Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.57k | proof stringlengths 5 7.36k | hint bool 2
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import Mathlib.Order.Lattice
import Mathlib.Data.List.Sort
import Mathlib.Logic.Equiv.Fin
import Mathlib.Logic.Equiv.Functor
import Mathlib.Data.Fintype.Card
import Mathlib.Order.RelSeries
#align_import order.jordan_holder from "leanprover-community/mathlib"@"91288e351d51b3f0748f0a38faa7613fb0ae2ada"
universe u
... | Mathlib/Order/JordanHolder.lean | 102 | 106 | theorem isMaximal_inf_right_of_isMaximal_sup {x y : X} (hxz : IsMaximal x (x ⊔ y))
(hyz : IsMaximal y (x ⊔ y)) : IsMaximal (x ⊓ y) y := by
rw [inf_comm] |
rw [inf_comm]
rw [sup_comm] at hxz hyz
exact isMaximal_inf_left_of_isMaximal_sup hyz hxz
| true |
import Mathlib.CategoryTheory.Subobject.Limits
#align_import algebra.homology.image_to_kernel from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a148bf9ff"
universe v u w
open CategoryTheory CategoryTheory.Limits
variable {ι : Type*}
variable {V : Type u} [Category.{v} V] [HasZeroMorphisms V]
o... | Mathlib/Algebra/Homology/ImageToKernel.lean | 95 | 98 | theorem imageToKernel_zero_left [HasKernels V] [HasZeroObject V] {w} :
imageToKernel (0 : A ⟶ B) g w = 0 := by
ext |
ext
simp
| true |
import Mathlib.Algebra.CharP.Two
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Periodic
import Mathlib.Data.ZMod.Basic
import Mathlib.Tactic.Monotonicity
#align_import data.nat.totient from "leanprover-community/mathlib"@"5cc2dfdd3e92f340411acea4427d701dc7ed26f8"
open Finset
namespace Nat
... | Mathlib/Data/Nat/Totient.lean | 129 | 135 | theorem totient_even {n : ℕ} (hn : 2 < n) : Even n.totient := by
haveI : Fact (1 < n) := ⟨one_lt_two.trans hn⟩ |
haveI : Fact (1 < n) := ⟨one_lt_two.trans hn⟩
haveI : NeZero n := NeZero.of_gt hn
suffices 2 = orderOf (-1 : (ZMod n)ˣ) by
rw [← ZMod.card_units_eq_totient, even_iff_two_dvd, this]
exact orderOf_dvd_card
rw [← orderOf_units, Units.coe_neg_one, orderOf_neg_one, ringChar.eq (ZMod n) n, if_neg hn.ne']
| true |
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.Tactic.Linarith
#align_import combinatorics.simple_graph.acyclic from "leanprover-community/mathlib"@"b07688016d62f81d14508ff339ea3415558d6353"
universe u v
namespace SimpleGraph
open Walk
variable {V : Type u} (G : SimpleGraph V)
def IsAcy... | Mathlib/Combinatorics/SimpleGraph/Acyclic.lean | 134 | 154 | theorem isTree_iff_existsUnique_path :
G.IsTree ↔ Nonempty V ∧ ∀ v w : V, ∃! p : G.Walk v w, p.IsPath := by
classical |
classical
rw [isTree_iff, isAcyclic_iff_path_unique]
constructor
· rintro ⟨hc, hu⟩
refine ⟨hc.nonempty, ?_⟩
intro v w
let q := (hc v w).some.toPath
use q
simp only [true_and_iff, Path.isPath]
intro p hp
specialize hu ⟨p, hp⟩ q
exact Subtype.ext_iff.mp hu
· rintro ⟨hV, h⟩
r... | true |
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 | 141 | 143 | theorem copy_nil {u u'} (hu : u = u') : (Walk.nil : G.Walk u u).copy hu hu = Walk.nil := by
subst_vars |
subst_vars
rfl
| true |
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.Ideal.Basic
#align_import data.polynomial.induction from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
noncomputable section
open Finsupp Finset
namespace Polynomial
open Polynomial
universe u v w x y z
variable {R ... | Mathlib/Algebra/Polynomial/Induction.lean | 75 | 78 | theorem span_le_of_C_coeff_mem (cf : ∀ i : ℕ, C (f.coeff i) ∈ I) :
Ideal.span { g | ∃ i, g = C (f.coeff i) } ≤ I := by
simp only [@eq_comm _ _ (C _)] |
simp only [@eq_comm _ _ (C _)]
exact (Ideal.span_le.trans range_subset_iff).mpr cf
| true |
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 | 58 | 62 | theorem List.untrop_prod [AddMonoid R] (l : List (Tropical R)) :
untrop l.prod = List.sum (l.map untrop) := by
induction' l with hd tl IH |
induction' l with hd tl IH
· simp
· simp [← IH]
| true |
import Mathlib.GroupTheory.Solvable
import Mathlib.FieldTheory.PolynomialGaloisGroup
import Mathlib.RingTheory.RootsOfUnity.Basic
#align_import field_theory.abel_ruffini from "leanprover-community/mathlib"@"e3f4be1fcb5376c4948d7f095bec45350bfb9d1a"
noncomputable section
open scoped Classical Polynomial Intermedi... | Mathlib/FieldTheory/AbelRuffini.lean | 53 | 53 | theorem gal_X_sub_C_isSolvable (x : F) : IsSolvable (X - C x).Gal := by | infer_instance
| true |
import Mathlib.Analysis.Normed.Group.Quotient
import Mathlib.Topology.Instances.AddCircle
#align_import analysis.normed.group.add_circle from "leanprover-community/mathlib"@"084f76e20c88eae536222583331abd9468b08e1c"
noncomputable section
open Set
open Int hiding mem_zmultiples_iff
open AddSubgroup
namespace A... | Mathlib/Analysis/Normed/Group/AddCircle.lean | 71 | 75 | theorem norm_neg_period (x : ℝ) : ‖(x : AddCircle (-p))‖ = ‖(x : AddCircle p)‖ := by
suffices ‖(↑(-1 * x) : AddCircle (-1 * p))‖ = ‖(x : AddCircle p)‖ by |
suffices ‖(↑(-1 * x) : AddCircle (-1 * p))‖ = ‖(x : AddCircle p)‖ by
rw [← this, neg_one_mul]
simp
simp only [norm_coe_mul, abs_neg, abs_one, one_mul]
| true |
import Mathlib.Data.Int.Bitwise
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import linear_algebra.matrix.zpow from "leanprover-community/mathlib"@"03fda9112aa6708947da13944a19310684bfdfcb"
open Matrix
namespace Matrix
variable {n' : Type*} [Decidab... | Mathlib/LinearAlgebra/Matrix/ZPow.lean | 50 | 54 | theorem pow_sub' (A : M) {m n : ℕ} (ha : IsUnit A.det) (h : n ≤ m) :
A ^ (m - n) = A ^ m * (A ^ n)⁻¹ := by
rw [← tsub_add_cancel_of_le h, pow_add, Matrix.mul_assoc, mul_nonsing_inv, |
rw [← tsub_add_cancel_of_le h, pow_add, Matrix.mul_assoc, mul_nonsing_inv,
tsub_add_cancel_of_le h, Matrix.mul_one]
simpa using ha.pow n
| true |
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Nat
import Mathlib.Init.Data.Nat.Lemmas
#align_import data.nat.psub from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
namespace Nat
def ppred : ℕ → Option ℕ
| 0 => none
| n + 1 => some n
#align nat.ppred Nat.ppred
@... | Mathlib/Data/Nat/PSub.lean | 118 | 122 | theorem psub'_eq_psub (m n) : psub' m n = psub m n := by
rw [psub'] |
rw [psub']
split_ifs with h
· exact (psub_eq_sub h).symm
· exact (psub_eq_none.2 (not_le.1 h)).symm
| true |
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Exponent
#align_import group_theory.specific_groups.dihedral from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
inductive DihedralGroup (n : ℕ) : Type
| r : ZMod n → DihedralGroup n
| sr : ZMod n → DihedralGroup n
derivin... | Mathlib/GroupTheory/SpecificGroups/Dihedral.lean | 170 | 184 | theorem orderOf_r_one : orderOf (r 1 : DihedralGroup n) = n := by
rcases eq_zero_or_neZero n with (rfl | hn) |
rcases eq_zero_or_neZero n with (rfl | hn)
· rw [orderOf_eq_zero_iff']
intro n hn
rw [r_one_pow, one_def]
apply mt r.inj
simpa using hn.ne'
· apply (Nat.le_of_dvd (NeZero.pos n) <|
orderOf_dvd_of_pow_eq_one <| @r_one_pow_n n).lt_or_eq.resolve_left
intro h
have h1 : (r 1 : DihedralGr... | true |
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 | 116 | 125 | theorem controlled_closure_range_of_complete {f : NormedAddGroupHom G H} {K : Type*}
[SeminormedAddCommGroup K] {j : NormedAddGroupHom K H} (hj : ∀ x, ‖j x‖ = ‖x‖) {C ε : ℝ}
(hC : 0 < C) (hε : 0 < ε) (hyp : ∀ k, ∃ g, f g = j k ∧ ‖g‖ ≤ C * ‖k‖) :
f.SurjectiveOnWith j.range.topologicalClosure (C + ε) := by
... |
replace hyp : ∀ h ∈ j.range, ∃ g, f g = h ∧ ‖g‖ ≤ C * ‖h‖ := by
intro h h_in
