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 |
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/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Batteries.Data.List.Perm
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.TakeWhile
import Mathlib.Order.Fin.Basic
/-!
# So... |
theorem sorted_replicate (n : ℕ) (a : α) : Sorted r (replicate n a) ↔ n ≤ 1 ∨ r a a :=
pairwise_replicate
| Mathlib/Data/List/Sort.lean | 145 | 148 |
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.GroupTheory.Index
/-!
# Complements
In this file we define the complement of a subgroup.
## Main definitions
- `Subgroup.IsComplement S T` where ... |
theorem equiv_snd_eq_one_of_mem_of_one_mem {g : G} (h1 : 1 ∈ T) (hg : g ∈ S) :
(hST.equiv g).snd = ⟨1, h1⟩ := by
| Mathlib/GroupTheory/Complement.lean | 514 | 516 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.Cover.Open
import Mathlib.AlgebraicGeometry.GammaSpecAdjunction
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Lim... | theorem basicOpen :
IsAffineOpen (X.basicOpen f) := by
rw [← hU.fromSpec_image_basicOpen, Scheme.Hom.isAffineOpen_iff_of_isOpenImmersion]
convert isAffineOpen_opensRange
(Spec.map (CommRingCat.ofHom <| algebraMap Γ(X, U) (Localization.Away f)))
exact Opens.ext (PrimeSpectrum.localization_away_comap_range ... | Mathlib/AlgebraicGeometry/AffineScheme.lean | 539 | 550 |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Module.Synonym
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
import Mathlib.Order.Mono... |
@[to_additive (attr := simp)]
lemma antivaryOn_inv_left : AntivaryOn f⁻¹ g s ↔ MonovaryOn f g s := by
| Mathlib/Algebra/Order/Monovary.lean | 35 | 37 |
/-
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 ... | intro h
simp [H.edge_vert h, H.edge_vert h.symm]
/-! ### Edge deletion -/
| Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 1,021 | 1,026 |
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, María Inés de Frutos-Fernández
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Algebra.DirectSum.Decomposition
import Mathlib.Algeb... | exfalso; exact h _ hd.1 hd.2
theorem weightedHomogeneousComponent_eq_zero [SemilatticeSup M] [OrderBot M]
(h : weightedTotalDegree w φ < n) : weightedHomogeneousComponent w n φ = 0 := by
classical
| Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean | 347 | 351 |
/-
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, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Div
import Mathlib.RingTheory... | Mathlib/Algebra/Polynomial/RingDivision.lean | 644 | 647 | |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Operations
import Mathlib.Order.Basic
import Mathlib.Order.BooleanAlgebra
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.B... | @[simp]
theorem ite_inter_self (t s s' : Set α) : t.ite s s' ∩ t = s ∩ t := by
| Mathlib/Data/Set/Basic.lean | 1,324 | 1,325 |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.DoldKan.FunctorN
/-!
# Comparison with the normalized Moore complex functor
In this file, we show that when the category `A` is abelian,
the... | dsimp [AlgebraicTopology.inclusionOfMooreComplexMap, NormalizedMooreComplex.objX]
rw [← factorThru_arrow _ _ (finset_inf_arrow_factors Finset.univ _ j
(by simp only [Finset.mem_univ])), assoc, kernelSubobject_arrow_comp, comp_zero]
theorem factors_normalizedMooreComplex_PInfty (n : ℕ) :
| Mathlib/AlgebraicTopology/DoldKan/Normalized.lean | 44 | 48 |
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.LinearOrder
/-!
# Monotone surjective functions are surjective on intervals
A monotone surjecti... | intro p hp
rcases h_surj p with ⟨x, rfl⟩
refine ⟨x, mem_Ioo.2 ?_, rfl⟩
contrapose! hp
exact fun h => h.2.not_le (h_mono <| hp <| h_mono.reflect_lt h.1)
theorem surjOn_Ico_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)
| Mathlib/Order/Interval/Set/SurjOn.lean | 26 | 32 |
/-
Copyright (c) 2021 Luke Kershaw. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Luke Kershaw, Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.Basic
import Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts
import Mathlib.CategoryThe... | lemma yoneda_exact₂ {X : C} (f : T.obj₂ ⟶ X) (hf : T.mor₁ ≫ f = 0) :
∃ (g : T.obj₃ ⟶ X), f = T.mor₂ ≫ g := by
obtain ⟨g, ⟨hg₁, _⟩⟩ := complete_distinguished_triangle_morphism T _ hT
(contractible_distinguished₁ X) 0 f (by aesop_cat)
exact ⟨g, by simpa using hg₁.symm⟩
lemma yoneda_exact₃ {X : C} (f : T.obj₃... | Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean | 232 | 239 |
/-
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.Basic
/-!
# Maps between real and extended non-negative real numbers
This file focuses on the functions `ENNReal.toReal... |
protected theorem dichotomy (p : ℝ≥0∞) [Fact (1 ≤ p)] : p = ∞ ∨ 1 ≤ p.toReal :=
haveI : p = ⊤ ∨ 0 < p.toReal ∧ 1 ≤ p.toReal := by
| Mathlib/Data/ENNReal/Real.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, Simon Hudon
-/
import Mathlib.CategoryTheory.Monoidal.Category
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.PEmpty
/-!
