Context stringlengths 57 92.3k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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import Mathlib.Algebra.Module.Submodule.EqLocus
import Mathlib.Algebra.Module.Submodule.RestrictScalars
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.LinearAlgebra.Basic
import Mathlib.Order.CompactlyGenerated.Basic
import Mathlib.Order.OmegaCompletePartialOrder
#align_... | Mathlib/LinearAlgebra/Span.lean | 561 | 562 | theorem span_insert (x) (s : Set M) : span R (insert x s) = (R ∙ x) ⊔ span R s := by |
rw [insert_eq, span_union]
|
import Mathlib.GroupTheory.GroupAction.Pointwise
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Analysis.LocallyConvex.BalancedCoreHull
import Mathlib.Analysis.Seminorm
import Mathlib.Topology.Bornology.Basic
import Mathlib.Topology.Algebra.UniformGroup
import Mathlib.Topology.UniformSpace.Cauchy
import Ma... | Mathlib/Analysis/LocallyConvex/Bounded.lean | 390 | 397 | theorem isVonNBounded_iff {s : Set E} : Bornology.IsVonNBounded 𝕜 s ↔ Bornology.IsBounded s := by |
refine ⟨fun h ↦ ?_, isVonNBounded_of_isBounded _⟩
rcases (h (Metric.ball_mem_nhds 0 zero_lt_one)).exists_pos with ⟨ρ, hρ, hρball⟩
rcases NormedField.exists_lt_norm 𝕜 ρ with ⟨a, ha⟩
specialize hρball a ha.le
rw [← ball_normSeminorm 𝕜 E, Seminorm.smul_ball_zero (norm_pos_iff.1 <| hρ.trans ha),
ball_normS... |
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 396 | 396 | theorem image_neg_Ioo : Neg.neg '' Ioo a b = Ioo (-b) (-a) := by | simp
|
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "leanprover-community/mathlib"@"996b0ff959da753a555053a480f36e5f264d4207"
variable {α β γ : Type*} {ι : Sort*} {π : α → Type*}
open Equiv Equiv.Perm Function
namespace Set
variable {s s₁ s₂ : Set α} {t ... | Mathlib/Data/Set/Function.lean | 1,659 | 1,663 | theorem apply_piecewise₂ {δ' δ'' : α → Sort*} (f' g' : ∀ i, δ' i) (h : ∀ i, δ i → δ' i → δ'' i)
{x : α} :
h x (s.piecewise f g x) (s.piecewise f' g' x) =
s.piecewise (fun x => h x (f x) (f' x)) (fun x => h x (g x) (g' x)) x := by |
by_cases hx : x ∈ s <;> simp [hx]
|
import Mathlib.Geometry.Manifold.Algebra.Monoid
#align_import geometry.manifold.algebra.lie_group from "leanprover-community/mathlib"@"f9ec187127cc5b381dfcf5f4a22dacca4c20b63d"
noncomputable section
open scoped Manifold
-- See note [Design choices about smooth algebraic structures]
class LieAddGroup {𝕜 : Type*... | Mathlib/Geometry/Manifold/Algebra/LieGroup.lean | 351 | 353 | theorem ContMDiffAt.div₀ (hf : ContMDiffAt I' I n f a) (hg : ContMDiffAt I' I n g a)
(h₀ : g a ≠ 0) : ContMDiffAt I' I n (f / g) a := by |
simpa [div_eq_mul_inv] using hf.mul (hg.inv₀ h₀)
|
import Mathlib.MeasureTheory.Constructions.Pi
import Mathlib.Probability.Kernel.Basic
open MeasureTheory MeasurableSpace
open scoped MeasureTheory ENNReal
namespace ProbabilityTheory.kernel
variable {α Ω ι : Type*}
section Indep
variable {_mα : MeasurableSpace α}
@[symm]
theorem IndepSets.symm {_mΩ : Measurab... | Mathlib/Probability/Independence/Kernel.lean | 193 | 197 | theorem indepSet_empty_right {_mΩ : MeasurableSpace Ω}
{κ : kernel α Ω} {μ : Measure α} [IsMarkovKernel κ] (s : Set Ω) :
IndepSet s ∅ κ μ := by |
simp only [IndepSet, generateFrom_singleton_empty];
exact indep_bot_right _
|
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib.Deprecated.Group
#align_import deprecated.submonoid from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
variable {M : Type*} [Monoid M] {s : Set M}
variable {A : Type*} [AddMonoi... | Mathlib/Deprecated/Submonoid.lean | 195 | 198 | theorem Range.isSubmonoid {γ : Type*} [Monoid γ] {f : M → γ} (hf : IsMonoidHom f) :
IsSubmonoid (Set.range f) := by |
rw [← Set.image_univ]
exact Univ.isSubmonoid.image hf
|
import Mathlib.RingTheory.Valuation.Integers
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Localization.Integer
import Mathlib.RingTheory.DiscreteValuationRing.Basic
import Mathlib.RingTheory.Bezout
import Mathlib.Tactic.FieldSimp
#align_import... | Mathlib/RingTheory/Valuation/ValuationRing.lean | 282 | 286 | theorem iff_dvd_total : ValuationRing R ↔ IsTotal R (· ∣ ·) := by |
classical
refine ⟨fun H => ⟨fun a b => ?_⟩, fun H => ⟨fun a b => ?_⟩⟩
· obtain ⟨c, rfl | rfl⟩ := ValuationRing.cond a b <;> simp
· obtain ⟨c, rfl⟩ | ⟨c, rfl⟩ := @IsTotal.total _ _ H a b <;> use c <;> simp
|
import Mathlib.Data.Matrix.Basis
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.Pi
#align_import linear_algebra.std_basis from "leanprover-community/mathlib"@"13bce9a6b6c44f6b4c91ac1c1d2a816e2533d395"
open Function Set Submodule
namespace Pi
open LinearMap
open Set
variable {R : Type*}
secti... | Mathlib/LinearAlgebra/StdBasis.lean | 278 | 280 | theorem basisFun_apply [DecidableEq η] (i) : basisFun R η i = stdBasis R (fun _ : η => R) i 1 := by |
simp only [basisFun, Basis.coe_ofEquivFun, LinearEquiv.refl_symm, LinearEquiv.refl_apply,
stdBasis_apply]
|
import Mathlib.Data.Real.Cardinality
import Mathlib.Topology.MetricSpace.Perfect
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
import Mathlib.Topology.CountableSeparatingOn
#align_import measure_theory.constructions.polish from "leanprover-community/mathlib"@"9f55d0d4363ae59948c33864cbc52e0b12e0e8ce"
... | Mathlib/MeasureTheory/Constructions/Polish.lean | 169 | 171 | theorem analyticSet_empty : AnalyticSet (∅ : Set α) := by |
rw [AnalyticSet]
exact Or.inl rfl
|
import Mathlib.Order.Filter.Interval
import Mathlib.Order.Interval.Set.Pi
import Mathlib.Tactic.TFAE
import Mathlib.Tactic.NormNum
import Mathlib.Topology.Order.LeftRight
import Mathlib.Topology.Order.OrderClosed
#align_import topology.order.basic from "leanprover-community/mathlib"@"3efd324a3a31eaa40c9d5bfc669c4fafe... | Mathlib/Topology/Order/Basic.lean | 353 | 357 | theorem tendsto_nhds_top_mono [TopologicalSpace β] [Preorder β] [OrderTop β] [OrderTopology β]
{l : Filter α} {f g : α → β} (hf : Tendsto f l (𝓝 ⊤)) (hg : f ≤ᶠ[l] g) : Tendsto g l (𝓝 ⊤) := by |
simp only [nhds_top_order, tendsto_iInf, tendsto_principal] at hf ⊢
intro x hx
filter_upwards [hf x hx, hg] with _ using lt_of_lt_of_le
|
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Tactic.NoncommRing
#align_import algebra.algebra.spectrum from "leanprover-community/mathlib"@"58a272265b5e05f258161260dd2c5d247213cbd3"
open Set
open scoped Pointwise
universe u v
namespace spectrum
section ScalarSemir... | Mathlib/Algebra/Algebra/Spectrum.lean | 198 | 207 | theorem inv_mem_resolventSet {r : Rˣ} {a : Aˣ} (h : (r : R) ∈ resolventSet R (a : A)) :
(↑r⁻¹ : R) ∈ resolventSet R (↑a⁻¹ : A) := by |
rw [mem_resolventSet_iff, Algebra.algebraMap_eq_smul_one, ← Units.smul_def] at h ⊢
rw [IsUnit.smul_sub_iff_sub_inv_smul, inv_inv, IsUnit.sub_iff]
have h₁ : (a : A) * (r • (↑a⁻¹ : A) - 1) = r • (1 : A) - a := by
rw [mul_sub, mul_smul_comm, a.mul_inv, mul_one]
have h₂ : (r • (↑a⁻¹ : A) - 1) * a = r • (1 : A)... |
import Batteries.Control.ForInStep.Lemmas
import Batteries.Data.List.Basic
import Batteries.Tactic.Init
import Batteries.Tactic.Alias
namespace List
open Nat
@[simp] theorem mem_toArray {a : α} {l : List α} : a ∈ l.toArray ↔ a ∈ l := by
simp [Array.mem_def]
@[simp]
theorem drop_one : ∀ l : List α, drop 1 l =... | .lake/packages/batteries/Batteries/Data/List/Lemmas.lean | 963 | 964 | theorem diff_erase (l₁ l₂ : List α) (a : α) : (l₁.diff l₂).erase a = (l₁.erase a).diff l₂ := by |
rw [← diff_cons_right, diff_cons]
|
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Calculus.Dslope
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Analytic.Uniqueness
#align_import analysis.analytic.isolated_zeros from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
open sco... | Mathlib/Analysis/Analytic/IsolatedZeros.lean | 146 | 148 | theorem frequently_eq_iff_eventually_eq (hf : AnalyticAt 𝕜 f z₀) (hg : AnalyticAt 𝕜 g z₀) :
(∃ᶠ z in 𝓝[≠] z₀, f z = g z) ↔ ∀ᶠ z in 𝓝 z₀, f z = g z := by |
simpa [sub_eq_zero] using frequently_zero_iff_eventually_zero (hf.sub hg)
|
