Context stringlengths 57 85k | file_name stringlengths 21 79 | start int64 14 2.42k | end int64 18 2.43k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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import Mathlib.Data.Set.Finite
import Mathlib.GroupTheory.GroupAction.FixedPoints
import Mathlib.GroupTheory.Perm.Support
open Equiv List MulAction Pointwise Set Subgroup
variable {G α : Type*} [Group G] [MulAction G α] [DecidableEq α]
theorem finite_compl_fixedBy_closure_iff {S : Set G} :
(∀ g ∈ closure S, ... | Mathlib/GroupTheory/Perm/ClosureSwap.lean | 74 | 88 | theorem swap_mem_closure_isSwap {S : Set (Perm α)} (hS : ∀ f ∈ S, f.IsSwap) {x y : α} :
swap x y ∈ closure S ↔ x ∈ orbit (closure S) y := by |
refine ⟨fun h ↦ ⟨⟨swap x y, h⟩, swap_apply_right x y⟩, fun hf ↦ ?_⟩
by_contra h
have := exists_smul_not_mem_of_subset_orbit_closure S {x | swap x y ∈ closure S}
(fun f hf ↦ ?_) (fun z hz ↦ ?_) h ⟨y, ?_⟩
· obtain ⟨σ, hσ, a, ha, hσa⟩ := this
obtain ⟨z, w, hzw, rfl⟩ := hS σ hσ
have := ne_of_mem_of_not... |
import Mathlib.CategoryTheory.ConcreteCategory.BundledHom
import Mathlib.Topology.ContinuousFunction.Basic
#align_import topology.category.Top.basic from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78ad6527b7e4"
open CategoryTheory
open TopologicalSpace
universe u
@[to_additive existing TopCat... | Mathlib/Topology/Category/TopCat/Basic.lean | 175 | 179 | theorem of_isoOfHomeo {X Y : TopCat.{u}} (f : X ≃ₜ Y) : homeoOfIso (isoOfHomeo f) = f := by |
-- Porting note: unfold some defs now
dsimp [homeoOfIso, isoOfHomeo]
ext
rfl
|
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 447 | 449 | theorem image_const_sub_Ioo : (fun x => a - x) '' Ioo b c = Ioo (a - c) (a - b) := by |
have := image_comp (fun x => a + x) fun x => -x; dsimp [Function.comp_def] at this
simp [sub_eq_add_neg, this, add_comm]
|
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.CategoryTheory.Groupoid.VertexGroup
import Mathlib.CategoryTheory.Groupoid.Basic
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Data.Set.Lattice
import Mathlib.Order.GaloisConnection
#align_import category_theory.groupoid.subgroupoid from "leanprover-c... | Mathlib/CategoryTheory/Groupoid/Subgroupoid.lean | 566 | 570 | theorem obj_surjective_of_im_eq_top (hφ : Function.Injective φ.obj) (hφ' : im φ hφ = ⊤) :
Function.Surjective φ.obj := by |
rintro d
rw [← mem_im_objs_iff, hφ']
apply mem_top_objs
|
import Mathlib.Init.Data.Sigma.Lex
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.Antichain
import Mathlib.Order.OrderIsoNat
import Mathlib.Order.WellFounded
import Mathlib.Tactic.TFAE
#align_import order.well_founded_set from "leanprover-community/mathlib"@"2c84c2c5496117349007d97104... | Mathlib/Order/WellFoundedSet.lean | 92 | 93 | theorem wellFoundedOn_univ : (univ : Set α).WellFoundedOn r ↔ WellFounded r := by |
simp [wellFoundedOn_iff]
|
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"@"b99e2d58a5e6861833fa8de11e51a81144258db4"
open Function
universe u v w
namespace SimpleGraph
variable {V : Type u} {V' : Type v} {V'' : Typ... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 752 | 753 | theorem map_fst_darts {u v : V} (p : G.Walk u v) : p.darts.map (·.fst) = p.support.dropLast := by |
simpa! using congr_arg List.dropLast (map_fst_darts_append p)
|
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 135 | 139 | theorem tan_oangle_add_left_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
Real.Angle.tan (o.oangle (x + y) y) = ‖x‖ / ‖y‖ := by |
rw [← neg_inj, oangle_rev, ← oangle_neg_orientation_eq_neg, neg_inj] at h ⊢
rw [add_comm]
exact (-o).tan_oangle_add_right_of_oangle_eq_pi_div_two h
|
import Mathlib.Algebra.CharP.Two
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Periodic
import Mathlib.Data.ZMod.Basic
import Mathlib.Tactic.Monotonicity
#align_import data.nat.totient from "leanprover-community/mathlib"@"5cc2dfdd3e92f340411acea4427d701dc7ed26f8"
open Finset
namespace Nat
... | Mathlib/Data/Nat/Totient.lean | 381 | 384 | theorem totient_mul_of_prime_of_not_dvd {p n : ℕ} (hp : p.Prime) (h : ¬p ∣ n) :
(p * n).totient = (p - 1) * n.totient := by |
rw [totient_mul _, totient_prime hp]
simpa [h] using coprime_or_dvd_of_prime hp n
|
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.Algebra.Order.Group.OrderIso
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.Order.UpperLower.Basic
#align_import algebra.order.upper_lower from "leanprover-community/mathlib"@"c0c52abb75074ed8b73a948341f50521fbf43b4c"
open Function Set
open Pointw... | Mathlib/Algebra/Order/UpperLower.lean | 70 | 72 | theorem IsUpperSet.mul_right (hs : IsUpperSet s) : IsUpperSet (s * t) := by |
rw [mul_comm]
exact hs.mul_left
|
import Mathlib.Order.Filter.Basic
import Mathlib.Topology.Bases
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.Basic
import Mathlib.Topology.LocallyFinite
open Set Filter Topology TopologicalSpace Classical Function
universe u v
variable {X : Type u} {Y : Type v} {ι : Type*}
variable [Topolog... | Mathlib/Topology/Compactness/Compact.lean | 281 | 285 | theorem IsCompact.inter_iInter_nonempty {ι : Type v} (hs : IsCompact s) (t : ι → Set X)
(htc : ∀ i, IsClosed (t i)) (hst : ∀ u : Finset ι, (s ∩ ⋂ i ∈ u, t i).Nonempty) :
(s ∩ ⋂ i, t i).Nonempty := by |
contrapose! hst
exact hs.elim_finite_subfamily_closed t htc hst
|
import Mathlib.Data.Rat.Encodable
import Mathlib.Data.Real.EReal
import Mathlib.Topology.Instances.ENNReal
import Mathlib.Topology.Order.MonotoneContinuity
#align_import topology.instances.ereal from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Class... | Mathlib/Topology/Instances/EReal.lean | 217 | 221 | theorem continuousAt_add_bot_bot : ContinuousAt (fun p : EReal × EReal => p.1 + p.2) (⊥, ⊥) := by |
simp only [ContinuousAt, tendsto_nhds_bot_iff_real, bot_add]
refine fun r ↦ ((gt_mem_nhds (bot_lt_coe 0)).prod_nhds
(gt_mem_nhds <| bot_lt_coe r)).mono fun _ h ↦ ?_
simpa only [coe_zero, zero_add] using add_lt_add h.1 h.2
|
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.matrix.transvection from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e... | Mathlib/LinearAlgebra/Matrix/Transvection.lean | 226 | 235 | theorem prod_mul_reverse_inv_prod (L : List (TransvectionStruct n R)) :
(L.map toMatrix).prod * (L.reverse.map (toMatrix ∘ TransvectionStruct.inv)).prod = 1 := by |
induction' L with t L IH
· simp
· suffices
t.toMatrix *
((L.map toMatrix).prod * (L.reverse.map (toMatrix ∘ TransvectionStruct.inv)).prod) *
t.inv.toMatrix = 1
by simpa [Matrix.mul_assoc]
simp_rw [IH, Matrix.mul_one, t.mul_inv]
|
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.MeasureTheory.Integral.Pi
import Mathlib.Analysis.Fourier.FourierTransform
open Real Set MeasureTheory Filter Asymptotics intervalIntegral
open scoped Real Topology FourierTransform Re... | Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean | 360 | 368 | theorem integral_rexp_neg_mul_sq_norm {b : ℝ} (hb : 0 < b) :
∫ v : V, rexp (- b * ‖v‖^2) = (π / b) ^ (FiniteDimensional.finrank ℝ V / 2 : ℝ) := by |
rw [← ofReal_inj]
convert integral_cexp_neg_mul_sq_norm (show 0 < (b : ℂ).re from hb) (V := V)
· change ofRealLI (∫ (v : V), rexp (-b * ‖v‖ ^ 2)) = ∫ (v : V), cexp (-↑b * ↑‖v‖ ^ 2)
rw [← ofRealLI.integral_comp_comm]
simp [ofRealLI]
· rw [← ofReal_div, ofReal_cpow (by positivity)]
simp
|
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.Spectrum
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Algebra.Star.StarAlgHom
#align_import analysis.normed_space.star.spectrum from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
l... | Mathlib/Analysis/NormedSpace/Star/Spectrum.lean | 72 | 86 | theorem IsStarNormal.spectralRadius_eq_nnnorm (a : A) [IsStarNormal a] :
spectralRadius ℂ a = ‖a‖₊ := by |
refine (ENNReal.pow_strictMono two_ne_zero).injective ?_
have heq :
(fun n : ℕ => (‖(a⋆ * a) ^ n‖₊ : ℝ≥0∞) ^ (1 / n : ℝ)) =
(fun x => x ^ 2) ∘ fun n : ℕ => (‖a ^ n‖₊ : ℝ≥0∞) ^ (1 / n : ℝ) := by
funext n
rw [Function.comp_apply, ← rpow_natCast, ← rpow_mul, mul_comm, rpow_mul, rpow_natCast, ←
... |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncompu... | Mathlib/FieldTheory/RatFunc/Basic.lean | 164 | 166 | theorem ofFractionRing_div (p q : FractionRing K[X]) :
ofFractionRing (p / q) = ofFractionRing p / ofFractionRing q := by |
simp only [Div.div, HDiv.hDiv, RatFunc.div]
|
import Mathlib.Algebra.Polynomial.Basic
#align_import data.polynomial.monomial from "leanprover-community/mathlib"@"220f71ba506c8958c9b41bd82226b3d06b0991e8"
noncomputable section
namespace Polynomial
open Polynomial
universe u
variable {R : Type u} {a b : R} {m n : ℕ}
variable [Semiring R] {p q r : R[X]}
| Mathlib/Algebra/Polynomial/Monomial.lean | 28 | 32 | theorem monomial_one_eq_iff [Nontrivial R] {i j : ℕ} :
(monomial i 1 : R[X]) = monomial j 1 ↔ i = j := by |
-- Porting note: `ofFinsupp.injEq` is required.
