Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | num_lines int64 1 150 |
|---|---|---|---|---|---|---|
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Convex.Uniform
import Mathlib.Analysis.NormedSpace.Completion
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
#align_import analysis.inner_product_space.basic from "leanprover-community/mathlib"@"3f655f5297b030... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 224 | 226 | theorem ofReal_normSq_eq_inner_self (x : F) : (normSqF x : 𝕜) = ⟪x, x⟫ := by |
rw [ext_iff]
exact ⟨by simp only [ofReal_re]; rfl, by simp only [inner_self_im, ofReal_im]⟩
| 2 |
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
#align_import order.symm_diff from "leanprover-community/mathlib"@"6eb334bd8f3433d5b08ba156b8ec3e6af47e1904"
open Function OrderDual
variable {ι α β : Type*} {π : ι → Type*}
def symmDiff [Sup α] [SDiff α] (a b : α) : α :=
a \ b ⊔ b \ a
#ali... | Mathlib/Order/SymmDiff.lean | 170 | 172 | theorem symmDiff_sdiff_inf : a ∆ b \ (a ⊓ b) = a ∆ b := by |
rw [symmDiff_sdiff]
simp [symmDiff]
| 2 |
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 | 265 | 268 | theorem cthickening_singleton {α : Type*} [PseudoMetricSpace α] (x : α) {δ : ℝ} (hδ : 0 ≤ δ) :
cthickening δ ({x} : Set α) = closedBall x δ := by |
ext y
simp [cthickening, edist_dist, ENNReal.ofReal_le_ofReal_iff hδ]
| 2 |
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Order.Partition.Finpartition
import Mathlib.Tactic.GCongr
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Positivity
import Mathlib.Tactic.Ring
#align_import combinatorics.simp... | Mathlib/Combinatorics/SimpleGraph/Density.lean | 109 | 112 | theorem interedges_biUnion_left (s : Finset ι) (t : Finset β) (f : ι → Finset α) :
interedges r (s.biUnion f) t = s.biUnion fun a ↦ interedges r (f a) t := by |
ext
simp only [mem_biUnion, mem_interedges_iff, exists_and_right, ← and_assoc]
| 2 |
set_option autoImplicit true
namespace Array
@[simp]
| Mathlib/Data/Array/ExtractLemmas.lean | 16 | 19 | theorem extract_eq_nil_of_start_eq_end {a : Array α} :
a.extract i i = #[] := by |
refine extract_empty_of_stop_le_start a ?h
exact Nat.le_refl i
| 2 |
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Infix
#align_import data.list.sort from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
open List.Perm
universe u
namespace List
section Sorted
variable {α : Type u} {r : α → α → Prop} {a : α} {l... | Mathlib/Data/List/Sort.lean | 123 | 126 | theorem sublist_of_subperm_of_sorted [IsAntisymm α r] {l₁ l₂ : List α} (hp : l₁ <+~ l₂)
(hs₁ : l₁.Sorted r) (hs₂ : l₂.Sorted r) : l₁ <+ l₂ := by |
let ⟨_, h, h'⟩ := hp
rwa [← eq_of_perm_of_sorted h (hs₂.sublist h') hs₁]
| 2 |
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
#align_import algebra.order.monoid.min_max from "leanprover-community/mathlib"@"de87d5053a9fe5cbde723172c0fb7e27e7436473"
open Function
variable {α β : Type*}
section CovariantClassMulLe
variable [LinearOrder α]
section Mul
variable [Mul α]
@[to_additive... | Mathlib/Algebra/Order/Monoid/Unbundled/MinMax.lean | 90 | 94 | theorem lt_or_lt_of_mul_lt_mul [CovariantClass α α (· * ·) (· ≤ ·)]
[CovariantClass α α (Function.swap (· * ·)) (· ≤ ·)] {a₁ a₂ b₁ b₂ : α} :
a₁ * b₁ < a₂ * b₂ → a₁ < a₂ ∨ b₁ < b₂ := by |
contrapose!
exact fun h => mul_le_mul' h.1 h.2
| 2 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section RelPrime
variable {α I} [Comm... | Mathlib/RingTheory/Coprime/Lemmas.lean | 299 | 301 | theorem pow_right (H : IsRelPrime x y) : IsRelPrime x (y ^ n) := by |
rw [← Finset.card_range n, ← Finset.prod_const]
exact IsRelPrime.prod_right fun _ _ ↦ H
| 2 |
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryTheory.IsomorphismClasses
import Mathlib.CategoryTheory.Limits.Shapes.ZeroObjects
#align_import category_theory.limits.shapes.zero_morphisms from "leanpr... | Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | 140 | 142 | theorem zero_of_comp_mono {X Y Z : C} {f : X ⟶ Y} (g : Y ⟶ Z) [Mono g] (h : f ≫ g = 0) : f = 0 := by |
rw [← zero_comp, cancel_mono] at h
exact h
| 2 |
import Mathlib.Algebra.Group.Semiconj.Defs
import Mathlib.Algebra.Group.Units
#align_import algebra.group.semiconj from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358ef64"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
open scoped Int
variable {M G : Type*}
namespace Sem... | Mathlib/Algebra/Group/Semiconj/Units.lean | 64 | 67 | theorem units_inv_symm_left {a : Mˣ} {x y : M} (h : SemiconjBy (↑a) x y) : SemiconjBy (↑a⁻¹) y x :=
calc
↑a⁻¹ * y = ↑a⁻¹ * (y * a * ↑a⁻¹) := by | rw [Units.mul_inv_cancel_right]
_ = x * ↑a⁻¹ := by rw [← h.eq, ← mul_assoc, Units.inv_mul_cancel_left]
| 2 |
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factors
import Mathlib.Order.Interval.Finset.Nat
#align_import number_theory.divisors from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open scoped Classical
open Finset
namespace Nat
variable (n : ℕ)
d... | Mathlib/NumberTheory/Divisors.lean | 84 | 86 | theorem insert_self_properDivisors (h : n ≠ 0) : insert n (properDivisors n) = divisors n := by |
rw [divisors, properDivisors, Ico_succ_right_eq_insert_Ico (one_le_iff_ne_zero.2 h),
Finset.filter_insert, if_pos (dvd_refl n)]
| 2 |
import Mathlib.Algebra.Order.Field.Power
import Mathlib.Data.Int.LeastGreatest
import Mathlib.Data.Rat.Floor
import Mathlib.Data.NNRat.Defs
#align_import algebra.order.archimedean from "leanprover-community/mathlib"@"6f413f3f7330b94c92a5a27488fdc74e6d483a78"
open Int Set
variable {α : Type*}
class Archimedean (... | Mathlib/Algebra/Order/Archimedean.lean | 84 | 87 | theorem existsUnique_zsmul_near_of_pos' {a : α} (ha : 0 < a) (g : α) :
∃! k : ℤ, 0 ≤ g - k • a ∧ g - k • a < a := by |
simpa only [sub_nonneg, add_zsmul, one_zsmul, sub_lt_iff_lt_add'] using
existsUnique_zsmul_near_of_pos ha g
| 2 |
import Mathlib.Analysis.InnerProductSpace.Rayleigh
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Algebra.DirectSum.Decomposition
import Mathlib.LinearAlgebra.Eigenspace.Minpoly
#align_import analysis.inner_product_space.spectrum from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da... | Mathlib/Analysis/InnerProductSpace/Spectrum.lean | 102 | 105 | theorem orthogonalComplement_iSup_eigenspaces_invariant ⦃v : E⦄ (hv : v ∈ (⨆ μ, eigenspace T μ)ᗮ) :
T v ∈ (⨆ μ, eigenspace T μ)ᗮ := by |
rw [← Submodule.iInf_orthogonal] at hv ⊢
exact T.iInf_invariant hT.invariant_orthogonalComplement_eigenspace v hv
| 2 |
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import set_theory.cardinal.continuum from "leanprover-community/mathlib"@"e08a42b2dd544cf11eba72e5fc7bf199d4349925"