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.ProdSigma
#align_import model_theory.semantics from "leanprover-community/mathlib"@"d565b3df44619c1498326936be16f1a935df0728"
universe u v w u' v'
namespace FirstOrder
namespace Language
variable {L : Language.{u, v}} {... | Mathlib/ModelTheory/Semantics.lean | 957 | 961 | theorem realize_iExs [Finite γ] {f : α → β ⊕ γ}
{φ : L.Formula α} {v : β → M} {v' : Fin 0 → M} :
BoundedFormula.Realize (φ.iExs f) v v' ↔
∃ (i : γ → M), φ.Realize (fun a => Sum.elim v i (f a)) := by |
rw [← Formula.realize_iExs, iff_iff_eq]; congr; simp [eq_iff_true_of_subsingleton]
|
import Mathlib.Order.SuccPred.Basic
import Mathlib.Topology.Order.Basic
import Mathlib.Topology.Metrizable.Uniformity
#align_import topology.instances.discrete from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
open Order Set TopologicalSpace Filter
variable {α : Type*} [TopologicalSp... | Mathlib/Topology/Instances/Discrete.lean | 80 | 108 | theorem LinearOrder.bot_topologicalSpace_eq_generateFrom [LinearOrder α] [PredOrder α]
[SuccOrder α] : (⊥ : TopologicalSpace α) = generateFrom { s | ∃ a, s = Ioi a ∨ s = Iio a } := by |
refine (eq_bot_of_singletons_open fun a => ?_).symm
have h_singleton_eq_inter : {a} = Iic a ∩ Ici a := by rw [inter_comm, Ici_inter_Iic, Icc_self a]
by_cases ha_top : IsTop a
· rw [ha_top.Iic_eq, inter_comm, inter_univ] at h_singleton_eq_inter
by_cases ha_bot : IsBot a
· rw [ha_bot.Ici_eq] at h_singlet... |
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section Real
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
theorem toReal_add (ha : a ≠ ∞) (hb : b ≠ ∞) : (a + b).toReal = a.toReal ... | Mathlib/Data/ENNReal/Real.lean | 454 | 456 | theorem toReal_top_mul (a : ℝ≥0∞) : ENNReal.toReal (∞ * a) = 0 := by |
rw [mul_comm]
exact toReal_mul_top _
|
import Mathlib.Topology.Maps
import Mathlib.Topology.NhdsSet
#align_import topology.constructions from "leanprover-community/mathlib"@"f7ebde7ee0d1505dfccac8644ae12371aa3c1c9f"
noncomputable section
open scoped Classical
open Topology TopologicalSpace Set Filter Function
universe u v
variable {X : Type u} {Y :... | Mathlib/Topology/Constructions.lean | 839 | 844 | theorem map_mem_closure₂ {f : X → Y → Z} {x : X} {y : Y} {s : Set X} {t : Set Y} {u : Set Z}
(hf : Continuous (uncurry f)) (hx : x ∈ closure s) (hy : y ∈ closure t)
(h : ∀ a ∈ s, ∀ b ∈ t, f a b ∈ u) : f x y ∈ closure u :=
have H₁ : (x, y) ∈ closure (s ×ˢ t) := by | simpa only [closure_prod_eq] using mk_mem_prod hx hy
have H₂ : MapsTo (uncurry f) (s ×ˢ t) u := forall_prod_set.2 h
H₂.closure hf H₁
|
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Set.Finite
import Mathlib.GroupTheory.GroupAction.BigOperators
#align_import data.dfinsupp.basic from "leanpr... | Mathlib/Data/DFinsupp/Basic.lean | 709 | 713 | theorem equivFunOnFintype_single [Fintype ι] (i : ι) (m : β i) :
(@DFinsupp.equivFunOnFintype ι β _ _) (DFinsupp.single i m) = Pi.single i m := by |
ext x
dsimp [Pi.single, Function.update]
simp [DFinsupp.single_eq_pi_single, @eq_comm _ i]
|
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Analytic.Composition
import Mathlib.Analysis.Analytic.Linear
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Geometry.Manifold.ChartedSpace
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.Analysis.Calculus.ContDiff.Basic
... | Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean | 1,257 | 1,263 | theorem contDiffWithinAt_extend_coord_change [ChartedSpace H M] (hf : f ∈ maximalAtlas I M)
(hf' : f' ∈ maximalAtlas I M) {x : E} (hx : x ∈ ((f'.extend I).symm ≫ f.extend I).source) :
ContDiffWithinAt 𝕜 ⊤ (f.extend I ∘ (f'.extend I).symm) (range I) x := by |
apply (contDiffOn_extend_coord_change I hf hf' x hx).mono_of_mem
rw [extend_coord_change_source] at hx ⊢
obtain ⟨z, hz, rfl⟩ := hx
exact I.image_mem_nhdsWithin ((PartialHomeomorph.open_source _).mem_nhds hz)
|
import Mathlib.RingTheory.WittVector.InitTail
#align_import ring_theory.witt_vector.truncated from "leanprover-community/mathlib"@"acbe099ced8be9c9754d62860110295cde0d7181"
open Function (Injective Surjective)
noncomputable section
variable {p : ℕ} [hp : Fact p.Prime] (n : ℕ) (R : Type*)
local notation "𝕎" =>... | Mathlib/RingTheory/WittVector/Truncated.lean | 259 | 261 | theorem truncateFun_mul (x y : 𝕎 R) :
truncateFun n (x * y) = truncateFun n x * truncateFun n y := by |
witt_truncateFun_tac
|
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Analytic.Composition
import Mathlib.Analysis.Analytic.Linear
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Geometry.Manifold.ChartedSpace
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.Analysis.Calculus.ContDiff.Basic
... | Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean | 1,520 | 1,523 | theorem extChartAt_preimage_mem_nhds {x : M} (ht : t ∈ 𝓝 x) :
(extChartAt I x).symm ⁻¹' t ∈ 𝓝 ((extChartAt I x) x) := by |
apply (continuousAt_extChartAt_symm I x).preimage_mem_nhds
rwa [(extChartAt I x).left_inv (mem_extChartAt_source _ _)]
|
import Mathlib.CategoryTheory.Localization.Construction
#align_import category_theory.localization.predicate from "leanprover-community/mathlib"@"8efef279998820353694feb6ff5631ed0d309ecc"
noncomputable section
namespace CategoryTheory
open Category
variable {C D : Type*} [Category C] [Category D] (L : C ⥤ D) (... | Mathlib/CategoryTheory/Localization/Predicate.lean | 263 | 267 | theorem natTrans_ext {F₁ F₂ : D ⥤ E} (τ τ' : F₁ ⟶ F₂)
(h : ∀ X : C, τ.app (L.obj X) = τ'.app (L.obj X)) : τ = τ' := by |
haveI := essSurj L W
ext Y
rw [← cancel_epi (F₁.map (L.objObjPreimageIso Y).hom), τ.naturality, τ'.naturality, h]
|
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.NumberTheory.Padics.PadicNorm
import Mathlib.Analysis.Normed.Field.Basic
#align_import number_theory.padics.padic_numbers from "leanprover-community/mathlib"@"b9b2114f7711fec1c1e055d507f082f8ceb2c3b7"
noncomputable section
open scoped Classical
open Nat m... | Mathlib/NumberTheory/Padics/PadicNumbers.lean | 845 | 848 | theorem norm_p_lt_one : ‖(p : ℚ_[p])‖ < 1 := by |
rw [norm_p]
apply inv_lt_one
exact mod_cast hp.1.one_lt
|
import Mathlib.Data.Matroid.Basic
open Set Matroid
variable {α : Type*} {I B X : Set α}
section IndepMatroid
structure IndepMatroid (α : Type*) where
(E : Set α)
(Indep : Set α → Prop)
(indep_empty : Indep ∅)
(indep_subset : ∀ ⦃I J⦄, Indep J → I ⊆ J → Indep I)
(indep_aug : ∀⦃I B⦄, Indep I → I ∉ ... | Mathlib/Data/Matroid/IndepAxioms.lean | 423 | 427 | theorem ofFinset_indep' [DecidableEq α] (E : Set α) Indep indep_empty indep_subset indep_aug
subset_ground {I : Set α} : (IndepMatroid.ofFinset
E Indep indep_empty indep_subset indep_aug subset_ground).Indep I ↔
∀ (J : Finset α), (J : Set α) ⊆ I → Indep J := by |
simp only [IndepMatroid.ofFinset, ofFinitary_indep]
|
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.Order.Circular
import Mathlib.Data.List.TFAE
import Mathlib.Data.Set.Lattice
#align_import a... | Mathlib/Algebra/Order/ToIntervalMod.lean | 222 | 224 | theorem toIcoMod_apply_right (a : α) : toIcoMod hp a (a + p) = a := by |
rw [toIcoMod_eq_iff hp, Set.left_mem_Ico]
exact ⟨lt_add_of_pos_right _ hp, 1, by simp⟩
|
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import analysis.special_functions.complex.log from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
namespace Complex
open Set Filter Bornology
open scop... | Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 36 | 36 | theorem log_im (x : ℂ) : x.log.im = x.arg := by | simp [log]
|
import Mathlib.MeasureTheory.Covering.Differentiation
import Mathlib.MeasureTheory.Covering.VitaliFamily
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.Data.Set.Pairwise.Lat... | Mathlib/MeasureTheory/Covering/Besicovitch.lean | 362 | 478 | theorem color_lt {i : Ordinal.{u}} (hi : i < p.lastStep) {N : ℕ}
(hN : IsEmpty (SatelliteConfig α N p.τ)) : p.color i < N := by |
/- By contradiction, consider the first ordinal `i` for which one would have `p.color i = N`.
