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.Dynamics.Ergodic.Ergodic
import Mathlib.MeasureTheory.Function.AEEqFun
open Function Set Filter MeasureTheory Topology TopologicalSpace
variable {α X : Type*} [MeasurableSpace α] {μ : MeasureTheory.Measure α}
theorem QuasiErgodic.ae_eq_const_of_ae_eq_comp_of_ae_range₀ [Nonempty X] [MeasurableSpace... | Mathlib/Dynamics/Ergodic/Function.lean | 77 | 82 | theorem ae_eq_const_of_ae_eq_comp_ae {g : α → X} (h : QuasiErgodic f μ)
(hgm : AEStronglyMeasurable g μ) (hg_eq : g ∘ f =ᵐ[μ] g) : ∃ c, g =ᵐ[μ] const α c := by |
borelize X
rcases hgm.isSeparable_ae_range with ⟨t, ht, hgt⟩
haveI := ht.secondCountableTopology
exact h.ae_eq_const_of_ae_eq_comp_of_ae_range₀ hgt hgm.aemeasurable.nullMeasurable hg_eq
| 4 |
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 | 109 | 113 | theorem mutuallySingular_singularPart (μ ν : Measure α) : μ.singularPart ν ⟂ₘ ν := by |
by_cases h : HaveLebesgueDecomposition μ ν
· exact (haveLebesgueDecomposition_spec μ ν).2.1
· rw [singularPart_of_not_haveLebesgueDecomposition h]
exact MutuallySingular.zero_left
| 4 |
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Group.Int
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Ring.Rat
import Mathlib.Data.PNat.Defs
#align_import data.rat.lemmas from "leanprover-community/mathlib"@"550b58538991c8977703fdeb7c9d51a5aa27df11"
namespace Rat
o... | Mathlib/Data/Rat/Lemmas.lean | 114 | 119 | theorem add_num_den (q r : ℚ) :
q + r = (q.num * r.den + q.den * r.num : ℤ) /. (↑q.den * ↑r.den : ℤ) := by |
have hqd : (q.den : ℤ) ≠ 0 := Int.natCast_ne_zero_iff_pos.2 q.den_pos
have hrd : (r.den : ℤ) ≠ 0 := Int.natCast_ne_zero_iff_pos.2 r.den_pos
conv_lhs => rw [← num_divInt_den q, ← num_divInt_den r, divInt_add_divInt _ _ hqd hrd]
rw [mul_comm r.num q.den]
| 4 |
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 | 98 | 102 | theorem commProb_def' : commProb G = Nat.card (ConjClasses G) / Nat.card G := by |
rw [commProb, card_comm_eq_card_conjClasses_mul_card, Nat.cast_mul, sq]
by_cases h : (Nat.card G : ℚ) = 0
· rw [h, zero_mul, div_zero, div_zero]
· exact mul_div_mul_right _ _ h
| 4 |
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Topology.Instances.NNReal
#align_import analysis.normed.group.infinite_sum from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd"
open Topology NNReal
open Finset Filter Metric
variabl... | Mathlib/Analysis/Normed/Group/InfiniteSum.lean | 40 | 46 | theorem cauchySeq_finset_iff_vanishing_norm {f : ι → E} :
(CauchySeq fun s : Finset ι => ∑ i ∈ s, f i) ↔
∀ ε > (0 : ℝ), ∃ s : Finset ι, ∀ t, Disjoint t s → ‖∑ i ∈ t, f i‖ < ε := by |
rw [cauchySeq_finset_iff_sum_vanishing, nhds_basis_ball.forall_iff]
· simp only [ball_zero_eq, Set.mem_setOf_eq]
· rintro s t hst ⟨s', hs'⟩
exact ⟨s', fun t' ht' => hst <| hs' _ ht'⟩
| 4 |
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Measurable
open MeasureTheory
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] [LocallyCompactSpace 𝕜]
{E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [MeasurableSpace E] [OpensMeasurableSpace E]
{F : Type*} [Norm... | Mathlib/Analysis/Calculus/LineDeriv/Measurable.lean | 40 | 45 | theorem measurable_lineDeriv [MeasurableSpace F] [BorelSpace F]
(hf : Continuous f) : Measurable (fun x ↦ lineDeriv 𝕜 f x v) := by |
borelize 𝕜
let g : E → 𝕜 → F := fun x t ↦ f (x + t • v)
have hg : Continuous g.uncurry := by apply hf.comp; continuity
exact (measurable_deriv_with_param hg).comp measurable_prod_mk_right
| 4 |
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 | 424 | 429 | theorem measurePreserving_mul_prod_inv_right [IsMulRightInvariant μ] [IsMulRightInvariant ν] :
MeasurePreserving (fun z : G × G => (z.1 * z.2, z.1⁻¹)) (μ.prod ν) (μ.prod ν) := by |
convert (measurePreserving_prod_div_swap ν μ).comp (measurePreserving_prod_mul_swap_right μ ν)
using 1
ext1 ⟨x, y⟩
simp_rw [Function.comp_apply, div_mul_eq_div_div_swap, div_self', one_div]
| 4 |
import Mathlib.Analysis.LocallyConvex.Basic
#align_import analysis.locally_convex.balanced_core_hull from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Pointwise Topology Filter
variable {𝕜 E ι : Type*}
section balancedHull
section SeminormedRing
variable [SeminormedRing ... | Mathlib/Analysis/LocallyConvex/BalancedCoreHull.lean | 85 | 90 | theorem smul_balancedCore_subset (s : Set E) {a : 𝕜} (ha : ‖a‖ ≤ 1) :
a • balancedCore 𝕜 s ⊆ balancedCore 𝕜 s := by |
rintro x ⟨y, hy, rfl⟩
rw [mem_balancedCore_iff] at hy
rcases hy with ⟨t, ht1, ht2, hy⟩
exact ⟨t, ⟨ht1, ht2⟩, ht1 a ha (smul_mem_smul_set hy)⟩
| 4 |
import Mathlib.CategoryTheory.Category.Grpd
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Homotopy.Path
import Mathlib.Data.Set.Subsingleton
#align_import algebraic_topology.fundamental_groupoid.basic from "leanprover-community/mathlib"@"3d7987cda72abc473... | Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean | 131 | 135 | theorem continuous_transReflReparamAux : Continuous transReflReparamAux := by |
refine continuous_if_le ?_ ?_ (Continuous.continuousOn ?_) (Continuous.continuousOn ?_) ?_ <;>
[continuity; continuity; continuity; continuity; skip]
intro x hx
simp [hx]
| 4 |
import Mathlib.Data.List.GetD
import Mathlib.Data.Nat.Bits
import Mathlib.Algebra.Ring.Nat
import Mathlib.Order.Basic
import Mathlib.Tactic.AdaptationNote
import Mathlib.Tactic.Common
#align_import data.nat.bitwise from "leanprover-community/mathlib"@"6afc9b06856ad973f6a2619e3e8a0a8d537a58f2"
open Function
names... | Mathlib/Data/Nat/Bitwise.lean | 75 | 81 | theorem binaryRec_of_ne_zero {C : Nat → Sort*} (z : C 0) (f : ∀ b n, C n → C (bit b n)) {n}
(h : n ≠ 0) :
binaryRec z f n = bit_decomp n ▸ f (bodd n) (div2 n) (binaryRec z f (div2 n)) := by |
rw [Eq.rec_eq_cast]
rw [binaryRec]
dsimp only
rw [dif_neg h, eq_mpr_eq_cast]
| 4 |
import Mathlib.AlgebraicTopology.DoldKan.FunctorGamma
import Mathlib.AlgebraicTopology.DoldKan.SplitSimplicialObject
import Mathlib.CategoryTheory.Idempotents.HomologicalComplex
#align_import algebraic_topology.dold_kan.gamma_comp_n from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
no... | Mathlib/AlgebraicTopology/DoldKan/GammaCompN.lean | 76 | 82 | theorem N₁Γ₀_app (K : ChainComplex C ℕ) :
N₁Γ₀.app K = (Γ₀.splitting K).toKaroubiNondegComplexIsoN₁.symm ≪≫
(toKaroubi _).mapIso (Γ₀NondegComplexIso K) := by |
ext1
dsimp [N₁Γ₀]
erw [id_comp, comp_id, comp_id]
rfl
| 4 |
import Mathlib.AlgebraicTopology.FundamentalGroupoid.InducedMaps
import Mathlib.Topology.Homotopy.Contractible
import Mathlib.CategoryTheory.PUnit
import Mathlib.AlgebraicTopology.FundamentalGroupoid.PUnit
#align_import algebraic_topology.fundamental_groupoid.simply_connected from "leanprover-community/mathlib"@"3834... | Mathlib/AlgebraicTopology/FundamentalGroupoid/SimplyConnected.lean | 42 | 48 | theorem simply_connected_iff_unique_homotopic (X : Type*) [TopologicalSpace X] :
SimplyConnectedSpace X ↔
Nonempty X ∧ ∀ x y : X, Nonempty (Unique (Path.Homotopic.Quotient x y)) := by |
simp only [simply_connected_def, equiv_punit_iff_unique,
FundamentalGroupoid.nonempty_iff X, and_congr_right_iff, Nonempty.forall]
intros
exact ⟨fun h _ _ => h _ _, fun h _ _ => h _ _⟩
| 4 |
import Mathlib.Algebra.Field.Subfield
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Topology.Algebra.GroupWithZero
import Mathlib.Topology.Algebra.Ring.Basic
import Mathlib.Topology.Order.LocalExtr
#align_import topology.algebra.field from "leanprover-community/mathlib"@"c10e724be91096453ee3db13862... | Mathlib/Topology/Algebra/Field.lean | 142 | 149 | theorem IsPreconnected.eq_or_eq_neg_of_sq_eq [Field 𝕜] [HasContinuousInv₀ 𝕜] [ContinuousMul 𝕜]
