Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k |
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import Mathlib.Algebra.Algebra.Quasispectrum
import Mathlib.FieldTheory.IsAlgClosed.Spectrum
import Mathlib.Analysis.Complex.Liouville
import Mathlib.Analysis.Complex.Polynomial
import Mathlib.Analysis.Analytic.RadiusLiminf
import Mathlib.Topology.Algebra.Module.CharacterSpace
import Mathlib.Analysis.NormedSpace.Expon... | Mathlib/Analysis/NormedSpace/Spectrum.lean | 79 | 80 | theorem SpectralRadius.of_subsingleton [Subsingleton A] (a : A) : spectralRadius 𝕜 a = 0 := by |
simp [spectralRadius]
|
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.TryThis
import Mathlib.Util.AtomM
set_option autoImplicit true
namespace Mathlib.Tactic.Abel
open Lean Elab Meta Tactic Qq
initialize registerTraceClass `abel
initialize registerTraceClass `abel.detail
structure Context where
α : Expr
univ :... | Mathlib/Tactic/Abel.lean | 144 | 146 | theorem term_add_term {α} [AddCommMonoid α] (n₁ x a₁ n₂ a₂ n' a') (h₁ : n₁ + n₂ = n')
(h₂ : a₁ + a₂ = a') : @term α _ n₁ x a₁ + @term α _ n₂ x a₂ = term n' x a' := by |
simp [h₁.symm, h₂.symm, term, add_nsmul, add_assoc, add_left_comm]
|
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Smooth.Basic
import Mathlib.RingTheory.Unramified.Basic
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
-- Porting note: added to make the syntax work below.
open scoped TensorProdu... | Mathlib/RingTheory/Etale/Basic.lean | 66 | 69 | theorem iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by |
rw [formallyUnramified_iff, formallySmooth_iff, formallyEtale_iff]
simp_rw [← forall_and, Function.Bijective]
|
import Mathlib.Algebra.IsPrimePow
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.Order.Hom.Bounded
import Mathlib.Algebra.GCDMonoid.Basic
#align_import ring_theory.chain_of_divisors from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
variable {M : Type*} [CancelCommMonoidWithZero... | Mathlib/RingTheory/ChainOfDivisors.lean | 111 | 132 | theorem eq_second_of_chain_of_prime_dvd {p q r : Associates M} {n : ℕ} (hn : n ≠ 0)
{c : Fin (n + 1) → Associates M} (h₁ : StrictMono c)
(h₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i) (hp : Prime p) (hr : r ∣ q) (hp' : p ∣ r) :
p = c 1 := by |
cases' n with n
· contradiction
obtain ⟨i, rfl⟩ := h₂.1 (dvd_trans hp' hr)
refine congr_arg c (eq_of_ge_of_not_gt ?_ fun hi => ?_)
· rw [Fin.le_iff_val_le_val, Fin.val_one, Nat.succ_le_iff, ← Fin.val_zero' (n.succ + 1), ←
Fin.lt_iff_val_lt_val, Fin.pos_iff_ne_zero]
rintro rfl
exact hp.not_unit ... |
import Mathlib.Algebra.Polynomial.Mirror
import Mathlib.Analysis.Complex.Polynomial
#align_import data.polynomial.unit_trinomial from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836"
namespace Polynomial
open scoped Polynomial
open Finset
section Semiring
variable {R : Type*} [Semirin... | Mathlib/Algebra/Polynomial/UnitTrinomial.lean | 61 | 64 | theorem trinomial_trailing_coeff' (hkm : k < m) (hmn : m < n) :
(trinomial k m n u v w).coeff k = u := by |
rw [trinomial_def, coeff_add, coeff_add, coeff_C_mul_X_pow, coeff_C_mul_X_pow, coeff_C_mul_X_pow,
