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 | eval_complexity float64 0 1 |
|---|---|---|---|---|---|---|
import Mathlib.LinearAlgebra.Matrix.DotProduct
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
#align_import data.matrix.rank from "leanprover-community/mathlib"@"17219820a8aa8abe85adf5dfde19af1dd1bd8ae7"
open Matrix
namespace Matrix
open FiniteDimensional
variable {l m n ... | Mathlib/Data/Matrix/Rank.lean | 96 | 99 | theorem rank_of_isUnit [StrongRankCondition R] [DecidableEq n] (A : Matrix n n R) (h : IsUnit A) :
A.rank = Fintype.card n := by |
obtain ⟨A, rfl⟩ := h
exact rank_unit A
| 0.4375 |
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 | 47 | 53 | theorem mirror_monomial (n : ℕ) (a : R) : (monomial n a).mirror = monomial n a := by |
classical
by_cases ha : a = 0
· rw [ha, monomial_zero_right, mirror_zero]
· rw [mirror, reverse, natDegree_monomial n a, if_neg ha, natTrailingDegree_monomial ha, ←
C_mul_X_pow_eq_monomial, reflect_C_mul_X_pow, revAt_le (le_refl n), tsub_self, pow_zero,
mul_one]
| 0.4375 |
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.Dual
import Mathlib.Data.Fin.FlagRange
open Set Submodule
namespace Basis
section Semiring
variable {R M : Type*} [Semiring R] [AddCommMonoid M] [Module R M] {n : ℕ}
def flag (b : Basis (Fin n) R M) (k : Fin (n + 1)) : Submodule R M :=
.span R <... | Mathlib/LinearAlgebra/Basis/Flag.lean | 35 | 36 | theorem flag_last (b : Basis (Fin n) R M) : b.flag (.last n) = ⊤ := by |
simp [flag, Fin.castSucc_lt_last]
| 0.4375 |
import Mathlib.Algebra.Order.EuclideanAbsoluteValue
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Polynomial.FieldDivision
#align_import data.polynomial.degree.card_pow_degree from "leanprover-community/mathlib"@"85d9f2189d9489f9983c0d01536575b0233bd305"
n... | Mathlib/Algebra/Polynomial/Degree/CardPowDegree.lean | 79 | 83 | theorem cardPowDegree_apply [DecidableEq Fq] (p : Fq[X]) :
cardPowDegree p = if p = 0 then 0 else (Fintype.card Fq : ℤ) ^ natDegree p := by |
rw [cardPowDegree]
dsimp
convert rfl
| 0.4375 |
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.GeomSum
import Mathlib.Data.Fintype.BigOperators
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.PowerSeries.WellKnown
import Mathlib.Tactic.FieldSimp
#align_import number_theory.bernoulli from "leanprover-community/mat... | Mathlib/NumberTheory/Bernoulli.lean | 78 | 80 | theorem bernoulli'_def (n : ℕ) :
bernoulli' n = 1 - ∑ k ∈ range n, n.choose k / (n - k + 1) * bernoulli' k := by |
rw [bernoulli'_def', ← Fin.sum_univ_eq_sum_range]
| 0.4375 |
import Mathlib.Algebra.Module.Equiv
import Mathlib.Algebra.Module.Submodule.Basic
import Mathlib.Algebra.PUnitInstances
import Mathlib.Data.Set.Subsingleton
#align_import algebra.module.submodule.lattice from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
universe v
variable {R S M : Ty... | Mathlib/Algebra/Module/Submodule/Lattice.lean | 131 | 132 | theorem nontrivial_iff_ne_bot : Nontrivial p ↔ p ≠ ⊥ := by |
rw [iff_not_comm, not_nontrivial_iff_subsingleton, subsingleton_iff_eq_bot]
| 0.4375 |
import Mathlib.Algebra.ContinuedFractions.Basic
import Mathlib.Algebra.GroupWithZero.Basic
#align_import algebra.continued_fractions.translations from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
section WithDivisionRing
variable {K : Type*}... | Mathlib/Algebra/ContinuedFractions/Translations.lean | 162 | 163 | theorem first_numerator_eq {gp : Pair K} (zeroth_s_eq : g.s.get? 0 = some gp) :
g.numerators 1 = gp.b * g.h + gp.a := by | simp [num_eq_conts_a, first_continuant_eq zeroth_s_eq]
| 0.4375 |
import Mathlib.Algebra.Group.Subsemigroup.Basic
#align_import group_theory.subsemigroup.membership from "leanprover-community/mathlib"@"6cb77a8eaff0ddd100e87b1591c6d3ad319514ff"
assert_not_exists MonoidWithZero
variable {ι : Sort*} {M A B : Type*}
section NonAssoc
variable [Mul M]
open Set
namespace Subsemigr... | Mathlib/Algebra/Group/Subsemigroup/Membership.lean | 123 | 128 | theorem iSup_induction (S : ι → Subsemigroup M) {C : M → Prop} {x₁ : M} (hx₁ : x₁ ∈ ⨆ i, S i)
(mem : ∀ i, ∀ x₂ ∈ S i, C x₂) (mul : ∀ x y, C x → C y → C (x * y)) : C x₁ := by |
rw [iSup_eq_closure] at hx₁
refine closure_induction hx₁ (fun x₂ hx₂ => ?_) mul
obtain ⟨i, hi⟩ := Set.mem_iUnion.mp hx₂
exact mem _ _ hi
| 0.4375 |
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Set.Subsingleton
#align_import combinatorics.composition from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
open List
variable {n : ℕ}
... | Mathlib/Combinatorics/Enumerative/Composition.lean | 647 | 649 | theorem splitWrtCompositionAux_cons (l : List α) (n ns) :
l.splitWrtCompositionAux (n::ns) = take n l::(drop n l).splitWrtCompositionAux ns := by |
simp [splitWrtCompositionAux]
| 0.4375 |
import Mathlib.Topology.Category.CompHaus.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Limits.Preserves.Finite
namespace CompHaus
attribute [local instance] CategoryTheory.ConcreteCategory.instFunLike
universe u w
open Categor... | Mathlib/Topology/Category/CompHaus/Limits.lean | 205 | 207 | theorem Sigma.ι_comp_toFiniteCoproduct (a : α) :
(Limits.Sigma.ι X a) ≫ (coproductIsoCoproduct X).inv = finiteCoproduct.ι X a := by |
simp [coproductIsoCoproduct]
| 0.4375 |
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.Asymptotics.Theta
import Mathlib.Analysis.Normed.Order.Basic
#align_import analysis.asymptotics.asymptotic_equivalent from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
namespace Asymptotics
open Filter Function
... | Mathlib/Analysis/Asymptotics/AsymptoticEquivalent.lean | 151 | 154 | theorem IsEquivalent.tendsto_const {c : β} (hu : u ~[l] const _ c) : Tendsto u l (𝓝 c) := by |
rcases em <| c = 0 with rfl | h
· exact (tendsto_congr' <| isEquivalent_zero_iff_eventually_zero.mp hu).mpr tendsto_const_nhds
· exact (isEquivalent_const_iff_tendsto h).mp hu
| 0.4375 |
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 | 216 | 217 | theorem fderivWithin_zero_of_isolated (h : 𝓝[s \ {x}] x = ⊥) : fderivWithin 𝕜 f s x = 0 := by |
rw [fderivWithin, if_pos h]
| 0.4375 |
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup
import Mathlib.LinearAlgebra.Matrix.SpecialLinearGroup
