Context stringlengths 57 6.04k | file_name stringlengths 21 79 | start int64 14 1.49k | end int64 18 1.5k | theorem stringlengths 25 1.55k | proof stringlengths 5 7.36k | eval_complexity float64 0 1 |
|---|---|---|---|---|---|---|
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.BigOperators
import Mathlib.Algebra.Polynomial.Lifts
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.RingTheory.RootsOfUnity.Complex
import Mathlib.NumberTheory.ArithmeticFunction
import Mathlib.RingTheory.RootsOfUnity.Basic
import Mathlib.FieldTh... | Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean | 78 | 80 | theorem cyclotomic'_one (R : Type*) [CommRing R] [IsDomain R] : cyclotomic' 1 R = X - 1 := by |
simp only [cyclotomic', Finset.prod_singleton, RingHom.map_one,
IsPrimitiveRoot.primitiveRoots_one]
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import Mathlib.Analysis.InnerProductSpace.Dual
#align_import analysis.inner_product_space.lax_milgram from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
noncomputable section
open RCLike LinearMap ContinuousLinearMap InnerProductSpace
open LinearMap (ker range)
open RealInnerProduct... | Mathlib/Analysis/InnerProductSpace/LaxMilgram.lean | 74 | 77 | theorem ker_eq_bot (coercive : IsCoercive B) : ker B♯ = ⊥ := by |
rw [LinearMapClass.ker_eq_bot]
rcases coercive.antilipschitz with ⟨_, _, antilipschitz⟩
exact antilipschitz.injective
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import Mathlib.LinearAlgebra.Matrix.DotProduct
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Diagonal
#align_import data.matrix.rank from "leanprover-community/mathlib"@"17219820a8aa8abe85adf5dfde19af1dd1bd8ae7"
open Matrix
namespace Matrix
open FiniteDimensional
variable {l m n ... | Mathlib/Data/Matrix/Rank.lean | 49 | 51 | theorem rank_one [StrongRankCondition R] [DecidableEq n] :
rank (1 : Matrix n n R) = Fintype.card n := by |
rw [rank, mulVecLin_one, LinearMap.range_id, finrank_top, finrank_pi]
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import Mathlib.Data.Int.Interval
import Mathlib.Data.Int.SuccPred
import Mathlib.Data.Int.ConditionallyCompleteOrder
import Mathlib.Topology.Instances.Discrete
import Mathlib.Topology.MetricSpace.Bounded
import Mathlib.Order.Filter.Archimedean
#align_import topology.instances.int from "leanprover-community/mathlib"@"... | Mathlib/Topology/Instances/Int.lean | 84 | 85 | theorem cofinite_eq : (cofinite : Filter ℤ) = atBot ⊔ atTop := by |
rw [← cocompact_eq_cofinite, cocompact_eq_atBot_atTop]
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import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
#align_import data.finset.locally_finite from "leanprover-community/mathlib"@"442a83d738cb208d3600056c489be16900ba701d"
assert_not_exists MonoidWithZero
assert_not_exists Finset.sum
open Function OrderDual
open FinsetInterval
variable {ι α : T... | Mathlib/Order/Interval/Finset/Basic.lean | 139 | 139 | theorem left_mem_Ico : a ∈ Ico a b ↔ a < b := by | simp only [mem_Ico, true_and_iff, le_refl]
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import Mathlib.Data.Multiset.Bind
#align_import data.multiset.pi from "leanprover-community/mathlib"@"b2c89893177f66a48daf993b7ba5ef7cddeff8c9"
namespace Multiset
section Pi
variable {α : Type*}
open Function
def Pi.empty (δ : α → Sort*) : ∀ a ∈ (0 : Multiset α), δ a :=
nofun
#align multiset.pi.empty Multi... | Mathlib/Data/Multiset/Pi.lean | 62 | 68 | theorem pi.cons_eta {m : Multiset α} {a : α} (f : ∀ a' ∈ a ::ₘ m, δ a') :
(Pi.cons m a (f _ (mem_cons_self _ _)) fun a' ha' => f a' (mem_cons_of_mem ha')) = f := by |
ext a' h'
by_cases h : a' = a
· subst h
rw [Pi.cons_same]
· rw [Pi.cons_ne _ h]
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import Mathlib.Analysis.NormedSpace.BoundedLinearMaps
import Mathlib.Topology.FiberBundle.Basic
#align_import topology.vector_bundle.basic from "leanprover-community/mathlib"@"e473c3198bb41f68560cab68a0529c854b618833"
noncomputable section
open scoped Classical
open Bundle Set
open scoped Topology
variable (R : ... | Mathlib/Topology/VectorBundle/Basic.lean | 120 | 123 | theorem coe_linearMapAt (e : Pretrivialization F (π F E)) [e.IsLinear R] (b : B) :
⇑(e.linearMapAt R b) = fun y => if b ∈ e.baseSet then (e ⟨b, y⟩).2 else 0 := by |
rw [Pretrivialization.linearMapAt]
split_ifs <;> rfl
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import Mathlib.Analysis.Calculus.ContDiff.Bounds
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysis.LocallyConvex.WithSeminorms
import Mathlib.Analysis.Normed.Group.ZeroAtInfty
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Ana... | Mathlib/Analysis/Distribution/SchwartzSpace.lean | 103 | 106 | theorem decay (f : 𝓢(E, F)) (k n : ℕ) :
∃ C : ℝ, 0 < C ∧ ∀ x, ‖x‖ ^ k * ‖iteratedFDeriv ℝ n f x‖ ≤ C := by |
rcases f.decay' k n with ⟨C, hC⟩
exact ⟨max C 1, by positivity, fun x => (hC x).trans (le_max_left _ _)⟩
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import Mathlib.Algebra.Quaternion
import Mathlib.Tactic.Ring
#align_import algebra.quaternion_basis from "leanprover-community/mathlib"@"3aa5b8a9ed7a7cabd36e6e1d022c9858ab8a8c2d"
open Quaternion
namespace QuaternionAlgebra
structure Basis {R : Type*} (A : Type*) [CommRing R] [Ring A] [Algebra R A] (c₁ c₂ : R) ... | Mathlib/Algebra/QuaternionBasis.lean | 94 | 95 | theorem k_mul_j : q.k * q.j = c₂ • q.i := by |
rw [← i_mul_j, mul_assoc, j_mul_j, mul_smul_comm, mul_one]
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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 | 54 | 58 | theorem nextOr_cons_of_ne (xs : List α) (y x d : α) (h : x ≠ y) :
nextOr (y :: xs) x d = nextOr xs x d := by |
cases' xs with z zs
· rfl
· exact if_neg h
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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 | 95 | 98 | theorem IsPositive.adjoint_conj {T : E →L[𝕜] E} (hT : T.IsPositive) (S : F →L[𝕜] E) :
(S† ∘L T ∘L S).IsPositive := by |
convert hT.conj_adjoint (S†)
rw [adjoint_adjoint]
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import Mathlib.Data.Finset.Prod
import Mathlib.Data.Set.Finite
#align_import data.finset.n_ary from "leanprover-community/mathlib"@"eba7871095e834365616b5e43c8c7bb0b37058d0"
open Function Set
variable {α α' β β' γ γ' δ δ' ε ε' ζ ζ' ν : Type*}
namespace Finset
variable [DecidableEq α'] [DecidableEq β'] [Decidabl... | Mathlib/Data/Finset/NAry.lean | 73 | 74 | theorem mem_image₂_iff (hf : Injective2 f) : f a b ∈ image₂ f s t ↔ a ∈ s ∧ b ∈ t := by |
rw [← mem_coe, coe_image₂, mem_image2_iff hf, mem_coe, mem_coe]
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import Mathlib.Data.Set.Lattice
import Mathlib.Data.Set.Pairwise.Basic
#align_import data.set.pairwise.lattice from "leanprover-community/mathlib"@"c4c2ed622f43768eff32608d4a0f8a6cec1c047d"
open Function Set Order
variable {α β γ ι ι' : Type*} {κ : Sort*} {r p q : α → α → Prop}
section Pairwise
