Context stringlengths 285 6.98k | file_name stringlengths 21 79 | start int64 14 184 | end int64 18 184 | theorem stringlengths 25 1.34k | proof stringlengths 5 3.43k |
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/-
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.ModelTheory.Ultraproducts
import Mathlib.ModelTheory.Bundled
import Mathlib.ModelTheory.Skolem
#align_import model_theory.satisfiability from "leanpro... | Mathlib/ModelTheory/Satisfiability.lean | 93 | 98 | theorem isSatisfiable_onTheory_iff {L' : Language.{w, w'}} {φ : L →ᴸ L'} (h : φ.Injective) :
(φ.onTheory T).IsSatisfiable ↔ T.IsSatisfiable := by |
classical
refine ⟨isSatisfiable_of_isSatisfiable_onTheory φ, fun h' => ?_⟩
haveI : Inhabited h'.some := Classical.inhabited_of_nonempty'
exact Model.isSatisfiable (h'.some.defaultExpansion h)
|
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Dynamics.BirkhoffSum.NormedSpace
/-!
# Von Neumann Mean Ergodic Theorem in a Hilbert Space
In ... | Mathlib/Analysis/InnerProductSpace/MeanErgodic.lean | 43 | 71 | theorem LinearMap.tendsto_birkhoffAverage_of_ker_subset_closure [NormedSpace 𝕜 E]
(f : E →ₗ[𝕜] E) (hf : LipschitzWith 1 f) (g : E →L[𝕜] LinearMap.eqLocus f 1)
(hg_proj : ∀ x : LinearMap.eqLocus f 1, g x = x)
(hg_ker : (LinearMap.ker g : Set E) ⊆ closure (LinearMap.range (f - 1))) (x : E) :
Tendsto (b... |
/- Any point can be represented as a sum of `y ∈ LinearMap.ker g` and a fixed point `z`. -/
obtain ⟨y, hy, z, hz, rfl⟩ : ∃ y, g y = 0 ∧ ∃ z, IsFixedPt f z ∧ x = y + z :=
⟨x - g x, by simp [hg_proj], g x, (g x).2, by simp⟩
/- For a fixed point, the theorem is trivial,
so it suffices to prove it for `y ∈ Lin... |
/-
Copyright (c) 2020 Filippo A. E. Nuccio. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Filippo A. E. Nuccio, Andrew Yang
-/
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
import Mathlib.Topology.NoetherianSpace
#align_import algebraic_geometry.prime_spectrum... | Mathlib/AlgebraicGeometry/PrimeSpectrum/Noetherian.lean | 60 | 97 | theorem exists_primeSpectrum_prod_le_and_ne_bot_of_domain (h_fA : ¬IsField A) {I : Ideal A}
(h_nzI : I ≠ ⊥) :
∃ Z : Multiset (PrimeSpectrum A),
Multiset.prod (Z.map asIdeal) ≤ I ∧ Multiset.prod (Z.map asIdeal) ≠ ⊥ := by |
revert h_nzI
-- Porting note: Need to specify `P` explicitly
refine IsNoetherian.induction (P := fun I => I ≠ ⊥ → ∃ Z : Multiset (PrimeSpectrum A),
Multiset.prod (Z.map asIdeal) ≤ I ∧ Multiset.prod (Z.map asIdeal) ≠ ⊥)
(fun (M : Ideal A) hgt => ?_) I
intro h_nzM
have hA_nont : Nontrivial A := IsDom... |
/-
Copyright (c) 2022 Yakov Pechersky. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yakov Pechersky
-/
import Mathlib.Data.List.Infix
#align_import data.list.rdrop from "leanprover-community/mathlib"@"26f081a2fb920140ed5bc5cc5344e84bcc7cb2b2"
/-!
# Dropping or taki... | Mathlib/Data/List/DropRight.lean | 78 | 78 | theorem rtake_zero : rtake l 0 = [] := by | simp [rtake]
|
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll
-/
import Mathlib.Analysis.SpecialFunctions.Integrals
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.MeasureTheory.Integral.Layercake
#align_import analy... | Mathlib/Analysis/SpecialFunctions/JapaneseBracket.lean | 49 | 59 | theorem rpow_neg_one_add_norm_sq_le {r : ℝ} (x : E) (hr : 0 < r) :
((1 : ℝ) + ‖x‖ ^ 2) ^ (-r / 2) ≤ (2 : ℝ) ^ (r / 2) * (1 + ‖x‖) ^ (-r) :=
calc
((1 : ℝ) + ‖x‖ ^ 2) ^ (-r / 2)
= (2 : ℝ) ^ (r / 2) * ((√2 * √((1 : ℝ) + ‖x‖ ^ 2)) ^ r)⁻¹ := by |
rw [rpow_div_two_eq_sqrt, rpow_div_two_eq_sqrt, mul_rpow, mul_inv, rpow_neg,
mul_inv_cancel_left₀] <;> positivity
_ ≤ (2 : ℝ) ^ (r / 2) * ((1 + ‖x‖) ^ r)⁻¹ := by
gcongr
apply one_add_norm_le_sqrt_two_mul_sqrt
_ = (2 : ℝ) ^ (r / 2) * (1 + ‖x‖) ^ (-r) := by rw [rpow_neg]; positivity
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Algebra.Subalgebra.Basic
import Mathlib.Data.Set.UnionLift
#align_import algebra.algebra.subalgebra.basic from "leanprover-community/mathlib"@"b915e... | Mathlib/Algebra/Algebra/Subalgebra/Directed.lean | 96 | 99 | 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 |
dsimp [iSupLift, inclusion]
rw [Set.iUnionLift_of_mem]
|
/-
Copyright (c) 2022 Rishikesh Vaishnav. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rishikesh Vaishnav
-/
import Mathlib.MeasureTheory.Measure.Typeclasses
#align_import probability.conditional_probability from "leanprover-community/mathlib"@"70fd9563a21e7b963887c... | Mathlib/Probability/ConditionalProbability.lean | 130 | 130 | theorem cond_empty : μ[|∅] = 0 := by | simp [cond]
|
/-
Copyright (c) 2023 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.LinearAlgebra.Dual
/-!
# Perfect pairings of modules
A perfect pairing of two (left) modules may be defined either as:
1. A bilinear map `M × N → R` such ... | Mathlib/LinearAlgebra/PerfectPairing.lean | 91 | 94 | theorem toDualLeft_of_toDualRight_symm (x : M) (f : Dual R M) :
(p.toDualLeft x) (p.toDualRight.symm f) = f x := by |
rw [@toDualLeft_apply]
exact apply_apply_toDualRight_symm p x f
|
/-
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
-/
import Mathlib.RingTheory.WittVector.Identities
#align_import ring_theory.witt_vector.domain from "leanprover-community/mathlib"@"b1d911acd60ab198808e853292106ee35... | Mathlib/RingTheory/WittVector/Domain.lean | 88 | 98 | theorem verschiebung_nonzero {x : 𝕎 R} (hx : x ≠ 0) :
∃ n : ℕ, ∃ x' : 𝕎 R, x'.coeff 0 ≠ 0 ∧ x = verschiebung^[n] x' := by |
have hex : ∃ k : ℕ, x.coeff k ≠ 0 := by
by_contra! hall
apply hx
ext i
simp only [hall, zero_coeff]
let n := Nat.find hex
use n, x.shift n
refine ⟨Nat.find_spec hex, eq_iterate_verschiebung fun i hi => not_not.mp ?_⟩
exact Nat.find_min hex hi
|
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.MetricSpace.HausdorffDistance
#align_import topology.metric_space.pi_nat from "leanprover-community/mathlib"@"49b7f94aab3a3bdca1f9f34c5... | Mathlib/Topology/MetricSpace/PiNat.lean | 112 | 115 | theorem cylinder_eq_pi (x : ∀ n, E n) (n : ℕ) :
cylinder x n = Set.pi (Finset.range n : Set ℕ) fun i : ℕ => {x i} := by |
ext y
simp [cylinder]
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Algebra.Order.Monoid.Unbundled.MinMax
#align_import algebra.or... | Mathlib/Algebra/Order/Group/MinMax.lean | 86 | 93 | theorem max_sub_max_le_max (a b c d : α) : max a b - max c d ≤ max (a - c) (b - d) := by |
simp only [sub_le_iff_le_add, max_le_iff]; constructor
· calc
a = a - c + c := (sub_add_cancel a c).symm
_ ≤ max (a - c) (b - d) + max c d := add_le_add (le_max_left _ _) (le_max_left _ _)
· calc
b = b - d + d := (sub_add_cancel b d).symm
_ ≤ max (a - c) (b - d) + max c d := add_le_add (le_max_ri... |
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Kenny Lau
-/
import Mathlib.Data.List.Forall2
#align_import data.list.zip from "leanprover-community/mathlib"@"134625f523e737f650a6ea7f0c82a6177e45e622"
/-!
