Context stringlengths 285 157k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
|---|---|---|---|---|---|
/-
Copyright (c) 2024 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.FieldTheory.Perfect
import Mathlib.LinearAlgebra.Basis.VectorSpace
import Mathlib.LinearAlgebra.AnnihilatingPolynomial
import Mathlib.Order.CompleteSublattic... | Mathlib/LinearAlgebra/Semisimple.lean | 171 | 211 | theorem IsSemisimple.of_mem_adjoin_pair {a : End K M} (ha : a ∈ Algebra.adjoin K {f, g}) :
a.IsSemisimple := by |
let R := K[X] ⧸ Ideal.span {minpoly K f}
let S := AdjoinRoot ((minpoly K g).map <| algebraMap K R)
have : Finite K R :=
(AdjoinRoot.powerBasis' <| minpoly.monic <| Algebra.IsIntegral.isIntegral f).finite
have : Finite R S :=
(AdjoinRoot.powerBasis' <| (minpoly.monic <| Algebra.IsIntegral.isIntegral g).... |
/-
Copyright (c) 2023 Bulhwi Cha. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bulhwi Cha, Mario Carneiro
-/
import Batteries.Data.Char
import Batteries.Data.List.Lemmas
import Batteries.Data.String.Basic
import Batteries.Tactic.Lint.Misc
import Batteries.Tactic.SeqF... | .lake/packages/batteries/Batteries/Data/String/Lemmas.lean | 134 | 143 | theorem utf8GetAux_add_right_cancel (s : List Char) (i p n : Nat) :
utf8GetAux s ⟨i + n⟩ ⟨p + n⟩ = utf8GetAux s ⟨i⟩ ⟨p⟩ := by |
apply utf8InductionOn s ⟨i⟩ ⟨p⟩ (motive := fun s i =>
utf8GetAux s ⟨i.byteIdx + n⟩ ⟨p + n⟩ = utf8GetAux s i ⟨p⟩) <;>
simp [utf8GetAux]
intro c cs ⟨i⟩ h ih
simp [Pos.ext_iff, Pos.addChar_eq] at h ⊢
simp [Nat.add_right_cancel_iff, h]
rw [Nat.add_right_comm]
exact ih
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.affine_subspace from "leanprover-community/mathlib"@"e96bdfbd1e8c98a09ff75... | Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | 913 | 918 | theorem direction_inf (s1 s2 : AffineSubspace k P) :
(s1 ⊓ s2).direction ≤ s1.direction ⊓ s2.direction := by |
simp only [direction_eq_vectorSpan, vectorSpan_def]
exact
le_inf (sInf_le_sInf fun p hp => trans (vsub_self_mono inter_subset_left) hp)
(sInf_le_sInf fun p hp => trans (vsub_self_mono inter_subset_right) hp)
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Eric Wieser
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.RowCol
import Mathlib.Data.Fin.VecNotation
import Mathlib.Tactic.FinCases
#align_import data.matri... | Mathlib/Data/Matrix/Notation.lean | 288 | 291 | theorem cons_vecMul (x : α) (v : Fin n → α) (B : Fin n.succ → o' → α) :
vecCons x v ᵥ* of B = x • vecHead B + v ᵥ* of (vecTail B) := by |
ext i
simp [vecMul]
|
/-
Copyright (c) 2023 François G. Dorais. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: François G. Dorais
-/
import Batteries.Data.Array.Lemmas
namespace ByteArray
@[ext] theorem ext : {a b : ByteArray} → a.data = b.data → a = b
| ⟨_⟩, ⟨_⟩, rfl => rfl
theorem ge... | .lake/packages/batteries/Batteries/Data/ByteArray.lean | 84 | 87 | theorem get_append_right {a b : ByteArray} (hle : a.size ≤ i) (h : i < (a ++ b).size)
(h' : i - a.size < b.size := Nat.sub_lt_left_of_lt_add hle (size_append .. ▸ h)) :
(a ++ b)[i] = b[i - a.size] := by |
simp [getElem_eq_data_getElem]; exact Array.get_append_right hle
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Limits.Pullbacks
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
#align_import algebraic_geometry.open_immersion.basic from ... | Mathlib/Geometry/RingedSpace/OpenImmersion.lean | 133 | 141 | theorem isoRestrict_hom_ofRestrict : H.isoRestrict.hom ≫ Y.ofRestrict _ = f := by |
-- Porting note: `ext` did not pick up `NatTrans.ext`
refine PresheafedSpace.Hom.ext _ _ rfl <| NatTrans.ext _ _ <| funext fun x => ?_
simp only [isoRestrict_hom_c_app, NatTrans.comp_app, eqToHom_refl,
ofRestrict_c_app, Category.assoc, whiskerRight_id']
erw [Category.comp_id, comp_c_app, f.c.naturality_ass... |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import topology.metric_space.pi_nat from "leanprover-community/mathlib"@"49b7f94aab3a3bdca1f9f34c5... | Mathlib/Topology/MetricSpace/PiNat.lean | 722 | 806 | theorem exists_nat_nat_continuous_surjective_of_completeSpace (α : Type*) [MetricSpace α]
[CompleteSpace α] [SecondCountableTopology α] [Nonempty α] :
∃ f : (ℕ → ℕ) → α, Continuous f ∧ Surjective f := by |
/- First, we define a surjective map from a closed subset `s` of `ℕ → ℕ`. Then, we compose
this map with a retraction of `ℕ → ℕ` onto `s` to obtain the desired map.
Let us consider a dense sequence `u` in `α`. Then `s` is the set of sequences `xₙ` such that the
balls `closedBall (u xₙ) (1/2^n)` have a no... |
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib... | Mathlib/Analysis/Calculus/ParametricIntegral.lean | 258 | 284 | theorem hasDerivAt_integral_of_dominated_loc_of_lip {F' : α → E} (ε_pos : 0 < ε)
(hF_meas : ∀ᶠ x in 𝓝 x₀, AEStronglyMeasurable (F x) μ) (hF_int : Integrable (F x₀) μ)
(hF'_meas : AEStronglyMeasurable F' μ)
(h_lipsch : ∀ᵐ a ∂μ, LipschitzOnWith (Real.nnabs <| bound a) (F · a) (ball x₀ ε))
(bound_integrab... |
set L : E →L[𝕜] 𝕜 →L[𝕜] E := ContinuousLinearMap.smulRightL 𝕜 𝕜 E 1
replace h_diff : ∀ᵐ a ∂μ, HasFDerivAt (F · a) (L (F' a)) x₀ :=
h_diff.mono fun x hx ↦ hx.hasFDerivAt
have hm : AEStronglyMeasurable (L ∘ F') μ := L.continuous.comp_aestronglyMeasurable hF'_meas
cases'
hasFDerivAt_integral_of_domin... |
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral
import Mathlib.Analysis.Complex.CauchyIntegral
import Mathlib.MeasureTheory.Integral.Pi
impor... | Mathlib/Analysis/SpecialFunctions/Gaussian/FourierTransform.lean | 307 | 315 | theorem integral_cexp_neg_sum_mul_add {ι : Type*} [Fintype ι] {b : ι → ℂ}
(hb : ∀ i, 0 < (b i).re) (c : ι → ℂ) :
∫ v : ι → ℝ, cexp (- ∑ i, b i * (v i : ℂ) ^ 2 + ∑ i, c i * v i)
= ∏ i, (π / b i) ^ (1 / 2 : ℂ) * cexp (c i ^ 2 / (4 * b i)) := by |