variable {f g : ... | Mathlib/Data/Set/Pairwise/Lattice.lean | 147 | 153 | theorem Set.PairwiseDisjoint.subset_of_biUnion_subset_biUnion (h₀ : (s ∪ t).PairwiseDisjoint f)
(h₁ : ∀ i ∈ s, (f i).Nonempty) (h : ⋃ i ∈ s, f i ⊆ ⋃ i ∈ t, f i) : s ⊆ t := by |
rintro i hi
obtain ⟨a, hai⟩ := h₁ i hi
obtain ⟨j, hj, haj⟩ := mem_iUnion₂.1 (h <| mem_iUnion₂_of_mem hi hai)
rwa [h₀.eq (subset_union_left hi) (subset_union_right hj)
(not_disjoint_iff.2 ⟨a, hai, haj⟩)]
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import Mathlib.Data.Finsupp.Defs
#align_import data.finsupp.fin from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
noncomputable section
namespace Finsupp
variable {n : ℕ} (i : Fin n) {M : Type*} [Zero M] (y : M) (t : Fin (n + 1) →₀ M) (s : Fin n →₀ M)
def tail (s : Fin (n + 1) →₀ ... | Mathlib/Data/Finsupp/Fin.lean | 68 | 73 | theorem cons_zero_zero : cons 0 (0 : Fin n →₀ M) = 0 := by |
ext a
by_cases c : a = 0
· simp [c]
· rw [← Fin.succ_pred a c, cons_succ]
simp
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import Mathlib.Order.Filter.CountableInter
set_option autoImplicit true
open Function Set Filter
class HasCountableSeparatingOn (α : Type*) (p : Set α → Prop) (t : Set α) : Prop where
exists_countable_separating : ∃ S : Set (Set α), S.Countable ∧ (∀ s ∈ S, p s) ∧
∀ x ∈ t, ∀ y ∈ t, (∀ s ∈ S, x ∈ s ↔ y ∈ s) ... | Mathlib/Order/Filter/CountableSeparatingOn.lean | 103 | 109 | theorem exists_seq_separating (α : Type*) {p : Set α → Prop} {s₀} (hp : p s₀) (t : Set α)
[HasCountableSeparatingOn α p t] :
∃ S : ℕ → Set α, (∀ n, p (S n)) ∧ ∀ x ∈ t, ∀ y ∈ t, (∀ n, x ∈ S n ↔ y ∈ S n) → x = y := by |
rcases exists_nonempty_countable_separating α hp t with ⟨S, hSne, hSc, hS⟩
rcases hSc.exists_eq_range hSne with ⟨S, rfl⟩
use S
simpa only [forall_mem_range] using hS
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import Mathlib.CategoryTheory.EqToHom
import Mathlib.CategoryTheory.Pi.Basic
import Mathlib.Data.ULift
#align_import category_theory.discrete_category from "leanprover-community/mathlib"@"369525b73f229ccd76a6ec0e0e0bf2be57599768"
namespace CategoryTheory
-- morphism levels before object levels. See note [Category... | Mathlib/CategoryTheory/DiscreteCategory.lean | 186 | 187 | theorem functor_map {I : Type u₁} (F : I → C) {i : Discrete I} (f : i ⟶ i) :
(Discrete.functor F).map f = 𝟙 (F i.as) := by | aesop_cat
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import Mathlib.Data.Finset.Fold
import Mathlib.Algebra.GCDMonoid.Multiset
#align_import algebra.gcd_monoid.finset from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
#align_import algebra.gcd_monoid.div from "leanprover-community/mathlib"@"b537794f8409bc9598febb79cd510b1df5f4539d"
variab... | Mathlib/Algebra/GCDMonoid/Finset.lean | 189 | 192 | theorem gcd_congr {f g : β → α} (hs : s₁ = s₂) (hfg : ∀ a ∈ s₂, f a = g a) :
s₁.gcd f = s₂.gcd g := by |
subst hs
exact Finset.fold_congr hfg
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import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.Polynomial.RingDivision
#align_import data.polynomial.mirror from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
namespace Polynomial
open Polynomial
section Semiring
variable {R : Type*} [Semiring R] (p q : R... | Mathlib/Algebra/Polynomial/Mirror.lean | 75 | 79 | theorem mirror_natTrailingDegree : p.mirror.natTrailingDegree = p.natTrailingDegree := by |
by_cases hp : p = 0
· rw [hp, mirror_zero]
· rw [mirror, natTrailingDegree_mul_X_pow ((mt reverse_eq_zero.mp) hp),
natTrailingDegree_reverse, zero_add]
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import Mathlib.Data.Fintype.Card
import Mathlib.Data.List.MinMax
import Mathlib.Data.Nat.Order.Lemmas
import Mathlib.Logic.Encodable.Basic
#align_import logic.denumerable from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
variable {α β : Type*}
class Denumerable (α : Type*) extends E... | Mathlib/Logic/Denumerable.lean | 65 | 67 | theorem encode_ofNat (n) : encode (ofNat α n) = n := by |
obtain ⟨a, h, e⟩ := decode_inv (α := α) n
rwa [ofNat_of_decode h]
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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 | 178 | 179 | theorem inv_mul_lt_iff_lt_mul : b⁻¹ * a < c ↔ a < b * c := by |
rw [← mul_lt_mul_iff_left b, mul_inv_cancel_left]
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import Mathlib.Algebra.ContinuedFractions.Basic
import Mathlib.Algebra.GroupWithZero.Basic
#align_import algebra.continued_fractions.translations from "leanprover-community/mathlib"@"a7e36e48519ab281320c4d192da6a7b348ce40ad"
namespace GeneralizedContinuedFraction
section WithDivisionRing
variable {K : Type*}... | Mathlib/Algebra/ContinuedFractions/Translations.lean | 180 | 181 | theorem convergents'Aux_succ_none {s : Stream'.Seq (Pair K)} (h : s.head = none) (n : ℕ) :
convergents'Aux s (n + 1) = 0 := by | simp [convergents'Aux, h, convergents'Aux.match_1]
| 0.15625 |
import Mathlib.Data.Complex.Exponential
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
#align_import data.complex.exponential_bounds from "leanprover-community/mathlib"@"402f8982dddc1864bd703da2d6e2ee304a866973"
namespace Real
open IsAbsoluteValue Finset CauSeq Complex
theorem exp_one_near_10 : |exp 1 - 224... | Mathlib/Data/Complex/ExponentialBounds.lean | 51 | 55 | theorem exp_neg_one_lt_d9 : exp (-1) < 0.3678794412 := by |
rw [exp_neg, inv_lt (exp_pos _)]
· refine lt_of_lt_of_le ?_ (sub_le_comm.1 (abs_sub_le_iff.1 exp_one_near_10).2)
norm_num
· norm_num
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import Mathlib.Data.Set.Lattice
import Mathlib.Order.Directed
#align_import data.set.Union_lift from "leanprover-community/mathlib"@"5a4ea8453f128345f73cc656e80a49de2a54f481"
variable {α : Type*} {ι β : Sort _}
namespace Set
section UnionLift
@[nolint unusedArguments]
noncomputable def iUnionLift (S : ι → Set... | Mathlib/Data/Set/UnionLift.lean | 75 | 76 | theorem iUnionLift_of_mem (x : T) {i : ι} (hx : (x : α) ∈ S i) :
iUnionLift S f hf T hT x = f i ⟨x, hx⟩ := by | cases' x with x hx; exact hf _ _ _ _ _
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import Mathlib.Data.Multiset.Bind
import Mathlib.Control.Traversable.Lemmas
import Mathlib.Control.Traversable.Instances
#align_import data.multiset.functor from "leanprover-community/mathlib"@"1f0096e6caa61e9c849ec2adbd227e960e9dff58"
universe u
namespace Multiset
open List
instance functor : Functor Multiset... | Mathlib/Data/Multiset/Functor.lean | 129 | 134 | theorem traverse_map {G : Type* → Type _} [Applicative G] [CommApplicative G] {α β γ : Type _}
(g : α → β) (h : β → G γ) (x : Multiset α) : traverse h (map g x) = traverse (h ∘ g) x := by |
refine Quotient.inductionOn x ?_
intro
simp only [traverse, quot_mk_to_coe, map_coe, lift_coe, Function.comp_apply]
rw [← Traversable.traverse_map h g, List.map_eq_map]
| 0.15625 |
import Mathlib.Data.List.Basic
#align_import data.list.lattice from "leanprover-community/mathlib"@"dd71334db81d0bd444af1ee339a29298bef40734"