namespace Cardinal
universe u v
open Cardinal
def continuum : Cardinal.{u} :=
2 ^ ℵ₀
#align cardinal.continuum Cardinal.continuum
scoped notat... | Mathlib/SetTheory/Cardinal/Continuum.lean | 58 | 60 | theorem continuum_lt_lift {c : Cardinal.{u}} : 𝔠 < lift.{v} c ↔ 𝔠 < c := by |
-- Porting note: added explicit universes
rw [← lift_continuum.{u,v}, lift_lt]
| 2 |
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.LinearAlgebra.Dual
import Mathlib.LinearAlgebra.Contraction
import Mathlib.RingTheory.TensorProduct.Basic
#align_import representation_... | Mathlib/RepresentationTheory/Basic.lean | 159 | 162 | theorem asModuleEquiv_symm_map_smul (r : k) (x : V) :
ρ.asModuleEquiv.symm (r • x) = algebraMap k (MonoidAlgebra k G) r • ρ.asModuleEquiv.symm x := by |
apply_fun ρ.asModuleEquiv
simp
| 2 |
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Combinatorics.SimpleGraph.AdjMatrix
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Set.Finite
#align_import combinatorics.simple_graph.strongly_regular from "leanprover-community/mathlib"@"2b35fc7bea4640cb75e477e83f32fbd5389208... | Mathlib/Combinatorics/SimpleGraph/StronglyRegular.lean | 137 | 140 | theorem IsSRGWith.compl_is_regular (h : G.IsSRGWith n k ℓ μ) :
Gᶜ.IsRegularOfDegree (n - k - 1) := by |
rw [← h.card, Nat.sub_sub, add_comm, ← Nat.sub_sub]
exact h.regular.compl
| 2 |
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
import Mathlib.CategoryTheory.Monoidal.Opposite
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.CommSq
#align_import category_... | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 125 | 128 | theorem braiding_naturality {X X' Y Y' : C} (f : X ⟶ Y) (g : X' ⟶ Y') :
(f ⊗ g) ≫ (braiding Y Y').hom = (braiding X X').hom ≫ (g ⊗ f) := by |
rw [tensorHom_def' f g, tensorHom_def g f]
simp_rw [Category.assoc, braiding_naturality_left, braiding_naturality_right_assoc]
| 2 |
import Mathlib.Topology.Sets.Closeds
#align_import topology.noetherian_space from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
variable (α β : Type*) [TopologicalSpace α] [TopologicalSpace β]
namespace TopologicalSpace
@[mk_iff]
class NoetherianSpace : Prop where
wellFounded_open... | Mathlib/Topology/NoetherianSpace.lean | 139 | 142 | theorem noetherianSpace_set_iff (s : Set α) :
NoetherianSpace s ↔ ∀ t, t ⊆ s → IsCompact t := by |
simp only [noetherianSpace_iff_isCompact, embedding_subtype_val.isCompact_iff,
Subtype.forall_set_subtype]
| 2 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.TensorProduct.Tower
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.LinearAlgebra.DirectSum.Finsupp
#align_import ring_theory.tensor_product from "leanprover-community/mathlib"@"88fcdc3da43943f5b01925deddaa5bf0c0e85e4e"
suppress_comp... | Mathlib/RingTheory/TensorProduct/Basic.lean | 90 | 92 | theorem baseChange_zero : baseChange A (0 : M →ₗ[R] N) = 0 := by |
ext
simp [baseChange_eq_ltensor]
| 2 |
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Ring.Defs
#align_import algebra.euclidean_domain.defs from "leanprover-community/mathlib"@"ee7b9f9a9ac2a8d9f04ea39bbfe6b1a3be053b38"
universe u
class EuclideanDomain (R : Type u) extends CommRing R, Nontrivial R ... | Mathlib/Algebra/EuclideanDomain/Defs.lean | 209 | 211 | theorem gcd_zero_left (a : R) : gcd 0 a = a := by |
rw [gcd]
exact if_pos rfl
| 2 |
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 | 136 | 139 | theorem discr_not_zero_of_basis [IsSeparable K L] (b : Basis ι K L) :
discr K b ≠ 0 := by |
rw [discr_def, traceMatrix_of_basis, ← LinearMap.BilinForm.nondegenerate_iff_det_ne_zero]
exact traceForm_nondegenerate _ _
| 2 |
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.Ideal.Prod
import Mathlib.RingTheory.Ideal.MinimalPrime
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.Topology.Sets.Closeds
import Mathlib.Topology.Sober
#a... | Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean | 116 | 119 | theorem primeSpectrumProd_symm_inl_asIdeal (x : PrimeSpectrum R) :
((primeSpectrumProd R S).symm <| Sum.inl x).asIdeal = Ideal.prod x.asIdeal ⊤ := by |
cases x
rfl
| 2 |
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 | 142 | 145 | theorem IsPreconnected.union' {s t : Set α} (H : (s ∩ t).Nonempty) (hs : IsPreconnected s)
(ht : IsPreconnected t) : IsPreconnected (s ∪ t) := by |
rcases H with ⟨x, hxs, hxt⟩
exact hs.union x hxs hxt ht
| 2 |
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.Transfer
#align_import group_theory.schreier from "leanprover-community/mathlib"@"8350c34a64b9bc3fc64335df8006bffcadc7baa6"
open scoped Pointwise
namespace Subgroup
open MemRightTransversals
variable {G : T... | Mathlib/GroupTheory/Schreier.lean | 85 | 89 | theorem closure_mul_image_eq_top (hR : R ∈ rightTransversals (H : Set G)) (hR1 : (1 : G) ∈ R)
(hS : closure S = ⊤) : closure ((R * S).image fun g =>
⟨g * (toFun hR g : G)⁻¹, mul_inv_toFun_mem hR g⟩ : Set H) = ⊤ := by |
rw [eq_top_iff, ← map_subtype_le_map_subtype, MonoidHom.map_closure, Set.image_image]
exact (map_subtype_le ⊤).trans (ge_of_eq (closure_mul_image_eq hR hR1 hS))
| 2 |
import Mathlib.CategoryTheory.Adjunction.Basic
import Mathlib.CategoryTheory.Limits.Cones
#align_import category_theory.limits.is_limit from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
noncomputable section
open CategoryTheory CategoryTheory.Category CategoryTheory.Functor Opposite
... | Mathlib/CategoryTheory/Limits/IsLimit.lean | 101 | 104 | theorem uniq_cone_morphism {s t : Cone F} (h : IsLimit t) {f f' : s ⟶ t} : f = f' :=
have : ∀ {g : s ⟶ t}, g = h.liftConeMorphism s := by |
intro g; apply ConeMorphism.ext; exact h.uniq _ _ g.w
this.trans this.symm
| 2 |
import Mathlib.Algebra.Order.CauSeq.Basic
#align_import data.real.cau_seq_completion from "leanprover-community/mathlib"@"cf4c49c445991489058260d75dae0ff2b1abca28"
namespace CauSeq.Completion
open CauSeq
section
variable {α : Type*} [LinearOrderedField α]
variable {β : Type*} [Ring β] (abv : β → α) [IsAbsolute... | Mathlib/Algebra/Order/CauSeq/Completion.lean | 73 | 75 | theorem mk_eq_zero {f : CauSeq _ abv} : mk f = 0 ↔ LimZero f := by |
have : mk f = 0 ↔ LimZero (f - 0) := Quotient.eq
rwa [sub_zero] at this
| 2 |
import Mathlib.Data.Finsupp.Defs
#align_import data.list.to_finsupp from "leanprover-community/mathlib"@"06a655b5fcfbda03502f9158bbf6c0f1400886f9"
namespace List
variable {M : Type*} [Zero M] (l : List M) [DecidablePred (getD l · 0 ≠ 0)] (n : ℕ)
def toFinsupp : ℕ →₀ M where
toFun i := getD l i 0
