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.LinearAlgebra.TensorProduct.Graded.External
import Mathlib.RingTheory.GradedAlgebra.Basic
import Mathlib.GroupTheory.GroupAction.Ring
suppress_compilation
open scoped TensorProduct
variable {R ι A B : Type*}
variable [CommSemiring ι] [Module ι (Additive ℤˣ)] [DecidableEq ι]
variable [CommRing R] [R... | Mathlib/LinearAlgebra/TensorProduct/Graded/Internal.lean | 133 | 135 | theorem auxEquiv_one : auxEquiv R 𝒜 ℬ 1 = 1 := by |
rw [← of_one, Algebra.TensorProduct.one_def, auxEquiv_tmul 𝒜 ℬ, DirectSum.decompose_one,
DirectSum.decompose_one, Algebra.TensorProduct.one_def]
| 2 |
import Mathlib.LinearAlgebra.Dual
open Function Module
variable (R M N : Type*) [CommRing R] [AddCommGroup M] [Module R M] [AddCommGroup N] [Module R N]
structure PerfectPairing :=
toLin : M →ₗ[R] N →ₗ[R] R
bijectiveLeft : Bijective toLin
bijectiveRight : Bijective toLin.flip
attribute [nolint docBlame] P... | Mathlib/LinearAlgebra/PerfectPairing.lean | 102 | 105 | theorem toDualRight_symm_comp_toDualLeft :
p.toDualRight.symm.dualMap ∘ₗ (p.toDualLeft : M →ₗ[R] Dual R N) = Dual.eval R M := by |
ext1 x
exact p.toDualRight_symm_toDualLeft x
| 2 |
import Mathlib.NumberTheory.Cyclotomic.Discriminant
import Mathlib.RingTheory.Polynomial.Eisenstein.IsIntegral
import Mathlib.RingTheory.Ideal.Norm
#align_import number_theory.cyclotomic.rat from "leanprover-community/mathlib"@"b353176c24d96c23f0ce1cc63efc3f55019702d9"
universe u
open Algebra IsCyclotomicExtensio... | Mathlib/NumberTheory/Cyclotomic/Rat.lean | 65 | 69 | theorem discr_prime_pow_eq_unit_mul_pow' [IsCyclotomicExtension {p ^ k} ℚ K]
(hζ : IsPrimitiveRoot ζ ↑(p ^ k)) :
∃ (u : ℤˣ) (n : ℕ), discr ℚ (hζ.subOnePowerBasis ℚ).basis = u * p ^ n := by |
rw [hζ.discr_zeta_eq_discr_zeta_sub_one.symm]
exact discr_prime_pow_eq_unit_mul_pow hζ (cyclotomic.irreducible_rat (p ^ k).pos)
| 2 |
import Mathlib.Data.ULift
import Mathlib.Data.ZMod.Defs
import Mathlib.SetTheory.Cardinal.PartENat
#align_import set_theory.cardinal.finite from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
set_option autoImplicit true
open Cardinal Function
noncomputable section
variable {α β : Typ... | Mathlib/SetTheory/Cardinal/Finite.lean | 144 | 146 | theorem card_of_subsingleton (a : α) [Subsingleton α] : Nat.card α = 1 := by |
letI := Fintype.ofSubsingleton a
rw [card_eq_fintype_card, Fintype.card_ofSubsingleton a]
| 2 |
import Mathlib.Order.Bounds.Basic
import Mathlib.Order.WellFounded
import Mathlib.Data.Set.Image
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Data.Set.Lattice
#align_import order.conditionally_complete_lattice.basic from "leanprover-community/mathlib"@"29cb56a7b35f72758b05a30490e1f10bd62c35c1"
open Func... | Mathlib/Order/ConditionallyCompleteLattice/Basic.lean | 110 | 113 | theorem WithTop.coe_iInf [Nonempty ι] [InfSet α] {f : ι → α} (hf : BddBelow (range f)) :
↑(⨅ i, f i) = (⨅ i, f i : WithTop α) := by |
rw [iInf, iInf, WithTop.coe_sInf' (range_nonempty f) hf, ← range_comp]
rfl
| 2 |
import Mathlib.Algebra.Group.Hom.Defs
#align_import algebra.group.ext from "leanprover-community/mathlib"@"e574b1a4e891376b0ef974b926da39e05da12a06"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
open Function
universe u
@[to_additive (attr := ext)]
theorem Monoid.ext {M : Type u} ⦃m₁ m₂ : Mo... | Mathlib/Algebra/Group/Ext.lean | 103 | 106 | theorem CancelMonoid.toLeftCancelMonoid_injective {M : Type u} :
Function.Injective (@CancelMonoid.toLeftCancelMonoid M) := by |
rintro ⟨⟩ ⟨⟩ h
congr
| 2 |
import Mathlib.Topology.Basic
#align_import topology.nhds_set from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Filter Topology
variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] {f : Filter X}
{s t s₁ s₂ t₁ t₂ : Set X} {x : X}
theorem nhdsSet_diagonal (X) [T... | Mathlib/Topology/NhdsSet.lean | 56 | 58 | theorem disjoint_principal_nhdsSet : Disjoint (𝓟 s) (𝓝ˢ t) ↔ Disjoint (closure s) t := by |
rw [disjoint_principal_left, ← subset_interior_iff_mem_nhdsSet, interior_compl,
subset_compl_iff_disjoint_left]
| 2 |
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Convex.Uniform
import Mathlib.Analysis.NormedSpace.Completion
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
#align_import analysis.inner_product_space.basic from "leanprover-community/mathlib"@"3f655f5297b030... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 239 | 241 | theorem inner_smul_right (x y : F) {r : 𝕜} : ⟪x, r • y⟫ = r * ⟪x, y⟫ := by |
rw [← inner_conj_symm, inner_smul_left];
simp only [conj_conj, inner_conj_symm, RingHom.map_mul]
| 2 |
import Mathlib.CategoryTheory.Abelian.Basic
import Mathlib.CategoryTheory.Preadditive.FunctorCategory
import Mathlib.CategoryTheory.Limits.Shapes.FunctorCategory
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Kernels
#align_import category_theory.abelian.functor_category from "leanprover-community/mathlib"@"8a... | Mathlib/CategoryTheory/Abelian/FunctorCategory.lean | 79 | 83 | theorem coimageImageComparison_app' :
(coimageImageComparison α).app X =
(coimageObjIso α X).hom ≫ coimageImageComparison (α.app X) ≫ (imageObjIso α X).inv := by |
simp only [coimageImageComparison_app, Iso.hom_inv_id_assoc, Iso.hom_inv_id, Category.assoc,
Category.comp_id]
| 2 |
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.OrdConnected
#align_import data.set.intervals.proj_Icc from "leanprover-community/mathlib"@"4e24c4bfcff371c71f7ba22050308aa17815626c"
variable {α β : Type*} [LinearOrder α]
open Function
namespace Set
def projIci (a x : α) : Ici a := ⟨max a x,... | Mathlib/Order/Interval/Set/ProjIcc.lean | 132 | 134 | theorem projIcc_val (x : Icc a b) : projIcc a b h x = x := by |
cases x
apply projIcc_of_mem
| 2 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analysis.inner_product_space.calculus from "leanprover-community/mathlib"@"f9dd3204df14a0749cd456fac1e6849dfe7d2b88"
noncomputable section
open RCLike Real ... | Mathlib/Analysis/InnerProductSpace/Calculus.lean | 359 | 362 | theorem contDiffOn_euclidean {n : ℕ∞} :
ContDiffOn 𝕜 n f t ↔ ∀ i, ContDiffOn 𝕜 n (fun x => f x i) t := by |
rw [← (EuclideanSpace.equiv ι 𝕜).comp_contDiffOn_iff, contDiffOn_pi]
rfl
| 2 |
import Mathlib.Data.Complex.Module
import Mathlib.LinearAlgebra.Determinant
#align_import data.complex.determinant from "leanprover-community/mathlib"@"65ec59902eb17e4ab7da8d7e3d0bd9774d1b8b99"
