Context stringlengths 285 157k | file_name stringlengths 21 79 | start int64 14 3.67k | end int64 18 3.69k | theorem stringlengths 25 2.71k | proof stringlengths 5 10.6k |
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/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.FieldTheory.RatFunc.Basic
import Mathlib.RingTheory.EuclideanDomain
import Mathlib.RingTheory.Localization.FractionRing
import Mathlib.RingTheory.Polynomial.... | Mathlib/FieldTheory/RatFunc/AsPolynomial.lean | 135 | 135 | theorem eval_X : eval f a X = a := by | simp [eval]
|
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Heather Macbeth
-/
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Constructions.BorelSpace.Metrizable
#align_import measu... | Mathlib/MeasureTheory/Function/SimpleFuncDense.lean | 95 | 99 | theorem nearestPtInd_le (e : ℕ → α) (N : ℕ) (x : α) : nearestPtInd e N x ≤ N := by |
induction' N with N ihN; · simp
simp only [nearestPtInd_succ]
split_ifs
exacts [le_rfl, ihN.trans N.le_succ]
|
/-
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.Init.ZeroOne
import Mathlib.Data.Set.Defs
import Mathlib.Order.Basic
import Mathlib.Order.SymmDiff
import Mathlib.Tactic.Tauto
import ... | Mathlib/Data/Set/Basic.lean | 2,028 | 2,031 | theorem insert_diff_singleton_comm (hab : a ≠ b) (s : Set α) :
insert a (s \ {b}) = insert a s \ {b} := by |
simp_rw [← union_singleton, union_diff_distrib,
diff_singleton_eq_self (mem_singleton_iff.not.2 hab.symm)]
|
/-
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 | 688 | 690 | theorem parallelogram_law {x y : E} : ⟪x + y, x + y⟫ + ⟪x - y, x - y⟫ = 2 * (⟪x, x⟫ + ⟪y, y⟫) := by |
simp only [inner_add_add_self, inner_sub_sub_self]
ring
|
/-
Copyright (c) 2022 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Analysis.NormedSpace.Dual
import Mathlib.Analysis.NormedSpace.Star.Basic
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.InnerProductSpac... | Mathlib/Analysis/VonNeumannAlgebra/Basic.lean | 137 | 139 | theorem coe_commutant (S : VonNeumannAlgebra H) :
↑S.commutant = Set.centralizer (S : Set (H →L[ℂ] H)) := by |
simp [commutant]
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Group.Nat
import Mathlib.Algebra.Order.Sub.Canonical
import Mathlib.Data.List.Perm
import Mathlib.Data.Set.List
import Mathlib.Init.Quot... | Mathlib/Data/Multiset/Basic.lean | 1,360 | 1,363 | theorem map_lt_map {f : α → β} {s t : Multiset α} (h : s < t) : s.map f < t.map f := by |
refine (map_le_map h.le).lt_of_not_le fun H => h.ne <| eq_of_le_of_card_le h.le ?_
rw [← s.card_map f, ← t.card_map f]
exact card_le_card H
|
/-
Copyright (c) 2021 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Eric Wieser
-/
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.RingTheory.GradedAlgebra.Basic... | Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean | 314 | 317 | theorem sInf {ℐ : Set (Ideal A)} (h : ∀ I ∈ ℐ, Ideal.IsHomogeneous 𝒜 I) :
(sInf ℐ).IsHomogeneous 𝒜 := by |
rw [sInf_eq_iInf]
exact iInf₂ h
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Defs
import Mathlib.GroupTheory.GroupAction.BigOperators
import Mathlib.LinearAlgebra.Prod
#align_import algebra.triv_sq_zero_ext f... | Mathlib/Algebra/TrivSqZeroExt.lean | 711 | 723 | theorem snd_list_prod [Semiring R] [AddCommMonoid M] [Module R M] [Module Rᵐᵒᵖ M]
[SMulCommClass R Rᵐᵒᵖ M] (l : List (tsze R M)) :
l.prod.snd =
(l.enum.map fun x : ℕ × tsze R M =>
((l.map fst).take x.1).prod •> x.snd.snd <• ((l.map fst).drop x.1.succ).prod).sum := by |
induction' l with x xs ih
· simp
· rw [List.enum_cons, ← List.map_fst_add_enum_eq_enumFrom]
simp_rw [List.map_cons, List.map_map, Function.comp, Prod.map_snd, Prod.map_fst, id,
List.take_zero, List.take_cons, List.prod_nil, List.prod_cons, snd_mul, one_smul, List.drop,
mul_smul, List.sum_cons, fs... |
/-
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.Set.Lattice
import Mathlib.Logic.Small.Basic
import Mathlib.Logic.Function.OfArity
import Mathlib.Order.WellFounded
#align_import set_theory.zfc.... | Mathlib/SetTheory/ZFC/Basic.lean | 1,546 | 1,548 | theorem classToCong_empty : classToCong ∅ = ∅ := by |
ext
simp [classToCong]
|
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.Topology.Algebra.Module.WeakDual
import Mathlib.MeasureTheory.Integral.BoundedContinuousFunction