support := ... | Mathlib/Data/List/ToFinsupp.lean | 139 | 143 | theorem toFinsupp_concat_eq_toFinsupp_add_single {R : Type*} [AddZeroClass R] (x : R) (xs : List R)
[DecidablePred fun i => getD (xs ++ [x]) i 0 ≠ 0] [DecidablePred fun i => getD xs i 0 ≠ 0] :
toFinsupp (xs ++ [x]) = toFinsupp xs + Finsupp.single xs.length x := by |
classical rw [toFinsupp_append, toFinsupp_singleton, Finsupp.embDomain_single,
addLeftEmbedding_apply, add_zero]
| 2 |
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 | 130 | 132 | theorem eq_cons {n m : ℕ} (h : n < m) : Ico n m = n :: Ico (n + 1) m := by |
rw [← append_consecutive (Nat.le_succ n) h, succ_singleton]
rfl
| 2 |
import Mathlib.Tactic.CategoryTheory.Elementwise
import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
import Mathlib.CategoryTheory.Limits.Preserves.Limits
import Mathlib.CategoryTheory.Limits.Shapes.Types
#align_import category_theory.glue_data from "l... | Mathlib/CategoryTheory/GlueData.lean | 108 | 111 | theorem t'_inv (i j k : D.J) :
D.t' i j k ≫ (pullbackSymmetry _ _).hom ≫ D.t' j i k ≫ (pullbackSymmetry _ _).hom = 𝟙 _ := by |
rw [← cancel_mono (pullback.fst : pullback (D.f i j) (D.f i k) ⟶ _)]
simp [t_fac, t_fac_assoc]
| 2 |
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 | 101 | 104 | theorem hasProd_subtype_iff_mulIndicator {s : Set β} :
HasProd (f ∘ (↑) : s → α) a ↔ HasProd (s.mulIndicator f) a := by |
rw [← Set.mulIndicator_range_comp, Subtype.range_coe,
hasProd_subtype_iff_of_mulSupport_subset Set.mulSupport_mulIndicator_subset]
| 2 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.AddTorsor
#align_import analysis.convex.normed from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052... | Mathlib/Analysis/Convex/Normed.lean | 53 | 55 | theorem convexOn_dist (z : E) (hs : Convex ℝ s) : ConvexOn ℝ s fun z' => dist z' z := by |
simpa [dist_eq_norm, preimage_preimage] using
(convexOn_norm (hs.translate (-z))).comp_affineMap (AffineMap.id ℝ E - AffineMap.const ℝ E z)
| 2 |
import Mathlib.NumberTheory.LegendreSymbol.Basic
import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum
#align_import number_theory.legendre_symbol.quadratic_reciprocity from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
open Nat
section Values
variable {p : ℕ} [Fact p.Pri... | Mathlib/NumberTheory/LegendreSymbol/QuadraticReciprocity.lean | 66 | 68 | theorem at_neg_two : legendreSym p (-2) = χ₈' p := by |
have : (-2 : ZMod p) = (-2 : ℤ) := by norm_cast
rw [legendreSym, ← this, quadraticChar_neg_two ((ringChar_zmod_n p).substr hp), card p]
| 2 |
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.add_torsor from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
class AddTorsor (G : outParam Type*) (P : Type*) [AddGroup G] extends AddAction G P,
VSub G P where
[nonempty : Nonempty P]
vsub_vadd' : ∀ p₁ p₂ : P, (p₁ ... | Mathlib/Algebra/AddTorsor.lean | 98 | 100 | theorem vadd_right_cancel {g₁ g₂ : G} (p : P) (h : g₁ +ᵥ p = g₂ +ᵥ p) : g₁ = g₂ := by |
-- Porting note: vadd_vsub g₁ → vadd_vsub g₁ p
rw [← vadd_vsub g₁ p, h, vadd_vsub]
| 2 |
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.projective_space.basic from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
variable (K V : Type*) [DivisionRing K] [AddCommGroup V] [Module K V]
def projectivizationSetoid : Setoid { v : V // v ≠ 0 } :=
(MulA... | Mathlib/LinearAlgebra/Projectivization/Basic.lean | 142 | 144 | theorem finrank_submodule (v : ℙ K V) : finrank K v.submodule = 1 := by |
rw [submodule_eq]
exact finrank_span_singleton v.rep_nonzero
| 2 |
import Mathlib.Algebra.Group.Hom.Defs
#align_import algebra.group.ext from "leanprover-community/mathlib"@"e574b1a4e891376b0ef974b926da39e05da12a06"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
open Function
universe u
@[to_additive (attr := ext)]
theorem Monoid.ext {M : Type u} ⦃m₁ m₂ : Mo... | Mathlib/Algebra/Group/Ext.lean | 87 | 90 | theorem RightCancelMonoid.toMonoid_injective {M : Type u} :
Function.Injective (@RightCancelMonoid.toMonoid M) := by |
rintro @⟨@⟨⟩⟩ @⟨@⟨⟩⟩ h
congr <;> injection h
| 2 |
import Mathlib.Analysis.Normed.Group.SemiNormedGroupCat
import Mathlib.Analysis.Normed.Group.Quotient
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
#align_import analysis.normed.group.SemiNormedGroup.kernels from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
open CategoryTheory C... | Mathlib/Analysis/Normed/Group/SemiNormedGroupCat/Kernels.lean | 242 | 245 | theorem comp_explicitCokernelπ {X Y : SemiNormedGroupCat.{u}} (f : X ⟶ Y) :
f ≫ explicitCokernelπ f = 0 := by |
convert (cokernelCocone f).w WalkingParallelPairHom.left
simp
| 2 |
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 | 131 | 133 | theorem comp_left_bot (r : Rel α β) : (⊥ : Rel γ α) • r = ⊥ := by |
ext x y
simp [comp, Bot.bot]
| 2 |
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Set.Subsingleton
#align_import combinatorics.composition from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open List
variable {n : ℕ}
... | Mathlib/Combinatorics/Enumerative/Composition.lean | 187 | 189 | theorem length_le : c.length ≤ n := by |
conv_rhs => rw [← c.blocks_sum]
exact length_le_sum_of_one_le _ fun i hi => c.one_le_blocks hi
| 2 |
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Factorial.DoubleFactorial
#align_import ring_theory.polynomial.hermite.basic from "leanprover-community/mathlib"@"938d3db9c278f8a52c0f964a405806f0f2b09b74"
noncomputable section
open Polynomial
namespace P... | Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | 86 | 89 | theorem coeff_hermite_succ_succ (n k : ℕ) : coeff (hermite (n + 1)) (k + 1) =
coeff (hermite n) k - (k + 2) * coeff (hermite n) (k + 2) := by |
rw [hermite_succ, coeff_sub, coeff_X_mul, coeff_derivative, mul_comm]
norm_cast
| 2 |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
open Topology InnerProductSpace Set
noncomputable section
variable {𝕜 F : Type*} [RCLike 𝕜]
variable [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F]
variabl... | Mathlib/Analysis/Calculus/Gradient/Basic.lean | 118 | 121 | theorem DifferentiableAt.hasGradientAt (h : DifferentiableAt 𝕜 f x) :
HasGradientAt f (∇ f x) x := by |
rw [hasGradientAt_iff_hasFDerivAt, gradient, (toDual 𝕜 F).apply_symm_apply (fderiv 𝕜 f x)]
exact h.hasFDerivAt
| 2 |
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Data.Finite.Card
import Mathlib.GroupTheory.Finiteness
import Mathlib.GroupTheory.GroupAction.Quotient
#align_import group_theory.index from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
