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) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Analysis.Convex.Hull
#align_import analysis.convex.join from "leanprover-community/mathlib"@"951bf1d9e98a2042979ced62c0620bcfb3587cf8"
/-!
# Convex join
... | Mathlib/Analysis/Convex/Join.lean | 70 | 71 | theorem convexJoin_singleton_right (s : Set E) (y : E) :
convexJoin 𝕜 s {y} = ⋃ x ∈ s, segment 𝕜 x y := by | simp [convexJoin]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Group.List
import Mathlib.Data.Vector.Defs
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.I... | Mathlib/Data/Vector/Basic.lean | 331 | 333 | theorem scanl_cons (x : α) : scanl f b (x ::ᵥ v) = b ::ᵥ scanl f (f b x) v := by |
simp only [scanl, toList_cons, List.scanl]; dsimp
simp only [cons]; rfl
|
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Probability.Variance
import Mathlib.MeasureTheory.Function.UniformIntegrable
#align_import probability.ident_distrib from "leanprover-community/... | Mathlib/Probability/IdentDistrib.lean | 132 | 135 | theorem measure_mem_eq (h : IdentDistrib f g μ ν) {s : Set γ} (hs : MeasurableSet s) :
μ (f ⁻¹' s) = ν (g ⁻¹' s) := by |
rw [← Measure.map_apply_of_aemeasurable h.aemeasurable_fst hs, ←
Measure.map_apply_of_aemeasurable h.aemeasurable_snd hs, h.map_eq]
|
/-
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, Patrick Massot
-/
import Mathlib.Algebra.GroupWithZero.Indicator
import Mathlib.Algebra.Module.Basic
import Mathlib.Topology.Separation
#align_import topology.supp... | Mathlib/Topology/Support.lean | 258 | 270 | theorem comp₂_left (hf : HasCompactMulSupport f)
(hf₂ : HasCompactMulSupport f₂) (hm : m 1 1 = 1) :
HasCompactMulSupport fun x => m (f x) (f₂ x) := by |
rw [hasCompactMulSupport_iff_eventuallyEq] at hf hf₂ ⊢
#adaptation_note /-- `nightly-2024-03-11`
If we *either* (1) remove the type annotations on the
binders in the following `fun` or (2) revert `simp only` to `simp_rw`, `to_additive` fails
because an `OfNat.ofNat 1` is not replaced with `0`. Notably, as of... |
/-
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 | 199 | 203 | theorem Convex.linear_preimage {s : Set F} (hs : Convex 𝕜 s) (f : E →ₗ[𝕜] F) :
Convex 𝕜 (f ⁻¹' s) := by |
intro x hx y hy a b ha hb hab
rw [mem_preimage, f.map_add, f.map_smul, f.map_smul]
exact hs hx hy ha hb hab
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fin.Fin2
import Mathlib.Init.Logic
import Mathlib.Mathport.Notation
import Mathlib.Tactic.TypeStar
#align_import data.vector3 from "leanprover-co... | Mathlib/Data/Vector3.lean | 280 | 291 | theorem vectorAllP_iff_forall (p : α → Prop) (v : Vector3 α n) :
VectorAllP p v ↔ ∀ i, p (v i) := by |
refine v.recOn ?_ ?_
· exact ⟨fun _ => Fin2.elim0, fun _ => trivial⟩
· simp only [vectorAllP_cons]
refine fun {n} a v IH =>
(and_congr_right fun _ => IH).trans
⟨fun ⟨pa, h⟩ i => by
refine i.cases' ?_ ?_
exacts [pa, h], fun h => ⟨?_, fun i => ?_⟩⟩
· simpa using h fz
·... |
/-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Order.ProjIcc
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.UnitInterval
#align_import topology.path_connected from "leanprover... | Mathlib/Topology/Connected/PathConnected.lean | 648 | 653 | theorem truncate_range {a b : X} (γ : Path a b) {t₀ t₁ : ℝ} :
range (γ.truncate t₀ t₁) ⊆ range γ := by |
rw [← γ.extend_range]
simp only [range_subset_iff, SetCoe.exists, SetCoe.forall]
intro x _hx
simp only [DFunLike.coe, Path.truncate, mem_range_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.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import n... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 274 | 279 | theorem normAtPlace_smul (w : InfinitePlace K) (x : E K) (c : ℝ) :
normAtPlace w (c • x) = |c| * normAtPlace w x := by |
rw [normAtPlace, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk]
split_ifs
· rw [Prod.smul_fst, Pi.smul_apply, norm_smul, Real.norm_eq_abs]
· rw [Prod.smul_snd, Pi.smul_apply, norm_smul, Real.norm_eq_abs, Complex.norm_eq_abs]
|
/-
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.Group.Measure
import Mathlib.Topology.Constructions
#align_import measure_theo... | Mathlib/MeasureTheory/Constructions/Pi.lean | 545 | 547 | theorem univ_pi_Ioo_ae_eq_Icc {f g : ∀ i, α i} :
(pi univ fun i => Ioo (f i) (g i)) =ᵐ[Measure.pi μ] Icc f g := by |
rw [← pi_univ_Icc]; exact pi_Ioo_ae_eq_pi_Icc
|
/-
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.FDeriv
/-!
# Differentiability of specific functions
In this file, we establish differentiability r... | Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean | 285 | 288 | theorem tangentMap_prod_fst {p : TangentBundle (I.prod I') (M × M')} :
tangentMap (I.prod I') I Prod.fst p = ⟨p.proj.1, p.2.1⟩ := by |
-- Porting note: `rfl` wasn't needed
simp [tangentMap]; rfl
|
/-
Copyright (c) 2014 Robert Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Lewis, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ring.Commute... | Mathlib/Algebra/Field/Basic.lean | 183 | 185 | theorem one_div_mul_sub_mul_one_div_eq_one_div_add_one_div (ha : a ≠ 0) (hb : b ≠ 0) :
1 / a * (b - a) * (1 / b) = 1 / a - 1 / b := by |
simpa only [one_div] using (inv_sub_inv' ha hb).symm
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Data.Finset.Fold
import Mathlib.Data.Finset.Option
import Mathlib.Data.Finset.Pi
import Mathlib.Data.... | Mathlib/Data/Finset/Lattice.lean | 533 | 537 | theorem sup_inf_distrib_left (s : Finset ι) (f : ι → α) (a : α) :
a ⊓ s.sup f = s.sup fun i => a ⊓ f i := by |
induction s using Finset.cons_induction with
| empty => simp_rw [Finset.sup_empty, inf_bot_eq]
| cons _ _ _ h => rw [sup_cons, sup_cons, inf_sup_left, h]
|
/-
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 | 232 | 235 | theorem HasDerivAtFilter.comp_of_eq (hh₂ : HasDerivAtFilter h₂ h₂' y L')
(hh : HasDerivAtFilter h h' x L) (hL : Tendsto h L L') (hy : y = h x) :
HasDerivAtFilter (h₂ ∘ h) (h₂' * h') x L := by |
rw [hy] at hh₂; exact hh₂.comp x hh hL
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Patrick Massot, Yury Kudryashov
-/
import Mathlib.Topology.Connected.Basic
/-!
