Context stringlengths 285 157k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen, Wen Yang
-/
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mat... | Mathlib/LinearAlgebra/Matrix/Block.lean | 253 | 257 | theorem BlockTriangular.det_fintype [DecidableEq α] [Fintype α] [LinearOrder α]
(h : BlockTriangular M b) : M.det = ∏ k : α, (M.toSquareBlock b k).det := by |
refine h.det.trans (prod_subset (subset_univ _) fun a _ ha => ?_)
have : IsEmpty { i // b i = a } := ⟨fun i => ha <| mem_image.2 ⟨i, mem_univ _, i.2⟩⟩
exact det_isEmpty
|
/-
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, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.RingTheory.Multiplicit... | Mathlib/Algebra/Polynomial/Div.lean | 491 | 495 | theorem not_isField : ¬IsField R[X] := by |
nontriviality R
intro h
letI := h.toField
simpa using congr_arg natDegree (monic_X.eq_one_of_isUnit <| monic_X (R := R).ne_zero.isUnit)
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Part
import Mathlib.Tactic.NormNum
#align_import data.nat.part_enat from "l... | Mathlib/Data/Nat/PartENat.lean | 526 | 531 | theorem add_one_le_of_lt {x y : PartENat} (h : x < y) : x + 1 ≤ y := by |
induction' y using PartENat.casesOn with n
· apply le_top
rcases ne_top_iff.mp (ne_top_of_lt h) with ⟨m, rfl⟩
-- Porting note: was `apply_mod_cast Nat.succ_le_of_lt; apply_mod_cast h`
norm_cast; apply Nat.succ_le_of_lt; norm_cast at h
|
/-
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.LinearAlgebra.QuadraticForm.TensorProduct
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.TensorProduct.Opposite
import... | Mathlib/LinearAlgebra/CliffordAlgebra/BaseChange.lean | 104 | 113 | theorem toBaseChange_comp_involute (Q : QuadraticForm R V) :
(toBaseChange A Q).comp (involute : CliffordAlgebra (Q.baseChange A) →ₐ[A] _) =
(Algebra.TensorProduct.map (AlgHom.id _ _) involute).comp (toBaseChange A Q) := by |
ext v
show toBaseChange A Q (involute (ι (Q.baseChange A) (1 ⊗ₜ[R] v)))
= (Algebra.TensorProduct.map (AlgHom.id _ _) involute :
A ⊗[R] CliffordAlgebra Q →ₐ[A] _)
(toBaseChange A Q (ι (Q.baseChange A) (1 ⊗ₜ[R] v)))
rw [toBaseChange_ι, involute_ι, map_neg (toBaseChange A Q), toBaseChange_ι,
A... |
/-
Copyright (c) 2020 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.Geometry.Manifold.ChartedSpace
#align_import geometry.manifold.local_invariant_properties from "leanprover-community/mathlib"@... | Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | 579 | 585 | theorem liftProp_inclusion {Q : (H → H) → Set H → H → Prop} (hG : LocalInvariantProp G G Q)
(hQ : ∀ y, Q id univ y) {U V : Opens M} (hUV : U ≤ V) :
LiftProp Q (Set.inclusion hUV : U → V) := by |
intro x
show LiftPropAt Q (id ∘ inclusion hUV) x
rw [← hG.liftPropAt_iff_comp_inclusion hUV]
apply hG.liftProp_id hQ
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker, Aaron Anderson
-/
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Algebra.GCDMonoid.Nat
#ali... | Mathlib/RingTheory/Int/Basic.lean | 111 | 118 | theorem prime_two_or_dvd_of_dvd_two_mul_pow_self_two {m : ℤ} {p : ℕ} (hp : Nat.Prime p)
(h : (p : ℤ) ∣ 2 * m ^ 2) : p = 2 ∨ p ∣ Int.natAbs m := by |
cases' Int.Prime.dvd_mul hp h with hp2 hpp
· apply Or.intro_left
exact le_antisymm (Nat.le_of_dvd zero_lt_two hp2) (Nat.Prime.two_le hp)
· apply Or.intro_right
rw [sq, Int.natAbs_mul] at hpp
exact or_self_iff.mp ((Nat.Prime.dvd_mul hp).mp hpp)
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury G. Kudryashov, Scott Morrison
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.Algebra.BigOperators.Finsupp
impor... | Mathlib/Algebra/MonoidAlgebra/Basic.lean | 1,943 | 1,944 | theorem opRingEquiv_symm_single [AddCommMonoid G] (r : kᵐᵒᵖ) (x : Gᵐᵒᵖ) :
AddMonoidAlgebra.opRingEquiv.symm (single x r) = op (single x r.unop) := by | simp
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.MeasureSpace
/-!
# Restricting a measure to a subset or a subtype
Given a measure `μ` on a type `α` and a subse... | Mathlib/MeasureTheory/Measure/Restrict.lean | 294 | 295 | theorem restrict_compl_add_restrict (hs : MeasurableSet s) : μ.restrict sᶜ + μ.restrict s = μ := by |
rw [add_comm, restrict_add_restrict_compl hs]
|
/-
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.Group.Indicator
import Mathlib.Data.Finset.Piecewise
import Mathlib.Data.Finset.Preimage
#align_import algebra.big_operators.basic from "leanp... | Mathlib/Algebra/BigOperators/Group/Finset.lean | 2,551 | 2,551 | theorem ofAdd_list_prod (s : List α) : ofAdd s.sum = (s.map ofAdd).prod := by | simp [ofAdd]; rfl
|
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Scott Morrison
-/
import Mathlib.RingTheory.OrzechProperty
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import linear_a... | Mathlib/LinearAlgebra/InvariantBasisNumber.lean | 158 | 164 | theorem card_le_of_injective [StrongRankCondition R] {α β : Type*} [Fintype α] [Fintype β]
(f : (α → R) →ₗ[R] β → R) (i : Injective f) : Fintype.card α ≤ Fintype.card β := by |
let P := LinearEquiv.funCongrLeft R R (Fintype.equivFin α)
let Q := LinearEquiv.funCongrLeft R R (Fintype.equivFin β)
exact
le_of_fin_injective R ((Q.symm.toLinearMap.comp f).comp P.toLinearMap)
(((LinearEquiv.symm Q).injective.comp i).comp (LinearEquiv.injective P))
|
/-
Copyright (c) 2021 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Tactic.NoncommRing
#align_import algebra.algebra.spectrum from "leanprover-c... | Mathlib/Algebra/Algebra/Spectrum.lean | 372 | 376 | theorem smul_eq_smul [Nontrivial A] (k : 𝕜) (a : A) (ha : (σ a).Nonempty) :
σ (k • a) = k • σ a := by |
rcases eq_or_ne k 0 with (rfl | h)
· simpa [ha, zero_smul_set] using (show {(0 : 𝕜)} = (0 : Set 𝕜) from rfl)
· exact unit_smul_eq_smul a (Units.mk0 k h)
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.CharP.Two
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Periodic
import Mathlib.Data.ZMod.Basic
import Mathlib.Tactic.Monoton... | Mathlib/Data/Nat/Totient.lean | 298 | 306 | theorem totient_mul_prod_primeFactors (n : ℕ) :