namespace Subgroup
open Ca... | Mathlib/GroupTheory/Index.lean | 146 | 148 | theorem relindex_inf_mul_relindex : H.relindex (K ⊓ L) * K.relindex L = (H ⊓ K).relindex L := by |
rw [← inf_relindex_right H (K ⊓ L), ← inf_relindex_right K L, ← inf_relindex_right (H ⊓ K) L,
inf_assoc, relindex_mul_relindex (H ⊓ (K ⊓ L)) (K ⊓ L) L inf_le_right inf_le_right]
| 2 |
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 | 74 | 76 | theorem gcdA_zero_left {s : ℕ} : gcdA 0 s = 0 := by |
unfold gcdA
rw [xgcd, xgcd_zero_left]
| 2 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Combinatorics.SimpleGraph.Density
import Mathlib.Data.Nat.Cast.Field
import Mathlib.Order.Partition.Equipartition
import Mathlib.SetTheory.Ordinal.Basic
#align_import combinatorics.simple_graph.regularity.uniform from "leanprover-community/mathlib"@"bf7ef0e83e5b... | Mathlib/Combinatorics/SimpleGraph/Regularity/Uniform.lean | 142 | 145 | theorem left_nonuniformWitnesses_card (h : ¬G.IsUniform ε s t) :
(s.card : 𝕜) * ε ≤ (G.nonuniformWitnesses ε s t).1.card := by |
rw [nonuniformWitnesses, dif_pos h]
exact (not_isUniform_iff.1 h).choose_spec.2.choose_spec.2.1
| 2 |
import Mathlib.Topology.FiberBundle.Trivialization
import Mathlib.Topology.Order.LeftRightNhds
#align_import topology.fiber_bundle.basic from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
variable {ι B F X : Type*} [TopologicalSpace X]
open TopologicalSpace Filter Set Bundle Topology
... | Mathlib/Topology/FiberBundle/Basic.lean | 474 | 477 | theorem mem_trivChange_source (i j : ι) (p : B × F) :
p ∈ (Z.trivChange i j).source ↔ p.1 ∈ Z.baseSet i ∩ Z.baseSet j := by |
erw [mem_prod]
simp
| 2 |
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Iterate
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.Algebra.InfiniteSum.Real
#align_import analysis.specific_limits.basic from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
n... | Mathlib/Analysis/SpecificLimits/Basic.lean | 51 | 54 | theorem NNReal.tendsto_inverse_atTop_nhds_zero_nat :
Tendsto (fun n : ℕ ↦ (n : ℝ≥0)⁻¹) atTop (𝓝 0) := by |
rw [← NNReal.tendsto_coe]
exact _root_.tendsto_inverse_atTop_nhds_zero_nat
| 2 |
import Mathlib.Analysis.Calculus.TangentCone
import Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics
#align_import analysis.calculus.fderiv.basic from "leanprover-community/mathlib"@"41bef4ae1254365bc190aee63b947674d2977f01"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open To... | Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 225 | 228 | theorem fderivWithin_zero_of_not_differentiableWithinAt (h : ¬DifferentiableWithinAt 𝕜 f s x) :
fderivWithin 𝕜 f s x = 0 := by |
have : ¬∃ f', HasFDerivWithinAt f f' s x := h
simp [fderivWithin, this]
| 2 |
import Mathlib.Algebra.Module.Submodule.Lattice
import Mathlib.Data.ZMod.Basic
import Mathlib.Order.OmegaCompletePartialOrder
variable {n : ℕ} {M M₁ : Type*}
abbrev AddCommMonoid.zmodModule [NeZero n] [AddCommMonoid M] (h : ∀ (x : M), n • x = 0) :
Module (ZMod n) M := by
have h_mod (c : ℕ) (x : M) : (c % n)... | Mathlib/Data/ZMod/Module.lean | 54 | 56 | theorem smul_mem (hx : x ∈ K) (c : ZMod n) : c • x ∈ K := by |
rw [← ZMod.intCast_zmod_cast c, ← zsmul_eq_smul_cast]
exact zsmul_mem hx (cast c)
| 2 |
import Mathlib.MeasureTheory.Measure.MeasureSpace
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
#align_import measure_theory.measure.open_pos from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology ENNReal MeasureTheory
open Set Function Filter
namespace Measur... | Mathlib/MeasureTheory/Measure/OpenPos.lean | 57 | 59 | theorem _root_.IsOpen.measure_eq_zero_iff (hU : IsOpen U) : μ U = 0 ↔ U = ∅ := by |
simpa only [not_lt, nonpos_iff_eq_zero, not_nonempty_iff_eq_empty] using
not_congr (hU.measure_pos_iff μ)
| 2 |
import Batteries.Classes.Order
namespace Batteries.PairingHeapImp
inductive Heap (α : Type u) where
| nil : Heap α
| node (a : α) (child sibling : Heap α) : Heap α
deriving Repr
def Heap.size : Heap α → Nat
| .nil => 0
| .node _ c s => c.size + 1 + s.size
def Heap.singleton (a : α) : Heap α := .... | .lake/packages/batteries/Batteries/Data/PairingHeap.lean | 90 | 93 | theorem Heap.noSibling_merge (le) (s₁ s₂ : Heap α) :
(s₁.merge le s₂).NoSibling := by |
unfold merge
(split <;> try split) <;> constructor
| 2 |
import Mathlib.Data.Fintype.Basic
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Defs
#align_import logic.equiv.fintype from "leanprover-community/mathlib"@"9407b03373c8cd201df99d6bc5514fc2db44054f"
namespace Equiv
variable {α β : Type*} [Finite α]
noncomputable def toCompl {p q : α → Prop} (e ... | Mathlib/Logic/Equiv/Fintype.lean | 132 | 135 | theorem extendSubtype_mem (e : { x // p x } ≃ { x // q x }) (x) (hx : p x) :
q (e.extendSubtype x) := by |
convert (e ⟨x, hx⟩).2
rw [e.extendSubtype_apply_of_mem _ hx]
| 2 |
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 support
v... | Mathlib/GroupTheory/Perm/Support.lean | 304 | 306 | theorem coe_support_eq_set_support (f : Perm α) : (f.support : Set α) = { x | f x ≠ x } := by |
ext
simp
| 2 |
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 | 290 | 292 | theorem linfty_opNNNorm_col (v : m → α) : ‖col v‖₊ = ‖v‖₊ := by |
rw [linfty_opNNNorm_def, Pi.nnnorm_def]
simp
| 2 |
import Mathlib.Analysis.NormedSpace.Multilinear.Curry
#align_import analysis.calculus.formal_multilinear_series from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open Set Fin Topology
-- Porting note: added explicit universes to fix compile
universe u u' v w x
... | Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean | 111 | 114 | theorem removeZero_of_pos (p : FormalMultilinearSeries 𝕜 E F) {n : ℕ} (h : 0 < n) :
p.removeZero n = p n := by |
rw [← Nat.succ_pred_eq_of_pos h]
rfl
| 2 |
import Mathlib.Order.Interval.Set.ProjIcc
import Mathlib.Topology.Algebra.Order.Field
import Mathlib.Topology.Bornology.Hom
import Mathlib.Topology.EMetricSpace.Lipschitz
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.Topology.MetricSpace.Bounded
#align_import topology.metric_space.lipschitz from "leanprove... | Mathlib/Topology/MetricSpace/Lipschitz.lean | 51 | 55 | theorem lipschitzOnWith_iff_dist_le_mul [PseudoMetricSpace α] [PseudoMetricSpace β] {K : ℝ≥0}
{s : Set α} {f : α → β} :
LipschitzOnWith K f s ↔ ∀ x ∈ s, ∀ y ∈ s, dist (f x) (f y) ≤ K * dist x y := by |
simp only [LipschitzOnWith, edist_nndist, dist_nndist]
norm_cast
| 2 |
import Mathlib.ModelTheory.Satisfiability
#align_import model_theory.types from "leanprover-community/mathlib"@"98bd247d933fb581ff37244a5998bd33d81dd46d"