# Totally disconnected and totally separated topological spaces
## Main definitions
We define the... | Mathlib/Topology/Connected/TotallyDisconnected.lean | 334 | 339 | theorem IsPreconnected.constant_of_mapsTo {S : Set α} (hS : IsPreconnected S)
{T : Set β} [DiscreteTopology T] {f : α → β} (hc : ContinuousOn f S) (hTm : MapsTo f S T)
{x y : α} (hx : x ∈ S) (hy : y ∈ S) : f x = f y := by |
let F : S → T := hTm.restrict f S T
suffices F ⟨x, hx⟩ = F ⟨y, hy⟩ by rwa [← Subtype.coe_inj] at this
exact (isPreconnected_iff_preconnectedSpace.mp hS).constant (hc.restrict_mapsTo _)
|
/-
Copyright (c) 2016 Leonardo de Moura. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Data.Set.Lattice
import Mathlib.Init.Set
import Mathlib.Control.Basic
import Mathlib.Lean.Expr.ExtraRecognizers
#align_import data.set.functor f... | Mathlib/Data/Set/Functor.lean | 135 | 139 | theorem coe_eq_image_val (t : Set s) :
@Lean.Internal.coeM Set s α _ Set.monad t = (t : Set α) := by |
change ⋃ (x ∈ t), {x.1} = _
ext
simp
|
/-
Copyright (c) 2020 Kexing Ying and Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.FiniteSupport
import Mathlib.Algebra... | Mathlib/Algebra/BigOperators/Finprod.lean | 1,174 | 1,179 | theorem finprod_eq_zero {M₀ : Type*} [CommMonoidWithZero M₀] (f : α → M₀) (x : α) (hx : f x = 0)
(hf : (mulSupport f).Finite) : ∏ᶠ x, f x = 0 := by |
nontriviality
rw [finprod_eq_prod f hf]
refine Finset.prod_eq_zero (hf.mem_toFinset.2 ?_) hx
simp [hx]
|
/-
Copyright (c) 2014 Robert Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.Order.Fiel... | Mathlib/Algebra/Order/Field/Power.lean | 155 | 159 | theorem Odd.zpow_neg_iff (hn : Odd n) : a ^ n < 0 ↔ a < 0 := by |
refine ⟨lt_imp_lt_of_le_imp_le (zpow_nonneg · _), fun ha ↦ ?_⟩
obtain ⟨k, rfl⟩ := hn
rw [zpow_add_one₀ ha.ne]
exact mul_neg_of_pos_of_neg (Even.zpow_pos (even_two_mul _) ha.ne) ha
|
/-
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.Circumcenter
#align_import geometry.euclidean.monge_point from "leanprover-community/mathlib"@"1a4df69ca1a9a0e5e26bfe12e2b92814216016d0... | Mathlib/Geometry/Euclidean/MongePoint.lean | 277 | 284 | theorem mongePoint_mem_mongePlane {n : ℕ} (s : Simplex ℝ P (n + 2)) {i₁ i₂ : Fin (n + 3)} :
s.mongePoint ∈ s.mongePlane i₁ i₂ := by |
rw [mongePlane_def, mem_inf_iff, ← vsub_right_mem_direction_iff_mem (self_mem_mk' _ _),
direction_mk', Submodule.mem_orthogonal']
refine ⟨?_, s.mongePoint_mem_affineSpan⟩
intro v hv
rcases Submodule.mem_span_singleton.mp hv with ⟨r, rfl⟩
rw [inner_smul_right, s.inner_mongePoint_vsub_face_centroid_vsub, m... |
/-
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.MeanValue
#align_import analysis.calculus.extend_deriv from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a... | Mathlib/Analysis/Calculus/FDeriv/Extend.lean | 111 | 140 | theorem has_deriv_at_interval_left_endpoint_of_tendsto_deriv {s : Set ℝ} {e : E} {a : ℝ} {f : ℝ → E}
(f_diff : DifferentiableOn ℝ f s) (f_lim : ContinuousWithinAt f s a) (hs : s ∈ 𝓝[>] a)
(f_lim' : Tendsto (fun x => deriv f x) (𝓝[>] a) (𝓝 e)) : HasDerivWithinAt f e (Ici a) a := by |
/- This is a specialization of `has_fderiv_at_boundary_of_tendsto_fderiv`. To be in the setting of
this theorem, we need to work on an open interval with closure contained in `s ∪ {a}`, that we
call `t = (a, b)`. Then, we check all the assumptions of this theorem and we apply it. -/
obtain ⟨b, ab : a < b, ... |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "... | Mathlib/Data/Set/Function.lean | 1,625 | 1,631 | theorem MapsTo.piecewise_ite {s s₁ s₂ : Set α} {t t₁ t₂ : Set β} {f₁ f₂ : α → β}
[∀ i, Decidable (i ∈ s)] (h₁ : MapsTo f₁ (s₁ ∩ s) (t₁ ∩ t))
(h₂ : MapsTo f₂ (s₂ ∩ sᶜ) (t₂ ∩ tᶜ)) :
MapsTo (s.piecewise f₁ f₂) (s.ite s₁ s₂) (t.ite t₁ t₂) := by |
refine (h₁.congr ?_).union_union (h₂.congr ?_)
exacts [(piecewise_eqOn s f₁ f₂).symm.mono inter_subset_right,
(piecewise_eqOn_compl s f₁ f₂).symm.mono inter_subset_right]
|
/-
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.Init.Core
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.affine_space.finite_d... | Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | 524 | 527 | theorem affineIndependent_iff_not_collinear_set {p₁ p₂ p₃ : P} :
AffineIndependent k ![p₁, p₂, p₃] ↔ ¬Collinear k ({p₁, p₂, p₃} : Set P) := by |
rw [affineIndependent_iff_not_collinear]
simp_rw [Matrix.range_cons, Matrix.range_empty, Set.singleton_union, insert_emptyc_eq]
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Convex.Uniform
import Mathlib.... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 1,072 | 1,075 | theorem norm_sub_mul_self_real (x y : F) :
‖x - y‖ * ‖x - y‖ = ‖x‖ * ‖x‖ - 2 * ⟪x, y⟫_ℝ + ‖y‖ * ‖y‖ := by |
have h := @norm_sub_mul_self ℝ _ _ _ _ x y
simpa using h
|
/-
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.Gluing
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.CategoryTheory.Limits.Sh... | Mathlib/AlgebraicGeometry/Pullbacks.lean | 143 | 148 | theorem t'_snd_fst_snd (i j k : 𝒰.J) :
t' 𝒰 f g i j k ≫ pullback.snd ≫ pullback.fst ≫ pullback.snd =
pullback.fst ≫ pullback.fst ≫ pullback.snd := by |
simp only [t', Category.assoc, pullbackSymmetry_hom_comp_snd_assoc,
pullbackRightPullbackFstIso_inv_fst_assoc, pullback.lift_fst_assoc, t_fst_snd,
pullbackRightPullbackFstIso_hom_fst_assoc]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Module.LinearMap.Basic
import ... | Mathlib/Data/DFinsupp/Basic.lean | 2,155 | 2,159 | theorem mapRange_add (f : ∀ i, β₁ i → β₂ i) (hf : ∀ i, f i 0 = 0)
(hf' : ∀ i x y, f i (x + y) = f i x + f i y) (g₁ g₂ : Π₀ i, β₁ i) :
mapRange f hf (g₁ + g₂) = mapRange f hf g₁ + mapRange f hf g₂ := by |
ext
simp only [mapRange_apply f, coe_add, Pi.add_apply, hf']
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.IsometricSMul
#align_import topology.metric_space.hausdorff_distance from "lea... | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 410 | 417 | theorem hausdorffEdist_closure₁ : hausdorffEdist (closure s) t = hausdorffEdist s t := by |
refine le_antisymm ?_ ?_
· calc
_ ≤ hausdorffEdist (closure s) s + hausdorffEdist s t := hausdorffEdist_triangle
_ = hausdorffEdist s t := by simp [hausdorffEdist_comm]
· calc
