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 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
/-!
# Right-angled triangles
This file proves ba... | theorem norm_div_sin_angle_sub_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x = 0 ∨ y ≠ 0) :
‖y‖ / Real.sin (angle x (x - y)) = ‖x - y‖ := by
rw [← neg_eq_zero, ← inner_neg_right] at h
rw [← neg_ne_zero] at h0
| Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean | 300 | 303 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus
deprecated_module (since := "2025-04-06")
| Mathlib/MeasureTheory/Integral/FundThmCalculus.lean | 1,586 | 1,593 | |
/-
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.Sites.Sieves
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.Mono
/-!
# The sheaf condition for a presieve
We define what it means fo... | apply and_congr (Iff.refl _)
constructor
· intro q x hx
apply Exists.imp _ (q _ (hx.restrict (le_generate R)))
intro t ht
simpa [hx] using isAmalgamation_sieveExtend _ _ ht
· intro q x hx
apply Exists.imp _ (q _ hx.sieveExtend)
intro t ht
simpa [hx] using isAmalgamation_restrict (le_gene... | Mathlib/CategoryTheory/Sites/IsSheafFor.lean | 573 | 586 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.IsPrimePow
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Algebra.Ring.CharZero
im... | theorem sum_properDivisors_eq_one_iff_prime : ∑ x ∈ n.properDivisors, x = 1 ↔ n.Prime := by
rcases n with - | n
· simp [Nat.not_prime_zero]
· cases n
· simp [Nat.not_prime_one]
· rw [← properDivisors_eq_singleton_one_iff_prime]
refine ⟨fun h => ?_, fun h => h.symm ▸ sum_singleton _ _⟩
rw [@eq_... | Mathlib/NumberTheory/Divisors.lean | 479 | 491 |
/-
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.Real
/-!
# Properties of addition, multiplication and subtraction on extended non-negative real numbers
In this file we... | Mathlib/Data/ENNReal/Operations.lean | 232 | 232 | |
/-
Copyright (c) 2021 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison
-/
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Monoidal.Functor
/-!
# Preadditive monoidal categories
A monoidal category is `M... | (leftDistributor X f).inv = ∑ j : J, biproduct.π _ j ≫ (X ◁ biproduct.ι f j) := by
classical
ext
dsimp [leftDistributor, Functor.mapBiproduct, Functor.mapBicone]
simp only [Preadditive.comp_sum, biproduct.ι_π_assoc, dite_comp, zero_comp,
Finset.sum_dite_eq, Finset.mem_univ, ite_true, eqToHom_refl, Categ... | Mathlib/CategoryTheory/Monoidal/Preadditive.lean | 151 | 158 |
/-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Trunc
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this fil... | theorem derivativeFun_add (f g : R⟦X⟧) :
derivativeFun (f + g) = derivativeFun f + derivativeFun g := by
ext
rw [coeff_derivativeFun, map_add, map_add, coeff_derivativeFun,
coeff_derivativeFun, add_mul]
| Mathlib/RingTheory/PowerSeries/Derivative.lean | 49 | 53 |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Topology.Homeomorph.Lemmas
import Mathlib.Topology.Sets.OpenCover
import Mathlib.Topology.LocallyClosed
/-!
# Properties of maps that are local at the targe... |
variable {α β : Type*} [TopologicalSpace α] [TopologicalSpace β] {f : α → β}
variable {ι : Type*} {U : ι → Opens β}
theorem Set.restrictPreimage_isInducing (s : Set β) (h : IsInducing f) :
IsInducing (s.restrictPreimage f) := by
| Mathlib/Topology/LocalAtTarget.lean | 29 | 34 |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Violeta Hernández Palacios
-/
import Mathlib.MeasureTheory.MeasurableSpace.Defs
import Mathlib.SetTheory.Cardinal.Regular
import Mathlib.SetTheory.Cardinal.Contin... | ⋃ n, f n ∈ generateMeasurableRec s i := by
unfold generateMeasurableRec
exact mem_union_right _ ⟨fun n => ⟨f n, let ⟨j, hj, hf⟩ := hf n; mem_iUnion₂.2 ⟨j, hj, hf⟩⟩, rfl⟩
| Mathlib/MeasureTheory/MeasurableSpace/Card.lean | 68 | 71 |
/-
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... | ∑' i, f i * g i ≤ (∑' i, f i ^ p) ^ (1 / p) * (∑' i, g i ^ q) ^ (1 / q) := by
lift f to ι → ℝ≥0 using hf
lift g to ι → ℝ≥0 using hg
-- After https://github.com/leanprover/lean4/pull/2734, `norm_cast` needs help with beta reduction.
beta_reduce at *
norm_cast at *
exact NNReal.inner_le_Lp_mul_Lq_tsum h... | Mathlib/Analysis/MeanInequalities.lean | 777 | 787 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibili... | classical
refine le_antisymm (coeffsIn_le.2 ?_) ?_
| Mathlib/Algebra/MvPolynomial/Basic.lean | 963 | 964 |
/-
Copyright (c) 2024 Mitchell Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Lee
-/
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
import Mathlib.Tactic.Linarith
import Mathlib.Tactic.Zify
/-!
