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 |
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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 | 53 | 54 | theorem coeSubmodule_top : coeSubmodule S (⊤ : Ideal R) = 1 := by |
rw [coeSubmodule, Submodule.map_top, Submodule.one_eq_range]
|
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Measurable
open MeasureTheory
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] [LocallyCompactSpace 𝕜]
{E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [MeasurableSpace E] [OpensMeasurableSpace E]
{F : Type*} [Norm... | Mathlib/Analysis/Calculus/LineDeriv/Measurable.lean | 33 | 38 | theorem measurableSet_lineDifferentiableAt (hf : Continuous f) :
MeasurableSet {x : E | LineDifferentiableAt 𝕜 f x v} := by |
borelize 𝕜
let g : E → 𝕜 → F := fun x t ↦ f (x + t • v)
have hg : Continuous g.uncurry := by apply hf.comp; continuity
exact measurable_prod_mk_right (measurableSet_of_differentiableAt_with_param 𝕜 hg)
|
import Mathlib.Topology.Instances.RealVectorSpace
import Mathlib.Analysis.NormedSpace.AffineIsometry
#align_import analysis.normed_space.mazur_ulam from "leanprover-community/mathlib"@"78261225eb5cedc61c5c74ecb44e5b385d13b733"
variable {E PE F PF : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [MetricSpace PE]
... | Mathlib/Analysis/NormedSpace/MazurUlam.lean | 87 | 96 | theorem map_midpoint (f : PE ≃ᵢ PF) (x y : PE) : f (midpoint ℝ x y) = midpoint ℝ (f x) (f y) := by |
set e : PE ≃ᵢ PE :=
((f.trans <| (pointReflection ℝ <| midpoint ℝ (f x) (f y)).toIsometryEquiv).trans f.symm).trans
(pointReflection ℝ <| midpoint ℝ x y).toIsometryEquiv
have hx : e x = x := by simp [e]
have hy : e y = y := by simp [e]
have hm := e.midpoint_fixed hx hy
simp only [e, trans_apply] at... |
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.Complement
open Monoid Coprod Multiplicative Subgroup Function
def HNNExtension.con (G : Type*) [Group G] (A B : Subgroup G) (φ : A ≃* B) :
Con (G ∗ Multiplicative ℤ) :=
conGen (fun x y => ∃ (a : A),
x = inr (ofAdd 1) * inl (a : G) ∧
... | Mathlib/GroupTheory/HNNExtension.lean | 97 | 99 | theorem lift_t (f : G →* H) (x : H) (hx : ∀ a : A, x * f ↑a = f (φ a : G) * x) :
lift f x hx t = x := by |
delta HNNExtension; simp [lift, t]
|
import Mathlib.CategoryTheory.Sites.Coherent.Comparison
import Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves
import Mathlib.CategoryTheory.Sites.Coherent.ReflectsPrecoherent
import Mathlib.CategoryTheory.Sites.Coherent.ReflectsPreregular
import Mathlib.CategoryTheory.Sites.InducedTopology
import Mathlib.Categ... | Mathlib/CategoryTheory/Sites/Coherent/SheafComparison.lean | 55 | 76 | theorem exists_effectiveEpiFamily_iff_mem_induced (X : C) (S : Sieve X) :
(∃ (α : Type) (_ : Finite α) (Y : α → C) (π : (a : α) → (Y a ⟶ X)),
EffectiveEpiFamily Y π ∧ (∀ a : α, (S.arrows) (π a)) ) ↔
(S ∈ F.inducedTopologyOfIsCoverDense (coherentTopology _) X) := by |
refine ⟨fun ⟨α, _, Y, π, ⟨H₁, H₂⟩⟩ ↦ ?_, fun hS ↦ ?_⟩
· apply (mem_sieves_iff_hasEffectiveEpiFamily (Sieve.functorPushforward _ S)).mpr
refine ⟨α, inferInstance, fun i => F.obj (Y i),
fun i => F.map (π i), ⟨?_,
fun a => Sieve.image_mem_functorPushforward F S (H₂ a)⟩⟩
exact F.map_finite_effectiv... |
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
import Mathlib.LinearAlgebra.Ray
import Mathlib.Tactic.GCongr
#align_import analysis.convex.segment from "leanprover-community/mathlib"@"c5773405394e073885e2a144c9ca14637e8eb963"
... | Mathlib/Analysis/Convex/Segment.lean | 62 | 65 | theorem segment_eq_image₂ (x y : E) :
[x -[𝕜] y] =
(fun p : 𝕜 × 𝕜 => p.1 • x + p.2 • y) '' { p | 0 ≤ p.1 ∧ 0 ≤ p.2 ∧ p.1 + p.2 = 1 } := by |
simp only [segment, image, Prod.exists, mem_setOf_eq, exists_prop, and_assoc]
|
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.Star.SelfAdjoint
#align_import algebra.star.unitary from "leanprover-community/mathlib"@"247a102b14f3cebfee126293341af5f6bed00237"
def unitary (R : Type*) [Monoid R] [StarMul R] : Submonoid R where
carrier := { U | star U * U = 1 ∧ U * st... | Mathlib/Algebra/Star/Unitary.lean | 141 | 145 | theorem _root_.IsUnit.mem_unitary_of_star_mul_self {u : R} (hu : IsUnit u)
(h_mul : star u * u = 1) : u ∈ unitary R := by |
refine unitary.mem_iff.mpr ⟨h_mul, ?_⟩
lift u to Rˣ using hu
exact left_inv_eq_right_inv h_mul u.mul_inv ▸ u.mul_inv
|
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 Sigma
variable {α : ι → Type*} {β : Type*} (s s₁ s₂ : Finset ι) (... | Mathlib/Data/Finset/Sigma.lean | 112 | 114 | theorem _root_.biSup_finsetSigma [CompleteLattice β] (s : Finset ι) (t : ∀ i, Finset (α i))
(f : Sigma α → β) : ⨆ ij ∈ s.sigma t, f ij = ⨆ (i ∈ s) (j ∈ t i), f ⟨i, j⟩ := by |
simp_rw [← Finset.iSup_coe, Finset.coe_sigma, biSup_sigma]
|
import Mathlib.CategoryTheory.Sites.Coherent.Comparison
import Mathlib.CategoryTheory.Sites.Coherent.ExtensiveSheaves
import Mathlib.CategoryTheory.Sites.Coherent.ReflectsPrecoherent
import Mathlib.CategoryTheory.Sites.Coherent.ReflectsPreregular
import Mathlib.CategoryTheory.Sites.InducedTopology
import Mathlib.Categ... | Mathlib/CategoryTheory/Sites/Coherent/SheafComparison.lean | 161 | 178 | theorem exists_effectiveEpi_iff_mem_induced (X : C) (S : Sieve X) :
(∃ (Y : C) (π : Y ⟶ X),
EffectiveEpi π ∧ S.arrows π) ↔
(S ∈ F.inducedTopologyOfIsCoverDense (regularTopology _) X) := by |
refine ⟨fun ⟨Y, π, ⟨H₁, H₂⟩⟩ ↦ ?_, fun hS ↦ ?_⟩
· apply (mem_sieves_iff_hasEffectiveEpi (Sieve.functorPushforward _ S)).mpr
refine ⟨F.obj Y, F.map π, ⟨?_, Sieve.image_mem_functorPushforward F S H₂⟩⟩
exact F.map_effectiveEpi _
· obtain ⟨Y, π, ⟨H₁, H₂⟩⟩ := (mem_sieves_iff_hasEffectiveEpi _).mp hS
let g... |
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Analysis.MeanInequalities
import Mathlib.Analysis.MeanInequalitiesPow
import Mathlib.MeasureTheory.Function.SpecialFunctions.Basic
#align_import measure_theory.integral.mean_inequalities from "leanprover-community/mathlib"@"13bf7613c96a9fd66a81b9020a82cad9... | Mathlib/MeasureTheory/Integral/MeanInequalities.lean | 87 | 90 | theorem fun_eq_funMulInvSnorm_mul_snorm {p : ℝ} (f : α → ℝ≥0∞) (hf_nonzero : (∫⁻ a, f a ^ p ∂μ) ≠ 0)
(hf_top : (∫⁻ a, f a ^ p ∂μ) ≠ ⊤) {a : α} :
f a = funMulInvSnorm f p μ a * (∫⁻ c, f c ^ p ∂μ) ^ (1 / p) := by |
simp [funMulInvSnorm, mul_assoc, ENNReal.inv_mul_cancel, hf_nonzero, hf_top]
|
import Mathlib.Combinatorics.SimpleGraph.AdjMatrix
import Mathlib.LinearAlgebra.Matrix.PosDef
open Finset Matrix
namespace SimpleGraph
variable {V : Type*} (R : Type*)
variable [Fintype V] [DecidableEq V] (G : SimpleGraph V) [DecidableRel G.Adj]
def degMatrix [AddMonoidWithOne R] : Matrix V V R := Matrix.diago... | Mathlib/Combinatorics/SimpleGraph/LapMatrix.lean | 117 | 120 | theorem lapMatrix_toLin'_apply_eq_zero_iff_forall_reachable (x : V → ℝ) :
Matrix.toLin' (G.lapMatrix ℝ) x = 0 ↔ ∀ i j : V, G.Reachable i j → x i = x j := by |
rw [← (posSemidef_lapMatrix ℝ G).toLinearMap₂'_zero_iff, star_trivial,
lapMatrix_toLinearMap₂'_apply'_eq_zero_iff_forall_reachable]
|
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.Calculus.Monotone
import Mathlib.Data.Set.Function
import Mathlib.Algebra.Group.Basic
import Mathlib.Tactic.WLOG