# The mo... | simp
theorem associator_naturality {X₁ X₂ X₃ Y₁ Y₂ Y₃ : C} (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (f₃ : X₃ ⟶ Y₃) :
tensorHom ℬ (tensorHom ℬ f₁ f₂) f₃ ≫ (BinaryFan.associatorOfLimitCone ℬ Y₁ Y₂ Y₃).hom =
(BinaryFan.associatorOfLimitCone ℬ X₁ X₂ X₃).hom ≫ tensorHom ℬ f₁ (tensorHom ℬ f₂ f₃) := by
| Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Basic.lean | 290 | 294 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Analysis.InnerProductSpace.LinearMap
import Mathlib.MeasureTheory.Function.LpSpace.ContinuousFunctions
import Mathlib.MeasureTheory.Function.StronglyMeasur... | suffices h_int_zero :
(∫ x in sᶜ, inner (indicatorConstLp 2 hs hμs c x) (f x) ∂μ) = ∫ _ in sᶜ, (0 : 𝕜) ∂μ by
rw [h_int_zero]
simp
exact setIntegral_congr_ae hs.compl h_ae_eq
have h_indicator : ∀ᵐ x : α ∂μ, x ∉ s → indicatorConstLp 2 hs hμs c x = 0 :=
indicatorConstLp_coe... | Mathlib/MeasureTheory/Function/L2Space.lean | 231 | 255 |
/-
Copyright (c) 2019 Rohan Mitta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rohan Mitta, Kevin Buzzard, Alistair Tucker, Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.Dynamics.Fixed... | · exact lt_top_iff_ne_top.2 hxy
· apply edist_efixedPoint_lt_top'
end ContractingWith
| Mathlib/Topology/MetricSpace/Contracting.lean | 223 | 227 |
/-
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... | theorem frequently_iff {f : Filter α} {P : α → Prop} :
(∃ᶠ x in f, P x) ↔ ∀ {U}, U ∈ f → ∃ x ∈ U, P x := by
| Mathlib/Order/Filter/Basic.lean | 750 | 751 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Lebesgue.Countable
import Mathlib.MeasureTheory.Integral.Le... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 1,234 | 1,250 | |
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.Ring.Int.Parity
/-!
# Results about powers in fields or... | obtain ⟨k, rfl⟩ := h
simp_rw [zpow_add' (.inr (.inl hn)), zpow_one, zpow_mul, zpow_two, neg_mul_neg,
neg_mul_eq_mul_neg]
theorem Odd.neg_one_zpow (h : Odd n) : (-1 : α) ^ n = -1 := by rw [h.neg_zpow, one_zpow]
| Mathlib/Algebra/Field/Power.lean | 26 | 30 |
/-
Copyright (c) 2024 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir
-/
import Mathlib.Algebra.Pointwise.Stabilizer
import Mathlib.Data.Setoid.Partition
import Mathlib.GroupTheory.GroupAction.Pointwise
import Mathlib.GroupTh... |
- `MulAction.BlockMem` : the type of blocks containing a given element
- `MulAction.BlockMem.instBoundedOrder` :
the type of blocks containing a given element is a bounded order.
| Mathlib/GroupTheory/GroupAction/Blocks.lean | 38 | 42 |
/-
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.Data.Fintype.Option
import Mathlib.GroupTheory.Perm.Sign
/-!
# Permutations of `Option α`
-/
open Equiv
@[simp]
theorem Equiv.optionCongr_one {α : Type*}... | · have hn : σ (some x) ≠ none := by simp [h]
have hσn : σ (some x) ≠ σ none := σ.injective.ne (by simp)
simp [removeNone_some _ ⟨_, h⟩, ← h, swap_apply_of_ne_of_ne hn hσn]
simpa using this
/-- Permutations of `Option α` are equivalent to fixing an
`Option α` and permuting the remaining with a `... | Mathlib/GroupTheory/Perm/Option.lean | 47 | 58 |
/-
Copyright (c) 2022 George Peter Banyard, Yaël Dillies, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: George Peter Banyard, Yaël Dillies, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Path
import Mathlib.Combinatorics.SimpleGraph.Metric
/-!
#... | variable (G H)
| Mathlib/Combinatorics/SimpleGraph/Prod.lean | 65 | 66 |
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Eric Wieser, Jeremy Avigad, Johan Commelin
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.Linear... | cases hD.nonempty_invertible
letI := fromBlocksZero₂₁Invertible A B D
simp_rw [← invOf_eq_nonsing_inv, invOf_fromBlocks_zero₂₁_eq]
· have hD := hAD.not.mp hA
| Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 194 | 197 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
/-!
