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 |
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import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.RingTheory.SimpleModule
#align_import representation_theory.maschke from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v w
noncomputable section
open Module MonoidAlgeb... | Mathlib/RepresentationTheory/Maschke.lean | 125 | 127 | theorem equivariantProjection_apply (v : W) :
π.equivariantProjection G v = ⅟(Fintype.card G : k) • ∑ g : G, π.conjugate g v := by |
simp only [equivariantProjection, smul_apply, sumOfConjugatesEquivariant_apply]
| 1 |
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Finset.Card
import Mathlib.Data.List.NodupEquivFin
import Mathlib.Data.Set.Image
#align_import data.fintype.card from "leanprover-community/mathlib"@"bf2428c9486c407ca38b5b3fb10b87dad0bc99fa"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
open Fu... | Mathlib/Data/Fintype/Card.lean | 139 | 140 | theorem card_of_finset' {p : Set α} (s : Finset α) (H : ∀ x, x ∈ s ↔ x ∈ p) [Fintype p] :
Fintype.card p = s.card := by | rw [← card_ofFinset s H]; congr; apply Subsingleton.elim
| 1 |
import Mathlib.CategoryTheory.EqToHom
#align_import category_theory.sums.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
namespace CategoryTheory
universe v₁ u₁
-- morphism levels before object levels. See note [category_theory universes].
open Sum
section
variable (C : Ty... | Mathlib/CategoryTheory/Sums/Basic.lean | 66 | 67 | theorem hom_inr_inl_false {X : C} {Y : D} (f : Sum.inr X ⟶ Sum.inl Y) : False := by |
cases f
| 1 |
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 | 106 | 108 | theorem trop_iInf [ConditionallyCompleteLinearOrder R] [Fintype S] (f : S → WithTop R) :
trop (⨅ i : S, f i) = ∑ i : S, trop (f i) := by |
rw [iInf, ← Set.image_univ, ← coe_univ, trop_sInf_image]
| 1 |
import Mathlib.Analysis.Complex.Basic
import Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle
#align_import analysis.complex.re_im_topology from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1e360b7e3cf6"
open Set
noncomputable section
namespace Complex
theorem isHomeomorphicTrivialFiber... | Mathlib/Analysis/Complex/ReImTopology.lean | 114 | 115 | theorem closure_setOf_re_lt (a : ℝ) : closure { z : ℂ | z.re < a } = { z | z.re ≤ a } := by |
simpa only [closure_Iio] using closure_preimage_re (Iio a)
| 1 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 389 | 389 | theorem inv_lt' : a⁻¹ < b ↔ b⁻¹ < a := by | rw [← inv_lt_inv_iff, inv_inv]
| 1 |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.NormedSpace.LinearIsometry
import Mathlib.Algebra.Star.SelfAdjoint
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.Algebra.Star.Unitary
import Mathlib.Topology.Algebra.Module.Star
#align_import analysis.no... | Mathlib/Analysis/NormedSpace/Star/Basic.lean | 149 | 150 | theorem mul_star_self_ne_zero_iff (x : E) : x * x⋆ ≠ 0 ↔ x ≠ 0 := by |
simp only [Ne, mul_star_self_eq_zero_iff]
| 1 |
import Mathlib.Algebra.ContinuedFractions.Basic
import Mathlib.Algebra.GroupWithZero.Basic
#align_import algebra.continued_fractions.translations from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
section General
variable {α : Type*} {g : Gen... | Mathlib/Algebra/ContinuedFractions/Translations.lean | 53 | 55 | theorem terminatedAt_iff_part_denom_none :
g.TerminatedAt n ↔ g.partialDenominators.get? n = none := by |
rw [terminatedAt_iff_s_none, part_denom_none_iff_s_none]
| 1 |
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : ℝ}
-- @... | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 308 | 309 | theorem logb_le_logb_of_base_lt_one (h : 0 < x) (h₁ : 0 < y) : logb b x ≤ logb b y ↔ y ≤ x := by |
rw [logb, logb, div_le_div_right_of_neg (log_neg b_pos b_lt_one), log_le_log_iff h₁ h]
| 1 |
import Mathlib.Combinatorics.SimpleGraph.Basic
namespace SimpleGraph
variable {V : Type*} (G : SimpleGraph V)
structure Dart extends V × V where
adj : G.Adj fst snd
deriving DecidableEq
#align simple_graph.dart SimpleGraph.Dart
initialize_simps_projections Dart (+toProd, -fst, -snd)
attribute [simp] Dart.a... | Mathlib/Combinatorics/SimpleGraph/Dart.lean | 33 | 34 | theorem Dart.ext_iff (d₁ d₂ : G.Dart) : d₁ = d₂ ↔ d₁.toProd = d₂.toProd := by |
cases d₁; cases d₂; simp
| 1 |
import Mathlib.Data.Nat.Bitwise
import Mathlib.SetTheory.Game.Birthday
import Mathlib.SetTheory.Game.Impartial
#align_import set_theory.game.nim from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
noncomputable section
universe u
namespace SetTheory
open scoped PGame
namespace PGame... | Mathlib/SetTheory/Game/Nim.lean | 73 | 75 | theorem moveLeft_nim_hEq (o : Ordinal) :
have : IsWellOrder (Quotient.out o).α (· < ·) := inferInstance
HEq (nim o).moveLeft fun i : o.out.α => nim (typein (· < ·) i) := by | rw [nim_def]; rfl
| 1 |
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
#align_import category_theory.monoidal.coherence_lemmas from "leanprover-community/mathlib"@"b8b8bf3ea0c625fa1f950034a184e07c67f7bcfe"
open CategoryTheory Category Iso
namespace CategoryTheory.MonoidalCategory
v... | Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean | 63 | 64 | theorem unitors_equal : (λ_ (𝟙_ C)).hom = (ρ_ (𝟙_ C)).hom := by |
coherence
| 1 |
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 | 286 | 288 | theorem normAtPlace_apply_isReal {w : InfinitePlace K} (hw : IsReal w) (x : E K):
normAtPlace w x = ‖x.1 ⟨w, hw⟩‖ := by |
rw [normAtPlace, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk, dif_pos]