import Mathlib.Tactic.AdaptationNote
#align_import number_theory.modular_forms.slash_actions from "leanprover-community/mathlib"@"738054fa93d43512da144ec45... | Mathlib/NumberTheory/ModularForms/SlashActions.lean | 67 | 70 | theorem SlashAction.smul_slash_of_tower {R β G α : Type*} (γ : Type*) [Group G] [AddGroup α]
[Monoid γ] [MulAction γ α] [SMul R γ] [SMul R α] [IsScalarTower R γ α] [SlashAction β G α γ]
(k : β) (g : G) (a : α) (r : R) : (r • a) ∣[k;γ] g = r • a ∣[k;γ] g := by |
rw [← smul_one_smul γ r a, SlashAction.smul_slash, smul_one_smul]
| 0.4375 |
import Mathlib.Algebra.Module.Submodule.EqLocus
import Mathlib.Algebra.Module.Submodule.RestrictScalars
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.Data.Set.Pointwise.SMul
import Mathlib.LinearAlgebra.Basic
import Mathlib.Order.CompactlyGenerated.Basic
import Mathlib.Order.OmegaCompletePartialOrder
#align_... | Mathlib/LinearAlgebra/Span.lean | 154 | 157 | theorem span_preimage_le (f : F) (s : Set M₂) :
span R (f ⁻¹' s) ≤ (span R₂ s).comap f := by |
rw [span_le, comap_coe]
exact preimage_mono subset_span
| 0.4375 |
import Mathlib.Data.Set.Image
#align_import data.nat.set from "leanprover-community/mathlib"@"cf9386b56953fb40904843af98b7a80757bbe7f9"
namespace Nat
section Set
open Set
| Mathlib/Data/Nat/Set.lean | 21 | 23 | theorem zero_union_range_succ : {0} ∪ range succ = univ := by |
ext n
cases n <;> simp
| 0.4375 |
import Mathlib.Algebra.TrivSqZeroExt
import Mathlib.Topology.Algebra.InfiniteSum.Basic
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.instances.triv_sq_zero_ext from "leanprover-community/mathlib"@"b8d2eaa69d69ce8f03179a5cda774fc0cde984e4"
open scoped Topology
variable {α S R M : Type*}
local... | Mathlib/Topology/Instances/TrivSqZeroExt.lean | 46 | 48 | theorem nhds_def (x : tsze R M) : 𝓝 x = (𝓝 x.fst).prod (𝓝 x.snd) := by |
cases x using Prod.rec
exact nhds_prod_eq
| 0.4375 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 382 | 384 | theorem inv_lt_inv_iff : a⁻¹ < b⁻¹ ↔ b < a := by |
rw [← mul_lt_mul_iff_left a, ← mul_lt_mul_iff_right b]
simp
| 0.4375 |
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 | 49 | 51 | theorem summable_iff_vanishing_norm [CompleteSpace E] {f : ι → E} :
Summable f ↔ ∀ ε > (0 : ℝ), ∃ s : Finset ι, ∀ t, Disjoint t s → ‖∑ i ∈ t, f i‖ < ε := by |
rw [summable_iff_cauchySeq_finset, cauchySeq_finset_iff_vanishing_norm]
| 0.4375 |
import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
import Mathlib.Topology.FiberBundle.Basic
#align_import topology.vector_bundle.basic from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
noncomputable section
open scoped Classical
open Bundle Set
open scoped Topology
variable (R : ... | Mathlib/Topology/VectorBundle/Basic.lean | 151 | 154 | theorem symmₗ_linearMapAt (e : Pretrivialization F (π F E)) [e.IsLinear R] {b : B}
(hb : b ∈ e.baseSet) (y : E b) : e.symmₗ R b (e.linearMapAt R b y) = y := by |
rw [e.linearMapAt_def_of_mem hb]
exact (e.linearEquivAt R b hb).left_inv y
| 0.4375 |
import Mathlib.RingTheory.WittVector.Frobenius
import Mathlib.RingTheory.WittVector.Verschiebung
import Mathlib.RingTheory.WittVector.MulP
#align_import ring_theory.witt_vector.identities from "leanprover-community/mathlib"@"0798037604b2d91748f9b43925fb7570a5f3256c"
namespace WittVector
variable {p : ℕ} {R : Typ... | Mathlib/RingTheory/WittVector/Identities.lean | 90 | 92 | theorem p_nonzero [Nontrivial R] [CharP R p] : (p : 𝕎 R) ≠ 0 := by |
intro h
simpa only [h, zero_coeff, zero_ne_one] using coeff_p_one p R
| 0.4375 |
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.RowCol
import Mathlib.Data.Fin.VecNotation
import Mathlib.Tactic.FinCases
#align_import data.matrix.notation from "leanprover-community/mathlib"@"a99f85220eaf38f14f94e04699943e185a5e1d1a"
namespace Matrix
universe u uₘ uₙ uₒ
variable {α : Type u} {o n m... | Mathlib/Data/Matrix/Notation.lean | 168 | 170 | theorem dotProduct_cons (v : Fin n.succ → α) (x : α) (w : Fin n → α) :
dotProduct v (vecCons x w) = vecHead v * x + dotProduct (vecTail v) w := by |
simp [dotProduct, Fin.sum_univ_succ, vecHead, vecTail]
| 0.4375 |
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 | 114 | 118 | theorem dirichletLSeries_eulerProduct_hasProd {N : ℕ} (χ : DirichletCharacter ℂ N)
(hs : 1 < s.re) :
HasProd (fun p : Primes ↦ (1 - χ p * (p : ℂ) ^ (-s))⁻¹) (L ↗χ s) := by |
rw [← tsum_dirichletSummand χ hs]
convert eulerProduct_completely_multiplicative_hasProd <| summable_dirichletSummand χ hs
| 0.4375 |
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
import Mathlib.Analysis.SpecificLimits.Normed
import Mathlib.Logic.Equiv.Fin
import Mathlib.Topology.Algebra.InfiniteSum.Module
#align_import analysis.analytic.basic from "leanprover-community/mathlib"@"32253a1a1071173b33dc7d6a218cf722c6feb514"
noncomputable... | Mathlib/Analysis/Analytic/Basic.lean | 102 | 105 | theorem partialSum_continuous (p : FormalMultilinearSeries 𝕜 E F) (n : ℕ) :
Continuous (p.partialSum n) := by |
unfold partialSum -- Porting note: added
continuity
| 0.4375 |
import Mathlib.Data.Fintype.Option
import Mathlib.Topology.Separation
import Mathlib.Topology.Sets.Opens
#align_import topology.alexandroff from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
open Set Filter Topology
variable {X : Type*}
def OnePoint (X : Type*) :=
Option X
#ali... | Mathlib/Topology/Compactification/OnePoint.lean | 152 | 153 | theorem not_mem_range_coe_iff {x : OnePoint X} : x ∉ range some ↔ x = ∞ := by |
rw [← mem_compl_iff, compl_range_coe, mem_singleton_iff]
| 0.4375 |
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
| 0.4375 |
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⟩
| 0.4375 |
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.Ring
#align_import data.nat.count from "leanprover-community/mathlib"@"dc6c365e751e34d100e80fe6e314c3c3e0fd2988"
open Finset
namespace Nat
variable (p : ℕ → Prop)
section Count
variable [DecidablePred p]
def count (n : ℕ) : ℕ :=
(List.range n).... | Mathlib/Data/Nat/Count.lean | 102 | 103 | theorem count_lt_count_succ_iff {n : ℕ} : count p n < count p (n + 1) ↔ p n := by |
by_cases h : p n <;> simp [count_succ, h]
| 0.4375 |
import Mathlib.Order.Filter.Basic
