Context stringlengths 285 157k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
|---|---|---|---|---|---|
/-
Copyright (c) 2022 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell, Eric Wieser, Yaël Dillies, Patrick Massot, Scott Morrison
-/
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.Order.Interval... | Mathlib/Algebra/Order/Interval/Set/Instances.lean | 266 | 268 | theorem coe_eq_one [Nontrivial α] {x : Ioc (0 : α) 1} : (x : α) = 1 ↔ x = 1 := by |
symm
exact Subtype.ext_iff
|
/-
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.Data.Finset.Sort
import Mathlib.Data.List.FinRange
import Mathlib.Data.Prod.Lex
import Mathlib.GroupTheory.Perm.Basic
import Mathlib.Order.Interval.Finset.Fi... | Mathlib/Data/Fin/Tuple/Sort.lean | 186 | 189 | theorem sort_eq_refl_iff_monotone : sort f = Equiv.refl _ ↔ Monotone f := by |
rw [eq_comm, eq_sort_iff, Equiv.coe_refl, Function.comp_id]
simp only [id, and_iff_left_iff_imp]
exact fun _ _ _ hij _ => hij
|
/-
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 | 493 | 496 | theorem basis_repr (h : IsAdjoinRootMonic S f) (x : S) (i : Fin (natDegree f)) :
h.basis.repr x i = (h.modByMonicHom x).coeff (i : ℕ) := by |
change (h.modByMonicHom x).toFinsupp.comapDomain _ Fin.val_injective.injOn i = _
rw [Finsupp.comapDomain_apply, Polynomial.toFinsupp_apply]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.function.simple_func from "leanprover-community/mathlib"@"bf... | Mathlib/MeasureTheory/Function/SimpleFunc.lean | 1,343 | 1,364 | theorem _root_.Measurable.simpleFunc_add
{E : Type*} {_ : MeasurableSpace α} [MeasurableSpace E] [AddGroup E] [MeasurableAdd E]
{g : α → E} (hg : Measurable g) (f : SimpleFunc α E) :
Measurable ((f : α → E) + g) := by |
classical
induction' f using SimpleFunc.induction with c s hs f f' hff' hf hf'
· simp only [SimpleFunc.const_zero, SimpleFunc.coe_piecewise, SimpleFunc.coe_const,
SimpleFunc.coe_zero]
change Measurable (s.piecewise (Function.const α c) (0 : α → E) + g)
rw [← s.piecewise_same g, ← piecewise_add]
... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Scott Morrison
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib.Data.Set.Finite
#align_import data.finsupp.defs fr... | Mathlib/Data/Finsupp/Defs.lean | 984 | 986 | theorem support_zipWith [D : DecidableEq α] {f : M → N → P} {hf : f 0 0 = 0} {g₁ : α →₀ M}
{g₂ : α →₀ N} : (zipWith f hf g₁ g₂).support ⊆ g₁.support ∪ g₂.support := by |
rw [Subsingleton.elim D] <;> exact support_onFinset_subset
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Set.Pairwise.Basic
import Mathlib.Data.Set.Lattice
import Mathlib.Data.SetLike.Basic
#align_import order.chain from "leanprover-community/mathlib... | Mathlib/Order/Chain.lean | 269 | 275 | theorem ChainClosure.isChain (hc : ChainClosure r c) : IsChain r c := by |
induction hc with
| succ _ h => exact h.succ
| union hs h =>
exact fun c₁ ⟨t₁, ht₁, (hc₁ : c₁ ∈ t₁)⟩ c₂ ⟨t₂, ht₂, (hc₂ : c₂ ∈ t₂)⟩ hneq =>
((hs _ ht₁).total <| hs _ ht₂).elim (fun ht => h t₂ ht₂ (ht hc₁) hc₂ hneq) fun ht =>
h t₁ ht₁ hc₁ (ht hc₂) hneq
|
/-
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 | 1,370 | 1,394 | theorem prod_eq_iff_prod_pow_moebius_eq_on_of_nonzero [CommGroupWithZero R]
(s : Set ℕ) (hs : ∀ m n, m ∣ n → n ∈ s → m ∈ s) {f g : ℕ → R}
(hf : ∀ n > 0, f n ≠ 0) (hg : ∀ n > 0, g n ≠ 0):
(∀ n > 0, n ∈ s → (∏ i ∈ n.divisors, f i) = g n) ↔
∀ n > 0, n ∈ s → (∏ x ∈ n.divisorsAntidiagonal, g x.snd ^ μ x.fs... |
refine
Iff.trans
(Iff.trans (forall_congr' fun n => ?_)
(@prod_eq_iff_prod_pow_moebius_eq_on Rˣ _
(fun n => if h : 0 < n then Units.mk0 (f n) (hf n h) else 1)
(fun n => if h : 0 < n then Units.mk0 (g n) (hg n h) else 1)
s hs) )
(forall_congr' fun n ... |
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Sébastien Gouëzel, Yury G. Kudryashov, Dylan MacKenzie, Patrick Massot
-/
import Mathlib.Algebra.BigOperators.Module
import Mathlib.Algebra.Order.Field.Basic
impor... | Mathlib/Analysis/SpecificLimits/Normed.lean | 293 | 301 | theorem hasSum_geometric_of_norm_lt_one (h : ‖ξ‖ < 1) : HasSum (fun n : ℕ ↦ ξ ^ n) (1 - ξ)⁻¹ := by |
have xi_ne_one : ξ ≠ 1 := by
contrapose! h
simp [h]
have A : Tendsto (fun n ↦ (ξ ^ n - 1) * (ξ - 1)⁻¹) atTop (𝓝 ((0 - 1) * (ξ - 1)⁻¹)) :=
((tendsto_pow_atTop_nhds_zero_of_norm_lt_one h).sub tendsto_const_nhds).mul tendsto_const_nhds
rw [hasSum_iff_tendsto_nat_of_summable_norm]
· simpa [geom_sum_eq... |
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc429200506... | Mathlib/Data/Set/Image.lean | 929 | 930 | theorem preimage_inr_range_inl : Sum.inr ⁻¹' range (Sum.inl : α → Sum α β) = ∅ := by |
rw [← image_univ, preimage_inr_image_inl]
|
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.Analysis.Complex.Basic
#align_import measure_theory.measure.vector_measure from "leanprover-community/mathl... | Mathlib/MeasureTheory/Measure/VectorMeasure.lean | 1,363 | 1,366 | theorem toMeasureOfZeroLE_toSignedMeasure (hs : 0 ≤[Set.univ] s) :
(s.toMeasureOfZeroLE Set.univ MeasurableSet.univ hs).toSignedMeasure = s := by |
ext i hi
simp [hi, toMeasureOfZeroLE_apply _ _ _ hi]
|
/-
Copyright (c) 2018 Michael Jendrusch. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Jendrusch, Scott Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Monoidal.Category
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheo... | Mathlib/CategoryTheory/Monoidal/Functor.lean | 243 | 246 | theorem OplaxMonoidalFunctor.right_unitality_hom (F : OplaxMonoidalFunctor C D) (X : C) :
F.δ X (𝟙_ C) ≫ F.obj X ◁ F.η ≫ (ρ_ (F.obj X)).hom = F.map (ρ_ X).hom := by |
rw [← Category.assoc, ← Iso.eq_comp_inv, F.right_unitality, ← Category.assoc,
← F.toFunctor.map_comp, Iso.hom_inv_id, F.toFunctor.map_id, id_comp]
|
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTh... | Mathlib/MeasureTheory/Function/LpSpace.lean | 1,920 | 1,924 | theorem hasSum_of_hasSum_Lp {β : Type*} [μ.IsOpenPosMeasure] [NormedField 𝕜] [NormedSpace 𝕜 E]
{g : β → C(α, E)} {f : C(α, E)} (hg : Summable g)
(hg2 : HasSum (toLp (E := E) p μ 𝕜 ∘ g) (toLp (E := E) p μ 𝕜 f)) : HasSum g f := by |
convert Summable.hasSum hg
exact toLp_injective μ (hg2.unique ((toLp p μ 𝕜).hasSum <| Summable.hasSum hg))
|
/-
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 | 62 | 65 | theorem empty_mem_generateMeasurableRec (s : Set (Set α)) (i : ω₁) :
∅ ∈ generateMeasurableRec s i := by |
unfold generateMeasurableRec
exact mem_union_left _ (mem_union_left _ (mem_union_right _ (mem_singleton ∅)))
|
/-
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 | 926 | 927 | theorem isBigOWith_neg_left : IsBigOWith c l (fun x => -f' x) g ↔ IsBigOWith c l f' g := by |
simp only [IsBigOWith_def, norm_neg]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Order.PartialSups
#align_import order.disjointed from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
/-!