namespace Complex
@[simp]
theorem det_conjAe : LinearMap.det conjAe.toLinearMap = -1 := by
rw [← LinearMap.det_toMat... | Mathlib/Data/Complex/Determinant.lean | 31 | 33 | theorem linearEquiv_det_conjAe : LinearEquiv.det conjAe.toLinearEquiv = -1 := by |
rw [← Units.eq_iff, LinearEquiv.coe_det, AlgEquiv.toLinearEquiv_toLinearMap, det_conjAe,
Units.coe_neg_one]
| 2 |
import Mathlib.AlgebraicTopology.SimplexCategory
import Mathlib.CategoryTheory.Comma.Arrow
import Mathlib.CategoryTheory.Limits.FunctorCategory
import Mathlib.CategoryTheory.Opposites
#align_import algebraic_topology.simplicial_object from "leanprover-community/mathlib"@"5ed51dc37c6b891b79314ee11a50adc2b1df6fd6"
o... | Mathlib/AlgebraicTopology/SimplicialObject.lean | 154 | 156 | theorem δ_comp_σ_self {n} {i : Fin (n + 1)} : X.σ i ≫ X.δ (Fin.castSucc i) = 𝟙 _ := by |
dsimp [δ, σ]
simp only [← X.map_comp, ← op_comp, SimplexCategory.δ_comp_σ_self, op_id, X.map_id]
| 2 |
import Mathlib.Data.Int.Bitwise
import Mathlib.Data.Int.Order.Lemmas
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.Basic
#align_import data.int.lemmas from "leanprover-community/mathlib"@"09597669f02422ed388036273d8848119699c22f"
open Nat
namespace Int
theorem le_natCast_sub (m n : ℕ) : (m ... | Mathlib/Data/Int/Lemmas.lean | 55 | 57 | theorem natAbs_le_iff_sq_le {a b : ℤ} : a.natAbs ≤ b.natAbs ↔ a ^ 2 ≤ b ^ 2 := by |
rw [sq, sq]
exact natAbs_le_iff_mul_self_le
| 2 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Fin
import Mathlib.GroupTheory.GroupAction.Pi
import Mathlib.Logic.Equiv.Fin
#align_import algebra.big_operators.fin from "leanprover-community/mathlib"@"cc5dd6244981976cc9da7afc4eee5682b037a013"
open Fins... | Mathlib/Algebra/BigOperators/Fin.lean | 143 | 146 | theorem prod_univ_five [CommMonoid β] (f : Fin 5 → β) :
∏ i, f i = f 0 * f 1 * f 2 * f 3 * f 4 := by |
rw [prod_univ_castSucc, prod_univ_four]
rfl
| 2 |
import Mathlib.Data.Finset.Pointwise
#align_import combinatorics.additive.e_transform from "leanprover-community/mathlib"@"207c92594599a06e7c134f8d00a030a83e6c7259"
open MulOpposite
open Pointwise
variable {α : Type*} [DecidableEq α]
namespace Finset
section Group
variable [Group α] (e : α) (x : Finset... | Mathlib/Combinatorics/Additive/ETransform.lean | 150 | 153 | theorem mulETransformRight.fst_mul_snd_subset :
(mulETransformRight e x).1 * (mulETransformRight e x).2 ⊆ x.1 * x.2 := by |
refine union_mul_inter_subset_union.trans (union_subset Subset.rfl ?_)
rw [op_smul_finset_mul_eq_mul_smul_finset, smul_inv_smul]
| 2 |
import Mathlib.Analysis.Convex.Gauge
import Mathlib.Analysis.Convex.Normed
open Metric Bornology Filter Set
open scoped NNReal Topology Pointwise
noncomputable section
section Module
variable {E : Type*} [AddCommGroup E] [Module ℝ E]
def gaugeRescale (s t : Set E) (x : E) : E := (gauge s x / gauge t x) • x
the... | Mathlib/Analysis/Convex/GaugeRescale.lean | 58 | 61 | theorem gauge_gaugeRescale' (s : Set E) {t : Set E} {x : E} (hx : gauge t x ≠ 0) :
gauge t (gaugeRescale s t x) = gauge s x := by |
rw [gaugeRescale, gauge_smul_of_nonneg (div_nonneg (gauge_nonneg _) (gauge_nonneg _)),
smul_eq_mul, div_mul_cancel₀ _ hx]
| 2 |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Topology.Order.ProjIcc
#align_import analysis.special_functions.trigonometric.inverse from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open Topology Filter
open S... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean | 45 | 47 | theorem range_arcsin : range arcsin = Icc (-(π / 2)) (π / 2) := by |
rw [arcsin, range_comp Subtype.val]
simp [Icc]
| 2 |
import Mathlib.Topology.Compactness.SigmaCompact
import Mathlib.Topology.Connected.TotallyDisconnected
import Mathlib.Topology.Inseparable
#align_import topology.separation from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c735d"
open Function Set Filter Topology TopologicalSpace
open scoped... | Mathlib/Topology/Separation.lean | 125 | 127 | theorem separatedNhds_iff_disjoint {s t : Set X} : SeparatedNhds s t ↔ Disjoint (𝓝ˢ s) (𝓝ˢ t) := by |
simp only [(hasBasis_nhdsSet s).disjoint_iff (hasBasis_nhdsSet t), SeparatedNhds, exists_prop, ←
exists_and_left, and_assoc, and_comm, and_left_comm]
| 2 |
import Mathlib.Algebra.EuclideanDomain.Defs
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Regular
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Basic
#align_import algebra.euclidean_domain.basic from "leanprover-community/mathlib"@"bf9bbbcf0c1c1ead18280b0d0... | Mathlib/Algebra/EuclideanDomain/Basic.lean | 136 | 138 | theorem gcd_zero_right (a : R) : gcd a 0 = a := by |
rw [gcd]
split_ifs with h <;> simp only [h, zero_mod, gcd_zero_left]
| 2 |
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 | 190 | 193 | theorem not_mem_sigmaLift_of_ne_right (f : ∀ ⦃i⦄, α i → β i → Finset (γ i)) {a : Sigma α}
(b : Sigma β) {x : Sigma γ} (h : b.1 ≠ x.1) : x ∉ sigmaLift f a b := by |
rw [mem_sigmaLift]
exact fun H => h H.snd.fst
| 2 |
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.LinearAlgebra.Matrix.Orthogonal
import Mathlib.Data.Matrix.Kronecker
#align_import linear_algebra.matrix.is_diag from "leanprover-community/mathlib"@"55e2dfde0cff928ce5c70926a3f2c7dee3e2dd99"
namespace Matrix
variable {α β R n m : Type*}
open Function... | Mathlib/LinearAlgebra/Matrix/IsDiag.lean | 76 | 79 | theorem IsDiag.map [Zero α] [Zero β] {A : Matrix n n α} (ha : A.IsDiag) {f : α → β} (hf : f 0 = 0) :
(A.map f).IsDiag := by |
intro i j h
simp [ha h, hf]
| 2 |
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.MvPolynomial.Rename
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp Ad... | Mathlib/Algebra/MvPolynomial/Degrees.lean | 95 | 98 | theorem degrees_monomial_eq (s : σ →₀ ℕ) (a : R) (ha : a ≠ 0) :
degrees (monomial s a) = toMultiset s := by |
classical
exact (supDegree_single s a).trans (if_neg ha)
| 2 |
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Data.Int.LeastGreatest
#align_import data.int.conditionally_complete_order from "leanprover-community/mathlib"@"1e05171a5e8cf18d98d9cf7b207540acb044acae"
open Int
noncomputable section
open scoped Classical
instance instConditionallyComplet... | Mathlib/Data/Int/ConditionallyCompleteOrder.lean | 94 | 96 | theorem csSup_mem {s : Set ℤ} (h1 : s.Nonempty) (h2 : BddAbove s) : sSup s ∈ s := by |