_ ≤ hausdorffEdist s (closure s) + hausdorffEdist (closure s) t := hausdorffEdist_triangle
_ = hausdorffEdist (cl... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Thomas Browning
-/
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.SetLike.Fintype
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory... | Mathlib/GroupTheory/Sylow.lean | 460 | 472 | theorem Sylow.normalizer_sup_eq_top {p : ℕ} [Fact p.Prime] {N : Subgroup G} [N.Normal]
[Finite (Sylow p N)] (P : Sylow p N) :
((↑P : Subgroup N).map N.subtype).normalizer ⊔ N = ⊤ := by |
refine top_le_iff.mp fun g _ => ?_
obtain ⟨n, hn⟩ := exists_smul_eq N ((MulAut.conjNormal g : MulAut N) • P) P
rw [← inv_mul_cancel_left (↑n) g, sup_comm]
apply mul_mem_sup (N.inv_mem n.2)
rw [Sylow.smul_def, ← mul_smul, ← MulAut.conjNormal_val, ← MulAut.conjNormal.map_mul,
Sylow.ext_iff, Sylow.pointwise... |
/-
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 | 150 | 152 | theorem le_nhds_mul (a b : M) : 𝓝 a * 𝓝 b ≤ 𝓝 (a * b) := by |
rw [← map₂_mul, ← map_uncurry_prod, ← nhds_prod_eq]
exact continuous_mul.tendsto _
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.BaseChange
import Mathlib.Algebra.Lie.Solvable
import Mathlib.Algebra.Lie.Quotient
import Mathlib.Algebra.Lie.Normalizer
import Mathlib.LinearAlg... | Mathlib/Algebra/Lie/Nilpotent.lean | 826 | 832 | theorem _root_.LieAlgebra.ad_nilpotent_of_nilpotent {a : A} (h : IsNilpotent a) :
IsNilpotent (LieAlgebra.ad R A a) := by |
rw [LieAlgebra.ad_eq_lmul_left_sub_lmul_right]
have hl : IsNilpotent (LinearMap.mulLeft R a) := by rwa [LinearMap.isNilpotent_mulLeft_iff]
have hr : IsNilpotent (LinearMap.mulRight R a) := by rwa [LinearMap.isNilpotent_mulRight_iff]
have := @LinearMap.commute_mulLeft_right R A _ _ _ _ _ a a
exact this.isNilp... |
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
/-!
# Noncomputable... | Mathlib/Data/Set/Card.lean | 1,102 | 1,106 | theorem ncard_eq_succ {n : ℕ} (hs : s.Finite := by | toFinite_tac) :
s.ncard = n + 1 ↔ ∃ a t, a ∉ t ∧ insert a t = s ∧ t.ncard = n := by
refine ⟨eq_insert_of_ncard_eq_succ, ?_⟩
rintro ⟨a, t, hat, h, rfl⟩
rw [← h, ncard_insert_of_not_mem hat (hs.subset ((subset_insert a t).trans_eq h))]
|
/-
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.Complex.Asymptotics
import Mathlib.Analysis.SpecificLimits.Normed
#align_import analysis.special_... | Mathlib/Analysis/SpecialFunctions/Exp.lean | 380 | 382 | theorem tendsto_exp_comp_nhds_zero {f : α → ℝ} :
Tendsto (fun x => exp (f x)) l (𝓝 0) ↔ Tendsto f l atBot := by |
simp_rw [← comp_apply (f := exp), ← tendsto_comap_iff, comap_exp_nhds_zero]
|
/-
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.CategoryTheory.Monoidal.Free.Coherence
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
imp... | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 681 | 691 | theorem associator_monoidal (X₁ X₂ X₃ Y₁ Y₂ Y₃ : C) :
tensor_μ C (X₁ ⊗ X₂, X₃) (Y₁ ⊗ Y₂, Y₃) ≫
(tensor_μ C (X₁, X₂) (Y₁, Y₂) ▷ (X₃ ⊗ Y₃)) ≫ (α_ (X₁ ⊗ Y₁) (X₂ ⊗ Y₂) (X₃ ⊗ Y₃)).hom =
((α_ X₁ X₂ X₃).hom ⊗ (α_ Y₁ Y₂ Y₃).hom) ≫
tensor_μ C (X₁, X₂ ⊗ X₃) (Y₁, Y₂ ⊗ Y₃) ≫ ((X₁ ⊗ Y₁) ◁ tensor_μ C (X₂, X... |
dsimp only [tensor_μ]
calc
_ = 𝟙 _ ⊗≫ X₁ ◁ X₂ ◁ (β_ X₃ Y₁).hom ▷ Y₂ ▷ Y₃ ⊗≫
X₁ ◁ ((X₂ ⊗ Y₁) ◁ (β_ X₃ Y₂).hom ≫
(β_ X₂ Y₁).hom ▷ (Y₂ ⊗ X₃)) ▷ Y₃ ⊗≫ 𝟙 _ := by simp; coherence
_ = _ := by rw [whisker_exchange]; simp; coherence
|
/-
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 | 966 | 972 | theorem eval₂_C_X_eq_coe : φ.eval₂ (PowerSeries.C R) PowerSeries.X = ↑φ := by |
nth_rw 2 [← eval₂_C_X (p := φ)]
rw [← coeToPowerSeries.ringHom_apply, eval₂_eq_sum_range, eval₂_eq_sum_range, map_sum]
apply Finset.sum_congr rfl
intros
rw [map_mul, map_pow, coeToPowerSeries.ringHom_apply,
coeToPowerSeries.ringHom_apply, coe_C, coe_X]
|
/-
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.Data.Int.Interval
import Mathlib.RingTheory.Binomial
import Mathlib.RingTheory.HahnSeries.PowerSeries
import Mathlib.RingTheory.HahnSeries.Summable
imp... | Mathlib/RingTheory/LaurentSeries.lean | 379 | 380 | theorem coe_smul (r : F) : ((r • f : RatFunc F) : LaurentSeries F) = r • (f : LaurentSeries F) := by |
rw [RatFunc.smul_eq_C_mul, ← C_mul_eq_smul, coe_mul, coe_C]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.Bounded
import Mathlib.SetTheory.Cardinal.PartENat
import Mathlib.SetTheor... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 1,189 | 1,193 | theorem mk_list_le_max (α : Type u) : #(List α) ≤ max ℵ₀ #α := by |
cases finite_or_infinite α
· exact mk_le_aleph0.trans (le_max_left _ _)
· rw [mk_list_eq_mk]
apply le_max_right
|
/-
Copyright (c) 2020 Fox Thomson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Fox Thomson, Markus Himmel
-/
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanp... | Mathlib/SetTheory/Game/Nim.lean | 266 | 267 | theorem grundyValue_eq_mex_left (G : PGame) :
grundyValue G = Ordinal.mex.{u, u} fun i => grundyValue (G.moveLeft i) := by | rw [grundyValue]
|
/-
Copyright (c) 2023 Scott Carnahan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Carnahan
-/
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.Ev... | Mathlib/Algebra/Polynomial/Smeval.lean | 61 | 63 | theorem smeval_monomial (n : ℕ) :
(monomial n r).smeval x = r • x ^ n := by |
simp only [smeval_eq_sum, smul_pow, zero_smul, sum_monomial_index]
|
/-
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.FunctorN
#align_import algebraic_topology.dold_kan.normalized from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f... | Mathlib/AlgebraicTopology/DoldKan/Normalized.lean | 83 | 86 | theorem PInftyToNormalizedMooreComplex_naturality {X Y : SimplicialObject A} (f : X ⟶ Y) :
AlternatingFaceMapComplex.map f ≫ PInftyToNormalizedMooreComplex Y =
PInftyToNormalizedMooreComplex X ≫ NormalizedMooreComplex.map f := by |
aesop_cat
|
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Algebra.Polynomial.Splits
import... | Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | 291 | 295 | theorem map_cyclotomic (n : ℕ) {R S : Type*} [Ring R] [Ring S] (f : R →+* S) :
map f (cyclotomic n R) = cyclotomic n S := by |
rw [← map_cyclotomic_int n R, ← map_cyclotomic_int n S, map_map]
have : Subsingleton (ℤ →+* S) := inferInstance
congr!
