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.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
open Set Filter Topology TopologicalSpace
universe u v
variable {X : Type u} {Y : Type v} {ι : Type*}
variable [TopologicalSpace X] [TopologicalSpace Y] {s t : Set X}
section Lindelof
def I... | Mathlib/Topology/Compactness/Lindelof.lean | 60 | 64 | theorem IsLindelof.compl_mem_sets_of_nhdsWithin (hs : IsLindelof s) {f : Filter X}
[CountableInterFilter f] (hf : ∀ x ∈ s, ∃ t ∈ 𝓝[s] x, tᶜ ∈ f) : sᶜ ∈ f := by |
refine hs.compl_mem_sets fun x hx ↦ ?_
rw [← disjoint_principal_right, disjoint_right_comm, (basis_sets _).disjoint_iff_left]
exact hf x hx
|
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_theory.perm.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Equiv Finset
namespace Equiv.Perm
variable {α : Type*}
section Disjoint
... | Mathlib/GroupTheory/Perm/Support.lean | 110 | 111 | theorem disjoint_inv_right_iff : Disjoint f g⁻¹ ↔ Disjoint f g := by |
rw [disjoint_comm, disjoint_inv_left_iff, disjoint_comm]
|
import Mathlib.Data.Int.Interval
import Mathlib.RingTheory.Binomial
import Mathlib.RingTheory.HahnSeries.PowerSeries
import Mathlib.RingTheory.HahnSeries.Summable
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.Localization.FractionRing
#align_import ring_theory.laurent_series from "leanprov... | Mathlib/RingTheory/LaurentSeries.lean | 87 | 89 | theorem coeff_coe_powerSeries (x : PowerSeries R) (n : ℕ) :
HahnSeries.coeff (x : LaurentSeries R) n = PowerSeries.coeff R n x := by |
rw [ofPowerSeries_apply_coeff]
|
import Mathlib.Topology.Baire.Lemmas
import Mathlib.Topology.Algebra.Group.Basic
open scoped Topology Pointwise
open MulAction Set Function
variable {G X : Type*} [TopologicalSpace G] [TopologicalSpace X]
[Group G] [TopologicalGroup G] [MulAction G X]
[SigmaCompactSpace G] [BaireSpace X] [T2Space X]
[Contin... | Mathlib/Topology/Algebra/Group/OpenMapping.lean | 96 | 107 | theorem isOpenMap_smul_of_sigmaCompact (x : X) : IsOpenMap (fun (g : G) ↦ g • x) := by |
/- We have already proved the theorem around the basepoint of the orbit, in
`smul_singleton_mem_nhds_of_sigmaCompact`. The general statement follows around an arbitrary
point by changing basepoints. -/
simp_rw [isOpenMap_iff_nhds_le, Filter.le_map_iff]
intro g U hU
have : (· • x) = (· • (g • x)) ∘ (· * g⁻¹... |
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.OrdConnected
#align_import data.set.intervals.proj_Icc from "leanprover-community/mathlib"@"4e24c4bfcff371c71f7ba22050308aa17815626c"
variable {α β : Type*} [LinearOrder α]
open Function
namespace Set
def projIci (a x : α) : Ici a := ⟨max a x,... | Mathlib/Order/Interval/Set/ProjIcc.lean | 77 | 78 | theorem projIcc_of_right_le (hx : b ≤ x) : projIcc a b h x = ⟨b, right_mem_Icc.2 h⟩ := by |
simp [projIcc, hx, h]
|
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Normed.Field.InfiniteSum
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.Data.Finset.NoncommProd
import Mathlib.Topology.Algebra.Algebra
#align_import analysis.normed_space.exponential from "leanprover-community/ma... | Mathlib/Analysis/NormedSpace/Exponential.lean | 160 | 162 | theorem star_exp [T2Space 𝔸] [StarRing 𝔸] [ContinuousStar 𝔸] (x : 𝔸) :
star (exp 𝕂 x) = exp 𝕂 (star x) := by |
simp_rw [exp_eq_tsum, ← star_pow, ← star_inv_natCast_smul, ← tsum_star]
|
import Mathlib.Order.Filter.Partial
import Mathlib.Topology.Basic
#align_import topology.partial from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Filter
open Topology
variable {X Y : Type*} [TopologicalSpace X]
theorem rtendsto_nhds {r : Rel Y X} {l : Filter Y} {x : X} :
... | Mathlib/Topology/Partial.lean | 30 | 34 | theorem rtendsto'_nhds {r : Rel Y X} {l : Filter Y} {x : X} :
RTendsto' r l (𝓝 x) ↔ ∀ s, IsOpen s → x ∈ s → r.preimage s ∈ l := by |
rw [rtendsto'_def]
apply all_mem_nhds_filter
apply Rel.preimage_mono
|
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.CategoryTheory.Monoidal.OfChosenFiniteProducts.Basic
#align_import category_theory.monoidal.of_chosen_finite_products.symmetric from "leanprover-community/mathlib"@"95a87616d63b3cb49d3fe678d416fbe9c4217bf4"
universe v u
namespace CategoryTheory
... | Mathlib/CategoryTheory/Monoidal/OfChosenFiniteProducts/Symmetric.lean | 34 | 39 | theorem braiding_naturality {X X' Y Y' : C} (f : X ⟶ Y) (g : X' ⟶ Y') :
tensorHom ℬ f g ≫ (Limits.BinaryFan.braiding (ℬ Y Y').isLimit (ℬ Y' Y).isLimit).hom =
(Limits.BinaryFan.braiding (ℬ X X').isLimit (ℬ X' X).isLimit).hom ≫ tensorHom ℬ g f := by |
dsimp [tensorHom, Limits.BinaryFan.braiding]
apply (ℬ _ _).isLimit.hom_ext
rintro ⟨⟨⟩⟩ <;> · dsimp [Limits.IsLimit.conePointUniqueUpToIso]; simp
|
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.RCLike.Lemmas
import Mathlib.Topology.TietzeExtension
import Mathlib.Analysis.NormedSpace.HomeomorphBall
import Mathlib.Analysis.NormedSpace.RCLike
universe u u₁ v w
-- this is not an instance because Lean cannot determine `𝕜`.
theorem TietzeExtension.o... | Mathlib/Analysis/Complex/Tietze.lean | 105 | 118 | theorem exists_norm_eq_restrict_eq (f : s →ᵇ E) :
∃ g : X →ᵇ E, ‖g‖ = ‖f‖ ∧ g.restrict s = f := by |
by_cases hf : ‖f‖ = 0; · exact ⟨0, by aesop⟩
have := Metric.instTietzeExtensionClosedBall.{u, v} 𝕜 (0 : E) (by aesop : 0 < ‖f‖)
have hf' x : f x ∈ Metric.closedBall 0 ‖f‖ := by simpa using f.norm_coe_le_norm x
obtain ⟨g, hg_mem, hg⟩ := (f : C(s, E)).exists_forall_mem_restrict_eq hs hf'
simp only [Metric.mem... |
import Mathlib.Analysis.Complex.UpperHalfPlane.Topology
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Geometry.Euclidean.Inversion.Basic
#align_import analysis.complex.upper_half_plane.metric from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c"
noncomputable section
ope... | Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | 60 | 63 | theorem tanh_half_dist (z w : ℍ) :
tanh (dist z w / 2) = dist (z : ℂ) w / dist (z : ℂ) (conj ↑w) := by |
rw [tanh_eq_sinh_div_cosh, sinh_half_dist, cosh_half_dist, div_div_div_comm, div_self, div_one]
positivity
|
import Mathlib.Algebra.Group.Defs
#align_import algebra.invertible from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
universe u
variable {α : Type u}
class Invertible [Mul α] [One α] (a : α) : Type u where
invOf... | Mathlib/Algebra/Group/Invertible/Defs.lean | 170 | 171 | theorem Invertible.congr [Monoid α] (a b : α) [Invertible a] [Invertible b] (h : a = b) :
⅟a = ⅟b := by | subst h; congr; apply Subsingleton.allEq
|
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov
import Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp
import Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality... | Mathlib/MeasureTheory/Function/LpSpace.lean | 177 | 178 | theorem mem_Lp_iff_memℒp {f : α →ₘ[μ] E} : f ∈ Lp E p μ ↔ Memℒp f p μ := by |
simp [mem_Lp_iff_snorm_lt_top, Memℒp, f.stronglyMeasurable.aestronglyMeasurable]
|
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.NthRewrite
#align_import data.nat.gcd.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
namespace Nat
