Context stringlengths 57 92.3k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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import Mathlib.Topology.UniformSpace.UniformConvergenceTopology
#align_import topology.uniform_space.equicontinuity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
section
open UniformSpace Filter Set Uniformity Topology UniformConvergence Function
variable {ΞΉ ΞΊ X X' Y Z Ξ± Ξ±' Ξ² Ξ²'... | Mathlib/Topology/UniformSpace/Equicontinuity.lean | 308 | 316 | theorem equicontinuousWithinAt_iff_pair {F : ΞΉ β X β Ξ±} {S : Set X} {xβ : X} (hxβ : xβ β S) :
EquicontinuousWithinAt F S xβ β
β U β π€ Ξ±, β V β π[S] xβ, β x β V, β y β V, β i, (F i x, F i y) β U := by |
constructor <;> intro H U hU
Β· rcases comp_symm_mem_uniformity_sets hU with β¨V, hV, hVsymm, hVUβ©
refine β¨_, H V hV, fun x hx y hy i => hVU (prod_mk_mem_compRel ?_ (hy i))β©
exact hVsymm.mk_mem_comm.mp (hx i)
Β· rcases H U hU with β¨V, hV, hVUβ©
filter_upwards [hV] using fun x hx i => hVU xβ (mem_of_mem_n... |
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.RowCol
import Mathlib.Data.Fin.VecNotation
import Mathlib.Tactic.FinCases
#align_import data.matrix.notation from "leanprover-community/mathlib"@"a99f85220eaf38f14f94e04699943e185a5e1d1a"
namespace Matrix
universe u uβ uβ uβ
variable {Ξ± : Type u} {o n m... | Mathlib/Data/Matrix/Notation.lean | 295 | 298 | theorem vecMul_cons (v : Fin n.succ β Ξ±) (w : o' β Ξ±) (B : Fin n β o' β Ξ±) :
v α΅₯* of (vecCons w B) = vecHead v β’ w + vecTail v α΅₯* of B := by |
ext i
simp [vecMul]
|
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
#align_import measure_theory.integral.set_to_l1 from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical Topology NNReal ENNReal MeasureTheory Pointwise
open Set Filter TopologicalSpace ENNR... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 528 | 536 | theorem setToSimpleFunc_nonneg {m : MeasurableSpace Ξ±} (T : Set Ξ± β G' βL[β] G'')
(hT_nonneg : β s x, 0 β€ x β 0 β€ T s x) (f : Ξ± ββ G') (hf : 0 β€ f) :
0 β€ setToSimpleFunc T f := by |
refine sum_nonneg fun i hi => hT_nonneg _ i ?_
rw [mem_range] at hi
obtain β¨y, hyβ© := Set.mem_range.mp hi
rw [β hy]
refine le_trans ?_ (hf y)
simp
|
import Mathlib.Topology.Algebra.Ring.Basic
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Topology.Algebra.UniformGroup
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.UniformSpace.UniformEmbedding
import Mathlib.Algebra.Algebra.Defs
import Mathlib.LinearAlgebra.Projection
import Mat... | Mathlib/Topology/Algebra/Module/Basic.lean | 1,534 | 1,539 | theorem smulRight_one_pow [TopologicalSpace R] [TopologicalRing R] (c : R) (n : β) :
smulRight (1 : R βL[R] R) c ^ n = smulRight (1 : R βL[R] R) (c ^ n) := by |
induction' n with n ihn
Β· ext
simp
Β· rw [pow_succ, ihn, mul_def, smulRight_comp, smul_eq_mul, pow_succ']
|
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.Module.Basic
import Mathlib.Algebra.Regular.SMul
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Rat.BigOperators
import Mathlib.GroupTheory.GroupAction.Hom
import Mathlib.Data.Set.Subsingleton
#align_import data.finsupp.basic from "leanprover... | Mathlib/Data/Finsupp/Basic.lean | 701 | 705 | theorem sum_comapDomain [Zero M] [AddCommMonoid N] (f : Ξ± β Ξ²) (l : Ξ² ββ M) (g : Ξ² β M β N)
(hf : Set.BijOn f (f β»ΒΉ' βl.support) βl.support) :
(comapDomain f l hf.injOn).sum (g β f) = l.sum g := by |
simp only [sum, comapDomain_apply, (Β· β Β·), comapDomain]
exact Finset.sum_preimage_of_bij f _ hf fun x => g x (l x)
|
import Mathlib.Analysis.Convex.Combination
import Mathlib.Analysis.Convex.Strict
import Mathlib.Topology.Connected.PathConnected
import Mathlib.Topology.Algebra.Affine
import Mathlib.Topology.Algebra.Module.Basic
#align_import analysis.convex.topology from "leanprover-community/mathlib"@"0e3aacdc98d25e0afe035c452d876... | Mathlib/Analysis/Convex/Topology.lean | 349 | 355 | theorem JoinedIn.of_segment_subset {E : Type*} [AddCommGroup E] [Module β E]
[TopologicalSpace E] [ContinuousAdd E] [ContinuousSMul β E]
{x y : E} {s : Set E} (h : [x -[β] y] β s) : JoinedIn s x y := by |
have A : Continuous (fun t β¦ (1 - t) β’ x + t β’ y : β β E) := by continuity
apply JoinedIn.ofLine A.continuousOn (by simp) (by simp)
convert h
rw [segment_eq_image β x y]
|
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Set.Finite
#align_import order.filter.basic from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
set_option autoImplicit true
open Function Set Order
open scoped Classical
universe u v w x y
structure Filter (Ξ± : Type*) where... | Mathlib/Order/Filter/Basic.lean | 829 | 845 | theorem iInf_sets_eq {f : ΞΉ β Filter Ξ±} (h : Directed (Β· β₯ Β·) f) [ne : Nonempty ΞΉ] :
(iInf f).sets = β i, (f i).sets :=
let β¨iβ© := ne
let u :=
{ sets := β i, (f i).sets
univ_sets := mem_iUnion.2 β¨i, univ_memβ©
sets_of_superset := by |
simp only [mem_iUnion, exists_imp]
exact fun i hx hxy => β¨i, mem_of_superset hx hxyβ©
inter_sets := by
simp only [mem_iUnion, exists_imp]
intro x y a hx b hy
rcases h a b with β¨c, ha, hbβ©
exact β¨c, inter_mem (ha hx) (hb hy)β© }
have : u = iInf f := eq_iInf_of_mem_i... |
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Analysis.Normed.MulAction
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.PartialHomeomorph
#align_import analysis.asymptotics.asymptotics from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open ... | Mathlib/Analysis/Asymptotics/Asymptotics.lean | 2,053 | 2,062 | theorem isBigO_iff_div_isBoundedUnder {Ξ± : Type*} {l : Filter Ξ±} {f g : Ξ± β π}
(hgf : βαΆ x in l, g x = 0 β f x = 0) :
f =O[l] g β IsBoundedUnder (Β· β€ Β·) l fun x => βf x / g xβ := by |
refine β¨div_isBoundedUnder_of_isBigO, fun h => ?_β©
obtain β¨c, hcβ© := h
simp only [eventually_map, norm_div] at hc
refine IsBigO.of_bound c (hc.mp <| hgf.mono fun x hxβ hxβ => ?_)
by_cases hgx : g x = 0
Β· simp [hxβ hgx, hgx]
Β· exact (div_le_iff (norm_pos_iff.2 hgx)).mp hxβ
|
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 | 206 | 208 | theorem map_pmap (g : Ξ² β Ξ³) (f : β a, p a β Ξ²) (x H) :
Option.map g (pmap f x H) = pmap (fun a h β¦ g (f a h)) x H := by |
cases x <;> simp only [map_none', map_some', pmap]
|
import Mathlib.CategoryTheory.Monoidal.Functor
#align_import category_theory.monoidal.End from "leanprover-community/mathlib"@"85075bccb68ab7fa49fb05db816233fb790e4fe9"
universe v u
namespace CategoryTheory
variable (C : Type u) [Category.{v} C]
def endofunctorMonoidalCategory : MonoidalCategory (C β₯€ C) where... | Mathlib/CategoryTheory/Monoidal/End.lean | 159 | 163 | theorem ΞΌ_naturalityβ {m n m' : M} (f : m βΆ m') (X : C) :
