Context stringlengths 57 92.3k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.ord_connected_component from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Interval Function OrderDual
namespace Set
variable {α : Type*} [LinearOrder α] {s t : Set α}... | Mathlib/Order/Interval/Set/OrdConnectedComponent.lean | 53 | 54 | theorem self_mem_ordConnectedComponent : x ∈ ordConnectedComponent s x ↔ x ∈ s := by |
rw [mem_ordConnectedComponent, uIcc_self, singleton_subset_iff]
|
import Mathlib.MeasureTheory.Measure.Sub
import Mathlib.MeasureTheory.Decomposition.SignedHahn
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
#align_import measure_theory.decomposition.lebesgue from "leanprover-community/mathlib"@"b2ff9a3d7a15fd5b0f060b135421d6a89a999c2f"
open scoped MeasureTheory NNReal ENN... | Mathlib/MeasureTheory/Decomposition/Lebesgue.lean | 102 | 106 | theorem measurable_rnDeriv (μ ν : Measure α) : Measurable <| μ.rnDeriv ν := by |
by_cases h : HaveLebesgueDecomposition μ ν
· exact (haveLebesgueDecomposition_spec μ ν).1
· rw [rnDeriv_of_not_haveLebesgueDecomposition h]
exact measurable_zero
|
import Mathlib.Analysis.Analytic.Composition
#align_import analysis.analytic.inverse from "leanprover-community/mathlib"@"284fdd2962e67d2932fa3a79ce19fcf92d38e228"
open scoped Classical Topology
open Finset Filter
namespace FormalMultilinearSeries
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} ... | Mathlib/Analysis/Analytic/Inverse.lean | 248 | 262 | theorem comp_rightInv (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F)
(h : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm i) (h0 : p 0 = 0) :
p.comp (rightInv p i) = id 𝕜 F := by |
ext (n v)
match n with
| 0 =>
simp only [h0, ContinuousMultilinearMap.zero_apply, id_apply_ne_one, Ne, not_false_iff,
zero_ne_one, comp_coeff_zero']
| 1 =>
simp only [comp_coeff_one, h, rightInv_coeff_one, ContinuousLinearEquiv.apply_symm_apply,
id_apply_one, ContinuousLinearEquiv.coe_apply... |
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.constructions.borel_space.basic from "leanprover-community/mathlib"@"9f55d0d4363ae59948c33864cbc52e0b12e0e8ce"
open Set Filter MeasureTheory MeasurableSpace
open scoped Classical Topology NNReal ENNReal MeasureTheory
univers... | Mathlib/MeasureTheory/Constructions/BorelSpace/Real.lean | 44 | 54 | theorem borel_eq_generateFrom_Iio_rat : borel ℝ = .generateFrom (⋃ a : ℚ, {Iio (a : ℝ)}) := by |
rw [borel_eq_generateFrom_Iio]
refine le_antisymm
(generateFrom_le ?_)
(generateFrom_mono <| iUnion_subset fun q ↦ singleton_subset_iff.mpr <| mem_range_self _)
rintro _ ⟨a, rfl⟩
have : IsLUB (range ((↑) : ℚ → ℝ) ∩ Iio a) a := by
simp [isLUB_iff_le_iff, mem_upperBounds, ← le_iff_forall_rat_lt_imp_l... |
import Mathlib.Order.CompleteLattice
import Mathlib.Order.Cover
import Mathlib.Order.Iterate
import Mathlib.Order.WellFounded
#align_import order.succ_pred.basic from "leanprover-community/mathlib"@"0111834459f5d7400215223ea95ae38a1265a907"
open Function OrderDual Set
variable {α β : Type*}
@[ext]
class SuccOr... | Mathlib/Order/SuccPred/Basic.lean | 343 | 344 | theorem Ico_succ_left_of_not_isMax (ha : ¬IsMax a) : Ico (succ a) b = Ioo a b := by |
rw [← Ici_inter_Iio, Ici_succ_of_not_isMax ha, Ioi_inter_Iio]
|
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.StrongEpi
import Mathlib.CategoryTheory.MorphismProperty.Factorization
#align_import category_theory.limits.shapes.images from "leanprover-community/mathlib"@"563aed... | Mathlib/CategoryTheory/Limits/Shapes/Images.lean | 1,020 | 1,023 | theorem image.isoStrongEpiMono_hom_comp_ι {I' : C} (e : X ⟶ I') (m : I' ⟶ Y) (comm : e ≫ m = f)
[StrongEpi e] [Mono m] : (image.isoStrongEpiMono e m comm).hom ≫ image.ι f = m := by |
dsimp [isoStrongEpiMono]
apply IsImage.lift_fac
|
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"@"b99e2d58a5e6861833fa8de11e51a81144258db4"
open Function
universe u v w
namespace SimpleGraph
variable {V : Type u} {V' : Type v} {V'' : Typ... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 979 | 982 | theorem isTrail_copy {u v u' v'} (p : G.Walk u v) (hu : u = u') (hv : v = v') :
(p.copy hu hv).IsTrail ↔ p.IsTrail := by |
subst_vars
rfl
|
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Int.Lemmas
import Mathlib.Data.Set.Subsingleton
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Order.GaloisConnection
import Mathlib.Tactic.Abel
import Mathlib.Tactic.Linarith... | Mathlib/Algebra/Order/Floor.lean | 577 | 589 | theorem subsingleton_floorSemiring {α} [LinearOrderedSemiring α] :
Subsingleton (FloorSemiring α) := by |
refine ⟨fun H₁ H₂ => ?_⟩
have : H₁.ceil = H₂.ceil := funext fun a => (H₁.gc_ceil.l_unique H₂.gc_ceil) fun n => rfl
have : H₁.floor = H₂.floor := by
ext a
cases' lt_or_le a 0 with h h
· rw [H₁.floor_of_neg, H₂.floor_of_neg] <;> exact h
· refine eq_of_forall_le_iff fun n => ?_
rw [H₁.gc_floor... |
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Regular.Pow
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Data.Finsupp.Antidiagonal
import Mathlib.Order.SymmDiff
import Mathlib.RingTheory.Adjoin.Basic
#align_import data.mv_polynomial.basic fr... | Mathlib/Algebra/MvPolynomial/Basic.lean | 600 | 601 | theorem sum_def {A} [AddCommMonoid A] {p : MvPolynomial σ R} {b : (σ →₀ ℕ) → R → A} :
p.sum b = ∑ m ∈ p.support, b m (p.coeff m) := by | simp [support, Finsupp.sum, coeff]
|
import Mathlib.Analysis.Complex.AbsMax
import Mathlib.Analysis.Complex.RemovableSingularity
#align_import analysis.complex.schwarz from "leanprover-community/mathlib"@"3f655f5297b030a87d641ad4e825af8d9679eb0b"
open Metric Set Function Filter TopologicalSpace
open scoped Topology
namespace Complex
section Space... | Mathlib/Analysis/Complex/Schwarz.lean | 154 | 160 | theorem dist_le_div_mul_dist_of_mapsTo_ball (hd : DifferentiableOn ℂ f (ball c R₁))
(h_maps : MapsTo f (ball c R₁) (ball (f c) R₂)) (hz : z ∈ ball c R₁) :
dist (f z) (f c) ≤ R₂ / R₁ * dist z c := by |
rcases eq_or_ne z c with (rfl | hne);
· simp only [dist_self, mul_zero, le_rfl]
simpa only [dslope_of_ne _ hne, slope_def_module, norm_smul, norm_inv, ← div_eq_inv_mul, ←
dist_eq_norm, div_le_iff (dist_pos.2 hne)] using norm_dslope_le_div_of_mapsTo_ball hd h_maps hz
|
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import number_theory.ramification_inertia from "leanprover-community/mathlib"@"039a089d2a4b93c761b234f3e5f5aeb752bac60f"
namespace Ideal
universe u v
variable {R : Type u} [CommRing R]
variable {S : Type v} [CommRing S] (f : R →+* S)
variable (p : Ideal R) (... | Mathlib/NumberTheory/RamificationInertia.lean | 242 | 272 | theorem FinrankQuotientMap.linearIndependent_of_nontrivial [IsDedekindDomain R]
(hRS : RingHom.ker (algebraMap R S) ≠ ⊤) (f : V'' →ₗ[R] V) (hf : Function.Injective f)
(f' : V'' →ₗ[R] V') {ι : Type*} {b : ι → V''} (hb' : LinearIndependent S (f' ∘ b)) :
LinearIndependent K (f ∘ b) := by |
contrapose! hb' with hb
-- Informally, if we have a nontrivial linear dependence with coefficients `g` in `K`,
-- then we can find a linear dependence with coefficients `I.Quotient.mk g'` in `R/I`,
-- where `I = ker (algebraMap R S)`.
