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 |
|---|---|---|---|---|
/-
Copyright (c) 2020 Kexing Ying and Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.Gro... | theorem single_le_finprod {M : Type*} [CommMonoid M] [PartialOrder M] [IsOrderedMonoid M]
(i : α) {f : α → M}
(hf : (mulSupport f).Finite) (h : ∀ j, 1 ≤ f j) : f i ≤ ∏ᶠ j, f j := by
classical calc
| Mathlib/Algebra/BigOperators/Finprod.lean | 994 | 997 |
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury Kudryashov
-/
import Mathlib.Order.UpperLower.Closure
import Mathlib.Order.UpperLower.Fibration
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
import Ma... | attribute [local instance] specializationPreorder
| Mathlib/Topology/Inseparable.lean | 201 | 202 |
/-
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 ucs_eq_top_iff (k : ℕ) : N.ucs k = ⊤ ↔ LieModule.lowerCentralSeries R L M k ≤ N := by
rw [eq_top_iff, ← lcs_le_iff]; rfl
variable (R) in
theorem _root_.LieModule.isNilpotent_iff_exists_ucs_eq_top :
LieModule.IsNilpotent L M ↔ ∃ k, (⊥ : LieSubmodule R L M).ucs k = ⊤ := by
rw [LieModule.isNilpotent_iff R... | Mathlib/Algebra/Lie/Nilpotent.lean | 593 | 601 |
/-
Copyright (c) 2017 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Mario Carneiro
-/
import Mathlib.Algebra.Ring.CharZero
import Mathlib.Algebra.Star.Basic
import Mathlib.Data.Real.Basic
import Mathlib.Order.Interval.Set.UnorderedInterva... | Mathlib/Data/Complex/Basic.lean | 935 | 935 | |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Divisibility.Hom
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.A... | element `b` with `c` iff `a` divides `b`. -/
| Mathlib/Algebra/Ring/Divisibility/Basic.lean | 129 | 129 |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.MvPowerSer... | intro x _
apply prod_congr rfl
intro i _
congr 2
simp only [AddEquiv.toEquiv_eq_coe, Finsupp.mapRange.addEquiv_toEquiv, AddEquiv.toEquiv_symm,
| Mathlib/RingTheory/PowerSeries/Basic.lean | 642 | 646 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Group.Multiset.Basic
/-!
# Bind operation for multisets
This file defines a few basic operations on `Multiset`, notably the mona... | Mathlib/Data/Multiset/Bind.lean | 142 | 142 | |
/-
Copyright (c) 2022 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Yury Kudryashov, Kevin H. Wilson, Heather Macbeth
-/
import Mathlib.Order.Filter.Tendsto
/-!
# Product and coproduct filters
In this file we define `F... |
theorem tendsto_snd : Tendsto Prod.snd (f ×ˢ g) g :=
tendsto_inf_right tendsto_comap
| Mathlib/Order/Filter/Prod.lean | 121 | 123 |
/-
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... | simp [sign_antiperiodic.sub_eq']
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 732 | 733 |
/-
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.Topology.ExtendFrom
import Mathlib.Topology.Order.DenselyOrdered
/-!
# Lemmas about `extendFrom` in an order topology... | (hab : a ≠ b) (hf : ContinuousOn f (Ioo a b)) (ha : Tendsto f (𝓝[>] a) (𝓝 la))
(hb : Tendsto f (𝓝[<] b) (𝓝 lb)) : ContinuousOn (extendFrom (Ioo a b) f) (Icc a b) := by
apply continuousOn_extendFrom
· rw [closure_Ioo hab]
· intro x x_in
rcases eq_endpoints_or_mem_Ioo_of_mem_Icc x_in with (rfl | rfl... | Mathlib/Topology/Order/ExtendFrom.lean | 19 | 29 |
/-
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, Huỳnh Trần Khanh, Stuart Presnell
-/
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Fintype.Sum
import Mathlib.Data.Fintype.Prod
import Math... | -- `hxy'` is used in `exact`
rintro (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩) <;> rw [mem_offDiag] <;> exact ⟨‹_›, ‹_›, ‹_›⟩
rw [this, card_insert_of_not_mem, card_singleton]
simp only [not_and, Prod.mk_inj, mem_singleton]
exact fun _ => hxy'
/-- The `offDiag` of `s : Finset α` is sent on a finset of `Sym2 α` of card `#... | Mathlib/Data/Sym/Card.lean | 143 | 161 |
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Edward Ayers
-/
import Mathlib.CategoryTheory.Limits.Shapes.Pullback.HasPullback
import Mathlib.Data.Set.BooleanAlgebra
/-!
