Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | num_lines int64 1 150 |
|---|---|---|---|---|---|---|
import Mathlib.Algebra.CharP.Defs
import Mathlib.RingTheory.Multiplicity
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Polynomial
open Finset (antidiagonal mem_anti... | Mathlib/RingTheory/PowerSeries/Order.lean | 47 | 51 | theorem exists_coeff_ne_zero_iff_ne_zero : (∃ n : ℕ, coeff R n φ ≠ 0) ↔ φ ≠ 0 := by |
refine not_iff_not.mp ?_
push_neg
-- FIXME: the `FunLike.coe` doesn't seem to be picked up in the expression after #8386?
simp [PowerSeries.ext_iff, (coeff R _).map_zero]
| 4 |
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.Data.ZMod.Quotient
import Mathlib.RingTheory.DedekindDomain.AdicValuation
#align_import ring_theory.dedekind_domain.selmer_group from "leanprover-community/mathlib"@"2032a878972d5672e7c27c957e7a6e297b044973"
set_option quotPrecheck false
local notation K "... | Mathlib/RingTheory/DedekindDomain/SelmerGroup.lean | 150 | 155 | theorem valuation_of_unit_mod_eq (n : ℕ) (x : Rˣ) :
v.valuationOfNeZeroMod n (Units.map (algebraMap R K : R →* K) x : K/n) = 1 := by |
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [valuationOfNeZeroMod, MonoidHom.comp_apply, ← QuotientGroup.coe_mk',
QuotientGroup.map_mk' (G := Kˣ) (N := MonoidHom.range (powMonoidHom n)),
valuation_of_unit_eq, QuotientGroup.mk_one, map_one]
| 4 |
import Mathlib.Data.Set.Image
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Clopen
import Mathlib.Topology.Irreducible
#align_import topology.connected from "leanprover-community/mathlib"@"d101e93197bb5f6ea89bd7ba386b7f7dff1f3903"
open Set Function Topology TopologicalSpace Relation
open scoped C... | Mathlib/Topology/Connected/Basic.lean | 148 | 153 | theorem IsConnected.union {s t : Set α} (H : (s ∩ t).Nonempty) (Hs : IsConnected s)
(Ht : IsConnected t) : IsConnected (s ∪ t) := by |
rcases H with ⟨x, hx⟩
refine ⟨⟨x, mem_union_left t (mem_of_mem_inter_left hx)⟩, ?_⟩
exact Hs.isPreconnected.union x (mem_of_mem_inter_left hx) (mem_of_mem_inter_right hx)
Ht.isPreconnected
| 4 |
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Adjunction.Reflective
#align_import algebraic_geometry.Gamma_Spec_adjunction from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
-- Explicit universe annotations were u... | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | 91 | 95 | theorem toΓSpec_continuous : Continuous X.toΓSpecFun := by |
rw [isTopologicalBasis_basic_opens.continuous_iff]
rintro _ ⟨r, rfl⟩
erw [X.toΓSpec_preim_basicOpen_eq r]
exact (X.toRingedSpace.basicOpen r).2
| 4 |
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.Convex.StrictConvexSpace
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
import Mathlib.MeasureTheory.Integral.Average
#align_import analysis.convex.integral from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Mea... | Mathlib/Analysis/Convex/Integral.lean | 112 | 119 | theorem ConvexOn.average_mem_epigraph [IsFiniteMeasure μ] [NeZero μ] (hg : ConvexOn ℝ s g)
(hgc : ContinuousOn g s) (hsc : IsClosed s) (hfs : ∀ᵐ x ∂μ, f x ∈ s)
(hfi : Integrable f μ) (hgi : Integrable (g ∘ f) μ) :
(⨍ x, f x ∂μ, ⨍ x, g (f x) ∂μ) ∈ {p : E × ℝ | p.1 ∈ s ∧ g p.1 ≤ p.2} := by |
have ht_mem : ∀ᵐ x ∂μ, (f x, g (f x)) ∈ {p : E × ℝ | p.1 ∈ s ∧ g p.1 ≤ p.2} :=
hfs.mono fun x hx => ⟨hx, le_rfl⟩
exact average_pair hfi hgi ▸
hg.convex_epigraph.average_mem (hsc.epigraph hgc) ht_mem (hfi.prod_mk hgi)
| 4 |
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.pointwise from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set
open Pointwise Topology
variable {𝕜 E : Type*}
variable [NormedField 𝕜]
sectio... | Mathlib/Analysis/NormedSpace/Pointwise.lean | 84 | 88 | theorem smul_ball {c : 𝕜} (hc : c ≠ 0) (x : E) (r : ℝ) : c • ball x r = ball (c • x) (‖c‖ * r) := by |
ext y
rw [mem_smul_set_iff_inv_smul_mem₀ hc]
conv_lhs => rw [← inv_smul_smul₀ hc x]
simp [← div_eq_inv_mul, div_lt_iff (norm_pos_iff.2 hc), mul_comm _ r, dist_smul₀]
| 4 |
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Data.Set.MemPartition
import Mathlib.Order.Filter.CountableSeparatingOn
open Set MeasureTheory
namespace MeasurableSpace
variable {α β : Type*}
class CountablyGenerated (α : Type*) [m : MeasurableSpace α] : Prop where
isCountablyGenerated : ∃ b... | Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean | 96 | 101 | theorem CountablyGenerated.comap [m : MeasurableSpace β] [h : CountablyGenerated β] (f : α → β) :
@CountablyGenerated α (.comap f m) := by |
rcases h with ⟨⟨b, hbc, rfl⟩⟩
rw [comap_generateFrom]
letI := generateFrom (preimage f '' b)
exact ⟨_, hbc.image _, rfl⟩
| 4 |
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Ring.Subring.Pointwise
import Mathlib.RingTheory.Adjoin.Basic
#align_import algebra.algebra.subalgebra.pointwise from "leanprover-community/mathlib"@"b2c707cd190a58ea0565c86695a19e99ccecc215"
namespace... | Mathlib/Algebra/Algebra/Subalgebra/Pointwise.lean | 27 | 32 | theorem mul_toSubmodule_le (S T : Subalgebra R A) :
(Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by |
rw [Submodule.mul_le]
intro y hy z hz
show y * z ∈ S ⊔ T
exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)
| 4 |
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Group.Int
import Mathlib.Algebra.GroupWithZero.Semiconj
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Order.Bounds.Basic
#align_import data.int.gcd from "leanprover-community/mathlib"@"47a1a73351de8dd6c8d3d32b569c8e434b03ca47"
namespace Nat
... | Mathlib/Data/Int/GCD.lean | 94 | 98 | theorem gcdB_zero_right {s : ℕ} (h : s ≠ 0) : gcdB s 0 = 0 := by |
unfold gcdB xgcd
obtain ⟨s, rfl⟩ := Nat.exists_eq_succ_of_ne_zero h
rw [xgcdAux]
simp
| 4 |
import Batteries.Data.DList
import Mathlib.Mathport.Rename
import Mathlib.Tactic.Cases
#align_import data.dlist from "leanprover-community/lean"@"855e5b74e3a52a40552e8f067169d747d48743fd"
universe u
#align dlist Batteries.DList
namespace Batteries.DList
open Function
variable {α : Type u}
#align dlist.of_list... | Mathlib/Data/DList/Defs.lean | 62 | 66 | theorem ofList_toList (l : DList α) : DList.ofList (DList.toList l) = l := by |
cases' l with app inv
simp only [ofList, toList, mk.injEq]
funext x
rw [(inv x)]
| 4 |
import Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity
#align_import number_theory.legendre_symbol.jacobi_symbol from "leanprover-community/mathlib"@"74a27133cf29446a0983779e37c8f829a85368f3"
section Jacobi
open Nat ZMod
-- Since we need the fact that the factors are prime, we use `List.pmap`.
def ... | Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean | 159 | 163 | theorem mul_left (a₁ a₂ : ℤ) (b : ℕ) : J(a₁ * a₂ | b) = J(a₁ | b) * J(a₂ | b) := by |
simp_rw [jacobiSym, List.pmap_eq_map_attach, legendreSym.mul _ _ _];
exact List.prod_map_mul (α := ℤ) (l := (factors b).attach)
(f := fun x ↦ @legendreSym x {out := prime_of_mem_factors x.2} a₁)
(g := fun x ↦ @legendreSym x {out := prime_of_mem_factors x.2} a₂)
| 4 |
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.GroupTheory.Solvable
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.Sylow
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Tactic.TFAE
#align_import group_theory.nilpotent from "leanprover-community/mathlib"@"2bbc7e3884ba234309d2a43b19144... | Mathlib/GroupTheory/Nilpotent.lean | 210 | 214 | theorem upperCentralSeries_mono : Monotone (upperCentralSeries G) := by |
refine monotone_nat_of_le_succ ?_
intro n x hx y
rw [mul_assoc, mul_assoc, ← mul_assoc y x⁻¹ y⁻¹]
exact mul_mem hx (Normal.conj_mem (upperCentralSeries_normal G n) x⁻¹ (inv_mem hx) y)
| 4 |
import Mathlib.AlgebraicTopology.SplitSimplicialObject
import Mathlib.AlgebraicTopology.DoldKan.Degeneracies
import Mathlib.AlgebraicTopology.DoldKan.FunctorN
#align_import algebraic_topology.dold_kan.split_simplicial_object from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open Categ... | Mathlib/AlgebraicTopology/DoldKan/SplitSimplicialObject.lean | 127 | 132 | theorem PInfty_comp_πSummand_id (n : ℕ) :
PInfty.f n ≫ s.πSummand (IndexSet.id (op [n])) = s.πSummand (IndexSet.id (op [n])) := by |
conv_rhs => rw [← id_comp (s.πSummand _)]
symm
rw [← sub_eq_zero, ← sub_comp, ← comp_PInfty_eq_zero_iff, sub_comp, id_comp, PInfty_f_idem,
sub_self]
| 4 |
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 | 29 | 34 | theorem Set.restrictPreimage_inducing (s : Set β) (h : Inducing f) :
Inducing (s.restrictPreimage f) := by |
simp_rw [← inducing_subtype_val.of_comp_iff, inducing_iff_nhds, restrictPreimage,
MapsTo.coe_restrict, restrict_eq, ← @Filter.comap_comap _ _ _ _ _ f, Function.comp_apply] at h ⊢
intro a
rw [← h, ← inducing_subtype_val.nhds_eq_comap]
| 4 |
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.Calculus.Deriv.Inv
#align_import analysis.calculus.dslope from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
open scoped Classical Topology Filter
open Function Set Filter
variable {𝕜 E : Type*} [NontriviallyNormed... | Mathlib/Analysis/Calculus/Dslope.lean | 106 | 111 | theorem continuousWithinAt_dslope_of_ne (h : b ≠ a) :
ContinuousWithinAt (dslope f a) s b ↔ ContinuousWithinAt f s b := by |
refine ⟨ContinuousWithinAt.of_dslope, fun hc => ?_⟩
simp only [dslope, continuousWithinAt_update_of_ne h]
exact ((continuousWithinAt_id.sub continuousWithinAt_const).inv₀ (sub_ne_zero.2 h)).smul
(hc.sub continuousWithinAt_const)
| 4 |
import Mathlib.Topology.Order.IsLUB
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section DenselyOrdered
variable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}
{s : Set α}
theorem closure_Ioi' {a : α} (h : (Io... | Mathlib/Topology/Order/DenselyOrdered.lean | 66 | 70 | theorem closure_Ioc {a b : α} (hab : a ≠ b) : closure (Ioc a b) = Icc a b := by |
apply Subset.antisymm
· exact closure_minimal Ioc_subset_Icc_self isClosed_Icc
· apply Subset.trans _ (closure_mono Ioo_subset_Ioc_self)
rw [closure_Ioo hab]
| 4 |
import Mathlib.Dynamics.Ergodic.Ergodic
import Mathlib.MeasureTheory.Function.AEEqFun
open Function Set Filter MeasureTheory Topology TopologicalSpace
variable {α X : Type*} [MeasurableSpace α] {μ : MeasureTheory.Measure α}
| Mathlib/Dynamics/Ergodic/Function.lean | 27 | 35 | theorem QuasiErgodic.ae_eq_const_of_ae_eq_comp_of_ae_range₀ [Nonempty X] [MeasurableSpace X]
{s : Set X} [MeasurableSpace.CountablySeparated s] {f : α → α} {g : α → X}
(h : QuasiErgodic f μ) (hs : ∀ᵐ x ∂μ, g x ∈ s) (hgm : NullMeasurable g μ)
(hg_eq : g ∘ f =ᵐ[μ] g) :
∃ c, g =ᵐ[μ] const α c := by |
refine exists_eventuallyEq_const_of_eventually_mem_of_forall_separating MeasurableSet hs ?_
refine fun U hU ↦ h.ae_mem_or_ae_nmem₀ (s := g ⁻¹' U) (hgm hU) ?_b
refine (hg_eq.mono fun x hx ↦ ?_).set_eq
rw [← preimage_comp, mem_preimage, mem_preimage, hx]
| 4 |
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factors
import Mathlib.Order.Interval.Finset.Nat
#align_import number_theory.divisors from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open scoped Classical
open Finset
namespace Nat
variable (n : ℕ)
d... | Mathlib/NumberTheory/Divisors.lean | 147 | 151 | theorem divisor_le {m : ℕ} : n ∈ divisors m → n ≤ m := by |
cases' m with m
· simp
· simp only [mem_divisors, Nat.succ_ne_zero m, and_true_iff, Ne, not_false_iff]
exact Nat.le_of_dvd (Nat.succ_pos m)
| 4 |
import Mathlib.Data.List.Sublists
import Mathlib.Data.Multiset.Bind
#align_import data.multiset.powerset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
open List
variable {α : Type*}
-- Porting note (#11215): TODO: Write a more efficient version
def powerset... | Mathlib/Data/Multiset/Powerset.lean | 132 | 137 | theorem revzip_powersetAux' {l : List α} ⦃x⦄ (h : x ∈ revzip (powersetAux' l)) :
x.1 + x.2 = ↑l := by |
rw [revzip, powersetAux', ← map_reverse, zip_map, ← revzip, List.mem_map] at h
simp only [Prod.map_apply, Prod.exists] at h
rcases h with ⟨l₁, l₂, h, rfl, rfl⟩
exact Quot.sound (revzip_sublists' _ _ _ h)
| 4 |
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 | 147 | 154 | theorem realize_restrictVarLeft [DecidableEq α] {γ : Type*} {t : L.Term (Sum α γ)} {s : Set α}
(h : ↑t.varFinsetLeft ⊆ s) {v : α → M} {xs : γ → M} :
(t.restrictVarLeft (Set.inclusion h)).realize (Sum.elim (v ∘ (↑)) xs) =
t.realize (Sum.elim v xs) := by |
induction' t with a _ _ _ ih
· cases a <;> rfl
· simp_rw [varFinsetLeft, Finset.coe_biUnion, Set.iUnion_subset_iff] at h
exact congr rfl (funext fun i => ih i (h i (Finset.mem_univ i)))
| 4 |
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.LinearAlgebra.Dual
import Mathlib.LinearAlgebra.Contraction
import Mathlib.RingTheory.TensorProduct.Basic
#align_import representation_... | Mathlib/RepresentationTheory/Basic.lean | 238 | 245 | theorem ofModule_asModule_act (g : G) (x : RestrictScalars k (MonoidAlgebra k G) ρ.asModule) :
ofModule (k := k) (G := G) ρ.asModule g x = -- Porting note: more help with implicit
(RestrictScalars.addEquiv _ _ _).symm
(ρ.asModuleEquiv.symm (ρ g (ρ.asModuleEquiv (RestrictScalars.addEquiv _ _ _ x)))) :=... |
apply_fun RestrictScalars.addEquiv _ _ ρ.asModule using
(RestrictScalars.addEquiv _ _ ρ.asModule).injective
dsimp [ofModule, RestrictScalars.lsmul_apply_apply]
simp
| 4 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Denumerable
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.SetTheory.Cardinal.Continuum
#align_import data.real.cardinality from "leanprover-community/mathlib"@"7e7aaccf9b0182576cabdde36cf1b5ad3585b70d"
open Nat Set
open Cardinal
no... | Mathlib/Data/Real/Cardinality.lean | 105 | 110 | theorem cantorFunction_le (h1 : 0 ≤ c) (h2 : c < 1) (h3 : ∀ n, f n → g n) :
cantorFunction c f ≤ cantorFunction c g := by |
apply tsum_le_tsum _ (summable_cantor_function f h1 h2) (summable_cantor_function g h1 h2)
intro n; cases h : f n
· simp [h, cantorFunctionAux_nonneg h1]
replace h3 : g n = true := h3 n h; simp [h, h3]
| 4 |
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 | 63 | 69 | theorem sum_smul_index_linearMap' [Semiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid M₂]
[Module R M₂] {v : α →₀ M} {c : R} {h : α → M →ₗ[R] M₂} :
((c • v).sum fun a => h a) = c • v.sum fun a => h a := by |
rw [Finsupp.sum_smul_index', Finsupp.smul_sum]
· simp only [map_smul]
· intro i
exact (h i).map_zero
| 4 |
import Mathlib.CategoryTheory.Sites.Sheaf
#align_import category_theory.sites.plus from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace CategoryTheory.GrothendieckTopology
open CategoryTheory
open CategoryTheory.Limits
open Opposite
universe w v u