if_pos rfl, if_neg hkm.ne, if_neg (hkm.trans hmn).ne, add_zero, add_zero]
|
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.RingTheory.Ideal.Quotient
#align_import linear_algebra.smodeq from "leanprover-community/mathlib"@"146d3d1fa59c091fedaad8a4afa09d6802886d24"
open Submodule
open Polynomial
variable {R : Type*} [Ring R]
variable {A : Type*} [CommRing A]
variable {M : Type*} [... | Mathlib/LinearAlgebra/SModEq.lean | 114 | 119 | theorem eval {R : Type*} [CommRing R] {I : Ideal R} {x y : R} (h : x ≡ y [SMOD I]) (f : R[X]) :
f.eval x ≡ f.eval y [SMOD I] := by |
rw [SModEq.def] at h ⊢
show Ideal.Quotient.mk I (f.eval x) = Ideal.Quotient.mk I (f.eval y)
replace h : Ideal.Quotient.mk I x = Ideal.Quotient.mk I y := h
rw [← Polynomial.eval₂_at_apply, ← Polynomial.eval₂_at_apply, h]
|
import Mathlib.RingTheory.Polynomial.Cyclotomic.Basic
import Mathlib.RingTheory.RootsOfUnity.Minpoly
#align_import ring_theory.polynomial.cyclotomic.roots from "leanprover-community/mathlib"@"7fdeecc0d03cd40f7a165e6cf00a4d2286db599f"
namespace Polynomial
variable {R : Type*} [CommRing R] {n : ℕ}
theorem isRoot_... | Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean | 107 | 113 | theorem roots_cyclotomic_nodup [NeZero (n : R)] : (cyclotomic n R).roots.Nodup := by |
obtain h | ⟨ζ, hζ⟩ := (cyclotomic n R).roots.empty_or_exists_mem
· exact h.symm ▸ Multiset.nodup_zero
rw [mem_roots <| cyclotomic_ne_zero n R, isRoot_cyclotomic_iff] at hζ
refine Multiset.nodup_of_le
(roots.le_of_dvd (X_pow_sub_C_ne_zero (NeZero.pos_of_neZero_natCast R) 1) <|
cyclotomic.dvd_X_pow_sub... |
import Mathlib.Topology.Instances.Real
import Mathlib.Order.Filter.Archimedean
#align_import analysis.subadditive from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open Set Filter Topology
def Subadditive (u : ℕ → ℝ) : Prop :=
∀ m n, u (m + n) ≤ u m + u n
#al... | Mathlib/Analysis/Subadditive.lean | 45 | 48 | theorem lim_le_div (hbdd : BddBelow (range fun n => u n / n)) {n : ℕ} (hn : n ≠ 0) :
h.lim ≤ u n / n := by |
rw [Subadditive.lim]
exact csInf_le (hbdd.mono <| image_subset_range _ _) ⟨n, hn.bot_lt, rfl⟩
|
import Mathlib.Algebra.Polynomial.Splits
#align_import algebra.cubic_discriminant from "leanprover-community/mathlib"@"930133160e24036d5242039fe4972407cd4f1222"
noncomputable section
@[ext]
structure Cubic (R : Type*) where
(a b c d : R)
#align cubic Cubic
namespace Cubic
open Cubic Polynomial
open Polynom... | Mathlib/Algebra/CubicDiscriminant.lean | 145 | 146 | theorem of_b_eq_zero (ha : P.a = 0) (hb : P.b = 0) : P.toPoly = C P.c * X + C P.d := by |
rw [of_a_eq_zero ha, hb, C_0, zero_mul, zero_add]
|
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.RCLike.Basic
#align_import analysis.convex.gauge from "leanprover-community/mathlib"@"373b03b5b9d0486534edbe94747f23cb3712f93d"
... | Mathlib/Analysis/Convex/Gauge.lean | 103 | 110 | theorem gauge_zero' : gauge (0 : Set E) = 0 := by |
ext x
rw [gauge_def']
obtain rfl | hx := eq_or_ne x 0
· simp only [csInf_Ioi, mem_zero, Pi.zero_apply, eq_self_iff_true, sep_true, smul_zero]
· simp only [mem_zero, Pi.zero_apply, inv_eq_zero, smul_eq_zero]
convert Real.sInf_empty
