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 | goals listlengths 0 224 | goals_before listlengths 0 220 |
|---|---|---|---|---|---|---|---|
import Mathlib.Computability.DFA
import Mathlib.Data.Fintype.Powerset
#align_import computability.NFA from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
open Set
open Computability
universe u v
-- Porting note: Required as `NFA` is used in mathlib3
set_option linter.uppercaseLean3 fa... | Mathlib/Computability/NFA.lean | 53 | 54 | theorem mem_stepSet (s : Ο) (S : Set Ο) (a : Ξ±) : s β M.stepSet S a β β t β S, s β M.step t a := by |
simp [stepSet]
| [
" s β M.stepSet S a β β t β S, s β M.step t a"
] | [] |
import Mathlib.Combinatorics.Quiver.Path
import Mathlib.Combinatorics.Quiver.Push
#align_import combinatorics.quiver.symmetric from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
universe v u w v'
namespace Quiver
-- Porting note: no hasNonemptyInstance linter yet
def Symmetrify (V : ... | Mathlib/Combinatorics/Quiver/Symmetric.lean | 158 | 163 | theorem Path.reverse_reverse [h : HasInvolutiveReverse V] {a b : V} (p : Path a b) :
p.reverse.reverse = p := by |
induction' p with _ _ _ _ h
Β· simp
Β· rw [Path.reverse, Path.reverse_comp, h, Path.reverse_toPath, Quiver.reverse_reverse]
rfl
| [
" reverse (reverse f) = f",
" reverse f = reverse g β f = g",
" reverse f = reverse g β f = g",
" f = g",
" f = g β reverse f = reverse g",
" reverse f = reverse g",
" f = reverse g β reverse f = g",
" (p.comp q).reverse = q.reverse.comp p.reverse",
" (p.comp nil).reverse = nil.reverse.comp p.revers... | [
" reverse (reverse f) = f",
" reverse f = reverse g β f = g",
" reverse f = reverse g β f = g",
" f = g",
" f = g β reverse f = reverse g",
" reverse f = reverse g",
" f = reverse g β reverse f = g",
" (p.comp q).reverse = q.reverse.comp p.reverse",
" (p.comp nil).reverse = nil.reverse.comp p.revers... |
import Mathlib.Topology.EMetricSpace.Basic
#align_import topology.metric_space.metric_separated from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
open EMetric Set
noncomputable section
def IsMetricSeparated {X : Type*} [EMetricSpace X] (s t : Set X) :=
β r, r β 0 β§ β x β s, β y β... | Mathlib/Topology/MetricSpace/MetricSeparated.lean | 106 | 109 | theorem finite_iUnion_left_iff {ΞΉ : Type*} {I : Set ΞΉ} (hI : I.Finite) {s : ΞΉ β Set X}
{t : Set X} : IsMetricSeparated (β i β I, s i) t β β i β I, IsMetricSeparated (s i) t := by |
refine Finite.induction_on hI (by simp) @fun i I _ _ hI => ?_
rw [biUnion_insert, forall_mem_insert, union_left_iff, hI]
| [
" r = 0",
" IsMetricSeparated (s βͺ s') t",
" min r r' β 0",
" 0 < min r r'",
" x β s β min r r' β€ edist x y",
" x β s' β min r r' β€ edist x y",
" IsMetricSeparated (β i β I, s i) t β β i β I, IsMetricSeparated (s i) t",
" IsMetricSeparated (β i β β
, s i) t β β i β β
, IsMetricSeparated (s i) t",
" Is... | [
" r = 0",
" IsMetricSeparated (s βͺ s') t",
" min r r' β 0",
" 0 < min r r'",
" x β s β min r r' β€ edist x y",
" x β s' β min r r' β€ edist x y"
] |
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
import Mathlib.Analysis.Complex.CauchyIntegral
#align_import analysis.special_functions.gamma.beta from "l... | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | 80 | 90 | theorem betaIntegral_convergent {u v : β} (hu : 0 < re u) (hv : 0 < re v) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 1 := by |
refine (betaIntegral_convergent_left hu v).trans ?_
rw [IntervalIntegrable.iff_comp_neg]
convert ((betaIntegral_convergent_left hv u).comp_add_right 1).symm using 1
Β· ext1 x
conv_lhs => rw [mul_comm]
congr 2 <;> Β· push_cast; ring
Β· norm_num
Β· norm_num
| [
" IntervalIntegrable (fun x => βx ^ (u - 1) * (1 - βx) ^ (v - 1)) volume 0 (1 / 2)",
" IntervalIntegrable (fun x => βx ^ (u - 1)) volume 0 (1 / 2)",
" -1 < (u - 1).re",
" ContinuousOn (fun x => (1 - βx) ^ (v - 1)) (uIcc 0 (1 / 2))",
" β x β uIcc 0 (1 / 2), ContinuousAt (fun x => (1 - βx) ^ (v - 1)) x",
" ... | [
" IntervalIntegrable (fun x => βx ^ (u - 1) * (1 - βx) ^ (v - 1)) volume 0 (1 / 2)",
" IntervalIntegrable (fun x => βx ^ (u - 1)) volume 0 (1 / 2)",
" -1 < (u - 1).re",
" ContinuousOn (fun x => (1 - βx) ^ (v - 1)) (uIcc 0 (1 / 2))",
" β x β uIcc 0 (1 / 2), ContinuousAt (fun x => (1 - βx) ^ (v - 1)) x",
" ... |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Data.Set.Lattice
#align_import group_theory.archimedean from "leanprover-community/mathlib"@"f93c11933efbc3c2f0299e47b8ff83e9b539cbf6"
open Set
variable {G : Type*} [LinearOrderedAddCommGroup G] [Archimedean G]
| Mathlib/GroupTheory/Archimedean.lean | 40 | 54 | theorem AddSubgroup.cyclic_of_min {H : AddSubgroup G} {a : G}
(ha : IsLeast { g : G | g β H β§ 0 < g } a) : H = AddSubgroup.closure {a} := by |
obtain β¨β¨a_in, a_posβ©, a_minβ© := ha
refine le_antisymm ?_ (H.closure_le.mpr <| by simp [a_in])
intro g g_in
obtain β¨k, β¨nonneg, ltβ©, _β© := existsUnique_zsmul_near_of_pos' a_pos g
have h_zero : g - k β’ a = 0 := by
by_contra h
have h : a β€ g - k β’ a := by
refine a_min β¨?_, ?_β©
Β· exact AddSu... | [
" H = closure {a}",
" {a} β βH",
" H β€ closure {a}",
" g β closure {a}",
" g - k β’ a = 0",
" False",
" a β€ g - k β’ a",
" g - k β’ a β H",
" 0 < g - k β’ a"
] | [] |
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.Analysis.RCLike.Basic
#align_import data.is_R_or_C.lemmas from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1e360b7e3cf6"
variable {K E : Type*} [RCLike K]
namespace RCLike
@[simp, rclike_simps]
| Mathlib/Analysis/RCLike/Lemmas.lean | 71 | 74 | theorem reCLM_norm : β(reCLM : K βL[β] β)β = 1 := by |
apply le_antisymm (LinearMap.mkContinuous_norm_le _ zero_le_one _)
convert ContinuousLinearMap.ratio_le_opNorm (reCLM : K βL[β] β) (1 : K)
simp
| [
" βreCLMβ = 1",
" 1 β€ βreLm.mkContinuous 1 β―β",
" 1 = βreCLM 1β / β1β"
] | [] |
import Mathlib.Order.Interval.Finset.Nat
#align_import data.fin.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
assert_not_exists MonoidWithZero
open Finset Fin Function
namespace Fin
variable (n : β)
instance instLocallyFiniteOrder : LocallyFiniteOrder (Fin n) :=
Orde... | Mathlib/Order/Interval/Finset/Fin.lean | 89 | 90 | theorem map_valEmbedding_Ioc : (Ioc a b).map Fin.valEmbedding = Ioc βa βb := by |
simp [Ioc_eq_finset_subtype, Finset.fin, Finset.map_map, Ioc_filter_lt_of_lt_right]
| [
" map valEmbedding (Icc a b) = Icc βa βb",
" map valEmbedding (Ico a b) = Ico βa βb",
" map valEmbedding (Ioc a b) = Ioc βa βb"
] | [
" map valEmbedding (Icc a b) = Icc βa βb",
" map valEmbedding (Ico a b) = Ico βa βb"
] |
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 {Ξ± Ξ² R : Type*}
namespace Fin... | Mathlib/Data/Finset/Card.lean | 111 | 111 | theorem card_insert_of_mem (h : a β s) : card (insert a s) = s.card := by | rw [insert_eq_of_mem h]
| [
" Monotone card",
" ({a} β© s).card β€ 1",
" (insert a s).card = s.card + 1",
" (insert a s).card = s.card"
] | [
" Monotone card",
" ({a} β© s).card β€ 1",
" (insert a s).card = s.card + 1"
] |
import Batteries.Tactic.Lint.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Order.ZeroLEOne
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Init.Data.Int.Order
set_option autoImplicit true
namespace Linarith
theorem lt_irrefl {Ξ± : Type u} ... | Mathlib/Tactic/Linarith/Lemmas.lean | 27 | 28 | theorem eq_of_eq_of_eq {Ξ±} [OrderedSemiring Ξ±] {a b : Ξ±} (ha : a = 0) (hb : b = 0) : a + b = 0 := by |
simp [*]
| [
" a + b = 0"
] | [] |
import Mathlib.Combinatorics.Enumerative.DoubleCounting
import Mathlib.Combinatorics.SimpleGraph.AdjMatrix
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Set.Finite
#align_import combinatorics.simple_graph.strongly_regular from "leanprover-community/mathlib"@"2b35fc7bea4640cb75e477e83f32fbd5389208... | Mathlib/Combinatorics/SimpleGraph/StronglyRegular.lean | 102 | 106 | theorem IsSRGWith.card_neighborFinset_union_of_not_adj {v w : V} (h : G.IsSRGWith n k β ΞΌ)
(hne : v β w) (ha : Β¬G.Adj v w) :
(G.neighborFinset v βͺ G.neighborFinset w).card = 2 * k - ΞΌ := by |
rw [β h.of_not_adj hne ha]
apply h.card_neighborFinset_union_eq
| [
" (fun v w => Β¬β₯.Adj v w β Fintype.card β(β₯.commonNeighbors v w) = 0) v w",
" filter (fun x => x β β₯.commonNeighbors v w) univ = β
",
" aβ β filter (fun x => x β β₯.commonNeighbors v w) univ β aβ β β
",
" Fintype.card β(β€.commonNeighbors v w) = Fintype.card V - 2",
" v β w",
" (G.neighborFinset v βͺ G.neighbo... | [
" (fun v w => Β¬β₯.Adj v w β Fintype.card β(β₯.commonNeighbors v w) = 0) v w",
" filter (fun x => x β β₯.commonNeighbors v w) univ = β
",
" aβ β filter (fun x => x β β₯.commonNeighbors v w) univ β aβ β β
",
" Fintype.card β(β€.commonNeighbors v w) = Fintype.card V - 2",
" v β w",
" (G.neighborFinset v βͺ G.neighbo... |
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.special_functions.log.basic from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
open Set Filter Function
open Topology
noncomputable ... | Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 118 | 119 | theorem cosh_log {x : β} (hx : 0 < x) : cosh (log x) = (x + xβ»ΒΉ) / 2 := by |
rw [cosh_eq, exp_neg, exp_log hx]
| [
" x.log = expOrderIso.symm β¨x, hxβ©",
" expOrderIso.symm β¨|x|, β―β© = expOrderIso.symm β¨x, hxβ©",
" |x| = x",
" rexp x.log = |x|",
" rexp x.log = x",
" rexp x.log = -x",
" |x| = -x",
" x β€ rexp x.log",
" 0 β€ 1",
" x β€ |x|",
" rexp (log 1) = rexp 0",
" |x|.log = x.log",
" (-x).log = x.log",
" x... | [
" x.log = expOrderIso.symm β¨x, hxβ©",
" expOrderIso.symm β¨|x|, β―β© = expOrderIso.symm β¨x, hxβ©",
" |x| = x",
" rexp x.log = |x|",
" rexp x.log = x",
" rexp x.log = -x",
" |x| = -x",
" x β€ rexp x.log",
" 0 β€ 1",
" x β€ |x|",
" rexp (log 1) = rexp 0",
" |x|.log = x.log",
" (-x).log = x.log",
" x... |
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 | 110 | 119 | theorem disjoint_or_exists_inter_eq_convexHull (hs : s β K.faces) (ht : t β K.faces) :
Disjoint (convexHull π (s : Set E)) (convexHull π βt) β¨
β u β K.faces, convexHull π (s : Set E) β© convexHull π βt = convexHull π βu := by |
classical
by_contra! h
refine h.2 (s β© t) (K.down_closed hs inter_subset_left fun hst => h.1 <|
disjoint_iff_inf_le.mpr <| (K.inter_subset_convexHull hs ht).trans ?_) ?_
Β· rw [β coe_inter, hst, coe_empty, convexHull_empty]
rfl
Β· rw [coe_inter, convexHull_inter_convexHull hs ht]
| [
" x β K.space β β s β K.faces, x β (convexHull π) βs",
" (convexHull π) βs β K.space",
" (convexHull π) βs = (convexHull π) βs",
" Disjoint ((convexHull π) βs) ((convexHull π) βt) β¨\n β u β K.faces, (convexHull π) βs β© (convexHull π) βt = (convexHull π) βu",
" False",
" (convexHull π) (βs β© β... | [
" x β K.space β β s β K.faces, x β (convexHull π) βs",
" (convexHull π) βs β K.space",
" (convexHull π) βs = (convexHull π) βs"
] |
import Mathlib.Analysis.Convex.Between
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.Topology.MetricSpace.Holder
import Mathlib.Topology.MetricSpace.MetricSeparated
#align_import measure_theory.measure.hausdorff from "leanprover-communit... | Mathlib/MeasureTheory/Measure/Hausdorff.lean | 270 | 271 | theorem le_pre : ΞΌ β€ pre m r β β s : Set X, diam s β€ r β ΞΌ s β€ m s := by |
simp only [pre, le_boundedBy, extend, le_iInf_iff]
| [
" ΞΌ β€ pre m r β β (s : Set X), diam s β€ r β ΞΌ s β€ m s"
] | [] |
import Mathlib.Algebra.Algebra.Subalgebra.Operations
import Mathlib.Algebra.Ring.Fin
import Mathlib.RingTheory.Ideal.Quotient
#align_import ring_theory.ideal.quotient_operations from "leanprover-community/mathlib"@"b88d81c84530450a8989e918608e5960f015e6c8"
universe u v w
namespace Ideal
open Function RingHom
var... | Mathlib/RingTheory/Ideal/QuotientOperations.lean | 136 | 138 | theorem map_mk_eq_bot_of_le {I J : Ideal R} (h : I β€ J) : I.map (Quotient.mk J) = β₯ := by |
rw [map_eq_bot_iff_le_ker, mk_ker]
exact h
| [
" ker (Quotient.mk I) = I",
" xβ β ker (Quotient.mk I) β xβ β I",
" map (Quotient.mk J) I = β₯",
" I β€ J"
] | [
" ker (Quotient.mk I) = I",
" xβ β ker (Quotient.mk I) β xβ β I"
] |
import Mathlib.MeasureTheory.Covering.VitaliFamily
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.MeasureTheory.Function.AEMeasurableOrder
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.MeasureTheory.Integral.Average
import Mathlib.MeasureTheory.Decomposition.Lebesgue
#align_import measure... | Mathlib/MeasureTheory/Covering/Differentiation.lean | 125 | 149 | theorem measure_le_of_frequently_le [SecondCountableTopology Ξ±] [BorelSpace Ξ±] {Ο : Measure Ξ±}
(Ξ½ : Measure Ξ±) [IsLocallyFiniteMeasure Ξ½] (hΟ : Ο βͺ ΞΌ) (s : Set Ξ±)
(hs : β x β s, βαΆ a in v.filterAt x, Ο a β€ Ξ½ a) : Ο s β€ Ξ½ s := by |
-- this follows from a covering argument using the sets satisfying `Ο a β€ Ξ½ a`.