# Theory of sieves
- For an object `X` of a ca... | @[simps]
def sieveOfSubfunctor {R} (f : R ⟶ yoneda.obj X) : Sieve X where
arrows Y g := ∃ t, f.app (Opposite.op Y) t = g
downward_closed := by
rintro Y Z _ ⟨t, rfl⟩ g
refine ⟨R.map g.op t, ?_⟩
rw [FunctorToTypes.naturality _ _ f]
simp
| Mathlib/CategoryTheory/Sites/Sieves.lean | 860 | 867 |
/-
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.... |
This is essentially the Lie module equivalent of the fact that a central
extension of nilpotent Lie algebras is nilpotent. See `LieAlgebra.nilpotent_of_nilpotent_quotient`
below for the corresponding result for Lie algebras. -/
theorem nilpotentOfNilpotentQuotient {N : LieSubmodule R L M} (h₁ : N ≤ maxTrivSubmodule R ... | Mathlib/Algebra/Lie/Nilpotent.lean | 357 | 362 |
/-
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... |
variable [SMulCommClass R S S]
theorem smeval_mul : (p * q).smeval x = p.smeval x * q.smeval x := by
induction p using Polynomial.induction_on' with
| add r s hr hs => simp only [add_comp, hr, hs, smeval_add, add_mul]
| Mathlib/Algebra/Polynomial/Smeval.lean | 264 | 269 |
/-
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.NatAntidiagonal
import Mathlib.Topology.Algebra.InfiniteSum.Constructions
import Mathlib.Topology.Algebra.Ring.Basic
/-!
# Infini... | theorem tsum_mul_left [T2Space α] : ∑' x, a * f x = a * ∑' x, f x := by
classical
| Mathlib/Topology/Algebra/InfiniteSum/Ring.lean | 93 | 94 |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Data.Set.Equitable
import Mathlib.Logic.Equiv.Fin.Basic
import Mathlib.Order.Partition.Fi... | Mathlib/Order/Partition/Equipartition.lean | 180 | 182 | |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | Mathlib/SetTheory/Cardinal/Basic.lean | 1,821 | 1,823 | |
/-
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.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Rat
import Mathlib.Algebra.Ring.Int.Parity
import Mathlib.Data.PNat.Defs... | (q₁ * q₂).num = q₁.num * q₂.num / Nat.gcd (q₁.num * q₂.num).natAbs (q₁.den * q₂.den) := by
rw [mul_def, normalize_eq]
theorem mul_den (q₁ q₂ : ℚ) :
| Mathlib/Data/Rat/Lemmas.lean | 87 | 90 |
/-
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
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.Special... | hf.rpow contDiff_const h
variable {m : ℕ}
| Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean | 565 | 567 |
/-
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... | hf (h₂ <| h₁ h) h (hf' (h₁ h))
theorem eqOn_of_leftInvOn_of_rightInvOn (h₁ : LeftInvOn f₁' f s) (h₂ : RightInvOn f₂' f t)
(h : MapsTo f₂' t s) : EqOn f₁' f₂' t := fun y hy =>
| Mathlib/Data/Set/Function.lean | 843 | 846 |
/-
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.Order.Filter.AtTopBot.Archimedean
import Mathlib.Order.Filter.AtTopBot.Finite
import Mathlib.Order.Filter.AtTopBot.Prod
import Mathlib.Topology.A... | | zero => simp only [Nat.cast_zero, zero_mul, zero_add]; rfl
| succ k IH =>
calc
u ((k + 1) * n + r) = u (n + (k * n + r)) := by congr 1; ring
_ ≤ u n + u (k * n + r) := h _ _
_ ≤ u n + (k * u n + u r) := add_le_add_left IH _
_ = (k + 1 : ℕ) * u n + u r := by simp; ring
include h in
| Mathlib/Analysis/Subadditive.lean | 51 | 59 |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.AlgebraicTopology.DoldKan.PInfty
/-!
# Decomposition of the Q endomorphisms
In this file, we obtain a lemma `decomposition_Q` which expresses
explicitly the p... | Mathlib/AlgebraicTopology/DoldKan/Decomposition.lean | 150 | 155 | |
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Data.Matrix.Mul
import Mathlib.Data.PEquiv
/-!
# partial equivalences for matrices
Using partial equivalences to represent matrices.
This file introduces... | simp only [transpose, mem_iff_mem f, toMatrix_apply]
congr
@[simp]
theorem toMatrix_refl [DecidableEq n] [Zero α] [One α] :
((PEquiv.refl n).toMatrix : Matrix n n α) = 1 := by
ext
simp [toMatrix_apply, one_apply]
@[simp]
| Mathlib/Data/Matrix/PEquiv.lean | 84 | 93 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
import Mathlib.LinearAlgebra.Orientation
/-!
# Orientations of real inner product spaces.
Th... | · ext
simp
end VolumeForm
end Orientation
| Mathlib/Analysis/InnerProductSpace/Orientation.lean | 315 | 328 |
/-
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.SetTheory.Cardinal.Finite
import Mathlib.Data.Set.Finite.Powerset
/-!
# Noncomputable Set Cardinality
We define the cardinality of set `s` as a term `Set... | convert (ncard_image_of_injective _ (@Subtype.coe_injective _ P)).symm
ext x
simp [← and_assoc, exists_eq_right]
| Mathlib/Data/Set/Card.lean | 710 | 713 |
/-
Copyright (c) 2023 Jeremy Tan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Tan
-/
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
import Mathlib.Combinatorics.SimpleGraph.Subgraph
/-!
# Local graph operations
... | simp only [incidenceFinset, neighborFinset, ← Set.toFinset_diff, ← Set.toFinset_image,
← Set.toFinset_union]
exact Set.toFinset_congr (G.edgeSet_replaceVertex_of_not_adj hn)
theorem edgeFinset_replaceVertex_of_adj (ha : G.Adj s t) : (G.replaceVertex s t).edgeFinset =
| Mathlib/Combinatorics/SimpleGraph/Operations.lean | 98 | 102 |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Operations
import Mathlib.Analysis.Normed.Module.FiniteDimension
/-!
