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) 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,
Scott Morrison
-/
import Mathlib.Data.List.Basic
#align_import data.list.lattice from "leanprov... | Mathlib/Data/List/Lattice.lean | 139 | 140 | theorem inter_cons_of_not_mem (l₁ : List α) (h : a ∉ l₂) : (a :: l₁) ∩ l₂ = l₁ ∩ l₂ := by |
simp [Inter.inter, List.inter, h]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Floris van Doorn, Sébastien Gouëzel, Alex J. Best
-/
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Int
import Mathlib.Algebra.Group.Nat
import ... | Mathlib/Algebra/BigOperators/Group/List.lean | 249 | 252 | theorem length_pos_of_prod_ne_one (L : List M) (h : L.prod ≠ 1) : 0 < L.length := by |
cases L
· simp at h
· simp
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Judith Ludwig, Christian Merten
-/
import Mathlib.Algebra.GeomSum
import Mathlib.LinearAlgebra.SModEq
import Mathlib.RingTheory.JacobsonIdeal
#align_import linear_algebra.adic_c... | Mathlib/RingTheory/AdicCompletion/Basic.lean | 313 | 315 | theorem transitionMap_eq (n : ℕ) : transitionMap I M (Nat.le_refl n) = LinearMap.id := by |
ext
simp
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
#align_import... | Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean | 36 | 41 | theorem zero_cpow_eq_nhds {b : ℂ} (hb : b ≠ 0) : (fun x : ℂ => (0 : ℂ) ^ x) =ᶠ[𝓝 b] 0 := by |
suffices ∀ᶠ x : ℂ in 𝓝 b, x ≠ 0 from
this.mono fun x hx ↦ by
dsimp only
rw [zero_cpow hx, Pi.zero_apply]
exact IsOpen.eventually_mem isOpen_ne hb
|
/-
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 | 2,342 | 2,343 | theorem splitOnP.go_ne_nil (xs acc : List α) : splitOnP.go p xs acc ≠ [] := by |
induction xs generalizing acc <;> simp [go]; split <;> simp [*]
|
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Data.Real.Sqrt
#al... | Mathlib/Analysis/Seminorm.lean | 891 | 895 | theorem closedBall_finset_sup_eq_iInter (p : ι → Seminorm 𝕜 E) (s : Finset ι) (x : E) {r : ℝ}
(hr : 0 ≤ r) : closedBall (s.sup p) x r = ⋂ i ∈ s, closedBall (p i) x r := by |
lift r to NNReal using hr
simp_rw [closedBall, iInter_setOf, finset_sup_apply, NNReal.coe_le_coe, Finset.sup_le_iff, ←
NNReal.coe_le_coe, NNReal.coe_mk]
|
/-
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.Order.Filter.SmallSets
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Compactness.Compact
import Mathlib.To... | Mathlib/Topology/UniformSpace/Basic.lean | 773 | 776 | theorem UniformSpace.ball_mem_nhdsWithin {x : α} {S : Set α} ⦃V : Set (α × α)⦄ (x_in : x ∈ S)
(V_in : V ∈ 𝓤 α ⊓ 𝓟 (S ×ˢ S)) : ball x V ∈ 𝓝[S] x := by |
rw [nhdsWithin_eq_comap_uniformity_of_mem x_in, mem_comap]
exact ⟨V, V_in, Subset.rfl⟩
|
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 618 | 619 | theorem toReal_eq_pi_div_two_iff {θ : Angle} : θ.toReal = π / 2 ↔ θ = (π / 2 : ℝ) := by |
rw [← toReal_inj, toReal_pi_div_two]
|
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Floris van Doorn
-/
import Mathlib.Tactic.Lemma
import Mathlib.Mathport.Attributes
import Mathlib.Mathport.Rename
import Mathlib.Tactic.Relatio... | Mathlib/Init/Logic.lean | 290 | 290 | theorem decide_False' (h : Decidable False) : decide False = false := by | simp
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker, Johan Commelin
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib... | Mathlib/Algebra/Polynomial/RingDivision.lean | 649 | 662 | theorem eq_leadingCoeff_mul_of_monic_of_dvd_of_natDegree_le {R} [CommRing R] {p q : R[X]}
(hp : p.Monic) (hdiv : p ∣ q) (hdeg : q.natDegree ≤ p.natDegree) :
q = C q.leadingCoeff * p := by |
obtain ⟨r, hr⟩ := hdiv
obtain rfl | hq := eq_or_ne q 0; · simp
have rzero : r ≠ 0 := fun h => by simp [h, hq] at hr
rw [hr, natDegree_mul'] at hdeg; swap
· rw [hp.leadingCoeff, one_mul, leadingCoeff_ne_zero]
exact rzero
rw [mul_comm, @eq_C_of_natDegree_eq_zero _ _ r] at hr
· convert hr
convert le... |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Poly... | Mathlib/FieldTheory/RatFunc/Degree.lean | 49 | 50 | theorem intDegree_one : intDegree (1 : RatFunc K) = 0 := by |
rw [intDegree, num_one, denom_one, sub_self]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Su... | Mathlib/RingTheory/Polynomial/Bernstein.lean | 222 | 231 | theorem iterate_derivative_at_1 (n ν : ℕ) (h : ν ≤ n) :
(Polynomial.derivative^[n - ν] (bernsteinPolynomial R n ν)).eval 1 =
(-1) ^ (n - ν) * (ascPochhammer R (n - ν)).eval (ν + 1 : R) := by |
rw [flip' _ _ _ h]
simp [Polynomial.eval_comp, h]
obtain rfl | h' := h.eq_or_lt
· simp
· norm_cast
congr
omega
|
/-
Copyright (c) 2021 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Tactic.NoncommRing
#align_import algebra.algebra.spectrum from "leanprover-c... | Mathlib/Algebra/Algebra/Spectrum.lean | 379 | 384 | theorem nonzero_mul_eq_swap_mul (a b : A) : σ (a * b) \ {0} = σ (b * a) \ {0} := by |
suffices h : ∀ x y : A, σ (x * y) \ {0} ⊆ σ (y * x) \ {0} from
Set.eq_of_subset_of_subset (h a b) (h b a)
rintro _ _ k ⟨k_mem, k_neq⟩
change ((Units.mk0 k k_neq) : 𝕜) ∈ _ at k_mem
exact ⟨unit_mem_mul_iff_mem_swap_mul.mp k_mem, k_neq⟩
|
/-
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.Whiskering
import Mathlib.CategoryTheory.Sites.Plus
#align_import category_theory.sites.compatible_plus from "leanprover-community/mathli... | Mathlib/CategoryTheory/Sites/CompatiblePlus.lean | 217 | 218 | theorem toPlus_comp_plusCompIso_inv :
