Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.57k | proof stringlengths 5 7.36k | hint bool 2
classes |
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import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Range
#align_import data.list.indexes from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
assert_not_exists MonoidWithZero
universe u v
open Function
namespace List
variable {α : Type u} {β : Type v}
section MapIdx
-- Porting n... | Mathlib/Data/List/Indexes.lean | 132 | 138 | theorem mapIdxGo_length : ∀ (f : ℕ → α → β) (l : List α) (arr : Array β),
length (mapIdx.go f l arr) = length l + arr.size := by
intro f l |
intro f l
induction' l with head tail ih
· intro; simp only [mapIdx.go, Array.toList_eq, length_nil, Nat.zero_add]
· intro; simp only [mapIdx.go]; rw [ih]; simp only [Array.size_push, length_cons];
simp only [Nat.add_succ, add_zero, Nat.add_comm]
| true |
import Mathlib.Logic.Function.Basic
import Mathlib.Logic.Relator
import Mathlib.Init.Data.Quot
import Mathlib.Tactic.Cases
import Mathlib.Tactic.Use
import Mathlib.Tactic.MkIffOfInductiveProp
import Mathlib.Tactic.SimpRw
#align_import logic.relation from "leanprover-community/mathlib"@"3365b20c2ffa7c35e47e5209b89ba9a... | Mathlib/Logic/Relation.lean | 360 | 369 | theorem total_of_right_unique (U : Relator.RightUnique r) (ab : ReflTransGen r a b)
(ac : ReflTransGen r a c) : ReflTransGen r b c ∨ ReflTransGen r c b := by
induction' ab with b d _ bd IH |
induction' ab with b d _ bd IH
· exact Or.inl ac
· rcases IH with (IH | IH)
· rcases cases_head IH with (rfl | ⟨e, be, ec⟩)
· exact Or.inr (single bd)
· cases U bd be
exact Or.inl ec
· exact Or.inr (IH.tail bd)
| true |
import Mathlib.Algebra.Group.Center
import Mathlib.Data.Int.Cast.Lemmas
#align_import group_theory.subsemigroup.center from "leanprover-community/mathlib"@"1ac8d4304efba9d03fa720d06516fac845aa5353"
variable {M : Type*}
namespace Set
variable (M)
@[simp]
| Mathlib/Algebra/Ring/Center.lean | 24 | 37 | theorem natCast_mem_center [NonAssocSemiring M] (n : ℕ) : (n : M) ∈ Set.center M where
comm _:= by rw [Nat.commute_cast] | rw [Nat.commute_cast]
left_assoc _ _ := by
induction n with
| zero => rw [Nat.cast_zero, zero_mul, zero_mul, zero_mul]
| succ n ihn => rw [Nat.cast_succ, add_mul, one_mul, ihn, add_mul, add_mul, one_mul]
mid_assoc _ _ := by
induction n with
| zero => rw [Nat.cast_zero, zero_mul, mul_zero, zero_... | true |
import Mathlib.Data.Int.Order.Units
import Mathlib.Data.ZMod.IntUnitsPower
import Mathlib.RingTheory.TensorProduct.Basic
import Mathlib.LinearAlgebra.DirectSum.TensorProduct
import Mathlib.Algebra.DirectSum.Algebra
suppress_compilation
open scoped TensorProduct DirectSum
variable {R ι A B : Type*}
namespace Tens... | Mathlib/LinearAlgebra/TensorProduct/Graded/External.lean | 85 | 90 | theorem gradedCommAux_lof_tmul (i j : ι) (a : 𝒜 i) (b : ℬ j) :
gradedCommAux R 𝒜 ℬ (lof R _ 𝒜ℬ (i, j) (a ⊗ₜ b)) =
(-1 : ℤˣ)^(j * i) • lof R _ ℬ𝒜 (j, i) (b ⊗ₜ a) := by
rw [gradedCommAux] |
rw [gradedCommAux]
dsimp
simp [mul_comm i j]
| true |
import Mathlib.Analysis.SpecialFunctions.Pow.Complex
import Qq
#align_import analysis.special_functions.pow.real from "leanprover-community/mathlib"@"4fa54b337f7d52805480306db1b1439c741848c8"
noncomputable section
open scoped Classical
open Real ComplexConjugate
open Finset Set
namespace Real
variable {x y z... | Mathlib/Analysis/SpecialFunctions/Pow/Real.lean | 85 | 87 | theorem rpow_eq_zero_iff_of_nonneg (hx : 0 ≤ x) : x ^ y = 0 ↔ x = 0 ∧ y ≠ 0 := by
simp only [rpow_def_of_nonneg hx] |
simp only [rpow_def_of_nonneg hx]
split_ifs <;> simp [*, exp_ne_zero]
| true |
import Mathlib.Tactic.ApplyFun
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.Separation
#align_import topology.uniform_space.separation from "leanprover-community/mathlib"@"0c1f285a9f6e608ae2bdffa3f993eafb01eba829"
open Filter Set Function Topology Uniformity UniformSpace
open scoped Classical... | Mathlib/Topology/UniformSpace/Separation.lean | 150 | 152 | theorem t0Space_iff_uniformity :
T0Space α ↔ ∀ x y, (∀ r ∈ 𝓤 α, (x, y) ∈ r) → x = y := by |
simp only [t0Space_iff_inseparable, inseparable_iff_ker_uniformity, mem_ker, id]
| true |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.RingTheory.PowerBasis
#align_import ring_theory.is_adjoin_root from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
open scoped Polynomial
open Polynomial
noncomputable sec... | Mathlib/RingTheory/IsAdjoinRoot.lean | 203 | 207 | theorem eval₂_repr_eq_eval₂_of_map_eq (h : IsAdjoinRoot S f) (z : S) (w : R[X])
(hzw : h.map w = z) : (h.repr z).eval₂ i x = w.eval₂ i x := by
rw [eq_comm, ← sub_eq_zero, ← h.map_repr z, ← map_sub, h.map_eq_zero_iff] at hzw |
rw [eq_comm, ← sub_eq_zero, ← h.map_repr z, ← map_sub, h.map_eq_zero_iff] at hzw
obtain ⟨y, hy⟩ := hzw
rw [← sub_eq_zero, ← eval₂_sub, hy, eval₂_mul, hx, zero_mul]
| true |
import Mathlib.Algebra.Homology.Linear
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Tactic.Abel
#align_import algebra.homology.homotopy from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a148bf9ff"
universe v u
open scoped Classical
noncomputable section
open ... | Mathlib/Algebra/Homology/Homotopy.lean | 124 | 130 | theorem prevD_nat (C D : CochainComplex V ℕ) (i : ℕ) (f : ∀ i j, C.X i ⟶ D.X j) :
prevD i f = f i (i - 1) ≫ D.d (i - 1) i := by
dsimp [prevD] |
dsimp [prevD]
cases i
· simp only [shape, CochainComplex.prev_nat_zero, ComplexShape.up_Rel, Nat.one_ne_zero,
not_false_iff, comp_zero]
· congr <;> simp
| true |
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Mathlib.Analysis.NormedSpace.FunctionSeries
#align_import analysis.calculus.series from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982... | Mathlib/Analysis/Calculus/SmoothSeries.lean | 72 | 84 | theorem hasFDerivAt_tsum_of_isPreconnected (hu : Summable u) (hs : IsOpen s)
(h's : IsPreconnected s) (hf : ∀ n x, x ∈ s → HasFDerivAt (f n) (f' n x) x)
(hf' : ∀ n x, x ∈ s → ‖f' n x‖ ≤ u n) (hx₀ : x₀ ∈ s) (hf0 : Summable fun n => f n x₀)
(hx : x ∈ s) : HasFDerivAt (fun y => ∑' n, f n y) (∑' n, f' n x) x :=... |
classical
have A :
∀ x : E, x ∈ s → Tendsto (fun t : Finset α => ∑ n ∈ t, f n x) atTop (𝓝 (∑' n, f n x)) := by
intro y hy
apply Summable.hasSum
exact summable_of_summable_hasFDerivAt_of_isPreconnected hu hs h's hf hf' hx₀ hf0 hy
refine hasFDerivAt_of_tendstoUniformlyOn hs (tendstoUni... | true |
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.GroupTheory.GroupAction.Pi
open Function Set
structure AddConstMap (G H : Type*) [Add G] [Add H] (a : G) (b : H) where
protected... | Mathlib/Algebra/AddConstMap/Basic.lean | 142 | 144 | theorem map_nat_add' [AddCommMonoidWithOne G] [AddMonoid H] [AddConstMapClass F G H 1 b]
(f : F) (n : ℕ) (x : G) : f (↑n + x) = f x + n • b := by |
simpa using map_nsmul_add f n x
| true |
import Mathlib.Data.Fintype.Prod
import Mathlib.Data.Fintype.Sum
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.fintype.units from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
variable {α : Type*}
instance UnitsInt.fintype : Fintype ℤˣ :=
⟨{1, -1}, fun x ↦ by cases Int... | Mathlib/Data/Fintype/Units.lean | 48 | 50 | theorem Fintype.card_units [GroupWithZero α] [Fintype α] [DecidableEq α] :
Fintype.card αˣ = Fintype.card α - 1 := by |
rw [@Fintype.card_eq_card_units_add_one α, Nat.add_sub_cancel]
| true |
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTheory.Function.LpSeminorm.ChebyshevMarkov
import Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp
import Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality... | Mathlib/MeasureTheory/Function/LpSpace.lean | 163 | 167 | theorem ext {f g : Lp E p μ} (h : f =ᵐ[μ] g) : f = g := by
cases f |
cases f
cases g
simp only [Subtype.mk_eq_mk]
exact AEEqFun.ext h
| true |
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.Module.Submodule.Basic
#align_import algebra.direct_sum.decomposition from "leanprover-community/mathlib"@"4e861f25ba5ceef42ba0712d8ffeb32f38ad6441"
variable {ι R M σ : Type*}
open DirectSum
namespace DirectSum
section AddCommMonoid
variable [Deci... | Mathlib/Algebra/DirectSum/Decomposition.lean | 140 | 142 | theorem decompose_of_mem_ne {x : M} {i j : ι} (hx : x ∈ ℳ i) (hij : i ≠ j) :
(decompose ℳ x j : M) = 0 := by |
rw [decompose_of_mem _ hx, DirectSum.of_eq_of_ne _ _ _ _ hij, ZeroMemClass.coe_zero]
| true |
import Mathlib.Geometry.Manifold.ContMDiff.Atlas
import Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear
import Mathlib.Topology.VectorBundle.Constructions
#align_import geometry.manifold.vector_bundle.basic from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
assert_not_exists mfde... | Mathlib/Geometry/Manifold/VectorBundle/Basic.lean | 108 | 114 | theorem FiberBundle.chartedSpace_chartAt (x : TotalSpace F E) :
chartAt (ModelProd HB F) x =
(trivializationAt F E x.proj).toPartialHomeomorph ≫ₕ
(chartAt HB x.proj).prod (PartialHomeomorph.refl F) := by
dsimp only [chartAt_comp, prodChartedSpace_chartAt, FiberBundle.chartedSpace'_chartAt, |
dsimp only [chartAt_comp, prodChartedSpace_chartAt, FiberBundle.chartedSpace'_chartAt,
chartAt_self_eq]
rw [Trivialization.coe_coe, Trivialization.coe_fst' _ (mem_baseSet_trivializationAt F E x.proj)]
| true |
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Geometry.Euclidean.PerpBisector
import Mathlib.Algebra.QuadraticDiscriminant
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
noncomputable section
open scoped Classical
open ... | Mathlib/Geometry/Euclidean/Basic.lean | 112 | 117 | theorem dist_smul_vadd_sq (r : ℝ) (v : V) (p₁ p₂ : P) :
dist (r • v +ᵥ p₁) p₂ * dist (r • v +ᵥ p₁) p₂ =
⟪v, v⟫ * r * r + 2 * ⟪v, p₁ -ᵥ p₂⟫ * r + ⟪p₁ -ᵥ p₂, p₁ -ᵥ p₂⟫ := by
rw [dist_eq_norm_vsub V _ p₂, ← real_inner_self_eq_norm_mul_norm, vadd_vsub_assoc, |
rw [dist_eq_norm_vsub V _ p₂, ← real_inner_self_eq_norm_mul_norm, vadd_vsub_assoc,
real_inner_add_add_self, real_inner_smul_left, real_inner_smul_left, real_inner_smul_right]
ring
| true |
import Mathlib.Data.Stream.Init
import Mathlib.Tactic.ApplyFun
import Mathlib.Control.Fix
import Mathlib.Order.OmegaCompletePartialOrder
#align_import control.lawful_fix from "leanprover-community/mathlib"@"92ca63f0fb391a9ca5f22d2409a6080e786d99f7"
universe u v
open scoped Classical
variable {α : Type*} {β : α →... | Mathlib/Control/LawfulFix.lean | 120 | 123 | theorem le_f_of_mem_approx {x} : x ∈ approxChain f → x ≤ f x := by
simp only [(· ∈ ·), forall_exists_index] |
simp only [(· ∈ ·), forall_exists_index]
rintro i rfl
apply approx_mono'
| true |
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.add from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
open Filter Asymptotics ContinuousLinearMap Set Metric
open scoped Classical
open Topology NNReal F... | Mathlib/Analysis/Calculus/FDeriv/Add.lean | 488 | 489 | theorem fderiv_neg : fderiv 𝕜 (fun y => -f y) x = -fderiv 𝕜 f x := by |
simp only [← fderivWithin_univ, fderivWithin_neg uniqueDiffWithinAt_univ]
| true |
import Mathlib.Probability.IdentDistrib
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.Analysis.SpecificLimits.FloorPow
import Mathlib.Analysis.PSeries
import Mathlib.Analysis.Asymptotics.SpecificAsymptotics
#align_import probability.strong_law from "leanprover-community/mathlib"@"f2ce60867... | Mathlib/Probability/StrongLaw.lean | 88 | 92 | theorem abs_truncation_le_bound (f : α → ℝ) (A : ℝ) (x : α) : |truncation f A x| ≤ |A| := by
simp only [truncation, Set.indicator, Set.mem_Icc, id, Function.comp_apply] |
simp only [truncation, Set.indicator, Set.mem_Icc, id, Function.comp_apply]
split_ifs with h
· exact abs_le_abs h.2 (neg_le.2 h.1.le)
· simp [abs_nonneg]
| true |
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.LinearAlgebra.Dual
import Mathlib.LinearAlgebra.Contraction
import Mathlib.RingTheory.TensorProduct.Basic
#align_import representation_... | Mathlib/RepresentationTheory/Basic.lean | 166 | 169 | theorem asModuleEquiv_symm_map_rho (g : G) (x : V) :
ρ.asModuleEquiv.symm (ρ g x) = MonoidAlgebra.of k G g • ρ.asModuleEquiv.symm x := by
apply_fun ρ.asModuleEquiv |
apply_fun ρ.asModuleEquiv
simp
| true |
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_import linear_algebra.dimension from "leanprover-community/mathlib"@"47a5f8186becdbc826190ced4312f8199f9db6a5"
noncomputable section
universe u v v' u₁' w w'
variable {R S : Type u} {M : Type v} {M' : Type v'} {M₁ : Type v}... | Mathlib/LinearAlgebra/Dimension/Constructions.lean | 544 | 547 | theorem subalgebra_top_finrank_eq_submodule_top_finrank :
finrank F (⊤ : Subalgebra F E) = finrank F (⊤ : Submodule F E) := by
rw [← Algebra.top_toSubmodule] |
rw [← Algebra.top_toSubmodule]
rfl
| true |
import Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity
#align_import number_theory.legendre_symbol.jacobi_symbol from "leanprover-community/mathlib"@"74a27133cf29446a0983779e37c8f829a85368f3"
section Jacobi
open Nat ZMod
-- Since we need the fact that the factors are prime, we use `List.pmap`.
def ... | Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean | 104 | 105 | theorem zero_right (a : ℤ) : J(a | 0) = 1 := by |
simp only [jacobiSym, factors_zero, List.prod_nil, List.pmap]
| true |
import Mathlib.Analysis.InnerProductSpace.Rayleigh
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Algebra.DirectSum.Decomposition
import Mathlib.LinearAlgebra.Eigenspace.Minpoly
#align_import analysis.inner_product_space.spectrum from "leanprover-community/mathlib"@"6b0169218d01f2837d79ea2784882009a0da... | Mathlib/Analysis/InnerProductSpace/Spectrum.lean | 110 | 115 | theorem orthogonalComplement_iSup_eigenspaces (μ : 𝕜) :
eigenspace (T.restrict hT.orthogonalComplement_iSup_eigenspaces_invariant) μ = ⊥ := by
set p : Submodule 𝕜 E := (⨆ μ, eigenspace T μ)ᗮ |
set p : Submodule 𝕜 E := (⨆ μ, eigenspace T μ)ᗮ
refine eigenspace_restrict_eq_bot hT.orthogonalComplement_iSup_eigenspaces_invariant ?_
have H₂ : eigenspace T μ ⟂ p := (Submodule.isOrtho_orthogonal_right _).mono_left (le_iSup _ _)
exact H₂.disjoint
| true |
import Mathlib.Algebra.CharP.LocalRing
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.Tactic.FieldSimp
#align_import algebra.char_p.mixed_char_zero from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
variable (R : Type*) [CommRing R]
class MixedCharZero (p : ℕ) : Prop where
... | Mathlib/Algebra/CharP/MixedCharZero.lean | 264 | 270 | theorem to_not_mixedCharZero (h : ∀ I : Ideal R, I ≠ ⊤ → CharZero (R ⧸ I)) :
∀ p > 0, ¬MixedCharZero R p := by
intro p p_pos |
intro p p_pos
by_contra hp_mixedChar
rcases hp_mixedChar.charP_quotient with ⟨I, hI_ne_top, hI_p⟩
replace hI_zero : CharP (R ⧸ I) 0 := @CharP.ofCharZero _ _ (h I hI_ne_top)
exact absurd (CharP.eq (R ⧸ I) hI_p hI_zero) (ne_of_gt p_pos)
| true |
import Mathlib.CategoryTheory.Monad.Types
import Mathlib.CategoryTheory.Monad.Limits
import Mathlib.CategoryTheory.Equivalence
import Mathlib.Topology.Category.CompHaus.Basic
import Mathlib.Topology.Category.Profinite.Basic
import Mathlib.Data.Set.Constructions
#align_import topology.category.Compactum from "leanprov... | Mathlib/Topology/Category/Compactum.lean | 173 | 185 | theorem isClosed_iff {X : Compactum} (S : Set X) :
IsClosed S ↔ ∀ F : Ultrafilter X, S ∈ F → X.str F ∈ S := by
rw [← isOpen_compl_iff] |
rw [← isOpen_compl_iff]
constructor
· intro cond F h
by_contra c
specialize cond F c
rw [compl_mem_iff_not_mem] at cond
contradiction
· intro h1 F h2
specialize h1 F
cases' F.mem_or_compl_mem S with h h
exacts [absurd (h1 h) h2, h]
| true |
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 | 52 | 55 | theorem _root_.Filter.cardinalInterFilter_aleph0 (l : Filter α) : CardinalInterFilter l aleph0 where
cardinal_sInter_mem := by
simp_all only [aleph_zero, lt_aleph0_iff_subtype_finite, setOf_mem_eq, sInter_mem, |
simp_all only [aleph_zero, lt_aleph0_iff_subtype_finite, setOf_mem_eq, sInter_mem,
implies_true, forall_const]
| true |
import Mathlib.LinearAlgebra.Dimension.DivisionRing
import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
noncomputable section
universe u v v' v''
variable {K : Type u} {V V₁ : Type v} {V' V'₁ : Type v'} {V'' : Type v''}
open Cardinal Basis Submodule Function Set
namespace LinearMap
section Ring
... | Mathlib/LinearAlgebra/Dimension/LinearMap.lean | 52 | 55 | theorem rank_comp_le_left (g : V →ₗ[K] V') (f : V' →ₗ[K] V'') : rank (f.comp g) ≤ rank f := by
refine rank_le_of_submodule _ _ ?_ |
refine rank_le_of_submodule _ _ ?_
rw [LinearMap.range_comp]
exact LinearMap.map_le_range
| true |
import Mathlib.RingTheory.FinitePresentation
import Mathlib.RingTheory.Localization.Away.Basic
import Mathlib.RingTheory.Localization.Away.AdjoinRoot
import Mathlib.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.TensorProduct.Basic
-- Porting note: added to make the syntax work below.
