Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | num_lines int64 1 150 |
|---|---|---|---|---|---|---|
import Mathlib.Data.Finset.Image
#align_import data.finset.card from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
assert_not_exists MonoidWithZero
-- TODO: After a lot more work,
-- assert_not_exists OrderedCommMonoid
open Function Multiset Nat
variable {α β R : Type*}
namespace Fin... | Mathlib/Data/Finset/Card.lean | 107 | 108 | theorem card_insert_of_not_mem (h : a ∉ s) : (insert a s).card = s.card + 1 := by |
rw [← cons_eq_insert _ _ h, card_cons]
| 1 |
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w
variable {ι : Sort u} {α : Type v} {β : Type w}
open Set
open OrderDual (toDual)
namespace Set
section Preorder
variable [Preorder α] {a b c... | Mathlib/Order/Interval/Set/Disjoint.lean | 117 | 118 | theorem iUnion_Ico_right [NoMaxOrder α] (a : α) : ⋃ b, Ico a b = Ici a := by |
simp only [← Ici_inter_Iio, ← inter_iUnion, iUnion_Iio, inter_univ]
| 1 |
import Mathlib.Data.Finsupp.Basic
import Mathlib.Data.Finsupp.Order
#align_import data.finsupp.multiset from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Finset
variable {α β ι : Type*}
namespace Finsupp
def toMultiset : (α →₀ ℕ) →+ Multiset α where
toFun f := Finsupp.sum f... | Mathlib/Data/Finsupp/Multiset.lean | 52 | 53 | theorem toMultiset_single (a : α) (n : ℕ) : toMultiset (single a n) = n • {a} := by |
rw [toMultiset_apply, sum_single_index]; apply zero_nsmul
| 1 |
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import computability.tm_to_partrec from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
open Function (update)
open Relation
namespa... | Mathlib/Computability/TMToPartrec.lean | 149 | 152 | theorem cons_eval (f fs) : (cons f fs).eval = fun v => do {
let n ← Code.eval f v
let ns ← Code.eval fs v
pure (n.headI :: ns) } := by | simp [eval]
| 1 |
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.GroupTheory.MonoidLocalization
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.GroupTheory.GroupAction.Ring
#align_import ring_theory.localization.basic from "leanprover-community/mathlib"@"b69c9a770ecf37eb21... | Mathlib/RingTheory/Localization/Basic.lean | 202 | 204 | theorem sec_spec' (z : S) :
algebraMap R S (IsLocalization.sec M z).1 = algebraMap R S (IsLocalization.sec M z).2 * z := by |
rw [mul_comm, sec_spec]
| 1 |
import Mathlib.AlgebraicTopology.DoldKan.Projections
import Mathlib.CategoryTheory.Idempotents.FunctorCategories
import Mathlib.CategoryTheory.Idempotents.FunctorExtension
#align_import algebraic_topology.dold_kan.p_infty from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open Category... | Mathlib/AlgebraicTopology/DoldKan/PInfty.lean | 104 | 105 | theorem PInfty_f_idem (n : ℕ) : (PInfty.f n : X _[n] ⟶ _) ≫ PInfty.f n = PInfty.f n := by |
simp only [PInfty_f, P_f_idem]
| 1 |
import Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.bind from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
universe v
variable {α : Type*} {β : Type v} {γ δ : Ty... | Mathlib/Data/Multiset/Bind.lean | 138 | 138 | theorem bind_zero : s.bind (fun _ => 0 : α → Multiset β) = 0 := by | simp [bind, join, nsmul_zero]
| 1 |
import Mathlib.Data.List.Infix
#align_import data.list.rdrop from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
-- Make sure we don't import algebra
assert_not_exists Monoid
variable {α : Type*} (p : α → Bool) (l : List α) (n : ℕ)
namespace List
def rdrop : List α :=
l.take (l.leng... | Mathlib/Data/List/DropRight.lean | 78 | 78 | theorem rtake_zero : rtake l 0 = [] := by | simp [rtake]
| 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 | 63 | 64 | theorem lt_length_left_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.Divisibility.Basic
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Ring.Defs
#align_import algebra.euclidean_domain.defs from "leanprover-community/mathlib"@"ee7b9f9a9ac2a8d9f04ea39bbfe6b1a3be053b38"
universe u
class EuclideanDomain (R : Type u) extends CommRing R, Nontrivial R ... | Mathlib/Algebra/EuclideanDomain/Defs.lean | 157 | 157 | theorem mod_zero (a : R) : a % 0 = a := by | simpa only [zero_mul, zero_add] using div_add_mod a 0
| 1 |
import Mathlib.Data.Fin.VecNotation
import Mathlib.Logic.Embedding.Set
#align_import logic.equiv.fin from "leanprover-community/mathlib"@"bd835ef554f37ef9b804f0903089211f89cb370b"
assert_not_exists MonoidWithZero
universe u
variable {m n : ℕ}
def finZeroEquiv : Fin 0 ≃ Empty :=
Equiv.equivEmpty _
#align fin_... | Mathlib/Logic/Equiv/Fin.lean | 121 | 123 | theorem finSuccEquiv'_below {i : Fin (n + 1)} {m : Fin n} (h : Fin.castSucc m < i) :
(finSuccEquiv' i) (Fin.castSucc m) = m := by |
rw [← Fin.succAbove_of_castSucc_lt _ _ h, finSuccEquiv'_succAbove]
| 1 |
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Data.List.Chain
#align_import data.bool.count from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