apply ENNReal.le_of_forall_pos_le_add fun Ξ΅ Ξ΅pos _ => ?_
obtain β¨U, sU, U_open, Ξ½Uβ© : β (U : Set Ξ±), s β U β§ IsOpen U β§ Ξ½ U β€ Ξ½ s + Ξ΅ :=
exists_isOpen_le_add s Ξ½ (ENNReal.coe_pos.2 Ξ΅pos).ne'
let f : Ξ± β Set (Set Ξ±) := fun _ => ... | [
" βα΅ (x : Ξ±) βΞΌ, βαΆ (a : Set Ξ±) in v.filterAt x, 0 < ΞΌ a",
" ΞΌ s = 0",
" v.FineSubfamilyOn f s",
" β a β v.setsAt x β© f x, a β closedBall x Ξ΅",
" ΞΌ s β€ 0",
" β' (x : βh.index), ΞΌ (h.covering βx) = β' (x : βh.index), 0",
" (fun x => ΞΌ (h.covering βx)) = fun x => 0",
" ΞΌ (h.covering βx) = 0",
" β' (x ... | [
" βα΅ (x : Ξ±) βΞΌ, βαΆ (a : Set Ξ±) in v.filterAt x, 0 < ΞΌ a",
" ΞΌ s = 0",
" v.FineSubfamilyOn f s",
" β a β v.setsAt x β© f x, a β closedBall x Ξ΅",
" ΞΌ s β€ 0",
" β' (x : βh.index), ΞΌ (h.covering βx) = β' (x : βh.index), 0",
" (fun x => ΞΌ (h.covering βx)) = fun x => 0",
" ΞΌ (h.covering βx) = 0",
" β' (x ... |
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 | 135 | 135 | theorem inv_neg : (-a)β»ΒΉ = -aβ»ΒΉ := by | rw [neg_inv]
| [
" -1 * -1 = 1",
" 1 / -a = 1 / (-1 * a)",
" 1 / (-1 * a) = 1 / a * (1 / -1)",
" 1 / a * (1 / -1) = 1 / a * -1",
" 1 / a * -1 = -(1 / a)",
" b / -a = b * (1 / -a)",
" b * (1 / -a) = b * -(1 / a)",
" b * -(1 / a) = -(b * (1 / a))",
" -(b * (1 / a)) = -(b / a)",
" -b / a = -(b / a)",
" -(b / a) = -... | [
" -1 * -1 = 1",
" 1 / -a = 1 / (-1 * a)",
" 1 / (-1 * a) = 1 / a * (1 / -1)",
" 1 / a * (1 / -1) = 1 / a * -1",
" 1 / a * -1 = -(1 / a)",
" b / -a = b * (1 / -a)",
" b * (1 / -a) = b * -(1 / a)",
" b * -(1 / a) = -(b * (1 / a))",
" -(b * (1 / a)) = -(b / a)",
" -b / a = -(b / a)",
" -(b / a) = -... |
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.constructions.borel_space.basic from "leanprover-community/mathlib"@"9f55d0d4363ae59948c33864cbc52e0b12e0e8ce"
open Set Filter MeasureTheory MeasurableSpace
open scoped Classical Topology NNReal ENNReal MeasureTheory
univers... | Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean | 101 | 104 | theorem isPiSystem_Iic_rat : IsPiSystem (β a : β, {Iic (a : β)}) := by |
convert isPiSystem_image_Iic (((β) : β β β) '' univ)
ext x
simp only [iUnion_singleton_eq_range, mem_range, image_univ, mem_image, exists_exists_eq_and]
| [
" borel β = generateFrom (β a, {Iio βa})",
" generateFrom (range Iio) = generateFrom (β a, {Iio βa})",
" β t β range Iio, MeasurableSet t",
" MeasurableSet (Iio a)",
" IsLUB (range Rat.cast β© Iio a) a",
" MeasurableSet (β y β Rat.cast β»ΒΉ' Iio a, Iio βy)",
" Iio βb β β a, {Iio βa}",
" borel β = generat... | [
" borel β = generateFrom (β a, {Iio βa})",
" generateFrom (range Iio) = generateFrom (β a, {Iio βa})",
" β t β range Iio, MeasurableSet t",
" MeasurableSet (Iio a)",
" IsLUB (range Rat.cast β© Iio a) a",
" MeasurableSet (β y β Rat.cast β»ΒΉ' Iio a, Iio βy)",
" Iio βb β β a, {Iio βa}",
" borel β = generat... |
import Mathlib.Algebra.GroupWithZero.Indicator
import Mathlib.Topology.ContinuousOn
import Mathlib.Topology.Instances.ENNReal
#align_import topology.semicontinuous from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology ENNReal
open Set Function Filter
variable {Ξ± : Type*} [... | Mathlib/Topology/Semicontinuous.lean | 213 | 220 | theorem IsOpen.lowerSemicontinuous_indicator (hs : IsOpen s) (hy : 0 β€ y) :
LowerSemicontinuous (indicator s fun _x => y) := by |
intro x z hz
by_cases h : x β s <;> simp [h] at hz
Β· filter_upwards [hs.mem_nhds h]
simp (config := { contextual := true }) [hz]
Β· refine Filter.eventually_of_forall fun x' => ?_
by_cases h' : x' β s <;> simp [h', hz.trans_le hy, hz]
| [
" LowerSemicontinuousWithinAt f univ x β LowerSemicontinuousAt f x",
" LowerSemicontinuousOn f univ β LowerSemicontinuous f",
" LowerSemicontinuous (s.indicator fun _x => y)",
" βαΆ (x' : Ξ±) in π x, z < s.indicator (fun _x => y) x'",
" β a β s, z < s.indicator (fun _x => y) a",
" z < s.indicator (fun _x =... | [
" LowerSemicontinuousWithinAt f univ x β LowerSemicontinuousAt f x",
" LowerSemicontinuousOn f univ β LowerSemicontinuous f"
] |
import Mathlib.Algebra.Order.CauSeq.Basic
#align_import data.real.cau_seq_completion from "leanprover-community/mathlib"@"cf4c49c445991489058260d75dae0ff2b1abca28"
variable {Ξ± : Type*} [LinearOrderedField Ξ±]
namespace CauSeq
section
variable (Ξ² : Type*) [Ring Ξ²] (abv : Ξ² β Ξ±) [IsAbsoluteValue abv]
class IsCo... | Mathlib/Algebra/Order/CauSeq/Completion.lean | 413 | 436 | theorem lim_inv {f : CauSeq Ξ² abv} (hf : Β¬LimZero f) : lim (inv f hf) = (lim f)β»ΒΉ :=
have hl : lim f β 0 := by | rwa [β lim_eq_zero_iff] at hf
lim_eq_of_equiv_const <|
show LimZero (inv f hf - const abv (lim f)β»ΒΉ) from
have hβ : β (g f : CauSeq Ξ² abv) (hf : Β¬LimZero f), LimZero (g - f * inv f hf * g) :=
fun g f hf => by
have hβ : g - f * inv f hf * g = 1 * g - f * inv f hf * g := by rw [one_mul g]
... | [
" (const abv (f.lim + g.lim) - (f + g)).LimZero",
" (const abv f.lim - f + (const abv g.lim - g)).LimZero",
" (const abv (f.lim * g.lim) - f * g).LimZero",
" const abv (f.lim * g.lim) - f * g = (const abv f.lim - f) * g + const abv f.lim * (const abv g.lim - g)",
" β(const abv (f.lim * g.lim) - f * g) = β((... | [
" (const abv (f.lim + g.lim) - (f + g)).LimZero",
" (const abv f.lim - f + (const abv g.lim - g)).LimZero",
" (const abv (f.lim * g.lim) - f * g).LimZero",
" const abv (f.lim * g.lim) - f * g = (const abv f.lim - f) * g + const abv f.lim * (const abv g.lim - g)",
" β(const abv (f.lim * g.lim) - f * g) = β((... |
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
import Mathlib.RepresentationTheory.GroupCohomology.Basic
import Mathlib.RepresentationTheory.Invariants
universe v u
noncomputable section
open CategoryTheory Limits Representation
variable {k G : Type u} [CommRing k] [Group G] (A : Rep k G)
namespace grou... | Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean | 546 | 551 | theorem smul_map_inv_div_map_inv_of_isMulTwoCocycle
{f : G Γ G β M} (hf : IsMulTwoCocycle f) (g : G) :
g β’ f (gβ»ΒΉ, g) / f (g, gβ»ΒΉ) = f (1, 1) / f (g, 1) := by |
have := hf g gβ»ΒΉ g
simp only [mul_right_inv, mul_left_inv, map_one_fst_of_isMulTwoCocycle hf g] at this
exact div_eq_div_iff_mul_eq_mul.2 this.symm
| [
" f 1 = 1",
" f (1, g) = f (1, 1)",
" f (g, 1) = g β’ f (1, 1)",
" g β’ f gβ»ΒΉ = (f g)β»ΒΉ",
" g β’ f (gβ»ΒΉ, g) / f (g, gβ»ΒΉ) = f (1, 1) / f (g, 1)"
] | [
" f 1 = 1",
" f (1, g) = f (1, 1)",
" f (g, 1) = g β’ f (1, 1)",
" g β’ f gβ»ΒΉ = (f g)β»ΒΉ"
] |
import Mathlib.Algebra.Order.Ring.Int
#align_import data.int.least_greatest from "leanprover-community/mathlib"@"3342d1b2178381196f818146ff79bc0e7ccd9e2d"
namespace Int
def leastOfBdd {P : β€ β Prop} [DecidablePred P] (b : β€) (Hb : β z : β€, P z β b β€ z)
(Hinh : β z : β€, P z) : { lb : β€ // P lb β§ β z : β€, P z... | Mathlib/Data/Int/LeastGreatest.lean | 61 | 68 | theorem exists_least_of_bdd
{P : β€ β Prop}
(Hbdd : β b : β€ , β z : β€ , P z β b β€ z)
(Hinh : β z : β€ , P z) : β lb : β€ , P lb β§ β z : β€ , P z β lb β€ z := by |
classical
let β¨b , Hbβ© := Hbdd
let β¨lb , Hβ© := leastOfBdd b Hb Hinh
exact β¨lb , Hβ©
| [
" β lb, P lb β§ β (z : β€), P z β lb β€ z"
] | [] |
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.GroupTheory.Solvable
import Mathlib.GroupTheory.PGroup
import Mathlib.GroupTheory.Sylow
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Tactic.TFAE
#align_import group_theory.nilpotent from "leanprover-community/mathlib"@"2bbc7e3884ba234309d2a43b19144... | Mathlib/GroupTheory/Nilpotent.lean | 112 | 119 | theorem upperCentralSeriesStep_eq_comap_center :
upperCentralSeriesStep H = Subgroup.comap (mk' H) (center (G β§Έ H)) := by |
ext
rw [mem_comap, mem_center_iff, forall_mk]
apply forall_congr'
intro y
rw [coe_mk', β QuotientGroup.mk_mul, β QuotientGroup.mk_mul, eq_comm, eq_iff_div_mem,
div_eq_mul_inv, mul_inv_rev, mul_assoc]
| [
" a * b * y * (a * b)β»ΒΉ * yβ»ΒΉ β H",
" a * b * y * (a * b)β»ΒΉ * yβ»ΒΉ = a * (b * y * bβ»ΒΉ) * aβ»ΒΉ * (b * y * bβ»ΒΉ)β»ΒΉ * (b * y * bβ»ΒΉ * yβ»ΒΉ)",
" 1 * y * 1β»ΒΉ * yβ»ΒΉ β H",
" xβ»ΒΉ * y * xβ»ΒΉβ»ΒΉ * yβ»ΒΉ β H",
" xβ»ΒΉ * y * (x * yβ»ΒΉ) β H",
" upperCentralSeriesStep H = comap (mk' H) (center (G β§Έ H))",
" xβ β upperCentralSerie... | [
" a * b * y * (a * b)β»ΒΉ * yβ»ΒΉ β H",
" a * b * y * (a * b)β»ΒΉ * yβ»ΒΉ = a * (b * y * bβ»ΒΉ) * aβ»ΒΉ * (b * y * bβ»ΒΉ)β»ΒΉ * (b * y * bβ»ΒΉ * yβ»ΒΉ)",
" 1 * y * 1β»ΒΉ * yβ»ΒΉ β H",
" xβ»ΒΉ * y * xβ»ΒΉβ»ΒΉ * yβ»ΒΉ β H",
" xβ»ΒΉ * y * (x * yβ»ΒΉ) β H"
] |
import Mathlib.Algebra.Group.Commute.Units
import Mathlib.Algebra.Group.Int
import Mathlib.Algebra.GroupWithZero.Semiconj
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Order.Bounds.Basic
#align_import data.int.gcd from "leanprover-community/mathlib"@"47a1a73351de8dd6c8d3d32b569c8e434b03ca47"
namespace Nat
... | Mathlib/Data/Int/GCD.lean | 146 | 154 | theorem exists_mul_emod_eq_gcd {k n : β} (hk : gcd n k < k) : β m, n * m % k = gcd n k := by |
have hk' := Int.ofNat_ne_zero.2 (ne_of_gt (lt_of_le_of_lt (zero_le (gcd n k)) hk))
have key := congr_arg (fun (m : β€) => (m % k).toNat) (gcd_eq_gcd_ab n k)
simp only at key
rw [Int.add_mul_emod_self_left, β Int.natCast_mod, Int.toNat_natCast, mod_eq_of_lt hk] at key
refine β¨(n.gcdA k % k).toNat, Eq.trans (In... | [
" (invImage\n (fun x =>\n PSigma.casesOn x fun a a_1 =>\n PSigma.casesOn a_1 fun a_2 a_3 =>\n PSigma.casesOn a_3 fun a_4 a_5 => PSigma.casesOn a_5 fun a_6 a_7 => PSigma.casesOn a_7 fun a_8 a_9 => a)\n instWellFoundedRelationOfSizeOf).1\n β¨r' % k.succ, β¨s' - βq * s, ... | [
" (invImage\n (fun x =>\n PSigma.casesOn x fun a a_1 =>\n PSigma.casesOn a_1 fun a_2 a_3 =>\n PSigma.casesOn a_3 fun a_4 a_5 => PSigma.casesOn a_5 fun a_6 a_7 => PSigma.casesOn a_7 fun a_8 a_9 => a)\n instWellFoundedRelationOfSizeOf).1\n β¨r' % k.succ, β¨s' - βq * s, ... |
import Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties
import Mathlib.RingTheory.RingHom.FiniteType
#align_import algebraic_geometry.morphisms.finite_type from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
open CategoryTheory CategoryTheory.Limits Opposite ... | Mathlib/AlgebraicGeometry/Morphisms/FiniteType.lean | 65 | 71 | theorem locallyOfFiniteTypeOfComp {X Y Z : Scheme} (f : X βΆ Y) (g : Y βΆ Z)
[hf : LocallyOfFiniteType (f β« g)] : LocallyOfFiniteType f := by |
revert hf
rw [locallyOfFiniteType_eq]
apply RingHom.finiteType_is_local.affineLocally_of_comp
introv H
exact RingHom.FiniteType.of_comp_finiteType H
| [
" @LocallyOfFiniteType = affineLocally @RingHom.FiniteType",
" LocallyOfFiniteType f β affineLocally (@RingHom.FiniteType) f",
" RingHom.RespectsIso @RingHom.FiniteType",
" LocallyOfFiniteType f",
" β [hf : LocallyOfFiniteType (f β« g)], LocallyOfFiniteType f",
" β [hf : affineLocally (@RingHom.FiniteType)... | [
" @LocallyOfFiniteType = affineLocally @RingHom.FiniteType",
" LocallyOfFiniteType f β affineLocally (@RingHom.FiniteType) f",
" RingHom.RespectsIso @RingHom.FiniteType"
] |
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.Transfer
#align_import group_theory.schreier from "leanprover-community/mathlib"@"8350c34a64b9bc3fc64335df8006bffcadc7baa6"
open scoped Pointwise
namespace Subgroup
open MemRightTransversals
variable {G : T... | Mathlib/GroupTheory/Schreier.lean | 64 | 79 | theorem closure_mul_image_eq (hR : R β rightTransversals (H : Set G)) (hR1 : (1 : G) β R)
(hS : closure S = β€) : closure ((R * S).image fun g => g * (toFun hR g : G)β»ΒΉ) = H := by |
have hU : closure ((R * S).image fun g => g * (toFun hR g : G)β»ΒΉ) β€ H := by
rw [closure_le]
rintro - β¨g, -, rflβ©
exact mul_inv_toFun_mem hR g
refine le_antisymm hU fun h hh => ?_
obtain β¨g, hg, r, hr, rflβ© :=
show h β _ from eq_top_iff.mp (closure_mul_image_mul_eq_top hR hR1 hS) (mem_top h)
suf... | [
" β(closure ((fun g => g * (β(toFun hR g))β»ΒΉ) '' (R * S))) * R = β€",
" β(closure U) * R = β€",
" g β β(closure U) * R",
" 1 β β(closure U) * R",
" β x β closure S, β y β S, x β β(closure U) * R β x * y β β(closure U) * R",
" (fun x x_1 => x * x_1) u r * s β β(closure U) * R",
" u * r * s = u * (r * s * (... | [
" β(closure ((fun g => g * (β(toFun hR g))β»ΒΉ) '' (R * S))) * R = β€",
" β(closure U) * R = β€",
" g β β(closure U) * R",
" 1 β β(closure U) * R",
" β x β closure S, β y β S, x β β(closure U) * R β x * y β β(closure U) * R",
" (fun x x_1 => x * x_1) u r * s β β(closure U) * R",
" u * r * s = u * (r * s * (... |
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
import Mathlib.Analysis.Analytic.IsolatedZeros
import Mathlib.Analysis.Complex.CauchyIntegral
#align_import analysis.special_functions.gamma.beta from "l... | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | 63 | 76 | theorem betaIntegral_convergent_left {u : β} (hu : 0 < re u) (v : β) :
IntervalIntegrable (fun x =>
(x : β) ^ (u - 1) * (1 - (x : β)) ^ (v - 1) : β β β) volume 0 (1 / 2) := by |
apply IntervalIntegrable.mul_continuousOn
Β· refine intervalIntegral.intervalIntegrable_cpow' ?_
rwa [sub_re, one_re, β zero_sub, sub_lt_sub_iff_right]
Β· apply ContinuousAt.continuousOn
intro x hx
rw [uIcc_of_le (by positivity : (0 : β) β€ 1 / 2)] at hx
apply ContinuousAt.cpow
Β· exact (continuo... | [
" IntervalIntegrable (fun x => βx ^ (u - 1) * (1 - βx) ^ (v - 1)) volume 0 (1 / 2)",
" IntervalIntegrable (fun x => βx ^ (u - 1)) volume 0 (1 / 2)",
" -1 < (u - 1).re",
" ContinuousOn (fun x => (1 - βx) ^ (v - 1)) (uIcc 0 (1 / 2))",
" β x β uIcc 0 (1 / 2), ContinuousAt (fun x => (1 - βx) ^ (v - 1)) x",
" ... | [] |
import Mathlib.Topology.Homotopy.Basic
import Mathlib.Topology.Connected.PathConnected
import Mathlib.Analysis.Convex.Basic
#align_import topology.homotopy.path from "leanprover-community/mathlib"@"bb9d1c5085e0b7ea619806a68c5021927cecb2a6"
universe u v
variable {X : Type u} {Y : Type v} [TopologicalSpace X] [Top... | Mathlib/Topology/Homotopy/Path.lean | 89 | 91 | theorem eval_one (F : Homotopy pβ pβ) : F.eval 1 = pβ := by |
ext t
simp [eval]
| [
" { toFun := β(F.curry t), continuous_toFun := β― }.toFun 0 = xβ",
