Context stringlengths 57 92.3k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import measure_theory.measure.haar.of_basis from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
open Set TopologicalSpace MeasureTheory MeasureTheory.Measure FiniteDimensional
open sco... | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean | 232 | 260 | theorem Basis.prod_parallelepiped (v : Basis ι ℝ E) (w : Basis ι' ℝ F) :
(v.prod w).parallelepiped = v.parallelepiped.prod w.parallelepiped := by |
ext x
simp only [Basis.coe_parallelepiped, TopologicalSpace.PositiveCompacts.coe_prod, Set.mem_prod,
mem_parallelepiped_iff]
constructor
· intro h
rcases h with ⟨t, ht1, ht2⟩
constructor
· use t ∘ Sum.inl
constructor
· exact ⟨(ht1.1 <| Sum.inl ·), (ht1.2 <| Sum.inl ·)⟩
simp [h... |
import Mathlib.RingTheory.WittVector.InitTail
#align_import ring_theory.witt_vector.truncated from "leanprover-community/mathlib"@"acbe099ced8be9c9754d62860110295cde0d7181"
open Function (Injective Surjective)
noncomputable section
variable {p : ℕ} [hp : Fact p.Prime] (n : ℕ) (R : Type*)
local notation "𝕎" =>... | Mathlib/RingTheory/WittVector/Truncated.lean | 502 | 503 | theorem truncate_comp_lift : (WittVector.truncate n).comp (lift _ f_compat) = f n := by |
ext1; rw [RingHom.comp_apply, truncate_lift]
|
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subsemigroup.Operations
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Data.Set.Image
import Mathlib.Order.Atoms
import Mathlib.Tactic.ApplyFun
#align_import g... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 923 | 925 | theorem bot_or_nontrivial (H : Subgroup G) : H = ⊥ ∨ Nontrivial H := by |
have := nontrivial_iff_ne_bot H
tauto
|
import Mathlib.Probability.Notation
import Mathlib.Probability.Integration
import Mathlib.MeasureTheory.Function.L2Space
#align_import probability.variance from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open MeasureTheory Filter Finset
noncomputable section
open scoped MeasureThe... | Mathlib/Probability/Variance.lean | 194 | 198 | theorem variance_smul' {A : Type*} [CommSemiring A] [Algebra A ℝ] (c : A) (X : Ω → ℝ)
(μ : Measure Ω) : variance (c • X) μ = c ^ 2 • variance X μ := by |
convert variance_smul (algebraMap A ℝ c) X μ using 1
· congr; simp only [algebraMap_smul]
· simp only [Algebra.smul_def, map_pow]
|
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.BigOperators.Group.Multiset
import Mathlib.Tactic.NormNum.Basic
import Mathlib.Tactic.Positivity.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
open Function
... | Mathlib/Algebra/Order/BigOperators/Group/Finset.lean | 531 | 541 | theorem prod_eq_prod_iff_of_le {f g : ι → M} (h : ∀ i ∈ s, f i ≤ g i) :
((∏ i ∈ s, f i) = ∏ i ∈ s, g i) ↔ ∀ i ∈ s, f i = g i := by |
classical
revert h
refine Finset.induction_on s (fun _ ↦ ⟨fun _ _ h ↦ False.elim (Finset.not_mem_empty _ h),
fun _ ↦ rfl⟩) fun a s ha ih H ↦ ?_
specialize ih fun i ↦ H i ∘ Finset.mem_insert_of_mem
rw [Finset.prod_insert ha, Finset.prod_insert ha, Finset.forall_mem_insert, ← ih]
exact
... |
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.Measure.Count
import Mathlib.Topology.IndicatorConstPointwise
import Mathlib.MeasureTheory.Constructions.BorelSpace.Real
#align_import m... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 266 | 280 | theorem lintegral_mono_ae {f g : α → ℝ≥0∞} (h : ∀ᵐ a ∂μ, f a ≤ g a) :
∫⁻ a, f a ∂μ ≤ ∫⁻ a, g a ∂μ := by |
rcases exists_measurable_superset_of_null h with ⟨t, hts, ht, ht0⟩
have : ∀ᵐ x ∂μ, x ∉ t := measure_zero_iff_ae_nmem.1 ht0
rw [lintegral, lintegral]
refine iSup_le fun s => iSup_le fun hfs => le_iSup_of_le (s.restrict tᶜ) <| le_iSup_of_le ?_ ?_
· intro a
by_cases h : a ∈ t <;>
simp only [restrict_a... |
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Order.Bounds.OrderIso
import Mathlib.Tactic.Positivity.Core
#align_import algebra.order.field.basic from "leanprover-community/mathlib"@"8477... | Mathlib/Algebra/Order/Field/Basic.lean | 379 | 379 | theorem one_div_lt (ha : 0 < a) (hb : 0 < b) : 1 / a < b ↔ 1 / b < a := by | simpa using inv_lt ha hb
|
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Polynomial.IntegralNormalization
#align_import ring_theory.algebraic from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
universe u v w
open scoped Classical
open Polynomi... | Mathlib/RingTheory/Algebraic.lean | 128 | 131 | theorem isAlgebraic_rat (R : Type u) {A : Type v} [DivisionRing A] [Field R] [Algebra R A] (n : ℚ) :
IsAlgebraic R (n : A) := by |
rw [← map_ratCast (algebraMap R A)]
exact isAlgebraic_algebraMap (Rat.cast n)
|
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.Factors
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Multiplicity
#align_import ring_theory.unique_factorization_domain from "leanprover-community/mathli... | Mathlib/RingTheory/UniqueFactorizationDomain.lean | 1,757 | 1,759 | theorem count_self [Nontrivial α] [DecidableEq (Associates α)] {p : Associates α}
(hp : Irreducible p) : p.count p.factors = 1 := by |
simp [factors_self hp, Associates.count_some hp]
|
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Algebra.Group.TypeTags
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.Algebra.Ring.Nat
#align_import data.nat.cast.basic from "leanprover-community/mathlib"@"acebd8d49928f6ed8920e502a6c90674e75bd441"
assert_not_e... | Mathlib/Data/Nat/Cast/Basic.lean | 199 | 202 | theorem NeZero.nat_of_neZero {R S} [Semiring R] [Semiring S]
{F} [FunLike F R S] [RingHomClass F R S] (f : F)
{n : ℕ} [hn : NeZero (n : S)] : NeZero (n : R) :=
.of_map (f := f) (neZero := by | simp only [map_natCast, hn])
|
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Data.Complex.Exponential
import Mathlib.Data.Complex.Module
import Mathlib.RingTheory.Polynomial.Chebyshev
#align_import analysis.special_functions.trigonometric.chebyshev from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
set_... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev.lean | 92 | 105 | theorem U_complex_cos (n : ℤ) : (U ℂ n).eval (cos θ) * sin θ = sin ((n + 1) * θ) := by |
induction n using Polynomial.Chebyshev.induct with
| zero => simp
| one => simp [one_add_one_eq_two, sin_two_mul]; ring
| add_two n ih1 ih2 =>
simp only [U_add_two, add_sub_cancel_right, eval_sub, eval_mul, eval_X, eval_ofNat, sub_mul,
mul_assoc, ih1, ih2, sub_eq_iff_eq_add, sin_add_sin]
push_cas... |
import Mathlib.Algebra.CharP.Two
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.GroupTheory.SpecificGroups.Cyclic
import Mathlib.NumberTheory.Divisors
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Tactic.Zify
#align_import rin... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 335 | 342 | theorem mk_of_lt (ζ : M) (hk : 0 < k) (h1 : ζ ^ k = 1) (h : ∀ l : ℕ, 0 < l → l < k → ζ ^ l ≠ 1) :
IsPrimitiveRoot ζ k := by |
refine ⟨h1, fun l hl => ?_⟩
suffices k.gcd l = k by exact this ▸ k.gcd_dvd_right l
rw [eq_iff_le_not_lt]