(F.obj n).map ((F.map f).app X) β« (F.ΞΌ m' n).app X =
(F.ΞΌ m n).app X β« (F.map (f β· n)).app X := by |
rw [β tensorHom_id, β ΞΌ_naturalityβ F f (π n) X]
simp
|
import Mathlib.Topology.Defs.Induced
import Mathlib.Topology.Basic
#align_import topology.order from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
open Function Set Filter Topology
universe u v w
namespace TopologicalSpace
variable {Ξ± : Type u}
inductive GenerateOpen (g : Set (Set ... | Mathlib/Topology/Order.lean | 639 | 641 | theorem le_nhdsAdjoint_iff {Ξ± : Type*} (a : Ξ±) (f : Filter Ξ±) (t : TopologicalSpace Ξ±) :
t β€ nhdsAdjoint a f β @nhds Ξ± t a β€ pure a β f β§ β b β a, IsOpen[t] {b} := by |
simp only [le_nhdsAdjoint_iff', @isOpen_singleton_iff_nhds_eq_pure Ξ± t]
|
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.RowCol
#align_import linear_algebra.matrix.trace from "leanprover-community/mathlib"@"32b08ef840dd25ca2e47e035c5da03ce16d2dc3c"
open Matrix
namespace Matrix
variable {ΞΉ m n p : Type*} {Ξ± R S : Type*}
variable [Fintype m] [Fintype n] [Fintype p]
sectio... | Mathlib/LinearAlgebra/Matrix/Trace.lean | 177 | 179 | theorem trace_mul_cycle' [NonUnitalCommSemiring R] (A : Matrix m n R) (B : Matrix n p R)
(C : Matrix p m R) : trace (A * (B * C)) = trace (C * (A * B)) := by |
rw [β Matrix.mul_assoc, trace_mul_comm]
|
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
#align_import geometry.euclidean.angle.oriented.basic from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped Real Rea... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 566 | 568 | theorem oangle_add_cyc3_neg_right {x y z : V} (hx : x β 0) (hy : y β 0) (hz : z β 0) :
o.oangle x (-y) + o.oangle y (-z) + o.oangle z (-x) = Ο := by |
simp_rw [β oangle_neg_left_eq_neg_right, o.oangle_add_cyc3_neg_left hx hy hz]
|
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 | 371 | 373 | theorem cons_eq_append {Ξ± : Type*} (x : Ξ±) (xs : Fin n β Ξ±) :
cons x xs = append (cons x Fin.elim0) xs β Fin.cast (Nat.add_comm ..) := by |
funext i; simp [append_left_eq_cons]
|
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Analysis.Convex.Star
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
#align_import analysis.convex.basic from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
variable {π E F Ξ² : Type*}
open LinearMap Set
open scope... | Mathlib/Analysis/Convex/Basic.lean | 683 | 690 | theorem convex_stdSimplex : Convex π (stdSimplex π ΞΉ) := by |
refine fun f hf g hg a b ha hb hab => β¨fun x => ?_, ?_β©
Β· apply_rules [add_nonneg, mul_nonneg, hf.1, hg.1]
Β· erw [Finset.sum_add_distrib]
simp only [Pi.smul_apply] -- Porting note: `erw` failed to rewrite with `β Finset.smul_sum`
rw [β Finset.smul_sum, β Finset.smul_sum, hf.2, hg.2, smul_eq_mul,
sm... |
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.pointwise from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set
open Pointwise Topology
variable {π E : Type*}
variable [NormedField π]
sectio... | Mathlib/Analysis/NormedSpace/Pointwise.lean | 312 | 316 | theorem thickening_cthickening (hΞ΅ : 0 < Ξ΅) (hΞ΄ : 0 β€ Ξ΄) (s : Set E) :
thickening Ξ΅ (cthickening Ξ΄ s) = thickening (Ξ΅ + Ξ΄) s := by |
obtain rfl | hΞ΄ := hΞ΄.eq_or_lt
Β· rw [cthickening_zero, thickening_closure, add_zero]
Β· rw [β closure_thickening hΞ΄, thickening_closure, thickening_thickening hΞ΅ hΞ΄]
|
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.MeasureTheory.Covering.OneDim
import Mathlib.Order.Monotone.Extension
#align_import analysis.calculus.monotone from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
open Set Filter Function Metric MeasureTheory MeasureTheory.Meas... | Mathlib/Analysis/Calculus/Monotone.lean | 250 | 253 | theorem MonotoneOn.ae_differentiableWithinAt {f : β β β} {s : Set β} (hf : MonotoneOn f s)
(hs : MeasurableSet s) : βα΅ x βvolume.restrict s, DifferentiableWithinAt β f s x := by |
rw [ae_restrict_iff' hs]
exact hf.ae_differentiableWithinAt_of_mem
|
import Mathlib.Tactic.FinCases
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Algebra.Field.IsField
#align_import ring_theory.ideal.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe u v w
variable {Ξ± : Type u} {Ξ² : Type v}
open ... | Mathlib/RingTheory/Ideal/Basic.lean | 581 | 583 | theorem pow_mem_of_pow_mem {m n : β} (ha : a ^ m β I) (h : m β€ n) : a ^ n β I := by |
rw [β Nat.add_sub_of_le h, pow_add]
exact I.mul_mem_right _ ha
|
import Mathlib.Order.MinMax
import Mathlib.Data.Set.Subsingleton
import Mathlib.Tactic.Says
#align_import data.set.intervals.basic from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
open Function
open OrderDual (toDual ofDual)
variable {Ξ± Ξ² : Type*}
namespace Set
theorem Icc_bot_top... | Mathlib/Order/Interval/Set/Basic.lean | 1,783 | 1,785 | theorem Ici_inter_Ici {a b : Ξ±} : Ici a β© Ici b = Ici (a β b) := by |
ext x
simp [Ici]
|
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 | 218 | 221 | theorem pmap_bind {Ξ± Ξ² Ξ³} {x : Option Ξ±} {g : Ξ± β Option Ξ²} {p : Ξ² β Prop} {f : β b, p b β Ξ³} (H)
(H' : β (a : Ξ±), β b β g a, b β x >>= g) :
pmap f (x >>= g) H = x >>= fun a β¦ pmap f (g a) fun b h β¦ H _ (H' a _ h) := by |
cases x <;> simp only [pmap, bind_eq_bind, none_bind, some_bind]
|
import Mathlib.MeasureTheory.Measure.Typeclasses
open scoped ENNReal
namespace MeasureTheory
variable {Ξ± : Type*}
noncomputable
def Measure.trim {m m0 : MeasurableSpace Ξ±} (ΞΌ : @Measure Ξ± m0) (hm : m β€ m0) : @Measure Ξ± m :=
@OuterMeasure.toMeasure Ξ± m ΞΌ.toOuterMeasure (hm.trans (le_toOuterMeasure_caratheodory... | Mathlib/MeasureTheory/Measure/Trim.lean | 37 | 38 | theorem trim_eq_self [MeasurableSpace Ξ±] {ΞΌ : Measure Ξ±} : ΞΌ.trim le_rfl = ΞΌ := by |
simp [Measure.trim]
|
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Int.ModEq
import Mathlib.Data.Nat.Bits
import Mathlib.Data.Nat.Log
import Mathlib.Data.List.Indexes
import Mathlib.Data.List.Palindrome
import Mathlib.Tactic.IntervalCases
import Mathlib.Tactic.Linarith
impo... | Mathlib/Data/Nat/Digits.lean | 427 | 436 | theorem ofDigits_lt_base_pow_length' {b : β} {l : List β} (hl : β x β l, x < b + 2) :
ofDigits (b + 2) l < (b + 2) ^ l.length := by |
induction' l with hd tl IH
Β· simp [ofDigits]
Β· rw [ofDigits, List.length_cons, pow_succ]
have : (ofDigits (b + 2) tl + 1) * (b + 2) β€ (b + 2) ^ tl.length * (b + 2) :=
mul_le_mul (IH fun x hx => hl _ (List.mem_cons_of_mem _ hx)) (by rfl) (by simp only [zero_le])
(Nat.zero_le _)
