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 |
|---|---|---|---|---|---|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Init.Logic
import Mathlib.Tactic.AdaptationNote
import Mathlib.Tactic.Coe
/-!
# Lemmas about booleans
These are the lemmas about booleans w... | Mathlib/Init/Data/Bool/Lemmas.lean | 86 | 88 | theorem or_eq_false_eq_eq_false_and_eq_false (a b : Bool) :
((a || b) = false) = (a = false ∧ b = false) := by |
cases a <;> cases b <;> simp
|
/-
Copyright (c) 2023 Jason Yuen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jason Yuen
-/
import Mathlib.Data.Real.ConjExponents
import Mathlib.Data.Real.Irrational
/-!
# Rayleigh's theorem on Beatty sequences
This file proves Rayleigh's theorem on Beatty sequen... | Mathlib/NumberTheory/Rayleigh.lean | 162 | 168 | theorem Irrational.beattySeq'_pos_eq {r : ℝ} (hr : Irrational r) :
{beattySeq' r k | k > 0} = {beattySeq r k | k > 0} := by |
dsimp only [beattySeq, beattySeq']
congr! 4; rename_i k; rw [and_congr_right_iff]; intro hk; congr!
rw [sub_eq_iff_eq_add, Int.ceil_eq_iff, Int.cast_add, Int.cast_one, add_sub_cancel_right]
refine ⟨(Int.floor_le _).lt_of_ne fun h ↦ ?_, (Int.lt_floor_add_one _).le⟩
exact (hr.int_mul hk.ne').ne_int ⌊k * r⌋ h.s... |
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Wen Yang
-/
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.RingT... | Mathlib/LinearAlgebra/Matrix/SpecialLinearGroup.lean | 531 | 532 | theorem T_mul_apply_one (g : SL(2, ℤ)) : ↑ₘ(T * g) 1 = ↑ₘg 1 := by |
simpa using T_pow_mul_apply_one 1 g
|
/-
Copyright (c) 2020 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Data.Multiset.Basic
import Mathlib.Data.Vector.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.Tactic.ApplyFun
#align_import data.sym.basic from "lean... | Mathlib/Data/Sym/Basic.lean | 684 | 694 | theorem encode_decode [DecidableEq α] (s : Sum (Sym (Option α) n) (Sym α n.succ)) :
encode (decode s) = s := by |
obtain s | s := s
· simp
· unfold SymOptionSuccEquiv.encode
split_ifs with h
· obtain ⟨a, _, ha⟩ := Multiset.mem_map.mp h
exact Option.some_ne_none _ ha
· refine congr_arg Sum.inr ?_
refine map_injective (Option.some_injective _) _ ?_
refine Eq.trans ?_ (.trans (SymOptionSuccEquiv.d... |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl
-/
import Mathlib.Topology.UniformSpace.AbstractCompletion
#align_import topology.uniform_space.completion from "leanprover-community/mathlib"@"dc6c365e... | Mathlib/Topology/UniformSpace/Completion.lean | 302 | 305 | theorem cauchyFilter_eq {α : Type*} [UniformSpace α] [CompleteSpace α] [T0Space α]
{f g : CauchyFilter α} :
haveI := f.2.1.nonempty; lim f.1 = lim g.1 ↔ Inseparable f g := by |
rw [← inseparable_iff_eq, inseparable_lim_iff]
|
/-
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.RingTheory.Ideal.Operations
#align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74"
/-!
# Map... | Mathlib/RingTheory/Ideal/Maps.lean | 558 | 560 | theorem IsRadical.comap (hK : K.IsRadical) : (comap f K).IsRadical := by |
rw [← hK.radical, comap_radical]
apply radical_isRadical
|
/-
Copyright (c) 2021 Apurva Nakade. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Apurva Nakade
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Localization.Basic
import... | Mathlib/SetTheory/Surreal/Dyadic.lean | 263 | 266 | theorem dyadicMap_apply (m : ℤ) (p : Submonoid.powers (2 : ℤ)) :
dyadicMap (IsLocalization.mk' (Localization (Submonoid.powers 2)) m p) =
m • powHalf (Submonoid.log p) := by |
rw [← Localization.mk_eq_mk']; rfl
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.convex.side from "lean... | Mathlib/Analysis/Convex/Side.lean | 744 | 748 | theorem sOppSide_smul_vsub_vadd_left {s : AffineSubspace R P} {x p₁ p₂ : P} (hx : x ∉ s)
(hp₁ : p₁ ∈ s) (hp₂ : p₂ ∈ s) {t : R} (ht : t < 0) : s.SOppSide (t • (x -ᵥ p₁) +ᵥ p₂) x := by |
refine ⟨wOppSide_smul_vsub_vadd_left x hp₁ hp₂ ht.le, fun h => hx ?_, hx⟩
rwa [vadd_mem_iff_mem_direction _ hp₂, s.direction.smul_mem_iff ht.ne,
vsub_right_mem_direction_iff_mem hp₁] at 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.CategoryTheory.Idempotents.Basic
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Equivalence
#align_import category_theo... | Mathlib/CategoryTheory/Idempotents/Karoubi.lean | 97 | 98 | theorem comp_proof {P Q R : Karoubi C} (g : Hom Q R) (f : Hom P Q) :
f.f ≫ g.f = P.p ≫ (f.f ≫ g.f) ≫ R.p := by | rw [assoc, comp_p, ← assoc, p_comp]
|
/-
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.Submodule
#align_import algebra.lie.ideal_operations from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
/-!