namespace List
@[simp]
theorem count_not_add_count (l : List Bool) (b : Bool) : count (!b) l + count b l = length l := by
-- Porting ... | Mathlib/Data/Bool/Count.lean | 100 | 102 | theorem two_mul_count_bool_of_even (hl : Chain' (· ≠ ·) l) (h2 : Even (length l)) (b : Bool) :
2 * count b l = length l := by |
rw [← count_not_add_count l b, hl.count_not_eq_count h2, two_mul]
| 1 |
import Mathlib.Data.Multiset.Dedup
#align_import data.multiset.finset_ops from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
namespace Multiset
open List
variable {α : Type*} [DecidableEq α] {s : Multiset α}
def ndinsert (a : α) (s : Multiset α) : Multiset α :=
Quot.liftOn s (... | Mathlib/Data/Multiset/FinsetOps.lean | 83 | 84 | theorem dedup_cons {a : α} {s : Multiset α} : dedup (a ::ₘ s) = ndinsert a (dedup s) := by |
by_cases h : a ∈ s <;> simp [h]
| 1 |
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 | 75 | 76 | theorem left_ne_of_oangle_ne_zero {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ ≠ 0) : p₁ ≠ p₂ := by |
rw [← @vsub_ne_zero V]; exact o.left_ne_zero_of_oangle_ne_zero h
| 1 |
import Mathlib.Analysis.Convex.Cone.Basic
import Mathlib.Analysis.InnerProductSpace.Projection
#align_import analysis.convex.cone.dual from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284a161a7f2a9e3d4"
open Set LinearMap
open scoped Classical
open Pointwise
variable {𝕜 E F G : Type*}
section Dua... | Mathlib/Analysis/Convex/Cone/InnerDual.lean | 119 | 121 | theorem innerDualCone_sUnion (S : Set (Set H)) :
(⋃₀ S).innerDualCone = sInf (Set.innerDualCone '' S) := by |
simp_rw [sInf_image, sUnion_eq_biUnion, innerDualCone_iUnion]
| 1 |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Ring.Opposite
import Mathlib.Tactic.Abel
#align_import algebra.geom_sum fro... | Mathlib/Algebra/GeomSum.lean | 60 | 60 | theorem geom_sum_one (x : α) : ∑ i ∈ range 1, x ^ i = 1 := by | simp [geom_sum_succ']
| 1 |
import Mathlib.Algebra.Order.Group.Defs
import Mathlib.Algebra.Order.Monoid.WithTop
#align_import algebra.order.group.with_top from "leanprover-community/mathlib"@"f178c0e25af359f6cbc72a96a243efd3b12423a3"
namespace WithTop
variable {α : Type*}
namespace LinearOrderedAddCommGroup
variable [LinearOrderedAddCommG... | Mathlib/Algebra/Order/Group/WithTop.lean | 65 | 65 | theorem sub_top {a : WithTop α} : a - ⊤ = ⊤ := by | cases a <;> rfl
| 1 |
import Mathlib.Init.Logic
import Mathlib.Init.Function
import Mathlib.Init.Algebra.Classes
import Batteries.Util.LibraryNote
import Batteries.Tactic.Lint.Basic
#align_import logic.basic from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
#align_import init.ite_simp from "leanprover-communit... | Mathlib/Logic/Basic.lean | 601 | 602 | theorem rec_heq_of_heq {C : α → Sort*} {x : C a} {y : β} (e : a = b) (h : HEq x y) :
HEq (e ▸ x) y := by | subst e; exact h
| 1 |
import Batteries.Classes.Order
namespace Batteries.PairingHeapImp
inductive Heap (α : Type u) where
| nil : Heap α
| node (a : α) (child sibling : Heap α) : Heap α
deriving Repr
def Heap.size : Heap α → Nat
| .nil => 0
| .node _ c s => c.size + 1 + s.size
def Heap.singleton (a : α) : Heap α := .... | .lake/packages/batteries/Batteries/Data/PairingHeap.lean | 103 | 105 | theorem Heap.noSibling_deleteMin {s : Heap α} (eq : s.deleteMin le = some (a, s')) :
s'.NoSibling := by |
cases s with cases eq | node a c => exact noSibling_combine _ _
| 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 | 47 | 51 | theorem birthday_def (x : PGame) :
birthday x =
max (lsub.{u, u} fun i => birthday (x.moveLeft i))
(lsub.{u, u} fun i => birthday (x.moveRight i)) := by |
cases x; rw [birthday]; rfl
| 1 |
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Data.Nat.Cast.NeZero
import Mathlib.Algebra.Order.Ring.Nat
#align_import data.nat.cast.basic from "leanprover-community/mathlib"@"acebd8d49928f6ed8920e502a6c90674e75bd441"
variable {α β : T... | Mathlib/Data/Nat/Cast/Order.lean | 88 | 88 | theorem cast_pos' {n : ℕ} : (0 : α) < n ↔ 0 < n := by | cases n <;> simp [cast_add_one_pos]
| 1 |
import Mathlib.Topology.Separation
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.UniformSpace.Cauchy
#align_import topology.uniform_space.uniform_convergence from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062a169c9"
noncomputable section
open Topology Uniformity Filter S... | Mathlib/Topology/UniformSpace/UniformConvergence.lean | 160 | 162 | theorem tendstoUniformly_iff_tendsto {F : ι → α → β} {f : α → β} {p : Filter ι} :
TendstoUniformly F f p ↔ Tendsto (fun q : ι × α => (f q.2, F q.1 q.2)) (p ×ˢ ⊤) (𝓤 β) := by |
simp [tendstoUniformly_iff_tendstoUniformlyOnFilter, tendstoUniformlyOnFilter_iff_tendsto]
| 1 |
import Mathlib.Data.Finite.Card
import Mathlib.GroupTheory.Commutator
import Mathlib.GroupTheory.Finiteness
#align_import group_theory.abelianization from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
universe u v w
-- Let G be a group.