-- We make use of the same principle but stay in `R` everywhere.
simp on... |
import Mathlib.LinearAlgebra.Alternating.Basic
import Mathlib.LinearAlgebra.Multilinear.TensorProduct
import Mathlib.GroupTheory.GroupAction.Quotient
#align_import linear_algebra.alternating from "leanprover-community/mathlib"@"0c1d80f5a86b36c1db32e021e8d19ae7809d5b79"
suppress_compilation
open TensorProduct
vari... | Mathlib/LinearAlgebra/Alternating/DomCoprod.lean | 96 | 132 | theorem domCoprod.summand_eq_zero_of_smul_invariant (a : Mᵢ [⋀^ιa]→ₗ[R'] N₁)
(b : Mᵢ [⋀^ιb]→ₗ[R'] N₂) (σ : Perm.ModSumCongr ιa ιb) {v : Sum ιa ιb → Mᵢ}
{i j : Sum ιa ιb} (hv : v i = v j) (hij : i ≠ j) :
swap i j • σ = σ → domCoprod.summand a b σ v = 0 := by |
refine Quotient.inductionOn' σ fun σ => ?_
dsimp only [Quotient.liftOn'_mk'', Quotient.map'_mk'', MultilinearMap.smul_apply,
MultilinearMap.domDomCongr_apply, MultilinearMap.domCoprod_apply, domCoprod.summand]
intro hσ
cases' hi : σ⁻¹ i with val val <;> cases' hj : σ⁻¹ j with val_1 val_1 <;>
rw [Perm.i... |
import Mathlib.Order.CompleteLattice
import Mathlib.Order.Cover
import Mathlib.Order.Iterate
import Mathlib.Order.WellFounded
#align_import order.succ_pred.basic from "leanprover-community/mathlib"@"0111834459f5d7400215223ea95ae38a1265a907"
open Function OrderDual Set
variable {α β : Type*}
@[ext]
class SuccOr... | Mathlib/Order/SuccPred/Basic.lean | 472 | 473 | theorem Icc_succ_right (h : a ≤ succ b) : Icc a (succ b) = insert (succ b) (Icc a b) := by |
simp_rw [← Ici_inter_Iic, Iic_succ, inter_insert_of_mem (mem_Ici.2 h)]
|
import Mathlib.Geometry.Manifold.PartitionOfUnity
import Mathlib.Geometry.Manifold.Metrizable
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
open MeasureTheory Filter Metric Function Set TopologicalSpace
open scoped Topology Manifold
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [FiniteDimen... | Mathlib/Analysis/Distribution/AEEqOfIntegralContDiff.lean | 156 | 169 | theorem ae_eq_of_integral_smooth_smul_eq
(hf : LocallyIntegrable f μ) (hf' : LocallyIntegrable f' μ) (h : ∀ (g : M → ℝ),
Smooth I 𝓘(ℝ) g → HasCompactSupport g → ∫ x, g x • f x ∂μ = ∫ x, g x • f' x ∂μ) :
∀ᵐ x ∂μ, f x = f' x := by |
have : ∀ᵐ x ∂μ, (f - f') x = 0 := by
apply ae_eq_zero_of_integral_smooth_smul_eq_zero I (hf.sub hf')
intro g g_diff g_supp
simp only [Pi.sub_apply, smul_sub]
rw [integral_sub, sub_eq_zero]
· exact h g g_diff g_supp
· exact hf.integrable_smul_left_of_hasCompactSupport g_diff.continuous g_supp
... |
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Relator
import Mathlib.Init.Data.Quot
import Mathlib.Tactic.Cases
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
#align_import logic.relation from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9a... | Mathlib/Logic/Relation.lean | 159 | 164 | theorem comp_assoc : (r ∘r p) ∘r q = r ∘r p ∘r q := by |
funext a d
apply propext
constructor
· exact fun ⟨c, ⟨b, hab, hbc⟩, hcd⟩ ↦ ⟨b, hab, c, hbc, hcd⟩
· exact fun ⟨b, hab, c, hbc, hcd⟩ ↦ ⟨c, ⟨b, hab, hbc⟩, hcd⟩
|
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 | 349 | 359 | theorem IsPoly.map [Fact p.Prime] {f} (hf : IsPoly p f) (g : R →+* S) (x : 𝕎 R) :
map g (f x) = f (map g x) := by |
-- this could be turned into a tactic “macro” (taking `hf` as parameter)
-- so that applications do not have to worry about the universe issue
-- see `IsPoly₂.map` for a slightly more general proof strategy
obtain ⟨φ, hf⟩ := hf
ext n
simp only [map_coeff, hf, map_aeval]
apply eval₂Hom_congr (RingHom.ext_... |
import Mathlib.Topology.Separation
open Set Filter
open OrderDual (toDual)
open scoped Topology
universe u v w
variable {α : Type u} {β : Type v} {γ : Type w}
class ClosedIicTopology (α : Type*) [TopologicalSpace α] [Preorder α] : Prop where
isClosed_Iic (a : α) : IsClosed (Iic a)
class ClosedIciTopology ... | Mathlib/Topology/Order/OrderClosed.lean | 261 | 262 | theorem Ioo_mem_nhdsWithin_Iio' (H : a < b) : Ioo a b ∈ 𝓝[<] b := by |
simpa only [← Iio_inter_Ioi] using inter_mem_nhdsWithin _ (Ioi_mem_nhds H)
|
import Mathlib.Probability.Notation
import Mathlib.Probability.Integration
import Mathlib.MeasureTheory.Function.L2Space
#align_import probability.variance from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open MeasureTheory Filter Finset
noncomputable section
open scoped MeasureThe... | Mathlib/Probability/Variance.lean | 128 | 139 | theorem _root_.MeasureTheory.Memℒp.variance_eq [IsFiniteMeasure μ] (hX : Memℒp X 2 μ) :
variance X μ = μ[(X - fun _ => μ[X] :) ^ (2 : Nat)] := by |
rw [variance, evariance_eq_lintegral_ofReal, ← ofReal_integral_eq_lintegral_ofReal,
ENNReal.toReal_ofReal (by positivity)]
· rfl
· -- Porting note: `μ[X]` without whitespace is ambiguous as it could be GetElem,
-- and `convert` cannot disambiguate based on typeclass inference failure.
convert (hX.sub... |
import Mathlib.Order.SuccPred.Basic
import Mathlib.Order.BoundedOrder
#align_import order.succ_pred.limit from "leanprover-community/mathlib"@"1e05171a5e8cf18d98d9cf7b207540acb044acae"
variable {α : Type*}
namespace Order
open Function Set OrderDual
section Preorder
variable [Preorder α] {a : α}
protected... | Mathlib/Order/SuccPred/Limit.lean | 360 | 360 | theorem not_isPredLimit_of_noMin [NoMinOrder α] [NoMaxOrder α] : ¬IsPredLimit a := by | simp
|
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 | 722 | 724 | theorem affineCombinationLineMapWeights_apply_of_ne [DecidableEq ι] {i j t : ι} (hi : t ≠ i)
(hj : t ≠ j) (c : k) : affineCombinationLineMapWeights i j c t = 0 := by |
simp [affineCombinationLineMapWeights, hi, hj]
|
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 | 293 | 298 | theorem trace_eq_sum_roots [FiniteDimensional K L] {x : L}
(hF : (minpoly K x).Splits (algebraMap K F)) :
algebraMap K F (Algebra.trace K L x) =
finrank K⟮x⟯ L • ((minpoly K x).aroots F).sum := by |
rw [trace_eq_trace_adjoin K x, Algebra.smul_def, RingHom.map_mul, ← Algebra.smul_def,
IntermediateField.AdjoinSimple.trace_gen_eq_sum_roots _ hF, IsScalarTower.algebraMap_smul]
|
import Mathlib.RingTheory.DedekindDomain.Ideal
#align_import number_theory.ramification_inertia from "leanprover-community/mathlib"@"039a089d2a4b93c761b234f3e5f5aeb752bac60f"
namespace Ideal
universe u v
variable {R : Type u} [CommRing R]
variable {S : Type v} [CommRing S] (f : R →+* S)
variable (p : Ideal R) (... | Mathlib/NumberTheory/RamificationInertia.lean | 159 | 169 | theorem ramificationIdx_ne_zero (hp0 : map f p ≠ ⊥) (hP : P.IsPrime) (le : map f p ≤ P) :
ramificationIdx f p P ≠ 0 := by |
have hP0 : P ≠ ⊥ := by
rintro rfl
have := le_bot_iff.mp le
contradiction
have hPirr := (Ideal.prime_of_isPrime hP0 hP).irreducible
rw [IsDedekindDomain.ramificationIdx_eq_normalizedFactors_count hp0 hP hP0]
obtain ⟨P', hP', P'_eq⟩ :=
exists_mem_normalizedFactors_of_dvd hp0 hPirr (Ideal.dvd_iff_... |
import Mathlib.Algebra.Homology.ShortComplex.Basic
import Mathlib.CategoryTheory.Limits.Constructions.FiniteProductsOfBinaryProducts
import Mathlib.CategoryTheory.Triangulated.TriangleShift
#align_import category_theory.triangulated.pretriangulated from "leanprover-community/mathlib"@"6876fa15e3158ff3e4a4e2af1fb6e194... | Mathlib/CategoryTheory/Triangulated/Pretriangulated.lean | 126 | 130 | theorem comp_distTriang_mor_zero₁₂ (T) (H : T ∈ (distTriang C)) : T.mor₁ ≫ T.mor₂ = 0 := by |
obtain ⟨c, hc⟩ :=
complete_distinguished_triangle_morphism _ _ (contractible_distinguished T.obj₁) H (𝟙 T.obj₁)
T.mor₁ rfl
simpa only [contractibleTriangle_mor₂, zero_comp] using hc.left.symm
|
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.Hom.CompleteLattice