# zip & u... | Mathlib/Data/List/Zip.lean | 63 | 64 | theorem lt_length_left_of_zipWith {f : α → β → γ} {i : ℕ} {l : List α} {l' : List β}
(h : i < (zipWith f l l').length) : i < l.length := by | rw [length_zipWith] at h; omega
|
/-
Copyright (c) 2022 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.Polynomial.UnitTrinomial
import Mathlib.RingTheory.Polynomial.GaussLemma
import Mathlib.Tactic.LinearCombination
#align_import ring_theory.p... | Mathlib/RingTheory/Polynomial/Selmer.lean | 31 | 45 | theorem X_pow_sub_X_sub_one_irreducible_aux (z : ℂ) : ¬(z ^ n = z + 1 ∧ z ^ n + z ^ 2 = 0) := by |
rintro ⟨h1, h2⟩
replace h3 : z ^ 3 = 1 := by
linear_combination (1 - z - z ^ 2 - z ^ n) * h1 + (z ^ n - 2) * h2
have key : z ^ n = 1 ∨ z ^ n = z ∨ z ^ n = z ^ 2 := by
rw [← Nat.mod_add_div n 3, pow_add, pow_mul, h3, one_pow, mul_one]
have : n % 3 < 3 := Nat.mod_lt n zero_lt_three
interval_cases n... |
/-
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.MeasureTheory.Function.LpSeminorm.Basic
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27dd... | Mathlib/MeasureTheory/Function/LpSeminorm/Trim.lean | 35 | 45 | theorem limsup_trim (hm : m ≤ m0) {f : α → ℝ≥0∞} (hf : Measurable[m] f) :
limsup f (ae (μ.trim hm)) = limsup f (ae μ) := by |
simp_rw [limsup_eq]
suffices h_set_eq : { a : ℝ≥0∞ | ∀ᵐ n ∂μ.trim hm, f n ≤ a } = { a : ℝ≥0∞ | ∀ᵐ n ∂μ, f n ≤ a } by
rw [h_set_eq]
ext1 a
suffices h_meas_eq : μ { x | ¬f x ≤ a } = μ.trim hm { x | ¬f x ≤ a } by
simp_rw [Set.mem_setOf_eq, ae_iff, h_meas_eq]
refine (trim_measurableSet_eq hm ?_).symm
r... |
/-
Copyright (c) 2022 Paul Reichert. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Paul Reichert
-/
import Mathlib.LinearAlgebra.AffineSpace.AffineSubspace
#align_import linear_algebra.affine_space.restrict from "leanprover-community/mathlib"@"09258fb7f75d741b7eda9fa... | Mathlib/LinearAlgebra/AffineSpace/Restrict.lean | 81 | 87 | theorem AffineMap.restrict.surjective (φ : P₁ →ᵃ[k] P₂) {E : AffineSubspace k P₁}
{F : AffineSubspace k P₂} [Nonempty E] [Nonempty F] (h : E.map φ = F) :
Function.Surjective (AffineMap.restrict φ (le_of_eq h)) := by |
rintro ⟨x, hx : x ∈ F⟩
rw [← h, AffineSubspace.mem_map] at hx
obtain ⟨y, hy, rfl⟩ := hx
exact ⟨⟨y, hy⟩, rfl⟩
|
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Data.List.Chain
#align_import data.bool.count from "leanprover-community/mathlib"@"8631e2d5ea77f6c13054d9151d8... | Mathlib/Data/Bool/Count.lean | 100 | 102 | theorem two_mul_count_bool_of_even (hl : Chain' (· ≠ ·) l) (h2 : Even (length l)) (b : Bool) :
2 * count b l = length l := by |
rw [← count_not_add_count l b, hl.count_not_eq_count h2, two_mul]
|
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Probability.Martingale.BorelCantelli
import Mathlib.Probability.ConditionalExpectation
import Mathlib.Probability.Independence.Basic
#align_import probabili... | Mathlib/Probability/BorelCantelli.lean | 43 | 48 | theorem iIndepFun.indep_comap_natural_of_lt (hf : ∀ i, StronglyMeasurable (f i))
(hfi : iIndepFun (fun _ => mβ) f μ) (hij : i < j) :
Indep (MeasurableSpace.comap (f j) mβ) (Filtration.natural f hf i) μ := by |
suffices Indep (⨆ k ∈ ({j} : Set ι), MeasurableSpace.comap (f k) mβ)
(⨆ k ∈ {k | k ≤ i}, MeasurableSpace.comap (f k) mβ) μ by rwa [iSup_singleton] at this
exact indep_iSup_of_disjoint (fun k => (hf k).measurable.comap_le) hfi (by simpa)
|
/-
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.Multiset.Dedup
#align_import data.multiset.finset_ops from "leanprover-community/mathlib"@"c227d107bbada5d0d9d20287e3282c0a7f1651a0"
/-!
# Prepa... | Mathlib/Data/Multiset/FinsetOps.lean | 83 | 84 | theorem dedup_cons {a : α} {s : Multiset α} : dedup (a ::ₘ s) = ndinsert a (dedup s) := by |
by_cases h : a ∈ s <;> simp [h]
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Units.Hom
import Mathlib.Algebra.GroupWithZero.Commute
import Mathlib.Algebra.GroupWithZero.Hom
import Mathlib.GroupTheory.GroupAction.Un... | Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean | 64 | 68 | theorem map_inv₀ : f a⁻¹ = (f a)⁻¹ := by |
by_cases h : a = 0
· simp [h, map_zero f]
· apply eq_inv_of_mul_eq_one_left
rw [← map_mul, inv_mul_cancel h, map_one]
|
/-
Copyright (c) 2023 Hanneke Wiersema. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard, Hanneke Wiersema
-/
import Mathlib.RingTheory.RootsOfUnity.Basic
/-!
