Context stringlengths 285 6.98k | file_name stringlengths 21 79 | start int64 14 184 | end int64 18 184 | theorem stringlengths 25 1.34k | proof stringlengths 5 3.43k |
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/-
Copyright (c) 2021 Shing Tak Lam. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Shing Tak Lam
-/
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.Algebra.... | Mathlib/LinearAlgebra/UnitaryGroup.lean | 76 | 80 | theorem det_of_mem_unitary {A : Matrix n n α} (hA : A ∈ Matrix.unitaryGroup n α) :
A.det ∈ unitary α := by |
constructor
· simpa [star, det_transpose] using congr_arg det hA.1
· simpa [star, det_transpose] using congr_arg det hA.2
|
/-
Copyright (c) 2024 Mitchell Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Lee
-/
import Mathlib.GroupTheory.Coxeter.Length
import Mathlib.Data.ZMod.Parity
/-!
# Reflections, inversions, and inversion sequences
Throughout this file, `B` is a type and... | Mathlib/GroupTheory/Coxeter/Inversion.lean | 102 | 105 | theorem conj (w : W) : cs.IsReflection (w * t * w⁻¹) := by |
obtain ⟨u, i, rfl⟩ := ht
use w * u, i
group
|
/-
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 | 78 | 78 | theorem edgeFinset_ssubset_edgeFinset : G₁.edgeFinset ⊂ G₂.edgeFinset ↔ G₁ < G₂ := by | simp
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Order.Filter.Bases
import Mathlib.Order.ConditionallyCompleteLattice.Basic
#align_import order.filter.lift from "leanprover-community/mathlib"@"8631e2... | Mathlib/Order/Filter/Lift.lean | 32 | 32 | theorem lift_top (g : Set α → Filter β) : (⊤ : Filter α).lift g = g univ := by | simp [Filter.lift]
|
/-
Copyright (c) 2022 Moritz Firsching. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Firsching, Fabian Kruse, Nikolas Kuhn
-/
import Mathlib.Analysis.PSeries
import Mathlib.Data.Real.Pi.Wallis
import Mathlib.Tactic.AdaptationNote
#align_import analysis.specia... | Mathlib/Analysis/SpecialFunctions/Stirling.lean | 61 | 62 | theorem stirlingSeq_one : stirlingSeq 1 = exp 1 / √2 := by |
rw [stirlingSeq, pow_one, factorial_one, cast_one, mul_one, mul_one_div, one_div_div]
|
/-
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.Interval.Set.Basic
import Mathlib.Order.Hom.Set
#align_import data.set.intervals.... | Mathlib/Order/Interval/Set/OrderIso.lean | 42 | 44 | theorem preimage_Ioi (e : α ≃o β) (b : β) : e ⁻¹' Ioi b = Ioi (e.symm b) := by |
ext x
simp [← e.lt_iff_lt]
|
/-
Copyright (c) 2021 Martin Dvorak. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Martin Dvorak, Kyle Miller, Eric Wieser
-/
import Mathlib.Data.Matrix.Notation
import Mathlib.LinearAlgebra.BilinearMap
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Math... | Mathlib/LinearAlgebra/CrossProduct.lean | 134 | 138 | theorem cross_dot_cross (u v w x : Fin 3 → R) :
u ×₃ v ⬝ᵥ w ×₃ x = u ⬝ᵥ w * v ⬝ᵥ x - u ⬝ᵥ x * v ⬝ᵥ w := by |
simp_rw [cross_apply, vec3_dotProduct]
norm_num
ring
|
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Floris van Doorn
-/
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.NormedSpace.Connected
import Mathlib.LinearAlgebra.AffineSpace.ContinuousAffineEqu... | Mathlib/Analysis/Convex/AmpleSet.lean | 79 | 86 | theorem image {s : Set E} (h : AmpleSet s) (L : E ≃ᵃL[ℝ] F) :
AmpleSet (L '' s) := forall_mem_image.mpr fun x hx ↦
calc (convexHull ℝ) (connectedComponentIn (L '' s) (L x))
_ = (convexHull ℝ) (L '' (connectedComponentIn s x)) :=
.symm <| congrArg _ <| L.toHomeomorph.image_connectedComponentIn hx
... | rw [h x hx, image_univ, L.surjective.range_eq]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.Order.BigOperators.Ring.... | Mathlib/Algebra/MvPolynomial/Degrees.lean | 84 | 85 | theorem degrees_def [DecidableEq σ] (p : MvPolynomial σ R) :
p.degrees = p.support.sup fun s : σ →₀ ℕ => Finsupp.toMultiset s := by | rw [degrees]; convert rfl
|
/-
Copyright (c) 2022 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.List.Infix
#align_import data.list.rdrop from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
/-!
# Dropping or taki... | Mathlib/Data/List/DropRight.lean | 105 | 108 | theorem rdropWhile_concat (x : α) :
rdropWhile p (l ++ [x]) = if p x then rdropWhile p l else l ++ [x] := by |
simp only [rdropWhile, dropWhile, reverse_append, reverse_singleton, singleton_append]
split_ifs with h <;> simp [h]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polyno... | Mathlib/RingTheory/Polynomial/IntegralNormalization.lean | 71 | 73 | theorem integralNormalization_coeff_ne_degree {f : R[X]} {i : ℕ} (hi : f.degree ≠ i) :
coeff (integralNormalization f) i = coeff f i * f.leadingCoeff ^ (f.natDegree - 1 - i) := by |
rw [integralNormalization_coeff, if_neg hi]
|
/-
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.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 69 | 76 | theorem monic_ascPochhammer (n : ℕ) [Nontrivial S] [NoZeroDivisors S] :
Monic <| ascPochhammer S n := by |
induction' n with n hn
· simp
· have : leadingCoeff (X + 1 : S[X]) = 1 := leadingCoeff_X_add_C 1
rw [ascPochhammer_succ_left, Monic.def, leadingCoeff_mul,
leadingCoeff_comp (ne_zero_of_eq_one <| natDegree_X_add_C 1 : natDegree (X + 1) ≠ 0), hn,
monic_X, one_mul, one_mul, this, one_pow]
|
/-
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.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factors
import Mathlib.Order.Interval.Finset.Nat
#align_import number_theory.divisors ... | Mathlib/NumberTheory/Divisors.lean | 68 | 72 | theorem filter_dvd_eq_properDivisors (h : n ≠ 0) :
(Finset.range n).filter (· ∣ n) = n.properDivisors := by |
ext
simp only [properDivisors, mem_filter, mem_range, mem_Ico, and_congr_left_iff, iff_and_self]
exact fun ha _ => succ_le_iff.mpr (pos_of_dvd_of_pos ha h.bot_lt)
|
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.GroupTheory.QuotientGroup
#align_import algebra.char_zero.quotient from "leanprover-community/mathlib"@"d90e4e186f1d18e375dcd4e5b5f6364b01cb3e46"
/-!
