Context stringlengths 57 92.3k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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import Mathlib.Data.Option.NAry
import Mathlib.Data.Seq.Computation
#align_import data.seq.seq from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace Stream'
universe u v w
def IsSeq {α : Type u} (s : Stream' (Option α)) : Prop :=
∀ {n : ℕ}, s n = none → s (n + 1) = none
#al... | Mathlib/Data/Seq/Seq.lean | 761 | 778 | theorem join_cons (a : α) (s S) : join (cons (a, s) S) = cons a (append s (join S)) := by |
apply
eq_of_bisim
(fun s1 s2 => s1 = s2 ∨ ∃ a s S, s1 = join (cons (a, s) S) ∧ s2 = cons a (append s (join S)))
_ (Or.inr ⟨a, s, S, rfl, rfl⟩)
intro s1 s2 h
exact
match s1, s2, h with
| s, _, Or.inl <| Eq.refl s => by
apply recOn s; · trivial
· intro x s
rw [destruct_c... |
import Mathlib.Tactic.CategoryTheory.Elementwise
import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
import Mathlib.CategoryTheory.Limits.Preserves.Limits
import Mathlib.CategoryTheory.Limits.Shapes.Types
#align_import category_theory.glue_data from "l... | Mathlib/CategoryTheory/GlueData.lean | 99 | 105 | theorem t_inv (i j : D.J) : D.t i j ≫ D.t j i = 𝟙 _ := by |
have eq : (pullbackSymmetry (D.f i i) (D.f i j)).hom = pullback.snd ≫ inv pullback.fst := by simp
have := D.cocycle i j i
rw [D.t'_iij, D.t'_jii, D.t'_iji, fst_eq_snd_of_mono_eq, eq] at this
simp only [Category.assoc, IsIso.inv_hom_id_assoc] at this
rw [← IsIso.eq_inv_comp, ← Category.assoc, IsIso.comp_inv_e... |
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Tactic.AdaptationNote
#align_import geometry.euclidean.inversion from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open Metric Function AffineMap Set Aff... | Mathlib/Geometry/Euclidean/Inversion/Basic.lean | 75 | 78 | theorem inversion_mul (c : P) (a R : ℝ) (x : P) :
inversion c (a * R) x = homothety c (a ^ 2) (inversion c R x) := by |
simp only [inversion_eq_lineMap, ← homothety_eq_lineMap, ← homothety_mul_apply, mul_div_assoc,
mul_pow]
|
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 | 839 | 847 | theorem dist_convolution_le {f : G → ℝ} {x₀ : G} {R ε : ℝ} {z₀ : E'} (hε : 0 ≤ ε)
(hf : support f ⊆ ball (0 : G) R) (hnf : ∀ x, 0 ≤ f x) (hintf : ∫ x, f x ∂μ = 1)
(hmg : AEStronglyMeasurable g μ) (hg : ∀ x ∈ ball x₀ R, dist (g x) z₀ ≤ ε) :
dist ((f ⋆[lsmul ℝ ℝ, μ] g : G → E') x₀) z₀ ≤ ε := by |
have hif : Integrable f μ := integrable_of_integral_eq_one hintf
convert (dist_convolution_le' (lsmul ℝ ℝ) hε hif hf hmg hg).trans _
· simp_rw [lsmul_apply, integral_smul_const, hintf, one_smul]
· simp_rw [Real.norm_of_nonneg (hnf _), hintf, mul_one]
exact (mul_le_mul_of_nonneg_right opNorm_lsmul_le hε).tr... |
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Convex.Uniform
import Mathlib.Analysis.NormedSpace.Completion
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
#align_import analysis.inner_product_space.basic from "leanprover-community/mathlib"@"3f655f5297b030... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 2,228 | 2,230 | theorem real_inner_I_smul_self (x : E) :
@Inner.inner ℝ E (Inner.rclikeToReal 𝕜 E) x ((I : 𝕜) • x) = 0 := by |
simp [real_inner_eq_re_inner 𝕜, inner_smul_right]
|
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Set.Subsingleton
#align_import combinatorics.composition from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open List
variable {n : ℕ}
... | Mathlib/Combinatorics/Enumerative/Composition.lean | 443 | 450 | theorem blocksFun_congr {n₁ n₂ : ℕ} (c₁ : Composition n₁) (c₂ : Composition n₂) (i₁ : Fin c₁.length)
(i₂ : Fin c₂.length) (hn : n₁ = n₂) (hc : c₁.blocks = c₂.blocks) (hi : (i₁ : ℕ) = i₂) :
c₁.blocksFun i₁ = c₂.blocksFun i₂ := by |
cases hn
rw [← Composition.ext_iff] at hc
cases hc
congr
rwa [Fin.ext_iff]
|
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.SetTheory.Cardinal.Cofinality
#align_import linear_algebra.basis from "leanprover-communit... | Mathlib/LinearAlgebra/Basis.lean | 280 | 283 | theorem ext' {f₁ f₂ : M ≃ₛₗ[σ] M₁} (h : ∀ i, f₁ (b i) = f₂ (b i)) : f₁ = f₂ := by |
ext x
rw [← b.total_repr x, Finsupp.total_apply, Finsupp.sum]
simp only [map_sum, LinearEquiv.map_smulₛₗ, h]
|
import Mathlib.MeasureTheory.PiSystem
import Mathlib.Order.OmegaCompletePartialOrder
import Mathlib.Topology.Constructions
import Mathlib.MeasureTheory.MeasurableSpace.Basic
open Set
namespace MeasureTheory
variable {ι : Type _} {α : ι → Type _}
section cylinder
def cylinder (s : Finset ι) (S : Set (∀ i : s, α... | Mathlib/MeasureTheory/Constructions/Cylinders.lean | 161 | 162 | theorem cylinder_empty (s : Finset ι) : cylinder s (∅ : Set (∀ i : s, α i)) = ∅ := by |
rw [cylinder, preimage_empty]
|
import Mathlib.CategoryTheory.Balanced
import Mathlib.CategoryTheory.Limits.EssentiallySmall
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.CategoryTheory.Subobject.WellPowered
import Mathlib.Data.S... | Mathlib/CategoryTheory/Generator.lean | 166 | 175 | theorem IsSeparating.isDetecting [Balanced C] {𝒢 : Set C} (h𝒢 : IsSeparating 𝒢) :
IsDetecting 𝒢 := by |
intro X Y f hf
refine
(isIso_iff_mono_and_epi _).2 ⟨⟨fun g h hgh => h𝒢 _ _ fun G hG i => ?_⟩, ⟨fun g h hgh => ?_⟩⟩
· obtain ⟨t, -, ht⟩ := hf G hG (i ≫ g ≫ f)
rw [ht (i ≫ g) (Category.assoc _ _ _), ht (i ≫ h) (hgh.symm ▸ Category.assoc _ _ _)]
· refine h𝒢 _ _ fun G hG i => ?_
obtain ⟨t, rfl, -⟩ :=... |
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.RingTheory.AlgebraTower
import Mathlib.Algebra.Algebra.Subalgebra.Tower
#align_import linear_algebra.matrix.to_lin from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e8491d6"
... | Mathlib/LinearAlgebra/Matrix/ToLin.lean | 927 | 930 | theorem leftMulMatrix_eq_repr_mul (x : S) (i j) : leftMulMatrix b x i j = b.repr (x * b j) i := by |
-- This is defeq to just `toMatrix_lmul' b x i j`,
-- but the unfolding goes a lot faster with this explicit `rw`.
