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) 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 | 245 | 260 | theorem LinearMap.continuous_of_finiteDimensional [T2Space E] [FiniteDimensional 𝕜 E]
(f : E →ₗ[𝕜] F') : Continuous f := by |
-- for the proof, go to a model vector space `b → 𝕜` thanks to `continuous_equivFun_basis`, and
-- argue that all linear maps there are continuous.
let b := Basis.ofVectorSpace 𝕜 E
have A : Continuous b.equivFun := continuous_equivFun_basis_aux b
have B : Continuous (f.comp (b.equivFun.symm : (Basis.ofVect... |
/-
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
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Set.Finite
#align_import order.filter.basic from "leanprover-community/mathlib"@"d4f691b9e5... | Mathlib/Order/Filter/Basic.lean | 2,424 | 2,426 | theorem le_comap_top (f : α → β) (l : Filter α) : l ≤ comap f ⊤ := by |
rw [comap_top]
exact le_top
|
/-
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 | 187 | 205 | theorem isRat_add {α} [Ring α] {f : α → α → α} {a b : α} {na nb nc : ℤ} {da db dc k : ℕ} :
f = HAdd.hAdd → IsRat a na da → IsRat b nb db →
Int.add (Int.mul na db) (Int.mul nb da) = 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 * db + nb * da = k * nc) (h₂ : da * db = k * dc)
have : Invertible (↑(da * db) : α) := by simpa using invertibleMul (da:α) db
have := invertibleOfMul' (α := α) h₂
use this
have H := (Nat.cast_commute (α := α) da db).invOf_left.invOf_right.right_comm
have h₁ := congr_ar... |
/-
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.Group.Commute.Defs
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Algebra.Order.Monoid.Unbundled.Bas... | Mathlib/Algebra/Regular/Basic.lean | 248 | 252 | theorem IsLeftRegular.ne_zero [Nontrivial R] (la : IsLeftRegular a) : a ≠ 0 := by |
rintro rfl
rcases exists_pair_ne R with ⟨x, y, xy⟩
refine xy (la (?_ : 0 * x = 0 * y)) -- Porting note: lean4 seems to need the type signature
rw [zero_mul, zero_mul]
|
/-
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.FieldTheory.PrimitiveElement
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.Matrix.Char... | Mathlib/RingTheory/Norm.lean | 214 | 221 | theorem _root_.IntermediateField.AdjoinSimple.norm_gen_eq_one {x : L} (hx : ¬IsIntegral K x) :
norm K (AdjoinSimple.gen K x) = 1 := by |
rw [norm_eq_one_of_not_exists_basis]
contrapose! hx
obtain ⟨s, ⟨b⟩⟩ := hx
refine .of_mem_of_fg K⟮x⟯.toSubalgebra ?_ x ?_
· exact (Submodule.fg_iff_finiteDimensional _).mpr (of_fintype_basis b)
· exact IntermediateField.subset_adjoin K _ (Set.mem_singleton x)
|
/-
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.Algebra.Squarefree.Basic
import Mathlib.Data.Nat.Factorization.PrimePow
#align_import data.nat.squarefree from "leanprover-community/mathlib"@"3c1368c... | Mathlib/Data/Nat/Squarefree.lean | 45 | 56 | theorem _root_.Squarefree.natFactorization_le_one {n : ℕ} (p : ℕ) (hn : Squarefree n) :
n.factorization p ≤ 1 := by |
rcases eq_or_ne n 0 with (rfl | hn')
· simp
rw [multiplicity.squarefree_iff_multiplicity_le_one] at hn
by_cases hp : p.Prime
· have := hn p
simp only [multiplicity_eq_factorization hp hn', Nat.isUnit_iff, hp.ne_one, or_false_iff]
at this
exact mod_cast this
· rw [factorization_eq_zero_of_non_... |
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Floris van Doorn
-/
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra.Opposites
import Mathlib.Algebra.Order.GroupW... | Mathlib/Data/Set/Pointwise/Basic.lean | 303 | 304 | theorem image_op_inv : op '' s⁻¹ = (op '' s)⁻¹ := by |
simp_rw [← image_inv, Function.Semiconj.set_image op_inv s]
|
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Felix Weilacher
-/
import Mathlib.Data.Real.Cardinality
import Mathlib.Topology.MetricSpace.Perfect
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metric
i... | Mathlib/MeasureTheory/Constructions/Polish.lean | 345 | 355 | theorem _root_.MeasurableSet.analyticSet_image {X Y : Type*} [MeasurableSpace X]
[StandardBorelSpace X] [TopologicalSpace Y] [MeasurableSpace Y]
[OpensMeasurableSpace Y] {f : X → Y} [SecondCountableTopology (range f)] {s : Set X}
(hs : MeasurableSet s) (hf : Measurable f) : AnalyticSet (f '' s) := by |
letI := upgradeStandardBorel X
rw [eq_borel_upgradeStandardBorel X] at hs
rcases hf.exists_continuous with ⟨τ', hle, hfc, hτ'⟩
letI m' : MeasurableSpace X := @borel _ τ'
haveI b' : BorelSpace X := ⟨rfl⟩
have hle := borel_anti hle
exact (hle _ hs).analyticSet.image_of_continuous hfc
|
/-
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.Polynomial.Eval
#align_import data.polynomial.degree.lemmas from "leanprover-community/mathlib"@"7... | Mathlib/Algebra/Polynomial/Degree/Lemmas.lean | 438 | 440 | theorem monic_mul_leadingCoeff_inv {p : K[X]} (h : p ≠ 0) : Monic (p * C (leadingCoeff p)⁻¹) := by |
rw [Monic, leadingCoeff_mul, leadingCoeff_C,
mul_inv_cancel (show leadingCoeff p ≠ 0 from mt leadingCoeff_eq_zero.1 h)]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Data.Bool.Set
import Mathlib.Data.Nat.Set
import Mathlib.Data.Set.Prod
import Mathlib.Data.ULift
import Mathlib.Order.Bounds.Basic
import Mathlib.Order... | Mathlib/Order/CompleteLattice.lean | 1,224 | 1,229 | theorem biSup_sup {p : ι → Prop} {f : ∀ i, p i → α} {a : α} (h : ∃ i, p i) :
(⨆ (i) (h : p i), f i h) ⊔ a = ⨆ (i) (h : p i), f i h ⊔ a := by |
haveI : Nonempty { i // p i } :=
let ⟨i, hi⟩ := h
⟨⟨i, hi⟩⟩
rw [iSup_subtype', iSup_subtype', iSup_sup]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Finset.Update
import Mathlib.Data.Prod.TProd
import Mathlib.GroupTheory.Coset
import Mathlib.Logic.Equiv.Fin
import Mathlib.Measur... | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | 1,046 | 1,053 | theorem measurableSet_pi_of_nonempty {s : Set δ} {t : ∀ i, Set (π i)} (hs : s.Countable)
(h : (pi s t).Nonempty) : MeasurableSet (pi s t) ↔ ∀ i ∈ s, MeasurableSet (t i) := by |
classical
rcases h with ⟨f, hf⟩
refine ⟨fun hst i hi => ?_, MeasurableSet.pi hs⟩
convert measurable_update f (a := i) hst
rw [update_preimage_pi hi]
exact fun j hj _ => hf j hj
|
/-
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 | 671 | 679 | theorem sizeOf_dedupKeys [DecidableEq α] [SizeOf (Sigma β)]
(xs : List (Sigma β)) : SizeOf.sizeOf (dedupKeys xs) ≤ SizeOf.sizeOf xs := by |
simp only [SizeOf.sizeOf, _sizeOf_1]
induction' xs with x xs
· simp [dedupKeys]
· simp only [dedupKeys_cons, kinsert_def, Nat.add_le_add_iff_left, Sigma.eta]
trans
· apply sizeOf_kerase
· assumption
|
/-
Copyright (c) 2020 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.Degree.Definitions
import Mathlib.Data.ENat.Basic
#align_import data.polynomial.degree.trailing_degree from "leanprover-community/mat... | Mathlib/Algebra/Polynomial/Degree/TrailingDegree.lean | 294 | 297 | theorem le_trailingDegree_C_mul_X_pow (n : ℕ) (a : R) :
(n : ℕ∞) ≤ trailingDegree (C a * X ^ n) := by |
rw [C_mul_X_pow_eq_monomial]
exact le_trailingDegree_monomial
|
/-