# Lem... | Mathlib/Algebra/CharZero/Quotient.lean | 67 | 71 | theorem zmultiples_nsmul_eq_nsmul_iff {ψ θ : R ⧸ AddSubgroup.zmultiples p} {n : ℕ} (hz : n ≠ 0) :
n • ψ = n • θ ↔ ∃ k : Fin n, ψ = θ + (k : ℕ) • (p / n : R) := by |
rw [← natCast_zsmul ψ, ← natCast_zsmul θ,
zmultiples_zsmul_eq_zsmul_iff (Int.natCast_ne_zero.mpr hz), Int.cast_natCast]
rfl
|
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.CategoryTheory.Monoidal.Transport
import Mathlib.Algebra.Category.AlgebraCat.Basic
import Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic
import Mathlib.Ri... | Mathlib/Algebra/Category/AlgebraCat/Monoidal.lean | 58 | 64 | theorem forget₂_map_associator_hom (X Y Z : AlgebraCat.{u} R) :
(forget₂ (AlgebraCat R) (ModuleCat R)).map (α_ X Y Z).hom =
(α_
(forget₂ _ (ModuleCat R) |>.obj X)
(forget₂ _ (ModuleCat R) |>.obj Y)
(forget₂ _ (ModuleCat R) |>.obj Z)).hom := by |
rfl
|
/-
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.Probability.Kernel.Basic
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.DominatedConvergence
#align_import pr... | Mathlib/Probability/Kernel/MeasurableIntegral.lean | 102 | 110 | theorem measurable_kernel_prod_mk_left [IsSFiniteKernel κ] {t : Set (α × β)}
(ht : MeasurableSet t) : Measurable fun a => κ a (Prod.mk a ⁻¹' t) := by |
rw [← kernel.kernel_sum_seq κ]
have : ∀ a, kernel.sum (kernel.seq κ) a (Prod.mk a ⁻¹' t) =
∑' n, kernel.seq κ n a (Prod.mk a ⁻¹' t) := fun a =>
kernel.sum_apply' _ _ (measurable_prod_mk_left ht)
simp_rw [this]
refine Measurable.ennreal_tsum fun n => ?_
exact measurable_kernel_prod_mk_left_of_finite... |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Topology.Instances.ENNReal
#align_import analysis.calculus.series from "leanprover-community/ma... | Mathlib/Analysis/NormedSpace/FunctionSeries.lean | 70 | 76 | theorem continuousOn_tsum [TopologicalSpace β] {f : α → β → F} {s : Set β}
(hf : ∀ i, ContinuousOn (f i) s) (hu : Summable u) (hfu : ∀ n x, x ∈ s → ‖f n x‖ ≤ u n) :
ContinuousOn (fun x => ∑' n, f n x) s := by |
classical
refine (tendstoUniformlyOn_tsum hu hfu).continuousOn (eventually_of_forall ?_)
intro t
exact continuousOn_finset_sum _ fun i _ => hf i
|
/-
Copyright (c) 2022 Pim Otte. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Pim Otte
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.Data.Nat.Factorial.BigOperators
import Mathlib.Data.Fin.VecNotation
import ... | Mathlib/Data/Nat/Choose/Multinomial.lean | 102 | 104 | theorem binomial_eq [DecidableEq α] (h : a ≠ b) :
multinomial {a, b} f = (f a + f b)! / ((f a)! * (f b)!) := by |
simp [multinomial, Finset.sum_pair h, Finset.prod_pair h]
|
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.pointwise from "leanp... | Mathlib/Analysis/NormedSpace/Pointwise.lean | 91 | 92 | theorem smul_unitBall {c : 𝕜} (hc : c ≠ 0) : c • ball (0 : E) (1 : ℝ) = ball (0 : E) ‖c‖ := by |
rw [_root_.smul_ball hc, smul_zero, mul_one]
|
/-
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.Abelian
import Mathlib.Algebra.Lie.IdealOperations
import Mathlib.Order.Hom.Basic
#align_import algebra.lie.solvable from "leanprover-community/... | Mathlib/Algebra/Lie/Solvable.lean | 82 | 85 | theorem derivedSeriesOfIdeal_add (k l : ℕ) : D (k + l) I = D k (D l I) := by |
induction' k with k ih
· rw [Nat.zero_add, derivedSeriesOfIdeal_zero]
· rw [Nat.succ_add k l, derivedSeriesOfIdeal_succ, derivedSeriesOfIdeal_succ, ih]
|
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.NumberTheory.ModularForms.JacobiTheta.TwoVariable
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
#align_import number_theory.modular_forms.jacob... | Mathlib/NumberTheory/ModularForms/JacobiTheta/OneVariable.lean | 75 | 86 | theorem hasSum_nat_jacobiTheta {τ : ℂ} (hτ : 0 < im τ) :
HasSum (fun n : ℕ => cexp (π * I * ((n : ℂ) + 1) ^ 2 * τ)) ((jacobiTheta τ - 1) / 2) := by |
have := hasSum_jacobiTheta₂_term 0 hτ
simp_rw [jacobiTheta₂_term, mul_zero, zero_add, ← jacobiTheta_eq_jacobiTheta₂] at this
have := this.nat_add_neg
rw [← hasSum_nat_add_iff' 1] at this
simp_rw [Finset.sum_range_one, Int.cast_neg, Int.cast_natCast, Nat.cast_zero, neg_zero,
Int.cast_zero, sq (0 : ℂ), mul... |
/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stephen Morgan, Scott Morrison
-/
import Mathlib.CategoryTheory.Products.Basic
#align_import category_theory.products.bifunctor from "leanprover-community/mathlib"@"dc6c365e751e34d100... | Mathlib/CategoryTheory/Products/Bifunctor.lean | 52 | 55 | theorem diagonal' (F : C × D ⥤ E) (X X' : C) (f : X ⟶ X') (Y Y' : D) (g : Y ⟶ Y') :
F.map ((f, 𝟙 Y) : (X, Y) ⟶ (X', Y)) ≫ F.map ((𝟙 X', g) : (X', Y) ⟶ (X', Y')) =
F.map ((f, g) : (X, Y) ⟶ (X', Y')) := by |
rw [← Functor.map_comp, prod_comp, Category.id_comp, Category.comp_id]
|
/-
Copyright (c) 2022 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012... | Mathlib/Order/Interval/Set/WithBotTop.lean | 102 | 104 | theorem image_coe_Iic : (some : α → WithTop α) '' Iic a = Iic (a : WithTop α) := by |
rw [← preimage_coe_Iic, image_preimage_eq_inter_range, range_coe,
inter_eq_self_of_subset_left (Iic_subset_Iio.2 <| coe_lt_top a)]
|
/-
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.Basic
#align_import analysis.calculus.fderiv.restrict_scalars from "leanprover-community/ma... | Mathlib/Analysis/Calculus/FDeriv/RestrictScalars.lean | 120 | 124 | theorem differentiableAt_iff_restrictScalars (hf : DifferentiableAt 𝕜 f x) :
DifferentiableAt 𝕜' f x ↔ ∃ g' : E →L[𝕜'] F, g'.restrictScalars 𝕜 = fderiv 𝕜 f x := by |
rw [← differentiableWithinAt_univ, ← fderivWithin_univ]
exact
differentiableWithinAt_iff_restrictScalars 𝕜 hf.differentiableWithinAt uniqueDiffWithinAt_univ
|
/-
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.Deprecated.Group
#align_import deprecated.ring from "leanprover-community/mathlib"@"5a3e819569b0f12cbec59d740a2613018e7b8eec"
/-!