open scoped TensorProd... | Mathlib/RingTheory/Unramified/Basic.lean | 155 | 163 | theorem of_comp [FormallyUnramified R B] : FormallyUnramified A B := by
constructor |
constructor
intro Q _ _ I e f₁ f₂ e'
letI := ((algebraMap A Q).comp (algebraMap R A)).toAlgebra
letI : IsScalarTower R A Q := IsScalarTower.of_algebraMap_eq' rfl
refine AlgHom.restrictScalars_injective R ?_
refine FormallyUnramified.ext I ⟨2, e⟩ ?_
intro x
exact AlgHom.congr_fun e' x
| true |
import Mathlib.MeasureTheory.Measure.VectorMeasure
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
#align_import measure_theory.measure.with_density_vector_measure from "leanprover-community/mathlib"@"d1bd9c5df2867c1cb463bc6364446d57bdd9f7f1"
noncomputable section
open scoped Classical MeasureTheory NNReal ... | Mathlib/MeasureTheory/Measure/WithDensityVectorMeasure.lean | 59 | 60 | theorem withDensityᵥ_apply (hf : Integrable f μ) {s : Set α} (hs : MeasurableSet s) :
μ.withDensityᵥ f s = ∫ x in s, f x ∂μ := by | rw [withDensityᵥ, dif_pos hf]; exact dif_pos hs
| true |
import Mathlib.CategoryTheory.Idempotents.Basic
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Equivalence
#align_import category_theory.idempotents.karoubi from "leanprover-community/mathlib"@"200eda15d8ff5669854ff6bcc10aaf37cb70498f"
noncomputable section
open CategoryT... | Mathlib/CategoryTheory/Idempotents/Karoubi.lean | 89 | 90 | theorem comp_p {P Q : Karoubi C} (f : Hom P Q) : f.f ≫ Q.p = f.f := by |
rw [f.comm, assoc, assoc, Q.idem]
| true |
import Mathlib.Data.Fin.VecNotation
import Mathlib.SetTheory.Cardinal.Basic
#align_import model_theory.basic from "leanprover-community/mathlib"@"369525b73f229ccd76a6ec0e0e0bf2be57599768"
set_option autoImplicit true
universe u v u' v' w w'
open Cardinal
open Cardinal
namespace FirstOrder
-- intended to b... | Mathlib/ModelTheory/Basic.lean | 174 | 178 | theorem card_eq_card_functions_add_card_relations :
L.card =
(Cardinal.sum fun l => Cardinal.lift.{v} #(L.Functions l)) +
Cardinal.sum fun l => Cardinal.lift.{u} #(L.Relations l) := by |
simp [card, Symbols]
| true |
import Mathlib.Analysis.Convex.Hull
#align_import analysis.convex.join from "leanprover-community/mathlib"@"951bf1d9e98a2042979ced62c0620bcfb3587cf8"
open Set
variable {ι : Sort*} {𝕜 E : Type*}
section OrderedSemiring
variable (𝕜) [OrderedSemiring 𝕜] [AddCommMonoid E] [Module 𝕜 E] {s t s₁ s₂ t₁ t₂ u : Set ... | Mathlib/Analysis/Convex/Join.lean | 65 | 66 | theorem convexJoin_singleton_left (t : Set E) (x : E) :
convexJoin 𝕜 {x} t = ⋃ y ∈ t, segment 𝕜 x y := by | simp [convexJoin]
| true |
import Mathlib.Algebra.Order.Interval.Set.Instances
import Mathlib.Order.Interval.Set.ProjIcc
import Mathlib.Topology.Instances.Real
#align_import topology.unit_interval from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open Topology Filter
... | Mathlib/Topology/UnitInterval.lean | 154 | 155 | theorem half_le_symm_iff (t : I) : 1 / 2 ≤ (σ t : ℝ) ↔ (t : ℝ) ≤ 1 / 2 := by |
rw [coe_symm_eq, le_sub_iff_add_le, add_comm, ← le_sub_iff_add_le, sub_half]
| true |
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
noncomputable section
open Set Function Finsupp AddMonoidAlgebra
universe u v w
variable {R : Type u} {S : Type v}
namespace MvPolynomial
varia... | Mathlib/Algebra/MvPolynomial/Variables.lean | 77 | 78 | theorem vars_0 : (0 : MvPolynomial σ R).vars = ∅ := by |
classical rw [vars_def, degrees_zero, Multiset.toFinset_zero]
| true |
import Mathlib.Analysis.Convex.Combination
import Mathlib.LinearAlgebra.AffineSpace.Independent
import Mathlib.Tactic.FieldSimp
#align_import analysis.convex.caratheodory from "leanprover-community/mathlib"@"e6fab1dc073396d45da082c644642c4f8bff2264"
open Set Finset
universe u
variable {𝕜 : Type*} {E : Type u} ... | Mathlib/Analysis/Convex/Caratheodory.lean | 119 | 121 | theorem minCardFinsetOfMemConvexHull_nonempty : (minCardFinsetOfMemConvexHull hx).Nonempty := by
rw [← Finset.coe_nonempty, ← @convexHull_nonempty_iff 𝕜] |
rw [← Finset.coe_nonempty, ← @convexHull_nonempty_iff 𝕜]
exact ⟨x, mem_minCardFinsetOfMemConvexHull hx⟩
| true |
import Mathlib.LinearAlgebra.CliffordAlgebra.Contraction
variable {R M : Type*}
variable [CommRing R] [AddCommGroup M] [Module R M] {Q : QuadraticForm R M}
namespace CliffordAlgebra
variable (Q)
def invertibleιOfInvertible (m : M) [Invertible (Q m)] : Invertible (ι Q m) where
invOf := ι Q (⅟ (Q m) • m)
invO... | Mathlib/LinearAlgebra/CliffordAlgebra/Inversion.lean | 37 | 40 | theorem isUnit_ι_of_isUnit {m : M} (h : IsUnit (Q m)) : IsUnit (ι Q m) := by
cases h.nonempty_invertible |
cases h.nonempty_invertible
letI := invertibleιOfInvertible Q m
exact isUnit_of_invertible (ι Q m)
| true |
import Mathlib.Algebra.Homology.Single
#align_import algebra.homology.augment from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
open CategoryTheory Limits HomologicalComplex
universe v u
variable {V : Type u} [Category.{v} V]
namespace ChainComplex
@[simps]... | Mathlib/Algebra/Homology/Augment.lean | 132 | 134 | theorem chainComplex_d_succ_succ_zero (C : ChainComplex V ℕ) (i : ℕ) : C.d (i + 2) 0 = 0 := by
rw [C.shape] |
rw [C.shape]
exact i.succ_succ_ne_one.symm
| true |
import Mathlib.Combinatorics.SimpleGraph.Connectivity
#align_import combinatorics.simple_graph.prod from "leanprover-community/mathlib"@"2985fa3c31a27274aed06c433510bc14b73d6488"
variable {α β γ : Type*}
namespace SimpleGraph
-- Porting note: pruned variables to keep things out of local contexts, which
-- can im... | Mathlib/Combinatorics/SimpleGraph/Prod.lean | 69 | 73 | theorem boxProd_neighborSet (x : α × β) :
(G □ H).neighborSet x = G.neighborSet x.1 ×ˢ {x.2} ∪ {x.1} ×ˢ H.neighborSet x.2 := by
ext ⟨a', b'⟩ |
ext ⟨a', b'⟩
simp only [mem_neighborSet, Set.mem_union, boxProd_adj, Set.mem_prod, Set.mem_singleton_iff]
simp only [eq_comm, and_comm]
| true |
import Mathlib.RingTheory.AdicCompletion.Basic
import Mathlib.Algebra.Module.Torsion
open Submodule
variable {R : Type*} [CommRing R] (I : Ideal R)
variable {M : Type*} [AddCommGroup M] [Module R M]
namespace AdicCompletion
attribute [-simp] smul_eq_mul Algebra.id.smul_eq_mul
@[local simp]
theorem transitionMap... | Mathlib/RingTheory/AdicCompletion/Algebra.lean | 127 | 131 | theorem Ideal.mk_eq_mk {m n : ℕ} (hmn : m ≤ n) (r : AdicCauchySequence I R) :
Ideal.Quotient.mk (I ^ m) (r.val n) = Ideal.Quotient.mk (I ^ m) (r.val m) := by
have h : I ^ m = I ^ m • ⊤ := by simp |
have h : I ^ m = I ^ m • ⊤ := by simp
rw [h, ← Ideal.Quotient.mk_eq_mk, ← Ideal.Quotient.mk_eq_mk]
exact (r.property hmn).symm
| true |
import Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.measure.haar.normed_space from "leanprover-community/mathlib"@"b84aee748341da06a6d78491367e2c0e9f15e8a5"
noncomputable sect... | Mathlib/MeasureTheory/Measure/Haar/NormedSpace.lean | 110 | 123 | theorem setIntegral_comp_smul (f : E → F) {R : ℝ} (s : Set E) (hR : R ≠ 0) :
∫ x in s, f (R • x) ∂μ = |(R ^ finrank ℝ E)⁻¹| • ∫ x in R • s, f x ∂μ := by
let e : E ≃ᵐ E := (Homeomorph.smul (Units.mk0 R hR)).toMeasurableEquiv |
let e : E ≃ᵐ E := (Homeomorph.smul (Units.mk0 R hR)).toMeasurableEquiv
calc
∫ x in s, f (R • x) ∂μ
= ∫ x in e ⁻¹' (e.symm ⁻¹' s), f (e x) ∂μ := by simp [← preimage_comp]; rfl
_ = ∫ y in e.symm ⁻¹' s, f y ∂map (fun x ↦ R • x) μ := (setIntegral_map_equiv _ _ _).symm
_ = |(R ^ finrank ℝ E)⁻¹| • ∫ y in e.sym... | true |