rcases (j.mem_range _).mp h_in with ⟨k, rfl⟩
rw [hj]
exact hyp k
exact controlled_closure_of_complete hC hε hyp
| true |
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Conj
import Mathlib.CategoryTheory.Functor.ReflectsIso
#align_import category_theory.adjunction.reflective from "leanprover-community/mathlib"@"239d882c4fb58361ee8b3b39fb2091320edef10a"
universe v₁ v₂ v₃ u₁ u₂ u₃
noncomputable s... | Mathlib/CategoryTheory/Adjunction/Reflective.lean | 62 | 67 | theorem unit_obj_eq_map_unit [Reflective i] (X : C) :
(reflectorAdjunction i).unit.app (i.obj ((reflector i).obj X)) =
i.map ((reflector i).map ((reflectorAdjunction i).unit.app X)) := by
rw [← cancel_mono (i.map ((reflectorAdjunction i).counit.app ((reflector i).obj X))), |
rw [← cancel_mono (i.map ((reflectorAdjunction i).counit.app ((reflector i).obj X))),
← i.map_comp]
simp
| true |
import Mathlib.Init.Core
import Mathlib.RingTheory.Polynomial.Cyclotomic.Roots
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.FieldTheory.Galois
#align_import number_theory.cyclotomic.basic from "leanprover-community/mathlib"@"4b05d3f4f0601dca8abf99c4ec99187682ed0bba"
open Polynomial Algebra FiniteD... | Mathlib/NumberTheory/Cyclotomic/Basic.lean | 154 | 168 | theorem subsingleton_iff [Subsingleton B] : IsCyclotomicExtension S A B ↔ S = { } ∨ S = {1} := by
have : Subsingleton (Subalgebra A B) := inferInstance |
have : Subsingleton (Subalgebra A B) := inferInstance
constructor
· rintro ⟨hprim, -⟩
rw [← subset_singleton_iff_eq]
intro t ht
obtain ⟨ζ, hζ⟩ := hprim ht
rw [mem_singleton_iff, ← PNat.coe_eq_one_iff]
exact mod_cast hζ.unique (IsPrimitiveRoot.of_subsingleton ζ)
· rintro (rfl | rfl)
-- Porti... | true |
import Mathlib.Data.List.Chain
#align_import data.list.destutter from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
variable {α : Type*} (l : List α) (R : α → α → Prop) [DecidableRel R] {a b : α}
namespace List
@[simp]
theorem destutter'_nil : destutter' R a [] = [a] :=
rfl
#align ... | Mathlib/Data/List/Destutter.lean | 92 | 98 | theorem destutter'_is_chain' (a) : (l.destutter' R a).Chain' R := by
induction' l with b l hl generalizing a |
induction' l with b l hl generalizing a
· simp
rw [destutter']
split_ifs with h
· exact destutter'_is_chain R l h
· exact hl a
| true |
import Mathlib.Algebra.ContinuedFractions.Computation.Approximations
import Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating
import Mathlib.Data.Rat.Floor
#align_import algebra.continued_fractions.computation.terminates_iff_rat from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b3... | Mathlib/Algebra/ContinuedFractions/Computation/TerminatesIffRat.lean | 315 | 331 | theorem exists_nth_stream_eq_none_of_rat (q : ℚ) : ∃ n : ℕ, IntFractPair.stream q n = none := by
let fract_q_num := (Int.fract q).num; let n := fract_q_num.natAbs + 1 |
let fract_q_num := (Int.fract q).num; let n := fract_q_num.natAbs + 1
cases' stream_nth_eq : IntFractPair.stream q n with ifp
· use n, stream_nth_eq
· -- arrive at a contradiction since the numerator decreased num + 1 times but every fractional
-- value is nonnegative.
have ifp_fr_num_le_q_fr_num_sub_n... | true |
import Mathlib.Probability.IdentDistrib
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.Analysis.SpecificLimits.FloorPow
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
#align_import probability.strong_law from "leanprover-community/mathlib"@"f2ce60867... | Mathlib/Probability/StrongLaw.lean | 96 | 96 | theorem truncation_zero (f : α → ℝ) : truncation f 0 = 0 := by | simp [truncation]; rfl
| true |
import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
import Mathlib.Analysis.SpecialFunctions.PolarCoord
import Mathlib.Analysis.Convex.Complex
#align_import analysis.special_functions.gaussian from "leanprover-community/mathlib"@"7982767093ae38cba236487f9c9dd9cd99f63c16"
noncomputable section
open Real Set Measu... | Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean | 45 | 48 | theorem exp_neg_mul_sq_isLittleO_exp_neg {b : ℝ} (hb : 0 < b) :
(fun x : ℝ => exp (-b * x ^ 2)) =o[atTop] fun x : ℝ => exp (-x) := by
simp_rw [← rpow_two] |
simp_rw [← rpow_two]
exact exp_neg_mul_rpow_isLittleO_exp_neg hb one_lt_two
| true |
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 | 110 | 117 | theorem trinomial_mirror (hkm : k < m) (hmn : m < n) (hu : u ≠ 0) (hw : w ≠ 0) :
(trinomial k m n u v w).mirror = trinomial k (n - m + k) n w v u := by
rw [mirror, trinomial_natTrailingDegree hkm hmn hu, reverse, trinomial_natDegree hkm hmn hw, |
rw [mirror, trinomial_natTrailingDegree hkm hmn hu, reverse, trinomial_natDegree hkm hmn hw,
trinomial_def, reflect_add, reflect_add, reflect_C_mul_X_pow, reflect_C_mul_X_pow,
reflect_C_mul_X_pow, revAt_le (hkm.trans hmn).le, revAt_le hmn.le, revAt_le le_rfl, add_mul,
add_mul, mul_assoc, mul_assoc, mul_a... | true |
import Mathlib.Data.Nat.Choose.Central
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Multiplicity
#align_import data.nat.choose.factorization from "leanprover-community/mathlib"@"dc9db541168768af03fe228703e758e649afdbfc"
namespace Nat
variable {p n k : ℕ}
theorem factorization_choose_le_l... | Mathlib/Data/Nat/Choose/Factorization.lean | 106 | 110 | theorem factorization_choose_eq_zero_of_lt (h : n < p) : (choose n k).factorization p = 0 := by
by_cases hnk : n < k; · simp [choose_eq_zero_of_lt hnk] |
by_cases hnk : n < k; · simp [choose_eq_zero_of_lt hnk]
rw [choose_eq_factorial_div_factorial (le_of_not_lt hnk),
factorization_div (factorial_mul_factorial_dvd_factorial (le_of_not_lt hnk)), Finsupp.coe_tsub,
Pi.sub_apply, factorization_factorial_eq_zero_of_lt h, zero_tsub]
| true |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.Topology.Instances.RealVectorSpace
#align_import analysis.normed_space.add_torsor from "leanprover-community/mathlib"@... | Mathlib/Analysis/NormedSpace/AddTorsor.lean | 109 | 111 | theorem dist_lineMap_right (p₁ p₂ : P) (c : 𝕜) :
dist (lineMap p₁ p₂ c) p₂ = ‖1 - c‖ * dist p₁ p₂ := by |
simpa only [lineMap_apply_one, dist_eq_norm'] using dist_lineMap_lineMap p₁ p₂ c 1
| true |
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 | 230 | 232 | theorem tail_transpose (A : Matrix m' (Fin n.succ) α) : vecTail (of.symm Aᵀ) = (vecTail ∘ A)ᵀ := by
ext i j |
ext i j
rfl
| true |
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.Topology.Sheaves.SheafCondition.Sites
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.RingTheory.LocalProperties
#align_import algebraic_geometry.properties from "leanprover-community/mathlib"@"88... | Mathlib/AlgebraicGeometry/Properties.lean | 61 | 68 | theorem isReducedOfStalkIsReduced [∀ x : X.carrier, _root_.IsReduced (X.presheaf.stalk x)] :
IsReduced X := by
refine ⟨fun U => ⟨fun s hs => ?_⟩⟩ |
refine ⟨fun U => ⟨fun s hs => ?_⟩⟩
apply Presheaf.section_ext X.sheaf U s 0
intro x
rw [RingHom.map_zero]
change X.presheaf.germ x s = 0
exact (hs.map _).eq_zero
| true |
import Mathlib.RingTheory.Flat.Basic
import Mathlib.LinearAlgebra.TensorProduct.Vanishing
import Mathlib.Algebra.Module.FinitePresentation
universe u
variable {R M : Type u} [CommRing R] [AddCommGroup M] [Module R M]
open Classical DirectSum LinearMap TensorProduct Finsupp
open scoped BigOperators
namespace Modu... | Mathlib/RingTheory/Flat/EquationalCriterion.lean | 88 | 92 | theorem sum_smul_eq_zero_of_isTrivialRelation (h : IsTrivialRelation f x) :
∑ i, f i • x i = 0 := by
simpa using |
simpa using
