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) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.Data.Nat.Factorization.PrimePow
import Mathlib.RingTheory.UniqueFactorizationDomain.Nat
/-!
# Lemmas about squ... | Set.InvOn (fun n : ℕ ↦ (factorization n).support) (fun s ↦ ∏ p ∈ s, p)
{s | ∀ p ∈ s, p.Prime} {n | Squarefree n} :=
⟨fun _s ↦ primeFactors_prod, fun _n ↦ prod_primeFactors_of_squarefree⟩
theorem prod_primeFactors_sdiff_of_squarefree {n : ℕ} (hn : Squarefree n) {t : Finset ℕ}
| Mathlib/Data/Nat/Squarefree.lean | 388 | 392 |
/-
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.Combinatorics.SimpleGraph.Path
import Mathlib.Combinatorics.SimpleGraph.Operations
import Mathlib.Data.Finset.Pairwise
import M... | @[simp]
theorem isNClique_bot_iff : (⊥ : SimpleGraph α).IsNClique n s ↔ n ≤ 1 ∧ #s = n := by
rw [isNClique_iff, isClique_bot_iff]
refine and_congr_left ?_
rintro rfl
| Mathlib/Combinatorics/SimpleGraph/Clique.lean | 227 | 231 |
/-
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.Data.ENNReal.Holder
/-!
# Real conjugate exponents
This file defines Hölder triple and Hölder conjugate exponents in `ℝ` and `... | /-- For `r`, instead of `p` -/
| Mathlib/Data/Real/ConjExponents.lean | 95 | 95 |
/-
Copyright (c) 2024 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.LSeries.HurwitzZetaEven
import Mathlib.NumberTheory.LSeries.HurwitzZetaOdd
import Mathlib.Analysis.SpecialFunctions.Gamma.Beta
/-!
# The... | lemma hasSum_expZeta_of_one_lt_re (a : ℝ) {s : ℂ} (hs : 1 < re s) :
HasSum (fun n : ℕ ↦ cexp (2 * π * I * a * n) / n ^ s) (expZeta a s) := by
convert (hasSum_nat_cosZeta a hs).add ((hasSum_nat_sinZeta a hs).mul_left I) using 1
ext1 n
simp only [mul_right_comm _ I, ← cos_add_sin_I, push_cast]
rw [add_div, mu... | Mathlib/NumberTheory/LSeries/HurwitzZeta.lean | 125 | 130 |
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.LinearOrder
/-!
# Monotone surjective functions are surjective on intervals
A monotone surjecti... | -- to see that the hypothesis `a ≤ b` is necessary, consider a constant function
theorem surjOn_Icc_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)
{a b : α} (hab : a ≤ b) : SurjOn f (Icc a b) (Icc (f a) (f b)) := by
| Mathlib/Order/Interval/Set/SurjOn.lean | 47 | 49 |
/-
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.Group
import Mathlib.NumberTheory.EllipticDivisibilitySequence
/-!
# Division polynomials of Weier... | rw [C_Ψ₂Sq, map_sub, map_mul, AdjoinRoot.mk_self, mul_zero, sub_zero, map_pow]
| Mathlib/AlgebraicGeometry/EllipticCurve/DivisionPolynomial/Basic.lean | 131 | 132 |
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yury Kudryashov, Sébastien Gouëzel, Chris Hughes
-/
import Mathlib.Data.Fin.Rev
import Mathlib.Data.Nat.Find
/-!
# Operation on tuples
We interpret maps `∀ i : Fi... | · split_ifs with hl
· exact Option.isSome_some
· have := (@isSome_find_iff n (fun x ↦ p (x.castLT (Nat.lt_succ_of_lt x.2))) _).2
| Mathlib/Data/Fin/Tuple/Basic.lean | 1,024 | 1,026 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Yaël Dillies
-/
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Int.Order.Basic
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Compare
impor... | exists_strictMono αᵒᵈ
lemma pow_self_mono : Monotone fun n : ℕ ↦ n ^ n := by
refine monotone_nat_of_le_succ fun n ↦ ?_
rw [Nat.pow_succ]
exact (Nat.pow_le_pow_left n.le_succ _).trans (Nat.le_mul_of_pos_right _ n.succ_pos)
| Mathlib/Order/Monotone/Basic.lean | 599 | 605 |
/-
Copyright (c) 2021 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis... | rw [gauge_def', gauge_def', ← Real.sInf_smul_of_nonneg ha]
congr 1
ext r
simp_rw [Set.mem_smul_set, Set.mem_sep_iff]
constructor
· rintro ⟨hr, hx⟩
simp_rw [mem_Ioi] at hr ⊢
rw [← mem_smul_set_iff_inv_smul_mem₀ hr.ne'] at hx
have := smul_pos (inv_pos.2 ha') hr
refine ⟨a⁻¹ • r, ⟨this, ?_⟩, smu... | Mathlib/Analysis/Convex/Gauge.lean | 233 | 245 |
/-
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.Deriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Normed.Operator.BoundedLinear... | rcases mem_nhdsGE_iff_exists_Ico_subset.1 (this (half_pos hε)) with ⟨m, xm, hm⟩
refine ⟨m - x, by linarith [show x < m from xm], fun r hr => ?_⟩
have : r ∈ Ioc (r / 2) r := ⟨half_lt_self hr.1, le_rfl⟩
refine ⟨r, this, fun y hy z hz => ?_⟩
calc
‖f z - f y - (z - y) • derivWithin f (Ici x) x‖ =
‖f z... | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 487 | 493 |
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.LineDeriv.Measurable
import Mathlib.Analysis.Normed.Module.FiniteDimension
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar... | have A := (Basis.ofVectorSpace ℝ F).equivFun.toContinuousLinearEquiv
suffices H : ∀ᵐ x ∂μ, x ∈ s → DifferentiableWithinAt ℝ (A ∘ f) s x by
filter_upwards [H] with x hx xs
have : f = (A.symm ∘ A) ∘ f := by
simp only [ContinuousLinearEquiv.symm_comp_self, Function.id_comp]
| Mathlib/Analysis/Calculus/Rademacher.lean | 368 | 372 |
/-
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
-/
import Mathlib.Algebra.Order.SuccPred
import Mathlib.Data.Sum.Order
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.PPWithUniv
/-!
