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 |
|---|---|---|---|---|---|
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.MeasureTheory.Group.FundamentalDomain
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.RingTheory.Localization.Module
#align_import algebra.module.zlattice from "leanprover-community/mathlib"@"a3e83f0fa4391c8740f7d773a7a9b74e311ae2a3"
n... | Mathlib/Algebra/Module/Zlattice/Basic.lean | 145 | 146 | theorem repr_fract_apply (m : E) (i : ΞΉ) : b.repr (fract b m) i = Int.fract (b.repr m i) := by |
rw [fract, map_sub, Finsupp.coe_sub, Pi.sub_apply, repr_floor_apply, Int.fract]
|
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 | 464 | 466 | theorem affineCombination_eq_linear_combination (s : Finset ΞΉ) (p : ΞΉ β V) (w : ΞΉ β k)
(hw : β i β s, w i = 1) : s.affineCombination k p w = β i β s, w i β’ p i := by |
simp [s.affineCombination_eq_weightedVSubOfPoint_vadd_of_sum_eq_one w p hw 0]
|
import Mathlib.Algebra.Polynomial.Cardinal
import Mathlib.RingTheory.Algebraic
#align_import algebra.algebraic_card from "leanprover-community/mathlib"@"40494fe75ecbd6d2ec61711baa630cf0a7b7d064"
universe u v
open Cardinal Polynomial Set
open Cardinal Polynomial
namespace Algebraic
theorem infinite_of_charZero... | Mathlib/Algebra/AlgebraicCard.lean | 45 | 54 | theorem cardinal_mk_lift_le_mul :
Cardinal.lift.{u} #{ x : A // IsAlgebraic R x } β€ Cardinal.lift.{v} #R[X] * β΅β := by |
rw [β mk_uLift, β mk_uLift]
choose g hgβ hgβ using fun x : { x : A | IsAlgebraic R x } => x.coe_prop
refine lift_mk_le_lift_mk_mul_of_lift_mk_preimage_le g fun f => ?_
rw [lift_le_aleph0, le_aleph0_iff_set_countable]
suffices MapsTo (β) (g β»ΒΉ' {f}) (f.rootSet A) from
this.countable_of_injOn Subtype.coe_i... |
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 | 1,163 | 1,167 | theorem ContDiff.contDiff_fderiv_apply {f : E β F} (hf : ContDiff π n f) (hmn : m + 1 β€ n) :
ContDiff π m fun p : E Γ E => (fderiv π f p.1 : E βL[π] F) p.2 := by |
rw [β contDiffOn_univ] at hf β’
rw [β fderivWithin_univ, β univ_prod_univ]
exact contDiffOn_fderivWithin_apply hf uniqueDiffOn_univ hmn
|
import Mathlib.Data.Finsupp.Basic
import Mathlib.Data.List.AList
#align_import data.finsupp.alist from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
namespace AList
variable {Ξ± M : Type*} [Zero M]
open List
noncomputable def lookupFinsupp (l : AList fun _x : Ξ± => M) : Ξ± ββ M where
... | Mathlib/Data/Finsupp/AList.lean | 124 | 132 | theorem _root_.Finsupp.toAList_lookupFinsupp (f : Ξ± ββ M) : f.toAList.lookupFinsupp = f := by |
ext a
classical
by_cases h : f a = 0
Β· suffices f.toAList.lookup a = none by simp [h, this]
simp [lookup_eq_none, h]
Β· suffices f.toAList.lookup a = some (f a) by simp [h, this]
apply mem_lookup_iff.2
simpa using h
|
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp Ad... | Mathlib/Algebra/MvPolynomial/Degrees.lean | 219 | 225 | theorem degrees_rename_of_injective {p : MvPolynomial Ο R} {f : Ο β Ο} (h : Function.Injective f) :
degrees (rename f p) = (degrees p).map f := by |
classical
simp only [degrees, Multiset.map_finset_sup p.support Finsupp.toMultiset f h,
support_rename_of_injective h, Finset.sup_image]
refine Finset.sup_congr rfl fun x _ => ?_
exact (Finsupp.toMultiset_map _ _).symm
|
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.LinearAlgebra.Isomorphisms
#align_import algebra.lie.quotient from "leanprover-community/mathlib"@"3d7987cda72abc473c7cdbbb075170e9ac620042"
universe u v w wβ wβ
namespace LieSubmodule
variable {R : Type u} {L : Type v}... | Mathlib/Algebra/Lie/Quotient.lean | 210 | 211 | theorem map_mk'_eq_bot_le : map (mk' N) N' = β₯ β N' β€ N := by |
rw [β LieModuleHom.le_ker_iff_map, mk'_ker]
|
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
import Mathlib.Algebra.PUnitInstances
#align_import category_theory.monoidal.Mon_ from "leanprover-community/... | Mathlib/CategoryTheory/Monoidal/Mon_.lean | 522 | 524 | theorem whiskerRight_hom (X : Mon_ C) {Y Z : Mon_ C} (f : Y βΆ Z) :
(X β f).hom = X.X β f.hom := by |
rw [β id_tensorHom]; rfl
|
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Mul
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.deriv.mul from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
universe u v w
noncomputable section
open scoped Classical... | Mathlib/Analysis/Calculus/Deriv/Mul.lean | 389 | 391 | theorem HasDerivAt.div_const (hc : HasDerivAt c c' x) (d : π') :
HasDerivAt (fun x => c x / d) (c' / d) x := by |
simpa only [div_eq_mul_inv] using hc.mul_const dβ»ΒΉ
|
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 | 217 | 220 | theorem ker_mulVecLin_conjTranspose_mul_self (A : Matrix m n R) :
LinearMap.ker (Aα΄΄ * A).mulVecLin = LinearMap.ker (mulVecLin A) := by |
ext x
simp only [LinearMap.mem_ker, mulVecLin_apply, conjTranspose_mul_self_mulVec_eq_zero]
|
import Mathlib.Algebra.EuclideanDomain.Instances
import Mathlib.RingTheory.Ideal.Colon
import Mathlib.RingTheory.UniqueFactorizationDomain
#align_import ring_theory.principal_ideal_domain from "leanprover-community/mathlib"@"6010cf523816335f7bae7f8584cb2edaace73940"
universe u v
variable {R : Type u} {M : Type v... | Mathlib/RingTheory/PrincipalIdealDomain.lean | 434 | 436 | theorem gcd_dvd_iff_exists (a b : R) {z} : gcd a b β£ z β β x y, z = a * x + b * y := by |
simp_rw [mul_comm a, mul_comm b, @eq_comm _ z, β Ideal.mem_span_pair, β span_gcd,
Ideal.mem_span_singleton]
|
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
#align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74"
assert_not_exists Basis -- See `RingTheory.Ideal.Basis`
assert_not_exists ... | Mathlib/RingTheory/Ideal/Operations.lean | 649 | 656 | theorem multiset_prod_le_inf {s : Multiset (Ideal R)} : s.prod β€ s.inf := by |
classical
refine s.induction_on ?_ ?_
Β· rw [Multiset.inf_zero]
exact le_top
intro a s ih
rw [Multiset.prod_cons, Multiset.inf_cons]
exact le_trans mul_le_inf (inf_le_inf le_rfl ih)
|
import Mathlib.Topology.MetricSpace.PiNat
import Mathlib.Topology.MetricSpace.Isometry
import Mathlib.Topology.MetricSpace.Gluing
import Mathlib.Topology.Sets.Opens
import Mathlib.Analysis.Normed.Field.Basic
#align_import topology.metric_space.polish from "leanprover-community/mathlib"@"bcfa726826abd57587355b4b5b7e78... | Mathlib/Topology/MetricSpace/Polish.lean | 409 | 423 | theorem IsClopenable.iUnion [t : TopologicalSpace Ξ±] [PolishSpace Ξ±] {s : β β Set Ξ±}
(hs : β n, IsClopenable (s n)) : IsClopenable (β n, s n) := by |
choose m mt m_polish _ m_open using hs
obtain β¨t', t'm, -, t'_polishβ© :
β t' : TopologicalSpace Ξ±, (β n : β, t' β€ m n) β§ t' β€ t β§ @PolishSpace Ξ± t' :=
exists_polishSpace_forall_le m mt m_polish
have A : IsOpen[t'] (β n, s n) := by
apply isOpen_iUnion
intro n
apply t'm n
exact m_open n
... |
import Mathlib.Algebra.Algebra.Hom
#align_import algebra.algebra.equiv from "leanprover-community/mathlib"@"bd9851ca476957ea4549eb19b40e7b5ade9428cc"
universe u v w uβ vβ
structure AlgEquiv (R : Type u) (A : Type v) (B : Type w) [CommSemiring R] [Semiring A] [Semiring B]