exact eq_empty_iff_forall_not_mem.2 fun r hr => hr.2.elim (ne_of_gt hr... |
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Preserves.Basic
#align_import category_theory.limits.preserves.shapes.binary_products from "leanprover-community/mathlib"@"024a4231815538ac739f52d08dd20a55da0d6b23"
noncomputable section
universe v₁ v₂ u₁ u₂
open Cat... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/BinaryProducts.lean | 100 | 103 | theorem PreservesLimitPair.iso_inv_fst :
(PreservesLimitPair.iso G X Y).inv ≫ G.map prod.fst = prod.fst := by |
rw [← Iso.cancel_iso_hom_left (PreservesLimitPair.iso G X Y), ← Category.assoc, Iso.hom_inv_id]
simp
|
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 | 221 | 234 | theorem ofModule_asAlgebraHom_apply_apply (r : MonoidAlgebra k G)
(m : RestrictScalars k (MonoidAlgebra k G) M) :
((ofModule M).asAlgebraHom r) m =
(RestrictScalars.addEquiv _ _ _).symm (r • RestrictScalars.addEquiv _ _ _ m) := by |
apply MonoidAlgebra.induction_on r
· intro g
simp only [one_smul, MonoidAlgebra.lift_symm_apply, MonoidAlgebra.of_apply,
Representation.asAlgebraHom_single, Representation.ofModule, AddEquiv.apply_eq_iff_eq,
RestrictScalars.lsmul_apply_apply]
· intro f g fw gw
simp only [fw, gw, map_add, add_... |
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct
universe u v
variable (K : Type u) [Field K]
var... | Mathlib/LinearAlgebra/Coevaluation.lean | 61 | 76 | theorem contractLeft_assoc_coevaluation :
(contractLeft K V).rTensor _ ∘ₗ
(TensorProduct.assoc K _ _ _).symm.toLinearMap ∘ₗ
(coevaluation K V).lTensor (Module.Dual K V) =
(TensorProduct.lid K _).symm.toLinearMap ∘ₗ (TensorProduct.rid K _).toLinearMap := by |
letI := Classical.decEq (Basis.ofVectorSpaceIndex K V)
apply TensorProduct.ext
apply (Basis.ofVectorSpace K V).dualBasis.ext; intro j; apply LinearMap.ext_ring
rw [LinearMap.compr₂_apply, LinearMap.compr₂_apply, TensorProduct.mk_apply]
simp only [LinearMap.coe_comp, Function.comp_apply, LinearEquiv.coe_toLin... |
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
#align_import analysis.special_functions.pow.asymptotics from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
set_option linter.uppercaseLean3 false
noncomputable section
open scoped Classical
open Real Topology NNReal ENNReal Filter C... | Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | 36 | 46 | theorem tendsto_rpow_atTop {y : ℝ} (hy : 0 < y) : Tendsto (fun x : ℝ => x ^ y) atTop atTop := by |
rw [tendsto_atTop_atTop]
intro b
use max b 0 ^ (1 / y)
intro x hx
exact
le_of_max_le_left
(by
convert rpow_le_rpow (rpow_nonneg (le_max_right b 0) (1 / y)) hx (le_of_lt hy)
using 1
rw [← rpow_mul (le_max_right b 0), (eq_div_iff (ne_of_gt hy)).mp rfl, Real.rpow_one])
|
import Mathlib.Logic.Encodable.Lattice
import Mathlib.MeasureTheory.MeasurableSpace.Defs
#align_import measure_theory.pi_system from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"
open MeasurableSpace Set
open scoped Classical
open MeasureTheory
def IsPiSystem {α} (C : Set (Set α)) :... | Mathlib/MeasureTheory/PiSystem.lean | 149 | 151 | theorem isPiSystem_image_Iic (s : Set α) : IsPiSystem (Iic '' s) := by |
rintro _ ⟨a, ha, rfl⟩ _ ⟨b, hb, rfl⟩ -