# ... | ⟨0⟩
instance one : One Ordinal :=
| Mathlib/SetTheory/Ordinal/Basic.lean | 137 | 139 |
/-
Copyright (c) 2020 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Lacker, Bryan Gin-ge Chen
-/
import Mathlib.Data.Nat.Prime.Basic
import Mathlib.Algebra.Group.Int.Defs
import Mathlib.Data.Int.Basic
/-!
# Lemmas about `Nat.Prime` using `Int... | {k l : ℕ} (hpm : ↑(p ^ k) ∣ m) (hpn : ↑(p ^ l) ∣ n) (hpmn : ↑(p ^ (k + l + 1)) ∣ m * n) :
↑(p ^ (k + 1)) ∣ m ∨ ↑(p ^ (l + 1)) ∣ n :=
have hpm' : p ^ k ∣ m.natAbs := Int.natCast_dvd_natCast.1 <| Int.dvd_natAbs.2 hpm
have hpn' : p ^ l ∣ n.natAbs := Int.natCast_dvd_natCast.1 <| Int.dvd_natAbs.2 hpn
have hpmn... | Mathlib/Data/Int/NatPrime.lean | 24 | 33 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro
-/
import Mathlib.Algebra.Module.Submodule.Bilinear
import Mathlib.Algebra.Module.Equiv.Basic
import Mathlib.GroupTheory.Congruence.Hom
import Mathlib.Tactic.Abel
... |
variable (f) in
/-- Constructing a linear map `M ⊗ N → P` given a bilinear map `M → N → P` with the property that
its composition with the canonical bilinear map `M → N → M ⊗ N` is
the given bilinear map `M → N → P`. -/
def lift : M ⊗[R] N →ₗ[R] P :=
{ liftAux f with map_smul' := liftAux.smul }
@[simp]
theorem lift... | Mathlib/LinearAlgebra/TensorProduct/Basic.lean | 503 | 514 |
/-
Copyright (c) 2021 Julian Kuelshammer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Kuelshammer
-/
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Data.Nat.Factorization.B... | exact Set.mem_range_self g
@[to_additive]
theorem exponent_eq_zero_iff_range_orderOf_infinite (h : ∀ g : G, 0 < orderOf g) :
exponent G = 0 ↔ (Set.range (orderOf : G → ℕ)).Infinite := by
have := exponent_ne_zero_iff_range_orderOf_finite h
rwa [Ne, not_iff_comm, Iff.comm] at this
@[to_additive]
theorem lcm... | Mathlib/GroupTheory/Exponent.lean | 323 | 337 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.MeasureTheory.Measure.Typeclasses.Probability
import... | lemma measure_Iic_of_tendsto_atBot_atBot (hf : Tendsto f atBot atBot) (x : ℝ) :
f.measure (Iic x) = ∞ := by
refine ENNReal.eq_top_of_forall_nnreal_le fun r ↦ ?_
obtain ⟨N, hN⟩ := eventually_atBot.mp (tendsto_atBot.mp hf (f x - r))
exact (f.measure_Ioc (min x N) x ▸ ENNReal.coe_nnreal_eq r ▸ (ENNReal.ofReal_le... | Mathlib/MeasureTheory/Measure/Stieltjes.lean | 472 | 481 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.BigOperators.Group.Multiset.Defs
import Mathlib.Algebra.Order.BigOperators.Group.List
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Alg... | lemma prod_min_le [CommMonoid α] [LinearOrder α] [IsOrderedMonoid α]
{s : Multiset ι} {f g : ι → α} :
| Mathlib/Algebra/Order/BigOperators/Group/Multiset.lean | 168 | 169 |
/-
Copyright (c) 2023 Scott Carnahan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Carnahan
-/
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Eval.SMul
/-!
# Scalar-multiple polynomial ev... | theorem eval₂_smulOneHom_eq_smeval (R : Type*) [Semiring R] {S : Type*} [Semiring S] [Module R S]
[IsScalarTower R S S] (p : R[X]) (x : S) :
p.eval₂ RingHom.smulOneHom x = p.smeval x := by
rw [smeval_eq_sum, eval₂_eq_sum]
congr 1 with e a
simp only [RingHom.smulOneHom_apply, smul_one_mul, smul_pow]
| Mathlib/Algebra/Polynomial/Smeval.lean | 69 | 74 |
/-
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.AlgebraicGeometry.Spec
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.CategoryTheory.Elementwise
/-!
# The category of schemes
A schem... | def fullyFaithfulForgetToLocallyRingedSpace :
forgetToLocallyRingedSpace.FullyFaithful where
preimage := Hom.mk
| Mathlib/AlgebraicGeometry/Scheme.lean | 205 | 207 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic... | continuous_iff_continuousAt.2 fun x => h₁.continuousAt.clog (h₂ x)
end LogDeriv
| Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 250 | 257 |
/-
Copyright (c) 2022 Kevin H. Wilson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin H. Wilson
-/
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Order.Filter.Curry
/-!