#align_import order.liminf_limsup from "leanprover-community/mathlib"@"ffde2d8a6e689149e44fd95fa862c23a57f8c780"
set_option autoImplicit true
open Filter Set Function
variable {α β γ ι ι' : Type*}
namespace Filter
theorem isCobounde... | Mathlib/Order/LiminfLimsup.lean | 1,235 | 1,236 | theorem cofinite.liminf_set_eq : liminf s cofinite = { x | { n | x ∉ s n }.Finite } := by |
simp only [← cofinite.bliminf_true s, cofinite.bliminf_set_eq, true_and_iff]
|
import Mathlib.Data.DFinsupp.Lex
import Mathlib.Order.GameAdd
import Mathlib.Order.Antisymmetrization
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Tactic.AdaptationNote
#align_import data.dfinsupp.well_founded from "leanprover-community/mathlib"@"e9b8651eb1ad354f4de6be35a38ef31efcd2cfaa"
variable {ι : Ty... | Mathlib/Data/DFinsupp/WellFounded.lean | 118 | 129 | theorem Lex.acc_of_single [DecidableEq ι] [∀ (i) (x : α i), Decidable (x ≠ 0)] (x : Π₀ i, α i) :
(∀ i ∈ x.support, Acc (DFinsupp.Lex r s) <| single i (x i)) → Acc (DFinsupp.Lex r s) x := by |
generalize ht : x.support = t; revert x
classical
induction' t using Finset.induction with b t hb ih
· intro x ht
rw [support_eq_empty.1 ht]
exact fun _ => Lex.acc_zero hbot
refine fun x ht h => Lex.acc_of_single_erase b (h b <| t.mem_insert_self b) ?_
refine ih _ (by rw [support_erase,... |
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Analysis.Normed.MulAction
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.PartialHomeomorph
#align_import analysis.asymptotics.asymptotics from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open ... | Mathlib/Analysis/Asymptotics/Asymptotics.lean | 113 | 114 | theorem isBigO_iff : f =O[l] g ↔ ∃ c : ℝ, ∀ᶠ x in l, ‖f x‖ ≤ c * ‖g x‖ := by |
simp only [IsBigO_def, IsBigOWith_def]
|
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.Decomposition.RadonNikodym
#align_import measure_theory.function.conditional_expectation.real from "leanprover-community/mathlib"@"b2ff9a3d7a15fd5b0f060b135421d6a... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean | 184 | 223 | theorem Integrable.uniformIntegrable_condexp {ι : Type*} [IsFiniteMeasure μ] {g : α → ℝ}
(hint : Integrable g μ) {ℱ : ι → MeasurableSpace α} (hℱ : ∀ i, ℱ i ≤ m0) :
UniformIntegrable (fun i => μ[g|ℱ i]) 1 μ := by |
let A : MeasurableSpace α := m0
have hmeas : ∀ n, ∀ C, MeasurableSet {x | C ≤ ‖(μ[g|ℱ n]) x‖₊} := fun n C =>
measurableSet_le measurable_const (stronglyMeasurable_condexp.mono (hℱ n)).measurable.nnnorm
have hg : Memℒp g 1 μ := memℒp_one_iff_integrable.2 hint
refine uniformIntegrable_of le_rfl ENNReal.one_n... |
import Mathlib.LinearAlgebra.LinearIndependent
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe w w' u u' v v'
variable {R : Type u} {R' : Type u'} {M M₁ : Type v} {M' : Type v'}
open Cardinal Submodule Function... | Mathlib/LinearAlgebra/Dimension/Basic.lean | 228 | 232 | theorem rank_le_of_surjective_injective
(i : R →+* R') (j : S →+* S') (hi : Surjective i) (hj : Injective j)
(hc : (algebraMap R' S').comp i = j.comp (algebraMap R S)) :
Module.rank R S ≤ Module.rank R' S' := by |
simpa only [lift_id] using lift_rank_le_of_surjective_injective i j hi hj hc
|
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Data.Real.Sqrt
#align_import analysis.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
open NormedField Set Filter
open scoped NNReal Pointwi... | Mathlib/Analysis/Seminorm.lean | 821 | 824 | theorem ball_comp (p : Seminorm 𝕜₂ E₂) (f : E →ₛₗ[σ₁₂] E₂) (x : E) (r : ℝ) :
(p.comp f).ball x r = f ⁻¹' p.ball (f x) r := by |
ext
simp_rw [ball, mem_preimage, comp_apply, Set.mem_setOf_eq, map_sub]
|
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.CategoryTheory.EssentiallySmall
import Mathlib.CategoryTheory.Simple
#align_import category_theory.noetherian from "leanprover-community/mathlib"@"7c4c90f422a4a4477e4d8bc4dc9f1e634e6b2349"
namespace CategoryTheory
open CategoryTheory.Limits
variable... | Mathlib/CategoryTheory/Noetherian.lean | 82 | 87 | theorem exists_simple_subobject {X : C} [ArtinianObject X] (h : ¬IsZero X) :
∃ Y : Subobject X, Simple (Y : C) := by |
haveI : Nontrivial (Subobject X) := nontrivial_of_not_isZero h
haveI := isAtomic_of_orderBot_wellFounded_lt (ArtinianObject.subobject_lt_wellFounded X)
obtain ⟨Y, s⟩ := (IsAtomic.eq_bot_or_exists_atom_le (⊤ : Subobject X)).resolve_left top_ne_bot
exact ⟨Y, (subobject_simple_iff_isAtom _).mpr s.1⟩
|
import Mathlib.Init.ZeroOne
import Mathlib.Data.Set.Defs
import Mathlib.Order.Basic
import Mathlib.Order.SymmDiff
import Mathlib.Tactic.Tauto
import Mathlib.Tactic.ByContra
import Mathlib.Util.Delaborators
#align_import data.set.basic from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
... | Mathlib/Data/Set/Basic.lean | 1,325 | 1,326 | theorem inter_singleton_eq_empty : s ∩ {a} = ∅ ↔ a ∉ s := by |
rw [inter_comm, singleton_inter_eq_empty]
|
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.matrix.basis from "leanprover-community/mathlib"@"6c263e4bfc2e6714de30f22178b4d0ca4d149a76"
noncomputable section
open LinearMap Matrix Set Submodule
open Matrix
section BasisToMatrix
variable {ι... | Mathlib/LinearAlgebra/Matrix/Basis.lean | 97 | 102 | theorem toMatrix_unitsSMul [DecidableEq ι] (e : Basis ι R₂ M₂) (w : ι → R₂ˣ) :
e.toMatrix (e.unitsSMul w) = diagonal ((↑) ∘ w) := by |
ext i j
by_cases h : i = j
· simp [h, toMatrix_apply, unitsSMul_apply, Units.smul_def]
· simp [h, toMatrix_apply, unitsSMul_apply, Units.smul_def, Ne.symm h]
|
import Mathlib.Algebra.Group.Indicator
import Mathlib.Data.Finset.Piecewise
import Mathlib.Data.Finset.Preimage
#align_import algebra.big_operators.basic from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
-- TODO
-- assert_not_exists AddCommMonoidWithOne
assert_not_exists MonoidWithZero... | Mathlib/Algebra/BigOperators/Group/Finset.lean | 431 | 435 | theorem prod_disjUnion (h) :
∏ x ∈ s₁.disjUnion s₂ h, f x = (∏ x ∈ s₁, f x) * ∏ x ∈ s₂, f x := by |
refine Eq.trans ?_ (fold_disjUnion h)
rw [one_mul]
rfl
|
import Mathlib.Analysis.NormedSpace.Multilinear.Curry
#align_import analysis.calculus.formal_multilinear_series from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open Set Fin Topology
-- Porting note: added explicit universes to fix compile
universe u u' v w x
... | Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean | 302 | 303 | theorem coeff_eq_zero : p.coeff n = 0 ↔ p n = 0 := by |
rw [← mkPiRing_coeff_eq p, ContinuousMultilinearMap.mkPiRing_eq_zero_iff]
|
import Mathlib.Data.List.Range
import Mathlib.Data.List.Perm
#align_import data.list.sigma from "leanprover-community/mathlib"@"f808feb6c18afddb25e66a71d317643cf7fb5fbb"
universe u v
namespace List
variable {α : Type u} {β : α → Type v} {l l₁ l₂ : List (Sigma β)}
def keys : List (Sigma β) → List α :=
map ... | Mathlib/Data/List/Sigma.lean | 510 | 523 | theorem dlookup_kerase_ne {a a'} {l : List (Sigma β)} (h : a ≠ a') :
dlookup a (kerase a' l) = dlookup a l := by |
induction l with
| nil => rfl
| cons hd tl ih =>
cases' hd with ah bh
by_cases h₁ : a = ah <;> by_cases h₂ : a' = ah
· substs h₁ h₂
cases Ne.irrefl h
· subst h₁
simp [h₂]
· subst h₂
simp [h]
· simp [h₁, h₂, ih]
|
import Mathlib.RingTheory.IntegralClosure
#align_import field_theory.minpoly.basic from "leanprover-community/mathlib"@"df0098f0db291900600f32070f6abb3e178be2ba"
open scoped Classical
open Polynomial Set Function
variable {A B B' : Type*}
namespace minpoly
section Ring
variable [CommRing A] [Ring B] [Ring B']... | Mathlib/FieldTheory/Minpoly/Basic.lean | 115 | 120 | theorem not_isUnit [Nontrivial B] : ¬IsUnit (minpoly A x) := by |