# Ideal... | Mathlib/Algebra/Lie/IdealOperations.lean | 196 | 198 | theorem inf_lie : ⁅I ⊓ J, N⁆ ≤ ⁅I, N⁆ ⊓ ⁅J, N⁆ := by |
rw [le_inf_iff]; constructor <;>
apply mono_lie_left <;> [exact inf_le_left; exact inf_le_right]
|
/-
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.Dynamics.Ergodic.MeasurePreserving
import Mathlib.GroupTheory.GroupAction.Hom
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.... | Mathlib/MeasureTheory/Group/Measure.lean | 365 | 368 | theorem eventually_mul_right_iff (μ : Measure G) [IsMulRightInvariant μ] (t : G) {p : G → Prop} :
(∀ᵐ x ∂μ, p (x * t)) ↔ ∀ᵐ x ∂μ, p x := by |
conv_rhs => rw [Filter.Eventually, ← map_mul_right_ae μ t]
rfl
|
/-
Copyright (c) 2022 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell, Eric Wieser, Yaël Dillies, Patrick Massot, Scott Morrison
-/
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.Order.Interval... | Mathlib/Algebra/Order/Interval/Set/Instances.lean | 165 | 167 | theorem one_sub_mem {t : β} (ht : t ∈ Icc (0 : β) 1) : 1 - t ∈ Icc (0 : β) 1 := by |
rw [mem_Icc] at *
exact ⟨sub_nonneg.2 ht.2, (sub_le_self_iff _).2 ht.1⟩
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.AlgebraicGeometry.Pullbacks
import Mathlib.CategoryTheory.MorphismProperty.Limits
import Mathlib.Data.List.TFAE... | Mathlib/AlgebraicGeometry/Morphisms/Basic.lean | 547 | 574 | theorem AffineTargetMorphismProperty.IsLocal.diagonal_affine_openCover_TFAE
{P : AffineTargetMorphismProperty} (hP : P.IsLocal) {X Y : Scheme.{u}} (f : X ⟶ Y) :
TFAE
[(targetAffineLocally P).diagonal f,
∃ (𝒰 : Scheme.OpenCover.{u} Y) (_ : ∀ i, IsAffine (𝒰.obj i)),
∀ i : 𝒰.J, P.diagona... |
tfae_have 1 → 4
· introv H hU hg _ _; apply P.diagonalOfTargetAffineLocally <;> assumption
tfae_have 4 → 3
· introv H h𝒰; apply H
tfae_have 3 → 2
· exact fun H => ⟨Y.affineCover, inferInstance, H Y.affineCover⟩
tfae_have 2 → 5
· rintro ⟨𝒰, h𝒰, H⟩
refine ⟨𝒰, inferInstance, fun _ => Scheme.affine... |
/-
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 | 451 | 457 | theorem ContinuousLinearMap.iteratedFDeriv_comp_right (g : G →L[𝕜] E) {f : E → F}
(hf : ContDiff 𝕜 n f) (x : G) {i : ℕ} (hi : (i : ℕ∞) ≤ n) :
iteratedFDeriv 𝕜 i (f ∘ g) x =
(iteratedFDeriv 𝕜 i f (g x)).compContinuousLinearMap fun _ => g := by |
simp only [← iteratedFDerivWithin_univ]
exact g.iteratedFDerivWithin_comp_right hf.contDiffOn uniqueDiffOn_univ uniqueDiffOn_univ
(mem_univ _) hi
|
/-
Copyright (c) 2024 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Order.Group.Instance... | Mathlib/Algebra/AddConstMap/Basic.lean | 98 | 100 | theorem map_const [AddZeroClass G] [Add H] [AddConstMapClass F G H a b] (f : F) :
f a = f 0 + b := by |
simpa using map_add_const f 0
|
/-
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.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Eq... | Mathlib/CategoryTheory/Abelian/NonPreadditive.lean | 374 | 375 | theorem neg_sub {X Y : C} (a b : X ⟶ Y) : -a - b = -b - a := by |
conv_lhs => rw [neg_def, ← sub_zero b, sub_sub_sub, sub_zero, ← neg_def]
|
/-
Copyright (c) 2022 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Devon Tuma
-/
import Mathlib.Data.Vector.Basic
#align_import data.vector.mem from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
/-!
# Theorems about membershi... | Mathlib/Data/Vector/Mem.lean | 81 | 82 | theorem not_mem_map_zero (b : β) (v : Vector α 0) (f : α → β) : b ∉ (v.map f).toList := by |
simpa only [Vector.eq_nil v, Vector.map_nil, Vector.toList_nil] using List.not_mem_nil b
|
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Measure.Sub
import Mathlib.MeasureTheory.Decomposition.SignedHahn
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
#align_import measure_t... | Mathlib/MeasureTheory/Decomposition/Lebesgue.lean | 752 | 761 | theorem sup_mem_measurableLE {f g : α → ℝ≥0∞} (hf : f ∈ measurableLE μ ν)
(hg : g ∈ measurableLE μ ν) : (fun a ↦ f a ⊔ g a) ∈ measurableLE μ ν := by |
simp_rw [ENNReal.sup_eq_max]
refine ⟨Measurable.max hf.1 hg.1, fun A hA ↦ ?_⟩
have h₁ := hA.inter (measurableSet_le hf.1 hg.1)
have h₂ := hA.inter (measurableSet_lt hg.1 hf.1)
rw [set_lintegral_max hf.1 hg.1]
refine (add_le_add (hg.2 _ h₁) (hf.2 _ h₂)).trans_eq ?_
simp only [← not_le, ← compl_setOf, ← di... |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.GammaSpecAdjunction
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.RingTheory.Loca... | Mathlib/AlgebraicGeometry/AffineScheme.lean | 101 | 102 | theorem isAffineOfIso {X Y : Scheme} (f : X ⟶ Y) [IsIso f] [h : IsAffine Y] : IsAffine X := by |
rw [← mem_Spec_essImage] at h ⊢; exact Functor.essImage.ofIso (asIso f).symm h
|
/-
Copyright (c) 2024 Newell Jensen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Newell Jensen, Mitchell Lee
-/
import Mathlib.Algebra.Ring.Int
import Mathlib.GroupTheory.PresentedGroup
import Mathlib.GroupTheory.Coxeter.Matrix
/-!
# Coxeter groups and Coxeter syst... | Mathlib/GroupTheory/Coxeter/Basic.lean | 253 | 262 | theorem simple_induction_left {p : W → Prop} (w : W) (one : p 1)
(mul_simple_left : ∀ (w : W) (i : B), p w → p (s i * w)) : p w := by |
let p' : (w : W) → w ∈ Submonoid.closure (Set.range cs.simple) → Prop :=
fun w _ ↦ p w
have := cs.submonoid_closure_range_simple.symm ▸ Submonoid.mem_top w
apply Submonoid.closure_induction_left (p := p')
· exact one
· rintro _ ⟨i, rfl⟩ y _
exact mul_simple_left y i
· exact this
|
/-
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.Measure.MeasureSpace
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.Topology.Sets.Compacts
#align_import measure_theory... | Mathlib/MeasureTheory/Measure/Content.lean | 304 | 309 | theorem outerMeasure_preimage (f : G ≃ₜ G) (h : ∀ ⦃K : Compacts G⦄, μ (K.map f f.continuous) = μ K)
(A : Set G) : μ.outerMeasure (f ⁻¹' A) = μ.outerMeasure A := by |
refine inducedOuterMeasure_preimage _ μ.innerContent_iUnion_nat μ.innerContent_mono _
(fun _ => f.isOpen_preimage) ?_
intro s hs
convert μ.innerContent_comap f h ⟨s, hs⟩
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attr
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Logic.Equiv.Set
import Math... | Mathlib/Data/Finset/Basic.lean | 3,467 | 3,477 | theorem pairwise_cons' {a : α} (ha : a ∉ s) (r : β → β → Prop) (f : α → β) :
Pairwise (r on fun a : s.cons a ha => f a) ↔
Pairwise (r on fun a : s => f a) ∧ ∀ b ∈ s, r (f a) (f b) ∧ r (f b) (f a) := by |
simp only [pairwise_subtype_iff_pairwise_finset', Finset.coe_cons, Set.pairwise_insert,
Finset.mem_coe, and_congr_right_iff]
exact fun _ =>
⟨fun h b hb =>
h b hb <| by
rintro rfl
contradiction,
fun h b hb _ => h b hb⟩
|
/-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/... | Mathlib/Algebra/Polynomial/Mirror.lean | 75 | 79 | theorem mirror_natTrailingDegree : p.mirror.natTrailingDegree = p.natTrailingDegree := by |
by_cases hp : p = 0
· rw [hp, mirror_zero]
· rw [mirror, natTrailingDegree_mul_X_pow ((mt reverse_eq_zero.mp) hp),
natTrailingDegree_reverse, zero_add]
|
/-
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, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Submodule.EqLocus
import Mathlib.Algebra.Module.Subm... | Mathlib/LinearAlgebra/Span.lean | 519 | 522 | theorem span_singleton_smul_le {S} [Monoid S] [SMul S R] [MulAction S M] [IsScalarTower S R M]
(r : S) (x : M) : (R ∙ r • x) ≤ R ∙ x := by |
rw [span_le, Set.singleton_subset_iff, SetLike.mem_coe]
exact smul_of_tower_mem _ _ (mem_span_singleton_self _)
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.Analysis.NormedSpace.IndicatorFunction
#align_import measure_theory.integral.integrable_on... | Mathlib/MeasureTheory/Integral/IntegrableOn.lean | 239 | 242 | theorem _root_.MeasurableEmbedding.integrableOn_map_iff [MeasurableSpace β] {e : α → β}
(he : MeasurableEmbedding e) {f : β → E} {μ : Measure α} {s : Set β} :
IntegrableOn f s (μ.map e) ↔ IntegrableOn (f ∘ e) (e ⁻¹' s) μ := by |
simp_rw [IntegrableOn, he.restrict_map, he.integrable_map_iff]
|
/-
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.MFDeriv.Basic
/-!