theorem gcd_greatest {a b d : ℕ} (hda : d ∣ a) (hdb : d ∣ b) (hd ... | Mathlib/Data/Nat/GCD/Basic.lean | 96 | 99 | theorem gcd_sub_self_left {m n : ℕ} (h : m ≤ n) : gcd (n - m) m = gcd n m := by |
calc
gcd (n - m) m = gcd (n - m + m) m := by rw [← gcd_add_self_left (n - m) m]
_ = gcd n m := by rw [Nat.sub_add_cancel h]
|
import Mathlib.Probability.Kernel.MeasurableIntegral
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import probability.kernel.with_density from "leanprover-community/mathlib"@"c0d694db494dd4f9aa57f2714b6e4c82b4ebc113"
open MeasureTheory ProbabilityTheory
open scoped MeasureTheory ENNReal NNReal
namesp... | Mathlib/Probability/Kernel/WithDensity.lean | 135 | 144 | theorem withDensity_kernel_sum [Countable ι] (κ : ι → kernel α β) (hκ : ∀ i, IsSFiniteKernel (κ i))
(f : α → β → ℝ≥0∞) :
@withDensity _ _ _ _ (kernel.sum κ) (isSFiniteKernel_sum hκ) f =
kernel.sum fun i => withDensity (κ i) f := by |
by_cases hf : Measurable (Function.uncurry f)
· ext1 a
simp_rw [sum_apply, kernel.withDensity_apply _ hf, sum_apply,
withDensity_sum (fun n => κ n a) (f a)]
· simp_rw [withDensity_of_not_measurable _ hf]
exact sum_zero.symm
|
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.degree.lemmas from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
noncomputable section
open Polynomial
open Finsupp Finset
namespace Polynomial
universe u v w
variable {R : Type u} {S : Type v} {ι : Type w} {a b ... | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 121 | 127 | theorem natDegree_mul_C_eq_of_mul_eq_one {ai : R} (au : a * ai = 1) :
(p * C a).natDegree = p.natDegree :=
le_antisymm (natDegree_mul_C_le p a)
(calc
p.natDegree = (p * 1).natDegree := by | nth_rw 1 [← mul_one p]
_ = (p * C a * C ai).natDegree := by rw [← C_1, ← au, RingHom.map_mul, ← mul_assoc]
_ ≤ (p * C a).natDegree := natDegree_mul_C_le (p * C a) ai)
|
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Algebra.Group.Aut
import Mathlib.Data.ZMod.Defs
import Mathlib.Tactic.Ring
#align_import algebra.quandle from "leanprover-community/mathlib"@"28aa996fc6fb4317f0083c4e6daf79878d81be33"
open MulOpposite
universe u v
class Shelf (α : Type u) where
act : ... | Mathlib/Algebra/Quandle.lean | 251 | 253 | theorem ad_conj {R : Type*} [Rack R] (x y : R) : act' (x ◃ y) = act' x * act' y * (act' x)⁻¹ := by |
rw [eq_mul_inv_iff_mul_eq]; ext z
apply self_distrib.symm
|
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Topology.Instances.NNReal
#align_import analysis.normed.group.infinite_sum from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd"
open Topology NNReal
open Finset Filter Metric
variabl... | Mathlib/Analysis/Normed/Group/InfiniteSum.lean | 113 | 116 | theorem Summable.of_norm_bounded [CompleteSpace E] {f : ι → E} (g : ι → ℝ) (hg : Summable g)
(h : ∀ i, ‖f i‖ ≤ g i) : Summable f := by |
rw [summable_iff_cauchySeq_finset]
exact cauchySeq_finset_of_norm_bounded g hg h
|
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Measure.Haar.Unique
open MeasureTheory Measure Set
open scoped ENNReal
variable {𝕜 E F : Type*}
[NontriviallyNormedField 𝕜] [CompleteSpace 𝕜]
[NormedAddCommGroup E] [MeasurableSp... | Mathlib/MeasureTheory/Measure/Haar/Disintegration.lean | 106 | 109 | theorem LinearMap.exists_map_addHaar_eq_smul_addHaar (h : Function.Surjective L) :
∃ (c : ℝ≥0∞), 0 < c ∧ μ.map L = c • ν := by |
rcases L.exists_map_addHaar_eq_smul_addHaar' μ ν h with ⟨c, c_pos, -, hc⟩
exact ⟨_, by simp [c_pos, NeZero.ne addHaar], hc⟩
|
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable ... | Mathlib/Algebra/Polynomial/RingDivision.lean | 444 | 445 | theorem pow_rootMultiplicity_not_dvd {p : R[X]} (p0 : p ≠ 0) (a : R) :
¬(X - C a) ^ (rootMultiplicity a p + 1) ∣ p := by | rw [← rootMultiplicity_le_iff p0]
|
import Mathlib.RingTheory.FractionalIdeal.Basic
import Mathlib.RingTheory.Ideal.Norm
namespace FractionalIdeal
open scoped Pointwise nonZeroDivisors
variable {R : Type*} [CommRing R] [IsDedekindDomain R] [Module.Free ℤ R] [Module.Finite ℤ R]
variable {K : Type*} [CommRing K] [Algebra R K] [IsFractionRing R K]
th... | Mathlib/RingTheory/FractionalIdeal/Norm.lean | 84 | 84 | theorem absNorm_nonneg (I : FractionalIdeal R⁰ K) : 0 ≤ absNorm I := by | dsimp [absNorm]; positivity
|
import Mathlib.Topology.Bases
import Mathlib.Topology.DenseEmbedding
#align_import topology.stone_cech from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
noncomputable section
open Filter Set
open Topology
universe u v
section Ultrafilter
def ultrafilterBasis (α : Type u) : Set ... | Mathlib/Topology/StoneCech.lean | 122 | 126 | theorem ultrafilter_pure_injective : Function.Injective (pure : α → Ultrafilter α) := by |
intro x y h
have : {x} ∈ (pure x : Ultrafilter α) := singleton_mem_pure
rw [h] at this
exact (mem_singleton_iff.mp (mem_pure.mp this)).symm
|
import Mathlib.Combinatorics.Hall.Finite
import Mathlib.CategoryTheory.CofilteredSystem
import Mathlib.Data.Rel
#align_import combinatorics.hall.basic from "leanprover-community/mathlib"@"8195826f5c428fc283510bc67303dd4472d78498"
open Finset CategoryTheory
universe u v
def hallMatchingsOn {ι : Type u} {α : Typ... | Mathlib/Combinatorics/Hall/Basic.lean | 77 | 86 | theorem hallMatchingsOn.nonempty {ι : Type u} {α : Type v} [DecidableEq α] (t : ι → Finset α)
(h : ∀ s : Finset ι, s.card ≤ (s.biUnion t).card) (ι' : Finset ι) :
Nonempty (hallMatchingsOn t ι') := by |
classical
refine ⟨Classical.indefiniteDescription _ ?_⟩
apply (all_card_le_biUnion_card_iff_existsInjective' fun i : ι' => t i).mp
intro s'
convert h (s'.image (↑)) using 1
· simp only [card_image_of_injective s' Subtype.coe_injective]
· rw [image_biUnion]
|
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.Data.Rat.Lemmas
import Mathlib.Data.Int.Sqrt
#align_import data.rat.sqrt from "leanprover-community/mathlib"@"46a64b5b4268c594af770c44d9e502afc6a515cb"
namespace Rat
-- @[pp_nodot] porting note: unknown attribute
def sqrt... | Mathlib/Data/Rat/Sqrt.lean | 30 | 31 | theorem sqrt_eq (q : ℚ) : Rat.sqrt (q * q) = |q| := by |
rw [sqrt, mul_self_num, mul_self_den, Int.sqrt_eq, Nat.sqrt_eq, abs_def, divInt_ofNat]
|
import Mathlib.Data.Set.Function
import Mathlib.Analysis.BoundedVariation
#align_import analysis.constant_speed from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open scoped NNReal ENNReal
open Set MeasureTheory Classical
variable {α : Type*} [LinearOrder α] {E : Type*} [PseudoEMetr... | Mathlib/Analysis/ConstantSpeed.lean | 64 | 68 | theorem hasConstantSpeedOnWith_of_subsingleton (f : ℝ → E) {s : Set ℝ} (hs : s.Subsingleton)
(l : ℝ≥0) : HasConstantSpeedOnWith f s l := by |
rintro x hx y hy; cases hs hx hy
rw [eVariationOn.subsingleton f (fun y hy z hz => hs hy.1 hz.1 : (s ∩ Icc x x).Subsingleton)]
simp only [sub_self, mul_zero, ENNReal.ofReal_zero]
|
import Mathlib.Data.List.Forall2
#align_import data.list.zip from "leanprover-community/mathlib"@"134625f523e737f650a6ea7f0c82a6177e45e622"
-- Make sure we don't import algebra
assert_not_exists Monoid
universe u
open Nat
namespace List