Copyright (c) 2020 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Order.ProjIcc
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.UnitInterval
#align_import topology.path_connected from "leanprover... | Mathlib/Topology/Connected/PathConnected.lean | 730 | 742 | theorem range_reparam (γ : Path x y) {f : I → I} (hfcont : Continuous f) (hf₀ : f 0 = 0)
(hf₁ : f 1 = 1) : range (γ.reparam f hfcont hf₀ hf₁) = range γ := by |
change range (γ ∘ f) = range γ
have : range f = univ := by
rw [range_iff_surjective]
intro t
have h₁ : Continuous (Set.IccExtend (zero_le_one' ℝ) f) := by continuity
have := intermediate_value_Icc (zero_le_one' ℝ) h₁.continuousOn
· rw [IccExtend_left, IccExtend_right, Icc.mk_zero, Icc.mk_one, h... |
/-
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.Associated
import Mathlib.Algebra.Ring.Regular
import Mathlib.Tactic.Common
#align_import algebra.gcd_monoid.basic from "leanp... | Mathlib/Algebra/GCDMonoid/Basic.lean | 538 | 540 | theorem dvd_mul_gcd_of_dvd_mul [GCDMonoid α] {m n k : α} (H : k ∣ m * n) : k ∣ m * gcd k n := by |
rw [mul_comm] at H ⊢
exact dvd_gcd_mul_of_dvd_mul H
|
/-
Copyright (c) 2023 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Topology.MetricSpace.Algebra
import Mathlib.Analysis.Normed.Field.Basic
#align_import analysis.normed.mul_action from "leanprover-community/mathlib"@"bc91ed... | Mathlib/Analysis/Normed/MulAction.lean | 29 | 30 | theorem norm_smul_le (r : α) (x : β) : ‖r • x‖ ≤ ‖r‖ * ‖x‖ := by |
simpa [smul_zero] using dist_smul_pair r 0 x
|
/-
Copyright (c) 2020 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Manuel Candales
-/
import Mathlib.Analysis.Convex.Between
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
import Mat... | Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean | 61 | 64 | theorem _root_.AffineIsometry.angle_map {V₂ P₂ : Type*} [NormedAddCommGroup V₂]
[InnerProductSpace ℝ V₂] [MetricSpace P₂] [NormedAddTorsor V₂ P₂]
(f : P →ᵃⁱ[ℝ] P₂) (p₁ p₂ p₃ : P) : ∠ (f p₁) (f p₂) (f p₃) = ∠ p₁ p₂ p₃ := by |
simp_rw [angle, ← AffineIsometry.map_vsub, LinearIsometry.angle_map]
|
/-
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 | 393 | 403 | theorem mem_rcons_iff {i j : ι} (p : Pair M i) (m : M j) :
⟨_, m⟩ ∈ (rcons p).toList ↔ ⟨_, m⟩ ∈ p.tail.toList ∨
m ≠ 1 ∧ (∃ h : i = j, m = h ▸ p.head) := by |
simp only [rcons, cons, ne_eq]
by_cases hij : i = j
· subst i
by_cases hm : m = p.head
· subst m
split_ifs <;> simp_all
· split_ifs <;> simp_all
· split_ifs <;> simp_all [Ne.symm hij]
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Kexing Ying, Eric Wieser
-/
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.LinearAlgebra.Matrix.SesquilinearForm
import Mathlib.LinearAlgebra.Matrix.Sym... | Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | 98 | 100 | theorem polar_add (f g : M → R) (x y : M) : polar (f + g) x y = polar f x y + polar g x y := by |
simp only [polar, Pi.add_apply]
abel
|
/-
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.SetTheory.Cardinal.ENat
#align_import set_theory.cardinal.basic from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
/-!
# ... | Mathlib/SetTheory/Cardinal/ToNat.lean | 60 | 61 | theorem cast_toNat_of_aleph0_le {c : Cardinal} (h : ℵ₀ ≤ c) : ↑(toNat c) = (0 : Cardinal) := by |
rw [toNat_apply_of_aleph0_le h, Nat.cast_zero]
|
/-
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.Algebra.Pi
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.RingTheory.Adjoin.Basic
#align_im... | Mathlib/Algebra/Polynomial/AlgebraMap.lean | 356 | 359 | theorem aeval_eq_zero_of_dvd_aeval_eq_zero [CommSemiring S] [CommSemiring T] [Algebra S T]
{p q : S[X]} (h₁ : p ∣ q) {a : T} (h₂ : aeval a p = 0) : aeval a q = 0 := by |
rw [aeval_def, ← eval_map] at h₂ ⊢
exact eval_eq_zero_of_dvd_of_eval_eq_zero (Polynomial.map_dvd (algebraMap S T) h₁) h₂
|
/-
Copyright (c) 2023 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Category.ModuleCat.ChangeOfRings
import Mathlib.Algebra.Category.Ring.Basic
/-!
# Presheaves of modules over a presheaf of rings.
We give a h... | Mathlib/Algebra/Category/ModuleCat/Presheaf.lean | 137 | 141 | theorem ext {f g : P ⟶ Q} (w : ∀ X, f.app X = g.app X) : f = g := by |
cases f; cases g
congr
ext X x
exact LinearMap.congr_fun (w X) x
|
/-
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.BigOperators.Fin
import Mathlib.Algebra.GeomSum
import Mathlib.LinearAlgebra.Matrix.Block
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
impor... | Mathlib/LinearAlgebra/Vandermonde.lean | 59 | 64 | theorem vandermonde_succ {n : ℕ} (v : Fin n.succ → R) :
vandermonde v =
Fin.cons (fun (j : Fin n.succ) => v 0 ^ (j : ℕ)) fun i =>
Fin.cons 1 fun j => v i.succ * vandermonde (Fin.tail v) i j := by |
conv_lhs => rw [← Fin.cons_self_tail v, vandermonde_cons]
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, Kevin Buzzard, Yury Kudryashov, Frédéric Dupuis,
Heather Macbeth
-/
import Mathlib.Algebra.Module.Submodule.Ker
#align_import linear_algebra.basic fr... | Mathlib/Algebra/Module/Submodule/Range.lean | 179 | 180 | theorem _root_.Submodule.map_comap_eq_self [RingHomSurjective τ₁₂] {f : F} {q : Submodule R₂ M₂}
(h : q ≤ range f) : map f (comap f q) = q := by | rwa [Submodule.map_comap_eq, inf_eq_right]
|
/-
Copyright (c) 2020 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker
-/
import Mathlib.Analysis.Calculus.MeanValue
import Mathlib.Analysis.Calculus.Deriv.Inv
#align_import analysis.calculus.lhopital from "leanprover-community/mathlib... | Mathlib/Analysis/Calculus/LHopital.lean | 231 | 241 | theorem lhopital_zero_left_on_Ioo (hdf : DifferentiableOn ℝ f (Ioo a b))
(hg' : ∀ x ∈ Ioo a b, (deriv g) x ≠ 0) (hfb : Tendsto f (𝓝[<] b) (𝓝 0))
(hgb : Tendsto g (𝓝[<] b) (𝓝 0))
(hdiv : Tendsto (fun x => (deriv f) x / (deriv g) x) (𝓝[<] b) l) :
Tendsto (fun x => f x / g x) (𝓝[<] b) l := by |
have hdf : ∀ x ∈ Ioo a b, DifferentiableAt ℝ f x := fun x hx =>
(hdf x hx).differentiableAt (Ioo_mem_nhds hx.1 hx.2)
have hdg : ∀ x ∈ Ioo a b, DifferentiableAt ℝ g x := fun x hx =>
by_contradiction fun h => hg' x hx (deriv_zero_of_not_differentiableAt h)
exact HasDerivAt.lhopital_zero_left_on_Ioo hab (fu... |
/-
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.NumberTheory.FLT.Basic
import Mathlib.NumberTheory.PythagoreanTriples
import Mathlib.RingTheory.Coprime.Lemmas
import Mathlib.Tactic.LinearCombinat... | Mathlib/NumberTheory/FLT/Four.lean | 159 | 162 | theorem Int.coprime_of_sq_sum' {r s : ℤ} (h : IsCoprime r s) :
IsCoprime (r ^ 2 + s ^ 2) (r * s) := by |
apply IsCoprime.mul_right (Int.coprime_of_sq_sum (isCoprime_comm.mp h))
rw [add_comm]; apply Int.coprime_of_sq_sum h
|
/-
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 | 700 | 719 | theorem normalizedFactors_mul {x y : α} (hx : x ≠ 0) (hy : y ≠ 0) :
normalizedFactors (x * y) = normalizedFactors x + normalizedFactors y := by |
have h : (normalize : α → α) = Associates.out ∘ Associates.mk := by
ext
rw [Function.comp_apply, Associates.out_mk]