congr_arg (TensorProduct.lid R M) <|
sum_tmul_eq_zero_of_vanishesTrivially R (isTrivialRelation_iff_vanishesTrivially.mp h)
| true |
import Mathlib.GroupTheory.Sylow
import Mathlib.GroupTheory.Transfer
#align_import group_theory.schur_zassenhaus from "leanprover-community/mathlib"@"d57133e49cf06508700ef69030cd099917e0f0de"
namespace Subgroup
section SchurZassenhausAbelian
open MulOpposite MulAction Subgroup.leftTransversals MemLeftTransversa... | Mathlib/GroupTheory/SchurZassenhaus.lean | 92 | 99 | theorem eq_one_of_smul_eq_one (hH : Nat.Coprime (Nat.card H) H.index) (α : H.QuotientDiff)
(h : H) : h • α = α → h = 1 :=
Quotient.inductionOn' α fun α hα =>
(powCoprime hH).injective <|
calc
h ^ H.index = diff (MonoidHom.id H) (op ((h⁻¹ : H) : G) • α) α := by
rw [← diff_inv, smul_diff... |
rw [← diff_inv, smul_diff', diff_self, one_mul, inv_pow, inv_inv]
_ = 1 ^ H.index := (Quotient.exact' hα).trans (one_pow H.index).symm
| true |
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.Logic.Equiv.Fin
#align_import category_theory.limits.... | Mathlib/CategoryTheory/Limits/Constructions/FiniteProductsOfBinaryProducts.lean | 106 | 110 | theorem hasFiniteProducts_of_has_binary_and_terminal : HasFiniteProducts C := by
refine ⟨fun n => ⟨fun K => ?_⟩⟩ |
refine ⟨fun n => ⟨fun K => ?_⟩⟩
letI := hasProduct_fin n fun n => K.obj ⟨n⟩
let that : (Discrete.functor fun n => K.obj ⟨n⟩) ≅ K := Discrete.natIso fun ⟨i⟩ => Iso.refl _
apply @hasLimitOfIso _ _ _ _ _ _ this that
| true |
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.Algebra.Module.ULift
#align_import ring_theory.is_tensor_product from "leanprover-community/mathlib"@"c4926d76bb9c5a4a62ed2f03d998081786132105"
universe u v₁ v₂ v₃ v₄
open TensorProduct
section IsTensorProduct
variable {R : Type*} [CommSemiring R]
va... | Mathlib/RingTheory/IsTensorProduct.lean | 115 | 127 | theorem IsTensorProduct.inductionOn (h : IsTensorProduct f) {C : M → Prop} (m : M) (h0 : C 0)
(htmul : ∀ x y, C (f x y)) (hadd : ∀ x y, C x → C y → C (x + y)) : C m := by
rw [← h.equiv.right_inv m] |
rw [← h.equiv.right_inv m]
generalize h.equiv.invFun m = y
change C (TensorProduct.lift f y)
induction y using TensorProduct.induction_on with
| zero => rwa [map_zero]
| tmul _ _ =>
rw [TensorProduct.lift.tmul]
apply htmul
| add _ _ _ _ =>
rw [map_add]
apply hadd <;> assumption
| true |
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Set.Finite
#align_import data.finset.n_ary from "leanprover-community/mathlib"@"eba7871095e834365616b5e43c8c7bb0b37058d0"
open Function Set
variable {α α' β β' γ γ' δ δ' ε ε' ζ ζ' ν : Type*}
namespace Finset
variable [DecidableEq α'] [DecidableEq β'] [Decidabl... | Mathlib/Data/Finset/NAry.lean | 43 | 44 | theorem mem_image₂ : c ∈ image₂ f s t ↔ ∃ a ∈ s, ∃ b ∈ t, f a b = c := by |
simp [image₂, and_assoc]
| true |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.GroupTheory.GroupAction.Pi
open Function Set
structure AddConstMap (G H : Type*) [Add G] [Add H] (a : G) (b : H) where
protected... | Mathlib/Algebra/AddConstMap/Basic.lean | 151 | 152 | theorem map_nat_add [AddCommMonoidWithOne G] [AddMonoidWithOne H] [AddConstMapClass F G H 1 1]
(f : F) (n : ℕ) (x : G) : f (↑n + x) = f x + n := by | simp
| true |
import Mathlib.CategoryTheory.Limits.Constructions.Pullbacks
import Mathlib.CategoryTheory.Preadditive.Biproducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
import Math... | Mathlib/CategoryTheory/Abelian/Basic.lean | 147 | 152 | theorem imageMonoFactorisation_e' {X Y : C} (f : X ⟶ Y) :
(imageMonoFactorisation f).e = cokernel.π _ ≫ Abelian.coimageImageComparison f := by
dsimp |
dsimp
ext
simp only [Abelian.coimageImageComparison, imageMonoFactorisation_e, Category.assoc,
cokernel.π_desc_assoc]
| true |
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Tactic.AdaptationNote
#align_import analysis.special_functions.log.deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
ope... | Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean | 42 | 47 | theorem hasStrictDerivAt_log (hx : x ≠ 0) : HasStrictDerivAt log x⁻¹ x := by
cases' hx.lt_or_lt with hx hx |
cases' hx.lt_or_lt with hx hx
· convert (hasStrictDerivAt_log_of_pos (neg_pos.mpr hx)).comp x (hasStrictDerivAt_neg x) using 1
· ext y; exact (log_neg_eq_log y).symm
· field_simp [hx.ne]
· exact hasStrictDerivAt_log_of_pos hx
| true |
import Mathlib.Data.List.Chain
import Mathlib.Data.List.Enum
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Pairwise
import Mathlib.Data.List.Zip
#align_import data.list.range from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
set_option autoImplicit true
universe u
open Nat... | Mathlib/Data/List/Range.lean | 79 | 80 | theorem pairwise_lt_range (n : ℕ) : Pairwise (· < ·) (range n) := by |
simp (config := {decide := true}) only [range_eq_range', pairwise_lt_range']
| true |
import Mathlib.Data.Multiset.Powerset
#align_import data.multiset.antidiagonal from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
assert_not_exists Ring
universe u
namespace Multiset
open List
variable {α β : Type*}
def antidiagonal (s : Multiset α) : Multiset (Multiset α × Multis... | Mathlib/Data/Multiset/Antidiagonal.lean | 103 | 105 | theorem card_antidiagonal (s : Multiset α) : card (antidiagonal s) = 2 ^ card s := by
have := card_powerset s |
have := card_powerset s
rwa [← antidiagonal_map_fst, card_map] at this
| true |
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Combinatorics.Hall.Basic
import Mathlib.Data.Fintype.BigOperators
import Mathlib.SetTheory.Cardinal.Finite
#align_import combinatorics.configuration from "leanprover-community/mathlib"@"d2d8742b0c21426362a9dacebc6005db895ca963"
open Finset
nam... | Mathlib/Combinatorics/Configuration.lean | 125 | 166 | theorem Nondegenerate.exists_injective_of_card_le [Nondegenerate P L] [Fintype P] [Fintype L]
(h : Fintype.card L ≤ Fintype.card P) : ∃ f : L → P, Function.Injective f ∧ ∀ l, f l ∉ l := by
classical |
classical
let t : L → Finset P := fun l => Set.toFinset { p | p ∉ l }
suffices ∀ s : Finset L, s.card ≤ (s.biUnion t).card by
-- Hall's marriage theorem
obtain ⟨f, hf1, hf2⟩ := (Finset.all_card_le_biUnion_card_iff_exists_injective t).mp this
exact ⟨f, hf1, fun l => Set.mem_toFinset.mp (hf2 ... | true |
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.Finset.Sym
import Mathlib.Data.Matrix.Basic
#align_import combinatorics.simple_graph.inc_matrix from "leanprover-community/mathlib"@"bb168510ef455e9280a152e7f31673cabd3d7496"
open Finset Matrix SimpleGraph Sym2
open Matrix
namespace SimpleGraph... | Mathlib/Combinatorics/SimpleGraph/IncMatrix.lean | 69 | 72 | theorem incMatrix_apply' [Zero R] [One R] [DecidableEq α] [DecidableRel G.Adj] {a : α}
{e : Sym2 α} : G.incMatrix R a e = if e ∈ G.incidenceSet a then 1 else 0 := by
unfold incMatrix Set.indicator |
unfold incMatrix Set.indicator
convert rfl
| true |
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open ENNReal MeasureTheory MeasureTheory.Measure Metric Set Filter TopologicalSpace Function
open scoped Topology ENNReal Convex
variable... | Mathlib/MeasureTheory/Integral/Average.lean | 149 | 150 | theorem setLaverage_congr (h : s =ᵐ[μ] t) : ⨍⁻ x in s, f x ∂μ = ⨍⁻ x in t, f x ∂μ := by |
simp only [setLaverage_eq, set_lintegral_congr h, measure_congr h]
| true |
import Mathlib.Probability.Notation
import Mathlib.Probability.Process.Stopping
