Context stringlengths 57 85k | file_name stringlengths 21 79 | start int64 14 2.42k | end int64 18 2.43k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
#align_import geometry.euclidean.angle.oriented.rotation from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 338 | 345 | theorem oangle_eq_iff_eq_pos_smul_rotation_of_ne_zero {x y : V} (hx : x ≠ 0) (hy : y ≠ 0)
(θ : Real.Angle) : o.oangle x y = θ ↔ ∃ r : ℝ, 0 < r ∧ y = r • o.rotation θ x := by |
constructor
· intro h
rw [o.oangle_eq_iff_eq_norm_div_norm_smul_rotation_of_ne_zero hx hy] at h
exact ⟨‖y‖ / ‖x‖, div_pos (norm_pos_iff.2 hy) (norm_pos_iff.2 hx), h⟩
· rintro ⟨r, hr, rfl⟩
rw [o.oangle_smul_right_of_pos _ _ hr, o.oangle_rotation_self_right hx]
|
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Polynomial.Degree.Lemmas
#align_import data.polynomial.erase_lead from "leanprover-community/mathlib"@"fa256f00ce018e7b40e1dc756e403c86680bf448"
noncomputable section
open Polynomial
open Polynomial Finset
namespace Polynomial
variable {R : Type*}... | Mathlib/Algebra/Polynomial/EraseLead.lean | 182 | 190 | theorem eraseLead_add_of_natDegree_lt_left {p q : R[X]} (pq : q.natDegree < p.natDegree) :
(p + q).eraseLead = p.eraseLead + q := by |
ext n
by_cases nd : n = p.natDegree
· rw [nd, eraseLead_coeff, if_pos (natDegree_add_eq_left_of_natDegree_lt pq).symm]
simpa using (coeff_eq_zero_of_natDegree_lt pq).symm
· rw [eraseLead_coeff, coeff_add, coeff_add, eraseLead_coeff, if_neg, if_neg nd]
rintro rfl
exact nd (natDegree_add_eq_left_of_n... |
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 | 182 | 184 | theorem Finset.coe_einfsep [DecidableEq α] {s : Finset α} :
(s : Set α).einfsep = s.offDiag.inf (uncurry edist) := by |
simp_rw [einfsep_of_fintype, ← Finset.coe_offDiag, Finset.toFinset_coe]
|
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import data.polynomial.expand from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
universe u v w
open Polynomial
open Finset
namespace Polynomial
section CommSemiring
variable (R : Type u) [... | Mathlib/Algebra/Polynomial/Expand.lean | 263 | 268 | theorem expand_char (f : R[X]) : map (frobenius R p) (expand R p f) = f ^ p := by |
refine f.induction_on' (fun a b ha hb => ?_) fun n a => ?_
· rw [AlgHom.map_add, Polynomial.map_add, ha, hb, add_pow_expChar]
· rw [expand_monomial, map_monomial, ← C_mul_X_pow_eq_monomial, ← C_mul_X_pow_eq_monomial,
mul_pow, ← C.map_pow, frobenius_def]
ring
|
import Mathlib.Data.Countable.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Order.Disjointed
import Mathlib.MeasureTheory.OuterMeasure.Defs
#align_import measure_theory.measure.outer_measure from "leanprover-community/mathlib"@"343e80208d29d2d15f8050b929aa50fe4ce71b55"
noncomputable section
open Set F... | Mathlib/MeasureTheory/OuterMeasure/Basic.lean | 93 | 94 | theorem measure_le_inter_add_diff (μ : F) (s t : Set α) : μ s ≤ μ (s ∩ t) + μ (s \ t) := by |
simpa using measure_union_le (s ∩ t) (s \ t)
|
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 | 203 | 207 | theorem not_dvd_of_natDegree_lt (hp : Monic p) (h0 : q ≠ 0) (hl : natDegree q < natDegree p) :
¬p ∣ q := by |
rintro ⟨r, rfl⟩
rw [hp.natDegree_mul' <| right_ne_zero_of_mul h0] at hl
exact hl.not_le (Nat.le_add_right _ _)
|
set_option autoImplicit true
namespace Array
@[simp]
theorem extract_eq_nil_of_start_eq_end {a : Array α} :
a.extract i i = #[] := by
refine extract_empty_of_stop_le_start a ?h
exact Nat.le_refl i
| Mathlib/Data/Array/ExtractLemmas.lean | 21 | 27 | theorem extract_append_left {a b : Array α} {i j : Nat} (h : j ≤ a.size) :
(a ++ b).extract i j = a.extract i j := by |
apply ext
· simp only [size_extract, size_append]
omega
· intro h1 h2 h3
rw [get_extract, get_append_left, get_extract]
|
import Mathlib.LinearAlgebra.Alternating.Basic
import Mathlib.LinearAlgebra.Multilinear.TensorProduct
import Mathlib.GroupTheory.GroupAction.Quotient
#align_import linear_algebra.alternating from "leanprover-community/mathlib"@"0c1d80f5a86b36c1db32e021e8d19ae7809d5b79"
suppress_compilation
open TensorProduct
vari... | Mathlib/LinearAlgebra/Alternating/DomCoprod.lean | 212 | 222 | theorem MultilinearMap.domCoprod_alternization_coe [DecidableEq ιa] [DecidableEq ιb]
(a : MultilinearMap R' (fun _ : ιa => Mᵢ) N₁) (b : MultilinearMap R' (fun _ : ιb => Mᵢ) N₂) :
MultilinearMap.domCoprod (MultilinearMap.alternatization a)
(MultilinearMap.alternatization b) =
∑ σa : Perm ιa, ∑ σb : P... |
simp_rw [← MultilinearMap.domCoprod'_apply, MultilinearMap.alternatization_coe]
simp_rw [TensorProduct.sum_tmul, TensorProduct.tmul_sum, _root_.map_sum,
← TensorProduct.smul_tmul', TensorProduct.tmul_smul]
rfl
|
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Data.Finset.Fin
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Int.Order.Units
import Mathlib.GroupTheory.Perm.Support
import Mathlib.Logic.Equiv.Fin
import Mathlib.Tactic.NormNum.Ineq
#align_import group_theory.perm.sign from "leanprover-community/math... | Mathlib/GroupTheory/Perm/Sign.lean | 524 | 548 | theorem prod_prodExtendRight {α : Type*} [DecidableEq α] (σ : α → Perm β) {l : List α}
(hl : l.Nodup) (mem_l : ∀ a, a ∈ l) :
(l.map fun a => prodExtendRight a (σ a)).prod = prodCongrRight σ := by |
ext ⟨a, b⟩ : 1
-- We'll use induction on the list of elements,
-- but we have to keep track of whether we already passed `a` in the list.
suffices a ∈ l ∧ (l.map fun a => prodExtendRight a (σ a)).prod (a, b) = (a, σ a b) ∨
a ∉ l ∧ (l.map fun a => prodExtendRight a (σ a)).prod (a, b) = (a, b) by
obtai... |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Fin
import Mathlib.GroupTheory.GroupAction.Pi
import Mathlib.Logic.Equiv.Fin
#align_import algebra.big_operators.fin from "leanprover-community/mathlib"@"cc5dd6244981976cc9da7afc4eee5682b037a013"
open Fins... | Mathlib/Algebra/BigOperators/Fin.lean | 167 | 170 | theorem prod_univ_eight [CommMonoid β] (f : Fin 8 → β) :
∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 * f 5 * f 6 * f 7 := by |
rw [prod_univ_castSucc, prod_univ_seven]
rfl
|
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Normed.Group.Lemmas
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Analysis.NormedSpace.RieszLemma
import Mathli... | Mathlib/Analysis/NormedSpace/FiniteDimension.lean | 198 | 220 | theorem LipschitzOnWith.extend_finite_dimension {α : Type*} [PseudoMetricSpace α] {E' : Type*}
[NormedAddCommGroup E'] [NormedSpace ℝ E'] [FiniteDimensional ℝ E'] {s : Set α} {f : α → E'}
{K : ℝ≥0} (hf : LipschitzOnWith K f s) :
∃ g : α → E', LipschitzWith (lipschitzExtensionConstant E' * K) g ∧ EqOn f g s ... |
/- This result is already known for spaces `ι → ℝ`. We use a continuous linear equiv between
`E'` and such a space to transfer the result to `E'`. -/
let ι : Type _ := Basis.ofVectorSpaceIndex ℝ E'
let A := (Basis.ofVectorSpace ℝ E').equivFun.toContinuousLinearEquiv
have LA : LipschitzWith ‖A.toContinuousL... |
import Mathlib.RingTheory.FiniteType
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Localization.Integer
import Mathlib.RingTheory.Localization.Submodule
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.RingTheory.RingHomProperties
im... | Mathlib/RingTheory/LocalProperties.lean | 166 | 176 | theorem RingHom.ofLocalizationSpanTarget_iff_finite :
RingHom.OfLocalizationSpanTarget @P ↔ RingHom.OfLocalizationFiniteSpanTarget @P := by |
delta RingHom.OfLocalizationSpanTarget RingHom.OfLocalizationFiniteSpanTarget
apply forall₅_congr
-- TODO: Using `refine` here breaks `resetI`.
