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) 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 | 361 | 367 | theorem rightDistributor_ext₂_right {J : Type} [Fintype J]
{f : J → C} {X Y Z : C} {g h : X ⟶ ((⨁ f) ⊗ Y) ⊗ Z}
(w : ∀ j, g ≫ ((biproduct.π f j ▷ Y) ▷ Z) = h ≫ ((biproduct.π f j ▷ Y) ▷ Z)) :
g = h := by |
apply (cancel_mono (α_ _ _ _).hom).mp
ext
simp [w]
|
/-
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, Yaël Dillies
-/
import Mathlib.Data.Finset.NAry
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Set.Pointwise.Finite
import Mathlib.Data.Set.Pointwise.SMul
... | Mathlib/Data/Finset/Pointwise.lean | 1,346 | 1,347 | theorem preimage_mul_left_one : preimage 1 (a * ·) (mul_right_injective _).injOn = {a⁻¹} := by |
classical rw [← image_mul_left', image_one, mul_one]
|
/-
Copyright (c) 2020 Pim Spelier, Daan van Gent. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Pim Spelier, Daan van Gent
-/
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Num.Lemmas
import Mathlib.Data.Option.Basic
import Mathlib.SetTheory.Cardinal.Basic
#al... | Mathlib/Computability/Encoding.lean | 143 | 149 | theorem decode_encodeNum : ∀ n, decodeNum (encodeNum n) = n := by |
intro n
cases' n with n <;> unfold encodeNum decodeNum
· rfl
rw [decode_encodePosNum n]
rw [PosNum.cast_to_num]
exact if_neg (encodePosNum_nonempty n)
|
/-
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.MeasureTheory.Measure.FiniteMeasure
import Mathlib.MeasureTheory.Integral.Average
#align_import measure_theory.measure.probability_measure from "leanprove... | Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean | 223 | 226 | theorem eq_of_forall_apply_eq (μ ν : ProbabilityMeasure Ω)
(h : ∀ s : Set Ω, MeasurableSet s → μ s = ν s) : μ = ν := by |
ext1 s s_mble
simpa [ennreal_coeFn_eq_coeFn_toMeasure] using congr_arg ((↑) : ℝ≥0 → ℝ≥0∞) (h s s_mble)
|
/-
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.Calculus.FDeriv.Bilinear
#align_import analysis.calculus.fderiv.mul from "leanprover-community/mathlib"@"d6... | Mathlib/Analysis/Calculus/FDeriv/Mul.lean | 820 | 825 | theorem HasStrictFDerivAt.finset_prod [DecidableEq ι] {x : E}
(hg : ∀ i ∈ u, HasStrictFDerivAt (g i) (g' i) x) :
HasStrictFDerivAt (∏ i ∈ u, g i ·) (∑ i ∈ u, (∏ j ∈ u.erase i, g j x) • g' i) x := by |
simpa [← Finset.prod_attach u] using .congr_fderiv
(hasStrictFDerivAt_finset_prod.comp x <| hasStrictFDerivAt_pi.mpr fun i ↦ hg i i.prop)
(by ext; simp [Finset.prod_erase_attach (g · x), ← u.sum_attach])
|
/-
Copyright (c) 2023 Scott Carnahan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Carnahan
-/
import Mathlib.Algebra.Group.NatPowAssoc
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Induction
import Mathlib.Algebra.Polynomial.Ev... | Mathlib/Algebra/Polynomial/Smeval.lean | 279 | 284 | theorem smeval_comp : (p.comp q).smeval x = p.smeval (q.smeval x) := by |
induction p using Polynomial.induction_on' with
| h_add r s hr hs =>
simp [add_comp, hr, hs, smeval_add]
| h_monomial n a =>
simp [smeval_monomial, smeval_C_mul, smeval_pow]
|
/-
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.Algebra.InfiniteSum.Basic
import Mathlib.Topology.Algebra.UniformGroup
/-!
# Infinite sums and products in topological groups
Lemmas on topo... | Mathlib/Topology/Algebra/InfiniteSum/Group.lean | 372 | 374 | theorem Multipliable.countable_mulSupport [FirstCountableTopology G] [T1Space G]
(hf : Multipliable f) : f.mulSupport.Countable := by |
simpa only [ker_nhds] using hf.tendsto_cofinite_one.countable_compl_preimage_ker
|
/-
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.Hull
#align_import analysis.convex.extreme from "leanprover-community/mathlib"@"c5773405394e073885e2a144c9ca14... | Mathlib/Analysis/Convex/Extreme.lean | 287 | 292 | theorem extremePoints_convexHull_subset : (convexHull 𝕜 A).extremePoints 𝕜 ⊆ A := by |
rintro x hx
rw [(convex_convexHull 𝕜 _).mem_extremePoints_iff_convex_diff] at hx
by_contra h
exact (convexHull_min (subset_diff.2 ⟨subset_convexHull 𝕜 _, disjoint_singleton_right.2 h⟩) hx.2
hx.1).2 rfl
|
/-
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
-/
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.ZornAtoms
#align_import order.filter.ultrafilter from "leanprover-community... | Mathlib/Order/Filter/Ultrafilter.lean | 176 | 177 | theorem union_mem_iff : s ∪ t ∈ f ↔ s ∈ f ∨ t ∈ f := by |
simp only [← mem_coe, ← le_principal_iff, ← sup_principal, le_sup_iff]
|
/-
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.MeasureTheory.Measure.NullMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Topology.Algebra.Order.LiminfLim... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 1,627 | 1,630 | theorem mk (h : ∀ ⦃s : Set α⦄, MeasurableSet s → ν s = 0 → μ s = 0) : μ ≪ ν := by |
intro s hs
rcases exists_measurable_superset_of_null hs with ⟨t, h1t, h2t, h3t⟩
exact measure_mono_null h1t (h h2t h3t)
|
/-
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 | 1,176 | 1,179 | theorem isClosed_C1 : IsClosed (C1 C ho) := by |
refine hC.inter ?_
have h : Continuous (fun (f : I → Bool) ↦ f (term I ho)) := continuous_apply (term I ho)
exact IsClosed.preimage h (t := {true}) (isClosed_discrete _)
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Scott Morrison, Jakob von Raumer
-/
import Mathlib.Algebra.Category.ModuleCat.Basic
import Mathlib.LinearAlgebra.TensorProduct.Basic
import Mathlib.CategoryTheory.Monoid... | Mathlib/Algebra/Category/ModuleCat/Monoidal/Basic.lean | 193 | 202 | theorem triangle (M N : ModuleCat.{u} R) :
(associator M (ModuleCat.of R R) N).hom ≫ tensorHom (𝟙 M) (leftUnitor N).hom =
tensorHom (rightUnitor M).hom (𝟙 N) := by |
apply TensorProduct.ext_threefold
intro x y z
change R at y
-- Porting note (#10934): used to be dsimp [tensorHom, associator]
change x ⊗ₜ[R] ((leftUnitor N).hom) (y ⊗ₜ[R] z) = ((rightUnitor M).hom) (x ⊗ₜ[R] y) ⊗ₜ[R] z
erw [TensorProduct.lid_tmul, TensorProduct.rid_tmul]
exact (TensorProduct.smul_tmul _ ... |
/-
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.CategoryTheory.Sites.Canonical
#align_import category_theory.sites.types from "leanprover-community/mathlib"@"9f9015c645d85695581237cc761981036be8bd37"
/-!