suffices βhd ... |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import analysis.special_functions.arsinh from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open Function Filter Set
open scoped Topology
name... | Mathlib/Analysis/SpecialFunctions/Arsinh.lean | 83 | 84 | theorem cosh_arsinh (x : β) : cosh (arsinh x) = β(1 + x ^ 2) := by |
rw [β sqrt_sq (cosh_pos _).le, cosh_sq', sinh_arsinh]
|
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Order.MonotoneContinuity
#align_import dynamics.circle.rotation_number.translation_number from "leanprover-... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 334 | 335 | theorem commute_nat_add (n : β) : Function.Commute f (n + Β·) := by |
simpa only [nsmul_one, add_left_iterate] using Function.Commute.iterate_right f.map_one_add n
|
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Order.Interval.Finset.Basic
#align_import data.int.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
open Finset Int
namespace Int
instance instLocallyFiniteOrder : LocallyFiniteOrder β€ where
finsetIcc a b :=
(Fins... | Mathlib/Data/Int/Interval.lean | 150 | 151 | theorem card_fintype_Icc : Fintype.card (Set.Icc a b) = (b + 1 - a).toNat := by |
rw [β card_Icc, Fintype.card_ofFinset]
|
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.UnitaryGroup
#align_import analysis.inner_product_space.pi_L2 from "leanprover-community/mathlib"@"13bce9a6b6c44f6b4c91ac1c1d2a816e2533d395"
set_... | Mathlib/Analysis/InnerProductSpace/PiL2.lean | 134 | 138 | theorem EuclideanSpace.ball_zero_eq {n : Type*} [Fintype n] (r : β) (hr : 0 β€ r) :
Metric.ball (0 : EuclideanSpace β n) r = {x | β i, x i ^ 2 < r ^ 2} := by |
ext x
have : (0 : β) β€ β i, x i ^ 2 := Finset.sum_nonneg fun _ _ => sq_nonneg _
simp_rw [mem_setOf, mem_ball_zero_iff, norm_eq, norm_eq_abs, sq_abs, sqrt_lt this hr]
|
import Mathlib.Order.RelIso.Set
import Mathlib.Data.Multiset.Sort
import Mathlib.Data.List.NodupEquivFin
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Fintype.Card
#align_import data.finset.sort from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
namespace Finset
open Multiset... | Mathlib/Data/Finset/Sort.lean | 219 | 226 | theorem orderEmbOfFin_unique {s : Finset Ξ±} {k : β} (h : s.card = k) {f : Fin k β Ξ±}
(hfs : β x, f x β s) (hmono : StrictMono f) : f = s.orderEmbOfFin h := by |
apply Fin.strictMono_unique hmono (s.orderEmbOfFin h).strictMono
rw [range_orderEmbOfFin, β Set.image_univ, β coe_univ, β coe_image, coe_inj]
refine eq_of_subset_of_card_le (fun x hx => ?_) ?_
Β· rcases mem_image.1 hx with β¨x, _, rflβ©
exact hfs x
Β· rw [h, card_image_of_injective _ hmono.injective, card_un... |
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 | 393 | 394 | theorem one_div_le_one_div_of_le (ha : 0 < a) (h : a β€ b) : 1 / b β€ 1 / a := by |
simpa using inv_le_inv_of_le ha h
|
import Mathlib.CategoryTheory.Limits.IsLimit
import Mathlib.CategoryTheory.Category.ULift
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.Logic.Equiv.Basic
#align_import category_theory.limits.has_limits from "leanprover-community/mathlib"@"2738d2ca56cbc63be80c3bd48e9ed90ad94e947d"
noncomputable sec... | Mathlib/CategoryTheory/Limits/HasLimits.lean | 309 | 310 | theorem limit.lift_extend {F : J β₯€ C} [HasLimit F] (c : Cone F) {X : C} (f : X βΆ c.pt) :
limit.lift F (c.extend f) = f β« limit.lift F c := by | aesop_cat
|
import Mathlib.MeasureTheory.Measure.Typeclasses
#align_import probability.conditional_probability from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
open ENNReal MeasureTheory MeasureTheory.Measure MeasurableSpace Set
variable {Ξ© Ξ©' Ξ± : Type*} {m : MeasurableSpa... | Mathlib/Probability/ConditionalProbability.lean | 147 | 148 | theorem cond_apply' {t : Set Ξ©} (hA : MeasurableSet t) : ΞΌ[t|s] = (ΞΌ s)β»ΒΉ * ΞΌ (s β© t) := by |
rw [cond, Measure.smul_apply, Measure.restrict_apply hA, Set.inter_comm, smul_eq_mul]
|
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Relator
import Mathlib.Init.Data.Quot
import Mathlib.Tactic.Cases
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
#align_import logic.relation from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9a... | Mathlib/Logic/Relation.lean | 192 | 196 | theorem _root_.Acc.of_fibration (fib : Fibration rΞ± rΞ² f) {a} (ha : Acc rΞ± a) : Acc rΞ² (f a) := by |
induction' ha with a _ ih
refine Acc.intro (f a) fun b hr β¦ ?_
obtain β¨a', hr', rflβ© := fib hr
exact ih a' hr'
|
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 | 512 | 519 | theorem nonTorsionWeight_of [NoZeroSMulDivisors β M] (hw : β i : Ο, w i β 0) :
NonTorsionWeight w := by |
intro n x
rw [smul_eq_zero]
intro hnx
cases' hnx with hn hx
Β· exact hn
Β· exact absurd hx (hw x)
|
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 | 730 | 731 | theorem coeff_X_of_ne_one {n : β} (hn : n β 1) : coeff (X : R[X]) n = 0 := by |
rw [coeff_X, if_neg hn.symm]
|
import Mathlib.MeasureTheory.Measure.Sub
import Mathlib.MeasureTheory.Decomposition.SignedHahn
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
#align_import measure_theory.decomposition.lebesgue from "leanprover-community/mathlib"@"b2ff9a3d7a15fd5b0f060b135421d6a89a999c2f"
open scoped MeasureTheory NNReal ENN... | Mathlib/MeasureTheory/Decomposition/Lebesgue.lean | 584 | 597 | theorem rnDeriv_smul_left (Ξ½ ΞΌ : Measure Ξ±) [IsFiniteMeasure Ξ½]
[Ξ½.HaveLebesgueDecomposition ΞΌ] (r : ββ₯0) :
(r β’ Ξ½).rnDeriv ΞΌ =α΅[ΞΌ] r β’ Ξ½.rnDeriv ΞΌ := by |
rw [β withDensity_eq_iff]
Β· simp_rw [ENNReal.smul_def]
rw [withDensity_smul _ (measurable_rnDeriv _ _)]
suffices (r β’ Ξ½).singularPart ΞΌ + withDensity ΞΌ (rnDeriv (r β’ Ξ½) ΞΌ)
= (r β’ Ξ½).singularPart ΞΌ + r β’ withDensity ΞΌ (rnDeriv Ξ½ ΞΌ) by
rwa [Measure.add_right_inj] at this
rw [β (r β’ Ξ½).haveL... |
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.SumOverResidueClass
#align_import analysis.p_series from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
def SuccDiffBounded (C : β) (u : β β β) : Prop :=... | Mathlib/Analysis/PSeries.lean | 381 | 382 | theorem summable_rpow {p : β} : Summable (fun n => (n : ββ₯0) ^ p : β β ββ₯0) β p < -1 := by |
simp [β NNReal.summable_coe]
|
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.matrix.basis from "leanprover-community/mathlib"@"6c263e4bfc2e6714de30f22178b4d0ca4d149a76"
noncomputable section
open LinearMap Matrix Set Submodule
open Matrix
section MulLinearMapToMatrix
vari... | Mathlib/LinearAlgebra/Matrix/Basis.lean | 286 | 289 | theorem Basis.toMatrix_map (b : Basis ΞΉ R M) (f : M ββ[R] N) (v : ΞΉ β N) :
