Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | num_lines int64 1 150 |
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import Mathlib.Topology.Sets.Closeds
import Mathlib.Topology.QuasiSeparated
#align_import topology.sets.compacts from "leanprover-community/mathlib"@"8c1b484d6a214e059531e22f1be9898ed6c1fd47"
open Set
variable {α β γ : Type*} [TopologicalSpace α] [TopologicalSpace β] [TopologicalSpace γ]
namespace TopologicalSp... | Mathlib/Topology/Sets/Compacts.lean | 125 | 129 | theorem coe_finset_sup {ι : Type*} {s : Finset ι} {f : ι → Compacts α} :
(↑(s.sup f) : Set α) = s.sup fun i => ↑(f i) := by |
refine Finset.cons_induction_on s rfl fun a s _ h => ?_
simp_rw [Finset.sup_cons, coe_sup, sup_eq_union]
congr
| 3 |
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Data.Set.Basic
import Mathlib.Data.Setoid.Basic
import Mathlib.GroupTheory.Coset
#align_import group_theory.double_coset from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
-- Porting note: removed import
-- import Mathlib.Tac... | Mathlib/GroupTheory/DoubleCoset.lean | 69 | 73 | theorem eq_of_not_disjoint {H K : Subgroup G} {a b : G}
(h : ¬Disjoint (doset a H K) (doset b H K)) : doset a H K = doset b H K := by |
rw [disjoint_comm] at h
have ha : a ∈ doset b H K := mem_doset_of_not_disjoint h
apply doset_eq_of_mem ha
| 3 |
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.RingTheory.PolynomialAlgebra
#align_import linear_algebra.matrix.charpoly.basic from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
noncomputable section
universe u v w
namespace Matrix
open Finset Matrix Polynomial
variable... | Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean | 79 | 83 | theorem charmatrix_reindex (e : n ≃ m) :
charmatrix (reindex e e M) = reindex e e (charmatrix M) := by |
ext i j x
by_cases h : i = j
all_goals simp [h]
| 3 |
import Mathlib.Algebra.Order.Group.Nat
import Mathlib.Data.Finset.Antidiagonal
import Mathlib.Data.Finset.Card
import Mathlib.Data.Multiset.NatAntidiagonal
#align_import data.finset.nat_antidiagonal from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function
namespace Finset
name... | Mathlib/Data/Finset/NatAntidiagonal.lean | 78 | 86 | theorem antidiagonal_succ' (n : ℕ) :
antidiagonal (n + 1) =
cons (n + 1, 0)
((antidiagonal n).map
(Embedding.prodMap (Embedding.refl _) ⟨Nat.succ, Nat.succ_injective⟩))
(by simp) := by |
apply eq_of_veq
rw [cons_val, map_val]
exact Multiset.Nat.antidiagonal_succ'
| 3 |
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.asymptotics.specific_asymptotics from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Filter Asymptotics
open Topology
sectio... | Mathlib/Analysis/Asymptotics/SpecificAsymptotics.lean | 49 | 53 | theorem pow_div_pow_eventuallyEq_atBot {p q : ℕ} :
(fun x : 𝕜 => x ^ p / x ^ q) =ᶠ[atBot] fun x => x ^ ((p : ℤ) - q) := by |
apply (eventually_lt_atBot (0 : 𝕜)).mono fun x hx => _
intro x hx
simp [zpow_sub₀ hx.ne]
| 3 |
import Mathlib.LinearAlgebra.Quotient
import Mathlib.LinearAlgebra.Prod
#align_import linear_algebra.projection from "leanprover-community/mathlib"@"6d584f1709bedbed9175bd9350df46599bdd7213"
noncomputable section Ring
variable {R : Type*} [Ring R] {E : Type*} [AddCommGroup E] [Module R E]
variable {F : Type*} [Ad... | Mathlib/LinearAlgebra/Projection.lean | 430 | 433 | theorem codRestrict_apply_cod {f : M →ₗ[S] M} (h : IsProj m f) (x : m) : h.codRestrict x = x := by |
ext
rw [codRestrict_apply]
exact h.map_id x x.2
| 3 |
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 | 104 | 109 | theorem summable_of_summable_hasFDerivAt (hu : Summable u)
(hf : ∀ n x, HasFDerivAt (f n) (f' n x) x) (hf' : ∀ n x, ‖f' n x‖ ≤ u n)
(hf0 : Summable fun n => f n x₀) (x : E) : Summable fun n => f n x := by |
let _ : NormedSpace ℝ E := NormedSpace.restrictScalars ℝ 𝕜 _
exact summable_of_summable_hasFDerivAt_of_isPreconnected hu isOpen_univ isPreconnected_univ
(fun n x _ => hf n x) (fun n x _ => hf' n x) (mem_univ _) hf0 (mem_univ _)
| 3 |
import Mathlib.Data.Finsupp.Lex
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.GameAdd
#align_import logic.hydra from "leanprover-community/mathlib"@"48085f140e684306f9e7da907cd5932056d1aded"
namespace Relation
open Multiset Prod
variable {α : Type*}
def CutExpand (r : α → α → Prop) (s' s : Multise... | Mathlib/Logic/Hydra.lean | 101 | 104 | theorem not_cutExpand_zero [IsIrrefl α r] (s) : ¬CutExpand r s 0 := by |
classical
rw [cutExpand_iff]
rintro ⟨_, _, _, ⟨⟩, _⟩
| 3 |
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique
import Mathlib.MeasureTheory.Function.L2Space
#align_import measure_theory.function.conditional_expectation.condexp_L2 from "leanprover-community/mathlib"@"d8bbb04e2d2a44596798a9207ceefc0fb236e41e"... | Mathlib/MeasureTheory/Function/ConditionalExpectation/CondexpL2.lean | 106 | 110 | theorem snorm_condexpL2_le (hm : m ≤ m0) (f : α →₂[μ] E) :
snorm (F := E) (condexpL2 E 𝕜 hm f) 2 μ ≤ snorm f 2 μ := by |
rw [lpMeas_coe, ← ENNReal.toReal_le_toReal (Lp.snorm_ne_top _) (Lp.snorm_ne_top _), ←
Lp.norm_def, ← Lp.norm_def, Submodule.norm_coe]
exact norm_condexpL2_le hm f
| 3 |
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.inner_product_space.adjoint from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open RCLike
open scoped ComplexConjugate
variable {𝕜 E F G : Type... | Mathlib/Analysis/InnerProductSpace/Adjoint.lean | 92 | 95 | theorem adjointAux_adjointAux (A : E →L[𝕜] F) : adjointAux (adjointAux A) = A := by |
ext v
refine ext_inner_left 𝕜 fun w => ?_
rw [adjointAux_inner_right, adjointAux_inner_left]
| 3 |
import Mathlib.Data.Fintype.List
#align_import data.list.cycle from "leanprover-community/mathlib"@"7413128c3bcb3b0818e3e18720abc9ea3100fb49"
assert_not_exists MonoidWithZero
namespace List
variable {α : Type*} [DecidableEq α]
def nextOr : ∀ (_ : List α) (_ _ : α), α
| [], _, default => default
| [_], _, d... | Mathlib/Data/List/Cycle.lean | 172 | 177 | theorem next_cons_concat (y : α) (hy : x ≠ y) (hx : x ∉ l)
(h : x ∈ y :: l ++ [x] := mem_append_right _ (mem_singleton_self x)) :
next (y :: l ++ [x]) x h = y := by |
rw [next, nextOr_concat]
· rfl
· simp [hy, hx]
| 3 |
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Algebra.Order.Group.Instances
import Mathlib.LinearAlgebra.AffineSpace.Slope
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
import Mathlib.Tactic.FieldSimp
#align_import li... | Mathlib/LinearAlgebra/AffineSpace/Ordered.lean | 127 | 130 | theorem lineMap_le_lineMap_iff_of_lt (h : r < r') : lineMap a b r ≤ lineMap a b r' ↔ a ≤ b := by |
simp only [lineMap_apply_module]
rw [← le_sub_iff_add_le, add_sub_assoc, ← sub_le_iff_le_add', ← sub_smul, ← sub_smul,
sub_sub_sub_cancel_left, smul_le_smul_iff_of_pos_left (sub_pos.2 h)]
| 3 |
import Mathlib.Algebra.MvPolynomial.Basic
import Mathlib.Data.Finset.PiAntidiagonal
import Mathlib.LinearAlgebra.StdBasis
import Mathlib.Tactic.Linarith
#align_import ring_theory.power_series.basic from "leanprover-community/mathlib"@"2d5739b61641ee4e7e53eca5688a08f66f2e6a60"
