Context stringlengths 57 85k | file_name stringlengths 21 79 | start int64 14 2.42k | end int64 18 2.43k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import computability.tm_to_partrec from "leanprover-community/mathlib"@"6155d4351090a6fad236e3d2e4e0e4e7342668e8"
open Function (update)
open Relation
namespa... | Mathlib/Computability/TMToPartrec.lean | 163 | 166 | theorem fix_eval (f) : (fix f).eval =
PFun.fix fun v => (f.eval v).map fun v =>
if v.headI = 0 then Sum.inl v.tail else Sum.inr v.tail := by |
simp [eval]
|
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/Variables.lean | 71 | 73 | theorem vars_def [DecidableEq σ] (p : MvPolynomial σ R) : p.vars = p.degrees.toFinset := by |
rw [vars]
convert rfl
|
import Mathlib.LinearAlgebra.Matrix.BilinearForm
import Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.LinearAlgebra.Trace
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosu... | Mathlib/RingTheory/Trace.lean | 169 | 176 | theorem trace_prod_apply [Module.Free R S] [Module.Free R T] [Module.Finite R S] [Module.Finite R T]
(x : S × T) : trace R (S × T) x = trace R S x.fst + trace R T x.snd := by |
nontriviality R
let f := (lmul R S).toLinearMap.prodMap (lmul R T).toLinearMap
have : (lmul R (S × T)).toLinearMap = (prodMapLinear R S T S T R).comp f :=
LinearMap.ext₂ Prod.mul_def
simp_rw [trace, this]
exact trace_prodMap' _ _
|
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 | 419 | 420 | theorem mul_le_of_one_le_right (ha : a ≤ 0) (h : 1 ≤ b) : a * b ≤ a := by |
simpa only [mul_one] using mul_le_mul_of_nonpos_left h ha
|
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 | 162 | 163 | theorem sameCycle_pow_right {n : ℕ} : SameCycle f x ((f ^ n) y) ↔ SameCycle f x y := by |
rw [← zpow_natCast, sameCycle_zpow_right]
|
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 | 440 | 444 | theorem ListBlank.map_modifyNth {Γ Γ'} [Inhabited Γ] [Inhabited Γ'] (F : PointedMap Γ Γ')
(f : Γ → Γ) (f' : Γ' → Γ') (H : ∀ x, F (f x) = f' (F x)) (n) (L : ListBlank Γ) :
(L.modifyNth f n).map F = (L.map F).modifyNth f' n := by |
induction' n with n IH generalizing L <;>
simp only [*, ListBlank.head_map, ListBlank.modifyNth, ListBlank.map_cons, ListBlank.tail_map]
|
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 | 1,021 | 1,022 | theorem oangle_sign_sub_left_swap (x y : V) : (o.oangle (x - y) x).sign = (o.oangle x y).sign := by |
rw [oangle_sign_sub_left_eq_neg, o.oangle_rev y x, Real.Angle.sign_neg]
|
import Mathlib.NumberTheory.Harmonic.Defs
import Mathlib.NumberTheory.Padics.PadicNumbers
theorem padicValRat_two_harmonic (n : ℕ) : padicValRat 2 (harmonic n) = -Nat.log 2 n := by
induction' n with n ih
· simp
· rcases eq_or_ne n 0 with rfl | hn
· simp
rw [harmonic_succ]
have key : padicValRat 2... | Mathlib/NumberTheory/Harmonic/Int.lean | 42 | 46 | theorem harmonic_not_int {n : ℕ} (h : 2 ≤ n) : ¬ (harmonic n).isInt := by |
apply padicNorm.not_int_of_not_padic_int 2
rw [padicNorm.eq_zpow_of_nonzero (harmonic_pos (ne_zero_of_lt h)).ne',
padicValRat_two_harmonic, neg_neg, zpow_natCast]
exact one_lt_pow one_lt_two (Nat.log_pos one_lt_two h).ne'
|
import Mathlib.Probability.Variance
#align_import probability.moments from "leanprover-community/mathlib"@"85453a2a14be8da64caf15ca50930cf4c6e5d8de"
open MeasureTheory Filter Finset Real
noncomputable section
open scoped MeasureTheory ProbabilityTheory ENNReal NNReal
namespace ProbabilityTheory
variable {Ω ι ... | Mathlib/Probability/Moments.lean | 210 | 210 | theorem mgf_neg : mgf (-X) μ t = mgf X μ (-t) := by | simp_rw [mgf, Pi.neg_apply, mul_neg, neg_mul]
|
import Mathlib.Logic.Pairwise
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathlib.Order.GaloisConnection
#align_import data.set.lattice from "leanprover-community/mathlib"@"b86832321b586c6ac23ef8cdef6a7a27e42b13bd"
open Function Set
universe u
variable {α β γ : Type*} {ι ι' ι... | Mathlib/Data/Set/Lattice.lean | 1,748 | 1,749 | theorem preimage_sInter {f : α → β} {s : Set (Set β)} : f ⁻¹' ⋂₀ s = ⋂ t ∈ s, f ⁻¹' t := by |
rw [sInter_eq_biInter, preimage_iInter₂]
|
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
#align_import measure_theory.function.egorov from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open MeasureTheory NNReal ENNReal Topology
namespace MeasureTheory
open Set Filt... | Mathlib/MeasureTheory/Function/Egorov.lean | 98 | 107 | theorem exists_notConvergentSeq_lt (hε : 0 < ε) (hf : ∀ n, StronglyMeasurable (f n))
(hg : StronglyMeasurable g) (hsm : MeasurableSet s) (hs : μ s ≠ ∞)
(hfg : ∀ᵐ x ∂μ, x ∈ s → Tendsto (fun n => f n x) atTop (𝓝 (g x))) (n : ℕ) :
∃ j : ι, μ (s ∩ notConvergentSeq f g n j) ≤ ENNReal.ofReal (ε * 2⁻¹ ^ n) := by |
have ⟨N, hN⟩ := (ENNReal.tendsto_atTop ENNReal.zero_ne_top).1
(measure_notConvergentSeq_tendsto_zero hf hg hsm hs hfg n) (ENNReal.ofReal (ε * 2⁻¹ ^ n)) (by
rw [gt_iff_lt, ENNReal.ofReal_pos]
exact mul_pos hε (pow_pos (by norm_num) n))
rw [zero_add] at hN
exact ⟨N, (hN N le_rfl).2⟩
|
import Mathlib.CategoryTheory.Monoidal.Free.Basic
import Mathlib.CategoryTheory.Groupoid
import Mathlib.CategoryTheory.DiscreteCategory
#align_import category_theory.monoidal.free.coherence from "leanprover-community/mathlib"@"f187f1074fa1857c94589cc653c786cadc4c35ff"
universe u
namespace CategoryTheory
open Mo... | Mathlib/CategoryTheory/Monoidal/Free/Coherence.lean | 91 | 95 | theorem inclusion_map {X Y : N C} (f : X ⟶ Y) :
inclusion.map f = eqToHom (congr_arg _ (Discrete.ext _ _ (Discrete.eq_of_hom f))) := by |
rcases f with ⟨⟨⟩⟩
cases Discrete.ext _ _ (by assumption)
apply inclusion.map_id
|
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Group.Int
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Ring.Rat
import Mathlib.Data.PNat.Defs
#align_import data.rat.lemmas from "leanprover-community/mathlib"@"550b58538991c8977703fdeb7c9d51a5aa27df11"
namespace Rat
o... | Mathlib/Data/Rat/Lemmas.lean | 255 | 257 | theorem inv_natCast_den_of_pos {a : ℕ} (ha0 : 0 < a) : (a : ℚ)⁻¹.den = a := by |
rw [← Int.ofNat_inj, ← Int.cast_natCast a, inv_intCast_den_of_pos]
rwa [Int.natCast_pos]
|
import Mathlib.Analysis.Complex.RemovableSingularity
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Analysis.NormedSpace.FunctionSeries
#align_import analysis.complex.locally_uniform_limit from "leanprover-community/mathlib"@"fe44cd36149e675eb5dec87acc7e8f1d6568e081"
open Set Metric MeasureThe... | Mathlib/Analysis/Complex/LocallyUniformLimit.lean | 79 | 86 | theorem norm_cderiv_lt (hr : 0 < r) (hfM : ∀ w ∈ sphere z r, ‖f w‖ < M)
(hf : ContinuousOn f (sphere z r)) : ‖cderiv r f z‖ < M / r := by |
obtain ⟨L, hL1, hL2⟩ : ∃ L < M, ∀ w ∈ sphere z r, ‖f w‖ ≤ L := by
have e1 : (sphere z r).Nonempty := NormedSpace.sphere_nonempty.mpr hr.le
have e2 : ContinuousOn (fun w => ‖f w‖) (sphere z r) := continuous_norm.comp_continuousOn hf
obtain ⟨x, hx, hx'⟩ := (isCompact_sphere z r).exists_isMaxOn e1 e2
ex... |
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.Order.Circular
import Mathlib.Data.List.TFAE
import Mathlib.Data.Set.Lattice
#align_import a... | Mathlib/Algebra/Order/ToIntervalMod.lean | 209 | 211 | theorem toIocMod_apply_left (a : α) : toIocMod hp a a = a + p := by |
rw [toIocMod_eq_iff hp, Set.right_mem_Ioc]
exact ⟨lt_add_of_pos_right _ hp, -1, by simp⟩
|
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Computability.Primrec
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Linarith
#align_import computability.ackermann from "leanprover-community/mathlib"@"9b2660e1b25419042c8da10bf411aa3c67f14383"
open Nat
def ack : ℕ → ℕ → ℕ
| 0, n => n + 1
| m + 1, 0 ... | Mathlib/Computability/Ackermann.lean | 251 | 256 | theorem ack_succ_right_le_ack_succ_left (m n : ℕ) : ack m (n + 1) ≤ ack (m + 1) n := by |
cases' n with n n
· simp
· rw [ack_succ_succ]
apply ack_mono_right m (le_trans _ <| add_add_one_le_ack _ n)
omega
|
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts
#align_import category_theory.limits.constructions.zero_objects from "leanprover-community/mathlib"@"52a270e2ea4e342c2587c106f8be904524214a4... | Mathlib/CategoryTheory/Limits/Constructions/ZeroObjects.lean | 170 | 173 | theorem pullbackZeroZeroIso_inv_fst (X Y : C) [HasBinaryProduct X Y] :
(pullbackZeroZeroIso X Y).inv ≫ pullback.fst = prod.fst := by |
dsimp [pullbackZeroZeroIso]
simp
|
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Topology.Order.ProjIcc
#align_import analysis.special_functions.trigonometric.inverse from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open Topology Filter
open S... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean | 436 | 436 | theorem arccos_le_pi_div_two {x} : arccos x ≤ π / 2 ↔ 0 ≤ x := by | simp [arccos]
|
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.MeasureTheory.Covering.OneDim
import Mathlib.Order.Monotone.Extension
#align_import analysis.calculus.monotone from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
open Set Filter Function Metric MeasureTheory MeasureTheory.Meas... | Mathlib/Analysis/Calculus/Monotone.lean | 67 | 131 | theorem StieltjesFunction.ae_hasDerivAt (f : StieltjesFunction) :
∀ᵐ x, HasDerivAt f (rnDeriv f.measure volume x).toReal x := by |
/- Denote by `μ` the Stieltjes measure associated to `f`.