exact ⟨a ⊓ b, inf_ind a b ha hb, Iic_inter_Iic.symm⟩
|
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 | 112 | 114 | theorem binomial_spec [DecidableEq α] (hab : a ≠ b) :
(f a)! * (f b)! * multinomial {a, b} f = (f a + f b)! := by |
simpa [Finset.sum_pair hab, Finset.prod_pair hab] using multinomial_spec {a, b} f
|
import Mathlib.LinearAlgebra.Dual
open Function Module
variable (R M N : Type*) [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N]
structure PerfectPairing :=
toLin : M →ₗ[R] N →ₗ[R] R
bijectiveLeft : Bijective toLin
bijectiveRight : Bijective toLin.flip
attribute [nolint docBlame] P... | Mathlib/LinearAlgebra/PerfectPairing.lean | 96 | 100 | theorem toDualRight_symm_toDualLeft (x : M) :
p.toDualRight.symm.dualMap (p.toDualLeft x) = Dual.eval R M x := by |
ext f
simp only [LinearEquiv.dualMap_apply, Dual.eval_apply]
exact toDualLeft_of_toDualRight_symm p x f
|
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.Separation
import Mathlib.Order.Filter.CountableInter
#align_import topology.G_delta from "leanprover-community/mathlib"@"b9e46fe101fc897fb2e7edaf0bf1f09ea49eb81a"
noncomputable section
open Topology TopologicalSpace Filter Encodable Set
open sco... | Mathlib/Topology/GDelta.lean | 136 | 138 | theorem IsGδ.inter {s t : Set X} (hs : IsGδ s) (ht : IsGδ t) : IsGδ (s ∩ t) := by |
rw [inter_eq_iInter]
exact .iInter (Bool.forall_bool.2 ⟨ht, hs⟩)
|
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Ring.Defs
#align_import algebra.euclidean_domain.defs from "leanprover-community/mathlib"@"ee7b9f9a9ac2a8d9f04ea39bbfe6b1a3be053b38"
universe u
class EuclideanDomain (R : Type u) extends CommRing R, Nontrivial R ... | Mathlib/Algebra/EuclideanDomain/Defs.lean | 131 | 133 | theorem mod_add_div' (m k : R) : m % k + m / k * k = m := by |
rw [mul_comm]
exact mod_add_div _ _
|
import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.IdealOperations
import Mathlib.Algebra.Lie.Quotient
#align_import algebra.lie.normalizer from "leanprover-community/mathlib"@"938fead7abdc0cbbca8eba7a1052865a169dc102"
variable {R L M M' : Type*}
variable [CommRing R] [LieRing L] [LieAlgebra R L]
varia... | Mathlib/Algebra/Lie/Normalizer.lean | 86 | 87 | theorem top_lie_le_iff_le_normalizer (N' : LieSubmodule R L M) :
⁅(⊤ : LieIdeal R L), N⁆ ≤ N' ↔ N ≤ N'.normalizer := by | rw [lie_le_iff]; tauto
|
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.NormedSpace.Complemented
#align_import analysis.calculus.implicit from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a21598... | Mathlib/Analysis/Calculus/Implicit.lean | 201 | 214 | theorem implicitFunction_hasStrictFDerivAt (g'inv : G →L[𝕜] E)
(hg'inv : φ.rightDeriv.comp g'inv = ContinuousLinearMap.id 𝕜 G)
(hg'invf : φ.leftDeriv.comp g'inv = 0) :
HasStrictFDerivAt (φ.implicitFunction (φ.leftFun φ.pt)) g'inv (φ.rightFun φ.pt) := by |
have := φ.hasStrictFDerivAt.to_localInverse
simp only [prodFun] at this
convert this.comp (φ.rightFun φ.pt) ((hasStrictFDerivAt_const _ _).prod (hasStrictFDerivAt_id _))
-- Porting note: added parentheses to help `simp`
simp only [ContinuousLinearMap.ext_iff, (ContinuousLinearMap.comp_apply)] at hg'inv hg'in... |
