Context stringlengths 285 157k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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/-
Copyright (c) 2022 Bhavik Mehta, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Alena Gusakov, Yaël Dillies
-/
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Field.Rat
import Mathlib.Algebra.Order.Field.Basic
import Mathl... | Mathlib/Combinatorics/SetFamily/LYM.lean | 228 | 245 | theorem IsAntichain.sperner [Fintype α] {𝒜 : Finset (Finset α)}
(h𝒜 : IsAntichain (· ⊆ ·) (𝒜 : Set (Finset α))) :
𝒜.card ≤ (Fintype.card α).choose (Fintype.card α / 2) := by |
classical
suffices (∑ r ∈ Iic (Fintype.card α),
((𝒜 # r).card : ℚ) / (Fintype.card α).choose (Fintype.card α / 2)) ≤ 1 by
rw [← sum_div, ← Nat.cast_sum, div_le_one] at this
· simp only [cast_le] at this
rwa [sum_card_slice] at this
simp only [cast_pos]
exact choose_pos (N... |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Topology.Order.Basic
#align_import analysis.convex.strict from "leanprove... | Mathlib/Analysis/Convex/Strict.lean | 358 | 362 | theorem StrictConvex.add_smul_sub_mem (h : StrictConvex 𝕜 s) (hx : x ∈ s) (hy : y ∈ s) (hxy : x ≠ y)
{t : 𝕜} (ht₀ : 0 < t) (ht₁ : t < 1) : x + t • (y - x) ∈ interior s := by |
apply h.openSegment_subset hx hy hxy
rw [openSegment_eq_image']
exact mem_image_of_mem _ ⟨ht₀, ht₁⟩
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov, Kexing Ying
-/
import Mathlib.Topology.Semicontinuous
import Mathlib.MeasureTheory.Function.AEMeasurableSequence
import Mathlib.MeasureTheory.Order.Lat... | Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean | 773 | 778 | theorem Measurable.iSup_Prop {α} [MeasurableSpace α] [ConditionallyCompleteLattice α]
(p : Prop) {f : δ → α} (hf : Measurable f) : Measurable fun b => ⨆ _ : p, f b := by |
simp_rw [ciSup_eq_ite]
split_ifs with h
· exact hf
· exact measurable_const
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
#align_import analysis.normed_space.multilinear from "leanprover-community/mathlib"@"f40476639bac089693a4... | Mathlib/Analysis/NormedSpace/Multilinear/Curry.lean | 155 | 160 | theorem ContinuousLinearMap.curry_uncurryLeft
(f : Ei 0 →L[𝕜] ContinuousMultilinearMap 𝕜 (fun i : Fin n => Ei i.succ) G) :
f.uncurryLeft.curryLeft = f := by |
ext m x
rw [ContinuousMultilinearMap.curryLeft_apply, ContinuousLinearMap.uncurryLeft_apply, tail_cons,
cons_zero]
|
/-
Copyright (c) 2021 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson
-/
import Mathlib.Algebra.Field.Opposite
import Mathlib.Algebra.Group.Subgroup.ZPowers
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Rin... | Mathlib/Algebra/Periodic.lean | 441 | 443 | theorem Antiperiodic.neg_eq [AddGroup α] [InvolutiveNeg β] (h : Antiperiodic f c) :
f (-c) = -f 0 := by |
simpa only [zero_add] using h.neg 0
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Topology.Constructions
import Mathlib.Topology.ContinuousOn
#align_import topology.bases from "leanprover-community/mathlib"@"bcfa7268... | Mathlib/Topology/Bases.lean | 914 | 920 | theorem IsTopologicalBasis.sigma {s : ∀ i : ι, Set (Set (E i))}
(hs : ∀ i, IsTopologicalBasis (s i)) :
IsTopologicalBasis (⋃ i : ι, (fun u => (Sigma.mk i '' u : Set (Σi, E i))) '' s i) := by |
refine .of_hasBasis_nhds fun a ↦ ?_
rw [Sigma.nhds_eq]
convert (((hs a.1).nhds_hasBasis).map _).to_image_id
aesop
|
/-
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.Data.Set.Lattice
#align_import data.nat.pairing from "leanprover-community/mathlib"@"207cf... | Mathlib/Data/Nat/Pairing.lean | 122 | 123 | theorem unpair_right_le (n : ℕ) : (unpair n).2 ≤ n := by |
simpa using right_le_pair n.unpair.1 n.unpair.2
|
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Scott Morrison
-/
import Mathlib.Data.Opposite
import Mathlib.Tactic.Cases
#align_import combinatorics.quiver.basic from "leanprover-community/mathlib"@"56adee5b5eef9e734d8227... | Mathlib/Combinatorics/Quiver/Basic.lean | 76 | 87 | theorem ext {V : Type u} [Quiver.{v₁} V] {W : Type u₂} [Quiver.{v₂} W] {F G : Prefunctor V W}
(h_obj : ∀ X, F.obj X = G.obj X)
(h_map : ∀ (X Y : V) (f : X ⟶ Y),
F.map f = Eq.recOn (h_obj Y).symm (Eq.recOn (h_obj X).symm (G.map f))) : F = G := by |
cases' F with F_obj _
cases' G with G_obj _
obtain rfl : F_obj = G_obj := by
ext X
apply h_obj
congr
funext X Y f
simpa using h_map X Y f
|
/-
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 | 362 | 364 | theorem fromSpec_map_basicOpen :
hU.fromSpec ⁻¹ᵁ X.basicOpen f = PrimeSpectrum.basicOpen f := by |
rw [fromSpec_map_basicOpen', ← basicOpen_eq_of_affine, NatIso.app_inv]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.Order.IsLUB
/-!