# Higher differentiability in ... | theorem contDiffOn_succ_iff_fderiv_apply [FiniteDimensional 𝕜 D] (hs : UniqueDiffOn 𝕜 s) :
ContDiffOn 𝕜 (n + 1) f s ↔
DifferentiableOn 𝕜 f s ∧ (n = ω → AnalyticOn 𝕜 f s) ∧
∀ y, ContDiffOn 𝕜 n (fun x => fderivWithin 𝕜 f s x y) s := by
rw [contDiffOn_succ_iff_fderivWithin hs, contDiffOn_clm_apply... | Mathlib/Analysis/Calculus/ContDiff/FiniteDimension.lean | 71 | 75 |
/-
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.Algebra.GCDMonoid.Basic
import Mathlib.RingTheory.IntegralClosure.IntegrallyClosed
import Mathlib.RingTheory.Polynomial.Eisenstein.Basic
/-!
# GCD domains ... | obtain ⟨x, ⟨y, hy⟩, rfl⟩ := IsLocalization.mk'_surjective M z
obtain ⟨x', y', hx', hy', hu⟩ := extract_gcd x y
use x', y', hu
rw [mul_comm, IsLocalization.mul_mk'_eq_mk'_of_mul]
convert IsLocalization.mk'_mul_cancel_left (M := M) (S := A) _ _ using 2
rw [Subtype.coe_mk, hy', ← mul_comm y', mul_assoc]; conv_... | Mathlib/Algebra/GCDMonoid/IntegrallyClosed.lean | 23 | 30 |
/-
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.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
/-!
# Triangle inequality for `Lp`-seminorm
In this file w... | Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean | 180 | 182 | |
/-
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.Topology.MetricSpace.Dilation
/-!
# Dilation equivalence
In this file we define `DilationEquiv X Y`, a type of bundled equivalences between `X` and... | · have hY : ∀ x y : Y, edist x y = 0 ∨ edist x y = ∞ := e.surjective.forall₂.2 fun x y ↦ by
refine (hX x y).imp (fun h ↦ ?_) fun h ↦ ?_ <;> simp [*, Dilation.ratio_ne_zero]
simp [Dilation.ratio_of_trivial, *]
push_neg at hX
exact (Dilation.ratio_comp' (g := e'.toDilation) (f := e.toDilation) hX).trans (... | Mathlib/Topology/MetricSpace/DilationEquiv.lean | 125 | 132 |
/-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Simon Hudon, Kenny Lau
-/
import Mathlib.Data.Multiset.Bind
import Mathlib.Control.Traversable.Lemmas
import Mathlib.Control.Traversable.Instances
/-!
# F... | (g : α → β) (h : β → G γ) (x : Multiset α) : traverse h (map g x) = traverse (h ∘ g) x := by
refine Quotient.inductionOn x ?_
intro
simp only [traverse, quot_mk_to_coe, map_coe, lift_coe, Function.comp_apply]
rw [← Traversable.traverse_map h g, List.map_eq_map]
theorem naturality {G H : Type _ → Type _} [A... | Mathlib/Data/Multiset/Functor.lean | 113 | 120 |
/-
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.CharP.Defs
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.P... | nth_rw 2 [(sum_C_mul_X_pow_eq f).symm]
refine Submodule.sum_mem _ fun n _hn => ?_
dsimp
have : C (coeff f n) ∈ p := by
apply subset_span
rw [mem_setOf_eq]
use n
have : monomial n (1 : R) • C (coeff f n) ∈ p := p.smul_mem _ this
| Mathlib/RingTheory/Polynomial/Basic.lean | 725 | 732 |
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Yaël Dillies
-/
import Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap
/-!
# Integral average of a function
In this file we define `MeasureTheory.average... | /-- **First moment method**. An integrable function is greater than its integral on a set of
positive measure. -/
theorem measure_integral_le_pos (hf : Integrable f μ) : 0 < μ {x | ∫ a, f a ∂μ ≤ f x} := by
simpa only [average_eq_integral] using
| Mathlib/MeasureTheory/Integral/Average.lean | 573 | 576 |
/-
Copyright (c) 2021 Martin Zinkevich. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Martin Zinkevich, Rémy Degenne
-/
import Mathlib.Logic.Encodable.Lattice
import Mathlib.MeasureTheory.MeasurableSpace.Defs
import Mathlib.Order.Disjointed
/-!
# Indu... | rintro _ ⟨a, ha, rfl⟩ _ ⟨b, hb, rfl⟩ -
exact ⟨a ⊓ b, inf_ind a b ha hb, Iio_inter_Iio.symm⟩
| Mathlib/MeasureTheory/PiSystem.lean | 132 | 134 |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Kim Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Degree.Domain
import Mathlib.Algebra.Polynomial.Degree.Support
import Mathlib.Algebra.Poly... | exact derivative_C
@[simp]
theorem derivative_ofNat (n : ℕ) [n.AtLeastTwo] :
derivative (ofNat(n) : R[X]) = 0 :=
derivative_natCast
| Mathlib/Algebra/Polynomial/Derivative.lean | 198 | 204 |
/-
Copyright (c) 2022 David Kurniadi Angdinata. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Kurniadi Angdinata
-/
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Tactic.IntervalCases
/-!