variable {C : Type u} [Ca... | Mathlib/CategoryTheory/Sites/Plus.lean | 90 | 95 | theorem diagramNatTrans_comp {P Q R : Cᵒᵖ ⥤ D} (η : P ⟶ Q) (γ : Q ⟶ R) (X : C) :
J.diagramNatTrans (η ≫ γ) X = J.diagramNatTrans η X ≫ J.diagramNatTrans γ X := by |
ext : 2
refine Multiequalizer.hom_ext _ _ _ (fun i => ?_)
dsimp
simp
| 4 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Constructions.Prod.Integral
import Mathlib.MeasureTheory.Function.LocallyIntegrable
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Group.Prod
import Mathlib.Measure... | Mathlib/Analysis/Convolution.lean | 239 | 246 | theorem ConvolutionExistsAt.ofNorm' {x₀ : G}
(h : ConvolutionExistsAt (fun x => ‖f x‖) (fun x => ‖g x‖) x₀ (mul ℝ ℝ) μ)
(hmf : AEStronglyMeasurable f μ) (hmg : AEStronglyMeasurable g <| map (fun t => x₀ - t) μ) :
ConvolutionExistsAt f g x₀ L μ := by |
refine (h.const_mul ‖L‖).mono'
(hmf.convolution_integrand_snd' L hmg) (eventually_of_forall fun x => ?_)
rw [mul_apply', ← mul_assoc]
apply L.le_opNorm₂
| 4 |
import Mathlib.Order.Filter.Bases
#align_import order.filter.pi from "leanprover-community/mathlib"@"ce64cd319bb6b3e82f31c2d38e79080d377be451"
open Set Function
open scoped Classical
open Filter
namespace Filter
variable {ι : Type*} {α : ι → Type*} {f f₁ f₂ : (i : ι) → Filter (α i)} {s : (i : ι) → Set (α i)}
... | Mathlib/Order/Filter/Pi.lean | 284 | 290 | theorem map_pi_map_coprodᵢ_le :
map (fun k : ∀ i, α i => fun i => m i (k i)) (Filter.coprodᵢ f) ≤
Filter.coprodᵢ fun i => map (m i) (f i) := by |
simp only [le_def, mem_map, mem_coprodᵢ_iff]
intro s h i
obtain ⟨t, H, hH⟩ := h i
exact ⟨{ x : α i | m i x ∈ t }, H, fun x hx => hH hx⟩
| 4 |
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 | 78 | 87 | theorem inner_weightedVSub {ι₁ : Type*} {s₁ : Finset ι₁} {w₁ : ι₁ → ℝ} (p₁ : ι₁ → P)
(h₁ : ∑ i ∈ s₁, w₁ i = 0) {ι₂ : Type*} {s₂ : Finset ι₂} {w₂ : ι₂ → ℝ} (p₂ : ι₂ → P)
(h₂ : ∑ i ∈ s₂, w₂ i = 0) :
⟪s₁.weightedVSub p₁ w₁, s₂.weightedVSub p₂ w₂⟫ =
(-∑ i₁ ∈ s₁, ∑ i₂ ∈ s₂, w₁ i₁ * w₂ i₂ * (dist (p₁ i₁) (p... |
rw [Finset.weightedVSub_apply, Finset.weightedVSub_apply,
inner_sum_smul_sum_smul_of_sum_eq_zero _ h₁ _ h₂]
simp_rw [vsub_sub_vsub_cancel_right]
rcongr (i₁ i₂) <;> rw [dist_eq_norm_vsub V (p₁ i₁) (p₂ i₂)]
| 4 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.RingTheory.Polynomial.Basic
#align_import linear_algebra.lagrange from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open Polynomial
section PolynomialDetermination
namespace Poly... | Mathlib/LinearAlgebra/Lagrange.lean | 55 | 60 | theorem eq_of_degree_sub_lt_of_eval_finset_eq (degree_fg_lt : (f - g).degree < s.card)
(eval_fg : ∀ x ∈ s, f.eval x = g.eval x) : f = g := by |
rw [← sub_eq_zero]
refine eq_zero_of_degree_lt_of_eval_finset_eq_zero _ degree_fg_lt ?_
simp_rw [eval_sub, sub_eq_zero]
exact eval_fg
| 4 |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 158 | 162 | theorem laverage_lt_top (hf : ∫⁻ x, f x ∂μ ≠ ∞) : ⨍⁻ x, f x ∂μ < ∞ := by |
obtain rfl | hμ := eq_or_ne μ 0
· simp
· rw [laverage_eq]
exact div_lt_top hf (measure_univ_ne_zero.2 hμ)
| 4 |
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.Algebra.Lie.IdealOperations
#align_import algebra.lie.abelian from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
universe u v w w₁ w₂
class LieModule.IsTrivial (L : Type v) (M : Type w) [Bracket L M] [Zero M] : Prop where
triv... | Mathlib/Algebra/Lie/Abelian.lean | 91 | 96 | theorem commutative_ring_iff_abelian_lie_ring {A : Type v} [Ring A] :
Std.Commutative (α := A) (· * ·) ↔ IsLieAbelian A := by |
have h₁ : Std.Commutative (α := A) (· * ·) ↔ ∀ a b : A, a * b = b * a :=
⟨fun h => h.1, fun h => ⟨h⟩⟩
have h₂ : IsLieAbelian A ↔ ∀ a b : A, ⁅a, b⁆ = 0 := ⟨fun h => h.1, fun h => ⟨h⟩⟩
simp only [h₁, h₂, LieRing.of_associative_ring_bracket, sub_eq_zero]
| 4 |
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section iInf
variable {ι : Sort*} {f g : ι → ℝ≥0∞}
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
theorem toNNReal_iInf (hf : ∀ i, f ... | Mathlib/Data/ENNReal/Real.lean | 548 | 553 | theorem toNNReal_sInf (s : Set ℝ≥0∞) (hs : ∀ r ∈ s, r ≠ ∞) :
(sInf s).toNNReal = sInf (ENNReal.toNNReal '' s) := by |
have hf : ∀ i, ((↑) : s → ℝ≥0∞) i ≠ ∞ := fun ⟨r, rs⟩ => hs r rs
-- Porting note: `← sInf_image'` had to be replaced by `← image_eq_range` as the lemmas are used
-- in a different order.
simpa only [← sInf_range, ← image_eq_range, Subtype.range_coe_subtype] using (toNNReal_iInf hf)
| 4 |
import Mathlib.Topology.Order.IsLUB
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section DenselyOrdered
variable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}
{s : Set α}
theorem closure_Ioi' {a : α} (h : (Io... | Mathlib/Topology/Order/DenselyOrdered.lean | 75 | 79 | theorem closure_Ico {a b : α} (hab : a ≠ b) : closure (Ico a b) = Icc a b := by |
apply Subset.antisymm
· exact closure_minimal Ico_subset_Icc_self isClosed_Icc
· apply Subset.trans _ (closure_mono Ioo_subset_Ico_self)
rw [closure_Ioo hab]
| 4 |
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
#align_import measure_theory.integral.set_to_l1 from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical Topology NNReal ENNReal MeasureTheory Pointwise
open Set Filter TopologicalSpace ENNR... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 122 | 127 | theorem of_smul_measure (c : ℝ≥0∞) (hc_ne_top : c ≠ ∞) (hT : FinMeasAdditive (c • μ) T) :
FinMeasAdditive μ T := by |
refine of_eq_top_imp_eq_top (fun s _ hμs => ?_) hT
rw [Measure.smul_apply, smul_eq_mul, ENNReal.mul_eq_top] at hμs
simp only [hc_ne_top, or_false_iff, Ne, false_and_iff] at hμs
exact hμs.2
| 4 |
import Mathlib.RingTheory.EisensteinCriterion
import Mathlib.RingTheory.Polynomial.ScaleRoots
#align_import ring_theory.polynomial.eisenstein.basic from "leanprover-community/mathlib"@"2032a878972d5672e7c27c957e7a6e297b044973"
universe u v w z
variable {R : Type u}
open Ideal Algebra Finset
open Polynomial
na... | Mathlib/RingTheory/Polynomial/Eisenstein/Basic.lean | 161 | 166 | theorem scaleRoots.isWeaklyEisensteinAt (p : R[X]) {x : R} {P : Ideal R} (hP : x ∈ P) :
(scaleRoots p x).IsWeaklyEisensteinAt P := by |
refine ⟨fun i => ?_⟩
rw [coeff_scaleRoots]
rw [natDegree_scaleRoots, ← tsub_pos_iff_lt] at i
exact Ideal.mul_mem_left _ _ (Ideal.pow_mem_of_mem P hP _ i)
| 4 |
import Mathlib.Data.Matrix.Basis
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.Pi
#align_import linear_algebra.std_basis from "leanprover-community/mathlib"@"13bce9a6b6c44f6b4c91ac1c1d2a816e2533d395"
open Function Set Submodule
namespace LinearMap
variable (R : Type*) {ι : Type*} [Semiring R] ... | Mathlib/LinearAlgebra/StdBasis.lean | 123 | 129 | theorem iSup_range_stdBasis_eq_iInf_ker_proj {I J : Set ι} (hd : Disjoint I J)
(hu : Set.univ ⊆ I ∪ J) (hI : Set.Finite I) :
⨆ i ∈ I, range (stdBasis R φ i) = ⨅ i ∈ J, ker (proj i : (∀ i, φ i) →ₗ[R] φ i) := by |
refine le_antisymm (iSup_range_stdBasis_le_iInf_ker_proj _ _ _ _ hd) ?_
have : Set.univ ⊆ ↑hI.toFinset ∪ J := by rwa [hI.coe_toFinset]
refine le_trans (iInf_ker_proj_le_iSup_range_stdBasis R φ this) (iSup_mono fun i => ?_)
rw [Set.Finite.mem_toFinset]
| 4 |
import Batteries.Data.List.Lemmas
import Batteries.Data.Array.Basic
import Batteries.Tactic.SeqFocus
import Batteries.Util.ProofWanted
namespace Array
theorem forIn_eq_data_forIn [Monad m]
(as : Array α) (b : β) (f : α → β → m (ForInStep β)) :
forIn as b f = forIn as.data b f := by
let rec loop : ∀ {i h b ... | .lake/packages/batteries/Batteries/Data/Array/Lemmas.lean | 121 | 125 | theorem size_shrink_loop (a : Array α) (n) : (shrink.loop n a).size = a.size - n := by |
induction n generalizing a with simp[shrink.loop]
| succ n ih =>
simp[ih]
omega
| 4 |
import Mathlib.SetTheory.Game.State
#align_import set_theory.game.domineering from "leanprover-community/mathlib"@"b134b2f5cf6dd25d4bbfd3c498b6e36c11a17225"
namespace SetTheory
namespace PGame
namespace Domineering
open Function
@[simps!]
