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 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... | Mathlib/CategoryTheory/CofilteredSystem.lean | 372 | 383 | |
/-
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.Algebra.Group.Support
import Mathlib.Algebra.GroupWithZero.Action.Pointwise.Set
/-!
# Support of a function composed with a scalar action
We sh... | attribute [to_additive existing support_comp_inv_smul₀] mulSupport_comp_inv_smul₀
end GroupWithZero
| Mathlib/Data/Set/Pointwise/Support.lean | 56 | 59 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Order.Filter.Tendsto
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.Basic
import Mathlib.Topolog... | neBot_of_le Filter.cocompact_le_coclosedCompact
theorem noncompactSpace_of_neBot (_ : NeBot (Filter.cocompact X)) : NoncompactSpace X :=
⟨fun h' => (Filter.nonempty_of_mem h'.compl_mem_cocompact).ne_empty compl_univ⟩
| Mathlib/Topology/Compactness/Compact.lean | 730 | 733 |
/-
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... |
@[norm_cast]
theorem coe_rpow_of_ne_zero {x : ℝ≥0} (h : x ≠ 0) (y : ℝ) : (↑(x ^ y) : ℝ≥0∞) = x ^ y := by
rw [← ENNReal.some_eq_coe]
dsimp only [(· ^ ·), Pow.pow, rpow]
simp [h]
@[norm_cast]
| Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 488 | 495 |
/-
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.Field.Basic
import Mathlib.Algebra.NoZeroSMulDivisors.Basic
import Mathlib.Data.Int.ModEq
import Mathlib.GroupTheory.QuotientGroup.Defs
import Math... | protected theorem add_iff_left :
a₁ ≡ b₁ [PMOD p] → (a₁ + a₂ ≡ b₁ + b₂ [PMOD p] ↔ a₂ ≡ b₂ [PMOD p]) := fun ⟨m, hm⟩ =>
| Mathlib/Algebra/ModEq.lean | 157 | 158 |
/-
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.Monoid.Unbundled.WithTop
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
/-! # Adjoi... | Mathlib/Algebra/Order/Monoid/WithTop.lean | 630 | 630 | |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Jujian Zhang, Yongle Hu
-/
import Mathlib.Algebra.Colimit.TensorProduct
import Mathlib.Algebra.DirectSum.Finsupp
import Mathlib.Algebra.DirectSum.Module
import Mathlib.... | EquivLike.injective_comp, lmap_injective]
constructor <;> (intro h; intros; apply h)
theorem dfinsupp_iff : Flat R (Π₀ i, M i) ↔ ∀ i, Flat R (M i) := directSum_iff ..
/-- A direct sum of flat `R`-modules is flat. -/
instance directSum [∀ i, Flat R (M i)] : Flat R (⨁ i, M i) := directSum_iff.mpr ‹_›
instance df... | Mathlib/RingTheory/Flat/Basic.lean | 209 | 218 |
/-
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 | 1,088 | 1,092 | |
/-
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.Iterate
import Aesop
import Mathlib.Tactic.Inhabit
/-!
# Extra facts about `Prod`
This file proves ... |
/-- Delaborator for `Prod.snd x` as `x.2`. -/
@[app_delab Prod.snd]
| Mathlib/Data/Prod/Basic.lean | 330 | 332 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Disintegration.Unique
import Mathlib.Probability.Notation
/-!
# Regular conditional probability distribution
We define the regular con... | rwa [integrable_map_measure hf (hX.aemeasurable.prodMk hY)]
condExp_prod_ae_eq_integral_condDistrib' hX hY hf_int'
@[deprecated (since := "2025-01-21")]
alias condexp_prod_ae_eq_integral_condDistrib₀ := condExp_prod_ae_eq_integral_condDistrib₀
/-- The conditional expectation of a function `f` of the product `(X... | Mathlib/Probability/Kernel/CondDistrib.lean | 265 | 272 |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin.Basic
imp... | rw [← hasFPowerSeriesWithinAt_univ] at hf
simpa using hf.isBigO_image_sub_norm_mul_norm_sub
/-- If a function admits a power series expansion within a set at `x`, then it is the uniform limit
of the partial sums of this power series on strict subdisks of the disk of convergence, i.e.,
| Mathlib/Analysis/Analytic/Basic.lean | 1,168 | 1,172 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Analytic.IteratedFDeriv
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.Ca... | f' (x + h • v + (t * h) • w) (h • w) - h • f' x w - h ^ 2 • f'' v w - (t * h ^ 2) • f'' w w
with hg'
-- check that `g'` is the derivative of `g`, by a straightforward computation
have g_deriv : ∀ t ∈ Icc (0 : ℝ) 1, HasDerivWithinAt g (g' t) (Icc 0 1) t := by
intro t ht
apply_rules [HasDerivWithinAt.... | Mathlib/Analysis/Calculus/FDeriv/Symmetric.lean | 234 | 254 |
/-
Copyright (c) 2022 Bolton Bailey. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bolton Bailey, Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib... | @[deprecated (since := "2025-03-18")]
alias tendsto_logb_nhdsWithin_zero := tendsto_logb_nhdsNE_zero
theorem tendsto_logb_nhdsNE_zero_of_base_lt_one (hb₀ : 0 < b) (hb : b < 1) :
Tendsto (logb b) (𝓝[≠] 0) atTop :=
tendsto_log_nhdsNE_zero.atBot_mul_const_of_neg (inv_lt_zero.2 (log_neg hb₀ hb))
@[deprecated (sinc... | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 424 | 434 |
/-
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
-/
import Mathlib.Analysis.Calculus.ContDiff.RCLike
import Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas
import Mathlib.Anal... | theorem DifferentiableWithinAt.exp (hf : DifferentiableWithinAt ℝ f s x) :
DifferentiableWithinAt ℝ (fun x => Real.exp (f x)) s x :=
hf.hasFDerivWithinAt.exp.differentiableWithinAt
| Mathlib/Analysis/SpecialFunctions/ExpDeriv.lean | 337 | 339 |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Geometry.Euclidean.Inversion.Basic
import Mathlib.Geometry.Euclidean.PerpBisector
/-!
