Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.57k | proof stringlengths 5 7.36k | hint bool 2
classes |
|---|---|---|---|---|---|---|
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.RingTheory.Coprime.Basic
import Mathlib.Tactic.AdaptationNote
#align_import ring_theory.polynomial.scale_roots from "leanprover-community/mathlib"@"40ac1b258344e0c2b4568dc37bfad937ec35a727"
variable {R... | Mathlib/RingTheory/Polynomial/ScaleRoots.lean | 48 | 50 | theorem zero_scaleRoots (s : R) : scaleRoots 0 s = 0 := by |
ext
simp
| false |
import Mathlib.Order.RelClasses
import Mathlib.Order.Interval.Set.Basic
#align_import order.bounded from "leanprover-community/mathlib"@"aba57d4d3dae35460225919dcd82fe91355162f9"
namespace Set
variable {α : Type*} {r : α → α → Prop} {s t : Set α}
theorem Bounded.mono (hst : s ⊆ t) (hs : Bounded r t) : Bounde... | Mathlib/Order/Bounded.lean | 116 | 118 | theorem unbounded_lt_iff_unbounded_le [Preorder α] [NoMaxOrder α] :
Unbounded (· < ·) s ↔ Unbounded (· ≤ ·) s := by |
simp_rw [← not_bounded_iff, bounded_le_iff_bounded_lt]
| false |
import Mathlib.AlgebraicTopology.DoldKan.Homotopies
import Mathlib.Tactic.Ring
#align_import algebraic_topology.dold_kan.faces from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open CategoryTheory CategoryTheory.Limits CategoryTheory.Category
CategoryTheory.Preadditive CategoryTheor... | Mathlib/AlgebraicTopology/DoldKan/Faces.lean | 69 | 139 | theorem comp_Hσ_eq {Y : C} {n a q : ℕ} {φ : Y ⟶ X _[n + 1]} (v : HigherFacesVanish q φ)
(hnaq : n = a + q) :
φ ≫ (Hσ q).f (n + 1) =
-φ ≫ X.δ ⟨a + 1, Nat.succ_lt_succ (Nat.lt_succ_iff.mpr (Nat.le.intro hnaq.symm))⟩ ≫
X.σ ⟨a, Nat.lt_succ_iff.mpr (Nat.le.intro hnaq.symm)⟩ := by |
have hnaq_shift : ∀ d : ℕ, n + d = a + d + q := by
intro d
rw [add_assoc, add_comm d, ← add_assoc, hnaq]
rw [Hσ, Homotopy.nullHomotopicMap'_f (c_mk (n + 2) (n + 1) rfl) (c_mk (n + 1) n rfl),
hσ'_eq hnaq (c_mk (n + 1) n rfl), hσ'_eq (hnaq_shift 1) (c_mk (n + 2) (n + 1) rfl)]
simp only [AlternatingFace... | false |
import Mathlib.Algebra.Lie.Matrix
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.Tactic.NoncommRing
#align_import algebra.lie.skew_adjoint from "leanprover-community/mathlib"@"075b3f7d19b9da85a0b54b3e33055a74fc388dec"
universe u v w w₁
section SkewAdjointEndomorphisms
open LinearMap (BilinF... | Mathlib/Algebra/Lie/SkewAdjoint.lean | 77 | 80 | theorem skewAdjointLieSubalgebraEquiv_apply
(f : skewAdjointLieSubalgebra (B.compl₁₂ (Qₗ := N) (Qₗ' := N) ↑e ↑e)) :
↑(skewAdjointLieSubalgebraEquiv B e f) = e.lieConj f := by |
simp [skewAdjointLieSubalgebraEquiv]
| false |
import Mathlib.NumberTheory.ModularForms.SlashInvariantForms
import Mathlib.NumberTheory.ModularForms.CongruenceSubgroups
noncomputable section
open ModularForm UpperHalfPlane Matrix
namespace SlashInvariantForm
| Mathlib/NumberTheory/ModularForms/Identities.lean | 22 | 32 | theorem vAdd_width_periodic (N : ℕ) (k n : ℤ) (f : SlashInvariantForm (Gamma N) k) (z : ℍ) :
f (((N * n) : ℝ) +ᵥ z) = f z := by |
norm_cast
rw [← modular_T_zpow_smul z (N * n)]
have Hn := (ModularGroup_T_pow_mem_Gamma N (N * n) (by simp))
simp only [zpow_natCast, Int.natAbs_ofNat] at Hn
convert (SlashInvariantForm.slash_action_eqn' k (Gamma N) f ⟨((ModularGroup.T ^ (N * n))), Hn⟩ z)
unfold SpecialLinearGroup.coeToGL
simp only [Fin.... | false |
import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Pi.Basic
import Mathlib.Data.ULift
#align_import category_theory.discrete_category from "leanprover-community/mathlib"@"369525b73f229ccd76a6ec0e0e0bf2be57599768"
namespace CategoryTheory
-- morphism levels before object levels. See note [Category... | Mathlib/CategoryTheory/DiscreteCategory.lean | 56 | 57 | theorem Discrete.mk_as {α : Type u₁} (X : Discrete α) : Discrete.mk X.as = X := by |
rfl
| false |
import Mathlib.Data.List.Infix
#align_import data.list.rdrop from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
-- Make sure we don't import algebra
assert_not_exists Monoid
variable {α : Type*} (p : α → Bool) (l : List α) (n : ℕ)
namespace List
def rdrop : List α :=
l.take (l.leng... | Mathlib/Data/List/DropRight.lean | 54 | 60 | theorem rdrop_eq_reverse_drop_reverse : l.rdrop n = reverse (l.reverse.drop n) := by |
rw [rdrop]
induction' l using List.reverseRecOn with xs x IH generalizing n
· simp
· cases n
· simp [take_append]
· simp [take_append_eq_append_take, IH]
| false |
import Mathlib.FieldTheory.SeparableDegree
import Mathlib.FieldTheory.IsSepClosed
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField Field
noncomputable section
universe u v w
variable (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E]
