Context stringlengths 57 92.3k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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import Mathlib.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]
... |
import Mathlib.RingTheory.WittVector.Basic
import Mathlib.RingTheory.WittVector.IsPoly
#align_import ring_theory.witt_vector.init_tail from "leanprover-community/mathlib"@"0798037604b2d91748f9b43925fb7570a5f3256c"
variable {p : β} [hp : Fact p.Prime] (n : β) {R : Type*} [CommRing R]
-- type as `\bbW`
local notat... | Mathlib/RingTheory/WittVector/InitTail.lean | 213 | 214 | theorem init_sub (x y : π R) (n : β) : init n (x - y) = init n (init n x - init n y) := by |
init_ring using wittSub_vars
|
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Iterate
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.Algebra.InfiniteSum.Real
#align_import analysis.specific_limits.basic from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
n... | Mathlib/Analysis/SpecificLimits/Basic.lean | 410 | 413 | theorem edist_le_of_edist_le_geometric_of_tendsto {a : Ξ±} (ha : Tendsto f atTop (π a)) (n : β) :
edist (f n) a β€ C * r ^ n / (1 - r) := by |
convert edist_le_tsum_of_edist_le_of_tendsto _ hu ha _
simp only [pow_add, ENNReal.tsum_mul_left, ENNReal.tsum_geometric, div_eq_mul_inv, mul_assoc]
|
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
|
import Mathlib.CategoryTheory.Functor.Flat
import Mathlib.CategoryTheory.Sites.Sheaf
import Mathlib.Tactic.ApplyFun
#align_import category_theory.sites.cover_preserving from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
universe w vβ vβ vβ uβ uβ uβ
noncomputable section
open CategoryT... | Mathlib/CategoryTheory/Sites/CoverPreserving.lean | 126 | 158 | theorem compatiblePreservingOfFlat {C : Type uβ} [Category.{vβ} C] {D : Type uβ} [Category.{vβ} D]
(K : GrothendieckTopology D) (G : C β₯€ D) [RepresentablyFlat G] : CompatiblePreserving K G := by |
constructor
intro β± Z T x hx Yβ Yβ X fβ fβ gβ gβ hgβ hgβ e
-- First, `fβ` and `fβ` form a cone over `cospan gβ gβ β u`.
let c : Cone (cospan gβ gβ β G) :=
(Cones.postcompose (diagramIsoCospan (cospan gβ gβ β G)).inv).obj (PullbackCone.mk fβ fβ e)
/-
This can then be viewed as a cospan of structured a... |
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Tactic.Positivity.Core
import Mathlib.Algebra.Ring.NegOnePow
#align_import analysis.special_functions.trigonometric.basic from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
noncomputable section
open scoped Classical
open Top... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 1,374 | 1,375 | theorem exp_mul_I_antiperiodic : Function.Antiperiodic (fun x => exp (x * I)) Ο := by |
simpa only [mul_inv_cancel_rightβ I_ne_zero] using exp_antiperiodic.mul_const I_ne_zero
|
import Mathlib.Order.Filter.SmallSets
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Compactness.Compact
import Mathlib.Topology.NhdsSet
import Mathlib.Algebra.Group.Defs
#align_import topology.uniform_space.basic from "leanprover-community/mathlib"@"195fcd60ff2bfe392543bceb0ec2adcdb472db4c"
open Set F... | Mathlib/Topology/UniformSpace/Basic.lean | 512 | 516 | theorem Filter.Tendsto.uniformity_trans {l : Filter Ξ²} {fβ fβ fβ : Ξ² β Ξ±}
(hββ : Tendsto (fun x => (fβ x, fβ x)) l (π€ Ξ±))
(hββ : Tendsto (fun x => (fβ x, fβ x)) l (π€ Ξ±)) : Tendsto (fun x => (fβ x, fβ x)) l (π€ Ξ±) := by |
refine le_trans (le_lift'.2 fun s hs => mem_map.2 ?_) comp_le_uniformity
filter_upwards [mem_map.1 (hββ hs), mem_map.1 (hββ hs)] with x hxββ hxββ using β¨_, hxββ, hxβββ©
|
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.SpecificLimits.Basic
#align_import analysis.calculus.tangent_cone from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
variable (π : Type*) [NontriviallyNormedField π]
open Filter... | Mathlib/Analysis/Calculus/TangentCone.lean | 98 | 100 | theorem tangentCone_mono (h : s β t) : tangentConeAt π s x β tangentConeAt π t x := by |
rintro y β¨c, d, ds, ctop, climβ©
exact β¨c, d, mem_of_superset ds fun n hn => h hn, ctop, climβ©
|
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©
|
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 {Ξ± Ξ² Ξ³ Ξ΄ : Type*} {s sβ sβ : Set Ξ±} {t tβ tβ : Set Ξ²} {a : Ξ±} {b : Ξ²}
theorem Subsingleton.pro... | Mathlib/Data/Set/Prod.lean | 771 | 772 | theorem disjoint_univ_pi : Disjoint (pi univ tβ) (pi univ tβ) β β i, Disjoint (tβ i) (tβ i) := by |
simp only [disjoint_iff_inter_eq_empty, β pi_inter_distrib, univ_pi_eq_empty_iff]
|
import Mathlib.LinearAlgebra.FinsuppVectorSpace
import Mathlib.LinearAlgebra.Matrix.Basis
import Mathlib.LinearAlgebra.Matrix.Nondegenerate
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.ToLinearEquiv
import Mathlib.LinearAlgebra.SesquilinearForm
import Mathlib.LinearAlgebra... | Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean | 456 | 461 | theorem LinearMap.toMatrixβ_mul_basis_toMatrix (cβ : Basis n' R Mβ) (cβ : Basis m' R Mβ)
(B : Mβ ββ[R] Mβ ββ[R] R) :
(bβ.toMatrix cβ)α΅ * LinearMap.toMatrixβ bβ bβ B * bβ.toMatrix cβ =
LinearMap.toMatrixβ cβ cβ B := by |
simp_rw [β LinearMap.toMatrix_id_eq_basis_toMatrix]
rw [β LinearMap.toMatrixβ_complββ, LinearMap.complββ_id_id]
|
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]
|
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... |
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
|
import Mathlib.Analysis.Convex.Basic
import Mathlib.Order.Filter.Extr
import Mathlib.Tactic.GCongr
#align_import analysis.convex.function from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open scoped Classical
