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
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import Mathlib.NumberTheory.Liouville.Basic
#align_import number_theory.liouville.liouville_number from "leanprover-community/mathlib"@"04e80bb7e8510958cd9aacd32fe2dc147af0b9f1"
noncomputable section
open scoped Nat
open Real Finset
def liouvilleNumber (m : ℝ) : ℝ :=
∑' i : ℕ, 1 / m ^ i !
#align liouville_n... | Mathlib/NumberTheory/Liouville/LiouvilleNumber.lean | 84 | 86 | theorem remainder_summable {m : ℝ} (hm : 1 < m) (k : ℕ) :
Summable fun i : ℕ => 1 / m ^ (i + (k + 1))! := by |
convert (summable_nat_add_iff (k + 1)).2 (LiouvilleNumber.summable hm)
| false |
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.Topology.Sheaves.SheafCondition.Sites
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.RingTheory.LocalProperties
#align_import algebraic_geometry.properties from "leanprover-community/mathlib"@"88... | Mathlib/AlgebraicGeometry/Properties.lean | 84 | 93 | theorem isReducedOfOpenImmersion {X Y : Scheme} (f : X ⟶ Y) [H : IsOpenImmersion f]
[IsReduced Y] : IsReduced X := by |
constructor
intro U
have : U = (Opens.map f.1.base).obj (H.base_open.isOpenMap.functor.obj U) := by
ext1; exact (Set.preimage_image_eq _ H.base_open.inj).symm
rw [this]
exact isReduced_of_injective (inv <| f.1.c.app (op <| H.base_open.isOpenMap.functor.obj U))
(asIso <| f.1.c.app (op <| H.base_open.i... | false |
import Mathlib.Algebra.Star.Basic
import Mathlib.Algebra.Order.CauSeq.Completion
#align_import data.real.basic from "leanprover-community/mathlib"@"cb42593171ba005beaaf4549fcfe0dece9ada4c9"
assert_not_exists Finset
assert_not_exists Module
assert_not_exists Submonoid
assert_not_exists FloorRing
structure Real w... | Mathlib/Data/Real/Basic.lean | 130 | 132 | theorem ofCauchy_sub (a b) : (⟨a - b⟩ : ℝ) = ⟨a⟩ - ⟨b⟩ := by |
rw [sub_eq_add_neg, ofCauchy_add, ofCauchy_neg]
rfl
| false |
import Mathlib.Algebra.DirectSum.Internal
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.MvPolynomial.Variables
import Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous
import Mathlib.Algebra.Polynomial.Roots
#align_i... | Mathlib/RingTheory/MvPolynomial/Homogeneous.lean | 57 | 61 | theorem weightedTotalDegree_one (φ : MvPolynomial σ R) :
weightedTotalDegree (1 : σ → ℕ) φ = φ.totalDegree := by |
simp only [totalDegree, weightedTotalDegree, weightedDegree, LinearMap.toAddMonoidHom_coe,
Finsupp.total, Pi.one_apply, Finsupp.coe_lsum, LinearMap.coe_smulRight, LinearMap.id_coe,
id, Algebra.id.smul_eq_mul, mul_one]
| false |
import Mathlib.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from "leanprover-community/mathlib"@"00638177efd1b2534fc5269363ebf42a7871df9a"
open Equiv Equiv.Perm List
variable {α : Type*}
namespace Equiv.Perm
secti... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 265 | 274 | theorem mem_toList_iff {y : α} : y ∈ toList p x ↔ SameCycle p x y ∧ x ∈ p.support := by |
simp only [toList, mem_range, mem_map]
constructor
· rintro ⟨n, hx, rfl⟩
refine ⟨⟨n, rfl⟩, ?_⟩
contrapose! hx
rw [← support_cycleOf_eq_nil_iff] at hx
simp [hx]
· rintro ⟨h, hx⟩
simpa using h.exists_pow_eq_of_mem_support hx
| false |
import Mathlib.Data.Set.Image
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Clopen
import Mathlib.Topology.Irreducible
#align_import topology.connected from "leanprover-community/mathlib"@"d101e93197bb5f6ea89bd7ba386b7f7dff1f3903"
open Set Function Topology TopologicalSpace Relation
open scoped C... | Mathlib/Topology/Connected/Basic.lean | 96 | 111 | theorem isPreconnected_of_forall {s : Set α} (x : α)
(H : ∀ y ∈ s, ∃ t, t ⊆ s ∧ x ∈ t ∧ y ∈ t ∧ IsPreconnected t) : IsPreconnected s := by |
rintro u v hu hv hs ⟨z, zs, zu⟩ ⟨y, ys, yv⟩
have xs : x ∈ s := by
rcases H y ys with ⟨t, ts, xt, -, -⟩
exact ts xt
-- Porting note (#11215): TODO: use `wlog xu : x ∈ u := hs xs using u v y z, v u z y`
cases hs xs with
| inl xu =>
rcases H y ys with ⟨t, ts, xt, yt, ht⟩
have := ht u v hu hv (ts... | false |
import Mathlib.Probability.ProbabilityMassFunction.Basic
#align_import probability.probability_mass_function.monad from "leanprover-community/mathlib"@"4ac69b290818724c159de091daa3acd31da0ee6d"
noncomputable section
variable {α β γ : Type*}
open scoped Classical
open NNReal ENNReal
open MeasureTheory
namespac... | Mathlib/Probability/ProbabilityMassFunction/Monad.lean | 54 | 54 | theorem mem_support_pure_iff : a' ∈ (pure a).support ↔ a' = a := by | simp
| false |
import Mathlib.Data.PFunctor.Univariate.Basic
#align_import data.pfunctor.univariate.M from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
universe u v w
open Nat Function
open List
variable (F : PFunctor.{u})
-- Porting note: the ♯ tactic is never used
-- local prefix:0 "♯" => cast... | Mathlib/Data/PFunctor/Univariate/M.lean | 66 | 67 | theorem approx_eta {n : ℕ} (x : CofixA F (n + 1)) : x = CofixA.intro (head' x) (children' x) := by |
cases x; rfl
| false |
import Mathlib.Data.Nat.Defs
import Mathlib.Tactic.GCongr.Core
import Mathlib.Tactic.Common
import Mathlib.Tactic.Monotonicity.Attr
#align_import data.nat.factorial.basic from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
namespace Nat
def factorial : ℕ → ℕ
| 0 => 1
| succ n => s... | Mathlib/Data/Nat/Factorial/Basic.lean | 73 | 76 | theorem factorial_dvd_factorial {m n} (h : m ≤ n) : m ! ∣ n ! := by |
induction' h with n _ ih
· exact Nat.dvd_refl _
· exact Nat.dvd_trans ih (Nat.dvd_mul_left _ _)
| false |
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Topology.Algebra.Module.Basic
import Mathlib.RingTheory.Adjoin.Basic
#align_import topology.algebra.algebra from "leanprover-community/mathlib"@"43afc5ad87891456c57b5a183e3e617d67c2b1db"
open scoped Classical
open Set TopologicalSpace Algebra
open sc... | Mathlib/Topology/Algebra/Algebra.lean | 47 | 51 | theorem continuous_algebraMap_iff_smul [TopologicalSemiring A] :
Continuous (algebraMap R A) ↔ Continuous fun p : R × A => p.1 • p.2 := by |
refine ⟨fun h => ?_, fun h => have : ContinuousSMul R A := ⟨h⟩; continuous_algebraMap _ _⟩
simp only [Algebra.smul_def]
exact (h.comp continuous_fst).mul continuous_snd
| false |
import Mathlib.Analysis.Calculus.FDeriv.Bilinear
#align_import analysis.calculus.fderiv.mul from "leanprover-community/mathlib"@"d608fc5d4e69d4cc21885913fb573a88b0deb521"
open scoped Classical
open Filter Asymptotics ContinuousLinearMap Set Metric Topology NNReal ENNReal
noncomputable section
section