#align_import analysis.bounded_variation from ... | Mathlib/Analysis/BoundedVariation.lean | 89 | 94 | theorem eq_of_edist_zero_on {f f' : α → E} {s : Set α} (h : ∀ ⦃x⦄, x ∈ s → edist (f x) (f' x) = 0) :
eVariationOn f s = eVariationOn f' s := by |
dsimp only [eVariationOn]
congr 1 with p : 1
congr 1 with i : 1
rw [edist_congr_right (h <| p.snd.prop.2 (i + 1)), edist_congr_left (h <| p.snd.prop.2 i)]
|
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Topology.Algebra.OpenSubgroup
import Mathlib.Topology.Algebra.Ring.Basic
#align_import topology.algebra.nonarchimedean.basic from "leanprover-community/mathlib"@"83f81aea33931a1edb94ce0f32b9a5d484de6978"
open scoped Pointwise Topology
class Nonarchimede... | Mathlib/Topology/Algebra/Nonarchimedean/Basic.lean | 69 | 75 | theorem nonarchimedean_of_emb (f : G →* H) (emb : OpenEmbedding f) : NonarchimedeanGroup H :=
{ is_nonarchimedean := fun U hU =>
have h₁ : f ⁻¹' U ∈ 𝓝 (1 : G) := by |
apply emb.continuous.tendsto
rwa [f.map_one]
let ⟨V, hV⟩ := is_nonarchimedean (f ⁻¹' U) h₁
⟨{ Subgroup.map f V with isOpen' := emb.isOpenMap _ V.isOpen }, Set.image_subset_iff.2 hV⟩ }
|
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Data.Nat.SuccPred
#align_import data.int.succ_pred from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Order
namespace Int
-- so that Lean reads `Int.succ` through `SuccOrder.succ`
@[instance] abbrev instSuccOrder : Su... | Mathlib/Data/Int/SuccPred.lean | 88 | 90 | theorem natCast_covBy {a b : ℕ} : (a : ℤ) ⋖ b ↔ a ⋖ b := by |
rw [Nat.covBy_iff_succ_eq, Int.covBy_iff_succ_eq]
exact Int.natCast_inj
|
import Mathlib.FieldTheory.Normal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.Integral
#align_import field_theory.is_alg_closed.basic from "leanprover-community/mathlib"@"00f91228655eecdcd3ac97a7fd8dbcb139fe990a"
universe u v w
open scoped Classical Polynomial
open Polynomial
vari... | Mathlib/FieldTheory/IsAlgClosed/Basic.lean | 149 | 162 | theorem of_ringEquiv (k' : Type u) [Field k'] (e : k ≃+* k')
[IsAlgClosed k] : IsAlgClosed k' := by |
apply IsAlgClosed.of_exists_root
intro p hmp hp
have hpe : degree (p.map e.symm.toRingHom) ≠ 0 := by
rw [degree_map]
exact ne_of_gt (degree_pos_of_irreducible hp)
rcases IsAlgClosed.exists_root (k := k) (p.map e.symm) hpe with ⟨x, hx⟩
use e x
rw [IsRoot] at hx
apply e.symm.injective
rw [map_zer... |
import Mathlib.Algebra.DirectSum.Finsupp
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
#align_import linear_algebra.direct_sum.finsupp from "leanprover-community/mathlib"@"9b9d125b7be0930f564a68f1d73ace10cf46064d"
noncomputable section
open DirectSum TensorProduct
ope... | Mathlib/LinearAlgebra/DirectSum/Finsupp.lean | 102 | 107 | theorem finsuppLeft_apply (t : (ι →₀ M) ⊗[R] N) (i : ι) :
finsuppLeft R M N ι t i = rTensor N (Finsupp.lapply i) t := by |
induction t using TensorProduct.induction_on with
| zero => simp
| tmul f n => simp only [finsuppLeft_apply_tmul_apply, rTensor_tmul, Finsupp.lapply_apply]
| add x y hx hy => simp [map_add, hx, hy]
|
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Finset.Pairwise
#align_import data.finsupp.big_operators from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
variable {ι M : Type*} [DecidableEq ι]
theorem List.support_sum_subset [Add... | Mathlib/Data/Finsupp/BigOperators.lean | 60 | 66 | theorem List.mem_foldr_sup_support_iff [Zero M] {l : List (ι →₀ M)} {x : ι} :
x ∈ l.foldr (Finsupp.support · ⊔ ·) ∅ ↔ ∃ f ∈ l, x ∈ f.support := by |
simp only [Finset.sup_eq_union, List.foldr_map, Finsupp.mem_support_iff, exists_prop]
induction' l with hd tl IH
· simp
· simp only [foldr, Function.comp_apply, Finset.mem_union, Finsupp.mem_support_iff, ne_eq, IH,
find?, mem_cons, exists_eq_or_imp]
|
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Preserves.Basic
#align_import category_theory.limits.preserves.shapes.pullbacks from "leanprover-community/mathlib"@"f11e306adb9f2a393539d2bb4293bf1b42caa7ac"
noncomputable section
universe v₁ v₂ u₁ u₂
-- Porting note: ne... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Pullbacks.lean | 126 | 128 | theorem PreservesPullback.iso_hom_snd :
(PreservesPullback.iso G f g).hom ≫ pullback.snd = G.map pullback.snd := by |
simp [PreservesPullback.iso]
|
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 | 101 | 103 | theorem χ₄_nat_three_mod_four {n : ℕ} (hn : n % 4 = 3) : χ₄ n = -1 := by |
rw [χ₄_nat_mod_four, hn]
rfl
|
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.Dual
import Mathlib.Data.Fin.FlagRange
open Set Submodule
namespace Basis
section Semiring
variable {R M : Type*} [Semiring R] [AddCommMonoid M] [Module R M] {n : ℕ}
def flag (b : Basis (Fin n) R M) (k : Fin (n + 1)) : Submodule R M :=
.span R <... | Mathlib/LinearAlgebra/Basis/Flag.lean | 42 | 45 | theorem flag_succ (b : Basis (Fin n) R M) (k : Fin n) :
b.flag k.succ = (R ∙ b k) ⊔ b.flag k.castSucc := by |
simp only [flag, Fin.castSucc_lt_castSucc_iff]
simp [Fin.castSucc_lt_iff_succ_le, le_iff_eq_or_lt, setOf_or, image_insert_eq, span_insert]
|
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Algebra.Regular.Basic
import Mathlib.Data.Nat.Choose.Sum
#align_import data.polynomial.coeff from "leanprover-community/mathlib"@"2651125b48fc5c170ab1111afd0817c903b1fc6c"
set_option linter.uppercaseLean3 false
no... | Mathlib/Algebra/Polynomial/Coeff.lean | 60 | 65 | theorem support_smul [SMulZeroClass S R] (r : S) (p : R[X]) :
support (r • p) ⊆ support p := by |
intro i hi
simp? [mem_support_iff] at hi ⊢ says simp only [mem_support_iff, coeff_smul, ne_eq] at hi ⊢
contrapose! hi
simp [hi]
|
import Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
import Mathlib.LinearAlgebra.CliffordAlgebra.Fold
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.Dual
#align_import linear_algebra.clifford_algebra.contraction from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2... | Mathlib/LinearAlgebra/CliffordAlgebra/Contraction.lean | 167 | 172 | theorem contractLeft_ι (x : M) : d⌋ι Q x = algebraMap R _ (d x) := by |
-- Porting note: Lean cannot figure out anymore the third argument
refine (foldr'_ι _ _ ?_ _ _).trans <| by
simp_rw [contractLeftAux_apply_apply, mul_zero, sub_zero,
Algebra.algebraMap_eq_smul_one]
exact fun m x fx ↦ contractLeftAux_contractLeftAux Q d m x fx
|
import Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.bind from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
universe v
variable {α : Type*} {β : Type v} {γ δ : Ty... | Mathlib/Data/Multiset/Bind.lean | 89 | 93 | theorem prod_join [CommMonoid α] {S : Multiset (Multiset α)} :
prod (join S) = prod (map prod S) := by |
induction S using Multiset.induction with
| empty => simp
| cons _ _ ih => simp [ih]
|
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Topology.Algebra.Constructions
#align_import topology.algebra.group.basic from "leanprover-community/mathlib"@"3b1890e71632be9e3... | Mathlib/Topology/Algebra/Group/Basic.lean | 146 | 154 | theorem discreteTopology_of_isOpen_singleton_one (h : IsOpen ({1} : Set G)) :
DiscreteTopology G := by |
rw [← singletons_open_iff_discrete]
intro g
suffices {g} = (g⁻¹ * ·) ⁻¹' {1} by
rw [this]
exact (continuous_mul_left g⁻¹).isOpen_preimage _ h
simp only [mul_one, Set.preimage_mul_left_singleton, eq_self_iff_true, inv_inv,
Set.singleton_eq_singleton_iff]
|