# GCD and LCM operations on finsets
## Main definitions
- `Finset.gcd` - the greatest... | @[local simp] -- This will later be provable by other `simp` lemmas.
theorem normalize_gcd : normalize (s.gcd f) = s.gcd f := by simp [gcd_def]
theorem gcd_union [DecidableEq β] : (s₁ ∪ s₂).gcd f = GCDMonoid.gcd (s₁.gcd f) (s₂.gcd f) :=
| Mathlib/Algebra/GCDMonoid/Finset.lean | 151 | 154 |
/-
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, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib... | [NoZeroSMulDivisors T S] {p : T[X]} {a : S} : a ∈ p.aroots S ↔ p ≠ 0 ∧ aeval a p = 0 := by
rw [mem_aroots', Polynomial.map_ne_zero_iff]
exact FaithfulSMul.algebraMap_injective T S
theorem aroots_mul [CommRing S] [IsDomain S] [Algebra T S]
| Mathlib/Algebra/Polynomial/Roots.lean | 401 | 405 |
/-
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... |
namespace ENat
theorem card_eq_coe_natCard (α : Type*) [Finite α] : card α = Nat.card α := by
| Mathlib/Data/Finite/Card.lean | 166 | 169 |
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subgroup.Ker
/-!
# Basic results on subgroups
We prove basic results... | rw [normal_subgroupOf_iff_le_normalizer le_sup_right]
exact sup_le hLE le_normalizer
| Mathlib/Algebra/Group/Subgroup/Basic.lean | 917 | 919 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Group.Defs
import Mathlib.Algebra.Order.Group.Unbundled.Abs
import Mathlib.Algebra.Order... | Mathlib/Algebra/Order/Group/Abs.lean | 267 | 271 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Real
/-!
# Power function... | rfl
@[gcongr] theorem rpow_le_rpow {x y : ℝ≥0∞} {z : ℝ} (h₁ : x ≤ y) (h₂ : 0 ≤ z) : x ^ z ≤ y ^ z :=
monotone_rpow_of_nonneg h₂ h₁
| Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 731 | 734 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.IsPrimePow
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Algebra.Ring.CharZero
im... | intro ne_n
have hlt : ∑ x ∈ n.succ.succ.properDivisors, x < n.succ.succ :=
lt_of_le_of_ne (Nat.le_of_dvd (Nat.succ_pos _) h) ne_n
symm
rw [← mem_singleton, eq_properDivisors_of_subset_of_sum_eq_sum (singleton_subset_iff.2
(mem_properDivisors.2 ⟨h, hlt⟩)) (sum_singleton _ _), mem_... | Mathlib/NumberTheory/Divisors.lean | 446 | 461 |
/-
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... | cons_injective2.left _
| Mathlib/Data/Stream/Init.lean | 94 | 94 |
/-
Copyright (c) 2023 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.Analysis.SpecialFunctions.PolarCoord
import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
/-!
# Integrals involving the Gamma function
In this file, we... | theorem integral_exp_neg_rpow {p : ℝ} (hp : 0 < p) :
∫ x in Ioi (0 : ℝ), exp (- x ^ p) = Gamma (1 / p + 1) := by
convert (integral_rpow_mul_exp_neg_rpow hp neg_one_lt_zero) using 1
· simp_rw [rpow_zero, one_mul]
· rw [zero_add, Gamma_add_one (one_div_ne_zero (ne_of_gt hp))]
| Mathlib/MeasureTheory/Integral/Gamma.lean | 59 | 63 |
/-
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.Bousfield
import Mathlib.CategoryTheory.Sites.Sheafification
/-!
# The sheaf category as a localized category
In this file, it is s... |
lemma W_eq_inverseImage_isomorphisms_of_adjunction (adj : G ⊣ sheafToPresheaf J A) :
J.W = (MorphismProperty.isomorphisms _).inverseImage G := by
rw [W_eq_W_range_sheafToPresheaf_obj,
| Mathlib/CategoryTheory/Sites/Localization.lean | 59 | 62 |
/-
Copyright (c) 2021 Justus Springer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Justus Springer
-/
import Mathlib.CategoryTheory.Sites.Spaces
import Mathlib.Topology.Sheaves.Sheaf
import Mathlib.CategoryTheory.Sites.DenseSubsite.Basic
/-!
# Coverings and sieves... | · constructor
· intro V W
exact ⟨⟨⟨PUnit.unit⟩, V.right ⊓ W.right, homOfLE <| le_inf V.hom.le W.hom.le⟩,
StructuredArrow.homMk (homOfLE inf_le_left),
StructuredArrow.homMk (homOfLE inf_le_right), trivial⟩
· exact fun _ _ _ _ ↦ ⟨_, 𝟙 _, by simp [eq_iff_true_of_subsingleton]⟩
| Mathlib/Topology/Sheaves/SheafCondition/Sites.lean | 180 | 185 |
/-
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... | ⟨fun _x => H₁.lt_iff.2 (H₂.1 _), fun o l _a =>
H₁.le_set' (· < o) ⟨0, l.pos⟩ g _ fun _c => H₂.2 _ l _⟩
| Mathlib/SetTheory/Ordinal/Arithmetic.lean | 434 | 435 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Joël Riou
-/
import Mathlib.Algebra.Homology.ExactSequence
import Mathlib.CategoryTheory.Abelian.Refinements
/-!
