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) 2022 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.ImproperIntegrals
import Mathlib.MeasureTheory.Integral.ExpDecay
/-!
# The Gamma function
This file defines the `Γ` functio... | rw [GammaAux]
dsimp only
| Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean | 362 | 363 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Algebra.Ring.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.Order.Cir... | theorem toIcoDiv_neg' (a b : α) : toIcoDiv hp (-a) b = -(toIocDiv hp a (-b) + 1) := by
simpa only [neg_neg] using toIcoDiv_neg hp (-a) (-b)
| Mathlib/Algebra/Order/ToIntervalMod.lean | 329 | 330 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Yury Kudryashov
-/
import Mathlib.Data.Finset.Fin
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Order.Interval.Set.Fin
/-!
# Finite intervals in `Fin n`
This fi... |
@[simp]
| Mathlib/Order/Interval/Finset/Fin.lean | 152 | 153 |
/-
Copyright (c) 2022 Rémy Degenne, Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.Egorov
import Mathlib.MeasureTheory.Function.LpSpace.Complete
/-!
# Convergence in measure
We define con... | · simp only [hδi, imp_true_iff, le_top, exists_const]
lift δ to ℝ≥0 using hδi
rw [gt_iff_lt, ENNReal.coe_pos, ← NNReal.coe_pos] at hδ
obtain ⟨t, _, ht, hunif⟩ := tendstoUniformlyOn_of_ae_tendsto' hf hg hfg hδ
rw [ENNReal.ofReal_coe_nnreal] at ht
rw [Metric.tendstoUniformlyOn_iff] at hunif
obtain ⟨N, hN⟩ :... | Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean | 128 | 138 |
/-
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, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ord... | Mathlib/Algebra/Order/Field/Basic.lean | 852 | 856 | |
/-
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.Order.Filter.SmallSets
import Mathlib.Topology.ContinuousOn
/-!
### Locally finite families of sets
We say that a family of sets in a topological s... | Mathlib/Topology/LocallyFinite.lean | 225 | 229 | |
/-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Filtered.Basic
import Mathlib.CategoryTheory.Limits.HasLimits
import Mathlib.CategoryTheory.Limits.Types.Yoneda
/-!
# Filtered categories... | ∃ (X : C), Nonempty (limit (F ⋙ coyoneda.obj (op X))) := by
rw [IsCofiltered.iff_cone_nonempty.{v}]
refine ⟨fun h J _ _ F => ?_, fun h J _ _ F => ?_⟩
· obtain ⟨c⟩ := h F
exact ⟨c.pt, ⟨(limitCompCoyonedaIsoCone F c.pt).inv c.π⟩⟩
· obtain ⟨pt, ⟨π⟩⟩ := h F
exact ⟨⟨pt, (limitCompCoyonedaIsoCone F pt).... | Mathlib/CategoryTheory/Limits/Filtered.lean | 52 | 60 |
/-
Copyright (c) 2024 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul Lezeau, Calle Sönne
-/
import Mathlib.CategoryTheory.Functor.Category
import Mathlib.CategoryTheory.CommSq
/-!
# HomLift
Given a functor `p : 𝒳 ⥤ 𝒮`, this file provides API for... |
@[simp]
lemma eqToHom_comp_lift_iff {R S : 𝒮} {a b b' : 𝒳} (f : R ⟶ S) (φ : a ⟶ b) (h : b = b') :
p.IsHomLift f (φ ≫ eqToHom h) ↔ p.IsHomLift f φ where
mp hφ' := by subst h; simpa using hφ'
| Mathlib/CategoryTheory/FiberedCategory/HomLift.lean | 188 | 192 |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Group.Nat.Even
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Data.Nat.Cast.Commute
import Mathlib.Data.Set.Operations
import Mathlib.Logic.Fu... |
lemma ne_of_odd_add (h : Odd (m + n)) : m ≠ n := by rintro rfl; simp [← not_even_iff_odd] at h
| Mathlib/Algebra/Ring/Parity.lean | 271 | 272 |
/-
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... | Mathlib/Order/Interval/Set/Basic.lean | 1,063 | 1,063 | |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.Algebra.Homology.Linear
import Mathlib.CategoryTheory.MorphismProperty.IsInvertedBy
import Mathlib.CategoryTheory.... | rw [← (quotient V c).map_comp, ← (quotient V c).map_id]
exact eq_of_homotopy _ _ f.homotopyHomInvId
| Mathlib/Algebra/Homology/HomotopyCategory.lean | 138 | 139 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.LeftHomology
import Mathlib.CategoryTheory.Limits.Opposites
/-!
# Right Homology of short complexes
In this file, we define the ... | @[simps]
noncomputable def ofEpiOfIsIsoOfMono' (φ : S₁ ⟶ S₂) (h : RightHomologyData S₂)
[Epi φ.τ₁] [IsIso φ.τ₂] [Mono φ.τ₃] :
RightHomologyMapData φ (RightHomologyData.ofEpiOfIsIsoOfMono' φ h) h where
| Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean | 1,155 | 1,158 |
/-
Copyright (c) 2022 Kyle Miller, Adam Topaz, Rémi Bottinelli, Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Adam Topaz, Rémi Bottinelli, Junyan Xu
-/
import Mathlib.Topology.Category.TopCat.Limits.Konig
/-!