| 1 |
import Mathlib.Control.Applicative
import Mathlib.Control.Traversable.Basic
import Mathlib.Data.List.Forall2
import Mathlib.Data.Set.Functor
#align_import control.traversable.instances from "leanprover-community/mathlib"@"18a5306c091183ac90884daa9373fa3b178e8607"
universe u v
section Option
open Functor
variab... | Mathlib/Control/Traversable/Instances.lean | 31 | 32 | theorem Option.id_traverse {α} (x : Option α) : Option.traverse (pure : α → Id α) x = x := by |
cases x <;> rfl
| 1 |
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.preserves.shapes.biproducts from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe w₁ w₂ v₁ v₂ u₁ u₂
noncomputable section
open ... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Biproducts.lean | 349 | 351 | theorem biprodComparison'_comp_biprodComparison :
biprodComparison' F X Y ≫ biprodComparison F X Y = 𝟙 (F.obj X ⊞ F.obj Y) := by |
ext <;> simp [← Functor.map_comp]
| 1 |
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/Variables.lean | 87 | 88 | theorem vars_C : (C r : MvPolynomial σ R).vars = ∅ := by |
classical rw [vars_def, degrees_C, Multiset.toFinset_zero]
| 1 |
import Mathlib.Data.Matrix.Basis
import Mathlib.RingTheory.TensorProduct.Basic
#align_import ring_theory.matrix_algebra from "leanprover-community/mathlib"@"6c351a8fb9b06e5a542fdf427bfb9f46724f9453"
suppress_compilation
universe u v w
open TensorProduct
open TensorProduct
open Algebra.TensorProduct
open Matri... | Mathlib/RingTheory/MatrixAlgebra.lean | 99 | 101 | theorem invFun_smul (a : A) (M : Matrix n n A) :
invFun R A n (a • M) = a ⊗ₜ 1 * invFun R A n M := by |
simp [invFun, Finset.mul_sum]
| 1 |
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Algebra.GCDMonoid.Nat
#align_import ring_theory.int.basic from "leanprover-community/mathlib"@"e655e4ea5c6d02854696f97494997ba4c31be802"
namespace Int
theorem gcd_eq_one_iff_coprime {a b : ℤ} : Int.gcd a b ... | Mathlib/RingTheory/Int/Basic.lean | 49 | 50 | theorem coprime_iff_nat_coprime {a b : ℤ} : IsCoprime a b ↔ Nat.Coprime a.natAbs b.natAbs := by |
rw [← gcd_eq_one_iff_coprime, Nat.coprime_iff_gcd_eq_one, gcd_eq_natAbs]
| 1 |
import Mathlib.Tactic.Ring
#align_import algebra.group_power.identities from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
variable {R : Type*} [CommRing R] {a b x₁ x₂ x₃ x₄ x₅ x₆ x₇ x₈ y₁ y₂ y₃ y₄ y₅ y₆ y₇ y₈ n : R}
theorem sq_add_sq_mul_sq_add_sq :
(x₁ ^ 2 + x₂ ^ 2) * (y₁ ^ 2 +... | Mathlib/Algebra/Ring/Identities.lean | 55 | 60 | theorem sum_four_sq_mul_sum_four_sq :
(x₁ ^ 2 + x₂ ^ 2 + x₃ ^ 2 + x₄ ^ 2) * (y₁ ^ 2 + y₂ ^ 2 + y₃ ^ 2 + y₄ ^ 2) =
(x₁ * y₁ - x₂ * y₂ - x₃ * y₃ - x₄ * y₄) ^ 2 + (x₁ * y₂ + x₂ * y₁ + x₃ * y₄ - x₄ * y₃) ^ 2 +
(x₁ * y₃ - x₂ * y₄ + x₃ * y₁ + x₄ * y₂) ^ 2 +
(x₁ * y₄ + x₂ * y₃ - x₃ * y₂ + x₄ * y₁) ... |
ring
| 1 |
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polynomial.eval from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
set_option linter.uppercaseLean3 false
noncomputable section
open Finset AddMonoidAlgebra
open Polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 77 | 78 | theorem eval₂_monomial {n : ℕ} {r : R} : (monomial n r).eval₂ f x = f r * x ^ n := by |
simp [eval₂_eq_sum]
| 1 |
import Mathlib.NumberTheory.BernoulliPolynomials
import Mathlib.MeasureTheory.Integral.IntervalIntegral
import Mathlib.Analysis.Calculus.Deriv.Polynomial
import Mathlib.Analysis.Fourier.AddCircle
import Mathlib.Analysis.PSeries
#align_import number_theory.zeta_values from "leanprover-community/mathlib"@"f0c8bf9245297... | Mathlib/NumberTheory/ZetaValues.lean | 49 | 50 | theorem bernoulliFun_eval_zero (k : ℕ) : bernoulliFun k 0 = bernoulli k := by |
rw [bernoulliFun, Polynomial.eval_zero_map, Polynomial.bernoulli_eval_zero, eq_ratCast]
| 1 |
import Mathlib.Algebra.BigOperators.Group.List
import Mathlib.Algebra.Group.Prod
import Mathlib.Data.Multiset.Basic
#align_import algebra.big_operators.multiset.basic from "leanprover-community/mathlib"@"6c5f73fd6f6cc83122788a80a27cdd54663609f4"
assert_not_exists MonoidWithZero
variable {F ι α β γ : Type*}
names... | Mathlib/Algebra/BigOperators/Group/Multiset.lean | 125 | 127 | theorem prod_filter_mul_prod_filter_not (p) [DecidablePred p] :
(s.filter p).prod * (s.filter (fun a ↦ ¬ p a)).prod = s.prod := by |
rw [← prod_add, filter_add_not]
| 1 |
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 Image
variable {f : α → β} {s t : Set... | Mathlib/Data/Set/Image.lean | 263 | 263 | theorem image_comp (f : β → γ) (g : α → β) (a : Set α) : f ∘ g '' a = f '' (g '' a) := by | aesop
| 1 |
import Mathlib.CategoryTheory.Balanced
import Mathlib.CategoryTheory.Limits.EssentiallySmall
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.CategoryTheory.Subobject.WellPowered
import Mathlib.Data.S... | Mathlib/CategoryTheory/Generator.lean | 109 | 110 | theorem isCoseparating_unop_iff (𝒢 : Set Cᵒᵖ) : IsCoseparating 𝒢.unop ↔ IsSeparating 𝒢 := by |
rw [← isSeparating_op_iff, Set.unop_op]
| 1 |
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Inverse
#align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputab... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 117 | 118 | theorem contDiffWithinAt_of_subsingleton [Subsingleton F] : ContDiffWithinAt 𝕜 n f s x := by |
rw [Subsingleton.elim f fun _ => 0]; exact contDiffWithinAt_const
| 1 |
import Mathlib.Init.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive
import Aesop
#align_import data.option.basic from "leanprover-community/mathlib"@"f340f229b1f461aa1c8ee11e0a172d0a3b301a4a"