# Unbundled semirin... | Mathlib/Deprecated/Ring.lean | 54 | 54 | theorem id : IsSemiringHom (@id α) := by | constructor <;> intros <;> rfl
|
/-
Copyright (c) 2023 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Order.Filter.Germ
import Mathlib.Topology.NhdsSet
import Mathlib.Topology.LocallyConstant.Basic
import Mathlib.Analysis.NormedSpace.Basic
/-! # Germs ... | Mathlib/Topology/Germ.lean | 94 | 102 | theorem Filter.Eventually.germ_congr_set
{P : ∀ x : X, Germ (𝓝 x) Y → Prop} (hf : ∀ᶠ x in 𝓝ˢ A, P x f)
(h : ∀ᶠ z in 𝓝ˢ A, g z = f z) : ∀ᶠ x in 𝓝ˢ A, P x g := by |
rw [eventually_nhdsSet_iff_forall] at *
intro x hx
apply ((hf x hx).and (h x hx).eventually_nhds).mono
intro y hy
convert hy.1 using 1
exact Germ.coe_eq.mpr hy.2
|
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Scott Morrison
-/
import Mathlib.Algebra.Order.Hom.Monoid
import Mathlib.SetTheory.Game.Ordinal
#align_import set_theory.surreal.basic from "leanprover-community/mathl... | Mathlib/SetTheory/Surreal/Basic.lean | 109 | 115 | theorem Relabelling.numeric_imp {x y : PGame} (r : x ≡r y) (ox : Numeric x) : Numeric y := by |
induction' x using PGame.moveRecOn with x IHl IHr generalizing y
apply Numeric.mk (fun i j => ?_) (fun i => ?_) fun j => ?_
· rw [← lt_congr (r.moveLeftSymm i).equiv (r.moveRightSymm j).equiv]
apply ox.left_lt_right
· exact IHl _ (r.moveLeftSymm i) (ox.moveLeft _)
· exact IHr _ (r.moveRightSymm j) (ox.mo... |
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Analytic.Composition
#align_import analysis.analytic.inverse from "leanprover-community/mathlib"@"284fdd2962e67d2932fa3a79ce19fcf92d38e... | Mathlib/Analysis/Analytic/Inverse.lean | 79 | 92 | theorem leftInv_removeZero (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) :
p.removeZero.leftInv i = p.leftInv i := by |
ext1 n
induction' n using Nat.strongRec' with n IH
match n with
| 0 => simp -- if one replaces `simp` with `refl`, the proof times out in the kernel.
| 1 => simp -- TODO: why?
| n + 2 =>
simp only [leftInv, neg_inj]
refine Finset.sum_congr rfl fun c cuniv => ?_
rcases c with ⟨c, hc⟩
ext v
... |
/-
Copyright (c) 2023 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Category.ModuleCat.ChangeOfRings
import Mathlib.Algebra.Category.Ring.Basic
/-!
# Presheaves of modules over a presheaf of rings.
We give a h... | Mathlib/Algebra/Category/ModuleCat/Presheaf.lean | 85 | 88 | theorem map_comp (P : PresheafOfModules R) {X Y Z : Cᵒᵖ} (f : X ⟶ Y) (g : Y ⟶ Z) :
P.map (f ≫ g) = (P.map g).comp (P.map f) := by |
ext
simp [map_apply]
|
/-
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.Basic
import Mathlib.Data.Set.Pointwise.Basic
/-!
# Neighborhoods to the left and to the right on an ... | Mathlib/Topology/Order/LeftRightNhds.lean | 40 | 60 | theorem TFAE_mem_nhdsWithin_Ioi {a b : α} (hab : a < b) (s : Set α) :
TFAE [s ∈ 𝓝[>] a,
s ∈ 𝓝[Ioc a b] a,
s ∈ 𝓝[Ioo a b] a,
∃ u ∈ Ioc a b, Ioo a u ⊆ s,
∃ u ∈ Ioi a, Ioo a u ⊆ s] := by |
tfae_have 1 ↔ 2
· rw [nhdsWithin_Ioc_eq_nhdsWithin_Ioi hab]
tfae_have 1 ↔ 3
· rw [nhdsWithin_Ioo_eq_nhdsWithin_Ioi hab]
tfae_have 4 → 5
· exact fun ⟨u, umem, hu⟩ => ⟨u, umem.1, hu⟩
tfae_have 5 → 1
· rintro ⟨u, hau, hu⟩
exact mem_of_superset (Ioo_mem_nhdsWithin_Ioi ⟨le_refl a, hau⟩) hu
tfae_have 1... |
/-
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.Quaternion
import Mathlib.Tactic.Ring
#align_import algebra.quaternion_basis from "leanprover-community/mathlib"@"3aa5b8a9ed7a7cabd36e6e1d022c9858ab... | Mathlib/Algebra/QuaternionBasis.lean | 117 | 117 | theorem lift_one : q.lift (1 : ℍ[R,c₁,c₂]) = 1 := by | simp [lift]
|
/-
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
-/
import Mathlib.Algebra.Group.Even
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.Order.Sub.Defs
#align_import algebra.order.sub.canoni... | Mathlib/Algebra/Order/Sub/Canonical.lean | 31 | 33 | theorem tsub_add_cancel_of_le (h : a ≤ b) : b - a + a = b := by |
rw [add_comm]
exact add_tsub_cancel_of_le h
|
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jireh Loreaux
-/
import Mathlib.Algebra.Group.Center
import Mathlib.Data.Int.Cast.Lemmas
#align_import group_theory.subsemigroup.center from "leanprover-community/mathlib"@"... | Mathlib/Algebra/Ring/Center.lean | 24 | 37 | theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _:= by | rw [Nat.commute_cast]
left_assoc _ _ := by
induction n with
| zero => rw [Nat.cast_zero, zero_mul, zero_mul, zero_mul]
| succ n ihn => rw [Nat.cast_succ, add_mul, one_mul, ihn, add_mul, add_mul, one_mul]
mid_assoc _ _ := by
induction n with
| zero => rw [Nat.cast_zero, zero_mul, mul_zero, zero_... |
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Data.Int.Order.Units
import Mathlib.Data.ZMod.IntUnitsPower
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.LinearAlgebra.DirectSum.TensorProduc... | Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean | 85 | 90 | theorem gradedCommAux_lof_tmul (i j : ι) (a : 𝒜 i) (b : ℬ j) :
gradedCommAux R 𝒜 ℬ (lof R _ 𝒜ℬ (i, j) (a ⊗ₜ b)) =
(-1 : ℤˣ)^(j * i) • lof R _ ℬ𝒜 (j, i) (b ⊗ₜ a) := by |
rw [gradedCommAux]
dsimp
simp [mul_comm i j]
|
/-
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.Complex
import Qq
#align_... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 85 | 87 | theorem rpow_eq_zero_iff_of_nonneg (hx : 0 ≤ x) : x ^ y = 0 ↔ x = 0 ∧ y ≠ 0 := by |
simp only [rpow_def_of_nonneg hx]
split_ifs <;> simp [*, exp_ne_zero]
|
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Ma... | Mathlib/Analysis/Calculus/SmoothSeries.lean | 72 | 84 | theorem hasFDerivAt_tsum_of_isPreconnected (hu : Summable u) (hs : IsOpen s)
(h's : IsPreconnected s) (hf : ∀ n x, x ∈ s → HasFDerivAt (f n) (f' n x) x)
(hf' : ∀ n x, x ∈ s → ‖f' n x‖ ≤ u n) (hx₀ : x₀ ∈ s) (hf0 : Summable fun n => f n x₀)
(hx : x ∈ s) : HasFDerivAt (fun y => ∑' n, f n y) (∑' n, f' n x) x :=... |
classical
have A :
∀ x : E, x ∈ s → Tendsto (fun t : Finset α => ∑ n ∈ t, f n x) atTop (𝓝 (∑' n, f n x)) := by
intro y hy
apply Summable.hasSum
exact summable_of_summable_hasFDerivAt_of_isPreconnected hu hs h's hf hf' hx₀ hf0 hy
refine hasFDerivAt_of_tendstoUniformlyOn hs (tendstoUni... |
/-
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.Prod
import Mathlib.Data.Fintype.Sum
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.fintype.units from "leanprover-community... | Mathlib/Data/Fintype/Units.lean | 48 | 50 | theorem Fintype.card_units [GroupWithZero α] [Fintype α] [DecidableEq α] :