variable (K : Type w) [Field K] [... | Mathlib/FieldTheory/SeparableClosure.lean | 186 | 192 | theorem IsSepClosed.separableClosure_eq_bot_iff [IsSepClosed E] :
separableClosure F E = ⊥ ↔ IsSepClosed F := by |
refine ⟨fun h ↦ IsSepClosed.of_exists_root _ fun p _ hirr hsep ↦ ?_,
fun _ ↦ IntermediateField.eq_bot_of_isSepClosed_of_isSeparable _⟩
obtain ⟨x, hx⟩ := IsSepClosed.exists_aeval_eq_zero E p (degree_pos_of_irreducible hirr).ne' hsep
obtain ⟨x, rfl⟩ := h ▸ mem_separableClosure_iff.2 (hsep.of_dvd <| minpoly.dvd... | false |
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.Algebra.Star.Unitary
#align_import linear_algebra.unitary_group from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062a169c9"
universe u ... | 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
| false |
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
namespace Set
variable {α β : Type*} {s t : Set α}
noncomputable def encard (s : Set α) : ℕ∞ := PartENat.withTopEquiv (PartENat.card s)
@[simp] theorem encard_uni... | Mathlib/Data/Set/Card.lean | 119 | 123 | theorem Finite.encard_lt_top (h : s.Finite) : s.encard < ⊤ := by |
refine h.induction_on (by simp) ?_
rintro a t hat _ ht'
rw [encard_insert_of_not_mem hat]
exact lt_tsub_iff_right.1 ht'
| false |
import Mathlib.GroupTheory.Coxeter.Length
import Mathlib.Data.ZMod.Parity
namespace CoxeterSystem
open List Matrix Function
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
local prefi... | 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
| false |
import Mathlib.Order.Filter.Bases
import Mathlib.Order.ConditionallyCompleteLattice.Basic
#align_import order.filter.lift from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
open Set Classical Filter Function
namespace Filter
variable {α β γ : Type*} {ι : Sort*}
section lift
protect... | 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]
| false |
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Extreme
#align_import analysis.convex.independent from "leanprover-community/mathlib"@"fefd8a38be7811574cd2ec2f77d3a393a407f112"
open scoped Classical
open Affine
open Finset Function
variable {𝕜 E ι : Type*}
section OrderedSemiring
va... | Mathlib/Analysis/Convex/Independent.lean | 133 | 141 | theorem convexIndependent_iff_not_mem_convexHull_diff {p : ι → E} :
ConvexIndependent 𝕜 p ↔ ∀ i s, p i ∉ convexHull 𝕜 (p '' (s \ {i})) := by |
refine ⟨fun hc i s h => ?_, fun h s i hi => ?_⟩
· rw [hc.mem_convexHull_iff] at h
exact h.2 (Set.mem_singleton _)
· by_contra H
refine h i s ?_
rw [Set.diff_singleton_eq_self H]
exact hi
| false |
import Mathlib.Algebra.Lie.Submodule
#align_import algebra.lie.ideal_operations from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
universe u v w w₁ w₂
namespace LieSubmodule
variable {R : Type u} {L : Type v} {M : Type w} {M₂ : Type w₁}
variable [CommRing R] [LieRing L] [LieAlgebra ... | Mathlib/Algebra/Lie/IdealOperations.lean | 124 | 124 | theorem lie_le_left : ⁅I, J⁆ ≤ I := by | rw [lie_comm]; exact lie_le_right I J
| false |
import Mathlib.Analysis.PSeries
import Mathlib.Data.Real.Pi.Wallis
import Mathlib.Tactic.AdaptationNote
#align_import analysis.special_functions.stirling from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open scoped Topology Real Nat Asymptotics
open Finset Filter Nat Real
namespace... | 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]
| false |
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
import Mathlib.Analysis.Complex.CauchyIntegral
#align_import analysis.special_functions.gamma.beta from "l... | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | 114 | 132 | theorem betaIntegral_scaled (s t : ℂ) {a : ℝ} (ha : 0 < a) :
∫ x in (0)..a, (x : ℂ) ^ (s - 1) * ((a : ℂ) - x) ^ (t - 1) =
(a : ℂ) ^ (s + t - 1) * betaIntegral s t := by |
have ha' : (a : ℂ) ≠ 0 := ofReal_ne_zero.mpr ha.ne'
rw [betaIntegral]
have A : (a : ℂ) ^ (s + t - 1) = a * ((a : ℂ) ^ (s - 1) * (a : ℂ) ^ (t - 1)) := by
rw [(by abel : s + t - 1 = 1 + (s - 1) + (t - 1)), cpow_add _ _ ha', cpow_add 1 _ ha', cpow_one,
mul_assoc]
rw [A, mul_assoc, ← intervalIntegral.int... | false |
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Hom.Set
#align_import data.set.intervals.order_iso from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
namespace OrderIso
section Preorder
variable {α β : Type*} [Preorder α] [Preorder β]
@[simp]
theorem preimage_I... | 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]
| false |
import Mathlib.CategoryTheory.Opposites
#align_import category_theory.eq_to_hom from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe v₁ v₂ v₃ u₁ u₂ u₃
-- morphism levels before object levels. See note [CategoryTheory universes].