[Algebra R A] [Algebra R B] extends A β... | Mathlib/Algebra/Algebra/Equiv.lean | 655 | 657 | theorem toLinearEquiv_ofLinearEquiv : toLinearEquiv (ofLinearEquiv l map_one map_mul) = l := by |
ext
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 | 680 | 680 | theorem bit0_comp : comp (bit0 p : R[X]) q = bit0 (p.comp q) := by | simp only [bit0, add_comp]
|
import Mathlib.RingTheory.FiniteType
import Mathlib.RingTheory.MvPolynomial.Tower
import Mathlib.RingTheory.Ideal.QuotientOperations
#align_import ring_theory.finite_presentation from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
set_option autoImplicit true
open Function (Surjective)
... | Mathlib/RingTheory/FinitePresentation.lean | 150 | 158 | theorem iff :
FinitePresentation R A β
β (n : _) (I : Ideal (MvPolynomial (Fin n) R)) (_ : (_ β§Έ I) ββ[R] A), I.FG := by |
constructor
Β· rintro β¨n, f, hfβ©
exact β¨n, RingHom.ker f.toRingHom, Ideal.quotientKerAlgEquivOfSurjective hf.1, hf.2β©
Β· rintro β¨n, I, e, hfgβ©
letI := (FinitePresentation.mvPolynomial R _).quotient hfg
exact equiv e
|
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 | 853 | 856 | theorem mem_biInf_of_directed {f : Ξ² β Filter Ξ±} {s : Set Ξ²} (h : DirectedOn (f β»ΒΉ'o (Β· β₯ Β·)) s)
(ne : s.Nonempty) {t : Set Ξ±} : (t β β¨
i β s, f i) β β i β s, t β f i := by |
haveI := ne.to_subtype
simp_rw [iInf_subtype', mem_iInf_of_directed h.directed_val, Subtype.exists, exists_prop]
|
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 | 320 | 326 | theorem cos_oangle_sub_right_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = β(Ο / 2)) :
Real.Angle.cos (o.oangle y (y - x)) = βyβ / βy - xβ := by |
have hs : (o.oangle y (y - x)).sign = 1 := by
rw [oangle_sign_sub_right_swap, h, Real.Angle.sign_coe_pi_div_two]
rw [o.oangle_eq_angle_of_sign_eq_one hs, Real.Angle.cos_coe,
InnerProductGeometry.cos_angle_sub_of_inner_eq_zero
(o.inner_rev_eq_zero_of_oangle_eq_pi_div_two h)]
|
import Mathlib.CategoryTheory.Limits.HasLimits
import Mathlib.CategoryTheory.Products.Basic
import Mathlib.CategoryTheory.Functor.Currying
import Mathlib.CategoryTheory.Products.Bifunctor
#align_import category_theory.limits.fubini from "leanprover-community/mathlib"@"59382264386afdbaf1727e617f5fdda511992eb9"
uni... | Mathlib/CategoryTheory/Limits/Fubini.lean | 423 | 429 | theorem colimitFlipCompColimIsoColimitCompColim_ΞΉ_ΞΉ_inv (k) (j) :
colimit.ΞΉ (F.obj j) k β« colimit.ΞΉ (F β colim) j β«
(colimitFlipCompColimIsoColimitCompColim F).inv =
(colimit.ΞΉ _ j β« colimit.ΞΉ (F.flip β colim) k : _ βΆ colimit (F.flip β colim)) := by |
dsimp [colimitFlipCompColimIsoColimitCompColim]
slice_lhs 1 3 => simp only []
simp
|
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
import Mathlib.MeasureTheory.Integral.MeanInequalities
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
open Filter
open scoped ENNReal Topology
namespace MeasureTheory
variable ... | Mathlib/MeasureTheory/Function/LpSeminorm/TriangleInequality.lean | 140 | 142 | theorem snorm_sub_le' {f g : Ξ± β E} (hf : AEStronglyMeasurable f ΞΌ) (hg : AEStronglyMeasurable g ΞΌ)
(p : ββ₯0β) : snorm (f - g) p ΞΌ β€ LpAddConst p * (snorm f p ΞΌ + snorm g p ΞΌ) := by |
simpa only [sub_eq_add_neg, snorm_neg] using snorm_add_le' hf hg.neg p
|
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Combinatorics.Quiver.Path
#align_import combinatorics.quiver.cast from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0e"
universe v vβ vβ u uβ uβ
variable {U : Type*} [Quiver.{u + 1} U]
namespace Quiver
def Hom.cast {u v u' v... | Mathlib/Combinatorics/Quiver/Cast.lean | 63 | 66 | theorem Hom.cast_eq_iff_heq {u v u' v' : U} (hu : u = u') (hv : v = v') (e : u βΆ v) (e' : u' βΆ v') :
e.cast hu hv = e' β HEq e e' := by |
rw [Hom.cast_eq_cast]
exact _root_.cast_eq_iff_heq
|
import Mathlib.Topology.Algebra.Ring.Basic
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Topology.Algebra.UniformGroup
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.UniformSpace.UniformEmbedding
import Mathlib.Algebra.Algebra.Defs
import Mathlib.LinearAlgebra.Projection
import Mat... | Mathlib/Topology/Algebra/Module/Basic.lean | 61 | 72 | theorem Submodule.eq_top_of_nonempty_interior' [NeBot (π[{ x : R | IsUnit x }] 0)]
(s : Submodule R M) (hs : (interior (s : Set M)).Nonempty) : s = β€ := by |
rcases hs with β¨y, hyβ©
refine Submodule.eq_top_iff'.2 fun x => ?_
rw [mem_interior_iff_mem_nhds] at hy
have : Tendsto (fun c : R => y + c β’ x) (π[{ x : R | IsUnit x }] 0) (π (y + (0 : R) β’ x)) :=
tendsto_const_nhds.add ((tendsto_nhdsWithin_of_tendsto_nhds tendsto_id).smul tendsto_const_nhds)
rw [zero_s... |
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.Algebra.Regular.Basic
import Mathlib.Data.Nat.Choose.Sum
#align_import data.polynomial.coeff from "leanprover-community/mathlib"@"2651125b48fc5c170ab1111afd0817c903b1fc6c"
set_option linter.uppercaseLean3 false
no... | Mathlib/Algebra/Polynomial/Coeff.lean | 233 | 240 | theorem support_trinomial {k m n : β} (hkm : k < m) (hmn : m < n) {x y z : R} (hx : x β 0)
(hy : y β 0) (hz : z β 0) :
support (C x * X ^ k + C y * X ^ m + C z * X ^ n) = {k, m, n} := by |
apply subset_antisymm (support_trinomial' k m n x y z)
simp_rw [insert_subset_iff, singleton_subset_iff, mem_support_iff, coeff_add, coeff_C_mul,
coeff_X_pow_self, mul_one, coeff_X_pow, if_neg hkm.ne, if_neg hkm.ne', if_neg hmn.ne,
if_neg hmn.ne', if_neg (hkm.trans hmn).ne, if_neg (hkm.trans hmn).ne', mul_... |
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.Topology.Semicontinuous
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.Topology.Instances.EReal
#align_import measure_theory.integral.vitali_caratheodory from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
open sc... | Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean | 231 | 260 | theorem exists_lt_lowerSemicontinuous_lintegral_ge_of_aemeasurable [SigmaFinite ΞΌ] (f : Ξ± β ββ₯0)
(fmeas : AEMeasurable f ΞΌ) {Ξ΅ : ββ₯0β} (Ξ΅0 : Ξ΅ β 0) :
β g : Ξ± β ββ₯0β,
(β x, (f x : ββ₯0β) < g x) β§ LowerSemicontinuous g β§ (β«β» x, g x βΞΌ) β€ (β«β» x, f x βΞΌ) + Ξ΅ := by |
have : Ξ΅ / 2 β 0 := (ENNReal.half_pos Ξ΅0).ne'
rcases exists_lt_lowerSemicontinuous_lintegral_ge ΞΌ (fmeas.mk f) fmeas.measurable_mk this with
β¨g0, f_lt_g0, g0_cont, g0_intβ©
rcases exists_measurable_superset_of_null fmeas.ae_eq_mk with β¨s, hs, smeas, ΞΌsβ©
rcases exists_le_lowerSemicontinuous_lintegral_ge ΞΌ (s... |
import Mathlib.Algebra.Module.Submodule.Map
#align_import linear_algebra.basic from "leanprover-community/mathlib"@"9d684a893c52e1d6692a504a118bfccbae04feeb"
open Function
open Pointwise
variable {R : Type*} {Rβ : Type*} {Rβ : Type*} {Rβ : Type*}
variable {K : Type*}
variable {M : Type*} {Mβ : Type*} {Mβ : Type*... | Mathlib/Algebra/Module/Submodule/Ker.lean | 107 | 109 | theorem disjoint_ker {f : F} {p : Submodule R M} :
Disjoint p (ker f) β β x β p, f x = 0 β x = 0 := by |
simp [disjoint_def]
|
import Mathlib.Data.Fin.Fin2
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Common