simp_rw [← Finset.sum_neg_distrib, ← Finset.sum_add_distrib, Complex.exp_sum, ← neg_mul]
rw [integral_fintype_prod_eq_prod (f := fun i (v : ℝ) ↦ cexp (-b i * v ^ 2 + c i * v))]
congr with i
have : (-b i).re < 0 := by simpa using hb i
convert integral_cexp_quadratic this (c i) 0 using 1 <;> simp [div_neg]
|
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Algebra.Valuation
import Mathlib.Topology.Algebra.WithZeroTopology
import Mathlib.Topology.Algebra.UniformField
#align_import topology.algebr... | Mathlib/Topology/Algebra/ValuedField.lean | 117 | 127 | theorem Valued.continuous_valuation [Valued K Γ₀] : Continuous (v : K → Γ₀) := by |
rw [continuous_iff_continuousAt]
intro x
rcases eq_or_ne x 0 with (rfl | h)
· rw [ContinuousAt, map_zero, WithZeroTopology.tendsto_zero]
intro γ hγ
rw [Filter.Eventually, Valued.mem_nhds_zero]
use Units.mk0 γ hγ; rfl
· have v_ne : (v x : Γ₀) ≠ 0 := (Valuation.ne_zero_iff _).mpr h
rw [Continuo... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Arg
import Mathlib.Analysis.SpecialFunctions.Log.Basic... | Mathlib/Analysis/SpecialFunctions/Complex/Log.lean | 131 | 132 | theorem log_conj (x : ℂ) (h : x.arg ≠ π) : log (conj x) = conj (log x) := by |
rw [log_conj_eq_ite, if_neg h]
|
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Yury Kudryashov
-/
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Analysis.Normed.MulAction
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mat... | Mathlib/Analysis/Asymptotics/Asymptotics.lean | 379 | 381 | theorem isLittleO_congr (hf : f₁ =ᶠ[l] f₂) (hg : g₁ =ᶠ[l] g₂) : f₁ =o[l] g₁ ↔ f₂ =o[l] g₂ := by |
simp only [IsLittleO_def]
exact forall₂_congr fun c _hc => isBigOWith_congr (Eq.refl c) hf hg
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Johan Commelin
-/
import Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic
import Mathlib.CategoryTheory.Monoidal.Functorial
import Mathlib.CategoryTheory.Monoidal.Type... | Mathlib/Algebra/Category/ModuleCat/Adjunctions.lean | 112 | 129 | theorem left_unitality (X : Type u) :
(λ_ ((free R).obj X)).hom =
(ε R ⊗ 𝟙 ((free R).obj X)) ≫ (μ R (𝟙_ (Type u)) X).hom ≫ map (free R).obj (λ_ X).hom := by |
-- Porting note (#11041): broken ext
apply TensorProduct.ext
apply LinearMap.ext_ring
apply Finsupp.lhom_ext'
intro x
apply LinearMap.ext_ring
apply Finsupp.ext
intro x'
-- Porting note (#10934): used to be dsimp [ε, μ]
let q : X →₀ R := ((λ_ (of R (X →₀ R))).hom) (1 ⊗ₜ[R] Finsupp.single x 1)
cha... |
/-
Copyright (c) 2014 Robert Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Fiel... | Mathlib/Algebra/Order/Field/Basic.lean | 904 | 906 | theorem sub_self_div_two (a : α) : a - a / 2 = a / 2 := by |
suffices a / 2 + a / 2 - a / 2 = a / 2 by rwa [add_halves] at this
rw [add_sub_cancel_right]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.MvPolynomial.Basic
#align_import data.mv_polynomial.rename from "leanprover-community/mathlib"@"2f5b500a507264... | Mathlib/Algebra/MvPolynomial/Rename.lean | 199 | 203 | theorem rename_eval₂ (g : τ → MvPolynomial σ R) :
rename k (p.eval₂ C (g ∘ k)) = (rename k p).eval₂ C (rename k ∘ g) := by |
apply MvPolynomial.induction_on p <;>
· intros
simp [*]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Finset.Update
import Mathlib.Data.Prod.TProd
import Mathlib.GroupTheory.Coset
import Mathlib.Logic.Equiv.Fin
import Mathlib.Measur... | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | 487 | 492 | theorem measurable_to_prop {f : α → Prop} (h : MeasurableSet (f ⁻¹' {True})) : Measurable f := by |
refine measurable_to_countable' fun x => ?_
by_cases hx : x
· simpa [hx] using h
· simpa only [hx, ← preimage_compl, Prop.compl_singleton, not_true, preimage_singleton_false]
using h.compl
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Order.RelIso.Set
import Mathlib.Data.Multiset.Sort
import Mathlib.Data.List.NodupEquivFin
import Mathlib.Data.Finset.Lattice
import Mathlib.Data.Fintyp... | Mathlib/Data/Finset/Sort.lean | 211 | 214 | theorem orderEmbOfFin_singleton (a : α) (i : Fin 1) :
orderEmbOfFin {a} (card_singleton a) i = a := by |
rw [Subsingleton.elim i ⟨0, Nat.zero_lt_one⟩, orderEmbOfFin_zero _ Nat.zero_lt_one,
min'_singleton]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Finset.Sort
import Mathlib.Data.Set.Subsingle... | Mathlib/Combinatorics/Enumerative/Composition.lean | 728 | 735 | theorem join_splitWrtCompositionAux {ns : List ℕ} :
∀ {l : List α}, ns.sum = l.length → (l.splitWrtCompositionAux ns).join = l := by |
induction' ns with n ns IH <;> intro l h <;> simp at h
· exact (length_eq_zero.1 h.symm).symm
simp only [splitWrtCompositionAux_cons]; dsimp
rw [IH]
· simp
· rw [length_drop, ← h, add_tsub_cancel_left]
|
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Analysis.NormedSpace.LinearIsometry
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.No... | Mathlib/Analysis/NormedSpace/AffineIsometry.lean | 452 | 454 | theorem range_eq_univ (e : P ≃ᵃⁱ[𝕜] P₂) : Set.range e = Set.univ := by |
rw [← coe_toIsometryEquiv]
exact IsometryEquiv.range_eq_univ _
|
/-
Copyright (c) 2022 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez, Eric Wieser
-/
import Mathlib.Data.List.Chain
#align_import data.list.destutter from "leanprover-community/mathlib"@"7b78d1776212a91ecc94cf601f83bdcc46b04213"
/-!
# D... | Mathlib/Data/List/Destutter.lean | 64 | 70 | theorem destutter'_sublist (a) : l.destutter' R a <+ a :: l := by |
induction' l with b l hl generalizing a
· simp
rw [destutter']
split_ifs
· exact Sublist.cons₂ a (hl b)
· exact (hl a).trans ((l.sublist_cons b).cons_cons a)
|
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.RCLike.Basic
import Mathlib.Dynamics.BirkhoffSum.Average
/-!