refine ⟨Nat.le_of_dvd hk (k.gcd_dvd_left l), ?_⟩
intro h'; apply h _ (Nat.gcd_pos_of_pos_left _ hk) h'
exact pow_gcd_eq_one _ h1 hl
|
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.FiniteSupport
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Set.Subsingleton
#align_import algebra.big_operators.finprod from "leanprover-community/mathlib"@"d6fad0e5bf... | Mathlib/Algebra/BigOperators/Finprod.lean | 368 | 369 | theorem finprod_mem_mulSupport (f : α → M) (a : α) : ∏ᶠ _ : f a ≠ 1, f a = f a := by |
rw [← mem_mulSupport, finprod_eq_mulIndicator_apply, mulIndicator_mulSupport]
|
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.IsometricSMul
#align_import topology.metric_space.hausdorff_distance from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
noncomputable section
open NNReal ENNReal Topology Set Filter Pointwise Bornology
u... | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 505 | 507 | theorem infEdist_ne_top (h : s.Nonempty) : infEdist x s ≠ ⊤ := by |
rcases h with ⟨y, hy⟩
exact ne_top_of_le_ne_top (edist_ne_top _ _) (infEdist_le_edist_of_mem hy)
|
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Set.Finite
#align_import combinatorics.hall.finite from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
open Finset
universe u v
namespace HallMarriageTheorem
variable {ι : Type u} {α : Type v} [DecidableEq α] {t : ι → Finset α}
v... | Mathlib/Combinatorics/Hall/Finite.lean | 226 | 243 | theorem hall_hard_inductive (ht : ∀ s : Finset ι, s.card ≤ (s.biUnion t).card) :
∃ f : ι → α, Function.Injective f ∧ ∀ x, f x ∈ t x := by |
cases nonempty_fintype ι
induction' hn : Fintype.card ι using Nat.strong_induction_on with n ih generalizing ι
rcases n with (_ | n)
· rw [Fintype.card_eq_zero_iff] at hn
exact ⟨isEmptyElim, isEmptyElim, isEmptyElim⟩
· have ih' : ∀ (ι' : Type u) [Fintype ι'] (t' : ι' → Finset α), Fintype.card ι' ≤ n →
... |
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 | 218 | 237 | theorem hasEigenvector_eigenvectorBasis (i : Fin n) :
HasEigenvector T (hT.eigenvalues hn i) (hT.eigenvectorBasis hn i) := by |
let v : E := hT.eigenvectorBasis hn i
let μ : 𝕜 :=
(hT.direct_sum_isInternal.subordinateOrthonormalBasisIndex hn i
hT.orthogonalFamily_eigenspaces').val
simp_rw [eigenvalues]
change HasEigenvector T (RCLike.re μ) v
have key : HasEigenvector T μ v := by
have H₁ : v ∈ eigenspace T μ := by
... |
import Mathlib.Logic.Pairwise
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#align_import data.set.lattice from "leanprover-community/mathlib"@"b86832321b586c6ac23ef8cdef6a7a27e42b13bd"
open Function Set
universe u
variable {α β γ : Type*} {ι ι' ι... | Mathlib/Data/Set/Lattice.lean | 896 | 897 | theorem iUnion_nonempty_self (s : Set α) : ⋃ _ : s.Nonempty, s = s := by |
rw [iUnion_nonempty_index, biUnion_self]
|
import Mathlib.Analysis.Convex.Between
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.Topology.MetricSpace.Holder
import Mathlib.Topology.MetricSpace.MetricSeparated
#align_import measure_theory.measure.hausdorff from "leanprover-communit... | Mathlib/MeasureTheory/Measure/Hausdorff.lean | 652 | 658 | theorem noAtoms_hausdorff {d : ℝ} (hd : 0 < d) : NoAtoms (hausdorffMeasure d : Measure X) := by |
refine ⟨fun x => ?_⟩
rw [← nonpos_iff_eq_zero, hausdorffMeasure_apply]
refine iSup₂_le fun ε _ => iInf₂_le_of_le (fun _ => {x}) ?_ <| iInf_le_of_le (fun _ => ?_) ?_
· exact subset_iUnion (fun _ => {x} : ℕ → Set X) 0
· simp only [EMetric.diam_singleton, zero_le]
· simp [hd]
|
import Mathlib.Algebra.Group.Hom.End
import Mathlib.Algebra.Ring.Invertible
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.GroupTheory.GroupAction.Units
#align_import algebra.module.basic from "leanprover-community/mathlib"@"30413fc89f202a090a54d78e540963ed3de0056e"
assert_n... | Mathlib/Algebra/Module/Defs.lean | 643 | 644 | theorem Int.smul_one_eq_cast {R : Type*} [NonAssocRing R] (m : ℤ) : m • (1 : R) = ↑m := by |
rw [zsmul_eq_mul, mul_one]
|
import Mathlib.GroupTheory.CoprodI
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.GroupTheory.Complement
namespace Monoid
open CoprodI Subgroup Coprod Function List
variable {ι : Type*} {G : ι → Type*} {H : Type*} {K : Type*} [Monoid K]
def PushoutI.con [∀ i, Mo... | Mathlib/GroupTheory/PushoutI.lean | 544 | 545 | theorem prod_empty : (empty : NormalWord d).prod = 1 := by |
simp [prod, empty]
|
import Mathlib.Init.Control.Combinators
import Mathlib.Init.Function
import Mathlib.Tactic.CasesM
import Mathlib.Tactic.Attr.Core
#align_import control.basic from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
universe u v w
variable {α β γ : Type u}
section Applicative
variable {F : ... | Mathlib/Control/Basic.lean | 68 | 70 | theorem map_seq (f : β → γ) (x : F (α → β)) (y : F α) :
f <$> (x <*> y) = (f ∘ ·) <$> x <*> y := by |
simp only [← pure_seq]; simp [seq_assoc]
|
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.class_group from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
variable {R K L : Type*} [CommRing R]
variable [Field K] [Field L] [DecidableEq L]
variable [Algebra R K] [Is... | Mathlib/RingTheory/ClassGroup.lean | 119 | 123 | theorem ClassGroup.mk_eq_mk {I J : (FractionalIdeal R⁰ <| FractionRing R)ˣ} :
ClassGroup.mk I = ClassGroup.mk J ↔
∃ x : (FractionRing R)ˣ, I * toPrincipalIdeal R (FractionRing R) x = J := by |
erw [QuotientGroup.mk'_eq_mk', canonicalEquiv_self, Units.map_id, Set.exists_range_iff]
rfl
|
import Mathlib.Topology.Order.MonotoneContinuity
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.EMetricSpace.Lipschitz
import Mathlib.Topology.Metrizable.Basic
import Mathlib.Topology.Order.T5
#align_import topology.instances.ennreal from "leanprove... | Mathlib/Topology/Instances/ENNReal.lean | 1,600 | 1,602 | theorem edist_le_tsum_of_edist_le_of_tendsto₀ {f : ℕ → α} (d : ℕ → ℝ≥0∞)
(hf : ∀ n, edist (f n) (f n.succ) ≤ d n) {a : α} (ha : Tendsto f atTop (𝓝 a)) :
edist (f 0) a ≤ ∑' m, d m := by | simpa using edist_le_tsum_of_edist_le_of_tendsto d hf ha 0
|
import Mathlib.Algebra.Group.Nat
import Mathlib.Algebra.Order.Sub.Canonical
import Mathlib.Data.List.Perm
import Mathlib.Data.Set.List
import Mathlib.Init.Quot
import Mathlib.Order.Hom.Basic
#align_import data.multiset.basic from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
universe v
... | Mathlib/Data/Multiset/Basic.lean | 742 | 744 | theorem nsmul_cons {s : Multiset α} (n : ℕ) (a : α) :
n • (a ::ₘ s) = n • ({a} : Multiset α) + n • s := by |
rw [← singleton_add, nsmul_add]
|
import Mathlib.Logic.Pairwise
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#align_import data.set.lattice from "leanprover-community/mathlib"@"b86832321b586c6ac23ef8cdef6a7a27e42b13bd"
open Function Set
universe u
variable {α β γ : Type*} {ι ι' ι... | Mathlib/Data/Set/Lattice.lean | 909 | 910 | theorem biInter_insert (a : α) (s : Set α) (t : α → Set β) :
⋂ x ∈ insert a s, t x = t a ∩ ⋂ x ∈ s, t x := by | simp
|
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Set.Sigma