(hS : IsPreconnected S) (hf : ContinuousOn f S) (hg : ContinuousOn g S)
(hsq : EqOn (f ^ 2) (g ^ 2) S) (hg_ne : ∀ {x : α}, x ∈ S → g x ≠ 0) :
EqOn f g S ∨ EqOn f (-g) S := by |
have hsq : EqOn ((f / g) ^ 2) 1 S := fun x hx => by
simpa [div_eq_one_iff_eq (pow_ne_zero _ (hg_ne hx))] using hsq hx
simpa (config := { contextual := true }) [EqOn, div_eq_iff (hg_ne _)]
using hS.eq_one_or_eq_neg_one_of_sq_eq (hf.div hg fun z => hg_ne) hsq
| 4 |
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
| 4 |
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
| 4 |
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⟩
| 4 |
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]
| 4 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
namespace Polynomial
open Polynomial
section Semiring
variable {R : Type*} [Semiring R] (p q : R... | Mathlib/Algebra/Polynomial/Mirror.lean | 75 | 79 | theorem mirror_natTrailingDegree : p.mirror.natTrailingDegree = p.natTrailingDegree := by |
by_cases hp : p = 0
· rw [hp, mirror_zero]
· rw [mirror, natTrailingDegree_mul_X_pow ((mt reverse_eq_zero.mp) hp),
natTrailingDegree_reverse, zero_add]
| 4 |
import Mathlib.FieldTheory.Fixed
import Mathlib.FieldTheory.NormalClosure
import Mathlib.FieldTheory.PrimitiveElement
import Mathlib.GroupTheory.GroupAction.FixingSubgroup
#align_import field_theory.galois from "leanprover-community/mathlib"@"9fb8964792b4237dac6200193a0d533f1b3f7423"
open scoped Polynomial Interm... | Mathlib/FieldTheory/Galois.lean | 93 | 100 | theorem IntermediateField.AdjoinSimple.card_aut_eq_finrank [FiniteDimensional F E] {α : E}
(hα : IsIntegral F α) (h_sep : (minpoly F α).Separable)
(h_splits : (minpoly F α).Splits (algebraMap F F⟮α⟯)) :
Fintype.card (F⟮α⟯ ≃ₐ[F] F⟮α⟯) = finrank F F⟮α⟯ := by |
letI : Fintype (F⟮α⟯ →ₐ[F] F⟮α⟯) := IntermediateField.fintypeOfAlgHomAdjoinIntegral F hα
rw [IntermediateField.adjoin.finrank hα]
rw [← IntermediateField.card_algHom_adjoin_integral F hα h_sep h_splits]
exact Fintype.card_congr (algEquivEquivAlgHom F F⟮α⟯)
| 4 |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' u₁' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}... | Mathlib/LinearAlgebra/Dimension/Constructions.lean | 439 | 443 | theorem rank_span_le (s : Set M) : Module.rank R (span R s) ≤ #s := by |
rw [Finsupp.span_eq_range_total, ← lift_strictMono.le_iff_le]
refine (lift_rank_range_le _).trans ?_
rw [rank_finsupp_self]
simp only [lift_lift, ge_iff_le, le_refl]
| 4 |
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Order.Partition.Finpartition
import Mathlib.Tactic.GCongr
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Positivity
import Mathlib.Tactic.Ring
#align_import combinatorics.simp... | Mathlib/Combinatorics/SimpleGraph/Density.lean | 115 | 120 | theorem interedges_biUnion_right (s : Finset α) (t : Finset ι) (f : ι → Finset β) :
interedges r s (t.biUnion f) = t.biUnion fun b ↦ interedges r s (f b) := by |
ext a
simp only [mem_interedges_iff, mem_biUnion]
exact ⟨fun ⟨x₁, ⟨x₂, x₃, x₄⟩, x₅⟩ ↦ ⟨x₂, x₃, x₁, x₄, x₅⟩,
fun ⟨x₂, x₃, x₁, x₄, x₅⟩ ↦ ⟨x₁, ⟨x₂, x₃, x₄⟩, x₅⟩⟩
| 4 |
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 | 39 | 44 | theorem HasFPowerSeriesAt.hasStrictFDerivAt (h : HasFPowerSeriesAt f p x) :
HasStrictFDerivAt f (continuousMultilinearCurryFin1 𝕜 E F (p 1)) x := by |
refine h.isBigO_image_sub_norm_mul_norm_sub.trans_isLittleO (IsLittleO.of_norm_right ?_)
refine isLittleO_iff_exists_eq_mul.2 ⟨fun y => ‖y - (x, x)‖, ?_, EventuallyEq.rfl⟩
refine (continuous_id.sub continuous_const).norm.tendsto' _ _ ?_
rw [_root_.id, sub_self, norm_zero]
| 4 |
import Mathlib.CategoryTheory.Limits.Final
import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
import Mathlib.CategoryTheory.Countable
import Mathlib.Data.Countable.Defs
open CategoryTheory Opposite CountableCategory
variable (C : Type*) [Category C] (J : Type*) [Countable J]
namespace CategoryTheory.Limits
... | Mathlib/CategoryTheory/Limits/Shapes/Countable.lean | 102 | 106 | theorem sequentialFunctor_initial_aux (j : J) : ∃ (n : ℕ), sequentialFunctor_obj J n ≤ j := by |
obtain ⟨m, h⟩ := (exists_surjective_nat _).choose_spec j
refine ⟨m + 1, ?_⟩
simpa [h] using leOfHom (IsCofilteredOrEmpty.cone_objs ((exists_surjective_nat _).choose m)
(sequentialFunctor_obj J m)).choose_spec.choose
| 4 |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Filter Metric Set
open scoped ComplexConjugate Real To... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 33 | 37 | theorem sin_arg (x : ℂ) : Real.sin (arg x) = x.im / abs x := by |
unfold arg; split_ifs <;>
simp [sub_eq_add_neg, arg,
Real.sin_arcsin (abs_le.1 (abs_im_div_abs_le_one x)).1 (abs_le.1 (abs_im_div_abs_le_one x)).2,
Real.sin_add, neg_div, Real.arcsin_neg, Real.sin_neg]
| 4 |
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 | 120 | 129 | theorem mul_eq_one_comm : A * B = 1 ↔ B * A = 1 :=
suffices ∀ A B : Matrix n n α, A * B = 1 → B * A = 1 from ⟨this A B, this B A⟩
fun A B h => by
letI : Invertible B.det := detInvertibleOfLeftInverse _ _ h
letI : Invertible B := invertibleOfDetInvertible B
calc
B * A = B * A * (B * ⅟ B) := by | rw [mul_invOf_self, Matrix.mul_one]
_ = B * (A * B * ⅟ B) := by simp only [Matrix.mul_assoc]
_ = B * ⅟ B := by rw [h, Matrix.one_mul]
_ = 1 := mul_invOf_self B
| 4 |
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
#align_import data.complex.exponential_bounds from "leanprover-community/mathlib"@"402f8982dddc1864bd703da2d6e2ee304a866973"
namespace Real
open IsAbsoluteValue Finset CauSeq Complex
theorem exp_one_near_10 : |exp 1 - 224... | Mathlib/Data/Complex/ExponentialBounds.lean | 51 | 55 | theorem exp_neg_one_lt_d9 : exp (-1) < 0.3678794412 := by |
rw [exp_neg, inv_lt (exp_pos _)]
· refine lt_of_lt_of_le ?_ (sub_le_comm.1 (abs_sub_le_iff.1 exp_one_near_10).2)
norm_num
· norm_num
| 4 |
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.Algebra.GCDMonoid.Nat
#align_import ring_theory.int.basic from "leanprover-community/mathlib"@"e655e4ea5c6d02854696f97494997ba4c31be802"
theorem Int.Prime.dvd_mul {m n : ℤ} {p : ℕ} (hp : Nat.Prime p) (h : (p ... | Mathlib/RingTheory/Int/Basic.lean | 147 | 152 | theorem eq_pow_of_mul_eq_pow_bit1_left {a b c : ℤ} (hab : IsCoprime a b) {k : ℕ}
(h : a * b = c ^ bit1 k) : ∃ d, a = d ^ bit1 k := by |
obtain ⟨d, hd⟩ := exists_associated_pow_of_mul_eq_pow' hab h
replace hd := hd.symm
rw [associated_iff_natAbs, natAbs_eq_natAbs_iff, ← neg_pow_bit1] at hd
obtain rfl | rfl := hd <;> exact ⟨_, rfl⟩
| 4 |
import Mathlib.Data.Finset.Image
#align_import data.finset.card from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
assert_not_exists MonoidWithZero
-- TODO: After a lot more work,
-- assert_not_exists OrderedCommMonoid
open Function Multiset Nat
variable {α β R : Type*}
namespace Fin... | Mathlib/Data/Finset/Card.lean | 114 | 118 | theorem card_insert_le (a : α) (s : Finset α) : card (insert a s) ≤ s.card + 1 := by |
by_cases h : a ∈ s
· rw [insert_eq_of_mem h]
exact Nat.le_succ _
· rw [card_insert_of_not_mem h]
| 4 |
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.Analysis.SumIntegralComparisons
import Mathlib.NumberTheory.Harmonic.Defs
theorem log_add_one_le_harmonic (n : ℕ) :
Real.log ↑(n+1) ≤ harmonic n := by
calc _ = ∫ x in (1:ℕ)..↑(n+1), x⁻¹ := ?_
_ ≤ ∑ d ∈ Finset.Icc 1 n, (d:ℝ)⁻¹ := ?_
... | Mathlib/NumberTheory/Harmonic/Bounds.lean | 64 | 69 | theorem harmonic_floor_le_one_add_log (y : ℝ) (hy : 1 ≤ y) :
harmonic ⌊y⌋₊ ≤ 1 + Real.log y := by |
refine (harmonic_le_one_add_log _).trans ?_