namespace CategoryTheory
open Opposite
variable ... | Mathlib/CategoryTheory/EqToHom.lean | 169 | 170 | theorem eqToIso_trans {X Y Z : C} (p : X = Y) (q : Y = Z) :
eqToIso p ≪≫ eqToIso q = eqToIso (p.trans q) := by | ext; simp
| false |
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
#align_import algebra.order.monoid.min_max from "leanprover-community/mathlib"@"de87d5053a9fe5cbde723172c0fb7e27e7436473"
open Function
variable {α β : Type*}
section CovariantClassMulLe
variable [LinearOrder α]
section Mul
variable [Mul α]
@[to_additive... | Mathlib/Algebra/Order/Monoid/Unbundled/MinMax.lean | 126 | 133 | theorem mul_lt_mul_iff_of_le_of_le [CovariantClass α α (· * ·) (· ≤ ·)]
[CovariantClass α α (Function.swap (· * ·)) (· ≤ ·)] [CovariantClass α α (· * ·) (· < ·)]
[CovariantClass α α (Function.swap (· * ·)) (· < ·)] {a₁ a₂ b₁ b₂ : α} (ha : a₁ ≤ a₂)
(hb : b₁ ≤ b₂) : a₁ * b₁ < a₂ * b₂ ↔ a₁ < a₂ ∨ b₁ < b₂ := by |
refine ⟨lt_or_lt_of_mul_lt_mul, fun h => ?_⟩
cases' h with ha' hb'
· exact mul_lt_mul_of_lt_of_le ha' hb
· exact mul_lt_mul_of_le_of_lt ha hb'
| false |
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import topology.metric_space.hausdorff_distance from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
noncomputable section
open NNReal ENNReal Topology Set Filter Bornology
universe u v w
variable {ι : Sort*} {α : Type u} {β :... | Mathlib/Topology/MetricSpace/Thickening.lean | 242 | 244 | theorem cthickening_of_nonpos {δ : ℝ} (hδ : δ ≤ 0) (E : Set α) : cthickening δ E = closure E := by |
ext x
simp [mem_closure_iff_infEdist_zero, cthickening, ENNReal.ofReal_eq_zero.2 hδ]
| false |
import Mathlib.Data.List.Sublists
import Mathlib.Data.Multiset.Bind
#align_import data.multiset.powerset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
open List
variable {α : Type*}
-- Porting note (#11215): TODO: Write a more efficient version
def powerset... | Mathlib/Data/Multiset/Powerset.lean | 125 | 129 | theorem revzip_powersetAux {l : List α} ⦃x⦄ (h : x ∈ revzip (powersetAux l)) : x.1 + x.2 = ↑l := by |
rw [revzip, powersetAux_eq_map_coe, ← map_reverse, zip_map, ← revzip, List.mem_map] at h
simp only [Prod.map_apply, Prod.exists] at h
rcases h with ⟨l₁, l₂, h, rfl, rfl⟩
exact Quot.sound (revzip_sublists _ _ _ h)
| false |
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.NormedSpace.Connected
import Mathlib.LinearAlgebra.AffineSpace.ContinuousAffineEquiv
open Set
variable {F : Type*} [AddCommGroup F] [Module ℝ F] [TopologicalSpace F]
def AmpleSet (s : Set F) : Prop :=
∀ x ∈ s, convexHull ℝ (connectedComponentIn s ... | 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]
| false |
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp Ad... | 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
| false |
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.RingTheory.FinitePresentation
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Localization.Away.AdjoinRoot
#align_import ring_theory.etale from ... | Mathlib/RingTheory/Smooth/Basic.lean | 121 | 131 | theorem liftOfSurjective_apply [FormallySmooth R A] (f : A →ₐ[R] C) (g : B →ₐ[R] C)
(hg : Function.Surjective g) (hg' : IsNilpotent <| RingHom.ker (g : B →+* C)) (x : A) :
g (FormallySmooth.liftOfSurjective f g hg hg' x) = f x := by |
apply (Ideal.quotientKerAlgEquivOfSurjective hg).symm.injective
change _ = ((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom.comp f) x
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [← FormallySmooth.mk_lift _ hg'
((Ideal.quotientKerAlgEquivOfSurjective hg).symm.toAlgHom... | false |
import Mathlib.Data.List.Infix
#align_import data.list.rdrop from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
-- Make sure we don't import algebra
assert_not_exists Monoid
variable {α : Type*} (p : α → Bool) (l : List α) (n : ℕ)
namespace List
def rdrop : List α :=
l.take (l.leng... | 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]
| false |
import Mathlib.MeasureTheory.Group.Action
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Group.Pointwise
#align_import measure_theory.group.fundamental_domain from "leanprover-community/mathlib"@"3b52265189f3fb43aa631edffce5d060fafaf82f"
open scoped ENNReal Pointwise Topology NNRea... | Mathlib/MeasureTheory/Group/FundamentalDomain.lean | 121 | 137 | theorem mk_of_measure_univ_le [IsFiniteMeasure μ] [Countable G] (h_meas : NullMeasurableSet s μ)
(h_ae_disjoint : ∀ g ≠ (1 : G), AEDisjoint μ (g • s) s)
(h_qmp : ∀ g : G, QuasiMeasurePreserving (g • · : α → α) μ μ)
(h_measure_univ_le : μ (univ : Set α) ≤ ∑' g : G, μ (g • s)) : IsFundamentalDomain G s μ :=
... |
replace h_meas : ∀ g : G, NullMeasurableSet (g • s) μ := fun g => by
rw [← inv_inv g, ← preimage_smul]; exact h_meas.preimage (h_qmp g⁻¹)
have h_meas' : NullMeasurableSet {a | ∃ g : G, g • a ∈ s} μ := by
rw [← iUnion_smul_eq_setOf_exists]; exact .iUnion h_meas
rw [ae_iff_measure_eq h_... | false |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Monic
#align_import data.polynomial.integral_normalization from "leanprover-community/mathlib"@"6f401acf4faec3ab9ab13a42789c4f68064a61cd"
open Polynomial
namespace Polynomial
universe u... | 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]
| false |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.RingTheory.Localization.Basic
#align_import data.polynomial.laurent from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
open Polynomial Func... | Mathlib/Algebra/Polynomial/Laurent.lean | 196 | 197 | theorem T_pow (m : ℤ) (n : ℕ) : (T m ^ n : R[T;T⁻¹]) = T (n * m) := by |
rw [T, T, single_pow n, one_pow, nsmul_eq_mul]
| false |
import Mathlib.MeasureTheory.Measure.Content
import Mathlib.MeasureTheory.Group.Prod
import Mathlib.Topology.Algebra.Group.Compact
#align_import measure_theory.measure.haar.basic from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open Set Inv Function Topological... | Mathlib/MeasureTheory/Measure/Haar/Basic.lean | 142 | 144 | theorem mem_prehaar_empty {K₀ : Set G} {f : Compacts G → ℝ} :