J.toPlus _ ≫ (J.plusCompIso F P).inv = whiskerRight (J.toPlus _) _ := by | simp [Iso.comp_inv_eq]
|
/-
Copyright (c) 2020 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo
-/
import Mathlib.Dynamics.Flow
import Mathlib.Tactic.Monotonicity
#align_import dynamics.omega_limit from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
... | Mathlib/Dynamics/OmegaLimit.lean | 333 | 337 | theorem isInvariant_omegaLimit (hf : ∀ t, Tendsto (t + ·) f f) : IsInvariant ϕ (ω f ϕ s) := by |
refine fun t ↦ MapsTo.mono_right ?_ (omegaLimit_subset_of_tendsto ϕ s (hf t))
exact
mapsTo_omegaLimit _ (mapsTo_id _) (fun t' x ↦ (ϕ.map_add _ _ _).symm)
(continuous_const.flow ϕ continuous_id)
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, 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 | 1,311 | 1,311 | theorem div_div_cancel_left (a b : G) : a / b / a = b⁻¹ := by | simp
|
/-
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.Topology.Sheaves.SheafCondition.OpensLeCover
import Mathlib.CategoryTheory.Limits.Final
import Mathlib.CategoryTheory.Limits.Preserves.Basic
import Mat... | Mathlib/Topology/Sheaves/SheafCondition/PairwiseIntersections.lean | 280 | 282 | theorem isSheaf_iff_isSheafPairwiseIntersections : F.IsSheaf ↔ F.IsSheafPairwiseIntersections := by |
rw [isSheaf_iff_isSheafOpensLeCover,
isSheafOpensLeCover_iff_isSheafPairwiseIntersections]
|
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris Van Doorn, Yury Kudryashov
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_imp... | Mathlib/MeasureTheory/Measure/Regular.lean | 349 | 353 | theorem _root_.Set.measure_eq_iInf_isOpen (A : Set α) (μ : Measure α) [OuterRegular μ] :
μ A = ⨅ (U : Set α) (_ : A ⊆ U) (_ : IsOpen U), μ U := by |
refine le_antisymm (le_iInf₂ fun s hs => le_iInf fun _ => μ.mono hs) ?_
refine le_of_forall_lt' fun r hr => ?_
simpa only [iInf_lt_iff, exists_prop] using A.exists_isOpen_lt_of_lt r hr
|
/-
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, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Prod
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Span
... | Mathlib/LinearAlgebra/Prod.lean | 625 | 629 | theorem fst_map_fst : (Submodule.fst R M M₂).map (LinearMap.fst R M M₂) = ⊤ := by |
-- Porting note (#10936): was `tidy`
rw [eq_top_iff]; rintro x -
simp only [fst, comap_bot, mem_map, mem_ker, snd_apply, fst_apply,
Prod.exists, exists_eq_left, exists_eq]
|
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Data.Real.Sqrt
#al... | Mathlib/Analysis/Seminorm.lean | 797 | 798 | theorem sub_mem_ball (p : Seminorm 𝕜 E) (x₁ x₂ y : E) (r : ℝ) :
x₁ - x₂ ∈ p.ball y r ↔ x₁ ∈ p.ball (x₂ + y) r := by | simp_rw [mem_ball, sub_sub]
|
/-
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.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.Topology.Sheaves.SheafCondition.PairwiseInt... | Mathlib/Topology/Sheaves/SheafCondition/EqualizerProducts.lean | 80 | 81 | theorem res_π (i : ι) : res F U ≫ limit.π _ ⟨i⟩ = F.map (Opens.leSupr U i).op := by |
rw [res, limit.lift_π, Fan.mk_π_app]
|
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.RingTh... | Mathlib/RingTheory/Polynomial/Basic.lean | 240 | 249 | theorem not_finite [Nontrivial R] : ¬ Module.Finite R R[X] := by |
rw [Module.finite_def, Submodule.fg_def]
push_neg
intro s hs contra
rcases span_le_degreeLE_of_finite hs with ⟨n,hn⟩
have : ((X : R[X]) ^ (n + 1)) ∈ Polynomial.degreeLE R ↑n := by
rw [contra] at hn
exact hn Submodule.mem_top
rw [mem_degreeLE, degree_X_pow, Nat.cast_le, add_le_iff_nonpos_right, nonp... |
/-
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.Logic.Basic
import Mathlib.Tactic.Positivity.Basic
#align_import algebra.order.hom.basic from "leanprover-community/mathlib"@"28aa996fc6fb4317f0083c4e6daf... | Mathlib/Algebra/Order/Hom/Basic.lean | 241 | 243 | theorem map_div_le_add : f (x / y) ≤ f x + f y := by |
rw [div_eq_mul_inv, ← map_inv_eq_map f y]
exact map_mul_le_add _ _ _
|
/-
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.InnerProductSpace.Adjoint
import Mathlib.Topology.Algebra.Module.Basic
#align_import analysis.inner_product_space.linear_pmap from "leanprover-comm... | Mathlib/Analysis/InnerProductSpace/LinearPMap.lean | 252 | 261 | theorem _root_.IsSelfAdjoint.dense_domain (hA : IsSelfAdjoint A) : Dense (A.domain : Set E) := by |
by_contra h
rw [isSelfAdjoint_def] at hA
have h' : A.domain = ⊤ := by
rw [← hA, Submodule.eq_top_iff']
intro x
rw [mem_adjoint_domain_iff, ← hA]
refine (innerSL 𝕜 x).cont.comp ?_
simp [adjoint, h, continuous_const]
simp [h'] at h
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.shapes.kernels from "leanprover-community/mathlib"@"95... | Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean | 1,020 | 1,022 | theorem MonoFactorisation.kernel_ι_comp [HasKernel f] (F : MonoFactorisation f) :
kernel.ι f ≫ F.e = 0 := by |
rw [← cancel_mono F.m, zero_comp, Category.assoc, F.fac, kernel.condition]
|
/-
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 | 407 | 409 | theorem isometryEquiv_map_mkMetric (m : ℝ≥0∞ → ℝ≥0∞) (f : X ≃ᵢ Y) :
map f (mkMetric m) = mkMetric m := by |
rw [← isometryEquiv_comap_mkMetric _ f, map_comap_of_surjective f.surjective]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.OuterMeasure.Operations
import Mathlib.Analysis.SpecificLimits.Basic
/-!