open LinearMap Set Convex Pointwise
variable {π E F Ξ± Ξ² ΞΉ : Type*}
secti... | Mathlib/Analysis/Convex/Function.lean | 565 | 573 | theorem ConvexOn.openSegment_subset_strict_epigraph (hf : ConvexOn π s f) (p q : E Γ Ξ²)
(hp : p.1 β s β§ f p.1 < p.2) (hq : q.1 β s β§ f q.1 β€ q.2) :
openSegment π p q β { p : E Γ Ξ² | p.1 β s β§ f p.1 < p.2 } := by |
rintro _ β¨a, b, ha, hb, hab, rflβ©
refine β¨hf.1 hp.1 hq.1 ha.le hb.le hab, ?_β©
calc
f (a β’ p.1 + b β’ q.1) β€ a β’ f p.1 + b β’ f q.1 := hf.2 hp.1 hq.1 ha.le hb.le hab
_ < a β’ p.2 + b β’ q.2 := add_lt_add_of_lt_of_le
(smul_lt_smul_of_pos_left hp.2 ha) (smul_le_smul_of_nonneg_left hq.2 hb.le)
|
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Order.Interval.Set.OrderIso
import Mathlib.Data.Set.Lattice
#align_import order.upper_lower.basic from "leanprover-community/mathlib"@"c0c52abb75074ed8b73a948341f50521fbf43b4c"
open Function OrderDual Set
variable {Ξ± Ξ²... | Mathlib/Order/UpperLower/Basic.lean | 748 | 748 | theorem coe_eq_empty : (s : Set Ξ±) = β
β s = β₯ := by | simp [SetLike.ext'_iff]
|
import Mathlib.Topology.Separation
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.UniformSpace.Cauchy
#align_import topology.uniform_space.uniform_convergence from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062a169c9"
noncomputable section
open Topology Uniformity Filter S... | Mathlib/Topology/UniformSpace/UniformConvergence.lean | 842 | 847 | theorem continuousAt_of_locally_uniform_approx_of_continuousAt
(L : β u β π€ Ξ², β t β π x, β F, ContinuousAt F x β§ β y β t, (f y, F y) β u) :
ContinuousAt f x := by |
rw [β continuousWithinAt_univ]
apply continuousWithinAt_of_locally_uniform_approx_of_continuousWithinAt (mem_univ _) _
simpa only [exists_prop, nhdsWithin_univ, continuousWithinAt_univ] using L
|
import Mathlib.AlgebraicGeometry.GammaSpecAdjunction
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.RingTheory.Localization.InvSubmonoid
#align_import algebraic_geometry.AffineScheme from "leanprover-community/mathlib"@"88474d1b5af6d37c2ab728b757771bced7f5194c"... | Mathlib/AlgebraicGeometry/AffineScheme.lean | 567 | 593 | theorem basicOpen_union_eq_self_iff (s : Set (X.presheaf.obj <| op U)) :
β¨ f : s, X.basicOpen (f : X.presheaf.obj <| op U) = U β Ideal.span s = β€ := by |
trans β i : s, (PrimeSpectrum.basicOpen i.1).1 = Set.univ
Β· trans
hU.fromSpec.1.base β»ΒΉ' (β¨ f : s, X.basicOpen (f : X.presheaf.obj <| op U)).1 =
hU.fromSpec.1.base β»ΒΉ' U.1
Β· refine β¨fun h => by rw [h], ?_β©
intro h
apply_fun Set.image hU.fromSpec.1.base at h
rw [Set.image_preimag... |
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 | 689 | 693 | theorem one_le_hausdorffMeasure_zero_of_nonempty {s : Set X} (h : s.Nonempty) : 1 β€ ΞΌH[0] s := by |
rcases h with β¨x, hxβ©
calc
(1 : ββ₯0β) = ΞΌH[0] ({x} : Set X) := (hausdorffMeasure_zero_singleton x).symm
_ β€ ΞΌH[0] s := measure_mono (singleton_subset_iff.2 hx)
|
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Order.Ring.Int
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.Nat.Cast.Order
#align_import algebra.order.ring.abs from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
#align_import data.nat.parity from "leanpr... | Mathlib/Algebra/Order/Ring/Abs.lean | 179 | 182 | theorem abs_sub_sq (a b : Ξ±) : |a - b| * |a - b| = a * a + b * b - (1 + 1) * a * b := by |
rw [abs_mul_abs_self]
simp only [mul_add, add_comm, add_left_comm, mul_comm, sub_eq_add_neg, mul_one, mul_neg,
neg_add_rev, neg_neg, add_assoc]
|
import Mathlib.LinearAlgebra.Dimension.Finrank
import Mathlib.LinearAlgebra.InvariantBasisNumber
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v w w'
variable {R : Type u} {M : Type v} [Ring R] [AddCommGroup... | Mathlib/LinearAlgebra/Dimension/StrongRankCondition.lean | 369 | 372 | theorem rank_span_set {s : Set M} (hs : LinearIndependent R (fun x => x : s β M)) :
Module.rank R β(span R s) = #s := by |
rw [β @setOf_mem_eq _ s, β Subtype.range_coe_subtype]
exact rank_span hs
|
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Sym.Basic
import Mathlib.Data.Sym.Sym2.Init
import Mathlib.Data.SetLike.Basic
#align_import data.sym.sym2 from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
assert_not_exists MonoidWithZero
open Finset Function Sym
universe u
variab... | Mathlib/Data/Sym/Sym2.lean | 377 | 378 | theorem other_spec {a : Ξ±} {z : Sym2 Ξ±} (h : a β z) : s(a, Mem.other h) = z := by |
erw [β Classical.choose_spec h]
|
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 | 335 | 349 | theorem condexp_condexp_of_le {mβ mβ m0 : MeasurableSpace Ξ±} {ΞΌ : Measure Ξ±} (hmββ : mβ β€ mβ)
(hmβ : mβ β€ m0) [SigmaFinite (ΞΌ.trim hmβ)] : ΞΌ[ΞΌ[f|mβ]|mβ] =α΅[ΞΌ] ΞΌ[f|mβ] := by |
by_cases hΞΌmβ : SigmaFinite (ΞΌ.trim (hmββ.trans hmβ))
swap; Β· simp_rw [condexp_of_not_sigmaFinite (hmββ.trans hmβ) hΞΌmβ]; rfl
haveI : SigmaFinite (ΞΌ.trim (hmββ.trans hmβ)) := hΞΌmβ
by_cases hf : Integrable f ΞΌ
swap; Β· simp_rw [condexp_undef hf, condexp_zero]; rfl
refine ae_eq_of_forall_setIntegral_eq_of_sig... |
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 | 1,817 | 1,820 | theorem ContDiffWithinAt.div [CompleteSpace π] {f g : E β π} {n} (hf : ContDiffWithinAt π n f s x)
(hg : ContDiffWithinAt π n g s x) (hx : g x β 0) :
ContDiffWithinAt π n (fun x => f x / g x) s x := by |
simpa only [div_eq_mul_inv] using hf.mul (hg.inv hx)
|
import Mathlib.Data.Finset.Finsupp
import Mathlib.Data.Finsupp.Order
import Mathlib.Order.Interval.Finset.Basic
#align_import data.finsupp.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
noncomputable section
open Finset Finsupp Function