# Cubics and discriminants
This file defines cubic polynomials ... | Mathlib/Algebra/CubicDiscriminant.lean | 578 | 582 | |
/-
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.Control.Combinators
import Mathlib.Logic.Function.Defs
import Mathlib.Tactic.CasesM
import Mathlib.Tactic.Attr.Core
/-!
Extends the theory on functors... | variable {β' γ' : Type v}
variable {m' : Type v → Type w} [Monad m']
/-- Takes a value `β` and `List α` and accumulates pairs according to a monadic function `f`.
Accumulation occurs from the right (i.e., starting from the tail of the list). -/
| Mathlib/Control/Basic.lean | 86 | 90 |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Polynomial.Degree.Domain
import Mathlib.Algebra.Ring.NonZeroDivisors
import Mathlib.RingTheory.Localization.FractionRing
/-!
# The field of rational... |
If `q = 0`, then `mk` returns 0.
This is an auxiliary definition used to define an `Algebra` structure on `RatFunc`;
the `simp` normal form of `mk p q` is `algebraMap _ _ p / algebraMap _ _ q`.
-/
protected irreducible_def mk (p q : K[X]) : RatFunc K :=
| Mathlib/FieldTheory/RatFunc/Defs.lean | 130 | 136 |
/-
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... | cos_antiperiodic.sub_eq x
theorem cos_sub_two_pi (x : ℂ) : cos (x - 2 * π) = cos x :=
| Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 1,074 | 1,076 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.Interval.Set.Defs
/-!
# Intervals
In any pr... | Mathlib/Order/Interval/Set/Basic.lean | 1,235 | 1,236 | |
/-
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 Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Init
import Mathlib.Data.Int.Init
import Mathlib.... | (hmul : ∀ {a b c}, r a b → r b c → p a → p b → p c → f a c = f a b * f b c) {a b c : α}
| Mathlib/Algebra/Group/Basic.lean | 1,041 | 1,041 |
/-
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.Closeds
/-!
# Noetherian space
A Noetherian space is a topological space that satisfies any of the ... |
/-- In a Noetherian space, every closed set is a finite union of irreducible closed sets. -/
theorem NoetherianSpace.exists_finite_set_closeds_irreducible [NoetherianSpace α] (s : Closeds α) :
∃ S : Set (Closeds α), S.Finite ∧ (∀ t ∈ S, IsIrreducible (t : Set α)) ∧ s = sSup S := by
apply wellFounded_lt.induction... | Mathlib/Topology/NoetherianSpace.lean | 150 | 155 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Group.Unbundled.Basic
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Or... | Mathlib/Algebra/Order/Group/Defs.lean | 910 | 911 | |
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Joël Riou
-/
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.ConeCategory
/-!
... |
variable [HasCoproduct I.left] [HasCoproduct I.right]
| Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean | 609 | 611 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Hom.Set
/-!
# Lemmas about images of inte... |
@[simp]
| Mathlib/Order/Interval/Set/OrderIso.lean | 68 | 69 |
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
/-!
# Oriented angles in right-angled triangles.
T... | have hs : (o.oangle y (y - x)).sign = 1 := by
rw [oangle_sign_sub_right_swap, h, Real.Angle.sign_coe_pi_div_two]
rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.sin_coe,
InnerProductGeometry.sin_angle_sub_mul_norm_of_inner_eq_zero
(o.inner_rev_eq_zero_of_oangle_eq_pi_div_two h)]
/-- The sine of a... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 359 | 365 |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Analytic.IsolatedZeros
import Mathlib.Analysis.SpecialFunctions.Complex.CircleMap
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
/-... | have : (∮ z in C(c, R), ((w - c) / (z - c)) ^ 0 * (z - c)⁻¹) = 2 * π * I := by simp [hR.ne']
refine this ▸ hasSum_single _ fun n hn => ?_
simp only [div_eq_mul_inv, mul_pow, integral_const_mul, mul_assoc]
rw [(integral_congr hR.le fun z hz => _).trans (H n hn), mul_zero]
intro z _
rw [← pow_succ, ← zpow_nat... | Mathlib/MeasureTheory/Integral/CircleIntegral.lean | 584 | 608 |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Batteries.Data.List.Perm
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.TakeWhile
import Mathlib.Order.Fin.Basic
/-!
# So... | @[simp]
theorem sorted_nil : Sorted r [] :=
Pairwise.nil
| Mathlib/Data/List/Sort.lean | 59 | 61 |
/-
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... |
end WellFounded
| Mathlib/Order/Monotone/Basic.lean | 255 | 256 |
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Set.Lattice
import Mathlib.Order.ConditionallyCompleteLattice.Defs
/-!