# G... | Mathlib/CategoryTheory/Sites/Types.lean | 130 | 132 | theorem eval_map (S : Type uᵒᵖ ⥤ Type u) (α β) (f : β ⟶ α) (s x) :
eval S β (S.map f.op s) x = eval S α s (f x) := by |
simp_rw [eval, ← FunctorToTypes.map_comp_apply, ← op_comp]; rfl
|
/-
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 | 801 | 803 | theorem UniformSpace.mem_closure_iff_ball {s : Set α} {x} :
x ∈ closure s ↔ ∀ {V}, V ∈ 𝓤 α → (ball x V ∩ s).Nonempty := by |
simp [mem_closure_iff_nhds_basis' (nhds_basis_uniformity' (𝓤 α).basis_sets)]
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Algebra.Group.Prod
import Mathlib.Algebra.Group.Units.Equiv
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Logic... | Mathlib/GroupTheory/Perm/Basic.lean | 537 | 538 | theorem mul_swap_eq_swap_mul (f : Perm α) (x y : α) : f * swap x y = swap (f x) (f y) * f := by |
rw [swap_mul_eq_mul_swap, Perm.inv_apply_self, Perm.inv_apply_self]
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Wrenna Robson
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.LinearAlgebra.Vandermonde
import Mathlib.RingTheory.Polynomial.Basic
#align_import linear_algebr... | Mathlib/LinearAlgebra/Lagrange.lean | 443 | 477 | theorem interpolate_eq_sum_interpolate_insert_sdiff (hvt : Set.InjOn v t) (hs : s.Nonempty)
(hst : s ⊆ t) :
interpolate t v r = ∑ i ∈ s, interpolate (insert i (t \ s)) v r * Lagrange.basis s v i := by |
symm
refine eq_interpolate_of_eval_eq _ hvt (lt_of_le_of_lt (degree_sum_le _ _) ?_) fun i hi => ?_
· simp_rw [Nat.cast_withBot, Finset.sup_lt_iff (WithBot.bot_lt_coe t.card), degree_mul]
intro i hi
have hs : 1 ≤ s.card := Nonempty.card_pos ⟨_, hi⟩
have hst' : s.card ≤ t.card := card_le_card hst
h... |
/-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Thomas Murrills
-/
import Mathlib.Tactic.NormNum.Core
import Mathlib.Tactic.HaveI
import Mathlib.Data.Nat.Cast.Commute
import Mathlib.Algebra.Ring.Int
import Mathlib.Al... | Mathlib/Tactic/NormNum/Basic.lean | 370 | 388 | theorem isRat_mul {α} [Ring α] {f : α → α → α} {a b : α} {na nb nc : ℤ} {da db dc k : ℕ} :
f = HMul.hMul → IsRat a na da → IsRat b nb db →
Int.mul na nb = Int.mul k nc →
Nat.mul da db = Nat.mul k dc →
IsRat (f a b) nc dc := by |
rintro rfl ⟨_, rfl⟩ ⟨_, rfl⟩ (h₁ : na * nb = k * nc) (h₂ : da * db = k * dc)
have : Invertible (↑(da * db) : α) := by simpa using invertibleMul (da:α) db
have := invertibleOfMul' (α := α) h₂
refine ⟨this, ?_⟩
have H := (Nat.cast_commute (α := α) da db).invOf_left.invOf_right.right_comm
have h₁ := congr_arg... |
/-
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.Analysis.Calculus.FDeriv.Prod
import Mathlib.Analysis.Calculus.InverseFunctionTheorem.FDeriv
import Mathlib.LinearAlgebra.Dual
#align_import analysi... | Mathlib/Analysis/Calculus/LagrangeMultipliers.lean | 132 | 142 | theorem IsLocalExtrOn.linear_dependent_of_hasStrictFDerivAt {ι : Type*} [Finite ι] {f : ι → E → ℝ}
{f' : ι → E →L[ℝ] ℝ} (hextr : IsLocalExtrOn φ {x | ∀ i, f i x = f i x₀} x₀)
(hf' : ∀ i, HasStrictFDerivAt (f i) (f' i) x₀) (hφ' : HasStrictFDerivAt φ φ' x₀) :
¬LinearIndependent ℝ (Option.elim' φ' f' : Option ... |
cases nonempty_fintype ι
rw [Fintype.linearIndependent_iff]; push_neg
rcases hextr.exists_multipliers_of_hasStrictFDerivAt hf' hφ' with ⟨Λ, Λ₀, hΛ, hΛf⟩
refine ⟨Option.elim' Λ₀ Λ, ?_, ?_⟩
· simpa [add_comm] using hΛf
· simpa only [Function.funext_iff, not_and_or, or_comm, Option.exists, Prod.mk_eq_zero, Ne... |
/-
Copyright (c) 2021 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.NormedSpace.BoundedLinearMaps
import Mathlib.MeasureTheory.Measure.WithDensity
import Mathlib.MeasureTheory.Function.SimpleFunc... | Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | 779 | 795 | theorem _root_.stronglyMeasurable_of_tendsto {ι : Type*} {m : MeasurableSpace α}
[TopologicalSpace β] [PseudoMetrizableSpace β] (u : Filter ι) [NeBot u] [IsCountablyGenerated u]
{f : ι → α → β} {g : α → β} (hf : ∀ i, StronglyMeasurable (f i)) (lim : Tendsto f u (𝓝 g)) :
StronglyMeasurable g := by |
borelize β
refine stronglyMeasurable_iff_measurable_separable.2 ⟨?_, ?_⟩
· exact measurable_of_tendsto_metrizable' u (fun i => (hf i).measurable) lim
· rcases u.exists_seq_tendsto with ⟨v, hv⟩
have : IsSeparable (closure (⋃ i, range (f (v i)))) :=
.closure <| .iUnion fun i => (hf (v i)).isSeparable_r... |
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Order.Monoid.Unbundled.Basic
import Mathlib.Order.Lattice
#align_import algebra.order.sub.defs from "le... | Mathlib/Algebra/Order/Sub/Defs.lean | 94 | 94 | theorem tsub_le_iff_left : a - b ≤ c ↔ a ≤ b + c := by | rw [tsub_le_iff_right, add_comm]
|
/-
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.Finsupp.Multiset
import Mathlib.Order.Bounded
import Mathlib.SetTheory.Cardinal.PartENat
import Mathlib.SetTheor... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 569 | 570 | theorem aleph_mul_aleph (o₁ o₂ : Ordinal) : aleph o₁ * aleph o₂ = aleph (max o₁ o₂) := by |
rw [Cardinal.mul_eq_max (aleph0_le_aleph o₁) (aleph0_le_aleph o₂), max_aleph_eq]
|
/-
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.InnerProductSpace.Orientation
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
#align_import measure_theory.measure.haar.inner_prod... | Mathlib/MeasureTheory/Measure/Haar/InnerProductSpace.lean | 48 | 54 | theorem Orientation.measure_eq_volume (o : Orientation ℝ F (Fin n)) :
o.volumeForm.measure = volume := by |
have A : o.volumeForm.measure (stdOrthonormalBasis ℝ F).toBasis.parallelepiped = 1 :=
Orientation.measure_orthonormalBasis o (stdOrthonormalBasis ℝ F)
rw [addHaarMeasure_unique o.volumeForm.measure
(stdOrthonormalBasis ℝ F).toBasis.parallelepiped, A, one_smul]
simp only [volume, Basis.addHaar]
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Data.Bundle
import Mathlib.Data.Set.Image
import Mathlib.Topology.PartialHomeomorph
import Mathlib.Topology.Order.Basic
#align_import topology.f... | Mathlib/Topology/FiberBundle/Trivialization.lean | 220 | 222 | theorem symm_trans_target_eq (e e' : Pretrivialization F proj) :