f ∈ haarProduct K₀ ↔ ∀ K : Compacts G, f K ∈ Icc (0 : ℝ) (index (K : Set G) K₀) := by |
simp only [haarProduct, Set.pi, forall_prop_of_true, mem_univ, mem_setOf_eq]
| false |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
#align_import analysis.special_functions.trigonometric.inverse_deriv from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classic... | Mathlib/Analysis/SpecialFunctions/Trigonometric/InverseDeriv.lean | 108 | 114 | theorem deriv_arcsin : deriv arcsin = fun x => 1 / √(1 - x ^ 2) := by |
funext x
by_cases h : x ≠ -1 ∧ x ≠ 1
· exact (hasDerivAt_arcsin h.1 h.2).deriv
· rw [deriv_zero_of_not_differentiableAt (mt differentiableAt_arcsin.1 h)]
simp only [not_and_or, Ne, Classical.not_not] at h
rcases h with (rfl | rfl) <;> simp
| false |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | 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]
| false |
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factors
import Mathlib.Order.Interval.Finset.Nat
#align_import number_theory.divisors from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open scoped Classical
open Finset
namespace Nat
variable (n : ℕ)
d... | 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)
| false |
import Mathlib.Init.Core
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
open Polynomial Algebra FiniteD... | Mathlib/NumberTheory/Cyclotomic/Basic.lean | 387 | 399 | theorem adjoin_roots_cyclotomic_eq_adjoin_root_cyclotomic {n : ℕ+} [IsDomain B] {ζ : B}
(hζ : IsPrimitiveRoot ζ n) : adjoin A ((cyclotomic n A).rootSet B) = adjoin A {ζ} := by |
refine le_antisymm (adjoin_le fun x hx => ?_) (adjoin_mono fun x hx => ?_)
· suffices hx : x ^ n.1 = 1 by
obtain ⟨i, _, rfl⟩ := hζ.eq_pow_of_pow_eq_one hx n.pos
exact SetLike.mem_coe.2 (Subalgebra.pow_mem _ (subset_adjoin <| mem_singleton ζ) _)
refine (isRoot_of_unity_iff n.pos B).2 ?_
refine ⟨... | false |
import Mathlib.MeasureTheory.Integral.IntegralEqImproper
#align_import measure_theory.integral.peak_function from "leanprover-community/mathlib"@"13b0d72fd8533ba459ac66e9a885e35ffabb32b2"
open Set Filter MeasureTheory MeasureTheory.Measure TopologicalSpace Metric
open scoped Topology ENNReal
open Set
variable... | Mathlib/MeasureTheory/Integral/PeakFunction.lean | 99 | 182 | theorem tendsto_setIntegral_peak_smul_of_integrableOn_of_tendsto_aux
(hs : MeasurableSet s) (ht : MeasurableSet t) (hts : t ⊆ s) (h'ts : t ∈ 𝓝[s] x₀)
(hnφ : ∀ᶠ i in l, ∀ x ∈ s, 0 ≤ φ i x)
(hlφ : ∀ u : Set α, IsOpen u → x₀ ∈ u → TendstoUniformlyOn φ 0 l (s \ u))
(hiφ : Tendsto (fun i ↦ ∫ x in t, φ i x ∂... |
refine Metric.tendsto_nhds.2 fun ε εpos => ?_
obtain ⟨δ, hδ, δpos, δone⟩ : ∃ δ, (δ * ∫ x in s, ‖g x‖ ∂μ) + 2 * δ < ε ∧ 0 < δ ∧ δ < 1:= by
have A :
Tendsto (fun δ => (δ * ∫ x in s, ‖g x‖ ∂μ) + 2 * δ) (𝓝[>] 0)
(𝓝 ((0 * ∫ x in s, ‖g x‖ ∂μ) + 2 * 0)) := by
apply Tendsto.mono_left _ nhdsWithin... | false |
import Mathlib.CategoryTheory.Adjunction.Opposites
import Mathlib.CategoryTheory.Comma.Presheaf
import Mathlib.CategoryTheory.Elements
import Mathlib.CategoryTheory.Limits.ConeCategory
import Mathlib.CategoryTheory.Limits.Final
import Mathlib.CategoryTheory.Limits.KanExtension
import Mathlib.CategoryTheory.Limits.Over... | Mathlib/CategoryTheory/Limits/Presheaf.lean | 486 | 503 | theorem final_toCostructuredArrow_comp_pre {c : Cocone (F ⋙ yoneda)} (hc : IsColimit c) :
Functor.Final (c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) := by |
apply Functor.cofinal_of_isTerminal_colimit_comp_yoneda
suffices IsTerminal (colimit ((c.toCostructuredArrow ⋙ CostructuredArrow.pre F yoneda c.pt) ⋙
CostructuredArrow.toOver yoneda c.pt)) by
apply IsTerminal.isTerminalOfObj (overEquivPresheafCostructuredArrow c.pt).inverse
apply IsTerminal.ofIso th... | false |
import Mathlib.Topology.Defs.Sequences
import Mathlib.Topology.UniformSpace.Cauchy
#align_import topology.sequences from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Function Filter TopologicalSpace Bornology
open scoped Topology Uniformity
variable {X Y : Type*}
section ... | Mathlib/Topology/Sequences.lean | 125 | 134 | theorem tendsto_nhds_iff_seq_tendsto [FrechetUrysohnSpace X] {f : X → Y} {a : X} {b : Y} :
Tendsto f (𝓝 a) (𝓝 b) ↔ ∀ u : ℕ → X, Tendsto u atTop (𝓝 a) → Tendsto (f ∘ u) atTop (𝓝 b) := by |
refine
⟨fun hf u hu => hf.comp hu, fun h =>
((nhds_basis_closeds _).tendsto_iff (nhds_basis_closeds _)).2 ?_⟩
rintro s ⟨hbs, hsc⟩
refine ⟨closure (f ⁻¹' s), ⟨mt ?_ hbs, isClosed_closure⟩, fun x => mt fun hx => subset_closure hx⟩
rw [← seqClosure_eq_closure]
rintro ⟨u, hus, hu⟩
exact hsc.mem_of_te... | false |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology.Instances.Sign
#align_import analysis.special_functions.trigonometric.angle from "leanprover-community/mathlib"@"213b0cff7bc5ab6696ee07cceec80829... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 141 | 142 | theorem two_nsmul_coe_div_two (θ : ℝ) : (2 : ℕ) • (↑(θ / 2) : Angle) = θ := by |
rw [← coe_nsmul, two_nsmul, add_halves]
| false |
import Mathlib.Probability.Kernel.Basic
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.DominatedConvergence
#align_import probability.kernel.measurable_integral from "leanprover-community/mathlib"@"28b2a92f2996d28e580450863c130955de0ed398"
open MeasureTheory Probabilit... | 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... | false |
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Topology.Instances.ENNReal
#align_import analysis.calculus.series from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Metric TopologicalSpace Function Filter
open scoped Topology NNReal
variable {α β F : Type*} [N... | 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
| false |
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.Finset.Sym
import Mathlib.Data.Finsupp.Multiset
#align_import data.nat.choose.multinomial from "leanprover-community/mathlib"@"2738d2ca56cbc... | 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]
| false |