# Theory of conditionally complete lattices
A conditionally complet... | s.Nonempty → BddAbove s → t.Nonempty → BddBelow t → sInf (image2 u s t) = u (sSup s) (sInf t) :=
csInf_image2_eq_csInf_csInf (α := αᵒᵈ) h₁ h₂
| Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | 858 | 860 |
/-
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... | classical exact IsSplitEpi.mk' { section_ := Pi.lift <| Pi.single b (𝟙 _) }
/-- In the presence of zero morphisms, coprojections into a coproduct are (split) monomorphisms. -/
instance isSplitMono_coprod_inl [HasZeroMorphisms C] {X Y : C} [HasColimit (pair X Y)] :
IsSplitMono (coprod.inl : X ⟶ X ⨿ Y) :=
IsSpl... | Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | 556 | 562 |
/-
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 | 455 | 455 | |
/-
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.Data.Nat.Gcd
import Mathlib.Algebra.Group.Nat.Units
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.GroupWi... | h.coprime_dvd_right hb
| Mathlib/Data/Nat/GCD/Basic.lean | 167 | 168 |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Yury Kudryashov, Neil Strickland
-/
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Opposites
import Mathlib.Algebra.GroupWit... | right_distrib := hf.forall₃.2 fun x y z => by simp only [← add, ← mul, right_distrib]
/-- Pushforward a `Distrib` instance along a surjective function. -/
| Mathlib/Algebra/Ring/InjSurj.lean | 211 | 213 |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Logic.Encodable.Pi
import Mathlib.Logic.Function.Iterate
/-!
# The primitive recursive functions
The primitive ... | -- I added it because it actually simplified the proofs
-- and because I couldn't understand the original proof
/-- A function is `PrimrecBounded` if its size is bounded by a primitive recursive function -/
def PrimrecBounded (f : α → β) : Prop :=
| Mathlib/Computability/Primrec.lean | 659 | 662 |
/-
Copyright (c) 2021 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.Module.Algebra
import Mathlib.Algebra.Ring.Subring.Units
import Mathlib.LinearAlgebra.LinearIndepende... | SameRay R (r • v) v ↔ 0 < r :=
SameRay.sameRay_comm.trans (sameRay_smul_right_iff_of_ne hv hr)
| Mathlib/LinearAlgebra/Ray.lean | 476 | 478 |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.BaseChange
import Mathlib.Algebra.Lie.IdealOperations
import Mathlib.Order.Hom.Basic
import Mathlib.RingTheory... | @[simp] theorem derivedSeriesOfIdeal_baseChange {A : Type*} [CommRing A] [Algebra R A] (k : ℕ) :
derivedSeriesOfIdeal A (A ⊗[R] L) k (I.baseChange A) =
| Mathlib/Algebra/Lie/Solvable.lean | 127 | 128 |
/-
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.Integral.Bochner.ContinuousLinearMap
import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed
import Mathlib.MeasureTheory.Measure.Prod
impo... | rw [coeFn_smul, Pi.smul_apply]
/-- Restrict a finite measure μ to a set A. -/
def restrict (μ : FiniteMeasure Ω) (A : Set Ω) : FiniteMeasure Ω where
val := (μ : Measure Ω).restrict A
| Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 254 | 258 |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Control.Basic
import Mathlib.Data.Nat.Basic
import Mathlib.Data.Option.Basic
im... | Mathlib/Data/List/Basic.lean | 1,376 | 1,377 | |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Bundle
import Mathlib.Data.Set.Image
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Topology.O... |
theorem target_inter_preimage_symm_source_eq (e f : Pretrivialization F proj) :
f.target ∩ f.toPartialEquiv.symm ⁻¹' e.source = (e.baseSet ∩ f.baseSet) ×ˢ univ := by
| Mathlib/Topology/FiberBundle/Trivialization.lean | 175 | 177 |
/-
Copyright (c) 2024 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nathaniel Thomas, Jeremy Avigad, Johannes Hölzl, Mario Carneiro, Andrew Yang,
Johannes Hölzl, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.Algebra.Module.Submodule.Lattice
import Ma... | def restrictScalarsLatticeHom : CompleteLatticeHom (Submodule R M) (Submodule S M) where
toFun := restrictScalars S
| Mathlib/Algebra/Module/Submodule/RestrictScalars.lean | 106 | 107 |
/-
Copyright (c) 2023 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.Canonical
import Mathlib.CategoryTheory.Sites.Coherent.Basic
import Mathlib.CategoryTheory.Sites.EffectiveEpimorphic
/-!
# Sheaves for th... | theorem isSheaf_yoneda_obj (W : C) : Presieve.IsSheaf (coherentTopology C) (yoneda.obj W) := by
rw [isSheaf_coherent]
intro X α _ Y π H
have h_colim := isColimitOfEffectiveEpiFamilyStruct Y π H.effectiveEpiFamily.some
rw [← Sieve.generateFamily_eq] at h_colim
intro x hx
let x_ext := Presieve.FamilyOfElement... | Mathlib/CategoryTheory/Sites/Coherent/CoherentSheaves.lean | 44 | 58 |
/-
Copyright (c) 2023 Bhavik Mehta, Rishi Mehta, Linus Sommer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Rishi Mehta, Linus Sommer
-/
import Mathlib.Algebra.GroupWithZero.Nat
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Combinatorics.SimpleG... | lemma IsHamiltonian.map {H : SimpleGraph β} (f : G →g H) (hf : Bijective f) (hp : p.IsHamiltonian) :
(p.map f).IsHamiltonian := by
simp [IsHamiltonian, hf.surjective.forall, hf.injective, hp _]
| Mathlib/Combinatorics/SimpleGraph/Hamiltonian.lean | 34 | 36 |
/-
Copyright (c) 2023 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Discriminant
import Mathlib.RingTheory.DedekindDomain.IntegralClosure
import Mathlib.NumberTheory.K... |
@[simp]
lemma traceDual_bot :
| Mathlib/RingTheory/DedekindDomain/Different.lean | 83 | 85 |
/-
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.Primrec
import Mathlib.Data.Nat.PSub
import Mathlib.Data.PFun
/-!