open scoped Classical
open Pointwis... | Mathlib/Data/Finsupp/Interval.lean | 145 | 147 | theorem card_Iic : (Iic f).card = β i β f.support, (Iic (f i)).card := by |
classical simp_rw [Iic_eq_Icc, card_Icc, Finsupp.bot_eq_zero, support_zero, empty_union,
zero_apply, bot_eq_zero]
|
import Mathlib.Analysis.Complex.Circle
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
#align_import analysis.complex.isometry from "leanprover-community/mathlib"@"ae690b0c236e488a0043f6faa8ce3546e7f2f9c5"
noncomputable section
open Complex
open ComplexConjugate
... | Mathlib/Analysis/Complex/Isometry.lean | 167 | 169 | theorem det_rotation (a : circle) : LinearMap.det ((rotation a).toLinearEquiv : β ββ[β] β) = 1 := by |
rw [β LinearMap.det_toMatrix basisOneI, toMatrix_rotation, Matrix.det_fin_two]
simp [β normSq_apply]
|
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... |
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... |
import Mathlib.CategoryTheory.CommSq
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.CategoryTheory.Limits.Shapes.Biproducts
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts
import Mathlib.CategoryTheory.Limits.Constructions.ZeroO... | Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean | 784 | 788 | theorem inl_snd' {b : BinaryBicone X Y} (h : b.IsBilimit) :
IsPushout b.inl (0 : X βΆ 0) b.snd (0 : 0 βΆ Y) := by |
apply flip
refine of_right ?_ (by simp) (of_isBilimit h)
simp
|
import Mathlib.Algebra.Divisibility.Basic
import Mathlib.Algebra.Group.Units
#align_import algebra.divisibility.units from "leanprover-community/mathlib"@"e574b1a4e891376b0ef974b926da39e05da12a06"
variable {Ξ± : Type*}
namespace Units
end IsUnit
section CommMonoid
variable [CommMonoid Ξ±]
theorem isUnit_iff_dvd... | Mathlib/Algebra/Divisibility/Units.lean | 208 | 209 | theorem isRelPrime_mul_unit_right_left : IsRelPrime (y * x) z β IsRelPrime y z := by |
rw [mul_comm, isRelPrime_mul_unit_left_left hu]
|
import Mathlib.Algebra.Category.ModuleCat.Free
import Mathlib.Topology.Category.Profinite.CofilteredLimit
import Mathlib.Topology.Category.Profinite.Product
import Mathlib.Topology.LocallyConstant.Algebra
import Mathlib.Init.Data.Bool.Lemmas
universe u
namespace Profinite
namespace NobelingProof
variable {I : Ty... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 1,363 | 1,385 | theorem union_succ : GoodProducts C = GoodProducts (Ο C (ord I Β· < o)) βͺ MaxProducts C ho := by |
ext l
simp only [GoodProducts, MaxProducts, Set.mem_union, Set.mem_setOf_eq]
refine β¨fun h β¦ ?_, fun h β¦ ?_β©
Β· by_cases hh : term I ho β l.val
Β· exact Or.inr β¨h, hhβ©
Β· left
intro he
apply h
have h' := Products.prop_of_isGood_of_contained C _ h hsC
simp only [Order.lt_succ_iff] a... |
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]
|
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2
#align_import measure_theory.function.conditional_expectation.condexp_L1 from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
noncomputable section
open TopologicalSpace MeasureTheory.Lp Filter ContinuousLinearMap
o... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL1.lean | 360 | 364 | theorem condexpInd_nonneg {E} [NormedLatticeAddCommGroup E] [NormedSpace β E] [OrderedSMul β E]
(hs : MeasurableSet s) (hΞΌs : ΞΌ s β β) (x : E) (hx : 0 β€ x) : 0 β€ condexpInd E hm ΞΌ s x := by |
rw [β coeFn_le]
refine EventuallyLE.trans_eq ?_ (condexpInd_ae_eq_condexpIndSMul hm hs hΞΌs x).symm
exact (coeFn_zero E 1 ΞΌ).trans_le (condexpIndSMul_nonneg hs hΞΌs x hx)
|
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.Tactic.NormNum.Inv
import Mathlib.Tactic.NormNum.Pow
import Mathlib.Util.AtomM
set_option autoImplicit true
namespace Mathlib.Tactic
namespace Ring
open Mathlib.Meta Qq NormNum Lean.Meta AtomM
open Lean (MetaM Expr mkRawNatLit)
def instCommSemiringNat : CommSe... | Mathlib/Tactic/Ring/Basic.lean | 586 | 586 | theorem pow_prod_atom (a : R) (b) : a ^ b = (a + 0) ^ b * (nat_lit 1).rawCast := by | simp
|
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©
|
import Mathlib.Analysis.Calculus.SmoothSeries
import Mathlib.Analysis.Calculus.BumpFunction.InnerProduct
import Mathlib.Analysis.Convolution
import Mathlib.Analysis.InnerProductSpace.EuclideanDist
import Mathlib.Data.Set.Pointwise.Support
import Mathlib.MeasureTheory.Measure.Haar.NormedSpace
import Mathlib.MeasureTheo... | Mathlib/Analysis/Calculus/BumpFunction/FiniteDimension.lean | 466 | 491 | theorem y_smooth : ContDiffOn β β€ (uncurry y) (Ioo (0 : β) 1 ΓΛ’ (univ : Set E)) := by |
have hs : IsOpen (Ioo (0 : β) (1 : β)) := isOpen_Ioo
have hk : IsCompact (closedBall (0 : E) 1) := ProperSpace.isCompact_closedBall _ _
refine contDiffOn_convolution_left_with_param (lsmul β β) hs hk ?_ ?_ ?_
Β· rintro p x hp hx
simp only [w, mul_inv_rev, Algebra.id.smul_eq_mul, mul_eq_zero, inv_eq_zero]
... |
import Mathlib.Analysis.Seminorm
import Mathlib.Topology.Algebra.Equicontinuity
import Mathlib.Topology.MetricSpace.Equicontinuity
import Mathlib.Topology.Algebra.FilterBasis
import Mathlib.Topology.Algebra.Module.LocallyConvex
#align_import analysis.locally_convex.with_seminorms from "leanprover-community/mathlib"@"... | Mathlib/Analysis/LocallyConvex/WithSeminorms.lean | 923 | 926 | theorem Inducing.withSeminorms [hΞΉ : Nonempty ΞΉ] {q : SeminormFamily πβ F ΞΉ} (hq : WithSeminorms q)
[TopologicalSpace E] {f : E βββ[Οββ] F} (hf : Inducing f) : WithSeminorms (q.comp f) := by |
rw [hf.induced]
exact f.withSeminorms_induced hq
|
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]
|
import Mathlib.Data.Finset.Attr