variable {α : Type u} {β γ δ ε : Type*}
#align list.zip_with_cons_cons Li... | Mathlib/Data/List/Zip.lean | 133 | 134 | theorem unzip_zip_right {l₁ : List α} {l₂ : List β} (h : length l₂ ≤ length l₁) :
(unzip (zip l₁ l₂)).2 = l₂ := by | rw [← zip_swap, unzip_swap]; exact unzip_zip_left h
|
import Mathlib.Algebra.GCDMonoid.Multiset
import Mathlib.Combinatorics.Enumerative.Partition
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Cycle.Factors
import Mathlib.GroupTheory.Perm.Closure
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Tactic.NormNum.GCD
#align_import group_theory.perm.cycl... | Mathlib/GroupTheory/Perm/Cycle/Type.lean | 79 | 80 | theorem cycleType_eq_zero {σ : Perm α} : σ.cycleType = 0 ↔ σ = 1 := by |
simp [cycleType_def, cycleFactorsFinset_eq_empty_iff]
|
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 | 146 | 149 | theorem copy_cons {u v w u' w'} (h : G.Adj u v) (p : G.Walk v w) (hu : u = u') (hw : w = w') :
(Walk.cons h p).copy hu hw = Walk.cons (hu ▸ h) (p.copy rfl hw) := by |
subst_vars
rfl
|
import Mathlib.Algebra.Order.Group.PiLex
import Mathlib.Data.DFinsupp.Order
import Mathlib.Data.DFinsupp.NeLocus
import Mathlib.Order.WellFoundedSet
#align_import data.dfinsupp.lex from "leanprover-community/mathlib"@"dde670c9a3f503647fd5bfdf1037bad526d3397a"
variable {ι : Type*} {α : ι → Type*}
namespace DFinsu... | Mathlib/Data/DFinsupp/Lex.lean | 51 | 58 | theorem lex_lt_of_lt_of_preorder [∀ i, Preorder (α i)] (r) [IsStrictOrder ι r] {x y : Π₀ i, α i}
(hlt : x < y) : ∃ i, (∀ j, r j i → x j ≤ y j ∧ y j ≤ x j) ∧ x i < y i := by |
obtain ⟨hle, j, hlt⟩ := Pi.lt_def.1 hlt
classical
have : (x.neLocus y : Set ι).WellFoundedOn r := (x.neLocus y).finite_toSet.wellFoundedOn
obtain ⟨i, hi, hl⟩ := this.has_min { i | x i < y i } ⟨⟨j, mem_neLocus.2 hlt.ne⟩, hlt⟩
refine ⟨i, fun k hk ↦ ⟨hle k, ?_⟩, hi⟩
exact of_not_not fun h ↦ hl ⟨k, mem_neLocus... |
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Localization.Integral
#align_import ring_theory.integrally_closed from "leanprover-community/mathlib"@"d35b4ff446f1421bd551fafa4b8efd98ac3ac408"
open scoped nonZeroDivisors Polynomial
open Polynomial
abbrev IsIntegrallyClosedIn (R A : Type*) [... | Mathlib/RingTheory/IntegrallyClosed.lean | 110 | 120 | theorem isIntegrallyClosedIn_iff {R A : Type*} [CommRing R] [CommRing A] [Algebra R A] :
IsIntegrallyClosedIn R A ↔
Function.Injective (algebraMap R A) ∧
∀ {x : A}, IsIntegral R x → ∃ y, algebraMap R A y = x := by |
constructor
· rintro ⟨_, cl⟩
aesop
· rintro ⟨inj, cl⟩
refine ⟨inj, by aesop, ?_⟩
rintro ⟨y, rfl⟩
apply isIntegral_algebraMap
|
import Mathlib.Order.Monotone.Union
import Mathlib.Algebra.Order.Group.Instances
#align_import order.monotone.odd from "leanprover-community/mathlib"@"9116dd6709f303dcf781632e15fdef382b0fc579"
open Set
variable {G H : Type*} [LinearOrderedAddCommGroup G] [OrderedAddCommGroup H]
| Mathlib/Order/Monotone/Odd.lean | 26 | 30 | theorem strictMono_of_odd_strictMonoOn_nonneg {f : G → H} (h₁ : ∀ x, f (-x) = -f x)
(h₂ : StrictMonoOn f (Ici 0)) : StrictMono f := by |
refine StrictMonoOn.Iic_union_Ici (fun x hx y hy hxy => neg_lt_neg_iff.1 ?_) h₂
rw [← h₁, ← h₁]
exact h₂ (neg_nonneg.2 hy) (neg_nonneg.2 hx) (neg_lt_neg hxy)
|
import Mathlib.Data.List.Infix
#align_import data.list.rdrop from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
-- Make sure we don't import algebra
assert_not_exists Monoid
variable {α : Type*} (p : α → Bool) (l : List α) (n : ℕ)
namespace List
def rdrop : List α :=
l.take (l.leng... | Mathlib/Data/List/DropRight.lean | 81 | 87 | theorem rtake_eq_reverse_take_reverse : l.rtake n = reverse (l.reverse.take n) := by |
rw [rtake]
induction' l using List.reverseRecOn with xs x IH generalizing n
· simp
· cases n
· exact drop_length _
· simp [drop_append_eq_append_drop, IH]
|
import Mathlib.Data.ENNReal.Basic
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.MetricSpace.Thickening
#align_import topology.metric_space.thickened_indicator from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classical
open NNReal ENNReal Topol... | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | 130 | 153 | theorem thickenedIndicatorAux_tendsto_indicator_closure {δseq : ℕ → ℝ}
(δseq_lim : Tendsto δseq atTop (𝓝 0)) (E : Set α) :
Tendsto (fun n => thickenedIndicatorAux (δseq n) E) atTop
(𝓝 (indicator (closure E) fun _ => (1 : ℝ≥0∞))) := by |
rw [tendsto_pi_nhds]
intro x
by_cases x_mem_closure : x ∈ closure E
· simp_rw [thickenedIndicatorAux_one_of_mem_closure _ E x_mem_closure]
rw [show (indicator (closure E) fun _ => (1 : ℝ≥0∞)) x = 1 by
simp only [x_mem_closure, indicator_of_mem]]
exact tendsto_const_nhds
· rw [show (closure E)... |
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
namespace Set
variable {α β : Type*} {s t : Set α}
noncomputable def encard (s : Set α) : ℕ∞ := PartENat.withTopEquiv (PartENat.card s)
@[simp] theorem encard_uni... | Mathlib/Data/Set/Card.lean | 286 | 288 | theorem encard_pair {x y : α} (hne : x ≠ y) : ({x, y} : Set α).encard = 2 := by |
rw [encard_insert_of_not_mem (by simpa), ← one_add_one_eq_two,
WithTop.add_right_cancel_iff WithTop.one_ne_top, encard_singleton]
|
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Data.Finset.Basic
import Mathlib.Order.Interval.Finset.Defs
open Function
namespace Finset
class HasAntidiagonal (A : Type*) [AddMonoid A] where
antidiagonal : A → Finset (A × A)
mem_antidiagonal {n} {a} : a ∈ antidiagonal n ↔ a.fst + a.snd = n
exp... | Mathlib/Data/Finset/Antidiagonal.lean | 131 | 133 | theorem antidiagonal_zero : antidiagonal (0 : A) = {(0, 0)} := by |
ext ⟨x, y⟩
simp
|
import Mathlib.Geometry.Euclidean.Inversion.Basic
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Tactic.AdaptationNote
open Metric Function AffineMap Set AffineSubspace
open scoped Topology RealInnerProductSpace
variable {E F : Type*} [NormedAddCommGrou... | Mathlib/Geometry/Euclidean/Inversion/Calculus.lean | 87 | 108 | theorem hasFDerivAt_inversion (hx : x ≠ c) :
HasFDerivAt (inversion c R)
((R / dist x c) ^ 2 • (reflection (ℝ ∙ (x - c))ᗮ : F →L[ℝ] F)) x := by |
rcases add_left_surjective c x with ⟨x, rfl⟩
have : HasFDerivAt (inversion c R) (?_ : F →L[ℝ] F) (c + x) := by
#adaptation_note /-- nightly-2024-03-16: simp was
simp (config := { unfoldPartialApp := true }) only [inversion] -/
simp only [inversion_def]
simp_rw [dist_eq_norm, div_pow, div_eq_mul_inv... |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecificLimits.Normed
#align_import analysis.normed.group.controlled_closure from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter Finset
open Topology
variable {G : Type*} [NormedAddCommGroup G] [CompleteSpace... | Mathlib/Analysis/Normed/Group/ControlledClosure.lean | 32 | 106 | theorem controlled_closure_of_complete {f : NormedAddGroupHom G H} {K : AddSubgroup H} {C ε : ℝ}
(hC : 0 < C) (hε : 0 < ε) (hyp : f.SurjectiveOnWith K C) :
f.SurjectiveOnWith K.topologicalClosure (C + ε) := by |
rintro (h : H) (h_in : h ∈ K.topologicalClosure)
-- We first get rid of the easy case where `h = 0`.