convert (greatestOfBdd _ (Classical.choose_spec h2) h1).2.1
exact dif_pos ⟨h1, h2⟩
| 2 |
import Mathlib.GroupTheory.Abelianization
import Mathlib.GroupTheory.Exponent
import Mathlib.GroupTheory.Transfer
#align_import group_theory.schreier from "leanprover-community/mathlib"@"8350c34a64b9bc3fc64335df8006bffcadc7baa6"
open scoped Pointwise
namespace Subgroup
open MemRightTransversals
variable {G : T... | Mathlib/GroupTheory/Schreier.lean | 95 | 100 | theorem closure_mul_image_eq_top' [DecidableEq G] {R S : Finset G}
(hR : (R : Set G) ∈ rightTransversals (H : Set G)) (hR1 : (1 : G) ∈ R)
(hS : closure (S : Set G) = ⊤) :
closure (((R * S).image fun g => ⟨_, mul_inv_toFun_mem hR g⟩ : Finset H) : Set H) = ⊤ := by |
rw [Finset.coe_image, Finset.coe_mul]
exact closure_mul_image_eq_top hR hR1 hS
| 2 |
import Mathlib.Topology.Category.LightProfinite.Basic
import Mathlib.Topology.Category.Profinite.Limits
namespace LightProfinite
universe u w
attribute [local instance] CategoryTheory.ConcreteCategory.instFunLike
open CategoryTheory Limits
section Pullbacks
variable {X Y B : LightProfinite.{u}} (f : X ⟶ B) (g ... | Mathlib/Topology/Category/LightProfinite/Limits.lean | 123 | 126 | theorem pullback_fst_eq :
LightProfinite.pullback.fst f g = (pullbackIsoPullback f g).hom ≫ Limits.pullback.fst := by |
dsimp [pullbackIsoPullback]
simp only [Limits.limit.conePointUniqueUpToIso_hom_comp, pullback.cone_pt, pullback.cone_π]
| 2 |
import Mathlib.Data.Fin.VecNotation
import Mathlib.Logic.Embedding.Set
#align_import logic.equiv.fin from "leanprover-community/mathlib"@"bd835ef554f37ef9b804f0903089211f89cb370b"
assert_not_exists MonoidWithZero
universe u
variable {m n : ℕ}
def finZeroEquiv : Fin 0 ≃ Empty :=
Equiv.equivEmpty _
#align fin_... | Mathlib/Logic/Equiv/Fin.lean | 56 | 60 | theorem Fin.preimage_apply_01_prod {α : Fin 2 → Type u} (s : Set (α 0)) (t : Set (α 1)) :
(fun f : ∀ i, α i => (f 0, f 1)) ⁻¹' s ×ˢ t =
Set.pi Set.univ (Fin.cons s <| Fin.cons t finZeroElim) := by |
ext f
simp [Fin.forall_fin_two]
| 2 |
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Polynomial.Degree.Lemmas
#align_import data.polynomial.erase_lead from "leanprover-community/mathlib"@"fa256f00ce018e7b40e1dc756e403c86680bf448"
noncomputable section
open Polynomial
open Polynomial Finset
namespace Polynomial
variable {R : Type*}... | Mathlib/Algebra/Polynomial/EraseLead.lean | 95 | 98 | theorem lt_natDegree_of_mem_eraseLead_support {a : ℕ} (h : a ∈ (eraseLead f).support) :
a < f.natDegree := by |
rw [eraseLead_support, mem_erase] at h
exact (le_natDegree_of_mem_supp a h.2).lt_of_ne h.1
| 2 |
import Mathlib.Control.Functor.Multivariate
import Mathlib.Data.PFunctor.Univariate.Basic
#align_import data.pfunctor.multivariate.basic from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f78155c6c27d62a621f4c152d"
universe u v
open MvFunctor
@[pp_with_univ]
structure MvPFunctor (n : ℕ) where
A : Type u
... | Mathlib/Data/PFunctor/Multivariate/Basic.lean | 106 | 108 | theorem const.get_map (f : α ⟹ β) (x : const n A α) : const.get (f <$$> x) = const.get x := by |
cases x
rfl
| 2 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Ring.Pi
import Mathlib.GroupTheory.GroupAction.Pi
#align_import algebra.big_operators.pi from "leanprover-community/mathlib"@"fa2309577c7009ea243cffdf990cd6c84f0ad497"
@[to_additive (attr := simp)]
theorem Finset.prod_apply {α : Type*} {β : α... | Mathlib/Algebra/BigOperators/Pi.lean | 69 | 72 | theorem pi_eq_sum_univ {ι : Type*} [Fintype ι] [DecidableEq ι] {R : Type*} [Semiring R]
(x : ι → R) : x = ∑ i, (x i) • fun j => if i = j then (1 : R) else 0 := by |
ext
simp
| 2 |
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 | 58 | 61 | theorem ghostComponent_zero_verschiebungFun (x : 𝕎 R) :
ghostComponent 0 (verschiebungFun x) = 0 := by |
rw [ghostComponent_apply, aeval_wittPolynomial, Finset.range_one, Finset.sum_singleton,
verschiebungFun_coeff_zero, pow_zero, pow_zero, pow_one, one_mul]
| 2 |
import Mathlib.Algebra.GroupWithZero.Hom
import Mathlib.Algebra.GroupWithZero.Units.Basic
import Mathlib.Algebra.Ring.Defs
import Mathlib.Data.Nat.Lattice
#align_import ring_theory.nilpotent from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
universe u v
open Function Set
variable {R ... | Mathlib/RingTheory/Nilpotent/Defs.lean | 81 | 85 | theorem IsNilpotent.map [MonoidWithZero R] [MonoidWithZero S] {r : R} {F : Type*}
[FunLike F R S] [MonoidWithZeroHomClass F R S] (hr : IsNilpotent r) (f : F) :
IsNilpotent (f r) := by |
use hr.choose
rw [← map_pow, hr.choose_spec, map_zero]
| 2 |
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Zip
import Mathlib.Data.Nat.Defs
import Mathlib.Data.List.Infix
#align_import data.list.rotate from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
universe u
variable {α : Type u}
open Nat Function
namespace List
theorem rotate... | Mathlib/Data/List/Rotate.lean | 93 | 100 | theorem rotate'_length_mul (l : List α) : ∀ n : ℕ, l.rotate' (l.length * n) = l
| 0 => by simp
| n + 1 =>
calc
l.rotate' (l.length * (n + 1)) =
(l.rotate' (l.length * n)).rotate' (l.rotate' (l.length * n)).length := by |
simp [-rotate'_length, Nat.mul_succ, rotate'_rotate']
_ = l := by rw [rotate'_length, rotate'_length_mul l n]
| 2 |
import Mathlib.Data.Int.Range
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.MulChar.Basic
#align_import number_theory.legendre_symbol.zmod_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace ZMod
section QuadCharModP
@[simps]
def χ₄ : MulChar (ZMod 4) ℤ... | Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean | 125 | 128 | theorem neg_one_pow_div_two_of_three_mod_four {n : ℕ} (hn : n % 4 = 3) :
(-1 : ℤ) ^ (n / 2) = -1 := by |
rw [← χ₄_eq_neg_one_pow (Nat.odd_of_mod_four_eq_three hn), ← natCast_mod, hn]
rfl
| 2 |
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]
| Mathlib/Logic/Encodable/Lattice.lean | 30 | 33 | theorem iSup_decode₂ [CompleteLattice α] (f : β → α) :
⨆ (i : ℕ) (b ∈ decode₂ β i), f b = (⨆ b, f b) := by |
rw [iSup_comm]
simp only [mem_decode₂, iSup_iSup_eq_right]
| 2 |
import Mathlib.CategoryTheory.Comma.Basic
#align_import category_theory.arrow from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
namespace CategoryTheory
universe v u
-- morphism levels before object levels. See note [CategoryTheory universes].