variable (G : Type u) [Group G]
open Subgroup (... | Mathlib/GroupTheory/Abelianization.lean | 49 | 50 | theorem commutator_eq_closure : commutator G = Subgroup.closure (commutatorSet G) := by |
simp [commutator, Subgroup.commutator_def, commutatorSet]
| 1 |
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.ord_connected_component from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Interval Function OrderDual
namespace Set
variable {α : Type*} [LinearOrder α] {s t : Set α}... | Mathlib/Order/Interval/Set/OrdConnectedComponent.lean | 63 | 64 | theorem ordConnectedComponent_eq_empty : ordConnectedComponent s x = ∅ ↔ x ∉ s := by |
rw [← not_nonempty_iff_eq_empty, nonempty_ordConnectedComponent]
| 1 |
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 | 33 | 34 | theorem range_list_map_coe (s : Set α) : range (map ((↑) : s → α)) = { l | ∀ x ∈ l, x ∈ s } := by |
rw [range_list_map, Subtype.range_coe]
| 1 |
import Mathlib.Data.List.Basic
#align_import data.list.join from "leanprover-community/mathlib"@"18a5306c091183ac90884daa9373fa3b178e8607"
-- Make sure we don't import algebra
assert_not_exists Monoid
variable {α β : Type*}
namespace List
attribute [simp] join
-- Porting note (#10618): simp can prove this
-- @... | Mathlib/Data/List/Join.lean | 65 | 66 | theorem join_join (l : List (List (List α))) : l.join.join = (l.map join).join := by |
induction l <;> simp [*]
| 1 |
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Order.Bounds.OrderIso
import Mathlib.Tactic.Positivity.Core
#align_import algebra.order.field.basic from "leanprover-community/mathlib"@"8477... | Mathlib/Algebra/Order/Field/Basic.lean | 630 | 631 | theorem div_pos_iff : 0 < a / b ↔ 0 < a ∧ 0 < b ∨ a < 0 ∧ b < 0 := by |
simp only [division_def, mul_pos_iff, inv_pos, inv_lt_zero]
| 1 |
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.GroupTheory.GroupAction.Units
#align_import data.int.absolute_value from "leanprover-community/mathlib"@"9aba7801eeecebb61f58a5763c2b6dd1b47dc6ef"
variable {R S : Type*} [Ring R] [Linea... | Mathlib/Data/Int/AbsoluteValue.lean | 41 | 42 | theorem AbsoluteValue.map_units_int_smul (abv : AbsoluteValue R S) (x : ℤˣ) (y : R) :
abv (x • y) = abv y := by | rcases Int.units_eq_one_or x with (rfl | rfl) <;> simp
| 1 |
import Mathlib.MeasureTheory.OuterMeasure.Basic
open Filter Set
open scoped ENNReal
namespace MeasureTheory
variable {α β F : Type*} [FunLike F (Set α) ℝ≥0∞] [OuterMeasureClass F α] {μ : F} {s t : Set α}
def ae (μ : F) : Filter α :=
.ofCountableUnion (μ · = 0) (fun _S hSc ↦ (measure_sUnion_null_iff hSc).2) fu... | Mathlib/MeasureTheory/OuterMeasure/AE.lean | 148 | 151 | theorem ae_le_set : s ≤ᵐ[μ] t ↔ μ (s \ t) = 0 :=
calc
s ≤ᵐ[μ] t ↔ ∀ᵐ x ∂μ, x ∈ s → x ∈ t := Iff.rfl
_ ↔ μ (s \ t) = 0 := by | simp [ae_iff]; rfl
| 1 |
import Mathlib.MeasureTheory.Decomposition.SignedHahn
import Mathlib.MeasureTheory.Measure.MutuallySingular
#align_import measure_theory.decomposition.jordan from "leanprover-community/mathlib"@"70a4f2197832bceab57d7f41379b2592d1110570"
noncomputable section
open scoped Classical MeasureTheory ENNReal NNReal
va... | Mathlib/MeasureTheory/Decomposition/Jordan.lean | 153 | 155 | theorem real_smul_negPart_nonneg (r : ℝ) (hr : 0 ≤ r) :
(r • j).negPart = r.toNNReal • j.negPart := by |
rw [real_smul_def, ← smul_negPart, if_pos hr]
| 1 |
import Mathlib.Order.UpperLower.Basic
import Mathlib.Data.Finset.Preimage
#align_import combinatorics.young.young_diagram from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Function
@[ext]
structure YoungDiagram where
cells : Finset (ℕ × ℕ)
isLowerSet : IsLowerSet (cel... | Mathlib/Combinatorics/Young/YoungDiagram.lean | 351 | 351 | theorem mk_mem_col_iff {μ : YoungDiagram} {i j : ℕ} : (i, j) ∈ μ.col j ↔ (i, j) ∈ μ := by | simp [col]
| 1 |
import Mathlib.RingTheory.Noetherian
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.DirectSum.Finsupp
import Mathlib.Algebra.Module.Projective
import Mathlib.Algebra.Module.Injective
import Mathlib.Algebra.Module.CharacterModule
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
import Mathlib.Linear... | Mathlib/RingTheory/Flat/Basic.lean | 112 | 114 | theorem iff_lTensor_injective :
Module.Flat R M ↔ ∀ ⦃I : Ideal R⦄ (_ : I.FG), Function.Injective (lTensor M I.subtype) := by |
simpa [← comm_comp_rTensor_comp_comm_eq] using Module.Flat.iff_rTensor_injective R M
| 1 |
import Mathlib.Order.BooleanAlgebra
import Mathlib.Tactic.Common
#align_import order.heyting.boundary from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
variable {α : Type*}
namespace Coheyting
variable [CoheytingAlgebra α] {a b : α}
def boundary (a : α) : α :=
a ⊓ ￢a
#align cohe... | Mathlib/Order/Heyting/Boundary.lean | 63 | 63 | theorem boundary_top : ∂ (⊤ : α) = ⊥ := by | rw [boundary, hnot_top, inf_bot_eq]
| 1 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.Block
#align_import analysis.inner_product_space.gram_schmidt_ortho from "leanprover-community/mathlib"@"1a4df69ca1a9a0e5e26bfe12e2b92814216016d0"
open Finset Submodule FiniteDimensional
variable (𝕜 : Type*) {E : Type*} [RCLike �... | Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean | 58 | 60 | theorem gramSchmidt_def (f : ι → E) (n : ι) :
gramSchmidt 𝕜 f n = f n - ∑ i ∈ Iio n, orthogonalProjection (𝕜 ∙ gramSchmidt 𝕜 f i) (f n) := by |
rw [← sum_attach, attach_eq_univ, gramSchmidt]
| 1 |
import Mathlib.MeasureTheory.OuterMeasure.Caratheodory
#align_import measure_theory.measure.outer_measure from "leanprover-community/mathlib"@"343e80208d29d2d15f8050b929aa50fe4ce71b55"