# Swapping limits and derivatives via u... | (hfg : ∀ x : 𝕜, x ∈ s → Tendsto (fun n => f n x) l (𝓝 (g x))) (hx : x ∈ s) :
HasDerivAt g (g' x) x :=
hasDerivAt_of_tendstoLocallyUniformlyOn hs hf'.tendstoLocallyUniformlyOn hf hfg hx
theorem hasDerivAt_of_tendstoUniformly [NeBot l] (hf' : TendstoUniformly f' g' l)
(hf : ∀ᶠ n in l, ∀ x : 𝕜, HasDerivA... | Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean | 539 | 544 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Function.L1Space.Integrable
import Mathlib.MeasureTheory.Function.LpSpace.Indicator
/-! # Functions integrable on a set an... | theorem integrableOn_Ioc_iff_integrableOn_Ioo :
IntegrableOn f (Ioc a b) μ ↔ IntegrableOn f (Ioo a b) μ :=
integrableOn_Ioc_iff_integrableOn_Ioo' (by rw [measure_singleton]; exact ENNReal.zero_ne_top)
theorem integrableOn_Icc_iff_integrableOn_Ioo :
IntegrableOn f (Icc a b) μ ↔ IntegrableOn f (Ioo a b) μ := b... | Mathlib/MeasureTheory/Integral/IntegrableOn.lean | 712 | 719 |
/-
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.Interval.Set.OrdConnected
import Mathlib.Data.Set.Lattice.Image
/-!
# Order connected components of a set
In this file we define `Set.ordConn... | · exact ha (Icc_subset_uIcc ⟨hab, hbx⟩) hbt
have h' : x ∈ ordSeparatingSet s t := ⟨mem_iUnion₂.2 ⟨a, has, ha⟩, mem_iUnion₂.2 ⟨b, hbt, hb⟩⟩
lift x to ordSeparatingSet s t using h'
suffices ordConnectedComponent (ordSeparatingSet s t) x ⊆ [[a, b]] from
| Mathlib/Order/Interval/Set/OrdConnectedComponent.lean | 188 | 191 |
/-
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... | { Equiv.setCongr <| congr_arg _ h with
map_mul' := fun _ _ => rfl
map_add' := fun _ _ => rfl }
/-- Restrict a ring homomorphism with a left inverse to a ring isomorphism to its
`RingHom.range`. -/
def ofLeftInverse {g : S → R} {f : R →+* S} (h : Function.LeftInverse g f) : R ≃+* f.range :=
{ f.rangeRestric... | Mathlib/Algebra/Ring/Subring/Basic.lean | 883 | 897 |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Finite.Prod
import Mathlib.Data.Matroid.Init
import Mathlib.Data.Set.Card
import Mathlib.Data.Set.Finite.Powerset
import Mathlib.Order.UpperLower.Clos... | (h : ∀ J, J ⊆ I → J.Finite → M.Indep J) : M.Indep I :=
Finitary.indep_of_forall_finite I h
theorem indep_iff_forall_finite_subset_indep {M : Matroid α} [Finitary M] :
| Mathlib/Data/Matroid/Basic.lean | 746 | 749 |
/-
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
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Algebra.Order.Star.Basic
import Mathlib.Data.C... | Mathlib/Data/Complex/Exponential.lean | 992 | 992 | |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Kim Morrison, Jakob von Raumer
-/
import Mathlib.Algebra.Category.ModuleCat.Basic
import Mathlib.LinearAlgebra.TensorProduct.Associator
import Mathlib.CategoryTheory.Monoi... |
theorem tensor_comp {X₁ Y₁ Z₁ X₂ Y₂ Z₂ : ModuleCat R} (f₁ : X₁ ⟶ Y₁) (f₂ : X₂ ⟶ Y₂) (g₁ : Y₁ ⟶ Z₁)
(g₂ : Y₂ ⟶ Z₂) : tensorHom (f₁ ≫ g₁) (f₂ ≫ g₂) = tensorHom f₁ f₂ ≫ tensorHom g₁ g₂ := by
ext : 1
| Mathlib/Algebra/Category/ModuleCat/Monoidal/Basic.lean | 74 | 77 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.SumOverResidueClass
/-!
# Conve... | convert summable_schlomilch_iff_of_nonneg h_nonneg h_mono (pow_pos zero_lt_two)
(pow_right_strictMono₀ one_lt_two) two_ne_zero h_succ_diff
simp [pow_succ, mul_two, two_mul]
section p_series
/-!
### Convergence of the `p`-series
| Mathlib/Analysis/PSeries.lean | 238 | 246 |
/-
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.Tape
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.PFun
import M... | Mathlib/Computability/TuringMachine.lean | 1,395 | 1,407 | |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.ConditionalProbability
import Mathlib.Probability.Kernel.Basic
import Mathlib.Probability.Kernel.Composition.MeasureComp
import Mathlib.Tactic.... | -/
variable {_mα : MeasurableSpace α}
theorem iIndepSet_iff_iIndepSets_singleton {_mΩ : MeasurableSpace Ω} {κ : Kernel α Ω}
{μ : Measure α} {f : ι → Set Ω} (hf : ∀ i, MeasurableSet (f i)) :
iIndepSet f κ μ ↔ iIndepSets (fun i ↦ {f i}) κ μ :=
⟨iIndep.iIndepSets fun _ ↦ rfl,
| Mathlib/Probability/Independence/Kernel.lean | 780 | 787 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Shapes.Reflexive
import Mathlib.CategoryTheory.Monad.Coequalizer
imp... | /--
If `G` is monadic, it creates colimits of `G`-split pairs. This is the "boring" direction of Beck's
monadicity theorem, the converse is given in `monadicOfCreatesGSplitCoequalizers`.
-/
def createsGSplitCoequalizersOfMonadic [MonadicRightAdjoint G] ⦃A B⦄ (f g : A ⟶ B)
[G.IsSplitPair f g] : CreatesColimit (paral... | Mathlib/CategoryTheory/Monad/Monadicity.lean | 230 | 237 |
/-
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.Data.Set.SymmDiff
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Irreducible
/-!