# The cyclotomic character
Let `L` be an integral domain and let `n : ℕ+` be a positive integer... | Mathlib/NumberTheory/Cyclotomic/CyclotomicCharacter.lean | 72 | 75 | theorem rootsOfUnity.integer_power_of_ringEquiv (g : L ≃+* L) :
∃ m : ℤ, ∀ t : rootsOfUnity n L, g (t : Lˣ) = (t ^ m : Lˣ) := by |
obtain ⟨m, hm⟩ := MonoidHom.map_cyclic ((g : L ≃* L).restrictRootsOfUnity n).toMonoidHom
exact ⟨m, fun t ↦ Units.ext_iff.1 <| SetCoe.ext_iff.2 <| hm t⟩
|
/-
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 | 75 | 76 | theorem left_ne_of_oangle_ne_zero {p₁ p₂ p₃ : P} (h : ∡ p₁ p₂ p₃ ≠ 0) : p₁ ≠ p₂ := by |
rw [← @vsub_ne_zero V]; exact o.left_ne_zero_of_oangle_ne_zero h
|
/-
Copyright (c) 2021 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Analysis.Convex.Cone.Basic
import Mathlib.Analysis.InnerProductSpace.Projection
#align_import analysis.convex.cone.dual from "leanprover-commu... | Mathlib/Analysis/Convex/Cone/InnerDual.lean | 119 | 121 | theorem innerDualCone_sUnion (S : Set (Set H)) :
(⋃₀ S).innerDualCone = sInf (Set.innerDualCone '' S) := by |
simp_rw [sInf_image, sUnion_eq_biUnion, innerDualCone_iUnion]
|
/-
Copyright (c) 2019 Neil Strickland. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Neil Strickland
-/
import Mathlib.Algebra.BigOperators.Intervals
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.... | Mathlib/Algebra/GeomSum.lean | 60 | 60 | theorem geom_sum_one (x : α) : ∑ i ∈ range 1, x ^ i = 1 := by | simp [geom_sum_succ']
|
/-
Copyright (c) 2022 Joël Riou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joël Riou
-/
import Mathlib.CategoryTheory.ConcreteCategory.Basic
import Mathlib.CategoryTheory.Limits.Preserves.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Shapes.RegularMon... | Mathlib/CategoryTheory/Limits/MonoCoprod.lean | 78 | 87 | theorem mono_inl_iff {A B : C} {c₁ c₂ : BinaryCofan A B} (hc₁ : IsColimit c₁) (hc₂ : IsColimit c₂) :
Mono c₁.inl ↔ Mono c₂.inl := by |
suffices
∀ (c₁ c₂ : BinaryCofan A B) (_ : IsColimit c₁) (_ : IsColimit c₂) (_ : Mono c₁.inl),
Mono c₂.inl
by exact ⟨fun h₁ => this _ _ hc₁ hc₂ h₁, fun h₂ => this _ _ hc₂ hc₁ h₂⟩
intro c₁ c₂ hc₁ hc₂
intro
simpa only [IsColimit.comp_coconePointUniqueUpToIso_hom] using
mono_comp c₁.inl (hc₁.coco... |
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Robert Y. Lewis
-/
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.MvPolynomial.Variables
import Mathlib.Algebra.MvPolynomial.CommRing
import Mathlib.Alg... | Mathlib/RingTheory/WittVector/WittPolynomial.lean | 125 | 132 | theorem constantCoeff_wittPolynomial [hp : Fact p.Prime] (n : ℕ) :
constantCoeff (wittPolynomial p R n) = 0 := by |
simp only [wittPolynomial, map_sum, constantCoeff_monomial]
rw [sum_eq_zero]
rintro i _
rw [if_neg]
rw [Finsupp.single_eq_zero]
exact ne_of_gt (pow_pos hp.1.pos _)
|
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot
-/
import Mathlib.Topology.Algebra.Nonarchimedean.Bases
import Mathlib.Topology.Algebra.UniformFilterBasis
import Mathlib.RingTheory.Valuation.ValuationSubring
#align_i... | Mathlib/Topology/Algebra/Valuation.lean | 119 | 121 | theorem hasBasis_nhds_zero :
(𝓝 (0 : R)).HasBasis (fun _ => True) fun γ : Γ₀ˣ => { x | v x < (γ : Γ₀) } := by |
simp [Filter.hasBasis_iff, is_topological_valuation]
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.List.Range
import Mathlib.Data.Multiset.Range
#align_import data.multiset.nodup from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f4... | Mathlib/Data/Multiset/Nodup.lean | 96 | 100 | theorem count_eq_of_nodup [DecidableEq α] {a : α} {s : Multiset α} (d : Nodup s) :
count a s = if a ∈ s then 1 else 0 := by |
split_ifs with h
· exact count_eq_one_of_mem d h
· exact count_eq_zero_of_not_mem h
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.NaturalTransformation
imp... | Mathlib/CategoryTheory/Monoidal/Braided/Basic.lean | 102 | 108 | theorem braiding_tensor_right (X Y Z : C) :
(β_ X (Y ⊗ Z)).hom =
(α_ X Y Z).inv ≫ (β_ X Y).hom ▷ Z ≫ (α_ Y X Z).hom ≫
Y ◁ (β_ X Z).hom ≫ (α_ Y Z X).inv := by |
apply (cancel_epi (α_ X Y Z).hom).1
apply (cancel_mono (α_ Y Z X).hom).1
simp [hexagon_forward]
|
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura, Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Algebra.Order.Group.Defs
import Mathlib.Algebra.Order.Monoid.WithTop
#align_import algebra.order.grou... | Mathlib/Algebra/Order/Group/WithTop.lean | 65 | 65 | theorem sub_top {a : WithTop α} : a - ⊤ = ⊤ := by | cases a <;> rfl
|
/-
Copyright (c) 2019 Yury Kudriashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudriashov
-/
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Analysis.Convex.Hull
import Mathlib.LinearAlgebra.AffineSpace.Basis
#align_import analysis.con... | Mathlib/Analysis/Convex/Combination.lean | 61 | 67 | theorem Finset.centerMass_insert (ha : i ∉ t) (hw : ∑ j ∈ t, w j ≠ 0) :
(insert i t).centerMass w z =
(w i / (w i + ∑ j ∈ t, w j)) • z i +
((∑ j ∈ t, w j) / (w i + ∑ j ∈ t, w j)) • t.centerMass w z := by |
simp only [centerMass, sum_insert ha, smul_add, (mul_smul _ _ _).symm, ← div_eq_inv_mul]
congr 2
rw [div_mul_eq_mul_div, mul_inv_cancel hw, one_div]
|
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Topology.Separation
import Mathlib.Topology.Bases
#align_import topology.dense_embedding from "leanprover-community/mathl... | Mathlib/Topology/DenseEmbedding.lean | 75 | 78 | theorem dense_image (di : DenseInducing i) {s : Set α} : Dense (i '' s) ↔ Dense s := by |
refine ⟨fun H x => ?_, di.dense.dense_image di.continuous⟩
rw [di.toInducing.closure_eq_preimage_closure_image, H.closure_eq, preimage_univ]
trivial
|
/-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Functor.Const
import Mathlib.CategoryTheory.DiscreteCategory
#align_import category_theory.punit from "leanprover-communi... | Mathlib/CategoryTheory/PUnit.lean | 73 | 102 | theorem equiv_punit_iff_unique :
Nonempty (C ≌ Discrete PUnit.{w + 1}) ↔ Nonempty C ∧ ∀ x y : C, Nonempty <| Unique (x ⟶ y) := by |
constructor
· rintro ⟨h⟩
refine ⟨⟨h.inverse.obj ⟨⟨⟩⟩⟩, fun x y => Nonempty.intro ?_⟩
let f : x ⟶ y := by
have hx : x ⟶ h.inverse.obj ⟨⟨⟩⟩ := by convert h.unit.app x
have hy : h.inverse.obj ⟨⟨⟩⟩ ⟶ y := by convert h.unitInv.app y
exact hx ≫ hy
suffices sub : Subsingleton (x ⟶ y) from un... |
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.MonoidAlgebra.Basic
import Mathlib.Data.Finset.Pointwise
#align_import algebra.monoid_algebra.support from "leanprover-community/mathlib"@"16749... | Mathlib/Algebra/MonoidAlgebra/Support.lean | 65 | 71 | theorem support_mul_single [Mul G] [IsRightCancelMul G] (f : MonoidAlgebra k G) (r : k)
(hr : ∀ y, y * r = 0 ↔ y = 0) (x : G) :
(f * single x r).support = f.support.map (mulRightEmbedding x) := by |
classical
ext
simp only [support_mul_single_eq_image f hr (IsRightRegular.all x),
mem_image, mem_map, mulRightEmbedding_apply]
|
/-
Copyright (c) 2022 Yuyang Zhao. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuyang Zhao
-/
import Batteries.Classes.Order
namespace Batteries.PairingHeapImp
/--
A `Heap` is the nodes of the pairing heap.