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Logic.Equiv.Set
import Mathlib.Order.Directed
import Mathlib.Order.Interval.Set.Basic
#align_import data.finset.basic from "leanprover-community/mathlib"@"442a83d738cb208d3600056c489be16900ba701d"
-- Assert that we define `Finset... | Mathlib/Data/Finset/Basic.lean | 786 | 787 | theorem subset_singleton_iff {s : Finset Ξ±} {a : Ξ±} : s β {a} β s = β
β¨ s = {a} := by |
rw [β coe_subset, coe_singleton, Set.subset_singleton_iff_eq, coe_eq_empty, coe_eq_singleton]
|
import Mathlib.Computability.Primrec
import Mathlib.Data.Nat.PSub
import Mathlib.Data.PFun
#align_import computability.partrec from "leanprover-community/mathlib"@"9ee02c6c2208fd7795005aa394107c0374906cca"
open Encodable Denumerable Part
attribute [-simp] not_forall
namespace Nat
def rfind (p : β β. Bool) : Pa... | Mathlib/Computability/Partrec.lean | 463 | 465 | theorem map {f : Ξ± β. Ξ²} {g : Ξ± β Ξ² β Ο} (hf : Partrec f) (hg : Computableβ g) :
Partrec fun a => (f a).map (g a) := by |
simpa [bind_some_eq_map] using @Partrec.bind _ _ _ _ _ _ _ (fun a => Part.some β (g a)) hf hg
|
import Mathlib.Data.Real.Sqrt
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.ContinuousLinearMap
import Mathlib.Analysis.NormedSpace.Basic
#align_import data.is_R_or_C.basic from "leanprover-community/mathlib"@"baa88307f3e699fa7054ef04ec79fa4f056169cb"
section
local notation "οΏ½... | Mathlib/Analysis/RCLike/Basic.lean | 598 | 601 | theorem inv_I : (I : K)β»ΒΉ = -I := by |
by_cases h : (I : K) = 0
Β· simp [h]
Β· field_simp [I_mul_I_of_nonzero h]
|
import Mathlib.Analysis.Complex.Asymptotics
import Mathlib.Analysis.SpecificLimits.Normed
#align_import analysis.special_functions.exp from "leanprover-community/mathlib"@"ba5ff5ad5d120fb0ef094ad2994967e9bfaf5112"
noncomputable section
open Finset Filter Metric Asymptotics Set Function Bornology
open scoped Cla... | Mathlib/Analysis/SpecialFunctions/Exp.lean | 413 | 417 | theorem isTheta_exp_comp_exp_comp {f g : Ξ± β β} :
((fun x => exp (f x)) =Ξ[l] fun x => exp (g x)) β
IsBoundedUnder (Β· β€ Β·) l fun x => |f x - g x| := by |
simp only [isBoundedUnder_le_abs, β isBoundedUnder_le_neg, neg_sub, IsTheta,
isBigO_exp_comp_exp_comp, Pi.sub_def]
|
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
|
import Mathlib.Analysis.LocallyConvex.Basic
#align_import analysis.locally_convex.balanced_core_hull from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Pointwise Topology Filter
variable {π E ΞΉ : Type*}
section balancedHull
section SeminormedRing
variable [SeminormedRing ... | Mathlib/Analysis/LocallyConvex/BalancedCoreHull.lean | 163 | 165 | theorem balancedCoreAux_empty : balancedCoreAux π (β
: Set E) = β
:= by |
simp_rw [balancedCoreAux, iInterβ_eq_empty_iff, smul_set_empty]
exact fun _ => β¨1, norm_one.ge, not_mem_empty _β©
|
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]
|
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.MvPolynomial.Degrees
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Data.Finsupp.Fin
import Mathlib.Logic.Equiv.Fin
#align_import data.mv_polynomial.equiv from "leanprover-community/mathlib"@"2f5b500... | Mathlib/Algebra/MvPolynomial/Equiv.lean | 505 | 515 | theorem degree_finSuccEquiv {f : MvPolynomial (Fin (n + 1)) R} (h : f β 0) :
(finSuccEquiv R n f).degree = degreeOf 0 f := by |
-- TODO: these should be lemmas
have hβ : β {Ξ± Ξ² : Type _} (f : Ξ± β Ξ²), (fun x => x) β f = f := fun f => rfl
have hβ : β {Ξ± Ξ² : Type _} (f : Ξ± β Ξ²), f β (fun x => x) = f := fun f => rfl
have hβ : WithBot.some = Nat.cast := rfl
have h' : ((finSuccEquiv R n f).support.sup fun x => x) = degreeOf 0 f := by
... |
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 | 1,585 | 1,586 | theorem map_comp_map (g : Ξ² β Ξ³) (f : Ξ± β Ξ²) : map g β map f = map (g β f) := by |
ext l; rw [comp_map, Function.comp_apply]
|
import Mathlib.Order.Heyting.Basic
#align_import order.boolean_algebra from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2f4"
open Function OrderDual
universe u v
variable {Ξ± : Type u} {Ξ² : Type*} {w x y z : Ξ±}
class GeneralizedBooleanAlgebra (Ξ± : Type u) extends DistribLattice Ξ±, S... | Mathlib/Order/BooleanAlgebra.lean | 586 | 586 | theorem compl_sup_eq_top : xαΆ β x = β€ := by | rw [sup_comm, sup_compl_eq_top]
|
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Finset.NoncommProd
import Mathlib.Data.Fintype.Perm
import Mathlib.Data.Int.ModEq
import Mathlib.GroupTheory.Perm.List
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Fintype
import Mathlib.GroupTheory.Perm.Cycle.Basic
#align_import grou... | Mathlib/GroupTheory/Perm/Cycle/Factors.lean | 107 | 109 | theorem cycleOf_apply_apply_pow_self (f : Perm Ξ±) (x : Ξ±) (k : β) :
cycleOf f x ((f ^ k) x) = (f ^ (k + 1) : Perm Ξ±) x := by |
convert cycleOf_apply_apply_zpow_self f x k using 1
|
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
|
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Ring.Nat
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.RingTheory.Fintype
import Mathlib.Tactic.IntervalCases
#align_import number_the... | Mathlib/NumberTheory/LucasLehmer.lean | 491 | 493 | theorem mersenne_coe_X (p : β) : (mersenne p : X (q p)) = 0 := by |
ext <;> simp [mersenne, q, ZMod.natCast_zmod_eq_zero_iff_dvd, -pow_pos]
apply Nat.minFac_dvd
|
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])... |
import Mathlib.MeasureTheory.Constructions.Prod.Integral
import Mathlib.MeasureTheory.Integral.CircleIntegral
#align_import measure_theory.integral.torus_integral from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
variable {n : β}