Fintype.card αˣ = Fintype.card α - 1 := by |
rw [@Fintype.card_eq_card_units_add_one α, Nat.add_sub_cancel]
|
/-
Copyright (c) 2022 Floris van Doorn, Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Heather Macbeth
-/
import Mathlib.Geometry.Manifold.ContMDiff.Atlas
import Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear
import Mathlib.To... | Mathlib/Geometry/Manifold/VectorBundle/Basic.lean | 108 | 114 | theorem FiberBundle.chartedSpace_chartAt (x : TotalSpace F E) :
chartAt (ModelProd HB F) x =
(trivializationAt F E x.proj).toPartialHomeomorph ≫ₕ
(chartAt HB x.proj).prod (PartialHomeomorph.refl F) := by |
dsimp only [chartAt_comp, prodChartedSpace_chartAt, FiberBundle.chartedSpace'_chartAt,
chartAt_self_eq]
rw [Trivialization.coe_coe, Trivialization.coe_fst' _ (mem_baseSet_trivializationAt F E x.proj)]
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Manuel Candales
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Geometry.Euclidean.PerpBisector
import Mathlib.Algebra.QuadraticDiscriminant
#align_... | Mathlib/Geometry/Euclidean/Basic.lean | 112 | 117 | theorem dist_smul_vadd_sq (r : ℝ) (v : V) (p₁ p₂ : P) :
dist (r • v +ᵥ p₁) p₂ * dist (r • v +ᵥ p₁) p₂ =
⟪v, v⟫ * r * r + 2 * ⟪v, p₁ -ᵥ p₂⟫ * r + ⟪p₁ -ᵥ p₂, p₁ -ᵥ p₂⟫ := by |
rw [dist_eq_norm_vsub V _ p₂, ← real_inner_self_eq_norm_mul_norm, vadd_vsub_assoc,
real_inner_add_add_self, real_inner_smul_left, real_inner_smul_left, real_inner_smul_right]
ring
|
/-
Copyright (c) 2020 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.ApplyFun
import Mathlib.Control.Fix
import Mathlib.Order.OmegaCompletePartialOrder
#align_import control.lawful_fix f... | Mathlib/Control/LawfulFix.lean | 120 | 123 | theorem le_f_of_mem_approx {x} : x ∈ approxChain f → x ≤ f x := by |
simp only [(· ∈ ·), forall_exists_index]
rintro i rfl
apply approx_mono'
|
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Probability.IdentDistrib
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.Analysis.SpecificLimits.FloorPow
import Mathli... | Mathlib/Probability/StrongLaw.lean | 88 | 92 | theorem abs_truncation_le_bound (f : α → ℝ) (A : ℝ) (x : α) : |truncation f A x| ≤ |A| := by |
simp only [truncation, Set.indicator, Set.mem_Icc, id, Function.comp_apply]
split_ifs with h
· exact abs_le_abs h.2 (neg_le.2 h.1.le)
· simp [abs_nonneg]
|
/-
Copyright (c) 2022 Antoine Labelle. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle
-/
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.Algebra.MonoidAlgebra.Ba... | Mathlib/RepresentationTheory/Basic.lean | 166 | 169 | theorem asModuleEquiv_symm_map_rho (g : G) (x : V) :
ρ.asModuleEquiv.symm (ρ g x) = MonoidAlgebra.of k G g • ρ.asModuleEquiv.symm x := by |
apply_fun ρ.asModuleEquiv
simp
|
/-
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.GaloisConnection
#align_import order.heyting.regular from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
/-!
# Heyting r... | Mathlib/Order/Heyting/Regular.lean | 63 | 63 | theorem isRegular_top : IsRegular (⊤ : α) := by | rw [IsRegular, compl_top, compl_bot]
|
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll, Thomas Zhu, Mario Carneiro
-/
import Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity
#align_import number_theory.legendre_symbol.jacobi_symbol from "leanprover-... | Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean | 104 | 105 | theorem zero_right (a : ℤ) : J(a | 0) = 1 := by |
simp only [jacobiSym, factors_zero, List.prod_nil, List.pmap]
|
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.Rayleigh
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Algebra.DirectSum.Decomposition
import Mathlib.Line... | Mathlib/Analysis/InnerProductSpace/Spectrum.lean | 110 | 115 | theorem orthogonalComplement_iSup_eigenspaces (μ : 𝕜) :
eigenspace (T.restrict hT.orthogonalComplement_iSup_eigenspaces_invariant) μ = ⊥ := by |
set p : Submodule 𝕜 E := (⨆ μ, eigenspace T μ)ᗮ
refine eigenspace_restrict_eq_bot hT.orthogonalComplement_iSup_eigenspaces_invariant ?_
have H₂ : eigenspace T μ ⟂ p := (Submodule.isOrtho_orthogonal_right _).mono_left (le_iSup _ _)
exact H₂.disjoint
|
/-
Copyright (c) 2024 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Order.Filter.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.SetTheory.Cardinal.Cofinality
/-!
#... | Mathlib/Order/Filter/CardinalInter.lean | 52 | 55 | theorem _root_.Filter.cardinalInterFilter_aleph0 (l : Filter α) : CardinalInterFilter l aleph0 where
cardinal_sInter_mem := by |
simp_all only [aleph_zero, lt_aleph0_iff_subtype_finite, setOf_mem_eq, sInter_mem,
implies_true, forall_const]
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov
-/
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Algebra.Group.Sub... | Mathlib/Algebra/Group/Submonoid/Membership.lean | 73 | 76 | theorem list_prod_mem {l : List M} (hl : ∀ x ∈ l, x ∈ S) : l.prod ∈ S := by |
lift l to List S using hl
rw [← coe_list_prod]
exact l.prod.coe_prop
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.LinearAlgebra.Dimension.DivisionRing
import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
/-!
# The rank of a linear map
## Main Definit... | Mathlib/LinearAlgebra/Dimension/LinearMap.lean | 52 | 55 | theorem rank_comp_le_left (g : V →ₗ[K] V') (f : V' →ₗ[K] V'') : rank (f.comp g) ≤ rank f := by |
refine rank_le_of_submodule _ _ ?_
rw [LinearMap.range_comp]
exact LinearMap.map_le_range
|
/-
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.VectorMeasure
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
#align_import measure_theory.measure.with_density_vector_measure fr... | Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean | 59 | 60 | theorem withDensityᵥ_apply (hf : Integrable f μ) {s : Set α} (hs : MeasurableSet s) :
μ.withDensityᵥ f s = ∫ x in s, f x ∂μ := by | rw [withDensityᵥ, dif_pos hf]; exact dif_pos hs
|
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Idempotents.Basic
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Equivalence
#align_import category_theo... | Mathlib/CategoryTheory/Idempotents/Karoubi.lean | 89 | 90 | theorem comp_p {P Q : Karoubi C} (f : Hom P Q) : f.f ≫ Q.p = f.f := by |
rw [f.comm, assoc, assoc, Q.idem]
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Analysis.Convex.Hull
#align_import analysis.convex.join from "leanprover-community/mathlib"@"951bf1d9e98a2042979ced62c0620bcfb3587cf8"
/-!