haveI : Nontrivial A := (algebraMap A B).domain_nontrivial
by_cases hx : IsIntegral A x
· exact mt (monic hx).eq_one_of_isUnit (ne_one A x)
· rw [eq_zero hx]
exact not_isUnit_zero
|
import Mathlib.Topology.ExtendFrom
import Mathlib.Topology.Order.DenselyOrdered
#align_import topology.algebra.order.extend_from from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
set_option autoImplicit true
open Filter Set TopologicalSpace
open scoped Classical
open Topology
theor... | Mathlib/Topology/Order/ExtendFrom.lean | 68 | 74 | theorem continuousOn_Ioc_extendFrom_Ioo [TopologicalSpace α] [LinearOrder α] [DenselyOrdered α]
[OrderTopology α] [TopologicalSpace β] [RegularSpace β] {f : α → β} {a b : α} {lb : β}
(hab : a < b) (hf : ContinuousOn f (Ioo a b)) (hb : Tendsto f (𝓝[<] b) (𝓝 lb)) :
ContinuousOn (extendFrom (Ioo a b) f) (Ioc... |
have := @continuousOn_Ico_extendFrom_Ioo αᵒᵈ _ _ _ _ _ _ _ f _ _ lb hab
erw [dual_Ico, dual_Ioi, dual_Ioo] at this
exact this hf hb
|
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 | 169 | 169 | theorem two_zsmul_coe_pi : (2 : ℤ) • (π : Angle) = 0 := by | simp [← intCast_mul_eq_zsmul]
|
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 | 938 | 940 | theorem exists_rat_pow_btwn_rat (hn : n ≠ 0) {x y : ℚ} (h : x < y) (hy : 0 < y) :
∃ q : ℚ, 0 < q ∧ x < q ^ n ∧ q ^ n < y := by |
apply_mod_cast exists_rat_pow_btwn_rat_aux hn x y <;> assumption
|
import Mathlib.Order.MinMax
import Mathlib.Data.Set.Subsingleton
import Mathlib.Tactic.Says
#align_import data.set.intervals.basic from "leanprover-community/mathlib"@"3ba15165bd6927679be7c22d6091a87337e3cd0c"
open Function
open OrderDual (toDual ofDual)
variable {α β : Type*}
namespace Set
theorem Icc_bot_top... | Mathlib/Order/Interval/Set/Basic.lean | 1,826 | 1,827 | theorem Ico_inter_Ico : Ico a₁ b₁ ∩ Ico a₂ b₂ = Ico (a₁ ⊔ a₂) (b₁ ⊓ b₂) := by |
simp only [Ici_inter_Iio.symm, Ici_inter_Ici.symm, Iio_inter_Iio.symm]; ac_rfl
|
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.SumOverResidueClass
#align_import analysis.p_series from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
def SuccDiffBounded (C : ℕ) (u : ℕ → ℕ) : Prop :=... | Mathlib/Analysis/PSeries.lean | 267 | 295 | theorem summable_nat_rpow_inv {p : ℝ} :
Summable (fun n => ((n : ℝ) ^ p)⁻¹ : ℕ → ℝ) ↔ 1 < p := by |
rcases le_or_lt 0 p with hp | hp
/- Cauchy condensation test applies only to antitone sequences, so we consider the
cases `0 ≤ p` and `p < 0` separately. -/
· rw [← summable_condensed_iff_of_nonneg]
· simp_rw [Nat.cast_pow, Nat.cast_two, ← rpow_natCast, ← rpow_mul zero_lt_two.le, mul_comm _ p,
rp... |
import Mathlib.FieldTheory.RatFunc.Defs
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.Content
#align_import field_theory.ratfunc from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d010e417b10abb1b6"
universe u v
noncompu... | Mathlib/FieldTheory/RatFunc/Basic.lean | 601 | 603 | theorem ofFractionRing_algebraMap (x : K[X]) :
ofFractionRing (algebraMap _ (FractionRing K[X]) x) = algebraMap _ _ x := by |
rw [← mk_one, mk_one']
|
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Order.LiminfLimsup
import Mathlib.Topology.Instances.Rat
import Mathlib.Topology.MetricSpace.Algebra
import Mathlib.Topology.MetricSpace.IsometricSMul
import Mathlib.Topology.Sequences
#align_import analysis.normed.group.basic from "leanprover-community/mat... | Mathlib/Analysis/Normed/Group/Basic.lean | 1,894 | 1,894 | theorem nnnorm_abs (r : ℝ) : ‖|r|‖₊ = ‖r‖₊ := by | simp [nnnorm]
|
import Mathlib.Analysis.BoxIntegral.Partition.SubboxInduction
import Mathlib.Analysis.BoxIntegral.Partition.Split
#align_import analysis.box_integral.partition.filter from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open Set Function Filter Metric Finset Bool
open scoped Classical
o... | Mathlib/Analysis/BoxIntegral/Partition/Filter.lean | 372 | 383 | theorem MemBaseSet.exists_common_compl (h₁ : l.MemBaseSet I c₁ r₁ π₁) (h₂ : l.MemBaseSet I c₂ r₂ π₂)
(hU : π₁.iUnion = π₂.iUnion) :
∃ π : Prepartition I, π.iUnion = ↑I \ π₁.iUnion ∧
(l.bDistortion → π.distortion ≤ c₁) ∧ (l.bDistortion → π.distortion ≤ c₂) := by |
wlog hc : c₁ ≤ c₂ with H
· simpa [hU, _root_.and_comm] using
@H _ _ I J c c₂ c₁ r r₂ r₁ π π₂ π₁ _ l₂ l₁ h₂ h₁ hU.symm (le_of_not_le hc)
by_cases hD : (l.bDistortion : Prop)
· rcases h₁.4 hD with ⟨π, hπU, hπc⟩
exact ⟨π, hπU, fun _ => hπc, fun _ => hπc.trans hc⟩
· exact ⟨π₁.toPrepartition.compl, π₁.t... |
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"@"b99e2d58a5e6861833fa8de11e51a81144258db4"
open Function
universe u v w
namespace SimpleGraph
variable {V : Type u} {V' : Type v} {V'' : Typ... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 1,194 | 1,203 | theorem take_spec {u v w : V} (p : G.Walk v w) (h : u ∈ p.support) :
(p.takeUntil u h).append (p.dropUntil u h) = p := by |
induction p
· rw [mem_support_nil_iff] at h
subst u
rfl
· cases h
· simp!
· simp! only
split_ifs with h' <;> subst_vars <;> simp [*]
|
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.convex.side from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052a5230f"
variable {R V V' P P' : Type*}
open AffineEquiv AffineMap
namespace Affine... | Mathlib/Analysis/Convex/Side.lean | 321 | 323 | theorem sOppSide_vadd_right_iff {s : AffineSubspace R P} {x y : P} {v : V} (hv : v ∈ s.direction) :
s.SOppSide x (v +ᵥ y) ↔ s.SOppSide x y := by |
rw [sOppSide_comm, sOppSide_vadd_left_iff hv, sOppSide_comm]
|
import Mathlib.Analysis.ODE.Gronwall
import Mathlib.Analysis.ODE.PicardLindelof
import Mathlib.Geometry.Manifold.InteriorBoundary
import Mathlib.Geometry.Manifold.MFDeriv.Atlas
open scoped Manifold Topology
open Function Set
variable
{E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] [CompleteSpace E]
{H : ... | Mathlib/Geometry/Manifold/IntegralCurve.lean | 433 | 469 | theorem isIntegralCurveOn_Ioo_eqOn_of_contMDiff (ht₀ : t₀ ∈ Ioo a b)
(hγt : ∀ t ∈ Ioo a b, I.IsInteriorPoint (γ t))
(hv : ContMDiff I I.tangent 1 (fun x ↦ (⟨x, v x⟩ : TangentBundle I M)))
(hγ : IsIntegralCurveOn γ v (Ioo a b)) (hγ' : IsIntegralCurveOn γ' v (Ioo a b))
(h : γ t₀ = γ' t₀) : EqOn γ γ' (Ioo ... |
set s := {t | γ t = γ' t} ∩ Ioo a b with hs
-- since `Ioo a b` is connected, we get `s = Ioo a b` by showing that `s` is clopen in `Ioo a b`
-- in the subtype topology (`s` is also non-empty by assumption)
-- here we use a slightly weaker alternative theorem
suffices hsub : Ioo a b ⊆ s from fun t ht ↦ mem_se... |
import Mathlib.Data.Set.Image
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Clopen
import Mathlib.Topology.Irreducible
#align_import topology.connected from "leanprover-community/mathlib"@"d101e93197bb5f6ea89bd7ba386b7f7dff1f3903"
open Set Function Topology TopologicalSpace Relation
open scoped C... | Mathlib/Topology/Connected/Basic.lean | 866 | 868 | theorem nonempty_inter [PreconnectedSpace α] {s t : Set α} :
IsOpen s → IsOpen t → s ∪ t = univ → s.Nonempty → t.Nonempty → (s ∩ t).Nonempty := by |
simpa only [univ_inter, univ_subset_iff] using @PreconnectedSpace.isPreconnected_univ α _ _ s t
|
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Group.Measure
#align_import measure_theory.group.prod from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open Set hiding prod_eq
open Function MeasureTheory
open Filter hiding ma... | Mathlib/MeasureTheory/Group/Prod.lean | 247 | 262 | theorem measure_mul_lintegral_eq [IsMulLeftInvariant ν] (sm : MeasurableSet s) (f : G → ℝ≥0∞)
(hf : Measurable f) : (μ s * ∫⁻ y, f y ∂ν) = ∫⁻ x, ν ((fun z => z * x) ⁻¹' s) * f x⁻¹ ∂μ := by |