The general theorem `VitaliFamily.ae_tendsto_rnDeriv` ensures that `μ [x, y] / (y - x)` tends
to the Radon-Nikodym derivative as `y` tends to `x` from the right. As `μ [x,y] = f y - f (x^-)`
and `f (x^-) = f x` almost everywhere, this gives differ... |
import Mathlib.Data.Multiset.Nodup
import Mathlib.Data.List.NatAntidiagonal
#align_import data.multiset.nat_antidiagonal from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
namespace Nat
def antidiagonal (n : ℕ) : Multiset (ℕ × ℕ) :=
List.Nat.antidiagonal n
#align... | Mathlib/Data/Multiset/NatAntidiagonal.lean | 42 | 43 | theorem card_antidiagonal (n : ℕ) : card (antidiagonal n) = n + 1 := by |
rw [antidiagonal, coe_card, List.Nat.length_antidiagonal]
|
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Init.Data.Ordering.Lemmas
import Mathlib.SetTheory.Ordinal.Principal
import Mathlib.Tactic.NormNum
#align_import set_theory.ordinal.notation from "leanprover-community/mathlib"@"b67044ba53af18680e1dd246861d9584e968495d"
set_option linter.uppercaseLean3 ... | Mathlib/SetTheory/Ordinal/Notation.lean | 224 | 225 | theorem NFBelow.fst {e n a b} (h : NFBelow (ONote.oadd e n a) b) : NF e := by |
cases' h with _ _ _ _ eb _ h₁ h₂ h₃; exact ⟨⟨_, h₁⟩⟩
|
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.BigOperators.Ring.Multiset
import Mathlib.Algebra.Field.Defs
import Mathlib.Data.Fintype.Powerset
import Mathlib.Data.Int.Cast.Lemmas
#align_import algebra.big_operators.ring from "leanprover-community/mathlib"@"b2c89893177f66a48daf993b7b... | Mathlib/Algebra/BigOperators/Ring.lean | 118 | 123 | theorem prod_add_prod_eq {s : Finset ι} {i : ι} {f g h : ι → α} (hi : i ∈ s)
(h1 : g i + h i = f i) (h2 : ∀ j ∈ s, j ≠ i → g j = f j) (h3 : ∀ j ∈ s, j ≠ i → h j = f j) :
(∏ i ∈ s, g i) + ∏ i ∈ s, h i = ∏ i ∈ s, f i := by |
classical
simp_rw [prod_eq_mul_prod_diff_singleton hi, ← h1, right_distrib]
congr 2 <;> apply prod_congr rfl <;> simpa
|
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
#align_import category_theory.limits.shapes.strict_initial from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe v u
namespace CategoryTheory
namespace Limits
open C... | Mathlib/CategoryTheory/Limits/Shapes/StrictInitial.lean | 74 | 77 | theorem IsInitial.strict_hom_ext (hI : IsInitial I) {A : C} (f g : A ⟶ I) : f = g := by |
haveI := hI.isIso_to f
haveI := hI.isIso_to g
exact eq_of_inv_eq_inv (hI.hom_ext (inv f) (inv g))
|
import Mathlib.AlgebraicGeometry.Spec
import Mathlib.Algebra.Category.Ring.Constructions
import Mathlib.CategoryTheory.Elementwise
#align_import algebraic_geometry.Scheme from "leanprover-community/mathlib"@"88474d1b5af6d37c2ab728b757771bced7f5194c"
-- Explicit universe annotations were used in this file to improv... | Mathlib/AlgebraicGeometry/Scheme.lean | 160 | 167 | theorem app_eq {X Y : Scheme} (f : X ⟶ Y) {U V : Opens Y.carrier} (e : U = V) :
f.val.c.app (op U) =
Y.presheaf.map (eqToHom e.symm).op ≫
f.val.c.app (op V) ≫
X.presheaf.map (eqToHom (congr_arg (Opens.map f.val.base).obj e)).op := by |
rw [← IsIso.inv_comp_eq, ← Functor.map_inv, f.val.c.naturality, Presheaf.pushforwardObj_map]
cases e
rfl
|
import Mathlib.MeasureTheory.Measure.FiniteMeasure
import Mathlib.MeasureTheory.Integral.Average
#align_import measure_theory.measure.probability_measure from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open MeasureTheory
open Set
open Filter
open BoundedCon... | Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean | 216 | 220 | theorem eq_of_forall_toMeasure_apply_eq (μ ν : ProbabilityMeasure Ω)
(h : ∀ s : Set Ω, MeasurableSet s → (μ : Measure Ω) s = (ν : Measure Ω) s) : μ = ν := by |
apply toMeasure_injective
ext1 s s_mble
exact h s s_mble
|
import Mathlib.Analysis.PSeries
import Mathlib.Data.Real.Pi.Wallis
import Mathlib.Tactic.AdaptationNote
#align_import analysis.special_functions.stirling from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open scoped Topology Real Nat Asymptotics
open Finset Filter Nat Real
namespace... | Mathlib/Analysis/SpecialFunctions/Stirling.lean | 56 | 57 | theorem stirlingSeq_zero : stirlingSeq 0 = 0 := by |
rw [stirlingSeq, cast_zero, mul_zero, Real.sqrt_zero, zero_mul, div_zero]
|
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import analysis.special_functions.arsinh from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open Function Filter Set
open scoped Topology
name... | Mathlib/Analysis/SpecialFunctions/Arsinh.lean | 65 | 65 | theorem arsinh_zero : arsinh 0 = 0 := by | simp [arsinh]
|
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Init.Data.Ordering.Lemmas
import Mathlib.SetTheory.Ordinal.Principal
import Mathlib.Tactic.NormNum
#align_import set_theory.ordinal.notation from "leanprover-community/mathlib"@"b67044ba53af18680e1dd246861d9584e968495d"
set_option linter.uppercaseLean3 ... | Mathlib/SetTheory/Ordinal/Notation.lean | 891 | 965 | theorem repr_opow_aux₂ {a0 a'} [N0 : NF a0] [Na' : NF a'] (m : ℕ) (d : ω ∣ repr a')
(e0 : repr a0 ≠ 0) (h : repr a' + m < (ω ^ repr a0)) (n : ℕ+) (k : ℕ) :
let R := repr (opowAux 0 a0 (oadd a0 n a' * ofNat m) k m)
(k ≠ 0 → R < ((ω ^ repr a0) ^ succ (k : Ordinal))) ∧
((ω ^ repr a0) ^ (k : Ordinal)) * (... |
intro R'
haveI No : NF (oadd a0 n a') :=
N0.oadd n (Na'.below_of_lt' <| lt_of_le_of_lt (le_add_right _ _) h)
induction' k with k IH
· cases m <;> simp [R', opowAux]
-- rename R => R'
let R := repr (opowAux 0 a0 (oadd a0 n a' * ofNat m) k m)
let ω0 := ω ^ repr a0
let α' := ω0 * n + repr a'
change ... |
import Mathlib.Geometry.Manifold.MFDeriv.FDeriv
noncomputable section
open scoped Manifold
open Bundle Set Topology
section SpecificFunctions
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [NormedAddCommGroup E]
[NormedSpace 𝕜 E] {H : Type*} [TopologicalSpace H] (I : ModelWithCorners 𝕜 E H)... | Mathlib/Geometry/Manifold/MFDeriv/SpecificFunctions.lean | 349 | 353 | theorem mfderivWithin_snd {s : Set (M × M')} {x : M × M'}
(hxs : UniqueMDiffWithinAt (I.prod I') s x) :
mfderivWithin (I.prod I') I' Prod.snd s x =
ContinuousLinearMap.snd 𝕜 (TangentSpace I x.1) (TangentSpace I' x.2) := by |
rw [MDifferentiable.mfderivWithin (mdifferentiableAt_snd I I') hxs]; exact mfderiv_snd I I'
|
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 | 43 | 73 | theorem exists_smooth_tsupport_subset {s : Set E} {x : E} (hs : s ∈ 𝓝 x) :
∃ f : E → ℝ,
tsupport f ⊆ s ∧ HasCompactSupport f ∧ ContDiff ℝ ⊤ f ∧ range f ⊆ Icc 0 1 ∧ f x = 1 := by |