# Convex join
... | Mathlib/Analysis/Convex/Join.lean | 65 | 66 | theorem convexJoin_singleton_left (t : Set E) (x : E) :
convexJoin 𝕜 {x} t = ⋃ y ∈ t, segment 𝕜 x y := by | simp [convexJoin]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a5... | Mathlib/Algebra/MvPolynomial/Variables.lean | 77 | 78 | theorem vars_0 : (0 : MvPolynomial σ R).vars = ∅ := by |
classical rw [vars_def, degrees_zero, Multiset.toFinset_zero]
|
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Contraction
/-! # Results about inverses in Clifford algebras
This contains some basic results about the inversion of vectors... | Mathlib/LinearAlgebra/CliffordAlgebra/Inversion.lean | 37 | 40 | theorem isUnit_ι_of_isUnit {m : M} (h : IsUnit (Q m)) : IsUnit (ι Q m) := by |
cases h.nonempty_invertible
letI := invertibleιOfInvertible Q m
exact isUnit_of_invertible (ι Q m)
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Homology.Single
#align_import algebra.homology.augment from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
/-!
# A... | Mathlib/Algebra/Homology/Augment.lean | 132 | 134 | theorem chainComplex_d_succ_succ_zero (C : ChainComplex V ℕ) (i : ℕ) : C.d (i + 2) 0 = 0 := by |
rw [C.shape]
exact i.succ_succ_ne_one.symm
|
/-
Copyright (c) 2022 George Peter Banyard, Yaël Dillies, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: George Peter Banyard, Yaël Dillies, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Connectivity
#align_import combinatorics.simple_graph.prod... | Mathlib/Combinatorics/SimpleGraph/Prod.lean | 69 | 73 | theorem boxProd_neighborSet (x : α × β) :
(G □ H).neighborSet x = G.neighborSet x.1 ×ˢ {x.2} ∪ {x.1} ×ˢ H.neighborSet x.2 := by |
ext ⟨a', b'⟩
simp only [mem_neighborSet, Set.mem_union, boxProd_adj, Set.mem_prod, Set.mem_singleton_iff]
simp only [eq_comm, and_comm]
|
/-
Copyright (c) 2024 Judith Ludwig, Christian Merten. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Judith Ludwig, Christian Merten
-/
import Mathlib.RingTheory.AdicCompletion.Basic
import Mathlib.Algebra.Module.Torsion
/-!
# Algebra instance on adic completion
In ... | Mathlib/RingTheory/AdicCompletion/Algebra.lean | 127 | 131 | theorem Ideal.mk_eq_mk {m n : ℕ} (hmn : m ≤ n) (r : AdicCauchySequence I R) :
Ideal.Quotient.mk (I ^ m) (r.val n) = Ideal.Quotient.mk (I ^ m) (r.val m) := by |
have h : I ^ m = I ^ m • ⊤ := by simp
rw [h, ← Ideal.Quotient.mk_eq_mk, ← Ideal.Quotient.mk_eq_mk]
exact (r.property hmn).symm
|
/-
Copyright (c) 2022 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.Algebra.IsPrimePow
import Mathlib.Data.Nat.Factorization.Basic
#align_import data.nat.factorization.prime_pow from "leanprover-community/mathlib"@"6ca1a09... | Mathlib/Data/Nat/Factorization/PrimePow.lean | 41 | 54 | theorem isPrimePow_iff_factorization_eq_single {n : ℕ} :
IsPrimePow n ↔ ∃ p k : ℕ, 0 < k ∧ n.factorization = Finsupp.single p k := by |
rw [isPrimePow_nat_iff]
refine exists₂_congr fun p k => ?_
constructor
· rintro ⟨hp, hk, hn⟩
exact ⟨hk, by rw [← hn, Nat.Prime.factorization_pow hp]⟩
· rintro ⟨hk, hn⟩
have hn0 : n ≠ 0 := by
rintro rfl
simp_all only [Finsupp.single_eq_zero, eq_comm, Nat.factorization_zero, hk.ne']
rw ... |
/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Functor.Hom
import Mathlib.CategoryTheory.Products.Basic
import Mathlib.Data.ULift
#align_import category_theory.yoneda from "leanprove... | Mathlib/CategoryTheory/Yoneda.lean | 59 | 62 | theorem obj_map_id {X Y : C} (f : op X ⟶ op Y) :
(yoneda.obj X).map f (𝟙 X) = (yoneda.map f.unop).app (op Y) (𝟙 Y) := by |
dsimp
simp
|
/-
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.LocallyConvex.Basic
#align_import analysis.locally_convex.balanced_core_hull from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547... | Mathlib/Analysis/LocallyConvex/BalancedCoreHull.lean | 114 | 118 | theorem Balanced.balancedHull_subset_of_subset (ht : Balanced 𝕜 t) (h : s ⊆ t) :
balancedHull 𝕜 s ⊆ t := by |
intros x hx
obtain ⟨r, hr, y, hy, rfl⟩ := mem_balancedHull_iff.1 hx
exact ht.smul_mem hr (h hy)
|
/-
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.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Data.Rat.Cast.Defs
#align_import data.rat.cast from "leanprover-community/mathlib"@"... | Mathlib/Data/Rat/Cast/CharZero.lean | 78 | 79 | theorem cast_bit1 [CharZero α] (n : ℚ) : ((bit1 n : ℚ) : α) = (bit1 n : α) := by |
rw [bit1, cast_add, cast_one, cast_bit0]; rfl
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.LinearAlgebra.Span
#align_import algebra.algebra.tower from "leanprover-community/mathlib"@"71150516f28d9826c7... | Mathlib/Algebra/Algebra/Tower.lean | 88 | 90 | theorem algebraMap_smul [SMul R M] [IsScalarTower R A M] (r : R) (x : M) :
algebraMap R A r • x = r • x := by |
rw [Algebra.algebraMap_eq_smul_one, smul_assoc, one_smul]
|
/-
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.Data.Multiset.Basic
#align_import data.multiset.range from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
/-! # `Mu... | Mathlib/Data/Multiset/Range.lean | 65 | 70 | theorem range_disjoint_map_add (a : ℕ) (m : Multiset ℕ) :
(range a).Disjoint (m.map (a + ·)) := by |
intro x hxa hxb
rw [range, mem_coe, List.mem_range] at hxa
obtain ⟨c, _, rfl⟩ := mem_map.1 hxb
exact (Nat.le_add_right _ _).not_lt hxa
|
/-
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.RepresentationTheory.Action.Limits
import Mathlib.RepresentationTheory.Action.Concrete
import Mathlib.CategoryTheory.Monoidal.FunctorCategory
import Ma... | Mathlib/RepresentationTheory/Action/Monoidal.lean | 105 | 107 | theorem leftUnitor_inv_hom {X : Action V G} : Hom.hom (λ_ X).inv = (λ_ X.V).inv := by |
dsimp
simp
|
/-
Copyright (c) 2023 Alex Keizer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Keizer
-/
import Mathlib.Data.Vector.Basic
import Mathlib.Data.Vector.Snoc
/-!