# C... | Mathlib/Order/Disjointed.lean | 114 | 118 | theorem partialSups_disjointed (f : ℕ → α) : partialSups (disjointed f) = partialSups f := by |
ext n
induction' n with k ih
· rw [partialSups_zero, partialSups_zero, disjointed_zero]
· rw [partialSups_succ, partialSups_succ, disjointed_succ, ih, sup_sdiff_self_right]
|
/-
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 | 132 | 135 | theorem fourier_zero {x : AddCircle T} : fourier 0 x = 1 := by |
induction x using QuotientAddGroup.induction_on'
simp only [fourier_coe_apply]
norm_num
|
/-
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, Alexander Bentkamp
-/
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Data.Fintype.BigOperators
i... | Mathlib/LinearAlgebra/Basis.lean | 908 | 910 | theorem Basis.equivFun_symm_apply [Fintype ι] (b : Basis ι R M) (x : ι → R) :
b.equivFun.symm x = ∑ i, x i • b i := by |
simp [Basis.equivFun, Finsupp.total_apply, Finsupp.sum_fintype, Finsupp.equivFunOnFinite]
|
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Group.Measure
#align_import measure_theory.group.integration from "leanprover-communit... | Mathlib/MeasureTheory/Group/Integral.lean | 40 | 43 | theorem integral_inv_eq_self (f : G → E) (μ : Measure G) [IsInvInvariant μ] :
∫ x, f x⁻¹ ∂μ = ∫ x, f x ∂μ := by |
have h : MeasurableEmbedding fun x : G => x⁻¹ := (MeasurableEquiv.inv G).measurableEmbedding
rw [← h.integral_map, map_inv_eq_self]
|
/-
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.Real
#align_import analys... | Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 551 | 551 | theorem rpow_neg_one (x : ℝ≥0∞) : x ^ (-1 : ℝ) = x⁻¹ := by | simp [rpow_neg]
|
/-
Copyright (c) 2019 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, François Dupuis
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Order.Filter.Extr
import Mathlib.Tactic.GCongr
#align_import analysis.convex.function fr... | Mathlib/Analysis/Convex/Function.lean | 520 | 526 | theorem StrictConvexOn.add (hf : StrictConvexOn 𝕜 s f) (hg : StrictConvexOn 𝕜 s g) :
StrictConvexOn 𝕜 s (f + g) :=
⟨hf.1, fun x hx y hy hxy a b ha hb hab =>
calc
f (a • x + b • y) + g (a • x + b • y) < a • f x + b • f y + (a • g x + b • g y) :=
add_lt_add (hf.2 hx hy hxy ha hb hab) (hg.2 hx h... | rw [smul_add, smul_add, add_add_add_comm]⟩
|
/-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang
-/
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.NormedSpace.LinearIsometry
#align_import analysis.normed_space.conformal_linear_map from "leanprover-co... | Mathlib/Analysis/NormedSpace/ConformalLinearMap.lean | 97 | 100 | theorem ne_zero [Nontrivial M'] {f' : M' →L[R] N} (hf' : IsConformalMap f') : f' ≠ 0 := by |
rintro rfl
rcases exists_ne (0 : M') with ⟨a, ha⟩
exact ha (hf'.injective rfl)
|
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.PSub
import Mathlib.Data.Nat... | Mathlib/Data/Num/Lemmas.lean | 145 | 147 | theorem cmp_swap (m) : ∀ n, (cmp m n).swap = cmp n m := by |
induction' m with m IH m IH <;> intro n <;> cases' n with n n <;> unfold cmp <;>
try { rfl } <;> rw [← IH] <;> cases cmp m n <;> rfl
|
/-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Order.Basic
import Mathlib.Order.Monotone.Basic
#align_import algebra.covariant_and_contravariant from "leanprover-co... | Mathlib/Algebra/Order/Monoid/Unbundled/Defs.lean | 323 | 324 | theorem contravariant_flip_iff [IsSymmOp N N mu] :
Contravariant N N (flip mu) r ↔ Contravariant N N mu r := by | rw [IsSymmOp.flip_eq]
|
/-
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 | 460 | 462 | theorem IsNormal.le_set' {f o} (H : IsNormal f) (p : Set α) (p0 : p.Nonempty) (g : α → Ordinal) (b)
(H₂ : ∀ o, b ≤ o ↔ ∀ a ∈ p, g a ≤ o) : f b ≤ o ↔ ∀ a ∈ p, f (g a) ≤ o := by |
simpa [H₂] using H.le_set (g '' p) (p0.image g) b
|
/-
Copyright (c) 2018 Violeta Hernández Palacios, Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios, Mario Carneiro
-/
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Exponential
#align_import set_th... | Mathlib/SetTheory/Ordinal/FixedPoint.lean | 229 | 239 | theorem derivFamily_eq_enumOrd (H : ∀ i, IsNormal (f i)) :
derivFamily.{u, v} f = enumOrd (⋂ i, Function.fixedPoints (f i)) := by |
rw [← eq_enumOrd _ (fp_family_unbounded.{u, v} H)]
use (derivFamily_isNormal f).strictMono
rw [Set.range_eq_iff]
refine ⟨?_, fun a ha => ?_⟩
· rintro a S ⟨i, hi⟩
rw [← hi]
exact derivFamily_fp (H i) a
rw [Set.mem_iInter] at ha
rwa [← fp_iff_derivFamily H]
|
/-
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.Algebra.RestrictScalars
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.LinearAlgebra.Quotient
import Mathlib.LinearAlgebra.StdB... | Mathlib/RingTheory/Finiteness.lean | 137 | 141 | theorem exists_mem_and_smul_eq_self_of_fg_of_le_smul {R : Type*} [CommRing R] {M : Type*}
[AddCommGroup M] [Module R M] (I : Ideal R) (N : Submodule R M) (hn : N.FG) (hin : N ≤ I • N) :
∃ r ∈ I, ∀ n ∈ N, r • n = n := by |
obtain ⟨r, hr, hr'⟩ := exists_sub_one_mem_and_smul_eq_zero_of_fg_of_le_smul I N hn hin
exact ⟨-(r - 1), I.neg_mem hr, fun n hn => by simpa [sub_smul] using hr' n hn⟩
|
/-
Copyright (c) 2020 Hanting Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hanting Zhang, Johan Commelin
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.MvPolynomial.CommRing
#align_import r... | Mathlib/RingTheory/MvPolynomial/Symmetric.lean | 272 | 276 | theorem support_esymm (n : ℕ) [DecidableEq σ] [Nontrivial R] :
(esymm σ R n).support =
(powersetCard n (univ : Finset σ)).image fun t => ∑ i ∈ t, Finsupp.single i 1 := by |
rw [support_esymm']
exact biUnion_singleton
|
/-
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 | 810 | 814 | theorem coeff_monomial_mul' (m) (s : σ →₀ ℕ) (r : R) (p : MvPolynomial σ R) :
coeff m (monomial s r * p) = if s ≤ m then r * coeff (m - s) p else 0 := by |
-- note that if we allow `R` to be non-commutative we will have to duplicate the proof above.