(e.toPartialEquiv.symm.trans e'.toPartialEquiv).target = (e.baseSet ∩ e'.baseSet) ×ˢ univ := by |
rw [← PartialEquiv.symm_source, symm_trans_symm, symm_trans_source_eq, inter_comm]
|
/-
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.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory... | Mathlib/MeasureTheory/Measure/Lebesgue/Integral.lean | 22 | 31 | theorem volume_regionBetween_eq_integral' [SigmaFinite μ] (f_int : IntegrableOn f s μ)
(g_int : IntegrableOn g s μ) (hs : MeasurableSet s) (hfg : f ≤ᵐ[μ.restrict s] g) :
μ.prod volume (regionBetween f g s) = ENNReal.ofReal (∫ y in s, (g - f) y ∂μ) := by |
have h : g - f =ᵐ[μ.restrict s] fun x => Real.toNNReal (g x - f x) :=
hfg.mono fun x hx => (Real.coe_toNNReal _ <| sub_nonneg.2 hx).symm
rw [volume_regionBetween_eq_lintegral f_int.aemeasurable g_int.aemeasurable hs,
integral_congr_ae h, lintegral_congr_ae,
lintegral_coe_eq_integral _ ((integrable_cong... |
/-
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 | 1,134 | 1,142 | theorem irreducible_mk {a : α} : Irreducible (Associates.mk a) ↔ Irreducible a := by |
simp only [irreducible_iff, isUnit_mk, forall_associated, isUnit_mk, mk_mul_mk,
mk_eq_mk_iff_associated, Associated.comm (x := a)]
apply Iff.rfl.and
constructor
· rintro h x y rfl
exact h _ _ <| .refl _
· rintro h x y ⟨u, rfl⟩
simpa using h x (y * u) (mul_assoc _ _ _)
|
/-
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, Kexing Ying
-/
import Mathlib.Topology.Semicontinuous
import Mathlib.MeasureTheory.Function.AEMeasurableSequence
import Mathlib.MeasureTheory.Order.Lat... | Mathlib/MeasureTheory/Constructions/BorelSpace/Order.lean | 473 | 475 | theorem Measurable.max {f g : δ → α} (hf : Measurable f) (hg : Measurable g) :
Measurable fun a => max (f a) (g a) := by |
simpa only [max_def'] using hf.piecewise (measurableSet_le hg hf) hg
|
/-
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
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.RingTheory.Ideal.Operations
import Mathlib.RingTheory.JacobsonIdeal
import Mathlib.Logic.Equiv.... | Mathlib/RingTheory/Ideal/LocalRing.lean | 297 | 319 | theorem local_hom_TFAE (f : R →+* S) :
List.TFAE
[IsLocalRingHom f, f '' (maximalIdeal R).1 ⊆ maximalIdeal S,
(maximalIdeal R).map f ≤ maximalIdeal S, maximalIdeal R ≤ (maximalIdeal S).comap f,
(maximalIdeal S).comap f = maximalIdeal R] := by |
tfae_have 1 → 2
· rintro _ _ ⟨a, ha, rfl⟩
exact map_nonunit f a ha
tfae_have 2 → 4
· exact Set.image_subset_iff.1
tfae_have 3 ↔ 4
· exact Ideal.map_le_iff_le_comap
tfae_have 4 → 1
· intro h
constructor
exact fun x => not_imp_not.1 (@h x)
tfae_have 1 → 5
· intro
ext
exact not_iff... |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Mario Carneiro
-/
import Mathlib.Data.List.Count
import Mathlib.Data.List.Dedup
import Mathlib.Data.List.InsertNth
import Mathlib.Data.List.Lat... | Mathlib/Data/List/Perm.lean | 178 | 185 | theorem rel_perm_imp (hr : RightUnique r) : (Forall₂ r ⇒ Forall₂ r ⇒ (· → ·)) Perm Perm :=
fun a b h₁ c d h₂ h =>
have : (flip (Forall₂ r) ∘r Perm ∘r Forall₂ r) b d := ⟨a, h₁, c, h, h₂⟩
have : ((flip (Forall₂ r) ∘r Forall₂ r) ∘r Perm) b d := by |
rwa [← forall₂_comp_perm_eq_perm_comp_forall₂, ← Relation.comp_assoc] at this
let ⟨b', ⟨c', hbc, hcb⟩, hbd⟩ := this
have : b' = b := right_unique_forall₂' hr hcb hbc
this ▸ hbd
|
/-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.GroupPower.IterateHom
import Mathlib.Algebra.Regular.Basic
#align_import algebra.regular.pow fr... | Mathlib/Algebra/Regular/Pow.lean | 47 | 50 | theorem IsLeftRegular.pow_iff {n : ℕ} (n0 : 0 < n) : IsLeftRegular (a ^ n) ↔ IsLeftRegular a := by |
refine ⟨?_, IsLeftRegular.pow n⟩
rw [← Nat.succ_pred_eq_of_pos n0, pow_succ]
exact IsLeftRegular.of_mul
|
/-
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 | 1,075 | 1,079 | theorem isIdempotentElem_apply_one_iff {f : Module.End R R} :
IsIdempotentElem (f 1) ↔ IsIdempotentElem f := by |
rw [IsIdempotentElem, ← smul_eq_mul, ← map_smul, smul_eq_mul, mul_one, IsIdempotentElem, ext_iff]
simp_rw [mul_apply]
exact ⟨fun h r ↦ by rw [← mul_one r, ← smul_eq_mul, map_smul, map_smul, h], (· 1)⟩
|
/-
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.GroupTheory.GroupAction.ConjAct
import Mathlib.GroupTheory.GroupAction.Quotient
import Mathlib.GroupTheory.QuotientGrou... | Mathlib/Topology/Algebra/Group/Basic.lean | 1,971 | 1,973 | theorem QuotientGroup.continuous_smul₁ (x : G ⧸ Γ) : Continuous fun g : G => g • x := by |
induction x using QuotientGroup.induction_on
exact continuous_quotient_mk'.comp (continuous_mul_right _)
|
/-
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 | 176 | 183 | theorem IsInvariantBlock.isFixedBlock {B : Set X} (hfB : IsInvariantBlock G B) :
IsFixedBlock G B := by |
intro g
apply le_antisymm (hfB g)
intro x hx
rw [Set.mem_smul_set_iff_inv_smul_mem]
apply hfB g⁻¹
rwa [Set.smul_mem_smul_set_iff]
|
/-
Copyright (c) 2020 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Valuation.PrimeMultiplicity
import Mathlib.RingTheory... | Mathlib/RingTheory/DiscreteValuationRing/Basic.lean | 310 | 315 | theorem ofHasUnitMulPowIrreducibleFactorization {R : Type u} [CommRing R] [IsDomain R]
(hR : HasUnitMulPowIrreducibleFactorization R) : DiscreteValuationRing R := by |
letI : UniqueFactorizationMonoid R := hR.toUniqueFactorizationMonoid
apply of_ufd_of_unique_irreducible _ hR.unique_irreducible
obtain ⟨p, hp, H⟩ := hR
exact ⟨p, hp⟩
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Sean Leather
-/
import Mathlib.Data.List.Range
import Mathlib.Data.List.Perm
#align_import data.list.sigma from "leanprover-community/mathlib"@"f808feb6c18afddb25e66a7... | Mathlib/Data/List/Sigma.lean | 123 | 125 | theorem NodupKeys.eq_of_mk_mem {a : α} {b b' : β a} {l : List (Sigma β)} (nd : NodupKeys l)
(h : Sigma.mk a b ∈ l) (h' : Sigma.mk a b' ∈ l) : b = b' := by |
cases nd.eq_of_fst_eq h h' rfl; rfl
|
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Data.ZMod.Quotient
#align_import group_theory.complement from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
/-!