" { toFun := β(F.curry t), continuous_toFun := β― }.toFun 1 = xβ",
" F.eval 0 = pβ",
" (F.eval 0) t = pβ t",
" F.eval 1 = pβ",
" (F.eval 1) t = pβ t"
] | [
" { toFun := β(F.curry t), continuous_toFun := β― }.toFun 0 = xβ",
" { toFun := β(F.curry t), continuous_toFun := β― }.toFun 1 = xβ",
" F.eval 0 = pβ",
" (F.eval 0) t = pβ t"
] |
import Mathlib.Analysis.Convex.Hull
#align_import analysis.convex.extreme from "leanprover-community/mathlib"@"c5773405394e073885e2a144c9ca14637e8eb963"
open Function Set
open scoped Classical
open Affine
variable {π E F ΞΉ : Type*} {Ο : ΞΉ β Type*}
section SMul
variable (π) [OrderedSemiring π] [AddCommMonoi... | Mathlib/Analysis/Convex/Extreme.lean | 97 | 103 | theorem IsExtreme.inter (hAB : IsExtreme π A B) (hAC : IsExtreme π A C) :
IsExtreme π A (B β© C) := by |
use Subset.trans inter_subset_left hAB.1
rintro xβ hxβA xβ hxβA x β¨hxB, hxCβ© hx
obtain β¨hxβB, hxβBβ© := hAB.2 hxβA hxβA hxB hx
obtain β¨hxβC, hxβCβ© := hAC.2 hxβA hxβA hxC hx
exact β¨β¨hxβB, hxβCβ©, hxβB, hxβCβ©
| [
" IsExtreme π A C",
" xβ β C β§ xβ β C",
" IsExtreme π A (B β© C)",
" β β¦xβ : Eβ¦, xβ β A β β β¦xβ : Eβ¦, xβ β A β β β¦x : Eβ¦, x β B β© C β x β openSegment π xβ xβ β xβ β B β© C β§ xβ β B β© C",
" xβ β B β© C β§ xβ β B β© C"
] | [
" IsExtreme π A C",
" xβ β C β§ xβ β C"
] |
import Mathlib.Algebra.BigOperators.Group.List
import Mathlib.Algebra.Group.Prod
import Mathlib.Data.Multiset.Basic
#align_import algebra.big_operators.multiset.basic from "leanprover-community/mathlib"@"6c5f73fd6f6cc83122788a80a27cdd54663609f4"
assert_not_exists MonoidWithZero
variable {F ΞΉ Ξ± Ξ² Ξ³ : Type*}
names... | Mathlib/Algebra/BigOperators/Group/Multiset.lean | 99 | 100 | theorem prod_singleton (a : Ξ±) : prod {a} = a := by |
simp only [mul_one, prod_cons, β cons_zero, eq_self_iff_true, prod_zero]
| [
" (fun x x_1 => x * x_1) x ((fun x x_1 => x * x_1) y z) = (fun x x_1 => x * x_1) y ((fun x x_1 => x * x_1) x z)",
" (fun x x_1 => x * x_1) ((fun x x_1 => x * x_1) x y) z = (fun x x_1 => x * x_1) ((fun x x_1 => x * x_1) x z) y",
" foldl (fun x y => y * x) β― 1 s = foldl (fun x x_1 => x * x_1) β― 1 s",
" s.toList... | [
" (fun x x_1 => x * x_1) x ((fun x x_1 => x * x_1) y z) = (fun x x_1 => x * x_1) y ((fun x x_1 => x * x_1) x z)",
" (fun x x_1 => x * x_1) ((fun x x_1 => x * x_1) x y) z = (fun x x_1 => x * x_1) ((fun x x_1 => x * x_1) x z) y",
" foldl (fun x y => y * x) β― 1 s = foldl (fun x x_1 => x * x_1) β― 1 s",
" s.toList... |
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
import Mathlib.Topology.Algebra.Module.Basic
open Function
structure ContinuousAffineEquiv (k Pβ Pβ : Type*) {Vβ Vβ : Type*} [Ring k]
[AddCommGroup Vβ] [Module k Vβ] [AddTorsor Vβ Pβ] [TopologicalSpace Pβ]
[AddCommGroup Vβ] [Module k Vβ] [AddTorsor Vβ P... | Mathlib/LinearAlgebra/AffineSpace/ContinuousAffineEquiv.lean | 65 | 67 | theorem toAffineEquiv_injective : Injective (toAffineEquiv : (Pβ βα΅L[k] Pβ) β Pβ βα΅[k] Pβ) := by |
rintro β¨e, econt, einv_contβ© β¨e', e'cont, e'inv_contβ© H
congr
| [
" Injective toAffineEquiv",
" { toAffineEquiv := e, continuous_toFun := econt, continuous_invFun := einv_cont } =\n { toAffineEquiv := e', continuous_toFun := e'cont, continuous_invFun := e'inv_cont }"
] | [] |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
import Mathlib.LinearAlgebra.Projection
import Mathlib.LinearAlgebra.SesquilinearForm
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.RingTheory.I... | Mathlib/LinearAlgebra/Dual.lean | 388 | 392 | theorem sum_dual_apply_smul_coord (f : Module.Dual R M) :
(β x, f (b x) β’ b.coord x) = f := by |
ext m
simp_rw [LinearMap.sum_apply, LinearMap.smul_apply, smul_eq_mul, mul_comm (f _), β smul_eq_mul, β
f.map_smul, β _root_.map_sum, Basis.coord_apply, Basis.sum_repr]
| [
" β x : ΞΉ, f (b x) β’ b.coord x = f",
" (β x : ΞΉ, f (b x) β’ b.coord x) m = f m"
] | [] |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.MvPowerSeries.Basic
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
... | Mathlib/RingTheory/PowerSeries/Basic.lean | 229 | 231 | theorem coeff_zero_eq_constantCoeff : β(coeff R 0) = constantCoeff R := by |
rw [coeff, Finsupp.single_zero]
rfl
| [
" Inhabited Rβ¦Xβ§",
" Inhabited (MvPowerSeries Unit R)",
" Zero Rβ¦Xβ§",
" Zero (MvPowerSeries Unit R)",
" AddMonoid Rβ¦Xβ§",
" AddMonoid (MvPowerSeries Unit R)",
" AddGroup Rβ¦Xβ§",
" AddGroup (MvPowerSeries Unit R)",
" AddCommMonoid Rβ¦Xβ§",
" AddCommMonoid (MvPowerSeries Unit R)",
" AddCommGroup Rβ¦Xβ§"... | [
" Inhabited Rβ¦Xβ§",
" Inhabited (MvPowerSeries Unit R)",
" Zero Rβ¦Xβ§",
" Zero (MvPowerSeries Unit R)",
" AddMonoid Rβ¦Xβ§",
" AddMonoid (MvPowerSeries Unit R)",
" AddGroup Rβ¦Xβ§",
" AddGroup (MvPowerSeries Unit R)",
" AddCommMonoid Rβ¦Xβ§",
" AddCommMonoid (MvPowerSeries Unit R)",
" AddCommGroup Rβ¦Xβ§"... |
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.MeasureTheory.Function.SimpleFuncDense
#align_import measure_theory.function.simple_func_dense_lp from "leanprover-community/mathlib"@"5a2df4cd59cb31e97a516d4603a14bed5c2f9425"
noncomputable section
set_option linter.uppercaseLean3 false
open Set Func... | Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | 296 | 322 | theorem measure_preimage_lt_top_of_memβp (hp_pos : p β 0) (hp_ne_top : p β β) (f : Ξ± ββ E)
(hf : Memβp f p ΞΌ) (y : E) (hy_ne : y β 0) : ΞΌ (f β»ΒΉ' {y}) < β := by |
have hp_pos_real : 0 < p.toReal := ENNReal.toReal_pos hp_pos hp_ne_top
have hf_snorm := Memβp.snorm_lt_top hf
rw [snorm_eq_snorm' hp_pos hp_ne_top, f.snorm'_eq, β
@ENNReal.lt_rpow_one_div_iff _ _ (1 / p.toReal) (by simp [hp_pos_real]),
@ENNReal.top_rpow_of_pos (1 / (1 / p.toReal)) (by simp [hp_pos_real])... | [
" snorm' (βf) p ΞΌ = (β y β f.range, ββyββ ^ p * ΞΌ (βf β»ΒΉ' {y})) ^ (1 / p)",
" (fun a => βββf aββ ^ p) = β(map (fun a => ββaββ ^ p) f)",
" (fun a => βββf aββ ^ p) = (fun a => ββaββ ^ p) β βf",
" ΞΌ (βf β»ΒΉ' {y}) < β€",
" 0 < 1 / p.toReal",
" 0 < 1 / (1 / p.toReal)",
" 0 < β€",
" βf β»ΒΉ' {y} = β
",
" x β βf... | [
" snorm' (βf) p ΞΌ = (β y β f.range, ββyββ ^ p * ΞΌ (βf β»ΒΉ' {y})) ^ (1 / p)",
" (fun a => βββf aββ ^ p) = β(map (fun a => ββaββ ^ p) f)",
" (fun a => βββf aββ ^ p) = (fun a => ββaββ ^ p) β βf"
] |
import Mathlib.Topology.Algebra.Module.Basic
import Mathlib.LinearAlgebra.Multilinear.Basic
#align_import topology.algebra.module.multilinear from "leanprover-community/mathlib"@"f40476639bac089693a489c9e354ebd75dc0f886"
open Function Fin Set
universe u v w wβ wβ' wβ wβ wβ
variable {R : Type u} {ΞΉ : Type v} {n ... | Mathlib/Topology/Algebra/Module/Multilinear/Basic.lean | 113 | 114 | theorem ext_iff {f f' : ContinuousMultilinearMap R Mβ Mβ} : f = f' β β x, f x = f' x := by |
rw [β toMultilinearMap_injective.eq_iff, MultilinearMap.ext_iff]; rfl
| [
" { toMultilinearMap := f, cont := hf } = { toMultilinearMap := g, cont := hg }",
" { toMultilinearMap := f, cont := hf } =\n { toMultilinearMap := { toMultilinearMap := f, cont := hf }.toMultilinearMap, cont := hg }",
" f = f' β β (x : (i : ΞΉ) β Mβ i), f x = f' x",
" (β (x : (i : ΞΉ) β Mβ i), f.toMultiline... | [
" { toMultilinearMap := f, cont := hf } = { toMultilinearMap := g, cont := hg }",
" { toMultilinearMap := f, cont := hf } =\n { toMultilinearMap := { toMultilinearMap := f, cont := hf }.toMultilinearMap, cont := hg }"
] |
import Mathlib.NumberTheory.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex Real Set
open scoped Nat
namespace HurwitzZeta
variable {k : β} {x : β}
theorem cosZeta_two_mul_nat (hk : k β 0) (hx : x β Icc 0 1) :
cosZeta x (2 * k) = (-1) ^ (k + 1) * (2 * Ο) ^ (2 * k) / 2 / (2 * k)! *
... | Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 126 | 146 | theorem hurwitzZetaEven_one_sub_two_mul_nat (hk : k β 0) (hx : x β Icc (0 : β) 1) :
hurwitzZetaEven x (1 - 2 * k) =
-1 / (2 * k) * ((Polynomial.bernoulli (2 * k)).map (algebraMap β β)).eval (x : β) := by |
have h1 (n : β) : (2 * k : β) β -n := by
rw [β Int.cast_ofNat, β Int.cast_natCast, β Int.cast_mul, β Int.cast_natCast n, β Int.cast_neg,
Ne, Int.cast_inj, β Ne]
refine ne_of_gt ((neg_nonpos_of_nonneg n.cast_nonneg).trans_lt (mul_pos two_pos ?_))
exact Nat.cast_pos.mpr (Nat.pos_of_ne_zero hk)
have... | [
" cosZeta (βx) (2 * βk) =\n (-1) ^ (k + 1) * (2 * βΟ) ^ (2 * k) / 2 / β(2 * k)! *\n Polynomial.eval (βx) (Polynomial.map (algebraMap β β) (Polynomial.bernoulli (2 * k)))",
" 1 < (2 * βk).re",
" β' (b : β), β(2 * Ο * x * βb).cos / βb ^ (2 * βk) = β(β' (b : β), 1 / βb ^ (2 * k) * (2 * Ο * βb * x).cos)",
... | [
" cosZeta (βx) (2 * βk) =\n (-1) ^ (k + 1) * (2 * βΟ) ^ (2 * k) / 2 / β(2 * k)! *\n Polynomial.eval (βx) (Polynomial.map (algebraMap β β) (Polynomial.bernoulli (2 * k)))",
" 1 < (2 * βk).re",
" β' (b : β), β(2 * Ο * x * βb).cos / βb ^ (2 * βk) = β(β' (b : β), 1 / βb ^ (2 * k) * (2 * Ο * βb * x).cos)",
... |
import Mathlib.Data.Set.Function
import Mathlib.Logic.Relation
import Mathlib.Logic.Pairwise
#align_import data.set.pairwise.basic from "leanprover-community/mathlib"@"c4c2ed622f43768eff32608d4a0f8a6cec1c047d"
open Function Order Set
variable {Ξ± Ξ² Ξ³ ΞΉ ΞΉ' : Type*} {r p q : Ξ± β Ξ± β Prop}
section Pairwise
variabl... | Mathlib/Data/Set/Pairwise/Basic.lean | 137 | 143 | theorem pairwise_union :
(s βͺ t).Pairwise r β
s.Pairwise r β§ t.Pairwise r β§ β a β s, β b β t, a β b β r a b β§ r b a := by |
simp only [Set.Pairwise, mem_union, or_imp, forall_and]
exact
β¨fun H => β¨H.1.1, H.2.2, H.1.2, fun x hx y hy hne => H.2.1 y hy x hx hne.symmβ©,
fun H => β¨β¨H.1, H.2.2.1β©, fun x hx y hy hne => H.2.2.2 y hy x hx hne.symm, H.2.1β©β©
| [
" Pairwise (r on fun c => bif c then a else b) β r a b",
" s.Pairwise (r on f) β β z, β x β s, r (f x) z",
" s.Pairwise (r on f) β β z, β x β s, r (f x) z",
" r (f x) (f y)",
" r (f x) (f x)",
" (β z, β x β s, r (f x) z) β s.Pairwise (r on f)",
" (r on f) x y",
" β
.Pairwise (r on f) β β z, β x β β
, r ... | [
" Pairwise (r on fun c => bif c then a else b) β r a b",
" s.Pairwise (r on f) β β z, β x β s, r (f x) z",
" s.Pairwise (r on f) β β z, β x β s, r (f x) z",
" r (f x) (f y)",
" r (f x) (f x)",
" (β z, β x β s, r (f x) z) β s.Pairwise (r on f)",
" (r on f) x y",
" β
.Pairwise (r on f) β β z, β x β β
, r ... |
import Mathlib.LinearAlgebra.Matrix.DotProduct
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
#align_import data.matrix.rank from "leanprover-community/mathlib"@"17219820a8aa8abe85adf5dfde19af1dd1bd8ae7"
open Matrix
namespace Matrix
open FiniteDimensional
variable {l m n ... | Mathlib/Data/Matrix/Rank.lean | 71 | 74 | theorem rank_mul_le_left [StrongRankCondition R] (A : Matrix m n R) (B : Matrix n o R) :
(A * B).rank β€ A.rank := by |
rw [rank, rank, mulVecLin_mul]
exact Cardinal.toNat_le_toNat (LinearMap.rank_comp_le_left _ _) (rank_lt_aleph0 _ _)
| [
" rank 1 = Fintype.card n",
" rank 0 = 0",
" A.rank β€ Fintype.card n",
" (A * B).rank β€ A.rank",
" finrank R β₯(LinearMap.range (A.mulVecLin ββ B.mulVecLin)) β€ finrank R β₯(LinearMap.range A.mulVecLin)"
] | [
" rank 1 = Fintype.card n",
" rank 0 = 0",
" A.rank β€ Fintype.card n"
] |
import Mathlib.Topology.Order.IsLUB
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {Ξ± Ξ² Ξ³ : Type*}
section DenselyOrdered
variable [TopologicalSpace Ξ±] [LinearOrder Ξ±] [OrderTopology Ξ±] [DenselyOrdered Ξ±] {a b : Ξ±}
{s : Set Ξ±}
theorem closure_Ioi' {a : Ξ±} (h : (Io... | Mathlib/Topology/Order/DenselyOrdered.lean | 125 | 126 | theorem Ioc_mem_nhds_iff [NoMaxOrder Ξ±] {a b x : Ξ±} : Ioc a b β π x β x β Ioo a b := by |
rw [β interior_Ioc, mem_interior_iff_mem_nhds]
| [
" closure (Ioi a) = Ici a",
" closure (Ioi a) β Ici a",
" Ici a β closure (Ioi a)",
" a β closure (Ioi a)",
" closure (Ioo a b) = Icc a b",
" closure (Ioo a b) β Icc a b",
" Icc a b β closure (Ioo a b)",
" {a, b} β closure (Ioo a b)",
" a β closure (Ioo a b) β§ b β closure (Ioo a b)",
" β
β closure... | [
" closure (Ioi a) = Ici a",
" closure (Ioi a) β Ici a",
" Ici a β closure (Ioi a)",
" a β closure (Ioi a)",
" closure (Ioo a b) = Icc a b",
" closure (Ioo a b) β Icc a b",
" Icc a b β closure (Ioo a b)",
" {a, b} β closure (Ioo a b)",
" a β closure (Ioo a b) β§ b β closure (Ioo a b)",
" β
β closure... |
import Mathlib.Order.Filter.AtTopBot
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.LinearCombination
import Mathlib.Tactic.Linarith.Frontend
#align_import algebra.quadratic_discriminant from "leanprover-community/mathlib"@"e085d1df33274f4b32f611f483aae678ba0b42df"
open Filter
section Ring
variable {R : ... | Mathlib/Algebra/QuadraticDiscriminant.lean | 63 | 70 | theorem quadratic_eq_zero_iff_discrim_eq_sq [NeZero (2 : R)] [NoZeroDivisors R]
(ha : a β 0) (x : R) :
a * x * x + b * x + c = 0 β discrim a b c = (2 * a * x + b) ^ 2 := by |
refine β¨discrim_eq_sq_of_quadratic_eq_zero, fun h β¦ ?_β©
rw [discrim] at h
have ha : 2 * 2 * a β 0 := mul_ne_zero (mul_ne_zero (NeZero.ne _) (NeZero.ne _)) ha
apply mul_left_cancelβ ha
linear_combination -h
| [
" discrim (-a) (-b) (-c) = discrim a b c",
" discrim a b c = (2 * a * x + b) ^ 2",
" b ^ 2 - 4 * a * c = (2 * a * x + b) ^ 2",
" a * x * x + b * x + c = 0 β discrim a b c = (2 * a * x + b) ^ 2",
" a * x * x + b * x + c = 0",
" 2 * 2 * a * (a * x * x + b * x + c) = 2 * 2 * a * 0"
] | [
" discrim (-a) (-b) (-c) = discrim a b c",
" discrim a b c = (2 * a * x + b) ^ 2",
" b ^ 2 - 4 * a * c = (2 * a * x + b) ^ 2"
] |
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Comp
#align_import analysis.calculus.deriv.pow from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
open scoped Classical
open Topology Filter ENNReal
open Filter Asymptotics Set
variable {... | Mathlib/Analysis/Calculus/Deriv/Pow.lean | 99 | 102 | theorem HasDerivAt.pow (hc : HasDerivAt c c' x) :
HasDerivAt (fun y => c y ^ n) ((n : π) * c x ^ (n - 1) * c') x := by |
rw [β hasDerivWithinAt_univ] at *
exact hc.pow n
| [
" HasStrictDerivAt (fun x => x ^ 0) (β0 * x ^ (0 - 1)) x",
" HasStrictDerivAt (fun x => x ^ 1) (β1 * x ^ (1 - 1)) x",