rw [mul_comm, mul_comm r]
exact coeff_mul_monomial' _ _ _ _
|
/-
Copyright (c) 2023 Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémi Bottinelli
-/
import Mathlib.Data.Set.Function
import Mathlib.Analysis.BoundedVariation
#align_import analysis.constant_speed from "leanprover-community/mathlib"@"f0c8bf9245297a... | Mathlib/Analysis/ConstantSpeed.lean | 140 | 153 | theorem HasConstantSpeedOnWith.Icc_Icc {x y z : ℝ} (hfs : HasConstantSpeedOnWith f (Icc x y) l)
(hft : HasConstantSpeedOnWith f (Icc y z) l) : HasConstantSpeedOnWith f (Icc x z) l := by |
rcases le_total x y with (xy | yx)
· rcases le_total y z with (yz | zy)
· rw [← Set.Icc_union_Icc_eq_Icc xy yz]
exact hfs.union hft (isGreatest_Icc xy) (isLeast_Icc yz)
· rintro u ⟨xu, uz⟩ v ⟨xv, vz⟩
rw [Icc_inter_Icc, sup_of_le_right xu, inf_of_le_right vz, ←
hfs ⟨xu, uz.trans zy⟩ ⟨xv,... |
/-
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.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 308 | 310 | theorem arg_of_re_neg_of_im_nonneg {x : ℂ} (hx_re : x.re < 0) (hx_im : 0 ≤ x.im) :
arg x = Real.arcsin ((-x).im / abs x) + π := by |
simp only [arg, hx_re.not_le, hx_im, if_true, if_false]
|
/-
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.Basic
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Algebra.Subalgebra.Basic
#align_import a... | Mathlib/Algebra/Lie/OfAssociative.lean | 346 | 350 | theorem LieSubalgebra.ad_comp_incl_eq (K : LieSubalgebra R L) (x : K) :
(ad R L ↑x).comp (K.incl : K →ₗ[R] L) = (K.incl : K →ₗ[R] L).comp (ad R K x) := by |
ext y
simp only [ad_apply, LieHom.coe_toLinearMap, LieSubalgebra.coe_incl, LinearMap.coe_comp,
LieSubalgebra.coe_bracket, Function.comp_apply]
|
/-
Copyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Fin
import Mathlib.GroupTheo... | Mathlib/Algebra/BigOperators/Fin.lean | 179 | 179 | theorem prod_const [CommMonoid α] (n : ℕ) (x : α) : ∏ _i : Fin n, x = x ^ n := by | 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 | 191 | 191 | theorem left_mem_Icc : a ∈ Icc a b ↔ a ≤ b := by | simp [le_refl]
|
/-
Copyright (c) 2021 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Eric Wieser
-/
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.GradedAlgebra.Basic... | Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean | 233 | 236 | theorem Ideal.IsHomogeneous.iff_exists :
I.IsHomogeneous 𝒜 ↔ ∃ S : Set (homogeneousSubmonoid 𝒜), I = Ideal.span ((↑) '' S) := by |
rw [Ideal.IsHomogeneous.iff_eq, eq_comm]
exact ((Set.image_preimage.compose (Submodule.gi _ _).gc).exists_eq_l _).symm
|
/-
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.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mat... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | 192 | 200 | theorem condexp_congr_ae (h : f =ᵐ[μ] g) : μ[f|m] =ᵐ[μ] μ[g|m] := by |
by_cases hm : m ≤ m0
swap; · simp_rw [condexp_of_not_le hm]; rfl
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · simp_rw [condexp_of_not_sigmaFinite hm hμm]; rfl
haveI : SigmaFinite (μ.trim hm) := hμm
exact (condexp_ae_eq_condexpL1 hm f).trans
(Filter.EventuallyEq.trans (by rw [condexpL1_congr_ae hm h])... |
/-
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.RingTheory.WittVector.Frobenius
import Mathlib.RingTheory.WittVector.Verschiebung
import Mathlib.RingTheory.WittVector.MulP
#align_import ring_theory.... | Mathlib/RingTheory/WittVector/Identities.lean | 87 | 87 | theorem coeff_p_one [CharP R p] : (p : 𝕎 R).coeff 1 = 1 := by | rw [coeff_p, if_pos rfl]
|
/-
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 | 244 | 252 | theorem map' {α β} [MeasurableSpace α] [MeasurableSpace β] {μ : Measure α} {pa qa : Set α → Prop}
(H : InnerRegularWRT μ pa qa) (f : α ≃ᵐ β) {pb qb : Set β → Prop}
(hAB : ∀ U, qb U → qa (f ⁻¹' U)) (hAB' : ∀ K, pa K → pb (f '' K)) :
InnerRegularWRT (map f μ) pb qb := by |
intro U hU r hr
rw [f.map_apply U] at hr
rcases H (hAB U hU) r hr with ⟨K, hKU, hKc, hK⟩
refine ⟨f '' K, image_subset_iff.2 hKU, hAB' _ hKc, ?_⟩
rwa [f.map_apply, f.preimage_image]
|
/-
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 | 473 | 488 | theorem aestronglyMeasurable'_condexpL1CLM (f : α →₁[μ] F') :
AEStronglyMeasurable' m (condexpL1CLM F' hm μ f) μ := by |
refine @Lp.induction _ _ _ _ _ _ _ ENNReal.one_ne_top
(fun f : α →₁[μ] F' => AEStronglyMeasurable' m (condexpL1CLM F' hm μ f) μ) ?_ ?_ ?_ f
· intro c s hs hμs
rw [condexpL1CLM_indicatorConst hs hμs.ne c]
exact aestronglyMeasurable'_condexpInd hs hμs.ne c
· intro f g hf hg _ hfm hgm
rw [(condexpL1... |
/-
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.RingTheory.WittVector.Frobenius
import Mathlib.RingTheory.WittVector.Verschiebung
import Mathlib.RingTheory.WittVector.MulP
#align_import ring_theory.... | Mathlib/RingTheory/WittVector/Identities.lean | 42 | 47 | theorem frobenius_verschiebung (x : 𝕎 R) : frobenius (verschiebung x) = x * p := by |
have : IsPoly p fun {R} [CommRing R] x ↦ frobenius (verschiebung x) :=
IsPoly.comp (hg := frobenius_isPoly p) (hf := verschiebung_isPoly)
have : IsPoly p fun {R} [CommRing R] x ↦ x * p := mulN_isPoly p p
ghost_calc x
ghost_simp [mul_comm]
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Rat.Cast.Defs
import Mathlib.Algebra.Field.Basic
#align_import data.rat.cast from "leanprover-community/mathlib"@"acebd8d49928f6e... | Mathlib/Data/Rat/Cast/Lemmas.lean | 37 | 40 | theorem cast_inv_int (n : ℤ) : ((n⁻¹ : ℚ) : α) = (n : α)⁻¹ := by |
cases' n with n n
· simp [ofInt_eq_cast, cast_inv_nat]
· simp only [ofInt_eq_cast, Int.cast_negSucc, ← Nat.cast_succ, cast_neg, inv_neg, cast_inv_nat]
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.Extension
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.GroupTheory.Solvable
#align_import fi... | Mathlib/FieldTheory/Normal.lean | 425 | 429 | theorem exists_algEquiv_of_root' [Normal K L]{x y : L} (hy : IsAlgebraic K y)
(h_ev : (Polynomial.aeval x) (minpoly K y) = 0) : ∃ σ : L ≃ₐ[K] L, σ y = x := by |
obtain ⟨σ, hσ⟩ := exists_algEquiv_of_root hy h_ev
use σ.symm
rw [← hσ, symm_apply_apply]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, James Gallicchio
-/
import Batteries.Data.List.Count
import Batteries.Data.Fin.Lemmas
/-!
