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) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Category.ModuleCat.Abelian
import Mathlib.CategoryTheory.Limits.Shapes.Images
#align_import algebra.category.Module.images from "leanprover-co... | Mathlib/Algebra/Category/ModuleCat/Images.lean | 117 | 119 | theorem imageIsoRange_hom_subtype {G H : ModuleCat.{v} R} (f : G ⟶ H) :
(imageIsoRange f).hom ≫ ModuleCat.ofHom f.range.subtype = Limits.image.ι f := by |
erw [← imageIsoRange_inv_image_ι f, Iso.hom_inv_id_assoc]
|
/-
Copyright (c) 2024 Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Christian Merten
-/
import Mathlib.CategoryTheory.Galois.Basic
import Mathlib.RepresentationTheory.Action.Basic
import Mathlib.RepresentationTheory.Action.Concrete
import Mathlib.Rep... | Mathlib/CategoryTheory/Galois/Examples.lean | 127 | 145 | theorem Action.isConnected_of_transitive (X : FintypeCat) [MulAction G X]
[MulAction.IsPretransitive G X] [h : Nonempty X] :
IsConnected (Action.FintypeCat.ofMulAction G X) where
notInitial := not_initial_of_inhabited (Action.forget _ _) h.some
noTrivialComponent Y i hm hni := by |
/- We show that the induced inclusion `i.hom` of finite sets is surjective, using the
transitivity of the `G`-action. -/
obtain ⟨(y : Y.V)⟩ := (not_initial_iff_fiber_nonempty (Action.forget _ _) Y).mp hni
have : IsIso i.hom := by
refine (ConcreteCategory.isIso_iff_bijective i.hom).mpr ⟨?_, fun x'... |
/-
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 | 361 | 368 | theorem map_iterate (f : α → α) (a : α) : iterate f (f a) = map f (iterate f a) := by |
funext n
induction' n with n' ih
· rfl
· unfold map iterate get
rw [map, get] at ih
rw [iterate]
exact congrArg f ih
|
/-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Joël Riou
-/
import Mathlib.CategoryTheory.CommSq
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.C... | Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean | 1,011 | 1,019 | theorem Functor.map_isPushout [PreservesColimit (span f g) F] (s : IsPushout f g h i) :
IsPushout (F.map f) (F.map g) (F.map h) (F.map i) := by |
refine
IsPushout.of_isColimit' (F.map_commSq s.toCommSq)
(IsColimit.equivOfNatIsoOfIso (spanCompIso F f g) _ _ (WalkingSpan.ext ?_ ?_ ?_)
(isColimitOfPreserves F s.isColimit))
· rfl
· simp
· simp
|
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Support
#align_import group_theory.perm.list from "leanprove... | Mathlib/GroupTheory/Perm/List.lean | 95 | 97 | theorem support_formPerm_le' : { x | formPerm l x ≠ x } ≤ l.toFinset := by |
refine (zipWith_swap_prod_support' l l.tail).trans ?_
simpa [Finset.subset_iff] using tail_subset l
|
/-
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.Sheaf
#align_import category_theory.sites.canonical from "leanprover-community/mathlib"@"9e7c80f638149bfb3504ba8ff48dfdbfc949fb1a"
/... | Mathlib/CategoryTheory/Sites/Canonical.lean | 197 | 202 | theorem le_finestTopology (Ps : Set (Cᵒᵖ ⥤ Type v)) (J : GrothendieckTopology C)
(hJ : ∀ P ∈ Ps, Presieve.IsSheaf J P) : J ≤ finestTopology Ps := by |
rintro X S hS _ ⟨⟨_, _, ⟨P, hP, rfl⟩, rfl⟩, rfl⟩
intro Y f
-- this can't be combined with the previous because the `subst` is applied at the end
exact hJ P hP (S.pullback f) (J.pullback_stable f hS)
|
/-
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.Data.Complex.Basic
import Mathlib.Data.Real.Cardinality
#align_import data.complex.cardinality from "leanprover-community/math... | Mathlib/Data/Complex/Cardinality.lean | 31 | 31 | theorem mk_univ_complex : #(Set.univ : Set ℂ) = 𝔠 := by | rw [mk_univ, mk_complex]
|
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Thomas Read, Andrew Yang, Dagur Asgeirsson, Joël Riou
-/
import Mathlib.CategoryTheory.Adjunction.Basic
/-!
# Uniqueness of adjoints
This file shows that adjoints are uni... | Mathlib/CategoryTheory/Adjunction/Unique.lean | 123 | 127 | theorem unit_leftAdjointUniq_hom {F F' : C ⥤ D} {G : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F' ⊣ G) :
adj1.unit ≫ whiskerRight (leftAdjointUniq adj1 adj2).hom G = adj2.unit := by |
ext x
rw [NatTrans.comp_app, ← homEquiv_leftAdjointUniq_hom_app adj1 adj2]
simp [← G.map_comp]
|
/-
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.Order.Group.Int
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.... | Mathlib/Data/Rat/Lemmas.lean | 248 | 252 | theorem inv_intCast_den_of_pos {a : ℤ} (ha0 : 0 < a) : ((a : ℚ)⁻¹.den : ℤ) = a := by |
rw [← ofInt_eq_cast, ofInt, mk_eq_divInt, Rat.inv_divInt', divInt_eq_div, Nat.cast_one]
apply den_div_eq_of_coprime ha0
rw [Int.natAbs_one]
exact Nat.coprime_one_left _
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Wrenna Robson
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.RingTheory.Polynomial.Basic
#align_import linear_algebr... | Mathlib/LinearAlgebra/Lagrange.lean | 262 | 271 | theorem natDegree_basis (hvs : Set.InjOn v s) (hi : i ∈ s) :
(Lagrange.basis s v i).natDegree = s.card - 1 := by |
have H : ∀ j, j ∈ s.erase i → basisDivisor (v i) (v j) ≠ 0 := by
simp_rw [Ne, mem_erase, basisDivisor_eq_zero_iff]
exact fun j ⟨hij₁, hj⟩ hij₂ => hij₁ (hvs hj hi hij₂.symm)
rw [← card_erase_of_mem hi, card_eq_sum_ones]
convert natDegree_prod _ _ H using 1
refine sum_congr rfl fun j hj => (natDegree_bas... |
/-
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, Yury Kudryashov, Patrick Massot
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.... | Mathlib/Order/Filter/AtTopBot.lean | 1,608 | 1,610 | theorem eventually_atBot_prod_self [SemilatticeInf α] [Nonempty α] {p : α × α → Prop} :
(∀ᶠ x in atBot, p x) ↔ ∃ a, ∀ k l, k ≤ a → l ≤ a → p (k, l) := by |
simp [← prod_atBot_atBot_eq, (@atBot_basis α _ _).prod_self.eventually_iff]
|
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.TangentCone
import Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics
#align_import analysis.ca... | Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 675 | 680 | theorem differentiableOn_of_locally_differentiableOn
(h : ∀ x ∈ s, ∃ u, IsOpen u ∧ x ∈ u ∧ DifferentiableOn 𝕜 f (s ∩ u)) :
DifferentiableOn 𝕜 f s := by |
intro x xs
rcases h x xs with ⟨t, t_open, xt, ht⟩
exact (differentiableWithinAt_inter (IsOpen.mem_nhds t_open xt)).1 (ht x ⟨xs, xt⟩)
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Anne Baanen
-/
import Mathlib.LinearAlgebra.Dimension.Basic
import Mathlib.SetTheory.Cardinal.ToNat
#align_import linear_algebra.finrank from "leanprover-community/mathlib... | Mathlib/LinearAlgebra/Dimension/Finrank.lean | 72 | 75 | theorem finrank_le_of_rank_le {n : ℕ} (h : Module.rank R M ≤ ↑n) : finrank R M ≤ n := by |