# The length function, reduced word... |
theorem exists_reduced_word (w : W) : ∃ ω, ω.length = ℓ w ∧ w = π ω := by
| Mathlib/GroupTheory/Coxeter/Length.lean | 75 | 76 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Operations
import Mathlib.Order.Basic
import Mathlib.Order.BooleanAlgebra
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.B... | /-! ### Distributivity laws -/
theorem inter_union_distrib_left (s t u : Set α) : s ∩ (t ∪ u) = s ∩ t ∪ s ∩ u :=
| Mathlib/Data/Set/Basic.lean | 853 | 855 |
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Algebra.Algebra.Field
import Mathlib.Algebra.BigOperators.Balance
import Mathlib.Algebra.Order.BigOperators.Expect
import Mathlib.Algebra.Order.Star.... | theorem abs_re_div_norm_le_one (z : K) : |re z / ‖z‖| ≤ 1 := by
| Mathlib/Analysis/RCLike/Basic.lean | 677 | 677 |
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Order.Floor.Semiring
import Mathlib.Data.Nat.Log
/-!
# Integer logarithms in a field with respect to a natural base
This file defines two `ℤ`-value... | theorem lt_zpow_succ_log_self {b : ℕ} (hb : 1 < b) (r : R) : r < (b : R) ^ (log b r + 1) := by
rcases le_or_lt r 0 with hr | hr
· rw [log_of_right_le_zero _ hr, zero_add, zpow_one]
exact hr.trans_lt (zero_lt_one.trans_le <| mod_cast hb.le)
rcases le_or_lt 1 r with hr1 | hr1
· rw [log_of_one_le_right _ hr1]
... | Mathlib/Data/Int/Log.lean | 99 | 105 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Option.NAry
import Mathlib.Data.Seq.Computation
import Mathlib.Tactic.ApplyFun
import Mathlib.Data.List.Basic
/-!
# Possibly infinite lists
This... | destruct (corec f b) = omap (corec f) (f b) := by
dsimp [corec, destruct, get]
rw [show Stream'.corec' (Corec.f f) (some b) 0 = (Corec.f f (some b)).1 from rfl]
dsimp [Corec.f]
induction' h : f b with s; · rfl
obtain ⟨a, b'⟩ := s; dsimp [Corec.f]
apply congr_arg fun b' => some (a, b')
apply Subtype.eq... | Mathlib/Data/Seq/Seq.lean | 339 | 352 |
/-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.Tactic.CategoryTheory.Monoidal.Basic
import Mathlib.CategoryTheory.Closed.Monoidal
import Mathlib.Tactic.ApplyFun
/-!
# Rigid (autonomous) monoida... | theorem tensorLeftHomEquiv_symm_coevaluation_comp_whiskerRight {X Y : C} [HasRightDual X]
[HasRightDual Y] (f : X ⟶ Y) :
(tensorLeftHomEquiv _ _ _ _).symm (η_ _ _ ≫ f ▷ (Xᘁ)) = (ρ_ _).hom ≫ fᘁ := by
dsimp [tensorLeftHomEquiv, rightAdjointMate]
simp
| Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean | 395 | 399 |
/-
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.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Order.Sub.WithTop
import Mathlib.Data.NNReal.Defs
import Mathlib.Order.Interval.Set.... |
theorem forall_ne_top {p : ℝ≥0∞ → Prop} : (∀ a, a ≠ ∞ → p a) ↔ ∀ r : ℝ≥0, p r :=
| Mathlib/Data/ENNReal/Basic.lean | 297 | 298 |
/-
Copyright (c) 2018 Sean Leather. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sean Leather, Mario Carneiro
-/
import Mathlib.Data.List.Sigma
/-!
# Association Lists
This file defines association lists. An association list is a list where every element consists o... | rw [AList.ext_iff]; infer_instance
| Mathlib/Data/List/AList.lean | 64 | 65 |
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Int.DivMod
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.... | variable {a b : ℕ}
| Mathlib/Data/Fin/Basic.lean | 392 | 393 |
/-
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.Constructions.BorelSpace.Metric
import Mathlib.MeasureTheory.Constructions.BorelSpace.Real
import Mathlib.Topology.Metrizable.Real
im... | set p : α → (ℕ → β) → Prop := fun x f' => Tendsto (fun n => f' n) atTop (𝓝 (g x))
have hp : ∀ᵐ x ∂μ, p x fun n => f (v n) x := by
filter_upwards [h_tendsto] with x hx using hx.comp hv
set aeSeqLim := fun x => ite (x ∈ aeSeqSet h'f p) (g x) (⟨f (v 0) x⟩ : Nonempty β).some
refine
⟨aeSeqLim,
measura... | Mathlib/MeasureTheory/Constructions/BorelSpace/Metrizable.lean | 57 | 78 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Joël Riou
-/
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.CategoryTheory.Shift.Basic
import Mathlib.Data.Set.Subsingleton
import Mathlib.Algebra.Grou... | F.map (e.inv j) ≫ F.map (e.hom j) = 𝟙 _ := by
rw [← F.map_comp, ← GradedObject.categoryOfGradedObjects_comp, e.inv_hom_id,
GradedObject.categoryOfGradedObjects_id, Functor.map_id]
@[reassoc (attr := simp)]
| Mathlib/CategoryTheory/GradedObject.lean | 126 | 130 |
/-
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.Algebra.MvPolynomial.Counit
import Mathlib.Algebra.MvPolynomial.Invertible
import Mathlib.RingTheory.WittVector.Defs
/-!