rw [← Multiset.map_id' (normalizedFactors (x * y)), ← Multiset.map_id' (normalizedFactors x), ←
Multiset.map_id' (normalizedFactors y), ← Multiset.map_congr rfl normalize_normalized_facto... |
/-
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.Degree.Definitions
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.Algebra.Polynomial.... | Mathlib/RingTheory/Polynomial/Pochhammer.lean | 64 | 66 | theorem ascPochhammer_succ_left (n : ℕ) :
ascPochhammer S (n + 1) = X * (ascPochhammer S n).comp (X + 1) := by |
rw [ascPochhammer]
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Simon Hudon, Mario Carneiro
-/
import Aesop
import Mathlib.Algebra.Group.Defs
import Mathlib.Data.Nat.Defs
import Mathlib.Data.Int.Defs
import Mathlib.... | Mathlib/Algebra/Group/Basic.lean | 1,337 | 1,337 | theorem div_mul_cancel_left (a b : G) : a / (a * b) = b⁻¹ := by | rw [← inv_div, mul_div_cancel_left]
|
/-
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 | 763 | 769 | theorem associated_iff_normalizedFactors_eq_normalizedFactors {x y : α} (hx : x ≠ 0) (hy : y ≠ 0) :
x ~ᵤ y ↔ normalizedFactors x = normalizedFactors y := by |
refine
⟨fun h => ?_, fun h =>
(normalizedFactors_prod hx).symm.trans (_root_.trans (by rw [h]) (normalizedFactors_prod hy))⟩
apply le_antisymm <;> rw [← dvd_iff_normalizedFactors_le_normalizedFactors]
all_goals simp [*, h.dvd, h.symm.dvd]
|
/-
Copyright (c) 2020 Thomas Browning, Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.FieldTheory.Fixed
import Mathlib.FieldTheory.NormalClosure
import Mathlib.FieldTheory.PrimitiveElement
import Mathlib.Gro... | Mathlib/FieldTheory/Galois.lean | 220 | 230 | theorem fixingSubgroup_fixedField [FiniteDimensional F E] : fixingSubgroup (fixedField H) = H := by |
have H_le : H ≤ fixingSubgroup (fixedField H) := (le_iff_le _ _).mp le_rfl
classical
suffices Fintype.card H = Fintype.card (fixingSubgroup (fixedField H)) by
exact SetLike.coe_injective (Set.eq_of_inclusion_surjective
((Fintype.bijective_iff_injective_and_card (Set.inclusion H_le)).mpr
⟨Set.in... |
/-
Copyright (c) 2020 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Utensil Song
-/
import Mathlib.Algebra.RingQuot
import Mathlib.LinearAlgebra.TensorAlgebra.Basic
import Mathlib.LinearAlgebra.QuadraticForm.Isometry
import Mathlib.LinearAlge... | Mathlib/LinearAlgebra/CliffordAlgebra/Basic.lean | 333 | 334 | theorem map_id : map (QuadraticForm.Isometry.id Q₁) = AlgHom.id R (CliffordAlgebra Q₁) := by |
ext m; exact map_apply_ι _ m
|
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying, Rémy Degenne
-/
import Mathlib.Probability.Process.Adapted
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import probability.process.stopping from "leanp... | Mathlib/Probability/Process/Stopping.lean | 788 | 789 | theorem stoppedProcess_eq_of_le {u : ι → Ω → β} {τ : Ω → ι} {i : ι} {ω : Ω} (h : i ≤ τ ω) :
stoppedProcess u τ i ω = u i ω := by | simp [stoppedProcess, min_eq_left h]
|
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris Van Doorn, Yury Kudryashov
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_imp... | Mathlib/MeasureTheory/Measure/Regular.lean | 369 | 374 | theorem _root_.MeasurableSet.exists_isOpen_diff_lt [OuterRegular μ] {A : Set α}
(hA : MeasurableSet A) (hA' : μ A ≠ ∞) {ε : ℝ≥0∞} (hε : ε ≠ 0) :
∃ U, U ⊇ A ∧ IsOpen U ∧ μ U < ∞ ∧ μ (U \ A) < ε := by |
rcases A.exists_isOpen_lt_add hA' hε with ⟨U, hAU, hUo, hU⟩
use U, hAU, hUo, hU.trans_le le_top
exact measure_diff_lt_of_lt_add hA hAU hA' hU
|
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, E. W. Ayers
-/
import Mathlib.CategoryTheory.Comma.Over
import Mathlib.CategoryTheory.Limits.Shapes.Pullbacks
import Mathlib.CategoryTheory.Yoneda
import Mathlib.Data.Set.L... | Mathlib/CategoryTheory/Sites/Sieves.lean | 449 | 452 | theorem generate_of_contains_isSplitEpi {R : Presieve X} (f : Y ⟶ X) [IsSplitEpi f] (hf : R f) :
generate R = ⊤ := by |
rw [← id_mem_iff_eq_top]
exact ⟨_, section_ f, f, hf, by simp⟩
|
/-
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.Order.Bounds.Basic
import Mathlib.Order.Hom.Set
#align_import order.bounds.order_iso from "leanprover-community/mathlib"@"a59dad53320... | Mathlib/Order/Bounds/OrderIso.lean | 41 | 42 | theorem isLUB_image' {s : Set α} {x : α} : IsLUB (f '' s) (f x) ↔ IsLUB s x := by |
rw [isLUB_image, f.symm_apply_apply]
|
/-
Copyright (c) 2022 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Fourier.FourierTransform
import Mathlib.Analysis.InnerProductSpace.Dual
import Mathlib.Analysis.InnerProductSpace.EuclideanDist
import Mathlib... | Mathlib/Analysis/Fourier/RiemannLebesgueLemma.lean | 68 | 92 | theorem fourierIntegral_half_period_translate {w : V} (hw : w ≠ 0) :
(∫ v : V, 𝐞 (-⟪v, w⟫) • f (v + i w)) = -∫ v : V, 𝐞 (-⟪v, w⟫) • f v := by |
have hiw : ⟪i w, w⟫ = 1 / 2 := by
rw [inner_smul_left, inner_self_eq_norm_sq_to_K, RCLike.ofReal_real_eq_id, id,
RCLike.conj_to_real, ← div_div, div_mul_cancel₀]
rwa [Ne, sq_eq_zero_iff, norm_eq_zero]
have :
(fun v : V => 𝐞 (-⟪v, w⟫) • f (v + i w)) =
fun v : V => (fun x : V => -(𝐞 (-⟪x, w... |
/-
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.Calculus.ContDiff.Basic
import Mathlib.Analysis.Calculus.UniformLimitsDeriv
import Mathlib.Topology.Algebra.InfiniteSum.Module
import Ma... | Mathlib/Analysis/Calculus/SmoothSeries.lean | 161 | 165 | theorem differentiable_tsum' (hu : Summable u) (hg : ∀ n y, HasDerivAt (g n) (g' n y) y)
(hg' : ∀ n y, ‖g' n y‖ ≤ u n) : Differentiable 𝕜 fun z => ∑' n, g n z := by |
simp_rw [hasDerivAt_iff_hasFDerivAt] at hg
refine differentiable_tsum hu hg ?_
simpa? says simpa only [ContinuousLinearMap.norm_smulRight_apply, norm_one, one_mul]
|
/-
Copyright (c) 2022 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Yaël Dillies
-/
import Mathlib.MeasureTheory.Integral.SetIntegral
#align_import measure_theory.integral.average from "leanprover-community/mathlib"@"c14c8fcde9... | Mathlib/MeasureTheory/Integral/Average.lean | 648 | 651 | theorem exists_not_mem_null_integral_le (hf : Integrable f μ) (hN : μ N = 0) :
∃ x, x ∉ N ∧ ∫ a, f a ∂μ ≤ f x := by |
simpa only [average_eq_integral] using
exists_not_mem_null_average_le (IsProbabilityMeasure.ne_zero μ) hf hN
|
/-
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 | 452 | 454 | theorem inner_add_right (x y z : E) : ⟪x, y + z⟫ = ⟪x, y⟫ + ⟪x, z⟫ := by |
rw [← inner_conj_symm, inner_add_left, RingHom.map_add]
simp only [inner_conj_symm]
|
/-
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 | 352 | 358 | theorem rightDistributor_ext₂_left {J : Type} [Fintype J]
{f : J → C} {X Y Z : C} {g h : ((⨁ f) ⊗ X) ⊗ Y ⟶ Z}
(w : ∀ j, ((biproduct.ι f j ▷ X) ▷ Y) ≫ g = ((biproduct.ι f j ▷ X) ▷ Y) ≫ h) :
g = h := by |
apply (cancel_epi (α_ _ _ _).inv).mp
ext
simp [w]
|
/-
Copyright (c) 2023 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.NonUnitalSubalgebra
import Mathlib.Algebra.Star.StarAlgHom
import Mathlib.Algebra.Star.Center
/-!