|
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align... | Mathlib/Computability/Ackermann.lean | 78 | 78 | theorem ack_succ_succ (m n : ℕ) : ack (m + 1) (n + 1) = ack m (ack (m + 1) n) := by | rw [ack]
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.IsometricSMul
#align_import topology.metric_space.hausdorff_distance from "lea... | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 451 | 458 | theorem empty_or_nonempty_of_hausdorffEdist_ne_top (fin : hausdorffEdist s t ≠ ⊤) :
(s = ∅ ∧ t = ∅) ∨ (s.Nonempty ∧ t.Nonempty) := by |
rcases s.eq_empty_or_nonempty with hs | hs
· rcases t.eq_empty_or_nonempty with ht | ht
· exact Or.inl ⟨hs, ht⟩
· rw [hausdorffEdist_comm] at fin
exact Or.inr ⟨nonempty_of_hausdorffEdist_ne_top ht fin, ht⟩
· exact Or.inr ⟨hs, nonempty_of_hausdorffEdist_ne_top hs fin⟩
|
/-
Copyright (c) 2022 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Integral.IntervalIntegral
#align_import measure_theory.integral.layercake from "leanprover-community/mathlib"@"08a4542bec7242a5c60f179e4e49d... | Mathlib/MeasureTheory/Integral/Layercake.lean | 482 | 491 | theorem lintegral_comp_eq_lintegral_meas_lt_mul (μ : Measure α) (f_nn : 0 ≤ᵐ[μ] f)
(f_mble : AEMeasurable f μ) (g_intble : ∀ t > 0, IntervalIntegrable g volume 0 t)
(g_nn : ∀ᵐ t ∂volume.restrict (Ioi 0), 0 ≤ g t) :
∫⁻ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) ∂μ =
∫⁻ t in Ioi 0, μ {a : α | t < f a} * E... |
rw [lintegral_comp_eq_lintegral_meas_le_mul μ f_nn f_mble g_intble g_nn]
apply lintegral_congr_ae
filter_upwards [meas_le_ae_eq_meas_lt μ (volume.restrict (Ioi 0)) f]
with t ht
rw [ht]
|
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Topology.Algebra.Module.Multilinear.Bounded
import Mathlib.Topology.Algebra.UniformConvergence
/-!
# Topology on continuous multilinear maps
In thi... | Mathlib/Topology/Algebra/Module/Multilinear/Topology.lean | 99 | 109 | theorem hasBasis_nhds_zero_of_basis {ι : Type*} {p : ι → Prop} {b : ι → Set F}
(h : (𝓝 (0 : F)).HasBasis p b) :
(𝓝 (0 : ContinuousMultilinearMap 𝕜 E F)).HasBasis
(fun Si : Set (Π i, E i) × ι => IsVonNBounded 𝕜 Si.1 ∧ p Si.2)
fun Si => { f | MapsTo f Si.1 (b Si.2) } := by |
letI : UniformSpace F := TopologicalAddGroup.toUniformSpace F
haveI : UniformAddGroup F := comm_topologicalAddGroup_is_uniform
rw [nhds_induced]
refine (UniformOnFun.hasBasis_nhds_zero_of_basis _ ?_ ?_ h).comap DFunLike.coe
· exact ⟨∅, isVonNBounded_empty _ _⟩
· exact directedOn_of_sup_mem fun _ _ => Borno... |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov
-/
import Mathlib.MeasureTheory.Integral.IntegrableOn
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Function.LocallyIntegrabl... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 1,440 | 1,443 | theorem snd_integral [CompleteSpace E] {f : X → E × F} (hf : Integrable f μ) :
(∫ x, f x ∂μ).2 = ∫ x, (f x).2 ∂μ := by |
rw [← Prod.fst_swap, swap_integral]
exact fst_integral <| hf.snd.prod_mk hf.fst
|
/-
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 | 1,713 | 1,717 | theorem ContDiffAt.prod_map {f : E → F} {g : E' → F'} {x : E} {y : E'} (hf : ContDiffAt 𝕜 n f x)
(hg : ContDiffAt 𝕜 n g y) : ContDiffAt 𝕜 n (Prod.map f g) (x, y) := by |
rw [ContDiffAt] at *
convert hf.prod_map hg
simp only [univ_prod_univ]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Scott Morrison
-/
import Mathlib.CategoryTheory.Subobject.Basic
import Mathlib.CategoryTheory.Preadditive.Basic
#align_import category_theory.subobject.factor_thru from ... | Mathlib/CategoryTheory/Subobject/FactorThru.lean | 188 | 192 | theorem factorThru_add {X Y : C} {P : Subobject Y} (f g : X ⟶ Y) (w : P.Factors (f + g))
(wf : P.Factors f) (wg : P.Factors g) :
P.factorThru (f + g) w = P.factorThru f wf + P.factorThru g wg := by |
ext
simp
|
/-
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.Order.MonotoneContinuity
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.E... | Mathlib/Topology/Instances/ENNReal.lean | 664 | 667 | theorem smul_iSup {ι : Sort*} {R} [SMul R ℝ≥0∞] [IsScalarTower R ℝ≥0∞ ℝ≥0∞] (f : ι → ℝ≥0∞)
(c : R) : (c • ⨆ i, f i) = ⨆ i, c • f i := by |
-- Porting note: replaced `iSup _` with `iSup f`
simp only [← smul_one_mul c (f _), ← smul_one_mul c (iSup f), ENNReal.mul_iSup]
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "... | Mathlib/Data/Set/Function.lean | 1,841 | 1,844 | theorem bijOn_range (h : Semiconj f fa fb) (ha : Bijective fa) (hf : Injective f) :
BijOn fb (range f) (range f) := by |
rw [← image_univ]
exact h.bijOn_image (bijective_iff_bijOn_univ.1 ha) hf.injOn
|
/-
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 | 321 | 326 | theorem image_connectedComponentIn {s : Set X} (h : X ≃ₜ Y) {x : X} (hx : x ∈ s) :
h '' connectedComponentIn s x = connectedComponentIn (h '' s) (h x) := by |
refine (h.continuous.image_connectedComponentIn_subset hx).antisymm ?_
have := h.symm.continuous.image_connectedComponentIn_subset (mem_image_of_mem h hx)
rwa [image_subset_iff, h.preimage_symm, h.image_symm, h.preimage_image, h.symm_apply_apply]
at this
|
/-
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.Data.Fin.VecNotation
import Mathlib.SetTheory.Cardinal.Basic
#align_import m... | Mathlib/ModelTheory/Basic.lean | 95 | 100 | theorem lift_mk {i : ℕ} :
Cardinal.lift.{v,u} #(Sequence₂ a₀ a₁ a₂ i)
= #(Sequence₂ (ULift.{v,u} a₀) (ULift.{v,u} a₁) (ULift.{v,u} a₂) i) := by |
rcases i with (_ | _ | _ | i) <;>
simp only [Sequence₂, mk_uLift, Nat.succ_ne_zero, IsEmpty.forall_iff, Nat.succ.injEq,
add_eq_zero, OfNat.ofNat_ne_zero, and_false, one_ne_zero, mk_eq_zero, lift_zero]
|
/-
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 | 587 | 660 | theorem sbtw_of_sbtw_of_sbtw_of_mem_affineSpan_pair [NoZeroSMulDivisors R V]
{t : Affine.Triangle R P} {i₁ i₂ i₃ : Fin 3} (h₁₂ : i₁ ≠ i₂) {p₁ p₂ p : P}
(h₁ : Sbtw R (t.points i₂) p₁ (t.points i₃)) (h₂ : Sbtw R (t.points i₁) p₂ (t.points i₃))
(h₁' : p ∈ line[R, t.points i₁, p₁]) (h₂' : p ∈ line[R, t.points i... |
-- Should not be needed; see comments on local instances in `Data.Sign`.