variable {T : Type u} [Category.{v} T]
... | Mathlib/CategoryTheory/Comma/Arrow.lean | 86 | 88 | theorem mk_eq (f : Arrow T) : Arrow.mk f.hom = f := by |
cases f
rfl
| 2 |
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis.LocallyConvex.Bounded
import Mathlib.Analysis.RCLike.Basic
#align_import analysis.convex.gauge from "leanprover-community/mathlib"@"373b03b5b9d0486534edbe94747f23cb3712f93d"
... | Mathlib/Analysis/Convex/Gauge.lean | 129 | 131 | theorem gauge_neg (symmetric : ∀ x ∈ s, -x ∈ s) (x : E) : gauge s (-x) = gauge s x := by |
have : ∀ x, -x ∈ s ↔ x ∈ s := fun x => ⟨fun h => by simpa using symmetric _ h, symmetric x⟩
simp_rw [gauge_def', smul_neg, this]
| 2 |
import Mathlib.Algebra.Algebra.Bilinear
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.Algebra.Algebra.Opposite
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Module.Opposites
import Mathlib.Algebra.Module.Submodule.Bilinear
import Mathlib.Algebra.Module.Submodule.Pointwise
import Mat... | Mathlib/Algebra/Algebra/Operations.lean | 88 | 90 | theorem le_one_toAddSubmonoid : 1 ≤ (1 : Submodule R A).toAddSubmonoid := by |
rintro x ⟨n, rfl⟩
exact ⟨n, map_natCast (algebraMap R A) n⟩
| 2 |
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Ideal.Operations
namespace Submodule
open Pointwise
variable {R M M' F G : Type*} [CommRing R] [AddCommGroup M] [Module R M]
variable {N N₁ N₂ P P₁ P₂ : Submodule R M}
def colon (N P : Submodule R M) : Ideal R :=
annihilator (P.map N.mkQ)
#align ... | Mathlib/RingTheory/Ideal/Colon.lean | 40 | 42 | theorem colon_top {I : Ideal R} : I.colon ⊤ = I := by |
simp_rw [SetLike.ext_iff, mem_colon, smul_eq_mul]
exact fun x ↦ ⟨fun h ↦ mul_one x ▸ h 1 trivial, fun h _ _ ↦ I.mul_mem_right _ h⟩
| 2 |
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
import Mathlib.Algebra.Star.Unitary
#align_import linear_algebra.unitary_group from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062a169c9"
universe u ... | Mathlib/LinearAlgebra/UnitaryGroup.lean | 71 | 73 | theorem mem_unitaryGroup_iff' : A ∈ Matrix.unitaryGroup n α ↔ star A * A = 1 := by |
refine ⟨And.left, fun hA => ⟨hA, ?_⟩⟩
rwa [mul_eq_one_comm] at hA
| 2 |
import Mathlib.Topology.Order
import Mathlib.Topology.Sets.Opens
import Mathlib.Topology.ContinuousFunction.Basic
#align_import topology.continuous_function.t0_sierpinski from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
noncomputable section
namespace TopologicalSpace
theorem eq_in... | Mathlib/Topology/ContinuousFunction/T0Sierpinski.lean | 50 | 52 | theorem productOfMemOpens_inducing : Inducing (productOfMemOpens X) := by |
convert inducing_iInf_to_pi fun (u : Opens X) (x : X) => x ∈ u
apply eq_induced_by_maps_to_sierpinski
| 2 |
import Mathlib.SetTheory.Cardinal.ENat
#align_import set_theory.cardinal.basic from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
universe u v
open Function Set
namespace Cardinal
variable {α : Type u} {c d : Cardinal.{u}}
noncomputable def toNat : Cardinal →*₀ ℕ :=
ENat.toNat.com... | Mathlib/SetTheory/Cardinal/ToNat.lean | 47 | 49 | theorem cast_toNat_of_lt_aleph0 {c : Cardinal} (h : c < ℵ₀) : ↑(toNat c) = c := by |
lift c to ℕ using h
rw [toNat_natCast]
| 2 |
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Shapes.Terminal
import Mathlib.CategoryTheory.Subobject.MonoOver
#align_import category_theory.subterminal from "leanprover-community/mathlib"@"bb103f356534a9a7d3596a672097e375290a4c3a"
universe v₁ v₂ u₁ u₂
noncomput... | Mathlib/CategoryTheory/Subterminal.lean | 107 | 110 | theorem isSubterminal_of_isIso_diag [HasBinaryProduct A A] [IsIso (diag A)] : IsSubterminal A :=
fun Z f g => by
have : (Limits.prod.fst : A ⨯ A ⟶ _) = Limits.prod.snd := by | simp [← cancel_epi (diag A)]
rw [← prod.lift_fst f g, this, prod.lift_snd]
| 2 |
import Mathlib.CategoryTheory.Subobject.Limits
#align_import algebra.homology.image_to_kernel from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a148bf9ff"
universe v u w
open CategoryTheory CategoryTheory.Limits
variable {ι : Type*}
variable {V : Type u} [Category.{v} V] [HasZeroMorphisms V]
o... | Mathlib/Algebra/Homology/ImageToKernel.lean | 112 | 116 | theorem imageToKernel_comp_right {D : V} (h : C ⟶ D) (w : f ≫ g = 0) :
imageToKernel f (g ≫ h) (by simp [reassoc_of% w]) =
imageToKernel f g w ≫ Subobject.ofLE _ _ (kernelSubobject_comp_le g h) := by |
ext
simp
| 2 |
import Mathlib.Geometry.Manifold.ContMDiff.Basic
open Set ChartedSpace SmoothManifoldWithCorners
open scoped Manifold
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
-- declare a smooth manifold `M` over the pair `(E, H)`.
{E : Type*}
[NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type*} [TopologicalSpace... | Mathlib/Geometry/Manifold/ContMDiff/Atlas.lean | 113 | 116 | theorem contMDiffOn_extChartAt_symm (x : M) :
ContMDiffOn 𝓘(𝕜, E) I n (extChartAt I x).symm (extChartAt I x).target := by |
convert contMDiffOn_extend_symm (chart_mem_maximalAtlas I x)
rw [extChartAt_target, I.image_eq]
| 2 |
import Mathlib.Topology.Connected.Basic
open Set Topology
universe u v
variable {α : Type u} {β : Type v} {ι : Type*} {π : ι → Type*} [TopologicalSpace α]
{s t u v : Set α}
section LocallyConnectedSpace
class LocallyConnectedSpace (α : Type*) [TopologicalSpace α] : Prop where
open_connected_basis : ∀ x,... | Mathlib/Topology/Connected/LocallyConnected.lean | 118 | 122 | theorem locallyConnectedSpace_iff_connected_basis :
LocallyConnectedSpace α ↔
∀ x, (𝓝 x).HasBasis (fun s : Set α => s ∈ 𝓝 x ∧ IsPreconnected s) id := by |
rw [locallyConnectedSpace_iff_connected_subsets]
exact forall_congr' fun x => Filter.hasBasis_self.symm
| 2 |
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Set.Subsingleton
#align_import combinatorics.double_counting from "leanprover-community/mathlib"@"1126441d6bccf98c81214a0780c73d499f6721fe"
open Finset Function Relator
variable {α β : Type*}
namespace Finset
section Bipartite
varia... | Mathlib/Combinatorics/Enumerative/DoubleCounting.lean | 79 | 82 | theorem sum_card_bipartiteAbove_eq_sum_card_bipartiteBelow [∀ a b, Decidable (r a b)] :
(∑ a ∈ s, (t.bipartiteAbove r a).card) = ∑ b ∈ t, (s.bipartiteBelow r b).card := by |
simp_rw [card_eq_sum_ones, bipartiteAbove, bipartiteBelow, sum_filter]
exact sum_comm
| 2 |
import Mathlib.Algebra.BigOperators.Group.List
import Mathlib.Algebra.Group.Prod
import Mathlib.Data.Multiset.Basic
#align_import algebra.big_operators.multiset.basic from "leanprover-community/mathlib"@"6c5f73fd6f6cc83122788a80a27cdd54663609f4"
assert_not_exists MonoidWithZero
variable {F ι α β γ : Type*}
names... | Mathlib/Algebra/BigOperators/Group/Multiset.lean | 91 | 94 | theorem prod_map_erase [DecidableEq ι] {a : ι} (h : a ∈ m) :
f a * ((m.erase a).map f).prod = (m.map f).prod := by |
rw [← m.coe_toList, coe_erase, map_coe, map_coe, prod_coe, prod_coe,
List.prod_map_erase f (mem_toList.2 h)]
| 2 |
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 | 165 | 167 | theorem sqrt_two_inv_mul_self : (√2)⁻¹ * (√2)⁻¹ = (2⁻¹ : ℝ) := by |
rw [← mul_inv]
norm_num
| 2 |
import Mathlib.LinearAlgebra.DFinsupp
import Mathlib.LinearAlgebra.StdBasis
#align_import linear_algebra.finsupp_vector_space from "leanprover-community/mathlib"@"59628387770d82eb6f6dd7b7107308aa2509ec95"
noncomputable section
open Set LinearMap Submodule
open scoped Cardinal
universe u v w
namespace Finsupp
... | Mathlib/LinearAlgebra/FinsuppVectorSpace.lean | 167 | 170 | theorem equivFun_symm_stdBasis [Finite n] (b : Basis n R M) (i : n) :
b.equivFun.symm (LinearMap.stdBasis R (fun _ => R) i 1) = b i := by |
cases nonempty_fintype n
simp
| 2 |
import Mathlib.Data.List.Infix
#align_import data.list.rdrop from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
-- Make sure we don't import algebra
assert_not_exists Monoid
variable {α : Type*} (p : α → Bool) (l : List α) (n : ℕ)
namespace List
def rdrop : List α :=
l.take (l.leng... | Mathlib/Data/List/DropRight.lean | 131 | 133 | theorem rdropWhile_prefix : l.rdropWhile p <+: l := by |
rw [← reverse_suffix, rdropWhile, reverse_reverse]
exact dropWhile_suffix _
| 2 |
import Mathlib.Algebra.BigOperators.Option
import Mathlib.Analysis.BoxIntegral.Box.Basic
import Mathlib.Data.Set.Pairwise.Lattice
#align_import analysis.box_integral.partition.basic from "leanprover-community/mathlib"@"84dc0bd6619acaea625086d6f53cb35cdd554219"
open Set Finset Function
open scoped Classical
open ... | Mathlib/Analysis/BoxIntegral/Partition/Basic.lean | 108 | 110 | theorem injective_boxes : Function.Injective (boxes : Prepartition I → Finset (Box ι)) := by |
rintro ⟨s₁, h₁, h₁'⟩ ⟨s₂, h₂, h₂'⟩ (rfl : s₁ = s₂)
rfl
| 2 |
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 | 72 | 74 | theorem hermite_one : hermite 1 = X := by |
rw [hermite_succ, hermite_zero]
simp only [map_one, mul_one, derivative_one, sub_zero]
| 2 |
import Mathlib.Geometry.Manifold.ContMDiff.Defs
open Set Filter Function
open scoped Topology Manifold
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
-- declare a smooth manifold `M` over the pair `(E, H)`.