# Witt vecto... | theorem neg : mapFun f (-x) = -mapFun f x := by map_fun_tac
| Mathlib/RingTheory/WittVector/Basic.lean | 108 | 108 |
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Order.ToIntervalMod
import Mathlib.Algebra.Ring.AddAut
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.Divisible
import Mathlib.Topology.C... | exact Int.emod_lt_of_pos _ (Int.ofNat_lt.2 h)
· have := (gcd_mul_addOrderOf_div_eq p (Int.toNat (a % ↑n)) h).trans
((congr_arg addOrderOf he).trans hk).symm
rw [he, Nat.mul_left_eq_self_iff] at this
· exact this
· rwa [hk]
| Mathlib/Topology/Instances/AddCircle.lean | 434 | 440 |
/-
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.LinearAlgebra.TensorAlgebra.Basic
import Mathlib.LinearAlgebra.TensorPower.Basic
/-!
# Tensor algebras as direct sums of tensor powers
In this file we show... | LinearMap.compl₂_apply, ← LinearMap.comp_apply]
refine LinearMap.congr_fun (LinearMap.congr_fun ?_ a) b
clear! a b
ext (a b)
-- Porting note: pulled the next two lines out of the long `simp only` below.
simp only [LinearMap.compMultilinearMap_apply]
rw [LinearMap.compr₂_apply, ← gMul_eq_coe_linearMap]
... | Mathlib/LinearAlgebra/TensorAlgebra/ToTensorPower.lean | 44 | 64 |
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Integral.Prod
import Mathlib.Meas... | exact L.le_opNorm (f t)
refine (integral_mono (L.integrable_comp hif).norm (hif.norm.const_mul _) h3).trans_eq ?_
rw [integral_const_mul]
variable [NormedSpace ℝ E] [NormedSpace ℝ E'] [CompleteSpace E']
/-- Approximate `f ⋆ g` if the support of the `f` is bounded within a ball, and `g` is near `g x₀`
on a bal... | Mathlib/Analysis/Convolution.lean | 772 | 783 |
/-
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.Multiset.Bind
/-!
# The cartesian product of multisets
## Main definitions
* `Multiset.pi`: Cartesian product of multisets indexed by a multise... | (cons m a (f _ (mem_cons_self _ _)) fun a' ha' => f a' (mem_cons_of_mem ha')) = f := by
ext a' h'
by_cases h : a' = a
· subst h
rw [Pi.cons_same]
· rw [Pi.cons_ne _ h]
| Mathlib/Data/Multiset/Pi.lean | 62 | 68 |
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Neil Strickland
-/
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Data.PNat.Basic
/-!
# Primality and GCD on pnat
This file extends the theory of `ℕ+` with ... | Mathlib/Data/PNat/Prime.lean | 302 | 305 | |
/-
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.Algebra.Polynomial.GroupRingAction
import Mathlib.Algebra.Ring.Action.Field
import Mathlib.Algebra.Ring.Action.Invariant
import Mathlib.FieldTheory.Finiteness
im... | [FiniteDimensional K V] : Fintype (V ≃ₐ[K] V) :=
Fintype.ofEquiv (V →ₐ[K] V) (algEquivEquivAlgHom K V).symm
| Mathlib/FieldTheory/Fixed.lean | 304 | 306 |
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.Group.Action.Units
import Mathlib.Algebra.Group.Nat.Units
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra... | ⟨b, a, by rw [add_comm, H]⟩
theorem isCoprime_comm : IsCoprime x y ↔ IsCoprime y x :=
| Mathlib/RingTheory/Coprime/Basic.lean | 47 | 49 |
/-
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.Topology.UniformSpace.CompleteSeparated
import Mathlib.Topology.EMetricSpace.Lipschitz
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.Topol... |
variable [PseudoEMetricSpace α] [PseudoEMetricSpace β] [PseudoEMetricSpace γ]
variable {K : ℝ≥0} {f : α → β}
| Mathlib/Topology/MetricSpace/Antilipschitz.lean | 77 | 79 |
/-
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.Algebra.Order.AddGroupWithTop
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Order.Sub.WithTop
import Mathlib.Data.ENat.Defs
import... |
@[simp] lemma toNat_eq_iff_eq_coe (n : ℕ∞) (m : ℕ) [NeZero m] :
| Mathlib/Data/ENat/Basic.lean | 233 | 234 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.RelIso.Basic
/-!
# Order continuity
We say that a function is *left o... | theorem comp (hg : LeftOrdContinuous g) (hf : LeftOrdContinuous f) : LeftOrdContinuous (g ∘ f) :=
fun s x h => by simpa only [image_image] using hg (hf h)
| Mathlib/Order/OrdContinuous.lean | 71 | 73 |
/-
Copyright (c) 2020 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Grading
import Mathlib.Algebra.Module.Opposite
/-!
# Conjugations
This file defines the grade reversal and grade involution f... | op_injective (map_one reverseOp)
| Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean | 111 | 111 |
/-
Copyright (c) 2014 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ord... |
theorem IsGLB.mul_left {s : Set α} (ha : 0 ≤ a) (hs : IsGLB s b) :
IsGLB ((fun b => a * b) '' s) (a * b) := by
rcases lt_or_eq_of_le ha with (ha | rfl)
| Mathlib/Algebra/Order/Field/Basic.lean | 285 | 288 |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.Cover
import Mathlib.Order.Iterate
/-!