#align_import probability.martingale.basic from "leanprover-community/mathlib"@"ba074af83b6cf54c3104e59402b39410ddbd6dca"
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory
namespace MeasureTheor... | Mathlib/Probability/Martingale/Basic.lean | 75 | 79 | theorem martingale_const_fun [OrderBot ι] (ℱ : Filtration ι m0) (μ : Measure Ω) [IsFiniteMeasure μ]
{f : Ω → E} (hf : StronglyMeasurable[ℱ ⊥] f) (hfint : Integrable f μ) :
Martingale (fun _ => f) ℱ μ := by
refine ⟨fun i => hf.mono <| ℱ.mono bot_le, fun i j _ => ?_⟩ |
refine ⟨fun i => hf.mono <| ℱ.mono bot_le, fun i j _ => ?_⟩
rw [condexp_of_stronglyMeasurable (ℱ.le _) (hf.mono <| ℱ.mono bot_le) hfint]
| true |
import Mathlib.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.oriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 80 | 81 | theorem right_ne_of_oangle_ne_zero {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ ≠ 0) : p₃ ≠ p₂ := by |
rw [← @vsub_ne_zero V]; exact o.right_ne_zero_of_oangle_ne_zero h
| true |
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.Data.Nat.Choose.Dvd
import Mathlib.Data.Nat.Prime
#align_import number_theory.primorial from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
open Finset
... | Mathlib/NumberTheory/Primorial.lean | 73 | 91 | theorem primorial_le_4_pow (n : ℕ) : n# ≤ 4 ^ n := by
induction' n using Nat.strong_induction_on with n ihn |
induction' n using Nat.strong_induction_on with n ihn
cases' n with n; · rfl
rcases n.even_or_odd with (⟨m, rfl⟩ | ho)
· rcases m.eq_zero_or_pos with (rfl | hm)
· decide
calc
(m + m + 1)# = (m + 1 + m)# := by rw [add_right_comm]
_ ≤ (m + 1)# * choose (m + 1 + m) (m + 1) := primorial_add_le ... | true |
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 | 126 | 128 | theorem comp_right_bot (r : Rel α β) : r • (⊥ : Rel β γ) = ⊥ := by
ext x y |
ext x y
simp [comp, Bot.bot]
| true |
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.matrix.basis from "leanprover-community/mathlib"@"6c263e4bfc2e6714de30f22178b4d0ca4d149a76"
noncomputable section
open LinearMap Matrix Set Submodule
open Matrix
section BasisToMatrix
variable {ι... | Mathlib/LinearAlgebra/Matrix/Basis.lean | 66 | 69 | theorem toMatrix_eq_toMatrix_constr [Fintype ι] [DecidableEq ι] (v : ι → M) :
e.toMatrix v = LinearMap.toMatrix e e (e.constr ℕ v) := by
ext |
ext
rw [Basis.toMatrix_apply, LinearMap.toMatrix_apply, Basis.constr_basis]
| true |
import Mathlib.RingTheory.LocalProperties
import Mathlib.RingTheory.Localization.InvSubmonoid
#align_import ring_theory.ring_hom.finite_type from "leanprover-community/mathlib"@"64fc7238fb41b1a4f12ff05e3d5edfa360dd768c"
namespace RingHom
open scoped Pointwise
theorem finiteType_stableUnderComposition : StableUn... | Mathlib/RingTheory/RingHom/FiniteType.lean | 29 | 35 | theorem finiteType_holdsForLocalizationAway : HoldsForLocalizationAway @FiniteType := by
introv R _ |
introv R _
suffices Algebra.FiniteType R S by
rw [RingHom.FiniteType]
convert this; ext;
rw [Algebra.smul_def]; rfl
exact IsLocalization.finiteType_of_monoid_fg (Submonoid.powers r) S
| true |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Constructions.Prod.Integral
import Mathlib.MeasureTheory.Function.LocallyIntegrable
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Group.Prod
import Mathlib.Measure... | Mathlib/Analysis/Convolution.lean | 131 | 136 | theorem _root_.HasCompactSupport.convolution_integrand_bound_right_of_subset
(hcg : HasCompactSupport g) (hg : Continuous g)
{x t : G} {s u : Set G} (hx : x ∈ s) (hu : -tsupport g + s ⊆ u) :
‖L (f t) (g (x - t))‖ ≤ u.indicator (fun t => ‖L‖ * ‖f t‖ * ⨆ i, ‖g i‖) t := by
refine convolution_integrand_bound_... |
refine convolution_integrand_bound_right_of_le_of_subset _ (fun i => ?_) hx hu
exact le_ciSup (hg.norm.bddAbove_range_of_hasCompactSupport hcg.norm) _
| true |
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.Dual
import Mathlib.Analysis.NormedSpace.Star.Basic
#align_import analysis.inner_product_space.dual from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped Classical
o... | Mathlib/Analysis/InnerProductSpace/Dual.lean | 157 | 159 | theorem toDual_symm_apply {x : E} {y : NormedSpace.Dual 𝕜 E} : ⟪(toDual 𝕜 E).symm y, x⟫ = y x := by
rw [← toDual_apply] |
rw [← toDual_apply]
simp only [LinearIsometryEquiv.apply_symm_apply]
| true |
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.Eval
namespace Polynomial
section MulActionWithZero
variable {R : Type*} [Semiring R] (r : R) (p : R[X]) {S : Type*} [AddCommMonoid S] [Pow S ℕ]
[Mu... | Mathlib/Algebra/Polynomial/Smeval.lean | 69 | 74 | theorem eval₂_eq_smeval (R : Type*) [Semiring R] {S : Type*} [Semiring S] (f : R →+* S) (p : R[X])
(x: S) : letI : Module R S := RingHom.toModule f
p.eval₂ f x = p.smeval x := by
letI : Module R S := RingHom.toModule f |
letI : Module R S := RingHom.toModule f
rw [smeval_eq_sum, eval₂_eq_sum]
rfl
| true |
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Measure.Haar.Unique
#align_import measure_theory.measure.lebesgue.integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open Set Filter MeasureTheory... | Mathlib/MeasureTheory/Measure/Lebesgue/Integral.lean | 85 | 91 | theorem integral_comp_neg_Iic {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]
(c : ℝ) (f : ℝ → E) : (∫ x in Iic c, f (-x)) = ∫ x in Ioi (-c), f x := by
have A : MeasurableEmbedding fun x : ℝ => -x := |
have A : MeasurableEmbedding fun x : ℝ => -x :=
(Homeomorph.neg ℝ).closedEmbedding.measurableEmbedding
have := MeasurableEmbedding.setIntegral_map (μ := volume) A f (Ici (-c))
rw [Measure.map_neg_eq_self (volume : Measure ℝ)] at this
simp_rw [← integral_Ici_eq_integral_Ioi, this, neg_preimage, preimage_neg... | true |
import Mathlib.Data.PFunctor.Univariate.Basic
#align_import data.pfunctor.univariate.M from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
universe u v w
open Nat Function
open List
variable (F : PFunctor.{u})
-- Porting note: the ♯ tactic is never used
-- local prefix:0 "♯" => cast... | Mathlib/Data/PFunctor/Univariate/M.lean | 101 | 115 | theorem truncate_eq_of_agree {n : ℕ} (x : CofixA F n) (y : CofixA F (succ n)) (h : Agree x y) :
truncate y = x := by
induction n <;> cases x <;> cases y |
induction n <;> cases x <;> cases y
· rfl
· -- cases' h with _ _ _ _ _ h₀ h₁
cases h
simp only [truncate, Function.comp, true_and_iff, eq_self_iff_true, heq_iff_eq]
-- Porting note: used to be `ext y`
rename_i n_ih a f y h₁
suffices (fun x => truncate (y x)) = f
by simp [this]
funex... | true |
import Mathlib.Combinatorics.SimpleGraph.DegreeSum
import Mathlib.Combinatorics.SimpleGraph.Subgraph
#align_import combinatorics.simple_graph.matching from "leanprover-community/mathlib"@"138448ae98f529ef34eeb61114191975ee2ca508"
universe u
namespace SimpleGraph
variable {V : Type u} {G : SimpleGraph V} (M : Su... | Mathlib/Combinatorics/SimpleGraph/Matching.lean | 122 | 124 | theorem isPerfectMatching_iff_forall_degree {M : Subgraph G} [∀ v, Fintype (M.neighborSet v)] :
M.IsPerfectMatching ↔ ∀ v, M.degree v = 1 := by |
simp [degree_eq_one_iff_unique_adj, isPerfectMatching_iff]
| true |
import Mathlib.CategoryTheory.Generator
import Mathlib.CategoryTheory.Limits.ConeCategory
import Mathlib.CategoryTheory.Limits.Constructions.WeaklyInitial
import Mathlib.CategoryTheory.Limits.FunctorCategory
import Mathlib.CategoryTheory.Subobject.Comma