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ring.Commute
import Mathlib.Algebra.Ring.Invertible
import Mathlib.Order.Synonym
#align_import algebra.field.basic from "leanprover-community/mathlib"@"05101c3df9d9cfe9430edc205860c79b6d660102"
open Function ... | Mathlib/Algebra/Field/Basic.lean | 132 | 132 | theorem div_neg (a : K) : a / -b = -(a / b) := by | rw [← div_neg_eq_neg_div]
| false |
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.pointwise from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set
open Pointwise Topology
variable {𝕜 E : Type*}
variable [NormedField 𝕜]
sectio... | 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]
| false |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Filter Metric Set
open scoped ComplexConjugate Real To... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 93 | 114 | theorem arg_mul_cos_add_sin_mul_I {r : ℝ} (hr : 0 < r) {θ : ℝ} (hθ : θ ∈ Set.Ioc (-π) π) :
arg (r * (cos θ + sin θ * I)) = θ := by |
simp only [arg, map_mul, abs_cos_add_sin_mul_I, abs_of_nonneg hr.le, mul_one]
simp only [re_ofReal_mul, im_ofReal_mul, neg_im, ← ofReal_cos, ← ofReal_sin, ←
mk_eq_add_mul_I, neg_div, mul_div_cancel_left₀ _ hr.ne', mul_nonneg_iff_right_nonneg_of_pos hr]
by_cases h₁ : θ ∈ Set.Icc (-(π / 2)) (π / 2)
· rw [if_... | false |
import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.IdealOperations
import Mathlib.Order.Hom.Basic
#align_import algebra.lie.solvable from "leanprover-community/mathlib"@"a50170a88a47570ed186b809ca754110590f9476"
universe u v w w₁ w₂
variable (R : Type u) (L : Type v) (M : Type w) {L' : Type w₁}
variab... | 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]
| false |
import Mathlib.CategoryTheory.Products.Basic
#align_import category_theory.products.bifunctor from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
open CategoryTheory
namespace CategoryTheory.Bifunctor
universe v₁ v₂ v₃ u₁ u₂ u₃
variable {C : Type u₁} {D : Type u₂} {E : Type u₃}
varia... | 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]
| false |
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 | 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)]
| false |
import Mathlib.Analysis.Calculus.FDeriv.Basic
#align_import analysis.calculus.fderiv.restrict_scalars from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open Topology NNReal Filter Asymptotics ENNReal
noncom... | 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
| false |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 274 | 281 | theorem deriv_mul_const_field (v : 𝕜') : deriv (fun y => u y * v) x = deriv u x * v := by |
by_cases hu : DifferentiableAt 𝕜 u x
· exact deriv_mul_const hu v
· rw [deriv_zero_of_not_differentiableAt hu, zero_mul]
rcases eq_or_ne v 0 with (rfl | hd)
· simp only [mul_zero, deriv_const]
· refine deriv_zero_of_not_differentiableAt (mt (fun H => ?_) hu)
simpa only [mul_inv_cancel_right₀ h... | false |
import Mathlib.Deprecated.Group
#align_import deprecated.ring from "leanprover-community/mathlib"@"5a3e819569b0f12cbec59d740a2613018e7b8eec"
universe u v w
variable {α : Type u}
structure IsSemiringHom {α : Type u} {β : Type v} [Semiring α] [Semiring β] (f : α → β) : Prop where
map_zero : f 0 = 0
map... | Mathlib/Deprecated/Ring.lean | 54 | 54 | theorem id : IsSemiringHom (@id α) := by | constructor <;> intros <;> rfl
| false |
import Mathlib.Order.Filter.Germ
import Mathlib.Topology.NhdsSet
import Mathlib.Topology.LocallyConstant.Basic
import Mathlib.Analysis.NormedSpace.Basic
variable {F G : Type*} [NormedAddCommGroup F] [NormedSpace ℝ F]
[NormedAddCommGroup G] [NormedSpace ℝ G]
open scoped Topology
open Filter Set
variable {X Y Z ... | 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
| false |
import Mathlib.Analysis.Analytic.Composition
#align_import analysis.analytic.inverse from "leanprover-community/mathlib"@"284fdd2962e67d2932fa3a79ce19fcf92d38e228"
open scoped Classical Topology
open Finset Filter
namespace FormalMultilinearSeries
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} ... | 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
... | false |
import Mathlib.Init.Classical
import Mathlib.Order.FixedPoints
import Mathlib.Order.Zorn
#align_import set_theory.cardinal.schroeder_bernstein from "leanprover-community/mathlib"@"1e05171a5e8cf18d98d9cf7b207540acb044acae"
open Set Function
open scoped Classical
universe u v
namespace Function
namespace Embedd... | Mathlib/SetTheory/Cardinal/SchroederBernstein.lean | 100 | 131 | theorem min_injective [I : Nonempty ι] : ∃ i, Nonempty (∀ j, β i ↪ β j) :=
let ⟨s, hs, ms⟩ :=
show ∃ s ∈ sets β, ∀ a ∈ sets β, s ⊆ a → a = s from
zorn_subset (sets β) fun c hc hcc =>
⟨⋃₀c, fun x ⟨p, hpc, hxp⟩ y ⟨q, hqc, hyq⟩ i hi =>
(hcc.total hpc hqc).elim (fun h => hc hqc x (h hxp) y hyq... |
simpa only [ne_eq, not_exists, not_forall, not_and] using h
let ⟨f, hf⟩ := Classical.axiom_of_choice h
have : f ∈ s :=
have : insert f s ∈ sets β := fun x hx y hy => by
cases' hx with hx hx <;> cases' hy with hy hy; · simp [hx, hy]
· subst x
exa... | false |
import Mathlib.Data.ZMod.Quotient
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Tactic.ByContra
import Mathlib.Tactic.Peel
#align_import group_... | Mathlib/GroupTheory/Exponent.lean | 145 | 148 | theorem exponent_eq_zero_iff_forall : exponent G = 0 ↔ ∀ n > 0, ∃ g : G, g ^ n ≠ 1 := by |
rw [exponent_eq_zero_iff, ExponentExists]
push_neg
rfl
| false |
import Mathlib.Topology.Order.Basic
import Mathlib.Data.Set.Pointwise.Basic
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section LinearOrder
variable [TopologicalSpace α] [LinearOrder α]
section OrderTopology
variable [OrderTopology α]
open List ... | 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... | false |
import Mathlib.Algebra.Quaternion
import Mathlib.Tactic.Ring
#align_import algebra.quaternion_basis from "leanprover-community/mathlib"@"3aa5b8a9ed7a7cabd36e6e1d022c9858ab8a8c2d"
open Quaternion
namespace QuaternionAlgebra