# Order topology on a densely ordered set
-/
open Set Filter TopologicalSpace Topology Func... | Mathlib/Topology/Order/DenselyOrdered.lean | 390 | 392 | theorem tendsto_Iio_atTop {f : β → Iio a} :
Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] a) := by |
rw [← comap_coe_Iio_nhdsWithin_Iio, tendsto_comap_iff]; rfl
|
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Analysis.InnerProductSpace.... | Mathlib/Analysis/Fourier/AddCircle.lean | 408 | 415 | theorem fourierBasis_repr (f : Lp ℂ 2 <| @haarAddCircle T hT) (i : ℤ) :
fourierBasis.repr f i = fourierCoeff f i := by |
trans ∫ t : AddCircle T, conj ((@fourierLp T hT 2 _ i : AddCircle T → ℂ) t) * f t ∂haarAddCircle
· rw [fourierBasis.repr_apply_apply f i, MeasureTheory.L2.inner_def, coe_fourierBasis]
simp only [RCLike.inner_apply]
· apply integral_congr_ae
filter_upwards [coeFn_fourierLp 2 i] with _ ht
rw [ht, ← fou... |
/-
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 | 103 | 104 | theorem setIntegral_congr_set_ae (hst : s =ᵐ[μ] t) : ∫ x in s, f x ∂μ = ∫ x in t, f x ∂μ := by |
rw [Measure.restrict_congr_set hst]
|
/-
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 | 136 | 141 | theorem leval_coe_eq_smeval {R : Type*} [Semiring R] (r : R) :
⇑(leval r) = fun p => p.smeval r := by |
rw [Function.funext_iff]
intro
rw [leval_apply, smeval_def, eval_eq_sum]
rfl
|
/-
Copyright (c) 2022 Moritz Firsching. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Firsching, Fabian Kruse, Nikolas Kuhn
-/
import Mathlib.Analysis.PSeries
import Mathlib.Data.Real.Pi.Wallis
import Mathlib.Tactic.AdaptationNote
#align_import analysis.specia... | Mathlib/Analysis/SpecialFunctions/Stirling.lean | 170 | 172 | theorem log_stirlingSeq_bounded_by_constant : ∃ c, ∀ n : ℕ, c ≤ log (stirlingSeq (n + 1)) := by |
obtain ⟨d, h⟩ := log_stirlingSeq_bounded_aux
exact ⟨log (stirlingSeq 1) - d, fun n => sub_le_comm.mp (h n)⟩
|
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Gabin Kolly
-/
import Mathlib.Order.Closure
import Mathlib.ModelTheory.Semantics
import Mathlib.ModelTheory.Encoding
#align_import model_theory.substructures from "lea... | Mathlib/ModelTheory/Substructures.lean | 781 | 785 | theorem subset_closure_withConstants : A ⊆ closure (L[[A]]) s := by |
intro a ha
simp only [SetLike.mem_coe]
let a' : L[[A]].Constants := Sum.inr ⟨a, ha⟩
exact constants_mem a'
|
/-
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.SpecialFunctions.Gamma.Beta
import Mathlib.NumberTheory.LSeries.HurwitzZeta
import Mathlib.Analysis.Complex.RemovableSingularity
import Mathli... | Mathlib/NumberTheory/LSeries/RiemannZeta.lean | 149 | 150 | theorem riemannZeta_zero : riemannZeta 0 = -1 / 2 := by |
simp_rw [riemannZeta, hurwitzZetaEven, Function.update_same, if_true]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Vector.Basic
import Mathlib.Data.PFun
import Ma... | Mathlib/Computability/TuringMachine.lean | 1,538 | 1,580 | theorem tr_supports {S : Finset Λ} (ss : TM1.Supports M S) :
TM0.Supports (tr M) ↑(trStmts M S) := by |
constructor
· apply Finset.mem_product.2
constructor
· simp only [default, TM1.stmts, Finset.mem_insertNone, Option.mem_def, Option.some_inj,
forall_eq', Finset.mem_biUnion]
exact ⟨_, ss.1, TM1.stmts₁_self⟩
· apply Finset.mem_univ
· intro q a q' s h₁ h₂
rcases q with ⟨_ | q, v⟩; · 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, Patrick Massot
-/
import Mathlib.Topology.Maps
import Mathlib.Topology.NhdsSet
#align_import topology.constructions from "leanprover-community/mathlib"... | Mathlib/Topology/Constructions.lean | 306 | 313 | theorem nhds_eq (x : CofiniteTopology X) : 𝓝 x = pure x ⊔ cofinite := by |
ext U
rw [mem_nhds_iff]
constructor
· rintro ⟨V, hVU, V_op, haV⟩
exact mem_sup.mpr ⟨hVU haV, mem_of_superset (V_op ⟨_, haV⟩) hVU⟩
· rintro ⟨hU : x ∈ U, hU' : Uᶜ.Finite⟩
exact ⟨U, Subset.rfl, fun _ => hU', hU⟩
|
/-
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 | 2,251 | 2,252 | theorem inner_map_complex [NormedAddCommGroup G] [InnerProductSpace ℝ G] (f : G ≃ₗᵢ[ℝ] ℂ)
(x y : G) : ⟪x, y⟫_ℝ = (conj (f x) * f y).re := by | rw [← Complex.inner, f.inner_map_map]
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.Perm
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.List
#a... | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 903 | 906 | theorem IsCycleOn.zpow_apply_eq_zpow_apply {s : Finset α} (hf : f.IsCycleOn s) (ha : a ∈ s)
{m n : ℤ} : (f ^ m) a = (f ^ n) a ↔ m ≡ n [ZMOD s.card] := by |
rw [Int.modEq_iff_dvd, ← hf.zpow_apply_eq ha]
simp [sub_eq_neg_add, zpow_add, eq_inv_iff_eq, eq_comm]
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Scott Morrison, Adam Topaz
-/
import Mathlib.AlgebraicTopology.SimplexCategory
import Mathlib.CategoryTheory.Comma.Arrow
import Mathlib.CategoryTheory.Limits.FunctorCat... | Mathlib/AlgebraicTopology/SimplicialObject.lean | 469 | 471 | theorem eqToIso_refl {n : ℕ} (h : n = n) : X.eqToIso h = Iso.refl _ := by |
ext
simp [eqToIso]
|
/-
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.NormedSpace.Multilinear.Curry
#align_import analysis.calculus.formal_multilinear_series from "leanprover-community/mathlib"@"f2ce608671... | Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean | 328 | 331 | theorem coeff_fslope : p.fslope.coeff n = p.coeff (n + 1) := by |
simp only [fslope, coeff, ContinuousMultilinearMap.curryLeft_apply]
congr 1
exact Fin.cons_self_tail 1
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.bind from "leanprover-community/mathlib"@"f694c7dea... | Mathlib/Data/Multiset/Bind.lean | 306 | 307 | theorem product_singleton : ({a} : Multiset α) ×ˢ ({b} : Multiset β) = {(a, b)} := by |
simp only [SProd.sprod, product, bind_singleton, map_singleton]
|
/-
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 | 278 | 284 | theorem seminormAux_smul_le (k n : ℕ) (c : 𝕜) (f : 𝓢(E, F)) :
(c • f).seminormAux k n ≤ ‖c‖ * f.seminormAux k n := by |
refine
(c • f).seminormAux_le_bound k n (mul_nonneg (norm_nonneg _) (seminormAux_nonneg _ _ _))
fun x => (decay_smul_aux k n f c x).le.trans ?_
rw [mul_assoc]
exact mul_le_mul_of_nonneg_left (f.le_seminormAux k n x) (norm_nonneg _)
|
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Data.Fintype.Units
import Mathlib.GroupTheory.OrderOfElement
#align_import number_theory.legendre_symbol.mul_characte... | Mathlib/NumberTheory/MulChar/Basic.lean | 507 | 514 | theorem IsQuadratic.pow_char {χ : MulChar R R'} (hχ : χ.IsQuadratic) (p : ℕ) [hp : Fact p.Prime]
[CharP R' p] : χ ^ p = χ := by |
ext x
rw [pow_apply_coe]
rcases hχ x with (hx | hx | hx) <;> rw [hx]
· rw [zero_pow (@Fact.out p.Prime).ne_zero]
· rw [one_pow]
· exact CharP.neg_one_pow_char R' p
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Fabian Glöckle, Kyle Miller
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModu... | Mathlib/LinearAlgebra/Dual.lean | 478 | 483 | theorem eval_ker {ι : Type*} (b : Basis ι R M) :
LinearMap.ker (Dual.eval R M) = ⊥ := by |
rw [ker_eq_bot']
intro m hm
simp_rw [LinearMap.ext_iff, Dual.eval_apply, zero_apply] at hm
exact (Basis.forall_coord_eq_zero_iff _).mp fun i => hm (b.coord i)
|
/-
Copyright (c) 2022 Chris Birkbeck. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Birkbeck, David Loeffler
-/
import Mathlib.Algebra.Module.Submodule.Basic
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.Order.Filter.ZeroAndBoundedAtFilter
... | Mathlib/Analysis/Complex/UpperHalfPlane/FunctionsBoundedAtInfty.lean | 71 | 74 | theorem bounded_mem (f : ℍ → ℂ) :
IsBoundedAtImInfty f ↔ ∃ M A : ℝ, ∀ z : ℍ, A ≤ im z → abs (f z) ≤ M := by |
simp [IsBoundedAtImInfty, BoundedAtFilter, Asymptotics.isBigO_iff, Filter.Eventually,
atImInfty_mem]
|
/-
Copyright (c) 2022 Daniel Roca González. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Daniel Roca González
-/
import Mathlib.Analysis.InnerProductSpace.Dual
#align_import analysis.inner_product_space.lax_milgram from "leanprover-community/mathlib"@"46b633fd842bef... | Mathlib/Analysis/InnerProductSpace/LaxMilgram.lean | 87 | 102 | theorem range_eq_top (coercive : IsCoercive B) : range B♯ = ⊤ := by |
haveI := coercive.isClosed_range.completeSpace_coe
rw [← (range B♯).orthogonal_orthogonal]
rw [Submodule.eq_top_iff']
intro v w mem_w_orthogonal
rcases coercive with ⟨C, C_pos, coercivity⟩
obtain rfl : w = 0 := by
rw [← norm_eq_zero, ← mul_self_eq_zero, ← mul_right_inj' C_pos.ne', mul_zero, ←
mul... |
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Analysis.RCLike.Lemmas
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import meas... | Mathlib/MeasureTheory/Function/L2Space.lean | 42 | 43 | theorem Memℒp.integrable_sq {f : α → ℝ} (h : Memℒp f 2 μ) : Integrable (fun x => f x ^ 2) μ := by |
simpa [← memℒp_one_iff_integrable] using h.norm_rpow two_ne_zero ENNReal.two_ne_top
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.ContinuousOn
import Mathlib.Order.Minimal
/-!