def shiftUp : ℤ × ℤ ≃ ℤ × ℤ :=
(Equiv.refl ℤ).prodCongr (Equiv.addRig... | Mathlib/SetTheory/Game/Domineering.lean | 109 | 114 | theorem moveLeft_card {b : Board} {m : ℤ × ℤ} (h : m ∈ left b) :
Finset.card (moveLeft b m) + 2 = Finset.card b := by |
dsimp [moveLeft]
rw [Finset.card_erase_of_mem (snd_pred_mem_erase_of_mem_left h)]
rw [Finset.card_erase_of_mem (Finset.mem_of_mem_inter_left h)]
exact tsub_add_cancel_of_le (card_of_mem_left h)
| 4 |
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 | 804 | 809 | theorem sum_centroidWeights_eq_one_of_card_ne_zero [CharZero k] (h : card s ≠ 0) :
∑ i ∈ s, s.centroidWeights k i = 1 := by |
-- Porting note: `simp` cannot find `mul_inv_cancel` and does not use `norm_cast`
simp only [centroidWeights_apply, sum_const, nsmul_eq_mul, ne_eq, Nat.cast_eq_zero, card_eq_zero]
refine mul_inv_cancel ?_
norm_cast
| 4 |
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
#align_import analysis.ODE.gronwall from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F]
[NormedSpace ℝ F]
open Metric Set Asymptotics Fil... | Mathlib/Analysis/ODE/Gronwall.lean | 79 | 83 | theorem gronwallBound_x0 (δ K ε : ℝ) : gronwallBound δ K ε 0 = δ := by |
by_cases hK : K = 0
· simp only [gronwallBound, if_pos hK, mul_zero, add_zero]
· simp only [gronwallBound, if_neg hK, mul_zero, exp_zero, sub_self, mul_one,
add_zero]
| 4 |
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.Multilinear.Basic
#align_import linear_algebra.multilinear.basis from "leanprover-community/mathlib"@"ce11c3c2a285bbe6937e26d9792fda4e51f3fe1a"
open MultilinearMap
variable {R : Type*} {ι : Type*} {n : ℕ} {M : Fin n → Type*} {M₂ : Type*} {M₃ : Type*... | Mathlib/LinearAlgebra/Multilinear/Basis.lean | 56 | 61 | theorem Basis.ext_multilinear [Finite ι] {f g : MultilinearMap R (fun _ : ι => M₂) M₃} {ι₁ : Type*}
(e : Basis ι₁ R M₂) (h : ∀ v : ι → ι₁, (f fun i => e (v i)) = g fun i => e (v i)) : f = g := by |
cases nonempty_fintype ι
exact
(domDomCongr_eq_iff (Fintype.equivFin ι) f g).mp
(Basis.ext_multilinear_fin (fun _ => e) fun i => h (i ∘ _))
| 4 |
import Mathlib.Algebra.MvPolynomial.Variables
#align_import data.mv_polynomial.supported from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
universe u v w
namespace MvPolynomial
variable {σ τ : Type*} {R : Type u} {S : Type v} {r : R} {e : ℕ} {n m : σ}
section CommSemiring
variable... | Mathlib/Algebra/MvPolynomial/Supported.lean | 123 | 127 | theorem supported_le_supported_iff [Nontrivial R] : supported R s ≤ supported R t ↔ s ⊆ t := by |
constructor
· intro h i
simpa using @h (X i)
· exact supported_mono
| 4 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Monic
#align_import data.polynomial.lifts from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
open Polynomial
noncomputable section
namespace Polynomial
universe u v w
section Semiring
variable {R : Type... | Mathlib/Algebra/Polynomial/Lifts.lean | 87 | 91 | theorem C'_mem_lifts {f : R →+* S} {s : S} (h : s ∈ Set.range f) : C s ∈ lifts f := by |
obtain ⟨r, rfl⟩ := Set.mem_range.1 h
use C r
simp only [coe_mapRingHom, map_C, Set.mem_univ, Subsemiring.coe_top, eq_self_iff_true,
and_self_iff]
| 4 |
import Mathlib.Algebra.Algebra.Subalgebra.Directed
import Mathlib.FieldTheory.IntermediateField
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.SplittingField.IsSplittingField
import Mathlib.RingTheory.TensorProduct.Basic
#align_import field_theory.adjoin from "leanprover-community/mathlib"@"df76f4335... | Mathlib/FieldTheory/Adjoin.lean | 62 | 67 | theorem mem_adjoin_simple_iff {α : E} (x : E) :
x ∈ adjoin F {α} ↔ ∃ r s : F[X], x = aeval α r / aeval α s := by |
simp only [adjoin, mem_mk, Subring.mem_toSubsemiring, Subfield.mem_toSubring,
Subfield.mem_closure_iff, ← Algebra.adjoin_eq_ring_closure, Subalgebra.mem_toSubring,
Algebra.adjoin_singleton_eq_range_aeval, AlgHom.mem_range, exists_exists_eq_and]
tauto
| 4 |
import Mathlib.Data.PFunctor.Univariate.M
#align_import data.qpf.univariate.basic from "leanprover-community/mathlib"@"14b69e9f3c16630440a2cbd46f1ddad0d561dee7"
universe u
class QPF (F : Type u → Type u) [Functor F] where
P : PFunctor.{u}
abs : ∀ {α}, P α → F α
repr : ∀ {α}, F α → P α
abs_repr : ∀ {α} (... | Mathlib/Data/QPF/Univariate/Basic.lean | 71 | 75 | theorem id_map {α : Type _} (x : F α) : id <$> x = x := by |
rw [← abs_repr x]
cases' repr x with a f
rw [← abs_map]
rfl
| 4 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Fintype.Sum
import Mathlib.Data.Fintype.Prod
#align_import data.sym.card from "leanprover-community/mathlib"@"0bd2ea37bcba5769e14866170f251c9bc64e35d7"
open Finset Fintype Function Sum Nat
variable {α β : Type*}
... | Mathlib/Data/Sym/Card.lean | 110 | 115 | theorem card_sym_eq_multichoose (α : Type*) (k : ℕ) [Fintype α] [Fintype (Sym α k)] :
card (Sym α k) = multichoose (card α) k := by |
rw [← card_sym_fin_eq_multichoose]
-- FIXME: Without the `Fintype` namespace, why does it complain about `Finset.card_congr` being
-- deprecated?
exact Fintype.card_congr (equivCongr (equivFin α))
| 4 |
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 | 300 | 304 | theorem eq_iSup_nat (m : Set X → ℝ≥0∞) : mkMetric' m = ⨆ n : ℕ, mkMetric'.pre m n⁻¹ := by |
ext1 s
rw [iSup_apply]
refine tendsto_nhds_unique (mkMetric'.tendsto_pre_nat m s)
(tendsto_atTop_iSup fun k l hkl => mkMetric'.mono_pre_nat m hkl s)
| 4 |
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.monoid from "leanprover-community/mathlib"@"aba57d4d3dae35460225919dcd82fe91355162f9"
namespace Set
variable {M : Type*} ... | Mathlib/Algebra/Order/Interval/Set/Monoid.lean | 65 | 69 | theorem Ico_add_bij : BijOn (· + d) (Ico a b) (Ico (a + d) (b + d)) := by |
rw [← Ici_inter_Iio, ← Ici_inter_Iio]
exact
(Ici_add_bij a d).inter_mapsTo (fun x hx => add_lt_add_right hx _) fun x hx =>
lt_of_add_lt_add_right hx.2
| 4 |
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
#align_import analysis.special_functions.pow.continuity from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
noncomputable section
open scoped Classical
open Real Topology NNReal ENNReal Filter ComplexConjugate
open Filter Finset... | Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean | 74 | 78 | theorem continuousAt_const_cpow' {a b : ℂ} (h : b ≠ 0) : ContinuousAt (fun x : ℂ => a ^ x) b := by |
by_cases ha : a = 0
· rw [ha, continuousAt_congr (zero_cpow_eq_nhds h)]
exact continuousAt_const
· exact continuousAt_const_cpow ha
| 4 |
import Mathlib.Analysis.NormedSpace.AddTorsorBases
#align_import analysis.convex.intrinsic from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open AffineSubspace Set
open scoped Pointwise
variable {𝕜 V W Q P : Type*}
section AddTorsor
variable (𝕜) [Ring 𝕜] [AddCommGroup V] [Modu... | Mathlib/Analysis/Convex/Intrinsic.lean | 157 | 161 | theorem intrinsicClosure_mono (h : s ⊆ t) : intrinsicClosure 𝕜 s ⊆ intrinsicClosure 𝕜 t := by |
refine image_subset_iff.2 fun x hx => ?_
refine ⟨Set.inclusion (affineSpan_mono _ h) x, ?_, rfl⟩
refine (continuous_inclusion (affineSpan_mono _ h)).closure_preimage_subset _ (closure_mono ?_ hx)
exact fun y hy => h hy
| 4 |
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Combinatorics.SimpleGraph.Density
import Mathlib.Data.Rat.BigOperators
#align_import combinatorics.simple_graph.regularity.energy from "leanprover-community/mathlib"@"bf7ef0... | Mathlib/Combinatorics/SimpleGraph/Regularity/Energy.lean | 46 | 57 | theorem energy_le_one : P.energy G ≤ 1 :=
div_le_of_nonneg_of_le_mul (sq_nonneg _) zero_le_one <|
calc
∑ uv ∈ P.parts.offDiag, G.edgeDensity uv.1 uv.2 ^ 2 ≤ P.parts.offDiag.card • (1 : ℚ) :=
sum_le_card_nsmul _ _ 1 fun uv _ =>
(sq_le_one_iff <| G.edgeDensity_nonneg _ _).2 <| G.edgeDensity_... |
rw [offDiag_card, one_mul]
norm_cast
rw [sq]
exact tsub_le_self
| 4 |
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products
import Mathlib.CategoryTheory.Limits.Yoneda
import Mathlib.CategoryTheory.Preadditive.FunctorCategory