letI : DecidableRel ((· < ·) : R → R → Prop) := LinearOrderedRing.decidableLT
have h₁₃ : i₁ ≠ i₃ := by
rintro rfl
simp at h₂
have h₂₃ : i₂ ≠ i₃ := by
rintro rfl
simp at h₁
have h3 : ∀ i : Fin 3, i = i₁ ∨ i = i₂ ∨ i = i₃ ... |
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux, Jon Bannon
-/
import Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness
import Mathlib.Analysis.NormedSpace.Star.Unitization
import Mathlib.Analysis.SpecialFunctions.... | Mathlib/Analysis/NormedSpace/Star/Multiplier.lean | 654 | 654 | theorem norm_snd (a : 𝓜(𝕜, A)) : ‖a.snd‖ = ‖a‖ := by | rw [← norm_fst, norm_fst_eq_snd]
|
/-
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 | 129 | 130 | theorem exists_range_iff {f : α →ₛ β} {p : β → Prop} : (∃ y ∈ f.range, p y) ↔ ∃ x, p (f x) := by |
simpa only [mem_range, exists_prop] using Set.exists_range_iff
|
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Orde... | Mathlib/Combinatorics/SimpleGraph/Density.lean | 171 | 176 | theorem card_interedges_finpartition_right [DecidableEq β] (s : Finset α) (P : Finpartition t) :
(interedges r s t).card = ∑ b ∈ P.parts, (interedges r s b).card := by |
classical
simp_rw [← P.biUnion_parts, interedges_biUnion_right, id]
rw [card_biUnion]
exact fun x hx y hy h ↦ interedges_disjoint_right r _ (P.disjoint hx hy h)
|
/-
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, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathli... | Mathlib/Data/Set/Lattice.lean | 1,681 | 1,682 | theorem iUnion_iUnion_eq' {f : ι → α} {g : α → Set β} :
⋃ (x) (y) (_ : f y = x), g x = ⋃ y, g (f y) := by | simpa using biUnion_range
|
/-
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 | 388 | 389 | theorem le_comap_mkQ (p' : Submodule R (M ⧸ p)) : p ≤ comap p.mkQ p' := by |
simpa using (comap_mono bot_le : ker p.mkQ ≤ comap p.mkQ p')
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Shing Tak Lam, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Int.ModEq
import Mathli... | Mathlib/Data/Nat/Digits.lean | 734 | 737 | theorem modEq_eleven_digits_sum (n : ℕ) :
n ≡ ((digits 10 n).map fun n : ℕ => (n : ℤ)).alternatingSum [ZMOD 11] := by |
have t := zmodeq_ofDigits_digits 11 10 (-1 : ℤ) (by unfold Int.ModEq; rfl) n
rwa [ofDigits_neg_one] at t
|
/-
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.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calc... | Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | 254 | 264 | theorem comp_right_fderivWithin {f : F → G} {s : Set F} {x : E}
(hxs : UniqueDiffWithinAt 𝕜 (iso ⁻¹' s) x) :
fderivWithin 𝕜 (f ∘ iso) (iso ⁻¹' s) x =
(fderivWithin 𝕜 f s (iso x)).comp (iso : E →L[𝕜] F) := by |
by_cases h : DifferentiableWithinAt 𝕜 f s (iso x)
· exact (iso.comp_right_hasFDerivWithinAt_iff.2 h.hasFDerivWithinAt).fderivWithin hxs
· have : ¬DifferentiableWithinAt 𝕜 (f ∘ iso) (iso ⁻¹' s) x := by
intro h'
exact h (iso.comp_right_differentiableWithinAt_iff.1 h')
rw [fderivWithin_zero_of_not... |
/-
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.FDeriv
/-!
# Differentiability of specific functions
In this file, we establish differentiability r... | Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean | 184 | 187 | theorem hasMFDerivAt_const (c : M') (x : M) :
HasMFDerivAt I I' (fun _ : M => c) x (0 : TangentSpace I x →L[𝕜] TangentSpace I' c) := by |
refine ⟨continuous_const.continuousAt, ?_⟩
simp only [writtenInExtChartAt, (· ∘ ·), hasFDerivWithinAt_const]
|
/-
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.Calculus.ContDiff.Bounds
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysi... | Mathlib/Analysis/Distribution/SchwartzSpace.lean | 1,265 | 1,265 | theorem integralCLM_dirac_eq_delta (x : E) : integralCLM 𝕜 (dirac x) = delta 𝕜 F x := by | aesop
|
/-
Copyright (c) 2023 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.Star.SelfAdjoint
#align_import algebra.star.order from "leanprover-community/mathlib"@"31c24... | Mathlib/Algebra/Star/Order.lean | 169 | 170 | theorem conjugate_nonneg' {a : R} (ha : 0 ≤ a) (c : R) : 0 ≤ c * a * star c := by |
simpa only [star_star] using conjugate_nonneg ha (star c)
|
/-
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, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Order.LiminfLimsup
import Mathlib.Topology.Instances.Rat
import Mathlib.Top... | Mathlib/Analysis/Normed/Group/Basic.lean | 652 | 655 | theorem norm_le_norm_add_norm_div' (u v : E) : ‖u‖ ≤ ‖v‖ + ‖u / v‖ := by |
rw [add_comm]
refine (norm_mul_le' _ _).trans_eq' ?_
rw [div_mul_cancel]
|
/-
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.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Limits.Shapes.CommSq
import Mathlib.Cat... | Mathlib/CategoryTheory/Limits/VanKampen.lean | 505 | 534 | theorem BinaryCofan.isVanKampen_iff (c : BinaryCofan X Y) :
IsVanKampenColimit c ↔
∀ {X' Y' : C} (c' : BinaryCofan X' Y') (αX : X' ⟶ X) (αY : Y' ⟶ Y) (f : c'.pt ⟶ c.pt)
(_ : αX ≫ c.inl = c'.inl ≫ f) (_ : αY ≫ c.inr = c'.inr ≫ f),
Nonempty (IsColimit c') ↔ IsPullback c'.inl αX f c.inl ∧ IsPullb... |
constructor
· introv H hαX hαY
rw [H c' (mapPair αX αY) f (by ext ⟨⟨⟩⟩ <;> dsimp <;> assumption) (mapPair_equifibered _)]
constructor
· intro H
exact ⟨H _, H _⟩
· rintro H ⟨⟨⟩⟩
exacts [H.1, H.2]
· introv H F' hα h
let X' := F'.obj ⟨WalkingPair.left⟩
let Y' := F'.obj ⟨WalkingPa... |
/-
Copyright (c) 2021 Vladimir Goryachev. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Vladimir Goryachev, Kyle Miller, Scott Morrison, Eric Rodriguez
-/
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.Ring
#align_import data.nat.count fr... | Mathlib/Data/Nat/Count.lean | 133 | 137 | theorem count_injective {m n : ℕ} (hm : p m) (hn : p n) (heq : count p m = count p n) : m = n := by |
by_contra! h : m ≠ n
wlog hmn : m < n
· exact this hn hm heq.symm h.symm (h.lt_or_lt.resolve_left hmn)
