Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | num_lines int64 1 150 |
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import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.BigOperators
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.GroupTheory.Torsion
import Mathlib.RingTheory.Coprime.Ideal
import Mathlib.RingTheory.Finiteness
import Mathlib.Data.Set.Lattice
#align_import algebra.module.torsion from "leanpr... | Mathlib/Algebra/Module/Torsion.lean | 99 | 105 | theorem torsionOf_eq_bot_iff_of_noZeroSMulDivisors [Nontrivial R] [NoZeroSMulDivisors R M] (m : M) :
torsionOf R M m = ⊥ ↔ m ≠ 0 := by |
refine ⟨fun h contra => ?_, fun h => (Submodule.eq_bot_iff _).mpr fun r hr => ?_⟩
· rw [contra, torsionOf_zero] at h
exact bot_ne_top.symm h
· rw [mem_torsionOf_iff, smul_eq_zero] at hr
tauto
| 5 |
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
#align_import measure_theory.integral.set_to_l1 from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical Topology NNReal ENNReal MeasureTheory Pointwise
open Set Filter TopologicalSpace ENNR... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 143 | 149 | theorem map_empty_eq_zero {β} [AddCancelMonoid β] {T : Set α → β} (hT : FinMeasAdditive μ T) :
T ∅ = 0 := by |
have h_empty : μ ∅ ≠ ∞ := (measure_empty.le.trans_lt ENNReal.coe_lt_top).ne
specialize hT ∅ ∅ MeasurableSet.empty MeasurableSet.empty h_empty h_empty (Set.inter_empty ∅)
rw [Set.union_empty] at hT
nth_rw 1 [← add_zero (T ∅)] at hT
exact (add_left_cancel hT).symm
| 5 |
import Mathlib.Topology.MetricSpace.HausdorffDistance
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.measure.regular from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
open Set Filter ENNReal Topology NNReal TopologicalSpace
namespace MeasureTh... | Mathlib/MeasureTheory/Measure/Regular.lean | 222 | 228 | theorem exists_subset_lt_add (H : InnerRegularWRT μ p q) (h0 : p ∅) (hU : q U) (hμU : μ U ≠ ∞)
(hε : ε ≠ 0) : ∃ K, K ⊆ U ∧ p K ∧ μ U < μ K + ε := by |
rcases eq_or_ne (μ U) 0 with h₀ | h₀
· refine ⟨∅, empty_subset _, h0, ?_⟩
rwa [measure_empty, h₀, zero_add, pos_iff_ne_zero]
· rcases H hU _ (ENNReal.sub_lt_self hμU h₀ hε) with ⟨K, hKU, hKc, hrK⟩
exact ⟨K, hKU, hKc, ENNReal.lt_add_of_sub_lt_right (Or.inl hμU) hrK⟩
| 5 |
import Mathlib.RingTheory.GradedAlgebra.HomogeneousIdeal
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Sets.Opens
import Mathlib.Data.Set.Subsingleton
#align_import algebraic_geometry.projective_spectrum.topology from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
... | Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean | 99 | 106 | theorem coe_vanishingIdeal (t : Set (ProjectiveSpectrum 𝒜)) :
(vanishingIdeal t : Set A) =
{ f | ∀ x : ProjectiveSpectrum 𝒜, x ∈ t → f ∈ x.asHomogeneousIdeal } := by |
ext f
rw [vanishingIdeal, SetLike.mem_coe, ← HomogeneousIdeal.mem_iff, HomogeneousIdeal.toIdeal_iInf,
Submodule.mem_iInf]
refine forall_congr' fun x => ?_
rw [HomogeneousIdeal.toIdeal_iInf, Submodule.mem_iInf, HomogeneousIdeal.mem_iff]
| 5 |
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products
import Mathlib.CategoryTheory.Limits.ConcreteCategory
import Mathlib.CategoryTheory.Limits.Shapes.Types
import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
import Mathlib.CategoryT... | Mathlib/CategoryTheory/Limits/Shapes/ConcreteCategory.lean | 252 | 259 | theorem multiequalizer_ext {I : MulticospanIndex.{w} C} [HasMultiequalizer I]
[PreservesLimit I.multicospan (forget C)] (x y : ↑(multiequalizer I))
(h : ∀ t : I.L, Multiequalizer.ι I t x = Multiequalizer.ι I t y) : x = y := by |
apply Concrete.limit_ext
rintro (a | b)
· apply h
· rw [← limit.w I.multicospan (WalkingMulticospan.Hom.fst b), comp_apply, comp_apply]
simp [h]
| 5 |
import Mathlib.Data.SetLike.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.ModelTheory.Semantics
#align_import model_theory.definability from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v w u₁
namespace Set
variable {M : Type w} (A : Set M) (L : FirstOrder.Lang... | Mathlib/ModelTheory/Definability.lean | 116 | 122 | theorem Definable.union {f g : Set (α → M)} (hf : A.Definable L f) (hg : A.Definable L g) :
A.Definable L (f ∪ g) := by |
rcases hf with ⟨φ, hφ⟩
rcases hg with ⟨θ, hθ⟩
refine ⟨φ ⊔ θ, ?_⟩
ext
rw [hφ, hθ, mem_setOf_eq, Formula.realize_sup, mem_union, mem_setOf_eq, mem_setOf_eq]
| 5 |
import Mathlib.Order.Filter.Lift
import Mathlib.Topology.Separation
import Mathlib.Order.Interval.Set.Monotone
#align_import topology.filter from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Set Filter TopologicalSpace
open Filter Topology
variable {ι : Sort*} {α β X Y : Type*}... | Mathlib/Topology/Filter.lean | 59 | 66 | theorem isTopologicalBasis_Iic_principal :
IsTopologicalBasis (range (Iic ∘ 𝓟 : Set α → Set (Filter α))) :=
{ exists_subset_inter := by |
rintro _ ⟨s, rfl⟩ _ ⟨t, rfl⟩ l hl
exact ⟨Iic (𝓟 s) ∩ Iic (𝓟 t), ⟨s ∩ t, by simp⟩, hl, Subset.rfl⟩
sUnion_eq := sUnion_eq_univ_iff.2 fun l => ⟨Iic ⊤, ⟨univ, congr_arg Iic principal_univ⟩,
mem_Iic.2 le_top⟩
eq_generateFrom := rfl }
| 5 |
import Mathlib.CategoryTheory.Abelian.Opposite
import Mathlib.CategoryTheory.Abelian.Homology
import Mathlib.Algebra.Homology.Additive
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
#align_import algebra.homology.opposite from "leanprover-community/mathlib"@"8c75ef3517d4106e89fe524e6281d0b0545f47fc"
... | Mathlib/Algebra/Homology/Opposite.lean | 53 | 63 | theorem imageToKernel_unop {X Y Z : Vᵒᵖ} (f : X ⟶ Y) (g : Y ⟶ Z) (w : f ≫ g = 0) :
imageToKernel g.unop f.unop (by rw [← unop_comp, w, unop_zero]) =
(imageSubobjectIso _ ≪≫ (imageUnopUnop _).symm).hom ≫
(cokernel.desc f (factorThruImage g)
(by rw [← cancel_mono (image.ι g), Category.asso... |
ext
dsimp only [imageUnopUnop]
simp only [Iso.trans_hom, Iso.symm_hom, Iso.trans_inv, kernelUnopUnop_inv, Category.assoc,
imageToKernel_arrow, kernelSubobject_arrow', kernel.lift_ι, cokernel.π_desc, Iso.unop_inv,
← unop_comp, factorThruImage_comp_imageUnopOp_inv, Quiver.Hom.unop_op, imageSubobject_arrow]... | 5 |
import Mathlib.Algebra.Polynomial.Monic
#align_import algebra.polynomial.big_operators from "leanprover-community/mathlib"@"47adfab39a11a072db552f47594bf8ed2cf8a722"
open Finset
open Multiset
open Polynomial
universe u w
variable {R : Type u} {ι : Type w}
namespace Polynomial
variable (s : Finset ι)
sectio... | Mathlib/Algebra/Polynomial/BigOperators.lean | 253 | 259 | theorem multiset_prod_X_sub_C_nextCoeff (t : Multiset R) :
nextCoeff (t.map fun x => X - C x).prod = -t.sum := by |
rw [nextCoeff_multiset_prod]
· simp only [nextCoeff_X_sub_C]
exact t.sum_hom (-AddMonoidHom.id R)
· intros
apply monic_X_sub_C
| 5 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Data.Real.Sqrt
import Mathlib.Tactic.Polyrith
#align_import algebra.star.chsh from "leanprover-community/mathlib"@"31c24aa72e7b3e5ed97a8412470e904f82b81004"
universe u
--@[nolint has_nonempty_instance] Porting note(#5171): linter not ported yet
structure Is... | Mathlib/Algebra/Star/CHSH.lean | 104 | 112 | theorem CHSH_id [CommRing R] {A₀ A₁ B₀ B₁ : R} (A₀_inv : A₀ ^ 2 = 1) (A₁_inv : A₁ ^ 2 = 1)
(B₀_inv : B₀ ^ 2 = 1) (B₁_inv : B₁ ^ 2 = 1) :
(2 - A₀ * B₀ - A₀ * B₁ - A₁ * B₀ + A₁ * B₁) * (2 - A₀ * B₀ - A₀ * B₁ - A₁ * B₀ + A₁ * B₁) =
4 * (2 - A₀ * B₀ - A₀ * B₁ - A₁ * B₀ + A₁ * B₁) := by |
-- polyrith suggests:
linear_combination
(2 * B₀ * B₁ + 2) * A₀_inv + (B₀ ^ 2 - 2 * B₀ * B₁ + B₁ ^ 2) * A₁_inv +
(A₀ ^ 2 + 2 * A₀ * A₁ + 1) * B₀_inv +
(A₀ ^ 2 - 2 * A₀ * A₁ + 1) * B₁_inv
| 5 |
import Mathlib.Data.Multiset.Bind
import Mathlib.Control.Traversable.Lemmas
import Mathlib.Control.Traversable.Instances
#align_import data.multiset.functor from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
universe u
namespace Multiset
open List
instance functor : Functor Multiset... | Mathlib/Data/Multiset/Functor.lean | 108 | 116 | theorem comp_traverse {G H : Type _ → Type _} [Applicative G] [Applicative H] [CommApplicative G]
[CommApplicative H] {α β γ : Type _} (g : α → G β) (h : β → H γ) (x : Multiset α) :
traverse (Comp.mk ∘ Functor.map h ∘ g) x =
Comp.mk (Functor.map (traverse h) (traverse g x)) := by |
refine Quotient.inductionOn x ?_
intro
simp only [traverse, quot_mk_to_coe, lift_coe, Coe.coe, Function.comp_apply, Functor.map_map,