(φ n * ∏ p ∈ n.primeFactors, p) = n * ∏ p ∈ n.primeFactors, (p - 1) := by |
by_cases hn : n = 0; · simp [hn]
rw [totient_eq_prod_factorization hn]
nth_rw 3 [← factorization_prod_pow_eq_self hn]
simp only [prod_primeFactors_prod_factorization, ← Finsupp.prod_mul]
refine Finsupp.prod_congr (M := ℕ) (N := ℕ) fun p hp => ?_
rw [Finsupp.mem_support_iff, ← zero_lt_iff] at hp
rw [mul_c... |
/-
Copyright (c) 2018 Rohan Mitta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rohan Mitta, Kevin Buzzard, Alistair Tucker, Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Order.Interval.Set.ProjIcc
import Mathlib.Topology.Algebra.Order.Field
import Mathlib.Topolo... | Mathlib/Topology/MetricSpace/Lipschitz.lean | 354 | 355 | theorem const_max (hf : LocallyLipschitz f) (a : ℝ) : LocallyLipschitz fun x => max a (f x) := by |
simpa [max_comm] using (hf.max_const a)
|
/-
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.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | Mathlib/Topology/Compactification/OnePoint.lean | 140 | 141 | theorem compl_image_coe (s : Set X) : ((↑) '' s : Set (OnePoint X))ᶜ = (↑) '' sᶜ ∪ {∞} := by |
rw [coe_injective.compl_image_eq, compl_range_coe]
|
/-
Copyright (c) 2022 Vincent Beffara. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Vincent Beffara
-/
import Mathlib.Analysis.Complex.RemovableSingularity
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Analysis.NormedSpace.FunctionSeries
#align_... | Mathlib/Analysis/Complex/LocallyUniformLimit.lean | 131 | 135 | theorem exists_cthickening_tendstoUniformlyOn (hf : TendstoLocallyUniformlyOn F f φ U)
(hF : ∀ᶠ n in φ, DifferentiableOn ℂ (F n) U) (hK : IsCompact K) (hU : IsOpen U) (hKU : K ⊆ U) :
∃ δ > 0, cthickening δ K ⊆ U ∧ TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K := by |
obtain ⟨δ, hδ, hKδ⟩ := hK.exists_cthickening_subset_open hU hKU
exact ⟨δ, hδ, hKδ, tendstoUniformlyOn_deriv_of_cthickening_subset hf hF hδ hK hU hKδ⟩
|
/-
Copyright (c) 2014 Robert Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Fiel... | Mathlib/Algebra/Order/Field/Basic.lean | 328 | 329 | theorem div_le_div_iff (b0 : 0 < b) (d0 : 0 < d) : a / b ≤ c / d ↔ a * d ≤ c * b := by |
rw [le_div_iff d0, div_mul_eq_mul_div, div_le_iff b0]
|
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Alex Meiburg
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Polynomial.Degree.Lemmas
#align_import data.polynomial.erase_lead from "leanprover-communi... | Mathlib/Algebra/Polynomial/EraseLead.lean | 110 | 112 | theorem eraseLead_support_card_lt (h : f ≠ 0) : (eraseLead f).support.card < f.support.card := by |
rw [eraseLead_support]
exact card_lt_card (erase_ssubset <| natDegree_mem_support_of_nonzero h)
|
/-
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.BigOperators.Ring
import Mathlib.Algebra.Module.BigOperators
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Squarefree
import Mat... | Mathlib/NumberTheory/ArithmeticFunction.lean | 577 | 580 | theorem pdiv_zeta [DivisionSemiring R] (f : ArithmeticFunction R) :
pdiv f zeta = f := by |
ext n
cases n <;> simp [succ_ne_zero]
|
/-
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.Group.Units.Equiv
import Mathlib.Algebra.Order.Group.Defs
import Mathlib.Order.Hom.Basic
#ali... | Mathlib/Algebra/Order/Group/OrderIso.lean | 104 | 106 | theorem OrderIso.mulRight_symm (a : α) : (OrderIso.mulRight a).symm = OrderIso.mulRight a⁻¹ := by |
ext x
rfl
|
/-
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.Topology.Maps
import Mathlib.Topology.NhdsSet
#align_import topology.constructions from "leanprover-community/mathlib"... | Mathlib/Topology/Constructions.lean | 1,527 | 1,542 | theorem pi_generateFrom_eq_finite {π : ι → Type*} {g : ∀ a, Set (Set (π a))} [Finite ι]
(hg : ∀ a, ⋃₀ g a = univ) :
(@Pi.topologicalSpace ι π fun a => generateFrom (g a)) =
generateFrom { t | ∃ s : ∀ a, Set (π a), (∀ a, s a ∈ g a) ∧ t = pi univ s } := by |
cases nonempty_fintype ι
rw [pi_generateFrom_eq]
refine le_antisymm (generateFrom_anti ?_) (le_generateFrom ?_)
· exact fun s ⟨t, ht, Eq⟩ => ⟨t, Finset.univ, by simp [ht, Eq]⟩
· rintro s ⟨t, i, ht, rfl⟩
letI := generateFrom { t | ∃ s : ∀ a, Set (π a), (∀ a, s a ∈ g a) ∧ t = pi univ s }
refine isOpen_... |
/-
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.Partrec
import Mathlib.Data.Option.Basic
#align_import computability.partrec_code from "leanprover-community/mathlib"@"6155d4351090a6fad... | Mathlib/Computability/PartrecCode.lean | 206 | 209 | theorem encode_lt_comp (cf cg) :
encode cf < encode (comp cf cg) ∧ encode cg < encode (comp cf cg) := by |
have : encode (pair cf cg) < encode (comp cf cg) := by simp [encodeCode_eq, encodeCode]
exact (encode_lt_pair cf cg).imp (fun h => lt_trans h this) fun h => lt_trans h this
|
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
#align_import order.symm_diff from "leanprover-community/mathlib... | Mathlib/Order/SymmDiff.lean | 457 | 458 | theorem le_symmDiff_iff_right : b ≤ a ∆ b ↔ Disjoint a b := by |
rw [symmDiff_comm, le_symmDiff_iff_left, disjoint_comm]
|
/-
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, Yaël Dillies
-/
import Mathlib.Data.Finset.NAry
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Set.Pointwise.Finite
import Mathlib.Data.Set.Pointwise.SMul
... | Mathlib/Data/Finset/Pointwise.lean | 1,287 | 1,289 | theorem image_mul_left' :
image (fun b => a⁻¹ * b) t = preimage t (fun b => a * b) (mul_right_injective _).injOn := by |
simp
|
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Nat.Choose.Vanderm... | Mathlib/Algebra/Polynomial/HasseDeriv.lean | 133 | 134 | theorem hasseDeriv_apply_one (hk : 0 < k) : hasseDeriv k (1 : R[X]) = 0 := by |
rw [← C_1, hasseDeriv_C k _ hk]
|
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Anal... | Mathlib/MeasureTheory/Integral/CircleIntegral.lean | 198 | 199 | theorem deriv_circleMap_eq_zero_iff {c : ℂ} {R : ℝ} {θ : ℝ} :