simp_rw [← ofFinsupp_single, ofFinsupp.injEq]
exact AddMonoidAlgebra.of_injective.eq_iff
|
import Mathlib.Topology.Category.TopCat.OpenNhds
import Mathlib.Topology.Sheaves.Presheaf
import Mathlib.Topology.Sheaves.SheafCondition.UniqueGluing
import Mathlib.CategoryTheory.Adjunction.Evaluation
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Limits.Preserves.Filtered
import Mathlib.Cat... | Mathlib/Topology/Sheaves/Stalks.lean | 337 | 340 | theorem stalkSpecializes_refl {C : Type*} [Category C] [Limits.HasColimits C] {X : TopCat}
(F : X.Presheaf C) (x : X) : F.stalkSpecializes (specializes_refl x) = 𝟙 _ := by |
ext
simp
|
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Order.Bounds.OrderIso
import Mathlib.Tactic.Positivity.Core
#align_import algebra.order.field.basic from "leanprover-community/mathlib"@"8477... | Mathlib/Algebra/Order/Field/Basic.lean | 107 | 107 | theorem mul_inv_le_iff (h : 0 < b) : a * b⁻¹ ≤ c ↔ a ≤ b * c := by | rw [mul_comm, inv_mul_le_iff h]
|
import Mathlib.Probability.IdentDistrib
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.Analysis.SpecificLimits.FloorPow
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
#align_import probability.strong_law from "leanprover-community/mathlib"@"f2ce60867... | Mathlib/Probability/StrongLaw.lean | 188 | 196 | theorem integral_truncation_le_integral_of_nonneg (hf : Integrable f μ) (h'f : 0 ≤ f) {A : ℝ} :
∫ x, truncation f A x ∂μ ≤ ∫ x, f x ∂μ := by |
apply integral_mono_of_nonneg
(eventually_of_forall fun x => ?_) hf (eventually_of_forall fun x => ?_)
· exact truncation_nonneg _ (h'f x)
· calc
truncation f A x ≤ |truncation f A x| := le_abs_self _
_ ≤ |f x| := abs_truncation_le_abs_self _ _ _
_ = f x := abs_of_nonneg (h'f x)
|
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.order.pointwise from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Set
open Pointwise
variable ... | Mathlib/Algebra/Order/Pointwise.lean | 137 | 139 | theorem csInf_inv (hs₀ : s.Nonempty) (hs₁ : BddAbove s) : sInf s⁻¹ = (sSup s)⁻¹ := by |
rw [← image_inv]
exact ((OrderIso.inv α).map_csSup' hs₀ hs₁).symm
|
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : ℝ}
-- @... | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 68 | 69 | theorem logb_neg_eq_logb (x : ℝ) : logb b (-x) = logb b x := by |
rw [← logb_abs x, ← logb_abs (-x), abs_neg]
|
import Mathlib.Analysis.Complex.UpperHalfPlane.Topology
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Geometry.Euclidean.Inversion.Basic
#align_import analysis.complex.upper_half_plane.metric from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c"
noncomputable section
ope... | Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | 50 | 57 | theorem cosh_half_dist (z w : ℍ) :
cosh (dist z w / 2) = dist (z : ℂ) (conj (w : ℂ)) / (2 * √(z.im * w.im)) := by |
rw [← sq_eq_sq, cosh_sq', sinh_half_dist, div_pow, div_pow, one_add_div, mul_pow, sq_sqrt]
· congr 1
simp only [Complex.dist_eq, Complex.sq_abs, Complex.normSq_sub, Complex.normSq_conj,
Complex.conj_conj, Complex.mul_re, Complex.conj_re, Complex.conj_im, coe_im]
ring
all_goals positivity
|
import Mathlib.Analysis.SpecialFunctions.Pow.Real
#align_import analysis.special_functions.pow.nnreal from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real NNReal ENNReal ComplexConjugate
open Finset Function Set
namespace NNReal
var... | Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean | 227 | 228 | theorem le_rpow_one_div_iff {x y : ℝ≥0} {z : ℝ} (hz : 0 < z) : x ≤ y ^ (1 / z) ↔ x ^ z ≤ y := by |
rw [← rpow_le_rpow_iff hz, rpow_self_rpow_inv hz.ne']
|
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
namespace Set
variable {α β : Type*} {s t : Set α}
noncomputable def encard (s : Set α) : ℕ∞ := PartENat.withTopEquiv (PartENat.card s)
@[simp] theorem encard_uni... | Mathlib/Data/Set/Card.lean | 140 | 141 | theorem finite_of_encard_le_coe {k : ℕ} (h : s.encard ≤ k) : s.Finite := by |
rw [← encard_lt_top_iff]; exact h.trans_lt (WithTop.coe_lt_top _)
|
import Mathlib.Topology.Homeomorph
import Mathlib.Topology.StoneCech
#align_import topology.extremally_disconnected from "leanprover-community/mathlib"@"7e281deff072232a3c5b3e90034bd65dde396312"
noncomputable section
open scoped Classical
open Function Set
universe u
section
variable (X : Type u) [TopologicalS... | Mathlib/Topology/ExtremallyDisconnected.lean | 83 | 92 | theorem StoneCech.projective [DiscreteTopology X] : CompactT2.Projective (StoneCech X) := by |
intro Y Z _tsY _tsZ _csY _t2Y _csZ _csZ f g hf hg g_sur
let s : Z → Y := fun z => Classical.choose <| g_sur z
have hs : g ∘ s = id := funext fun z => Classical.choose_spec (g_sur z)
let t := s ∘ f ∘ stoneCechUnit
have ht : Continuous t := continuous_of_discreteTopology
let h : StoneCech X → Y := stoneCechE... |
import Mathlib.MeasureTheory.Measure.FiniteMeasure
import Mathlib.MeasureTheory.Integral.Average
#align_import measure_theory.measure.probability_measure from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open MeasureTheory
open Set
open Filter
open BoundedCon... | Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean | 199 | 201 | theorem apply_mono (μ : ProbabilityMeasure Ω) {s₁ s₂ : Set Ω} (h : s₁ ⊆ s₂) : μ s₁ ≤ μ s₂ := by |
rw [← coeFn_comp_toFiniteMeasure_eq_coeFn]
exact MeasureTheory.FiniteMeasure.apply_mono _ h
|
import Mathlib.SetTheory.Game.Short
#align_import set_theory.game.state from "leanprover-community/mathlib"@"b134b2f5cf6dd25d4bbfd3c498b6e36c11a17225"
universe u
namespace SetTheory
namespace PGame
class State (S : Type u) where
turnBound : S → ℕ
l : S → Finset S
r : S → Finset S
left_bound : ∀ {s t : S... | Mathlib/SetTheory/Game/State.lean | 57 | 61 | theorem turnBound_ne_zero_of_right_move {s t : S} (m : t ∈ r s) : turnBound s ≠ 0 := by |
intro h
have t := right_bound m
rw [h] at t
exact Nat.not_succ_le_zero _ t
|
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Order.OrderClosed
#align_import topology.algebra.with_zero_topology from "leanprover-community/mathlib"@"3e0c4d76b6ebe9dfafb67d16f7286d2731ed6064"
open Topology Filter TopologicalSpace Filter Set Function
namespace WithZeroTopology