obtain ⟨d : ℝ, d_pos : 0 < d, hd : Euclidean.closedBall x d ⊆ s⟩ :=
Euclidean.nhds_basis_closedBall.mem_iff.1 hs
let c : ContDiffBump (toEuclidean x) :=
{ rIn := d / 2
rOut := d
rIn_pos := half_pos d_pos
rIn_lt_rOut := half_lt_self d_pos }
let f : E → ℝ := c ∘ toEuclidean
have f_supp ... |
import Mathlib.CategoryTheory.NatIso
#align_import category_theory.bicategory.basic from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
namespace CategoryTheory
universe w v u
open Category Iso
-- intended to be used with explicit universe parameters
@[nolint checkUnivs]
class Bicate... | Mathlib/CategoryTheory/Bicategory/Basic.lean | 354 | 355 | theorem associator_naturality_middle (f : a ⟶ b) {g g' : b ⟶ c} (η : g ⟶ g') (h : c ⟶ d) :
(f ◁ η) ▷ h ≫ (α_ f g' h).hom = (α_ f g h).hom ≫ f ◁ η ▷ h := by | simp
|
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.LinearAlgebra.Matrix.Orthogonal
import Mathlib.Data.Matrix.Kronecker
#align_import linear_algebra.matrix.is_diag from "leanprover-community/mathlib"@"55e2dfde0cff928ce5c70926a3f2c7dee3e2dd99"
namespace Matrix
variable {α β R n m : Type*}
open Function... | Mathlib/LinearAlgebra/Matrix/IsDiag.lean | 82 | 84 | theorem IsDiag.neg [AddGroup α] {A : Matrix n n α} (ha : A.IsDiag) : (-A).IsDiag := by |
intro i j h
simp [ha h]
|
import Mathlib.Analysis.LocallyConvex.Basic
#align_import analysis.locally_convex.balanced_core_hull from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Pointwise Topology Filter
variable {𝕜 E ι : Type*}
section balancedHull
section SeminormedRing
variable [SeminormedRing ... | Mathlib/Analysis/LocallyConvex/BalancedCoreHull.lean | 85 | 90 | theorem smul_balancedCore_subset (s : Set E) {a : 𝕜} (ha : ‖a‖ ≤ 1) :
a • balancedCore 𝕜 s ⊆ balancedCore 𝕜 s := by |
rintro x ⟨y, hy, rfl⟩
rw [mem_balancedCore_iff] at hy
rcases hy with ⟨t, ht1, ht2, hy⟩
exact ⟨t, ⟨ht1, ht2⟩, ht1 a ha (smul_mem_smul_set hy)⟩
|
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 | 830 | 837 | theorem _root_.TopologicalSpace.IsSeparable.exists_countable_dense_subset
{s : Set α} (hs : IsSeparable s) : ∃ t, t ⊆ s ∧ t.Countable ∧ s ⊆ closure t := by |
have : ∀ ε > 0, ∃ t : Set α, t.Countable ∧ s ⊆ ⋃ x ∈ t, closedBall x ε := fun ε ε0 => by
rcases hs with ⟨t, htc, hst⟩
refine ⟨t, htc, hst.trans fun x hx => ?_⟩
rcases mem_closure_iff.1 hx ε ε0 with ⟨y, hyt, hxy⟩
exact mem_iUnion₂.2 ⟨y, hyt, mem_closedBall.2 hxy.le⟩
exact subset_countable_closure_of... |
import Mathlib.Order.Zorn
import Mathlib.Order.Atoms
#align_import order.zorn_atoms from "leanprover-community/mathlib"@"9aba7801eeecebb61f58a5763c2b6dd1b47dc6ef"
open Set
| Mathlib/Order/ZornAtoms.lean | 24 | 36 | theorem IsCoatomic.of_isChain_bounded {α : Type*} [PartialOrder α] [OrderTop α]
(h :
∀ c : Set α,
IsChain (· ≤ ·) c → c.Nonempty → ⊤ ∉ c → ∃ x ≠ ⊤, x ∈ upperBounds c) :
IsCoatomic α := by |
refine ⟨fun x => le_top.eq_or_lt.imp_right fun hx => ?_⟩
have : ∃ y ∈ Ico x ⊤, x ≤ y ∧ ∀ z ∈ Ico x ⊤, y ≤ z → z = y := by
refine zorn_nonempty_partialOrder₀ (Ico x ⊤) (fun c hxc hc y hy => ?_) x (left_mem_Ico.2 hx)
rcases h c hc ⟨y, hy⟩ fun h => (hxc h).2.ne rfl with ⟨z, hz, hcz⟩
exact ⟨z, ⟨le_trans (h... |
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.LinearAlgebra.Determinant
variable {K n : Type*} [NormedField K] [Fintype n] [DecidableEq n] {A : Matrix n n K}
theorem eigenvalue_mem_ball {μ : K} (hμ : Module.End.HasEigenvalue (Matrix.toLin' A) μ) :
∃ k,... | Mathlib/LinearAlgebra/Matrix/Gershgorin.lean | 59 | 66 | theorem det_ne_zero_of_sum_row_lt_diag (h : ∀ k, ∑ j ∈ Finset.univ.erase k, ‖A k j‖ < ‖A k k‖) :
A.det ≠ 0 := by |
contrapose! h
suffices ∃ k, 0 ∈ Metric.closedBall (A k k) (∑ j ∈ Finset.univ.erase k, ‖A k j‖) by
exact this.imp (fun a h ↦ by rwa [mem_closedBall_iff_norm', sub_zero] at h)
refine eigenvalue_mem_ball ?_
rw [Module.End.HasEigenvalue, Module.End.eigenspace_zero, ne_comm]
exact ne_of_lt (LinearMap.bot_lt_... |
import Mathlib.Topology.Algebra.Order.Compact
import Mathlib.Topology.MetricSpace.PseudoMetric
open Set Filter
universe u v w
variable {α : Type u} {β : Type v} {X ι : Type*}
section ProperSpace
open Metric
class ProperSpace (α : Type u) [PseudoMetricSpace α] : Prop where
isCompact_closedBall : ∀ x : α, ∀ r... | Mathlib/Topology/MetricSpace/ProperSpace.lean | 134 | 144 | theorem exists_pos_lt_subset_ball (hr : 0 < r) (hs : IsClosed s) (h : s ⊆ ball x r) :
∃ r' ∈ Ioo 0 r, s ⊆ ball x r' := by |
rcases eq_empty_or_nonempty s with (rfl | hne)
· exact ⟨r / 2, ⟨half_pos hr, half_lt_self hr⟩, empty_subset _⟩
have : IsCompact s :=
(isCompact_closedBall x r).of_isClosed_subset hs (h.trans ball_subset_closedBall)
obtain ⟨y, hys, hy⟩ : ∃ y ∈ s, s ⊆ closedBall x (dist y x) :=
this.exists_isMaxOn hne (c... |
import Mathlib.MeasureTheory.Constructions.Cylinders
import Mathlib.MeasureTheory.Measure.Typeclasses
open Set
namespace MeasureTheory
variable {ι : Type*} {α : ι → Type*} [∀ i, MeasurableSpace (α i)]
{P : ∀ J : Finset ι, Measure (∀ j : J, α j)}
def IsProjectiveMeasureFamily (P : ∀ J : Finset ι, Measure (∀ j ... | Mathlib/MeasureTheory/Constructions/Projective.lean | 143 | 150 | theorem unique [∀ i, IsFiniteMeasure (P i)]
(hμ : IsProjectiveLimit μ P) (hν : IsProjectiveLimit ν P) :
μ = ν := by |
haveI : IsFiniteMeasure μ := hμ.isFiniteMeasure
refine ext_of_generate_finite (measurableCylinders α) generateFrom_measurableCylinders.symm
isPiSystem_measurableCylinders (fun s hs ↦ ?_) (hμ.measure_univ_unique hν)
obtain ⟨I, S, hS, rfl⟩ := (mem_measurableCylinders _).mp hs
rw [hμ.measure_cylinder _ hS, hν... |
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.SetTheory.Cardinal.Cofinality
#align_import linear_algebra.basis from "leanprover-communit... | Mathlib/LinearAlgebra/Basis.lean | 240 | 246 | theorem coe_sumCoords_of_fintype [Fintype ι] : (b.sumCoords : M → R) = ∑ i, b.coord i := by |
ext m
-- Porting note: - `eq_self_iff_true`
-- + `comp_apply` `LinearMap.coeFn_sum`
simp only [sumCoords, Finsupp.sum_fintype, LinearMap.id_coe, LinearEquiv.coe_coe, coord_apply,
id, Fintype.sum_apply, imp_true_iff, Finsupp.coe_lsum, LinearMap.coe_comp, comp_apply,
LinearMap.coeFn_sum]
|
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 | 62 | 65 | theorem hasStrictDerivAt_of_bilinear (hu : HasStrictDerivAt u u' x) (hv : HasStrictDerivAt v v' x) :
HasStrictDerivAt (fun x ↦ B (u x) (v x)) (B (u x) v' + B u' (v x)) x := by |
simpa using
(B.hasStrictFDerivAt_of_bilinear hu.hasStrictFDerivAt hv.hasStrictFDerivAt).hasStrictDerivAt
|