# The partial recursive functions
The partial recursive functions are... | theorem unpair : Computable Nat.unpair :=
Primrec.unpair.to_comp
theorem succ : Computable Nat.succ :=
| Mathlib/Computability/Partrec.lean | 291 | 294 |
/-
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.GroupWithZero.Hom
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.Algebra.Ring.Defs
import Mathlib.Data.Nat.Lattice
/-!
# Definition... |
lemma pow_pred_nilpotencyClass [Nontrivial R] (hx : IsNilpotent x) :
x ^ (nilpotencyClass x - 1) ≠ 0 :=
(nilpotencyClass_eq_succ_iff.mp <| Nat.eq_add_of_sub_eq (pos_nilpotencyClass_iff.mpr hx) rfl).2
lemma eq_zero_of_nilpotencyClass_eq_one (hx : nilpotencyClass x = 1) :
| Mathlib/RingTheory/Nilpotent/Defs.lean | 147 | 152 |
/-
Copyright (c) 2022 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.FiniteDimensional
import Mathlib.Da... | · simpa [real_inner_comm] using h
· simp_all
| Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 143 | 144 |
/-
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 ... |
@[deprecated (since := "2025-01-21")] alias condexpIndL1Fin_smul := condExpIndL1Fin_smul
theorem condExpIndL1Fin_smul' [NormedSpace ℝ F] [SMulCommClass ℝ 𝕜 F] (hs : MeasurableSet s)
(hμs : μ s ≠ ∞) (c : 𝕜) (x : F) :
condExpIndL1Fin hm hs hμs (c • x) = c • condExpIndL1Fin hm hs hμs x := by
ext1
refine (M... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean | 116 | 125 |
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.Data.ENNReal.Action
import Mathlib.Topology.Algebra.UniformMulAction
import Mathlib.Topology.MetricSpace.Algebra
... | IsBoundedSMul α β :=
{ dist_smul_pair' := fun a b₁ b₂ => by simpa [smul_sub, dist_eq_norm] using h a (b₁ - b₂)
dist_pair_smul' := fun a₁ a₂ b => by simpa [sub_smul, dist_eq_norm] using h (a₁ - a₂) b }
| Mathlib/Analysis/Normed/MulAction.lean | 74 | 77 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Notation.Pi
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Filter.Defs
/-!
# Theory of... | Mathlib/Order/Filter/Basic.lean | 2,797 | 2,799 | |
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.Data.Fintype.Order
import Mathlib.MeasureTheory.Function.AEEqFun
import Mathlib.Me... | by_cases hc0 : c = 0
· simp [hc0]
rw [eLpNorm_smul_measure_of_ne_zero hc0, smul_eq_mul]
refine ENNReal.mul_lt_top (Ne.lt_top ?_) hf.2
simp [hc, hc0]
@[deprecated (since := "2025-02-21")]
alias Memℒp.of_measure_le_smul := MemLp.of_measure_le_smul
theorem MemLp.smul_measure {f : α → ε} {c : ℝ≥0∞} (hf : MemLp ... | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 894 | 904 |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Algebra.Group.Fin.Tuple
import Mathlib.Data.Matrix.RowCol
import Mathlib.Data.Fin.VecNotation
import Mathlib.Tactic.FinCases
import Mathlib.Alge... |
@[simp]
theorem mul_empty [Fintype n'] (A : Matrix m' n' α) (B : Matrix n' (Fin 0) α) :
A * B = of fun _ => ![] :=
funext fun _ => empty_eq _
| Mathlib/Data/Matrix/Notation.lean | 266 | 271 |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Jeremy Avigad, Simon Hudon
-/
import Mathlib.Algebra.Notation.Defs
import Mathlib.Data.Set.Subsingleton
import Mathlib.Logic.Equiv.Defs
/-!
# Partial values of a type
... | theorem bind_le {α} (x : Part α) (f : α → Part β) (y : Part β) :
x >>= f ≤ y ↔ ∀ a, a ∈ x → f a ≤ y := by
| Mathlib/Data/Part.lean | 525 | 526 |
/-
Copyright (c) 2023 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.BigOperators.Group.Finset.Piecewise
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Pi
import Mathlib.Algebra.O... | (mul_nonneg (hf₀ _) <| hg₀ _) <| mul_nonneg (hμ₀ _) <| hμ₀ _
end DistribLattice
open Booleanisation
variable [DecidableEq α] [GeneralizedBooleanAlgebra α]
/-- A slight generalisation of the **Marica-Schönheim Inequality**. -/
lemma Finset.le_card_diffs_mul_card_diffs (s t : Finset α) :
#s * #t ≤ #(s \\ t) *... | Mathlib/Combinatorics/SetFamily/FourFunctions.lean | 352 | 362 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Order.Filter.SmallSets
import Mathlib.Topology.UniformSpace.Defs
import Mathlib.Topology.ContinuousOn
/-!