deriv (circleMap c R) θ = 0 ↔ R = 0 := by | simp [I_ne_zero]
|
/-
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.Group.Subgroup.Actions
import Mathlib.Algebra.Order.Module.Algebra
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.Algebra.Ring.Subri... | Mathlib/LinearAlgebra/Ray.lean | 731 | 735 | theorem exists_pos_right_iff_sameRay_and_ne_zero (hy : y ≠ 0) :
(∃ r : R, 0 < r ∧ x = r • y) ↔ SameRay R x y ∧ x ≠ 0 := by |
rw [SameRay.sameRay_comm]
simp_rw [eq_comm (a := x)]
exact exists_pos_left_iff_sameRay_and_ne_zero hy
|
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Order.Filter.Basic
import Mathlib.Data.PFun
#align_import order.filter.partial from "leanprover-community/mathlib"@"b363547b3113d350d053abdf2884e9850a56... | Mathlib/Order/Filter/Partial.lean | 248 | 251 | theorem tendsto_iff_ptendsto_univ (l₁ : Filter α) (l₂ : Filter β) (f : α → β) :
Tendsto f l₁ l₂ ↔ PTendsto (PFun.res f Set.univ) l₁ l₂ := by |
rw [← tendsto_iff_ptendsto]
simp [principal_univ]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Category.MonCat.Limits
import Mathlib.Algebra.Category.GroupCat.ForgetCorepresentable
import Mathlib.Algebra.Category.GroupCat.Preadditive
impo... | Mathlib/Algebra/Category/GroupCat/Limits.lean | 544 | 545 | theorem kernelIsoKer_hom_comp_subtype {G H : AddCommGroupCat.{u}} (f : G ⟶ H) :
(kernelIsoKer f).hom ≫ AddSubgroup.subtype f.ker = kernel.ι f := by | ext; rfl
|
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 783 | 783 | theorem dotProduct_zero : v ⬝ᵥ 0 = 0 := by | simp [dotProduct]
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Julian Kuelshammer
-/
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Da... | Mathlib/GroupTheory/OrderOfElement.lean | 927 | 931 | theorem exists_zpow_eq_one (x : G) : ∃ (i : ℤ) (_ : i ≠ 0), x ^ (i : ℤ) = 1 := by |
obtain ⟨w, hw1, hw2⟩ := isOfFinOrder_of_finite x
refine ⟨w, Int.natCast_ne_zero.mpr (_root_.ne_of_gt hw1), ?_⟩
rw [zpow_natCast]
exact (isPeriodicPt_mul_iff_pow_eq_one _).mp hw2
|
/-
Copyright (c) 2023 Bulhwi Cha. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bulhwi Cha, Mario Carneiro
-/
import Batteries.Data.Char
import Batteries.Data.List.Lemmas
import Batteries.Data.String.Basic
import Batteries.Tactic.Lint.Misc
import Batteries.Tactic.SeqF... | .lake/packages/batteries/Batteries/Data/String/Lemmas.lean | 166 | 175 | theorem utf8GetAux?_of_valid (cs cs' : List Char) {i p : Nat} (hp : i + utf8Len cs = p) :
utf8GetAux? (cs ++ cs') ⟨i⟩ ⟨p⟩ = cs'.head? := by |
match cs, cs' with
| [], [] => rfl
| [], c::cs' => simp [← hp, utf8GetAux?]
| c::cs, cs' =>
simp [utf8GetAux?]; rw [if_neg]
case hnc => simp [← hp, Pos.ext_iff]; exact ne_self_add_add_csize
refine utf8GetAux?_of_valid cs cs' ?_
simpa [Nat.add_assoc, Nat.add_comm] using hp
|
/-
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.Minpoly.Field
import Mathli... | Mathlib/FieldTheory/Separable.lean | 259 | 264 | theorem nodup_of_separable_prod [Nontrivial R] {s : Multiset R}
(hs : Separable (Multiset.map (fun a => X - C a) s).prod) : s.Nodup := by |
rw [Multiset.nodup_iff_ne_cons_cons]
rintro a t rfl
refine not_isUnit_X_sub_C a (isUnit_of_self_mul_dvd_separable hs ?_)
simpa only [Multiset.map_cons, Multiset.prod_cons] using mul_dvd_mul_left _ (dvd_mul_right _ _)
|
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Data.Set.Lattice
import Mathlib.Order.ModularLattice
import Mathlib.Order.WellFounded
import Mathlib.Tactic.Nontriviality
#align_import order.atoms fr... | Mathlib/Order/Atoms.lean | 107 | 108 | theorem bot_covBy_iff : ⊥ ⋖ a ↔ IsAtom a := by |
simp only [CovBy, bot_lt_iff_ne_bot, IsAtom, not_imp_not]
|
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Scott Morrison, Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Functor.Trifunctor
import Mathlib.CategoryTheory.Products.Basic
#align_import cat... | Mathlib/CategoryTheory/Monoidal/Category.lean | 568 | 571 | theorem pentagon_inv_hom_hom_hom_hom :
(α_ W X Y).inv ▷ Z ≫ (α_ (W ⊗ X) Y Z).hom ≫ (α_ W X (Y ⊗ Z)).hom =
(α_ W (X ⊗ Y) Z).hom ≫ W ◁ (α_ X Y Z).hom := by |
simp [← cancel_epi ((α_ W X Y).hom ▷ Z)]
|
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.Tactic.Abel
#align_import set_theory.ordinal.natural_ops from "leanprover-communit... | Mathlib/SetTheory/Ordinal/NaturalOps.lean | 547 | 548 | theorem nmul_le_iff : a ⨳ b ≤ c ↔ ∀ a' < a, ∀ b' < b, a' ⨳ b ♯ a ⨳ b' < c ♯ a' ⨳ b' := by |
rw [← not_iff_not]; simp [lt_nmul_iff]
|
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Matroid.IndepAxioms
/-!
# Matroid Duality
For a matroid `M` on ground set `E`, the collection of complements of the bases of `M` is the
collection o... | Mathlib/Data/Matroid/Dual.lean | 209 | 211 | theorem base_iff_dual_base_compl (hB : B ⊆ M.E := by | aesop_mat) :
M.Base B ↔ M✶.Base (M.E \ B) := by
rw [dual_base_iff, diff_diff_cancel_left hB]
|
/-
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.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.Option
import Mathlib.Logic.Equiv.Fin
import Mathlib.Logic.Equiv.Fintype
#align_import group_the... | Mathlib/GroupTheory/Perm/Fin.lean | 128 | 136 | theorem isCycle_finRotate {n : ℕ} : IsCycle (finRotate (n + 2)) := by |
refine ⟨0, by simp, fun x hx' => ⟨x, ?_⟩⟩
clear hx'
cases' x with x hx
rw [zpow_natCast, Fin.ext_iff, Fin.val_mk]
induction' x with x ih; · rfl
rw [pow_succ', Perm.mul_apply, coe_finRotate_of_ne_last, ih (lt_trans x.lt_succ_self hx)]
rw [Ne, Fin.ext_iff, ih (lt_trans x.lt_succ_self hx), Fin.val_last]
e... |
/-
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.Init.Data.Quot
import Mathlib.Logic.Relator
import Mathlib.Logic.Unique
import Mathlib.Mathport.Notation
#align_import data.quot from "leanprover-comm... | Mathlib/Data/Quot.lean | 398 | 401 | theorem Quotient.eq_mk_iff_out [s : Setoid α] {x : Quotient s} {y : α} :
x = ⟦y⟧ ↔ Quotient.out x ≈ y := by |
refine Iff.trans ?_ Quotient.eq
rw [Quotient.out_eq x]
|
/-
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.Sheaf
/-!