Choose for each `k < N` a ball with color `k` that intersects the ball at color `i`
(there is such a ball, otherwise one would have used the color `k` and not `N`).
Then this family of `N+1` balls forms a satell... |
import Mathlib.Algebra.Module.Submodule.Bilinear
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.Tactic.SuppressCompilation
#align_import linear_algebra.tensor_product from "leanprover-community/mathlib"@"88fcdc3da43943f5b01925deddaa5bf0c0e85e4e"
suppress_compilation
... | Mathlib/LinearAlgebra/TensorProduct/Basic.lean | 899 | 901 | theorem map_id : map (id : M →ₗ[R] M) (id : N →ₗ[R] N) = .id := by |
ext
simp only [mk_apply, id_coe, compr₂_apply, _root_.id, map_tmul]
|
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
namespace Polynomial
open Polynomial
section Semiring
variable {R : Type*} [Semiring R] (p q : R... | Mathlib/Algebra/Polynomial/Mirror.lean | 161 | 169 | theorem coeff_mul_mirror :
(p * p.mirror).coeff (p.natDegree + p.natTrailingDegree) = p.sum fun n => (· ^ 2) := by |
rw [coeff_mul, Finset.Nat.sum_antidiagonal_eq_sum_range_succ_mk]
refine
(Finset.sum_congr rfl fun n hn => ?_).trans
(p.sum_eq_of_subset (fun _ ↦ (· ^ 2)) (fun _ ↦ zero_pow two_ne_zero) fun n hn ↦
Finset.mem_range_succ_iff.mpr
((le_natDegree_of_mem_supp n hn).trans (Nat.le_add_right ... |
import Mathlib.LinearAlgebra.LinearPMap
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.module.linear_pmap from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology
variable {R E F : Type*}
variable [CommRing R] [AddCommGroup E] [AddCommGroup F]
vari... | Mathlib/Topology/Algebra/Module/LinearPMap.lean | 77 | 85 | theorem IsClosable.leIsClosable {f g : E →ₗ.[R] F} (hf : f.IsClosable) (hfg : g ≤ f) :
g.IsClosable := by |
cases' hf with f' hf
have : g.graph.topologicalClosure ≤ f'.graph := by
rw [← hf]
exact Submodule.topologicalClosure_mono (le_graph_of_le hfg)
use g.graph.topologicalClosure.toLinearPMap
rw [Submodule.toLinearPMap_graph_eq]
exact fun _ hx hx' => f'.graph_fst_eq_zero_snd (this hx) hx'
|
import Mathlib.MeasureTheory.Group.Measure
assert_not_exists NormedSpace
namespace MeasureTheory
open Measure TopologicalSpace
open scoped ENNReal
variable {G : Type*} [MeasurableSpace G] {μ : Measure G} {g : G}
section MeasurableMul
variable [Group G] [MeasurableMul G]
@[to_additive
"Translating a fu... | Mathlib/MeasureTheory/Group/LIntegral.lean | 34 | 37 | theorem lintegral_mul_left_eq_self [IsMulLeftInvariant μ] (f : G → ℝ≥0∞) (g : G) :
(∫⁻ x, f (g * x) ∂μ) = ∫⁻ x, f x ∂μ := by |
convert (lintegral_map_equiv f <| MeasurableEquiv.mulLeft g).symm
simp [map_mul_left_eq_self μ g]
|
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Order.SupIndep
import Mathlib.Order.Atoms
#align_import order.partition.finpartition from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
open Finset Function
variable {α : Type*}
@[ext]
structure Finpartition [Lattice α]... | Mathlib/Order/Partition/Finpartition.lean | 689 | 701 | theorem card_filter_atomise_le_two_pow (ht : t ∈ F) :
((atomise s F).parts.filter fun u ↦ u ⊆ t ∧ u.Nonempty).card ≤ 2 ^ (F.card - 1) := by |
suffices h :
((atomise s F).parts.filter fun u ↦ u ⊆ t ∧ u.Nonempty) ⊆
(F.erase t).powerset.image fun P ↦ s.filter fun i ↦ ∀ x ∈ F, x ∈ insert t P ↔ i ∈ x by
refine (card_le_card h).trans (card_image_le.trans ?_)
rw [card_powerset, card_erase_of_mem ht]
rw [subset_iff]
simp_rw [mem_image, mem_p... |
import Mathlib.Algebra.Group.Prod
import Mathlib.Order.Cover
#align_import algebra.support from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
assert_not_exists MonoidWithZero
open Set
namespace Function
variable {α β A B M N P G : Type*}
section One
variable [One M] [One N] [One P]
... | Mathlib/Algebra/Group/Support.lean | 119 | 122 | theorem mulSupport_disjoint_iff {f : α → M} {s : Set α} :
Disjoint (mulSupport f) s ↔ EqOn f 1 s := by |
simp_rw [← subset_compl_iff_disjoint_right, mulSupport_subset_iff', not_mem_compl_iff, EqOn,
Pi.one_apply]
|
import Mathlib.Tactic.Linarith
import Mathlib.CategoryTheory.Skeletal
import Mathlib.Data.Fintype.Sort
import Mathlib.Order.Category.NonemptyFinLinOrd
import Mathlib.CategoryTheory.Functor.ReflectsIso
#align_import algebraic_topology.simplex_category from "leanprover-community/mathlib"@"e8ac6315bcfcbaf2d19a046719c3b5... | Mathlib/AlgebraicTopology/SimplexCategory.lean | 572 | 575 | theorem len_le_of_mono {x y : SimplexCategory} {f : x ⟶ y} : Mono f → x.len ≤ y.len := by |
intro hyp_f_mono
have f_inj : Function.Injective f.toOrderHom.toFun := mono_iff_injective.1 hyp_f_mono
simpa using Fintype.card_le_of_injective f.toOrderHom.toFun f_inj
|
import Mathlib.AlgebraicGeometry.OpenImmersion
-- Explicit universe annotations were used in this file to improve perfomance #12737
set_option linter.uppercaseLean3 false
noncomputable section
open TopologicalSpace CategoryTheory Opposite
open CategoryTheory.Limits
namespace AlgebraicGeometry
universe v v₁ v₂... | Mathlib/AlgebraicGeometry/Restrict.lean | 131 | 135 | theorem Scheme.restrictFunctor_map_base {U V : Opens X} (i : U ⟶ V) :
(X.restrictFunctor.map i).1.1.base = (Opens.toTopCat _).map i := by |
ext a; refine Subtype.ext ?_ -- Porting note: `ext` did not pick up `Subtype.ext`
exact (congr_arg (fun f : X.restrict U.openEmbedding ⟶ X => f.1.base a)
(X.restrictFunctor_map_ofRestrict i))
|
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Adjunction.Reflective
#align_import algebraic_geometry.Gamma_Spec_adjunction from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
-- Explicit universe annotations were u... | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | 468 | 488 | theorem adjunction_unit_app_app_top (X : Scheme.{u}) :
(ΓSpec.adjunction.unit.app X).1.c.app (op ⊤) =
SpecΓIdentity.hom.app (X.presheaf.obj (op ⊤)) := by |
have := congr_app ΓSpec.adjunction.left_triangle X
dsimp at this
-- Porting note: Slightly changed some rewrites.