# Non-unital Star Subalgebras
In this... | Mathlib/Algebra/Star/NonUnitalSubalgebra.lean | 1,019 | 1,023 | theorem iSupLift_of_mem {i : ι} (x : T) (hx : (x : A) ∈ K i) :
iSupLift K dir f hf T hT x = f i ⟨x, hx⟩ := by |
subst hT
dsimp [iSupLift]
apply Set.iUnionLift_of_mem
|
/-
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.DoldKan.Projections
import Mathlib.CategoryTheory.Idempotents.FunctorCategories
import Mathlib.CategoryTheory.Idempotents.FunctorExtension
#al... | Mathlib/AlgebraicTopology/DoldKan/PInfty.lean | 36 | 42 | theorem P_is_eventually_constant {q n : ℕ} (hqn : n ≤ q) :
((P (q + 1)).f n : X _[n] ⟶ _) = (P q).f n := by |
rcases n with (_|n)
· simp only [Nat.zero_eq, P_f_0_eq]
· simp only [P_succ, add_right_eq_self, comp_add, HomologicalComplex.comp_f,
HomologicalComplex.add_f_apply, comp_id]
exact (HigherFacesVanish.of_P q n).comp_Hσ_eq_zero (Nat.succ_le_iff.mp hqn)
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Order.Lattice
#align_import order.min_max from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd516abe4025"
/-!
# `max` and `min`
This ... | Mathlib/Order/MinMax.lean | 196 | 197 | theorem min_lt_min_right_iff : min a b < min a c ↔ b < c ∧ b < a := by |
simp_rw [min_comm a, min_lt_min_left_iff]
|
/-
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.Analysis.Convex.Side
import Mathlib.Geometry.Euclidean.Angle.Oriented.Rotation
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Affine
#align_import geo... | Mathlib/Geometry/Euclidean/Angle/Oriented/Affine.lean | 585 | 587 | theorem oangle_midpoint_left (p₁ p₂ p₃ : P) : ∡ (midpoint ℝ p₁ p₂) p₂ p₃ = ∡ p₁ p₂ p₃ := by |
by_cases h : p₁ = p₂; · simp [h]
exact (sbtw_midpoint_of_ne ℝ h).symm.oangle_eq_left
|
/-
Copyright (c) 2019 Calle Sönne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic
import Mathlib.Analysis.Normed.Group.AddCircle
import Mathlib.Algebra.CharZero.Quotient
import Mathlib.Topology... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | 352 | 354 | theorem sin_eq_iff_eq_or_add_eq_pi {θ ψ : Angle} : sin θ = sin ψ ↔ θ = ψ ∨ θ + ψ = π := by |
induction ψ using Real.Angle.induction_on
exact sin_eq_real_sin_iff_eq_or_add_eq_pi
|
/-
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.Lie.Submodule
#align_import algebra.lie.ideal_operations from "leanprover-community/mathlib"@"8983bec7cdf6cb2dd1f21315c8a34ab00d7b2f6d"
/-!
# Ideal... | Mathlib/Algebra/Lie/IdealOperations.lean | 131 | 131 | theorem lie_bot : ⁅I, (⊥ : LieSubmodule R L M)⁆ = ⊥ := by | rw [eq_bot_iff]; apply lie_le_right
|
/-
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.Order.BooleanAlgebra
import Mathlib.Tactic.Common
#align_import order.heyting.boundary from "leanprover-community/mathlib"@"70d50ecfd4900dd6d328da39ab7ebd... | Mathlib/Order/Heyting/Boundary.lean | 132 | 132 | theorem boundary_idem (a : α) : ∂ ∂ a = ∂ a := by | rw [boundary, hnot_boundary, inf_top_eq]
|
/-
Copyright (c) 2021 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Algebra.Quasispectrum
import Mathlib.FieldTheory.IsAlgClosed.Spectrum
import Mathlib.Analysis.Complex.Liouville
import Mathlib.Analysis.Complex.P... | Mathlib/Analysis/NormedSpace/Spectrum.lean | 252 | 257 | theorem hasDerivAt_resolvent {a : A} {k : 𝕜} (hk : k ∈ ρ a) :
HasDerivAt (resolvent a) (-resolvent a k ^ 2) k := by |
have H₁ : HasFDerivAt Ring.inverse _ (↑ₐ k - a) := hasFDerivAt_ring_inverse (𝕜 := 𝕜) hk.unit
have H₂ : HasDerivAt (fun k => ↑ₐ k - a) 1 k := by
simpa using (Algebra.linearMap 𝕜 A).hasDerivAt.sub_const a
simpa [resolvent, sq, hk.unit_spec, ← Ring.inverse_unit hk.unit] using H₁.comp_hasDerivAt k H₂
|
/-
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 | 338 | 347 | theorem IsSFiniteKernel.finset_sum {κs : ι → kernel α β} (I : Finset ι)
(h : ∀ i ∈ I, IsSFiniteKernel (κs i)) : IsSFiniteKernel (∑ i ∈ I, κs i) := by |
classical
induction' I using Finset.induction with i I hi_nmem_I h_ind h
· rw [Finset.sum_empty]; infer_instance
· rw [Finset.sum_insert hi_nmem_I]
haveI : IsSFiniteKernel (κs i) := h i (Finset.mem_insert_self _ _)
have : IsSFiniteKernel (∑ x ∈ I, κs x) :=
h_ind fun i hiI => h i (Finset.mem_inser... |
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Integral.PeakFunction
#align_import analysis.special_functions.trigonometric.euler_si... | Mathlib/Analysis/SpecialFunctions/Trigonometric/EulerSineProd.lean | 278 | 295 | theorem tendsto_integral_cos_pow_mul_div {f : ℝ → ℂ} (hf : ContinuousOn f (Icc 0 (π / 2))) :
Tendsto
(fun n : ℕ => (∫ x in (0 : ℝ)..π / 2, (cos x : ℂ) ^ n * f x) /
(∫ x in (0 : ℝ)..π / 2, cos x ^ n : ℝ))
atTop (𝓝 <| f 0) := by |
simp_rw [div_eq_inv_mul (α := ℂ), ← Complex.ofReal_inv, integral_of_le pi_div_two_pos.le,
← MeasureTheory.integral_Icc_eq_integral_Ioc, ← Complex.ofReal_pow, ← Complex.real_smul]
have c_lt : ∀ y : ℝ, y ∈ Icc 0 (π / 2) → y ≠ 0 → cos y < cos 0 := fun y hy hy' =>
cos_lt_cos_of_nonneg_of_le_pi_div_two (le_refl... |
/-
Copyright (c) 2023 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir
-/
import Mathlib.GroupTheory.Perm.Cycle.Concrete
/-! # Possible cycle types of permutations
* For `m : Multiset ℕ`, `Equiv.Perm.exists_with_cycleType_i... | Mathlib/GroupTheory/Perm/Cycle/PossibleTypes.lean | 21 | 74 | theorem Equiv.Perm.exists_with_cycleType_iff {m : Multiset ℕ} :
(∃ g : Equiv.Perm α, g.cycleType = m) ↔
(m.sum ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a) := by |
constructor
· -- empty case
intro h
obtain ⟨g, hg⟩ := h
constructor
· rw [← hg, Equiv.Perm.sum_cycleType]
exact (Equiv.Perm.support g).card_le_univ
· intro a
rw [← hg]
exact Equiv.Perm.two_le_of_mem_cycleType
· rintro ⟨hc, h2c⟩
have hc' : m.toList.sum ≤ Fintype.card α :=... |
/-
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.Topology.Instances.ENNReal
#align_import order.filter.ennreal from "leanprover-community/mathlib"@"52932b3a083d4142e78a15dc928084a22fea9ba0"
/-!