import Mathlib.Order.Filter.CountableInter
import Mathlib.SetTheory.Cardinal.Ordinal
import Mathlib.SetTheory.Cardinal.Cofinality
open Set Filter Cardinal
universe u
variable {ι : Type u} {α β : Type u} {c : Cardinal.{u}}
class CardinalInterFilter (l : Filter α) (c : Cardinal.{... | Mathlib/Order/Filter/CardinalInter.lean | 96 | 100 | theorem cardinal_bInter_mem {S : Set ι} (hS : #S < c)
{s : ∀ i ∈ S, Set α} :
(⋂ i, ⋂ hi : i ∈ S, s i ‹_›) ∈ l ↔ ∀ i, ∀ hi : i ∈ S, s i ‹_› ∈ l := by |
rw [biInter_eq_iInter]
exact (cardinal_iInter_mem hS).trans Subtype.forall
| 0.4375 |
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
#align_import analysis.special_functions.pow.continuity from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
noncomputable section
open scoped Classical
open Real Topology NNReal ENNReal Filter ComplexConjugate
open Filter Finset... | Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean | 36 | 41 | theorem zero_cpow_eq_nhds {b : ℂ} (hb : b ≠ 0) : (fun x : ℂ => (0 : ℂ) ^ x) =ᶠ[𝓝 b] 0 := by |
suffices ∀ᶠ x : ℂ in 𝓝 b, x ≠ 0 from
this.mono fun x hx ↦ by
dsimp only
rw [zero_cpow hx, Pi.zero_apply]
exact IsOpen.eventually_mem isOpen_ne hb
| 0.4375 |
import Mathlib.Algebra.Field.Opposite
import Mathlib.Algebra.Group.Subgroup.ZPowers
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Ring.NegOnePow
import Mathlib.Algebra.Order.Archimedean
import Mathlib.GroupTheory.Coset
#align_import algebra.periodic from "leanprover-community/mathlib"@"3041... | Mathlib/Algebra/Periodic.lean | 143 | 144 | theorem Periodic.mul_const' [DivisionSemiring α] (h : Periodic f c) (a : α) :
Periodic (fun x => f (x * a)) (c / a) := by | simpa only [div_eq_mul_inv] using h.mul_const a
| 0.4375 |
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]
| 0.4375 |
import Mathlib.RingTheory.FractionalIdeal.Basic
import Mathlib.RingTheory.Ideal.Norm
namespace FractionalIdeal
open scoped Pointwise nonZeroDivisors
variable {R : Type*} [CommRing R] [IsDedekindDomain R] [Module.Free ℤ R] [Module.Finite ℤ R]
variable {K : Type*} [CommRing K] [Algebra R K] [IsFractionRing R K]
th... | Mathlib/RingTheory/FractionalIdeal/Norm.lean | 84 | 84 | theorem absNorm_nonneg (I : FractionalIdeal R⁰ K) : 0 ≤ absNorm I := by | dsimp [absNorm]; positivity
| 0.4375 |
import Mathlib.Analysis.Calculus.Deriv.Inv
import Mathlib.Analysis.NormedSpace.BallAction
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Geometry.Manifold.Algebra.LieGroup
import Mathlib.Geometry.Manifol... | Mathlib/Geometry/Manifold/Instances/Sphere.lean | 385 | 386 | theorem sphere_ext_iff (u v : sphere (0 : E) 1) : u = v ↔ ⟪(u : E), v⟫_ℝ = 1 := by |
simp [Subtype.ext_iff, inner_eq_one_iff_of_norm_one]
| 0.4375 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Fintype.Option
import Mathlib.Data.Fintype.Pi
import Mathlib.Data.Fintype.Sum
#align_import combinatorics.hales_jewett from "leanprover-community/mathlib"@"1126441d6bccf98c81214a0780c73d499f6721fe"
open scoped Classical
universe u v
namespace ... | Mathlib/Combinatorics/HalesJewett.lean | 175 | 176 | theorem apply_none {α ι} (l : Line α ι) (x : α) (i : ι) (h : l.idxFun i = none) : l x i = x := by |
simp only [Option.getD_none, h, l.apply]
| 0.4375 |
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import topology.metric_space.hausdorff_distance from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e69df601fc33dde156"
noncomputable section
open NNReal ENNReal Topology Set Filter Bornology
universe u v w
variable {ι : Sort*} {α : Type u} {β :... | Mathlib/Topology/MetricSpace/Thickening.lean | 219 | 223 | theorem mem_cthickening_of_dist_le {α : Type*} [PseudoMetricSpace α] (x y : α) (δ : ℝ) (E : Set α)
(h : y ∈ E) (h' : dist x y ≤ δ) : x ∈ cthickening δ E := by |
apply mem_cthickening_of_edist_le x y δ E h
rw [edist_dist]
exact ENNReal.ofReal_le_ofReal h'
| 0.4375 |
import Mathlib.CategoryTheory.Limits.Preserves.Basic
#align_import category_theory.limits.preserves.limits from "leanprover-community/mathlib"@"e97cf15cd1aec9bd5c193b2ffac5a6dc9118912b"
universe w' w v₁ v₂ u₁ u₂
noncomputable section
namespace CategoryTheory
open Category Limits
variable {C : Type u₁} [Catego... | Mathlib/CategoryTheory/Limits/Preserves/Limits.lean | 69 | 73 | theorem lift_comp_preservesLimitsIso_hom (t : Cone F) :
G.map (limit.lift _ t) ≫ (preservesLimitIso G F).hom =
limit.lift (F ⋙ G) (G.mapCone _) := by |
ext
simp [← G.map_comp]
| 0.4375 |
import Mathlib.Order.UpperLower.Basic
import Mathlib.Data.Finset.Preimage
#align_import combinatorics.young.young_diagram from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Function
@[ext]
structure YoungDiagram where
cells : Finset (ℕ × ℕ)
isLowerSet : IsLowerSet (cel... | Mathlib/Combinatorics/Young/YoungDiagram.lean | 219 | 221 | theorem transpose_transpose (μ : YoungDiagram) : μ.transpose.transpose = μ := by |
ext x
simp
| 0.4375 |
import Mathlib.Algebra.Module.Equiv
import Mathlib.Data.DFinsupp.Basic
import Mathlib.Data.Finsupp.Basic
#align_import data.finsupp.to_dfinsupp from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
variable {ι : Type*} {R : Type*} {M : Type*}
section Defs
def Finsupp.toDFinsupp [Zer... | Mathlib/Data/Finsupp/ToDFinsupp.lean | 117 | 119 | theorem DFinsupp.toFinsupp_support (f : Π₀ _ : ι, M) : f.toFinsupp.support = f.support := by |
ext
simp
| 0.4375 |
import Mathlib.Data.Rat.Sqrt
import Mathlib.Data.Real.Sqrt
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.Int.Basic
import Mathlib.Tactic.IntervalCases
#align_import data.real.irrational from "leanprover-community/mathlib"@"7e7aaccf9b0182576cabdde36cf1b5ad3585b70d"
open Rat Real multiplicity
def ... | Mathlib/Data/Real/Irrational.lean | 38 | 40 | theorem Transcendental.irrational {r : ℝ} (tr : Transcendental ℚ r) : Irrational r := by |
rintro ⟨a, rfl⟩
exact tr (isAlgebraic_algebraMap a)
| 0.4375 |
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
local macro:max "local_hAdd[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HAdd.hAdd : $type → $type → $type))
local macro:max "local_hMul[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HMul.hMul : $type → $typ... | Mathlib/Algebra/Ring/Ext.lean | 497 | 499 | theorem toSemiring_injective :