intros
constructor
· intro h s; exact h s
· intro h s hs hs'
obtain ⟨s', h₁, h₂⟩ := (Ideal.span_eq_top_iff_finite s).mp hs
exact h s' h₂ fun x => hs' ⟨_,... |
import Mathlib.Analysis.Complex.Basic
import Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle
#align_import analysis.complex.re_im_topology from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1e360b7e3cf6"
open Set
noncomputable section
namespace Complex
theorem isHomeomorphicTrivialFiber... | Mathlib/Analysis/Complex/ReImTopology.lean | 149 | 150 | theorem frontier_setOf_le_im (a : ℝ) : frontier { z : ℂ | a ≤ z.im } = { z | z.im = a } := by |
simpa only [frontier_Ici] using frontier_preimage_im (Ici a)
|
import Mathlib.Topology.UniformSpace.UniformConvergenceTopology
#align_import topology.uniform_space.equicontinuity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
section
open UniformSpace Filter Set Uniformity Topology UniformConvergence Function
variable {ι κ X X' Y Z α α' β β'... | Mathlib/Topology/UniformSpace/Equicontinuity.lean | 709 | 715 | theorem Filter.HasBasis.uniformEquicontinuousOn_iff_right {p : κ → Prop}
{s : κ → Set (α × α)} {F : ι → β → α} {S : Set β} (hα : (𝓤 α).HasBasis p s) :
UniformEquicontinuousOn F S ↔
∀ k, p k → ∀ᶠ xy : β × β in 𝓤 β ⊓ 𝓟 (S ×ˢ S), ∀ i, (F i xy.1, F i xy.2) ∈ s k := by |
rw [uniformEquicontinuousOn_iff_uniformContinuousOn, UniformContinuousOn,
(UniformFun.hasBasis_uniformity_of_basis ι α hα).tendsto_right_iff]
rfl
|
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.special_functions.log.basic from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
open Set Filter Function
open Topology
noncomputable ... | Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 309 | 312 | theorem log_zpow (x : ℝ) (n : ℤ) : log (x ^ n) = n * log x := by |
induction n
· rw [Int.ofNat_eq_coe, zpow_natCast, log_pow, Int.cast_natCast]
rw [zpow_negSucc, log_inv, log_pow, Int.cast_negSucc, Nat.cast_add_one, neg_mul_eq_neg_mul]
|
import Mathlib.MeasureTheory.Integral.FundThmCalculus
import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv
import Mathlib.Analysis.SpecialFunctions.NonIntegrable
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
#align_import analysis.special_functions.integrals from "leanprover-community/mathlib"@"011... | Mathlib/Analysis/SpecialFunctions/Integrals.lean | 269 | 271 | theorem intervalIntegrable_inv_one_add_sq :
IntervalIntegrable (fun x : ℝ => (↑1 + x ^ 2)⁻¹) μ a b := by |
field_simp; exact mod_cast intervalIntegrable_one_div_one_add_sq
|
import Mathlib.Order.Filter.FilterProduct
import Mathlib.Analysis.SpecificLimits.Basic
#align_import data.real.hyperreal from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classical
open Filter Germ Topology
def Hyperreal : Type :=
Germ (hyperfilter ℕ : Filter ℕ) ℝ deri... | Mathlib/Data/Real/Hyperreal.lean | 842 | 845 | theorem infiniteNeg_mul_of_infinitePos_not_infinitesimal_neg {x y : ℝ*} :
InfinitePos x → ¬Infinitesimal y → y < 0 → InfiniteNeg (x * y) := by |
rw [← infinitePos_neg, ← neg_pos, neg_mul_eq_mul_neg, ← infinitesimal_neg]
exact infinitePos_mul_of_infinitePos_not_infinitesimal_pos
|
import Mathlib.Data.Set.Pairwise.Basic
import Mathlib.Data.Set.Lattice
import Mathlib.Data.SetLike.Basic
#align_import order.chain from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
open scoped Classical
open Set
variable {α β : Type*}
section Chain
variable (r : α → α → Prop)
... | Mathlib/Order/Chain.lean | 210 | 212 | theorem chainClosure_empty : ChainClosure r ∅ := by |
have : ChainClosure r (⋃₀∅) := ChainClosure.union fun a h => False.rec h
simpa using this
|
import Mathlib.Analysis.Seminorm
import Mathlib.Topology.Algebra.Equicontinuity
import Mathlib.Topology.MetricSpace.Equicontinuity
import Mathlib.Topology.Algebra.FilterBasis
import Mathlib.Topology.Algebra.Module.LocallyConvex
#align_import analysis.locally_convex.with_seminorms from "leanprover-community/mathlib"@"... | Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | 79 | 81 | theorem basisSets_iff {U : Set E} :
U ∈ p.basisSets ↔ ∃ (i : Finset ι) (r : ℝ), 0 < r ∧ U = ball (i.sup p) 0 r := by |
simp only [basisSets, mem_iUnion, exists_prop, mem_singleton_iff]
|
import Mathlib.Topology.Category.TopCat.Limits.Pullbacks
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
#align_import algebraic_geometry.open_immersion.basic from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
-- Porting note: due to `PresheafedSpace`, `SheafedSpace` and `Locally... | Mathlib/Geometry/RingedSpace/OpenImmersion.lean | 233 | 243 | theorem invApp_app (U : Opens X) :
H.invApp U ≫ f.c.app (op (H.openFunctor.obj U)) =
X.presheaf.map (eqToHom (by
-- Porting note: was just `simp [opens.map, Set.preimage_image_eq _ H.base_open.inj]`
-- See https://github.com/leanprover-community/mathlib4/issues/5026
-- I think this is ... |
rw [invApp, Category.assoc, IsIso.inv_hom_id, Category.comp_id]
|
import Mathlib.Data.Nat.Factorial.Basic
import Mathlib.Order.Monotone.Basic
#align_import data.nat.choose.basic from "leanprover-community/mathlib"@"2f3994e1b117b1e1da49bcfb67334f33460c3ce4"
open Nat
namespace Nat
def choose : ℕ → ℕ → ℕ
| _, 0 => 1
| 0, _ + 1 => 0
| n + 1, k + 1 => choose n k + choose n ... | Mathlib/Data/Nat/Choose/Basic.lean | 145 | 161 | theorem choose_mul {n k s : ℕ} (hkn : k ≤ n) (hsk : s ≤ k) :
n.choose k * k.choose s = n.choose s * (n - s).choose (k - s) :=
have h : 0 < (n - k)! * (k - s)! * s ! := by | apply_rules [factorial_pos, Nat.mul_pos]
Nat.mul_right_cancel h <|
calc
n.choose k * k.choose s * ((n - k)! * (k - s)! * s !) =
n.choose k * (k.choose s * s ! * (k - s)!) * (n - k)! := by
rw [Nat.mul_assoc, Nat.mul_assoc, Nat.mul_assoc, Nat.mul_assoc _ s !, Nat.mul_assoc,
Nat.mul_comm (n ... |
import Mathlib.Analysis.Complex.Basic
import Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle
#align_import analysis.complex.re_im_topology from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1e360b7e3cf6"
open Set
noncomputable section
namespace Complex
theorem isHomeomorphicTrivialFiber... | Mathlib/Analysis/Complex/ReImTopology.lean | 119 | 120 | theorem closure_setOf_im_lt (a : ℝ) : closure { z : ℂ | z.im < a } = { z | z.im ≤ a } := by |
simpa only [closure_Iio] using closure_preimage_im (Iio a)
|
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 | 290 | 296 | theorem rename_isHomogeneous_iff {f : σ → τ} (hf : f.Injective) :
(rename f φ).IsHomogeneous n ↔ φ.IsHomogeneous n := by |
refine ⟨fun h d hd ↦ ?_, rename_isHomogeneous⟩
convert ← @h (d.mapDomain f) _
· simp only [weightedDegree_apply, Pi.one_apply, smul_eq_mul, mul_one]
exact Finsupp.sum_mapDomain_index_inj (h := fun _ ↦ id) hf
· rwa [coeff_rename_mapDomain f hf]
|
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import analysis.normed_space.lp_space from "leanprover-community/mathlib"@"de83b43717abe3... | Mathlib/Analysis/NormedSpace/lpSpace.lean | 99 | 103 | theorem memℓp_gen_iff (hp : 0 < p.toReal) {f : ∀ i, E i} :
Memℓp f p ↔ Summable fun i => ‖f i‖ ^ p.toReal := by |
rw [ENNReal.toReal_pos_iff] at hp
dsimp [Memℓp]
rw [if_neg hp.1.ne', if_neg hp.2.ne]
|