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Combinatorics.Additive.AP.Three.Defs
import Mathlib.Combinatorics.Pigeonhole
import Mathlib.Data.Complex.ExponentialBounds
#align_import combinatorics.additive.behrend from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
open N... | Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | 163 | 164 | theorem map_mod (a : Fin n.succ → ℕ) : map d a % d = a 0 % d := by |
rw [map_succ, Nat.add_mul_mod_self_right]
|
import Mathlib.Combinatorics.Quiver.Path
import Mathlib.Combinatorics.Quiver.Push
#align_import combinatorics.quiver.symmetric from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
universe v u w v'
namespace Quiver
-- Porting note: no hasNonemptyInstance linter yet
def Symmetrify (V : ... | Mathlib/Combinatorics/Quiver/Symmetric.lean | 66 | 72 | theorem reverse_inj [h : HasInvolutiveReverse V] {a b : V}
(f g : a ⟶ b) : reverse f = reverse g ↔ f = g := by |
constructor
· rintro h
simpa using congr_arg Quiver.reverse h
· rintro h
congr
|
import Mathlib.MeasureTheory.Covering.Differentiation
import Mathlib.MeasureTheory.Covering.VitaliFamily
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.Data.Set.Pairwise.Lat... | Mathlib/MeasureTheory/Covering/Besicovitch.lean | 187 | 192 | theorem inter' (i : Fin N.succ) : dist (a.c i) (a.c (last N)) ≤ a.r i + a.r (last N) := by |
rcases lt_or_le i (last N) with (H | H)
· exact a.inter i H
· have I : i = last N := top_le_iff.1 H
have := (a.rpos (last N)).le
simp only [I, add_nonneg this this, dist_self]
|
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.List.MinMax
import Mathlib.Algebra.Tropical.Basic
import Mathlib.Order.ConditionallyCompleteLattice.Finset
#align_import algebra.tropical.big_operators from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
variable {R S :... | Mathlib/Algebra/Tropical/BigOperators.lean | 92 | 96 | theorem Finset.trop_inf [LinearOrder R] [OrderTop R] (s : Finset S) (f : S → R) :
trop (s.inf f) = ∑ i ∈ s, trop (f i) := by |
convert Multiset.trop_inf (s.val.map f)
simp only [Multiset.map_map, Function.comp_apply]
rfl
|
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.asymptotics.specific_asymptotics from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter Asymptotics
open Topology
sectio... | Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean | 140 | 152 | theorem Filter.Tendsto.cesaro_smul {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {u : ℕ → E}
{l : E} (h : Tendsto u atTop (𝓝 l)) :
Tendsto (fun n : ℕ => (n⁻¹ : ℝ) • ∑ i ∈ range n, u i) atTop (𝓝 l) := by |
rw [← tendsto_sub_nhds_zero_iff, ← isLittleO_one_iff ℝ]
have := Asymptotics.isLittleO_sum_range_of_tendsto_zero (tendsto_sub_nhds_zero_iff.2 h)
apply ((isBigO_refl (fun n : ℕ => (n : ℝ)⁻¹) atTop).smul_isLittleO this).congr' _ _
· filter_upwards [Ici_mem_atTop 1] with n npos
have nposℝ : (0 : ℝ) < n := Nat.... |
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
noncomputable section
universe u
namespace SetTheory
open scoped PGame
namespace PGame... | Mathlib/SetTheory/Game/Nim.lean | 67 | 67 | theorem leftMoves_nim (o : Ordinal) : (nim o).LeftMoves = o.out.α := by | rw [nim_def]; rfl
|
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.HasseDeriv
#align_import data.polynomial.taylor from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
namespace Polynomial
open Polynomial... | Mathlib/Algebra/Polynomial/Taylor.lean | 66 | 66 | theorem taylor_one : taylor r (1 : R[X]) = C 1 := by | rw [← C_1, taylor_C]
|
import Mathlib.MeasureTheory.Function.Jacobian
import Mathlib.MeasureTheory.Measure.Lebesgue.Complex
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
#align_import analysis.special_functions.polar_coord from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
noncomputable section... | Mathlib/Analysis/SpecialFunctions/PolarCoord.lean | 110 | 123 | theorem polarCoord_source_ae_eq_univ : polarCoord.source =ᵐ[volume] univ := by |
have A : polarCoord.sourceᶜ ⊆ LinearMap.ker (LinearMap.snd ℝ ℝ ℝ) := by
intro x hx
simp only [polarCoord_source, compl_union, mem_inter_iff, mem_compl_iff, mem_setOf_eq, not_lt,
Classical.not_not] at hx
exact hx.2
have B : volume (LinearMap.ker (LinearMap.snd ℝ ℝ ℝ) : Set (ℝ × ℝ)) = 0 := by
a... |
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u
variable (R : Type u)
section Semiring
variable [Semiring R]
class inductive Ex... | Mathlib/Algebra/CharP/ExpChar.lean | 93 | 97 | theorem char_eq_expChar_iff (p q : ℕ) [hp : CharP R p] [hq : ExpChar R q] : p = q ↔ p.Prime := by |
cases' hq with q hq_one hq_prime hq_hchar
· rw [(CharP.eq R hp inferInstance : p = 0)]
decide
· exact ⟨fun hpq => hpq.symm ▸ hq_prime, fun _ => CharP.eq R hp hq_hchar⟩
|
import Mathlib.Analysis.Convex.Basic
import Mathlib.Order.Closure
#align_import analysis.convex.hull from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
open Set
open Pointwise
variable {𝕜 E F : Type*}
section convexHull
section OrderedSemiring
variable [OrderedSemiring 𝕜]
secti... | Mathlib/Analysis/Convex/Hull.lean | 127 | 131 | theorem convexHull_pair (x y : E) : convexHull 𝕜 {x, y} = segment 𝕜 x y := by |
refine (convexHull_min ?_ <| convex_segment _ _).antisymm
(segment_subset_convexHull (mem_insert _ _) <| subset_insert _ _ <| mem_singleton _)
rw [insert_subset_iff, singleton_subset_iff]
exact ⟨left_mem_segment _ _ _, right_mem_segment _ _ _⟩
|
import Mathlib.CategoryTheory.Functor.Hom
import Mathlib.CategoryTheory.Products.Basic
import Mathlib.Data.ULift
#align_import category_theory.yoneda from "leanprover-community/mathlib"@"369525b73f229ccd76a6ec0e0e0bf2be57599768"
namespace CategoryTheory
open Opposite
universe v₁ u₁ u₂
-- morphism levels before ... | Mathlib/CategoryTheory/Yoneda.lean | 255 | 258 | theorem coreprW_app_hom (X : C) (f : F.coreprX ⟶ X) :
(F.coreprW.app X).hom f = F.map f F.coreprx := by |
simp only [coyoneda_obj_obj, unop_op, Iso.app_hom, coreprx, ← FunctorToTypes.naturality,
coyoneda_obj_map, Category.id_comp]
|
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.MeasureTheory.Group.LIntegral
import Mathlib.MeasureTheory.Integral.Marginal
import Mathlib.MeasureTheory.Measure.Stiel... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 449 | 450 | theorem regionBetween_subset (f g : α → ℝ) (s : Set α) : regionBetween f g s ⊆ s ×ˢ univ := by |
simpa only [prod_univ, regionBetween, Set.preimage, setOf_subset_setOf] using fun a => And.left
|
import Mathlib.Order.CompleteLattice
import Mathlib.Order.GaloisConnection
import Mathlib.Data.Set.Lattice
import Mathlib.Tactic.AdaptationNote
#align_import data.rel from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
variable {α β γ : Type*}
def Rel (α β : Type*) :=
α → β → Prop --... | Mathlib/Data/Rel.lean | 112 | 115 | theorem comp_right_id (r : Rel α β) : r • @Eq β = r := by |
unfold comp
ext y
simp
|
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import topology.metric_space.hausdorff_distance from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
noncomputable section
open NNReal ENNReal Topology Set Filter Bornology
universe u v w
variable {ι : Sort*} {α : Type u} {β :... | Mathlib/Topology/MetricSpace/Thickening.lean | 114 | 122 | theorem frontier_thickening_disjoint (A : Set α) :
Pairwise (Disjoint on fun r : ℝ => frontier (thickening r A)) := by |