# Connected subsets ... | show ((↑) : G → α) ∘ inclusion h = (↑) from rfl, image_comp]
exact image_subset _ ((continuous_inclusion h).image_connectedComponent_subset ⟨x, hx⟩)
· rw [connectedComponentIn_eq_empty hx]
exact Set.empty_subset _
| Mathlib/Topology/Connected/Basic.lean | 605 | 608 |
/-
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.... | · simpa only [mul_comm, add_comm, and_comm,
isCoprime_comm] using this g f h.symm (h.resolve_left hf)
rw [hf, zero_mul, natSepDegree_zero, zero_add, isCoprime_zero_left, isUnit_iff, eq_comm,
natSepDegree_eq_zero_iff, natDegree_eq_zero]
refine ⟨fun ⟨x, h⟩ ↦ ?_, ?_⟩
· by_cases hx : x = 0
... | Mathlib/FieldTheory/SeparableDegree.lean | 433 | 440 |
/-
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.MeasureTheory.Measure.FiniteMeasure
import Mathlib.MeasureTheory.Integral.Average
import Mathlib.MeasureTheory.Measure.Prod
/-!
# Probability measures
Th... | ProbabilityMeasure.coeFn_comp_toFiniteMeasure_eq_coeFn]
theorem normalize_eq_of_nonzero (nonzero : μ ≠ 0) (s : Set Ω) : μ.normalize s = μ.mass⁻¹ * μ s := by
simp only [μ.self_eq_mass_mul_normalize, μ.mass_nonzero_iff.mpr nonzero, inv_mul_cancel_left₀,
Ne, not_false_iff]
| Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean | 400 | 405 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
/-!
# (Generalized) Boolean algebras
A Boolean algebra is a bounded distributive lattice with a complement ope... | lt_iff_lt_of_le_iff_le' compl_le_compl_iff_le compl_le_compl_iff_le
theorem compl_le_of_compl_le (h : yᶜ ≤ x) : xᶜ ≤ y := by
simpa only [compl_compl] using compl_le_compl h
| Mathlib/Order/BooleanAlgebra.lean | 640 | 643 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Alena Gusakov, Yaël Dillies
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Finset.Slice
import Mathlib.Data.Nat.BitIndices
import Mathlib.Order.SupClosed
import Math... | @[simp] lemma toColex_sdiff_le_toColex_sdiff' :
toColex (s \ t) ≤ toColex (t \ s) ↔ toColex s ≤ toColex t := by
simpa using toColex_sdiff_le_toColex_sdiff (inter_subset_left (s₁ := s)) inter_subset_right
| Mathlib/Combinatorics/Colex.lean | 243 | 245 |
/-
Copyright (c) 2016 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Set.Defs
import Mathlib.Logic.Basic
import Mathlib.Logic.ExistsUnique
import Mathlib.Logic.Nonempty
import Mathlib.Logic.Nontrivia... | /-- Composition by a surjective function on the left is itself surjective. -/
theorem Surjective.comp_left {g : β → γ} (hg : Surjective g) :
Surjective (g ∘ · : (α → β) → α → γ) :=
.piMap fun _ ↦ hg
theorem surjective_comp_left_iff [Nonempty α] {g : β → γ} :
Surjective (g ∘ · : (α → β) → α → γ) ↔ Surjective ... | Mathlib/Logic/Function/Basic.lean | 453 | 459 |
/-
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.Asymptotics
/-!
# Continu... | exact IsOpen.eventually_mem isOpen_ne ha
theorem cpow_eq_nhds' {p : ℂ × ℂ} (hp_fst : p.fst ≠ 0) :
(fun x => x.1 ^ x.2) =ᶠ[𝓝 p] fun x => exp (log x.1 * x.2) := by
suffices ∀ᶠ x : ℂ × ℂ in 𝓝 p, x.1 ≠ 0 from
this.mono fun x hx ↦ by
dsimp only
| Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean | 44 | 50 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau
-/
import Mathlib.Data.DFinsupp.Submonoid
import Mathlib.Data.Finsupp.ToDFinsupp
import Mathlib.LinearAlgebra.Finsupp.SumProd
import Mathlib.LinearAlgebra.LinearIn... |
namespace Submodule
variable [Semiring R] [AddCommMonoid N] [Module R N]
open DFinsupp
section DecidableEq
| Mathlib/LinearAlgebra/DFinsupp.lean | 307 | 315 |
/-
Copyright (c) 2021 Rémy Degenne. 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.FinMeasAdditive
/-!
# Extension of a linear function from indicators to L1
Give... | SimpleFunc.setToSimpleFunc_congr T h_zero h_add (SimpleFunc.integrable f) h
theorem setToL1S_congr_left (T T' : Set α → E →L[ℝ] F)
(h : ∀ s, MeasurableSet s → μ s < ∞ → T s = T' s) (f : α →₁ₛ[μ] E) :
setToL1S T f = setToL1S T' f :=
SimpleFunc.setToSimpleFunc_congr_left T T' h (simpleFunc.toSimpleFunc f) (S... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 122 | 127 |
/-
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.ContDiff.Operations
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Topology.Algebra.InfiniteSum.Module
impo... | haveI A :
∀ᶠ i in (Filter.cofinite : Filter α),
∀ k : ℕ, k ∈ Finset.range (m + 1) → ∀ x : E, ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i := by
rw [eventually_all_finset]
intro i hi
apply h'f
simp only [Finset.mem_range_succ_iff] at hi
exact (WithTop.coe_le_coe.2... | Mathlib/Analysis/Calculus/SmoothSeries.lean | 257 | 294 |
/-
Copyright (c) 2021 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Log.NegMulLog
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv... | @[simp]
theorem intervalIntegrable_sin : IntervalIntegrable sin μ a b :=
continuous_sin.intervalIntegrable a b
@[simp]
| Mathlib/Analysis/SpecialFunctions/Integrals.lean | 261 | 265 |
/-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.DeleteEdges
import Mathlib.Data.Fintype.Powerset
/-!