#align_import category_theory.adjunction.adjoint_functor_theorem... | Mathlib/CategoryTheory/Adjunction/AdjointFunctorTheorems.lean | 69 | 75 | theorem solutionSetCondition_of_isRightAdjoint [G.IsRightAdjoint] : SolutionSetCondition G := by
intro A |
intro A
refine
⟨PUnit, fun _ => G.leftAdjoint.obj A, fun _ => (Adjunction.ofIsRightAdjoint G).unit.app A, ?_⟩
intro B h
refine ⟨PUnit.unit, ((Adjunction.ofIsRightAdjoint G).homEquiv _ _).symm h, ?_⟩
rw [← Adjunction.homEquiv_unit, Equiv.apply_symm_apply]
| true |
import Mathlib.MeasureTheory.Group.Measure
assert_not_exists NormedSpace
namespace MeasureTheory
open Measure TopologicalSpace
open scoped ENNReal
variable {G : Type*} [MeasurableSpace G] {μ : Measure G} {g : G}
section MeasurableMul
variable [Group G] [MeasurableMul G]
@[to_additive
"Translating a fu... | Mathlib/MeasureTheory/Group/LIntegral.lean | 54 | 56 | theorem lintegral_div_right_eq_self [IsMulRightInvariant μ] (f : G → ℝ≥0∞) (g : G) :
(∫⁻ x, f (x / g) ∂μ) = ∫⁻ x, f x ∂μ := by |
simp_rw [div_eq_mul_inv, lintegral_mul_right_eq_self f g⁻¹]
| true |
import Mathlib.Order.Ideal
import Mathlib.Data.Finset.Lattice
#align_import order.countable_dense_linear_order from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062a169c9"
noncomputable section
open scoped Classical
namespace Order
theorem exists_between_finsets {α : Type*} [LinearOrder α] [... | Mathlib/Order/CountableDenseLinearOrder.lean | 94 | 122 | theorem exists_across [DenselyOrdered β] [NoMinOrder β] [NoMaxOrder β] [Nonempty β]
(f : PartialIso α β) (a : α) :
∃ b : β, ∀ p ∈ f.val, cmp (Prod.fst p) a = cmp (Prod.snd p) b := by
by_cases h : ∃ b, (a, b) ∈ f.val |
by_cases h : ∃ b, (a, b) ∈ f.val
· cases' h with b hb
exact ⟨b, fun p hp ↦ f.prop _ hp _ hb⟩
have :
∀ x ∈ (f.val.filter fun p : α × β ↦ p.fst < a).image Prod.snd,
∀ y ∈ (f.val.filter fun p : α × β ↦ a < p.fst).image Prod.snd, x < y := by
intro x hx y hy
rw [Finset.mem_image] at hx hy
rc... | true |
import Batteries.Data.Fin.Basic
namespace Fin
attribute [norm_cast] val_last
protected theorem le_antisymm_iff {x y : Fin n} : x = y ↔ x ≤ y ∧ y ≤ x :=
Fin.ext_iff.trans Nat.le_antisymm_iff
protected theorem le_antisymm {x y : Fin n} (h1 : x ≤ y) (h2 : y ≤ x) : x = y :=
Fin.le_antisymm_iff.2 ⟨h1, h2⟩
@[simp... | .lake/packages/batteries/Batteries/Data/Fin/Lemmas.lean | 59 | 61 | theorem foldl_loop_lt (f : α → Fin n → α) (x) (h : m < n) :
foldl.loop n f x m = foldl.loop n f (f x ⟨m, h⟩) (m+1) := by |
rw [foldl.loop, dif_pos h]
| true |
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
section Preimage
variable {f : α → β} {g : β → γ... | Mathlib/Data/Set/Image.lean | 133 | 136 | theorem preimage_const (b : β) (s : Set β) [Decidable (b ∈ s)] :
(fun _ : α => b) ⁻¹' s = if b ∈ s then univ else ∅ := by
split_ifs with hb |
split_ifs with hb
exacts [preimage_const_of_mem hb, preimage_const_of_not_mem hb]
| true |
import Mathlib.Data.Set.Image
import Mathlib.Data.List.GetD
#align_import data.set.list from "leanprover-community/mathlib"@"2ec920d35348cb2d13ac0e1a2ad9df0fdf1a76b4"
open List
variable {α β : Type*} (l : List α)
namespace Set
theorem range_list_map (f : α → β) : range (map f) = { l | ∀ x ∈ l, x ∈ range f } :=... | Mathlib/Data/Set/List.lean | 44 | 48 | theorem range_list_get? : range l.get? = insert none (some '' { x | x ∈ l }) := by
rw [← range_list_get, ← range_comp] |
rw [← range_list_get, ← range_comp]
refine (range_subset_iff.2 fun n => ?_).antisymm (insert_subset_iff.2 ⟨?_, ?_⟩)
exacts [(le_or_lt l.length n).imp get?_eq_none.2 (fun hlt => ⟨⟨_, hlt⟩, (get?_eq_get hlt).symm⟩),
⟨_, get?_eq_none.2 le_rfl⟩, range_subset_iff.2 fun k => ⟨_, get?_eq_get _⟩]
| true |
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Group.Invertible.Defs
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Logic.Equiv.Defs
#align_import algebra.invertible from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
assert_not_exists MonoidWithZero
assert_not_ex... | Mathlib/Algebra/Group/Invertible/Basic.lean | 77 | 82 | theorem Commute.invOf_left [Monoid α] {a b : α} [Invertible b] (h : Commute b a) :
Commute (⅟ b) a :=
calc
⅟ b * a = ⅟ b * (a * b * ⅟ b) := by simp [mul_assoc] | simp [mul_assoc]
_ = ⅟ b * (b * a * ⅟ b) := by rw [h.eq]
_ = a * ⅟ b := by simp [mul_assoc]
| true |
import Mathlib.SetTheory.Game.Basic
import Mathlib.SetTheory.Ordinal.NaturalOps
#align_import set_theory.game.ordinal from "leanprover-community/mathlib"@"b90e72c7eebbe8de7c8293a80208ea2ba135c834"
universe u
open SetTheory PGame
open scoped NaturalOps PGame
namespace Ordinal
noncomputable def toPGame : Ordin... | Mathlib/SetTheory/Game/Ordinal.lean | 83 | 87 | theorem toPGame_moveLeft_hEq {o : Ordinal} :
have : IsWellOrder o.out.α (· < ·) := isWellOrder_out_lt o
HEq o.toPGame.moveLeft fun x : o.out.α => (typein (· < ·) x).toPGame := by
rw [toPGame] |
rw [toPGame]
rfl
| true |
import Mathlib.RingTheory.Nilpotent.Basic
import Mathlib.RingTheory.UniqueFactorizationDomain
#align_import algebra.squarefree from "leanprover-community/mathlib"@"00d163e35035c3577c1c79fa53b68de17781ffc1"
variable {R : Type*}
def Squarefree [Monoid R] (r : R) : Prop :=
∀ x : R, x * x ∣ r → IsUnit x
#align sq... | Mathlib/Algebra/Squarefree/Basic.lean | 120 | 126 | theorem squarefree_iff_multiplicity_le_one (r : R) :
Squarefree r ↔ ∀ x : R, multiplicity x r ≤ 1 ∨ IsUnit x := by
refine forall_congr' fun a => ?_ |
refine forall_congr' fun a => ?_
rw [← sq, pow_dvd_iff_le_multiplicity, or_iff_not_imp_left, not_le, imp_congr _ Iff.rfl]
norm_cast
rw [← one_add_one_eq_two]
simpa using PartENat.add_one_le_iff_lt (PartENat.natCast_ne_top 1)
| true |
import Mathlib.Tactic.Ring
import Mathlib.Data.PNat.Prime
#align_import data.pnat.xgcd from "leanprover-community/mathlib"@"6afc9b06856ad973f6a2619e3e8a0a8d537a58f2"
open Nat
namespace PNat
structure XgcdType where
wp : ℕ
x : ℕ
y : ℕ
zp : ℕ
ap : ℕ
bp : ℕ
deriving Inhabited
#alig... | Mathlib/Data/PNat/Xgcd.lean | 222 | 224 | theorem flip_isSpecial : (flip u).IsSpecial ↔ u.IsSpecial := by
dsimp [IsSpecial, flip] |
dsimp [IsSpecial, flip]
rw [mul_comm u.x, mul_comm u.zp, add_comm u.zp]
| true |
import Mathlib.Analysis.SpecialFunctions.Pow.Real
#align_import analysis.special_functions.log.monotone from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {x y : ℝ}
theorem log_mul_self_monotoneOn... | Mathlib/Analysis/SpecialFunctions/Log/Monotone.lean | 85 | 88 | theorem log_div_sqrt_antitoneOn : AntitoneOn (fun x : ℝ => log x / √x) { x | exp 2 ≤ x } := by
simp_rw [sqrt_eq_rpow] |
simp_rw [sqrt_eq_rpow]
convert @log_div_self_rpow_antitoneOn (1 / 2) (by norm_num)
norm_num
| true |
import Mathlib.CategoryTheory.Balanced
import Mathlib.CategoryTheory.LiftingProperties.Basic
#align_import category_theory.limits.shapes.strong_epi from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
universe v u
namespace CategoryTheory
variable {C : Type u} [Category.{v} C]
variable... | Mathlib/CategoryTheory/Limits/Shapes/StrongEpi.lean | 114 | 123 | theorem strongEpi_of_strongEpi [StrongEpi (f ≫ g)] : StrongEpi g :=
{ epi := epi_of_epi f g
llp := fun {X Y} z _ => by
constructor
intro u v sq
have h₀ : (f ≫ u) ≫ z = (f ≫ g) ≫ v := by simp only [Category.assoc, sq.w] | simp only [Category.assoc, sq.w]
exact
CommSq.HasLift.mk'
⟨(CommSq.mk h₀).lift, by
simp only [← cancel_mono z, Category.assoc, CommSq.fac_right, sq.w], by simp⟩ }
| true |
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology MeasureTheory
names... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | 159 | 165 | theorem condexp_undef (hf : ¬Integrable f μ) : μ[f|m] = 0 := by