variable {E : Type*} [NormedAddCommGroup E]
noncomputa... | Mathlib/MeasureTheory/Integral/TorusIntegral.lean | 170 | 173 | theorem torusIntegral_add (hf : TorusIntegrable f c R) (hg : TorusIntegrable g c R) :
(β― x in T(c, R), f x + g x) = (β― x in T(c, R), f x) + β― x in T(c, R), g x := by |
simpa only [torusIntegral, smul_add, Pi.add_apply] using
integral_add hf.function_integrable hg.function_integrable
|
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
|
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.RingTheory.RootsOfUnity.Complex
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.RingTheory.RootsOfUnity.Basic
import Mathlib.FieldTh... | Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | 387 | 393 | theorem prod_cyclotomic_eq_geom_sum {n : β} (h : 0 < n) (R) [CommRing R] :
β i β n.divisors.erase 1, cyclotomic i R = β i β Finset.range n, X ^ i := by |
suffices (β i β n.divisors.erase 1, cyclotomic i β€) = β i β Finset.range n, X ^ i by
simpa only [Polynomial.map_prod, map_cyclotomic_int, Polynomial.map_sum, Polynomial.map_pow,
Polynomial.map_X] using congr_arg (map (Int.castRingHom R)) this
rw [β mul_left_inj' (cyclotomic_ne_zero 1 β€), prod_erase_mul _... |
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.Measure.Count
import Mathlib.Topology.IndicatorConstPointwise
import Mathlib.MeasureTheory.Constructions.BorelSpace.Real
#align_import m... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 167 | 168 | theorem lintegral_const_lt_top [IsFiniteMeasure ΞΌ] {c : ββ₯0β} (hc : c β β) : β«β» _, c βΞΌ < β := by |
simpa only [Measure.restrict_univ] using set_lintegral_const_lt_top (univ : Set Ξ±) hc
|
import Mathlib.LinearAlgebra.TensorProduct.Tower
import Mathlib.Algebra.DirectSum.Module
#align_import linear_algebra.direct_sum.tensor_product from "leanprover-community/mathlib"@"9b9d125b7be0930f564a68f1d73ace10cf46064d"
suppress_compilation
universe u vβ vβ wβ wβ' wβ wβ'
section Ring
namespace TensorProduct
... | Mathlib/LinearAlgebra/DirectSum/TensorProduct.lean | 189 | 192 | theorem directSumRight_symm_lof_tmul (x : Mβ') (i : ΞΉβ) (y : Mβ i) :
(directSumRight R Mβ' Mβ).symm (DirectSum.lof R _ _ i (x ββ[R] y)) =
x ββ[R] DirectSum.lof R _ _ i y := by |
rw [LinearEquiv.symm_apply_eq, directSumRight_tmul_lof]
|
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.PSub
import Mathlib.Data.Nat.Size
import Mathlib.Data.Num.Bitwise
#align_import data.num.lemmas from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
set_opti... | Mathlib/Data/Num/Lemmas.lean | 238 | 238 | theorem ofNat'_zero : Num.ofNat' 0 = 0 := by | simp [Num.ofNat']
|
import Mathlib.CategoryTheory.Limits.IsLimit
import Mathlib.CategoryTheory.Category.ULift
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.Logic.Equiv.Basic
#align_import category_theory.limits.has_limits from "leanprover-community/mathlib"@"2738d2ca56cbc63be80c3bd48e9ed90ad94e947d"
noncomputable sec... | Mathlib/CategoryTheory/Limits/HasLimits.lean | 1,046 | 1,049 | theorem colimit.ΞΉ_post (j : J) :
colimit.ΞΉ (F β G) j β« colimit.post F G = G.map (colimit.ΞΉ F j) := by |
erw [IsColimit.fac]
rfl
|
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.Order.Filter.Germ
import Mathlib.Topology.ContinuousFunction.Algebra
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
#align_import measure_theory.function.ae_eq_fun from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a21598... | Mathlib/MeasureTheory/Function/AEEqFun.lean | 178 | 180 | theorem coeFn_mk (f : Ξ± β Ξ²) (hf) : (mk f hf : Ξ± ββ[ΞΌ] Ξ²) =α΅[ΞΌ] f := by |
apply (AEStronglyMeasurable.ae_eq_mk _).symm.trans
exact @Quotient.mk_out' _ (ΞΌ.aeEqSetoid Ξ²) (β¨f, hfβ© : { f // AEStronglyMeasurable f ΞΌ })
|
import Mathlib.Algebra.Category.Ring.FilteredColimits
import Mathlib.Geometry.RingedSpace.SheafedSpace
import Mathlib.Topology.Sheaves.Stalks
import Mathlib.Algebra.Category.Ring.Colimits
import Mathlib.Algebra.Category.Ring.Limits
#align_import algebraic_geometry.ringed_space from "leanprover-community/mathlib"@"5dc... | Mathlib/Geometry/RingedSpace/Basic.lean | 226 | 232 | theorem basicOpen_of_isUnit {U : Opens X} {f : X.presheaf.obj (op U)} (hf : IsUnit f) :
X.basicOpen f = U := by |
apply le_antisymm
Β· exact X.basicOpen_le f
intro x hx
erw [X.mem_basicOpen f (β¨x, hxβ© : U)]
exact RingHom.isUnit_map _ hf
|
import Mathlib.RingTheory.WittVector.Basic
import Mathlib.RingTheory.WittVector.IsPoly
#align_import ring_theory.witt_vector.init_tail from "leanprover-community/mathlib"@"0798037604b2d91748f9b43925fb7570a5f3256c"
variable {p : β} [hp : Fact p.Prime] (n : β) {R : Type*} [CommRing R]
-- type as `\bbW`
local notat... | Mathlib/RingTheory/WittVector/InitTail.lean | 217 | 218 | theorem init_nsmul (m : β) (x : π R) (n : β) : init n (m β’ x) = init n (m β’ init n x) := by |
init_ring using fun p [Fact (Nat.Prime p)] n => wittNSMul_vars p m n
|
import Mathlib.Algebra.Group.Int
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.CategoryTheory.Shift.Basic
import Mathlib.Data.Set.Subsingleton
#align_import category_theory.graded_object from "leanprover-community/mathlib"@"6876fa15e3158ff3e4a4e2af1fb6e1945c6e8803"
namespace CategoryTheory
o... | Mathlib/CategoryTheory/GradedObject.lean | 167 | 170 | theorem eqToHom_proj {I : Type*} {x x' : GradedObject I C} (h : x = x') (i : I) :
(eqToHom h : x βΆ x') i = eqToHom (Function.funext_iff.mp h i) := by |
subst h
rfl
|
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... |
import Mathlib.Geometry.Manifold.Algebra.Monoid
#align_import geometry.manifold.algebra.lie_group from "leanprover-community/mathlib"@"f9ec187127cc5b381dfcf5f4a22dacca4c20b63d"