import Mathlib.MeasureTheory.Measure.HasOuterApproxClosed
... | Mathlib/MeasureTheory/Measure/FiniteMeasure.lean | 282 | 284 | theorem smul_apply [IsScalarTower R ℝ≥0 ℝ≥0] (c : R) (μ : FiniteMeasure Ω) (s : Set Ω) :
(c • μ) s = c • μ s := by |
rw [coeFn_smul, Pi.smul_apply]
|
/-
Copyright (c) 2022 Jake Levinson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jake Levinson
-/
import Mathlib.Order.UpperLower.Basic
import Mathlib.Data.Finset.Preimage
#align_import combinatorics.young.young_diagram from "leanprover-community/mathlib"@"59694bd0... | Mathlib/Combinatorics/Young/YoungDiagram.lean | 379 | 384 | theorem col_eq_prod {μ : YoungDiagram} {j : ℕ} : μ.col j = Finset.range (μ.colLen j) ×ˢ {j} := by |
ext ⟨a, b⟩
simp only [Finset.mem_product, Finset.mem_singleton, Finset.mem_range, mem_col_iff,
mem_iff_lt_colLen, and_comm, and_congr_right_iff]
rintro rfl
rfl
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Alexander Bentkamp, Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
import Ma... | Mathlib/LinearAlgebra/LinearIndependent.lean | 930 | 933 | theorem LinearIndependent.repr_eq_single (i) (x : span R (range v)) (hx : ↑x = v i) :
hv.repr x = Finsupp.single i 1 := by |
apply hv.repr_eq
simp [Finsupp.total_single, hx]
|
/-
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 | 385 | 394 | theorem isSolvable' {α : E} {q : F[X]} (q_irred : Irreducible q) (q_aeval : aeval α q = 0)
(hα : IsSolvableByRad F α) : IsSolvable q.Gal := by |
have : _root_.IsSolvable (q * C q.leadingCoeff⁻¹).Gal := by
rw [minpoly.eq_of_irreducible q_irred q_aeval, ←
show minpoly F (⟨α, hα⟩ : solvableByRad F E) = minpoly F α from
(minpoly.algebraMap_eq (RingHom.injective _) _).symm]
exact isSolvable ⟨α, hα⟩
refine solvable_of_surjective (Gal.restri... |
/-
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 | 1,881 | 1,884 | theorem EventuallyLE.mul_le_mul' [Mul β] [Preorder β] [CovariantClass β β (· * ·) (· ≤ ·)]
[CovariantClass β β (swap (· * ·)) (· ≤ ·)] {l : Filter α} {f₁ f₂ g₁ g₂ : α → β}
(hf : f₁ ≤ᶠ[l] f₂) (hg : g₁ ≤ᶠ[l] g₂) : f₁ * g₁ ≤ᶠ[l] f₂ * g₂ := by |
filter_upwards [hf, hg] with x hfx hgx using _root_.mul_le_mul' hfx hgx
|
/-
Copyright (c) 2022 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.SetTheory.Cardinal.Finite
#align_import data.finite.card from "leanprover-community/mathlib"@"3ff3f2d6a3118b8711063de7111a0d77a53219a8"
/-!
# Cardinality ... | Mathlib/Data/Finite/Card.lean | 185 | 188 | theorem card_subtype_lt [Finite α] {p : α → Prop} {x : α} (hx : ¬p x) :
Nat.card { x // p x } < Nat.card α := by |
haveI := Fintype.ofFinite α
simpa only [Nat.card_eq_fintype_card, gt_iff_lt] using Fintype.card_subtype_lt hx
|
/-
Copyright (c) 2014 Robert Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Fiel... | Mathlib/Algebra/Order/Field/Basic.lean | 235 | 236 | theorem inv_le_one (ha : 1 ≤ a) : a⁻¹ ≤ 1 := by |
rwa [inv_le (zero_lt_one.trans_le ha) zero_lt_one, inv_one]
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau, Johan Commelin, Mario Carneiro, Kevin Buzzard,
Amelia Livingston, Yury Kudryashov
-/
import Mathlib.Algebra.FreeMonoid.Basic
import Mathlib.Algebra.Group.Sub... | Mathlib/Algebra/Group/Submonoid/Membership.lean | 400 | 402 | theorem exists_list_of_mem_closure {s : Set M} {x : M} (hx : x ∈ closure s) :
∃ l : List M, (∀ y ∈ l, y ∈ s) ∧ l.prod = x := by |
rwa [← SetLike.mem_coe, closure_eq_image_prod, Set.mem_image] at hx
|
/-
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 | 325 | 340 | theorem models_iff_not_satisfiable (φ : L.Sentence) : T ⊨ᵇ φ ↔ ¬IsSatisfiable (T ∪ {φ.not}) := by |
rw [models_sentence_iff, IsSatisfiable]
refine
⟨fun h1 h2 =>
(Sentence.realize_not _).1
(realize_sentence_of_mem (T ∪ {Formula.not φ})
(Set.subset_union_right (Set.mem_singleton _)))
(h1 (h2.some.subtheoryModel Set.subset_union_left)),
fun h M => ?_⟩
contrapose! h
rw [... |
/-
Copyright (c) 2021 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 810 | 812 | theorem dart_snd_mem_support_of_mem_darts {u v : V} (p : G.Walk u v) {d : G.Dart}
(h : d ∈ p.darts) : d.snd ∈ p.support := by |
simpa using p.reverse.dart_fst_mem_support_of_mem_darts (by simp [h] : d.symm ∈ p.reverse.darts)
|
/-
Copyright (c) 2023 Peter Nelson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Peter Nelson
-/
import Mathlib.Data.Matroid.Dual
/-!