" HasStrictDerivAt (fun x => x ^ (n + 1 + 1)) (β(n + 1 + 1) * x ^ (n + 1 + 1 - 1)) x",
" HasDerivAt (fun y => c y ^ n) (βn * c x ^ (n - 1) * c') x",
" HasDerivWithinAt (fun y => c y ^ n) (βn... | [
" HasStrictDerivAt (fun x => x ^ 0) (β0 * x ^ (0 - 1)) x",
" HasStrictDerivAt (fun x => x ^ 1) (β1 * x ^ (1 - 1)) x",
" HasStrictDerivAt (fun x => x ^ (n + 1 + 1)) (β(n + 1 + 1) * x ^ (n + 1 + 1 - 1)) x"
] |
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Set.Finite
import Mathlib.GroupTheory.GroupAction.BigOperators
#align_import data.dfinsupp.basic from "leanpr... | Mathlib/Data/DFinsupp/Basic.lean | 158 | 161 | theorem mapRange_zero (f : β i, Ξ²β i β Ξ²β i) (hf : β i, f i 0 = 0) :
mapRange f hf (0 : Ξ β i, Ξ²β i) = 0 := by |
ext
simp only [mapRange_apply, coe_zero, Pi.zero_apply, hf]
| [
" { toFun := fβ, support' := sββ } = { toFun := fβ, support' := sβ }",
" { toFun := fβ, support' := sββ } = { toFun := fβ, support' := sβ }",
" sββ = sβ",
" (fun i => f i (x i)) i = 0",
" mapRange (fun i => id) h g = g",
" (mapRange (fun i => id) h g) iβ = g iβ",
" mapRange (fun i => f i β fβ i) h g = m... | [
" { toFun := fβ, support' := sββ } = { toFun := fβ, support' := sβ }",
" { toFun := fβ, support' := sββ } = { toFun := fβ, support' := sβ }",
" sββ = sβ",
" (fun i => f i (x i)) i = 0",
" mapRange (fun i => id) h g = g",
" (mapRange (fun i => id) h g) iβ = g iβ",
" mapRange (fun i => f i β fβ i) h g = m... |
import Mathlib.Algebra.Order.Interval.Set.Instances
import Mathlib.Order.Interval.Set.ProjIcc
import Mathlib.Topology.Instances.Real
#align_import topology.unit_interval from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open Topology Filter
... | Mathlib/Topology/UnitInterval.lean | 323 | 324 | theorem affineHomeomorph_image_I (a b : π) (h : 0 < a) :
affineHomeomorph a b h.ne.symm '' Set.Icc 0 1 = Set.Icc b (a + b) := by | simp [h]
| [
" β(affineHomeomorph a b β―) '' Icc 0 1 = Icc b (a + b)"
] | [] |
import Mathlib.LinearAlgebra.QuadraticForm.IsometryEquiv
#align_import linear_algebra.quadratic_form.prod from "leanprover-community/mathlib"@"9b2755b951bc323c962bd072cd447b375cf58101"
universe u v w
variable {ΞΉ : Type*} {R : Type*} {Mβ Mβ Nβ Nβ : Type*} {Mα΅’ Nα΅’ : ΞΉ β Type*}
namespace QuadraticForm
section Pro... | Mathlib/LinearAlgebra/QuadraticForm/Prod.lean | 150 | 160 | theorem nonneg_prod_iff {R} [OrderedCommRing R] [Module R Mβ] [Module R Mβ]
{Qβ : QuadraticForm R Mβ} {Qβ : QuadraticForm R Mβ} :
(β x, 0 β€ (Qβ.prod Qβ) x) β (β x, 0 β€ Qβ x) β§ β x, 0 β€ Qβ x := by |
simp_rw [Prod.forall, prod_apply]
constructor
Β· intro h
constructor
Β· intro x; simpa only [add_zero, map_zero] using h x 0
Β· intro x; simpa only [zero_add, map_zero] using h 0 x
Β· rintro β¨hβ, hββ© xβ xβ
exact add_nonneg (hβ xβ) (hβ xβ)
| [
" (Qβ.prod Qβ) ((LinearMap.inl R Mβ Mβ).toFun mβ) = Qβ mβ",
" (Qβ.prod Qβ) ((LinearMap.inr R Mβ Mβ).toFun mβ) = Qβ mβ",
" Qβ ((LinearMap.fst R Mβ Mβ).toFun mβ) = (Qβ.prod 0) mβ",
" Qβ ((LinearMap.snd R Mβ Mβ).toFun mβ) = (prod 0 Qβ) mβ",
" Qβ.Anisotropic β§ Qβ.Anisotropic",
" Qβ.Anisotropic",
" x = 0",
... | [
" (Qβ.prod Qβ) ((LinearMap.inl R Mβ Mβ).toFun mβ) = Qβ mβ",
" (Qβ.prod Qβ) ((LinearMap.inr R Mβ Mβ).toFun mβ) = Qβ mβ",
" Qβ ((LinearMap.fst R Mβ Mβ).toFun mβ) = (Qβ.prod 0) mβ",
" Qβ ((LinearMap.snd R Mβ Mβ).toFun mβ) = (prod 0 Qβ) mβ",
" Qβ.Anisotropic β§ Qβ.Anisotropic",
" Qβ.Anisotropic",
" x = 0",
... |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.HasseDeriv
#align_import data.polynomial.taylor from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
namespace Polynomial
open Polynomial... | Mathlib/Algebra/Polynomial/Taylor.lean | 46 | 46 | theorem taylor_X : taylor r X = X + C r := by | simp only [taylor_apply, X_comp]
| [
" { toFun := fun f => f.comp (X + C r), map_add' := β― }.toFun (c β’ f) =\n (RingHom.id R) c β’ { toFun := fun f => f.comp (X + C r), map_add' := β― }.toFun f",
" (taylor r) X = X + C r"
] | [
" { toFun := fun f => f.comp (X + C r), map_add' := β― }.toFun (c β’ f) =\n (RingHom.id R) c β’ { toFun := fun f => f.comp (X + C r), map_add' := β― }.toFun f"
] |
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 | 82 | 101 | theorem isUnit_den (r : β) (h : β(r : β_[p])β β€ 1) : IsUnit (r.den : β€_[p]) := by |
rw [isUnit_iff]
apply le_antisymm (r.den : β€_[p]).2
rw [β not_lt, coe_natCast]
intro norm_denom_lt
have hr : β(r * r.den : β_[p])β = β(r.num : β_[p])β := by
congr
rw_mod_cast [@Rat.mul_den_eq_num r]
rw [padicNormE.mul] at hr
have key : β(r.num : β_[p])β < 1 := by
calc
_ = _ := hr.symm
... | [
" modPart p r < βp",
" βp = |βp|",
" βp β 0",
" IsUnit βr.den",
" ββr.denβ = 1",
" 1 β€ βββr.denβ",
" Β¬ββr.denβ < 1",
" False",
" ββr * βr.denβ = ββr.numβ",
" βr * βr.den = βr.num",
" ββr.numβ < 1",
" βp β£ r.num β§ βp β£ βr.den",
" βββr.numβ < 1 β§ ββββr.denβ < 1",
" p β£ 1"
] | [
" modPart p r < βp",
" βp = |βp|",
" βp β 0"
] |
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Valuation.PrimeMultiplicity
import Mathlib.RingTheory.AdicCompletion.Basic
#align_import ring_theory.discrete_valuation_ring.basic from "leanprover-community/mathlib"@"c163ec99dfc664628ca15d215fce0a5b9c2... | Mathlib/RingTheory/DiscreteValuationRing/Basic.lean | 227 | 245 | theorem of_ufd_of_unique_irreducible [UniqueFactorizationMonoid R] (hβ : β p : R, Irreducible p)
(hβ : β β¦p q : Rβ¦, Irreducible p β Irreducible q β Associated p q) :
HasUnitMulPowIrreducibleFactorization R := by |
obtain β¨p, hpβ© := hβ
refine β¨p, hp, ?_β©
intro x hx
cases' WfDvdMonoid.exists_factors x hx with fx hfx
refine β¨Multiset.card fx, ?_β©
have H := hfx.2
rw [β Associates.mk_eq_mk_iff_associated] at H β’
rw [β H, β Associates.prod_mk, Associates.mk_pow, β Multiset.prod_replicate]
congr 1
symm
rw [Multis... | [
" Associated p q",
" β {p : R}, Irreducible p β Associated p Ο",
" Associated p Ο",
" n = 0",
" β n < 1, n = 0",
" β f, (β b β f, Prime b) β§ Associated f.prod x",
" (β b β Multiset.replicate (Classical.choose β―) p, Prime b) β§\n Associated (Multiset.replicate (Classical.choose β―) p).prod x",
" β b β... | [
" Associated p q",
" β {p : R}, Irreducible p β Associated p Ο",
" Associated p Ο",
" n = 0",
" β n < 1, n = 0",
" β f, (β b β f, Prime b) β§ Associated f.prod x",
" (β b β Multiset.replicate (Classical.choose β―) p, Prime b) β§\n Associated (Multiset.replicate (Classical.choose β―) p).prod x",
" β b β... |
import Mathlib.Topology.Separation
import Mathlib.Algebra.Group.Defs
#align_import topology.algebra.semigroup from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
@[to_additive
"Any nonempty compact Hausdorff additive semigroup where right-addition is continuous
contains an ... | Mathlib/Topology/Algebra/Semigroup.lean | 82 | 95 | theorem exists_idempotent_in_compact_subsemigroup {M} [Semigroup M] [TopologicalSpace M] [T2Space M]
(continuous_mul_left : β r : M, Continuous (Β· * r)) (s : Set M) (snemp : s.Nonempty)
(s_compact : IsCompact s) (s_add : βα΅ (x β s) (y β s), x * y β s) :
β m β s, m * m = m := by |
let M' := { m // m β s }
letI : Semigroup M' :=
{ mul := fun p q => β¨p.1 * q.1, s_add _ p.2 _ q.2β©
mul_assoc := fun p q r => Subtype.eq (mul_assoc _ _ _) }
haveI : CompactSpace M' := isCompact_iff_compactSpace.mp s_compact
haveI : Nonempty M' := nonempty_subtype.mpr snemp
have : β p : M', Continuou... | [
" β m, m * m = m",
" m * m = m",
" (fun x => x * m) '' N = N",
" (fun x => x * m) '' N β S",
" β m_1 β (fun x => x * m) '' N, β m' β (fun x => x * m) '' N, m_1 * m' β (fun x => x * m) '' N",
" (fun x => x * m) m'' * (fun x => x * m) m' β (fun x => x * m) '' N",
" (fun x => x * m) '' N β N",
" (fun x =... | [
" β m, m * m = m",
" m * m = m",
" (fun x => x * m) '' N = N",
" (fun x => x * m) '' N β S",
" β m_1 β (fun x => x * m) '' N, β m' β (fun x => x * m) '' N, m_1 * m' β (fun x => x * m) '' N",
" (fun x => x * m) m'' * (fun x => x * m) m' β (fun x => x * m) '' N",
" (fun x => x * m) '' N β N",
" (fun x =... |
import Mathlib.Probability.ProbabilityMassFunction.Basic
import Mathlib.Probability.ProbabilityMassFunction.Constructions
import Mathlib.MeasureTheory.Integral.Bochner
namespace PMF
open MeasureTheory ENNReal TopologicalSpace
section General
variable {Ξ± : Type*} [MeasurableSpace Ξ±] [MeasurableSingletonClass Ξ±]
v... | Mathlib/Probability/ProbabilityMassFunction/Integrals.lean | 43 | 47 | theorem integral_eq_sum [Fintype Ξ±] (p : PMF Ξ±) (f : Ξ± β E) :
β« a, f a β(p.toMeasure) = β a, (p a).toReal β’ f a := by |
rw [integral_fintype _ (.of_finite _ f)]
congr with x; congr 2
exact PMF.toMeasure_apply_singleton p x (MeasurableSet.singleton _)
| [
" β« (a : Ξ±), f a βp.toMeasure = β« (a : Ξ±) in p.support, f a βp.toMeasure",
" β« (a : Ξ±) in p.support, f a βp.toMeasure = β' (a : βp.support), (p.toMeasure {βa}).toReal β’ f βa",
" Integrable f (p.toMeasure.restrict p.support)",
" β' (a : βp.support), (p.toMeasure {βa}).toReal β’ f βa = β' (a : βp.support), (p βa... | [
" β« (a : Ξ±), f a βp.toMeasure = β« (a : Ξ±) in p.support, f a βp.toMeasure",
" β« (a : Ξ±) in p.support, f a βp.toMeasure = β' (a : βp.support), (p.toMeasure {βa}).toReal β’ f βa",
" Integrable f (p.toMeasure.restrict p.support)",
" β' (a : βp.support), (p.toMeasure {βa}).toReal β’ f βa = β' (a : βp.support), (p βa... |
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
#align_import order.symm_diff from "leanprover-community/mathlib"@"6eb334bd8f3433d5b08ba156b8ec3e6af47e1904"
open Function OrderDual
variable {ΞΉ Ξ± Ξ² : Type*} {Ο : ΞΉ β Type*}
def symmDiff [Sup Ξ±] [SDiff Ξ±] (a b : Ξ±) : Ξ± :=
a \ b β b \ a
#ali... | Mathlib/Order/SymmDiff.lean | 161 | 162 | theorem Disjoint.symmDiff_eq_sup {a b : Ξ±} (h : Disjoint a b) : a β b = a β b := by |
rw [symmDiff, h.sdiff_eq_left, h.sdiff_eq_right]
| [
" β (p q : Bool), p β q = xor p q",
" a β b = b β a",
" a β a = β₯",
" a β β₯ = a",
" β₯ β a = a",
" a β b = β₯ β a = b",
" a β b = b \\ a",
" a β b = a \\ b",
" a β b β€ c β a β€ b β c β§ b β€ a β c",
" a β b = (a β b) \\ (a β b)",
" a β b = a β b"
] | [
" β (p q : Bool), p β q = xor p q",
" a β b = b β a",
" a β a = β₯",
" a β β₯ = a",
" β₯ β a = a",
" a β b = β₯ β a = b",
" a β b = b \\ a",
" a β b = a \\ b",
" a β b β€ c β a β€ b β c β§ b β€ a β c",
" a β b = (a β b) \\ (a β b)"
] |
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.Topology.Semicontinuous
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.Topology.Instances.EReal
#align_import measure_theory.integral.vitali_caratheodory from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
open sc... | Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean | 93 | 152 | theorem SimpleFunc.exists_le_lowerSemicontinuous_lintegral_ge (f : Ξ± ββ ββ₯0) {Ξ΅ : ββ₯0β}
(Ξ΅0 : Ξ΅ β 0) :
β g : Ξ± β ββ₯0, (β x, f x β€ g x) β§ LowerSemicontinuous g β§
(β«β» x, g x βΞΌ) β€ (β«β» x, f x βΞΌ) + Ξ΅ := by |
induction' f using MeasureTheory.SimpleFunc.induction with c s hs fβ fβ _ hβ hβ generalizing Ξ΅
Β· let f := SimpleFunc.piecewise s hs (SimpleFunc.const Ξ± c) (SimpleFunc.const Ξ± 0)
by_cases h : β«β» x, f x βΞΌ = β€
Β· refine
β¨fun _ => c, fun x => ?_, lowerSemicontinuous_const, by
simp only [_root... | [
" β g, (β (x : Ξ±), βf x β€ g x) β§ LowerSemicontinuous g β§ β«β» (x : Ξ±), β(g x) βΞΌ β€ β«β» (x : Ξ±), β(βf x) βΞΌ + Ξ΅",
" β g,\n (β (x : Ξ±), β(piecewise s hs (const Ξ± c) (const Ξ± 0)) x β€ g x) β§\n LowerSemicontinuous g β§ β«β» (x : Ξ±), β(g x) βΞΌ β€ β«β» (x : Ξ±), β(β(piecewise s hs (const Ξ± c) (const Ξ± 0)) x) βΞΌ + Ξ΅",
" ... | [] |
import Mathlib.Probability.Kernel.Disintegration.Unique
import Mathlib.Probability.Notation
#align_import probability.kernel.cond_distrib from "leanprover-community/mathlib"@"00abe0695d8767201e6d008afa22393978bb324d"
open MeasureTheory Set Filter TopologicalSpace
open scoped ENNReal MeasureTheory ProbabilityTheo... | Mathlib/Probability/Kernel/CondDistrib.lean | 145 | 148 | theorem _root_.MeasureTheory.Integrable.condDistrib_ae_map
(hY : AEMeasurable Y ΞΌ) (hf_int : Integrable f (ΞΌ.map fun a => (X a, Y a))) :
βα΅ b βΞΌ.map X, Integrable (fun Ο => f (b, Ο)) (condDistrib Y X ΞΌ b) := by |
rw [condDistrib, β Measure.fst_map_prod_mkβ (X := X) hY]; exact hf_int.condKernel_ae
| [
" IsMarkovKernel (condDistrib Y X ΞΌ)",
" IsMarkovKernel (Measure.map (fun a => (X a, Y a)) ΞΌ).condKernel",
" ((condDistrib Y X ΞΌ) x) s = ((Measure.map X ΞΌ) {x})β»ΒΉ * (Measure.map (fun a => (X a, Y a)) ΞΌ) ({x} ΓΛ’ s)",
" ((Measure.map (fun a => (X a, Y a)) ΞΌ).fst {x})β»ΒΉ * (Measure.map (fun a => (X a, Y a)) ΞΌ) ({... | [
" IsMarkovKernel (condDistrib Y X ΞΌ)",
" IsMarkovKernel (Measure.map (fun a => (X a, Y a)) ΞΌ).condKernel",
" ((condDistrib Y X ΞΌ) x) s = ((Measure.map X ΞΌ) {x})β»ΒΉ * (Measure.map (fun a => (X a, Y a)) ΞΌ) ({x} ΓΛ’ s)",
" ((Measure.map (fun a => (X a, Y a)) ΞΌ).fst {x})β»ΒΉ * (Measure.map (fun a => (X a, Y a)) ΞΌ) ({... |
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Topology.Algebra.UniformGroup
noncomputable section
open Filter Finset Function
open scoped Topology
variable {Ξ± Ξ² Ξ³ Ξ΄ : Type*}
section TopologicalGroup
variable [CommGroup Ξ±] [TopologicalSpace Ξ±] [TopologicalGroup Ξ±]
variable {f g : Ξ² β Ξ±} {a aβ... | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | 63 | 65 | theorem Multipliable.trans_div (hg : Multipliable g) (hfg : Multipliable fun b β¦ f b / g b) :
Multipliable f := by |
simpa only [div_mul_cancel] using hfg.mul hg
| [
" HasProd (fun b => (f b)β»ΒΉ) aβ»ΒΉ",
" Multipliable f",
" HasProd (fun b => f b / g b) (aβ / aβ)",
" HasProd (fun b => f b * (g b)β»ΒΉ) (aβ * aββ»ΒΉ)"
] | [
" HasProd (fun b => (f b)β»ΒΉ) aβ»ΒΉ",
" Multipliable f",
" HasProd (fun b => f b / g b) (aβ / aβ)",
" HasProd (fun b => f b * (g b)β»ΒΉ) (aβ * aββ»ΒΉ)"
] |
import Mathlib.Analysis.NormedSpace.Exponential
#align_import analysis.normed_space.star.exponential from "leanprover-community/mathlib"@"1e3201306d4d9eb1fd54c60d7c4510ad5126f6f9"
open NormedSpace -- For `NormedSpace.exp`.