# Order ... | Mathlib/Order/Filter/ENNReal.lean | 86 | 93 | theorem limsup_liminf_le_liminf_limsup {β} [Countable β] {f : Filter α} [CountableInterFilter f]
{g : Filter β} (u : α → β → ℝ≥0∞) :
(f.limsup fun a : α => g.liminf fun b : β => u a b) ≤
g.liminf fun b => f.limsup fun a => u a b :=
have h1 : ∀ᶠ a in f, ∀ b, u a b ≤ f.limsup fun a' => u a' b := by |
rw [eventually_countable_forall]
exact fun b => ENNReal.eventually_le_limsup fun a => u a b
sInf_le <| h1.mono fun x hx => Filter.liminf_le_liminf (Filter.eventually_of_forall hx)
|
/-
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.Algebra.BigOperators.Finsupp
import Mathlib.Data.Finset.Pointwise
import Mathlib.Data.Finsupp.Indicator
import Mathlib.Data.Fintype.BigOperators
#align_im... | Mathlib/Data/Finset/Finsupp.lean | 62 | 74 | theorem mem_finsupp_iff_of_support_subset {t : ι →₀ Finset α} (ht : t.support ⊆ s) :
f ∈ s.finsupp t ↔ ∀ i, f i ∈ t i := by |
refine
mem_finsupp_iff.trans
(forall_and.symm.trans <|
forall_congr' fun i =>
⟨fun h => ?_, fun h =>
⟨fun hi => ht <| mem_support_iff.2 fun H => mem_support_iff.1 hi ?_, fun _ => h⟩⟩)
· by_cases hi : i ∈ s
· exact h.2 hi
· rw [not_mem_support_iff.1 (mt h.1 hi), not_m... |
/-
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.LinearAlgebra.ExteriorAlgebra.Basic
import Mathlib.LinearAlgebra.CliffordAlgebra.Fold
import Mathlib.LinearAlgebra.CliffordAlgebra.Conjugation
import Mathlib... | Mathlib/LinearAlgebra/CliffordAlgebra/Contraction.lean | 249 | 253 | theorem changeFormAux_changeFormAux (B : BilinForm R M) (v : M) (x : CliffordAlgebra Q) :
changeFormAux Q B v (changeFormAux Q B v x) = (Q v - B v v) • x := by |
simp only [changeFormAux_apply_apply]
rw [mul_sub, ← mul_assoc, ι_sq_scalar, map_sub, contractLeft_ι_mul, ← sub_add, sub_sub_sub_comm,
← Algebra.smul_def, sub_self, sub_zero, contractLeft_contractLeft, add_zero, sub_smul]
|
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.SpecialFunctions.Gamma.Basic
import Mathlib.Analysis.SpecialFunctions.PolarCoord
import Mathlib.Analysis.Convex.Complex
#align_import a... | Mathlib/Analysis/SpecialFunctions/Gaussian/GaussianIntegral.lean | 183 | 201 | theorem integral_mul_cexp_neg_mul_sq {b : ℂ} (hb : 0 < b.re) :
∫ r : ℝ in Ioi 0, (r : ℂ) * cexp (-b * (r : ℂ) ^ 2) = (2 * b)⁻¹ := by |
have hb' : b ≠ 0 := by contrapose! hb; rw [hb, zero_re]
have A : ∀ x : ℂ, HasDerivAt (fun x => -(2 * b)⁻¹ * cexp (-b * x ^ 2))
(x * cexp (-b * x ^ 2)) x := by
intro x
convert ((hasDerivAt_pow 2 x).const_mul (-b)).cexp.const_mul (-(2 * b)⁻¹) using 1
field_simp [hb']
ring
have B : Tendsto (fun ... |
/-
Copyright (c) 2021 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Joël Riou
-/
import Mathlib.Algebra.Homology.Homotopy
import Mathlib.Algebra.Homology.SingleHomology
import Mathlib.CategoryTheory.Abelian.Homology
#align_import algeb... | Mathlib/Algebra/Homology/QuasiIso.lean | 122 | 127 | theorem to_single₀_epi_at_zero [hf : QuasiIso' f] : Epi (f.f 0) := by |
constructor
intro Z g h Hgh
rw [← cokernel.π_desc (X.d 1 0) (f.f 0) (by rw [← f.2 1 0 rfl]; exact comp_zero),
← toSingle₀CokernelAtZeroIso_hom_eq] at Hgh
rw [(@cancel_epi _ _ _ _ _ _ (epi_comp _ _) _ _).1 Hgh]
|
/-
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 | 165 | 167 | theorem measure_sUnion₀ {S : Set (Set α)} (hs : S.Countable) (hd : S.Pairwise (AEDisjoint μ))
(h : ∀ s ∈ S, NullMeasurableSet s μ) : μ (⋃₀ S) = ∑' s : S, μ s := by |
rw [sUnion_eq_biUnion, measure_biUnion₀ hs hd h]
|
/-
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.BoxIntegral.Box.SubboxInduction
import Mathlib.Analysis.BoxIntegral.Partition.Tagged
#align_import analysis.box_integral.partition.subbox_i... | Mathlib/Analysis/BoxIntegral/Partition/SubboxInduction.lean | 249 | 252 | theorem distortion_unionComplToSubordinate (π₁ : TaggedPrepartition I) (π₂ : Prepartition I)
(hU : π₂.iUnion = ↑I \ π₁.iUnion) (r : (ι → ℝ) → Ioi (0 : ℝ)) :
(π₁.unionComplToSubordinate π₂ hU r).distortion = max π₁.distortion π₂.distortion := by |
simp [unionComplToSubordinate]
|
/-
Copyright (c) 2021 Filippo A. E. Nuccio. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Filippo A. E. Nuccio, Eric Wieser
-/
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Matrix.Block
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
import Mathlib.Linear... | Mathlib/Data/Matrix/Kronecker.lean | 410 | 418 | theorem det_kronecker [Fintype m] [Fintype n] [DecidableEq m] [DecidableEq n] [CommRing R]
(A : Matrix m m R) (B : Matrix n n R) :
det (A ⊗ₖ B) = det A ^ Fintype.card n * det B ^ Fintype.card m := by |
refine (det_kroneckerMapBilinear (Algebra.lmul ℕ R).toLinearMap mul_mul_mul_comm _ _).trans ?_
congr 3
· ext i j
exact mul_one _
· ext i j
exact one_mul _
|
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Analytic.Composition
#align_import analysis.analytic.inverse from "leanprover-community/mathlib"@"284fdd2962e67d2932fa3a79ce19fcf92d38e... | Mathlib/Analysis/Analytic/Inverse.lean | 188 | 199 | theorem rightInv_removeZero (p : FormalMultilinearSeries 𝕜 E F) (i : E ≃L[𝕜] F) :
p.removeZero.rightInv i = p.rightInv i := by |
ext1 n
induction' n using Nat.strongRec' with n IH
match n with
| 0 => simp only [rightInv_coeff_zero]
| 1 => simp only [rightInv_coeff_one]
| n + 2 =>
simp only [rightInv, neg_inj]
rw [removeZero_comp_of_pos _ _ (add_pos_of_nonneg_of_pos n.zero_le zero_lt_two)]
congr (config := { closePost := ... |
/-
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.Data.Fin.Tuple.Basic
import Mathlib.Data.List.Join
#align_import data.list.of_fn from "leanprover-community/mathlib"@"bf27744463e9620ca4e4ebe951fe8353... | Mathlib/Data/List/OfFn.lean | 135 | 136 | theorem ofFn_eq_nil_iff {n : ℕ} {f : Fin n → α} : ofFn f = [] ↔ n = 0 := by |
cases n <;> simp only [ofFn_zero, ofFn_succ, eq_self_iff_true, Nat.succ_ne_zero]
|
/-
Copyright (c) 2024 Markus Himmel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Markus Himmel
-/
import Mathlib.CategoryTheory.Filtered.Connected
import Mathlib.CategoryTheory.Limits.TypesFiltered
import Mathlib.CategoryTheory.Limits.Final
/-!