(b.map f).toMatrix v = b.toMatrix (f.symm β v) := by |
ext
simp only [Basis.toMatrix_apply, Basis.map, LinearEquiv.trans_apply, (Β· β Β·)]
|
import Mathlib.Topology.Order.LeftRightNhds
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {Ξ± Ξ² Ξ³ : Type*}
section OrderTopology
variable [TopologicalSpace Ξ±] [TopologicalSpace Ξ²] [LinearOrder Ξ±] [LinearOrder Ξ²] [OrderTopology Ξ±]
[OrderTopology Ξ²]
| Mathlib/Topology/Order/IsLUB.lean | 24 | 32 | theorem IsLUB.frequently_mem {a : Ξ±} {s : Set Ξ±} (ha : IsLUB s a) (hs : s.Nonempty) :
βαΆ x in π[β€] a, x β s := by |
rcases hs with β¨a', ha'β©
intro h
rcases (ha.1 ha').eq_or_lt with (rfl | ha'a)
Β· exact h.self_of_nhdsWithin le_rfl ha'
Β· rcases (mem_nhdsWithin_Iic_iff_exists_Ioc_subset' ha'a).1 h with β¨b, hba, hbβ©
rcases ha.exists_between hba with β¨b', hb's, hb'β©
exact hb hb' hb's
|
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Preserves.Basic
#align_import category_theory.limits.preserves.shapes.pullbacks from "leanprover-community/mathlib"@"f11e306adb9f2a393539d2bb4293bf1b42caa7ac"
noncomputable section
universe vβ vβ uβ uβ
-- Porting note: ne... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Pullbacks.lean | 245 | 247 | theorem PreservesPushout.inr_iso_inv :
G.map pushout.inr β« (PreservesPushout.iso G f g).inv = pushout.inr := by |
simp [PreservesPushout.iso, Iso.comp_inv_eq]
|
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Finset.Option
#align_import data.fintype.option from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
open Function
open Nat
universe u v
variable {Ξ± Ξ² Ξ³ : Type*}
open Fin... | Mathlib/Data/Fintype/Option.lean | 94 | 106 | theorem induction_empty_option {P : β (Ξ± : Type u) [Fintype Ξ±], Prop}
(of_equiv : β (Ξ± Ξ²) [Fintype Ξ²] (e : Ξ± β Ξ²), @P Ξ± (@Fintype.ofEquiv Ξ± Ξ² βΉ_βΊ e.symm) β @P Ξ² βΉ_βΊ)
(h_empty : P PEmpty) (h_option : β (Ξ±) [Fintype Ξ±], P Ξ± β P (Option Ξ±)) (Ξ± : Type u)
[h_fintype : Fintype Ξ±] : P Ξ± := by |
obtain β¨pβ© :=
let f_empty := fun i => by convert h_empty
let h_option : β {Ξ± : Type u} [Fintype Ξ±] [DecidableEq Ξ±],
(β (h : Fintype Ξ±), P Ξ±) β β (h : Fintype (Option Ξ±)), P (Option Ξ±) := by
rintro Ξ± hΞ± - PΞ± hΞ±'
convert h_option Ξ± (PΞ± _)
@truncRecEmptyOption (fun Ξ± => β h, @P Ξ± h) (... |
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.GroupTheory.Coset
#align_import group_theory.quotient_group from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Function
open scope... | Mathlib/GroupTheory/QuotientGroup.lean | 108 | 113 | theorem sound (U : Set (G β§Έ N)) (g : N.op) :
g β’ (mk' N) β»ΒΉ' U = (mk' N) β»ΒΉ' U := by |
ext x
simp only [Set.mem_preimage, Set.mem_smul_set_iff_inv_smul_mem]
congr! 1
exact Quotient.sound β¨gβ»ΒΉ, rflβ©
|
import Mathlib.Algebra.Lie.BaseChange
import Mathlib.Algebra.Lie.Solvable
import Mathlib.Algebra.Lie.Quotient
import Mathlib.Algebra.Lie.Normalizer
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.Order.Filter.AtTopBot
import Mathlib.RingTheory.Artinian
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mat... | Mathlib/Algebra/Lie/Nilpotent.lean | 353 | 356 | theorem nilpotencyLength_eq_one_iff [Nontrivial M] :
nilpotencyLength R L M = 1 β IsTrivial L M := by |
rw [nilpotencyLength_eq_succ_iff, β trivial_iff_lower_central_eq_bot]
simp
|
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.Complement
open Monoid Coprod Multiplicative Subgroup Function
def HNNExtension.con (G : Type*) [Group G] (A B : Subgroup G) (Ο : A β* B) :
Con (G β Multiplicative β€) :=
conGen (fun x y => β (a : A),
x = inr (ofAdd 1) * inl (a : G) β§
... | Mathlib/GroupTheory/HNNExtension.lean | 164 | 170 | theorem toSubgroupEquiv_neg_apply (u : β€Λ£) (a : toSubgroup A B u) :
(toSubgroupEquiv Ο (-u) (toSubgroupEquiv Ο u a) : G) = a := by |
rcases Int.units_eq_one_or u with rfl | rfl
Β· -- This used to be `simp` before leanprover/lean4#2644
simp; erw [MulEquiv.symm_apply_apply]
Β· simp only [toSubgroup_neg_one, toSubgroupEquiv_neg_one, SetLike.coe_eq_coe]
exact Ο.apply_symm_apply a
|
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.OrdConnected
#align_import data.set.intervals.proj_Icc from "leanprover-community/mathlib"@"4e24c4bfcff371c71f7ba22050308aa17815626c"
variable {Ξ± Ξ² : Type*} [LinearOrder Ξ±]
open Function
namespace Set
def projIci (a x : Ξ±) : Ici a := β¨max a x,... | Mathlib/Order/Interval/Set/ProjIcc.lean | 116 | 116 | theorem projIic_of_mem (hx : x β Iic b) : projIic b x = β¨x, hxβ© := by | simpa [projIic]
|
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.Algebra.Category.Ring.Colimits
import Mathlib.Algebra.Category.Ring.Limits
import Mathlib.Topology.Sheaves.LocalPredicate
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.Algebra.Ring.Subring.Basic
#align_import algebraic_geometry.struct... | Mathlib/AlgebraicGeometry/StructureSheaf.lean | 393 | 394 | theorem const_congr {fβ fβ gβ gβ : R} {U hu} (hf : fβ = fβ) (hg : gβ = gβ) :
const R fβ gβ U hu = const R fβ gβ U (hg βΈ hu) := by | substs hf hg; rfl
|
import Mathlib.Data.PFunctor.Multivariate.Basic
#align_import data.qpf.multivariate.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe u
open MvFunctor
class MvQPF {n : β} (F : TypeVec.{u} n β Type*) [MvFunctor F] where
P : MvPFunctor.{u} n
abs : β {Ξ±}, P Ξ± β F Ξ±
... | Mathlib/Data/QPF/Multivariate/Basic.lean | 236 | 248 | theorem liftP_iff_of_isUniform (h : q.IsUniform) {Ξ± : TypeVec n} (x : F Ξ±) (p : β i, Ξ± i β Prop) :
LiftP p x β β (i), β u β supp x i, p i u := by |
rw [liftP_iff, β abs_repr x]
cases' repr x with a f; constructor
Β· rintro β¨a', f', abseq, hfβ© u
rw [supp_eq_of_isUniform h, h _ _ _ _ abseq]
rintro b β¨i, _, hiβ©
rw [β hi]
apply hf
intro h'
refine β¨a, f, rfl, fun _ i => h' _ _ ?_β©
rw [supp_eq_of_isUniform h]
exact β¨i, mem_univ i, rflβ©
|
import Mathlib.MeasureTheory.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
assert_not_exists Differentiable
noncomputable section
open scoped Topology NNReal ENNReal MeasureTheory
open Set Filter TopologicalSpace EN... | Mathlib/MeasureTheory/Integral/Bochner.lean | 1,581 | 1,599 | theorem integral_add_measure {f : Ξ± β G} (hΞΌ : Integrable f ΞΌ) (hΞ½ : Integrable f Ξ½) :
β« x, f x β(ΞΌ + Ξ½) = β« x, f x βΞΌ + β« x, f x βΞ½ := by |
by_cases hG : CompleteSpace G; swap
Β· simp [integral, hG]
have hfi := hΞΌ.add_measure hΞ½
simp_rw [integral_eq_setToFun]
have hΞΌ_dfma : DominatedFinMeasAdditive (ΞΌ + Ξ½) (weightedSMul ΞΌ : Set Ξ± β G βL[β] G) 1 :=
DominatedFinMeasAdditive.add_measure_right ΞΌ Ξ½ (dominatedFinMeasAdditive_weightedSMul ΞΌ)
z... |
