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 |
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/-
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.Data.Fin.Fin2
import Mathlib.Init.Logic
import Mathlib.Mathport.Notation
import Mathlib.Tactic.TypeStar
#align_import data.vector3 from "leanprover-co... | Mathlib/Data/Vector3.lean | 189 | 190 | theorem insert_fz (a : α) (v : Vector3 α n) : insert a v fz = a :: v := by |
refine funext fun j => j.cases' ?_ ?_ <;> intros <;> rfl
|
/-
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.Data.ENNReal.Real
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Topol... | Mathlib/Topology/EMetricSpace/Basic.lean | 882 | 889 | theorem secondCountable_of_almost_dense_set
(hs : ∀ ε > 0, ∃ t : Set α, t.Countable ∧ ⋃ x ∈ t, closedBall x ε = univ) :
SecondCountableTopology α := by |
suffices SeparableSpace α from UniformSpace.secondCountable_of_separable α
have : ∀ ε > 0, ∃ t : Set α, Set.Countable t ∧ univ ⊆ ⋃ x ∈ t, closedBall x ε := by
simpa only [univ_subset_iff] using hs
rcases subset_countable_closure_of_almost_dense_set (univ : Set α) this with ⟨t, -, htc, ht⟩
exact ⟨⟨t, htc, f... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Operations
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520... | Mathlib/Data/ENNReal/Inv.lean | 588 | 592 | theorem coe_zpow (hr : r ≠ 0) (n : ℤ) : (↑(r ^ n) : ℝ≥0∞) = (r : ℝ≥0∞) ^ n := by |
cases' n with n n
· simp only [Int.ofNat_eq_coe, coe_pow, zpow_natCast]
· have : r ^ n.succ ≠ 0 := pow_ne_zero (n + 1) hr
simp only [zpow_negSucc, coe_inv this, coe_pow]
|
/-
Copyright (c) 2022 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno
-/
import Mathlib.CategoryTheory.Bicategory.Functor.Oplax
#align_import category_theory.bicategory.natural_transformation from "leanprover-community/mathlib"@"4ff75f5b8502275... | Mathlib/CategoryTheory/Bicategory/NaturalTransformation.lean | 111 | 114 | theorem whiskerRight_naturality_naturality {f g : a ⟶ b} (β : f ⟶ g) (h : G.obj b ⟶ a') :
F.map₂ β ▷ η.app b ▷ h ≫ η.naturality g ▷ h =
η.naturality f ▷ h ≫ (α_ _ _ _).hom ≫ η.app a ◁ G.map₂ β ▷ h ≫ (α_ _ _ _).inv := by |
rw [← comp_whiskerRight, naturality_naturality, comp_whiskerRight, whisker_assoc]
|
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.Module.LinearMap.Basic
import Mathlib.RingTheory.HahnSeries.Basic
#align_import ring_theory.hahn_series from "leanprover-community/mathlib"@"a... | Mathlib/RingTheory/HahnSeries/Addition.lean | 176 | 179 | theorem order_neg [Zero Γ] {f : HahnSeries Γ R} : (-f).order = f.order := by |
by_cases hf : f = 0
· simp only [hf, neg_zero]
simp only [order, support_neg, neg_eq_zero]
|
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Measure.MeasureSpace
import Mathlib.MeasureTheory.Measure.Regular
import Mathlib.Topology.Sets.Compacts
#align_import measure_theory... | Mathlib/MeasureTheory/Measure/Content.lean | 289 | 293 | theorem outerMeasure_exists_compact {U : Opens G} (hU : μ.outerMeasure U ≠ ∞) {ε : ℝ≥0}
(hε : ε ≠ 0) : ∃ K : Compacts G, (K : Set G) ⊆ U ∧ μ.outerMeasure U ≤ μ.outerMeasure K + ε := by |
rw [μ.outerMeasure_opens] at hU ⊢
rcases μ.innerContent_exists_compact hU hε with ⟨K, h1K, h2K⟩
exact ⟨K, h1K, le_trans h2K <| add_le_add_right (μ.le_outerMeasure_compacts K) _⟩
|
/-
Copyright (c) 2020 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.List.Basic
/-!
# Properties of `List.reduceOption`
In this file we prove basic lemmas about `List.reduceOption`.
-/
namespace List
variable ... | Mathlib/Data/List/ReduceOption.lean | 55 | 57 | theorem length_eq_reduceOption_length_add_filter_none {l : List (Option α)} :
l.length = l.reduceOption.length + (l.filter Option.isNone).length := by |
simp_rw [reduceOption_length_eq, l.length_eq_length_filter_add Option.isSome, Option.bnot_isSome]
|
/-
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.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.GroupTh... | Mathlib/RingTheory/Localization/Basic.lean | 1,080 | 1,081 | theorem mk_eq_mk'_apply (x y) : mk x y = IsLocalization.mk' (Localization M) x y := by |
rw [mk_eq_monoidOf_mk'_apply, mk', toLocalizationMap_eq_monoidOf]
|
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Simon Hudon
-/
import Mathlib.Data.Fin.Fin2
import Mathlib.Logic.Function.Basic
import Mathlib.Tactic.Common
#align_import data.typevec from "leanprover-... | Mathlib/Data/TypeVec.lean | 770 | 777 | theorem subtypeVal_diagSub {α : TypeVec n} : subtypeVal (repeatEq α) ⊚ diagSub = prod.diag := by |
ext i x
induction' i with _ _ _ i_ih
· simp [comp, diagSub, subtypeVal, prod.diag]
· simp [prod.diag]
simp? [comp, diagSub, subtypeVal, prod.diag] at * says
simp only [comp, subtypeVal, diagSub] at *
apply @i_ih (drop _)
|
/-
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.LinearAlgebra.Ray
import Mathlib.LinearAlgebra.Determinant
#align_import linear_algebra.orientation from "leanprover-community/mathlib"@"0c1d80f5a86b36c1d... | Mathlib/LinearAlgebra/Orientation.lean | 249 | 254 | theorem orientation_eq_iff_det_pos (e₁ e₂ : Basis ι R M) :
e₁.orientation = e₂.orientation ↔ 0 < e₁.det e₂ :=
calc
e₁.orientation = e₂.orientation ↔ SameRay R e₁.det e₂.det := ray_eq_iff _ _
_ ↔ SameRay R (e₁.det e₂ • e₂.det) e₂.det := by | rw [← e₁.det.eq_smul_basis_det e₂]
_ ↔ 0 < e₁.det e₂ := sameRay_smul_left_iff_of_ne e₂.det_ne_zero (e₁.isUnit_det e₂).ne_zero
|
/-
Copyright (c) 2018 Andreas Swerdlow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andreas Swerdlow
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Deprecated.Subring
#align_import deprecated.subfield from "leanprover-community/mathlib"@"bd9851ca476957ea4549e... | Mathlib/Deprecated/Subfield.lean | 93 | 99 | theorem closure.isSubmonoid : IsSubmonoid (closure S) :=
{ mul_mem := by |
rintro _ _ ⟨p, hp, q, hq, hq0, rfl⟩ ⟨r, hr, s, hs, hs0, rfl⟩
exact ⟨p * r, IsSubmonoid.mul_mem Ring.closure.isSubring.toIsSubmonoid hp hr, q * s,
IsSubmonoid.mul_mem Ring.closure.isSubring.toIsSubmonoid hq hs,
(div_mul_div_comm _ _ _ _).symm⟩
one_mem := ring_closure_subset <| IsSubmonoi... |
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.SplittingField.IsSplittingField
import Mathlib.Algebra.CharP.Algebra
#align_import field_theory.splitting_field.construction from "leanprover-... | Mathlib/FieldTheory/SplittingField/Construction.lean | 191 | 218 | theorem adjoin_rootSet (n : ℕ) :
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (SplittingFieldAux n f)) = ⊤ :=
Nat.recOn (motive := fun n =>
∀ {K : Type u} [Field K],
∀ (f : K[X]) (_hfn : f.natDegree = n),
Algebra.adjoin K (f.rootSet (... | intro h; rw [h] at hfn; cases hfn
have hfn0 : f ≠ 0 := by intro h; rw [h] at hndf; exact hndf rfl
have hmf0 : map (algebraMap K (SplittingFieldAux n.succ f)) f ≠ 0 := map_ne_zero hfn0
rw [rootSet_def, aroots_def]
rw [algebraMap_succ, ← map_map, ← X_sub_C_mul_removeFactor _ hndf, Polynomial.map_mul] at ... |
/-
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
-/
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.LinearAlgebra.Span
#align_import linear_algebra.quotient from... | Mathlib/LinearAlgebra/Quotient.lean | 436 | 442 | theorem mapQ_comp {R₃ M₃ : Type*} [Ring R₃] [AddCommGroup M₃] [Module R₃ M₃] (p₂ : Submodule R₂ M₂)
(p₃ : Submodule R₃ M₃) {τ₂₃ : R₂ →+* R₃} {τ₁₃ : R →+* R₃} [RingHomCompTriple τ₁₂ τ₂₃ τ₁₃]
(f : M →ₛₗ[τ₁₂] M₂) (g : M₂ →ₛₗ[τ₂₃] M₃) (hf : p ≤ p₂.comap f) (hg : p₂ ≤ p₃.comap g)
(h := hf.trans (comap_mono hg)) ... |
ext
simp
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.DirectLimit
import Mathlib.Algebra.CharP.Algebra
import Mathlib.FieldTheory.IsAlgClosed.Basic
import Mathlib.FieldTheory.SplittingField.Construction
#al... | Mathlib/FieldTheory/IsAlgClosed/AlgebraicClosure.lean | 138 | 148 | theorem AdjoinMonic.isIntegral (z : AdjoinMonic k) : IsIntegral k z := by |
let ⟨p, hp⟩ := Ideal.Quotient.mk_surjective z
rw [← hp]
induction p using MvPolynomial.induction_on generalizing z with
| h_C => exact isIntegral_algebraMap
| h_add _ _ ha hb => exact (ha _ rfl).add (hb _ rfl)
| h_X p f ih =>
refine @IsIntegral.mul k _ _ _ _ _ (Ideal.Quotient.mk (maxIdeal k) _)... |
/-
Copyright (c) 2019 Gabriel Ebner. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Gabriel Ebner, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.FDeriv.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
#align_import analysis.calculus.deriv.bas... | Mathlib/Analysis/Calculus/Deriv/Basic.lean | 481 | 481 | theorem deriv_fderiv : smulRight (1 : 𝕜 →L[𝕜] 𝕜) (deriv f x) = fderiv 𝕜 f x := by | simp [deriv]
|
/-
Copyright (c) 2023 Michael Rothgang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Rothgang
-/
import Mathlib.Geometry.Manifold.Diffeomorph
import Mathlib.Topology.IsLocalHomeomorph
/-!