# Cofiltered systems
This file de... | ⟨_, i, hi, hf.coe_toFinset⟩
refine ⟨i, f, fun k g =>
(directedOn_range.mp <| F.ranges_directed j).is_bot_of_is_min ⟨⟨i, f⟩, rfl⟩ ?_ _ ⟨⟨k, g⟩, rfl⟩⟩
rintro _ ⟨⟨k', g'⟩, rfl⟩ hl
refine (eq_of_le_of_not_lt hl ?_).ge
have := hmin _ ⟨k', g', (m.finite_toSet.subset <| hm.substr hl).coe_toFinset⟩
rwa [Finse... | Mathlib/CategoryTheory/CofilteredSystem.lean | 238 | 256 |
/-
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.Filter
import Mathlib.Analysis.BoxIntegral.Partition.Measure
import Mathlib.Analysis.Oscillation
import Mathlib.Data.B... |
- if `l.bRiemann`, then `r` is a constant;
- if `ε > 0`, then for any tagged partition `π` of `I` subordinate to `r` (more precisely,
satisfying the predicate `l.mem_base_set I c r`), the integral sum of `f` over `π` differs from
the integral of `f` over `I` by at most `ε`.
| Mathlib/Analysis/BoxIntegral/Basic.lean | 376 | 381 |
/-
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 | 2,567 | 2,569 | |
/-
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... | exact IsWellFounded.wf
refine Subrelation.wf (fun h => ?_) <| InvImage.wf (mapRange e fun _ ↦ rfl) this
have := IsStrictOrder.swap (@WellOrderingRel ι)
obtain ⟨i, he, hl⟩ := lex_lt_of_lt_of_preorder (Function.swap WellOrderingRel) h
exact ⟨i, fun j hj ↦ Quot.sound (he j hj), hl⟩⟩
instance DFins... | Mathlib/Data/DFinsupp/WellFounded.lean | 216 | 235 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Degree.Domain
import Mathlib.Algebra.Polynomial.Degree.Support
import Mathlib.Algebra.Poly... | theorem derivative_X_sq : derivative (X ^ 2 : R[X]) = C 2 * X := by
rw [derivative_X_pow, Nat.cast_two, pow_one]
| Mathlib/Algebra/Polynomial/Derivative.lean | 109 | 110 |
/-
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... | rw [sub_eq_add_neg, ← neg_smul, toIcoDiv_add_zsmul', sub_neg_eq_add]
@[simp]
theorem toIocDiv_sub_zsmul (a b : α) (m : ℤ) : toIocDiv hp a (b - m • p) = toIocDiv hp a b - m := by
rw [sub_eq_add_neg, ← neg_smul, toIocDiv_add_zsmul, sub_eq_add_neg]
| Mathlib/Algebra/Order/ToIntervalMod.lean | 239 | 243 |
/-
Copyright (c) 2024 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Category.ModuleCat.ChangeOfRings
import Mathlib.Algebra.Category.Ring.Basic
import Mathlib.RingTheory.Kaehler.Basic
/-!
# The differentials of a morphis... | { toFun := fun x ↦ _root_.KaehlerDifferential.map A A' B B' x
map_add' := by simp
map_smul' := by simp }
@[simp]
lemma map_d (b : B) : map fac (d b) = d (g' b) := by
algebraize [f.hom, f'.hom, g.hom, g'.hom, f'.hom.comp g.hom]
have := IsScalarTower.of_algebraMap_eq' (congrArg Hom.hom fac)
exact _root_.... | Mathlib/Algebra/Category/ModuleCat/Differentials/Basic.lean | 140 | 149 |
/-
Copyright (c) 2021 Apurva Nakade. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Apurva Nakade
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.GroupTheory.MonoidLocalization.Away
impo... | (m * 2 ^ n) * powHalf (n + k) = m * powHalf k := by
rw [mul_assoc]
| Mathlib/SetTheory/Surreal/Dyadic.lean | 182 | 183 |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
/-!
# Neighborhoods and continuity relative to a subset
This file develops API on the relative versions
* `nhdsWithin` ... |
/- `continuousWithinAt_insert` gives the same equivalence but at a point `y` possibly different
from `x`. As this requires the space to be T1, and this property is not available in this file,
| Mathlib/Topology/ContinuousOn.lean | 757 | 759 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Group.Multiset.Basic
/-!
# Bind operation for multisets
This file defines a few basic operations on `Multiset`, notably the mona... | Mathlib/Data/Multiset/Bind.lean | 364 | 365 | |
/-
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.Basic
/-!
# Compact subsets of products as limits in `Profinite`
This file exhibits a compact subset `C` of a product... | theorem surjective_π_app :
Function.Surjective (π_app C J) := by
intro x
obtain ⟨y, hy⟩ := x.prop
exact ⟨⟨y, hy.1⟩, Subtype.ext hy.2⟩
| Mathlib/Topology/Category/Profinite/Product.lean | 58 | 62 |
/-
Copyright (c) 2021 David Wärn,. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Kim Morrison
-/
import Mathlib.Combinatorics.Quiver.Prefunctor
import Mathlib.Logic.Lemmas
import Batteries.Data.List.Basic
/-!