universe u
namespace Op... | Mathlib/Data/Option/Basic.lean | 108 | 110 | theorem bind_congr {f g : α → Option β} {x : Option α}
(h : ∀ a ∈ x, f a = g a) : x.bind f = x.bind g := by |
cases x <;> simp only [some_bind, none_bind, mem_def, h]
| 1 |
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Covering.Besicovitch
import Mathlib.Tactic.AdaptationNote
#align_import measure_theory.covering.besicovitch_vector_space from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
universe u
open Metric Set Fini... | Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean | 87 | 89 | theorem centerAndRescale_radius {N : ℕ} {τ : ℝ} (a : SatelliteConfig E N τ) :
a.centerAndRescale.r (last N) = 1 := by |
simp [SatelliteConfig.centerAndRescale, inv_mul_cancel (a.rpos _).ne']
| 1 |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} =... | Mathlib/Order/Interval/Set/WithBotTop.lean | 59 | 59 | theorem preimage_coe_Icc : (some : α → WithTop α) ⁻¹' Icc a b = Icc a b := by | simp [← Ici_inter_Iic]
| 1 |
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.Polynomial.CancelLeads
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.FieldDivision
#align_import ring_theory.polynomial.content from "leanprover-community/mathlib"@"7a030ab8eb5d99f05a891dccc49c5b5b90c947d3"
namespace Po... | Mathlib/RingTheory/Polynomial/Content.lean | 106 | 106 | theorem content_one : content (1 : R[X]) = 1 := by | rw [← C_1, content_C, normalize_one]
| 1 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.HasseDeriv
#align_import data.polynomial.taylor from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
namespace Polynomial
open Polynomial... | Mathlib/Algebra/Polynomial/Taylor.lean | 51 | 51 | theorem taylor_C (x : R) : taylor r (C x) = C x := by | simp only [taylor_apply, C_comp]
| 1 |
import Mathlib.Algebra.Group.Even
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.Order.Sub.Defs
#align_import algebra.order.sub.canonical from "leanprover-community/mathlib"@"62a5626868683c104774de8d85b9855234ac807c"
variable {α : Type*}
section ExistsAddOfLE
variable [AddCommSemigrou... | Mathlib/Algebra/Order/Sub/Canonical.lean | 48 | 49 | theorem tsub_left_inj (h1 : c ≤ a) (h2 : c ≤ b) : a - c = b - c ↔ a = b := by |
simp_rw [le_antisymm_iff, tsub_le_tsub_iff_right h1, tsub_le_tsub_iff_right h2]
| 1 |
import Mathlib.Tactic.Ring
import Mathlib.Tactic.FailIfNoProgress
import Mathlib.Algebra.Group.Commutator
#align_import tactic.group from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
namespace Mathlib.Tactic.Group
open Lean
open Lean.Meta
open Lean.Parser.Tactic
open Lean.Elab.Tactic
... | Mathlib/Tactic/Group.lean | 37 | 38 | theorem zpow_trick {G : Type*} [Group G] (a b : G) (n m : ℤ) :
a * b ^ n * b ^ m = a * b ^ (n + m) := by | rw [mul_assoc, ← zpow_add]
| 1 |
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 | 501 | 504 | theorem dualTensorHom_comm (g : G) :
dualTensorHom k V W ∘ₗ TensorProduct.map (ρV.dual g) (ρW g) =
(linHom ρV ρW) g ∘ₗ dualTensorHom k V W := by |
ext; simp [Module.Dual.transpose_apply]
| 1 |
import Mathlib.SetTheory.Ordinal.Arithmetic
#align_import set_theory.ordinal.exponential from "leanprover-community/mathlib"@"b67044ba53af18680e1dd246861d9584e968495d"
noncomputable section
open Function Cardinal Set Equiv Order
open scoped Classical
open Cardinal Ordinal
universe u v w
namespace Ordinal
in... | Mathlib/SetTheory/Ordinal/Exponential.lean | 72 | 74 | theorem lt_opow_of_limit {a b c : Ordinal} (b0 : b ≠ 0) (h : IsLimit c) :
a < b ^ c ↔ ∃ c' < c, a < b ^ c' := by |
rw [← not_iff_not, not_exists]; simp only [not_lt, opow_le_of_limit b0 h, exists_prop, not_and]
| 1 |
import Mathlib.Algebra.ContinuedFractions.Basic
import Mathlib.Algebra.GroupWithZero.Basic
#align_import algebra.continued_fractions.translations from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
section General
variable {α : Type*} {g : Gen... | Mathlib/Algebra/ContinuedFractions/Translations.lean | 71 | 74 | theorem exists_s_b_of_part_denom {b : α}
(nth_part_denom_eq : g.partialDenominators.get? n = some b) :
∃ gp, g.s.get? n = some gp ∧ gp.b = b := by |
simpa [partialDenominators, Stream'.Seq.map_get?] using nth_part_denom_eq
| 1 |
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.Ring.Int
#align_import data.int.cast.field from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
namespace Int
open Nat
variable {α : Type*}
@[norm_cast]
| Mathlib/Data/Int/Cast/Field.lean | 34 | 34 | theorem cast_neg_natCast {R} [DivisionRing R] (n : ℕ) : ((-n : ℤ) : R) = -n := by | simp
| 1 |
import Mathlib.Data.List.Forall2
#align_import data.list.zip from "leanprover-community/mathlib"@"134625f523e737f650a6ea7f0c82a6177e45e622"
-- Make sure we don't import algebra
assert_not_exists Monoid
universe u
open Nat
namespace List
variable {α : Type u} {β γ δ ε : Type*}
#align list.zip_with_cons_cons Li... | Mathlib/Data/List/Zip.lean | 67 | 68 | theorem lt_length_right_of_zipWith {f : α → β → γ} {i : ℕ} {l : List α} {l' : List β}
(h : i < (zipWith f l l').length) : i < l'.length := by | rw [length_zipWith] at h; omega
| 1 |
import Mathlib.Algebra.BigOperators.WithTop
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.ENNReal.Basic
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
variable {a b c d : ℝ≥0∞} {r p q... | Mathlib/Data/ENNReal/Operations.lean | 200 | 200 | theorem not_lt_top {x : ℝ≥0∞} : ¬x < ∞ ↔ x = ∞ := by | rw [lt_top_iff_ne_top, Classical.not_not]
| 1 |
import Mathlib.MeasureTheory.Group.GeometryOfNumbers