# Local diffeomorphisms between manifolds
In this file, we define `C... | Mathlib/Geometry/Manifold/LocalDiffeomorph.lean | 218 | 222 | theorem IsLocalDiffeomorph.isLocalHomeomorph (hf : IsLocalDiffeomorph I J n f) :
IsLocalHomeomorph f := by |
rw [isLocalHomeomorph_iff_isLocalHomeomorphOn_univ]
rw [isLocalDiffeomorph_iff_isLocalDiffeomorphOn_univ] at hf
exact hf.isLocalHomeomorphOn
|
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Composition
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import probability.kernel.integral_comp_prod from "leanprover-comm... | Mathlib/Probability/Kernel/IntegralCompProd.lean | 158 | 164 | theorem kernel.integral_fn_integral_add ⦃f g : β × γ → E⦄ (F : E → E')
(hf : Integrable f ((κ ⊗ₖ η) a)) (hg : Integrable g ((κ ⊗ₖ η) a)) :
∫ x, F (∫ y, f (x, y) + g (x, y) ∂η (a, x)) ∂κ a =
∫ x, F (∫ y, f (x, y) ∂η (a, x) + ∫ y, g (x, y) ∂η (a, x)) ∂κ a := by |
refine integral_congr_ae ?_
filter_upwards [hf.compProd_mk_left_ae, hg.compProd_mk_left_ae] with _ h2f h2g
simp [integral_add h2f h2g]
|
/-
Copyright (c) 2024 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir
-/
import Mathlib.Data.Setoid.Partition
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.GroupTheory.GroupAction.Pointwise
import Mathlib.Group... | Mathlib/GroupTheory/GroupAction/Blocks.lean | 375 | 380 | theorem IsBlockSystem.of_normal {N : Subgroup G} [N.Normal] :
IsBlockSystem G (Set.range fun a : X => orbit N a) := by |
constructor
· apply IsPartition.of_orbits
· intro b; rintro ⟨a, rfl⟩
exact orbit.isBlock_of_normal a
|
/-
Copyright (c) 2021 Paul Lezeau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Paul Lezeau
-/
import Mathlib.Algebra.IsPrimePow
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.Order.Hom.Bounded
import Mathlib.Algebra.GCDMonoid.Basic
#align_impor... | Mathlib/RingTheory/ChainOfDivisors.lean | 392 | 422 | theorem mem_normalizedFactors_factor_dvd_iso_of_mem_normalizedFactors {m p : M} {n : N} (hm : m ≠ 0)
(hn : n ≠ 0) (hp : p ∈ normalizedFactors m) {d : { l : M // l ∣ m } ≃ { l : N // l ∣ n }}
(hd : ∀ l l', (d l : N) ∣ d l' ↔ (l : M) ∣ (l' : M)) :
↑(d ⟨p, dvd_of_mem_normalizedFactors hp⟩) ∈ normalizedFactors ... |
suffices
Prime (d ⟨associatesEquivOfUniqueUnits (associatesEquivOfUniqueUnits.symm p), by
simp [dvd_of_mem_normalizedFactors hp]⟩ : N) by
simp only [associatesEquivOfUniqueUnits_apply, out_mk, normalize_eq,
associatesEquivOfUniqueUnits_symm_apply] at this
obtain ⟨q, hq, hq'⟩ :=
ex... |
/-
Copyright (c) 2020 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.LinearAlgebra.Projection
import Mathlib.Order.JordanHolder
import Mathlib.Order.CompactlyGenerated.Intervals
... | Mathlib/RingTheory/SimpleModule.lean | 202 | 203 | theorem sSup_simples_eq_top : sSup { m : Submodule R M | IsSimpleModule R m } = ⊤ := by |
simpa only [isSimpleModule_iff_isAtom] using sSup_atoms_eq_top
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.NormedSpace.HomeomorphBall
#align_import analy... | Mathlib/Analysis/InnerProductSpace/Calculus.lean | 179 | 182 | theorem ContDiffAt.dist (hf : ContDiffAt ℝ n f x) (hg : ContDiffAt ℝ n g x) (hne : f x ≠ g x) :
ContDiffAt ℝ n (fun y => dist (f y) (g y)) x := by |
simp only [dist_eq_norm]
exact (hf.sub hg).norm 𝕜 (sub_ne_zero.2 hne)
|
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Data.PFunctor.Univariate.M
#align_import data.qpf.univariate.basic from "leanprover-community/mathlib"@"14b69e9f3c16630440a2cbd46f1ddad0d561dee7"
/-!