{E : Type*}
[NormedAddCommGroup E] [NormedSpace 𝕜 E] {H : Type*} [TopologicalSpace H]
(I : Mode... | Mathlib/Geometry/Manifold/ContMDiff/Basic.lean | 119 | 122 | theorem ContMDiff.comp {g : M' → M''} (hg : ContMDiff I' I'' n g) (hf : ContMDiff I I' n f) :
ContMDiff I I'' n (g ∘ f) := by |
rw [← contMDiffOn_univ] at hf hg ⊢
exact hg.comp hf subset_preimage_univ
| 2 |
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analysis.inner_product_space.calculus from "leanprover-community/mathlib"@"f9dd3204df14a0749cd456fac1e6849dfe7d2b88"
noncomputable section
open RCLike Real ... | Mathlib/Analysis/InnerProductSpace/Calculus.lean | 328 | 330 | theorem differentiable_euclidean : Differentiable 𝕜 f ↔ ∀ i, Differentiable 𝕜 fun x => f x i := by |
rw [← (EuclideanSpace.equiv ι 𝕜).comp_differentiable_iff, differentiable_pi]
rfl
| 2 |
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 | 126 | 129 | theorem nonMemberSubfamily_memberSubfamily :
(𝒜.memberSubfamily a).nonMemberSubfamily a = 𝒜.memberSubfamily a := by |
ext
simp
| 2 |
import Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion
import Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated
import Mathlib.AlgebraicGeometry.Pullbacks
import Mathlib.CategoryTheory.MorphismProperty.Limits
noncomputable section
open CategoryTheory CategoryTheory.Limits Opposite TopologicalSpace
universe ... | Mathlib/AlgebraicGeometry/Morphisms/Separated.lean | 49 | 52 | theorem isSeparated_eq_diagonal_isClosedImmersion :
@IsSeparated = MorphismProperty.diagonal @IsClosedImmersion := by |
ext
exact isSeparated_iff _
| 2 |
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.add_torsor from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
class AddTorsor (G : outParam Type*) (P : Type*) [AddGroup G] extends AddAction G P,
VSub G P where
[nonempty : Nonempty P]
vsub_vadd' : ∀ p₁ p₂ : P, (p₁ ... | Mathlib/Algebra/AddTorsor.lean | 154 | 156 | theorem neg_vsub_eq_vsub_rev (p₁ p₂ : P) : -(p₁ -ᵥ p₂) = p₂ -ᵥ p₁ := by |
refine neg_eq_of_add_eq_zero_right (vadd_right_cancel p₁ ?_)
rw [vsub_add_vsub_cancel, vsub_self]
| 2 |
import Mathlib.Topology.Basic
#align_import topology.nhds_set from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Filter Topology
variable {X Y : Type*} [TopologicalSpace X] [TopologicalSpace Y] {f : Filter X}
{s t s₁ s₂ t₁ t₂ : Set X} {x : X}
| Mathlib/Topology/NhdsSet.lean | 35 | 38 | theorem nhdsSet_diagonal (X) [TopologicalSpace (X × X)] :
𝓝ˢ (diagonal X) = ⨆ (x : X), 𝓝 (x, x) := by |
rw [nhdsSet, ← range_diag, ← range_comp]
rfl
| 2 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Combinatorics.SimpleGraph.Dart
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Data.ZMod.Parity
#align_import combinatorics.simple_graph.degree_sum from "leanprover-community/mathlib"@"90659cbe25e59ec302e2fb92b00e9732160cc620"
open Finset
nam... | Mathlib/Combinatorics/SimpleGraph/DegreeSum.lean | 67 | 70 | theorem dart_fst_fiber_card_eq_degree [DecidableEq V] (v : V) :
(univ.filter fun d : G.Dart => d.fst = v).card = G.degree v := by |
simpa only [dart_fst_fiber, Finset.card_univ, card_neighborSet_eq_degree] using
card_image_of_injective univ (G.dartOfNeighborSet_injective v)
| 2 |
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u
variable (R : Type u)
section Semiring
variable [Semiring R]
class inductive Ex... | Mathlib/Algebra/CharP/ExpChar.lean | 148 | 150 | theorem expChar_pos (q : ℕ) [ExpChar R q] : 0 < q := by |
rcases expChar_is_prime_or_one R q with h | rfl
exacts [Nat.Prime.pos h, Nat.one_pos]
| 2 |
import Mathlib.Algebra.CharP.Two
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.GroupTheory.SpecificGroups.Cyclic
import Mathlib.NumberTheory.Divisors
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Tactic.Zify
#align_import rin... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 125 | 127 | theorem map_rootsOfUnity (f : Mˣ →* Nˣ) (k : ℕ+) : (rootsOfUnity k M).map f ≤ rootsOfUnity k N := by |
rintro _ ⟨ζ, h, rfl⟩
simp_all only [← map_pow, mem_rootsOfUnity, SetLike.mem_coe, MonoidHom.map_one]
| 2 |
import Mathlib.Data.Nat.Lattice
import Mathlib.Logic.Denumerable
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Hom.Basic
import Mathlib.Data.Set.Subsingleton
#align_import order.order_iso_nat from "leanprover-community/mathlib"@"210657c4ea4a4a7b234392f70a3a2a83346dfa90"
variable {α : Type*}
namespa... | Mathlib/Order/OrderIsoNat.lean | 99 | 101 | theorem not_wellFounded_of_decreasing_seq (f : ((· > ·) : ℕ → ℕ → Prop) ↪r r) : ¬WellFounded r := by |
rw [wellFounded_iff_no_descending_seq, not_isEmpty_iff]
exact ⟨f⟩
| 2 |
import Mathlib.LinearAlgebra.Basis
import Mathlib.Algebra.Module.LocalizedModule
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Localization.Integer
#align_import ring_theory.localization.module from "leanprover-community/mathlib"@"2e59a6de168f95d16b16d217b808a36290398c0a"
open nonZ... | Mathlib/RingTheory/Localization/Module.lean | 73 | 76 | theorem LinearIndependent.localization {ι : Type*} {b : ι → M} (hli : LinearIndependent R b) :
LinearIndependent Rₛ b := by |
have := isLocalizedModule_id S M Rₛ
exact hli.of_isLocalizedModule Rₛ S .id
| 2 |
import Mathlib.Analysis.NormedSpace.PiTensorProduct.ProjectiveSeminorm
import Mathlib.LinearAlgebra.Isomorphisms
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)]
... | Mathlib/Analysis/NormedSpace/PiTensorProduct/InjectiveSeminorm.lean | 144 | 150 | theorem injectiveSeminorm_apply (x : ⨂[𝕜] i, E i) :
injectiveSeminorm x = ⨆ p : {p | ∃ (G : Type (max uι u𝕜 uE))
(_ : SeminormedAddCommGroup G) (_ : NormedSpace 𝕜 G), p = Seminorm.comp (normSeminorm 𝕜
(ContinuousMultilinearMap 𝕜 E G →L[𝕜] G))
(toDualContinuousMultilinearMap G (𝕜 := 𝕜) (E := E))}... |
simp [injectiveSeminorm]
exact Seminorm.sSup_apply dualSeminorms_bounded
| 2 |
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 | 109 | 111 | theorem mem_roots' : a ∈ p.roots ↔ p ≠ 0 ∧ IsRoot p a := by |
classical
rw [← count_pos, count_roots p, rootMultiplicity_pos']
| 2 |
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
import Mathlib.Tactic.AdaptationNote