import Mathlib.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.oriented.affine from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 687 | 763 | theorem _root_.Collinear.oangle_sign_of_sameRay_vsub {p₁ p₂ p₃ p₄ : P} (p₅ : P) (hp₁p₂ : p₁ ≠ p₂)
(hp₃p₄ : p₃ ≠ p₄) (hc : Collinear ℝ ({p₁, p₂, p₃, p₄} : Set P))
(hr : SameRay ℝ (p₂ -ᵥ p₁) (p₄ -ᵥ p₃)) : (∡ p₁ p₅ p₂).sign = (∡ p₃ p₅ p₄).sign := by |
by_cases hc₅₁₂ : Collinear ℝ ({p₅, p₁, p₂} : Set P)
· have hc₅₁₂₃₄ : Collinear ℝ ({p₅, p₁, p₂, p₃, p₄} : Set P) :=
(hc.collinear_insert_iff_of_ne (Set.mem_insert _ _)
(Set.mem_insert_of_mem _ (Set.mem_insert _ _)) hp₁p₂).2 hc₅₁₂
have hc₅₃₄ : Collinear ℝ ({p₅, p₃, p₄} : Set P) :=
(hc.colline... |
import Mathlib.CategoryTheory.Subobject.Limits
#align_import algebra.homology.image_to_kernel from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a148bf9ff"
universe v u w
open CategoryTheory CategoryTheory.Limits
variable {ι : Type*}
variable {V : Type u} [Category.{v} V] [HasZeroMorphisms V]
o... | Mathlib/Algebra/Homology/ImageToKernel.lean | 147 | 152 | theorem imageToKernel_comp_hom_inv_comp [HasEqualizers V] [HasImages V] {Z : V} {i : B ≅ Z} (w) :
imageToKernel (f ≫ i.hom) (i.inv ≫ g) w =
(imageSubobjectCompIso _ _).hom ≫
imageToKernel f g (by simpa using w) ≫ (kernelSubobjectIsoComp i.inv g).inv := by |
ext
simp
|
import Mathlib.Analysis.Complex.UpperHalfPlane.Topology
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Geometry.Euclidean.Inversion.Basic
#align_import analysis.complex.upper_half_plane.metric from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c"
noncomputable section
ope... | Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | 71 | 73 | theorem cosh_dist (z w : ℍ) : cosh (dist z w) = 1 + dist (z : ℂ) w ^ 2 / (2 * z.im * w.im) := by |
rw [dist_eq, cosh_two_mul, cosh_sq', add_assoc, ← two_mul, sinh_arsinh, div_pow, mul_pow,
sq_sqrt, sq (2 : ℝ), mul_assoc, ← mul_div_assoc, mul_assoc, mul_div_mul_left] <;> positivity
|
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Logic.Embedding.Basic
import Mathlib.Data.Fintype.CardEmbedding
import Mathlib.Topology.Algebra.Module.Multilinear.Topology
#align_import analysis.normed_space.multilinear from "leanprover-community/mathlib"@"f40476639bac089693a489c9e354ebd75... | Mathlib/Analysis/NormedSpace/Multilinear/Basic.lean | 113 | 123 | theorem bound_of_shell_of_norm_map_coord_zero (hf₀ : ∀ {m i}, ‖m i‖ = 0 → ‖f m‖ = 0)
{ε : ι → ℝ} {C : ℝ} (hε : ∀ i, 0 < ε i) {c : ι → 𝕜} (hc : ∀ i, 1 < ‖c i‖)
(hf : ∀ m : ∀ i, E i, (∀ i, ε i / ‖c i‖ ≤ ‖m i‖) → (∀ i, ‖m i‖ < ε i) → ‖f m‖ ≤ C * ∏ i, ‖m i‖)
(m : ∀ i, E i) : ‖f m‖ ≤ C * ∏ i, ‖m i‖ := by |
rcases em (∃ i, ‖m i‖ = 0) with (⟨i, hi⟩ | hm)
· rw [hf₀ hi, prod_eq_zero (mem_univ i) hi, mul_zero]
push_neg at hm
choose δ hδ0 hδm_lt hle_δm _ using fun i => rescale_to_shell_semi_normed (hc i) (hε i) (hm i)
have hδ0 : 0 < ∏ i, ‖δ i‖ := prod_pos fun i _ => norm_pos_iff.2 (hδ0 i)
simpa [map_smul_univ, nor... |
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 | 310 | 311 | theorem coeff_X_pow (m n : ℕ) : coeff R m ((X : R⟦X⟧) ^ n) = if m = n then 1 else 0 := by |
rw [X_pow_eq, coeff_monomial]
|
import Mathlib.Data.Nat.Prime
import Mathlib.Tactic.NormNum.Basic
#align_import data.nat.prime_norm_num from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
open Nat Qq Lean Meta
namespace Mathlib.Meta.NormNum
theorem not_prime_mul_of_ble (a b n : ℕ) (h : a * b = n) (h₁ : a.ble 1 = fals... | Mathlib/Tactic/NormNum/Prime.lean | 58 | 65 | theorem minFacHelper_0 (n : ℕ)
(h1 : Nat.ble (nat_lit 2) n = true) (h2 : nat_lit 1 = n % (nat_lit 2)) :
MinFacHelper n (nat_lit 3) := by |
refine ⟨by norm_num, by norm_num, ?_⟩
refine (le_minFac'.mpr λ p hp hpn ↦ ?_).resolve_left (Nat.ne_of_gt (Nat.le_of_ble_eq_true h1))
rcases hp.eq_or_lt with rfl|h
· simp [(Nat.dvd_iff_mod_eq_zero ..).1 hpn] at h2
· exact h
|
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.JacobsonIdeal
#align_import ring_theory.jacobson from "leanprover-community/mathlib"@"a7c017d750512a352b623b1824d75da5998457d0"
set_option autoImplicit true
universe u
namespace Ideal
open Polynomial
... | Mathlib/RingTheory/Jacobson.lean | 70 | 78 | theorem isJacobson_iff_prime_eq : IsJacobson R ↔ ∀ P : Ideal R, IsPrime P → P.jacobson = P := by |
refine isJacobson_iff.trans ⟨fun h I hI => h I hI.isRadical, ?_⟩
refine fun h I hI ↦ le_antisymm (fun x hx ↦ ?_) (fun x hx ↦ mem_sInf.mpr fun _ hJ ↦ hJ.left hx)
rw [← hI.radical, radical_eq_sInf I, mem_sInf]
intro P hP
rw [Set.mem_setOf_eq] at hP
erw [mem_sInf] at hx
erw [← h P hP.right, mem_sInf]
exac... |
import Mathlib.CategoryTheory.Category.Grpd
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Homotopy.Path
import Mathlib.Data.Set.Subsingleton
#align_import algebraic_topology.fundamental_groupoid.basic from "leanprover-community/mathlib"@"3d7987cda72abc473... | Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean | 131 | 135 | theorem continuous_transReflReparamAux : Continuous transReflReparamAux := by |
refine continuous_if_le ?_ ?_ (Continuous.continuousOn ?_) (Continuous.continuousOn ?_) ?_ <;>
[continuity; continuity; continuity; continuity; skip]
intro x hx
simp [hx]
|
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.Data.Complex.Orientation
import Mathlib.Tactic.LinearCombination
#align_import analysis.inner_product_space.two_dim from "leanprover-community/mathlib"@"cd8fafa2fac98e1a67097e8a91ad9901cfde48af"
non... | Mathlib/Analysis/InnerProductSpace/TwoDim.lean | 445 | 471 | theorem nonneg_inner_and_areaForm_eq_zero_iff_sameRay (x y : E) :
0 ≤ ⟪x, y⟫ ∧ ω x y = 0 ↔ SameRay ℝ x y := by |
by_cases hx : x = 0
· simp [hx]
constructor
· let a : ℝ := (o.basisRightAngleRotation x hx).repr y 0
let b : ℝ := (o.basisRightAngleRotation x hx).repr y 1
suffices ↑0 ≤ a * ‖x‖ ^ 2 ∧ b * ‖x‖ ^ 2 = 0 → SameRay ℝ x (a • x + b • J x) by
rw [← (o.basisRightAngleRotation x hx).sum_repr y]
simp ... |
import Mathlib.RingTheory.HahnSeries.Addition
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Data.Finset.MulAntidiagonal
#align_import ring_theory.hahn_series from "leanprover-community/mathlib"@"a484a7d0eade4e1268f4fb402859b6686037f965"
set_option linter.uppercaseLean3 false
open Finset Function
... | Mathlib/RingTheory/HahnSeries/Multiplication.lean | 65 | 68 | theorem order_one [MulZeroOneClass R] : order (1 : HahnSeries Γ R) = 0 := by |
cases subsingleton_or_nontrivial R
· rw [Subsingleton.elim (1 : HahnSeries Γ R) 0, order_zero]