#align_import data.typevec from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
universe u v w
@[pp_with_univ]
def TypeVec (n : β) :=
Fin2 n β Type*
#align typevec TypeVec
instance {n} : Inh... | Mathlib/Data/TypeVec.lean | 789 | 793 | theorem subtypeVal_toSubtype {Ξ± : TypeVec n} (p : Ξ± βΉ Β«repeatΒ» n Prop) :
subtypeVal p β toSubtype p = fun _ => Subtype.val := by |
ext i x
induction i <;> dsimp only [toSubtype, comp, subtypeVal] at *
simp [*]
|
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 | 435 | 437 | theorem proj_map_nth {ΞΉ : Type*} {Ξ : ΞΉ β Type*} [β i, Inhabited (Ξ i)] (i : ΞΉ) (L n) :
(ListBlank.map (@proj ΞΉ Ξ _ i) L).nth n = L.nth n i := by |
rw [ListBlank.nth_map]; rfl
|
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.RingTheory.AlgebraTower
import Mathlib.Algebra.Algebra.Subalgebra.Tower
#align_import linear_algebra.matrix.to_lin from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e8491d6"
... | Mathlib/LinearAlgebra/Matrix/ToLin.lean | 827 | 830 | theorem LinearMap.toMatrixAlgEquiv_comp (f g : Mβ ββ[R] Mβ) :
LinearMap.toMatrixAlgEquiv vβ (f.comp g) =
LinearMap.toMatrixAlgEquiv vβ f * LinearMap.toMatrixAlgEquiv vβ g := by |
simp [LinearMap.toMatrixAlgEquiv, LinearMap.toMatrix_comp vβ vβ vβ f g]
|
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Analytic.Composition
import Mathlib.Analysis.Analytic.Linear
import Mathlib.Analysis.Calculus.FDeriv.Analytic
import Mathlib.Geometry.Manifold.ChartedSpace
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.Analysis.Calculus.ContDiff.Basic
... | Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean | 1,394 | 1,396 | theorem isOpen_extChartAt_target [I.Boundaryless] (x : M) : IsOpen (extChartAt I x).target := by |
simp_rw [extChartAt_target, I.range_eq_univ, inter_univ]
exact (PartialHomeomorph.open_target _).preimage I.continuous_symm
|
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.Normed.Group.Completion
#align_import analysis.normed.group.hom_completion from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
noncomputable section
open Set NormedAddGroupHom UniformSpace
section Completion
variable {G... | Mathlib/Analysis/Normed/Group/HomCompletion.lean | 171 | 193 | theorem NormedAddGroupHom.ker_completion {f : NormedAddGroupHom G H} {C : β}
(h : f.SurjectiveOnWith f.range C) :
(f.completion.ker : Set <| Completion G) = closure (toCompl.comp <| incl f.ker).range := by |
refine le_antisymm ?_ (closure_minimal f.ker_le_ker_completion f.completion.isClosed_ker)
rintro hatg (hatg_in : f.completion hatg = 0)
rw [SeminormedAddCommGroup.mem_closure_iff]
intro Ξ΅ Ξ΅_pos
rcases h.exists_pos with β¨C', C'_pos, hC'β©
rcases exists_pos_mul_lt Ξ΅_pos (1 + C' * βfβ) with β¨Ξ΄, Ξ΄_pos, hΞ΄β©
ob... |
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Equiv.Fin
import Mathlib.ModelTheory.LanguageMap
#align_import model_theory.syntax from "leanprover-community/mathlib"@"d565b3df44619c1498326936be16f1a935df0728"
universe u v w u' v'
namespace FirstOrder
namespace Language
variable (L : Language.{u, v}) {L' : L... | Mathlib/ModelTheory/Syntax.lean | 630 | 638 | theorem relabel_sum_inl (Ο : L.BoundedFormula Ξ± n) :
(Ο.relabel Sum.inl : L.BoundedFormula Ξ± (0 + n)) = Ο.castLE (ge_of_eq (zero_add n)) := by |
simp only [relabel, relabelAux_sum_inl]
induction' Ο with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih3
Β· rfl
Β· simp [Fin.natAdd_zero, castLE_of_eq, mapTermRel]
Β· simp [Fin.natAdd_zero, castLE_of_eq, mapTermRel]; rfl
Β· simp [mapTermRel, ih1, ih2]
Β· simp [mapTermRel, ih3, castLE]
|
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.schur_complement from "leanprover-community/mathlib"@"a176cb1219e300e85793d44583dede42377b51af"
variable {l m n Ξ± : Type*}
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/SchurComplement.lean | 202 | 215 | theorem inv_fromBlocks_zeroββ_of_isUnit_iff (A : Matrix m m Ξ±) (B : Matrix m n Ξ±) (D : Matrix n n Ξ±)
(hAD : IsUnit A β IsUnit D) :
(fromBlocks A B 0 D)β»ΒΉ = fromBlocks Aβ»ΒΉ (-(Aβ»ΒΉ * B * Dβ»ΒΉ)) 0 Dβ»ΒΉ := by |
by_cases hA : IsUnit A
Β· have hD := hAD.mp hA
cases hA.nonempty_invertible
cases hD.nonempty_invertible
letI := fromBlocksZeroββInvertible A B D
simp_rw [β invOf_eq_nonsing_inv, invOf_fromBlocks_zeroββ_eq]
Β· have hD := hAD.not.mp hA
have : Β¬IsUnit (fromBlocks A B 0 D) :=
isUnit_fromBloc... |
import Mathlib.Algebra.Group.Opposite
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Control.Traversable.Instances
import Mathlib.Control.Traversable.Lemmas
import Mathlib.CategoryTheory.Endomorphism
import Mathlib.CategoryTheory.Types
import Mathlib.CategoryTheory.Category.KleisliCat
import Mathlib.Tactic.Ada... | Mathlib/Control/Fold.lean | 334 | 349 | theorem toList_spec (xs : t Ξ±) : toList xs = FreeMonoid.toList (foldMap FreeMonoid.of xs) :=
Eq.symm <|
calc
FreeMonoid.toList (foldMap FreeMonoid.of xs) =
FreeMonoid.toList (foldMap FreeMonoid.of xs).reverse.reverse := by |
simp only [List.reverse_reverse]
_ = FreeMonoid.toList (List.foldr cons [] (foldMap FreeMonoid.of xs).reverse).reverse := by
simp only [List.foldr_eta]
_ = (unop (Foldl.ofFreeMonoid (flip cons) (foldMap FreeMonoid.of xs)) []).reverse := by
#adaptation_note /-- nightly-2024-0... |
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.FiniteSupport
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Set.Subsingleton
#align_import algebra.big_operators.finprod from "leanprover-community/mathlib"@"d6fad0e5bf... | Mathlib/Algebra/BigOperators/Finprod.lean | 1,286 | 1,292 | theorem finprod_curryβ {Ξ³ : Type*} (f : Ξ± Γ Ξ² Γ Ξ³ β M) (h : (mulSupport f).Finite) :
βαΆ abc, f abc = βαΆ (a) (b) (c), f (a, b, c) := by |
rw [finprod_curry f h]
congr
ext a
rw [finprod_curry]
simp [h]
|
import Mathlib.Algebra.Homology.ExactSequence
import Mathlib.CategoryTheory.Abelian.Refinements
#align_import category_theory.abelian.diagram_lemmas.four from "leanprover-community/mathlib"@"d34cbcf6c94953e965448c933cd9cc485115ebbd"
namespace CategoryTheory
open Category Limits Preadditive
namespace Abelian
va... | Mathlib/CategoryTheory/Abelian/DiagramLemmas/Four.lean | 62 | 83 | theorem mono_of_epi_of_mono_of_mono' (hRβ : Rβ.map' 0 2 = 0)
(hRβ' : (mkβ (Rβ.map' 1 2) (Rβ.map' 2 3)).Exact)
(hRβ : (mkβ (Rβ.map' 0 1) (Rβ.map' 1 2)).Exact)
(hβ : Epi (app' Ο 0)) (hβ : Mono (app' Ο 1)) (hβ : Mono (app' Ο 3)) :
Mono (app' Ο 2) := by |
apply mono_of_cancel_zero
intro A fβ hβ
have hβ : fβ β« Rβ.map' 2 3 = 0 := by
rw [β cancel_mono (app' Ο 3 _), assoc, NatTrans.naturality, reassoc_of% hβ,
zero_comp, zero_comp]
obtain β¨Aβ, Οβ, _, fβ, hfββ© := (hRβ'.exact 0).exact_up_to_refinements fβ hβ
dsimp at hfβ
have hβ : (fβ β« app' Ο 1) β« Rβ.ma... |
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.GroupWithZero.NeZero
import Mathlib.Logic.Unique
#align_import algebra.group_with_zero.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
assert_not_exists DenselyOrdered
open scoped Classical
open Function
variable {Ξ± M... | Mathlib/Algebra/GroupWithZero/Basic.lean | 411 | 411 | theorem inv_eq_zero {a : Gβ} : aβ»ΒΉ = 0 β a = 0 := by | rw [inv_eq_iff_eq_inv, inv_zero]