by_cases hm : m ≤ m0 |
by_cases hm : m ≤ m0
swap; · rw [condexp_of_not_le hm]
by_cases hμm : SigmaFinite (μ.trim hm)
swap; · rw [condexp_of_not_sigmaFinite hm hμm]
haveI : SigmaFinite (μ.trim hm) := hμm
rw [condexp_of_sigmaFinite, if_neg hf]
| true |
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 | 113 | 116 | theorem charZero_of_expChar_one' [hq : ExpChar R 1] : CharZero R := by
cases hq |
cases hq
· assumption
· exact False.elim (CharP.char_ne_one R 1 rfl)
| true |
import Mathlib.Algebra.Quotient
import Mathlib.Algebra.Group.Subgroup.Actions
import Mathlib.Algebra.Group.Subgroup.MulOpposite
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.SetTheory.Cardinal.Finite
#align_import group_theory.coset from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce4... | Mathlib/GroupTheory/Coset.lean | 117 | 118 | theorem leftCoset_rightCoset (s : Set α) (a b : α) : op b • a • s = a • (op b • s) := by |
simp [← image_smul, (image_comp _ _ _).symm, Function.comp, mul_assoc]
| true |
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.BilinearMap
#align_import linear_algebra.sesquilinear_form from "leanprover-community/mathlib"@"87c54600fe3cdc7d32ff5b50873ac724d86aef8d"
variable {R R₁ R₂ R₃ M M₁ M₂ M₃... | Mathlib/LinearAlgebra/SesquilinearForm.lean | 64 | 66 | theorem isOrtho_zero_left (B : M₁ →ₛₗ[I₁] M₂ →ₛₗ[I₂] M) (x) : IsOrtho B (0 : M₁) x := by
dsimp only [IsOrtho] |
dsimp only [IsOrtho]
rw [map_zero B, zero_apply]
| true |
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 | 87 | 92 | theorem map_eq_sum [Countable β] [MeasurableSingletonClass β] (μ : Measure α) (f : α → β)
(hf : Measurable f) : μ.map f = sum fun b : β => μ (f ⁻¹' {b}) • dirac b := by
ext s |
ext s
have : ∀ y ∈ s, MeasurableSet (f ⁻¹' {y}) := fun y _ => hf (measurableSet_singleton _)
simp [← tsum_measure_preimage_singleton (to_countable s) this, *,
tsum_subtype s fun b => μ (f ⁻¹' {b}), ← indicator_mul_right s fun b => μ (f ⁻¹' {b})]
| true |
import Mathlib.Analysis.Quaternion
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Series
#align_import analysis.normed_space.quaternion_exponential from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open scoped Quaternion Nat
open... | Mathlib/Analysis/NormedSpace/QuaternionExponential.lean | 98 | 103 | theorem exp_of_re_eq_zero (q : Quaternion ℝ) (hq : q.re = 0) :
exp ℝ q = ↑(Real.cos ‖q‖) + (Real.sin ‖q‖ / ‖q‖) • q := by
rw [exp_eq_tsum] |
rw [exp_eq_tsum]
refine HasSum.tsum_eq ?_
simp_rw [← expSeries_apply_eq]
exact hasSum_expSeries_of_imaginary hq (Real.hasSum_cos _) (Real.hasSum_sin _)
| true |
import Mathlib.Data.List.Join
#align_import data.list.permutation from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
-- Make sure we don't import algebra
assert_not_exists Monoid
open Nat
variable {α β : Type*}
namespace List
theorem permutationsAux2_fst (t : α) (ts : List α) (r : L... | Mathlib/Data/List/Permutation.lean | 69 | 73 | theorem permutationsAux2_snd_cons (t : α) (ts : List α) (r : List β) (y : α) (ys : List α)
(f : List α → β) :
(permutationsAux2 t ts r (y :: ys) f).2 =
f (t :: y :: ys ++ ts) :: (permutationsAux2 t ts r ys fun x : List α => f (y :: x)).2 := by |
simp [permutationsAux2, permutationsAux2_fst t _ _ ys]
| true |
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Calculus.Dslope
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Analysis.Analytic.Uniqueness
#align_import analysis.analytic.isolated_zeros from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
open sco... | Mathlib/Analysis/Analytic/IsolatedZeros.lean | 69 | 80 | theorem has_fpower_series_dslope_fslope (hp : HasFPowerSeriesAt f p z₀) :
HasFPowerSeriesAt (dslope f z₀) p.fslope z₀ := by
have hpd : deriv f z₀ = p.coeff 1 := hp.deriv |
have hpd : deriv f z₀ = p.coeff 1 := hp.deriv
have hp0 : p.coeff 0 = f z₀ := hp.coeff_zero 1
simp only [hasFPowerSeriesAt_iff, apply_eq_pow_smul_coeff, coeff_fslope] at hp ⊢
refine hp.mono fun x hx => ?_
by_cases h : x = 0
· convert hasSum_single (α := E) 0 _ <;> intros <;> simp [*]
· have hxx : ∀ n : ℕ,... | true |
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 | 113 | 114 | theorem asAlgebraHom_of (g : G) : asAlgebraHom ρ (of k G g) = ρ g := by |
simp only [MonoidAlgebra.of_apply, asAlgebraHom_single, one_smul]
| true |
import Mathlib.Data.Finset.Sum
import Mathlib.Data.Sum.Order
import Mathlib.Order.Interval.Finset.Defs
#align_import data.sum.interval from "leanprover-community/mathlib"@"48a058d7e39a80ed56858505719a0b2197900999"
open Function Sum
namespace Finset
variable {α₁ α₂ β₁ β₂ γ₁ γ₂ : Type*}
section SumLift₂
variabl... | Mathlib/Data/Sum/Interval.lean | 76 | 88 | theorem sumLift₂_eq_empty :
sumLift₂ f g a b = ∅ ↔
(∀ a₁ b₁, a = inl a₁ → b = inl b₁ → f a₁ b₁ = ∅) ∧
∀ a₂ b₂, a = inr a₂ → b = inr b₂ → g a₂ b₂ = ∅ := by
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩ |
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
· constructor <;>
· rintro a b rfl rfl
exact map_eq_empty.1 h
cases a <;> cases b
· exact map_eq_empty.2 (h.1 _ _ rfl rfl)
· rfl
· rfl
· exact map_eq_empty.2 (h.2 _ _ rfl rfl)
| true |
import Mathlib.RingTheory.WittVector.Basic
import Mathlib.RingTheory.WittVector.IsPoly
#align_import ring_theory.witt_vector.verschiebung from "leanprover-community/mathlib"@"32b08ef840dd25ca2e47e035c5da03ce16d2dc3c"
namespace WittVector
open MvPolynomial
variable {p : ℕ} {R S : Type*} [hp : Fact p.Prime] [Comm... | Mathlib/RingTheory/WittVector/Verschiebung.lean | 42 | 44 | theorem verschiebungFun_coeff (x : 𝕎 R) (n : ℕ) :
(verschiebungFun x).coeff n = if n = 0 then 0 else x.coeff (n - 1) := by |
simp only [verschiebungFun, ge_iff_le]
| true |
import Mathlib.Algebra.Polynomial.Degree.Definitions
#align_import ring_theory.polynomial.opposites from "leanprover-community/mathlib"@"63417e01fbc711beaf25fa73b6edb395c0cfddd0"
open Polynomial
open Polynomial MulOpposite
variable {R : Type*} [Semiring R]
noncomputable section
namespace Polynomial
def opRi... | Mathlib/RingTheory/Polynomial/Opposites.lean | 95 | 99 | theorem coeff_opRingEquiv (p : R[X]ᵐᵒᵖ) (n : ℕ) :
(opRingEquiv R p).coeff n = op ((unop p).coeff n) := by
induction' p using MulOpposite.rec' with p |
induction' p using MulOpposite.rec' with p
cases p
rfl
| true |
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.Common
#align_import data.seq.computation from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
open Function
universe u v w
def Computation (α : Type u) : Type u :=
{ f : Stream' (Option α) // ∀ ⦃n a⦄, f n = some a → f (n + 1) = ... | Mathlib/Data/Seq/Computation.lean | 114 | 123 | theorem destruct_eq_pure {s : Computation α} {a : α} : destruct s = Sum.inl a → s = pure a := by
dsimp [destruct] |
dsimp [destruct]
induction' f0 : s.1 0 with _ <;> intro h
· contradiction
· apply Subtype.eq
funext n
induction' n with n IH
· injection h with h'
rwa [h'] at f0
· exact s.2 IH
| true |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Topology.Algebra.Module.FiniteDimension
variable {𝕜 E A B : Type*} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedSpace 𝕜 E]
[CommSemiring A] {z : E} {... | Mathlib/Analysis/Analytic/Polynomial.lean | 26 | 32 | theorem AnalyticAt.aeval_polynomial (hf : AnalyticAt 𝕜 f z) (p : A[X]) :
AnalyticAt 𝕜 (fun x ↦ aeval (f x) p) z := by
refine p.induction_on (fun k ↦ ?_) (fun p q hp hq ↦ ?_) fun p i hp ↦ ?_ |