by_cases hyp_h : h = 0
· rw [hyp_h]
use 0
simp
/- The desired preimage will be constructed as the sum of a series. Convergence of
the series will be guaranteed by completeness of `G`. We first wri... |
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
#align_import geometry.euclidean.angle.oriented.rotation from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 145 | 147 | theorem rotation_pi : o.rotation π = LinearIsometryEquiv.neg ℝ := by |
ext x
simp [rotation]
|
import Mathlib.Control.Functor
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6eb4348fb1749c302d8abc303d34a"
universe u₀ u₁ u₂ v₀ v₁ v₂
open Function
class Bifunctor (F : Type u₀ → Type u₁ → Type u₂) where
bimap : ∀ {α α' β β'}, (α → α') → (β → β'... | Mathlib/Control/Bifunctor.lean | 98 | 99 | theorem snd_fst {α₀ α₁ β₀ β₁} (f : α₀ → α₁) (f' : β₀ → β₁) (x : F α₀ β₀) :
snd f' (fst f x) = bimap f f' x := by | simp [snd, bimap_bimap]
|
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2f4"
open Function OrderDual
universe u v
variable {α : Type u} {β : Type*} {w x y z : α}
class GeneralizedBooleanAlgebra (α : Type u) extends DistribLattice α, S... | Mathlib/Order/BooleanAlgebra.lean | 149 | 160 | theorem sdiff_inf_sdiff : x \ y ⊓ y \ x = ⊥ :=
Eq.symm <|
calc
⊥ = x ⊓ y ⊓ x \ y := by | rw [inf_inf_sdiff]
_ = x ⊓ (y ⊓ x ⊔ y \ x) ⊓ x \ y := by rw [sup_inf_sdiff]
_ = (x ⊓ (y ⊓ x) ⊔ x ⊓ y \ x) ⊓ x \ y := by rw [inf_sup_left]
_ = (y ⊓ (x ⊓ x) ⊔ x ⊓ y \ x) ⊓ x \ y := by ac_rfl
_ = (y ⊓ x ⊔ x ⊓ y \ x) ⊓ x \ y := by rw [inf_idem]
_ = x ⊓ y ⊓ x \ y ⊔ x ⊓ y \ x ⊓ x \ y := by rw [... |
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 | 98 | 101 | theorem lapMatrix_toLinearMap₂'_apply'_eq_zero_iff_forall_adj [LinearOrderedField R] (x : V → R) :
Matrix.toLinearMap₂' (G.lapMatrix R) x x = 0 ↔ ∀ i j : V, G.Adj i j → x i = x j := by |
simp (disch := intros; positivity)
[lapMatrix_toLinearMap₂', sum_eq_zero_iff_of_nonneg, sub_eq_zero]
|
import Mathlib.LinearAlgebra.TensorAlgebra.Basic
import Mathlib.LinearAlgebra.TensorPower
#align_import linear_algebra.tensor_algebra.to_tensor_power from "leanprover-community/mathlib"@"d97a0c9f7a7efe6d76d652c5a6b7c9c634b70e0a"
suppress_compilation
open scoped DirectSum TensorProduct
variable {R M : Type*} [Com... | Mathlib/LinearAlgebra/TensorAlgebra/ToTensorPower.lean | 68 | 72 | theorem toTensorAlgebra_galgebra_toFun (r : R) :
TensorPower.toTensorAlgebra (DirectSum.GAlgebra.toFun (R := R) (A := fun n => ⨂[R]^n M) r) =
algebraMap _ _ r := by |
rw [TensorPower.galgebra_toFun_def, TensorPower.algebraMap₀_eq_smul_one, LinearMap.map_smul,
TensorPower.toTensorAlgebra_gOne, Algebra.algebraMap_eq_smul_one]
|
import Mathlib.CategoryTheory.Category.Grpd
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Homotopy.Path
import Mathlib.Data.Set.Subsingleton
#align_import algebraic_topology.fundamental_groupoid.basic from "leanprover-community/mathlib"@"3d7987cda72abc473... | Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean | 56 | 79 | theorem reflTransSymmAux_mem_I (x : I × I) : reflTransSymmAux x ∈ I := by |
dsimp only [reflTransSymmAux]
split_ifs
· constructor
· apply mul_nonneg
· apply mul_nonneg
· unit_interval
· norm_num
· unit_interval
· rw [mul_assoc]
apply mul_le_one
· unit_interval
· apply mul_nonneg
· norm_num
· unit_interval
· lina... |
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 | 133 | 136 | theorem char_prime_of_ne_zero {p : ℕ} [hp : CharP R p] (p_ne_zero : p ≠ 0) : Nat.Prime p := by |
cases' CharP.char_is_prime_or_zero R p with h h
· exact h
· contradiction
|
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.Tactic.ComputeDegree
#align_import linear_algebra.matrix.polynomial from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
set_... | Mathlib/LinearAlgebra/Matrix/Polynomial.lean | 89 | 102 | theorem leadingCoeff_det_X_one_add_C (A : Matrix n n α) :
leadingCoeff (det ((X : α[X]) • (1 : Matrix n n α[X]) + A.map C)) = 1 := by |
cases subsingleton_or_nontrivial α
· simp [eq_iff_true_of_subsingleton]
rw [← @det_one n, ← coeff_det_X_add_C_card _ A, leadingCoeff]
simp only [Matrix.map_one, C_eq_zero, RingHom.map_one]
rcases (natDegree_det_X_add_C_le 1 A).eq_or_lt with h | h
· simp only [RingHom.map_one, Matrix.map_one, C_eq_zero] at ... |
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
#align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74"
assert_not_exists Basis -- See `RingTheory.Ideal.Basis`
assert_not_exists ... | Mathlib/RingTheory/Ideal/Operations.lean | 82 | 96 | theorem mem_annihilator_span (s : Set M) (r : R) :
r ∈ (Submodule.span R s).annihilator ↔ ∀ n : s, r • (n : M) = 0 := by |
rw [Submodule.mem_annihilator]
constructor
· intro h n
exact h _ (Submodule.subset_span n.prop)
· intro h n hn
refine Submodule.span_induction hn ?_ ?_ ?_ ?_
· intro x hx
exact h ⟨x, hx⟩
· exact smul_zero _
· intro x y hx hy
rw [smul_add, hx, hy, zero_add]
· intro a x hx
... |
import Mathlib.Topology.Algebra.InfiniteSum.Defs
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Topology.Algebra.Monoid
noncomputable section
open Filter Finset Function
open scoped Topology
variable {α β γ δ : Type*}
section HasProd
variable [CommMonoid α] [TopologicalSpace α]
variable {f g : β → α} ... | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | 132 | 133 | theorem multipliable_of_finite_mulSupport (h : (mulSupport f).Finite) : Multipliable f := by |
apply multipliable_of_ne_finset_one (s := h.toFinset); simp
|
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 | 64 | 70 | theorem ext {f g : E →ₗ.[R] F} (h : f.domain = g.domain)
(h' : ∀ ⦃x : f.domain⦄ ⦃y : g.domain⦄ (_h : (x : E) = y), f x = g y) : f = g := by |
rcases f with ⟨f_dom, f⟩
rcases g with ⟨g_dom, g⟩
obtain rfl : f_dom = g_dom := h
obtain rfl : f = g := LinearMap.ext fun x => h' rfl
rfl
|
import Mathlib.Analysis.Convex.Slope
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Tactic.LinearCombination
#align_import analysis.convex.specific_functions.basic from "leanprover-community/mathlib"@"8f9fea08977f7e450770933ee6abb20733b47c92"
open Real Set NNReal
theorem strictConvexOn_exp : St... | Mathlib/Analysis/Convex/SpecificFunctions/Basic.lean | 138 | 163 | theorem rpow_one_add_lt_one_add_mul_self {s : ℝ} (hs : -1 ≤ s) (hs' : s ≠ 0) {p : ℝ} (hp1 : 0 < p)
(hp2 : p < 1) : (1 + s) ^ p < 1 + p * s := by |
rcases eq_or_lt_of_le hs with rfl | hs
· rwa [add_right_neg, zero_rpow hp1.ne', mul_neg_one, lt_add_neg_iff_add_lt, zero_add]
have hs1 : 0 < 1 + s := neg_lt_iff_pos_add'.mp hs
have hs2 : 0 < 1 + p * s := by
rw [← neg_lt_iff_pos_add']
rcases lt_or_gt_of_ne hs' with h | h
· exact hs.trans (lt_mul_of_... |
import Mathlib.Algebra.CharP.Defs
import Mathlib.RingTheory.Multiplicity
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Polynomial
open Finset (antidiagonal mem_anti... | Mathlib/RingTheory/PowerSeries/Order.lean | 47 | 51 | theorem exists_coeff_ne_zero_iff_ne_zero : (∃ n : ℕ, coeff R n φ ≠ 0) ↔ φ ≠ 0 := by |
refine not_iff_not.mp ?_
push_neg
-- FIXME: the `FunLike.coe` doesn't seem to be picked up in the expression after #8386?