# Coverages
A coverage `K` on a category `C` is a set of presieves associated to every object `X : C`,
called "covering pres... | Mathlib/CategoryTheory/Sites/Coverage.lean | 416 | 422 | theorem isSheaf_iff_isLimit_coverage (K : Coverage C) (P : Cᵒᵖ ⥤ D) :
Presheaf.IsSheaf (toGrothendieck _ K) P ↔ ∀ ⦃X : C⦄ (R : Presieve X),
R ∈ K.covering X →
Nonempty (IsLimit (P.mapCone (Sieve.generate R).arrows.cocone.op)) := by |
simp only [Presheaf.IsSheaf, Presieve.isSheaf_coverage, isLimit_iff_isSheafFor,
← Presieve.isSheafFor_iff_generate]
aesop
|
/-
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
#align_import category_theory.sites.sheaf_of_types from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e622... | Mathlib/CategoryTheory/Sites/IsSheafFor.lean | 328 | 335 | theorem FamilyOfElements.Compatible.pullback (f : Y ⟶ X) {x : FamilyOfElements P S}
(h : x.Compatible) : (x.pullback f).Compatible := by |
simp only [compatible_iff_sieveCompatible] at h ⊢
intro W Z f₁ f₂ hf
unfold FamilyOfElements.pullback
rw [← h (f₁ ≫ f) f₂ hf]
congr 1
simp only [assoc]
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou
-/
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a... | Mathlib/MeasureTheory/Function/L1Space.lean | 133 | 139 | theorem HasFiniteIntegral.mono {f : α → β} {g : α → γ} (hg : HasFiniteIntegral g μ)
(h : ∀ᵐ a ∂μ, ‖f a‖ ≤ ‖g a‖) : HasFiniteIntegral f μ := by |
simp only [hasFiniteIntegral_iff_norm] at *
calc
(∫⁻ a, ENNReal.ofReal ‖f a‖ ∂μ) ≤ ∫⁻ a : α, ENNReal.ofReal ‖g a‖ ∂μ :=
lintegral_mono_ae (h.mono fun a h => ofReal_le_ofReal h)
_ < ∞ := hg
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.NumberTheory.LegendreSymbol.Basic
import Mathlib.Analysis.Normed.Field.Basic
#align_import number_theory.legendre_symbol.gauss_eisenstein_lemmas from "lea... | Mathlib/NumberTheory/LegendreSymbol/GaussEisensteinLemmas.lean | 30 | 60 | theorem Ico_map_valMinAbs_natAbs_eq_Ico_map_id (p : ℕ) [hp : Fact p.Prime] (a : ZMod p)
(hap : a ≠ 0) : ((Ico 1 (p / 2).succ).1.map fun (x : ℕ) => (a * x).valMinAbs.natAbs) =
(Ico 1 (p / 2).succ).1.map fun a => a := by |
have he : ∀ {x}, x ∈ Ico 1 (p / 2).succ → x ≠ 0 ∧ x ≤ p / 2 := by
simp (config := { contextual := true }) [Nat.lt_succ_iff, Nat.succ_le_iff, pos_iff_ne_zero]
have hep : ∀ {x}, x ∈ Ico 1 (p / 2).succ → x < p := fun hx =>
lt_of_le_of_lt (he hx).2 (Nat.div_lt_self hp.1.pos (by decide))
have hpe : ∀ {x}, x ∈... |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Arithmetic
#align_import set_theory.ordinal.exponential from "leanprover-community/mat... | Mathlib/SetTheory/Ordinal/Exponential.lean | 165 | 173 | theorem left_le_opow (a : Ordinal) {b : Ordinal} (b1 : 0 < b) : a ≤ a ^ b := by |
nth_rw 1 [← opow_one a]
cases' le_or_gt a 1 with a1 a1
· rcases lt_or_eq_of_le a1 with a0 | a1
· rw [lt_one_iff_zero] at a0
rw [a0, zero_opow Ordinal.one_ne_zero]
exact Ordinal.zero_le _
rw [a1, one_opow, one_opow]
rwa [opow_le_opow_iff_right a1, one_le_iff_pos]
|
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.IntegrableOn
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Function.LocallyIntegrabl... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 460 | 465 | theorem setIntegral_eq_integral_of_ae_compl_eq_zero (h : ∀ᵐ x ∂μ, x ∉ s → f x = 0) :
∫ x in s, f x ∂μ = ∫ x, f x ∂μ := by |
symm
nth_rw 1 [← integral_univ]
apply setIntegral_eq_of_subset_of_ae_diff_eq_zero nullMeasurableSet_univ (subset_univ _)
filter_upwards [h] with x hx h'x using hx h'x.2
|
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.List.Perm
import Mathlib.Data.List.Range
#align_import data.list.sublists from "leanprover-community/mathlib... | Mathlib/Data/List/Sublists.lean | 189 | 190 | theorem sublists_eq_sublists' (l : List α) : sublists l = map reverse (sublists' (reverse l)) := by |
rw [← sublists_reverse, reverse_reverse]
|
/-
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.Logic.Equiv.List
import Mathlib.Logic.Function.Iterate
#align_import computability.primrec from "leanprover-community/mathlib"@"2738d2ca56cbc63be80c3b... | Mathlib/Computability/Primrec.lean | 1,604 | 1,608 | theorem unpair₂ {n f} (hf : @Primrec' n f) : @Primrec' n fun v => (f v).unpair.2 := by |
have s := sqrt.comp₁ _ hf
have fss := sub.comp₂ _ hf (mul.comp₂ _ s s)
refine (if_lt fss s s (sub.comp₂ _ fss s)).of_eq fun v => ?_
simp [Nat.unpair]; split_ifs <;> rfl
|
/-
Copyright (c) 2021 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Analysis.Convex.Cone.Basic
import Mathlib.Analysis.InnerProductSpace.Projection
#align_import analysis.convex.cone.dual from "leanprover-commu... | Mathlib/Analysis/Convex/Cone/InnerDual.lean | 170 | 200 | theorem ConvexCone.hyperplane_separation_of_nonempty_of_isClosed_of_nmem (K : ConvexCone ℝ H)
(ne : (K : Set H).Nonempty) (hc : IsClosed (K : Set H)) {b : H} (disj : b ∉ K) :
∃ y : H, (∀ x : H, x ∈ K → 0 ≤ ⟪x, y⟫_ℝ) ∧ ⟪y, b⟫_ℝ < 0 := by |
-- let `z` be the point in `K` closest to `b`
obtain ⟨z, hzK, infi⟩ := exists_norm_eq_iInf_of_complete_convex ne hc.isComplete K.convex b
-- for any `w` in `K`, we have `⟪b - z, w - z⟫_ℝ ≤ 0`
have hinner := (norm_eq_iInf_iff_real_inner_le_zero K.convex hzK).1 infi
-- set `y := z - b`
use z - b
constructo... |
/-
Copyright (c) 2021 Jakob Scholbach. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob Scholbach