· exact order_single one_ne_zero
|
import Mathlib.Algebra.MvPolynomial.Funext
import Mathlib.Algebra.Ring.ULift
import Mathlib.RingTheory.WittVector.Basic
#align_import ring_theory.witt_vector.is_poly from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
namespace WittVector
universe u
variable {p : ℕ} {R S : Type u} {σ id... | Mathlib/RingTheory/WittVector/IsPoly.lean | 296 | 298 | theorem bind₁_zero_wittPolynomial [Fact p.Prime] (n : ℕ) :
bind₁ (0 : ℕ → MvPolynomial ℕ R) (wittPolynomial p R n) = 0 := by |
rw [← aeval_eq_bind₁, aeval_zero, constantCoeff_wittPolynomial, RingHom.map_zero]
|
import Mathlib.CategoryTheory.Subobject.MonoOver
import Mathlib.CategoryTheory.Skeletal
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.Tactic.ApplyFun
import Mathlib.Tactic.CategoryTheory.Elementwise
#align_import category_theory.subobject.basic from "leanprover-community/mathlib"@"70fd9563a21e7b... | Mathlib/CategoryTheory/Subobject/Basic.lean | 380 | 381 | theorem ofMkLEMk_comp {B A₁ A₂ : C} {f : A₁ ⟶ B} {g : A₂ ⟶ B} [Mono f] [Mono g] (h : mk f ≤ mk g) :
ofMkLEMk f g h ≫ g = f := by | simp [ofMkLEMk]
|
import Mathlib.Data.List.GetD
import Mathlib.Data.Nat.Bits
import Mathlib.Algebra.Ring.Nat
import Mathlib.Order.Basic
import Mathlib.Tactic.AdaptationNote
import Mathlib.Tactic.Common
#align_import data.nat.bitwise from "leanprover-community/mathlib"@"6afc9b06856ad973f6a2619e3e8a0a8d537a58f2"
open Function
names... | Mathlib/Data/Nat/Bitwise.lean | 75 | 81 | theorem binaryRec_of_ne_zero {C : Nat → Sort*} (z : C 0) (f : ∀ b n, C n → C (bit b n)) {n}
(h : n ≠ 0) :
binaryRec z f n = bit_decomp n ▸ f (bodd n) (div2 n) (binaryRec z f (div2 n)) := by |
rw [Eq.rec_eq_cast]
rw [binaryRec]
dsimp only
rw [dif_neg h, eq_mpr_eq_cast]
|
import Mathlib.AlgebraicTopology.DoldKan.FunctorGamma
import Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject
import Mathlib.CategoryTheory.Idempotents.HomologicalComplex
#align_import algebraic_topology.dold_kan.gamma_comp_n from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
no... | Mathlib/AlgebraicTopology/DoldKan/GammaCompN.lean | 76 | 82 | theorem N₁Γ₀_app (K : ChainComplex C ℕ) :
N₁Γ₀.app K = (Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.symm ≪≫
(toKaroubi _).mapIso (Γ₀NondegComplexIso K) := by |
ext1
dsimp [N₁Γ₀]
erw [id_comp, comp_id, comp_id]
rfl
|
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Data.Real.Basic
import Mathlib.Order.Interval.Set.Disjoint
#align_import data.real.basic from "leanprover-community/mathlib"@"cb42593171ba005beaaf4549fcfe0dece9ada4c9"
open scoped Classical
open Pointwise CauSeq
namespace Real
... | Mathlib/Data/Real/Archimedean.lean | 58 | 106 | theorem exists_isLUB {S : Set ℝ} (hne : S.Nonempty) (hbdd : BddAbove S) : ∃ x, IsLUB S x := by |
rcases hne, hbdd with ⟨⟨L, hL⟩, ⟨U, hU⟩⟩
have : ∀ d : ℕ, BddAbove { m : ℤ | ∃ y ∈ S, (m : ℝ) ≤ y * d } := by
cases' exists_int_gt U with k hk
refine fun d => ⟨k * d, fun z h => ?_⟩
rcases h with ⟨y, yS, hy⟩
refine Int.cast_le.1 (hy.trans ?_)
push_cast
exact mul_le_mul_of_nonneg_right ((hU y... |
import Mathlib.FieldTheory.PrimitiveElement
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly
import Mathlib.LinearAlgebra.Matrix.ToLinearEquiv
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.G... | Mathlib/RingTheory/Norm.lean | 91 | 97 | theorem norm_algebraMap_of_basis [Fintype ι] (b : Basis ι R S) (x : R) :
norm R (algebraMap R S x) = x ^ Fintype.card ι := by |
haveI := Classical.decEq ι
rw [norm_apply, ← det_toMatrix b, lmul_algebraMap]
convert @det_diagonal _ _ _ _ _ fun _ : ι => x
· ext (i j); rw [toMatrix_lsmul]
· rw [Finset.prod_const, Finset.card_univ]
|
import Mathlib.Order.BooleanAlgebra
import Mathlib.Tactic.Common
#align_import order.heyting.boundary from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
variable {α : Type*}
namespace Coheyting
variable [CoheytingAlgebra α] {a b : α}
def boundary (a : α) : α :=
a ⊓ ￢a
#align cohe... | Mathlib/Order/Heyting/Boundary.lean | 135 | 137 | theorem hnot_hnot_sup_boundary (a : α) : ￢￢a ⊔ ∂ a = a := by |
rw [boundary, sup_inf_left, hnot_sup_self, inf_top_eq, sup_eq_right]
exact hnot_hnot_le
|
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.Topology.MetricSpace.Bounded
import Mathlib.Topology.Order.MonotoneConvergence
#align_import analysis.box_integral.box.basic from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Fu... | Mathlib/Analysis/BoxIntegral/Box/Basic.lean | 504 | 515 | theorem distortion_eq_of_sub_eq_div {I J : Box ι} {r : ℝ}
(h : ∀ i, I.upper i - I.lower i = (J.upper i - J.lower i) / r) :
distortion I = distortion J := by |
simp only [distortion, nndist_pi_def, Real.nndist_eq', h, map_div₀]
congr 1 with i
have : 0 < r := by
by_contra hr
have := div_nonpos_of_nonneg_of_nonpos (sub_nonneg.2 <| J.lower_le_upper i) (not_lt.1 hr)
rw [← h] at this
exact this.not_lt (sub_pos.2 <| I.lower_lt_upper i)
have hn0 := (map_ne_z... |
import Mathlib.Topology.Algebra.Module.WeakDual
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.Analysis.LocallyConvex.WithSeminorms
#align_import analysis.locally_convex.weak_dual from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
variable {𝕜 E F ι : Type*}
open Topology
... | Mathlib/Analysis/LocallyConvex/WeakDual.lean | 68 | 70 | theorem toSeminorm_ball_zero {f : E →ₗ[𝕜] 𝕜} {r : ℝ} :
Seminorm.ball f.toSeminorm 0 r = { x : E | ‖f x‖ < r } := by |
simp only [Seminorm.ball_zero_eq, toSeminorm_apply]
|
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 | 322 | 379 | theorem SimpleFunc.exists_upperSemicontinuous_le_lintegral_le (f : α →ₛ ℝ≥0)
(int_f : (∫⁻ x, f x ∂μ) ≠ ∞) {ε : ℝ≥0∞} (ε0 : ε ≠ 0) :
∃ g : α → ℝ≥0, (∀ x, g x ≤ f x) ∧ UpperSemicontinuous g ∧
(∫⁻ x, f x ∂μ) ≤ (∫⁻ x, g x ∂μ) + ε := by |
induction' f using MeasureTheory.SimpleFunc.induction with c s hs f₁ f₂ _ h₁ h₂ generalizing ε
· by_cases hc : c = 0
· refine ⟨fun _ => 0, ?_, upperSemicontinuous_const, ?_⟩
· classical
simp only [hc, Set.indicator_zero', Pi.zero_apply, SimpleFunc.const_zero, imp_true_iff,
eq_self_iff_t... |
import Mathlib.Topology.MetricSpace.Basic
#align_import topology.metric_space.infsep from "leanprover-community/mathlib"@"5316314b553dcf8c6716541851517c1a9715e22b"
variable {α β : Type*}
namespace Set
section Einfsep
open ENNReal
open Function
noncomputable def einfsep [EDist α] (s : Set α) : ℝ≥0∞ :=
⨅ (x... | Mathlib/Topology/MetricSpace/Infsep.lean | 176 | 179 | theorem Finite.einfsep (hs : s.Finite) : s.einfsep = hs.offDiag.toFinset.inf (uncurry edist) := by |