rw [leftMulMatrix_apply, toMatrix_lmul' b x i j]
|
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.FiniteSupport
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Set.Subsingleton
#align_import algebra.big_operators.finprod from "leanprover-community/mathlib"@"d6fad0e5bf... | Mathlib/Algebra/BigOperators/Finprod.lean | 1,183 | 1,198 | theorem finprod_prod_comm (s : Finset β) (f : α → β → M)
(h : ∀ b ∈ s, (mulSupport fun a => f a b).Finite) :
(∏ᶠ a : α, ∏ b ∈ s, f a b) = ∏ b ∈ s, ∏ᶠ a : α, f a b := by |
have hU :
(mulSupport fun a => ∏ b ∈ s, f a b) ⊆
(s.finite_toSet.biUnion fun b hb => h b (Finset.mem_coe.1 hb)).toFinset := by
rw [Finite.coe_toFinset]
intro x hx
simp only [exists_prop, mem_iUnion, Ne, mem_mulSupport, Finset.mem_coe]
contrapose! hx
rw [mem_mulSupport, not_not, Finset.p... |
import Mathlib.CategoryTheory.Functor.FullyFaithful
import Mathlib.CategoryTheory.FullSubcategory
import Mathlib.CategoryTheory.Whiskering
import Mathlib.CategoryTheory.EssentialImage
import Mathlib.Tactic.CategoryTheory.Slice
#align_import category_theory.equivalence from "leanprover-community/mathlib"@"9aba7801eeec... | Mathlib/CategoryTheory/Equivalence.lean | 517 | 517 | theorem changeFunctor_refl (e : C ≌ D) : e.changeFunctor (Iso.refl _) = e := by | aesop_cat
|
import Mathlib.CategoryTheory.Closed.Cartesian
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
#align_import category_theory.closed.functor from "leanprover-community/mathlib"@"cea27692b3fdeb328a2ddba6aabf181754543184"
noncomputable secti... | Mathlib/CategoryTheory/Closed/Functor.lean | 166 | 168 | theorem expComparison_iso_of_frobeniusMorphism_iso (h : L ⊣ F) (A : C)
[i : IsIso (frobeniusMorphism F h A)] : IsIso (expComparison F A) := by |
rw [← frobeniusMorphism_mate F h]; infer_instance
|
import Mathlib.Probability.Process.Adapted
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import probability.process.stopping from "leanprover-community/mathlib"@"ba074af83b6cf54c3104e59402b39410ddbd6dca"
open Filter Order TopologicalSpace
open scoped Classical MeasureTheory NNReal ENNReal Top... | Mathlib/Probability/Process/Stopping.lean | 150 | 155 | theorem IsStoppingTime.measurableSet_gt (hτ : IsStoppingTime f τ) (i : ι) :
MeasurableSet[f i] {ω | i < τ ω} := by |
have : {ω | i < τ ω} = {ω | τ ω ≤ i}ᶜ := by
ext1 ω; simp only [Set.mem_setOf_eq, Set.mem_compl_iff, not_le]
rw [this]
exact (hτ.measurableSet_le i).compl
|
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 | 1,185 | 1,185 | theorem lt_inv_self_iff : a < a⁻¹ ↔ a < 1 := by | simp [← not_iff_not]
|
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
#align_import number_theory.modular_forms.congruence_subgroups from "leanprover-community/mathlib"@"ae690b0c236e488a0043f6faa8ce3546e7f2f... | Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean | 73 | 75 | theorem Gamma_one_top : Gamma 1 = ⊤ := by |
ext
simp [eq_iff_true_of_subsingleton]
|
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Set.Image
import Mathlib.GroupTheory.Perm.Support
import Mathlib.Order.Monotone.Monovary
import Mathlib.Tactic.Abel
#align_impo... | Mathlib/Algebra/Order/Rearrangement.lean | 143 | 147 | theorem MonovaryOn.sum_smul_comp_perm_lt_sum_smul_iff (hfg : MonovaryOn f g s)
(hσ : { x | σ x ≠ x } ⊆ s) :
((∑ i ∈ s, f i • g (σ i)) < ∑ i ∈ s, f i • g i) ↔ ¬MonovaryOn f (g ∘ σ) s := by |
simp [← hfg.sum_smul_comp_perm_eq_sum_smul_iff hσ, lt_iff_le_and_ne,
hfg.sum_smul_comp_perm_le_sum_smul hσ]
|
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#align_import data.complex.exponential from "leanprover-community/mathlib"@"a8b2226cfb0a79f5986492053fc49b1a0c6aeffb"
open CauSeq Finset IsAbsoluteValue
open ... | Mathlib/Data/Complex/Exponential.lean | 546 | 546 | theorem cos_neg : cos (-x) = cos x := by | simp [cos, sub_eq_add_neg, exp_neg, add_comm]
|
import Mathlib.Analysis.BoxIntegral.Partition.Split
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
#align_import analysis.box_integral.partition.additive from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
open scoped Classical
open Function Set
namespace B... | Mathlib/Analysis/BoxIntegral/Partition/Additive.lean | 159 | 175 | theorem sum_boxes_congr [Finite ι] (f : ι →ᵇᵃ[I₀] M) (hI : ↑I ≤ I₀) {π₁ π₂ : Prepartition I}
(h : π₁.iUnion = π₂.iUnion) : ∑ J ∈ π₁.boxes, f J = ∑ J ∈ π₂.boxes, f J := by |
rcases exists_splitMany_inf_eq_filter_of_finite {π₁, π₂} ((finite_singleton _).insert _) with
⟨s, hs⟩
simp only [inf_splitMany] at hs
rcases hs _ (Or.inl rfl), hs _ (Or.inr rfl) with ⟨h₁, h₂⟩; clear hs
rw [h] at h₁
calc
∑ J ∈ π₁.boxes, f J = ∑ J ∈ π₁.boxes, ∑ J' ∈ (splitMany J s).boxes, f J' :=
... |
import Mathlib.Data.ENat.Lattice
import Mathlib.Order.OrderIsoNat
import Mathlib.Tactic.TFAE
#align_import order.height from "leanprover-community/mathlib"@"bf27744463e9620ca4e4ebe951fe83530ae6949b"
open List hiding le_antisymm
open OrderDual
universe u v
variable {α β : Type*}
namespace Set
section LT
varia... | Mathlib/Order/Height.lean | 162 | 184 | theorem chainHeight_add_le_chainHeight_add (s : Set α) (t : Set β) (n m : ℕ) :
s.chainHeight + n ≤ t.chainHeight + m ↔
∀ l ∈ s.subchain, ∃ l' ∈ t.subchain, length l + n ≤ length l' + m := by |
refine
⟨fun e l h ↦
le_chainHeight_add_nat_iff.1
((add_le_add_right (length_le_chainHeight_of_mem_subchain h) _).trans e),
fun H ↦ ?_⟩
by_cases h : s.chainHeight = ⊤
· suffices t.chainHeight = ⊤ by
rw [this, top_add]
exact le_top
rw [chainHeight_eq_top_iff] at h ⊢
intr... |
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 | 546 | 549 | theorem uniformEquicontinuous_iff_uniformContinuous {F : ι → β → α} :
UniformEquicontinuous F ↔ UniformContinuous (ofFun ∘ Function.swap F : β → ι →ᵤ α) := by |
rw [UniformContinuous, (UniformFun.hasBasis_uniformity ι α).tendsto_right_iff]
rfl
|
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Localization.Ideal
import Mathlib.RingTheory.Noetherian
#align_import ring_theory.localization.submodule from "leanprover-community/mathlib"@"1ebb20602a8caef435ce47f6373e1aa40851a177"
variable {R : Type*} [CommRing R] (M : Submonoid R) ... | Mathlib/RingTheory/Localization/Submodule.lean | 125 | 133 | theorem coeSubmodule_isPrincipal {I : Ideal R} (h : M ≤ nonZeroDivisors R) :
(coeSubmodule S I).IsPrincipal ↔ I.IsPrincipal := by |
constructor <;> rintro ⟨⟨x, hx⟩⟩
· have x_mem : x ∈ coeSubmodule S I := hx.symm ▸ Submodule.mem_span_singleton_self x
obtain ⟨x, _, rfl⟩ := (mem_coeSubmodule _ _).mp x_mem
refine ⟨⟨x, coeSubmodule_injective S h ?_⟩⟩
rw [Ideal.submodule_span_eq, hx, coeSubmodule_span_singleton]
· refine ⟨⟨algebraMap R... |
import Mathlib.Order.Filter.Basic
#align_import order.filter.prod from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
open Set
open Filter
namespace Filter
variable {α β γ δ : Type*} {ι : Sort*}
section Prod
variable {s : Set α} {t : Set β} {f : Filter α} {g : Filter β}
protected ... | Mathlib/Order/Filter/Prod.lean | 295 | 296 | theorem map_snd_prod (f : Filter α) (g : Filter β) [NeBot f] : map Prod.snd (f ×ˢ g) = g := by |
rw [prod_comm, map_map]; apply map_fst_prod
|
import Mathlib.RingTheory.Ideal.Cotangent
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.RingTheory.FinitePresentation
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Localization.Away.AdjoinRoot
#align_import ring_theory.etale from ... | Mathlib/RingTheory/Smooth/Basic.lean | 188 | 196 | theorem comp [FormallySmooth R A] [FormallySmooth A B] : FormallySmooth R B := by |
constructor
intro C _ _ I hI f
obtain ⟨f', e⟩ := FormallySmooth.comp_surjective I hI (f.comp (IsScalarTower.toAlgHom R A B))
letI := f'.toRingHom.toAlgebra
obtain ⟨f'', e'⟩ :=
FormallySmooth.comp_surjective I hI { f.toRingHom with commutes' := AlgHom.congr_fun e.symm }
apply_fun AlgHom.restrictScalars ... |
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 | 233 | 235 | theorem compl_mem_coprodᵢ {s : Set (∀ i, α i)} :
sᶜ ∈ Filter.coprodᵢ f ↔ ∀ i, (eval i '' s)ᶜ ∈ f i := by |
simp only [Filter.coprodᵢ, mem_iSup, compl_mem_comap]
|
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.MeasureTheory.Integral.Average
#align_import measure_theory.integral.interval_average from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open MeasureTheory Set TopologicalSpace
open scoped Interval
variable {E : Ty... | Mathlib/MeasureTheory/Integral/IntervalAverage.lean | 39 | 40 | theorem interval_average_symm (f : ℝ → E) (a b : ℝ) : (⨍ x in a..b, f x) = ⨍ x in b..a, f x := by |
rw [setAverage_eq, setAverage_eq, uIoc_comm]
|
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Real.Sqrt
#align_import data.complex.basic from "leanprover-community/mathlib"@"31c24aa72e7b3e5ed97a8412470e904f82b81004"
open Set ComplexConjugate
namespace Complex
theorem abs_def : (Complex.abs : ℂ → ℝ) = fun z => (normSq z).sqrt :=
rfl
#align complex.... | Mathlib/Data/Complex/Abs.lean | 249 | 250 | theorem normSq_eq_abs (x : ℂ) : normSq x = (Complex.abs x) ^ 2 := by |
simp [abs, sq, abs_def, Real.mul_self_sqrt (normSq_nonneg _)]
|
import Mathlib.GroupTheory.Coxeter.Length
import Mathlib.Data.ZMod.Parity
namespace CoxeterSystem
open List Matrix Function
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
local prefi... | Mathlib/GroupTheory/Coxeter/Inversion.lean | 227 | 229 | theorem rightInvSeq_reverse (ω : List B) :
ris (ω.reverse) = (lis ω).reverse := by |
simp [leftInvSeq_eq_reverse_rightInvSeq_reverse]
|
import Mathlib.Algebra.Algebra.Subalgebra.Unitization
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.StarSubalgebra
import Mathlib.Topology.ContinuousFunction.ContinuousMapZero
import Mathlib.Topology.ContinuousFunction.Weierstrass
#align_import topology.continuous_function.stone_weierstrass fro... | Mathlib/Topology/ContinuousFunction/StoneWeierstrass.lean | 94 | 113 | theorem comp_attachBound_mem_closure (A : Subalgebra ℝ C(X, ℝ)) (f : A)
(p : C(Set.Icc (-‖f‖) ‖f‖, ℝ)) : p.comp (attachBound (f : C(X, ℝ))) ∈ A.topologicalClosure := by |
-- `p` itself is in the closure of polynomials, by the Weierstrass theorem,
have mem_closure : p ∈ (polynomialFunctions (Set.Icc (-‖f‖) ‖f‖)).topologicalClosure :=
continuousMap_mem_polynomialFunctions_closure _ _ p
-- and so there are polynomials arbitrarily close.