simp [PowerSeries.ext_iff, (coeff R _).map_zero]
|
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.Data.ZMod.Quotient
import Mathlib.RingTheory.DedekindDomain.AdicValuation
#align_import ring_theory.dedekind_domain.selmer_group from "leanprover-community/mathlib"@"2032a878972d5672e7c27c957e7a6e297b044973"
set_option quotPrecheck false
local notation K "... | Mathlib/RingTheory/DedekindDomain/SelmerGroup.lean | 150 | 155 | theorem valuation_of_unit_mod_eq (n : ℕ) (x : Rˣ) :
v.valuationOfNeZeroMod n (Units.map (algebraMap R K : R →* K) x : K/n) = 1 := by |
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [valuationOfNeZeroMod, MonoidHom.comp_apply, ← QuotientGroup.coe_mk',
QuotientGroup.map_mk' (G := Kˣ) (N := MonoidHom.range (powMonoidHom n)),
valuation_of_unit_eq, QuotientGroup.mk_one, map_one]
|
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.Choose.Central
import Mathlib.Data.Tree.Basic
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.GCongr
import Mathlib... | Mathlib/Combinatorics/Enumerative/Catalan.lean | 116 | 137 | theorem catalan_eq_centralBinom_div (n : ℕ) : catalan n = n.centralBinom / (n + 1) := by |
suffices (catalan n : ℚ) = Nat.centralBinom n / (n + 1) by
have h := Nat.succ_dvd_centralBinom n
exact mod_cast this
induction' n using Nat.case_strong_induction_on with d hd
· simp
· simp_rw [catalan_succ, Nat.cast_sum, Nat.cast_mul]
trans (∑ i : Fin d.succ, Nat.centralBinom i / (i + 1) *
... |
import Mathlib.Data.Set.Image
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Clopen
import Mathlib.Topology.Irreducible
#align_import topology.connected from "leanprover-community/mathlib"@"d101e93197bb5f6ea89bd7ba386b7f7dff1f3903"
open Set Function Topology TopologicalSpace Relation
open scoped C... | Mathlib/Topology/Connected/Basic.lean | 148 | 153 | theorem IsConnected.union {s t : Set α} (H : (s ∩ t).Nonempty) (Hs : IsConnected s)
(Ht : IsConnected t) : IsConnected (s ∪ t) := by |
rcases H with ⟨x, hx⟩
refine ⟨⟨x, mem_union_left t (mem_of_mem_inter_left hx)⟩, ?_⟩
exact Hs.isPreconnected.union x (mem_of_mem_inter_left hx) (mem_of_mem_inter_right hx)
Ht.isPreconnected
|
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.Algebra.Category.ModuleCat.Abelian
import Mathlib.Algebra.Category.ModuleCat.Subobject
import Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory
#align_import algebra.homology.Module from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225... | Mathlib/Algebra/Homology/ModuleCat.lean | 37 | 49 | theorem homology'_ext {L M N K : ModuleCat.{u} R} {f : L ⟶ M} {g : M ⟶ N} (w : f ≫ g = 0)
{h k : homology' f g w ⟶ K}
(w :
∀ x : LinearMap.ker g,
h (cokernel.π (imageToKernel _ _ w) (toKernelSubobject x)) =
k (cokernel.π (imageToKernel _ _ w) (toKernelSubobject x))) :
h = k := by |
refine Concrete.cokernel_funext fun n => ?_
-- Porting note: as `equiv_rw` was not ported, it was replaced by `Equiv.surjective`
-- Gosh it would be nice if `equiv_rw` could directly use an isomorphism, or an enriched `≃`.
obtain ⟨n, rfl⟩ := (kernelSubobjectIso g ≪≫
ModuleCat.kernelIsoKer g).toLinearEquiv.... |
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Adjunction.Reflective
#align_import algebraic_geometry.Gamma_Spec_adjunction from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
-- Explicit universe annotations were u... | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | 91 | 95 | theorem toΓSpec_continuous : Continuous X.toΓSpecFun := by |
rw [isTopologicalBasis_basic_opens.continuous_iff]
rintro _ ⟨r, rfl⟩
erw [X.toΓSpec_preim_basicOpen_eq r]
exact (X.toRingedSpace.basicOpen r).2
|
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.NormedSpace.BallAction
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Geometry.Manifold.Algebra.LieGroup
import Mathlib.Geometry.Manifol... | Mathlib/Geometry/Manifold/Instances/Sphere.lean | 385 | 386 | theorem sphere_ext_iff (u v : sphere (0 : E) 1) : u = v ↔ ⟪(u : E), v⟫_ℝ = 1 := by |
simp [Subtype.ext_iff, inner_eq_one_iff_of_norm_one]
|
import Mathlib.Topology.Category.Profinite.Basic
import Mathlib.Topology.LocallyConstant.Basic
import Mathlib.Topology.DiscreteQuotient
import Mathlib.Topology.Category.TopCat.Limits.Cofiltered
import Mathlib.Topology.Category.TopCat.Limits.Konig
#align_import topology.category.Profinite.cofiltered_limit from "leanpr... | Mathlib/Topology/Category/Profinite/CofilteredLimit.lean | 116 | 126 | theorem exists_locallyConstant_fin_two (hC : IsLimit C) (f : LocallyConstant C.pt (Fin 2)) :
∃ (j : J) (g : LocallyConstant (F.obj j) (Fin 2)), f = g.comap (C.π.app _) := by |
let U := f ⁻¹' {0}
have hU : IsClopen U := f.isLocallyConstant.isClopen_fiber _
obtain ⟨j, V, hV, h⟩ := exists_isClopen_of_cofiltered C hC hU
use j, LocallyConstant.ofIsClopen hV
apply LocallyConstant.locallyConstant_eq_of_fiber_zero_eq
simp only [Fin.isValue, Functor.const_obj_obj, LocallyConstant.coe_com... |
import Mathlib.NumberTheory.FLT.Basic
import Mathlib.NumberTheory.PythagoreanTriples
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.Tactic.LinearCombination
#align_import number_theory.fermat4 from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
noncomputable section
open scope... | Mathlib/NumberTheory/FLT/Four.lean | 89 | 105 | theorem coprime_of_minimal {a b c : ℤ} (h : Minimal a b c) : IsCoprime a b := by |
apply Int.gcd_eq_one_iff_coprime.mp
by_contra hab
obtain ⟨p, hp, hpa, hpb⟩ := Nat.Prime.not_coprime_iff_dvd.mp hab
obtain ⟨a1, rfl⟩ := Int.natCast_dvd.mpr hpa
obtain ⟨b1, rfl⟩ := Int.natCast_dvd.mpr hpb
have hpc : (p : ℤ) ^ 2 ∣ c := by
rw [← Int.pow_dvd_pow_iff two_ne_zero, ← h.1.2.2]
apply Dvd.int... |
import Mathlib.LinearAlgebra.Ray
import Mathlib.LinearAlgebra.Determinant
#align_import linear_algebra.orientation from "leanprover-community/mathlib"@"0c1d80f5a86b36c1db32e021e8d19ae7809d5b79"
noncomputable section
section OrderedCommSemiring
variable (R : Type*) [StrictOrderedCommSemiring R]
variable (M : Typ... | Mathlib/LinearAlgebra/Orientation.lean | 74 | 75 | theorem Orientation.map_refl : (Orientation.map ι <| LinearEquiv.refl R M) = Equiv.refl _ := by |
rw [Orientation.map, AlternatingMap.domLCongr_refl, Module.Ray.map_refl]
|
import Mathlib.Algebra.Polynomial.Splits
#align_import algebra.cubic_discriminant from "leanprover-community/mathlib"@"930133160e24036d5242039fe4972407cd4f1222"
noncomputable section
@[ext]
structure Cubic (R : Type*) where
(a b c d : R)
#align cubic Cubic
namespace Cubic
open Cubic Polynomial
open Polynom... | Mathlib/Algebra/CubicDiscriminant.lean | 130 | 130 | theorem d_of_eq (h : P.toPoly = Q.toPoly) : P.d = Q.d := by | rw [← coeff_eq_d, h, coeff_eq_d]
|
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.MvPolynomial.Variables
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Expand
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.ZMod.Basic
#align_import ring_theory.witt_vector.witt_polynomial from "leanprover-c... | Mathlib/RingTheory/WittVector/WittPolynomial.lean | 218 | 234 | theorem constantCoeff_xInTermsOfW [hp : Fact p.Prime] [Invertible (p : R)] (n : ℕ) :
constantCoeff (xInTermsOfW p R n) = 0 := by |
apply Nat.strongInductionOn n; clear n
intro n IH
rw [xInTermsOfW_eq, mul_comm, RingHom.map_mul, RingHom.map_sub, map_sum, constantCoeff_C,
constantCoeff_X, zero_sub, mul_neg, neg_eq_zero]
-- Porting note: here, we should be able to do `rw [sum_eq_zero]`, but the goal that
-- is created is not what we ex... |
import Mathlib.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from "leanprover-community/mathlib"@"00638177efd1b2534fc5269363ebf42a7871df9a"
open Equiv Equiv.Perm List