# Paths in quivers
Given a quiver `V`, we def... | | _, nil, nil, _ => rfl
| _, nil, @cons _ _ _ c _ p f, h => by cases h
| _, @cons _ _ _ c _ p f, nil, h => by cases h
| _, @cons _ _ _ c _ p f, @cons _ _ _ t _ C D, h => by
simp only [toList, List.cons.injEq] at h
obtain ⟨rfl, hAC⟩ := h
simp [toList_injective _ _ hAC, eq_iff_true_of_subsingleton]
| Mathlib/Combinatorics/Quiver/Path.lean | 184 | 191 |
/-
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.Algebra.Group.Subgroup.Lattice
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra... |
protected theorem ne_bot_iff (p : Submodule R M) : p ≠ ⊥ ↔ ∃ x ∈ p, x ≠ (0 : M) := by
simp only [ne_eq, p.eq_bot_iff, not_forall, exists_prop]
| Mathlib/Algebra/Module/Submodule/Lattice.lean | 91 | 94 |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 1,840 | 1,844 | |
/-
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.Geometry.Euclidean.Projection
import Mathlib.Geometry.Euclidean.Sphere.Basic
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tact... | Mathlib/Geometry/Euclidean/Circumcenter.lean | 833 | 839 | |
/-
Copyright (c) 2023 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Heather Macbeth
-/
import Mathlib.MeasureTheory.Constructions.Pi
/-!
# Marginals of multivariate functions
In this file, we define a convenient way to compute int... | (hi : i ∉ s) :
∫⋯∫⁻_insert i s, f ∂μ = ∫⋯∫⁻_s, (fun x ↦ ∫⁻ xᵢ, f (Function.update x i xᵢ) ∂μ i) ∂μ := by
rw [Finset.insert_eq, Finset.union_comm,
lmarginal_union (s := s) μ f hf (Finset.disjoint_singleton_right.mpr hi), lmarginal_singleton]
| Mathlib/MeasureTheory/Integral/Marginal.lean | 179 | 182 |
/-
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.Tactic.NormNum.Basic
import Mathlib.Data.Nat.Prime.Basic
/-!
# `norm_num` extensions on natural numbers
This... | | n, _, k, ⟨rfl⟩, h1, h2 => by
refine ⟨(Nat.prime_def_minFac.mp ?_).2⟩
rw [Nat.prime_def_le_sqrt]
refine ⟨h1.one_lt, fun m hm hmn h2mn ↦ ?_⟩
exact lt_irrefl m <| calc
m ≤ sqrt n := hmn
_ < k := sqrt_lt.mpr (ble_eq_false.mp h2)
_ ≤ n.minFac := h1.2.2
_ ≤ m := Nat... | Mathlib/Tactic/NormNum/Prime.lean | 109 | 119 |
/-
Copyright (c) 2024 Judith Ludwig, Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Judith Ludwig, Christian Merten
-/
import Mathlib.RingTheory.AdicCompletion.Basic
import Mathlib.RingTheory.AdicCompletion.Algebra
import Mathlib.Algebra.DirectSum.Bas... |
/-- Equality of maps out of an adic completion can be checked on Cauchy sequences. -/
theorem map_ext {N} {f g : AdicCompletion I M → N}
(h : ∀ (a : AdicCauchySequence I M),
f (AdicCompletion.mk I M a) = g (AdicCompletion.mk I M a)) :
f = g := by
| Mathlib/RingTheory/AdicCompletion/Functoriality.lean | 141 | 146 |
/-
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... | noncomputable def toPGameEmbedding : Ordinal.{u} ↪o PGame.{u} where
toFun := Ordinal.toPGame
| Mathlib/SetTheory/Game/Ordinal.lean | 149 | 150 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.FieldTheory.Finiteness
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
/-!
# Finite-dimensional subsp... | classical
have hn : Nonempty ι := by simp [← Fintype.card_pos_iff, hc]
obtain ⟨i₁⟩ := hn
rw [affineIndependent_iff_linearIndependent_vsub _ _ i₁,
linearIndependent_iff_card_eq_finrank_span, eq_comm,
vectorSpan_range_eq_span_range_vsub_right_ne k p i₁, Set.finrank]
rw [← Finset.card_univ] at hc
rw [F... | Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | 180 | 191 |
/-
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.Ordinal.Family
import Mathlib.Tactic.Abel
/-!
# Natural operations on ordinals
The goal of this file is to define n... |
theorem nmul_eq_mul (a b) : a ⨳ b = toOrdinal (toNatOrdinal a * toNatOrdinal b) :=
rfl
theorem nmul_nadd_one : ∀ a b, a ⨳ (b ♯ 1) = a ⨳ b ♯ a :=
@mul_add_one NatOrdinal _ _ _
| Mathlib/SetTheory/Ordinal/NaturalOps.lean | 708 | 713 |
/-
Copyright (c) 2024 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
/-!