# Image of a hyperplane under inversion
In this file we prove... |
theorem image_inversion_perpBisector (hR : R ≠ 0) (hy : y ≠ c) :
inversion c R '' perpBisector c y = sphere (inversion c R y) (R ^ 2 / dist y c) \ {c} := by
rw [image_eq_preimage_of_inverse (inversion_involutive _ hR) (inversion_involutive _ hR),
| Mathlib/Geometry/Euclidean/Inversion/ImageHyperplane.lean | 61 | 64 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yakov Pechersky, Eric Wieser
-/
import Mathlib.Data.List.Basic
/-!
# Properties of `List.enum`
## Deprecation note
Many lemmas in this file have been replaced by the... | Mathlib/Data/List/Enum.lean | 165 | 170 | |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou
-/
import Mathlib.Data.Set.Order
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.Interval.Set.Image
import Mathlib.Order.Interval.Set.LinearOrder
import Mathlib.Tac... | rw [uIoc_eq_union, mem_union, mem_Ioc, mem_Ioc]
lemma not_mem_uIoc : a ∉ Ι b c ↔ a ≤ b ∧ a ≤ c ∨ c < a ∧ b < a := by
simp only [uIoc_eq_union, mem_union, mem_Ioc, not_lt, ← not_le]
tauto
@[simp] lemma left_mem_uIoc : a ∈ Ι a b ↔ b < a := by simp [mem_uIoc]
@[simp] lemma right_mem_uIoc : b ∈ Ι a b ↔ a < b := by ... | Mathlib/Order/Interval/Set/UnorderedInterval.lean | 256 | 265 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Sébastien Gouëzel, Jean Lo, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Opposite
import Mathlib.Topology.Algebra.Group.Qu... | Mathlib/Topology/Algebra/Module/Basic.lean | 2,054 | 2,056 | |
/-
Copyright (c) 2023 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Prod
/-!
# Products of finite measures and probability measures
This file i... | lemma prod_swap : (μ.prod ν).map Prod.swap = ν.prod μ := by
apply Subtype.ext
simp [Measure.prod_swap]
| Mathlib/MeasureTheory/Measure/FiniteMeasureProd.lean | 80 | 82 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Chris Hughes
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.Data.Fintype.Inv
import Mathlib.GroupTheory.SpecificGroups.Cyclic
... | /-- A finite subgroup of the unit group of an integral domain is cyclic. -/
theorem isCyclic_of_subgroup_isDomain [Finite G] (f : G →* R) (hf : Injective f) : IsCyclic G := by
classical
cases nonempty_fintype G
apply isCyclic_of_card_pow_eq_one_le
intro n hn
exact le_trans (card_nthRoots_subgroup_unit... | Mathlib/RingTheory/IntegralDomain.lean | 122 | 128 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.PartrecCode
import Mathlib.Data.Set.Subsingleton
/-!
# Computability theory and the halting problem
A universal partial recursive funct... |
/-- Analogous to `Nat.Partrec'` for `ℕ`-valued functions, a predicate for partial recursive
vector-valued functions. -/
| Mathlib/Computability/Halting.lean | 327 | 329 |
/-
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.Ring.Int.Units
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Sum
import Mathlib.SetTheory.Cardinal.Finite
/-!
# fintype instanc... | Nat.card αˣ = Nat.card α - 1 := by
classical
rw [Nat.card_congr unitsEquivNeZero, eq_comm, ← Nat.card_congr (Equiv.sumCompl (· = (0 : α)))]
rcases finite_or_infinite {a : α // a ≠ 0}
· rw [Nat.card_sum, Nat.card_unique, add_tsub_cancel_left]
| Mathlib/Data/Fintype/Units.lean | 36 | 40 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.SuccPred.Initial... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,458 | 1,461 | |
/-
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.Subalgebra
import Mathlib.LinearAlgebra.Finsupp.Span
/-!
# Lie submodules of a Lie algebra
In this file we define Lie submodules, we construct ... | lemma isCompactElement_lieSpan_singleton (m : M) :
CompleteLattice.IsCompactElement (lieSpan R L {m}) := by
rw [CompleteLattice.isCompactElement_iff_le_of_directed_sSup_le]
| Mathlib/Algebra/Lie/Submodule.lean | 743 | 745 |
/-
Copyright (c) 2022 María Inés de Frutos-Fernández, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: María Inés de Frutos-Fernández, Yaël Dillies
-/
import Mathlib.Data.NNReal.Defs
import Mathlib.Order.ConditionallyCompleteLattice.Group
import Mathlib.Tac... |
namespace GroupSeminorm
| Mathlib/Analysis/Normed/Group/Seminorm.lean | 526 | 527 |
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Anatole Dedecker
-/
import Mathlib.Analysis.Calculus.LocalExtr.Basic
import Mathlib.Topology.Order.Rolle
/-!