import Mathlib.CategoryTheory.Sites.SheafOfTypes
import Mathlib.CategoryTheory.Sites.Equa... | Mathlib/CategoryTheory/Sites/Sheaf.lean | 248 | 255 | theorem IsSheaf.amalgamate_map {A : Type u₂} [Category.{v₂} A] {E : A} {X : C} {P : Cᵒᵖ ⥤ A}
(hP : Presheaf.IsSheaf J P) (S : J.Cover X) (x : ∀ I : S.Arrow, E ⟶ P.obj (op I.Y))
(hx : ∀ I : S.Relation, x I.fst ≫ P.map I.g₁.op = x I.snd ≫ P.map I.g₂.op) (I : S.Arrow) :
hP.amalgamate S x hx ≫ P.map I.f.op = x ... |
rcases I with ⟨Y, f, hf⟩
apply
@Presieve.IsSheafFor.valid_glue _ _ _ _ _ _ (hP _ _ S.condition) (fun Y f hf => x ⟨Y, f, hf⟩)
(fun Y₁ Y₂ Z g₁ g₂ f₁ f₂ h₁ h₂ w => hx ⟨Y₁, Y₂, Z, g₁, g₂, f₁, f₂, h₁, h₂, w⟩) f hf
| 4 |
import Mathlib.SetTheory.Game.Short
#align_import set_theory.game.state from "leanprover-community/mathlib"@"b134b2f5cf6dd25d4bbfd3c498b6e36c11a17225"
universe u
namespace SetTheory
namespace PGame
class State (S : Type u) where
turnBound : S → ℕ
l : S → Finset S
r : S → Finset S
left_bound : ∀ {s t : S... | Mathlib/SetTheory/Game/State.lean | 50 | 54 | theorem turnBound_ne_zero_of_left_move {s t : S} (m : t ∈ l s) : turnBound s ≠ 0 := by |
intro h
have t := left_bound m
rw [h] at t
exact Nat.not_succ_le_zero _ t
| 4 |
import Mathlib.Analysis.RCLike.Lemmas
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.function.l2_space from "leanprover-community/mathlib"@"83a66c8775fa14ee5180c85cab98e970956401ad"
set_option linter.uppercaseLean3 false... | Mathlib/MeasureTheory/Function/L2Space.lean | 46 | 51 | theorem memℒp_two_iff_integrable_sq_norm {f : α → F} (hf : AEStronglyMeasurable f μ) :
Memℒp f 2 μ ↔ Integrable (fun x => ‖f x‖ ^ 2) μ := by |
rw [← memℒp_one_iff_integrable]
convert (memℒp_norm_rpow_iff hf two_ne_zero ENNReal.two_ne_top).symm
· simp
· rw [div_eq_mul_inv, ENNReal.mul_inv_cancel two_ne_zero ENNReal.two_ne_top]
| 4 |
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 | 176 | 181 | theorem mk_mem_sigmaLift (f : ∀ ⦃i⦄, α i → β i → Finset (γ i)) (i : ι) (a : α i) (b : β i)
(x : γ i) : (⟨i, x⟩ : Sigma γ) ∈ sigmaLift f ⟨i, a⟩ ⟨i, b⟩ ↔ x ∈ f a b := by |
rw [sigmaLift, dif_pos rfl, mem_map]
refine ⟨?_, fun hx => ⟨_, hx, rfl⟩⟩
rintro ⟨x, hx, _, rfl⟩
exact hx
| 4 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polynomial.eval from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
set_option linter.uppercaseLean3 false
noncomputable section
open Finset AddMonoidAlgebra
open Polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 135 | 139 | theorem eval₂_natCast (n : ℕ) : (n : R[X]).eval₂ f x = n := by |
induction' n with n ih
-- Porting note: `Nat.zero_eq` is required.
· simp only [eval₂_zero, Nat.cast_zero, Nat.zero_eq]
· rw [n.cast_succ, eval₂_add, ih, eval₂_one, n.cast_succ]
| 4 |
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Topology.ContinuousFunction.Algebra
import Mathlib.Topology.Compactness.Paracompact
import Mathlib.Topology.ShrinkingLemma
import Mathlib.Topology.UrysohnsLemma
#align_import topology.partition_of_unity from "leanprover-... | Mathlib/Topology/PartitionOfUnity.lean | 244 | 249 | theorem eventually_fintsupport_subset :
∀ᶠ y in 𝓝 x₀, ρ.fintsupport y ⊆ ρ.fintsupport x₀ := by |
apply (ρ.locallyFinite.closure.eventually_subset (fun _ ↦ isClosed_closure) x₀).mono
intro y hy z hz
rw [PartitionOfUnity.mem_fintsupport_iff] at *
exact hy hz
| 4 |
import Mathlib.Data.List.Chain
#align_import data.list.destutter from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
variable {α : Type*} (l : List α) (R : α → α → Prop) [DecidableRel R] {a b : α}
namespace List
@[simp]
theorem destutter'_nil : destutter' R a [] = [a] :=
rfl
#align ... | Mathlib/Data/List/Destutter.lean | 101 | 105 | theorem destutter'_of_chain (h : l.Chain R a) : l.destutter' R a = a :: l := by |
induction' l with b l hb generalizing a
· simp
obtain ⟨h, hc⟩ := chain_cons.mp h
rw [l.destutter'_cons_pos h, hb hc]
| 4 |
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 | 49 | 53 | theorem nnnorm_def (f : E →SL[σ₁₂] F) : ‖f‖₊ = sInf { c | ∀ x, ‖f x‖₊ ≤ c * ‖x‖₊ } := by |
ext
rw [NNReal.coe_sInf, coe_nnnorm, norm_def, NNReal.coe_image]
simp_rw [← NNReal.coe_le_coe, NNReal.coe_mul, coe_nnnorm, mem_setOf_eq, NNReal.coe_mk,
exists_prop]
| 4 |
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 | 145 | 149 | theorem bodd_subNatNat (m n : ℕ) : bodd (subNatNat m n) = xor m.bodd n.bodd := by |
apply subNatNat_elim m n fun m n i => bodd i = xor m.bodd n.bodd <;>
intros i j <;>
simp only [Int.bodd, Int.bodd_coe, Nat.bodd_add] <;>
cases Nat.bodd i <;> simp
| 4 |
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct
universe u v
variable (K : Type u) [Field K]
var... | Mathlib/LinearAlgebra/Coevaluation.lean | 47 | 54 | theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
∑ i : Basis.ofVectorSpaceIndex K V, bV i ⊗ₜ[K] bV.coord i := by |
simp only [coevaluation, id]
rw [(Basis.singleton Unit K).constr_apply_fintype K]
simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,
Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton]
| 4 |
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.FieldTheory.Minpoly.Field
#align_import linear_algebra.eigenspace.minpoly from "leanprover-community/mathlib"@"c3216069e5f9369e6be586ccbfcde2592b3cec92"
universe u v w
namespace Module
namespace End
open Polynomial FiniteDimensional
open scoped Poly... | Mathlib/LinearAlgebra/Eigenspace/Minpoly.lean | 46 | 51 | theorem ker_aeval_ring_hom'_unit_polynomial (f : End K V) (c : K[X]ˣ) :
LinearMap.ker (aeval f (c : K[X])) = ⊥ := by |
rw [Polynomial.eq_C_of_degree_eq_zero (degree_coe_units c)]
simp only [aeval_def, eval₂_C]
apply ker_algebraMap_end
apply coeff_coe_units_zero_ne_zero c
| 4 |
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 IsSwap
va... | Mathlib/GroupTheory/Perm/Support.lean | 248 | 253 | theorem ne_and_ne_of_swap_mul_apply_ne_self {f : Perm α} {x y : α} (hy : (swap x (f x) * f) y ≠ y) :
f y ≠ y ∧ y ≠ x := by |
simp only [swap_apply_def, mul_apply, f.injective.eq_iff] at *
by_cases h : f y = x
· constructor <;> intro <;> simp_all only [if_true, eq_self_iff_true, not_true, Ne]
· split_ifs at hy with h h <;> try { simp [*] at * }
| 4 |
import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic
import Mathlib.NumberTheory.GaussSum
#align_import number_theory.legendre_symbol.quadratic_char.gauss_sum from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
section SpecialValues
open ZMod MulChar
variable {F : Type*} ... | Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/GaussSum.lean | 119 | 125 | theorem quadraticChar_odd_prime [DecidableEq F] (hF : ringChar F ≠ 2) {p : ℕ} [Fact p.Prime]
(hp₁ : p ≠ 2) (hp₂ : ringChar F ≠ p) :
quadraticChar F p = quadraticChar (ZMod p) (χ₄ (Fintype.card F) * Fintype.card F) := by |
rw [← quadraticChar_neg_one hF]
have h := quadraticChar_card_card hF (ne_of_eq_of_ne (ringChar_zmod_n p) hp₁)
(ne_of_eq_of_ne (ringChar_zmod_n p) hp₂.symm)
rwa [card p] at h
| 4 |
import Mathlib.Computability.Halting
#align_import computability.reduce from "leanprover-community/mathlib"@"d13b3a4a392ea7273dfa4727dbd1892e26cfd518"
universe u v w
open Function
def ManyOneReducible {α β} [Primcodable α] [Primcodable β] (p : α → Prop) (q : β → Prop) :=
∃ f, Computable f ∧ ∀ a, p a ↔ q (f a... | Mathlib/Computability/Reduce.lean | 131 | 136 | theorem computable_of_manyOneReducible {p : α → Prop} {q : β → Prop} (h₁ : p ≤₀ q)
(h₂ : ComputablePred q) : ComputablePred p := by |
rcases h₁ with ⟨f, c, hf⟩
rw [show p = fun a => q (f a) from Set.ext hf]
rcases computable_iff.1 h₂ with ⟨g, hg, rfl⟩
exact ⟨by infer_instance, by simpa using hg.comp c⟩
| 4 |
import Mathlib.LinearAlgebra.Dual
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.contraction from "leanprover-community/mathlib"@"657df4339ae6ceada048c8a2980fb10e393143ec"
suppress_compilation
-- Porting note: universe metavariables behave oddly
universe w u v₁ v₂ v₃ v₄