import Mathlib.Algebra.IsPrimePow
import Mathlib.Data.Nat.Factorization.Basic
#align_import data.nat.factorization.prime_pow from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
variable {R : Type*} [CommMonoidWithZero R] (n p : R) (k : ℕ)
theorem IsPrimePow.minFac_pow_factorization_eq ... | Mathlib/Data/Nat/Factorization/PrimePow.lean | 41 | 54 | theorem isPrimePow_iff_factorization_eq_single {n : ℕ} :
IsPrimePow n ↔ ∃ p k : ℕ, 0 < k ∧ n.factorization = Finsupp.single p k := by
rw [isPrimePow_nat_iff] |
rw [isPrimePow_nat_iff]
refine exists₂_congr fun p k => ?_
constructor
· rintro ⟨hp, hk, hn⟩
exact ⟨hk, by rw [← hn, Nat.Prime.factorization_pow hp]⟩
· rintro ⟨hk, hn⟩
have hn0 : n ≠ 0 := by
rintro rfl
simp_all only [Finsupp.single_eq_zero, eq_comm, Nat.factorization_zero, hk.ne']
rw ... | true |
import Mathlib.CategoryTheory.Functor.Hom
import Mathlib.CategoryTheory.Products.Basic
import Mathlib.Data.ULift
#align_import category_theory.yoneda from "leanprover-community/mathlib"@"369525b73f229ccd76a6ec0e0e0bf2be57599768"
namespace CategoryTheory
open Opposite
universe v₁ u₁ u₂
-- morphism levels before ... | Mathlib/CategoryTheory/Yoneda.lean | 59 | 62 | theorem obj_map_id {X Y : C} (f : op X ⟶ op Y) :
(yoneda.obj X).map f (𝟙 X) = (yoneda.map f.unop).app (op Y) (𝟙 Y) := by
dsimp |
dsimp
simp
| true |
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Algebra.Group.Aut
import Mathlib.Data.ZMod.Defs
import Mathlib.Tactic.Ring
#align_import algebra.quandle from "leanprover-community/mathlib"@"28aa996fc6fb4317f0083c4e6daf79878d81be33"
open MulOpposite
universe u v
class Shelf (α : Type u) where
act : ... | Mathlib/Algebra/Quandle.lean | 287 | 289 | theorem self_invAct_invAct_eq {x y : R} : (x ◃⁻¹ x) ◃⁻¹ y = x ◃⁻¹ y := by
have h := @self_act_act_eq _ _ (op x) (op y) |
have h := @self_act_act_eq _ _ (op x) (op y)
simpa using h
| true |
import Mathlib.Analysis.LocallyConvex.Basic
#align_import analysis.locally_convex.balanced_core_hull from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Pointwise Topology Filter
variable {𝕜 E ι : Type*}
section balancedHull
section SeminormedRing
variable [SeminormedRing ... | Mathlib/Analysis/LocallyConvex/BalancedCoreHull.lean | 114 | 118 | theorem Balanced.balancedHull_subset_of_subset (ht : Balanced 𝕜 t) (h : s ⊆ t) :
balancedHull 𝕜 s ⊆ t := by
intros x hx |
intros x hx
obtain ⟨r, hr, y, hy, rfl⟩ := mem_balancedHull_iff.1 hx
exact ht.smul_mem hr (h hy)
| true |
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Data.Rat.Cast.Defs
#align_import data.rat.cast from "leanprover-community/mathlib"@"acebd8d49928f6ed8920e502a6c90674e75bd441"
variable {F ι α β : Type*}
namespace Rat
open Rat
section WithDivRing
variable [DivisionRing α]
@[simp, norm_cast]
th... | Mathlib/Data/Rat/Cast/CharZero.lean | 78 | 79 | theorem cast_bit1 [CharZero α] (n : ℚ) : ((bit1 n : ℚ) : α) = (bit1 n : α) := by |
rw [bit1, cast_add, cast_one, cast_bit0]; rfl
| true |
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.LinearAlgebra.Span
#align_import algebra.algebra.tower from "leanprover-community/mathlib"@"71150516f28d9826c7341f8815b31f7d8770c212"
open Pointwise
universe u v w u₁ v₁
variable (R : Type u) (S : Type v) (A : Type w) (B : Type u₁) (M : Type v₁)
namespace IsS... | Mathlib/Algebra/Algebra/Tower.lean | 88 | 90 | theorem algebraMap_smul [SMul R M] [IsScalarTower R A M] (r : R) (x : M) :
algebraMap R A r • x = r • x := by |
rw [Algebra.algebraMap_eq_smul_one, smul_assoc, one_smul]
| true |
import Mathlib.Data.Multiset.Basic
#align_import data.multiset.range from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
open List Nat
namespace Multiset
-- range
def range (n : ℕ) : Multiset ℕ :=
List.range n
#align multiset.range Multiset.range
theorem coe_range (n : ℕ) : ↑(List... | Mathlib/Data/Multiset/Range.lean | 65 | 70 | theorem range_disjoint_map_add (a : ℕ) (m : Multiset ℕ) :
(range a).Disjoint (m.map (a + ·)) := by
intro x hxa hxb |
intro x hxa hxb
rw [range, mem_coe, List.mem_range] at hxa
obtain ⟨c, _, rfl⟩ := mem_map.1 hxb
exact (Nat.le_add_right _ _).not_lt hxa
| true |
import Mathlib.RepresentationTheory.Action.Limits
import Mathlib.RepresentationTheory.Action.Concrete
import Mathlib.CategoryTheory.Monoidal.FunctorCategory
import Mathlib.CategoryTheory.Monoidal.Transport
import Mathlib.CategoryTheory.Monoidal.Rigid.OfEquivalence
import Mathlib.CategoryTheory.Monoidal.Rigid.FunctorCa... | Mathlib/RepresentationTheory/Action/Monoidal.lean | 105 | 107 | theorem leftUnitor_inv_hom {X : Action V G} : Hom.hom (λ_ X).inv = (λ_ X.V).inv := by
dsimp |
dsimp
simp
| true |
import Mathlib.MeasureTheory.Measure.Content
import Mathlib.MeasureTheory.Group.Prod
import Mathlib.Topology.Algebra.Group.Compact
#align_import measure_theory.measure.haar.basic from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open Set Inv Function Topological... | Mathlib/MeasureTheory/Measure/Haar/Basic.lean | 185 | 196 | theorem le_index_mul (K₀ : PositiveCompacts G) (K : Compacts G) {V : Set G}
(hV : (interior V).Nonempty) :
index (K : Set G) V ≤ index (K : Set G) K₀ * index (K₀ : Set G) V := by
obtain ⟨s, h1s, h2s⟩ := index_elim K.isCompact K₀.interior_nonempty |
obtain ⟨s, h1s, h2s⟩ := index_elim K.isCompact K₀.interior_nonempty
obtain ⟨t, h1t, h2t⟩ := index_elim K₀.isCompact hV
rw [← h2s, ← h2t, mul_comm]
refine le_trans ?_ Finset.card_mul_le
apply Nat.sInf_le; refine ⟨_, ?_, rfl⟩; rw [mem_setOf_eq]; refine Subset.trans h1s ?_
apply iUnion₂_subset; intro g₁ hg₁; ... | true |
import Mathlib.Data.Vector.Basic
import Mathlib.Data.Vector.Snoc
set_option autoImplicit true
namespace Vector
section Fold
section Unary
variable (xs : Vector α n) (f₁ : β → σ₁ → σ₁ × γ) (f₂ : α → σ₂ → σ₂ × β)
@[simp]
| Mathlib/Data/Vector/MapLemmas.lean | 27 | 35 | theorem mapAccumr_mapAccumr :
mapAccumr f₁ (mapAccumr f₂ xs s₂).snd s₁
= let m := (mapAccumr (fun x s =>
let r₂ := f₂ x s.snd
let r₁ := f₁ r₂.snd s.fst
((r₁.fst, r₂.fst), r₁.snd)
) xs (s₁, s₂))
(m.fst.fst, m.snd) := by |
induction xs using Vector.revInductionOn generalizing s₁ s₂ <;> simp_all
| true |
import Mathlib.Order.ConditionallyCompleteLattice.Finset
import Mathlib.Order.Interval.Finset.Nat
#align_import data.nat.lattice from "leanprover-community/mathlib"@"52fa514ec337dd970d71d8de8d0fd68b455a1e54"
assert_not_exists MonoidWithZero
open Set
namespace Nat
open scoped Classical
noncomputable instance : ... | Mathlib/Data/Nat/Lattice.lean | 66 | 67 | theorem iInf_of_empty {ι : Sort*} [IsEmpty ι] (f : ι → ℕ) : iInf f = 0 := by |
rw [iInf_of_isEmpty, sInf_empty]
| true |
import Mathlib.FieldTheory.PrimitiveElement
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.Matrix.Charpoly.Minpoly
import Mathlib.LinearAlgebra.Matrix.ToLinearEquiv
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.G... | Mathlib/RingTheory/Norm.lean | 126 | 135 | theorem PowerBasis.norm_gen_eq_prod_roots [Algebra R F] (pb : PowerBasis R S)
(hf : (minpoly R pb.gen).Splits (algebraMap R F)) :
algebraMap R F (norm R pb.gen) = ((minpoly R pb.gen).aroots F).prod := by
haveI := Module.nontrivial R F |
haveI := Module.nontrivial R F
have := minpoly.monic pb.isIntegral_gen
rw [PowerBasis.norm_gen_eq_coeff_zero_minpoly, ← pb.natDegree_minpoly, RingHom.map_mul,
← coeff_map,
prod_roots_eq_coeff_zero_of_monic_of_split (this.map _) ((splits_id_iff_splits _).2 hf),