· simpa [heq] using count_strict_mono hm hmn
|
/-
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.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.Affine... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 1,116 | 1,124 | theorem eq_affineCombination_of_mem_affineSpan_of_fintype [Fintype ι] {p1 : P} {p : ι → P}
(h : p1 ∈ affineSpan k (Set.range p)) :
∃ w : ι → k, ∑ i, w i = 1 ∧ p1 = Finset.univ.affineCombination k p w := by |
classical
obtain ⟨s, w, hw, rfl⟩ := eq_affineCombination_of_mem_affineSpan h
refine
⟨(s : Set ι).indicator w, ?_, Finset.affineCombination_indicator_subset w p s.subset_univ⟩
simp only [Finset.mem_coe, Set.indicator_apply, ← hw]
rw [Fintype.sum_extend_by_zero s w]
|
/-
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 Batteries.Control.ForInStep.Lemmas
import Batteries.Data.List.Basic
import Batteries.Ta... | .lake/packages/batteries/Batteries/Data/List/Lemmas.lean | 694 | 700 | theorem findIdx?_eq_some_iff (xs : List α) (p : α → Bool) :
xs.findIdx? p = some i ↔ (xs.take (i + 1)).map p = replicate i false ++ [true] := by |
induction xs generalizing i with
| nil => simp
| cons x xs ih =>
simp only [findIdx?_cons, Nat.zero_add, findIdx?_succ, take_succ_cons, map_cons]
split <;> cases i <;> simp_all
|
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Yury Kudryashov
-/
import Mathlib.Algebra.Star.Order
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.Order.MonotoneContinuity
#align... | Mathlib/Data/Real/Sqrt.lean | 194 | 197 | theorem sqrt_eq_one : √x = 1 ↔ x = 1 :=
calc
√x = 1 ↔ 1 * 1 = x := sqrt_eq_iff_mul_self_eq_of_pos zero_lt_one
_ ↔ x = 1 := by | rw [eq_comm, mul_one]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
... | Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean | 53 | 54 | theorem pos_of_pos {x : ℝ} (hx : 0 < x) : 0 < expNegInvGlue x := by |
simp [expNegInvGlue, not_le.2 hx, exp_pos]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Topology.Order
#align_import topology.maps from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c73... | Mathlib/Topology/Maps.lean | 338 | 340 | theorem isOpen_range (hf : IsOpenMap f) : IsOpen (range f) := by |
rw [← image_univ]
exact hf _ isOpen_univ
|
/-
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 | 399 | 402 | theorem mem_range_embedding (j : Fin n) : j ∈ Set.range (c.embedding (c.index j)) := by |
have : c.embedding (c.index j) (c.invEmbedding j) ∈ Set.range (c.embedding (c.index j)) :=
Set.mem_range_self _
rwa [c.embedding_comp_inv j] at this
|
/-
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.CharP.Invertible
import Mathlib.Algebra.MvPolynomial.Variables
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Alg... | Mathlib/RingTheory/WittVector/WittPolynomial.lean | 146 | 148 | theorem aeval_wittPolynomial {A : Type*} [CommRing A] [Algebra R A] (f : ℕ → A) (n : ℕ) :
aeval f (W_ R n) = ∑ i ∈ range (n + 1), (p : A) ^ i * f i ^ p ^ (n - i) := by |
simp [wittPolynomial, AlgHom.map_sum, aeval_monomial, Finsupp.prod_single_index]
|
/-
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.Algebra.GroupPower.IterateHom
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 930 | 937 | theorem exists_eq_add_translationNumber (hf : Continuous f) : ∃ x, f x = x + τ f := by |
obtain ⟨a, ha⟩ : ∃ x, f x ≤ x + τ f := by
by_contra! H
exact lt_irrefl _ (f.lt_translationNumber_of_forall_add_lt hf H)
obtain ⟨b, hb⟩ : ∃ x, x + τ f ≤ f x := by
by_contra! H
exact lt_irrefl _ (f.translationNumber_lt_of_forall_lt_add hf H)
exact intermediate_value_univ₂ hf (continuous_id.add cont... |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Algebra.Group.Prod
import Mathlib.Algebra.Group.Units.Equiv
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Logic... | Mathlib/GroupTheory/Perm/Basic.lean | 541 | 542 | theorem swap_apply_apply (f : Perm α) (x y : α) : swap (f x) (f y) = f * swap x y * f⁻¹ := by |
rw [mul_swap_eq_swap_mul, mul_inv_cancel_right]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Patrick Stevens
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.BigOperators.Ring
import Mathlib... | Mathlib/Data/Nat/Choose/Sum.lean | 220 | 224 | theorem sum_choose_succ_mul (f : ℕ → ℕ → R) (n : ℕ) :
(∑ i ∈ range (n + 2), ((n + 1).choose i : R) * f i (n + 1 - i)) =
(∑ i ∈ range (n + 1), (n.choose i : R) * f i (n + 1 - i)) +
∑ i ∈ range (n + 1), (n.choose i : R) * f (i + 1) (n - i) := by |
simpa only [nsmul_eq_mul] using sum_choose_succ_nsmul f n
|
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.NatAntidiagonal
#align_import algebra.big_operators.nat_antidiagonal from "leanprover-com... | Mathlib/Algebra/BigOperators/NatAntidiagonal.lean | 42 | 45 | theorem prod_antidiagonal_succ' {n : ℕ} {f : ℕ × ℕ → M} : (∏ p ∈ antidiagonal (n + 1), f p) =
f (n + 1, 0) * ∏ p ∈ antidiagonal n, f (p.1, p.2 + 1) := by |
rw [← prod_antidiagonal_swap, prod_antidiagonal_succ, ← prod_antidiagonal_swap]
rfl
|
/-
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 | 137 | 138 | theorem mem_map_of_involutive {f : α → α} (hf : Involutive f) {a : α} {l : List α} :
a ∈ map f l ↔ f a ∈ l := by | rw [mem_map, hf.exists_mem_and_apply_eq_iff]
|
/-
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.Order.Interval.Set.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
import Mathlib.Tactic.AdaptationNote
... | Mathlib/Probability/Martingale/Upcrossing.lean | 236 | 239 | theorem stoppedValue_upperCrossingTime (h : upperCrossingTime a b f N (n + 1) ω ≠ N) :
b ≤ stoppedValue f (upperCrossingTime a b f N (n + 1)) ω := by |
obtain ⟨j, hj₁, hj₂⟩ := (hitting_le_iff_of_lt _ (lt_of_le_of_ne upperCrossingTime_le h)).1 le_rfl
exact stoppedValue_hitting_mem ⟨j, ⟨hj₁.1, le_trans hj₁.2 (hitting_le _)⟩, hj₂⟩
|
/-
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.MeasureTheory.Integral.Lebesgue