open Nat
namespace List
variable {α : Type*} {l l₁ l₂ : List α} {p : α → Prop} {a : α}
variable [DecidableEq α]
section BagInter
@[simp]
theorem nil_bagInt... | Mathlib/Data/List/Lattice.lean | 199 | 199 | theorem bagInter_nil (l : List α) : l.bagInter [] = [] := by | cases l <;> rfl
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import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.Group.Indicator
import Mathlib.Order.LiminfLimsup
import Mathlib.Order.Filter.Archimedean
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Data.Set.La... | Mathlib/Topology/Algebra/Order/LiminfLimsup.lean | 476 | 479 | theorem iInf_eq_of_forall_le_of_tendsto {x : R} {as : ι → R} (x_le : ∀ i, x ≤ as i) {F : Filter ι}
[Filter.NeBot F] (as_lim : Filter.Tendsto as F (𝓝 x)) : ⨅ i, as i = x := by |
refine iInf_eq_of_forall_ge_of_forall_gt_exists_lt (fun i ↦ x_le i) ?_
apply fun w x_lt_w ↦ ‹Filter.NeBot F›.nonempty_of_mem (eventually_lt_of_tendsto_lt x_lt_w as_lim)
| 0.15625 |
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 | 260 | 264 | theorem trans {q' : Set α → Prop} (H : InnerRegularWRT μ p q) (H' : InnerRegularWRT μ q q') :
InnerRegularWRT μ p q' := by |
intro U hU r hr
rcases H' hU r hr with ⟨F, hFU, hqF, hF⟩; rcases H hqF _ hF with ⟨K, hKF, hpK, hrK⟩
exact ⟨K, hKF.trans hFU, hpK, hrK⟩
| 0.15625 |
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.RingTheory.Polynomial.Bernstein
import Mathlib.Topology.ContinuousFunction.Polynomial
import Mathlib.Topology.ContinuousFunction.Compact
#align_import analysis.special_functions.bernstein from "leanprover-community/mathlib"@"2c1d8ca2812b64f88992a5294ea3dba14... | Mathlib/Analysis/SpecialFunctions/Bernstein.lean | 61 | 64 | theorem bernstein_apply (n ν : ℕ) (x : I) :
bernstein n ν x = (n.choose ν : ℝ) * (x : ℝ) ^ ν * (1 - (x : ℝ)) ^ (n - ν) := by |
dsimp [bernstein, Polynomial.toContinuousMapOn, Polynomial.toContinuousMap, bernsteinPolynomial]
simp
| 0.15625 |
import Mathlib.LinearAlgebra.Dual
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.contraction from "leanprover-community/mathlib"@"657df4339ae6ceada048c8a2980fb10e393143ec"
suppress_compilation
-- Porting note: universe metavariables behave oddly
universe w u v₁ v₂ v₃ v₄
variable {ι : Type... | Mathlib/LinearAlgebra/Contraction.lean | 96 | 101 | theorem dualTensorHom_prodMap_zero (f : Module.Dual R M) (p : P) :
((dualTensorHom R M P) (f ⊗ₜ[R] p)).prodMap (0 : N →ₗ[R] Q) =
dualTensorHom R (M × N) (P × Q) ((f ∘ₗ fst R M N) ⊗ₜ inl R P Q p) := by |
ext <;>
simp only [coe_comp, coe_inl, Function.comp_apply, prodMap_apply, dualTensorHom_apply,
fst_apply, Prod.smul_mk, LinearMap.zero_apply, smul_zero]
| 0.15625 |
import Mathlib.Data.Fintype.Basic
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Defs
#align_import logic.equiv.fintype from "leanprover-community/mathlib"@"9407b03373c8cd201df99d6bc5514fc2db44054f"
namespace Equiv
variable {α β : Type*} [Finite α]
noncomputable def toCompl {p q : α → Prop} (e ... | Mathlib/Logic/Equiv/Fintype.lean | 125 | 129 | theorem extendSubtype_apply_of_mem (e : { x // p x } ≃ { x // q x }) (x) (hx : p x) :
e.extendSubtype x = e ⟨x, hx⟩ := by |
dsimp only [extendSubtype]
simp only [subtypeCongr, Equiv.trans_apply, Equiv.sumCongr_apply]
rw [sumCompl_apply_symm_of_pos _ _ hx, Sum.map_inl, sumCompl_apply_inl]
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import Mathlib.SetTheory.Ordinal.Arithmetic
import Mathlib.SetTheory.Ordinal.Exponential
#align_import set_theory.ordinal.fixed_point from "leanprover-community/mathlib"@"0dd4319a17376eda5763cd0a7e0d35bbaaa50e83"
noncomputable section
universe u v
open Function Order
namespace Ordinal
section
variable {ι ... | Mathlib/SetTheory/Ordinal/FixedPoint.lean | 128 | 134 | theorem apply_le_nfpFamily [hι : Nonempty ι] {f : ι → Ordinal → Ordinal} (H : ∀ i, IsNormal (f i))
{a b} : (∀ i, f i b ≤ nfpFamily.{u, v} f a) ↔ b ≤ nfpFamily.{u, v} f a := by |
refine ⟨fun h => ?_, fun h i => ?_⟩
· cases' hι with i
exact ((H i).self_le b).trans (h i)
rw [← nfpFamily_fp (H i)]
exact (H i).monotone h
| 0.15625 |
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 | 200 | 201 | theorem sum_finsupport (hx₀ : x₀ ∈ s) : ∑ i ∈ ρ.finsupport x₀, ρ i x₀ = 1 := by |
rw [← ρ.sum_eq_one hx₀, finsum_eq_sum_of_support_subset _ (ρ.coe_finsupport x₀).superset]
| 0.15625 |
import Mathlib.Data.Fintype.Basic
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Defs
#align_import logic.equiv.fintype from "leanprover-community/mathlib"@"9407b03373c8cd201df99d6bc5514fc2db44054f"
section Fintype
variable {α β : Type*} [Fintype α] [DecidableEq β] (e : Equiv.Perm α) (f : α ↪ β)
... | Mathlib/Logic/Equiv/Fintype.lean | 91 | 93 | theorem Equiv.Perm.viaFintypeEmbedding_sign [DecidableEq α] [Fintype β] :
Equiv.Perm.sign (e.viaFintypeEmbedding f) = Equiv.Perm.sign e := by |
simp [Equiv.Perm.viaFintypeEmbedding]
| 0.15625 |
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
#align_import linear_algebra.affine_space.restrict from "leanprover-community/mathlib"@"09258fb7f75d741b7eda9fa18d5c869e2135d9f1"
variable {k V₁ P₁ V₂ P₂ : Type*} [Ring k] [AddCommGroup V₁] [AddCommGroup V₂] [Module k V₁]
[Module k V₂] [AddTorsor V₁ P₁] [A... | Mathlib/LinearAlgebra/AffineSpace/Restrict.lean | 73 | 78 | theorem AffineMap.restrict.injective {φ : P₁ →ᵃ[k] P₂} (hφ : Function.Injective φ)
{E : AffineSubspace k P₁} {F : AffineSubspace k P₂} [Nonempty E] [Nonempty F]
(hEF : E.map φ ≤ F) : Function.Injective (AffineMap.restrict φ hEF) := by |
intro x y h
simp only [Subtype.ext_iff, Subtype.coe_mk, AffineMap.restrict.coe_apply] at h ⊢
exact hφ h
| 0.15625 |
import Mathlib.Data.Finset.Prod
import Mathlib.Data.Set.Finite
#align_import data.finset.n_ary from "leanprover-community/mathlib"@"eba7871095e834365616b5e43c8c7bb0b37058d0"
open Function Set
variable {α α' β β' γ γ' δ δ' ε ε' ζ ζ' ν : Type*}
namespace Finset
variable [DecidableEq α'] [DecidableEq β'] [Decidabl... | Mathlib/Data/Finset/NAry.lean | 112 | 113 | theorem image₂_subset_iff_right : image₂ f s t ⊆ u ↔ ∀ b ∈ t, (s.image fun a => f a b) ⊆ u := by |
simp_rw [image₂_subset_iff, image_subset_iff, @forall₂_swap α]
| 0.15625 |
import Mathlib.Topology.UniformSpace.UniformEmbedding
#align_import topology.uniform_space.pi from "leanprover-community/mathlib"@"2705404e701abc6b3127da906f40bae062a169c9"
noncomputable section
open scoped Uniformity Topology
open Filter UniformSpace Function Set
universe u
variable {ι ι' β : Type*} (α : ι → ... | Mathlib/Topology/UniformSpace/Pi.lean | 46 | 49 | theorem uniformContinuous_pi {β : Type*} [UniformSpace β] {f : β → ∀ i, α i} :
UniformContinuous f ↔ ∀ i, UniformContinuous fun x => f x i := by |
-- Porting note: required `Function.comp` to close
simp only [UniformContinuous, Pi.uniformity, tendsto_iInf, tendsto_comap_iff, Function.comp]
| 0.15625 |
import Mathlib.Probability.Notation
import Mathlib.Probability.Process.Stopping