this.natDegree_map, map_pow, ← mul_assoc... | true |
import Mathlib.MeasureTheory.Measure.Typeclasses
open scoped ENNReal
namespace MeasureTheory
variable {α : Type*}
noncomputable
def Measure.trim {m m0 : MeasurableSpace α} (μ : @Measure α m0) (hm : m ≤ m0) : @Measure α m :=
@OuterMeasure.toMeasure α m μ.toOuterMeasure (hm.trans (le_toOuterMeasure_caratheodory... | Mathlib/MeasureTheory/Measure/Trim.lean | 57 | 59 | theorem le_trim (hm : m ≤ m0) : μ s ≤ μ.trim hm s := by
simp_rw [Measure.trim] |
simp_rw [Measure.trim]
exact @le_toMeasure_apply _ m _ _ _
| true |
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.LinearAlgebra.SesquilinearForm
#align_import analysis.inner_product_space.orthogonal from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
variable {𝕜 E F : Type*} [RCLike 𝕜]
variable [NormedAddCommGroup E] [InnerProductSpace... | Mathlib/Analysis/InnerProductSpace/Orthogonal.lean | 111 | 111 | theorem orthogonal_disjoint : Disjoint K Kᗮ := by | simp [disjoint_iff, K.inf_orthogonal_eq_bot]
| true |
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 | 120 | 122 | theorem memberSubfamily_nonMemberSubfamily : (𝒜.nonMemberSubfamily a).memberSubfamily a = ∅ := by
ext |
ext
simp
| true |
import Mathlib.Algebra.Star.Basic
import Mathlib.Algebra.Star.Pointwise
import Mathlib.Algebra.Group.Centralizer
variable {R : Type*} [Mul R] [StarMul R] {a : R} {s : Set R}
| Mathlib/Algebra/Star/Center.lean | 14 | 34 | theorem Set.star_mem_center (ha : a ∈ Set.center R) : star a ∈ Set.center R where
comm := by simpa only [star_mul, star_star] using fun g => | simpa only [star_mul, star_star] using fun g =>
congr_arg star (((Set.mem_center_iff R).mp ha).comm <| star g).symm
left_assoc b c := calc
star a * (b * c) = star a * (star (star b) * star (star c)) := by rw [star_star, star_star]
_ = star a * star (star c * star b) := by rw [star_mul]
_ = star ((sta... | true |
import Mathlib.Order.Filter.Bases
import Mathlib.Order.Filter.Ultrafilter
open Set
variable {α β : Type*} {l : Filter α}
namespace Filter
protected def Subsingleton (l : Filter α) : Prop := ∃ s ∈ l, Set.Subsingleton s
theorem HasBasis.subsingleton_iff {ι : Sort*} {p : ι → Prop} {s : ι → Set α} (h : l.HasBasis p ... | Mathlib/Order/Filter/Subsingleton.lean | 51 | 55 | theorem Subsingleton.exists_eq_pure [l.NeBot] (hl : l.Subsingleton) : ∃ a, l = pure a := by
rcases hl with ⟨s, hsl, hs⟩ |
rcases hl with ⟨s, hsl, hs⟩
rcases exists_eq_singleton_iff_nonempty_subsingleton.2 ⟨nonempty_of_mem hsl, hs⟩ with ⟨a, rfl⟩
refine ⟨a, (NeBot.le_pure_iff ‹_›).1 ?_⟩
rwa [le_pure_iff]
| true |
import Mathlib.Data.List.Basic
namespace List
variable {α β : Type*}
@[simp]
theorem reduceOption_cons_of_some (x : α) (l : List (Option α)) :
reduceOption (some x :: l) = x :: l.reduceOption := by
simp only [reduceOption, filterMap, id, eq_self_iff_true, and_self_iff]
#align list.reduce_option_cons_of_some... | Mathlib/Data/List/ReduceOption.lean | 77 | 77 | theorem reduceOption_singleton (x : Option α) : [x].reduceOption = x.toList := by | cases x <;> rfl
| true |
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]
| true |
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.FieldTheory.Minpoly.Field
#align_import linear_algebra.eigenspace.minpoly from "leanprover-community/mathlib"@"c3216069e5f9369e6be586ccbfcde2592b3cec92"
universe u v w
namespace Module
namespace End
open Polynomial FiniteDimensional
open scoped Poly... | Mathlib/LinearAlgebra/Eigenspace/Minpoly.lean | 75 | 91 | theorem hasEigenvalue_of_isRoot (h : (minpoly K f).IsRoot μ) : f.HasEigenvalue μ := by
cases' dvd_iff_isRoot.2 h with p hp |
cases' dvd_iff_isRoot.2 h with p hp
rw [HasEigenvalue, eigenspace]
intro con
cases' (LinearMap.isUnit_iff_ker_eq_bot _).2 con with u hu
have p_ne_0 : p ≠ 0 := by
intro con
apply minpoly.ne_zero (Algebra.IsIntegral.isIntegral (R := K) f)
rw [hp, con, mul_zero]
have : (aeval f) p = 0 := by
ha... | true |
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Mathlib.Order.Interval.Finset.Nat
#align_import data.nat.factorization.basic from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"
... | Mathlib/Data/Nat/Factorization/Basic.lean | 67 | 81 | theorem factors_count_eq {n p : ℕ} : n.factors.count p = n.factorization p := by
rcases n.eq_zero_or_pos with (rfl | hn0) |
rcases n.eq_zero_or_pos with (rfl | hn0)
· simp [factorization, count]
if pp : p.Prime then ?_ else
rw [count_eq_zero_of_not_mem (mt prime_of_mem_factors pp)]
simp [factorization, pp]
simp only [factorization_def _ pp]
apply _root_.le_antisymm
· rw [le_padicValNat_iff_replicate_subperm_factors pp h... | true |
import Mathlib.Order.Interval.Finset.Fin
#align_import data.fintype.fin from "leanprover-community/mathlib"@"759575657f189ccb424b990164c8b1fa9f55cdfe"
open Finset
open Fintype
namespace Fin
variable {α β : Type*} {n : ℕ}
| Mathlib/Data/Fintype/Fin.lean | 25 | 27 | theorem map_valEmbedding_univ : (Finset.univ : Finset (Fin n)).map Fin.valEmbedding = Iio n := by
ext |
ext
simp [orderIsoSubtype.symm.surjective.exists, OrderIso.symm]
| true |
import Mathlib.Order.Filter.Bases
#align_import order.filter.pi from "leanprover-community/mathlib"@"ce64cd319bb6b3e82f31c2d38e79080d377be451"
open Set Function
open scoped Classical
open Filter
namespace Filter
variable {ι : Type*} {α : ι → Type*} {f f₁ f₂ : (i : ι) → Filter (α i)} {s : (i : ι) → Set (α i)}
... | Mathlib/Order/Filter/Pi.lean | 244 | 245 | theorem coprodᵢ_neBot_iff [∀ i, Nonempty (α i)] : NeBot (Filter.coprodᵢ f) ↔ ∃ d, NeBot (f d) := by |
simp [coprodᵢ_neBot_iff', *]
| true |
import Mathlib.Data.ZMod.Quotient
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Algebra.GCDMonoid.Finset
import Mathlib.Algebra.GCDMonoid.Nat
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Tactic.ByContra
import Mathlib.Tactic.Peel
#align_import group_... | Mathlib/GroupTheory/Exponent.lean | 176 | 180 | theorem exponent_min' (n : ℕ) (hpos : 0 < n) (hG : ∀ g : G, g ^ n = 1) : exponent G ≤ n := by
rw [exponent, dif_pos] |
rw [exponent, dif_pos]
· apply Nat.find_min'
exact ⟨hpos, hG⟩
· exact ⟨n, hpos, hG⟩
| true |
import Mathlib.Order.Filter.Cofinite
#align_import topology.bornology.basic from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
open Set Filter
variable {ι α β : Type*}
class Bornology (α : Type*) where
cobounded' : Filter α
le_cofinite' : cobounded' ≤ cofinite
#align borno... | Mathlib/Topology/Bornology/Basic.lean | 166 | 169 | theorem nonempty_of_not_isBounded (h : ¬IsBounded s) : s.Nonempty := by
rw [nonempty_iff_ne_empty] |
rw [nonempty_iff_ne_empty]
rintro rfl
exact h isBounded_empty
| true |
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.Matrix.Transvection
import Mathlib.MeasureTheory.Group.LIntegral
import Mathlib.MeasureTheory.Integral.Marginal
import Mathlib.MeasureTheory.Measure.Stiel... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 88 | 88 | theorem volume_Ioo {a b : ℝ} : volume (Ioo a b) = ofReal (b - a) := by | simp [volume_val]
| true |
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 | 56 | 57 | theorem rotate'_cons_succ (l : List α) (a : α) (n : ℕ) :
(a :: l : List α).rotate' n.succ = (l ++ [a]).rotate' n := by | simp [rotate']
| true |
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.NthRewrite
#align_import data.nat.gcd.basic from "leanprover-community/mathlib"@"e8638a0fcaf73e4500469f368ef9494e495099b3"
