ` (with values in a metric space) converges uniformly on a se... | Mathlib/Topology/UniformSpace/UniformConvergence.lean | 764 | 773 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Order.Filter.Bases.Finit... | ### Topological groups
A topological group is a group in which the multiplication and inversion operations are
| Mathlib/Topology/Algebra/Group/Basic.lean | 344 | 346 |
/-
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.Fin.Fin2
import Mathlib.Data.PFun
import Mathlib.Data.Vector3
import Mathlib.NumberTheory.PellMatiyasevic
/-!
# Diophantine functions and Matiyas... | @[inherit_doc]
scoped notation:35 x " D∨ " y => Dioph.union x y
@[inherit_doc]
scoped notation:30 "D∃" => Dioph.vec_ex1_dioph
/-- Local abbreviation for `Fin2.ofNat'` -/
| Mathlib/NumberTheory/Dioph.lean | 504 | 510 |
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.LinearAlgebra.DirectSum.Finsupp
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
/-!
# Some results on tensor product of s... | (mulMap _ N).rangeRestrict
| Mathlib/LinearAlgebra/TensorProduct/Submodule.lean | 137 | 138 |
/-
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.Notation.Prod
import Mathlib.Data.Nat.Sqrt
import Mathlib.Data.Set.Lattice.Image
/-!
# Naturals pairing function
Th... | · simpa [pair, h] using le_trans (le_mul_self _) (Nat.le_add_right _ _)
· simp [pair, h]
| Mathlib/Data/Nat/Pairing.lean | 102 | 104 |
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Ring.Int.Defs
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Cast.Order.Basic
import Math... | cases e : pred' n
· have : (1 : ℕ) ≤ Nat.pred n := Nat.pred_le_pred ((@cast_lt ℕ _ _ _).2 h)
| Mathlib/Data/Num/Lemmas.lean | 694 | 695 |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll, Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Subspace
import Mathlib.Analysis.Normed.Operator.Banach
import Mathlib.LinearAlgebra.Sesqui... |
section Complex
variable {V : Type*} [SeminormedAddCommGroup V] [InnerProductSpace ℂ V]
| Mathlib/Analysis/InnerProductSpace/Symmetric.lean | 129 | 133 |
/-
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.Group.Action.Pi
import Mathlib.Algebra.Order.AbsoluteValue.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.Mi... | rfl
theorem sup_limZero {f g : CauSeq α abs} (hf : LimZero f) (hg : LimZero g) : LimZero (f ⊔ g)
| ε, ε0 =>
| Mathlib/Algebra/Order/CauSeq/Basic.lean | 733 | 736 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
/-! # P... | pure (.positive q(Real.rpow_zero_pos $a))
| _, _, _ => throwError "not Real.rpow"
/-- Extension for the `positivity` tactic: exponentiation by a real number is nonnegative when
the base is nonnegative and positive when the base is positive. -/
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 363 | 367 |
/-
Copyright (c) 2022 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Yury Kudryashov, Kevin H. Wilson, Heather Macbeth
-/
import Mathlib.Order.Filter.Tendsto
/-!
# Product and coproduct filters
In this file we define `F... |
theorem le_prod {f : Filter (α × β)} {g : Filter α} {g' : Filter β} :
| Mathlib/Order/Filter/Prod.lean | 422 | 423 |
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.MeasureTheory.Group.Measure
/-!