# Compl... | Mathlib/GroupTheory/Complement.lean | 447 | 451 | theorem equiv_snd_eq_self_of_mem_of_one_mem {g : G} (h1 : 1 ∈ S) (hg : g ∈ T) :
(hST.equiv g).snd = ⟨g, hg⟩ := by |
have : hST.equiv.symm (⟨1, h1⟩, ⟨g, hg⟩) = g := by
rw [equiv, Equiv.ofBijective]; simp
conv_lhs => rw [← this, Equiv.apply_symm_apply]
|
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Fin.VecNotation
import Mathlib.Logic.Equiv.Fin
#align_import data.fi... | Mathlib/Data/Fin/Tuple/NatAntidiagonal.lean | 96 | 119 | theorem nodup_antidiagonalTuple (k n : ℕ) : List.Nodup (antidiagonalTuple k n) := by |
induction' k with k ih generalizing n
· cases n
· simp
· simp [eq_comm]
simp_rw [antidiagonalTuple, List.nodup_bind]
constructor
· intro i _
exact (ih i.snd).map (Fin.cons_right_injective (α := fun _ => ℕ) i.fst)
induction' n with n n_ih
· exact List.pairwise_singleton _ _
· rw [List.Nat.an... |
/-
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.Algebra.Homology.Linear
import Mathlib.Algebra.Homology.ShortComplex.HomologicalComplex
import Mathlib.Tactic.Abel
#align_import algebra.homology.homo... | Mathlib/Algebra/Homology/Homotopy.lean | 394 | 400 | theorem nullHomotopicMap'_f {k₂ k₁ k₀ : ι} (r₂₁ : c.Rel k₂ k₁) (r₁₀ : c.Rel k₁ k₀)
(h : ∀ i j, c.Rel j i → (C.X i ⟶ D.X j)) :
(nullHomotopicMap' h).f k₁ = C.d k₁ k₀ ≫ h k₀ k₁ r₁₀ + h k₁ k₂ r₂₁ ≫ D.d k₂ k₁ := by |
simp only [nullHomotopicMap']
rw [nullHomotopicMap_f r₂₁ r₁₀]
split_ifs
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.Topology.Order.MonotoneContinuity
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.E... | Mathlib/Topology/Instances/ENNReal.lean | 1,038 | 1,041 | theorem tsum_iUnion_le {ι : Type*} [Fintype ι] (f : α → ℝ≥0∞) (t : ι → Set α) :
∑' x : ⋃ i, t i, f x ≤ ∑ i, ∑' x : t i, f x := by |
rw [← tsum_fintype]
exact tsum_iUnion_le_tsum f t
|
/-
Copyright (c) 2022 Aaron Anderson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson
-/
import Mathlib.ModelTheory.Satisfiability
#align_import model_theory.types from "leanprover-community/mathlib"@"98bd247d933fb581ff37244a5998bd33d81dd46d"
/-!
# Type... | Mathlib/ModelTheory/Types.lean | 195 | 196 | theorem formula_mem_typeOf {φ : L.Formula α} :
Formula.equivSentence φ ∈ T.typeOf v ↔ φ.Realize v := by | simp
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Ring.Divisibility.Lemmas
import Mathlib.Algebra.Lie.Nilpotent
import Mathlib.Algebra.Lie.Engel
import Mathlib.LinearAlgebra.Eigenspace.Triangularizab... | Mathlib/Algebra/Lie/Weights/Basic.lean | 168 | 170 | theorem coe_weightSpaceOf_zero (x : L) :
↑(weightSpaceOf M (0 : R) x) = ⨆ k, LinearMap.ker (toEnd R L M x ^ k) := by |
simp [weightSpaceOf, Module.End.maxGenEigenspace]
|
/-
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.GroupWithZero.Divisibility
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.Data.Finset.So... | Mathlib/Algebra/Polynomial/Basic.lean | 281 | 282 | theorem ofFinsupp_eq_zero {a} : (⟨a⟩ : R[X]) = 0 ↔ a = 0 := by |
rw [← ofFinsupp_zero, ofFinsupp_inj]
|
/-
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 | 1,754 | 1,762 | theorem dualAnnihilator_dualAnnihilator_eq_map (W : Subspace K V) [FiniteDimensional K W] :
W.dualAnnihilator.dualAnnihilator = W.map (Dual.eval K V) := by |
let e1 := (Free.chooseBasis K W).toDualEquiv ≪≫ₗ W.quotAnnihilatorEquiv.symm
haveI := e1.finiteDimensional
let e2 := (Free.chooseBasis K _).toDualEquiv ≪≫ₗ W.dualAnnihilator.dualQuotEquivDualAnnihilator
haveI := LinearEquiv.finiteDimensional (V₂ := W.dualAnnihilator.dualAnnihilator) e2
rw [FiniteDimensional.... |
/-
Copyright (c) 2020 Yury Kudryashov, Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.Fin
import Mathlib.GroupTheo... | Mathlib/Algebra/BigOperators/Fin.lean | 218 | 222 | theorem prod_trunc {M : Type*} [CommMonoid M] {a b : ℕ} (f : Fin (a + b) → M)
(hf : ∀ j : Fin b, f (natAdd a j) = 1) :
(∏ i : Fin (a + b), f i) = ∏ i : Fin a, f (castLE (Nat.le.intro rfl) i) := by |
rw [prod_univ_add, Fintype.prod_eq_one _ hf, mul_one]
rfl
|
/-
Copyright (c) 2022 Yaël Dillies, Sara Rousta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Sara Rousta
-/
import Mathlib.Data.SetLike.Basic
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Order.Interval.Set.OrderIso
import Mathlib.Data.... | Mathlib/Order/UpperLower/Basic.lean | 1,209 | 1,211 | theorem map_Ioi (f : α ≃o β) (a : α) : map f (Ioi a) = Ioi (f a) := by |
ext
simp
|
/-
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.List.Basic
#align_import data.list.infix from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
/-!
# Prefixes, suffixes... | Mathlib/Data/List/Infix.lean | 239 | 244 | theorem concat_get_prefix {x y : List α} (h : x <+: y) (hl : x.length < y.length) :
x ++ [y.get ⟨x.length, hl⟩] <+: y := by |
use y.drop (x.length + 1)
nth_rw 1 [List.prefix_iff_eq_take.mp h]
convert List.take_append_drop (x.length + 1) y using 2
rw [← List.take_concat_get, List.concat_eq_append]; rfl
|
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Nat.Defs
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.Co... | Mathlib/Data/Fin/Basic.lean | 1,207 | 1,209 | theorem lt_castPred_iff {j : Fin n} {i : Fin (n + 1)} (hi : i ≠ last n) :
j < castPred i hi ↔ castSucc j < i := by |
rw [← castSucc_lt_castSucc_iff, castSucc_castPred]
|
/-
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.MeasureTheory.Constructions.Pi
import Mathlib.Probability.Kernel.Basic
/-!
# Independence with respect to a kernel and a measure
A family of sets of sets... | Mathlib/Probability/Independence/Kernel.lean | 838 | 930 | theorem iIndepFun.indepFun_finset [IsMarkovKernel κ] (S T : Finset ι) (hST : Disjoint S T)
(hf_Indep : iIndepFun m f κ μ) (hf_meas : ∀ i, Measurable (f i)) :
IndepFun (fun a (i : S) => f i a) (fun a (i : T) => f i a) κ μ := by |
-- We introduce π-systems, built from the π-system of boxes which generates `MeasurableSpace.pi`.