refine (pairwise_disjoint_on _).2 fun r₁ r₂ hr => ?_
rcases le_total r₁ 0 with h₁ | h₁
· simp [thickening_of_nonpos h₁]
refine ((disjoint_singleton.2 fun h => hr.ne ?_).preimage _).mono (frontier_thickening_subset _)
(frontier_thickening_subset _)
apply_fun ENNReal.toReal at h
rwa [ENNReal.toReal_ofRea... |
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
#align_import algebra.gcd_monoid.multiset from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
namespace Multiset
variable {α : Type*} [CancelCommMonoidWithZero α] [NormalizedG... | Mathlib/Algebra/GCDMonoid/Multiset.lean | 207 | 209 | theorem gcd_ndunion (s₁ s₂ : Multiset α) : (ndunion s₁ s₂).gcd = GCDMonoid.gcd s₁.gcd s₂.gcd := by |
rw [← gcd_dedup, dedup_ext.2, gcd_dedup, gcd_add]
simp
|
import Mathlib.Control.Applicative
import Mathlib.Control.Traversable.Basic
#align_import control.traversable.lemmas from "leanprover-community/mathlib"@"3342d1b2178381196f818146ff79bc0e7ccd9e2d"
universe u
open LawfulTraversable
open Function hiding comp
open Functor
attribute [functor_norm] LawfulTraversabl... | Mathlib/Control/Traversable/Lemmas.lean | 89 | 90 | theorem id_sequence (x : t α) : sequence (f := Id) (pure <$> x) = pure x := by |
simp [sequence, traverse_map, id_traverse]
|
import Mathlib.Data.SetLike.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.ModelTheory.Semantics
#align_import model_theory.definability from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v w u₁
namespace Set
variable {M : Type w} (A : Set M) (L : FirstOrder.Lang... | Mathlib/ModelTheory/Definability.lean | 141 | 144 | theorem definable_finset_biInter {ι : Type*} {f : ι → Set (α → M)}
(hf : ∀ i, A.Definable L (f i)) (s : Finset ι) : A.Definable L (⋂ i ∈ s, f i) := by |
rw [← Finset.inf_set_eq_iInter]
exact definable_finset_inf hf s
|
import Mathlib.Algebra.Group.Commute.Defs
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Set.Function
import Mathlib.Logic.Pairwise
#align_import algebra.group.pi from "leanprover-community/mathlib"@"e4bc74cbaf429d706cb9140902f7ca6c431e75a4"
assert_not_exists AddMonoidWithOne
assert_not_exists Mono... | Mathlib/Algebra/Group/Pi/Lemmas.lean | 552 | 555 | theorem uncurry_mulSingle_mulSingle [DecidableEq α] [∀ a, DecidableEq (β a)] [∀ a b, One (γ a b)]
(a : α) (b : β a) (x : γ a b) :
Sigma.uncurry (Pi.mulSingle a (Pi.mulSingle b x)) = Pi.mulSingle (Sigma.mk a b) x := by |
rw [← curry_mulSingle ⟨a, b⟩, uncurry_curry]
|
import Mathlib.Data.SetLike.Fintype
import Mathlib.Algebra.Divisibility.Prod
import Mathlib.RingTheory.Nakayama
import Mathlib.RingTheory.SimpleModule
import Mathlib.Tactic.RSuffices
#align_import ring_theory.artinian from "leanprover-community/mathlib"@"210657c4ea4a4a7b234392f70a3a2a83346dfa90"
open Set Filter Po... | Mathlib/RingTheory/Artinian.lean | 175 | 195 | theorem IsArtinian.finite_of_linearIndependent [Nontrivial R] [IsArtinian R M] {s : Set M}
(hs : LinearIndependent R ((↑) : s → M)) : s.Finite := by |
refine by_contradiction fun hf => (RelEmbedding.wellFounded_iff_no_descending_seq.1
(wellFounded_submodule_lt (R := R) (M := M))).elim' ?_
have f : ℕ ↪ s := Set.Infinite.natEmbedding s hf
have : ∀ n, (↑) ∘ f '' { m | n ≤ m } ⊆ s := by
rintro n x ⟨y, _, rfl⟩
exact (f y).2
have : ∀ a b : ℕ, a ≤ b ↔
... |
import Mathlib.Algebra.Module.Submodule.Map
#align_import linear_algebra.basic from "leanprover-community/mathlib"@"9d684a893c52e1d6692a504a118bfccbae04feeb"
open Function
open Pointwise
variable {R : Type*} {R₁ : Type*} {R₂ : Type*} {R₃ : Type*}
variable {K : Type*}
variable {M : Type*} {M₁ : Type*} {M₂ : Type*... | Mathlib/Algebra/Module/Submodule/Ker.lean | 129 | 132 | theorem ker_restrict [AddCommMonoid M₁] [Module R M₁] {p : Submodule R M} {q : Submodule R M₁}
{f : M →ₗ[R] M₁} (hf : ∀ x : M, x ∈ p → f x ∈ q) :
ker (f.restrict hf) = LinearMap.ker (f.domRestrict p) := by |
rw [restrict_eq_codRestrict_domRestrict, ker_codRestrict]
|
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 | 232 | 238 | theorem const_isBounded (ι : Type*) [Nonempty ι] {p : Seminorm 𝕜 E} {q : ι' → Seminorm 𝕜₂ F}
(f : E →ₛₗ[σ₁₂] F) : IsBounded (fun _ : ι => p) q f ↔ ∀ i, ∃ C : ℝ≥0, (q i).comp f ≤ C • p := by |
constructor <;> intro h i
· rcases h i with ⟨s, C, h⟩
exact ⟨C, le_trans h (smul_le_smul (Finset.sup_le fun _ _ => le_rfl) le_rfl)⟩
use {Classical.arbitrary ι}
simp only [h, Finset.sup_singleton]
|
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 | 117 | 119 | theorem top_prod : (⊤ : Filter α) ×ˢ g = g.comap Prod.snd := by |
dsimp only [SProd.sprod]
rw [Filter.prod, comap_top, top_inf_eq]
|
import Mathlib.Algebra.Group.Units.Equiv
import Mathlib.CategoryTheory.Endomorphism
#align_import category_theory.conj from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
universe v u
namespace CategoryTheory
namespace Iso
variable {C : Type u} [Category.{v} C]
def homCongr {X Y X₁... | Mathlib/CategoryTheory/Conj.lean | 114 | 115 | theorem refl_conj (f : End X) : (Iso.refl X).conj f = f := by |
rw [conj_apply, Iso.refl_inv, Iso.refl_hom, Category.id_comp, Category.comp_id]
|
import Mathlib.Analysis.Normed.Group.Quotient
import Mathlib.Topology.Instances.AddCircle
#align_import analysis.normed.group.add_circle from "leanprover-community/mathlib"@"084f76e20c88eae536222583331abd9468b08e1c"
noncomputable section
open Set
open Int hiding mem_zmultiples_iff
open AddSubgroup
namespace A... | Mathlib/Analysis/Normed/Group/AddCircle.lean | 86 | 117 | theorem norm_eq {x : ℝ} : ‖(x : AddCircle p)‖ = |x - round (p⁻¹ * x) * p| := by |
suffices ∀ x : ℝ, ‖(x : AddCircle (1 : ℝ))‖ = |x - round x| by
rcases eq_or_ne p 0 with (rfl | hp)
· simp
have hx := norm_coe_mul p x p⁻¹
rw [abs_inv, eq_inv_mul_iff_mul_eq₀ ((not_congr abs_eq_zero).mpr hp)] at hx
rw [← hx, inv_mul_cancel hp, this, ← abs_mul, mul_sub, mul_inv_cancel_left₀ hp, mul... |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.RingTheory.MatrixAlgebra
#align_import ring_theory.polynomial_algebra from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
universe u v w
open Polynomial Tensor... | Mathlib/RingTheory/PolynomialAlgebra.lean | 109 | 111 | theorem toFunLinear_one_tmul_one :
toFunLinear R A (1 ⊗ₜ[R] 1) = 1 := by |
rw [toFunLinear_tmul_apply, toFunBilinear_apply_apply, Polynomial.aeval_one, one_smul]
|
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.GroupAction.Units
import Mathlib.Logic.Basic
import Mathlib.Tactic.Ring
#align_import ring_theory.coprime.basic from "leanprover-community/mathlib"@"a95b16cb... | Mathlib/RingTheory/Coprime/Basic.lean | 102 | 105 | theorem IsCoprime.dvd_of_dvd_mul_right (H1 : IsCoprime x z) (H2 : x ∣ y * z) : x ∣ y := by |
let ⟨a, b, H⟩ := H1
rw [← mul_one y, ← H, mul_add, ← mul_assoc, mul_left_comm]
exact dvd_add (dvd_mul_left _ _) (H2.mul_left _)
|
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 | 341 | 347 | theorem measure_smul_average [IsFiniteMeasure μ] (f : α → E) :
(μ univ).toReal • ⨍ x, f x ∂μ = ∫ x, f x ∂μ := by |
rcases eq_or_ne μ 0 with hμ | hμ
· rw [hμ, integral_zero_measure, average_zero_measure, smul_zero]
· rw [average_eq, smul_inv_smul₀]
refine (ENNReal.toReal_pos ?_ <| measure_ne_top _ _).ne'
rwa [Ne, measure_univ_eq_zero]
|
import Mathlib.MeasureTheory.Decomposition.Lebesgue