# Matroid Restriction
Given `M : Matroid α` and `R : Set α`, the independent sets of `M` that are contained in `R`
are the independ... | Mathlib/Data/Matroid/Restrict.lean | 290 | 291 | theorem Restriction.of_subset {R' : Set α} (M : Matroid α) (h : R ⊆ R') : (M ↾ R) ≤r (M ↾ R') := by |
rw [← restrict_restrict_eq M h]; exact restrict_restriction _ _ h
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Fabian Glöckle, Kyle Miller
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModu... | Mathlib/LinearAlgebra/Dual.lean | 839 | 844 | theorem coe_basis : ⇑h.basis = e := by |
ext i
rw [Basis.apply_eq_iff]
ext j
rw [h.basis_repr_apply, coeffs_apply, h.eval, Finsupp.single_apply]
convert if_congr (eq_comm (a := j) (b := i)) rfl rfl
|
/-
Copyright (c) 2021 Arthur Paulino. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Arthur Paulino, Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Clique
import Mathlib.Data.ENat.Lattice
import Mathlib.Data.Nat.Lattice
import Mathlib.Data.Setoid.Partition
imp... | Mathlib/Combinatorics/SimpleGraph/Coloring.lean | 415 | 417 | theorem chromaticNumber_bot [Nonempty V] : (⊥ : SimpleGraph V).chromaticNumber = 1 := by |
have : (⊥ : SimpleGraph V).Colorable 1 := ⟨.mk 0 $ by simp⟩
exact this.chromaticNumber_le.antisymm $ ENat.one_le_iff_pos.2 $ chromaticNumber_pos this
|
/-
Copyright (c) 2022 Alexander Bentkamp. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp
-/
import Mathlib.Analysis.InnerProductSpace.Spectrum
import Mathlib.Data.Matrix.Rank
import Mathlib.LinearAlgebra.Matrix.Diagonal
import Mathlib.LinearAlgebra.M... | Mathlib/LinearAlgebra/Matrix/Spectrum.lean | 87 | 100 | theorem star_mul_self_mul_eq_diagonal :
(star (eigenvectorUnitary hA : Matrix n n 𝕜)) * A * (eigenvectorUnitary hA : Matrix n n 𝕜)
= diagonal (RCLike.ofReal ∘ hA.eigenvalues) := by |
apply Matrix.toEuclideanLin.injective
apply Basis.ext (EuclideanSpace.basisFun n 𝕜).toBasis
intro i
simp only [toEuclideanLin_apply, OrthonormalBasis.coe_toBasis, EuclideanSpace.basisFun_apply,
WithLp.equiv_single, ← mulVec_mulVec, eigenvectorUnitary_mulVec, ← mulVec_mulVec,
mulVec_eigenvectorBasis, M... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Floris van Doorn, Violeta Hernández Palacios
-/
import Mathlib.SetTheory.Ordinal.Basic
import Mathlib.Data.Nat.SuccPred
#align_import set_theory.ordinal.arithmetic fro... | Mathlib/SetTheory/Ordinal/Arithmetic.lean | 382 | 384 | theorem mk_initialSeg (o : Ordinal.{u}) :
#{ o' : Ordinal | o' < o } = Cardinal.lift.{u + 1} o.card := by |
rw [lift_card, ← type_subrel_lt, card_type]
|
/-
Copyright (c) 2014 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Gabriel Ebner
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Tactic.SplitIfs
#align_import data.nat.cast.defs from "leanprover-community/mathlib"@"a148d797a1094ab... | Mathlib/Data/Nat/Cast/Defs.lean | 242 | 246 | theorem three_add_one_eq_four [AddMonoidWithOne R] : 3 + 1 = (4 : R) := by |
rw [← two_add_one_eq_three, ← one_add_one_eq_two, ← Nat.cast_one,
← Nat.cast_add, ← Nat.cast_add, ← Nat.cast_add]
apply congrArg
decide
|
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Data.ENNReal.Real
import Mathlib.Order.Interval.Finset.Nat
import Mathlib.Topol... | Mathlib/Topology/EMetricSpace/Basic.lean | 960 | 961 | theorem diam_pair : diam ({x, y} : Set α) = edist x y := by |
simp only [iSup_singleton, diam_insert, diam_singleton, ENNReal.max_zero_right]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl, Patrick Massot
-/
import Mathlib.Data.Set.Image
import Mathlib.Data.SProd
#align_import data.set.prod from "leanprover-community/mathlib"@"48fb5b5280e7... | Mathlib/Data/Set/Prod.lean | 132 | 134 | theorem prod_union : s ×ˢ (t₁ ∪ t₂) = s ×ˢ t₁ ∪ s ×ˢ t₂ := by |
ext ⟨x, y⟩
simp [and_or_left]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johan Commelin, Amelia Livingston, Anne Baanen
-/
import Mathlib.Algebra.Algebra.Tower
import Mathlib.Algebra.GroupWithZero.NonZeroDivisors
import Mathlib.GroupTh... | Mathlib/RingTheory/Localization/Basic.lean | 1,206 | 1,212 | theorem sub_mk (a c) (b d) : (mk a b : Localization M) - mk c d =
mk ((d : R) * a - b * c) (b * d) :=
calc
mk a b - mk c d = mk a b + -mk c d := sub_eq_add_neg _ _
_ = mk a b + mk (-c) d := by | rw [neg_mk]
_ = mk (b * -c + d * a) (b * d) := add_mk _ _ _ _
_ = mk (d * a - b * c) (b * d) := by congr; ring
|
/-
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.Topology.UniformSpace.UniformConvergence
import Mathlib.Topology.UniformSpace.UniformEmbedding
import Mathlib.Topology.UniformSpace.Com... | Mathlib/Topology/Algebra/UniformGroup.lean | 722 | 732 | theorem tendsto_div_comap_self (x₀ : α) :
Tendsto (fun t : β × β => t.2 / t.1) ((comap fun p : β × β => (e p.1, e p.2)) <| 𝓝 (x₀, x₀))
(𝓝 1) := by |
have comm : ((fun x : α × α => x.2 / x.1) ∘ fun t : β × β => (e t.1, e t.2)) =
e ∘ fun t : β × β => t.2 / t.1 := by
ext t
change e t.2 / e t.1 = e (t.2 / t.1)
rw [← map_div e t.2 t.1]
have lim : Tendsto (fun x : α × α => x.2 / x.1) (𝓝 (x₀, x₀)) (𝓝 (e 1)) := by
simpa using (continuous_div'.c... |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attr
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Logic.Equiv.Set
import Math... | Mathlib/Data/Finset/Basic.lean | 2,344 | 2,345 | theorem erase_union_distrib (s t : Finset α) (a : α) : (s ∪ t).erase a = s.erase a ∪ t.erase a := by |
simp_rw [erase_eq, union_sdiff_distrib]
|
/-
Copyright (c) 2020 Zhangir Azerbayev. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser, Zhangir Azerbayev
-/
import Mathlib.GroupTheory.Perm.Sign
import Mathlib.Data.Fintype.Perm
import Mathlib.LinearAlgebra.Multilinear.Basis
#align_import linear_algebra.... | Mathlib/LinearAlgebra/Alternating/Basic.lean | 830 | 845 | theorem map_linearDependent {K : Type*} [Ring K] {M : Type*} [AddCommGroup M] [Module K M]
{N : Type*} [AddCommGroup N] [Module K N] [NoZeroSMulDivisors K N] (f : M [⋀^ι]→ₗ[K] N)
(v : ι → M) (h : ¬LinearIndependent K v) : f v = 0 := by |
obtain ⟨s, g, h, i, hi, hz⟩ := not_linearIndependent_iff.mp h
letI := Classical.decEq ι
suffices f (update v i (g i • v i)) = 0 by
rw [f.map_smul, Function.update_eq_self, smul_eq_zero] at this
exact Or.resolve_left this hz
-- Porting note: Was `conv at h in .. => ..`.