let πSβ := Set.pi (Set.univ : Set S) ''
Set.pi (Set.univ : Set S) fun i => { s : Set (β i) | MeasurableSet[m i] s }
let πS := { s : Set Ω | ∃ t ∈ πSβ, (fun a (i : S) => f i a) ⁻¹' t = s }
have hπS_pi : IsPiSys... |
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Order.LiminfLimsup
import Mathlib.Topology.Instances.Rat
import Mathlib.Top... | Mathlib/Analysis/Normed/Group/Basic.lean | 836 | 838 | theorem NormedCommGroup.cauchySeq_iff [Nonempty α] [SemilatticeSup α] {u : α → E} :
CauchySeq u ↔ ∀ ε > 0, ∃ N, ∀ m, N ≤ m → ∀ n, N ≤ n → ‖u m / u n‖ < ε := by |
simp [Metric.cauchySeq_iff, dist_eq_norm_div]
|
/-
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.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
imp... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean | 282 | 285 | theorem mul_lt_sin {x : ℝ} (hx : 0 < x) (hx' : x < π / 2) : 2 / π * x < sin x := by |
rw [← inv_div]
simpa [-inv_div, mul_inv_cancel_left₀ pi_div_two_pos.ne'] using @lt_sin_mul ((π / 2)⁻¹ * x)
(mul_pos (inv_pos.2 pi_div_two_pos) hx) (by rwa [← div_eq_inv_mul, div_lt_one pi_div_two_pos])
|
/-
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.Algebra.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Fintype.... | Mathlib/LinearAlgebra/Multilinear/Basic.lean | 1,643 | 1,648 | theorem MultilinearMap.curry_uncurryRight
(f : MultilinearMap R (fun i : Fin n => M (castSucc i)) (M (last n) →ₗ[R] M₂)) :
f.uncurryRight.curryRight = f := by |
ext m x
simp only [snoc_last, MultilinearMap.curryRight_apply, MultilinearMap.uncurryRight_apply]
rw [init_snoc]
|
/-
Copyright (c) 2021 Bryan Gin-ge Chen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Bryan Gin-ge Chen, Yaël Dillies
-/
import Mathlib.Order.BooleanAlgebra
import Mathlib.Logic.Equiv.Basic
#align_import order.symm_diff from "leanprover-community/mathlib... | Mathlib/Order/SymmDiff.lean | 639 | 639 | theorem bihimp_right_comm : a ⇔ b ⇔ c = a ⇔ c ⇔ b := by | simp_rw [bihimp_assoc, bihimp_comm]
|
/-
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.Algebra.Algebra.Equiv
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearA... | Mathlib/Algebra/Quaternion.lean | 515 | 515 | theorem coe_mul : ((x * y : R) : ℍ[R,c₁,c₂]) = x * y := by | ext <;> 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, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle
import Mathlib.Analysis.SpecialFunctions.T... | Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | 434 | 435 | theorem arg_conj_coe_angle (x : ℂ) : (arg (conj x) : Real.Angle) = -arg x := by |
by_cases h : arg x = π <;> simp [arg_conj, h]
|
/-
Copyright (c) 2022 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.InnerProductSpace.PiL2
import Mathlib.Combinatorics.Additive.AP.Three.Defs
import Mathlib.Combinatorics.Pigeonhole
imp... | Mathlib/Combinatorics/Additive/AP/Three/Behrend.lean | 105 | 105 | theorem box_zero : box (n + 1) 0 = ∅ := by | simp [box]
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Data.Option.Defs
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Sigma.Basic
import Mathlib... | Mathlib/Logic/Equiv/Basic.lean | 1,857 | 1,858 | theorem piCongrLeft'_symm (P : Sort*) (e : α ≃ β) :
(piCongrLeft' (fun _ => P) e).symm = piCongrLeft' _ e.symm := by | ext; simp [piCongrLeft']
|
/-
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, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Order.MinMax
import Mathlib.Data.Set.Subsingleton
import Mathlib.Tactic.Says
#align_imp... | Mathlib/Order/Interval/Set/Basic.lean | 1,394 | 1,400 | theorem Icc_union_Ici (h : c ≤ max a b) : Icc a b ∪ Ici c = Ici (min a c) := by |
rcases le_or_lt a b with hab | hab <;> simp [hab] at h
· exact Icc_union_Ici' h
· cases' h with h h
· simp [*]
· have hca : c ≤ a := h.trans hab.le
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.Hull
#align_import analysis.convex.extreme from "leanprover-community/mathlib"@"c5773405394e073885e2a144c9ca14... | Mathlib/Analysis/Convex/Extreme.lean | 126 | 127 | theorem isExtreme_sInter {F : Set (Set E)} (hF : F.Nonempty) (hAF : ∀ B ∈ F, IsExtreme 𝕜 A B) :
IsExtreme 𝕜 A (⋂₀ F) := by | simpa [sInter_eq_biInter] using isExtreme_biInter hF hAF
|
/-
Copyright (c) 2019 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann, Kyle Miller, Mario Carneiro
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Finset.NatAntidiagonal
import Mathlib.Data.Nat.GCD.Basic
import ... | Mathlib/Data/Nat/Fib/Basic.lean | 265 | 278 | theorem gcd_fib_add_self (m n : ℕ) : gcd (fib m) (fib (n + m)) = gcd (fib m) (fib n) := by |
rcases Nat.eq_zero_or_pos n with h | h
· rw [h]
simp
replace h := Nat.succ_pred_eq_of_pos h; rw [← h, succ_eq_add_one]
calc
gcd (fib m) (fib (n.pred + 1 + m)) =
gcd (fib m) (fib n.pred * fib m + fib (n.pred + 1) * fib (m + 1)) := by
rw [← fib_add n.pred _]
ring_nf
_ = gcd (f... |
/-
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 | 1,703 | 1,704 | theorem atTop_Ici_eq [SemilatticeSup α] (a : α) : atTop = comap ((↑) : Ici a → α) atTop := by |
rw [← map_val_Ici_atTop a, comap_map Subtype.coe_injective]
|
/-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Regular.Basic
import Mathlib.GroupTheory.GroupAction.Hom
#align_import algebra.regular.smul from "leanprover-community/mathlib"@"550b58538991c89... | Mathlib/Algebra/Regular/SMul.lean | 156 | 160 | theorem pow (n : ℕ) (ra : IsSMulRegular M a) : IsSMulRegular M (a ^ n) := by |
induction' n with n hn
· rw [pow_zero]; simp only [one]
· rw [pow_succ']
exact (ra.smul_iff (a ^ n)).mpr hn
|
/-
Copyright (c) 2021 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Data.Array.Lemmas
import Batteries.Tactic.Lint.Misc
namespace Batteries
/-- Union-find node type -/
structure UFNode where
/-- Parent of node -/
... | .lake/packages/batteries/Batteries/Data/UnionFind/Basic.lean | 137 | 139 | theorem rank'_lt (self : UnionFind) (i : Fin self.size) : (self.arr.get i).parent ≠ i →
self.rank i < self.rank (self.arr.get i).parent := by |
simpa only [← parentD_eq] using self.rankD_lt
|
/-
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.Algebra.Basic
import Mathlib.Algebra.Periodic
import Mathlib.Topology.Algebra.Order.Field
import Mathlib.Topology.Algebra.Unifo... | Mathlib/Topology/Instances/Real.lean | 154 | 155 | theorem Real.totallyBounded_ball (x ε : ℝ) : TotallyBounded (ball x ε) := by |
rw [Real.ball_eq_Ioo]; apply totallyBounded_Ioo
|
/-
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.Calculus.FDeriv.Linear
import Mathlib.Analysis.Calculus.FDeriv.Comp