... | Mathlib/Data/QPF/Univariate/Basic.lean | 626 | 649 | theorem has_good_supp_iff {α : Type u} (x : F α) :
(∀ p, Liftp p x ↔ ∀ u ∈ supp x, p u) ↔
∃ a f, abs ⟨a, f⟩ = x ∧ ∀ a' f', abs ⟨a', f'⟩ = x → f '' univ ⊆ f' '' univ := by |
constructor
· intro h
have : Liftp (supp x) x := by rw [h]; intro u; exact id
rw [liftp_iff] at this
rcases this with ⟨a, f, xeq, h'⟩
refine ⟨a, f, xeq.symm, ?_⟩
intro a' f' h''
rintro u ⟨i, _, hfi⟩
have : u ∈ supp x := by rw [← hfi]; apply h'
exact (mem_supp x u).mp this _ _ h''
... |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.MeasureTheory.Measure.Typeclasses
import Mathlib.MeasureTheory.Measure.MutuallySingular
import Mathlib.MeasureTheory.MeasurableSpace.CountablyGenerated... | Mathlib/MeasureTheory/Measure/Dirac.lean | 117 | 118 | theorem mem_ae_dirac_iff {a : α} (hs : MeasurableSet s) : s ∈ ae (dirac a) ↔ a ∈ s := by |
by_cases a ∈ s <;> simp [mem_ae_iff, dirac_apply', hs.compl, indicator_apply, *]
|
/-
Copyright (c) 2021 Alex J. Best. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Yaël Dillies
-/
import Mathlib.Algebra.Bounds
import Mathlib.Algebra.Order.Field.Basic -- Porting note: `LinearOrderedField`, etc
import Mathlib.Data.Set.Pointwise.SMul
#a... | Mathlib/Algebra/Order/Pointwise.lean | 265 | 275 | theorem smul_Ici : r • Ici a = Ici (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_le_mul_left hr).mpr a_h_left
· rintro h
use x / r
constructor
· exact (le_div_iff' hr).mpr h
· exact mul_div_cancel₀ _ (ne_of_gt hr)
|
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.MeasureTheory.Integral.Bochner
import Mathlib.MeasureTheory.Measure.GiryMonad
#align_import probability.kernel.basic from "leanprover-community/mathlib"@"... | Mathlib/Probability/Kernel/Basic.lean | 569 | 570 | theorem lintegral_restrict (κ : kernel α β) (hs : MeasurableSet s) (a : α) (f : β → ℝ≥0∞) :
∫⁻ b, f b ∂kernel.restrict κ hs a = ∫⁻ b in s, f b ∂κ a := by | rw [restrict_apply]
|
/-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Computation.Basic
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.comp... | Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean | 263 | 270 | theorem of_s_head_aux (v : K) : (of v).s.get? 0 = (IntFractPair.stream v 1).bind (some ∘ fun p =>
{ a := 1
b := p.b }) := by |
rw [of, IntFractPair.seq1]
simp only [of, Stream'.Seq.map_tail, Stream'.Seq.map, Stream'.Seq.tail, Stream'.Seq.head,
Stream'.Seq.get?, Stream'.map]
rw [← Stream'.get_succ, Stream'.get, Option.map]
split <;> simp_all only [Option.some_bind, Option.none_bind, Function.comp_apply]
|
/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Stephen Morgan, Scott Morrison, Johannes Hölzl, Reid Barton
-/
import Mathlib.CategoryTheory.Category.Init
import Mathlib.Combinatorics.Quiver.Basic
import Mathlib.Tactic.PPWithUniv
im... | Mathlib/CategoryTheory/Category/Basic.lean | 356 | 362 | theorem epi_of_epi {X Y Z : C} (f : X ⟶ Y) (g : Y ⟶ Z) [Epi (f ≫ g)] : Epi g := by |
constructor
intro Z a b w
replace w := congr_arg (fun k => f ≫ k) w
dsimp at w
rw [← Category.assoc, ← Category.assoc] at w
exact (cancel_epi _).1 w
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad, Yury Kudryashov, Patrick Massot
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.... | Mathlib/Order/Filter/AtTopBot.lean | 1,642 | 1,651 | theorem map_atTop_eq_of_gc [SemilatticeSup α] [SemilatticeSup β] {f : α → β} (g : β → α) (b' : β)
(hf : Monotone f) (gc : ∀ a, ∀ b ≥ b', f a ≤ b ↔ a ≤ g b) (hgi : ∀ b ≥ b', b ≤ f (g b)) :
map f atTop = atTop := by |
refine
le_antisymm
(hf.tendsto_atTop_atTop fun b => ⟨g (b ⊔ b'), le_sup_left.trans <| hgi _ le_sup_right⟩) ?_
rw [@map_atTop_eq _ _ ⟨g b'⟩]
refine le_iInf fun a => iInf_le_of_le (f a ⊔ b') <| principal_mono.2 fun b hb => ?_
rw [mem_Ici, sup_le_iff] at hb
exact ⟨g b, (gc _ _ hb.2).1 hb.1, le_antisym... |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Anatole Dedecker
-/
import Mathlib.Analysis.LocallyConvex.BalancedCoreHull
import Mathlib.LinearAlgebra.FreeModule.Finite.Matrix
import Mathlib.Topology.Algebra.Mo... | Mathlib/Topology/Algebra/Module/FiniteDimension.lean | 185 | 192 | theorem LinearMap.continuous_of_nonzero_on_open (l : E →ₗ[𝕜] 𝕜) (s : Set E) (hs₁ : IsOpen s)
(hs₂ : s.Nonempty) (hs₃ : ∀ x ∈ s, l x ≠ 0) : Continuous l := by |
refine l.continuous_of_isClosed_ker (l.isClosed_or_dense_ker.resolve_right fun hl => ?_)
rcases hs₂ with ⟨x, hx⟩
have : x ∈ interior (LinearMap.ker l : Set E)ᶜ := by
rw [mem_interior_iff_mem_nhds]
exact mem_of_superset (hs₁.mem_nhds hx) hs₃
rwa [hl.interior_compl] at this
|
/-
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics
import Mathlib.NumberTheory.Liouville.Basic
import Mathlib.Topology.Instances.Irrational
#align_impo... | Mathlib/NumberTheory/Liouville/LiouvilleWith.lean | 341 | 364 | theorem frequently_exists_num (hx : Liouville x) (n : ℕ) :
∃ᶠ b : ℕ in atTop, ∃ a : ℤ, x ≠ a / b ∧ |x - a / b| < 1 / (b : ℝ) ^ n := by |
refine Classical.not_not.1 fun H => ?_
simp only [Liouville, not_forall, not_exists, not_frequently, not_and, not_lt,
eventually_atTop] at H
rcases H with ⟨N, hN⟩
have : ∀ b > (1 : ℕ), ∀ᶠ m : ℕ in atTop, ∀ a : ℤ, 1 / (b : ℝ) ^ m ≤ |x - a / b| := by
intro b hb
replace hb : (1 : ℝ) < b := Nat.one_lt_... |
/-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Mario Carneiro, Reid Barton, Andrew Yang
-/
import Mathlib.CategoryTheory.Limits.KanExtension
import Mathlib.Topology.Category.TopCat.Opens
import Mathlib.CategoryTheor... | Mathlib/Topology/Sheaves/Presheaf.lean | 154 | 158 | theorem map_restrict {X : TopCat} {C : Type*} [Category C] [ConcreteCategory C]
{F G : X.Presheaf C} (e : F ⟶ G) {U V : Opens X} (h : U ≤ V) (x : F.obj (op V)) :
e.app _ (x |_ U) = e.app _ x |_ U := by |
delta restrictOpen restrict
rw [← comp_apply, NatTrans.naturality, comp_apply]
|
/-
Copyright (c) 2021 David Wärn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Wärn, Joachim Breitner
-/
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.GroupTheory.Congruence.Basic
import Mathlib.GroupTh... | Mathlib/GroupTheory/CoprodI.lean | 280 | 284 | theorem range_eq_iSup {N} [Group N] (f : ∀ i, G i →* N) : (lift f).range = ⨆ i, (f i).range := by |
apply le_antisymm (lift_range_le _ f fun i => le_iSup (fun i => MonoidHom.range (f i)) i)
apply iSup_le _
rintro i _ ⟨x, rfl⟩
exact ⟨of x, by simp only [lift_of]⟩
|
/-
Copyright (c) 2022 Pierre-Alexandre Bazin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pierre-Alexandre Bazin
-/
import Mathlib.Algebra.Module.DedekindDomain
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.Algebra.Module.Projective
import Mathlib.Algeb... | Mathlib/Algebra/Module/PID.lean | 266 | 280 | theorem equiv_free_prod_directSum [h' : Module.Finite R N] :
∃ (n : ℕ) (ι : Type u) (_ : Fintype ι) (p : ι → R) (_ : ∀ i, Irreducible <| p i) (e : ι → ℕ),
Nonempty <| N ≃ₗ[R] (Fin n →₀ R) × ⨁ i : ι, R ⧸ R ∙ p i ^ e i := by |
haveI := isNoetherian_submodule' (torsion R N)
haveI := Module.Finite.of_surjective _ (torsion R N).mkQ_surjective