### Relations between vector space derivative and manifold derivative
The manifold deriva... | Mathlib/Geometry/Manifold/MFDeriv/FDeriv.lean | 26 | 28 | theorem uniqueMDiffWithinAt_iff_uniqueDiffWithinAt :
UniqueMDiffWithinAt 𝓘(𝕜, E) s x ↔ UniqueDiffWithinAt 𝕜 s x := by |
simp only [UniqueMDiffWithinAt, mfld_simps]
|
/-
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.BigOperators.Ring.List
import Mathlib.Data.Nat.Prime
import Mathlib.Data.List.Prime
import Mathlib.Dat... | Mathlib/Data/Nat/Factors.lean | 285 | 290 | theorem coprime_factors_disjoint {a b : ℕ} (hab : a.Coprime b) :
List.Disjoint a.factors b.factors := by |
intro q hqa hqb
apply not_prime_one
rw [← eq_one_of_dvd_coprimes hab (dvd_of_mem_factors hqa) (dvd_of_mem_factors hqb)]
exact prime_of_mem_factors hqa
|
/-
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 | 780 | 781 | theorem toIcoMod_eq_sub (a b : α) : toIcoMod hp a b = toIcoMod hp 0 (b - a) + a := by |
rw [toIcoMod_sub_eq_sub, zero_add, sub_add_cancel]
|
/-
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.CharP.Invertible
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Analysis.Convex.Segment
import Mathlib.LinearAlgebra.AffineSpace.Fi... | Mathlib/Analysis/Convex/Between.lean | 443 | 446 | theorem wbtw_zero_one_iff {x : R} : Wbtw R 0 x 1 ↔ x ∈ Set.Icc (0 : R) 1 := by |
rw [Wbtw, affineSegment, Set.mem_image]
simp_rw [lineMap_apply_ring]
simp
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.... | Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean | 789 | 794 | theorem biproduct.fromSubtype_π_subtype (j : Subtype p) :
biproduct.fromSubtype f p ≫ biproduct.π f j = biproduct.π (Subtype.restrict p f) j := by |
ext
rw [biproduct.fromSubtype, biproduct.ι_desc_assoc, biproduct.ι_π, biproduct.ι_π]
split_ifs with h₁ h₂ h₂
exacts [rfl, False.elim (h₂ (Subtype.ext h₁)), False.elim (h₁ (congr_arg Subtype.val h₂)), rfl]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Michael Howes
-/
import Mathlib.Data.Finite.Card
import Mathlib.GroupTheory.Commutator
import Mathlib.GroupTheory.Finiteness
#align_import group_theory.abelianization from "lean... | Mathlib/GroupTheory/Abelianization.lean | 132 | 135 | theorem commutator_subset_ker : commutator G ≤ f.ker := by |
rw [commutator_eq_closure, Subgroup.closure_le]
rintro x ⟨p, q, rfl⟩
simp [MonoidHom.mem_ker, mul_right_comm (f p) (f q), commutatorElement_def]
|
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv
import Mathlib.LinearAlgebra.Dual
#align_import analysi... | Mathlib/Analysis/Calculus/LagrangeMultipliers.lean | 60 | 78 | theorem IsLocalExtrOn.exists_linear_map_of_hasStrictFDerivAt
(hextr : IsLocalExtrOn φ {x | f x = f x₀} x₀) (hf' : HasStrictFDerivAt f f' x₀)
(hφ' : HasStrictFDerivAt φ φ' x₀) :
∃ (Λ : Module.Dual ℝ F) (Λ₀ : ℝ), (Λ, Λ₀) ≠ 0 ∧ ∀ x, Λ (f' x) + Λ₀ • φ' x = 0 := by |
rcases Submodule.exists_le_ker_of_lt_top _
(lt_top_iff_ne_top.2 <| hextr.range_ne_top_of_hasStrictFDerivAt hf' hφ') with
⟨Λ', h0, hΛ'⟩
set e : ((F →ₗ[ℝ] ℝ) × ℝ) ≃ₗ[ℝ] F × ℝ →ₗ[ℝ] ℝ :=
((LinearEquiv.refl ℝ (F →ₗ[ℝ] ℝ)).prod (LinearMap.ringLmapEquivSelf ℝ ℝ ℝ).symm).trans
(LinearMap.coprodEquiv ℝ... |
/-
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 | 59 | 63 | theorem rank_le_card_width [StrongRankCondition R] (A : Matrix m n R) :
A.rank ≤ Fintype.card n := by |
haveI : Module.Finite R (n → R) := Module.Finite.pi
haveI : Module.Free R (n → R) := Module.Free.pi _ _
exact A.mulVecLin.finrank_range_le.trans_eq (finrank_pi _)
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.Fixed
import Mathlib.FieldTheory.NormalClosure
import Mathlib.FieldTheory.PrimitiveElement
import Mathlib.Gro... | Mathlib/FieldTheory/Galois.lean | 93 | 100 | theorem IntermediateField.AdjoinSimple.card_aut_eq_finrank [FiniteDimensional F E] {α : E}
(hα : IsIntegral F α) (h_sep : (minpoly F α).Separable)
(h_splits : (minpoly F α).Splits (algebraMap F F⟮α⟯)) :
Fintype.card (F⟮α⟯ ≃ₐ[F] F⟮α⟯) = finrank F F⟮α⟯ := by |
letI : Fintype (F⟮α⟯ →ₐ[F] F⟮α⟯) := IntermediateField.fintypeOfAlgHomAdjoinIntegral F hα
rw [IntermediateField.adjoin.finrank hα]
rw [← IntermediateField.card_algHom_adjoin_integral F hα h_sep h_splits]
exact Fintype.card_congr (algEquivEquivAlgHom F F⟮α⟯)
|
/-
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 | 1,098 | 1,100 | theorem exists_measurableEmbedding_real : ∃ f : α → ℝ, MeasurableEmbedding f := by |
obtain ⟨s, hs, ⟨e⟩⟩ := exists_subset_real_measurableEquiv α
exact ⟨(↑) ∘ e, (MeasurableEmbedding.subtype_coe hs).comp e.measurableEmbedding⟩
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.function.simple_func from "leanprover-community/mathlib"@"bf... | Mathlib/MeasureTheory/Function/SimpleFunc.lean | 762 | 762 | theorem restrict_empty (f : α →ₛ β) : restrict f ∅ = 0 := by | simp [restrict]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Grou... | Mathlib/RingTheory/Localization/Integral.lean | 55 | 58 | theorem coeffIntegerNormalization_of_not_mem_support (p : S[X]) (i : ℕ) (h : coeff p i = 0) :
coeffIntegerNormalization M p i = 0 := by |
simp only [coeffIntegerNormalization, h, mem_support_iff, eq_self_iff_true, not_true, Ne,
dif_neg, not_false_iff]
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc429200506... | Mathlib/Data/Set/Image.lean | 1,037 | 1,038 | theorem image_compl_preimage {f : α → β} {s : Set β} : f '' (f ⁻¹' s)ᶜ = range f \ s := by |