noncomputable section
open Set Function Filter
open scoped Classical NNReal Topology ENNReal
namespace MeasureTheory
open Outer... | Mathlib/MeasureTheory/OuterMeasure/Induced.lean | 52 | 52 | theorem extend_eq_top {s : α} (h : ¬P s) : extend m s = ∞ := by | simp [extend, h]
| 1 |
import Mathlib.Algebra.Quaternion
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Topology.Algebra.Algebra
#align_import analysis.quaternion from "leanprover-community/mathlib"@"07992a1d1f7a4176c6d3f160209608be4e198566"
@[inherit_doc] scoped[Quaternion... | Mathlib/Analysis/Quaternion.lean | 73 | 74 | theorem norm_coe (a : ℝ) : ‖(a : ℍ)‖ = ‖a‖ := by |
rw [norm_eq_sqrt_real_inner, inner_self, normSq_coe, Real.sqrt_sq_eq_abs, Real.norm_eq_abs]
| 1 |
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Measure.GiryMonad
#align_import probability.kernel.basic from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open MeasureTheory
open scoped MeasureTheory ENNReal NNReal
namespace ProbabilityTheory
noncomputab... | Mathlib/Probability/Kernel/Basic.lean | 117 | 118 | theorem finset_sum_apply' (I : Finset ι) (κ : ι → kernel α β) (a : α) (s : Set β) :
(∑ i ∈ I, κ i) a s = ∑ i ∈ I, κ i a s := by | rw [finset_sum_apply, Measure.finset_sum_apply]
| 1 |
import Mathlib.Data.Multiset.Nodup
#align_import data.multiset.dedup from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
open List
variable {α β : Type*} [DecidableEq α]
def dedup (s : Multiset α) : Multiset α :=
Quot.liftOn s (fun l => (l.dedup : Multiset α)... | Mathlib/Data/Multiset/Dedup.lean | 112 | 113 | theorem le_dedup_self {s : Multiset α} : s ≤ dedup s ↔ Nodup s := by |
rw [le_dedup, and_iff_right le_rfl]
| 1 |
import Mathlib.Topology.Separation
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.UniformSpace.Cauchy
#align_import topology.uniform_space.uniform_convergence from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062a169c9"
noncomputable section
open Topology Uniformity Filter S... | Mathlib/Topology/UniformSpace/UniformConvergence.lean | 138 | 139 | theorem tendstoUniformlyOn_univ : TendstoUniformlyOn F f p univ ↔ TendstoUniformly F f p := by |
simp [TendstoUniformlyOn, TendstoUniformly]
| 1 |
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.GroupTheory.FreeGroup.IsFreeGroup
import Mathlib.Data.List.Chain
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Data.Set.Pointwise.SMul
#align_import group_theor... | Mathlib/GroupTheory/CoprodI.lean | 199 | 200 | theorem of_injective (i : ι) : Function.Injective (of : M i →* _) := by |
classical exact (of_leftInverse i).injective
| 1 |
import Mathlib.Order.Filter.AtTopBot
import Mathlib.Order.Filter.Pi
#align_import order.filter.cofinite from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
open Set Function
variable {ι α β : Type*} {l : Filter α}
namespace Filter
def cofinite : Filter α :=
comk Set.Finite finite_e... | Mathlib/Order/Filter/Cofinite.lean | 57 | 58 | theorem cofinite_eq_bot_iff : @cofinite α = ⊥ ↔ Finite α := by |
simp [← empty_mem_iff_bot, finite_univ_iff]
| 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 | 75 | 75 | theorem properDivisors.not_self_mem : ¬n ∈ properDivisors n := by | simp [properDivisors]
| 1 |
import Mathlib.CategoryTheory.Monoidal.Mon_
#align_import category_theory.monoidal.Mod_ from "leanprover-community/mathlib"@"33085c9739c41428651ac461a323fde9a2688d9b"
universe v₁ v₂ u₁ u₂
open CategoryTheory MonoidalCategory
variable (C : Type u₁) [Category.{v₁} C] [MonoidalCategory.{v₁} C]
variable {C}
struc... | Mathlib/CategoryTheory/Monoidal/Mod_.lean | 37 | 38 | theorem assoc_flip :
(A.X ◁ M.act) ≫ M.act = (α_ A.X A.X M.X).inv ≫ (A.mul ▷ M.X) ≫ M.act := by | simp
| 1 |
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.NthRewrite
#align_import data.nat.gcd.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
namespace Nat
theorem gcd_greatest {a b d : ℕ} (hda : d ∣ a) (hdb : d ∣ b) (hd ... | Mathlib/Data/Nat/GCD/Basic.lean | 35 | 36 | theorem gcd_add_mul_right_right (m n k : ℕ) : gcd m (n + k * m) = gcd m n := by |
simp [gcd_rec m (n + k * m), gcd_rec m n]
| 1 |
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.IsAdjoinRoot
#align_import number_theory.kummer_dedekind from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
variable (R : Type*) {S : Type*} [CommRing R] [CommRing S] [Algebra R S]
open Ideal Polynomial DoubleQuo... | Mathlib/NumberTheory/KummerDedekind.lean | 85 | 86 | theorem conductor_eq_top_of_adjoin_eq_top (h : R<x> = ⊤) : conductor R x = ⊤ := by |
simp only [Ideal.eq_top_iff_one, mem_conductor_iff, h, mem_top, forall_const]
| 1 |
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.matrix from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped NNReal Matrix
namespace Matrix
variable {R l m n α β : Type*} [Fintype l] [Fintyp... | Mathlib/Analysis/Matrix.lean | 151 | 152 | theorem nnnorm_col (v : m → α) : ‖col v‖₊ = ‖v‖₊ := by |
simp [nnnorm_def, Pi.nnnorm_def]
| 1 |
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.matrix.nonsingular_inverse from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
namespace Matrix
universe u u' v
variable {l : ... | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 151 | 152 | theorem isUnit_iff_isUnit_det : IsUnit A ↔ IsUnit A.det := by |
simp only [← nonempty_invertible_iff_isUnit, (invertibleEquivDetInvertible A).nonempty_congr]
| 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 | 46 | 47 | theorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by |
simp [prod_eq_multiset_prod]
| 1 |
import Mathlib.Algebra.Algebra.Hom