#align_import probability.martingale.basic from "leanprover-community/mathlib"@"ba074af83b6cf54c3104e59402b39410ddbd6dca"
open TopologicalSpace Filter
open scoped NNReal ENNReal MeasureTheory ProbabilityTheory
namespace MeasureTheor... | Mathlib/Probability/Martingale/Basic.lean | 75 | 79 | theorem martingale_const_fun [OrderBot ι] (ℱ : Filtration ι m0) (μ : Measure Ω) [IsFiniteMeasure μ]
{f : Ω → E} (hf : StronglyMeasurable[ℱ ⊥] f) (hfint : Integrable f μ) :
Martingale (fun _ => f) ℱ μ := by |
refine ⟨fun i => hf.mono <| ℱ.mono bot_le, fun i j _ => ?_⟩
rw [condexp_of_stronglyMeasurable (ℱ.le _) (hf.mono <| ℱ.mono bot_le) hfint]
| 0.15625 |
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u
variable (R : Type u)
section Semiring
variable [Semiring R]
class inductive Ex... | Mathlib/Algebra/CharP/ExpChar.lean | 113 | 116 | theorem charZero_of_expChar_one' [hq : ExpChar R 1] : CharZero R := by |
cases hq
· assumption
· exact False.elim (CharP.char_ne_one R 1 rfl)
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import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated
open Function Set
open scoped ENNReal Classical
noncomputable section
variable {α β δ : Type*} [MeasurableSpace α] [MeasurableSpace β] {s : Set α} ... | Mathlib/MeasureTheory/Measure/Dirac.lean | 87 | 92 | theorem map_eq_sum [Countable β] [MeasurableSingletonClass β] (μ : Measure α) (f : α → β)
(hf : Measurable f) : μ.map f = sum fun b : β => μ (f ⁻¹' {b}) • dirac b := by |
ext s
have : ∀ y ∈ s, MeasurableSet (f ⁻¹' {y}) := fun y _ => hf (measurableSet_singleton _)
simp [← tsum_measure_preimage_singleton (to_countable s) this, *,
tsum_subtype s fun b => μ (f ⁻¹' {b}), ← indicator_mul_right s fun b => μ (f ⁻¹' {b})]
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import Mathlib.Data.Finsupp.Basic
import Mathlib.Data.Finsupp.Order
#align_import data.finsupp.multiset from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
open Finset
variable {α β ι : Type*}
namespace Finsupp
def toMultiset : (α →₀ ℕ) →+ Multiset α where
toFun f := Finsupp.sum f... | Mathlib/Data/Finsupp/Multiset.lean | 105 | 114 | theorem count_toMultiset [DecidableEq α] (f : α →₀ ℕ) (a : α) : (toMultiset f).count a = f a :=
calc
(toMultiset f).count a = Finsupp.sum f (fun x n => (n • {x} : Multiset α).count a) := by |
rw [toMultiset_apply]; exact map_sum (Multiset.countAddMonoidHom a) _ f.support
_ = f.sum fun x n => n * ({x} : Multiset α).count a := by simp only [Multiset.count_nsmul]
_ = f a * ({a} : Multiset α).count a :=
sum_eq_single _
(fun a' _ H => by simp only [Multiset.count_singleton, if_false,... | 0.15625 |
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
universe u
variable (R : Type u)
section Semiring
variable [Semiring R]
class inductive Ex... | Mathlib/Algebra/CharP/ExpChar.lean | 133 | 136 | theorem char_prime_of_ne_zero {p : ℕ} [hp : CharP R p] (p_ne_zero : p ≠ 0) : Nat.Prime p := by |
cases' CharP.char_is_prime_or_zero R p with h h
· exact h
· contradiction
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import Mathlib.Algebra.DualNumber
import Mathlib.Algebra.QuaternionBasis
import Mathlib.Data.Complex.Module
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib.LinearAlgebra.CliffordAlgebra.Star
import Mathlib.LinearAlgebra.QuadraticForm.Prod
#align_import linear_algebra.clifford_algebra.equivs fr... | Mathlib/LinearAlgebra/CliffordAlgebra/Equivs.lean | 90 | 96 | theorem reverse_apply (x : CliffordAlgebra (0 : QuadraticForm R Unit)) :
reverse (R := R) x = x := by |
induction x using CliffordAlgebra.induction with
| algebraMap r => exact reverse.commutes _
| ι x => rw [ι_eq_zero, LinearMap.zero_apply, reverse.map_zero]
| mul x₁ x₂ hx₁ hx₂ => rw [reverse.map_mul, mul_comm, hx₁, hx₂]
| add x₁ x₂ hx₁ hx₂ => rw [reverse.map_add, hx₁, hx₂]
| 0.15625 |
import Mathlib.CategoryTheory.Category.Grpd
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Homotopy.Path
import Mathlib.Data.Set.Subsingleton
#align_import algebraic_topology.fundamental_groupoid.basic from "leanprover-community/mathlib"@"3d7987cda72abc473... | Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean | 144 | 145 | theorem transReflReparamAux_zero : transReflReparamAux 0 = 0 := by |
set_option tactic.skipAssignedInstances false in norm_num [transReflReparamAux]
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import Mathlib.CategoryTheory.Sites.Sieves
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.Multiequalizer
import Mathlib.CategoryTheory.Category.Preorder
import Mathlib.Order.Copy
import Mathlib.Data.Set.Subsingleton
#align_import category_theory.sites.grothendieck fr... | Mathlib/CategoryTheory/Sites/Grothendieck.lean | 187 | 187 | theorem covering_iff_covers_id (S : Sieve X) : S ∈ J X ↔ J.Covers S (𝟙 X) := by | simp [covers_iff]
| 0.15625 |
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w
variable {ι : Sort u} {α : Type v} {β : Type w}
open Set
open OrderDual (toDual)
namespace Set
section LinearOrder
variable [LinearOrder α] ... | Mathlib/Order/Interval/Set/Disjoint.lean | 170 | 172 | theorem iUnion_Ico_eq_Iio_self_iff {f : ι → α} {a : α} :
⋃ i, Ico (f i) a = Iio a ↔ ∀ x < a, ∃ i, f i ≤ x := by |
simp [← Ici_inter_Iio, ← iUnion_inter, subset_def]
| 0.15625 |
import Mathlib.CategoryTheory.Category.Grpd
import Mathlib.CategoryTheory.Groupoid
import Mathlib.Topology.Category.TopCat.Basic
import Mathlib.Topology.Homotopy.Path
import Mathlib.Data.Set.Subsingleton
#align_import algebraic_topology.fundamental_groupoid.basic from "leanprover-community/mathlib"@"3d7987cda72abc473... | Mathlib/AlgebraicTopology/FundamentalGroupoid/Basic.lean | 138 | 140 | theorem transReflReparamAux_mem_I (t : I) : transReflReparamAux t ∈ I := by |
unfold transReflReparamAux
split_ifs <;> constructor <;> linarith [unitInterval.le_one t, unitInterval.nonneg t]
| 0.15625 |
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.midpoint from "leanprover-community/mathlib"@"2196ab363eb097c008d4497125e0dde23fb36db2"
open AffineMap AffineEquiv
section
variable (R : Type*) {V V' P P' : Type*} [Ring R] [Invertible (2 : R)] [AddCommGroup V]
[Modu... | Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | 61 | 64 | theorem AffineEquiv.pointReflection_midpoint_left (x y : P) :
pointReflection R (midpoint R x y) x = y := by |
rw [midpoint, pointReflection_apply, lineMap_apply, vadd_vsub, vadd_vadd, ← add_smul, ← two_mul,
mul_invOf_self, one_smul, vsub_vadd]
| 0.15625 |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Set.Sigma
#align_import data.finset.sigma from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Multiset
variable {ι : Type*}
namespace Finset
section Sigma
variable {α : ι → Type*} {β : Type*} (s s₁ s₂ : Finset ι) (... | Mathlib/Data/Finset/Sigma.lean | 91 | 94 | theorem sigma_eq_biUnion [DecidableEq (Σi, α i)] (s : Finset ι) (t : ∀ i, Finset (α i)) :
s.sigma t = s.biUnion fun i => (t i).map <| Embedding.sigmaMk i := by |
ext ⟨x, y⟩
simp [and_left_comm]
| 0.15625 |
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Digits