noncomputable section
open Finset (... | Mathlib/RingTheory/MvPowerSeries/Basic.lean | 150 | 153 | theorem coeff_monomial_ne {m n : σ →₀ ℕ} (h : m ≠ n) (a : R) : coeff R m (monomial R n a) = 0 := by |
classical
rw [monomial_def]
exact LinearMap.stdBasis_ne R (fun _ ↦ R) _ _ h a
| 3 |
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Ring.Divisibility.Basic
import Mathlib.Algebra.Ring.Hom.Defs
import Mathlib.GroupTheory.GroupAction.Units
import Mathlib.Logic.Basic
import Mathlib.Tactic.Ring
#align_import ring_theory.coprime.basic from "leanprover-community/mathlib"@"a95b16cb... | Mathlib/RingTheory/Coprime/Basic.lean | 108 | 111 | theorem IsCoprime.dvd_of_dvd_mul_left (H1 : IsCoprime x y) (H2 : x ∣ y * z) : x ∣ z := by |
let ⟨a, b, H⟩ := H1
rw [← one_mul z, ← H, add_mul, mul_right_comm, mul_assoc b]
exact dvd_add (dvd_mul_left _ _) (H2.mul_left _)
| 3 |
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Topology.ContinuousFunction.Algebra
import Mathlib.Topology.Compactness.Paracompact
import Mathlib.Topology.ShrinkingLemma
import Mathlib.Topology.UrysohnsLemma
#align_import topology.partition_of_unity from "leanprover-... | Mathlib/Topology/PartitionOfUnity.lean | 161 | 164 | theorem exists_pos {x : X} (hx : x ∈ s) : ∃ i, 0 < f i x := by |
have H := f.sum_eq_one hx
contrapose! H
simpa only [fun i => (H i).antisymm (f.nonneg i x), finsum_zero] using zero_ne_one
| 3 |
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 | 356 | 359 | theorem map_modByMonic (h : IsAdjoinRootMonic S f) (g : R[X]) : h.map (g %ₘ f) = h.map g := by |
rw [← RingHom.sub_mem_ker_iff, mem_ker_map, modByMonic_eq_sub_mul_div _ h.Monic, sub_right_comm,
sub_self, zero_sub, dvd_neg]
exact ⟨_, rfl⟩
| 3 |
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 | 164 | 168 | theorem rank_finsupp (ι : Type w) :
Module.rank R (ι →₀ M) = Cardinal.lift.{v} #ι * Cardinal.lift.{w} (Module.rank R M) := by |
obtain ⟨⟨_, bs⟩⟩ := Module.Free.exists_basis (R := R) (M := M)
rw [← bs.mk_eq_rank'', ← (Finsupp.basis fun _ : ι => bs).mk_eq_rank'', Cardinal.mk_sigma,
Cardinal.sum_const]
| 3 |
import Mathlib.Data.Set.Pairwise.Basic
import Mathlib.Data.Set.Lattice
import Mathlib.Data.SetLike.Basic
#align_import order.chain from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
open scoped Classical
open Set
variable {α β : Type*}
section Chain
variable (r : α → α → Prop)
... | Mathlib/Order/Chain.lean | 95 | 98 | theorem isChain_univ_iff : IsChain r (univ : Set α) ↔ IsTrichotomous α r := by |
refine ⟨fun h => ⟨fun a b => ?_⟩, fun h => @isChain_of_trichotomous _ _ h univ⟩
rw [or_left_comm, or_iff_not_imp_left]
exact h trivial trivial
| 3 |
import Mathlib.Combinatorics.Young.YoungDiagram
#align_import combinatorics.young.semistandard_tableau from "leanprover-community/mathlib"@"b363547b3113d350d053abdf2884e9850a56b205"
structure SemistandardYoungTableau (μ : YoungDiagram) where
entry : ℕ → ℕ → ℕ
row_weak' : ∀ {i j1 j2 : ℕ}, j1 < j2 → (i, ... | Mathlib/Combinatorics/Young/SemistandardTableau.lean | 136 | 140 | theorem col_weak {μ : YoungDiagram} (T : SemistandardYoungTableau μ) {i1 i2 j : ℕ} (hi : i1 ≤ i2)
(cell : (i2, j) ∈ μ) : T i1 j ≤ T i2 j := by |
cases' eq_or_lt_of_le hi with h h
· rw [h]
· exact le_of_lt (T.col_strict h cell)
| 3 |
import Mathlib.AlgebraicTopology.SplitSimplicialObject
import Mathlib.AlgebraicTopology.DoldKan.PInfty
#align_import algebraic_topology.dold_kan.functor_gamma from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504"
noncomputable section
open CategoryTheory CategoryTheory.Category CategoryT... | Mathlib/AlgebraicTopology/DoldKan/FunctorGamma.lean | 64 | 68 | theorem eq_δ₀ {n : ℕ} {i : ([n] : SimplexCategory) ⟶ [n + 1]} [Mono i] (hi : Isδ₀ i) :
i = SimplexCategory.δ 0 := by |
obtain ⟨j, rfl⟩ := SimplexCategory.eq_δ_of_mono i
rw [iff] at hi
rw [hi]
| 3 |
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Algebra.Order.Ring.Abs
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.Data.Nat.Choose.Dvd
import Mathlib.Data.Nat.Prime
#align_import number_theory.primorial from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
open Finset
... | Mathlib/NumberTheory/Primorial.lean | 45 | 48 | theorem primorial_succ {n : ℕ} (hn1 : n ≠ 1) (hn : Odd n) : (n + 1)# = n# := by |
refine prod_congr ?_ fun _ _ ↦ rfl
rw [range_succ, filter_insert, if_neg fun h ↦ odd_iff_not_even.mp hn _]
exact fun h ↦ h.even_sub_one <| mt succ.inj hn1
| 3 |
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 | 86 | 90 | theorem trim_trim {m₁ m₂ : MeasurableSpace α} {hm₁₂ : m₁ ≤ m₂} {hm₂ : m₂ ≤ m0} :
(μ.trim hm₂).trim hm₁₂ = μ.trim (hm₁₂.trans hm₂) := by |
refine @Measure.ext _ m₁ _ _ (fun t ht => ?_)
rw [trim_measurableSet_eq hm₁₂ ht, trim_measurableSet_eq (hm₁₂.trans hm₂) ht,
trim_measurableSet_eq hm₂ (hm₁₂ t ht)]
| 3 |
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
open Set Filter Topology TopologicalSpace
universe u v
variable {X : Type u} {Y : Type v} {ι : Type*}
variable [TopologicalSpace X] [TopologicalSpace Y] {s t : Set X}
section Lindelof
def I... | Mathlib/Topology/Compactness/Lindelof.lean | 52 | 56 | theorem IsLindelof.compl_mem_sets (hs : IsLindelof s) {f : Filter X} [CountableInterFilter f]
(hf : ∀ x ∈ s, sᶜ ∈ 𝓝 x ⊓ f) : sᶜ ∈ f := by |
contrapose! hf
simp only [not_mem_iff_inf_principal_compl, compl_compl, inf_assoc] at hf ⊢
exact hs inf_le_right
| 3 |
import Mathlib.Topology.Homotopy.Path
import Mathlib.Topology.Homotopy.Equiv
#align_import topology.homotopy.contractible from "leanprover-community/mathlib"@"16728b3064a1751103e1dc2815ed8d00560e0d87"
noncomputable section
namespace ContinuousMap
variable {X Y Z : Type*} [TopologicalSpace X] [TopologicalSpace Y]... | Mathlib/Topology/Homotopy/Contractible.lean | 32 | 36 | theorem Nullhomotopic.comp_right {f : C(X, Y)} (hf : f.Nullhomotopic) (g : C(Y, Z)) :
(g.comp f).Nullhomotopic := by |
cases' hf with y hy
use g y
exact Homotopic.hcomp hy (Homotopic.refl g)
| 3 |
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Data.Nat.Cast.NeZero
import Mathlib.Algebra.Order.Ring.Nat
#align_import data.nat.cast.basic from "leanprover-community/mathlib"@"acebd8d49928f6ed8920e502a6c90674e75bd441"
variable {α β : T... | Mathlib/Data/Nat/Cast/Order.lean | 80 | 83 | theorem cast_add_one_pos (n : ℕ) : 0 < (n : α) + 1 := by |
apply zero_lt_one.trans_le
convert (@mono_cast α _).imp (?_ : 1 ≤ n + 1)
<;> simp
| 3 |
import Mathlib.Data.Matrix.Kronecker
import Mathlib.LinearAlgebra.Matrix.ToLin
import Mathlib.LinearAlgebra.TensorProduct.Basis
#align_import linear_algebra.tensor_product.matrix from "leanprover-community/mathlib"@"f784cc6142443d9ee623a20788c282112c322081"
variable {R : Type*} {M N P M' N' : Type*} {ι κ τ ι' κ' ... | Mathlib/LinearAlgebra/TensorProduct/Matrix.lean | 39 | 44 | theorem TensorProduct.toMatrix_map (f : M →ₗ[R] M') (g : N →ₗ[R] N') :
toMatrix (bM.tensorProduct bN) (bM'.tensorProduct bN') (TensorProduct.map f g) =
toMatrix bM bM' f ⊗ₖ toMatrix bN bN' g := by |
ext ⟨i, j⟩ ⟨i', j'⟩