gcongr
· exact_mod_cast Nat.floor_pos.mpr hy
· exact Nat.floor_le <| zero_le_one.trans hy
| 4 |
import Mathlib.Tactic.CategoryTheory.Elementwise
import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
import Mathlib.CategoryTheory.Limits.Preserves.Limits
import Mathlib.CategoryTheory.Limits.Shapes.Types
#align_import category_theory.glue_data from "l... | Mathlib/CategoryTheory/GlueData.lean | 123 | 129 | theorem t'_comp_eq_pullbackSymmetry (i j k : D.J) :
D.t' j k i ≫ D.t' k i j =
(pullbackSymmetry _ _).hom ≫ D.t' j i k ≫ (pullbackSymmetry _ _).hom := by |
trans inv (D.t' i j k)
· exact IsIso.eq_inv_of_hom_inv_id (D.cocycle _ _ _)
· rw [← cancel_mono (pullback.fst : pullback (D.f i j) (D.f i k) ⟶ _)]
simp [t_fac, t_fac_assoc]
| 4 |
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Algebra.MonoidAlgebra.Support
import Mathlib.Algebra.DirectSum.Internal
import Mathlib.RingTheory.GradedAlgebra.Basic
#align_import algebra.monoid_algebra.grading from "leanprover-community/mathlib"@"feb99064803fd3108e37c18b0f77d0a8344677a3"
noncomputable sectio... | Mathlib/Algebra/MonoidAlgebra/Grading.lean | 72 | 78 | theorem mem_grade_iff' (m : M) (a : R[M]) :
a ∈ grade R m ↔ a ∈ (LinearMap.range (Finsupp.lsingle m : R →ₗ[R] M →₀ R) :
Submodule R R[M]) := by |
rw [mem_grade_iff, Finsupp.support_subset_singleton']
apply exists_congr
intro r
constructor <;> exact Eq.symm
| 4 |
import Mathlib.Algebra.ContinuedFractions.Computation.ApproximationCorollaries
import Mathlib.Algebra.ContinuedFractions.Computation.Translations
import Mathlib.Data.Real.Irrational
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.Tactic.Basic
#align_import number_theory.diophantine_approximation from "leanpro... | Mathlib/NumberTheory/DiophantineApproximation.lean | 139 | 144 | theorem exists_nat_abs_mul_sub_round_le (ξ : ℝ) {n : ℕ} (n_pos : 0 < n) :
∃ k : ℕ, 0 < k ∧ k ≤ n ∧ |↑k * ξ - round (↑k * ξ)| ≤ 1 / (n + 1) := by |
obtain ⟨j, k, hk₀, hk₁, h⟩ := exists_int_int_abs_mul_sub_le ξ n_pos
have hk := toNat_of_nonneg hk₀.le
rw [← hk] at hk₀ hk₁ h
exact ⟨k.toNat, natCast_pos.mp hk₀, Nat.cast_le.mp hk₁, (round_le (↑k.toNat * ξ) j).trans h⟩
| 4 |
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysis.Calculus.FDeriv.Measurable
open MeasureTheory
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] [LocallyCompactSpace 𝕜]
{E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E] [MeasurableSpace E] [OpensMeasurableSpace E]
{F : Type*} [Norm... | Mathlib/Analysis/Calculus/LineDeriv/Measurable.lean | 47 | 52 | theorem stronglyMeasurable_lineDeriv [SecondCountableTopologyEither E F] (hf : Continuous f) :
StronglyMeasurable (fun x ↦ lineDeriv 𝕜 f x v) := by |
borelize 𝕜
let g : E → 𝕜 → F := fun x t ↦ f (x + t • v)
have hg : Continuous g.uncurry := by apply hf.comp; continuity
exact (stronglyMeasurable_deriv_with_param hg).comp_measurable measurable_prod_mk_right
| 4 |
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 | 129 | 134 | theorem traverse_map {G : Type* → Type _} [Applicative G] [CommApplicative G] {α β γ : Type _}
(g : α → β) (h : β → G γ) (x : Multiset α) : traverse h (map g x) = traverse (h ∘ g) x := by |
refine Quotient.inductionOn x ?_
intro
simp only [traverse, quot_mk_to_coe, map_coe, lift_coe, Function.comp_apply]
rw [← Traversable.traverse_map h g, List.map_eq_map]
| 4 |
import Mathlib.CategoryTheory.Sites.Coherent.SheafComparison
import Mathlib.CategoryTheory.Sites.Equivalence
namespace CategoryTheory
variable {C : Type*} [Category C]
open GrothendieckTopology
namespace Equivalence
variable {D : Type*} [Category D]
variable (e : C ≌ D)
section Regular
variable [Preregular C]
... | Mathlib/CategoryTheory/Sites/Coherent/Equivalence.lean | 101 | 106 | theorem preregular_isSheaf_iff (F : Cᵒᵖ ⥤ A) : haveI := e.preregular
IsSheaf (regularTopology C) F ↔ IsSheaf (regularTopology D) (e.inverse.op ⋙ F) := by |
refine ⟨fun hF ↦ ((e.sheafCongrPreregular A).functor.obj ⟨F, hF⟩).cond, fun hF ↦ ?_⟩
rw [isSheaf_of_iso_iff (P' := e.functor.op ⋙ e.inverse.op ⋙ F)]
· exact (e.sheafCongrPreregular A).inverse.obj ⟨e.inverse.op ⋙ F, hF⟩ |>.cond
· exact isoWhiskerRight e.op.unitIso F
| 4 |
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
#align_import data.set.intervals.monoid from "leanprover-community/mathlib"@"aba57d4d3dae35460225919dcd82fe91355162f9"
namespace Set
variable {M : Type*} ... | Mathlib/Algebra/Order/Interval/Set/Monoid.lean | 44 | 48 | theorem Icc_add_bij : BijOn (· + d) (Icc a b) (Icc (a + d) (b + d)) := by |
rw [← Ici_inter_Iic, ← Ici_inter_Iic]
exact
(Ici_add_bij a d).inter_mapsTo (fun x hx => add_le_add_right hx _) fun x hx =>
le_of_add_le_add_right hx.2
| 4 |
import Mathlib.Algebra.GCDMonoid.Multiset
import Mathlib.Combinatorics.Enumerative.Partition
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Cycle.Factors
import Mathlib.GroupTheory.Perm.Closure
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Tactic.NormNum.GCD
#align_import group_theory.perm.cycl... | Mathlib/GroupTheory/Perm/Cycle/Type.lean | 57 | 64 | theorem cycleType_eq' {σ : Perm α} (s : Finset (Perm α)) (h1 : ∀ f : Perm α, f ∈ s → f.IsCycle)
(h2 : (s : Set (Perm α)).Pairwise Disjoint)
(h0 : s.noncommProd id (h2.imp fun _ _ => Disjoint.commute) = σ) :
σ.cycleType = s.1.map (Finset.card ∘ support) := by |
rw [cycleType_def]
congr
rw [cycleFactorsFinset_eq_finset]
exact ⟨h1, h2, h0⟩
| 4 |
import Mathlib.NumberTheory.FLT.Basic
import Mathlib.NumberTheory.PythagoreanTriples
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.Tactic.LinearCombination
#align_import number_theory.fermat4 from "leanprover-community/mathlib"@"10b4e499f43088dd3bb7b5796184ad5216648ab1"
noncomputable section
open scope... | Mathlib/NumberTheory/FLT/Four.lean | 58 | 62 | theorem ne_zero {a b c : ℤ} (h : Fermat42 a b c) : c ≠ 0 := by |
apply ne_zero_pow two_ne_zero _; apply ne_of_gt
rw [← h.2.2, (by ring : a ^ 4 + b ^ 4 = (a ^ 2) ^ 2 + (b ^ 2) ^ 2)]
exact
add_pos (sq_pos_of_ne_zero (pow_ne_zero 2 h.1)) (sq_pos_of_ne_zero (pow_ne_zero 2 h.2.1))
| 4 |
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Shapes.Kernels
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Subobject.Lattice
import Mathlib.Order.Atoms
#align_import category_theory.simple from "leanprover-community/mathlib"@"4ed0bcaef698011... | Mathlib/CategoryTheory/Simple.lean | 84 | 89 | theorem kernel_zero_of_nonzero_from_simple {X Y : C} [Simple X] {f : X ⟶ Y} [HasKernel f]
(w : f ≠ 0) : kernel.ι f = 0 := by |
classical
by_contra h
haveI := isIso_of_mono_of_nonzero h
exact w (eq_zero_of_epi_kernel f)
| 4 |
import Mathlib.Data.Fintype.List
#align_import data.list.cycle from "leanprover-community/mathlib"@"7413128c3bcb3b0818e3e18720abc9ea3100fb49"
assert_not_exists MonoidWithZero
namespace List
variable {α : Type*} [DecidableEq α]
def nextOr : ∀ (_ : List α) (_ _ : α), α
| [], _, default => default
| [_], _, d... | Mathlib/Data/List/Cycle.lean | 87 | 91 | theorem nextOr_concat {xs : List α} {x : α} (d : α) (h : x ∉ xs) : nextOr (xs ++ [x]) x d = d := by |
induction' xs with z zs IH
· simp
· obtain ⟨hz, hzs⟩ := not_or.mp (mt mem_cons.2 h)
rw [cons_append, nextOr_cons_of_ne _ _ _ _ hz, IH hzs]
| 4 |
import Mathlib.Topology.Algebra.UniformConvergence
#align_import topology.algebra.module.strong_topology from "leanprover-community/mathlib"@"8905e5ed90859939681a725b00f6063e65096d95"
open scoped Topology UniformConvergence
section General
variable {𝕜₁ 𝕜₂ : Type*} [NormedField 𝕜₁] [NormedField 𝕜₂] (σ : 𝕜... | Mathlib/Topology/Algebra/Module/StrongTopology.lean | 152 | 157 | theorem t2Space [TopologicalSpace F] [TopologicalAddGroup F] [T2Space F]