variable {α... | Mathlib/Topology/Algebra/WithZeroTopology.lean | 136 | 139 | theorem isOpen_iff {s : Set Γ₀} : IsOpen s ↔ (0 : Γ₀) ∉ s ∨ ∃ γ, γ ≠ 0 ∧ Iio γ ⊆ s := by |
rw [isOpen_iff_mem_nhds, ← and_forall_ne (0 : Γ₀)]
simp (config := { contextual := true }) [nhds_of_ne_zero, imp_iff_not_or,
hasBasis_nhds_zero.mem_iff]
|
import Mathlib.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.oriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 555 | 559 | theorem _root_.Wbtw.oangle_eq_left {p₁ p₁' p₂ p₃ : P} (h : Wbtw ℝ p₂ p₁ p₁') (hp₁p₂ : p₁ ≠ p₂) :
∡ p₁ p₂ p₃ = ∡ p₁' p₂ p₃ := by |
by_cases hp₃p₂ : p₃ = p₂; · simp [hp₃p₂]
by_cases hp₁'p₂ : p₁' = p₂; · rw [hp₁'p₂, wbtw_self_iff] at h; exact False.elim (hp₁p₂ h)
rw [← oangle_add hp₁'p₂ hp₁p₂ hp₃p₂, h.oangle₃₁₂_eq_zero, zero_add]
|
import Mathlib.MeasureTheory.Measure.MeasureSpace
open scoped ENNReal NNReal Topology
open Set MeasureTheory Measure Filter MeasurableSpace ENNReal Function
variable {R α β δ γ ι : Type*}
namespace MeasureTheory
variable {m0 : MeasurableSpace α} [MeasurableSpace β] [MeasurableSpace γ]
variable {μ μ₁ μ₂ μ₃ ν ν' ν... | Mathlib/MeasureTheory/Measure/Restrict.lean | 255 | 258 | theorem restrict_union_add_inter₀ (s : Set α) (ht : NullMeasurableSet t μ) :
μ.restrict (s ∪ t) + μ.restrict (s ∩ t) = μ.restrict s + μ.restrict t := by |
rw [← restrict_inter_add_diff₀ (s ∪ t) ht, union_inter_cancel_right, union_diff_right, ←
restrict_inter_add_diff₀ s ht, add_comm, ← add_assoc, add_right_comm]
|
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Basis
#align_import analysis.convex.combination from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
open Set Function
open scoped Classical
open Pointwise
... | Mathlib/Analysis/Convex/Combination.lean | 168 | 188 | theorem Convex.centerMass_mem (hs : Convex R s) :
(∀ i ∈ t, 0 ≤ w i) → (0 < ∑ i ∈ t, w i) → (∀ i ∈ t, z i ∈ s) → t.centerMass w z ∈ s := by |
induction' t using Finset.induction with i t hi ht
· simp [lt_irrefl]
intro h₀ hpos hmem
have zi : z i ∈ s := hmem _ (mem_insert_self _ _)
have hs₀ : ∀ j ∈ t, 0 ≤ w j := fun j hj => h₀ j <| mem_insert_of_mem hj
rw [sum_insert hi] at hpos
by_cases hsum_t : ∑ j ∈ t, w j = 0
· have ws : ∀ j ∈ t, w j = 0 :... |
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.MeasureTheory.Integral.Pi
import Mathlib.Analysis.Fourier.FourierTransform
open Real Set MeasureTheory Filter Asymptotics intervalIntegral
open scoped Real Topology FourierTransform Re... | Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean | 370 | 381 | theorem _root_.fourierIntegral_gaussian_innerProductSpace' (hb : 0 < b.re) (x w : V) :
𝓕 (fun v ↦ cexp (- b * ‖v‖^2 + 2 * π * Complex.I * ⟪x, v⟫)) w =
(π / b) ^ (FiniteDimensional.finrank ℝ V / 2 : ℂ) * cexp (-π ^ 2 * ‖x - w‖ ^ 2 / b) := by |
simp only [neg_mul, fourierIntegral_eq', ofReal_neg, ofReal_mul, ofReal_ofNat,
smul_eq_mul, ← Complex.exp_add, real_inner_comm w]
convert integral_cexp_neg_mul_sq_norm_add hb (2 * π * Complex.I) (x - w) using 3 with v
· congr 1
simp [inner_sub_left]
ring
· have : b ≠ 0 := by contrapose! hb; rw [hb,... |
import Mathlib.AlgebraicTopology.DoldKan.Faces
import Mathlib.CategoryTheory.Idempotents.Basic
#align_import algebraic_topology.dold_kan.projections from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Pread... | Mathlib/AlgebraicTopology/DoldKan/Projections.lean | 61 | 65 | theorem P_f_0_eq (q : ℕ) : ((P q).f 0 : X _[0] ⟶ X _[0]) = 𝟙 _ := by |
induction' q with q hq
· rfl
· simp only [P_succ, HomologicalComplex.add_f_apply, HomologicalComplex.comp_f,
HomologicalComplex.id_f, id_comp, hq, Hσ_eq_zero, add_zero]
|
import Mathlib.Init.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive
import Aesop
#align_import data.option.basic from "leanprover-community/mathlib"@"f340f229b1f461aa1c8ee11e0a172d0a3b301a4a"
universe u
namespace Op... | Mathlib/Data/Option/Basic.lean | 195 | 198 | theorem mem_pmem {a : α} (h : ∀ a ∈ x, p a) (ha : a ∈ x) : f a (h a ha) ∈ pmap f x h := by |
rw [mem_def] at ha ⊢
subst ha
rfl
|
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Int.Basic
import Mathlib.Tactic.Ring
import Mathlib.Tactic.FieldSimp
import Mathlib.Data.Int.NatPrime
import Mathlib.Data.ZMod.Basic
#align_import number_theory.pythagorean_tri... | Mathlib/NumberTheory/PythagoreanTriples.lean | 255 | 263 | theorem coprime_of_coprime (hc : Int.gcd x y = 1) : Int.gcd y z = 1 := by |
by_contra H
obtain ⟨p, hp, hpy, hpz⟩ := Nat.Prime.not_coprime_iff_dvd.mp H
apply hp.not_dvd_one
rw [← hc]
apply Nat.dvd_gcd (Int.Prime.dvd_natAbs_of_coe_dvd_sq hp _ _) hpy
rw [sq, eq_sub_of_add_eq h]
rw [← Int.natCast_dvd] at hpy hpz
exact dvd_sub (hpz.mul_right _) (hpy.mul_right _)
|
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.List.Defs
import Mathlib.Init.Data.List.Basic
import Mathlib.Init.Data.List.Instances
import Mathlib.Init.Data.List.Lemmas
import Mathlib.Logic.Unique
import Mathlib.Order.Basic
import Mathlib.Tactic.Common
#align_import data.list.basic... | Mathlib/Data/List/Basic.lean | 990 | 994 | theorem bidirectionalRec_singleton {motive : List α → Sort*}
(nil : motive []) (singleton : ∀ a : α, motive [a])
(cons_append : ∀ (a : α) (l : List α) (b : α), motive l → motive (a :: (l ++ [b]))) (a : α):
bidirectionalRec nil singleton cons_append [a] = singleton a := by |
simp [bidirectionalRec]
|
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Data.Set.Lattice
import Mathlib.Data.SetLike.Basic
#align_import group_theory.subsemigroup.basic from "leanprover-community/mathlib"@"feb99064803fd3108e37c18b0f77d0a8344677a3"
assert_not_exists MonoidWithZero
-- Only needed for notation
variable {M : Type*} {N ... | Mathlib/Algebra/Group/Subsemigroup/Basic.lean | 446 | 447 | theorem closure_singleton_le_iff_mem (m : M) (p : Subsemigroup M) : closure {m} ≤ p ↔ m ∈ p := by |
rw [closure_le, singleton_subset_iff, SetLike.mem_coe]
|
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Tactic.FinCases
#align_import linear_algebra.affine_space.combination from ... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 96 | 104 | theorem weightedVSubOfPoint_eq_of_weights_eq (p : ι → P) (j : ι) (w₁ w₂ : ι → k)
(hw : ∀ i, i ≠ j → w₁ i = w₂ i) :
s.weightedVSubOfPoint p (p j) w₁ = s.weightedVSubOfPoint p (p j) w₂ := by |
simp only [Finset.weightedVSubOfPoint_apply]
congr