import Mathlib.Algebra.Exact
import Mathlib.RingTheory.TensorProduct.Basic
section Modules
open TensorProduct LinearMap
section Semiring
variable {R : Type*} [CommSemiring R] {M N P Q: Type*}
[AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q]
[Module R M] [Module R N] [Module R P] [... | Mathlib/LinearAlgebra/TensorProduct/RightExactness.lean | 111 | 122 | theorem LinearMap.lTensor_surjective (hg : Function.Surjective g) :
Function.Surjective (lTensor Q g) := by |
intro z
induction z using TensorProduct.induction_on with
| zero => exact ⟨0, map_zero _⟩
| tmul q p =>
obtain ⟨n, rfl⟩ := hg p
exact ⟨q ⊗ₜ[R] n, rfl⟩
| add x y hx hy =>
obtain ⟨x, rfl⟩ := hx
obtain ⟨y, rfl⟩ := hy
exact ⟨x + y, map_add _ _ _⟩
|
import Mathlib.SetTheory.Game.Ordinal
import Mathlib.SetTheory.Ordinal.NaturalOps
#align_import set_theory.game.birthday from "leanprover-community/mathlib"@"a347076985674932c0e91da09b9961ed0a79508c"
universe u
open Ordinal
namespace SetTheory
open scoped NaturalOps PGame
namespace PGame
noncomputable def b... | Mathlib/SetTheory/Game/Birthday.lean | 107 | 107 | theorem birthday_one : birthday 1 = 1 := by | rw [birthday_def]; simp
|
import Mathlib.Probability.Kernel.CondDistrib
#align_import probability.kernel.condexp from "leanprover-community/mathlib"@"00abe0695d8767201e6d008afa22393978bb324d"
open MeasureTheory Set Filter TopologicalSpace
open scoped ENNReal MeasureTheory ProbabilityTheory
namespace ProbabilityTheory
section AuxLemmas
... | Mathlib/Probability/Kernel/Condexp.lean | 52 | 57 | theorem _root_.MeasureTheory.Integrable.comp_snd_map_prod_id [NormedAddCommGroup F] (hm : m ≤ mΩ)
(hf : Integrable f μ) : Integrable (fun x : Ω × Ω => f x.2)
(@Measure.map Ω (Ω × Ω) (m.prod mΩ) mΩ (fun ω => (id ω, id ω)) μ) := by |
rw [← integrable_comp_snd_map_prod_mk_iff (measurable_id'' hm)] at hf
simp_rw [id] at hf ⊢
exact hf
|
import Mathlib.Algebra.Algebra.Spectrum
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.Nilpotent.Basic
#align_import linear_algebra.eigenspace.basic from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da1aa1"
universe u v... | Mathlib/LinearAlgebra/Eigenspace/Basic.lean | 98 | 101 | theorem hasEigenvalue_of_hasEigenvector {f : End R M} {μ : R} {x : M} (h : HasEigenvector f μ x) :
HasEigenvalue f μ := by |
rw [HasEigenvalue, Submodule.ne_bot_iff]
use x; exact h
|
import Mathlib.Topology.Separation
import Mathlib.Topology.Bases
#align_import topology.dense_embedding from "leanprover-community/mathlib"@"148aefbd371a25f1cff33c85f20c661ce3155def"
noncomputable section
open Set Filter
open scoped Topology
variable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}
structure D... | Mathlib/Topology/DenseEmbedding.lean | 83 | 90 | theorem interior_compact_eq_empty [T2Space β] (di : DenseInducing i) (hd : Dense (range i)ᶜ)
{s : Set α} (hs : IsCompact s) : interior s = ∅ := by |
refine eq_empty_iff_forall_not_mem.2 fun x hx => ?_
rw [mem_interior_iff_mem_nhds] at hx
have := di.closure_image_mem_nhds hx
rw [(hs.image di.continuous).isClosed.closure_eq] at this
rcases hd.inter_nhds_nonempty this with ⟨y, hyi, hys⟩
exact hyi (image_subset_range _ _ hys)