obtain ⟨I, fI, p, hp, e, ⟨h⟩⟩ :=
equiv_directSum_of_isTorsion.{u, v} (@torsion_isTorsion R N _ _ _)
obtain ⟨n, ⟨g⟩⟩ := @Module.basisOfFiniteTypeTorsionFree' R _ _ _ (N ⧸ torsion R N) _ _ _ _
... |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Fabian Glöckle, Kyle Miller
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModu... | Mathlib/LinearAlgebra/Dual.lean | 1,120 | 1,122 | theorem dualRestrict_comp_dualLift (W : Subspace K V) : W.dualRestrict.comp W.dualLift = 1 := by |
ext φ x
simp
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.CategoryTheory.Adjunction.FullyFaithful
import Mathlib.CategoryTheory.Adjunction.Limits
import Mathlib.CategoryTheory.Limits.Shapes.CommSq
import Mathlib.Cat... | Mathlib/CategoryTheory/Limits/VanKampen.lean | 729 | 755 | theorem isPullback_of_cofan_isVanKampen [HasInitial C] {ι : Type*} {X : ι → C}
{c : Cofan X} (hc : IsVanKampenColimit c) (i j : ι) [DecidableEq ι] :
IsPullback (P := (if j = i then X i else ⊥_ C))
(if h : j = i then eqToHom (if_pos h) else eqToHom (if_neg h) ≫ initial.to (X i))
(if h : j = i then eq... |
refine (hc (Cofan.mk (X i) (f := fun k ↦ if k = i then X i else ⊥_ C)
(fun k ↦ if h : k = i then (eqToHom <| if_pos h) else (eqToHom <| if_neg h) ≫ initial.to _))
(Discrete.natTrans (fun k ↦ if h : k.1 = i then (eqToHom <| (if_pos h).trans
(congr_arg X h.symm)) else (eqToHom <| if_neg h) ≫ initial.to _... |
/-
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 | 1,369 | 1,377 | theorem snorm_exponent_top_lim_eq_essSup_liminf {ι} [Nonempty ι] [LinearOrder ι] {f : ι → α → G}
{f_lim : α → G} (h_lim : ∀ᵐ x : α ∂μ, Tendsto (fun n => f n x) atTop (𝓝 (f_lim x))) :
snorm f_lim ∞ μ = essSup (fun x => atTop.liminf fun m => (‖f m x‖₊ : ℝ≥0∞)) μ := by |
rw [snorm_exponent_top, snormEssSup]
refine essSup_congr_ae (h_lim.mono fun x hx => ?_)
dsimp only
apply (Tendsto.liminf_eq ..).symm
rw [ENNReal.tendsto_coe]
exact (continuous_nnnorm.tendsto (f_lim x)).comp hx
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Joey van Langen, Casper Putz
-/
import Mathlib.FieldTheory.Separable
import Mathlib.RingTheory.IntegralDomain
import Mathlib.Algebra.CharP.Reduced
import Mathlib.Tactic.App... | Mathlib/FieldTheory/Finite/Basic.lean | 179 | 210 | theorem sum_subgroup_pow_eq_zero [CommRing K] [NoZeroDivisors K]
{G : Subgroup Kˣ} [Fintype G] {k : ℕ} (k_pos : k ≠ 0) (k_lt_card_G : k < Fintype.card G) :
∑ x : G, ((x : Kˣ) : K) ^ k = 0 := by |
nontriviality K
have := NoZeroDivisors.to_isDomain K
rcases (exists_pow_ne_one_of_isCyclic k_pos k_lt_card_G) with ⟨a, ha⟩
rw [Finset.sum_eq_multiset_sum]
have h_multiset_map :
Finset.univ.val.map (fun x : G => ((x : Kˣ) : K) ^ k) =
Finset.univ.val.map (fun x : G => ((x : Kˣ) : K) ^ k * ((a : Kˣ) :... |
/-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel, Bhavik Mehta, Andrew Yang, Emily Riehl
-/
import Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks
import Mathlib.CategoryTheory.Limits.Shapes.BinaryPro... | Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean | 2,620 | 2,628 | theorem inl_inl_pushoutAssoc_hom [HasPushout (g₃ ≫ (pushout.inr : X₂ ⟶ Y₁)) g₄]
[HasPushout g₁ (g₂ ≫ (pushout.inl : X₂ ⟶ Y₂))] :
pushout.inl ≫ pushout.inl ≫ (pushoutAssoc g₁ g₂ g₃ g₄).hom = pushout.inl := by |
trans f₁ ≫ l₁
· congr 1
exact
(pushoutPushoutLeftIsPushout g₁ g₂ g₃ g₄).comp_coconePointUniqueUpToIso_hom _
WalkingCospan.left
· exact pushout.inl_desc _ _ _
|
/-
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 | 1,247 | 1,253 | theorem support_update (f : Π₀ i, β i) (i : ι) (b : β i) [Decidable (b = 0)] :
support (f.update i b) = if b = 0 then support (f.erase i) else insert i f.support := by |
ext j
split_ifs with hb
· subst hb
simp [update_eq_erase, support_erase]
· rw [support_update_ne_zero f _ hb]
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.FieldTheory.Finiteness
import Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition
import Mathlib.LinearAlgebra.Dimension.DivisionRing
#align_import... | Mathlib/LinearAlgebra/FiniteDimensional.lean | 294 | 299 | theorem exists_relation_sum_zero_pos_coefficient_of_finrank_succ_lt_card [FiniteDimensional L W]
{t : Finset W} (h : finrank L W + 1 < t.card) :
∃ f : W → L, ∑ e ∈ t, f e • e = 0 ∧ ∑ e ∈ t, f e = 0 ∧ ∃ x ∈ t, 0 < f x := by |
obtain ⟨f, sum, total, nonzero⟩ :=
Module.exists_nontrivial_relation_sum_zero_of_finrank_succ_lt_card h
exact ⟨f, sum, total, exists_pos_of_sum_zero_of_exists_nonzero f total nonzero⟩
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
imp... | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 347 | 349 | theorem rightUnitor_inv_braiding (X : C) : (ρ_ X).inv ≫ (β_ X (𝟙_ C)).hom = (λ_ X).inv := by |
apply (cancel_mono (λ_ X).hom).1
simp only [assoc, braiding_leftUnitor, Iso.inv_hom_id]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Johannes Hölzl, Scott Morrison, Jens Wagemaker
-/
import Mathlib.Algebra.MonoidAlgebra.Degree
import Mathlib.Algebra.Polynomial.Coeff
import Mathlib.Algebra.Polynomial.Mono... | Mathlib/Algebra/Polynomial/Degree/Definitions.lean | 866 | 867 | theorem leadingCoeff_C_mul_X_pow (a : R) (n : ℕ) : leadingCoeff (C a * X ^ n) = a := by |
rw [C_mul_X_pow_eq_monomial, leadingCoeff_monomial]
|
/-
Copyright (c) 2022 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johanes Hölzl, Patrick Massot, Yury Kudryashov, Kevin Wilson, Heather Macbeth
-/
import Mathlib.Order.Filter.Basic
#align_import order.filter.prod from "leanprover-community/mathlib"@... | Mathlib/Order/Filter/Prod.lean | 196 | 199 | theorem Eventually.curry {la : Filter α} {lb : Filter β} {p : α × β → Prop}
(h : ∀ᶠ x in la ×ˢ lb, p x) : ∀ᶠ x in la, ∀ᶠ y in lb, p (x, y) := by |
rcases eventually_prod_iff.1 h with ⟨pa, ha, pb, hb, h⟩
exact ha.mono fun a ha => hb.mono fun b hb => h ha hb
|
/-
Copyright (c) 2021 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Geometry.Manifold.Algebra.Structures
import Mathlib.Geometry.Manifold.BumpFunction
import Mathlib.Topology.MetricSpace.PartitionOfUnity
import ... | Mathlib/Geometry/Manifold/PartitionOfUnity.lean | 713 | 722 | theorem Metric.exists_smooth_forall_closedBall_subset {M} [MetricSpace M] [ChartedSpace H M]
[SmoothManifoldWithCorners I M] [SigmaCompactSpace M] {K : ι → Set M} {U : ι → Set M}
(hK : ∀ i, IsClosed (K i)) (hU : ∀ i, IsOpen (U i)) (hKU : ∀ i, K i ⊆ U i)
(hfin : LocallyFinite K) :
∃ δ : C^∞⟮I, M; 𝓘(ℝ, ℝ... |
rcases Emetric.exists_smooth_forall_closedBall_subset I hK hU hKU hfin with ⟨δ, hδ0, hδ⟩
refine ⟨δ, hδ0, fun i x hx => ?_⟩
rw [← Metric.emetric_closedBall (hδ0 _).le]
exact hδ i x hx
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
... | Mathlib/Algebra/Order/ToIntervalMod.lean | 419 | 421 | theorem toIocMod_add_zsmul (a b : α) (m : ℤ) : toIocMod hp a (b + m • p) = toIocMod hp a b := by |
rw [toIocMod, toIocDiv_add_zsmul, toIocMod, add_smul]
abel
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Interval.Finset.Nat
#align_import data.fin.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd33b5cfeecfc2a27a5a8a29"
/-!