# Irreducibility in topological spaces
## Main definitions
* ... | Mathlib/Topology/Irreducible.lean | 282 | 292 | theorem isIrreducible_iff_sUnion_closed :
IsIrreducible s ↔
∀ t : Finset (Set X), (∀ z ∈ t, IsClosed z) → (s ⊆ ⋃₀ ↑t) → ∃ z ∈ t, s ⊆ z := by |
simp only [isIrreducible_iff_sInter]
refine ((@compl_involutive (Set X) _).toPerm _).finsetCongr.forall_congr fun {t} => ?_
simp_rw [Equiv.finsetCongr_apply, Finset.forall_mem_map, Finset.mem_map, Finset.coe_map,
sUnion_image, Equiv.coe_toEmbedding, Function.Involutive.coe_toPerm, isClosed_compl_iff,
exi... |
/-
Copyright (c) 2020 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Analysis.InnerProductSpace.Basic
#align_import analysis.inner_product_space.of_norm from "leanprover-communi... | Mathlib/Analysis/InnerProductSpace/OfNorm.lean | 139 | 161 | theorem inner_.conj_symm (x y : E) : conj (inner_ 𝕜 y x) = inner_ 𝕜 x y := by |
simp only [inner_]
have h4 : conj (4⁻¹ : 𝕜) = 4⁻¹ := by norm_num
rw [map_mul, h4]
congr 1
simp only [map_sub, map_add, algebraMap_eq_ofReal, ← ofReal_mul, conj_ofReal, map_mul, conj_I]
rw [add_comm y x, norm_sub_rev]
by_cases hI : (I : 𝕜) = 0
· simp only [hI, neg_zero, zero_mul]
-- Porting note: th... |
/-
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 | 458 | 460 | theorem toIcoMod_sub_zsmul' (a b : α) (m : ℤ) :
toIcoMod hp (a - m • p) b = toIcoMod hp a b - m • p := by |
simp_rw [sub_eq_add_neg, ← neg_smul, toIcoMod_add_zsmul']
|
/-
Copyright (c) 2024 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Order.Filter.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.SetTheory.Cardinal.Cofinality
/-!
#... | Mathlib/Order/Filter/CardinalInter.lean | 107 | 111 | theorem eventually_cardinal_ball {S : Set ι} (hS : #S < c)
{p : α → ∀ i ∈ S, Prop} :
(∀ᶠ x in l, ∀ i hi, p x i hi) ↔ ∀ i hi, ∀ᶠ x in l, p x i hi := by |
simp only [Filter.Eventually, setOf_forall]
exact cardinal_bInter_mem hS
|
/-
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 | 190 | 191 | theorem eqOn_univ (f₁ f₂ : α → β) : EqOn f₁ f₂ univ ↔ f₁ = f₂ := by |
simp [EqOn, funext_iff]
|
/-
Copyright (c) 2019 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov
-/
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Basis
#align_import analysis.con... | Mathlib/Analysis/Convex/Combination.lean | 329 | 350 | theorem convexHull_eq (s : Set E) : convexHull R s =
{ x : E | ∃ (ι : Type) (t : Finset ι) (w : ι → R) (z : ι → E), (∀ i ∈ t, 0 ≤ w i) ∧
∑ i ∈ t, w i = 1 ∧ (∀ i ∈ t, z i ∈ s) ∧ t.centerMass w z = x } := by |
refine Subset.antisymm (convexHull_min ?_ ?_) ?_
· intro x hx
use PUnit, {PUnit.unit}, fun _ => 1, fun _ => x, fun _ _ => zero_le_one, sum_singleton _ _,
fun _ _ => hx
simp only [Finset.centerMass, Finset.sum_singleton, inv_one, one_smul]
· rintro x ⟨ι, sx, wx, zx, hwx₀, hwx₁, hzx, rfl⟩ y ⟨ι', sy, ... |
/-
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 | 533 | 539 | theorem rootMultiplicity_eq_nat_find_of_nonzero [DecidableEq R] {p : R[X]} (p0 : p ≠ 0) {a : R} :
letI : DecidablePred fun n : ℕ => ¬(X - C a) ^ (n + 1) ∣ p := fun n =>
@Not.decidable _ (decidableDvdMonic p ((monic_X_sub_C a).pow (n + 1)))
rootMultiplicity a p = Nat.find (multiplicity_X_sub_C_finite a p0)... |
dsimp [rootMultiplicity]
cases Subsingleton.elim ‹DecidableEq R› (Classical.decEq R)
rw [dif_neg p0]
|
/-
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, Sander Dahmen, Scott Morrison, Chris Hughes, Anne Baanen
-/
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_im... | Mathlib/LinearAlgebra/Dimension/Constructions.lean | 241 | 246 | theorem finrank_directSum {ι : Type v} [Fintype ι] (M : ι → Type w) [∀ i : ι, AddCommGroup (M i)]
[∀ i : ι, Module R (M i)] [∀ i : ι, Module.Free R (M i)] [∀ i : ι, Module.Finite R (M i)] :
finrank R (⨁ i, M i) = ∑ i, finrank R (M i) := by |
letI := nontrivial_of_invariantBasisNumber R
simp only [finrank, fun i => rank_eq_card_chooseBasisIndex R (M i), rank_directSum, ← mk_sigma,
mk_toNat_eq_card, card_sigma]
|
/-
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.Data.Set.Finite
#align_import data.finset.preimage from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
/-!... | Mathlib/Data/Finset/Preimage.lean | 119 | 123 | theorem sigma_preimage_mk {β : α → Type*} [DecidableEq α] (s : Finset (Σa, β a)) (t : Finset α) :
(t.sigma fun a => s.preimage (Sigma.mk a) sigma_mk_injective.injOn) =
s.filter fun a => a.1 ∈ t := by |
ext x
simp [and_comm]
|
/-
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.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction... | Mathlib/Algebra/Polynomial/Derivative.lean | 97 | 99 | theorem derivative_C_mul_X_pow (a : R) (n : ℕ) :
derivative (C a * X ^ n) = C (a * n) * X ^ (n - 1) := by |
rw [C_mul_X_pow_eq_monomial, C_mul_X_pow_eq_monomial, derivative_monomial]
|
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Floris van Doorn
-/
import Mathlib.Order.Filter.AtTopBot
import Mathlib.Order.Filter.Subsingleton
/-!