structure Basis {R : Type*} (A : Type*) [CommRing R] [Ring A] [Algebra R A] (c₁ c₂ : R) ... | Mathlib/Algebra/QuaternionBasis.lean | 117 | 117 | theorem lift_one : q.lift (1 : ℍ[R,c₁,c₂]) = 1 := by | simp [lift]
| false |
import Mathlib.Algebra.Group.Even
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.Order.Sub.Defs
#align_import algebra.order.sub.canonical from "leanprover-community/mathlib"@"62a5626868683c104774de8d85b9855234ac807c"
variable {α : Type*}
section ExistsAddOfLE
variable [AddCommSemigrou... | 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
| false |
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Range
#align_import data.list.indexes from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
assert_not_exists MonoidWithZero
universe u v
open Function
namespace List
variable {α : Type u} {β : Type v}
section MapIdx
-- Porting n... | Mathlib/Data/List/Indexes.lean | 132 | 138 | theorem mapIdxGo_length : ∀ (f : ℕ → α → β) (l : List α) (arr : Array β),
length (mapIdx.go f l arr) = length l + arr.size := by |
intro f l
induction' l with head tail ih
· intro; simp only [mapIdx.go, Array.toList_eq, length_nil, Nat.zero_add]
· intro; simp only [mapIdx.go]; rw [ih]; simp only [Array.size_push, length_cons];
simp only [Nat.add_succ, add_zero, Nat.add_comm]
| false |
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Relator
import Mathlib.Init.Data.Quot
import Mathlib.Tactic.Cases
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
#align_import logic.relation from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9a... | Mathlib/Logic/Relation.lean | 360 | 369 | theorem total_of_right_unique (U : Relator.RightUnique r) (ab : ReflTransGen r a b)
(ac : ReflTransGen r a c) : ReflTransGen r b c ∨ ReflTransGen r c b := by |
induction' ab with b d _ bd IH
· exact Or.inl ac
· rcases IH with (IH | IH)
· rcases cases_head IH with (rfl | ⟨e, be, ec⟩)
· exact Or.inr (single bd)
· cases U bd be
exact Or.inl ec
· exact Or.inr (IH.tail bd)
| false |
import Mathlib.Algebra.Group.Center
import Mathlib.Data.Int.Cast.Lemmas
#align_import group_theory.subsemigroup.center from "leanprover-community/mathlib"@"1ac8d4304efba9d03fa720d06516fac845aa5353"
variable {M : Type*}
namespace Set
variable (M)
@[simp]
| 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_... | false |
import Mathlib.Data.Int.Order.Units
import Mathlib.Data.ZMod.IntUnitsPower
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
import Mathlib.Algebra.DirectSum.Algebra
suppress_compilation
open scoped TensorProduct DirectSum
variable {R ι A B : Type*}
namespace Tens... | 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]
| false |
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
#align_import analysis.special_functions.pow.real from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real ComplexConjugate
open Finset Set
namespace Real
variable {x y z... | 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]
| false |
import Mathlib.Tactic.ApplyFun
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.Separation
#align_import topology.uniform_space.separation from "leanprover-community/mathlib"@"0c1f285a9f6e608ae2bdffa3f993eafb01eba829"
open Filter Set Function Topology Uniformity UniformSpace
open scoped Classical... | Mathlib/Topology/UniformSpace/Separation.lean | 150 | 152 | theorem t0Space_iff_uniformity :
T0Space α ↔ ∀ x y, (∀ r ∈ 𝓤 α, (x, y) ∈ r) → x = y := by |
simp only [t0Space_iff_inseparable, inseparable_iff_ker_uniformity, mem_ker, id]
| false |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.RingTheory.PowerBasis
#align_import ring_theory.is_adjoin_root from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
open scoped Polynomial
open Polynomial
noncomputable sec... | Mathlib/RingTheory/IsAdjoinRoot.lean | 203 | 207 | theorem eval₂_repr_eq_eval₂_of_map_eq (h : IsAdjoinRoot S f) (z : S) (w : R[X])
(hzw : h.map w = z) : (h.repr z).eval₂ i x = w.eval₂ i x := by |
rw [eq_comm, ← sub_eq_zero, ← h.map_repr z, ← map_sub, h.map_eq_zero_iff] at hzw
obtain ⟨y, hy⟩ := hzw
rw [← sub_eq_zero, ← eval₂_sub, hy, eval₂_mul, hx, zero_mul]
| false |
import Mathlib.Algebra.Homology.Linear
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Tactic.Abel
#align_import algebra.homology.homotopy from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a148bf9ff"
universe v u
open scoped Classical
noncomputable section
open ... | Mathlib/Algebra/Homology/Homotopy.lean | 124 | 130 | theorem prevD_nat (C D : CochainComplex V ℕ) (i : ℕ) (f : ∀ i j, C.X i ⟶ D.X j) :
prevD i f = f i (i - 1) ≫ D.d (i - 1) i := by |
dsimp [prevD]
cases i
· simp only [shape, CochainComplex.prev_nat_zero, ComplexShape.up_Rel, Nat.one_ne_zero,
not_false_iff, comp_zero]
· congr <;> simp
| false |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Mathlib.Analysis.NormedSpace.FunctionSeries
#align_import analysis.calculus.series from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982... | 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... | false |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.GroupTheory.GroupAction.Pi
open Function Set
structure AddConstMap (G H : Type*) [Add G] [Add H] (a : G) (b : H) where
protected... | Mathlib/Algebra/AddConstMap/Basic.lean | 142 | 144 | theorem map_nat_add' [AddCommMonoidWithOne G] [AddMonoid H] [AddConstMapClass F G H 1 b]
(f : F) (n : ℕ) (x : G) : f (↑n + x) = f x + n • b := by |
simpa using map_nsmul_add f n x
| false |
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Sum
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.fintype.units from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
variable {α : Type*}
instance UnitsInt.fintype : Fintype ℤˣ :=
⟨{1, -1}, fun x ↦ by cases Int... | 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]
| false |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov
import Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp
import Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality... | Mathlib/MeasureTheory/Function/LpSpace.lean | 163 | 167 | theorem ext {f g : Lp E p μ} (h : f =ᵐ[μ] g) : f = g := by |