Function.Injective (@toSemiring R) := by |
rintro ⟨⟩ ⟨⟩ _; congr
| 0.4375 |
import Mathlib.Algebra.CharP.Two
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Data.Nat.Periodic
import Mathlib.Data.ZMod.Basic
import Mathlib.Tactic.Monotonicity
#align_import data.nat.totient from "leanprover-community/mathlib"@"5cc2dfdd3e92f340411acea4427d701dc7ed26f8"
open Finset
namespace Nat
... | Mathlib/Data/Nat/Totient.lean | 51 | 57 | theorem totient_eq_card_lt_and_coprime (n : ℕ) : φ n = Nat.card { m | m < n ∧ n.Coprime m } := by |
let e : { m | m < n ∧ n.Coprime m } ≃ Finset.filter n.Coprime (Finset.range n) :=
{ toFun := fun m => ⟨m, by simpa only [Finset.mem_filter, Finset.mem_range] using m.property⟩
invFun := fun m => ⟨m, by simpa only [Finset.mem_filter, Finset.mem_range] using m.property⟩
left_inv := fun m => by simp onl... | 0.4375 |
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 | 97 | 99 | theorem map_comp_coe {α β} (h : α → β) :
Functor.map h ∘ Coe.coe = (Coe.coe ∘ Functor.map h : List α → Multiset β) := by |
funext; simp only [Function.comp_apply, Coe.coe, fmap_def, map_coe, List.map_eq_map]
| 0.4375 |
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Analysis.SumOverResidueClass
#align_import analysis.p_series from "leanprover-community/mathlib"@"0b9eaaa7686280fad8cce467f5c3c57ee6ce77f8"
def SuccDiffBounded (C : ℕ) (u : ℕ → ℕ) : Prop :=... | Mathlib/Analysis/PSeries.lean | 100 | 104 | theorem sum_condensed_le' (hf : ∀ ⦃m n⦄, 1 < m → m ≤ n → f n ≤ f m) (n : ℕ) :
(∑ k ∈ range n, 2 ^ k • f (2 ^ (k + 1))) ≤ ∑ k ∈ Ico 2 (2 ^ n + 1), f k := by |
convert sum_schlomilch_le' hf (fun n => pow_pos zero_lt_two n)
(fun m n hm => pow_le_pow_right one_le_two hm) n using 2
simp [pow_succ, mul_two, two_mul]
| 0.4375 |
import Mathlib.Data.List.Range
import Mathlib.Data.List.Perm
#align_import data.list.sigma from "leanprover-community/mathlib"@"f808feb6c18afddb25e66a71d317643cf7fb5fbb"
universe u v
namespace List
variable {α : Type u} {β : α → Type v} {l l₁ l₂ : List (Sigma β)}
def keys : List (Sigma β) → List α :=
map ... | Mathlib/Data/List/Sigma.lean | 123 | 125 | theorem NodupKeys.eq_of_mk_mem {a : α} {b b' : β a} {l : List (Sigma β)} (nd : NodupKeys l)
(h : Sigma.mk a b ∈ l) (h' : Sigma.mk a b' ∈ l) : b = b' := by |
cases nd.eq_of_fst_eq h h' rfl; rfl
| 0.4375 |
import Mathlib.Geometry.Manifold.MFDeriv.Defs
#align_import geometry.manifold.mfderiv from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
noncomputable section
open scoped Topology Manifold
open Set Bundle
section DerivativesProperties
variable
{𝕜 : Type*} [NontriviallyNormedFiel... | Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | 46 | 49 | theorem uniqueMDiffWithinAt_univ : UniqueMDiffWithinAt I univ x := by |
unfold UniqueMDiffWithinAt
simp only [preimage_univ, univ_inter]
exact I.unique_diff _ (mem_range_self _)
| 0.4375 |
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Combinatorics.Quiver.Path
#align_import combinatorics.quiver.cast from "leanprover-community/mathlib"@"fc2ed6f838ce7c9b7c7171e58d78eaf7b438fb0e"
universe v v₁ v₂ u u₁ u₂
variable {U : Type*} [Quiver.{u + 1} U]
namespace Quiver
def Hom.cast {u v u' v... | Mathlib/Combinatorics/Quiver/Cast.lean | 99 | 103 | theorem Path.cast_cast {u v u' v' u'' v'' : U} (p : Path u v) (hu : u = u') (hv : v = v')
(hu' : u' = u'') (hv' : v' = v'') :
(p.cast hu hv).cast hu' hv' = p.cast (hu.trans hu') (hv.trans hv') := by |
subst_vars
rfl
| 0.4375 |
import Mathlib.Data.Real.NNReal
import Mathlib.Tactic.GCongr.Core
#align_import analysis.normed.group.seminorm from "leanprover-community/mathlib"@"09079525fd01b3dda35e96adaa08d2f943e1648c"
open Set
open NNReal
variable {ι R R' E F G : Type*}
structure AddGroupSeminorm (G : Type*) [AddGroup G] where
-- Port... | Mathlib/Analysis/Normed/Group/Seminorm.lean | 148 | 150 | theorem map_sub_le_max : f (x - y) ≤ max (f x) (f y) := by |
rw [sub_eq_add_neg, ← NonarchAddGroupSeminormClass.map_neg_eq_map' f y]
exact map_add_le_max _ _ _
| 0.4375 |
import Mathlib.Analysis.InnerProductSpace.Calculus
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.inner_product_space.euclidean_dist from "leanprover-community/mathlib"@"9425b6f8220e53b059f5a4904786c3c4b50fc057"
open scoped Topology
open Set
variable {E : Type*} [AddCommGroup E] [Topologi... | Mathlib/Analysis/InnerProductSpace/EuclideanDist.lean | 108 | 110 | theorem nhds_basis_closedBall {x : E} : (𝓝 x).HasBasis (fun r : ℝ => 0 < r) (closedBall x) := by |
rw [toEuclidean.toHomeomorph.nhds_eq_comap x]
exact Metric.nhds_basis_closedBall.comap _
| 0.4375 |
import Mathlib.MeasureTheory.Function.AEEqFun.DomAct
import Mathlib.MeasureTheory.Function.LpSpace
set_option autoImplicit true
open MeasureTheory Filter
open scoped ENNReal
namespace DomMulAct
variable {M N α E : Type*} [MeasurableSpace M] [MeasurableSpace N]
[MeasurableSpace α] [NormedAddCommGroup E] {μ : Me... | Mathlib/MeasureTheory/Function/LpSpace/DomAct/Basic.lean | 70 | 71 | theorem smul_Lp_add (c : Mᵈᵐᵃ) : ∀ f g : Lp E p μ, c • (f + g) = c • f + c • g := by |
rintro ⟨⟨⟩, _⟩ ⟨⟨⟩, _⟩; rfl
| 0.4375 |
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.RingTheory.AlgebraTower
import Mathlib.Algebra.Algebra.Subalgebra.Tower
#align_import linear_algebra.matrix.to_lin from "leanprover-community/mathlib"@"0e2aab2b0d521f060f62a14d2cf2e2c54e8491d6"
... | Mathlib/LinearAlgebra/Matrix/ToLin.lean | 173 | 176 | theorem Matrix.toLinearMapRight'_one :
Matrix.toLinearMapRight' (1 : Matrix m m R) = LinearMap.id := by |
ext
simp [LinearMap.one_apply, stdBasis_apply]
| 0.4375 |
import Mathlib.AlgebraicTopology.DoldKan.Faces
import Mathlib.CategoryTheory.Idempotents.Basic
#align_import algebraic_topology.dold_kan.projections from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Pread... | Mathlib/AlgebraicTopology/DoldKan/Projections.lean | 75 | 77 | theorem P_add_Q (q : ℕ) : P q + Q q = 𝟙 K[X] := by |
rw [Q]
abel
| 0.4375 |
import Mathlib.AlgebraicTopology.DoldKan.Faces
import Mathlib.CategoryTheory.Idempotents.Basic
#align_import algebraic_topology.dold_kan.projections from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Pread... | Mathlib/AlgebraicTopology/DoldKan/Projections.lean | 61 | 65 | theorem P_f_0_eq (q : ℕ) : ((P q).f 0 : X _[0] ⟶ X _[0]) = 𝟙 _ := by |