set_option linter.uppercaseLean3 false
universe u v w w'
open Cardinal Set
open scoped Classical
open Cardinal FirstOrder
namespace FirstOrder
namespace La... | Mathlib/ModelTheory/Types.lean | 129 | 132 | theorem setOf_mem_eq_univ_iff (φ : L[[α]].Sentence) :
{ p : T.CompleteType α | φ ∈ p } = Set.univ ↔ (L.lhomWithConstants α).onTheory T ⊨ᵇ φ := by |
rw [models_iff_not_satisfiable, ← compl_empty_iff, compl_setOf_mem, ← setOf_subset_eq_empty_iff]
simp
| 2 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 92 | 95 | theorem HasDerivAt.smul (hc : HasDerivAt c c' x) (hf : HasDerivAt f f' x) :
HasDerivAt (fun y => c y • f y) (c x • f' + c' • f x) x := by |
rw [← hasDerivWithinAt_univ] at *
exact hc.smul hf
| 2 |
import Mathlib.Analysis.NormedSpace.PiTensorProduct.ProjectiveSeminorm
import Mathlib.LinearAlgebra.Isomorphisms
universe uι u𝕜 uE uF
variable {ι : Type uι} [Fintype ι]
variable {𝕜 : Type u𝕜} [NontriviallyNormedField 𝕜]
variable {E : ι → Type uE} [∀ i, SeminormedAddCommGroup (E i)] [∀ i, NormedSpace 𝕜 (E i)]
... | Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean | 116 | 119 | theorem toDualContinuousMultilinearMap_le_projectiveSeminorm (x : ⨂[𝕜] i, E i) :
‖toDualContinuousMultilinearMap F x‖ ≤ projectiveSeminorm x := by |
simp only [toDualContinuousMultilinearMap, LinearMap.coe_mk, AddHom.coe_mk]
apply LinearMap.mkContinuous_norm_le _ (apply_nonneg _ _)
| 2 |
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Sym.Sym2
namespace List
variable {α : Type*}
section Sym2
protected def sym2 : List α → List (Sym2 α)
| [] => []
| x :: xs => (x :: xs).map (fun y => s(x, y)) ++ xs.sym2
| Mathlib/Data/List/Sym.lean | 40 | 43 | theorem mem_sym2_cons_iff {x : α} {xs : List α} {z : Sym2 α} :
z ∈ (x :: xs).sym2 ↔ z = s(x, x) ∨ (∃ y, y ∈ xs ∧ z = s(x, y)) ∨ z ∈ xs.sym2 := by |
simp only [List.sym2, map_cons, cons_append, mem_cons, mem_append, mem_map]
simp only [eq_comm]
| 2 |
import Mathlib.Algebra.Algebra.RestrictScalars
import Mathlib.Analysis.NormedSpace.OperatorNorm.Basic
import Mathlib.Analysis.RCLike.Basic
#align_import analysis.normed_space.extend from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open RCLike
open ComplexConjugate
variable {𝕜 : Ty... | Mathlib/Analysis/NormedSpace/Extend.lean | 88 | 90 | theorem extendTo𝕜'_apply_re (fr : F →ₗ[ℝ] ℝ) (x : F) : re (fr.extendTo𝕜' x : 𝕜) = fr x := by |
simp only [extendTo𝕜'_apply, map_sub, zero_mul, mul_zero, sub_zero,
rclike_simps]
| 2 |
import Mathlib.Algebra.Group.Support
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Nat.Cast.Field
#align_import algebra.char_zero.lemmas from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
open Function Set
section AddMonoidWithOne
variable {α M : Type*} [AddMonoidWith... | Mathlib/Algebra/CharZero/Lemmas.lean | 166 | 168 | theorem one_eq_bit1 {a : R} : 1 = bit1 a ↔ a = 0 := by |
rw [eq_comm]
exact bit1_eq_one
| 2 |
import Mathlib.CategoryTheory.Limits.Preserves.Basic
#align_import category_theory.limits.preserves.limits from "leanprover-community/mathlib"@"e97cf15cd1aec9bd5c193b2ffac5a6dc9118912b"
universe w' w v₁ v₂ u₁ u₂
noncomputable section
namespace CategoryTheory
open Category Limits
variable {C : Type u₁} [Catego... | Mathlib/CategoryTheory/Limits/Preserves/Limits.lean | 142 | 146 | theorem preservesColimitsIso_inv_comp_desc (t : Cocone F) :
(preservesColimitIso G F).inv ≫ G.map (colimit.desc _ t) =
colimit.desc _ (G.mapCocone t) := by |
ext
simp [← G.map_comp]
| 2 |
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 | 613 | 615 | theorem frobenius_norm_row (v : m → α) : ‖row v‖ = ‖(WithLp.equiv 2 _).symm v‖ := by |
rw [frobenius_norm_def, Fintype.sum_unique, PiLp.norm_eq_of_L2, Real.sqrt_eq_rpow]
simp only [row_apply, Real.rpow_two, WithLp.equiv_symm_pi_apply]
| 2 |
import Mathlib.CategoryTheory.Equivalence
#align_import algebraic_topology.dold_kan.compatibility from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open CategoryTheory CategoryTheory.Category
namespace AlgebraicTopology
namespace DoldKan
namespace Compatibility
variable {A A' B B'... | Mathlib/AlgebraicTopology/DoldKan/Compatibility.lean | 86 | 88 | theorem equivalence₁CounitIso_eq : (equivalence₁ hF).counitIso = equivalence₁CounitIso hF := by |
ext Y
simp [equivalence₁, equivalence₀]
| 2 |
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 | 98 | 102 | theorem natDegree_mul_C_le (f : R[X]) (a : R) : (f * C a).natDegree ≤ f.natDegree :=
calc
(f * C a).natDegree ≤ f.natDegree + (C a).natDegree := natDegree_mul_le
_ = f.natDegree + 0 := by | rw [natDegree_C a]
_ = f.natDegree := add_zero _
| 2 |
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Regular.SMul
#align_import data.polynomial.monic from "leanprover-community/mathlib"@"cbdf7b565832144d024caa5a550117c6df0204a5"
noncomputable section
open Finset
open Polynomial
namespace Polynomial
universe u v y
variable {R : Type u} {S : Typ... | Mathlib/Algebra/Polynomial/Monic.lean | 117 | 125 | theorem Monic.mul (hp : Monic p) (hq : Monic q) : Monic (p * q) :=
letI := Classical.decEq R
if h0 : (0 : R) = 1 then
haveI := subsingleton_of_zero_eq_one h0
Subsingleton.elim _ _
else by
have : p.leadingCoeff * q.leadingCoeff ≠ 0 := by |
simp [Monic.def.1 hp, Monic.def.1 hq, Ne.symm h0]
rw [Monic.def, leadingCoeff_mul' this, Monic.def.1 hp, Monic.def.1 hq, one_mul]
| 2 |
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.Ideal.Prod
import Mathlib.RingTheory.Ideal.MinimalPrime
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.Topology.Sets.Closeds
import Mathlib.Topology.Sober
#a... | Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean | 147 | 149 | theorem zeroLocus_span (s : Set R) : zeroLocus (Ideal.span s : Set R) = zeroLocus s := by |
ext x
exact (Submodule.gi R R).gc s x.asIdeal
| 2 |
import Mathlib.Algebra.Order.Field.Power
import Mathlib.NumberTheory.Padics.PadicVal
#align_import number_theory.padics.padic_norm from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
def padicNorm (p : ℕ) (q : ℚ) : ℚ :=
if q = 0 then 0 else (p : ℚ) ^ (-padicValRat p q)
#align padic_n... | Mathlib/NumberTheory/Padics/PadicNorm.lean | 104 | 106 | theorem padicNorm_p_lt_one (hp : 1 < p) : padicNorm p p < 1 := by |
rw [padicNorm_p hp, inv_lt_one_iff]
exact mod_cast Or.inr hp
| 2 |
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Smooth.Basic
import Mathlib.RingTheory.Unramified.Basic
#align_import ring_theory.etale from "leanprover-community/mathlib"@"73f96237417835f148a1f7bc1ff55f67119b7166"
-- Porting note: added to make the syntax work below.