rw [← (funext fun x => ite_self... |
/-
Copyright (c) 2023 Antoine Chambert-Loir and María Inés de Frutos-Fernández. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, María Inés de Frutos-Fernández, Eric Wieser, Bhavik Mehta
-/
import Mathlib.Data.Finset.Antidiagonal
import Mathlib.Dat... | Mathlib/Data/Finset/PiAntidiagonal.lean | 161 | 168 | theorem finsuppAntidiag_empty_zero :
finsuppAntidiag (∅ : Finset ι) (0 : μ) = {0} := by |
ext f
rw [mem_finsuppAntidiag]
simp only [mem_singleton, subset_empty]
rw [support_eq_empty, and_iff_left_iff_imp]
intro hf
rw [hf, sum_zero_index]
|
/-
Copyright (c) 2021 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Yury Kudryashov
-/
import Mathlib.Topology.Order.LocalExtr
import Mathlib.Topology.Order.IntermediateValue
import Mathlib.Topology.Support
import Mathlib.Topology.Order... | Mathlib/Topology/Algebra/Order/Compact.lean | 227 | 238 | theorem atBot_le_cocompact [NoMinOrder α] [ClosedIicTopology α] :
atBot ≤ cocompact α := by |
refine fun s hs ↦ ?_
obtain ⟨t, ht, hts⟩ := mem_cocompact.mp hs
refine (Set.eq_empty_or_nonempty t).casesOn (fun h_empty ↦ ?_) (fun h_nonempty ↦ ?_)
· rewrite [compl_univ_iff.mpr h_empty, univ_subset_iff] at hts
convert univ_mem
· haveI := h_nonempty.nonempty
obtain ⟨a, ha⟩ := ht.exists_isLeast h_non... |
/-
Copyright (c) 2022 Yaël Dillies, Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Bhavik Mehta
-/
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Combinatorics.SimpleGraph.Basic
import Mathlib.Data.Rat.Cast.Order
import Mathlib.Orde... | Mathlib/Combinatorics/SimpleGraph/Density.lean | 196 | 205 | theorem edgeDensity_sub_edgeDensity_le_one_sub_mul (hs : s₂ ⊆ s₁) (ht : t₂ ⊆ t₁) (hs₂ : s₂.Nonempty)
(ht₂ : t₂.Nonempty) :
edgeDensity r s₂ t₂ - edgeDensity r s₁ t₁ ≤ 1 - s₂.card / s₁.card * (t₂.card / t₁.card) := by |
refine (sub_le_sub_left (mul_edgeDensity_le_edgeDensity r hs ht hs₂ ht₂) _).trans ?_
refine le_trans ?_ (mul_le_of_le_one_right ?_ (edgeDensity_le_one r s₂ t₂))
· rw [sub_mul, one_mul]
refine sub_nonneg_of_le (mul_le_one ?_ ?_ ?_)
· exact div_le_one_of_le ((@Nat.cast_le ℚ).2 (card_le_card hs)) (Nat.cast_nonn... |
/-
Copyright (c) 2022 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed
import Mathlib.RingTheory.PowerBasis
#align_import ring_theory.is_adjoin... | Mathlib/RingTheory/IsAdjoinRoot.lean | 136 | 137 | theorem map_eq_zero_iff (h : IsAdjoinRoot S f) {p} : h.map p = 0 ↔ f ∣ p := by |
rw [← h.mem_ker_map, RingHom.mem_ker]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.MeasureTheory.Function.SimpleFunc
import Mathlib.MeasureTheory.Measure.MutuallySingul... | Mathlib/MeasureTheory/Integral/Lebesgue.lean | 1,590 | 1,592 | theorem lintegral_singleton [MeasurableSingletonClass α] (f : α → ℝ≥0∞) (a : α) :
∫⁻ x in {a}, f x ∂μ = f a * μ {a} := by |
simp only [restrict_singleton, lintegral_smul_measure, lintegral_dirac, mul_comm]
|
/-
Copyright (c) 2021 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn, Sébastien Gouëzel
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.MeasureTheory.Group.Pointwise
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic... | Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | 402 | 404 | theorem addHaar_smul_of_nonneg {r : ℝ} (hr : 0 ≤ r) (s : Set E) :
μ (r • s) = ENNReal.ofReal (r ^ finrank ℝ E) * μ s := by |
rw [addHaar_smul, abs_pow, abs_of_nonneg hr]
|
/-
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.Algebraic
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.RingTheory.Localization.Integral
#align_import ring_theory.ideal.over fro... | Mathlib/RingTheory/Ideal/Over.lean | 390 | 408 | theorem exists_ideal_over_prime_of_isIntegral_of_isPrime
[Algebra.IsIntegral R S] (P : Ideal R) [IsPrime P]
(I : Ideal S) [IsPrime I] (hIP : I.comap (algebraMap R S) ≤ P) :
∃ Q ≥ I, IsPrime Q ∧ Q.comap (algebraMap R S) = P := by |
obtain ⟨Q' : Ideal (S ⧸ I), ⟨Q'_prime, hQ'⟩⟩ :=
@exists_ideal_over_prime_of_isIntegral_of_isDomain (R ⧸ I.comap (algebraMap R S)) _ (S ⧸ I) _
Ideal.quotientAlgebra _ _
(map (Ideal.Quotient.mk (I.comap (algebraMap R S))) P)
(map_isPrime_of_surjective Quotient.mk_surjective (by simp [hIP]))
... |
/-
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.LinearAlgebra.Finsupp
import Mathlib.RingTheory.Ideal.Over
import Mathlib.RingTheory.Ideal.Prod
import Mathlib.RingTheory.Ideal.MinimalPrime
import Mat... | Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean | 280 | 281 | theorem vanishingIdeal_univ : vanishingIdeal (∅ : Set (PrimeSpectrum R)) = ⊤ := by |
simpa using (gc R).u_top
|
/-
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 | 992 | 994 | theorem inner_self_eq_norm_mul_norm (x : E) : re ⟪x, x⟫ = ‖x‖ * ‖x‖ := by |
rw [@norm_eq_sqrt_inner 𝕜, ← sqrt_mul inner_self_nonneg (re ⟪x, x⟫),
sqrt_mul_self inner_self_nonneg]
|
/-
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, Kyle Miller
-/