section Star
variable {A : Type*} [NormedRing A] [NormedAlgebra β A] [StarRing A] [Continu... | Mathlib/Analysis/NormedSpace/Star/Exponential.lean | 51 | 56 | theorem Commute.expUnitary {a b : selfAdjoint A} (h : Commute (a : A) (b : A)) :
Commute (expUnitary a) (expUnitary b) :=
calc
selfAdjoint.expUnitary a * selfAdjoint.expUnitary b =
selfAdjoint.expUnitary b * selfAdjoint.expUnitary a := by |
rw [β h.expUnitary_add, β h.symm.expUnitary_add, add_comm]
| [
" expUnitary (a + b) = expUnitary a * expUnitary b",
" β(expUnitary (a + b)) = β(expUnitary a * expUnitary b)",
" Commute (I β’ βa) (I β’ βb)",
" I β’ βa * I β’ βb = I β’ βb * I β’ βa",
" selfAdjoint.expUnitary a * selfAdjoint.expUnitary b = selfAdjoint.expUnitary b * selfAdjoint.expUnitary a"
] | [
" expUnitary (a + b) = expUnitary a * expUnitary b",
" β(expUnitary (a + b)) = β(expUnitary a * expUnitary b)",
" Commute (I β’ βa) (I β’ βb)",
" I β’ βa * I β’ βb = I β’ βb * I β’ βa"
] |
import Mathlib.Algebra.Polynomial.Degree.Lemmas
open Polynomial
namespace Mathlib.Tactic.ComputeDegree
section recursion_lemmas
variable {R : Type*}
section semiring
variable [Semiring R]
theorem natDegree_C_le (a : R) : natDegree (C a) β€ 0 := (natDegree_C a).le
theorem natDegree_natCast_le (n : β) : natDeg... | Mathlib/Tactic/ComputeDegree.lean | 150 | 155 | theorem natDegree_eq_of_le_of_coeff_ne_zero' {deg m o : β} {c : R} {p : R[X]}
(h_natDeg_le : natDegree p β€ m) (coeff_eq : coeff p o = c)
(coeff_ne_zero : c β 0) (deg_eq_deg : m = deg) (coeff_eq_deg : o = deg) :
natDegree p = deg := by |
subst coeff_eq deg_eq_deg coeff_eq_deg
exact natDegree_eq_of_le_of_coeff_ne_zero βΉ_βΊ βΉ_βΊ
| [
" (f + g).coeff n = a + b",
" (f + g).coeff n = f.coeff n + g.coeff n",
" (f * g).coeff d = if d = df + dg then a * b else 0",
" (f * g).coeff d = a * b",
" (f * g).coeff (df + dg) = f.coeff df * g.coeff dg",
" (f * g).coeff d = 0",
" (f * g).natDegree < d",
" (f * g).natDegree β€ df + dg",
" df + dg... | [
" (f + g).coeff n = a + b",
" (f + g).coeff n = f.coeff n + g.coeff n",
" (f * g).coeff d = if d = df + dg then a * b else 0",
" (f * g).coeff d = a * b",
" (f * g).coeff (df + dg) = f.coeff df * g.coeff dg",
" (f * g).coeff d = 0",
" (f * g).natDegree < d",
" (f * g).natDegree β€ df + dg",
" df + dg... |
import Mathlib.Analysis.Complex.Basic
import Mathlib.FieldTheory.IntermediateField
import Mathlib.Topology.Algebra.Field
import Mathlib.Topology.Algebra.UniformRing
#align_import topology.instances.complex from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
section ComplexSubfield
open... | Mathlib/Topology/Instances/Complex.lean | 25 | 44 | theorem Complex.subfield_eq_of_closed {K : Subfield β} (hc : IsClosed (K : Set β)) :
K = ofReal.fieldRange β¨ K = β€ := by |
suffices range (ofReal' : β β β) β K by
rw [range_subset_iff, β coe_algebraMap] at this
have :=
(Subalgebra.isSimpleOrder_of_finrank finrank_real_complex).eq_bot_or_eq_top
(Subfield.toIntermediateField K this).toSubalgebra
simp_rw [β SetLike.coe_set_eq, IntermediateField.coe_toSubalgebra] a... | [
" K = ofReal.fieldRange β¨ K = β€",
" βK = βofReal.fieldRange β¨ βK = ββ€",
" range ofReal' β βK",
" closure (range (ofReal' β Rat.cast)) β βK",
" closure (range (ofReal' β Rat.cast)) β closure βK",
" range (ofReal' β Rat.cast) β βK",
" (ofReal' β Rat.cast) wβ β βK",
" range ofReal' β closure (range (ofRe... | [] |
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Topology.Algebra.Nonarchimedean.Bases
import Mathlib.Topology.Algebra.UniformRing
#align_import topology.algebra.nonarchimedean.adic_topology from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variable {R : Type*} [CommRing R]
open S... | Mathlib/Topology/Algebra/Nonarchimedean/AdicTopology.lean | 92 | 103 | theorem hasBasis_nhds_zero_adic (I : Ideal R) :
HasBasis (@nhds R I.adicTopology (0 : R)) (fun _n : β => True) fun n =>
((I ^ n : Ideal R) : Set R) :=
β¨by
intro U
rw [I.ringFilterBasis.toAddGroupFilterBasis.nhds_zero_hasBasis.mem_iff]
constructor
Β· rintro β¨-, β¨i, rflβ©, hβ©
replace h : β... | simpa using h
exact β¨i, trivial, hβ©
Β· rintro β¨i, -, hβ©
exact β¨(I ^ i : Ideal R), β¨i, by simpβ©, hβ©β©
| [
" β (i j : β), β k, I ^ k β’ β€ β€ I ^ i β’ β€ β I ^ j β’ β€",
" β (i j : β), β k, I ^ k β€ I ^ i β§ I ^ k β€ I ^ j",
" β k, I ^ k β€ I ^ i β§ I ^ k β€ I ^ j",
" β (a : R) (i : β), β j, a β’ I ^ j β’ β€ β€ I ^ i β’ β€",
" β (a : R) (i : β), β j, a β’ I ^ j β€ I ^ i",
" β j, r β’ I ^ j β€ I ^ n",
" r β’ I ^ n β€ I ^ n",
" (Dis... | [
" β (i j : β), β k, I ^ k β’ β€ β€ I ^ i β’ β€ β I ^ j β’ β€",
" β (i j : β), β k, I ^ k β€ I ^ i β§ I ^ k β€ I ^ j",
" β k, I ^ k β€ I ^ i β§ I ^ k β€ I ^ j",
" β (a : R) (i : β), β j, a β’ I ^ j β’ β€ β€ I ^ i β’ β€",
" β (a : R) (i : β), β j, a β’ I ^ j β€ I ^ i",
" β j, r β’ I ^ j β€ I ^ n",
" r β’ I ^ n β€ I ^ n",
" (Dis... |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Set.Image
import Mathlib.GroupTheory.Perm.Support
import Mathlib.Order.Monotone.Monovary
import Mathlib.Tactic.Abel
#align_impo... | Mathlib/Algebra/Order/Rearrangement.lean | 114 | 137 | theorem MonovaryOn.sum_smul_comp_perm_eq_sum_smul_iff (hfg : MonovaryOn f g s)
(hΟ : { x | Ο x β x } β s) :
((β i β s, f i β’ g (Ο i)) = β i β s, f i β’ g i) β MonovaryOn f (g β Ο) s := by |
classical
refine β¨not_imp_not.1 fun h β¦ ?_, fun h β¦ (hfg.sum_smul_comp_perm_le_sum_smul hΟ).antisymm ?_β©
Β· rw [MonovaryOn] at h
push_neg at h
obtain β¨x, hx, y, hy, hgxy, hfxyβ© := h
set Ο : Perm ΞΉ := (Equiv.swap x y).trans Ο
have hΟs : { x | Ο x β x } β s := by
refine (set_supp... | [
" β i β s, f i β’ g (Ο i) β€ β i β s, f i β’ g i",
" β {Ο : Perm ΞΉ}, MonovaryOn f g βs β {x | Ο x β x} β βs β β i β s, f i β’ g (Ο i) β€ β i β s, f i β’ g i",
" β {Ο : Perm ΞΉ}, MonovaryOn f g ββ
β {x | Ο x β x} β ββ
β β i β β
, f i β’ g (Ο i) β€ β i β β
, f i β’ g i",
" β (a : ΞΉ) (s : Finset ΞΉ),\n a β s β\n (β x... | [
" β i β s, f i β’ g (Ο i) β€ β i β s, f i β’ g i",
" β {Ο : Perm ΞΉ}, MonovaryOn f g βs β {x | Ο x β x} β βs β β i β s, f i β’ g (Ο i) β€ β i β s, f i β’ g i",
" β {Ο : Perm ΞΉ}, MonovaryOn f g ββ
β {x | Ο x β x} β ββ
β β i β β
, f i β’ g (Ο i) β€ β i β β
, f i β’ g i",
" β (a : ΞΉ) (s : Finset ΞΉ),\n a β s β\n (β x... |
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.terminated_stable from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
variable {K : Type*} {g : GeneralizedContinuedFraction K} {n m : β}
theorem te... | Mathlib/Algebra/ContinuedFractions/TerminatedStable.lean | 31 | 34 | theorem continuantsAux_stable_step_of_terminated (terminated_at_n : g.TerminatedAt n) :
g.continuantsAux (n + 2) = g.continuantsAux (n + 1) := by |
rw [terminatedAt_iff_s_none] at terminated_at_n
simp only [continuantsAux, Nat.add_eq, Nat.add_zero, terminated_at_n]
| [
" g.continuantsAux (n + 2) = g.continuantsAux (n + 1)"
] | [] |
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 | 32 | 33 | theorem All_def {t : RBNode Ξ±} : t.All p β β x β t, p x := by |
induction t <;> simp [or_imp, forall_and, *]
| [
" t.reverse.min? = t.max?",
" t.reverse.min? =\n match t with\n | nil => none\n | node c l v nil => some v\n | node c l v r => r.max?",
" nil.reverse.min? = none",
" (node cβ lβ vβ nil).reverse.min? = some vβ",
" (node cβ lβ vβ rβ).reverse.min? = rβ.max?",
" (node cβ rβ.reverse vβ lβ.reverse).... | [
" t.reverse.min? = t.max?",
" t.reverse.min? =\n match t with\n | nil => none\n | node c l v nil => some v\n | node c l v r => r.max?",
" nil.reverse.min? = none",
" (node cβ lβ vβ nil).reverse.min? = some vβ",
" (node cβ lβ vβ rβ).reverse.min? = rβ.max?",
" (node cβ rβ.reverse vβ lβ.reverse).... |
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.Order.Monoid.OrderDual
import Mathlib.Algebra.Order.ZeroLEOne
import Mathlib.Data.Nat.Cast.Defs
import Mathlib.Order.WithBot
#align_import algebra.order.monoid.with_top ... | Mathlib/Algebra/Order/Monoid/WithTop.lean | 128 | 128 | theorem add_top (a : WithTop Ξ±) : a + β€ = β€ := by | cases a <;> rfl
| [
" a + β€ = β€",
" β€ + β€ = β€",
" βaβ + β€ = β€"
] | [] |
import Mathlib.RingTheory.RootsOfUnity.Basic
import Mathlib.RingTheory.AdjoinRoot
import Mathlib.FieldTheory.Galois
import Mathlib.LinearAlgebra.Eigenspace.Minpoly
import Mathlib.RingTheory.Norm
universe u
variable {K : Type u} [Field K]
open Polynomial IntermediateField AdjoinRoot
section Splits
lemma root_X_pow... | Mathlib/FieldTheory/KummerExtension.lean | 74 | 82 | theorem X_pow_sub_C_splits_of_isPrimitiveRoot
{n : β} {ΞΆ : K} (hΞΆ : IsPrimitiveRoot ΞΆ n) {Ξ± a : K} (e : Ξ± ^ n = a) :
(X ^ n - C a).Splits (RingHom.id _) := by |
cases n.eq_zero_or_pos with
| inl hn =>
rw [hn, pow_zero, β C.map_one, β map_sub]
exact splits_C _ _
| inr hn =>
rw [splits_iff_card_roots, β nthRoots, hΞΆ.card_nthRoots, natDegree_X_pow_sub_C, if_pos β¨Ξ±, eβ©]
| [
" root (X ^ n - C a) ^ n = (of (X ^ n - C a)) a",
" X.natDegree < (X ^ n - C a).natDegree",
" root (X ^ n - C a) β 0",
" root (X ^ Nat.succ 0 - C a) β 0",
" root (X - C a) β 0",
" False",
" (C a).natDegree < (X - C a).natDegree",
" (AdjoinRoot.mk (X - C a)) (C a) = 0",
" (AdjoinRoot.mk (X - C a)) (C... | [
" root (X ^ n - C a) ^ n = (of (X ^ n - C a)) a",
" X.natDegree < (X ^ n - C a).natDegree",
" root (X ^ n - C a) β 0",
" root (X ^ Nat.succ 0 - C a) β 0",
" root (X - C a) β 0",
" False",
" (C a).natDegree < (X - C a).natDegree",
" (AdjoinRoot.mk (X - C a)) (C a) = 0",
" (AdjoinRoot.mk (X - C a)) (C... |
import Mathlib.ModelTheory.Ultraproducts
import Mathlib.ModelTheory.Bundled
import Mathlib.ModelTheory.Skolem
#align_import model_theory.satisfiability from "leanprover-community/mathlib"@"d565b3df44619c1498326936be16f1a935df0728"
set_option linter.uppercaseLean3 false
universe u v w w'
open Cardinal CategoryTh... | Mathlib/ModelTheory/Satisfiability.lean | 129 | 135 | theorem isSatisfiable_directed_union_iff {ΞΉ : Type*} [Nonempty ΞΉ] {T : ΞΉ β L.Theory}
(h : Directed (Β· β Β·) T) : Theory.IsSatisfiable (β i, T i) β β i, (T i).IsSatisfiable := by |
refine β¨fun h' i => h'.mono (Set.subset_iUnion _ _), fun h' => ?_β©
rw [isSatisfiable_iff_isFinitelySatisfiable, IsFinitelySatisfiable]
intro T0 hT0
obtain β¨i, hiβ© := h.exists_mem_subset_of_finset_subset_biUnion hT0
exact (h' i).mono hi
| [
" (Ο.onTheory T).IsSatisfiable β T.IsSatisfiable",
" (Ο.onTheory T).IsSatisfiable",
" T.IsSatisfiable",
" M' β¨ T",
" M' β¨ Ο",
" βαΆ (a : Finset βT) in β(Ultrafilter.of Filter.atTop), M a β¨ Ο",
" Ο β β(Finset.map (Function.Embedding.subtype fun x => x β T) s)",
" β (x : Ο β T), β¨Ο, β―β© β s",
" IsSatisf... | [
" (Ο.onTheory T).IsSatisfiable β T.IsSatisfiable",
" (Ο.onTheory T).IsSatisfiable",
" T.IsSatisfiable",
" M' β¨ T",
" M' β¨ Ο",
" βαΆ (a : Finset βT) in β(Ultrafilter.of Filter.atTop), M a β¨ Ο",
" Ο β β(Finset.map (Function.Embedding.subtype fun x => x β T) s)",
" β (x : Ο β T), β¨Ο, β―β© β s"
] |
import Mathlib.Combinatorics.SetFamily.HarrisKleitman
import Mathlib.Combinatorics.SetFamily.Intersecting
#align_import combinatorics.set_family.kleitman from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Finset
open Fintype (card)
variable {ΞΉ Ξ± : Type*} [Fintype Ξ±] [DecidableEq... | Mathlib/Combinatorics/SetFamily/Kleitman.lean | 37 | 85 | theorem Finset.card_biUnion_le_of_intersecting (s : Finset ΞΉ) (f : ΞΉ β Finset (Finset Ξ±))
(hf : β i β s, (f i : Set (Finset Ξ±)).Intersecting) :
(s.biUnion f).card β€ 2 ^ Fintype.card Ξ± - 2 ^ (Fintype.card Ξ± - s.card) := by |
have : DecidableEq ΞΉ := by
classical
infer_instance
obtain hs | hs := le_total (Fintype.card Ξ±) s.card
Β· rw [tsub_eq_zero_of_le hs, pow_zero]
refine (card_le_card <| biUnion_subset.2 fun i hi a ha β¦
mem_compl.2 <| not_mem_singleton.2 <| (hf _ hi).ne_bot ha).trans_eq ?_
rw [card_compl, Finty... | [
" (s.biUnion f).card β€ 2 ^ Fintype.card Ξ± - 2 ^ (Fintype.card Ξ± - s.card)",
" DecidableEq ΞΉ",
" (s.biUnion f).card β€ 2 ^ Fintype.card Ξ± - 1",
" {β₯}αΆ.card = 2 ^ Fintype.card Ξ± - 1",
" (β
.biUnion f).card β€ 2 ^ Fintype.card Ξ± - 2 ^ (Fintype.card Ξ± - β
.card)",
" ((cons i s hi).biUnion f).card β€ 2 ^ Fintype.ca... | [] |
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w
variable {ΞΉ : Sort u} {Ξ± : Type v} {Ξ² : Type w}
open Set
open OrderDual (toDual)
namespace Set
section LinearOrder
variable [LinearOrder Ξ±] ... | Mathlib/Order/Interval/Set/Disjoint.lean | 176 | 178 | theorem iUnion_Ioc_eq_Ioi_self_iff {f : ΞΉ β Ξ±} {a : Ξ±} :
β i, Ioc a (f i) = Ioi a β β x, a < x β β i, x β€ f i := by |
simp [β Ioi_inter_Iic, β inter_iUnion, subset_def]
| [
" Disjoint (Ico aβ aβ) (Ico bβ bβ) β min aβ bβ β€ max aβ bβ",
" Disjoint (Ioc aβ aβ) (Ioc bβ bβ) β min aβ bβ β€ max aβ bβ",
" Disjoint (Ioo aβ aβ) (Ioo bβ bβ) β min aβ bβ β€ max aβ bβ",
" yβ = xβ",
" xβ β€ yβ",
" β i, Ico (f i) a = Iio a β β x < a, β i, f i β€ x",