# Functions that are eventually constant along a filter
In this... | Mathlib/Order/Filter/EventuallyConst.lean | 73 | 75 | theorem EventuallyEq.eventuallyConst_iff {g : α → β} (h : f =ᶠ[l] g) :
EventuallyConst f l ↔ EventuallyConst g l := by |
simp only [EventuallyConst, map_congr h]
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
#align_import measure_theory.integral.set_to_l1 from "leanprov... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 379 | 388 | theorem setToSimpleFunc_congr (T : Set α → E →L[ℝ] F)
(h_zero : ∀ s, MeasurableSet s → μ s = 0 → T s = 0) (h_add : FinMeasAdditive μ T) {f g : α →ₛ E}
(hf : Integrable f μ) (h : f =ᵐ[μ] g) : f.setToSimpleFunc T = g.setToSimpleFunc T := by |
refine setToSimpleFunc_congr' T h_add hf ((integrable_congr h).mp hf) ?_
refine fun x y hxy => h_zero _ ((measurableSet_fiber f x).inter (measurableSet_fiber g y)) ?_
rw [EventuallyEq, ae_iff] at h
refine measure_mono_null (fun z => ?_) h
simp_rw [Set.mem_inter_iff, Set.mem_setOf_eq, Set.mem_preimage, Set.me... |
/-
Copyright (c) 2022 George Peter Banyard, Yaël Dillies, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: George Peter Banyard, Yaël Dillies, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Connectivity
#align_import combinatorics.simple_graph.prod... | Mathlib/Combinatorics/SimpleGraph/Prod.lean | 233 | 241 | theorem boxProd_neighborFinset (x : α × β)
[Fintype (G.neighborSet x.1)] [Fintype (H.neighborSet x.2)] [Fintype ((G □ H).neighborSet x)] :
(G □ H).neighborFinset x =
(G.neighborFinset x.1 ×ˢ {x.2}).disjUnion ({x.1} ×ˢ H.neighborFinset x.2)
(Finset.disjoint_product.mpr <| Or.inl <| neighborFinset_d... |
-- swap out the fintype instance for the canonical one
letI : Fintype ((G □ H).neighborSet x) := SimpleGraph.boxProdFintypeNeighborSet _
convert_to (G □ H).neighborFinset x = _ using 2
exact Eq.trans (Finset.map_map _ _ _) Finset.attach_map_val
|
/-
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.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Regular.Pow
import Mathl... | Mathlib/Algebra/MvPolynomial/Basic.lean | 326 | 327 | theorem X_pow_eq_monomial : X n ^ e = monomial (Finsupp.single n e) (1 : R) := by |
simp [X, monomial_pow]
|
/-
Copyright (c) 2021 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno
-/
import Mathlib.CategoryTheory.NatIso
#align_import category_theory.bicategory.basic from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
/-!
# B... | Mathlib/CategoryTheory/Bicategory/Basic.lean | 422 | 426 | theorem leftUnitor_whiskerRight (f : a ⟶ b) (g : b ⟶ c) :
(λ_ f).hom ▷ g = (α_ (𝟙 a) f g).hom ≫ (λ_ (f ≫ g)).hom := by |
rw [← whiskerLeft_iff, whiskerLeft_comp, ← cancel_epi (α_ _ _ _).hom, ←
cancel_epi ((α_ _ _ _).hom ▷ _), pentagon_assoc, triangle, ← associator_naturality_middle, ←
comp_whiskerRight_assoc, triangle, associator_naturality_left]
|
/-
Copyright (c) 2019 Abhimanyu Pallavi Sudhir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Abhimanyu Pallavi Sudhir
-/
import Mathlib.Order.Filter.FilterProduct
import Mathlib.Analysis.SpecificLimits.Basic
#align_import data.real.hyperreal from "leanprover-communi... | Mathlib/Data/Real/Hyperreal.lean | 260 | 262 | theorem isSt_iff_tendsto {x : ℝ*} {r : ℝ} : IsSt x r ↔ x.Tendsto (𝓝 r) := by |
rcases ofSeq_surjective x with ⟨f, rfl⟩
exact isSt_ofSeq_iff_tendsto
|
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Data.List.Basic
/-!