# Pairwise relations on a list
This file provides basic results about `List.... | .lake/packages/batteries/Batteries/Data/List/Pairwise.lean | 128 | 142 | theorem pairwise_filterMap (f : β → Option α) {l : List β} :
Pairwise R (filterMap f l) ↔ Pairwise (fun a a' : β => ∀ b ∈ f a, ∀ b' ∈ f a', R b b') l := by |
let _S (a a' : β) := ∀ b ∈ f a, ∀ b' ∈ f a', R b b'
simp only [Option.mem_def]
induction l with
| nil => simp only [filterMap, Pairwise.nil]
| cons a l IH => ?_
match e : f a with
| none =>
rw [filterMap_cons_none _ _ e, pairwise_cons]
simp only [e, false_implies, implies_true, true_and, IH]
| ... |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Data.Fin.VecNotation
import Mathlib.Logic.Embedding.Set
#align_import logic.equiv.fin from "leanprover-community/mathlib"@"bd835ef554f37ef9b804f0903089211f89cb3... | Mathlib/Logic/Equiv/Fin.lean | 126 | 128 | theorem finSuccEquiv'_above {i : Fin (n + 1)} {m : Fin n} (h : i ≤ Fin.castSucc m) :
(finSuccEquiv' i) m.succ = some m := by |
rw [← Fin.succAbove_of_le_castSucc _ _ h, finSuccEquiv'_succAbove]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Set
import Mathlib.Data.Nat.Set
import Mathlib.Data.Set.Prod
import Mathlib.Data.ULift
import Mathlib.Order.Bounds.Basic
import Mathlib.Order... | Mathlib/Order/CompleteLattice.lean | 486 | 490 | theorem eq_singleton_bot_of_sSup_eq_bot_of_nonempty {s : Set α} (h_sup : sSup s = ⊥)
(hne : s.Nonempty) : s = {⊥} := by |
rw [Set.eq_singleton_iff_nonempty_unique_mem]
rw [sSup_eq_bot] at h_sup
exact ⟨hne, h_sup⟩
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.SmoothManifoldWithCorners
import Mathlib.Geometry.Manifold.LocalInvariantProperties
#align_import geometry.m... | Mathlib/Geometry/Manifold/ContMDiff/Defs.lean | 494 | 497 | theorem contMDiffAt_iff_source_of_mem_source {x' : M} (hx' : x' ∈ (chartAt H x).source) :
ContMDiffAt I I' n f x' ↔
ContMDiffWithinAt 𝓘(𝕜, E) I' n (f ∘ (extChartAt I x).symm) (range I) (extChartAt I x x') := by |
simp_rw [ContMDiffAt, contMDiffWithinAt_iff_source_of_mem_source hx', preimage_univ, univ_inter]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Yury G. Kudryashov, Patrick Massot
-/
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Iterate
import Math... | Mathlib/Analysis/SpecificLimits/Basic.lean | 709 | 710 | theorem tendsto_nat_ceil_div_atTop : Tendsto (fun x ↦ (⌈x⌉₊ : R) / x) atTop (𝓝 1) := by |
simpa using tendsto_nat_ceil_mul_div_atTop (zero_le_one' R)
|
/-
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Alex Kontorovich
-/
import Mathlib.Order.Filter.Bases
#align_import order.filter.pi from "leanprover-community/mathlib"@"ce64cd319bb6b3e82f31c2d38e79080d377be4... | Mathlib/Order/Filter/Pi.lean | 248 | 250 | theorem coprodᵢ_eq_bot_iff' : Filter.coprodᵢ f = ⊥ ↔ (∃ i, IsEmpty (α i)) ∨ f = ⊥ := by |
simpa only [not_neBot, not_and_or, funext_iff, not_forall, not_exists, not_nonempty_iff]
using coprodᵢ_neBot_iff'.not
|
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Jujian Zhang
-/
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.RingTheory.Localization.Basic
#align_import algebra.module.localized_module from "leanprover-communit... | Mathlib/Algebra/Module/LocalizedModule.lean | 1,102 | 1,105 | theorem mk'_surjective : Function.Surjective (Function.uncurry <| mk' f : M × S → M') := by |
intro x
obtain ⟨⟨m, s⟩, e : s • x = f m⟩ := IsLocalizedModule.surj S f x
exact ⟨⟨m, s⟩, mk'_eq_iff.mpr e.symm⟩
|
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Algebra.PUnitInstances
import Mathlib.Tactic.Abel
import Mathlib.Tactic.Ring
import Mathlib.Order.Hom.Lattice
#align_import algebr... | Mathlib/Algebra/Ring/BooleanRing.lean | 230 | 232 | theorem le_sup_inf (a b c : α) : (a ⊔ b) ⊓ (a ⊔ c) ⊔ (a ⊔ b ⊓ c) = a ⊔ b ⊓ c := by |
dsimp only [(· ⊔ ·), (· ⊓ ·)]
rw [le_sup_inf_aux, add_self, mul_self, zero_add]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.MFDeriv.Defs
#align_import geometry.manifold.mfderiv from "leanprover-community/mathlib"@"e473c3198bb41f6856... | Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | 204 | 208 | theorem hasMFDerivWithinAt_inter' (h : t ∈ 𝓝[s] x) :
HasMFDerivWithinAt I I' f (s ∩ t) x f' ↔ HasMFDerivWithinAt I I' f s x f' := by |
rw [HasMFDerivWithinAt, HasMFDerivWithinAt, extChartAt_preimage_inter_eq,
hasFDerivWithinAt_inter', continuousWithinAt_inter' h]
exact extChartAt_preimage_mem_nhdsWithin I h
|
/-
Copyright (c) 2022 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Balanced
import Mathlib.CategoryTheory.Limits.EssentiallySmall
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.CategoryTheor... | Mathlib/CategoryTheory/Generator.lean | 182 | 184 | theorem IsCoseparating.isCodetecting [Balanced C] {𝒢 : Set C} :
IsCoseparating 𝒢 → IsCodetecting 𝒢 := by |
simpa only [← isDetecting_op_iff, ← isSeparating_op_iff] using IsSeparating.isDetecting
|
/-
Copyright (c) 2022 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.BernoulliPolynomials
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathl... | Mathlib/NumberTheory/ZetaValues.lean | 332 | 348 | theorem hasSum_zeta_nat {k : ℕ} (hk : k ≠ 0) :
HasSum (fun n : ℕ => 1 / (n : ℝ) ^ (2 * k))
((-1 : ℝ) ^ (k + 1) * (2 : ℝ) ^ (2 * k - 1) * π ^ (2 * k) *
bernoulli (2 * k) / (2 * k)!) := by |
convert hasSum_one_div_nat_pow_mul_cos hk (left_mem_Icc.mpr zero_le_one) using 1
· ext1 n; rw [mul_zero, Real.cos_zero, mul_one]
rw [Polynomial.eval_zero_map, Polynomial.bernoulli_eval_zero, eq_ratCast]