import Mathlib.Topology.Gluing
import Mathlib.Geometry.RingedSpace.OpenImmersion
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits
#align_import algebraic_geometry.presheafed_space.gluing from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
set_option linter.uppercaseLean... | Mathlib/Geometry/RingedSpace/PresheafedSpace/Gluing.lean | 186 | 232 | theorem snd_invApp_t_app' (i j k : D.J) (U : Opens (pullback (D.f i j) (D.f i k)).carrier) :
∃ eq,
(π₂⁻¹ i, j, k) U ≫ (D.t k i).c.app _ ≫ (D.V (k, i)).presheaf.map (eqToHom eq) =
(D.t' k i j).c.app _ ≫ (π₁⁻¹ k, j, i) (unop _) := by |
fconstructor
-- Porting note: I don't know what the magic was in Lean3 proof, it just skipped the proof of `eq`
· delta IsOpenImmersion.openFunctor
dsimp only [Functor.op, Opens.map, IsOpenMap.functor, unop_op, Opens.coe_mk]
congr
have := (𝖣.t_fac k i j).symm
rw [← IsIso.inv_comp_eq] at this
... |
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 | 339 | 342 | theorem coe_lt_iff (n : ℕ) (x : PartENat) : (n : PartENat) < x ↔ ∀ h : x.Dom, n < x.get h := by |
rw [← some_eq_natCast]
simp only [lt_def, exists_prop_of_true, dom_some, forall_true_iff]
rfl
|
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_theory.perm.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Equiv Finset
namespace Equiv.Perm
variable {α : Type*}
section Disjoint
... | Mathlib/GroupTheory/Perm/Support.lean | 183 | 188 | theorem Disjoint.mul_apply_eq_iff {σ τ : Perm α} (hστ : Disjoint σ τ) {a : α} :
(σ * τ) a = a ↔ σ a = a ∧ τ a = a := by |
refine ⟨fun h => ?_, fun h => by rw [mul_apply, h.2, h.1]⟩
cases' hστ a with hσ hτ
· exact ⟨hσ, σ.injective (h.trans hσ.symm)⟩
· exact ⟨(congr_arg σ hτ).symm.trans h, hτ⟩
|
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.Tactic.NormNum.Inv
import Mathlib.Tactic.NormNum.Pow
import Mathlib.Util.AtomM
set_option autoImplicit true
namespace Mathlib.Tactic
namespace Ring
open Mathlib.Meta Qq NormNum Lean.Meta AtomM
open Lean (MetaM Expr mkRawNatLit)
def instCommSemiringNat : CommSe... | Mathlib/Tactic/Ring/Basic.lean | 872 | 872 | theorem atom_pf (a : R) : a = a ^ (nat_lit 1).rawCast * (nat_lit 1).rawCast + 0 := by | simp
|
import Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction
import Mathlib.Analysis.BoxIntegral.Partition.Split
#align_import analysis.box_integral.partition.filter from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Set Function Filter Metric Finset Bool
open scoped Classical
o... | Mathlib/Analysis/BoxIntegral/Partition/Filter.lean | 280 | 280 | theorem henstock_le_mcShane : Henstock ≤ McShane := by | trivial
|
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable ... | Mathlib/Algebra/Polynomial/RingDivision.lean | 319 | 320 | theorem root_mul : IsRoot (p * q) a ↔ IsRoot p a ∨ IsRoot q a := by |
simp_rw [IsRoot, eval_mul, mul_eq_zero]
|
import Mathlib.Data.List.Range
import Mathlib.Data.List.Perm
#align_import data.list.sigma from "leanprover-community/mathlib"@"f808feb6c18afddb25e66a71d317643cf7fb5fbb"
universe u v
namespace List
variable {α : Type u} {β : α → Type v} {l l₁ l₂ : List (Sigma β)}
def keys : List (Sigma β) → List α :=
map ... | Mathlib/Data/List/Sigma.lean | 326 | 327 | theorem lookupAll_nodup (a : α) {l : List (Sigma β)} (h : l.NodupKeys) : (lookupAll a l).Nodup := by |
(rw [lookupAll_eq_dlookup a h]; apply Option.toList_nodup)
|
import Mathlib.Topology.Separation
import Mathlib.Topology.NoetherianSpace
#align_import topology.quasi_separated from "leanprover-community/mathlib"@"5dc6092d09e5e489106865241986f7f2ad28d4c8"
open TopologicalSpace
variable {α β : Type*} [TopologicalSpace α] [TopologicalSpace β] {f : α → β}
def IsQuasiSeparate... | Mathlib/Topology/QuasiSeparated.lean | 99 | 103 | theorem isQuasiSeparated_iff_quasiSeparatedSpace (s : Set α) (hs : IsOpen s) :
IsQuasiSeparated s ↔ QuasiSeparatedSpace s := by |
rw [← isQuasiSeparated_univ_iff]
convert (hs.openEmbedding_subtype_val.isQuasiSeparated_iff (s := Set.univ)).symm
simp
|
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
#align_import linear_algebra.matrix.reindex from "leanprover-community/mathlib"@"1cfdf5f34e1044ecb65d10be753008baaf118edf"
namespace Matrix
open Equiv Matrix
variable {l m n o : Type*} {l' m' n' o' : Type*} {m'' n'' : Type*}
variable (R A : Type*)
section A... | Mathlib/LinearAlgebra/Matrix/Reindex.lean | 66 | 70 | theorem reindexLinearEquiv_trans (e₁ : m ≃ m') (e₂ : n ≃ n') (e₁' : m' ≃ m'') (e₂' : n' ≃ n'') :
(reindexLinearEquiv R A e₁ e₂).trans (reindexLinearEquiv R A e₁' e₂') =
(reindexLinearEquiv R A (e₁.trans e₁') (e₂.trans e₂') : _ ≃ₗ[R] _) := by |
ext
rfl
|
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Logic.Function.Iterate
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Zify
#align_import data.nat.fib from "leanprover-community/mathlib"@"... | Mathlib/Data/Nat/Fib/Basic.lean | 94 | 94 | theorem fib_le_fib_succ {n : ℕ} : fib n ≤ fib (n + 1) := by | cases n <;> simp [fib_add_two]
|
import Mathlib.Geometry.Manifold.ContMDiff.Product
import Mathlib.Analysis.NormedSpace.OperatorNorm.Prod
open Set ChartedSpace SmoothManifoldWithCorners
open scoped Topology Manifold
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
-- declare a smooth manifold `M` over the pair `(E, H)`.
{E : Type*}
[Norme... | Mathlib/Geometry/Manifold/ContMDiff/NormedSpace.lean | 63 | 65 | theorem contMDiffAt_iff_contDiffAt {f : E → E'} {x : E} :
ContMDiffAt 𝓘(𝕜, E) 𝓘(𝕜, E') n f x ↔ ContDiffAt 𝕜 n f x := by |
rw [← contMDiffWithinAt_univ, contMDiffWithinAt_iff_contDiffWithinAt, contDiffWithinAt_univ]
|
import Mathlib.Init.Function
import Mathlib.Init.Order.Defs
#align_import data.bool.basic from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
namespace Bool
@[deprecated (since := "2024-06-07")] alias decide_True := decide_true_eq_true
#align bool.to_bool_true decide_true_eq_true
@[dep... | Mathlib/Data/Bool/Basic.lean | 224 | 224 | theorem and_le_right : ∀ x y : Bool, (x && y) ≤ y := by | decide
|
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.RingTheory.Polynomial.Bernstein
import Mathlib.Topology.ContinuousFunction.Polynomial
import Mathlib.Topology.ContinuousFunction.Compact
#align_import analysis.special_functions.bernstein from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba14... | Mathlib/Analysis/SpecialFunctions/Bernstein.lean | 109 | 114 | theorem probability (n : ℕ) (x : I) : (∑ k : Fin (n + 1), bernstein n k x) = 1 := by |
have := bernsteinPolynomial.sum ℝ n
apply_fun fun p => Polynomial.aeval (x : ℝ) p at this
simp? [AlgHom.map_sum, Finset.sum_range] at this says
simp only [Finset.sum_range, map_sum, Polynomial.coe_aeval_eq_eval, map_one] at this
exact this
|
import Mathlib.Order.Filter.Basic
import Mathlib.Data.Set.Countable
#align_import order.filter.countable_Inter from "leanprover-community/mathlib"@"b9e46fe101fc897fb2e7edaf0bf1f09ea49eb81a"
open Set Filter
open Filter
variable {ι : Sort*} {α β : Type*}
class CountableInterFilter (l : Filter α) : Prop where
... | Mathlib/Order/Filter/CountableInter.lean | 71 | 75 | theorem eventually_countable_ball {ι : Type*} {S : Set ι} (hS : S.Countable)