cases f
cases g
simp only [Subtype.mk_eq_mk]
exact AEEqFun.ext h
| false |
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.Submodule.Basic
#align_import algebra.direct_sum.decomposition from "leanprover-community/mathlib"@"4e861f25ba5ceef42ba0712d8ffeb32f38ad6441"
variable {ι R M σ : Type*}
open DirectSum
namespace DirectSum
section AddCommMonoid
variable [Deci... | Mathlib/Algebra/DirectSum/Decomposition.lean | 140 | 142 | theorem decompose_of_mem_ne {x : M} {i j : ι} (hx : x ∈ ℳ i) (hij : i ≠ j) :
(decompose ℳ x j : M) = 0 := by |
rw [decompose_of_mem _ hx, DirectSum.of_eq_of_ne _ _ _ _ hij, ZeroMemClass.coe_zero]
| false |
import Mathlib.Geometry.Manifold.ContMDiff.Atlas
import Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear
import Mathlib.Topology.VectorBundle.Constructions
#align_import geometry.manifold.vector_bundle.basic from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
assert_not_exists mfde... | 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)]
| false |
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Geometry.Euclidean.PerpBisector
import Mathlib.Algebra.QuadraticDiscriminant
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
noncomputable section
open scoped Classical
open ... | 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
| false |
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.ApplyFun
import Mathlib.Control.Fix
import Mathlib.Order.OmegaCompletePartialOrder
#align_import control.lawful_fix from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
universe u v
open scoped Classical
variable {α : Type*} {β : α →... | 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'
| false |
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.add from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open Topology NNReal F... | Mathlib/Analysis/Calculus/FDeriv/Add.lean | 488 | 489 | theorem fderiv_neg : fderiv 𝕜 (fun y => -f y) x = -fderiv 𝕜 f x := by |
simp only [← fderivWithin_univ, fderivWithin_neg uniqueDiffWithinAt_univ]
| false |
import Mathlib.Probability.IdentDistrib
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.Analysis.SpecificLimits.FloorPow
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
#align_import probability.strong_law from "leanprover-community/mathlib"@"f2ce60867... | 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]
| false |
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.LinearAlgebra.Dual
import Mathlib.LinearAlgebra.Contraction
import Mathlib.RingTheory.TensorProduct.Basic
#align_import representation_... | 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
| false |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' u₁' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}... | Mathlib/LinearAlgebra/Dimension/Constructions.lean | 544 | 547 | theorem subalgebra_top_finrank_eq_submodule_top_finrank :
finrank F (⊤ : Subalgebra F E) = finrank F (⊤ : Submodule F E) := by |
rw [← Algebra.top_toSubmodule]
rfl
| false |
import Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity
#align_import number_theory.legendre_symbol.jacobi_symbol from "leanprover-community/mathlib"@"74a27133cf29446a0983779e37c8f829a85368f3"
section Jacobi
open Nat ZMod
-- Since we need the fact that the factors are prime, we use `List.pmap`.
def ... | 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]
| false |
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 | 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
| false |
import Mathlib.Algebra.CharP.LocalRing
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.Tactic.FieldSimp
#align_import algebra.char_p.mixed_char_zero from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
variable (R : Type*) [CommRing R]
class MixedCharZero (p : ℕ) : Prop where
... | Mathlib/Algebra/CharP/MixedCharZero.lean | 264 | 270 | theorem to_not_mixedCharZero (h : ∀ I : Ideal R, I ≠ ⊤ → CharZero (R ⧸ I)) :
∀ p > 0, ¬MixedCharZero R p := by |
intro p p_pos
by_contra hp_mixedChar
rcases hp_mixedChar.charP_quotient with ⟨I, hI_ne_top, hI_p⟩
replace hI_zero : CharP (R ⧸ I) 0 := @CharP.ofCharZero _ _ (h I hI_ne_top)
exact absurd (CharP.eq (R ⧸ I) hI_p hI_zero) (ne_of_gt p_pos)
| false |
import Mathlib.CategoryTheory.Monad.Types
import Mathlib.CategoryTheory.Monad.Limits
import Mathlib.CategoryTheory.Equivalence
import Mathlib.Topology.Category.CompHaus.Basic
import Mathlib.Topology.Category.Profinite.Basic
import Mathlib.Data.Set.Constructions
#align_import topology.category.Compactum from "leanprov... | Mathlib/Topology/Category/Compactum.lean | 173 | 185 | theorem isClosed_iff {X : Compactum} (S : Set X) :
IsClosed S ↔ ∀ F : Ultrafilter X, S ∈ F → X.str F ∈ S := by |
rw [← isOpen_compl_iff]
constructor
· intro cond F h
by_contra c
specialize cond F c
rw [compl_mem_iff_not_mem] at cond
contradiction
· intro h1 F h2
specialize h1 F
cases' F.mem_or_compl_mem S with h h
exacts [absurd (h1 h) h2, h]
| false |
import Mathlib.Order.Filter.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.SetTheory.Cardinal.Cofinality
open Set Filter Cardinal
universe u
variable {ι : Type u} {α β : Type u} {c : Cardinal.{u}}
class CardinalInterFilter (l : Filter α) (c : Cardinal.{... | 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]
| false |
import Mathlib.LinearAlgebra.Dimension.DivisionRing
import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
noncomputable section
universe u v v' v''
variable {K : Type u} {V V₁ : Type v} {V' V'₁ : Type v'} {V'' : Type v''}
open Cardinal Basis Submodule Function Set
namespace LinearMap
section Ring
... | 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
| false |
import Mathlib.RingTheory.FinitePresentation
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Localization.Away.AdjoinRoot
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.TensorProduct.Basic
-- Porting note: added to make the syntax work below.