(𝔖 : Set (Set E)) (h𝔖 : ⋃₀ 𝔖 = Set.univ) : T2Space (UniformConvergenceCLM σ F 𝔖) := by |
letI : UniformSpace F := TopologicalAddGroup.toUniformSpace F
haveI : UniformAddGroup F := comm_topologicalAddGroup_is_uniform
haveI : T2Space (E →ᵤ[𝔖] F) := UniformOnFun.t2Space_of_covering h𝔖
exact (embedding_coeFn σ F 𝔖).t2Space
| 4 |
import Mathlib.CategoryTheory.Generator
import Mathlib.CategoryTheory.Preadditive.Yoneda.Basic
#align_import category_theory.preadditive.generator from "leanprover-community/mathlib"@"09f981f72d43749f1fa072deade828d9c1e185bb"
universe v u
open CategoryTheory Opposite
namespace CategoryTheory
variable {C : Type... | Mathlib/CategoryTheory/Preadditive/Generator.lean | 69 | 74 | theorem isCoseparator_iff_faithful_preadditiveYoneda (G : C) :
IsCoseparator G ↔ (preadditiveYoneda.obj G).Faithful := by |
rw [isCoseparator_iff_faithful_yoneda_obj, ← whiskering_preadditiveYoneda, Functor.comp_obj,
whiskeringRight_obj_obj]
exact ⟨fun h => Functor.Faithful.of_comp _ (forget AddCommGroupCat),
fun h => Functor.Faithful.comp _ _⟩
| 4 |
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Data.Set.MemPartition
import Mathlib.Order.Filter.CountableSeparatingOn
open Set MeasureTheory
namespace MeasurableSpace
variable {α β : Type*}
class CountablyGenerated (α : Type*) [m : MeasurableSpace α] : Prop where
isCountablyGenerated : ∃ b... | Mathlib/MeasureTheory/MeasurableSpace/CountablyGenerated.lean | 157 | 163 | theorem separating_of_generateFrom (S : Set (Set α))
[h : @SeparatesPoints α (generateFrom S)] :
∀ x y : α, (∀ s ∈ S, x ∈ s ↔ y ∈ s) → x = y := by |
letI := generateFrom S
intros x y hxy
rw [← forall_generateFrom_mem_iff_mem_iff] at hxy
exact separatesPoints_def $ fun _ hs ↦ (hxy _ hs).mp
| 4 |
import Mathlib.Analysis.Analytic.Composition
import Mathlib.Analysis.Analytic.Constructions
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.Analysis.SpecialFunctions.Complex.LogDeriv
open Complex Set
open scoped Topology
variable {E : Type} [NormedAddCommGroup E] [NormedSpace ℂ E]
variable {f g : E →... | Mathlib/Analysis/SpecialFunctions/Complex/Analytic.lean | 40 | 44 | theorem analyticAt_clog (m : z ∈ slitPlane) : AnalyticAt ℂ log z := by |
rw [analyticAt_iff_eventually_differentiableAt]
filter_upwards [isOpen_slitPlane.eventually_mem m]
intro z m
exact differentiableAt_id.clog m
| 4 |
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Field.Rat
import Mathlib.GroupTheory.GroupAction.Group
import Mathlib.GroupTheory.GroupAction.Pi
#align_import algebra.module.basic from "leanprover-community/mathlib"@"30413fc89f202a090a54d78e540963ed3de0056e"
o... | Mathlib/Algebra/Module/Basic.lean | 49 | 55 | theorem map_inv_intCast_smul [AddCommGroup M] [AddCommGroup M₂] {F : Type*} [FunLike F M M₂]
[AddMonoidHomClass F M M₂] (f : F) (R S : Type*) [DivisionRing R] [DivisionRing S] [Module R M]
[Module S M₂] (z : ℤ) (x : M) : f ((z⁻¹ : R) • x) = (z⁻¹ : S) • f x := by |
obtain ⟨n, rfl | rfl⟩ := z.eq_nat_or_neg
· rw [Int.cast_natCast, Int.cast_natCast, map_inv_natCast_smul _ R S]
· simp_rw [Int.cast_neg, Int.cast_natCast, inv_neg, neg_smul, map_neg,
map_inv_natCast_smul _ R S]
| 4 |
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Range
#align_import data.list.nat_antidiagonal from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
open List Function Nat
namespace List
namespace Nat
def antidiagonal (n : ℕ) : List (ℕ × ℕ) :=
(range (n + 1)).map fun i ↦ (i,... | Mathlib/Data/List/NatAntidiagonal.lean | 68 | 73 | theorem antidiagonal_succ {n : ℕ} :
antidiagonal (n + 1) = (0, n + 1) :: (antidiagonal n).map (Prod.map Nat.succ id) := by |
simp only [antidiagonal, range_succ_eq_map, map_cons, true_and_iff, Nat.add_succ_sub_one,
Nat.add_zero, id, eq_self_iff_true, Nat.sub_zero, map_map, Prod.map_mk]
apply congr rfl (congr rfl _)
ext; simp
| 4 |
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 | 259 | 265 | theorem Monic.irreducible_iff_natDegree (hp : p.Monic) :
Irreducible p ↔
p ≠ 1 ∧ ∀ f g : R[X], f.Monic → g.Monic → f * g = p → f.natDegree = 0 ∨ g.natDegree = 0 := by |
by_cases hp1 : p = 1; · simp [hp1]
rw [irreducible_of_monic hp hp1, and_iff_right hp1]
refine forall₄_congr fun a b ha hb => ?_
rw [ha.natDegree_eq_zero_iff_eq_one, hb.natDegree_eq_zero_iff_eq_one]
| 4 |
import Mathlib.Order.Interval.Finset.Fin
#align_import data.fintype.fin from "leanprover-community/mathlib"@"759575657f189ccb424b990164c8b1fa9f55cdfe"
open Finset
open Fintype
namespace Fin
variable {α β : Type*} {n : ℕ}
theorem map_valEmbedding_univ : (Finset.univ : Finset (Fin n)).map Fin.valEmbedding = Iio ... | Mathlib/Data/Fintype/Fin.lean | 73 | 78 | theorem card_filter_univ_eq_vector_get_eq_count [DecidableEq α] (a : α) (v : Vector α n) :
(univ.filter fun i => a = v.get i).card = v.toList.count a := by |
induction' v with n x xs hxs
· simp
· simp_rw [card_filter_univ_succ', Vector.get_cons_zero, Vector.toList_cons, Function.comp,
Vector.get_cons_succ, hxs, List.count_cons, add_comm (ite (a = x) 1 0)]
| 4 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Degree.Lemmas
import Mathlib.Algebra.Polynomial.HasseDeriv
#align_import data.polynomial.taylor from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
namespace Polynomial
open Polynomial... | Mathlib/Algebra/Polynomial/Taylor.lean | 98 | 102 | theorem natDegree_taylor (p : R[X]) (r : R) : natDegree (taylor r p) = natDegree p := by |
refine map_natDegree_eq_natDegree _ ?_
nontriviality R
intro n c c0
simp [taylor_monomial, natDegree_C_mul_eq_of_mul_ne_zero, natDegree_pow_X_add_C, c0]
| 4 |
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.PEquiv
#align_import data.matrix.pequiv from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
namespace PEquiv
open Matrix
universe u v
variable {k l m n : Type*}
variable {α : Type v}
open Matrix
def toMatrix [DecidableEq n] [Zer... | Mathlib/Data/Matrix/PEquiv.lean | 109 | 114 | theorem toMatrix_trans [Fintype m] [DecidableEq m] [DecidableEq n] [Semiring α] (f : l ≃. m)
(g : m ≃. n) : ((f.trans g).toMatrix : Matrix l n α) = f.toMatrix * g.toMatrix := by |
ext i j
rw [mul_matrix_apply]
dsimp [toMatrix, PEquiv.trans]
cases f i <;> simp
| 4 |
import Mathlib.Data.Opposite
import Mathlib.Data.Set.Defs
#align_import data.set.opposite from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0e"
variable {α : Type*}
open Opposite
namespace Set
protected def op (s : Set α) : Set αᵒᵖ :=
unop ⁻¹' s
#align set.op Set.op
protected def u... | Mathlib/Data/Set/Opposite.lean | 76 | 80 | theorem singleton_op (x : α) : ({x} : Set α).op = {op x} := by |
ext
constructor
· apply unop_injective
· apply op_injective
| 4 |
import Mathlib.Order.CompleteLattice
import Mathlib.Data.Finset.Lattice
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Category.Preorder
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.FiniteLimits
#align_import category_theory.limi... | Mathlib/CategoryTheory/Limits/Lattice.lean | 122 | 128 | theorem prod_eq_inf [SemilatticeInf α] [OrderTop α] (x y : α) : Limits.prod x y = x ⊓ y :=
calc
Limits.prod x y = limit (pair x y) := rfl
_ = Finset.univ.inf (pair x y).obj := by | rw [finite_limit_eq_finset_univ_inf (pair.{u} x y)]
_ = x ⊓ (y ⊓ ⊤) := rfl
-- Note: finset.inf is realized as a fold, hence the definitional equality
_ = x ⊓ y := by rw [inf_top_eq]
| 4 |
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
open Order.PFilter
namespace Order
variable {P : Type*}
namespace Ideal
-- Porting note(#5171): this linter isn't ported yet.