rw [compl_eq_univ_diff, image_diff_preimage, image_univ]
|
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yury Kudryashov, Sébastien Gouëzel, Chris Hughes
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Order.Fin
import Mathlib... | Mathlib/Data/Fin/Tuple/Basic.lean | 493 | 496 | theorem init_snoc : init (snoc p x) = p := by |
ext i
simp only [init, snoc, coe_castSucc, is_lt, cast_eq, dite_true]
convert cast_eq rfl (p i)
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Polynomial.IntegralNormalization
#align_import rin... | Mathlib/RingTheory/Algebraic.lean | 83 | 85 | theorem isAlgebraic_iff_not_injective {x : A} :
IsAlgebraic R x ↔ ¬Function.Injective (Polynomial.aeval x : R[X] →ₐ[R] A) := by |
simp only [IsAlgebraic, injective_iff_map_eq_zero, not_forall, and_comm, exists_prop]
|
/-
Copyright (c) 2019 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.Archimedean
import Mathlib.Order.Filter.AtTopBot
import Mathlib.Tactic.GCongr
#align_import order.filter.archimedean fr... | Mathlib/Order/Filter/Archimedean.lean | 235 | 237 | theorem Tendsto.atTop_mul_const_of_neg' (hr : r < 0) (hf : Tendsto f l atTop) :
Tendsto (fun x => f x * r) l atBot := by |
simpa only [tendsto_neg_atTop_iff, mul_neg] using hf.atTop_mul_const' (neg_pos.mpr hr)
|
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl
-/
import Mathlib.Topology.UniformSpace.AbstractCompletion
#align_import topology.uniform_space.completion from "leanprover-community/mathlib"@"dc6c365e... | Mathlib/Topology/UniformSpace/Completion.lean | 265 | 268 | theorem extend_pureCauchy {f : α → β} (hf : UniformContinuous f) (a : α) :
extend f (pureCauchy a) = f a := by |
rw [extend, if_pos hf]
exact uniformly_extend_of_ind uniformInducing_pureCauchy denseRange_pureCauchy hf _
|
/-
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.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.I... | Mathlib/Algebra/Ring/Defs.lean | 223 | 226 | theorem ite_add_ite {α} [Add α] (P : Prop) [Decidable P] (a b c d : α) :
((if P then a else b) + if P then c else d) = if P then a + c else b + d := by |
split
repeat rfl
|
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Analysis.NormedSpace.Star.GelfandDuality
import Mathlib.Topology.Algebra.StarSubalgebra
#align_import analysis.normed_space.star.continuous_functional_c... | Mathlib/Analysis/NormedSpace/Star/ContinuousFunctionalCalculus.lean | 243 | 255 | theorem elementalStarAlgebra.bijective_characterSpaceToSpectrum :
Function.Bijective (elementalStarAlgebra.characterSpaceToSpectrum a) := by |
refine ⟨fun φ ψ h => starAlgHomClass_ext ℂ ?_ ?_ ?_, ?_⟩
· exact (map_continuous φ)
· exact (map_continuous ψ)
· simpa only [elementalStarAlgebra.characterSpaceToSpectrum, Subtype.mk_eq_mk,
ContinuousMap.coe_mk] using h
· rintro ⟨z, hz⟩
have hz' := (StarSubalgebra.spectrum_eq (elementalStarAlgebra.... |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.RingTheory.WittVector.InitTail
#align_import ring_theory.witt_vector.truncated from "leanprover-community/mathlib"@"acbe099ced8be9c97... | Mathlib/RingTheory/WittVector/Truncated.lean | 393 | 397 | theorem truncate_truncate {n₁ n₂ n₃ : ℕ} (h1 : n₁ ≤ n₂) (h2 : n₂ ≤ n₃)
(x : TruncatedWittVector p n₃ R) :
(truncate h1) (truncate h2 x) = truncate (h1.trans h2) x := by |
obtain ⟨x, rfl⟩ := @WittVector.truncate_surjective p _ n₃ R _ x
simp only [truncate_wittVector_truncate]
|
/-
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 | 1,535 | 1,541 | theorem eventually_constant_prod {u : ℕ → β} {N : ℕ} (hu : ∀ n ≥ N, u n = 1) {n : ℕ} (hn : N ≤ n) :
(∏ k ∈ range n, u k) = ∏ k ∈ range N, u k := by |
obtain ⟨m, rfl : n = N + m⟩ := Nat.exists_eq_add_of_le hn
clear hn
induction' m with m hm
· simp
· simp [← add_assoc, prod_range_succ, hm, hu]
|
/-
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.SmoothManifoldWithCorners
import Mathlib.Geometry.Manifold.LocalInvariantProperties
#align_import geometry.m... | Mathlib/Geometry/Manifold/ContMDiff/Defs.lean | 450 | 463 | theorem contMDiffWithinAt_iff_target_of_mem_source {x : M} {y : M'}
(hy : f x ∈ (chartAt H' y).source) :
ContMDiffWithinAt I I' n f s x ↔
ContinuousWithinAt f s x ∧ ContMDiffWithinAt I 𝓘(𝕜, E') n (extChartAt I' y ∘ f) s x := by |
simp_rw [ContMDiffWithinAt]
rw [(contDiffWithinAt_localInvariantProp I I' n).liftPropWithinAt_indep_chart_target
(chart_mem_maximalAtlas I' y) hy,
and_congr_right]
intro hf
simp_rw [StructureGroupoid.liftPropWithinAt_self_target]
simp_rw [((chartAt H' y).continuousAt hy).comp_continuousWithinAt hf]... |
/-
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.Init.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive... | Mathlib/Data/Option/Basic.lean | 53 | 55 | theorem mem_map_of_injective {f : α → β} (H : Function.Injective f) {a : α} {o : Option α} :
f a ∈ o.map f ↔ a ∈ o := by |
aesop
|
/-
Copyright (c) 2019 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.Analysis.SpecificLimits.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.Dynamics.Fixed... | Mathlib/Topology/MetricSpace/Contracting.lean | 143 | 146 | theorem edist_efixedPoint_le (hf : ContractingWith K f) {x : α} (hx : edist x (f x) ≠ ∞) :
edist x (efixedPoint f hf x hx) ≤ edist x (f x) / (1 - K) := by |
convert hf.apriori_edist_iterate_efixedPoint_le hx 0
simp only [pow_zero, mul_one]
|
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.Regular.Basic