{p : α → ∀ i ∈ S, Prop} :
(∀ᶠ x in l, ∀ i hi, p x i hi) ↔ ∀ i hi, ∀ᶠ x in l, p x i hi := by |
simpa only [Filter.Eventually, setOf_forall] using
@countable_bInter_mem _ l _ _ _ hS fun i hi => { x | p x i hi }
|
import Mathlib.LinearAlgebra.Pi
import Mathlib.LinearAlgebra.Quotient
#align_import linear_algebra.quotient_pi from "leanprover-community/mathlib"@"398f60f60b43ef42154bd2bdadf5133daf1577a4"
namespace Submodule
open LinearMap
variable {ι R : Type*} [CommRing R]
variable {Ms : ι → Type*} [∀ i, AddCommGroup (Ms i)... | Mathlib/LinearAlgebra/QuotientPi.lean | 50 | 60 | theorem piQuotientLift_single [Fintype ι] [DecidableEq ι] (p : ∀ i, Submodule R (Ms i))
(q : Submodule R N) (f : ∀ i, Ms i →ₗ[R] N) (hf : ∀ i, p i ≤ q.comap (f i)) (i)
(x : Ms i ⧸ p i) : piQuotientLift p q f hf (Pi.single i x) = mapQ _ _ (f i) (hf i) x := by |
simp_rw [piQuotientLift, lsum_apply, sum_apply, comp_apply, proj_apply]
rw [Finset.sum_eq_single i]
· rw [Pi.single_eq_same]
· rintro j - hj
rw [Pi.single_eq_of_ne hj, _root_.map_zero]
· intros
have := Finset.mem_univ i
contradiction
|
import Mathlib.Algebra.DirectSum.Basic
import Mathlib.LinearAlgebra.DFinsupp
import Mathlib.LinearAlgebra.Basis
#align_import algebra.direct_sum.module from "leanprover-community/mathlib"@"6623e6af705e97002a9054c1c05a980180276fc1"
universe u v w u₁
namespace DirectSum
open DirectSum
section General
variable {... | Mathlib/Algebra/DirectSum/Module.lean | 320 | 323 | theorem IsInternal.ofBijective_coeLinearMap_same (h : IsInternal A)
{i : ι} (x : A i) :
(LinearEquiv.ofBijective (coeLinearMap A) h).symm x i = x := by |
rw [← coeLinearMap_of, LinearEquiv.ofBijective_symm_apply_apply, of_eq_same]
|
import Mathlib.CategoryTheory.Sites.Coherent.Comparison
import Mathlib.Topology.Category.CompHaus.Limits
universe u
attribute [local instance] CategoryTheory.ConcreteCategory.instFunLike
open CategoryTheory Limits
namespace CompHaus
noncomputable
def struct {B X : CompHaus.{u}} (π : X ⟶ B) (hπ : Function.Surje... | Mathlib/Topology/Category/CompHaus/EffectiveEpi.lean | 102 | 142 | theorem effectiveEpiFamily_tfae
{α : Type} [Finite α] {B : CompHaus.{u}}
(X : α → CompHaus.{u}) (π : (a : α) → (X a ⟶ B)) :
TFAE
[ EffectiveEpiFamily X π
, Epi (Sigma.desc π)
, ∀ b : B, ∃ (a : α) (x : X a), π a x = b
] := by |
tfae_have 2 → 1
· intro
simpa [← effectiveEpi_desc_iff_effectiveEpiFamily, (effectiveEpi_tfae (Sigma.desc π)).out 0 1]
tfae_have 1 → 2
· intro; infer_instance
tfae_have 3 → 2
· intro e
rw [epi_iff_surjective]
intro b
obtain ⟨t, x, h⟩ := e b
refine ⟨Sigma.ι X t x, ?_⟩
change (Sigma.ι... |
import Mathlib.SetTheory.Ordinal.Arithmetic
namespace OrdinalApprox
universe u
variable {α : Type u}
variable [CompleteLattice α] (f : α →o α) (x : α)
open Function fixedPoints Cardinal Order OrderHom
set_option linter.unusedVariables false in
def lfpApprox (a : Ordinal.{u}) : α :=
sSup ({ f (lfpApprox b) | ... | Mathlib/SetTheory/Ordinal/FixedPointApproximants.lean | 116 | 133 | theorem lfpApprox_eq_of_mem_fixedPoints {a b : Ordinal} (h_init : x ≤ f x) (h_ab : a ≤ b)
(h: lfpApprox f x a ∈ fixedPoints f) : lfpApprox f x b = lfpApprox f x a := by |
rw [mem_fixedPoints_iff] at h
induction b using Ordinal.induction with | h b IH =>
apply le_antisymm
· conv => left; unfold lfpApprox
apply sSup_le
simp only [exists_prop, Set.union_singleton, Set.mem_insert_iff, Set.mem_setOf_eq,
forall_eq_or_imp, forall_exists_index, and_imp, forall_apply_eq_im... |
import Mathlib.Algebra.Lie.Abelian
import Mathlib.Algebra.Lie.Solvable
import Mathlib.LinearAlgebra.Dual
#align_import algebra.lie.character from "leanprover-community/mathlib"@"132328c4dd48da87adca5d408ca54f315282b719"
universe u v w w₁
namespace LieAlgebra
variable (R : Type u) (L : Type v) [CommRing R] [LieR... | Mathlib/Algebra/Lie/Character.lean | 49 | 50 | theorem lieCharacter_apply_lie' (χ : LieCharacter R L) (x y : L) : ⁅χ x, χ y⁆ = 0 := by |
rw [LieRing.of_associative_ring_bracket, mul_comm, sub_self]
|
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.special_functions.log.basic from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
open Set Filter Function
open Topology
noncomputable ... | Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 256 | 260 | theorem strictAntiOn_log : StrictAntiOn log (Set.Iio 0) := by |
rintro x (hx : x < 0) y (hy : y < 0) hxy
rw [← log_abs y, ← log_abs x]
refine log_lt_log (abs_pos.2 hy.ne) ?_
rwa [abs_of_neg hy, abs_of_neg hx, neg_lt_neg_iff]
|
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.MeasureTheory.Covering.OneDim
import Mathlib.Order.Monotone.Extension
#align_import analysis.calculus.monotone from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
open Set Filter Function Metric MeasureTheory MeasureTheory.Meas... | Mathlib/Analysis/Calculus/Monotone.lean | 218 | 220 | theorem Monotone.ae_differentiableAt {f : ℝ → ℝ} (hf : Monotone f) :
∀ᵐ x, DifferentiableAt ℝ f x := by |
filter_upwards [hf.ae_hasDerivAt] with x hx using hx.differentiableAt
|
import Mathlib.LinearAlgebra.Contraction
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
#align_import linear_algebra.trace from "leanprover-community/mathlib"@"4cf7ca0e69e048b006674cf4499e5c7d296a89e0"
noncomputable section
universe u v w
namespace LinearMap
open Matrix
open FiniteDimensional
open Tensor... | Mathlib/LinearAlgebra/Trace.lean | 172 | 174 | theorem trace_eq_contract_apply (x : Module.Dual R M ⊗[R] M) :
(LinearMap.trace R M) ((dualTensorHom R M M) x) = contractLeft R M x := by |
rw [← comp_apply, trace_eq_contract]
|
import Mathlib.MeasureTheory.Measure.GiryMonad
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.MeasureTheory.Measure.OpenPos
#align_import measure_theory.constructions.prod.basic from "leanprover-community/mathlib"@"00abe0695d8767201e6d008afa22393978bb32... | Mathlib/MeasureTheory/Constructions/Prod/Basic.lean | 631 | 641 | theorem prod_eq_generateFrom {μ : Measure α} {ν : Measure β} {C : Set (Set α)} {D : Set (Set β)}
(hC : generateFrom C = ‹_›) (hD : generateFrom D = ‹_›) (h2C : IsPiSystem C)
(h2D : IsPiSystem D) (h3C : μ.FiniteSpanningSetsIn C) (h3D : ν.FiniteSpanningSetsIn D)
{μν : Measure (α × β)} (h₁ : ∀ s ∈ C, ∀ t ∈ D, ... |
refine
(h3C.prod h3D).ext
(generateFrom_eq_prod hC hD h3C.isCountablySpanning h3D.isCountablySpanning).symm
(h2C.prod h2D) ?_
rintro _ ⟨s, hs, t, ht, rfl⟩
haveI := h3D.sigmaFinite
rw [h₁ s hs t ht, prod_prod]
|
import Mathlib.Geometry.Manifold.Diffeomorph
import Mathlib.Geometry.Manifold.Instances.Real
import Mathlib.Geometry.Manifold.PartitionOfUnity
#align_import geometry.manifold.whitney_embedding from "leanprover-community/mathlib"@"86c29aefdba50b3f33e86e52e3b2f51a0d8f0282"
universe uι uE uH uM
variable {ι : Type u... | Mathlib/Geometry/Manifold/WhitneyEmbedding.lean | 83 | 98 | theorem comp_embeddingPiTangent_mfderiv (x : M) (hx : x ∈ s) :
((ContinuousLinearMap.fst ℝ E ℝ).comp
(@ContinuousLinearMap.proj ℝ _ ι (fun _ => E × ℝ) _ _ (fun _ => inferInstance)
(f.ind x hx))).comp
(mfderiv I 𝓘(ℝ, ι → E × ℝ) f.embeddingPiTangent x) =
mfderiv I I (chartAt H... |
set L :=
(ContinuousLinearMap.fst ℝ E ℝ).comp