Each node have two pointers: `child` going to the first c... | .lake/packages/batteries/Batteries/Data/PairingHeap.lean | 103 | 105 | theorem Heap.noSibling_deleteMin {s : Heap α} (eq : s.deleteMin le = some (a, s')) :
s'.NoSibling := by |
cases s with cases eq | node a c => exact noSibling_combine _ _
|
/-
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.FreeNonUnitalNonAssocAlgebra
import Mathlib.Algebra.Lie.NonUnitalNonAssocAlgebra
import Mathlib.Algebra.Lie.UniversalEnveloping
import Mathlib.GroupT... | Mathlib/Algebra/Lie/Free.lean | 103 | 106 | theorem Rel.smulOfTower {S : Type*} [Monoid S] [DistribMulAction S R] [IsScalarTower S R R] (t : S)
(a b : lib R X) (h : Rel R X a b) : Rel R X (t • a) (t • b) := by |
rw [← smul_one_smul R t a, ← smul_one_smul R t b]
exact h.smul _
|
/-
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 | 65 | 67 | theorem eval_eq_smeval : p.eval r = p.smeval r := by |
rw [eval_eq_sum, smeval_eq_sum]
rfl
|
/-
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.Init.Data.Sigma.Lex
import Mathlib.Data.Prod.Lex
import Mathlib.Data.Sigma.Lex
import Mathlib.Order.Antichain
import Mathlib.Order.OrderIsoNat
import M... | Mathlib/Order/WellFoundedSet.lean | 101 | 108 | theorem wellFoundedOn_range : (range f).WellFoundedOn r ↔ WellFounded (r on f) := by |
let f' : β → range f := fun c => ⟨f c, c, rfl⟩
refine ⟨fun h => (InvImage.wf f' h).mono fun c c' => id, fun h => ⟨?_⟩⟩
rintro ⟨_, c, rfl⟩
refine Acc.of_downward_closed f' ?_ _ ?_
· rintro _ ⟨_, c', rfl⟩ -
exact ⟨c', rfl⟩
· exact h.apply _
|
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Variance
#align_import probability.moments from "leanprover-community/mathlib"@"85453a2a14be8da64caf15ca50930cf4c6e5d8de"
/-!
# Moments and m... | Mathlib/Probability/Moments.lean | 68 | 70 | theorem centralMoment_zero (hp : p ≠ 0) : centralMoment 0 p μ = 0 := by |
simp only [centralMoment, hp, Pi.zero_apply, integral_const, smul_eq_mul,
mul_zero, zero_sub, Pi.pow_apply, Pi.neg_apply, neg_zero, zero_pow, Ne, not_false_iff]
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.LinearAlgebra.Dual
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.matrix.dual from "... | Mathlib/LinearAlgebra/Matrix/Dual.lean | 41 | 44 | theorem Matrix.toLin_transpose (M : Matrix ι₁ ι₂ K) : Matrix.toLin B₁.dualBasis B₂.dualBasis Mᵀ =
Module.Dual.transpose (R := K) (Matrix.toLin B₂ B₁ M) := by |
apply (LinearMap.toMatrix B₁.dualBasis B₂.dualBasis).injective
rw [LinearMap.toMatrix_toLin, LinearMap.toMatrix_transpose, LinearMap.toMatrix_toLin]
|
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Game.Ordinal
import Mathlib.SetTheory.Ordinal.NaturalOps
#align_import set_theory.game.birthday from "leanprover-com... | Mathlib/SetTheory/Game/Birthday.lean | 47 | 51 | theorem birthday_def (x : PGame) :
birthday x =
max (lsub.{u, u} fun i => birthday (x.moveLeft i))
(lsub.{u, u} fun i => birthday (x.moveRight i)) := by |
cases x; rw [birthday]; rfl
|
/-
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.Fintype.Card
import Mathlib.GroupTheory.Perm.Basic
import Mathlib.Tactic.Ring
#align_import data.fintype.perm from "leanprover-community/mathlib"... | Mathlib/Data/Fintype/Perm.lean | 77 | 94 | theorem mem_of_mem_permsOfList :
-- Porting note: was `∀ {x}` but need to capture the `x`
∀ {l : List α} {f : Perm α}, f ∈ permsOfList l → (x :α ) → f x ≠ x → x ∈ l
| [], f, h, heq_iff_eq => by
have : f = 1 := by | simpa [permsOfList] using h
rw [this]; simp
| a :: l, f, h, x =>
(mem_append.1 h).elim (fun h hx => mem_cons_of_mem _ (mem_of_mem_permsOfList h x hx))
fun h hx =>
let ⟨y, hy, hy'⟩ := List.mem_bind.1 h
let ⟨g, hg₁, hg₂⟩ := List.mem_map.1 hy'
-- Porting note: Seems like the implicit var... |
/-
Copyright (c) 2014 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Cast.Basic
import Mathlib.Algebra.CharZero.Defs
import Mathlib.Algebra.Order.Group.Abs
import Mathlib.Data.Nat.Cast.NeZero
import Mathlib.Alge... | Mathlib/Data/Nat/Cast/Order.lean | 88 | 88 | theorem cast_pos' {n : ℕ} : (0 : α) < n ↔ 0 < n := by | cases n <;> simp [cast_add_one_pos]
|
/-
Copyright (c) 2023 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.LinearAlgebra.FreeModule.PID
import Mathlib.MeasureTheory.Group.FundamentalDomain
import Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar
import Mathlib.Rin... | Mathlib/Algebra/Module/Zlattice/Basic.lean | 66 | 71 | theorem map_fundamentalDomain {F : Type*} [NormedAddCommGroup F] [NormedSpace K F] (f : E ≃ₗ[K] F) :
f '' (fundamentalDomain b) = fundamentalDomain (b.map f) := by |
ext x
rw [mem_fundamentalDomain, Basis.map_repr, LinearEquiv.trans_apply, ← mem_fundamentalDomain,
show f.symm x = f.toEquiv.symm x by rfl, ← Set.mem_image_equiv]
rfl
|
/-
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 | 68 | 72 | theorem contractLeftAux_contractLeftAux (v : M) (x : CliffordAlgebra Q) (fx : CliffordAlgebra Q) :
contractLeftAux Q d v (ι Q v * x, contractLeftAux Q d v (x, fx)) = Q v • fx := by |
simp only [contractLeftAux_apply_apply]
rw [mul_sub, ← mul_assoc, ι_sq_scalar, ← Algebra.smul_def, ← sub_add, mul_smul_comm, sub_self,
zero_add]
|
/-
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 | 119 | 124 | theorem maxPowDiv_eq_multiplicity {p n : ℕ} (hp : 1 < p) (hn : 0 < n) :
p.maxPowDiv n = multiplicity p n := by |
apply multiplicity.unique <| pow_dvd p n
intro h
apply Nat.not_lt.mpr <| le_of_dvd hp hn h
simp
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Michael Howes
-/
import Mathlib.Data.Finite.Card
import Mathlib.GroupTheory.Commutator
import Mathlib.GroupTheory.Finiteness
#align_import group_theory.abelianization from "lean... | Mathlib/GroupTheory/Abelianization.lean | 49 | 50 | theorem commutator_eq_closure : commutator G = Subgroup.closure (commutatorSet G) := by |
simp [commutator, Subgroup.commutator_def, commutatorSet]
|
/-
Copyright (c) 2022 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Data.Set.Lattice
#align_import data.set.intervals.ord_connected_component from "leanprover-community/... | Mathlib/Order/Interval/Set/OrdConnectedComponent.lean | 63 | 64 | theorem ordConnectedComponent_eq_empty : ordConnectedComponent s x = ∅ ↔ x ∉ s := by |
rw [← not_nonempty_iff_eq_empty, nonempty_ordConnectedComponent]
|
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Peter Pfaffelhuber
-/
import Mathlib.MeasureTheory.PiSystem
import Mathlib.Order.OmegaCompletePartialOrder
import Mathlib.Topology.Constructions