#align_import measure_theory.measure.giry_monad from "leanprover-community/mathlib"@"56f4cd1ef396e9fd389b5d8371ee9ad91... | Mathlib/MeasureTheory/Measure/GiryMonad.lean | 205 | 208 | theorem dirac_bind {m : Measure α} : bind m dirac = m := by |
ext1 s hs
simp only [bind_apply hs measurable_dirac, dirac_apply' _ hs, lintegral_indicator 1 hs,
Pi.one_apply, lintegral_one, restrict_apply, MeasurableSet.univ, univ_inter]
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Joël Riou
-/
import Mathlib.CategoryTheory.Limits.Shapes.CommSq
import Mathlib.CategoryTheory.Limits.Shapes.Diagonal
import Mathlib.CategoryTheory.MorphismProperty.Compositio... | Mathlib/CategoryTheory/MorphismProperty/Limits.lean | 285 | 287 | theorem universally_le (P : MorphismProperty C) : P.universally ≤ P := by |
intro X Y f hf
exact hf (𝟙 _) (𝟙 _) _ (IsPullback.of_vert_isIso ⟨by rw [Category.comp_id, Category.id_comp]⟩)
|
/-
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.Convex.Between
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathli... | Mathlib/MeasureTheory/Measure/Hausdorff.lean | 397 | 399 | theorem isometry_map_mkMetric (m : ℝ≥0∞ → ℝ≥0∞) {f : X → Y} (hf : Isometry f)
(H : Monotone m ∨ Surjective f) : map f (mkMetric m) = restrict (range f) (mkMetric m) := by |
rw [← isometry_comap_mkMetric _ hf H, map_comap]
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Andrew Zipperer, Haitao Zhang, Minchao Wu, Yury Kudryashov
-/
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "... | Mathlib/Data/Set/Function.lean | 139 | 143 | theorem range_extend {f : α → β} (hf : Injective f) (g : α → γ) (g' : β → γ) :
range (extend f g g') = range g ∪ g' '' (range f)ᶜ := by |
refine (range_extend_subset _ _ _).antisymm ?_
rintro z (⟨x, rfl⟩ | ⟨y, hy, rfl⟩)
exacts [⟨f x, hf.extend_apply _ _ _⟩, ⟨y, extend_apply' _ _ _ hy⟩]
|
/-
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 | 1,048 | 1,049 | theorem mem_disjUnion {α s t h a} : a ∈ @disjUnion α s t h ↔ a ∈ s ∨ a ∈ t := by |
rcases s with ⟨⟨s⟩⟩; rcases t with ⟨⟨t⟩⟩; apply List.mem_append
|
/-
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.Algebra.Group.Equiv.Basic
import Mathlib.Algebra.Group.Aut
import Mathlib.Data.ZMod.Defs
import Mathlib.Tactic.Ring
#align_import algebra.quandle from "lean... | Mathlib/Algebra/Quandle.lean | 481 | 484 | theorem dihedralAct.inv (n : ℕ) (a : ZMod n) : Function.Involutive (dihedralAct n a) := by |
intro b
dsimp only [dihedralAct]
simp
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Init.Function
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Inhabit
#align_import data.prod.basic from "leanprover-community/mathlib"@"d07... | Mathlib/Data/Prod/Basic.lean | 122 | 123 | theorem ext_iff {p q : α × β} : p = q ↔ p.1 = q.1 ∧ p.2 = q.2 := by |
rw [mk.inj_iff]
|
/-
Copyright (c) 2022 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.Biproducts
import Mathlib.GroupTheory.EckmannHilton
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_the... | Mathlib/CategoryTheory/Preadditive/OfBiproducts.lean | 129 | 132 | theorem add_comp (f g : X ⟶ Y) (h : Y ⟶ Z) : (f + g) ≫ h = f ≫ h + g ≫ h := by |
simp only [add_eq_right_addition, Category.assoc]
congr
ext <;> simp
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Order.Filter.Basic
import Mathlib.Topology.Bases
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.... | Mathlib/Topology/Compactness/Compact.lean | 970 | 980 | theorem isClosedMap_snd_of_compactSpace [CompactSpace X] :
IsClosedMap (Prod.snd : X × Y → Y) := fun s hs => by
rw [← isOpen_compl_iff, isOpen_iff_mem_nhds]
intro y hy
have : univ ×ˢ {y} ⊆ sᶜ := by |
exact fun (x, y') ⟨_, rfl⟩ hs => hy ⟨(x, y'), hs, rfl⟩
rcases generalized_tube_lemma isCompact_univ isCompact_singleton hs.isOpen_compl this
with ⟨U, V, -, hVo, hU, hV, hs⟩
refine mem_nhds_iff.2 ⟨V, ?_, hVo, hV rfl⟩
rintro _ hzV ⟨z, hzs, rfl⟩
exact hs ⟨hU trivial, hzV⟩ hzs
|
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Algebra.Category.ModuleCat.Free
import Mathlib.Topology.Category.Profinite.CofilteredLimit
import Mathlib.Topology.Category.Profinite.Product
impor... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 1,017 | 1,019 | theorem smaller_factorization (o : Ordinal) :
(fun (p : smaller C o) ↦ p.1) ∘ (range_equiv_smaller C o).toFun =
(πs C o) ∘ (fun (p : range (π C (ord I · < o))) ↦ p.1) := by | rfl
|
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Alex J. Best
-/
import Mathlib.Algebra.CharP.Quotient
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Data.Finsupp.Fintype
import Mathlib.Data.Int.Absolut... | Mathlib/RingTheory/Ideal/Norm.lean | 677 | 678 | theorem relNorm_top : relNorm R (⊤ : Ideal S) = ⊤ := by |
simpa only [one_eq_top] using map_one (relNorm R : Ideal S →*₀ _)
|
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Complex.Circle
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Meas... | Mathlib/Analysis/Fourier/FourierTransform.lean | 191 | 230 | theorem integral_bilin_fourierIntegral_eq_flip
{f : V → E} {g : W → F} (M : E →L[ℂ] F →L[ℂ] G) (he : Continuous e)
(hL : Continuous fun p : V × W ↦ L p.1 p.2) (hf : Integrable f μ) (hg : Integrable g ν) :
∫ ξ, M (fourierIntegral e μ L f ξ) (g ξ) ∂ν =
∫ x, M (f x) (fourierIntegral e ν L.flip g x) ∂μ :=... |
by_cases hG : CompleteSpace G; swap; · simp [integral, hG]
calc
_ = ∫ ξ, M.flip (g ξ) (∫ x, e (-L x ξ) • f x ∂μ) ∂ν := rfl
_ = ∫ ξ, (∫ x, M.flip (g ξ) (e (-L x ξ) • f x) ∂μ) ∂ν := by
congr with ξ
apply (ContinuousLinearMap.integral_comp_comm _ _).symm
exact (fourierIntegral_convergent_iff he hL _).... |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Finsupp.Basic
import Mathlib.Data.Finsupp.Order
#align_import data.finsupp.multiset from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34... | Mathlib/Data/Finsupp/Multiset.lean | 61 | 63 | theorem toMultiset_sum_single (s : Finset ι) (n : ℕ) :