variable {ι : Type... | Mathlib/LinearAlgebra/Contraction.lean | 122 | 128 | theorem comp_dualTensorHom (f : Module.Dual R M) (n : N) (g : Module.Dual R N) (p : P) :
dualTensorHom R N P (g ⊗ₜ[R] p) ∘ₗ dualTensorHom R M N (f ⊗ₜ[R] n) =
g n • dualTensorHom R M P (f ⊗ₜ p) := by |
ext m
simp only [coe_comp, Function.comp_apply, dualTensorHom_apply, LinearMap.map_smul,
RingHom.id_apply, LinearMap.smul_apply]
rw [smul_comm]
| 4 |
import Mathlib.Topology.Separation
import Mathlib.Algebra.BigOperators.Finprod
#align_import topology.algebra.infinite_sum.basic from "leanprover-community/mathlib"@"3b52265189f3fb43aa631edffce5d060fafaf82f"
noncomputable section
open Filter Function
open scoped Topology
variable {α β γ : Type*}
section HasP... | Mathlib/Topology/Algebra/InfiniteSum/Defs.lean | 166 | 170 | theorem Multipliable.hasProd (ha : Multipliable f) : HasProd f (∏' b, f b) := by |
simp only [tprod_def, ha, dite_true]
by_cases H : (mulSupport f).Finite
· simp [H, hasProd_prod_of_ne_finset_one, finprod_eq_prod]
· simpa [H] using ha.choose_spec
| 4 |
import Mathlib.Data.Countable.Basic
import Mathlib.Logic.Encodable.Basic
import Mathlib.Order.SuccPred.Basic
import Mathlib.Order.Interval.Finset.Defs
#align_import order.succ_pred.linear_locally_finite from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062a169c9"
open Order
variable {ι : Type*}... | Mathlib/Order/SuccPred/LinearLocallyFinite.lean | 108 | 112 | theorem le_of_lt_succFn (j i : ι) (hij : j < succFn i) : j ≤ i := by |
rw [lt_isGLB_iff (succFn_spec i)] at hij
obtain ⟨k, hk_lb, hk⟩ := hij
rw [mem_lowerBounds] at hk_lb
exact not_lt.mp fun hi_lt_j ↦ not_le.mpr hk (hk_lb j hi_lt_j)
| 4 |
import Mathlib.Combinatorics.SimpleGraph.Coloring
#align_import combinatorics.simple_graph.partition from "leanprover-community/mathlib"@"2303b3e299f1c75b07bceaaac130ce23044d1386"
universe u v
namespace SimpleGraph
variable {V : Type u} (G : SimpleGraph V)
structure Partition where
parts : Set (Set V)
... | Mathlib/Combinatorics/SimpleGraph/Partition.lean | 98 | 102 | theorem partOfVertex_ne_of_adj {v w : V} (h : G.Adj v w) : P.partOfVertex v ≠ P.partOfVertex w := by |
intro hn
have hw := P.mem_partOfVertex w
rw [← hn] at hw
exact P.independent _ (P.partOfVertex_mem v) (P.mem_partOfVertex v) hw (G.ne_of_adj h) h
| 4 |
import Mathlib.Probability.ConditionalProbability
import Mathlib.MeasureTheory.Measure.Count
#align_import probability.cond_count from "leanprover-community/mathlib"@"117e93f82b5f959f8193857370109935291f0cc4"
noncomputable section
open ProbabilityTheory
open MeasureTheory MeasurableSpace
namespace ProbabilityT... | Mathlib/Probability/CondCount.lean | 110 | 115 | theorem condCount_eq_one_of (hs : s.Finite) (hs' : s.Nonempty) (ht : s ⊆ t) :
condCount s t = 1 := by |
haveI := condCount_isProbabilityMeasure hs hs'
refine eq_of_le_of_not_lt prob_le_one ?_
rw [not_lt, ← condCount_self hs hs']
exact measure_mono ht
| 4 |
namespace Nat
@[reducible] def Coprime (m n : Nat) : Prop := gcd m n = 1
instance (m n : Nat) : Decidable (Coprime m n) := inferInstanceAs (Decidable (_ = 1))
theorem coprime_iff_gcd_eq_one : Coprime m n ↔ gcd m n = 1 := .rfl
theorem Coprime.gcd_eq_one : Coprime m n → gcd m n = 1 := id
theorem Coprime.symm ... | .lake/packages/batteries/Batteries/Data/Nat/Gcd.lean | 87 | 91 | theorem Coprime.coprime_dvd_left (H1 : m ∣ k) (H2 : Coprime k n) : Coprime m n := by |
apply eq_one_of_dvd_one
rw [Coprime] at H2
have := Nat.gcd_dvd_gcd_of_dvd_left n H1
rwa [← H2]
| 4 |
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
#align_import linear_algebra.affine_space.pointwise from "leanprover-community/mathlib"@"e96bdfbd1e8c98a09ff75f7ac6204d142debc840"
open Affine Pointwise
open Set
namespace AffineSubspace
variable {k : Type*} [Ring k]
variable {V P V₁ P₁ V₂ P₂ : Type*}
var... | Mathlib/LinearAlgebra/AffineSpace/Pointwise.lean | 74 | 79 | theorem map_pointwise_vadd (f : P₁ →ᵃ[k] P₂) (v : V₁) (s : AffineSubspace k P₁) :
(v +ᵥ s).map f = f.linear v +ᵥ s.map f := by |
erw [pointwise_vadd_eq_map, pointwise_vadd_eq_map, map_map, map_map]
congr 1
ext
exact f.map_vadd _ _
| 4 |
import Mathlib.LinearAlgebra.Matrix.BilinearForm
import Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.LinearAlgebra.Trace
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosu... | Mathlib/RingTheory/Trace.lean | 115 | 119 | theorem trace_algebraMap_of_basis (x : R) : trace R S (algebraMap R S x) = Fintype.card ι • x := by |
haveI := Classical.decEq ι
rw [trace_apply, LinearMap.trace_eq_matrix_trace R b, Matrix.trace]
convert Finset.sum_const x
simp [-coe_lmul_eq_mul]
| 4 |
import Mathlib.Topology.Separation
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.UniformSpace.Cauchy
#align_import topology.uniform_space.uniform_convergence from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062a169c9"
noncomputable section
open Topology Uniformity Filter S... | Mathlib/Topology/UniformSpace/UniformConvergence.lean | 106 | 111 | theorem tendstoUniformlyOn_iff_tendstoUniformlyOnFilter :
TendstoUniformlyOn F f p s ↔ TendstoUniformlyOnFilter F f p (𝓟 s) := by |
simp only [TendstoUniformlyOn, TendstoUniformlyOnFilter]
apply forall₂_congr
simp_rw [eventually_prod_principal_iff]
simp
| 4 |
import Mathlib.CategoryTheory.ConcreteCategory.BundledHom
import Mathlib.Topology.ContinuousFunction.Basic
#align_import topology.category.Top.basic from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
open CategoryTheory
open TopologicalSpace
universe u
@[to_additive existing TopCat... | Mathlib/Topology/Category/TopCat/Basic.lean | 175 | 179 | theorem of_isoOfHomeo {X Y : TopCat.{u}} (f : X ≃ₜ Y) : homeoOfIso (isoOfHomeo f) = f := by |
-- Porting note: unfold some defs now
dsimp [homeoOfIso, isoOfHomeo]
ext
rfl
| 4 |
import Mathlib.SetTheory.Game.Short
#align_import set_theory.game.state from "leanprover-community/mathlib"@"b134b2f5cf6dd25d4bbfd3c498b6e36c11a17225"
universe u
namespace SetTheory
namespace PGame
class State (S : Type u) where
turnBound : S → ℕ
l : S → Finset S
r : S → Finset S
left_bound : ∀ {s t : S... | Mathlib/SetTheory/Game/State.lean | 57 | 61 | theorem turnBound_ne_zero_of_right_move {s t : S} (m : t ∈ r s) : turnBound s ≠ 0 := by |
intro h
have t := right_bound m
rw [h] at t
exact Nat.not_succ_le_zero _ t
| 4 |
import Mathlib.AlgebraicTopology.DoldKan.Faces
import Mathlib.CategoryTheory.Idempotents.Basic
#align_import algebraic_topology.dold_kan.projections from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Pread... | Mathlib/AlgebraicTopology/DoldKan/Projections.lean | 61 | 65 | theorem P_f_0_eq (q : ℕ) : ((P q).f 0 : X _[0] ⟶ X _[0]) = 𝟙 _ := by |
induction' q with q hq
· rfl
· simp only [P_succ, HomologicalComplex.add_f_apply, HomologicalComplex.comp_f,
HomologicalComplex.id_f, id_comp, hq, Hσ_eq_zero, add_zero]
| 4 |
import Mathlib.Data.Int.Range
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.MulChar.Basic
#align_import number_theory.legendre_symbol.zmod_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace ZMod
section QuadCharModP
@[simps]
def χ₄ : MulChar (ZMod 4) ℤ... | Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean | 142 | 146 | theorem isQuadratic_χ₈ : χ₈.IsQuadratic := by |
intro a
-- Porting note: was `decide!`
fin_cases a
all_goals decide
| 4 |
import Mathlib.Topology.Compactness.Compact
open Set Filter Topology TopologicalSpace Classical
variable {X : Type*} {Y : Type*} {ι : Type*}
variable [TopologicalSpace X] [TopologicalSpace Y] {s t : Set X}
instance [WeaklyLocallyCompactSpace X] [WeaklyLocallyCompactSpace Y] :
WeaklyLocallyCompactSpace (X × Y) ... | Mathlib/Topology/Compactness/LocallyCompact.lean | 40 | 45 | theorem exists_compact_superset [WeaklyLocallyCompactSpace X] {K : Set X} (hK : IsCompact K) :
∃ K', IsCompact K' ∧ K ⊆ interior K' := by |
choose s hc hmem using fun x : X ↦ exists_compact_mem_nhds x
rcases hK.elim_nhds_subcover _ fun x _ ↦ interior_mem_nhds.2 (hmem x) with ⟨I, -, hIK⟩