# Outer measures from functions
Given an arbit... | Mathlib/MeasureTheory/OuterMeasure/OfFunction.lean | 260 | 262 | theorem boundedBy_eq (s : Set α) (m_empty : m ∅ = 0) (m_mono : ∀ ⦃t : Set α⦄, s ⊆ t → m s ≤ m t)
(m_subadd : ∀ s : ℕ → Set α, m (⋃ i, s i) ≤ ∑' i, m (s i)) : boundedBy m s = m s := by |
rw [boundedBy_eq_ofFunction m_empty, ofFunction_eq s m_mono m_subadd]
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Jakob von Raumer
-/
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.... | Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean | 92 | 93 | theorem bicone_ι_π_ne {F : J → C} (B : Bicone F) {j j' : J} (h : j ≠ j') : B.ι j ≫ B.π j' = 0 := by |
simpa [h] using B.ι_π j j'
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Andrew Yang
-/
import Mathlib.CategoryTheory.Monoidal.Functor
#align_import category_theory.monoidal.End from "leanprover-community/mathlib"@"85075bccb68ab7fa49fb05db8... | Mathlib/CategoryTheory/Monoidal/End.lean | 211 | 214 | theorem obj_ε_inv_app (n : M) (X : C) :
(F.obj n).map (F.εIso.inv.app X) = (F.μ (𝟙_ M) n).app X ≫ (F.map (λ_ n).hom).app X := by |
rw [← cancel_mono ((F.obj n).map (F.ε.app X)), ← Functor.map_comp]
simp
|
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Yuyang Zhao
-/
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Algebra.Order.Monoid.Unbundled.Defs
import Mathlib.Tactic.GCongr.Core
#align_import algebra.order... | Mathlib/Algebra/Order/GroupWithZero/Unbundled.lean | 728 | 729 | theorem le_mul_of_one_le_left [MulPosMono α] (hb : 0 ≤ b) (h : 1 ≤ a) : b ≤ a * b := by |
simpa only [one_mul] using mul_le_mul_of_nonneg_right h hb
|
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Violeta Hernández Palacios
-/
import Mathlib.MeasureTheory.MeasurableSpace.Defs
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.SetTheory.Cardinal.Con... | Mathlib/MeasureTheory/MeasurableSpace/Card.lean | 117 | 151 | theorem generateMeasurable_eq_rec (s : Set (Set α)) :
{ t | GenerateMeasurable s t } =
⋃ (i : (Quotient.out (aleph 1).ord).α), generateMeasurableRec s i := by |
ext t; refine ⟨fun ht => ?_, fun ht => ?_⟩
· inhabit ω₁
induction' ht with u hu u _ IH f _ IH
· exact mem_iUnion.2 ⟨default, self_subset_generateMeasurableRec s _ hu⟩
· exact mem_iUnion.2 ⟨default, empty_mem_generateMeasurableRec s _⟩
· rcases mem_iUnion.1 IH with ⟨i, hi⟩
obtain ⟨j, hj⟩ := ex... |
/-
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 | 1,079 | 1,081 | theorem cons_isPath_iff {u v w : V} (h : G.Adj u v) (p : G.Walk v w) :
(cons h p).IsPath ↔ p.IsPath ∧ u ∉ p.support := by |
constructor <;> simp (config := { contextual := true }) [isPath_def]
|
/-
Copyright (c) 2021 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.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Extreme
#align_import analysis.convex.independent from "leanprover-... | Mathlib/Analysis/Convex/Independent.lean | 176 | 195 | theorem convexIndependent_iff_finset {p : ι → E} :
ConvexIndependent 𝕜 p ↔
∀ (s : Finset ι) (x : ι), p x ∈ convexHull 𝕜 (s.image p : Set E) → x ∈ s := by |
refine ⟨fun hc s x hx => hc s x ?_, fun h s x hx => ?_⟩
· rwa [Finset.coe_image] at hx
have hp : Injective p := by
rintro a b hab
rw [← mem_singleton]
refine h {b} a ?_
rw [hab, image_singleton, coe_singleton, convexHull_singleton]
exact Set.mem_singleton _
rw [convexHull_eq_union_convexHul... |
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Measure.GiryMonad
#align_import probability.kernel.basic from "leanprover-community/mathlib"@"... | Mathlib/Probability/Kernel/Basic.lean | 391 | 395 | theorem deterministic_apply' {f : α → β} (hf : Measurable f) (a : α) {s : Set β}
(hs : MeasurableSet s) : deterministic f hf a s = s.indicator (fun _ => 1) (f a) := by |
rw [deterministic]
change Measure.dirac (f a) s = s.indicator 1 (f a)
simp_rw [Measure.dirac_apply' _ hs]
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.RingTheory.IntegrallyClosed
import Mathlib.RingTheory.Localization.NumDen
import Mathlib.RingTheory.Polynomial.ScaleRoots
#align_import ring_theory.polynomi... | Mathlib/RingTheory/Polynomial/RationalRoot.lean | 39 | 44 | theorem scaleRoots_aeval_eq_zero_of_aeval_mk'_eq_zero {p : A[X]} {r : A} {s : M}
(hr : aeval (mk' S r s) p = 0) : aeval (algebraMap A S r) (scaleRoots p s) = 0 := by |
convert scaleRoots_eval₂_eq_zero (algebraMap A S) hr
-- Porting note: added
funext
rw [aeval_def, mk'_spec' _ r s]
|
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Set.Card
import Mathlib.Order.Minimal
import Mathlib.Data.Matroid.Init
/-!