import Mathlib.MeasureTheor... | Mathlib/MeasureTheory/Constructions/Cylinders.lean | 57 | 61 | theorem squareCylinders_eq_iUnion_image (C : ∀ i, Set (Set (α i))) :
squareCylinders C = ⋃ s : Finset ι, (fun t ↦ (s : Set ι).pi t) '' univ.pi C := by |
ext1 f
simp only [squareCylinders, mem_iUnion, mem_image, mem_univ_pi, exists_prop, mem_setOf_eq,
eq_comm (a := f)]
|
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.List.GetD
#align_import data.set.list from "leanprover-community/mathlib"@"2ec920d35348cb2d13ac0e1a2ad9df0fdf1a76... | Mathlib/Data/Set/List.lean | 33 | 34 | theorem range_list_map_coe (s : Set α) : range (map ((↑) : s → α)) = { l | ∀ x ∈ l, x ∈ s } := by |
rw [range_list_map, Subtype.range_coe]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn, Mario Carneiro, Martin Dvorak
-/
import Mathlib.Data.List.Basic
#align_import data.list.join from "leanprover-community/mathlib"@"18a5306c091183ac... | Mathlib/Data/List/Join.lean | 65 | 66 | theorem join_join (l : List (List (List α))) : l.join.join = (l.map join).join := by |
induction l <;> simp [*]
|
/-
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.Logic.Pairwise
import Mathlib.Logic.Relation
import Mathlib.Data.List.Basic
#align_import data.list.pairwise from "leanprover-community/mathlib"@"f694... | Mathlib/Data/List/Pairwise.lean | 136 | 141 | theorem Pairwise.pmap {l : List α} (hl : Pairwise R l) {p : α → Prop} {f : ∀ a, p a → β}
(h : ∀ x ∈ l, p x) {S : β → β → Prop}
(hS : ∀ ⦃x⦄ (hx : p x) ⦃y⦄ (hy : p y), R x y → S (f x hx) (f y hy)) :
Pairwise S (l.pmap f h) := by |
refine (pairwise_pmap h).2 (Pairwise.imp_of_mem ?_ hl)
intros; apply hS; assumption
|
/-
Copyright (c) 2021 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Basic
import Mathlib.Topology.MetricSpace.Thickening
import Mathlib.Topology.MetricSpace.IsometricSMul
#alig... | Mathlib/Analysis/Normed/Group/Pointwise.lean | 30 | 35 | theorem Bornology.IsBounded.mul (hs : IsBounded s) (ht : IsBounded t) : IsBounded (s * t) := by |
obtain ⟨Rs, hRs⟩ : ∃ R, ∀ x ∈ s, ‖x‖ ≤ R := hs.exists_norm_le'
obtain ⟨Rt, hRt⟩ : ∃ R, ∀ x ∈ t, ‖x‖ ≤ R := ht.exists_norm_le'
refine isBounded_iff_forall_norm_le'.2 ⟨Rs + Rt, ?_⟩
rintro z ⟨x, hx, y, hy, rfl⟩
exact norm_mul_le_of_le (hRs x hx) (hRt y hy)
|
/-
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 | 78 | 80 | theorem QInfty_f_0 : (QInfty.f 0 : X _[0] ⟶ X _[0]) = 0 := by |
dsimp [QInfty]
simp only [sub_self]
|
/-
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.Module.Defs
import Mathlib.Algebra.Order.AbsoluteValue
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.GroupTheory.GroupAction.Units
#align_impor... | Mathlib/Data/Int/AbsoluteValue.lean | 41 | 42 | theorem AbsoluteValue.map_units_int_smul (abv : AbsoluteValue R S) (x : ℤˣ) (y : R) :
abv (x • y) = abv y := by | rcases Int.units_eq_one_or x with (rfl | rfl) <;> simp
|
/-
Copyright (c) 2022 Jireh Loreaux. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jireh Loreaux
-/
import Mathlib.Algebra.Star.Basic
import Mathlib.Data.Set.Finite
import Mathlib.Data.Set.Pointwise.Basic
#align_import algebra.star.pointwise from "leanprover-communit... | Mathlib/Algebra/Star/Pointwise.lean | 70 | 72 | theorem image_star [InvolutiveStar α] : Star.star '' s = s⋆ := by |
simp only [← star_preimage]
rw [image_eq_preimage_of_inverse] <;> intro <;> simp only [star_star]
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Topology.Algebra.Group.Basic
import Mathlib.Topology.Order.Basic
#align_import analysis.convex.strict from "leanprove... | Mathlib/Analysis/Convex/Strict.lean | 95 | 98 | theorem DirectedOn.strictConvex_sUnion {S : Set (Set E)} (hdir : DirectedOn (· ⊆ ·) S)
(hS : ∀ s ∈ S, StrictConvex 𝕜 s) : StrictConvex 𝕜 (⋃₀ S) := by |
rw [sUnion_eq_iUnion]
exact (directedOn_iff_directed.1 hdir).strictConvex_iUnion fun s => hS _ s.2
|
/-
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.Init.Logic
import Mathlib.Init.Function
import Mathlib.Tactic.TypeStar
#align_import logic.nontrivial from "leanprover-community/mathlib"@"48fb5... | Mathlib/Logic/Nontrivial/Defs.lean | 99 | 101 | theorem subsingleton_or_nontrivial (α : Type*) : Subsingleton α ∨ Nontrivial α := by |
rw [← not_nontrivial_iff_subsingleton, or_comm]
exact Classical.em _
|
/-
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.RingTheory.QuotientNilpotent
import Mathlib.RingTheory.Smooth.Basic
import Mathlib.RingTheory.Unramified.Basic
#align_import ring_theory.etale from "leanpro... | Mathlib/RingTheory/Etale/Basic.lean | 148 | 151 | theorem localization_map [FormallyEtale R S] : FormallyEtale Rₘ Sₘ := by |
haveI : FormallyEtale S Sₘ := FormallyEtale.of_isLocalization (M.map (algebraMap R S))
haveI : FormallyEtale R Sₘ := FormallyEtale.comp R S Sₘ
exact FormallyEtale.localization_base M
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.InnerProductSpace.TwoDim
import Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic
#align_import geometry.euclidean.angle.oriente... | Mathlib/Geometry/Euclidean/Angle/Oriented/Basic.lean | 78 | 82 | theorem oangle_self (x : V) : o.oangle x x = 0 := by |
rw [oangle, kahler_apply_self, ← ofReal_pow]
convert QuotientAddGroup.mk_zero (AddSubgroup.zmultiples (2 * π))
apply arg_ofReal_of_nonneg
positivity
|
/-
Copyright (c) 2024 Lawrence Wu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Lawrence Wu
-/
import Mathlib.Analysis.Fourier.Inversion
/-!
# Mellin inversion formula
We derive the Mellin inversion formula as a consequence of the Fourier inversion formula.
## Mai... | Mathlib/Analysis/MellinInversion.lean | 69 | 84 | theorem mellinInv_eq_fourierIntegralInv (σ : ℝ) (f : ℂ → E) {x : ℝ} (hx : 0 < x) :
mellinInv σ f x =
(x : ℂ) ^ (-σ : ℂ) • 𝓕⁻ (fun (y : ℝ) ↦ f (σ + 2 * π * y * I)) (-Real.log x) := calc
mellinInv σ f x
= (x : ℂ) ^ (-σ : ℂ) •
(∫ (y : ℝ), Complex.exp (2 * π * (y * (-Real.log x)) * I) • f (σ + 2 * π * ... |
rw [mellinInv, one_div, ← abs_of_pos (show 0 < (2 * π)⁻¹ by norm_num; exact pi_pos)]
have hx0 : (x : ℂ) ≠ 0 := ofReal_ne_zero.mpr (ne_of_gt hx)
simp_rw [neg_add, cpow_add _ _ hx0, mul_smul, integral_smul]
rw [smul_comm, ← Measure.integral_comp_mul_left]
congr! 3
rw [cpow_def_of_ne_zero hx0, ← C... |
/-
Copyright (c) 2024 Jeremy Tan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Tan
-/
import Mathlib.Combinatorics.SimpleGraph.Clique
/-!
# The Turán graph
This file defines the Turán graph and proves some of its basic properties.