have frequently_mem_polynomials := mem_c... |
import Mathlib.Algebra.Lie.Subalgebra
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Artinian
#align_import algebra.lie.submodule from "leanprover-community/mathlib"@"9822b65bfc4ac74537d77ae318d27df1df662471"
universe u v w w₁ w₂
variable {R M}
theorem Submodule.exists_lieSubmodule_coe_eq_iff (p... | Mathlib/Algebra/Lie/Submodule.lean | 591 | 592 | theorem mem_sup (x : M) : x ∈ N ⊔ N' ↔ ∃ y ∈ N, ∃ z ∈ N', y + z = x := by |
rw [← mem_coeSubmodule, sup_coe_toSubmodule, Submodule.mem_sup]; exact Iff.rfl
|
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Data.Finset.Fold
import Mathlib.Data.Finset.Option
import Mathlib.Data.Finset.Pi
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Multiset.Lattice
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Hom.Lattice
import Mathlib.Order.Nat
#align_import... | Mathlib/Data/Finset/Lattice.lean | 913 | 917 | theorem sup'_congr {t : Finset β} {f g : β → α} (h₁ : s = t) (h₂ : ∀ x ∈ s, f x = g x) :
s.sup' H f = t.sup' (h₁ ▸ H) g := by |
subst s
refine eq_of_forall_ge_iff fun c => ?_
simp (config := { contextual := true }) only [sup'_le_iff, h₂]
|
import Mathlib.LinearAlgebra.Prod
#align_import linear_algebra.linear_pmap from "leanprover-community/mathlib"@"8b981918a93bc45a8600de608cde7944a80d92b9"
universe u v w
structure LinearPMap (R : Type u) [Ring R] (E : Type v) [AddCommGroup E] [Module R E] (F : Type w)
[AddCommGroup F] [Module R F] where
domai... | Mathlib/LinearAlgebra/LinearPMap.lean | 1,025 | 1,033 | theorem toLinearPMap_apply_aux {g : Submodule R (E × F)}
(hg : ∀ (x : E × F) (_hx : x ∈ g) (_hx' : x.fst = 0), x.snd = 0)
(x : g.map (LinearMap.fst R E F)) :
g.toLinearPMap x = valFromGraph hg x.2 := by |
classical
change (if hg : _ then g.toLinearPMapAux hg else 0) x = _
rw [dif_pos]
· rfl
· exact hg
|
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 | 71 | 73 | theorem OrderedCommGroup.to_contravariantClass_left_le (α : Type u) [OrderedCommGroup α] :
ContravariantClass α α (· * ·) (· ≤ ·) where
elim a b c bc := by | simpa using mul_le_mul_left' bc a⁻¹
|
import Mathlib.Algebra.DirectSum.Algebra
import Mathlib.Algebra.DirectSum.Decomposition
import Mathlib.Algebra.DirectSum.Internal
import Mathlib.Algebra.DirectSum.Ring
#align_import ring_theory.graded_algebra.basic from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open DirectSum
vari... | Mathlib/RingTheory/GradedAlgebra/Basic.lean | 115 | 117 | theorem GradedRing.proj_recompose (a : ⨁ i, 𝒜 i) (i : ι) :
GradedRing.proj 𝒜 i ((decompose 𝒜).symm a) = (decompose 𝒜).symm (DirectSum.of _ i (a i)) := by |
rw [GradedRing.proj_apply, decompose_symm_of, Equiv.apply_symm_apply]
|
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.RingTheory.RootsOfUnity.Complex
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.RingTheory.RootsOfUnity.Basic
import Mathlib.FieldTh... | Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | 107 | 114 | theorem natDegree_cyclotomic' {ζ : R} {n : ℕ} (h : IsPrimitiveRoot ζ n) :
(cyclotomic' n R).natDegree = Nat.totient n := by |
rw [cyclotomic']
rw [natDegree_prod (primitiveRoots n R) fun z : R => X - C z]
· simp only [IsPrimitiveRoot.card_primitiveRoots h, mul_one, natDegree_X_sub_C, Nat.cast_id,
Finset.sum_const, nsmul_eq_mul]
intro z _
exact X_sub_C_ne_zero z
|
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.Factors
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Multiplicity
#align_import ring_theory.unique_factorization_domain from "leanprover-community/mathli... | Mathlib/RingTheory/UniqueFactorizationDomain.lean | 1,658 | 1,663 | theorem mem_factors_iff_dvd {a p : α} (ha0 : a ≠ 0) (hp : Irreducible p) :
Associates.mk p ∈ factors (Associates.mk a) ↔ p ∣ a := by |
constructor
· rw [← mk_dvd_mk]
apply dvd_of_mem_factors
· apply mem_factors_of_dvd ha0 hp
|
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 | 1,005 | 1,017 | theorem EquicontinuousAt.tendsto_of_mem_closure {l : Filter ι} {F : ι → X → α} {f : X → α}
{s : Set X} {x : X} {z : α} (hF : EquicontinuousAt F x) (hf : Tendsto f (𝓝[s] x) (𝓝 z))
(hs : ∀ y ∈ s, Tendsto (F · y) l (𝓝 (f y))) (hx : x ∈ closure s) :
Tendsto (F · x) l (𝓝 z) := by |
rw [(nhds_basis_uniformity (𝓤 α).basis_sets).tendsto_right_iff] at hf ⊢
intro U hU
rcases comp_comp_symm_mem_uniformity_sets hU with ⟨V, hV, hVs, hVU⟩
rw [mem_closure_iff_nhdsWithin_neBot] at hx
have : ∀ᶠ y in 𝓝[s] x, y ∈ s ∧ (∀ i, (F i x, F i y) ∈ V) ∧ (f y, z) ∈ V :=
eventually_mem_nhdsWithin.and <| ... |
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Nat.Dist
import Mathlib.Data.Ordmap.Ordnode
import Mathlib.Tactic.Abel
import Mathlib.Tactic.Linarith
#align_import data.ordmap.ordset from "leanprover-community/mathlib"@"47b51515e69f59bca5cf34ef456e6000fe205a69"
variable... | Mathlib/Data/Ordmap/Ordset.lean | 1,159 | 1,160 | theorem Valid'.node4L_lemma₅ {a b c d : ℕ} (lr₂ : 3 * (b + c + 1 + d) ≤ 16 * a + 9)
(mr₁ : 2 * d ≤ b + c + 1) (mm₂ : c ≤ 3 * b) : c + d + 1 ≤ 3 * (a + b + 1) := by | omega
|
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Algebra.Order.Group.MinMax
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.Order.Int... | Mathlib/Order/Filter/AtTopBot.lean | 957 | 960 | theorem Tendsto.atTop_mul_atBot (hf : Tendsto f l atTop) (hg : Tendsto g l atBot) :
Tendsto (fun x => f x * g x) l atBot := by |
have := hf.atTop_mul_atTop <| tendsto_neg_atBot_atTop.comp hg
simpa only [(· ∘ ·), neg_mul_eq_mul_neg, neg_neg] using tendsto_neg_atTop_atBot.comp this
|
import Mathlib.Order.CompleteLattice
import Mathlib.Order.Directed
import Mathlib.Logic.Equiv.Set
#align_import order.complete_boolean_algebra from "leanprover-community/mathlib"@"71b36b6f3bbe3b44e6538673819324d3ee9fcc96"
set_option autoImplicit true
open Function Set
universe u v w
variable {α : Type u} {β : ... | Mathlib/Order/CompleteBooleanAlgebra.lean | 205 | 207 | theorem inf_iSup₂_eq {f : ∀ i, κ i → α} (a : α) :
(a ⊓ ⨆ (i) (j), f i j) = ⨆ (i) (j), a ⊓ f i j := by |
simp only [inf_iSup_eq]
|
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.TensorProduct.Tower
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.LinearAlgebra.DirectSum.Finsupp
#align_import ring_theory.tensor_product from "leanprover-community/mathlib"@"88fcdc3da43943f5b01925deddaa5bf0c0e85e4e"
suppress_comp... | Mathlib/RingTheory/TensorProduct/Basic.lean | 198 | 199 | theorem natCast_def' (n : ℕ) : (n : A ⊗[R] B) = (1 : A) ⊗ₜ (n : B) := by |
rw [natCast_def, ← nsmul_one, smul_tmul, nsmul_one]
|
import Mathlib.FieldTheory.Tower
import Mathlib.RingTheory.Algebraic
import Mathlib.FieldTheory.Minpoly.Basic
#align_import field_theory.intermediate_field from "leanprover-community/mathlib"@"c596622fccd6e0321979d94931c964054dea2d26"
open FiniteDimensional Polynomial
open Polynomial
variable (K L L' : Type*) [... | Mathlib/FieldTheory/IntermediateField.lean | 606 | 610 | theorem map_injective (f : L →ₐ[K] L'):