(@ContinuousLinearMap.proj ℝ _ ι (fun _ => E × ℝ) _ _ (fun _ => inferInstance) (f.ind x hx))
have := L.hasMFDerivAt.comp x f.embeddingPiTangent.smooth.mdifferentiableAt.hasMFDerivAt
convert hasMFDerivAt_unique this _
refine (hasMFDerivAt_extChartAt I (f.m... |
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 | 710 | 712 | theorem affineCombinationLineMapWeights_apply_left [DecidableEq ι] {i j : ι} (h : i ≠ j) (c : k) :
affineCombinationLineMapWeights i j c i = 1 - c := by |
simp [affineCombinationLineMapWeights, h.symm, sub_eq_neg_add]
|
import Mathlib.Algebra.Algebra.Defs
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Localization.Basic
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Surreal.Basic
#align_import set_theory.surreal.dyadic from "leanprover-community/mathlib"@"9... | Mathlib/SetTheory/Surreal/Dyadic.lean | 116 | 117 | theorem powHalf_pos (n : ℕ) : 0 < powHalf n := by |
rw [← lf_iff_lt numeric_zero (numeric_powHalf n), zero_lf_le]; simp
|
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Group.Int
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Ring.Rat
import Mathlib.Data.PNat.Defs
#align_import data.rat.lemmas from "leanprover-community/mathlib"@"550b58538991c8977703fdeb7c9d51a5aa27df11"
namespace Rat
o... | Mathlib/Data/Rat/Lemmas.lean | 81 | 84 | theorem add_den_dvd (q₁ q₂ : ℚ) : (q₁ + q₂).den ∣ q₁.den * q₂.den := by |
rw [add_def, normalize_eq]
apply Nat.div_dvd_of_dvd
apply Nat.gcd_dvd_right
|
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 | 941 | 943 | theorem setLintegral_pos_iff {f : α → ℝ≥0∞} (hf : Measurable f) {s : Set α} :
0 < ∫⁻ a in s, f a ∂μ ↔ 0 < μ (Function.support f ∩ s) := by |
rw [lintegral_pos_iff_support hf, Measure.restrict_apply (measurableSet_support hf)]
|
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.CategoryTheory.Groupoid.VertexGroup
import Mathlib.CategoryTheory.Groupoid.Basic
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Data.Set.Lattice
import Mathlib.Order.GaloisConnection
#align_import category_theory.groupoid.subgroupoid from "leanprover-c... | Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | 358 | 360 | theorem sInf_isNormal (s : Set <| Subgroupoid C) (sn : ∀ S ∈ s, IsNormal S) : IsNormal (sInf s) :=
{ wide := by | simp_rw [sInf, mem_iInter₂]; exact fun c S Ss => (sn S Ss).wide c
conj := by simp_rw [sInf, mem_iInter₂]; exact fun p γ hγ S Ss => (sn S Ss).conj p (hγ S Ss) }
|
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 | 532 | 534 | theorem oangle_sub_left {x y z : V} (hx : x ≠ 0) (hy : y ≠ 0) (hz : z ≠ 0) :
o.oangle x z - o.oangle x y = o.oangle y z := by |
rw [sub_eq_iff_eq_add, o.oangle_add_swap hx hy hz]
|
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Fin
import Mathlib.GroupTheory.GroupAction.Pi
import Mathlib.Logic.Equiv.Fin
#align_import algebra.big_operators.fin from "leanprover-community/mathlib"@"cc5dd6244981976cc9da7afc4eee5682b037a013"
open Fins... | Mathlib/Algebra/BigOperators/Fin.lean | 69 | 72 | theorem prod_univ_succAbove [CommMonoid β] {n : ℕ} (f : Fin (n + 1) → β) (x : Fin (n + 1)) :
∏ i, f i = f x * ∏ i : Fin n, f (x.succAbove i) := by |
rw [univ_succAbove, prod_cons, Finset.prod_map _ x.succAboveEmb]
rfl
|
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Algebra.Group.Submonoid.MulOpposite
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Finset.NoncommProd
import Mathlib.Data.Int.Order.Lemmas
#align_import group_theory.submonoid.membership fro... | Mathlib/Algebra/Group/Submonoid/Membership.lean | 332 | 333 | theorem mem_closure_singleton {x y : M} : y ∈ closure ({x} : Set M) ↔ ∃ n : ℕ, x ^ n = y := by |
rw [closure_singleton_eq, mem_mrange]; rfl
|
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.RingTheory.SimpleModule
#align_import representation_theory.maschke from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v w
noncomputable section
open Module MonoidAlgeb... | Mathlib/RepresentationTheory/Maschke.lean | 125 | 127 | theorem equivariantProjection_apply (v : W) :
π.equivariantProjection G v = ⅟(Fintype.card G : k) • ∑ g : G, π.conjugate g v := by |
simp only [equivariantProjection, smul_apply, sumOfConjugatesEquivariant_apply]
|
import Mathlib.Probability.Notation
import Mathlib.Probability.Density
import Mathlib.Probability.ConditionalProbability
import Mathlib.Probability.ProbabilityMassFunction.Constructions
open scoped Classical MeasureTheory NNReal ENNReal
-- TODO: We can't `open ProbabilityTheory` without opening the `ProbabilityThe... | Mathlib/Probability/Distributions/Uniform.lean | 172 | 180 | theorem integral_eq (huX : IsUniform X s ℙ) :
∫ x, X x ∂ℙ = (volume s)⁻¹.toReal * ∫ x in s, x := by |
rw [← smul_eq_mul, ← integral_smul_measure]
dsimp only [IsUniform, ProbabilityTheory.cond] at huX
rw [← huX]
by_cases hX : AEMeasurable X ℙ
· exact (integral_map hX aestronglyMeasurable_id).symm
· rw [map_of_not_aemeasurable hX, integral_zero_measure, integral_non_aestronglyMeasurable]
rwa [aestronglyM... |
import Mathlib.Analysis.Complex.RealDeriv
import Mathlib.Analysis.Calculus.ContDiff.RCLike
import Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas
#align_import analysis.special_functions.exp_deriv from "leanprover-community/mathlib"@"6a5c85000ab93fe5dcfdf620676f614ba8e18c26"
noncomputable section
open Filter Asym... | Mathlib/Analysis/SpecialFunctions/ExpDeriv.lean | 64 | 73 | theorem contDiff_exp : ∀ {n}, ContDiff 𝕜 n exp := by |
-- Porting note: added `@` due to `∀ {n}` weirdness above
refine @(contDiff_all_iff_nat.2 fun n => ?_)
have : ContDiff ℂ (↑n) exp := by
induction' n with n ihn
· exact contDiff_zero.2 continuous_exp
· rw [contDiff_succ_iff_deriv]
use differentiable_exp
rwa [deriv_exp]
exact this.restric... |
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.Polynomial.CancelLeads
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.FieldDivision
#align_import ring_theory.polynomial.content from "leanprover-community/mathlib"@"7a030ab8eb5d99f05a891dccc49c5b5b90c947d3"
namespace Po... | Mathlib/RingTheory/Polynomial/Content.lean | 109 | 129 | theorem content_X_mul {p : R[X]} : content (X * p) = content p := by |
rw [content, content, Finset.gcd_def, Finset.gcd_def]
refine congr rfl ?_
have h : (X * p).support = p.support.map ⟨Nat.succ, Nat.succ_injective⟩ := by
ext a
simp only [exists_prop, Finset.mem_map, Function.Embedding.coeFn_mk, Ne, mem_support_iff]
cases' a with a
· simp [coeff_X_mul_zero, Nat.suc... |
import Mathlib.Data.List.Range
import Mathlib.Data.List.Perm
#align_import data.list.sigma from "leanprover-community/mathlib"@"f808feb6c18afddb25e66a71d317643cf7fb5fbb"
universe u v
namespace List
variable {α : Type u} {β : α → Type v} {l l₁ l₂ : List (Sigma β)}
def keys : List (Sigma β) → List α :=
map ... | Mathlib/Data/List/Sigma.lean | 595 | 597 | theorem dlookup_kinsert {a} {b : β a} (l : List (Sigma β)) :
dlookup a (kinsert a b l) = some b := by |
simp only [kinsert, dlookup_cons_eq]
|
import Mathlib.Algebra.Category.GroupCat.Basic
import Mathlib.CategoryTheory.Limits.Shapes.ZeroObjects
#align_import algebra.category.Group.zero from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open CategoryTheory