|
import Mathlib.SetTheory.Game.Basic
import Mathlib.SetTheory.Ordinal.NaturalOps
#align_import set_theory.game.ordinal from "leanprover-community/mathlib"@"b90e72c7eebbe8de7c8293a80208ea2ba135c834"
universe u
open SetTheory PGame
open scoped NaturalOps PGame
namespace Ordinal
noncomputable def toPGame : Ordin... | Mathlib/SetTheory/Game/Ordinal.lean | 134 | 137 | theorem toPGame_le {a b : Ordinal} (h : a ≤ b) : a.toPGame ≤ b.toPGame := by |
refine le_iff_forall_lf.2 ⟨fun i => ?_, isEmptyElim⟩
rw [toPGame_moveLeft']
exact toPGame_lf ((toLeftMovesToPGame_symm_lt i).trans_le h)
|
import Mathlib.FieldTheory.Galois
import Mathlib.Topology.Algebra.FilterBasis
import Mathlib.Topology.Algebra.OpenSubgroup
import Mathlib.Tactic.ByContra
#align_import field_theory.krull_topology from "leanprover-community/mathlib"@"039a089d2a4b93c761b234f3e5f5aeb752bac60f"
open scoped Classical Pointwise
theore... | Mathlib/FieldTheory/KrullTopology.lean | 93 | 100 | theorem IntermediateField.fixingSubgroup.bot {K L : Type*} [Field K] [Field L] [Algebra K L] :
IntermediateField.fixingSubgroup (⊥ : IntermediateField K L) = ⊤ := by |
ext f
refine ⟨fun _ => Subgroup.mem_top _, fun _ => ?_⟩
rintro ⟨x, hx : x ∈ (⊥ : IntermediateField K L)⟩
rw [IntermediateField.mem_bot] at hx
rcases hx with ⟨y, rfl⟩
exact f.commutes y
|
import Mathlib.Combinatorics.SetFamily.Shadow
#align_import combinatorics.set_family.compression.uv from "leanprover-community/mathlib"@"6f8ab7de1c4b78a68ab8cf7dd83d549eb78a68a1"
open Finset
variable {α : Type*}
theorem sup_sdiff_injOn [GeneralizedBooleanAlgebra α] (u v : α) :
{ x | Disjoint u x ∧ v ≤ x }.... | Mathlib/Combinatorics/SetFamily/Compression/UV.lean | 107 | 110 | theorem compress_sdiff_sdiff (a b : α) : compress (a \ b) (b \ a) b = a := by |
refine (compress_of_disjoint_of_le disjoint_sdiff_self_left sdiff_le).trans ?_
rw [sup_sdiff_self_right, sup_sdiff, disjoint_sdiff_self_right.sdiff_eq_left, sup_eq_right]
exact sdiff_sdiff_le
|
import Mathlib.Probability.Variance
#align_import probability.moments from "leanprover-community/mathlib"@"85453a2a14be8da64caf15ca50930cf4c6e5d8de"
open MeasureTheory Filter Finset Real
noncomputable section
open scoped MeasureTheory ProbabilityTheory ENNReal NNReal
namespace ProbabilityTheory
variable {Ω ι ... | Mathlib/Probability/Moments.lean | 156 | 157 | theorem mgf_zero' : mgf X μ 0 = (μ Set.univ).toReal := by |
simp only [mgf, zero_mul, exp_zero, integral_const, smul_eq_mul, mul_one]
|
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.Group.Indicator
import Mathlib.Order.LiminfLimsup
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Data.Set.La... | Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | 487 | 498 | theorem iUnion_Ici_eq_Ioi_of_lt_of_tendsto (x : R) {as : ι → R} (x_lt : ∀ i, x < as i)
{F : Filter ι} [Filter.NeBot F] (as_lim : Filter.Tendsto as F (𝓝 x)) :
⋃ i : ι, Ici (as i) = Ioi x := by |
have obs : x ∉ range as := by
intro maybe_x_is
rcases mem_range.mp maybe_x_is with ⟨i, hi⟩
simpa only [hi, lt_self_iff_false] using x_lt i
-- Porting note: `rw at *` was too destructive. Let's only rewrite `obs` and the goal.