Function.Injective (map f) := by |
intro _ _ h
rwa [← toSubalgebra_injective.eq_iff, toSubalgebra_map, toSubalgebra_map,
(Subalgebra.map_injective f.injective).eq_iff, toSubalgebra_injective.eq_iff] at h
|
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.Analysis.NormedSpace.IndicatorFunction
#align_import measure_theory.integral.integrable_on from "leanprover-community/mathlib"@"8b8ba04e2f326f3f7cf24ad129beda58531ada61"
noncomputable section
open Set Filter TopologicalSpace MeasureTheory Function
ope... | Mathlib/MeasureTheory/Integral/IntegrableOn.lean | 268 | 271 | theorem integrable_indicator_iff (hs : MeasurableSet s) :
Integrable (indicator s f) μ ↔ IntegrableOn f s μ := by |
simp [IntegrableOn, Integrable, HasFiniteIntegral, nnnorm_indicator_eq_indicator_nnnorm,
ENNReal.coe_indicator, lintegral_indicator _ hs, aestronglyMeasurable_indicator_iff hs]
|
import Mathlib.Algebra.Ring.Int
import Mathlib.Data.ZMod.Basic
import Mathlib.FieldTheory.Finite.Basic
import Mathlib.Data.Fintype.BigOperators
#align_import number_theory.sum_four_squares from "leanprover-community/mathlib"@"bd9851ca476957ea4549eb19b40e7b5ade9428cc"
open Finset Polynomial FiniteField Equiv
the... | Mathlib/NumberTheory/SumFourSquares.lean | 224 | 234 | theorem sum_four_squares (n : ℕ) : ∃ a b c d : ℕ, a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = n := by |
-- The proof is by induction on prime factorization. The case of prime `n` was proved above,
-- the inductive step follows from `Nat.euler_four_squares`.
induction n using Nat.recOnMul with
| h0 => exact ⟨0, 0, 0, 0, rfl⟩
| h1 => exact ⟨1, 0, 0, 0, rfl⟩
| hp p hp => exact hp.sum_four_squares
| h m n hm h... |
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.Order.Circular
import Mathlib.Data.List.TFAE
import Mathlib.Data.Set.Lattice
#align_import a... | Mathlib/Algebra/Order/ToIntervalMod.lean | 138 | 139 | theorem toIocMod_sub_self (a b : α) : toIocMod hp a b - b = -toIocDiv hp a b • p := by |
rw [toIocMod, sub_sub_cancel_left, neg_smul]
|
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Order.LiminfLimsup
import Mathlib.Topology.Instances.Rat
import Mathlib.Topology.MetricSpace.Algebra
import Mathlib.Topology.MetricSpace.IsometricSMul
import Mathlib.Topology.Sequences
#align_import analysis.normed.group.basic from "leanprover-community/mat... | Mathlib/Analysis/Normed/Group/Basic.lean | 1,160 | 1,162 | theorem tendsto_iff_norm_div_tendsto_zero {f : α → E} {a : Filter α} {b : E} :
Tendsto f a (𝓝 b) ↔ Tendsto (fun e => ‖f e / b‖) a (𝓝 0) := by |
simp only [← dist_eq_norm_div, ← tendsto_iff_dist_tendsto_zero]
|
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.Topology.MetricSpace.Isometry
import Mathlib.Topology.MetricSpace.Lipschitz
#align_import topology.metric_space.isometric_smul from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Set
open ENNReal Pointwise
universe u v w
vari... | Mathlib/Topology/MetricSpace/IsometricSMul.lean | 392 | 394 | theorem nndist_div_right [DivInvMonoid M] [PseudoMetricSpace M] [IsometricSMul Mᵐᵒᵖ M]
(a b c : M) : nndist (a / c) (b / c) = nndist a b := by |
simp only [div_eq_mul_inv, nndist_mul_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
theorem extract_append_left {a b : Array α} {i j : Nat} (h : j ≤ a.size) :
(a ++ b).extract i j = a.extrac... | Mathlib/Data/Array/ExtractLemmas.lean | 40 | 42 | theorem extract_eq_of_size_le_end {a : Array α} (h : a.size ≤ l) :
a.extract p l = a.extract p a.size := by |
simp only [extract, Nat.min_eq_right h, Nat.sub_eq, mkEmpty_eq, Nat.min_self]
|
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.Orientation
import Mathlib.Tactic.LinearCombination
#align_import analysis.inner_product_space.two_dim from "leanprover-community/mathlib"@"cd8fafa2fac98e1a67097e8a91ad9901cfde48af"
non... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 161 | 168 | theorem areaForm_map {F : Type*} [NormedAddCommGroup F] [InnerProductSpace ℝ F]
[hF : Fact (finrank ℝ F = 2)] (φ : E ≃ₗᵢ[ℝ] F) (x y : F) :
(Orientation.map (Fin 2) φ.toLinearEquiv o).areaForm x y =
o.areaForm (φ.symm x) (φ.symm y) := by |
have : φ.symm ∘ ![x, y] = ![φ.symm x, φ.symm y] := by
ext i
fin_cases i <;> rfl
simp [areaForm_to_volumeForm, volumeForm_map, this]
|
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"@"b99e2d58a5e6861833fa8de11e51a81144258db4"
open Function
universe u v w
namespace SimpleGraph
variable {V : Type u} {V' : Type v} {V'' : Typ... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 743 | 744 | theorem map_snd_darts {u v : V} (p : G.Walk u v) : p.darts.map (·.snd) = p.support.tail := by |
simpa using congr_arg List.tail (cons_map_snd_darts p)
|
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.UnitaryGroup
#align_import analysis.inner_product_space.pi_L2 from "leanprover-community/mathlib"@"13bce9a6b6c44f6b4c91ac1c1d2a816e2533d395"
set_... | Mathlib/Analysis/InnerProductSpace/PiL2.lean | 813 | 830 | theorem Orthonormal.exists_orthonormalBasis_extension_of_card_eq {ι : Type*} [Fintype ι]
(card_ι : finrank 𝕜 E = Fintype.card ι) {v : ι → E} {s : Set ι}
(hv : Orthonormal 𝕜 (s.restrict v)) : ∃ b : OrthonormalBasis ι 𝕜 E, ∀ i ∈ s, b i = v i := by |
have hsv : Injective (s.restrict v) := hv.linearIndependent.injective
have hX : Orthonormal 𝕜 ((↑) : Set.range (s.restrict v) → E) := by
rwa [orthonormal_subtype_range hsv]
obtain ⟨Y, b₀, hX, hb₀⟩ := hX.exists_orthonormalBasis_extension
have hιY : Fintype.card ι = Y.card := by
refine card_ι.symm.trans... |
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Logic.Embedding.Basic
import Mathlib.Data.Fintype.CardEmbedding
import Mathlib.Topology.Algebra.Module.Multilinear.Topology
#align_import analysis.normed_space.multilinear from "leanprover-community/mathlib"@"f40476639bac089693a489c9e354ebd75... | Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean | 453 | 457 | theorem opNorm_neg : ‖-f‖ = ‖f‖ := by |
rw [norm_def]
apply congr_arg
ext
simp
|
import Mathlib.Data.Real.Sqrt
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.ContinuousLinearMap
import Mathlib.Analysis.NormedSpace.Basic
#align_import data.is_R_or_C.basic from "leanprover-community/mathlib"@"baa88307f3e699fa7054ef04ec79fa4f056169cb"
section
local notation "�... | Mathlib/Analysis/RCLike/Basic.lean | 606 | 606 | theorem div_I (z : K) : z / I = -(z * I) := by | rw [div_eq_mul_inv, inv_I, mul_neg]
|
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.GroupTheory.FreeGroup.IsFreeGroup
import Mathlib.Data.List.Chain
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Data.Set.Pointwise.SMul
#align_import group_theor... | Mathlib/GroupTheory/CoprodI.lean | 584 | 586 | theorem mem_smul_iff_of_ne {i j : ι} (hij : i ≠ j) {m₁ : M i} {m₂ : M j} {w : Word M} :
⟨_, m₁⟩ ∈ (of m₂ • w).toList ↔ ⟨i, m₁⟩ ∈ w.toList := by |
simp [mem_smul_iff, *]
|
import Mathlib.Order.SuccPred.Basic
import Mathlib.Order.BoundedOrder