import Mathlib.Data.Nat.MaxPowDiv
import Mathlib.Data.Nat.Multiplicity
import Mathlib.Tactic.IntervalCases
#align_import number_theory.padics.padic_val from "leanprover-community/mathlib"@"60fa54e778c9e85d930efae172435f42fb0d71f7"
universe u
ope... | Mathlib/NumberTheory/Padics/PadicVal.lean | 162 | 169 | theorem of_ne_one_ne_zero {z : ℤ} (hp : p ≠ 1) (hz : z ≠ 0) :
padicValInt p z =
(multiplicity (p : ℤ) z).get
(by
apply multiplicity.finite_int_iff.2
simp [hp, hz]) := by |
rw [padicValInt, padicValNat, dif_pos (And.intro hp (Int.natAbs_pos.mpr hz))]
simp only [multiplicity.Int.natAbs p z]
| 0.15625 |
import Mathlib.Algebra.Group.Semiconj.Defs
import Mathlib.Algebra.Group.Units
#align_import algebra.group.semiconj from "leanprover-community/mathlib"@"a148d797a1094ab554ad4183a4ad6f130358ef64"
assert_not_exists MonoidWithZero
assert_not_exists DenselyOrdered
open scoped Int
variable {M G : Type*}
namespace Sem... | Mathlib/Algebra/Group/Semiconj/Units.lean | 48 | 51 | theorem units_inv_right {a : M} {x y : Mˣ} (h : SemiconjBy a x y) : SemiconjBy a ↑x⁻¹ ↑y⁻¹ :=
calc
a * ↑x⁻¹ = ↑y⁻¹ * (y * a) * ↑x⁻¹ := by | rw [Units.inv_mul_cancel_left]
_ = ↑y⁻¹ * a := by rw [← h.eq, mul_assoc, Units.mul_inv_cancel_right]
| 0.15625 |
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.disjoint from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w
variable {ι : Sort u} {α : Type v} {β : Type w}
open Set
open OrderDual (toDual)
namespace Set
section Preorder
variable [Preorder α] {a b c... | Mathlib/Order/Interval/Set/Disjoint.lean | 60 | 61 | theorem Ici_disjoint_Iic : Disjoint (Ici a) (Iic b) ↔ ¬a ≤ b := by |
rw [Set.disjoint_iff_inter_eq_empty, Ici_inter_Iic, Icc_eq_empty_iff]
| 0.15625 |
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Fintype.Vector
import Mathlib.Data.Multiset.Sym
#align_import data.finset.sym from "leanprover-community/mathlib"@"02ba8949f486ebecf93fe7460f1ed0564b5e442c"
namespace Finset
variable {α : Type*}
@[simps]
protected def sym2 (s : Finset α) : Finset (Sym2 α) :... | Mathlib/Data/Finset/Sym.lean | 77 | 80 | theorem injective_sym2 : Function.Injective (Finset.sym2 : Finset α → _) := by |
intro s t h
ext x
simpa using congr(s(x, x) ∈ $h)
| 0.15625 |
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.lpSpace
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.inner_product_space.l2_space from "leanprover-community/mathlib"@"46b633fd842bef9469441c0209906f6dddd2b4f5"
open RCLike Submodule Filter
open scop... | Mathlib/Analysis/InnerProductSpace/l2Space.lean | 174 | 175 | theorem inner_single_right (i : ι) (a : G i) (f : lp G 2) : ⟪f, lp.single 2 i a⟫ = ⟪f i, a⟫ := by |
simpa [inner_conj_symm] using congr_arg conj (@inner_single_left _ 𝕜 _ _ _ _ i a f)
| 0.15625 |
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Algebra.Order.Ring.Pow
import Mathlib.Algebra.Ring.Int
#align_import algebra.order.field.power from "leanprover-community/mathlib"@"acb3d204d4ee883eb686f45d486a2a6811a01329"
... | Mathlib/Algebra/Order/Field/Power.lean | 114 | 116 | theorem zpow_le_max_iff_min_le {x : α} (hx : 1 < x) {a b c : ℤ} :
x ^ (-c) ≤ max (x ^ (-a)) (x ^ (-b)) ↔ min a b ≤ c := by |
simp_rw [le_max_iff, min_le_iff, zpow_le_iff_le hx, neg_le_neg_iff]
| 0.15625 |
import Mathlib.CategoryTheory.Subobject.Lattice
#align_import category_theory.subobject.limits from "leanprover-community/mathlib"@"956af7c76589f444f2e1313911bad16366ea476d"
universe v u
noncomputable section
open CategoryTheory CategoryTheory.Category CategoryTheory.Limits CategoryTheory.Subobject Opposite
var... | Mathlib/CategoryTheory/Subobject/Limits.lean | 175 | 177 | theorem kernel_map_comp_kernelSubobjectIso_inv (sq : Arrow.mk f ⟶ Arrow.mk f') :
kernel.map f f' sq.1 sq.2 sq.3.symm ≫ (kernelSubobjectIso _).inv =
(kernelSubobjectIso _).inv ≫ kernelSubobjectMap sq := by | aesop_cat
| 0.15625 |
import Mathlib.Data.List.Chain
#align_import data.list.destutter from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
variable {α : Type*} (l : List α) (R : α → α → Prop) [DecidableRel R] {a b : α}
namespace List
@[simp]
theorem destutter'_nil : destutter' R a [] = [a] :=
rfl
#align ... | Mathlib/Data/List/Destutter.lean | 101 | 105 | theorem destutter'_of_chain (h : l.Chain R a) : l.destutter' R a = a :: l := by |
induction' l with b l hb generalizing a
· simp
obtain ⟨h, hc⟩ := chain_cons.mp h
rw [l.destutter'_cons_pos h, hb hc]
| 0.15625 |
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 | 127 | 131 | theorem monomial_def [DecidableEq σ] (n : σ →₀ ℕ) :
(monomial R n) = LinearMap.stdBasis R (fun _ ↦ R) n := by |
rw [monomial]
-- unify the `Decidable` arguments
convert rfl
| 0.15625 |
import Mathlib.LinearAlgebra.Contraction
#align_import linear_algebra.coevaluation from "leanprover-community/mathlib"@"d6814c584384ddf2825ff038e868451a7c956f31"
noncomputable section
section coevaluation
open TensorProduct FiniteDimensional
open TensorProduct
universe u v
variable (K : Type u) [Field K]
var... | Mathlib/LinearAlgebra/Coevaluation.lean | 47 | 54 | theorem coevaluation_apply_one :
(coevaluation K V) (1 : K) =
let bV := Basis.ofVectorSpace K V
∑ i : Basis.ofVectorSpaceIndex K V, bV i ⊗ₜ[K] bV.coord i := by |
simp only [coevaluation, id]
rw [(Basis.singleton Unit K).constr_apply_fintype K]
simp only [Fintype.univ_punit, Finset.sum_const, one_smul, Basis.singleton_repr,
Basis.equivFun_apply, Basis.coe_ofVectorSpace, one_nsmul, Finset.card_singleton]
| 0.15625 |
import Mathlib.Algebra.Polynomial.Degree.TrailingDegree
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.reverse from "leanprover-community/mathlib"@"44de64f183393284a16016dfb2a48ac97382f2bd"
namespace Polynomial
open Polynomial Finsupp Finset
open... | Mathlib/Algebra/Polynomial/Reverse.lean | 105 | 109 | theorem reflect_support (N : ℕ) (f : R[X]) :
(reflect N f).support = Finset.image (revAt N) f.support := by |
rcases f with ⟨⟩
ext1
simp only [reflect, support_ofFinsupp, support_embDomain, Finset.mem_map, Finset.mem_image]
| 0.15625 |
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.matrix.charpoly.linear_map from "leanprover-community/mathlib"@"62c0a4ef1441edb463095ea02a06e87f3dfe135c"
variable {ι : Type*} [Fintype ι]
variable {M : Type*} [AddCommGroup M] (R : Type*) [Co... | Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean | 43 | 46 | theorem PiToModule.fromMatrix_apply_single_one [DecidableEq ι] (A : Matrix ι ι R) (j : ι) :
PiToModule.fromMatrix R b A (Pi.single j 1) = ∑ i : ι, A i j • b i := by |
rw [PiToModule.fromMatrix_apply, Fintype.total_apply, Matrix.mulVec_single]
simp_rw [mul_one]
| 0.15625 |
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import geometry.euclidean.basic from "leanprover-community/mathlib"@"2de9c37fa71dde2f1c6feff19876dd6a7b1519f0"
open Set
open scoped RealInnerProductSpace
variable {V P : Type*} [NormedAddCommGroup V] [InnerP... | Mathlib/Geometry/Euclidean/PerpBisector.lean | 59 | 63 | theorem mem_perpBisector_pointReflection_iff_inner_eq_zero :