# The four and five lemmas
Consider the following commu... | end Five
/-! The following "three lemmas" for morphisms in `ComposableArrows C 2` are
special cases of "four lemmas" applied to diagrams where some of the
leftmost or rightmost maps (or objects) are zero. -/
section Three
variable {R₁ R₂ : ComposableArrows C 2} (φ : R₁ ⟶ R₂)
| Mathlib/CategoryTheory/Abelian/DiagramLemmas/Four.lean | 143 | 152 |
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
/-!
# Symmetric difference and bi-implication
This file defines... | Mathlib/Order/SymmDiff.lean | 796 | 800 | |
/-
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.Abelian
import Mathlib.Algebra.Lie.BaseChange
import Mathlib.Algebra.Lie.IdealOperations
import Mathlib.Order.Hom.Basic
import Mathlib.RingTheory... | @[simp] theorem derivedSeries_baseChange {A : Type*} [CommRing A] [Algebra R A] (k : ℕ) :
derivedSeries A (A ⊗[R] L) k = (derivedSeries R L k).baseChange A := by
rw [derivedSeries_def, derivedSeries_def, ← derivedSeriesOfIdeal_baseChange,
| Mathlib/Algebra/Lie/Solvable.lean | 136 | 138 |
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subgroup.Ker
/-!
# Basic results on subgroups
We prove basic results... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 2,566 | 2,568 | |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.IsolatedZeros
import Mathlib.Analysis.SpecialFunctions.Complex.CircleMap
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
/-... | hf.norm.mul_continuousOn continuousOn_const
· refine Eventually.of_forall fun θ _ => HasSum.const_smul _ ?_
simp only [smul_smul]
refine HasSum.smul_const ?_ _
have : ‖w / (circleMap c R θ - c)‖ < 1 := by simpa [abs_of_pos hR] using hwR.2
convert (hasSum_geometric_of_norm_lt_one this).mul_right ... | Mathlib/MeasureTheory/Integral/CircleIntegral.lean | 533 | 538 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
/-!
# Oriented angles in right-angled triangles.
T... | theorem norm_div_tan_oangle_sub_right_of_oangle_eq_pi_div_two {x y : V}
(h : o.oangle x y = ↑(π / 2)) : ‖x‖ / Real.Angle.tan (o.oangle y (y - x)) = ‖y‖ := by
have hs : (o.oangle y (y - x)).sign = 1 := by
rw [oangle_sign_sub_right_swap, h, Real.Angle.sign_coe_pi_div_two]
| Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 428 | 431 |
/-
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... | rwa [IsThreeCycle, cycleType_inv]
@[simp]
theorem inv_iff {f : Perm α} : IsThreeCycle f⁻¹ ↔ IsThreeCycle f :=
| Mathlib/GroupTheory/Perm/Cycle/Type.lean | 622 | 625 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen, Antoine Labelle
-/
import Mathlib.LinearAlgebra.Contraction
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.... | trace R P (g ∘ₗ f ∘ₗ h) = trace R N (f ∘ₗ h ∘ₗ g) := by
rw [trace_comp_comm', comp_assoc]
| Mathlib/LinearAlgebra/Trace.lean | 265 | 267 |
/-
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
-/
import Mathlib.Algebra.Module.Submodule.Bilinear
import Mathlib.Algebra.Module.Equiv.Basic
import Mathlib.GroupTheory.Congruence.Hom
import Mathlib.Tactic.Abel
... |
theorem rTensor_def : f.rTensor M = TensorProduct.map f LinearMap.id := rfl
@[simp]
| Mathlib/LinearAlgebra/TensorProduct/Basic.lean | 932 | 935 |
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Mario Carneiro
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds
/-!
# Pi
This file contains lemmas which establish bounds on `Real.pi`.
Notably, t... | theorem pi_upper_bound_start (n : ℕ) {a}
(h : (2 : ℝ) - ((a - 1 / (4 : ℝ) ^ n) / (2 : ℝ) ^ (n + 1)) ^ 2 ≤
sqrtTwoAddSeries ((0 : ℕ) / (1 : ℕ)) n)
(h₂ : (1 : ℝ) / (4 : ℝ) ^ n ≤ a) : π < a := by
refine lt_of_lt_of_le (pi_lt_sqrtTwoAddSeries n) ?_
rw [← le_sub_iff_add_le, ← le_div_iff₀', sqrt_le_left, ... | Mathlib/Data/Real/Pi/Bounds.lean | 77 | 82 |
/-
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... | @[simp]
theorem transpose_empty_cols (A : Matrix (Fin 0) m' α) : Aᵀ = of fun _ => ![] :=
funext fun _ => empty_eq _
| Mathlib/Data/Matrix/Notation.lean | 233 | 235 |
/-
Copyright (c) 2020 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.GroupTheory.Complement
/-!
# Semidirect product
This file defines semidirect products of groups, and the canonical maps in and out of the
semidirect prod... | Mathlib/GroupTheory/SemidirectProduct.lean | 240 | 240 | |
/-
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.LinearAlgebra.CliffordAlgebra.Conjugation
/-!