# Rolle's Theorem
In this file we prove Rolle's Theorem... | push_neg at h; exact h
exact ⟨c, hc, deriv_zero_of_not_differentiableAt hcdiff⟩
| Mathlib/Analysis/Calculus/LocalExtr/Rolle.lean | 78 | 84 |
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Yury Kudryashov
-/
import Mathlib.Topology.Instances.NNReal.Lemmas
import Mathlib.Topology.Order.MonotoneContinuity
/-!
# Square root of a real numbe... | rw [Real.sqrt, Real.sqrt, NNReal.coe_le_coe, NNReal.sqrt_le_sqrt]
exact toNNReal_le_toNNReal h
| Mathlib/Data/Real/Sqrt.lean | 189 | 190 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Analytic.Composition
import Mathlib.Analysis.Analytic.Linear
import Mathlib.Tactic.Positivity
/-!
# Inverse of analytic functions
We ... | p c.length (applyComposition (fun k : ℕ => ite (k < n + 2) (p.rightInv i x k) 0) c v) =
∑ c ∈ {c : Composition (n + 2) | 1 < c.length}.toFinset,
p c.length ((p.rightInv i x).applyComposition c v) := by
have N : 0 < n + 2 := by norm_num
refine sum_congr rfl fun c hc => p.congr rfl fun j hj1 hj2... | Mathlib/Analysis/Analytic/Inverse.lean | 231 | 243 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kevin Buzzard
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.PowerSeries.WellKnown
/-!
# Bernoull... | ext q : 1
let f a b := bernoulli a / a ! * coeff ℚ (b + 1) (exp ℚ ^ n)
-- key step: use `PowerSeries.coeff_mul` and then rewrite sums
simp only [f, coeff_mul, coeff_mk, cast_mul, sum_antidiagonal_eq_sum_range_succ f]
apply sum_congr rfl
intros m h
simp only [f, exp_pow_eq_rescale_exp, rescal... | Mathlib/NumberTheory/Bernoulli.lean | 276 | 296 |
/-
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... | alias sInter_mono := sInter_subset_sInter
| Mathlib/Data/Set/Lattice.lean | 985 | 986 |
/-
Copyright (c) 2023 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.Algebra.Group.Nat.Defs
import Mathlib.CategoryTheory.Category.Preorder
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Functor.Const
import M... | obtain ⟨i, hi⟩ := i
obtain ⟨j, hj⟩ := j
obtain ⟨k, hk⟩ := k
cases i
· obtain _ | _ | j := j
| Mathlib/CategoryTheory/ComposableArrows.lean | 343 | 347 |
/-
Copyright (c) 2023 Moritz Firsching. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Firsching
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.LinearAlgebra.Vanderm... | | n + 1 => by
rw [← Ico_succ_right 1 n.succ, prod_Ico_succ_top <| Nat.succ_le_succ <| Nat.zero_le n,
Nat.factorial_succ, Ico_succ_right 1 n, prod_Icc_factorial n, superFactorial, factorial,
Nat.succ_eq_add_one, mul_comm]
@[simp]
| Mathlib/Data/Nat/Factorial/SuperFactorial.lean | 56 | 61 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno
import Mathlib.Analysis.Calculus.FDeriv.Ad... | exact ((hi hmn).fderivWithin_right hs le_rfl hx₀s).continuousLinearMap_comp
((continuousMultilinearCurryLeftEquiv 𝕜 (fun _ : Fin (i+1) ↦ E) F).symm :
_ →L[𝕜] E [×(i+1)]→L[𝕜] F)
| Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 1,156 | 1,159 |
/-
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 ... | rw [← cancel_mono S₁.op.iCycles, assoc, assoc, cyclesOpIso_inv_op_iCycles, cyclesMap_i,
cyclesOpIso_inv_op_iCycles_assoc, ← op_comp, p_opcyclesMap, op_comp, opMap_τ₂]
| Mathlib/Algebra/Homology/ShortComplex/RightHomology.lean | 1,023 | 1,024 |
/-
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.MeasureTheory.Measure.MeasureSpace
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.Topology.Sets.Compacts
/-!
# Contents
In this file... | biSup_mono fun _ hK => hK.trans h2
section OuterMeasure
/-- Extending a content on compact sets to an outer measure on all sets. -/
protected def outerMeasure : OuterMeasure G :=
inducedOuterMeasure (fun U hU => μ.innerContent ⟨U, hU⟩) isOpen_empty μ.innerContent_bot
variable [R1Space G]
theorem outerMeasure_op... | Mathlib/MeasureTheory/Measure/Content.lean | 228 | 240 |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.Connected.Basic
/-!
# Locally connected topological spaces
A topological space is **locally connected** if each neighborhood filter admi... | theorem isOpen_connectedComponent [LocallyConnectedSpace α] {x : α} :
IsOpen (connectedComponent x) := by
rw [← connectedComponentIn_univ]
exact isOpen_univ.connectedComponentIn
| Mathlib/Topology/Connected/LocallyConnected.lean | 78 | 81 |
/-
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... | have D2 : dist x y ≤ δ := by
rw [edist_dist] at D1
exact (ENNReal.ofReal_le_ofReal_iff hδ).1 D1
exact mem_biUnion yE D2
theorem cthickening_eq_biUnion_closedBall {α : Type*} [PseudoMetricSpace α] [ProperSpace α]
(E : Set α) (hδ : 0 ≤ δ) : cthickening δ E = ⋃ x ∈ closure E, closedBall x δ := by
rcases... | Mathlib/Topology/MetricSpace/Thickening.lean | 543 | 553 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Finset.Card
import Mathlib.Data.Fintype.Basic
/-!