c ∈ perpBisector p₁ (Equiv.pointReflection p₂ p₁) ↔ ⟪c -ᵥ p₂, p₁ -ᵥ p₂⟫ = 0 := by |
rw [mem_perpBisector_iff_inner_eq_zero, midpoint_pointReflection_right,
Equiv.pointReflection_apply, vadd_vsub_assoc, inner_add_right, add_self_eq_zero,
← neg_eq_zero, ← inner_neg_right, neg_vsub_eq_vsub_rev]
| 0.15625 |
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Monoidal.Functor
#align_import category_theory.monoidal.preadditive from "leanprover-community/mathlib"@"986c4d5761f938b2e1c43c01f001b6d9d88c2055"
noncomputable section
open scoped Classical
namespace CategoryTheory
open Cat... | Mathlib/CategoryTheory/Monoidal/Preadditive.lean | 113 | 115 | theorem tensor_sum {P Q R S : C} {J : Type*} (s : Finset J) (f : P ⟶ Q) (g : J → (R ⟶ S)) :
(f ⊗ ∑ j ∈ s, g j) = ∑ j ∈ s, f ⊗ g j := by |
simp only [tensorHom_def, whiskerLeft_sum, Preadditive.comp_sum]
| 0.15625 |
import Mathlib.SetTheory.Game.Basic
import Mathlib.Tactic.NthRewrite
#align_import set_theory.game.impartial from "leanprover-community/mathlib"@"2e0975f6a25dd3fbfb9e41556a77f075f6269748"
universe u
namespace SetTheory
open scoped PGame
namespace PGame
def ImpartialAux : PGame → Prop
| G => (G ≈ -G) ∧ (∀ i... | Mathlib/SetTheory/Game/Impartial.lean | 50 | 52 | theorem impartial_def {G : PGame} :
G.Impartial ↔ (G ≈ -G) ∧ (∀ i, Impartial (G.moveLeft i)) ∧ ∀ j, Impartial (G.moveRight j) := by |
simpa only [impartial_iff_aux] using impartialAux_def
| 0.15625 |
import Mathlib.Algebra.Homology.ComplexShape
import Mathlib.CategoryTheory.Subobject.Limits
import Mathlib.CategoryTheory.GradedObject
import Mathlib.Algebra.Homology.ShortComplex.Basic
#align_import algebra.homology.homological_complex from "leanprover-community/mathlib"@"88bca0ce5d22ebfd9e73e682e51d60ea13b48347"
... | Mathlib/Algebra/Homology/HomologicalComplex.lean | 717 | 719 | theorem of_d (j : α) : (of X d sq).d (j + 1) j = d j := by |
dsimp [of]
rw [if_pos rfl, Category.id_comp]
| 0.15625 |
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 | 87 | 88 | theorem preperfect_iff_nhds : Preperfect C ↔ ∀ x ∈ C, ∀ U ∈ 𝓝 x, ∃ y ∈ U ∩ C, y ≠ x := by |
simp only [Preperfect, accPt_iff_nhds]
| 0.15625 |
import Mathlib.Data.Int.Range
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.MulChar.Basic
#align_import number_theory.legendre_symbol.zmod_char from "leanprover-community/mathlib"@"70fd9563a21e7b963887c9360bd29b2393e6225a"
namespace ZMod
section QuadCharModP
@[simps]
def χ₄ : MulChar (ZMod 4) ℤ... | Mathlib/NumberTheory/LegendreSymbol/ZModChar.lean | 142 | 146 | theorem isQuadratic_χ₈ : χ₈.IsQuadratic := by |
intro a
-- Porting note: was `decide!`
fin_cases a
all_goals decide
| 0.15625 |
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 | 94 | 94 | theorem p_comm {P Q : Karoubi C} (f : Hom P Q) : P.p ≫ f.f = f.f ≫ Q.p := by | rw [p_comp, comp_p]
| 0.15625 |
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryTheory.Limits.Constructions.EpiMono
#align_import category_theory.limits.preserves.shapes.images from "leanprover-community/mathlib"@"fc78e3c190c72a109699385da6be2725e88df841"
noncomputable section
namespace CategoryTheory
namespace Prese... | Mathlib/CategoryTheory/Limits/Preserves/Shapes/Images.lean | 62 | 63 | theorem inv_comp_image_ι_map {X Y : A} (f : X ⟶ Y) :
(iso L f).inv ≫ image.ι (L.map f) = L.map (image.ι f) := by | simp
| 0.15625 |
import Mathlib.Data.Nat.Defs
import Mathlib.Tactic.GCongr.Core
import Mathlib.Tactic.Common
import Mathlib.Tactic.Monotonicity.Attr
#align_import data.nat.factorial.basic from "leanprover-community/mathlib"@"d012cd09a9b256d870751284dd6a29882b0be105"
namespace Nat
def factorial : ℕ → ℕ
| 0 => 1
| succ n => s... | Mathlib/Data/Nat/Factorial/Basic.lean | 150 | 155 | theorem add_factorial_succ_lt_factorial_add_succ {i : ℕ} (n : ℕ) (hi : 2 ≤ i) :
i + (n + 1)! < (i + n + 1)! := by |
rw [factorial_succ (i + _), Nat.add_mul, Nat.one_mul]
have := (i + n).self_le_factorial
refine Nat.add_lt_add_of_lt_of_le (Nat.lt_of_le_of_lt ?_ ((Nat.lt_mul_iff_one_lt_right ?_).2 ?_))
(factorial_le ?_) <;> omega
| 0.15625 |
import Mathlib.Data.Fintype.Basic
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Logic.Equiv.Defs
#align_import logic.equiv.fintype from "leanprover-community/mathlib"@"9407b03373c8cd201df99d6bc5514fc2db44054f"
section Fintype
variable {α β : Type*} [Fintype α] [DecidableEq β] (e : Equiv.Perm α) (f : α ↪ β)
... | Mathlib/Logic/Equiv/Fintype.lean | 78 | 82 | theorem Equiv.Perm.viaFintypeEmbedding_apply_mem_range {b : β} (h : b ∈ Set.range f) :
e.viaFintypeEmbedding f b = f (e (f.invOfMemRange ⟨b, h⟩)) := by |
simp only [viaFintypeEmbedding, Function.Embedding.invOfMemRange]
rw [Equiv.Perm.extendDomain_apply_subtype]
congr
| 0.15625 |
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.CategoryTheory.Limits.Shapes.Products
import Mathlib.CategoryTheory.Limits.Shapes.Images
import Mathlib.CategoryTheory.IsomorphismClasses
import Mathlib.CategoryTheory.Limits.Shapes.ZeroObjects
#align_import category_theory.limits.shapes.zero_morphisms from "leanpr... | Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | 145 | 147 | theorem zero_of_epi_comp {X Y Z : C} (f : X ⟶ Y) {g : Y ⟶ Z} [Epi f] (h : f ≫ g = 0) : g = 0 := by |
rw [← comp_zero, cancel_epi] at h
exact h
| 0.15625 |
import Mathlib.Order.ConditionallyCompleteLattice.Basic
import Mathlib.Order.RelIso.Basic
#align_import order.ord_continuous from "leanprover-community/mathlib"@"207cfac9fcd06138865b5d04f7091e46d9320432"
universe u v w x
variable {α : Type u} {β : Type v} {γ : Type w} {ι : Sort x}
open Function OrderDual Set
... | Mathlib/Order/OrdContinuous.lean | 135 | 137 | theorem map_iSup (hf : LeftOrdContinuous f) (g : ι → α) : f (⨆ i, g i) = ⨆ i, f (g i) := by |
simp only [iSup, hf.map_sSup', ← range_comp]
rfl
| 0.15625 |
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.RingTheory.MatrixAlgebra
#align_import ring_theory.polynomial_algebra from "leanprover-community/mathlib"@"565eb991e264d0db702722b4bde52ee5173c9950"
universe u v w
open Polynomial Tensor... | Mathlib/RingTheory/PolynomialAlgebra.lean | 109 | 111 | theorem toFunLinear_one_tmul_one :
toFunLinear R A (1 ⊗ₜ[R] 1) = 1 := by |
rw [toFunLinear_tmul_apply, toFunBilinear_apply_apply, Polynomial.aeval_one, one_smul]
| 0.15625 |
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_import ring_theory.coprime.lemmas from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
universe u v
section IsCoprime
variable {R : Type ... | Mathlib/RingTheory/Coprime/Lemmas.lean | 42 | 43 | theorem Nat.isCoprime_iff_coprime {m n : ℕ} : IsCoprime (m : ℤ) n ↔ Nat.Coprime m n := by |
rw [Int.isCoprime_iff_gcd_eq_one, Int.gcd_natCast_natCast]
| 0.15625 |
import Batteries.Data.RBMap.Alter
import Batteries.Data.List.Lemmas
namespace Batteries
namespace RBNode
open RBColor
attribute [simp] fold foldl foldr Any forM foldlM Ordered
@[simp] theorem min?_reverse (t : RBNode α) : t.reverse.min? = t.max? := by
unfold RBNode.max?; split <;> simp [RBNode.min?]