This file establishes a set of normalization lemmas for `map`/`mapAccumr` operations on vectors
-/
... | Mathlib/Data/Vector/MapLemmas.lean | 27 | 35 | theorem mapAccumr_mapAccumr :
mapAccumr f₁ (mapAccumr f₂ xs s₂).snd s₁
= let m := (mapAccumr (fun x s =>
let r₂ := f₂ x s.snd
let r₁ := f₁ r₂.snd s.fst
((r₁.fst, r₂.fst), r₁.snd)
) xs (s₁, s₂))
(m.fst.fst, m.snd) := by |
induction xs using Vector.revInductionOn generalizing s₁ s₂ <;> simp_all
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Floris van Doorn, Gabriel Ebner, Yury Kudryashov
-/
import Mathlib.Order.ConditionallyCompleteLattice.Finset
import Mathlib.Order.Interval.Finset.Nat
#align_import dat... | Mathlib/Data/Nat/Lattice.lean | 66 | 67 | theorem iInf_of_empty {ι : Sort*} [IsEmpty ι] (f : ι → ℕ) : iInf f = 0 := by |
rw [iInf_of_isEmpty, sInf_empty]
|
/-
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.PrimitiveElement
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.Matrix.Char... | Mathlib/RingTheory/Norm.lean | 126 | 135 | theorem PowerBasis.norm_gen_eq_prod_roots [Algebra R F] (pb : PowerBasis R S)
(hf : (minpoly R pb.gen).Splits (algebraMap R F)) :
algebraMap R F (norm R pb.gen) = ((minpoly R pb.gen).aroots F).prod := by |
haveI := Module.nontrivial R F
have := minpoly.monic pb.isIntegral_gen
rw [PowerBasis.norm_gen_eq_coeff_zero_minpoly, ← pb.natDegree_minpoly, RingHom.map_mul,
← coeff_map,
prod_roots_eq_coeff_zero_of_monic_of_split (this.map _) ((splits_id_iff_splits _).2 hf),
this.natDegree_map, map_pow, ← mul_assoc... |
/-
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.Measure.Typeclasses
/-!
# Restriction of a measure to a sub-σ-algebra
## Main definitions
* `MeasureTheory.Measure.trim`: restriction of ... | Mathlib/MeasureTheory/Measure/Trim.lean | 57 | 59 | theorem le_trim (hm : m ≤ m0) : μ s ≤ μ.trim hm s := by |
simp_rw [Measure.trim]
exact @le_toMeasure_apply _ m _ _ _
|
/-
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.Analysis.InnerProductSpace.Basic
import Mathlib.LinearAlgebra.SesquilinearForm
#align_import analysis.inner_product_... | Mathlib/Analysis/InnerProductSpace/Orthogonal.lean | 111 | 111 | theorem orthogonal_disjoint : Disjoint K Kᗮ := by | simp [disjoint_iff, K.inf_orthogonal_eq_bot]
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Lattice
#align_import combinatorics.set_family.compression.down from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"... | Mathlib/Combinatorics/SetFamily/Compression/Down.lean | 120 | 122 | theorem memberSubfamily_nonMemberSubfamily : (𝒜.nonMemberSubfamily a).memberSubfamily a = ∅ := by |
ext
simp
|
/-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Star.Basic
import Mathlib.Algebra.Star.Pointwise
import Mathlib.Algebra.Group.Centralizer
/-! # `Set.center`, `Set.centralizer` and the `star` o... | Mathlib/Algebra/Star/Center.lean | 14 | 34 | theorem Set.star_mem_center (ha : a ∈ Set.center R) : star a ∈ Set.center R where
comm := by | simpa only [star_mul, star_star] using fun g =>
congr_arg star (((Set.mem_center_iff R).mp ha).comm <| star g).symm
left_assoc b c := calc
star a * (b * c) = star a * (star (star b) * star (star c)) := by rw [star_star, star_star]
_ = star a * star (star c * star b) := by rw [star_mul]
_ = star ((sta... |
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Order.Filter.Bases
import Mathlib.Order.Filter.Ultrafilter
/-!
# Subsingleton filters
We say that a filter `l` is a *subsingleton* if there exists a... | Mathlib/Order/Filter/Subsingleton.lean | 51 | 55 | theorem Subsingleton.exists_eq_pure [l.NeBot] (hl : l.Subsingleton) : ∃ a, l = pure a := by |
rcases hl with ⟨s, hsl, hs⟩
rcases exists_eq_singleton_iff_nonempty_subsingleton.2 ⟨nonempty_of_mem hsl, hs⟩ with ⟨a, rfl⟩
refine ⟨a, (NeBot.le_pure_iff ‹_›).1 ?_⟩
rwa [le_pure_iff]
|
/-
Copyright (c) 2020 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.List.Basic
/-!
# Properties of `List.reduceOption`
In this file we prove basic lemmas about `List.reduceOption`.
-/
namespace List
variable ... | Mathlib/Data/List/ReduceOption.lean | 77 | 77 | theorem reduceOption_singleton (x : Option α) : [x].reduceOption = x.toList := by | cases x <;> rfl
|
/-
Copyright (c) 2020 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.FieldTheory.Minpoly.Field
#align_import linear_algebra.eigenspace.minpoly from "leanprover-commu... | Mathlib/LinearAlgebra/Eigenspace/Minpoly.lean | 75 | 91 | theorem hasEigenvalue_of_isRoot (h : (minpoly K f).IsRoot μ) : f.HasEigenvalue μ := by |
cases' dvd_iff_isRoot.2 h with p hp
rw [HasEigenvalue, eigenspace]
intro con
cases' (LinearMap.isUnit_iff_ker_eq_bot _).2 con with u hu
have p_ne_0 : p ≠ 0 := by
intro con
apply minpoly.ne_zero (Algebra.IsIntegral.isIntegral (R := K) f)
rw [hp, con, mul_zero]
have : (aeval f) p = 0 := by
ha... |
/-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Ma... | Mathlib/Data/Nat/Factorization/Basic.lean | 67 | 81 | theorem factors_count_eq {n p : ℕ} : n.factors.count p = n.factorization p := by |
rcases n.eq_zero_or_pos with (rfl | hn0)
· simp [factorization, count]
if pp : p.Prime then ?_ else
rw [count_eq_zero_of_not_mem (mt prime_of_mem_factors pp)]
simp [factorization, pp]
simp only [factorization_def _ pp]
apply _root_.le_antisymm
· rw [le_padicValNat_iff_replicate_subperm_factors pp h... |
/-
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.Order.Interval.Finset.Fin
#align_import data.fintype.fin from "leanprover-community/mathlib"@"759575657f189ccb424b990164c8b1fa9f55cdfe"
/-!