simp_rw [Matrix.kroneckerMap_apply, toMatrix_apply, Basis.tensorProduct_apply,
TensorProduct.map_tmul, Basis.tensorProduct_repr_tmul_apply]
| 3 |
import Mathlib.CategoryTheory.Limits.Types
import Mathlib.CategoryTheory.Filtered.Basic
#align_import category_theory.limits.types from "leanprover-community/mathlib"@"4aa2a2e17940311e47007f087c9df229e7f12942"
open CategoryTheory CategoryTheory.Limits
universe v u w
namespace CategoryTheory.Limits.Types.Filtered... | Mathlib/CategoryTheory/Limits/TypesFiltered.lean | 112 | 117 | theorem colimit_eq_iff_aux {i j : J} {xi : F.obj i} {xj : F.obj j} :
(colimitCocone F).ι.app i xi = (colimitCocone F).ι.app j xj ↔
FilteredColimit.Rel.{v, u} F ⟨i, xi⟩ ⟨j, xj⟩ := by |
dsimp
rw [← (equivShrink _).symm.injective.eq_iff, Equiv.symm_apply_apply, Equiv.symm_apply_apply,
Quot.eq, FilteredColimit.rel_eq_eqvGen_quot_rel]
| 3 |
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearAlgebra.LinearPMap
import Mathlib.LinearAlgebra.Projection
#align_import linear_algebra.basis from "leanprover-community/mathlib"@"13bce9a6b6c44f6b4c91ac1c1d2a816e2533d395"
open Function Set Submodule
set_option ... | Mathlib/LinearAlgebra/Basis/VectorSpace.lean | 127 | 131 | theorem ofVectorSpaceIndex.linearIndependent :
LinearIndependent K ((↑) : ofVectorSpaceIndex K V → V) := by |
convert (ofVectorSpace K V).linearIndependent
ext x
rw [ofVectorSpace_apply_self]
| 3 |
import Mathlib.CategoryTheory.Sites.Canonical
#align_import category_theory.sites.types from "leanprover-community/mathlib"@"9f9015c645d85695581237cc761981036be8bd37"
universe u
namespace CategoryTheory
--open scoped CategoryTheory.Type -- Porting note: unknown namespace
def typesGrothendieckTopology : Grothe... | Mathlib/CategoryTheory/Sites/Types.lean | 102 | 105 | theorem eval_typesGlue {S hs α} (f) : eval.{u} S α (typesGlue S hs α f) = f := by |
funext x
apply (IsSheafFor.valid_glue _ _ _ <| ⟨PUnit.unit, fun _ => Subsingleton.elim _ _⟩).trans
convert FunctorToTypes.map_id_apply S _
| 3 |
import Mathlib.Algebra.Category.ModuleCat.Basic
import Mathlib.LinearAlgebra.TensorProduct.Basic
import Mathlib.CategoryTheory.Monoidal.Linear
#align_import algebra.category.Module.monoidal.basic from "leanprover-community/mathlib"@"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2"
-- Porting note: Module
set_option linte... | Mathlib/Algebra/Category/ModuleCat/Monoidal/Basic.lean | 75 | 78 | theorem tensor_id (M N : ModuleCat R) : tensorHom (𝟙 M) (𝟙 N) = 𝟙 (ModuleCat.of R (M ⊗ N)) := by |
-- Porting note: even with high priority ext fails to find this
apply TensorProduct.ext
rfl
| 3 |
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.Algebra.MvPolynomial.Supported
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.RingTheory.Algebraic
import Mathlib.RingTheory.MvPolynomial.Basic
#align_import ring_theory.algebraic_independent from "leanprove... | Mathlib/RingTheory/AlgebraicIndependent.lean | 156 | 160 | theorem of_comp (f : A →ₐ[R] A') (hfv : AlgebraicIndependent R (f ∘ x)) :
AlgebraicIndependent R x := by |
have : aeval (f ∘ x) = f.comp (aeval x) := by ext; simp
rw [AlgebraicIndependent, this, AlgHom.coe_comp] at hfv
exact hfv.of_comp
| 3 |
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic
#align_import algebra.category.Module.monoidal.symmetric from "leanprover-community/mathlib"@"74403a3b2551b0970855e14ef5e8fd0d6af1bfc2"
suppress_compilation
universe v w x u
open CategoryTheory MonoidalC... | Mathlib/Algebra/Category/ModuleCat/Monoidal/Symmetric.lean | 55 | 60 | theorem hexagon_forward (X Y Z : ModuleCat.{u} R) :
(α_ X Y Z).hom ≫ (braiding X _).hom ≫ (α_ Y Z X).hom =
(braiding X Y).hom ▷ Z ≫ (α_ Y X Z).hom ≫ Y ◁ (braiding X Z).hom := by |
apply TensorProduct.ext_threefold
intro x y z
rfl
| 3 |
import Mathlib.Algebra.Polynomial.Reverse
import Mathlib.Algebra.Regular.SMul
#align_import data.polynomial.monic from "leanprover-community/mathlib"@"cbdf7b565832144d024caa5a550117c6df0204a5"
noncomputable section
open Finset
open Polynomial
namespace Polynomial
universe u v y
variable {R : Type u} {S : Typ... | Mathlib/Algebra/Polynomial/Monic.lean | 84 | 88 | theorem monic_mul_C_of_leadingCoeff_mul_eq_one {b : R} (hp : p.leadingCoeff * b = 1) :
Monic (p * C b) := by |
unfold Monic
nontriviality
rw [leadingCoeff_mul' _] <;> simp [leadingCoeff_C b, hp]
| 3 |
import Mathlib.GroupTheory.GroupAction.Prod
import Mathlib.Algebra.Ring.Int
import Mathlib.Data.Nat.Cast.Basic
assert_not_exists DenselyOrdered
variable {M : Type*}
class NatPowAssoc (M : Type*) [MulOneClass M] [Pow M ℕ] : Prop where
protected npow_add : ∀ (k n: ℕ) (x : M), x ^ (k + n) = x ^ k * x ^ n
... | Mathlib/Algebra/Group/NatPowAssoc.lean | 72 | 75 | theorem npow_mul (x : M) (m n : ℕ) : x ^ (m * n) = (x ^ m) ^ n := by |
induction n with
| zero => rw [npow_zero, Nat.mul_zero, npow_zero]
| succ n ih => rw [mul_add, npow_add, ih, mul_one, npow_add, npow_one]
| 3 |
import Mathlib.Probability.Kernel.Composition
#align_import probability.kernel.invariance from "leanprover-community/mathlib"@"3b92d54a05ee592aa2c6181a4e76b1bb7cc45d0b"
open MeasureTheory
open scoped MeasureTheory ENNReal ProbabilityTheory
namespace ProbabilityTheory
variable {α β γ : Type*} {mα : MeasurableSp... | Mathlib/Probability/Kernel/Invariance.lean | 51 | 54 | theorem bind_smul (κ : kernel α β) (μ : Measure α) (r : ℝ≥0∞) : (r • μ).bind κ = r • μ.bind κ := by |
ext1 s hs
rw [Measure.bind_apply hs (kernel.measurable _), lintegral_smul_measure, Measure.coe_smul,
Pi.smul_apply, Measure.bind_apply hs (kernel.measurable _), smul_eq_mul]
| 3 |
import Mathlib.Algebra.Regular.Basic
import Mathlib.LinearAlgebra.Matrix.MvPolynomial
import Mathlib.LinearAlgebra.Matrix.Polynomial
import Mathlib.RingTheory.Polynomial.Basic
#align_import linear_algebra.matrix.adjugate from "leanprover-community/mathlib"@"a99f85220eaf38f14f94e04699943e185a5e1d1a"
namespace Matr... | Mathlib/LinearAlgebra/Matrix/Adjugate.lean | 82 | 85 | theorem cramer_is_linear : IsLinearMap α (cramerMap A) := by |
constructor <;> intros <;> ext i
· apply (cramerMap_is_linear A i).1
· apply (cramerMap_is_linear A i).2
| 3 |
import Mathlib.Algebra.Order.Floor
import Mathlib.Algebra.Order.Field.Power
import Mathlib.Data.Nat.Log
#align_import data.int.log from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
variable {R : Type*} [LinearOrderedSemifield R] [FloorSemiring R]
namespace Int
def log (b : ℕ) (r : ... | Mathlib/Data/Int/Log.lean | 93 | 96 | theorem log_of_right_le_zero (b : ℕ) {r : R} (hr : r ≤ 0) : log b r = 0 := by |
rw [log_of_right_le_one _ (hr.trans zero_le_one),
Nat.clog_of_right_le_one ((Nat.ceil_eq_zero.mpr <| inv_nonpos.2 hr).trans_le zero_le_one),
Int.ofNat_zero, neg_zero]
| 3 |
import Mathlib.Geometry.Manifold.ContMDiff.Product
import Mathlib.Analysis.NormedSpace.OperatorNorm.Prod
open Set ChartedSpace SmoothManifoldWithCorners
open scoped Topology Manifold
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
-- declare a smooth manifold `M` over the pair `(E, H)`.