have := iInf_eq_of_forall_le_of_tendsto (fun i ↦ (x_lt i).le) as_lim
rw [... |
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.RingTheory.WittVector.Truncated
#align_import ring_theory.witt_vector.mul_coeff from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
namespace WittVector
variable (p : ℕ) [hp : Fact p.Prime]
variable {k ... | Mathlib/RingTheory/WittVector/MulCoeff.lean | 69 | 85 | theorem wittPolyProdRemainder_vars (n : ℕ) :
(wittPolyProdRemainder p n).vars ⊆ univ ×ˢ range n := by |
rw [wittPolyProdRemainder]
refine Subset.trans (vars_sum_subset _ _) ?_
rw [biUnion_subset]
intro x hx
apply Subset.trans (vars_mul _ _)
refine union_subset ?_ ?_
· apply Subset.trans (vars_pow _ _)
have : (p : 𝕄) = C (p : ℤ) := by simp only [Int.cast_natCast, eq_intCast]
rw [this, vars_C]
a... |
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 | 108 | 113 | theorem bounded_le_iff_bounded_lt [Preorder α] [NoMaxOrder α] :
Bounded (· ≤ ·) s ↔ Bounded (· < ·) s := by |
refine ⟨fun h => ?_, bounded_le_of_bounded_lt⟩
cases' h with a ha
cases' exists_gt a with b hb
exact ⟨b, fun c hc => lt_of_le_of_lt (ha c hc) hb⟩
|
import Mathlib.Algebra.Associated
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.Data.Nat.Lattice
import Mathlib.RingTheory.Nilpotent.Defs
#align_import ring_th... | Mathlib/RingTheory/Nilpotent/Basic.lean | 58 | 62 | theorem IsNilpotent.isUnit_sub_one [Ring R] {r : R} (hnil : IsNilpotent r) : IsUnit (r - 1) := by |
obtain ⟨n, hn⟩ := hnil
refine ⟨⟨r - 1, -∑ i ∈ Finset.range n, r ^ i, ?_, ?_⟩, rfl⟩
· simp [mul_geom_sum, hn]
· simp [geom_sum_mul, hn]
|
import Mathlib.GroupTheory.GroupAction.BigOperators
import Mathlib.Logic.Equiv.Fin
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.Module.Prod
import Mathlib.Algebra.Module.Submodule.Ker
#align_import linear_algebra.pi from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
un... | Mathlib/LinearAlgebra/Pi.lean | 69 | 69 | theorem pi_zero : pi (fun i => 0 : (i : ι) → M₂ →ₗ[R] φ i) = 0 := by | ext; rfl
|
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : ℝ}
-- @... | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 97 | 98 | theorem logb_mul_base {a b : ℝ} (h₁ : a ≠ 0) (h₂ : b ≠ 0) (c : ℝ) :
logb (a * b) c = ((logb a c)⁻¹ + (logb b c)⁻¹)⁻¹ := by | rw [← inv_logb_mul_base h₁ h₂ c, inv_inv]
|
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.AlgebraicTopology.DoldKan.Notations
#align_import algebraic_topology.dold_kan.homotopies from "leanprover-community/mathlib"@"b12099d3b7febf4209824444dd836ef5ad96db55"
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Preadditi... | Mathlib/AlgebraicTopology/DoldKan/Homotopies.lean | 156 | 166 | theorem hσ'_naturality (q : ℕ) (n m : ℕ) (hnm : c.Rel m n) {X Y : SimplicialObject C} (f : X ⟶ Y) :