rwa [← Cardinal.toNat_le_iff_le_of_lt_aleph0, toNat_natCast] at h
· exact h.trans_lt (nat_lt_aleph0 n)
· exact nat_lt_aleph0 n
|
/-
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, Sébastien Gouëzel
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic... | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 522 | 527 | theorem addHaar_closedBall_eq_addHaar_ball [Nontrivial E] (x : E) (r : ℝ) :
μ (closedBall x r) = μ (ball x r) := by |
by_cases h : r < 0
· rw [Metric.closedBall_eq_empty.mpr h, Metric.ball_eq_empty.mpr h.le]
push_neg at h
rw [addHaar_closedBall μ x h, addHaar_ball μ x h]
|
/-
Copyright (c) 2021 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson
-/
import Mathlib.Algebra.Field.Opposite
import Mathlib.Algebra.Group.Subgroup.ZPowers
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Rin... | Mathlib/Algebra/Periodic.lean | 156 | 157 | theorem Periodic.add_period [AddSemigroup α] (h1 : Periodic f c₁) (h2 : Periodic f c₂) :
Periodic f (c₁ + c₂) := by | simp_all [← add_assoc]
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Order.Filter.Bases
import Mathlib.Order.ConditionallyCompleteLattice.Basic
#align_import order.filter.lift from "leanprover-community/mathlib"@"8631e2... | Mathlib/Order/Filter/Lift.lean | 322 | 323 | theorem lift'_pure {a : α} (hh : Monotone h) : (pure a : Filter α).lift' h = 𝓟 (h {a}) := by |
rw [← principal_singleton, lift'_principal hh]
|
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Yuyang Zhao
-/
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Algebra.Order.Monoid.Unbundled.Defs
import Mathlib.Tactic.GCongr.Core
#align_import algebra.order... | Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean | 1,089 | 1,090 | theorem posMulReflectLT_iff_mulPosReflectLT : PosMulReflectLT α ↔ MulPosReflectLT α := by |
simp only [PosMulReflectLT, MulPosReflectLT, IsSymmOp.symm_op]
|
/-
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.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.MvPower... | Mathlib/RingTheory/PowerSeries/Basic.lean | 150 | 151 | theorem coeff_def {s : Unit →₀ ℕ} {n : ℕ} (h : s () = n) : coeff R n = MvPowerSeries.coeff R s := by |
erw [coeff, ← h, ← Finsupp.unique_single s]
|
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Isabel Longbottom, Scott Morrison
-/
import Mathlib.Algebra.Order.ZeroLEOne
import Mathlib.Data.List.InsertNth
import Mathlib.Logic.Relation
import Mathlib... | Mathlib/SetTheory/Game/PGame.lean | 1,356 | 1,357 | theorem moveLeft_neg_symm' {x : PGame} (i) :
x.moveLeft i = -(-x).moveRight (toRightMovesNeg i) := by | simp
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Init.Logic
import Mathlib.Init.Function
import Mathlib.Init.Algebra.Classes
import Batteries.Util.LibraryNote
import Batteries.Tactic.... | Mathlib/Logic/Basic.lean | 1,223 | 1,224 | theorem dite_ne_right_iff : (dite P A fun _ ↦ b) ≠ b ↔ ∃ h, A h ≠ b := by |
simp only [Ne, dite_eq_right_iff, not_forall]
|
/-
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 | 356 | 359 | theorem mul_int (h : Irrational x) {m : ℤ} (hm : m ≠ 0) : Irrational (x * m) := by |
rw [← cast_intCast]
refine h.mul_rat ?_
rwa [Int.cast_ne_zero]
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Devon Tuma
-/
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.RingTheory.Coprime.Basic
import Mathlib.Tactic.... | Mathlib/RingTheory/Polynomial/ScaleRoots.lean | 170 | 177 | theorem scaleRoots_eval₂_eq_zero_of_eval₂_div_eq_zero {p : S[X]} {f : S →+* K}
(hf : Function.Injective f) {r s : S} (hr : eval₂ f (f r / f s) p = 0)
(hs : s ∈ nonZeroDivisors S) : eval₂ f (f r) (scaleRoots p s) = 0 := by |
set_option tactic.skipAssignedInstances false in
nontriviality S using Subsingleton.eq_zero
convert @scaleRoots_eval₂_eq_zero _ _ _ _ p f _ s hr
rw [← mul_div_assoc, mul_comm, mul_div_cancel_right₀]
exact map_ne_zero_of_mem_nonZeroDivisors _ hf hs
|
/-
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
-/
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Ri... | Mathlib/LinearAlgebra/Matrix/ToLin.lean | 272 | 274 | theorem Matrix.ker_mulVecLin_eq_bot_iff {M : Matrix m n R} :
(LinearMap.ker M.mulVecLin) = ⊥ ↔ ∀ v, M *ᵥ v = 0 → v = 0 := by |
simp only [Submodule.eq_bot_iff, LinearMap.mem_ker, Matrix.mulVecLin_apply]
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Algebra.Ring.Int
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Size
#align_import data.int.bitwise from "leanprover-community/mathlib"@"0743cc... | Mathlib/Data/Int/Bitwise.lean | 159 | 167 | theorem bodd_neg (n : ℤ) : bodd (-n) = bodd n := by |
cases n with
| ofNat =>
rw [← negOfNat_eq, bodd_negOfNat]
simp
| negSucc n =>
rw [neg_negSucc, bodd_coe, Nat.bodd_succ]
change (!Nat.bodd n) = !(bodd n)
rw [bodd_coe]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau
-/
import Mathlib.Data.Finsupp.ToDFinsupp
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.LinearIndependent
#align_import linear_algebra.dfinsup... | Mathlib/LinearAlgebra/DFinsupp.lean | 495 | 511 | theorem Independent.dfinsupp_lsum_injective {p : ι → Submodule R N} (h : Independent p) :
Function.Injective (lsum ℕ (M := fun i ↦ ↥(p i)) fun i => (p i).subtype) := by |
-- simplify everything down to binders over equalities in `N`
rw [independent_iff_forall_dfinsupp] at h
suffices LinearMap.ker (lsum ℕ (M := fun i ↦ ↥(p i)) fun i => (p i).subtype) = ⊥ by
-- Lean can't find this without our help
letI thisI : AddCommGroup (Π₀ i, p i) := inferInstance
rw [LinearMap.ker... |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Scott Morrison
-/
import Mathlib.CategoryTheory.FinCategory.Basic
import Mathlib.CategoryTheory.Limits.Cones
import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
import M... | Mathlib/CategoryTheory/Filtered/Basic.lean | 461 | 463 | theorem coeq₃_condition₁ {j₁ j₂ : C} (f g h : j₁ ⟶ j₂) :
f ≫ coeq₃Hom f g h = g ≫ coeq₃Hom f g h := by |
simp only [coeq₃Hom, ← Category.assoc, coeq_condition f g]
|
/-
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.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Conj
import Mathlib.CategoryTheory.Functor.ReflectsIso
#align_import category_theory... | Mathlib/CategoryTheory/Adjunction/Reflective.lean | 159 | 162 | theorem unitCompPartialBijective_symm_natural [Reflective i] (A : C) {B B' : C} (h : B ⟶ B')
(hB : B ∈ i.essImage) (hB' : B' ∈ i.essImage) (f : i.obj ((reflector i).obj A) ⟶ B) :
(unitCompPartialBijective A hB').symm (f ≫ h) = (unitCompPartialBijective A hB).symm f ≫ h := by |
simp
|
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.SeparableClosure
import Mathlib.Algebra.CharP.IntermediateField
/-!