#align_import probability.martingale.upcrossing from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Topological... | Mathlib/Probability/Martingale/Upcrossing.lean | 173 | 176 | theorem upperCrossingTime_succ_eq (ω : Ω) : upperCrossingTime a b f N (n + 1) ω =
hitting f (Set.Ici b) (lowerCrossingTime a b f N n ω) N ω := by |
simp only [upperCrossingTime_succ]
rfl
| 2 |
import Mathlib.CategoryTheory.Subobject.Lattice
#align_import category_theory.subobject.limits from "leanprover-community/mathlib"@"956af7c76589f444f2e1313911bad16366ea476d"
universe v u
noncomputable section
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Subobject Opposite
var... | Mathlib/CategoryTheory/Subobject/Limits.lean | 134 | 137 | theorem factorThruKernelSubobject_comp_arrow {W : C} (h : W ⟶ X) (w : h ≫ f = 0) :
factorThruKernelSubobject f h w ≫ (kernelSubobject f).arrow = h := by |
dsimp [factorThruKernelSubobject]
simp
| 2 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib.Deprecated.Group
#align_import deprecated.submonoid from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
variable {M : Type*} [Monoid M] {s : Set M}
variable {A : Type*} [AddMonoi... | Mathlib/Deprecated/Submonoid.lean | 232 | 237 | theorem list_prod_mem (hs : IsSubmonoid s) : ∀ {l : List M}, (∀ x ∈ l, x ∈ s) → l.prod ∈ s
| [], _ => hs.one_mem
| a :: l, h =>
suffices a * l.prod ∈ s by simpa
have : a ∈ s ∧ ∀ x ∈ l, x ∈ s := by | simpa using h
hs.mul_mem this.1 (list_prod_mem hs this.2)
| 2 |
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Group.Measure
#align_import measure_theory.group.integration from "leanprover-community/mathlib"@"ec247d43814751ffceb33b758e8820df2372bf6f"
namespace MeasureTheory
open Measure TopologicalSpace
open scoped ENNReal
variable {𝕜 M α G E F ... | Mathlib/MeasureTheory/Group/Integral.lean | 165 | 168 | theorem integral_smul_eq_self {μ : Measure α} [SMulInvariantMeasure G α μ] (f : α → E) {g : G} :
(∫ x, f (g • x) ∂μ) = ∫ x, f x ∂μ := by |
have h : MeasurableEmbedding fun x : α => g • x := (MeasurableEquiv.smul g).measurableEmbedding
rw [← h.integral_map, map_smul]
| 2 |
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.Tactic.FieldSimp
#align_import linear_algebra.affine_space.slope from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
open AffineMap
variable {k E PE : Type*} [Field k] [AddCommGroup E] [Module k E] [AddTorsor E PE]
def ... | Mathlib/LinearAlgebra/AffineSpace/Slope.lean | 121 | 124 | theorem lineMap_slope_slope_sub_div_sub (f : k → PE) (a b c : k) (h : a ≠ c) :
lineMap (slope f a b) (slope f b c) ((c - b) / (c - a)) = slope f a c := by |
field_simp [sub_ne_zero.2 h.symm, ← sub_div_sub_smul_slope_add_sub_div_sub_smul_slope f a b c,
lineMap_apply_module]
| 2 |
import Mathlib.Probability.Notation
import Mathlib.Probability.Integration
import Mathlib.MeasureTheory.Function.L2Space
#align_import probability.variance from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
open MeasureTheory Filter Finset
noncomputable section
open scoped MeasureThe... | Mathlib/Probability/Variance.lean | 100 | 103 | theorem _root_.MeasureTheory.Memℒp.ofReal_variance_eq [IsFiniteMeasure μ] (hX : Memℒp X 2 μ) :
ENNReal.ofReal (variance X μ) = evariance X μ := by |
rw [variance, ENNReal.ofReal_toReal]
exact hX.evariance_lt_top.ne
| 2 |
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Hom.Defs
#align_import algebra.hom.group from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358ef64"
-- `NeZero` cannot be additivised, hence its theory should be developed outside of the
-- `Algebra.Group` folder.
assert_not_exists... | Mathlib/Algebra/Group/Hom/Basic.lean | 110 | 113 | theorem comp_mul [Mul M] [CommSemigroup N] [CommSemigroup P] (g : N →ₙ* P) (f₁ f₂ : M →ₙ* N) :
g.comp (f₁ * f₂) = g.comp f₁ * g.comp f₂ := by |
ext
simp only [mul_apply, Function.comp_apply, map_mul, coe_comp]
| 2 |
import Mathlib.CategoryTheory.Subobject.Lattice
#align_import category_theory.subobject.limits from "leanprover-community/mathlib"@"956af7c76589f444f2e1313911bad16366ea476d"
universe v u
noncomputable section
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Subobject Opposite
var... | Mathlib/CategoryTheory/Subobject/Limits.lean | 350 | 353 | theorem factorThruImageSubobject_comp_self_assoc {W W' : C} (k : W ⟶ W') (k' : W' ⟶ X) (h) :
(imageSubobject f).factorThru (k ≫ k' ≫ f) h = k ≫ k' ≫ factorThruImageSubobject f := by |
ext
simp
| 2 |
import Batteries.Data.List.Basic
import Batteries.Data.List.Lemmas
open Nat
namespace List
section countP
variable (p q : α → Bool)
@[simp] theorem countP_nil : countP p [] = 0 := rfl
protected theorem countP_go_eq_add (l) : countP.go p l n = n + countP.go p l 0 := by
induction l generalizing n with
| nil... | .lake/packages/batteries/Batteries/Data/List/Count.lean | 68 | 70 | theorem countP_le_length : countP p l ≤ l.length := by |
simp only [countP_eq_length_filter]
apply length_filter_le
| 2 |
namespace Nat
@[reducible] def Coprime (m n : Nat) : Prop := gcd m n = 1
instance (m n : Nat) : Decidable (Coprime m n) := inferInstanceAs (Decidable (_ = 1))
theorem coprime_iff_gcd_eq_one : Coprime m n ↔ gcd m n = 1 := .rfl
theorem Coprime.gcd_eq_one : Coprime m n → gcd m n = 1 := id
theorem Coprime.symm ... | .lake/packages/batteries/Batteries/Data/Nat/Gcd.lean | 32 | 34 | theorem Coprime.dvd_of_dvd_mul_right (H1 : Coprime k n) (H2 : k ∣ m * n) : k ∣ m := by |
let t := dvd_gcd (Nat.dvd_mul_left k m) H2
rwa [gcd_mul_left, H1.gcd_eq_one, Nat.mul_one] at t
| 2 |
import Mathlib.CategoryTheory.Galois.GaloisObjects
import Mathlib.CategoryTheory.Limits.Shapes.CombinedProducts
universe u₁ u₂ w
namespace CategoryTheory
open Limits Functor
variable {C : Type u₁} [Category.{u₂} C]
namespace PreGaloisCategory
variable [GaloisCategory C]
section Decomposition
private lemma... | Mathlib/CategoryTheory/Galois/Decomposition.lean | 111 | 115 | theorem has_decomp_connected_components (X : C) :
∃ (ι : Type) (f : ι → C) (g : (i : ι) → f i ⟶ X) (_ : IsColimit (Cofan.mk X g)),
(∀ i, IsConnected (f i)) ∧ Finite ι := by |
let F := GaloisCategory.getFiberFunctor C
exact has_decomp_connected_components_aux F (Nat.card <| F.obj X) X rfl