# Successor and predecessor
This file defines succes... | (pred_lt_of_not_isMin ha).trans_le
theorem le_pred_iff_of_not_isMin (ha : ¬IsMin a) : b ≤ pred a ↔ b < a :=
⟨fun h => h.trans_lt <| pred_lt_of_not_isMin ha, le_pred_of_lt⟩
lemma pred_lt_pred_of_not_isMin (h : a < b) (ha : ¬ IsMin a) : pred a < pred b :=
| Mathlib/Order/SuccPred/Basic.lean | 597 | 602 |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Callum Sutton, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Equiv.Opposite
import Mathlib.Algebra.GroupWithZero.Equiv
import Mathlib.Algebra.GroupWithZero.InjSurj
im... |
@[simp]
theorem toNonUnitalRingHom_refl :
(RingEquiv.refl R).toNonUnitalRingHom = NonUnitalRingHom.id R :=
| Mathlib/Algebra/Ring/Equiv.lean | 670 | 673 |
/-
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.Group.Finset.Sigma
import Mathlib.Algebra.Order.Interval.Finset.Basic
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Tact... |
@[to_additive (attr := simp)]
theorem prod_Ico_add_right_sub_eq [AddCommMonoid α] [PartialOrder α] [IsOrderedCancelAddMonoid α]
[ExistsAddOfLE α] [LocallyFiniteOrder α] [Sub α] [OrderedSub α] (a b c : α) :
| Mathlib/Algebra/BigOperators/Intervals.lean | 103 | 106 |
/-
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... | _ = ‖x‖ ^ n * ∑ m ∈ range j with n ≤ m, (‖x‖ ^ (m - n) / (m - n).factorial) := by
rw [← mul_sum]
| Mathlib/Data/Complex/Exponential.lean | 490 | 491 |
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Bhavik Mehta
-/
import Mathlib.Analysis.Calculus.Deriv.Support
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.MeasureTheory.Function.Jacobian
imp... |
variable {E : Type*} [NormedAddCommGroup E]
/-- The substitution `y = x ^ p` in integrals over `Ioi 0` preserves integrability. -/
theorem integrableOn_Ioi_comp_rpow_iff [NormedSpace ℝ E] (f : ℝ → E) {p : ℝ} (hp : p ≠ 0) :
IntegrableOn (fun x => (|p| * x ^ (p - 1)) • f (x ^ p)) (Ioi 0) ↔ IntegrableOn f (Ioi 0) :=... | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | 1,121 | 1,128 |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.IsPrimePow
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Algebra.Ring.CharZero
im... |
@[simp high]
lemma swap_mem_divisorsAntidiagonalList {a : ℕ × ℕ} :
a.swap ∈ n.divisorsAntidiagonalList ↔ a ∈ n.divisorsAntidiagonalList := by simp [mul_comm]
| Mathlib/NumberTheory/Divisors.lean | 169 | 173 |
/-
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.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Covering.Besicovitch
import Mathlib.Tactic.AdaptationNote
import Mathlib.Algeb... | _ ≤ 1 := by linarith only [sq_nonneg δ]
have J : 1 - δ ≤ 1 - δ / 4 := by linarith only [δnonneg]
have K : 1 - δ / 4 ≤ τ⁻¹ := by rw [inv_eq_one_div, le_div_iff₀ τpos]; exact I
suffices L : τ⁻¹ ≤ ‖a.c i - a.c j‖ by linarith only [J, K, L]
have hτ' : ∀ k, τ⁻¹ ≤ a.r k := by
intro k
rw [inv_eq_one_div,... | Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean | 322 | 349 |
/-
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, Damiano Testa,
Yuyang Zhao
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Defs
import Mathlib.Data.Ordering.Basic
imp... | Mathlib/Algebra/Order/Monoid/Unbundled/Basic.lean | 1,656 | 1,658 | |
/-
Copyright (c) 2019 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Yury Kudryashov
-/
import Mathlib.Algebra.Group.Action.Defs
import Mathlib.Algebra.Group.Units.Defs
import Mathlib.Algebra.BigOperators.Group.List.Basic
import Mathlib.Algebr... | def casesOn {C : FreeMonoid α → Sort*} (xs : FreeMonoid α) (h0 : C 1)
(ih : ∀ x xs, C (of x * xs)) : C xs := List.casesOn xs h0 ih
| Mathlib/Algebra/FreeMonoid/Basic.lean | 267 | 269 |
/-
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.Control.Traversable.Instances
import Mathlib.Order.Filter.Map
/-!
# Properties of `Traversable.traverse` on `List`s and `Filter`s
In this file we prov... | · induction fs generalizing t with
| nil =>
simp only [sequence, mem_pure, imp_self, forall₂_nil_left_iff, exists_eq_left, Set.pure_def,
singleton_subset_iff, traverse_nil]
| cons b fs ih =>
intro ht
rcases mem_seq_iff.1 ht with ⟨u, hu, v, hv, ht⟩
rcases mem_map_iff_exists_imag... | Mathlib/Order/Filter/ListTraverse.lean | 38 | 53 |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Topology.Bases
import Mathlib.Topology.Separation.Regular
/-!