# Subgraphs of a simple graph
A subgraph of ... | simp [funext_iff]
· classical
exact ⟨⟨(H.verts.toFinset, fun a b ↦ H.Adj a b), fun a b ↦ by simpa using H.adj_sub,
fun a b ↦ by simpa using H.edge_vert, by simp [H.adj_comm]⟩, by simp⟩
instance [Finite V] : Finite G.Subgraph := by classical cases nonempty_fintype V; infer_instance
/-- Given two subgra... | Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 696 | 708 |
/-
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 | 1,931 | 1,932 | |
/-
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.Order.Filter.Prod
/-!
# N-ary maps of filter
This file defines the binary and ternary maps of filters. This is mostly useful to define pointwise
operatio... | map₂ m f (g.map n) = (map₂ m' g f).map n' :=
(map_map₂_antidistrib_right fun a b => (h_right_anticomm b a).symm).symm
/-- If `a` is a left identity for `f : α → β → β`, then `pure a` is a left identity for
`Filter.map₂ f`. -/
theorem map₂_left_identity {f : α → β → β} {a : α} (h : ∀ b, f a b = b) (l : Filter β) ... | Mathlib/Order/Filter/NAry.lean | 261 | 268 |
/-
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... | def equivFunOnFinite [Finite α] : (α →₀ M) ≃ (α → M) where
toFun := (⇑)
invFun f := mk (Function.support f).toFinite.toFinset f fun _a => Set.Finite.mem_toFinset _
left_inv _f := ext fun _x => rfl
right_inv _f := rfl
@[simp]
theorem equivFunOnFinite_symm_coe {α} [Finite α] (f : α →₀ M) : equivFunOnFinite.symm ... | Mathlib/Data/Finsupp/Defs.lean | 188 | 195 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Data.Nat.Choose.Sum
impo... | coeff_mul_monomial p 0 d r
-- This can already be proved by `simp`.
| Mathlib/Algebra/Polynomial/Coeff.lean | 270 | 272 |
/-
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.Data.Set.Image
import Mathlib.Data.List.Defs
/-!
# Lemmas about `List`s and `Set.range`
In this file we prove lemmas about range of some operations... | rw [range_list_map, Subtype.range_coe]
| Mathlib/Data/Set/List.lean | 33 | 34 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.FractionalIdeal.Basic
import Mathlib.RingTheory.IntegralClosure.IsIntegral.Basi... | (haJ : I = spanSingleton R⁰ ((algebraMap R K) a)⁻¹ * ↑J) :
(Ideal.span {a} : Ideal R) ≠ 0 := fun h ↦ by
rw [Ideal.zero_eq_bot, Ideal.span_singleton_eq_bot] at h
rw [h, RingHom.map_zero, inv_zero, spanSingleton_zero, MulZeroClass.zero_mul] at haJ
exact hI haJ
instance isPrincipal {R} [CommRing R] [IsDomai... | Mathlib/RingTheory/FractionalIdeal/Operations.lean | 775 | 790 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Operations
import Mathlib.Order.Basic
import Mathlib.Order.BooleanAlgebra
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.B... | Mathlib/Data/Set/Basic.lean | 2,081 | 2,082 | |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Order.Ring.Unbundled.Rat
/-!
# The rational numbers form a linear ordered field
This f... | Mathlib/Algebra/Order/Ring/Rat.lean | 47 | 48 | |
/-
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... | @[reassoc]
theorem leftUnitor_tensor_inv (X Y : C) :
(λ_ (X ⊗ Y)).inv = (λ_ X).inv ▷ Y ≫ (α_ (𝟙_ C) X Y).hom := by simp
@[reassoc]
| Mathlib/CategoryTheory/Monoidal/Category.lean | 601 | 605 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Julian Kuelshammer
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.Group.Pointwise.Set.Finite
import Mathlib.Alge... | @[to_additive]
lemma Subgroup.orderOf_le_card {G : Type*} [Group G] (s : Subgroup G) (hs : (s : Set G).Finite)
{x} (hx : x ∈ s) : orderOf x ≤ Nat.card s :=
le_of_dvd (Nat.card_pos_iff.2 <| ⟨s.coe_nonempty.to_subtype, hs.to_subtype⟩) <|
s.orderOf_dvd_natCard hx
| Mathlib/GroupTheory/OrderOfElement.lean | 899 | 903 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.FieldTheory.Finite.Polynomial
import Mathlib.NumberTheory.Basic
import Mathlib.RingTheory.WittVector.WittPolynomial
/-!
# Witt struct... | simpa only [map_bind₁, ← eval₂Hom_map_hom, eval₂Hom_C_left, map_rename, map_wittPolynomial,
AlgHom.coe_toRingHom] using h
· intro n; apply wittStructureRat_prop
theorem wittStructureInt_existsUnique (Φ : MvPolynomial idx ℤ) :
∃! φ : ℕ → MvPolynomial (idx × ℕ) ℤ,
∀ n : ℕ,
bind₁ φ (wittPoly... | Mathlib/RingTheory/WittVector/StructurePolynomial.lean | 311 | 326 |
/-
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.Group.Submonoid.Operations
import Mathlib.Algebra.MonoidAlgebra.Defs
import Mathlib.Algebra.Order.Mon... |
@[simp]
theorem toFinsupp_eq_zero {a : R[X]} : a.toFinsupp = 0 ↔ a = 0 := by
| Mathlib/Algebra/Polynomial/Basic.lean | 246 | 248 |
/-
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... | prod (join S) = prod (map prod S) := by
induction S using Multiset.induction with
| Mathlib/Data/Multiset/Bind.lean | 77 | 78 |
/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Satisfiability
import Mathlib.Combinatorics.SimpleGraph.Basic
/-!