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.Tactic.NormNum.Inv
import Mathlib.Tactic.NormNum.Pow
import Mathlib.Util.AtomM
set_option autoImplicit true
namespace Mathlib.Tactic
namespace Ring
open Mathlib.Meta Qq NormNum Lean.Meta AtomM
open Lean (MetaM Expr mkRawNatLit)
def instCommSemiringNat : CommSe... | Mathlib/Tactic/Ring/Basic.lean | 532 | 533 | theorem neg_mul {R} [Ring R] (aβ : R) (aβ) {aβ b : R}
(_ : -aβ = b) : -(aβ ^ aβ * aβ) = aβ ^ aβ * b := by | subst_vars; simp
|
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.Measure.Count
import Mathlib.Topology.IndicatorConstPointwise
import Mathlib.MeasureTheory.Constructions.BorelSpace.Real
#align_import m... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 1,585 | 1,587 | theorem lintegral_singleton' {f : Ξ± β ββ₯0β} (hf : Measurable f) (a : Ξ±) :
β«β» x in {a}, f x βΞΌ = f a * ΞΌ {a} := by |
simp only [restrict_singleton, lintegral_smul_measure, lintegral_dirac' _ hf, mul_comm]
|
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable
#align_import measure_theory.function.simple_func_dense from "leanprover-community/mathlib"@"7317149f12f55affbc900fc873d0d422485122b9"
open Set Function Filter TopologicalSpace ENNReal EMetric Finset
... | Mathlib/MeasureTheory/Function/SimpleFuncDense.lean | 102 | 113 | theorem edist_nearestPt_le (e : β β Ξ±) (x : Ξ±) {k N : β} (hk : k β€ N) :
edist (nearestPt e N x) x β€ edist (e k) x := by |
induction' N with N ihN generalizing k
Β· simp [nonpos_iff_eq_zero.1 hk, le_refl]
Β· simp only [nearestPt, nearestPtInd_succ, map_apply]
split_ifs with h
Β· rcases hk.eq_or_lt with (rfl | hk)
exacts [le_rfl, (h k (Nat.lt_succ_iff.1 hk)).le]
Β· push_neg at h
rcases h with β¨l, hlN, hxlβ©
r... |
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.ZMod
#align_import number_theory.padics.ring_homs from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
noncomputable section
open scoped Classical
open Nat LocalRing Padic
namespace PadicInt
variable {p : β} [h... | Mathlib/NumberTheory/Padics/RingHoms.lean | 124 | 134 | theorem norm_sub_modPart (h : β(r : β_[p])β β€ 1) : β(β¨r, hβ© - modPart p r : β€_[p])β < 1 := by |
let n := modPart p r
rw [norm_lt_one_iff_dvd, β (isUnit_den r h).dvd_mul_right]
suffices βp β£ r.num - n * r.den by
convert (Int.castRingHom β€_[p]).map_dvd this
simp only [sub_mul, Int.cast_natCast, eq_intCast, Int.cast_mul, sub_left_inj, Int.cast_sub]
apply Subtype.coe_injective
simp only [coe_mu... |
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.Tactic.NormNum.Inv
import Mathlib.Tactic.NormNum.Pow
import Mathlib.Util.AtomM
set_option autoImplicit true
namespace Mathlib.Tactic
namespace Ring
open Mathlib.Meta Qq NormNum Lean.Meta AtomM
open Lean (MetaM Expr mkRawNatLit)
def instCommSemiringNat : CommSe... | Mathlib/Tactic/Ring/Basic.lean | 980 | 981 | theorem neg_congr {R} [Ring R] {a a' b : R} (_ : a = a')
(_ : -a' = b) : (-a : R) = b := by | subst_vars; rfl
|
import Mathlib.Analysis.Calculus.TangentCone
import Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics
#align_import analysis.calculus.fderiv.basic from "leanprover-community/mathlib"@"41bef4ae1254365bc190aee63b947674d2977f01"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open To... | Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 225 | 228 | theorem fderivWithin_zero_of_not_differentiableWithinAt (h : Β¬DifferentiableWithinAt π f s x) :
fderivWithin π f s x = 0 := by |
have : Β¬β f', HasFDerivWithinAt f f' s x := h
simp [fderivWithin, this]
|
import Mathlib.AlgebraicGeometry.GammaSpecAdjunction
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.RingTheory.Localization.InvSubmonoid
#align_import algebraic_geometry.AffineScheme from "leanprover-community/mathlib"@"88474d1b5af6d37c2ab728b757771bced7f5194c"... | Mathlib/AlgebraicGeometry/AffineScheme.lean | 550 | 557 | theorem isLocalization_stalk (x : U) :
IsLocalization.AtPrime (X.presheaf.stalk x) (hU.primeIdealOf x).asIdeal := by |
rcases x with β¨x, hxβ©
set y := hU.primeIdealOf β¨x, hxβ© with hy
have : hU.fromSpec.val.base y = x := hy βΈ hU.fromSpec_primeIdealOf β¨x, hxβ©
clear_value y
subst this
exact hU.isLocalization_stalk' y hx
|
import Mathlib.Topology.Algebra.InfiniteSum.Defs
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Topology.Algebra.Monoid
noncomputable section
open Filter Finset Function
open scoped Topology
variable {Ξ± Ξ² Ξ³ Ξ΄ : Type*}
section tprod
variable [CommMonoid Ξ±] [TopologicalSpace Ξ±] {f g : Ξ² β Ξ±} {a aβ aβ : ... | Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | 412 | 412 | theorem tprod_one : β' _ : Ξ², (1 : Ξ±) = 1 := by | rw [tprod_eq_finprod] <;> simp
|
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Nat.Choose.Sum
#align_import data.complex.exponential from "leanprover-community/mathlib"@"a8b2226cfb0a79f5986492053fc49b1a0c6aeffb"
open CauSeq Finset IsAbsoluteValue
open ... | Mathlib/Data/Complex/Exponential.lean | 1,070 | 1,071 | theorem cosh_abs : cosh |x| = cosh x := by |
cases le_total x 0 <;> simp [*, _root_.abs_of_nonneg, abs_of_nonpos]
|
import Mathlib.Algebra.Module.Submodule.Lattice
import Mathlib.Data.ZMod.Basic
import Mathlib.Order.OmegaCompletePartialOrder
variable {n : β} {M Mβ : Type*}
abbrev AddCommMonoid.zmodModule [NeZero n] [AddCommMonoid M] (h : β (x : M), n β’ x = 0) :
Module (ZMod n) M := by
have h_mod (c : β) (x : M) : (c % n)... | Mathlib/Data/ZMod/Module.lean | 54 | 56 | theorem smul_mem (hx : x β K) (c : ZMod n) : c β’ x β K := by |
rw [β ZMod.intCast_zmod_cast c, β zsmul_eq_smul_cast]
exact zsmul_mem hx (cast c)
|
import Mathlib.MeasureTheory.Measure.MeasureSpace
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
#align_import measure_theory.measure.open_pos from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology ENNReal MeasureTheory
open Set Function Filter
namespace Measur... | Mathlib/MeasureTheory/Measure/OpenPos.lean | 57 | 59 | theorem _root_.IsOpen.measure_eq_zero_iff (hU : IsOpen U) : ΞΌ U = 0 β U = β
:= by |
simpa only [not_lt, nonpos_iff_eq_zero, not_nonempty_iff_eq_empty] using
not_congr (hU.measure_pos_iff ΞΌ)
|
import Mathlib.Algebra.Algebra.Basic
import Mathlib.Algebra.Periodic
import Mathlib.Topology.Algebra.Order.Field
import Mathlib.Topology.Algebra.UniformMulAction
import Mathlib.Topology.Algebra.Star
import Mathlib.Topology.Instances.Int
import Mathlib.Topology.Order.Bornology
#align_import topology.instances.real fro... | Mathlib/Topology/Instances/Real.lean | 236 | 239 | theorem tendsto_coe_cofinite : Tendsto ((β) : β€ β β) cofinite (cocompact β) := by |