functor_norm]
simp only [Function.comp, lift_coe]
| 5 |
import Mathlib.FieldTheory.SeparableDegree
import Mathlib.FieldTheory.IsSepClosed
open scoped Classical Polynomial
open FiniteDimensional Polynomial IntermediateField Field
noncomputable section
universe u v w
variable (F : Type u) (E : Type v) [Field F] [Field E] [Algebra F E]
variable (K : Type w) [Field K] [... | Mathlib/FieldTheory/SeparableClosure.lean | 186 | 192 | theorem IsSepClosed.separableClosure_eq_bot_iff [IsSepClosed E] :
separableClosure F E = ⊥ ↔ IsSepClosed F := by |
refine ⟨fun h ↦ IsSepClosed.of_exists_root _ fun p _ hirr hsep ↦ ?_,
fun _ ↦ IntermediateField.eq_bot_of_isSepClosed_of_isSeparable _⟩
obtain ⟨x, hx⟩ := IsSepClosed.exists_aeval_eq_zero E p (degree_pos_of_irreducible hirr).ne' hsep
obtain ⟨x, rfl⟩ := h ▸ mem_separableClosure_iff.2 (hsep.of_dvd <| minpoly.dvd... | 5 |
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Range
#align_import data.list.indexes from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
assert_not_exists MonoidWithZero
universe u v
open Function
namespace List
variable {α : Type u} {β : Type v}
section MapIdx
-- Porting n... | Mathlib/Data/List/Indexes.lean | 132 | 138 | theorem mapIdxGo_length : ∀ (f : ℕ → α → β) (l : List α) (arr : Array β),
length (mapIdx.go f l arr) = length l + arr.size := by |
intro f l
induction' l with head tail ih
· intro; simp only [mapIdx.go, Array.toList_eq, length_nil, Nat.zero_add]
· intro; simp only [mapIdx.go]; rw [ih]; simp only [Array.size_push, length_cons];
simp only [Nat.add_succ, add_zero, Nat.add_comm]
| 5 |
import Mathlib.Algebra.Homology.Linear
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Tactic.Abel
#align_import algebra.homology.homotopy from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a148bf9ff"
universe v u
open scoped Classical
noncomputable section
open ... | Mathlib/Algebra/Homology/Homotopy.lean | 124 | 130 | theorem prevD_nat (C D : CochainComplex V ℕ) (i : ℕ) (f : ∀ i j, C.X i ⟶ D.X j) :
prevD i f = f i (i - 1) ≫ D.d (i - 1) i := by |
dsimp [prevD]
cases i
· simp only [shape, CochainComplex.prev_nat_zero, ComplexShape.up_Rel, Nat.one_ne_zero,
not_false_iff, comp_zero]
· congr <;> simp
| 5 |
import Mathlib.Algebra.CharP.LocalRing
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.Tactic.FieldSimp
#align_import algebra.char_p.mixed_char_zero from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
variable (R : Type*) [CommRing R]
class MixedCharZero (p : ℕ) : Prop where
... | Mathlib/Algebra/CharP/MixedCharZero.lean | 264 | 270 | theorem to_not_mixedCharZero (h : ∀ I : Ideal R, I ≠ ⊤ → CharZero (R ⧸ I)) :
∀ p > 0, ¬MixedCharZero R p := by |
intro p p_pos
by_contra hp_mixedChar
rcases hp_mixedChar.charP_quotient with ⟨I, hI_ne_top, hI_p⟩
replace hI_zero : CharP (R ⧸ I) 0 := @CharP.ofCharZero _ _ (h I hI_ne_top)
exact absurd (CharP.eq (R ⧸ I) hI_p hI_zero) (ne_of_gt p_pos)
| 5 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.MeasureTheory.Measure.Doubling
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
#align_import measu... | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 95 | 104 | theorem Basis.map_addHaar {ι E F : Type*} [Fintype ι] [NormedAddCommGroup E] [NormedAddCommGroup F]
[NormedSpace ℝ E] [NormedSpace ℝ F] [MeasurableSpace E] [MeasurableSpace F] [BorelSpace E]
[BorelSpace F] [SecondCountableTopology F] [SigmaCompactSpace F]
(b : Basis ι ℝ E) (f : E ≃L[ℝ] F) :
map f b.addH... |
have : IsAddHaarMeasure (map f b.addHaar) :=
AddEquiv.isAddHaarMeasure_map b.addHaar f.toAddEquiv f.continuous f.symm.continuous
rw [eq_comm, Basis.addHaar_eq_iff, Measure.map_apply f.continuous.measurable
(PositiveCompacts.isCompact _).measurableSet, Basis.coe_parallelepiped, Basis.coe_map]
erw [← image... | 5 |
import Mathlib.RingTheory.OrzechProperty
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import linear_algebra.invariant_basis_number from "leanprover-community/mathlib"@"5fd3186f1ec30a75d5f65732e3ce5e623382556f"
noncomputable section
open Function
universe u v w
... | Mathlib/LinearAlgebra/InvariantBasisNumber.lean | 167 | 173 | theorem card_le_of_injective' [StrongRankCondition R] {α β : Type*} [Fintype α] [Fintype β]
(f : (α →₀ R) →ₗ[R] β →₀ R) (i : Injective f) : Fintype.card α ≤ Fintype.card β := by |
let P := Finsupp.linearEquivFunOnFinite R R β
let Q := (Finsupp.linearEquivFunOnFinite R R α).symm
exact
card_le_of_injective R ((P.toLinearMap.comp f).comp Q.toLinearMap)
((P.injective.comp i).comp Q.injective)
| 5 |
import Mathlib.Data.Matrix.Basic
import Mathlib.LinearAlgebra.Matrix.Trace
#align_import data.matrix.basis from "leanprover-community/mathlib"@"320df450e9abeb5fc6417971e75acb6ae8bc3794"
variable {l m n : Type*}
variable {R α : Type*}
namespace Matrix
open Matrix
variable [DecidableEq l] [DecidableEq m] [Decida... | Mathlib/Data/Matrix/Basis.lean | 57 | 63 | theorem mulVec_stdBasisMatrix [Fintype m] (i : n) (j : m) (c : α) (x : m → α) :
mulVec (stdBasisMatrix i j c) x = Function.update (0 : n → α) i (c * x j) := by |
ext i'
simp [stdBasisMatrix, mulVec, dotProduct]
rcases eq_or_ne i i' with rfl|h
· simp
simp [h, h.symm]
| 5 |
import Mathlib.MeasureTheory.Group.GeometryOfNumbers
import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
import Mathlib.NumberTheory.NumberField.CanonicalEmbedding.Basic
#align_import number_theory.number_field.canonical_embedding from "leanprover-community/mathlib"@"60da01b41bbe4206f05d34fd70c8dd7498717a30"
... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | 337 | 343 | theorem convexBodySumFun_continuous :
Continuous (convexBodySumFun : (E K) → ℝ) := by |
refine continuous_finset_sum Finset.univ fun w ↦ ?_
obtain hw | hw := isReal_or_isComplex w
all_goals
· simp only [normAtPlace_apply_isReal, normAtPlace_apply_isComplex, hw]
fun_prop
| 5 |
import Mathlib.Algebra.Polynomial.Roots
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
#align_import analysis.special_functions.polynomials from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter Finset Asymptotic... | Mathlib/Analysis/SpecialFunctions/Polynomials.lean | 91 | 97 | theorem abs_isBoundedUnder_iff :
(IsBoundedUnder (· ≤ ·) atTop fun x => |eval x P|) ↔ P.degree ≤ 0 := by |
refine ⟨fun h => ?_, fun h => ⟨|P.coeff 0|, eventually_map.mpr (eventually_of_forall
(forall_imp (fun _ => le_of_eq) fun x => congr_arg abs <| _root_.trans (congr_arg (eval x)
(eq_C_of_degree_le_zero h)) eval_C))⟩⟩
contrapose! h
exact not_isBoundedUnder_of_tendsto_atTop (abs_tendsto_atTop P h)
| 5 |
import Mathlib.RepresentationTheory.Basic
import Mathlib.RepresentationTheory.FdRep
#align_import representation_theory.invariants from "leanprover-community/mathlib"@"55b3f8206b8596db8bb1804d8a92814a0b6670c9"
suppress_compilation
open MonoidAlgebra
open Representation
namespace GroupAlgebra
variable (k G : Ty... | Mathlib/RepresentationTheory/Invariants.lean | 43 | 48 | theorem mul_average_left (g : G) : ↑(Finsupp.single g 1) * average k G = average k G := by |
simp only [mul_one, Finset.mul_sum, Algebra.mul_smul_comm, average, MonoidAlgebra.of_apply,
Finset.sum_congr, MonoidAlgebra.single_mul_single]
set f : G → MonoidAlgebra k G := fun x => Finsupp.single x 1
show ⅟ (Fintype.card G : k) • ∑ x : G, f (g * x) = ⅟ (Fintype.card G : k) • ∑ x : G, f x
rw [Function.B... | 5 |
import Mathlib.Algebra.Homology.ComplexShape
import Mathlib.CategoryTheory.Subobject.Limits
import Mathlib.CategoryTheory.GradedObject
import Mathlib.Algebra.Homology.ShortComplex.Basic
#align_import algebra.homology.homological_complex from "leanprover-community/mathlib"@"88bca0ce5d22ebfd9e73e682e51d60ea13b48347"
... | Mathlib/Algebra/Homology/HomologicalComplex.lean | 71 | 76 | theorem d_comp_d (C : HomologicalComplex V c) (i j k : ι) : C.d i j ≫ C.d j k = 0 := by |
by_cases hij : c.Rel i j
· by_cases hjk : c.Rel j k
· exact C.d_comp_d' i j k hij hjk
· rw [C.shape j k hjk, comp_zero]
· rw [C.shape i j hij, zero_comp]
| 5 |
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 | 167 | 172 | theorem flip_comp : flip (r ∘r p) = flip p ∘r flip r := by |
funext c a
apply propext
constructor
· exact fun ⟨b, hab, hbc⟩ ↦ ⟨b, hbc, hab⟩
· exact fun ⟨b, hbc, hab⟩ ↦ ⟨b, hab, hbc⟩
| 5 |