refine ⟨⋃ x ∈ I, s x, I.isCompact_biUnion fun _ _ ↦ hc _, hIK.trans ?_⟩
exact iUnion₂_subset fun x hx ↦ interior_mono <| subset_iUnion₂ (s := fun x _ ↦ s x) x hx
| 4 |
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Limits.Shapes.CommSq
import Mathlib.CategoryTheory.Limits.Shapes.StrictInitial
import Mathlib.CategoryTheory.Limits.FunctorCategory
import Mathlib.CategoryTheory.Limits.Constructions.Fin... | Mathlib/CategoryTheory/Limits/VanKampen.lean | 75 | 80 | theorem mapPair_equifibered {F F' : Discrete WalkingPair ⥤ C} (α : F ⟶ F') :
NatTrans.Equifibered α := by |
rintro ⟨⟨⟩⟩ ⟨j⟩ ⟨⟨rfl : _ = j⟩⟩
all_goals
dsimp; simp only [Discrete.functor_map_id]
exact IsPullback.of_horiz_isIso ⟨by simp only [Category.comp_id, Category.id_comp]⟩
| 4 |
import Mathlib.Data.Set.Pairwise.Basic
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.Directed
import Mathlib.Order.Hom.Set
#align_import order.antichain from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
open Function Set
section General
variable {α β : Type*} {r r₁ r₂ : α →... | Mathlib/Order/Antichain.lean | 120 | 124 | theorem image_relEmbedding (hs : IsAntichain r s) (φ : r ↪r r') : IsAntichain r' (φ '' s) := by |
intro b hb b' hb' h₁ h₂
rw [Set.mem_image] at hb hb'
obtain ⟨⟨a, has, rfl⟩, ⟨a', has', rfl⟩⟩ := hb, hb'
exact hs has has' (fun haa' => h₁ (by rw [haa'])) (φ.map_rel_iff.mp h₂)
| 4 |
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Topology.MetricSpace.CauSeqFilter
#align_import analysis.special_functions.exponential from "leanprover-community/mathlib"@"e1a18cad9cd462973d760af7de36b05776b8811c"
open Filter RCLike ContinuousMultili... | Mathlib/Analysis/SpecialFunctions/Exponential.lean | 67 | 72 | theorem hasStrictFDerivAt_exp_zero_of_radius_pos (h : 0 < (expSeries 𝕂 𝔸).radius) :
HasStrictFDerivAt (exp 𝕂) (1 : 𝔸 →L[𝕂] 𝔸) 0 := by |
convert (hasFPowerSeriesAt_exp_zero_of_radius_pos h).hasStrictFDerivAt
ext x
change x = expSeries 𝕂 𝔸 1 fun _ => x
simp [expSeries_apply_eq, Nat.factorial]
| 4 |
import Mathlib.Algebra.Group.Basic
import Mathlib.Order.Basic
import Mathlib.Order.Monotone.Basic
#align_import algebra.covariant_and_contravariant from "leanprover-community/mathlib"@"2258b40dacd2942571c8ce136215350c702dc78f"
-- TODO: convert `ExistsMulOfLE`, `ExistsAddOfLE`?
-- TODO: relationship with `Con/AddC... | Mathlib/Algebra/Order/Monoid/Unbundled/Defs.lean | 281 | 286 | theorem covariant_le_of_covariant_lt [PartialOrder N] :
Covariant M N μ (· < ·) → Covariant M N μ (· ≤ ·) := by |
intro h a b c bc
rcases bc.eq_or_lt with (rfl | bc)
· exact le_rfl
· exact (h _ bc).le
| 4 |
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Convex.Strict
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.NormedSpace.Ray
#align_import analysis.convex.strict_convex_space from "leanprover-... | Mathlib/Analysis/Convex/StrictConvexSpace.lean | 76 | 81 | theorem strictConvex_closedBall [StrictConvexSpace 𝕜 E] (x : E) (r : ℝ) :
StrictConvex 𝕜 (closedBall x r) := by |
rcases le_or_lt r 0 with hr | hr
· exact (subsingleton_closedBall x hr).strictConvex
rw [← vadd_closedBall_zero]
exact (StrictConvexSpace.strictConvex_closedBall r hr).vadd _
| 4 |
import Mathlib.Algebra.Lie.Submodule
#align_import algebra.lie.ideal_operations from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
universe u v w w₁ w₂
namespace LieSubmodule
variable {R : Type u} {L : Type v} {M : Type w} {M₂ : Type w₁}
variable [CommRing R] [LieRing L] [LieAlgebra ... | Mathlib/Algebra/Lie/IdealOperations.lean | 96 | 100 | theorem lie_le_iff : ⁅I, N⁆ ≤ N' ↔ ∀ x ∈ I, ∀ m ∈ N, ⁅x, m⁆ ∈ N' := by |
rw [lieIdeal_oper_eq_span, LieSubmodule.lieSpan_le]
refine ⟨fun h x hx m hm => h ⟨⟨x, hx⟩, ⟨m, hm⟩, rfl⟩, ?_⟩
rintro h _ ⟨⟨x, hx⟩, ⟨m, hm⟩, rfl⟩
exact h x hx m hm
| 4 |
import Mathlib.Data.Nat.Defs
import Mathlib.Tactic.GCongr.Core
import Mathlib.Tactic.Common
import Mathlib.Tactic.Monotonicity.Attr
#align_import data.nat.factorial.basic from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
namespace Nat
def factorial : ℕ → ℕ
| 0 => 1
| succ n => s... | Mathlib/Data/Nat/Factorial/Basic.lean | 150 | 155 | theorem add_factorial_succ_lt_factorial_add_succ {i : ℕ} (n : ℕ) (hi : 2 ≤ i) :
i + (n + 1)! < (i + n + 1)! := by |
rw [factorial_succ (i + _), Nat.add_mul, Nat.one_mul]
have := (i + n).self_le_factorial
refine Nat.add_lt_add_of_lt_of_le (Nat.lt_of_le_of_lt ?_ ((Nat.lt_mul_iff_one_lt_right ?_).2 ?_))
(factorial_le ?_) <;> omega
| 4 |
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 | 93 | 100 | theorem Finset.centerMass_segment' (s : Finset ι) (t : Finset ι') (ws : ι → R) (zs : ι → E)
(wt : ι' → R) (zt : ι' → E) (hws : ∑ i ∈ s, ws i = 1) (hwt : ∑ i ∈ t, wt i = 1) (a b : R)
(hab : a + b = 1) : a • s.centerMass ws zs + b • t.centerMass wt zt = (s.disjSum t).centerMass
(Sum.elim (fun i => a * ws i) f... |
rw [s.centerMass_eq_of_sum_1 _ hws, t.centerMass_eq_of_sum_1 _ hwt, smul_sum, smul_sum, ←
Finset.sum_sum_elim, Finset.centerMass_eq_of_sum_1]
· congr with ⟨⟩ <;> simp only [Sum.elim_inl, Sum.elim_inr, mul_smul]
· rw [sum_sum_elim, ← mul_sum, ← mul_sum, hws, hwt, mul_one, mul_one, hab]
| 4 |
import Mathlib.CategoryTheory.Idempotents.Basic
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Equivalence
#align_import category_theory.idempotents.karoubi from "leanprover-community/mathlib"@"200eda15d8ff5669854ff6bcc10aaf37cb70498f"
noncomputable section
open CategoryT... | Mathlib/CategoryTheory/Idempotents/Karoubi.lean | 108 | 112 | theorem hom_ext_iff {P Q : Karoubi C} {f g : P ⟶ Q} : f = g ↔ f.f = g.f := by |
constructor
· intro h
rw [h]
· apply Hom.ext
| 4 |
import Mathlib.CategoryTheory.Preadditive.Injective
import Mathlib.Algebra.Category.ModuleCat.EpiMono
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.LinearAlgebra.LinearPMap
import Mathlib.Logic.Equiv.TransferInstance
#align_import algebra.module.injective from "leanprover-community/mathlib"@"f8d8465c3c392a93b9... | Mathlib/Algebra/Module/Injective.lean | 112 | 119 | theorem ExtensionOf.ext {a b : ExtensionOf i f} (domain_eq : a.domain = b.domain)
(to_fun_eq :
∀ ⦃x : a.domain⦄ ⦃y : b.domain⦄, (x : N) = y → a.toLinearPMap x = b.toLinearPMap y) :
a = b := by |
rcases a with ⟨a, a_le, e1⟩
rcases b with ⟨b, b_le, e2⟩
congr
exact LinearPMap.ext domain_eq to_fun_eq
| 4 |
import Mathlib.CategoryTheory.Filtered.Connected
import Mathlib.CategoryTheory.Limits.TypesFiltered
import Mathlib.CategoryTheory.Limits.Final
universe v₁ v₂ u₁ u₂
namespace CategoryTheory
open CategoryTheory.Limits CategoryTheory.Functor Opposite
section ArbitraryUniverses
variable {C : Type u₁} [Category.{v₁}... | Mathlib/CategoryTheory/Filtered/Final.lean | 121 | 126 | theorem IsCofilteredOrEmpty.of_exists_of_isCofiltered_of_fullyFaithful [IsCofilteredOrEmpty D]
[F.Full] [F.Faithful] (h : ∀ d, ∃ c, Nonempty (F.obj c ⟶ d)) : IsCofilteredOrEmpty C := by |
suffices IsFilteredOrEmpty Cᵒᵖ from isCofilteredOrEmpty_of_isFilteredOrEmpty_op _
refine IsFilteredOrEmpty.of_exists_of_isFiltered_of_fullyFaithful F.op (fun d => ?_)
obtain ⟨c, ⟨f⟩⟩ := h d.unop
exact ⟨op c, ⟨f.op⟩⟩
| 4 |
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable
#align_import measure_theory.function.simple_func_dense from "leanprover-community/mathlib"@"7317149f12f55affbc900fc873d0d422485122b9"
open Set Function Filter TopologicalSpace ENNReal EMetric Finset
... | Mathlib/MeasureTheory/Function/SimpleFuncDense.lean | 140 | 145 | theorem approxOn_mem {f : β → α} (hf : Measurable f) {s : Set α} {y₀ : α} (h₀ : y₀ ∈ s)
[SeparableSpace s] (n : ℕ) (x : β) : approxOn f hf s y₀ h₀ n x ∈ s := by |
haveI : Nonempty s := ⟨⟨y₀, h₀⟩⟩
suffices ∀ n, (Nat.casesOn n y₀ ((↑) ∘ denseSeq s) : α) ∈ s by apply this
rintro (_ | n)
exacts [h₀, Subtype.mem _]
| 4 |
import Mathlib.Geometry.Manifold.ContMDiff.Basic
open Set ChartedSpace SmoothManifoldWithCorners
open scoped Manifold
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
-- declare a smooth manifold `M` over the pair `(E, H)`.