have : (2 : ℝ) ^ (2 * k - 1) = (2 : ℝ) ^ (2 * k) / 2 := by
rw [eq_div_iff (two_ne_zero' ℝ)]
conv_lhs... |
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.ContinuousOn
#align_import topology.algebra.order.left_right from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e... | Mathlib/Topology/Order/LeftRight.lean | 111 | 112 | theorem nhds_left_sup_nhds_right (a : α) : 𝓝[≤] a ⊔ 𝓝[≥] a = 𝓝 a := by |
rw [← nhdsWithin_union, Iic_union_Ici, nhdsWithin_univ]
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.RCLike.Basic
import Mathlib... | Mathlib/Analysis/Complex/Basic.lean | 137 | 139 | theorem dist_conj_self (z : ℂ) : dist (conj z) z = 2 * |z.im| := by |
rw [dist_of_re_eq (conj_re z), conj_im, dist_comm, Real.dist_eq, sub_neg_eq_add, ← two_mul,
_root_.abs_mul, abs_of_pos (zero_lt_two' ℝ)]
|
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Module.Equiv
import Mathlib.Data.DFinsupp.Basic
import Mathlib.Data.Finsupp.Basic
#align_import data.finsupp.to_dfinsupp from "leanprover-community/... | Mathlib/Data/Finsupp/ToDFinsupp.lean | 359 | 363 | theorem sigmaFinsuppEquivDFinsupp_smul {R} [Monoid R] [AddMonoid N] [DistribMulAction R N] (r : R)
(f : (Σ i, η i) →₀ N) :
sigmaFinsuppEquivDFinsupp (r • f) = r • sigmaFinsuppEquivDFinsupp f := by |
ext
rfl
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.MvPolynomial.Monad
#align_import data.mv_polynomial.expand from "leanprover-community/mathlib"@"5da451b4c96b4c2e122c0325a7fce... | Mathlib/Algebra/MvPolynomial/Expand.lean | 88 | 92 | theorem rename_comp_expand (f : σ → τ) (p : ℕ) :
(rename f).comp (expand p) =
(expand p).comp (rename f : MvPolynomial σ R →ₐ[R] MvPolynomial τ R) := by |
ext1 φ
simp only [rename_expand, AlgHom.comp_apply]
|
/-
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.Algebra.Algebra.Equiv
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearA... | Mathlib/Algebra/Quaternion.lean | 1,309 | 1,310 | theorem normSq_coe : normSq (x : ℍ[R]) = x ^ 2 := by |
rw [normSq_def, star_coe, ← coe_mul, coe_re, sq]
|
/-
Copyright (c) 2021 Yourong Zang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yourong Zang, Yury Kudryashov
-/
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanpr... | Mathlib/Topology/Compactification/OnePoint.lean | 334 | 338 | theorem hasBasis_nhds_infty :
(𝓝 (∞ : OnePoint X)).HasBasis (fun s : Set X => IsClosed s ∧ IsCompact s) fun s =>
(↑) '' sᶜ ∪ {∞} := by |
rw [nhds_infty_eq]
exact (hasBasis_coclosedCompact.map _).sup_pure _
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Submodule.EqLocus
import Mathlib.Algebra.Module.Subm... | Mathlib/LinearAlgebra/Span.lean | 847 | 847 | theorem prod_bot : (prod ⊥ ⊥ : Submodule R (M × M')) = ⊥ := by | ext ⟨x, y⟩; simp [Prod.zero_eq_mk]
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathli... | Mathlib/Data/Set/Lattice.lean | 913 | 914 | theorem biInter_pair (a b : α) (s : α → Set β) : ⋂ x ∈ ({a, b} : Set α), s x = s a ∩ s b := by |
rw [biInter_insert, biInter_singleton]
|
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Eric Wieser
-/
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.matrix from "leanprover-community/mathl... | Mathlib/Analysis/Matrix.lean | 102 | 104 | theorem nnnorm_lt_iff {r : ℝ≥0} (hr : 0 < r) {A : Matrix m n α} :
‖A‖₊ < r ↔ ∀ i j, ‖A i j‖₊ < r := by |
simp_rw [nnnorm_def, pi_nnnorm_lt_iff hr]
|
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Group.Subsemigroup.Basic
#align_import group_theory.subsemigroup.membership from "leanprover-community/mathlib"@"6cb77a8eaff0ddd100e87b1591c6d3a... | Mathlib/Algebra/Group/Subsemigroup/Membership.lean | 67 | 70 | theorem mem_sSup_of_directed_on {S : Set (Subsemigroup M)} (hS : DirectedOn (· ≤ ·) S) {x : M} :
x ∈ sSup S ↔ ∃ s ∈ S, x ∈ s := by |
simp only [sSup_eq_iSup', mem_iSup_of_directed hS.directed_val, SetCoe.exists, Subtype.coe_mk,
exists_prop]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Devon Tuma
-/
import Mathlib.Probability.ProbabilityMassFunction.Monad
#align_import probability.probability_mass_function.constructions from "leanprover-community/mat... | Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | 316 | 321 | theorem support_bernoulli : (bernoulli p h).support = { b | cond b (p ≠ 0) (p ≠ 1) } := by |
refine Set.ext fun b => ?_
induction b
· simp_rw [mem_support_iff, bernoulli_apply, Bool.cond_false, Ne, tsub_eq_zero_iff_le, not_le]
exact ⟨ne_of_lt, lt_of_le_of_ne h⟩
· simp only [mem_support_iff, bernoulli_apply, Bool.cond_true, Set.mem_setOf_eq]
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Sébastien Gouëzel, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.LinearAlgebra.FiniteDimen... | Mathlib/Analysis/InnerProductSpace/PiL2.lean | 317 | 322 | theorem EuclideanSpace.orthonormal_single :
Orthonormal 𝕜 fun i : ι => EuclideanSpace.single i (1 : 𝕜) := by |
simp_rw [orthonormal_iff_ite, EuclideanSpace.inner_single_left, map_one, one_mul,
EuclideanSpace.single_apply]
intros
trivial
|
/-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Set.Pointwise.... | Mathlib/GroupTheory/Finiteness.lean | 173 | 179 | theorem Monoid.fg_of_surjective {M' : Type*} [Monoid M'] [Monoid.FG M] (f : M →* M')
(hf : Function.Surjective f) : Monoid.FG M' := by |
classical
obtain ⟨s, hs⟩ := Monoid.fg_def.mp ‹_›
use s.image f
rwa [Finset.coe_image, ← MonoidHom.map_mclosure, hs, ← MonoidHom.mrange_eq_map,
MonoidHom.mrange_top_iff_surjective]
|
/-
Copyright (c) 2021 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.MvPolynomial.Variables