{E : Type*}
[Norme... | Mathlib/Geometry/Manifold/ContMDiff/NormedSpace.lean | 51 | 55 | theorem contMDiffWithinAt_iff_contDiffWithinAt {f : E → E'} {s : Set E} {x : E} :
ContMDiffWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E') n f s x ↔ ContDiffWithinAt 𝕜 n f s x := by |
simp (config := { contextual := true }) only [ContMDiffWithinAt, liftPropWithinAt_iff',
ContDiffWithinAtProp, iff_def, mfld_simps]
exact ContDiffWithinAt.continuousWithinAt
| 3 |
import Mathlib.Analysis.InnerProductSpace.Adjoint
#align_import analysis.inner_product_space.positive from "leanprover-community/mathlib"@"caa58cbf5bfb7f81ccbaca4e8b8ac4bc2b39cc1c"
open InnerProductSpace RCLike ContinuousLinearMap
open scoped InnerProduct ComplexConjugate
namespace ContinuousLinearMap
variable... | Mathlib/Analysis/InnerProductSpace/Positive.lean | 88 | 92 | theorem IsPositive.conj_adjoint {T : E →L[𝕜] E} (hT : T.IsPositive) (S : E →L[𝕜] F) :
(S ∘L T ∘L S†).IsPositive := by |
refine ⟨hT.isSelfAdjoint.conj_adjoint S, fun x => ?_⟩
rw [reApplyInnerSelf, comp_apply, ← adjoint_inner_right]
exact hT.inner_nonneg_left _
| 3 |
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.TensorProduct.Tower
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.LinearAlgebra.DirectSum.Finsupp
#align_import ring_theory.tensor_product from "leanprover-community/mathlib"@"88fcdc3da43943f5b01925deddaa5bf0c0e85e4e"
suppress_comp... | Mathlib/RingTheory/TensorProduct/Basic.lean | 83 | 86 | theorem baseChange_add : (f + g).baseChange A = f.baseChange A + g.baseChange A := by |
ext
-- Porting note: added `-baseChange_tmul`
simp [baseChange_eq_ltensor, -baseChange_tmul]
| 3 |
import Mathlib.LinearAlgebra.Quotient
import Mathlib.LinearAlgebra.Prod
#align_import linear_algebra.projection from "leanprover-community/mathlib"@"6d584f1709bedbed9175bd9350df46599bdd7213"
noncomputable section Ring
variable {R : Type*} [Ring R] {E : Type*} [AddCommGroup E] [Module R E]
variable {F : Type*} [Ad... | Mathlib/LinearAlgebra/Projection.lean | 139 | 143 | theorem prodEquivOfIsCompl_symm_apply_snd_eq_zero (h : IsCompl p q) {x : E} :
((prodEquivOfIsCompl p q h).symm x).2 = 0 ↔ x ∈ p := by |
conv_rhs => rw [← (prodEquivOfIsCompl p q h).apply_symm_apply x]
rw [coe_prodEquivOfIsCompl', Submodule.add_mem_iff_right _ (Submodule.coe_mem _),
mem_left_iff_eq_zero_of_disjoint h.disjoint]
| 3 |
import Mathlib.Analysis.Convex.Between
import Mathlib.MeasureTheory.Constructions.BorelSpace.Basic
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.Topology.MetricSpace.Holder
import Mathlib.Topology.MetricSpace.MetricSeparated
#align_import measure_theory.measure.hausdorff from "leanprover-communit... | Mathlib/MeasureTheory/Measure/Hausdorff.lean | 293 | 297 | theorem tendsto_pre_nat (m : Set X → ℝ≥0∞) (s : Set X) :
Tendsto (fun n : ℕ => pre m n⁻¹ s) atTop (𝓝 <| mkMetric' m s) := by |
refine (tendsto_pre m s).comp (tendsto_inf.2 ⟨ENNReal.tendsto_inv_nat_nhds_zero, ?_⟩)
refine tendsto_principal.2 (eventually_of_forall fun n => ?_)
simp
| 3 |
import Mathlib.LinearAlgebra.Quotient
import Mathlib.LinearAlgebra.Prod
#align_import linear_algebra.projection from "leanprover-community/mathlib"@"6d584f1709bedbed9175bd9350df46599bdd7213"
noncomputable section Ring
variable {R : Type*} [Ring R] {E : Type*} [AddCommGroup E] [Module R E]
variable {F : Type*} [Ad... | Mathlib/LinearAlgebra/Projection.lean | 41 | 45 | theorem ker_id_sub_eq_of_proj {f : E →ₗ[R] p} (hf : ∀ x : p, f x = x) :
ker (id - p.subtype.comp f) = p := by |
ext x
simp only [comp_apply, mem_ker, subtype_apply, sub_apply, id_apply, sub_eq_zero]
exact ⟨fun h => h.symm ▸ Submodule.coe_mem _, fun hx => by erw [hf ⟨x, hx⟩, Subtype.coe_mk]⟩
| 3 |
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 | 104 | 107 | theorem Ioi_mul_Ici_subset' (a b : α) : Ioi a * Ici b ⊆ Ioi (a * b) := by |
haveI := covariantClass_le_of_lt
rintro x ⟨y, hya, z, hzb, rfl⟩
exact mul_lt_mul_of_lt_of_le hya hzb
| 3 |
import Mathlib.MeasureTheory.Integral.Lebesgue
#align_import measure_theory.measure.giry_monad from "leanprover-community/mathlib"@"56f4cd1ef396e9fd389b5d8371ee9ad91d163625"
noncomputable section
open scoped Classical
open ENNReal
open scoped Classical
open Set Filter
variable {α β : Type*}
namespace MeasureT... | Mathlib/MeasureTheory/Measure/GiryMonad.lean | 85 | 88 | theorem measurable_dirac : Measurable (Measure.dirac : α → Measure α) := by |
refine measurable_of_measurable_coe _ fun s hs => ?_
simp_rw [dirac_apply' _ hs]
exact measurable_one.indicator hs
| 3 |
import Mathlib.Algebra.BigOperators.Group.Multiset
import Mathlib.Algebra.BigOperators.Ring.List
import Mathlib.Data.Multiset.Antidiagonal
import Mathlib.Data.Multiset.Sections
#align_import algebra.big_operators.multiset.lemmas from "leanprover-community/mathlib"@"0a0ec35061ed9960bf0e7ffb0335f44447b58977"
variab... | Mathlib/Algebra/BigOperators/Ring/Multiset.lean | 99 | 102 | theorem multiset_sum_right (a : α) (h : ∀ b ∈ s, Commute a b) : Commute a s.sum := by |
induction s using Quotient.inductionOn
rw [quot_mk_to_coe, sum_coe]
exact Commute.list_sum_right _ _ h
| 3 |
import Mathlib.Logic.Relation
import Mathlib.Data.List.Forall2
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Infix
#align_import data.list.chain from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
-- Make sure we haven't imported `Data.Nat.Order.Basic`
assert_not_exists OrderedSu... | Mathlib/Data/List/Chain.lean | 101 | 106 | theorem chain_pmap_of_chain {S : β → β → Prop} {p : α → Prop} {f : ∀ a, p a → β}
(H : ∀ a b ha hb, R a b → S (f a ha) (f b hb)) {a : α} {l : List α} (hl₁ : Chain R a l)
(ha : p a) (hl₂ : ∀ a ∈ l, p a) : Chain S (f a ha) (List.pmap f l hl₂) := by |
induction' l with lh lt l_ih generalizing a
· simp
· simp [H _ _ _ _ (rel_of_chain_cons hl₁), l_ih (chain_of_chain_cons hl₁)]
| 3 |
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subsemigroup.Operations
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Data.Set.Image
import Mathlib.Order.Atoms
import Mathlib.Tactic.ApplyFun
#align_import g... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 169 | 173 | theorem exists_inv_mem_iff_exists_mem {P : G → Prop} :
(∃ x : G, x ∈ H ∧ P x⁻¹) ↔ ∃ x ∈ H, P x := by |
constructor <;>
· rintro ⟨x, x_in, hx⟩
exact ⟨x⁻¹, inv_mem x_in, by simp [hx]⟩
| 3 |
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import analysis.special_functions.complex.log from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
namespace Complex
open Set Filter Bornology
open scop... | Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 86 | 90 | theorem log_ofReal_mul {r : ℝ} (hr : 0 < r) {x : ℂ} (hx : x ≠ 0) :
log (r * x) = Real.log r + log x := by |
replace hx := Complex.abs.ne_zero_iff.mpr hx
simp_rw [log, map_mul, abs_ofReal, arg_real_mul _ hr, abs_of_pos hr, Real.log_mul hr.ne' hx,
ofReal_add, add_assoc]
| 3 |
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 | 58 | 61 | theorem subsingleton_iff_bot_or_pure : l.Subsingleton ↔ l = ⊥ ∨ ∃ a, l = pure a := by |
refine ⟨fun hl ↦ ?_, ?_⟩
· exact (eq_or_neBot l).imp_right (@Subsingleton.exists_eq_pure _ _ · hl)
· rintro (rfl | ⟨a, rfl⟩) <;> simp
| 3 |
import Mathlib.Logic.Relation
import Mathlib.Data.List.Forall2
import Mathlib.Data.List.Lex
import Mathlib.Data.List.Infix
#align_import data.list.chain from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
-- Make sure we haven't imported `Data.Nat.Order.Basic`
assert_not_exists OrderedSu... | Mathlib/Data/List/Chain.lean | 109 | 114 | theorem chain_of_chain_pmap {S : β → β → Prop} {p : α → Prop} (f : ∀ a, p a → β) {l : List α}
(hl₁ : ∀ a ∈ l, p a) {a : α} (ha : p a) (hl₂ : Chain S (f a ha) (List.pmap f l hl₁))
(H : ∀ a b ha hb, S (f a ha) (f b hb) → R a b) : Chain R a l := by |
induction' l with lh lt l_ih generalizing a
· simp
· simp [H _ _ _ _ (rel_of_chain_cons hl₂), l_ih _ _ (chain_of_chain_cons hl₂)]
| 3 |
import Mathlib.Combinatorics.Quiver.Cast
import Mathlib.Combinatorics.Quiver.Symmetric
#align_import combinatorics.quiver.single_obj from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
namespace Quiver
-- Porting note: Removed `deriving Unique`.