# Bochner Integration on Groups
We develop properties of inte... | @[to_additive]
theorem integral_div_right_eq_self [IsMulRightInvariant μ] (f : G → E) (g : G) :
(∫ x, f (x / g) ∂μ) = ∫ x, f x ∂μ := by
| Mathlib/MeasureTheory/Group/Integral.lean | 103 | 105 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Amelia Livingston, Yury Kudryashov,
Neil Strickland, Aaron Anderson
-/
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algeb... |
theorem isRelPrime_mul_unit_left_right : IsRelPrime y (x * z) ↔ IsRelPrime y z := by
| Mathlib/Algebra/Divisibility/Units.lean | 187 | 188 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.GroupTheory.Perm.Basic
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.Lis... |
namespace Equiv.Perm
theorem subtypePerm_apply_pow_of_mem {g : Perm α} {s : Finset α}
| Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 972 | 975 |
/-
Copyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Group.Action.Pi
import Mathlib.Data.Fintype.BigOperators
import Mat... | rw [Fintype.prod_equiv finSumFinEquiv.symm f fun i => f (finSumFinEquiv.toFun i)]
| Mathlib/Algebra/BigOperators/Fin.lean | 179 | 179 |
/-
Copyright (c) 2020 Kexing Ying and Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.Gro... | Mathlib/Algebra/BigOperators/Finprod.lean | 1,267 | 1,272 | |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
import Mathlib.LinearAlgebra.Orientation
/-!
# Orientations of real inner product spaces.
Th... | theorem abs_volumeForm_apply_of_orthonormal (v : OrthonormalBasis (Fin n) ℝ E) :
|o.volumeForm v| = 1 := by
simpa [o.volumeForm_robust' v v] using congr_arg abs v.toBasis.det_self
theorem volumeForm_map {F : Type*} [NormedAddCommGroup F] [InnerProductSpace ℝ F]
[Fact (finrank ℝ F = n)] (φ : E ≃ₗᵢ[ℝ] F) (x : ... | Mathlib/Analysis/InnerProductSpace/Orientation.lean | 285 | 305 |
/-
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... | · subst j
| Mathlib/Logic/Function/Basic.lean | 611 | 611 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Data.Subtype
import Mathlib.Order.Defs.LinearOrder
import Mathlib.Order.Notation
import Mathlib.Tactic.GCongr.Core
import Mathlib.Tactic.... | Mathlib/Order/Basic.lean | 1,559 | 1,560 | |
/-
Copyright (c) 2024 Theodore Hwa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kim Morrison, Violeta Hernández Palacios, Junyan Xu, Theodore Hwa
-/
import Mathlib.Logic.Hydra
import Mathlib.SetTheory.Surreal.Basic
/-!
### Surreal multiplication
In... | (hs : ∀ i, IH3 (-x₂) ((-x₁).moveLeft i) (-x₁) y₁ y₂) (hl : x₁ < x₂) :
P3 x₁ x₂ y₁ y₂ := by
obtain (⟨i,hi⟩|⟨i,hi⟩) := lf_iff_exists_le.1 (lf_of_lt hl)
· exact P3_of_le_left i (h i) hi
· apply P3_neg.2 <| P3_of_le_left _ (hs (toLeftMovesNeg i)) _
simpa
/-- The main chunk of Theorem 8 in [Conway2001] / ... | Mathlib/SetTheory/Surreal/Multiplication.lean | 412 | 419 |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.FieldTheory.Finite.Basic
/-!
# The Chevalley–Warning theorem
This file contains a proof of the Chevalley–Warning theorem.
Throughout most of this fil... | /-- The **Chevalley–Warning theorem**, binary version.
Let `f₁`, `f₂` be two multivariate polynomials in finitely many variables (`X s`, `s : σ`) over a
finite field of characteristic `p`.
Assume that the sum of the total degrees of `f₁` and `f₂` is less than the cardinality of `σ`.