# Recursive computation rules for the Clifford algebra
This file provides API for a special case `CliffordAlg... | Mathlib/LinearAlgebra/CliffordAlgebra/Fold.lean | 224 | 234 | |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 398 | 398 | |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Set.Lattice
import Mathlib.Order.ConditionallyCompleteLattice.Defs
/-!
# Theory of conditionally complete lattices
A conditionally complet... | Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | 963 | 970 | |
/-
Copyright (c) 2023 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Adam Topaz
-/
import Mathlib.Algebra.Category.Ring.Colimits
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.Algebra.Category.Ring.Limits
import M... | simp_rw [← CategoryTheory.Iso.eq_inv_comp]
rw [← smoothSheafCommRing.forgetStalk_inv_comp_eval]
lemma smoothSheafCommRing.eval_surjective (x) :
Function.Surjective (smoothSheafCommRing.eval IM I M R x) := by
intro r
| Mathlib/Geometry/Manifold/Sheaf/Smooth.lean | 363 | 368 |
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Data.Setoid.Basic
import Mathlib.GroupTheory.Congruence.Hom
/-!
# Congruence relations
This f... | Mathlib/GroupTheory/Congruence/Basic.lean | 593 | 597 | |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Algebra.Order.Group.Pointwise.Bounds
import Mathlib.Data.Real.Basic
import Mathlib.Ord... |
noncomputable instance : InfSet ℝ :=
⟨fun s => -sSup (-s)⟩
| Mathlib/Data/Real/Archimedean.lean | 125 | 128 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.Order.Floor.Defs
import Mathlib.Algebra.Order.Floor.Ring
import Mathlib.Algebra.Order.Floor.Semiring
deprecated_module (sinc... | Mathlib/Algebra/Order/Floor.lean | 441 | 443 | |
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: María Inés de Frutos-Fernández
-/
import Mathlib.Order.Filter.Cofinite
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.UniqueFactorizationDomai... | · rw [hI, MulZeroClass.zero_mul, count, dif_pos (Eq.refl _)]
· rw [h hI, MulZeroClass.mul_zero, count, dif_pos (Eq.refl _)]
/-- val_v(1) = 0. -/
theorem count_one : count K v (1 : FractionalIdeal R⁰ K) = 0 := by
have h1 : (1 : FractionalIdeal R⁰ K) =
| Mathlib/RingTheory/DedekindDomain/Factorization.lean | 363 | 368 |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau, María Inés de Frutos-Fernández, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.Polynomial.FieldDivision
import Mathlib.RingTheory.DiscreteValuationRing.Basi... |
instance : IsLocalRing R⟦X⟧ :=
{ inferInstanceAs <| IsLocalRing <| MvPowerSeries Unit R with }
| Mathlib/RingTheory/PowerSeries/Inverse.lean | 270 | 273 |
/-
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.... |
/-! ### Inverses of permutated matrices
| Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 647 | 649 |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Game.Basic
import Mathlib.SetTheory.Ordinal.NaturalOps
/-!
# Ordinals as games
We define the canonical map `Ordinal... | theorem toLeftMovesToPGame_symm_lt {o : Ordinal} (i : o.toPGame.LeftMoves) :
↑(toLeftMovesToPGame.symm i) < o :=
| Mathlib/SetTheory/Game/Ordinal.lean | 58 | 59 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.BoxIntegral.Partition.Basic
/-!
# Split a box along one or more hyperplanes
## Main definitions
A hyperplane `{x : ι → ℝ | x i = a}` spli... | pairs `(i, r)`. Suppose that this set contains all faces of a box `J`. The hyperplanes of `s` split
a box `I` into subboxes. Let `Js` be one of them. If `J` and `Js` have nonempty intersection, then
`Js` is a subbox of `J`. -/
| Mathlib/Analysis/BoxIntegral/Partition/Split.lean | 254 | 256 |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | Mathlib/Data/List/Basic.lean | 2,442 | 2,452 | |
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Data.Num.ZNum
import Mathlib.Tactic.Ring
/-!
# Primality for binary natural numbers
This file defines versions of ... | rw [minFacAux_to_nat]
· rfl
simp only [cast_one, cast_bit1]
rw [Nat.sqrt_lt]
calc
(n : ℕ) + (n : ℕ) + 1 ≤ (n : ℕ) + (n : ℕ) + (n : ℕ) := by simp
_ = (n : ℕ) * (1 + 1 + 1) := by simp only [mul_add, mul_one]
_ < _ := by
set_option simprocs false in simp [mul_lt_mul]
· rw [m... | Mathlib/Data/Num/Prime.lean | 65 | 83 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Edward Ayers
-/
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.HasPullback
import Mathlib.Data.Set.BooleanAlgebra
/-!