import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | 302 | 304 | theorem convexBodySumFun_neg (x : E K) :
convexBodySumFun (- x) = convexBodySumFun x := by |
simp_rw [convexBodySumFun, normAtPlace_neg]
| 1 |
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.GroupTheory.OrderOfElement
#align_import algebra.char_p.two from "leanprover-community/mathlib"@"7f1ba1a333d66eed531ecb4092493cd1b6715450"
variable {R ι : Type*}
namespace CharTwo
section CommSemiring
variable [CommSemiring R] [CharP R 2]
theorem add_sq (x y... | Mathlib/Algebra/CharP/Two.lean | 107 | 108 | theorem multiset_sum_mul_self (l : Multiset R) :
l.sum * l.sum = (Multiset.map (fun x => x * x) l).sum := by | simp_rw [← pow_two, multiset_sum_sq]
| 1 |
import Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing
import Mathlib.AlgebraicGeometry.OpenImmersion
#align_import algebraic_geometry.gluing from "leanprover-community/mathlib"@"533f62f4dd62a5aad24a04326e6e787c8f7e98b1"
set_option linter.uppercaseLean3 false
noncomputable section
universe u
open Topologica... | Mathlib/AlgebraicGeometry/Gluing.lean | 296 | 298 | theorem gluedCoverT'_fst_fst (x y z : 𝒰.J) :
𝒰.gluedCoverT' x y z ≫ pullback.fst ≫ pullback.fst = pullback.fst ≫ pullback.snd := by |
delta gluedCoverT'; simp
| 1 |
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 680 | 681 | theorem preimage_mul_const_Icc_of_neg (a b : α) {c : α} (h : c < 0) :
(fun x => x * c) ⁻¹' Icc a b = Icc (b / c) (a / c) := by | simp [← Ici_inter_Iic, h, inter_comm]
| 1 |
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.Data.Finset.Sort
#align_import data.polynomial.basic from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
set_option linter.uppercaseLean3 false
noncomputable section
structure ... | Mathlib/Algebra/Polynomial/Basic.lean | 98 | 98 | theorem eta (f : R[X]) : Polynomial.ofFinsupp f.toFinsupp = f := by | cases f; rfl
| 1 |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Set.Sigma
#align_import data.finset.sigma from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Multiset
variable {ι : Type*}
namespace Finset
section Sigma
variable {α : ι → Type*} {β : Type*} (s s₁ s₂ : Finset ι) (... | Mathlib/Data/Finset/Sigma.lean | 60 | 60 | theorem sigma_nonempty : (s.sigma t).Nonempty ↔ ∃ i ∈ s, (t i).Nonempty := by | simp [Finset.Nonempty]
| 1 |
import Mathlib.Control.Functor
import Mathlib.Tactic.Common
#align_import control.bifunctor from "leanprover-community/mathlib"@"dc1525fb3ef6eb4348fb1749c302d8abc303d34a"
universe u₀ u₁ u₂ v₀ v₁ v₂
open Function
class Bifunctor (F : Type u₀ → Type u₁ → Type u₂) where
bimap : ∀ {α α' β β'}, (α → α') → (β → β'... | Mathlib/Control/Bifunctor.lean | 104 | 105 | theorem comp_snd {α β₀ β₁ β₂} (g : β₀ → β₁) (g' : β₁ → β₂) (x : F α β₀) :
snd g' (snd g x) = snd (g' ∘ g) x := by | simp [snd, bimap_bimap]
| 1 |
import Mathlib.Probability.ProbabilityMassFunction.Constructions
import Mathlib.Tactic.FinCases
namespace PMF
open ENNReal
noncomputable
def binomial (p : ℝ≥0∞) (h : p ≤ 1) (n : ℕ) : PMF (Fin (n + 1)) :=
.ofFintype (fun i => p^(i : ℕ) * (1-p)^((Fin.last n - i) : ℕ) * (n.choose i : ℕ)) (by
convert (add_pow ... | Mathlib/Probability/ProbabilityMassFunction/Binomial.lean | 45 | 47 | theorem binomial_apply_last (p : ℝ≥0∞) (h : p ≤ 1) (n : ℕ) :
binomial p h n (.last n) = p^n := by |
simp [binomial_apply]
| 1 |
import Mathlib.Data.Rat.Sqrt
import Mathlib.Data.Real.Sqrt
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Int.Basic
import Mathlib.Tactic.IntervalCases
#align_import data.real.irrational from "leanprover-community/mathlib"@"7e7aaccf9b0182576cabdde36cf1b5ad3585b70d"
open Rat Real multiplicity
def ... | Mathlib/Data/Real/Irrational.lean | 103 | 104 | theorem irrational_sqrt_two : Irrational (√2) := by |
simpa using Nat.prime_two.irrational_sqrt
| 1 |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.Orientation
import Mathlib.Tactic.LinearCombination
#align_import analysis.inner_product_space.two_dim from "leanprover-community/mathlib"@"cd8fafa2fac98e1a67097e8a91ad9901cfde48af"
non... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 142 | 143 | theorem abs_areaForm_le (x y : E) : |ω x y| ≤ ‖x‖ * ‖y‖ := by |
simpa [areaForm_to_volumeForm, Fin.prod_univ_succ] using o.abs_volumeForm_apply_le ![x, y]
| 1 |
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Analysis.LocallyConvex.Polar
#align_import analysis.normed_space.dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open ... | Mathlib/Analysis/NormedSpace/Dual.lean | 87 | 88 | theorem double_dual_bound (x : E) : ‖(inclusionInDoubleDual 𝕜 E) x‖ ≤ ‖x‖ := by |
simpa using ContinuousLinearMap.le_of_opNorm_le _ (inclusionInDoubleDual_norm_le 𝕜 E) x
| 1 |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology.Instances.Sign
#align_import analysis.special_functions.trigonometric.angle from "leanprover-community/mathlib"@"213b0cff7bc5ab6696ee07cceec80829... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 146 | 147 | theorem two_zsmul_coe_div_two (θ : ℝ) : (2 : ℤ) • (↑(θ / 2) : Angle) = θ := by |
rw [← coe_zsmul, two_zsmul, add_halves]
| 1 |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.List.Sublists
import Mathlib.Data.List.InsertNth
#align_import group_theory.free_group from "leanprover-community/mathlib"@"f93c11933efbc3c2f0299e47b8ff83e9b539cbf6"
open Relation
universe u v w
variable {α : Type u... | Mathlib/GroupTheory/FreeGroup/Basic.lean | 115 | 116 | theorem Step.not_rev {x b} : Step (L₁ ++ (x, !b) :: (x, b) :: L₂) (L₁ ++ L₂) := by |