refine p.induction_on (fun k ↦ ?_) (fun p q hp hq ↦ ?_) fun p i hp ↦ ?_
· simp_rw [aeval_C]; apply analyticAt_const
· simp_rw [aeval_add]; exact hp.add hq
· convert hp.mul hf
simp_rw [pow_succ, aeval_mul, ← mul_assoc, aeval_X]
| true |
import Mathlib.CategoryTheory.Filtered.Connected
import Mathlib.CategoryTheory.Limits.TypesFiltered
import Mathlib.CategoryTheory.Limits.Final
universe v₁ v₂ u₁ u₂
namespace CategoryTheory
open CategoryTheory.Limits CategoryTheory.Functor Opposite
section ArbitraryUniverses
variable {C : Type u₁} [Category.{v₁}... | Mathlib/CategoryTheory/Filtered/Final.lean | 56 | 72 | theorem isFiltered_structuredArrow_of_isFiltered_of_exists [IsFilteredOrEmpty C]
(h₁ : ∀ d, ∃ c, Nonempty (d ⟶ F.obj c)) (h₂ : ∀ {d : D} {c : C} (s s' : d ⟶ F.obj c),
∃ (c' : C) (t : c ⟶ c'), s ≫ F.map t = s' ≫ F.map t) (d : D) :
IsFiltered (StructuredArrow d F) := by
have : Nonempty (StructuredArrow d ... |
have : Nonempty (StructuredArrow d F) := by
obtain ⟨c, ⟨f⟩⟩ := h₁ d
exact ⟨.mk f⟩
suffices IsFilteredOrEmpty (StructuredArrow d F) from IsFiltered.mk
refine ⟨fun f g => ?_, fun f g η μ => ?_⟩
· obtain ⟨c, ⟨t, ht⟩⟩ := h₂ (f.hom ≫ F.map (IsFiltered.leftToMax f.right g.right))
(g.hom ≫ F.map (IsFi... | true |
import Mathlib.Topology.MetricSpace.PseudoMetric
#align_import topology.metric_space.basic from "leanprover-community/mathlib"@"c8f305514e0d47dfaa710f5a52f0d21b588e6328"
open Set Filter Bornology
open scoped NNReal Uniformity
universe u v w
variable {α : Type u} {β : Type v} {X ι : Type*}
variable [PseudoMetricS... | Mathlib/Topology/MetricSpace/Basic.lean | 102 | 103 | theorem nndist_eq_zero {x y : γ} : nndist x y = 0 ↔ x = y := by |
simp only [← NNReal.eq_iff, ← dist_nndist, imp_self, NNReal.coe_zero, dist_eq_zero]
| false |
import Mathlib.RingTheory.Ideal.Operations
#align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74"
assert_not_exists Basis -- See `RingTheory.Ideal.Basis`
assert_not_exists Submodule.hasQuotient -- See `RingTheory.Ideal.QuotientOperations`
universe... | Mathlib/RingTheory/Ideal/Maps.lean | 90 | 95 | theorem map_le_comap_of_inv_on (g : G) (I : Ideal R) (hf : Set.LeftInvOn g f I) :
I.map f ≤ I.comap g := by |
refine Ideal.span_le.2 ?_
rintro x ⟨x, hx, rfl⟩
rw [SetLike.mem_coe, mem_comap, hf hx]
exact hx
| false |
import Mathlib.Topology.Algebra.FilterBasis
import Mathlib.Topology.Algebra.UniformGroup
#align_import topology.algebra.uniform_filter_basis from "leanprover-community/mathlib"@"531db2ef0fdddf8b3c8dcdcd87138fe969e1a81a"
open uniformity Filter
open Filter
namespace AddGroupFilterBasis
variable {G : Type*} [AddC... | Mathlib/Topology/Algebra/UniformFilterBasis.lean | 42 | 51 | theorem cauchy_iff {F : Filter G} :
@Cauchy G B.uniformSpace F ↔
F.NeBot ∧ ∀ U ∈ B, ∃ M ∈ F, ∀ᵉ (x ∈ M) (y ∈ M), y - x ∈ U := by |
letI := B.uniformSpace
haveI := B.uniformAddGroup
suffices F ×ˢ F ≤ uniformity G ↔ ∀ U ∈ B, ∃ M ∈ F, ∀ᵉ (x ∈ M) (y ∈ M), y - x ∈ U by
constructor <;> rintro ⟨h', h⟩ <;> refine ⟨h', ?_⟩ <;> [rwa [← this]; rwa [this]]
rw [uniformity_eq_comap_nhds_zero G, ← map_le_iff_le_comap]
change Tendsto _ _ _ ↔ _
si... | false |
import Mathlib.Algebra.CharP.Two
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.GroupTheory.SpecificGroups.Cyclic
import Mathlib.NumberTheory.Divisors
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Tactic.Zify
#align_import rin... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 98 | 98 | theorem rootsOfUnity_one (M : Type*) [CommMonoid M] : rootsOfUnity 1 M = ⊥ := by | ext; simp
| false |
import Mathlib.Algebra.Ring.Defs
import Mathlib.Data.Rat.Init
#align_import algebra.field.defs from "leanprover-community/mathlib"@"2651125b48fc5c170ab1111afd0817c903b1fc6c"
-- `NeZero` should not be needed in the basic algebraic hierarchy.
assert_not_exists NeZero
-- Check that we have not imported `Mathlib.Tact... | Mathlib/Algebra/Field/Defs.lean | 226 | 227 | theorem smul_one_eq_cast (A : Type*) [DivisionRing A] (m : ℚ) : m • (1 : A) = ↑m := by |
rw [Rat.smul_def, mul_one]
| false |
import Mathlib.Analysis.InnerProductSpace.Dual
#align_import analysis.inner_product_space.lax_milgram from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open RCLike LinearMap ContinuousLinearMap InnerProductSpace
open LinearMap (ker range)
open RealInnerProduct... | Mathlib/Analysis/InnerProductSpace/LaxMilgram.lean | 80 | 82 | theorem isClosed_range (coercive : IsCoercive B) : IsClosed (range B♯ : Set V) := by |
rcases coercive.antilipschitz with ⟨_, _, antilipschitz⟩
exact antilipschitz.isClosed_range B♯.uniformContinuous
| false |
import Mathlib.Algebra.IsPrimePow
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.Tactic.WLOG
#align_import set_theory.cardinal.divisibility from "leanprover-community/mathlib"@"ea050b44c0f9aba9d16a948c7cc7d2e7c8493567"
namespace Cardinal
open Cardinal
universe u
variable {a b : Cardinal.{u}} {n m : ℕ... | Mathlib/SetTheory/Cardinal/Divisibility.lean | 92 | 96 | theorem not_irreducible_of_aleph0_le (ha : ℵ₀ ≤ a) : ¬Irreducible a := by |
rw [irreducible_iff, not_and_or]
refine Or.inr fun h => ?_
simpa [mul_aleph0_eq ha, isUnit_iff, (one_lt_aleph0.trans_le ha).ne', one_lt_aleph0.ne'] using
h a ℵ₀
| false |
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 65 | 69 | theorem oangle_add_left_eq_arcsin_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
o.oangle (x + y) y = Real.arcsin (‖x‖ / ‖x + y‖) := by |
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
rw [add_comm]
exact (-o).oangle_add_right_eq_arcsin_of_oangle_eq_pi_div_two h
| false |
import Mathlib.Algebra.Associated
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.Data.Nat.Lattice
import Mathlib.RingTheory.Nilpotent.Defs
#align_import ring_th... | Mathlib/RingTheory/Nilpotent/Basic.lean | 95 | 97 | theorem zero_isRadical_iff [MonoidWithZero R] : IsRadical (0 : R) ↔ IsReduced R := by |
simp_rw [isReduced_iff, IsNilpotent, exists_imp, ← zero_dvd_iff]
exact forall_swap
| false |
import Mathlib.Deprecated.Group
#align_import deprecated.ring from "leanprover-community/mathlib"@"5a3e819569b0f12cbec59d740a2613018e7b8eec"
universe u v w
variable {α : Type u}
structure IsSemiringHom {α : Type u} {β : Type v} [Semiring α] [Semiring β] (f : α → β) : Prop where
map_zero : f 0 = 0
map... | Mathlib/Deprecated/Ring.lean | 124 | 127 | theorem comp (hf : IsRingHom f) {γ} [Ring γ] {g : β → γ} (hg : IsRingHom g) : IsRingHom (g ∘ f) :=
{ map_add := fun x y => by simp only [Function.comp_apply, map_add hf, map_add hg]
map_mul := fun x y => by simp only [Function.comp_apply, map_mul hf, map_mul hg]
map_one := by | simp only [Function.comp_apply, map_one hf, map_one hg] }
| false |
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 | 162 | 167 | theorem exists_dist_eq (x z : E) {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b = 1) :
∃ y, dist x y = b * dist x z ∧ dist y z = a * dist x z := by |
use a • x + b • z
nth_rw 1 [← one_smul ℝ x]
nth_rw 4 [← one_smul ℝ z]
simp [dist_eq_norm, ← hab, add_smul, ← smul_sub, norm_smul_of_nonneg, ha, hb]
| false |
import Mathlib.Data.Int.Order.Units
import Mathlib.Data.ZMod.IntUnitsPower
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
import Mathlib.Algebra.DirectSum.Algebra
suppress_compilation
open scoped TensorProduct DirectSum
variable {R ι A B : Type*}
namespace Tens... | Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean | 137 | 145 | theorem gradedComm_of_zero_tmul (a : 𝒜 0) (b : ⨁ i, ℬ i) :