# Final functors wit... | Mathlib/CategoryTheory/Filtered/Final.lean | 150 | 156 | theorem Functor.final_of_exists_of_isFiltered_of_fullyFaithful [IsFilteredOrEmpty D] [F.Full]
[F.Faithful] (h : ∀ d, ∃ c, Nonempty (d ⟶ F.obj c)) : Final F := by |
have := IsFilteredOrEmpty.of_exists_of_isFiltered_of_fullyFaithful F h
refine Functor.final_of_exists_of_isFiltered F h (fun {d c} s s' => ?_)
obtain ⟨c₀, ⟨f⟩⟩ := h (IsFiltered.coeq s s')
refine ⟨c₀, F.preimage (IsFiltered.coeqHom s s' ≫ f), ?_⟩
simp [reassoc_of% (IsFiltered.coeq_condition s s')]
|
/-
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
-/
import Mathlib.Analysis.NormedSpace.Multilinear.Basic
import Mathlib.Analysis.NormedSpace.Units
import Mathlib.Analysis.NormedSpace.OperatorNorm.Compl... | Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | 389 | 400 | theorem IsBoundedBilinearMap.continuous (h : IsBoundedBilinearMap 𝕜 f) : Continuous f := by |
refine continuous_iff_continuousAt.2 fun x ↦ tendsto_sub_nhds_zero_iff.1 ?_
suffices Tendsto (fun y : E × F ↦ f (y.1 - x.1, y.2) + f (x.1, y.2 - x.2)) (𝓝 x) (𝓝 (0 + 0)) by
simpa only [h.map_sub_left, h.map_sub_right, sub_add_sub_cancel, zero_add] using this
apply Tendsto.add
· rw [← isLittleO_one_iff ℝ, ... |
/-
Copyright (c) 2018 Ellen Arlt. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Ellen Arlt, Blair Shi, Sean Leather, Mario Carneiro, Johan Commelin
-/
import Mathlib.Data.Matrix.Basic
#align_import data.matrix.block from "leanprover-community/mathlib"@"c060baa79af5ca... | Mathlib/Data/Matrix/Block.lean | 149 | 152 | theorem fromBlocks_transpose (A : Matrix n l α) (B : Matrix n m α) (C : Matrix o l α)
(D : Matrix o m α) : (fromBlocks A B C D)ᵀ = fromBlocks Aᵀ Cᵀ Bᵀ Dᵀ := by |
ext i j
rcases i with ⟨⟩ <;> rcases j with ⟨⟩ <;> simp [fromBlocks]
|
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.Regular.Basic
import Mathlib.GroupTheory.Congruence.... | Mathlib/GroupTheory/MonoidLocalization.lean | 894 | 895 | theorem mul_mk'_one_eq_mk' (x) (y : S) : f.toMap x * f.mk' 1 y = f.mk' x y := by |
rw [mul_mk'_eq_mk'_of_mul, mul_one]
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Kenny Lau
-/
import Mathlib.RingTheory.MvPowerSeries.Basic
import Mathlib.RingTheory.Ideal.LocalRing
#align_import ring_theory.power_series.basic from "leanprover-co... | Mathlib/RingTheory/MvPowerSeries/Inverse.lean | 90 | 97 | theorem coeff_invOfUnit [DecidableEq σ] (n : σ →₀ ℕ) (φ : MvPowerSeries σ R) (u : Rˣ) :
coeff R n (invOfUnit φ u) =
if n = 0 then ↑u⁻¹
else
-↑u⁻¹ *
∑ x ∈ antidiagonal n,
if x.2 < n then coeff R x.1 φ * coeff R x.2 (invOfUnit φ u) else 0 := by |
convert coeff_inv_aux n (↑u⁻¹) φ
|
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Johannes Hölzl, Rémy Degenne
-/
import Mathlib.Order.Filter.Cofinite
import Mathlib.Order.Hom.CompleteLattice
#align_import order.liminf_limsup from "leanprover-... | Mathlib/Order/LiminfLimsup.lean | 501 | 502 | theorem blimsup_true (f : Filter β) (u : β → α) : (blimsup u f fun _ => True) = limsup u f := by |
simp [blimsup_eq, limsup_eq]
|
/-
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.Algebra.Polynomial.Cardinal
import Mathlib.RingTheory.Algebraic
#align_import algebra.algebraic_card from "leanprover-communit... | Mathlib/Algebra/AlgebraicCard.lean | 98 | 100 | theorem cardinal_mk_le_max : #{ x : A // IsAlgebraic R x } ≤ max #R ℵ₀ := by |
rw [← lift_id #_, ← lift_id #R]
exact cardinal_mk_lift_le_max R A
|
/-
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 | 1,716 | 1,717 | theorem mem_smul_finset {x : β} : x ∈ a • s ↔ ∃ y, y ∈ s ∧ a • y = x := by |
simp only [Finset.smul_finset_def, and_assoc, mem_image, exists_prop, Prod.exists, mem_product]
|
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Xavier Roblot
-/
import Mathlib.Analysis.Complex.Polynomial
import Mathlib.NumberTheory.NumberField.Norm
import Mathlib.NumberTheory.NumberField.Basic
import Mathlib.RingT... | Mathlib/NumberTheory/NumberField/Embeddings.lean | 622 | 626 | theorem card_add_two_mul_card_eq_rank :
NrRealPlaces K + 2 * NrComplexPlaces K = finrank ℚ K := by |
rw [← card_real_embeddings, ← card_complex_embeddings, Fintype.card_subtype_compl,
← Embeddings.card K ℂ, Nat.add_sub_of_le]
exact Fintype.card_subtype_le _
|
/-
Copyright (c) 2018 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.lpSpace
import Mathlib.Topology.Sets.Compacts
#align_import topology.metric_space.kuratowski from "leanprover-community/mat... | Mathlib/Topology/MetricSpace/Kuratowski.lean | 61 | 87 | theorem embeddingOfSubset_isometry (H : DenseRange x) : Isometry (embeddingOfSubset x) := by |
refine Isometry.of_dist_eq fun a b => ?_
refine (embeddingOfSubset_dist_le x a b).antisymm (le_of_forall_pos_le_add fun e epos => ?_)
-- First step: find n with dist a (x n) < e
rcases Metric.mem_closure_range_iff.1 (H a) (e / 2) (half_pos epos) with ⟨n, hn⟩
-- Second step: use the norm control at index n to... |
/-
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.Finset.Image
import Mathlib.Data.List.FinRange
#align_import data.fintype.basic from "leanprover-community/mathlib"@"d78597269638367c3863d40d4510... | Mathlib/Data/Fintype/Basic.lean | 813 | 815 | theorem Fin.univ_image_def {n : ℕ} [DecidableEq α] (f : Fin n → α) :
Finset.univ.image f = (List.ofFn f).toFinset := by |
simp [Finset.image]
|
/-
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.RingTheory.Ideal.Operations
#align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74"
/-!