ext i
rcases eq_or_ne i j with h | h
· simp [h]
· simp [hw i h]
|
import Mathlib.Order.Filter.FilterProduct
import Mathlib.Analysis.SpecificLimits.Basic
#align_import data.real.hyperreal from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classical
open Filter Germ Topology
def Hyperreal : Type :=
Germ (hyperfilter ℕ : Filter ℕ) ℝ deri... | Mathlib/Data/Real/Hyperreal.lean | 283 | 286 | theorem IsSt.st_eq {x : ℝ*} {r : ℝ} (hxr : IsSt x r) : st x = r := by |
have h : ∃ r, IsSt x r := ⟨r, hxr⟩
rw [st, dif_pos h]
exact (Classical.choose_spec h).unique hxr
|
import Batteries.Data.UInt
@[ext] theorem Char.ext : {a b : Char} → a.val = b.val → a = b
| ⟨_,_⟩, ⟨_,_⟩, rfl => rfl
theorem Char.ext_iff {x y : Char} : x = y ↔ x.val = y.val := ⟨congrArg _, Char.ext⟩
theorem Char.le_antisymm_iff {x y : Char} : x = y ↔ x ≤ y ∧ y ≤ x :=
Char.ext_iff.trans UInt32.le_antisymm_iff
... | .lake/packages/batteries/Batteries/Data/Char.lean | 30 | 31 | theorem csize_pos (c) : 0 < csize c := by |
rcases csize_eq c with _|_|_|_ <;> simp_all (config := {decide := true})
|
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Shapes.Reflexive
import Mathlib.CategoryTheory.Monad.Coequalizer
import Mathlib.CategoryTheory.Monad.Limits
#align_import category_theory.monad.monadicity from "leanprover-community/mathlib"@"4bd8c855d6ba8f0d5eefbf80... | Mathlib/CategoryTheory/Monad/Monadicity.lean | 178 | 188 | theorem comparisonAdjunction_unit_f
[∀ A : adj.toMonad.Algebra, HasCoequalizer (F.map A.a)
(adj.counit.app (F.obj A.A))]
(A : adj.toMonad.Algebra) :
((comparisonAdjunction adj).unit.app A).f = (beckCoequalizer A).desc (unitCofork A) := by |
apply Limits.Cofork.IsColimit.hom_ext (beckCoequalizer A)
rw [Cofork.IsColimit.π_desc]
dsimp only [beckCofork_π, unitCofork_π]
rw [comparisonAdjunction_unit_f_aux, ← adj.homEquiv_naturality_left A.a, coequalizer.condition,
adj.homEquiv_naturality_right, adj.homEquiv_unit, Category.assoc]
apply adj.right_... |
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section iInf
variable {ι : Sort*} {f g : ι → ℝ≥0∞}
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
theorem toNNReal_iInf (hf : ∀ i, f ... | Mathlib/Data/ENNReal/Real.lean | 609 | 610 | theorem add_iInf {a : ℝ≥0∞} : a + iInf f = ⨅ b, a + f b := by |
rw [add_comm, iInf_add]; simp [add_comm]
|
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Fin.VecNotation
import Mathlib.Data.Sign
import Mathlib.LinearAlgebra.AffineSpace.Combination
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
import Mathlib.LinearAlgebra.Basis.VectorSpace
#align_import linear_algebra.affine_space.independent from "leanprover-c... | Mathlib/LinearAlgebra/AffineSpace/Independent.lean | 360 | 369 | theorem AffineIndependent.of_comp {p : ι → P} (f : P →ᵃ[k] P₂) (hai : AffineIndependent k (f ∘ p)) :
AffineIndependent k p := by |
cases' isEmpty_or_nonempty ι with h h;
· haveI := h
apply affineIndependent_of_subsingleton
obtain ⟨i⟩ := h
rw [affineIndependent_iff_linearIndependent_vsub k p i]
simp_rw [affineIndependent_iff_linearIndependent_vsub k (f ∘ p) i, Function.comp_apply, ←
f.linearMap_vsub] at hai
exact LinearIndepend... |
import Mathlib.Data.ENNReal.Real
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Topology.UniformSpace.Pi
import Mathlib.Topology.UniformSpace.UniformConvergence
import Mathlib.Topology.UniformSpace.UniformEmbedding
#align_import topology.metric_space.emetric_space from "leanprover-community/mathlib"@"c8f3055... | Mathlib/Topology/EMetricSpace/Basic.lean | 586 | 587 | theorem mem_closedBall_comm : x ∈ closedBall y ε ↔ y ∈ closedBall x ε := by |
rw [mem_closedBall', mem_closedBall]
|
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
#align_import category_theory.limits.shapes.kernels from "leanprover-community/mathlib"@"956af7c76589f444f2e1313911bad16366ea476d"
noncomputable section
universe v v₂ u u' u₂
open CategoryTheory
open CategoryTheory.Limits.WalkingParallelPair
namespace... | Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean | 388 | 390 | theorem kernelIsoOfEq_hom_comp_ι {f g : X ⟶ Y} [HasKernel f] [HasKernel g] (h : f = g) :
(kernelIsoOfEq h).hom ≫ kernel.ι g = kernel.ι f := by |
cases h; simp
|
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
#align_import geometry.euclidean.angle.oriented.basic from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped Real Rea... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 418 | 419 | theorem oangle_eq_pi_iff_oangle_rev_eq_pi {x y : V} : o.oangle x y = π ↔ o.oangle y x = π := by |
rw [oangle_rev, neg_eq_iff_eq_neg, Real.Angle.neg_coe_pi]
|
import Mathlib.Data.ENNReal.Operations
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal
namespace ENNReal
noncomputable section Inv
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
protected theorem div_eq_inv_mul : a / b = b⁻¹ * a := by rw [... | Mathlib/Data/ENNReal/Inv.lean | 226 | 227 | theorem inv_le_iff_inv_le : a⁻¹ ≤ b ↔ b⁻¹ ≤ a := by |
simpa only [inv_inv] using @ENNReal.inv_le_inv a b⁻¹
|
import Mathlib.Order.Filter.FilterProduct
import Mathlib.Analysis.SpecificLimits.Basic
#align_import data.real.hyperreal from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classical
open Filter Germ Topology
def Hyperreal : Type :=
Germ (hyperfilter ℕ : Filter ℕ) ℝ deri... | Mathlib/Data/Real/Hyperreal.lean | 584 | 592 | theorem infinitePos_of_tendsto_top {f : ℕ → ℝ} (hf : Tendsto f atTop atTop) :
InfinitePos (ofSeq f) := fun r =>
have hf' := tendsto_atTop_atTop.mp hf
let ⟨i, hi⟩ := hf' (r + 1)
have hi' : ∀ a : ℕ, f a < r + 1 → a < i := fun a => lt_imp_lt_of_le_imp_le (hi a)
have hS : { a : ℕ | r < f a }ᶜ ⊆ { a : ℕ | a ≤ i ... |
simp only [Set.compl_setOf, not_lt]
exact fun a har => le_of_lt (hi' a (lt_of_le_of_lt har (lt_add_one _)))
Germ.coe_lt.2 <| mem_hyperfilter_of_finite_compl <| (Set.finite_le_nat _).subset hS
|
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.RCLike.Basic
#align_import analysis.convex.gauge from "leanprover-community/mathlib"@"373b03b5b9d0486534edbe94747f23cb3712f93d"
... | Mathlib/Analysis/Convex/Gauge.lean | 66 | 68 | theorem gauge_def' : gauge s x = sInf {r ∈ Set.Ioi (0 : ℝ) | r⁻¹ • x ∈ s} := by |
congrm sInf {r | ?_}
exact and_congr_right fun hr => mem_smul_set_iff_inv_smul_mem₀ hr.ne' _ _