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-commun... | Mathlib/Algebra/CharP/ExpChar.lean | 184 | 189 | theorem expChar_of_injective_ringHom {R A : Type*}
[Semiring R] [Semiring A] {f : R →+* A} (h : Function.Injective f)
(q : ℕ) [hR : ExpChar R q] : ExpChar A q := by |
cases' hR with _ _ hprime _
· haveI := charZero_of_injective_ringHom h; exact .zero
haveI := charP_of_injective_ringHom h q; exact .prime hprime
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.Separation
import Mathlib.Order.Filter.CountableInter
#align_import topolog... | Mathlib/Topology/GDelta.lean | 130 | 131 | theorem IsGδ.sInter {S : Set (Set X)} (h : ∀ s ∈ S, IsGδ s) (hS : S.Countable) : IsGδ (⋂₀ S) := by |
simpa only [sInter_eq_biInter] using IsGδ.biInter hS h
|
/-
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.BaseChange
import Mathlib.Algebra.Lie.Solvable
import Mathlib.Algebra.Lie.Quotient
import Mathlib.Algebra.Lie.Normalizer
import Mathlib.LinearAlg... | Mathlib/Algebra/Lie/Nilpotent.lean | 450 | 454 | theorem isNilpotent_range_toEnd_iff :
IsNilpotent R (toEnd R L M).range M ↔ IsNilpotent R L M := by |
constructor <;> rintro ⟨k, hk⟩ <;> use k <;>
rw [← LieSubmodule.coe_toSubmodule_eq_iff] at hk ⊢ <;>
simpa using hk
|
/-
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.Associated
import Mathlib.Algebra.Star.Unitary
import Mathlib.RingTheory.Int.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathli... | Mathlib/NumberTheory/Zsqrtd/Basic.lean | 326 | 326 | theorem decompose {x y : ℤ} : (⟨x, y⟩ : ℤ√d) = x + sqrtd (d := d) * y := by | ext <;> simp
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Data.Stream.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Init.Data.List.Basic
import Mathlib.Data.List.Basic
#align_import data.s... | Mathlib/Data/Stream/Init.lean | 426 | 427 | theorem interleave_tail_tail (s₁ s₂ : Stream' α) : tail s₁ ⋈ tail s₂ = tail (tail (s₁ ⋈ s₂)) := by |
rw [interleave_eq s₁ s₂]; rfl
|
/-
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
#align_import category_theory.sites.sheaf_of_types from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e622... | Mathlib/CategoryTheory/Sites/IsSheafFor.lean | 281 | 284 | theorem FamilyOfElements.comp_of_compatible (S : Sieve X) {x : FamilyOfElements P S}
(t : x.Compatible) {f : Y ⟶ X} (hf : S f) {Z} (g : Z ⟶ Y) :
x (g ≫ f) (S.downward_closed hf g) = P.map g.op (x f hf) := by |
simpa using t (𝟙 _) g (S.downward_closed hf g) hf (id_comp _)
|
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Kexing Ying
-/
import Mathlib.Probability.Notation
import Mathlib.Probability.Integration
import Mathlib.MeasureTheory.Function.L2Space
#align_import probability... | Mathlib/Probability/Variance.lean | 218 | 237 | theorem variance_le_expectation_sq [@IsProbabilityMeasure Ω _ ℙ] {X : Ω → ℝ}
(hm : AEStronglyMeasurable X ℙ) : Var[X] ≤ 𝔼[X ^ 2] := by |
by_cases hX : Memℒp X 2
· rw [variance_def' hX]
simp only [sq_nonneg, sub_le_self_iff]
rw [variance, evariance_eq_lintegral_ofReal, ← integral_eq_lintegral_of_nonneg_ae]
· by_cases hint : Integrable X; swap
· simp only [integral_undef hint, Pi.pow_apply, Pi.sub_apply, sub_zero]
exact le_rfl
·... |
/-
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... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 487 | 490 | theorem abs_sin_eq_of_two_zsmul_eq {θ ψ : Angle} (h : (2 : ℤ) • θ = (2 : ℤ) • ψ) :
|sin θ| = |sin ψ| := by |
simp_rw [two_zsmul, ← two_nsmul] at h
exact abs_sin_eq_of_two_nsmul_eq h
|
/-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Data.Array.Lemmas
import Batteries.Tactic.Lint.Misc
namespace Batteries
/-- Union-find node type -/
structure UFNode where
/-- Parent of node -/
... | .lake/packages/batteries/Batteries/Data/UnionFind/Basic.lean | 338 | 348 | theorem parentD_findAux_or (self : UnionFind) (x : Fin self.size) (i) :
parentD (findAux self x).s i = self.rootD i ∧ self.rootD i = self.rootD x ∨
parentD (findAux self x).s i = self.parent i := by |
if h' : (self.arr.get x).parent = x then
rw [findAux_s, if_pos h']; exact .inr rfl
else
rw [parentD_findAux]; split <;> [simp [*]; skip]
have := Nat.sub_lt_sub_left (self.lt_rankMax x) (self.rank'_lt _ ‹_›)
exact (parentD_findAux_or self ⟨_, self.parent'_lt x⟩ i).imp_left <| .imp_right fun h => by
... |
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.... | Mathlib/Algebra/GeomSum.lean | 205 | 208 | theorem sub_one_dvd_pow_sub_one [Ring α] (x : α) (n : ℕ) :
x - 1 ∣ x ^ n - 1 := by |
conv_rhs => rw [← one_pow n]
exact (Commute.one_right x).sub_dvd_pow_sub_pow n
|
/-
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.BaseChange
import Mathlib.Algebra.Lie.Solvable
import Mathlib.Algebra.Lie.Quotient
import Mathlib.Algebra.Lie.Normalizer
import Mathlib.LinearAlg... | Mathlib/Algebra/Lie/Nilpotent.lean | 306 | 311 | theorem isNilpotent_quotient_iff :
IsNilpotent R L (M ⧸ N) ↔ ∃ k, lowerCentralSeries R L M k ≤ N := by |
rw [LieModule.isNilpotent_iff]
refine exists_congr fun k ↦ ?_
rw [← LieSubmodule.Quotient.map_mk'_eq_bot_le, map_lowerCentralSeries_eq k
(LieSubmodule.Quotient.surjective_mk' N)]
|
/-
Copyright (c) 2024 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Filtered.Connected
import Mathlib.CategoryTheory.Limits.TypesFiltered
import Mathlib.CategoryTheory.Limits.Final
/-!