" β i, Ioc a (f i) = Ioi a β β (x : Ξ±), a < ... | [
" Disjoint (Ico aβ aβ) (Ico bβ bβ) β min aβ bβ β€ max aβ bβ",
" Disjoint (Ioc aβ aβ) (Ioc bβ bβ) β min aβ bβ β€ max aβ bβ",
" Disjoint (Ioo aβ aβ) (Ioo bβ bβ) β min aβ bβ β€ max aβ bβ",
" yβ = xβ",
" xβ β€ yβ",
" β i, Ico (f i) a = Iio a β β x < a, β i, f i β€ x"
] |
import Mathlib.Data.Finset.Basic
variable {ΞΉ : Sort _} {Ο : ΞΉ β Sort _} {x : β i, Ο i} [DecidableEq ΞΉ]
namespace Function
def updateFinset (x : β i, Ο i) (s : Finset ΞΉ) (y : β i : β₯s, Ο i) (i : ΞΉ) : Ο i :=
if hi : i β s then y β¨i, hiβ© else x i
open Finset Equiv
theorem updateFinset_def {s : Finset ΞΉ} {y} :
... | Mathlib/Data/Finset/Update.lean | 43 | 50 | theorem update_eq_updateFinset {i y} :
Function.update x i y = updateFinset x {i} (uniqueElim y) := by |
congr with j
by_cases hj : j = i
Β· cases hj
simp only [dif_pos, Finset.mem_singleton, update_same, updateFinset]
exact uniqueElim_default (Ξ± := fun j : ({i} : Finset ΞΉ) => Ο j) y
Β· simp [hj, updateFinset]
| [
" updateFinset x {i} y = update x i (y β¨i, β―β©)",
" updateFinset x {i} y j = update x i (y β¨i, β―β©) j",
" updateFinset x {i} y i = update x i (y β¨i, β―β©) i",
" update x i y = updateFinset x {i} (uniqueElim y)",
" update x i y j = updateFinset x {i} (uniqueElim y) j",
" update x i y i = updateFinset x {i} (un... | [
" updateFinset x {i} y = update x i (y β¨i, β―β©)",
" updateFinset x {i} y j = update x i (y β¨i, β―β©) j",
" updateFinset x {i} y i = update x i (y β¨i, β―β©) i"
] |
import Mathlib.RingTheory.Localization.Module
import Mathlib.RingTheory.Norm
import Mathlib.RingTheory.Discriminant
#align_import ring_theory.localization.norm from "leanprover-community/mathlib"@"2e59a6de168f95d16b16d217b808a36290398c0a"
open scoped nonZeroDivisors
variable (R : Type*) {S : Type*} [CommRing R] ... | Mathlib/RingTheory/Localization/NormTrace.lean | 101 | 109 | theorem Algebra.traceMatrix_localizationLocalization (b : Basis ΞΉ R S) :
Algebra.traceMatrix Rβ (b.localizationLocalization Rβ M Sβ) =
(algebraMap R Rβ).mapMatrix (Algebra.traceMatrix R b) := by |
have : Module.Finite R S := Module.Finite.of_basis b
have : Module.Free R S := Module.Free.of_basis b
ext i j : 2
simp_rw [RingHom.mapMatrix_apply, Matrix.map_apply, traceMatrix_apply, traceForm_apply,
Basis.localizationLocalization_apply, β map_mul]
exact Algebra.trace_localization R M _
| [
" (algebraMap R Rβ).mapMatrix ((leftMulMatrix b) a) =\n (leftMulMatrix (Basis.localizationLocalization Rβ M Sβ b)) ((algebraMap S Sβ) a)",
" (algebraMap R Rβ).mapMatrix ((leftMulMatrix b) a) i j =\n (leftMulMatrix (Basis.localizationLocalization Rβ M Sβ b)) ((algebraMap S Sβ) a) i j",
" (norm Rβ) ((algebr... | [
" (algebraMap R Rβ).mapMatrix ((leftMulMatrix b) a) =\n (leftMulMatrix (Basis.localizationLocalization Rβ M Sβ b)) ((algebraMap S Sβ) a)",
" (algebraMap R Rβ).mapMatrix ((leftMulMatrix b) a) i j =\n (leftMulMatrix (Basis.localizationLocalization Rβ M Sβ b)) ((algebraMap S Sβ) a) i j",
" (norm Rβ) ((algebr... |
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.Algebra.Lie.IdealOperations
#align_import algebra.lie.abelian from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
universe u v w wβ wβ
class LieModule.IsTrivial (L : Type v) (M : Type w) [Bracket L M] [Zero M] : Prop where
triv... | Mathlib/Algebra/Lie/Abelian.lean | 136 | 141 | theorem ideal_oper_maxTrivSubmodule_eq_bot (I : LieIdeal R L) :
β
I, maxTrivSubmodule R L Mβ = β₯ := by |
rw [β LieSubmodule.coe_toSubmodule_eq_iff, LieSubmodule.lieIdeal_oper_eq_linear_span,
LieSubmodule.bot_coeSubmodule, Submodule.span_eq_bot]
rintro m β¨β¨x, hxβ©, β¨β¨m, hmβ©, rflβ©β©
exact hm x
| [
" β
x, mβ = 0",
" β
x, yβ = 0",
" β
f u, yβ = 0",
" β
f u, f vβ = 0",
" (Std.Commutative fun x x_1 => x * x_1) β IsLieAbelian A",
" x β LieModule.ker R L M β β (m : M), β
x, mβ = 0",
" β
z, x + yβ = 0",
" β
y, c β’ xβ = 0",
" β
y, β
x, mββ = 0",
" β
I, maxTrivSubmodule R L Mβ = β₯",
" β x β {m | β x n, β
βx,... | [
" β
x, mβ = 0",
" β
x, yβ = 0",
" β
f u, yβ = 0",
" β
f u, f vβ = 0",
" (Std.Commutative fun x x_1 => x * x_1) β IsLieAbelian A",
" x β LieModule.ker R L M β β (m : M), β
x, mβ = 0",
" β
z, x + yβ = 0",
" β
y, c β’ xβ = 0",
" β
y, β
x, mββ = 0"
] |
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import analysis.special_functions.complex.log from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
namespace Complex
open Set Filter Bornology
open scop... | Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 106 | 106 | theorem log_zero : log 0 = 0 := by | simp [log]
| [
" x.log.re = (abs x).log",
" x.log.im = x.arg",
" -Ο < x.log.im",
" x.log.im β€ Ο",
" cexp x.log = x",
" x β Set.range cexp β x β {0}αΆ",
" cexp x β {0}αΆ",
" (cexp x).log = x",
" x = y",
" (βx.log).re = (βx).log.re",
" (βx.log).im = (βx).log.im",
" (βx).log.re = x.log",
" (βr * x).log = βr.log... | [
" x.log.re = (abs x).log",
" x.log.im = x.arg",
" -Ο < x.log.im",
" x.log.im β€ Ο",
" cexp x.log = x",
" x β Set.range cexp β x β {0}αΆ",
" cexp x β {0}αΆ",
" (cexp x).log = x",
" x = y",
" (βx.log).re = (βx).log.re",
" (βx.log).im = (βx).log.im",
" (βx).log.re = x.log",
" (βr * x).log = βr.log... |
import Mathlib.Topology.Algebra.Module.Basic
import Mathlib.Analysis.Normed.MulAction
#align_import analysis.normed_space.continuous_linear_map from "leanprover-community/mathlib"@"fe18deda804e30c594e75a6e5fe0f7d14695289f"
open Metric ContinuousLinearMap
open Set Real
open NNReal
variable {π πβ E F G : Type*}... | Mathlib/Analysis/NormedSpace/ContinuousLinearMap.lean | 198 | 205 | theorem ContinuousLinearEquiv.homothety_inverse (a : β) (ha : 0 < a) (f : E βββ[Ο] F) :
(β x : E, βf xβ = a * βxβ) β β y : F, βf.symm yβ = aβ»ΒΉ * βyβ := by |
intro hf y
calc
βf.symm yβ = aβ»ΒΉ * (a * βf.symm yβ) := by
rw [β mul_assoc, inv_mul_cancel (ne_of_lt ha).symm, one_mul]
_ = aβ»ΒΉ * βf (f.symm y)β := by rw [hf]
_ = aβ»ΒΉ * βyβ := by simp
| [
" (β (x : E), βf xβ = a * βxβ) β β (y : F), βf.symm yβ = aβ»ΒΉ * βyβ",
" βf.symm yβ = aβ»ΒΉ * βyβ",
" βf.symm yβ = aβ»ΒΉ * (a * βf.symm yβ)",
" aβ»ΒΉ * (a * βf.symm yβ) = aβ»ΒΉ * βf (f.symm y)β",
" aβ»ΒΉ * βf (f.symm y)β = aβ»ΒΉ * βyβ"
] | [] |
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 | 201 | 201 | theorem zero_eval (v) : zero.eval v = pure [0] := by | simp [zero]
| [
" zero'.eval = fun v => pure (0 :: v)",
" succ.eval = fun v => pure [v.headI.succ]",
" tail.eval = fun v => pure v.tail",
" (f.cons fs).eval = fun v => do\n let n β f.eval v\n let ns β fs.eval v\n pure (n.headI :: ns)",
" (f.comp g).eval = fun v => g.eval v >>= f.eval",
" (f.case g).eval = fun v ... | [
" zero'.eval = fun v => pure (0 :: v)",
" succ.eval = fun v => pure [v.headI.succ]",
" tail.eval = fun v => pure v.tail",
" (f.cons fs).eval = fun v => do\n let n β f.eval v\n let ns β fs.eval v\n pure (n.headI :: ns)",
" (f.comp g).eval = fun v => g.eval v >>= f.eval",
" (f.case g).eval = fun v ... |
import Mathlib.Topology.Instances.Irrational
import Mathlib.Topology.Instances.Rat
import Mathlib.Topology.Compactification.OnePoint
#align_import topology.instances.rat_lemmas from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Set Metric Filter TopologicalSpace
open Topology One... | Mathlib/Topology/Instances/RatLemmas.lean | 56 | 62 | theorem not_countably_generated_cocompact : Β¬IsCountablyGenerated (cocompact β) := by |
intro H
rcases exists_seq_tendsto (cocompact β β π 0) with β¨x, hxβ©
rw [tendsto_inf] at hx; rcases hx with β¨hxc, hx0β©
obtain β¨n, hnβ© : β n : β, x n β insert (0 : β) (range x) :=
(hxc.eventually hx0.isCompact_insert_range.compl_mem_cocompact).exists
exact hn (Or.inr β¨n, rflβ©)
| [
" (cocompact β β π p).NeBot",
" β {i : Set β Γ Set β}, IsCompact i.1 β§ p β i.2 β§ IsOpen i.2 β (i.1αΆ β© i.2).Nonempty",
" ((s, o).1αΆ β© (s, o).2).Nonempty",
" ((s, o).2 β© (s, o).1αΆ).Nonempty",
" Β¬(cocompact β).IsCountablyGenerated",
" False"
] | [
" (cocompact β β π p).NeBot",
" β {i : Set β Γ Set β}, IsCompact i.1 β§ p β i.2 β§ IsOpen i.2 β (i.1αΆ β© i.2).Nonempty",
" ((s, o).1αΆ β© (s, o).2).Nonempty",
" ((s, o).2 β© (s, o).1αΆ).Nonempty"
] |
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology MeasureTheory
names... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | 152 | 155 | theorem condexp_ae_eq_condexpL1CLM (hm : m β€ m0) [SigmaFinite (ΞΌ.trim hm)] (hf : Integrable f ΞΌ) :
ΞΌ[f|m] =α΅[ΞΌ] condexpL1CLM F' hm ΞΌ (hf.toL1 f) := by |
refine (condexp_ae_eq_condexpL1 hm f).trans (eventually_of_forall fun x => ?_)
rw [condexpL1_eq hf]
| [
" ΞΌ[f|m] = 0",
" Β¬(SigmaFinite (ΞΌ.trim hm) β§ Integrable f ΞΌ)",
" SigmaFinite (ΞΌ.trim hm) β Β¬Integrable f ΞΌ",
" ΞΌ[f|m] =\n if Integrable f ΞΌ then if StronglyMeasurable f then f else AEStronglyMeasurable'.mk ββ(condexpL1 hm ΞΌ f) β― else 0",
" (if h : SigmaFinite (ΞΌ.trim hm) β§ Integrable f ΞΌ then\n if S... | [
" ΞΌ[f|m] = 0",
" Β¬(SigmaFinite (ΞΌ.trim hm) β§ Integrable f ΞΌ)",
" SigmaFinite (ΞΌ.trim hm) β Β¬Integrable f ΞΌ",
" ΞΌ[f|m] =\n if Integrable f ΞΌ then if StronglyMeasurable f then f else AEStronglyMeasurable'.mk ββ(condexpL1 hm ΞΌ f) β― else 0",
" (if h : SigmaFinite (ΞΌ.trim hm) β§ Integrable f ΞΌ then\n if S... |
import Mathlib.Control.Functor.Multivariate
import Mathlib.Data.PFunctor.Univariate.Basic
#align_import data.pfunctor.multivariate.basic from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f78155c6c27d62a621f4c152d"
universe u v
open MvFunctor
@[pp_with_univ]
structure MvPFunctor (n : β) where
A : Type u
... | Mathlib/Data/PFunctor/Multivariate/Basic.lean | 116 | 119 | theorem const.mk_get (x : const n A Ξ±) : const.mk n (const.get x) = x := by |
cases x
dsimp [const.get, const.mk]
congr with (_β¨β©)
| [
" get (f <$$> x) = get x",
" get (f <$$> β¨fstβ, sndββ©) = get β¨fstβ, sndββ©",
" mk n (get x) = x",
" mk n (get β¨fstβ, sndββ©) = β¨fstβ, sndββ©",
" β¨fstβ, fun x a => PEmpty.elim aβ© = β¨fstβ, sndββ©"
] | [
" get (f <$$> x) = get x",
" get (f <$$> β¨fstβ, sndββ©) = get β¨fstβ, sndββ©"
] |
import Mathlib.Algebra.Module.Torsion
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import algebra.module.dedekind_domain from "leanprover-community/mathlib"@"cdc34484a07418af43daf8198beaf5c00324bca8"
universe u v
variable {R : Type u} [CommRing R] [IsDomain R] {M : Type v} [AddCommGroup M] [Module R M]
... | Mathlib/Algebra/Module/DedekindDomain.lean | 65 | 72 | theorem isInternal_prime_power_torsion [Module.Finite R M] (hM : Module.IsTorsion R M) :
DirectSum.IsInternal fun p : (factors (β€ : Submodule R M).annihilator).toFinset =>
torsionBySet R M (p ^ (factors (β€ : Submodule R M).annihilator).count βp : Ideal R) := by |
have hM' := Module.isTorsionBySet_annihilator_top R M
have hI := Submodule.annihilator_top_inter_nonZeroDivisors hM
refine isInternal_prime_power_torsion_of_is_torsion_by_ideal ?_ hM'
rw [β Set.nonempty_iff_ne_empty] at hI; rw [Submodule.ne_bot_iff]
obtain β¨x, H, hxβ© := hI; exact β¨x, H, nonZeroDivisors.ne_ze... | [
" DirectSum.IsInternal fun p => torsionBySet R M β(βp ^ Multiset.count (βp) (factors I))",
" Module.IsTorsionBySet R M β(β¨
i β (factors I).toFinset, i ^ Multiset.count i P)",
" β¨
i β (factors I).toFinset, i ^ Multiset.count i P = I",
" Associated (factors I).prod I",
" β i β (factors I).toFinset, Prime i",
... | [
" DirectSum.IsInternal fun p => torsionBySet R M β(βp ^ Multiset.count (βp) (factors I))",
" Module.IsTorsionBySet R M β(β¨
i β (factors I).toFinset, i ^ Multiset.count i P)",
" β¨
i β (factors I).toFinset, i ^ Multiset.count i P = I",
" Associated (factors I).prod I",
" β i β (factors I).toFinset, Prime i",
... |
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Laurent
import Mathlib.LinearAlgebra.Matrix.Charpoly.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.RingTheory.Polynomial.Nilpotent
#align_import linear_algebra.matrix.charpoly.coeff from "leanprover-community/mathlib"@"9745b... | Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean | 61 | 78 | theorem charpoly_sub_diagonal_degree_lt :
(M.charpoly - β i : n, (X - C (M i i))).degree < β(Fintype.card n - 1) := by |
rw [charpoly, det_apply', β insert_erase (mem_univ (Equiv.refl n)),
sum_insert (not_mem_erase (Equiv.refl n) univ), add_comm]
simp only [charmatrix_apply_eq, one_mul, Equiv.Perm.sign_refl, id, Int.cast_one,
Units.val_one, add_sub_cancel_right, Equiv.coe_refl]
rw [β mem_degreeLT]
apply Submodule.sum_mem... | [
" (M.charmatrix i j).natDegree = if i = j then 1 else 0",
" (M.charmatrix i j).natDegree β€ if i = j then 1 else 0",
" (M.charmatrix i j).natDegree β€ 1",
" (M.charmatrix i j).natDegree β€ 0",
" (M.charpoly - β i : n, (X - C (M i i))).degree < β(Fintype.card n - 1)",
" (β x β univ.erase (Equiv.refl n), ββ(Eq... | [
" (M.charmatrix i j).natDegree = if i = j then 1 else 0",