variable {α : Type*}
namespace Equiv.Perm
secti... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 259 | 260 | theorem toList_nthLe_zero (h : x ∈ p.support) :
(toList p x).nthLe 0 (length_toList_pos_of_mem_support _ _ h) = x := by | simp [toList]
|
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 | 237 | 244 | theorem disjoint_cylinder_iff [Nonempty (∀ i, α i)] {s t : Finset ι} {S : Set (∀ i : s, α i)}
{T : Set (∀ i : t, α i)} [DecidableEq ι] :
Disjoint (cylinder s S) (cylinder t T) ↔
Disjoint
((fun f : ∀ i : (s ∪ t : Finset ι), α i
↦ fun j : s ↦ f ⟨j, Finset.mem_union_left t j.prop⟩) ⁻¹' S)
... |
simp_rw [Set.disjoint_iff, subset_empty_iff, inter_cylinder, cylinder_eq_empty_iff]
|
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.matrix from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped NNReal Matrix
namespace Matrix
variable {R l m n α β : Type*} [Fintype l] [Fintyp... | Mathlib/Analysis/Matrix.lean | 161 | 162 | theorem nnnorm_row (v : n → α) : ‖row v‖₊ = ‖v‖₊ := by |
simp [nnnorm_def, Pi.nnnorm_def]
|
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.Block
#align_import analysis.inner_product_space.gram_schmidt_ortho from "leanprover-community/mathlib"@"1a4df69ca1a9a0e5e26bfe12e2b92814216016d0"
open Finset Submodule FiniteDimensional
variable (𝕜 : Type*) {E : Type*} [RCLike �... | Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean | 133 | 139 | theorem mem_span_gramSchmidt (f : ι → E) {i j : ι} (hij : i ≤ j) :
f i ∈ span 𝕜 (gramSchmidt 𝕜 f '' Set.Iic j) := by |
rw [gramSchmidt_def' 𝕜 f i]
simp_rw [orthogonalProjection_singleton]
exact Submodule.add_mem _ (subset_span <| mem_image_of_mem _ hij)
(Submodule.sum_mem _ fun k hk => smul_mem (span 𝕜 (gramSchmidt 𝕜 f '' Set.Iic j)) _ <|
subset_span <| mem_image_of_mem (gramSchmidt 𝕜 f) <| (Finset.mem_Iio.1 hk).le... |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Rat.Encodable
import Mathlib.Topology.GDelta
#align_import topology.instances.irrational from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Filter Metric
open Filter Topology
protected theorem IsGδ.setOf_irrational : Is... | Mathlib/Topology/Instances/Irrational.lean | 45 | 51 | theorem dense_irrational : Dense { x : ℝ | Irrational x } := by |
refine Real.isTopologicalBasis_Ioo_rat.dense_iff.2 ?_
simp only [gt_iff_lt, Rat.cast_lt, not_lt, ge_iff_le, Rat.cast_le, mem_iUnion, mem_singleton_iff,
exists_prop, forall_exists_index, and_imp]
rintro _ a b hlt rfl _
rw [inter_comm]
exact exists_irrational_btwn (Rat.cast_lt.2 hlt)
|
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import number_theory.ramification_inertia from "leanprover-community/mathlib"@"039a089d2a4b93c761b234f3e5f5aeb752bac60f"
namespace Ideal
universe u v
variable {R : Type u} [CommRing R]
variable {S : Type v} [CommRing S] (f : R →+* S)
variable (p : Ideal R) (... | Mathlib/NumberTheory/RamificationInertia.lean | 143 | 150 | theorem ramificationIdx_eq_normalizedFactors_count (hp0 : map f p ≠ ⊥) (hP : P.IsPrime)
(hP0 : P ≠ ⊥) : ramificationIdx f p P = (normalizedFactors (map f p)).count P := by |
have hPirr := (Ideal.prime_of_isPrime hP0 hP).irreducible
refine ramificationIdx_spec (Ideal.le_of_dvd ?_) (mt Ideal.dvd_iff_le.mpr ?_) <;>
rw [dvd_iff_normalizedFactors_le_normalizedFactors (pow_ne_zero _ hP0) hp0,
normalizedFactors_pow, normalizedFactors_irreducible hPirr, normalize_eq,
Multiset.... |
import Mathlib.Data.Multiset.Basic
#align_import data.multiset.range from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
open List Nat
namespace Multiset
-- range
def range (n : ℕ) : Multiset ℕ :=
List.range n
#align multiset.range Multiset.range
theorem coe_range (n : ℕ) : ↑(List... | Mathlib/Data/Multiset/Range.lean | 34 | 35 | theorem range_succ (n : ℕ) : range (succ n) = n ::ₘ range n := by |
rw [range, List.range_succ, ← coe_add, add_comm]; rfl
|
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.Convex.StrictConvexSpace
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
import Mathlib.MeasureTheory.Integral.Average
#align_import analysis.convex.integral from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Mea... | Mathlib/Analysis/Convex/Integral.lean | 112 | 119 | theorem ConvexOn.average_mem_epigraph [IsFiniteMeasure μ] [NeZero μ] (hg : ConvexOn ℝ s g)
(hgc : ContinuousOn g s) (hsc : IsClosed s) (hfs : ∀ᵐ x ∂μ, f x ∈ s)
(hfi : Integrable f μ) (hgi : Integrable (g ∘ f) μ) :
(⨍ x, f x ∂μ, ⨍ x, g (f x) ∂μ) ∈ {p : E × ℝ | p.1 ∈ s ∧ g p.1 ≤ p.2} := by |
have ht_mem : ∀ᵐ x ∂μ, (f x, g (f x)) ∈ {p : E × ℝ | p.1 ∈ s ∧ g p.1 ≤ p.2} :=
hfs.mono fun x hx => ⟨hx, le_rfl⟩
exact average_pair hfi hgi ▸
hg.convex_epigraph.average_mem (hsc.epigraph hgc) ht_mem (hfi.prod_mk hgi)
|
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.pointwise from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set
open Pointwise Topology
variable {𝕜 E : Type*}
variable [NormedField 𝕜]
sectio... | Mathlib/Analysis/NormedSpace/Pointwise.lean | 84 | 88 | theorem smul_ball {c : 𝕜} (hc : c ≠ 0) (x : E) (r : ℝ) : c • ball x r = ball (c • x) (‖c‖ * r) := by |
ext y
rw [mem_smul_set_iff_inv_smul_mem₀ hc]
conv_lhs => rw [← inv_smul_smul₀ hc x]
simp [← div_eq_inv_mul, div_lt_iff (norm_pos_iff.2 hc), mul_comm _ r, dist_smul₀]
|
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.NthRewrite
#align_import data.nat.gcd.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
namespace Nat
theorem gcd_greatest {a b d : ℕ} (hda : d ∣ a) (hdb : d ∣ b) (hd ... | Mathlib/Data/Nat/GCD/Basic.lean | 53 | 54 | theorem gcd_add_mul_right_left (m n k : ℕ) : gcd (m + k * n) n = gcd m n := by |
rw [gcd_comm, gcd_add_mul_right_right, gcd_comm]
|
import Batteries.Data.List.Lemmas
namespace List
universe u v
variable {α : Type u} {β : Type v}
@[simp] theorem eraseIdx_zero (l : List α) : eraseIdx l 0 = tail l := by cases l <;> rfl
theorem eraseIdx_eq_take_drop_succ :
∀ (l : List α) (i : Nat), l.eraseIdx i = l.take i ++ l.drop (i + 1)
| nil, _ => by s... | .lake/packages/batteries/Batteries/Data/List/EraseIdx.lean | 43 | 47 | theorem eraseIdx_append_of_lt_length {l : List α} {k : Nat} (hk : k < length l) (l' : List α) :
eraseIdx (l ++ l') k = eraseIdx l k ++ l' := by |
rw [eraseIdx_eq_take_drop_succ, take_append_of_le_length, drop_append_of_le_length,
eraseIdx_eq_take_drop_succ, append_assoc]
all_goals omega
|
import Mathlib.Init.Algebra.Classes
import Mathlib.Logic.Nontrivial.Basic
import Mathlib.Order.BoundedOrder
import Mathlib.Data.Option.NAry
import Mathlib.Tactic.Lift
import Mathlib.Data.Option.Basic
#align_import order.with_bot from "leanprover-community/mathlib"@"0111834459f5d7400215223ea95ae38a1265a907"
variabl... | Mathlib/Order/WithBot.lean | 143 | 145 | theorem unbot'_eq_unbot'_iff {d : α} {x y : WithBot α} :
unbot' d x = unbot' d y ↔ x = y ∨ x = d ∧ y = ⊥ ∨ x = ⊥ ∧ y = d := by |
induction y <;> simp [unbot'_eq_iff, or_comm]
|
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.Algebra.Polynomial.RingDivision
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
noncomputable section
open scoped Classical
open scoped nonZeroDivisors Polynomial
universe u v
va... | Mathlib/FieldTheory/RatFunc/Defs.lean | 154 | 155 | theorem mk_eq_div' (p q : K[X]) :
RatFunc.mk p q = ofFractionRing (algebraMap _ _ p / algebraMap _ _ q) := by | rw [RatFunc.mk]
|
import Mathlib.Algebra.Algebra.Hom
import Mathlib.RingTheory.Ideal.Quotient
#align_import algebra.ring_quot from "leanprover-community/mathlib"@"e5820f6c8fcf1b75bcd7738ae4da1c5896191f72"
universe uR uS uT uA u₄
variable {R : Type uR} [Semiring R]
variable {S : Type uS} [CommSemiring S]
variable {T : Type uT}