# The structure... | Mathlib/Data/Fintype/Fin.lean | 25 | 27 | theorem map_valEmbedding_univ : (Finset.univ : Finset (Fin n)).map Fin.valEmbedding = Iio n := by |
ext
simp [orderIsoSubtype.symm.surjective.exists, OrderIso.symm]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Dia... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 88 | 88 | theorem volume_Ioo {a b : ℝ} : volume (Ioo a b) = ofReal (b - a) := by | simp [volume_val]
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yakov Pechersky
-/
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Zip
import Mathlib.Data.Nat.Defs
import Mathlib.Data.List.Infix
#align_import data.list.rotate f... | Mathlib/Data/List/Rotate.lean | 56 | 57 | theorem rotate'_cons_succ (l : List α) (a : α) (n : ℕ) :
(a :: l : List α).rotate' n.succ = (l ++ [a]).rotate' n := by | simp [rotate']
|
/-
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.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.NthRewrite
#align_import data.nat.gcd.... | Mathlib/Data/Nat/GCD/Basic.lean | 58 | 59 | theorem gcd_add_mul_left_left (m n k : ℕ) : gcd (m + n * k) n = gcd m n := by |
rw [gcd_comm, gcd_add_mul_left_right, gcd_comm]
|
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang
-/
import Mathlib.Algebra.Category.GroupCat.EquivalenceGroupAddGroup
import Mathlib.GroupTheory.QuotientGroup
#align_import algebra.category.Group.epi_mono from "leanprover... | Mathlib/Algebra/Category/GroupCat/EpiMono.lean | 47 | 56 | theorem range_eq_top_of_cancel {f : A →* B}
(h : ∀ u v : B →* B ⧸ f.range, u.comp f = v.comp f → u = v) : f.range = ⊤ := by |
specialize h 1 (QuotientGroup.mk' _) _
· ext1 x
simp only [one_apply, coe_comp, coe_mk', Function.comp_apply]
rw [show (1 : B ⧸ f.range) = (1 : B) from QuotientGroup.mk_one _, QuotientGroup.eq, inv_one,
one_mul]
exact ⟨x, rfl⟩
replace h : (QuotientGroup.mk' f.range).ker = (1 : B →* B ⧸ f.range)... |
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.GroupTheory.Perm.Fin
import Mathlib.GroupTheory.Subgroup.Simple
import Mathlib.Tactic.IntervalCases
#align_imp... | Mathlib/GroupTheory/SpecificGroups/Alternating.lean | 77 | 80 | theorem prod_list_swap_mem_alternatingGroup_iff_even_length {l : List (Perm α)}
(hl : ∀ g ∈ l, IsSwap g) : l.prod ∈ alternatingGroup α ↔ Even l.length := by |
rw [mem_alternatingGroup, sign_prod_list_swap hl, neg_one_pow_eq_one_iff_even]
decide
|
/-
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.Fin.Basic
namespace Fin
attribute [norm_cast] val_last
protected theorem le_antisymm_iff {x y : Fin n} : x = y ↔ x ≤ y ∧ y ≤ x :=
Fin.ext_i... | .lake/packages/batteries/Batteries/Data/Fin/Lemmas.lean | 41 | 47 | theorem list_succ_last (n) : list (n+1) = (list n).map castSucc ++ [last n] := by |
rw [list_succ]
induction n with
| zero => rfl
| succ n ih =>
rw [list_succ, List.map_cons castSucc, ih]
simp [Function.comp_def, succ_castSucc]
|
/-
Copyright (c) 2023 Antoine Chambert-Loir and 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, Bhavik Mehta, Eric Wieser
-/
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Data... | Mathlib/Data/Finset/Antidiagonal.lean | 100 | 104 | theorem antidiagonal_congr (hp : p ∈ antidiagonal n) (hq : q ∈ antidiagonal n) :
p = q ↔ p.1 = q.1 := by |
refine ⟨congr_arg Prod.fst, fun h ↦ Prod.ext h ((add_right_inj q.fst).mp ?_)⟩
rw [mem_antidiagonal] at hp hq
rw [hq, ← h, hp]
|
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.Matrix
import Mathlib.LinearAlgebra.Matrix.ZPow
import Mathlib.LinearAlgebra.Matrix.Hermitian
import... | Mathlib/Analysis/NormedSpace/MatrixExponential.lean | 84 | 86 | theorem exp_blockDiagonal (v : m → Matrix n n 𝔸) :
exp 𝕂 (blockDiagonal v) = blockDiagonal (exp 𝕂 v) := by |
simp_rw [exp_eq_tsum, ← blockDiagonal_pow, ← blockDiagonal_smul, ← blockDiagonal_tsum]
|
/-
Copyright (c) 2022 Felix Weilacher. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Felix Weilacher
-/
import Mathlib.Topology.Separation
/-!
# Perfect Sets
In this file we define perfect subsets of a topological space, and prove some basic properties,
including a... | Mathlib/Topology/Perfect.lean | 111 | 115 | theorem Preperfect.open_inter {U : Set α} (hC : Preperfect C) (hU : IsOpen U) :
Preperfect (U ∩ C) := by |
rintro x ⟨xU, xC⟩
apply (hC _ xC).nhds_inter
exact hU.mem_nhds xU
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Anne Baanen
-/
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Matrix.RowCol
import Mathlib.GroupTheory.GroupAction.Ring
im... | Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean | 98 | 100 | theorem coe_det_isEmpty [IsEmpty n] : (det : Matrix n n R → R) = Function.const _ 1 := by |
ext
exact det_isEmpty
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Data.Finset.Lattice
#align_import data.finset.pairwise from "leanprover-community/mathlib"@"c4c2ed622f43768eff32608d4a0f8a6cec1c047d"
/-!