# Birkhoff average in a normed space
In this file we prove some lemmas about ... | Mathlib/Dynamics/BirkhoffSum/NormedSpace.lean | 42 | 44 | theorem dist_birkhoffSum_apply_birkhoffSum (f : α → α) (g : α → E) (n : ℕ) (x : α) :
dist (birkhoffSum f g n (f x)) (birkhoffSum f g n x) = dist (g (f^[n] x)) (g x) := by |
simp only [dist_eq_norm, birkhoffSum_apply_sub_birkhoffSum]
|
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Data.ENat.Basic
#align_import data.polynomial.degree.trailing_degree from "leanprover-community/mat... | Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean | 447 | 458 | theorem natTrailingDegree_mul' (h : p.trailingCoeff * q.trailingCoeff ≠ 0) :
(p * q).natTrailingDegree = p.natTrailingDegree + q.natTrailingDegree := by |
have hp : p ≠ 0 := fun hp => h (by rw [hp, trailingCoeff_zero, zero_mul])
have hq : q ≠ 0 := fun hq => h (by rw [hq, trailingCoeff_zero, mul_zero])
-- Porting note: Needed to account for different coercion behaviour & add the lemmas below
have aux1 : ∀ n, Nat.cast n = WithTop.some (n) := fun n ↦ rfl
have aux... |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.GroupTheory.Submonoid.Inverses
import Mathlib.RingTheory.FiniteType
import Mathlib.RingTheory.Loc... | Mathlib/RingTheory/Localization/InvSubmonoid.lean | 87 | 91 | theorem surj'' (z : S) : ∃ (r : R) (m : M), z = r • (toInvSubmonoid M S m : S) := by |
rcases IsLocalization.surj M z with ⟨⟨r, m⟩, e : z * _ = algebraMap R S r⟩
refine ⟨r, m, ?_⟩
rw [Algebra.smul_def, ← e, mul_assoc]
simp
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Module.LinearMap.Basic
import ... | Mathlib/Data/DFinsupp/Basic.lean | 784 | 787 | theorem filter_ne_eq_erase' (f : Π₀ i, β i) (i : ι) : f.filter (i ≠ ·) = f.erase i := by |
rw [← filter_ne_eq_erase f i]
congr with j
exact ne_comm
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.CharP.Two
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Polynomial.RingDivision
import Mathlib.Grou... | Mathlib/RingTheory/RootsOfUnity/Basic.lean | 131 | 133 | theorem rootsOfUnity.coe_pow [CommMonoid R] (ζ : rootsOfUnity k R) (m : ℕ) :
(((ζ ^ m :) : Rˣ) : R) = ((ζ : Rˣ) : R) ^ m := by |
rw [Subgroup.coe_pow, Units.val_pow_eq_pow_val]
|
/-
Copyright (c) 2022 Eric Rodriguez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Rodriguez
-/
import Mathlib.Algebra.GroupWithZero.Units.Lemmas
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.BigOperators
#align_import data.... | Mathlib/Data/Sign.lean | 365 | 369 | theorem sign_eq_neg_one_iff : sign a = -1 ↔ a < 0 := by |
refine ⟨fun h => ?_, fun h => sign_neg h⟩
rw [sign_apply] at h
split_ifs at h
assumption
|
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Batteries.Control.ForInStep.Lemmas
import Batteries.Data.List.Basic
import Batteries.Ta... | .lake/packages/batteries/Batteries/Data/List/Lemmas.lean | 350 | 352 | theorem modifyNth_eq_take_cons_drop (f : α → α) {n l} (h) :
modifyNth f n l = take n l ++ f (get l ⟨n, h⟩) :: drop (n + 1) l := by |
rw [modifyNth_eq_take_drop, drop_eq_get_cons h]; rfl
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Devon Tuma
-/
import Mathlib.Topology.Instances.ENNReal
import Mathlib.MeasureTheory.Measure.Dirac
#align_import probability.probability_mass_function.basic from "lean... | Mathlib/Probability/ProbabilityMassFunction/Basic.lean | 403 | 404 | theorem toPMF_eq_iff_toMeasure_eq (μ : Measure α) [IsProbabilityMeasure μ] :
μ.toPMF = p ↔ μ = p.toMeasure := by | rw [← toMeasure_inj, Measure.toPMF_toMeasure]
|
/-
Copyright (c) 2021 Stuart Presnell. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stuart Presnell
-/
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Data.Nat.GCD.BigOperators
import Mathlib.Data.Nat.PrimeFin
import Mathlib.NumberTheory.Padics.PadicVal
import Ma... | Mathlib/Data/Nat/Factorization/Basic.lean | 219 | 227 | theorem factorization_prod {α : Type*} {S : Finset α} {g : α → ℕ} (hS : ∀ x ∈ S, g x ≠ 0) :
(S.prod g).factorization = S.sum fun x => (g x).factorization := by |
classical
ext p
refine Finset.induction_on' S ?_ ?_
· simp
· intro x T hxS hTS hxT IH
have hT : T.prod g ≠ 0 := prod_ne_zero_iff.mpr fun x hx => hS x (hTS hx)
simp [prod_insert hxT, sum_insert hxT, ← IH, factorization_mul (hS x hxS) hT]
|
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Asymptotics.AsymptoticEquivalent
import Mathlib.Analysis.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calc... | Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | 391 | 410 | theorem HasStrictFDerivAt.of_local_left_inverse {f : E → F} {f' : E ≃L[𝕜] F} {g : F → E} {a : F}
(hg : ContinuousAt g a) (hf : HasStrictFDerivAt f (f' : E →L[𝕜] F) (g a))
(hfg : ∀ᶠ y in 𝓝 a, f (g y) = y) : HasStrictFDerivAt g (f'.symm : F →L[𝕜] E) a := by |
replace hg := hg.prod_map' hg
replace hfg := hfg.prod_mk_nhds hfg
have :
(fun p : F × F => g p.1 - g p.2 - f'.symm (p.1 - p.2)) =O[𝓝 (a, a)] fun p : F × F =>
f' (g p.1 - g p.2) - (p.1 - p.2) := by
refine ((f'.symm : F →L[𝕜] E).isBigO_comp _ _).congr (fun x => ?_) fun _ => rfl
simp
refine th... |
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn, Mario Carneiro, Martin Dvorak
-/
import Mathlib.Data.List.Basic
#align_import data.list.join from "leanprover-community/mathlib"@"18a5306c091183ac... | Mathlib/Data/List/Join.lean | 105 | 109 | theorem take_sum_join' (L : List (List α)) (i : ℕ) :
L.join.take (Nat.sum ((L.map length).take i)) = (L.take i).join := by |
induction L generalizing i
· simp
· cases i <;> simp [take_append, *]
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Yury Kudryashov
-/
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
import Mathlib.LinearAlgebra.AffineSpace.Midpoint
#al... | Mathlib/Analysis/Normed/Group/AddTorsor.lean | 197 | 201 | theorem edist_vsub_vsub_le (p₁ p₂ p₃ p₄ : P) :
edist (p₁ -ᵥ p₂) (p₃ -ᵥ p₄) ≤ edist p₁ p₃ + edist p₂ p₄ := by |
simp only [edist_nndist]
norm_cast -- Porting note: was apply_mod_cast
apply dist_vsub_vsub_le
|
/-
Copyright (c) 2019 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Mario Carneiro, Isabel Longbottom, Scott Morrison
-/
import Mathlib.Algebra.Order.ZeroLEOne
import Mathlib.Data.List.InsertNth
import Mathlib.Logic.Relation
import Mathlib... | Mathlib/SetTheory/Game/PGame.lean | 1,364 | 1,365 | theorem moveRight_neg_symm' {x : PGame} (i) :
x.moveRight i = -(-x).moveLeft (toLeftMovesNeg i) := by | simp
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.FractionalIdeal.Basic