# Cardinalities of finite types
This file defines the cardinality `Fintype.card α` as the numb... | Mathlib/Data/Fintype/Card.lean | 925 | 930 | |
/-
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.CategoryTheory.GradedObject.Bifunctor
import Mathlib.CategoryTheory.Functor.Trifunctor
/-!
# The action of trifunctors on graded objects
Given a trifunctor `F. ... | (G.map (ιMapBifunctorMapObj F₁₂ ρ₁₂.p X₁ X₂ i₁ i₂ _ rfl)).app (X₃ i₃) ≫
ιMapBifunctorMapObj G ρ₁₂.q (mapBifunctorMapObj F₁₂ ρ₁₂.p X₁ X₂) X₃ (ρ₁₂.p ⟨i₁, i₂⟩) i₃ j
(by rw [← h, ← ρ₁₂.hpq])
@[reassoc]
lemma ιMapBifunctor₁₂BifunctorMapObj_eq (i₁ : I₁) (i₂ : I₂) (i₃ : I₃) (j : J)
(h : r (i₁, i₂, i₃) = j) (i... | Mathlib/CategoryTheory/GradedObject/Trifunctor.lean | 287 | 295 |
/-
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.CategoryTheory.Localization.Construction
/-!
# Predicate for localized categories
In this file, a predicate `L.IsLocalization W` is introduced for a functor `... |
theorem IsLocalization.for_id (hW : W ≤ MorphismProperty.isomorphisms C) : (𝟭 C).IsLocalization W :=
IsLocalization.mk' _ _ (Localization.strictUniversalPropertyFixedTargetId W _ hW)
(Localization.strictUniversalPropertyFixedTargetId W _ hW)
end Functor
namespace Localization
| Mathlib/CategoryTheory/Localization/Predicate.lean | 121 | 129 |
/-
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.Algebra.Algebra.Pi
import Mathlib.LinearAlgebra.Finsupp.SumProd
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.LinearIndepe... |
@[simp]
theorem basis_apply [DecidableEq η] (s : ∀ j, Basis (ιs j) R (Ms j)) (ji) :
Pi.basis s ji = Pi.single ji.1 (s ji.1 ji.2) :=
Basis.apply_eq_iff.mpr (by simp)
| Mathlib/LinearAlgebra/StdBasis.lean | 132 | 137 |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Comma.Basic
/-!
# The category of arrows
The category of arrows, with morphisms commutative squares.
We set this up as a specialization ... | simp only [← Category.assoc, h]
instance epi_right [Epi sq] : Epi sq.right where
| Mathlib/CategoryTheory/Comma/Arrow.lean | 251 | 253 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Bundle
import Mathlib.Data.Set.Image
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Topology.O... | theorem symm_trans_source_eq (e e' : Pretrivialization F proj) :
(e.toPartialEquiv.symm.trans e'.toPartialEquiv).source = (e.baseSet ∩ e'.baseSet) ×ˢ univ := by
rw [PartialEquiv.trans_source, e'.source_eq, PartialEquiv.symm_source, e.target_eq, inter_comm,
e.preimage_symm_proj_inter, inter_comm]
theorem symm... | Mathlib/Topology/FiberBundle/Trivialization.lean | 189 | 195 |
/-
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... |
variable [Preorder β] {l : Filter α} {f g h : α → β}
theorem EventuallyEq.le (h : f =ᶠ[l] g) : f ≤ᶠ[l] g :=
h.mono fun _ => le_of_eq
| Mathlib/Order/Filter/Basic.lean | 1,056 | 1,060 |
/-
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.AlgebraicTopology.SimplicialSet.Monoidal
import Mathlib.CategoryTheory.Enriched.Ordinary.Basic
/-!
# Simplicial categories
A simplicial category is a category ... | Mathlib/AlgebraicTopology/SimplicialCategory/Basic.lean | 91 | 93 | |
/-
Copyright (c) 2021 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.PiL2
/-!
# Adjoint of operators on Hilbert spaces
Given ... | instance : InvolutiveStar (E →ₗ[𝕜] E) :=
⟨adjoint_adjoint⟩
instance : StarMul (E →ₗ[𝕜] E) :=
⟨adjoint_comp⟩
| Mathlib/Analysis/InnerProductSpace/Adjoint.lean | 407 | 411 |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Sébastien Gouëzel, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.Normed.Lp.PiLp
import Mathlib.LinearAlgebra.FiniteDimensi... | simp_rw [mem_setOf, mem_closedBall_zero_iff, norm_eq, norm_eq_abs, sq_abs, sqrt_le_left hr]
| Mathlib/Analysis/InnerProductSpace/PiL2.lean | 166 | 167 |
/-
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, Eric Wieser
-/
import Mathlib.Algebra.Group.Fin.Tuple
import Mathlib.Algebra.BigOperators.GroupWithZero.Action
import Ma... |
theorem single_eq_pi_diag (i : ι) : single R φ i = pi (diag i) := by
ext x j
convert (update_apply 0 x i j _).symm
| Mathlib/LinearAlgebra/Pi.lean | 311 | 314 |
/-
Copyright (c) 2021 Vladimir Goryachev. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Vladimir Goryachev, Kyle Miller, Kim Morrison, Eric Rodriguez
-/
import Mathlib.Algebra.Group.Nat.Range
import Mathlib.Data.Set.Finite.Basic
/-!