import Mathlib.RingTheory.Ideal.Quotient
#align_import algebra.ring_quot from "leanprover-community/mathlib"@"e5820f6c8fcf1b75bcd7738ae4da1c5896191f72"
universe uR uS uT uA u₄
variable {R : Type uR} [Semiring R]
variable {S : Type uS} [CommSemiring S]
variable {T : Type uT}
vari... | Mathlib/Algebra/RingQuot.lean | 67 | 68 | theorem Rel.neg {R : Type uR} [Ring R] {r : R → R → Prop} ⦃a b : R⦄ (h : Rel r a b) :
Rel r (-a) (-b) := by | simp only [neg_eq_neg_one_mul a, neg_eq_neg_one_mul b, Rel.mul_right h]
| 1 |
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 | 153 | 153 | theorem sr_mul_self (i : ZMod n) : sr i * sr i = 1 := by | rw [sr_mul_sr, sub_self, one_def]
| 1 |
import Mathlib.Data.ENNReal.Real
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Topology.UniformSpace.Pi
import Mathlib.Topology.UniformSpace.UniformConvergence
import Mathlib.Topology.UniformSpace.UniformEmbedding
#align_import topology.metric_space.emetric_space from "leanprover-community/mathlib"@"c8f3055... | Mathlib/Topology/EMetricSpace/Basic.lean | 110 | 111 | theorem edist_triangle_left (x y z : α) : edist x y ≤ edist z x + edist z y := by |
rw [edist_comm z]; apply edist_triangle
| 1 |
import Mathlib.Data.ENNReal.Basic
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.MetricSpace.Thickening
#align_import topology.metric_space.thickened_indicator from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classical
open NNReal ENNReal Topol... | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | 69 | 71 | theorem thickenedIndicatorAux_le_one (δ : ℝ) (E : Set α) (x : α) :
thickenedIndicatorAux δ E x ≤ 1 := by |
apply @tsub_le_self _ _ _ _ (1 : ℝ≥0∞)
| 1 |
import Mathlib.SetTheory.Cardinal.ToNat
import Mathlib.Data.Nat.PartENat
#align_import set_theory.cardinal.basic from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
universe u v
open Function
variable {α : Type u}
namespace Cardinal
noncomputable def toPartENat : Cardinal →+o PartEN... | Mathlib/SetTheory/Cardinal/PartENat.lean | 43 | 44 | theorem toPartENat_apply_of_lt_aleph0 {c : Cardinal} (h : c < ℵ₀) : toPartENat c = toNat c := by |
lift c to ℕ using h; simp
| 1 |
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 TopologicalGroup
variable [TopologicalSpace G] [Group G] [TopologicalGroup G] [BorelSpace ... | Mathlib/MeasureTheory/Group/LIntegral.lean | 71 | 73 | theorem lintegral_eq_zero_of_isMulLeftInvariant [Regular μ] [NeZero μ] {f : G → ℝ≥0∞}
(hf : Continuous f) : ∫⁻ x, f x ∂μ = 0 ↔ f = 0 := by |
rw [lintegral_eq_zero_iff hf.measurable, hf.ae_eq_iff_eq μ continuous_zero]
| 1 |
import Mathlib.Data.List.Basic
#align_import data.list.lattice from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
open Nat
namespace List
variable {α : Type*} {l l₁ l₂ : List α} {p : α → Prop} {a : α}
variable [DecidableEq α]
section Inter
@[simp]
theorem inter_nil (l : L... | Mathlib/Data/List/Lattice.lean | 183 | 184 | theorem inter_reverse {xs ys : List α} : xs.inter ys.reverse = xs.inter ys := by |
simp only [List.inter, elem_eq_mem, mem_reverse]
| 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 | 118 | 119 | theorem prod_univ_two [CommMonoid β] (f : Fin 2 → β) : ∏ i, f i = f 0 * f 1 := by |
simp [prod_univ_succ]
| 1 |
import Mathlib.Geometry.Manifold.VectorBundle.Tangent
#align_import geometry.manifold.mfderiv from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
noncomputable section
open scoped Classical Topology Manifold
open Set ChartedSpace
section DerivativesDefinitions
variable {𝕜 : Type*} ... | Mathlib/Geometry/Manifold/MFDeriv/Defs.lean | 203 | 207 | theorem mdifferentiableWithinAt_iff' (f : M → M') (s : Set M) (x : M) :
MDifferentiableWithinAt I I' f s x ↔ ContinuousWithinAt f s x ∧
DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I' x f)
((extChartAt I x).symm ⁻¹' s ∩ range I) ((extChartAt I x) x) := by |
rw [MDifferentiableWithinAt, liftPropWithinAt_iff']; rfl
| 1 |
import Mathlib.Order.RelIso.Basic
import Mathlib.Logic.Embedding.Set
import Mathlib.Logic.Equiv.Set
#align_import order.rel_iso.set from "leanprover-community/mathlib"@"ee0c179cd3c8a45aa5bffbf1b41d8dbede452865"
open Function
universe u v w
variable {α β γ δ : Type*} {r : α → α → Prop} {s : β → β → Prop} {t : γ ... | Mathlib/Order/RelIso/Set.lean | 111 | 112 | theorem RelIso.preimage_eq_image_symm (e : r ≃r s) (t : Set β) : e ⁻¹' t = e.symm '' t := by |
rw [e.symm.image_eq_preimage_symm]; rfl
| 1 |
import Mathlib.LinearAlgebra.Dimension.LinearMap
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
#align_import linear_algebra.free_module.finite.matrix from "leanprover-community/mathlib"@"b1c23399f01266afe392a0d8f71f599a0dad4f7b"
universe u u' v w
variable (R : Type u) (S : Type u') (M : Type v) (N ... | Mathlib/LinearAlgebra/FreeModule/Finite/Matrix.lean | 66 | 68 | theorem FiniteDimensional.rank_linearMap_self :
Module.rank S (M →ₗ[R] S) = lift.{u'} (Module.rank R M) := by |
rw [rank_linearMap, rank_self, lift_one, mul_one]
| 1 |
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.NthRewrite
#align_import data.nat.gcd.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
namespace Nat
theorem gcd_greatest {a b d : ℕ} (hda : d ∣ a) (hdb : d ∣ b) (hd ... | Mathlib/Data/Nat/GCD/Basic.lean | 49 | 49 | theorem gcd_mul_left_add_right (m n k : ℕ) : gcd m (m * k + n) = gcd m n := by | simp [add_comm _ n]
| 1 |
import Mathlib.Data.Option.NAry
import Mathlib.Data.Seq.Computation
#align_import data.seq.seq from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace Stream'
universe u v w