# First-Order Structures in Graph Theory
This file defines first-order ... | | 2, .adj =>
rw [iff_eq_eq]
change RelMap adj ![xs 0, xs 1] = _
refine congr rfl (funext ?_)
| Mathlib/ModelTheory/Graph.lean | 107 | 110 |
/-
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... |
end PartialHomeomorph
| Mathlib/Topology/PartialHomeomorph.lean | 1,449 | 1,451 |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Calculus.FDe... | HasStrictFDerivAt (fun x ↦ (u.map (g · x)).prod)
(u.map fun i ↦ ((u.erase i).map (g · x)).prod • g' i).sum x := by
simp only [← Multiset.attach_map_val u, Multiset.map_map]
exact .congr_fderiv
(hasStrictFDerivAt_multiset_prod.comp x <|
hasStrictFDerivAt_pi.mpr fun i ↦ h (Subtype.val i) i.prop :)... | Mathlib/Analysis/Calculus/FDeriv/Mul.lean | 720 | 729 |
/-
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... | · exact hδ₁ ⟨n, hn⟩ s hs hμs
· exact (eLpNorm_indicator_le _).trans (hN n (not_lt.1 hn))
/-- Convergence in Lp implies uniform integrability. -/
theorem unifIntegrable_of_tendsto_Lp (hp : 1 ≤ p) (hp' : p ≠ ∞) (hf : ∀ n, MemLp (f n) p μ)
(hg : MemLp g p μ) (hfg : Tendsto (fun n => eLpNorm (f n - g) p μ) atTop (... | Mathlib/MeasureTheory/Function/UniformIntegrable.lean | 577 | 588 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive
impo... | Mathlib/Data/Option/Basic.lean | 476 | 479 | |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Operations
import Mathlib.Order.Basic
import Mathlib.Order.BooleanAlgebra
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.B... | Mathlib/Data/Set/Basic.lean | 2,093 | 2,098 | |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Lu-Ming Zhang
-/
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity.WalkCounting
import Math... | (M * G.adjMatrix α) v w = ∑ u ∈ G.neighborFinset w, M v u := by
simp [mul_apply, neighborFinset_eq_filter, sum_filter, adj_comm]
variable (α) in
@[simp]
| Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 216 | 220 |
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Calculus.FDe... | theorem fderiv_inverse (x : Rˣ) : fderiv 𝕜 (@Ring.inverse R _) x = -mulLeftRight 𝕜 R ↑x⁻¹ ↑x⁻¹ :=
(hasFDerivAt_ringInverse x).fderiv
theorem hasStrictFDerivAt_ringInverse (x : Rˣ) :
HasStrictFDerivAt Ring.inverse (-mulLeftRight 𝕜 R ↑x⁻¹ ↑x⁻¹) x := by
convert (analyticAt_inverse (𝕜 := 𝕜) x).hasStrictFDeriv... | Mathlib/Analysis/Calculus/FDeriv/Mul.lean | 834 | 840 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kim Morrison
-/
import Mathlib.Analysis.Convex.Combination
import Mathlib.LinearAlgebra.AffineSpace.Independent
import Mathlib.Tactic.FieldSimp
/-!
# Carathéodory's conv... |
theorem minCardFinsetOfMemConvexHull_card_le_card {t : Finset E} (ht₁ : ↑t ⊆ s)
(ht₂ : x ∈ convexHull 𝕜 (t : Set E)) : #(minCardFinsetOfMemConvexHull hx) ≤ #t :=
| Mathlib/Analysis/Convex/Caratheodory.lean | 119 | 121 |
/-
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.Algebra.GroupWithZero.Indicator
import Mathlib.Topology.Piecewise
import Mathlib.Topology.Instances.ENNReal.Lemmas
/-!
# Semicontinuous maps
A ... | theorem lowerSemicontinuousOn_biSup {p : ι → Prop} {f : ∀ i, p i → α → δ}
(h : ∀ i hi, LowerSemicontinuousOn (f i hi) s) :
LowerSemicontinuousOn (fun x' => ⨆ (i) (hi), f i hi x') s :=
lowerSemicontinuousOn_iSup fun i => lowerSemicontinuousOn_iSup fun hi => h i hi
| Mathlib/Topology/Semicontinuous.lean | 616 | 620 |
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | · exact tan_add_pi _
theorem tan_eq_of_two_zsmul_eq {θ ψ : Angle} (h : (2 : ℤ) • θ = (2 : ℤ) • ψ) : tan θ = tan ψ := by
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 676 | 678 |
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yury Kudryashov
-/
import Mathlib.Data.Set.Lattice.Image
import Mathlib.Order.Interval.Set.LinearOrder
/-!
# Extra lemmas about intervals
This file contains lemma... | theorem iUnion_Ioi [NoMinOrder α] : ⋃ a : α, Ioi a = univ :=
iUnion_eq_univ_iff.2 exists_lt
| Mathlib/Order/Interval/Set/Disjoint.lean | 102 | 103 |
/-
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... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 80 | 80 | |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.Set.BooleanAlgebra
/-!
# Sets in sigma types
This file defines `Set.sigma`, the indexed sum of sets.
-/
namespace Set... | Mathlib/Data/Set/Sigma.lean | 245 | 248 | |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.InitTail
/-!
# Truncated Witt vectors
The ring of truncated Witt vectors (of length `n`) is a quotient of the... | theorem truncateFun_out (x : TruncatedWittVector p n R) : x.out.truncateFun n = x := by
simp only [WittVector.truncateFun, coeff_out, mk_coeff]
| Mathlib/RingTheory/WittVector/Truncated.lean | 145 | 146 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryTheor... | theorem zero_of_target_iso_zero {X Y : C} (f : X ⟶ Y) (i : Y ≅ 0) : f = 0 := by
| Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | 332 | 332 |
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Eric Wieser
-/
import Mathlib.Data.Real.Basic
import Mathlib.Tactic.NormNum.Inv
/-!
# Real sign function
This file introduces and contains some results about `Real.sign` wh... | Mathlib/Data/Real/Sign.lean | 120 | 124 | |
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Basis
/-!