" (M.charmatrix i j).natDegree β€ if i = j then 1 else 0",
" (M.charmatrix i j).natDegree β€ 1",
" (M.charmatrix i j).natDegree β€ 0"
] |
import Mathlib.FieldTheory.Galois
#align_import field_theory.polynomial_galois_group from "leanprover-community/mathlib"@"e3f4be1fcb5376c4948d7f095bec45350bfb9d1a"
noncomputable section
open scoped Polynomial
open FiniteDimensional
namespace Polynomial
variable {F : Type*} [Field F] (p q : F[X]) (E : Type*) [... | Mathlib/FieldTheory/PolynomialGaloisGroup.lean | 74 | 79 | theorem ext {Ο Ο : p.Gal} (h : β x β p.rootSet p.SplittingField, Ο x = Ο x) : Ο = Ο := by |
refine
AlgEquiv.ext fun x =>
(AlgHom.mem_equalizer Ο.toAlgHom Ο.toAlgHom x).mp
((SetLike.ext_iff.mp ?_ x).mpr Algebra.mem_top)
rwa [eq_top_iff, β SplittingField.adjoin_rootSet, Algebra.adjoin_le_iff]
| [
" Ο = Ο",
" (βΟ).equalizer βΟ = β€"
] | [] |
import Mathlib.Algebra.Regular.Basic
import Mathlib.LinearAlgebra.Matrix.MvPolynomial
import Mathlib.LinearAlgebra.Matrix.Polynomial
import Mathlib.RingTheory.Polynomial.Basic
#align_import linear_algebra.matrix.adjugate from "leanprover-community/mathlib"@"a99f85220eaf38f14f94e04699943e185a5e1d1a"
namespace Matr... | Mathlib/LinearAlgebra/Matrix/Adjugate.lean | 160 | 170 | theorem sum_cramer_apply {Ξ²} (s : Finset Ξ²) (f : n β Ξ² β Ξ±) (i : n) :
(β x β s, cramer A (fun j => f j x) i) = cramer A (fun j : n => β x β s, f j x) i :=
calc
(β x β s, cramer A (fun j => f j x) i) = (β x β s, cramer A fun j => f j x) i :=
(Finset.sum_apply i s _).symm
_ = cramer A (fun j : n => β ... |
rw [sum_cramer, cramer_apply, cramer_apply]
simp only [updateColumn]
congr with j
congr
apply Finset.sum_apply
| [
" IsLinearMap Ξ± A.cramerMap",
" β (x y : n β Ξ±), A.cramerMap (x + y) = A.cramerMap x + A.cramerMap y",
" β (c : Ξ±) (x : n β Ξ±), A.cramerMap (c β’ x) = c β’ A.cramerMap x",
" A.cramerMap (xβ + yβ) = A.cramerMap xβ + A.cramerMap yβ",
" A.cramerMap (cβ β’ xβ) = cβ β’ A.cramerMap xβ",
" A.cramerMap (xβ + yβ) i = ... | [
" IsLinearMap Ξ± A.cramerMap",
" β (x y : n β Ξ±), A.cramerMap (x + y) = A.cramerMap x + A.cramerMap y",
" β (c : Ξ±) (x : n β Ξ±), A.cramerMap (c β’ x) = c β’ A.cramerMap x",
" A.cramerMap (xβ + yβ) = A.cramerMap xβ + A.cramerMap yβ",
" A.cramerMap (cβ β’ xβ) = cβ β’ A.cramerMap xβ",
" A.cramerMap (xβ + yβ) i = ... |
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 | 404 | 408 | theorem fderivWithin.comp_derivWithin_of_eq {t : Set F} (hl : DifferentiableWithinAt π l t y)
(hf : DifferentiableWithinAt π f s x) (hs : MapsTo f s t) (hxs : UniqueDiffWithinAt π s x)
(hy : y = f x) :
derivWithin (l β f) s x = (fderivWithin π l t (f x) : F β E) (derivWithin f s x) := by |
rw [hy] at hl; exact fderivWithin.comp_derivWithin x hl hf hs hxs
| [
" HasDerivWithinAt (l β f) (l' f') s x",
" HasDerivAt (l β f) (l' f') x",
" HasStrictDerivAt (l β f) (l' f') x",
" derivWithin (l β f) s x = (fderivWithin π l t (f x)) (derivWithin f s x)"
] | [
" HasDerivWithinAt (l β f) (l' f') s x",
" HasDerivAt (l β f) (l' f') x",
" HasStrictDerivAt (l β f) (l' f') x"
] |
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.MeasureTheory.Group.LIntegral
import Mathlib.MeasureTheory.Integral.Marginal
import Mathlib.MeasureTheory.Measure.Stiel... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 506 | 508 | theorem measurableSet_graph (hf : Measurable f) :
MeasurableSet { p : Ξ± Γ β | p.snd = f p.fst } := by |
simpa using measurableSet_region_between_cc hf hf MeasurableSet.univ
| [
" regionBetween f g s β s ΓΛ’ univ",
" MeasurableSet (regionBetween f g s)",
" MeasurableSet ({a | a.1 β s} β© {a | a.2 β {a_1 | f a.1 < a_1} β© {a_1 | a_1 < g a.1}})",
" MeasurableSet {a | a.1 β s}",
" MeasurableSet {p | p.1 β s β§ p.2 β Ioc (f p.1) (g p.1)}",
" MeasurableSet ({a | a.1 β s} β© {a | a.2 β {a_1... | [
" regionBetween f g s β s ΓΛ’ univ",
" MeasurableSet (regionBetween f g s)",
" MeasurableSet ({a | a.1 β s} β© {a | a.2 β {a_1 | f a.1 < a_1} β© {a_1 | a_1 < g a.1}})",
" MeasurableSet {a | a.1 β s}",
" MeasurableSet {p | p.1 β s β§ p.2 β Ioc (f p.1) (g p.1)}",
" MeasurableSet ({a | a.1 β s} β© {a | a.2 β {a_1... |
import Mathlib.Logic.Function.Iterate
import Mathlib.Init.Data.Int.Order
import Mathlib.Order.Compare
import Mathlib.Order.Max
import Mathlib.Order.RelClasses
import Mathlib.Tactic.Choose
#align_import order.monotone.basic from "leanprover-community/mathlib"@"554bb38de8ded0dafe93b7f18f0bfee6ef77dc5d"
open Functio... | Mathlib/Order/Monotone/Basic.lean | 1,014 | 1,018 | theorem Nat.rel_of_forall_rel_succ_of_le_of_lt (r : Ξ² β Ξ² β Prop) [IsTrans Ξ² r] {f : β β Ξ²} {a : β}
(h : β n, a β€ n β r (f n) (f (n + 1))) β¦b c : ββ¦ (hab : a β€ b) (hbc : b < c) :
r (f b) (f c) := by |
induction' hbc with k b_lt_k r_b_k
exacts [h _ hab, _root_.trans r_b_k (h _ (hab.trans_lt b_lt_k).le)]
| [
" r (f b) (f c)",
" r (f b) (f k.succ)"
] | [] |
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.Dimension.Constructions
open Cardinal Submodule Set FiniteDimensional
universe u v
namespace Subalgebra
variable {F E : Type*} [CommRing F] [StrongRankCondition F] [Ring E] [Algebra F E]
{S : Subalgebra F E}
theorem eq_bot_of_rank_le_o... | Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean | 277 | 280 | theorem eq_bot_of_finrank_one (h : finrank F S = 1) [Module.Free F S] : S = β₯ := by |
refine Subalgebra.eq_bot_of_rank_le_one ?_
rw [finrank, toNat_eq_one] at h
rw [h]
| [
" S = β₯",
" x β range β(algebraMap F E)",
" Module.rank F β₯S β€ 1"
] | [
" S = β₯",
" x β range β(algebraMap F E)"
] |
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks
#align_import category_theory.limits.constructions.epi_mono from "leanprover-community/mathlib"@"f7baecbb54bd0f24f228576f97b1752fc3c9b318"
... | Mathlib/CategoryTheory/Limits/Constructions/EpiMono.lean | 58 | 62 | theorem preserves_epi_of_preservesColimit {X Y : C} (f : X βΆ Y) [PreservesColimit (span f f) F]
[Epi f] : Epi (F.map f) := by |
have := isColimitPushoutCoconeMapOfIsColimit F _ (PushoutCocone.isColimitMkIdId f)
simp_rw [F.map_id] at this
apply PushoutCocone.epi_of_isColimitMkIdId _ this
| [
" Mono (F.map f)",
" Mono f",
" Epi (F.map f)"
] | [
" Mono (F.map f)",
" Mono f"
] |
import Mathlib.Data.Int.Interval
import Mathlib.Data.Int.SuccPred
import Mathlib.Data.Int.ConditionallyCompleteOrder
import Mathlib.Topology.Instances.Discrete
import Mathlib.Topology.MetricSpace.Bounded
import Mathlib.Order.Filter.Archimedean
#align_import topology.instances.int from "leanprover-community/mathlib"@"... | Mathlib/Topology/Instances/Int.lean | 84 | 85 | theorem cofinite_eq : (cofinite : Filter β€) = atBot β atTop := by |
rw [β cocompact_eq_cofinite, cocompact_eq_atBot_atTop]
| [
" dist m n = β|m - n|",
" |βm - βn| = β|m - n|",
" Pairwise fun m n => 1 β€ dist m n",
" 1 β€ dist m n",
" 1 β€ |βm - βn|",
" 1 β€ |m - n|",
" ball x r = Ioo ββx - rβ ββx + rβ",
" closedBall x r = Icc ββx - rβ ββx + rβ",
" IsCompact (closedBall x r)",
" IsCompact (Icc ββx - rβ ββx + rβ)",
" Bornolog... | [
" dist m n = β|m - n|",
" |βm - βn| = β|m - n|",
" Pairwise fun m n => 1 β€ dist m n",
" 1 β€ dist m n",
" 1 β€ |βm - βn|",
" 1 β€ |m - n|",
" ball x r = Ioo ββx - rβ ββx + rβ",
" closedBall x r = Icc ββx - rβ ββx + rβ",
" IsCompact (closedBall x r)",
" IsCompact (Icc ββx - rβ ββx + rβ)",
" Bornolog... |
import Mathlib.AlgebraicTopology.DoldKan.PInfty
#align_import algebraic_topology.dold_kan.decomposition from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open CategoryTheory CategoryTheory.Category CategoryTheory.Preadditive
Opposite Simplicial
noncomputable section
namespace Alge... | Mathlib/AlgebraicTopology/DoldKan/Decomposition.lean | 120 | 124 | theorem id_Ο : (id X n).Ο = π _ := by |
simp only [β P_add_Q_f (n + 1) (n + 1), Ο]
congr 1
Β· simp only [id, PInfty_f, P_f_idem]
Β· exact Eq.trans (by congr; simp) (decomposition_Q n (n + 1)).symm
| [
" (Q q).f (n + 1) = β i β Finset.filter (fun i => βi < q) Finset.univ, (P βi).f (n + 1) β« X.Ξ΄ i.rev.succ β« X.Ο i.rev",
" (Q 0).f (n + 1) = β i β Finset.filter (fun i => βi < 0) Finset.univ, (P βi).f (n + 1) β« X.Ξ΄ i.rev.succ β« X.Ο i.rev",
" (Q (q + 1)).f (n + 1) =\n β i β Finset.filter (fun i => βi < q + 1) F... | [
" (Q q).f (n + 1) = β i β Finset.filter (fun i => βi < q) Finset.univ, (P βi).f (n + 1) β« X.Ξ΄ i.rev.succ β« X.Ο i.rev",
" (Q 0).f (n + 1) = β i β Finset.filter (fun i => βi < 0) Finset.univ, (P βi).f (n + 1) β« X.Ξ΄ i.rev.succ β« X.Ο i.rev",
" (Q (q + 1)).f (n + 1) =\n β i β Finset.filter (fun i => βi < q + 1) F... |
import Mathlib.CategoryTheory.Sites.Coherent.ReflectsPreregular
import Mathlib.Topology.Category.CompHaus.EffectiveEpi
import Mathlib.Topology.Category.Stonean.Limits
import Mathlib.Topology.Category.CompHaus.EffectiveEpi
universe u
open CategoryTheory Limits
namespace Stonean
noncomputable
def struct {B X : St... | Mathlib/Topology/Category/Stonean/EffectiveEpi.lean | 103 | 121 | theorem effectiveEpiFamily_tfae
{Ξ± : Type} [Finite Ξ±] {B : Stonean.{u}}
(X : Ξ± β Stonean.{u}) (Ο : (a : Ξ±) β (X a βΆ B)) :
TFAE
[ EffectiveEpiFamily X Ο
, Epi (Sigma.desc Ο)
, β b : B, β (a : Ξ±) (x : X a), Ο a x = b
] := by |
tfae_have 2 β 1
Β· intro
simpa [β effectiveEpi_desc_iff_effectiveEpiFamily, (effectiveEpi_tfae (Sigma.desc Ο)).out 0 1]
tfae_have 1 β 2
Β· intro; infer_instance
tfae_have 3 β 1
Β· erw [((CompHaus.effectiveEpiFamily_tfae
(fun a β¦ Stonean.toCompHaus.obj (X a)) (fun a β¦ Stonean.toCompHaus.map (Ο a))).o... | [
" { toFun := fun x => a, continuous_toFun := β― } β« Ο = { toFun := fun x => b, continuous_toFun := β― } β« Ο",
" ({ toFun := fun x => a, continuous_toFun := β― } β« Ο) xβ = ({ toFun := fun x => b, continuous_toFun := β― } β« Ο) xβ",
" g = (fun {W} e h => β―.lift e β―) e β―",
" g = β―.liftEquiv β¨e, β―β©",
" β―.liftEquiv.s... | [
" { toFun := fun x => a, continuous_toFun := β― } β« Ο = { toFun := fun x => b, continuous_toFun := β― } β« Ο",
" ({ toFun := fun x => a, continuous_toFun := β― } β« Ο) xβ = ({ toFun := fun x => b, continuous_toFun := β― } β« Ο) xβ",
" g = (fun {W} e h => β―.lift e β―) e β―",
" g = β―.liftEquiv β¨e, β―β©",
" β―.liftEquiv.s... |
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 | 63 | 67 | theorem digitsAux_def (b : β) (h : 2 β€ b) (n : β) (w : 0 < n) :
digitsAux b h n = (n % b) :: digitsAux b h (n / b) := by |
cases n
Β· cases w
Β· rw [digitsAux]
| [
" (invImage (fun x => x) instWellFoundedRelationOfSizeOf).1 ((n + 1) / b) n.succ",
" b.digitsAux h 0 = []",
" b.digitsAux h n = n % b :: b.digitsAux h (n / b)",
" b.digitsAux h 0 = 0 % b :: b.digitsAux h (0 / b)",
" b.digitsAux h (nβ + 1) = (nβ + 1) % b :: b.digitsAux h ((nβ + 1) / b)"
] | [
" (invImage (fun x => x) instWellFoundedRelationOfSizeOf).1 ((n + 1) / b) n.succ",
" b.digitsAux h 0 = []"
] |
import Mathlib.Data.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
#align_import algebra.gcd_monoid.finset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
#align_import algebra.gcd_monoid.div from "leanprover-community/mathlib"@"b537794f8409bc9598febb79cd510b1df5f4539d"
variab... | Mathlib/Algebra/GCDMonoid/Finset.lean | 100 | 103 | theorem lcm_congr {f g : Ξ² β Ξ±} (hs : sβ = sβ) (hfg : β a β sβ, f a = g a) :
sβ.lcm f = sβ.lcm g := by |
subst hs
exact Finset.fold_congr hfg
| [
" s.lcm f β£ a β β b β s, f b β£ a",
" (β b β Multiset.map f s.val, b β£ a) β β b β s, f b β£ a",
" (β (b : Ξ±), β x β s.val, f x = b β b β£ a) β β b β s, f b β£ a",
" (insert b s).lcm f = GCDMonoid.lcm (f b) (s.lcm f)",
" normalize (s.lcm f) = s.lcm f",
" (β
βͺ sβ).lcm f = GCDMonoid.lcm (β
.lcm f) (sβ.lcm f)",
... | [
" s.lcm f β£ a β β b β s, f b β£ a",
" (β b β Multiset.map f s.val, b β£ a) β β b β s, f b β£ a",
" (β (b : Ξ±), β x β s.val, f x = b β b β£ a) β β b β s, f b β£ a",
" (insert b s).lcm f = GCDMonoid.lcm (f b) (s.lcm f)",
" normalize (s.lcm f) = s.lcm f",
" (β
βͺ sβ).lcm f = GCDMonoid.lcm (β
.lcm f) (sβ.lcm f)",
... |
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
section Preorder
v... | Mathlib/Order/Interval/Set/Basic.lean | 181 | 181 | theorem left_mem_Ioo : a β Ioo a b β False := by | simp [lt_irrefl]
| [
" Decidable (x β Ioo a b)",
" Decidable (x β Ico a b)",
" Decidable (x β Iio b)",
" Decidable (x β Icc a b)",
" Decidable (x β Iic b)",
" Decidable (x β Ioc a b)",
" Decidable (x β Ici a)",
" Decidable (x β Ioi a)",
" a β Ioo a b β False"
] | [
" Decidable (x β Ioo a b)",
" Decidable (x β Ico a b)",
" Decidable (x β Iio b)",
" Decidable (x β Icc a b)",
" Decidable (x β Iic b)",
" Decidable (x β Ioc a b)",
" Decidable (x β Ici a)",
" Decidable (x β Ioi a)"
] |
import Mathlib.NumberTheory.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex Real Set
open scoped Nat
open HurwitzZeta
theorem riemannZeta_two_mul_nat {k : β} (hk : k β 0) :