#align_import data.finset.sigma from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Multiset
variable {ι : Type*}
namespace Finset
section SigmaLift
variable {α β γ : ι → Type*} [DecidableEq ι]
def sigm... | Mathlib/Data/Finset/Sigma.lean | 198 | 201 | theorem sigmaLift_nonempty :
(sigmaLift f a b).Nonempty ↔ ∃ h : a.1 = b.1, (f (h ▸ a.2) b.2).Nonempty := by |
simp_rw [nonempty_iff_ne_empty, sigmaLift]
split_ifs with h <;> simp [h]
|
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.RingTheory.PolynomialAlgebra
#align_import linear_algebra.matrix.charpoly.basic from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
universe u v w
namespace Matrix
open Finset Matrix Polynomial
variable... | Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean | 55 | 57 | theorem charmatrix_apply_eq : charmatrix M i i = (X : R[X]) - C (M i i) := by |
simp only [charmatrix, RingHom.mapMatrix_apply, sub_apply, scalar_apply, map_apply,
diagonal_apply_eq]
|
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.BigOperators.RingEquiv
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.Algebra.Module.Pi
import Mathlib.Algebra.Star.BigOperators... | Mathlib/Data/Matrix/Basic.lean | 1,956 | 1,958 | theorem mulVec_neg [Fintype n] (v : n → α) (A : Matrix m n α) : A *ᵥ (-v) = - (A *ᵥ v) := by |
ext
apply dotProduct_neg
|
import Mathlib.CategoryTheory.Adjunction.Basic
open CategoryTheory
variable {C D : Type*} [Category C] [Category D]
namespace CategoryTheory.Adjunction
@[simps]
def natTransEquiv {F F' : C ⥤ D} {G G' : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F' ⊣ G') :
(G ⟶ G') ≃ (F' ⟶ F) where
toFun f := {
app := fun X ↦ F'.map... | Mathlib/CategoryTheory/Adjunction/Unique.lean | 197 | 203 | theorem unit_rightAdjointUniq_hom_app {F : C ⥤ D} {G G' : D ⥤ C} (adj1 : F ⊣ G) (adj2 : F ⊣ G')
(x : C) : adj1.unit.app x ≫ (rightAdjointUniq adj1 adj2).hom.app (F.obj x) =
adj2.unit.app x := by |
simp only [Functor.id_obj, Functor.comp_obj, rightAdjointUniq, natIsoEquiv_symm_apply_hom,
Iso.refl_hom, natTransEquiv_symm_apply_app, NatTrans.id_app, Category.id_comp]
rw [← adj2.unit_naturality_assoc, ← G'.map_comp]
simp
|
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 | 340 | 343 | theorem closure_subset_thickening {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) :
closure E ⊆ thickening δ E := by |
rw [← cthickening_zero]
exact cthickening_subset_thickening' δ_pos δ_pos E
|
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Basis
#align_import analysis.convex.combination from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
open Set Function
open scoped Classical
open Pointwise
... | Mathlib/Analysis/Convex/Combination.lean | 82 | 84 | theorem Finset.centerMass_eq_of_sum_1 (hw : ∑ i ∈ t, w i = 1) :
t.centerMass w z = ∑ i ∈ t, w i • z i := by |
simp only [Finset.centerMass, hw, inv_one, one_smul]
|
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
#align_import measure_theory.integral.interval_integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open scoped... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 333 | 342 | theorem comp_add_right (hf : IntervalIntegrable f volume a b) (c : ℝ) :
IntervalIntegrable (fun x => f (x + c)) volume (a - c) (b - c) := by |
wlog h : a ≤ b generalizing a b
· exact IntervalIntegrable.symm (this hf.symm (le_of_not_le h))
rw [intervalIntegrable_iff'] at hf ⊢
have A : MeasurableEmbedding fun x => x + c :=
(Homeomorph.addRight c).closedEmbedding.measurableEmbedding
rw [← map_add_right_eq_self volume c] at hf
convert (Measurable... |
import Mathlib.Algebra.ContinuedFractions.Computation.Basic
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.computation.translations from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
open Generali... | Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean | 87 | 92 | theorem succ_nth_stream_eq_some_iff {ifp_succ_n : IntFractPair K} :
IntFractPair.stream v (n + 1) = some ifp_succ_n ↔
∃ ifp_n : IntFractPair K,
IntFractPair.stream v n = some ifp_n ∧
ifp_n.fr ≠ 0 ∧ IntFractPair.of ifp_n.fr⁻¹ = ifp_succ_n := by |
simp [IntFractPair.stream, ite_eq_iff, Option.bind_eq_some]
|
import Mathlib.Algebra.CharP.Two
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.GroupTheory.SpecificGroups.Cyclic
import Mathlib.NumberTheory.Divisors
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Tactic.Zify
#align_import rin... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 1,018 | 1,022 | theorem nthRoots_one_eq_biUnion_primitiveRoots {ζ : R} {n : ℕ} (h : IsPrimitiveRoot ζ n) :
nthRootsFinset n R = (Nat.divisors n).biUnion fun i => primitiveRoots i R := by |
by_cases hn : n = 0
· simp [hn]
exact nthRoots_one_eq_biUnion_primitiveRoots' (n := ⟨n, Nat.pos_of_ne_zero hn⟩) h
|
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.class_group from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
variable {R K L : Type*} [CommRing R]
variable [Field K] [Field L] [DecidableEq L]
variable [Algebra R K] [Is... | Mathlib/RingTheory/ClassGroup.lean | 245 | 249 | theorem FractionalIdeal.canonicalEquiv_mk0 [IsDedekindDomain R] (K' : Type*) [Field K']
[Algebra R K'] [IsFractionRing R K'] (I : (Ideal R)⁰) :
FractionalIdeal.canonicalEquiv R⁰ K K' (FractionalIdeal.mk0 K I) =
FractionalIdeal.mk0 K' I := by |
simp only [FractionalIdeal.coe_mk0, FractionalIdeal.canonicalEquiv_coeIdeal]
|
import Mathlib.Computability.PartrecCode
import Mathlib.Data.Set.Subsingleton
#align_import computability.halting from "leanprover-community/mathlib"@"a50170a88a47570ed186b809ca754110590f9476"
open Encodable Denumerable
def ComputablePred {α} [Primcodable α] (p : α → Prop) :=
∃ _ : DecidablePred p, Computable... | Mathlib/Computability/Halting.lean | 191 | 198 | theorem to_re {p : α → Prop} (hp : ComputablePred p) : RePred p := by |
obtain ⟨f, hf, rfl⟩ := computable_iff.1 hp
unfold RePred
dsimp only []
refine
(Partrec.cond hf (Decidable.Partrec.const' (Part.some ())) Partrec.none).of_eq fun n =>
Part.ext fun a => ?_
cases a; cases f n <;> simp
|
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Vector.Basic
import Mathlib.Data.PFun
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Basic
import Mathlib.Tactic.ApplyFun
#align_import computability.turing_machine from "leanprover-commu... | Mathlib/Computability/TuringMachine.lean | 1,696 | 1,702 | theorem stepAux_move (d : Dir) (q : Stmt'₁) (v : σ) (T : Tape Bool) :
stepAux (moveₙ d q) v T = stepAux q v ((Tape.move d)^[n] T) := by |
suffices ∀ i, stepAux ((Stmt.move d)^[i] q) v T = stepAux q v ((Tape.move d)^[i] T) from this n
intro i; induction' i with i IH generalizing T; · rfl
rw [iterate_succ', iterate_succ]
simp only [stepAux, Function.comp_apply]
rw [IH]
|
import Mathlib.Data.Finset.Attr
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.Directed
import Mathlib.Order.Interval.Set.Basic