# Finite int... | Mathlib/Order/Interval/Finset/Fin.lean | 253 | 254 | theorem card_fintypeIic : Fintype.card (Set.Iic b) = b + 1 := by |
rw [Fintype.card_ofFinset, card_Iic]
|
/-
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 | 2,133 | 2,141 | theorem mk_le_iff_forall_finset_subset_card_le {α : Type u} {n : ℕ} {t : Set α} :
#t ≤ n ↔ ∀ s : Finset α, (s : Set α) ⊆ t → s.card ≤ n := by |
refine ⟨fun H s hs ↦ by simpa using (mk_le_mk_of_subset hs).trans H, fun H ↦ ?_⟩
apply card_le_of (fun s ↦ ?_)
let u : Finset α := s.image Subtype.val
have : u.card = s.card := Finset.card_image_of_injOn Subtype.coe_injective.injOn
rw [← this]
apply H
simp only [u, Finset.coe_image, image_subset_iff, Sub... |
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.Equivalence
#align_import algebraic_topology.dold_kan.compatibility from "leanprover-community/mathlib"@"32a7e535287f9c73f2e4d2aef306a39190f0b504... | Mathlib/AlgebraicTopology/DoldKan/Compatibility.lean | 215 | 227 | theorem equivalenceCounitIso_eq : (equivalence hF hG).counitIso = equivalenceCounitIso η := by |
ext1; apply NatTrans.ext; ext Y
dsimp [equivalence]
simp only [comp_id, id_comp, Functor.map_comp, equivalence₂CounitIso_eq,
equivalence₂CounitIso_hom_app, assoc, equivalenceCounitIso_hom_app]
simp only [← eB.inverse.map_comp_assoc, ← τ₀_hom_app, hη, τ₁_hom_app]
erw [hF.inv.naturality_assoc, hF.inv.natur... |
/-
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.Geometry.Euclidean.Angle.Oriented.Affine
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
import Mathlib.Tactic.IntervalCases
#a... | Mathlib/Geometry/Euclidean/Triangle.lean | 300 | 308 | theorem angle_add_angle_add_angle_eq_pi {p1 p2 p3 : P} (h2 : p2 ≠ p1) (h3 : p3 ≠ p1) :
∠ p1 p2 p3 + ∠ p2 p3 p1 + ∠ p3 p1 p2 = π := by |
rw [add_assoc, add_comm, add_comm (∠ p2 p3 p1), angle_comm p2 p3 p1]
unfold angle
rw [← angle_neg_neg (p1 -ᵥ p3), ← angle_neg_neg (p1 -ᵥ p2), neg_vsub_eq_vsub_rev,
neg_vsub_eq_vsub_rev, neg_vsub_eq_vsub_rev, neg_vsub_eq_vsub_rev, ←
vsub_sub_vsub_cancel_right p3 p2 p1, ← vsub_sub_vsub_cancel_right p2 p3 p... |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Sébastien Gouëzel, Frédéric Dupuis
-/
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Convex.Uniform
import Mathlib.... | Mathlib/Analysis/InnerProductSpace/Basic.lean | 1,561 | 1,565 | theorem real_inner_div_norm_mul_norm_eq_one_of_ne_zero_of_pos_mul {x : F} {r : ℝ} (hx : x ≠ 0)
(hr : 0 < r) : ⟪x, r • x⟫_ℝ / (‖x‖ * ‖r • x‖) = 1 := by |
rw [real_inner_smul_self_right, norm_smul, Real.norm_eq_abs, ← mul_assoc ‖x‖, mul_comm _ |r|,
mul_assoc, abs_of_nonneg hr.le, div_self]
exact mul_ne_zero hr.ne' (mul_self_ne_zero.2 (norm_ne_zero_iff.2 hx))
|
/-
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.CategoryTheory.Limits.ConeCategory
import Mathlib.CategoryTheory.Limits.FilteredColimitCommutesFiniteLimit
import Mathlib.CategoryTheory.Limits.Preserves.Fil... | Mathlib/CategoryTheory/Functor/Flat.lean | 97 | 106 | theorem flat_of_preservesFiniteLimits [HasFiniteLimits C] (F : C ⥤ D) [PreservesFiniteLimits F] :
RepresentablyFlat F :=
⟨fun X =>
haveI : HasFiniteLimits (StructuredArrow X F) := by |
apply hasFiniteLimits_of_hasFiniteLimits_of_size.{v₁} (StructuredArrow X F)
intro J sJ fJ
constructor
-- Porting note: instance was inferred automatically in Lean 3
infer_instance
IsCofiltered.of_hasFiniteLimits _⟩
|
/-
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.Option
import Mathlib.Analysis.BoxIntegral.Box.Basic
import Mathlib.Data.Set.Pairwise.Lattice
#align_import analysis.box_integr... | Mathlib/Analysis/BoxIntegral/Partition/Basic.lean | 226 | 228 | theorem mem_iUnion : x ∈ π.iUnion ↔ ∃ J ∈ π, x ∈ J := by |
convert Set.mem_iUnion₂
rw [Box.mem_coe, exists_prop]
|
/-
Copyright (c) 2020 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.CliffordAlgebra.Grading
import Mathlib.Algebra.Module.Opposites
#align_import linear_algebra.clifford_algebra.conjugation from "leanprover-com... | Mathlib/LinearAlgebra/CliffordAlgebra/Conjugation.lean | 308 | 310 | theorem evenOdd_comap_reverse (n : ZMod 2) :
(evenOdd Q n).comap (reverse : CliffordAlgebra Q →ₗ[R] CliffordAlgebra Q) = evenOdd Q n := by |
rw [← submodule_map_reverse_eq_comap, evenOdd_map_reverse]
|
/-
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 | 166 | 172 | theorem gauge_lt_eq' (absorbs : Absorbent ℝ s) (a : ℝ) :
{ x | gauge s x < a } = ⋃ (r : ℝ) (_ : 0 < r) (_ : r < a), r • s := by |
ext
simp_rw [mem_setOf, mem_iUnion, exists_prop]
exact
⟨exists_lt_of_gauge_lt absorbs, fun ⟨r, hr₀, hr₁, hx⟩ =>
(gauge_le_of_mem hr₀.le hx).trans_lt hr₁⟩
|
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Star.Basic
import Mathlib.Data.Set.Finite
import Mathlib.Data.Set.Pointwise.Basic
#align_import algebra.star.pointwise from "leanprover-communit... | Mathlib/Algebra/Star/Pointwise.lean | 62 | 62 | theorem star_mem_star [InvolutiveStar α] : a⋆ ∈ s⋆ ↔ a ∈ s := by | simp only [mem_star, star_star]
|
/-
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 | 187 | 189 | theorem sum_smul_vsub_const_eq_weightedVSubOfPoint_sub (w : ι → k) (p₁ : ι → P) (p₂ b : P) :
(∑ i ∈ s, w i • (p₁ i -ᵥ p₂)) = s.weightedVSubOfPoint p₁ b w - (∑ i ∈ s, w i) • (p₂ -ᵥ b) := by |
rw [sum_smul_vsub_eq_weightedVSubOfPoint_sub, weightedVSubOfPoint_apply_const]
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Data.List.Lattice
import Mathlib.Data.List.Range
import Mathlib.Data.Bool.Basic
#align_import data.list.intervals from "leanprover-community/mathlib"@... | Mathlib/Data/List/Intervals.lean | 198 | 203 | theorem filter_le_of_le {n m l : ℕ} (hnl : n ≤ l) :
((Ico n m).filter fun x => l ≤ x) = Ico l m := by |
rcases le_total l m with hlm | hml
· rw [← append_consecutive hnl hlm, filter_append, filter_le_of_top_le (le_refl l),
filter_le_of_le_bot (le_refl l), nil_append]
· rw [eq_nil_of_le hml, filter_le_of_top_le hml]
|
/-
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 | 1,935 | 1,938 | theorem dist_pi_le_iff {f g : ∀ b, π b} {r : ℝ} (hr : 0 ≤ r) :
dist f g ≤ r ↔ ∀ b, dist (f b) (g b) ≤ r := by |
lift r to ℝ≥0 using hr
exact nndist_pi_le_iff
|
/-
Copyright (c) 2023 María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: María Inés de Frutos-Fernández
-/
import Mathlib.RingTheory.DedekindDomain.AdicValuation
import Mathlib.RingTheory.DedekindDomain.Factorization
import Mathlib.Algebra.O... | Mathlib/RingTheory/DedekindDomain/FiniteAdeleRing.lean | 281 | 306 | theorem algebraMap' (k : K) : (_root_.algebraMap K (K_hat R K) k).IsFiniteAdele := by |
rw [IsFiniteAdele, Filter.eventually_cofinite]
simp_rw [mem_adicCompletionIntegers, ProdAdicCompletions.algebraMap_apply',
Valued.valuedCompletion_apply, not_le]
change {v : HeightOneSpectrum R | 1 < v.valuation k}.Finite
-- The goal currently: if k ∈ K = field of fractions of a Dedekind domain R,
-- the... |
/-
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.Order.Disjoint
#align_import order.prop_instances from "leanprover-community/mathlib"@"6623e6af705e97002a9054c1c05a980180276fc1"
/-!