# Map... | Mathlib/RingTheory/Ideal/Maps.lean | 660 | 661 | theorem ker_equiv {F' : Type*} [EquivLike F' R S] [RingEquivClass F' R S] (f : F') : ker f = ⊥ := by |
simpa only [← injective_iff_ker_eq_bot] using EquivLike.injective f
|
/-
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.LinearAlgebra.CliffordAlgebra.Fold
import Mathlib.LinearAlgebra.ExteriorAlgebra.Basic
#align_import linear_algebra.exterior_algebra.of_alternating from "lea... | Mathlib/LinearAlgebra/ExteriorAlgebra/OfAlternating.lean | 79 | 85 | theorem liftAlternating_ι_mul (f : ∀ i, M [⋀^Fin i]→ₗ[R] N) (m : M)
(x : ExteriorAlgebra R M) :
liftAlternating (R := R) (M := M) (N := N) f (ι R m * x) =
liftAlternating (R := R) (M := M) (N := N) (fun i => (f i.succ).curryLeft m) x := by |
dsimp [liftAlternating]
rw [foldl_mul, foldl_ι]
rfl
|
/-
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.Topology.ContinuousFunction.Bounded
import Mathlib.Topology.UniformSpace.Compact
import Mathlib.Topology.CompactOpen
import Mathlib.Topology.Sets.Compa... | Mathlib/Topology/ContinuousFunction/Compact.lean | 137 | 138 | theorem dist_le (C0 : (0 : ℝ) ≤ C) : dist f g ≤ C ↔ ∀ x : α, dist (f x) (g x) ≤ C := by |
simp only [← dist_mkOfCompact, BoundedContinuousFunction.dist_le C0, mkOfCompact_apply]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Aurélien Saue, Anne Baanen
-/
import Mathlib.Algebra.Order.Ring.Rat
import Mathlib.Tactic.NormNum.Inv
import Mathlib.Tactic.NormNum.Pow
import Mathlib.Util.AtomM
/-!
#... | Mathlib/Tactic/Ring/Basic.lean | 434 | 435 | theorem add_mul (_ : (a₁ : R) * b = c₁) (_ : a₂ * b = c₂) (_ : c₁ + c₂ = d) :
(a₁ + a₂) * b = d := by | subst_vars; simp [_root_.add_mul]
|
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.Regular.Basic
import Mathlib.GroupTheory.Congruence.... | Mathlib/GroupTheory/MonoidLocalization.lean | 652 | 654 | theorem mul_inv_right {f : M →* N} (h : ∀ y : S, IsUnit (f y)) (y : S) (w z : N) :
z = w * (IsUnit.liftRight (f.restrict S) h y)⁻¹ ↔ z * f y = w := by |
rw [eq_comm, mul_inv_left h, mul_comm, eq_comm]
|
/-
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.Polynomial.Inductions
import Mathlib.Algebra.Polynomial.Monic
import Mathlib.RingTheory.Multiplicit... | Mathlib/Algebra/Polynomial/Div.lean | 685 | 687 | theorem rootMultiplicity_pos' {p : R[X]} {x : R} :
0 < rootMultiplicity x p ↔ p ≠ 0 ∧ IsRoot p x := by |
rw [pos_iff_ne_zero, Ne, rootMultiplicity_eq_zero_iff, Classical.not_imp, and_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, Sander Dahmen, Scott Morrison, Chris Hughes, Anne Baanen
-/
import Mathlib.LinearAlgebra.Dimension.Free
import Mathlib.Algebra.Module.Torsion
#align_im... | Mathlib/LinearAlgebra/Dimension/Constructions.lean | 66 | 75 | theorem rank_quotient_add_rank_le [Nontrivial R] (M' : Submodule R M) :
Module.rank R (M ⧸ M') + Module.rank R M' ≤ Module.rank R M := by |
conv_lhs => simp only [Module.rank_def]
have := nonempty_linearIndependent_set R (M ⧸ M')
have := nonempty_linearIndependent_set R M'
rw [Cardinal.ciSup_add_ciSup _ (bddAbove_range.{v, v} _) _ (bddAbove_range.{v, v} _)]
refine ciSup_le fun ⟨s, hs⟩ ↦ ciSup_le fun ⟨t, ht⟩ ↦ ?_
choose f hf using Quotient.mk_s... |
/-
Copyright (c) 2020 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Yury Kudryashov
-/
import Mathlib.Algebra.Star.Order
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.Order.MonotoneContinuity
#align... | Mathlib/Data/Real/Sqrt.lean | 189 | 190 | theorem sqrt_eq_iff_mul_self_eq_of_pos (h : 0 < y) : √x = y ↔ y * y = x := by |
simp [sqrt_eq_cases, h.ne', h.le]
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Julian Kuelshammer, Heather Macbeth, Mitchell Lee
-/
import Mathlib.Algebra.Polynomial.Derivative
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.pol... | Mathlib/RingTheory/Polynomial/Chebyshev.lean | 175 | 175 | theorem U_neg_one : U R (-1) = 0 := by | simpa using U_sub_one R 0
|
/-
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 | 1,754 | 1,782 | theorem controlled_prod_of_mem_closure {s : Subgroup E} (hg : a ∈ closure (s : Set E)) {b : ℕ → ℝ}
(b_pos : ∀ n, 0 < b n) :
∃ v : ℕ → E,
Tendsto (fun n => ∏ i ∈ range (n + 1), v i) atTop (𝓝 a) ∧
(∀ n, v n ∈ s) ∧ ‖v 0 / a‖ < b 0 ∧ ∀ n, 0 < n → ‖v n‖ < b n := by |
obtain ⟨u : ℕ → E, u_in : ∀ n, u n ∈ s, lim_u : Tendsto u atTop (𝓝 a)⟩ :=
mem_closure_iff_seq_limit.mp hg
obtain ⟨n₀, hn₀⟩ : ∃ n₀, ∀ n ≥ n₀, ‖u n / a‖ < b 0 :=
haveI : { x | ‖x / a‖ < b 0 } ∈ 𝓝 a := by
simp_rw [← dist_eq_norm_div]
exact Metric.ball_mem_nhds _ (b_pos _)
Filter.tendsto_atTo... |
/-
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 | 299 | 300 | theorem toIcoDiv_add_right (a b : α) : toIcoDiv hp a (b + p) = toIcoDiv hp a b + 1 := by |
simpa only [one_zsmul] using toIcoDiv_add_zsmul hp a b 1
|
/-
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 | 208 | 220 | theorem card_mul_pow_le (hAB : ∀ A' ⊆ A, (A * B).card * A'.card ≤ (A' * B).card * A.card)
(n : ℕ) : (A * B ^ n).card ≤ ((A * B).card / A.card : ℚ≥0) ^ n * A.card := by |
obtain rfl | hA := A.eq_empty_or_nonempty
· simp
induction' n with n ih
· simp
rw [_root_.pow_succ', ← mul_assoc, _root_.pow_succ', @mul_assoc ℚ≥0, ← mul_div_right_comm,
le_div_iff, ← cast_mul]
swap
· exact cast_pos.2 hA.card_pos
refine (Nat.cast_le.2 <| mul_pluennecke_petridis _ hAB).trans ?_
rw... |
/-
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 | 207 | 210 | theorem filter_le (n m l : ℕ) : ((Ico n m).filter fun x => l ≤ x) = Ico (max n l) m := by |
rcases le_total n l with hnl | hln
· rw [max_eq_right hnl, filter_le_of_le hnl]
· rw [max_eq_left hln, filter_le_of_le_bot hln]
|
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Bhavik Mehta
-/
import Mathlib.Analysis.Calculus.Deriv.Support
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.MeasureTheory.Integral.FundThmCalcu... | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | 829 | 856 | theorem integrableOn_Ioi_deriv_of_nonneg (hcont : ContinuousWithinAt g (Ici a) a)
(hderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x) (g'pos : ∀ x ∈ Ioi a, 0 ≤ g' x)
(hg : Tendsto g atTop (𝓝 l)) : IntegrableOn g' (Ioi a) := by |
have hcont : ContinuousOn g (Ici a) := by
intro x hx
rcases hx.out.eq_or_lt with rfl|hx
· exact hcont
· exact (hderiv x hx).continuousAt.continuousWithinAt
refine integrableOn_Ioi_of_intervalIntegral_norm_tendsto (l - g a) a (fun x => ?_) tendsto_id ?_
· exact intervalIntegral.integrableOn_deriv_... |
/-
Copyright (c) 2020 Thomas Browning and Patrick Lutz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning, Patrick Lutz
-/
import Mathlib.GroupTheory.Solvable
import Mathlib.FieldTheory.PolynomialGaloisGroup
import Mathlib.RingTheory.RootsOfUnity.Basic