induction' q with q hq
· rfl
· simp only [P_succ, HomologicalComplex.add_f_apply, HomologicalComplex.comp_f,
HomologicalComplex.id_f, id_comp, hq, Hσ_eq_zero, add_zero]
| 0.4375 |
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 | 66 | 68 | theorem gauge_def' : gauge s x = sInf {r ∈ Set.Ioi (0 : ℝ) | r⁻¹ • x ∈ s} := by |
congrm sInf {r | ?_}
exact and_congr_right fun hr => mem_smul_set_iff_inv_smul_mem₀ hr.ne' _ _
| 0.4375 |
import Mathlib.Algebra.Order.Field.Power
import Mathlib.Data.Int.LeastGreatest
import Mathlib.Data.Rat.Floor
import Mathlib.Data.NNRat.Defs
#align_import algebra.order.archimedean from "leanprover-community/mathlib"@"6f413f3f7330b94c92a5a27488fdc74e6d483a78"
open Int Set
variable {α : Type*}
class Archimedean (... | Mathlib/Algebra/Order/Archimedean.lean | 138 | 148 | theorem add_one_pow_unbounded_of_pos (x : α) (hy : 0 < y) : ∃ n : ℕ, x < (y + 1) ^ n :=
have : 0 ≤ 1 + y := add_nonneg zero_le_one hy.le
(Archimedean.arch x hy).imp fun n h ↦
calc
x ≤ n • y := h
_ = n * y := nsmul_eq_mul _ _
_ < 1 + n * y := lt_one_add _
_ ≤ (1 + y) ^ n :=
one_ad... | rw [add_comm]
| 0.4375 |
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.Algebra.CharP.Reduced
open Function Polynomial
class PerfectRing (R : Type*) (p : ℕ) [CommSemiring R] [ExpChar R p] : Prop where
bijective_frobenius : Bijective <| frobenius R p
section PerfectRing
va... | Mathlib/FieldTheory/Perfect.lean | 168 | 171 | theorem not_irreducible_expand (R p) [CommSemiring R] [Fact p.Prime] [CharP R p] [PerfectRing R p]
(f : R[X]) : ¬ Irreducible (expand R p f) := by |
rw [polynomial_expand_eq]
exact not_irreducible_pow (Fact.out : p.Prime).ne_one
| 0.4375 |
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.Matrix.NonsingularInverse
#align_import linear_algebra.affine_space.matrix from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
open Affine Matrix
open Set
universe u₁ u₂ u₃ u₄
variable {ι : Type u₁} {k : Type... | Mathlib/LinearAlgebra/AffineSpace/Matrix.lean | 55 | 56 | theorem toMatrix_row_sum_one [Fintype ι] (q : ι' → P) (i : ι') : ∑ j, b.toMatrix q i j = 1 := by |
simp
| 0.4375 |
import Mathlib.Data.Matrix.Basis
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.Pi
#align_import linear_algebra.std_basis from "leanprover-community/mathlib"@"13bce9a6b6c44f6b4c91ac1c1d2a816e2533d395"
open Function Set Submodule
namespace LinearMap
variable (R : Type*) {ι : Type*} [Semiring R] ... | Mathlib/LinearAlgebra/StdBasis.lean | 55 | 57 | theorem stdBasis_apply' (i i' : ι) : (stdBasis R (fun _x : ι => R) i) 1 i' = ite (i = i') 1 0 := by |
rw [LinearMap.stdBasis_apply, Function.update_apply, Pi.zero_apply]
congr 1; rw [eq_iff_iff, eq_comm]
| 0.4375 |
import Mathlib.Order.Interval.Set.Image
import Mathlib.Order.CompleteLatticeIntervals
import Mathlib.Topology.Order.DenselyOrdered
import Mathlib.Topology.Order.Monotone
#align_import topology.algebra.order.intermediate_value from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
open Filt... | Mathlib/Topology/Order/IntermediateValue.lean | 115 | 124 | theorem IsPreconnected.intermediate_value₂_eventually₂ {s : Set X} (hs : IsPreconnected s)
{l₁ l₂ : Filter X} [NeBot l₁] [NeBot l₂] (hl₁ : l₁ ≤ 𝓟 s) (hl₂ : l₂ ≤ 𝓟 s) {f g : X → α}
(hf : ContinuousOn f s) (hg : ContinuousOn g s) (he₁ : f ≤ᶠ[l₁] g) (he₂ : g ≤ᶠ[l₂] f) :
∃ x ∈ s, f x = g x := by |
rw [continuousOn_iff_continuous_restrict] at hf hg
obtain ⟨b, h⟩ :=
@intermediate_value_univ₂_eventually₂ _ _ _ _ _ _ (Subtype.preconnectedSpace hs) _ _
(comap_coe_neBot_of_le_principal hl₁) (comap_coe_neBot_of_le_principal hl₂) _ _ hf hg
(he₁.comap _) (he₂.comap _)
exact ⟨b, b.prop, h⟩
| 0.4375 |
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 Sigma
variable {α : ι → Type*} {β : Type*} (s s₁ s₂ : Finset ι) (... | Mathlib/Data/Finset/Sigma.lean | 112 | 114 | theorem _root_.biSup_finsetSigma [CompleteLattice β] (s : Finset ι) (t : ∀ i, Finset (α i))
(f : Sigma α → β) : ⨆ ij ∈ s.sigma t, f ij = ⨆ (i ∈ s) (j ∈ t i), f ⟨i, j⟩ := by |
simp_rw [← Finset.iSup_coe, Finset.coe_sigma, biSup_sigma]
| 0.4375 |
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.Algebra.CharP.Reduced
open Function Polynomial
class PerfectRing (R : Type*) (p : ℕ) [CommSemiring R] [ExpChar R p] : Prop where
bijective_frobenius : Bijective <| frobenius R p
section PerfectRing
va... | Mathlib/FieldTheory/Perfect.lean | 131 | 133 | theorem iterateFrobeniusEquiv_symm :
(iterateFrobeniusEquiv R p n).symm = (frobeniusEquiv R p).symm ^ n := by |
rw [iterateFrobeniusEquiv_eq_pow]; exact (inv_pow _ _).symm
| 0.4375 |
import Mathlib.Data.Finset.Image
import Mathlib.Data.List.FinRange
#align_import data.fintype.basic from "leanprover-community/mathlib"@"d78597269638367c3863d40d45108f52207e03cf"
assert_not_exists MonoidWithZero
assert_not_exists MulAction
open Function
open Nat
universe u v
variable {α β γ : Type*}
class Fi... | Mathlib/Data/Fintype/Basic.lean | 150 | 151 | theorem codisjoint_left : Codisjoint s t ↔ ∀ ⦃a⦄, a ∉ s → a ∈ t := by |
classical simp [codisjoint_iff, eq_univ_iff_forall, or_iff_not_imp_left]
| 0.4375 |
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.BigOperators
#align_import data.sign from "leanprover-community/mathlib"@"2445c98ae4b87eabebdde552593519b9b6dc350c"
-- Porting note (#11081): cannot automatically derive Fintype, adde... | Mathlib/Data/Sign.lean | 177 | 177 | theorem lt_one_iff {a : SignType} : a < 1 ↔ a ≤ 0 := by | cases a <;> decide
| 0.4375 |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Basic
open Topology InnerProductSpace Set
noncomputable section
variable {𝕜 F : Type*} [RCLike 𝕜]
variable [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F]
variabl... | Mathlib/Analysis/Calculus/Gradient/Basic.lean | 110 | 111 | theorem gradient_eq_zero_of_not_differentiableAt (h : ¬DifferentiableAt 𝕜 f x) : ∇ f x = 0 := by |
rw [gradient, fderiv_zero_of_not_differentiableAt h, map_zero]
| 0.4375 |
import Mathlib.Algebra.DirectSum.Basic
import Mathlib.LinearAlgebra.DFinsupp
import Mathlib.LinearAlgebra.Basis
#align_import algebra.direct_sum.module from "leanprover-community/mathlib"@"6623e6af705e97002a9054c1c05a980180276fc1"