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 | 149 | 154 | theorem exists_lt_subset_ball (hs : IsClosed s) (h : s ⊆ ball x r) : ∃ r' < r, s ⊆ ball x r' := by |
rcases le_or_lt r 0 with hr | hr
· rw [ball_eq_empty.2 hr, subset_empty_iff] at h
subst s
exact (exists_lt r).imp fun r' hr' => ⟨hr', empty_subset _⟩
· exact (exists_pos_lt_subset_ball hr hs h).imp fun r' hr' => ⟨hr'.1.2, hr'.2⟩
| 5 |
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 | 95 | 101 | theorem angle_add_eq_arctan_of_inner_eq_zero {x y : V} (h : ⟪x, y⟫ = 0) (h0 : x ≠ 0) :
angle x (x + y) = Real.arctan (‖y‖ / ‖x‖) := by |
rw [angle_add_eq_arcsin_of_inner_eq_zero h (Or.inl h0), Real.arctan_eq_arcsin, ←
div_mul_eq_div_div, norm_add_eq_sqrt_iff_real_inner_eq_zero.2 h]
nth_rw 3 [← Real.sqrt_sq (norm_nonneg x)]
rw_mod_cast [← Real.sqrt_mul (sq_nonneg _), div_pow, pow_two, pow_two, mul_add, mul_one, mul_div,
mul_comm (‖x‖ * ‖x‖... | 5 |
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Normed.Group.Lemmas
import Mathlib.Analysis.NormedSpace.AddTorsor
import Mathlib.Analysis.NormedSpace.AffineIsometry
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Analysis.NormedSpace.RieszLemma
import Mathli... | Mathlib/Analysis/NormedSpace/FiniteDimension.lean | 235 | 241 | theorem LinearMap.injective_iff_antilipschitz [FiniteDimensional 𝕜 E] (f : E →ₗ[𝕜] F) :
Injective f ↔ ∃ K > 0, AntilipschitzWith K f := by |
constructor
· rw [← LinearMap.ker_eq_bot]
exact f.exists_antilipschitzWith
· rintro ⟨K, -, H⟩
exact H.injective
| 5 |
import Mathlib.Order.Filter.Basic
import Mathlib.Data.PFun
#align_import order.filter.partial from "leanprover-community/mathlib"@"b363547b3113d350d053abdf2884e9850a56b205"
universe u v w
namespace Filter
variable {α : Type u} {β : Type v} {γ : Type w}
open Filter
def rmap (r : Rel α β) (l : Filter α) : F... | Mathlib/Order/Filter/Partial.lean | 130 | 136 | theorem rtendsto_iff_le_rcomap (r : Rel α β) (l₁ : Filter α) (l₂ : Filter β) :
RTendsto r l₁ l₂ ↔ l₁ ≤ l₂.rcomap r := by |
rw [rtendsto_def]
simp_rw [← l₂.mem_sets]
simp [Filter.le_def, rcomap, Rel.mem_image]; constructor
· exact fun h s t tl₂ => mem_of_superset (h t tl₂)
· exact fun h t tl₂ => h _ t tl₂ Set.Subset.rfl
| 5 |
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.LinearAlgebra.SesquilinearForm
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variable {𝕜 E F : Type*} [RCLike 𝕜]
variable [NormedAddCommGroup E] [InnerProductSpace... | Mathlib/Analysis/InnerProductSpace/Orthogonal.lean | 73 | 78 | theorem mem_orthogonal_singleton_iff_inner_right {u v : E} : v ∈ (𝕜 ∙ u)ᗮ ↔ ⟪u, v⟫ = 0 := by |
refine ⟨inner_right_of_mem_orthogonal (mem_span_singleton_self u), ?_⟩
intro hv w hw
rw [mem_span_singleton] at hw
obtain ⟨c, rfl⟩ := hw
simp [inner_smul_left, hv]
| 5 |
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 | 220 | 225 | theorem smul_eq_C_smul (x : RatFunc K) (r : K) : r • x = Polynomial.C r • x := by |
cases' x with x
-- Porting note: had to specify the induction principle manually
induction x using Localization.induction_on
rw [← ofFractionRing_smul, ← ofFractionRing_smul, Localization.smul_mk,
Localization.smul_mk, smul_eq_mul, Polynomial.smul_eq_C_mul]
| 5 |
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.Function
#align_import data.set.intervals.surj_on from "leanprover-community/mathlib"@"a59dad53320b73ef180174aae867addd707ef00e"
variable {α : Type*} {β : Type*} [LinearOrder α] [PartialOrder β] {f : α → β}
open Set Function
open OrderDual (toDual)... | Mathlib/Order/Interval/Set/SurjOn.lean | 26 | 32 | theorem surjOn_Ioo_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)
(a b : α) : SurjOn f (Ioo a b) (Ioo (f a) (f b)) := by |
intro p hp
rcases h_surj p with ⟨x, rfl⟩
refine ⟨x, mem_Ioo.2 ?_, rfl⟩
contrapose! hp
exact fun h => h.2.not_le (h_mono <| hp <| h_mono.reflect_lt h.1)
| 5 |
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated
open Function Set
open scoped ENNReal Classical
noncomputable section
variable {α β δ : Type*} [MeasurableSpace α] [MeasurableSpace β] {s : Set α} ... | Mathlib/MeasureTheory/Measure/Dirac.lean | 53 | 59 | theorem dirac_apply [MeasurableSingletonClass α] (a : α) (s : Set α) :
dirac a s = s.indicator 1 a := by |
by_cases h : a ∈ s; · rw [dirac_apply_of_mem h, indicator_of_mem h, Pi.one_apply]
rw [indicator_of_not_mem h, ← nonpos_iff_eq_zero]
calc
dirac a s ≤ dirac a {a}ᶜ := measure_mono (subset_compl_comm.1 <| singleton_subset_iff.2 h)
_ = 0 := by simp [dirac_apply' _ (measurableSet_singleton _).compl]
| 5 |
import Mathlib.Algebra.Lie.Submodule
#align_import algebra.lie.ideal_operations from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
universe u v w w₁ w₂
namespace LieSubmodule
variable {R : Type u} {L : Type v} {M : Type w} {M₂ : Type w₁}
variable [CommRing R] [LieRing L] [LieAlgebra ... | Mathlib/Algebra/Lie/IdealOperations.lean | 111 | 116 | theorem lie_comm : ⁅I, J⁆ = ⁅J, I⁆ := by |
suffices ∀ I J : LieIdeal R L, ⁅I, J⁆ ≤ ⁅J, I⁆ by exact le_antisymm (this I J) (this J I)
clear! I J; intro I J
rw [lieIdeal_oper_eq_span, lieSpan_le]; rintro x ⟨y, z, h⟩; rw [← h]
rw [← lie_skew, ← lie_neg, ← LieSubmodule.coe_neg]
apply lie_coe_mem_lie
| 5 |
import Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk
import Mathlib.Combinatorics.SimpleGraph.Regularity.Energy
#align_import combinatorics.simple_graph.regularity.increment from "leanprover-community/mathlib"@"bf7ef0e83e5b7e6c1169e97f055e58a2e4e9d52d"
open Finset Fintype SimpleGraph SzemerediRegularity
ope... | Mathlib/Combinatorics/SimpleGraph/Regularity/Increment.lean | 82 | 87 | theorem increment_isEquipartition : (increment hP G ε).IsEquipartition := by |
simp_rw [IsEquipartition, Set.equitableOn_iff_exists_eq_eq_add_one]
refine ⟨m, fun A hA => ?_⟩
rw [mem_coe, increment, mem_bind] at hA
obtain ⟨U, hU, hA⟩ := hA
exact card_eq_of_mem_parts_chunk hA
| 5 |
import Mathlib.Data.Fin.Fin2
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Common
#align_import data.typevec from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
universe u v w
@[pp_with_univ]
def TypeVec (n : ℕ) :=
Fin2 n → Type*
#align typevec TypeVec
instance {n} : Inh... | Mathlib/Data/TypeVec.lean | 171 | 177 | theorem eq_of_drop_last_eq {α β : TypeVec (n + 1)} {f g : α ⟹ β} (h₀ : dropFun f = dropFun g)
(h₁ : lastFun f = lastFun g) : f = g := by |
-- Porting note: FIXME: congr_fun h₀ <;> ext1 ⟨⟩ <;> apply_assumption
refine funext (fun x => ?_)
cases x
· apply h₁
· apply congr_fun h₀
| 5 |
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.calculus.diff_cont_on_cl from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
open Set Filter Metric
open scoped Topology
variable (𝕜 : Type*) {E F G : Type*} [NontriviallyNormed... | Mathlib/Analysis/Calculus/DiffContOnCl.lean | 64 | 70 | theorem continuousOn_ball [NormedSpace ℝ E] {x : E} {r : ℝ} (h : DiffContOnCl 𝕜 f (ball x r)) :
ContinuousOn f (closedBall x r) := by |
rcases eq_or_ne r 0 with (rfl | hr)
· rw [closedBall_zero]
exact continuousOn_singleton f x
· rw [← closure_ball x hr]
exact h.continuousOn
| 5 |
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section Real
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
theorem toReal_add (ha : a ≠ ∞) (hb : b ≠ ∞) : (a + b).toReal = a.toReal ... | Mathlib/Data/ENNReal/Real.lean | 50 | 55 | theorem le_toReal_sub {a b : ℝ≥0∞} (hb : b ≠ ∞) : a.toReal - b.toReal ≤ (a - b).toReal := by |
lift b to ℝ≥0 using hb
induction a
· simp
· simp only [← coe_sub, NNReal.sub_def, Real.coe_toNNReal', coe_toReal]
exact le_max_left _ _
| 5 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Mul
import Mathlib.Analysis.Calculus.Deriv.Shift
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
variable
{𝕜 : Type*} [NontriviallyNormedField 𝕜]
{F : Type*} [NormedAddCommGroup F] [NormedSpace 𝕜 F]
{R : Type*} [Semi... | Mathlib/Analysis/Calculus/IteratedDeriv/Lemmas.lean | 40 | 46 | theorem iteratedDerivWithin_const_add (hn : 0 < n) (c : F) :
iteratedDerivWithin n (fun z => c + f z) s x = iteratedDerivWithin n f s x := by |
obtain ⟨n, rfl⟩ := n.exists_eq_succ_of_ne_zero hn.ne'
rw [iteratedDerivWithin_succ' h hx, iteratedDerivWithin_succ' h hx]
refine iteratedDerivWithin_congr h ?_ hx
intro y hy
exact derivWithin_const_add (h.uniqueDiffWithinAt hy) _