# Theory of sieves
- For an object `X` of a ca... | theorem pullback_inter {f : Y ⟶ X} (S R : Sieve X) :
(S ⊓ R).pullback f = S.pullback f ⊓ R.pullback f := by simp [Sieve.ext_iff]
| Mathlib/CategoryTheory/Sites/Sieves.lean | 539 | 540 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.Order.Hom.Monoid
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Tac... |
theorem map_sum_lt' {ι : Type*} {s : Finset ι} {f : ι → R} {g : Γ₀} (hg : 0 < g)
(hf : ∀ i ∈ s, v (f i) < g) : v (∑ i ∈ s, f i) < g :=
v.map_sum_lt (ne_of_gt hg) hf
protected theorem map_pow : ∀ (x) (n : ℕ), v (x ^ n) = v x ^ n :=
v.toMonoidWithZeroHom.toMonoidHom.map_pow
| Mathlib/RingTheory/Valuation/Basic.lean | 187 | 193 |
/-
Copyright (c) 2022 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Algebra.ZMod
import Mathlib.Algebra.Field.ZMod
import Mathlib.Algebra.MvPolynomial.Cardinal
import Mathlib.FieldTheory.IsAlgClosed.Basic
import Mat... | _ = Cardinal.lift.{v} (max #(MvPolynomial ι R) ℵ₀) := by
rw [lift_max, ← Cardinal.lift_mk_eq.2 ⟨hv.1.aevalEquiv.toEquiv⟩, lift_aleph0,
← lift_aleph0.{max u v w, max u w}, ← lift_max, lift_umax.{max u w, v}]
_ ≤ Cardinal.lift.{v} (max (max (max (Cardinal.lift #R) (Cardinal.lift #ι)) ℵ₀) ℵ₀) :=
... | Mathlib/FieldTheory/IsAlgClosed/Classification.lean | 96 | 100 |
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Integral.Prod
import Mathlib.Meas... | (quasiMeasurePreserving_sub_left_of_right_invariant μ x).tendsto_ae).fun_comp ↿fun x y ↦ L x y
theorem support_convolution_subset_swap : support (f ⋆[L, μ] g) ⊆ support g + support f := by
intro x h2x
| Mathlib/Analysis/Convolution.lean | 512 | 515 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Between
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathli... | simp only [zero_le, measure_singleton]
-- Now assume `C ≠ 0`
· have hCd0 : (C : ℝ≥0∞) ^ d ≠ 0 := by simp [hC0.ne']
have hCd : (C : ℝ≥0∞) ^ d ≠ ∞ := by simp [hd]
simp only [hausdorffMeasure_apply, ENNReal.mul_iSup, ENNReal.mul_iInf_of_ne hCd0 hCd,
| Mathlib/MeasureTheory/Measure/Hausdorff.lean | 689 | 693 |
/-
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... | open OrthonormalBasis
/-- An orthonormal basis, indexed by `Fin n`, with the given orientation. -/
| Mathlib/Analysis/InnerProductSpace/Orientation.lean | 141 | 143 |
/-
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 ... | Mathlib/CategoryTheory/Sites/CoverPreserving.lean | 215 | 235 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Michael Stoll
-/
import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic
/-!
# Legendre symbol
This file contains results about Legendre symbols.
We define the Le... | have ha : (a : ZMod p) ≠ 0 := by
intro hf
rw [(eq_zero_iff p a).mpr hf] at h
simp at h
by_contra hf
rcases imp_iff_or_not.mp (not_and'.mp hf) with hx | hy
· rw [eq_one_of_sq_sub_mul_sq_eq_zero' ha hx hxy, CharZero.eq_neg_self_iff] at h
| Mathlib/NumberTheory/LegendreSymbol/Basic.lean | 232 | 238 |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.C... | instance : Add (RatFunc K) :=
⟨RatFunc.add⟩
| Mathlib/FieldTheory/RatFunc/Basic.lean | 75 | 76 |
/-
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.Group.Basic
import Mathlib.Tactic.Commo... | section CommMonoid
| Mathlib/Algebra/Divisibility/Basic.lean | 183 | 184 |
/-
Copyright (c) 2020 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Monad.Types
import Mathlib.CategoryTheory.Monad.Limits
import Mathlib.CategoryTheory.Equivalence
import Mathlib.Topology.Category.CompHaus.Basic... | let ssu := Set (Set (Ultrafilter X))
let ι : fsu → ssu := fun x ↦ ↑x
let C0 : ssu := { Z | ∃ B ∈ F, X.str ⁻¹' B = Z }
let AA := { G : Ultrafilter X | A ∈ G }
let C1 := insert AA C0
let C2 := finiteInterClosure C1
-- C0 is closed under intersections.
have claim1 : ∀ (B) (_ : B ∈ C0) (C) (_ : C ∈ C0), B ∩... | Mathlib/Topology/Category/Compactum.lean | 204 | 211 |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.Bornology.Constructions
import Mathlib.Topology.MetricSpace.Pseudo.Def... |
@[continuity]
lemma continuous_dist : Continuous fun p : α × α ↦ dist p.1 p.2 := uniformContinuous_dist.continuous
@[continuity, fun_prop]
protected lemma Continuous.dist [TopologicalSpace β] {f g : β → α} (hf : Continuous f)
(hg : Continuous g) : Continuous fun b => dist (f b) (g b) :=
continuous_dist.comp₂ hf... | Mathlib/Topology/MetricSpace/Pseudo/Constructions.lean | 199 | 209 |
/-
Copyright (c) 2018 Guy Leroy. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sangwoo Jo (aka Jason), Guy Leroy, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.GroupWithZero.Semiconj
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Data.Nat.GCD.Bas... | let ⟨m', n', h⟩ := exists_gcd_one H
⟨_, m', n', H, h⟩
| Mathlib/Data/Int/GCD.lean | 281 | 282 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | Mathlib/SetTheory/Cardinal/Basic.lean | 1,096 | 1,098 | |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis, Eric Wieser
-/
import Mathlib.LinearAlgebra.Multilinear.TensorProduct
import Mathlib.Tactic.AdaptationNote
import Mathlib.LinearAlgebra.Multilinear.Curry
/-!