#align_import data.mv_polynomial.supported from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
... | Mathlib/Algebra/MvPolynomial/Supported.lean | 117 | 118 | theorem X_mem_supported [Nontrivial R] {i : σ} : X i ∈ supported R s ↔ i ∈ s := by |
simp [mem_supported]
|
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Data.Nat.Gcd
import Batteries.Data.Int.DivMod
import Batteries.Lean.Float
/-! # Basics for the Rational Numbers -/
/--
Rational numbers, implemente... | .lake/packages/batteries/Batteries/Data/Rat/Basic.lean | 239 | 245 | theorem sub.aux (a b : Rat) {g ad bd} (hg : g = a.den.gcd b.den)
(had : ad = a.den / g) (hbd : bd = b.den / g) :
let den := ad * b.den; let num := a.num * bd - b.num * ad
num.natAbs.gcd g = num.natAbs.gcd den := by |
have := add.aux a (-b) hg had hbd
simp only [show (-b).num = -b.num from rfl, Int.neg_mul] at this
exact this
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Order.Filter.SmallSets
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Compactness.Compact
import Mathlib.To... | Mathlib/Topology/UniformSpace/Basic.lean | 1,404 | 1,407 | theorem uniformContinuous_iInf_rng {f : α → β} {u₁ : UniformSpace α} {u₂ : ι → UniformSpace β} :
UniformContinuous[u₁, iInf u₂] f ↔ ∀ i, UniformContinuous[u₁, u₂ i] f := by |
delta UniformContinuous
rw [iInf_uniformity, tendsto_iInf]
|
/-
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 | 76 | 90 | theorem Set.OrdConnected.image_hasDerivWithinAt {s : Set ℝ} (hs : OrdConnected s)
(hf : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x) : OrdConnected (f' '' s) := by |
apply ordConnected_of_Ioo
rintro _ ⟨a, ha, rfl⟩ _ ⟨b, hb, rfl⟩ - m ⟨hma, hmb⟩
rcases le_total a b with hab | hab
· have : Icc a b ⊆ s := hs.out ha hb
rcases exists_hasDerivWithinAt_eq_of_gt_of_lt hab (fun x hx => (hf x <| this hx).mono this) hma
hmb with
⟨c, cmem, hc⟩
exact ⟨c, this <| Io... |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.AlgebraicGeometry.Pullbacks
import Mathlib.CategoryTheory.MorphismProperty.Limits
import Mathlib.Data.List.TFAE... | Mathlib/AlgebraicGeometry/Morphisms/Basic.lean | 415 | 422 | theorem PropertyIsLocalAtTarget.openCover_iff {P : MorphismProperty Scheme}
(hP : PropertyIsLocalAtTarget P) {X Y : Scheme.{u}} (f : X ⟶ Y) (𝒰 : Scheme.OpenCover.{u} Y) :
P f ↔ ∀ i, P (pullback.snd : pullback f (𝒰.map i) ⟶ _) := by |
-- Porting note: couldn't get the term mode proof work
refine ⟨fun H => let h := ((hP.openCover_TFAE f).out 0 2).mp H; fun i => ?_,
fun H => let h := ((hP.openCover_TFAE f).out 1 0).mp; ?_⟩
· exact h 𝒰 i
· exact h ⟨𝒰, H⟩
|
/-
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 | 53 | 53 | theorem factors_two : factors 2 = [2] := by | simp [factors]
|
/-
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 | 340 | 341 | theorem card_finset_fin_le {n : ℕ} (s : Finset (Fin n)) : s.card ≤ n := by |
simpa only [Fintype.card_fin] using s.card_le_univ
|
/-
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.Order.Antichain
import Mathlib.Order.UpperLower.Basic
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.RelIso.Set
#align_import order.minimal ... | Mathlib/Order/Minimal.lean | 489 | 490 | theorem minimals_Ico (hab : a < b) : minimals (· ≤ ·) (Ico a b) = {a} := by |
simp_rw [Ico, and_comm (a := _ ≤ _)]; exact maximals_Ioc (α := αᵒᵈ) hab
|
/-
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.LinearAlgebra.Matrix.BilinearForm
import Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.... | Mathlib/RingTheory/Trace.lean | 109 | 111 | theorem trace_eq_matrix_trace [DecidableEq ι] (b : Basis ι R S) (s : S) :
trace R S s = Matrix.trace (Algebra.leftMulMatrix b s) := by |
rw [trace_apply, LinearMap.trace_eq_matrix_trace _ b, ← toMatrix_lmul_eq]; rfl
|
/-
Copyright (c) 2019 Zhouhang Zhou. 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.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-communit... | Mathlib/MeasureTheory/Integral/Bochner.lean | 714 | 717 | theorem integral_smul (c : 𝕜) (f : α →₁[μ] E) : integral (c • f) = c • integral f := by |
simp only [integral]
show (integralCLM' (E := E) 𝕜) (c • f) = c • (integralCLM' (E := E) 𝕜) f
exact map_smul (integralCLM' (E := E) 𝕜) c f
|
/-
Copyright (c) 2022 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johanes Hölzl, Patrick Massot, Yury Kudryashov, Kevin Wilson, Heather Macbeth
-/
import Mathlib.Order.Filter.Basic
#align_import order.filter.prod from "leanprover-community/mathlib"@... | Mathlib/Order/Filter/Prod.lean | 208 | 211 | theorem Eventually.diag_of_prod {p : α × α → Prop} (h : ∀ᶠ i in f ×ˢ f, p i) :
∀ᶠ i in f, p (i, i) := by |
obtain ⟨t, ht, s, hs, hst⟩ := eventually_prod_iff.1 h
apply (ht.and hs).mono fun x hx => hst hx.1 hx.2
|
/-
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.Convex.Topology
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.SpecificLimits.Basic
#align_import analysis.calculus.... | Mathlib/Analysis/Calculus/TangentCone.lean | 333 | 340 | theorem UniqueDiffWithinAt.pi (ι : Type*) [Finite ι] (E : ι → Type*)
[∀ i, NormedAddCommGroup (E i)] [∀ i, NormedSpace 𝕜 (E i)] (s : ∀ i, Set (E i)) (x : ∀ i, E i)
(I : Set ι) (h : ∀ i ∈ I, UniqueDiffWithinAt 𝕜 (s i) (x i)) :
UniqueDiffWithinAt 𝕜 (Set.pi I s) x := by |
classical
rw [← Set.univ_pi_piecewise_univ]
refine UniqueDiffWithinAt.univ_pi ι E _ _ fun i => ?_
by_cases hi : i ∈ I <;> simp [*, uniqueDiffWithinAt_univ]
|
/-