# Counting on ℕ
Thi... | theorem count_le {n : ℕ} : count p n ≤ n := by
rw [count_eq_card_filter_range]
exact (card_filter_le _ _).trans_eq (card_range _)
| Mathlib/Data/Nat/Count.lean | 60 | 62 |
/-
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... | theorem xgcd_zero_left {s t r' s' t'} : xgcdAux 0 s t r' s' t' = (r', s', t') := by simp [xgcdAux]
| Mathlib/Data/Int/GCD.lean | 48 | 48 |
/-
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_apply : coeAlgHom F f = f :=
rfl
-- avoids a diamond, see https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/compiling.20behaviour.20within.20one.20file
| Mathlib/RingTheory/LaurentSeries.lean | 399 | 402 |
/-
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
-/
import Mathlib.Tactic.Attr.Register
import Mathlib.Tactic.Basic
import Batteries.Logic
import Batteries.Tactic.Trans
import Batteries.Util.LibraryNot... | Mathlib/Logic/Basic.lean | 721 | 721 | |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.BigOperators.Finsupp.Basic
import Mathlib.Data.Finsupp.Indicator
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Algebra.Group.Pointwise.Fi... | mem_finsupp_iff.trans
(forall_and.symm.trans <|
forall_congr' fun i =>
⟨fun h => ?_, fun h =>
⟨fun hi => ht <| mem_support_iff.2 fun H => mem_support_iff.1 hi ?_, fun _ => h⟩⟩)
· by_cases hi : i ∈ s
· exact h.2 hi
· rw [not_mem_support_iff.1 (mt h.1 hi), not_mem_support... | Mathlib/Data/Finset/Finsupp.lean | 62 | 74 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Group.Action.Pi
import Mathlib.Algebra.Order.AbsoluteValue.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.Mi... | let i := max N (max N1 N2)
have hN' := hN i (le_max_left _ _)
| Mathlib/Algebra/Order/CauSeq/Basic.lean | 482 | 483 |
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.Algebra.BigOperators.Expect
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Convex.Jensen
import M... | by_cases H : (∑ i ∈ s, f i ^ p) ^ (1 / p) = 0 ∨ (∑ i ∈ s, g i ^ q) ^ (1 / q) = 0
· replace H : (∀ i ∈ s, f i = 0) ∨ ∀ i ∈ s, g i = 0 := by
simpa [ENNReal.rpow_eq_zero_iff, hpq.pos, hpq.symm.pos, asymm hpq.pos, asymm hpq.symm.pos,
sum_eq_zero_iff_of_nonneg] using H
have : ∀ i ∈ s, f i * g i = 0 := ... | Mathlib/Analysis/MeanInequalities.lean | 891 | 903 |
/-
Copyright (c) 2021 Vladimir Goryachev. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Vladimir Goryachev, Kyle Miller, Kim Morrison, Eric Rodriguez
-/
import Mathlib.Data.List.GetD
import Mathlib.Data.Nat.Count
import Mathlib.Data.Nat.SuccPred
import M... | rintro m (hm : m < _) n (hn : n < _) h
simp only [nth_eq_orderEmbOfFin, *]
exact OrderEmbedding.strictMono _ h
theorem nth_lt_nth_of_lt_card (hf : (setOf p).Finite) {m n : ℕ} (h : m < n)
| Mathlib/Data/Nat/Nth.lean | 76 | 80 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Batteries.Tactic.Congr
import Mathlib.Data.Option.Basic
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Set.Subsingleton
import Mathlib.Dat... | theorem Surjective.nonempty_preimage (hf : Surjective f) {s : Set β} :
(f ⁻¹' s).Nonempty ↔ s.Nonempty := by rw [← image_nonempty, hf.image_preimage]
theorem Injective.image_injective (hf : Injective f) : Injective (image f) := by
intro s t h
rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, h]
lemma In... | Mathlib/Data/Set/Image.lean | 1,097 | 1,104 |
/-
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... | theorem gcd_eq_gcd_filter_ne_zero [DecidablePred fun x : β ↦ f x = 0] :
s.gcd f = {x ∈ s | f x ≠ 0}.gcd f := by
classical
trans ({x ∈ s | f x = 0} ∪ {x ∈ s | f x ≠ 0}).gcd f
| Mathlib/Algebra/GCDMonoid/Finset.lean | 189 | 192 |
/-
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... |
theorem uniformity_basis_dist_inv_nat_pos :
(𝓤 α).HasBasis (fun n : ℕ => 0 < n) fun n : ℕ => { p : α × α | dist p.1 p.2 < 1 / ↑n } :=
Metric.mk_uniformity_basis (fun _ hn => div_pos zero_lt_one <| Nat.cast_pos.2 hn) fun _ ε0 =>
let ⟨n, hn⟩ := exists_nat_one_div_lt ε0
| Mathlib/Topology/MetricSpace/Pseudo/Defs.lean | 644 | 648 |
/-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Andrew Yang
-/
import Mathlib.RingTheory.Derivation.ToSquareZero
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.IsTensorProduct
import Mathlib.... |
theorem KaehlerDifferential.linearCombination_surjective :
Function.Surjective (Finsupp.linearCombination S (KaehlerDifferential.D R S)) := by
| Mathlib/RingTheory/Kaehler/Basic.lean | 577 | 579 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.SuccPred.Initial... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 2,092 | 2,097 | |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sort
/-!