refine eq_of_forall_le_iff fun _ => ?_
simp_rw [le_einfsep_iff, imp_forall_iff, Finset.le_inf_iff, Finite.mem_toFinset, mem_offDiag,
Prod.forall, uncurry_apply_pair, and_imp]
|
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Basic
import Mathlib.RingTheory.Localization.FractionRing
#align_import ring_theory.localization.localization_localization from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
open Function
namespace ... | Mathlib/RingTheory/Localization/LocalizationLocalization.lean | 73 | 89 | theorem localization_localization_surj [IsLocalization N T] (x : T) :
∃ y : R × localizationLocalizationSubmodule M N,
x * algebraMap R T y.2 = algebraMap R T y.1 := by |
rcases IsLocalization.surj N x with ⟨⟨y, s⟩, eq₁⟩
-- x = y / s
rcases IsLocalization.surj M y with ⟨⟨z, t⟩, eq₂⟩
-- y = z / t
rcases IsLocalization.surj M (s : S) with ⟨⟨z', t'⟩, eq₃⟩
-- s = z' / t'
dsimp only at eq₁ eq₂ eq₃
refine ⟨⟨z * t', z' * t, ?_⟩, ?_⟩ -- x = y / s = (z * t') / (z' * t)
· rw [m... |
import Mathlib.Algebra.Algebra.Spectrum
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.Nilpotent.Basic
#align_import linear_algebra.eigenspace.basic from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da1aa1"
universe u v... | Mathlib/LinearAlgebra/Eigenspace/Basic.lean | 104 | 105 | theorem mem_eigenspace_iff {f : End R M} {μ : R} {x : M} : x ∈ eigenspace f μ ↔ f x = μ • x := by |
rw [eigenspace, LinearMap.mem_ker, LinearMap.sub_apply, algebraMap_end_apply, sub_eq_zero]
|
import Mathlib.Topology.ContinuousFunction.Basic
#align_import topology.compact_open from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Set Filter TopologicalSpace
open scoped Topology
namespace ContinuousMap
section CompactOpen
variable {α X Y Z T : Type*}
variable [Topologica... | Mathlib/Topology/CompactOpen.lean | 358 | 364 | theorem continuous_coev : Continuous (coev X Y) := by |
have : ∀ {a K U}, MapsTo (coev X Y a) K U ↔ {a} ×ˢ K ⊆ U := by simp [mapsTo']
simp only [continuous_iff_continuousAt, ContinuousAt, tendsto_nhds_compactOpen, this]
intro x K hK U hU hKU
rcases generalized_tube_lemma isCompact_singleton hK hU hKU with ⟨V, W, hV, -, hxV, hKW, hVWU⟩
filter_upwards [hV.mem_nhds ... |
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 | 207 | 211 | theorem norm_div_sin_angle_add_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x = 0 ∨ y ≠ 0) :
‖y‖ / Real.sin (angle x (x + y)) = ‖x + y‖ := by |
rcases h0 with (h0 | h0); · simp [h0]
rw [sin_angle_add_of_inner_eq_zero h (Or.inr h0), div_div_eq_mul_div, mul_comm, div_eq_mul_inv,
mul_inv_cancel_right₀ (norm_ne_zero_iff.2 h0)]
|
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.continuous_affine_map from "leanprover-community/mathlib"@"bd1fc183335ea95a9519a1630bcf901fe9326d83"
structure ContinuousAffineMap (R : T... | Mathlib/Topology/Algebra/ContinuousAffineMap.lean | 53 | 57 | theorem to_affineMap_injective {f g : P →ᴬ[R] Q} (h : (f : P →ᵃ[R] Q) = (g : P →ᵃ[R] Q)) :
f = g := by |
cases f
cases g
congr
|
import Mathlib.RingTheory.Polynomial.Pochhammer
#align_import data.nat.factorial.cast from "leanprover-community/mathlib"@"d50b12ae8e2bd910d08a94823976adae9825718b"
open Nat
variable (S : Type*)
namespace Nat
section Semiring
variable [Semiring S] (a b : ℕ)
-- Porting note: added type ascription around a + 1... | Mathlib/Data/Nat/Factorial/Cast.lean | 51 | 52 | theorem cast_factorial : (a ! : S) = (ascPochhammer S a).eval 1 := by |
rw [← one_ascFactorial, cast_ascFactorial, cast_one]
|
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.NormedSpace.BallAction
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Geometry.Manifold.Algebra.LieGroup
import Mathlib.Geometry.Manifol... | Mathlib/Geometry/Manifold/Instances/Sphere.lean | 98 | 104 | theorem contDiffOn_stereoToFun :
ContDiffOn ℝ ⊤ (stereoToFun v) {x : E | innerSL _ v x ≠ (1 : ℝ)} := by |
refine ContDiffOn.smul ?_ (orthogonalProjection (ℝ ∙ v)ᗮ).contDiff.contDiffOn
refine contDiff_const.contDiffOn.div ?_ ?_
· exact (contDiff_const.sub (innerSL ℝ v).contDiff).contDiffOn
· intro x h h'
exact h (sub_eq_zero.mp h').symm
|
import Mathlib.Data.ENNReal.Basic
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.MetricSpace.Thickening
#align_import topology.metric_space.thickened_indicator from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classical
open NNReal ENNReal Topol... | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | 58 | 66 | theorem continuous_thickenedIndicatorAux {δ : ℝ} (δ_pos : 0 < δ) (E : Set α) :
Continuous (thickenedIndicatorAux δ E) := by |
unfold thickenedIndicatorAux
let f := fun x : α => (⟨1, infEdist x E / ENNReal.ofReal δ⟩ : ℝ≥0 × ℝ≥0∞)
let sub := fun p : ℝ≥0 × ℝ≥0∞ => (p.1 : ℝ≥0∞) - p.2
rw [show (fun x : α => (1 : ℝ≥0∞) - infEdist x E / ENNReal.ofReal δ) = sub ∘ f by rfl]
apply (@ENNReal.continuous_nnreal_sub 1).comp
apply (ENNReal.cont... |
import Mathlib.Data.Multiset.Bind
#align_import data.multiset.fold from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
variable {α β : Type*}
section Fold
variable (op : α → α → α) [hc : Std.Commutative op] [ha : Std.Associative op]
local notation a " * " b => ... | Mathlib/Data/Multiset/Fold.lean | 82 | 87 | theorem fold_bind {ι : Type*} (s : Multiset ι) (t : ι → Multiset α) (b : ι → α) (b₀ : α) :
(s.bind t).fold op ((s.map b).fold op b₀) =
(s.map fun i => (t i).fold op (b i)).fold op b₀ := by |
induction' s using Multiset.induction_on with a ha ih
· rw [zero_bind, map_zero, map_zero, fold_zero]
· rw [cons_bind, map_cons, map_cons, fold_cons_left, fold_cons_left, fold_add, ih]
|
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.NormedSpace.Banach
import Mathlib.LinearAlgebra.SesquilinearForm
#align_import analysis.inner_product_space.symmetric from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open RCLike
open ComplexConjugate
variable ... | Mathlib/Analysis/InnerProductSpace/Symmetric.lean | 71 | 72 | theorem IsSymmetric.conj_inner_sym {T : E →ₗ[𝕜] E} (hT : IsSymmetric T) (x y : E) :
conj ⟪T x, y⟫ = ⟪T y, x⟫ := by | rw [hT x y, inner_conj_symm]
|
import Mathlib.Data.List.Basic
#align_import data.list.forall2 from "leanprover-community/mathlib"@"5a3e819569b0f12cbec59d740a2613018e7b8eec"
open Nat Function
namespace List
variable {α β γ δ : Type*} {R S : α → β → Prop} {P : γ → δ → Prop} {Rₐ : α → α → Prop}
open Relator