namespace Nat
theorem gcd_greatest {a b d : ℕ} (hda : d ∣ a) (hdb : d ∣ b) (hd ... | Mathlib/Data/Nat/GCD/Basic.lean | 58 | 59 | theorem gcd_add_mul_left_left (m n k : ℕ) : gcd (m + n * k) n = gcd m n := by |
rw [gcd_comm, gcd_add_mul_left_right, gcd_comm]
| true |
import Mathlib.Algebra.Category.GroupCat.EquivalenceGroupAddGroup
import Mathlib.GroupTheory.QuotientGroup
#align_import algebra.category.Group.epi_mono from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
open scoped Pointwise
universe u v
namespace MonoidHom
o... | Mathlib/Algebra/Category/GroupCat/EpiMono.lean | 47 | 56 | theorem range_eq_top_of_cancel {f : A →* B}
(h : ∀ u v : B →* B ⧸ f.range, u.comp f = v.comp f → u = v) : f.range = ⊤ := by
specialize h 1 (QuotientGroup.mk' _) _ |
specialize h 1 (QuotientGroup.mk' _) _
· ext1 x
simp only [one_apply, coe_comp, coe_mk', Function.comp_apply]
rw [show (1 : B ⧸ f.range) = (1 : B) from QuotientGroup.mk_one _, QuotientGroup.eq, inv_one,
one_mul]
exact ⟨x, rfl⟩
replace h : (QuotientGroup.mk' f.range).ker = (1 : B →* B ⧸ f.range)... | true |
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Analysis.NormedSpace.OperatorNorm.Completeness
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
#align_import analysis.normed_space.bounded_linear_maps from "leanprover-community/mathlib"@"ce11c3c2a285b... | Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | 289 | 290 | theorem map_zero₂ (f : M →SL[ρ₁₂] F →SL[σ₁₂] G') (y : F) : f 0 y = 0 := by |
rw [f.map_zero, zero_apply]
| true |
import Mathlib.Algebra.Group.ConjFinite
import Mathlib.GroupTheory.Perm.Fin
import Mathlib.GroupTheory.Subgroup.Simple
import Mathlib.Tactic.IntervalCases
#align_import group_theory.specific_groups.alternating from "leanprover-community/mathlib"@"0f6670b8af2dff699de1c0b4b49039b31bc13c46"
-- An example on how to de... | Mathlib/GroupTheory/SpecificGroups/Alternating.lean | 77 | 80 | theorem prod_list_swap_mem_alternatingGroup_iff_even_length {l : List (Perm α)}
(hl : ∀ g ∈ l, IsSwap g) : l.prod ∈ alternatingGroup α ↔ Even l.length := by
rw [mem_alternatingGroup, sign_prod_list_swap hl, neg_one_pow_eq_one_iff_even] |
rw [mem_alternatingGroup, sign_prod_list_swap hl, neg_one_pow_eq_one_iff_even]
decide
| true |
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 Image
variable {f : α → β} {s t : Set... | Mathlib/Data/Set/Image.lean | 227 | 228 | theorem exists_mem_image {f : α → β} {s : Set α} {p : β → Prop} :
(∃ y ∈ f '' s, p y) ↔ ∃ x ∈ s, p (f x) := by | simp
| true |
import Batteries.Data.Fin.Basic
namespace Fin
attribute [norm_cast] val_last
protected theorem le_antisymm_iff {x y : Fin n} : x = y ↔ x ≤ y ∧ y ≤ x :=
Fin.ext_iff.trans Nat.le_antisymm_iff
protected theorem le_antisymm {x y : Fin n} (h1 : x ≤ y) (h2 : y ≤ x) : x = y :=
Fin.le_antisymm_iff.2 ⟨h1, h2⟩
@[simp... | .lake/packages/batteries/Batteries/Data/Fin/Lemmas.lean | 41 | 47 | theorem list_succ_last (n) : list (n+1) = (list n).map castSucc ++ [last n] := by
rw [list_succ] |
rw [list_succ]
induction n with
| zero => rfl
| succ n ih =>
rw [list_succ, List.map_cons castSucc, ih]
simp [Function.comp_def, succ_castSucc]
| true |
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)
| true |
import Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1
#align_import measure_theory.function.conditional_expectation.basic from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"
open TopologicalSpace MeasureTheory.Lp Filter
open scoped ENNReal Topology MeasureTheory
names... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Basic.lean | 126 | 128 | theorem condexp_of_stronglyMeasurable (hm : m ≤ m0) [hμm : SigmaFinite (μ.trim hm)] {f : α → F'}
(hf : StronglyMeasurable[m] f) (hfi : Integrable f μ) : μ[f|m] = f := by |
rw [condexp_of_sigmaFinite hm, if_pos hfi, if_pos hf]
| true |
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Data.Finset.Basic
import Mathlib.Order.Interval.Finset.Defs
open Function
namespace Finset
class HasAntidiagonal (A : Type*) [AddMonoid A] where
antidiagonal : A → Finset (A × A)
mem_antidiagonal {n} {a} : a ∈ antidiagonal n ↔ a.fst + a.snd = n
exp... | Mathlib/Data/Finset/Antidiagonal.lean | 100 | 104 | theorem antidiagonal_congr (hp : p ∈ antidiagonal n) (hq : q ∈ antidiagonal n) :
p = q ↔ p.1 = q.1 := by
refine ⟨congr_arg Prod.fst, fun h ↦ Prod.ext h ((add_right_inj q.fst).mp ?_)⟩ |
refine ⟨congr_arg Prod.fst, fun h ↦ Prod.ext h ((add_right_inj q.fst).mp ?_)⟩
rw [mem_antidiagonal] at hp hq
rw [hq, ← h, hp]
| true |
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.Matrix
import Mathlib.LinearAlgebra.Matrix.ZPow
import Mathlib.LinearAlgebra.Matrix.Hermitian
import Mathlib.LinearAlgebra.Matrix.Symmetric
import Mathlib.Topology.UniformSpace.Matrix
#align_import analysis.normed_space.matrix_exponential from "l... | Mathlib/Analysis/NormedSpace/MatrixExponential.lean | 84 | 86 | theorem exp_blockDiagonal (v : m → Matrix n n 𝔸) :
exp 𝕂 (blockDiagonal v) = blockDiagonal (exp 𝕂 v) := by |
simp_rw [exp_eq_tsum, ← blockDiagonal_pow, ← blockDiagonal_smul, ← blockDiagonal_tsum]
| true |
import Mathlib.Topology.Separation
open Topology Filter Set TopologicalSpace
section Basic
variable {α : Type*} [TopologicalSpace α] {C : Set α}
theorem AccPt.nhds_inter {x : α} {U : Set α} (h_acc : AccPt x (𝓟 C)) (hU : U ∈ 𝓝 x) :
AccPt x (𝓟 (U ∩ C)) := by
have : 𝓝[≠] x ≤ 𝓟 U := by
rw [le_princ... | Mathlib/Topology/Perfect.lean | 111 | 115 | theorem Preperfect.open_inter {U : Set α} (hC : Preperfect C) (hU : IsOpen U) :
Preperfect (U ∩ C) := by
rintro x ⟨xU, xC⟩ |
rintro x ⟨xU, xC⟩
apply (hC _ xC).nhds_inter
exact hU.mem_nhds xU
| true |
import Mathlib.Data.Matrix.Block
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Matrix.RowCol
import Mathlib.GroupTheory.GroupAction.Ring
import Mathlib.GroupTheory.Perm.Fin
import Mathlib.LinearAlgebra.Alternating.Basic
#align_import linear_algebra.matrix.determinant from "leanprover-community/mathlib"@"c30... | Mathlib/LinearAlgebra/Matrix/Determinant/Basic.lean | 98 | 100 | theorem coe_det_isEmpty [IsEmpty n] : (det : Matrix n n R → R) = Function.const _ 1 := by
ext |
ext
exact det_isEmpty
| true |
import Mathlib.LinearAlgebra.LinearPMap
import Mathlib.Topology.Algebra.Module.Basic
#align_import topology.algebra.module.linear_pmap from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Topology
variable {R E F : Type*}
variable [CommRing R] [AddCommGroup E] [AddCommGroup F]
vari... | Mathlib/Topology/Algebra/Module/LinearPMap.lean | 103 | 104 | theorem closure_def {f : E →ₗ.[R] F} (hf : f.IsClosable) : f.closure = hf.choose := by |
simp [closure, hf]
| true |
import Mathlib.NumberTheory.Liouville.Basic
import Mathlib.Topology.Baire.Lemmas
import Mathlib.Topology.Baire.LocallyCompactRegular
import Mathlib.Topology.Instances.Irrational
#align_import number_theory.liouville.residual from "leanprover-community/mathlib"@"32b08ef840dd25ca2e47e035c5da03ce16d2dc3c"
open scope... | Mathlib/NumberTheory/Liouville/Residual.lean | 44 | 55 | theorem setOf_liouville_eq_irrational_inter_iInter_iUnion :
{ x | Liouville x } =
{ x | Irrational x } ∩ ⋂ n : ℕ, ⋃ (a : ℤ) (b : ℤ) (hb : 1 < b),
ball (a / b) (1 / (b : ℝ) ^ n) := by
refine Subset.antisymm ?_ ?_ |
refine Subset.antisymm ?_ ?_
· refine subset_inter (fun x hx => hx.irrational) ?_