open scoped TensorProd... | Mathlib/RingTheory/Unramified/Basic.lean | 155 | 163 | theorem of_comp [FormallyUnramified R B] : FormallyUnramified A B := by |
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine AlgHom.restrictScalars_injective R ?_
refine FormallyUnramified.ext I ⟨2, e⟩ ?_
intro x
exact AlgHom.congr_fun e' x
| false |
import Mathlib.MeasureTheory.Measure.VectorMeasure
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
#align_import measure_theory.measure.with_density_vector_measure from "leanprover-community/mathlib"@"d1bd9c5df2867c1cb463bc6364446d57bdd9f7f1"
noncomputable section
open scoped Classical MeasureTheory NNReal ... | 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
| false |
import Mathlib.CategoryTheory.Idempotents.Basic
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Equivalence
#align_import category_theory.idempotents.karoubi from "leanprover-community/mathlib"@"200eda15d8ff5669854ff6bcc10aaf37cb70498f"
noncomputable section
open CategoryT... | 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]
| false |
import Mathlib.Data.Fin.VecNotation
import Mathlib.SetTheory.Cardinal.Basic
#align_import model_theory.basic from "leanprover-community/mathlib"@"369525b73f229ccd76a6ec0e0e0bf2be57599768"
set_option autoImplicit true
universe u v u' v' w w'
open Cardinal
open Cardinal
namespace FirstOrder
-- intended to b... | Mathlib/ModelTheory/Basic.lean | 174 | 178 | theorem card_eq_card_functions_add_card_relations :
L.card =
(Cardinal.sum fun l => Cardinal.lift.{v} #(L.Functions l)) +
Cardinal.sum fun l => Cardinal.lift.{u} #(L.Relations l) := by |
simp [card, Symbols]
| false |
import Mathlib.Analysis.Convex.Hull
#align_import analysis.convex.join from "leanprover-community/mathlib"@"951bf1d9e98a2042979ced62c0620bcfb3587cf8"
open Set
variable {ι : Sort*} {𝕜 E : Type*}
section OrderedSemiring
variable (𝕜) [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {s t s₁ s₂ t₁ t₂ u : Set ... | 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]
| false |
import Mathlib.Algebra.Order.Interval.Set.Instances
import Mathlib.Order.Interval.Set.ProjIcc
import Mathlib.Topology.Instances.Real
#align_import topology.unit_interval from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open Topology Filter
... | Mathlib/Topology/UnitInterval.lean | 154 | 155 | theorem half_le_symm_iff (t : I) : 1 / 2 ≤ (σ t : ℝ) ↔ (t : ℝ) ≤ 1 / 2 := by |
rw [coe_symm_eq, le_sub_iff_add_le, add_comm, ← le_sub_iff_add_le, sub_half]
| false |
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | 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]
| false |
import Mathlib.Analysis.Convex.Combination
import Mathlib.LinearAlgebra.AffineSpace.Independent
import Mathlib.Tactic.FieldSimp
#align_import analysis.convex.caratheodory from "leanprover-community/mathlib"@"e6fab1dc073396d45da082c644642c4f8bff2264"
open Set Finset
universe u
variable {𝕜 : Type*} {E : Type u} ... | Mathlib/Analysis/Convex/Caratheodory.lean | 119 | 121 | theorem minCardFinsetOfMemConvexHull_nonempty : (minCardFinsetOfMemConvexHull hx).Nonempty := by |
rw [← Finset.coe_nonempty, ← @convexHull_nonempty_iff 𝕜]
exact ⟨x, mem_minCardFinsetOfMemConvexHull hx⟩
| false |
import Mathlib.LinearAlgebra.CliffordAlgebra.Contraction
variable {R M : Type*}
variable [CommRing R] [AddCommGroup M] [Module R M] {Q : QuadraticForm R M}
namespace CliffordAlgebra
variable (Q)
def invertibleιOfInvertible (m : M) [Invertible (Q m)] : Invertible (ι Q m) where
invOf := ι Q (⅟ (Q m) • m)
invO... | 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)
| false |
import Mathlib.Algebra.Homology.Single
#align_import algebra.homology.augment from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
open CategoryTheory Limits HomologicalComplex
universe v u
variable {V : Type u} [Category.{v} V]
namespace ChainComplex
@[simps]... | 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
| false |
import Mathlib.Combinatorics.SimpleGraph.Connectivity
#align_import combinatorics.simple_graph.prod from "leanprover-community/mathlib"@"2985fa3c31a27274aed06c433510bc14b73d6488"
variable {α β γ : Type*}
namespace SimpleGraph
-- Porting note: pruned variables to keep things out of local contexts, which
-- can im... | 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]
| false |
import Mathlib.RingTheory.AdicCompletion.Basic
import Mathlib.Algebra.Module.Torsion
open Submodule
variable {R : Type*} [CommRing R] (I : Ideal R)
variable {M : Type*} [AddCommGroup M] [Module R M]
namespace AdicCompletion
attribute [-simp] smul_eq_mul Algebra.id.smul_eq_mul
@[local simp]
theorem transitionMap... | 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
| false |
import Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.measure.haar.normed_space from "leanprover-community/mathlib"@"b84aee748341da06a6d78491367e2c0e9f15e8a5"
noncomputable sect... | Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean | 110 | 123 | theorem setIntegral_comp_smul (f : E → F) {R : ℝ} (s : Set E) (hR : R ≠ 0) :
∫ x in s, f (R • x) ∂μ = |(R ^ finrank ℝ E)⁻¹| • ∫ x in R • s, f x ∂μ := by |
let e : E ≃ᵐ E := (Homeomorph.smul (Units.mk0 R hR)).toMeasurableEquiv
calc
∫ x in s, f (R • x) ∂μ
= ∫ x in e ⁻¹' (e.symm ⁻¹' s), f (e x) ∂μ := by simp [← preimage_comp]; rfl
_ = ∫ y in e.symm ⁻¹' s, f y ∂map (fun x ↦ R • x) μ := (setIntegral_map_equiv _ _ _).symm
_ = |(R ^ finrank ℝ E)⁻¹| • ∫ y in e.sym... | false |
import Mathlib.Algebra.IsPrimePow
import Mathlib.Data.Nat.Factorization.Basic
#align_import data.nat.factorization.prime_pow from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
variable {R : Type*} [CommMonoidWithZero R] (n p : R) (k : ℕ)
theorem IsPrimePow.minFac_pow_factorization_eq ... | 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 ... | false |
import Mathlib.CategoryTheory.Functor.Hom
import Mathlib.CategoryTheory.Products.Basic
import Mathlib.Data.ULift
#align_import category_theory.yoneda from "leanprover-community/mathlib"@"369525b73f229ccd76a6ec0e0e0bf2be57599768"
namespace CategoryTheory
open Opposite
universe v₁ u₁ u₂
-- morphism levels before ... | 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
| false |
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Algebra.Group.Aut
import Mathlib.Data.ZMod.Defs
import Mathlib.Tactic.Ring
#align_import algebra.quandle from "leanprover-community/mathlib"@"28aa996fc6fb4317f0083c4e6daf79878d81be33"
open MulOpposite
universe u v
class Shelf (α : Type u) where
act : ... | Mathlib/Algebra/Quandle.lean | 287 | 289 | theorem self_invAct_invAct_eq {x y : R} : (x ◃⁻¹ x) ◃⁻¹ y = x ◃⁻¹ y := by |
have h := @self_act_act_eq _ _ (op x) (op y)
simpa using h
| false |
import Mathlib.Analysis.LocallyConvex.Basic
#align_import analysis.locally_convex.balanced_core_hull from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Pointwise Topology Filter
variable {𝕜 E ι : Type*}
section balancedHull
section SeminormedRing
variable [SeminormedRing ... | 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)
| false |
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Data.Rat.Cast.Defs
#align_import data.rat.cast from "leanprover-community/mathlib"@"acebd8d49928f6ed8920e502a6c90674e75bd441"
variable {F ι α β : Type*}
namespace Rat
open Rat
section WithDivRing
variable [DivisionRing α]
@[simp, norm_cast]
th... | 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
| false |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.