-- Originally:
-- rw [← is_iso.eq_comp_inv] at this
-- simp only [Γ_Spec.LocallyRingedSpace_adjunction_counit, nat_trans.op_app, category.id_comp,
-- Γ_Spec.adjunction_counit_app] at thi... |
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import measure_theory.measure.haar.of_basis from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
open Set TopologicalSpace MeasureTheory MeasureTheory.Measure FiniteDimensional
open sco... | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean | 297 | 301 | theorem Basis.prod_addHaar (v : Basis ι ℝ E) (w : Basis ι' ℝ F) :
(v.prod w).addHaar = v.addHaar.prod w.addHaar := by |
have : FiniteDimensional ℝ E := FiniteDimensional.of_fintype_basis v
have : FiniteDimensional ℝ F := FiniteDimensional.of_fintype_basis w
simp [(v.prod w).addHaar_eq_iff, Basis.prod_parallelepiped, Basis.addHaar_self]
|
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.matrix.nonsingular_inverse from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
namespace Matrix
universe u u' v
variable {l : ... | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 772 | 780 | theorem inv_submatrix_equiv (A : Matrix m m α) (e₁ e₂ : n ≃ m) :
(A.submatrix e₁ e₂)⁻¹ = A⁻¹.submatrix e₂ e₁ := by |
by_cases h : IsUnit A
· cases h.nonempty_invertible
letI := submatrixEquivInvertible A e₁ e₂
rw [← invOf_eq_nonsing_inv, ← invOf_eq_nonsing_inv, invOf_submatrix_equiv_eq A]
· have := (isUnit_submatrix_equiv e₁ e₂).not.mpr h
simp_rw [nonsing_inv_eq_ring_inverse, Ring.inverse_non_unit _ h, Ring.inverse... |
import Mathlib.NumberTheory.LegendreSymbol.Basic
import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum
#align_import number_theory.legendre_symbol.quadratic_reciprocity from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
open Nat
section Values
variable {p : ℕ} [Fact p.Pri... | Mathlib/NumberTheory/LegendreSymbol/QuadraticReciprocity.lean | 173 | 178 | theorem exists_sq_eq_prime_iff_of_mod_four_eq_one (hp1 : p % 4 = 1) (hq1 : q ≠ 2) :
IsSquare (q : ZMod p) ↔ IsSquare (p : ZMod q) := by |
rcases eq_or_ne p q with h | h
· subst p; rfl
· rw [← eq_one_iff' p (prime_ne_zero p q h), ← eq_one_iff' q (prime_ne_zero q p h.symm),
quadratic_reciprocity_one_mod_four hp1 hq1]
|
import Mathlib.MeasureTheory.Covering.Differentiation
import Mathlib.MeasureTheory.Covering.VitaliFamily
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.Data.Set.Pairwise.Lat... | Mathlib/MeasureTheory/Covering/Besicovitch.lean | 1,119 | 1,123 | theorem ae_tendsto_rnDeriv (ρ μ : Measure β) [IsLocallyFiniteMeasure μ] [IsLocallyFiniteMeasure ρ] :
∀ᵐ x ∂μ,
Tendsto (fun r => ρ (closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 (ρ.rnDeriv μ x)) := by |
filter_upwards [VitaliFamily.ae_tendsto_rnDeriv (Besicovitch.vitaliFamily μ) ρ] with x hx
exact hx.comp (tendsto_filterAt μ x)
|
import Mathlib.Topology.Separation
#align_import topology.sober from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
open Set
variable {α β : Type*} [TopologicalSpace α] [TopologicalSpace β]
section genericPoint
def IsGenericPoint (x : α) (S : Set α) : Prop :=
closure ({x} : Set α)... | Mathlib/Topology/Sober.lean | 53 | 54 | theorem isGenericPoint_iff_specializes : IsGenericPoint x S ↔ ∀ y, x ⤳ y ↔ y ∈ S := by |
simp only [specializes_iff_mem_closure, IsGenericPoint, Set.ext_iff]
|
import Mathlib.Analysis.NormedSpace.Real
import Mathlib.Analysis.Seminorm
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import analysis.normed_space.riesz_lemma from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Metric
open Topology
variable {𝕜 : Type*} [Norm... | Mathlib/Analysis/NormedSpace/RieszLemma.lean | 41 | 70 | theorem riesz_lemma {F : Subspace 𝕜 E} (hFc : IsClosed (F : Set E)) (hF : ∃ x : E, x ∉ F) {r : ℝ}
(hr : r < 1) : ∃ x₀ : E, x₀ ∉ F ∧ ∀ y ∈ F, r * ‖x₀‖ ≤ ‖x₀ - y‖ := by |
classical
obtain ⟨x, hx⟩ : ∃ x : E, x ∉ F := hF
let d := Metric.infDist x F
have hFn : (F : Set E).Nonempty := ⟨_, F.zero_mem⟩
have hdp : 0 < d :=
lt_of_le_of_ne Metric.infDist_nonneg fun heq =>
hx ((hFc.mem_iff_infDist_zero hFn).2 heq.symm)
let r' := max r 2⁻¹
have hr' : r' < 1... |
import Mathlib.Algebra.Polynomial.Degree.Definitions
#align_import ring_theory.polynomial.opposites from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
open Polynomial
open Polynomial MulOpposite
variable {R : Type*} [Semiring R]
noncomputable section
namespace Polynomial
def opRi... | Mathlib/RingTheory/Polynomial/Opposites.lean | 85 | 87 | theorem opRingEquiv_symm_C_mul_X_pow (r : Rᵐᵒᵖ) (n : ℕ) :
(opRingEquiv R).symm (C r * X ^ n : Rᵐᵒᵖ[X]) = op (C (unop r) * X ^ n) := by |
rw [C_mul_X_pow_eq_monomial, opRingEquiv_symm_monomial, C_mul_X_pow_eq_monomial]
|
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.FiniteSupport
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Set.Subsingleton
#align_import algebra.big_operators.finprod from "leanprover-community/mathlib"@"d6fad0e5bf... | Mathlib/Algebra/BigOperators/Finprod.lean | 561 | 565 | theorem finprod_mem_inter_mulSupport_eq' (f : α → M) (s t : Set α)
(h : ∀ x ∈ mulSupport f, x ∈ s ↔ x ∈ t) : ∏ᶠ i ∈ s, f i = ∏ᶠ i ∈ t, f i := by |
apply finprod_mem_inter_mulSupport_eq
ext x
exact and_congr_left (h x)
|
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.MeasureTheory.Measure.Doubling
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
#align_import measu... | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 120 | 124 | theorem addHaarMeasure_eq_volume_pi (ι : Type*) [Fintype ι] :
addHaarMeasure (piIcc01 ι) = volume := by |
convert (addHaarMeasure_unique volume (piIcc01 ι)).symm
simp only [piIcc01, volume_pi_pi fun _ => Icc (0 : ℝ) 1, PositiveCompacts.coe_mk,
Compacts.coe_mk, Finset.prod_const_one, ENNReal.ofReal_one, Real.volume_Icc, one_smul, sub_zero]
|
import Mathlib.MeasureTheory.Group.GeometryOfNumbers
import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | 70 | 75 | theorem convexBodyLT_neg_mem (x : E K) (hx : x ∈ (convexBodyLT K f)) :
-x ∈ (convexBodyLT K f) := by |
simp only [Set.mem_prod, Prod.fst_neg, Set.mem_pi, Set.mem_univ, Pi.neg_apply,
mem_ball_zero_iff, norm_neg, Real.norm_eq_abs, forall_true_left, Subtype.forall,
Prod.snd_neg, Complex.norm_eq_abs] at hx ⊢
exact hx
|
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Covering.Besicovitch
import Mathlib.Tactic.AdaptationNote
#align_import measure_theory.covering.besicovitch_vector_space from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
universe u
open Metric Set Fini... | Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean | 83 | 84 | theorem centerAndRescale_center : a.centerAndRescale.c (last N) = 0 := by |
simp [SatelliteConfig.centerAndRescale]
|
import Mathlib.CategoryTheory.Filtered.Connected
import Mathlib.CategoryTheory.Limits.TypesFiltered
import Mathlib.CategoryTheory.Limits.Final
universe v₁ v₂ u₁ u₂
namespace CategoryTheory
open CategoryTheory.Limits CategoryTheory.Functor Opposite
section ArbitraryUniverses
variable {C : Type u₁} [Category.{v₁}... | Mathlib/CategoryTheory/Filtered/Final.lean | 91 | 95 | theorem Functor.final_of_exists_of_isFiltered [IsFilteredOrEmpty C]
(h₁ : ∀ d, ∃ c, Nonempty (d ⟶ F.obj c)) (h₂ : ∀ {d : D} {c : C} (s s' : d ⟶ F.obj c),
∃ (c' : C) (t : c ⟶ c'), s ≫ F.map t = s' ≫ F.map t) : Functor.Final F := by |
suffices ∀ d, IsFiltered (StructuredArrow d F) from final_of_isFiltered_structuredArrow F
exact isFiltered_structuredArrow_of_isFiltered_of_exists F h₁ h₂
|
import Mathlib.Topology.Maps
import Mathlib.Topology.NhdsSet
#align_import topology.constructions from "leanprover-community/mathlib"@"f7ebde7ee0d1505dfccac8644ae12371aa3c1c9f"
noncomputable section
open scoped Classical
open Topology TopologicalSpace Set Filter Function
universe u v
variable {X : Type u} {Y :... | Mathlib/Topology/Constructions.lean | 1,305 | 1,306 | theorem nhds_pi {a : ∀ i, π i} : 𝓝 a = pi fun i => 𝓝 (a i) := by |
simp only [nhds_iInf, nhds_induced, Filter.pi]
|
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.degree.lemmas from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
noncomputable section
open Polynomial
open Finsupp Finset
namespace Polynomial
universe u v w
variable {R : Type u} {S : Type v} {ι : Type w} {a b ... | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 304 | 312 | theorem natDegree_map_eq_iff {f : R →+* S} {p : Polynomial R} :
natDegree (map f p) = natDegree p ↔ f (p.leadingCoeff) ≠ 0 ∨ natDegree p = 0 := by |
rcases eq_or_ne (natDegree p) 0 with h|h
· simp_rw [h, ne_eq, or_true, iff_true, ← Nat.le_zero, ← h, natDegree_map_le f p]
have h2 : p ≠ 0 := by rintro rfl; simp at h
have h3 : degree p ≠ (0 : ℕ) := degree_ne_of_natDegree_ne h
simp_rw [h, or_false, natDegree, WithBot.unbot'_eq_unbot'_iff, degree_map_eq_iff]... |
import Mathlib.Data.Set.Lattice
import Mathlib.Logic.Small.Basic
import Mathlib.Logic.Function.OfArity
import Mathlib.Order.WellFounded
#align_import set_theory.zfc.basic from "leanprover-community/mathlib"@"f0b3759a8ef0bd8239ecdaa5e1089add5feebe1a"
-- Porting note: Lean 3 uses `Set` for `ZFSet`.