# Purely inseparable extension and relative perfect closure
This file contains basic... | Mathlib/FieldTheory/PurelyInseparable.lean | 230 | 243 | theorem isPurelyInseparable_iff_pow_mem (q : ℕ) [ExpChar F q] :
IsPurelyInseparable F E ↔ ∀ x : E, ∃ n : ℕ, x ^ q ^ n ∈ (algebraMap F E).range := by |
rw [isPurelyInseparable_iff]
refine ⟨fun h x ↦ ?_, fun h x ↦ ?_⟩
· obtain ⟨g, h1, n, h2⟩ := (minpoly.irreducible (h x).1).hasSeparableContraction q
exact ⟨n, (h _).2 <| h1.of_dvd <| minpoly.dvd F _ <| by
simpa only [expand_aeval, minpoly.aeval] using congr_arg (aeval x) h2⟩
have hdeg := (minpoly.natS... |
/-
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.Control.Functor.Multivariate
import Mathlib.Data.PFunctor.Univariate.Basic
#align_import data.pfunctor.multivariate.basic from "leanprover-... | Mathlib/Data/PFunctor/Multivariate/Basic.lean | 160 | 170 | theorem liftP_iff {α : TypeVec n} (p : ∀ ⦃i⦄, α i → Prop) (x : P α) :
LiftP p x ↔ ∃ a f, x = ⟨a, f⟩ ∧ ∀ i j, p (f i j) := by |
constructor
· rintro ⟨y, hy⟩
cases' h : y with a f
refine ⟨a, fun i j => (f i j).val, ?_, fun i j => (f i j).property⟩
rw [← hy, h, map_eq]
rfl
rintro ⟨a, f, xeq, pf⟩
use ⟨a, fun i j => ⟨f i j, pf i j⟩⟩
rw [xeq]; rfl
|
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Felix Weilacher
-/
import Mathlib.Data.Real.Cardinality
import Mathlib.Topology.MetricSpace.Perfect
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
i... | Mathlib/MeasureTheory/Constructions/Polish.lean | 839 | 848 | theorem _root_.MeasurableSet.image_of_continuousOn_injOn [OpensMeasurableSpace β]
[tγ : TopologicalSpace γ] [PolishSpace γ] [MeasurableSpace γ] [BorelSpace γ]
(hs : MeasurableSet s)
(f_cont : ContinuousOn f s) (f_inj : InjOn f s) : MeasurableSet (f '' s) := by |
obtain ⟨t', t't, t'_polish, s_closed, _⟩ :
∃ t' : TopologicalSpace γ, t' ≤ tγ ∧ @PolishSpace γ t' ∧ IsClosed[t'] s ∧ IsOpen[t'] s :=
hs.isClopenable
exact
@IsClosed.measurableSet_image_of_continuousOn_injOn γ t' t'_polish β _ _ _ _ s s_closed f
(f_cont.mono_dom t't) f_inj
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264... | Mathlib/Algebra/MvPolynomial/Rename.lean | 284 | 288 | theorem eval₂_cast_comp (f : σ → τ) (c : ℤ →+* R) (g : τ → R) (p : MvPolynomial σ ℤ) :
eval₂ c (g ∘ f) p = eval₂ c g (rename f p) := by |
apply MvPolynomial.induction_on p (fun n => by simp only [eval₂_C, rename_C])
(fun p q hp hq => by simp only [hp, hq, rename, eval₂_add, AlgHom.map_add])
fun p n hp => by simp only [eval₂_mul, hp, eval₂_X, comp_apply, map_mul, rename_X, eval₂_mul]
|
/-
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.Topology.Algebra.InfiniteSum.Group
import Mathlib.Logic.Encodable.Lattice
/-!
# Infinite sums and products over `ℕ` and `ℤ`
This file contains lemma... | Mathlib/Topology/Algebra/InfiniteSum/NatInt.lean | 155 | 160 | theorem rel_iSup_tprod [CompleteLattice α] (m : α → M) (m0 : m ⊥ = 1) (R : M → M → Prop)
(m_iSup : ∀ s : ℕ → α, R (m (⨆ i, s i)) (∏' i, m (s i))) (s : β → α) :
R (m (⨆ b : β, s b)) (∏' b : β, m (s b)) := by |
cases nonempty_encodable β
rw [← iSup_decode₂, ← tprod_iSup_decode₂ _ m0 s]
exact m_iSup _
|
/-
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.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Su... | Mathlib/RingTheory/Polynomial/Bernstein.lean | 289 | 294 | theorem sum (n : ℕ) : (∑ ν ∈ Finset.range (n + 1), bernsteinPolynomial R n ν) = 1 :=
calc
(∑ ν ∈ Finset.range (n + 1), bernsteinPolynomial R n ν) = (X + (1 - X)) ^ n := by |
rw [add_pow]
simp only [bernsteinPolynomial, mul_comm, mul_assoc, mul_left_comm]
_ = 1 := by simp
|
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import comp... | Mathlib/Computability/TMToPartrec.lean | 1,370 | 1,390 | theorem move₂_ok {p k₁ k₂ q s L₁ o L₂} {S : K' → List Γ'} (h₁ : k₁ ≠ rev ∧ k₂ ≠ rev ∧ k₁ ≠ k₂)
(h₂ : S rev = []) (e : splitAtPred p (S k₁) = (L₁, o, L₂)) :
Reaches₁ (TM2.step tr) ⟨some (move₂ p k₁ k₂ q), s, S⟩
⟨some q, none, update (update S k₁ (o.elim id List.cons L₂)) k₂ (L₁ ++ S k₂)⟩ := by |
refine (move_ok h₁.1 e).trans (TransGen.head rfl ?_)
simp only [TM2.step, Option.mem_def, TM2.stepAux, id_eq, ne_eq, Option.elim]
cases o <;> simp only [Option.elim, id]
· simp only [TM2.stepAux, Option.isSome, cond_false]
convert move_ok h₁.2.1.symm (splitAtPred_false _) using 2
simp only [Function.up... |
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.ModelTheory.Basic
#align_import model_theory.language_map from "leanprover-c... | Mathlib/ModelTheory/LanguageMap.lean | 521 | 522 | theorem LHom.map_constants_comp_sumInl {f : α → β} :
(L.lhomWithConstantsMap f).comp LHom.sumInl = L.lhomWithConstants β := by | ext <;> rfl
|