| 5 |
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Int.Basic
import Mathlib.Tactic.Ring
import Mathlib.Tactic.FieldSimp
import Mathlib.Data.Int.NatPrime
import Mathlib.Data.ZMod.Basic
#align_import number_theory.pythagorean_tri... | Mathlib/NumberTheory/PythagoreanTriples.lean | 87 | 93 | theorem mul_iff (k : ℤ) (hk : k ≠ 0) :
PythagoreanTriple (k * x) (k * y) (k * z) ↔ PythagoreanTriple x y z := by |
refine ⟨?_, fun h => h.mul k⟩
simp only [PythagoreanTriple]
intro h
rw [← mul_left_inj' (mul_ne_zero hk hk)]
convert h using 1 <;> ring
| 5 |
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Factorial.DoubleFactorial
#align_import ring_theory.polynomial.hermite.basic from "leanprover-community/mathlib"@"938d3db9c278f8a52c0f964a405806f0f2b09b74"
noncomputable section
open Polynomial
namespace P... | Mathlib/RingTheory/Polynomial/Hermite/Basic.lean | 111 | 116 | theorem degree_hermite (n : ℕ) : (hermite n).degree = n := by |
rw [degree_eq_of_le_of_coeff_ne_zero]
· simp_rw [degree_le_iff_coeff_zero, Nat.cast_lt]
rintro m hnm
exact coeff_hermite_of_lt hnm
· simp [coeff_hermite_self n]
| 5 |
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.RingTheory.MvPolynomial.Basic
#align_import field_theory.mv_polynomial from "leanprover-community/mathlib"@"039a089d2a4b93c761b234f3e5f5aeb752bac60f"
noncomputable section
open scoped Classical
... | Mathlib/FieldTheory/MvPolynomial.lean | 34 | 40 | theorem quotient_mk_comp_C_injective (I : Ideal (MvPolynomial σ K)) (hI : I ≠ ⊤) :
Function.Injective ((Ideal.Quotient.mk I).comp MvPolynomial.C) := by |
refine (injective_iff_map_eq_zero _).2 fun x hx => ?_
rw [RingHom.comp_apply, Ideal.Quotient.eq_zero_iff_mem] at hx
refine _root_.by_contradiction fun hx0 => absurd (I.eq_top_iff_one.2 ?_) hI
have := I.mul_mem_left (MvPolynomial.C x⁻¹) hx
rwa [← MvPolynomial.C.map_mul, inv_mul_cancel hx0, MvPolynomial.C_1] a... | 5 |
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 36 | 42 | theorem oangle_add_right_eq_arccos_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
o.oangle x (x + y) = Real.arccos (‖x‖ / ‖x + y‖) := by |
have hs : (o.oangle x (x + y)).sign = 1 := by
rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two]
rw [o.oangle_eq_angle_of_sign_eq_one hs,
InnerProductGeometry.angle_add_eq_arccos_of_inner_eq_zero
(o.inner_eq_zero_of_oangle_eq_pi_div_two h)]
| 5 |
import Mathlib.Algebra.Group.Embedding
import Mathlib.Data.Fin.Basic
import Mathlib.Data.Finset.Union
#align_import data.finset.image from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
-- TODO
-- assert_not_exists OrderedCommMonoid
assert_not_exists MonoidWithZero
assert_not_exists MulA... | Mathlib/Data/Finset/Image.lean | 81 | 86 | theorem mem_map_equiv {f : α ≃ β} {b : β} : b ∈ s.map f.toEmbedding ↔ f.symm b ∈ s := by |
rw [mem_map]
exact
⟨by
rintro ⟨a, H, rfl⟩
simpa, fun h => ⟨_, h, by simp⟩⟩
| 5 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section IsCoprime
variable {R : Type ... | Mathlib/RingTheory/Coprime/Lemmas.lean | 61 | 66 | theorem IsCoprime.prod_left : (∀ i ∈ t, IsCoprime (s i) x) → IsCoprime (∏ i ∈ t, s i) x := by |
classical
refine Finset.induction_on t (fun _ ↦ isCoprime_one_left) fun b t hbt ih H ↦ ?_
rw [Finset.prod_insert hbt]
rw [Finset.forall_mem_insert] at H
exact H.1.mul_left (ih H.2)
| 5 |
import Mathlib.Topology.Bornology.Basic
#align_import topology.bornology.constructions from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f78155c6c27d62a621f4c152d"
open Set Filter Bornology Function
open Filter
variable {α β ι : Type*} {π : ι → Type*} [Bornology α] [Bornology β]
[∀ i, Bornology (π i)]
inst... | Mathlib/Topology/Bornology/Constructions.lean | 126 | 131 | theorem isBounded_pi : IsBounded (pi univ S) ↔ (∃ i, S i = ∅) ∨ ∀ i, IsBounded (S i) := by |
by_cases hne : ∃ i, S i = ∅
· simp [hne, univ_pi_eq_empty_iff.2 hne]
· simp only [hne, false_or_iff]
simp only [not_exists, ← Ne.eq_def, ← nonempty_iff_ne_empty, ← univ_pi_nonempty_iff] at hne
exact isBounded_pi_of_nonempty hne
| 5 |
import Mathlib.Geometry.Manifold.VectorBundle.Tangent
#align_import geometry.manifold.mfderiv from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
noncomputable section
open scoped Classical Topology Manifold
open Set ChartedSpace
section DerivativesDefinitions
variable {𝕜 : Type*} ... | Mathlib/Geometry/Manifold/MFDeriv/Defs.lean | 239 | 246 | theorem mdifferentiableAt_iff (f : M → M') (x : M) :
MDifferentiableAt I I' f x ↔ ContinuousAt f x ∧
DifferentiableWithinAt 𝕜 (writtenInExtChartAt I I' x f) (range I) ((extChartAt I x) x) := by |
rw [MDifferentiableAt, liftPropAt_iff]
congrm _ ∧ ?_
simp [DifferentiableWithinAtProp, Set.univ_inter]
-- Porting note: `rfl` wasn't needed
rfl
| 5 |
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.Module.Basic
import Mathlib.Algebra.Regular.SMul
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Rat.BigOperators
import Mathlib.GroupTheory.GroupAction.Hom
import Mathlib.Data.Set.Subsingleton
#align_import data.finsupp.basic from "leanprover... | Mathlib/Data/Finsupp/Basic.lean | 101 | 106 | theorem graph_injective (α M) [Zero M] : Injective (@graph α M _) := by |
intro f g h
classical
have hsup : f.support = g.support := by rw [← image_fst_graph, h, image_fst_graph]
refine ext_iff'.2 ⟨hsup, fun x hx => apply_eq_of_mem_graph <| h.symm ▸ ?_⟩
exact mk_mem_graph _ (hsup ▸ hx)
| 5 |
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_theory.perm.support from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Equiv Finset
namespace Equiv.Perm
variable {α : Type*}
section support
v... | Mathlib/GroupTheory/Perm/Support.lean | 324 | 329 | theorem support_congr (h : f.support ⊆ g.support) (h' : ∀ x ∈ g.support, f x = g x) : f = g := by |
ext x
by_cases hx : x ∈ g.support
· exact h' x hx
· rw [not_mem_support.mp hx, ← not_mem_support]
exact fun H => hx (h H)
| 5 |
import Mathlib.Data.SetLike.Basic
import Mathlib.Data.Finset.Preimage
import Mathlib.ModelTheory.Semantics
#align_import model_theory.definability from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u v w u₁
namespace Set
variable {M : Type w} (A : Set M) (L : FirstOrder.Lang... | Mathlib/ModelTheory/Definability.lean | 106 | 112 | theorem Definable.inter {f g : Set (α → M)} (hf : A.Definable L f) (hg : A.Definable L g) :
A.Definable L (f ∩ g) := by |
rcases hf with ⟨φ, rfl⟩
rcases hg with ⟨θ, rfl⟩
refine ⟨φ ⊓ θ, ?_⟩
ext
simp
| 5 |
import Mathlib.Order.UpperLower.Basic
import Mathlib.Data.Finset.Preimage
#align_import combinatorics.young.young_diagram from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Function
@[ext]
structure YoungDiagram where
cells : Finset (ℕ × ℕ)
isLowerSet : IsLowerSet (cel... | Mathlib/Combinatorics/Young/YoungDiagram.lean | 313 | 318 | theorem row_eq_prod {μ : YoungDiagram} {i : ℕ} : μ.row i = {i} ×ˢ Finset.range (μ.rowLen i) := by |
ext ⟨a, b⟩
simp only [Finset.mem_product, Finset.mem_singleton, Finset.mem_range, mem_row_iff,
mem_iff_lt_rowLen, and_comm, and_congr_right_iff]
rintro rfl
rfl
| 5 |
import Mathlib.Data.Int.AbsoluteValue
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
#align_import linear_algebra.matrix.absolute_value from "leanprover-community/mathlib"@"ab0a2959c83b06280ef576bc830d4aa5fe8c8e61"
open Matrix
namespace Matrix
open Equiv Finset
variable {R S : Type*} [CommRing R] [Nontr... | Mathlib/LinearAlgebra/Matrix/AbsoluteValue.lean | 64 | 73 | theorem det_sum_smul_le {ι : Type*} (s : Finset ι) {c : ι → R} {A : ι → Matrix n n R}
{abv : AbsoluteValue R S} {x : S} (hx : ∀ k i j, abv (A k i j) ≤ x) {y : S}
(hy : ∀ k, abv (c k) ≤ y) :
abv (det (∑ k ∈ s, c k • A k)) ≤
Nat.factorial (Fintype.card n) • (Finset.card s • y * x) ^ Fintype.card n := by |
simpa only [smul_mul_assoc] using
det_sum_le s fun k i j =>
calc
abv (c k * A k i j) = abv (c k) * abv (A k i j) := abv.map_mul _ _
_ ≤ y * x := mul_le_mul (hy k) (hx k i j) (abv.nonneg _) ((abv.nonneg _).trans (hy k))
| 5 |
import Mathlib.Data.List.Basic
namespace List
variable {α β : Type*}
@[simp]
theorem reduceOption_cons_of_some (x : α) (l : List (Option α)) :
reduceOption (some x :: l) = x :: l.reduceOption := by
simp only [reduceOption, filterMap, id, eq_self_iff_true, and_self_iff]