cases b <;> exact Step.not
| 1 |
import Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.measure.haar.normed_space from "leanprover-community/mathlib"@"b84aee748341da06a6d78491367e2c0e9f15e8a5"
noncomputable sect... | Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean | 89 | 91 | theorem integral_comp_smul_of_nonneg (f : E → F) (R : ℝ) {hR : 0 ≤ R} :
∫ x, f (R • x) ∂μ = (R ^ finrank ℝ E)⁻¹ • ∫ x, f x ∂μ := by |
rw [integral_comp_smul μ f R, abs_of_nonneg (inv_nonneg.2 (pow_nonneg hR _))]
| 1 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Fin
import Mathlib.GroupTheory.GroupAction.Pi
import Mathlib.Logic.Equiv.Fin
#align_import algebra.big_operators.fin from "leanprover-community/mathlib"@"cc5dd6244981976cc9da7afc4eee5682b037a013"
open Fins... | Mathlib/Algebra/BigOperators/Fin.lean | 106 | 108 | theorem prod_cons [CommMonoid β] {n : ℕ} (x : β) (f : Fin n → β) :
(∏ i : Fin n.succ, (cons x f : Fin n.succ → β) i) = x * ∏ i : Fin n, f i := by |
simp_rw [prod_univ_succ, cons_zero, cons_succ]
| 1 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.convex.side from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052a5230f"
variable {R V V' P P' : Type*}
open AffineEquiv AffineMap
namespace Affine... | Mathlib/Analysis/Convex/Side.lean | 83 | 86 | theorem _root_.Function.Injective.sSameSide_map_iff {s : AffineSubspace R P} {x y : P}
{f : P →ᵃ[R] P'} (hf : Function.Injective f) :
(s.map f).SSameSide (f x) (f y) ↔ s.SSameSide x y := by |
simp_rw [SSameSide, hf.wSameSide_map_iff, mem_map_iff_mem_of_injective hf]
| 1 |
import Mathlib.Algebra.Category.GroupCat.Basic
import Mathlib.CategoryTheory.SingleObj
import Mathlib.CategoryTheory.Limits.FunctorCategory
import Mathlib.CategoryTheory.Limits.Preserves.Basic
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Conj
#align_import representation_theory.Action... | Mathlib/RepresentationTheory/Action/Basic.lean | 50 | 50 | theorem ρ_one {G : MonCat.{u}} (A : Action V G) : A.ρ 1 = 𝟙 A.V := by | rw [MonoidHom.map_one]; rfl
| 1 |
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.monoid from "leanprover-community/mathlib"@"aba57d4d3dae35460225919dcd82fe91355162f9"
namespace Set
variable {M : Type*} ... | Mathlib/Algebra/Order/Interval/Set/Monoid.lean | 118 | 119 | theorem image_const_add_Ioi : (fun x => a + x) '' Ioi b = Ioi (a + b) := by |
simp only [add_comm a, image_add_const_Ioi]
| 1 |
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.Algebra.CharP.Reduced
open Function Polynomial
class PerfectRing (R : Type*) (p : ℕ) [CommSemiring R] [ExpChar R p] : Prop where
bijective_frobenius : Bijective <| frobenius R p
section PerfectRing
va... | Mathlib/FieldTheory/Perfect.lean | 113 | 114 | theorem iterateFrobeniusEquiv_zero_apply (x : R) : iterateFrobeniusEquiv R p 0 x = x := by |
rw [iterateFrobeniusEquiv_def, pow_zero, pow_one]
| 1 |
import Mathlib.Logic.Equiv.Nat
import Mathlib.Logic.Equiv.Fin
import Mathlib.Data.Countable.Defs
#align_import data.countable.basic from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
universe u v w
open Function
instance : Countable ℤ :=
Countable.of_equiv ℕ Equiv.intEquivNat.symm
... | Mathlib/Data/Countable/Basic.lean | 38 | 39 | theorem uncountable_iff_isEmpty_embedding : Uncountable α ↔ IsEmpty (α ↪ ℕ) := by |
rw [← not_countable_iff, countable_iff_nonempty_embedding, not_nonempty_iff]
| 1 |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.GroupTheory.GroupAction.Ring
#align_import data.polynomial.derivative from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
noncomputable section
open Finset
open Polynomial
namespace Pol... | Mathlib/Algebra/Polynomial/Derivative.lean | 149 | 150 | theorem derivative_X_add_C (c : R) : derivative (X + C c) = 1 := by |
rw [derivative_add, derivative_X, derivative_C, add_zero]
| 1 |
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Convert
#align_import control.equiv_functor from "leanprover-community/mathlib"@"d6aae1bcbd04b8de2022b9b83a5b5b10e10c777d"
universe u₀ u₁ u₂ v₀ v₁ v₂
open Function
class EquivFunctor (f : Type u₀ → Type u₁) where
map : ∀ {α β}, α ≃ β → f α → f β
m... | Mathlib/Control/EquivFunctor.lean | 70 | 71 | theorem mapEquiv_refl (α) : mapEquiv f (Equiv.refl α) = Equiv.refl (f α) := by |
simp only [mapEquiv, map_refl', Equiv.refl_symm]; rfl
| 1 |
import Mathlib.MeasureTheory.Measure.MeasureSpace
open scoped ENNReal NNReal Topology
open Set MeasureTheory Measure Filter MeasurableSpace ENNReal Function
variable {R α β δ γ ι : Type*}
namespace MeasureTheory
variable {m0 : MeasurableSpace α} [MeasurableSpace β] [MeasurableSpace γ]
variable {μ μ₁ μ₂ μ₃ ν ν' ν... | Mathlib/MeasureTheory/Measure/Restrict.lean | 140 | 141 | theorem restrict_apply_univ (s : Set α) : μ.restrict s univ = μ s := by |
rw [restrict_apply MeasurableSet.univ, Set.univ_inter]
| 1 |
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Data.Complex.Basic
import Mathlib.Data.Real.Archimedean
#align_import number_theory.zsqrtd.gaussian_int from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
open Zsqrtd Complex
open sc... | Mathlib/NumberTheory/Zsqrtd/GaussianInt.lean | 141 | 142 | theorem toComplex_inj {x y : ℤ[i]} : (x : ℂ) = y ↔ x = y := by |
cases x; cases y; simp [toComplex_def₂]
| 1 |
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.terminated_stable from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
variable {K : Type*} {g : GeneralizedContinuedFraction K} {n m : ℕ}