rw [← set_lintegral_one, ← lintegral_indicator _ sm,
← lintegral_lintegral_mul (measurable_const.indicator sm).aemeasurable hf.aemeasurable,
← lintegral_lintegral_mul_inv μ ν]
swap
· exact (((measurable_const.indicator sm).comp measurable_fst).mul
(hf.comp measurable_snd)).aemeasurable
have ms :
... |
import Mathlib.Logic.Relation
import Mathlib.Data.Option.Basic
import Mathlib.Data.Seq.Seq
#align_import data.seq.wseq from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace Stream'
open Function
universe u v w
def WSeq (α) :=
Seq (Option α)
#align stream.wseq Stream'.WSeq
... | Mathlib/Data/Seq/WSeq.lean | 1,694 | 1,709 | theorem join_map_ret (s : WSeq α) : join (map ret s) ~ʷ s := by |
refine ⟨fun s1 s2 => join (map ret s2) = s1, rfl, ?_⟩
intro s' s h; rw [← h]
apply liftRel_rec fun c1 c2 => ∃ s, c1 = destruct (join (map ret s)) ∧ c2 = destruct s
· exact fun {c1 c2} h =>
match c1, c2, h with
| _, _, ⟨s, rfl, rfl⟩ => by
clear h
-- Porting note: `ret` is simplified ... |
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 | 185 | 193 | theorem memℒp_finset_sum {ι} (s : Finset ι) {f : ι → α → E} (hf : ∀ i ∈ s, Memℒp (f i) p μ) :
Memℒp (fun a => ∑ i ∈ s, f i a) p μ := by |
haveI : DecidableEq ι := Classical.decEq _
revert hf
refine Finset.induction_on s ?_ ?_
· simp only [zero_mem_ℒp', Finset.sum_empty, imp_true_iff]
· intro i s his ih hf
simp only [his, Finset.sum_insert, not_false_iff]
exact (hf i (s.mem_insert_self i)).add (ih fun j hj => hf j (Finset.mem_insert_of_... |
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Support
#align_import algebra.indicator_function from "leanprover-community/mathlib"@"2445c98ae4b87eabebdde552593519b9b6dc350c"
assert_not_exists MonoidWithZero
open Function
variable {α β ι M N : Type*}
namespace Set
section Monoid
variable ... | Mathlib/Algebra/Group/Indicator.lean | 429 | 433 | theorem mulIndicator_mul_eq_left {f g : α → M} (h : Disjoint (mulSupport f) (mulSupport g)) :
(mulSupport f).mulIndicator (f * g) = f := by |
refine (mulIndicator_congr fun x hx => ?_).trans mulIndicator_mulSupport
have : g x = 1 := nmem_mulSupport.1 (disjoint_left.1 h hx)
rw [Pi.mul_apply, this, mul_one]
|
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.ModEq
import Mathlib.Data.Set.Finite
#align_import combinatorics.pigeonhole from "leanprover-community/mathlib"@"d6fad0e5bf2d6f48da9175d25c3dc5706b3834ce"
un... | Mathlib/Combinatorics/Pigeonhole.lean | 291 | 296 | theorem exists_card_fiber_le_of_card_le_nsmul (ht : t.Nonempty) (hb : ↑s.card ≤ t.card • b) :
∃ y ∈ t, ↑(s.filter fun x => f x = y).card ≤ b := by |
simp_rw [cast_card] at hb ⊢
refine
exists_sum_fiber_le_of_sum_fiber_nonneg_of_sum_le_nsmul
(fun _ _ => sum_nonneg fun _ _ => zero_le_one) ht hb
|
import Mathlib.Logic.Equiv.Nat
import Mathlib.Data.PNat.Basic
import Mathlib.Order.Directed
import Mathlib.Data.Countable.Defs
import Mathlib.Order.RelIso.Basic
import Mathlib.Data.Fin.Basic
#align_import logic.encodable.basic from "leanprover-community/mathlib"@"7c523cb78f4153682c2929e3006c863bfef463d0"
open Opt... | Mathlib/Logic/Encodable/Basic.lean | 207 | 209 | theorem decode₂_encode [Encodable α] (a : α) : decode₂ α (encode a) = some a := by |
ext
simp [mem_decode₂, eq_comm, decode₂_eq_some]
|
import Mathlib.Analysis.NormedSpace.Basic
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Data.Real.Sqrt
import Mathlib.RingTheory.Ideal.QuotientOperations
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import analysis.normed.group.quotient from "leanprover-community/mathlib"@"2196ab363eb097c008... | Mathlib/Analysis/Normed/Group/Quotient.lean | 365 | 372 | theorem lift_unique {N : Type*} [SeminormedAddCommGroup N] (S : AddSubgroup M)
(f : NormedAddGroupHom M N) (hf : ∀ s ∈ S, f s = 0) (g : NormedAddGroupHom (M ⧸ S) N)
(h : g.comp S.normedMk = f) : g = lift S f hf := by |
ext x
rcases AddSubgroup.surjective_normedMk _ x with ⟨x, rfl⟩
change g.comp S.normedMk x = _
simp only [h]
rfl
|
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.MeasureTheory.Constructio... | Mathlib/MeasureTheory/Function/Jacobian.lean | 1,207 | 1,215 | theorem det_one_smulRight {𝕜 : Type*} [NormedField 𝕜] (v : 𝕜) :
((1 : 𝕜 →L[𝕜] 𝕜).smulRight v).det = v := by |
have : (1 : 𝕜 →L[𝕜] 𝕜).smulRight v = v • (1 : 𝕜 →L[𝕜] 𝕜) := by
ext1
simp only [ContinuousLinearMap.smulRight_apply, ContinuousLinearMap.one_apply,
Algebra.id.smul_eq_mul, one_mul, ContinuousLinearMap.coe_smul', Pi.smul_apply, mul_one]
rw [this, ContinuousLinearMap.det, ContinuousLinearMap.coe_s... |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Data.List.Cycle
import Mathlib.Data.Nat.Prime
import Mathlib.Data.PNat.Basic
import Mathlib.Dynamics.FixedPoints.Basic
import Mathlib.GroupTheory.GroupAction.Group
#align_import dynamics.periodic_pts from "leanp... | Mathlib/Dynamics/PeriodicPts.lean | 422 | 425 | theorem minimalPeriod_eq_prime_pow {p k : ℕ} [hp : Fact p.Prime] (hk : ¬IsPeriodicPt f (p ^ k) x)
(hk1 : IsPeriodicPt f (p ^ (k + 1)) x) : minimalPeriod f x = p ^ (k + 1) := by |
apply Nat.eq_prime_pow_of_dvd_least_prime_pow hp.out <;>
rwa [← isPeriodicPt_iff_minimalPeriod_dvd]
|
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Dynamics.PeriodicPts
import Mathlib.GroupTheory.Commutator
import Mathlib.GroupTheory.Coset
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Hom
#align_import group_theory.group_action.... | Mathlib/GroupTheory/GroupAction/Quotient.lean | 210 | 213 | theorem card_orbit_mul_card_stabilizer_eq_card_group (b : β) [Fintype α] [Fintype <| orbit α b]
[Fintype <| stabilizer α b] :
Fintype.card (orbit α b) * Fintype.card (stabilizer α b) = Fintype.card α := by |
rw [← Fintype.card_prod, Fintype.card_congr (orbitProdStabilizerEquivGroup α b)]
|
import Mathlib.MeasureTheory.Integral.IntegrableOn
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Function.LocallyIntegrable
import Mathlib.Topology.MetricSpace.ThickenedIndicator
import Mathlib.Topology.ContinuousFunction.Compact
import Mathlib.Analysis.NormedSpace.HahnBanach.SeparatingDua... | Mathlib/MeasureTheory/Integral/SetIntegral.lean | 631 | 642 | theorem norm_setIntegral_le_of_norm_le_const_ae' {C : ℝ} (hs : μ s < ∞)
(hC : ∀ᵐ x ∂μ, x ∈ s → ‖f x‖ ≤ C) (hfm : AEStronglyMeasurable f (μ.restrict s)) :
‖∫ x in s, f x ∂μ‖ ≤ C * (μ s).toReal := by |
apply norm_setIntegral_le_of_norm_le_const_ae hs
have A : ∀ᵐ x : X ∂μ, x ∈ s → ‖AEStronglyMeasurable.mk f hfm x‖ ≤ C := by
filter_upwards [hC, hfm.ae_mem_imp_eq_mk] with _ h1 h2 h3
rw [← h2 h3]
exact h1 h3
have B : MeasurableSet {x | ‖hfm.mk f x‖ ≤ C} :=
hfm.stronglyMeasurable_mk.norm.measurable ... |
import Mathlib.AlgebraicGeometry.GammaSpecAdjunction
import Mathlib.AlgebraicGeometry.Restrict
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.RingTheory.Localization.InvSubmonoid
#align_import algebraic_geometry.AffineScheme from "leanprover-community/mathlib"@"88474d1b5af6d37c2ab728b757771bced7f5194c"... | Mathlib/AlgebraicGeometry/AffineScheme.lean | 295 | 300 | theorem imageIsOpenImmersion (f : X ⟶ Y) [H : IsOpenImmersion f] :
IsAffineOpen (f.opensFunctor.obj U) := by |
have : IsAffine _ := hU
convert rangeIsAffineOpenOfOpenImmersion (X.ofRestrict U.openEmbedding ≫ f)
ext1
exact Set.image_eq_range _ _
|
import Mathlib.RingTheory.LocalProperties
#align_import ring_theory.ring_hom.surjective from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
namespace RingHom
open scoped TensorProduct
open TensorProduct Algebra.TensorProduct