# Matroids
A `Matroid` is a structure that combinatorially abstracts
the ... | Mathlib/Data/Matroid/Basic.lean | 744 | 746 | theorem basis_iff (hX : X ⊆ M.E := by | aesop_mat) :
M.Basis I X ↔ (M.Indep I ∧ I ⊆ X ∧ ∀ J, M.Indep J → I ⊆ J → J ⊆ X → I = J) := by
rw [basis_iff', and_iff_left hX]
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Real.Archimedean
#align_imp... | Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean | 89 | 89 | theorem to_real_re (x : ℤ[i]) : ((x.re : ℤ) : ℝ) = (x : ℂ).re := by | simp [toComplex_def]
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
#align_import geometry.euclidean.angle.or... | Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 260 | 265 | theorem oangle_rotation_oangle_left (x y : V) : o.oangle (o.rotation (o.oangle x y) x) y = 0 := by |
by_cases hx : x = 0
· simp [hx]
· by_cases hy : y = 0
· simp [hy]
· simp [hx, hy]
|
/-
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.Countable.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Order.Disjointed
import Mathlib.MeasureTheory.OuterMeasure.Defs... | Mathlib/MeasureTheory/OuterMeasure/Basic.lean | 162 | 167 | theorem exists_mem_forall_mem_nhdsWithin_pos_measure [TopologicalSpace α]
[SecondCountableTopology α] {s : Set α} (hs : μ s ≠ 0) :
∃ x ∈ s, ∀ t ∈ 𝓝[s] x, 0 < μ t := by |
contrapose! hs
simp only [nonpos_iff_eq_zero] at hs
exact measure_null_of_locally_null s hs
|
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Lu-Ming Zhang
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algeb... | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 103 | 105 | theorem det_invOf [Invertible A] [Invertible A.det] : (⅟ A).det = ⅟ A.det := by |
letI := detInvertibleOfInvertible A
convert (rfl : _ = ⅟ A.det)
|
/-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel, Bhavik Mehta, Andrew Yang, Emily Riehl
-/
import Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks
import Mathlib.CategoryTheory.Limits.Shapes.BinaryPro... | Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean | 474 | 475 | theorem spanExt_hom_app_left : (spanExt iX iY iZ wf wg).hom.app WalkingSpan.left = iY.hom := by |
dsimp [spanExt]
|
/-
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.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.... | Mathlib/ModelTheory/Semantics.lean | 396 | 400 | theorem realize_relabel {m n : ℕ} {φ : L.BoundedFormula α n} {g : α → Sum β (Fin m)} {v : β → M}
{xs : Fin (m + n) → M} :
(φ.relabel g).Realize v xs ↔
φ.Realize (Sum.elim v (xs ∘ Fin.castAdd n) ∘ g) (xs ∘ Fin.natAdd m) := by |
rw [relabel, realize_mapTermRel_add_castLe] <;> intros <;> simp
|
/-
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 | 714 | 716 | theorem ord_term_aux {o : Ordinal} (ho : o < Ordinal.type ((·<·) : I → I → Prop)) :
ord I (term I ho) = o := by |
simp only [ord, term, Ordinal.typein_enum]
|
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Sébastien Gouëzel
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic... | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 115 | 116 | theorem addHaarMeasure_eq_volume : addHaarMeasure Icc01 = volume := by |
convert (addHaarMeasure_unique volume Icc01).symm; simp [Icc01]
|
/-
Copyright (c) 2022 Kevin H. Wilson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin H. Wilson
-/
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Order.Filter.Curry
#align_import analysis.calculus.uniform_lim... | Mathlib/Analysis/Calculus/UniformLimitsDeriv.lean | 396 | 405 | theorem hasFDerivAt_of_tendstoLocallyUniformlyOn [NeBot l] {s : Set E} (hs : IsOpen s)
(hf' : TendstoLocallyUniformlyOn f' g' l s) (hf : ∀ n, ∀ x ∈ s, HasFDerivAt (f n) (f' n x) x)
(hfg : ∀ x ∈ s, Tendsto (fun n => f n x) l (𝓝 (g x))) (hx : x ∈ s) :
HasFDerivAt g (g' x) x := by |
have h1 : s ∈ 𝓝 x := hs.mem_nhds hx
have h3 : Set.univ ×ˢ s ∈ l ×ˢ 𝓝 x := by simp only [h1, prod_mem_prod_iff, univ_mem, and_self_iff]
have h4 : ∀ᶠ n : ι × E in l ×ˢ 𝓝 x, HasFDerivAt (f n.1) (f' n.1 n.2) n.2 :=
eventually_of_mem h3 fun ⟨n, z⟩ ⟨_, hz⟩ => hf n z hz
refine hasFDerivAt_of_tendstoUniformlyOn... |
/-
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 | 879 | 882 | theorem Fintype.card_subtype [Fintype α] (p : α → Prop) [DecidablePred p] :
Fintype.card { x // p x } = ((Finset.univ : Finset α).filter p).card := by |
refine Fintype.card_of_subtype _ ?_
simp
|
/-
Copyright (c) 2023 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
/-!
# Lindelöf sets and Lindelöf spaces
## Mai... | Mathlib/Topology/Compactness/Lindelof.lean | 440 | 446 | theorem hasBasis_coclosedLindelof :
(Filter.coclosedLindelof X).HasBasis (fun s => IsClosed s ∧ IsLindelof s) compl := by |
simp only [Filter.coclosedLindelof, iInf_and']
refine hasBasis_biInf_principal' ?_ ⟨∅, isClosed_empty, isLindelof_empty⟩
rintro s ⟨hs₁, hs₂⟩ t ⟨ht₁, ht₂⟩
exact ⟨s ∪ t, ⟨⟨hs₁.union ht₁, hs₂.union ht₂⟩, compl_subset_compl.2 subset_union_left,
compl_subset_compl.2 subset_union_right⟩⟩
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Nat.Prime
import Mathlib.Data.List.Prime
import Mathlib.Dat... | Mathlib/Data/Nat/Factors.lean | 55 | 65 | theorem prime_of_mem_factors {n : ℕ} : ∀ {p : ℕ}, (h : p ∈ factors n) → Prime p := by |
match n with
| 0 => simp
| 1 => simp
| k + 2 =>
intro p h
let m := minFac (k + 2)
have : (k + 2) / m < (k + 2) := factors_lemma
have h₁ : p = m ∨ p ∈ factors ((k + 2) / m) :=
List.mem_cons.1 (by rwa [factors] at h)
exact Or.casesOn h₁ (fun h₂ => h₂.symm ▸ minFac_prime (by ... |
/-
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 | 247 | 250 | theorem encard_diff_singleton_of_mem (h : a ∈ s) :
(s \ {a}).encard = s.encard - 1 := by |
rw [← encard_diff_singleton_add_one h, ← WithTop.add_right_cancel_iff WithTop.one_ne_top,
tsub_add_cancel_of_le (self_le_add_left _ _)]
|
/-
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 | 333 | 335 | theorem limitRecOn_limit {C} (o H₁ H₂ H₃ h) :
@limitRecOn C o H₁ H₂ H₃ = H₃ o h fun x _h => @limitRecOn C x H₁ H₂ H₃ := by |
simp_rw [limitRecOn, SuccOrder.limitRecOn_limit _ _ h.isSuccLimit, dif_neg h.1]
|
/-
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 | 404 | 407 | theorem preimage_closure_subset_closure_preimage (hf : IsOpenMap f) {s : Set Y} :
f ⁻¹' closure s ⊆ closure (f ⁻¹' s) := by |
rw [← compl_subset_compl]
simp only [← interior_compl, ← preimage_compl, hf.interior_preimage_subset_preimage_interior]
|
/-
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 | 356 | 358 | theorem commute_sub_int (n : ℤ) : Function.Commute f (· - n) := by |
simpa only [sub_eq_add_neg] using
(f.commute_add_int n).inverses_right (Equiv.addRight _).right_inv (Equiv.addRight _).left_inv
|
/-
Copyright (c) 2020 Johan Commelin, Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.MvPolynomial.Variables
#align_import data.mv_polynomial.monad fro... | Mathlib/Algebra/MvPolynomial/Monad.lean | 298 | 300 | theorem eval₂Hom_bind₁ (f : R →+* S) (g : τ → S) (h : σ → MvPolynomial τ R) (φ : MvPolynomial σ R) :
eval₂Hom f g (bind₁ h φ) = eval₂Hom f (fun i => eval₂Hom f g (h i)) φ := by |
rw [hom_bind₁, eval₂Hom_comp_C]
|
/-
Copyright (c) 2017 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Data.PFunctor.Univariate.Basic
#align_import data.pfunctor.univariate.M from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
/-!