# insertNth
Proves various lemmas about `List.insertNth`.... | Mathlib/Data/List/InsertNth.lean | 138 | 148 | theorem get_insertNth_of_lt (l : List α) (x : α) (n k : ℕ) (hn : k < n) (hk : k < l.length)
(hk' : k < (insertNth n x l).length := hk.trans_le (length_le_length_insertNth _ _ _)) :
(insertNth n x l).get ⟨k, hk'⟩ = l.get ⟨k, hk⟩ := by |
induction' n with n IH generalizing k l
· simp at hn
· cases' l with hd tl
· simp
· cases k
· simp [get]
· rw [Nat.succ_lt_succ_iff] at hn
simpa using IH _ _ hn _
|
/-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Applicative
import Mathlib.Control.Traversable.Basic
#align_import control.traversable.lemmas from "leanprover-community/mathlib"@"3342d1b2178381196... | Mathlib/Control/Traversable/Lemmas.lean | 116 | 119 | theorem traverse_eq_map_id' (f : β → γ) :
traverse (m := Id) (pure ∘ f) = pure ∘ (map f : t β → t γ) := by |
ext
exact traverse_eq_map_id _ _
|
/-
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.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
i... | Mathlib/Data/Set/Pointwise/Interval.lean | 51 | 53 | theorem Iic_mul_Iic_subset' (a b : α) : Iic a * Iic b ⊆ Iic (a * b) := by |
rintro x ⟨y, hya, z, hzb, rfl⟩
exact mul_le_mul' hya hzb
|
/-
Copyright (c) 2021 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 711 | 713 | theorem darts_concat {u v w : V} (p : G.Walk u v) (h : G.Adj v w) :
(p.concat h).darts = p.darts.concat ⟨(v, w), h⟩ := by |
induction p <;> simp [*, concat_nil]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yury Kudryashov
-/
import Mathlib.Algebra.Algebra.Operations
#align_import algebra.algebra.subalgebra.basic from "leanprover-community/mathlib"@"b915e9392ecb2a861e1e766f0e1df6ac... | Mathlib/Algebra/Algebra/Subalgebra/Basic.lean | 1,085 | 1,086 | theorem range_comp_val : (f.comp S.val).range = S.map f := by |
rw [AlgHom.range_comp, range_val]
|
/-
Copyright (c) 2021 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Sites.Plus
import Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory
#align_import category_theory.sites.sheafification from "leanprover-com... | Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean | 130 | 134 | theorem equiv_symm_eq_apply {X : C} {P : Cᵒᵖ ⥤ D} {S : J.Cover X} [HasMultiequalizer (S.index P)]
(x : Meq P S) (I : S.Arrow) :
Multiequalizer.ι (S.index P) I ((Meq.equiv P S).symm x) = x I := by |
rw [← equiv_apply]
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, Patrick Massot, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Order.MinMax
import Mathlib.Data.Set.Subsingleton
import Mathlib.Tactic.Says
#align_imp... | Mathlib/Order/Interval/Set/Basic.lean | 850 | 851 | theorem Icc_diff_Ico_same (h : a ≤ b) : Icc a b \ Ico a b = {b} := by |
rw [← Icc_diff_right, diff_diff_cancel_left (singleton_subset_iff.2 <| right_mem_Icc.2 h)]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.RingTheory.EuclideanDo... | Mathlib/Algebra/Polynomial/FieldDivision.lean | 259 | 261 | theorem leadingCoeff_map [Semiring S] [Nontrivial S] (f : R →+* S) :
leadingCoeff (p.map f) = f (leadingCoeff p) := by |
simp only [← coeff_natDegree, coeff_map f, natDegree_map]
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.GroupWithZero.Basic
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Contrapose
import Mathlib.Tactic.N... | Mathlib/Algebra/GroupWithZero/Units/Basic.lean | 152 | 154 | theorem inverse_zero : inverse (0 : M₀) = 0 := by |
nontriviality
exact inverse_non_unit _ not_isUnit_zero
|
/-
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 | 168 | 169 | theorem exp_eq_exp_iff_exp_sub_eq_one {x y : ℂ} : exp x = exp y ↔ exp (x - y) = 1 := by |
rw [exp_sub, div_eq_one_iff_eq (exp_ne_zero _)]
|
/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stephen Morgan, Scott Morrison
-/
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Functor.Const
import Mathlib.CategoryTheory.Opposites
import Mathlib.Data.Prod.Bas... | Mathlib/CategoryTheory/Products/Basic.lean | 64 | 75 | theorem isIso_prod_iff {P Q : C} {S T : D} {f : (P, S) ⟶ (Q, T)} :
IsIso f ↔ IsIso f.1 ∧ IsIso f.2 := by |
constructor
· rintro ⟨g, hfg, hgf⟩
simp? at hfg hgf says simp only [prod_Hom, prod_comp, prod_id, Prod.mk.injEq] at hfg hgf
rcases hfg with ⟨hfg₁, hfg₂⟩
rcases hgf with ⟨hgf₁, hgf₂⟩
exact ⟨⟨⟨g.1, hfg₁, hgf₁⟩⟩, ⟨⟨g.2, hfg₂, hgf₂⟩⟩⟩
· rintro ⟨⟨g₁, hfg₁, hgf₁⟩, ⟨g₂, hfg₂, hgf₂⟩⟩
dsimp at hfg₁ hg... |
/-
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 | 1,080 | 1,081 | theorem edist_eq_coe_nnnorm_div (a b : E) : edist a b = ‖a / b‖₊ := by |
rw [edist_dist, dist_eq_norm_div, ofReal_norm_eq_coe_nnnorm']
|
/-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
#align_import combinatorics.simple_graph.subg... | Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 721 | 724 | theorem inclusion.injective {x y : Subgraph G} (h : x ≤ y) : Function.Injective (inclusion h) := by |
intro v w h
rw [inclusion, DFunLike.coe, Subtype.mk_eq_mk] at h
exact Subtype.ext h
|
/-
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 | 271 | 273 | theorem weightedVSub_apply_const (w : ι → k) (p : P) (h : ∑ i ∈ s, w i = 0) :
s.weightedVSub (fun _ => p) w = 0 := by |
rw [weightedVSub, weightedVSubOfPoint_apply_const, h, zero_smul]
|
/-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri, Sebastien Gouezel, Heather Macbeth, Patrick Massot, Floris van Doorn
-/
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
import Mathlib.Topology.FiberBundle.Ba... | Mathlib/Topology/VectorBundle/Basic.lean | 792 | 796 | theorem localTriv_continuousLinearMapAt {b : B} (hb : b ∈ Z.baseSet i) :
(Z.localTriv i).continuousLinearMapAt R b = Z.coordChange (Z.indexAt b) i b := by |
ext1 v
rw [(Z.localTriv i).continuousLinearMapAt_apply R, (Z.localTriv i).coe_linearMapAt_of_mem]
exacts [rfl, hb]
|
/-
Copyright (c) 2022 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.Transfer
#align_import group_theory.schreier from "leanpro... | Mathlib/GroupTheory/Schreier.lean | 150 | 178 | theorem card_commutator_dvd_index_center_pow [Finite (commutatorSet G)] :
Nat.card (_root_.commutator G) ∣
(center G).index ^ ((center G).index * Nat.card (commutatorSet G) + 1) := by |
-- First handle the case when `Z(G)` has infinite index and `[G : Z(G)]` is defined to be `0`
by_cases hG : (center G).index = 0
· simp_rw [hG, zero_mul, zero_add, pow_one, dvd_zero]
haveI : FiniteIndex (center G) := ⟨hG⟩
-- Rewrite as `|Z(G) ∩ G'| * [G' : Z(G) ∩ G'] ∣ [G : Z(G)] ^ ([G : Z(G)] * n) * [G : Z(... |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
/-!