#align_import order.succ_pred.limit from "leanprover-community/mathlib"@"1e05171a5e8cf18d98d9cf7b207540acb044acae"
variable {α : Type*}
namespace Order
open Function Set OrderDual
section LT
variable [LT α]
def IsSuccLimit (a : α) : Pr... | Mathlib/Order/SuccPred/Limit.lean | 46 | 47 | theorem not_isSuccLimit_iff_exists_covBy (a : α) : ¬IsSuccLimit a ↔ ∃ b, b ⋖ a := by |
simp [IsSuccLimit]
|
import Mathlib.Algebra.BigOperators.Group.Finset
#align_import data.nat.gcd.big_operators from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
namespace Nat
variable {ι : Type*}
theorem coprime_list_prod_left_iff {l : List ℕ} {k : ℕ} :
Coprime l.prod k ↔ ∀ n ∈ l, Coprime n k := by
... | Mathlib/Data/Nat/GCD/BigOperators.lean | 56 | 58 | theorem coprime_fintype_prod_right_iff [Fintype ι] {x : ℕ} {s : ι → ℕ} :
Coprime x (∏ i, s i) ↔ ∀ i, Coprime x (s i) := by |
simp [coprime_prod_right_iff]
|
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Order.LiminfLimsup
import Mathlib.Topology.Instances.Rat
import Mathlib.Topology.MetricSpace.Algebra
import Mathlib.Topology.MetricSpace.IsometricSMul
import Mathlib.Topology.Sequences
#align_import analysis.normed.group.basic from "leanprover-community/mat... | Mathlib/Analysis/Normed/Group/Basic.lean | 1,723 | 1,724 | theorem mul_mem_closedBall_mul_iff {c : E} : a * c ∈ closedBall (b * c) r ↔ a ∈ closedBall b r := by |
simp only [mem_closedBall, dist_eq_norm_div, mul_div_mul_right_eq_div]
|
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 | 358 | 360 | theorem tendsto_comp_coe_Ioi_atBot :
Tendsto (fun x : Ioi a => f x) atBot l ↔ Tendsto f (𝓝[>] a) l := by |
rw [← map_coe_Ioi_atBot, tendsto_map'_iff]; rfl
|
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Polynomial.Inductions
import Mathlib.RingTheory.Localization.Basic
#align_import data.polynomial.laurent from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
open Polynomial Func... | Mathlib/Algebra/Polynomial/Laurent.lean | 522 | 526 | theorem degree_C_mul_T_ite [DecidableEq R] (n : ℤ) (a : R) :
degree (C a * T n) = if a = 0 then ⊥ else ↑n := by |
split_ifs with h <;>
simp only [h, map_zero, zero_mul, degree_zero, degree_C_mul_T, Ne,
not_false_iff]
|
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.GroupTheory.MonoidLocalization
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.GroupTheory.GroupAction.Ring
#align_import ring_theory.localization.basic from "leanprover-community/mathlib"@"b69c9a770ecf37eb21... | Mathlib/RingTheory/Localization/Basic.lean | 135 | 144 | theorem of_le (N : Submonoid R) (h₁ : M ≤ N) (h₂ : ∀ r ∈ N, IsUnit (algebraMap R S r)) :
IsLocalization N S where
map_units' r := h₂ r r.2
surj' s :=
have ⟨⟨x, y, hy⟩, H⟩ := IsLocalization.surj M s
⟨⟨x, y, h₁ hy⟩, H⟩
exists_of_eq {x y} := by |
rw [IsLocalization.eq_iff_exists M]
rintro ⟨c, hc⟩
exact ⟨⟨c, h₁ c.2⟩, hc⟩
|
import Mathlib.LinearAlgebra.FinsuppVectorSpace
import Mathlib.LinearAlgebra.Matrix.Basis
import Mathlib.LinearAlgebra.Matrix.Nondegenerate
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.ToLinearEquiv
import Mathlib.LinearAlgebra.SesquilinearForm
import Mathlib.LinearAlgebra... | Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean | 292 | 295 | theorem LinearMap.toMatrix₂'_comp (B : (n → R) →ₗ[R] (m → R) →ₗ[R] R) (f : (n' → R) →ₗ[R] n → R) :
toMatrix₂' (B.comp f) = (toMatrix' f)ᵀ * toMatrix₂' B := by |
rw [← LinearMap.compl₂_id (B.comp f), ← LinearMap.compl₁₂]
simp
|
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.unoriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
... | Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean | 408 | 413 | theorem angle_le_pi_div_two_of_angle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∠ p₁ p₂ p₃ = π / 2) :
∠ p₂ p₃ p₁ ≤ π / 2 := by |
rw [angle, ← inner_eq_zero_iff_angle_eq_pi_div_two, real_inner_comm, ← neg_eq_zero, ←
inner_neg_left, neg_vsub_eq_vsub_rev] at h
rw [angle, ← vsub_add_vsub_cancel p₁ p₂ p₃, add_comm]
exact angle_add_le_pi_div_two_of_inner_eq_zero h
|
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.Perm
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open Equiv Function Finset
variable {... | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 237 | 243 | theorem SameCycle.exists_pow_eq'' [Finite α] (h : SameCycle f x y) :
∃ i : ℕ, 0 < i ∧ i ≤ orderOf f ∧ (f ^ i) x = y := by |
classical
obtain ⟨_ | i, hi, rfl⟩ := h.exists_pow_eq'
· refine ⟨orderOf f, orderOf_pos f, le_rfl, ?_⟩
rw [pow_orderOf_eq_one, pow_zero]
· exact ⟨i.succ, i.zero_lt_succ, hi.le, by rfl⟩
|
import Mathlib.MeasureTheory.Measure.VectorMeasure
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
#align_import measure_theory.measure.with_density_vector_measure from "leanprover-community/mathlib"@"d1bd9c5df2867c1cb463bc6364446d57bdd9f7f1"
noncomputable section
open scoped Classical MeasureTheory NNReal ... | Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean | 64 | 65 | theorem withDensityᵥ_zero : μ.withDensityᵥ (0 : α → E) = 0 := by |
ext1 s hs; erw [withDensityᵥ_apply (integrable_zero α E μ) hs]; simp
|
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.Module.Basic
import Mathlib.Algebra.Regular.SMul
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Rat.BigOperators
import Mathlib.GroupTheory.GroupAction.Hom
import Mathlib.Data.Set.Subsingleton
#align_import data.finsupp.basic from "leanprover... | Mathlib/Data/Finsupp/Basic.lean | 785 | 789 | theorem mapDomain_comapDomain (hf : Function.Injective f) (l : β →₀ M)
(hl : ↑l.support ⊆ Set.range f) :
mapDomain f (comapDomain f l hf.injOn) = l := by |
conv_rhs => rw [← embDomain_comapDomain (f := ⟨f, hf⟩) hl (M := M), embDomain_eq_mapDomain]
rfl
|
import Mathlib.ModelTheory.Satisfiability
import Mathlib.Combinatorics.SimpleGraph.Basic
#align_import model_theory.graph from "leanprover-community/mathlib"@"e56b8fea84d60fe434632b9d3b829ee685fb0c8f"
set_option linter.uppercaseLean3 false
universe u v w w'
namespace FirstOrder
namespace Language
open FirstOr... | Mathlib/ModelTheory/Graph.lean | 107 | 110 | theorem _root_.SimpleGraph.simpleGraphOfStructure (G : SimpleGraph V) :
@simpleGraphOfStructure V G.structure _ = G := by |
ext
rfl
|
import Mathlib.Data.List.GetD
import Mathlib.Data.Nat.Bits
import Mathlib.Algebra.Ring.Nat
import Mathlib.Order.Basic
import Mathlib.Tactic.AdaptationNote
import Mathlib.Tactic.Common
#align_import data.nat.bitwise from "leanprover-community/mathlib"@"6afc9b06856ad973f6a2619e3e8a0a8d537a58f2"
open Function
names... | Mathlib/Data/Nat/Bitwise.lean | 285 | 294 | theorem testBit_two_pow_of_ne {n m : ℕ} (hm : n ≠ m) : testBit (2 ^ n) m = false := by |
rw [testBit, shiftRight_eq_div_pow]
cases' hm.lt_or_lt with hm hm
· rw [Nat.div_eq_of_lt]
· simp
· exact Nat.pow_lt_pow_right Nat.one_lt_two hm
· rw [Nat.pow_div hm.le Nat.two_pos, ← Nat.sub_add_cancel (succ_le_of_lt <| Nat.sub_pos_of_lt hm)]
-- Porting note: XXX why does this make it work?