# The order on ... | Mathlib/Order/PropInstances.lean | 106 | 108 | theorem Prop.isCompl_iff {P Q : Prop} : IsCompl P Q ↔ ¬(P ↔ Q) := by |
rw [_root_.isCompl_iff, Prop.disjoint_iff, Prop.codisjoint_iff, not_iff]
by_cases P <;> by_cases Q <;> 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.Data.Finsupp.ToDFinsupp
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.LinearIndependent
#align_import linear_algebra.dfinsup... | Mathlib/LinearAlgebra/DFinsupp.lean | 543 | 560 | theorem Independent.linearIndependent [NoZeroSMulDivisors R N] (p : ι → Submodule R N)
(hp : Independent p) {v : ι → N} (hv : ∀ i, v i ∈ p i) (hv' : ∀ i, v i ≠ 0) :
LinearIndependent R v := by |
let _ := Classical.decEq ι
let _ := Classical.decEq R
rw [linearIndependent_iff]
intro l hl
let a :=
DFinsupp.mapRange.linearMap (fun i => LinearMap.toSpanSingleton R (p i) ⟨v i, hv i⟩)
l.toDFinsupp
have ha : a = 0 := by
apply hp.dfinsupp_lsum_injective
rwa [← lsum_comp_mapRange_toSpanSin... |
/-
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.Analysis.Convex.Hull
#align_import analysis.convex.join from "leanprover-community/mathlib"@"951bf1d9e98a2042979ced62c0620bcfb3587cf8"
/-!
# Convex join
... | Mathlib/Analysis/Convex/Join.lean | 85 | 87 | theorem convexJoin_union_right (s t₁ t₂ : Set E) :
convexJoin 𝕜 s (t₁ ∪ t₂) = convexJoin 𝕜 s t₁ ∪ convexJoin 𝕜 s t₂ := by |
simp_rw [convexJoin_comm s, convexJoin_union_left]
|
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Preadditive.AdditiveFunctor
import Mathlib.CategoryTheory.Monoidal.Functor
#align_import category_theory.monoidal.preadditive from "lea... | Mathlib/CategoryTheory/Monoidal/Preadditive.lean | 230 | 232 | theorem rightDistributor_hom_comp_biproduct_π {J : Type} [Fintype J] (f : J → C) (X : C) (j : J) :
(rightDistributor f X).hom ≫ biproduct.π _ j = biproduct.π _ j ▷ X := by |
simp [rightDistributor_hom, Preadditive.sum_comp, biproduct.ι_π, comp_dite]
|
/-
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 | 1,387 | 1,394 | theorem exists_boundary_dart {u v : V} (p : G.Walk u v) (S : Set V) (uS : u ∈ S) (vS : v ∉ S) :
∃ d : G.Dart, d ∈ p.darts ∧ d.fst ∈ S ∧ d.snd ∉ S := by |
induction' p with _ x y w a p' ih
· cases vS uS
· by_cases h : y ∈ S
· obtain ⟨d, hd, hcd⟩ := ih h vS
exact ⟨d, List.Mem.tail _ hd, hcd⟩
· exact ⟨⟨(x, y), a⟩, List.Mem.head _, uS, h⟩
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.GroupWithZero.Invertible
import Mathlib.Algebra.Ring.Defs
#align_import algebra.invertible from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c... | Mathlib/Algebra/Ring/Invertible.lean | 65 | 73 | theorem Ring.inverse_sub_inverse [Ring α] {a b : α} (h : IsUnit a ↔ IsUnit b) :
Ring.inverse a - Ring.inverse b = Ring.inverse a * (b - a) * Ring.inverse b := by |
by_cases ha : IsUnit a
· have hb := h.mp ha
obtain ⟨ia⟩ := ha.nonempty_invertible
obtain ⟨ib⟩ := hb.nonempty_invertible
simp_rw [inverse_invertible, invOf_sub_invOf]
· have hb := h.not.mp ha
simp [inverse_non_unit, ha, hb]
|
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Disintegration.Unique
import Mathlib.Probability.Notation
#align_import probability.kernel.cond_distrib from "leanprover-community/math... | Mathlib/Probability/Kernel/CondDistrib.lean | 218 | 229 | theorem condDistrib_ae_eq_condexp (hX : Measurable X) (hY : Measurable Y) (hs : MeasurableSet s) :
(fun a => (condDistrib Y X μ (X a) s).toReal) =ᵐ[μ] μ⟦Y ⁻¹' s|mβ.comap X⟧ := by |
refine ae_eq_condexp_of_forall_setIntegral_eq hX.comap_le ?_ ?_ ?_ ?_
· exact (integrable_const _).indicator (hY hs)
· exact fun t _ _ => (integrable_toReal_condDistrib hX.aemeasurable hs).integrableOn
· intro t ht _
rw [integral_toReal ((measurable_condDistrib hs).mono hX.comap_le le_rfl).aemeasurable
... |
/-
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.Order
#align_import topology.maps from "leanprover-community/mathlib"@"d91e7f7a7f1c7e9f0e18fdb6bde4f652004c73... | Mathlib/Topology/Maps.lean | 156 | 157 | theorem isClosed_iff' (hf : Inducing f) {s : Set X} :
IsClosed s ↔ ∀ x, f x ∈ closure (f '' s) → x ∈ s := by | rw [hf.induced, isClosed_induced_iff']
|
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Kexing Ying
-/
import Mathlib.MeasureTheory.Function.ConditionalExpectation.Indicator
import Mathlib.MeasureTheory.Function.UniformIntegrable
import Mathlib.MeasureTheory.D... | Mathlib/MeasureTheory/Function/ConditionalExpectation/Real.lean | 259 | 300 | theorem condexp_stronglyMeasurable_mul_of_bound (hm : m ≤ m0) [IsFiniteMeasure μ] {f g : α → ℝ}
(hf : StronglyMeasurable[m] f) (hg : Integrable g μ) (c : ℝ) (hf_bound : ∀ᵐ x ∂μ, ‖f x‖ ≤ c) :
μ[f * g|m] =ᵐ[μ] f * μ[g|m] := by |
let fs := hf.approxBounded c
have hfs_tendsto : ∀ᵐ x ∂μ, Tendsto (fs · x) atTop (𝓝 (f x)) :=
hf.tendsto_approxBounded_ae hf_bound
by_cases hμ : μ = 0
· simp only [hμ, ae_zero]; norm_cast
have : (ae μ).NeBot := ae_neBot.2 hμ
have hc : 0 ≤ c := by
rcases hf_bound.exists with ⟨_x, hx⟩
exact (norm... |
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Data.Matrix.Basis
import Mathlib.Data.Matrix.DMatrix
import Mathlib.RingTheory.MatrixAlgebra
#align_impor... | Mathlib/RingTheory/PolynomialAlgebra.lean | 138 | 139 | theorem invFun_add {p q} : invFun R A (p + q) = invFun R A p + invFun R A q := by |
simp only [invFun, eval₂_add]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Group.List
import Mathlib.Data.Vector.Defs
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.I... | Mathlib/Data/Vector/Basic.lean | 210 | 212 | theorem toList_singleton (v : Vector α 1) : v.toList = [v.head] := by |
rw [← v.cons_head_tail]
simp only [toList_cons, toList_nil, head_cons, eq_self_iff_true, and_self_iff, singleton_tail]
|
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Alexey Soloyev, Junyan Xu, Kamila Szewczyk
-/
import Mathlib.Data.Real.Irrational
import Mathlib.Data.Nat.Fib.Basic
import Mathlib.Data.Fin.VecNotation
import Mathl... | Mathlib/Data/Real/GoldenRatio.lean | 112 | 114 | theorem one_lt_gold : 1 < φ := by |
refine lt_of_mul_lt_mul_left ?_ (le_of_lt gold_pos)
simp [← sq, gold_pos, zero_lt_one, - div_pow] -- Porting note: Added `- div_pow`
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Order.Filter.Bases
import Mathlib.Order.ConditionallyCompleteLattice.Basic
#align_import order.filter.lift from "leanprover-community/mathlib"@"8631e2... | Mathlib/Order/Filter/Lift.lean | 422 | 425 | theorem prod_def {f : Filter α} {g : Filter β} :
f ×ˢ g = f.lift fun s => g.lift' fun t => s ×ˢ t := by |
simpa only [Filter.lift', Filter.lift, (f.basis_sets.prod g.basis_sets).eq_biInf,
iInf_prod, iInf_and] using iInf_congr fun i => iInf_comm
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingul... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 317 | 321 | theorem set_lintegral_congr_fun {f g : α → ℝ≥0∞} {s : Set α} (hs : MeasurableSet s)
(hfg : ∀ᵐ x ∂μ, x ∈ s → f x = g x) : ∫⁻ x in s, f x ∂μ = ∫⁻ x in s, g x ∂μ := by |
rw [lintegral_congr_ae]
rw [EventuallyEq]