open scoped TensorProdu... | Mathlib/RingTheory/Etale/Basic.lean | 66 | 69 | theorem iff_unramified_and_smooth :
FormallyEtale R A ↔ FormallyUnramified R A ∧ FormallySmooth R A := by |
rw [formallyUnramified_iff, formallySmooth_iff, formallyEtale_iff]
simp_rw [← forall_and, Function.Bijective]
| 2 |
import Mathlib.Algebra.Polynomial.Mirror
import Mathlib.Analysis.Complex.Polynomial
#align_import data.polynomial.unit_trinomial from "leanprover-community/mathlib"@"302eab4f46abb63de520828de78c04cb0f9b5836"
namespace Polynomial
open scoped Polynomial
open Finset
section Semiring
variable {R : Type*} [Semirin... | Mathlib/Algebra/Polynomial/UnitTrinomial.lean | 61 | 64 | theorem trinomial_trailing_coeff' (hkm : k < m) (hmn : m < n) :
(trinomial k m n u v w).coeff k = u := by |
rw [trinomial_def, coeff_add, coeff_add, coeff_C_mul_X_pow, coeff_C_mul_X_pow, coeff_C_mul_X_pow,
if_pos rfl, if_neg hkm.ne, if_neg (hkm.trans hmn).ne, add_zero, add_zero]
| 2 |
import Mathlib.Topology.Instances.Real
import Mathlib.Order.Filter.Archimedean
#align_import analysis.subadditive from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open Set Filter Topology
def Subadditive (u : ℕ → ℝ) : Prop :=
∀ m n, u (m + n) ≤ u m + u n
#al... | Mathlib/Analysis/Subadditive.lean | 45 | 48 | theorem lim_le_div (hbdd : BddBelow (range fun n => u n / n)) {n : ℕ} (hn : n ≠ 0) :
h.lim ≤ u n / n := by |
rw [Subadditive.lim]
exact csInf_le (hbdd.mono <| image_subset_range _ _) ⟨n, hn.bot_lt, rfl⟩
| 2 |
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Data.Set.UnionLift
#align_import algebra.algebra.subalgebra.basic from "leanprover-community/mathlib"@"b915e9392ecb2a861e1e766f0e1df6ac481188ca"
namespace Subalgebra
open Algebra
variable {R A B : Type*} [CommSemiring R] [Semiring A] [Algebra R A] [... | Mathlib/Algebra/Algebra/Subalgebra/Directed.lean | 78 | 81 | theorem iSupLift_inclusion {i : ι} (x : K i) (h : K i ≤ T) :
iSupLift K dir f hf T hT (inclusion h x) = f i x := by |
dsimp [iSupLift, inclusion]
rw [Set.iUnionLift_inclusion]
| 2 |
import Mathlib.Logic.Encodable.Lattice
import Mathlib.MeasureTheory.MeasurableSpace.Defs
#align_import measure_theory.pi_system from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"
open MeasurableSpace Set
open scoped Classical
open MeasureTheory
def IsPiSystem {α} (C : Set (Set α)) :... | Mathlib/MeasureTheory/PiSystem.lean | 149 | 151 | theorem isPiSystem_image_Iic (s : Set α) : IsPiSystem (Iic '' s) := by |
rintro _ ⟨a, ha, rfl⟩ _ ⟨b, hb, rfl⟩ -
exact ⟨a ⊓ b, inf_ind a b ha hb, Iic_inter_Iic.symm⟩
| 2 |
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Ring.Defs
#align_import algebra.euclidean_domain.defs from "leanprover-community/mathlib"@"ee7b9f9a9ac2a8d9f04ea39bbfe6b1a3be053b38"
universe u
class EuclideanDomain (R : Type u) extends CommRing R, Nontrivial R ... | Mathlib/Algebra/EuclideanDomain/Defs.lean | 131 | 133 | theorem mod_add_div' (m k : R) : m % k + m / k * k = m := by |
rw [mul_comm]
exact mod_add_div _ _
| 2 |
import Mathlib.Algebra.Algebra.Spectrum
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.Nilpotent.Basic
#align_import linear_algebra.eigenspace.basic from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da1aa1"
universe u v... | Mathlib/LinearAlgebra/Eigenspace/Basic.lean | 98 | 101 | theorem hasEigenvalue_of_hasEigenvector {f : End R M} {μ : R} {x : M} (h : HasEigenvector f μ x) :
HasEigenvalue f μ := by |
rw [HasEigenvalue, Submodule.ne_bot_iff]
use x; exact h
| 2 |
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.GroupTheory.Coset
#align_import group_theory.quotient_group from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Function
open scope... | Mathlib/GroupTheory/QuotientGroup.lean | 149 | 152 | theorem eq_iff_div_mem {N : Subgroup G} [nN : N.Normal] {x y : G} :
(x : G ⧸ N) = y ↔ x / y ∈ N := by |
refine eq_comm.trans (QuotientGroup.eq.trans ?_)
rw [nN.mem_comm_iff, div_eq_mul_inv]
| 2 |
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Tactic.Common
#align_import data.nat.cast.field from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
namespace Nat
variable {α : Type*}
@[simp]
theorem cast_div [DivisionSemiring α] {m n : ℕ} (n_... | Mathlib/Data/Nat/Cast/Field.lean | 76 | 79 | theorem one_div_lt_one_div {n m : ℕ} (h : n < m) : 1 / ((m : α) + 1) < 1 / ((n : α) + 1) := by |
refine one_div_lt_one_div_of_lt ?_ ?_
· exact Nat.cast_add_one_pos _
· simpa
| 3 |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Tactic.NormNum.NatFib
import Mathlib.Tactic.NormNum.Prime
#align_import data.real.golden_ratio from "leanprover-community/mathlib"@"2196ab363eb097c008d449712... | Mathlib/Data/Real/GoldenRatio.lean | 98 | 101 | theorem goldConj_sq : ψ ^ 2 = ψ + 1 := by |
rw [goldenConj, ← sub_eq_zero]
ring_nf
rw [Real.sq_sqrt] <;> norm_num
| 3 |
import Mathlib.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
variable (K : Type*) [F... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 296 | 300 | theorem normAtPlace_apply (w : InfinitePlace K) (x : K) :
normAtPlace w (mixedEmbedding K x) = w x := by |
simp_rw [normAtPlace, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk, mixedEmbedding,
RingHom.prod_apply, Pi.ringHom_apply, norm_embedding_of_isReal, norm_embedding_eq, dite_eq_ite,
ite_id]
| 3 |
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.ProdSigma
#align_import model_theory.semantics from "leanprover-community/mathlib"@"d565b3df44619c1498326936be16f1a935df0728"
universe u v w u' v'
namespace FirstOrder