#align list.reduce_option_cons_of_some... | Mathlib/Data/List/ReduceOption.lean | 35 | 41 | theorem reduceOption_map {l : List (Option α)} {f : α → β} :
reduceOption (map (Option.map f) l) = map f (reduceOption l) := by |
induction' l with hd tl hl
· simp only [reduceOption_nil, map_nil]
· cases hd <;>
simpa [true_and_iff, Option.map_some', map, eq_self_iff_true,
reduceOption_cons_of_some] using hl
| 5 |
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.LinearAlgebra.PiTensorProduct
universe uι u𝕜 uE uF
variable {ι : Type uι} [Fintype ι]
variable {𝕜 : Type u𝕜} [NontriviallyNormedField 𝕜]
variable {E : ι → Type uE} [∀ i, SeminormedAddCommGroup (E i)] [∀ i, NormedSpace 𝕜 (E i)]
variable {F : ... | Mathlib/Analysis/NormedSpace/PiTensorProduct/ProjectiveSeminorm.lean | 126 | 132 | theorem projectiveSeminorm_tprod_le (m : Π i, E i) :
projectiveSeminorm (⨂ₜ[𝕜] i, m i) ≤ ∏ i, ‖m i‖ := by |
rw [projectiveSeminorm_apply]
convert ciInf_le (bddBelow_projectiveSemiNormAux _) ⟨[((1 : 𝕜), m)] ,?_⟩
· simp only [projectiveSeminormAux, Function.comp_apply, List.map_cons, norm_one, one_mul,
List.map_nil, List.sum_cons, List.sum_nil, add_zero]
· rw [mem_lifts_iff, List.map_singleton, List.sum_singleton... | 5 |
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Sym.Sym2
namespace List
variable {α : Type*}
section Sym2
protected def sym2 : List α → List (Sym2 α)
| [] => []
| x :: xs => (x :: xs).map (fun y => s(x, y)) ++ xs.sym2
theorem mem_sym2_cons_iff {x : α} {xs : List α} {z : Sym2 α} :
z ∈ (x :: xs)... | Mathlib/Data/List/Sym.lean | 146 | 151 | theorem length_sym2 {xs : List α} : xs.sym2.length = Nat.choose (xs.length + 1) 2 := by |
induction xs with
| nil => rfl
| cons x xs ih =>
rw [List.sym2, length_append, length_map, length_cons,
Nat.choose_succ_succ, ← ih, Nat.choose_one_right]
| 5 |
import Mathlib.Algebra.ContinuedFractions.Computation.Basic
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.computation.translations from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
open Generali... | Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean | 250 | 258 | theorem get?_of_eq_some_of_get?_intFractPair_stream_fr_ne_zero {ifp_n : IntFractPair K}
(stream_nth_eq : IntFractPair.stream v n = some ifp_n) (nth_fr_ne_zero : ifp_n.fr ≠ 0) :
(of v).s.get? n = some ⟨1, (IntFractPair.of ifp_n.fr⁻¹).b⟩ :=
have : IntFractPair.stream v (n + 1) = some (IntFractPair.of ifp_n.fr⁻¹... |
cases ifp_n
simp only [IntFractPair.stream, Nat.add_eq, add_zero, stream_nth_eq, Option.some_bind,
ite_eq_right_iff]
intro; contradiction
get?_of_eq_some_of_succ_get?_intFractPair_stream this
| 5 |
import Mathlib.MeasureTheory.Measure.VectorMeasure
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
#align_import measure_theory.measure.with_density_vector_measure from "leanprover-community/mathlib"@"d1bd9c5df2867c1cb463bc6364446d57bdd9f7f1"
noncomputable section
open scoped Classical MeasureTheory NNReal ... | Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean | 131 | 138 | theorem withDensityᵥ_smul_eq_withDensityᵥ_withDensity {f : α → ℝ≥0} {g : α → E}
(hf : AEMeasurable f μ) (hfg : Integrable (f • g) μ) :
μ.withDensityᵥ (f • g) = (μ.withDensity (fun x ↦ f x)).withDensityᵥ g := by |
ext s hs
rw [withDensityᵥ_apply hfg hs,
withDensityᵥ_apply ((integrable_withDensity_iff_integrable_smul₀ hf).mpr hfg) hs,
setIntegral_withDensity_eq_setIntegral_smul₀ hf.restrict _ hs]
rfl
| 5 |
import Mathlib.Topology.Order.LeftRight
import Mathlib.Topology.Order.Monotone
#align_import topology.algebra.order.left_right_lim from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
open Set Filter
open Topology
section
variable {α β : Type*} [LinearOrder α] [TopologicalSpace β]
n... | Mathlib/Topology/Order/LeftRightLim.lean | 65 | 72 | theorem leftLim_eq_of_tendsto [hα : TopologicalSpace α] [h'α : OrderTopology α] [T2Space β]
{f : α → β} {a : α} {y : β} (h : 𝓝[<] a ≠ ⊥) (h' : Tendsto f (𝓝[<] a) (𝓝 y)) :
leftLim f a = y := by |
have h'' : ∃ y, Tendsto f (𝓝[<] a) (𝓝 y) := ⟨y, h'⟩
rw [h'α.topology_eq_generate_intervals] at h h' h''
simp only [leftLim, h, h'', not_true, or_self_iff, if_false]
haveI := neBot_iff.2 h
exact lim_eq h'
| 5 |
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 | 221 | 227 | theorem HasStrictDerivAt.mul (hc : HasStrictDerivAt c c' x) (hd : HasStrictDerivAt d d' x) :
HasStrictDerivAt (fun y => c y * d y) (c' * d x + c x * d') x := by |
have := (HasStrictFDerivAt.mul' hc hd).hasStrictDerivAt
rwa [ContinuousLinearMap.add_apply, ContinuousLinearMap.smul_apply,
ContinuousLinearMap.smulRight_apply, ContinuousLinearMap.smulRight_apply,
ContinuousLinearMap.smulRight_apply, ContinuousLinearMap.one_apply, one_smul, one_smul,
add_comm] at this... | 5 |
import Mathlib.Data.Matrix.Basis
import Mathlib.RingTheory.TensorProduct.Basic
#align_import ring_theory.matrix_algebra from "leanprover-community/mathlib"@"6c351a8fb9b06e5a542fdf427bfb9f46724f9453"
suppress_compilation
universe u v w
open TensorProduct
open TensorProduct
open Algebra.TensorProduct
open Matri... | Mathlib/RingTheory/MatrixAlgebra.lean | 105 | 110 | theorem invFun_algebraMap (M : Matrix n n R) : invFun R A n (M.map (algebraMap R A)) = 1 ⊗ₜ M := by |
dsimp [invFun]
simp only [Algebra.algebraMap_eq_smul_one, smul_tmul, ← tmul_sum, mul_boole]
congr
conv_rhs => rw [matrix_eq_sum_std_basis M]
convert Finset.sum_product (β := Matrix n n R); simp
| 5 |
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.Index
import Mathlib.GroupTheory.SpecificGroups.Dihedral
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Line... | Mathlib/GroupTheory/CommutingProbability.lean | 47 | 52 | theorem commProb_prod (M' : Type*) [Mul M'] : commProb (M × M') = commProb M * commProb M' := by |
simp_rw [commProb_def, div_mul_div_comm, Nat.card_prod, Nat.cast_mul, mul_pow, ← Nat.cast_mul,
← Nat.card_prod, Commute, SemiconjBy, Prod.ext_iff]
congr 2
exact Nat.card_congr ⟨fun x => ⟨⟨⟨x.1.1.1, x.1.2.1⟩, x.2.1⟩, ⟨⟨x.1.1.2, x.1.2.2⟩, x.2.2⟩⟩,
fun x => ⟨⟨⟨x.1.1.1, x.2.1.1⟩, ⟨x.1.1.2, x.2.1.2⟩⟩, ⟨x.1.2,... | 5 |
import Mathlib.AlgebraicGeometry.Morphisms.Basic
import Mathlib.RingTheory.LocalProperties
#align_import algebraic_geometry.morphisms.ring_hom_properties from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
-- Explicit universe annotations were used in this file to improve perfomance #127... | Mathlib/AlgebraicGeometry/Morphisms/RingHomProperties.lean | 73 | 80 | theorem RespectsIso.basicOpen_iff_localization (hP : RespectsIso @P) {X Y : Scheme.{u}} [IsAffine X]
[IsAffine Y] (f : X ⟶ Y) (r : Y.presheaf.obj (Opposite.op ⊤)) :
P (Scheme.Γ.map (f ∣_ Y.basicOpen r).op) ↔ P (Localization.awayMap (Scheme.Γ.map f.op) r) := by |
refine (hP.basicOpen_iff _ _).trans ?_
-- Porting note: was a one line term mode proof, but this `dsimp` is vital so the term mode
-- one liner is not possible
dsimp
rw [← hP.is_localization_away_iff]
| 5 |
import Mathlib.Algebra.Lie.Matrix
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.Tactic.NoncommRing
#align_import algebra.lie.skew_adjoint from "leanprover-community/mathlib"@"075b3f7d19b9da85a0b54b3e33055a74fc388dec"
universe u v w w₁
section SkewAdjointEndomorphisms
open LinearMap (BilinF... | Mathlib/Algebra/Lie/SkewAdjoint.lean | 46 | 53 | theorem LinearMap.BilinForm.isSkewAdjoint_bracket {f g : Module.End R M}
(hf : f ∈ B.skewAdjointSubmodule) (hg : g ∈ B.skewAdjointSubmodule) :
⁅f, g⁆ ∈ B.skewAdjointSubmodule := by |
rw [mem_skewAdjointSubmodule] at *
have hfg : IsAdjointPair B B (f * g) (g * f) := by rw [← neg_mul_neg g f]; exact hf.mul hg
have hgf : IsAdjointPair B B (g * f) (f * g) := by rw [← neg_mul_neg f g]; exact hg.mul hf
change IsAdjointPair B B (f * g - g * f) (-(f * g - g * f)); rw [neg_sub]
exact hfg.sub hgf
| 5 |
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Order.Monoid.WithTop
#align_import data.nat.with_bot from "leanprover-community/mathlib"@"966e0cf0685c9cedf8a3283ac69eef4d5f2eaca2"
namespace Nat
namespace WithBot
instance : WellFoundedRelation (WithBot ℕ) where
rel := (· < ·)
wf := IsWellFounde... | Mathlib/Data/Nat/WithBot.lean | 43 | 49 | theorem add_eq_two_iff {n m : WithBot ℕ} :
n + m = 2 ↔ n = 0 ∧ m = 2 ∨ n = 1 ∧ m = 1 ∨ n = 2 ∧ m = 0 := by |
rcases n, m with ⟨_ | _, _ | _⟩
repeat refine ⟨fun h => Option.noConfusion h, fun h => ?_⟩;