#align_import data.finset.basic from "leanprover-community/mathlib"@"442a83d738cb208d3600056c489be16900ba701d"
-- Assert that we define `Finset... | Mathlib/Data/Finset/Basic.lean | 1,146 | 1,147 | theorem mem_insert_coe {s : Finset α} {x y : α} : x ∈ insert y s ↔ x ∈ insert y (s : Set α) := by |
simp
|
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Set.Finite
#align_import order.filter.basic from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
set_option autoImplicit true
open Function Set Order
open scoped Classical
universe u v w x y
structure Filter (α : Type*) where... | Mathlib/Order/Filter/Basic.lean | 1,187 | 1,189 | theorem eventually_all {ι : Sort*} [Finite ι] {l} {p : ι → α → Prop} :
(∀ᶠ x in l, ∀ i, p i x) ↔ ∀ i, ∀ᶠ x in l, p i x := by |
simpa only [Filter.Eventually, setOf_forall] using iInter_mem
|
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Order.Partition.Finpartition
import Mathlib.Tactic.GCongr
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Positivity
import Mathlib.Tactic.Ring
#align_import combinatorics.simp... | Mathlib/Combinatorics/SimpleGraph/Density.lean | 136 | 137 | theorem edgeDensity_nonneg (s : Finset α) (t : Finset β) : 0 ≤ edgeDensity r s t := by |
apply div_nonneg <;> exact mod_cast Nat.zero_le _
|
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.Order.Circular
import Mathlib.Data.List.TFAE
import Mathlib.Data.Set.Lattice
#align_import a... | Mathlib/Algebra/Order/ToIntervalMod.lean | 282 | 284 | theorem toIcoDiv_sub_zsmul' (a b : α) (m : ℤ) :
toIcoDiv hp (a - m • p) b = toIcoDiv hp a b + m := by |
rw [sub_eq_add_neg, ← neg_smul, toIcoDiv_add_zsmul', sub_neg_eq_add]
|
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 | 431 | 435 | theorem imageToKernel'_kernelSubobjectIso (w : f ≫ g = 0) :
imageToKernel' f g w ≫ (kernelSubobjectIso g).inv =
(imageSubobjectIso f).inv ≫ imageToKernel f g w := by |
ext
simp [imageToKernel']
|
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 | 339 | 343 | theorem homology'.map_comp (p₁ : α₁.right = β₁.left) (p₂ : α₂.right = β₂.left) :
homology'.map w₁ w₂ α₁ β₁ p₁ ≫ homology'.map w₂ w₃ α₂ β₂ p₂ =
homology'.map w₁ w₃ (α₁ ≫ α₂) (β₁ ≫ β₂) (homology'.comp_right_eq_comp_left p₁ p₂) := by |
ext
simp only [kernelSubobjectMap_comp, homology'.π_map_assoc, homology'.π_map, Category.assoc]
|
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 | 1,049 | 1,053 | theorem ncard_le_one_iff_subset_singleton [Nonempty α]
(hs : s.Finite := by | toFinite_tac) :
s.ncard ≤ 1 ↔ ∃ x : α, s ⊆ {x} := by
simp_rw [ncard_eq_toFinset_card _ hs, Finset.card_le_one_iff_subset_singleton,
Finite.toFinset_subset, Finset.coe_singleton]
|
import Mathlib.MeasureTheory.Measure.Sub
import Mathlib.MeasureTheory.Decomposition.SignedHahn
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
#align_import measure_theory.decomposition.lebesgue from "leanprover-community/mathlib"@"b2ff9a3d7a15fd5b0f060b135421d6a89a999c2f"
open scoped MeasureTheory NNReal ENN... | Mathlib/MeasureTheory/Decomposition/Lebesgue.lean | 430 | 442 | theorem singularPart_smul (μ ν : Measure α) (r : ℝ≥0) :
(r • μ).singularPart ν = r • μ.singularPart ν := by |
by_cases hr : r = 0
· rw [hr, zero_smul, zero_smul, singularPart_zero]
by_cases hl : HaveLebesgueDecomposition μ ν
· refine (eq_singularPart ((measurable_rnDeriv μ ν).const_smul (r : ℝ≥0∞))
(MutuallySingular.smul r (mutuallySingular_singularPart _ _)) ?_).symm
rw [withDensity_smul _ (measurable_r... |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.convex.side from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052a5230f"
variable {R V V' P P' : Type*}
open AffineEquiv AffineMap
namespace Affine... | Mathlib/Analysis/Convex/Side.lean | 891 | 900 | theorem isPreconnected_setOf_sSameSide (s : AffineSubspace ℝ P) (x : P) :
IsPreconnected { y | s.SSameSide x y } := by |
rcases Set.eq_empty_or_nonempty (s : Set P) with (h | h)
· rw [coe_eq_bot_iff] at h
simp only [h, not_sSameSide_bot]
exact isPreconnected_empty
· by_cases hx : x ∈ s
· simp only [hx, SSameSide, not_true, false_and_iff, and_false_iff]
exact isPreconnected_empty
· exact (isConnected_setOf_sSa... |
import Mathlib.Algebra.Ring.InjSurj
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra.Ring.Hom.Defs
#align_import algebra.ring.units from "leanprover-community/mathlib"@"2ed7e4aec72395b6a7c3ac4ac7873a7a43ead17c"
universe u v w x
variable {α : Type u} {β : Type v} {γ : Type w} {R : Type x}
open Funct... | Mathlib/Algebra/Ring/Units.lean | 133 | 135 | theorem divp_sub_divp [CommRing α] (a b : α) (u₁ u₂ : αˣ) :
a /ₚ u₁ - b /ₚ u₂ = (a * u₂ - u₁ * b) /ₚ (u₁ * u₂) := by |
simp only [sub_eq_add_neg, neg_divp, divp_add_divp, mul_neg]
|
import Mathlib.Topology.Category.TopCat.Limits.Products
#align_import topology.category.Top.limits.pullbacks from "leanprover-community/mathlib"@"178a32653e369dce2da68dc6b2694e385d484ef1"
-- Porting note: every ML3 decl has an uppercase letter
set_option linter.uppercaseLean3 false
open TopologicalSpace
open Cat... | Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean | 158 | 173 | theorem range_pullback_to_prod {X Y Z : TopCat} (f : X ⟶ Z) (g : Y ⟶ Z) :
Set.range (prod.lift pullback.fst pullback.snd : pullback f g ⟶ X ⨯ Y) =
{ x | (Limits.prod.fst ≫ f) x = (Limits.prod.snd ≫ g) x } := by |
ext x
constructor
· rintro ⟨y, rfl⟩
change (_ ≫ _ ≫ f) _ = (_ ≫ _ ≫ g) _ -- new `change` after #13170
simp [pullback.condition]
· rintro (h : f (_, _).1 = g (_, _).2)
use (pullbackIsoProdSubtype f g).inv ⟨⟨_, _⟩, h⟩
change (forget TopCat).map _ _ = _ -- new `change` after #13170
apply Concr... |
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 | 1,059 | 1,060 | theorem div_mul_div_cancel' (a b c : G) : a / b * (b / c) = a / c := by |
rw [← mul_div_assoc, div_mul_cancel]
|
import Mathlib.Data.Finsupp.Encodable
import Mathlib.LinearAlgebra.Pi
import Mathlib.LinearAlgebra.Span
import Mathlib.Data.Set.Countable
#align_import linear_algebra.finsupp from "leanprover-community/mathlib"@"9d684a893c52e1d6692a504a118bfccbae04feeb"
noncomputable section
open Set LinearMap Submodule
namespa... | Mathlib/LinearAlgebra/Finsupp.lean | 314 | 316 | theorem mem_supported' {s : Set α} (p : α →₀ M) :
p ∈ supported M R s ↔ ∀ x ∉ s, p x = 0 := by |
haveI := Classical.decPred fun x : α => x ∈ s; simp [mem_supported, Set.subset_def, not_imp_comm]
|
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.RingTheory.EuclideanDomain
#align_import data.polynomial.field_division from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
noncomputable section
open Polynomial
namespace Polynomial
u... | Mathlib/Algebra/Polynomial/FieldDivision.lean | 385 | 388 | theorem degree_div_le (p q : R[X]) : degree (p / q) ≤ degree p := by |
by_cases hq : q = 0
· simp [hq]
· rw [div_def, mul_comm, degree_mul_leadingCoeff_inv _ hq]; exact degree_divByMonic_le _ _
|
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.Topology.Instances.Matrix
import Mathlib.Topology.Algebra.Module.FiniteDimension