def IsSeq {α : Type u} (s : Stream' (Option α)) : Prop :=
∀ {n : ℕ}, s n = none → s (n + 1) = none
#al... | Mathlib/Data/Seq/Seq.lean | 129 | 130 | theorem not_terminates_iff {s : Seq α} : ¬s.Terminates ↔ ∀ n, (s.get? n).isSome := by |
simp only [Terminates, TerminatedAt, ← Ne.eq_def, Option.ne_none_iff_isSome, not_exists, iff_self]
| 1 |
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Cases
import Mathlib.Tactic.SimpRw
import Mathlib.Tactic.SplitIfs
#align_import algebra.group.basic from "leanprover-community/mathlib"@"a07d750983b94c530a... | Mathlib/Algebra/Group/Basic.lean | 208 | 209 | theorem mul_rotate' (a b c : G) : a * (b * c) = b * (c * a) := by |
simp only [mul_left_comm, mul_comm]
| 1 |
import Mathlib.Init.Logic
import Mathlib.Init.Function
import Mathlib.Init.Algebra.Classes
import Batteries.Util.LibraryNote
import Batteries.Tactic.Lint.Basic
#align_import logic.basic from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
#align_import init.ite_simp from "leanprover-communit... | Mathlib/Logic/Basic.lean | 1,131 | 1,131 | theorem exists₂_imp : (∃ x h, P x h) → b ↔ ∀ x h, P x h → b := by | simp
| 1 |
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
#align_import data.multiset.lattice from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
namespace Multiset
variable {α : Type*}
section Sup
-- can be defined with just `[Bot α]` where some lemmas hold without... | Mathlib/Data/Multiset/Lattice.lean | 84 | 85 | theorem sup_union (s₁ s₂ : Multiset α) : (s₁ ∪ s₂).sup = s₁.sup ⊔ s₂.sup := by |
rw [← sup_dedup, dedup_ext.2, sup_dedup, sup_add]; simp
| 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 | 175 | 175 | theorem sub_lie : ⁅x - y, m⁆ = ⁅x, m⁆ - ⁅y, m⁆ := by | simp [sub_eq_add_neg]
| 1 |
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Ring.Commute
import Mathlib.Algebra.Ring.Invertible
import Mathlib.Order.Synonym
#align_import algebra.field.basic from "leanprover-community/mathlib"@"05101c3df9d9cfe9430edc205860c79b6d660102"
open Function ... | Mathlib/Algebra/Field/Basic.lean | 117 | 118 | theorem neg_div (a b : K) : -b / a = -(b / a) := by |
rw [neg_eq_neg_one_mul, mul_div_assoc, ← neg_eq_neg_one_mul]
| 1 |
import Mathlib.LinearAlgebra.Quotient
import Mathlib.LinearAlgebra.Prod
#align_import linear_algebra.projection from "leanprover-community/mathlib"@"6d584f1709bedbed9175bd9350df46599bdd7213"
noncomputable section Ring
variable {R : Type*} [Ring R] {E : Type*} [AddCommGroup E] [Module R E]
variable {F : Type*} [Ad... | Mathlib/LinearAlgebra/Projection.lean | 160 | 161 | theorem linearProjOfIsCompl_apply_left (h : IsCompl p q) (x : p) :
linearProjOfIsCompl p q h x = x := by | simp [linearProjOfIsCompl]
| 1 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 74 | 77 | theorem HasDerivAtFilter.scomp (hg : HasDerivAtFilter g₁ g₁' (h x) L')
(hh : HasDerivAtFilter h h' x L) (hL : Tendsto h L L') :
HasDerivAtFilter (g₁ ∘ h) (h' • g₁') x L := by |
simpa using ((hg.restrictScalars 𝕜).comp x hh hL).hasDerivAtFilter
| 1 |
import Batteries.Tactic.Init
import Batteries.Tactic.Alias
import Batteries.Tactic.Lint.Misc
instance {f : α → β} [DecidablePred p] : DecidablePred (p ∘ f) :=
inferInstanceAs <| DecidablePred fun x => p (f x)
@[deprecated] alias proofIrrel := proof_irrel
theorem Function.id_def : @id α = fun x => x := rfl
al... | .lake/packages/batteries/Batteries/Logic.lean | 100 | 103 | theorem eqRec_heq_iff_heq {α : Sort _} {a : α} {motive : (a' : α) → a = a' → Sort _}
(x : motive a (rfl : a = a)) {a' : α} (e : a = a') {β : Sort _} (y : β) :
HEq (@Eq.rec α a motive x a' e) y ↔ HEq x y := by |
subst e; rfl
| 1 |
import Mathlib.Tactic.Qify
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation
import Mathlib.NumberTheory.Zsqrtd.Basic
#align_import number_theory.pell from "leanprover-community/mathlib"@"7ad820c4997738e2f542f8a20f32911f52020e26"
namespace Pell
open Zsqrtd
theorem is_pell_s... | Mathlib/NumberTheory/Pell.lean | 137 | 137 | theorem prop_y (a : Solution₁ d) : d * a.y ^ 2 = a.x ^ 2 - 1 := by | rw [← a.prop]; ring
| 1 |
import Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction
import Mathlib.Analysis.BoxIntegral.Partition.Split
#align_import analysis.box_integral.partition.filter from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Set Function Filter Metric Finset Bool
open scoped Classical
o... | Mathlib/Analysis/BoxIntegral/Partition/Filter.lean | 347 | 349 | theorem rCond_of_bRiemann_eq_false {ι} (l : IntegrationParams) (hl : l.bRiemann = false)
{r : (ι → ℝ) → Ioi (0 : ℝ)} : l.RCond r := by |
simp [RCond, hl]
| 1 |
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.Star.SelfAdjoint
#align_import algebra.star.order from "leanprover-community/mathlib"@"31c24aa72e7b3e5ed97a8412470e904f82b81004"
open Set
open scoped NNRat
universe u
variable {R : Type u}
class StarOrderedRing (R : Type u) [NonUnitalSemi... | Mathlib/Algebra/Star/Order.lean | 137 | 139 | theorem nonneg_iff [NonUnitalSemiring R] [PartialOrder R] [StarRing R] [StarOrderedRing R] {x : R} :
0 ≤ x ↔ x ∈ AddSubmonoid.closure (Set.range fun s : R => star s * s) := by |
simp only [le_iff, zero_add, exists_eq_right']
| 1 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Comp
import Mathlib.Analysis.Calculus.FDeriv.RestrictScalars
#align_import analysis.calculus.deriv.comp from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Top... | Mathlib/Analysis/Calculus/Deriv/Comp.lean | 357 | 361 | theorem HasFDerivWithinAt.comp_hasDerivWithinAt_of_eq {t : Set F}
(hl : HasFDerivWithinAt l l' t y)