# Compositions
A compositio... | | _, ⟨blocks, blocks_pos, rfl⟩ =>
match blocks with
| [] => zero
| 0 :: _ => by simp at blocks_pos
| (k + 1) :: l =>
| Mathlib/Combinatorics/Enumerative/Composition.lean | 652 | 656 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.Typeclasses.Finite
import Mathlib.MeasureTheory.Measure.Typeclasses.NoAtoms
import Mathlib.MeasureTheory.Measure.... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 1,335 | 1,347 | |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kexing Ying, Eric Wieser
-/
import Mathlib.Data.Finset.Sym
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
import Mathlib.Linea... | protected theorem map_add (f : M → N) (x y : M) :
f (x + y) = f x + f y + polar f x y := by
rw [polar]
| Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 98 | 100 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Covering.VitaliFamily
import Mathlib.MeasureTheory.Function.AEMeasurableOrder
import Mathlib.MeasureTheory.Integral.Average
import ... | refine ENNReal.Tendsto.mul_const (ENNReal.tendsto_coe.2 nhdsWithin_le_nhds) ?_
simp only [ENNReal.coe_one, true_or, Ne, not_false_iff, one_ne_zero]
simp only [one_mul, ENNReal.coe_one] at this
refine ge_of_tendsto this ?_
filter_upwards [self_mem_nhdsWithin] with _ ht
exact v.le_mul_withDens... | Mathlib/MeasureTheory/Covering/Differentiation.lean | 659 | 680 |
/-
Copyright (c) 2024 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Basic
import Mathlib.Tactic.ComputeDegree
/-!
# Division polynomials of Weierst... |
@[simp]
lemma leadingCoeff_Φ [Nontrivial R] (n : ℤ) : (W.Φ n).leadingCoeff = 1 := by
rw [leadingCoeff, natDegree_Φ, coeff_Φ]
| Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Degree.lean | 440 | 443 |
/-
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.Projection
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique
import Mathlib.MeasureTheory.Function.L2Space
/-... |
theorem inner_condExpL2_left_eq_right (hm : m ≤ m0) {f g : α →₂[μ] E} :
⟪(condExpL2 E 𝕜 hm f : α →₂[μ] E), g⟫₂ = ⟪f, (condExpL2 E 𝕜 hm g : α →₂[μ] E)⟫₂ :=
haveI : Fact (m ≤ m0) := ⟨hm⟩
Submodule.inner_orthogonalProjection_left_eq_right _ f g
@[deprecated (since := "2025-01-21")]
alias inner_condexpL2_left_e... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean | 126 | 137 |
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Isabel Longbottom, Kim Morrison, Apurva Nakade, Yuyang Zhao
-/
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.SetTheory.PGame.Algebra
import Mathl... |
instance isOrderedAddMonoid : IsOrderedAddMonoid Game :=
{ add_le_add_left := @add_le_add_left _ _ _ Game.addLeftMono }
| Mathlib/SetTheory/Game/Basic.lean | 191 | 193 |
/-
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... | · have := (sec i).prop
simp only [mem_iInter, mem_preimage, mem_singleton_iff] at this
have := this (𝟙 i)
rwa [map_id_apply] at this
| Mathlib/CategoryTheory/CofilteredSystem.lean | 327 | 330 |
/-
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
-/
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finset.Image
/-!
# Cardinality of a finite set
This defines the cardinality of a `Fins... | card_bij (fun a _ ↦ i a) hi i_inj (by simpa using i_surj)
/-- Reorder a finset.
The difference with `Finset.card_nbij` is that the bijection is specified with an inverse, rather
than as a surjective injection.
The difference with `Finset.card_bij'` is that the bijection and its inverse are non-dependent
functions,... | Mathlib/Data/Finset/Card.lean | 367 | 389 |
/-
Copyright (c) 2022 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.Star.Module
import Mathlib.Algebra.Star.NonUn... |
namespace StarAlgHom
| Mathlib/Algebra/Star/Subalgebra.lean | 705 | 707 |
/-
Copyright (c) 2023 Wrenna Robson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wrenna Robson
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.Group.Submonoid.Pointwise
import Mathlib.Algebra.Group.Subgroup.Lattice
/-!
# Submonoid of un... | @[to_additive (attr := simp)]
lemma val_mem_ofUnits_iff_mem (H : Subgroup Gˣ) (x : Gˣ) : (x : G) ∈ H.ofUnits ↔ x ∈ H := by
simp_rw [mem_ofUnits_iff_toUnits_mem, toUnits_val_apply]
| Mathlib/Algebra/Group/Submonoid/Units.lean | 322 | 324 |
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Function.LocallyIntegrable
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Integral.Prod
import Mathlib.Me... | Otherwise, the family is uncountable, and each intersection with `s` has positive measure by
the everywhere positivity assumption, so `ν s = ∞`, and `μ' s = ∞` in the same way. -/
obtain ⟨k, k_comp, k_closed, k_mem⟩ : ∃ k, IsCompact k ∧ IsClosed k ∧ k ∈ 𝓝 (1 : G) := by
rcases exists_compact_mem_nhds (1 : G) ... | Mathlib/MeasureTheory/Measure/Haar/Unique.lean | 695 | 794 |
/-
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 := (ringChar_zmod_n p).symm.trans hc
-- p = 2
subst p
rw [legendreSym, quadraticChar_eq_one_of_char_two hc ha]
revert ha
push_cast
generalize (a : ZMod 2) = b; fin_cases b
· tauto
· simp
· convert quadraticChar_eq_pow_of_char_ne_two' hc (a : ZMod p)
exact... | Mathlib/NumberTheory/LegendreSymbol/Basic.lean | 116 | 132 |
/-
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... | elim _a ha _b₁ _b₂ hb := smul_le_smul_of_nonneg_left (β := β) hb ha
instance instPosSMulStrictMono [PosSMulStrictMono α β] : PosSMulStrictMono α βᵒᵈ where
| Mathlib/Algebra/Order/Module/Defs.lean | 761 | 762 |
/-
Copyright (c) 2019 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.UniformSpace.UniformEmbedding
/-!