|
import Mathlib.Algebra.Group.Fin
import Mathlib.Algebra.NeZero
import Mathlib.Data.Nat.ModEq
import Mathlib.Data.Fintype.Card
#align_import data.zmod.defs from "leanprover-community/mathlib"@"3a2b5524a138b5d0b818b858b516d4ac8a484b03"
def ZMod : ℕ → Type
| 0 => ℤ
| n + 1 => Fin (n + 1)
#align zmod ZMod
insta... | Mathlib/Data/ZMod/Defs.lean | 124 | 127 | theorem card (n : ℕ) [Fintype (ZMod n)] : Fintype.card (ZMod n) = n := by |
cases n with
| zero => exact (not_finite (ZMod 0)).elim
| succ n => convert Fintype.card_fin (n + 1) using 2
|
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Inverse
#align_import analysis.calculus.cont_diff from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
noncomputab... | Mathlib/Analysis/Calculus/ContDiff/Basic.lean | 1,022 | 1,039 | theorem ContDiffWithinAt.fderivWithin'' {f : E → F → G} {g : E → F} {t : Set F} {n : ℕ∞}
(hf : ContDiffWithinAt 𝕜 n (Function.uncurry f) (insert x₀ s ×ˢ t) (x₀, g x₀))
(hg : ContDiffWithinAt 𝕜 m g s x₀)
(ht : ∀ᶠ x in 𝓝[insert x₀ s] x₀, UniqueDiffWithinAt 𝕜 t (g x)) (hmn : m + 1 ≤ n)
(hgt : t ∈ 𝓝[g ... |
have : ∀ k : ℕ, (k : ℕ∞) ≤ m →
ContDiffWithinAt 𝕜 k (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀ := fun k hkm ↦ by
obtain ⟨v, hv, -, f', hvf', hf'⟩ :=
(hf.of_le <| (add_le_add_right hkm 1).trans hmn).hasFDerivWithinAt_nhds (hg.of_le hkm) hgt
refine hf'.congr_of_eventuallyEq_insert ?_
filter... |
import Mathlib.Algebra.Ring.Prod
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Tactic.FinCases
#align_import data.zmod.basic from "leanprover-community/mathlib"@"74ad1c88c77e799d2fea62801d1dbbd698cff1b7"
assert_not_exists Submodule
open Function
namespace ZMod
instance charZero : CharZero (ZMod 0) :=... | Mathlib/Data/ZMod/Basic.lean | 183 | 186 | theorem cast_eq_val [NeZero n] (a : ZMod n) : (cast a : R) = a.val := by |
cases n
· cases NeZero.ne 0 rfl
rfl
|
import Mathlib.Data.Real.Basic
import Mathlib.Data.ENNReal.Real
import Mathlib.Data.Sign
#align_import data.real.ereal from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
open Function ENNReal NNReal Set
noncomputable section
def EReal := WithBot (WithTop ℝ)
deriving Bot, Zero, One,... | Mathlib/Data/Real/EReal.lean | 411 | 413 | theorem coe_toReal {x : EReal} (hx : x ≠ ⊤) (h'x : x ≠ ⊥) : (x.toReal : EReal) = x := by |
lift x to ℝ using ⟨hx, h'x⟩
rfl
|
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.GroupWithZero.Basic
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Contrapose
import Mathlib.Tactic.Nontriviality
import Mathlib.Tactic.Spread
import Mathlib.Util.AssertExists
#align_import algebra.group_with_zero.units.basic from "leanprover-community/... | Mathlib/Algebra/GroupWithZero/Units/Basic.lean | 118 | 119 | theorem mul_inverse_cancel_right (x y : M₀) (h : IsUnit x) : y * x * inverse x = y := by |
rw [mul_assoc, mul_inverse_cancel x h, mul_one]
|
import Mathlib.MeasureTheory.Constructions.Pi
import Mathlib.MeasureTheory.Integral.Lebesgue
open scoped Classical ENNReal
open Set Function Equiv Finset
noncomputable section
namespace MeasureTheory
section LMarginal
variable {δ δ' : Type*} {π : δ → Type*} [∀ x, MeasurableSpace (π x)]
variable {μ : ∀ i, Measu... | Mathlib/MeasureTheory/Integral/Marginal.lean | 244 | 249 | theorem lintegral_le_of_lmarginal_le [Fintype δ] (s : Finset δ) {f g : (∀ i, π i) → ℝ≥0∞}
(hf : Measurable f) (hg : Measurable g) (hfg : ∫⋯∫⁻_s, f ∂μ ≤ ∫⋯∫⁻_s, g ∂μ) :
∫⁻ x, f x ∂Measure.pi μ ≤ ∫⁻ x, g x ∂Measure.pi μ := by |
rcases isEmpty_or_nonempty (∀ i, π i) with h|⟨⟨x⟩⟩
· simp_rw [lintegral_of_isEmpty, le_rfl]
simp_rw [lintegral_eq_lmarginal_univ x, lmarginal_le_of_subset (Finset.subset_univ s) hf hg hfg x]
|
import Mathlib.CategoryTheory.Limits.Shapes.CommSq
import Mathlib.CategoryTheory.Limits.Shapes.StrictInitial
import Mathlib.CategoryTheory.Limits.Shapes.Types
import Mathlib.Topology.Category.TopCat.Limits.Pullbacks
import Mathlib.CategoryTheory.Limits.FunctorCategory
import Mathlib.CategoryTheory.Limits.Constructions... | Mathlib/CategoryTheory/Extensive.lean | 102 | 112 | theorem FinitaryExtensive.vanKampen [FinitaryExtensive C] {F : Discrete WalkingPair ⥤ C}
(c : Cocone F) (hc : IsColimit c) : IsVanKampenColimit c := by |
let X := F.obj ⟨WalkingPair.left⟩
let Y := F.obj ⟨WalkingPair.right⟩
have : F = pair X Y := by
apply Functor.hext
· rintro ⟨⟨⟩⟩ <;> rfl
· rintro ⟨⟨⟩⟩ ⟨j⟩ ⟨⟨rfl : _ = j⟩⟩ <;> simp
clear_value X Y
subst this
exact FinitaryExtensive.van_kampen' c hc
|
import Mathlib.Analysis.Analytic.Composition
#align_import analysis.analytic.inverse from "leanprover-community/mathlib"@"284fdd2962e67d2932fa3a79ce19fcf92d38e228"
open scoped Classical Topology
open Finset Filter
namespace FormalMultilinearSeries
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} ... | Mathlib/Analysis/Analytic/Inverse.lean | 97 | 148 | theorem leftInv_comp (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F)
(h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm i) : (leftInv p i).comp p = id 𝕜 E := by |
ext (n v)
match n with
| 0 =>
simp only [leftInv_coeff_zero, ContinuousMultilinearMap.zero_apply, id_apply_ne_one, Ne,
not_false_iff, zero_ne_one, comp_coeff_zero']
| 1 =>
simp only [leftInv_coeff_one, comp_coeff_one, h, id_apply_one, ContinuousLinearEquiv.coe_apply,
ContinuousLinearEquiv.s... |
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Data.Int.Log
#align_import analysis.special_functions.log.base from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
open Set Filter Function
open Topology
noncomputable section
namespace Real
variable {b x y : ℝ}
-- @... | Mathlib/Analysis/SpecialFunctions/Log/Base.lean | 119 | 120 | theorem logb_pow {k : ℕ} (hx : 0 < x) : logb b (x ^ k) = k * logb b x := by |
rw [← rpow_natCast, logb_rpow_eq_mul_logb_of_pos hx]
|
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Filter Metric Set
open scoped ComplexConjugate Real To... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 173 | 181 | theorem arg_nonneg_iff {z : ℂ} : 0 ≤ arg z ↔ 0 ≤ z.im := by |
rcases eq_or_ne z 0 with (rfl | h₀); · simp
calc
0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) :=
⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by
contrapose!