#a... | Mathlib/FieldTheory/AbelRuffini.lean | 308 | 331 | theorem induction3 {α : solvableByRad F E} {n : ℕ} (hn : n ≠ 0) (hα : P (α ^ n)) : P α := by |
let p := minpoly F (α ^ n)
have hp : p.comp (X ^ n) ≠ 0 := by
intro h
cases' comp_eq_zero_iff.mp h with h' h'
· exact minpoly.ne_zero (isIntegral (α ^ n)) h'
· exact hn (by rw [← @natDegree_C F, ← h'.2, natDegree_X_pow])
apply gal_isSolvable_of_splits
· exact ⟨splits_of_splits_of_dvd _ hp (Spli... |
/-
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 | 1,055 | 1,067 | theorem sInf_caratheodory (s : Set α) (hs : MeasurableSet s) :
MeasurableSet[(sInf (toOuterMeasure '' m)).caratheodory] s := by |
rw [OuterMeasure.sInf_eq_boundedBy_sInfGen]
refine OuterMeasure.boundedBy_caratheodory fun t => ?_
simp only [OuterMeasure.sInfGen, le_iInf_iff, forall_mem_image, measure_eq_iInf t,
coe_toOuterMeasure]
intro μ hμ u htu _hu
have hm : ∀ {s t}, s ⊆ t → OuterMeasure.sInfGen (toOuterMeasure '' m) s ≤ μ t := b... |
/-
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.AlgebraicGeometry.Gluing
import Mathlib.CategoryTheory.Limits.Opposites
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.CategoryTheory.Limits.Sh... | Mathlib/AlgebraicGeometry/Pullbacks.lean | 159 | 162 | theorem cocycle_fst_fst_fst (i j k : 𝒰.J) :
t' 𝒰 f g i j k ≫ t' 𝒰 f g j k i ≫ t' 𝒰 f g k i j ≫ pullback.fst ≫ pullback.fst ≫ pullback.fst =
pullback.fst ≫ pullback.fst ≫ pullback.fst := by |
simp only [t'_fst_fst_fst, t'_fst_snd, t'_snd_snd]
|
/-
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.DirectSum.Module
import Mathlib.Algebra.Module.BigOperators
import Mathlib.LinearAlgebra.Isomorphisms
import Mathlib.GroupTheor... | Mathlib/Algebra/Module/Torsion.lean | 465 | 468 | theorem supIndep_torsionBy : S.SupIndep fun i => torsionBy R M <| q i := by |
convert supIndep_torsionBySet_ideal (M := M) fun i hi j hj ij =>
(Ideal.sup_eq_top_iff_isCoprime (q i) _).mpr <| hq hi hj ij
exact (torsionBySet_span_singleton_eq (R := R) (M := M) _).symm
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Violeta Hernández Palacios, Grayson Burton, Floris van Doorn
-/
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Order.Antisymmetrization
#align_import order.... | Mathlib/Order/Cover.lean | 162 | 165 | theorem WCovBy.eq_or_eq (h : a ⩿ b) (h2 : a ≤ c) (h3 : c ≤ b) : c = a ∨ c = b := by |
rcases h2.eq_or_lt with (h2 | h2); · exact Or.inl h2.symm
rcases h3.eq_or_lt with (h3 | h3); · exact Or.inr h3
exact (h.2 h2 h3).elim
|
/-
Copyright (c) 2021 Aaron Anderson, Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Kevin Buzzard, Yaël Dillies, Eric Wieser
-/
import Mathlib.Data.Finset.Sigma
import Mathlib.Data.Finset.Pairwise
import Mathlib.Data.Finset.Powerset
impor... | Mathlib/Order/SupIndep.lean | 400 | 405 | theorem Independent.comp' {ι ι' : Sort*} {t : ι → α} {f : ι' → ι} (ht : Independent <| t ∘ f)
(hf : Surjective f) : Independent t := by |
intro i
obtain ⟨i', rfl⟩ := hf i
rw [← hf.iSup_comp]
exact (ht i').mono_right (biSup_mono fun j' hij => mt (congr_arg f) hij)
|
/-
Copyright (c) 2023 Kim Liesinger. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kim Liesinger
-/
import Mathlib.Algebra.Order.Monoid.Canonical.Defs
import Mathlib.Data.List.Infix
import Mathlib.Data.List.MinMax
import Mathlib.Data.List.EditDistance.Defs
/-!
# Lowe... | Mathlib/Data/List/EditDistance/Bounds.lean | 26 | 56 | theorem suffixLevenshtein_minimum_le_levenshtein_cons (xs : List α) (y ys) :
(suffixLevenshtein C xs ys).1.minimum ≤ levenshtein C xs (y :: ys) := by |
induction xs with
| nil =>
simp only [suffixLevenshtein_nil', levenshtein_nil_cons,
List.minimum_singleton, WithTop.coe_le_coe]
exact le_add_of_nonneg_left (by simp)
| cons x xs ih =>
suffices
(suffixLevenshtein C (x :: xs) ys).1.minimum ≤ (C.delete x + levenshtein C xs (y :: ys)) ∧... |
/-
Copyright (c) 2018 Simon Hudon. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Simon Hudon
-/
import Mathlib.Control.Applicative
import Mathlib.Control.Traversable.Basic
#align_import control.traversable.lemmas from "leanprover-community/mathlib"@"3342d1b2178381196... | Mathlib/Control/Traversable/Lemmas.lean | 135 | 138 | theorem naturality_pf (η : ApplicativeTransformation F G) (f : α → F β) :
traverse (@η _ ∘ f) = @η _ ∘ (traverse f : t α → F (t β)) := by |
ext
rw [comp_apply, naturality]
|
/-
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.Dynamics.Ergodic.MeasurePreserving
import Mathlib.GroupTheory.GroupAction.Hom
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.... | Mathlib/MeasureTheory/Group/Measure.lean | 180 | 187 | theorem forall_measure_preimage_mul_iff (μ : Measure G) :
(∀ (g : G) (A : Set G), MeasurableSet A → μ ((fun h => g * h) ⁻¹' A) = μ A) ↔
IsMulLeftInvariant μ := by |
trans ∀ g, map (g * ·) μ = μ
· simp_rw [Measure.ext_iff]
refine forall_congr' fun g => forall_congr' fun A => forall_congr' fun hA => ?_
rw [map_apply (measurable_const_mul g) hA]
exact ⟨fun h => ⟨h⟩, fun h => h.1⟩
|
/-
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.Subobject.Limits
#align_import algebra.homology.image_to_kernel from "leanprover-community/mathlib"@"618ea3d5c99240cd7000d8376924906a14... | Mathlib/Algebra/Homology/ImageToKernel.lean | 423 | 427 | theorem imageSubobjectIso_imageToKernel' (w : f ≫ g = 0) :
(imageSubobjectIso f).hom ≫ imageToKernel' f g w =
imageToKernel f g w ≫ (kernelSubobjectIso g).hom := by |
ext
simp [imageToKernel']
|
/-
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.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.function.simple_func from "leanprover-community/mathlib"@"bf... | Mathlib/MeasureTheory/Function/SimpleFunc.lean | 729 | 733 | theorem finset_sup_apply [SemilatticeSup β] [OrderBot β] {f : γ → α →ₛ β} (s : Finset γ) (a : α) :
s.sup f a = s.sup fun c => f c a := by |
refine Finset.induction_on s rfl ?_
intro a s _ ih
rw [Finset.sup_insert, Finset.sup_insert, sup_apply, ih]
|
/-
Copyright (c) 2019 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.Analysis.Calculus.ContDiff.Basic
import Mathlib.Analysis.NormedSpace.FiniteDimension
#align_import analysis.calculus.cont_dif... | Mathlib/Analysis/Calculus/ContDiff/FiniteDimension.lean | 60 | 62 | theorem contDiff_succ_iff_fderiv_apply [FiniteDimensional 𝕜 E] {n : ℕ} {f : E → F} :
ContDiff 𝕜 (n + 1 : ℕ) f ↔ Differentiable 𝕜 f ∧ ∀ y, ContDiff 𝕜 n fun x => fderiv 𝕜 f x y := by |
rw [contDiff_succ_iff_fderiv, contDiff_clm_apply_iff]
|
/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Scott Morrison, Floris van Doorn
-/
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_theory.isomorphism from "leanprover-community/math... | Mathlib/CategoryTheory/Iso.lean | 414 | 416 | theorem Iso.inv_inv (f : X ≅ Y) : inv f.inv = f.hom := by |
apply inv_eq_of_hom_inv_id
simp
|
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