apply (castAddHom β).tendsto_coe_cofinite_of_discrete cast_injective
rw [range_castAddHom]
infer_instance
|
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"@"b99e2d58a5e6861833fa8de11e51a81144258db4"
open Function
universe u v w
namespace SimpleGraph
variable {V : Type u} {V' : Type v} {V'' : Typ... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 1,859 | 1,860 | theorem length_transfer (hp) : (p.transfer H hp).length = p.length := by |
induction p <;> simp [*]
|
import Mathlib.Logic.Relation
import Mathlib.Data.List.Forall2
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Infix
#align_import data.list.chain from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
-- Make sure we haven't imported `Data.Nat.Order.Basic`
assert_not_exists OrderedSu... | Mathlib/Data/List/Chain.lean | 310 | 312 | theorem Chain'.infix (h : Chain' R l) (h' : lβ <:+: l) : Chain' R lβ := by |
rcases h' with β¨lβ, lβ, rflβ©
exact h.left_of_append.right_of_append
|
import Mathlib.MeasureTheory.Measure.FiniteMeasure
import Mathlib.MeasureTheory.Integral.Average
#align_import measure_theory.measure.probability_measure from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open MeasureTheory
open Set
open Filter
open BoundedCon... | Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean | 317 | 324 | theorem tendsto_iff_forall_integral_tendsto {Ξ³ : Type*} {F : Filter Ξ³}
{ΞΌs : Ξ³ β ProbabilityMeasure Ξ©} {ΞΌ : ProbabilityMeasure Ξ©} :
Tendsto ΞΌs F (π ΞΌ) β
β f : Ξ© βα΅ β,
Tendsto (fun i => β« Ο, f Ο β(ΞΌs i : Measure Ξ©)) F (π (β« Ο, f Ο β(ΞΌ : Measure Ξ©))) := by |
rw [tendsto_nhds_iff_toFiniteMeasure_tendsto_nhds]
rw [FiniteMeasure.tendsto_iff_forall_integral_tendsto]
rfl
|
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
#align_import analysis.special_functions.gamma.bohr_mollerup from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
set_option linter.uppercaseLean3 false
noncomputable section
open Filter Set MeasureTheory
open scoped Na... | Mathlib/Analysis/SpecialFunctions/Gamma/BohrMollerup.lean | 452 | 454 | theorem doublingGamma_one : doublingGamma 1 = 1 := by |
simp_rw [doublingGamma, Gamma_one_half_eq, add_halves (1 : β), sub_self, Gamma_one, mul_one,
rpow_zero, mul_one, div_self (sqrt_ne_zero'.mpr pi_pos)]
|
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.Calculus.Deriv.Inv
#align_import analysis.calculus.deriv.zpow from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Topology Filter
open Filter Asymptotics Set
variable {π : Typ... | Mathlib/Analysis/Calculus/Deriv/ZPow.lean | 86 | 92 | theorem deriv_zpow (m : β€) (x : π) : deriv (fun x => x ^ m) x = m * x ^ (m - 1) := by |
by_cases H : x β 0 β¨ 0 β€ m
Β· exact (hasDerivAt_zpow m x H).deriv
Β· rw [deriv_zero_of_not_differentiableAt (mt differentiableAt_zpow.1 H)]
push_neg at H
rcases H with β¨rfl, hmβ©
rw [zero_zpow _ ((sub_one_lt _).trans hm).ne, mul_zero]
|
import Mathlib.Data.Finset.Basic
import Mathlib.ModelTheory.Syntax
import Mathlib.Data.List.ProdSigma
#align_import model_theory.semantics from "leanprover-community/mathlib"@"d565b3df44619c1498326936be16f1a935df0728"
universe u v w u' v'
namespace FirstOrder
namespace Language
variable {L : Language.{u, v}} {... | Mathlib/ModelTheory/Semantics.lean | 441 | 445 | theorem realize_liftAt_one_self {n : β} {Ο : L.BoundedFormula Ξ± n} {v : Ξ± β M}
{xs : Fin (n + 1) β M} : (Ο.liftAt 1 n).Realize v xs β Ο.Realize v (xs β castSucc) := by |
rw [realize_liftAt_one (refl n), iff_eq_eq]
refine congr rfl (congr rfl (funext fun i => ?_))
rw [if_pos i.is_lt]
|
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finite.Basic
import Mathlib.Data.Set.Functor
import Mathlib.Data.Set.Lattice
#align_import data.set.finite from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
assert_not_exists OrderedRing
assert_not_exists MonoidWithZero
open Set Fun... | Mathlib/Data/Set/Finite.lean | 1,031 | 1,035 | theorem Finite.finite_subsets {Ξ± : Type u} {a : Set Ξ±} (h : a.Finite) : { b | b β a }.Finite := by |
convert ((Finset.powerset h.toFinset).map Finset.coeEmb.1).finite_toSet
ext s
simpa [β @exists_finite_iff_finset Ξ± fun t => t β a β§ t = s, Finite.subset_toFinset,
β and_assoc, Finset.coeEmb] using h.subset
|
import Batteries.Classes.Order
namespace Batteries.PairingHeapImp
inductive Heap (Ξ± : Type u) where
| nil : Heap Ξ±
| node (a : Ξ±) (child sibling : Heap Ξ±) : Heap Ξ±
deriving Repr
def Heap.size : Heap Ξ± β Nat
| .nil => 0
| .node _ c s => c.size + 1 + s.size
def Heap.singleton (a : Ξ±) : Heap Ξ± := .... | .lake/packages/batteries/Batteries/Data/PairingHeap.lean | 90 | 93 | theorem Heap.noSibling_merge (le) (sβ sβ : Heap Ξ±) :
(sβ.merge le sβ).NoSibling := by |
unfold merge
(split <;> try split) <;> constructor
|
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Algebra.Group.Subsemigroup.Basic
import Mathlib.Algebra.Group.Units
#align_import group_theory.submonoid.basic from "leanprover-community/mathlib"@"feb99064803fd3108e37c18b0f77d0a8344677a3"
assert_not_exists MonoidWithZero
-- Only needed for notation
-- Only ne... | Mathlib/Algebra/Group/Submonoid/Basic.lean | 567 | 570 | theorem mem_iSup {ΞΉ : Sort*} (p : ΞΉ β Submonoid M) {m : M} :
(m β β¨ i, p i) β β N, (β i, p i β€ N) β m β N := by |
rw [β closure_singleton_le_iff_mem, le_iSup_iff]
simp only [closure_singleton_le_iff_mem]
|
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Localization.FractionRing
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable section
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Ty... | Mathlib/Algebra/Polynomial/Roots.lean | 380 | 388 | theorem ne_zero_of_mem_nthRootsFinset {Ξ· : R} (hΞ· : Ξ· β nthRootsFinset n R) : Ξ· β 0 := by |
nontriviality R
rintro rfl
cases n with
| zero =>
simp only [Nat.zero_eq, nthRootsFinset_zero, not_mem_empty] at hΞ·
| succ n =>
rw [mem_nthRootsFinset n.succ_pos, zero_pow n.succ_ne_zero] at hΞ·
exact zero_ne_one hΞ·
|
import Mathlib.CategoryTheory.Limits.IsLimit
import Mathlib.CategoryTheory.Category.ULift
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.Logic.Equiv.Basic
#align_import category_theory.limits.has_limits from "leanprover-community/mathlib"@"2738d2ca56cbc63be80c3bd48e9ed90ad94e947d"
noncomputable sec... | Mathlib/CategoryTheory/Limits/HasLimits.lean | 252 | 255 | theorem limit.isoLimitCone_hom_Ο {F : J β₯€ C} [HasLimit F] (t : LimitCone F) (j : J) :
(limit.isoLimitCone t).hom β« t.cone.Ο.app j = limit.Ο F j := by |
dsimp [limit.isoLimitCone, IsLimit.conePointUniqueUpToIso]
aesop_cat
|
import Mathlib.Geometry.Euclidean.Sphere.Basic