-- @[nolint has_... | Mathlib/Order/PrimeIdeal.lean | 124 | 128 | theorem IsPrime.mem_or_mem (hI : IsPrime I) {x y : P} : x ⊓ y ∈ I → x ∈ I ∨ y ∈ I := by |
contrapose!
let F := hI.compl_filter.toPFilter
show x ∈ F ∧ y ∈ F → x ⊓ y ∈ F
exact fun h => inf_mem h.1 h.2
| 4 |
import Mathlib.Topology.Order.IsLUB
open Set Filter TopologicalSpace Topology Function
open OrderDual (toDual ofDual)
variable {α β γ : Type*}
section DenselyOrdered
variable [TopologicalSpace α] [LinearOrder α] [OrderTopology α] [DenselyOrdered α] {a b : α}
{s : Set α}
| Mathlib/Topology/Order/DenselyOrdered.lean | 25 | 29 | theorem closure_Ioi' {a : α} (h : (Ioi a).Nonempty) : closure (Ioi a) = Ici a := by |
apply Subset.antisymm
· exact closure_minimal Ioi_subset_Ici_self isClosed_Ici
· rw [← diff_subset_closure_iff, Ici_diff_Ioi_same, singleton_subset_iff]
exact isGLB_Ioi.mem_closure h
| 4 |
import Mathlib.Algebra.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Laurent
import Mathlib.LinearAlgebra.Matrix.Charpoly.Basic
import Mathlib.LinearAlgebra.Matrix.Reindex
import Mathlib.RingTheory.Polynomial.Nilpotent
#align_import linear_algebra.matrix.charpoly.coeff from "leanprover-community/mathlib"@"9745b... | Mathlib/LinearAlgebra/Matrix/Charpoly/Coeff.lean | 89 | 93 | theorem det_of_card_zero (h : Fintype.card n = 0) (M : Matrix n n R) : M.det = 1 := by |
rw [Fintype.card_eq_zero_iff] at h
suffices M = 1 by simp [this]
ext i
exact h.elim i
| 4 |
import Mathlib.MeasureTheory.Integral.SetToL1
#align_import measure_theory.integral.bochner from "leanprover-community/mathlib"@"48fb5b5280e7c81672afc9524185ae994553ebf4"
assert_not_exists Differentiable
noncomputable section
open scoped Topology NNReal ENNReal MeasureTheory
open Set Filter TopologicalSpace EN... | Mathlib/MeasureTheory/Integral/Bochner.lean | 282 | 286 | theorem posPart_sub_negPart (f : α →ₛ ℝ) : f.posPart - f.negPart = f := by |
simp only [posPart, negPart]
ext a
rw [coe_sub]
exact max_zero_sub_eq_self (f a)
| 4 |
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Ring.Nat
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.RingTheory.Fintype
import Mathlib.Tactic.IntervalCases
#align_import number_the... | Mathlib/NumberTheory/LucasLehmer.lean | 154 | 158 | theorem sZMod_eq_s (p' : ℕ) (i : ℕ) : sZMod (p' + 2) i = (s i : ZMod (2 ^ (p' + 2) - 1)) := by |
induction' i with i ih
· dsimp [s, sZMod]
norm_num
· push_cast [s, sZMod, ih]; rfl
| 4 |
import Mathlib.CategoryTheory.Closed.Cartesian
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
#align_import category_theory.closed.functor from "leanprover-community/mathlib"@"cea27692b3fdeb328a2ddba6aabf181754543184"
noncomputable secti... | Mathlib/CategoryTheory/Closed/Functor.lean | 91 | 97 | theorem coev_expComparison (A B : C) :
F.map ((exp.coev A).app B) ≫ (expComparison F A).app (A ⨯ B) =
(exp.coev _).app (F.obj B) ≫ (exp (F.obj A)).map (inv (prodComparison F A B)) := by |
convert unit_transferNatTrans _ _ (prodComparisonNatIso F A).inv B using 3
apply IsIso.inv_eq_of_hom_inv_id -- Porting note: was `ext`
dsimp
simp
| 4 |
import Mathlib.Tactic.FinCases
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Algebra.Field.IsField
#align_import ring_theory.ideal.basic from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
universe u v w
variable {α : Type u} {β : Type v}
open ... | Mathlib/RingTheory/Ideal/Basic.lean | 106 | 110 | theorem unit_mul_mem_iff_mem {x y : α} (hy : IsUnit y) : y * x ∈ I ↔ x ∈ I := by |
refine ⟨fun h => ?_, fun h => I.mul_mem_left y h⟩
obtain ⟨y', hy'⟩ := hy.exists_left_inv
have := I.mul_mem_left y' h
rwa [← mul_assoc, hy', one_mul] at this
| 4 |
import Mathlib.Data.List.Basic
open Function
open Nat hiding one_pos
assert_not_exists Set.range
namespace List
universe u v w
variable {ι : Type*} {α : Type u} {β : Type v} {γ : Type w} {l₁ l₂ : List α}
section InsertNth
variable {a : α}
@[simp]
theorem insertNth_zero (s : List α) (x : α) : insertNth 0 x s... | Mathlib/Data/List/InsertNth.lean | 122 | 127 | theorem length_le_length_insertNth (l : List α) (x : α) (n : ℕ) :
l.length ≤ (insertNth n x l).length := by |
rcases le_or_lt n l.length with hn | hn
· rw [length_insertNth _ _ hn]
exact (Nat.lt_succ_self _).le
· rw [insertNth_of_length_lt _ _ _ hn]
| 4 |
import Mathlib.Algebra.Star.Subalgebra
import Mathlib.Topology.Algebra.Algebra
import Mathlib.Topology.Algebra.Star
#align_import topology.algebra.star_subalgebra from "leanprover-community/mathlib"@"b7f5a77fa29ad9a3ccc484109b0d7534178e7ecd"
open scoped Classical
open Set TopologicalSpace
open scoped Classical
... | Mathlib/Topology/Algebra/StarSubalgebra.lean | 122 | 127 | theorem _root_.Subalgebra.topologicalClosure_star_comm (s : Subalgebra R A) :
(star s).topologicalClosure = star s.topologicalClosure := by |
suffices ∀ t : Subalgebra R A, (star t).topologicalClosure ≤ star t.topologicalClosure from
le_antisymm (this s) (by simpa only [star_star] using Subalgebra.star_mono (this (star s)))
exact fun t => (star t).topologicalClosure_minimal (Subalgebra.star_mono subset_closure)
(isClosed_closure.preimage continu... | 4 |
import Mathlib.Data.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
#align_import algebra.gcd_monoid.finset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
#align_import algebra.gcd_monoid.div from "leanprover-community/mathlib"@"b537794f8409bc9598febb79cd510b1df5f4539d"
variab... | Mathlib/Algebra/GCDMonoid/Finset.lean | 166 | 171 | theorem gcd_insert [DecidableEq β] {b : β} :
(insert b s : Finset β).gcd f = GCDMonoid.gcd (f b) (s.gcd f) := by |
by_cases h : b ∈ s
· rw [insert_eq_of_mem h,
(gcd_eq_right_iff (f b) (s.gcd f) (Multiset.normalize_gcd (s.1.map f))).2 (gcd_dvd h)]
apply fold_insert h
| 4 |
import Mathlib.CategoryTheory.Filtered.Basic
import Mathlib.Data.Set.Finite
import Mathlib.Data.Set.Subsingleton
import Mathlib.Topology.Category.TopCat.Limits.Konig
import Mathlib.Tactic.AdaptationNote
#align_import category_theory.cofiltered_system from "leanprover-community/mathlib"@"178a32653e369dce2da68dc6b2694e... | Mathlib/CategoryTheory/CofilteredSystem.lean | 166 | 171 | theorem eventualRange_eq_range_precomp (f : i ⟶ j) (g : j ⟶ k)
(h : F.eventualRange k = range (F.map g)) : F.eventualRange k = range (F.map <| f ≫ g) := by |
apply subset_antisymm
· apply iInter₂_subset
· rw [h, F.map_comp]
apply range_comp_subset_range
| 4 |
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Order.Partition.Finpartition
import Mathlib.Tactic.GCongr
import Mathlib.Tactic.NormNum
import Mathlib.Tactic.Positivity
import Mathlib.Tactic.Ring
#align_import combinatorics.simp... | Mathlib/Combinatorics/SimpleGraph/Density.lean | 85 | 90 | theorem interedges_disjoint_left {s s' : Finset α} (hs : Disjoint s s') (t : Finset β) :
Disjoint (interedges r s t) (interedges r s' t) := by |
rw [Finset.disjoint_left] at hs ⊢
intro _ hx hy
rw [mem_interedges_iff] at hx hy
exact hs hx.1 hy.1
| 4 |
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Covering.Besicovitch
import Mathlib.Tactic.AdaptationNote