set_option linter... | Mathlib/SetTheory/ZFC/Basic.lean | 1,325 | 1,325 | theorem mem_funs {x y f : ZFSet.{u}} : f ∈ funs x y ↔ IsFunc x y f := by | simp [funs, IsFunc]
|
import Mathlib.Data.Set.Function
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Says
#align_import logic.equiv.set from "leanprover-community/mathlib"@"aba57d4d3dae35460225919dcd82fe91355162f9"
open Function Set
universe u v w z
variable {α : Sort u} {β : Sort v} {γ : Sort w}
namespace Equiv
@[simp]
th... | Mathlib/Logic/Equiv/Set.lean | 711 | 728 | theorem dite_comp_equiv_update {α : Type*} {β : Sort*} {γ : Sort*} {p : α → Prop}
(e : β ≃ Subtype p)
(v : β → γ) (w : α → γ) (j : β) (x : γ) [DecidableEq β] [DecidableEq α]
[∀ j, Decidable (p j)] :
(fun i : α => if h : p i then (Function.update v j x) (e.symm ⟨i, h⟩) else w i) =
Function.update (... |
ext i
by_cases h : p i
· rw [dif_pos h, Function.update_apply_equiv_apply, Equiv.symm_symm,
Function.update_apply, Function.update_apply, dif_pos h]
have h_coe : (⟨i, h⟩ : Subtype p) = e j ↔ i = e j :=
Subtype.ext_iff.trans (by rw [Subtype.coe_mk])
simp [h_coe]
· have : i ≠ e j := by
... |
import Mathlib.Algebra.Group.Indicator
import Mathlib.Data.Finset.Piecewise
import Mathlib.Data.Finset.Preimage
#align_import algebra.big_operators.basic from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
-- TODO
-- assert_not_exists AddCommMonoidWithOne
assert_not_exists MonoidWithZero... | Mathlib/Algebra/BigOperators/Group/Finset.lean | 2,450 | 2,452 | theorem disjoint_finset_sum_right {β : Type*} {i : Finset β} {f : β → Multiset α}
{a : Multiset α} : Multiset.Disjoint a (i.sum f) ↔ ∀ b ∈ i, Multiset.Disjoint a (f b) := by |
simpa only [disjoint_comm] using disjoint_finset_sum_left
|
import Batteries.Control.ForInStep.Lemmas
import Batteries.Data.List.Basic
import Batteries.Tactic.Init
import Batteries.Tactic.Alias
namespace List
open Nat
@[simp] theorem mem_toArray {a : α} {l : List α} : a ∈ l.toArray ↔ a ∈ l := by
simp [Array.mem_def]
@[simp]
theorem drop_one : ∀ l : List α, drop 1 l =... | .lake/packages/batteries/Batteries/Data/List/Lemmas.lean | 1,371 | 1,372 | theorem range_sublist {m n : Nat} : range m <+ range n ↔ m ≤ n := by |
simp only [range_eq_range', range'_sublist_right]
|
import Mathlib.Data.Matrix.Basic
import Mathlib.LinearAlgebra.Matrix.Trace
#align_import data.matrix.basis from "leanprover-community/mathlib"@"320df450e9abeb5fc6417971e75acb6ae8bc3794"
variable {l m n : Type*}
variable {R α : Type*}
namespace Matrix
open Matrix
variable [DecidableEq l] [DecidableEq m] [Decida... | Mathlib/Data/Matrix/Basis.lean | 45 | 48 | theorem stdBasisMatrix_zero (i : m) (j : n) : stdBasisMatrix i j (0 : α) = 0 := by |
unfold stdBasisMatrix
ext
simp
|
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.Order.Circular
import Mathlib.Data.List.TFAE
import Mathlib.Data.Set.Lattice
#align_import a... | Mathlib/Algebra/Order/ToIntervalMod.lean | 431 | 432 | theorem toIcoMod_zsmul_add (a b : α) (m : ℤ) : toIcoMod hp a (m • p + b) = toIcoMod hp a b := by |
rw [add_comm, toIcoMod_add_zsmul]
|
import Mathlib.Algebra.Category.GroupCat.Basic
import Mathlib.CategoryTheory.ConcreteCategory.ReflectsIso
import Mathlib.Algebra.Ring.Equiv
#align_import algebra.category.Ring.basic from "leanprover-community/mathlib"@"34b2a989ad80bce3a5de749d935a4f23726e26e9"
universe u v
open CategoryTheory
abbrev SemiRingCat... | Mathlib/Algebra/Category/Ring/Basic.lean | 665 | 668 | theorem commRingIsoToRingEquiv_toRingHom {X Y : CommRingCat} (i : X ≅ Y) :
i.commRingCatIsoToRingEquiv.toRingHom = i.hom := by |
ext
rfl
|
import Mathlib.Data.PNat.Defs
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Data.Set.Basic
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Positive.Ring
import Mathlib.Order.Hom.Basic
#align_import data.pnat.basic from "leanprover-community/mathlib"@"172bf2812857f5e56938cc148b7a5... | Mathlib/Data/PNat/Basic.lean | 33 | 34 | theorem one_add_natPred (n : ℕ+) : 1 + n.natPred = n := by |
rw [natPred, add_tsub_cancel_iff_le.mpr <| show 1 ≤ (n : ℕ) from n.2]
|
import Mathlib.Data.Finset.Update
import Mathlib.Data.Prod.TProd
import Mathlib.GroupTheory.Coset
import Mathlib.Logic.Equiv.Fin
import Mathlib.MeasureTheory.MeasurableSpace.Defs
import Mathlib.Order.Filter.SmallSets
import Mathlib.Order.LiminfLimsup
import Mathlib.Data.Set.UnionLift
#align_import measure_theory.meas... | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | 836 | 842 | theorem measurableSet_prod_of_nonempty {s : Set α} {t : Set β} (h : (s ×ˢ t).Nonempty) :
MeasurableSet (s ×ˢ t) ↔ MeasurableSet s ∧ MeasurableSet t := by |
rcases h with ⟨⟨x, y⟩, hx, hy⟩
refine ⟨fun hst => ?_, fun h => h.1.prod h.2⟩
have : MeasurableSet ((fun x => (x, y)) ⁻¹' s ×ˢ t) := measurable_prod_mk_right hst
have : MeasurableSet (Prod.mk x ⁻¹' s ×ˢ t) := measurable_prod_mk_left hst
simp_all
|
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Valuation.PrimeMultiplicity
import Mathlib.RingTheory.AdicCompletion.Basic
#align_import ring_theory.discrete_valuation_ring.basic from "leanprover-community/mathlib"@"c163ec99dfc664628ca15d215fce0a5b9c2... | Mathlib/RingTheory/DiscreteValuationRing/Basic.lean | 342 | 347 | theorem eq_unit_mul_pow_irreducible {x : R} (hx : x ≠ 0) {ϖ : R} (hirr : Irreducible ϖ) :
∃ (n : ℕ) (u : Rˣ), x = u * ϖ ^ n := by |
obtain ⟨n, hn⟩ := associated_pow_irreducible hx hirr
obtain ⟨u, rfl⟩ := hn.symm
use n, u
apply mul_comm
|
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
#align_import category_theory.limits.shapes.biproducts from "leanprover-community/mathlib"@"ac3ae212f394f508df43e37aa093722fa9b65d31"
noncomputab... | Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean | 619 | 630 | theorem biproduct.map_eq_map' {f g : J → C} [HasBiproduct f] [HasBiproduct g] (p : ∀ b, f b ⟶ g b) :
biproduct.map p = biproduct.map' p := by |
ext
dsimp
simp only [Discrete.natTrans_app, Limits.IsColimit.ι_map_assoc, Limits.IsLimit.map_π,
Category.assoc, ← Bicone.toCone_π_app_mk, ← biproduct.bicone_π, ← Bicone.toCocone_ι_app_mk,
← biproduct.bicone_ι]
dsimp
rw [biproduct.ι_π_assoc, biproduct.ι_π]
split_ifs with h
· subst h; rw [eqToHom_r... |
import Mathlib.MeasureTheory.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
assert_not_exists Differentiable
noncomputable section
open scoped Topology NNReal ENNReal MeasureTheory
open Set Filter TopologicalSpace EN... | Mathlib/MeasureTheory/Integral/Bochner.lean | 1,204 | 1,209 | theorem ofReal_integral_eq_lintegral_ofReal {f : α → ℝ} (hfi : Integrable f μ) (f_nn : 0 ≤ᵐ[μ] f) :
ENNReal.ofReal (∫ x, f x ∂μ) = ∫⁻ x, ENNReal.ofReal (f x) ∂μ := by |
have : f =ᵐ[μ] (‖f ·‖) := f_nn.mono fun _x hx ↦ (abs_of_nonneg hx).symm
simp_rw [integral_congr_ae this, ofReal_integral_norm_eq_lintegral_nnnorm hfi,
← ofReal_norm_eq_coe_nnnorm]
exact lintegral_congr_ae (this.symm.fun_comp ENNReal.ofReal)
|
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Set.Sigma