Copyright (c) 2022 Jon Eugster. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jon Eugster
-/
import Mathlib.Algebra.CharP.LocalRing
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.Tactic.FieldSimp
#align_import algebra.char_p.mixed_char_zero from "leanprov... | Mathlib/Algebra/CharP/MixedCharZero.lean | 85 | 105 | theorem reduce_to_p_prime {P : Prop} :
(∀ p > 0, MixedCharZero R p → P) ↔ ∀ p : ℕ, p.Prime → MixedCharZero R p → P := by |
constructor
· intro h q q_prime q_mixedChar
exact h q (Nat.Prime.pos q_prime) q_mixedChar
· intro h q q_pos q_mixedChar
rcases q_mixedChar.charP_quotient with ⟨I, hI_ne_top, _⟩
-- Krull's Thm: There exists a prime ideal `P` such that `I ≤ P`
rcases Ideal.exists_le_maximal I hI_ne_top with ⟨M, hM_... |
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.Analysis.Complex.Basic
#align_import measure_theory.measure.vector_measure from "leanprover-community/mathl... | Mathlib/MeasureTheory/Measure/VectorMeasure.lean | 1,264 | 1,266 | theorem zero_trim (hle : m ≤ n) : (0 : VectorMeasure α M).trim hle = 0 := by |
ext i hi
exact if_pos hi
|
/-
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Winston Yin
-/
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.Topology.MetricSpace.Contracting
#align_import analysis.ODE.picard_lindelof fr... | Mathlib/Analysis/ODE/PicardLindelof.lean | 226 | 229 | theorem continuous_vComp : Continuous f.vComp := by |
have := (continuous_subtype_val.prod_mk f.continuous).comp v.continuous_proj
refine ContinuousOn.comp_continuous v.continuousOn this fun x => ?_
exact ⟨(v.proj x).2, f.mem_closedBall _⟩
|
/-
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.Data.Finset.Sort
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Sign
import Mathlib.LinearAlgebra.AffineSpace.Combination
import Mathlib.LinearAlg... | Mathlib/LinearAlgebra/AffineSpace/Independent.lean | 286 | 304 | theorem AffineIndependent.comp_embedding {ι2 : Type*} (f : ι2 ↪ ι) {p : ι → P}
(ha : AffineIndependent k p) : AffineIndependent k (p ∘ f) := by |
classical
intro fs w hw hs i0 hi0
let fs' := fs.map f
let w' i := if h : ∃ i2, f i2 = i then w h.choose else 0
have hw' : ∀ i2 : ι2, w' (f i2) = w i2 := by
intro i2
have h : ∃ i : ι2, f i = f i2 := ⟨i2, rfl⟩
have hs : h.choose = i2 := f.injective h.choose_spec
simp_rw [w', dif... |
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Data.Rat.Basic
import Batteries.Tactic.SeqFocus
/-! # Additional lemmas about the Rational Numbers -/
namespace Rat
theorem ext : {p q : Rat} → p.... | .lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean | 121 | 123 | theorem mkRat_mul_left {a : Nat} (a0 : a ≠ 0) : mkRat (↑a * n) (a * d) = mkRat n d := by |
if d0 : d = 0 then simp [d0] else
rw [← normalize_eq_mkRat d0, ← normalize_mul_left d0 a0, normalize_eq_mkRat]
|
/-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Junyan Xu
-/
import Mathlib.Data.DFinsupp.Basic
#align_import data.dfinsupp.ne_locus from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
/-!
# Lo... | Mathlib/Data/DFinsupp/NeLocus.lean | 165 | 166 | theorem neLocus_self_add_left : neLocus (f + g) f = g.support := by |
rw [neLocus_comm, neLocus_self_add_right]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingul... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 1,595 | 1,601 | theorem lintegral_countable [MeasurableSingletonClass α] (f : α → ℝ≥0∞) {s : Set α}
(hs : s.Countable) : ∫⁻ a in s, f a ∂μ = ∑' a : s, f a * μ {(a : α)} :=
calc
∫⁻ a in s, f a ∂μ = ∫⁻ a in ⋃ x ∈ s, {x}, f a ∂μ := by | rw [biUnion_of_singleton]
_ = ∑' a : s, ∫⁻ x in {(a : α)}, f x ∂μ :=
(lintegral_biUnion hs (fun _ _ => measurableSet_singleton _) (pairwiseDisjoint_fiber id s) _)
_ = ∑' a : s, f a * μ {(a : α)} := by simp only [lintegral_singleton]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon, Scott Morrison
-/
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.NatIso
import Mathlib.CategoryTheory.Products.Basic
#align_import category_theory.pi... | Mathlib/CategoryTheory/Pi/Basic.lean | 237 | 242 | theorem pi'_eval (f : ∀ i, A ⥤ C i) (i : I) : pi' f ⋙ Pi.eval C i = f i := by |
apply Functor.ext
· intro _ _ _
simp
· intro _
rfl
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.GroupTh... | Mathlib/RingTheory/Localization/Basic.lean | 1,270 | 1,282 | theorem noZeroDivisors_of_le_nonZeroDivisors [Algebra A S] {M : Submonoid A} [IsLocalization M S]
(hM : M ≤ nonZeroDivisors A) : NoZeroDivisors S :=
{ eq_zero_or_eq_zero_of_mul_eq_zero := by |
intro z w h
cases' surj M z with x hx
cases' surj M w with y hy
have :
z * w * algebraMap A S y.2 * algebraMap A S x.2 = algebraMap A S x.1 * algebraMap A S y.1 :=
by rw [mul_assoc z, hy, ← hx]; ring
rw [h, zero_mul, zero_mul, ← (algebraMap A S).map_mul] at this
case... |
/-
Copyright (c) 2023 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.MFDeriv.Basic
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Geometry.Manifold.Algebra.LieGroup
... | Mathlib/Geometry/Manifold/VectorBundle/SmoothSection.lean | 201 | 204 | theorem coe_nsmul (s : Cₛ^n⟮I; F, V⟯) (k : ℕ) : ⇑(k • s : Cₛ^n⟮I; F, V⟯) = k • ⇑s := by |
induction' k with k ih
· simp_rw [zero_smul]; rfl
simp_rw [succ_nsmul, ← ih]; rfl
|
/-
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.Data.Set.Pointwise.Interval
import Mathlib.LinearAlgebra.AffineSpace.Basic
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.Pi
import ... | Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean | 570 | 573 | theorem lineMap_eq_lineMap_iff [NoZeroSMulDivisors k V1] {p₀ p₁ : P1} {c₁ c₂ : k} :