#align_import measure_theory.covering.besicovitch_vector_space from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
universe u
open Metric Set Fini... | Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean | 153 | 157 | theorem multiplicity_le : multiplicity E ≤ 5 ^ finrank ℝ E := by |
apply csSup_le
· refine ⟨0, ⟨∅, by simp⟩⟩
· rintro _ ⟨s, ⟨rfl, h⟩⟩
exact Besicovitch.card_le_of_separated s h.1 h.2
| 4 |
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Nat.Lattice
import Mathlib.Data.Setoid.Partition
import Mathlib.Order.Antichain
#align_import combinatorics.simple_graph.coloring from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open ... | Mathlib/Combinatorics/SimpleGraph/Coloring.lean | 114 | 119 | theorem Coloring.card_colorClasses_le [Fintype α] [Fintype C.colorClasses] :
Fintype.card C.colorClasses ≤ Fintype.card α := by |
simp [colorClasses]
-- Porting note: brute force instance declaration `[Fintype (Setoid.classes (Setoid.ker C))]`
haveI : Fintype (Setoid.classes (Setoid.ker C)) := by assumption
convert Setoid.card_classes_ker_le C
| 4 |
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
#align_import data.complex.exponential_bounds from "leanprover-community/mathlib"@"402f8982dddc1864bd703da2d6e2ee304a866973"
namespace Real
open IsAbsoluteValue Finset CauSeq Complex
theorem exp_one_near_10 : |exp 1 - 224... | Mathlib/Data/Complex/ExponentialBounds.lean | 44 | 48 | theorem exp_neg_one_gt_d9 : 0.36787944116 < exp (-1) := by |
rw [exp_neg, lt_inv _ (exp_pos _)]
· refine lt_of_le_of_lt (sub_le_iff_le_add.1 (abs_sub_le_iff.1 exp_one_near_10).1) ?_
norm_num
· norm_num
| 4 |
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 | 75 | 80 | theorem surjOn_Ici_of_monotone_surjective (h_mono : Monotone f) (h_surj : Function.Surjective f)
(a : α) : SurjOn f (Ici a) (Ici (f a)) := by |
rw [← Ioi_union_left, ← Ioi_union_left]
exact
(surjOn_Ioi_of_monotone_surjective h_mono h_surj a).union_union
(@image_singleton _ _ f a ▸ surjOn_image _ _)
| 4 |
import Mathlib.RingTheory.Ideal.Operations
#align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74"
assert_not_exists Basis -- See `RingTheory.Ideal.Basis`
assert_not_exists Submodule.hasQuotient -- See `RingTheory.Ideal.QuotientOperations`
universe... | Mathlib/RingTheory/Ideal/Maps.lean | 90 | 95 | theorem map_le_comap_of_inv_on (g : G) (I : Ideal R) (hf : Set.LeftInvOn g f I) :
I.map f ≤ I.comap g := by |
refine Ideal.span_le.2 ?_
rintro x ⟨x, hx, rfl⟩
rw [SetLike.mem_coe, mem_comap, hf hx]
exact hx
| 4 |
import Mathlib.Algebra.IsPrimePow
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.Tactic.WLOG
#align_import set_theory.cardinal.divisibility from "leanprover-community/mathlib"@"ea050b44c0f9aba9d16a948c7cc7d2e7c8493567"
namespace Cardinal
open Cardinal
universe u
variable {a b : Cardinal.{u}} {n m : ℕ... | Mathlib/SetTheory/Cardinal/Divisibility.lean | 92 | 96 | theorem not_irreducible_of_aleph0_le (ha : ℵ₀ ≤ a) : ¬Irreducible a := by |
rw [irreducible_iff, not_and_or]
refine Or.inr fun h => ?_
simpa [mul_aleph0_eq ha, isUnit_iff, (one_lt_aleph0.trans_le ha).ne', one_lt_aleph0.ne'] using
h a ℵ₀
| 4 |
import Mathlib.Analysis.Normed.Group.Pointwise
import Mathlib.Analysis.NormedSpace.Real
#align_import analysis.normed_space.pointwise from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
open Metric Set
open Pointwise Topology
variable {𝕜 E : Type*}
variable [NormedField 𝕜]
sectio... | Mathlib/Analysis/NormedSpace/Pointwise.lean | 162 | 167 | theorem exists_dist_eq (x z : E) {a b : ℝ} (ha : 0 ≤ a) (hb : 0 ≤ b) (hab : a + b = 1) :
∃ y, dist x y = b * dist x z ∧ dist y z = a * dist x z := by |
use a • x + b • z
nth_rw 1 [← one_smul ℝ x]
nth_rw 4 [← one_smul ℝ z]
simp [dist_eq_norm, ← hab, add_smul, ← smul_sub, norm_smul_of_nonneg, ha, hb]
| 4 |
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}
open Finset
-- The namespace is here to distinguish fro... | Mathlib/Combinatorics/SetFamily/Compression/Down.lean | 264 | 268 | theorem mem_compression_of_insert_mem_compression (h : insert a s ∈ 𝓓 a 𝒜) : s ∈ 𝓓 a 𝒜 := by |
by_cases ha : a ∈ s
· rwa [insert_eq_of_mem ha] at h
· rw [← erase_insert ha]
exact erase_mem_compression_of_mem_compression h
| 4 |
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.Order.Monoid.OrderDual
import Mathlib.Algebra.Order.ZeroLEOne
import Mathlib.Data.Nat.Cast.Defs
import Mathlib.Order.WithBot
#align_import algebra.order.monoid.with_top ... | Mathlib/Algebra/Order/Monoid/WithTop.lean | 132 | 136 | theorem add_eq_top : a + b = ⊤ ↔ a = ⊤ ∨ b = ⊤ := by |
match a, b with
| ⊤, _ => simp
| _, ⊤ => simp
| (a : α), (b : α) => simp only [← coe_add, coe_ne_top, or_false]
| 4 |
import Mathlib.AlgebraicTopology.DoldKan.Homotopies
import Mathlib.Tactic.Ring
#align_import algebraic_topology.dold_kan.faces from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open CategoryTheory CategoryTheory.Limits CategoryTheory.Category
CategoryTheory.Preadditive CategoryTheor... | Mathlib/AlgebraicTopology/DoldKan/Faces.lean | 53 | 58 | theorem comp_δ_eq_zero {Y : C} {n : ℕ} {q : ℕ} {φ : Y ⟶ X _[n + 1]} (v : HigherFacesVanish q φ)
(j : Fin (n + 2)) (hj₁ : j ≠ 0) (hj₂ : n + 2 ≤ (j : ℕ) + q) : φ ≫ X.δ j = 0 := by |
obtain ⟨i, rfl⟩ := Fin.eq_succ_of_ne_zero hj₁
apply v i
simp only [Fin.val_succ] at hj₂
omega
| 4 |
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.NthRewrite
#align_import data.nat.gcd.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
namespace Nat
theorem gcd_greatest {a b d : ℕ} (hda : d ∣ a) (hdb : d ∣ b) (hd ... | Mathlib/Data/Nat/GCD/Basic.lean | 133 | 137 | theorem lcm_mul_left {m n k : ℕ} : (m * n).lcm (m * k) = m * n.lcm k := by |
apply dvd_antisymm
· exact lcm_dvd (mul_dvd_mul_left m (dvd_lcm_left n k)) (mul_dvd_mul_left m (dvd_lcm_right n k))
· have h : m ∣ lcm (m * n) (m * k) := (dvd_mul_right m n).trans (dvd_lcm_left (m * n) (m * k))
rw [← dvd_div_iff h, lcm_dvd_iff, dvd_div_iff h, dvd_div_iff h, ← lcm_dvd_iff]
| 4 |
import Mathlib.Init.Data.Ordering.Basic
import Mathlib.Order.Synonym
#align_import order.compare from "leanprover-community/mathlib"@"c4658a649d216f57e99621708b09dcb3dcccbd23"
variable {α β : Type*}
def cmpLE {α} [LE α] [@DecidableRel α (· ≤ ·)] (x y : α) : Ordering :=
if x ≤ y then if y ≤ x then Ordering.eq ... | Mathlib/Order/Compare.lean | 67 | 71 | theorem compares_swap [LT α] {a b : α} {o : Ordering} : o.swap.Compares a b ↔ o.Compares b a := by |
cases o
· exact Iff.rfl
· exact eq_comm
· exact Iff.rfl
| 4 |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.Deriv.Linear
import Mathlib.Analysis.Complex.Conformal
import Mathlib.Analysis.Calculus.Conformal.NormedSpace
#align_import analysis.complex.real_deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
se... | Mathlib/Analysis/Complex/RealDeriv.lean | 84 | 89 | theorem ContDiffAt.real_of_complex {n : ℕ∞} (h : ContDiffAt ℂ n e z) :
ContDiffAt ℝ n (fun x : ℝ => (e x).re) z := by |
have A : ContDiffAt ℝ n ((↑) : ℝ → ℂ) z := ofRealCLM.contDiff.contDiffAt
have B : ContDiffAt ℝ n e z := h.restrict_scalars ℝ