variable ... | Mathlib/Analysis/Calculus/FDeriv/Mul.lean | 307 | 309 | theorem HasStrictFDerivAt.smul_const (hc : HasStrictFDerivAt c c' x) (f : F) :
HasStrictFDerivAt (fun y => c y • f) (c'.smulRight f) x := by |
simpa only [smul_zero, zero_add] using hc.smul (hasStrictFDerivAt_const f x)
| false |
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.GroupTheory.GroupAction.Hom
open Set Pointwise
theorem MulAction.smul_bijective_of_is_unit
{M : Type*} [Monoid M] {α : Type*} [MulAction M α] {m : M} (hm : IsUnit m) :
Function.Bijective (fun (a : α) ↦ m • a) := by
lift m to Mˣ using hm
rw [Functio... | Mathlib/GroupTheory/GroupAction/Pointwise.lean | 87 | 91 | theorem preimage_smul_setₛₗ_of_units (hc : IsUnit c) (hc' : IsUnit (σ c)) :
h ⁻¹' (σ c • t) = c • h ⁻¹' t := by |
apply preimage_smul_setₛₗ'
· exact (MulAction.smul_bijective_of_is_unit hc).surjective
· exact (MulAction.smul_bijective_of_is_unit hc').injective
| false |
import Mathlib.CategoryTheory.CofilteredSystem
import Mathlib.Combinatorics.SimpleGraph.Subgraph
#align_import combinatorics.simple_graph.finsubgraph from "leanprover-community/mathlib"@"c6ef6387ede9983aee397d442974e61f89dfd87b"
open Set CategoryTheory
universe u v
variable {V : Type u} {W : Type v} {G : Simple... | Mathlib/Combinatorics/SimpleGraph/Finsubgraph.lean | 98 | 100 | theorem singletonFinsubgraph_le_adj_right {u v : V} {e : G.Adj u v} :
singletonFinsubgraph v ≤ finsubgraphOfAdj e := by |
simp [singletonFinsubgraph, finsubgraphOfAdj]
| false |
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Real.Archimedean
#align_import number_theory.zsqrtd.gaussian_int from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
open Zsqrtd Complex
open sc... | Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean | 149 | 150 | theorem toComplex_eq_zero {x : ℤ[i]} : (x : ℂ) = 0 ↔ x = 0 := by |
rw [← toComplex_zero, toComplex_inj]
| false |
import Mathlib.Analysis.Calculus.Deriv.Comp
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Slope
noncomputable section
open scoped Topology Filter ENNReal NNReal
open Filter Asymptotics Set
variable {𝕜 : Type*} [NontriviallyNormedFiel... | Mathlib/Analysis/Calculus/LineDeriv/Basic.lean | 147 | 150 | theorem hasLineDerivAt_iff_isLittleO_nhds_zero :
HasLineDerivAt 𝕜 f f' x v ↔
(fun t : 𝕜 => f (x + t • v) - f x - t • f') =o[𝓝 0] fun t => t := by |
simp only [HasLineDerivAt, hasDerivAt_iff_isLittleO_nhds_zero, zero_add, zero_smul, add_zero]
| false |
import Mathlib.Data.Fin.VecNotation
#align_import data.fin.tuple.monotone from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f78155c6c27d62a621f4c152d"
open Set Fin Matrix Function
variable {α : Type*}
| Mathlib/Data/Fin/Tuple/Monotone.lean | 21 | 24 | theorem liftFun_vecCons {n : ℕ} (r : α → α → Prop) [IsTrans α r] {f : Fin (n + 1) → α} {a : α} :
((· < ·) ⇒ r) (vecCons a f) (vecCons a f) ↔ r a (f 0) ∧ ((· < ·) ⇒ r) f f := by |
simp only [liftFun_iff_succ r, forall_fin_succ, cons_val_succ, cons_val_zero, ← succ_castSucc,
castSucc_zero]
| false |
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
#align_import linear_algebra.matrix.mv_polynomial from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
set_option linter.uppercaseLean3 false
varia... | Mathlib/LinearAlgebra/Matrix/MvPolynomial.lean | 75 | 80 | theorem det_mvPolynomialX_ne_zero [DecidableEq m] [Fintype m] [CommRing R] [Nontrivial R] :
det (mvPolynomialX m m R) ≠ 0 := by |
intro h_det
have := congr_arg Matrix.det (mvPolynomialX_mapMatrix_eval (1 : Matrix m m R))
rw [det_one, ← RingHom.map_det, h_det, RingHom.map_zero] at this
exact zero_ne_one this
| 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 | 131 | 131 | theorem zero_rpow {x : ℝ} (h : x ≠ 0) : (0 : ℝ) ^ x = 0 := by | simp [rpow_def, *]
| false |
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Tactic.Spread
import Mathlib.Util.AssertExists
#align_import algebra.ring.defs from "leanprover-community/mathlib"@"76de8ae01554c3b37d66544866659ff174e66e1f"
universe u v w x
variable {α : ... | Mathlib/Algebra/Ring/Defs.lean | 249 | 250 | theorem boole_mul {α} [MulZeroOneClass α] (P : Prop) [Decidable P] (a : α) :
(if P then 1 else 0) * a = if P then a else 0 := by | simp
| false |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
open Topology InnerProductSpace Set
noncomputable section
variable {𝕜 F : Type*} [RCLike 𝕜]
variable [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F]
variabl... | Mathlib/Analysis/Calculus/Gradient/Basic.lean | 261 | 263 | theorem HasGradientAtFilter.congr_of_eventuallyEq (h : HasGradientAtFilter f f' x L)
(hL : f₁ =ᶠ[L] f) (hx : f₁ x = f x) : HasGradientAtFilter f₁ f' x L := by |
rwa [hL.hasGradientAtFilter_iff hx rfl]
| false |
import Mathlib.Algebra.DirectSum.Finsupp
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
#align_import linear_algebra.direct_sum.finsupp from "leanprover-community/mathlib"@"9b9d125b7be0930f564a68f1d73ace10cf46064d"
noncomputable section
open DirectSum TensorProduct
ope... | Mathlib/LinearAlgebra/DirectSum/Finsupp.lean | 263 | 277 | theorem finsuppTensorFinsupp_apply (f : ι →₀ M) (g : κ →₀ N) (i : ι) (k : κ) :
finsuppTensorFinsupp R S M N ι κ (f ⊗ₜ g) (i, k) = f i ⊗ₜ g k := by |
apply Finsupp.induction_linear f
· simp
· intro f₁ f₂ hf₁ hf₂
simp [add_tmul, hf₁, hf₂]
intro i' m
apply Finsupp.induction_linear g
· simp
· intro g₁ g₂ hg₁ hg₂
simp [tmul_add, hg₁, hg₂]
intro k' n
classical
simp_rw [finsuppTensorFinsupp_single, Finsupp.single_apply, Prod.mk.inj_iff, ite_an... | false |
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.ContinuousFunction.Compact
import Mathlib.Topology.UrysohnsLemma
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Topology.Algebra.Module.CharacterSpace
#align_import topology.continuous_function.ideals from "... | Mathlib/Topology/ContinuousFunction/Ideals.lean | 94 | 98 | theorem idealOfSet_closed [T2Space R] (s : Set X) :
IsClosed (idealOfSet R s : Set C(X, R)) := by |
simp only [idealOfSet, Submodule.coe_set_mk, Set.setOf_forall]
exact isClosed_iInter fun x => isClosed_iInter fun _ =>
isClosed_eq (continuous_eval_const x) continuous_const
| false |
import Mathlib.Algebra.Algebra.Defs
import Mathlib.RingTheory.Ideal.Operations
import Mathlib.RingTheory.JacobsonIdeal
import Mathlib.Logic.Equiv.TransferInstance
import Mathlib.Tactic.TFAE
#align_import ring_theory.ideal.local_ring from "leanprover-community/mathlib"@"ec1c7d810034d4202b0dd239112d1792be9f6fdc"
un... | Mathlib/RingTheory/Ideal/LocalRing.lean | 93 | 96 | theorem isUnit_or_isUnit_of_isUnit_add {a b : R} (h : IsUnit (a + b)) : IsUnit a ∨ IsUnit b := by |