gradedComm R 𝒜 ℬ (lof R _ 𝒜 0 a ⊗ₜ b) = b ⊗ₜ lof R _ 𝒜 _ a := by |
suffices
(gradedComm R 𝒜 ℬ).toLinearMap ∘ₗ (TensorProduct.mk R (⨁ i, 𝒜 i) (⨁ i, ℬ i)) (lof R _ 𝒜 0 a) =
(TensorProduct.mk R _ _).flip (lof R _ 𝒜 0 a) from
DFunLike.congr_fun this b
ext i b
dsimp
rw [gradedComm_of_tmul_of, mul_zero, uzpow_zero, one_smul]
| false |
import Mathlib.Data.Finset.Lattice
#align_import combinatorics.set_family.compression.down from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
variable {α : Type*} [DecidableEq α] {𝒜 ℬ : Finset (Finset α)} {s : Finset α} {a : α}
open Finset
-- The namespace is here to distinguish fro... | Mathlib/Combinatorics/SetFamily/Compression/Down.lean | 264 | 268 | theorem mem_compression_of_insert_mem_compression (h : insert a s ∈ 𝓓 a 𝒜) : s ∈ 𝓓 a 𝒜 := by |
by_cases ha : a ∈ s
· rwa [insert_eq_of_mem ha] at h
· rw [← erase_insert ha]
exact erase_mem_compression_of_mem_compression h
| false |
import Mathlib.Combinatorics.Quiver.Basic
#align_import combinatorics.quiver.push from "leanprover-community/mathlib"@"2258b40dacd2942571c8ce136215350c702dc78f"
namespace Quiver
universe v v₁ v₂ u u₁ u₂
variable {V : Type*} [Quiver V] {W : Type*} (σ : V → W)
@[nolint unusedArguments]
def Push (_ : V → W) :=
... | Mathlib/Combinatorics/Quiver/Push.lean | 92 | 102 | theorem lift_unique (Φ : Push σ ⥤q W') (Φ₀ : Φ.obj = τ) (Φcomp : (of σ ⋙q Φ) = φ) :
Φ = lift σ φ τ h := by |
dsimp only [of, lift]
fapply Prefunctor.ext
· intro X
simp only
rw [Φ₀]
· rintro _ _ ⟨⟩
subst_vars
simp only [Prefunctor.comp_map, cast_eq]
rfl
| false |
import Mathlib.Data.Finset.Option
import Mathlib.Data.PFun
import Mathlib.Data.Part
#align_import data.finset.pimage from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable {α β : Type*}
namespace Part
def toFinset (o : Part α) [Decidable o.Dom] : Finset α :=
o.toOption.toFins... | Mathlib/Data/Finset/PImage.lean | 44 | 45 | theorem toFinset_some {a : α} [Decidable (some a).Dom] : (some a).toFinset = {a} := by |
simp [toFinset]
| false |
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.BigOperators
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.GroupTheory.Torsion
import Mathlib.RingTheory.Coprime.Ideal
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Set.Lattice
#align_import algebra.module.torsion from "leanpr... | Mathlib/Algebra/Module/Torsion.lean | 92 | 95 | theorem torsionOf_eq_top_iff (m : M) : torsionOf R M m = ⊤ ↔ m = 0 := by |
refine ⟨fun h => ?_, fun h => by simp [h]⟩
rw [← one_smul R m, ← mem_torsionOf_iff m (1 : R), h]
exact Submodule.mem_top
| false |
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.NormedSpace.Dual
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.function.ae_eq_of_integral from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284... | Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean | 57 | 67 | theorem ae_eq_zero_of_forall_inner [NormedAddCommGroup E] [InnerProductSpace 𝕜 E]
[SecondCountableTopology E] {f : α → E} (hf : ∀ c : E, (fun x => (inner c (f x) : 𝕜)) =ᵐ[μ] 0) :
f =ᵐ[μ] 0 := by |
let s := denseSeq E
have hs : DenseRange s := denseRange_denseSeq E
have hf' : ∀ᵐ x ∂μ, ∀ n : ℕ, inner (s n) (f x) = (0 : 𝕜) := ae_all_iff.mpr fun n => hf (s n)
refine hf'.mono fun x hx => ?_
rw [Pi.zero_apply, ← @inner_self_eq_zero 𝕜]
have h_closed : IsClosed {c : E | inner c (f x) = (0 : 𝕜)} :=
is... | false |
import Mathlib.FieldTheory.PrimitiveElement
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly
import Mathlib.LinearAlgebra.Matrix.ToLinearEquiv
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.G... | Mathlib/RingTheory/Norm.lean | 85 | 87 | theorem norm_eq_matrix_det [Fintype ι] [DecidableEq ι] (b : Basis ι R S) (s : S) :
norm R s = Matrix.det (Algebra.leftMulMatrix b s) := by |
rw [norm_apply, ← LinearMap.det_toMatrix b, ← toMatrix_lmul_eq]; rfl
| false |
import Mathlib.Data.List.Basic
namespace List
variable {α β : Type*}
@[simp]
theorem reduceOption_cons_of_some (x : α) (l : List (Option α)) :
reduceOption (some x :: l) = x :: l.reduceOption := by
simp only [reduceOption, filterMap, id, eq_self_iff_true, and_self_iff]
#align list.reduce_option_cons_of_some... | Mathlib/Data/List/ReduceOption.lean | 97 | 99 | theorem reduceOption_get?_iff {l : List (Option α)} {x : α} :
(∃ i, l.get? i = some (some x)) ↔ ∃ i, l.reduceOption.get? i = some x := by |
rw [← mem_iff_get?, ← mem_iff_get?, reduceOption_mem_iff]
| false |
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Exponential
#align_import set_theory.ordinal.cantor_normal_form from "leanprover-community/mathlib"@"991ff3b5269848f6dd942ae8e9dd3c946035dc8b"
noncomputable section
universe u
open List
namespace Ordinal
@[elab_as_elim]
noncomputabl... | Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | 97 | 97 | theorem one_CNF {o : Ordinal} (ho : o ≠ 0) : CNF 1 o = [⟨0, o⟩] := by | simp [CNF_ne_zero ho]
| false |
import Mathlib.MeasureTheory.Integral.ExpDecay
import Mathlib.Analysis.MellinTransform
#align_import analysis.special_functions.gamma.basic from "leanprover-community/mathlib"@"cca40788df1b8755d5baf17ab2f27dacc2e17acb"
noncomputable section
set_option linter.uppercaseLean3 false
open Filter intervalIntegral Set... | Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean | 56 | 67 | theorem Gamma_integrand_isLittleO (s : ℝ) :
(fun x : ℝ => exp (-x) * x ^ s) =o[atTop] fun x : ℝ => exp (-(1 / 2) * x) := by |
refine isLittleO_of_tendsto (fun x hx => ?_) ?_
· exfalso; exact (exp_pos (-(1 / 2) * x)).ne' hx
have : (fun x : ℝ => exp (-x) * x ^ s / exp (-(1 / 2) * x)) =
(fun x : ℝ => exp (1 / 2 * x) / x ^ s)⁻¹ := by
ext1 x
field_simp [exp_ne_zero, exp_neg, ← Real.exp_add]
left
ring
rw [this]
exac... | false |
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 | 168 | 171 | theorem HasGradientAt.hasDerivAt (h : HasGradientAt g g' u) :
HasDerivAt g (starRingEnd 𝕜 g') u := by |
rw [hasGradientAt_iff_hasFDerivAt, hasFDerivAt_iff_hasDerivAt] at h
simpa using h
| false |
import Mathlib.Algebra.Polynomial.Cardinal
import Mathlib.Algebra.MvPolynomial.Cardinal
import Mathlib.Data.ZMod.Algebra
import Mathlib.FieldTheory.IsAlgClosed.Basic
import Mathlib.RingTheory.AlgebraicIndependent
#align_import field_theory.is_alg_closed.classification from "leanprover-community/mathlib"@"0723536a0522... | Mathlib/FieldTheory/IsAlgClosed/Classification.lean | 64 | 76 | theorem cardinal_mk_le_max : #L ≤ max #R ℵ₀ :=
calc
#L ≤ #(Σ p : R[X], { x : L // x ∈ p.aroots L }) :=
cardinal_mk_le_sigma_polynomial R L
_ = Cardinal.sum fun p : R[X] => #{x : L | x ∈ p.aroots L} := by |
rw [← mk_sigma]; rfl
_ ≤ Cardinal.sum.{u, u} fun _ : R[X] => ℵ₀ :=
(sum_le_sum _ _ fun p => (Multiset.finite_toSet _).lt_aleph0.le)
_ = #(R[X]) * ℵ₀ := sum_const' _ _
_ ≤ max (max #(R[X]) ℵ₀) ℵ₀ := mul_le_max _ _
_ ≤ max (max (max #R ℵ₀) ℵ₀) ℵ₀ :=
(max_le_max (max_le_max Polynomial.ca... | false |
import Mathlib.Data.Nat.Bits
import Mathlib.Order.Lattice
#align_import data.nat.size from "leanprover-community/mathlib"@"18a5306c091183ac90884daa9373fa3b178e8607"
namespace Nat
section
set_option linter.deprecated false
theorem shiftLeft_eq_mul_pow (m) : ∀ n, m <<< n = m * 2 ^ n := shiftLeft_eq _
#align nat.... | Mathlib/Data/Nat/Size.lean | 137 | 138 | theorem lt_size {m n : ℕ} : m < size n ↔ 2 ^ m ≤ n := by |
rw [← not_lt, Decidable.iff_not_comm, not_lt, size_le]
| false |
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