have C : ContDiffAt ℝ n re (e z) := reCLM.contDiff.contDiffAt
exact C.comp z (B.comp z A)
| 4 |
import Mathlib.LinearAlgebra.Quotient
#align_import linear_algebra.isomorphisms from "leanprover-community/mathlib"@"2738d2ca56cbc63be80c3bd48e9ed90ad94e947d"
universe u v
variable {R M M₂ M₃ : Type*}
variable [Ring R] [AddCommGroup M] [AddCommGroup M₂] [AddCommGroup M₃]
variable [Module R M] [Module R M₂] [Modul... | Mathlib/LinearAlgebra/Isomorphisms.lean | 88 | 93 | theorem quotientInfEquivSupQuotient_surjective (p p' : Submodule R M) :
Function.Surjective (quotientInfToSupQuotient p p') := by |
rw [← range_eq_top, quotientInfToSupQuotient, range_liftQ, eq_top_iff']
rintro ⟨x, hx⟩; rcases mem_sup.1 hx with ⟨y, hy, z, hz, rfl⟩
use ⟨y, hy⟩; apply (Submodule.Quotient.eq _).2
simp only [mem_comap, map_sub, coeSubtype, coe_inclusion, sub_add_cancel_left, neg_mem_iff, hz]
| 4 |
import Mathlib.Analysis.Normed.Group.Quotient
import Mathlib.Topology.Instances.AddCircle
#align_import analysis.normed.group.add_circle from "leanprover-community/mathlib"@"084f76e20c88eae536222583331abd9468b08e1c"
noncomputable section
open Set
open Int hiding mem_zmultiples_iff
open AddSubgroup
namespace A... | Mathlib/Analysis/Normed/Group/AddCircle.lean | 71 | 75 | theorem norm_neg_period (x : ℝ) : ‖(x : AddCircle (-p))‖ = ‖(x : AddCircle p)‖ := by |
suffices ‖(↑(-1 * x) : AddCircle (-1 * p))‖ = ‖(x : AddCircle p)‖ by
rw [← this, neg_one_mul]
simp
simp only [norm_coe_mul, abs_neg, abs_one, one_mul]
| 4 |
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Nat
import Mathlib.Init.Data.Nat.Lemmas
#align_import data.nat.psub from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
namespace Nat
def ppred : ℕ → Option ℕ
| 0 => none
| n + 1 => some n
#align nat.ppred Nat.ppred
@... | Mathlib/Data/Nat/PSub.lean | 118 | 122 | theorem psub'_eq_psub (m n) : psub' m n = psub m n := by |
rw [psub']
split_ifs with h
· exact (psub_eq_sub h).symm
· exact (psub_eq_none.2 (not_le.1 h)).symm
| 4 |
import Mathlib.Data.Nat.Choose.Central
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Multiplicity
#align_import data.nat.choose.factorization from "leanprover-community/mathlib"@"dc9db541168768af03fe228703e758e649afdbfc"
namespace Nat
variable {p n k : ℕ}
theorem factorization_choose_le_l... | Mathlib/Data/Nat/Choose/Factorization.lean | 106 | 110 | theorem factorization_choose_eq_zero_of_lt (h : n < p) : (choose n k).factorization p = 0 := by |
by_cases hnk : n < k; · simp [choose_eq_zero_of_lt hnk]
rw [choose_eq_factorial_div_factorial (le_of_not_lt hnk),
factorization_div (factorial_mul_factorial_dvd_factorial (le_of_not_lt hnk)), Finsupp.coe_tsub,
Pi.sub_apply, factorization_factorial_eq_zero_of_lt h, zero_tsub]
| 4 |
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Shapes.FiniteProducts
import Mathlib.Logic.Equiv.Fin
#align_import category_theory.limits.... | Mathlib/CategoryTheory/Limits/Constructions/FiniteProductsOfBinaryProducts.lean | 106 | 110 | theorem hasFiniteProducts_of_has_binary_and_terminal : HasFiniteProducts C := by |
refine ⟨fun n => ⟨fun K => ?_⟩⟩
letI := hasProduct_fin n fun n => K.obj ⟨n⟩
let that : (Discrete.functor fun n => K.obj ⟨n⟩) ≅ K := Discrete.natIso fun ⟨i⟩ => Iso.refl _
apply @hasLimitOfIso _ _ _ _ _ _ this that
| 4 |
import Mathlib.CategoryTheory.Limits.Constructions.Pullbacks
import Mathlib.CategoryTheory.Preadditive.Biproducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryTheory.Limits.Constructions.LimitsOfProductsAndEqualizers
import Math... | Mathlib/CategoryTheory/Abelian/Basic.lean | 147 | 152 | theorem imageMonoFactorisation_e' {X Y : C} (f : X ⟶ Y) :
(imageMonoFactorisation f).e = cokernel.π _ ≫ Abelian.coimageImageComparison f := by |
dsimp
ext
simp only [Abelian.coimageImageComparison, imageMonoFactorisation_e, Category.assoc,
cokernel.π_desc_assoc]
| 4 |
import Mathlib.Data.Set.Image
import Mathlib.Data.List.GetD
#align_import data.set.list from "leanprover-community/mathlib"@"2ec920d35348cb2d13ac0e1a2ad9df0fdf1a76b4"
open List
variable {α β : Type*} (l : List α)
namespace Set
theorem range_list_map (f : α → β) : range (map f) = { l | ∀ x ∈ l, x ∈ range f } :=... | Mathlib/Data/Set/List.lean | 44 | 48 | theorem range_list_get? : range l.get? = insert none (some '' { x | x ∈ l }) := by |
rw [← range_list_get, ← range_comp]
refine (range_subset_iff.2 fun n => ?_).antisymm (insert_subset_iff.2 ⟨?_, ?_⟩)
exacts [(le_or_lt l.length n).imp get?_eq_none.2 (fun hlt => ⟨⟨_, hlt⟩, (get?_eq_get hlt).symm⟩),
⟨_, get?_eq_none.2 le_rfl⟩, range_subset_iff.2 fun k => ⟨_, get?_eq_get _⟩]
| 4 |
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 | 87 | 92 | theorem map_eq_sum [Countable β] [MeasurableSingletonClass β] (μ : Measure α) (f : α → β)
(hf : Measurable f) : μ.map f = sum fun b : β => μ (f ⁻¹' {b}) • dirac b := by |
ext s
have : ∀ y ∈ s, MeasurableSet (f ⁻¹' {y}) := fun y _ => hf (measurableSet_singleton _)
simp [← tsum_measure_preimage_singleton (to_countable s) this, *,
tsum_subtype s fun b => μ (f ⁻¹' {b}), ← indicator_mul_right s fun b => μ (f ⁻¹' {b})]
| 4 |
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 | 98 | 103 | theorem exp_of_re_eq_zero (q : Quaternion ℝ) (hq : q.re = 0) :
exp ℝ q = ↑(Real.cos ‖q‖) + (Real.sin ‖q‖ / ‖q‖) • q := by |
rw [exp_eq_tsum]
refine HasSum.tsum_eq ?_
simp_rw [← expSeries_apply_eq]
exact hasSum_expSeries_of_imaginary hq (Real.hasSum_cos _) (Real.hasSum_sin _)
| 4 |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Set.Sigma
#align_import data.finset.sigma from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Multiset
variable {ι : Type*}
namespace Finset
section SigmaLift
variable {α β γ : ι → Type*} [DecidableEq ι]
def sigm... | Mathlib/Data/Finset/Sigma.lean | 211 | 216 | theorem sigmaLift_mono (h : ∀ ⦃i⦄ ⦃a : α i⦄ ⦃b : β i⦄, f a b ⊆ g a b) (a : Σi, α i) (b : Σi, β i) :
sigmaLift f a b ⊆ sigmaLift g a b := by |
rintro x hx
rw [mem_sigmaLift] at hx ⊢
obtain ⟨ha, hb, hx⟩ := hx
exact ⟨ha, hb, h hx⟩
| 4 |
import Mathlib.Data.ZMod.Quotient
#align_import group_theory.complement from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
open Set
open scoped Pointwise
namespace Subgroup
variable {G : Type*} [Group G] (H K : Subgroup G) (S T : Set G)
@[to_additive "`S` and `T` are complements if ... | Mathlib/GroupTheory/Complement.lean | 124 | 128 | theorem isComplement_singleton_left {g : G} : IsComplement {g} S ↔ S = univ := by |
refine
⟨fun h => top_le_iff.mp fun x _ => ?_, fun h => (congr_arg _ h).mpr isComplement_singleton_univ⟩
obtain ⟨⟨⟨z, rfl : z = g⟩, y, _⟩, hy⟩ := h.2 (g * x)
rwa [← mul_left_cancel hy]
| 4 |
import Mathlib.Geometry.Euclidean.Sphere.Basic
#align_import geometry.euclidean.sphere.second_inter from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open RealInnerProductSpace
namespace EuclideanGeometry
variable {V : Type*} {P : Type*} [NormedAddCommGroup V]... | Mathlib/Geometry/Euclidean/Sphere/SecondInter.lean | 44 | 49 | theorem Sphere.secondInter_dist (s : Sphere P) (p : P) (v : V) :
dist (s.secondInter p v) s.center = dist p s.center := by |