rw ... |
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Order.Filter.Germ
import Mathlib.Topology.ContinuousFunction.Algebra
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
#align_import measure_theory.function.ae_eq_fun from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a21598... | Mathlib/MeasureTheory/Function/AEEqFun.lean | 170 | 171 | theorem ext {f g : α →ₘ[μ] β} (h : f =ᵐ[μ] g) : f = g := by |
rwa [← f.mk_coeFn, ← g.mk_coeFn, mk_eq_mk]
|
import Mathlib.Topology.MetricSpace.Algebra
import Mathlib.Analysis.Normed.Field.Basic
#align_import analysis.normed.mul_action from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
variable {α β : Type*}
instance NonUnitalSeminormedRing.to_boundedSMul [NonUnitalSeminormedRing α] : Boun... | Mathlib/Analysis/Normed/MulAction.lean | 119 | 120 | theorem edist_smul₀ (s : α) (x y : β) : edist (s • x) (s • y) = ‖s‖₊ • edist x y := by |
simp only [edist_nndist, nndist_smul₀, ENNReal.coe_mul, ENNReal.smul_def, smul_eq_mul]
|
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 | 611 | 613 | theorem connectedComponentIn_subset (F : Set α) (x : α) : connectedComponentIn F x ⊆ F := by |
rw [connectedComponentIn]
split_ifs <;> simp
|
import Mathlib.Geometry.Manifold.MFDeriv.Defs
#align_import geometry.manifold.mfderiv from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
noncomputable section
open scoped Topology Manifold
open Set Bundle
section DerivativesProperties
variable
{𝕜 : Type*} [NontriviallyNormedFiel... | Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | 297 | 300 | theorem mdifferentiableWithinAt_inter (ht : t ∈ 𝓝 x) :
MDifferentiableWithinAt I I' f (s ∩ t) x ↔ MDifferentiableWithinAt I I' f s x := by |
rw [MDifferentiableWithinAt, MDifferentiableWithinAt,
(differentiable_within_at_localInvariantProp I I').liftPropWithinAt_inter ht]
|
import Mathlib.Topology.UniformSpace.Cauchy
import Mathlib.Topology.UniformSpace.Separation
import Mathlib.Topology.DenseEmbedding
#align_import topology.uniform_space.uniform_embedding from "leanprover-community/mathlib"@"195fcd60ff2bfe392543bceb0ec2adcdb472db4c"
open Filter Function Set Uniformity Topology
sec... | Mathlib/Topology/UniformSpace/UniformEmbedding.lean | 326 | 329 | theorem completeSpace_congr {e : α ≃ β} (he : UniformEmbedding e) :
CompleteSpace α ↔ CompleteSpace β := by |
rw [completeSpace_iff_isComplete_range he.toUniformInducing, e.range_eq_univ,
completeSpace_iff_isComplete_univ]
|
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_theory.isomorphism from "leanprover-community/mathlib"@"8350c34a64b9bc3fc64335df8006bffcadc7baa6"
universe v u
-- morphism levels before object levels. See note [CategoryTheory universes].
namespace CategoryTheory
open Category
structure Iso {... | Mathlib/CategoryTheory/Iso.lean | 475 | 477 | theorem eq_of_inv_eq_inv {f g : X ⟶ Y} [IsIso f] [IsIso g] (p : inv f = inv g) : f = g := by |
apply (cancel_epi (inv f)).1
erw [inv_hom_id, p, inv_hom_id]
|
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 | 657 | 688 | theorem norm_const_smul_le (hp : p ≠ 0) (c : 𝕜) (f : lp E p) : ‖c • f‖ ≤ ‖c‖ * ‖f‖ := by |
rcases p.trichotomy with (rfl | rfl | hp)
· exact absurd rfl hp
· cases isEmpty_or_nonempty α
· simp [lp.eq_zero' f]
have hcf := lp.isLUB_norm (c • f)
have hfc := (lp.isLUB_norm f).mul_left (norm_nonneg c)
simp_rw [← Set.range_comp, Function.comp] at hfc
-- TODO: some `IsLUB` API should make ... |
import Mathlib.CategoryTheory.Limits.Shapes.NormalMono.Basic
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
#align_import category_theory.limits.shapes.normal_mono.equalizers from "leanprover-community/mathlib"@"3a061790136d13594ec10c7c90d202335ac5d854"
noncomputable section
open CategoryTheory
open... | Mathlib/CategoryTheory/Limits/Shapes/NormalMono/Equalizers.lean | 315 | 328 | theorem mono_of_zero_kernel {X Y : C} (f : X ⟶ Y) (Z : C)
(l : IsLimit (KernelFork.ofι (0 : Z ⟶ X) (show 0 ≫ f = 0 by simp))) : Mono f :=
⟨fun u v huv => by
obtain ⟨W, w, hw, hl⟩ := normalEpiOfEpi (coequalizer.π u v)
obtain ⟨m, hm⟩ := coequalizer.desc' f huv
have reassoced {W : C} (h : coequalizer u v... |
rw [← Category.assoc, eq_whisker hw]
have hwf : w ≫ f = 0 := by rw [← hm, reassoced, zero_comp]
obtain ⟨n, hn⟩ := KernelFork.IsLimit.lift' l _ hwf
rw [Fork.ι_ofι, HasZeroMorphisms.comp_zero] at hn
have : IsIso (coequalizer.π u v) := by
apply isIso_colimit_cocone_parallelPair_of_eq hn.symm h... |
import Mathlib.Topology.UniformSpace.UniformConvergence
import Mathlib.Topology.UniformSpace.Equicontinuity
import Mathlib.Topology.Separation
import Mathlib.Topology.Support
#align_import topology.uniform_space.compact from "leanprover-community/mathlib"@"735b22f8f9ff9792cf4212d7cb051c4c994bc685"
open scoped Cla... | Mathlib/Topology/UniformSpace/Compact.lean | 215 | 224 | theorem HasCompactMulSupport.is_one_at_infty {f : α → γ} [TopologicalSpace γ] [One γ]
(h : HasCompactMulSupport f) : Tendsto f (cocompact α) (𝓝 1) := by |
-- Porting note: move to src/topology/support.lean once the port is over
intro N hN
rw [mem_map, mem_cocompact']
refine ⟨mulTSupport f, h.isCompact, ?_⟩
rw [compl_subset_comm]
intro v hv
rw [mem_preimage, image_eq_one_of_nmem_mulTSupport hv]
exact mem_of_mem_nhds hN
|
import Mathlib.Analysis.Convex.Topology
#align_import topology.algebra.module.locally_convex from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open TopologicalSpace Filter Set
open Topology Pointwise
section LinearOrderedField
variable (𝕜 E : Type*) [LinearOrderedField 𝕜] [AddCom... | Mathlib/Topology/Algebra/Module/LocallyConvex.lean | 130 | 142 | theorem Disjoint.exists_open_convexes [LocallyConvexSpace 𝕜 E] {s t : Set E} (disj : Disjoint s t)
(hs₁ : Convex 𝕜 s) (hs₂ : IsCompact s) (ht₁ : Convex 𝕜 t) (ht₂ : IsClosed t) :
∃ u v, IsOpen u ∧ IsOpen v ∧ Convex 𝕜 u ∧ Convex 𝕜 v ∧ s ⊆ u ∧ t ⊆ v ∧ Disjoint u v := by |
letI : UniformSpace E := TopologicalAddGroup.toUniformSpace E
haveI : UniformAddGroup E := comm_topologicalAddGroup_is_uniform
have := (LocallyConvexSpace.convex_open_basis_zero 𝕜 E).comap fun x : E × E => x.2 - x.1
rw [← uniformity_eq_comap_nhds_zero] at this
rcases disj.exists_uniform_thickening_of_basis ... |
import Mathlib.CategoryTheory.Bicategory.Basic
import Mathlib.CategoryTheory.Monoidal.Mon_
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equalizers
#align_import category_theory.monoidal.Bimod from "leanprover-community/mathlib"@"4698e35ca56a0d4fa53aa5639c3364e0a77f4eba"
universe v₁ v₂ u₁ u₂