intro h
exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩
_ ↔ _ := by rw [sin_arg, le_div_iff (abs.pos h₀), zero... |
import Mathlib.Control.Applicative
import Mathlib.Control.Traversable.Basic
#align_import control.traversable.lemmas from "leanprover-community/mathlib"@"3342d1b2178381196f818146ff79bc0e7ccd9e2d"
universe u
open LawfulTraversable
open Function hiding comp
open Functor
attribute [functor_norm] LawfulTraversabl... | Mathlib/Control/Traversable/Lemmas.lean | 103 | 105 | theorem traverse_id : traverse pure = (pure : t α → Id (t α)) := by |
ext
exact id_traverse _
|
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
#align_import data.set.prod from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
open Function
namespace Set
section Prod
variable {α β γ δ : Type*} {s s₁ s₂ : Set α} {t t₁ t₂ : Set β} {a : α} {b : β}
theorem Subsingleton.pro... | Mathlib/Data/Set/Prod.lean | 137 | 139 | theorem inter_prod : (s₁ ∩ s₂) ×ˢ t = s₁ ×ˢ t ∩ s₂ ×ˢ t := by |
ext ⟨x, y⟩
simp only [← and_and_right, mem_inter_iff, mem_prod]
|
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryTheory.Limits.HasLimits
#align_import category_theory.limits.shapes.equalizers from "leanprover-community/mathlib"@"4698e35ca56a0d4fa53aa5639c3364e0a77f4eba"
section
open CategoryTheory Opposite
namespace CategoryTheory.Limits
-- attribute [local tid... | Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean | 347 | 348 | theorem Cofork.app_zero_eq_comp_π_left (s : Cofork f g) : s.ι.app zero = f ≫ s.π := by |
rw [← s.app_one_eq_π, ← s.w left, parallelPair_map_left]
|
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
#align_import geometry.euclidean.angle.oriented.basic from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped Real Rea... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 940 | 942 | theorem oangle_sign_sub_smul_right (x y : V) (r : ℝ) :
(o.oangle x (y - r • x)).sign = (o.oangle x y).sign := by |
rw [sub_eq_add_neg, ← neg_smul, add_comm, oangle_sign_smul_add_right]
|
import Mathlib.NumberTheory.ModularForms.SlashInvariantForms
import Mathlib.NumberTheory.ModularForms.CongruenceSubgroups
noncomputable section
open ModularForm UpperHalfPlane Matrix
namespace SlashInvariantForm
theorem vAdd_width_periodic (N : ℕ) (k n : ℤ) (f : SlashInvariantForm (Gamma N) k) (z : ℍ) :
f ... | Mathlib/NumberTheory/ModularForms/Identities.lean | 34 | 37 | theorem T_zpow_width_invariant (N : ℕ) (k n : ℤ) (f : SlashInvariantForm (Gamma N) k) (z : ℍ) :
f (((ModularGroup.T ^ (N * n))) • z) = f z := by |
rw [modular_T_zpow_smul z (N * n)]
simpa only [Int.cast_mul, Int.cast_natCast] using vAdd_width_periodic N k n f z
|
import Mathlib.RepresentationTheory.Action.Limits
import Mathlib.RepresentationTheory.Action.Concrete
import Mathlib.CategoryTheory.Monoidal.FunctorCategory
import Mathlib.CategoryTheory.Monoidal.Transport
import Mathlib.CategoryTheory.Monoidal.Rigid.OfEquivalence
import Mathlib.CategoryTheory.Monoidal.Rigid.FunctorCa... | Mathlib/RepresentationTheory/Action/Monoidal.lean | 229 | 233 | theorem functorCategoryMonoidalEquivalence.ε_app :
(functorCategoryMonoidalEquivalence V G).ε.app PUnit.unit = 𝟙 _ := by |
dsimp only [functorCategoryMonoidalEquivalence]
simp only [Monoidal.fromTransported_toLaxMonoidalFunctor_ε]
rfl
|
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analysis.inner_product_space.calculus from "leanprover-community/mathlib"@"f9dd3204df14a0749cd456fac1e6849dfe7d2b88"
noncomputable section
open RCLike Real ... | Mathlib/Analysis/InnerProductSpace/Calculus.lean | 333 | 337 | theorem hasStrictFDerivAt_euclidean :
HasStrictFDerivAt f f' y ↔
∀ i, HasStrictFDerivAt (fun x => f x i) (EuclideanSpace.proj i ∘L f') y := by |
rw [← (EuclideanSpace.equiv ι 𝕜).comp_hasStrictFDerivAt_iff, hasStrictFDerivAt_pi']
rfl
|
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic from "leanprover-community/mathlib"@"31b269b60935483943542d547a6dd83a66b37dc7"
assert_not_exists Field
assert_not_exists Module
noncomputable section
open Function Cardinal Set Equiv Order
open sc... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 1,064 | 1,068 | theorem mod_eq_zero_of_dvd {a b : Ordinal} (H : b ∣ a) : a % b = 0 := by |
rcases H with ⟨c, rfl⟩
rcases eq_or_ne b 0 with (rfl | hb)
· simp
· simp [mod_def, hb]
|
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.FinCases
import Mathlib.Tactic.LinearCombination
import Mathlib.Lean.Expr.ExtraRecognizers
import Mathlib.Data.Set.Subsingleton
#align_import lin... | Mathlib/LinearAlgebra/LinearIndependent.lean | 435 | 450 | theorem linearIndependent_comp_subtype {s : Set ι} :
LinearIndependent R (v ∘ (↑) : s → M) ↔
∀ l ∈ Finsupp.supported R R s, (Finsupp.total ι M R v) l = 0 → l = 0 := by |
simp only [linearIndependent_iff, (· ∘ ·), Finsupp.mem_supported, Finsupp.total_apply,
Set.subset_def, Finset.mem_coe]
constructor
· intro h l hl₁ hl₂
have := h (l.subtypeDomain s) ((Finsupp.sum_subtypeDomain_index hl₁).trans hl₂)
exact (Finsupp.subtypeDomain_eq_zero_iff hl₁).1 this
· intro h l hl
... |
import Mathlib.Algebra.Module.Submodule.Basic
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Algebra.Algebra.Pi
#align_import order.filter.zero_and_bounded_at_filter from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
namespace Filt... | Mathlib/Order/Filter/ZeroAndBoundedAtFilter.lean | 84 | 87 | theorem ZeroAtFilter.boundedAtFilter [NormedAddCommGroup β] {l : Filter α} {f : α → β}
(hf : ZeroAtFilter l f) : BoundedAtFilter l f := by |
rw [ZeroAtFilter, ← Asymptotics.isLittleO_const_iff (one_ne_zero' ℝ)] at hf
exact hf.isBigO
|
import Mathlib.Geometry.Manifold.MFDeriv.Basic
noncomputable section
open scoped Manifold
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [NormedAddCommGroup E]
[NormedSpace 𝕜 E] {E' : Type*} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {f : E → E'}
{s : Set E} {x : E}
section MFDerivFderiv
t... | Mathlib/Geometry/Manifold/MFDeriv/FDeriv.lean | 131 | 133 | theorem mfderiv_eq_fderiv : mfderiv 𝓘(𝕜, E) 𝓘(𝕜, E') f x = fderiv 𝕜 f x := by |
rw [← mfderivWithin_univ, ← fderivWithin_univ]
exact mfderivWithin_eq_fderivWithin
|
import Mathlib.ModelTheory.Syntax
import Mathlib.ModelTheory.Semantics
import Mathlib.ModelTheory.Algebra.Ring.Basic
import Mathlib.Algebra.Field.MinimalAxioms
variable {K : Type*}
namespace FirstOrder
namespace Field
open Language Ring Structure BoundedFormula
inductive FieldAxiom : Type
| addAssoc : Field... | Mathlib/ModelTheory/Algebra/Field/Basic.lean | 81 | 86 | theorem FieldAxiom.realize_toSentence_iff_toProp {K : Type*}
[Add K] [Mul K] [Neg K] [Zero K] [One K] [CompatibleRing K]
(ax : FieldAxiom) :
(K ⊨ (ax.toSentence : Sentence Language.ring)) ↔ ax.toProp K := by |
cases ax <;>
simp [Sentence.Realize, Formula.Realize, Fin.snoc]
|
import Mathlib.LinearAlgebra.TensorProduct.Graded.External
import Mathlib.RingTheory.GradedAlgebra.Basic
import Mathlib.GroupTheory.GroupAction.Ring
suppress_compilation
open scoped TensorProduct
variable {R ι A B : Type*}
variable [CommSemiring ι] [Module ι (Additive ℤˣ)] [DecidableEq ι]
variable [CommRing R] [R... | Mathlib/LinearAlgebra/TensorProduct/Graded/Internal.lean | 201 | 204 | theorem tmul_zero_coe_mul_coe_tmul {i₂ : ι} (a₁ : A) (b₁ : ℬ 0) (a₂ : 𝒜 i₂) (b₂ : B) :
(a₁ ᵍ⊗ₜ[R] (b₁ : B) * (a₂ : A) ᵍ⊗ₜ[R] b₂ : 𝒜 ᵍ⊗[R] ℬ) =
((a₁ * a₂ : A) ᵍ⊗ₜ (b₁ * b₂ : B)) := by |
rw [tmul_coe_mul_coe_tmul, zero_mul, uzpow_zero, one_smul]
|
import Mathlib.Topology.Order.Basic
#align_import topology.algebra.order.monotone_convergence from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Filter Set Function
open scoped Classical
open Filter Topology
variable {α β : Type*}
class SupConvergenceClass (α : Type*) [Preorde... | Mathlib/Topology/Order/MonotoneConvergence.lean | 96 | 100 | theorem tendsto_atTop_isLUB (h_mono : Monotone f) (ha : IsLUB (Set.range f) a) :