mk_iff_of_inductive_prop List.Foral... | Mathlib/Data/List/Forall2.lean | 61 | 69 | theorem forall₂_eq_eq_eq : Forall₂ ((· = ·) : α → α → Prop) = Eq := by |
funext a b; apply propext
constructor
· intro h
induction h
· rfl
simp only [*]
· rintro rfl
exact forall₂_refl _
|
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
open Finset
namespace SimpleGraph
variable {V : Type*} [DecidableEq V] (G : SimpleGraph V) (s t : V)
section ReplaceVertex
def replaceVertex : SimpleGraph V where
Adj v w := if v = t then if w = t then False else G... | Mathlib/Combinatorics/SimpleGraph/Operations.lean | 115 | 124 | theorem card_edgeFinset_replaceVertex_of_not_adj (hn : ¬G.Adj s t) :
(G.replaceVertex s t).edgeFinset.card = G.edgeFinset.card + G.degree s - G.degree t := by |
have inc : G.incidenceFinset t ⊆ G.edgeFinset := by simp [incidenceFinset, incidenceSet_subset]
rw [G.edgeFinset_replaceVertex_of_not_adj hn,
card_union_of_disjoint G.disjoint_sdiff_neighborFinset_image, card_sdiff inc,
← Nat.sub_add_comm <| card_le_card inc, card_incidenceFinset_eq_degree]
congr 2
rw ... |
import Mathlib.Analysis.Complex.UpperHalfPlane.Topology
import Mathlib.Analysis.SpecialFunctions.Arsinh
import Mathlib.Geometry.Euclidean.Inversion.Basic
#align_import analysis.complex.upper_half_plane.metric from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c"
noncomputable section
ope... | Mathlib/Analysis/Complex/UpperHalfPlane/Metric.lean | 91 | 93 | theorem dist_le_iff_le_sinh :
dist z w ≤ r ↔ dist (z : ℂ) w / (2 * √(z.im * w.im)) ≤ sinh (r / 2) := by |
rw [← div_le_div_right (zero_lt_two' ℝ), ← sinh_le_sinh, sinh_half_dist]
|
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Order.Bounds.OrderIso
import Mathlib.Tactic.Positivity.Core
#align_import algebra.order.field.basic from "leanprover-community/mathlib"@"8477... | Mathlib/Algebra/Order/Field/Basic.lean | 385 | 385 | theorem lt_one_div (ha : 0 < a) (hb : 0 < b) : a < 1 / b ↔ b < 1 / a := by | simpa using lt_inv ha hb
|
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 | 616 | 620 | theorem getLast_cons {a : α} {l : List α} :
∀ h : l ≠ nil, getLast (a :: l) (cons_ne_nil a l) = getLast l h := by |
induction l <;> intros
· contradiction
· rfl
|
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
open Set
open scoped RealInnerProductSpace
variable {V P : Type*} [NormedAddCommGroup V] [InnerP... | Mathlib/Geometry/Euclidean/PerpBisector.lean | 53 | 57 | theorem mem_perpBisector_iff_inner_pointReflection_vsub_eq_zero :
c ∈ perpBisector p₁ p₂ ↔ ⟪Equiv.pointReflection c p₁ -ᵥ p₂, p₂ -ᵥ p₁⟫ = 0 := by |
rw [mem_perpBisector_iff_inner_eq_zero, Equiv.pointReflection_apply,
vsub_midpoint, invOf_eq_inv, ← smul_add, real_inner_smul_left, vadd_vsub_assoc]
simp
|
import Mathlib.Data.Complex.Module
import Mathlib.Data.Complex.Order
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Mathlib.Topology.Instances.RealVectorSpace
#align_import analysis.complex.basic from "leanprover-community/mathlib... | Mathlib/Analysis/Complex/Basic.lean | 225 | 226 | theorem lipschitz_equivRealProd : LipschitzWith 1 equivRealProd := by |
simpa using AddMonoidHomClass.lipschitz_of_bound equivRealProdLm 1 equivRealProd_apply_le'
|
import Mathlib.AlgebraicTopology.DoldKan.Projections
import Mathlib.CategoryTheory.Idempotents.FunctorCategories
import Mathlib.CategoryTheory.Idempotents.FunctorExtension
#align_import algebraic_topology.dold_kan.p_infty from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open Category... | Mathlib/AlgebraicTopology/DoldKan/PInfty.lean | 130 | 133 | theorem PInfty_f_comp_QInfty_f (n : ℕ) : (PInfty.f n : X _[n] ⟶ _) ≫ QInfty.f n = 0 := by |
dsimp only [QInfty]
simp only [HomologicalComplex.sub_f_apply, HomologicalComplex.id_f, comp_sub, comp_id,
PInfty_f_idem, sub_self]
|
import Mathlib.Topology.Category.Profinite.Basic
import Mathlib.Topology.LocallyConstant.Basic
import Mathlib.Topology.DiscreteQuotient
import Mathlib.Topology.Category.TopCat.Limits.Cofiltered
import Mathlib.Topology.Category.TopCat.Limits.Konig
#align_import topology.category.Profinite.cofiltered_limit from "leanpr... | Mathlib/Topology/Category/Profinite/CofilteredLimit.lean | 45 | 112 | theorem exists_isClopen_of_cofiltered {U : Set C.pt} (hC : IsLimit C) (hU : IsClopen U) :
∃ (j : J) (V : Set (F.obj j)), IsClopen V ∧ U = C.π.app j ⁻¹' V := by |
-- First, we have the topological basis of the cofiltered limit obtained by pulling back
-- clopen sets from the factors in the limit. By continuity, all such sets are again clopen.
have hB := TopCat.isTopologicalBasis_cofiltered_limit.{u, v} (F ⋙ Profinite.toTopCat)
(Profinite.toTopCat.mapCone C) (isLimit... |
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.Analysis.NormedSpace.IndicatorFunction
#align_import measure_theory.integral.integrable_on from "leanprover-community/mathlib"@"8b8ba04e2f326f3f7cf24ad129beda58531ada61"
noncomputable section
open Set Filter TopologicalSpace MeasureTheory Function
ope... | Mathlib/MeasureTheory/Integral/IntegrableOn.lean | 251 | 253 | theorem integrableOn_map_equiv [MeasurableSpace β] (e : α ≃ᵐ β) {f : β → E} {μ : Measure α}
{s : Set β} : IntegrableOn f s (μ.map e) ↔ IntegrableOn (f ∘ e) (e ⁻¹' s) μ := by |
simp only [IntegrableOn, e.restrict_map, integrable_map_equiv e]
|
import Mathlib.Analysis.InnerProductSpace.Spectrum
import Mathlib.Data.Matrix.Rank
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Hermitian
#align_import linear_algebra.matrix.spectrum from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
namespace Matrix
... | Mathlib/LinearAlgebra/Matrix/Spectrum.lean | 123 | 126 | theorem det_eq_prod_eigenvalues : det A = ∏ i, (hA.eigenvalues i : 𝕜) := by |
convert congr_arg det hA.spectral_theorem
rw [det_mul_right_comm]
simp
|
import Mathlib.Geometry.Manifold.MFDeriv.UniqueDifferential
import Mathlib.Geometry.Manifold.ContMDiffMap
#align_import geometry.manifold.cont_mdiff_mfderiv from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
open Set Function Filter ChartedSpace SmoothManifoldWithCorners Bundle
open sc... | Mathlib/Geometry/Manifold/ContMDiffMFDeriv.lean | 344 | 518 | theorem ContMDiffOn.contMDiffOn_tangentMapWithin (hf : ContMDiffOn I I' n f s) (hmn : m + 1 ≤ n)
(hs : UniqueMDiffOn I s) :
ContMDiffOn I.tangent I'.tangent m (tangentMapWithin I I' f s)
(π E (TangentSpace I) ⁻¹' s) := by |
/- The strategy of the proof is to avoid unfolding the definitions, and reduce by functoriality
to the case of functions on the model spaces, where we have already proved the result.