# Angle between complex numbers
This file rel... | rw [← div_le_iff₀' <| by positivity, div_eq_inv_mul, inv_div]; exact mul_angle_le_norm_sub hx hy
end Complex
| Mathlib/Analysis/Complex/Angle.lean | 118 | 120 |
/-
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... |
theorem prime_X_sub_C (r : R) : Prime (X - C r) :=
⟨X_sub_C_ne_zero r, not_isUnit_X_sub_C r, fun _ _ => by
simp_rw [dvd_iff_isRoot, IsRoot.def, eval_mul, mul_eq_zero]
exact id⟩
| Mathlib/Algebra/Polynomial/RingDivision.lean | 190 | 195 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Data.FunLike.Equiv
import Mathlib.Data.Quot
import Mathlib.Data.Subtype
import Mathlib.Logic.U... |
theorem symm_symm_apply (f : α ≃ β) (b : α) : f.symm.symm b = f b := rfl
| Mathlib/Logic/Equiv/Defs.lean | 262 | 263 |
/-
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... | refine ⟨fun h ↦ ?_, image_le_image_of_le f⟩
rw [← preimage_image_eq f U, ← preimage_image_eq f U']
apply preimage_le_preimage_of_le f h
| Mathlib/AlgebraicGeometry/OpenImmersion.lean | 123 | 126 |
/-
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... | rfl
@[to_additive]
| Mathlib/Algebra/BigOperators/Fin.lean | 129 | 131 |
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.Order.Monoid.Unbundled.OrderDual
import Mathlib.Algebra.BigOperators.Group.List.Basic
/-!... | @[to_additive sum_lt_sum]
lemma prod_lt_prod' [Preorder M] [MulLeftStrictMono M]
[MulLeftMono M] [MulRightStrictMono M]
[MulRightMono M] {l : List ι} (f g : ι → M)
(h₁ : ∀ i ∈ l, f i ≤ g i) (h₂ : ∃ i ∈ l, f i < g i) : (l.map f).prod < (l.map g).prod := by
| Mathlib/Algebra/Order/BigOperators/Group/List.lean | 64 | 68 |
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Subalgebra.Prod
import Mathlib.Algebra.Algebra.Subalgebra.Tower
import Mathlib.LinearAlgebra.Basis.Basic
import Mathlib.LinearAlgebra.Prod
/-!
# ... | Mathlib/RingTheory/Adjoin/Basic.lean | 532 | 536 | |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.Module.End
import Mathlib.Algebra.Ring.Prod
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.GroupAc... | @[simp]
lemma castHom_comp {m d : ℕ} (hm : n ∣ m) (hd : m ∣ d) :
| Mathlib/Data/ZMod/Basic.lean | 1,101 | 1,102 |
/-
Copyright (c) 2024 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Combinatorics.Additive.FreimanHom
/-!
# Corners
This file defines corners, namely triples of the form `(x, y), (x, y + d), (x + d, y)`, and the
property ... | @[simp] lemma not_isCorner_empty : ¬ IsCorner ∅ x₁ y₁ x₂ y₂ := by simp [isCorner_iff]
| Mathlib/Combinatorics/Additive/Corner/Defs.lean | 62 | 63 |
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
/-!
# Thickenings in pseudo-metric spaces
## Main definitions
* `Metric.thickening δ s`, the open thickening by ra... | Mathlib/Topology/MetricSpace/Thickening.lean | 706 | 709 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Sébastien Gouëzel, Patrick Massot
-/
import Mathlib.Topology.UniformSpace.Cauchy
import Mathlib.Topology.UniformSpace.Separation
import Mathlib.Topology.DenseEmbedding
... | dsimp only [UniformContinuous, Tendsto]
simp only [← hg.comap_uniformity, ← map_le_iff_le_comap, Filter.map_map, Function.comp_def]
protected theorem IsUniformInducing.isUniformInducing_comp_iff {f : α → β} {g : β → γ}
(hg : IsUniformInducing g) : IsUniformInducing (g ∘ f) ↔ IsUniformInducing f := by
| Mathlib/Topology/UniformSpace/UniformEmbedding.lean | 93 | 97 |
/-
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.Basic
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Eval.Algebra
impo... | simpa [descPochhammer_map, Polynomial.map_mul, Polynomial.map_add, map_X,
| Mathlib/RingTheory/Polynomial/Pochhammer.lean | 295 | 295 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kim Morrison
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.InjSurj
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Tactic.FastInstance
impo... | Mathlib/Data/Finsupp/Defs.lean | 696 | 698 | |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Covering.Bes... | (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) :
(μ.withDensity (fun x ↦ ENNReal.ofReal (g x))).map f.symm s
= ENNReal.ofReal (∫ x in s, |(f' x).det| * g (f x) ∂μ) := by
rw [MeasurableEquiv.map_symm,
MeasurableEmbedding.withDensity_ofReal_comap_apply_eq_integral_abs_det_fderiv_mul μ hs
f.meas... | Mathlib/MeasureTheory/Function/Jacobian.lean | 1,233 | 1,238 |
/-
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.Algebra.Polynomial.Degree.CardPowDegree
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.NumberTheory.ClassNumber.AdmissibleAbsoluteValue
imp... | induction n with | zero => refine ⟨finZeroElim, finZeroElim⟩ | succ n ih =>
-- Show `anti_archimedean` also holds for real distances.
have anti_archim' : ∀ {i j k} {ε : ℝ},
(cardPowDegree (A i % b - A j % b) : ℝ) < ε →
(cardPowDegree (A j % b - A k % b) : ℝ) < ε →
(cardPowDegree (A i % b - A k %... | Mathlib/NumberTheory/ClassNumber/AdmissibleCardPowDegree.lean | 178 | 243 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
/-!