Tendsto f atTop (𝓝 a) := by |
suffices Tendsto (rangeFactorization f) atTop atTop from
(SupConvergenceClass.tendsto_coe_atTop_isLUB _ _ ha).comp this
exact h_mono.rangeFactorization.tendsto_atTop_atTop fun b => b.2.imp fun a ha => ha.ge
|
import Mathlib.CategoryTheory.Subobject.MonoOver
import Mathlib.CategoryTheory.Skeletal
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.Tactic.ApplyFun
import Mathlib.Tactic.CategoryTheory.Elementwise
#align_import category_theory.subobject.basic from "leanprover-community/mathlib"@"70fd9563a21e7b... | Mathlib/CategoryTheory/Subobject/Basic.lean | 210 | 213 | theorem arrow_congr {A : C} (X Y : Subobject A) (h : X = Y) :
eqToHom (congr_arg (fun X : Subobject A => (X : C)) h) ≫ Y.arrow = X.arrow := by |
induction h
simp
|
import Mathlib.Topology.Separation
import Mathlib.Topology.Bases
#align_import topology.dense_embedding from "leanprover-community/mathlib"@"148aefbd371a25f1cff33c85f20c661ce3155def"
noncomputable section
open Set Filter
open scoped Topology
variable {α : Type*} {β : Type*} {γ : Type*} {δ : Type*}
structure D... | Mathlib/Topology/DenseEmbedding.lean | 176 | 184 | theorem extend_unique_at [T2Space γ] {b : β} {f : α → γ} {g : β → γ} (di : DenseInducing i)
(hf : ∀ᶠ x in comap i (𝓝 b), g (i x) = f x) (hg : ContinuousAt g b) : di.extend f b = g b := by |
refine di.extend_eq_of_tendsto fun s hs => mem_map.2 ?_
suffices ∀ᶠ x : α in comap i (𝓝 b), g (i x) ∈ s from
hf.mp (this.mono fun x hgx hfx => hfx ▸ hgx)
clear hf f
refine eventually_comap.2 ((hg.eventually hs).mono ?_)
rintro _ hxs x rfl
exact hxs
|
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels
import Mathlib.CategoryTheory.Preadditive.LeftExact
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.Algebra.Homology.Exact
import Mathli... | Mathlib/CategoryTheory/Abelian/Exact.lean | 295 | 299 | theorem Exact.op (h : Exact f g) : Exact g.op f.op := by |
rw [exact_iff]
refine ⟨by simp [← op_comp, h.w], Quiver.Hom.unop_inj ?_⟩
simp only [unop_comp, cokernel.π_op, eqToHom_refl, kernel.ι_op, Category.id_comp,
Category.assoc, kernel_comp_cokernel_assoc _ _ h, zero_comp, comp_zero, unop_zero]
|
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Arctan
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.unoriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
... | Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean | 225 | 228 | theorem angle_sub_eq_arccos_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) :
angle x (x - y) = Real.arccos (‖x‖ / ‖x - y‖) := by |
rw [← neg_eq_zero, ← inner_neg_right] at h
rw [sub_eq_add_neg, angle_add_eq_arccos_of_inner_eq_zero h]
|
import Mathlib.Algebra.Order.Hom.Monoid
import Mathlib.SetTheory.Game.Ordinal
#align_import set_theory.surreal.basic from "leanprover-community/mathlib"@"8900d545017cd21961daa2a1734bb658ef52c618"
universe u
namespace SetTheory
open scoped PGame
namespace PGame
def Numeric : PGame → Prop
| ⟨_, _, L, R⟩ => (... | Mathlib/SetTheory/Surreal/Basic.lean | 71 | 75 | theorem numeric_def {x : PGame} :
Numeric x ↔
(∀ i j, x.moveLeft i < x.moveRight j) ∧
(∀ i, Numeric (x.moveLeft i)) ∧ ∀ j, Numeric (x.moveRight j) := by |
cases x; rfl
|
import Mathlib.RingTheory.DedekindDomain.Dvr
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import ring_theory.dedekind_domain.pid from "leanprover-community/mathlib"@"6010cf523816335f7bae7f8584cb2edaace73940"
variable {R : Type*} [CommRing R]
open Ideal
open UniqueFactorizationMonoid
open scoped nonZer... | Mathlib/RingTheory/DedekindDomain/PID.lean | 248 | 262 | theorem IsDedekindDomain.isPrincipalIdealRing_localization_over_prime [IsDomain S] :
IsPrincipalIdealRing Sₚ := by |
letI := Classical.decEq (Ideal Sₚ)
letI := Classical.decPred fun P : Ideal Sₚ => P.IsPrime
refine
IsPrincipalIdealRing.of_finite_primes
(Set.Finite.ofFinset
(Finset.filter (fun P => P.IsPrime)
({⊥} ∪ (normalizedFactors (Ideal.map (algebraMap R Sₚ) p)).toFinset))
fun P => ?_)
... |
import Mathlib.Data.Int.Range
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.MulChar.Basic
#align_import number_theory.legendre_symbol.zmod_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace ZMod
section QuadCharModP
@[simps]
def χ₄ : MulChar (ZMod 4) ℤ... | Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean | 142 | 146 | theorem isQuadratic_χ₈ : χ₈.IsQuadratic := by |
intro a
-- Porting note: was `decide!`
fin_cases a
all_goals decide
|
import Mathlib.Topology.Maps
import Mathlib.Topology.NhdsSet
#align_import topology.constructions from "leanprover-community/mathlib"@"f7ebde7ee0d1505dfccac8644ae12371aa3c1c9f"
noncomputable section
open scoped Classical
open Topology TopologicalSpace Set Filter Function
universe u v
variable {X : Type u} {Y :... | Mathlib/Topology/Constructions.lean | 243 | 246 | theorem nhdsWithin_subtype_eq_bot_iff {s t : Set X} {x : s} :
𝓝[((↑) : s → X) ⁻¹' t] x = ⊥ ↔ 𝓝[t] (x : X) ⊓ 𝓟 s = ⊥ := by |
rw [inf_principal_eq_bot_iff_comap, nhdsWithin, nhdsWithin, comap_inf, comap_principal,
nhds_induced]
|
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 | 144 | 147 | theorem mapRange_id (h : ∀ i, id (0 : β₁ i) = 0 := fun i => rfl) (g : Π₀ i : ι, β₁ i) :
mapRange (fun i => (id : β₁ i → β₁ i)) h g = g := by |
ext
rfl
|
import Mathlib.CategoryTheory.Abelian.Basic
#align_import category_theory.idempotents.basic from "leanprover-community/mathlib"@"3a061790136d13594ec10c7c90d202335ac5d854"
open CategoryTheory
open CategoryTheory.Category
open CategoryTheory.Limits
open CategoryTheory.Preadditive
open Opposite
namespace Catego... | Mathlib/CategoryTheory/Idempotents/Basic.lean | 143 | 154 | theorem split_iff_of_iso {X X' : C} (φ : X ≅ X') (p : X ⟶ X) (p' : X' ⟶ X')
(hpp' : p ≫ φ.hom = φ.hom ≫ p') :
(∃ (Y : C) (i : Y ⟶ X) (e : X ⟶ Y), i ≫ e = 𝟙 Y ∧ e ≫ i = p) ↔
∃ (Y' : C) (i' : Y' ⟶ X') (e' : X' ⟶ Y'), i' ≫ e' = 𝟙 Y' ∧ e' ≫ i' = p' := by |
constructor
· exact split_imp_of_iso φ p p' hpp'
· apply split_imp_of_iso φ.symm p' p
rw [← comp_id p, ← φ.hom_inv_id]
slice_rhs 2 3 => rw [hpp']
slice_rhs 1 2 => erw [φ.inv_hom_id]
simp only [id_comp]
rfl
|
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import geometry.euclidean.angle.unoriented.basic from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
assert_not_exists HasFDerivAt
assert_not_exists ConformalAt
noncom... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean | 177 | 178 | theorem angle_smul_left_of_neg (x y : V) {r : ℝ} (hr : r < 0) : angle (r • x) y = angle (-x) y := by |
rw [angle_comm, angle_smul_right_of_neg y x hr, angle_comm]
|
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 | 181 | 181 | theorem normalize_gcd : normalize (s.gcd f) = s.gcd f := by | simp [gcd_def]
|
import Mathlib.Topology.Compactness.Compact
open Set Filter Topology TopologicalSpace Classical
variable {X : Type*} {Y : Type*} {ι : Type*}
variable [TopologicalSpace X] [TopologicalSpace Y] {s t : Set X}
instance [WeaklyLocallyCompactSpace X] [WeaklyLocallyCompactSpace Y] :
WeaklyLocallyCompactSpace (X × Y) ... | Mathlib/Topology/Compactness/LocallyCompact.lean | 40 | 45 | theorem exists_compact_superset [WeaklyLocallyCompactSpace X] {K : Set X} (hK : IsCompact K) :
∃ K', IsCompact K' ∧ K ⊆ interior K' := by |
choose s hc hmem using fun x : X ↦ exists_compact_mem_nhds x
rcases hK.elim_nhds_subcover _ fun x _ ↦ interior_mem_nhds.2 (hmem x) with ⟨I, -, hIK⟩
refine ⟨⋃ x ∈ I, s x, I.isCompact_biUnion fun _ _ ↦ hc _, hIK.trans ?_⟩
exact iUnion₂_subset fun x hx ↦ interior_mono <| subset_iUnion₂ (s := fun x _ ↦ s x) x hx
|
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Vector.Basic
import Mathlib.Data.PFun
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Basic