namespace Language
variable {L : Language.{u, v}} {... | Mathlib/ModelTheory/Semantics.lean | 109 | 113 | theorem realize_functions_apply₁ {f : L.Functions 1} {t : L.Term α} {v : α → M} :
(f.apply₁ t).realize v = funMap f ![t.realize v] := by |
rw [Functions.apply₁, Term.realize]
refine congr rfl (funext fun i => ?_)
simp only [Matrix.cons_val_fin_one]
| 3 |
import Mathlib.Topology.Order.IsLUB
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section ConditionallyCompleteLinearOrder
variable [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α]
[ConditionallyCompleteLinearOrder β] [Top... | Mathlib/Topology/Order/Monotone.lean | 75 | 79 | theorem Antitone.map_iInf_of_continuousAt' {ι : Sort*} [Nonempty ι] {f : α → β} {g : ι → α}
(Cf : ContinuousAt f (iInf g)) (Af : Antitone f)
(bdd : BddBelow (range g) := by | bddDefault) : f (⨅ i, g i) = ⨆ i, f (g i) := by
rw [iInf, Antitone.map_sInf_of_continuousAt' Cf Af (range_nonempty g) bdd, ← range_comp, iSup]
rfl
| 3 |
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Tactic.FinCases
#align_import linear_algebra.affine_space.combination from ... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 141 | 145 | theorem weightedVSubOfPoint_erase [DecidableEq ι] (w : ι → k) (p : ι → P) (i : ι) :
(s.erase i).weightedVSubOfPoint p (p i) w = s.weightedVSubOfPoint p (p i) w := by |
rw [weightedVSubOfPoint_apply, weightedVSubOfPoint_apply]
apply sum_erase
rw [vsub_self, smul_zero]
| 3 |
import Mathlib.Analysis.SpecialFunctions.Pow.Real
#align_import analysis.special_functions.pow.nnreal from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real NNReal ENNReal ComplexConjugate
open Finset Function Set
namespace NNReal
var... | Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 124 | 127 | theorem sqrt_eq_rpow (x : ℝ≥0) : sqrt x = x ^ (1 / (2 : ℝ)) := by |
refine NNReal.eq ?_
push_cast
exact Real.sqrt_eq_rpow x.1
| 3 |
import Mathlib.Algebra.MvPolynomial.PDeriv
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Polynomial.Pochhammer
#align_import ring_theory.polynomial.bernstein from "le... | Mathlib/RingTheory/Polynomial/Bernstein.lean | 134 | 138 | theorem derivative_succ (n ν : ℕ) : Polynomial.derivative (bernsteinPolynomial R n (ν + 1)) =
n * (bernsteinPolynomial R (n - 1) ν - bernsteinPolynomial R (n - 1) (ν + 1)) := by |
cases n
· simp [bernsteinPolynomial]
· rw [Nat.cast_succ]; apply derivative_succ_aux
| 3 |
import Mathlib.Algebra.ContinuedFractions.Basic
import Mathlib.Algebra.GroupWithZero.Basic
#align_import algebra.continued_fractions.translations from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
section WithDivisionRing
variable {K : Type*}... | Mathlib/Algebra/ContinuedFractions/Translations.lean | 155 | 159 | theorem first_continuant_eq {gp : Pair K} (zeroth_s_eq : g.s.get? 0 = some gp) :
g.continuants 1 = ⟨gp.b * g.h + gp.a, gp.b⟩ := by |
simp [nth_cont_eq_succ_nth_cont_aux]
-- Porting note (#10959): simp used to work here, but now it can't figure out that 1 + 1 = 2
convert second_continuant_aux_eq zeroth_s_eq
| 3 |
import Mathlib.Data.List.Basic
#align_import data.list.lattice from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
open Nat
namespace List
variable {α : Type*} {l l₁ l₂ : List α} {p : α → Prop} {a : α}
variable [DecidableEq α]
section BagInter
@[simp]
theorem nil_bagInt... | Mathlib/Data/List/Lattice.lean | 203 | 207 | theorem cons_bagInter_of_pos (l₁ : List α) (h : a ∈ l₂) :
(a :: l₁).bagInter l₂ = a :: l₁.bagInter (l₂.erase a) := by |
cases l₂
· exact if_pos h
· simp only [List.bagInter, if_pos (elem_eq_true_of_mem h)]
| 3 |
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 | 902 | 906 | theorem SeminormFamily.finset_sup_comp (q : SeminormFamily 𝕜₂ F ι) (s : Finset ι)
(f : E →ₛₗ[σ₁₂] F) : (s.sup q).comp f = s.sup (q.comp f) := by |
ext x
rw [Seminorm.comp_apply, Seminorm.finset_sup_apply, Seminorm.finset_sup_apply]
rfl
| 3 |
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
#align_import analysis.special_functions.pow.asymptotics from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
set_option linter.uppercaseLean3 false
noncomputable section
open scoped Classical
open Real Topology NNReal ENNReal Filter C... | Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | 279 | 283 | theorem IsLittleO.rpow (hr : 0 < r) (hg : 0 ≤ᶠ[l] g) (h : f =o[l] g) :
(fun x => f x ^ r) =o[l] fun x => g x ^ r := by |
refine .of_isBigOWith fun c hc ↦ ?_
rw [← rpow_inv_rpow hc.le hr.ne']
refine (h.forall_isBigOWith ?_).rpow ?_ ?_ hg <;> positivity
| 3 |
import Mathlib.Data.Finset.Pointwise
#align_import combinatorics.additive.e_transform from "leanprover-community/mathlib"@"207c92594599a06e7c134f8d00a030a83e6c7259"
open MulOpposite
open Pointwise
variable {α : Type*} [DecidableEq α]
namespace Finset
section CommGroup
variable [CommGroup α] (e : α) (x : F... | Mathlib/Combinatorics/Additive/ETransform.lean | 66 | 70 | theorem mulDysonETransform.card :
(mulDysonETransform e x).1.card + (mulDysonETransform e x).2.card = x.1.card + x.2.card := by |
dsimp
rw [← card_smul_finset e (_ ∩ _), smul_finset_inter, smul_inv_smul, inter_comm,
card_union_add_card_inter, card_smul_finset]
| 3 |
import Mathlib.Data.ZMod.Quotient
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Tactic.ByContra
import Mathlib.Tactic.Peel
#align_import group_... | Mathlib/GroupTheory/Exponent.lean | 94 | 97 | theorem _root_.MulOpposite.exponent : exponent (MulOpposite G) = exponent G := by |
simp only [Monoid.exponent, ExponentExists]
congr!