import Mathlib.Data.Finset.Basic
import Mathlib.Data.Finite.Basic
import Mathlib.Data.Set.Functor
import Mathlib.Data.Set.Lattice
#align... | Mathlib/Data/Set/Finite.lean | 931 | 934 | theorem Finite.surjOn_iff_bijOn_of_mapsTo : SurjOn f s s ↔ BijOn f s s := by |
refine ⟨fun h ↦ ⟨hm, ?_, h⟩, BijOn.surjOn⟩
have : Finite s := finite_coe_iff.mpr hs
exact hm.restrict_inj.mp (Finite.injective_iff_surjective.mpr <| hm.restrict_surjective_iff.mpr h)
|
/-
Copyright (c) 2023 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Sym.Sym2
/-! # Unordered tuples of elements of a list
Defines `List.sym` and the specialized `List.sym2` for comp... | Mathlib/Data/List/Sym.lean | 63 | 66 | theorem right_mem_of_mk_mem_sym2 {xs : List α} {a b : α}
(h : s(a, b) ∈ xs.sym2) : b ∈ xs := by |
rw [Sym2.eq_swap] at h
exact left_mem_of_mk_mem_sym2 h
|
/-
Copyright (c) 2021 Justus Springer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Justus Springer
-/
import Mathlib.Algebra.Category.GroupCat.Basic
import Mathlib.Algebra.Category.MonCat.FilteredColimits
#align_import algebra.category.Group.filtered_colimits from ... | Mathlib/Algebra/Category/GroupCat/FilteredColimits.lean | 84 | 91 | theorem colimitInvAux_eq_of_rel (x y : Σ j, F.obj j)
(h : Types.FilteredColimit.Rel (F ⋙ forget GroupCat) x y) :
colimitInvAux.{v, u} F x = colimitInvAux F y := by |
apply G.mk_eq
obtain ⟨k, f, g, hfg⟩ := h
use k, f, g
rw [MonoidHom.map_inv, MonoidHom.map_inv, inv_inj]
exact hfg
|
/-
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 | 3,210 | 3,212 | theorem tendsto_iInf {f : α → β} {x : Filter α} {y : ι → Filter β} :
Tendsto f x (⨅ i, y i) ↔ ∀ i, Tendsto f x (y i) := by |
simp only [Tendsto, iff_self_iff, le_iInf_iff]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl
-/
import Mathlib.Topology.Order.MonotoneContinuity
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mathlib.Topology.Instances.NNReal
import Mathlib.Topology.E... | Mathlib/Topology/Instances/ENNReal.lean | 1,113 | 1,117 | theorem hasSum_toReal {f : α → ℝ≥0∞} (hsum : ∑' x, f x ≠ ∞) :
HasSum (fun x => (f x).toReal) (∑' x, (f x).toReal) := by |
lift f to α → ℝ≥0 using ENNReal.ne_top_of_tsum_ne_top hsum
simp only [coe_toReal, ← NNReal.coe_tsum, NNReal.hasSum_coe]
exact (tsum_coe_ne_top_iff_summable.1 hsum).hasSum
|
/-
Copyright (c) 2022 Alex J. Best. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex J. Best, Yaël Dillies
-/
import Mathlib.Algebra.Order.Hom.Ring
import Mathlib.Algebra.Order.Pointwise
import Mathlib.Analysis.SpecialFunctions.Pow.Real
#align_import algebra.order.... | Mathlib/Algebra/Order/CompleteField.lean | 191 | 199 | theorem inducedMap_rat (q : ℚ) : inducedMap α β (q : α) = q := by |
refine csSup_eq_of_forall_le_of_forall_lt_exists_gt
(cutMap_nonempty β (q : α)) (fun x h => ?_) fun w h => ?_
· rw [cutMap_coe] at h
obtain ⟨r, h, rfl⟩ := h
exact le_of_lt h
· obtain ⟨q', hwq, hq⟩ := exists_rat_btwn h
rw [cutMap_coe]
exact ⟨q', ⟨_, hq, rfl⟩, hwq⟩
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Kenny Lau
-/
import Mathlib.Algebra.BigOperators.GroupWithZero.Finset
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Module.LinearMap.Basic
import ... | Mathlib/Data/DFinsupp/Basic.lean | 1,166 | 1,167 | theorem support_subset_iff {s : Set ι} {f : Π₀ i, β i} : ↑f.support ⊆ s ↔ ∀ i ∉ s, f i = 0 := by |
simp [Set.subset_def]; exact forall_congr' fun i => not_imp_comm
|
/-
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.Subalgebra
import Mathlib.RingTheory.Noetherian
import Mathlib.RingTheory.Artinian
#align_import algebra.lie.submodule from "leanprover-communit... | Mathlib/Algebra/Lie/Submodule.lean | 1,153 | 1,155 | theorem map_le_idealRange : I.map f ≤ f.idealRange := by |
rw [f.idealRange_eq_map]
exact LieIdeal.map_mono le_top
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne
-/
import Mathlib.Analysis.SpecialFunctions.Exp
import Mathlib.Data.Nat.Factorization.Basic
import Mathlib.Analysis.NormedSpa... | Mathlib/Analysis/SpecialFunctions/Log/Basic.lean | 49 | 52 | theorem log_of_pos (hx : 0 < x) : log x = expOrderIso.symm ⟨x, hx⟩ := by |
rw [log_of_ne_zero hx.ne']
congr
exact abs_of_pos hx
|
/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Order.Ring.Cast
import Mathlib.Data.Int.Cast.Lemmas
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.PSub
import Mathlib.Data.Nat... | Mathlib/Data/Num/Lemmas.lean | 1,686 | 1,697 | theorem gcd_to_nat : ∀ a b, (gcd a b : ℕ) = Nat.gcd a b := by |
have : ∀ a b : Num, (a * b).natSize ≤ a.natSize + b.natSize := by
intros
simp only [natSize_to_nat, cast_mul]
rw [Nat.size_le, pow_add]
exact mul_lt_mul'' (Nat.lt_size_self _) (Nat.lt_size_self _) (Nat.zero_le _) (Nat.zero_le _)
intros
unfold gcd
split_ifs with h
· exact gcd_to_nat_aux h (thi... |
/-
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 | 977 | 978 | theorem pow_congr (_ : a = a') (_ : b = b')