rcases h with ⟨u, hu⟩
rw [← Units.inv_mul_eq_one, mul_add] at hu
apply Or.imp _ _ (isUnit_or_isUnit_of_add_one hu) <;> exact isUnit_of_mul_isUnit_right
| false |
import Mathlib.Analysis.Complex.Circle
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
#align_import analysis.complex.isometry from "leanprover-community/mathlib"@"ae690b0c236e488a0043f6faa8ce3546e7f2f9c5"
noncomputable section
open Complex
open ComplexConjugate
... | Mathlib/Analysis/Complex/Isometry.lean | 119 | 122 | theorem LinearIsometry.re_apply_eq_re {f : ℂ →ₗᵢ[ℝ] ℂ} (h : f 1 = 1) (z : ℂ) : (f z).re = z.re := by |
apply LinearIsometry.re_apply_eq_re_of_add_conj_eq
intro z
apply LinearIsometry.im_apply_eq_im h
| false |
import Mathlib.CategoryTheory.Balanced
import Mathlib.CategoryTheory.LiftingProperties.Basic
#align_import category_theory.limits.shapes.strong_epi from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
universe v u
namespace CategoryTheory
variable {C : Type u} [Category.{v} C]
variable... | Mathlib/CategoryTheory/Limits/Shapes/StrongEpi.lean | 161 | 169 | theorem StrongMono.of_arrow_iso {A B A' B' : C} {f : A ⟶ B} {g : A' ⟶ B'}
(e : Arrow.mk f ≅ Arrow.mk g) [h : StrongMono f] : StrongMono g :=
{ mono := by |
rw [Arrow.iso_w' e]
haveI := mono_comp f e.hom.right
apply mono_comp
rlp := fun {X Y} z => by
intro
apply HasLiftingProperty.of_arrow_iso_right z e }
| false |
import Mathlib.Algebra.MvPolynomial.Monad
#align_import data.mv_polynomial.expand from "leanprover-community/mathlib"@"5da451b4c96b4c2e122c0325a7fce17d62ee46c6"
namespace MvPolynomial
variable {σ τ R S : Type*} [CommSemiring R] [CommSemiring S]
noncomputable def expand (p : ℕ) : MvPolynomial σ R →ₐ[R] MvPolyno... | Mathlib/Algebra/MvPolynomial/Expand.lean | 82 | 84 | theorem rename_expand (f : σ → τ) (p : ℕ) (φ : MvPolynomial σ R) :
rename f (expand p φ) = expand p (rename f φ) := by |
simp [expand, bind₁_rename, rename_bind₁, Function.comp]
| false |
import Mathlib.NumberTheory.Cyclotomic.Discriminant
import Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral
import Mathlib.RingTheory.Ideal.Norm
#align_import number_theory.cyclotomic.rat from "leanprover-community/mathlib"@"b353176c24d96c23f0ce1cc63efc3f55019702d9"
universe u
open Algebra IsCyclotomicExtensio... | Mathlib/NumberTheory/Cyclotomic/Rat.lean | 38 | 43 | theorem discr_prime_pow_ne_two' [IsCyclotomicExtension {p ^ (k + 1)} ℚ K]
(hζ : IsPrimitiveRoot ζ ↑(p ^ (k + 1))) (hk : p ^ (k + 1) ≠ 2) :
discr ℚ (hζ.subOnePowerBasis ℚ).basis =
(-1) ^ ((p ^ (k + 1) : ℕ).totient / 2) * p ^ ((p : ℕ) ^ k * ((p - 1) * (k + 1) - 1)) := by |
rw [← discr_prime_pow_ne_two hζ (cyclotomic.irreducible_rat (p ^ (k + 1)).pos) hk]
exact hζ.discr_zeta_eq_discr_zeta_sub_one.symm
| false |
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Topology.MetricSpace.ThickenedIndicator
open MeasureTheory Topology Metric Filter Set ENNReal NNReal
open scoped Topology ENNReal NNReal BoundedContinuousFunction
section auxiliary
namespace MeasureTheory
variable {Ω : Type*} [TopologicalSpace Ω] [Mea... | Mathlib/MeasureTheory/Measure/HasOuterApproxClosed.lean | 56 | 65 | theorem tendsto_lintegral_nn_filter_of_le_const {ι : Type*} {L : Filter ι} [L.IsCountablyGenerated]
(μ : Measure Ω) [IsFiniteMeasure μ] {fs : ι → Ω →ᵇ ℝ≥0} {c : ℝ≥0}
(fs_le_const : ∀ᶠ i in L, ∀ᵐ ω : Ω ∂μ, fs i ω ≤ c) {f : Ω → ℝ≥0}
(fs_lim : ∀ᵐ ω : Ω ∂μ, Tendsto (fun i ↦ fs i ω) L (𝓝 (f ω))) :
Tendsto (... |
refine tendsto_lintegral_filter_of_dominated_convergence (fun _ ↦ c)
(eventually_of_forall fun i ↦ (ENNReal.continuous_coe.comp (fs i).continuous).measurable) ?_
(@lintegral_const_lt_top _ _ μ _ _ (@ENNReal.coe_ne_top c)).ne ?_
· simpa only [Function.comp_apply, ENNReal.coe_le_coe] using fs_le_const
· si... | false |
import Mathlib.Combinatorics.Quiver.Path
import Mathlib.Combinatorics.Quiver.Push
#align_import combinatorics.quiver.symmetric from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
universe v u w v'
namespace Quiver
-- Porting note: no hasNonemptyInstance linter yet
def Symmetrify (V : ... | Mathlib/Combinatorics/Quiver/Symmetric.lean | 75 | 77 | theorem eq_reverse_iff [h : HasInvolutiveReverse V] {a b : V} (f : a ⟶ b)
(g : b ⟶ a) : f = reverse g ↔ reverse f = g := by |
rw [← reverse_inj, reverse_reverse]
| false |
import Mathlib.Data.Finset.Grade
import Mathlib.Order.Interval.Finset.Basic
#align_import data.finset.interval from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"
variable {α β : Type*}
namespace Finset
section Decidable
variable [DecidableEq α] (s t : Finset α)
instance instLocally... | Mathlib/Data/Finset/Interval.lean | 120 | 121 | theorem card_Ioo_finset (h : s ⊆ t) : (Ioo s t).card = 2 ^ (t.card - s.card) - 2 := by |
rw [card_Ioo_eq_card_Icc_sub_two, card_Icc_finset h]
| false |
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.FieldTheory.IsAlgClosed.Spectrum
#align_import linear_algebra.eigenspace.is_alg_closed from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da1aa1"
open Set Function Module FiniteDimensional
variable {K V : Type*} [Field K] [AddCommGro... | Mathlib/LinearAlgebra/Eigenspace/Triangularizable.lean | 194 | 197 | theorem eq_iSup_inf_genEigenspace [FiniteDimensional K V]
(h : ∀ x ∈ p, f x ∈ p) (h' : ⨆ μ, ⨆ k, f.genEigenspace μ k = ⊤) :
p = ⨆ μ, ⨆ k, p ⊓ f.genEigenspace μ k := by |
rw [← inf_iSup_genEigenspace h, h', inf_top_eq]
| false |
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.RingTheory.Polynomial.Basic
#align_import algebraic_geometry.prime_spectrum.is_open_comap_C from "leanprover-community/mathlib"@"052f6013363326d50cb99c6939814a4b8eb7b301"
open Ideal Polynomial PrimeSpectrum Set
namespace AlgebraicGeometry
names... | Mathlib/AlgebraicGeometry/PrimeSpectrum/IsOpenComapC.lean | 38 | 40 | theorem isOpen_imageOfDf : IsOpen (imageOfDf f) := by |
rw [imageOfDf, setOf_exists fun i (x : PrimeSpectrum R) => coeff f i ∉ x.asIdeal]
exact isOpen_iUnion fun i => isOpen_basicOpen
| false |
import Mathlib.Algebra.Algebra.Unitization
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
suppress_compilation
variable (𝕜 A : Type*) [NontriviallyNormedField 𝕜] [NonUnitalNormedRing A]
variable [NormedSpace 𝕜 A] [IsScalarTower 𝕜 A A] [SMulCommClass 𝕜 A A]
open ContinuousLinearMap
namespace Unitizati... | Mathlib/Analysis/NormedSpace/Unitization.lean | 167 | 180 | theorem antilipschitzWith_addEquiv :
AntilipschitzWith 2 (addEquiv 𝕜 A) := by |