# Neig... | Mathlib/Topology/ContinuousOn.lean | 996 | 1,004 | theorem Set.LeftInvOn.map_nhdsWithin_eq {f : α → β} {g : β → α} {x : β} {s : Set β}
(h : LeftInvOn f g s) (hx : f (g x) = x) (hf : ContinuousWithinAt f (g '' s) (g x))
(hg : ContinuousWithinAt g s x) : map g (𝓝[s] x) = 𝓝[g '' s] g x := by |
apply le_antisymm
· exact hg.tendsto_nhdsWithin (mapsTo_image _ _)
· have A : g ∘ f =ᶠ[𝓝[g '' s] g x] id :=
h.rightInvOn_image.eqOn.eventuallyEq_of_mem self_mem_nhdsWithin
refine le_map_of_right_inverse A ?_
simpa only [hx] using hf.tendsto_nhdsWithin (h.mapsTo (surjOn_image _ _))
|
/-
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, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.FDeriv.Measurable
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calc... | Mathlib/MeasureTheory/Integral/FundThmCalculus.lean | 273 | 288 | theorem measure_integral_sub_linear_isLittleO_of_tendsto_ae' [IsMeasurablyGenerated l']
[TendstoIxxClass Ioc l l'] (hfm : StronglyMeasurableAtFilter f l' μ)
(hf : Tendsto f (l' ⊓ ae μ) (𝓝 c)) (hl : μ.FiniteAtFilter l') (hu : Tendsto u lt l)
(hv : Tendsto v lt l) :
(fun t => (∫ x in u t..v t, f x ∂μ) - ... |
by_cases hE : CompleteSpace E; swap
· simp [intervalIntegral, integral, hE]
have A := hf.integral_sub_linear_isLittleO_ae hfm hl (hu.Ioc hv)
have B := hf.integral_sub_linear_isLittleO_ae hfm hl (hv.Ioc hu)
simp_rw [integral_const', sub_smul]
refine ((A.trans_le fun t ↦ ?_).sub (B.trans_le fun t ↦ ?_)).cong... |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Nat.Dist
import Mathlib.Data.Ordmap.Ordnode
import Mathlib.Tactic.Abel
imp... | Mathlib/Data/Ordmap/Ordset.lean | 508 | 510 | theorem all_node4L {P l x m y r} :
@All α P (node4L l x m y r) ↔ All P l ∧ P x ∧ All P m ∧ P y ∧ All P r := by |
cases m <;> simp [node4L, all_node', All, all_node3L, and_assoc]
|
/-
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.ContDiff.RCLike
import Mathlib.MeasureTheory.Measure.Hausdorff
#align_import topology.metric_space.hausdorff_dimension from "leanp... | Mathlib/Topology/MetricSpace/HausdorffDimension.lean | 432 | 438 | theorem dimH_preimage_le (hf : AntilipschitzWith K f) (s : Set Y) : dimH (f ⁻¹' s) ≤ dimH s := by |
borelize X Y
refine dimH_le fun d hd => le_dimH_of_hausdorffMeasure_eq_top ?_
have := hf.hausdorffMeasure_preimage_le d.coe_nonneg s
rw [hd, top_le_iff] at this
contrapose! this
exact ENNReal.mul_ne_top (by simp) this
|
/-
Copyright (c) 2022 Rishikesh Vaishnav. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rishikesh Vaishnav
-/
import Mathlib.MeasureTheory.Measure.Typeclasses
#align_import probability.conditional_probability from "leanprover-community/mathlib"@"70fd9563a21e7b963887c... | Mathlib/Probability/ConditionalProbability.lean | 199 | 203 | theorem cond_add_cond_compl_eq [IsFiniteMeasure μ] (hms : MeasurableSet s) :
μ[t|s] * μ s + μ[t|sᶜ] * μ sᶜ = μ t := by |
rw [cond_mul_eq_inter μ hms, cond_mul_eq_inter μ hms.compl, Set.inter_comm _ t,
Set.inter_comm _ t]
exact measure_inter_add_diff t hms
|
/-
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.NumberTheory.ModularForms.JacobiTheta.TwoVariable
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
#align_import number_theory.modular_forms.jacob... | Mathlib/NumberTheory/ModularForms/JacobiTheta/OneVariable.lean | 120 | 131 | theorem isBigO_at_im_infty_jacobiTheta_sub_one :
(fun τ => jacobiTheta τ - 1) =O[comap im atTop] fun τ => rexp (-π * τ.im) := by |
simp_rw [IsBigO, IsBigOWith, Filter.eventually_comap, Filter.eventually_atTop]
refine ⟨2 / (1 - rexp (-π)), 1, fun y hy τ hτ =>
(norm_jacobiTheta_sub_one_le (hτ.symm ▸ zero_lt_one.trans_le hy : 0 < im τ)).trans ?_⟩
rw [Real.norm_eq_abs, Real.abs_exp]
refine mul_le_mul_of_nonneg_right ?_ (exp_pos _).le
rw... |
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Isabel Longbottom, Scott Morrison
-/
import Mathlib.Algebra.Order.ZeroLEOne
import Mathlib.Data.List.InsertNth
import Mathlib.Logic.Relation
import Mathlib... | Mathlib/SetTheory/Game/PGame.lean | 1,441 | 1,441 | theorem zero_le_neg_iff {x : PGame} : 0 ≤ -x ↔ x ≤ 0 := by | rw [le_neg_iff, neg_zero]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad, Yury Kudryashov, Patrick Massot
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.... | Mathlib/Order/Filter/AtTopBot.lean | 265 | 268 | theorem Tendsto.eventually_forall_ge_atTop {α β : Type*} [Preorder β] {l : Filter α}
{p : β → Prop} {f : α → β} (hf : Tendsto f l atTop) (h_evtl : ∀ᶠ x in atTop, p x) :
∀ᶠ x in l, ∀ y, f x ≤ y → p y := by |
rw [← Filter.eventually_forall_ge_atTop] at h_evtl; exact (h_evtl.comap f).filter_mono hf.le_comap
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.MvPolynomial.Degrees
import ... | Mathlib/Algebra/MvPolynomial/Equiv.lean | 351 | 355 | theorem finSuccEquiv_apply (p : MvPolynomial (Fin (n + 1)) R) :
finSuccEquiv R n p =
eval₂Hom (Polynomial.C.comp (C : R →+* MvPolynomial (Fin n) R))
(fun i : Fin (n + 1) => Fin.cases Polynomial.X (fun k => Polynomial.C (X k)) i) p := by |
rw [← finSuccEquiv_eq, RingHom.coe_coe]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Topology.Maps
import Mathlib.Topology.NhdsSet
#align_import topology.constructions from "leanprover-community/mathlib"... | Mathlib/Topology/Constructions.lean | 1,587 | 1,588 | theorem isClosed_sigma_iff {s : Set (Sigma σ)} : IsClosed s ↔ ∀ i, IsClosed (Sigma.mk i ⁻¹' s) := by |
simp only [← isOpen_compl_iff, isOpen_sigma_iff, preimage_compl]
|
/-
Copyright (c) 2021 Lu-Ming Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lu-Ming Zhang
-/
import Mathlib.LinearAlgebra.Matrix.Trace
#align_import data.matrix.hadamard from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
/-!