all_goals exact ⟨(op_injective <| · <| op ·), (unop_injective <| · <| unop ·)⟩
| 3 |
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 | 126 | 130 | theorem untrop_sum_eq_sInf_image [ConditionallyCompleteLinearOrder R] (s : Finset S)
(f : S → Tropical (WithTop R)) : untrop (∑ i ∈ s, f i) = sInf (untrop ∘ f '' s) := by |
rcases s.eq_empty_or_nonempty with (rfl | h)
· simp only [Set.image_empty, coe_empty, sum_empty, WithTop.sInf_empty, untrop_zero]
· rw [← inf'_eq_csInf_image _ h, inf'_eq_inf, Finset.untrop_sum']
| 3 |
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 | 131 | 136 | theorem Asymptotics.isLittleO_sum_range_of_tendsto_zero {α : Type*} [NormedAddCommGroup α]
{f : ℕ → α} (h : Tendsto f atTop (𝓝 0)) :
(fun n => ∑ i ∈ range n, f i) =o[atTop] fun n => (n : ℝ) := by |
have := ((isLittleO_one_iff ℝ).2 h).sum_range fun i => zero_le_one
simp only [sum_const, card_range, Nat.smul_one_eq_cast] at this
exact this tendsto_natCast_atTop_atTop
| 3 |
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 | 135 | 138 | theorem antidiagonal.fst_le {n : A} {kl : A × A} (hlk : kl ∈ antidiagonal n) : kl.1 ≤ n := by |
rw [le_iff_exists_add]
use kl.2
rwa [mem_antidiagonal, eq_comm] at hlk
| 3 |
import Batteries.Classes.Order
namespace Batteries.PairingHeapImp
inductive Heap (α : Type u) where
| nil : Heap α
| node (a : α) (child sibling : Heap α) : Heap α
deriving Repr
def Heap.size : Heap α → Nat
| .nil => 0
| .node _ c s => c.size + 1 + s.size
def Heap.singleton (a : α) : Heap α := .... | .lake/packages/batteries/Batteries/Data/PairingHeap.lean | 142 | 146 | theorem Heap.size_tail? {s : Heap α} (h : s.NoSibling) : s.tail? le = some s' →
s.size = s'.size + 1 := by |
simp only [Heap.tail?]; intro eq
match eq₂ : s.deleteMin le, eq with
| some (a, tl), rfl => exact size_deleteMin h eq₂
| 3 |
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated
open Function Set
open scoped ENNReal Classical
noncomputable section
variable {α β δ : Type*} [MeasurableSpace α] [MeasurableSpace β] {s : Set α} ... | Mathlib/MeasureTheory/Measure/Dirac.lean | 103 | 110 | theorem tsum_indicator_apply_singleton [Countable α] [MeasurableSingletonClass α] (μ : Measure α)
(s : Set α) (hs : MeasurableSet s) : (∑' x : α, s.indicator (fun x => μ {x}) x) = μ s :=
calc
(∑' x : α, s.indicator (fun x => μ {x}) x) =
Measure.sum (fun a => μ {a} • Measure.dirac a) s := by |
simp only [Measure.sum_apply _ hs, Measure.smul_apply, smul_eq_mul, Measure.dirac_apply,
Set.indicator_apply, mul_ite, Pi.one_apply, mul_one, mul_zero]
_ = μ s := by rw [μ.sum_smul_dirac]
| 3 |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex
#align_import analysis.special_functions.trigonometric.complex_deriv from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
noncomputable section
namespace Complex
open Set Filter
open scoped Real
| Mathlib/Analysis/SpecialFunctions/Trigonometric/ComplexDeriv.lean | 25 | 28 | theorem hasStrictDerivAt_tan {x : ℂ} (h : cos x ≠ 0) : HasStrictDerivAt tan (1 / cos x ^ 2) x := by |
convert (hasStrictDerivAt_sin x).div (hasStrictDerivAt_cos x) h using 1
rw_mod_cast [← sin_sq_add_cos_sq x]
ring
| 3 |
import Mathlib.Computability.NFA
#align_import computability.epsilon_NFA from "leanprover-community/mathlib"@"28aa996fc6fb4317f0083c4e6daf79878d81be33"
open Set
open Computability
-- "ε_NFA"
set_option linter.uppercaseLean3 false
universe u v
structure εNFA (α : Type u) (σ : Type v) where
step : σ → Opt... | Mathlib/Computability/EpsilonNFA.lean | 116 | 119 | theorem evalFrom_empty (x : List α) : M.evalFrom ∅ x = ∅ := by |
induction' x using List.reverseRecOn with x a ih
· rw [evalFrom_nil, εClosure_empty]
· rw [evalFrom_append_singleton, ih, stepSet_empty]
| 3 |
import Mathlib.Algebra.MvPolynomial.Monad
#align_import data.mv_polynomial.expand from "leanprover-community/mathlib"@"5da451b4c96b4c2e122c0325a7fce17d62ee46c6"
namespace MvPolynomial
variable {σ τ R S : Type*} [CommSemiring R] [CommSemiring S]
noncomputable def expand (p : ℕ) : MvPolynomial σ R →ₐ[R] MvPolyno... | Mathlib/Algebra/MvPolynomial/Expand.lean | 64 | 68 | theorem expand_comp_bind₁ (p : ℕ) (f : σ → MvPolynomial τ R) :
(expand p).comp (bind₁ f) = bind₁ fun i ↦ expand p (f i) := by |
apply algHom_ext
intro i
simp only [AlgHom.comp_apply, bind₁_X_right]
| 3 |
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factors
import Mathlib.Order.Interval.Finset.Nat
#align_import number_theory.divisors from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open scoped Classical
open Finset
namespace Nat
variable (n : ℕ)
d... | Mathlib/NumberTheory/Divisors.lean | 61 | 64 | theorem filter_dvd_eq_divisors (h : n ≠ 0) : (Finset.range n.succ).filter (· ∣ n) = n.divisors := by |
ext
simp only [divisors, mem_filter, mem_range, mem_Ico, and_congr_left_iff, iff_and_self]
exact fun ha _ => succ_le_iff.mpr (pos_of_dvd_of_pos ha h.bot_lt)
| 3 |
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 | 163 | 167 | theorem hasFDerivAt_stereoInvFunAux_comp_coe (v : E) :
HasFDerivAt (stereoInvFunAux v ∘ ((↑) : (ℝ ∙ v)ᗮ → E)) (ℝ ∙ v)ᗮ.subtypeL 0 := by |
have : HasFDerivAt (stereoInvFunAux v) (ContinuousLinearMap.id ℝ E) ((ℝ ∙ v)ᗮ.subtypeL 0) :=
hasFDerivAt_stereoInvFunAux v
convert this.comp (0 : (ℝ ∙ v)ᗮ) (by apply ContinuousLinearMap.hasFDerivAt)
| 3 |
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.Order.Atoms
#align_import category_theory.simple from "leanprover-community/mathlib"@"4ed0bcaef698011... | Mathlib/CategoryTheory/Simple.lean | 103 | 107 | theorem mono_to_simple_zero_of_not_iso {X Y : C} [Simple Y] {f : X ⟶ Y} [Mono f]
(w : IsIso f → False) : f = 0 := by |
classical
by_contra h
exact w (isIso_of_mono_of_nonzero h)
| 3 |
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