|
import Mathlib.LinearAlgebra.Contraction
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
#align_import linear_algebra.trace from "leanprover-community/mathlib"@"4cf7ca0e69e048b006674cf4499e5c7d296a89e0"
noncomputable section
universe u v w
namespace LinearMap
open Matrix
open FiniteDimensional
open Tensor... | Mathlib/LinearAlgebra/Trace.lean | 278 | 282 | theorem trace_comp_comm' (f : M ββ[R] N) (g : N ββ[R] M) :
trace R M (g ββ f) = trace R N (f ββ g) := by |
have h := ext_iff.1 (ext_iff.1 (trace_comp_comm R M N) g) f
simp only [llcomp_apply', comprβ_apply, flip_apply] at h
exact h
|
import Mathlib.MeasureTheory.Measure.MeasureSpaceDef
#align_import measure_theory.measure.ae_disjoint from "leanprover-community/mathlib"@"bc7d81beddb3d6c66f71449c5bc76c38cb77cf9e"
open Set Function
namespace MeasureTheory
variable {ΞΉ Ξ± : Type*} {m : MeasurableSpace Ξ±} (ΞΌ : Measure Ξ±)
def AEDisjoint (s t : Se... | Mathlib/MeasureTheory/Measure/AEDisjoint.lean | 100 | 102 | theorem iUnion_right_iff [Countable ΞΉ] {t : ΞΉ β Set Ξ±} :
AEDisjoint ΞΌ s (β i, t i) β β i, AEDisjoint ΞΌ s (t i) := by |
simp only [AEDisjoint, inter_iUnion, measure_iUnion_null_iff]
|
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 | 118 | 124 | theorem edist_congr_right {x y z : Ξ±} (h : edist x y = 0) : edist x z = edist y z := by |
apply le_antisymm
Β· rw [β zero_add (edist y z), β h]
apply edist_triangle
Β· rw [edist_comm] at h
rw [β zero_add (edist x z), β h]
apply edist_triangle
|
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.Perm
import Mathlib.GroupTheory.Perm.Finite
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
open Equiv Function Finset
variable {... | Mathlib/GroupTheory/Perm/Cycle/Basic.lean | 914 | 924 | theorem IsCycleOn.exists_pow_eq {s : Finset Ξ±} (hf : f.IsCycleOn s) (ha : a β s) (hb : b β s) :
β n < s.card, (f ^ n) a = b := by |
classical
obtain β¨n, rflβ© := hf.2 ha hb
obtain β¨k, hkβ© := (Int.mod_modEq n s.card).symm.dvd
refine β¨n.natMod s.card, Int.natMod_lt (Nonempty.card_pos β¨a, haβ©).ne', ?_β©
rw [β zpow_natCast, Int.natMod,
Int.toNat_of_nonneg (Int.emod_nonneg _ <| Nat.cast_ne_zero.2
(Nonempty.card_pos β¨a, haβ©... |
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.NormedSpace.Dual
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Lp
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.function.ae_eq_of_integral from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284... | Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean | 514 | 526 | theorem AEFinStronglyMeasurable.ae_eq_of_forall_setIntegral_eq {f g : Ξ± β E}
(hf_int_finite : β s, MeasurableSet s β ΞΌ s < β β IntegrableOn f s ΞΌ)
(hg_int_finite : β s, MeasurableSet s β ΞΌ s < β β IntegrableOn g s ΞΌ)
(hfg_eq : β s : Set Ξ±, MeasurableSet s β ΞΌ s < β β β« x in s, f x βΞΌ = β« x in s, g x βΞΌ)
... |
rw [β sub_ae_eq_zero]
have hfg : β s : Set Ξ±, MeasurableSet s β ΞΌ s < β β (β« x in s, (f - g) x βΞΌ) = 0 := by
intro s hs hΞΌs
rw [integral_sub' (hf_int_finite s hs hΞΌs) (hg_int_finite s hs hΞΌs),
sub_eq_zero.mpr (hfg_eq s hs hΞΌs)]
have hfg_int : β s, MeasurableSet s β ΞΌ s < β β IntegrableOn (f - g) s ... |
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 | 1,680 | 1,681 | theorem inter_union_self (s t : Finset Ξ±) : s β© (t βͺ s) = s := by |
rw [inter_comm, union_inter_cancel_right]
|
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.Algebra.Star.Unitary
#align_import linear_algebra.unitary_group from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062a169c9"
universe u ... | Mathlib/LinearAlgebra/UnitaryGroup.lean | 66 | 68 | theorem mem_unitaryGroup_iff : A β Matrix.unitaryGroup n Ξ± β A * star A = 1 := by |
refine β¨And.right, fun hA => β¨?_, hAβ©β©
simpa only [mul_eq_one_comm] using hA
|
import Mathlib.Init.Function
#align_import data.option.n_ary from "leanprover-community/mathlib"@"995b47e555f1b6297c7cf16855f1023e355219fb"
universe u
open Function
namespace Option
variable {Ξ± Ξ² Ξ³ Ξ΄ : Type*} {f : Ξ± β Ξ² β Ξ³} {a : Option Ξ±} {b : Option Ξ²} {c : Option Ξ³}
def mapβ (f : Ξ± β Ξ² β Ξ³) (a : Option Ξ±) ... | Mathlib/Data/Option/NAry.lean | 170 | 172 | theorem mapβ_map_left_comm {f : Ξ±' β Ξ² β Ξ³} {g : Ξ± β Ξ±'} {f' : Ξ± β Ξ² β Ξ΄} {g' : Ξ΄ β Ξ³}
(h_left_comm : β a b, f (g a) b = g' (f' a b)) : mapβ f (a.map g) b = (mapβ f' a b).map g' := by |
cases a <;> cases b <;> simp [h_left_comm]
|
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.MeasureTheory.Function.Egorov
import Mathlib.MeasureTheory.Function.LpSpace
#align_import measure_theory.function.convergence_in_measure from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
open TopologicalSpace Filter
ope... | Mathlib/MeasureTheory/Function/ConvergenceInMeasure.lean | 107 | 124 | theorem tendstoInMeasure_of_tendsto_ae_of_stronglyMeasurable [IsFiniteMeasure ΞΌ]
(hf : β n, StronglyMeasurable (f n)) (hg : StronglyMeasurable g)
(hfg : βα΅ x βΞΌ, Tendsto (fun n => f n x) atTop (π (g x))) : TendstoInMeasure ΞΌ f atTop g := by |
refine fun Ξ΅ hΞ΅ => ENNReal.tendsto_atTop_zero.mpr fun Ξ΄ hΞ΄ => ?_
by_cases hΞ΄i : Ξ΄ = β
Β· simp only [hΞ΄i, imp_true_iff, le_top, exists_const]
lift Ξ΄ to ββ₯0 using hΞ΄i
rw [gt_iff_lt, ENNReal.coe_pos, β NNReal.coe_pos] at hΞ΄
obtain β¨t, _, ht, hunifβ© := tendstoUniformlyOn_of_ae_tendsto' hf hg hfg hΞ΄
rw [ENNRea... |
import Mathlib.Data.PFunctor.Univariate.M
#align_import data.qpf.univariate.basic from "leanprover-community/mathlib"@"14b69e9f3c16630440a2cbd46f1ddad0d561dee7"
universe u
class QPF (F : Type u β Type u) [Functor F] where
P : PFunctor.{u}
abs : β {Ξ±}, P Ξ± β F Ξ±
repr : β {Ξ±}, F Ξ± β P Ξ±
abs_repr : β {Ξ±} (... | Mathlib/Data/QPF/Univariate/Basic.lean | 348 | 357 | theorem Fix.ind (p : Fix F β Prop) (h : β x : F (Fix F), Liftp p x β p (Fix.mk x)) : β x, p x := by |
apply Quot.ind
intro x
induction' x with a f ih
change p β¦β¨a, fβ©β§
rw [β Fix.ind_aux a f]
apply h
rw [liftp_iff]
refine β¨_, _, rfl, ?_β©
convert ih
|
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.ContinuousFunction.Compact
import Mathlib.Topology.UrysohnsLemma
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Topology.Algebra.Module.CharacterSpace
#align_import topology.continuous_function.ideals from "... | Mathlib/Topology/ContinuousFunction/Ideals.lean | 154 | 156 | theorem mem_idealOfSet_compl_singleton (x : X) (f : C(X, R)) :
f β idealOfSet R ({x}αΆ : Set X) β f x = 0 := by |
simp only [mem_idealOfSet, compl_compl, Set.mem_singleton_iff, forall_eq]
|
import Mathlib.Analysis.Convex.Basic
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Topology.Order.Basic
#align_import analysis.convex.strict from "leanprover-community/mathlib"@"84dc0bd6619acaea625086d6f53cb35cdd554219"
open Set
open Convex Pointwise
variable {π π E F Ξ² : Type*}
open Function Se... | Mathlib/Analysis/Convex/Strict.lean | 366 | 371 | theorem StrictConvex.affine_preimage {s : Set F} (hs : StrictConvex π s) {f : E βα΅[π] F}