#align_import ring_theory.fractional_ideal from "leanprover... | Mathlib/RingTheory/FractionalIdeal/Operations.lean | 795 | 800 | theorem spanSingleton_mul_coeIdeal_eq_coeIdeal {I J : Ideal R₁} {z : K} :
spanSingleton R₁⁰ z * (I : FractionalIdeal R₁⁰ K) = J ↔
Ideal.span {((IsLocalization.sec R₁⁰ z).1 : R₁)} * I =
Ideal.span {((IsLocalization.sec R₁⁰ z).2 : R₁)} * J := by |
rw [← mk'_mul_coeIdeal_eq_coeIdeal K (IsLocalization.sec R₁⁰ z).2.prop,
IsLocalization.mk'_sec K z]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Image
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Clopen
import Mathlib.Topology.Irreducib... | Mathlib/Topology/Connected/Basic.lean | 503 | 508 | theorem isConnected_univ_pi [∀ i, TopologicalSpace (π i)] {s : ∀ i, Set (π i)} :
IsConnected (pi univ s) ↔ ∀ i, IsConnected (s i) := by |
simp only [IsConnected, ← univ_pi_nonempty_iff, forall_and, and_congr_right_iff]
refine fun hne => ⟨fun hc i => ?_, isPreconnected_univ_pi⟩
rw [← eval_image_univ_pi hne]
exact hc.image _ (continuous_apply _).continuousOn
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Analysis.SpecificLimits.Basic
import Mathlib.Order.Iterate
import Mathlib.Order.SemiconjSup
import... | Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | 606 | 608 | theorem le_iterate_pos_iff {x : ℝ} {m : ℤ} {n : ℕ} (hn : 0 < n) :
x + n * m ≤ f^[n] x ↔ x + m ≤ f x := by |
simpa only [not_lt] using not_congr (f.iterate_pos_lt_iff hn)
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Manuel Candales
-/
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
import Mat... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean | 268 | 275 | theorem angle_left_midpoint_eq_pi_div_two_of_dist_eq {p1 p2 p3 : P} (h : dist p3 p1 = dist p3 p2) :
∠ p3 (midpoint ℝ p1 p2) p1 = π / 2 := by |
let m : P := midpoint ℝ p1 p2
have h1 : p3 -ᵥ p1 = p3 -ᵥ m - (p1 -ᵥ m) := (vsub_sub_vsub_cancel_right p3 p1 m).symm
have h2 : p3 -ᵥ p2 = p3 -ᵥ m + (p1 -ᵥ m) := by
rw [left_vsub_midpoint, ← midpoint_vsub_right, vsub_add_vsub_cancel]
rw [dist_eq_norm_vsub V p3 p1, dist_eq_norm_vsub V p3 p2, h1, h2] at h
ex... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Data.Set... | Mathlib/SetTheory/Cardinal/Basic.lean | 2,274 | 2,278 | theorem three_le {α : Type*} (h : 3 ≤ #α) (x : α) (y : α) : ∃ z : α, z ≠ x ∧ z ≠ y := by |
have : ↑(3 : ℕ) ≤ #α := by simpa using h
have : ↑(2 : ℕ) < #α := by rwa [← succ_le_iff, ← Cardinal.nat_succ]
have := exists_not_mem_of_length_lt [x, y] this
simpa [not_or] using this
|
/-
Copyright (c) 2020 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Yury G. Kudryashov
-/
import Mathlib.Data.Nat.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relation
import Mathlib.Order.Basic
#align_import or... | Mathlib/Order/RelClasses.lean | 552 | 553 | theorem not_bounded_iff {r : α → α → Prop} (s : Set α) : ¬Bounded r s ↔ Unbounded r s := by |
simp only [Bounded, Unbounded, not_forall, not_exists, exists_prop, not_and, not_not]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Order.Filter.SmallSets
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Compactness.Compact
import Mathlib.To... | Mathlib/Topology/UniformSpace/Basic.lean | 1,826 | 1,830 | theorem lebesgue_number_lemma_sUnion {S : Set (Set α)}
(hK : IsCompact K) (hopen : ∀ s ∈ S, IsOpen s) (hcover : K ⊆ ⋃₀ S) :
∃ V ∈ 𝓤 α, ∀ x ∈ K, ∃ s ∈ S, ball x V ⊆ s := by |
rw [sUnion_eq_iUnion] at hcover
simpa using lebesgue_number_lemma hK (by simpa) hcover
|
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import comp... | Mathlib/Computability/TMToPartrec.lean | 1,749 | 1,750 | theorem trStmts₁_self (q) : q ∈ trStmts₁ q := by |
induction q <;> · first |apply Finset.mem_singleton_self|apply Finset.mem_insert_self
|
/-
Copyright (c) 2021 Jakob Scholbach. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob Scholbach
-/
import Mathlib.Algebra.CharP.Basic
import Mathlib.Algebra.CharP.Algebra
import Mathlib.Data.Nat.Prime
#align_import algebra.char_p.exp_char from "leanprover-commun... | Mathlib/Algebra/CharP/ExpChar.lean | 82 | 83 | theorem ringExpChar.eq_one (R : Type*) [NonAssocSemiring R] [CharZero R] : ringExpChar R = 1 := by |
rw [ringExpChar, ringChar.eq_zero, max_eq_right zero_le_one]
|
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Game.Basic
import Mathlib.SetTheory.Ordinal.NaturalOps
#align_import set_theory.game.ordinal from "leanprover-commun... | Mathlib/SetTheory/Game/Ordinal.lean | 58 | 59 | theorem toPGame_rightMoves (o : Ordinal) : o.toPGame.RightMoves = PEmpty := by |
rw [toPGame, RightMoves]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Submodule.EqLocus
import Mathlib.Algebra.Module.Subm... | Mathlib/LinearAlgebra/Span.lean | 627 | 640 | theorem span_singleton_eq_span_singleton {R M : Type*} [Ring R] [AddCommGroup M] [Module R M]
[NoZeroSMulDivisors R M] {x y : M} : ((R ∙ x) = R ∙ y) ↔ ∃ z : Rˣ, z • x = y := by |
constructor
· simp only [le_antisymm_iff, span_singleton_le_iff_mem, mem_span_singleton]
rintro ⟨⟨a, rfl⟩, b, hb⟩
rcases eq_or_ne y 0 with rfl | hy; · simp
refine ⟨⟨b, a, ?_, ?_⟩, hb⟩
· apply smul_left_injective R hy
simpa only [mul_smul, one_smul]
· rw [← hb] at hy
apply smul_left_... |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Floris van Doorn, Gabriel Ebner, Yury Kudryashov
-/
import Mathlib.Order.ConditionallyCompleteLattice.Finset
import Mathlib.Order.Interval.Finset.Nat
#align_import dat... | Mathlib/Data/Nat/Lattice.lean | 59 | 62 | theorem sInf_empty : sInf ∅ = 0 := by |
rw [sInf_eq_zero]
right
rfl
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Group.Nat
import Mathlib.Algebra.Order.Sub.Canonical
import Mathlib.Data.List.Perm
import Mathlib.Data.Set.List
import Mathlib.Init.Quot... | Mathlib/Data/Multiset/Basic.lean | 600 | 611 | theorem le_cons_of_not_mem (m : a ∉ s) : s ≤ a ::ₘ t ↔ s ≤ t := by |
refine ⟨?_, fun h => le_trans h <| le_cons_self _ _⟩
suffices ∀ {t'}, s ≤ t' → a ∈ t' → a ::ₘ s ≤ t' by
exact fun h => (cons_le_cons_iff a).1 (this h (mem_cons_self _ _))
introv h
revert m
refine leInductionOn h ?_
introv s m₁ m₂
rcases append_of_mem m₂ with ⟨r₁, r₂, rfl⟩
exact
perm_middle.subp... |
/-
Copyright (c) 2020 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz
-/
import Mathlib.CategoryTheory.Monad.Types
import Mathlib.CategoryTheory.Monad.Limits
import Mathlib.CategoryTheory.Equivalence
import Mathlib.Topology.Category.CompHaus.Basic... | Mathlib/Topology/Category/Compactum.lean | 369 | 376 | theorem cl_eq_closure {X : Compactum} (A : Set X) : cl A = closure A := by |
ext
rw [mem_closure_iff_ultrafilter]
constructor
· rintro ⟨F, h1, h2⟩