Then the number of common solutions ... | Mathlib/FieldTheory/ChevalleyWarning.lean | 180 | 188 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Yury Kudryashov
-/
import Mathlib.Analysis.Normed.Group.Submodule
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
import Mathlib.LinearAlgebra.AffineSpace.Mid... | /-- The distance defines a metric space structure on the torsor. This
is not an instance because it depends on `V` to define a `MetricSpace P`. -/
def metricSpaceOfNormedAddCommGroupOfAddTorsor (V P : Type*) [NormedAddCommGroup V]
| Mathlib/Analysis/Normed/Group/AddTorsor.lean | 185 | 187 |
/-
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.MeasurableSpace.MeasurablyGenerated
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.Order.Interval.Set... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 2,230 | 2,231 | |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Filippo A. E. Nuccio, Sam van Gool
-/
import Mathlib.Data.Fintype.Order
import Mathlib.Order.Interval.Finset.Basic
import Mathlib.Order.Irreducible
import Mathlib.Order.Upp... | @[simp] lemma infIrred_iff_of_finite : InfIrred s ↔ ∃ a, Ici a = s := by
refine ⟨fun hs ↦ ?_, ?_⟩
· obtain ⟨a, ha, has⟩ := (s : Set α).toFinite.exists_minimal_wrt id _ (coe_nonempty.2 hs.ne_top)
exact ⟨a, (hs.2 <| erase_inf_Ici ha <| by simpa [eq_comm] using has).resolve_left
(lt_erase.2 ha).ne'⟩
· rint... | Mathlib/Order/Birkhoff.lean | 70 | 76 |
/-
Copyright (c) 2020 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.Tactic.Linarith.Datatypes
import Mathlib.Tactic.Zify
import Mathlib.Tactic.CancelDenoms.Core
import Mathlib.Control.Basic
import Mathlib.Util.AtomM
... | match e.getAppFnArgs with
| Mathlib/Tactic/Linarith/Preprocessing.lean | 272 | 272 |
/-
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.Module.Submodule.Ker
/-!
# The submodule of elements `x : M` such that `f x = g x`
## Main declarations
* `LinearMap.eqLocus`: the submodule of el... | include τ₁₂ in
theorem ext_on_codisjoint {f g : F} {S T : Submodule R M} (hST : Codisjoint S T)
(hS : Set.EqOn f g S) (hT : Set.EqOn f g T) : f = g :=
DFunLike.ext _ _ fun _ ↦ eqOn_sup hS hT <| hST.eq_top.symm ▸ trivial
| Mathlib/Algebra/Module/Submodule/EqLocus.lean | 73 | 76 |
/-
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.Notation.Prod
import Mathlib.Data.Nat.Sqrt
import Mathlib.Data.Set.Lattice.Image
/-!
# Naturals pairing function
Th... | · rw [if_pos h, max_eq_right h.le, min_eq_left h.le, Nat.pow_two]
rw [if_neg h.not_lt, max_eq_left h, min_eq_right h, Nat.pow_two, Nat.add_assoc,
Nat.add_le_add_iff_left]
exact Nat.le_add_left _ _
theorem add_le_pair (m n : ℕ) : m + n ≤ pair m n := by
simp only [pair, Nat.add_assoc]
split_ifs
· have :=... | Mathlib/Data/Nat/Pairing.lean | 138 | 146 |
/-
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 | 3,048 | 3,049 | |
/-
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... |
lemma toMvPolynomial_eval_eq_apply (f : M₁ →ₗ[R] M₂) (i : ι₂) (c : ι₁ →₀ R) :
eval c (f.toMvPolynomial b₁ b₂ i) = b₂.repr (f (b₁.repr.symm c)) i := by
rw [toMvPolynomial, Matrix.toMvPolynomial_eval_eq_apply,
| Mathlib/Algebra/Module/LinearMap/Polynomial.lean | 160 | 163 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.WithBot
/-!
# Intervals in `WithTop α` and `WithBot α`
In this file we ... | @[simp]
| Mathlib/Order/Interval/Set/WithBotTop.lean | 67 | 67 |
/-
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.Topology.MetricSpace.PiNat
import Mathlib.Topology.Metrizable.CompletelyMetrizable
import Mathlib.Topology.Sets.Opens
/-!
# Polish spaces
A top... | Mathlib/Topology/MetricSpace/Polish.lean | 408 | 422 | |
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.ZPow
import Mathlib.Data.Matrix.ConjTranspose
/-! # Hermitian matrices
Th... | (A + B).IsHermitian :=
IsSelfAdjoint.add hA hB
end AddMonoid
| Mathlib/LinearAlgebra/Matrix/Hermitian.lean | 143 | 146 |
/-
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... | · simp [(coprime_zero_right _).mp hab]
rw [mem_primeFactorsList_mul ha.ne' hb.ne', List.mem_union_iff]
open List
| Mathlib/Data/Nat/Factors.lean | 278 | 281 |
/-
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
-/
import Mathlib.Algebra.Field.Subfield.Defs
import Mathlib.Algebra.Order.Group.Pointwise.Interval
import Mathlib.Analysis.Normed.Ring.Basic
/-!