## Main declarations
* `S... | Mathlib/Combinatorics/SimpleGraph/Turan.lean | 66 | 75 | theorem turanGraph_cliqueFree : (turanGraph n r).CliqueFree (r + 1) := by |
rw [cliqueFree_iff]
by_contra h
rw [not_isEmpty_iff] at h
obtain ⟨f, ha⟩ := h
simp only [turanGraph, top_adj] at ha
obtain ⟨x, y, d, c⟩ := Fintype.exists_ne_map_eq_of_card_lt (fun x ↦
(⟨(f x).1 % r, Nat.mod_lt _ hr⟩ : Fin r)) (by simp)
simp only [Fin.mk.injEq] at c
exact absurd c ((@ha x y).mpr d)
|
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.MeasureTheory.Integral.Lebesgue
#align_import measure_theory.measure.giry_monad from "leanprover-community/mathlib"@"56f4cd1ef396e9fd389b5d8371ee9ad91... | Mathlib/MeasureTheory/Measure/GiryMonad.lean | 118 | 120 | theorem join_zero : (0 : Measure (Measure α)).join = 0 := by |
ext1 s hs
simp only [hs, join_apply, lintegral_zero_measure, coe_zero, Pi.zero_apply]
|
/-
Copyright (c) 2020 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Jujian Zhang
-/
import Mathlib.RingTheory.Noetherian
import Mathlib.Algebra.DirectSum.Module
import Mathlib.Algebra.DirectSum.Finsupp
import Mathlib.Algebra.Module.Proj... | Mathlib/RingTheory/Flat/Basic.lean | 112 | 114 | theorem iff_lTensor_injective :
Module.Flat R M ↔ ∀ ⦃I : Ideal R⦄ (_ : I.FG), Function.Injective (lTensor M I.subtype) := by |
simpa [← comm_comp_rTensor_comp_comm_eq] using Module.Flat.iff_rTensor_injective R M
|
/-
Copyright (c) 2022 Devon Tuma. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Devon Tuma
-/
import Mathlib.Data.Vector.Basic
#align_import data.vector.mem from "leanprover-community/mathlib"@"509de852e1de55e1efa8eacfa11df0823f26f226"
/-!
# Theorems about membershi... | Mathlib/Data/Vector/Mem.lean | 31 | 35 | theorem mem_iff_get (v : Vector α n) : a ∈ v.toList ↔ ∃ i, v.get i = a := by |
simp only [List.mem_iff_get, Fin.exists_iff, Vector.get_eq_get]
exact
⟨fun ⟨i, hi, h⟩ => ⟨i, by rwa [toList_length] at hi, h⟩, fun ⟨i, hi, h⟩ =>
⟨i, by rwa [toList_length], h⟩⟩
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Yury Kudryashov
-/
import Mathlib.Data.Rat.Sqrt
import Mathlib.Data.Real.Sqrt
import Mathlib.RingTheory.Algebraic
import... | Mathlib/Data/Real/Irrational.lean | 70 | 85 | theorem irrational_nrt_of_n_not_dvd_multiplicity {x : ℝ} (n : ℕ) {m : ℤ} (hm : m ≠ 0) (p : ℕ)
[hp : Fact p.Prime] (hxr : x ^ n = m)
(hv : (multiplicity (p : ℤ) m).get (finite_int_iff.2 ⟨hp.1.ne_one, hm⟩) % n ≠ 0) :
Irrational x := by |
rcases Nat.eq_zero_or_pos n with (rfl | hnpos)
· rw [eq_comm, pow_zero, ← Int.cast_one, Int.cast_inj] at hxr
simp [hxr, multiplicity.one_right (mt isUnit_iff_dvd_one.1
(mt Int.natCast_dvd_natCast.1 hp.1.not_dvd_one)), Nat.zero_mod] at hv
refine irrational_nrt_of_notint_nrt _ _ hxr ?_ hnpos
rintro ⟨y,... |
/-
Copyright (c) 2024 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.LinearAlgebra.Dimension.Finite
import Mathlib.LinearAlgebra.Dimension.Constructions
/-!
# Some results on free modules over rings satisfying strong rank condition
T... | Mathlib/LinearAlgebra/Dimension/FreeAndStrongRankCondition.lean | 76 | 100 | theorem rank_le_one_iff [Module.Free K V] :
Module.rank K V ≤ 1 ↔ ∃ v₀ : V, ∀ v, ∃ r : K, r • v₀ = v := by |
obtain ⟨κ, b⟩ := Module.Free.exists_basis (R := K) (M := V)
constructor
· intro hd
rw [← b.mk_eq_rank'', le_one_iff_subsingleton] at hd
rcases isEmpty_or_nonempty κ with hb | ⟨⟨i⟩⟩
· use 0
have h' : ∀ v : V, v = 0 := by
simpa [range_eq_empty, Submodule.eq_bot_iff] using b.span_eq.symm
... |
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Function.LpSeminorm.Basic
#align_import measure_theory.function.lp_seminorm from "leanprover-community/mathlib"@"c4015acc0a223449d44061e27ddac... | Mathlib/MeasureTheory/Function/LpSeminorm/ChebyshevMarkov.lean | 44 | 49 | theorem mul_meas_ge_le_pow_snorm' (hp_ne_zero : p ≠ 0) (hp_ne_top : p ≠ ∞)
(hf : AEStronglyMeasurable f μ) (ε : ℝ≥0∞) :
ε ^ p.toReal * μ { x | ε ≤ ‖f x‖₊ } ≤ snorm f p μ ^ p.toReal := by |
convert mul_meas_ge_le_pow_snorm μ hp_ne_zero hp_ne_top hf (ε ^ p.toReal) using 4
ext x
rw [ENNReal.rpow_le_rpow_iff (ENNReal.toReal_pos hp_ne_zero hp_ne_top)]
|
/-
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 | 63 | 63 | theorem boundary_top : ∂ (⊤ : α) = ⊥ := by | rw [boundary, hnot_top, inf_bot_eq]
|
/-
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.Between
import Mathlib.Analysis.Convex.Normed
import Mathlib.Analysis.Normed.Group.AddTorsor
#align_import analysis.convex.side from "lean... | Mathlib/Analysis/Convex/Side.lean | 101 | 106 | theorem WOppSide.map {s : AffineSubspace R P} {x y : P} (h : s.WOppSide x y) (f : P →ᵃ[R] P') :
(s.map f).WOppSide (f x) (f y) := by |
rcases h with ⟨p₁, hp₁, p₂, hp₂, h⟩
refine ⟨f p₁, mem_map_of_mem f hp₁, f p₂, mem_map_of_mem f hp₂, ?_⟩
simp_rw [← linearMap_vsub]
exact h.map f.linear
|
/-
Copyright (c) 2022 Jiale Miao. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jiale Miao, Kevin Buzzard, Alexander Bentkamp
-/
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.LinearAlgebra.Matrix.Block
#align_import analysis.inner_product_space.gram_s... | Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean | 58 | 60 | theorem gramSchmidt_def (f : ι → E) (n : ι) :
gramSchmidt 𝕜 f n = f n - ∑ i ∈ Iio n, orthogonalProjection (𝕜 ∙ gramSchmidt 𝕜 f i) (f n) := by |
rw [← sum_attach, attach_eq_univ, gramSchmidt]
|
/-
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.Data.Complex.Abs
/-!
# The partial order on the complex numbers
This order is defined by `z ≤ w ↔ z.re ≤ w.re ∧ z.im = w.im`.