(hf : HasDerivWithinAt f f' s x) (hst : MapsTo f s t) (hy : y = f x) :
HasDerivWithinAt (l ∘ f) (l' f') s x := by |
rw [hy] at hl; exact hl.comp_hasDerivWithinAt x hf hst
| 1 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
namespace Polynomial
open Polynomial
section Semiring
variable {R : Type*} [Semiring R] (p q : R... | Mathlib/Algebra/Polynomial/Mirror.lean | 44 | 44 | theorem mirror_zero : (0 : R[X]).mirror = 0 := by | simp [mirror]
| 1 |
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Adjunction.Reflective
#align_import algebraic_geometry.Gamma_Spec_adjunction from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
-- Explicit universe annotations were u... | Mathlib/AlgebraicGeometry/GammaSpecAdjunction.lean | 77 | 79 | theorem not_mem_prime_iff_unit_in_stalk (r : Γ.obj (op X)) (x : X) :
r ∉ (X.toΓSpecFun x).asIdeal ↔ IsUnit (X.ΓToStalk x r) := by |
erw [LocalRing.mem_maximalIdeal, Classical.not_not]
| 1 |
import Mathlib.Data.Set.Subsingleton
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Group.Nat
import Mathlib.Data.Set.Basic
#align_import data.set.equitable from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
variable {α β : Type*}
namespace Set
def Equ... | Mathlib/Data/Set/Equitable.lean | 57 | 59 | theorem equitableOn_iff_exists_image_subset_icc {s : Set α} {f : α → ℕ} :
s.EquitableOn f ↔ ∃ b, f '' s ⊆ Icc b (b + 1) := by |
simpa only [image_subset_iff] using equitableOn_iff_exists_le_le_add_one
| 1 |
import Mathlib.Computability.Halting
#align_import computability.reduce from "leanprover-community/mathlib"@"d13b3a4a392ea7273dfa4727dbd1892e26cfd518"
universe u v w
open Function
def ManyOneReducible {α β} [Primcodable α] [Primcodable β] (p : α → Prop) (q : β → Prop) :=
∃ f, Computable f ∧ ∀ a, p a ↔ q (f a... | Mathlib/Computability/Reduce.lean | 111 | 113 | theorem OneOneReducible.of_equiv_symm {α β} [Primcodable α] [Primcodable β] {e : α ≃ β}
(q : β → Prop) (h : Computable e.symm) : q ≤₁ (q ∘ e) := by |
convert OneOneReducible.of_equiv _ h; funext; simp
| 1 |
import Mathlib.Algebra.Group.Units.Equiv
import Mathlib.CategoryTheory.Endomorphism
#align_import category_theory.conj from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
universe v u
namespace CategoryTheory
namespace Iso
variable {C : Type u} [Category.{v} C]
def homCongr {X Y X₁... | Mathlib/CategoryTheory/Conj.lean | 124 | 125 | theorem symm_self_conj (f : End X) : α.symm.conj (α.conj f) = f := by |
rw [← trans_conj, α.self_symm_id, refl_conj]
| 1 |
import Mathlib.Algebra.Group.Fin
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import linear_algebra.matrix.circulant from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
variable {α β m n R : Type*}
namespace Matrix
open Function
open Matrix
def circulant [Sub n] (v : n → α)... | Mathlib/LinearAlgebra/Matrix/Circulant.lean | 85 | 86 | theorem conjTranspose_circulant [Star α] [AddGroup n] (v : n → α) :
(circulant v)ᴴ = circulant (star fun i => v (-i)) := by | ext; simp
| 1 |
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib.Data.Set.Finite
#align_import data.finsupp.defs from "leanprover-community/mathlib"@"842328d9df7e96fd90fc424e115679c15fb23a71"
noncomputable section
open Finset Function
variable {α β γ ι M M' N P G H R S : Type*}... | Mathlib/Data/Finsupp/Defs.lean | 203 | 204 | theorem support_nonempty_iff {f : α →₀ M} : f.support.Nonempty ↔ f ≠ 0 := by |
simp only [Finsupp.support_eq_empty, Finset.nonempty_iff_ne_empty, Ne]
| 1 |
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Order.Fin
import Mathlib.Order.PiLex
import Mathlib.Order.Interval.Set.Basic
#align_import data.fin.tuple.basic from "leanprover-community/mathlib"@"ef997baa41b5c428be3fb50089a7139bf4ee886b"
assert_not_exists MonoidWithZero
un... | Mathlib/Data/Fin/Tuple/Basic.lean | 78 | 78 | theorem cons_succ : cons x p i.succ = p i := by | simp [cons]
| 1 |
import Mathlib.Topology.MetricSpace.ProperSpace
import Mathlib.Topology.MetricSpace.Cauchy
open Set Filter Bornology
open scoped ENNReal Uniformity Topology Pointwise
universe u v w
variable {α : Type u} {β : Type v} {X ι : Type*}
variable [PseudoMetricSpace α]
namespace Metric
#align metric.bounded Bornology.I... | Mathlib/Topology/MetricSpace/Bounded.lean | 133 | 134 | theorem comap_dist_left_atTop (c : α) : comap (dist c) atTop = cobounded α := by |
simpa only [dist_comm _ c] using comap_dist_right_atTop c
| 1 |
import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.Basic
#align_import number_theory.legendre_symbol.basic from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
open Nat
section Euler
section Legendre
open ZMod
variable (p : ℕ) [Fact p.Prime]
def legendreSym (a : ℤ) : ℤ :=
... | Mathlib/NumberTheory/LegendreSymbol/Basic.lean | 207 | 208 | theorem eq_neg_one_iff' {a : ℕ} : legendreSym p a = -1 ↔ ¬IsSquare (a : ZMod p) := by |
rw [eq_neg_one_iff]; norm_cast
| 1 |
import Mathlib.CategoryTheory.Subobject.Lattice
#align_import category_theory.subobject.limits from "leanprover-community/mathlib"@"956af7c76589f444f2e1313911bad16366ea476d"
universe v u
noncomputable section
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Subobject Opposite
var... | Mathlib/CategoryTheory/Subobject/Limits.lean | 56 | 58 | theorem equalizerSubobject_arrow' :
(equalizerSubobjectIso f g).inv ≫ (equalizerSubobject f g).arrow = equalizer.ι f g := by |
simp [equalizerSubobjectIso]
| 1 |
import Mathlib.Init.Logic
import Mathlib.Tactic.AdaptationNote
import Mathlib.Tactic.Coe
set_option autoImplicit true
-- We align Lean 3 lemmas with lemmas in `Init.SimpLemmas` in Lean 4.