refine AddMonoidHomClass.antilipschitz_of_bound (addEquiv 𝕜 A) fun x => ?_
rw [norm_eq_sup, Prod.norm_def, NNReal.coe_two]
refine max_le ?_ ?_
· rw [mul_max_of_nonneg _ _ (zero_le_two : (0 : ℝ) ≤ 2)]
exact le_max_of_le_left ((le_add_of_nonneg_left (norm_nonneg _)).trans_eq (two_mul _).symm)
· nontrivial... | false |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n α : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 482 | 491 | theorem schur_complement_eq₁₁ [Fintype m] [DecidableEq m] [Fintype n] {A : Matrix m m 𝕜}
(B : Matrix m n 𝕜) (D : Matrix n n 𝕜) (x : m → 𝕜) (y : n → 𝕜) [Invertible A]
(hA : A.IsHermitian) :
(star (x ⊕ᵥ y)) ᵥ* (fromBlocks A B Bᴴ D) ⬝ᵥ (x ⊕ᵥ y) =
(star (x + (A⁻¹ * B) *ᵥ y)) ᵥ* A ⬝ᵥ (x + (A⁻¹ * B) *ᵥ... |
simp [Function.star_sum_elim, fromBlocks_mulVec, vecMul_fromBlocks, add_vecMul,
dotProduct_mulVec, vecMul_sub, Matrix.mul_assoc, vecMul_mulVec, hA.eq,
conjTranspose_nonsing_inv, star_mulVec]
abel
| false |
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.dedekind_domain.factorization from "leanprover-community/mathlib"@"2f588be38bb5bec02f218ba14f82fc82eb663f87"
noncomputable section
open scoped Classical nonZeroDivisors
open Set Function UniqueFactorizationMonoid IsDedekindDomain IsDedekind... | Mathlib/RingTheory/DedekindDomain/Factorization.lean | 97 | 107 | theorem finite_mulSupport {I : Ideal R} (hI : I ≠ 0) :
(mulSupport fun v : HeightOneSpectrum R => v.maxPowDividing I).Finite :=
haveI h_subset : {v : HeightOneSpectrum R | v.maxPowDividing I ≠ 1} ⊆
{v : HeightOneSpectrum R |
((Associates.mk v.asIdeal).count (Associates.mk I).factors : ℤ) ≠ 0} := by |
intro v hv h_zero
have hv' : v.maxPowDividing I = 1 := by
rw [IsDedekindDomain.HeightOneSpectrum.maxPowDividing, Int.natCast_eq_zero.mp h_zero,
pow_zero _]
exact hv hv'
Finite.subset (Filter.eventually_cofinite.mp (Associates.finite_factors hI)) h_subset
| false |
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
#align_import category_theory.monoidal.coherence_lemmas from "leanprover-community/mathlib"@"b8b8bf3ea0c625fa1f950034a184e07c67f7bcfe"
open CategoryTheory Category Iso
namespace CategoryTheory.MonoidalCategory
v... | Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean | 79 | 82 | theorem pentagon_inv_hom (W X Y Z : C) :
(α_ (W ⊗ X) Y Z).inv ≫ ((α_ W X Y).hom ⊗ 𝟙 Z) =
(α_ W X (Y ⊗ Z)).hom ≫ (𝟙 W ⊗ (α_ X Y Z).inv) ≫ (α_ W (X ⊗ Y) Z).inv := by |
coherence
| false |
import Mathlib.Data.DFinsupp.Basic
import Mathlib.Data.Finset.Pointwise
import Mathlib.LinearAlgebra.Basis.VectorSpace
#align_import algebra.group.unique_prods from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
@[to_additive
"Let `G` be a Type with addition, let `A B : Finset G` ... | Mathlib/Algebra/Group/UniqueProds.lean | 121 | 129 | theorem iff_card_le_one [DecidableEq G] (ha0 : a0 ∈ A) (hb0 : b0 ∈ B) :
UniqueMul A B a0 b0 ↔ ((A ×ˢ B).filter (fun p ↦ p.1 * p.2 = a0 * b0)).card ≤ 1 := by |
simp_rw [card_le_one_iff, mem_filter, mem_product]
refine ⟨fun h p1 p2 ⟨⟨ha1, hb1⟩, he1⟩ ⟨⟨ha2, hb2⟩, he2⟩ ↦ ?_, fun h a b ha hb he ↦ ?_⟩
· have h1 := h ha1 hb1 he1; have h2 := h ha2 hb2 he2
ext
· rw [h1.1, h2.1]
· rw [h1.2, h2.2]
· exact Prod.ext_iff.1 (@h (a, b) (a0, b0) ⟨⟨ha, hb⟩, he⟩ ⟨⟨ha0, hb0... | false |
import Mathlib.Data.Set.Lattice
#align_import data.semiquot from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
-- Porting note: removed universe parameter
structure Semiquot (α : Type*) where mk' ::
s : Set α
val : Trunc s
#align semiquot Semiquot
namespace Semiquot
vari... | Mathlib/Data/Semiquot.lean | 136 | 137 | theorem mem_bind (q : Semiquot α) (f : α → Semiquot β) (b : β) :
b ∈ bind q f ↔ ∃ a ∈ q, b ∈ f a := by | simp_rw [← exists_prop]; exact Set.mem_iUnion₂
| false |
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.Tactic.ByContra
import Mathlib.Topology.Algebra.Polynomial
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.Analysis.Complex.Arg
#align_import ring_theory.polynomial.cyclotomic.eval from "leanprover-community/mathlib"@"5bfbcca0a7ffdd21cf16... | Mathlib/RingTheory/Polynomial/Cyclotomic/Eval.lean | 41 | 44 | theorem eval_one_cyclotomic_prime_pow {R : Type*} [CommRing R] {p : ℕ} (k : ℕ)
[hn : Fact p.Prime] : eval 1 (cyclotomic (p ^ (k + 1)) R) = p := by |
simp only [cyclotomic_prime_pow_eq_geom_sum hn.out, eval_X, one_pow, Finset.sum_const, eval_pow,
eval_finset_sum, Finset.card_range, smul_one_eq_cast]
| false |
import Mathlib.Topology.MetricSpace.Basic
#align_import topology.metric_space.infsep from "leanprover-community/mathlib"@"5316314b553dcf8c6716541851517c1a9715e22b"
variable {α β : Type*}
namespace Set
section Einfsep
open ENNReal
open Function
noncomputable def einfsep [EDist α] (s : Set α) : ℝ≥0∞ :=
⨅ (x... | Mathlib/Topology/MetricSpace/Infsep.lean | 59 | 61 | theorem einfsep_pos : 0 < s.einfsep ↔ ∃ C > 0, ∀ x ∈ s, ∀ y ∈ s, x ≠ y → C ≤ edist x y := by |
rw [pos_iff_ne_zero, Ne, einfsep_zero]
simp only [not_forall, not_exists, not_lt, exists_prop, not_and]
| false |
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Range
#align_import data.list.nat_antidiagonal from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
open List Function Nat
namespace List
namespace Nat
def antidiagonal (n : ℕ) : List (ℕ × ℕ) :=
(range (n + 1)).map fun i ↦ (i,... | Mathlib/Data/List/NatAntidiagonal.lean | 76 | 82 | theorem antidiagonal_succ' {n : ℕ} :
antidiagonal (n + 1) = (antidiagonal n).map (Prod.map id Nat.succ) ++ [(n + 1, 0)] := by |
simp only [antidiagonal, range_succ, Nat.add_sub_cancel_left, map_append, append_assoc,
Nat.sub_self, singleton_append, map_map, map]
congr 1
apply map_congr
simp (config := { contextual := true }) [le_of_lt, Nat.succ_eq_add_one, Nat.sub_add_comm]
| false |
import Mathlib.Data.ENNReal.Basic
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.MetricSpace.Thickening
#align_import topology.metric_space.thickened_indicator from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classical
open NNReal ENNReal Topol... | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | 84 | 86 | theorem thickenedIndicatorAux_one (δ : ℝ) (E : Set α) {x : α} (x_in_E : x ∈ E) :
thickenedIndicatorAux δ E x = 1 := by |
simp [thickenedIndicatorAux, infEdist_zero_of_mem x_in_E, tsub_zero]
| false |
import Mathlib.Geometry.Manifold.ContMDiff.Basic
open Set Function Filter ChartedSpace SmoothManifoldWithCorners
open scoped Topology Manifold
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
-- declare a smooth manifold `M` over the pair `(E, H)`.