@[nolint unusedArguments]
def SingleObj ... | Mathlib/Combinatorics/Quiver/SingleObj.lean | 139 | 142 | theorem pathToList_listToPath (l : List α) : pathToList (listToPath l) = l := by |
induction' l with a l ih
· rfl
· change a :: pathToList (listToPath l) = a :: l; rw [ih]
| 3 |
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 | 49 | 53 | theorem reduceOption_length_eq {l : List (Option α)} :
l.reduceOption.length = (l.filter Option.isSome).length := by |
induction' l with hd tl hl
· simp_rw [reduceOption_nil, filter_nil, length]
· cases hd <;> simp [hl]
| 3 |
import Mathlib.MeasureTheory.Measure.MeasureSpace
open scoped ENNReal NNReal Topology
open Set MeasureTheory Measure Filter MeasurableSpace ENNReal Function
variable {R α β δ γ ι : Type*}
namespace MeasureTheory
variable {m0 : MeasurableSpace α} [MeasurableSpace β] [MeasurableSpace γ]
variable {μ μ₁ μ₂ μ₃ ν ν' ν... | Mathlib/MeasureTheory/Measure/Restrict.lean | 104 | 107 | theorem restrict_apply' (hs : MeasurableSet s) : μ.restrict s t = μ (t ∩ s) := by |
rw [← toOuterMeasure_apply,
Measure.restrict_toOuterMeasure_eq_toOuterMeasure_restrict hs,
OuterMeasure.restrict_apply s t _, toOuterMeasure_apply]
| 3 |
import Mathlib.Algebra.Polynomial.Div
import Mathlib.RingTheory.Polynomial.Basic
import Mathlib.RingTheory.Ideal.QuotientOperations
#align_import ring_theory.polynomial.quotient from "leanprover-community/mathlib"@"4f840b8d28320b20c87db17b3a6eef3d325fca87"
set_option linter.uppercaseLean3 false
open Polynomial
... | Mathlib/RingTheory/Polynomial/Quotient.lean | 87 | 91 | theorem quotient_map_C_eq_zero {I : Ideal R} :
∀ a ∈ I, ((Quotient.mk (map (C : R →+* R[X]) I : Ideal R[X])).comp C) a = 0 := by |
intro a ha
rw [RingHom.comp_apply, Quotient.eq_zero_iff_mem]
exact mem_map_of_mem _ ha
| 3 |
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.GCD.Basic
import Mathlib.Init.Data.Nat.Lemmas
import Mathlib.Logic.Function.Iterate
import Mathlib.Tactic.Ring
import Mathlib.Tactic.Zify
#align_import data.nat.fib from "leanprover-community/mathlib"@"... | Mathlib/Data/Nat/Fib/Basic.lean | 114 | 117 | theorem fib_lt_fib_succ {n : ℕ} (hn : 2 ≤ n) : fib n < fib (n + 1) := by |
rcases exists_add_of_le hn with ⟨n, rfl⟩
rw [← tsub_pos_iff_lt, add_comm 2, add_right_comm, fib_add_two, add_tsub_cancel_right, fib_pos]
exact succ_pos n
| 3 |
import Batteries.Data.HashMap.Basic
import Batteries.Data.Array.Lemmas
import Batteries.Data.Nat.Lemmas
namespace Batteries.HashMap
namespace Imp
attribute [-simp] Bool.not_eq_true
namespace Buckets
@[ext] protected theorem ext : ∀ {b₁ b₂ : Buckets α β}, b₁.1.data = b₂.1.data → b₁ = b₂
| ⟨⟨_⟩, _⟩, ⟨⟨_⟩, _⟩, rfl ... | .lake/packages/batteries/Batteries/Data/HashMap/WF.lean | 29 | 33 | theorem update_update (self : Buckets α β) (i d d' h h') :
(self.update i d h).update i d' h' = self.update i d' h := by |
simp only [update, Array.uset, Array.data_length]
congr 1
rw [Array.set_set]
| 3 |
import Mathlib.Algebra.Polynomial.Monic
#align_import algebra.polynomial.big_operators from "leanprover-community/mathlib"@"47adfab39a11a072db552f47594bf8ed2cf8a722"
open Finset
open Multiset
open Polynomial
universe u w
variable {R : Type u} {ι : Type w}
namespace Polynomial
variable (s : Finset ι)
sectio... | Mathlib/Algebra/Polynomial/BigOperators.lean | 80 | 83 | theorem natDegree_list_prod_le (l : List S[X]) : natDegree l.prod ≤ (l.map natDegree).sum := by |
induction' l with hd tl IH
· simp
· simpa using natDegree_mul_le.trans (add_le_add_left IH _)
| 3 |
import Mathlib.Topology.MetricSpace.HausdorffDistance
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.measure.regular from "leanprover-community/mathlib"@"bf6a01357ff5684b1ebcd0f1a13be314fc82c0bf"
open Set Filter ENNReal Topology NNReal TopologicalSpace
namespace MeasureTh... | Mathlib/MeasureTheory/Measure/Regular.lean | 369 | 374 | theorem _root_.MeasurableSet.exists_isOpen_diff_lt [OuterRegular μ] {A : Set α}
(hA : MeasurableSet A) (hA' : μ A ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
∃ U, U ⊇ A ∧ IsOpen U ∧ μ U < ∞ ∧ μ (U \ A) < ε := by |
rcases A.exists_isOpen_lt_add hA' hε with ⟨U, hAU, hUo, hU⟩
use U, hAU, hUo, hU.trans_le le_top
exact measure_diff_lt_of_lt_add hA hAU hA' hU
| 3 |
import Mathlib.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
open Filter ENNReal
namespace ENNReal
variable {α : Type*} {f : Filter α}
theorem eventually_le_limsup [CountableInterFilter f] (u : α → ℝ≥0∞) :
∀ᶠ y i... | Mathlib/Order/Filter/ENNReal.lean | 86 | 93 | theorem limsup_liminf_le_liminf_limsup {β} [Countable β] {f : Filter α} [CountableInterFilter f]
{g : Filter β} (u : α → β → ℝ≥0∞) :
(f.limsup fun a : α => g.liminf fun b : β => u a b) ≤
g.liminf fun b => f.limsup fun a => u a b :=
have h1 : ∀ᶠ a in f, ∀ b, u a b ≤ f.limsup fun a' => u a' b := by |
rw [eventually_countable_forall]
exact fun b => ENNReal.eventually_le_limsup fun a => u a b
sInf_le <| h1.mono fun x hx => Filter.liminf_le_liminf (Filter.eventually_of_forall hx)
| 3 |
import Mathlib.LinearAlgebra.CliffordAlgebra.Fold
import Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
#align_import linear_algebra.exterior_algebra.of_alternating from "leanprover-community/mathlib"@"ce11c3c2a285bbe6937e26d9792fda4e51f3fe1a"
variable {R M N N' : Type*}
variable [CommRing R] [AddCommGroup M] [AddCo... | Mathlib/LinearAlgebra/ExteriorAlgebra/OfAlternating.lean | 79 | 85 | theorem liftAlternating_ι_mul (f : ∀ i, M [⋀^Fin i]→ₗ[R] N) (m : M)
(x : ExteriorAlgebra R M) :
liftAlternating (R := R) (M := M) (N := N) f (ι R m * x) =
liftAlternating (R := R) (M := M) (N := N) (fun i => (f i.succ).curryLeft m) x := by |
dsimp [liftAlternating]
rw [foldl_mul, foldl_ι]
rfl
| 3 |
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Int.Basic
import Mathlib.Tactic.Ring
import Mathlib.Tactic.FieldSimp
import Mathlib.Data.Int.NatPrime
import Mathlib.Data.ZMod.Basic
#align_import number_theory.pythagorean_tri... | Mathlib/NumberTheory/PythagoreanTriples.lean | 78 | 82 | theorem mul (k : ℤ) : PythagoreanTriple (k * x) (k * y) (k * z) :=
calc
k * x * (k * x) + k * y * (k * y) = k ^ 2 * (x * x + y * y) := by | ring
_ = k ^ 2 * (z * z) := by rw [h.eq]
_ = k * z * (k * z) := by ring
| 3 |
import Mathlib.FieldTheory.Finite.Basic
import Mathlib.Order.Filter.Cofinite
#align_import number_theory.fermat_psp from "leanprover-community/mathlib"@"c0439b4877c24a117bfdd9e32faf62eee9b115eb"
namespace Nat
def ProbablePrime (n b : ℕ) : Prop :=
n ∣ b ^ (n - 1) - 1