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.MeasureTheory.Integral.SetIntegral... | Mathlib/MeasureTheory/Constructions/Prod/Integral.lean | 454 | 472 | theorem integral_prod (f : α × β → E) (hf : Integrable f (μ.prod ν)) :
∫ z, f z ∂μ.prod ν = ∫ x, ∫ y, f (x, y) ∂ν ∂μ := by |
by_cases hE : CompleteSpace E; swap; · simp only [integral, dif_neg hE]
revert f
apply Integrable.induction
· intro c s hs h2s
simp_rw [integral_indicator hs, ← indicator_comp_right, Function.comp,
integral_indicator (measurable_prod_mk_left hs), setIntegral_const, integral_smul_const,
integral... |
/-
Copyright (c) 2022 Siddhartha Prasad, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Siddhartha Prasad, Yaël Dillies
-/
import Mathlib.Algebra.Ring.Pi
import Mathlib.Algebra.Ring.Prod
import Mathlib.Algebra.Ring.InjSurj
import Mathlib.Tactic.Monotonici... | Mathlib/Algebra/Order/Kleene.lean | 154 | 154 | theorem add_eq_right_iff_le : a + b = b ↔ a ≤ b := by | simp
|
/-
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.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTh... | Mathlib/MeasureTheory/Function/LpSpace.lean | 290 | 291 | theorem coe_nnnorm_toLp {f : α → E} (hf : Memℒp f p μ) : (‖hf.toLp f‖₊ : ℝ≥0∞) = snorm f p μ := by |
rw [nnnorm_toLp f hf, ENNReal.coe_toNNReal hf.2.ne]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Set.Subsingle... | Mathlib/Combinatorics/Enumerative/Composition.lean | 870 | 872 | theorem length_lt_card_boundaries : c.length < c.boundaries.card := by |
rw [c.card_boundaries_eq_succ_length]
exact lt_add_one _
|
/-
Copyright (c) 2023 Antoine Labelle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle
-/
import Mathlib.Combinatorics.Quiver.Cast
import Mathlib.Combinatorics.Quiver.Symmetric
#align_import combinatorics.quiver.single_obj from "leanprover-community/mat... | Mathlib/Combinatorics/Quiver/SingleObj.lean | 110 | 112 | theorem toPrefunctor_symm_comp (f : SingleObj α ⥤q SingleObj β) (g : SingleObj β ⥤q SingleObj γ) :
toPrefunctor.symm (f ⋙q g) = toPrefunctor.symm g ∘ toPrefunctor.symm f := by |
simp only [Equiv.symm_apply_eq, toPrefunctor_comp, Equiv.apply_symm_apply]
|
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Analysis.InnerProductSpace.... | Mathlib/Analysis/Fourier/AddCircle.lean | 210 | 220 | theorem fourierSubalgebra_coe :
Subalgebra.toSubmodule (@fourierSubalgebra T).toSubalgebra = span ℂ (range (@fourier T)) := by |
apply adjoin_eq_span_of_subset
refine Subset.trans ?_ Submodule.subset_span
intro x hx
refine Submonoid.closure_induction hx (fun _ => id) ⟨0, ?_⟩ ?_
· ext1 z; exact fourier_zero
· rintro _ _ ⟨m, rfl⟩ ⟨n, rfl⟩
refine ⟨m + n, ?_⟩
ext1 z
exact fourier_add
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nathaniel Thomas, Jeremy Avigad, Johannes Hölzl, Mario Carneiro, Anne Baanen,
Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Algebra.Module.Hom
import Mathlib.Algebra.Module.LinearM... | Mathlib/Algebra/Module/Equiv.lean | 447 | 451 | theorem eq_toLinearMap_symm_comp (f : M₃ →ₛₗ[σ₃₁] M₁) (g : M₃ →ₛₗ[σ₃₂] M₂) :
f = e₁₂.symm.toLinearMap.comp g ↔ e₁₂.toLinearMap.comp f = g := by |
constructor <;> intro H <;> ext
· simp [H, e₁₂.toEquiv.eq_symm_comp f g]
· simp [← H, ← e₁₂.toEquiv.eq_symm_comp f g]
|
/-
Copyright (c) 2024 Lawrence Wu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lawrence Wu
-/
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.MeasureTheory.Integral.IntegrableOn
import Mathlib.MeasureTheory.Function.LocallyIntegrable
/-!
# Bounding of int... | Mathlib/MeasureTheory/Integral/Asymptotics.lean | 119 | 132 | theorem LocallyIntegrable.integrable_of_isBigO_atTop_of_norm_isNegInvariant
[IsMeasurablyGenerated (atTop (α := α))] [MeasurableNeg α] [μ.IsNegInvariant]
(hf : LocallyIntegrable f μ) (hsymm : norm ∘ f =ᵐ[μ] norm ∘ f ∘ Neg.neg) (ho : f =O[atTop] g)
(hg : IntegrableAtFilter g atTop μ) : Integrable f μ := by |
have h_int := (hf.locallyIntegrableOn (Ici 0)).integrableOn_of_isBigO_atTop ho hg
rw [← integrableOn_univ, ← Iic_union_Ici_of_le le_rfl, integrableOn_union]
refine ⟨?_, h_int⟩
have h_map_neg : (μ.restrict (Ici 0)).map Neg.neg = μ.restrict (Iic 0) := by
conv => rhs; rw [← Measure.map_neg_eq_self μ, measurab... |
/-
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 | 924 | 930 | theorem sign_toReal {θ : Angle} (h : θ ≠ π) : SignType.sign θ.toReal = θ.sign := by |
rcases lt_trichotomy θ.toReal 0 with (ht | ht | ht)
· simp [ht, toReal_neg_iff_sign_neg.1 ht]
· simp [sign, ht, ← sin_toReal]
· rw [sign, ← sin_toReal, sign_pos ht,
sign_pos
(sin_pos_of_pos_of_lt_pi ht ((toReal_le_pi θ).lt_of_ne (toReal_eq_pi_iff.not.2 h)))]
|
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Data.Matrix.Basic
/-!