import Mathlib.GroupTheory.Congruence.... | Mathlib/GroupTheory/MonoidLocalization.lean | 1,012 | 1,013 | theorem lift_mk'_spec (x v) (y : S) : f.lift hg (f.mk' x y) = v ↔ g x = g y * v := by |
rw [f.lift_mk' hg]; exact mul_inv_left hg _ _ _
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community... | Mathlib/LinearAlgebra/AffineSpace/Slope.lean | 129 | 135 | theorem lineMap_slope_lineMap_slope_lineMap (f : k → PE) (a b r : k) :
lineMap (slope f (lineMap a b r) b) (slope f a (lineMap a b r)) r = slope f a b := by |
obtain rfl | hab : a = b ∨ a ≠ b := Classical.em _; · simp
rw [slope_comm _ a, slope_comm _ a, slope_comm _ _ b]
convert lineMap_slope_slope_sub_div_sub f b (lineMap a b r) a hab.symm using 2
rw [lineMap_apply_ring, eq_div_iff (sub_ne_zero.2 hab), sub_mul, one_mul, mul_sub, ← sub_sub,
sub_sub_cancel]
|
/-
Copyright (c) 2020 Johan Commelin, Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.ZMod
#align_import number_theory.padics.ring_homs from "... | Mathlib/NumberTheory/Padics/RingHoms.lean | 547 | 560 | theorem nthHomSeq_add (r s : R) :
nthHomSeq f_compat (r + s) ≈ nthHomSeq f_compat r + nthHomSeq f_compat s := by |
intro ε hε
obtain ⟨n, hn⟩ := exists_pow_neg_lt_rat p hε
use n
intro j hj
dsimp [nthHomSeq]
apply lt_of_le_of_lt _ hn
rw [← Int.cast_add, ← Int.cast_sub, ← padicNorm.dvd_iff_norm_le, ←
ZMod.intCast_zmod_eq_zero_iff_dvd]
dsimp [nthHom]
simp only [ZMod.natCast_val, RingHom.map_add, Int.cast_sub, ZMo... |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#ali... | Mathlib/Algebra/Order/Group/Defs.lean | 937 | 939 | theorem div_lt_div_iff_left (a : α) : a / b < a / c ↔ c < b := by |
rw [div_eq_mul_inv, div_eq_mul_inv, ← mul_lt_mul_iff_left a⁻¹, inv_mul_cancel_left,
inv_mul_cancel_left, inv_lt_inv_iff]
|
/-
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 | 2,031 | 2,032 | theorem prod_pow_boole [DecidableEq α] (s : Finset α) (f : α → β) (a : α) :
(∏ x ∈ s, f x ^ ite (a = x) 1 0) = ite (a ∈ s) (f a) 1 := by | simp
|
/-
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 | 327 | 340 | theorem sizeUpTo_index_le (j : Fin n) : c.sizeUpTo (c.index j) ≤ j := by |
by_contra H
set i := c.index j
push_neg at H
have i_pos : (0 : ℕ) < i := by
by_contra! i_pos
revert H
simp [nonpos_iff_eq_zero.1 i_pos, c.sizeUpTo_zero]
let i₁ := (i : ℕ).pred
have i₁_lt_i : i₁ < i := Nat.pred_lt (ne_of_gt i_pos)
have i₁_succ : i₁ + 1 = i := Nat.succ_pred_eq_of_pos i_pos
ha... |
/-
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 | 1,061 | 1,069 | theorem colimit.post_post {E : Type u''} [Category.{v''} E] (H : D ⥤ E)
-- H G (colimit F) ⟶ H (colimit (F ⋙ G)) ⟶ colimit ((F ⋙ G) ⋙ H) equals
-- H G (colimit F) ⟶ colimit (F ⋙ (G ⋙ H))
[h : HasColimit ((F ⋙ G) ⋙ H)] : haveI : HasColimit (F ⋙ G ⋙ H) := h
colimit.post (F ⋙ G) H ≫ H.map (colimit.post F G... |
ext j
rw [← assoc, colimit.ι_post, ← H.map_comp, colimit.ι_post]
haveI : HasColimit (F ⋙ G ⋙ H) := h
exact (colimit.ι_post F (G ⋙ H) j).symm
|
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.MonoidAlgebra.Basic
#align_import algebra.monoid_algebra.division from "leanprover-community/mathlib"@"72c366d0475675f1309d3027d3d7d47ee4423951"
/-... | Mathlib/Algebra/MonoidAlgebra/Division.lean | 86 | 88 | theorem divOf_add (x : k[G]) (a b : G) : x /ᵒᶠ (a + b) = x /ᵒᶠ a /ᵒᶠ b := by |
refine Finsupp.ext fun _ => ?_ -- Porting note: `ext` doesn't work
simp only [AddMonoidAlgebra.divOf_apply, add_assoc]
|
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä, Yury Kudryashov
-/
import Mathlib.Analysis.NormedSpace.Dual
import Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness
#align_import analysis.normed_space.weak_dual fro... | Mathlib/Analysis/NormedSpace/WeakDual.lean | 141 | 145 | theorem dual_norm_topology_le_weak_dual_topology :
(UniformSpace.toTopologicalSpace : TopologicalSpace (Dual 𝕜 E)) ≤
(WeakDual.instTopologicalSpace : TopologicalSpace (WeakDual 𝕜 E)) := by |
convert (@toWeakDual_continuous _ _ _ _ (by assumption)).le_induced
exact induced_id.symm
|
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Sébastien Gouëzel, Zhouhang Zhou, Reid Barton
-/
import Mathlib.Logic.Equiv.Fin
import Mathlib.Topology.DenseEmbedding
import Mathlib.Topology.Support
impo... | Mathlib/Topology/Homeomorph.lean | 472 | 475 | theorem locallyCompactSpace_iff (h : X ≃ₜ Y) :
LocallyCompactSpace X ↔ LocallyCompactSpace Y := by |
exact ⟨fun _ => h.symm.openEmbedding.locallyCompactSpace,
fun _ => h.closedEmbedding.locallyCompactSpace⟩
|
/-
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.Topology.UniformSpace.UniformConvergenceTopology
#align_import topology.uniform_space.equicontinuity from "leanprover-community/mathlib"@"f2ce6086... | Mathlib/Topology/UniformSpace/Equicontinuity.lean | 656 | 661 | theorem Filter.HasBasis.equicontinuousWithinAt_iff_right {p : κ → Prop}
{s : κ → Set (α × α)} {F : ι → X → α} {S : Set X} {x₀ : X} (hα : (𝓤 α).HasBasis p s) :
EquicontinuousWithinAt F S x₀ ↔ ∀ k, p k → ∀ᶠ x in 𝓝[S] x₀, ∀ i, (F i x₀, F i x) ∈ s k := by |
rw [equicontinuousWithinAt_iff_continuousWithinAt, ContinuousWithinAt,
(UniformFun.hasBasis_nhds_of_basis ι α _ hα).tendsto_right_iff]
rfl
|
/-
Copyright (c) 2023 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.GroupTheory.CoprodI
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.GroupTheory.Complement
/-!