exact ⟨F, h1, le_nhds_of_str_eq _ _ h2⟩
· rintro ⟨F, h1, h2⟩
exact ⟨F, h1, str_eq_of_le_nhds _ _ h2⟩
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic fro... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 2,282 | 2,284 | theorem range_enumOrd (hS : Unbounded (· < ·) S) : range (enumOrd S) = S := by |
rw [range_eq_iff]
exact ⟨enumOrd_mem hS, enumOrd_surjective hS⟩
|
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Topology.Constructions
import Mathlib.Topology.Algebra.Monoid
import Mathlib.Order.Filter.ListTraverse
import Mathlib.Tactic.AdaptationNote
#align_import... | Mathlib/Topology/List.lean | 119 | 126 | theorem continuousAt_length : ∀ l : List α, ContinuousAt List.length l := by |
simp only [ContinuousAt, nhds_discrete]
refine tendsto_nhds ?_ ?_
· exact tendsto_pure_pure _ _
· intro l a ih
dsimp only [List.length]
refine Tendsto.comp (tendsto_pure_pure (fun x => x + 1) _) ?_
exact Tendsto.comp ih tendsto_snd
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Patrick Massot
-/
import Mathlib.Data.Set.Function
import Mathlib.Order.Interval.Set.OrdConnected
#align_import data.set.intervals.proj_Icc from "leanprover-co... | Mathlib/Order/Interval/Set/ProjIcc.lean | 109 | 110 | theorem projIcc_eq_right (h : a < b) : projIcc a b h.le x = ⟨b, right_mem_Icc.2 h.le⟩ ↔ b ≤ x := by |
simp [projIcc, Subtype.ext_iff, max_min_distrib_left, h.le, h.not_le]
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Data.Stream.Defs
import Mathlib.Logic.Function.Basic
import Mathlib.Init.Data.List.Basic
import Mathlib.Data.List.Basic
#align_import data.s... | Mathlib/Data/Stream/Init.lean | 94 | 94 | theorem head_drop (a : Stream' α) (n : ℕ) : (a.drop n).head = a.get n := by | simp
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir
-/
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Na... | Mathlib/Data/Complex/Exponential.lean | 1,489 | 1,493 | theorem exp_approx_end' {n} {x a b : ℝ} (m : ℕ) (e₁ : n + 1 = m) (rm : ℝ) (er : ↑m = rm)
(h : |x| ≤ 1) (e : |1 - a| ≤ b - |x| / rm * ((rm + 1) / rm)) :
|exp x - expNear n x a| ≤ |x| ^ n / n.factorial * b := by |
subst er
exact exp_approx_succ _ e₁ _ _ (by simpa using e) (exp_approx_end _ _ _ e₁ h)
|
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Yury Kudryashov
-/
import Mathlib.Algebra.Star.Order
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.Order.MonotoneContinuity
#align... | Mathlib/Data/Real/Sqrt.lean | 468 | 472 | theorem sqrt_one_add_le (h : -1 ≤ x) : √(1 + x) ≤ 1 + x / 2 := by |
refine sqrt_le_iff.mpr ⟨by linarith, ?_⟩
calc 1 + x
_ ≤ 1 + x + (x / 2) ^ 2 := le_add_of_nonneg_right <| sq_nonneg _
_ = _ := by ring
|
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Kevin Buzzard, Jujian Zhang
-/
import Mathlib.Algebra.DirectSum.Algebra
import Mathlib.Algebra.DirectSum.Decomposition
import Mathlib.Algebra.DirectSum.Internal
import Mathli... | Mathlib/RingTheory/GradedAlgebra/Basic.lean | 241 | 243 | theorem GradedAlgebra.proj_recompose (a : ⨁ i, 𝒜 i) (i : ι) :
GradedAlgebra.proj 𝒜 i ((decompose 𝒜).symm a) = (decompose 𝒜).symm (of _ i (a i)) := by |
rw [GradedAlgebra.proj_apply, decompose_symm_of, Equiv.apply_symm_apply]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Topology.Order.Basic
import Mathlib.Data.Set.Pointwise.Basic
/-!
# Neighborhoods to the left and to the right on an ... | Mathlib/Topology/Order/LeftRightNhds.lean | 376 | 379 | theorem nhds_basis_Ioo_pos [NoMaxOrder α] (a : α) :
(𝓝 a).HasBasis (fun ε : α => (0 : α) < ε) fun ε => Ioo (a - ε) (a + ε) := by |
convert nhds_basis_abs_sub_lt a
simp only [Ioo, abs_lt, ← sub_lt_iff_lt_add, neg_lt_sub_iff_lt_add, sub_lt_comm]
|
/-
Copyright (c) 2021 Noam Atar. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Noam Atar
-/
import Mathlib.Order.Ideal
import Mathlib.Order.PFilter
#align_import order.prime_ideal from "leanprover-community/mathlib"@"740acc0e6f9adf4423f92a485d0456fc271482da"
/-!
# P... | Mathlib/Order/PrimeIdeal.lean | 231 | 235 | theorem _root_.Order.Ideal.PrimePair.F_isPrime (IF : Ideal.PrimePair P) : IsPrime IF.F :=
{
compl_ideal := by |
rw [IF.compl_F_eq_I]
exact IF.I.isIdeal }
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.NthRewrite
#align_import data.nat.gcd.... | Mathlib/Data/Nat/GCD/Basic.lean | 211 | 212 | theorem coprime_add_mul_left_left (m n k : ℕ) : Coprime (m + n * k) n ↔ Coprime m n := by |
rw [Coprime, Coprime, gcd_add_mul_left_left]
|
/-
Copyright (c) 2019 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.CharP.ExpChar
import Mathlib.Algebra.GeomSum
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Algebra.MvPolynomial.Equiv
import Mathlib.RingTh... | Mathlib/RingTheory/Polynomial/Basic.lean | 186 | 191 | theorem degreeLT_succ_eq_degreeLE {n : ℕ} : degreeLT R (n + 1) = degreeLE R n := by |
ext x
by_cases x_zero : x = 0
· simp_rw [x_zero, Submodule.zero_mem]
· rw [mem_degreeLT, mem_degreeLE, ← natDegree_lt_iff_degree_lt (by rwa [ne_eq]),
← natDegree_le_iff_degree_le, Nat.lt_succ]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Data.Finset.Piecewise
import Mathlib.Data.Finset.Preimage
#align_import algebra.big_operators.basic from "leanp... | Mathlib/Algebra/BigOperators/Group/Finset.lean | 1,036 | 1,052 | theorem prod_eq_mul {s : Finset α} {f : α → β} (a b : α) (hn : a ≠ b)
(h₀ : ∀ c ∈ s, c ≠ a ∧ c ≠ b → f c = 1) (ha : a ∉ s → f a = 1) (hb : b ∉ s → f b = 1) :
∏ x ∈ s, f x = f a * f b := by |
haveI := Classical.decEq α; by_cases h₁ : a ∈ s <;> by_cases h₂ : b ∈ s
· exact prod_eq_mul_of_mem a b h₁ h₂ hn h₀
· rw [hb h₂, mul_one]
apply prod_eq_single_of_mem a h₁
exact fun c hc hca => h₀ c hc ⟨hca, ne_of_mem_of_not_mem hc h₂⟩
· rw [ha h₁, one_mul]
apply prod_eq_single_of_mem b h₂
exact ... |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Fintype.BigOperators
import Mathlib.LinearAlgebra.AffineSpace.AffineMap
import Mathlib.LinearAlgebra.Affine... | Mathlib/LinearAlgebra/AffineSpace/Combination.lean | 504 | 508 | theorem sum_smul_vsub_eq_affineCombination_vsub (w : ι → k) (p₁ p₂ : ι → P) :
(∑ i ∈ s, w i • (p₁ i -ᵥ p₂ i)) =
s.affineCombination k p₁ w -ᵥ s.affineCombination k p₂ w := by |
simp_rw [affineCombination_apply, vadd_vsub_vadd_cancel_right]
exact s.sum_smul_vsub_eq_weightedVSubOfPoint_sub _ _ _ _
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta
-/
import Mathlib.Algebra.Field.Defs
import Mathlib.Algebra.Ring.Int
#align_import data.int.cast.field from "leanprover-community/mathlib"@"acee671f47b8e7972a1eb6f4eed74b... | Mathlib/Data/Int/Cast/Field.lean | 38 | 42 | theorem cast_div [DivisionRing α] {m n : ℤ} (n_dvd : n ∣ m) (hn : (n : α) ≠ 0) :