rw [setOf_liouville_eq_iInter_iUnion]
exact iInter_mono fun n => iUnion₂_mono fun a b => iUnion_mono fun _hb => diff_subset
· simp only [inter_iInter, inter_iUnion, setOf_liouville_eq_iInter_iUnion]
refine iInter_mono f... | true |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.HausdorffDistance
import Mathlib.Topology.Sets.Compacts
#align_import topology.metric_space.closeds from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open scoped Classical
open Topo... | Mathlib/Topology/MetricSpace/Closeds.lean | 56 | 69 | theorem continuous_infEdist_hausdorffEdist :
Continuous fun p : α × Closeds α => infEdist p.1 p.2 := by
refine continuous_of_le_add_edist 2 (by simp) ?_ |
refine continuous_of_le_add_edist 2 (by simp) ?_
rintro ⟨x, s⟩ ⟨y, t⟩
calc
infEdist x s ≤ infEdist x t + hausdorffEdist (t : Set α) s :=
infEdist_le_infEdist_add_hausdorffEdist
_ ≤ infEdist y t + edist x y + hausdorffEdist (t : Set α) s :=
(add_le_add_right infEdist_le_infEdist_add_edist _)
... | true |
import Mathlib.CategoryTheory.Preadditive.Yoneda.Basic
import Mathlib.CategoryTheory.Preadditive.Projective
import Mathlib.Algebra.Category.GroupCat.EpiMono
#align_import category_theory.preadditive.yoneda.projective from "leanprover-community/mathlib"@"f8d8465c3c392a93b9ed226956e26dee00975946"
universe v u
open... | Mathlib/CategoryTheory/Preadditive/Yoneda/Projective.lean | 31 | 39 | theorem projective_iff_preservesEpimorphisms_preadditiveCoyoneda_obj (P : C) :
Projective P ↔ (preadditiveCoyoneda.obj (op P)).PreservesEpimorphisms := by
rw [projective_iff_preservesEpimorphisms_coyoneda_obj] |
rw [projective_iff_preservesEpimorphisms_coyoneda_obj]
refine ⟨fun h : (preadditiveCoyoneda.obj (op P) ⋙
forget AddCommGroupCat).PreservesEpimorphisms => ?_, ?_⟩
· exact Functor.preservesEpimorphisms_of_preserves_of_reflects (preadditiveCoyoneda.obj (op P))
(forget _)
· intro
exact (inferInst... | true |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Measure.Haar.Basic
import Mathlib.MeasureTheory.Measure.Doubling
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
#align_import measu... | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 95 | 104 | theorem Basis.map_addHaar {ι E F : Type*} [Fintype ι] [NormedAddCommGroup E] [NormedAddCommGroup F]
[NormedSpace ℝ E] [NormedSpace ℝ F] [MeasurableSpace E] [MeasurableSpace F] [BorelSpace E]
[BorelSpace F] [SecondCountableTopology F] [SigmaCompactSpace F]
(b : Basis ι ℝ E) (f : E ≃L[ℝ] F) :
map f b.addH... |
have : IsAddHaarMeasure (map f b.addHaar) :=
AddEquiv.isAddHaarMeasure_map b.addHaar f.toAddEquiv f.continuous f.symm.continuous
rw [eq_comm, Basis.addHaar_eq_iff, Measure.map_apply f.continuous.measurable
(PositiveCompacts.isCompact _).measurableSet, Basis.coe_parallelepiped, Basis.coe_map]
erw [← image... | true |
import Mathlib.Data.List.Sublists
import Mathlib.Data.Multiset.Bind
#align_import data.multiset.powerset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
namespace Multiset
open List
variable {α : Type*}
-- Porting note (#11215): TODO: Write a more efficient version
def powerset... | Mathlib/Data/Multiset/Powerset.lean | 45 | 46 | theorem powersetAux_perm_powersetAux' {l : List α} : powersetAux l ~ powersetAux' l := by |
rw [powersetAux_eq_map_coe]; exact (sublists_perm_sublists' _).map _
| true |
import Mathlib.RingTheory.Derivation.Basic
import Mathlib.RingTheory.Ideal.QuotientOperations
#align_import ring_theory.derivation.to_square_zero from "leanprover-community/mathlib"@"b608348ffaeb7f557f2fd46876037abafd326ff3"
section ToSquareZero
universe u v w
variable {R : Type u} {A : Type v} {B : Type w} [Co... | Mathlib/RingTheory/Derivation/ToSquareZero.lean | 106 | 110 | theorem liftOfDerivationToSquareZero_mk_apply (d : Derivation R A I) (x : A) :
Ideal.Quotient.mk I (liftOfDerivationToSquareZero I hI d x) = algebraMap A (B ⧸ I) x := by
rw [liftOfDerivationToSquareZero_apply, map_add, Ideal.Quotient.eq_zero_iff_mem.mpr (d x).prop, |
rw [liftOfDerivationToSquareZero_apply, map_add, Ideal.Quotient.eq_zero_iff_mem.mpr (d x).prop,
zero_add]
rfl
| true |
import Mathlib.RingTheory.OrzechProperty
import Mathlib.RingTheory.Ideal.Quotient
import Mathlib.RingTheory.PrincipalIdealDomain
#align_import linear_algebra.invariant_basis_number from "leanprover-community/mathlib"@"5fd3186f1ec30a75d5f65732e3ce5e623382556f"
noncomputable section
open Function
universe u v w
... | Mathlib/LinearAlgebra/InvariantBasisNumber.lean | 167 | 173 | theorem card_le_of_injective' [StrongRankCondition R] {α β : Type*} [Fintype α] [Fintype β]
(f : (α →₀ R) →ₗ[R] β →₀ R) (i : Injective f) : Fintype.card α ≤ Fintype.card β := by
let P := Finsupp.linearEquivFunOnFinite R R β |
let P := Finsupp.linearEquivFunOnFinite R R β
let Q := (Finsupp.linearEquivFunOnFinite R R α).symm
exact
card_le_of_injective R ((P.toLinearMap.comp f).comp Q.toLinearMap)
((P.injective.comp i).comp Q.injective)
| true |
import Mathlib.Analysis.Convex.Cone.Basic
import Mathlib.Analysis.InnerProductSpace.Projection
#align_import analysis.convex.cone.dual from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284a161a7f2a9e3d4"
open Set LinearMap
open scoped Classical
open Pointwise
variable {𝕜 E F G : Type*}
section Dua... | Mathlib/Analysis/Convex/Cone/InnerDual.lean | 144 | 161 | theorem ConvexCone.pointed_of_nonempty_of_isClosed (K : ConvexCone ℝ H) (ne : (K : Set H).Nonempty)
(hc : IsClosed (K : Set H)) : K.Pointed := by
obtain ⟨x, hx⟩ := ne |
obtain ⟨x, hx⟩ := ne
let f : ℝ → H := (· • x)
-- f (0, ∞) is a subset of K
have fI : f '' Set.Ioi 0 ⊆ (K : Set H) := by
rintro _ ⟨_, h, rfl⟩
exact K.smul_mem (Set.mem_Ioi.1 h) hx
-- closure of f (0, ∞) is a subset of K
have clf : closure (f '' Set.Ioi 0) ⊆ (K : Set H) := hc.closure_subset_iff.2 fI
... | true |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.Option
#align_import algebra.big_operators.option from "leanprover-community/mathlib"@"008205aa645b3f194c1da47025c5f110c8406eab"
open Function
namespace Finset
variable {α M : Type*} [CommMonoid M]
@[to_additive (attr := simp)]
| Mathlib/Algebra/BigOperators/Option.lean | 25 | 26 | theorem prod_insertNone (f : Option α → M) (s : Finset α) :
∏ x ∈ insertNone s, f x = f none * ∏ x ∈ s, f (some x) := by | simp [insertNone]
| true |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse
#align_import analysis.special_functions.complex.arg from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba144755cd1"
open Filter Metric Set
open scoped ComplexConjugate Real To... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 76 | 83 | theorem abs_eq_one_iff (z : ℂ) : abs z = 1 ↔ ∃ θ : ℝ, exp (θ * I) = z := by
refine ⟨fun hz => ⟨arg z, ?_⟩, ?_⟩ |
refine ⟨fun hz => ⟨arg z, ?_⟩, ?_⟩
· calc
exp (arg z * I) = abs z * exp (arg z * I) := by rw [hz, ofReal_one, one_mul]
_ = z := abs_mul_exp_arg_mul_I z
· rintro ⟨θ, rfl⟩
exact Complex.abs_exp_ofReal_mul_I θ
| true |
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Group.Measure
import Mathlib.Topology.Constructions
#align_import measure_theory.constructions.pi from "leanprover-community/mathlib"@"fd5edc43dc4f10b85abfe544b88f82cf13c5f844"
noncomputable section
open Function Set MeasureTheory... | Mathlib/MeasureTheory/Constructions/Pi.lean | 197 | 201 | theorem pi_pi_le (m : ∀ i, OuterMeasure (α i)) (s : ∀ i, Set (α i)) :
OuterMeasure.pi m (pi univ s) ≤ ∏ i, m i (s i) := by