# Dense embeddings
This file defines three properties of f... | variable {f : α → β}
/-- Two continuous functions to a t2-space that agree on the dense range of a function are equal. -/
theorem DenseRange.equalizer (hfd : DenseRange f) {g h : β → γ} (hg : Continuous g)
(hh : Continuous h) (H : g ∘ f = h ∘ f) : g = h :=
funext fun y => hfd.induction_on y (isClosed_eq hg hh) <... | Mathlib/Topology/DenseEmbedding.lean | 364 | 382 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Algebra.Group.TypeTags.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Finset.Piecewise
import Mathlib.Or... | Mathlib/Topology/Constructions.lean | 1,778 | 1,781 | |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.Algebra.Ring.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.Order.Cir... | simpa only [one_zsmul] using toIcoDiv_sub_zsmul' hp a b 1
| Mathlib/Algebra/Order/ToIntervalMod.lean | 288 | 289 |
/-
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.... | Filter.eventually_true, forall_true_left]
· intro a S ha_notin_S h_rec hf_m
have hf_m_S : ∀ x ∈ S, MeasurableSet[m x] (f x) := fun x hx => hf_m x (by simp [hx])
let p := piiUnionInter π S
set m_p := generateFrom p with hS_eq_generate
have h_indep : Indep m_p (m a) η μ := by
have hp : IsPiS... | Mathlib/Probability/Independence/Kernel.lean | 749 | 778 |
/-
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.Algebra.Field.NegOnePow
import Mathlib.Algebra.Field.Periodic
import Mathlib.Algebra.Qua... | rw [cos_pi_div_six, div_pow, sq_sqrt] <;> norm_num
/-- The sine of `π / 6` is `1 / 2`. -/
@[simp]
theorem sin_pi_div_six : sin (π / 6) = 1 / 2 := by
rw [← cos_pi_div_two_sub, ← cos_pi_div_three]
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 771 | 776 |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Peter Pfaffelhuber, Yaël Dillies, Kin Yau James Wong
-/
import Mathlib.MeasureTheory.MeasurableSpace.Constructions
import Mathlib.MeasureTheory.PiSystem
import Mathlib.Topo... | @[simp]
theorem cylinder_empty (s : Finset ι) : cylinder s (∅ : Set (∀ i : s, α i)) = ∅ := by
| Mathlib/MeasureTheory/Constructions/Cylinders.lean | 161 | 162 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
/-!
# Oriented angles.
This file defines orie... | @[simp]
theorem two_zsmul_oangle_neg_right (x y : V) :
| Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 199 | 200 |
/-
Copyright (c) 2024 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.CategoryTheory.Sites.Coherent.Comparison
import Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves
import Mathlib.CategoryTheory.Sites.Coherent... |
noncomputable instance [Preregular C] [FinitaryExtensive C]
(F : Sheaf (coherentTopology C) A) : PreservesFiniteProducts F.val :=
(Presheaf.isSheaf_iff_preservesFiniteProducts F.val).1
((Presheaf.isSheaf_coherent_iff_regular_and_extensive F.val).mp F.cond).1
theorem isSheaf_iff_preservesFiniteProducts_of_pr... | Mathlib/CategoryTheory/Sites/Coherent/SheafComparison.lean | 235 | 242 |
/-
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.Calculus.FDeriv.Basic
/-!
# The derivative of a composition (chain rule)
For detailed documentation of the... | /-- Ternary version of `fderivWithin_comp`, with equality assumptions of basepoints added, in
order to apply more easily as a rewrite from right-to-left. -/
theorem fderivWithin_comp₃ {g' : G → G'} {g : F → G} {t : Set F} {u : Set G} {y : F} {y' : G}
(hg' : DifferentiableWithinAt 𝕜 g' u y') (hg : DifferentiableW... | Mathlib/Analysis/Calculus/FDeriv/Comp.lean | 149 | 157 |
/-
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.CategoryTheory.Category.Pairwise
import Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts
import Mathlib.CategoryTheory.Limits.Final
import Mathli... | /-- The sheaf condition in terms of an equalizer diagram is equivalent
to the reformulation in terms of the presheaf preserving the limit of the diagram
consisting of the `U i` and `U i ⊓ U j`.
| Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean | 280 | 282 |
/-
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.Real
/-!
# Properties of addition, multiplication and subtraction on extended non-negative real numbers
In this file we... |
-- TODO: generalize to `WithTop`
| Mathlib/Data/ENNReal/Operations.lean | 239 | 240 |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Rémy Degenne
-/
import Mathlib.Order.ConditionallyCompleteLattice.Indexed
import Mathlib.Order.Filter.IsBounded
import Mathlib.Order.Hom.CompleteL... | Mathlib/Order/LiminfLimsup.lean | 1,406 | 1,426 | |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Multiset.ZeroCons
/-!
# Basic results on multisets
-/
-- No algebra should be required
assert_not_exists Monoid
universe v
open List S... | Mathlib/Data/Multiset/Basic.lean | 770 | 771 | |
/-
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... | simpa only [infDist_closure] using isClosed_closure.exists_infDist_eq_dist hne.closure x
| Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 597 | 598 |
/-
Copyright (c) 2020 Jannis Limperg. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jannis Limperg
-/
import Mathlib.Data.List.Induction
/-!
# Lemmas about List.*Idx functions.
Some specification lemmas for `List.mapIdx`, `List.mapIdxM`, `List.foldlIdx` and `List.fo... | Mathlib/Data/List/Indexes.lean | 284 | 290 | |
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Algebra.Order.Group.OrderIso
import Mathlib.SetTheory.Game.Ordinal
import Mathlib.SetTheory.Ordinal.NaturalOps
/-!
# Birthdays... | Mathlib/SetTheory/Game/Birthday.lean | 111 | 111 | |
/-
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.Nonneg.Field
import Mathlib.Data.Rat.Cast.Defs
import Mathlib.Tactic.Positivity.Basic
/-!