{E : Type*}
[NormedAddCommGroup E] [NormedSpace 𝕜 E] {H ... | Mathlib/Geometry/Manifold/ContMDiff/Product.lean | 149 | 162 | theorem contMDiffWithinAt_fst {s : Set (M × N)} {p : M × N} :
ContMDiffWithinAt (I.prod J) I n Prod.fst s p := by |
/- porting note: `simp` fails to apply lemmas to `ModelProd`. Was
rw [contMDiffWithinAt_iff']
refine' ⟨continuousWithinAt_fst, _⟩
refine' contDiffWithinAt_fst.congr (fun y hy => _) _
· simp only [mfld_simps] at hy
simp only [hy, mfld_simps]
· simp only [mfld_simps]
-/
rw [contMDiffWithinAt_iff']
... | 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 | 87 | 89 | theorem evalₐ_mk (n : ℕ) (x : AdicCauchySequence I R) :
evalₐ I n (mk I R x) = Ideal.Quotient.mk (I ^ n) (x.val n) := by |
simp [evalₐ]
| false |
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Nat.Choose.Vandermonde
import Mathlib.Tactic.FieldSimp
#align_import data.polynomial.hasse_deriv from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358... | Mathlib/Algebra/Polynomial/HasseDeriv.lean | 67 | 80 | theorem hasseDeriv_coeff (n : ℕ) :
(hasseDeriv k f).coeff n = (n + k).choose k * f.coeff (n + k) := by |
rw [hasseDeriv_apply, coeff_sum, sum_def, Finset.sum_eq_single (n + k), coeff_monomial]
· simp only [if_true, add_tsub_cancel_right, eq_self_iff_true]
· intro i _hi hink
rw [coeff_monomial]
by_cases hik : i < k
· simp only [Nat.choose_eq_zero_of_lt hik, ite_self, Nat.cast_zero, zero_mul]
· push_n... | false |
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Polynomial
open Finset (antid... | Mathlib/RingTheory/PowerSeries/Trunc.lean | 88 | 95 | theorem natDegree_trunc_lt (f : R⟦X⟧) (n) : (trunc (n + 1) f).natDegree < n + 1 := by |
rw [Nat.lt_succ_iff, natDegree_le_iff_coeff_eq_zero]
intros
rw [coeff_trunc]
split_ifs with h
· rw [lt_succ, ← not_lt] at h
contradiction
· rfl
| false |
import Mathlib.RingTheory.AdjoinRoot
import Mathlib.FieldTheory.Minpoly.Field
import Mathlib.RingTheory.Polynomial.GaussLemma
#align_import field_theory.minpoly.is_integrally_closed from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open scoped Classical Polynomial
open Polynomial Set... | Mathlib/FieldTheory/Minpoly/IsIntegrallyClosed.lean | 103 | 108 | theorem ker_eval {s : S} (hs : IsIntegral R s) :
RingHom.ker ((Polynomial.aeval s).toRingHom : R[X] →+* S) =
Ideal.span ({minpoly R s} : Set R[X]) := by |
ext p
simp_rw [RingHom.mem_ker, AlgHom.toRingHom_eq_coe, AlgHom.coe_toRingHom,
isIntegrallyClosed_dvd_iff hs, ← Ideal.mem_span_singleton]
| false |
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Hom.Lattice
#align_import order.hom.complete_lattice from "leanprover-community/mathlib"@"9d684a893c52e1d6692a504a118bfccbae04feeb"
open Function OrderDual Set
variable {F α β γ δ : Type*} {ι : Sort*} {κ : ι → Sort*}
-- Porting note: mathport made this & sInf... | Mathlib/Order/Hom/CompleteLattice.lean | 134 | 135 | theorem map_iSup₂ [SupSet α] [SupSet β] [sSupHomClass F α β] (f : F) (g : ∀ i, κ i → α) :
f (⨆ (i) (j), g i j) = ⨆ (i) (j), f (g i j) := by | simp_rw [map_iSup]
| false |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
#align_import analysis.calculus.iterated_deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open scoped Classical Topology
open Filter Asymptotics Set
variable {𝕜... | Mathlib/Analysis/Calculus/IteratedDeriv/Defs.lean | 142 | 146 | theorem contDiffOn_of_differentiableOn_deriv {n : ℕ∞}
(h : ∀ m : ℕ, (m : ℕ∞) ≤ n → DifferentiableOn 𝕜 (iteratedDerivWithin m f s) s) :
ContDiffOn 𝕜 n f s := by |
apply contDiffOn_of_differentiableOn
simpa only [iteratedFDerivWithin_eq_equiv_comp, LinearIsometryEquiv.comp_differentiableOn_iff]
| false |
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Combinatorics.SimpleGraph.AdjMatrix
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Set.Finite
#align_import combinatorics.simple_graph.strongly_regular from "leanprover-community/mathlib"@"2b35fc7bea4640cb75e477e83f32fbd5389208... | Mathlib/Combinatorics/SimpleGraph/StronglyRegular.lean | 84 | 95 | theorem IsSRGWith.card_neighborFinset_union_eq {v w : V} (h : G.IsSRGWith n k ℓ μ) :
(G.neighborFinset v ∪ G.neighborFinset w).card =
2 * k - Fintype.card (G.commonNeighbors v w) := by |
apply Nat.add_right_cancel (m := Fintype.card (G.commonNeighbors v w))
rw [Nat.sub_add_cancel, ← Set.toFinset_card]
-- Porting note: Set.toFinset_inter needs workaround to use unification to solve for one of the
-- instance arguments:
· simp [commonNeighbors, @Set.toFinset_inter _ _ _ _ _ _ (_),
← neig... | false |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.well_known from "leanprover-community/mathlib"@"8199f6717c150a7fe91c4534175f4cf99725978f"
namespace PowerS... | Mathlib/RingTheory/PowerSeries/WellKnown.lean | 187 | 189 | theorem coeff_sin_bit1 : coeff A (bit1 n) (sin A) = (-1) ^ n * coeff A (bit1 n) (exp A) := by |
rw [sin, coeff_mk, if_neg n.not_even_bit1, Nat.bit1_div_two, ← mul_one_div, map_mul, map_pow,
map_neg, map_one, coeff_exp]
| 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 | 88 | 89 | theorem image_Ioo (e : α ≃o β) (a b : α) : e '' Ioo a b = Ioo (e a) (e b) := by |
rw [e.image_eq_preimage, e.symm.preimage_Ioo, e.symm_symm]
| false |
import Mathlib.Data.Finset.Card
#align_import data.finset.prod from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
assert_not_exists MonoidWithZero
open Multiset
variable {α β γ : Type*}
namespace Finset
section Prod
variable {s s' : Finset α} {t t' : Finset β} {a : α} {b : β}
... | Mathlib/Data/Finset/Prod.lean | 76 | 78 | theorem product_image_fst [DecidableEq α] (ht : t.Nonempty) : (s ×ˢ t).image Prod.fst = s := by |
ext i
simp [mem_image, ht.exists_mem]
| false |
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 | 259 | 266 | theorem IsBigOWith.rpow (h : IsBigOWith c l f g) (hc : 0 ≤ c) (hr : 0 ≤ r) (hg : 0 ≤ᶠ[l] g) :
IsBigOWith (c ^ r) l (fun x => f x ^ r) fun x => g x ^ r := by |
apply IsBigOWith.of_bound
filter_upwards [hg, h.bound] with x hgx hx
calc
|f x ^ r| ≤ |f x| ^ r := abs_rpow_le_abs_rpow _ _
_ ≤ (c * |g x|) ^ r := rpow_le_rpow (abs_nonneg _) hx hr
_ = c ^ r * |g x ^ r| := by rw [mul_rpow hc (abs_nonneg _), abs_rpow_of_nonneg hgx]
| false |
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.NormedSpace.Dual
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.function.ae_eq_of_integral from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284... | Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean | 74 | 106 | theorem ae_eq_zero_of_forall_dual_of_isSeparable [NormedAddCommGroup E] [NormedSpace 𝕜 E]
{t : Set E} (ht : TopologicalSpace.IsSeparable t) {f : α → E}
(hf : ∀ c : Dual 𝕜 E, (fun x => ⟪f x, c⟫) =ᵐ[μ] 0) (h't : ∀ᵐ x ∂μ, f x ∈ t) : f =ᵐ[μ] 0 := by |
rcases ht with ⟨d, d_count, hd⟩
haveI : Encodable d := d_count.toEncodable
have : ∀ x : d, ∃ g : E →L[𝕜] 𝕜, ‖g‖ ≤ 1 ∧ g x = ‖(x : E)‖ :=
fun x => exists_dual_vector'' 𝕜 (x : E)
choose s hs using this
have A : ∀ a : E, a ∈ t → (∀ x, ⟪a, s x⟫ = (0 : 𝕜)) → a = 0 := by
intro a hat ha
contrapose! ... | false |
import Mathlib.Algebra.MonoidAlgebra.Division
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Order.Interval.Finset.Nat
#align_import data.polynomial.inductions from "leanprover-community/mathlib"@"57e09a1296bf... | Mathlib/Algebra/Polynomial/Inductions.lean | 116 | 117 | theorem divX_C_mul_X_pow : divX (C a * X ^ n) = if n = 0 then 0 else C a * X ^ (n - 1) := by |
simp only [divX_C_mul, divX_X_pow, mul_ite, mul_zero]
| false |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 118 | 119 | theorem laverage_eq (f : α → ℝ≥0∞) : ⨍⁻ x, f x ∂μ = (∫⁻ x, f x ∂μ) / μ univ := by |
rw [laverage_eq', lintegral_smul_measure, ENNReal.div_eq_inv_mul]
| false |
import Mathlib.Algebra.Lie.Nilpotent
import Mathlib.Algebra.Lie.Normalizer
#align_import algebra.lie.engel from "leanprover-community/mathlib"@"210657c4ea4a4a7b234392f70a3a2a83346dfa90"
universe u₁ u₂ u₃ u₄
variable {R : Type u₁} {L : Type u₂} {L₂ : Type u₃} {M : Type u₄}
variable [CommRing R] [LieRing L] [LieAl... | Mathlib/Algebra/Lie/Engel.lean | 173 | 183 | theorem Function.Surjective.isEngelian {f : L →ₗ⁅R⁆ L₂} (hf : Function.Surjective f)
(h : LieAlgebra.IsEngelian.{u₁, u₂, u₄} R L) : LieAlgebra.IsEngelian.{u₁, u₃, u₄} R L₂ := by |
intro M _i1 _i2 _i3 _i4 h'
letI : LieRingModule L M := LieRingModule.compLieHom M f
letI : LieModule R L M := compLieHom M f
have hnp : ∀ x, IsNilpotent (toEnd R L M x) := fun x => h' (f x)
have surj_id : Function.Surjective (LinearMap.id : M →ₗ[R] M) := Function.surjective_id
haveI : LieModule.IsNilpotent... | false |
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Tactic.FinCases
#align_import linear_algebra.affine_space.combination from ... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 796 | 797 | theorem sum_centroidWeights_eq_one_of_cast_card_ne_zero (h : (card s : k) ≠ 0) :
∑ i ∈ s, s.centroidWeights k i = 1 := by | simp [h]
| false |
import Mathlib.Data.Finite.Card
import Mathlib.GroupTheory.Commutator
import Mathlib.GroupTheory.Finiteness
#align_import group_theory.abelianization from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
universe u v w
-- Let G be a group.