rwa [ae_restrict_iff' hs]
|
/-
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, Sébastien Gouëzel,
Rémy Degenne, David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Pow.NNReal
#align_import anal... | Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | 317 | 319 | theorem isLittleO_pow_exp_pos_mul_atTop (k : ℕ) {b : ℝ} (hb : 0 < b) :
(fun x : ℝ => x ^ k) =o[atTop] fun x => exp (b * x) := by |
simpa using isLittleO_zpow_exp_pos_mul_atTop k hb
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Robert Y. Lewis
-/
import Mathlib.Algebra.Order.CauSeq.Basic
#align_import data.real.cau_seq_completion from "leanprover-community/mathlib"@"cf4c49c445991489058260d75d... | Mathlib/Algebra/Order/CauSeq/Completion.lean | 385 | 386 | theorem lim_mul (f : CauSeq β abv) (x : β) : lim f * x = lim (f * const abv x) := by |
rw [← lim_mul_lim, lim_const]
|
/-
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 | 188 | 191 | theorem col_cons (x : α) (u : Fin m → α) :
col (vecCons x u) = of (vecCons (fun _ => x) (col u)) := by |
ext i j
refine Fin.cases ?_ ?_ i <;> simp [vecHead, vecTail]
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Logic.Relation
import Mathlib.Data.Option.Basic
import Mathlib.Data.Seq.Seq
#align_import data.seq.wseq from "leanprover-community/mathlib"@"a7... | Mathlib/Data/Seq/WSeq.lean | 842 | 850 | theorem destruct_some_of_destruct_tail_some {s : WSeq α} {a} (h : some a ∈ destruct (tail s)) :
∃ a', some a' ∈ destruct s := by |
unfold tail Functor.map at h; simp only [destruct_flatten] at h
rcases exists_of_mem_bind h with ⟨t, tm, td⟩; clear h
rcases Computation.exists_of_mem_map tm with ⟨t', ht', ht2⟩; clear tm
cases' t' with t' <;> rw [← ht2] at td <;> simp only [destruct_nil] at td
· have := mem_unique td (ret_mem _)
contrad... |
/-
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.Algebra.BigOperators.Finprod
import Mathlib.Order.Filter.Pointwise
import Mathlib.Topology.Algebra.MulAction
import Mathlib.Algebra.Big... | Mathlib/Topology/Algebra/Monoid.lean | 883 | 886 | theorem continuousMul_iInf {ts : ι' → TopologicalSpace M}
(h' : ∀ i, @ContinuousMul M (ts i) _) : @ContinuousMul M (⨅ i, ts i) _ := by |
rw [← sInf_range]
exact continuousMul_sInf (Set.forall_mem_range.mpr h')
|
/-
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.Order.SuccPred.Basic
import Mathlib.Order.BoundedOrder
#align_import order.succ_pred.limit from "leanprover-community/mathlib"... | Mathlib/Order/SuccPred/Limit.lean | 135 | 137 | theorem mem_range_succ_of_not_isSuccLimit (h : ¬IsSuccLimit a) : a ∈ range (@succ α _ _) := by |
cases' not_isSuccLimit_iff.1 h with b hb
exact ⟨b, hb.2⟩
|
/-
Copyright (c) 2021 Hunter Monroe. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Hunter Monroe, Kyle Miller, Alena Gusakov
-/
import Mathlib.Combinatorics.SimpleGraph.Finite
import Mathlib.Combinatorics.SimpleGraph.Maps
#align_import combinatorics.simple_graph.subg... | Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | 439 | 443 | theorem verts_spanningCoe_injective :
(fun G' : Subgraph G => (G'.verts, G'.spanningCoe)).Injective := by |
intro G₁ G₂ h
rw [Prod.ext_iff] at h
exact Subgraph.ext _ _ h.1 (spanningCoe_inj.1 h.2)
|
/-
Copyright (c) 2022 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson, Devon Tuma, Eric Rodriguez, Oliver Nash
-/
import Mathlib.Data.Set.Pointwise.Interval
import Mathlib.Topology.Algebra.Field
import Mathlib.Topology.Algebra.Order.... | Mathlib/Topology/Algebra/Order/Field.lean | 102 | 105 | theorem Filter.Tendsto.atBot_mul_neg {C : 𝕜} (hC : C < 0) (hf : Tendsto f l atBot)
(hg : Tendsto g l (𝓝 C)) : Tendsto (fun x => f x * g x) l atTop := by |
have := (tendsto_neg_atBot_atTop.comp hf).atTop_mul_neg hC hg
simpa [(· ∘ ·)] using tendsto_neg_atBot_atTop.comp this
|
/-
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.Analysis.SpecialFunctions.Trigonometric.Arctan
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geometry.euclidean.angle.unoriented... | Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean | 43 | 46 | theorem norm_add_sq_eq_norm_sq_add_norm_sq_iff_angle_eq_pi_div_two (x y : V) :
‖x + y‖ * ‖x + y‖ = ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖ ↔ angle x y = π / 2 := by |
rw [norm_add_sq_eq_norm_sq_add_norm_sq_iff_real_inner_eq_zero]
exact inner_eq_zero_iff_angle_eq_pi_div_two x y
|
/-
Copyright (c) 2022 Yaël Dillies, George Shakan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, George Shakan
-/
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Combinatorics.Enumerative.DoubleCounting
imp... | Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean | 227 | 243 | theorem card_pow_div_pow_le (hA : A.Nonempty) (B : Finset α) (m n : ℕ) :
((B ^ m / B ^ n).card) ≤ ((A * B).card / A.card : ℚ≥0) ^ (m + n) * A.card := by |
have hA' : A ∈ A.powerset.erase ∅ := mem_erase_of_ne_of_mem hA.ne_empty (mem_powerset_self _)
obtain ⟨C, hC, hCA⟩ :=
exists_min_image (A.powerset.erase ∅) (fun C ↦ (C * B).card / C.card : _ → ℚ≥0) ⟨A, hA'⟩
rw [mem_erase, mem_powerset, ← nonempty_iff_ne_empty] at hC
refine (mul_le_mul_right <| cast_pos.2 hC... |
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import topology.metric_space.hausdorff_distance from "leanprover-community/mathlib"@"bc91ed7093bf098d253401e... | Mathlib/Topology/MetricSpace/Thickening.lean | 365 | 367 | theorem thickening_union (δ : ℝ) (s t : Set α) :
thickening δ (s ∪ t) = thickening δ s ∪ thickening δ t := by |
simp_rw [thickening, infEdist_union, inf_eq_min, min_lt_iff, setOf_or]
|
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Variance
#align_import probability.moments from "leanprover-community/mathlib"@"85453a2a14be8da64caf15ca50930cf4c6e5d8de"
/-!
# Moments and m... | Mathlib/Probability/Moments.lean | 122 | 122 | theorem mgf_zero_measure : mgf X (0 : Measure Ω) t = 0 := by | simp only [mgf, integral_zero_measure]
|
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.Data.Fintype.Parity
import Mathlib.NumberTheory.LegendreSymbol.ZModChar
import Mathlib.FieldTheory.Finite.Basic
#align_import number_theory.legendre_sym... | Mathlib/NumberTheory/LegendreSymbol/QuadraticChar/Basic.lean | 75 | 76 | theorem quadraticCharFun_zero : quadraticCharFun F 0 = 0 := by |
simp only [quadraticCharFun, eq_self_iff_true, if_true, id]
|
/-
Copyright (c) 2023 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Richard Hill
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Algebra.Polynomial.Module.AEval
import Mathlib.Ri... | Mathlib/Algebra/Polynomial/Derivation.lean | 131 | 136 | theorem compAEval_eq (d : Derivation R A M) (f : R[X]) :
d.compAEval a f = derivative f • (AEval.of R M a (d a)) := by |
rw [← mkDerivation_apply]
congr
apply derivation_ext
simp
|
/-
Copyright (c) 2017 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Data.PFunctor.Univariate.Basic
#align_import data.pfunctor.univariate.M from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d82b83069680cb1"
/-!