# Some exiled lemmas about cas... | theorem cast_pow {K} [DivisionSemiring K] (q : ℚ≥0) (n : ℕ) :
NNRat.cast (q ^ n) = (NNRat.cast q : K) ^ n := by
rw [cast_def, cast_def, den_pow, num_pow, Nat.cast_pow, Nat.cast_pow, div_eq_mul_inv, ← inv_pow,
← (Nat.cast_commute _ _).mul_pow, ← div_eq_mul_inv]
| Mathlib/Data/Rat/Cast/Lemmas.lean | 64 | 67 |
/-
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... |
theorem Pi.uniformity :
𝓤 (∀ i, α i) = ⨅ i : ι, (Filter.comap fun a => (a.1 i, a.2 i)) (𝓤 (α i)) :=
| Mathlib/Topology/UniformSpace/Pi.lean | 30 | 32 |
/-
Copyright (c) 2024 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.RingTheory.AdjoinRoot
/-!
# Bivariate polynomials
This file introduces the notation `R[X][Y]` for the polynomial ring `R[X][X]` in two variables,
and the notat... | simp only [evalEval, eval_smul]
lemma evalEval_surjective (x y : R) : Function.Surjective <| evalEval x y :=
| Mathlib/Algebra/Polynomial/Bivariate.lean | 82 | 84 |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Heather Macbeth, Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Analysis.Normed.Group.Uniform
import Mathlib.Topo... | assume that `∑' i, f i` is summable, and it might not be the case if `α` is not a complete space. -/
theorem tsum_of_norm_bounded {f : ι → E} {g : ι → ℝ} {a : ℝ} (hg : HasSum g a)
(h : ∀ i, ‖f i‖ ≤ g i) : ‖∑' i : ι, f i‖ ≤ a := by
by_cases hf : Summable f
| Mathlib/Analysis/Normed/Group/InfiniteSum.lean | 113 | 116 |
/-
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.Order.Bounds.Defs
import Mathlib.Order.Directed
import Mathlib.Order.BoundedOrder.Monotone
import Mathlib.Order.Interval.Set.Basic
/-... | theorem isGLB_pair [SemilatticeInf γ] {a b : γ} : IsGLB {a, b} (a ⊓ b) :=
isGLB_singleton.insert _
| Mathlib/Order/Bounds/Basic.lean | 802 | 804 |
/-
Copyright (c) 2023 Moritz Firsching. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Firsching, Ashvni Narayanan, Michael Stoll
-/
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Data.ZMod.Basic
import Mathlib.RingTheory.Coprime.Lemmas
/-!
# Lem... | ↑(unitsMap h a) = ((a : ZMod m).cast : ZMod n) := rfl
lemma isUnit_cast_of_dvd (hm : n ∣ m) (a : Units (ZMod m)) : IsUnit (cast (a : ZMod m) : ZMod n) :=
Units.isUnit (unitsMap hm a)
@[deprecated (since := "2024-12-16")] alias IsUnit_cast_of_dvd := isUnit_cast_of_dvd
theorem unitsMap_surjective [hm : NeZero m] ... | Mathlib/Data/ZMod/Units.lean | 38 | 63 |
/-
Copyright (c) 2024 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Matroid.Constructions
import Mathlib.Data.Set.Notation
/-!
# Maps between matroids
This file defines maps and comaps, which move a matroid on one ty... | instance [M.RankFinite] {f : α ↪ β} : (M.mapEmbedding f).RankFinite :=
inferInstanceAs (M.map f f.injective.injOn).RankFinite
| Mathlib/Data/Matroid/Map.lean | 576 | 578 |
/-
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... | lemma exists_subsuperset_card_eq (hst : s ⊆ t) (hsn : #s ≤ n) (hnt : n ≤ #t) :
∃ u, s ⊆ u ∧ u ⊆ t ∧ #u = n := by
classical
| Mathlib/Data/Finset/Card.lean | 562 | 564 |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.MeasureTheory.Covering.OneDim
import Mathlib.Order.Monotone.Extension
/-!
# Differentiability of mo... | filter_upwards [hf.stieltjesFunction.ae_hasDerivAt,
hf.countable_not_continuousAt.ae_not_mem volume] with x hx h'x
have A : hf.stieltjesFunction x = f x := by
rw [Classical.not_not, hf.continuousAt_iff_leftLim_eq_rightLim] at h'x
apply le_antisymm _ (hf.le_rightLim (le_refl _))
rw [← h'x]
exact ... | Mathlib/Analysis/Calculus/Monotone.lean | 136 | 214 |
/-
Copyright (c) 2020 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Johan Commelin
-/
import Mathlib.Algebra.Algebra.RestrictScalars
import Mathlib.Algebra.Algebra.Subalgebra.Lattice
import Mathlib.Algebra.Module.Rat
import Mathlib.GroupThe... |
This is just a special case of `Algebra.TensorProduct.lift` for when `C` is commutative. -/
abbrev productLeftAlgHom (f : A →ₐ[S] C) (g : B →ₐ[R] C) : A ⊗[R] B →ₐ[S] C :=
lift f g (fun _ _ => Commute.all _ _)
| Mathlib/RingTheory/TensorProduct/Basic.lean | 1,159 | 1,163 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Patrick Massot, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic
import Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalcul... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 359 | 361 | |
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Defs
/-!
# Basic kernels
This file contains basic results about kernels in general and definitions of some particular
kernels.
## Mai... | Mathlib/Probability/Kernel/Basic.lean | 511 | 512 | |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Sophie Morel, Yury Kudryashov
-/
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Logic.Embedding.Basic
import Mathlib.Data.Fintype.Car... | Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean | 1,407 | 1,408 | |
/-
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.Semisimple.Defs
import Mathlib.Order.BooleanGenerators
/-!