aesop (simp_config := { decide := true })
repeat erw [WithBot.coe_eq_coe]
exact Nat.add_eq_two_iff
| 5 |
import Mathlib.FieldTheory.Finite.Polynomial
import Mathlib.NumberTheory.Basic
import Mathlib.RingTheory.WittVector.WittPolynomial
#align_import ring_theory.witt_vector.structure_polynomial from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open MvPolynomial Set
open Finset (range)
o... | Mathlib/RingTheory/WittVector/StructurePolynomial.lean | 179 | 188 | theorem wittStructureRat_rec (Φ : MvPolynomial idx ℚ) (n : ℕ) :
wittStructureRat p Φ n =
C (1 / (p : ℚ) ^ n) *
(bind₁ (fun b => rename (fun i => (b, i)) (W_ ℚ n)) Φ -
∑ i ∈ range n, C ((p : ℚ) ^ i) * wittStructureRat p Φ i ^ p ^ (n - i)) := by |
calc
wittStructureRat p Φ n = C (1 / (p : ℚ) ^ n) * (wittStructureRat p Φ n * C ((p : ℚ) ^ n)) := ?_
_ = _ := by rw [wittStructureRat_rec_aux]
rw [mul_left_comm, ← C_mul, div_mul_cancel₀, C_1, mul_one]
exact pow_ne_zero _ (Nat.cast_ne_zero.2 hp.1.ne_zero)
| 5 |
import Mathlib.Topology.Order.LeftRightNhds
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section OrderTopology
variable [TopologicalSpace α] [TopologicalSpace β] [LinearOrder α] [LinearOrder β] [OrderTopology α]
[OrderTopology β]
theorem IsLUB.fr... | Mathlib/Topology/Order/IsLUB.lean | 93 | 100 | theorem IsLUB.mem_upperBounds_of_tendsto [Preorder γ] [TopologicalSpace γ] [OrderClosedTopology γ]
{f : α → γ} {s : Set α} {a : α} {b : γ} (hf : MonotoneOn f s) (ha : IsLUB s a)
(hb : Tendsto f (𝓝[s] a) (𝓝 b)) : b ∈ upperBounds (f '' s) := by |
rintro _ ⟨x, hx, rfl⟩
replace ha := ha.inter_Ici_of_mem hx
haveI := ha.nhdsWithin_neBot ⟨x, hx, le_rfl⟩
refine ge_of_tendsto (hb.mono_left (nhdsWithin_mono a (inter_subset_left (t := Ici x)))) ?_
exact mem_of_superset self_mem_nhdsWithin fun y hy => hf hx hy.1 hy.2
| 5 |
import Mathlib.Algebra.Homology.Linear
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Tactic.Abel
#align_import algebra.homology.homotopy from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a148bf9ff"
universe v u
open scoped Classical
noncomputable section
open ... | Mathlib/Algebra/Homology/Homotopy.lean | 115 | 121 | theorem dNext_nat (C D : ChainComplex V ℕ) (i : ℕ) (f : ∀ i j, C.X i ⟶ D.X j) :
dNext i f = C.d i (i - 1) ≫ f (i - 1) i := by |
dsimp [dNext]
cases i
· simp only [shape, ChainComplex.next_nat_zero, ComplexShape.down_Rel, Nat.one_ne_zero,
not_false_iff, zero_comp]
· congr <;> simp
| 5 |
import Mathlib.AlgebraicTopology.DoldKan.PInfty
#align_import algebraic_topology.dold_kan.decomposition from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open CategoryTheory CategoryTheory.Category CategoryTheory.Preadditive
Opposite Simplicial
noncomputable section
namespace Alge... | Mathlib/AlgebraicTopology/DoldKan/Decomposition.lean | 150 | 155 | theorem preComp_φ : (f.preComp g).φ = g.app (op [n + 1]) ≫ f.φ := by |
unfold φ preComp
simp only [PInfty_f, comp_add]
congr 1
· simp only [P_f_naturality_assoc]
· simp only [comp_sum, P_f_naturality_assoc, SimplicialObject.δ_naturality_assoc]
| 5 |
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.RingTheory.WittVector.Truncated
#align_import ring_theory.witt_vector.mul_coeff from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
namespace WittVector
variable (p : ℕ) [hp : Fact p.Prime]
variable {k ... | Mathlib/RingTheory/WittVector/MulCoeff.lean | 56 | 61 | theorem wittPolyProd_vars (n : ℕ) : (wittPolyProd p n).vars ⊆ univ ×ˢ range (n + 1) := by |
rw [wittPolyProd]
apply Subset.trans (vars_mul _ _)
refine union_subset ?_ ?_ <;>
· refine Subset.trans (vars_rename _ _) ?_
simp [wittPolynomial_vars, image_subset_iff]
| 5 |
import Mathlib.RepresentationTheory.Basic
import Mathlib.RepresentationTheory.FdRep
#align_import representation_theory.invariants from "leanprover-community/mathlib"@"55b3f8206b8596db8bb1804d8a92814a0b6670c9"
suppress_compilation
open MonoidAlgebra
open Representation
namespace GroupAlgebra
variable (k G : Ty... | Mathlib/RepresentationTheory/Invariants.lean | 54 | 59 | theorem mul_average_right (g : G) : average k G * ↑(Finsupp.single g 1) = average k G := by |
simp only [mul_one, Finset.sum_mul, Algebra.smul_mul_assoc, average, MonoidAlgebra.of_apply,
Finset.sum_congr, MonoidAlgebra.single_mul_single]
set f : G → MonoidAlgebra k G := fun x => Finsupp.single x 1
show ⅟ (Fintype.card G : k) • ∑ x : G, f (x * g) = ⅟ (Fintype.card G : k) • ∑ x : G, f x
rw [Function.... | 5 |
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Functor.Currying
import Mathlib.CategoryTheory.Limits.FunctorCategory
#align_import category_theory.limits.colimit_limit from "leanprover-community/mathlib"@"59382264386afdbaf1727e617f5fdda511992eb9"
universe v₁ v₂ v u₁ u₂ u
open CategoryTh... | Mathlib/CategoryTheory/Limits/ColimitLimit.lean | 97 | 105 | theorem ι_colimitLimitToLimitColimit_π_apply [Small.{v} J] [Small.{v} K] (F : J × K ⥤ Type v)
(j : J) (k : K) (f) : limit.π (curry.obj F ⋙ colim) j
(colimitLimitToLimitColimit F (colimit.ι (curry.obj (Prod.swap K J ⋙ F) ⋙ lim) k f)) =
colimit.ι ((curry.obj F).obj j) k (limit.π ((curry.obj (Prod.swap K... |
dsimp [colimitLimitToLimitColimit]
rw [Types.Limit.lift_π_apply]
dsimp only
rw [Types.Colimit.ι_desc_apply]
dsimp
| 5 |
import Mathlib.MeasureTheory.Covering.Differentiation
import Mathlib.MeasureTheory.Covering.VitaliFamily
import Mathlib.MeasureTheory.Integral.Lebesgue
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.Topology.MetricSpace.Basic
import Mathlib.Data.Set.Pairwise.Lat... | Mathlib/MeasureTheory/Covering/Besicovitch.lean | 195 | 200 | theorem hlast' (i : Fin N.succ) (h : 1 ≤ τ) : a.r (last N) ≤ τ * a.r i := by |
rcases lt_or_le i (last N) with (H | H)
· exact (a.hlast i H).2
· have : i = last N := top_le_iff.1 H
rw [this]
exact le_mul_of_one_le_left (a.rpos _).le h
| 5 |
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Algebra.DirectSum.Algebra
#align_import algebra.direct_sum.internal from "leanprover-community/mathlib"@"9936c3dfc04e5876f4368aeb2e60f8d8358d095a"
open DirectSum
variable {ι : Type*} {σ S R : Type*}
instance... | Mathlib/Algebra/DirectSum/Internal.lean | 74 | 80 | theorem SetLike.intCast_mem_graded [Zero ι] [AddGroupWithOne R] [SetLike σ R]
[AddSubgroupClass σ R] (A : ι → σ) [SetLike.GradedOne A] (z : ℤ) : (z : R) ∈ A 0 := by |
induction z
· rw [Int.ofNat_eq_coe, Int.cast_natCast]
exact SetLike.natCast_mem_graded _ _
· rw [Int.cast_negSucc]
exact neg_mem (SetLike.natCast_mem_graded _ _)
| 5 |
import Mathlib.Data.Set.Prod
import Mathlib.Logic.Function.Conjugate
#align_import data.set.function from "leanprover-community/mathlib"@"996b0ff959da753a555053a480f36e5f264d4207"
variable {α β γ : Type*} {ι : Sort*} {π : α → Type*}
open Equiv Equiv.Perm Function
namespace Set
section restrict
def restrict (... | Mathlib/Data/Set/Function.lean | 130 | 136 | theorem range_extend_subset (f : α → β) (g : α → γ) (g' : β → γ) :
range (extend f g g') ⊆ range g ∪ g' '' (range f)ᶜ := by |
classical
rintro _ ⟨y, rfl⟩
rw [extend_def]
split_ifs with h
exacts [Or.inl (mem_range_self _), Or.inr (mem_image_of_mem _ h)]
| 5 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Data.Nat.Choose.Cast
import Mathlib.NumberTheory.Bernoulli
#align_import number_theory.bernoulli_polynomials from "leanprover-community/mathlib"@"ca3d21f7f4fd613c2a3c54ac7871163e1e5ecb3a"
noncomputable section... | Mathlib/NumberTheory/BernoulliPolynomials.lean | 57 | 63 | theorem bernoulli_def (n : ℕ) : bernoulli n =
∑ i ∈ range (n + 1), Polynomial.monomial i (_root_.bernoulli (n - i) * choose n i) := by |
rw [← sum_range_reflect, add_succ_sub_one, add_zero, bernoulli]
apply sum_congr rfl
rintro x hx
rw [mem_range_succ_iff] at hx
rw [choose_symm hx, tsub_tsub_cancel_of_le hx]
| 5 |
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
#align_import analysis.ODE.gronwall from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F]
[NormedSpace ℝ F]
open Metric Set Asymptotics Fil... | Mathlib/Analysis/ODE/Gronwall.lean | 96 | 101 | theorem gronwallBound_continuous_ε (δ K x : ℝ) : Continuous fun ε => gronwallBound δ K ε x := by |
by_cases hK : K = 0
· simp only [gronwallBound_K0, hK]
exact continuous_const.add (continuous_id.mul continuous_const)
· simp only [gronwallBound_of_K_ne_0 hK]