(hf : Continuous f) (hfinj : Injective f) : StrictConvex π (f β»ΒΉ' s) := by |
intro x hx y hy hxy a b ha hb hab
refine preimage_interior_subset_interior_preimage hf ?_
rw [mem_preimage, Convex.combo_affine_apply hab]
exact hs hx hy (hfinj.ne hxy) ha hb hab
|
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 | 397 | 400 | theorem ListBlank.head_map {Ξ Ξ'} [Inhabited Ξ] [Inhabited Ξ'] (f : PointedMap Ξ Ξ')
(l : ListBlank Ξ) : (l.map f).head = f l.head := by |
conv => lhs; rw [β ListBlank.cons_head_tail l]
exact Quotient.inductionOn' l fun a β¦ rfl
|
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.LinearAlgebra.Matrix.Symmetric
#align_import linear_algebra.quadratic_form.basic from "leanprover-community/mathlib"@"d11f435d4e34a6cea0a1797d6b625b0c170be845"
universe u v w
variable {S T : ... | Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 222 | 227 | theorem map_add_add_add_map (x y z : M) :
Q (x + y + z) + (Q x + Q y + Q z) = Q (x + y) + Q (y + z) + Q (z + x) := by |
obtain β¨B, hβ© := Q.exists_companion
rw [add_comm z x]
simp only [h, map_add, LinearMap.add_apply]
abel
|
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 | 306 | 308 | theorem oangle_smul_left_of_neg (x y : V) {r : β} (hr : r < 0) :
o.oangle (r β’ x) y = o.oangle (-x) y := by |
rw [β neg_neg r, neg_smul, β smul_neg, o.oangle_smul_left_of_pos _ _ (neg_pos_of_neg hr)]
|
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.GroupTheory.GroupAction.Pi
open Function Set
structure AddConstMap (G H : Type*) [Add G] [Add H] (a : G) (b : H) where
protected... | Mathlib/Algebra/AddConstMap/Basic.lean | 191 | 193 | theorem map_zsmul_const [AddGroup G] [AddGroup H] [AddConstMapClass F G H a b]
(f : F) (n : β€) : f (n β’ a) = f 0 + n β’ b := by |
simpa using map_add_zsmul f 0 n
|
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.Tactic.TFAE
#align_import ring_theory.valuation.basic from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
open scoped Classical
open Function Ideal
nonco... | Mathlib/RingTheory/Valuation/Basic.lean | 386 | 393 | theorem map {v' : Valuation R Ξβ} (f : Ξβ β*β Ξ'β) (hf : Monotone f) (inf : Injective f)
(h : v.IsEquiv v') : (v.map f hf).IsEquiv (v'.map f hf) :=
let H : StrictMono f := hf.strictMono_of_injective inf
fun r s =>
calc
f (v r) β€ f (v s) β v r β€ v s := by | rw [H.le_iff_le]
_ β v' r β€ v' s := h r s
_ β f (v' r) β€ f (v' s) := by rw [H.le_iff_le]
|
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.inner_product_space.euclidean_dist from "leanprover-community/mathlib"@"9425b6f8220e53b059f5a4904786c3c4b50fc057"
open scoped Topology
open Set
variable {E : Type*} [AddCommGroup E] [Topologi... | Mathlib/Analysis/InnerProductSpace/EuclideanDist.lean | 117 | 119 | theorem nhds_basis_ball {x : E} : (π x).HasBasis (fun r : β => 0 < r) (ball x) := by |
rw [toEuclidean.toHomeomorph.nhds_eq_comap x]
exact Metric.nhds_basis_ball.comap _
|
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
#align_import analysis.normed.group.add_torsor from "leanprover-community/mathlib"@"837f72de63ad6cd96519cde5f1ffd5ed8d280ad0"
noncomputable section
open NNReal Topo... | Mathlib/Analysis/Normed/Group/AddTorsor.lean | 190 | 194 | theorem edist_vadd_vadd_le (v v' : V) (p p' : P) :
edist (v +α΅₯ p) (v' +α΅₯ p') β€ edist v v' + edist p p' := by |
simp only [edist_nndist]
norm_cast -- Porting note: was apply_mod_cast
apply dist_vadd_vadd_le
|
import Mathlib.Analysis.BoxIntegral.Partition.Basic
#align_import analysis.box_integral.partition.tagged from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
noncomputable section
open scoped Classical
open ENNReal NNReal
open Set Function
namespace BoxIntegral
variable {ΞΉ : Type*}
... | Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean | 384 | 390 | theorem IsSubordinate.disjUnion [Fintype ΞΉ] (hβ : IsSubordinate Οβ r) (hβ : IsSubordinate Οβ r)
(h : Disjoint Οβ.iUnion Οβ.iUnion) : IsSubordinate (Οβ.disjUnion Οβ h) r := by |
refine fun J hJ => (Finset.mem_union.1 hJ).elim (fun hJ => ?_) fun hJ => ?_
Β· rw [disjUnion_tag_of_mem_left _ hJ]
exact hβ _ hJ
Β· rw [disjUnion_tag_of_mem_right _ hJ]
exact hβ _ hJ
|
import Mathlib.Algebra.Lie.CartanSubalgebra
import Mathlib.Algebra.Lie.Weights.Basic
suppress_compilation
open Set
variable {R L : Type*} [CommRing R] [LieRing L] [LieAlgebra R L]
(H : LieSubalgebra R L) [LieAlgebra.IsNilpotent R H]
{M : Type*} [AddCommGroup M] [Module R M] [LieRingModule L M] [LieModule R L ... | Mathlib/Algebra/Lie/Weights/Cartan.lean | 239 | 242 | theorem is_cartan_of_zeroRootSubalgebra_eq (h : zeroRootSubalgebra R L H = H) :
H.IsCartanSubalgebra :=
{ nilpotent := inferInstance
self_normalizing := by | rw [β h]; exact zeroRootSubalgebra_normalizer_eq_self R L H }
|
import Mathlib.RingTheory.IntegrallyClosed
import Mathlib.RingTheory.Trace
import Mathlib.RingTheory.Norm
#align_import ring_theory.discriminant from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
universe u v w z
open scoped Matrix
open Matrix FiniteDimensional Fintype Polynomial Fin... | Mathlib/RingTheory/Discriminant.lean | 321 | 329 | theorem discr_eq_discr [Fintype ΞΉ] (b : Basis ΞΉ β€ A) (b' : Basis ΞΉ β€ A) :
Algebra.discr β€ b = Algebra.discr β€ b' := by |
convert Algebra.discr_of_matrix_vecMul b' (b'.toMatrix b)
Β· rw [Basis.toMatrix_map_vecMul]
Β· suffices IsUnit (b'.toMatrix b).det by
rw [Int.isUnit_iff, β sq_eq_one_iff] at this
rw [this, one_mul]
rw [β LinearMap.toMatrix_id_eq_basis_toMatrix b b']
exact LinearEquiv.isUnit_det (LinearEquiv.ref... |
import Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
#align_import category_theory.limits.shapes.pullbacks from "leanprover-community/mathlib"@"7316286ff2942aa14e540add9058c6b0aa1c8070"
noncomputable section
open CategoryTheory
universe w vβ vβ v u... | Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean | 1,608 | 1,609 | theorem inl_comp_pushoutSymmetry_inv [HasPushout f g] :
pushout.inl β« (pushoutSymmetry f g).inv = pushout.inr := by | simp [Iso.comp_inv_eq]
|
import Mathlib.MeasureTheory.Measure.Typeclasses
#align_import probability.conditional_probability from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
open ENNReal MeasureTheory MeasureTheory.Measure MeasurableSpace Set
variable {Ξ© Ξ©' Ξ± : Type*} {m : MeasurableSpa... | Mathlib/Probability/ConditionalProbability.lean | 134 | 135 | theorem cond_univ [IsProbabilityMeasure ΞΌ] : ΞΌ[|Set.univ] = ΞΌ := by |
simp [cond, measure_univ, Measure.restrict_univ]
|
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Tactic.Spread
import Mathlib.Util.AssertExists
#align_import algebra.ring.defs from "leanprover-community/mathlib"@"76de8ae01554c3b37d66544866659ff174e66e1f"
universe u v w x
variable {Ξ± : ... | Mathlib/Algebra/Ring/Defs.lean | 156 | 157 | theorem add_one_mul [RightDistribClass Ξ±] (a b : Ξ±) : (a + 1) * b = a * b + b := by |
rw [add_mul, one_mul]
|
import Mathlib.Order.Interval.Set.Image
import Mathlib.Order.CompleteLatticeIntervals
import Mathlib.Topology.Order.DenselyOrdered
import Mathlib.Topology.Order.Monotone