# Semisimple Lie algebras
The famous Cartan-Dynkin-Killing classification of semisim... |
theorem abelian_radical_iff_solvable_is_abelian [IsNoetherian R L] :
IsLieAbelian (radical R L) ↔ ∀ I : LieIdeal R L, IsSolvable I → IsLieAbelian I := by
| Mathlib/Algebra/Lie/Semisimple/Basic.lean | 308 | 310 |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.Order.Compact
import Mathlib.Topology.MetricSpace.ProperSpace
import M... | theorem isBounded_range_of_tendsto_cofinite {f : β → α} {a : α} (hf : Tendsto f cofinite (𝓝 a)) :
IsBounded (range f) :=
isBounded_range_of_tendsto_cofinite_uniformity <|
| Mathlib/Topology/MetricSpace/Bounded.lean | 205 | 207 |
/-
Copyright (c) 2020 Jalex Stark. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jalex Stark, Kim Morrison, Eric Wieser, Oliver Nash, Wen Yang
-/
import Mathlib.Data.Matrix.Basic
/-!
# Matrices with a single non-zero element.
This file provides `Matrix.stdBasisMatri... | end
| Mathlib/Data/Matrix/Basis.lean | 190 | 191 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Data.Set.Restrict
/-!
# Functions over sets
This file contains... | Mathlib/Data/Set/Function.lean | 1,870 | 1,873 | |
/-
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... | refine Complex.ext_iff.trans <| Iff.trans ?_ <| arg_mul_eq_add_arg_iff hx₀ hy₀
| Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 83 | 83 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Algebra.Notation.Prod
import Mathlib.Data.Nat.Sqrt
import Mathlib.Data.Set.Lattice.Image
/-!
# Naturals pairing function
Th... |
theorem unpair_add_le (n : ℕ) : (unpair n).1 + (unpair n).2 ≤ n :=
(add_le_pair _ _).trans_eq (pair_unpair _)
| Mathlib/Data/Nat/Pairing.lean | 149 | 151 |
/-
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.Combinatorics.SetFamily.Compression.Down
import Mathlib.Data.Fintype.Powerset
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Algebra.BigOperators.... | lemma shatterer_compress_subset_shatterer (a : α) (𝒜 : Finset (Finset α)) :
(𝓓 a 𝒜).shatterer ⊆ 𝒜.shatterer := by
simp only [subset_iff, mem_shatterer]; exact fun s hs ↦ hs.of_compression
| Mathlib/Combinatorics/SetFamily/Shatter.lean | 175 | 177 |
/-
Copyright (c) 2023 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Set.Finite.Lattice
import Mathlib.Order.Interval.Finset.Defs
import Mathlib.Order.UpperLower.Closure
/-!
# Upper and lower sets in a loc... | rw [coe_lowerClosure]
exact hs.biUnion fun _ _ => finite_Iic _
end Set
| Mathlib/Order/UpperLower/LocallyFinite.lean | 28 | 31 |
/-
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.Solvable
import Mathlib.Algebra.Lie.Quotient
import Mathlib.Algebra.Lie.Normalizer
import Mathlib.Algebra.Order.Archimedean.Basic
import Mathlib.... | theorem nilpotencyLength_eq_zero_iff [IsNilpotent L M] :
nilpotencyLength L M = 0 ↔ Subsingleton M := by
let s := {k | lowerCentralSeries ℤ L M k = ⊥}
have hs : s.Nonempty := by
obtain ⟨k, hk⟩ := IsNilpotent.nilpotent ℤ L M
exact ⟨k, hk⟩
change sInf s = 0 ↔ _
rw [← LieSubmodule.subsingleton_iff ℤ L ... | Mathlib/Algebra/Lie/Nilpotent.lean | 395 | 404 |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.RingTheory.LocalProperties.Reduced
/-!
# Basic properties of schemes
We provide some basic properties of sche... | class IsIntegral : Prop where
nonempty : Nonempty X := by infer_instance
component_integral : ∀ (U : X.Opens) [Nonempty U], IsDomain Γ(X, U) := by infer_instance
attribute [instance] IsIntegral.component_integral IsIntegral.nonempty
| Mathlib/AlgebraicGeometry/Properties.lean | 188 | 192 |
/-
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, Simon Hudon, Mario Carneiro
-/
import Mathlib.Algebra.Notation.Defs
import Mathlib.Data.Int.Notation
import Mathlib.Data.Nat.BinaryRec
import Mathlib.L... | Main examples are `ℕ` and groups. This is the right typeclass for many sum lemmas, as having a zero
is useful to define the sum over the empty set, so `AddRightCancelMonoid` is not enough. -/
class AddCancelMonoid (M : Type u) extends AddLeftCancelMonoid M, AddRightCancelMonoid M
| Mathlib/Algebra/Group/Defs.lean | 696 | 698 |
/-
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.Real
/-!
# Properties of addition, multiplication and subtraction on extended non-negative real numbers
In this file we... | Mathlib/Data/ENNReal/Operations.lean | 526 | 529 | |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
/-!