#align_import analysis.calculus.fderiv.... | Mathlib/Analysis/Calculus/FDeriv/Prod.lean | 411 | 417 | theorem hasStrictFDerivAt_apply (i : ι) (f : ∀ i, F' i) :
HasStrictFDerivAt (𝕜:=𝕜) (fun f : ∀ i, F' i => f i) (proj i) f := by |
let id' := ContinuousLinearMap.id 𝕜 (∀ i, F' i)
have h := ((hasStrictFDerivAt_pi'
(Φ := fun (f : ∀ i, F' i) (i' : ι) => f i') (Φ':=id') (x:=f))).1
have h' : comp (proj i) id' = proj i := by rfl
rw [← h']; apply h; apply hasStrictFDerivAt_id
|
/-
Copyright (c) 2020 Bhavik Mehta, E. W. Ayers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, E. W. Ayers
-/
import Mathlib.CategoryTheory.Sites.Sieves
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Limits.Shapes.Mul... | Mathlib/CategoryTheory/Sites/Grothendieck.lean | 206 | 212 | theorem arrow_trans (f : Y ⟶ X) (S R : Sieve X) (h : J.Covers S f) :
(∀ {Z : C} (g : Z ⟶ X), S g → J.Covers R g) → J.Covers R f := by |
intro k
apply J.transitive h
intro Z g hg
rw [← Sieve.pullback_comp]
apply k (g ≫ f) hg
|
/-
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.AlgebraicTopology.SimplicialObject
import Mathlib.CategoryTheory.Limits.Shapes.Products
#align_import algebraic_topology.split_simplicial_object from "leanprove... | Mathlib/AlgebraicTopology/SplitSimplicialObject.lean | 352 | 363 | theorem Hom.ext {S₁ S₂ : Split C} (Φ₁ Φ₂ : Hom S₁ S₂) (h : ∀ n : ℕ, Φ₁.f n = Φ₂.f n) : Φ₁ = Φ₂ := by |
rcases Φ₁ with ⟨F₁, f₁, c₁⟩
rcases Φ₂ with ⟨F₂, f₂, c₂⟩
have h' : f₁ = f₂ := by
ext
apply h
subst h'
simp only [mk.injEq, and_true]
apply S₁.s.hom_ext
intro n
dsimp
rw [c₁, c₂]
|
/-
Copyright (c) 2022 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Polynomial.Cardinal
import Mathlib.Algebra.MvPolynomial.Cardinal
import Mathlib.Data.ZMod.Algebra
import Mathlib.FieldTheory.IsAlgClosed.Basic
impo... | Mathlib/FieldTheory/IsAlgClosed/Classification.lean | 96 | 100 | theorem isAlgClosure_of_transcendence_basis [IsAlgClosed K] (hv : IsTranscendenceBasis R v) :
IsAlgClosure (Algebra.adjoin R (Set.range v)) K :=
letI := RingHom.domain_nontrivial (algebraMap R K)
{ alg_closed := by | infer_instance
algebraic := hv.isAlgebraic }
|
/-
Copyright (c) 2018 Jan-David Salchow. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jan-David Salchow, Patrick Massot, Yury Kudryashov
-/
import Mathlib.Topology.Defs.Sequences
import Mathlib.Topology.UniformSpace.Cauchy
#align_import topology.sequences from "lean... | Mathlib/Topology/Sequences.lean | 113 | 116 | theorem mem_closure_iff_seq_limit [FrechetUrysohnSpace X] {s : Set X} {a : X} :
a ∈ closure s ↔ ∃ x : ℕ → X, (∀ n : ℕ, x n ∈ s) ∧ Tendsto x atTop (𝓝 a) := by |
rw [← seqClosure_eq_closure]
rfl
|
/-
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 | 249 | 256 | theorem vadd_mem_of_mem_direction {s : AffineSubspace k P} {v : V} (hv : v ∈ s.direction) {p : P}
(hp : p ∈ s) : v +ᵥ p ∈ s := by |
rw [mem_direction_iff_eq_vsub ⟨p, hp⟩] at hv
rcases hv with ⟨p1, hp1, p2, hp2, hv⟩
rw [hv]
convert s.smul_vsub_vadd_mem 1 hp1 hp2 hp
rw [one_smul]
exact s.mem_coe k P _
|
/-
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.NormedSpace.AddTorsor
import Mathlib.LinearAlgebra.AffineSpace.Ordered
import Mathlib.Topology.ContinuousFunction.Basic
import Mathlib... | Mathlib/Topology/UrysohnsLemma.lean | 247 | 248 | theorem lim_of_mem_C (c : CU P) (x : X) (h : x ∈ c.C) : c.lim x = 0 := by |
simp only [CU.lim, approx_of_mem_C, h, ciSup_const]
|
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov, Sébastien Gouëzel
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
import Mathlib.Topology.Order.LeftRightLim
#align_import measure_t... | Mathlib/MeasureTheory/Measure/Stieltjes.lean | 71 | 73 | theorem rightLim_eq (f : StieltjesFunction) (x : ℝ) : Function.rightLim f x = f x := by |
rw [← f.mono.continuousWithinAt_Ioi_iff_rightLim_eq, continuousWithinAt_Ioi_iff_Ici]
exact f.right_continuous' x
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kevin Kappelmann
-/
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Algebra.Group.Int
import Mathlib.Data.Int.Lemm... | Mathlib/Algebra/Order/Floor.lean | 1,260 | 1,260 | theorem ceil_add_nat (a : α) (n : ℕ) : ⌈a + n⌉ = ⌈a⌉ + n := by | rw [← Int.cast_natCast, ceil_add_int]
|
/-
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.Topology.Algebra.Algebra
import Mathlib.Topology.ContinuousFunction.Compact
import Mathlib.Topology.UrysohnsLemma
import Mathlib.Analysis.RCLike.Basic
im... | Mathlib/Topology/ContinuousFunction/Ideals.lean | 108 | 109 | theorem not_mem_idealOfSet {s : Set X} {f : C(X, R)} : f ∉ idealOfSet R s ↔ ∃ x ∈ sᶜ, f x ≠ 0 := by |
simp_rw [mem_idealOfSet]; push_neg; rfl
|
/-
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
-/
import Mathlib.Tactic.FinCases
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Algebra.Field.IsField
#alig... | Mathlib/RingTheory/Ideal/Basic.lean | 554 | 557 | theorem span_singleton_lt_span_singleton [IsDomain α] {x y : α} :
span ({x} : Set α) < span ({y} : Set α) ↔ DvdNotUnit y x := by |
rw [lt_iff_le_not_le, span_singleton_le_span_singleton, span_singleton_le_span_singleton,
dvd_and_not_dvd_iff]
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.LinearAlgebra.Span
#align_import algebra.algebra.tower from "leanprover-community/mathlib"@"71150516f28d9826c7... | Mathlib/Algebra/Algebra/Tower.lean | 318 | 320 | theorem smul_mem_span_smul' {s : Set S} (hs : span R s = ⊤) {t : Set A} {k : S} {x : A}
(hx : x ∈ span R (s • t)) : k • x ∈ span R (s • t) := by |
rw [span_smul_of_span_eq_top hs] at hx ⊢; exact (span S t).smul_mem k hx
|
/-
Copyright (c) 2021 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov, Malo Jaffré
-/
import Mathlib.Analysis.Convex.Function
import Mathlib.Tactic.AdaptationNote
import Mathlib.Tactic.FieldSimp
import Mathlib.Tactic.Linarith
#align_imp... | Mathlib/Analysis/Convex/Slope.lean | 335 | 341 | theorem StrictConcaveOn.secant_strict_mono (hf : StrictConcaveOn 𝕜 s f) {a x y : 𝕜} (ha : a ∈ s)
(hx : x ∈ s) (hy : y ∈ s) (hxa : x ≠ a) (hya : y ≠ a) (hxy : x < y) :
(f y - f a) / (y - a) < (f x - f a) / (x - a) := by |
have key := hf.neg.secant_strict_mono ha hx hy hxa hya hxy
simp only [Pi.neg_apply] at key
rw [← neg_lt_neg_iff]
convert key using 1 <;> field_simp <;> ring
|
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Data.Finite.Card
import Mathlib.GroupTheory.Finiteness
import Mathlib.GroupTheory.GroupActio... | Mathlib/GroupTheory/Index.lean | 89 | 91 | theorem relindex_comap {G' : Type*} [Group G'] (f : G' →* G) (K : Subgroup G') :
relindex (comap f H) K = relindex H (map f K) := by |