# Final functors wit... | Mathlib/CategoryTheory/Filtered/Final.lean | 217 | 221 | theorem Functor.final_iff_isFiltered_structuredArrow [IsFilteredOrEmpty C] :
Final F ↔ ∀ d, IsFiltered (StructuredArrow d F) := by |
refine ⟨?_, fun h => final_of_isFiltered_structuredArrow F⟩
rw [final_iff_of_isFiltered]
exact fun h => isFiltered_structuredArrow_of_isFiltered_of_exists F h.1 h.2
|
/-
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.MvPolynomial.Basic
import Mathlib.Data.Finset.PiAntidiagonal
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Tactic.Linarith
... | Mathlib/RingTheory/MvPowerSeries/Basic.lean | 604 | 604 | theorem map_X (s : σ) : map σ f (X s) = X s := by | simp [MvPowerSeries.X]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic fro... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,939 | 1,947 | theorem bsup_comp {o o' : Ordinal.{max u v}} {f : ∀ a < o, Ordinal.{max u v w}}
(hf : ∀ {i j} (hi) (hj), i ≤ j → f i hi ≤ f j hj) {g : ∀ a < o', Ordinal.{max u v}}
(hg : blsub.{_, u} o' g = o) :
(bsup.{_, w} o' fun a ha => f (g a ha) (by rw [← hg]; apply lt_blsub)) = bsup.{_, w} o f := by |
apply le_antisymm <;> refine bsup_le fun i hi => ?_
· apply le_bsup
· rw [← hg, lt_blsub_iff] at hi
rcases hi with ⟨j, hj, hj'⟩
exact (hf _ _ hj').trans (le_bsup _ _ _)
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Yury Kudryashov
-/
import Mathlib.Data.Rat.Sqrt
import Mathlib.Data.Real.Sqrt
import Mathlib.RingTheory.Algebraic
import... | Mathlib/Data/Real/Irrational.lean | 606 | 607 | theorem irrational_int_mul_iff : Irrational (m * x) ↔ m ≠ 0 ∧ Irrational x := by |
rw [← cast_intCast, irrational_rat_mul_iff, Int.cast_ne_zero]
|
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calcul... | Mathlib/Analysis/Calculus/LineDeriv/Basic.lean | 542 | 543 | theorem lineDeriv_neg : lineDeriv 𝕜 f x (-v) = - lineDeriv 𝕜 f x v := by |
rw [← neg_one_smul (R := 𝕜) v, lineDeriv_smul, neg_one_smul]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
#align_import ring_theory.ideal.operations from "leanpro... | Mathlib/RingTheory/Ideal/Operations.lean | 985 | 990 | theorem disjoint_powers_iff_not_mem (y : R) (hI : I.IsRadical) :
Disjoint (Submonoid.powers y : Set R) ↑I ↔ y ∉ I.1 := by |
refine ⟨fun h => Set.disjoint_left.1 h (Submonoid.mem_powers _),
fun h => disjoint_iff.mpr (eq_bot_iff.mpr ?_)⟩
rintro x ⟨⟨n, rfl⟩, hx'⟩
exact h (hI <| mem_radical_of_pow_mem <| le_radical hx')
|
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Simon Hudon
-/
import Mathlib.Data.PFunctor.Multivariate.W
import Mathlib.Data.QPF.Multivariate.Basic
#align_import data.qpf.multivariate.constructions.fix from "leanpro... | Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean | 71 | 75 | theorem recF_eq' {α : TypeVec n} {β : Type u} (g : F (α.append1 β) → β) (x : q.P.W α) :
recF g x = g (abs (appendFun id (recF g) <$$> q.P.wDest' x)) := by |
apply q.P.w_cases _ x
intro a f' f
rw [recF_eq, q.P.wDest'_wMk, MvPFunctor.map_eq, appendFun_comp_splitFun, TypeVec.id_comp]
|
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow, Kexing Ying
-/
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.LinearAlgebra.BilinearForm.Properties
/-!
# Bilinear form
This file defines ort... | Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean | 192 | 200 | theorem orthogonal_span_singleton_eq_toLin_ker {B : BilinForm K V} (x : V) :
B.orthogonal (K ∙ x) = LinearMap.ker (LinearMap.BilinForm.toLinHomAux₁ B x) := by |
ext y
simp_rw [mem_orthogonal_iff, LinearMap.mem_ker, Submodule.mem_span_singleton]
constructor
· exact fun h => h x ⟨1, one_smul _ _⟩
· rintro h _ ⟨z, rfl⟩
rw [IsOrtho, smul_left, mul_eq_zero]
exact Or.intro_right _ h
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.GroupWithZero.NeZero
import Mathlib.Logic.Unique
#align_import algebra.group_with_zero.basic from "leanprov... | Mathlib/Algebra/GroupWithZero/Basic.lean | 399 | 402 | theorem div_self_mul_self' (a : G₀) : a / (a * a) = a⁻¹ :=
calc
a / (a * a) = a⁻¹⁻¹ * a⁻¹ * a⁻¹ := by | simp [mul_inv_rev]
_ = a⁻¹ := inv_mul_mul_self _
|
/-
Copyright (c) 2023 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
/-!
# Lindelöf sets and Lindelöf spaces
## Mai... | Mathlib/Topology/Compactness/Lindelof.lean | 448 | 450 | theorem mem_coclosedLindelof : s ∈ coclosedLindelof X ↔
∃ t, IsClosed t ∧ IsLindelof t ∧ tᶜ ⊆ s := by |
simp only [hasBasis_coclosedLindelof.mem_iff, and_assoc]
|
/-
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.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Re... | Mathlib/LinearAlgebra/Matrix/Transvection.lean | 464 | 506 | theorem mul_listTransvecRow_last_row (hM : M (inr unit) (inr unit) ≠ 0) (i : Fin r) :
(M * (listTransvecRow M).prod) (inr unit) (inl i) = 0 := by |
suffices H :
∀ k : ℕ,
k ≤ r →
(M * ((listTransvecRow M).take k).prod) (inr unit) (inl i) =
if k ≤ i then M (inr unit) (inl i) else 0 by
have A : (listTransvecRow M).length = r := by simp [listTransvecRow]
rw [← List.take_length (listTransvecRow M), A]
have : ¬r ≤ i := by simp
... |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Algebra.Ring.Int
import Ma... | Mathlib/Data/Nat/Prime.lean | 544 | 550 | theorem coprime_of_dvd {m n : ℕ} (H : ∀ k, Prime k → k ∣ m → ¬k ∣ n) : Coprime m n := by |
rw [coprime_iff_gcd_eq_one]
by_contra g2
obtain ⟨p, hp, hpdvd⟩ := exists_prime_and_dvd g2
apply H p hp <;> apply dvd_trans hpdvd
· exact gcd_dvd_left _ _
· exact gcd_dvd_right _ _
|
/-
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Alistair Tucker, Wen Yang
-/
import Mathlib.Order.Interval.Set.Image
import Mathlib.Order.CompleteLatticeIntervals
import Mathlib.Topology.Order.DenselyOrdered
... | Mathlib/Topology/Order/IntermediateValue.lean | 218 | 220 | theorem IsPreconnected.Icc_subset {s : Set α} (hs : IsPreconnected s) {a b : α} (ha : a ∈ s)
(hb : b ∈ s) : Icc a b ⊆ s := by |
simpa only [image_id] using hs.intermediate_value ha hb continuousOn_id
|
/-
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.Semiconj
import Mathlib.Algebra.Ring.Units
import Mathlib.Algebra.Gro... | Mathlib/Algebra/Ring/Commute.lean | 304 | 308 | theorem inv_eq_self_iff [Ring R] [NoZeroDivisors R] (u : Rˣ) : u⁻¹ = u ↔ u = 1 ∨ u = -1 := by |
rw [inv_eq_iff_mul_eq_one]
simp only [ext_iff]
push_cast
exact mul_self_eq_one_iff
|
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Covering.VitaliFamily
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.MeasureTheory.Function.AEMeasurableOrder
import M... | Mathlib/MeasureTheory/Covering/Differentiation.lean | 160 | 201 | theorem ae_eventually_measure_zero_of_singular (hρ : ρ ⟂ₘ μ) :
∀ᵐ x ∂μ, Tendsto (fun a => ρ a / μ a) (v.filterAt x) (𝓝 0) := by |
have A : ∀ ε > (0 : ℝ≥0), ∀ᵐ x ∂μ, ∀ᶠ a in v.filterAt x, ρ a < ε * μ a := by
intro ε εpos
set s := {x | ¬∀ᶠ a in v.filterAt x, ρ a < ε * μ a} with hs
change μ s = 0
obtain ⟨o, _, ρo, μo⟩ : ∃ o : Set α, MeasurableSet o ∧ ρ o = 0 ∧ μ oᶜ = 0 := hρ
apply le_antisymm _ bot_le
calc
μ s ≤ μ (s... |
/-
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, Scott Morrison, Jens Wagemaker, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib... | Mathlib/Algebra/Polynomial/RingDivision.lean | 558 | 563 | theorem rootMultiplicity_mul' {p q : R[X]} {x : R}
(hpq : (p /ₘ (X - C x) ^ p.rootMultiplicity x).eval x *
(q /ₘ (X - C x) ^ q.rootMultiplicity x).eval x ≠ 0) :
rootMultiplicity x (p * q) = rootMultiplicity x p + rootMultiplicity x q := by |
simp_rw [eval_divByMonic_eq_trailingCoeff_comp] at hpq
simp_rw [rootMultiplicity_eq_natTrailingDegree, mul_comp, natTrailingDegree_mul' hpq]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.iterated_deriv from "leanprover-com... | Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean | 196 | 201 | theorem iteratedDerivWithin_eq_iterate {x : 𝕜} (hs : UniqueDiffOn 𝕜 s) (hx : x ∈ s) :
iteratedDerivWithin n f s x = (fun g : 𝕜 → F => derivWithin g s)^[n] f x := by |
induction' n with n IH generalizing x
· simp
· rw [iteratedDerivWithin_succ (hs x hx), Function.iterate_succ']
exact derivWithin_congr (fun y hy => IH hy) (IH hx)
|
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin, Lu-Ming Zhang
-/
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOp... | Mathlib/Data/Matrix/Basic.lean | 622 | 623 | theorem bit1_apply_eq (M : Matrix n n α) (i : n) : (bit1 M) i i = bit1 (M i i) := by |
simp [bit1_apply]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Aurélien Saue, Anne Baanen
-/
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.Tactic.NormNum.Inv
import Mathlib.Tactic.NormNum.Pow
import Mathlib.Util.AtomM
/-!