rcases (pi univ s).eq_empty_or_nonempty with h | h |
rcases (pi univ s).eq_empty_or_nonempty with h | h
· simp [h]
exact (boundedBy_le _).trans_eq (piPremeasure_pi h)
| true |
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.NormedSpace.Spectrum
import Mathlib.Analysis.SpecialFunctions.Exponential
import Mathlib.Algebra.Star.StarAlgHom
#align_import analysis.normed_space.star.spectrum from "leanprover-community/mathlib"@"f0c8bf9245297a541f468be517f1bde6195105e9"
l... | Mathlib/Analysis/NormedSpace/Star/Spectrum.lean | 31 | 41 | theorem unitary.spectrum_subset_circle (u : unitary E) :
spectrum 𝕜 (u : E) ⊆ Metric.sphere 0 1 := by
nontriviality E |
nontriviality E
refine fun k hk => mem_sphere_zero_iff_norm.mpr (le_antisymm ?_ ?_)
· simpa only [CstarRing.norm_coe_unitary u] using norm_le_norm_of_mem hk
· rw [← unitary.val_toUnits_apply u] at hk
have hnk := ne_zero_of_mem_of_unit hk
rw [← inv_inv (unitary.toUnits u), ← spectrum.map_inv, Set.mem_in... | true |
import Mathlib.Analysis.NormedSpace.Banach
import Mathlib.Topology.Algebra.Module.FiniteDimension
#align_import analysis.normed_space.complemented from "leanprover-community/mathlib"@"3397560e65278e5f31acefcdea63138bd53d1cd4"
variable {𝕜 E F G : Type*} [NontriviallyNormedField 𝕜] [NormedAddCommGroup E] [NormedS... | Mathlib/Analysis/NormedSpace/Complemented.lean | 39 | 43 | theorem ker_closedComplemented_of_finiteDimensional_range (f : E →L[𝕜] F)
[FiniteDimensional 𝕜 (range f)] : (ker f).ClosedComplemented := by
set f' : E →L[𝕜] range f := f.codRestrict _ (LinearMap.mem_range_self (f : E →ₗ[𝕜] F)) |
set f' : E →L[𝕜] range f := f.codRestrict _ (LinearMap.mem_range_self (f : E →ₗ[𝕜] F))
rcases f'.exists_right_inverse_of_surjective (f : E →ₗ[𝕜] F).range_rangeRestrict with ⟨g, hg⟩
simpa only [f', ker_codRestrict] using f'.closedComplemented_ker_of_rightInverse g (ext_iff.1 hg)
| true |
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 | 130 | 133 | theorem Path.cast_cons {u v w u' w' : U} (p : Path u v) (e : v ⟶ w) (hu : u = u') (hw : w = w') :
(p.cons e).cast hu hw = (p.cast hu rfl).cons (e.cast rfl hw) := by
subst_vars |
subst_vars
rfl
| true |
import Mathlib.Data.Matrix.Basic
import Mathlib.LinearAlgebra.Matrix.Trace
#align_import data.matrix.basis from "leanprover-community/mathlib"@"320df450e9abeb5fc6417971e75acb6ae8bc3794"
variable {l m n : Type*}
variable {R α : Type*}
namespace Matrix
open Matrix
variable [DecidableEq l] [DecidableEq m] [Decida... | Mathlib/Data/Matrix/Basis.lean | 57 | 63 | theorem mulVec_stdBasisMatrix [Fintype m] (i : n) (j : m) (c : α) (x : m → α) :
mulVec (stdBasisMatrix i j c) x = Function.update (0 : n → α) i (c * x j) := by
ext i' |
ext i'
simp [stdBasisMatrix, mulVec, dotProduct]
rcases eq_or_ne i i' with rfl|h
· simp
simp [h, h.symm]
| true |
import Mathlib.Analysis.Convex.Cone.Basic
import Mathlib.Data.Real.Archimedean
import Mathlib.LinearAlgebra.LinearPMap
#align_import analysis.convex.cone.basic from "leanprover-community/mathlib"@"915591b2bb3ea303648db07284a161a7f2a9e3d4"
open Set LinearMap
variable {𝕜 E F G : Type*}
variable [AddCommGroup E... | Mathlib/Analysis/Convex/Cone/Extension.lean | 64 | 112 | theorem step (nonneg : ∀ x : f.domain, (x : E) ∈ s → 0 ≤ f x)
(dense : ∀ y, ∃ x : f.domain, (x : E) + y ∈ s) (hdom : f.domain ≠ ⊤) :
∃ g, f < g ∧ ∀ x : g.domain, (x : E) ∈ s → 0 ≤ g x := by
obtain ⟨y, -, hy⟩ : ∃ y ∈ ⊤, y ∉ f.domain := SetLike.exists_of_lt (lt_top_iff_ne_top.2 hdom) |
obtain ⟨y, -, hy⟩ : ∃ y ∈ ⊤, y ∉ f.domain := SetLike.exists_of_lt (lt_top_iff_ne_top.2 hdom)
obtain ⟨c, le_c, c_le⟩ :
∃ c, (∀ x : f.domain, -(x : E) - y ∈ s → f x ≤ c) ∧
∀ x : f.domain, (x : E) + y ∈ s → c ≤ f x := by
set Sp := f '' { x : f.domain | (x : E) + y ∈ s }
set Sn := f '' { x : f.do... | true |
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.Group.Prod
import Mathlib.Data.PNat.Basic
import Mathlib.GroupTheory.GroupAction.Prod
variable {M : Type*}
class PNatPowAssoc (M : Type*) [Mul M] [Pow M ℕ+] : Prop where
protected ppow_add : ∀ (k n : ℕ+) (x : M), x ^ (k + n) = x ^ k * x ^ n
prote... | Mathlib/Algebra/Group/PNatPowAssoc.lean | 60 | 62 | theorem ppow_mul_assoc (k m n : ℕ+) (x : M) :
(x ^ k * x ^ m) * x ^ n = x ^ k * (x ^ m * x ^ n) := by |
simp only [← ppow_add, add_assoc]
| true |
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 | 638 | 642 | theorem cos_oangle_right_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
Real.Angle.cos (∡ p₂ p₃ p₁) = dist p₃ p₂ / dist p₁ p₃ := by
have hs : (∡ p₂ p₃ p₁).sign = 1 := by rw [oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two] |
have hs : (∡ p₂ p₃ p₁).sign = 1 := by rw [oangle_rotate_sign, h, Real.Angle.sign_coe_pi_div_two]
rw [oangle_eq_angle_of_sign_eq_one hs, Real.Angle.cos_coe,
cos_angle_of_angle_eq_pi_div_two (angle_eq_pi_div_two_of_oangle_eq_pi_div_two h)]
| true |
import Mathlib.Probability.Kernel.Basic
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.DominatedConvergence
#align_import probability.kernel.measurable_integral from "leanprover-community/mathlib"@"28b2a92f2996d28e580450863c130955de0ed398"
open MeasureTheory Probabilit... | Mathlib/Probability/Kernel/MeasurableIntegral.lean | 42 | 99 | theorem measurable_kernel_prod_mk_left_of_finite {t : Set (α × β)} (ht : MeasurableSet t)
(hκs : ∀ a, IsFiniteMeasure (κ a)) : Measurable fun a => κ a (Prod.mk a ⁻¹' t) := by
-- `t` is a measurable set in the product `α × β`: we use that the product σ-algebra is generated |
-- `t` is a measurable set in the product `α × β`: we use that the product σ-algebra is generated
-- by boxes to prove the result by induction.
-- Porting note: added motive
refine MeasurableSpace.induction_on_inter
(C := fun t => Measurable fun a => κ a (Prod.mk a ⁻¹' t))
generateFrom_prod.symm isPiSy... | true |
import Mathlib.AlgebraicGeometry.Properties
#align_import algebraic_geometry.function_field from "leanprover-community/mathlib"@"d39590fc8728fbf6743249802486f8c91ffe07bc"
-- Explicit universe annotations were used in this file to improve perfomance #12737
set_option linter.uppercaseLean3 false
universe u v
open... | Mathlib/AlgebraicGeometry/FunctionField.lean | 83 | 93 | theorem genericPoint_eq_of_isOpenImmersion {X Y : Scheme} (f : X ⟶ Y) [H : IsOpenImmersion f]
[hX : IrreducibleSpace X.carrier] [IrreducibleSpace Y.carrier] :
f.1.base (genericPoint X.carrier : _) = (genericPoint Y.carrier : _) := by
apply ((genericPoint_spec Y).eq _).symm |
apply ((genericPoint_spec Y).eq _).symm
convert (genericPoint_spec X.carrier).image (show Continuous f.1.base by continuity)
symm
rw [eq_top_iff, Set.top_eq_univ, Set.top_eq_univ]
convert subset_closure_inter_of_isPreirreducible_of_isOpen _ H.base_open.isOpen_range _
· rw [Set.univ_inter, Set.image_univ]
... | true |
import Mathlib.GroupTheory.Coxeter.Length
import Mathlib.Data.ZMod.Parity
namespace CoxeterSystem
open List Matrix Function
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
local prefi... | Mathlib/GroupTheory/Coxeter/Inversion.lean | 82 | 86 | theorem odd_length : Odd (ℓ t) := by
suffices cs.lengthParity t = Multiplicative.ofAdd 1 by |
suffices cs.lengthParity t = Multiplicative.ofAdd 1 by
simpa [lengthParity_eq_ofAdd_length, ZMod.eq_one_iff_odd]
rcases ht with ⟨w, i, rfl⟩
simp [lengthParity_simple]
| true |
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