theorem te... | Mathlib/Algebra/ContinuedFractions/TerminatedStable.lean | 80 | 82 | theorem denominators_stable_of_terminated (n_le_m : n ≤ m) (terminated_at_n : g.TerminatedAt n) :
g.denominators m = g.denominators n := by |
simp only [denom_eq_conts_b, continuants_stable_of_terminated n_le_m terminated_at_n]
| 1 |
import Mathlib.Control.Monad.Basic
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.List.ProdSigma
#align_import data.fin_enum from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
universe u v
open Finset
class FinEnum (α : Sort*) where
card : ℕ
equiv : α ≃ Fin card
[... | Mathlib/Data/FinEnum.lean | 69 | 70 | theorem mem_toList [FinEnum α] (x : α) : x ∈ toList α := by |
simp [toList]; exists equiv x; simp
| 1 |
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Combinatorics.SimpleGraph.Connectivity
import Mathlib.LinearAlgebra.Matrix.Trace
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import combinatorics.simple_graph.adj_matrix from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1... | Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | 64 | 65 | theorem apply_diag_ne [MulZeroOneClass α] [Nontrivial α] (h : IsAdjMatrix A) (i : V) :
¬A i i = 1 := by | simp [h.apply_diag i]
| 1 |
import Mathlib.Order.Filter.Bases
#align_import order.filter.pi from "leanprover-community/mathlib"@"ce64cd319bb6b3e82f31c2d38e79080d377be451"
open Set Function
open scoped Classical
open Filter
namespace Filter
variable {ι : Type*} {α : ι → Type*} {f f₁ f₂ : (i : ι) → Filter (α i)} {s : (i : ι) → Set (α i)}
... | Mathlib/Order/Filter/Pi.lean | 121 | 125 | theorem hasBasis_pi {ι' : ι → Type} {s : ∀ i, ι' i → Set (α i)} {p : ∀ i, ι' i → Prop}
(h : ∀ i, (f i).HasBasis (p i) (s i)) :
(pi f).HasBasis (fun If : Set ι × ∀ i, ι' i => If.1.Finite ∧ ∀ i ∈ If.1, p i (If.2 i))
fun If : Set ι × ∀ i, ι' i => If.1.pi fun i => s i <| If.2 i := by |
simpa [Set.pi_def] using hasBasis_iInf' fun i => (h i).comap (eval i : (∀ j, α j) → α i)
| 1 |
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.Perm
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open Equiv Function Finset
variable {... | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 90 | 90 | theorem sameCycle_one : SameCycle 1 x y ↔ x = y := by | simp [SameCycle]
| 1 |
import Mathlib.Data.Bracket
import Mathlib.LinearAlgebra.Basic
#align_import algebra.lie.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe u v w w₁ w₂
open Function
class LieRing (L : Type v) extends AddCommGroup L, Bracket L L where
protected add_lie : ∀ x y z ... | Mathlib/Algebra/Lie/Basic.lean | 179 | 179 | theorem lie_sub : ⁅x, m - n⁆ = ⁅x, m⁆ - ⁅x, n⁆ := by | simp [sub_eq_add_neg]
| 1 |
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.GroupTheory.OrderOfElement
#align_import algebra.char_p.two from "leanprover-community/mathlib"@"7f1ba1a333d66eed531ecb4092493cd1b6715450"
variable {R ι : Type*}
namespace CharTwo
section CommSemiring
variable [CommSemiring R] [CharP R 2]
theorem add_sq (x y... | Mathlib/Algebra/CharP/Two.lean | 91 | 92 | theorem add_mul_self (x y : R) : (x + y) * (x + y) = x * x + y * y := by |
rw [← pow_two, ← pow_two, ← pow_two, add_sq]
| 1 |
import Mathlib.SetTheory.Game.Ordinal
import Mathlib.SetTheory.Ordinal.NaturalOps
#align_import set_theory.game.birthday from "leanprover-community/mathlib"@"a347076985674932c0e91da09b9961ed0a79508c"
universe u
open Ordinal
namespace SetTheory
open scoped NaturalOps PGame
namespace PGame
noncomputable def b... | Mathlib/SetTheory/Game/Birthday.lean | 142 | 143 | theorem neg_birthday_le : -x.birthday.toPGame ≤ x := by |
simpa only [neg_birthday, ← neg_le_iff] using le_birthday (-x)
| 1 |
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 | 96 | 98 | theorem isMatching_iff_forall_degree {M : Subgraph G} [∀ v : V, Fintype (M.neighborSet v)] :
M.IsMatching ↔ ∀ v : V, v ∈ M.verts → M.degree v = 1 := by |
simp only [degree_eq_one_iff_unique_adj, IsMatching]
| 1 |
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Ideal.Quotient
#align_import algebra.char_p.quotient from "leanprover-community/mathlib"@"85e3c05a94b27c84dc6f234cf88326d5e0096ec3"
universe u v
namespace CharP
theorem quotient (R : Type u) [CommRing R] (p ... | Mathlib/Algebra/CharP/Quotient.lean | 54 | 56 | theorem quotient_iff_le_ker_natCast {R : Type*} [CommRing R] (n : ℕ) [CharP R n] (I : Ideal R) :
CharP (R ⧸ I) n ↔ I.comap (Nat.castRingHom R) ≤ RingHom.ker (Nat.castRingHom R) := by |
rw [CharP.quotient_iff, RingHom.ker_eq_comap_bot]; rfl
| 1 |
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tactic.FinCases
#align_import ring_theory.ideal.quotient from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
universe u v w
namespace Ideal
open Set
variabl... | Mathlib/RingTheory/Ideal/Quotient.lean | 129 | 130 | theorem eq_zero_iff_dvd (x y : R) : Ideal.Quotient.mk (Ideal.span ({x} : Set R)) y = 0 ↔ x ∣ y := by |
rw [Ideal.Quotient.eq_zero_iff_mem, Ideal.mem_span_singleton]
| 1 |
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.terminated_stable from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
variable {K : Type*} {g : GeneralizedContinuedFraction K} {n m : ℕ}
theorem te... | Mathlib/Algebra/ContinuedFractions/TerminatedStable.lean | 91 | 93 | theorem convergents'_stable_of_terminated (n_le_m : n ≤ m) (terminated_at_n : g.TerminatedAt n) :
g.convergents' m = g.convergents' n := by |
simp only [convergents', convergents'Aux_stable_of_terminated n_le_m terminated_at_n]
| 1 |
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Mathlib.Topology.Instances.RealVectorSpace