... | Mathlib/Data/PFunctor/Univariate/M.lean | 86 | 86 | theorem agree_trival {x : CofixA F 0} {y : CofixA F 1} : Agree x y := by | constructor
|
/-
Copyright (c) 2022 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Yury Kudryashov
-/
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Convex.Strict
import Mathlib.Analysis.Normed.Order.Basic
import Mathlib.Analysis.NormedSpac... | Mathlib/Analysis/Convex/StrictConvexSpace.lean | 230 | 233 | theorem norm_midpoint_lt_iff (h : ‖x‖ = ‖y‖) : ‖(1 / 2 : ℝ) • (x + y)‖ < ‖x‖ ↔ x ≠ y := by |
rw [norm_smul, Real.norm_of_nonneg (one_div_nonneg.2 zero_le_two), ← inv_eq_one_div, ←
div_eq_inv_mul, div_lt_iff (zero_lt_two' ℝ), mul_two, ← not_sameRay_iff_of_norm_eq h,
not_sameRay_iff_norm_add_lt, h]
|
/-
Copyright (c) 2021 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.List.Cycle
import Mathlib.GroupTheory.Perm.Cycle.Type
import Mathlib.GroupTheory.Perm.List
#align_import group_theory.perm.cycle.concrete from ... | Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | 253 | 254 | theorem nthLe_toList (n : ℕ) (hn : n < length (toList p x)) :
(toList p x).nthLe n hn = (p ^ n) x := by | simp [toList]
|
/-
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 | 306 | 316 | theorem exists_ne_zero_mem_isInteger [Nontrivial R] (hI : I ≠ 0) :
∃ x, x ≠ 0 ∧ algebraMap R K x ∈ I := by |
obtain ⟨y : K, y_mem, y_not_mem⟩ :=
SetLike.exists_of_lt (by simpa only using bot_lt_iff_ne_bot.mpr hI)
have y_ne_zero : y ≠ 0 := by simpa using y_not_mem
obtain ⟨z, ⟨x, hx⟩⟩ := exists_integer_multiple R⁰ y
refine ⟨x, ?_, ?_⟩
· rw [Ne, ← @IsFractionRing.to_map_eq_zero_iff R _ K, hx, Algebra.smul_def]
... |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro, Anne Baanen
-/
import Mathlib.LinearAlgebra.Quotient
import Mathlib.RingTheory.Congruence
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.Tacti... | Mathlib/RingTheory/Ideal/Quotient.lean | 235 | 244 | theorem maximal_of_isField (I : Ideal R) (hqf : IsField (R ⧸ I)) : I.IsMaximal := by |
apply Ideal.isMaximal_iff.2
constructor
· intro h
rcases hqf.exists_pair_ne with ⟨⟨x⟩, ⟨y⟩, hxy⟩
exact hxy (Ideal.Quotient.eq.2 (mul_one (x - y) ▸ I.mul_mem_left _ h))
· intro J x hIJ hxnI hxJ
rcases hqf.mul_inv_cancel (mt Ideal.Quotient.eq_zero_iff_mem.1 hxnI) with ⟨⟨y⟩, hy⟩
rw [← zero_add (1 ... |
/-
Copyright (c) 2020 Johan Commelin, Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.NumberTheory.Padics.PadicIntegers
import Mathlib.RingTheory.ZMod
#align_import number_theory.padics.ring_homs from "... | Mathlib/NumberTheory/Padics/RingHoms.lean | 649 | 653 | theorem lift_spec (n : ℕ) : (toZModPow n).comp (lift f_compat) = f n := by |
ext r
rw [RingHom.comp_apply, ← ZMod.natCast_zmod_val (f n r), ← map_natCast <| toZModPow n, ←
sub_eq_zero, ← RingHom.map_sub, ← RingHom.mem_ker, ker_toZModPow]
apply lift_sub_val_mem_span
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
#align_import ring_theory.ideal.operations from "leanpro... | Mathlib/RingTheory/Ideal/Operations.lean | 712 | 713 | theorem iInf_sup_eq_top {s : Finset ι} {J : ι → Ideal R} (h : ∀ i, i ∈ s → J i ⊔ I = ⊤) :
(⨅ i ∈ s, J i) ⊔ I = ⊤ := by | rw [sup_comm, sup_iInf_eq_top]; intro i hi; rw [sup_comm, h i hi]
|
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Alex J. Best
-/
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
#align_import linear_algebra.free_module.determinant from "lean... | Mathlib/LinearAlgebra/FreeModule/Determinant.lean | 25 | 29 | theorem LinearMap.det_zero'' {R M : Type*} [CommRing R] [AddCommGroup M] [Module R M]
[Module.Free R M] [Module.Finite R M] [Nontrivial M] : LinearMap.det (0 : M →ₗ[R] M) = 0 := by |
letI : Nonempty (Module.Free.ChooseBasisIndex R M) := (Module.Free.chooseBasis R M).index_nonempty
nontriviality R
exact LinearMap.det_zero' (Module.Free.chooseBasis R M)
|
/-
Copyright (c) 2021 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Finprod
import Mathlib.Algebra.Order.Group.WithTop
import Mathlib.RingTheory.HahnSeries.Multiplication
import Mathlib.RingTheory.V... | Mathlib/RingTheory/HahnSeries/Summable.lean | 556 | 569 | theorem isUnit_iff {x : HahnSeries Γ R} : IsUnit x ↔ IsUnit (x.coeff x.order) := by |
constructor
· rintro ⟨⟨u, i, ui, iu⟩, rfl⟩
refine
isUnit_of_mul_eq_one (u.coeff u.order) (i.coeff i.order)
((mul_coeff_order_add_order u i).symm.trans ?_)
rw [ui, one_coeff, if_pos]
rw [← order_mul (left_ne_zero_of_mul_eq_one ui) (right_ne_zero_of_mul_eq_one ui), ui, order_one]
· rintro... |
/-
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.Init.Core
import Mathlib.LinearAlgebra.AffineSpace.Basis
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algebra.affine_space.finite_d... | Mathlib/LinearAlgebra/AffineSpace/FiniteDimensional.lean | 493 | 502 | theorem collinear_pair (p₁ p₂ : P) : Collinear k ({p₁, p₂} : Set P) := by |
rw [collinear_iff_exists_forall_eq_smul_vadd]
use p₁, p₂ -ᵥ p₁
intro p hp
rw [Set.mem_insert_iff, Set.mem_singleton_iff] at hp
cases' hp with hp hp
· use 0
simp [hp]
· use 1
simp [hp]
|
/-
Copyright (c) 2020 Paul van Wamelen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul van Wamelen
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.RingTheory.Int.Basic
import Mat... | Mathlib/NumberTheory/PythagoreanTriples.lean | 681 | 703 | theorem classification :
PythagoreanTriple x y z ↔
∃ k m n,
(x = k * (m ^ 2 - n ^ 2) ∧ y = k * (2 * m * n) ∨
x = k * (2 * m * n) ∧ y = k * (m ^ 2 - n ^ 2)) ∧
(z = k * (m ^ 2 + n ^ 2) ∨ z = -k * (m ^ 2 + n ^ 2)) := by |
clear h
constructor
· intro h
obtain ⟨k, m, n, H⟩ := h.classified
use k, m, n
rcases H with (⟨rfl, rfl⟩ | ⟨rfl, rfl⟩)
· refine ⟨Or.inl ⟨rfl, rfl⟩, ?_⟩
have : z ^ 2 = (k * (m ^ 2 + n ^ 2)) ^ 2 := by
rw [sq, ← h.eq]
ring
simpa using eq_or_eq_neg_of_sq_eq_sq _ _ this
... |
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import n... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 593 | 605 | theorem det_basisOfFractionalIdeal_eq_norm
(e : (ChooseBasisIndex ℤ (𝓞 K)) ≃ (ChooseBasisIndex ℤ I)) :
|Basis.det (latticeBasis K) ((mixedEmbedding K ∘ (basisOfFractionalIdeal K I) ∘ e))| =
FractionalIdeal.absNorm I.1 := by |
suffices Basis.det (latticeBasis K) ((mixedEmbedding K ∘ (basisOfFractionalIdeal K I) ∘ e)) =
(algebraMap ℚ ℝ) ((Basis.det (integralBasis K)) ((basisOfFractionalIdeal K I) ∘ e)) by
rw [this, eq_ratCast, ← Rat.cast_abs, ← Equiv.symm_symm e, ← Basis.coe_reindex,
det_basisOfFractionalIdeal_eq_absNorm K ... |
/-
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, Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Finsupp.Multiset
import Math... | Mathlib/RingTheory/UniqueFactorizationDomain.lean | 152 | 158 | theorem WfDvdMonoid.max_power_factor' [CommMonoidWithZero α] [WfDvdMonoid α] {a₀ x : α}
(h : a₀ ≠ 0) (hx : ¬IsUnit x) : ∃ (n : ℕ) (a : α), ¬x ∣ a ∧ a₀ = x ^ n * a := by |
obtain ⟨a, ⟨n, rfl⟩, hm⟩ := wellFounded_dvdNotUnit.has_min
{a | ∃ n, x ^ n * a = a₀} ⟨a₀, 0, by rw [pow_zero, one_mul]⟩
refine ⟨n, a, ?_, rfl⟩; rintro ⟨d, rfl⟩
exact hm d ⟨n + 1, by rw [pow_succ, mul_assoc]⟩
⟨(right_ne_zero_of_mul <| right_ne_zero_of_mul h), x, hx, mul_comm _ _⟩
|
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