#align_import number_theory.modular from "leanprover-community/mat... | Mathlib/NumberTheory/Modular.lean | 306 | 318 | theorem exists_row_one_eq_and_min_re {cd : Fin 2 → ℤ} (hcd : IsCoprime (cd 0) (cd 1)) :
∃ g : SL(2, ℤ), (↑ₘg) 1 = cd ∧ ∀ g' : SL(2, ℤ), (↑ₘg) 1 = (↑ₘg') 1 →
|(g • z).re| ≤ |(g' • z).re| := by |
haveI : Nonempty { g : SL(2, ℤ) // (↑ₘg) 1 = cd } :=
let ⟨x, hx⟩ := bottom_row_surj hcd
⟨⟨x, hx.2⟩⟩
obtain ⟨g, hg⟩ := Filter.Tendsto.exists_forall_le (tendsto_abs_re_smul z hcd)
refine ⟨g, g.2, ?_⟩
intro g1 hg1
have : g1 ∈ (fun g : SL(2, ℤ) => (↑ₘg) 1) ⁻¹' {cd} := by
rw [Set.mem_preimage, Set.mem... |
import Mathlib.Data.Vector.Basic
#align_import data.vector.mem from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
namespace Vector
variable {α β : Type*} {n : ℕ} (a a' : α)
@[simp]
theorem get_mem (i : Fin n) (v : Vector α n) : v.get i ∈ v.toList := by
rw [get_eq_get]
exact List.... | Mathlib/Data/Vector/Mem.lean | 52 | 54 | theorem mem_succ_iff (v : Vector α (n + 1)) : a ∈ v.toList ↔ a = v.head ∨ a ∈ v.tail.toList := by |
obtain ⟨a', v', h⟩ := exists_eq_cons v
simp_rw [h, Vector.mem_cons_iff, Vector.head_cons, Vector.tail_cons]
|
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.RingTheory.Localization.Basic
#align_import algebra.module.localized_module from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
instance localizedModuleIsLocalizedModule :
IsLocalizedModule S (LocalizedModule.mkLinearMap S M) wh... | Mathlib/Algebra/Module/LocalizedModule.lean | 980 | 980 | theorem mk'_zero (s : S) : mk' f 0 s = 0 := by | rw [← zero_smul R (0 : M), mk'_smul, zero_smul]
|
import Mathlib.Probability.ProbabilityMassFunction.Monad
#align_import probability.probability_mass_function.constructions from "leanprover-community/mathlib"@"4ac69b290818724c159de091daa3acd31da0ee6d"
universe u
namespace PMF
noncomputable section
variable {α β γ : Type*}
open scoped Classical
open NNReal ENN... | Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | 324 | 324 | theorem mem_support_bernoulli_iff : b ∈ (bernoulli p h).support ↔ cond b (p ≠ 0) (p ≠ 1) := by | simp
|
import Mathlib.NumberTheory.DirichletCharacter.Bounds
import Mathlib.NumberTheory.EulerProduct.Basic
import Mathlib.NumberTheory.LSeries.Basic
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex
variable {s : ℂ}
noncomputable
def riemannZetaSummandHom (hs : s ≠ 0) : ℕ →*₀ ℂ where
toFun n := (n : ℂ) ^ ... | Mathlib/NumberTheory/EulerProduct/DirichletLSeries.lean | 129 | 133 | theorem dirichletLSeries_eulerProduct {N : ℕ} (χ : DirichletCharacter ℂ N) (hs : 1 < s.re) :
Tendsto (fun n : ℕ ↦ ∏ p ∈ primesBelow n, (1 - χ p * (p : ℂ) ^ (-s))⁻¹) atTop
(𝓝 (L ↗χ s)) := by |
rw [← tsum_dirichletSummand χ hs]
apply eulerProduct_completely_multiplicative <| summable_dirichletSummand χ hs
|
import Mathlib.CategoryTheory.Preadditive.Basic
#align_import category_theory.preadditive.functor_category from "leanprover-community/mathlib"@"829895f162a1f29d0133f4b3538f4cd1fb5bffd3"
namespace CategoryTheory
open CategoryTheory.Limits Preadditive
variable {C D : Type*} [Category C] [Category D] [Preadditive D... | Mathlib/CategoryTheory/Preadditive/FunctorCategory.lean | 127 | 129 | theorem app_sum {ι : Type*} (s : Finset ι) (X : C) (α : ι → (F ⟶ G)) :
(∑ i ∈ s, α i).app X = ∑ i ∈ s, (α i).app X := by |
simp only [← appHom_apply, map_sum]
|
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Regular.SMul
#align_import data.polynomial.monic from "leanprover-community/mathlib"@"cbdf7b565832144d024caa5a550117c6df0204a5"
noncomputable section
open Finset
open Polynomial
namespace Polynomial
universe u v y
variable {R : Type u} {S : Typ... | Mathlib/Algebra/Polynomial/Monic.lean | 223 | 226 | theorem nextCoeff_pow (hp : p.Monic) (n : ℕ) : (p ^ n).nextCoeff = n • p.nextCoeff := by |
induction n with
| zero => rw [pow_zero, zero_smul, ← map_one (f := C), nextCoeff_C_eq_zero]
| succ n ih => rw [pow_succ, (hp.pow n).nextCoeff_mul hp, ih, succ_nsmul]
|
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev from "leanprover-community/mathlib"@"d774451114d6045faeb6751c396bea1eb9058946"
namespace Polynomial.Chebyshev
set_option linter.uppercaseLean3 false -- `T` `U` `X`
open Polynomial
v... | Mathlib/RingTheory/Polynomial/Chebyshev.lean | 310 | 323 | theorem mul_T (m k : ℤ) : 2 * T R m * T R k = T R (m + k) + T R (m - k) := by |
induction k using Polynomial.Chebyshev.induct with
| zero => simp [two_mul]
| one => rw [T_add_one, T_one]; ring
| add_two k ih1 ih2 =>
have h₁ := T_add_two R (m + k)
have h₂ := T_sub_two R (m - k)
have h₃ := T_add_two R k
linear_combination (norm := ring_nf) 2 * T R m * h₃ - h₂ - h₁ - ih2 + 2 ... |
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib.Data.Set.Finite
#align_import data.finsupp.defs from "leanprover-community/mathlib"@"842328d9df7e96fd90fc424e115679c15fb23a71"
noncomputable section
open Finset Function
variable {α β γ ι M M' N P G H R S : Type*}... | Mathlib/Data/Finsupp/Defs.lean | 914 | 920 | theorem embDomain_mapRange (f : α ↪ β) (g : M → N) (p : α →₀ M) (hg : g 0 = 0) :
embDomain f (mapRange g hg p) = mapRange g hg (embDomain f p) := by |
ext a
by_cases h : a ∈ Set.range f
· rcases h with ⟨a', rfl⟩
rw [mapRange_apply, embDomain_apply, embDomain_apply, mapRange_apply]
· rw [mapRange_apply, embDomain_notin_range, embDomain_notin_range, ← hg] <;> assumption
|
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
import Mathlib.Analysis.NormedSpace.AffineIsometry
#align_import geometry.euclidean.angle.unoriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean | 280 | 282 | theorem angle_right_midpoint_eq_pi_div_two_of_dist_eq {p1 p2 p3 : P} (h : dist p3 p1 = dist p3 p2) :
∠ p3 (midpoint ℝ p1 p2) p2 = π / 2 := by |
rw [midpoint_comm p1 p2, angle_left_midpoint_eq_pi_div_two_of_dist_eq h.symm]
|
import Mathlib.SetTheory.Ordinal.FixedPoint
#align_import set_theory.ordinal.principal from "leanprover-community/mathlib"@"31b269b60935483943542d547a6dd83a66b37dc7"
universe u v w
noncomputable section
open Order
namespace Ordinal
-- Porting note: commented out, doesn't seem necessary
--local infixr:0 "^" => ... | Mathlib/SetTheory/Ordinal/Principal.lean | 400 | 414 | theorem mul_eq_opow_log_succ {a b : Ordinal.{u}} (ha : a ≠ 0) (hb : Principal (· * ·) b)
(hb₂ : 2 < b) : a * b = (b^succ (log b a)) := by |
apply le_antisymm
· have hbl := principal_mul_isLimit hb₂ hb
rw [← IsNormal.bsup_eq.{u, u} (mul_isNormal (Ordinal.pos_iff_ne_zero.2 ha)) hbl, bsup_le_iff]
intro c hcb
have hb₁ : 1 < b := (lt_succ 1).trans (by simpa using hb₂)
have hbo₀ : (b^b.log a) ≠ 0 := Ordinal.pos_iff_ne_zero.1 (opow_pos _ (zer... |
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.Complement
open Monoid Coprod Multiplicative Subgroup Function
def HNNExtension.con (G : Type*) [Group G] (A B : Subgroup G) (φ : A ≃* B) :
Con (G ∗ Multiplicative ℤ) :=
conGen (fun x y => ∃ (a : A),
x = inr (ofAdd 1) * inl (a : G) ∧