((m / n : ℤ) : α) = m / n := by |
rcases n_dvd with ⟨k, rfl⟩
have : n ≠ 0 := by rintro rfl; simp at hn
rw [Int.mul_ediv_cancel_left _ this, mul_comm n, Int.cast_mul, mul_div_cancel_right₀ _ hn]
|
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Rémy Degenne
-/
import Mathlib.Probability.Process.Filtration
import Mathlib.Topology.Instances.Discrete
#align_import probability.process.adapted from "leanprover-community... | Mathlib/Probability/Process/Adapted.lean | 99 | 105 | theorem Filtration.adapted_natural [MetrizableSpace β] [mβ : MeasurableSpace β] [BorelSpace β]
{u : ι → Ω → β} (hum : ∀ i, StronglyMeasurable[m] (u i)) :
Adapted (Filtration.natural u hum) u := by |
intro i
refine StronglyMeasurable.mono ?_ (le_iSup₂_of_le i (le_refl i) le_rfl)
rw [stronglyMeasurable_iff_measurable_separable]
exact ⟨measurable_iff_comap_le.2 le_rfl, (hum i).isSeparable_range⟩
|
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTh... | Mathlib/MeasureTheory/Function/LpSpace.lean | 1,165 | 1,173 | theorem add_compLp (L L' : E →L[𝕜] F) (f : Lp E p μ) :
(L + L').compLp f = L.compLp f + L'.compLp f := by |
ext1
refine (coeFn_compLp' (L + L') f).trans ?_
refine EventuallyEq.trans ?_ (Lp.coeFn_add _ _).symm
refine
EventuallyEq.trans ?_ (EventuallyEq.add (L.coeFn_compLp' f).symm (L'.coeFn_compLp' f).symm)
filter_upwards with x
rw [coe_add', Pi.add_def]
|
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.ExpDeriv
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Analysis.InnerProductSpace.... | Mathlib/Analysis/Fourier/AddCircle.lean | 428 | 439 | theorem tsum_sq_fourierCoeff (f : Lp ℂ 2 <| @haarAddCircle T hT) :
∑' i : ℤ, ‖fourierCoeff f i‖ ^ 2 = ∫ t : AddCircle T, ‖f t‖ ^ 2 ∂haarAddCircle := by |
simp_rw [← fourierBasis_repr]
have H₁ : ‖fourierBasis.repr f‖ ^ 2 = ∑' i, ‖fourierBasis.repr f i‖ ^ 2 := by
apply_mod_cast lp.norm_rpow_eq_tsum ?_ (fourierBasis.repr f)
norm_num
have H₂ : ‖fourierBasis.repr f‖ ^ 2 = ‖f‖ ^ 2 := by simp
have H₃ := congr_arg RCLike.re (@L2.inner_def (AddCircle T) ℂ ℂ _ _ ... |
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Module.Defs
#align_import group_theory.subgroup.saturated from "leanprover-community/mathlib"@"f7f... | Mathlib/GroupTheory/Subgroup/Saturated.lean | 42 | 56 | theorem saturated_iff_zpow {H : Subgroup G} :
Saturated H ↔ ∀ (n : ℤ) (g : G), g ^ n ∈ H → n = 0 ∨ g ∈ H := by |
constructor
· intros hH n g hgn
induction' n with n n
· simp only [Int.natCast_eq_zero, Int.ofNat_eq_coe, zpow_natCast] at hgn ⊢
exact hH hgn
· suffices g ^ (n + 1) ∈ H by
refine (hH this).imp ?_ id
simp only [IsEmpty.forall_iff, Nat.succ_ne_zero]
simpa only [inv_mem_iff, zp... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Submodule.EqLocus
import Mathlib.Algebra.Module.Subm... | Mathlib/LinearAlgebra/Span.lean | 510 | 512 | theorem span_zero_singleton : (R ∙ (0 : M)) = ⊥ := by |
ext
simp [mem_span_singleton, eq_comm]
|
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.List.Defs
im... | Mathlib/Data/List/Basic.lean | 490 | 490 | theorem mem_pure (x y : α) : x ∈ (pure y : List α) ↔ x = y := by | simp
|
/-
Copyright (c) 2021 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Analysis.Convex.Topology
import Mathlib.Analysis.NormedSpace.Pointwise
import Mathlib.Analysis.Seminorm
import Mathlib.Analysis... | Mathlib/Analysis/Convex/Gauge.lean | 358 | 364 | theorem gauge_eq_zero (hs : Absorbent ℝ s) (hb : Bornology.IsVonNBounded ℝ s) :
gauge s x = 0 ↔ x = 0 := by |
refine ⟨fun h₀ ↦ by_contra fun (hne : x ≠ 0) ↦ ?_, fun h ↦ h.symm ▸ gauge_zero⟩
have : {x}ᶜ ∈ comap (gauge s) (𝓝 0) :=
comap_gauge_nhds_zero_le hs hb (isOpen_compl_singleton.mem_nhds hne.symm)
rcases ((nhds_basis_zero_abs_sub_lt _).comap _).mem_iff.1 this with ⟨r, hr₀, hr⟩
exact hr (by simpa [h₀]) rfl
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Set.Function
import Mathlib.Logic.Relation
import Mathlib.Logic.Pairwise
#align_import data.set.pairwise.basic from "leanprover-community/mathlib... | Mathlib/Data/Set/Pairwise/Basic.lean | 41 | 42 | theorem pairwise_on_bool (hr : Symmetric r) {a b : α} :
Pairwise (r on fun c => cond c a b) ↔ r a b := by | simpa [Pairwise, Function.onFun] using @hr a b
|
/-
Copyright (c) 2021 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.Group.Subgroup.Actions
import Mathlib.Algebra.Order.Module.Algebra
import Mathlib.LinearAlgebra.LinearIndependent
import Mathlib.Algebra.Ring.Subri... | Mathlib/LinearAlgebra/Ray.lean | 738 | 742 | theorem exists_nonneg_right_iff_sameRay (hy : y ≠ 0) :
(∃ r : R, 0 ≤ r ∧ x = r • y) ↔ SameRay R x y := by |
rw [SameRay.sameRay_comm]
simp_rw [eq_comm (a := x)]
exact exists_nonneg_left_iff_sameRay (R := R) hy
|
/-
Copyright (c) 2022 Wrenna Robson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wrenna Robson
-/
import Mathlib.Analysis.Normed.Group.Basic
#align_import information_theory.hamming from "leanprover-community/mathlib"@"17ef379e997badd73e5eabb4d38f11919ab3c4b3"
/-!... | Mathlib/InformationTheory/Hamming.lean | 122 | 123 | theorem hammingDist_lt_one {x y : ∀ i, β i} : hammingDist x y < 1 ↔ x = y := by |
rw [Nat.lt_one_iff, hammingDist_eq_zero]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Topology.Order.MonotoneContinuity
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.E... | Mathlib/Topology/Instances/ENNReal.lean | 1,161 | 1,165 | theorem hasSum_iff_tendsto_nat {f : ℕ → ℝ≥0} {r : ℝ≥0} :
HasSum f r ↔ Tendsto (fun n : ℕ => ∑ i ∈ Finset.range n, f i) atTop (𝓝 r) := by |
rw [← ENNReal.hasSum_coe, ENNReal.hasSum_iff_tendsto_nat]
simp only [← ENNReal.coe_finset_sum]
exact ENNReal.tendsto_coe
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Aaron Anderson, Yakov Pechersky
-/
import Mathlib.Algebra.Group.Commute.Basic
import Mathlib.Data.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
#align_import group_th... | Mathlib/GroupTheory/Perm/Support.lean | 316 | 316 | theorem support_one : (1 : Perm α).support = ∅ := by | rw [support_eq_empty_iff]
|
/-
Copyright (c) 2021 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 793 | 794 | theorem length_darts {u v : V} (p : G.Walk u v) : p.darts.length = p.length := by |
induction p <;> simp [*]
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel
-/
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shape... | Mathlib/CategoryTheory/Limits/Shapes/Images.lean | 1,053 | 1,068 | theorem hasStrongEpiMonoFactorisations_imp_of_isEquivalence (F : C ⥤ D) [IsEquivalence F]