#align_import data.finset.sigma from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Multiset
variable {ι : Type*}
namespace Finset
section SigmaLift
variable {α β γ : ι → Type*} [DecidableEq ι]
def sigm... | Mathlib/Data/Finset/Sigma.lean | 221 | 224 | theorem card_sigmaLift :
(sigmaLift f a b).card = dite (a.1 = b.1) (fun h => (f (h ▸ a.2) b.2).card) fun _ => 0 := by |
simp_rw [sigmaLift]
split_ifs with h <;> simp [h]
|
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
#align_import measure_theory.integral.interval_integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open scoped... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 108 | 110 | theorem intervalIntegrable_iff_integrableOn_Ico_of_le [NoAtoms μ] (hab : a ≤ b) :
IntervalIntegrable f μ a b ↔ IntegrableOn f (Ico a b) μ := by |
rw [intervalIntegrable_iff_integrableOn_Icc_of_le hab, integrableOn_Icc_iff_integrableOn_Ico]
|
import Mathlib.Order.Filter.Basic
#align_import order.filter.prod from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
open Set
open Filter
namespace Filter
variable {α β γ δ : Type*} {ι : Sort*}
section Prod
variable {s : Set α} {t : Set β} {f : Filter α} {g : Filter β}
protected ... | Mathlib/Order/Filter/Prod.lean | 272 | 274 | theorem prod_comm : f ×ˢ g = map (fun p : β × α => (p.2, p.1)) (g ×ˢ f) := by |
rw [prod_comm', ← map_swap_eq_comap_swap]
rfl
|
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Vector.Basic
import Mathlib.Data.PFun
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Basic
import Mathlib.Tactic.ApplyFun
#align_import computability.turing_machine from "leanprover-commu... | Mathlib/Computability/TuringMachine.lean | 901 | 906 | theorem tr_reaches {σ₁ σ₂ f₁ f₂} {tr : σ₁ → σ₂ → Prop} (H : Respects f₁ f₂ tr) {a₁ a₂}
(aa : tr a₁ a₂) {b₁} (ab : Reaches f₁ a₁ b₁) : ∃ b₂, tr b₁ b₂ ∧ Reaches f₂ a₂ b₂ := by |
rcases reflTransGen_iff_eq_or_transGen.1 ab with (rfl | ab)
· exact ⟨_, aa, ReflTransGen.refl⟩
· have ⟨b₂, bb, h⟩ := tr_reaches₁ H aa ab
exact ⟨b₂, bb, h.to_reflTransGen⟩
|
import Mathlib.RingTheory.FiniteType
#align_import ring_theory.rees_algebra from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R)
open Polynomial
open Polynomial
def reesAlgebra : Subalgebra... | Mathlib/RingTheory/ReesAlgebra.lean | 76 | 79 | theorem reesAlgebra.monomial_mem {I : Ideal R} {i : ℕ} {r : R} :
monomial i r ∈ reesAlgebra I ↔ r ∈ I ^ i := by |
simp (config := { contextual := true }) [mem_reesAlgebra_iff_support, coeff_monomial, ←
imp_iff_not_or]
|
import Mathlib.CategoryTheory.NatTrans
import Mathlib.CategoryTheory.Iso
#align_import category_theory.functor.category from "leanprover-community/mathlib"@"63721b2c3eba6c325ecf8ae8cca27155a4f6306f"
namespace CategoryTheory
-- declare the `v`'s first; see note [CategoryTheory universes].
universe v₁ v₂ v₃ u₁ u₂ u... | Mathlib/CategoryTheory/Functor/Category.lean | 132 | 134 | theorem exchange {I J K : D ⥤ E} (α : F ⟶ G) (β : G ⟶ H) (γ : I ⟶ J) (δ : J ⟶ K) :
(α ≫ β) ◫ (γ ≫ δ) = (α ◫ γ) ≫ β ◫ δ := by |
aesop_cat
|
import Mathlib.Algebra.Homology.Homology
import Mathlib.Algebra.Homology.Single
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
#align_import algebra.homology.additive from "leanprover-community/mathlib"@"200eda15d8ff5669854ff6bcc10aaf37cb70498f"
universe v u
open CategoryTheory CategoryTheory.Category... | Mathlib/Algebra/Homology/Additive.lean | 331 | 334 | theorem singleMapHomologicalComplex_inv_app_self (j : ι) (X : W₁) :
((singleMapHomologicalComplex F c j).inv.app X).f j =
(singleObjXSelf c j (F.obj X)).hom ≫ F.map (singleObjXSelf c j X).inv := by |
simp [singleMapHomologicalComplex, singleObjXSelf, singleObjXIsoOfEq, eqToHom_map]
|
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Regular.SMul
#align_import data.polynomial.monic from "leanprover-community/mathlib"@"cbdf7b565832144d024caa5a550117c6df0204a5"
noncomputable section
open Finset
open Polynomial
namespace Polynomial
universe u v y
variable {R : Type u} {S : Typ... | Mathlib/Algebra/Polynomial/Monic.lean | 496 | 507 | theorem degree_smul_of_smul_regular {S : Type*} [Monoid S] [DistribMulAction S R] {k : S}
(p : R[X]) (h : IsSMulRegular R k) : (k • p).degree = p.degree := by |
refine le_antisymm ?_ ?_
· rw [degree_le_iff_coeff_zero]
intro m hm
rw [degree_lt_iff_coeff_zero] at hm
simp [hm m le_rfl]
· rw [degree_le_iff_coeff_zero]
intro m hm
rw [degree_lt_iff_coeff_zero] at hm
refine h ?_
simpa using hm m le_rfl
|
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
import Mathlib.RepresentationTheory.GroupCohomology.Basic
import Mathlib.RepresentationTheory.Invariants
universe v u
noncomputable section
open CategoryTheory Limits Representation
variable {k G : Type u} [CommRing k] [Group G] (A : Rep k G)
namespace grou... | Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean | 166 | 173 | theorem dOne_comp_eq : dOne A ∘ₗ oneCochainsLequiv A =
twoCochainsLequiv A ∘ₗ (inhomogeneousCochains A).d 1 2 := by |
ext x y
show A.ρ y.1 (x _) - x _ + x _ = _ + _
rw [Fin.sum_univ_two]
simp only [Fin.val_zero, zero_add, pow_one, neg_smul, one_smul, Fin.val_one,
Nat.one_add, neg_one_sq, sub_eq_add_neg, add_assoc]
rcongr i <;> rw [Subsingleton.elim i 0] <;> rfl
|
import Mathlib.Algebra.MvPolynomial.Counit
import Mathlib.Algebra.MvPolynomial.Invertible
import Mathlib.RingTheory.WittVector.Defs
#align_import ring_theory.witt_vector.basic from "leanprover-community/mathlib"@"9556784a5b84697562e9c6acb40500d4a82e675a"
noncomputable section
open MvPolynomial Function
variable... | Mathlib/RingTheory/WittVector/Basic.lean | 102 | 102 | theorem zero : mapFun f (0 : 𝕎 R) = 0 := by | map_fun_tac
|
import Mathlib.Topology.Algebra.Constructions
import Mathlib.Topology.Bases
import Mathlib.Topology.UniformSpace.Basic
#align_import topology.uniform_space.cauchy from "leanprover-community/mathlib"@"22131150f88a2d125713ffa0f4693e3355b1eb49"
universe u v
open scoped Classical
open Filter TopologicalSpace Set Uni... | Mathlib/Topology/UniformSpace/Cauchy.lean | 70 | 72 | theorem cauchy_map_iff {l : Filter β} {f : β → α} :
Cauchy (l.map f) ↔ NeBot l ∧ Tendsto (fun p : β × β => (f p.1, f p.2)) (l ×ˢ l) (𝓤 α) := by |
rw [Cauchy, map_neBot_iff, prod_map_map_eq, Tendsto]
|
import Mathlib.Data.Finset.Grade
import Mathlib.Data.Finset.Sups
import Mathlib.Logic.Function.Iterate
#align_import combinatorics.set_family.shadow from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
open Finset Nat
variable {α : Type*}
namespace Finset
open FinsetFamily
section Up... | Mathlib/Combinatorics/SetFamily/Shadow.lean | 297 | 322 | theorem mem_upShadow_iff_exists_mem_card_add :
s ∈ ∂⁺ ^[k] 𝒜 ↔ ∃ t ∈ 𝒜, t ⊆ s ∧ t.card + k = s.card := by |
induction' k with k ih generalizing 𝒜 s
· refine ⟨fun hs => ⟨s, hs, Subset.refl _, rfl⟩, ?_⟩
rintro ⟨t, ht, hst, hcard⟩
rwa [← eq_of_subset_of_card_le hst hcard.ge]