local notation "surjective" => fun {X Y : Type _} [CommRing... | Mathlib/RingTheory/RingHom/Surjective.lean | 26 | 27 | theorem surjective_stableUnderComposition : StableUnderComposition surjective := by |
introv R hf hg; exact hg.comp hf
|
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.Convex.Slope
open Metric Set Asymptotics ContinuousLinearMap Filter
open scoped Classical Topology NNReal
theorem MonotoneOn.convexOn_of_deriv {D : Set ℝ} (hD : Convex ℝ D) {f : ℝ → ℝ}
(hf : ContinuousOn f D) (hf' : DifferentiableOn ℝ f (in... | Mathlib/Analysis/Convex/Deriv.lean | 65 | 75 | theorem StrictMonoOn.exists_slope_lt_deriv_aux {x y : ℝ} {f : ℝ → ℝ} (hf : ContinuousOn f (Icc x y))
(hxy : x < y) (hf'_mono : StrictMonoOn (deriv f) (Ioo x y)) (h : ∀ w ∈ Ioo x y, deriv f w ≠ 0) :
∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a := by |
have A : DifferentiableOn ℝ f (Ioo x y) := fun w wmem =>
(differentiableAt_of_deriv_ne_zero (h w wmem)).differentiableWithinAt
obtain ⟨a, ⟨hxa, hay⟩, ha⟩ : ∃ a ∈ Ioo x y, deriv f a = (f y - f x) / (y - x) :=
exists_deriv_eq_slope f hxy hf A
rcases nonempty_Ioo.2 hay with ⟨b, ⟨hab, hby⟩⟩
refine ⟨b, ⟨hxa... |
import Mathlib.Algebra.Algebra.RestrictScalars
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.LinearAlgebra.Quotient
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.GroupTheory.Finiteness
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.Nilpotent.Defs
#align_import ring_theory.finit... | Mathlib/RingTheory/Finiteness.lean | 335 | 344 | theorem fg_induction (R M : Type*) [Semiring R] [AddCommMonoid M] [Module R M]
(P : Submodule R M → Prop) (h₁ : ∀ x, P (Submodule.span R {x}))
(h₂ : ∀ M₁ M₂, P M₁ → P M₂ → P (M₁ ⊔ M₂)) (N : Submodule R M) (hN : N.FG) : P N := by |
classical
obtain ⟨s, rfl⟩ := hN
induction s using Finset.induction
· rw [Finset.coe_empty, Submodule.span_empty, ← Submodule.span_zero_singleton]
apply h₁
· rw [Finset.coe_insert, Submodule.span_insert]
apply h₂ <;> apply_assumption
|
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 | 680 | 681 | theorem weightedVSubVSubWeights_apply_right [DecidableEq ι] {i j : ι} (h : i ≠ j) :
weightedVSubVSubWeights k i j j = -1 := by | simp [weightedVSubVSubWeights, h.symm]
|
import Mathlib.Algebra.Group.Commute.Defs
import Mathlib.Algebra.Group.Hom.Instances
import Mathlib.Data.Set.Function
import Mathlib.Logic.Pairwise
#align_import algebra.group.pi from "leanprover-community/mathlib"@"e4bc74cbaf429d706cb9140902f7ca6c431e75a4"
assert_not_exists AddMonoidWithOne
assert_not_exists Mono... | Mathlib/Algebra/Group/Pi/Lemmas.lean | 546 | 549 | theorem curry_mulSingle [DecidableEq α] [∀ a, DecidableEq (β a)] [∀ a b, One (γ a b)]
(i : Σ a, β a) (x : γ i.1 i.2) :
Sigma.curry (Pi.mulSingle i x) = Pi.mulSingle i.1 (Pi.mulSingle i.2 x) := by |
simp only [Pi.mulSingle, Sigma.curry_update, Sigma.curry_one, Pi.one_apply]
|
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"@"b99e2d58a5e6861833fa8de11e51a81144258db4"
open Function
universe u v w
namespace SimpleGraph
variable {V : Type u} {V' : Type v} {V'' : Typ... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 1,959 | 1,964 | theorem map_toDeleteEdges_eq (s : Set (Sym2 V)) {p : G.Walk v w} (hp) :
Walk.map (Hom.mapSpanningSubgraphs (G.deleteEdges_le s)) (p.toDeleteEdges s hp) = p := by |
rw [← transfer_eq_map_of_le, transfer_transfer, transfer_self]
intros e
rw [edges_transfer]
apply edges_subset_edgeSet p
|
import Mathlib.Algebra.Order.Floor
import Mathlib.Algebra.Order.Field.Power
import Mathlib.Data.Nat.Log
#align_import data.int.log from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
variable {R : Type*} [LinearOrderedSemifield R] [FloorSemiring R]
namespace Int
def log (b : ℕ) (r : ... | Mathlib/Data/Int/Log.lean | 260 | 264 | theorem zpow_pred_clog_lt_self {b : ℕ} {r : R} (hb : 1 < b) (hr : 0 < r) :
(b : R) ^ (clog b r - 1) < r := by |
rw [← neg_log_inv_eq_clog, ← neg_add', zpow_neg, inv_lt _ hr]
· exact lt_zpow_succ_log_self hb _
· exact zpow_pos_of_pos (Nat.cast_pos.mpr <| zero_le_one.trans_lt hb) _
|
import Mathlib.Algebra.GeomSum
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Iterate
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.Algebra.InfiniteSum.Real
#align_import analysis.specific_limits.basic from "leanprover-community/mathlib"@"57ac39bd365c2f80589a700f9fbb664d3a1a30c2"
n... | Mathlib/Analysis/SpecificLimits/Basic.lean | 195 | 198 | theorem geom_le {u : ℕ → ℝ} {c : ℝ} (hc : 0 ≤ c) (n : ℕ) (h : ∀ k < n, c * u k ≤ u (k + 1)) :
c ^ n * u 0 ≤ u n := by |
apply (monotone_mul_left_of_nonneg hc).seq_le_seq n _ _ h <;>
simp [_root_.pow_succ', mul_assoc, le_refl]
|
import Mathlib.CategoryTheory.MorphismProperty.Composition
import Mathlib.CategoryTheory.MorphismProperty.IsInvertedBy
import Mathlib.CategoryTheory.Category.Quiv
#align_import category_theory.localization.construction from "leanprover-community/mathlib"@"1a5e56f2166e4e9d0964c71f4273b1d39227678d"
noncomputable se... | Mathlib/CategoryTheory/Localization/Construction.lean | 231 | 255 | theorem morphismProperty_is_top (P : MorphismProperty W.Localization)
[P.IsStableUnderComposition] (hP₁ : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), P (W.Q.map f))
(hP₂ : ∀ ⦃X Y : C⦄ (w : X ⟶ Y) (hw : W w), P (winv w hw)) :
P = ⊤ := by |
funext X Y f
ext
constructor
· intro
apply MorphismProperty.top_apply
· intro
let G : _ ⥤ W.Localization := Quotient.functor _
haveI : G.Full := Quotient.full_functor _
suffices ∀ (X₁ X₂ : Paths (LocQuiver W)) (f : X₁ ⟶ X₂), P (G.map f) by
rcases X with ⟨⟨X⟩⟩
rcases Y with ⟨⟨Y⟩⟩
... |
import Mathlib.Data.Set.Image
import Mathlib.Data.Set.Lattice
#align_import data.set.sigma from "leanprover-community/mathlib"@"2258b40dacd2942571c8ce136215350c702dc78f"
namespace Set
variable {ι ι' : Type*} {α β : ι → Type*} {s s₁ s₂ : Set ι} {t t₁ t₂ : ∀ i, Set (α i)}
{u : Set (Σ i, α i)} {x : Σ i, α i} {i j ... | Mathlib/Data/Set/Sigma.lean | 131 | 133 | theorem sigma_inter_sigma : s₁.sigma t₁ ∩ s₂.sigma t₂ = (s₁ ∩ s₂).sigma fun i ↦ t₁ i ∩ t₂ i := by |
ext ⟨x, y⟩
simp [and_assoc, and_left_comm]
|
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 | 103 | 104 | theorem polar_neg (f : M → R) (x y : M) : polar (-f) x y = -polar f x y := by |
simp only [polar, Pi.neg_apply, sub_eq_add_neg, neg_add]
|
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.Block
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.LinearAlgebra.TensorProduct.Basic
import Mathlib.RingTheory.TensorProduct.Basic
#align_import data.matrix.kronecker from "leanpr... | Mathlib/Data/Matrix/Kronecker.lean | 132 | 136 | theorem kroneckerMap_diagonal_right [Zero β] [Zero γ] [DecidableEq n] (f : α → β → γ)
(hf : ∀ a, f a 0 = 0) (A : Matrix l m α) (b : n → β) :
kroneckerMap f A (diagonal b) = blockDiagonal fun i => A.map fun a => f a (b i) := by |
ext ⟨i₁, i₂⟩ ⟨j₁, j₂⟩
simp [diagonal, blockDiagonal, apply_ite (f (A i₁ j₁)), hf]
|
import Mathlib.Algebra.Module.Submodule.EqLocus
import Mathlib.Algebra.Module.Submodule.RestrictScalars
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.LinearAlgebra.Basic
import Mathlib.Order.CompactlyGenerated.Basic
import Mathlib.Order.OmegaCompletePartialOrder
#align_... | Mathlib/LinearAlgebra/Span.lean | 1,063 | 1,065 | theorem toSpanSingleton_zero : toSpanSingleton R M 0 = 0 := by |
ext
simp
|
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.Deriv
import Mathlib.Analysis.SpecialFunctions.Complex.Log
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
#align_import analysis.special_functions.complex.log_deriv from "leanprover-community/mathlib"@"6a5c85000ab93fe5dcfdf620676f614ba8e18c26"