f.app (op [n]) ≫ hσ' q n m hnm = hσ' q n m hnm ≫ f.app (op [m]) := by |
have h : n + 1 = m := hnm
subst h
simp only [hσ', eqToHom_refl, comp_id]
unfold hσ
split_ifs
· rw [zero_comp, comp_zero]
· simp only [zsmul_comp, comp_zsmul]
erw [f.naturality]
rfl
|
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Shift
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
variable
{𝕜 : Type*} [NontriviallyNormedField 𝕜]
{F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
{R : Type*} [Semi... | Mathlib/Analysis/Calculus/IteratedDeriv/Lemmas.lean | 48 | 56 | theorem iteratedDerivWithin_const_neg (hn : 0 < n) (c : F) :
iteratedDerivWithin n (fun z => c - f z) s x = iteratedDerivWithin n (fun z => -f z) s x := by |
obtain ⟨n, rfl⟩ := n.exists_eq_succ_of_ne_zero hn.ne'
rw [iteratedDerivWithin_succ' h hx, iteratedDerivWithin_succ' h hx]
refine iteratedDerivWithin_congr h ?_ hx
intro y hy
have : UniqueDiffWithinAt 𝕜 s y := h.uniqueDiffWithinAt hy
rw [derivWithin.neg this]
exact derivWithin_const_sub this _
|
import Mathlib.Topology.Algebra.Module.WeakDual
import Mathlib.Algebra.Algebra.Spectrum
import Mathlib.Topology.ContinuousFunction.Algebra
import Mathlib.Data.Set.Lattice
#align_import topology.algebra.module.character_space from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358ef64"
namespace ... | Mathlib/Topology/Algebra/Module/CharacterSpace.lean | 128 | 134 | theorem union_zero_isClosed [T2Space 𝕜] [ContinuousMul 𝕜] :
IsClosed (characterSpace 𝕜 A ∪ {0}) := by |
simp only [union_zero, Set.setOf_forall]
exact
isClosed_iInter fun x =>
isClosed_iInter fun y =>
isClosed_eq (eval_continuous _) <| (eval_continuous _).mul (eval_continuous _)
|
import Mathlib.RingTheory.Localization.Basic
#align_import ring_theory.localization.integer from "leanprover-community/mathlib"@"9556784a5b84697562e9c6acb40500d4a82e675a"
variable {R : Type*} [CommSemiring R] {M : Submonoid R} {S : Type*} [CommSemiring S]
variable [Algebra R S] {P : Type*} [CommSemiring P]
open ... | Mathlib/RingTheory/Localization/Integer.lean | 63 | 66 | theorem isInteger_smul {a : R} {b : S} (hb : IsInteger R b) : IsInteger R (a • b) := by |
rcases hb with ⟨b', hb⟩
use a * b'
rw [← hb, (algebraMap R S).map_mul, Algebra.smul_def]
|
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Data.ZMod.Basic
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Chebyshev
import Mathlib.RingTheory.Ideal.LocalRing
#align_import ring_theory.polynomial.dickson from "leanprover-community/mathlib"@"70fd9563a21e7b963887c936... | Mathlib/RingTheory/Polynomial/Dickson.lean | 82 | 83 | theorem dickson_add_two (n : ℕ) :
dickson k a (n + 2) = X * dickson k a (n + 1) - C a * dickson k a n := by | rw [dickson]
|
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