simp only [exists_prop, Function.comp_apply, Function.iterate_succ]
refine ih.trans ?_
clear ih
constructor
· rintro ⟨t, ht, hts, h... |
import Mathlib.Data.DFinsupp.Interval
import Mathlib.Data.DFinsupp.Multiset
import Mathlib.Order.Interval.Finset.Nat
#align_import data.multiset.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
open Finset DFinsupp Function
open Pointwise
variable {α : Type*}
namespace Mu... | Mathlib/Data/Multiset/Interval.lean | 62 | 64 | theorem card_Ico :
(Finset.Ico s t).card = ∏ i ∈ s.toFinset ∪ t.toFinset, (t.count i + 1 - s.count i) - 1 := by |
rw [Finset.card_Ico_eq_card_Icc_sub_one, card_Icc]
|
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.NonUnitalHom
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Finsupp.Basic
import Mathlib.LinearAlgebra.Finsupp
#align_import algebra.monoid_algebra.basic from "leanprover-community/mathlib"@... | Mathlib/Algebra/MonoidAlgebra/Basic.lean | 2,100 | 2,104 | theorem mapDomain_algebraMap (A : Type*) {H F : Type*} [CommSemiring k] [Semiring A] [Algebra k A]
[AddMonoid G] [AddMonoid H] [FunLike F G H] [AddMonoidHomClass F G H]
(f : F) (r : k) :
mapDomain f (algebraMap k A[G] r) = algebraMap k A[H] r := by |
simp only [Function.comp_apply, mapDomain_single, AddMonoidAlgebra.coe_algebraMap, map_zero]
|
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#align_import data.complex.exponential from "leanprover-community/mathlib"@"a8b2226cfb0a79f5986492053fc49b1a0c6aeffb"
open CauSeq Finset IsAbsoluteValue
open ... | Mathlib/Data/Complex/Exponential.lean | 1,046 | 1,046 | theorem sinh_neg : sinh (-x) = -sinh x := by | simp [sinh, exp_neg, (neg_div _ _).symm, add_mul]
|
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Sign
import Mathlib.LinearAlgebra.AffineSpace.Combination
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
import Mathlib.LinearAlgebra.Basis.VectorSpace
#align_import linear_algebra.affine_space.independent from "leanprover-c... | Mathlib/LinearAlgebra/AffineSpace/Independent.lean | 572 | 602 | theorem exists_subset_affineIndependent_affineSpan_eq_top {s : Set P}
(h : AffineIndependent k (fun p => p : s → P)) :
∃ t : Set P, s ⊆ t ∧ AffineIndependent k (fun p => p : t → P) ∧ affineSpan k t = ⊤ := by |
rcases s.eq_empty_or_nonempty with (rfl | ⟨p₁, hp₁⟩)
· have p₁ : P := AddTorsor.nonempty.some
let hsv := Basis.ofVectorSpace k V
have hsvi := hsv.linearIndependent
have hsvt := hsv.span_eq
rw [Basis.coe_ofVectorSpace] at hsvi hsvt
have h0 : ∀ v : V, v ∈ Basis.ofVectorSpaceIndex k V → v ≠ 0 := b... |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.Dynamics.FixedPoints.Basic
#align_import data.set.pointwise.iterate from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Pointwise
open Set Function
@[to_additive
"Let `n : ℤ`... | Mathlib/Data/Set/Pointwise/Iterate.lean | 31 | 42 | theorem smul_eq_self_of_preimage_zpow_eq_self {G : Type*} [CommGroup G] {n : ℤ} {s : Set G}
(hs : (fun x => x ^ n) ⁻¹' s = s) {g : G} {j : ℕ} (hg : g ^ n ^ j = 1) : g • s = s := by |
suffices ∀ {g' : G} (_ : g' ^ n ^ j = 1), g' • s ⊆ s by
refine le_antisymm (this hg) ?_
conv_lhs => rw [← smul_inv_smul g s]
replace hg : g⁻¹ ^ n ^ j = 1 := by rw [inv_zpow, hg, inv_one]
simpa only [le_eq_subset, set_smul_subset_set_smul_iff] using this hg
rw [(IsFixedPt.preimage_iterate hs j : (zp... |
import Mathlib.Data.List.Range
import Mathlib.Data.List.Perm
#align_import data.list.sigma from "leanprover-community/mathlib"@"f808feb6c18afddb25e66a71d317643cf7fb5fbb"
universe u v
namespace List
variable {α : Type u} {β : α → Type v} {l l₁ l₂ : List (Sigma β)}
def keys : List (Sigma β) → List α :=
map ... | Mathlib/Data/List/Sigma.lean | 102 | 103 | theorem nodupKeys_cons {s : Sigma β} {l : List (Sigma β)} :
NodupKeys (s :: l) ↔ s.1 ∉ l.keys ∧ NodupKeys l := by | simp [keys, NodupKeys]
|
import Mathlib.Probability.Notation
import Mathlib.Probability.Process.Stopping
#align_import probability.martingale.basic from "leanprover-community/mathlib"@"ba074af83b6cf54c3104e59402b39410ddbd6dca"
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory
namespace MeasureTheor... | Mathlib/Probability/Martingale/Basic.lean | 179 | 184 | theorem setIntegral_le [SigmaFiniteFiltration μ ℱ] {f : ι → Ω → ℝ} (hf : Supermartingale f ℱ μ)
{i j : ι} (hij : i ≤ j) {s : Set Ω} (hs : MeasurableSet[ℱ i] s) :
∫ ω in s, f j ω ∂μ ≤ ∫ ω in s, f i ω ∂μ := by |
rw [← setIntegral_condexp (ℱ.le i) (hf.integrable j) hs]
refine setIntegral_mono_ae integrable_condexp.integrableOn (hf.integrable i).integrableOn ?_
filter_upwards [hf.2.1 i j hij] with _ heq using heq
|
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Set.Finite
#align_import order.filter.basic from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
set_option autoImplicit true
open Function Set Order
open scoped Classical
universe u v w x y
structure Filter (α : Type*) where... | Mathlib/Order/Filter/Basic.lean | 662 | 666 | theorem mem_iInf_of_finite {ι : Type*} [Finite ι] {α : Type*} {f : ι → Filter α} (s) :
(s ∈ ⨅ i, f i) ↔ ∃ t : ι → Set α, (∀ i, t i ∈ f i) ∧ s = ⋂ i, t i := by |
refine ⟨exists_iInter_of_mem_iInf, ?_⟩
rintro ⟨t, ht, rfl⟩
exact iInter_mem.2 fun i => mem_iInf_of_mem i (ht i)
|
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.GroupTheory.Coset
#align_import group_theory.quotient_group from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Function
open scope... | Mathlib/GroupTheory/QuotientGroup.lean | 724 | 729 | theorem comap_comap_center {H₁ : Subgroup G} [H₁.Normal] {H₂ : Subgroup (G ⧸ H₁)} [H₂.Normal] :
((Subgroup.center ((G ⧸ H₁) ⧸ H₂)).comap (mk' H₂)).comap (mk' H₁) =
(Subgroup.center (G ⧸ H₂.comap (mk' H₁))).comap (mk' (H₂.comap (mk' H₁))) := by |
ext x
simp only [mk'_apply, Subgroup.mem_comap, Subgroup.mem_center_iff, forall_mk, ← mk_mul,
eq_iff_div_mem, mk_div]
|
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.Option
import Mathlib.Logic.Equiv.Fin
import Mathlib.Logic.Equiv.Fintype
#align_import group_theory.perm.fin from "leanprover-community/mathlib"@"7e1c1263b6a25eb90bf16e80d8f47a657e403c4c"
open Equiv
def Equiv.Perm.decomposeFin {n : ℕ} : ... | Mathlib/GroupTheory/Perm/Fin.lean | 209 | 210 | theorem coe_cycleRange_of_lt {n : ℕ} {i j : Fin n.succ} (h : j < i) :
(cycleRange i j : ℕ) = j + 1 := by | rw [coe_cycleRange_of_le h.le, if_neg h.ne]
|
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.equiv from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMa... | Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | 378 | 381 | theorem comp_fderiv' {f : G → E} :
fderiv 𝕜 (iso ∘ f) = fun x ↦ (iso : E →L[𝕜] F).comp (fderiv 𝕜 f x) := by |
ext x : 1
exact LinearIsometryEquiv.comp_fderiv iso
|
import Mathlib.Data.Finite.Card
import Mathlib.GroupTheory.Commutator
import Mathlib.GroupTheory.Finiteness
#align_import group_theory.abelianization from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
universe u v w
-- Let G be a group.