... | Mathlib/Data/PFunctor/Univariate/M.lean | 541 | 542 | theorem children_mk {a} (x : F.B a → M F) (i : F.B (head (M.mk ⟨a, x⟩))) :
children (M.mk ⟨a, x⟩) i = x (cast (by rw [head_mk]) i) := by | apply ext'; intro n; rfl
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
imp... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean | 187 | 192 | theorem cos_eq_iff_quadratic {z w : ℂ} :
cos z = w ↔ exp (z * I) ^ 2 - 2 * w * exp (z * I) + 1 = 0 := by |
rw [← sub_eq_zero]
field_simp [cos, exp_neg, exp_ne_zero]
refine Eq.congr ?_ rfl
ring
|
/-
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 | 574 | 575 | theorem concat_eq_reverse_cons (a : α) (l : List α) : concat l a = reverse (a :: reverse l) := by |
simp only [concat_eq_append, reverse_cons, reverse_reverse]
|
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.class_group from "leanprover-community/mathlib"@"565eb991... | Mathlib/RingTheory/ClassGroup.lean | 126 | 144 | theorem ClassGroup.mk_eq_mk_of_coe_ideal {I J : (FractionalIdeal R⁰ <| FractionRing R)ˣ}
{I' J' : Ideal R} (hI : (I : FractionalIdeal R⁰ <| FractionRing R) = I')
(hJ : (J : FractionalIdeal R⁰ <| FractionRing R) = J') :
ClassGroup.mk I = ClassGroup.mk J ↔
∃ x y : R, x ≠ 0 ∧ y ≠ 0 ∧ Ideal.span {x} * I' ... |
rw [ClassGroup.mk_eq_mk]
constructor
· rintro ⟨x, rfl⟩
rw [Units.val_mul, hI, coe_toPrincipalIdeal, mul_comm,
spanSingleton_mul_coeIdeal_eq_coeIdeal] at hJ
exact ⟨_, _, sec_fst_ne_zero (R := R) le_rfl x.ne_zero,
sec_snd_ne_zero (R := R) le_rfl (x : FractionRing R), hJ⟩
· rintro ⟨x, y, hx, 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.MonoidAlgebra.Degree
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Mono... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 1,410 | 1,413 | theorem leadingCoeff_sub_of_degree_lt (h : Polynomial.degree q < Polynomial.degree p) :
(p - q).leadingCoeff = p.leadingCoeff := by |
rw [← q.degree_neg] at h
rw [sub_eq_add_neg, leadingCoeff_add_of_degree_lt' h]
|
/-
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.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.GroupTheory.Subgroup.Si... | Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | 136 | 141 | theorem Subgroup.eq_bot_or_eq_top_of_prime_card {G : Type*} [Group G] {_ : Fintype G}
(H : Subgroup G) [hp : Fact (Fintype.card G).Prime] : H = ⊥ ∨ H = ⊤ := by |
classical
have := card_subgroup_dvd_card H
rwa [Nat.card_eq_fintype_card (α := G), Nat.dvd_prime hp.1, ← Nat.card_eq_fintype_card,
← eq_bot_iff_card, card_eq_iff_eq_top] at this
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Algebra.Order.Ring.Abs
#align_import data.int.order.units from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
/-!
# Lemmas a... | Mathlib/Data/Int/Order/Units.lean | 45 | 46 | theorem units_coe_mul_self (u : ℤˣ) : (u * u : ℤ) = 1 := by |
rw [← Units.val_mul, units_mul_self, Units.val_one]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Morenikeji Neri
-/
import Mathlib.Algebra.EuclideanDomain.Instances
import Mathlib.RingTheory.Ideal.Colon
import Mathlib.RingTheory.UniqueFactorizationDomain
#align_import... | Mathlib/RingTheory/PrincipalIdealDomain.lean | 141 | 144 | theorem generator_map_dvd_of_mem {N : Submodule R M} (ϕ : M →ₗ[R] R) [(N.map ϕ).IsPrincipal] {x : M}
(hx : x ∈ N) : generator (N.map ϕ) ∣ ϕ x := by |
rw [← mem_iff_generator_dvd, Submodule.mem_map]
exact ⟨x, hx, rfl⟩
|
/-
Copyright (c) 2020 Alena Gusakov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alena Gusakov, Arthur Paulino, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.DegreeSum
import Mathlib.Combinatorics.SimpleGraph.Subgraph
#align_import combinatorics.simple_gr... | Mathlib/Combinatorics/SimpleGraph/Matching.lean | 114 | 119 | theorem isPerfectMatching_iff : M.IsPerfectMatching ↔ ∀ v, ∃! w, M.Adj v w := by |
refine ⟨?_, fun hm => ⟨fun v _ => hm v, fun v => ?_⟩⟩
· rintro ⟨hm, hs⟩ v
exact hm (hs v)
· obtain ⟨w, hw, -⟩ := hm v
exact M.edge_vert hw
|
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov, Hunter Monroe
-/
import Mathlib.Combinatorics.SimpleGraph.Init
import Mathlib.Data.Rel
import Mathlib... | Mathlib/Combinatorics/SimpleGraph/Basic.lean | 520 | 522 | theorem edgeSet_sup : (G₁ ⊔ G₂).edgeSet = G₁.edgeSet ∪ G₂.edgeSet := by |
ext ⟨x, y⟩
rfl
|
/-
Copyright (c) 2022 Wrenna Robson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wrenna Robson
-/
import Mathlib.Topology.MetricSpace.Basic
#align_import topology.metric_space.infsep from "leanprover-community/mathlib"@"5316314b553dcf8c6716541851517c1a9715e22b"
/-... | Mathlib/Topology/MetricSpace/Infsep.lean | 233 | 239 | theorem le_einfsep_pi_of_le {π : β → Type*} [Fintype β] [∀ b, PseudoEMetricSpace (π b)]
{s : ∀ b : β, Set (π b)} {c : ℝ≥0∞} (h : ∀ b, c ≤ einfsep (s b)) :
c ≤ einfsep (Set.pi univ s) := by |
refine le_einfsep fun x hx y hy hxy => ?_
rw [mem_univ_pi] at hx hy
rcases Function.ne_iff.mp hxy with ⟨i, hi⟩
exact le_trans (le_einfsep_iff.1 (h i) _ (hx _) _ (hy _) hi) (edist_le_pi_edist _ _ i)
|
/-
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.Set.Subsingleton
import Mathlib.Logic.Equiv.Defs
import Mathlib.Algebra.Group.Defs
#align_import data.part from "lean... | Mathlib/Data/Part.lean | 839 | 840 | theorem some_inter_some [Inter α] (a b : α) : some a ∩ some b = some (a ∩ b) := by |
simp [inter_def]
|
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.List.Perm
import Mathlib.Data.List.Range
#align_import data.list.sublists from "leanprover-community/mathlib... | Mathlib/Data/List/Sublists.lean | 438 | 456 | theorem revzip_sublists (l : List α) : ∀ l₁ l₂, (l₁, l₂) ∈ revzip l.sublists → l₁ ++ l₂ ~ l := by |