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tactic.DeriveFintype
#align_import geometry.euclidean.circumcenter from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
noncomputable section
open scoped Classical
o... | Mathlib/Geometry/Euclidean/Circumcenter.lean | 886 | 891 | theorem circumsphere_eq_of_cospherical {ps : Set P} {n : β} [FiniteDimensional β V]
(hd : finrank β V = n) (hc : Cospherical ps) {sxβ sxβ : Simplex β P n}
(hsxβ : Set.range sxβ.points β ps) (hsxβ : Set.range sxβ.points β ps) :
sxβ.circumsphere = sxβ.circumsphere := by |
rcases exists_circumsphere_eq_of_cospherical hd hc with β¨r, hrβ©
rw [hr sxβ hsxβ, hr sxβ hsxβ]
|
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 | 198 | 206 | theorem exists_inv {I : Ideal R} [hI : I.IsMaximal] :
β {a : R β§Έ I}, a β 0 β β b : R β§Έ I, a * b = 1 := by |
rintro β¨aβ© h
rcases hI.exists_inv (mt eq_zero_iff_mem.2 h) with β¨b, c, hc, abcβ©
rw [mul_comm] at abc
refine β¨mk _ b, Quot.sound ?_β©
simp only [Submodule.quotientRel_r_def]
rw [β eq_sub_iff_add_eq'] at abc
rwa [abc, β neg_mem_iff (G := R) (H := I), neg_sub] at hc
|
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 {Ξ± : Type*}
section ExistsAddOfLE
variable [AddCommSemigrou... | Mathlib/Algebra/Order/Sub/Canonical.lean | 57 | 60 | theorem lt_of_tsub_lt_tsub_right_of_le (h : c β€ b) (h2 : a - c < b - c) : a < b := by |
refine ((tsub_le_tsub_iff_right h).mp h2.le).lt_of_ne ?_
rintro rfl
exact h2.false
|
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import number_theory.ramification_inertia from "leanprover-community/mathlib"@"039a089d2a4b93c761b234f3e5f5aeb752bac60f"
namespace Ideal
universe u v
variable {R : Type u} [CommRing R]
variable {S : Type v} [CommRing S] (f : R β+* S)
variable (p : Ideal R) (... | Mathlib/NumberTheory/RamificationInertia.lean | 646 | 656 | theorem rank_prime_pow_ramificationIdx [IsDedekindDomain S] [p.IsMaximal] [P.IsPrime]
(hP0 : P β β₯) (he : e β 0) :
Module.rank (R β§Έ p) (S β§Έ P ^ e) =
e β’
@Module.rank (R β§Έ p) (S β§Έ P) _ _
(@Algebra.toModule _ _ _ _ <|
@Quotient.algebraQuotientOfRamificationIdxNeZero _ _ _ _ _ _... |
letI : NeZero e := β¨heβ©
have := rank_pow_quot f p P hP0 0 (Nat.zero_le e)
rw [pow_zero, Nat.sub_zero, Ideal.one_eq_top, Ideal.map_top] at this
exact (rank_top (R β§Έ p) _).symm.trans this
|
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 | 265 | 266 | theorem polar_zero_left (y : M) : polar Q 0 y = 0 := by |
simp only [polar, zero_add, QuadraticForm.map_zero, sub_zero, sub_self]
|
import Mathlib.Topology.EMetricSpace.Basic
import Mathlib.Topology.Bornology.Constructions
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Topology.Order.DenselyOrdered
open Set Filter TopologicalSpace Bornology
open scoped ENNReal NNReal Uniformity Topology
universe u v w
variable {Ξ± : Type u} {Ξ² : Typ... | Mathlib/Topology/MetricSpace/PseudoMetric.lean | 433 | 433 | theorem ball_zero : ball x 0 = β
:= by | rw [ball_eq_empty]
|
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.Order.Interval.Finset.Nat
#align_import data.nat.factorization.basic from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
... | Mathlib/Data/Nat/Factorization/Basic.lean | 430 | 434 | theorem pow_succ_factorization_not_dvd {n p : β} (hn : n β 0) (hp : p.Prime) :
Β¬p ^ (n.factorization p + 1) β£ n := by |
intro h
rw [β factorization_le_iff_dvd (pow_pos hp.pos _).ne' hn] at h
simpa [hp.factorization] using h p
|
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Defs.Filter
#align_import topology.basic from "leanprover-community/mathlib"@"e354e865255654389cc46e6032160238df2e0f40"
noncomputable section
open Set Filter
universe u v w x
def TopologicalSpace.ofClosed {X : Type u} (T : Set (Set X)) (empty_mem : β
β... | Mathlib/Topology/Basic.lean | 184 | 185 | theorem isClosed_sInter {s : Set (Set X)} : (β t β s, IsClosed t) β IsClosed (ββ s) := by |
simpa only [β isOpen_compl_iff, compl_sInter, sUnion_image] using isOpen_biUnion
|
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 (β Γ β)
isLowerSet : IsLowerSet (cel... | Mathlib/Combinatorics/Young/YoungDiagram.lean | 387 | 388 | theorem colLen_eq_card (ΞΌ : YoungDiagram) {j : β} : ΞΌ.colLen j = (ΞΌ.col j).card := by |
simp [col_eq_prod]
|
import Mathlib.Analysis.Calculus.TangentCone
import Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics
#align_import analysis.calculus.fderiv.basic from "leanprover-community/mathlib"@"41bef4ae1254365bc190aee63b947674d2977f01"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open To... | Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 469 | 475 | theorem HasStrictFDerivAt.exists_lipschitzOnWith_of_nnnorm_lt (hf : HasStrictFDerivAt f f' x)
(K : ββ₯0) (hK : βf'ββ < K) : β s β π x, LipschitzOnWith K f s := by |
have := hf.add_isBigOWith (f'.isBigOWith_comp _ _) hK
simp only [sub_add_cancel, IsBigOWith] at this
rcases exists_nhds_square this with β¨U, Uo, xU, hUβ©
exact
β¨U, Uo.mem_nhds xU, lipschitzOnWith_iff_norm_sub_le.2 fun x hx y hy => hU (mk_mem_prod hx hy)β©
|
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Combinatorics.Additive.AP.Three.Defs
import Mathlib.Combinatorics.Pigeonhole
import Mathlib.Data.Complex.ExponentialBounds
#align_import combinatorics.additive.behrend from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
open N... | Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | 400 | 409 | theorem three_le_nValue (hN : 64 β€ N) : 3 β€ nValue N := by |
rw [nValue, β lt_iff_add_one_le, lt_ceil, cast_two]
apply lt_sqrt_of_sq_lt
have : (2 : β) ^ ((6 : β) : β) β€ N := by
rw [rpow_natCast]
exact (cast_le.2 hN).trans' (by norm_num1)
apply lt_of_lt_of_le _ (log_le_log (rpow_pos_of_pos zero_lt_two _) this)
rw [log_rpow zero_lt_two, β div_lt_iff']
Β· exact ... |
import Mathlib.Algebra.Group.Embedding
import Mathlib.Data.Fin.Basic
import Mathlib.Data.Finset.Union
#align_import data.finset.image from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
-- TODO
-- assert_not_exists OrderedCommMonoid
assert_not_exists MonoidWithZero
assert_not_exists MulA... | Mathlib/Data/Finset/Image.lean | 365 | 366 | theorem forall_image {p : Ξ² β Prop} : (β b β s.image f, p b) β β a β s, p (f a) := by |
simp only [mem_image, forall_exists_index, and_imp, forall_apply_eq_imp_iffβ]
|
import Batteries.Data.RBMap.Alter
import Batteries.Data.List.Lemmas
namespace Batteries
namespace RBNode
open RBColor
attribute [simp] fold foldl foldr Any forM foldlM Ordered
@[simp] theorem min?_reverse (t : RBNode Ξ±) : t.reverse.min? = t.max? := by
unfold RBNode.max?; split <;> simp [RBNode.min?]