unfold RB... | .lake/packages/batteries/Batteries/Data/RBMap/Lemmas.lean | 67 | 68 | theorem isOrdered_iff [@TransCmp α cmp] {t : RBNode α} :
isOrdered cmp t ↔ Ordered cmp t := by | simp [isOrdered_iff']
| 0.15625 |
import Batteries.Data.List.Basic
import Batteries.Data.List.Lemmas
open Nat
namespace List
section countP
variable (p q : α → Bool)
@[simp] theorem countP_nil : countP p [] = 0 := rfl
protected theorem countP_go_eq_add (l) : countP.go p l n = n + countP.go p l 0 := by
induction l generalizing n with
| nil... | .lake/packages/batteries/Batteries/Data/List/Count.lean | 75 | 76 | theorem countP_pos : 0 < countP p l ↔ ∃ a ∈ l, p a := by |
simp only [countP_eq_length_filter, length_pos_iff_exists_mem, mem_filter, exists_prop]
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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 | 584 | 588 | theorem oangle_right_eq_arccos_of_oangle_eq_pi_div_two {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ = ↑(π / 2)) :
∡ p₂ p₃ p₁ = Real.arccos (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]
rw [oangle_eq_angle_of_sign_eq_one hs,
angle_eq_arccos_of_angle_eq_pi_div_two (angle_eq_pi_div_two_of_oangle_eq_pi_div_two h)]
| 0.15625 |
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
#align_import order.symm_diff from "leanprover-community/mathlib"@"6eb334bd8f3433d5b08ba156b8ec3e6af47e1904"
open Function OrderDual
variable {ι α β : Type*} {π : ι → Type*}
def symmDiff [Sup α] [SDiff α] (a b : α) : α :=
a \ b ⊔ b \ a
#ali... | Mathlib/Order/SymmDiff.lean | 133 | 134 | theorem symmDiff_eq_bot {a b : α} : a ∆ b = ⊥ ↔ a = b := by |
simp_rw [symmDiff, sup_eq_bot_iff, sdiff_eq_bot_iff, le_antisymm_iff]
| 0.15625 |
import Mathlib.CategoryTheory.Closed.Cartesian
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
#align_import category_theory.closed.functor from "leanprover-community/mathlib"@"cea27692b3fdeb328a2ddba6aabf181754543184"
noncomputable secti... | Mathlib/CategoryTheory/Closed/Functor.lean | 100 | 103 | theorem uncurry_expComparison (A B : C) :
CartesianClosed.uncurry ((expComparison F A).app B) =
inv (prodComparison F _ _) ≫ F.map ((exp.ev _).app _) := by |
rw [uncurry_eq, expComparison_ev]
| 0.15625 |
import Mathlib.Probability.Kernel.MeasurableIntegral
#align_import probability.kernel.composition from "leanprover-community/mathlib"@"3b92d54a05ee592aa2c6181a4e76b1bb7cc45d0b"
open MeasureTheory
open scoped ENNReal
namespace ProbabilityTheory
namespace kernel
variable {α β ι : Type*} {mα : MeasurableSpace α}... | Mathlib/Probability/Kernel/Composition.lean | 151 | 154 | theorem compProdFun_eq_tsum (κ : kernel α β) [IsSFiniteKernel κ] (η : kernel (α × β) γ)
[IsSFiniteKernel η] (a : α) (hs : MeasurableSet s) :
compProdFun κ η a s = ∑' (n) (m), compProdFun (seq κ n) (seq η m) a s := by |
simp_rw [compProdFun_tsum_left κ η a s, compProdFun_tsum_right _ η a hs]
| 0.15625 |
import Mathlib.Algebra.Polynomial.Degree.TrailingDegree
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Algebra.Polynomial.Eval
#align_import data.polynomial.reverse from "leanprover-community/mathlib"@"44de64f183393284a16016dfb2a48ac97382f2bd"
namespace Polynomial
open Polynomial Finsupp Finset
open... | Mathlib/Algebra/Polynomial/Reverse.lean | 128 | 129 | theorem reflect_eq_zero_iff {N : ℕ} {f : R[X]} : reflect N (f : R[X]) = 0 ↔ f = 0 := by |
rw [ofFinsupp_eq_zero, reflect, embDomain_eq_zero, ofFinsupp_eq_zero]
| 0.15625 |
import Mathlib.Topology.Separation
import Mathlib.Algebra.BigOperators.Finprod
#align_import topology.algebra.infinite_sum.basic from "leanprover-community/mathlib"@"3b52265189f3fb43aa631edffce5d060fafaf82f"
noncomputable section
open Filter Function
open scoped Topology
variable {α β γ : Type*}
section HasP... | Mathlib/Topology/Algebra/InfiniteSum/Defs.lean | 129 | 131 | theorem Function.Injective.hasProd_iff {g : γ → β} (hg : Injective g)
(hf : ∀ x, x ∉ Set.range g → f x = 1) : HasProd (f ∘ g) a ↔ HasProd f a := by |
simp only [HasProd, Tendsto, comp_apply, hg.map_atTop_finset_prod_eq hf]
| 0.15625 |
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Topology.Algebra.Field
import Mathlib.Topology.Algebra.Order.Group
#align_import topology.algebra.order.field from "leanprover-community/mathlib"@"9a59dcb7a2d06bf55da57b9030169219980660cd"
open Set Filter TopologicalSpace Function
open scoped Pointwise Top... | Mathlib/Topology/Algebra/Order/Field.lean | 87 | 89 | theorem Filter.Tendsto.neg_mul_atTop {C : 𝕜} (hC : C < 0) (hf : Tendsto f l (𝓝 C))
(hg : Tendsto g l atTop) : Tendsto (fun x => f x * g x) l atBot := by |
simpa only [mul_comm] using hg.atTop_mul_neg hC hf
| 0.15625 |
import Mathlib.Algebra.Group.Even
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Algebra.Order.Sub.Defs
#align_import algebra.order.sub.canonical from "leanprover-community/mathlib"@"62a5626868683c104774de8d85b9855234ac807c"
variable {α : Type*}
section ExistsAddOfLE
variable [AddCommSemigrou... | Mathlib/Algebra/Order/Sub/Canonical.lean | 63 | 65 | theorem tsub_add_tsub_cancel (hab : b ≤ a) (hcb : c ≤ b) : a - b + (b - c) = a - c := by |
convert tsub_add_cancel_of_le (tsub_le_tsub_right hab c) using 2
rw [tsub_tsub, add_tsub_cancel_of_le hcb]
| 0.15625 |
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Deriv
import Mathlib.Analysis.SpecialFunctions.Log.Basic
#align_import analysis.special_functions.arsinh from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
noncomputable section
open Function Filter Set
open scoped Topology
name... | Mathlib/Analysis/SpecialFunctions/Arsinh.lean | 78 | 79 | theorem sinh_arsinh (x : ℝ) : sinh (arsinh x) = x := by |
rw [sinh_eq, ← arsinh_neg, exp_arsinh, exp_arsinh, neg_sq]; field_simp
| 0.15625 |
import Mathlib.CategoryTheory.Elementwise
import Mathlib.CategoryTheory.Adjunction.Evaluation
import Mathlib.Tactic.CategoryTheory.Elementwise
import Mathlib.CategoryTheory.Adhesive
import Mathlib.CategoryTheory.Sites.ConcreteSheafification
#align_import category_theory.sites.subsheaf from "leanprover-community/mathl... | Mathlib/CategoryTheory/Sites/Subsheaf.lean | 146 | 149 | theorem Subpresheaf.lift_ι (f : F' ⟶ F) (hf : ∀ U x, f.app U x ∈ G.obj U) :
G.lift f hf ≫ G.ι = f := by |
ext
rfl
| 0.15625 |
import Mathlib.Analysis.Calculus.Deriv.Add
import Mathlib.Analysis.Calculus.Deriv.Linear
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
variable {𝕜 : Type*} [NontriviallyNormedField 𝕜]
{E : Type*} [NormedAddCommGroup E] [NormedSpace 𝕜 E]
(f : 𝕜 →ᵃ[𝕜] E) {a b : E} {L : Filter 𝕜} {s : Set 𝕜} {x : 𝕜}
n... | Mathlib/Analysis/Calculus/Deriv/AffineMap.lean | 64 | 65 | theorem hasStrictDerivAt_lineMap : HasStrictDerivAt (lineMap a b) (b - a) x := by |
simpa using (lineMap a b : 𝕜 →ᵃ[𝕜] E).hasStrictDerivAt
| 0.15625 |
import Mathlib.Algebra.Module.Defs
import Mathlib.Data.Fintype.BigOperators
import Mathlib.GroupTheory.GroupAction.BigOperators
#align_import algebra.module.big_operators from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
variable {ι κ α β R M : Type*}
section AddCommMonoid