## Pushou... | Mathlib/GroupTheory/PushoutI.lean | 570 | 573 | theorem prod_injective : Function.Injective (prod : NormalWord d → PushoutI φ) := by |
letI := Classical.decEq ι
letI := fun i => Classical.decEq (G i)
classical exact equiv.symm.injective
|
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Limits.HasLimits
import Mathlib.CategoryTheory.Thin
#align_import category_theory.limits.shapes.wide_pullbacks from "lean... | Mathlib/CategoryTheory/Limits/Shapes/WidePullbacks.lean | 358 | 359 | theorem lift_π (j : J) : lift f fs w ≫ π arrows j = fs _ := by |
simp only [limit.lift_π, WidePullbackShape.mkCone_pt, WidePullbackShape.mkCone_π_app]
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Data.List.Lattice
import Mathlib.Data.List.Range
import Mathlib.Data.Bool.Basic
#align_import data.list.intervals from "leanprover-community/mathlib"@... | Mathlib/Data/List/Intervals.lean | 136 | 139 | theorem pred_singleton {m : ℕ} (h : 0 < m) : Ico (m - 1) m = [m - 1] := by |
dsimp [Ico]
rw [Nat.sub_sub_self (succ_le_of_lt h)]
simp [← Nat.one_eq_succ_zero]
|
/-
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.Subalgebra
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Artinian
#align_import algebra.lie.submodule from "leanprover-communit... | Mathlib/Algebra/Lie/Submodule.lean | 565 | 567 | theorem iSup_eq_top_iff_coe_toSubmodule {ι : Sort*} {N : ι → LieSubmodule R L M} :
⨆ i, N i = ⊤ ↔ ⨆ i, (N i : Submodule R M) = ⊤ := by |
rw [← iSup_coe_toSubmodule, ← top_coeSubmodule (L := L), coe_toSubmodule_eq_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.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
#align_import analysis.calculus.cont_diff_def from "lean... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 1,595 | 1,601 | theorem fderiv_iteratedFDeriv {n : ℕ} :
fderiv 𝕜 (iteratedFDeriv 𝕜 n f) =
(continuousMultilinearCurryLeftEquiv 𝕜 (fun _ : Fin (n + 1) => E) F).symm ∘
iteratedFDeriv 𝕜 (n + 1) f := by |
rw [iteratedFDeriv_succ_eq_comp_left]
ext1 x
simp only [Function.comp_apply, LinearIsometryEquiv.symm_apply_apply]
|
/-
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 | 1,041 | 1,056 | theorem fixed_point {f : Code → Code} (hf : Computable f) : ∃ c : Code, eval (f c) = eval c :=
let g (x y : ℕ) : Part ℕ := eval (ofNat Code x) x >>= fun b => eval (ofNat Code b) y
have : Partrec₂ g :=
(eval_part.comp ((Computable.ofNat _).comp fst) fst).bind
(eval_part.comp ((Computable.ofNat _).comp snd)... | simp [eg]
let F (x : ℕ) : Code := f (curry cg x)
have : Computable F :=
hf.comp (curry_prim.comp (_root_.Primrec.const cg) _root_.Primrec.id).to_comp
let ⟨cF, eF⟩ := exists_code.1 this
have eF' : eval cF (encode cF) = Part.some (encode (F (encode cF))) := by simp [eF]
⟨curry cg (encode cF),
funext fu... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic... | Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 195 | 205 | theorem tendsto_log_nhdsWithin_im_neg_of_re_neg_of_im_zero {z : ℂ} (hre : z.re < 0)
(him : z.im = 0) : Tendsto log (𝓝[{ z : ℂ | z.im < 0 }] z) (𝓝 <| Real.log (abs z) - π * I) := by |
convert
(continuous_ofReal.continuousAt.comp_continuousWithinAt
(continuous_abs.continuousWithinAt.log _)).tendsto.add
(((continuous_ofReal.tendsto _).comp <|
tendsto_arg_nhdsWithin_im_neg_of_re_neg_of_im_zero hre him).mul
tendsto_const_nhds) using 1
· simp [sub_eq_add_neg... |
/-
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 | 327 | 344 | theorem natDegree_divByMonic (f : R[X]) {g : R[X]} (hg : g.Monic) :
natDegree (f /ₘ g) = natDegree f - natDegree g := by |
nontriviality R
by_cases hfg : f /ₘ g = 0
· rw [hfg, natDegree_zero]
rw [divByMonic_eq_zero_iff hg] at hfg
rw [tsub_eq_zero_iff_le.mpr (natDegree_le_natDegree <| le_of_lt hfg)]
have hgf := hfg
rw [divByMonic_eq_zero_iff hg] at hgf
push_neg at hgf
have := degree_add_divByMonic hg hgf
have hf : f... |
/-
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.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from ... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 232 | 234 | theorem toList_ne_singleton (y : α) : toList p x ≠ [y] := by |
intro H
simpa [card_support_ne_one] using congr_arg length H
|
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Variance
#align_import probability.moments from "leanprover-community/mathlib"@"85453a2a14be8da64caf15ca50930cf4c6e5d8de"
/-!
# Moments and m... | Mathlib/Probability/Moments.lean | 126 | 127 | theorem cgf_zero_measure : cgf X (0 : Measure Ω) t = 0 := by |
simp only [cgf, log_zero, mgf_zero_measure]
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.Algebra.Category.Ring.Colimits
import Mathlib.CategoryTheory.Iso
import Mathlib.RingTheory.Localization.Aw... | Mathlib/RingTheory/RingHomProperties.lean | 168 | 180 | theorem StableUnderBaseChange.pushout_inl (hP : RingHom.StableUnderBaseChange @P)
(hP' : RingHom.RespectsIso @P) {R S T : CommRingCat} (f : R ⟶ S) (g : R ⟶ T) (H : P g) :
P (pushout.inl : S ⟶ pushout f g) := by |
rw [←
show _ = pushout.inl from
colimit.isoColimitCocone_ι_inv ⟨_, CommRingCat.pushoutCoconeIsColimit f g⟩
WalkingSpan.left,
hP'.cancel_right_isIso]
letI := f.toAlgebra
letI := g.toAlgebra
dsimp only [CommRingCat.pushoutCocone_inl, PushoutCocone.ι_app_left]
apply hP R T S (TensorProduct... |
/-
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.Algebra.GradedMonoid
import Mathlib.Algebra.DirectSum.Basic
#align_import algebra.direct_sum.ring from "leanprover-community/mathlib"@"70fd9563a21e7b963887c... | Mathlib/Algebra/DirectSum/Ring.lean | 307 | 323 | theorem mul_eq_dfinsupp_sum [∀ (i : ι) (x : A i), Decidable (x ≠ 0)] (a a' : ⨁ i, A i) :
a * a'
= a.sum fun i ai => a'.sum fun j aj => DirectSum.of _ _ <| GradedMonoid.GMul.mul ai aj := by |
change mulHom _ a a' = _
-- Porting note: I have no idea how the proof from ml3 worked it used to be
-- simpa only [mul_hom, to_add_monoid, dfinsupp.lift_add_hom_apply, dfinsupp.sum_add_hom_apply,
-- add_monoid_hom.dfinsupp_sum_apply, flip_apply, add_monoid_hom.dfinsupp_sum_add_hom_apply],
rw [mulHom, toAddM... |
/-
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 | 504 | 505 | theorem rightUnitor_inv_naturality {X X' : C} (f : X ⟶ X') :
f ≫ (ρ_ X').inv = (ρ_ X).inv ≫ f ▷ _ := 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, 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 | 453 | 454 | theorem natCast_mul (a₂) (_ : ((a₁ : ℕ) : R) = b₁) (_ : ((a₃ : ℕ) : R) = b₃) :
((a₁ ^ a₂ * a₃ : ℕ) : R) = b₁ ^ a₂ * b₃ := by | subst_vars; simp
|
/-