[h : HasStrongEpiMonoFactorisations C] : HasStrongEpiMonoFactorisations D :=
⟨fun {X} {Y} f => by
let em : StrongEpiMonoFactorisation (F.inv.map f) :=
(HasStrongEpiMonoFactorisations.has_fac (F.inv.map f)).some
... |
simp only [asEquivalence_functor, Category.assoc, ← F.map_comp_assoc,
MonoFactorisation.fac, fun_inv_map, id_obj, Iso.inv_hom_id_app, Category.comp_id,
Iso.inv_hom_id_app_assoc] }⟩
|
/-
Copyright (c) 2021 Alena Gusakov, Bhavik Mehta, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alena Gusakov, Bhavik Mehta, Kyle Miller
-/
import Mathlib.Combinatorics.Hall.Finite
import Mathlib.CategoryTheory.CofilteredSystem
import Mathlib.Data.Rel
#... | Mathlib/Combinatorics/Hall/Basic.lean | 123 | 163 | theorem Finset.all_card_le_biUnion_card_iff_exists_injective {ι : Type u} {α : Type v}
[DecidableEq α] (t : ι → Finset α) :
(∀ s : Finset ι, s.card ≤ (s.biUnion t).card) ↔
∃ f : ι → α, Function.Injective f ∧ ∀ x, f x ∈ t x := by |
constructor
· intro h
-- Set up the functor
haveI : ∀ ι' : (Finset ι)ᵒᵖ, Nonempty ((hallMatchingsFunctor t).obj ι') := fun ι' =>
hallMatchingsOn.nonempty t h ι'.unop
classical
haveI : ∀ ι' : (Finset ι)ᵒᵖ, Finite ((hallMatchingsFunctor t).obj ι') := by
intro ι'
rw [hallMatchi... |
/-
Copyright (c) 2023 Dagur Asgeirsson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Dagur Asgeirsson
-/
import Mathlib.Algebra.Category.ModuleCat.Free
import Mathlib.Topology.Category.Profinite.CofilteredLimit
import Mathlib.Topology.Category.Profinite.Product
impor... | Mathlib/Topology/Category/Profinite/Nobeling.lean | 1,344 | 1,349 | theorem succ_exact :
(ShortComplex.mk (ModuleCat.ofHom (πs C o)) (ModuleCat.ofHom (Linear_CC' C hsC ho))
(by ext; apply CC_comp_zero)).Exact := by |
rw [ShortComplex.moduleCat_exact_iff]
intro f
exact CC_exact C hC hsC ho
|
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.RingTheory.PowerBasis
#align_import ring_theory.is_adjoin... | Mathlib/RingTheory/IsAdjoinRoot.lean | 253 | 257 | theorem apply_eq_lift (h : IsAdjoinRoot S f) (g : S →+* T) (hmap : ∀ a, g (algebraMap R S a) = i a)
(hroot : g h.root = x) (a : S) : g a = h.lift i x hx a := by |
rw [← h.map_repr a, Polynomial.as_sum_range_C_mul_X_pow (h.repr a)]
simp only [map_sum, map_mul, map_pow, h.map_X, hroot, ← h.algebraMap_apply, hmap, lift_root,
lift_algebraMap]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Dia... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 193 | 197 | theorem volume_le_diam (s : Set ℝ) : volume s ≤ EMetric.diam s := by |
by_cases hs : Bornology.IsBounded s
· rw [Real.ediam_eq hs, ← volume_Icc]
exact volume.mono hs.subset_Icc_sInf_sSup
· rw [Metric.ediam_of_unbounded hs]; exact le_top
|
/-
Copyright (c) 2021 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Limits.Pullbacks
import Mathlib.Geometry.RingedSpace.LocallyRingedSpace
#align_import algebraic_geometry.open_immersion.basic from ... | Mathlib/Geometry/RingedSpace/OpenImmersion.lean | 200 | 211 | theorem inv_naturality {U V : (Opens X)ᵒᵖ} (i : U ⟶ V) :
X.presheaf.map i ≫ H.invApp (unop V) =
H.invApp (unop U) ≫ Y.presheaf.map (H.openFunctor.op.map i) := by |
simp only [invApp, ← Category.assoc]
rw [IsIso.comp_inv_eq]
-- Porting note: `simp` can't pick up `f.c.naturality`
-- See https://github.com/leanprover-community/mathlib4/issues/5026
simp only [Category.assoc, ← X.presheaf.map_comp]
erw [f.c.naturality]
simp only [IsIso.inv_hom_id_assoc, ← X.presheaf.map... |
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineEquiv
#align_import linear_algebra.affine_space.affine_subspace from "leanprover-community/mathlib"@"e96bdfbd1e8c98a09ff75... | Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | 1,825 | 1,830 | theorem Parallel.trans {s₁ s₂ s₃ : AffineSubspace k P} (h₁₂ : s₁ ∥ s₂) (h₂₃ : s₂ ∥ s₃) :
s₁ ∥ s₃ := by |
rcases h₁₂ with ⟨v₁₂, rfl⟩
rcases h₂₃ with ⟨v₂₃, rfl⟩
refine ⟨v₂₃ + v₁₂, ?_⟩
rw [map_map, ← coe_trans_to_affineMap, ← constVAdd_add]
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.MvPolynomial.Counit
import Mathlib.Algebra.MvPolynomial.Invertible
import Mathlib.RingTheory.WittVector.Defs
#align_import ri... | Mathlib/RingTheory/WittVector/Basic.lean | 123 | 123 | theorem zsmul (z : ℤ) (x : WittVector p R) : mapFun f (z • x) = z • mapFun f x := by | map_fun_tac
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConvergenceInMeasure
import Mathlib.MeasureTheory.Function.L1Space
#align_import measure_theory.function.uniform_integrable from "lea... | Mathlib/MeasureTheory/Function/UniformIntegrable.lean | 635 | 696 | theorem unifIntegrable_of' (hp : 1 ≤ p) (hp' : p ≠ ∞) {f : ι → α → β}
(hf : ∀ i, StronglyMeasurable (f i))
(h : ∀ ε : ℝ, 0 < ε → ∃ C : ℝ≥0, 0 < C ∧
∀ i, snorm ({ x | C ≤ ‖f i x‖₊ }.indicator (f i)) p μ ≤ ENNReal.ofReal ε) :
UnifIntegrable f p μ := by |
have hpzero := (lt_of_lt_of_le zero_lt_one hp).ne.symm
by_cases hμ : μ Set.univ = 0
· rw [Measure.measure_univ_eq_zero] at hμ
exact hμ.symm ▸ unifIntegrable_zero_meas
intro ε hε
obtain ⟨C, hCpos, hC⟩ := h (ε / 2) (half_pos hε)
refine ⟨(ε / (2 * C)) ^ ENNReal.toReal p,
Real.rpow_pos_of_pos (div_pos ... |
/-
Copyright (c) 2021 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 618 | 620 | theorem end_mem_tail_support_of_ne {u v : V} (h : u ≠ v) (p : G.Walk u v) : v ∈ p.support.tail := by |
obtain ⟨_, _, _, rfl⟩ := exists_eq_cons_of_ne h p
simp
|
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Topology.ContinuousFunction.Algebra
import Mathlib.Topology.Compac... | Mathlib/Topology/PartitionOfUnity.lean | 440 | 444 | theorem exists_isSubordinate [NormalSpace X] [ParacompactSpace X] (hs : IsClosed s) (U : ι → Set X)
(ho : ∀ i, IsOpen (U i)) (hU : s ⊆ ⋃ i, U i) : ∃ f : BumpCovering ι X s, f.IsSubordinate U := by |
rcases precise_refinement_set hs _ ho hU with ⟨V, hVo, hsV, hVf, hVU⟩
rcases exists_isSubordinate_of_locallyFinite hs V hVo hVf hsV with ⟨f, hf⟩
exact ⟨f, hf.mono hVU⟩
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Data.Prod.PProd
import Mathlib.Data.Set.Countable
import Mathlib.Order.Filter.Prod
import Mathlib.Ord... | Mathlib/Order/Filter/Bases.lean | 1,174 | 1,177 | theorem isCountablyGenerated_seq [Countable ι'] (x : ι' → Set α) :
IsCountablyGenerated (⨅ i, 𝓟 (x i)) := by |
use range x, countable_range x
rw [generate_eq_biInf, iInf_range]
|
/-
Copyright (c) 2021 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno
-/
import Mathlib.CategoryTheory.NatIso
#align_import category_theory.bicategory.basic from "leanprover-community/mathlib"@"4c19a16e4b705bf135cf9a80ac18fcc99c438514"
/-!