#align_import topology.algebra.order.intermediate_value from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Filt... | Mathlib/Topology/Order/IntermediateValue.lean | 312 | 321 | theorem setOf_isPreconnected_subset_of_ordered :
{ s : Set Ξ± | IsPreconnected s } β
-- bounded intervals
(range (uncurry Icc) βͺ range (uncurry Ico) βͺ range (uncurry Ioc) βͺ range (uncurry Ioo)) βͺ
-- unbounded intervals and `univ`
(range Ici βͺ range Ioi βͺ range Iic βͺ range Iio βͺ {univ, β
}) := ... |
intro s hs
rcases hs.mem_intervals with (hs | hs | hs | hs | hs | hs | hs | hs | hs | hs) <;> rw [hs] <;>
simp only [union_insert, union_singleton, mem_insert_iff, mem_union, mem_range, Prod.exists,
uncurry_apply_pair, exists_apply_eq_apply, true_or, or_true, exists_apply_eq_apply2]
|
import Mathlib.Data.Set.Lattice
import Mathlib.Order.Directed
#align_import data.set.Union_lift from "leanprover-community/mathlib"@"5a4ea8453f128345f73cc656e80a49de2a54f481"
variable {Ξ± : Type*} {ΞΉ Ξ² : Sort _}
namespace Set
section UnionLift
@[nolint unusedArguments]
noncomputable def iUnionLift (S : ΞΉ β Set... | Mathlib/Data/Set/UnionLift.lean | 79 | 90 | theorem preimage_iUnionLift (t : Set Ξ²) :
iUnionLift S f hf T hT β»ΒΉ' t =
inclusion hT β»ΒΉ' (β i, inclusion (subset_iUnion S i) '' (f i β»ΒΉ' t)) := by |
ext x
simp only [mem_preimage, mem_iUnion, mem_image]
constructor
Β· rcases mem_iUnion.1 (hT x.prop) with β¨i, hiβ©
refine fun h => β¨i, β¨x, hiβ©, ?_, rflβ©
rwa [iUnionLift_of_mem x hi] at h
Β· rintro β¨i, β¨y, hiβ©, h, hxyβ©
obtain rfl : y = x := congr_arg Subtype.val hxy
rwa [iUnionLift_of_mem x hi]
|
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import ring_theory.adjoin.fg from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
universe u v w
open Subsemiring Ring Submodule
open Pointwise
na... | Mathlib/RingTheory/Adjoin/FG.lean | 170 | 179 | theorem induction_on_adjoin [IsNoetherian R A] (P : Subalgebra R A β Prop) (base : P β₯)
(ih : β (S : Subalgebra R A) (x : A), P S β P (Algebra.adjoin R (insert x S)))
(S : Subalgebra R A) : P S := by |
classical
obtain β¨t, rflβ© := S.fg_of_noetherian
refine Finset.induction_on t ?_ ?_
Β· simpa using base
intro x t _ h
rw [Finset.coe_insert]
simpa only [Algebra.adjoin_insert_adjoin] using ih _ x h
|
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 | 1,401 | 1,407 | theorem iSup_ord {ΞΉ} {f : ΞΉ β Cardinal} (hf : BddAbove (range f)) :
(iSup f).ord = β¨ i, (f i).ord := by |
unfold iSup
convert sSup_ord hf
-- Porting note: `change` is required.
conv_lhs => change range (ord β f)
rw [range_comp]
|
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 | 92 | 102 | theorem condexpIndL1Fin_add (hs : MeasurableSet s) (hΞΌs : ΞΌ s β β) (x y : G) :
condexpIndL1Fin hm hs hΞΌs (x + y) =
condexpIndL1Fin hm hs hΞΌs x + condexpIndL1Fin hm hs hΞΌs y := by |
ext1
refine (Memβp.coeFn_toLp q).trans ?_
refine EventuallyEq.trans ?_ (Lp.coeFn_add _ _).symm
refine EventuallyEq.trans ?_
(EventuallyEq.add (Memβp.coeFn_toLp q).symm (Memβp.coeFn_toLp q).symm)
rw [condexpIndSMul_add]
refine (Lp.coeFn_add _ _).trans (eventually_of_forall fun a => ?_)
rfl
|
import Mathlib.Data.Nat.Bits
import Mathlib.Order.Lattice
#align_import data.nat.size from "leanprover-community/mathlib"@"18a5306c091183ac90884daa9373fa3b178e8607"
namespace Nat
section
set_option linter.deprecated false
theorem shiftLeft_eq_mul_pow (m) : β n, m <<< n = m * 2 ^ n := shiftLeft_eq _
#align nat.... | Mathlib/Data/Nat/Size.lean | 144 | 145 | theorem size_eq_zero {n : β} : size n = 0 β n = 0 := by |
simpa [Nat.pos_iff_ne_zero, not_iff_not] using size_pos
|
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
open Set Filter Function Topology Filter
variable {Ξ± : Type*} {Ξ² : Type*} {Ξ³ : Type*} {Ξ΄ : Type*}
variable [TopologicalSpace Ξ±]
@[simp]
theorem nhds_bind_nhdsW... | Mathlib/Topology/ContinuousOn.lean | 256 | 258 | theorem nhdsWithin_inter (a : Ξ±) (s t : Set Ξ±) : π[s β© t] a = π[s] a β π[t] a := by |
delta nhdsWithin
rw [inf_left_comm, inf_assoc, inf_principal, β inf_assoc, inf_idem]
|
import Mathlib.GroupTheory.FreeGroup.Basic
import Mathlib.GroupTheory.QuotientGroup
#align_import group_theory.presented_group from "leanprover-community/mathlib"@"d90e4e186f1d18e375dcd4e5b5f6364b01cb3e46"
variable {Ξ± : Type*}
def PresentedGroup (rels : Set (FreeGroup Ξ±)) :=
FreeGroup Ξ± β§Έ Subgroup.normalClosu... | Mathlib/GroupTheory/PresentedGroup.lean | 101 | 104 | theorem ext {Ο Ο : PresentedGroup rels β* G} (hx : β (x : Ξ±), Ο (.of x) = Ο (.of x)) : Ο = Ο := by |
unfold PresentedGroup
ext
apply hx
|
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 | 829 | 832 | theorem of_is_bilimit' {b : BinaryBicone X Y} (h : b.IsBilimit) :
IsPushout b.fst b.snd (0 : X βΆ 0) (0 : Y βΆ 0) := by |
refine IsPushout.of_right ?_ (by simp) (IsPushout.inl_snd' h)
simp
|
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Monomial
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Nat.WithBot
import Mathlib.Data.Nat.Cast.WithTop
import Mathlib.Data.Nat.SuccPred
#align_import data.polynomial.degree.definitions... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 316 | 317 | theorem degree_C_mul_X_le (a : R) : degree (C a * X) β€ 1 := by |
simpa only [pow_one] using degree_C_mul_X_pow_le 1 a
|
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Deprecated.Submonoid
#align_import deprecated.subgroup from "leanprover-community/mathlib"@"f93c11933efbc3c2f0299e47b8ff83e9b539cbf6"
open Set Function
variable {G : Type*} {H : Type*} {A : Type*} {a aβ aβ b c : G}
structure IsNormalAddSubgroup [AddGro... | Mathlib/Deprecated/Subgroup.lean | 274 | 287 | theorem center_normal : IsNormalSubgroup (center G) :=
{ one_mem := by | simp [center]
mul_mem := fun ha hb g => by
rw [β mul_assoc, mem_center.2 ha g, mul_assoc, mem_center.2 hb g, β mul_assoc]
inv_mem := fun {a} ha g =>
calc
g * aβ»ΒΉ = aβ»ΒΉ * (g * a) * aβ»ΒΉ := by simp [ha g]
_ = aβ»ΒΉ * g := by rw [β mul_assoc, mul_assoc]; simp
normal := fun n ha g h =>... |
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.SetTheory.Cardinal.Ordinal
#align_import data.polynomial.cardinal from "leanprover-community/mathlib"@"62c0a4ef1441edb463095ea02a06e87f3dfe135c"
universe u
open Cardinal Polynomial
open Cardinal
namespace Polynomial
@[simp]
theorem cardinal_mk_eq_max {R :... | Mathlib/Algebra/Polynomial/Cardinal.lean | 34 | 37 | theorem cardinal_mk_le_max {R : Type u} [Semiring R] : #(R[X]) β€ max #R β΅β := by |
cases subsingleton_or_nontrivial R
Β· exact (mk_eq_one _).trans_le (le_max_of_le_right one_le_aleph0)
Β· exact cardinal_mk_eq_max.le
|
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
variab... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 508 | 510 | theorem pow_rpow_inv_natCast (hx : 0 β€ x) (hn : n β 0) : (x ^ n) ^ (nβ»ΒΉ : β) = x := by |
have hn0 : (n : β) β 0 := Nat.cast_ne_zero.2 hn
rw [β rpow_natCast, β rpow_mul hx, mul_inv_cancel hn0, rpow_one]
|
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 | 123 | 125 | theorem lcm_eq_zero_iff [Nontrivial Ξ±] : s.lcm f = 0 β 0 β f '' s := by |