(_ : a' ^ b' = c) : (a ^ b : R) = c := by | subst_vars; rfl
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.NumberTheory.Zsqrtd.GaussianInt
import Mathlib.NumberTheory.LegendreSymbol.Basic
import Mathlib.Analysis.Normed.Field.Basic
#align_import number_theory.zs... | Mathlib/NumberTheory/Zsqrtd/QuadraticReciprocity.lean | 33 | 83 | theorem mod_four_eq_three_of_nat_prime_of_prime (p : ℕ) [hp : Fact p.Prime]
(hpi : Prime (p : ℤ[i])) : p % 4 = 3 :=
hp.1.eq_two_or_odd.elim
(fun hp2 =>
absurd hpi
(mt irreducible_iff_prime.2 fun ⟨_, h⟩ => by
have := h ⟨1, 1⟩ ⟨1, -1⟩ (hp2.symm ▸ rfl)
rw [← norm_eq_one_iff, ← n... |
rw [← Nat.mod_mul_left_mod p 2 2, show 2 * 2 = 4 from rfl] at hp1
have := Nat.mod_lt p (show 0 < 4 by decide)
revert this hp3 hp1
generalize p % 4 = m
intros; interval_cases m <;> simp_all -- Porting note (#11043): was `decide!`
let ⟨k, hk⟩ := (ZMod.exists_sq_eq_neg_one_if... |
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Mario Carneiro
-/
import Batteries.Tactic.Alias
import Batteries.Data.List.Init.Attach
import Batteries.Data.List.Pairwise
-- Adaptation note: ... | .lake/packages/batteries/Batteries/Data/List/Perm.lean | 575 | 576 | theorem isSubperm_iff {l₁ l₂ : List α} : l₁.isSubperm l₂ ↔ l₁ <+~ l₂ := by |
simp [isSubperm, subperm_ext_iff]
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, Minchao Wu, Mario Carneiro
-/
import Mathlib.Data.Finset.Attr
import Mathlib.Data.Multiset.FinsetOps
import Mathlib.Logic.Equiv.Set
import Math... | Mathlib/Data/Finset/Basic.lean | 3,011 | 3,015 | theorem range_filter_eq {n m : ℕ} : (range n).filter (· = m) = if m < n then {m} else ∅ := by |
convert filter_eq (range n) m using 2
· ext
rw [eq_comm]
· simp
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Fintype.BigOperators
import Mathlib.Data.Finsupp.Defs
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Data.Set... | Mathlib/SetTheory/Cardinal/Basic.lean | 2,065 | 2,068 | theorem mk_biUnion_le_lift {α : Type u} {ι : Type v} (A : ι → Set α) (s : Set ι) :
lift.{v} #(⋃ x ∈ s, A x) ≤ lift.{u} #s * ⨆ x : s, lift.{v} #(A x.1) := by |
rw [biUnion_eq_iUnion]
apply mk_iUnion_le_lift
|
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Decomposition.RadonNikodym
import Mathlib.MeasureTheory.Measure.Haar.OfBasis
import Mathlib.Probability.Independence.Basic
#align_import proba... | Mathlib/Probability/Density.lean | 183 | 186 | theorem map_eq_withDensity_pdf {m : MeasurableSpace Ω} (X : Ω → E) (ℙ : Measure Ω)
(μ : Measure E := by | volume_tac) [hX : HasPDF X ℙ μ] :
map X ℙ = μ.withDensity (pdf X ℙ μ) := by
rw [pdf_def, withDensity_rnDeriv_eq _ _ hX.absolutelyContinuous]
|
/-
Copyright (c) 2022 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.Algebra.Module.Zlattice.Basic
import Mathlib.NumberTheory.NumberField.Embeddings
import Mathlib.NumberTheory.NumberField.FractionalIdeal
#align_import n... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/Basic.lean | 269 | 272 | theorem normAtPlace_add_le (w : InfinitePlace K) (x y : E K) :
normAtPlace w (x + y) ≤ normAtPlace w x + normAtPlace w y := by |
rw [normAtPlace, MonoidWithZeroHom.coe_mk, ZeroHom.coe_mk]
split_ifs <;> exact norm_add_le _ _
|
/-
Copyright (c) 2021 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.MeasureTheory.Measure.Sub
import Mathlib.MeasureTheory.Decomposition.SignedHahn
import Mathlib.MeasureTheory.Function.AEEqOfIntegral
#align_import measure_t... | Mathlib/MeasureTheory/Decomposition/Lebesgue.lean | 374 | 381 | theorem rnDeriv_lt_top (μ ν : Measure α) [SigmaFinite μ] : ∀ᵐ x ∂ν, μ.rnDeriv ν x < ∞ := by |
suffices ∀ n, ∀ᵐ x ∂ν, x ∈ spanningSets μ n → μ.rnDeriv ν x < ∞ by
filter_upwards [ae_all_iff.2 this] with _ hx using hx _ (mem_spanningSetsIndex _ _)
intro n
rw [← ae_restrict_iff' (measurable_spanningSets _ _)]
apply ae_lt_top (measurable_rnDeriv _ _)
refine (lintegral_rnDeriv_lt_top_of_measure_ne_top ... |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll, Thomas Zhu, Mario Carneiro
-/
import Mathlib.NumberTheory.LegendreSymbol.QuadraticReciprocity
#align_import number_theory.legendre_symbol.jacobi_symbol from "leanprover-... | Mathlib/NumberTheory/LegendreSymbol/JacobiSymbol.lean | 406 | 407 | theorem sq_eq_one {m n : ℕ} (hm : Odd m) (hn : Odd n) : qrSign m n ^ 2 = 1 := by |
rw [neg_one_pow hm hn, ← pow_mul, mul_comm, pow_mul, neg_one_sq, one_pow]
|
/-
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 | 319 | 323 | theorem card_commutator_closureCommutatorRepresentatives :
Nat.card (commutator (closureCommutatorRepresentatives G)) = Nat.card (commutator G) := by |
rw [commutator_eq_closure G, ← image_commutatorSet_closureCommutatorRepresentatives, ←
MonoidHom.map_closure, ← commutator_eq_closure]
exact Nat.card_congr (Equiv.Set.image _ _ (subtype_injective _))
|
/-
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.Topology.Connected.Basic
import Mathlib.Topology.Separation
/-!