This is a natural orde... | Mathlib/Data/Complex/Order.lean | 91 | 92 | theorem not_lt_iff {z w : ℂ} : ¬z < w ↔ w.re ≤ z.re ∨ z.im ≠ w.im := by |
rw [lt_def, not_and_or, not_lt]
|
/-
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 | 82 | 102 | theorem monoidalPreadditive_of_faithful {D} [Category D] [Preadditive D] [MonoidalCategory D]
(F : MonoidalFunctor D C) [F.Faithful] [F.Additive] :
MonoidalPreadditive D :=
{ whiskerLeft_zero := by |
intros
apply F.toFunctor.map_injective
simp [F.map_whiskerLeft]
zero_whiskerRight := by
intros
apply F.toFunctor.map_injective
simp [F.map_whiskerRight]
whiskerLeft_add := by
intros
apply F.toFunctor.map_injective
simp only [F.map_whiskerLeft, Functor.map_a... |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Jujian Zhang
-/
import Mathlib.Algebra.Group.Subgroup.Pointwise
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Algebra.Order.Group.Action
import Mathlib.LinearAlge... | Mathlib/Algebra/Module/Submodule/Pointwise.lean | 115 | 120 | theorem closure_neg (s : Set M) : span R (-s) = -span R s := by |
apply le_antisymm
· rw [span_le, coe_set_neg, ← Set.neg_subset, neg_neg]
exact subset_span
· rw [neg_le, span_le, coe_set_neg, ← Set.neg_subset]
exact subset_span
|
/-
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.Topology.Sheaves.LocalPredicate
import Mathlib.Topology.Sheaves.Stalks
#align_import topology.sheaves.sheafify from "leanprover-community/mathlib"@"bb... | Mathlib/Topology/Sheaves/Sheafify.lean | 100 | 121 | theorem stalkToFiber_injective (x : X) : Function.Injective (F.stalkToFiber x) := by |
apply TopCat.stalkToFiber_injective
intro U V fU hU fV hV e
rcases hU ⟨x, U.2⟩ with ⟨U', mU, iU, gU, wU⟩
rcases hV ⟨x, V.2⟩ with ⟨V', mV, iV, gV, wV⟩
have wUx := wU ⟨x, mU⟩
dsimp at wUx; erw [wUx] at e; clear wUx
have wVx := wV ⟨x, mV⟩
dsimp at wVx; erw [wVx] at e; clear wVx
rcases F.germ_eq x mU mV ... |
/-
Copyright (c) 2022 Moritz Doll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Moritz Doll, Kalle Kytölä
-/
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.LinearAlgebra.SesquilinearForm
import Mathlib.Topology.Algebra.Module.WeakDual
#align_import analys... | Mathlib/Analysis/LocallyConvex/Polar.lean | 123 | 127 | theorem polar_weak_closed (s : Set E) : IsClosed[WeakBilin.instTopologicalSpace B.flip]
(B.polar s) := by |
rw [polar_eq_iInter]
refine isClosed_iInter fun x => isClosed_iInter fun _ => ?_
exact isClosed_le (WeakBilin.eval_continuous B.flip x).norm continuous_const
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.CategoryTheory.Extensive
import Mathlib.CategoryTheory.Limits.Shapes.KernelPair
#align_import category_theory.adhesive from "leanprover-community/mathlib"@"... | Mathlib/CategoryTheory/Adhesive.lean | 66 | 110 | theorem IsPushout.isVanKampen_iff (H : IsPushout f g h i) :
H.IsVanKampen ↔ IsVanKampenColimit (PushoutCocone.mk h i H.w) := by |
constructor
· intro H F' c' α fα eα hα
refine Iff.trans ?_
((H (F'.map WalkingSpan.Hom.fst) (F'.map WalkingSpan.Hom.snd) (c'.ι.app _) (c'.ι.app _)
(α.app _) (α.app _) (α.app _) fα (by convert hα WalkingSpan.Hom.fst)
(by convert hα WalkingSpan.Hom.snd) ?_ ?_ ?_).trans ?_)
· have ... |
/-
Copyright (c) 2020 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou
-/
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Support
#align_import algebra.indicator_function from "leanprover-community/mathlib"@"2445c98ae4b87... | Mathlib/Algebra/Group/Indicator.lean | 118 | 120 | theorem mulIndicator_eq_one : (mulIndicator s f = fun x => 1) ↔ Disjoint (mulSupport f) s := by |
simp only [funext_iff, mulIndicator_apply_eq_one, Set.disjoint_left, mem_mulSupport,
not_imp_not]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.OuterMeasure.Caratheodory
/-!
# Induced Outer Measure
We can extend a function defined on a subset of `Set α` to an out... | Mathlib/MeasureTheory/OuterMeasure/Induced.lean | 52 | 52 | theorem extend_eq_top {s : α} (h : ¬P s) : extend m s = ∞ := by | simp [extend, h]
|
/-
Copyright (c) 2022 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Batteries.Tactic.SeqFocus
/-! ## Ordering -/
namespace Ordering
@[simp] theorem swap_swap {o : Ordering} : o.swap.swap = o := by cases o <;> rfl
@[simp] th... | .lake/packages/batteries/Batteries/Classes/Order.lean | 46 | 47 | theorem cmp_eq_gt [OrientedCmp cmp] : cmp x y = .gt ↔ cmp y x = .lt := by |
rw [← Ordering.swap_inj, symm]; exact .rfl
|
/-
Copyright (c) 2021 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Data.Finset.Preimage
import Mathlib.Data.Set.Pointwise.... | Mathlib/GroupTheory/Finiteness.lean | 74 | 75 | theorem AddSubmonoid.fg_iff_mul_fg (P : AddSubmonoid N) : P.FG ↔ P.toSubmonoid.FG := by |
convert (Submonoid.fg_iff_add_fg (toSubmonoid P)).symm
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.Polynomial.IntegralNormalization
#align_import rin... | Mathlib/RingTheory/Algebraic.lean | 67 | 74 | theorem Subalgebra.isAlgebraic_iff (S : Subalgebra R A) :
S.IsAlgebraic ↔ @Algebra.IsAlgebraic R S _ _ S.algebra := by |
delta Subalgebra.IsAlgebraic
rw [Subtype.forall', Algebra.isAlgebraic_def]
refine forall_congr' fun x => exists_congr fun p => and_congr Iff.rfl ?_
have h : Function.Injective S.val := Subtype.val_injective
conv_rhs => rw [← h.eq_iff, AlgHom.map_zero]
rw [← aeval_algHom_apply, S.val_apply]
|
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Polynomial.Eval
import Mathlib.Analysis.SpecialFunctions.Exp
/-!
# Limits of `P(x) / e ^ x` for a polynomial `P`
In this file we prove that... | Mathlib/Analysis/SpecialFunctions/PolynomialExp.lean | 27 | 31 | theorem tendsto_div_exp_atTop (p : ℝ[X]) : Tendsto (fun x ↦ p.eval x / exp x) atTop (𝓝 0) := by |
induction p using Polynomial.induction_on' with
| h_monomial n c => simpa [exp_neg, div_eq_mul_inv, mul_assoc]
using tendsto_const_nhds.mul (tendsto_pow_mul_exp_neg_atTop_nhds_zero n)
| h_add p q hp hq => simpa [add_div] using hp.add hq
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Eric Wieser
-/
import Mathlib.Algebra.Quaternion
import Mathlib.Analysis.InnerProductSpace.Basic
import Mathlib.Analysis.InnerProductSpace.PiL2
import Mathlib.Topolog... | Mathlib/Analysis/Quaternion.lean | 73 | 74 | theorem norm_coe (a : ℝ) : ‖(a : ℍ)‖ = ‖a‖ := by |
rw [norm_eq_sqrt_real_inner, inner_self, normSq_coe, Real.sqrt_sq_eq_abs, Real.norm_eq_abs]
|
/-
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 | 117 | 118 | theorem finset_sum_apply' (I : Finset ι) (κ : ι → kernel α β) (a : α) (s : Set β) :
(∑ i ∈ I, κ i) a s = ∑ i ∈ I, κ i a s := by | rw [finset_sum_apply, Measure.finset_sum_apply]
|
/-
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.Multiset.Nodup
#align_import data.multiset.dedup from "leanprover-community/mathlib"@"9003f28797c0664a49e4179487267c494477d853"
/-!