noncomputable section
open scoped Manifold
-- See note [Design choices about smooth algebraic structures]
class LieAddGroup {π : Type*... | Mathlib/Geometry/Manifold/Algebra/LieGroup.lean | 342 | 345 | theorem ContMDiffWithinAt.divβ
(hf : ContMDiffWithinAt I' I n f s a) (hg : ContMDiffWithinAt I' I n g s a) (hβ : g a β 0) :
ContMDiffWithinAt I' I n (f / g) s a := by |
simpa [div_eq_mul_inv] using hf.mul (hg.invβ hβ)
|
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Algebra.Subalgebra.Prod
import Mathlib.Algebra.Algebra.Subalgebra.Tower
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.Prod
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
#align_import ring_theory.adjoin.basic fr... | Mathlib/RingTheory/Adjoin/Basic.lean | 381 | 385 | theorem adjoin_union_coe_submodule :
Subalgebra.toSubmodule (adjoin R (s βͺ t)) =
Subalgebra.toSubmodule (adjoin R s) * Subalgebra.toSubmodule (adjoin R t) := by |
rw [adjoin_eq_span, adjoin_eq_span, adjoin_eq_span, span_mul_span]
congr 1 with z; simp [Submonoid.closure_union, Submonoid.mem_sup, Set.mem_mul]
|
import Mathlib.CategoryTheory.Equivalence
#align_import category_theory.adjunction.basic from "leanprover-community/mathlib"@"d101e93197bb5f6ea89bd7ba386b7f7dff1f3903"
namespace CategoryTheory
open Category
-- declare the `v`'s first; see `CategoryTheory.Category` for an explanation
universe vβ vβ vβ uβ uβ uβ
... | Mathlib/CategoryTheory/Adjunction/Basic.lean | 148 | 148 | theorem homEquiv_symm_id (X : D) : (adj.homEquiv _ X).symm (π _) = adj.counit.app X := by | simp
|
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib.Data.Set.Finite
#align_import data.finsupp.defs from "leanprover-community/mathlib"@"842328d9df7e96fd90fc424e115679c15fb23a71"
noncomputable section
open Finset Function
variable {Ξ± Ξ² Ξ³ ΞΉ M M' N P G H R S : Type*}... | Mathlib/Data/Finsupp/Defs.lean | 675 | 678 | theorem erase_of_not_mem_support {f : Ξ± ββ M} {a} (haf : a β f.support) : erase a f = f := by |
ext b; by_cases hab : b = a
Β· rwa [hab, erase_same, eq_comm, β not_mem_support_iff]
Β· rw [erase_ne hab]
|
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
#align_import analysis.special_functions.pow.continuity from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
noncomputable section
open scoped Classical
open Real Topology NNReal ENNReal Filter ComplexConjugate
open Filter Finset... | Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean | 368 | 393 | theorem continuousAt_ofReal_cpow (x : β) (y : β) (h : 0 < y.re β¨ x β 0) :
ContinuousAt (fun p => (p.1 : β) ^ p.2 : β Γ β β β) (x, y) := by |
rcases lt_trichotomy (0 : β) x with (hx | rfl | hx)
Β· -- x > 0 : easy case
have : ContinuousAt (fun p => β¨βp.1, p.2β© : β Γ β β β Γ β) (x, y) :=
continuous_ofReal.continuousAt.prod_map continuousAt_id
refine (continuousAt_cpow (Or.inl ?_)).comp this
rwa [ofReal_re]
Β· -- x = 0 : reduce to continu... |
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.FractionalIdeal.Basic
#align_import ring_theory.fractional_ideal from "leanprover-community/mathlib"@"ed90a7d327c3a5caf65a6faf7e8a0d63c4605df7"
open IsLocalization Pointwise nonZeroDivisors
namespace FractionalIdeal
open Set Submodule
variable... | Mathlib/RingTheory/FractionalIdeal/Operations.lean | 775 | 790 | theorem mk'_mul_coeIdeal_eq_coeIdeal {I J : Ideal Rβ} {x y : Rβ} (hy : y β Rββ°) :
spanSingleton Rββ° (IsLocalization.mk' K x β¨y, hyβ©) * I = (J : FractionalIdeal Rββ° K) β
Ideal.span {x} * I = Ideal.span {y} * J := by |
have :
spanSingleton Rββ° (IsLocalization.mk' _ (1 : Rβ) β¨y, hyβ©) *
spanSingleton Rββ° (algebraMap Rβ K y) =
1 := by
rw [spanSingleton_mul_spanSingleton, mul_comm, β IsLocalization.mk'_eq_mul_mk'_one,
IsLocalization.mk'_self, spanSingleton_one]
let y' : (FractionalIdeal Rββ° K)Λ£ := Units.m... |
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 | 417 | 419 | theorem image_const_sub_Ioi : (fun x => a - x) '' Ioi b = Iio (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.Analysis.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.IsometricSMul
#align_import topology.metric_space.hausdorff_distance from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
noncomputable section
open NNReal ENNReal Topology Set Filter Pointwise Bornology
u... | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 404 | 405 | theorem hausdorffEdist_self_closure : hausdorffEdist s (closure s) = 0 := by |
rw [hausdorffEdist_zero_iff_closure_eq_closure, closure_closure]
|
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Set.Finite
#align_import order.filter.basic from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
set_option autoImplicit true
open Function Set Order
open scoped Classical
universe u v w x y
structure Filter (Ξ± : Type*) where... | Mathlib/Order/Filter/Basic.lean | 2,481 | 2,482 | theorem map_comap_of_mem {f : Filter Ξ²} {m : Ξ± β Ξ²} (hf : range m β f) : (f.comap m).map m = f := by |
rw [map_comap, inf_eq_left.2 (le_principal_iff.2 hf)]
|
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©
|
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Algebra.Group.ULift
#align_import topology.algebra.monoid from "leanprover-community/mathli... | Mathlib/Topology/Algebra/Monoid.lean | 673 | 677 | theorem Filter.tendsto_cocompact_mul_right {a b : M} (ha : a * b = 1) :
Filter.Tendsto (fun x : M => x * a) (Filter.cocompact M) (Filter.cocompact M) := by |
refine Filter.Tendsto.of_tendsto_comp ?_ (Filter.comap_cocompact_le (continuous_mul_right b))
simp only [comp_mul_right, ha, mul_one]
exact Filter.tendsto_id