open Categor... | Mathlib/CategoryTheory/Monoidal/Bimod.lean | 1,000 | 1,039 | theorem pentagon_bimod {V W X Y Z : Mon_ C} (M : Bimod V W) (N : Bimod W X) (P : Bimod X Y)
(Q : Bimod Y Z) :
whiskerRight (associatorBimod M N P).hom Q ≫
(associatorBimod M (N.tensorBimod P) Q).hom ≫
whiskerLeft M (associatorBimod N P Q).hom =
(associatorBimod (M.tensorBimod N) P Q).hom ≫
... |
dsimp [associatorBimod]
ext
apply coequalizer.hom_ext
dsimp
dsimp only [AssociatorBimod.hom]
slice_lhs 1 2 => rw [ι_colimMap, parallelPairHom_app_one]
slice_lhs 2 3 => rw [coequalizer.π_desc]
slice_rhs 1 2 => rw [coequalizer.π_desc]
dsimp [AssociatorBimod.homAux]
refine (cancel_epi ((tensorRight _)... |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Set.Pointwise.Finite
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.SetTheory.Cardinal.Finite
#align_import group_theory.finiteness from "leanprover-community/... | Mathlib/GroupTheory/Finiteness.lean | 443 | 453 | theorem rank_closure_finset_le_card (s : Finset G) : Group.rank (closure (s : Set G)) ≤ s.card := by |
classical
let t : Finset (closure (s : Set G)) := s.preimage Subtype.val Subtype.coe_injective.injOn
have ht : closure (t : Set (closure (s : Set G))) = ⊤ := by
rw [Finset.coe_preimage]
exact closure_preimage_eq_top (s : Set G)
apply (Group.rank_le (closure (s : Set G)) ht).trans
suffices H : Set.Inj... |
import Mathlib.MeasureTheory.Function.LpOrder
#align_import measure_theory.function.l1_space from "leanprover-community/mathlib"@"ccdbfb6e5614667af5aa3ab2d50885e0ef44a46f"
noncomputable section
open scoped Classical
open Topology ENNReal MeasureTheory NNReal
open Set Filter TopologicalSpace ENNReal EMetric Meas... | Mathlib/MeasureTheory/Function/L1Space.lean | 505 | 508 | theorem Memℒp.integrable_norm_rpow {f : α → β} {p : ℝ≥0∞} (hf : Memℒp f p μ) (hp_ne_zero : p ≠ 0)
(hp_ne_top : p ≠ ∞) : Integrable (fun x : α => ‖f x‖ ^ p.toReal) μ := by |
rw [← memℒp_one_iff_integrable]
exact hf.norm_rpow hp_ne_zero hp_ne_top
|
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
import Mathlib.CategoryTheory.Monoidal.Opposite
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.CommSq
#align_import category_... | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 295 | 296 | theorem braiding_leftUnitor (X : C) : (β_ X (𝟙_ C)).hom ≫ (λ_ X).hom = (ρ_ X).hom := by |
rw [← whiskerRight_iff, comp_whiskerRight, braiding_leftUnitor_aux₂]
|
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 | 143 | 153 | theorem basisSets_smul_right (v : E) (U : Set E) (hU : U ∈ p.basisSets) :
∀ᶠ x : 𝕜 in 𝓝 0, x • v ∈ U := by |
rcases p.basisSets_iff.mp hU with ⟨s, r, hr, hU⟩
rw [hU, Filter.eventually_iff]
simp_rw [(s.sup p).mem_ball_zero, map_smul_eq_mul]
by_cases h : 0 < (s.sup p) v
· simp_rw [(lt_div_iff h).symm]
rw [← _root_.ball_zero_eq]
exact Metric.ball_mem_nhds 0 (div_pos hr h)
simp_rw [le_antisymm (not_lt.mp h) (... |
import Mathlib.Topology.Homotopy.Equiv
import Mathlib.CategoryTheory.Equivalence
import Mathlib.AlgebraicTopology.FundamentalGroupoid.Product
#align_import algebraic_topology.fundamental_groupoid.induced_maps from "leanprover-community/mathlib"@"e5470580a62bf043e10976760edfe73c913eb71e"
noncomputable section
uni... | Mathlib/AlgebraicTopology/FundamentalGroupoid/InducedMaps.lean | 205 | 216 | theorem eq_diag_path : (πₘ f).map p ≫ ⟦H.evalAt x₁⟧ = H.diagonalPath' p ∧
(⟦H.evalAt x₀⟧ ≫ (πₘ g).map p : fromTop (f x₀) ⟶ fromTop (g x₁)) = H.diagonalPath' p := by |
rw [H.apply_zero_path, H.apply_one_path, H.evalAt_eq]
erw [H.evalAt_eq] -- Porting note: `rw` didn't work, so using `erw`
dsimp only [prodToProdTopI]
constructor
· slice_lhs 2 4 => rw [eqToHom_trans, eqToHom_refl] -- Porting note: this ↓ `simp` didn't do this
slice_lhs 2 4 => simp [← CategoryTheory.Funct... |
import Mathlib.Analysis.NormedSpace.ContinuousAffineMap
import Mathlib.Analysis.Calculus.ContDiff.Basic
#align_import analysis.calculus.affine_map from "leanprover-community/mathlib"@"839b92fedff9981cf3fe1c1f623e04b0d127f57c"
namespace ContinuousAffineMap
variable {𝕜 V W : Type*} [NontriviallyNormedField 𝕜]
va... | Mathlib/Analysis/Calculus/AffineMap.lean | 30 | 33 | theorem contDiff {n : ℕ∞} (f : V →ᴬ[𝕜] W) : ContDiff 𝕜 n f := by |
rw [f.decomp]
apply f.contLinear.contDiff.add
exact contDiff_const
|
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.Algebra.Pi
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.BigOperators.RingEquiv
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.Algebra.Module.Pi
import Mathlib.Algebra.Star.BigOperators... | Mathlib/Data/Matrix/Basic.lean | 849 | 852 | theorem dotProduct_diagonal (i : m) : v ⬝ᵥ diagonal w i = v i * w i := by |
have : ∀ j ≠ i, v j * diagonal w i j = 0 := fun j hij => by
simp [diagonal_apply_ne' _ hij]
convert Finset.sum_eq_single i (fun j _ => this j) _ using 1 <;> simp
|
import Mathlib.Algebra.Polynomial.Degree.TrailingDegree
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.reverse from "leanprover-community/mathlib"@"44de64f183393284a16016dfb2a48ac97382f2bd"
namespace Polynomial
open Polynomial Finsupp Finset
open... | Mathlib/Algebra/Polynomial/Reverse.lean | 295 | 296 | theorem reverse_natDegree (f : R[X]) : f.reverse.natDegree = f.natDegree - f.natTrailingDegree := by |
rw [f.natDegree_eq_reverse_natDegree_add_natTrailingDegree, add_tsub_cancel_right]
|
import Mathlib.Algebra.Order.Field.Power
import Mathlib.Data.Int.LeastGreatest
import Mathlib.Data.Rat.Floor
import Mathlib.Data.NNRat.Defs
#align_import algebra.order.archimedean from "leanprover-community/mathlib"@"6f413f3f7330b94c92a5a27488fdc74e6d483a78"
open Int Set
variable {α : Type*}
class Archimedean (... | Mathlib/Algebra/Order/Archimedean.lean | 242 | 249 | theorem exists_pow_lt_of_lt_one (hx : 0 < x) (hy : y < 1) : ∃ n : ℕ, y ^ n < x := by |
by_cases y_pos : y ≤ 0
· use 1
simp only [pow_one]
exact y_pos.trans_lt hx
rw [not_le] at y_pos
rcases pow_unbounded_of_one_lt x⁻¹ (one_lt_inv y_pos hy) with ⟨q, hq⟩
exact ⟨q, by rwa [inv_pow, inv_lt_inv hx (pow_pos y_pos _)] at hq⟩
|
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.RingTheory.RootsOfUnity.Complex
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.RingTheory.RootsOfUnity.Basic
import Mathlib.FieldTh... | Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | 412 | 412 | theorem cyclotomic_two (R : Type*) [Ring R] : cyclotomic 2 R = X + 1 := by | simp [cyclotomic_prime]
|
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 | 520 | 521 | theorem nodal_empty : nodal ∅ v = 1 := by |