# Erasing du... | Mathlib/Data/Multiset/Dedup.lean | 112 | 113 | theorem le_dedup_self {s : Multiset α} : s ≤ dedup s ↔ Nodup s := by |
rw [le_dedup, and_iff_right le_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 | 117 | 123 | theorem spanPoints_nonempty (s : Set P) : (spanPoints k s).Nonempty ↔ s.Nonempty := by |
constructor
· contrapose
rw [Set.not_nonempty_iff_eq_empty, Set.not_nonempty_iff_eq_empty]
intro h
simp [h, spanPoints]
· exact fun h => h.mono (subset_spanPoints _ _)
|
/-
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.AtTopBot
import Mathlib.Order.Filter.Pi
#align_import order.filter.cofinite from "leanprover-community/ma... | Mathlib/Order/Filter/Cofinite.lean | 57 | 58 | theorem cofinite_eq_bot_iff : @cofinite α = ⊥ ↔ Finite α := by |
simp [← empty_mem_iff_bot, finite_univ_iff]
|
/-
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.Order.BigOperators.Group.Finset
import Mathlib.Data.Nat.Factors
import Mathlib.Order.Interval.Finset.Nat
#align_import number_theory.divisors ... | Mathlib/NumberTheory/Divisors.lean | 75 | 75 | theorem properDivisors.not_self_mem : ¬n ∈ properDivisors n := by | simp [properDivisors]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Mon_
#align_import category_theory.monoidal.Mod_ from "leanprover-community/mathlib"@"33085c9739c41428651ac461a323fde9a2688d9b... | Mathlib/CategoryTheory/Monoidal/Mod_.lean | 37 | 38 | theorem assoc_flip :
(A.X ◁ M.act) ≫ M.act = (α_ A.X A.X M.X).inv ≫ (A.mul ▷ M.X) ≫ M.act := by | 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.Basic
/-!
### Relations between vector space derivative and manifold derivative
The manifold deriva... | Mathlib/Geometry/Manifold/MFDeriv/FDeriv.lean | 71 | 74 | theorem mdifferentiableWithinAt_iff_differentiableWithinAt :
MDifferentiableWithinAt 𝓘(𝕜, E) 𝓘(𝕜, E') f s x ↔ DifferentiableWithinAt 𝕜 f s x := by |
simp only [mdifferentiableWithinAt_iff', mfld_simps]
exact ⟨fun H => H.2, fun H => ⟨H.continuousWithinAt, H⟩⟩
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Leonardo de Moura
-/
import Mathlib.Algebra.GroupWithZero.Divisibility
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Tactic.NthRewrite
#align_import data.nat.gcd.... | Mathlib/Data/Nat/GCD/Basic.lean | 35 | 36 | theorem gcd_add_mul_right_right (m n k : ℕ) : gcd m (n + k * m) = gcd m n := by |
simp [gcd_rec m (n + k * m), gcd_rec m n]
|
/-
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.RingTheory.IntegrallyClosed
import Mathlib.RingTheory.Localization.NumDen
import Mathlib.RingTheory.Polynomial.ScaleRoots
#align_import ring_theory.polynomi... | Mathlib/RingTheory/Polynomial/RationalRoot.lean | 69 | 86 | theorem num_dvd_of_is_root {p : A[X]} {r : K} (hr : aeval r p = 0) : num A r ∣ p.coeff 0 := by |
suffices num A r ∣ (scaleRoots p (den A r)).coeff 0 by
simp only [coeff_scaleRoots, tsub_zero] at this
haveI inst := Classical.propDecidable
by_cases hr : num A r = 0
· obtain ⟨u, hu⟩ := (isUnit_den_of_num_eq_zero hr).pow p.natDegree
rw [← hu] at this
exact Units.dvd_mul_right.mp this
... |
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Paul Lezeau
-/
import Mathlib.RingTheory.DedekindDomain.Ideal
import Mathlib.RingTheory.IsAdjoinRoot
#align_import number_theory.kummer_dedekind from "leanprover-community/m... | Mathlib/NumberTheory/KummerDedekind.lean | 85 | 86 | theorem conductor_eq_top_of_adjoin_eq_top (h : R<x> = ⊤) : conductor R x = ⊤ := by |
simp only [Ideal.eq_top_iff_one, mem_conductor_iff, h, mem_top, forall_const]
|
/-
Copyright (c) 2020 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Algebra.Star.Basic
import Mathlib.Algebra.FreeAlgebra
#align_import algebra.star.free from "leanprover-community/mathlib"@"07c3cf2d851866ff7198219ed3fedf42e... | Mathlib/Algebra/Star/Free.lean | 72 | 73 | theorem star_algebraMap (r : R) : star (algebraMap R (FreeAlgebra R X) r) = algebraMap R _ r := by |
simp [star, Star.star]
|
/-
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.Analysis.InnerProductSpace.Basic
import Mathlib.LinearAlgebra.SesquilinearForm
#align_import analysis.inner_product_... | Mathlib/Analysis/InnerProductSpace/Orthogonal.lean | 116 | 123 | theorem orthogonal_eq_inter : Kᗮ = ⨅ v : K, LinearMap.ker (innerSL 𝕜 (v : E)) := by |
apply le_antisymm
· rw [le_iInf_iff]
rintro ⟨v, hv⟩ w hw
simpa using hw _ hv
· intro v hv w hw
simp only [mem_iInf] at hv
exact hv ⟨w, hw⟩
|
/-
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 | 136 | 141 | theorem imageToKernel_epi_comp {Z : V} (h : Z ⟶ A) [Epi h] (w) :
imageToKernel (h ≫ f) g w =
Subobject.ofLE _ _ (imageSubobject_comp_le h f) ≫
imageToKernel f g ((cancel_epi h).mp (by simpa using w : h ≫ f ≫ g = h ≫ 0)) := by |
ext
simp
|
/-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Translations
#align_import algebra.continued_fractions.terminated_stable from "leanprover-community/mathlib"@"a7e36e485... | Mathlib/Algebra/ContinuedFractions/TerminatedStable.lean | 69 | 72 | theorem continuants_stable_of_terminated (n_le_m : n ≤ m) (terminated_at_n : g.TerminatedAt n) :
g.continuants m = g.continuants n := by |
simp only [nth_cont_eq_succ_nth_cont_aux,
continuantsAux_stable_of_terminated (Nat.pred_le_iff.mp n_le_m) terminated_at_n]
|
/-
Copyright (c) 2023 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Junyan Xu
-/
import Mathlib.Logic.UnivLE
import Mathlib.SetTheory.Ordinal.Basic
/-!
# UnivLE and cardinals
-/
set_option autoImplicit true
noncomputable section
open Cardinal
| Mathlib/SetTheory/Cardinal/UnivLE.lean | 19 | 27 | theorem univLE_iff_cardinal_le : UnivLE.{u, v} ↔ univ.{u, v+1} ≤ univ.{v, u+1} := by |
rw [← not_iff_not, UnivLE]; simp_rw [small_iff_lift_mk_lt_univ]; push_neg
-- strange: simp_rw [univ_umax.{v,u}] doesn't work
refine ⟨fun ⟨α, le⟩ ↦ ?_, fun h ↦ ?_⟩
· rw [univ_umax.{v,u}, ← lift_le.{u+1}, lift_univ, lift_lift] at le
exact le.trans_lt (lift_lt_univ'.{u,v+1} #α)
· obtain ⟨⟨α⟩, h⟩ := lt_univ'... |
/-
Copyright (c) 2023 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.HahnBanach.Extension
import Mathlib.Analysis.NormedSpace.HahnBanach.Separation
import Mathlib.LinearAlgebra.Dual
import Math... | Mathlib/Analysis/NormedSpace/HahnBanach/SeparatingDual.lean | 57 | 60 | theorem exists_separating_of_ne {x y : V} (h : x ≠ y) :
∃ f : V →L[R] R, f x ≠ f y := by |
rcases exists_ne_zero (R := R) (sub_ne_zero_of_ne h) with ⟨f, hf⟩
exact ⟨f, by simpa [sub_ne_zero] using hf⟩
|
/-
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 | 46 | 47 | theorem prod_ofFn [CommMonoid β] {n : ℕ} (f : Fin n → β) : (List.ofFn f).prod = ∏ i, f i := by |
simp [prod_eq_multiset_prod]
|
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