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 | 294 | 305 | theorem mem_of_span_eq_top_of_smul_pow_mem (M' : Submodule R M) (s : Set R) (hs : Ideal.span s = ⊤)
(x : M) (H : ∀ r : s, ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M') : x ∈ M' := by |
obtain ⟨s', hs₁, hs₂⟩ := (Ideal.span_eq_top_iff_finite _).mp hs
replace H : ∀ r : s', ∃ n : ℕ, ((r : R) ^ n : R) • x ∈ M' := fun r => H ⟨_, hs₁ r.2⟩
choose n₁ n₂ using H
let N := s'.attach.sup n₁
have hs' := Ideal.span_pow_eq_top (s' : Set R) hs₂ N
apply M'.mem_of_span_top_of_smul_mem _ hs'
rintro ⟨_, r,... |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Sébastien Gouëzel, Yury G. Kudryashov, Dylan MacKenzie, Patrick Massot
-/
import Mathlib.Algebra.BigOperators.Module
import Mathlib.Algebra.Order.Field.Basic
impor... | Mathlib/Analysis/SpecificLimits/Normed.lean | 711 | 714 | theorem Monotone.cauchySeq_alternating_series_of_tendsto_zero (hfa : Monotone f)
(hf0 : Tendsto f atTop (𝓝 0)) : CauchySeq fun n ↦ ∑ i ∈ range n, (-1) ^ i * f i := by |
simp_rw [mul_comm]
exact hfa.cauchySeq_series_mul_of_tendsto_zero_of_bounded hf0 norm_sum_neg_one_pow_le
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Set.Finite
#align_import data.finset.n_ary from "leanprover-community/mathlib"@"eba7871095e834365616b5e43c8c7bb0b3705... | Mathlib/Data/Finset/NAry.lean | 239 | 240 | theorem card_image₂_singleton_right (hf : Injective fun a => f a b) :
(image₂ f s {b}).card = s.card := by | rw [image₂_singleton_right, card_image_of_injective _ hf]
|
/-
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 | 1,753 | 1,756 | theorem smul_mulVec_assoc [Fintype n] [Monoid R] [DistribMulAction R α] [IsScalarTower R α α]
(a : R) (A : Matrix m n α) (b : n → α) : (a • A) *ᵥ b = a • A *ᵥ b := by |
ext
apply smul_dotProduct
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Analysis.NormedSpa... | Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 151 | 152 | theorem log_lt_log (hx : 0 < x) (h : x < y) : log x < log y := by |
rwa [← exp_lt_exp, exp_log hx, exp_log (lt_trans hx h)]
|
/-
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.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.Order.BigOperators.Ring.... | Mathlib/Algebra/MvPolynomial/Degrees.lean | 295 | 300 | theorem degreeOf_mul_le (i : σ) (f g : MvPolynomial σ R) :
degreeOf i (f * g) ≤ degreeOf i f + degreeOf i g := by |
classical
repeat' rw [degreeOf]
convert Multiset.count_le_of_le i (degrees_mul f g)
rw [Multiset.count_add]
|
/-
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 Batteries.Tactic.Alias
import Batteries.Data.List.Init.Attach
import Batteries.Data.List.Pairwise
-- Adaptation note: ... | .lake/packages/batteries/Batteries/Data/List/Perm.lean | 326 | 355 | theorem perm_inv_core {a : α} {l₁ l₂ r₁ r₂ : List α} :
l₁ ++ a :: r₁ ~ l₂ ++ a :: r₂ → l₁ ++ r₁ ~ l₂ ++ r₂ := by |
-- Necessary generalization for `induction`
suffices ∀ s₁ s₂ (_ : s₁ ~ s₂) {l₁ l₂ r₁ r₂},
l₁ ++ a :: r₁ = s₁ → l₂ ++ a :: r₂ = s₂ → l₁ ++ r₁ ~ l₂ ++ r₂ from (this _ _ · rfl rfl)
intro s₁ s₂ p
induction p using Perm.recOnSwap' with intro l₁ l₂ r₁ r₂ e₁ e₂
| nil =>
simp at e₁
| cons x p IH =>
c... |
/-
Copyright (c) 2022 Floris van Doorn, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Heather Macbeth
-/
import Mathlib.Geometry.Manifold.ContMDiff.Atlas
import Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear
import Mathlib.To... | Mathlib/Geometry/Manifold/VectorBundle/Basic.lean | 117 | 121 | theorem FiberBundle.chartedSpace_chartAt_symm_fst (x : TotalSpace F E) (y : ModelProd HB F)
(hy : y ∈ (chartAt (ModelProd HB F) x).target) :
((chartAt (ModelProd HB F) x).symm y).proj = (chartAt HB x.proj).symm y.1 := by |
simp only [FiberBundle.chartedSpace_chartAt, mfld_simps] at hy ⊢
exact (trivializationAt F E x.proj).proj_symm_apply hy.2
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Geometry.Euclidean.Sphere.Basic
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tactic.DeriveFintype
#align_import geometry.eucl... | Mathlib/Geometry/Euclidean/Circumcenter.lean | 580 | 584 | theorem sum_pointWeightsWithCircumcenter {n : ℕ} (i : Fin (n + 1)) :
∑ j, pointWeightsWithCircumcenter i j = 1 := by |
convert sum_ite_eq' univ (pointIndex i) (Function.const _ (1 : ℝ)) with j
· cases j <;> simp [pointWeightsWithCircumcenter]
· simp
|
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson, Filippo A. E. Nuccio, Riccardo Brasca
-/
import Mathlib.CategoryTheory.EffectiveEpi.Preserves
import Mathlib.CategoryTheory.Limits.Final.ParallelPair
import Mathlib... | Mathlib/CategoryTheory/Sites/Coherent/RegularSheaves.lean | 102 | 120 | theorem EqualizerCondition.mk (P : Cᵒᵖ ⥤ Type*)
(hP : ∀ (X B : C) (π : X ⟶ B) [EffectiveEpi π] [HasPullback π π], Function.Bijective
(MapToEqualizer P π (pullback.fst (f := π) (g := π)) (pullback.snd (f := π) (g := π))
pullback.condition)) : EqualizerCondition P := by |
intro X B π _ c hc
have : HasPullback π π := ⟨c, hc⟩
specialize hP X B π
rw [Types.type_equalizer_iff_unique]
rw [Function.bijective_iff_existsUnique] at hP
intro b hb
have h₁ : ((pullbackIsPullback π π).conePointUniqueUpToIso hc).hom ≫ c.fst =
pullback.fst (f := π) (g := π) := by simp
have hb' : P... |
/-
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 | 693 | 697 | theorem fderivWithin_inter (ht : t ∈ 𝓝 x) : fderivWithin 𝕜 f (s ∩ t) x = fderivWithin 𝕜 f s x := by |
have A : 𝓝[(s ∩ t) \ {x}] x = 𝓝[s \ {x}] x := by
have : (s ∩ t) \ {x} = (s \ {x}) ∩ t := by rw [inter_comm, inter_diff_assoc, inter_comm]
rw [this, ← nhdsWithin_restrict' _ ht]
simp [fderivWithin, A, hasFDerivWithinAt_inter ht]
|
/-
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.Data.ENNReal.Real
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Topol... | Mathlib/Topology/EMetricSpace/Basic.lean | 1,015 | 1,019 | theorem diam_pi_le_of_le {π : β → Type*} [Fintype β] [∀ b, PseudoEMetricSpace (π b)]
{s : ∀ b : β, Set (π b)} {c : ℝ≥0∞} (h : ∀ b, diam (s b) ≤ c) : diam (Set.pi univ s) ≤ c := by |
refine diam_le fun x hx y hy => edist_pi_le_iff.mpr ?_
rw [mem_univ_pi] at hx hy
exact fun b => diam_le_iff.1 (h b) (x b) (hx b) (y b) (hy b)
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.BigOperators.Group.Multiset
import Mathlib.Tactic.NormNum.Basic
import Mathlib.Tactic.Po... | Mathlib/Algebra/Order/BigOperators/Group/Finset.lean | 326 | 330 | theorem sum_card_le [Fintype α] (h : ∀ a, (B.filter (a ∈ ·)).card ≤ n) :
∑ s ∈ B, s.card ≤ Fintype.card α * n :=
calc
∑ s ∈ B, s.card = ∑ s ∈ B, (univ ∩ s).card := by | simp_rw [univ_inter]
_ ≤ Fintype.card α * n := sum_card_inter_le fun a _ ↦ h a