rw [revzip]
induction' l using List.reverseRecOn with l' a ih
· intro l₁ l₂ h
simp? at h says
simp only [sublists_nil, reverse_cons, reverse_nil, nil_append, zip_cons_cons, zip_nil_right,
mem_singleton, Prod.mk.injEq] at h
simp [h]
· intro l₁ l₂ h
rw [sublists_concat, reverse_append, ... |
/-
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.Algebra.Polynomial.AlgebraMap
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.RingTheory.PowerBasis
#align_import ring_theory.is_adjoin... | Mathlib/RingTheory/IsAdjoinRoot.lean | 387 | 390 | theorem map_modByMonicHom (h : IsAdjoinRootMonic S f) (x : S) : h.map (h.modByMonicHom x) = x := by |
rw [modByMonicHom, LinearMap.coe_mk]
dsimp -- Porting note (#11227):added a `dsimp`
rw [map_modByMonic, map_repr]
|
/-
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 | 292 | 299 | theorem get_zipWith {f : α → β → γ} {l : List α} {l' : List β} {i : Fin (zipWith f l l').length} :
(zipWith f l l').get i =
f (l.get ⟨i, lt_length_left_of_zipWith i.isLt⟩)
(l'.get ⟨i, lt_length_right_of_zipWith i.isLt⟩) := by |
rw [← Option.some_inj, ← get?_eq_get, get?_zip_with_eq_some]
exact
⟨l.get ⟨i, lt_length_left_of_zipWith i.isLt⟩, l'.get ⟨i, lt_length_right_of_zipWith i.isLt⟩,
by rw [get?_eq_get], by rw [get?_eq_get]; exact ⟨rfl, rfl⟩⟩
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.Tower
import Mathlib.RingTheory.Algebraic
import Mathlib.FieldTheory.Minpoly.Basic
#align_import field_theory.intermediate_field from "leanprove... | Mathlib/FieldTheory/IntermediateField.lean | 290 | 293 | theorem toSubalgebra_toIntermediateField (S : Subalgebra K L) (inv_mem : ∀ x ∈ S, x⁻¹ ∈ S) :
(S.toIntermediateField inv_mem).toSubalgebra = S := by |
ext
rfl
|
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Anne Baanen
-/
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Hom.Basic
import Mathlib.Algebra.Orde... | Mathlib/Algebra/Order/AbsoluteValue.lean | 403 | 404 | theorem abv_sub_le (a b c : R) : abv (a - c) ≤ abv (a - b) + abv (b - c) := by |
simpa [sub_eq_add_neg, add_assoc] using abv_add abv (a - b) (b - c)
|
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
#align_import order.symm_diff from "leanprover-community/mathlib... | Mathlib/Order/SymmDiff.lean | 200 | 200 | theorem inf_sup_symmDiff : a ⊓ b ⊔ a ∆ b = a ⊔ b := by | rw [sup_comm, symmDiff_sup_inf]
|
/-
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 | 147 | 148 | theorem preimage_const_add_Icc : (fun x => a + x) ⁻¹' Icc b c = Icc (b - a) (c - a) := by |
simp [← Ici_inter_Iic]
|
/-
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
import Mathlib.Data.Set.Pairwise.Basic
import Mathlib.Init.Data.Fin.Basic
#align_import data.list.nodup from "leanprover-... | Mathlib/Data/List/Nodup.lean | 328 | 329 | theorem Nodup.mem_erase_iff [DecidableEq α] (d : Nodup l) : a ∈ l.erase b ↔ a ≠ b ∧ a ∈ l := by |
rw [d.erase_eq_filter, mem_filter, and_comm, decide_eq_true_iff]
|
/-
Copyright (c) 2022 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, María Inés de Frutos-Fernández
-/
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.M... | Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean | 288 | 299 | theorem prod {ι : Type*} (s : Finset ι) (φ : ι → MvPolynomial σ R) (n : ι → M) {w : σ → M} :
(∀ i ∈ s, IsWeightedHomogeneous w (φ i) (n i)) →
IsWeightedHomogeneous w (∏ i ∈ s, φ i) (∑ i ∈ s, n i) := by |
classical
refine Finset.induction_on s ?_ ?_
· intro
simp only [isWeightedHomogeneous_one, Finset.sum_empty, Finset.prod_empty]
· intro i s his IH h
simp only [his, Finset.prod_insert, Finset.sum_insert, not_false_iff]
apply (h i (Finset.mem_insert_self _ _)).mul (IH _)
intro j hjs
exact h ... |
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import comp... | Mathlib/Computability/TMToPartrec.lean | 1,932 | 1,944 | theorem ret_supports {S k} (H₁ : contSupp k ⊆ S) : TM2.SupportsStmt S (tr (Λ'.ret k)) := by |
have W := fun {q} => trStmts₁_self q
cases k with
| halt => trivial
| cons₁ => rw [contSupp_cons₁, Finset.union_subset_iff] at H₁; exact fun _ => H₁.1 W
| cons₂ => rw [contSupp_cons₂, Finset.union_subset_iff] at H₁; exact fun _ => H₁.1 W
| comp => rw [contSupp_comp] at H₁; exact fun _ => H₁ (codeSupp_self ... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Bryan Gin-ge Chen
-/
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2... | Mathlib/Order/BooleanAlgebra.lean | 403 | 403 | theorem sdiff_sdiff_left' : (x \ y) \ z = x \ y ⊓ x \ z := by | rw [sdiff_sdiff_left, sdiff_sup]
|
/-
Copyright (c) 2020 Paul van Wamelen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul van Wamelen
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Int.Basic
import Mat... | Mathlib/NumberTheory/PythagoreanTriples.lean | 583 | 595 | theorem classified : h.IsClassified := by |
by_cases h0 : Int.gcd x y = 0
· have hx : x = 0 := by
apply Int.natAbs_eq_zero.mp
apply Nat.eq_zero_of_gcd_eq_zero_left h0
have hy : y = 0 := by
apply Int.natAbs_eq_zero.mp
apply Nat.eq_zero_of_gcd_eq_zero_right h0
use 0, 1, 0
field_simp [hx, hy]
apply h.isClassified_of_normal... |
/-
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 | 235 | 241 | theorem smeval_X_pow_assoc (m n : ℕ) :
x ^ m * x ^ n * p.smeval x = x ^ m * (x ^ n * p.smeval x) := by |
induction p using Polynomial.induction_on' with
| h_add p q ph qh =>
simp only [smeval_add, ph, qh, mul_add]
| h_monomial n a =>
simp only [smeval_monomial, mul_smul_comm, npow_mul_assoc]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Scott Morrison
-/
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.TryThis
import Mathlib.Util.AtomM
/-!