# Row and column matrices
This file provides results about row and column matrices
## Main definitions
* `Matrix.row r... | Mathlib/Data/Matrix/RowCol.lean | 301 | 305 | theorem updateRow_submatrix_equiv [DecidableEq l] [DecidableEq m] (A : Matrix m n α) (i : l)
(r : o → α) (e : l ≃ m) (f : o ≃ n) :
updateRow (A.submatrix e f) i r = (A.updateRow (e i) fun j => r (f.symm j)).submatrix e f := by |
ext i' j
simp only [submatrix_apply, updateRow_apply, Equiv.apply_eq_iff_eq, Equiv.symm_apply_apply]
|
/-
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 | 218 | 221 | theorem natDegree_pos_of_eraseLead_ne_zero (h : f.eraseLead ≠ 0) : 0 < f.natDegree := by |
by_contra h₂
rw [eq_C_of_natDegree_eq_zero (Nat.eq_zero_of_not_pos h₂)] at h
simp at h
|
/-
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 | 140 | 145 | theorem natDegree_pow (p : R[X]) (n : ℕ) : natDegree (p ^ n) = n * natDegree p := by |
classical
obtain rfl | hp := eq_or_ne p 0
· obtain rfl | hn := eq_or_ne n 0 <;> simp [*]
exact natDegree_pow' $ by
rw [← leadingCoeff_pow, Ne, leadingCoeff_eq_zero]; exact pow_ne_zero _ hp
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
... | Mathlib/Algebra/Order/ToIntervalMod.lean | 304 | 305 | theorem toIcoDiv_add_right' (a b : α) : toIcoDiv hp (a + p) b = toIcoDiv hp a b - 1 := by |
simpa only [one_zsmul] using toIcoDiv_add_zsmul' hp a b 1
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Algebra.Order.Ring.WithTop
import Mathlib.Algebra.Order.Sub.WithTop
import Mathlib.Data.Real.NNReal
import Mathlib.Order.Interval.Set.... | Mathlib/Data/ENNReal/Basic.lean | 438 | 440 | theorem toReal_eq_toReal_iff (x y : ℝ≥0∞) :
x.toReal = y.toReal ↔ x = y ∨ x = 0 ∧ y = ⊤ ∨ x = ⊤ ∧ y = 0 := by |
simp only [ENNReal.toReal, NNReal.coe_inj, toNNReal_eq_toNNReal_iff]
|
/-
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.Set.Function
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Core
import Mathlib.Tactic.Attr.Core
#align_import logic.equiv.local_equ... | Mathlib/Logic/Equiv/PartialEquiv.lean | 418 | 419 | theorem iff_preimage_eq : e.IsImage s t ↔ e.source ∩ e ⁻¹' t = e.source ∩ s := by |
simp only [IsImage, ext_iff, mem_inter_iff, mem_preimage, and_congr_right_iff]
|
/-
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.Data.List.Join
#align_import data.list.permutation from "leanprover-community/... | Mathlib/Data/List/Permutation.lean | 77 | 79 | theorem permutationsAux2_append (t : α) (ts : List α) (r : List β) (ys : List α) (f : List α → β) :
(permutationsAux2 t ts nil ys f).2 ++ r = (permutationsAux2 t ts r ys f).2 := by |
induction ys generalizing f <;> simp [*]
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Logic.Relation
import Mathlib.Data.Option.Basic
import Mathlib.Data.Seq.Seq
#align_import data.seq.wseq from "leanprover-community/mathlib"@"a7... | Mathlib/Data/Seq/WSeq.lean | 1,303 | 1,327 | theorem toList'_map (l : List α) (s : WSeq α) :
Computation.corec (fun ⟨l, s⟩ =>
match Seq.destruct s with
| none => Sum.inl l.reverse
| some (none, s') => Sum.inr (l, s')
| some (some a, s') => Sum.inr (a :: l, s')) (l, s) = (l.reverse ++ ·) <$> toList s := by |
refine
Computation.eq_of_bisim
(fun c1 c2 =>
∃ (l' : List α) (s : WSeq α),
c1 = Computation.corec (fun ⟨l, s⟩ =>
match Seq.destruct s with
| none => Sum.inl l.reverse
| some (none, s') => Sum.inr (l, s')
| some (some a, s') => Sum.inr (a::l,... |
/-
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.ContMDiff.Defs
/-!
## Basic properties of smooth functions between manifolds
In this file, we show that sta... | Mathlib/Geometry/Manifold/ContMDiff/Basic.lean | 262 | 262 | theorem smooth_one [One M'] : Smooth I I' (1 : M → M') := by | simp only [Pi.one_def, smooth_const]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johan Commelin
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Combinatorics.Enumerative.Composition
#align_import analysis.analytic.composition from "l... | Mathlib/Analysis/Analytic/Composition.lean | 520 | 533 | theorem le_comp_radius_of_summable (q : FormalMultilinearSeries 𝕜 F G)
(p : FormalMultilinearSeries 𝕜 E F) (r : ℝ≥0)
(hr : Summable fun i : Σ n, Composition n => ‖q.compAlongComposition p i.2‖₊ * r ^ i.1) :
(r : ℝ≥0∞) ≤ (q.comp p).radius := by |
refine
le_radius_of_bound_nnreal _
(∑' i : Σ n, Composition n, ‖compAlongComposition q p i.snd‖₊ * r ^ i.fst) fun n => ?_
calc
‖FormalMultilinearSeries.comp q p n‖₊ * r ^ n ≤
∑' c : Composition n, ‖compAlongComposition q p c‖₊ * r ^ n := by
rw [tsum_fintype, ← Finset.sum_mul]
exac... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Image
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Clopen
import Mathlib.Topology.Irreducib... | Mathlib/Topology/Connected/Basic.lean | 1,112 | 1,124 | theorem isPreconnected_iff_subset_of_fully_disjoint_closed {s : Set α} (hs : IsClosed s) :
IsPreconnected s ↔
∀ u v, IsClosed u → IsClosed v → s ⊆ u ∪ v → Disjoint u v → s ⊆ u ∨ s ⊆ v := by |
refine isPreconnected_iff_subset_of_disjoint_closed.trans ⟨?_, ?_⟩ <;> intro H u v hu hv hss huv
· apply H u v hu hv hss
rw [huv.inter_eq, inter_empty]
have H1 := H (u ∩ s) (v ∩ s)
rw [subset_inter_iff, subset_inter_iff] at H1
simp only [Subset.refl, and_true] at H1
apply H1 (hu.inter hs) (hv.inter hs)... |
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Lu-Ming Zhang
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algeb... | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 546 | 547 | theorem left_inv_eq_left_inv (h : B * A = 1) (g : C * A = 1) : B = C := by |
rw [← inv_eq_left_inv h, ← inv_eq_left_inv g]
|
/-
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 | 2,508 | 2,521 | theorem isLimit_iff_omega_dvd {a : Ordinal} : IsLimit a ↔ a ≠ 0 ∧ ω ∣ a := by |
refine ⟨fun l => ⟨l.1, ⟨a / ω, le_antisymm ?_ (mul_div_le _ _)⟩⟩, fun h => ?_⟩
· refine (limit_le l).2 fun x hx => le_of_lt ?_
rw [← div_lt omega_ne_zero, ← succ_le_iff, le_div omega_ne_zero, mul_succ,
add_le_of_limit omega_isLimit]
intro b hb
rcases lt_omega.1 hb with ⟨n, rfl⟩
exact
(a... |
/-
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 | 326 | 331 | theorem continuousMul_of_comm_of_nhds_one (M : Type u) [CommMonoid M] [TopologicalSpace M]
(hmul : Tendsto (uncurry ((· * ·) : M → M → M)) (𝓝 1 ×ˢ 𝓝 1) (𝓝 1))