open Set Fi... | Mathlib/Analysis/SpecialFunctions/Complex/LogDeriv.lean | 105 | 107 | theorem HasDerivAt.clog {f : ℂ → ℂ} {f' x : ℂ} (h₁ : HasDerivAt f f' x)
(h₂ : f x ∈ slitPlane) : HasDerivAt (fun t => log (f t)) (f' / f x) x := by |
rw [div_eq_inv_mul]; exact (hasStrictDerivAt_log h₂).hasDerivAt.comp x h₁
|
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib.Data.Set.Finite
#align_import data.finsupp.defs from "leanprover-community/mathlib"@"842328d9df7e96fd90fc424e115679c15fb23a71"
noncomputable section
open Finset Function
variable {α β γ ι M M' N P G H R S : Type*}... | Mathlib/Data/Finsupp/Defs.lean | 591 | 596 | theorem support_update_subset [DecidableEq α] [DecidableEq M] :
support (f.update a b) ⊆ insert a f.support := by |
rw [support_update]
split_ifs
· exact (erase_subset _ _).trans (subset_insert _ _)
· rfl
|
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.Tactic.Abel
#align_import set_theory.ordinal.natural_ops from "leanprover-community/mathlib"@"31b269b60935483943542d547a6dd83a66b37dc7"
set_option autoImplicit true
universe u v
open Function Order
noncomputable section
def NatOrdinal : Type _ :=
... | Mathlib/SetTheory/Ordinal/NaturalOps.lean | 237 | 239 | theorem lt_nadd_iff : a < b ♯ c ↔ (∃ b' < b, a ≤ b' ♯ c) ∨ ∃ c' < c, a ≤ b ♯ c' := by |
rw [nadd_def]
simp [lt_blsub_iff]
|
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.Div
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable ... | Mathlib/Algebra/Polynomial/RingDivision.lean | 547 | 550 | theorem rootMultiplicity_add {p q : R[X]} (a : R) (hzero : p + q ≠ 0) :
min (rootMultiplicity a p) (rootMultiplicity a q) ≤ rootMultiplicity a (p + q) := by |
rw [le_rootMultiplicity_iff hzero]
exact min_pow_dvd_add (pow_rootMultiplicity_dvd p a) (pow_rootMultiplicity_dvd q a)
|
import Batteries.Data.Sum.Basic
import Batteries.Logic
open Function
namespace Sum
@[simp] protected theorem «forall» {p : α ⊕ β → Prop} :
(∀ x, p x) ↔ (∀ a, p (inl a)) ∧ ∀ b, p (inr b) :=
⟨fun h => ⟨fun _ => h _, fun _ => h _⟩, fun ⟨h₁, h₂⟩ => Sum.rec h₁ h₂⟩
@[simp] protected theorem «exists» {p : α ⊕ β ... | .lake/packages/batteries/Batteries/Data/Sum/Lemmas.lean | 75 | 75 | theorem not_isLeft {x : α ⊕ β} : ¬x.isLeft ↔ x.isRight := by | simp
|
import Mathlib.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
variable (K : Type*) [F... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 361 | 363 | theorem norm_real (c : ℝ) :
mixedEmbedding.norm ((fun _ ↦ c, fun _ ↦ c) : (E K)) = |c| ^ finrank ℚ K := by |
rw [show ((fun _ ↦ c, fun _ ↦ c) : (E K)) = c • 1 by ext <;> simp, norm_smul, map_one, mul_one]
|
import Mathlib.Analysis.Complex.Asymptotics
import Mathlib.Analysis.SpecificLimits.Normed
#align_import analysis.special_functions.exp from "leanprover-community/mathlib"@"ba5ff5ad5d120fb0ef094ad2994967e9bfaf5112"
noncomputable section
open Finset Filter Metric Asymptotics Set Function Bornology
open scoped Cla... | Mathlib/Analysis/SpecialFunctions/Exp.lean | 399 | 401 | theorem isLittleO_pow_exp_atTop {n : ℕ} : (fun x : ℝ => x ^ n) =o[atTop] Real.exp := by |
simpa [isLittleO_iff_tendsto fun x hx => ((exp_pos x).ne' hx).elim] using
tendsto_div_pow_mul_exp_add_atTop 1 0 n zero_ne_one
|
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.TensorProduct.Tower
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.LinearAlgebra.DirectSum.Finsupp
#align_import ring_theory.tensor_product from "leanprover-community/mathlib"@"88fcdc3da43943f5b01925deddaa5bf0c0e85e4e"
suppress_comp... | Mathlib/RingTheory/TensorProduct/Basic.lean | 903 | 911 | theorem map_range (f : A →ₐ[R] B) (g : C →ₐ[R] D) :
(map f g).range = (includeLeft.comp f).range ⊔ (includeRight.comp g).range := by |
apply le_antisymm
· rw [← map_top, ← adjoin_tmul_eq_top, ← adjoin_image, adjoin_le_iff]
rintro _ ⟨_, ⟨a, b, rfl⟩, rfl⟩
rw [map_tmul, ← _root_.mul_one (f a), ← _root_.one_mul (g b), ← tmul_mul_tmul]
exact mul_mem_sup (AlgHom.mem_range_self _ a) (AlgHom.mem_range_self _ b)
· rw [← map_comp_includeLeft ... |
import Mathlib.Topology.UniformSpace.UniformConvergenceTopology
#align_import topology.uniform_space.equicontinuity from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
section
open UniformSpace Filter Set Uniformity Topology UniformConvergence Function
variable {ι κ X X' Y Z α α' β β'... | Mathlib/Topology/UniformSpace/Equicontinuity.lean | 648 | 653 | theorem Filter.HasBasis.equicontinuousAt_iff_right {p : κ → Prop} {s : κ → Set (α × α)}
{F : ι → X → α} {x₀ : X} (hα : (𝓤 α).HasBasis p s) :
EquicontinuousAt F x₀ ↔ ∀ k, p k → ∀ᶠ x in 𝓝 x₀, ∀ i, (F i x₀, F i x) ∈ s k := by |
rw [equicontinuousAt_iff_continuousAt, ContinuousAt,
(UniformFun.hasBasis_nhds_of_basis ι α _ hα).tendsto_right_iff]
rfl
|
import Mathlib.NumberTheory.ZetaValues
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex Real Set
open scoped Nat
open HurwitzZeta
theorem riemannZeta_two_mul_nat {k : ℕ} (hk : k ≠ 0) :
riemannZeta (2 * k) = (-1) ^ (k + 1) * (2 : ℂ) ^ (2 * k - 1)
* (π : ℂ) ^ (2 * k) * bernoulli (2 * k) / (... | Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean | 227 | 232 | theorem riemannZeta_four : riemannZeta 4 = π ^ 4 / 90 := by |
convert congr_arg ((↑) : ℝ → ℂ) hasSum_zeta_four.tsum_eq
· rw [← Nat.cast_one, show (4 : ℂ) = (4 : ℕ) by norm_num,
zeta_nat_eq_tsum_of_gt_one (by norm_num : 1 < 4)]
simp only [push_cast]
· norm_cast
|
import Mathlib.Data.Nat.Squarefree
import Mathlib.NumberTheory.Zsqrtd.QuadraticReciprocity
import Mathlib.Tactic.LinearCombination
#align_import number_theory.sum_two_squares from "leanprover-community/mathlib"@"5b2fe80501ff327b9109fb09b7cc8c325cd0d7d9"
section Main
| Mathlib/NumberTheory/SumTwoSquares.lean | 214 | 240 | theorem Nat.eq_sq_add_sq_iff {n : ℕ} :
(∃ x y : ℕ, n = x ^ 2 + y ^ 2) ↔ ∀ {q : ℕ}, q.Prime → q % 4 = 3 → Even (padicValNat q n) := by |
rcases n.eq_zero_or_pos with (rfl | hn₀)
· exact ⟨fun _ q _ _ => (@padicValNat.zero q).symm ▸ even_zero, fun _ => ⟨0, 0, rfl⟩⟩
-- now `0 < n`
rw [Nat.eq_sq_add_sq_iff_eq_sq_mul]
refine ⟨fun H q hq h => ?_, fun H => ?_⟩
· obtain ⟨a, b, h₁, h₂⟩ := H
have hqb := padicValNat.eq_zero_of_not_dvd fun hf =>
... |
import Mathlib.Algebra.Group.Nat
import Mathlib.Algebra.Order.Sub.Canonical
import Mathlib.Data.List.Perm
import Mathlib.Data.Set.List
import Mathlib.Init.Quot
import Mathlib.Order.Hom.Basic
#align_import data.multiset.basic from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
universe v
... | Mathlib/Data/Multiset/Basic.lean | 431 | 432 | theorem cons_subset {a : α} {s t : Multiset α} : a ::ₘ s ⊆ t ↔ a ∈ t ∧ s ⊆ t := by |
simp [subset_iff, or_imp, forall_and]
|
import Mathlib.CategoryTheory.Sites.Sieves
#align_import category_theory.sites.sheaf_of_types from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe w v₁ v₂ u₁ u₂
namespace CategoryTheory
open Opposite CategoryTheory Category Limits Sieve
namespace Presieve
variable {C : Type ... | Mathlib/CategoryTheory/Sites/IsSheafFor.lean | 386 | 391 | theorem is_compatible_of_exists_amalgamation (x : FamilyOfElements P R)
(h : ∃ t, x.IsAmalgamation t) : x.Compatible := by |
cases' h with t ht
intro Y₁ Y₂ Z g₁ g₂ f₁ f₂ h₁ h₂ comm
rw [← ht _ h₁, ← ht _ h₂, ← FunctorToTypes.map_comp_apply, ← op_comp, comm]
simp
|
import Mathlib.Algebra.BigOperators.Group.List
import Mathlib.Algebra.Group.InjSurj
import Mathlib.Data.List.FinRange
import Mathlib.Algebra.Group.Action.Defs
import Mathlib.Data.SetLike.Basic
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib.Data.Sigma.Basic
import Lean.Elab.Tactic
#align_import algebra.gr... | Mathlib/Algebra/GradedMonoid.lean | 608 | 618 | theorem list_prod_map_mem_graded {ι'} (l : List ι') (i : ι' → ι) (r : ι' → R)