# The `abel` tactic
Evaluate expressions in the language o... | Mathlib/Tactic/Abel.lean | 128 | 130 | theorem const_add_term {α} [AddCommMonoid α] (k n x a a') (h : k + a = a') :
k + @term α _ n x a = term n x a' := by |
simp [h.symm, term, add_comm, add_assoc]
|
/-
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 | 1,502 | 1,506 | theorem foldl_induction' (f : β → α → β) (H : RightCommutative f) (x : β) (q : α → Prop)
(p : β → Prop) (s : Multiset α) (hpqf : ∀ a b, q a → p b → p (f b a)) (px : p x)
(q_s : ∀ a ∈ s, q a) : p (foldl f H x s) := by |
rw [foldl_swap]
exact foldr_induction' (fun x y => f y x) (fun x y z => (H _ _ _).symm) x q p s hpqf px q_s
|
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Nat.Defs
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.Co... | Mathlib/Data/Fin/Basic.lean | 599 | 601 | theorem le_val_last (i : Fin (n + 1)) : i ≤ n := by |
rw [Fin.natCast_eq_last]
exact Fin.le_last i
|
/-
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.Set.Subsingleton
import Mathlib.Logic.Equiv.Defs
import Mathlib.Algebra.Group.Defs
#align_import data.part from "lean... | Mathlib/Data/Part.lean | 378 | 378 | theorem to_ofOption (o : Option α) : toOption (ofOption o) = o := by | cases o <;> rfl
|
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Yury Kudryashov
-/
import Mathlib.Topology.Order.LocalExtr
import Mathlib.Topology.Order.IntermediateValue
import Mathlib.Topology.Support
import Mathlib.Topology.Order... | Mathlib/Topology/Algebra/Order/Compact.lean | 640 | 644 | theorem image_uIcc (h : ContinuousOn f <| [[a, b]]) :
f '' [[a, b]] = [[sInf (f '' [[a, b]]), sSup (f '' [[a, b]])]] := by |
refine h.image_uIcc_eq_Icc.trans (uIcc_of_le ?_).symm
refine csInf_le_csSup ?_ ?_ (nonempty_uIcc.image _) <;> rw [h.image_uIcc_eq_Icc]
exacts [bddBelow_Icc, bddAbove_Icc]
|
/-
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.MeasureTheory.Integral.FundThmCalculus
import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
import Mathlib.Analysis.SpecialFunction... | Mathlib/Analysis/SpecialFunctions/Integrals.lean | 823 | 826 | theorem integral_sin_mul_cos_sq :
∫ x in a..b, sin x * cos x ^ 2 = (cos a ^ 3 - cos b ^ 3) / 3 := by |
have := @integral_sin_pow_odd_mul_cos_pow a b 0 2
norm_num at this; exact this
|
/-
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.MeasureTheory.Function.ConditionalExpectation.CondexpL2
#align_import measure_theory.function.conditional_expectation.condexp_L1 from "leanprover-communit... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean | 435 | 467 | theorem setIntegral_condexpL1CLM (f : α →₁[μ] F') (hs : MeasurableSet[m] s) :
∫ x in s, condexpL1CLM F' hm μ f x ∂μ = ∫ x in s, f x ∂μ := by |
let S := spanningSets (μ.trim hm)
have hS_meas : ∀ i, MeasurableSet[m] (S i) := measurable_spanningSets (μ.trim hm)
have hS_meas0 : ∀ i, MeasurableSet (S i) := fun i => hm _ (hS_meas i)
have hs_eq : s = ⋃ i, S i ∩ s := by
simp_rw [Set.inter_comm]
rw [← Set.inter_iUnion, iUnion_spanningSets (μ.trim hm),... |
/-
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 | 105 | 114 | theorem count_toMultiset [DecidableEq α] (f : α →₀ ℕ) (a : α) : (toMultiset f).count a = f a :=
calc
(toMultiset f).count a = Finsupp.sum f (fun x n => (n • {x} : Multiset α).count a) := by |
rw [toMultiset_apply]; exact map_sum (Multiset.countAddMonoidHom a) _ f.support
_ = f.sum fun x n => n * ({x} : Multiset α).count a := by simp only [Multiset.count_nsmul]
_ = f a * ({a} : Multiset α).count a :=
sum_eq_single _
(fun a' _ H => by simp only [Multiset.count_singleton, if_false,... |
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Constructions.Pi
import Mathlib.Probability.Kernel.Basic
/-!
# Independence with respect to a kernel and a measure
A family of sets of sets... | Mathlib/Probability/Independence/Kernel.lean | 615 | 644 | theorem iIndepSets.iIndep [IsMarkovKernel κ] (m : ι → MeasurableSpace Ω)
(h_le : ∀ i, m i ≤ _mΩ) (π : ι → Set (Set Ω)) (h_pi : ∀ n, IsPiSystem (π n))
(h_generate : ∀ i, m i = generateFrom (π i)) (h_ind : iIndepSets π κ μ) :
iIndep m κ μ := by |
classical
intro s f
refine Finset.induction ?_ ?_ s
· simp only [Finset.not_mem_empty, Set.mem_setOf_eq, IsEmpty.forall_iff, implies_true,
Set.iInter_of_empty, Set.iInter_univ, measure_univ, Finset.prod_empty,
Filter.eventually_true, forall_true_left]
· intro a S ha_notin_S h_rec hf_m
have hf... |
/-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel, Bhavik Mehta, Andrew Yang, Emily Riehl
-/
import Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks
import Mathlib.CategoryTheory.Limits.Shapes.BinaryPro... | Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean | 2,006 | 2,007 | theorem pullback_symmetry_hom_of_epi_eq [Epi f] : (pushoutSymmetry f f).hom = 𝟙 _ := by |
ext <;> simp [inl_eq_inr_of_epi_eq]
|
/-
Copyright (c) 2021 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Adjunction.Reflective
#align_import algebraic_geometry.... | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | 498 | 506 | theorem SpecΓIdentity_hom_app_presheaf_obj {X : Scheme.{u}} (U : Opens X) :
SpecΓIdentity.hom.app (X.presheaf.obj (op U)) =
Scheme.Γ.map (Scheme.Spec.map (X.presheaf.map (eqToHom U.openEmbedding_obj_top).op).op).op ≫
(ΓSpec.adjunction.unit.app (X ∣_ᵤ U)).val.c.app (op ⊤) ≫
X.presheaf.map (eqToHom ... |
rw [ΓSpec.adjunction_unit_app_app_top]
dsimp [-SpecΓIdentity_hom_app]
rw [SpecΓIdentity_naturality_assoc, ← Functor.map_comp, ← op_comp, eqToHom_trans, eqToHom_refl,
op_id, CategoryTheory.Functor.map_id, Category.comp_id]
|
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris Van Doorn, Yury Kudryashov
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_imp... | Mathlib/MeasureTheory/Measure/Regular.lean | 215 | 219 | theorem measure_eq_iSup (H : InnerRegularWRT μ p q) (hU : q U) :
μ U = ⨆ (K) (_ : K ⊆ U) (_ : p K), μ K := by |
refine
le_antisymm (le_of_forall_lt fun r hr => ?_) (iSup₂_le fun K hK => iSup_le fun _ => μ.mono hK)
simpa only [lt_iSup_iff, exists_prop] using H hU r hr
|
/-
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.Connected.Basic
/-!