(hleft : ∀ x₀ : M, 𝓝 x₀ = map (fun x => x₀ * x) (𝓝 1)) : ContinuousMul M := by |
apply ContinuousMul.of_nhds_one hmul hleft
intro x₀
simp_rw [mul_comm, hleft x₀]
|
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.InnerProductSpace.Adjoint
#align_import analysis.inner_product_space.positive from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca... | Mathlib/Analysis/InnerProductSpace/Positive.lean | 67 | 68 | theorem IsPositive.inner_nonneg_right {T : E →L[𝕜] E} (hT : IsPositive T) (x : E) :
0 ≤ re ⟪x, T x⟫ := by | rw [inner_re_symm]; exact hT.inner_nonneg_left x
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Relator
import Mathlib.Init.Data.Quot
import Mathlib.Tactic.Cases
import Mathlib.Tactic.Use
import Mathlib.Ta... | Mathlib/Logic/Relation.lean | 559 | 566 | theorem reflTransGen_iff_eq_or_transGen : ReflTransGen r a b ↔ b = a ∨ TransGen r a b := by |
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
· cases' h with c _ hac hcb
· exact Or.inl rfl
· exact Or.inr (TransGen.tail' hac hcb)
· rcases h with (rfl | h)
· rfl
· exact h.to_reflTransGen
|
/-
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.Group.Multiset
import Mathlib.Data.PNat.Prime
import Mathlib.Data.Nat.Factors
import Mathlib.Data.Multiset.Sort
#align_import d... | Mathlib/Data/PNat/Factors.lean | 195 | 198 | theorem prod_ofNatList (l : List ℕ) (h) : ((ofNatList l h).prod : ℕ) = l.prod := by |
have := prod_ofNatMultiset (l : Multiset ℕ) h
rw [Multiset.prod_coe] at this
exact this
|
/-
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.LinearAlgebra.AffineSpace.Independent
import Mathlib.LinearAlgebra.Basis
#align_import linear_algebra.affine_space.basis from "leanprover-community/mathlib"... | Mathlib/LinearAlgebra/AffineSpace/Basis.lean | 168 | 170 | theorem coord_apply_eq (i : ι) : b.coord i (b i) = 1 := by |
simp only [coord, Basis.coe_sumCoords, LinearEquiv.map_zero, LinearEquiv.coe_coe, sub_zero,
AffineMap.coe_mk, Finsupp.sum_zero_index, vsub_self]
|
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.MeasureTheory.Group.GeometryOfNumbers
import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
import Mathlib.NumberTheory.NumberField.CanonicalEmbedd... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | 316 | 324 | theorem convexBodySumFun_eq_zero_iff (x : E K) :
convexBodySumFun x = 0 ↔ x = 0 := by |
rw [← normAtPlace_eq_zero, convexBodySumFun, Finset.sum_eq_zero_iff_of_nonneg fun _ _ =>
mul_nonneg (Nat.cast_pos.mpr mult_pos).le (normAtPlace_nonneg _ _)]
conv =>
enter [1, w, hw]
rw [mul_left_mem_nonZeroDivisors_eq_zero_iff
(mem_nonZeroDivisors_iff_ne_zero.mpr <| Nat.cast_ne_zero.mpr mult_ne_z... |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.Algebra.Category.ModuleCat.Kernels
import Mathlib.Algebra.Category.ModuleCat.Limits
import Mathlib.CategoryTheo... | Mathlib/Algebra/Category/ModuleCat/Abelian.lean | 123 | 127 | theorem exact_iff : Exact f g ↔ LinearMap.range f = LinearMap.ker g := by |
rw [abelian.exact_iff' f g (kernelIsLimit _) (cokernelIsColimit _)]
exact
⟨fun h => le_antisymm (range_le_ker_iff.2 h.1) (ker_le_range_iff.2 h.2), fun h =>
⟨range_le_ker_iff.1 <| le_of_eq h, ker_le_range_iff.1 <| le_of_eq h.symm⟩⟩
|
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.PSub
import Mathlib.Data.Nat... | Mathlib/Data/Num/Lemmas.lean | 246 | 246 | theorem ofNat'_one : Num.ofNat' 1 = 1 := by | erw [ofNat'_bit true 0, cond, ofNat'_zero]; rfl
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.MvPolynomial.Degrees
import ... | Mathlib/Algebra/MvPolynomial/Equiv.lean | 377 | 378 | theorem finSuccEquiv_X_succ {j : Fin n} : finSuccEquiv R n (X j.succ) = Polynomial.C (X j) := by |
simp
|
/-
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 | 514 | 518 | theorem eval_prec_succ (cf cg : Code) (a k : ℕ) :
eval (prec cf cg) (Nat.pair a (Nat.succ k)) =
do {let ih ← eval (prec cf cg) (Nat.pair a k); eval cg (Nat.pair a (Nat.pair k ih))} := by |
rw [eval, Nat.unpaired, Part.bind_eq_bind, Nat.unpair_pair]
simp
|
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Integral.Layercake
#align_import analy... | Mathlib/Analysis/SpecialFunctions/JapaneseBracket.lean | 62 | 65 | theorem le_rpow_one_add_norm_iff_norm_le {r t : ℝ} (hr : 0 < r) (ht : 0 < t) (x : E) :
t ≤ (1 + ‖x‖) ^ (-r) ↔ ‖x‖ ≤ t ^ (-r⁻¹) - 1 := by |
rw [le_sub_iff_add_le', neg_inv]
exact (Real.le_rpow_inv_iff_of_neg (by positivity) ht (neg_lt_zero.mpr hr)).symm
|
/-
Copyright (c) 2021 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.Tactic.ByContra
import Mathlib.Topology.Algebra.Polynomial
import Mathlib.NumberTheory.Padics.Pad... | Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean | 36 | 37 | theorem eval₂_one_cyclotomic_prime {R S : Type*} [CommRing R] [Semiring S] (f : R →+* S) {p : ℕ}
[Fact p.Prime] : eval₂ f 1 (cyclotomic p R) = p := by | simp
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Vector.Basic
import Mathlib.Data.PFun
import Ma... | Mathlib/Computability/TuringMachine.lean | 2,382 | 2,384 | theorem addBottom_nth_snd (L : ListBlank (∀ k, Option (Γ k))) (n : ℕ) :
((addBottom L).nth n).2 = L.nth n := by |
conv => rhs; rw [← addBottom_map L, ListBlank.nth_map]
|
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.PSub
import Mathlib.Data.Nat... | Mathlib/Data/Num/Lemmas.lean | 1,138 | 1,138 | theorem add_zero (n : ZNum) : n + 0 = n := by | cases n <;> rfl
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
#align_... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 125 | 125 | theorem rpow_zero (x : ℝ) : x ^ (0 : ℝ) = 1 := by | simp [rpow_def]
|
/-
Copyright (c) 2022 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Dynamics.Ergodic.AddCircle
import Mathlib.MeasureTheory.Covering.LiminfLimsup
#align_import number_theory.well_approximable from "leanprover-community/mathl... | Mathlib/NumberTheory/WellApproximable.lean | 147 | 166 | theorem smul_eq_of_mul_dvd (hn : 0 < n) (han : orderOf a ^ 2 ∣ n) :
a • approxOrderOf A n δ = approxOrderOf A n δ := by |
simp_rw [approxOrderOf, thickening_eq_biUnion_ball, ← image_smul, image_iUnion₂, image_smul,
smul_ball'', smul_eq_mul, mem_setOf_eq]
replace han : ∀ {b : A}, orderOf b = n → orderOf (a * b) = n := by
intro b hb
rw [← hb] at han hn
rw [sq] at han