# Indexed product of uniform spaces
-/
noncomputable section
open scoped Uniformity Topology
open Filter... | Filter.comap_comap, comp_def, const, Prod.eta, comap_id']
-- Hence, it suffices to show that its range, the diagonal, is closed in `Π i, (X, u i)`.
simp_rw [@completeSpace_iff_isComplete_range _ _ (_) (_) _ this, range_const_eq_diagonal,
setOf_forall]
-- The separation of `t` ensures that this is the ca... | Mathlib/Topology/UniformSpace/Pi.lean | 126 | 147 |
/-
Copyright (c) 2023 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Ring.CharZero
import Mathlib.Algebra.Ring.Int.Units
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.Complement
/-!
## HNN Exte... | simp [hx])
@[simp]
| Mathlib/GroupTheory/HNNExtension.lean | 97 | 99 |
/-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.Normal.... | `(X - x) ^ (q ^ n)` for some `n : ℕ`. -/
theorem natSepDegree_eq_one_iff_eq_X_sub_C_pow : (minpoly F x).natSepDegree = 1 ↔
∃ n : ℕ, (minpoly F x).map (algebraMap F E) = (X - C x) ^ q ^ n := by
haveI := expChar_of_injective_algebraMap (algebraMap F E).injective q
| Mathlib/FieldTheory/SeparableDegree.lean | 661 | 664 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions.SumProd
/-!
# Maps equivariantly-homeomorphic to projection in a product
This file contains the definition `IsHomeomorph... | protected theorem isQuotientMap_proj [Nonempty F] (h : IsHomeomorphicTrivialFiberBundle F proj) :
IsQuotientMap proj :=
h.isOpenMap_proj.isQuotientMap h.continuous_proj h.surjective_proj
| Mathlib/Topology/FiberBundle/IsHomeomorphicTrivialBundle.lean | 56 | 58 |
/-
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.IntegralClosure.IntegrallyClosed
import Mathlib.RingTheory.Trace.Basic
import Mathlib.RingTheory.Norm.Basic
/-!
# Discriminant of a famil... | theorem discr_of_matrix_mulVec (b : ι → B) (P : Matrix ι ι A) :
discr A (P.map (algebraMap A B) *ᵥ b) = P.det ^ 2 * discr A b := by
rw [discr_def, traceMatrix_of_matrix_mulVec, det_mul, det_mul, det_transpose, mul_comm, ←
mul_assoc, discr_def, pow_two]
| Mathlib/RingTheory/Discriminant.lean | 113 | 116 |
/-
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 | 3,121 | 3,124 | |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.BigOperators.Group.List.Lemmas
import Mathlib.Algebra.Group.Action.Hom
import Mathlib.Algebra.Group.Submonoid.Defs
import Mathlib.Data.List.FinRange
... | ⟨List.prod (l.map fun a => fA a),
(l.dProdIndex_eq_map_sum fι).symm ▸
list_prod_map_mem_graded l _ _ fun i _ => (fA i).prop⟩ :=
Subtype.ext <| SetLike.coe_list_dProd _ _ _ _
end DProd
end Subobjects
| Mathlib/Algebra/GradedMonoid.lean | 635 | 642 |
/-
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.CategoryTheory.GradedObject.Bifunctor
import Mathlib.CategoryTheory.Functor.Trifunctor
/-!
# The action of trifunctors on graded objects
Given a trifunctor `F. ... | fun ⟨⟨i₁, i₂₃⟩, hi⟩ => by
refine (Cocones.precompose (Iso.hom ?_)).obj ((Cocones.whiskeringEquivalence
(Discrete.equivalence (e i₁ i₂₃))).functor.obj (c₂₃' ⟨⟨i₁, i₂₃⟩, hi⟩))
refine Discrete.natIso (fun ⟨⟨i₁', i₂, i₃⟩, hi⟩ => eqToIso ?_)
obtain rfl : i₁' = i₁ := congr_arg _root_.Prod.fst hi... | Mathlib/CategoryTheory/GradedObject/Trifunctor.lean | 513 | 520 |
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Star.Basic
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Data.Set.Lattice.Image
import Mathlib.Algebra.Group.Pointwise.Set.Basic
/-!
# Poi... | protected theorem star_inv' [GroupWithZero α] [StarMul α] (s : Set α) : s⁻¹⋆ = s⋆⁻¹ := by
ext
simp only [mem_star, mem_inv, star_inv₀]
| Mathlib/Algebra/Star/Pointwise.lean | 120 | 122 |
/-
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... |
@[to_additive]
| Mathlib/GroupTheory/GroupAction/Blocks.lean | 201 | 202 |
/-
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... | withBotCoe_subset_iff.1 <| by simp
@[simp]
theorem splitUpper_eq_bot {i x} : I.splitUpper i x = ⊥ ↔ I.upper i ≤ x := by
classical
rw [splitUpper, mk'_eq_bot, exists_update_iff I.lower fun j y => I.upper j ≤ y]
simp [(I.lower_lt_upper _).not_le]
| Mathlib/Analysis/BoxIntegral/Partition/Split.lean | 106 | 112 |
/-
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
-/
import Mathlib.Algebra.Order.Antidiag.Finsupp
import Mathlib.Data.Finsupp.Weight
import Mathlib.Tactic.Linarith
import Mathlib.LinearAlgebra.Pi
import Mat... | add_decl_doc MvPowerSeries.ext_iff
theorem monomial_def [DecidableEq σ] (n : σ →₀ ℕ) :
(monomial R n) = LinearMap.single R (fun _ ↦ R) n := by
| Mathlib/RingTheory/MvPowerSeries/Basic.lean | 149 | 152 |
/-
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.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.IsometricSMul
/-!