... | Mathlib/GroupTheory/HNNExtension.lean | 508 | 512 | theorem prod_cons (g : G) (u : ℤˣ) (w : NormalWord d) (h1 : w.head ∈ d.set u)
(h2 : ∀ u' ∈ Option.map Prod.fst w.toList.head?,
w.head ∈ toSubgroup A B u → u = u') :
(cons g u w h1 h2).prod φ = of g * (t ^ (u : ℤ) * w.prod φ) := by |
simp [ReducedWord.prod, cons, smul_def, mul_assoc]
|
import Mathlib.CategoryTheory.Comma.StructuredArrow
import Mathlib.CategoryTheory.IsConnected
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Terminal
import Mathlib.CategoryTheory.Limits.Shapes.Types
import Mathlib.CategoryTheory.Filtered.Basic
import Mathlib.CategoryTheory.Limits.Yoneda
import Mathlib.Category... | Mathlib/CategoryTheory/Limits/Final.lean | 386 | 404 | theorem zigzag_of_eqvGen_quot_rel {F : C ⥤ D} {d : D} {f₁ f₂ : ΣX, d ⟶ F.obj X}
(t : EqvGen (Types.Quot.Rel.{v, v} (F ⋙ coyoneda.obj (op d))) f₁ f₂) :
Zigzag (StructuredArrow.mk f₁.2) (StructuredArrow.mk f₂.2) := by |
induction t with
| rel x y r =>
obtain ⟨f, w⟩ := r
fconstructor
swap
· fconstructor
left; fconstructor
exact StructuredArrow.homMk f
| refl => fconstructor
| symm x y _ ih =>
apply zigzag_symmetric
exact ih
| trans x y z _ _ ih₁ ih₂ =>
apply Relation.ReflTransGen.trans
... |
import Mathlib.Analysis.InnerProductSpace.Dual
#align_import analysis.inner_product_space.lax_milgram from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open RCLike LinearMap ContinuousLinearMap InnerProductSpace
open LinearMap (ker range)
open RealInnerProduct... | Mathlib/Analysis/InnerProductSpace/LaxMilgram.lean | 74 | 77 | theorem ker_eq_bot (coercive : IsCoercive B) : ker B♯ = ⊥ := by |
rw [LinearMapClass.ker_eq_bot]
rcases coercive.antilipschitz with ⟨_, _, antilipschitz⟩
exact antilipschitz.injective
|
import Mathlib.Algebra.Group.Indicator
import Mathlib.Data.Finset.Piecewise
import Mathlib.Data.Finset.Preimage
#align_import algebra.big_operators.basic from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
-- TODO
-- assert_not_exists AddCommMonoidWithOne
assert_not_exists MonoidWithZero... | Mathlib/Algebra/BigOperators/Group/Finset.lean | 2,476 | 2,478 | theorem toFinset_sum_count_nsmul_eq (s : Multiset α) :
∑ a ∈ s.toFinset, s.count a • {a} = s := by |
rw [← Finset.sum_multiset_map_count, Multiset.sum_map_singleton]
|
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 | 152 | 174 | theorem head_succ' (n m : ℕ) (x : ∀ n, CofixA F n) (Hconsistent : AllAgree x) :
head' (x (succ n)) = head' (x (succ m)) := by |
suffices ∀ n, head' (x (succ n)) = head' (x 1) by simp [this]
clear m n
intro n
cases' h₀ : x (succ n) with _ i₀ f₀
cases' h₁ : x 1 with _ i₁ f₁
dsimp only [head']
induction' n with n n_ih
· rw [h₁] at h₀
cases h₀
trivial
· have H := Hconsistent (succ n)
cases' h₂ : x (succ n) with _ i₂ f... |
import Mathlib.CategoryTheory.Limits.Filtered
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.DiscreteCategory
#align_import category_theory.limits.opposites from "leanprover-community/mathlib"@"ac3ae212f394f508df43e37aa09372... | Mathlib/CategoryTheory/Limits/Opposites.lean | 624 | 625 | theorem unop_fst {X Y Z : Cᵒᵖ} {f : X ⟶ Y} {g : X ⟶ Z} (c : PushoutCocone f g) :
c.unop.fst = c.inl.unop := by | simp
|
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.RowCol
import Mathlib.Data.Fin.VecNotation
import Mathlib.Tactic.FinCases
#align_import data.matrix.notation from "leanprover-community/mathlib"@"a99f85220eaf38f14f94e04699943e185a5e1d1a"
namespace Matrix
universe u uₘ uₙ uₒ
variable {α : Type u} {o n m... | Mathlib/Data/Matrix/Notation.lean | 516 | 517 | theorem smul_vec2 {R : Type*} [SMul R α] (x : R) (a₀ a₁ : α) :
x • ![a₀, a₁] = ![x • a₀, x • a₁] := by | rw [smul_cons, smul_cons, smul_empty]
|
import Mathlib.Algebra.Associated
import Mathlib.Algebra.Star.Unitary
import Mathlib.RingTheory.Int.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Tactic.Ring
#align_import number_theory.zsqrtd.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
@[ext]
struct... | Mathlib/NumberTheory/Zsqrtd/Basic.lean | 446 | 459 | theorem sqLe_mul {d x y z w : ℕ} :
(SqLe x 1 y d → SqLe z 1 w d → SqLe (x * w + y * z) d (x * z + d * y * w) 1) ∧
(SqLe x 1 y d → SqLe w d z 1 → SqLe (x * z + d * y * w) 1 (x * w + y * z) d) ∧
(SqLe y d x 1 → SqLe z 1 w d → SqLe (x * z + d * y * w) 1 (x * w + y * z) d) ∧
(SqLe y d x 1 → SqLe... |
refine ⟨?_, ?_, ?_, ?_⟩ <;>
· intro xy zw
have :=
Int.mul_nonneg (sub_nonneg_of_le (Int.ofNat_le_ofNat_of_le xy))
(sub_nonneg_of_le (Int.ofNat_le_ofNat_of_le zw))
refine Int.le_of_ofNat_le_ofNat (le_of_sub_nonneg ?_)
convert this using 1
simp only [one_mul, Int.ofNat_add... |
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.List.Defs
import Mathlib.Init.Data.List.Basic
import Mathlib.Init.Data.List.Instances
import Mathlib.Init.Data.List.Lemmas
import Mathlib.Logic.Unique
import Mathlib.Order.Basic
import Mathlib.Tactic.Common
#align_import data.list.basic... | Mathlib/Data/List/Basic.lean | 2,604 | 2,610 | theorem get?_pmap {p : α → Prop} (f : ∀ a, p a → β) {l : List α} (h : ∀ a ∈ l, p a) (n : ℕ) :
get? (pmap f l h) n = Option.pmap f (get? l n) fun x H => h x (get?_mem H) := by |
induction' l with hd tl hl generalizing n
· simp
· cases' n with n
· simp
· simp [hl]
|
import Mathlib.Algebra.Group.Ext
import Mathlib.CategoryTheory.Simple
import Mathlib.CategoryTheory.Linear.Basic
import Mathlib.CategoryTheory.Endomorphism
import Mathlib.FieldTheory.IsAlgClosed.Spectrum
#align_import category_theory.preadditive.schur from "leanprover-community/mathlib"@"58a272265b5e05f258161260dd2c5... | Mathlib/CategoryTheory/Preadditive/Schur.lean | 206 | 210 | theorem finrank_hom_simple_simple (X Y : C) [∀ X Y : C, FiniteDimensional 𝕜 (X ⟶ Y)] [Simple X]
[Simple Y] : finrank 𝕜 (X ⟶ Y) = if Nonempty (X ≅ Y) then 1 else 0 := by |
split_ifs with h
· exact (finrank_hom_simple_simple_eq_one_iff 𝕜 X Y).2 h
· exact (finrank_hom_simple_simple_eq_zero_iff 𝕜 X Y).2 (not_nonempty_iff.mp h)
|
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.ZMod
#align_import number_theory.padics.ring_homs from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
noncomputable section
open scoped Classical
open Nat LocalRing Padic
namespace PadicInt
variable {p : ℕ} [h... | Mathlib/NumberTheory/Padics/RingHoms.lean | 450 | 459 | theorem denseRange_natCast : DenseRange (Nat.cast : ℕ → ℤ_[p]) := by |
intro x
rw [Metric.mem_closure_range_iff]
intro ε hε
obtain ⟨n, hn⟩ := exists_pow_neg_lt p hε
use x.appr n
rw [dist_eq_norm]
apply lt_of_le_of_lt _ hn
rw [norm_le_pow_iff_mem_span_pow]
apply appr_spec
|
import Mathlib.Order.Ideal
#align_import order.pfilter from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
open OrderDual
namespace Order
structure PFilter (P : Type*) [Preorder P] where
dual : Ideal Pᵒᵈ
#align order.pfilter Order.PFilter
variable {P : Type*}
def IsPFilter [Preor... | Mathlib/Order/PFilter.lean | 120 | 120 | theorem principal_le_principal_iff {p q : P} : principal q ≤ principal p ↔ p ≤ q := by | simp