open CategoryTheory.Limits
universe u
namespace GroupCat
@[to_addi... | Mathlib/Algebra/Category/GroupCat/Zero.lean | 28 | 34 | theorem isZero_of_subsingleton (G : GroupCat) [Subsingleton G] : IsZero G := by |
refine ⟨fun X => ⟨⟨⟨1⟩, fun f => ?_⟩⟩, fun X => ⟨⟨⟨1⟩, fun f => ?_⟩⟩⟩
· ext x
have : x = 1 := Subsingleton.elim _ _
rw [this, map_one, map_one]
· ext
apply Subsingleton.elim
|
import Mathlib.Algebra.Quaternion
import Mathlib.Tactic.Ring
#align_import algebra.quaternion_basis from "leanprover-community/mathlib"@"3aa5b8a9ed7a7cabd36e6e1d022c9858ab8a8c2d"
open Quaternion
namespace QuaternionAlgebra
structure Basis {R : Type*} (A : Type*) [CommRing R] [Ring A] [Algebra R A] (c₁ c₂ : R) ... | Mathlib/Algebra/QuaternionBasis.lean | 104 | 106 | theorem k_mul_k : q.k * q.k = -((c₁ * c₂) • (1 : A)) := by |
rw [← i_mul_j, mul_assoc, ← mul_assoc q.j _ _, j_mul_i, ← i_mul_j, ← mul_assoc, mul_neg, ←
mul_assoc, i_mul_i, smul_mul_assoc, one_mul, neg_mul, smul_mul_assoc, j_mul_j, smul_smul]
|
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 | 62 | 66 | theorem principal_one_iff {op : Ordinal → Ordinal → Ordinal} : Principal op 1 ↔ op 0 0 = 0 := by |
refine ⟨fun h => ?_, fun h a b ha hb => ?_⟩
· rw [← lt_one_iff_zero]
exact h zero_lt_one zero_lt_one
· rwa [lt_one_iff_zero, ha, hb] at *
|
import Mathlib.Algebra.DualNumber
import Mathlib.Algebra.QuaternionBasis
import Mathlib.Data.Complex.Module
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Star
import Mathlib.LinearAlgebra.QuadraticForm.Prod
#align_import linear_algebra.clifford_algebra.equivs fr... | Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean | 223 | 228 | theorem reverse_apply (x : CliffordAlgebra Q) : reverse (R := ℝ) x = x := by |
induction x using CliffordAlgebra.induction with
| algebraMap r => exact reverse.commutes _
| ι x => rw [reverse_ι]
| mul x₁ x₂ hx₁ hx₂ => rw [reverse.map_mul, mul_comm, hx₁, hx₂]
| add x₁ x₂ hx₁ hx₂ => rw [reverse.map_add, hx₁, hx₂]
|
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Set.Finite
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.Data.Setoid.Basic
import Mathlib.GroupTheory.GroupAction.Defs
import Mathlib.GroupTheory.GroupAction.Group
#align_import group_theory.group_action.basic fro... | Mathlib/GroupTheory/GroupAction/Basic.lean | 312 | 317 | theorem smul_cancel_of_non_zero_divisor {M R : Type*} [Monoid M] [NonUnitalNonAssocRing R]
[DistribMulAction M R] (k : M) (h : ∀ x : R, k • x = 0 → x = 0) {a b : R} (h' : k • a = k • b) :
a = b := by |
rw [← sub_eq_zero]
refine h _ ?_
rw [smul_sub, h', sub_self]
|
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Finset.Card
import Mathlib.Data.List.NodupEquivFin
import Mathlib.Data.Set.Image
#align_import data.fintype.card from "leanprover-community/mathlib"@"bf2428c9486c407ca38b5b3fb10b87dad0bc99fa"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
open Fu... | Mathlib/Data/Fintype/Card.lean | 139 | 140 | theorem card_of_finset' {p : Set α} (s : Finset α) (H : ∀ x, x ∈ s ↔ x ∈ p) [Fintype p] :
Fintype.card p = s.card := by | rw [← card_ofFinset s H]; congr; apply Subsingleton.elim
|
import Mathlib.Data.W.Basic
#align_import data.pfunctor.univariate.basic from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
-- "W", "Idx"
set_option linter.uppercaseLean3 false
universe u v v₁ v₂ v₃
@[pp_with_univ]
structure PFunctor where
A : Type u
B : A → Type u
#align p... | Mathlib/Data/PFunctor/Univariate/Basic.lean | 158 | 162 | theorem iget_map [DecidableEq P.A] [Inhabited α] [Inhabited β] (x : P α)
(f : α → β) (i : P.Idx) (h : i.1 = x.1) : (P.map f x).iget i = f (x.iget i) := by |
simp only [Obj.iget, fst_map, *, dif_pos, eq_self_iff_true]
cases x
rfl
|
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.MeasureTheory.Constructio... | Mathlib/MeasureTheory/Function/Jacobian.lean | 1,178 | 1,187 | theorem integrableOn_image_iff_integrableOn_abs_det_fderiv_smul (hs : MeasurableSet s)
(hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x) (hf : InjOn f s) (g : E → F) :
IntegrableOn g (f '' s) μ ↔ IntegrableOn (fun x => |(f' x).det| • g (f x)) s μ := by |
rw [IntegrableOn, ← restrict_map_withDensity_abs_det_fderiv_eq_addHaar μ hs hf' hf,
(measurableEmbedding_of_fderivWithin hs hf' hf).integrable_map_iff]
simp only [Set.restrict_eq, ← Function.comp.assoc, ENNReal.ofReal]
rw [← (MeasurableEmbedding.subtype_coe hs).integrable_map_iff, map_comap_subtype_coe hs,
... |
import Mathlib.Probability.Martingale.Upcrossing
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.Constructions.Polish
#align_import probability.martingale.convergence from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open TopologicalSpace Filter Me... | Mathlib/Probability/Martingale/Convergence.lean | 110 | 127 | theorem not_frequently_of_upcrossings_lt_top (hab : a < b) (hω : upcrossings a b f ω ≠ ∞) :
¬((∃ᶠ n in atTop, f n ω < a) ∧ ∃ᶠ n in atTop, b < f n ω) := by |
rw [← lt_top_iff_ne_top, upcrossings_lt_top_iff] at hω
replace hω : ∃ k, ∀ N, upcrossingsBefore a b f N ω < k := by
obtain ⟨k, hk⟩ := hω
exact ⟨k + 1, fun N => lt_of_le_of_lt (hk N) k.lt_succ_self⟩
rintro ⟨h₁, h₂⟩
rw [frequently_atTop] at h₁ h₂
refine Classical.not_not.2 hω ?_
push_neg
intro k
... |
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.sums.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
namespace CategoryTheory
universe v₁ u₁
-- morphism levels before object levels. See note [category_theory universes].
open Sum
section
variable (C : Ty... | Mathlib/CategoryTheory/Sums/Basic.lean | 66 | 67 | theorem hom_inr_inl_false {X : C} {Y : D} (f : Sum.inr X ⟶ Sum.inl Y) : False := by |
cases f
|
import Mathlib.Combinatorics.Quiver.Basic
#align_import combinatorics.quiver.push from "leanprover-community/mathlib"@"2258b40dacd2942571c8ce136215350c702dc78f"
namespace Quiver
universe v v₁ v₂ u u₁ u₂
variable {V : Type*} [Quiver V] {W : Type*} (σ : V → W)
@[nolint unusedArguments]
def Push (_ : V → W) :=
... | Mathlib/Combinatorics/Quiver/Push.lean | 73 | 89 | theorem lift_comp : (of σ ⋙q lift σ φ τ h) = φ := by |
fapply Prefunctor.ext
· rintro X
simp only [Prefunctor.comp_obj]
apply Eq.symm
exact h X
· rintro X Y f
simp only [Prefunctor.comp_map]
apply eq_of_heq
iterate 2 apply (cast_heq _ _).trans
apply HEq.symm
apply (eqRec_heq _ _).trans
have : ∀ {α γ} {β : α → γ → Sort _} {a a'} (p... |
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic from "leanprover-community/mathlib"@"31b269b60935483943542d547a6dd83a66b37dc7"
assert_not_exists Field
assert_not_exists Module
noncomputable section
open Function Cardinal Set Equiv Order
open sc... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,076 | 1,079 | theorem mul_add_mod_self (x y z : Ordinal) : (x * y + z) % x = z % x := by |
rcases eq_or_ne x 0 with rfl | hx
· simp
· rwa [mod_def, mul_add_div, mul_add, ← sub_sub, add_sub_cancel, mod_def]
|
import Mathlib.CategoryTheory.Opposites
#align_import category_theory.eq_to_hom from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe v₁ v₂ v₃ u₁ u₂ u₃
-- morphism levels before object levels. See note [CategoryTheory universes].