# Neighborhoods and continuity relative to a subset
This file develops API on the relative versions
* `nhdsWithin` ... | 𝓝[⋃ i ∈ I, s i] a = ⨆ i ∈ I, 𝓝[s i] a := by
induction I, hI using Set.Finite.induction_on with
| empty => simp
| Mathlib/Topology/ContinuousOn.lean | 231 | 233 |
/-
Copyright (c) 2023 Yaël Dillies, Vladimir Ivanov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Vladimir Ivanov
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.BigOperators.... | rintro b ⟨hb, hba⟩
exact (hs.eq hb ha hba).ge
variable [DecidableEq α]
| Mathlib/Combinatorics/SetFamily/AhlswedeZhang.lean | 222 | 226 |
/-
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.... | classical
rw [aroots, ← (SplittingField.lift f h).comp_algebraMap, ← map_map,
| Mathlib/FieldTheory/SeparableDegree.lean | 368 | 369 |
/-
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.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.IntermediateField.Basic
imp... | theorem separable_gcd_right {F : Type*} [Field F] [DecidableEq F[X]]
{g : F[X]} (f : F[X]) (hg : g.Separable) :
(EuclideanDomain.gcd f g).Separable :=
| Mathlib/FieldTheory/Separable.lean | 97 | 99 |
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Batteries.Tactic.Init
import Mathlib.Logic.Function.Defs
/-!
# Binary map of options
This file defines the binary map of `Option`. This is mostly useful to defin... | theorem map_map₂_right_anticomm {f : α → β' → γ} {g : β → β'} {f' : β → α → δ} {g' : δ → γ}
(h_right_anticomm : ∀ a b, f a (g b) = g' (f' b a)) :
map₂ f a (b.map g) = (map₂ f' b a).map g' := by cases a <;> cases b <;> simp [h_right_anticomm]
| Mathlib/Data/Option/NAry.lean | 181 | 184 |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2
import Mathlib.MeasureTheory.Measure.Real
/-! # Conditional expectation in L1
This file contains ... | have h_right : Tendsto (fun i => ∫ x in S i ∩ s, f x ∂μ) atTop (𝓝 (∫ x in s, f x ∂μ)) := by
have h :=
tendsto_setIntegral_of_monotone (fun i => (hS_meas0 i).inter (hm s hs)) h_mono
(L1.integrable_coeFn f).integrableOn
rwa [← hs_eq] at h
have h_left : Tendsto (fun i => ∫ x in S i ∩ s, condExpL... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean | 491 | 512 |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Frédéric Dupuis
-/
import Mathlib.Analysis.Convex.Hull
/-!
# Convex cones
In a `𝕜`-module `E`, we define a convex cone as a set `s` such that `a • x + b • y ∈ s` w... |
theorem blunt_iff_not_pointed (S : ConvexCone 𝕜 E) : S.Blunt ↔ ¬S.Pointed := by
rw [pointed_iff_not_blunt, Classical.not_not]
theorem Pointed.mono {S T : ConvexCone 𝕜 E} (h : S ≤ T) : S.Pointed → T.Pointed :=
@h _
| Mathlib/Analysis/Convex/Cone/Basic.lean | 303 | 308 |
/-
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.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Asymptotics.SpecificAsympt... | Mathlib/Analysis/SpecialFunctions/CompareExp.lean | 215 | 217 | |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Group.Subsemigroup.Operations
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Order.Filter.Extr
/-!
# Lemmas about the `sup` and `in... | (single a r).supDegree D = if r = 0 then ⊥ else D a := by
split_ifs with hr <;> simp [supDegree_single_ne_zero, hr]
| Mathlib/Algebra/MonoidAlgebra/Degree.lean | 252 | 254 |
/-
Copyright (c) 2023 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Topology.Connected.Basic
import Mathlib.Topology.Separation.Hausdorff
import Mathlib.Topology.Connected.Clopen
/-!
# Separated maps and locally injective maps ou... | ⟨IsClosedEmbedding.isClosedMap, .of_continuous_injective_isClosedMap
(IsEmbedding.toPullbackDiag f).continuous (injective_toPullbackDiag f)⟩
open Function.Pullback in
theorem IsSeparatedMap.pullback {f : X → Y} (sep : IsSeparatedMap f) (g : A → Y) :
IsSeparatedMap (@snd X Y A f g) := by
| Mathlib/Topology/SeparatedMap.lean | 101 | 106 |
/-
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.Topology.Sheaves.SheafCondition.Sites
/-!
# Another version of the sheaf condition.
Given a family of open sets `U : ι → Opens X` we can form the subcate... | end TopCat
| Mathlib/Topology/Sheaves/SheafCondition/OpensLeCover.lean | 236 | 247 |
/-
Copyright (c) 2021 Gabriel Moise. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Moise, Yaël Dillies, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Matrix.Mul
/-!
# Incidence matrix of a si... | simp only [incMatrix_apply, Set.indicator_apply_eq_zero, Pi.one_apply, one_ne_zero]
| Mathlib/Combinatorics/SimpleGraph/IncMatrix.lean | 92 | 93 |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Algebra.Group.Action.Defs
import Mathlib.Algebra.Group.End
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.Common
/-!
#... | Mathlib/GroupTheory/Perm/Basic.lean | 278 | 282 | |
/-
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.Basic
/-!
# Lemmas about `min` and `max` in an ordered monoid.
-/
op... | theorem lt_or_le_of_mul_le_mul [MulLeftStrictMono α] [MulRightMono α] {a₁ a₂ b₁ b₂ : α} :
a₁ * b₁ ≤ a₂ * b₂ → a₁ < a₂ ∨ b₁ ≤ b₂ := by
contrapose!
exact fun h => mul_lt_mul_of_le_of_lt h.1 h.2
| Mathlib/Algebra/Order/Monoid/Unbundled/MinMax.lean | 90 | 94 |
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