unfold RB... | .lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean | 219 | 220 | theorem toStream_toList' {t : RBNode Ξ±} {s} : (t.toStream s).toList = t.toList ++ s.toList := by |
induction t generalizing s <;> simp [*, toStream]
|
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"@"b99e2d58a5e6861833fa8de11e51a81144258db4"
open Function
universe u v w
namespace SimpleGraph
variable {V : Type u} {V' : Type v} {V'' : Typ... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 2,470 | 2,472 | theorem set_walk_self_length_zero_eq (u : V) : {p : G.Walk u u | p.length = 0} = {Walk.nil} := by |
ext p
simp
|
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Order.LiminfLimsup
import Mathlib.Topology.Instances.Rat
import Mathlib.Topology.MetricSpace.Algebra
import Mathlib.Topology.MetricSpace.IsometricSMul
import Mathlib.Topology.Sequences
#align_import analysis.normed.group.basic from "leanprover-community/mat... | Mathlib/Analysis/Normed/Group/Basic.lean | 693 | 693 | theorem mem_ball_iff_norm'' : b β ball a r β βb / aβ < r := by | rw [mem_ball, dist_eq_norm_div]
|
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Nat.Dist
import Mathlib.Data.Ordmap.Ordnode
import Mathlib.Tactic.Abel
import Mathlib.Tactic.Linarith
#align_import data.ordmap.ordset from "leanprover-community/mathlib"@"47b51515e69f59bca5cf34ef456e6000fe205a69"
variable... | Mathlib/Data/Ordmap/Ordset.lean | 498 | 500 | theorem all_node3L {P l x m y r} :
@All Ξ± P (node3L l x m y r) β All P l β§ P x β§ All P m β§ P y β§ All P r := by |
simp [node3L, all_node', and_assoc]
|
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.CategoryTheory.Functor.ReflectsIso
#align_import category_theory.monoidal.center from "leanprover-community/mathlib"@"14b69e9f3c16630440a2cbd46f1ddad0d561dee7"
open CategoryTheory
open CategoryTheory.MonoidalCategory
universe v vβ vβ vβ u uβ uβ... | Mathlib/CategoryTheory/Monoidal/Center.lean | 312 | 314 | theorem leftUnitor_inv_f (X : Center C) : Hom.f (Ξ»_ X).inv = (Ξ»_ X.1).inv := by |
apply Iso.inv_ext' -- Porting note: Originally `ext`
rw [β leftUnitor_hom_f, β comp_f, Iso.hom_inv_id]; rfl
|
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.NumberTheory.Bernoulli
#align_import number_theory.bernoulli_polynomials from "leanprover-community/mathlib"@"ca3d21f7f4fd613c2a3c54ac7871163e1e5ecb3a"
noncomputable section... | Mathlib/NumberTheory/BernoulliPolynomials.lean | 76 | 82 | theorem bernoulli_eval_zero (n : β) : (bernoulli n).eval 0 = _root_.bernoulli n := by |
rw [bernoulli, eval_finset_sum, sum_range_succ]
have : β x β range n, _root_.bernoulli x * n.choose x * 0 ^ (n - x) = 0 := by
apply sum_eq_zero fun x hx => _
intros x hx
simp [tsub_eq_zero_iff_le, mem_range.1 hx]
simp [this]
|
import Mathlib.RingTheory.Localization.Integer
import Mathlib.RingTheory.Localization.Submodule
#align_import ring_theory.fractional_ideal from "leanprover-community/mathlib"@"ed90a7d327c3a5caf65a6faf7e8a0d63c4605df7"
open IsLocalization Pointwise nonZeroDivisors
namespace FractionalIdeal
open Set Submodule
va... | Mathlib/RingTheory/FractionalIdeal/Basic.lean | 125 | 130 | theorem den_mul_self_eq_num (I : FractionalIdeal S P) :
I.den β’ (I : Submodule R P) = Submodule.map (Algebra.linearMap R P) I.num := by |
rw [den, num, Submodule.map_comap_eq]
refine (inf_of_le_right ?_).symm
rintro _ β¨a, ha, rflβ©
exact I.2.choose_spec.2 a ha
|
import Mathlib.Order.Antichain
import Mathlib.Order.UpperLower.Basic
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.RelIso.Set
#align_import order.minimal from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Function Set
variable {Ξ± : Type*} (r rβ rβ : Ξ± β Ξ± β Prop) (s... | Mathlib/Order/Minimal.lean | 482 | 483 | theorem minimals_Icc (hab : a β€ b) : minimals (Β· β€ Β·) (Icc a b) = {a} := by |
simp_rw [Icc, and_comm (a := (a β€ _))]; exact maximals_Icc (Ξ± := Ξ±α΅α΅) hab
|
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.Measure.Count
import Mathlib.Topology.IndicatorConstPointwise
import Mathlib.MeasureTheory.Constructions.BorelSpace.Real
#align_import m... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 643 | 644 | theorem set_lintegral_univ (f : Ξ± β ββ₯0β) : β«β» x in univ, f x βΞΌ = β«β» x, f x βΞΌ := by |
rw [Measure.restrict_univ]
|
import Mathlib.Algebra.Quotient
import Mathlib.Algebra.Group.Subgroup.Actions
import Mathlib.Algebra.Group.Subgroup.MulOpposite
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.SetTheory.Cardinal.Finite
#align_import group_theory.coset from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce4... | Mathlib/GroupTheory/Coset.lean | 163 | 164 | theorem mem_leftCoset_leftCoset {a : Ξ±} (ha : a β’ (s : Set Ξ±) = s) : a β s := by |
rw [β SetLike.mem_coe, β ha]; exact mem_own_leftCoset s a
|
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.unoriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
... | Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean | 283 | 287 | theorem sin_angle_sub_of_inner_eq_zero {x y : V} (h : βͺx, yβ« = 0) (h0 : x β 0 β¨ y β 0) :
Real.sin (angle x (x - y)) = βyβ / βx - yβ := by |
rw [β neg_eq_zero, β inner_neg_right] at h
rw [or_comm, β neg_ne_zero, or_comm] at h0
rw [sub_eq_add_neg, sin_angle_add_of_inner_eq_zero h h0, norm_neg]
|
import Mathlib.Data.Fintype.Basic
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Defs
#align_import logic.equiv.fintype from "leanprover-community/mathlib"@"9407b03373c8cd201df99d6bc5514fc2db44054f"
namespace Equiv
variable {Ξ± Ξ² : Type*} [Finite Ξ±]
noncomputable def toCompl {p q : Ξ± β Prop} (e ... | Mathlib/Logic/Equiv/Fintype.lean | 132 | 135 | theorem extendSubtype_mem (e : { x // p x } β { x // q x }) (x) (hx : p x) :
q (e.extendSubtype x) := by |
convert (e β¨x, hxβ©).2
rw [e.extendSubtype_apply_of_mem _ hx]
|
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 | 79 | 87 | theorem count_not_le_count_add_one (hl : Chain' (Β· β Β·) l) (b : Bool) :
count (!b) l β€ count b l + 1 := by |
cases' l with x l
Β· exact zero_le _
obtain rfl | rfl : b = x β¨ b = !x := by simp only [Bool.eq_not_iff, em]
Β· rw [count_cons_of_ne b.not_ne_self, count_cons_self, hl.count_not, add_assoc]
exact add_le_add_left (Nat.mod_lt _ two_pos).le _
Β· rw [Bool.not_not, count_cons_self, count_cons_of_ne x.not_ne_self... |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_theory.perm.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Equiv Finset
namespace Equiv.Perm
variable {Ξ± : Type*}
section support
v... | Mathlib/GroupTheory/Perm/Support.lean | 304 | 306 | theorem coe_support_eq_set_support (f : Perm Ξ±) : (f.support : Set Ξ±) = { x | f x β x } := by |
ext
simp
|
import Mathlib.Tactic.ApplyFun
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.Separation
#align_import topology.uniform_space.separation from "leanprover-community/mathlib"@"0c1f285a9f6e608ae2bdffa3f993eafb01eba829"
open Filter Set Function Topology Uniformity UniformSpace
open scoped Classical... | Mathlib/Topology/UniformSpace/Separation.lean | 310 | 314 | theorem uniformContinuous_lift' [T0Space Ξ²] (f : Ξ± β Ξ²) : UniformContinuous (lift' f) := by |
by_cases hf : UniformContinuous f
Β· rwa [lift', dif_pos hf, uniformContinuous_lift]
Β· rw [lift', dif_neg hf]
exact uniformContinuous_of_const fun a _ => rfl
|
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 | 357 | 400 | theorem content_mul {p q : R[X]} : (p * q).content = p.content * q.content := by |
classical
suffices h :
β (n : β) (p q : R[X]), (p * q).degree < n β (p * q).content = p.content * q.content by
apply h
apply lt_of_le_of_lt degree_le_natDegree (WithBot.coe_lt_coe.2 (Nat.lt_succ_self _))
intro n
induction' n with n ih
Β· intro p q hpq
rw [Nat.cast_zero,
... |
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Independent
#align_import analysis.convex.simplicial_complex.basic from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open Finset Set
variable (π E : Type*) {ΞΉ : Type*} [OrderedRing π] [AddCommGroup E] [Mod... | Mathlib/Analysis/Convex/SimplicialComplex/Basic.lean | 158 | 162 | theorem vertices_eq : K.vertices = β k β K.faces, (k : Set E) := by |
ext x
refine β¨fun h => mem_biUnion h <| mem_coe.2 <| mem_singleton_self x, fun h => ?_β©
obtain β¨s, hs, hxβ© := mem_iUnionβ.1 h
exact K.down_closed hs (Finset.singleton_subset_iff.2 <| mem_coe.1 hx) (singleton_ne_empty _)
|
import Mathlib.Data.Finset.NAry
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Set.Pointwise.Finite
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.Data.Set.Pointwise.ListOfFn
import Mathlib.GroupTheory.GroupAction.Pi
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.finset.pointwise from "le... | Mathlib/Data/Finset/Pointwise.lean | 2,114 | 2,117 | theorem smul_finset_subset_iff : a β’ s β t β s β aβ»ΒΉ β’ t := by |
simp_rw [β coe_subset]
push_cast
exact Set.set_smul_subset_iff
|
import Mathlib.Algebra.Associated
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.SMulWithZero
import Mathlib.Data.Nat.PartENat
import Mathlib.Tactic.Linarith
#align_import ring_theory.multiplicity from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
variable {Ξ± Ξ²... | Mathlib/RingTheory/Multiplicity.lean | 219 | 220 | theorem lt_top_iff_finite {a b : Ξ±} : multiplicity a b < β€ β Finite a b := by |
rw [lt_top_iff_ne_top, ne_top_iff_finite]
|
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