# Relations hol... | Mathlib/Data/Finset/Pairwise.lean | 80 | 83 | theorem pairwise_of_coe_toFinset_pairwise (hl : (l.toFinset : Set α).Pairwise r) (hn : l.Nodup) :
l.Pairwise r := by |
rw [coe_toFinset] at hl
exact hn.pairwise_of_set_pairwise hl
|
/-
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.LinearAlgebra.LinearPMap
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.module.linear_pmap from "leanprover-community/mathlib"@... | Mathlib/Topology/Algebra/Module/LinearPMap.lean | 103 | 104 | theorem closure_def {f : E →ₗ.[R] F} (hf : f.IsClosable) : f.closure = hf.choose := by |
simp [closure, hf]
|
/-
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.NumberTheory.Liouville.Basic
import Mathlib.Topology.Baire.Lemmas
import Mathlib.Topology.Baire.LocallyCompactRegular
import Mathlib.Topology.Instanc... | Mathlib/NumberTheory/Liouville/Residual.lean | 44 | 55 | theorem setOf_liouville_eq_irrational_inter_iInter_iUnion :
{ x | Liouville x } =
{ x | Irrational x } ∩ ⋂ n : ℕ, ⋃ (a : ℤ) (b : ℤ) (hb : 1 < b),
ball (a / b) (1 / (b : ℝ) ^ n) := by |
refine Subset.antisymm ?_ ?_
· refine subset_inter (fun x hx => hx.irrational) ?_
rw [setOf_liouville_eq_iInter_iUnion]
exact iInter_mono fun n => iUnion₂_mono fun a b => iUnion_mono fun _hb => diff_subset
· simp only [inter_iInter, inter_iUnion, setOf_liouville_eq_iInter_iUnion]
refine iInter_mono f... |
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.HausdorffDistance
import Mathlib.Topology.Sets.Compacts
#align_import topology... | Mathlib/Topology/MetricSpace/Closeds.lean | 56 | 69 | theorem continuous_infEdist_hausdorffEdist :
Continuous fun p : α × Closeds α => infEdist p.1 p.2 := by |
refine continuous_of_le_add_edist 2 (by simp) ?_
rintro ⟨x, s⟩ ⟨y, t⟩
calc
infEdist x s ≤ infEdist x t + hausdorffEdist (t : Set α) s :=
infEdist_le_infEdist_add_hausdorffEdist
_ ≤ infEdist y t + edist x y + hausdorffEdist (t : Set α) s :=
(add_le_add_right infEdist_le_infEdist_add_edist _)
... |
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Scott Morrison
-/
import Mathlib.CategoryTheory.Preadditive.Yoneda.Basic
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.Algebra.Category.GroupCat.Epi... | Mathlib/CategoryTheory/Preadditive/Yoneda/Projective.lean | 31 | 39 | theorem projective_iff_preservesEpimorphisms_preadditiveCoyoneda_obj (P : C) :
Projective P ↔ (preadditiveCoyoneda.obj (op P)).PreservesEpimorphisms := by |
rw [projective_iff_preservesEpimorphisms_coyoneda_obj]
refine ⟨fun h : (preadditiveCoyoneda.obj (op P) ⋙
forget AddCommGroupCat).PreservesEpimorphisms => ?_, ?_⟩
· exact Functor.preservesEpimorphisms_of_preserves_of_reflects (preadditiveCoyoneda.obj (op P))
(forget _)
· intro
exact (inferInst... |
/-
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 | 95 | 104 | theorem Basis.map_addHaar {ι E F : Type*} [Fintype ι] [NormedAddCommGroup E] [NormedAddCommGroup F]
[NormedSpace ℝ E] [NormedSpace ℝ F] [MeasurableSpace E] [MeasurableSpace F] [BorelSpace E]
[BorelSpace F] [SecondCountableTopology F] [SigmaCompactSpace F]
(b : Basis ι ℝ E) (f : E ≃L[ℝ] F) :
map f b.addH... |
have : IsAddHaarMeasure (map f b.addHaar) :=
AddEquiv.isAddHaarMeasure_map b.addHaar f.toAddEquiv f.continuous f.symm.continuous
rw [eq_comm, Basis.addHaar_eq_iff, Measure.map_apply f.continuous.measurable
(PositiveCompacts.isCompact _).measurableSet, Basis.coe_parallelepiped, Basis.coe_map]
erw [← image... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.List.Sublists
import Mathlib.Data.Multiset.Bind
#align_import data.multiset.powerset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179... | Mathlib/Data/Multiset/Powerset.lean | 45 | 46 | theorem powersetAux_perm_powersetAux' {l : List α} : powersetAux l ~ powersetAux' l := by |
rw [powersetAux_eq_map_coe]; exact (sublists_perm_sublists' _).map _
|
/-
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.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.Option
#align_import algebra.big_operators.option from "leanprover-community/mathlib"@"... | Mathlib/Algebra/BigOperators/Option.lean | 25 | 26 | theorem prod_insertNone (f : Option α → M) (s : Finset α) :
∏ x ∈ insertNone s, f x = f none * ∏ x ∈ s, f (some x) := by | simp [insertNone]
|
/-
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 | 76 | 83 | theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by |
refine ⟨fun hz => ⟨arg z, ?_⟩, ?_⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
|
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.Spectrum
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Algebra.... | Mathlib/Analysis/NormedSpace/Star/Spectrum.lean | 31 | 41 | theorem unitary.spectrum_subset_circle (u : unitary E) :
spectrum 𝕜 (u : E) ⊆ Metric.sphere 0 1 := by |
nontriviality E
refine fun k hk => mem_sphere_zero_iff_norm.mpr (le_antisymm ?_ ?_)
· simpa only [CstarRing.norm_coe_unitary u] using norm_le_norm_of_mem hk
· rw [← unitary.val_toUnits_apply u] at hk
have hnk := ne_zero_of_mem_of_unit hk
rw [← inv_inv (unitary.toUnits u), ← spectrum.map_inv, Set.mem_in... |
/-
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.NormedSpace.Banach
import Mathlib.Topology.Algebra.Module.FiniteDimension
#align_import analysis.normed_space.complemented from "leanprover... | Mathlib/Analysis/NormedSpace/Complemented.lean | 39 | 43 | theorem ker_closedComplemented_of_finiteDimensional_range (f : E →L[𝕜] F)
[FiniteDimensional 𝕜 (range f)] : (ker f).ClosedComplemented := by |
set f' : E →L[𝕜] range f := f.codRestrict _ (LinearMap.mem_range_self (f : E →ₗ[𝕜] F))
rcases f'.exists_right_inverse_of_surjective (f : E →ₗ[𝕜] F).range_rangeRestrict with ⟨g, hg⟩
simpa only [f', ker_codRestrict] using f'.closedComplemented_ker_of_rightInverse g (ext_iff.1 hg)
|
/-
Copyright (c) 2022 Antoine Labelle, Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Labelle, Rémi Bottinelli
-/
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Combinatorics.Quiver.Path
#align_import combinatorics.quiver.cast from "... | Mathlib/Combinatorics/Quiver/Cast.lean | 130 | 133 | theorem Path.cast_cons {u v w u' w' : U} (p : Path u v) (e : v ⟶ w) (hu : u = u') (hw : w = w') :
(p.cons e).cast hu hw = (p.cast hu rfl).cons (e.cast rfl hw) := by |
subst_vars
rfl
|
/-
Copyright (c) 2020 Jalex Stark. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jalex Stark, Scott Morrison, Eric Wieser, Oliver Nash, Wen Yang
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.LinearAlgebra.Matrix.Trace
#align_import data.matrix.basis from "leanpr... | Mathlib/Data/Matrix/Basis.lean | 57 | 63 | theorem mulVec_stdBasisMatrix [Fintype m] (i : n) (j : m) (c : α) (x : m → α) :
mulVec (stdBasisMatrix i j c) x = Function.update (0 : n → α) i (c * x j) := by |
ext i'
simp [stdBasisMatrix, mulVec, dotProduct]
rcases eq_or_ne i i' with rfl|h
· simp
simp [h, h.symm]
|
/-
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.Defs
import Mathlib.Algebra.Group.Prod
import Mathlib.Data.PNat.Basic
import Mathlib.GroupTheory.GroupAction.Prod
/-!
# Typeclasses for... | Mathlib/Algebra/Group/PNatPowAssoc.lean | 60 | 62 | theorem ppow_mul_assoc (k m n : ℕ+) (x : M) :
(x ^ k * x ^ m) * x ^ n = x ^ k * (x ^ m * x ^ n) := by |
simp only [← ppow_add, add_assoc]
|
/-
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.Probability.Kernel.Basic
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.DominatedConvergence
#align_import pr... | Mathlib/Probability/Kernel/MeasurableIntegral.lean | 42 | 99 | theorem measurable_kernel_prod_mk_left_of_finite {t : Set (α × β)} (ht : MeasurableSet t)
(hκs : ∀ a, IsFiniteMeasure (κ a)) : Measurable fun a => κ a (Prod.mk a ⁻¹' t) := by |
-- `t` is a measurable set in the product `α × β`: we use that the product σ-algebra is generated
-- by boxes to prove the result by induction.
-- Porting note: added motive
refine MeasurableSpace.induction_on_inter
(C := fun t => Measurable fun a => κ a (Prod.mk a ⁻¹' t))
generateFrom_prod.symm isPiSy... |
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