simp only [Multiset.mem_map, lcm_def, Multiset.lcm_eq_zero_iff, Set.mem_image, mem_coe, β
Finset.mem_def]
|
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_series.well_known from "leanprover-community/mathlib"@"8199f6717c150a7fe91c4534175f4cf99725978f"
namespace PowerS... | Mathlib/RingTheory/PowerSeries/WellKnown.lean | 52 | 55 | theorem invUnitsSub_mul_X (u : RΛ£) : invUnitsSub u * X = invUnitsSub u * C R u - 1 := by |
ext (_ | n)
Β· simp
Β· simp [n.succ_ne_zero, pow_succ']
|
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.affine_subspace from "leanprover-community/mathlib"@"e96bdfbd1e8c98a09ff75f7ac6204d142debc840"
noncomputable section
open Affine
open Set
section
variable (k : Type*) {V : Type*} {P : Type*} [Ring k] [AddCommGroup V]... | Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | 815 | 818 | theorem card_pos_of_affineSpan_eq_top {ΞΉ : Type*} [Fintype ΞΉ] {p : ΞΉ β P}
(h : affineSpan k (range p) = β€) : 0 < Fintype.card ΞΉ := by |
obtain β¨-, β¨i, -β©β© := nonempty_of_affineSpan_eq_top k V P h
exact Fintype.card_pos_iff.mpr β¨iβ©
|
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section iInf
variable {ΞΉ : Sort*} {f g : ΞΉ β ββ₯0β}
variable {a b c d : ββ₯0β} {r p q : ββ₯0}
theorem toNNReal_iInf (hf : β i, f ... | Mathlib/Data/ENNReal/Real.lean | 556 | 561 | theorem toNNReal_iSup (hf : β i, f i β β) : (iSup f).toNNReal = β¨ i, (f i).toNNReal := by |
lift f to ΞΉ β ββ₯0 using hf
simp_rw [toNNReal_coe]
by_cases h : BddAbove (range f)
Β· rw [β coe_iSup h, toNNReal_coe]
Β· rw [NNReal.iSup_of_not_bddAbove h, iSup_coe_eq_top.2 h, top_toNNReal]
|
import Mathlib.Combinatorics.SimpleGraph.DegreeSum
import Mathlib.Combinatorics.SimpleGraph.Subgraph
#align_import combinatorics.simple_graph.matching from "leanprover-community/mathlib"@"138448ae98f529ef34eeb61114191975ee2ca508"
universe u
namespace SimpleGraph
variable {V : Type u} {G : SimpleGraph V} (M : Su... | Mathlib/Combinatorics/SimpleGraph/Matching.lean | 77 | 80 | theorem IsMatching.toEdge_eq_toEdge_of_adj {M : Subgraph G} {v w : V} (h : M.IsMatching)
(hv : v β M.verts) (hw : w β M.verts) (ha : M.Adj v w) :
h.toEdge β¨v, hvβ© = h.toEdge β¨w, hwβ© := by |
rw [h.toEdge_eq_of_adj hv ha, h.toEdge_eq_of_adj hw (M.symm ha), Subtype.mk_eq_mk, Sym2.eq_swap]
|
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Exponential
#align_import set_theory.ordinal.cantor_normal_form from "leanprover-community/mathlib"@"991ff3b5269848f6dd942ae8e9dd3c946035dc8b"
noncomputable section
universe u
open List
namespace Ordinal
@[elab_as_elim]
noncomputabl... | Mathlib/SetTheory/Ordinal/CantorNormalForm.lean | 108 | 109 | theorem CNF_of_lt {b o : Ordinal} (ho : o β 0) (hb : o < b) : CNF b o = [β¨0, oβ©] := by |
simp only [CNF_ne_zero ho, log_eq_zero hb, opow_zero, div_one, mod_one, CNF_zero]
|
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.LinearAlgebra.Projection
import Mathlib.Order.JordanHolder
import Mathlib.Order.CompactlyGenerated.Intervals
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import ring_theory.simple_module from "leanprover-community/mathlib"@"cce7f68a7eaadadf74c82bbac207... | Mathlib/RingTheory/SimpleModule.lean | 193 | 195 | theorem sSup_simples_le (N : Submodule R M) :
sSup { m : Submodule R M | IsSimpleModule R m β§ m β€ N } = N := by |
simpa only [isSimpleModule_iff_isAtom] using sSup_atoms_le_eq _
|
import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.ldl from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
variable {π : Type*} [RCLike π]
variable {n : Type*} [LinearOrder n] [IsWellOrder n (Β· < Β·)... | Mathlib/LinearAlgebra/Matrix/LDL.lean | 123 | 127 | theorem LDL.lower_conj_diag : LDL.lower hS * LDL.diag hS * (LDL.lower hS)α΄΄ = S := by |
rw [LDL.lower, conjTranspose_nonsing_inv, Matrix.mul_assoc,
Matrix.inv_mul_eq_iff_eq_mul_of_invertible (LDL.lowerInv hS),
Matrix.mul_inv_eq_iff_eq_mul_of_invertible]
exact LDL.diag_eq_lowerInv_conj hS
|
import Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho
import Mathlib.LinearAlgebra.Matrix.PosDef
#align_import linear_algebra.matrix.ldl from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
variable {π : Type*} [RCLike π]
variable {n : Type*} [LinearOrder n] [IsWellOrder n (Β· < Β·)... | Mathlib/LinearAlgebra/Matrix/LDL.lean | 57 | 66 | theorem LDL.lowerInv_eq_gramSchmidtBasis :
LDL.lowerInv hS =
((Pi.basisFun π n).toMatrix
(@gramSchmidtBasis π (n β π) _ (_ : _) (InnerProductSpace.ofMatrix hS.transpose) n _ _ _
(Pi.basisFun π n)))α΅ := by |
letI := NormedAddCommGroup.ofMatrix hS.transpose
letI := InnerProductSpace.ofMatrix hS.transpose
ext i j
rw [LDL.lowerInv, Basis.coePiBasisFun.toMatrix_eq_transpose, coe_gramSchmidtBasis]
rfl
|
import Mathlib.Data.Complex.Basic
import Mathlib.MeasureTheory.Integral.CircleIntegral
#align_import measure_theory.integral.circle_transform from "leanprover-community/mathlib"@"d11893b411025250c8e61ff2f12ccbd7ee35ab15"
open Set MeasureTheory Metric Filter Function
open scoped Interval Real
noncomputable secti... | Mathlib/MeasureTheory/Integral/CircleTransform.lean | 48 | 55 | theorem circleTransformDeriv_periodic (f : β β E) :
Periodic (circleTransformDeriv R z w f) (2 * Ο) := by |
have := periodic_circleMap
simp_rw [Periodic] at *
intro x
simp_rw [circleTransformDeriv, this]
congr 2
simp [this]
|
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Nat.Prime
import Mathlib.Data.List.Prime
import Mathlib.Data.List.Sort
import Mathlib.Data.List.Chain
#align_import data.nat.factors from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
open Bool Subtype
open Nat
namespac... | Mathlib/Data/Nat/Factors.lean | 45 | 45 | theorem factors_zero : factors 0 = [] := by | rw [factors]
|
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology.Instances.Sign
#align_import analysis.special_functions.trigonometric.angle from "leanprover-community/mathlib"@"213b0cff7bc5ab6696ee07cceec80829... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 501 | 504 | theorem abs_cos_eq_of_two_zsmul_eq {ΞΈ Ο : Angle} (h : (2 : β€) β’ ΞΈ = (2 : β€) β’ Ο) :
|cos ΞΈ| = |cos Ο| := by |
simp_rw [two_zsmul, β two_nsmul] at h
exact abs_cos_eq_of_two_nsmul_eq h
|
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 | 112 | 113 | theorem rpow_self_rpow_inv {y : β} (hy : y β 0) (x : ββ₯0) : (x ^ (1 / y)) ^ y = x := by |
field_simp [β rpow_mul]
|
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.BigOperators.Group.Multiset
import Mathlib.Tactic.NormNum.Basic
import Mathlib.Tactic.Positivity.Core
#align_import algebra.big_operators.order from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
open Function
... | Mathlib/Algebra/Order/BigOperators/Group/Finset.lean | 281 | 287 | theorem card_le_mul_card_image_of_maps_to {f : Ξ± β Ξ²} {s : Finset Ξ±} {t : Finset Ξ²}
(Hf : β a β s, f a β t) (n : β) (hn : β a β t, (s.filter fun x β¦ f x = a).card β€ n) :
s.card β€ n * t.card :=
calc
s.card = β a β t, (s.filter fun x β¦ f x = a).card := card_eq_sum_card_fiberwise Hf
_ β€ β _a β t, n := su... | simp [mul_comm]
|