#align fermat_psp.probable_prime Nat.Probabl... | Mathlib/NumberTheory/FermatPsp.lean | 127 | 130 | theorem fermatPsp_base_one {n : ℕ} (h₁ : 1 < n) (h₂ : ¬n.Prime) : FermatPsp n 1 := by |
refine ⟨show n ∣ 1 ^ (n - 1) - 1 from ?_, h₂, h₁⟩
exact show 0 = 1 ^ (n - 1) - 1 by
set_option tactic.skipAssignedInstances false in norm_num ▸ dvd_zero n
| 3 |
import Mathlib.Data.List.Nodup
#align_import data.prod.tprod from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
open List Function
universe u v
variable {ι : Type u} {α : ι → Type v} {i j : ι} {l : List ι} {f : ∀ i, α i}
namespace List
variable (α)
abbrev TProd (l : List ι) : Type v... | Mathlib/Data/Prod/TProd.lean | 99 | 103 | theorem elim_of_mem (hl : (i :: l).Nodup) (hj : j ∈ l) (v : TProd α (i :: l)) :
v.elim (mem_cons_of_mem _ hj) = TProd.elim v.2 hj := by |
apply elim_of_ne
rintro rfl
exact hl.not_mem hj
| 3 |
import Mathlib.RingTheory.Ideal.Operations
import Mathlib.Algebra.Module.Torsion
import Mathlib.Algebra.Ring.Idempotents
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Filtration
import Mathlib.RingTheory.Nakayama
#align_import ring_theory.ideal.cota... | Mathlib/RingTheory/Ideal/Cotangent.lean | 132 | 136 | theorem to_quotient_square_range :
LinearMap.range I.cotangentToQuotientSquare = I.cotangentIdeal.restrictScalars R := by |
trans LinearMap.range (I.cotangentToQuotientSquare.comp I.toCotangent)
· rw [LinearMap.range_comp, I.toCotangent_range, Submodule.map_top]
· rw [to_quotient_square_comp_toCotangent, LinearMap.range_comp, I.range_subtype]; ext; rfl
| 3 |
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 | 111 | 114 | theorem encard_union_eq (h : Disjoint s t) : (s ∪ t).encard = s.encard + t.encard := by |
classical
have e := (Equiv.Set.union (by rwa [subset_empty_iff, ← disjoint_iff_inter_eq_empty])).symm
simp [encard, ← PartENat.card_congr e, PartENat.card_sum, PartENat.withTopEquiv]
| 3 |
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathlib.Algebra.LinearRecurrence
import Mathlib.Tactic.NormNum.NatFib
import Mathlib.Tactic.NormNum.Prime
#align_import data.real.golden_ratio from "leanprover-community/mathlib"@"2196ab363eb097c008d449712... | Mathlib/Data/Real/GoldenRatio.lean | 44 | 47 | theorem inv_gold : φ⁻¹ = -ψ := by |
have : 1 + √5 ≠ 0 := ne_of_gt (add_pos (by norm_num) <| Real.sqrt_pos.mpr (by norm_num))
field_simp [sub_mul, mul_add]
norm_num
| 3 |
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.Analysis.Calculus.Deriv.Slope
import Mathlib.Analysis.NormedSpace.FiniteDimension
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Basic
#align_import analysis.calculus.fderiv_... | Mathlib/Analysis/Calculus/FDeriv/Measurable.lean | 148 | 151 | theorem A_mono (L : E →L[𝕜] F) (r : ℝ) {ε δ : ℝ} (h : ε ≤ δ) : A f L r ε ⊆ A f L r δ := by |
rintro x ⟨r', r'r, hr'⟩
refine ⟨r', r'r, fun y hy z hz => (hr' y hy z hz).trans_le (mul_le_mul_of_nonneg_right h ?_)⟩
linarith [mem_ball.1 hy, r'r.2, @dist_nonneg _ _ y x]
| 3 |
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#align_import data.finset.fold from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
-- TODO:
-- assert_not_exists OrderedComm... | Mathlib/Data/Finset/Fold.lean | 116 | 120 | theorem fold_union_inter [DecidableEq α] {s₁ s₂ : Finset α} {b₁ b₂ : β} :
((s₁ ∪ s₂).fold op b₁ f * (s₁ ∩ s₂).fold op b₂ f) = s₁.fold op b₂ f * s₂.fold op b₁ f := by |
unfold fold
rw [← fold_add op, ← Multiset.map_add, union_val, inter_val, union_add_inter, Multiset.map_add,
hc.comm, fold_add]
| 3 |
import Mathlib.Combinatorics.SimpleGraph.DegreeSum
import Mathlib.Combinatorics.SimpleGraph.Subgraph
#align_import combinatorics.simple_graph.matching from "leanprover-community/mathlib"@"138448ae98f529ef34eeb61114191975ee2ca508"
universe u
namespace SimpleGraph
variable {V : Type u} {G : SimpleGraph V} (M : Su... | Mathlib/Combinatorics/SimpleGraph/Matching.lean | 63 | 67 | theorem IsMatching.toEdge_eq_of_adj {M : Subgraph G} (h : M.IsMatching) {v w : V} (hv : v ∈ M.verts)
(hvw : M.Adj v w) : h.toEdge ⟨v, hv⟩ = ⟨s(v, w), hvw⟩ := by |
simp only [IsMatching.toEdge, Subtype.mk_eq_mk]
congr
exact ((h (M.edge_vert hvw)).choose_spec.2 w hvw).symm
| 3 |
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FormalMultilinearSeries
#align_import analysis.calculus.cont_diff_def from "leanprover-community/mathlib"@"3a69562db5a458db8322b190ec8d9a8bbd8a5b14"
noncomputable section
open scoped Classical
open NNReal Topology Filter
local notatio... | Mathlib/Analysis/Calculus/ContDiff/Defs.lean | 204 | 208 | theorem HasFTaylorSeriesUpToOn.congr (h : HasFTaylorSeriesUpToOn n f p s)
(h₁ : ∀ x ∈ s, f₁ x = f x) : HasFTaylorSeriesUpToOn n f₁ p s := by |
refine ⟨fun x hx => ?_, h.fderivWithin, h.cont⟩
rw [h₁ x hx]
exact h.zero_eq x hx
| 3 |
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.BilinearMap
#align_import linear_algebra.basis.bilinear from "leanprover-community/mathlib"@"87c54600fe3cdc7d32ff5b50873ac724d86aef8d"
namespace LinearMap
variable {ι₁ ι₂ : Type*}
variable {R R₂ S S₂ M N P Rₗ : Type*}
variable {Mₗ Nₗ Pₗ : Type*}
--... | Mathlib/LinearAlgebra/Basis/Bilinear.lean | 44 | 49 | theorem sum_repr_mul_repr_mulₛₗ {B : M →ₛₗ[ρ₁₂] N →ₛₗ[σ₁₂] P} (x y) :
((b₁.repr x).sum fun i xi => (b₂.repr y).sum fun j yj => ρ₁₂ xi • σ₁₂ yj • B (b₁ i) (b₂ j)) =
B x y := by |
conv_rhs => rw [← b₁.total_repr x, ← b₂.total_repr y]
simp_rw [Finsupp.total_apply, Finsupp.sum, map_sum₂, map_sum, LinearMap.map_smulₛₗ₂,
LinearMap.map_smulₛₗ]
| 3 |
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 | 124 | 127 | theorem verschiebung_frobenius [CharP R p] (x : 𝕎 R) : verschiebung (frobenius x) = x * p := by |
ext ⟨i⟩
· rw [mul_charP_coeff_zero, verschiebung_coeff_zero]
· rw [mul_charP_coeff_succ, verschiebung_coeff_succ, coeff_frobenius_charP]
| 3 |
import Mathlib.Algebra.Module.Defs
import Mathlib.SetTheory.Cardinal.Basic
open Function
universe u v
namespace Cardinal
| Mathlib/Algebra/Module/Card.lean | 24 | 29 | theorem mk_le_of_module (R : Type u) (E : Type v)
[AddCommGroup E] [Ring R] [Module R E] [Nontrivial E] [NoZeroSMulDivisors R E] :
Cardinal.lift.{v} (#R) ≤ Cardinal.lift.{u} (#E) := by |
obtain ⟨x, hx⟩ : ∃ (x : E), x ≠ 0 := exists_ne 0
have : Injective (fun k ↦ k • x) := smul_left_injective R hx
exact lift_mk_le_lift_mk_of_injective this
| 3 |
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 | 139 | 143 | theorem ClosedComplemented.of_quotient_finiteDimensional [CompleteSpace 𝕜]
[FiniteDimensional 𝕜 (E ⧸ p)] (hp : IsClosed (p : Set E)) : p.ClosedComplemented := by |
obtain ⟨q, hq⟩ : ∃ q, IsCompl p q := p.exists_isCompl