|
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import linear_algebra.quadratic_form.basic from "leanprover-community/mathlib"@"d11f435d4e34a6cea0a1797d6b625b0c170be845"
universe u v w
variable {S T : ... | Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 257 | 258 | theorem map_neg (x : M) : Q (-x) = Q x := by |
rw [← @neg_one_smul R _ _ _ _ x, map_smul, neg_one_mul, neg_neg, one_mul]
|
import Mathlib.Data.Fin.Tuple.Basic
import Mathlib.Data.List.Join
#align_import data.list.of_fn from "leanprover-community/mathlib"@"bf27744463e9620ca4e4ebe951fe83530ae6949b"
universe u
variable {α : Type u}
open Nat
namespace List
#noalign list.length_of_fn_aux
@[simp]
theorem length_ofFn_go {n} (f : Fin n ... | Mathlib/Data/List/OfFn.lean | 139 | 141 | theorem last_ofFn {n : ℕ} (f : Fin n → α) (h : ofFn f ≠ [])
(hn : n - 1 < n := Nat.pred_lt <| ofFn_eq_nil_iff.not.mp h) :
getLast (ofFn f) h = f ⟨n - 1, hn⟩ := by | simp [getLast_eq_get]
|
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Int.Lemmas
import Mathlib.Data.Set.Subsingleton
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Order.GaloisConnection
import Mathlib.Tactic.Abel
import Mathlib.Tactic.Linarith... | Mathlib/Algebra/Order/Floor.lean | 1,525 | 1,526 | theorem round_nat_add (x : α) (y : ℕ) : round ((y : α) + x) = y + round x := by |
rw [add_comm, round_add_nat, add_comm]
|
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 | 1,008 | 1,030 | theorem affineCombination_mem_affineSpan [Nontrivial k] {s : Finset ι} {w : ι → k}
(h : ∑ i ∈ s, w i = 1) (p : ι → P) :
s.affineCombination k p w ∈ affineSpan k (Set.range p) := by |
classical
have hnz : ∑ i ∈ s, w i ≠ 0 := h.symm ▸ one_ne_zero
have hn : s.Nonempty := Finset.nonempty_of_sum_ne_zero hnz
cases' hn with i1 hi1
let w1 : ι → k := Function.update (Function.const ι 0) i1 1
have hw1 : ∑ i ∈ s, w1 i = 1 := by
simp only [Function.const_zero, Finset.sum_update_of_... |
import Mathlib.Algebra.Ring.Int
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Size
#align_import data.int.bitwise from "leanprover-community/mathlib"@"0743cc5d9d86bcd1bba10f480e948a257d65056f"
#align_import init.data.int.bitwise from "leanprover-community/lean"@"855e5b74e3a52a40552e8f067169d747d48743fd"
... | Mathlib/Data/Int/Bitwise.lean | 289 | 289 | theorem bit1_ne_zero (m : ℤ) : bit1 m ≠ 0 := by | simpa only [bit0_zero] using bit1_ne_bit0 m 0
|
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.constructions.borel_space.basic from "leanprover-community/mathlib"@"9f55d0d4363ae59948c33864cbc52e0b12e0e8ce"
open Set Filter MeasureTheory MeasurableSpace
open scoped Classical Topology NNReal ENNReal MeasureTheory
univers... | Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean | 524 | 551 | theorem exists_spanning_measurableSet_le {m : MeasurableSpace α} {f : α → ℝ≥0}
(hf : Measurable f) (μ : Measure α) [SigmaFinite μ] :
∃ s : ℕ → Set α,
(∀ n, MeasurableSet (s n) ∧ μ (s n) < ∞ ∧ ∀ x ∈ s n, f x ≤ n) ∧
⋃ i, s i = Set.univ := by |
let sigma_finite_sets := spanningSets μ
let norm_sets := fun n : ℕ => { x | f x ≤ n }
have norm_sets_spanning : ⋃ n, norm_sets n = Set.univ := by
ext1 x
simp only [Set.mem_iUnion, Set.mem_setOf_eq, Set.mem_univ, iff_true_iff]
exact exists_nat_ge (f x)
let sets n := sigma_finite_sets n ∩ norm_sets n... |
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.Order.Circular
import Mathlib.Data.List.TFAE
import Mathlib.Data.Set.Lattice
#align_import a... | Mathlib/Algebra/Order/ToIntervalMod.lean | 309 | 310 | theorem toIocDiv_add_right (a b : α) : toIocDiv hp a (b + p) = toIocDiv hp a b + 1 := by |
simpa only [one_zsmul] using toIocDiv_add_zsmul hp a b 1
|
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
variab... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 923 | 935 | theorem exists_rat_pow_btwn_rat_aux (hn : n ≠ 0) (x y : ℝ) (h : x < y) (hy : 0 < y) :
∃ q : ℚ, 0 < q ∧ x < (q : ℝ) ^ n ∧ (q : ℝ) ^ n < y := by |
have hn' : 0 < (n : ℝ) := mod_cast hn.bot_lt
obtain ⟨q, hxq, hqy⟩ :=
exists_rat_btwn (rpow_lt_rpow (le_max_left 0 x) (max_lt hy h) <| inv_pos.mpr hn')
have := rpow_nonneg (le_max_left 0 x) n⁻¹
have hq := this.trans_lt hxq
replace hxq := rpow_lt_rpow this hxq hn'
replace hqy := rpow_lt_rpow hq.le hqy hn... |
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.ProdSigma
#align_import model_theory.semantics from "leanprover-community/mathlib"@"d565b3df44619c1498326936be16f1a935df0728"
universe u v w u' v'
namespace FirstOrder
namespace Language
variable {L : Language.{u, v}} {... | Mathlib/ModelTheory/Semantics.lean | 875 | 880 | theorem realize_iff_of_model_completeTheory [N ⊨ L.completeTheory M] (φ : L.Sentence) :
N ⊨ φ ↔ M ⊨ φ := by |
refine ⟨fun h => ?_, (L.completeTheory M).realize_sentence_of_mem⟩
contrapose! h
rw [← Sentence.realize_not] at *
exact (L.completeTheory M).realize_sentence_of_mem (mem_completeTheory.2 h)
|
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
#align_import analysis.calculus.fderiv_... | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 388 | 391 | theorem measurableSet_of_differentiableAt : MeasurableSet { x | DifferentiableAt 𝕜 f x } := by |
have : IsComplete (univ : Set (E →L[𝕜] F)) := complete_univ
convert measurableSet_of_differentiableAt_of_isComplete 𝕜 f this
simp
|
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
#align_import analysis.normed_space.multilinear from "leanprover-community/mathlib"@"f40476639bac089693a489c9e354ebd75dc0f886"
suppress_compilation
noncomputable section
open NNReal Finset Metric ContinuousMultilinearMap Fin Function
universe u v v' wE wE... | Mathlib/Analysis/NormedSpace/Multilinear/Curry.lean | 531 | 535 | theorem norm_domDomCongr (σ : ι ≃ ι') (f : ContinuousMultilinearMap 𝕜 (fun _ : ι => G) G') :
‖domDomCongr σ f‖ = ‖f‖ := by |
simp only [norm_def, LinearEquiv.coe_mk, ← σ.prod_comp,
(σ.arrowCongr (Equiv.refl G)).surjective.forall, domDomCongr_apply, Equiv.arrowCongr_apply,
Equiv.coe_refl, id_comp, comp_apply, Equiv.symm_apply_apply, id]
|
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finite.Basic
import Mathlib.Data.Set.Functor
import Mathlib.Data.Set.Lattice
#align_import data.set.finite from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
assert_not_exists OrderedRing
assert_not_exists MonoidWithZero
open Set Fun... | Mathlib/Data/Set/Finite.lean | 181 | 182 | theorem coeSort_toFinset : ↥hs.toFinset = ↥s := by |
rw [← Finset.coe_sort_coe _, hs.coe_toFinset]
|
import Mathlib.Data.Finset.Attr
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.Directed
import Mathlib.Order.Interval.Set.Basic
#align_import data.finset.basic from "leanprover-community/mathlib"@"442a83d738cb208d3600056c489be16900ba701d"
-- Assert that we define `Finset... | Mathlib/Data/Finset/Basic.lean | 1,937 | 1,939 | theorem eq_of_mem_of_not_mem_erase (hs : b ∈ s) (hsa : b ∉ s.erase a) : b = a := by |
rw [mem_erase, not_and] at hsa
exact not_imp_not.mp hsa hs