# Tenso... | fun z f i r ↦ by simp [φ.map_update_smul, smul_smul, mul_comm]
theorem liftAux_tprod (φ : MultilinearMap R s E) (f : Π i, s i) : liftAux φ (tprod R f) = φ f := by
simp only [liftAux, liftAddHom, tprod_eq_tprodCoeff_one, tprodCoeff, AddCon.coe_mk']
-- The end of this proof was very different before https://gith... | Mathlib/LinearAlgebra/PiTensorProduct.lean | 402 | 407 |
/-
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.Ring.Semiconj
import Mathlib.Algebra.Ring.Units
import Mathlib.Algebra.Gro... | SemiconjBy.neg_left_iff
end
| Mathlib/Algebra/Ring/Commute.lean | 72 | 74 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Lattice
import Mathlib.Logic.Denumerable
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Hom.Basic
import Mathlib.Data.Set.Subsing... | /-- A value is accessible iff it isn't contained in any infinite decreasing sequence. -/
theorem acc_iff_no_decreasing_seq {x} :
Acc r x ↔ IsEmpty { f : ((· > ·) : ℕ → ℕ → Prop) ↪r r // x ∈ Set.range f } := by
constructor
· refine fun h => h.recOn fun x _ IH => ?_
| Mathlib/Order/OrderIsoNat.lean | 58 | 62 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
/-!
# Lie algeb... |
@[simp]
theorem toEnd_restrict_eq_toEnd (h := N.toEnd_comp_subtype_mem x) :
(toEnd R L M x).restrict h = toEnd R L N x := by
| Mathlib/Algebra/Lie/OfAssociative.lean | 332 | 335 |
/-
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, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.LinearAlgebra.Quotient.Basic
/-!
# Isomorphism theorems for modules.
* The Noether's first, second, a... | Note that in the following declaration the type of the domain is expressed using
``comap p.subtype p ⊓ comap p.subtype p'`
instead of
`comap p.subtype (p ⊓ p')`
| Mathlib/LinearAlgebra/Isomorphisms.lean | 67 | 70 |
/-
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... | QuotientAddGroup.equivIocMod hp a 0 = ⟨toIocMod hp a 0, toIocMod_mem_Ioc hp a _⟩ :=
rfl
end
| Mathlib/Algebra/Order/ToIntervalMod.lean | 708 | 710 |
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Gamma.Deriv
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
/-! # Convexity properties of the Gamma funct... | filter_upwards [eventually_ne_atTop 0] with n hn using
le_sub_iff_add_le'.mpr (ge_logGammaSeq hf_conv hf_feq hx hn)
theorem tendsto_logGammaSeq (hf_conv : ConvexOn ℝ (Ioi 0) f)
(hf_feq : ∀ {y : ℝ}, 0 < y → f (y + 1) = f y + log y) (hx : 0 < x) :
Tendsto (logGammaSeq x) atTop (𝓝 <| f x - f 1) := by
... | Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean | 236 | 249 |
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Topology.SeparatedMap
/-!
# Local homeomorphisms
This file defines local homeomorphisms.
## Main definit... | protected theorem continuous (hf : IsLocalHomeomorph f) : Continuous f :=
continuous_iff_continuousOn_univ.mpr hf.isLocalHomeomorphOn.continuousOn
| Mathlib/Topology/IsLocalHomeomorph.lean | 183 | 184 |
/-
Copyright (c) 2022 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.Algebra.Order.Ring.Nat
import Mathlib.Data.Set.Equitable
import Mathlib.Logic.Equiv.Fin.Basic
import Mathlib.Order.Partition.Fi... | ext p
simp only [mem_filter, and_congr_right_iff]
exact fun hp ↦ (hP.card_part_eq_average_iff hp).symm
| Mathlib/Order/Partition/Equipartition.lean | 74 | 77 |
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Data.Set.Prod
/-!
# N-ary images of sets
This file defines `Set.image2`, the binary image of sets.
This is mostly useful to define pointwise oper... | (image2 f s t).image g = image2 f' s (t.image g') :=
(image_image2_distrib h_distrib).trans <| by rw [image_id']
/-- Symmetric statement to `Set.image_image2_distrib_left`. -/
| Mathlib/Data/Set/NAry.lean | 238 | 241 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Sander Dahmen, Kim Morrison, Chris Hughes, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.LinearAlgebra.Basis.Prod
impo... | Module.rank R (M ⊗[S] M₁) = Module.rank R M * Module.rank S M₁ := by simp
| Mathlib/LinearAlgebra/Dimension/Constructions.lean | 376 | 377 |
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Gamma.Deriv
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
/-! # Convexity properties of the Gamma funct... | Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean | 445 | 449 | |
/-
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.Multilinear.Basic
import Mathlib.LinearAlgebra.PiTensorProduct
/-!