variable (G : Type u) [Group G]
open Subgroup (... | Mathlib/GroupTheory/Abelianization.lean | 313 | 316 | theorem card_commutatorSet_closureCommutatorRepresentatives :
Nat.card (commutatorSet (closureCommutatorRepresentatives G)) = Nat.card (commutatorSet G) := by |
rw [← image_commutatorSet_closureCommutatorRepresentatives G]
exact Nat.card_congr (Equiv.Set.image _ _ (subtype_injective _))
|
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Topology.SeparatedMap
#align_import topology.is_locally_homeomorph from "leanprover-community/mathlib"@"e97cf15cd1aec9bd5c193b2ffac5a6dc9118912b"
open Topology
variable {X Y Z : Type*} [TopologicalSpace X] [TopologicalSpace Y] [TopologicalSpace Z] (g : Y →... | Mathlib/Topology/IsLocalHomeomorph.lean | 66 | 77 | theorem mk (h : ∀ x ∈ s, ∃ e : PartialHomeomorph X Y, x ∈ e.source ∧ Set.EqOn f e e.source) :
IsLocalHomeomorphOn f s := by |
intro x hx
obtain ⟨e, hx, he⟩ := h x hx
exact
⟨{ e with
toFun := f
map_source' := fun _x hx ↦ by rw [he hx]; exact e.map_source' hx
left_inv' := fun _x hx ↦ by rw [he hx]; exact e.left_inv' hx
right_inv' := fun _y hy ↦ by rw [he (e.map_target' hy)]; exact e.right_inv' hy
... |
import Mathlib.Analysis.InnerProductSpace.Rayleigh
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Algebra.DirectSum.Decomposition
import Mathlib.LinearAlgebra.Eigenspace.Minpoly
#align_import analysis.inner_product_space.spectrum from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da... | Mathlib/Analysis/InnerProductSpace/Spectrum.lean | 182 | 191 | theorem diagonalization_apply_self_apply (v : E) (μ : Eigenvalues T) :
hT.diagonalization (T v) μ = (μ : 𝕜) • hT.diagonalization v μ := by |
suffices
∀ w : PiLp 2 fun μ : Eigenvalues T => eigenspace T μ,
T (hT.diagonalization.symm w) = hT.diagonalization.symm fun μ => (μ : 𝕜) • w μ by
simpa only [LinearIsometryEquiv.symm_apply_apply, LinearIsometryEquiv.apply_symm_apply] using
congr_arg (fun w => hT.diagonalization w μ) (this (hT.dia... |
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.Data.Finset.PiAntidiagonal
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Tactic.Linarith
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Finset (... | Mathlib/RingTheory/MvPowerSeries/Basic.lean | 251 | 257 | theorem commute_monomial {a : R} {n} :
Commute φ (monomial R n a) ↔ ∀ m, Commute (coeff R m φ) a := by |
refine ext_iff.trans ⟨fun h m => ?_, fun h m => ?_⟩
· have := h (m + n)
rwa [coeff_add_mul_monomial, add_comm, coeff_add_monomial_mul] at this
· rw [coeff_mul_monomial, coeff_monomial_mul]
split_ifs <;> [apply h; rfl]
|
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Ring.Action.Subobjects
import Mathlib.Algebra.Ring.Equiv
import Mathlib.Algebra.Ring.Prod
import Mathlib.Data.Set.Finite
import Mathlib.GroupTheory.Submonoid.Centralizer
import Mathlib.RingTheory.NonUnitalSubsem... | Mathlib/Algebra/Ring/Subsemiring/Basic.lean | 900 | 925 | theorem mem_closure_iff_exists_list {R} [Semiring R] {s : Set R} {x} :
x ∈ closure s ↔ ∃ L : List (List R), (∀ t ∈ L, ∀ y ∈ t, y ∈ s) ∧ (L.map List.prod).sum = x := by |
constructor
· intro hx
-- Porting note: needed explicit `p`
let p : R → Prop := fun x =>
∃ (L : List (List R)),
(∀ (t : List R), t ∈ L → ∀ (y : R), y ∈ t → y ∈ s) ∧ (List.map List.prod L).sum = x
exact AddSubmonoid.closure_induction (p := p) (mem_closure_iff.1 hx)
(fun x hx =>
... |
import Mathlib.Topology.EMetricSpace.Basic
import Mathlib.Topology.Bornology.Constructions
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Topology.Order.DenselyOrdered
open Set Filter TopologicalSpace Bornology
open scoped ENNReal NNReal Uniformity Topology
universe u v w
variable {α : Type u} {β : Typ... | Mathlib/Topology/MetricSpace/PseudoMetric.lean | 2,039 | 2,042 | theorem Fin.dist_insertNth_insertNth {n : ℕ} {α : Fin (n + 1) → Type*}
[∀ i, PseudoMetricSpace (α i)] (i : Fin (n + 1)) (x y : α i) (f g : ∀ j, α (i.succAbove j)) :
dist (i.insertNth x f) (i.insertNth y g) = max (dist x y) (dist f g) := by |
simp only [dist_nndist, Fin.nndist_insertNth_insertNth, NNReal.coe_max]
|
import Mathlib.Data.Set.Basic
#align_import order.circular from "leanprover-community/mathlib"@"213b0cff7bc5ab6696ee07cceec80829ce42efec"
class Btw (α : Type*) where
btw : α → α → α → Prop
#align has_btw Btw
export Btw (btw)
class SBtw (α : Type*) where
sbtw : α → α → α → Prop
#align has_sbtw SBtw
... | Mathlib/Order/Circular.lean | 374 | 376 | theorem compl_cIoo {a b : α} : (cIoo a b)ᶜ = cIcc b a := by |
ext
rw [Set.mem_cIcc, btw_iff_not_sbtw, cIoo, mem_compl_iff, mem_setOf]
|
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} =... | Mathlib/Order/Interval/Set/WithBotTop.lean | 97 | 99 | theorem image_coe_Iio : (some : α → WithTop α) '' Iio a = Iio (a : WithTop α) := by |
rw [← preimage_coe_Iio, image_preimage_eq_inter_range, range_coe,
inter_eq_self_of_subset_left (Iio_subset_Iio le_top)]
|
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Ring.Int
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.RingTheory.Ideal.Quotient
#align_import number_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open I... | Mathlib/NumberTheory/Multiplicity.lean | 181 | 189 | theorem pow_prime_pow_sub_pow_prime_pow (a : ℕ) :
multiplicity (↑p) (x ^ p ^ a - y ^ p ^ a) = multiplicity (↑p) (x - y) + a := by |
induction' a with a h_ind
· rw [Nat.cast_zero, add_zero, pow_zero, pow_one, pow_one]
rw [Nat.cast_add, Nat.cast_one, ← add_assoc, ← h_ind, pow_succ, pow_mul, pow_mul]
apply pow_prime_sub_pow_prime hp hp1
· rw [← geom_sum₂_mul]
exact dvd_mul_of_dvd_right hxy _
· exact fun h => hx (hp.dvd_of_dvd_pow h)
|
import Mathlib.Topology.EMetricSpace.Basic
import Mathlib.Topology.Bornology.Constructions
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Topology.Order.DenselyOrdered
open Set Filter TopologicalSpace Bornology
open scoped ENNReal NNReal Uniformity Topology
universe u v w
variable {α : Type u} {β : Typ... | Mathlib/Topology/MetricSpace/PseudoMetric.lean | 576 | 577 | theorem closedBall_diff_ball : closedBall x ε \ ball x ε = sphere x ε := by |
rw [← ball_union_sphere, Set.union_diff_cancel_left sphere_disjoint_ball.symm.le_bot]
|
import Mathlib.Order.Interval.Multiset
#align_import data.nat.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
-- TODO
-- assert_not_exists Ring
open Finset Nat
variable (a b c : ℕ)
namespace Nat
instance instLocallyFiniteOrder : LocallyFiniteOrder ℕ where
finsetIcc a b... | Mathlib/Order/Interval/Finset/Nat.lean | 214 | 217 | theorem Ico_succ_left_eq_erase_Ico : Ico a.succ b = erase (Ico a b) a := by |
ext x
rw [Ico_succ_left, mem_erase, mem_Ico, mem_Ioo, ← and_assoc, ne_comm, @and_comm (a ≠ x),
lt_iff_le_and_ne]
|
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
#align_import data.finset.locally_finite from "leanprover-community/mathlib"@"442a83d738cb208d3600056c489be16900ba701d"
assert_not_exists MonoidWithZero
assert_not_exists Finset.sum
open Function OrderDual
open FinsetInterval
variable {ι α : T... | Mathlib/Order/Interval/Finset/Basic.lean | 915 | 916 | theorem uIcc_of_le (h : a ≤ b) : [[a, b]] = Icc a b := by |
rw [uIcc, inf_eq_left.2 h, sup_eq_right.2 h]
|
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.Finset.Antidiagonal
import Mathlib.Data.Finset.Card
import Mathlib.Data.Multiset.NatAntidiagonal
#align_import data.finset.nat_antidiagonal from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function
namespace Finset
name... | Mathlib/Data/Finset/NatAntidiagonal.lean | 67 | 75 | theorem antidiagonal_succ (n : ℕ) :
antidiagonal (n + 1) =
cons (0, n + 1)
((antidiagonal n).map
(Embedding.prodMap ⟨Nat.succ, Nat.succ_injective⟩ (Embedding.refl _)))
(by simp) := by |
apply eq_of_veq
rw [cons_val, map_val]
apply Multiset.Nat.antidiagonal_succ
|
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
theorem powerset_insert (s : Set α) (a : α)... | Mathlib/Data/Set/Image.lean | 1,335 | 1,336 | theorem LeftInverse.preimage_preimage {g : β → α} (h : LeftInverse g f) (s : Set α) :
f ⁻¹' (g ⁻¹' s) = s := by | rw [← preimage_comp, h.comp_eq_id, preimage_id]
|
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