#... | Mathlib/Tactic/Ring/Basic.lean | 986 | 987 | theorem inv_congr {R} [DivisionRing R] {a a' b : R} (_ : a = a')
(_ : a'⁻¹ = b) : (a⁻¹ : R) = b := by | subst_vars; rfl
|
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Algebra.Group.Even
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.Order.Sub.Defs
#align_import algebra.order.sub.canoni... | Mathlib/Algebra/Order/Sub/Canonical.lean | 509 | 512 | theorem tsub_min : a - min a b = a - b := by |
rcases le_total a b with h | h
· rw [min_eq_left h, tsub_self, tsub_eq_zero_of_le h]
· rw [min_eq_right h]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
#align_import measure_theory.integral.set_to_l1 from "leanprov... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 634 | 637 | theorem setToSimpleFunc_const' [Nonempty α] (T : Set α → F →L[ℝ] F') (x : F)
{m : MeasurableSpace α} : SimpleFunc.setToSimpleFunc T (SimpleFunc.const α x) = T univ x := by |
simp only [setToSimpleFunc, range_const, Set.mem_singleton, preimage_const_of_mem,
sum_singleton, ← Function.const_def, coe_const]
|
/-
Copyright (c) 2020 Paul van Wamelen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul van Wamelen
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Int.Basic
import Mat... | Mathlib/NumberTheory/PythagoreanTriples.lean | 32 | 34 | theorem sq_ne_two_fin_zmod_four (z : ZMod 4) : z * z ≠ 2 := by |
change Fin 4 at z
fin_cases z <;> decide
|
/-
Copyright (c) 2018 Violeta Hernández Palacios, Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios, Mario Carneiro
-/
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Exponential
#align_import set_th... | Mathlib/SetTheory/Ordinal/FixedPoint.lean | 424 | 432 | theorem sup_iterate_eq_nfp (f : Ordinal.{u} → Ordinal.{u}) :
(fun a => sup fun n : ℕ => f^[n] a) = nfp f := by |
refine funext fun a => le_antisymm ?_ (sup_le fun l => ?_)
· rw [sup_le_iff]
intro n
rw [← List.length_replicate n Unit.unit, ← List.foldr_const f a]
apply le_sup
· rw [List.foldr_const f a l]
exact le_sup _ _
|
/-
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.Interval.Set.Basic
import Mathlib.Data.Set.NAry
import Mathlib.Order.Directed
#align_import order.bounds.basic from "leanprover... | Mathlib/Order/Bounds/Basic.lean | 139 | 141 | theorem not_bddAbove_iff {α : Type*} [LinearOrder α] {s : Set α} :
¬BddAbove s ↔ ∀ x, ∃ y ∈ s, x < y := by |
simp only [not_bddAbove_iff', not_le]
|
/-
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
#align_import geometry.euclidean.angle.oriente... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 992 | 994 | theorem oangle_sign_smul_sub_left (x y : V) (r : ℝ) :
(o.oangle (r • y - x) y).sign = -(o.oangle x y).sign := by |
rw [← oangle_sign_neg_left, sub_eq_neg_add, oangle_sign_add_smul_left]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Int.Lemm... | Mathlib/Algebra/Order/Floor.lean | 1,461 | 1,461 | theorem round_natCast (n : ℕ) : round (n : α) = n := by | simp [round]
|
/-
Copyright (c) 2021 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 587 | 588 | theorem support_eq_cons {u v : V} (p : G.Walk u v) : p.support = u :: p.support.tail := by |
cases p <;> simp
|
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import number_theory.ramification_inertia from "leanprover-community/mathlib"@"039a089d2a4b93c761b234f3e5f5aeb752bac6... | Mathlib/NumberTheory/RamificationInertia.lean | 741 | 743 | theorem Factors.inertiaDeg_ne_zero [IsNoetherian R S] [p.IsMaximal]
(P : (factors (map (algebraMap R S) p)).toFinset) : inertiaDeg (algebraMap R S) p P ≠ 0 := by |
rw [inertiaDeg_algebraMap]; exact (FiniteDimensional.finrank_pos_iff.mpr inferInstance).ne'
|
/-
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.Defs
import Mathlib.Data.Int.Defs
import Mathlib.... | Mathlib/Algebra/Group/Basic.lean | 731 | 731 | theorem inv_div' : (a / b)⁻¹ = a⁻¹ / b⁻¹ := by | simp
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | Mathlib/Topology/Algebra/Monoid.lean | 764 | 768 | theorem tendsto_multiset_prod {f : ι → α → M} {x : Filter α} {a : ι → M} (s : Multiset ι) :
(∀ i ∈ s, Tendsto (f i) x (𝓝 (a i))) →
Tendsto (fun b => (s.map fun c => f c b).prod) x (𝓝 (s.map a).prod) := by |
rcases s with ⟨l⟩
simpa using tendsto_list_prod l
|
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathli... | Mathlib/Combinatorics/SimpleGraph/Finite.lean | 449 | 452 | theorem card_commonNeighbors_le_degree_left [DecidableRel G.Adj] (v w : V) :
Fintype.card (G.commonNeighbors v w) ≤ G.degree v := by |
rw [← card_neighborSet_eq_degree]
exact Set.card_le_card Set.inter_subset_left
|
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Regular.Basic
import Mathlib.LinearAlgebra.Matrix.MvPolynomial
import Mathlib.LinearAlgebra.Matrix.Polynomial
import Mathlib.RingTheory.Polynomial.Ba... | Mathlib/LinearAlgebra/Matrix/Adjugate.lean | 311 | 312 | theorem mulVec_cramer (A : Matrix n n α) (b : n → α) : A *ᵥ cramer A b = A.det • b := by |
rw [cramer_eq_adjugate_mulVec, mulVec_mulVec, mul_adjugate, smul_mulVec_assoc, one_mulVec]
|
/-
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.Algebra.Category.ModuleCat.Free
import Mathlib.Topology.Category.Profinite.CofilteredLimit