|
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
#align_import geometry.euclidean.angle.oriented.basic from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped Real Rea... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 848 | 850 | theorem oangle_sign_smul_right (x y : V) (r : ℝ) :
(o.oangle x (r • y)).sign = SignType.sign r * (o.oangle x y).sign := by |
rcases lt_trichotomy r 0 with (h | h | h) <;> simp [h]
|
import Mathlib.Algebra.CharP.Defs
import Mathlib.Data.Nat.Multiplicity
import Mathlib.Data.Nat.Choose.Sum
#align_import algebra.char_p.basic from "leanprover-community/mathlib"@"47a1a73351de8dd6c8d3d32b569c8e434b03ca47"
assert_not_exists orderOf
universe u v
open Finset
variable {R : Type*}
variable (R)
theor... | Mathlib/Algebra/CharP/Basic.lean | 147 | 152 | theorem CharP.neg_one_pow_char_pow [Ring R] (p n : ℕ) [CharP R p] [Fact p.Prime] :
(-1 : R) ^ p ^ n = -1 := by |
rw [eq_neg_iff_add_eq_zero]
nth_rw 2 [← one_pow (p ^ n)]
rw [← add_pow_char_pow_of_commute R _ _ (Commute.one_right _), add_left_neg,
zero_pow (pow_ne_zero _ (Fact.out (p := Nat.Prime p)).ne_zero)]
|
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Topology.Instances.NNReal
#align_import analysis.normed.group.infinite_sum from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd"
open Topology NNReal
open Finset Filter Metric
variabl... | Mathlib/Analysis/Normed/Group/InfiniteSum.lean | 127 | 132 | theorem tsum_of_norm_bounded {f : ι → E} {g : ι → ℝ} {a : ℝ} (hg : HasSum g a)
(h : ∀ i, ‖f i‖ ≤ g i) : ‖∑' i : ι, f i‖ ≤ a := by |
by_cases hf : Summable f
· exact hf.hasSum.norm_le_of_bounded hg h
· rw [tsum_eq_zero_of_not_summable hf, norm_zero]
classical exact ge_of_tendsto' hg fun s => sum_nonneg fun i _hi => (norm_nonneg _).trans (h i)
|
import Mathlib.Order.Interval.Finset.Basic
#align_import data.multiset.locally_finite from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
variable {α : Type*}
namespace Multiset
section LinearOrder
variable [LinearOrder α] [LocallyFiniteOrder α] {a b c d : α}
theorem Ico_subset_Ico_... | Mathlib/Order/Interval/Multiset.lean | 354 | 355 | theorem Ico_filter_lt (a b c : α) : ((Ico a b).filter fun x => x < c) = Ico a (min b c) := by |
rw [Ico, Ico, ← Finset.filter_val, Finset.Ico_filter_lt]
|
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 | 612 | 617 | theorem dist_reflection_eq_of_mem (s : AffineSubspace ℝ P) [Nonempty s]
[HasOrthogonalProjection s.direction] {p₁ : P} (hp₁ : p₁ ∈ s) (p₂ : P) :
dist p₁ (reflection s p₂) = dist p₁ p₂ := by |
rw [← reflection_eq_self_iff p₁] at hp₁
convert (reflection s).dist_map p₁ p₂
rw [hp₁]
|
import Mathlib.LinearAlgebra.Matrix.DotProduct
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
#align_import data.matrix.rank from "leanprover-community/mathlib"@"17219820a8aa8abe85adf5dfde19af1dd1bd8ae7"
open Matrix
namespace Matrix
open FiniteDimensional
variable {l m n ... | Mathlib/Data/Matrix/Rank.lean | 49 | 51 | theorem rank_one [StrongRankCondition R] [DecidableEq n] :
rank (1 : Matrix n n R) = Fintype.card n := by |
rw [rank, mulVecLin_one, LinearMap.range_id, finrank_top, finrank_pi]
|
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.Algebra.Group.Semiconj.Units
import Mathlib.Init.Classical
#align_import algebra.group_with_zero.semiconj from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
assert_not_exists DenselyOrdered
variable {α M₀ G₀ M₀' G₀' F F' :... | Mathlib/Algebra/GroupWithZero/Semiconj.lean | 29 | 30 | theorem zero_left [MulZeroClass G₀] (x y : G₀) : SemiconjBy 0 x y := by |
simp only [SemiconjBy, mul_zero, zero_mul]
|
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 | 194 | 196 | theorem sum_smul_const_vsub_eq_sub_weightedVSubOfPoint (w : ι → k) (p₂ : ι → P) (p₁ b : P) :
(∑ i ∈ s, w i • (p₁ -ᵥ p₂ i)) = (∑ i ∈ s, w i) • (p₁ -ᵥ b) - s.weightedVSubOfPoint p₂ b w := by |
rw [sum_smul_vsub_eq_weightedVSubOfPoint_sub, weightedVSubOfPoint_apply_const]
|
import Mathlib.Analysis.BoxIntegral.Box.SubboxInduction
import Mathlib.Analysis.BoxIntegral.Partition.Tagged
#align_import analysis.box_integral.partition.subbox_induction from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
namespace BoxIntegral
open Set Metric
open scoped Classical
o... | Mathlib/Analysis/BoxIntegral/Partition/SubboxInduction.lean | 154 | 163 | theorem exists_tagged_le_isHenstock_isSubordinate_iUnion_eq {I : Box ι} (r : (ι → ℝ) → Ioi (0 : ℝ))
(π : Prepartition I) :
∃ π' : TaggedPrepartition I, π'.toPrepartition ≤ π ∧ π'.IsHenstock ∧ π'.IsSubordinate r ∧
π'.distortion = π.distortion ∧ π'.iUnion = π.iUnion := by |
have := fun J => Box.exists_taggedPartition_isHenstock_isSubordinate_homothetic J r
choose! πi πip πiH πir _ πid using this
refine ⟨π.biUnionTagged πi, biUnion_le _ _, isHenstock_biUnionTagged.2 fun J _ => πiH J,
isSubordinate_biUnionTagged.2 fun J _ => πir J, ?_, π.iUnion_biUnion_partition fun J _ => πip J⟩... |
import Mathlib.Topology.Sets.Opens
#align_import topology.local_at_target from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open TopologicalSpace Set Filter
open Topology Filter
variable {α β : Type*} [TopologicalSpace α] [TopologicalSpace β] {f : α → β}
variable {s : Set β} {ι : Ty... | Mathlib/Topology/LocalAtTarget.lean | 111 | 113 | theorem isClosed_iff_coe_preimage_of_iSup_eq_top (s : Set β) :
IsClosed s ↔ ∀ i, IsClosed ((↑) ⁻¹' s : Set (U i)) := by |
simpa using isOpen_iff_coe_preimage_of_iSup_eq_top hU sᶜ
|
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 381 | 381 | theorem image_neg_Ioi : Neg.neg '' Ioi a = Iio (-a) := by | simp
|
import Mathlib.Algebra.GroupWithZero.Prod
import Mathlib.Algebra.Ring.Opposite
import Mathlib.GroupTheory.GroupAction.Opposite
import Mathlib.GroupTheory.GroupAction.Prod
#align_import algebra.smul_with_zero from "leanprover-community/mathlib"@"966e0cf0685c9cedf8a3283ac69eef4d5f2eaca2"
variable {R R' M M' : Type*... | Mathlib/Algebra/SMulWithZero.lean | 209 | 216 | theorem smul_inv₀ [SMulCommClass α β β] [IsScalarTower α β β] (c : α) (x : β) :
(c • x)⁻¹ = c⁻¹ • x⁻¹ := by |
obtain rfl | hc := eq_or_ne c 0
· simp only [inv_zero, zero_smul]
obtain rfl | hx := eq_or_ne x 0
· simp only [inv_zero, smul_zero]
· refine inv_eq_of_mul_eq_one_left ?_
rw [smul_mul_smul, inv_mul_cancel hc, inv_mul_cancel hx, one_smul]
|
import Mathlib.Data.Set.Function
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Core
import Mathlib.Tactic.Attr.Core
#align_import logic.equiv.local_equiv from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
open Lean Meta Elab Tactic
def mfld_cfg : Simps.Config where
attrs :=... | Mathlib/Logic/Equiv/PartialEquiv.lean | 305 | 310 | theorem copy_eq (e : PartialEquiv α β) (f : α → β) (hf : ⇑e = f) (g : β → α) (hg : ⇑e.symm = g)
(s : Set α) (hs : e.source = s) (t : Set β) (ht : e.target = t) :
e.copy f hf g hg s hs t ht = e := by |
substs f g s t
cases e
rfl
|
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