import Mathlib.Tactic.ApplyFun
#align_import computability.turing_machine from "leanprover-commu... | Mathlib/Computability/TuringMachine.lean | 1,380 | 1,384 | theorem stmts_supportsStmt {M : Λ → Stmt₁} {S : Finset Λ} {q : Stmt₁} (ss : Supports M S) :
some q ∈ stmts M S → SupportsStmt S q := by |
simp only [stmts, Finset.mem_insertNone, Finset.mem_biUnion, Option.mem_def, Option.some.injEq,
forall_eq', exists_imp, and_imp]
exact fun l ls h ↦ stmts₁_supportsStmt_mono h (ss.2 _ ls)
|
import Mathlib.Algebra.Group.Subgroup.Finite
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.GroupTheory.Coset
#align_import group_theory.quotient_group from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Function
open scope... | Mathlib/GroupTheory/QuotientGroup.lean | 434 | 437 | theorem rangeKerLift_surjective : Surjective (rangeKerLift φ) := by |
rintro ⟨_, g, rfl⟩
use mk g
rfl
|
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polynomial.eval from "leanprover-community/mathlib"@"728baa2f54e6062c5879a3e397ac6bac323e506f"
set_option linter.uppercaseLean3 false
noncomputable section
open Finset AddMonoidAlgebra
open Polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 422 | 424 | theorem eval_smul [Monoid S] [DistribMulAction S R] [IsScalarTower S R R] (s : S) (p : R[X])
(x : R) : (s • p).eval x = s • p.eval x := by |
rw [← smul_one_smul R s p, eval, eval₂_smul, RingHom.id_apply, smul_one_mul]
|
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.PEquiv
#align_import data.matrix.pequiv from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
namespace PEquiv
open Matrix
universe u v
variable {k l m n : Type*}
variable {α : Type v}
open Matrix
def toMatrix [DecidableEq n] [Zer... | Mathlib/Data/Matrix/PEquiv.lean | 142 | 148 | theorem toMatrix_swap [DecidableEq n] [Ring α] (i j : n) :
(Equiv.swap i j).toPEquiv.toMatrix =
(1 : Matrix n n α) - (single i i).toMatrix - (single j j).toMatrix + (single i j).toMatrix +
(single j i).toMatrix := by |
ext
dsimp [toMatrix, single, Equiv.swap_apply_def, Equiv.toPEquiv, one_apply]
split_ifs <;> simp_all
|
import Mathlib.Data.List.Join
#align_import data.list.permutation from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
-- Make sure we don't import algebra
assert_not_exists Monoid
open Nat
variable {α β : Type*}
namespace List
theorem permutationsAux2_fst (t : α) (ts : List α) (r : L... | Mathlib/Data/List/Permutation.lean | 149 | 164 | theorem mem_permutationsAux2 {t : α} {ts : List α} {ys : List α} {l l' : List α} :
l' ∈ (permutationsAux2 t ts [] ys (l ++ ·)).2 ↔
∃ l₁ l₂, l₂ ≠ [] ∧ ys = l₁ ++ l₂ ∧ l' = l ++ l₁ ++ t :: l₂ ++ ts := by |
induction' ys with y ys ih generalizing l
· simp (config := { contextual := true })
rw [permutationsAux2_snd_cons,
show (fun x : List α => l ++ y :: x) = (l ++ [y] ++ ·) by funext _; simp, mem_cons, ih]
constructor
· rintro (rfl | ⟨l₁, l₂, l0, rfl, rfl⟩)
· exact ⟨[], y :: ys, by simp⟩
· exact ⟨y ... |
import Mathlib.Init.Function
import Mathlib.Logic.Function.Basic
#align_import data.sigma.basic from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358ef64"
open Function
section Sigma
variable {α α₁ α₂ : Type*} {β : α → Type*} {β₁ : α₁ → Type*} {β₂ : α₂ → Type*}
namespace Sigma
instance inst... | Mathlib/Data/Sigma/Basic.lean | 70 | 71 | theorem ext_iff {x₀ x₁ : Sigma β} : x₀ = x₁ ↔ x₀.1 = x₁.1 ∧ HEq x₀.2 x₁.2 := by |
cases x₀; cases x₁; exact Sigma.mk.inj_iff
|
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
#align_import analysis.special_functions.pow.real from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real ComplexConjugate
open Finset Set
namespace Real
variable {x y z... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 115 | 117 | theorem rpow_def_of_nonpos {x : ℝ} (hx : x ≤ 0) (y : ℝ) :
x ^ y = if x = 0 then if y = 0 then 1 else 0 else exp (log x * y) * cos (y * π) := by |
split_ifs with h <;> simp [rpow_def, *]; exact rpow_def_of_neg (lt_of_le_of_ne hx h) _
|
import Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.measure.haar.normed_space from "leanprover-community/mathlib"@"b84aee748341da06a6d78491367e2c0e9f15e8a5"
noncomputable sect... | Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean | 105 | 107 | theorem integral_comp_inv_smul_of_nonneg (f : E → F) {R : ℝ} (hR : 0 ≤ R) :
∫ x, f (R⁻¹ • x) ∂μ = R ^ finrank ℝ E • ∫ x, f x ∂μ := by |
rw [integral_comp_inv_smul μ f R, abs_of_nonneg (pow_nonneg hR _)]
|
import Mathlib.Probability.Notation
import Mathlib.Probability.Integration
import Mathlib.MeasureTheory.Function.L2Space
#align_import probability.variance from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open MeasureTheory Filter Finset
noncomputable section
open scoped MeasureThe... | Mathlib/Probability/Variance.lean | 240 | 257 | theorem evariance_def' [@IsProbabilityMeasure Ω _ ℙ] {X : Ω → ℝ} (hX : AEStronglyMeasurable X ℙ) :
eVar[X] = (∫⁻ ω, (‖X ω‖₊ ^ 2 :)) - ENNReal.ofReal (𝔼[X] ^ 2) := by |
by_cases hℒ : Memℒp X 2
· rw [← hℒ.ofReal_variance_eq, variance_def' hℒ, ENNReal.ofReal_sub _ (sq_nonneg _)]
congr
rw [lintegral_coe_eq_integral]
· congr 2 with ω
simp only [Pi.pow_apply, NNReal.coe_pow, coe_nnnorm, Real.norm_eq_abs, Even.pow_abs even_two]
· exact hℒ.abs.integrable_sq
· sym... |
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Limits.Shapes.CommSq
import Mathlib.CategoryTheory.Limits.Shapes.StrictInitial
import Mathlib.CategoryTheory.Limits.FunctorCategory
import Mathlib.CategoryTheory.Limits.Constructions.Fin... | Mathlib/CategoryTheory/Limits/VanKampen.lean | 75 | 80 | theorem mapPair_equifibered {F F' : Discrete WalkingPair ⥤ C} (α : F ⟶ F') :
NatTrans.Equifibered α := by |
rintro ⟨⟨⟩⟩ ⟨j⟩ ⟨⟨rfl : _ = j⟩⟩
all_goals
dsimp; simp only [Discrete.functor_map_id]
exact IsPullback.of_horiz_isIso ⟨by simp only [Category.comp_id, Category.id_comp]⟩
|
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analysis.inner_product_space.calculus from "leanprover-community/mathlib"@"f9dd3204df14a0749cd456fac1e6849dfe7d2b88"
noncomputable section
open RCLike Real ... | Mathlib/Analysis/InnerProductSpace/Calculus.lean | 151 | 153 | theorem contDiff_norm_sq : ContDiff ℝ n fun x : E => ‖x‖ ^ 2 := by |
convert (reCLM : 𝕜 →L[ℝ] ℝ).contDiff.comp ((contDiff_id (E := E)).inner 𝕜 (contDiff_id (E := E)))
exact (inner_self_eq_norm_sq _).symm
|
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.TryThis
import Mathlib.Util.AtomM
set_option autoImplicit true
namespace Mathlib.Tactic.Abel
open Lean Elab Meta Tactic Qq
initialize registerTraceClass `abel
initialize registerTraceClass `abel.detail
structure Context where
α : Expr
univ :... | Mathlib/Tactic/Abel.lean | 238 | 238 | theorem term_atomg {α} [AddCommGroup α] (x : α) : x = termg 1 x 0 := by | simp [termg]
|
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Data.ZMod.Basic
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Chebyshev
import Mathlib.RingTheory.Ideal.LocalRing
#align_import ring_theory.polynomial.dickson from "leanprover-community/mathlib"@"70fd9563a21e7b963887c936... | Mathlib/RingTheory/Polynomial/Dickson.lean | 265 | 269 | theorem dickson_one_one_charP (p : ℕ) [Fact p.Prime] [CharP R p] : dickson 1 (1 : R) p = X ^ p := by |
have h : (1 : R) = ZMod.castHom (dvd_refl p) R 1 := by
simp only [ZMod.castHom_apply, ZMod.cast_one']
rw [h, ← map_dickson (ZMod.castHom (dvd_refl p) R), dickson_one_one_zmod_p, Polynomial.map_pow,
map_X]
|
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 | 414 | 417 | theorem isLittleO_pow_log_id_atTop {n : ℕ} : (fun x => log x ^ n) =o[atTop] id := by |
rw [Asymptotics.isLittleO_iff_tendsto']
· simpa using tendsto_pow_log_div_mul_add_atTop 1 0 n one_ne_zero
filter_upwards [eventually_ne_atTop (0 : ℝ)] with x h₁ h₂ using (h₁ h₂).elim
|
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