rw [Sphere.secondInter]
by_cases hv : v = 0; · simp [hv]
rw [dist_smul_vadd_eq_dist _ _ hv]
exact Or.inr rfl
| 4 |
import Mathlib.MeasureTheory.Measure.Typeclasses
open scoped ENNReal
namespace MeasureTheory
variable {α : Type*}
noncomputable
def Measure.trim {m m0 : MeasurableSpace α} (μ : @Measure α m0) (hm : m ≤ m0) : @Measure α m :=
@OuterMeasure.toMeasure α m μ.toOuterMeasure (hm.trans (le_toOuterMeasure_caratheodory... | Mathlib/MeasureTheory/Measure/Trim.lean | 124 | 128 | theorem sigmaFinite_trim_bot_iff : SigmaFinite (μ.trim bot_le) ↔ IsFiniteMeasure μ := by |
rw [sigmaFinite_bot_iff]
refine ⟨fun h => ⟨?_⟩, fun h => ⟨?_⟩⟩ <;> have h_univ := h.measure_univ_lt_top
· rwa [trim_measurableSet_eq bot_le MeasurableSet.univ] at h_univ
· rwa [trim_measurableSet_eq bot_le MeasurableSet.univ]
| 4 |
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.Valuation.ExtendToLocalization
import Mathlib.RingTheory.Valuation.ValuationSubring
import Mathlib.Topology.Algebra.ValuedField
import Mathlib.Algebra.Order.Group.TypeTags
#align_import ring_theory.dedekind_domain.adic_valuation from "leanprover... | Mathlib/RingTheory/DedekindDomain/AdicValuation.lean | 156 | 160 | theorem IntValuation.map_one' : v.intValuationDef 1 = 1 := by |
rw [v.intValuationDef_if_neg (zero_ne_one.symm : (1 : R) ≠ 0), Ideal.span_singleton_one, ←
Ideal.one_eq_top, Associates.mk_one, Associates.factors_one,
Associates.count_zero (by apply v.associates_irreducible), Int.ofNat_zero, neg_zero, ofAdd_zero,
WithZero.coe_one]
| 4 |
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 | 45 | 51 | theorem eq_lim_at_right_extendFrom_Ioo [TopologicalSpace α] [LinearOrder α] [DenselyOrdered α]
[OrderTopology α] [TopologicalSpace β] [T2Space β] {f : α → β} {a b : α} {lb : β} (hab : a < b)
(hb : Tendsto f (𝓝[<] b) (𝓝 lb)) : extendFrom (Ioo a b) f b = lb := by |
apply extendFrom_eq
· rw [closure_Ioo hab.ne]
simp only [le_of_lt hab, left_mem_Icc, right_mem_Icc]
· simpa [hab]
| 4 |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.List.Sublists
import Mathlib.Data.List.InsertNth
#align_import group_theory.free_group from "leanprover-community/mathlib"@"f93c11933efbc3c2f0299e47b8ff83e9b539cbf6"
open Relation
universe u v w
variable {α : Type u... | Mathlib/GroupTheory/FreeGroup/Basic.lean | 151 | 155 | theorem not_step_nil : ¬Step [] L := by |
generalize h' : [] = L'
intro h
cases' h with L₁ L₂
simp [List.nil_eq_append] at h'
| 4 |
import Mathlib.RingTheory.IntegrallyClosed
import Mathlib.RingTheory.Localization.NumDen
import Mathlib.RingTheory.Polynomial.ScaleRoots
#align_import ring_theory.polynomial.rational_root from "leanprover-community/mathlib"@"62c0a4ef1441edb463095ea02a06e87f3dfe135c"
open scoped Polynomial
section ScaleRoots
var... | Mathlib/RingTheory/Polynomial/RationalRoot.lean | 39 | 44 | theorem scaleRoots_aeval_eq_zero_of_aeval_mk'_eq_zero {p : A[X]} {r : A} {s : M}
(hr : aeval (mk' S r s) p = 0) : aeval (algebraMap A S r) (scaleRoots p s) = 0 := by |
convert scaleRoots_eval₂_eq_zero (algebraMap A S) hr
-- Porting note: added
funext
rw [aeval_def, mk'_spec' _ r s]
| 4 |
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Data.Nat.Totient
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.GroupTheory.Subgroup.Simple
import Mathlib.Tactic.Group
import Mathlib.GroupTheory.Exponent
#align_import group_theory.specific_groups.cyclic from "leanprover-community/mathli... | Mathlib/GroupTheory/SpecificGroups/Cyclic.lean | 136 | 141 | theorem Subgroup.eq_bot_or_eq_top_of_prime_card {G : Type*} [Group G] {_ : Fintype G}
(H : Subgroup G) [hp : Fact (Fintype.card G).Prime] : H = ⊥ ∨ H = ⊤ := by |
classical
have := card_subgroup_dvd_card H
rwa [Nat.card_eq_fintype_card (α := G), Nat.dvd_prime hp.1, ← Nat.card_eq_fintype_card,
← eq_bot_iff_card, card_eq_iff_eq_top] at this
| 4 |
import Mathlib.Algebra.Polynomial.Module.Basic
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.ReesAlgebra
import Mathlib.RingTheory.Finiteness
import Mathlib.Order.Basic
import Mathlib.Order.Hom.Lattice
#align_import rin... | Mathlib/RingTheory/Filtration.lean | 67 | 71 | theorem pow_smul_le (i j : ℕ) : I ^ i • F.N j ≤ F.N (i + j) := by |
induction' i with _ ih
· simp
· rw [pow_succ', mul_smul, add_assoc, add_comm 1, ← add_assoc]
exact (smul_mono_right _ ih).trans (F.smul_le _)
| 4 |
import Mathlib.Analysis.InnerProductSpace.Orientation
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import measure_theory.measure.haar.inner_product_space from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
open FiniteDimensional MeasureTheory MeasureTheory.Measure Set
var... | Mathlib/MeasureTheory/Measure/Haar/InnerProductSpace.lean | 84 | 89 | theorem OrthonormalBasis.measurePreserving_measurableEquiv (b : OrthonormalBasis ι ℝ F) :
MeasurePreserving b.measurableEquiv volume volume := by |
convert (b.measurableEquiv.symm.measurable.measurePreserving _).symm
rw [← (EuclideanSpace.basisFun ι ℝ).addHaar_eq_volume]
erw [MeasurableEquiv.coe_toEquiv_symm, Basis.map_addHaar _ b.repr.symm.toContinuousLinearEquiv]
exact b.addHaar_eq_volume.symm
| 4 |
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Even
import Mathlib.LinearAlgebra.QuadraticForm.Prod
import Mathlib.Tactic.LiftLets
#align_import linear_algebra.clifford_algebra.even_equiv from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36d... | Mathlib/LinearAlgebra/CliffordAlgebra/EvenEquiv.lean | 82 | 86 | theorem neg_e0_mul_v (m : M) : -(e0 Q * v Q m) = v Q m * e0 Q := by |
refine neg_eq_of_add_eq_zero_right ((ι_mul_ι_add_swap _ _).trans ?_)
dsimp [QuadraticForm.polar]
simp only [add_zero, mul_zero, mul_one, zero_add, neg_zero, QuadraticForm.map_zero,
add_sub_cancel_right, sub_self, map_zero, zero_sub]
| 4 |
import Mathlib.Data.ENNReal.Basic
import Mathlib.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.MetricSpace.Thickening
#align_import topology.metric_space.thickened_indicator from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open scoped Classical
open NNReal ENNReal Topol... | Mathlib/Topology/MetricSpace/ThickenedIndicator.lean | 110 | 115 | theorem indicator_le_thickenedIndicatorAux (δ : ℝ) (E : Set α) :
(E.indicator fun _ => (1 : ℝ≥0∞)) ≤ thickenedIndicatorAux δ E := by |
intro a
by_cases h : a ∈ E
· simp only [h, indicator_of_mem, thickenedIndicatorAux_one δ E h, le_refl]
· simp only [h, indicator_of_not_mem, not_false_iff, zero_le]
| 4 |
import Mathlib.Topology.Bases
import Mathlib.Topology.DenseEmbedding
#align_import topology.stone_cech from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
noncomputable section
open Filter Set
open Topology
universe u v
section Ultrafilter
def ultrafilterBasis (α : Type u) : Set ... | Mathlib/Topology/StoneCech.lean | 122 | 126 | theorem ultrafilter_pure_injective : Function.Injective (pure : α → Ultrafilter α) := by |
intro x y h
have : {x} ∈ (pure x : Ultrafilter α) := singleton_mem_pure
rw [h] at this
exact (mem_singleton_iff.mp (mem_pure.mp this)).symm
| 4 |
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