exact continuous_const.add ((continuous_id.mul continuous_const).mul continuous_const)
| 5 |
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
#align_import geometry.euclidean.angle.oriented.rotation from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 114 | 119 | theorem det_rotation (θ : Real.Angle) : LinearMap.det (o.rotation θ).toLinearMap = 1 := by |
haveI : Nontrivial V :=
FiniteDimensional.nontrivial_of_finrank_eq_succ (@Fact.out (finrank ℝ V = 2) _)
obtain ⟨x, hx⟩ : ∃ x, x ≠ (0 : V) := exists_ne (0 : V)
rw [o.rotation_eq_matrix_toLin θ hx]
simpa [sq] using θ.cos_sq_add_sin_sq
| 5 |
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products
import Mathlib.CategoryTheory.Limits.ConcreteCategory
import Mathlib.CategoryTheory.Limits.Shapes.Types
import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
import Mathlib.CategoryT... | Mathlib/CategoryTheory/Limits/Shapes/ConcreteCategory.lean | 336 | 343 | theorem widePushout_exists_rep' {B : C} {α : Type _} [Nonempty α] {X : α → C}
(f : ∀ j : α, B ⟶ X j) [HasWidePushout.{v} B X f] [PreservesColimit (wideSpan B X f) (forget C)]
(x : ↑(widePushout B X f)) : ∃ (i : α) (y : X i), ι f i y = x := by |
rcases Concrete.widePushout_exists_rep f x with (⟨y, rfl⟩ | ⟨i, y, rfl⟩)
· inhabit α
use default, f _ y
simp only [← arrow_ι _ default, comp_apply]
· use i, y
| 5 |
import Mathlib.Algebra.IsPrimePow
import Mathlib.Data.Nat.Factorization.Basic
#align_import data.nat.factorization.prime_pow from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
variable {R : Type*} [CommMonoidWithZero R] (n p : R) (k : ℕ)
theorem IsPrimePow.minFac_pow_factorization_eq ... | Mathlib/Data/Nat/Factorization/PrimePow.lean | 27 | 33 | theorem isPrimePow_of_minFac_pow_factorization_eq {n : ℕ}
(h : n.minFac ^ n.factorization n.minFac = n) (hn : n ≠ 1) : IsPrimePow n := by |
rcases eq_or_ne n 0 with (rfl | hn')
· simp_all
refine ⟨_, _, (Nat.minFac_prime hn).prime, ?_, h⟩
simp [pos_iff_ne_zero, ← Finsupp.mem_support_iff, Nat.support_factorization, hn',
Nat.minFac_prime hn, Nat.minFac_dvd]
| 5 |
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 | 36 | 42 | theorem P_is_eventually_constant {q n : ℕ} (hqn : n ≤ q) :
((P (q + 1)).f n : X _[n] ⟶ _) = (P q).f n := by |
rcases n with (_|n)
· simp only [Nat.zero_eq, P_f_0_eq]
· simp only [P_succ, add_right_eq_self, comp_add, HomologicalComplex.comp_f,
HomologicalComplex.add_f_apply, comp_id]
exact (HigherFacesVanish.of_P q n).comp_Hσ_eq_zero (Nat.succ_le_iff.mp hqn)
| 5 |
import Mathlib.Algebra.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
variable {σ τ α R S : Type*} [CommSemiring R] [CommSemiring S]
namespace MvPolynomial... | Mathlib/Algebra/MvPolynomial/Rename.lean | 93 | 99 | theorem rename_monomial (f : σ → τ) (d : σ →₀ ℕ) (r : R) :
rename f (monomial d r) = monomial (d.mapDomain f) r := by |
rw [rename, aeval_monomial, monomial_eq (s := Finsupp.mapDomain f d),
Finsupp.prod_mapDomain_index]
· rfl
· exact fun n => pow_zero _
· exact fun n i₁ i₂ => pow_add _ _ _
| 5 |
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Convex.Jensen
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.AddTorsor
#align_import analysis.convex.normed from "leanprover-community/mathlib"@"a63928c34ec358b5edcda2bf7513c50052... | Mathlib/Analysis/Convex/Normed.lean | 75 | 80 | theorem Convex.cthickening (hs : Convex ℝ s) (δ : ℝ) : Convex ℝ (cthickening δ s) := by |
obtain hδ | hδ := le_total 0 δ
· rw [cthickening_eq_iInter_thickening hδ]
exact convex_iInter₂ fun _ _ => hs.thickening _
· rw [cthickening_of_nonpos hδ]
exact hs.closure
| 5 |
import Mathlib.Algebra.Homology.ComplexShape
import Mathlib.CategoryTheory.Subobject.Limits
import Mathlib.CategoryTheory.GradedObject
import Mathlib.Algebra.Homology.ShortComplex.Basic
#align_import algebra.homology.homological_complex from "leanprover-community/mathlib"@"88bca0ce5d22ebfd9e73e682e51d60ea13b48347"
... | Mathlib/Algebra/Homology/HomologicalComplex.lean | 316 | 321 | theorem isZero_zero [HasZeroObject V] : IsZero (zero : HomologicalComplex V c) := by |
refine ⟨fun X => ⟨⟨⟨0⟩, fun f => ?_⟩⟩, fun X => ⟨⟨⟨0⟩, fun f => ?_⟩⟩⟩
all_goals
ext
dsimp [zero]
apply Subsingleton.elim
| 5 |
import Mathlib.NumberTheory.Liouville.Basic
#align_import number_theory.liouville.liouville_number from "leanprover-community/mathlib"@"04e80bb7e8510958cd9aacd32fe2dc147af0b9f1"
noncomputable section
open scoped Nat
open Real Finset
def liouvilleNumber (m : ℝ) : ℝ :=
∑' i : ℕ, 1 / m ^ i !
#align liouville_n... | Mathlib/NumberTheory/Liouville/LiouvilleNumber.lean | 110 | 134 | theorem remainder_lt' (n : ℕ) {m : ℝ} (m1 : 1 < m) :
remainder m n < (1 - 1 / m)⁻¹ * (1 / m ^ (n + 1)!) :=
-- two useful inequalities
have m0 : 0 < m := zero_lt_one.trans m1
have mi : 1 / m < 1 := (div_lt_one m0).mpr m1
-- to show the strict inequality between these series, we prove that:
calc
(∑' i, ... |
simp only [pow_add, one_div, mul_inv, inv_pow]
-- factor the constant `(1 / m ^ (n + 1)!)` out of the series
_ = (∑' i, (1 / m) ^ i) * (1 / m ^ (n + 1)!) := tsum_mul_right
-- the series is the geometric series
_ = (1 - 1 / m)⁻¹ * (1 / m ^ (n + 1)!) := by rw [tsum_geometric_of_lt_one (by positivit... | 5 |
import Mathlib.GroupTheory.CoprodI
import Mathlib.GroupTheory.Coprod.Basic
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.GroupTheory.Complement
namespace Monoid
open CoprodI Subgroup Coprod Function List
variable {ι : Type*} {G : ι → Type*} {H : Type*} {K : Type*} [Monoid K]
def PushoutI.con [∀ i, Mo... | Mathlib/GroupTheory/PushoutI.lean | 88 | 93 | theorem of_comp_eq_base (i : ι) : (of i).comp (φ i) = (base φ) := by |
ext x
apply (Con.eq _).2
refine ConGen.Rel.of _ _ ?_
simp only [MonoidHom.comp_apply, Set.mem_iUnion, Set.mem_range]
exact ⟨_, _, rfl, rfl⟩
| 5 |
import Mathlib.Data.Finset.Lattice
#align_import combinatorics.set_family.compression.down from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
variable {α : Type*} [DecidableEq α] {𝒜 ℬ : Finset (Finset α)} {s : Finset α} {a : α}
namespace Finset
def nonMemberSubfamily (a : α) (𝒜 : ... | Mathlib/Combinatorics/SetFamily/Compression/Down.lean | 61 | 66 | theorem mem_memberSubfamily : s ∈ 𝒜.memberSubfamily a ↔ insert a s ∈ 𝒜 ∧ a ∉ s := by |
simp_rw [memberSubfamily, mem_image, mem_filter]
refine ⟨?_, fun h => ⟨insert a s, ⟨h.1, by simp⟩, erase_insert h.2⟩⟩
rintro ⟨s, ⟨hs1, hs2⟩, rfl⟩
rw [insert_erase hs2]
exact ⟨hs1, not_mem_erase _ _⟩
| 5 |
import Mathlib.Analysis.Convex.Between
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.Topology.MetricSpace.Holder
import Mathlib.Topology.MetricSpace.MetricSeparated
#align_import measure_theory.measure.hausdorff from "leanprover-communit... | Mathlib/MeasureTheory/Measure/Hausdorff.lean | 309 | 315 | theorem trim_pre [MeasurableSpace X] [OpensMeasurableSpace X] (m : Set X → ℝ≥0∞)
(hcl : ∀ s, m (closure s) = m s) (r : ℝ≥0∞) : (pre m r).trim = pre m r := by |
refine le_antisymm (le_pre.2 fun s hs => ?_) (le_trim _)
rw [trim_eq_iInf]
refine iInf_le_of_le (closure s) <| iInf_le_of_le subset_closure <|
iInf_le_of_le measurableSet_closure ((pre_le ?_).trans_eq (hcl _))
rwa [diam_closure]
| 5 |
import Mathlib.RingTheory.WittVector.Basic
import Mathlib.RingTheory.WittVector.IsPoly
#align_import ring_theory.witt_vector.verschiebung from "leanprover-community/mathlib"@"32b08ef840dd25ca2e47e035c5da03ce16d2dc3c"
namespace WittVector
open MvPolynomial
variable {p : ℕ} {R S : Type*} [hp : Fact p.Prime] [Comm... | Mathlib/RingTheory/WittVector/Verschiebung.lean | 86 | 92 | theorem aeval_verschiebung_poly' (x : 𝕎 R) (n : ℕ) :
aeval x.coeff (verschiebungPoly n) = (verschiebungFun x).coeff n := by |
cases' n with n
· simp only [verschiebungPoly, Nat.zero_eq, ge_iff_le, tsub_eq_zero_of_le, ite_true, map_zero,
verschiebungFun_coeff_zero]
· rw [verschiebungPoly, verschiebungFun_coeff_succ, if_neg n.succ_ne_zero, aeval_X,
add_tsub_cancel_right]
| 5 |
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Analytic.CPolynomial
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.ContDiff.Defs
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.fderiv_analytic from "leanprover-community/mathlib"@"3bce8d800a6f2... | Mathlib/Analysis/Calculus/FDeriv/Analytic.lean | 469 | 474 | theorem iteratedFDeriv_zero_apply_diag : iteratedFDeriv 𝕜 0 f x = p 0 := by |