| 2 |
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.GroupWithZero.Basic
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Contrapose
import Mathlib.Tactic.Nontriviality
import Mathlib.Tactic.Spread
import Mathlib.Util.AssertExists
#align_import algebra.group_with_zero.units.basic from "leanprover-community/... | Mathlib/Algebra/GroupWithZero/Units/Basic.lean | 108 | 110 | theorem mul_inverse_cancel (x : M₀) (h : IsUnit x) : x * inverse x = 1 := by |
rcases h with ⟨u, rfl⟩
rw [inverse_unit, Units.mul_inv]
| 2 |
import Mathlib.Data.Set.Subsingleton
import Mathlib.Order.WithBot
#align_import data.set.image from "leanprover-community/mathlib"@"001ffdc42920050657fd45bd2b8bfbec8eaaeb29"
universe u v
open Function Set
namespace Set
variable {α β γ : Type*} {ι ι' : Sort*}
section Preimage
variable {f : α → β} {g : β → γ... | Mathlib/Data/Set/Image.lean | 53 | 55 | theorem preimage_congr {f g : α → β} {s : Set β} (h : ∀ x : α, f x = g x) : f ⁻¹' s = g ⁻¹' s := by |
congr with x
simp [h]
| 2 |
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Vector.Basic
import Mathlib.Data.PFun
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Basic
import Mathlib.Tactic.ApplyFun
#align_import computability.turing_machine from "leanprover-commu... | Mathlib/Computability/TuringMachine.lean | 85 | 88 | theorem BlankExtends.trans {Γ} [Inhabited Γ] {l₁ l₂ l₃ : List Γ} :
BlankExtends l₁ l₂ → BlankExtends l₂ l₃ → BlankExtends l₁ l₃ := by |
rintro ⟨i, rfl⟩ ⟨j, rfl⟩
exact ⟨i + j, by simp [List.replicate_add]⟩
| 2 |
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 | 99 | 103 | theorem HasStrictDerivAt.complexToReal_fderiv' {f : ℂ → E} {x : ℂ} {f' : E}
(h : HasStrictDerivAt f f' x) :
HasStrictFDerivAt f (reCLM.smulRight f' + I • imCLM.smulRight f') x := by |
simpa only [Complex.restrictScalars_one_smulRight'] using
h.hasStrictFDerivAt.restrictScalars ℝ
| 2 |
import Mathlib.MeasureTheory.MeasurableSpace.Defs
import Mathlib.SetTheory.Cardinal.Cofinality
import Mathlib.SetTheory.Cardinal.Continuum
#align_import measure_theory.card_measurable_space from "leanprover-community/mathlib"@"f2b108e8e97ba393f22bf794989984ddcc1da89b"
universe u
variable {α : Type u}
open Cardi... | Mathlib/MeasureTheory/MeasurableSpace/Card.lean | 74 | 78 | theorem iUnion_mem_generateMeasurableRec {s : Set (Set α)} {i : ω₁} {f : ℕ → Set α}
(hf : ∀ n, ∃ j < i, f n ∈ generateMeasurableRec s j) :
(⋃ n, f n) ∈ generateMeasurableRec s i := by |
unfold generateMeasurableRec
exact mem_union_right _ ⟨fun n => ⟨f n, let ⟨j, hj, hf⟩ := hf n; mem_iUnion.2 ⟨⟨j, hj⟩, hf⟩⟩, rfl⟩
| 2 |
import Mathlib.Analysis.Calculus.TangentCone
import Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics
#align_import analysis.calculus.fderiv.basic from "leanprover-community/mathlib"@"41bef4ae1254365bc190aee63b947674d2977f01"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open To... | Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 231 | 233 | theorem fderiv_zero_of_not_differentiableAt (h : ¬DifferentiableAt 𝕜 f x) : fderiv 𝕜 f x = 0 := by |
have : ¬∃ f', HasFDerivAt f f' x := h
simp [fderiv, this]
| 2 |
import Mathlib.CategoryTheory.Sites.Sieves
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
import Mathlib.CategoryTheory.Category.Preorder
import Mathlib.Order.Copy
import Mathlib.Data.Set.Subsingleton
#align_import category_theory.sites.grothendieck fr... | Mathlib/CategoryTheory/Sites/Grothendieck.lean | 191 | 193 | theorem arrow_max (f : Y ⟶ X) (S : Sieve X) (hf : S f) : J.Covers S f := by |
rw [Covers, (Sieve.pullback_eq_top_iff_mem f).1 hf]
apply J.top_mem
| 2 |
import Mathlib.Analysis.Normed.Group.Basic
#align_import information_theory.hamming from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
section HammingDistNorm
open Finset Function
variable {α ι : Type*} {β : ι → Type*} [Fintype ι] [∀ i, DecidableEq (β i)]
variable {γ : ι → Type*} [∀ ... | Mathlib/InformationTheory/Hamming.lean | 91 | 93 | theorem eq_of_hammingDist_eq_zero {x y : ∀ i, β i} : hammingDist x y = 0 → x = y := by |
simp_rw [hammingDist, card_eq_zero, filter_eq_empty_iff, Classical.not_not, funext_iff, mem_univ,
forall_true_left, imp_self]
| 2 |
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts
#align_import category_theory.limits.constructions.zero_objects from "leanprover-community/mathlib"@"52a270e2ea4e342c2587c106f8be904524214a4... | Mathlib/CategoryTheory/Limits/Constructions/ZeroObjects.lean | 58 | 60 | theorem zeroProdIso_inv_snd (X : C) : (zeroProdIso X).inv ≫ prod.snd = 𝟙 X := by |
dsimp [zeroProdIso, binaryFanZeroLeft]
simp
| 2 |
import Mathlib.MeasureTheory.Decomposition.RadonNikodym
import Mathlib.MeasureTheory.Measure.Haar.OfBasis
import Mathlib.Probability.Independence.Basic
#align_import probability.density from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open scoped Classical MeasureTheory NNReal ENNRea... | Mathlib/Probability/Density.lean | 152 | 155 | theorem aemeasurable_of_pdf_ne_zero {m : MeasurableSpace Ω} {ℙ : Measure Ω} {μ : Measure E}
(X : Ω → E) (h : ¬pdf X ℙ μ =ᵐ[μ] 0) : AEMeasurable X ℙ := by |
contrapose! h
exact pdf_of_not_aemeasurable h
| 2 |
import Mathlib.NumberTheory.DirichletCharacter.Bounds
import Mathlib.NumberTheory.EulerProduct.Basic
import Mathlib.NumberTheory.LSeries.Basic
import Mathlib.NumberTheory.LSeries.RiemannZeta
open Complex
variable {s : ℂ}
noncomputable
def riemannZetaSummandHom (hs : s ≠ 0) : ℕ →*₀ ℂ where
toFun n := (n : ℂ) ^ ... | Mathlib/NumberTheory/EulerProduct/DirichletLSeries.lean | 91 | 94 | theorem riemannZeta_eulerProduct_hasProd (hs : 1 < s.re) :
HasProd (fun p : Primes ↦ (1 - (p : ℂ) ^ (-s))⁻¹) (riemannZeta s) := by |
rw [← tsum_riemannZetaSummand hs]
apply eulerProduct_completely_multiplicative_hasProd <| summable_riemannZetaSummand hs
| 2 |
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Data.Countable.Basic
import Mathlib.Data.Set.Image
import Mathlib.Data.Set.Subsingleton
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.GroupTheory.Subgroup.Centralizer
#align_import group_theory.subgroup.zpowers from "leanprover-community/mathlib"@"4be589... | Mathlib/Algebra/Group/Subgroup/ZPowers.lean | 47 | 49 | theorem zpowers_eq_closure (g : G) : zpowers g = closure {g} := by |