import Mathlib.Topology.Category.Profinite.Product
impor... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 1,445 | 1,449 | theorem linearIndependent_iff_sum :
LinearIndependent ℤ (eval C) ↔ LinearIndependent ℤ (SumEval C ho) := by |
rw [← linearIndependent_equiv (sum_equiv C hsC ho), SumEval,
← sum_equiv_comp_eval_eq_elim C hsC ho]
exact Iff.rfl
|
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä, Moritz Doll
-/
import Mathlib.Topology.Algebra.Module.Basic
import Mathlib.LinearAlgebra.BilinearMap
#align_import topology.algebra.module.weak_dual from "leanprover-commu... | Mathlib/Topology/Algebra/Module/WeakDual.lean | 133 | 137 | theorem tendsto_iff_forall_eval_tendsto {l : Filter α} {f : α → WeakBilin B} {x : WeakBilin B}
(hB : Function.Injective B) :
Tendsto f l (𝓝 x) ↔ ∀ y, Tendsto (fun i => B (f i) y) l (𝓝 (B x y)) := by |
rw [← tendsto_pi_nhds, Embedding.tendsto_nhds_iff (embedding hB)]
rfl
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Module.LinearMap.Basic
import ... | Mathlib/Data/DFinsupp/Basic.lean | 778 | 780 | theorem filter_ne_eq_erase (f : Π₀ i, β i) (i : ι) : f.filter (· ≠ i) = f.erase i := by |
ext1 j
simp only [DFinsupp.filter_apply, DFinsupp.erase_apply, ite_not]
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.FreeAlgebra
import Mathlib.GroupTheory.Finiteness
import Mathlib.RingTheory.Adjoin.Tower
import Mathlib.RingTheory.Finiteness
import Mathlib.Ri... | Mathlib/RingTheory/FiniteType.lean | 577 | 582 | theorem support_gen_of_gen' {S : Set (MonoidAlgebra R M)} (hS : Algebra.adjoin R S = ⊤) :
Algebra.adjoin R (of R M '' ⋃ f ∈ S, (f.support : Set M)) = ⊤ := by |
suffices (of R M '' ⋃ f ∈ S, (f.support : Set M)) = ⋃ f ∈ S, of R M '' (f.support : Set M) by
rw [this]
exact support_gen_of_gen hS
simp only [Set.image_iUnion]
|
/-
Copyright (c) 2018 Louis Carlin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Louis Carlin, Mario Carneiro
-/
import Mathlib.Algebra.EuclideanDomain.Defs
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.Algebra.Grou... | Mathlib/Algebra/EuclideanDomain/Basic.lean | 306 | 322 | theorem lcm_eq_zero_iff {x y : R} : lcm x y = 0 ↔ x = 0 ∨ y = 0 := by |
constructor
· intro hxy
rw [lcm, mul_div_assoc _ (gcd_dvd_right _ _), mul_eq_zero] at hxy
apply Or.imp_right _ hxy
intro hy
by_cases hgxy : gcd x y = 0
· rw [EuclideanDomain.gcd_eq_zero_iff] at hgxy
exact hgxy.2
· rcases gcd_dvd x y with ⟨⟨r, hr⟩, ⟨s, hs⟩⟩
generalize gcd x y = g... |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Mul
import ... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 306 | 310 | theorem LinearIsometry.norm_iteratedFDeriv_comp_left {f : E → F} (g : F →ₗᵢ[𝕜] G)
(hf : ContDiff 𝕜 n f) (x : E) {i : ℕ} (hi : (i : ℕ∞) ≤ n) :
‖iteratedFDeriv 𝕜 i (g ∘ f) x‖ = ‖iteratedFDeriv 𝕜 i f x‖ := by |
simp only [← iteratedFDerivWithin_univ]
exact g.norm_iteratedFDerivWithin_comp_left hf.contDiffOn uniqueDiffOn_univ (mem_univ x) hi
|
/-
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.Constructions.Prod.Integral
impor... | Mathlib/Analysis/Convolution.lean | 474 | 475 | theorem smul_convolution [SMulCommClass ℝ 𝕜 F] {y : 𝕜} : y • f ⋆[L, μ] g = y • (f ⋆[L, μ] g) := by |
ext; simp only [Pi.smul_apply, convolution_def, ← integral_smul, L.map_smul₂]
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison, Adam Topaz
-/
import Mathlib.AlgebraicTopology.SimplexCategory
import Mathlib.CategoryTheory.Comma.Arrow
import Mathlib.CategoryTheory.Limits.FunctorCat... | Mathlib/AlgebraicTopology/SimplicialObject.lean | 182 | 185 | theorem δ_comp_σ_of_gt {n} {i : Fin (n + 2)} {j : Fin (n + 1)} (H : Fin.castSucc j < i) :
X.σ (Fin.castSucc j) ≫ X.δ i.succ = X.δ i ≫ X.σ j := by |
dsimp [δ, σ]
simp only [← X.map_comp, ← op_comp, SimplexCategory.δ_comp_σ_of_gt H]
|
/-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.Unitization
import Mathlib.Algebra.Star.NonUnitalSubalgebra
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.GroupTheory.GroupAction... | Mathlib/Algebra/Algebra/Subalgebra/Unitization.lean | 102 | 109 | theorem lift_range_le {f : A →ₙₐ[R] C} {S : Subalgebra R C} :
(lift f).range ≤ S ↔ NonUnitalAlgHom.range f ≤ S.toNonUnitalSubalgebra := by |
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
· rintro - ⟨x, rfl⟩
exact @h (f x) ⟨x, by simp⟩
· rintro - ⟨x, rfl⟩
induction x with
| _ r a => simpa using add_mem (algebraMap_mem S r) (h ⟨a, rfl⟩)
|
/-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Algebra.Ring.Parity
import Mathlib.Combinatorics.SimpleGraph.Connectivity
#align_import combinatorics.simple_graph.trails from "leanprover-community/mathlib... | Mathlib/Combinatorics/SimpleGraph/Trails.lean | 53 | 81 | theorem IsTrail.even_countP_edges_iff {u v : V} {p : G.Walk u v} (ht : p.IsTrail) (x : V) :
Even (p.edges.countP fun e => x ∈ e) ↔ u ≠ v → x ≠ u ∧ x ≠ v := by |
induction' p with u u v w huv p ih
· simp
· rw [cons_isTrail_iff] at ht
specialize ih ht.1
simp only [List.countP_cons, Ne, edges_cons, Sym2.mem_iff]
split_ifs with h
· rw [decide_eq_true_eq] at h
obtain (rfl | rfl) := h
· rw [Nat.even_add_one, ih]
simp only [huv.ne, imp_false... |
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