variable [... | Mathlib/Algebra/Module/BigOperators.lean | 41 | 45 | theorem Finset.sum_smul_sum {f : α → R} {g : β → M} {s : Finset α} {t : Finset β} :
((∑ i ∈ s, f i) • ∑ i ∈ t, g i) = ∑ p ∈ s ×ˢ t, f p.fst • g p.snd := by |
rw [Finset.sum_product, Finset.sum_smul, Finset.sum_congr rfl]
intros
rw [Finset.smul_sum]
| 0.15625 |
import Mathlib.Tactic.Ring
import Mathlib.Tactic.FailIfNoProgress
import Mathlib.Algebra.Group.Commutator
#align_import tactic.group from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
namespace Mathlib.Tactic.Group
open Lean
open Lean.Meta
open Lean.Parser.Tactic
open Lean.Elab.Tactic
... | Mathlib/Tactic/Group.lean | 49 | 50 | theorem zpow_trick_one' {G : Type*} [Group G] (a b : G) (n : ℤ) :
a * b ^ n * b = a * b ^ (n + 1) := by | rw [mul_assoc, mul_zpow_self]
| 0.15625 |
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#align_import algebra.order.pointwise from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
open Function Set
open Pointwise
variable ... | Mathlib/Algebra/Order/Pointwise.lean | 239 | 249 | theorem smul_Ioi : r • Ioi a = Ioi (r • a) := by |
ext x
simp only [mem_smul_set, smul_eq_mul, mem_Ioi]
constructor
· rintro ⟨a_w, a_h_left, rfl⟩
exact (mul_lt_mul_left hr).mpr a_h_left
· rintro h
use x / r
constructor
· exact (lt_div_iff' hr).mpr h
· exact mul_div_cancel₀ _ (ne_of_gt hr)
| 0.15625 |
import Mathlib.Analysis.NormedSpace.ConformalLinearMap
import Mathlib.Analysis.Calculus.FDeriv.Add
#align_import analysis.calculus.conformal.normed_space from "leanprover-community/mathlib"@"e3fb84046afd187b710170887195d50bada934ee"
noncomputable section
variable {X Y Z : Type*} [NormedAddCommGroup X] [NormedAdd... | Mathlib/Analysis/Calculus/Conformal/NormedSpace.lean | 73 | 82 | theorem conformalAt_iff_isConformalMap_fderiv {f : X → Y} {x : X} :
ConformalAt f x ↔ IsConformalMap (fderiv ℝ f x) := by |
constructor
· rintro ⟨f', hf, hf'⟩
rwa [hf.fderiv]
· intro H
by_cases h : DifferentiableAt ℝ f x
· exact ⟨fderiv ℝ f x, h.hasFDerivAt, H⟩
· nontriviality X
exact absurd (fderiv_zero_of_not_differentiableAt h) H.ne_zero
| 0.15625 |
import Mathlib.CategoryTheory.Abelian.Subobject
import Mathlib.CategoryTheory.Limits.EssentiallySmall
import Mathlib.CategoryTheory.Preadditive.Injective
import Mathlib.CategoryTheory.Preadditive.Generator
import Mathlib.CategoryTheory.Abelian.Opposite
#align_import category_theory.abelian.generator from "leanprover-... | Mathlib/CategoryTheory/Abelian/Generator.lean | 55 | 58 | theorem has_projective_separator [HasColimits C] [EnoughProjectives C] (G : C)
(hG : IsCoseparator G) : ∃ G : C, Projective G ∧ IsSeparator G := by |
obtain ⟨T, hT₁, hT₂⟩ := has_injective_coseparator (op G) ((isSeparator_op_iff _).2 hG)
exact ⟨unop T, inferInstance, (isSeparator_unop_iff _).2 hT₂⟩
| 0.15625 |
import Mathlib.MeasureTheory.Measure.Restrict
open scoped ENNReal NNReal Topology
open Set MeasureTheory Measure Filter Function MeasurableSpace ENNReal
variable {α β δ ι : Type*}
namespace MeasureTheory
variable {m0 : MeasurableSpace α} [MeasurableSpace β] {μ ν ν₁ ν₂: Measure α}
{s t : Set α}
section NoAtoms... | Mathlib/MeasureTheory/Measure/Typeclasses.lean | 378 | 379 | theorem Measure.restrict_singleton' {a : α} : μ.restrict {a} = 0 := by |
simp only [measure_singleton, Measure.restrict_eq_zero]
| 0.15625 |
import Mathlib.Algebra.MonoidAlgebra.Division
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.EraseLead
import Mathlib.Order.Interval.Finset.Nat
#align_import data.polynomial.inductions from "leanprover-community/mathlib"@"57e09a1296bf... | Mathlib/Algebra/Polynomial/Inductions.lean | 45 | 46 | theorem coeff_divX : (divX p).coeff n = p.coeff (n + 1) := by |
rw [add_comm]; cases p; rfl
| 0.15625 |
import Mathlib.Topology.ContinuousOn
import Mathlib.Data.Set.BoolIndicator
open Set Filter Topology TopologicalSpace Classical
universe u v
variable {X : Type u} {Y : Type v} {ι : Type*}
variable [TopologicalSpace X] [TopologicalSpace Y] {s t : Set X}
section Clopen
protected theorem IsClopen.isOpen (hs : IsClo... | Mathlib/Topology/Clopen.lean | 30 | 34 | theorem isClopen_iff_frontier_eq_empty : IsClopen s ↔ frontier s = ∅ := by |
rw [IsClopen, ← closure_eq_iff_isClosed, ← interior_eq_iff_isOpen, frontier, diff_eq_empty]
refine ⟨fun h => (h.1.trans h.2.symm).subset, fun h => ?_⟩
exact ⟨(h.trans interior_subset).antisymm subset_closure,
interior_subset.antisymm (subset_closure.trans h)⟩
| 0.15625 |
import Mathlib.Topology.MetricSpace.Basic
#align_import topology.metric_space.metrizable from "leanprover-community/mathlib"@"f2ce6086713c78a7f880485f7917ea547a215982"
open Set Filter Metric
open scoped Filter Topology
namespace TopologicalSpace
variable {ι X Y : Type*} {π : ι → Type*} [TopologicalSpace X] [Top... | Mathlib/Topology/Metrizable/Basic.lean | 133 | 137 | theorem IsSeparable.secondCountableTopology [PseudoMetrizableSpace X] {s : Set X}
(hs : IsSeparable s) : SecondCountableTopology s := by |
letI := pseudoMetrizableSpacePseudoMetric X
have := hs.separableSpace
exact UniformSpace.secondCountable_of_separable s
| 0.15625 |
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 | 223 | 226 | theorem HasFTaylorSeriesUpToOn.continuousOn (h : HasFTaylorSeriesUpToOn n f p s) :
ContinuousOn f s := by |
have := (h.cont 0 bot_le).congr fun x hx => (h.zero_eq' hx).symm
rwa [← (continuousMultilinearCurryFin0 𝕜 E F).symm.comp_continuousOn_iff]
| 0.15625 |
import Mathlib.Order.RelClasses
import Mathlib.Order.Interval.Set.Basic
#align_import order.bounded from "leanprover-community/mathlib"@"aba57d4d3dae35460225919dcd82fe91355162f9"
namespace Set
variable {α : Type*} {r : α → α → Prop} {s t : Set α}
theorem Bounded.mono (hst : s ⊆ t) (hs : Bounded r t) : Bounde... | Mathlib/Order/Bounded.lean | 44 | 45 | theorem unbounded_le_iff [LinearOrder α] : Unbounded (· ≤ ·) s ↔ ∀ a, ∃ b ∈ s, a < b := by |
simp only [Unbounded, not_le]
| 0.15625 |
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Tree.Basic
import Mathlib.Logic.Basic
import Mathlib.Tactic.NormNum.Core
import Mathlib.Util.SynthesizeUsing
import Mathlib.Util.Qq
open Lean Parser Tactic Mathlib Meta NormNum Qq
initialize registerTraceClass `CancelDen... | Mathlib/Tactic/CancelDenoms/Core.lean | 39 | 42 | theorem mul_subst {α} [CommRing α] {n1 n2 k e1 e2 t1 t2 : α}
(h1 : n1 * e1 = t1) (h2 : n2 * e2 = t2) (h3 : n1 * n2 = k) : k * (e1 * e2) = t1 * t2 := by |
rw [← h3, mul_comm n1, mul_assoc n2, ← mul_assoc n1, h1,
← mul_assoc n2, mul_comm n2, mul_assoc, h2]
| 0.15625 |
import Mathlib.Algebra.Ring.Prod
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Tactic.FinCases
#align_import data.zmod.basic from "leanprover-community/mathlib"@"74ad1c88c77e799d2fea62801d1dbbd698cff1b7"
assert_not_exists Submodule
open Function
namespace ZMod
instance charZero : CharZero (ZMod 0) :=... | Mathlib/Data/ZMod/Basic.lean | 132 | 133 | theorem addOrderOf_coe' {a : ℕ} (n : ℕ) (a0 : a ≠ 0) : addOrderOf (a : ZMod n) = n / n.gcd a := by |
rw [← Nat.smul_one_eq_cast, addOrderOf_nsmul' _ a0, ZMod.addOrderOf_one]
| 0.15625 |
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