import Mathlib.MeasureTheory.Measure.Complex
import Mathlib.MeasureTheory.Decomposition.Jordan
import Mathlib.MeasureTheory.Measure.WithDensityVectorMeasure
noncomputable section
open scoped Classical MeasureTheory NNReal ENNReal
open Set
variable {α β : Type*... | Mathlib/MeasureTheory/Decomposition/SignedLebesgue.lean | 148 | 158 | theorem singularPart_totalVariation (s : SignedMeasure α) (μ : Measure α) :
(s.singularPart μ).totalVariation =
s.toJordanDecomposition.posPart.singularPart μ +
s.toJordanDecomposition.negPart.singularPart μ := by |
have :
(s.singularPart μ).toJordanDecomposition =
⟨s.toJordanDecomposition.posPart.singularPart μ,
s.toJordanDecomposition.negPart.singularPart μ, singularPart_mutuallySingular s μ⟩ := by
refine JordanDecomposition.toSignedMeasure_injective ?_
rw [toSignedMeasure_toJordanDecomposition, sing... |
import Mathlib.Algebra.Homology.QuasiIso
#align_import category_theory.preadditive.projective_resolution from "leanprover-community/mathlib"@"324a7502510e835cdbd3de1519b6c66b51fb2467"
universe v u
namespace CategoryTheory
open Category Limits ChainComplex HomologicalComplex
variable {C : Type u} [Category.{v} ... | Mathlib/CategoryTheory/Preadditive/ProjectiveResolution.lean | 95 | 97 | theorem complex_d_comp_π_f_zero :
P.complex.d 1 0 ≫ P.π.f 0 = 0 := by |
rw [← P.π.comm 1 0, single_obj_d, comp_zero]
|
import Mathlib.Algebra.Category.GroupCat.FilteredColimits
import Mathlib.Algebra.Category.ModuleCat.Basic
#align_import algebra.category.Module.filtered_colimits from "leanprover-community/mathlib"@"806bbb0132ba63b93d5edbe4789ea226f8329979"
universe v u
noncomputable section
open scoped Classical
open Category... | Mathlib/Algebra/Category/ModuleCat/FilteredColimits.lean | 72 | 79 | theorem colimitSMulAux_eq_of_rel (r : R) (x y : Σ j, F.obj j)
(h : Types.FilteredColimit.Rel (F ⋙ forget (ModuleCat R)) x y) :
colimitSMulAux F r x = colimitSMulAux F r y := by |
apply M.mk_eq
obtain ⟨k, f, g, hfg⟩ := h
use k, f, g
simp only [Functor.comp_obj, Functor.comp_map, forget_map] at hfg
simp [hfg]
|
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 | 87 | 89 | theorem toIcoMod_mem_Ico' (b : α) : toIcoMod hp 0 b ∈ Set.Ico 0 p := by |
convert toIcoMod_mem_Ico hp 0 b
exact (zero_add p).symm
|
import Mathlib.Data.Int.ModEq
import Mathlib.GroupTheory.QuotientGroup
#align_import algebra.modeq from "leanprover-community/mathlib"@"a07d750983b94c530ab69a726862c2ab6802b38c"
namespace AddCommGroup
variable {α : Type*}
section AddCommGroup
variable [AddCommGroup α] {p a a₁ a₂ b b₁ b₂ c : α} {n : ℕ} {z : ℤ}
... | Mathlib/Algebra/ModEq.lean | 311 | 312 | theorem modEq_iff_int_modEq {a b z : ℤ} : a ≡ b [PMOD z] ↔ a ≡ b [ZMOD z] := by |
simp [ModEq, dvd_iff_exists_eq_mul_left, Int.modEq_iff_dvd]
|
import Mathlib.Algebra.Group.Semiconj.Defs
import Mathlib.Algebra.Ring.Defs
#align_import algebra.ring.semiconj from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
universe u v w x
variable {α : Type u} {β : Type v} {γ : Type w} {R : Type x}
open Function
namespace SemiconjBy
@[simp... | Mathlib/Algebra/Ring/Semiconj.lean | 89 | 91 | theorem sub_right (h : SemiconjBy a x y) (h' : SemiconjBy a x' y') :
SemiconjBy a (x - x') (y - y') := by |
simpa only [sub_eq_add_neg] using h.add_right h'.neg_right
|
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.NumberTheory.Bernoulli
#align_import number_theory.bernoulli_polynomials from "leanprover-community/mathlib"@"ca3d21f7f4fd613c2a3c54ac7871163e1e5ecb3a"
noncomputable section... | Mathlib/NumberTheory/BernoulliPolynomials.lean | 72 | 72 | theorem bernoulli_zero : bernoulli 0 = 1 := by | simp [bernoulli]
|
import Mathlib.Topology.EMetricSpace.Paracompact
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.Topology.MetricSpace.ProperSpace
import Mathlib.Topology.ShrinkingLemma
#align_import topology.metric_space.shrinking_lemma from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
unive... | Mathlib/Topology/MetricSpace/ShrinkingLemma.lean | 39 | 46 | theorem exists_subset_iUnion_ball_radius_lt {r : ι → ℝ} (hs : IsClosed s)
(uf : ∀ x ∈ s, { i | x ∈ ball (c i) (r i) }.Finite) (us : s ⊆ ⋃ i, ball (c i) (r i)) :
∃ r' : ι → ℝ, (s ⊆ ⋃ i, ball (c i) (r' i)) ∧ ∀ i, r' i < r i := by |
rcases exists_subset_iUnion_closed_subset hs (fun i => @isOpen_ball _ _ (c i) (r i)) uf us with
⟨v, hsv, hvc, hcv⟩
have := fun i => exists_lt_subset_ball (hvc i) (hcv i)
choose r' hlt hsub using this
exact ⟨r', hsv.trans <| iUnion_mono <| hsub, hlt⟩
|
import Mathlib.FieldTheory.Separable
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Tactic.ApplyFun
#align_import field_theory.finite.basic from "leanprover-community/mathlib"@"12a85fac627bea918960da036049d611b1a3ee43"
variable {K : Type*} {R : Type*}
local notation ... | Mathlib/FieldTheory/Finite/Basic.lean | 168 | 176 | theorem sum_subgroup_units [Ring K] [NoZeroDivisors K]
{G : Subgroup Kˣ} [Fintype G] [Decidable (G = ⊥)] :
∑ x : G, (x.val : K) = if G = ⊥ then 1 else 0 := by |
by_cases G_bot : G = ⊥
· subst G_bot
simp only [ite_true, Subgroup.mem_bot, Fintype.card_ofSubsingleton, Nat.cast_ite, Nat.cast_one,
Nat.cast_zero, univ_unique, Set.default_coe_singleton, sum_singleton, Units.val_one]
· simp only [G_bot, ite_false]
exact sum_subgroup_units_eq_zero G_bot
|
import Mathlib.Analysis.SpecificLimits.Basic
#align_import analysis.hofer from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classical
open Topology
open Filter Finset
local notation "d" => dist
#noalign pos_div_pow_pos
| Mathlib/Analysis/Hofer.lean | 33 | 104 | theorem hofer {X : Type*} [MetricSpace X] [CompleteSpace X] (x : X) (ε : ℝ) (ε_pos : 0 < ε)
{ϕ : X → ℝ} (cont : Continuous ϕ) (nonneg : ∀ y, 0 ≤ ϕ y) : ∃ ε' > 0, ∃ x' : X,
ε' ≤ ε ∧ d x' x ≤ 2 * ε ∧ ε * ϕ x ≤ ε' * ϕ x' ∧ ∀ y, d x' y ≤ ε' → ϕ y ≤ 2 * ϕ x' := by |
by_contra H
have reformulation : ∀ (x') (k : ℕ), ε * ϕ x ≤ ε / 2 ^ k * ϕ x' ↔ 2 ^ k * ϕ x ≤ ϕ x' := by
intro x' k
rw [div_mul_eq_mul_div, le_div_iff, mul_assoc, mul_le_mul_left ε_pos, mul_comm]
positivity
-- Now let's specialize to `ε/2^k`
replace H : ∀ k : ℕ, ∀ x', d x' x ≤ 2 * ε ∧ 2 ^ k * ϕ x ≤ ϕ... |
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Topology.Instances.ENNReal
#align_import analysis.calculus.series from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Metric TopologicalSpace Function Filter
open scoped Topology NNReal
variable {α β F : Type*} [N... | Mathlib/Analysis/NormedSpace/FunctionSeries.lean | 28 | 39 | theorem tendstoUniformlyOn_tsum {f : α → β → F} (hu : Summable u) {s : Set β}
(hfu : ∀ n x, x ∈ s → ‖f n x‖ ≤ u n) :
TendstoUniformlyOn (fun t : Finset α => fun x => ∑ n ∈ t, f n x) (fun x => ∑' n, f n x) atTop
s := by |
refine tendstoUniformlyOn_iff.2 fun ε εpos => ?_
filter_upwards [(tendsto_order.1 (tendsto_tsum_compl_atTop_zero u)).2 _ εpos] with t ht x hx
have A : Summable fun n => ‖f n x‖ :=
.of_nonneg_of_le (fun _ ↦ norm_nonneg _) (fun n => hfu n x hx) hu
rw [dist_eq_norm, ← sum_add_tsum_subtype_compl A.of_norm t, a... |
import Mathlib.RingTheory.Ideal.IsPrimary
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.Polynomial.Quotient
#align_import ring_theory.jacobson_ideal from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
universe u v
namespace Ideal