rfl
|
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Iterate
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.Algebra.InfiniteSum.Real
#align_import analysis.specific_limits.basic from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
n... | Mathlib/Analysis/SpecificLimits/Basic.lean | 675 | 688 | theorem tendsto_nat_floor_mul_div_atTop {a : R} (ha : 0 ≤ a) :
Tendsto (fun x ↦ (⌊a * x⌋₊ : R) / x) atTop (𝓝 a) := by |
have A : Tendsto (fun x : R ↦ a - x⁻¹) atTop (𝓝 (a - 0)) :=
tendsto_const_nhds.sub tendsto_inv_atTop_zero
rw [sub_zero] at A
apply tendsto_of_tendsto_of_tendsto_of_le_of_le' A tendsto_const_nhds
· refine eventually_atTop.2 ⟨1, fun x hx ↦ ?_⟩
simp only [le_div_iff (zero_lt_one.trans_le hx), _root_.sub_... |
import Mathlib.Init.Core
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
open Polynomial Algebra FiniteD... | Mathlib/NumberTheory/Cyclotomic/Basic.lean | 174 | 188 | theorem union_right [h : IsCyclotomicExtension (S ∪ T) A B] :
IsCyclotomicExtension T (adjoin A {b : B | ∃ a : ℕ+, a ∈ S ∧ b ^ (a : ℕ) = 1}) B := by |
have : {b : B | ∃ n : ℕ+, n ∈ S ∪ T ∧ b ^ (n : ℕ) = 1} =
{b : B | ∃ n : ℕ+, n ∈ S ∧ b ^ (n : ℕ) = 1} ∪
{b : B | ∃ n : ℕ+, n ∈ T ∧ b ^ (n : ℕ) = 1} := by
refine le_antisymm ?_ ?_
· rintro x ⟨n, hn₁ | hn₂, hnpow⟩
· left; exact ⟨n, hn₁, hnpow⟩
· right; exact ⟨n, hn₂, hnpow⟩
· rintr... |
import Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol
#align_import number_theory.legendre_symbol.norm_num from "leanprover-community/mathlib"@"e2621d935895abe70071ab828a4ee6e26a52afe4"
section Lemmas
namespace Mathlib.Meta.NormNum
def jacobiSymNat (a b : ℕ) : ℤ :=
jacobiSym a b
#align norm_num.jacobi_sym_... | Mathlib/Tactic/NormNum/LegendreSymbol.lean | 121 | 131 | theorem jacobiSymNat.odd_even (a b c : ℕ) (r : ℤ) (ha : a % 2 = 1) (hb : b % 2 = 0) (hc : b / 2 = c)
(hr : jacobiSymNat a c = r) : jacobiSymNat a b = r := by |
have ha' : legendreSym 2 a = 1 := by
simp only [legendreSym.mod 2 a, Int.ofNat_mod_ofNat, ha]
decide
rcases eq_or_ne c 0 with (rfl | hc')
· rw [← hr, Nat.eq_zero_of_dvd_of_div_eq_zero (Nat.dvd_of_mod_eq_zero hb) hc]
· haveI : NeZero c := ⟨hc'⟩
-- for `jacobiSym.mul_right`
rwa [← Nat.mod_add_div... |
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
#align_import data.finset.locally_finite from "leanprover-community/mathlib"@"442a83d738cb208d3600056c489be16900ba701d"
assert_not_exists MonoidWithZero
assert_not_exists Finset.sum
open Function OrderDual
open FinsetInterval
variable {ι α : T... | Mathlib/Order/Interval/Finset/Basic.lean | 724 | 726 | theorem Iic_erase [DecidableEq α] (b : α) : (Iic b).erase b = Iio b := by |
ext
simp_rw [Finset.mem_erase, mem_Iic, mem_Iio, lt_iff_le_and_ne, and_comm]
|
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Set.Finite
#align_import order.filter.basic from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
set_option autoImplicit true
open Function Set Order
open scoped Classical
universe u v w x y
structure Filter (α : Type*) where... | Mathlib/Order/Filter/Basic.lean | 233 | 240 | theorem exists_mem_and_iff {P : Set α → Prop} {Q : Set α → Prop} (hP : Antitone P)
(hQ : Antitone Q) : ((∃ u ∈ f, P u) ∧ ∃ u ∈ f, Q u) ↔ ∃ u ∈ f, P u ∧ Q u := by |
constructor
· rintro ⟨⟨u, huf, hPu⟩, v, hvf, hQv⟩
exact
⟨u ∩ v, inter_mem huf hvf, hP inter_subset_left hPu, hQ inter_subset_right hQv⟩
· rintro ⟨u, huf, hPu, hQu⟩
exact ⟨⟨u, huf, hPu⟩, u, huf, hQu⟩
|
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.Topology.Order.LeftRightLim
#align_import measure_theory.measure.stieltjes from "leanprover-community/mathlib"@"20d5763051978e9bc6428578ed070445df6a18b3"
noncomputable section
open scoped Classical
open Set Filter Function ENNReal NNReal T... | Mathlib/MeasureTheory/Measure/Stieltjes.lean | 76 | 80 | theorem iInf_Ioi_eq (f : StieltjesFunction) (x : ℝ) : ⨅ r : Ioi x, f r = f x := by |
suffices Function.rightLim f x = ⨅ r : Ioi x, f r by rw [← this, f.rightLim_eq]
rw [f.mono.rightLim_eq_sInf, sInf_image']
rw [← neBot_iff]
infer_instance
|
import Mathlib.Data.Set.Finite
import Mathlib.Order.Partition.Finpartition
#align_import data.setoid.partition from "leanprover-community/mathlib"@"b363547b3113d350d053abdf2884e9850a56b205"
namespace Setoid
variable {α : Type*}
theorem eq_of_mem_eqv_class {c : Set (Set α)} (H : ∀ a, ∃! b ∈ c, a ∈ b) {x b b'}
... | Mathlib/Data/Setoid/Partition.lean | 382 | 384 | theorem iUnion : ⋃ i, s i = univ := by |
ext x
simp [hs.exists_mem x]
|
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.Data.Finset.Sort
#align_import data.polynomial.basic from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
set_option linter.uppercaseLean3 false
noncomputable section
structure ... | Mathlib/Algebra/Polynomial/Basic.lean | 708 | 709 | theorem coeff_one_zero : coeff (1 : R[X]) 0 = 1 := by |
simp [coeff_one]
|
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Equiv.Fin
import Mathlib.ModelTheory.LanguageMap
#align_import model_theory.syntax from "leanprover-community/mathlib"@"d565b3df44619c1498326936be16f1a935df0728"
universe u v w u' v'
namespace FirstOrder
namespace Language
variable (L : Language.{u, v}) {L' : L... | Mathlib/ModelTheory/Syntax.lean | 566 | 573 | theorem sum_elim_comp_relabelAux {m : ℕ} {g : α → Sum β (Fin n)} {v : β → M}
{xs : Fin (n + m) → M} : Sum.elim v xs ∘ relabelAux g m =
Sum.elim (Sum.elim v (xs ∘ castAdd m) ∘ g) (xs ∘ natAdd n) := by |
ext x
cases' x with x x
· simp only [BoundedFormula.relabelAux, Function.comp_apply, Sum.map_inl, Sum.elim_inl]
cases' g x with l r <;> simp
· simp [BoundedFormula.relabelAux]
|
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.PolynomialExp
#align_import analysis.calculus.bump_function_inner from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9... | Mathlib/Analysis/SpecialFunctions/SmoothTransition.lean | 46 | 46 | theorem zero_of_nonpos {x : ℝ} (hx : x ≤ 0) : expNegInvGlue x = 0 := by | simp [expNegInvGlue, hx]
|
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.List.Perm
import Mathlib.Data.List.Range
#align_import data.list.sublists from "leanprover-community/mathlib"@"ccad6d5093bd2f5c6ca621fc74674cce51355af6"
universe u v w
variable {α : Type u} {β : Type v} {γ : Type w}
open Nat
namespace List
@[simp]
theo... | Mathlib/Data/List/Sublists.lean | 197 | 198 | theorem sublists'_eq_sublists (l : List α) : sublists' l = map reverse (sublists (reverse l)) := by |
rw [← sublists'_reverse, reverse_reverse]
|
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