Let `l` and `r` be the charts to the left and to the right, so that we have
```
l^{-1} f r
H -------... |
import Mathlib.Algebra.CharP.Two
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.GroupTheory.SpecificGroups.Cyclic
import Mathlib.NumberTheory.Divisors
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Tactic.Zify
#align_import rin... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 119 | 122 | theorem rootsOfUnity_le_of_dvd (h : k ∣ l) : rootsOfUnity k M ≤ rootsOfUnity l M := by |
obtain ⟨d, rfl⟩ := h
intro ζ h
simp_all only [mem_rootsOfUnity, PNat.mul_coe, pow_mul, one_pow]
|
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.special_functions.log.basic from "leanprover-community/mathlib"@"f23a09ce6d3f367220dc3cecad6b7eb69eb01690"
open Set Filter Function
open Topology
noncomputable ... | Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 180 | 183 | theorem log_pos_of_lt_neg_one (hx : x < -1) : 0 < log x := by |
rw [← neg_neg x, log_neg_eq_log]
have : 1 < -x := by linarith
exact log_pos this
|
import Mathlib.Order.CompleteLattice
import Mathlib.Order.Synonym
import Mathlib.Order.Hom.Set
import Mathlib.Order.Bounds.Basic
#align_import order.galois_connection from "leanprover-community/mathlib"@"c5c7e2760814660967bc27f0de95d190a22297f3"
open Function OrderDual Set
universe u v w x
variable {α : Type u}... | Mathlib/Order/GaloisConnection.lean | 283 | 284 | theorem l_iSup₂ {f : ∀ i, κ i → α} : l (⨆ (i) (j), f i j) = ⨆ (i) (j), l (f i j) := by |
simp_rw [gc.l_iSup]
|
import Mathlib.MeasureTheory.Function.L1Space
import Mathlib.Analysis.NormedSpace.IndicatorFunction
#align_import measure_theory.integral.integrable_on from "leanprover-community/mathlib"@"8b8ba04e2f326f3f7cf24ad129beda58531ada61"
noncomputable section
open Set Filter TopologicalSpace MeasureTheory Function
ope... | Mathlib/MeasureTheory/Integral/IntegrableOn.lean | 403 | 409 | theorem _root_.MeasurableEmbedding.integrableAtFilter_map_iff [MeasurableSpace β] {e : α → β}
(he : MeasurableEmbedding e) {f : β → E} :
IntegrableAtFilter f (l.map e) (μ.map e) ↔ IntegrableAtFilter (f ∘ e) l μ := by |
simp_rw [IntegrableAtFilter, he.integrableOn_map_iff]
constructor <;> rintro ⟨s, hs⟩
· exact ⟨_, hs⟩
· exact ⟨e '' s, by rwa [mem_map, he.injective.preimage_image]⟩
|
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Ring.Opposite
import Mathlib.Tactic.Abel
#align_import algebra.geom_sum fro... | Mathlib/Algebra/GeomSum.lean | 64 | 64 | theorem geom_sum_two {x : α} : ∑ i ∈ range 2, x ^ i = x + 1 := by | simp [geom_sum_succ']
|
import Mathlib.Topology.Order.IsLUB
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section DenselyOrdered
variable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}
{s : Set α}
theorem closure_Ioi' {a : α} (h : (Io... | Mathlib/Topology/Order/DenselyOrdered.lean | 344 | 346 | theorem tendsto_comp_coe_Ioo_atTop (h : a < b) :
Tendsto (fun x : Ioo a b => f x) atTop l ↔ Tendsto f (𝓝[<] b) l := by |
rw [← map_coe_Ioo_atTop h, tendsto_map'_iff]; rfl
|
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 | 114 | 116 | theorem lcm_image [DecidableEq β] {g : γ → β} (s : Finset γ) :
(s.image g).lcm f = s.lcm (f ∘ g) := by |
classical induction' s using Finset.induction with c s _ ih <;> simp [*]
|
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.Analysis.Complex.Basic
#align_import measure_theory.measure.vector_measure from "leanprover-community/mathlib"@"70a4f2197832bceab57d7f41379b2592d1110570"
noncomputable section
open scoped Classical
open NNReal ENNReal MeasureTheory
namespace Measur... | Mathlib/MeasureTheory/Measure/VectorMeasure.lean | 128 | 136 | theorem ext_iff (v w : VectorMeasure α M) : v = w ↔ ∀ i : Set α, MeasurableSet i → v i = w i := by |
constructor
· rintro rfl _ _
rfl
· rw [ext_iff']
intro h i
by_cases hi : MeasurableSet i
· exact h i hi
· simp_rw [not_measurable _ hi]
|
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 | 366 | 387 | theorem HasFTaylorSeriesUpToOn.compContinuousLinearMap (hf : HasFTaylorSeriesUpToOn n f p s)
(g : G →L[𝕜] E) :
HasFTaylorSeriesUpToOn n (f ∘ g) (fun x k => (p (g x) k).compContinuousLinearMap fun _ => g)
(g ⁻¹' s) := by |
let A : ∀ m : ℕ, (E[×m]→L[𝕜] F) → G[×m]→L[𝕜] F := fun m h => h.compContinuousLinearMap fun _ => g
have hA : ∀ m, IsBoundedLinearMap 𝕜 (A m) := fun m =>
isBoundedLinearMap_continuousMultilinearMap_comp_linear g
constructor
· intro x hx
simp only [(hf.zero_eq (g x) hx).symm, Function.comp_apply]
c... |
import Mathlib.Logic.Encodable.Lattice
import Mathlib.MeasureTheory.MeasurableSpace.Defs
#align_import measure_theory.pi_system from "leanprover-community/mathlib"@"98e83c3d541c77cdb7da20d79611a780ff8e7d90"
open MeasurableSpace Set
open scoped Classical
open MeasureTheory
def IsPiSystem {α} (C : Set (Set α)) :... | Mathlib/MeasureTheory/PiSystem.lean | 480 | 488 | theorem subset_piiUnionInter {π : ι → Set (Set α)} {S : Set ι} {i : ι} (his : i ∈ S) :
π i ⊆ piiUnionInter π S := by |
have h_ss : {i} ⊆ S := by
intro j hj
rw [mem_singleton_iff] at hj
rwa [hj]
refine Subset.trans ?_ (piiUnionInter_mono_right h_ss)
rw [piiUnionInter_singleton]
exact subset_union_left
|
import Mathlib.Data.Stream.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Init.Data.List.Basic
import Mathlib.Data.List.Basic
#align_import data.stream.init from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
set_option autoImplicit true
open Nat Function Option
namespace Stre... | Mathlib/Data/Stream/Init.lean | 274 | 279 | theorem tail_iterate (f : α → α) (a : α) : tail (iterate f a) = iterate f (f a) := by |
ext n
rw [get_tail]
induction' n with n' ih
· rfl
· rw [get_succ_iterate', ih, get_succ_iterate']
|
import Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol
#align_import number_theory.legendre_symbol.norm_num from "leanprover-community/mathlib"@"e2621d935895abe70071ab828a4ee6e26a52afe4"
section Lemmas
namespace Mathlib.Meta.NormNum
def jacobiSymNat (a b : ℕ) : ℤ :=
jacobiSym a b
#align norm_num.jacobi_sym_... | Mathlib/Tactic/NormNum/LegendreSymbol.lean | 158 | 162 | theorem jacobiSymNat.even_odd₃ (a b c : ℕ) (r : ℤ) (ha : a % 2 = 0) (hb : b % 8 = 3)
(hc : a / 2 = c) (hr : jacobiSymNat c b = r) : jacobiSymNat a b = -r := by |
simp only [jacobiSymNat, ← hr, ← hc, Int.ofNat_ediv, Nat.cast_ofNat]
rw [← jacobiSym.even_odd (mod_cast ha), if_pos (by simp [hb])]
rw [← Nat.mod_mod_of_dvd, hb]; norm_num
|
import Mathlib.Algebra.Polynomial.Splits
#align_import algebra.cubic_discriminant from "leanprover-community/mathlib"@"930133160e24036d5242039fe4972407cd4f1222"
noncomputable section
@[ext]
structure Cubic (R : Type*) where
(a b c d : R)
#align cubic Cubic
namespace Cubic
open Cubic Polynomial
open Polynom... | Mathlib/Algebra/CubicDiscriminant.lean | 121 | 121 | theorem a_of_eq (h : P.toPoly = Q.toPoly) : P.a = Q.a := by | rw [← coeff_eq_a, h, coeff_eq_a]
|
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.midpoint from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
open AffineMap AffineEquiv
section
variable (R : Type*) {V V' P P' : Type*} [Ring R] [Invertible (2 : R)] [AddCommGroup V]
[Modu... | Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | 169 | 170 | theorem midpoint_eq_left_iff {x y : P} : midpoint R x y = x ↔ x = y := by |
rw [midpoint_eq_iff, pointReflection_self]
|
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.Order.Interval.Finset.Nat
#align_import data.nat.factorization.basic from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
... | Mathlib/Data/Nat/Factorization/Basic.lean | 133 | 135 | theorem factorization_eq_zero_iff (n p : ℕ) :
n.factorization p = 0 ↔ ¬p.Prime ∨ ¬p ∣ n ∨ n = 0 := by |
simp_rw [← not_mem_support_iff, support_factorization, mem_primeFactors, not_and_or, not_ne_iff]
|
import Mathlib.Algebra.CharP.Pi
import Mathlib.Algebra.CharP.Quotient
import Mathlib.Algebra.CharP.Subring
import Mathlib.Algebra.Ring.Pi
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.FieldTheory.Perfect
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.Algebra.Ring.Subring.Basic
... | Mathlib/RingTheory/Perfection.lean | 420 | 427 | theorem preVal_mul {x y : ModP K v O hv p} (hxy0 : x * y ≠ 0) :
preVal K v O hv p (x * y) = preVal K v O hv p x * preVal K v O hv p y := by |
have hx0 : x ≠ 0 := mt (by rintro rfl; rw [zero_mul]) hxy0
have hy0 : y ≠ 0 := mt (by rintro rfl; rw [mul_zero]) hxy0
obtain ⟨r, rfl⟩ := Ideal.Quotient.mk_surjective x
obtain ⟨s, rfl⟩ := Ideal.Quotient.mk_surjective y
rw [← map_mul (Ideal.Quotient.mk (Ideal.span {↑p})) r s] at hxy0 ⊢
rw [preVal_mk hx0, pre... |
import Mathlib.CategoryTheory.Sites.Plus
import Mathlib.CategoryTheory.Limits.Shapes.ConcreteCategory
#align_import category_theory.sites.sheafification from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace CategoryTheory
open CategoryTheory.Limits Opposite
universe w v u
var... | Mathlib/CategoryTheory/Sites/ConcreteSheafification.lean | 490 | 493 | theorem toSheafify_naturality {P Q : Cᵒᵖ ⥤ D} (η : P ⟶ Q) :
η ≫ J.toSheafify _ = J.toSheafify _ ≫ J.sheafifyMap η := by |
dsimp [sheafifyMap, sheafify, toSheafify]
simp
|
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