variable (G : Type u) [Group G]
open Subgroup (... | Mathlib/GroupTheory/Abelianization.lean | 71 | 79 | theorem commutator_centralizer_commutator_le_center :
⁅centralizer (commutator G : Set G), centralizer (commutator G)⁆ ≤ Subgroup.center G := by |
rw [← Subgroup.centralizer_univ, ← Subgroup.coe_top, ←
Subgroup.commutator_eq_bot_iff_le_centralizer]
suffices ⁅⁅⊤, centralizer (commutator G : Set G)⁆, centralizer (commutator G : Set G)⁆ = ⊥ by
refine Subgroup.commutator_commutator_eq_bot_of_rotate ?_ this
rwa [Subgroup.commutator_comm (centralizer (... | false |
import Mathlib.RingTheory.Ideal.Maps
#align_import ring_theory.ideal.prod from "leanprover-community/mathlib"@"052f6013363326d50cb99c6939814a4b8eb7b301"
universe u v
variable {R : Type u} {S : Type v} [Semiring R] [Semiring S] (I I' : Ideal R) (J J' : Ideal S)
namespace Ideal
def prod : Ideal (R × S) where
... | Mathlib/RingTheory/Ideal/Prod.lean | 72 | 78 | theorem map_snd_prod (I : Ideal R) (J : Ideal S) : map (RingHom.snd R S) (prod I J) = J := by |
ext x
rw [mem_map_iff_of_surjective (RingHom.snd R S) Prod.snd_surjective]
exact
⟨by
rintro ⟨x, ⟨h, rfl⟩⟩
exact h.2, fun h => ⟨⟨0, x⟩, ⟨⟨Ideal.zero_mem _, h⟩, rfl⟩⟩⟩
| false |
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.Bicategory.Coherence
namespace CategoryTheory
namespace Bicategory
open Category
open scoped Bicategory
open Mathlib.Tactic.BicategoryCoherence (bicategoricalComp bicategoricalIsoComp)
universe w v u
variable {B : Type u} [Bicategory... | Mathlib/CategoryTheory/Bicategory/Adjunction.lean | 151 | 164 | theorem comp_right_triangle_aux (adj₁ : f₁ ⊣ g₁) (adj₂ : f₂ ⊣ g₂) :
rightZigzag (compUnit adj₁ adj₂) (compCounit adj₁ adj₂) = (ρ_ _).hom ≫ (λ_ _).inv := by |
calc
_ = 𝟙 _ ⊗≫
(g₂ ≫ g₁) ◁ adj₁.unit ⊗≫
g₂ ◁ ((g₁ ≫ f₁) ◁ adj₂.unit ≫ adj₁.counit ▷ (f₂ ≫ g₂)) ▷ g₁ ⊗≫
adj₂.counit ▷ (g₂ ≫ g₁) ⊗≫ 𝟙 _ := by
simp [bicategoricalComp]; coherence
_ = 𝟙 _ ⊗≫
g₂ ◁ (rightZigzag adj₁.unit adj₁.counit) ⊗≫
(rightZigz... | false |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Tactic.NormNum.NatFib
import Mathlib.Tactic.NormNum.Prime
#align_import data.real.golden_ratio from "leanprover-community/mathlib"@"2196ab363eb097c008d449712... | Mathlib/Data/Real/GoldenRatio.lean | 64 | 66 | theorem goldConj_mul_gold : ψ * φ = -1 := by |
rw [mul_comm]
exact gold_mul_goldConj
| false |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncompu... | Mathlib/FieldTheory/RatFunc/Basic.lean | 145 | 147 | theorem ofFractionRing_mul (p q : FractionRing K[X]) :
ofFractionRing (p * q) = ofFractionRing p * ofFractionRing q := by |
simp only [Mul.mul, HMul.hMul, RatFunc.mul]
| false |
import Mathlib.CategoryTheory.Sites.Whiskering
import Mathlib.CategoryTheory.Sites.Plus
#align_import category_theory.sites.compatible_plus from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
namespace CategoryTheory.GrothendieckTopology
open CategoryTheory Limits... | Mathlib/CategoryTheory/Sites/CompatiblePlus.lean | 61 | 66 | theorem diagramCompIso_hom_ι (X : C) (W : (J.Cover X)ᵒᵖ) (i : W.unop.Arrow) :
(J.diagramCompIso F P X).hom.app W ≫ Multiequalizer.ι ((unop W).index (P ⋙ F)) i =
F.map (Multiequalizer.ι _ _) := by |
delta diagramCompIso
dsimp
simp
| false |
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "leanprover-community/mathlib"@"996b0ff959da753a555053a480f36e5f264d4207"
variable {α β γ : Type*} {ι : Sort*} {π : α → Type*}
open Equiv Equiv.Perm Function
namespace Set
section Order
variable {s : Se... | Mathlib/Data/Set/Function.lean | 274 | 278 | theorem _root_.StrictMonoOn.congr (h₁ : StrictMonoOn f₁ s) (h : s.EqOn f₁ f₂) :
StrictMonoOn f₂ s := by |
intro a ha b hb hab
rw [← h ha, ← h hb]
exact h₁ ha hb hab
| false |
import Mathlib.Algebra.Module.Equiv
import Mathlib.Algebra.Module.Hom
import Mathlib.Algebra.Module.Prod
import Mathlib.Algebra.Module.Submodule.Range
import Mathlib.Data.Set.Finite
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Tactic.Abel
#align_import linear_algebra.basic from "leanprover-c... | Mathlib/LinearAlgebra/Basic.lean | 169 | 173 | theorem ofSubmodule'_toLinearMap [Module R M] [Module R₂ M₂] (f : M ≃ₛₗ[σ₁₂] M₂)
(U : Submodule R₂ M₂) :
(f.ofSubmodule' U).toLinearMap = (f.toLinearMap.domRestrict _).codRestrict _ Subtype.prop := by |
ext
rfl
| 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 | 369 | 370 | theorem rank_tensorProduct' :
Module.rank S (M ⊗[S] M₁) = Module.rank S M * Module.rank S M₁ := by | 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 | 256 | 256 | theorem descPochhammer_one : descPochhammer R 1 = X := by | simp [descPochhammer]
| false |
import Mathlib.NumberTheory.Liouville.Basic
#align_import number_theory.liouville.liouville_number from "leanprover-community/mathlib"@"04e80bb7e8510958cd9aacd32fe2dc147af0b9f1"
noncomputable section
open scoped Nat
open Real Finset
def liouvilleNumber (m : ℝ) : ℝ :=
∑' i : ℕ, 1 / m ^ i !
#align liouville_n... | Mathlib/NumberTheory/Liouville/LiouvilleNumber.lean | 137 | 160 | theorem aux_calc (n : ℕ) {m : ℝ} (hm : 2 ≤ m) :
(1 - 1 / m)⁻¹ * (1 / m ^ (n + 1)!) ≤ 1 / (m ^ n !) ^ n :=
calc
(1 - 1 / m)⁻¹ * (1 / m ^ (n + 1)!) ≤ 2 * (1 / m ^ (n + 1)!) :=
-- the second factors coincide (and are non-negative),
-- the first factors satisfy the inequality `sub_one_div_inv_le_two`
... |
-- [NB: in this block, I do not follow the brace convention for subgoals -- I wait until
-- I solve all extraneous goals at once with `exact pow_pos (zero_lt_two.trans_le hm) _`.]