# Projective seminorm on the tensor of a finite family of normed s... | List.forall₂_map_left_iff, map_smul, lift.tprod, ContinuousMultilinearMap.coe_coe,
List.forall₂_same, Prod.forall]
intro a m _
rw [norm_smul, ← mul_assoc, mul_comm ‖f‖ _, mul_assoc]
exact mul_le_mul_of_nonneg_left (f.le_opNorm _) (norm_nonneg _)
end PiTensorProduct
| Mathlib/Analysis/NormedSpace/PiTensorProduct/ProjectiveSeminorm.lean | 134 | 153 |
/-
Copyright (c) 2020 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.NormedSpace.IndicatorFunction
import Mathlib.Data.Fintype.Order
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.Me... |
/-- See `eLpNorm_smul_measure_of_ne_zero'` for a version with scalar multiplication by `ℝ≥0`. -/
theorem eLpNorm_smul_measure_of_ne_zero {c : ℝ≥0∞} (hc : c ≠ 0) (f : α → ε) (p : ℝ≥0∞)
(μ : Measure α) : eLpNorm f p (c • μ) = c ^ (1 / p).toReal • eLpNorm f p μ := by
by_cases hp0 : p = 0
| Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 853 | 857 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Indicator
import Mathlib.Algebra.Module.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic
import... | that base point will cancel out later); a more typical use case for
`weightedVSub` would involve selecting a preferred base point with
`weightedVSub_eq_weightedVSubOfPoint_of_sum_eq_zero` and then
| Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 237 | 239 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Action.Faithful
import Mathlib.Algebra.Grou... | Mathlib/Algebra/Group/Submonoid/Operations.lean | 1,522 | 1,525 | |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.Finite.Basic
/-!
# Lagrange's four square theorem
The main result in this file is `sum_four_squares`,
a proof that every natural number is th... | Mathlib/NumberTheory/SumFourSquares.lean | 216 | 226 | |
/-
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.ENNReal.Action
import Mathlib.MeasureTheory.MeasurableSpace.Constructions
import Mathlib.MeasureTheory.OuterMeasure.Caratheodory
... | refine le_antisymm (ge_of_tendsto' this fun n => ?_) ?_
· exact le_trans (measure_mono <| iInter_subset t n) (hm' n).le
· refine iInf_le_of_le (⋂ n, t n) ?_
refine iInf_le_of_le (subset_iInter hsub) ?_
exact iInf_le _ (MeasurableSet.iInter hm)
| Mathlib/MeasureTheory/OuterMeasure/Induced.lean | 391 | 395 |
/-
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... | theorem pi_congr (h : s₁ = s₂) (h' : ∀ i ∈ s₁, t₁ i = t₂ i) : s₁.pi t₁ = s₂.pi t₂ :=
| Mathlib/Data/Set/Prod.lean | 647 | 647 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.Group.Pointwise.Set.Basic
import Mathlib.Algebra.G... | a * b = c * d → f a * f b = f c * f d where
mp hf := ⟨hf.mapsTo, fun _ ha _ hb _ hc _ hd ↦ hf.mul_eq_mul ha hb hc hd⟩
mpr hf := ⟨hf.1, by aesop (add simp card_eq_two)⟩
/-- Characterisation of `2`-Freiman homs. -/
@[to_additive "Characterisation of `2`-Freiman isomorphisms."]
lemma isMulFreimanIso_two :
I... | Mathlib/Combinatorics/Additive/FreimanHom.lean | 151 | 159 |
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Algebra.Order.GroupWithZero.Canonical
import Mathlib.Algebra.Order.Nonneg.Basic
import Mathlib.Algebra.Order.Nonneg.Lattice
import Mathlib.Algebra.... | Mathlib/Algebra/Order/Nonneg/Ring.lean | 371 | 373 | |
/-
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... | rw [← rpow_natCast]
simp only [Nat.cast_ofNat]
| Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 451 | 452 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive
impo... | Mathlib/Data/Option/Basic.lean | 466 | 469 | |
/-
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
/-!
# Typeclasses for measurability of operations
In this file we define classes `MeasurableMul` etc and prove d... | Mathlib/MeasureTheory/Group/Arithmetic.lean | 969 | 971 | |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Algebra.Hom
import Mathlib.RingTheory.Congruence.Basic
import Mathlib.RingTheory.Ideal.Quotient.Defs
import Mathlib.RingTheory.Ideal.Span
/-!
# Qu... | simp only [Algebra.smul_def, Rel.mul_right h]
| Mathlib/Algebra/RingQuot.lean | 73 | 74 |
/-
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.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
/-!
# Rotations by oriented angles.
This... | zero. -/
@[simp]
theorem inner_rotation_pi_div_two_right_smul (x : V) (r : ℝ) :
⟪r • x, o.rotation (π / 2 : ℝ) x⟫ = 0 := by
| Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 403 | 406 |
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