namespace CategoryTheory
open Opposite
variable ... | Mathlib/CategoryTheory/EqToHom.lean | 245 | 250 | theorem congr_inv_of_congr_hom (F G : C ⥤ D) {X Y : C} (e : X ≅ Y) (hX : F.obj X = G.obj X)
(hY : F.obj Y = G.obj Y)
(h₂ : F.map e.hom = eqToHom (by rw [hX]) ≫ G.map e.hom ≫ eqToHom (by rw [hY])) :
F.map e.inv = eqToHom (by rw [hY]) ≫ G.map e.inv ≫ eqToHom (by rw [hX]) := by |
simp only [← IsIso.Iso.inv_hom e, Functor.map_inv, h₂, IsIso.inv_comp, inv_eqToHom,
Category.assoc]
|
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Sign
import Mathlib.LinearAlgebra.AffineSpace.Combination
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
import Mathlib.LinearAlgebra.Basis.VectorSpace
#align_import linear_algebra.affine_space.independent from "leanprover-c... | Mathlib/LinearAlgebra/AffineSpace/Independent.lean | 234 | 248 | theorem affineIndependent_iff_eq_of_fintype_affineCombination_eq [Fintype ι] (p : ι → P) :
AffineIndependent k p ↔ ∀ w1 w2 : ι → k, ∑ i, w1 i = 1 → ∑ i, w2 i = 1 →
Finset.univ.affineCombination k p w1 = Finset.univ.affineCombination k p w2 → w1 = w2 := by |
rw [affineIndependent_iff_indicator_eq_of_affineCombination_eq]
constructor
· intro h w1 w2 hw1 hw2 hweq
simpa only [Set.indicator_univ, Finset.coe_univ] using h _ _ w1 w2 hw1 hw2 hweq
· intro h s1 s2 w1 w2 hw1 hw2 hweq
have hw1' : (∑ i, (s1 : Set ι).indicator w1 i) = 1 := by
rwa [Finset.sum_indi... |
import Mathlib.Algebra.Associated
import Mathlib.NumberTheory.Divisors
#align_import algebra.is_prime_pow from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable {R : Type*} [CommMonoidWithZero R] (n p : R) (k : ℕ)
def IsPrimePow : Prop :=
∃ (p : R) (k : ℕ), Prime p ∧ 0 < k ∧ p... | Mathlib/Algebra/IsPrimePow.lean | 97 | 104 | theorem IsPrimePow.dvd {n m : ℕ} (hn : IsPrimePow n) (hm : m ∣ n) (hm₁ : m ≠ 1) : IsPrimePow m := by |
rw [isPrimePow_nat_iff] at hn ⊢
rcases hn with ⟨p, k, hp, _hk, rfl⟩
obtain ⟨i, hik, rfl⟩ := (Nat.dvd_prime_pow hp).1 hm
refine ⟨p, i, hp, ?_, rfl⟩
apply Nat.pos_of_ne_zero
rintro rfl
simp only [pow_zero, ne_eq, not_true_eq_false] at hm₁
|
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.MeanValue
#align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputable section
open Set Fin Filter Function
open scoped NNReal Topology
section Real
variab... | Mathlib/Analysis/Calculus/ContDiff/RCLike.lean | 87 | 101 | theorem HasFTaylorSeriesUpToOn.exists_lipschitzOnWith_of_nnnorm_lt {E F : Type*}
[NormedAddCommGroup E] [NormedSpace ℝ E] [NormedAddCommGroup F] [NormedSpace ℝ F] {f : E → F}
{p : E → FormalMultilinearSeries ℝ E F} {s : Set E} {x : E}
(hf : HasFTaylorSeriesUpToOn 1 f p (insert x s)) (hs : Convex ℝ s) (K : ℝ... |
set f' := fun y => continuousMultilinearCurryFin1 ℝ E F (p y 1)
have hder : ∀ y ∈ s, HasFDerivWithinAt f (f' y) s y := fun y hy =>
(hf.hasFDerivWithinAt le_rfl (subset_insert x s hy)).mono (subset_insert x s)
have hcont : ContinuousWithinAt f' s x :=
(continuousMultilinearCurryFin1 ℝ E F).continuousAt.co... |
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Data.Real.Basic
import Mathlib.Order.Interval.Set.Disjoint
#align_import data.real.basic from "leanprover-community/mathlib"@"cb42593171ba005beaaf4549fcfe0dece9ada4c9"
open scoped Classical
open Pointwise CauSeq
namespace Real
... | Mathlib/Data/Real/Archimedean.lean | 196 | 199 | theorem ciSup_const_zero {α : Sort*} : ⨆ _ : α, (0 : ℝ) = 0 := by |
cases isEmpty_or_nonempty α
· exact Real.iSup_of_isEmpty _
· exact ciSup_const
|
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts
#align_import category_theory.limits.constructions.zero_objects from "leanprover-community/mathlib"@"52a270e2ea4e342c2587c106f8be904524214a4... | Mathlib/CategoryTheory/Limits/Constructions/ZeroObjects.lean | 226 | 227 | theorem inr_pushoutZeroZeroIso_inv (X Y : C) [HasBinaryCoproduct X Y] :
coprod.inr ≫ (pushoutZeroZeroIso X Y).inv = pushout.inr := by | simp [Iso.comp_inv_eq]
|
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.NumberTheory.Padics.PadicNorm
import Mathlib.Analysis.Normed.Field.Basic
#align_import number_theory.padics.padic_numbers from "leanprover-community/mathlib"@"b9b2114f7711fec1c1e055d507f082f8ceb2c3b7"
noncomputable section
open scoped Classical
open Nat m... | Mathlib/NumberTheory/Padics/PadicNumbers.lean | 185 | 192 | theorem lift_index_left {f : PadicSeq p} (hf : ¬f ≈ 0) (v1 v3 : ℕ) :
padicNorm p (f (stationaryPoint hf)) =
padicNorm p (f (max v1 (max (stationaryPoint hf) v3))) := by |
apply stationaryPoint_spec hf
· apply le_trans
· apply le_max_left _ v3
· apply le_max_right
· exact le_rfl
|
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.MkIffOfInductiveProp
#align_import data.sum.basic from "leanprover-community/mathlib"@"bd9851ca476957ea4549eb19b40e7b5ade9428cc"
universe u v w x
variable {α : Type u} {α' : Type w} {β : Type v} {β' : Type x} {γ δ : Type*}
namespace Sum
#align sum.foral... | Mathlib/Data/Sum/Basic.lean | 215 | 215 | theorem isRight_left (h : LiftRel r s x (inr d)) : x.isRight := by | cases h; rfl
|
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 | 360 | 362 | theorem comp_sub_left (hf : IntervalIntegrable f volume a b) (c : ℝ) :
IntervalIntegrable (fun x => f (c - x)) volume (c - a) (c - b) := by |
simpa only [neg_sub, ← sub_eq_add_neg] using iff_comp_neg.mp (hf.comp_add_left c)
|
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.List.MinMax
import Mathlib.Algebra.Tropical.Basic
import Mathlib.Order.ConditionallyCompleteLattice.Finset
#align_import algebra.tropical.big_operators from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
variable {R S :... | Mathlib/Algebra/Tropical/BigOperators.lean | 106 | 108 | theorem trop_iInf [ConditionallyCompleteLinearOrder R] [Fintype S] (f : S → WithTop R) :
trop (⨅ i : S, f i) = ∑ i : S, trop (f i) := by |
rw [iInf, ← Set.image_univ, ← coe_univ, trop_sInf_image]
|
import Mathlib.Analysis.Complex.Basic
import Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle
#align_import analysis.complex.re_im_topology from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1e360b7e3cf6"
open Set
noncomputable section
namespace Complex
theorem isHomeomorphicTrivialFiber... | Mathlib/Analysis/Complex/ReImTopology.lean | 114 | 115 | theorem closure_setOf_re_lt (a : ℝ) : closure { z : ℂ | z.re < a } = { z | z.re ≤ a } := by |
simpa only [closure_Iio] using closure_preimage_re (Iio a)
|
import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
suppress_compilation
open Bornology
open Filter hiding map_smul
open scoped Classical NNReal Topology Uniformity
-- the `ₗ` subscript variables are for special cases about linear (as opposed to semilinear) maps
variable {𝕜 𝕜₂ 𝕜₃ E Eₗ F Fₗ G Gₗ 𝓕 : Type*}
... | Mathlib/Analysis/NormedSpace/OperatorNorm/NNNorm.lean | 139 | 143 | theorem exists_mul_lt_apply_of_lt_opNNNorm (f : E →SL[σ₁₂] F) {r : ℝ≥0} (hr : r < ‖f‖₊) :
∃ x, r * ‖x‖₊ < ‖f x‖₊ := by |
simpa only [not_forall, not_le, Set.mem_setOf] using
not_mem_of_lt_csInf (nnnorm_def f ▸ hr : r < sInf { c : ℝ≥0 | ∀ x, ‖f x‖₊ ≤ c * ‖x‖₊ })
(OrderBot.bddBelow _)
|
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
noncomputable section
universe u
namespace SetTheory
open scoped PGame
namespace PGame... | Mathlib/SetTheory/Game/Nim.lean | 187 | 187 | theorem nim_one_moveLeft (x) : (nim 1).moveLeft x = nim 0 := by | simp
|
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.RingTheory.Localization.Basic
#align_import ring_theory.localization.ideal from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74"
namespace IsLocalization
section CommRing
variable {R : Type*} [CommRing R] (M : Submonoid R... | Mathlib/RingTheory/Localization/Ideal.lean | 171 | 204 | theorem surjective_quotientMap_of_maximal_of_localization {I : Ideal S} [I.IsPrime] {J : Ideal R}
{H : J ≤ I.comap (algebraMap R S)} (hI : (I.comap (algebraMap R S)).IsMaximal) :
Function.Surjective (Ideal.quotientMap I (algebraMap R S) H) := by |
intro s
obtain ⟨s, rfl⟩ := Ideal.Quotient.mk_surjective s