# Intervals as finsets
This file provides basic results about all the `Finset.Ixx... |
theorem Ico_filter_le_of_right_le {a b : α} [DecidablePred (b ≤ ·)] :
{x ∈ Ico a b | b ≤ x} = ∅ :=
filter_false_of_mem fun _ hx => (mem_Ico.1 hx).2.not_le
| Mathlib/Order/Interval/Finset/Basic.lean | 282 | 285 |
/-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Computation.Approximations
import Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating
import Mathlib.D... | theorem terminates_iff_rat (v : K) : (of v).Terminates ↔ ∃ q : ℚ, v = (q : K) :=
Iff.intro
(fun terminates_v : (of v).Terminates =>
show ∃ q : ℚ, v = (q : K) from exists_rat_eq_of_terminates terminates_v)
fun exists_q_eq_v : ∃ q : ℚ, v = (↑q : K) =>
Exists.elim exists_q_eq_v fun q => fun v_eq_q : v ... | Mathlib/Algebra/ContinuedFractions/Computation/TerminatesIffRat.lean | 315 | 331 |
/-
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.Data.ENNReal.Real
import Mathlib.Tactic.Bound.Attribute
import Mathlib.Topology... | Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | 1,499 | 1,502 | |
/-
Copyright (c) 2017 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Data.PFunctor.Univariate.Basic
/-!
# M-types
M types are potentially infinite tree-like structures. They are defined
as the greatest fixpoint of a polynomi... | congr
theorem ext_aux [Inhabited (M F)] [DecidableEq F.A] {n : ℕ} (x y z : M F) (hx : Agree' n z x)
(hy : Agree' n z y) (hrec : ∀ ps : Path F, n = ps.length → iselect ps x = iselect ps y) :
x.approx (n + 1) = y.approx (n + 1) := by
induction' n with n n_ih generalizing x y z
| Mathlib/Data/PFunctor/Univariate/M.lean | 469 | 474 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johan Commelin
-/
import Mathlib.Algebra.Group.Equiv.Defs
import Mathlib.Algebra.Group.WithOne.Defs
import Mathlib.Algebra.GroupWithZero.Equiv
import Mathlib.Algebra.Gr... | simp only [← coe_inj, coe_mul, coe_unzero]
instance instNoZeroDivisors : NoZeroDivisors (WithZero α) := ⟨Option.map₂_eq_none_iff.1⟩
| Mathlib/Algebra/GroupWithZero/WithZero.lean | 52 | 54 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner.Basic
import Mathlib.MeasureTheory.Integral.Bochner.L1
import Mathlib.Me... | Mathlib/MeasureTheory/Integral/Bochner.lean | 974 | 978 | |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathli... | theorem dist_conj_comm (z w : ℂ) : dist (conj z) w = dist z (conj w) := by
rw [← dist_conj_conj, conj_conj]
| Mathlib/Analysis/Complex/Basic.lean | 220 | 221 |
/-
Copyright (c) 2020 Ashvni Narayanan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ashvni Narayanan
-/
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Ring.Subring.Defs
import Mathlib.Algebra.Ring.Subsemiring.Bas... | Mathlib/Algebra/Ring/Subring/Basic.lean | 1,120 | 1,124 | |
/-
Copyright (c) 2021 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.GroupWithZero.Action.Defs
import Mathlib.Algebra.Order.Interval.Finset.Basic
import Mathlib.Combinatorics.Additive.Frei... | Mathlib/Combinatorics/Additive/AP/Three/Defs.lean | 506 | 512 | |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll, Kalle Kytölä
-/
import Mathlib.Analysis.Normed.Module.Basic
import Mathlib.LinearAlgebra.SesquilinearForm
import Mathlib.Topology.Algebra.Module.WeakBilin
/-!
# Polar set
I... |
theorem polar_nonempty (s : Set E) : Set.Nonempty (B.polar s) := by
use 0
| Mathlib/Analysis/LocallyConvex/Polar.lean | 73 | 75 |
/-
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.GCDMonoid.Finset
import Mathlib.Algebra.Polynomial.CancelLeads
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Fi... | Mathlib/RingTheory/Polynomial/Content.lean | 471 | 478 | |
/-
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.Order.Filter.SmallSets
import Mathlib.Topology.UniformSpace.Defs
import Mathlib.Topology.ContinuousOn
/-!
# Basic resu... | comp := le_trans
(by
rw [comap_lift'_eq, comap_lift'_eq2]
· exact lift'_mono' fun s _ ⟨a₁, a₂⟩ ⟨x, h₁, h₂⟩ => ⟨f x, h₁, h₂⟩
· exact monotone_id.compRel monotone_id)
| Mathlib/Topology/UniformSpace/Basic.lean | 381 | 385 |
/-
Copyright (c) 2023 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.Exact
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Ideal.Quotient.Defs
import Mathlib.RingTheory.TensorProduct... | lift.tmul, flip_apply, coe_mk, AddHom.coe_mk, mk_apply, Submodule.mkQ_apply]
rw [Submodule.Quotient.eq, ← TensorProduct.tmul_sub]
apply le_comap_range_lTensor f n
rw [← hfg, mem_ker, map_sub, sub_eq_zero, hgh]
lemma lTensor.inverse_of_rightInverse_comp_lTensor
{h : P → N} (hgh : Function.RightInverse h... | Mathlib/LinearAlgebra/TensorProduct/RightExactness.lean | 226 | 233 |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, David Loeffler, Heather Macbeth, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.Analysis.Calculus.ContDiff.CPolynomial
import... | ContinuousLinearMap.mul_apply', one_mul, ContinuousLinearMap.map_smul]
exact congr_arg (fun x ↦ v • x) (one_smul ℝ (f v)).symm
rw [← VectorFourier.fourierIntegral_convergent_iff continuous_fourierChar L.continuous₂ w]
at h_int
convert (VectorFourier.hasFDerivAt_fourierIntegral L hf hf'' w).hasDerivAt ... | Mathlib/Analysis/Fourier/FourierTransformDeriv.lean | 769 | 779 |
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Floris van Doorn
-/
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Analysis.Normed.Module.Convex
import Mathlib.Analysis.NormedSpace.Connected
import Mathli... | /-- The union of two ample sets is ample. -/
theorem union {s t : Set F} (hs : AmpleSet s) (ht : AmpleSet t) : AmpleSet (s ∪ t) := by
intro x hx
rcases hx with (h | h) <;>
-- The connected component of `x ∈ s` in `s ∪ t` contains the connected component of `x` in `s`,
-- hence is also full; similarly for `t`.