|
import Mathlib.RingTheory.Localization.LocalizationLocalization
import Mathlib.RingTheory.Localization.Submodule
import Mathlib.RingTheory.DiscreteValuationRing.TFAE
#align_import ring_theory.dedekind_domain.dvr from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variable (R A K : Type*... | Mathlib/RingTheory/DedekindDomain/Dvr.lean | 118 | 130 | theorem IsLocalization.AtPrime.not_isField {P : Ideal A} (hP : P β β₯) [pP : P.IsPrime] (Aβ : Type*)
[CommRing Aβ] [Algebra A Aβ] [IsLocalization.AtPrime Aβ P] : Β¬IsField Aβ := by |
intro h
letI := h.toField
obtain β¨x, x_mem, x_neβ© := P.ne_bot_iff.mp hP
exact
(LocalRing.maximalIdeal.isMaximal _).ne_top
(Ideal.eq_top_of_isUnit_mem _
((IsLocalization.AtPrime.to_map_mem_maximal_iff Aβ P _).mpr x_mem)
(isUnit_iff_ne_zero.mpr
((map_ne_zero_iff (algebraMap A ... |
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib.Data.Set.Finite
#align_import data.finsupp.defs from "leanprover-community/mathlib"@"842328d9df7e96fd90fc424e115679c15fb23a71"
noncomputable section
open Finset Function
variable {Ξ± Ξ² Ξ³ ΞΉ M M' N P G H R S : Type*}... | Mathlib/Data/Finsupp/Defs.lean | 328 | 335 | theorem single_of_single_apply (a a' : Ξ±) (b : M) :
single a ((single a' b) a) = single a' (single a' b) a := by |
classical
rw [single_apply, single_apply]
ext
split_ifs with h
Β· rw [h]
Β· rw [zero_apply, single_apply, ite_self]
|
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 _ _)
|
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Regular.Pow
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Data.Finsupp.Antidiagonal
import Mathlib.Order.SymmDiff
import Mathlib.RingTheory.Adjoin.Basic
#align_import data.mv_polynomial.basic fr... | Mathlib/Algebra/MvPolynomial/Basic.lean | 1,411 | 1,413 | theorem eval_map (f : R β+* Sβ) (g : Ο β Sβ) (p : MvPolynomial Ο R) :
eval g (map f p) = evalβ f g p := by |
apply MvPolynomial.induction_on p <;> Β· simp (config := { contextual := true })
|
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]
|
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
|
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
|
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.MeasureTheory.Constructio... | Mathlib/MeasureTheory/Function/Jacobian.lean | 1,048 | 1,087 | theorem lintegral_abs_det_fderiv_le_addHaar_image (hs : MeasurableSet s)
(hf' : β x β s, HasFDerivWithinAt f (f' x) s x) (hf : InjOn f s) :
(β«β» x in s, ENNReal.ofReal |(f' x).det| βΞΌ) β€ ΞΌ (f '' s) := by |
/- We already know the result for finite-measure sets. We cover `s` by finite-measure sets using
`spanningSets ΞΌ`, and apply the previous result to each of these parts. -/
let u n := disjointed (spanningSets ΞΌ) n
have u_meas : β n, MeasurableSet (u n) := by
intro n
apply MeasurableSet.disjointed fun ... |
import Mathlib.MeasureTheory.Measure.GiryMonad
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.MeasureTheory.Measure.OpenPos
#align_import measure_theory.constructions.prod.basic from "leanprover-community/mathlib"@"00abe0695d8767201e6d008afa22393978bb32... | Mathlib/MeasureTheory/Constructions/Prod/Basic.lean | 518 | 520 | theorem quasiMeasurePreserving_fst : QuasiMeasurePreserving Prod.fst (ΞΌ.prod Ξ½) ΞΌ := by |
refine β¨measurable_fst, AbsolutelyContinuous.mk fun s hs h2s => ?_β©
rw [map_apply measurable_fst hs, β prod_univ, prod_prod, h2s, zero_mul]
|
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.Tactic.NormNum.Inv
import Mathlib.Tactic.NormNum.Pow
import Mathlib.Util.AtomM
set_option autoImplicit true
namespace Mathlib.Tactic
namespace Ring
open Mathlib.Meta Qq NormNum Lean.Meta AtomM
open Lean (MetaM Expr mkRawNatLit)
def instCommSemiringNat : CommSe... | Mathlib/Tactic/Ring/Basic.lean | 413 | 413 | theorem mul_zero (a : R) : a * 0 = 0 := by | simp
|
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
#align_import analysis.calculus.fderiv_... | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 184 | 203 | theorem norm_sub_le_of_mem_A {c : π} (hc : 1 < βcβ) {r Ξ΅ : β} (hΞ΅ : 0 < Ξ΅) (hr : 0 < r) {x : E}
{Lβ Lβ : E βL[π] F} (hβ : x β A f Lβ r Ξ΅) (hβ : x β A f Lβ r Ξ΅) : βLβ - Lββ β€ 4 * βcβ * Ξ΅ := by |
refine opNorm_le_of_shell (half_pos hr) (by positivity) hc ?_
intro y ley ylt
rw [div_div, div_le_iff' (mul_pos (by norm_num : (0 : β) < 2) (zero_lt_one.trans hc))] at ley
calc
β(Lβ - Lβ) yβ = βf (x + y) - f x - Lβ (x + y - x) - (f (x + y) - f x - Lβ (x + y - x))β := by
simp
_ β€ βf (x + y) - f x ... |
import Mathlib.Tactic.CategoryTheory.Elementwise
import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
import Mathlib.CategoryTheory.Limits.Preserves.Limits
import Mathlib.CategoryTheory.Limits.Shapes.Types
#align_import category_theory.glue_data from "l... | Mathlib/CategoryTheory/GlueData.lean | 391 | 399 | theorem ΞΉ_jointly_surjective (F : C β₯€ Type v) [PreservesColimit D.diagram.multispan F]
[β i j k : D.J, PreservesLimit (cospan (D.f i j) (D.f i k)) F] (x : F.obj D.glued) :
β (i : _) (y : F.obj (D.U i)), F.map (D.ΞΉ i) y = x := by |
let e := D.gluedIso F
obtain β¨i, y, eqβ© := (D.mapGlueData F).types_ΞΉ_jointly_surjective (e.hom x)
replace eq := congr_arg e.inv eq
change ((D.mapGlueData F).ΞΉ i β« e.inv) y = (e.hom β« e.inv) x at eq
rw [e.hom_inv_id, D.ΞΉ_gluedIso_inv] at eq
exact β¨i, y, eqβ©
|
import Mathlib.Data.Set.Lattice
import Mathlib.Logic.Small.Basic
import Mathlib.Logic.Function.OfArity
import Mathlib.Order.WellFounded
#align_import set_theory.zfc.basic from "leanprover-community/mathlib"@"f0b3759a8ef0bd8239ecdaa5e1089add5feebe1a"
-- Porting note: Lean 3 uses `Set` for `ZFSet`.