-- Clear denominators and massage*
apply (div_le_div_iff _ _).mpr
focus
conv_rhs => rw [one_mul, mul_... | false |
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.Ideal.Prod
import Mathlib.RingTheory.Ideal.MinimalPrime
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.Topology.Sets.Closeds
import Mathlib.Topology.Sober
#a... | Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean | 172 | 174 | theorem mem_vanishingIdeal (t : Set (PrimeSpectrum R)) (f : R) :
f ∈ vanishingIdeal t ↔ ∀ x : PrimeSpectrum R, x ∈ t → f ∈ x.asIdeal := by |
rw [← SetLike.mem_coe, coe_vanishingIdeal, Set.mem_setOf_eq]
| false |
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Real.Archimedean
#align_import number_theory.zsqrtd.gaussian_int from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
open Zsqrtd Complex
open sc... | Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean | 135 | 137 | theorem toComplex_star (x : ℤ[i]) : ((star x : ℤ[i]) : ℂ) = conj (x : ℂ) := by |
rw [toComplex_def₂, toComplex_def₂]
exact congr_arg₂ _ rfl (Int.cast_neg _)
| 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 | 68 | 69 | theorem tsub_tsub_tsub_cancel_right (h : c ≤ b) : a - c - (b - c) = a - b := by |
rw [tsub_tsub, add_tsub_cancel_of_le h]
| false |
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
#align_import category_theory.limits.preserves.shapes.images from "leanprover-community/mathlib"@"fc78e3c190c72a109699385da6be2725e88df841"
noncomputable section
namespace CategoryTheory
namespace Prese... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Images.lean | 52 | 53 | theorem factorThruImage_comp_hom {X Y : A} (f : X ⟶ Y) :
factorThruImage (L.map f) ≫ (iso L f).hom = L.map (factorThruImage f) := by | simp
| false |
import Mathlib.Data.Finsupp.Defs
#align_import data.finsupp.ne_locus from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable {α M N P : Type*}
namespace Finsupp
variable [DecidableEq α]
section NHasZero
variable [DecidableEq N] [Zero N] (f g : α →₀ N)
def neLocus (f g : α →₀ ... | Mathlib/Data/Finsupp/NeLocus.lean | 42 | 44 | theorem mem_neLocus {f g : α →₀ N} {a : α} : a ∈ f.neLocus g ↔ f a ≠ g a := by |
simpa only [neLocus, Finset.mem_filter, Finset.mem_union, mem_support_iff,
and_iff_right_iff_imp] using Ne.ne_or_ne _
| false |
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.LinearAlgebra.Projection
import Mathlib.Order.JordanHolder
import Mathlib.Order.CompactlyGenerated.Intervals
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import ring_theory.simple_module from "leanprover-community/mathlib"@"cce7f68a7eaadadf74c82bbac207... | Mathlib/RingTheory/SimpleModule.lean | 86 | 88 | theorem isSimpleModule_iff_isAtom : IsSimpleModule R m ↔ IsAtom m := by |
rw [← Set.isSimpleOrder_Iic_iff_isAtom]
exact m.mapIic.isSimpleOrder_iff
| false |
import Mathlib.CategoryTheory.Subobject.Limits
#align_import algebra.homology.image_to_kernel from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a148bf9ff"
universe v u w
open CategoryTheory CategoryTheory.Limits
variable {ι : Type*}
variable {V : Type u} [Category.{v} V] [HasZeroMorphisms V]
o... | Mathlib/Algebra/Homology/ImageToKernel.lean | 119 | 123 | theorem imageToKernel_comp_left {Z : V} (h : Z ⟶ A) (w : f ≫ g = 0) :
imageToKernel (h ≫ f) g (by simp [w]) =
Subobject.ofLE _ _ (imageSubobject_comp_le h f) ≫ imageToKernel f g w := by |
ext
simp
| false |
import Mathlib.RingTheory.IntegrallyClosed
import Mathlib.RingTheory.Trace
import Mathlib.RingTheory.Norm
#align_import ring_theory.discriminant from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
universe u v w z
open scoped Matrix
open Matrix FiniteDimensional Fintype Polynomial Fin... | Mathlib/RingTheory/Discriminant.lean | 161 | 167 | theorem discr_powerBasis_eq_prod (e : Fin pb.dim ≃ (L →ₐ[K] E)) [IsSeparable K L] :
algebraMap K E (discr K pb.basis) =
∏ i : Fin pb.dim, ∏ j ∈ Ioi i, (e j pb.gen - e i pb.gen) ^ 2 := by |
rw [discr_eq_det_embeddingsMatrixReindex_pow_two K E pb.basis e,
embeddingsMatrixReindex_eq_vandermonde, det_transpose, det_vandermonde, ← prod_pow]
congr; ext i
rw [← prod_pow]
| false |
import Mathlib.Algebra.Field.Subfield
import Mathlib.Topology.Algebra.Field
import Mathlib.Topology.Algebra.UniformRing
#align_import topology.algebra.uniform_field from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open uniformity Topology
... | Mathlib/Topology/Algebra/UniformField.lean | 126 | 153 | theorem mul_hatInv_cancel {x : hat K} (x_ne : x ≠ 0) : x * hatInv x = 1 := by |
haveI : T1Space (hat K) := T2Space.t1Space
let f := fun x : hat K => x * hatInv x
let c := (fun (x : K) => (x : hat K))
change f x = 1
have cont : ContinuousAt f x := by
letI : TopologicalSpace (hat K × hat K) := instTopologicalSpaceProd
have : ContinuousAt (fun y : hat K => ((y, hatInv y) : hat K × ... | false |
import Mathlib.LinearAlgebra.Dimension.Finrank
import Mathlib.LinearAlgebra.InvariantBasisNumber
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v w w'
variable {R : Type u} {M : Type v} [Ring R] [AddCommGroup... | Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 244 | 258 | theorem linearIndependent_le_infinite_basis {ι : Type w} (b : Basis ι R M) [Infinite ι] {κ : Type w}
(v : κ → M) (i : LinearIndependent R v) : #κ ≤ #ι := by |
classical
by_contra h
rw [not_le, ← Cardinal.mk_finset_of_infinite ι] at h
let Φ := fun k : κ => (b.repr (v k)).support
obtain ⟨s, w : Infinite ↑(Φ ⁻¹' {s})⟩ := Cardinal.exists_infinite_fiber Φ h (by infer_instance)
let v' := fun k : Φ ⁻¹' {s} => v k
have i' : LinearIndependent R v' := i.comp _ Subtype.v... | false |
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic
#align_import measure_theory.function.conditional_expectation.indicator from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open TopologicalSpace MeasureTheory.Lp Filter ContinuousLinearMap
open s... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Indicator.lean | 115 | 140 | theorem condexp_restrict_ae_eq_restrict (hm : m ≤ m0) [SigmaFinite (μ.trim hm)]
(hs_m : MeasurableSet[m] s) (hf_int : Integrable f μ) :
(μ.restrict s)[f|m] =ᵐ[μ.restrict s] μ[f|m] := by |
have : SigmaFinite ((μ.restrict s).trim hm) := by rw [← restrict_trim hm _ hs_m]; infer_instance
rw [ae_eq_restrict_iff_indicator_ae_eq (hm _ hs_m)]
refine EventuallyEq.trans ?_ (condexp_indicator hf_int hs_m)
refine ae_eq_condexp_of_forall_setIntegral_eq hm (hf_int.indicator (hm _ hs_m)) ?_ ?_ ?_
· intro t ... | 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 | 134 | 140 | theorem vars_pow (φ : MvPolynomial σ R) (n : ℕ) : (φ ^ n).vars ⊆ φ.vars := by |
classical
induction' n with n ih
· simp
· rw [pow_succ']
apply Finset.Subset.trans (vars_mul _ _)
exact Finset.union_subset (Finset.Subset.refl _) ih
| false |
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Real.Cardinality
#align_import data.complex.cardinality from "leanprover-community/mathlib"@"1c4e18434eeb5546b212e830b2b39de6a83c473c"
-- Porting note: the lemmas `mk_complex` and `mk_univ_complex` should be in the namespace `Cardinal`
-- like their real counter... | Mathlib/Data/Complex/Cardinality.lean | 35 | 37 | theorem not_countable_complex : ¬(Set.univ : Set ℂ).Countable := by |
rw [← le_aleph0_iff_set_countable, not_le, mk_univ_complex]
apply cantor
| false |
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Algebra.Order.Group.OrderIso
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.Order.UpperLower.Basic
#align_import algebra.order.upper_lower from "leanprover-community/mathlib"@"c0c52abb75074ed8b73a948341f50521fbf43b4c"
open Function Set
open Pointw... | Mathlib/Algebra/Order/UpperLower.lean | 104 | 106 | theorem IsUpperSet.div_right (hs : IsUpperSet s) : IsUpperSet (s / t) := by |
rw [div_eq_mul_inv]
exact hs.mul_right
| false |
import Mathlib.Analysis.NormedSpace.AddTorsorBases
#align_import analysis.convex.intrinsic from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open AffineSubspace Set
open scoped Pointwise
variable {𝕜 V W Q P : Type*}