(h : ∀ j ∈ l, r j ∈ A (i j)) : (l.map r).prod ∈ A (l.map i).sum := by |
match l with
| [] =>
rw [List.map_nil, List.map_nil, List.prod_nil, List.sum_nil]
exact one_mem_graded _
| head::tail =>
rw [List.map_cons, List.map_cons, List.prod_cons, List.sum_cons]
exact
mul_mem_graded (h _ <| List.mem_cons_self _ _)
(list_prod_map_mem_graded tail _ _ fun j hj ... |
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Star.Unitary
import Mathlib.Data.Nat.ModEq
import Mathlib.NumberTheory.Zsqrtd.Basic
import Mathlib.Tactic.Monotonicity
#align_import number_theory.pell_matiyasevic from "leanprover-community/mathlib"@"795b501869b9f... | Mathlib/NumberTheory/PellMatiyasevic.lean | 520 | 526 | theorem xy_succ_succ (n) :
xn a1 (n + 2) + xn a1 n =
2 * a * xn a1 (n + 1) ∧ yn a1 (n + 2) + yn a1 n = 2 * a * yn a1 (n + 1) := by |
have := pellZd_succ_succ a1 n; unfold pellZd at this
erw [Zsqrtd.smul_val (2 * a : ℕ)] at this
injection this with h₁ h₂
constructor <;> apply Int.ofNat.inj <;> [simpa using h₁; simpa using h₂]
|
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 | 71 | 103 | theorem _root_.NumberField.mixedEmbedding.volume_fundamentalDomain_latticeBasis :
volume (fundamentalDomain (latticeBasis K)) =
(2 : ℝ≥0∞)⁻¹ ^ NrComplexPlaces K * sqrt ‖discr K‖₊ := by |
let f : Module.Free.ChooseBasisIndex ℤ (𝓞 K) ≃ (K →+* ℂ) :=
(canonicalEmbedding.latticeBasis K).indexEquiv (Pi.basisFun ℂ _)
let e : (index K) ≃ Module.Free.ChooseBasisIndex ℤ (𝓞 K) := (indexEquiv K).trans f.symm
let M := (mixedEmbedding.stdBasis K).toMatrix ((latticeBasis K).reindex e.symm)
let N := Alg... |
import Mathlib.Analysis.Quaternion
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Series
#align_import analysis.normed_space.quaternion_exponential from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open scoped Quaternion Nat
open... | Mathlib/Analysis/NormedSpace/QuaternionExponential.lean | 107 | 111 | theorem exp_eq (q : Quaternion ℝ) :
exp ℝ q = exp ℝ q.re • (↑(Real.cos ‖q.im‖) + (Real.sin ‖q.im‖ / ‖q.im‖) • q.im) := by |
rw [← exp_of_re_eq_zero q.im q.im_re, ← coe_mul_eq_smul, ← exp_coe, ← exp_add_of_commute,
re_add_im]
exact Algebra.commutes q.re (_ : ℍ[ℝ])
|
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 | 2,303 | 2,305 | theorem one_add_natCast (m : ℕ) : 1 + (m : Ordinal) = succ m := by |
rw [← Nat.cast_one, ← Nat.cast_add, add_comm]
rfl
|
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Order.LiminfLimsup
import Mathlib.Topology.Instances.Rat
import Mathlib.Topology.MetricSpace.Algebra
import Mathlib.Topology.MetricSpace.IsometricSMul
import Mathlib.Topology.Sequences
#align_import analysis.normed.group.basic from "leanprover-community/mat... | Mathlib/Analysis/Normed/Group/Basic.lean | 1,744 | 1,747 | theorem smul_ball'' : a • ball b r = ball (a • b) r := by |
ext
simp [mem_ball, Set.mem_smul_set, dist_eq_norm_div, _root_.div_eq_inv_mul,
← eq_inv_mul_iff_mul_eq, mul_assoc]
|
import Mathlib.RingTheory.WittVector.StructurePolynomial
#align_import ring_theory.witt_vector.defs from "leanprover-community/mathlib"@"f1944b30c97c5eb626e498307dec8b022a05bd0a"
noncomputable section
structure WittVector (p : ℕ) (R : Type*) where mk' ::
coeff : ℕ → R
#align witt_vector WittVector
-- Port... | Mathlib/RingTheory/WittVector/Defs.lean | 286 | 288 | theorem constantCoeff_wittAdd (n : ℕ) : constantCoeff (wittAdd p n) = 0 := by |
apply constantCoeff_wittStructureInt p _ _ n
simp only [add_zero, RingHom.map_add, constantCoeff_X]
|
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.matrix.nonsingular_inverse from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
namespace Matrix
universe u u' v
variable {l : ... | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 301 | 302 | theorem nonsing_inv_mul_cancel_left (B : Matrix n m α) (h : IsUnit A.det) : A⁻¹ * (A * B) = B := by |
simp [← Matrix.mul_assoc, nonsing_inv_mul A h]
|
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 | 876 | 885 | theorem le_restrict_univ_iff_le : v ≤[Set.univ] w ↔ v ≤ w := by |
constructor
· intro h s hs
have := h s hs
rwa [restrict_apply _ MeasurableSet.univ hs, Set.inter_univ,
restrict_apply _ MeasurableSet.univ hs, Set.inter_univ] at this
· intro h s hs
rw [restrict_apply _ MeasurableSet.univ hs, Set.inter_univ,
restrict_apply _ MeasurableSet.univ hs, Set.int... |
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Finset.Pairwise
import Mathlib.Data.Finset.Powerset
import Mathlib.Data.Fintype.Basic
import Mathlib.Order.CompleteLatticeIntervals
#align_import order.sup_indep from "leanprover-community/mathlib"@"c4c2ed622f43768eff32608d4a0f8a6cec1c047d"
variable {α β ι ι' :... | Mathlib/Order/SupIndep.lean | 130 | 148 | theorem supIndep_pair [DecidableEq ι] {i j : ι} (hij : i ≠ j) :
({i, j} : Finset ι).SupIndep f ↔ Disjoint (f i) (f j) :=
⟨fun h => h.pairwiseDisjoint (by simp) (by simp) hij,
fun h => by
rw [supIndep_iff_disjoint_erase]
intro k hk
rw [Finset.mem_insert, Finset.mem_singleton] at hk
obtain rfl | ... |
ext
rw [mem_erase, mem_insert, mem_singleton, mem_singleton, and_or_left, Ne,
not_and_self_iff, or_false_iff, and_iff_right_of_imp]
rintro rfl
exact hij
rw [this, Finset.sup_singleton]⟩
|
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Set.Pointwise.Basic
#align_import data.set.pointwise.big_operators from "leanprover-community/mathlib"@"fa2cb8a9e2b987db233e4e6eb47645feafba8861"
namespace Set
open Pointwise Function
variable {ι α β F : Type*} [FunLike F α β]
section CommMon... | Mathlib/Data/Set/Pointwise/BigOperators.lean | 85 | 88 | theorem mem_fintype_prod [Fintype ι] (f : ι → Set α) (a : α) :
(a ∈ ∏ i, f i) ↔ ∃ (g : ι → α) (_ : ∀ i, g i ∈ f i), ∏ i, g i = a := by |
rw [mem_finset_prod]
simp
|
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 | 125 | 157 | theorem ae_const_le_iff_forall_lt_measure_zero {β} [LinearOrder β] [TopologicalSpace β]
[OrderTopology β] [FirstCountableTopology β] (f : α → β) (c : β) :
(∀ᵐ x ∂μ, c ≤ f x) ↔ ∀ b < c, μ {x | f x ≤ b} = 0 := by |
rw [ae_iff]
push_neg
constructor
· intro h b hb
exact measure_mono_null (fun y hy => (lt_of_le_of_lt hy hb : _)) h
intro hc
by_cases h : ∀ b, c ≤ b
· have : {a : α | f a < c} = ∅ := by
apply Set.eq_empty_iff_forall_not_mem.2 fun x hx => ?_
exact (lt_irrefl _ (lt_of_lt_of_le hx (h (f x))))... |
import Mathlib.Order.Interval.Finset.Basic
#align_import data.multiset.locally_finite from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
variable {α : Type*}
namespace Multiset
section Preorder
variable [Preorder α] [LocallyFiniteOrder α] {a b c : α}
theorem nodup_Icc : (Icc a b).N... | Mathlib/Order/Interval/Multiset.lean | 153 | 154 | theorem Ioo_eq_zero_iff [DenselyOrdered α] : Ioo a b = 0 ↔ ¬a < b := by |
rw [Ioo, Finset.val_eq_zero, Finset.Ioo_eq_empty_iff]
|
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Analysis.Normed.MulAction
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.PartialHomeomorph
#align_import analysis.asymptotics.asymptotics from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open ... | Mathlib/Analysis/Asymptotics/Asymptotics.lean | 1,441 | 1,446 | theorem isBigO_const_iff {c : F''} : (f'' =O[l] fun _x => c) ↔
(c = 0 → f'' =ᶠ[l] 0) ∧ IsBoundedUnder (· ≤ ·) l fun x => ‖f'' x‖ := by |
refine ⟨fun h => ⟨fun hc => isBigO_zero_right_iff.1 (by rwa [← hc]), h.isBoundedUnder_le⟩, ?_⟩
rintro ⟨hcf, hf⟩
rcases eq_or_ne c 0 with (hc | hc)
exacts [(hcf hc).trans_isBigO (isBigO_zero _ _), hf.isBigO_const hc]
|
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