... | Mathlib/Analysis/Convex/AmpleSet.lean | 65 | 74 |
/-
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... | obtain ⟨⟨_, bs⟩⟩ := Module.Free.exists_basis (R := R) (M := M)
| Mathlib/LinearAlgebra/Dimension/Constructions.lean | 183 | 183 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yury Kudryashov
-/
import Mathlib.Algebra.Algebra.Hom
import Mathlib.Algebra.Ring.Action.Group
/-!
# Isomorphisms of `R`-algebras
This file defines bundled isomorphisms of `R`-... | @[simp]
theorem coe_ofBijective {f : A₁ →ₐ[R] A₂} {hf : Function.Bijective f} :
(AlgEquiv.ofBijective f hf : A₁ → A₂) = f :=
| Mathlib/Algebra/Algebra/Equiv.lean | 476 | 478 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Yaël Dillies
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
/-!
# Integral average of a function
In this file we define `MeasureTheory.average... |
theorem ofReal_setAverage {f : α → ℝ} (hf : IntegrableOn f s μ) (hf₀ : 0 ≤ᵐ[μ.restrict s] f) :
ENNReal.ofReal (⨍ x in s, f x ∂μ) = (∫⁻ x in s, ENNReal.ofReal (f x) ∂μ) / μ s := by
simpa using ofReal_average hf hf₀
theorem toReal_laverage {f : α → ℝ≥0∞} (hf : AEMeasurable f μ) (hf' : ∀ᵐ x ∂μ, f x ≠ ∞) :
(⨍⁻ ... | Mathlib/MeasureTheory/Integral/Average.lean | 459 | 465 |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.MeasureTheory.Integral.Pi
impor... |
theorem _root_.fourierIntegral_gaussian_pi' (hb : 0 < b.re) (c : ℂ) :
(𝓕 fun x : ℝ => cexp (-π * b * x ^ 2 + 2 * π * c * x)) = fun t : ℝ =>
1 / b ^ (1 / 2 : ℂ) * cexp (-π / b * (t + I * c) ^ 2) := by
| Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean | 212 | 215 |
/-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois.Basic
/-!
# Cyclotomic extens... |
/-- Any two `n`-th cyclotomic extensions are isomorphic. -/
def algEquiv (L' : Type*) [Field L'] [Algebra K L'] [IsCyclotomicExtension {n} K L'] :
L ≃ₐ[K] L' :=
let h₁ := isSplittingField_X_pow_sub_one n K L
let h₂ := isSplittingField_X_pow_sub_one n K L'
| Mathlib/NumberTheory/Cyclotomic/Basic.lean | 440 | 445 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Logic.Equiv.PartialEquiv
import Mathlib.Topology.Homeomorph.Lemmas
import Mathlib.Topology.Sets.Opens
/-!
# Partial homeomorphisms
This file de... | /-- An open embedding of `X` into `Y`, with `X` nonempty, defines a partial homeomorphism
whose source is all of `X`. The converse is also true; see
`PartialHomeomorph.to_isOpenEmbedding`. -/
@[simps! (config := mfld_cfg) apply source target]
noncomputable def toPartialHomeomorph [Nonempty X] : PartialHomeomorph X Y :=... | Mathlib/Topology/PartialHomeomorph.lean | 1,202 | 1,211 |
/-
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.Function.Defs
import Mathlib.Logic.Function.Basic
/-!
# Sigma types
This file proves basic results about sigma types.
A sigma type is a depend... | simpa [funext_iff] using h
| Mathlib/Data/Sigma/Basic.lean | 70 | 71 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathli... | Mathlib/Data/Finset/Image.lean | 830 | 835 | |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Batteries.Data.Rat.Lemmas
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Rat.Init
import Mathlib.Order.Basic
import Mathlib.Tactic.Commo... | Mathlib/Data/Rat/Defs.lean | 530 | 534 | |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Data.Complex.Norm
/-!
# The partial order on the complex numbers
This order is defined by `z ≤ w ↔ z.re ≤ w.re ∧ z.im = w.im`.
This is a natural order o... | theorem real_lt_real {x y : ℝ} : (x : ℂ) < (y : ℂ) ↔ x < y := by simp [lt_def, ofReal]
| Mathlib/Data/Complex/Order.lean | 74 | 74 |
/-
Copyright (c) 2020 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Grading
import Mathlib.Algebra.Module.Opposite
/-!