#align band_self Bool.and_self
#align band_tt Bool.and_true
#align band_ff Bool.and_false
#align tt_band Bool.true_and
#align f... | Mathlib/Init/Data/Bool/Lemmas.lean | 48 | 48 | theorem true_eq_false_eq_False : ¬true = false := by | decide
| 1 |
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w
variable {ι : Sort u} {α : Type v} {β : Type w}
open Set
open OrderDual (toDual)
namespace Set
section Preorder
variable [Preorder α] {a b c... | Mathlib/Order/Interval/Set/Disjoint.lean | 132 | 133 | theorem iUnion_Ioo_left [NoMinOrder α] (b : α) : ⋃ a, Ioo a b = Iio b := by |
simp only [← Ioi_inter_Iio, ← iUnion_inter, iUnion_Ioi, univ_inter]
| 1 |
import Mathlib.Data.Fintype.Basic
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Defs
#align_import logic.equiv.fintype from "leanprover-community/mathlib"@"9407b03373c8cd201df99d6bc5514fc2db44054f"
section Fintype
variable {α β : Type*} [Fintype α] [DecidableEq β] (e : Equiv.Perm α) (f : α ↪ β)
... | Mathlib/Logic/Equiv/Fintype.lean | 50 | 51 | theorem Function.Embedding.toEquivRange_symm_apply_self (a : α) :
f.toEquivRange.symm ⟨f a, Set.mem_range_self a⟩ = a := by | simp [Equiv.symm_apply_eq]
| 1 |
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Tactic.Spread
import Mathlib.Util.AssertExists
#align_import algebra.ring.defs from "leanprover-community/mathlib"@"76de8ae01554c3b37d66544866659ff174e66e1f"
universe u v w x
variable {α : ... | Mathlib/Algebra/Ring/Defs.lean | 168 | 169 | theorem mul_one_add [LeftDistribClass α] (a b : α) : a * (1 + b) = a + a * b := by |
rw [mul_add, mul_one]
| 1 |
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import data.polynomial.expand from "leanprover-community/mathlib"@"bbeb185db4ccee8ed07dc48449414ebfa39cb821"
universe u v w
open Polynomial
open Finset
namespace Polynomial
section CommSemiring
variable (R : Type u) [... | Mathlib/Algebra/Polynomial/Expand.lean | 127 | 128 | theorem coeff_expand_mul' {p : ℕ} (hp : 0 < p) (f : R[X]) (n : ℕ) :
(expand R p f).coeff (p * n) = f.coeff n := by | rw [mul_comm, coeff_expand_mul hp]
| 1 |
import Mathlib.Algebra.Group.Defs
import Mathlib.Control.Functor
#align_import control.applicative from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
universe u v w
section Lemmas
open Function
variable {F : Type u → Type v}
variable [Applicative F] [LawfulApplicative F]
variable {α ... | Mathlib/Control/Applicative.lean | 31 | 33 | theorem Applicative.map_seq_map (f : α → β → γ) (g : σ → β) (x : F α) (y : F σ) :
f <$> x <*> g <$> y = ((· ∘ g) ∘ f) <$> x <*> y := by |
simp [flip, functor_norm]
| 1 |
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Factorial.DoubleFactorial
#align_import ring_theory.polynomial.hermite.basic from "leanprover-community/mathlib"@"938d3db9c278f8a52c0f964a405806f0f2b09b74"
noncomputable section
open Polynomial
namespace P... | Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | 125 | 126 | theorem leadingCoeff_hermite (n : ℕ) : (hermite n).leadingCoeff = 1 := by |
rw [← coeff_natDegree, natDegree_hermite, coeff_hermite_self]
| 1 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.Dynamics.FixedPoints.Topology
import Mathlib.Topology.MetricSpace.Lipschitz
#align_import topology.metric_space.contracting from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classi... | Mathlib/Topology/MetricSpace/Contracting.lean | 53 | 53 | theorem one_sub_K_pos' (hf : ContractingWith K f) : (0 : ℝ≥0∞) < 1 - K := by | simp [hf.1]
| 1 |
import Mathlib.Algebra.Group.Hom.End
import Mathlib.Algebra.Ring.Invertible
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.GroupTheory.GroupAction.Units
#align_import algebra.module.basic from "leanprover-community/mathlib"@"30413fc89f202a090a54d78e540963ed3de0056e"
assert_n... | Mathlib/Algebra/Module/Defs.lean | 104 | 104 | theorem two_smul : (2 : R) • x = x + x := by | rw [← one_add_one_eq_two, add_smul, one_smul]
| 1 |
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v'... | Mathlib/LinearAlgebra/Dimension/Free.lean | 55 | 58 | theorem rank_mul_rank (A : Type v) [AddCommGroup A]
[Module K A] [Module F A] [IsScalarTower F K A] [Module.Free K A] :
Module.rank F K * Module.rank K A = Module.rank F A := by |
convert lift_rank_mul_lift_rank F K A <;> rw [lift_id]
| 1 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.AddTorsor
#align_import analysis.convex.normed from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052... | Mathlib/Analysis/Convex/Normed.lean | 62 | 63 | theorem convex_ball (a : E) (r : ℝ) : Convex ℝ (Metric.ball a r) := by |
simpa only [Metric.ball, sep_univ] using (convexOn_univ_dist a).convex_lt r
| 1 |
import Mathlib.Dynamics.Ergodic.MeasurePreserving
#align_import dynamics.ergodic.ergodic from "leanprover-community/mathlib"@"809e920edfa343283cea507aedff916ea0f1bd88"
open Set Function Filter MeasureTheory MeasureTheory.Measure
open ENNReal
variable {α : Type*} {m : MeasurableSpace α} (f : α → α) {s : Set α}
... | Mathlib/Dynamics/Ergodic/Ergodic.lean | 74 | 76 | theorem prob_eq_zero_or_one [IsProbabilityMeasure μ] (hf : PreErgodic f μ) (hs : MeasurableSet s)
(hs' : f ⁻¹' s = s) : μ s = 0 ∨ μ s = 1 := by |
simpa [hs] using hf.measure_self_or_compl_eq_zero hs hs'
| 1 |
import Mathlib.Order.Interval.Multiset
#align_import data.nat.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
-- TODO
-- assert_not_exists Ring
open Finset Nat
variable (a b c : ℕ)
namespace Nat
instance instLocallyFiniteOrder : LocallyFiniteOrder ℕ where
finsetIcc a b... | Mathlib/Order/Interval/Finset/Nat.lean | 109 | 109 | theorem card_Iio : (Iio b).card = b := by | rw [Iio_eq_Ico, card_Ico, Nat.bot_eq_zero, Nat.sub_zero]
| 1 |
import Mathlib.Init.ZeroOne
import Mathlib.Data.Set.Defs
import Mathlib.Order.Basic
import Mathlib.Order.SymmDiff
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.ByContra
import Mathlib.Util.Delaborators
#align_import data.set.basic from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
... | Mathlib/Data/Set/Basic.lean | 224 | 225 | theorem forall_in_swap {p : α → β → Prop} : (∀ a ∈ s, ∀ (b), p a b) ↔ ∀ (b), ∀ a ∈ s, p a b := by |
tauto
| 1 |
import Mathlib.Data.List.Basic
namespace List
variable {α β : Type*}
#align list.length_enum_from List.enumFrom_length
#align list.length_enum List.enum_length
@[simp]
theorem get?_enumFrom :
∀ n (l : List α) m, get? (enumFrom n l) m = (get? l m).map fun a => (n + m, a)
| n, [], m => rfl
| n, a :: l, 0 =... | Mathlib/Data/List/Enum.lean | 59 | 61 | theorem mk_add_mem_enumFrom_iff_get? {n i : ℕ} {x : α} {l : List α} :
(n + i, x) ∈ enumFrom n l ↔ l.get? i = x := by |
simp [mem_iff_get?]
| 1 |
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.MvPolynomial.Basic
#align_import ring_theory.mv_polynomial.weighted_homogeneous from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Fins... | Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean | 120 | 122 | theorem weightedTotalDegree_zero (w : σ → M) :
weightedTotalDegree w (0 : MvPolynomial σ R) = ⊥ := by |
simp only [weightedTotalDegree, support_zero, Finset.sup_empty]
| 1 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.