# Hada... | Mathlib/Data/Matrix/Hadamard.lean | 116 | 118 | theorem hadamard_one : M ⊙ (1 : Matrix n n α) = diagonal fun i => M i i := by |
ext i j
by_cases h: i = j <;> simp [h]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Jeremy Avigad, Simon Hudon
-/
import Mathlib.Data.Part
import Mathlib.Data.Rel
#align_import data.pfun from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f70... | Mathlib/Data/PFun.lean | 488 | 491 | theorem core_res (f : α → β) (s : Set α) (t : Set β) : (res f s).core t = sᶜ ∪ f ⁻¹' t := by |
ext x
rw [mem_core_res]
by_cases h : x ∈ s <;> simp [h]
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Manuel Candales
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Geometry.Euclidean.PerpBisector
import Mathlib.Algebra.QuadraticDiscriminant
#align_... | Mathlib/Geometry/Euclidean/Basic.lean | 604 | 607 | theorem dist_reflection (s : AffineSubspace ℝ P) [Nonempty s] [HasOrthogonalProjection s.direction]
(p₁ p₂ : P) : dist p₁ (reflection s p₂) = dist (reflection s p₁) p₂ := by |
conv_lhs => rw [← reflection_reflection s p₁]
exact (reflection s).dist_map _ _
|
/-
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.Fintype.Basic
import Mathlib.Data.Finset.Card
import Mathlib.Data.List.NodupEquivFin
import Mathlib.Data.Set.Image
#align_import data.fintype.car... | Mathlib/Data/Fintype/Card.lean | 305 | 307 | theorem Fintype.card_compl_set [Fintype α] (s : Set α) [Fintype s] [Fintype (↥sᶜ : Sort _)] :
Fintype.card (↥sᶜ : Sort _) = Fintype.card α - Fintype.card s := by |
classical rw [← Set.toFinset_card, ← Set.toFinset_card, ← Finset.card_compl, Set.toFinset_compl]
|
/-
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.GCDMonoid.Multiset
import Mathlib.Combinatorics.Enumerative.Partition
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Cycle.F... | Mathlib/GroupTheory/Perm/Cycle/Type.lean | 518 | 529 | theorem subgroup_eq_top_of_swap_mem [DecidableEq α] {H : Subgroup (Perm α)}
[d : DecidablePred (· ∈ H)] {τ : Perm α} (h0 : (Fintype.card α).Prime)
(h1 : Fintype.card α ∣ Fintype.card H) (h2 : τ ∈ H) (h3 : IsSwap τ) : H = ⊤ := by |
haveI : Fact (Fintype.card α).Prime := ⟨h0⟩
obtain ⟨σ, hσ⟩ := exists_prime_orderOf_dvd_card (Fintype.card α) h1
have hσ1 : orderOf (σ : Perm α) = Fintype.card α := (Subgroup.orderOf_coe σ).trans hσ
have hσ2 : IsCycle ↑σ := isCycle_of_prime_order'' h0 hσ1
have hσ3 : (σ : Perm α).support = ⊤ :=
Finset.eq_u... |
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Justus Springer
-/
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
import Mathlib.AlgebraicGeometry.StructureSheaf
import Mathlib.RingTheory.Localization.Localiz... | Mathlib/AlgebraicGeometry/Spec.lean | 339 | 343 | theorem Spec_Γ_naturality {R S : CommRingCat.{u}} (f : R ⟶ S) :
f ≫ toSpecΓ S = toSpecΓ R ≫ Γ.map (Spec.toLocallyRingedSpace.map f.op).op := by |
-- Porting note: `ext` failed to pick up one of the three lemmas
refine RingHom.ext fun x => Subtype.ext <| funext fun x' => ?_; symm;
apply Localization.localRingHom_to_map
|
/-
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.Algebra.UniformConvergence
#align_import topology.algebra.equicontinuity from "leanprover-community/mathlib"@"01ad394a11bf06b950232720cf7... | Mathlib/Topology/Algebra/Equicontinuity.lean | 36 | 47 | theorem uniformEquicontinuous_of_equicontinuousAt_one {ι G M hom : Type*} [UniformSpace G]
[UniformSpace M] [Group G] [Group M] [UniformGroup G] [UniformGroup M]
[FunLike hom G M] [MonoidHomClass hom G M]
(F : ι → hom) (hf : EquicontinuousAt ((↑) ∘ F) (1 : G)) :
UniformEquicontinuous ((↑) ∘ F) := by |
rw [uniformEquicontinuous_iff_uniformContinuous]
rw [equicontinuousAt_iff_continuousAt] at hf
let φ : G →* (ι →ᵤ M) :=
{ toFun := swap ((↑) ∘ F)
map_one' := by dsimp [UniformFun]; ext; exact map_one _
map_mul' := fun a b => by dsimp [UniformFun]; ext; exact map_mul _ _ _ }
exact uniformContinuo... |
/-
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.Limits.Shapes.KernelPair
import Mathlib.CategoryTheory.Limits.Shapes.CommSq
import Mathlib.CategoryTheory.Adjunction.Over
#align_import categ... | Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean | 266 | 269 | theorem pullbackDiagonalMapIdIso_inv_snd_snd :
(pullbackDiagonalMapIdIso f g i).inv ≫ pullback.snd ≫ pullback.snd = pullback.snd := by |
rw [Iso.inv_comp_eq]
simp
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.CharP.Two
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Grou... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 251 | 258 | theorem map_rootsOfUnity_eq_pow_self [FunLike F R R] [RingHomClass F R R] (σ : F)
(ζ : rootsOfUnity k R) :
∃ m : ℕ, σ (ζ : Rˣ) = ((ζ : Rˣ) : R) ^ m := by |
obtain ⟨m, hm⟩ := MonoidHom.map_cyclic (restrictRootsOfUnity σ k)
rw [← restrictRootsOfUnity_coe_apply, hm, ← zpow_mod_orderOf, ← Int.toNat_of_nonneg
(m.emod_nonneg (Int.natCast_ne_zero.mpr (pos_iff_ne_zero.mp (orderOf_pos ζ)))),
zpow_natCast, rootsOfUnity.coe_pow]
exact ⟨(m % orderOf ζ).toNat, rfl⟩
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic fro... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 881 | 882 | theorem lt_mul_succ_div (a) {b : Ordinal} (h : b ≠ 0) : a < b * succ (a / b) := by |
rw [div_def a h]; exact csInf_mem (div_nonempty h)
|
/-
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.Finset.Prod
import Mathlib.Data.Sym.Basic
import Mathlib.Data.Sym.Sym2.Init
import Mathlib.Data.SetLike.Basic
#align_import data.sym.sym2 from "leanpro... | Mathlib/Data/Sym/Sym2.lean | 180 | 181 | theorem congr_right {a b c : α} : s(a, b) = s(a, c) ↔ b = c := by |
simp (config := {contextual := true})
|
/-
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
-/
import Mathlib.Data.List.Forall2
#align_import data.list.zip from "leanprover-community/mathlib"@"134625f523e737f650a6ea7f0c82a6177e45e622"
/-!