# Basic resu... | Mathlib/Topology/UniformSpace/Basic.lean | 1,870 | 1,880 | |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
import Mathlib.Analysis.SpecialFunctions.PolarCoord
import Mathlib.Analysis.Complex.Convex
import Mathlib.D... | refine intervalIntegral_tendsto_integral_Iic _ ?_ tendsto_neg_atTop_atBot
apply (integrable_cexp_neg_mul_sq hb).integrableOn
exact tendsto_nhds_unique t2 t1
-- The Gaussian integral on the half-line, `∫ x in Ioi 0, exp (-b * x^2)`, for real `b`.
theorem integral_gaussian_Ioi (b : ℝ) :
∫ x in Ioi (0 : ℝ),... | Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean | 312 | 334 |
/-
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.Algebra.Tower
import Mathlib.Algebra.Field.IsField
import Mathlib.Algebra.GroupWithZero.N... | Mathlib/RingTheory/Localization/Basic.lean | 1,178 | 1,181 | |
/-
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... |
theorem log_le_log_iff (h : 0 < x) (h₁ : 0 < y) : log x ≤ log y ↔ x ≤ y := by
rw [← exp_le_exp, exp_log h, exp_log h₁]
| Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 137 | 139 |
/-
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... | simp
theorem image_id (s : Set α) : id '' s = s := by simp
| Mathlib/Data/Set/Image.lean | 309 | 311 |
/-
Copyright (c) 2018 . All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.SpecificGroups.Cyclic
/-!
# p-groups
This file contains a proof that if `G` is a `p`-group ac... |
namespace ZModModule
variable {n : ℕ} {G : Type*} [AddCommGroup G] [Module (ZMod n) G]
lemma isPGroup_multiplicative : IsPGroup n (Multiplicative G) := by
simpa [IsPGroup, Multiplicative.forall] using
fun _ ↦ ⟨1, by simp [← ofAdd_nsmul, ZModModule.char_nsmul_eq_zero]⟩
end ZModModule
| Mathlib/GroupTheory/PGroup.lean | 377 | 392 |
/-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux, Yaël Dillies
-/
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.Order.Monoid.Submonoid
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Or... | closure (range fun x : ℚ≥0 ↦ x ^ n) = ⊤ := by
refine (eq_top_iff' _).2 fun x ↦ ?_
suffices x = (x.num * x.den ^ (n - 1)) • (x.den : ℚ≥0)⁻¹ ^ n by
rw [this]
exact nsmul_mem (subset_closure <| mem_range_self _) _
rw [nsmul_eq_mul]
push_cast
rw [mul_assoc, pow_sub₀, pow_one, mul_right_comm, ← mul_pow... | Mathlib/Data/Rat/Star.lean | 26 | 36 |
/-
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.Ring.Nat
import Mathlib.Data.List.Chain
/-!
# List of booleans
In this file we prove lemmas about the number of `false`s and `true`s ... | Mathlib/Data/Bool/Count.lean | 132 | 135 | |
/-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Homeomorph.Lemmas
import Mathlib.Topology.Sets.Opens
/-!
# The OnePoint Compactification
We ... | @[simp]
theorem isClosed_image_coe {s : Set X} :
IsClosed ((↑) '' s : Set (OnePoint X)) ↔ IsClosed s ∧ IsCompact s := by
rw [← isOpen_compl_iff, isOpen_compl_image_coe]
| Mathlib/Topology/Compactification/OnePoint.lean | 248 | 251 |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth, Mitchell Lee
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Ri... |
theorem U_eq (n : ℤ) : U R n = 2 * X * U R (n - 1) - U R (n - 2) := by
| Mathlib/RingTheory/Polynomial/Chebyshev.lean | 184 | 185 |
/-
Copyright (c) 2024 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Order.Hom.Ring
import Mathlib.Data.ENat.Basic
import Mathlib.SetTheory.Cardinal.Basic
/-!
# Conversion between `Cardinal` and `ℕ∞`
In this ... | Mathlib/SetTheory/Cardinal/ENat.lean | 94 | 94 | |
/-
Copyright (c) 2022 Antoine Labelle, Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle, Rémi Bottinelli
-/
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Combinatorics.Quiver.Path
/-!
# Rewriting arrows and paths along vertex... |
open Path
| Mathlib/Combinatorics/Quiver/Cast.lean | 69 | 72 |
/-
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 | 1,085 | 1,089 | |
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.BoxIntegral.Basic
import Mathlib.Analysis.BoxIntegral.Partition.Additive
import Mathlib.Analysis.Calculus.FDeriv.Prod
/-!
# Divergence inte... | (hε : ∀ y ∈ (Box.Icc I), ‖f y - a - f' (y - x)‖ ≤ ε * ‖y - x‖) {c : ℝ≥0}
(hc : I.distortion ≤ c) :
‖(∏ j, (I.upper j - I.lower j)) • f' (Pi.single i 1) -
(integral (I.face i) ⊥ (f ∘ i.insertNth (α := fun _ ↦ ℝ) (I.upper i)) BoxAdditiveMap.volume -
integral (I.face i) ⊥ (f ∘ i.insertNth (α := f... | Mathlib/Analysis/BoxIntegral/DivergenceTheorem.lean | 65 | 136 |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Countable.Small
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Powerset
import Mathlib.Dat... | theorem one_le_iff_pos {c : Cardinal} : 1 ≤ c ↔ 0 < c := by
rw [← succ_zero, succ_le_iff]
theorem one_le_iff_ne_zero {c : Cardinal} : 1 ≤ c ↔ c ≠ 0 := by
| Mathlib/SetTheory/Cardinal/Basic.lean | 304 | 307 |
/-
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... | · exact not_cliqueFree_of_infinite V f
· have : Fintype ι := fintypeOfNotInfinite h
rw [ENat.card_eq_coe_fintype_card, Nat.cast_le] at hc
exact not_cliqueFree_of_le_card V f hc
end completeMultipartiteGraph
/-- Clique-freeness is preserved by `replaceVertex`. -/
protected theorem CliqueFree.replaceVertex ... | Mathlib/Combinatorics/SimpleGraph/Clique.lean | 427 | 454 |
/-
Copyright (c) 2019 Kim Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
/-!