universe u v w u₁
namespace DirectSum
open DirectSum
section General
variable {... | Mathlib/Algebra/DirectSum/Module.lean | 164 | 168 | theorem linearEquivFunOnFintype_lof [Fintype ι] [DecidableEq ι] (i : ι) (m : M i) :
(linearEquivFunOnFintype R ι M) (lof R ι M i m) = Pi.single i m := by |
ext a
change (DFinsupp.equivFunOnFintype (lof R ι M i m)) a = _
convert _root_.congr_fun (DFinsupp.equivFunOnFintype_single i m) a
| 0.4375 |
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
open Set NNReal ENNReal
namespace ENNReal
section Real
variable {a b c d : ℝ≥0∞} {r p q : ℝ≥0}
theorem toReal_add (ha : a ≠ ∞) (hb : b ≠ ∞) : (a + b).toReal = a.toReal ... | Mathlib/Data/ENNReal/Real.lean | 132 | 133 | theorem toNNReal_strict_mono (hb : b ≠ ∞) (h : a < b) : a.toNNReal < b.toNNReal := by |
simpa [← ENNReal.coe_lt_coe, hb, h.ne_top]
| 0.4375 |
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 | 93 | 98 | theorem interedges_disjoint_right (s : Finset α) {t t' : Finset β} (ht : Disjoint t t') :
Disjoint (interedges r s t) (interedges r s t') := by |
rw [Finset.disjoint_left] at ht ⊢
intro _ hx hy
rw [mem_interedges_iff] at hx hy
exact ht hx.2.1 hy.2.1
| 0.4375 |
import Mathlib.Geometry.Manifold.MFDeriv.Basic
noncomputable section
open scoped Manifold
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜] {E : Type*} [NormedAddCommGroup E]
[NormedSpace 𝕜 E] {E' : Type*} [NormedAddCommGroup E'] [NormedSpace 𝕜 E'] {f : E → E'}
{s : Set E} {x : E}
section MFDerivFderiv
t... | Mathlib/Geometry/Manifold/MFDeriv/FDeriv.lean | 49 | 52 | theorem hasMFDerivWithinAt_iff_hasFDerivWithinAt {f'} :
HasMFDerivWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E') f s x f' ↔ HasFDerivWithinAt f f' s x := by |
simpa only [HasMFDerivWithinAt, and_iff_right_iff_imp, mfld_simps] using
HasFDerivWithinAt.continuousWithinAt
| 0.4375 |
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Pullbacks
#align_import category_theory.limits.constructions.epi_mono from "leanprover-community/mathlib"@"f7baecbb54bd0f24f228576f97b1752fc3c9b318"
... | Mathlib/CategoryTheory/Limits/Constructions/EpiMono.lean | 32 | 36 | theorem preserves_mono_of_preservesLimit {X Y : C} (f : X ⟶ Y) [PreservesLimit (cospan f f) F]
[Mono f] : Mono (F.map f) := by |
have := isLimitPullbackConeMapOfIsLimit F _ (PullbackCone.isLimitMkIdId f)
simp_rw [F.map_id] at this
apply PullbackCone.mono_of_isLimitMkIdId _ this
| 0.4375 |
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
#align_import analysis.ODE.gronwall from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
variable {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E] {F : Type*} [NormedAddCommGroup F]
[NormedSpace ℝ F]
open Metric Set Asymptotics Fil... | Mathlib/Analysis/ODE/Gronwall.lean | 73 | 76 | theorem hasDerivAt_gronwallBound_shift (δ K ε x a : ℝ) :
HasDerivAt (fun y => gronwallBound δ K ε (y - a)) (K * gronwallBound δ K ε (x - a) + ε) x := by |
convert (hasDerivAt_gronwallBound δ K ε _).comp x ((hasDerivAt_id x).sub_const a) using 1
rw [id, mul_one]
| 0.4375 |
import Mathlib.LinearAlgebra.AffineSpace.Independent
import Mathlib.LinearAlgebra.Basis
#align_import linear_algebra.affine_space.basis from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
open Affine
open Set
universe u₁ u₂ u₃ u₄
structure AffineBasis (ι : Type u₁) (k : Type u₂) {V ... | Mathlib/LinearAlgebra/AffineSpace/Basis.lean | 139 | 143 | theorem basisOf_reindex (i : ι') :
(b.reindex e).basisOf i =
(b.basisOf <| e.symm i).reindex (e.subtypeEquiv fun _ => e.eq_symm_apply.not) := by |
ext j
simp
| 0.4375 |
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Ring.Pi
import Mathlib.Data.Finsupp.Defs
#align_import data.finsupp.pointwise from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
noncomputable section
open Finset
universe u₁ u₂ u₃ u₄ u₅
variable {α : Type u₁} {β : Type u₂} {... | Mathlib/Data/Finsupp/Pointwise.lean | 57 | 65 | theorem support_mul [DecidableEq α] {g₁ g₂ : α →₀ β} :
(g₁ * g₂).support ⊆ g₁.support ∩ g₂.support := by |
intro a h
simp only [mul_apply, mem_support_iff] at h
simp only [mem_support_iff, mem_inter, Ne]
rw [← not_or]
intro w
apply h
cases' w with w w <;> (rw [w]; simp)
| 0.4375 |
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 | 41 | 42 | theorem mem_nhdsSet_iff_forall : s ∈ 𝓝ˢ t ↔ ∀ x : X, x ∈ t → s ∈ 𝓝 x := by |
simp_rw [nhdsSet, Filter.mem_sSup, forall_mem_image]
| 0.4375 |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology.Instances.Sign
#align_import analysis.special_functions.trigonometric.angle from "leanprover-community/mathlib"@"213b0cff7bc5ab6696ee07cceec80829... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 112 | 113 | theorem intCast_mul_eq_zsmul (x : ℝ) (n : ℤ) : ↑((n : ℝ) * x : ℝ) = n • (↑x : Angle) := by |
simpa only [zsmul_eq_mul] using coeHom.map_zsmul x n
| 0.4375 |
import Mathlib.Data.ZMod.Basic
import Mathlib.GroupTheory.Coxeter.Basic
namespace CoxeterSystem
open List Matrix Function Classical
variable {B : Type*}
variable {W : Type*} [Group W]
variable {M : CoxeterMatrix B} (cs : CoxeterSystem M W)
local prefix:100 "s" => cs.simple
local prefix:100 "π" => cs.wordProd
... | Mathlib/GroupTheory/Coxeter/Length.lean | 81 | 88 | theorem length_eq_zero_iff {w : W} : ℓ w = 0 ↔ w = 1 := by |
constructor
· intro h
rcases cs.exists_reduced_word w with ⟨ω, hω, rfl⟩
have : ω = [] := eq_nil_of_length_eq_zero (hω.trans h)
rw [this, wordProd_nil]
· rintro rfl
exact cs.length_one
| 0.4375 |
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.RingTheory.Ideal.QuotientOperations
#align_import ring_theory.valuation.quotient from "leanprover-community/mathlib"@"da420a8c6dd5bdfb85c4ced85c34388f633bc6ff"
namespace Valuation
variable {R Γ₀ : Type*} [CommRing R] [LinearOrderedCommMonoidWithZero Γ₀]
va... | Mathlib/RingTheory/Valuation/Quotient.lean | 77 | 79 | theorem supp_quot_supp : supp (v.onQuot le_rfl) = 0 := by |
rw [supp_quot]
exact Ideal.map_quotient_self _
| 0.4375 |
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 | 98 | 101 | theorem IsDiag.sub [AddGroup α] {A B : Matrix n n α} (ha : A.IsDiag) (hb : B.IsDiag) :
(A - B).IsDiag := by |
intro i j h
simp [ha h, hb h]
| 0.4375 |