Finsupp.toMultiset (∑ i ∈ s, single i n) = n • s.val := by |
simp_rw [toMultiset_sum, Finsupp.toMultiset_single, sum_nsmul, sum_multiset_singleton]
|
/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Quotients
import Mathlib.Order.Filter.Germ
import Mathlib.Order.Filter.Ultrafilter
#align_import model_theory.ultraproducts from "leanprov... | Mathlib/ModelTheory/Ultraproducts.lean | 156 | 160 | theorem sentence_realize (φ : L.Sentence) :
(u : Filter α).Product M ⊨ φ ↔ ∀ᶠ a : α in u, M a ⊨ φ := by |
simp_rw [Sentence.Realize]
erw [← realize_formula_cast φ, iff_eq_eq]
exact congr rfl (Subsingleton.elim _ _)
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Yury G. Kudryashov
-/
import Mathlib.Tactic.TFAE
import Mathlib.Topology.ContinuousOn
#align_import topology.inseparable from "leanprover-community/mathlib"@"bcfa726826abd57... | Mathlib/Topology/Inseparable.lean | 569 | 573 | theorem tendsto_lift₂_nhds {f : X → Y → α} {hf : ∀ a b c d, (a ~ᵢ c) → (b ~ᵢ d) → f a b = f c d}
{x : X} {y : Y} {l : Filter α} :
Tendsto (uncurry <| lift₂ f hf) (𝓝 (mk x, mk y)) l ↔ Tendsto (uncurry f) (𝓝 (x, y)) l := by |
rw [← map_prod_map_mk_nhds, tendsto_map'_iff]
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.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a5... | Mathlib/Algebra/MvPolynomial/Variables.lean | 115 | 119 | theorem vars_add_of_disjoint [DecidableEq σ] (h : Disjoint p.vars q.vars) :
(p + q).vars = p.vars ∪ q.vars := by |
refine (vars_add_subset p q).antisymm fun x hx => ?_
simp only [vars_def, Multiset.disjoint_toFinset] at h hx ⊢
rwa [degrees_add_of_disjoint h, Multiset.toFinset_union]
|
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.FixedPoint
#align_import set_theory.ordinal.principal from "leanprover-community/mathlib"@"31b269b6093548394... | Mathlib/SetTheory/Ordinal/Principal.lean | 378 | 393 | theorem principal_mul_iff_le_two_or_omega_opow_opow {o : Ordinal} :
Principal (· * ·) o ↔ o ≤ 2 ∨ ∃ a : Ordinal, o = (omega^omega^a) := by |
refine ⟨fun ho => ?_, ?_⟩
· rcases le_or_lt o 2 with ho₂ | ho₂
· exact Or.inl ho₂
rcases principal_add_iff_zero_or_omega_opow.1 (principal_add_of_principal_mul ho ho₂.ne') with
(rfl | ⟨a, rfl⟩)
· exact (Ordinal.not_lt_zero 2 ho₂).elim
rcases principal_add_iff_zero_or_omega_opow.1
(pri... |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.... | Mathlib/Algebra/Group/Basic.lean | 521 | 522 | theorem eq_one_div_of_mul_eq_one_right (h : a * b = 1) : b = 1 / a := by |
rw [eq_inv_of_mul_eq_one_right h, one_div]
|
/-
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.Order.Lattice
import Mathlib.Order.ULift
import Mathlib.Tactic.PushNeg
#align_import order.bounded_order from "leanprover-community/mathlib"@"70d50ecf... | Mathlib/Order/BoundedOrder.lean | 393 | 398 | theorem OrderBot.ext_bot {α} {hA : PartialOrder α} (A : OrderBot α) {hB : PartialOrder α}
(B : OrderBot α) (H : ∀ x y : α, (haveI := hA; x ≤ y) ↔ x ≤ y) :
(@Bot.bot α (@OrderBot.toBot α hA.toLE A)) = (@Bot.bot α (@OrderBot.toBot α hB.toLE B)) := by |
cases PartialOrder.ext H
apply bot_unique
exact @bot_le _ _ A _
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Init.Order.LinearOrder
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Subtype
import Mathlib.Tactic.Spread
import Mathlib.Tactic.Conv... | Mathlib/Order/Basic.lean | 641 | 644 | theorem Preorder.ext {A B : Preorder α}
(H : ∀ x y : α, (haveI := A; x ≤ y) ↔ x ≤ y) : A = B := by |
ext x y
exact H x y
|
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import number_theory.ramification_inertia from "leanprover-community/mathlib"@"039a089d2a4b93c761b234f3e5f5aeb752bac6... | Mathlib/NumberTheory/RamificationInertia.lean | 568 | 595 | theorem quotientToQuotientRangePowQuotSucc_surjective [IsDedekindDomain S]
(hP0 : P ≠ ⊥) [hP : P.IsPrime] {i : ℕ} (hi : i < e) {a : S} (a_mem : a ∈ P ^ i)
(a_not_mem : a ∉ P ^ (i + 1)) :
Function.Surjective (quotientToQuotientRangePowQuotSucc f p P a_mem) := by |
rintro ⟨⟨⟨x⟩, hx⟩⟩
have Pe_le_Pi : P ^ e ≤ P ^ i := Ideal.pow_le_pow_right hi.le
rw [Submodule.Quotient.quot_mk_eq_mk, Ideal.Quotient.mk_eq_mk, Ideal.mem_quotient_iff_mem_sup,
sup_eq_left.mpr Pe_le_Pi] at hx
suffices hx' : x ∈ Ideal.span {a} ⊔ P ^ (i + 1) by
obtain ⟨y', hy', z, hz, rfl⟩ := Submodule.me... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau, Scott Morrison
-/
import Mathlib.Data.List.Chain
import Mathlib.Data.List.Enum
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Pairwise
import Mathli... | Mathlib/Data/List/Range.lean | 182 | 183 | theorem unzip_enum_eq_prod (l : List α) : l.enum.unzip = (range l.length, l) := by |
simp only [enum_eq_zip_range, unzip_zip, length_range]
|
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.Category.GroupCat.EquivalenceGroupAddGroup
import Mathlib.GroupTheory.QuotientGroup
#align_import algebra.category.Group.epi_mono from "leanprover... | Mathlib/Algebra/Category/GroupCat/EpiMono.lean | 288 | 308 | theorem agree : f.range = { x | h x = g x } := by |
refine Set.ext fun b => ⟨?_, fun hb : h b = g b => by_contradiction fun r => ?_⟩
· rintro ⟨a, rfl⟩
change h (f a) = g (f a)
ext ⟨⟨_, ⟨y, rfl⟩⟩⟩
· rw [g_apply_fromCoset]
by_cases m : y ∈ f.range
· rw [h_apply_fromCoset' _ _ _ m, fromCoset_eq_of_mem_range _ m]
change fromCoset _ = fro... |
/-
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, Patrick Massot
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Data.Set.Function
import Mathlib.Order.In... | Mathlib/Algebra/Order/Interval/Set/Monoid.lean | 58 | 62 | theorem Ioc_add_bij : BijOn (· + d) (Ioc a b) (Ioc (a + d) (b + d)) := by |
rw [← Ioi_inter_Iic, ← Ioi_inter_Iic]
exact
(Ioi_add_bij a d).inter_mapsTo (fun x hx => add_le_add_right hx _) fun x hx =>
le_of_add_le_add_right hx.2
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.