rw [relindex, subgroupOf, comap_comap, index_comap, ← f.map_range, K.subtype_range]
|
/-
Copyright (c) 2022 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.ParametricIntegral
import Mathlib.MeasureTheory.Constructions.Prod.Integral
impor... | Mathlib/Analysis/Convolution.lean | 379 | 388 | theorem _root_.BddAbove.convolutionExistsAt [MeasurableAdd₂ G] [SigmaFinite μ] {x₀ : G} {s : Set G}
(hbg : BddAbove ((fun i => ‖g i‖) '' ((fun t => x₀ - t) ⁻¹' s))) (hs : MeasurableSet s)
(h2s : (support fun t => L (f t) (g (x₀ - t))) ⊆ s) (hf : IntegrableOn f s μ)
(hmg : AEStronglyMeasurable g μ) : Convolu... |
refine BddAbove.convolutionExistsAt' L ?_ hs h2s hf ?_
· simp_rw [← sub_eq_neg_add, hbg]
· have : AEStronglyMeasurable g (map (fun t : G => x₀ - t) μ) :=
hmg.mono_ac (quasiMeasurePreserving_sub_left_of_right_invariant μ x₀).absolutelyContinuous
apply this.mono_measure
exact map_mono restrict_le_sel... |
/-
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.Module.LinearMap.Basic
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Basis
import Mathlib.LinearAlgebra.BilinearMap
#ali... | Mathlib/LinearAlgebra/SesquilinearForm.lean | 73 | 74 | theorem isOrtho_flip {B : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₁'] M} {x y} : B.IsOrtho x y ↔ B.flip.IsOrtho y x := by |
simp_rw [isOrtho_def, flip_apply]
|
/-
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.RepresentationTheory.Action.Limits
import Mathlib.RepresentationTheory.Action.Concrete
import Mathlib.CategoryTheory.Monoidal.FunctorCategory
import Ma... | Mathlib/RepresentationTheory/Action/Monoidal.lean | 90 | 93 | theorem associator_inv_hom {X Y Z : Action V G} :
Hom.hom (α_ X Y Z).inv = (α_ X.V Y.V Z.V).inv := by |
dsimp
simp
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn
-/
import Mathlib.Geometry.Manifold.MFDeriv.Defs
#align_import geometry.manifold.mfderiv from "leanprover-community/mathlib"@"e473c3198bb41f6856... | Mathlib/Geometry/Manifold/MFDeriv/Basic.lean | 46 | 49 | theorem uniqueMDiffWithinAt_univ : UniqueMDiffWithinAt I univ x := by |
unfold UniqueMDiffWithinAt
simp only [preimage_univ, univ_inter]
exact I.unique_diff _ (mem_range_self _)
|
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll, Thomas Zhu, Mario Carneiro
-/
import Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity
#align_import number_theory.legendre_symbol.jacobi_symbol from "leanprover-... | Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean | 371 | 380 | theorem even_odd {a : ℤ} {b : ℕ} (ha2 : a % 2 = 0) (hb2 : b % 2 = 1) :
(if b % 8 = 3 ∨ b % 8 = 5 then -J(a / 2 | b) else J(a / 2 | b)) = J(a | b) := by |
obtain ⟨a, rfl⟩ := Int.dvd_of_emod_eq_zero ha2
rw [Int.mul_ediv_cancel_left _ (by decide), jacobiSym.mul_left,
jacobiSym.at_two (Nat.odd_iff.mpr hb2), ZMod.χ₈_nat_eq_if_mod_eight,
if_neg (Nat.mod_two_ne_zero.mpr hb2)]
have := Nat.mod_lt b (by decide : 0 < 8)
interval_cases h : b % 8 <;> simp_all <;>
... |
/-
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 | 470 | 476 | theorem assert_pos {p : Prop} {f : p → Part α} (h : p) : assert p f = f h := by |
dsimp [assert]
cases h' : f h
simp only [h', mk.injEq, h, exists_prop_of_true, true_and]
apply Function.hfunext
· simp only [h, h', exists_prop_of_true]
· aesop
|
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Lu-Ming Zhang
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algeb... | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 609 | 613 | theorem invOf_diagonal_eq {α} [Semiring α] (v : n → α) [Invertible v] [Invertible (diagonal v)] :
⅟ (diagonal v) = diagonal (⅟ v) := by |
letI := diagonalInvertible v
-- Porting note: no longer need `haveI := Invertible.subsingleton (diagonal v)`
convert (rfl : ⅟ (diagonal v) = _)
|
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Mohanad Ahmed
-/
import Mathlib.LinearAlgebra.Matrix.Spectrum
import Mathlib.LinearAlgebra.QuadraticForm.Basic
#align_import linear_algebra.matrix.pos_def from... | Mathlib/LinearAlgebra/Matrix/PosDef.lean | 90 | 93 | theorem transpose {M : Matrix n n R} (hM : M.PosSemidef) : Mᵀ.PosSemidef := by |
refine ⟨IsHermitian.transpose hM.1, fun x => ?_⟩
convert hM.2 (star x) using 1
rw [mulVec_transpose, Matrix.dotProduct_mulVec, star_star, dotProduct_comm]
|
/-
Copyright (c) 2022 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Heather Macbeth
-/
import Mathlib.Data.Nat.Cast.WithTop
import Mathlib.FieldTheory.IsAlgClosed.Basic
import Mathlib.RingTheory.WittVector.DiscreteValuationRing
#alig... | Mathlib/RingTheory/WittVector/FrobeniusFractionField.lean | 241 | 267 | theorem exists_frobenius_solution_fractionRing_aux (m n : ℕ) (r' q' : 𝕎 k) (hr' : r'.coeff 0 ≠ 0)
(hq' : q'.coeff 0 ≠ 0) (hq : (p : 𝕎 k) ^ n * q' ∈ nonZeroDivisors (𝕎 k)) :
let b : 𝕎 k := frobeniusRotation p hr' hq'
IsFractionRing.fieldEquivOfRingEquiv (frobeniusEquiv p k)
(algebraMap (𝕎 k) (... |
intro b
have key : WittVector.frobenius b * (p : 𝕎 k) ^ m * r' * (p : 𝕎 k) ^ n =
(p : 𝕎 k) ^ m * b * ((p : 𝕎 k) ^ n * q') := by
have H := congr_arg (fun x : 𝕎 k => x * (p : 𝕎 k) ^ m * (p : 𝕎 k) ^ n)
(frobenius_frobeniusRotation p hr' hq')
dsimp at H
refine (Eq.trans ?_ H).trans ?_ <;... |
/-
Copyright (c) 2022 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Topology.UniformSpace.UniformConvergenceTopology
#align_import topology.uniform_space.equicontinuity from "leanprover-community/mathlib"@"f2ce6086... | Mathlib/Topology/UniformSpace/Equicontinuity.lean | 556 | 559 | theorem uniformEquicontinuousOn_iff_uniformContinuousOn {F : ι → β → α} {S : Set β} :
UniformEquicontinuousOn F S ↔ UniformContinuousOn (ofFun ∘ Function.swap F : β → ι →ᵤ α) S := by |
rw [UniformContinuousOn, (UniformFun.hasBasis_uniformity ι α).tendsto_right_iff]
rfl
|
/-
Copyright (c) 2022 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.Algebra.Module.Defs
import Mathlib.Algebra.Order.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Ring.Basic
import Mathl... | Mathlib/Combinatorics/SimpleGraph/Regularity/Energy.lean | 46 | 57 | theorem energy_le_one : P.energy G ≤ 1 :=
div_le_of_nonneg_of_le_mul (sq_nonneg _) zero_le_one <|
calc
∑ uv ∈ P.parts.offDiag, G.edgeDensity uv.1 uv.2 ^ 2 ≤ P.parts.offDiag.card • (1 : ℚ) :=
sum_le_card_nsmul _ _ 1 fun uv _ =>
(sq_le_one_iff <| G.edgeDensity_nonneg _ _).2 <| G.edgeDensity_... |
rw [offDiag_card, one_mul]
norm_cast
rw [sq]
exact tsub_le_self
|
/-
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.MeasureTheory.Measure.NullMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Topology.Algebra.Order.LiminfLim... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 170 | 172 | theorem measure_sUnion {S : Set (Set α)} (hs : S.Countable) (hd : S.Pairwise Disjoint)