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.LinearAlgebra.BilinearForm.Properties
open LinearMap (BilinForm)
universe u v w
variable {R : Type*} {M : Type*} [CommSemiring R] [AddCommMonoid M] [Module R M]
variable {R₁ : Type*} {M₁ : Type*} [CommRing R₁] [AddCommGroup M₁] [Module R₁ M₁]
va... | Mathlib/LinearAlgebra/BilinearForm/Orthogonal.lean | 109 | 114 | theorem isOrtho_smul_right {x y : M₄} {a : R₄} (ha : a ≠ 0) :
IsOrtho G x (a • y) ↔ IsOrtho G x y := by |
dsimp only [IsOrtho]
rw [map_smul]
simp only [smul_eq_mul, mul_eq_zero, or_iff_right_iff_imp]
exact fun a ↦ (ha a).elim
| 0.4375 |
import Mathlib.Algebra.Order.Ring.Abs
#align_import data.int.order.units from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
namespace Int
theorem isUnit_iff_abs_eq {x : ℤ} : IsUnit x ↔ abs x = 1 := by
rw [isUnit_iff_natAbs_eq, abs_eq_natAbs, ← Int.ofNat_one, natCast_inj]
#align int.... | Mathlib/Data/Int/Order/Units.lean | 63 | 67 | theorem units_pow_eq_pow_mod_two (u : ℤˣ) (n : ℕ) : u ^ n = u ^ (n % 2) := by |
conv =>
lhs
rw [← Nat.mod_add_div n 2];
rw [pow_add, pow_mul, units_sq, one_pow, mul_one]
| 0.4375 |
import Mathlib.MeasureTheory.MeasurableSpace.Defs
import Mathlib.Order.Filter.CountableInter
open Filter Set MeasurableSpace
variable {α : Type*} (m : MeasurableSpace α) (l : Filter α) [CountableInterFilter l] {s t : Set α}
def EventuallyMeasurableSpace : MeasurableSpace α where
MeasurableSet' s := ∃ t, Measu... | Mathlib/MeasureTheory/Constructions/EventuallyMeasurable.lean | 67 | 70 | theorem EventuallyMeasurableSet.congr
(ht : EventuallyMeasurableSet m l t) (hst : s =ᶠ[l] t) : EventuallyMeasurableSet m l s := by |
rcases ht with ⟨t', ht', htt'⟩
exact ⟨t', ht', hst.trans htt'⟩
| 0.4375 |
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.RingTheory.Coprime.Basic
import Mathlib.Tactic.AdaptationNote
#align_import ring_theory.polynomial.scale_roots from "leanprover-community/mathlib"@"40ac1b258344e0c2b4568dc37bfad937ec35a727"
variable {R... | Mathlib/RingTheory/Polynomial/ScaleRoots.lean | 94 | 95 | theorem monic_scaleRoots_iff {p : R[X]} (s : R) : Monic (scaleRoots p s) ↔ Monic p := by |
simp only [Monic, leadingCoeff, natDegree_scaleRoots, coeff_scaleRoots_natDegree]
| 0.4375 |
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.set.ncard from "leanprover-community/mathlib"@"74c2af38a828107941029b03839882c5c6f87a04"
namespace Set
variable {α β : Type*} {s t : Set α}
noncomputable def encard (s : Set α) : ℕ∞ := PartENat.withTopEquiv (PartENat.card s)
@[simp] theorem encard_uni... | Mathlib/Data/Set/Card.lean | 119 | 123 | theorem Finite.encard_lt_top (h : s.Finite) : s.encard < ⊤ := by |
refine h.induction_on (by simp) ?_
rintro a t hat _ ht'
rw [encard_insert_of_not_mem hat]
exact lt_tsub_iff_right.1 ht'
| 0.4375 |
import Mathlib.Algebra.Order.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#align_import algebra.order.group.defs from "leanprover-community/mathlib"@"b599f4e4e5cf1fbcb4194503671d3d9e569c1fce"
open Function
universe u
variable {α : Type u}
class OrderedAddCommGroup (α : Ty... | Mathlib/Algebra/Order/Group/Defs.lean | 287 | 288 | theorem Right.one_lt_inv_iff : 1 < a⁻¹ ↔ a < 1 := by |
rw [← mul_lt_mul_iff_right a, inv_mul_self, one_mul]
| 0.4375 |
import Mathlib.Algebra.Field.Rat
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.Field.Rat
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Rat.Lemmas
#align_import data.rat.cast from "leanprover-community/mathlib"@"acebd8d49928f6ed8920e... | Mathlib/Data/Rat/Cast/Defs.lean | 143 | 144 | theorem cast_commute (r : ℚ) (a : α) : Commute (↑r) a := by |
simpa only [cast_def] using (r.1.cast_commute a).div_left (r.2.cast_commute a)
| 0.4375 |
import Mathlib.Data.Finset.Option
import Mathlib.Data.PFun
import Mathlib.Data.Part
#align_import data.finset.pimage from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
variable {α β : Type*}
namespace Part
def toFinset (o : Part α) [Decidable o.Dom] : Finset α :=
o.toOption.toFins... | Mathlib/Data/Finset/PImage.lean | 34 | 35 | theorem mem_toFinset {o : Part α} [Decidable o.Dom] {x : α} : x ∈ o.toFinset ↔ x ∈ o := by |
simp [toFinset]
| 0.4375 |
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Group.MinMax
#align_import data.set.pointwise.interval from "leanprover-community/mathlib"@"2196ab363eb097c... | Mathlib/Data/Set/Pointwise/Interval.lean | 157 | 158 | theorem preimage_const_add_Ioc : (fun x => a + x) ⁻¹' Ioc b c = Ioc (b - a) (c - a) := by |
simp [← Ioi_inter_Iic]
| 0.4375 |
import Mathlib.Data.List.Forall2
#align_import data.list.zip from "leanprover-community/mathlib"@"134625f523e737f650a6ea7f0c82a6177e45e622"
-- Make sure we don't import algebra
assert_not_exists Monoid
universe u
open Nat
namespace List
variable {α : Type u} {β γ δ ε : Type*}
#align list.zip_with_cons_cons Li... | Mathlib/Data/List/Zip.lean | 63 | 64 | theorem lt_length_left_of_zipWith {f : α → β → γ} {i : ℕ} {l : List α} {l' : List β}
(h : i < (zipWith f l l').length) : i < l.length := by | rw [length_zipWith] at h; omega
| 0.4375 |
import Mathlib.AlgebraicTopology.DoldKan.Projections
import Mathlib.CategoryTheory.Idempotents.FunctorCategories
import Mathlib.CategoryTheory.Idempotents.FunctorExtension
#align_import algebraic_topology.dold_kan.p_infty from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
open Category... | Mathlib/AlgebraicTopology/DoldKan/PInfty.lean | 78 | 80 | theorem QInfty_f_0 : (QInfty.f 0 : X _[0] ⟶ X _[0]) = 0 := by |
dsimp [QInfty]
simp only [sub_self]
| 0.4375 |
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac1835a3852b9"
open scoped ENNReal
namespace MeasureTheory
variable {α E : Type*} {m0 : MeasurableSpace α} [NormedAddCommGroup E]
{p : ℝ≥0∞} (μ... | Mathlib/MeasureTheory/Function/LpSeminorm/ChebyshevMarkov.lean | 31 | 40 | theorem mul_meas_ge_le_pow_snorm (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
(hf : AEStronglyMeasurable f μ) (ε : ℝ≥0∞) :
ε * μ { x | ε ≤ (‖f x‖₊ : ℝ≥0∞) ^ p.toReal } ≤ snorm f p μ ^ p.toReal := by |
have : 1 / p.toReal * p.toReal = 1 := by
refine one_div_mul_cancel ?_
rw [Ne, ENNReal.toReal_eq_zero_iff]
exact not_or_of_not hp_ne_zero hp_ne_top