variable {R : Type u} {S : Type v}... | Mathlib/RingTheory/JacobsonIdeal.lean | 125 | 129 | theorem exists_mul_sub_mem_of_sub_one_mem_jacobson {I : Ideal R} (r : R) (h : r - 1 ∈ jacobson I) :
∃ s, s * r - 1 ∈ I := by |
cases' mem_jacobson_iff.1 h 1 with s hs
use s
simpa [mul_sub] using hs
|
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n α : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 100 | 104 | theorem invOf_fromBlocks_zero₂₁_eq (A : Matrix m m α) (B : Matrix m n α) (D : Matrix n n α)
[Invertible A] [Invertible D] [Invertible (fromBlocks A B 0 D)] :
⅟ (fromBlocks A B 0 D) = fromBlocks (⅟ A) (-(⅟ A * B * ⅟ D)) 0 (⅟ D) := by |
letI := fromBlocksZero₂₁Invertible A B D
convert (rfl : ⅟ (fromBlocks A B 0 D) = _)
|
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Polynomial.Degree.Lemmas
#align_import data.polynomial.erase_lead from "leanprover-community/mathlib"@"fa256f00ce018e7b40e1dc756e403c86680bf448"
noncomputable section
open Polynomial
open Polynomial Finset
namespace Polynomial
variable {R : Type*}... | Mathlib/Algebra/Polynomial/EraseLead.lean | 42 | 43 | theorem eraseLead_support (f : R[X]) : f.eraseLead.support = f.support.erase f.natDegree := by |
simp only [eraseLead, support_erase]
|
import Mathlib.Topology.Instances.Irrational
import Mathlib.Topology.Instances.Rat
import Mathlib.Topology.Compactification.OnePoint
#align_import topology.instances.rat_lemmas from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Set Metric Filter TopologicalSpace
open Topology One... | Mathlib/Topology/Instances/RatLemmas.lean | 72 | 74 | theorem not_firstCountableTopology_opc : ¬FirstCountableTopology ℚ∞ := by |
intro
exact not_countably_generated_nhds_infty_opc inferInstance
|
import Mathlib.Analysis.Complex.Liouville
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathlib.FieldTheory.PolynomialGaloisGroup
import Mathlib.Topology.Algebra.Polynomial
#align_import analysis.complex.polynomial from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
open Polyn... | Mathlib/Analysis/Complex/Polynomial.lean | 34 | 45 | theorem exists_root {f : ℂ[X]} (hf : 0 < degree f) : ∃ z : ℂ, IsRoot f z := by |
by_contra! hf'
/- Since `f` has no roots, `f⁻¹` is differentiable. And since `f` is a polynomial, it tends to
infinity at infinity, thus `f⁻¹` tends to zero at infinity. By Liouville's theorem, `f⁻¹ = 0`. -/
have (z : ℂ) : (f.eval z)⁻¹ = 0 :=
(f.differentiable.inv hf').apply_eq_of_tendsto_cocompact z <|
... |
import Mathlib.Data.Nat.Squarefree
import Mathlib.NumberTheory.Zsqrtd.QuadraticReciprocity
import Mathlib.Tactic.LinearCombination
#align_import number_theory.sum_two_squares from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
section NegOneSquare
-- This could be formulated for ... | Mathlib/NumberTheory/SumTwoSquares.lean | 77 | 81 | theorem ZMod.isSquare_neg_one_of_dvd {m n : ℕ} (hd : m ∣ n) (hs : IsSquare (-1 : ZMod n)) :
IsSquare (-1 : ZMod m) := by |
let f : ZMod n →+* ZMod m := ZMod.castHom hd _
rw [← RingHom.map_one f, ← RingHom.map_neg]
exact hs.map f
|
import Mathlib.Data.SetLike.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.ModelTheory.Semantics
#align_import model_theory.definability from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v w u₁
namespace Set
variable {M : Type w} (A : Set M) (L : FirstOrder.Lang... | Mathlib/ModelTheory/Definability.lean | 147 | 150 | theorem definable_finset_biUnion {ι : Type*} {f : ι → Set (α → M)}
(hf : ∀ i, A.Definable L (f i)) (s : Finset ι) : A.Definable L (⋃ i ∈ s, f i) := by |
rw [← Finset.sup_set_eq_biUnion]
exact definable_finset_sup hf s
|
import Mathlib.RingTheory.IntegrallyClosed
import Mathlib.RingTheory.Trace
import Mathlib.RingTheory.Norm
#align_import ring_theory.discriminant from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
universe u v w z
open scoped Matrix
open Matrix FiniteDimensional Fintype Polynomial Fin... | Mathlib/RingTheory/Discriminant.lean | 171 | 176 | theorem discr_powerBasis_eq_prod' [IsSeparable K L] (e : Fin pb.dim ≃ (L →ₐ[K] E)) :
algebraMap K E (discr K pb.basis) =
∏ i : Fin pb.dim, ∏ j ∈ Ioi i, -((e j pb.gen - e i pb.gen) * (e i pb.gen - e j pb.gen)) := by |
rw [discr_powerBasis_eq_prod _ _ _ e]
congr; ext i; congr; ext j
ring
|
import Mathlib.Data.List.Lattice
import Mathlib.Data.List.Range
import Mathlib.Data.Bool.Basic
#align_import data.list.intervals from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
open Nat
namespace List
def Ico (n m : ℕ) : List ℕ :=
range' n (m - n)
#align list.Ico List.Ico
names... | Mathlib/Data/List/Intervals.lean | 42 | 42 | theorem zero_bot (n : ℕ) : Ico 0 n = range n := by | rw [Ico, Nat.sub_zero, range_eq_range']
|
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.pow from "leanprover-community/mathlib"@"46a64b5b4268c594af770c44d9e502afc6a515cb"
variable {R : Type*} {a b : R}
section Monoid
variable [Monoid R]
... | Mathlib/Algebra/Regular/Pow.lean | 36 | 38 | theorem IsRightRegular.pow (n : ℕ) (rra : IsRightRegular a) : IsRightRegular (a ^ n) := by |
rw [IsRightRegular, ← mul_right_iterate]
exact rra.iterate n
|
import Mathlib.Order.Interval.Finset.Basic
import Mathlib.Data.Fintype.BigOperators
#align_import data.pi.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
open Finset Fintype
variable {ι : Type*} {α : ι → Type*} [Fintype ι] [DecidableEq ι] [∀ i, DecidableEq (α i)]
namespac... | Mathlib/Data/Pi/Interval.lean | 86 | 87 | theorem card_Ioi : (Ioi a).card = (∏ i, (Ici (a i)).card) - 1 := by |
rw [card_Ioi_eq_card_Ici_sub_one, card_Ici]
|
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.NumberTheory.Liouville.Residual
import Mathlib.NumberTheory.Liouville.LiouvilleWith
import Mathlib.Analysis.PSeries
#align_import number_theory.liouville.measure from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open sc... | Mathlib/NumberTheory/Liouville/Measure.lean | 34 | 71 | theorem setOf_liouvilleWith_subset_aux :
{ x : ℝ | ∃ p > 2, LiouvilleWith p x } ⊆
⋃ m : ℤ, (· + (m : ℝ)) ⁻¹' ⋃ n > (0 : ℕ),
{ x : ℝ | ∃ᶠ b : ℕ in atTop, ∃ a ∈ Finset.Icc (0 : ℤ) b,
|x - (a : ℤ) / b| < 1 / (b : ℝ) ^ (2 + 1 / n : ℝ) } := by |
rintro x ⟨p, hp, hxp⟩
rcases exists_nat_one_div_lt (sub_pos.2 hp) with ⟨n, hn⟩
rw [lt_sub_iff_add_lt'] at hn
suffices ∀ y : ℝ, LiouvilleWith p y → y ∈ Ico (0 : ℝ) 1 → ∃ᶠ b : ℕ in atTop,
∃ a ∈ Finset.Icc (0 : ℤ) b, |y - a / b| < 1 / (b : ℝ) ^ (2 + 1 / (n + 1 : ℕ) : ℝ) by
simp only [mem_iUnion, mem_pre... |
import Mathlib.RingTheory.Adjoin.FG
#align_import ring_theory.adjoin.tower from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open Pointwise
universe u v w u₁
variable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁)
namespace Algebra
| Mathlib/RingTheory/Adjoin/Tower.lean | 30 | 46 | theorem adjoin_restrictScalars (C D E : Type*) [CommSemiring C] [CommSemiring D] [CommSemiring E]
[Algebra C D] [Algebra C E] [Algebra D E] [IsScalarTower C D E] (S : Set E) :
(Algebra.adjoin D S).restrictScalars C =
(Algebra.adjoin ((⊤ : Subalgebra C D).map (IsScalarTower.toAlgHom C D E)) S).restrictScal... |
suffices
Set.range (algebraMap D E) =
Set.range (algebraMap ((⊤ : Subalgebra C D).map (IsScalarTower.toAlgHom C D E)) E) by
ext x
change x ∈ Subsemiring.closure (_ ∪ S) ↔ x ∈ Subsemiring.closure (_ ∪ S)
rw [this]
ext x
constructor
· rintro ⟨y, hy⟩