# Intervals as finsets
This file provides basic results about all the `Finset.Ixx... | theorem uIcc_subset_uIcc_iff_le :
[[a₁, b₁]] ⊆ [[a₂, b₂]] ↔ min a₂ b₂ ≤ min a₁ b₁ ∧ max a₁ b₁ ≤ max a₂ b₂ :=
uIcc_subset_uIcc_iff_le'
| Mathlib/Order/Interval/Finset/Basic.lean | 1,062 | 1,065 |
/-
Copyright (c) 2022 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Leonardo de Moura
-/
import Mathlib.Data.Set.Basic
/-!
# Indicator function valued in bool
See also `Set.indicator` and `Set.piecewise`.
-/
assert_not_exists Rel... | Mathlib/Data/Set/BoolIndicator.lean | 54 | 58 | |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique
import Mathlib.MeasureTheory.Function.L2Space
/-... |
@[deprecated (since := "2025-01-21")]
alias condexpL2_indicator_eq_toSpanSingleton_comp := condExpL2_indicator_eq_toSpanSingleton_comp
variable {𝕜}
theorem setLIntegral_nnnorm_condExpL2_indicator_le (hm : m ≤ m0) (hs : MeasurableSet s)
(hμs : μ s ≠ ∞) (x : E') {t : Set α} (ht : MeasurableSet[m] t) (hμt : μ t ≠ ... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean | 357 | 365 |
/-
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... | theorem sigma_subset_preimage_fst (s : Set ι) (t : ∀ i, Set (α i)) : s.sigma t ⊆ Sigma.fst ⁻¹' s :=
fun _ ↦ And.left
| Mathlib/Data/Set/Sigma.lean | 223 | 225 |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.BigOperators.Field
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.InnerProductSpace.Defs
impor... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 1,562 | 1,566 | |
/-
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.Data.Countable.Basic
import Mathlib.Data.Set.Finite.Basic
import Mathlib.Data.Set.Subsingleton
import Mathlib.Logic.Equiv.List
/-!
# Countable sets
I... | theorem Finite.countable {s : Set α} (hs : s.Finite) : s.Countable :=
have := hs.to_subtype; s.to_countable
| Mathlib/Data/Set/Countable.lean | 241 | 242 |
/-
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... | instances of `HasZeroMorphisms` to exist at all.
See, particularly, the note on `zeroMorphismsOfZeroObject` below.
-/
theorem ext (I J : HasZeroMorphisms C) : I = J := by
apply ext_aux
intro X Y
have : (I.zero X Y).zero ≫ (J.zero Y Y).zero = (I.zero X Y).zero := by
apply I.zero_comp X (J.zero Y Y).zero
hav... | Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | 88 | 97 |
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Analysis.InnerProductSpace.Spectrum
import Mathlib.Data.Matrix.Rank
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.M... | (hA.eigenvalues i) = RCLike.re (dotProduct (star ⇑(hA.eigenvectorBasis i))
(A *ᵥ ⇑(hA.eigenvectorBasis i))) := by
rw [dotProduct_comm]
simp only [mulVec_eigenvectorBasis, smul_dotProduct, ← EuclideanSpace.inner_eq_star_dotProduct,
inner_self_eq_norm_sq_to_K, RCLike.smul_re, hA.eigenvectorBasis.orthonorm... | Mathlib/LinearAlgebra/Matrix/Spectrum.lean | 123 | 128 |
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Mario Carneiro, Johan Commelin
-/
import Mathlib.NumberTheory.Padics.PadicNumbers
import Mathlib.RingTheory.DiscreteValuationRing.Basic
/-!
# p-adic integers
This f... | def ofIntSeq (seq : ℕ → ℤ) (h : IsCauSeq (padicNorm p) fun n => seq n) : ℤ_[p] :=
| Mathlib/NumberTheory/Padics/PadicIntegers.lean | 145 | 145 |
/-
Copyright (c) 2023 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta, Doga Can Sertbas
-/
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.ModEq
import Mathlib.Data.Nat.Prime.Defs
import Mathlib.Data.Real.Archimedea... | /--
If `k` is omitted from the set, its Schnirelmann density is upper bounded by `1 - k⁻¹`.
-/
lemma schnirelmannDensity_le_of_not_mem {k : ℕ} (hk : k ∉ A) :
schnirelmannDensity A ≤ 1 - (k⁻¹ : ℝ) := by
rcases k.eq_zero_or_pos with rfl | hk'
· simpa using schnirelmannDensity_le_one
apply schnirelmannDensity_le... | Mathlib/Combinatorics/Schnirelmann.lean | 88 | 101 |
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