ext
convert (h.hasSum <| EMetric.mem_ball_self h.r_pos).tsum_eq.symm
· rw [iteratedFDeriv_zero_apply, add_zero]
· rw [tsum_eq_single 0 fun n hn ↦ by haveI := NeZero.mk hn; exact (p n).map_zero]
exact congr(p 0 $(Subsingleton.elim _ _))
| 5 |
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.Algebra.Module.ULift
#align_import ring_theory.is_tensor_product from "leanprover-community/mathlib"@"c4926d76bb9c5a4a62ed2f03d998081786132105"
universe u v₁ v₂ v₃ v₄
open TensorProduct
section IsTensorProduct
variable {R : Type*} [CommSemiring R]
va... | Mathlib/RingTheory/IsTensorProduct.lean | 60 | 65 | theorem TensorProduct.isTensorProduct : IsTensorProduct (TensorProduct.mk R M N) := by |
delta IsTensorProduct
convert_to Function.Bijective (LinearMap.id : M ⊗[R] N →ₗ[R] M ⊗[R] N) using 2
· apply TensorProduct.ext'
simp
· exact Function.bijective_id
| 5 |
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 | 69 | 74 | theorem le_exp_log (x : ℝ) : x ≤ exp (log x) := by |
by_cases h_zero : x = 0
· rw [h_zero, log, dif_pos rfl, exp_zero]
exact zero_le_one
· rw [exp_log_eq_abs h_zero]
exact le_abs_self _
| 5 |
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Lie.OfAssociative
import Mathlib.Algebra.Lie.Submodule
import Mathlib.Algebra.Lie.Basic
#align_import algebra.lie.direct_sum from "leanprover-community/mathlib"@"c0cc689babd41c0e9d5f02429211ffbe2403472a"
universe u v w w₁
namespace DirectSum
open DF... | Mathlib/Algebra/Lie/DirectSum.lean | 130 | 136 | theorem lie_of_of_ne [DecidableEq ι] {i j : ι} (hij : i ≠ j) (x : L i) (y : L j) :
⁅of L i x, of L j y⁆ = 0 := by |
refine DFinsupp.ext fun k => ?_
rw [bracket_apply]
obtain rfl | hik := Decidable.eq_or_ne i k
· rw [of_eq_of_ne _ _ _ _ hij.symm, lie_zero, zero_apply]
· rw [of_eq_of_ne _ _ _ _ hik, zero_lie, zero_apply]
| 5 |
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.GroupTheory.Perm.Fin
import Mathlib.GroupTheory.Subgroup.Simple
import Mathlib.Tactic.IntervalCases
#align_import group_theory.specific_groups.alternating from "leanprover-community/mathlib"@"0f6670b8af2dff699de1c0b4b49039b31bc13c46"
-- An example on how to de... | Mathlib/GroupTheory/SpecificGroups/Alternating.lean | 219 | 224 | theorem nontrivial_of_three_le_card (h3 : 3 ≤ card α) : Nontrivial (alternatingGroup α) := by |
haveI := Fintype.one_lt_card_iff_nontrivial.1 (lt_trans (by decide) h3)
rw [← Fintype.one_lt_card_iff_nontrivial]
refine lt_of_mul_lt_mul_left ?_ (le_of_lt Nat.prime_two.pos)
rw [two_mul_card_alternatingGroup, card_perm, ← Nat.succ_le_iff]
exact le_trans h3 (card α).self_le_factorial
| 5 |
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
local macro:max "local_hAdd[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HAdd.hAdd : $type → $type → $type))
local macro:max "local_hMul[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HMul.hMul : $type → $typ... | Mathlib/Algebra/Ring/Ext.lean | 195 | 201 | theorem toNonUnitalNonAssocSemiring_injective :
Function.Injective (@toNonUnitalNonAssocSemiring R) := by |
intro _ _ h
-- Use above extensionality lemma to prove injectivity by showing that `h_add` and `h_mul` hold.
ext x y
· exact congrArg (·.toAdd.add x y) h
· exact congrArg (·.toMul.mul x y) h
| 5 |
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.GradedAlgebra.Basic
#align_import ring_theory.graded_algebra.homogeneous_localization from "leanprover-community/mathlib"@"831c494092374cfe9f50591ed0ac81a25efc5b86"
noncomputable section
open DirectSum Pointwise
open DirectSum SetLike
var... | Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean | 109 | 115 | theorem ext {c1 c2 : NumDenSameDeg 𝒜 x} (hdeg : c1.deg = c2.deg) (hnum : (c1.num : A) = c2.num)
(hden : (c1.den : A) = c2.den) : c1 = c2 := by |
rcases c1 with ⟨i1, ⟨n1, hn1⟩, ⟨d1, hd1⟩, h1⟩
rcases c2 with ⟨i2, ⟨n2, hn2⟩, ⟨d2, hd2⟩, h2⟩
dsimp only [Subtype.coe_mk] at *
subst hdeg hnum hden
congr
| 5 |
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.RingTheory.Localization.FractionRing
#align_import data.polynomial.ring_division from "leanprover-community/mathlib"@"8efcf8022aac8e01df8d302dcebdbc25d6a886c8"
noncomputable section
namespace Polynomial
universe u v w z
variable {R : Type u} {S : Ty... | Mathlib/Algebra/Polynomial/Roots.lean | 100 | 105 | theorem count_roots [DecidableEq R] (p : R[X]) : p.roots.count a = rootMultiplicity a p := by |
classical
by_cases hp : p = 0
· simp [hp]
rw [roots_def, dif_neg hp]
exact (Classical.choose_spec (exists_multiset_roots hp)).2 a
| 5 |
import Mathlib.Data.Multiset.Nodup
#align_import data.multiset.sum from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Sum
namespace Multiset
variable {α β : Type*} (s : Multiset α) (t : Multiset β)
def disjSum : Multiset (Sum α β) :=
s.map inl + t.map inr
#align multiset.dis... | Mathlib/Data/Multiset/Sum.lean | 55 | 60 | theorem inl_mem_disjSum : inl a ∈ s.disjSum t ↔ a ∈ s := by |
rw [mem_disjSum, or_iff_left]
-- Porting note: Previous code for L62 was: simp only [exists_eq_right]
· simp only [inl.injEq, exists_eq_right]
rintro ⟨b, _, hb⟩
exact inr_ne_inl hb
| 5 |
import Mathlib.CategoryTheory.Balanced
import Mathlib.CategoryTheory.Limits.EssentiallySmall
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.CategoryTheory.Subobject.WellPowered
import Mathlib.Data.S... | Mathlib/CategoryTheory/Generator.lean | 93 | 98 | theorem isSeparating_op_iff (𝒢 : Set C) : IsSeparating 𝒢.op ↔ IsCoseparating 𝒢 := by |
refine ⟨fun h𝒢 X Y f g hfg => ?_, fun h𝒢 X Y f g hfg => ?_⟩
· refine Quiver.Hom.op_inj (h𝒢 _ _ fun G hG h => Quiver.Hom.unop_inj ?_)
simpa only [unop_comp, Quiver.Hom.unop_op] using hfg _ (Set.mem_op.1 hG) _
· refine Quiver.Hom.unop_inj (h𝒢 _ _ fun G hG h => Quiver.Hom.op_inj ?_)
simpa only [op_comp,... | 5 |
import Mathlib.Logic.Encodable.Basic
import Mathlib.Logic.Pairwise
import Mathlib.Data.Set.Subsingleton
#align_import logic.encodable.lattice from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Set
namespace Encodable
variable {α : Type*} {β : Type*} [Encodable β]
theorem iSup_de... | Mathlib/Logic/Encodable/Lattice.lean | 53 | 59 | theorem iUnion_decode₂_disjoint_on {f : β → Set α} (hd : Pairwise (Disjoint on f)) :
Pairwise (Disjoint on fun i => ⋃ b ∈ decode₂ β i, f b) := by |
rintro i j ij
refine disjoint_left.mpr fun x => ?_
suffices ∀ a, encode a = i → x ∈ f a → ∀ b, encode b = j → x ∉ f b by simpa [decode₂_eq_some]
rintro a rfl ha b rfl hb
exact (hd (mt (congr_arg encode) ij)).le_bot ⟨ha, hb⟩
| 5 |
import Mathlib.LinearAlgebra.Dual
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.contraction from "leanprover-community/mathlib"@"657df4339ae6ceada048c8a2980fb10e393143ec"
suppress_compilation
-- Porting note: universe metavariables behave oddly
universe w u v₁ v₂ v₃ v₄
variable {ι : Type... | Mathlib/LinearAlgebra/Contraction.lean | 133 | 140 | theorem toMatrix_dualTensorHom {m : Type*} {n : Type*} [Fintype m] [Finite n] [DecidableEq m]
[DecidableEq n] (bM : Basis m R M) (bN : Basis n R N) (j : m) (i : n) :
toMatrix bM bN (dualTensorHom R M N (bM.coord j ⊗ₜ bN i)) = stdBasisMatrix i j 1 := by |
ext i' j'
by_cases hij : i = i' ∧ j = j' <;>
simp [LinearMap.toMatrix_apply, Finsupp.single_eq_pi_single, hij]
rw [and_iff_not_or_not, Classical.not_not] at hij
cases' hij with hij hij <;> simp [hij]
| 5 |
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 | 148 | 153 | theorem degree_le_mul_left (p : R[X]) (hq : q ≠ 0) : degree p ≤ degree (p * q) := by |
classical
exact if hp : p = 0 then by simp only [hp, zero_mul, le_refl]
else by
rw [degree_mul, degree_eq_natDegree hp, degree_eq_natDegree hq];
exact WithBot.coe_le_coe.2 (Nat.le_add_right _ _)
| 5 |
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 | 49 | 55 | theorem continuousAt_oangle {x : P × P × P} (hx12 : x.1 ≠ x.2.1) (hx32 : x.2.2 ≠ x.2.1) :
ContinuousAt (fun y : P × P × P => ∡ y.1 y.2.1 y.2.2) x := by |
let f : P × P × P → V × V := fun y => (y.1 -ᵥ y.2.1, y.2.2 -ᵥ y.2.1)
have hf1 : (f x).1 ≠ 0 := by simp [hx12]
have hf2 : (f x).2 ≠ 0 := by simp [hx32]
exact (o.continuousAt_oangle hf1 hf2).comp ((continuous_fst.vsub continuous_snd.fst).prod_mk
(continuous_snd.snd.vsub continuous_snd.fst)).continuousAt
| 5 |
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