exact ⟨⟨algebraMap D E y, ⟨y, ⟨Algebr... |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.well_known from "leanprover-community/mathlib"@"8199f6717c150a7fe91c4534175f4cf99725978f"
namespace PowerS... | Mathlib/RingTheory/PowerSeries/WellKnown.lean | 84 | 89 | theorem mk_one_mul_one_sub_eq_one : (mk 1 : S⟦X⟧) * (1 - X) = 1 := by |
rw [mul_comm, ext_iff]
intro n
cases n with
| zero => simp
| succ n => simp [sub_mul]
|
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.inner_product_space.adjoint from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open RCLike
open scoped ComplexConjugate
variable {𝕜 E F G : Type... | Mathlib/Analysis/InnerProductSpace/Adjoint.lean | 150 | 152 | theorem apply_norm_eq_sqrt_inner_adjoint_left (A : E →L[𝕜] F) (x : E) :
‖A x‖ = √(re ⟪(A† ∘L A) x, x⟫) := by |
rw [← apply_norm_sq_eq_inner_adjoint_left, Real.sqrt_sq (norm_nonneg _)]
|
import Mathlib.Algebra.Homology.Linear
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Tactic.Abel
#align_import algebra.homology.homotopy from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a148bf9ff"
universe v u
open scoped Classical
noncomputable section
open ... | Mathlib/Algebra/Homology/Homotopy.lean | 51 | 54 | theorem dNext_eq (f : ∀ i j, C.X i ⟶ D.X j) {i i' : ι} (w : c.Rel i i') :
dNext i f = C.d i i' ≫ f i' i := by |
obtain rfl := c.next_eq' w
rfl
|
import Mathlib.Analysis.NormedSpace.Star.GelfandDuality
import Mathlib.Topology.Algebra.StarSubalgebra
#align_import analysis.normed_space.star.continuous_functional_calculus from "leanprover-community/mathlib"@"31c24aa72e7b3e5ed97a8412470e904f82b81004"
open scoped Pointwise ENNReal NNReal ComplexOrder
open Weak... | Mathlib/Analysis/NormedSpace/Star/ContinuousFunctionalCalculus.lean | 103 | 174 | theorem elementalStarAlgebra.isUnit_of_isUnit_of_isStarNormal (h : IsUnit a) :
IsUnit (⟨a, self_mem ℂ a⟩ : elementalStarAlgebra ℂ a) := by |
/- Sketch of proof: Because `a` is normal, it suffices to prove that `star a * a` is invertible
in `elementalStarAlgebra ℂ a`. For this it suffices to prove that it is sufficiently close to a
unit, namely `algebraMap ℂ _ ‖star a * a‖`, and in this case the required distance is
`‖star a * a‖`. So one must... |
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 | 222 | 225 | theorem eq_zero_of_fst_eq_zero {z x} {y : M} (h : z * algebraMap R S y = algebraMap R S x)
(hx : x = 0) : z = 0 := by |
rw [hx, (algebraMap R S).map_zero] at h
exact (IsUnit.mul_left_eq_zero (IsLocalization.map_units S y)).1 h
|
import Mathlib.Algebra.Group.Hom.End
import Mathlib.Algebra.Ring.Invertible
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.GroupTheory.GroupAction.Units
#align_import algebra.module.basic from "leanprover-community/mathlib"@"30413fc89f202a090a54d78e540963ed3de0056e"
assert_n... | Mathlib/Algebra/Module/Defs.lean | 241 | 245 | theorem Module.ext' {R : Type*} [Semiring R] {M : Type*} [AddCommMonoid M] (P Q : Module R M)
(w : ∀ (r : R) (m : M), (haveI := P; r • m) = (haveI := Q; r • m)) :
P = Q := by |
ext
exact w _ _
|
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.Complement
open Monoid Coprod Multiplicative Subgroup Function
def HNNExtension.con (G : Type*) [Group G] (A B : Subgroup G) (φ : A ≃* B) :
Con (G ∗ Multiplicative ℤ) :=
conGen (fun x y => ∃ (a : A),
x = inr (ofAdd 1) * inl (a : G) ∧
... | Mathlib/GroupTheory/HNNExtension.lean | 77 | 79 | theorem equiv_symm_eq_conj (b : B) :
(of (φ.symm b : G) : HNNExtension G A B φ) = t⁻¹ * of (b : G) * t := by |
rw [mul_assoc, of_mul_t]; simp
|
import Mathlib.Order.Filter.Cofinite
#align_import topology.bornology.basic from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
open Set Filter
variable {ι α β : Type*}
class Bornology (α : Type*) where
cobounded' : Filter α
le_cofinite' : cobounded' ≤ cofinite
#align borno... | Mathlib/Topology/Bornology/Basic.lean | 161 | 163 | theorem isBounded_empty : IsBounded (∅ : Set α) := by |
rw [isBounded_def, compl_empty]
exact univ_mem
|
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 | 43 | 49 | theorem interval_average_eq (f : ℝ → E) (a b : ℝ) :
(⨍ x in a..b, f x) = (b - a)⁻¹ • ∫ x in a..b, f x := by |
rcases le_or_lt a b with h | h
· rw [setAverage_eq, uIoc_of_le h, Real.volume_Ioc, intervalIntegral.integral_of_le h,
ENNReal.toReal_ofReal (sub_nonneg.2 h)]
· rw [setAverage_eq, uIoc_of_lt h, Real.volume_Ioc, intervalIntegral.integral_of_ge h.le,
ENNReal.toReal_ofReal (sub_nonneg.2 h.le), smul_neg, ... |
import Mathlib.Algebra.Group.Opposite
import Mathlib.Algebra.Group.Units.Hom
#align_import algebra.group.prod from "leanprover-community/mathlib"@"cd391184c85986113f8c00844cfe6dda1d34be3d"
assert_not_exists MonoidWithZero
-- TODO:
-- assert_not_exists AddMonoidWithOne
assert_not_exists DenselyOrdered
variable {A ... | Mathlib/Algebra/Group/Prod.lean | 86 | 88 | theorem mk_one_mul_mk_one [Mul M] [Monoid N] (a₁ a₂ : M) :
(a₁, (1 : N)) * (a₂, 1) = (a₁ * a₂, 1) := by |
rw [mk_mul_mk, mul_one]
|
import Mathlib.Algebra.GroupWithZero.Hom
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Algebra.Order.GroupWithZero.Canonical
import Mathlib.Order.Hom.Basic
#align_import algebra.order.hom.monoid from "leanprover-community/mathlib"@"3342d1b2178381196f818146ff79bc0e7ccd9e2d"
open Function
variable {F... | Mathlib/Algebra/Order/Hom/Monoid.lean | 182 | 184 | theorem map_nonpos (ha : a ≤ 0) : f a ≤ 0 := by |
rw [← map_zero f]
exact OrderHomClass.mono _ ha
|
import Mathlib.CategoryTheory.Sites.Spaces
import Mathlib.Topology.Sheaves.Sheaf
import Mathlib.CategoryTheory.Sites.DenseSubsite
#align_import topology.sheaves.sheaf_condition.sites from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
noncomputable section
set_option linter.uppercaseLe... | Mathlib/Topology/Sheaves/SheafCondition/Sites.lean | 90 | 94 | theorem covering_presieve_eq_self {Y : Opens X} (R : Presieve Y) :
presieveOfCoveringAux (coveringOfPresieve Y R) Y = R := by |
funext Z
ext f
exact ⟨fun ⟨⟨_, f', h⟩, rfl⟩ => by rwa [Subsingleton.elim f f'], fun h => ⟨⟨Z, f, h⟩, rfl⟩⟩
|
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Basic
import Mathlib.RingTheory.Localization.FractionRing
#align_import ring_theory.localization.localization_localization from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
open Function
namespace ... | Mathlib/RingTheory/Localization/LocalizationLocalization.lean | 53 | 61 | theorem mem_localizationLocalizationSubmodule {x : R} :
x ∈ localizationLocalizationSubmodule M N ↔
∃ (y : N) (z : M), algebraMap R S x = y * algebraMap R S z := by |
rw [localizationLocalizationSubmodule, Submonoid.mem_comap, Submonoid.mem_sup]
constructor
· rintro ⟨y, hy, _, ⟨z, hz, rfl⟩, e⟩
exact ⟨⟨y, hy⟩, ⟨z, hz⟩, e.symm⟩
· rintro ⟨y, z, e⟩
exact ⟨y, y.prop, _, ⟨z, z.prop, rfl⟩, e.symm⟩
|
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Data.Complex.Exponential
import Mathlib.Data.Complex.Module
import Mathlib.RingTheory.Polynomial.Chebyshev
#align_import analysis.special_functions.trigonometric.chebyshev from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
set_... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Chebyshev.lean | 45 | 47 | theorem algebraMap_eval_U (x : R) (n : ℤ) :
algebraMap R A ((U R n).eval x) = (U A n).eval (algebraMap R A x) := by |