# Convex combinations
... | simp only [Finset.centerMass, hw, inv_one, one_smul]
theorem Finset.centerMass_smul : (t.centerMass w fun i => c • z i) = c • t.centerMass w z := by
simp only [Finset.centerMass, Finset.smul_sum, (mul_smul _ _ _).symm, mul_comm c, mul_assoc]
| Mathlib/Analysis/Convex/Combination.lean | 77 | 80 |
/-
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 | 306 | 309 | |
/-
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.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.Symmetric
/-!
# Integer powers of square matrices
In this file, we defi... | | hz => simp
| hp z =>
rw [← Int.natCast_succ, zpow_natCast, det_pow, isUnit_pow_succ_iff, ← Int.ofNat_zero,
Int.ofNat_inj]
| Mathlib/LinearAlgebra/Matrix/ZPow.lean | 112 | 115 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Nat.Totient
import Mathlib.Data.ZMod.Aut
import Mathlib.Data.ZMod.QuotientGroup
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.Sub... |
@[to_additive]
theorem IsCyclic.iff_exponent_eq_card [CommGroup α] [Finite α] :
IsCyclic α ↔ exponent α = Nat.card α :=
⟨fun _ => IsCyclic.exponent_eq_card, IsCyclic.of_exponent_eq_card⟩
@[to_additive]
theorem IsCyclic.exponent_eq_zero_of_infinite [Group α] [IsCyclic α] [Infinite α] :
| Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | 665 | 672 |
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Sébastien Gouëzel, Yury Kudryashov, Yuyang Zhao
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calcu... | with `h₂` taking values in a vector space, use `HasDerivAt.scomp_of_eq`. -/
theorem HasDerivAt.comp_of_eq
(hh₂ : HasDerivAt h₂ h₂' y) (hh : HasDerivAt h h' x) (hy : y = h x) :
HasDerivAt (h₂ ∘ h) (h₂' * h') x := by
| Mathlib/Analysis/Calculus/Deriv/Comp.lean | 244 | 247 |
/-
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... |
variable (H)
| Mathlib/Algebra/Group/Subgroup/Basic.lean | 334 | 336 |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Topology.Category.Profinite.Nobeling.Basic
import Mathlib.Topology.Category.Profinite.Nobeling.Induction
import Mathlib.Topology.Category.Profinite... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 654 | 662 | |
/-
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.BigOperators.Finprod
import Mathlib.Order.Filter.AtTopBot.BigOperators
import Mathlib.Topology.Separation.Hausdorff
/-!
# Infinite sum and pro... | @[to_additive]
theorem Multipliable.hasProd_iff (h : Multipliable f) : HasProd f a ↔ ∏' b, f b = a :=
| Mathlib/Topology/Algebra/InfiniteSum/Defs.lean | 174 | 175 |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Floris van Doorn
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Order.Filter.AtTopBot.Basic
import Mathlib.Order.Filter.Subsingleton
/-!
# Functions that ar... |
@[to_additive (attr := simp)]
theorem mulIndicator_const_iff :
EventuallyConst (s.mulIndicator fun _ ↦ c) l ↔ c = 1 ∨ EventuallyConst s l := by
| Mathlib/Order/Filter/EventuallyConst.lean | 151 | 154 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.RingTheory.Valuation.Basic
/-!
# Ring of integers under a given valuation
The elements with valuation less than or equal to 1.
TODO: Define characteristic pre... | rw [← mul_one (v (algebraMap O R x)), hz, RingHom.map_mul, v.map_mul]
exact mul_le_mul_left' (hv.2 z) _
end Integers
| Mathlib/RingTheory/Valuation/Integers.lean | 96 | 99 |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.Algebra.Field.Equiv
import Mathlib.Algebra.Field.Subfield.Basic
import Mathlib.Algebra.Order.Ring... | (hf : ∀ x : A, f1 (algebraMap A K x) = f2 (algebraMap A K x)) : f1 = f2 := by
ext z
obtain ⟨x, y, hy, rfl⟩ := IsFractionRing.div_surjective (A := A) z
rw [map_div₀, map_div₀, hf, hf]
theorem injective_comp_algebraMap :
Function.Injective fun (f : K →+* L) => f.comp (algebraMap A K) :=
fun _ _ h => ring... | Mathlib/RingTheory/Localization/FractionRing.lean | 249 | 265 |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Field.Rat
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.GroupWithZero.Un... | rw [div_def', cast_divInt_of_ne_zero, cast_def, cast_def, div_eq_mul_inv (_ / _), inv_div,
(Int.commute_cast _ _).div_mul_div_comm (Nat.commute_cast _ _)]
| Mathlib/Data/Rat/Cast/Defs.lean | 193 | 194 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Basic
/-!
# Maps between real and extended non-negative real numbers
This file focuses on the functions `ENNReal.toReal... | lift b to ℝ≥0 using hb
simpa [ENNReal.ofReal, ENNReal.toReal] using Real.toNNReal_lt_iff_lt_coe ha
| Mathlib/Data/ENNReal/Real.lean | 253 | 255 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Data.Subtype
import Mathlib.Order.Defs.LinearOrder
import Mathlib.Order.Notation
import Mathlib.Tactic.GCongr.Core
import Mathlib.Tactic.... |
end Sum
section Pi
/-- A function `a` is strongly less than a function `b` if `a i < b i` for all `i`. -/
| Mathlib/Order/Basic.lean | 777 | 782 |
/-
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 Injective.subsingleton_image_iff (hf : Injective f) {s : Set α} :
(f '' s).Subsingleton ↔ s.Subsingleton :=
⟨subsingleton_of_image hf s, fun h => h.image f⟩
| Mathlib/Data/Set/Image.lean | 1,083 | 1,087 |
/-
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 | 1,633 | 1,634 | |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.InnerProductSpace.Sy... | theorem orthogonalProjection_map_apply {E E' : Type*} [NormedAddCommGroup E]
[NormedAddCommGroup E'] [InnerProductSpace 𝕜 E] [InnerProductSpace 𝕜 E'] (f : E ≃ₗᵢ[𝕜] E')
| Mathlib/Analysis/InnerProductSpace/Projection.lean | 571 | 572 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
/-!