ext
exact mem_closure_singleton.symm
| 2 |
import Mathlib.Algebra.Group.Support
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Nat.Cast.Field
#align_import algebra.char_zero.lemmas from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
open Function Set
namespace Nat
variable {R : Type*} [AddMonoidWithOne R] [Char... | Mathlib/Algebra/CharZero/Lemmas.lean | 39 | 42 | theorem cast_pow_eq_one {R : Type*} [Semiring R] [CharZero R] (q : ℕ) (n : ℕ) (hn : n ≠ 0) :
(q : R) ^ n = 1 ↔ q = 1 := by |
rw [← cast_pow, cast_eq_one]
exact pow_eq_one_iff hn
| 2 |
import Mathlib.Analysis.Calculus.FDeriv.Basic
#align_import analysis.calculus.fderiv.restrict_scalars from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open Topology NNReal Filter Asymptotics ENNReal
noncom... | Mathlib/Analysis/Calculus/FDeriv/RestrictScalars.lean | 92 | 95 | theorem HasFDerivWithinAt.of_restrictScalars {g' : E →L[𝕜] F} (h : HasFDerivWithinAt f g' s x)
(H : f'.restrictScalars 𝕜 = g') : HasFDerivWithinAt f f' s x := by |
rw [← H] at h
exact .of_isLittleO h.1
| 2 |
import Mathlib.Analysis.SpecialFunctions.Exponential
#align_import analysis.special_functions.trigonometric.series from "leanprover-community/mathlib"@"ccf84e0d918668460a34aa19d02fe2e0e2286da0"
open NormedSpace
open scoped Nat
section SinCos
theorem Complex.hasSum_cos' (z : ℂ) :
HasSum (fun n : ℕ => (z *... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Series.lean | 68 | 71 | theorem Complex.hasSum_cos (z : ℂ) :
HasSum (fun n : ℕ => (-1) ^ n * z ^ (2 * n) / ↑(2 * n)!) (Complex.cos z) := by |
convert Complex.hasSum_cos' z using 1
simp_rw [mul_pow, pow_mul, Complex.I_sq, mul_comm]
| 2 |
import Mathlib.Data.Matrix.Basic
#align_import data.matrix.block from "leanprover-community/mathlib"@"c060baa79af5ca092c54b8bf04f0f10592f59489"
variable {l m n o p q : Type*} {m' n' p' : o → Type*}
variable {R : Type*} {S : Type*} {α : Type*} {β : Type*}
open Matrix
namespace Matrix
theorem dotProduct_block [F... | Mathlib/Data/Matrix/Block.lean | 97 | 100 | theorem fromBlocks_toBlocks (M : Matrix (Sum n o) (Sum l m) α) :
fromBlocks M.toBlocks₁₁ M.toBlocks₁₂ M.toBlocks₂₁ M.toBlocks₂₂ = M := by |
ext i j
rcases i with ⟨⟩ <;> rcases j with ⟨⟩ <;> rfl
| 2 |
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms
import Mathlib.CategoryTheory.Limits.Constructions.BinaryProducts
#align_import category_theory.limits.constructions.zero_objects from "leanprover-community/mathlib"@"52a270e2ea4e342c2587c106f8be904524214a4... | Mathlib/CategoryTheory/Limits/Constructions/ZeroObjects.lean | 89 | 91 | theorem prodZeroIso_iso_inv_snd (X : C) : (prodZeroIso X).inv ≫ prod.fst = 𝟙 X := by |
dsimp [prodZeroIso, binaryFanZeroRight]
simp
| 2 |
import Mathlib.Probability.Kernel.Composition
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import probability.kernel.integral_comp_prod from "leanprover-community/mathlib"@"c0d694db494dd4f9aa57f2714b6e4c82b4ebc113"
noncomputable section
open scoped Topology ENNReal MeasureTheory ProbabilityTheory
op... | Mathlib/Probability/Kernel/IntegralCompProd.lean | 78 | 82 | theorem _root_.MeasureTheory.AEStronglyMeasurable.compProd_mk_left {δ : Type*} [TopologicalSpace δ]
{f : β × γ → δ} (hf : AEStronglyMeasurable f ((κ ⊗ₖ η) a)) :
∀ᵐ x ∂κ a, AEStronglyMeasurable (fun y => f (x, y)) (η (a, x)) := by |
filter_upwards [ae_ae_of_ae_compProd hf.ae_eq_mk] with x hx using
⟨fun y => hf.mk f (x, y), hf.stronglyMeasurable_mk.comp_measurable measurable_prod_mk_left, hx⟩
| 2 |
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 | 108 | 110 | theorem int_valuation_zero_le (x : nonZeroDivisors R) : 0 < v.intValuationDef x := by |
rw [v.intValuationDef_if_neg (nonZeroDivisors.coe_ne_zero x)]
exact WithZero.zero_lt_coe _
| 2 |
import Mathlib.Data.Set.Prod
#align_import data.set.n_ary from "leanprover-community/mathlib"@"5e526d18cea33550268dcbbddcb822d5cde40654"
open Function
namespace Set
variable {α α' β β' γ γ' δ δ' ε ε' ζ ζ' ν : Type*} {f f' : α → β → γ} {g g' : α → β → γ → δ}
variable {s s' : Set α} {t t' : Set β} {u u' : Set γ} {v... | Mathlib/Data/Set/NAry.lean | 96 | 98 | theorem image2_swap (s : Set α) (t : Set β) : image2 f s t = image2 (fun a b => f b a) t s := by |
ext
constructor <;> rintro ⟨a, ha, b, hb, rfl⟩ <;> exact ⟨b, hb, a, ha, rfl⟩
| 2 |
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.LinearAlgebra.Determinant
variable {K n : Type*} [NormedField K] [Fintype n] [DecidableEq n] {A : Matrix n n K}
theorem eigenvalue_mem_ball {μ : K} (hμ : Module.End.HasEigenvalue (Matrix.toLin' A) μ) :
∃ k,... | Mathlib/LinearAlgebra/Matrix/Gershgorin.lean | 69 | 72 | theorem det_ne_zero_of_sum_col_lt_diag (h : ∀ k, ∑ i ∈ Finset.univ.erase k, ‖A i k‖ < ‖A k k‖) :
A.det ≠ 0 := by |
rw [← Matrix.det_transpose]
exact det_ne_zero_of_sum_row_lt_diag (by simp_rw [Matrix.transpose_apply]; exact h)
| 2 |
import Mathlib.Algebra.Ring.Int
import Mathlib.SetTheory.Game.PGame
import Mathlib.Tactic.Abel
#align_import set_theory.game.basic from "leanprover-community/mathlib"@"8900d545017cd21961daa2a1734bb658ef52c618"
-- Porting note: many definitions here are noncomputable as the compiler does not support PGame.rec
nonco... | Mathlib/SetTheory/Game/Basic.lean | 118 | 120 | theorem not_lf : ∀ {x y : Game}, ¬x ⧏ y ↔ y ≤ x := by |
rintro ⟨x⟩ ⟨y⟩
exact PGame.not_lf
| 2 |
import Mathlib.Data.Int.Interval
import Mathlib.RingTheory.Binomial
import Mathlib.RingTheory.HahnSeries.PowerSeries
import Mathlib.RingTheory.HahnSeries.Summable
import Mathlib.FieldTheory.RatFunc.AsPolynomial
import Mathlib.RingTheory.Localization.FractionRing
#align_import ring_theory.laurent_series from "leanprov... | Mathlib/RingTheory/LaurentSeries.lean | 143 | 146 | theorem ofPowerSeries_powerSeriesPart (x : LaurentSeries R) :
ofPowerSeries ℤ R x.powerSeriesPart = single (-x.order) 1 * x := by |
refine Eq.trans ?_ (congr rfl x.single_order_mul_powerSeriesPart)
rw [← mul_assoc, single_mul_single, neg_add_self, mul_one, ← C_apply, C_one, one_mul]
| 2 |
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