rwa [(Commute.all a b).orderOf_mul_eq_right_of_fo... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.LinearAlgebra.Span
#align_import linear_algebra.quotient from... | Mathlib/LinearAlgebra/Quotient.lean | 334 | 335 | theorem mkQ_surjective (A : Submodule R M) : Function.Surjective A.mkQ := by |
rintro ⟨x⟩; exact ⟨x, rfl⟩
|
/-
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.Data.Fin.VecNotation
import Mathlib.Logic.Embedding.Set
#align_import logic.equiv.fin from "leanprover-community/mathlib"@"bd835ef554f37ef9b804f0903089211f89cb3... | Mathlib/Logic/Equiv/Fin.lean | 452 | 454 | theorem coe_finRotate (i : Fin n.succ) :
(finRotate n.succ i : ℕ) = if i = Fin.last n then (0 : ℕ) else i + 1 := by |
rw [finRotate_succ_apply, Fin.val_add_one i]
|
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Ultraproducts
import Mathlib.ModelTheory.Bundled
import Mathlib.ModelTheory.Skolem
#align_import model_theory.satisfiability from "leanpro... | Mathlib/ModelTheory/Satisfiability.lean | 389 | 399 | theorem models_not_iff (h : T.IsComplete) (φ : L.Sentence) : T ⊨ᵇ φ.not ↔ ¬T ⊨ᵇ φ := by |
cases' h.2 φ with hφ hφn
· simp only [hφ, not_true, iff_false_iff]
rw [models_sentence_iff, not_forall]
refine ⟨h.1.some, ?_⟩
simp only [Sentence.realize_not, Classical.not_not]
exact models_sentence_iff.1 hφ _
· simp only [hφn, true_iff_iff]
intro hφ
rw [models_sentence_iff] at *
exa... |
/-
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 | 280 | 286 | theorem meas_ge_le_variance_div_sq [@IsFiniteMeasure Ω _ ℙ] {X : Ω → ℝ} (hX : Memℒp X 2) {c : ℝ}
(hc : 0 < c) : ℙ {ω | c ≤ |X ω - 𝔼[X]|} ≤ ENNReal.ofReal (Var[X] / c ^ 2) := by |
rw [ENNReal.ofReal_div_of_pos (sq_pos_of_ne_zero hc.ne.symm), hX.ofReal_variance_eq]
convert @meas_ge_le_evariance_div_sq _ _ _ hX.1 c.toNNReal (by simp [hc]) using 1
· simp only [Real.coe_toNNReal', max_le_iff, abs_nonneg, and_true_iff]
· rw [ENNReal.ofReal_pow hc.le]
rfl
|
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.EMetricSpace.Basic
import Mathlib.Topology.Bornology.Constructions
imp... | Mathlib/Topology/MetricSpace/PseudoMetric.lean | 1,393 | 1,396 | theorem Real.closedBall_eq_Icc {x r : ℝ} : closedBall x r = Icc (x - r) (x + r) := by |
ext y
rw [mem_closedBall, dist_comm, Real.dist_eq, abs_sub_le_iff, mem_Icc, ← sub_le_iff_le_add',
sub_le_comm]
|
/-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Scott Morrison, Floris van Doorn
-/
import Mathlib.CategoryTheory.Limits.IsLimit
import Mathlib.CategoryTheory.Category.ULift
import Mathlib.CategoryTheory... | Mathlib/CategoryTheory/Limits/HasLimits.lean | 224 | 225 | theorem limit.coneMorphism_π {F : J ⥤ C} [HasLimit F] (c : Cone F) (j : J) :
(limit.coneMorphism c).hom ≫ limit.π F j = c.π.app j := by | simp
|
/-
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 | 2,086 | 2,089 | theorem le_bmex_of_forall {o : Ordinal} (f : ∀ a < o, Ordinal) {a : Ordinal}
(H : ∀ b < a, ∃ i hi, f i hi = b) : a ≤ bmex o f := by |
by_contra! h
exact bmex_not_mem_brange f (H _ h)
|
/-
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.Data.Nat.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.List.Defs
im... | Mathlib/Data/List/Basic.lean | 892 | 901 | theorem nthLe_cons {l : List α} {a : α} {n} (hl) :
(a :: l).nthLe n hl = if hn : n = 0 then a else l.nthLe (n - 1) (nthLe_cons_aux hn hl) := by |
split_ifs with h
· simp [nthLe, h]
cases l
· rw [length_singleton, Nat.lt_succ_iff] at hl
omega
cases n
· contradiction
rfl
|
/-
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.Units.Lemmas
import Mathlib.Data.Rat.Cast.Defs
#align_import data.rat.cast from "leanprover-community/mathlib"@"... | Mathlib/Data/Rat/Cast/CharZero.lean | 46 | 46 | theorem cast_eq_zero [CharZero α] {n : ℚ} : (n : α) = 0 ↔ n = 0 := by | rw [← cast_zero, cast_inj]
|
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FDeriv.Bilinear
#align_import analysis.calculus.fderiv.mul from "leanprover-community/mathlib"@"d6... | Mathlib/Analysis/Calculus/FDeriv/Mul.lean | 230 | 233 | theorem fderiv_continuousMultilinear_apply_const_apply (hc : DifferentiableAt 𝕜 c x)
(u : ∀ i, M i) (m : E) :
(fderiv 𝕜 (fun y ↦ (c y) u) x) m = (fderiv 𝕜 c x) m u := by |
simp [fderiv_continuousMultilinear_apply_const hc]
|
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Yuyang Zhao
-/
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Algebra.Order.Monoid.Unbundled.Defs
import Mathlib.Tactic.GCongr.Core
#align_import algebra.order... | Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean | 402 | 403 | theorem Left.mul_pos [PosMulStrictMono α] (ha : 0 < a) (hb : 0 < b) : 0 < a * b := by |
simpa only [mul_zero] using mul_lt_mul_of_pos_left hb ha
|
/-
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.Algebra.BigOperators.Finprod
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Topology.ContinuousFunction.Algebra
import Mathlib.Topology.Compac... | Mathlib/Topology/PartitionOfUnity.lean | 474 | 475 | theorem toPOUFun_zero_of_zero {i : ι} {x : X} (h : f i x = 0) : f.toPOUFun i x = 0 := by |
rw [toPOUFun, h, zero_mul]
|
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Eric Wieser
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Normed.Field.InfiniteSum
import Mathlib.Data.Nat... | Mathlib/Analysis/NormedSpace/Exponential.lean | 226 | 231 | theorem norm_expSeries_summable_of_mem_ball' (x : 𝔸)
(hx : x ∈ EMetric.ball (0 : 𝔸) (expSeries 𝕂 𝔸).radius) :
Summable fun n => ‖(n !⁻¹ : 𝕂) • x ^ n‖ := by |
change Summable (norm ∘ _)
rw [← expSeries_apply_eq']
exact norm_expSeries_summable_of_mem_ball x hx
|
/-
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 | 1,394 | 1,397 | theorem edist_def (f g : α →₁[μ] β) : edist f g = ∫⁻ a, edist (f a) (g a) ∂μ := by |
simp only [Lp.edist_def, snorm, one_ne_zero, snorm', Pi.sub_apply, one_toReal, ENNReal.rpow_one,
ne_eq, not_false_eq_true, div_self, ite_false]
simp [edist_eq_coe_nnnorm_sub]
|
/-
Copyright (c) 2022 Wrenna Robson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wrenna Robson
-/
import Mathlib.Topology.MetricSpace.Basic
#align_import topology.metric_space.infsep from "leanprover-community/mathlib"@"5316314b553dcf8c6716541851517c1a9715e22b"
/-... | Mathlib/Topology/MetricSpace/Infsep.lean | 348 | 351 | theorem nontrivial_of_infsep_pos (hs : 0 < s.infsep) : s.Nontrivial := by |