rw [← aeval_algebraMap_apply_eq_algebraMap_eval, aeval_U]
|
import Mathlib.Order.Filter.Basic
import Mathlib.Topology.Bases
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.Basic
import Mathlib.Topology.LocallyFinite
open Set Filter Topology TopologicalSpace Classical Function
universe u v
variable {X : Type u} {Y : Type v} {ι : Type*}
variable [Topolog... | Mathlib/Topology/Compactness/Compact.lean | 48 | 52 | theorem IsCompact.compl_mem_sets (hs : IsCompact s) {f : Filter X} (hf : ∀ x ∈ s, sᶜ ∈ 𝓝 x ⊓ f) :
sᶜ ∈ f := by |
contrapose! hf
simp only [not_mem_iff_inf_principal_compl, compl_compl, inf_assoc] at hf ⊢
exact @hs _ hf inf_le_right
|
import Mathlib.Topology.MetricSpace.Antilipschitz
#align_import topology.metric_space.isometry from "leanprover-community/mathlib"@"b1859b6d4636fdbb78c5d5cefd24530653cfd3eb"
noncomputable section
universe u v w
variable {ι : Type*} {α : Type u} {β : Type v} {γ : Type w}
open Function Set
open scoped Topology ... | Mathlib/Topology/MetricSpace/Isometry.lean | 144 | 147 | theorem preimage_emetric_ball (h : Isometry f) (x : α) (r : ℝ≥0∞) :
f ⁻¹' EMetric.ball (f x) r = EMetric.ball x r := by |
ext y
simp [h.edist_eq]
|
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Data.Finsupp.Fintype
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import data.mv_polynomial.cardinal from "leanprover-community/mathlib"@"3cd7b577c6acf365f59a6376c5867533124eff6b"
universe u v
open Cardinal
open Cardinal
namespace MvPolynomial
secti... | Mathlib/Algebra/MvPolynomial/Cardinal.lean | 45 | 51 | theorem cardinal_lift_mk_le_max {σ : Type u} {R : Type v} [CommSemiring R] : #(MvPolynomial σ R) ≤
max (max (Cardinal.lift.{u} #R) <| Cardinal.lift.{v} #σ) ℵ₀ := by |
cases subsingleton_or_nontrivial R
· exact (mk_eq_one _).trans_le (le_max_of_le_right one_le_aleph0)
cases isEmpty_or_nonempty σ
· exact cardinal_mk_eq_lift.trans_le (le_max_of_le_left <| le_max_left _ _)
· exact cardinal_mk_eq_max_lift.le
|
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Topology.Algebra.Module.Basic
import Mathlib.RingTheory.Adjoin.Basic
#align_import topology.algebra.algebra from "leanprover-community/mathlib"@"43afc5ad87891456c57b5a183e3e617d67c2b1db"
open scoped Classical
open Set TopologicalSpace Algebra
open sc... | Mathlib/Topology/Algebra/Algebra.lean | 110 | 111 | theorem Subalgebra.isClosed_topologicalClosure (s : Subalgebra R A) :
IsClosed (s.topologicalClosure : Set A) := by | convert @isClosed_closure A s _
|
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Order.Filter.IndicatorFunction
open MeasureTheory
section DominatedConvergenceTheorem
open Set Filter TopologicalSpace ENNReal
open scoped Topology
namespace MeasureTheory
variable {α E G: Type*}
[NormedAddCommGroup E] [NormedSpace ℝ E] [C... | Mathlib/MeasureTheory/Integral/DominatedConvergence.lean | 53 | 62 | theorem tendsto_integral_of_dominated_convergence {F : ℕ → α → G} {f : α → G} (bound : α → ℝ)
(F_measurable : ∀ n, AEStronglyMeasurable (F n) μ) (bound_integrable : Integrable bound μ)
(h_bound : ∀ n, ∀ᵐ a ∂μ, ‖F n a‖ ≤ bound a)
(h_lim : ∀ᵐ a ∂μ, Tendsto (fun n => F n a) atTop (𝓝 (f a))) :
Tendsto (fun... |
by_cases hG : CompleteSpace G
· simp only [integral, hG, L1.integral]
exact tendsto_setToFun_of_dominated_convergence (dominatedFinMeasAdditive_weightedSMul μ)
bound F_measurable bound_integrable h_bound h_lim
· simp [integral, hG]
|
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Topology.Algebra.Order.LiminfLimsup
#align_import measure_theory.measure.measure_space from "leanprover-community/mathlib"@"343e80208d29d2d15f8050b929aa50fe4ce71b55"
noncomputable section
open Set... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 152 | 157 | theorem measure_biUnion₀ {s : Set β} {f : β → Set α} (hs : s.Countable)
(hd : s.Pairwise (AEDisjoint μ on f)) (h : ∀ b ∈ s, NullMeasurableSet (f b) μ) :
μ (⋃ b ∈ s, f b) = ∑' p : s, μ (f p) := by |
haveI := hs.toEncodable
rw [biUnion_eq_iUnion]
exact measure_iUnion₀ (hd.on_injective Subtype.coe_injective fun x => x.2) fun x => h x x.2
|
import Mathlib.MeasureTheory.Measure.Doubling
import Mathlib.MeasureTheory.Covering.Vitali
import Mathlib.MeasureTheory.Covering.Differentiation
#align_import measure_theory.covering.density_theorem from "leanprover-community/mathlib"@"5f6e827d81dfbeb6151d7016586ceeb0099b9655"
noncomputable section
open Set Filt... | Mathlib/MeasureTheory/Covering/DensityTheorem.lean | 71 | 109 | theorem closedBall_mem_vitaliFamily_of_dist_le_mul {K : ℝ} {x y : α} {r : ℝ} (h : dist x y ≤ K * r)
(rpos : 0 < r) : closedBall y r ∈ (vitaliFamily μ K).setsAt x := by |
let R := scalingScaleOf μ (max (4 * K + 3) 3)
simp only [vitaliFamily, VitaliFamily.enlarge, Vitali.vitaliFamily, mem_union, mem_setOf_eq,
isClosed_ball, true_and_iff, (nonempty_ball.2 rpos).mono ball_subset_interior_closedBall,
measurableSet_closedBall]
/- The measure is doubling on scales smaller than ... |
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.independence from "leanprover-community/mathlib"@"1e82f5ec4645f6a92bb9e02fce51e44e3bc3e1fe"
open scoped LinearAlgebra.Projectivization
variable {ι K V : Type*} [DivisionRing K] [AddCommGroup V] [Module K V] {f : ι → ... | Mathlib/LinearAlgebra/Projectivization/Independence.lean | 84 | 94 | theorem dependent_iff : Dependent f ↔ ¬LinearIndependent K (Projectivization.rep ∘ f) := by |
refine ⟨?_, fun h => ?_⟩
· rintro ⟨ff, hff, hh1⟩
contrapose! hh1
choose a ha using fun i : ι => exists_smul_eq_mk_rep K (ff i) (hff i)
convert hh1.units_smul a⁻¹
ext i
simp only [← ha, inv_smul_smul, Pi.smul_apply', Pi.inv_apply, Function.comp_apply]
· convert Dependent.mk _ _ h
· simp on... |
import Mathlib.CategoryTheory.Functor.ReflectsIso
import Mathlib.CategoryTheory.MorphismProperty.Basic
universe w v v' u u'
namespace CategoryTheory
namespace MorphismProperty
variable {C : Type u} [Category.{v} C] {D : Type u'} [Category.{v'} D]
def IsInvertedBy (P : MorphismProperty C) (F : C ⥤ D) : Prop :=
... | Mathlib/CategoryTheory/MorphismProperty/IsInvertedBy.lean | 128 | 131 | theorem IsInvertedBy.iff_of_iso (W : MorphismProperty C) {F₁ F₂ : C ⥤ D} (e : F₁ ≅ F₂) :
W.IsInvertedBy F₁ ↔ W.IsInvertedBy F₂ := by |
dsimp [IsInvertedBy]
simp only [NatIso.isIso_map_iff e]
|
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.Ring
#align_import data.nat.count from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
open Finset
namespace Nat
variable (p : ℕ → Prop)
section Count
variable [DecidablePred p]
def count (n : ℕ) : ℕ :=
(List.range n).... | Mathlib/Data/Nat/Count.lean | 106 | 107 | theorem count_succ_eq_succ_count_iff {n : ℕ} : count p (n + 1) = count p n + 1 ↔ p n := by |
by_cases h : p n <;> simp [h, count_succ]
|
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 | 93 | 102 | theorem valuationOfNeZeroToFun_eq (x : Kˣ) :
(v.valuationOfNeZeroToFun x : ℤₘ₀) = v.valuation (x : K) := by |
rw [show v.valuation (x : K) = _ * _ by rfl]
rw [Units.val_inv_eq_inv_val]
change _ = ite _ _ _ * (ite _ _ _)⁻¹
simp_rw [IsLocalization.toLocalizationMap_sec, SubmonoidClass.coe_subtype,
if_neg <| IsLocalization.sec_fst_ne_zero le_rfl x.ne_zero,
if_neg (nonZeroDivisors.coe_ne_zero _),
valuationOfNe... |
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