rw [← ENNReal.rpow_one (ε * μ { x | ε ≤ (‖f x‖₊ : ℝ≥0∞) ^ p.toReal }), ← this, ENNReal.rpow_mul]
gcongr
exact pow_mul_meas_ge_le_snorm μ hp_ne_zero hp_n... | 0.4375 |
import Mathlib.LinearAlgebra.Dimension.LinearMap
import Mathlib.LinearAlgebra.FreeModule.StrongRankCondition
#align_import linear_algebra.free_module.finite.matrix from "leanprover-community/mathlib"@"b1c23399f01266afe392a0d8f71f599a0dad4f7b"
universe u u' v w
variable (R : Type u) (S : Type u') (M : Type v) (N ... | Mathlib/LinearAlgebra/FreeModule/Finite/Matrix.lean | 66 | 68 | theorem FiniteDimensional.rank_linearMap_self :
Module.rank S (M →ₗ[R] S) = lift.{u'} (Module.rank R M) := by |
rw [rank_linearMap, rank_self, lift_one, mul_one]
| 0.4375 |
import Mathlib.Init.Logic
import Mathlib.Init.Function
import Mathlib.Init.Algebra.Classes
import Batteries.Util.LibraryNote
import Batteries.Tactic.Lint.Basic
#align_import logic.basic from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9abdddf3ffe"
#align_import init.ite_simp from "leanprover-communit... | Mathlib/Logic/Basic.lean | 1,131 | 1,131 | theorem exists₂_imp : (∃ x h, P x h) → b ↔ ∀ x h, P x h → b := by | simp
| 0.4375 |
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Data.Multiset.Fold
#align_import data.multiset.lattice from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
namespace Multiset
variable {α : Type*}
section Sup
-- can be defined with just `[Bot α]` where some lemmas hold without... | Mathlib/Data/Multiset/Lattice.lean | 84 | 85 | theorem sup_union (s₁ s₂ : Multiset α) : (s₁ ∪ s₂).sup = s₁.sup ⊔ s₂.sup := by |
rw [← sup_dedup, dedup_ext.2, sup_dedup, sup_add]; simp
| 0.4375 |
import Mathlib.Data.Real.Basic
#align_import data.real.sign from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Real
noncomputable def sign (r : ℝ) : ℝ :=
if r < 0 then -1 else if 0 < r then 1 else 0
#align real.sign Real.sign
theorem sign_of_neg {r : ℝ} (hr : r < 0) : si... | Mathlib/Data/Real/Sign.lean | 108 | 115 | theorem inv_sign (r : ℝ) : (sign r)⁻¹ = sign r := by |
obtain hn | hz | hp := sign_apply_eq r
· rw [hn]
norm_num
· rw [hz]
exact inv_zero
· rw [hp]
exact inv_one
| 0.4375 |
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Data.Int.Cast.Defs
import Mathlib.Tactic.Spread
import Mathlib.Util.AssertExists
#align_import algebra.ring.defs from "leanprover-community/mathlib"@"76de8ae01554c3b37d66544866659ff174e66e1f"
universe u v w x
variable {α : ... | Mathlib/Algebra/Ring/Defs.lean | 168 | 169 | theorem mul_one_add [LeftDistribClass α] (a b : α) : a * (1 + b) = a + a * b := by |
rw [mul_add, mul_one]
| 0.4375 |
import Mathlib.Order.Interval.Multiset
#align_import data.nat.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
-- TODO
-- assert_not_exists Ring
open Finset Nat
variable (a b c : ℕ)
namespace Nat
instance instLocallyFiniteOrder : LocallyFiniteOrder ℕ where
finsetIcc a b... | Mathlib/Order/Interval/Finset/Nat.lean | 109 | 109 | theorem card_Iio : (Iio b).card = b := by | rw [Iio_eq_Ico, card_Ico, Nat.bot_eq_zero, Nat.sub_zero]
| 0.4375 |
import Mathlib.Algebra.GradedMonoid
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.MvPolynomial.Basic
#align_import ring_theory.mv_polynomial.weighted_homogeneous from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Fins... | Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean | 120 | 122 | theorem weightedTotalDegree_zero (w : σ → M) :
weightedTotalDegree w (0 : MvPolynomial σ R) = ⊥ := by |
simp only [weightedTotalDegree, support_zero, Finset.sup_empty]
| 0.4375 |
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import measure_theory.measure.haar.of_basis from "leanprover-community/mathlib"@"92bd7b1ffeb306a89f450bee126ddd8a284c259d"
open Set TopologicalSpace MeasureTheory MeasureTheory.Measure FiniteDimensional
open sco... | Mathlib/MeasureTheory/Measure/Haar/OfBasis.lean | 52 | 54 | theorem mem_parallelepiped_iff (v : ι → E) (x : E) :
x ∈ parallelepiped v ↔ ∃ t ∈ Icc (0 : ι → ℝ) 1, x = ∑ i, t i • v i := by |
simp [parallelepiped, eq_comm]
| 0.4375 |
import Mathlib.Algebra.Ring.Defs
import Mathlib.Algebra.Group.Ext
local macro:max "local_hAdd[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HAdd.hAdd : $type → $type → $type))
local macro:max "local_hMul[" type:term ", " inst:term "]" : term =>
`(term| (letI := $inst; HMul.hMul : $type → $typ... | Mathlib/Algebra/Ring/Ext.lean | 90 | 92 | theorem toNonUnitalNonAssocSemiring_injective :
Function.Injective (@toNonUnitalNonAssocSemiring R) := by |
rintro ⟨⟩ ⟨⟩ _; congr
| 0.4375 |
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
#align_import geometry.euclidean.angle.oriented.rotation from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
noncomputable section
open FiniteDimensional Complex
open scoped ... | Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 140 | 140 | theorem rotation_zero : o.rotation 0 = LinearIsometryEquiv.refl ℝ V := by | ext; simp [rotation]
| 0.4375 |
import Mathlib.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.RightAngle
#align_import geometry.euclidean.angle.oriented.right_angle from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open scoped EuclideanGeometry
ope... | Mathlib/Geometry/Euclidean/Angle/Oriented/RightAngle.lean | 36 | 42 | theorem oangle_add_right_eq_arccos_of_oangle_eq_pi_div_two {x y : V} (h : o.oangle x y = ↑(π / 2)) :
o.oangle x (x + y) = Real.arccos (‖x‖ / ‖x + y‖) := by |
have hs : (o.oangle x (x + y)).sign = 1 := by
rw [oangle_sign_add_right, h, Real.Angle.sign_coe_pi_div_two]
rw [o.oangle_eq_angle_of_sign_eq_one hs,
InnerProductGeometry.angle_add_eq_arccos_of_inner_eq_zero
(o.inner_eq_zero_of_oangle_eq_pi_div_two h)]
| 0.4375 |
import Mathlib.SetTheory.Cardinal.ToNat
import Mathlib.Data.Nat.PartENat
#align_import set_theory.cardinal.basic from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
universe u v
open Function
variable {α : Type u}
namespace Cardinal
noncomputable def toPartENat : Cardinal →+o PartEN... | Mathlib/SetTheory/Cardinal/PartENat.lean | 104 | 105 | theorem toPartENat_lift (c : Cardinal.{v}) : toPartENat (lift.{u, v} c) = toPartENat c := by |
simp only [← partENatOfENat_toENat, toENat_lift]
| 0.4375 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.