# Locally connected topological spaces
A topological space is **locally connected** if each neighborhood filter admi... | Mathlib/Topology/Connected/LocallyConnected.lean | 63 | 67 | theorem connectedComponentIn_mem_nhds [LocallyConnectedSpace α] {F : Set α} {x : α} (h : F ∈ 𝓝 x) :
connectedComponentIn F x ∈ 𝓝 x := by |
rw [(LocallyConnectedSpace.open_connected_basis x).mem_iff] at h
rcases h with ⟨s, ⟨h1s, hxs, h2s⟩, hsF⟩
exact mem_nhds_iff.mpr ⟨s, h2s.isPreconnected.subset_connectedComponentIn hxs hsF, h1s, hxs⟩
|
/-
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 | 2,165 | 2,180 | theorem OrthogonalFamily.independent {V : ι → Submodule 𝕜 E}
(hV : OrthogonalFamily 𝕜 (fun i => V i) fun i => (V i).subtypeₗᵢ) :
CompleteLattice.Independent V := by |
classical
apply CompleteLattice.independent_of_dfinsupp_lsum_injective
refine LinearMap.ker_eq_bot.mp ?_
rw [Submodule.eq_bot_iff]
intro v hv
rw [LinearMap.mem_ker] at hv
ext i
suffices ⟪(v i : E), v i⟫ = 0 by simpa only [inner_self_eq_zero] using this
calc
⟪(v i : E), v i⟫ = ⟪(v i : E), DFinsupp... |
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Artinian
#align_import algebra.lie.submodule from "leanprover-communit... | Mathlib/Algebra/Lie/Submodule.lean | 547 | 550 | theorem disjoint_iff_coe_toSubmodule :
Disjoint N N' ↔ Disjoint (N : Submodule R M) (N' : Submodule R M) := by |
rw [disjoint_iff, disjoint_iff, ← coe_toSubmodule_eq_iff, inf_coe_toSubmodule, bot_coeSubmodule,
← disjoint_iff]
|
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.... | Mathlib/Analysis/Calculus/FDeriv/Add.lean | 707 | 709 | theorem fderivWithin_const_sub (hxs : UniqueDiffWithinAt 𝕜 s x) (c : F) :
fderivWithin 𝕜 (fun y => c - f y) s x = -fderivWithin 𝕜 f s x := by |
simp only [sub_eq_add_neg, fderivWithin_const_add, fderivWithin_neg, hxs]
|
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Module.BigOperators
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Squarefree
import Mat... | Mathlib/NumberTheory/ArithmeticFunction.lean | 931 | 936 | theorem IsMultiplicative.ppow [CommSemiring R] {f : ArithmeticFunction R} (hf : f.IsMultiplicative)
{k : ℕ} : IsMultiplicative (f.ppow k) := by |
induction' k with k hi
· exact isMultiplicative_zeta.natCast
· rw [ppow_succ']
apply hf.pmul hi
|
/-
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 | 399 | 400 | theorem rightUnitor_naturality {f g : a ⟶ b} (η : f ⟶ g) :
η ▷ 𝟙 b ≫ (ρ_ g).hom = (ρ_ f).hom ≫ η := by | simp
|
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Order.CompleteLattice
import Mathlib.Order.GaloisConnection
import Mathlib.Data.Set.Lattice
import Mathlib.Tactic.AdaptationNote
#align_import data.rel ... | Mathlib/Data/Rel.lean | 347 | 350 | theorem core_comp (s : Rel β γ) (t : Set γ) : core (r • s) t = core r (core s t) := by |
ext x; simp only [core, comp, forall_exists_index, and_imp, Set.mem_setOf_eq]; constructor
· exact fun h y rxy z => h z y rxy
· exact fun h z y rzy => h y rzy z
|
/-
Copyright (c) 2022 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson, Devon Tuma, Eric Rodriguez, Oliver Nash
-/
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Topology.Algebra.Field
import Mathlib.Topology.Algebra.Order.... | Mathlib/Topology/Algebra/Order/Field.lean | 236 | 239 | theorem eventually_nhdsWithin_pos_mul_left {x : 𝕜} (hx : 0 < x) {p : 𝕜 → Prop}
(h : ∀ᶠ ε in 𝓝[>] 0, p ε) : ∀ᶠ ε in 𝓝[>] 0, p (x * ε) := by |
rw [← nhdsWithin_pos_comap_mul_left hx]
exact h.comap fun ε => x * ε
|
/-
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.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.List.Cycle
import Mathlib.Data.Nat.Prime
impor... | Mathlib/Dynamics/PeriodicPts.lean | 331 | 340 | theorem minimalPeriod_apply_iterate (hx : x ∈ periodicPts f) (n : ℕ) :
minimalPeriod f (f^[n] x) = minimalPeriod f x := by |
apply
(IsPeriodicPt.minimalPeriod_le (minimalPeriod_pos_of_mem_periodicPts hx) _).antisymm
((isPeriodicPt_of_mem_periodicPts_of_isPeriodicPt_iterate hx
(isPeriodicPt_minimalPeriod f _)).minimalPeriod_le
(minimalPeriod_pos_of_mem_periodicPts _))
· exact (isPeriodicPt_minimalPeriod f x)... |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.LocalExtr.Basic
#align_import analysis.calcu... | Mathlib/Analysis/Calculus/Darboux.lean | 148 | 154 | theorem hasDerivWithinAt_forall_lt_or_forall_gt_of_forall_ne {s : Set ℝ} (hs : Convex ℝ s)
(hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) {m : ℝ} (hf' : ∀ x ∈ s, f' x ≠ m) :
(∀ x ∈ s, f' x < m) ∨ ∀ x ∈ s, m < f' x := by |
contrapose! hf'
rcases hf' with ⟨⟨b, hb, hmb⟩, ⟨a, ha, hma⟩⟩
exact (hs.ordConnected.image_hasDerivWithinAt hf).out (mem_image_of_mem f' ha)
(mem_image_of_mem f' hb) ⟨hma, hmb⟩
|
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