set_option linter... | Mathlib/SetTheory/ZFC/Basic.lean | 1,022 | 1,025 | theorem mem_sInter {x y : ZFSet} (h : x.Nonempty) : y β ββ x β β z β x, y β z := by |
rw [sInter, dif_pos h]
simp only [mem_toSet, mem_sep, and_iff_right_iff_imp]
exact fun H => H _ h.some_mem
|
import Mathlib.FieldTheory.Separable
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Tactic.ApplyFun
#align_import field_theory.finite.basic from "leanprover-community/mathlib"@"12a85fac627bea918960da036049d611b1a3ee43"
variable {K : Type*} {R : Type*}
local notation ... | Mathlib/FieldTheory/Finite/Basic.lean | 242 | 252 | theorem card (p : β) [CharP K p] : β n : β+, Nat.Prime p β§ q = p ^ (n : β) := by |
haveI hp : Fact p.Prime := β¨CharP.char_is_prime K pβ©
letI : Module (ZMod p) K := { (ZMod.castHom dvd_rfl K : ZMod p β+* _).toModule with }
obtain β¨n, hβ© := VectorSpace.card_fintype (ZMod p) K
rw [ZMod.card] at h
refine β¨β¨n, ?_β©, hp.1, hβ©
apply Or.resolve_left (Nat.eq_zero_or_pos n)
rintro rfl
rw [pow_z... |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.Data.Set.Finite
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.Data.Setoid.Basic
import Mathlib.GroupTheory.GroupAction.Defs
import Mathlib.GroupTheory.GroupAction.Group
#align_import group_theory.group_action.basic fro... | Mathlib/GroupTheory/GroupAction/Basic.lean | 826 | 829 | theorem mem_stabilizer_of_finite_iff_le_smul (s : Set Ξ±) (hs : s.Finite) (g : G) :
g β stabilizer G s β s β g β’ s := by |
rw [β @inv_mem_iff, mem_stabilizer_of_finite_iff_smul_le s hs]
exact Set.subset_set_smul_iff.symm
|
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Convex.Uniform
import Mathlib.Analysis.NormedSpace.Completion
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
#align_import analysis.inner_product_space.basic from "leanprover-community/mathlib"@"3f655f5297b030... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 2,063 | 2,067 | theorem OrthogonalFamily.norm_sum (l : β i, G i) (s : Finset ΞΉ) :
ββ i β s, V i (l i)β ^ 2 = β i β s, βl iβ ^ 2 := by |
have : ((ββ i β s, V i (l i)β : β) : π) ^ 2 = β i β s, ((βl iβ : β) : π) ^ 2 := by
simp only [β inner_self_eq_norm_sq_to_K, hV.inner_sum]
exact mod_cast this
|
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Limits.Shapes.KernelPair
#align_import category_theory.adhesive from "leanprover-community/mathlib"@"afff1f24a6b68d0077c9d63782a1d093e337758c"
namespace CategoryTheory
open Limits
universe v' u' v u
variable {J : Type v'} [Category.{u'} J] {... | Mathlib/CategoryTheory/Adhesive.lean | 306 | 316 | theorem adhesive_of_preserves_and_reflects_isomorphism (F : C β₯€ D)
[Adhesive D] [HasPullbacks C] [HasPushouts C]
[PreservesLimitsOfShape WalkingCospan F]
[PreservesColimitsOfShape WalkingSpan F]
[F.ReflectsIsomorphisms] :
Adhesive C := by |
haveI : ReflectsLimitsOfShape WalkingCospan F :=
reflectsLimitsOfShapeOfReflectsIsomorphisms
haveI : ReflectsColimitsOfShape WalkingSpan F :=
reflectsColimitsOfShapeOfReflectsIsomorphisms
exact adhesive_of_preserves_and_reflects F
|
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 | 1,354 | 1,357 | theorem le_sup_shrink_equiv {s : Set Ordinal.{u}} (hs : Small.{u} s) (a) (ha : a β s) :
a β€ sup.{u, u} fun x => ((@equivShrink s hs).symm x).val := by |
convert le_sup.{u, u} (fun x => ((@equivShrink s hs).symm x).val) ((@equivShrink s hs) β¨a, haβ©)
rw [symm_apply_apply]
|
import Mathlib.SetTheory.Game.Basic
import Mathlib.Tactic.NthRewrite
#align_import set_theory.game.impartial from "leanprover-community/mathlib"@"2e0975f6a25dd3fbfb9e41556a77f075f6269748"
universe u
namespace SetTheory
open scoped PGame
namespace PGame
def ImpartialAux : PGame β Prop
| G => (G β -G) β§ (β i... | Mathlib/SetTheory/Game/Impartial.lean | 183 | 184 | theorem lf_zero_iff {G : PGame} [G.Impartial] : G β§ 0 β 0 β§ G := by |
rw [β zero_lf_neg_iff, lf_congr_right (neg_equiv_self G)]
|
import Mathlib.Topology.Order.MonotoneContinuity
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.EMetricSpace.Lipschitz
import Mathlib.Topology.Metrizable.Basic
import Mathlib.Topology.Order.T5
#align_import topology.instances.ennreal from "leanprove... | Mathlib/Topology/Instances/ENNReal.lean | 193 | 195 | theorem tendsto_coe_nhds_top {f : Ξ± β ββ₯0} {l : Filter Ξ±} :
Tendsto (fun x => (f x : ββ₯0β)) l (π β) β Tendsto f l atTop := by |
rw [tendsto_nhds_top_iff_nnreal, atTop_basis_Ioi.tendsto_right_iff]; simp
|
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.Algebra.Polynomial.RingDivision
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
noncomputable section
open scoped Classical
open scoped nonZeroDivisors Polynomial
universe u v
va... | Mathlib/FieldTheory/RatFunc/Defs.lean | 228 | 232 | theorem liftOn'_mk {P : Sort v} (p q : K[X]) (f : K[X] β K[X] β P) (f0 : β p, f p 0 = f 0 1)
(H : β {p q a} (_hq : q β 0) (_ha : a β 0), f (a * p) (a * q) = f p q) :
(RatFunc.mk p q).liftOn' f @H = f p q := by |
rw [RatFunc.liftOn', RatFunc.liftOn_mk _ _ _ f0]
apply liftOn_condition_of_liftOn'_condition H
|
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.GroupWithZero.NeZero
import Mathlib.Algebra.Order.Group.Defs
import Mathlib.Algebra.Order.GroupWithZero.Unbundled
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebr... | Mathlib/Algebra/Order/Ring/Defs.lean | 697 | 698 | theorem mul_lt_of_one_lt_left (hb : b < 0) (h : 1 < a) : a * b < b := by |
simpa only [one_mul] using mul_lt_mul_of_neg_right h hb
|
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.UnitaryGroup
#align_import analysis.inner_product_space.pi_L2 from "leanprover-community/mathlib"@"13bce9a6b6c44f6b4c91ac1c1d2a816e2533d395"
set_... | Mathlib/Analysis/InnerProductSpace/PiL2.lean | 701 | 706 | theorem Complex.isometryOfOrthonormal_apply (v : OrthonormalBasis (Fin 2) β F) (z : β) :
Complex.isometryOfOrthonormal v z = z.re β’ v 0 + z.im β’ v 1 := by |
-- Porting note: was
-- simp [Complex.isometryOfOrthonormal, β v.sum_repr_symm]
rw [Complex.isometryOfOrthonormal, LinearIsometryEquiv.trans_apply]
simp [β v.sum_repr_symm]
|
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.Data.Finset.Sort
#align_import data.polynomial.basic from "leanprover-community/mathlib"@"949dc57e616a621462062668c9f39e4e17b64b69"
set_option linter.uppercaseLean3 false
noncomputable section
structure ... | Mathlib/Algebra/Polynomial/Basic.lean | 246 | 248 | theorem toFinsupp_pow (a : R[X]) (n : β) : (a ^ n).toFinsupp = a.toFinsupp ^ n := by |
cases a
rw [β ofFinsupp_pow]
|
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]
|
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