haveI : FiniteDimensional 𝕜 q := (p.quotientEquivOfIsCompl q hq).finiteDimensional
exact .of_isCompl_isClosed hq hp q.closed_of_finiteDimensional
| 3 |
import Mathlib.Topology.Algebra.InfiniteSum.Group
import Mathlib.Topology.Algebra.Star
noncomputable section
open Filter Finset Function
open scoped Topology
variable {α β γ δ : Type*}
section ProdDomain
variable [CommMonoid α] [TopologicalSpace α]
@[to_additive]
theorem hasProd_pi_single [DecidableEq β] (... | Mathlib/Topology/Algebra/InfiniteSum/Constructions.lean | 39 | 42 | theorem tprod_pi_single [DecidableEq β] (b : β) (a : α) : ∏' b', Pi.mulSingle b a b' = a := by |
rw [tprod_eq_mulSingle b]
· simp
· intro b' hb'; simp [hb']
| 3 |
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 | 78 | 82 | theorem card_interedges_add_card_interedges_compl (s : Finset α) (t : Finset β) :
(interedges r s t).card + (interedges (fun x y ↦ ¬r x y) s t).card = s.card * t.card := by |
classical
rw [← card_product, interedges, interedges, ← card_union_of_disjoint, filter_union_filter_neg_eq]
exact disjoint_filter.2 fun _ _ ↦ Classical.not_not.2
| 3 |
import Mathlib.RingTheory.Polynomial.Cyclotomic.Basic
import Mathlib.RingTheory.RootsOfUnity.Minpoly
#align_import ring_theory.polynomial.cyclotomic.roots from "leanprover-community/mathlib"@"7fdeecc0d03cd40f7a165e6cf00a4d2286db599f"
namespace Polynomial
variable {R : Type*} [CommRing R] {n : ℕ}
theorem isRoot_... | Mathlib/RingTheory/Polynomial/Cyclotomic/Roots.lean | 63 | 67 | theorem _root_.IsPrimitiveRoot.isRoot_cyclotomic (hpos : 0 < n) {μ : R} (h : IsPrimitiveRoot μ n) :
IsRoot (cyclotomic n R) μ := by |
rw [← mem_roots (cyclotomic_ne_zero n R), cyclotomic_eq_prod_X_sub_primitiveRoots h,
roots_prod_X_sub_C, ← Finset.mem_def]
rwa [← mem_primitiveRoots hpos] at h
| 3 |
import Mathlib.Data.Multiset.Bind
#align_import data.multiset.sections from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
assert_not_exists Ring
namespace Multiset
variable {α : Type*}
section Sections
def Sections (s : Multiset (Multiset α)) : Multiset (Multiset α) :=
Multiset.... | Mathlib/Data/Multiset/Sections.lean | 60 | 64 | theorem mem_sections {s : Multiset (Multiset α)} :
∀ {a}, a ∈ Sections s ↔ s.Rel (fun s a => a ∈ s) a := by |
induction s using Multiset.induction_on with
| empty => simp
| cons _ _ ih => simp [ih, rel_cons_left, eq_comm]
| 3 |
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.PEquiv
#align_import data.matrix.pequiv from "leanprover-community/mathlib"@"3e068ece210655b7b9a9477c3aff38a492400aa1"
namespace PEquiv
open Matrix
universe u v
variable {k l m n : Type*}
variable {α : Type v}
open Matrix
def toMatrix [DecidableEq n] [Zer... | Mathlib/Data/Matrix/PEquiv.lean | 70 | 74 | theorem toMatrix_symm [DecidableEq m] [DecidableEq n] [Zero α] [One α] (f : m ≃. n) :
(f.symm.toMatrix : Matrix n m α) = f.toMatrixᵀ := by |
ext
simp only [transpose, mem_iff_mem f, toMatrix_apply]
congr
| 3 |
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
open Order.PFilter
namespace Order
variable {P : Type*}
namespace Ideal
-- Porting note(#5171): this linter isn't ported yet.
-- @[nolint has_... | Mathlib/Order/PrimeIdeal.lean | 68 | 71 | theorem I_isProper : IsProper IF.I := by |
cases' IF.F.nonempty with w h
apply isProper_of_not_mem (_ : w ∉ IF.I)
rwa [← IF.compl_I_eq_F] at h
| 3 |
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 | 224 | 227 | theorem transpose_eq_iff_eq_transpose {μ ν : YoungDiagram} : μ.transpose = ν ↔ μ = ν.transpose := by |
constructor <;>
· rintro rfl
simp
| 3 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Data.Rat.Denumerable
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.SetTheory.Cardinal.Continuum
#align_import data.real.cardinality from "leanprover-community/mathlib"@"7e7aaccf9b0182576cabdde36cf1b5ad3585b70d"
open Nat Set
open Cardinal
no... | Mathlib/Data/Real/Cardinality.lean | 113 | 117 | theorem cantorFunction_succ (f : ℕ → Bool) (h1 : 0 ≤ c) (h2 : c < 1) :
cantorFunction c f = cond (f 0) 1 0 + c * cantorFunction c fun n => f (n + 1) := by |
rw [cantorFunction, tsum_eq_zero_add (summable_cantor_function f h1 h2)]
rw [cantorFunctionAux_succ, tsum_mul_left, cantorFunctionAux, _root_.pow_zero]
rfl
| 3 |
import Mathlib.Analysis.Calculus.FDeriv.Add
import Mathlib.Analysis.Calculus.FDeriv.Equiv
import Mathlib.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.Calculus.Monotone
import Mathlib.Data.Set.Function
import Mathlib.Algebra.Group.Basic
import Mathlib.Tactic.WLOG
#align_import analysis.bounded_variation from ... | Mathlib/Analysis/BoundedVariation.lean | 133 | 136 | theorem mono (f : α → E) {s t : Set α} (hst : t ⊆ s) : eVariationOn f t ≤ eVariationOn f s := by |
apply iSup_le _
rintro ⟨n, ⟨u, hu, ut⟩⟩
exact sum_le f n hu fun i => hst (ut i)
| 3 |
import Mathlib.Order.CompleteLattice
import Mathlib.Order.GaloisConnection
import Mathlib.Data.Set.Lattice
import Mathlib.Tactic.AdaptationNote
#align_import data.rel from "leanprover-community/mathlib"@"706d88f2b8fdfeb0b22796433d7a6c1a010af9f2"
variable {α β γ : Type*}
def Rel (α β : Type*) :=
α → β → Prop --... | Mathlib/Data/Rel.lean | 119 | 122 | theorem comp_left_id (r : Rel α β) : @Eq α • r = r := by |
unfold comp
ext x
simp
| 3 |
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 | 131 | 135 | theorem lengthParity_eq_ofAdd_length (w : W) :
cs.lengthParity w = Multiplicative.ofAdd (↑(ℓ w)) := by |
rcases cs.exists_reduced_word w with ⟨ω, hω, rfl⟩
rw [← hω, wordProd, map_list_prod, List.map_map, lengthParity_comp_simple, map_const',
prod_replicate, ← ofAdd_nsmul, nsmul_one]
| 3 |
import Mathlib.SetTheory.Ordinal.Arithmetic
#align_import set_theory.ordinal.exponential from "leanprover-community/mathlib"@"b67044ba53af18680e1dd246861d9584e968495d"
noncomputable section
open Function Cardinal Set Equiv Order
open scoped Classical
open Cardinal Ordinal
universe u v w
namespace Ordinal
in... | Mathlib/SetTheory/Ordinal/Exponential.lean | 51 | 54 | theorem opow_zero (a : Ordinal) : a ^ (0 : Ordinal) = 1 := by |
by_cases h : a = 0
· simp only [opow_def, if_pos h, sub_zero]
· simp only [opow_def, if_neg h, limitRecOn_zero]
| 3 |
import Mathlib.Algebra.Group.Support
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Nat.Cast.Field
#align_import algebra.char_zero.lemmas from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b4b3abce829"
open Function Set
namespace Nat
variable {R : Type*} [AddMonoidWithOne R] [Char... | Mathlib/Algebra/CharZero/Lemmas.lean | 46 | 50 | theorem cast_div_charZero {k : Type*} [DivisionSemiring k] [CharZero k] {m n : ℕ} (n_dvd : n ∣ m) :