riemannZeta (2 * k) = (-1) ^ (k + 1) * (2 : β) ^ (2 * k - 1)
* (Ο : β) ^ (2 * k) * bernoulli (2 * k) / (... | Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 220 | 224 | theorem riemannZeta_two : riemannZeta 2 = (Ο : β) ^ 2 / 6 := by |
convert congr_arg ((β) : β β β) hasSum_zeta_two.tsum_eq
Β· rw [β Nat.cast_two, zeta_nat_eq_tsum_of_gt_one one_lt_two]
simp only [push_cast]
Β· norm_cast
| [
" riemannZeta (2 * βk) = (-1) ^ (k + 1) * 2 ^ (2 * k - 1) * βΟ ^ (2 * k) * β(bernoulli (2 * k)) / β(2 * k)!",
" riemannZeta (2 * βk) = β(β' (b : β), 1 / βb ^ (2 * k))",
" 1 < 2 * k",
" β' (n : β), 1 / βn ^ (2 * k) = β(β' (b : β), 1 / βb ^ (2 * k))",
" (-1) ^ (k + 1) * 2 ^ (2 * k - 1) * βΟ ^ (2 * k) * β(bern... | [
" riemannZeta (2 * βk) = (-1) ^ (k + 1) * 2 ^ (2 * k - 1) * βΟ ^ (2 * k) * β(bernoulli (2 * k)) / β(2 * k)!",
" riemannZeta (2 * βk) = β(β' (b : β), 1 / βb ^ (2 * k))",
" 1 < 2 * k",
" β' (n : β), 1 / βn ^ (2 * k) = β(β' (b : β), 1 / βb ^ (2 * k))",
" (-1) ^ (k + 1) * 2 ^ (2 * k - 1) * βΟ ^ (2 * k) * β(bern... |
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.JacobsonIdeal
#align_import ring_theory.jacobson from "leanprover-community/mathlib"@"a7c017d750512a352b623b1824d75da5998457d0"
set_option autoImplicit true
universe u
namespace Ideal
open Polynomial
... | Mathlib/RingTheory/Jacobson.lean | 108 | 117 | theorem isJacobson_of_surjective [H : IsJacobson R] :
(β f : R β+* S, Function.Surjective βf) β IsJacobson S := by |
rintro β¨f, hfβ©
rw [isJacobson_iff_sInf_maximal]
intro p hp
use map f '' { J : Ideal R | comap f p β€ J β§ J.IsMaximal }
use fun j β¨J, hJ, hmapβ© => hmap βΈ (map_eq_top_or_isMaximal_of_surjective f hf hJ.right).symm
have : p = map f (comap f p).jacobson :=
(IsJacobson.out' _ <| hp.isRadical.comap f).symm βΈ ... | [
" IsJacobson R β β (P : Ideal R), P.IsPrime β P.jacobson = P",
" (β (P : Ideal R), P.IsPrime β P.jacobson = P) β β (I : Ideal R), I.IsRadical β I.jacobson = I",
" x β I",
" β β¦I_1 : Ideal Rβ¦, I_1 β {J | I β€ J β§ J.IsPrime} β x β I_1",
" x β P",
" β β¦I : Ideal Rβ¦, I β {J | P β€ J β§ J.IsMaximal} β x β I",
"... | [
" IsJacobson R β β (P : Ideal R), P.IsPrime β P.jacobson = P",
" (β (P : Ideal R), P.IsPrime β P.jacobson = P) β β (I : Ideal R), I.IsRadical β I.jacobson = I",
" x β I",
" β β¦I_1 : Ideal Rβ¦, I_1 β {J | I β€ J β§ J.IsPrime} β x β I_1",
" x β P",
" β β¦I : Ideal Rβ¦, I β {J | P β€ J β§ J.IsMaximal} β x β I",
"... |
import Mathlib.GroupTheory.Subgroup.Center
import Mathlib.GroupTheory.Submonoid.Centralizer
#align_import group_theory.subgroup.basic from "leanprover-community/mathlib"@"4be589053caf347b899a494da75410deb55fb3ef"
open Function
open Int
variable {G : Type*} [Group G]
namespace Subgroup
variable {H K : Subgroup ... | Mathlib/GroupTheory/Subgroup/Centralizer.lean | 42 | 44 | theorem mem_centralizer_iff_commutator_eq_one {g : G} {s : Set G} :
g β centralizer s β β h β s, h * g * hβ»ΒΉ * gβ»ΒΉ = 1 := by |
simp only [mem_centralizer_iff, mul_inv_eq_iff_eq_mul, one_mul]
| [
" g β centralizer s β β h β s, h * g * hβ»ΒΉ * gβ»ΒΉ = 1"
] | [] |
import Mathlib.AlgebraicTopology.DoldKan.FunctorGamma
import Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject
import Mathlib.CategoryTheory.Idempotents.HomologicalComplex
#align_import algebraic_topology.dold_kan.gamma_comp_n from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
no... | Mathlib/AlgebraicTopology/DoldKan/GammaCompN.lean | 105 | 108 | theorem NβΞβ_hom_app_f_f (K : ChainComplex C β) (n : β) :
(NβΞβ.hom.app K).f.f n = (Ξβ.splitting K).toKaroubiNondegComplexIsoNβ.inv.f.f n := by |
rw [NβΞβ_hom_app]
apply comp_id
| [
" β (i j : β),\n (ComplexShape.down β).Rel i j β\n ((fun n => Iso.refl ((Ξβ.splitting K).nondegComplex.X n)) i).hom β« K.d i j =\n (Ξβ.splitting K).nondegComplex.d i j β« ((fun n => Iso.refl ((Ξβ.splitting K).nondegComplex.X n)) j).hom",
" ((fun n => Iso.refl ((Ξβ.splitting K).nondegComplex.X n)) (n ... | [
" β (i j : β),\n (ComplexShape.down β).Rel i j β\n ((fun n => Iso.refl ((Ξβ.splitting K).nondegComplex.X n)) i).hom β« K.d i j =\n (Ξβ.splitting K).nondegComplex.d i j β« ((fun n => Iso.refl ((Ξβ.splitting K).nondegComplex.X n)) j).hom",
" ((fun n => Iso.refl ((Ξβ.splitting K).nondegComplex.X n)) (n ... |
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 | 132 | 134 | theorem PreservesPullback.iso_inv_fst :
(PreservesPullback.iso G f g).inv β« G.map pullback.fst = pullback.fst := by |
simp [PreservesPullback.iso, Iso.inv_comp_eq]
| [
" G.map h β« G.map f = G.map k β« G.map g",
" β (j : WalkingCospan),\n ((Cones.postcompose (diagramIsoCospan (cospan f g β G)).hom).obj (G.mapCone (PullbackCone.mk h k comm))).Ο.app j =\n (Iso.refl\n ((Cones.postcompose (diagramIsoCospan (cospan f g β G)).hom).obj\n (G.mapCone (Pul... | [
" G.map h β« G.map f = G.map k β« G.map g",
" β (j : WalkingCospan),\n ((Cones.postcompose (diagramIsoCospan (cospan f g β G)).hom).obj (G.mapCone (PullbackCone.mk h k comm))).Ο.app j =\n (Iso.refl\n ((Cones.postcompose (diagramIsoCospan (cospan f g β G)).hom).obj\n (G.mapCone (Pul... |
import Mathlib.Topology.Sets.Opens
#align_import topology.local_at_target from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open TopologicalSpace Set Filter
open Topology Filter
variable {Ξ± Ξ² : Type*} [TopologicalSpace Ξ±] [TopologicalSpace Ξ²] {f : Ξ± β Ξ²}
variable {s : Set Ξ²} {ΞΉ : Ty... | Mathlib/Topology/LocalAtTarget.lean | 78 | 84 | theorem IsOpenMap.restrictPreimage (H : IsOpenMap f) (s : Set Ξ²) :
IsOpenMap (s.restrictPreimage f) := by |
intro t
suffices β u, IsOpen u β Subtype.val β»ΒΉ' u = t β
β v, IsOpen v β§ Subtype.val β»ΒΉ' v = s.restrictPreimage f '' t by
simpa [isOpen_induced_iff]
exact fun u hu e => β¨f '' u, H u hu, by simp [β e, image_restrictPreimage]β©
| [
" Inducing (s.restrictPreimage f)",
" β (x : β(f β»ΒΉ' s)), π x = comap Subtype.val (comap f (π (f βx)))",
" π a = comap Subtype.val (comap f (π (f βa)))",
" IsClosedMap (s.restrictPreimage f)",
" IsClosed t β IsClosed (s.restrictPreimage f '' t)",
" β (u : Set Ξ±), IsClosed u β Subtype.val β»ΒΉ' u = t β β... | [
" Inducing (s.restrictPreimage f)",
" β (x : β(f β»ΒΉ' s)), π x = comap Subtype.val (comap f (π (f βx)))",
" π a = comap Subtype.val (comap f (π (f βa)))",
" IsClosedMap (s.restrictPreimage f)",
" IsClosed t β IsClosed (s.restrictPreimage f '' t)",
" β (u : Set Ξ±), IsClosed u β Subtype.val β»ΒΉ' u = t β β... |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Rat.Encodable
import Mathlib.Topology.GDelta
#align_import topology.instances.irrational from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Filter Metric
open Filter Topology
protected theorem IsGΞ΄.setOf_irrational : Is... | Mathlib/Topology/Instances/Irrational.lean | 78 | 89 | theorem eventually_forall_le_dist_cast_div (hx : Irrational x) (n : β) :
βαΆ Ξ΅ : β in π 0, β m : β€, Ξ΅ β€ dist x (m / n) := by |
have A : IsClosed (range (fun m => (n : β)β»ΒΉ * m : β€ β β)) :=
((isClosedMap_smulβ (nβ»ΒΉ : β)).comp Int.closedEmbedding_coe_real.isClosedMap).isClosed_range
have B : x β range (fun m => (n : β)β»ΒΉ * m : β€ β β) := by
rintro β¨m, rflβ©
simp at hx
rcases Metric.mem_nhds_iff.1 (A.isOpen_compl.mem_nhds B) with... | [
" Dense {x | Irrational x}",
" β o β β a, β b, β (_ : a < b), {Ioo βa βb}, o.Nonempty β (o β© {x | Irrational x}).Nonempty",
" β (o : Set β) (x x_1 : β), x < x_1 β o = Ioo βx βx_1 β o.Nonempty β (o β© {x | Irrational x}).Nonempty",
" (Ioo βa βb β© {x | Irrational x}).Nonempty",
" ({x | Irrational x} β© Ioo βa β... | [
" Dense {x | Irrational x}",
" β o β β a, β b, β (_ : a < b), {Ioo βa βb}, o.Nonempty β (o β© {x | Irrational x}).Nonempty",
" β (o : Set β) (x x_1 : β), x < x_1 β o = Ioo βx βx_1 β o.Nonempty β (o β© {x | Irrational x}).Nonempty",
" (Ioo βa βb β© {x | Irrational x}).Nonempty",
" ({x | Irrational x} β© Ioo βa β... |
import Mathlib.Analysis.Analytic.Linear
import Mathlib.Analysis.Analytic.Composition
import Mathlib.Analysis.NormedSpace.Completion
#align_import analysis.analytic.uniqueness from "leanprover-community/mathlib"@"a3209ddf94136d36e5e5c624b10b2a347cc9d090"
variable {π : Type*} [NontriviallyNormedField π] {E : Type... | Mathlib/Analysis/Analytic/Uniqueness.lean | 32 | 70 | theorem eqOn_zero_of_preconnected_of_eventuallyEq_zero_aux [CompleteSpace F] {f : E β F} {U : Set E}
(hf : AnalyticOn π f U) (hU : IsPreconnected U) {zβ : E} (hβ : zβ β U) (hfzβ : f =αΆ [π zβ] 0) :
EqOn f 0 U := by |
/- Let `u` be the set of points around which `f` vanishes. It is clearly open. We have to show
that its limit points in `U` still belong to it, from which the inclusion `U β u` will follow
by connectedness. -/
let u := {x | f =αΆ [π x] 0}
suffices main : closure u β© U β u by
have Uu : U β u :=
h... | [
" EqOn f 0 U",
" f z = 0 z",
" closure u β© U β u",
" x β u",
" HasFPowerSeriesOnBall f q y (r / 2)",
" ββy - xββ < r / 2",
" r / 2 β€ r - ββy - xββ",
" ββy - xββ + r / 2 β€ r",
" r / 2 + r / 2 = r",
" f z = OfNat.ofNat 0 z",
" HasSum (fun n => (q n) fun x => z - y) 0",
" (fun n => (q n) fun x =>... | [] |
import Mathlib.Order.BooleanAlgebra
import Mathlib.Tactic.Common
#align_import order.heyting.boundary from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
variable {Ξ± : Type*}
namespace Coheyting
variable [CoheytingAlgebra Ξ±] {a b : Ξ±}
def boundary (a : Ξ±) : Ξ± :=
a β οΏ’a
#align cohe... | Mathlib/Order/Heyting/Boundary.lean | 120 | 122 | theorem boundary_le_boundary_sup_sup_boundary_inf_right : β b β€ β (a β b) β β (a β b) := by |
rw [sup_comm a, inf_comm]
exact boundary_le_boundary_sup_sup_boundary_inf_left
| [
" β β€ = β₯",
" β (οΏ’οΏ’a) = β (οΏ’a)",
" οΏ’β a = β€",
" β (a β b) = β a β b β a β β b",
" a β b β οΏ’(a β b) = a β οΏ’a β b β a β (b β οΏ’b)",
" β (a β b) β€ β a β β b",
" a β οΏ’(a β b) β b β οΏ’(a β b) β€ β a β β b",
" (a β§ b β¨ Β¬(a β§ b)) β§ ((a β¨ b) β¨ Β¬(a β¨ b)) β a β¨ Β¬a",
" a β¨ Β¬a",
" β a β€ β (a β b) β β (a β b)",
... | [
" β β€ = β₯",
" β (οΏ’οΏ’a) = β (οΏ’a)",
" οΏ’β a = β€",
" β (a β b) = β a β b β a β β b",
" a β b β οΏ’(a β b) = a β οΏ’a β b β a β (b β οΏ’b)",
" β (a β b) β€ β a β β b",
" a β οΏ’(a β b) β b β οΏ’(a β b) β€ β a β β b",
" (a β§ b β¨ Β¬(a β§ b)) β§ ((a β¨ b) β¨ Β¬(a β¨ b)) β a β¨ Β¬a",
" a β¨ Β¬a",
" β a β€ β (a β b) β β (a β b)",
... |
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.Analysis.SumIntegralComparisons
import Mathlib.NumberTheory.Harmonic.Defs
| Mathlib/NumberTheory/Harmonic/Bounds.lean | 17 | 24 | theorem log_add_one_le_harmonic (n : β) :
Real.log β(n+1) β€ harmonic n := by |
calc _ = β« x in (1:β)..β(n+1), xβ»ΒΉ := ?_
_ β€ β d β Finset.Icc 1 n, (d:β)β»ΒΉ := ?_
_ = harmonic n := ?_
Β· rw [Nat.cast_one, integral_inv (by simp [(show Β¬ (1 : β) β€ 0 by norm_num)]), div_one]
Β· exact (inv_antitoneOn_Icc_right <| by norm_num).integral_le_sum_Ico (Nat.le_add_left 1 n)
Β· simp only [ha... | [
" (β(n + 1)).log β€ β(harmonic n)",
" (β(n + 1)).log = β« (x : β) in β1 ..β(n + 1), xβ»ΒΉ",
" 0 β Set.uIcc 1 β(n + 1)",
" Β¬1 β€ 0",
" β« (x : β) in β1 ..β(n + 1), xβ»ΒΉ β€ β d β Finset.Icc 1 n, (βd)β»ΒΉ",
" 0 < β1",
" β d β Finset.Icc 1 n, (βd)β»ΒΉ = β(harmonic n)"
] | [] |
import Mathlib.Algebra.MvPolynomial.Derivation
import Mathlib.Algebra.MvPolynomial.Variables
#align_import data.mv_polynomial.pderiv from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
universe u v
namespace MvPolynomial
open Set Function Finsupp
variable {R : ... | Mathlib/Algebra/MvPolynomial/PDeriv.lean | 125 | 126 | theorem pderiv_C_mul {f : MvPolynomial Ο R} {i : Ο} : pderiv i (C a * f) = C a * pderiv i f := by |
rw [C_mul', Derivation.map_smul, C_mul']
| [
" pderiv i = mkDerivation R (Pi.single i 1)",
" mkDerivation R (Pi.single i 1) = mkDerivation R (Pi.single i 1)",
" (pderiv i) ((monomial s) a) = (monomial (s - single i 1)) (a * β(s i))",
" (s.sum fun a_1 b => (monomial (s - single a_1 1)) (a * βb) * Pi.single i 1 a_1) =\n (monomial (s - single i 1)) (a *... | [
" pderiv i = mkDerivation R (Pi.single i 1)",
" mkDerivation R (Pi.single i 1) = mkDerivation R (Pi.single i 1)",
" (pderiv i) ((monomial s) a) = (monomial (s - single i 1)) (a * β(s i))",
" (s.sum fun a_1 b => (monomial (s - single a_1 1)) (a * βb) * Pi.single i 1 a_1) =\n (monomial (s - single i 1)) (a *... |
import Mathlib.Algebra.Group.Defs
import Mathlib.Logic.Relation
#align_import algebra.homology.complex_shape from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
noncomputable section
open scoped Classical
@[ext]
structure ComplexShape (ΞΉ : Type*) where
Rel : ΞΉ β ΞΉ β Prop
nex... | Mathlib/Algebra/Homology/ComplexShape.lean | 100 | 102 | theorem symm_symm (c : ComplexShape ΞΉ) : c.symm.symm = c := by |
ext
simp
| [
" c.symm.symm = c",
" c.symm.symm.Rel xβΒΉ xβ β c.Rel xβΒΉ xβ"
] | [] |
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