lineMap p₀ p₁ c₁ = lineMap p₀ p₁ c₂ ↔ p₀ = p₁ ∨ c₁ = c₂ := by |
rw [lineMap_apply, lineMap_apply, ← @vsub_eq_zero_iff_eq V1, vadd_vsub_vadd_cancel_right, ←
sub_smul, smul_eq_zero, sub_eq_zero, vsub_eq_zero_iff_eq, or_comm, eq_comm]
|
/-
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
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathli... | Mathlib/Combinatorics/SimpleGraph/Finite.lean | 260 | 263 | theorem card_incidenceSet_eq_degree [DecidableEq V] :
Fintype.card (G.incidenceSet v) = G.degree v := by |
rw [Fintype.card_congr (G.incidenceSetEquivNeighborSet v)]
simp
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Convex.Uniform
import Mathlib.... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 1,677 | 1,686 | theorem real_inner_div_norm_mul_norm_eq_one_iff (x y : F) :
⟪x, y⟫_ℝ / (‖x‖ * ‖y‖) = 1 ↔ x ≠ 0 ∧ ∃ r : ℝ, 0 < r ∧ y = r • x := by |
constructor
· intro h
have hx₀ : x ≠ 0 := fun h₀ => by simp [h₀] at h
have hy₀ : y ≠ 0 := fun h₀ => by simp [h₀] at h
refine ⟨hx₀, ‖y‖ / ‖x‖, div_pos (norm_pos_iff.2 hy₀) (norm_pos_iff.2 hx₀), ?_⟩
exact ((inner_eq_norm_mul_iff_div hx₀).1 (eq_of_div_eq_one h)).symm
· rintro ⟨hx, ⟨r, ⟨hr, rfl⟩⟩⟩
... |
/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Semantics
#align_import model_theory.order from "leanprover-community/mathlib"@"1ed3a113dbc6f5b33eae3b96211d4e26ca3a5e9d"
/-!
# Ordered F... | Mathlib/ModelTheory/Order.lean | 219 | 222 | theorem Term.realize_lt [Preorder M] [L.OrderedStructure M] {t₁ t₂ : L.Term (Sum α (Fin n))}
{v : α → M} {xs : Fin n → M} :
(t₁.lt t₂).Realize v xs ↔ t₁.realize (Sum.elim v xs) < t₂.realize (Sum.elim v xs) := by |
simp [Term.lt, lt_iff_le_not_le]
|
/-
Copyright (c) 2020 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Data.Countable.Basic
import Mathlib.Data.Set.Image
import Mathlib.Data.Set.Subsingleton
import Mathlib.Data.Int... | Mathlib/Algebra/Group/Subgroup/ZPowers.lean | 173 | 175 | theorem MonoidHom.map_zpowers (f : G →* N) (x : G) :
(Subgroup.zpowers x).map f = Subgroup.zpowers (f x) := by |
rw [Subgroup.zpowers_eq_closure, Subgroup.zpowers_eq_closure, f.map_closure, Set.image_singleton]
|
/-
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 | 775 | 778 | theorem reduct_withConstants :
(L.lhomWithConstants A).substructureReduct (S.withConstants h) = S := by |
ext
simp
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, 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 | 372 | 374 | theorem tendsto_Ioo_atTop {f : β → Ioo a b} :
Tendsto f l atTop ↔ Tendsto (fun x => (f x : α)) l (𝓝[<] b) := by |
rw [← comap_coe_Ioo_nhdsWithin_Iio, tendsto_comap_iff]; rfl
|
/-
Copyright (c) 2017 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Mario Carneiro
-/
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Real.Sqrt
#align_import data.complex.basic from "leanprover-community/mathlib"@"31c24aa72e7b3e5ed... | Mathlib/Data/Complex/Abs.lean | 109 | 110 | theorem sq_abs_sub_sq_im (z : ℂ) : Complex.abs z ^ 2 - z.im ^ 2 = z.re ^ 2 := by |
rw [← sq_abs_sub_sq_re, sub_sub_cancel]
|
/-
Copyright (c) 2020 Nicolò Cavalleri. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Nicolò Cavalleri
-/
import Mathlib.Geometry.Manifold.Algebra.Monoid
#align_import geometry.manifold.algebra.lie_group from "leanprover-community/mathlib"@"f9ec187127cc5b381dfcf5f4a2... | Mathlib/Geometry/Manifold/Algebra/LieGroup.lean | 171 | 174 | theorem ContMDiffWithinAt.div {f g : M → G} {s : Set M} {x₀ : M}
(hf : ContMDiffWithinAt I' I n f s x₀) (hg : ContMDiffWithinAt I' I n g s x₀) :
ContMDiffWithinAt I' I n (fun x => f x / g x) s x₀ := by |
simp_rw [div_eq_mul_inv]; exact hf.mul hg.inv
|
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from ... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 385 | 397 | theorem formPerm_toList (f : Perm α) (x : α) : formPerm (toList f x) = f.cycleOf x := by |
by_cases hx : f x = x
· rw [(cycleOf_eq_one_iff f).mpr hx, toList_eq_nil_iff.mpr (not_mem_support.mpr hx),
formPerm_nil]
ext y
by_cases hy : SameCycle f x y
· obtain ⟨k, _, rfl⟩ := hy.exists_pow_eq_of_mem_support (mem_support.mpr hx)
rw [cycleOf_apply_apply_pow_self, List.formPerm_apply_mem_eq_next... |
/-
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 | 820 | 849 | theorem hausdorffMeasure_preimage_le (hf : AntilipschitzWith K f) (hd : 0 ≤ d) (s : Set Y) :
μH[d] (f ⁻¹' s) ≤ (K : ℝ≥0∞) ^ d * μH[d] s := by |
rcases eq_or_ne K 0 with (rfl | h0)
· rcases eq_empty_or_nonempty (f ⁻¹' s) with (hs | ⟨x, hx⟩)
· simp only [hs, measure_empty, zero_le]
have : f ⁻¹' s = {x} := by
haveI : Subsingleton X := hf.subsingleton
have : (f ⁻¹' s).Subsingleton := subsingleton_univ.anti (subset_univ _)
exact (subs... |
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.EMetricSpace.Basic
import Mathlib.Topology.Bornology.Constructions
imp... | Mathlib/Topology/MetricSpace/PseudoMetric.lean | 329 | 330 | theorem edist_le_coe {x y : α} {c : ℝ≥0} : edist x y ≤ c ↔ nndist x y ≤ c := by |
rw [edist_nndist, ENNReal.coe_le_coe]
|
/-
Copyright (c) 2019 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Yury Kudryashov, Sébastien Gouëzel, Chris Hughes
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Order.Fin
import Mathlib... | Mathlib/Data/Fin/Tuple/Basic.lean | 601 | 603 | theorem init_update_last : init (update q (last n) z) = init q := by |
ext j
simp [init, ne_of_lt, castSucc_lt_last]
|
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