((m / n : ℕ) : k) = m / n := by |
rcases eq_or_ne n 0 with (rfl | hn)
· simp
· exact cast_div n_dvd (cast_ne_zero.2 hn)
| 3 |
import Mathlib.Data.Finset.Image
#align_import data.finset.card from "leanprover-community/mathlib"@"65a1391a0106c9204fe45bc73a039f056558cb83"
assert_not_exists MonoidWithZero
-- TODO: After a lot more work,
-- assert_not_exists OrderedCommMonoid
open Function Multiset Nat
variable {α β R : Type*}
namespace Fin... | Mathlib/Data/Finset/Card.lean | 143 | 146 | theorem card_insert_eq_ite : card (insert a s) = if a ∈ s then s.card else s.card + 1 := by |
by_cases h : a ∈ s
· rw [card_insert_of_mem h, if_pos h]
· rw [card_insert_of_not_mem h, if_neg h]
| 3 |
import Mathlib.Data.Finsupp.Basic
import Mathlib.Data.List.AList
#align_import data.finsupp.alist from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
namespace AList
variable {α M : Type*} [Zero M]
open List
noncomputable def lookupFinsupp (l : AList fun _x : α => M) : α →₀ M where
... | Mathlib/Data/Finsupp/AList.lean | 82 | 86 | theorem lookupFinsupp_support [DecidableEq α] [DecidableEq M] (l : AList fun _x : α => M) :
l.lookupFinsupp.support = (l.1.filter fun x => Sigma.snd x ≠ 0).keys.toFinset := by |
convert rfl; congr
· apply Subsingleton.elim
· funext; congr
| 3 |
import Mathlib.Data.Finset.Lattice
#align_import combinatorics.set_family.compression.down from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
variable {α : Type*} [DecidableEq α] {𝒜 ℬ : Finset (Finset α)} {s : Finset α} {a : α}
open Finset
-- The namespace is here to distinguish fro... | Mathlib/Combinatorics/SetFamily/Compression/Down.lean | 251 | 254 | theorem erase_mem_compression (hs : s ∈ 𝒜) : s.erase a ∈ 𝓓 a 𝒜 := by |
simp_rw [mem_compression, erase_idem, and_self_iff]
refine (em _).imp_right fun h => ⟨h, ?_⟩
rwa [insert_erase (erase_ne_self.1 (ne_of_mem_of_not_mem hs h).symm)]
| 3 |
import Mathlib.Data.List.Infix
#align_import data.list.rdrop from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
-- Make sure we don't import algebra
assert_not_exists Monoid
variable {α : Type*} (p : α → Bool) (l : List α) (n : ℕ)
namespace List
def rdrop : List α :=
l.take (l.leng... | Mathlib/Data/List/DropRight.lean | 125 | 128 | theorem rdropWhile_last_not (hl : l.rdropWhile p ≠ []) : ¬p ((rdropWhile p l).getLast hl) := by |
simp_rw [rdropWhile]
rw [getLast_reverse]
exact dropWhile_nthLe_zero_not _ _ _
| 3 |
import Mathlib.Analysis.Calculus.Deriv.Pow
import Mathlib.Analysis.SpecialFunctions.Log.Basic
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Tactic.AdaptationNote
#align_import analysis.special_functions.log.deriv from "leanprover-community/mathlib"@"3bce8d800a6f2b8f63fe1e588fd76a9ff4adcebe"
ope... | Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean | 78 | 81 | theorem contDiffOn_log {n : ℕ∞} : ContDiffOn ℝ n log {0}ᶜ := by |
suffices ContDiffOn ℝ ⊤ log {0}ᶜ from this.of_le le_top
refine (contDiffOn_top_iff_deriv_of_isOpen isOpen_compl_singleton).2 ?_
simp [differentiableOn_log, contDiffOn_inv]
| 3 |
import Mathlib.Data.Complex.Module
import Mathlib.RingTheory.Norm
import Mathlib.RingTheory.Trace
#align_import ring_theory.complex from "leanprover-community/mathlib"@"9015c511549dc77a0f8d6eba021d8ac4bba20c82"
open Complex
theorem Algebra.leftMulMatrix_complex (z : ℂ) :
Algebra.leftMulMatrix Complex.basisOn... | Mathlib/RingTheory/Complex.lean | 37 | 40 | theorem Algebra.norm_complex_apply (z : ℂ) : Algebra.norm ℝ z = Complex.normSq z := by |
rw [Algebra.norm_eq_matrix_det Complex.basisOneI, Algebra.leftMulMatrix_complex,
Matrix.det_fin_two, normSq_apply]
simp
| 3 |
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 | 306 | 309 | theorem head (hab : r a b) (hbc : ReflTransGen r b c) : ReflTransGen r a c := by |
induction hbc with
| refl => exact refl.tail hab
| tail _ hcd hac => exact hac.tail hcd
| 3 |
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 | 147 | 150 | theorem span_insert_zero : span R (insert (0 : M) s) = span R s := by |
refine le_antisymm ?_ (Submodule.span_mono (Set.subset_insert 0 s))
rw [span_le, Set.insert_subset_iff]
exact ⟨by simp only [SetLike.mem_coe, Submodule.zero_mem], Submodule.subset_span⟩
| 3 |
import Mathlib.LinearAlgebra.Projectivization.Basic
#align_import linear_algebra.projective_space.subspace from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
variable (K V : Type*) [Field K] [AddCommGroup V] [Module K V]
namespace Projectivization
open scoped LinearAlgebra.Projectivi... | Mathlib/LinearAlgebra/Projectivization/Subspace.lean | 155 | 158 | theorem span_univ : span (Set.univ : Set (ℙ K V)) = ⊤ := by |
rw [eq_top_iff, SetLike.le_def]
intro x _hx
exact subset_span _ (Set.mem_univ x)
| 3 |
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 | 87 | 90 | theorem foldl_eq_foldl_list (f : α → Fin n → α) (x) : foldl n f x = (list n).foldl f x := by |
induction n generalizing x with
| zero => rw [foldl_zero, list_zero, List.foldl_nil]
| succ n ih => rw [foldl_succ, ih, list_succ, List.foldl_cons, List.foldl_map]
| 3 |
import Mathlib.Data.Fin.Tuple.Basic
import Mathlib.Data.List.Join
#align_import data.list.of_fn from "leanprover-community/mathlib"@"bf27744463e9620ca4e4ebe951fe83530ae6949b"
universe u
variable {α : Type u}
open Nat
namespace List
#noalign list.length_of_fn_aux
@[simp]
theorem length_ofFn_go {n} (f : Fin n ... | Mathlib/Data/List/OfFn.lean | 50 | 54 | theorem get_ofFn_go {n} (f : Fin n → α) (i j h) (k) (hk) :
get (ofFn.go f i j h) ⟨k, hk⟩ = f ⟨j + k, by simp at hk; omega⟩ := by |
let i+1 := i
cases k <;> simp [ofFn.go, get_ofFn_go (i := i)]
congr 2; omega
| 3 |
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.AlgebraicGeometry.Pullbacks
import Mathlib.CategoryTheory.MorphismProperty.Limits
import Mathlib.Data.List.TFAE
#align_import algebraic_geometry.morphisms.basic from "leanprover-community/mathlib"@"434e2fd21c1900747afc6d13d8be7f4eedba7218"
set_option lin... | Mathlib/AlgebraicGeometry/Morphisms/Basic.lean | 109 | 116 | theorem AffineTargetMorphismProperty.respectsIso_mk {P : AffineTargetMorphismProperty}
(h₁ : ∀ {X Y Z} (e : X ≅ Y) (f : Y ⟶ Z) [IsAffine Z], P f → P (e.hom ≫ f))
(h₂ : ∀ {X Y Z} (e : Y ≅ Z) (f : X ⟶ Y) [h : IsAffine Y],
P f → @P _ _ (f ≫ e.hom) (isAffineOfIso e.inv)) :
P.toProperty.RespectsIso := by |
constructor
· rintro X Y Z e f ⟨a, h⟩; exact ⟨a, h₁ e f h⟩
· rintro X Y Z e f ⟨a, h⟩; exact ⟨isAffineOfIso e.inv, h₂ e f h⟩
| 3 |
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