# Norm... | Mathlib/Analysis/Normed/Field/Basic.lean | 1,120 | 1,123 | |
/-
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.CharP.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Module.Synonym
import Mathlib.LinearAlgebra.Aff... |
theorem lineMap_lt_left_iff_lt (h : 0 < r) : lineMap a b r < a ↔ b < a :=
left_lt_lineMap_iff_lt (E := Eᵒᵈ) h
| Mathlib/LinearAlgebra/AffineSpace/Ordered.lean | 83 | 86 |
/-
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.CondexpL1
/-! # Conditional expectation
We build the conditional expectation of an integrable function `f` ... | theorem setIntegral_condExp (hm : m ≤ m₀) [SigmaFinite (μ.trim hm)] (hf : Integrable f μ)
(hs : MeasurableSet[m] s) : ∫ x in s, (μ[f|m]) x ∂μ = ∫ x in s, f x ∂μ := by
rw [setIntegral_congr_ae (hm s hs) ((condExp_ae_eq_condExpL1 hm f).mono fun x hx _ => hx)]
exact setIntegral_condExpL1 hf hs
@[deprecated (since... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | 247 | 255 |
/-
Copyright (c) 2023 Scott Carnahan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Carnahan
-/
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Eval.SMul
/-!
# Scalar-multiple polynomial ev... | theorem smeval_C : (C r).smeval x = r • x ^ 0 := by
simp only [smeval_eq_sum, smul_pow, zero_smul, sum_C_index]
| Mathlib/Algebra/Polynomial/Smeval.lean | 57 | 58 |
/-
Copyright (c) 2022 Felix Weilacher. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Felix Weilacher
-/
import Mathlib.Topology.Separation.Regular
/-!
# Perfect Sets
In this file we define perfect subsets of a topological space, and prove some basic properties,
inc... | theorem AccPt.nhds_inter {x : α} {U : Set α} (h_acc : AccPt x (𝓟 C)) (hU : U ∈ 𝓝 x) :
AccPt x (𝓟 (U ∩ C)) := by
have : 𝓝[≠] x ≤ 𝓟 U := by
rw [le_principal_iff]
exact mem_nhdsWithin_of_mem_nhds hU
rw [AccPt, ← inf_principal, ← inf_assoc, inf_of_le_left this]
exact h_acc
| Mathlib/Topology/Perfect.lean | 62 | 68 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Logic.Nontrivial.Basic
import Mathlib.Order.TypeTags
import Mathlib.Data.Option.NAry
import Mathlib.Tactic.Contrapose
import Mathlib.Tactic.Lift
import... | obtain _ | y := y
· simp [WithBot.none_eq_bot, eq_bot_iff_forall_le]
· exact ⟨fun h ↦ h _ le_rfl, fun hmn a ham ↦ hmn.trans ham⟩
end Preorder
| Mathlib/Order/WithBot.lean | 342 | 346 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou, Kim Morrison, Adam Topaz
-/
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products
import Mathlib.Cat... | open WidePushout
open WidePushoutShape
variable {FC : C → C → Type*} {CC : C → Type v} [∀ X Y, FunLike (FC X Y) (CC X) (CC Y)]
variable [ConcreteCategory.{v} C FC]
theorem widePushout_exists_rep {B : C} {α : Type _} {X : α → C} (f : ∀ j : α, B ⟶ X j)
[HasWidePushout.{v} B X f] [PreservesColimit (wideSpan B X f) ... | Mathlib/CategoryTheory/Limits/Shapes/ConcreteCategory.lean | 326 | 335 |
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