#align_import analysis.complex.basic from "leanprover-community/mathlib... | Mathlib/Analysis/Complex/Basic.lean | 149 | 150 | theorem nndist_self_conj (z : ℂ) : nndist z (conj z) = 2 * Real.nnabs z.im := by |
rw [nndist_comm, nndist_conj_self]
| 1 |
import Mathlib.Algebra.MvPolynomial.Counit
import Mathlib.Algebra.MvPolynomial.Invertible
import Mathlib.RingTheory.WittVector.Defs
#align_import ring_theory.witt_vector.basic from "leanprover-community/mathlib"@"9556784a5b84697562e9c6acb40500d4a82e675a"
noncomputable section
open MvPolynomial Function
variable... | Mathlib/RingTheory/WittVector/Basic.lean | 114 | 114 | theorem mul : mapFun f (x * y) = mapFun f x * mapFun f y := by | map_fun_tac
| 1 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Logic.Function.Iterate
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Zify
#align_import data.nat.fib from "leanprover-community/mathlib"@"... | Mathlib/Data/Nat/Fib/Basic.lean | 110 | 111 | theorem fib_add_two_sub_fib_add_one {n : ℕ} : fib (n + 2) - fib (n + 1) = fib n := by |
rw [fib_add_two, add_tsub_cancel_right]
| 1 |
import Mathlib.SetTheory.Ordinal.Arithmetic
#align_import set_theory.ordinal.exponential from "leanprover-community/mathlib"@"b67044ba53af18680e1dd246861d9584e968495d"
noncomputable section
open Function Cardinal Set Equiv Order
open scoped Classical
open Cardinal Ordinal
universe u v w
namespace Ordinal
in... | Mathlib/SetTheory/Ordinal/Exponential.lean | 68 | 69 | theorem opow_le_of_limit {a b c : Ordinal} (a0 : a ≠ 0) (h : IsLimit b) :
a ^ b ≤ c ↔ ∀ b' < b, a ^ b' ≤ c := by | rw [opow_limit a0 h, bsup_le_iff]
| 1 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' u₁' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}... | Mathlib/LinearAlgebra/Dimension/Constructions.lean | 211 | 213 | theorem rank_matrix' (m n : Type v) [Finite m] [Finite n] :
Module.rank R (Matrix m n R) = Cardinal.lift.{u} (#m * #n) := by |
rw [rank_matrix, lift_mul, lift_umax.{v, u}]
| 1 |
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.polynomial.chebyshev from "leanprover-community/mathlib"@"d774451114d6045faeb6751c396bea1eb9058946"
namespace Polynomial.Chebyshev
set_option linter.uppercaseLean3 false -- `T` `U` `X`
open Polynomial
v... | Mathlib/RingTheory/Polynomial/Chebyshev.lean | 159 | 160 | theorem U_sub_one (n : ℤ) : U R (n - 1) = 2 * X * U R n - U R (n + 1) := by |
linear_combination (norm := ring_nf) U_add_two R (n - 1)
| 1 |
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import linear_algebra.quadratic_form.basic from "leanprover-community/mathlib"@"d11f435d4e34a6cea0a1797d6b625b0c170be845"
universe u v w
variable {S T : ... | Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 107 | 108 | theorem polar_smul [Monoid S] [DistribMulAction S R] (f : M → R) (s : S) (x y : M) :
polar (s • f) x y = s • polar f x y := by | simp only [polar, Pi.smul_apply, smul_sub]
| 1 |
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 | 89 | 91 | theorem cons_self_properDivisors (h : n ≠ 0) :
cons n (properDivisors n) properDivisors.not_self_mem = divisors n := by |
rw [cons_eq_insert, insert_self_properDivisors h]
| 1 |
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.with_bot_top from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
open Set
variable {α : Type*}
namespace WithTop
@[simp]
theorem preimage_coe_top : (some : α → WithTop α) ⁻¹' {⊤} =... | Mathlib/Order/Interval/Set/WithBotTop.lean | 89 | 90 | theorem image_coe_Ioi : (some : α → WithTop α) '' Ioi a = Ioo (a : WithTop α) ⊤ := by |
rw [← preimage_coe_Ioi, image_preimage_eq_inter_range, range_coe, Ioi_inter_Iio]
| 1 |
import Mathlib.Data.Set.Image
#align_import order.directed from "leanprover-community/mathlib"@"ffde2d8a6e689149e44fd95fa862c23a57f8c780"
open Function
universe u v w
variable {α : Type u} {β : Type v} {ι : Sort w} (r r' s : α → α → Prop)
local infixl:50 " ≼ " => r
def Directed (f : ι → α) :=
∀ x y, ∃ z, ... | Mathlib/Order/Directed.lean | 66 | 67 | theorem directedOn_range {f : ι → α} : Directed r f ↔ DirectedOn r (Set.range f) := by |
simp_rw [Directed, DirectedOn, Set.forall_mem_range, Set.exists_range_iff]
| 1 |
import Mathlib.Probability.ProbabilityMassFunction.Monad
#align_import probability.probability_mass_function.constructions from "leanprover-community/mathlib"@"4ac69b290818724c159de091daa3acd31da0ee6d"
universe u
namespace PMF
noncomputable section
variable {α β γ : Type*}
open scoped Classical
open NNReal ENN... | Mathlib/Probability/ProbabilityMassFunction/Constructions.lean | 60 | 60 | theorem mem_support_map_iff : b ∈ (map f p).support ↔ ∃ a ∈ p.support, f a = b := by | simp
| 1 |
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Index
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.SpecificGroups.Cyclic
import Mathlib.Tactic.IntervalCases
#align_import group_theory.p_gr... | Mathlib/GroupTheory/PGroup.lean | 123 | 124 | theorem powEquiv_symm_apply {n : ℕ} (hn : p.Coprime n) (g : G) :
(hG.powEquiv hn).symm g = g ^ (orderOf g).gcdB n := by | rw [← Nat.card_zpowers]; rfl
| 1 |
import Mathlib.Algebra.Group.Prod
import Mathlib.Order.Cover
#align_import algebra.support from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
assert_not_exists MonoidWithZero
open Set
namespace Function
variable {α β A B M N P G : Type*}
section One
variable [One M] [One N] [One P]
... | Mathlib/Algebra/Group/Support.lean | 127 | 129 | theorem disjoint_mulSupport_iff {f : α → M} {s : Set α} :
Disjoint s (mulSupport f) ↔ EqOn f 1 s := by |