obtain ⟨r, ⟨m, hm⟩, rfl⟩ := mk'_surjective M s
by_cases hM : (Ideal.Quotient.mk (I.comap (algebraMap R S))) m = 0
· have : I = ⊤ := by
rw [Ideal.eq_top_iff_one]
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_comap] at hM
convert I.mul... |
import Mathlib.Init.ZeroOne
import Mathlib.Data.Set.Defs
import Mathlib.Order.Basic
import Mathlib.Order.SymmDiff
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.ByContra
import Mathlib.Util.Delaborators
#align_import data.set.basic from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
... | Mathlib/Data/Set/Basic.lean | 2,197 | 2,199 | theorem powerset_singleton (x : α) : 𝒫({x} : Set α) = {∅, {x}} := by |
ext y
rw [mem_powerset_iff, subset_singleton_iff_eq, mem_insert_iff, mem_singleton_iff]
|
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.degree.lemmas from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
noncomputable section
open Polynomial
open Finsupp Finset
namespace Polynomial
universe u v w
variable {R : Type u} {S : Type v} {ι : Type w} {a b ... | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 426 | 431 | theorem irreducible_mul_leadingCoeff_inv {p : K[X]} :
Irreducible (p * C (leadingCoeff p)⁻¹) ↔ Irreducible p := by |
by_cases hp0 : p = 0
· simp [hp0]
exact irreducible_mul_isUnit
(isUnit_C.mpr (IsUnit.mk0 _ (inv_ne_zero (leadingCoeff_ne_zero.mpr hp0))))
|
import Mathlib.Algebra.ContinuedFractions.Basic
import Mathlib.Algebra.GroupWithZero.Basic
#align_import algebra.continued_fractions.translations from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
section WithDivisionRing
variable {K : Type*}... | Mathlib/Algebra/ContinuedFractions/Translations.lean | 184 | 186 | theorem convergents'Aux_succ_some {s : Stream'.Seq (Pair K)} {p : Pair K} (h : s.head = some p)
(n : ℕ) : convergents'Aux s (n + 1) = p.a / (p.b + convergents'Aux s.tail n) := by |
simp [convergents'Aux, h, convergents'Aux.match_1]
|
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.GroupTheory.MonoidLocalization
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.FractionRing
import M... | Mathlib/RingTheory/Localization/Integral.lean | 80 | 90 | theorem integerNormalization_spec (p : S[X]) :
∃ b : M, ∀ i, algebraMap R S ((integerNormalization M p).coeff i) = (b : R) • p.coeff i := by |
use Classical.choose (exist_integer_multiples_of_finset M (p.support.image p.coeff))
intro i
rw [integerNormalization_coeff, coeffIntegerNormalization]
split_ifs with hi
· exact
Classical.choose_spec
(Classical.choose_spec (exist_integer_multiples_of_finset M (p.support.image p.coeff))
... |
import Mathlib.CategoryTheory.Adjunction.Unique
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Sites.Sheaf
import Mathlib.CategoryTheory.Limits.Preserves.Finite
universe v₁ v₂ u₁ u₂
namespace CategoryTheory
open Limits
variable {C : Type u₁} [Category.{v₁} C] (J : Grothendiec... | Mathlib/CategoryTheory/Sites/Sheafification.lean | 195 | 199 | theorem sheafifyMap_sheafifyLift {P Q R : Cᵒᵖ ⥤ D} (η : P ⟶ Q) (γ : Q ⟶ R)
(hR : Presheaf.IsSheaf J R) :
sheafifyMap J η ≫ sheafifyLift J γ hR = sheafifyLift J (η ≫ γ) hR := by |
apply sheafifyLift_unique
rw [← Category.assoc, ← toSheafify_naturality, Category.assoc, toSheafify_sheafifyLift]
|
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 389 | 389 | theorem inv_lt' : a⁻¹ < b ↔ b⁻¹ < a := by | rw [← inv_lt_inv_iff, inv_inv]
|
import Mathlib.CategoryTheory.Opposites
#align_import category_theory.eq_to_hom from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe v₁ v₂ v₃ u₁ u₂ u₃
-- morphism levels before object levels. See note [CategoryTheory universes].
namespace CategoryTheory
open Opposite
variable ... | Mathlib/CategoryTheory/EqToHom.lean | 86 | 89 | theorem eqToHom_iso_hom_naturality {f g : β → C} (z : ∀ b, f b ≅ g b) {j j' : β} (w : j = j') :
(z j).hom ≫ eqToHom (by simp [w]) = eqToHom (by simp [w]) ≫ (z j').hom := by |
cases w
simp
|
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Mathlib.Analysis.NormedSpace.FunctionSeries
#align_import analysis.calculus.series from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982... | Mathlib/Analysis/Calculus/SmoothSeries.lean | 145 | 154 | theorem differentiable_tsum (hu : Summable u) (hf : ∀ n x, HasFDerivAt (f n) (f' n x) x)
(hf' : ∀ n x, ‖f' n x‖ ≤ u n) : Differentiable 𝕜 fun y => ∑' n, f n y := by |
by_cases h : ∃ x₀, Summable fun n => f n x₀
· rcases h with ⟨x₀, hf0⟩
intro x
exact (hasFDerivAt_tsum hu hf hf' hf0 x).differentiableAt
· push_neg at h
have : (fun x => ∑' n, f n x) = 0 := by ext1 x; exact tsum_eq_zero_of_not_summable (h x)
rw [this]
exact differentiable_const 0
|
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.NormedSpace.LinearIsometry
import Mathlib.Algebra.Star.SelfAdjoint
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.Algebra.Star.Unitary
import Mathlib.Topology.Algebra.Module.Star
#align_import analysis.no... | Mathlib/Analysis/NormedSpace/Star/Basic.lean | 149 | 150 | theorem mul_star_self_ne_zero_iff (x : E) : x * x⋆ ≠ 0 ↔ x ≠ 0 := by |
simp only [Ne, mul_star_self_eq_zero_iff]
|
import Mathlib.MeasureTheory.Function.ConvergenceInMeasure
import Mathlib.MeasureTheory.Function.L1Space
#align_import measure_theory.function.uniform_integrable from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
noncomputable section
open scoped Classical MeasureTheory NNReal ENNReal... | Mathlib/MeasureTheory/Function/UniformIntegrable.lean | 760 | 775 | theorem uniformIntegrable_finite [Finite ι] (hp_one : 1 ≤ p) (hp_top : p ≠ ∞)
(hf : ∀ i, Memℒp (f i) p μ) : UniformIntegrable f p μ := by |
cases nonempty_fintype ι
refine ⟨fun n => (hf n).1, unifIntegrable_finite hp_one hp_top hf, ?_⟩
by_cases hι : Nonempty ι
· choose _ hf using hf
set C := (Finset.univ.image fun i : ι => snorm (f i) p μ).max'
⟨snorm (f hι.some) p μ, Finset.mem_image.2 ⟨hι.some, Finset.mem_univ _, rfl⟩⟩
refine ⟨C.to... |
import Mathlib.Algebra.ContinuedFractions.Basic
import Mathlib.Algebra.GroupWithZero.Basic
#align_import algebra.continued_fractions.translations from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
section General
variable {α : Type*} {g : Gen... | Mathlib/Algebra/ContinuedFractions/Translations.lean | 53 | 55 | theorem terminatedAt_iff_part_denom_none :
g.TerminatedAt n ↔ g.partialDenominators.get? n = none := by |
rw [terminatedAt_iff_s_none, part_denom_none_iff_s_none]
|
import Mathlib.RingTheory.Valuation.Integers
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Localization.Integer
import Mathlib.RingTheory.DiscreteValuationRing.Basic
import Mathlib.RingTheory.Bezout
import Mathlib.Tactic.FieldSimp
#align_import... | Mathlib/RingTheory/Valuation/ValuationRing.lean | 303 | 307 | theorem unique_irreducible [ValuationRing R] ⦃p q : R⦄ (hp : Irreducible p) (hq : Irreducible q) :
Associated p q := by |
have := dvd_total p q
rw [Irreducible.dvd_comm hp hq, or_self_iff] at this
exact associated_of_dvd_dvd (Irreducible.dvd_symm hq hp this) this
|
import Mathlib.Analysis.BoxIntegral.DivergenceTheorem
import Mathlib.Analysis.BoxIntegral.Integrability
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.MeasureTheory.Constructions.Prod.Integral
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Analysis.Calculus.FDeriv.Equiv
#align_impo... | Mathlib/MeasureTheory/Integral/DivergenceTheorem.lean | 143 | 245 | theorem integral_divergence_of_hasFDerivWithinAt_off_countable_aux₂ (I : Box (Fin (n + 1)))
(f : ℝⁿ⁺¹ → Eⁿ⁺¹)
(f' : ℝⁿ⁺¹ → ℝⁿ⁺¹ →L[ℝ] Eⁿ⁺¹)
(s : Set ℝⁿ⁺¹) (hs : s.Countable) (Hc : ContinuousOn f (Box.Icc I))
(Hd : ∀ x ∈ Box.Ioo I \ s, HasFDerivAt f (f' x) x)
(Hi : IntegrableOn (∑ i, f' · (e i) i) (B... |
/- Choose a monotone sequence `J k` of subboxes that cover the interior of `I` and prove that
these boxes satisfy the assumptions of the previous lemma. -/
rcases I.exists_seq_mono_tendsto with ⟨J, hJ_sub, hJl, hJu⟩
have hJ_sub' : ∀ k, Box.Icc (J k) ⊆ Box.Icc I := fun k => (hJ_sub k).trans I.Ioo_subset_Icc
... |
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