# Conjugations
This file defines the grade reversal and grade involution f... | section Submodule
variable (Q)
| Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean | 186 | 189 |
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConvergenceInMeasure
import Mathlib.MeasureTheory.Function.L1Space.Integrable
/-!
# Uniform integrability
This file contains the def... | protected theorem ae_eq (hf : UnifIntegrable f p μ) (hfg : ∀ n, f n =ᵐ[μ] g n) :
UnifIntegrable g p μ := by
classical
intro ε hε
obtain ⟨δ, hδ_pos, hfδ⟩ := hf hε
| Mathlib/MeasureTheory/Function/UniformIntegrable.lean | 130 | 134 |
/-
Copyright (c) 2024 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.Cardinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Principal
/-!
# Ordinal arithmetic with cardinals
This file co... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 464 | 472 | |
/-
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... | theorem primeFactorsList_two : primeFactorsList 2 = [2] := by simp [primeFactorsList]
| Mathlib/Data/Nat/Factors.lean | 49 | 49 |
/-
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.Set.Finite.Lattice
import Mathlib.Order.Interval.Finset.Defs
import Mathlib.Order.Atoms
/-!
# Atoms, Coatoms, Simple Lattices, and Finiteness
Th... | variable [LE α] [BoundedOrder α] [IsSimpleOrder α] [DecidableEq α]
theorem univ : (Finset.univ : Finset α) = {⊤, ⊥} := by
change Finset.map _ (Finset.univ : Finset Bool) = _
rw [Fintype.univ_bool]
| Mathlib/Order/Atoms/Finite.lean | 45 | 49 |
/-
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... | fromBlocks A B C D =
fromBlocks 1 0 (C * ⅟ A) 1 * fromBlocks A 0 0 (D - C * ⅟ A * B) *
fromBlocks 1 (⅟ A * B) 0 1 := by
simp only [fromBlocks_multiply, Matrix.mul_zero, Matrix.zero_mul, add_zero, zero_add,
Matrix.one_mul, Matrix.mul_one, invOf_mul_self, Matrix.mul_invOf_cancel_left,
Matrix.i... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 52 | 59 |
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Constructions.Polish.Basic
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.Probability.Martingale.Upcrossing
/-!
# Mar... | have hlimint : Integrable (ℱ.limitProcess (fun n => μ[g|ℱ n]) μ) μ :=
(memLp_limitProcess_of_eLpNorm_bdd hunif.1 hR).integrable le_rfl
suffices g =ᵐ[μ] ℱ.limitProcess (fun n x => (μ[g|ℱ n]) x) μ by
filter_upwards [this, (martingale_condExp g ℱ μ).submartingale.ae_tendsto_limitProcess hR] with
x heq ht... | Mathlib/Probability/Martingale/Convergence.lean | 371 | 418 |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, María Inés de Frutos-Fernández, Filippo A. E. Nuccio
-/
import Mathlib.Data.Int.Interval
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.Binom... |
theorem coe_def : (f : F⸨X⸩) = coeAlgHom F f :=
rfl
attribute [-instance] RatFunc.instCoePolynomial in
| Mathlib/RingTheory/LaurentSeries.lean | 384 | 388 |
/-
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 | 1,783 | 1,784 | |
/-
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... | /-- An auxiliary lemma for manipulating sequence indices. -/
theorem lift_index_left {f : PadicSeq p} (hf : ¬f ≈ 0) (v1 v3 : ℕ) :
| Mathlib/NumberTheory/Padics/PadicNumbers.lean | 166 | 167 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Manuel Candales
-/
import Mathlib.Geometry.Euclidean.PerpBisector
import Mathlib.Algebra.QuadraticDiscriminant
/-!
# Euclidean spaces
This file makes some definitions and... | Mathlib/Geometry/Euclidean/Basic.lean | 322 | 326 | |
/-
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, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Data.Set.BooleanAlgebra
/-!