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Na... | Mathlib/Data/Complex/Exponential.lean | 1,738 | 1,740 | theorem abs_exp_ofReal (x : ℝ) : abs (exp x) = Real.exp x := by |
rw [← ofReal_exp]
exact abs_of_nonneg (le_of_lt (Real.exp_pos _))
|
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Data.Set.Equitable
import Mathlib.Logic.Equiv.Fin
import Mathlib.Order.Partition.Finpartition
#align_import order.partition.eq... | Mathlib/Order/Partition/Equipartition.lean | 80 | 85 | theorem IsEquipartition.filter_ne_average_add_one_eq_average (hP : P.IsEquipartition) :
P.parts.filter (fun p ↦ ¬p.card = s.card / P.parts.card + 1) =
P.parts.filter (fun p ↦ p.card = s.card / P.parts.card) := by |
ext p
simp only [mem_filter, and_congr_right_iff]
exact fun hp ↦ (hP.card_part_eq_average_iff hp).symm
|
/-
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.Algebra.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.... | Mathlib/LinearAlgebra/Multilinear/Basic.lean | 367 | 369 | theorem snoc_smul (f : MultilinearMap R M M₂) (m : ∀ i : Fin n, M (castSucc i)) (c : R)
(x : M (last n)) : f (snoc m (c • x)) = c • f (snoc m x) := by |
simp_rw [← update_snoc_last x m (c • x), f.map_smul, update_snoc_last]
|
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Data.Countable.Basic
import Mathlib.Logic.Encodable.Basic
import Mathlib.Order.SuccPred.Basic
import Mathlib.Order.Interval.Finset.Defs
#align_import orde... | Mathlib/Order/SuccPred/LinearLocallyFinite.lean | 87 | 99 | theorem isMax_of_succFn_le [LocallyFiniteOrder ι] (i : ι) (hi : succFn i ≤ i) : IsMax i := by |
refine fun j _ ↦ not_lt.mp fun hij_lt ↦ ?_
have h_succFn_eq : succFn i = i := le_antisymm hi (le_succFn i)
have h_glb : IsGLB (Finset.Ioc i j : Set ι) i := by
rw [Finset.coe_Ioc]
have h := succFn_spec i
rw [h_succFn_eq] at h
exact isGLB_Ioc_of_isGLB_Ioi hij_lt h
have hi_mem : i ∈ Finset.Ioc i j... |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Johan Commelin, Patrick Massot
-/
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Ta... | Mathlib/RingTheory/Valuation/Basic.lean | 678 | 681 | theorem map_add' : ∀ (x y : R), v x ≤ v (x + y) ∨ v y ≤ v (x + y) := by |
intro x y
rw [← @min_le_iff _ _ (v x) (v y) (v (x+y)), ← ge_iff_le]
apply map_add
|
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Yury Kudryashov
-/
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Analysis.Normed.MulAction
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mat... | Mathlib/Analysis/Asymptotics/Asymptotics.lean | 2,215 | 2,218 | theorem isLittleO_pi {ι : Type*} [Fintype ι] {E' : ι → Type*} [∀ i, NormedAddCommGroup (E' i)]
{f : α → ∀ i, E' i} : f =o[l] g' ↔ ∀ i, (fun x => f x i) =o[l] g' := by |
simp (config := { contextual := true }) only [IsLittleO_def, isBigOWith_pi, le_of_lt]
exact ⟨fun h i c hc => h hc i, fun h c hc i => h i hc⟩
|
/-
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 | 564 | 578 | theorem le_radius_cauchyPowerSeries (f : ℂ → E) (c : ℂ) (R : ℝ≥0) :
↑R ≤ (cauchyPowerSeries f c R).radius := by |
refine
(cauchyPowerSeries f c R).le_radius_of_bound
((2 * π)⁻¹ * ∫ θ : ℝ in (0)..2 * π, ‖f (circleMap c R θ)‖) fun n => ?_
refine (mul_le_mul_of_nonneg_right (norm_cauchyPowerSeries_le _ _ _ _)
(pow_nonneg R.coe_nonneg _)).trans ?_
rw [_root_.abs_of_nonneg R.coe_nonneg]
rcases eq_or_ne (R ^ n : ℝ... |
/-
Copyright (c) 2022 Kyle Miller, Adam Topaz, Rémi Bottinelli, Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Adam Topaz, Rémi Bottinelli, Junyan Xu
-/
import Mathlib.CategoryTheory.Filtered.Basic
import Mathlib.Data.Set.Finite
import Mathlib.D... | Mathlib/CategoryTheory/CofilteredSystem.lean | 259 | 271 | theorem isMittagLeffler_of_exists_finite_range
(h : ∀ j : J, ∃ (i : _) (f : i ⟶ j), (range <| F.map f).Finite) : F.IsMittagLeffler := by |
intro j
obtain ⟨i, hi, hf⟩ := h j
obtain ⟨m, ⟨i, f, hm⟩, hmin⟩ := Finset.wellFoundedLT.wf.has_min
{ s : Finset (F.obj j) | ∃ (i : _) (f : i ⟶ j), ↑s = range (F.map f) }
⟨_, i, hi, hf.coe_toFinset⟩
refine ⟨i, f, fun k g =>
(directedOn_range.mp <| F.ranges_directed j).is_bot_of_is_min ⟨⟨i, f⟩, rfl⟩ ?... |
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Sébastien Gouëzel, Yury Kudryashov, Yuyang Zhao
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calcu... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 262 | 265 | theorem HasDerivAt.comp_of_eq
(hh₂ : HasDerivAt h₂ h₂' y) (hh : HasDerivAt h h' x) (hy : y = h x) :
HasDerivAt (h₂ ∘ h) (h₂' * h') x := by |
rw [hy] at hh₂; exact hh₂.comp x hh
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Basic
import Mathlib.RingT... | Mathlib/RingTheory/Localization/LocalizationLocalization.lean | 140 | 144 | theorem isLocalization_isLocalization_atPrime_isLocalization (p : Ideal S) [Hp : p.IsPrime]
[IsLocalization.AtPrime T p] : IsLocalization.AtPrime T (p.comap (algebraMap R S)) := by |
apply localization_localization_isLocalization_of_has_all_units M p.primeCompl T
intro x hx hx'
exact (Hp.1 : ¬_) (p.eq_top_of_isUnit_mem hx' hx)
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.MvPolynomial.Counit
import Mathlib.Algebra.MvPolynomial.Invertible
import Mathlib.RingTheory.WittVector.Defs
#align_import ri... | Mathlib/RingTheory/WittVector/Basic.lean | 108 | 108 | theorem add : mapFun f (x + y) = mapFun f x + mapFun f y := by | map_fun_tac
|
/-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Init.Order.Defs
#align_import init.algebra.functions from "leanprover-community/lean"@"c2bcdbcbe741ed37c361a30d38e179182b989f... | Mathlib/Init/Order/LinearOrder.lean | 133 | 134 | theorem max_eq_left {a b : α} (h : b ≤ a) : max a b = a := by |
apply Eq.symm; apply eq_max (le_refl _) h; intros; assumption
|
/-
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, Heather Macbeth
-/
import Mathlib.Geometry.Manifold.VectorBundle.Basic
import Mathlib.Analysis.Convex.Normed
#align_import geometry.manifold.vector_bundle.tangent ... | Mathlib/Geometry/Manifold/VectorBundle/Tangent.lean | 374 | 385 | theorem tangentBundle_model_space_chartAt (p : TangentBundle I H) :
(chartAt (ModelProd H E) p).toPartialEquiv = (TotalSpace.toProd H E).toPartialEquiv := by |
ext x : 1
· ext; · rfl
exact (tangentBundleCore I H).coordChange_self (achart _ x.1) x.1 (mem_achart_source H x.1) x.2
· ext; · rfl
apply heq_of_eq
exact (tangentBundleCore I H).coordChange_self (achart _ x.1) x.1 (mem_achart_source H x.1) x.2
simp_rw [TangentBundle.chartAt, FiberBundleCore.localTr... |
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