section AddTorsor
variable (𝕜) [Ring 𝕜] [AddCommGroup V] [Modu... | Mathlib/Analysis/Convex/Intrinsic.lean | 112 | 112 | theorem intrinsicInterior_empty : intrinsicInterior 𝕜 (∅ : Set P) = ∅ := by | simp [intrinsicInterior]
| 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 | 151 | 154 | theorem mem_thickening_iff {E : Set X} {x : X} : x ∈ thickening δ E ↔ ∃ z ∈ E, dist x z < δ := by |
have key_iff : ∀ z : X, edist x z < ENNReal.ofReal δ ↔ dist x z < δ := fun z ↦ by
rw [dist_edist, lt_ofReal_iff_toReal_lt (edist_ne_top _ _)]
simp_rw [mem_thickening_iff_exists_edist_lt, key_iff]
| false |
import Mathlib.Order.Interval.Multiset
#align_import data.nat.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
-- TODO
-- assert_not_exists Ring
open Finset Nat
variable (a b c : ℕ)
namespace Nat
instance instLocallyFiniteOrder : LocallyFiniteOrder ℕ where
finsetIcc a b... | Mathlib/Order/Interval/Finset/Nat.lean | 132 | 133 | theorem card_fintypeIoo : Fintype.card (Set.Ioo a b) = b - a - 1 := by |
rw [Fintype.card_ofFinset, card_Ioo]
| false |
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Cases
import Mathlib.Tactic.SimpRw
import Mathlib.Tactic.SplitIfs
#align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530a... | Mathlib/Algebra/Group/Basic.lean | 450 | 451 | theorem mul_one_div (x y : G) : x * (1 / y) = x / y := by |
rw [div_eq_mul_inv, one_mul, div_eq_mul_inv]
| false |
import Mathlib.Data.Finset.Image
#align_import data.finset.card from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
assert_not_exists MonoidWithZero
-- TODO: After a lot more work,
-- assert_not_exists OrderedCommMonoid
open Function Multiset Nat
variable {α β R : Type*}
namespace Fin... | Mathlib/Data/Finset/Card.lean | 91 | 94 | theorem card_singleton_inter [DecidableEq α] : ({a} ∩ s).card ≤ 1 := by |
cases' Finset.decidableMem a s with h h
· simp [Finset.singleton_inter_of_not_mem h]
· simp [Finset.singleton_inter_of_mem h]
| false |
import Mathlib.Data.List.Count
import Mathlib.Data.List.Dedup
import Mathlib.Data.List.InsertNth
import Mathlib.Data.List.Lattice
import Mathlib.Data.List.Permutation
import Mathlib.Data.Nat.Factorial.Basic
#align_import data.list.perm from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
... | Mathlib/Data/List/Perm.lean | 167 | 175 | theorem forall₂_comp_perm_eq_perm_comp_forall₂ : Forall₂ r ∘r Perm = Perm ∘r Forall₂ r := by |
funext l₁ l₃; apply propext
constructor
· intro h
rcases h with ⟨l₂, h₁₂, h₂₃⟩
have : Forall₂ (flip r) l₂ l₁ := h₁₂.flip
rcases perm_comp_forall₂ h₂₃.symm this with ⟨l', h₁, h₂⟩
exact ⟨l', h₂.symm, h₁.flip⟩
· exact fun ⟨l₂, h₁₂, h₂₃⟩ => perm_comp_forall₂ h₁₂ h₂₃
| false |
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Set.Pointwise.Iterate
import Mathlib.Dynamics.Ergodic.Ergodic
import Mathlib.MeasureTheory.Covering.DensityTheorem
import Mathlib.MeasureTheory.Group.AddCircle
import Mathlib.MeasureTheory.Measure.Haar.Unique
#align_import dynamics.ergodic.add_circle from "lea... | Mathlib/Dynamics/Ergodic/AddCircle.lean | 104 | 120 | theorem ergodic_zsmul {n : ℤ} (hn : 1 < |n|) : Ergodic fun y : AddCircle T => n • y :=
{ measurePreserving_zsmul volume (abs_pos.mp <| lt_trans zero_lt_one hn) with
ae_empty_or_univ := fun s hs hs' => by
let u : ℕ → AddCircle T := fun j => ↑((↑1 : ℝ) / ↑(n.natAbs ^ j) * T)
replace hn : 1 < n.natAbs :=... | rwa [Int.abs_eq_natAbs, Nat.one_lt_cast] at hn
have hu₀ : ∀ j, addOrderOf (u j) = n.natAbs ^ j := fun j => by
convert addOrderOf_div_of_gcd_eq_one (p := T) (m := 1)
(pow_pos (pos_of_gt hn) j) (gcd_one_left _)
norm_cast
have hnu : ∀ j, n ^ j • u j = 0 := fun j => by
rw [← a... | false |
import Mathlib.Init.Function
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
universe u
open Function
namespace Option
variable {α β γ δ : Type*} {f : α → β → γ} {a : Option α} {b : Option β} {c : Option γ}
def map₂ (f : α → β → γ) (a : Option α) ... | Mathlib/Data/Option/NAry.lean | 130 | 131 | theorem map₂_comm {g : β → α → γ} (h_comm : ∀ a b, f a b = g b a) : map₂ f a b = map₂ g b a := by |
cases a <;> cases b <;> simp [h_comm]
| false |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Order.MonotoneContinuity
#align_import dynamics.circle.rotation_number.translation_number from "leanprover-... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 167 | 167 | theorem map_one_add (x : ℝ) : f (1 + x) = 1 + f x := by | rw [add_comm, map_add_one, add_comm 1]
| false |
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.NormedSpace.WithLp
open Real Set Filter RCLike Bornology Uniformity Topology NNReal ENNReal
noncomputable section
variable (p : ℝ≥0∞) (𝕜 α β : Type*)
namespace WithLp
section DistNorm
section EDist
variable [EDist α] [EDist β]
open scope... | Mathlib/Analysis/NormedSpace/ProdLp.lean | 161 | 164 | theorem prod_edist_eq_card (f g : WithLp 0 (α × β)) :
edist f g =
(if edist f.fst g.fst = 0 then 0 else 1) + (if edist f.snd g.snd = 0 then 0 else 1) := by |
convert if_pos rfl
| false |
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing
import Mathlib.CategoryTheory.Adjunction.Evaluation
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Preserves.Filtered
import Mathlib.Cat... | Mathlib/Topology/Sheaves/Stalks.lean | 113 | 114 | theorem germ_res_apply (F : X.Presheaf C) {U V : Opens X} (i : U ⟶ V) (x : U) [ConcreteCategory C]
(s) : germ F x (F.map i.op s) = germ F (i x) s := by | rw [← comp_apply, germ_res]
| false |
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
#align_import data.finset.locally_finite from "leanprover-community/mathlib"@"442a83d738cb208d3600056c489be16900ba701d"
assert_not_exists MonoidWithZero
assert_not_exists Finset.sum
open Function OrderDual
open FinsetInterval
variable {ι α : T... | Mathlib/Order/Interval/Finset/Basic.lean | 73 | 74 | theorem nonempty_Ioo [DenselyOrdered α] : (Ioo a b).Nonempty ↔ a < b := by |
rw [← coe_nonempty, coe_Ioo, Set.nonempty_Ioo]
| false |
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Data.List.Chain
#align_import data.bool.count from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
namespace List
@[simp]
theorem count_not_add_count (l : List Bool) (b : Bool) : count (!b) l + count b l = length l := by
-- Porting ... | Mathlib/Data/Bool/Count.lean | 60 | 71 | theorem count_not_eq_count (hl : Chain' (· ≠ ·) l) (h2 : Even (length l)) (b : Bool) :
count (!b) l = count b l := by |
cases' l with x l
· rfl
rw [length_cons, Nat.even_add_one, Nat.not_even_iff] at h2
suffices count (!x) (x :: l) = count x (x :: l) by
-- Porting note: old proof is
-- cases b <;> cases x <;> try exact this;
cases b <;> cases x <;>
revert this <;> simp only [Bool.not_false, Bool.not_true] <;> in... | false |
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv
import Mathlib.Analysis.Calculus.FDeriv.Extend
import Mathlib.Analysis.Calculus.Deriv.Prod
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
#... | Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean | 338 | 340 | theorem hasStrictDerivAt_const_rpow {a : ℝ} (ha : 0 < a) (x : ℝ) :
HasStrictDerivAt (fun x => a ^ x) (a ^ x * log a) x := by |
simpa using (hasStrictDerivAt_const _ _).rpow (hasStrictDerivAt_id x) ha
| false |
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Part
import Mathlib.Tactic.NormNum
#align_import data.nat.part_enat from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
open Part hiding some
def PartENat : Type :=
Part ℕ
#align part_enat ... | Mathlib/Data/Nat/PartENat.lean | 179 | 180 | theorem natCast_get {x : PartENat} (h : x.Dom) : (x.get h : PartENat) = x := by |
exact Part.ext' (iff_of_true trivial h) fun _ _ => rfl
| false |
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.Order.Interval.Finset.Nat
#align_import data.nat.factorization.basic from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
... | Mathlib/Data/Nat/Factorization/Basic.lean | 116 | 116 | theorem factorization_zero : factorization 0 = 0 := by | ext; simp [factorization]
| false |
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