# Separated maps and locally injective maps out of a topological space.
This module introduces a... | Mathlib/Topology/SeparatedMap.lean | 101 | 106 | theorem IsSeparatedMap.pullback {f : X → Y} (sep : IsSeparatedMap f) (g : A → Y) :
IsSeparatedMap (@snd X Y A f g) := by |
rw [isSeparatedMap_iff_isClosed_diagonal] at sep ⊢
rw [← preimage_map_fst_pullbackDiagonal]
refine sep.preimage (Continuous.mapPullback ?_ ?_) <;>
apply_rules [continuous_fst, continuous_subtype_val, Continuous.comp]
|
/-
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.Topology.Defs.Induced
import Mathlib.Topology.Basic
#align_import topology.order from "leanprover-community/mathlib"@"bcfa726826abd575... | Mathlib/Topology/Order.lean | 78 | 90 | theorem nhds_generateFrom {g : Set (Set α)} {a : α} :
@nhds α (generateFrom g) a = ⨅ s ∈ { s | a ∈ s ∧ s ∈ g }, 𝓟 s := by |
letI := generateFrom g
rw [nhds_def]
refine le_antisymm (biInf_mono fun s ⟨as, sg⟩ => ⟨as, .basic _ sg⟩) <| le_iInf₂ ?_
rintro s ⟨ha, hs⟩
induction hs with
| basic _ hs => exact iInf₂_le _ ⟨ha, hs⟩
| univ => exact le_top.trans_eq principal_univ.symm
| inter _ _ _ _ hs ht => exact (le_inf (hs ha.1) (ht ... |
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Constructions
#align_import topology.continuous_on from "leanprover-community/mathlib"@"d4f691b9e5f94cfc64639973f3544c95f8d5d494"
/-!
# Neig... | Mathlib/Topology/ContinuousOn.lean | 210 | 212 | theorem nhdsWithin_eq_nhdsWithin {a : α} {s t u : Set α} (h₀ : a ∈ s) (h₁ : IsOpen s)
(h₂ : t ∩ s = u ∩ s) : 𝓝[t] a = 𝓝[u] a := by |
rw [nhdsWithin_restrict t h₀ h₁, nhdsWithin_restrict u h₀ h₁, h₂]
|
/-
Copyright (c) 2020 Kevin Buzzard. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Buzzard
-/
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.RingTheory.Ideal.LocalRing
import Mathlib.RingTheory.Valuation.PrimeMultiplicity
import Mathlib.RingTheory... | Mathlib/RingTheory/DiscreteValuationRing/Basic.lean | 426 | 428 | theorem addVal_uniformizer {ϖ : R} (hϖ : Irreducible ϖ) : addVal R ϖ = 1 := by |
simpa only [one_mul, eq_self_iff_true, Units.val_one, pow_one, forall_true_left, Nat.cast_one]
using addVal_def ϖ 1 hϖ 1
|
/-
Copyright (c) 2021 Martin Zinkevich. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Martin Zinkevich, Rémy Degenne
-/
import Mathlib.Logic.Encodable.Lattice
import Mathlib.MeasureTheory.MeasurableSpace.Defs
#align_import measure_theory.pi_system fro... | Mathlib/MeasureTheory/PiSystem.lean | 314 | 344 | theorem mem_generatePiSystem_iUnion_elim' {α β} {g : β → Set (Set α)} {s : Set β}
(h_pi : ∀ b ∈ s, IsPiSystem (g b)) (t : Set α) (h_t : t ∈ generatePiSystem (⋃ b ∈ s, g b)) :
∃ (T : Finset β) (f : β → Set α), ↑T ⊆ s ∧ (t = ⋂ b ∈ T, f b) ∧ ∀ b ∈ T, f b ∈ g b := by |
have : t ∈ generatePiSystem (⋃ b : Subtype s, (g ∘ Subtype.val) b) := by
suffices h1 : ⋃ b : Subtype s, (g ∘ Subtype.val) b = ⋃ b ∈ s, g b by rwa [h1]
ext x
simp only [exists_prop, Set.mem_iUnion, Function.comp_apply, Subtype.exists, Subtype.coe_mk]
rfl
rcases @mem_generatePiSystem_iUnion_elim α (S... |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Group.Pi.Basic
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Data.Set.Lattice
#... | Mathlib/Order/Interval/Set/Pi.lean | 184 | 185 | theorem image_update_Ioc_right (f : ∀ i, α i) (i : ι) (b : α i) :
update f i '' Ioc (f i) b = Ioc f (update f i b) := by | simpa using image_update_Ioc f i (f i) b
|
/-
Copyright (c) 2018 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Data.PFunctor.Univariate.M
#align_import data.qpf.univariate.basic from "leanprover-community/mathlib"@"14b69e9f3c16630440a2cbd46f1ddad0d561dee7"
/-!