(h : ∀ s ∈ S, MeasurableSet s) : μ (⋃₀ S) = ∑' s : S, μ s := by |
rw [sUnion_eq_biUnion, measure_biUnion hs hd h]
|
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Polynomial.Degree.CardPowDegree
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.NumberTheory.ClassNumber.AdmissibleAbsoluteValue
imp... | Mathlib/NumberTheory/ClassNumber/AdmissibleCardPowDegree.lean | 106 | 149 | theorem exists_approx_polynomial {b : Fq[X]} (hb : b ≠ 0) {ε : ℝ} (hε : 0 < ε)
(A : Fin (Fintype.card Fq ^ ⌈-log ε / log (Fintype.card Fq)⌉₊).succ → Fq[X]) :
∃ i₀ i₁, i₀ ≠ i₁ ∧ (cardPowDegree (A i₁ % b - A i₀ % b) : ℝ) < cardPowDegree b • ε := by |
have hbε : 0 < cardPowDegree b • ε := by
rw [Algebra.smul_def, eq_intCast]
exact mul_pos (Int.cast_pos.mpr (AbsoluteValue.pos _ hb)) hε
have one_lt_q : 1 < Fintype.card Fq := Fintype.one_lt_card
have one_lt_q' : (1 : ℝ) < Fintype.card Fq := by assumption_mod_cast
have q_pos : 0 < Fintype.card Fq := by ... |
/-
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 | 82 | 87 | theorem nhdsWithin_Ioi_eq_bot_iff {a : α} : 𝓝[>] a = ⊥ ↔ IsTop a ∨ ∃ b, a ⋖ b := by |
by_cases ha : IsTop a
· simp [ha, ha.isMax.Ioi_eq]
· simp only [ha, false_or]
rw [isTop_iff_isMax, not_isMax_iff] at ha
simp only [(nhdsWithin_Ioi_basis' ha).eq_bot_iff, covBy_iff_Ioo_eq]
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Slope
import Mathlib.Analysis.SpecialFunctions.Pow.Real
import Mathlib.Tactic.LinearCombination
... | Mathlib/Analysis/Convex/SpecificFunctions/Basic.lean | 167 | 175 | theorem rpow_one_add_le_one_add_mul_self {s : ℝ} (hs : -1 ≤ s) {p : ℝ} (hp1 : 0 ≤ p) (hp2 : p ≤ 1) :
(1 + s) ^ p ≤ 1 + p * s := by |
rcases eq_or_lt_of_le hp1 with (rfl | hp1)
· simp
rcases eq_or_lt_of_le hp2 with (rfl | hp2)
· simp
by_cases hs' : s = 0
· simp [hs']
exact (rpow_one_add_lt_one_add_mul_self hs hs' hp1 hp2).le
|
/-
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 | 970 | 974 | theorem card_multiples (n p : ℕ) : card ((Finset.range n).filter fun e => p ∣ e + 1) = n / p := by |
induction' n with n hn
· simp
simp [Nat.succ_div, add_ite, add_zero, Finset.range_succ, filter_insert, apply_ite card,
card_insert_of_not_mem, hn]
|
/-
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 | 1,231 | 1,239 | theorem mem_ideal_of_coeff_mem_ideal (I : Ideal (MvPolynomial σ R)) (p : MvPolynomial σ R)
(hcoe : ∀ m : σ →₀ ℕ, p.coeff m ∈ I.comap (C : R →+* MvPolynomial σ R)) : p ∈ I := by |
rw [as_sum p]
suffices ∀ m ∈ p.support, monomial m (MvPolynomial.coeff m p) ∈ I by
exact Submodule.sum_mem I this
intro m _
rw [← mul_one (coeff m p), ← C_mul_monomial]
suffices C (coeff m p) ∈ I by exact I.mul_mem_right (monomial m 1) this
simpa [Ideal.mem_comap] using hcoe m
|
/-
Copyright (c) 2020 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel, Jakob von Raumer
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Group.Hom.Defs
import Mathlib.Algebra.Module.Defs
import Mathlib.CategoryTheor... | Mathlib/CategoryTheory/Preadditive/Basic.lean | 161 | 161 | theorem neg_comp_neg : (-f) ≫ (-g) = f ≫ g := by | simp
|
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Matthew Robert Ballard
-/
import Mathlib.NumberTheory.Divisors
import Mathlib.Data.Nat.Digits
import Mathlib.Data.Nat.MaxPowDiv
import Mathlib.Data.Nat.Multiplicity
i... | Mathlib/NumberTheory/Padics/PadicVal.lean | 108 | 110 | theorem eq_zero_iff {n : ℕ} : padicValNat p n = 0 ↔ p = 1 ∨ n = 0 ∨ ¬p ∣ n := by |
simp only [padicValNat, dite_eq_right_iff, PartENat.get_eq_iff_eq_coe, Nat.cast_zero,
multiplicity_eq_zero, and_imp, pos_iff_ne_zero, Ne, ← or_iff_not_imp_left]
|
/-
Copyright (c) 2022 Julian Kuelshammer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Kuelshammer
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.BigOperators.NatAntidiagonal
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Data.Finset.... | Mathlib/Combinatorics/Enumerative/Catalan.lean | 181 | 187 | theorem treesOfNumNodesEq_succ (n : ℕ) :
treesOfNumNodesEq (n + 1) =
(antidiagonal n).biUnion fun ij =>
pairwiseNode (treesOfNumNodesEq ij.1) (treesOfNumNodesEq ij.2) := by |
rw [treesOfNumNodesEq]
ext
simp
|
/-
Copyright (c) 2020 Kexing Ying and Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.FiniteSupport
import Mathlib.Algebra... | Mathlib/Algebra/BigOperators/Finprod.lean | 455 | 460 | theorem finprod_cond_ne (f : α → M) (a : α) [DecidableEq α] (hf : (mulSupport f).Finite) :
(∏ᶠ (i) (_ : i ≠ a), f i) = ∏ i ∈ hf.toFinset.erase a, f i := by |
apply finprod_cond_eq_prod_of_cond_iff
intro x hx
rw [Finset.mem_erase, Finite.mem_toFinset, mem_mulSupport]
exact ⟨fun h => And.intro h hx, fun h => h.1⟩
|
/-
Copyright (c) 2020 Kevin Buzzard, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.Products
import Mathl... | Mathlib/CategoryTheory/Sites/Sheaf.lean | 248 | 255 | theorem IsSheaf.amalgamate_map {A : Type u₂} [Category.{v₂} A] {E : A} {X : C} {P : Cᵒᵖ ⥤ A}
(hP : Presheaf.IsSheaf J P) (S : J.Cover X) (x : ∀ I : S.Arrow, E ⟶ P.obj (op I.Y))
(hx : ∀ I : S.Relation, x I.fst ≫ P.map I.g₁.op = x I.snd ≫ P.map I.g₂.op) (I : S.Arrow) :
hP.amalgamate S x hx ≫ P.map I.f.op = x ... |
rcases I with ⟨Y, f, hf⟩
apply
@Presieve.IsSheafFor.valid_glue _ _ _ _ _ _ (hP _ _ S.condition) (fun Y f hf => x ⟨Y, f, hf⟩)
(fun Y₁ Y₂ Z g₁ g₂ f₁ f₂ h₁ h₂ w => hx ⟨Y₁, Y₂, Z, g₁, g₂, f₁, f₂, h₁, h₂, w⟩) f hf
|
/-
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.SetTheory.Game.Short
#align_import set_theory.game.state from "leanprover-community/mathlib"@"b134b2f5cf6dd25d4bbfd3c498b6e36c11a17225"
/-!
# Games d... | Mathlib/SetTheory/Game/State.lean | 50 | 54 | theorem turnBound_ne_zero_of_left_move {s t : S} (m : t ∈ l s) : turnBound s ≠ 0 := by |
intro h
have t := left_bound m
rw [h] at t
exact Nat.not_succ_le_zero _ t
|
/-
Copyright (c) 2021 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Analysis.RCLike.Lemmas
import Mathlib.MeasureTheory.Function.StronglyMeasurable.Inner
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import meas... | Mathlib/MeasureTheory/Function/L2Space.lean | 46 | 51 | theorem memℒp_two_iff_integrable_sq_norm {f : α → F} (hf : AEStronglyMeasurable f μ) :
Memℒp f 2 μ ↔ Integrable (fun x => ‖f x‖ ^ 2) μ := by |
rw [← memℒp_one_iff_integrable]
convert (memℒp_norm_rpow_iff hf two_ne_zero ENNReal.two_ne_top).symm
· simp
· rw [div_eq_mul_inv, ENNReal.mul_inv_cancel two_ne_zero ENNReal.two_ne_top]
|
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