# Hausdorff distance
The Hausdorff distance on subsets of a... | ENNReal.toReal (infEdist x s)
theorem infDist_eq_iInf : infDist x s = ⨅ y : s, dist x y := by
| Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 433 | 435 |
/-
Copyright (c) 2019 Johannes Hölzl, Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Zhouhang Zhou
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.StronglyMeasurable.AEStronglyMeasurable
import M... |
/-- The class of `x ↦ (f x, g x)`. -/
def pair (f : α →ₘ[μ] β) (g : α →ₘ[μ] γ) : α →ₘ[μ] β × γ :=
Quotient.liftOn₂' f g (fun f g => mk (fun x => (f.1 x, g.1 x)) (f.2.prodMk g.2))
| Mathlib/MeasureTheory/Function/AEEqFun.lean | 320 | 323 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.InjSurj
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.GroupWithZero.WithZero
impo... |
#adaptation_note /-- 2025-03-29 lean4#7717 Needed to add `dsimp only` -/
-- See note [lower instance priority]
instance (priority := 100) LinearOrderedCommGroupWithZero.toMulPosStrictMono :
| Mathlib/Algebra/Order/GroupWithZero/Canonical.lean | 136 | 139 |
/-
Copyright (c) 2023 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.GroupTheory.CoprodI
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.Complement
/-!
## Pushouts of Monoids and Groups
This file defin... |
theorem base_smul_def (h : H) (w : NormalWord d) :
base φ h • w = { w with head := h * w.head } := by
| Mathlib/GroupTheory/PushoutI.lean | 467 | 469 |
/-
Copyright (c) 2022 Praneeth Kolichala. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Praneeth Kolichala
-/
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Nat.BinaryRec
import Mathlib.Data.List.Defs
import Mathlib.Tactic.Convert
import Mathlib.Tactic.GeneralizePr... | cases bodd m <;> cases bodd n <;> rfl
lemma mod_two_of_bodd (n : ℕ) : n % 2 = (bodd n).toNat := by
have := congr_arg bodd (mod_add_div n 2)
simp? [not] at this says
| Mathlib/Data/Nat/Bits.lean | 74 | 78 |
/-
Copyright (c) 2023 Moritz Firsching. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Firsching
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.LinearAlgebra.Vanderm... | (∏ i : Fin n, Nat.factorial i) *
(Matrix.of (fun (i j : Fin n) => (Nat.choose (v i) (j : ℕ) : ℤ))).det := by
convert Matrix.det_mul_row (fun (i : Fin n) => ((Nat.factorial (i : ℕ)) : ℤ)) _
· rw [Matrix.of_apply, descPochhammer_eval_eq_descFactorial ℤ _ _]
congr
exact Nat.descFactorial_eq_factorial... | Mathlib/Data/Nat/Factorial/SuperFactorial.lean | 104 | 112 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Jeremy Avigad, Simon Hudon
-/
import Mathlib.Algebra.Notation.Defs
import Mathlib.Data.Set.Subsingleton
import Mathlib.Logic.Equiv.Defs
/-!
# Partial values of a type
... | constructor
· rw [Ne, eq_none_iff', not_not]
| Mathlib/Data/Part.lean | 180 | 181 |
/-
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... | map_one (coeAlgHom F)
@[simp, norm_cast]
theorem coe_add : ((f + g : RatFunc F) : F⸨X⸩) = f + g :=
map_add (coeAlgHom F) _ _
@[simp, norm_cast]
| Mathlib/RingTheory/LaurentSeries.lean | 415 | 421 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Nat.SuccPred
import Mathlib.Order.SuccPred.Initial... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,477 | 1,481 | |
/-
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.Ring.Associated
import Mathlib.Algebra.Star.Unitary
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Tactic.Ring
import Mathlib.Al... | rw [smul_val, Zsqrtd.ext_iff]
| Mathlib/NumberTheory/Zsqrtd/Basic.lean | 293 | 293 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Kim Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.ObjectProperty.FullSubcategory
import Mat... | e.unit.app (e.inverse.obj Y) = e.inverse.map (e.counitInv.app Y) := by
simpa using Iso.comp_hom_eq_id (e.inverse.mapIso (e.counitIso.app Y)) (f := e.unit.app _)
theorem unitInv_app_inverse (e : C ≌ D) (Y : D) :
e.unitInv.app (e.inverse.obj Y) = e.inverse.map (e.counit.app Y) := by
| Mathlib/CategoryTheory/Equivalence.lean | 220 | 224 |
/-
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.Data.Finset.Attach
import Mathlib.Data.Finset.Disjoint
import Mathlib.Data.Finset.Erase
import Mat... | Mathlib/Data/Finset/Basic.lean | 2,335 | 2,336 | |
/-
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... | rwa [Finset.exists_mem_insert]
rcases s.eq_empty_or_nonempty with hse | hsne
· subst hse
| Mathlib/RingTheory/Ideal/Operations.lean | 1,093 | 1,095 |
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