# zip & u... | Mathlib/Data/List/Zip.lean | 148 | 152 | theorem map_prod_left_eq_zip {l : List α} (f : α → β) :
(l.map fun x => (x, f x)) = l.zip (l.map f) := by |
rw [← zip_map']
congr
exact map_id _
|
/-
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.Calculus.InverseFunctionTheorem.Deriv
import Mathlib.Analysis.SpecialFunctions.... | Mathlib/Analysis/SpecialFunctions/Complex/LogDeriv.lean | 90 | 92 | theorem HasStrictDerivAt.clog {f : ℂ → ℂ} {f' x : ℂ} (h₁ : HasStrictDerivAt f f' x)
(h₂ : f x ∈ slitPlane) : HasStrictDerivAt (fun t => log (f t)) (f' / f x) x := by |
rw [div_eq_inv_mul]; exact (hasStrictDerivAt_log h₂).comp x h₁
|
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Alexey Soloyev, Junyan Xu, Kamila Szewczyk
-/
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathl... | Mathlib/Data/Real/GoldenRatio.lean | 57 | 60 | theorem gold_mul_goldConj : φ * ψ = -1 := by |
field_simp
rw [← sq_sub_sq]
norm_num
|
/-
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 | 384 | 390 | theorem HasLimit.isoOfEquivalence_inv_π {F : J ⥤ C} [HasLimit F] {G : K ⥤ C} [HasLimit G]
(e : J ≌ K) (w : e.functor ⋙ G ≅ F) (j : J) :
(HasLimit.isoOfEquivalence e w).inv ≫ limit.π F j =
limit.π G (e.functor.obj j) ≫ w.hom.app j := by |
simp only [HasLimit.isoOfEquivalence, IsLimit.conePointsIsoOfEquivalence_hom]
dsimp
simp
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Homology.Linear
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Tactic.Abel
#align_import algebra.homology.homo... | Mathlib/Algebra/Homology/Homotopy.lean | 624 | 628 | theorem prevD_succ_cochainComplex (f : ∀ i j, P.X i ⟶ Q.X j) (i : ℕ) :
prevD (i + 1) f = f (i + 1) _ ≫ Q.d i (i + 1) := by |
dsimp [prevD]
have : (ComplexShape.up ℕ).prev (i + 1) = i := CochainComplex.prev_nat_succ i
congr 2
|
/-
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 | 93 | 94 | theorem coe_subset : (γ : Set α) ⊆ β := by |
intro _ ⟨_, ⟨⟨⟨_, ha⟩, rfl⟩, _, ⟨_, rfl⟩, _⟩⟩; convert ha
|
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 214 | 215 | theorem eq_neg_self_iff {θ : Angle} : θ = -θ ↔ θ = 0 ∨ θ = π := by |
rw [← add_eq_zero_iff_eq_neg, ← two_nsmul, two_nsmul_eq_zero_iff]
|
/-
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 | 1,315 | 1,325 | theorem isLittleO_principal {s : Set α} : f'' =o[𝓟 s] g' ↔ ∀ x ∈ s, f'' x = 0 := by |
refine ⟨fun h x hx ↦ norm_le_zero_iff.1 ?_, fun h ↦ ?_⟩
· simp only [isLittleO_iff, isBigOWith_principal] at h
have : Tendsto (fun c : ℝ => c * ‖g' x‖) (𝓝[>] 0) (𝓝 0) :=
((continuous_id.mul continuous_const).tendsto' _ _ (zero_mul _)).mono_left
inf_le_left
apply le_of_tendsto_of_tendsto ten... |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Infix
#align_import data.list.sort from "leanprover-community/mathlib"@"f694c7dea... | Mathlib/Data/List/Sort.lean | 333 | 336 | theorem sublist_orderedInsert (x : α) (xs : List α) : xs <+ xs.orderedInsert r x := by |
rw [orderedInsert_eq_take_drop]
refine Sublist.trans ?_ (.append_left (.cons _ (.refl _)) _)
rw [takeWhile_append_dropWhile]
|
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Support
#align_import group_theory.perm.list from "leanprove... | Mathlib/GroupTheory/Perm/List.lean | 286 | 289 | theorem formPerm_eq_of_isRotated {l l' : List α} (hd : Nodup l) (h : l ~r l') :
formPerm l = formPerm l' := by |
obtain ⟨n, rfl⟩ := h
exact (formPerm_rotate l hd n).symm
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Topology.Defs.Induced
import Mathlib.Topology.Basic
#align_import topology.order from "leanprover-community/mathlib"@"bcfa726826abd575... | Mathlib/Topology/Order.lean | 706 | 708 | theorem continuous_coinduced_dom {g : β → γ} {t₁ : TopologicalSpace α} {t₂ : TopologicalSpace γ} :
Continuous[coinduced f t₁, t₂] g ↔ Continuous[t₁, t₂] (g ∘ f) := by |
simp only [continuous_iff_coinduced_le, coinduced_compose]
|
/-
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 | 703 | 703 | theorem mem_ball_one_iff : a ∈ ball (1 : E) r ↔ ‖a‖ < r := by | rw [mem_ball, dist_one_right]
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Group.Nat
import Mathlib.Algebra.Order.Sub.Canonical
import Mathlib.Data.List.Perm
import Mathlib.Data.Set.List
import Mathlib.Init.Quot... | Mathlib/Data/Multiset/Basic.lean | 3,074 | 3,075 | theorem inter_eq_zero_iff_disjoint [DecidableEq α] {s t : Multiset α} :
s ∩ t = 0 ↔ Disjoint s t := by | rw [← subset_zero]; simp [subset_iff, Disjoint]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Set.Finite
#align_import order.filter.basic from "leanprover-community/mathlib"@"d4f691b9e5... | Mathlib/Order/Filter/Basic.lean | 1,897 | 1,899 | theorem EventuallyLE.sup [SemilatticeSup β] {l : Filter α} {f₁ f₂ g₁ g₂ : α → β} (hf : f₁ ≤ᶠ[l] f₂)
(hg : g₁ ≤ᶠ[l] g₂) : f₁ ⊔ g₁ ≤ᶠ[l] f₂ ⊔ g₂ := by |
filter_upwards [hf, hg] with x hfx hgx using sup_le_sup hfx hgx
|
/-
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 | 721 | 723 | theorem pow_sup_eq_top {n : ℕ} (h : I ⊔ J = ⊤) : I ^ n ⊔ J = ⊤ := by |
rw [← Finset.card_range n, ← Finset.prod_const]
exact prod_sup_eq_top fun _ _ => h
|
/-
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.Fintype.Basic
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Defs
#align_import logic.equiv.fintype from "leanprover-community... | Mathlib/Logic/Equiv/Fintype.lean | 91 | 93 | theorem Equiv.Perm.viaFintypeEmbedding_sign [DecidableEq α] [Fintype β] :
Equiv.Perm.sign (e.viaFintypeEmbedding f) = Equiv.Perm.sign e := by |
simp [Equiv.Perm.viaFintypeEmbedding]
|
/-
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.Computability.Encoding
import Mathlib.Logic.Small.List
import Mathlib.ModelTheory.Syntax
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import model... | Mathlib/ModelTheory/Encoding.lean | 235 | 287 | theorem listDecode_encode_list (l : List (Σn, L.BoundedFormula α n)) :
(listDecode (l.bind fun φ => φ.2.listEncode)).1 = l.headI := by |
suffices h : ∀ (φ : Σn, L.BoundedFormula α n) (l),
(listDecode (listEncode φ.2 ++ l)).1 = φ ∧ (listDecode (listEncode φ.2 ++ l)).2.1 = l by
induction' l with φ l _
· rw [List.nil_bind]
simp [listDecode]
· rw [cons_bind, (h φ _).1, headI_cons]
rintro ⟨n, φ⟩
induction' φ with _ _ _ _ φ_n φ_... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov, Kexing Ying
-/
import Mathlib.Topology.Semicontinuous
import Mathlib.MeasureTheory.Function.AEMeasurableSequence
import Mathlib.MeasureTheory.Order.Lat... | Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean | 988 | 1,029 | theorem measure_eq_measure_preimage_add_measure_tsum_Ico_zpow {α : Type*} [MeasurableSpace α]
(μ : Measure α) {f : α → ℝ≥0∞} (hf : Measurable f) {s : Set α} (hs : MeasurableSet s)
{t : ℝ≥0} (ht : 1 < t) :
μ s =
μ (s ∩ f ⁻¹' {0}) + μ (s ∩ f ⁻¹' {∞}) +
∑' n : ℤ, μ (s ∩ f ⁻¹' Ico ((t : ℝ≥0∞) ^ n) (... |
have A : μ s = μ (s ∩ f ⁻¹' {0}) + μ (s ∩ f ⁻¹' Ioi 0) := by
rw [← measure_union]
· rw [← inter_union_distrib_left, ← preimage_union, singleton_union, Ioi_insert,
← _root_.bot_eq_zero, Ici_bot, preimage_univ, inter_univ]
· exact disjoint_singleton_left.mpr not_mem_Ioi_self
|>.preimage f |... |
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