vari... | Mathlib/Algebra/RingQuot.lean | 79 | 80 | theorem Rel.smul {r : A → A → Prop} (k : S) ⦃a b : A⦄ (h : Rel r a b) : Rel r (k • a) (k • b) := by |
simp only [Algebra.smul_def, Rel.mul_right h]
|
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 | 135 | 138 | theorem one_le_chainHeight_iff : 1 ≤ s.chainHeight ↔ s.Nonempty := by |
rw [← Nat.cast_one, Set.le_chainHeight_iff]
simp only [length_eq_one, @and_comm (_ ∈ _), @eq_comm _ _ [_], exists_exists_eq_and,
singleton_mem_subchain_iff, Set.Nonempty]
|
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.finite.card from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
noncomputable section
open scoped Classical
variable {α β γ : Type*}
def Finite.equivFin (α : Type*) [Finite α] : α ≃ Fin (Nat.card α) := by
have := (Finite.... | Mathlib/Data/Finite/Card.lean | 78 | 80 | theorem card_le_one_iff_subsingleton [Finite α] : Nat.card α ≤ 1 ↔ Subsingleton α := by |
haveI := Fintype.ofFinite α
simp only [Nat.card_eq_fintype_card, Fintype.card_le_one_iff_subsingleton]
|
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Tactic.Abel
#align_import ring_theory.polynomial.pochhammer from "leanprover-community/mathlib"@"53b216bcc1146df1c4a0a868778... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 104 | 107 | theorem ascPochhammer_eval_cast (n k : ℕ) :
(((ascPochhammer ℕ n).eval k : ℕ) : S) = ((ascPochhammer S n).eval k : S) := by |
rw [← ascPochhammer_map (algebraMap ℕ S), eval_map, ← eq_natCast (algebraMap ℕ S),
eval₂_at_natCast,Nat.cast_id]
|
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Data.Set.MemPartition
import Mathlib.Order.Filter.CountableSeparatingOn
open Set MeasureTheory
namespace MeasurableSpace
variable {α β : Type*}
class CountablyGenerated (α : Type*) [m : MeasurableSpace α] : Prop where
isCountablyGenerated : ∃ b... | Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean | 96 | 101 | theorem CountablyGenerated.comap [m : MeasurableSpace β] [h : CountablyGenerated β] (f : α → β) :
@CountablyGenerated α (.comap f m) := by |
rcases h with ⟨⟨b, hbc, rfl⟩⟩
rw [comap_generateFrom]
letI := generateFrom (preimage f '' b)
exact ⟨_, hbc.image _, rfl⟩
|
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.Ring.Subring.Pointwise
import Mathlib.RingTheory.Adjoin.Basic
#align_import algebra.algebra.subalgebra.pointwise from "leanprover-community/mathlib"@"b2c707cd190a58ea0565c86695a19e99ccecc215"
namespace... | Mathlib/Algebra/Algebra/Subalgebra/Pointwise.lean | 27 | 32 | theorem mul_toSubmodule_le (S T : Subalgebra R A) :
(Subalgebra.toSubmodule S)* (Subalgebra.toSubmodule T) ≤ Subalgebra.toSubmodule (S ⊔ T) := by |
rw [Submodule.mul_le]
intro y hy z hz
show y * z ∈ S ⊔ T
exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)
|
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Group.Int
import Mathlib.Algebra.GroupWithZero.Semiconj
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Order.Bounds.Basic
#align_import data.int.gcd from "leanprover-community/mathlib"@"47a1a73351de8dd6c8d3d32b569c8e434b03ca47"
namespace Nat
... | Mathlib/Data/Int/GCD.lean | 94 | 98 | theorem gcdB_zero_right {s : ℕ} (h : s ≠ 0) : gcdB s 0 = 0 := by |
unfold gcdB xgcd
obtain ⟨s, rfl⟩ := Nat.exists_eq_succ_of_ne_zero h
rw [xgcdAux]
simp
|
import Mathlib.Data.Bundle
import Mathlib.Data.Set.Image
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Topology.Order.Basic
#align_import topology.fiber_bundle.trivialization from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
open TopologicalSpace Filter Set Bundle Function
... | Mathlib/Topology/FiberBundle/Trivialization.lean | 145 | 147 | theorem proj_symm_apply {x : B × F} (hx : x ∈ e.target) : proj (e.toPartialEquiv.symm x) = x.1 := by |
have := (e.coe_fst (e.map_target hx)).symm
rwa [← e.coe_coe, e.right_inv hx] at this
|
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 | 128 | 132 | theorem measure_union_add_inter (s : Set α) (ht : MeasurableSet t) :
μ (s ∪ t) + μ (s ∩ t) = μ s + μ t := by |
rw [← measure_inter_add_diff (s ∪ t) ht, Set.union_inter_cancel_right, union_diff_right, ←
measure_inter_add_diff s ht]
ac_rfl
|
import Mathlib.Algebra.MvPolynomial.Rename
#align_import data.mv_polynomial.comap from "leanprover-community/mathlib"@"aba31c938d3243cc671be7091b28a1e0814647ee"
namespace MvPolynomial
variable {σ : Type*} {τ : Type*} {υ : Type*} {R : Type*} [CommSemiring R]
noncomputable def comap (f : MvPolynomial σ R →ₐ[R] M... | Mathlib/Algebra/MvPolynomial/Comap.lean | 62 | 74 | theorem comap_comp_apply (f : MvPolynomial σ R →ₐ[R] MvPolynomial τ R)
(g : MvPolynomial τ R →ₐ[R] MvPolynomial υ R) (x : υ → R) :
comap (g.comp f) x = comap f (comap g x) := by |
funext i
trans aeval x (aeval (fun i => g (X i)) (f (X i)))
· apply eval₂Hom_congr rfl rfl
rw [AlgHom.comp_apply]
suffices g = aeval fun i => g (X i) by rw [← this]
exact aeval_unique g
· simp only [comap, aeval_eq_eval₂Hom, map_eval₂Hom, AlgHom.comp_apply]
refine eval₂Hom_congr ?_ rfl rfl
... |
import Mathlib.Order.Filter.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.SetTheory.Cardinal.Cofinality
open Set Filter Cardinal
universe u
variable {ι : Type u} {α β : Type u} {c : Cardinal.{u}}
class CardinalInterFilter (l : Filter α) (c : Cardinal.{... | Mathlib/Order/Filter/CardinalInter.lean | 123 | 127 | theorem EventuallyLE.cardinal_bUnion {S : Set ι} (hS : #S < c)
{s t : ∀ i ∈ S, Set α} (h : ∀ i hi, s i hi ≤ᶠ[l] t i hi) :
⋃ i ∈ S, s i ‹_› ≤ᶠ[l] ⋃ i ∈ S, t i ‹_› := by |
simp only [biUnion_eq_iUnion]
exact EventuallyLE.cardinal_iUnion hS fun i => h i i.2
|
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.class_group from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
variable {R K L : Type*} [CommRing R]
variable [Field K] [Field L] [DecidableEq L]
variable [Algebra R K] [Is... | Mathlib/RingTheory/ClassGroup.lean | 111 | 117 | theorem ClassGroup.Quot_mk_eq_mk (I : (FractionalIdeal R⁰ (FractionRing R))ˣ) :
Quot.mk _ I = ClassGroup.mk I := by |
rw [ClassGroup.mk, canonicalEquiv_self, RingEquiv.coe_monoidHom_refl, Units.map_id]
-- This used to be `rw`, but we need `erw` after leanprover/lean4#2644
erw [MonoidHom.comp_apply]
rw [MonoidHom.id_apply, QuotientGroup.mk'_apply]
rfl
|
import Mathlib.LinearAlgebra.Matrix.BilinearForm
import Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.LinearAlgebra.Trace
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosu... | Mathlib/RingTheory/Trace.lean | 149 | 153 | theorem trace_comp_trace_of_basis [Algebra S T] [IsScalarTower R S T] {ι κ : Type*} [Finite ι]
[Finite κ] (b : Basis ι R S) (c : Basis κ S T) :
(trace R S).comp ((trace S T).restrictScalars R) = trace R T := by |
ext
rw [LinearMap.comp_apply, LinearMap.restrictScalars_apply, trace_trace_of_basis b c]
|
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.MvPolynomial.Degrees
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.LinearAlgebra.FinsuppVectorSpace
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
#align_import ring_theory.mv_polynomial.basic from "leanprover-community/mathlib"@"2f5b500a507... | Mathlib/RingTheory/MvPolynomial/Basic.lean | 68 | 72 | theorem mapRange_eq_map {R S : Type*} [CommSemiring R] [CommSemiring S] (p : MvPolynomial σ R)
(f : R →+* S) : Finsupp.mapRange f f.map_zero p = map f p := by |
rw [p.as_sum, Finsupp.mapRange_finset_sum, map_sum (map f)]
refine Finset.sum_congr rfl fun n _ => ?_
rw [map_monomial, ← single_eq_monomial, Finsupp.mapRange_single, single_eq_monomial]
|
import Mathlib.MeasureTheory.Measure.GiryMonad
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.MeasureTheory.Measure.OpenPos