# The set lattice
This file is a collectio... | simp [Set.ext_iff]
| Mathlib/Data/Set/Lattice.lean | 963 | 964 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Dia... | theorem volume_real_Icc {a b : ℝ} : volume.real (Icc a b) = max (b - a) 0 := by
| Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 88 | 88 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Analysis.InnerProductSpace.Continuous
import Mathlib.Analysis.Normed.Module.Dual
import Mathlib.MeasureTheory.Function.AEEqOfLIntegral
import Mathlib.Measu... | (hf_int_finite s hs hμs).sub (hg_int_finite s hs hμs)
exact (hf.sub hg).ae_eq_zero_of_forall_setIntegral_eq_zero hfg_int hfg
theorem Lp.ae_eq_zero_of_forall_setIntegral_eq_zero (f : Lp E p μ) (hp_ne_zero : p ≠ 0)
(hp_ne_top : p ≠ ∞) (hf_int_finite : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn f s μ)
(h... | Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean | 308 | 319 |
/-
Copyright (c) 2019 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.Data.EReal.Basic
deprecated_module (since := "2025-04-13")
| Mathlib/Data/Real/EReal.lean | 1,389 | 1,389 | |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib... | instance [NoZeroDivisors R] : NoZeroDivisors (Ideal R) where
eq_zero_or_eq_zero_of_mul_eq_zero := mul_eq_bot.1
instance {S A : Type*} [Semiring S] [SMul R S] [AddCommMonoid A] [Module R A] [Module S A]
| Mathlib/RingTheory/Ideal/Operations.lean | 366 | 369 |
/-
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... | ⟨inv_injective, measurable_inv, fun s hs ↦ s.image_inv_eq_inv ▸ hs.inv⟩
end Inv
@[to_additive]
theorem Measurable.mul_iff_right {G : Type*} [MeasurableSpace G] [MeasurableSpace α] [CommGroup G]
| Mathlib/MeasureTheory/Group/Arithmetic.lean | 419 | 424 |
/-
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.Field.Basic
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Orde... |
theorem mul_edgeDensity_le_edgeDensity (hs : s₂ ⊆ s₁) (ht : t₂ ⊆ t₁) (hs₂ : s₂.Nonempty)
(ht₂ : t₂.Nonempty) :
(#s₂ : ℚ) / #s₁ * (#t₂ / #t₁) * edgeDensity r s₂ t₂ ≤ edgeDensity r s₁ t₁ := by
have hst : (#s₂ : ℚ) * #t₂ ≠ 0 := by simp [hs₂.ne_empty, ht₂.ne_empty]
rw [edgeDensity, edgeDensity, div_mul_div_com... | Mathlib/Combinatorics/SimpleGraph/Density.lean | 163 | 168 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.CharP.Two
import Mathlib.Data.Nat.Cast.Field
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Factorization.Induction
import Mat... | theorem prime_iff_card_units (p : ℕ) [Fintype (ZMod p)ˣ] :
p.Prime ↔ Fintype.card (ZMod p)ˣ = p - 1 := by
rcases eq_zero_or_neZero p with hp | hp
· subst hp
simp only [ZMod, not_prime_zero, false_iff, zero_tsub]
-- the subst created a non-defeq but subsingleton instance diamond; resolve it
suffices ... | Mathlib/Data/Nat/Totient.lean | 227 | 239 |
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Kim Morrison, Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Functor.Trifunctor
import Mathlib.CategoryTheo... | theorem tensor_inv_hom_id' {V W X Y Z : C} (f : V ⟶ W) [IsIso f] (g : X ⟶ Y) (h : Y ⟶ Z) :
(g ⊗ inv f) ≫ (h ⊗ f) = (g ⊗ 𝟙 W) ≫ (h ⊗ 𝟙 W) := by
rw [← tensor_comp, IsIso.inv_hom_id]; simp [tensorHom_id]
| Mathlib/CategoryTheory/Monoidal/Category.lean | 678 | 680 |
/-
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.Data.Finset.Sort
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Sign
import Mathlib.LinearAlgebra.AffineSpace.Combination
import Mathlib.LinearAlg... | theorem AffineIndependent.not_mem_affineSpan_diff [Nontrivial k] {p : ι → P}
(ha : AffineIndependent k p) (i : ι) (s : Set ι) : p i ∉ affineSpan k (p '' (s \ {i})) := by
simp [ha]
theorem exists_nontrivial_relation_sum_zero_of_not_affine_ind {t : Finset V}
(h : ¬AffineIndependent k ((↑) : t → V)) :
∃ f :... | Mathlib/LinearAlgebra/AffineSpace/Independent.lean | 446 | 454 |
/-
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.... | def diagonalInvertibleEquivInvertible (v : n → α) : Invertible (diagonal v) ≃ Invertible v where
toFun := @invertibleOfDiagonalInvertible _ _ _ _ _ _
| Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 546 | 547 |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Topology.Homeomorph.Lemmas
import Mathlib.Topology.StoneCech
/-!
# Extremally disconnected spaces
An extremally disconnected topological space is a s... | let π₁ : D → A := Prod.fst ∘ Subtype.val
have π₁_cont : Continuous π₁ := continuous_fst.comp continuous_subtype_val
have π₁_surj : π₁.Surjective := fun a => ⟨⟨⟨a, _⟩, (f_surj <| φ a).choose_spec.symm⟩, rfl⟩
rcases exists_compact_surjective_zorn_subset π₁_cont π₁_surj with ⟨E, _, E_onto, E_min⟩
-- apply Lemma ... | Mathlib/Topology/ExtremallyDisconnected.lean | 258 | 284 |
/-
Copyright (c) 2022 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Devon Tuma
-/
import Mathlib.Data.Vector.Basic
/-!
# Theorems about membership of elements in vectors
This file contains theorems for membership in a `v.toList` for a vector `v`.
Having ... | Mathlib/Data/Vector/Mem.lean | 85 | 87 | |
/-
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.NoZeroSMulDivisors.Basic
import Mathlib.Algebra.Order.GroupWithZero.Action.Synonym
import Mathlib.Tactic.GCongr
import Mathlib.Tactic.Positivity.Co... | simpa only [zero_smul] using smul_lt_smul_of_pos_right ha hb
| Mathlib/Algebra/Order/Module/Defs.lean | 496 | 497 |
/-
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.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Valuation.ExtendToLocalization
import Mathlib.Topology.Algebr... | /-- The `v`-adic valuation of a product equals the product of the valuations. -/
theorem intValuation.map_mul' (x y : R) :
v.intValuationDef (x * y) = v.intValuationDef x * v.intValuationDef y := by
classical
simp only [intValuationDef]
| Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | 156 | 160 |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yakov Pechersky
-/
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Infix
import Mathlib.Data.Quot
/-!
# List rotation
This file proves basic results about `List.r... | Mathlib/Data/List/Rotate.lean | 635 | 636 |
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