contrapose hs
rw [not_nontrivial_iff] at hs
exact hs.infsep_zero ▸ lt_irrefl _
|
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Eric Wieser
-/
import Mathlib.LinearAlgebra.Matrix.DotProduct
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
#align_import data.... | Mathlib/Data/Matrix/Rank.lean | 181 | 182 | theorem rank_eq_finrank_span_cols (A : Matrix m n R) :
A.rank = finrank R (Submodule.span R (Set.range Aᵀ)) := by | rw [rank, Matrix.range_mulVecLin]
|
/-
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.Algebra.BigOperators.Group.List
import Mathlib.Algebra.Group.Prod
import Mathlib.Data.Multiset.Basic
#align_import algebra.big_operators.multis... | Mathlib/Algebra/BigOperators/Group/Multiset.lean | 246 | 248 | theorem prod_dvd_prod_of_le (h : s ≤ t) : s.prod ∣ t.prod := by |
obtain ⟨z, rfl⟩ := exists_add_of_le h
simp only [prod_add, dvd_mul_right]
|
/-
Copyright (c) 2022 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Amelia Livingston
-/
import Mathlib.Algebra.Homology.Additive
import Mathlib.CategoryTheory.Abelian.Pseudoelements
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels... | Mathlib/CategoryTheory/Abelian/Homology.lean | 157 | 159 | theorem condition_ι : ι f g w ≫ cokernel.desc f g w = 0 := by |
dsimp [ι]
simp
|
/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.EpiMono
import Mathlib.Topology.Category.TopCat.Limits.Basic
import Mathlib.Categ... | Mathlib/Topology/Category/TopCat/Limits/Products.lean | 82 | 86 | theorem piIsoPi_hom_apply {ι : Type v} (α : ι → TopCat.{max v u}) (i : ι)
(x : (∏ᶜ α : TopCat.{max v u})) : (piIsoPi α).hom x i = (Pi.π α i : _) x := by |
have := piIsoPi_inv_π α i
rw [Iso.inv_comp_eq] at this
exact ConcreteCategory.congr_hom this x
|
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Yury Kudriashov, Yaël Dillies
-/
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Analysis.Convex.Star
import Mathlib.LinearAlgebra.AffineSpace.Af... | Mathlib/Analysis/Convex/Basic.lean | 121 | 128 | theorem Directed.convex_iUnion {ι : Sort*} {s : ι → Set E} (hdir : Directed (· ⊆ ·) s)
(hc : ∀ ⦃i : ι⦄, Convex 𝕜 (s i)) : Convex 𝕜 (⋃ i, s i) := by |
rintro x hx y hy a b ha hb hab
rw [mem_iUnion] at hx hy ⊢
obtain ⟨i, hx⟩ := hx
obtain ⟨j, hy⟩ := hy
obtain ⟨k, hik, hjk⟩ := hdir i j
exact ⟨k, hc (hik hx) (hjk hy) ha hb hab⟩
|
/-
Copyright (c) 2022 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.CategoryTheory.Abelian.Exact
import Mathlib.CategoryTheory.Preadditive.Injective
import Mathlib.CategoryTheory.Preadditive.Yoneda.Limits
import Mat... | Mathlib/CategoryTheory/Abelian/Injective.lean | 45 | 48 | theorem injective_of_preservesFiniteColimits_preadditiveYonedaObj (J : C)
[hP : PreservesFiniteColimits (preadditiveYonedaObj J)] : Injective J := by |
rw [injective_iff_preservesEpimorphisms_preadditive_yoneda_obj']
infer_instance
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Probability.Martingale.Convergence
import Mathlib.Probability.Martingale.OptionalStopping
import Mathlib.Probability.Martingale.Centering
#align_import prob... | Mathlib/Probability/Martingale/BorelCantelli.lean | 303 | 308 | theorem predictablePart_process_ae_eq (ℱ : Filtration ℕ m0) (μ : Measure Ω) (s : ℕ → Set Ω)
(n : ℕ) : predictablePart (process s) ℱ μ n =
∑ k ∈ Finset.range n, μ[(s (k + 1)).indicator (1 : Ω → ℝ)|ℱ k] := by |
have := martingalePart_process_ae_eq ℱ μ s n
simp_rw [martingalePart, process, Finset.sum_sub_distrib] at this
exact sub_right_injective this
|
/-
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.NullMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Topology.Algebra.Order.LiminfLim... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 2,027 | 2,033 | theorem ae_map_mem_range {m0 : MeasurableSpace α} (f : α → β) (hf : MeasurableSet (range f))
(μ : Measure α) : ∀ᵐ x ∂μ.map f, x ∈ range f := by |
by_cases h : AEMeasurable f μ
· change range f ∈ ae (μ.map f)
rw [mem_ae_map_iff h hf]
filter_upwards using mem_range_self
· simp [map_of_not_aemeasurable h]
|
/-
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.Decomposition
import Mathlib.Tactic.FinCases
#align_import algebraic_topology.dold_kan.degeneracies from "leanprover-community/mathlib... | Mathlib/AlgebraicTopology/DoldKan/Degeneracies.lean | 128 | 140 | theorem degeneracy_comp_PInfty (X : SimplicialObject C) (n : ℕ) {Δ' : SimplexCategory}
(θ : ([n] : SimplexCategory) ⟶ Δ') (hθ : ¬Mono θ) : X.map θ.op ≫ PInfty.f n = 0 := by |
rw [SimplexCategory.mono_iff_injective] at hθ
cases n
· exfalso
apply hθ
intro x y h
fin_cases x
fin_cases y
rfl
· obtain ⟨i, α, h⟩ := SimplexCategory.eq_σ_comp_of_not_injective θ hθ
rw [h, op_comp, X.map_comp, assoc, show X.map (SimplexCategory.σ i).op = X.σ i by rfl,
σ_comp_PInf... |
/-
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.Topology.Constructions
import Mathlib.Topology.ContinuousOn
#align_import topology.bases from "leanprover-community/mathlib"@"bcfa7268... | Mathlib/Topology/Bases.lean | 864 | 872 | theorem isOpen_iUnion_countable [SecondCountableTopology α] {ι} (s : ι → Set α)
(H : ∀ i, IsOpen (s i)) : ∃ T : Set ι, T.Countable ∧ ⋃ i ∈ T, s i = ⋃ i, s i := by |
let B := { b ∈ countableBasis α | ∃ i, b ⊆ s i }
choose f hf using fun b : B => b.2.2
haveI : Countable B := ((countable_countableBasis α).mono (sep_subset _ _)).to_subtype
refine ⟨_, countable_range f, (iUnion₂_subset_iUnion _ _).antisymm (sUnion_subset ?_)⟩
rintro _ ⟨i, rfl⟩ x xs
rcases (isBasis_countabl... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Star.Basic
import Mathlib.Algebra.Order.CauSeq.Completion
#align_import data.real.basic from "leanprover-community/mathlib"@... | Mathlib/Data/Real/Basic.lean | 358 | 363 | theorem add_lt_add_iff_left {a b : ℝ} (c : ℝ) : c + a < c + b ↔ a < b := by |
induction a using Real.ind_mk
induction b using Real.ind_mk
induction c using Real.ind_mk
simp only [mk_lt, ← mk_add]
show Pos _ ↔ Pos _; rw [add_sub_add_left_eq_sub]
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shape... | Mathlib/CategoryTheory/Limits/Shapes/Images.lean | 218 | 218 | theorem isoExt_hom_m : (isoExt hF hF').hom ≫ F'.m = F.m := by | simp
|
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