rw [disjoint_comm, mulSupport_disjoint_iff]
| 1 |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncompu... | Mathlib/FieldTheory/RatFunc/Basic.lean | 117 | 118 | theorem ofFractionRing_neg (p : FractionRing K[X]) :
ofFractionRing (-p) = -ofFractionRing p := by | simp only [Neg.neg, RatFunc.neg]
| 1 |
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.SetTheory.Cardinal.Cofinality
#align_import linear_algebra.basis from "leanprover-communit... | Mathlib/LinearAlgebra/Basis.lean | 149 | 150 | theorem repr_self_apply (j) [Decidable (i = j)] : b.repr (b i) j = if i = j then 1 else 0 := by |
rw [repr_self, Finsupp.single_apply]
| 1 |
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Int.Basic
import Mathlib.Tactic.Ring
import Mathlib.Tactic.FieldSimp
import Mathlib.Data.Int.NatPrime
import Mathlib.Data.ZMod.Basic
#align_import number_theory.pythagorean_tri... | Mathlib/NumberTheory/PythagoreanTriples.lean | 60 | 61 | theorem PythagoreanTriple.zero : PythagoreanTriple 0 0 0 := by |
simp only [PythagoreanTriple, zero_mul, zero_add]
| 1 |
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Order.Filter.AtTopBot
import Mathlib.Tactic.GCongr
#align_import order.filter.archimedean from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
variable {α R : Type*}
open Filter Set Function
@[simp]
theorem Nat.comap_cast_atTop [S... | Mathlib/Order/Filter/Archimedean.lean | 100 | 102 | theorem Filter.Eventually.intCast_atBot [StrictOrderedRing R] [Archimedean R] {p : R → Prop}
(h : ∀ᶠ (x:R) in atBot, p x) : ∀ᶠ (n:ℤ) in atBot, p n := by |
rw [← Int.comap_cast_atBot (R := R)]; exact h.comap _
| 1 |
import Mathlib.Data.Fintype.List
#align_import data.list.cycle from "leanprover-community/mathlib"@"7413128c3bcb3b0818e3e18720abc9ea3100fb49"
assert_not_exists MonoidWithZero
open List
def Cycle (α : Type*) : Type _ :=
Quotient (IsRotated.setoid α)
#align cycle Cycle
namespace Cycle
variable {α : Type*}
--... | Mathlib/Data/List/Cycle.lean | 601 | 602 | theorem subsingleton_reverse_iff {s : Cycle α} : s.reverse.Subsingleton ↔ s.Subsingleton := by |
simp [length_subsingleton_iff]
| 1 |
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 | 95 | 97 | theorem trinomial_leadingCoeff (hkm : k < m) (hmn : m < n) (hw : w ≠ 0) :
(trinomial k m n u v w).leadingCoeff = w := by |
rw [leadingCoeff, trinomial_natDegree hkm hmn hw, trinomial_leading_coeff' hkm hmn]
| 1 |
import Mathlib.Analysis.Convex.Hull
#align_import analysis.convex.join from "leanprover-community/mathlib"@"951bf1d9e98a2042979ced62c0620bcfb3587cf8"
open Set
variable {ι : Sort*} {𝕜 E : Type*}
section OrderedSemiring
variable (𝕜) [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {s t s₁ s₂ t₁ t₂ u : Set ... | Mathlib/Analysis/Convex/Join.lean | 36 | 37 | theorem mem_convexJoin : x ∈ convexJoin 𝕜 s t ↔ ∃ a ∈ s, ∃ b ∈ t, x ∈ segment 𝕜 a b := by |
simp [convexJoin]
| 1 |
import Mathlib.AlgebraicTopology.DoldKan.Faces
import Mathlib.CategoryTheory.Idempotents.Basic
#align_import algebraic_topology.dold_kan.projections from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Pread... | Mathlib/AlgebraicTopology/DoldKan/Projections.lean | 100 | 101 | theorem Q_f_0_eq (q : ℕ) : ((Q q).f 0 : X _[0] ⟶ X _[0]) = 0 := by |
simp only [HomologicalComplex.sub_f_apply, HomologicalComplex.id_f, Q, P_f_0_eq, sub_self]
| 1 |
import Mathlib.AlgebraicTopology.DoldKan.FunctorN
#align_import algebraic_topology.dold_kan.normalized from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits
CategoryTheory.Subobject CategoryTheory.Idempotents DoldKan
non... | Mathlib/AlgebraicTopology/DoldKan/Normalized.lean | 77 | 78 | theorem PInftyToNormalizedMooreComplex_comp_inclusionOfMooreComplexMap (X : SimplicialObject A) :
PInftyToNormalizedMooreComplex X ≫ inclusionOfMooreComplexMap X = PInfty := by | aesop_cat
| 1 |
import Mathlib.Dynamics.Flow
import Mathlib.Tactic.Monotonicity
#align_import dynamics.omega_limit from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Function Filter Topology
section omegaLimit
variable {τ : Type*} {α : Type*} {β : Type*} {ι : Type*}
def omegaLimit [Topol... | Mathlib/Dynamics/OmegaLimit.lean | 142 | 144 | theorem mem_omegaLimit_iff_frequently₂ (y : β) :
y ∈ ω f ϕ s ↔ ∀ n ∈ 𝓝 y, ∃ᶠ t in f, (ϕ t '' s ∩ n).Nonempty := by |
simp_rw [mem_omegaLimit_iff_frequently, image_inter_nonempty_iff]
| 1 |
import Mathlib.Topology.Order.IsLUB
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section DenselyOrdered
variable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}
{s : Set α}
theorem closure_Ioi' {a : α} (h : (Io... | Mathlib/Topology/Order/DenselyOrdered.lean | 111 | 112 | theorem interior_Ico [NoMinOrder α] {a b : α} : interior (Ico a b) = Ioo a b := by |
rw [← Ici_inter_Iio, interior_inter, interior_Ici, interior_Iio, Ioi_inter_Iio]
| 1 |
import Mathlib.Algebra.Star.Order
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.Order.MonotoneContinuity
#align_import data.real.sqrt from "leanprover-community/mathlib"@"31c24aa72e7b3e5ed97a8412470e904f82b81004"
open Set Filter
open scoped Filter NNReal Topology
namespace NNReal
variable {x y... | Mathlib/Data/Real/Sqrt.lean | 97 | 98 | theorem sqrt_mul (x y : ℝ≥0) : sqrt (x * y) = sqrt x * sqrt y := by |
rw [sqrt_eq_iff_eq_sq, mul_pow, sq_sqrt, sq_sqrt]
| 1 |
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 207 | 208 | theorem preimage_add_const_Ioo : (fun x => x + a) ⁻¹' Ioo b c = Ioo (b - a) (c - a) := by |
simp [← Ioi_inter_Iio]
| 1 |
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