{E : Type*}
[NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type*} [TopologicalSpace... | Mathlib/Geometry/Manifold/ContMDiff/Atlas.lean | 45 | 49 | theorem contMDiffOn_model_symm : ContMDiffOn 𝓘(𝕜, E) I n I.symm (range I) := by |
rw [contMDiffOn_iff]
refine ⟨I.continuousOn_symm, fun x y => ?_⟩
simp only [mfld_simps]
exact contDiffOn_id.congr fun x' => I.right_inv
| 4 |
import Mathlib.Algebra.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
variable {σ τ α R S : Type*} [CommSemiring R] [CommSemiring S]
namespace MvPolynomial... | Mathlib/Algebra/MvPolynomial/Rename.lean | 67 | 72 | theorem map_rename (f : R →+* S) (g : σ → τ) (p : MvPolynomial σ R) :
map f (rename g p) = rename g (map f p) := by |
apply MvPolynomial.induction_on p
(fun a => by simp only [map_C, rename_C])
(fun p q hp hq => by simp only [hp, hq, AlgHom.map_add, RingHom.map_add]) fun p n hp => by
simp only [hp, rename_X, map_X, RingHom.map_mul, AlgHom.map_mul]
| 4 |
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Data.Nat.Lattice
#align_import combinatorics.simple_graph.metric from "leanprover-community/mathlib"@"352ecfe114946c903338006dd3287cb5a9955ff2"
namespace SimpleGraph
variable {V : Type*} (G : SimpleGraph V)
noncomputable def dist (u v : V)... | Mathlib/Combinatorics/SimpleGraph/Metric.lean | 118 | 122 | theorem dist_comm {u v : V} : G.dist u v = G.dist v u := by |
by_cases h : G.Reachable u v
· apply le_antisymm (dist_comm_aux h) (dist_comm_aux h.symm)
· have h' : ¬G.Reachable v u := fun h' => absurd h'.symm h
simp [h, h', dist_eq_zero_of_not_reachable]
| 4 |
import Mathlib.Algebra.Group.Hom.Defs
#align_import algebra.group.ext from "leanprover-community/mathlib"@"e574b1a4e891376b0ef974b926da39e05da12a06"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
open Function
universe u
@[to_additive (attr := ext)]
theorem Monoid.ext {M : Type u} ⦃m₁ m₂ : Mo... | Mathlib/Algebra/Group/Ext.lean | 119 | 124 | theorem CancelCommMonoid.toCommMonoid_injective {M : Type u} :
Function.Injective (@CancelCommMonoid.toCommMonoid M) := by |
rintro @⟨@⟨@⟨⟩⟩⟩ @⟨@⟨@⟨⟩⟩⟩ h
congr <;> {
injection h with h'
injection h' }
| 4 |
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.LinearAlgebra.PiTensorProduct
universe uι u𝕜 uE uF
variable {ι : Type uι} [Fintype ι]
variable {𝕜 : Type u𝕜} [NontriviallyNormedField 𝕜]
variable {E : ι → Type uE} [∀ i, SeminormedAddCommGroup (E i)] [∀ i, NormedSpace 𝕜 (E i)]
variable {F : ... | Mathlib/Analysis/NormedSpace/PiTensorProduct/ProjectiveSeminorm.lean | 66 | 71 | theorem projectiveSeminormAux_add_le (p q : FreeAddMonoid (𝕜 × Π i, E i)) :
projectiveSeminormAux (p + q) ≤ projectiveSeminormAux p + projectiveSeminormAux q := by |
simp only [projectiveSeminormAux, Function.comp_apply, Multiset.map_coe, Multiset.sum_coe]
erw [List.map_append]
rw [List.sum_append]
rfl
| 4 |
import Mathlib.Algebra.GCDMonoid.Multiset
import Mathlib.Combinatorics.Enumerative.Partition
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Cycle.Factors
import Mathlib.GroupTheory.Perm.Closure
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Tactic.NormNum.GCD
#align_import group_theory.perm.cycl... | Mathlib/GroupTheory/Perm/Cycle/Type.lean | 94 | 98 | theorem two_le_of_mem_cycleType {σ : Perm α} {n : ℕ} (h : n ∈ σ.cycleType) : 2 ≤ n := by |
simp only [cycleType_def, ← Finset.mem_def, Function.comp_apply, Multiset.mem_map,
mem_cycleFactorsFinset_iff] at h
obtain ⟨_, ⟨hc, -⟩, rfl⟩ := h
exact hc.two_le_card_support
| 4 |
import Mathlib.Topology.GDelta
#align_import topology.metric_space.baire from "leanprover-community/mathlib"@"b9e46fe101fc897fb2e7edaf0bf1f09ea49eb81a"
noncomputable section
open scoped Topology
open Filter Set TopologicalSpace
variable {X α : Type*} {ι : Sort*}
section BaireTheorem
variable [TopologicalSpace... | Mathlib/Topology/Baire/Lemmas.lean | 50 | 55 | theorem dense_sInter_of_isOpen {S : Set (Set X)} (ho : ∀ s ∈ S, IsOpen s) (hS : S.Countable)
(hd : ∀ s ∈ S, Dense s) : Dense (⋂₀ S) := by |
rcases S.eq_empty_or_nonempty with h | h
· simp [h]
· rcases hS.exists_eq_range h with ⟨f, rfl⟩
exact dense_iInter_of_isOpen_nat (forall_mem_range.1 ho) (forall_mem_range.1 hd)
| 4 |
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Extreme
#align_import analysis.convex.independent from "leanprover-community/mathlib"@"fefd8a38be7811574cd2ec2f77d3a393a407f112"
open scoped Classical
open Affine
open Finset Function
variable {𝕜 E ι : Type*}
section OrderedSemiring
va... | Mathlib/Analysis/Convex/Independent.lean | 65 | 69 | theorem Subsingleton.convexIndependent [Subsingleton ι] (p : ι → E) : ConvexIndependent 𝕜 p := by |
intro s x hx
have : (convexHull 𝕜 (p '' s)).Nonempty := ⟨p x, hx⟩
rw [convexHull_nonempty_iff, Set.image_nonempty] at this
rwa [Subsingleton.mem_iff_nonempty]
| 4 |
import Mathlib.Dynamics.Ergodic.MeasurePreserving
#align_import dynamics.ergodic.ergodic from "leanprover-community/mathlib"@"809e920edfa343283cea507aedff916ea0f1bd88"
open Set Function Filter MeasureTheory MeasureTheory.Measure
open ENNReal
variable {α : Type*} {m : MeasurableSpace α} (f : α → α) {s : Set α}
... | Mathlib/Dynamics/Ergodic/Ergodic.lean | 89 | 96 | theorem preErgodic_of_preErgodic_conjugate (hg : MeasurePreserving g μ μ') (hf : PreErgodic f μ)
{f' : β → β} (h_comm : g ∘ f = f' ∘ g) : PreErgodic f' μ' :=
⟨by
intro s hs₀ hs₁
replace hs₁ : f ⁻¹' (g ⁻¹' s) = g ⁻¹' s := by | rw [← preimage_comp, h_comm, preimage_comp, hs₁]
cases' hf.ae_empty_or_univ (hg.measurable hs₀) hs₁ with hs₂ hs₂ <;> [left; right]
· simpa only [ae_eq_empty, hg.measure_preimage hs₀] using hs₂
· simpa only [ae_eq_univ, ← preimage_compl, hg.measure_preimage hs₀.compl] using hs₂⟩
| 4 |
import Mathlib.Algebra.Polynomial.Module.Basic
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
import Mathlib.Analysis.Calculus.MeanValue
#align_import analysis.calculus.taylor from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
open scoped... | Mathlib/Analysis/Calculus/Taylor.lean | 114 | 120 | theorem taylor_within_apply (f : ℝ → E) (n : ℕ) (s : Set ℝ) (x₀ x : ℝ) :
taylorWithinEval f n s x₀ x =
∑ k ∈ Finset.range (n + 1), ((k ! : ℝ)⁻¹ * (x - x₀) ^ k) • iteratedDerivWithin k f s x₀ := by |
induction' n with k hk
· simp
rw [taylorWithinEval_succ, Finset.sum_range_succ, hk]
simp [Nat.factorial]
| 4 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.