# Oriented an... |
/-- If the sign of the oriented angle at `p` between two points is zero, either one of the points
equals `p` or the unoriented angle is 0 or π. -/
theorem eq_zero_or_angle_eq_zero_or_pi_of_sign_oangle_eq_zero {p p₁ p₂ : P}
(h : (∡ p₁ p p₂).sign = 0) : p₁ = p ∨ p₂ = p ∨ ∠ p₁ p p₂ = 0 ∨ ∠ p₁ p p₂ = π := by
| Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 316 | 320 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark
-/
import Mathlib.Algebra.Polynomial.Monic
/-!
# Lemmas for the interaction between polynomials and `∑` and `∏`.
Recall that `∑` and `∏` are notation for ... | assumption
· simp
theorem degree_multiset_prod_of_monic [Nontrivial R] (h : ∀ f ∈ t, Monic f) :
t.prod.degree = (t.map degree).sum := by
have : t.prod ≠ 0 := Monic.ne_zero <| by simpa using monic_multiset_prod_of_monic _ _ h
rw [degree_eq_natDegree this, natDegree_multiset_prod_of_monic _ h, Nat.cast_mul... | Mathlib/Algebra/Polynomial/BigOperators.lean | 198 | 211 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Finset.Card
import Mathlib.Data.Finset.Lattice.Union
import Mathlib.Data.Multiset.Powerset
import Mathlib.Data.Set.Pairwise.Lattice
/-!
# The pow... | · rintro ⟨a, ⟨has, rfl⟩, rfl⟩
simp only [map_subset_map, has, card_map, and_self]
end powersetCard
end Finset
| Mathlib/Data/Finset/Powerset.lean | 313 | 326 |
/-
Copyright (c) 2020 Zhangir Azerbayev. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Zhangir Azerbayev
-/
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.LinearAlgebra.LinearIndependent.Defs
import Mathlib.LinearAlgebra.Multilinear.Basis
/-!
# Alte... | toFun f := f.curryLeft
map_add' := curryLeft_add
map_smul' := curryLeft_smul
/-- Currying with the same element twice gives the zero map. -/
@[simp]
theorem curryLeft_same {n : ℕ} (f : M'' [⋀^Fin n.succ.succ]→ₗ[R'] N'') (m : M'') :
| Mathlib/LinearAlgebra/Alternating/Basic.lean | 924 | 930 |
/-
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.DirectSum.Basic
import Mathlib.LinearAlgebra.DFinsupp
import Mathlib.LinearAlgebra.Basis.Defs
/-!
# Direct sum of modules
The first part of the file pr... |
/-- `curry` as a linear map. -/
def sigmaLcurry : (⨁ i : Σ_, _, δ i.1 i.2) →ₗ[R] ⨁ (i) (j), δ i j :=
{ sigmaCurry with map_smul' := fun r ↦ by convert DFinsupp.sigmaCurry_smul (δ := δ) r }
| Mathlib/Algebra/DirectSum/Module.lean | 326 | 329 |
/-
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.Analysis.Convex.Topology
import Mathlib.Topology.Connected.LocPathConnected
/-!
# Locally convex topological modules
A `LocallyConvexSpace` is a ... | refine locallyConvexSpace_sInf ?_
rwa [forall_mem_range]
theorem locallyConvexSpace_inf {t₁ t₂ : TopologicalSpace E}
(h₁ : @LocallyConvexSpace 𝕜 E _ _ _ _ _ t₁)
| Mathlib/Topology/Algebra/Module/LocallyConvex.lean | 171 | 175 |
/-
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.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Defs
import Mathlib.Analysis.NormedSpac... | exact abs_of_pos hx
theorem exp_log_eq_abs (hx : x ≠ 0) : exp (log x) = |x| := by
rw [log_of_ne_zero hx, ← coe_expOrderIso_apply, OrderIso.apply_symm_apply, Subtype.coe_mk]
| Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 49 | 52 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Order.Hom.CompleteLattice
import Mathlib.Topology.Compactness.Bases
import Mathlib.Topology.ContinuousMap.Basic
impor... | rw [← SetLike.mem_coe]
simp
| Mathlib/Topology/Sets/Opens.lean | 230 | 231 |
/-
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.Order.Atoms
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.RelIso.Set
import Mathlib.Order.SupClosed
import Mathlib.Order.SupIndep
import Mathlib.Orde... | simp [hs])
iSup_inf_le_inf_sSup
| Mathlib/Order/CompactlyGenerated/Basic.lean | 389 | 391 |
/-
Copyright (c) 2017 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stephen Morgan, Kim Morrison
-/
import Mathlib.CategoryTheory.Functor.Const
import Mathlib.CategoryTheory.Opposites
import Mathlib.Data.Prod.Basic
/-!
# Cartesian products of categories... | simp? at hfg hgf says simp only [prod_Hom, prod_comp, prod_id, Prod.mk.injEq] at hfg hgf
rcases hfg with ⟨hfg₁, hfg₂⟩
rcases hgf with ⟨hgf₁, hgf₂⟩
exact ⟨⟨⟨g.1, hfg₁, hgf₁⟩⟩, ⟨⟨g.2, hfg₂, hgf₂⟩⟩⟩
· rintro ⟨⟨g₁, hfg₁, hgf₁⟩, ⟨g₂, hfg₂, hgf₂⟩⟩
dsimp at hfg₁ hgf₁ hfg₂ hgf₂
refine ⟨⟨(g₁, g₂), ?_, ... | Mathlib/CategoryTheory/Products/Basic.lean | 64 | 75 |
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