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section Real
variable {a b c d : ββ₯0β} {r p q : ββ₯0}
theorem toReal_add (ha : a β β) (hb : b β β) : (a + b).toReal = a.toReal ... | Mathlib/Data/ENNReal/Real.lean | 509 | 510 | theorem toNNReal_div (a b : ββ₯0β) : (a / b).toNNReal = a.toNNReal / b.toNNReal := by |
rw [div_eq_mul_inv, toNNReal_mul, toNNReal_inv, div_eq_mul_inv]
|
import Mathlib.Algebra.Polynomial.Monic
#align_import algebra.polynomial.big_operators from "leanprover-community/mathlib"@"47adfab39a11a072db552f47594bf8ed2cf8a722"
open Finset
open Multiset
open Polynomial
universe u w
variable {R : Type u} {ΞΉ : Type w}
namespace Polynomial
variable (s : Finset ΞΉ)
sectio... | Mathlib/Algebra/Polynomial/BigOperators.lean | 145 | 156 | theorem leadingCoeff_multiset_prod' (h : (t.map leadingCoeff).prod β 0) :
t.prod.leadingCoeff = (t.map leadingCoeff).prod := by |
induction' t using Multiset.induction_on with a t ih; Β· simp
simp only [Multiset.map_cons, Multiset.prod_cons] at h β’
rw [Polynomial.leadingCoeff_mul']
Β· rw [ih]
simp only [ne_eq]
apply right_ne_zero_of_mul h
Β· rw [ih]
Β· exact h
simp only [ne_eq, not_false_eq_true]
apply right_ne_zero_of_... |
import Mathlib.Algebra.MonoidAlgebra.Basic
#align_import algebra.monoid_algebra.division from "leanprover-community/mathlib"@"72c366d0475675f1309d3027d3d7d47ee4423951"
variable {k G : Type*} [Semiring k]
namespace AddMonoidAlgebra
section
variable [AddCancelCommMonoid G]
noncomputable def divOf (x : k[G]) (g... | Mathlib/Algebra/MonoidAlgebra/Division.lean | 105 | 109 | theorem of'_mul_divOf (a : G) (x : k[G]) : of' k G a * x /α΅αΆ a = x := by |
refine Finsupp.ext fun _ => ?_ -- Porting note: `ext` doesn't work
rw [AddMonoidAlgebra.divOf_apply, of'_apply, single_mul_apply_aux, one_mul]
intro c
exact add_right_inj _
|
import Mathlib.Analysis.Complex.Basic
import Mathlib.Topology.FiberBundle.IsHomeomorphicTrivialBundle
#align_import analysis.complex.re_im_topology from "leanprover-community/mathlib"@"468b141b14016d54b479eb7a0fff1e360b7e3cf6"
open Set
noncomputable section
namespace Complex
theorem isHomeomorphicTrivialFiber... | Mathlib/Analysis/Complex/ReImTopology.lean | 164 | 165 | theorem frontier_setOf_lt_re (a : β) : frontier { z : β | a < z.re } = { z | z.re = a } := by |
simpa only [frontier_Ioi] using frontier_preimage_re (Ioi a)
|
import Mathlib.Algebra.CharP.Defs
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Data.Int.ModEq
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Dynamics.PeriodicPts
import Mathlib.GroupTheory.Index
import Mathlib.Order.Interval.Finset.Nat
import Mat... | Mathlib/GroupTheory/OrderOfElement.lean | 227 | 228 | theorem IsOfFinOrder.mono [Monoid Ξ²] {y : Ξ²} (hx : IsOfFinOrder x) (h : orderOf y β£ orderOf x) :
IsOfFinOrder y := by | rw [β orderOf_pos_iff] at hx β’; exact Nat.pos_of_dvd_of_pos h hx
|
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.MvPolynomial.Variables
#align_import data.mv_polynomial.monad from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
namespace MvPolynomial
open Finsupp
variable {Ο : Type*} {Ο : Type*}
variable {R S... | Mathlib/Algebra/MvPolynomial/Monad.lean | 339 | 342 | theorem bindβ_monomial (f : R β+* MvPolynomial Ο S) (d : Ο ββ β) (r : R) :
bindβ f (monomial d r) = f r * monomial d 1 := by |
simp only [monomial_eq, RingHom.map_mul, bindβ_C_right, Finsupp.prod, map_prod,
map_pow, bindβ_X_right, C_1, one_mul]
|
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
#align_import measure_theory.integral.set_to_l1 from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical Topology NNReal ENNReal MeasureTheory Pointwise
open Set Filter TopologicalSpace ENNR... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 1,198 | 1,207 | theorem norm_setToL1_le_norm_setToL1SCLM (hT : DominatedFinMeasAdditive ΞΌ T C) :
βsetToL1 hTβ β€ βsetToL1SCLM Ξ± E ΞΌ hTβ :=
calc
βsetToL1 hTβ β€ (1 : ββ₯0) * βsetToL1SCLM Ξ± E ΞΌ hTβ := by |
refine
ContinuousLinearMap.opNorm_extend_le (setToL1SCLM Ξ± E ΞΌ hT) (coeToLp Ξ± E β)
(simpleFunc.denseRange one_ne_top) fun x => le_of_eq ?_
rw [NNReal.coe_one, one_mul]
rfl
_ = βsetToL1SCLM Ξ± E ΞΌ hTβ := by rw [NNReal.coe_one, one_mul]
|
import Mathlib.MeasureTheory.Group.Action
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Group.Pointwise
#align_import measure_theory.group.fundamental_domain from "leanprover-community/mathlib"@"3b52265189f3fb43aa631edffce5d060fafaf82f"
open scoped ENNReal Pointwise Topology NNRea... | Mathlib/MeasureTheory/Group/FundamentalDomain.lean | 656 | 658 | theorem fundamentalInterior_smul [Group H] [MulAction H Ξ±] [SMulCommClass H G Ξ±] (g : H) :
fundamentalInterior G (g β’ s) = g β’ fundamentalInterior G s := by |
simp_rw [fundamentalInterior, smul_set_sdiff, smul_set_iUnion, smul_comm g (_ : G) (_ : Set Ξ±)]
|
import Mathlib.Tactic.FinCases
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Algebra.Field.IsField
#align_import ring_theory.ideal.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe u v w
variable {Ξ± : Type u} {Ξ² : Type v}
open ... | Mathlib/RingTheory/Ideal/Basic.lean | 274 | 280 | theorem not_isPrime_iff {I : Ideal Ξ±} :
Β¬I.IsPrime β I = β€ β¨ β (x : Ξ±) (_hx : x β I) (y : Ξ±) (_hy : y β I), x * y β I := by |
simp_rw [Ideal.isPrime_iff, not_and_or, Ne, Classical.not_not, not_forall, not_or]
exact
or_congr Iff.rfl
β¨fun β¨x, y, hxy, hx, hyβ© => β¨x, hx, y, hy, hxyβ©, fun β¨x, hx, y, hy, hxyβ© =>
β¨x, y, hxy, hx, hyβ©β©
|
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import topology.metric_space.hausdorff_distance from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
noncomputable section
open NNReal ENNReal Topology Set Filter Bornology
universe u v w
variable {ΞΉ : Sort*} {Ξ± : Type u} {Ξ² :... | Mathlib/Topology/MetricSpace/Thickening.lean | 425 | 426 | theorem thickening_closure : thickening Ξ΄ (closure s) = thickening Ξ΄ s := by |
simp_rw [thickening, infEdist_closure]
|
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 | 466 | 471 | theorem sin_angle_mul_dist_of_angle_eq_pi_div_two {pβ pβ pβ : P} (h : β pβ pβ pβ = Ο / 2) :
Real.sin (β pβ pβ pβ) * dist pβ pβ = dist pβ pβ := by |
rw [angle, β inner_eq_zero_iff_angle_eq_pi_div_two, real_inner_comm, β neg_eq_zero, β
inner_neg_left, neg_vsub_eq_vsub_rev] at h
rw [angle, dist_eq_norm_vsub V pβ pβ, dist_eq_norm_vsub V pβ pβ, β vsub_add_vsub_cancel pβ pβ pβ,
add_comm, sin_angle_add_mul_norm_of_inner_eq_zero h]
|
import Mathlib.Data.Real.Pi.Bounds
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.ConvexBody
-- TODO. Rewrite some of the FLT results on the disciminant using the definitions and results of
-- this file
namespace NumberField
open FiniteDimensional NumberField NumberField.InfinitePlace Matrix
open sco... | Mathlib/NumberTheory/NumberField/Discriminant.lean | 46 | 48 | theorem discr_ne_zero : discr K β 0 := by |
rw [β (Int.cast_injective (Ξ± := β)).ne_iff, coe_discr]
exact Algebra.discr_not_zero_of_basis β (integralBasis K)
|
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