# B... | Mathlib/CategoryTheory/Bicategory/Basic.lean | 274 | 277 | theorem pentagon_hom_inv_inv_inv_inv (f : a ⟶ b) (g : b ⟶ c) (h : c ⟶ d) (i : d ⟶ e) :
f ◁ (α_ g h i).hom ≫ (α_ f g (h ≫ i)).inv ≫ (α_ (f ≫ g) h i).inv =
(α_ f (g ≫ h) i).inv ≫ (α_ f g h).inv ▷ i := by |
simp [← cancel_epi (f ◁ (α_ g h i).inv)]
|
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Analysis.NormedSpace.Exponential
import Mathlib.Analysis.NormedSpace.ProdLp
import Mathlib.Topology.Instances.TrivSqZeroExt
#align_import analysis.normed_sp... | Mathlib/Analysis/NormedSpace/TrivSqZeroExt.lean | 135 | 137 | theorem exp_inr (m : M) : exp 𝕜 (inr m : tsze R M) = 1 + inr m := by |
rw [exp_def_of_smul_comm, snd_inr, fst_inr, exp_zero, one_smul, inl_one]
rw [snd_inr, fst_inr, MulOpposite.op_zero, zero_smul, zero_smul]
|
/-
Copyright (c) 2022 Jake Levinson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jake Levinson
-/
import Mathlib.Combinatorics.Young.YoungDiagram
#align_import combinatorics.young.semistandard_tableau from "leanprover-community/mathlib"@"b363547b3113d350d053abdf288... | Mathlib/Combinatorics/Young/SemistandardTableau.lean | 129 | 133 | theorem row_weak_of_le {μ : YoungDiagram} (T : SemistandardYoungTableau μ) {i j1 j2 : ℕ}
(hj : j1 ≤ j2) (cell : (i, j2) ∈ μ) : T i j1 ≤ T i j2 := by |
cases' eq_or_lt_of_le hj with h h
· rw [h]
· exact T.row_weak h cell
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Topology.Maps
import Mathlib.Topology.NhdsSet
#align_import topology.constructions from "leanprover-community/mathlib"... | Mathlib/Topology/Constructions.lean | 1,717 | 1,720 | theorem openEmbedding_sigma_map {f₁ : ι → κ} {f₂ : ∀ i, σ i → τ (f₁ i)} (h : Injective f₁) :
OpenEmbedding (Sigma.map f₁ f₂) ↔ ∀ i, OpenEmbedding (f₂ i) := by |
simp only [openEmbedding_iff_embedding_open, isOpenMap_sigma_map, embedding_sigma_map h,
forall_and]
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker
-/
import Mathlib.Algebra.Group.Even
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.GroupWithZero.Hom
import Mathlib.Algebra.Gr... | Mathlib/Algebra/Associated.lean | 148 | 151 | theorem Prime.pow_dvd_of_dvd_mul_right [CancelCommMonoidWithZero α] {p a b : α} (hp : Prime p)
(n : ℕ) (h : ¬p ∣ b) (h' : p ^ n ∣ a * b) : p ^ n ∣ a := by |
rw [mul_comm] at h'
exact hp.pow_dvd_of_dvd_mul_left n h h'
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Logic.Pairwise
import Mathlib.Order.CompleteBooleanAlgebra
import Mathlib.Order.Directed
import Mathli... | Mathlib/Data/Set/Lattice.lean | 963 | 964 | theorem biUnion_insert (a : α) (s : Set α) (t : α → Set β) :
⋃ x ∈ insert a s, t x = t a ∪ ⋃ x ∈ s, t x := by | simp
|
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Computability.Halting
import Mathlib.Computability.TuringMachine
import Mathlib.Data.Num.Lemmas
import Mathlib.Tactic.DeriveFintype
#align_import comp... | Mathlib/Computability/TMToPartrec.lean | 1,919 | 1,920 | theorem supports_union {K₁ K₂ S} : Supports (K₁ ∪ K₂) S ↔ Supports K₁ S ∧ Supports K₂ S := by |
simp [Supports, or_imp, forall_and]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Jeremy Avigad, Simon Hudon
-/
import Mathlib.Data.Set.Subsingleton
import Mathlib.Logic.Equiv.Defs
import Mathlib.Algebra.Group.Defs
#align_import data.part from "lean... | Mathlib/Data/Part.lean | 843 | 844 | theorem union_mem_union [Union α] (a b : Part α) (ma mb : α) (ha : ma ∈ a) (hb : mb ∈ b) :
ma ∪ mb ∈ a ∪ b := by | simp [union_def]; aesop
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Scott Morrison
-/
import Mathlib.Algebra.Group.Indicator
import Mathlib.Algebra.Group.Submonoid.Basic
import Mathlib.Data.Set.Finite
#align_import data.finsupp.defs fr... | Mathlib/Data/Finsupp/Defs.lean | 185 | 185 | theorem coe_eq_zero {f : α →₀ M} : (f : α → M) = 0 ↔ f = 0 := by | rw [← coe_zero, DFunLike.coe_fn_eq]
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Logic.Basic
import Mathlib.Tactic.Positivity.Basic
#align_import algebra.order.hom.basic from "leanprover-community/mathlib"@"28aa996fc6fb4317f0083c4e6daf... | Mathlib/Algebra/Order/Hom/Basic.lean | 264 | 267 | theorem abs_sub_map_le_div [Group α] [LinearOrderedAddCommGroup β] [GroupSeminormClass F α β]
(f : F) (x y : α) : |f x - f y| ≤ f (x / y) := by |
rw [abs_sub_le_iff, sub_le_iff_le_add', sub_le_iff_le_add']
exact ⟨le_map_add_map_div _ _ _, le_map_add_map_div' _ _ _⟩
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.FractionalIdeal.Basic
#align_import ring_theory.fractional_ideal from "leanprover... | Mathlib/RingTheory/FractionalIdeal/Operations.lean | 283 | 286 | theorem canonicalEquiv_self : canonicalEquiv S P P = RingEquiv.refl _ := by |
rw [← canonicalEquiv_trans_canonicalEquiv S P P]
convert (canonicalEquiv S P P).symm_trans_self
exact (canonicalEquiv_symm S P P).symm
|
/-
Copyright (c) 2024 Jeremy Tan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Tan
-/
import Mathlib.Data.Int.Interval
import Mathlib.Data.Int.ModEq
import Mathlib.Data.Nat.Count
import Mathlib.Data.Rat.Floor
import Mathlib.Order.Interval.Finset.Nat
/-!
# Cou... | Mathlib/Data/Int/CardIntervalMod.lean | 42 | 44 | theorem Ico_filter_dvd_card : ((Ico a b).filter (r ∣ ·)).card =
max (⌈b / (r : ℚ)⌉ - ⌈a / (r : ℚ)⌉) 0 := by |
rw [Ico_filter_dvd_eq _ _ hr, card_map, card_Ico, toNat_eq_max]
|
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.EMetricSpace.Basic
import Mathlib.Topology.Bornology.Constructions
imp... | Mathlib/Topology/MetricSpace/PseudoMetric.lean | 580 | 580 | theorem mem_ball_comm : x ∈ ball y ε ↔ y ∈ ball x ε := by | rw [mem_ball', mem_ball]
|
/-
Copyright (c) 2021 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson
-/
import Mathlib.Algebra.Field.Opposite
import Mathlib.Algebra.Group.Subgroup.ZPowers
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Rin... | Mathlib/Algebra/Periodic.lean | 509 | 512 | theorem Antiperiodic.nat_mul_sub_eq [Ring α] [Ring β] (h : Antiperiodic f c) (n : ℕ) :
f (n * c - x) = (-1) ^ n * f (-x) := by |
simpa only [nsmul_eq_mul, zsmul_eq_mul, Int.cast_pow, Int.cast_neg,
Int.cast_one] using h.nsmul_sub_eq n
|
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