#align_import measure_theory.constructions.prod.basic from "leanprover-community/mathlib"@"00abe0695d8767201e6d008afa22393978bb32... | Mathlib/MeasureTheory/Constructions/Prod/Basic.lean | 81 | 85 | theorem IsCountablySpanning.prod {C : Set (Set α)} {D : Set (Set β)} (hC : IsCountablySpanning C)
(hD : IsCountablySpanning D) : IsCountablySpanning (image2 (· ×ˢ ·) C D) := by |
rcases hC, hD with ⟨⟨s, h1s, h2s⟩, t, h1t, h2t⟩
refine ⟨fun n => s n.unpair.1 ×ˢ t n.unpair.2, fun n => mem_image2_of_mem (h1s _) (h1t _), ?_⟩
rw [iUnion_unpair_prod, h2s, h2t, univ_prod_univ]
|
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Valuation.PrimeMultiplicity
import Mathlib.RingTheory.AdicCompletion.Basic
#align_import ring_theory.discrete_valuation_ring.basic from "leanprover-community/mathlib"@"c163ec99dfc664628ca15d215fce0a5b9c2... | Mathlib/RingTheory/DiscreteValuationRing/Basic.lean | 75 | 88 | theorem irreducible_of_span_eq_maximalIdeal {R : Type*} [CommRing R] [LocalRing R] [IsDomain R]
(ϖ : R) (hϖ : ϖ ≠ 0) (h : maximalIdeal R = Ideal.span {ϖ}) : Irreducible ϖ := by |
have h2 : ¬IsUnit ϖ := show ϖ ∈ maximalIdeal R from h.symm ▸ Submodule.mem_span_singleton_self ϖ
refine ⟨h2, ?_⟩
intro a b hab
by_contra! h
obtain ⟨ha : a ∈ maximalIdeal R, hb : b ∈ maximalIdeal R⟩ := h
rw [h, mem_span_singleton'] at ha hb
rcases ha with ⟨a, rfl⟩
rcases hb with ⟨b, rfl⟩
rw [show a * ... |
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.RowCol
import Mathlib.Data.Fin.VecNotation
import Mathlib.Tactic.FinCases
#align_import data.matrix.notation from "leanprover-community/mathlib"@"a99f85220eaf38f14f94e04699943e185a5e1d1a"
namespace Matrix
universe u uₘ uₙ uₒ
variable {α : Type u} {o n m... | Mathlib/Data/Matrix/Notation.lean | 393 | 396 | theorem submatrix_cons_row (A : Matrix m' n' α) (i : m') (row : Fin m → m') (col : o' → n') :
submatrix A (vecCons i row) col = vecCons (fun j => A i (col j)) (submatrix A row col) := by |
ext i j
refine Fin.cases ?_ ?_ i <;> simp [submatrix]
|
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv
import Mathlib.Analysis.Calculus.FDeriv.Extend
import Mathlib.Analysis.Calculus.Deriv.Prod
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
#... | Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean | 52 | 60 | theorem hasStrictDerivAt_const_cpow {x y : ℂ} (h : x ≠ 0 ∨ y ≠ 0) :
HasStrictDerivAt (fun y => x ^ y) (x ^ y * log x) y := by |
rcases em (x = 0) with (rfl | hx)
· replace h := h.neg_resolve_left rfl
rw [log_zero, mul_zero]
refine (hasStrictDerivAt_const _ 0).congr_of_eventuallyEq ?_
exact (isOpen_ne.eventually_mem h).mono fun y hy => (zero_cpow hy).symm
· simpa only [cpow_def_of_ne_zero hx, mul_one] using
((hasStrictDe... |
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
import Mathlib.Tactic.IntervalCases
#align_import geometry.euclidean.triangle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped Classica... | Mathlib/Geometry/Euclidean/Triangle.lean | 109 | 143 | theorem cos_angle_sub_add_angle_sub_rev_eq_neg_cos_angle {x y : V} (hx : x ≠ 0) (hy : y ≠ 0) :
Real.cos (angle x (x - y) + angle y (y - x)) = -Real.cos (angle x y) := by |
by_cases hxy : x = y
· rw [hxy, angle_self hy]
simp
· rw [Real.cos_add, cos_angle, cos_angle, cos_angle]
have hxn : ‖x‖ ≠ 0 := fun h => hx (norm_eq_zero.1 h)
have hyn : ‖y‖ ≠ 0 := fun h => hy (norm_eq_zero.1 h)
have hxyn : ‖x - y‖ ≠ 0 := fun h => hxy (eq_of_sub_eq_zero (norm_eq_zero.1 h))
app... |
import Mathlib.RingTheory.MvPowerSeries.Basic
import Mathlib.Data.Finsupp.Interval
noncomputable section
open Finset (antidiagonal mem_antidiagonal)
namespace MvPowerSeries
open Finsupp
variable {σ R : Type*}
section Trunc
variable [CommSemiring R] (n : σ →₀ ℕ)
def truncFun (φ : MvPowerSeries σ R) : MvPol... | Mathlib/RingTheory/MvPowerSeries/Trunc.lean | 71 | 73 | theorem coeff_trunc (m : σ →₀ ℕ) (φ : MvPowerSeries σ R) :
(trunc R n φ).coeff m = if m < n then coeff R m φ else 0 := by |
classical simp [trunc, coeff_truncFun]
|
import Mathlib.SetTheory.Game.Ordinal
import Mathlib.SetTheory.Ordinal.NaturalOps
#align_import set_theory.game.birthday from "leanprover-community/mathlib"@"a347076985674932c0e91da09b9961ed0a79508c"
universe u
open Ordinal
namespace SetTheory
open scoped NaturalOps PGame
namespace PGame
noncomputable def b... | Mathlib/SetTheory/Game/Birthday.lean | 64 | 78 | theorem lt_birthday_iff {x : PGame} {o : Ordinal} :
o < x.birthday ↔
(∃ i : x.LeftMoves, o ≤ (x.moveLeft i).birthday) ∨
∃ i : x.RightMoves, o ≤ (x.moveRight i).birthday := by |
constructor
· rw [birthday_def]
intro h
cases' lt_max_iff.1 h with h' h'
· left
rwa [lt_lsub_iff] at h'
· right
rwa [lt_lsub_iff] at h'
· rintro (⟨i, hi⟩ | ⟨i, hi⟩)
· exact hi.trans_lt (birthday_moveLeft_lt i)
· exact hi.trans_lt (birthday_moveRight_lt i)
|
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Integral
#align_import ring_theory.ideal.over from "leanprover-community/mathlib"@"198cb64d5c961e1a8d0d3e219feb7058d5353861"
variable {R : Type*} [CommRing R]
namespace Ideal
open Polynomial... | Mathlib/RingTheory/Ideal/Over.lean | 139 | 149 | theorem comap_eq_of_scalar_tower_quotient [Algebra R S] [Algebra (R ⧸ p) (S ⧸ P)]
[IsScalarTower R (R ⧸ p) (S ⧸ P)] (h : Function.Injective (algebraMap (R ⧸ p) (S ⧸ P))) :
comap (algebraMap R S) P = p := by |
ext x
rw [mem_comap, ← Quotient.eq_zero_iff_mem, ← Quotient.eq_zero_iff_mem, Quotient.mk_algebraMap,
IsScalarTower.algebraMap_apply R (R ⧸ p) (S ⧸ P), Quotient.algebraMap_eq]
constructor
· intro hx
exact (injective_iff_map_eq_zero (algebraMap (R ⧸ p) (S ⧸ P))).mp h _ hx
· intro hx
rw [hx, RingHom... |
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
import Mathlib.RepresentationTheory.GroupCohomology.Basic
import Mathlib.RepresentationTheory.Invariants
universe v u
noncomputable section
open CategoryTheory Limits Representation
variable {k G : Type u} [CommRing k] [Group G] (A : Rep k G)
namespace grou... | Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean | 405 | 407 | theorem map_one_fst_of_isTwoCocycle {f : G × G → A} (hf : IsTwoCocycle f) (g : G) :
f (1, g) = f (1, 1) := by |
simpa only [one_smul, one_mul, mul_one, add_right_inj] using (hf 1 1 g).symm
|
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "leanprover-community/mathlib"@"996b0ff959da753a555053a480f36e5f264d4207"
variable {α β γ : Type*} {ι : Sort*} {π : α → Type*}
open Equiv Equiv.Perm Function
namespace Set
section restrict
def restrict (... | Mathlib/Data/Set/Function.lean | 117 | 120 | theorem restrict_extend_range (f : α → β) (g : α → γ) (g' : β → γ) :
(range f).restrict (extend f g g') = fun x => g x.coe_prop.choose := by |
classical
exact restrict_dite _ _
|
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 | 56 | 59 | theorem taylor_zero' : taylor (0 : R) = LinearMap.id := by |
ext
simp only [taylor_apply, add_zero, comp_X, _root_.map_zero, LinearMap.id_comp,
Function.comp_apply, LinearMap.coe_comp]
|
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open ... | Mathlib/Analysis/NormedSpace/Dual.lean | 82 | 84 | theorem inclusionInDoubleDual_norm_le : ‖inclusionInDoubleDual 𝕜 E‖ ≤ 1 := by |
rw [inclusionInDoubleDual_norm_eq]
exact ContinuousLinearMap.norm_id_le
|
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 | 127 | 132 | theorem volume_emetric_closedBall (a : ℝ) (r : ℝ≥0∞) : volume (EMetric.closedBall a r) = 2 * r := by |
rcases eq_or_ne r ∞ with (rfl | hr)
· rw [EMetric.closedBall_top, volume_univ, two_mul, _root_.top_add]
· lift r to ℝ≥0 using hr
rw [Metric.emetric_closedBall_nnreal, volume_closedBall, two_mul, ← NNReal.coe_add,
ENNReal.ofReal_coe_nnreal, ENNReal.coe_add, two_mul]
|
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