... | Mathlib/Data/QPF/Univariate/Basic.lean | 693 | 696 | theorem supp_map (h : q.IsUniform) {α β : Type u} (g : α → β) (x : F α) :
supp (g <$> x) = g '' supp x := by |
rw [← abs_repr x]; cases' repr x with a f; rw [← abs_map, PFunctor.map_eq]
rw [supp_eq_of_isUniform h, supp_eq_of_isUniform h, image_comp]
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Alexander Bentkamp, Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.LinearAlgebra.Prod
import Ma... | Mathlib/LinearAlgebra/LinearIndependent.lean | 1,410 | 1,412 | theorem linearIndependent_fin_succ' {n} {v : Fin (n + 1) → V} : LinearIndependent K v ↔
LinearIndependent K (Fin.init v) ∧ v (Fin.last _) ∉ Submodule.span K (range <| Fin.init v) := by |
rw [← linearIndependent_fin_snoc, Fin.snoc_init_self]
|
/-
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 | 184 | 185 | theorem U_neg_two : U R (-2) = -1 := by |
simpa [zero_sub, Int.reduceNeg, U_neg_one, mul_zero, U_zero] using U_sub_two R 0
|
/-
Copyright (c) 2021 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Data.ZMod.Quotient
#align_import group_theory.complement from "leanprover-community/mathlib"@"6ca1a09bc9aa75824bf97388c9e3b441fc4ccf3f"
/-!
# Compl... | Mathlib/GroupTheory/Complement.lean | 441 | 445 | theorem equiv_fst_eq_self_of_mem_of_one_mem {g : G} (h1 : 1 ∈ T) (hg : g ∈ S) :
(hST.equiv g).fst = ⟨g, hg⟩ := by |
have : hST.equiv.symm (⟨g, hg⟩, ⟨1, h1⟩) = g := by
rw [equiv, Equiv.ofBijective]; simp
conv_lhs => rw [← this, Equiv.apply_symm_apply]
|
/-
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.Algebra.CharP.Defs
import Mathlib.RingTheory.Multiplicity
import Mathlib.RingTheory.PowerSeries.Basic
#align_import ring_theory.power_seri... | Mathlib/RingTheory/PowerSeries/Order.lean | 368 | 387 | theorem divided_by_X_pow_orderMul {f g : R⟦X⟧} (hf : f ≠ 0) (hg : g ≠ 0) :
divided_by_X_pow_order hf * divided_by_X_pow_order hg =
divided_by_X_pow_order (mul_ne_zero hf hg) := by |
set df := f.order.get (order_finite_iff_ne_zero.mpr hf)
set dg := g.order.get (order_finite_iff_ne_zero.mpr hg)
set dfg := (f * g).order.get (order_finite_iff_ne_zero.mpr (mul_ne_zero hf hg)) with hdfg
have H_add_d : df + dg = dfg := by simp_all only [PartENat.get_add, order_mul f g]
have H := self_eq_X_pow_... |
/-
Copyright (c) 2014 Robert Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Lewis, Leonardo de Moura, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.GroupWithZero.Units.Equiv
import Mathlib.Algebra.Order.Fiel... | Mathlib/Algebra/Order/Field/Basic.lean | 367 | 367 | theorem div_le_one (hb : 0 < b) : a / b ≤ 1 ↔ a ≤ b := by | rw [div_le_iff hb, one_mul]
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.BigOperators.Pi
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Algebra.Order.BigOperators.Ring.Finset
import Mathlib.Algebra.BigOperators.Fin
im... | Mathlib/Algebra/BigOperators/Finsupp.lean | 131 | 134 | theorem prod_ite_eq' [DecidableEq α] (f : α →₀ M) (a : α) (b : α → M → N) :
(f.prod fun x v => ite (x = a) (b x v) 1) = ite (a ∈ f.support) (b a (f a)) 1 := by |
dsimp [Finsupp.prod]
rw [f.support.prod_ite_eq']
|
/-
Copyright (c) 2020 Alexander Bentkamp, Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alexander Bentkamp, Sébastien Gouëzel, Eric Wieser
-/
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Algebra.Order.Module.OrderedSMul
import Mathlib.Data... | Mathlib/Data/Complex/Module.lean | 442 | 444 | theorem skewAdjoint.I_smul_neg_I (a : skewAdjoint A) : I • (skewAdjoint.negISMul a : A) = a := by |
simp only [smul_smul, skewAdjoint.negISMul_apply_coe, neg_smul, smul_neg, I_mul_I, one_smul,
neg_neg]
|
/-
Copyright (c) 2019 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Lu-Ming Zhang
-/
import Mathlib.Data.Matrix.Invertible
import Mathlib.LinearAlgebra.Matrix.Adjugate
import Mathlib.LinearAlgebra.FiniteDimensional
#align_import linear_algeb... | Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | 596 | 598 | theorem inv_adjugate (A : Matrix n n α) (h : IsUnit A.det) : (adjugate A)⁻¹ = h.unit⁻¹ • A := by |
refine inv_eq_left_inv ?_
rw [smul_mul, mul_adjugate, Units.smul_def, smul_smul, h.val_inv_mul, one_smul]
|
/-
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.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Basic
import Mathlib.RingTheory.Ideal.Maps
import Mathlib.RingTheory.MvPower... | Mathlib/RingTheory/PowerSeries/Basic.lean | 453 | 464 | theorem coeff_X_pow_mul (p : R⟦X⟧) (n d : ℕ) :
coeff R (d + n) (X ^ n * p) = coeff R d p := by |
rw [coeff_mul, Finset.sum_eq_single (n, d), coeff_X_pow, if_pos rfl, one_mul]
· rintro ⟨i, j⟩ h1 h2
rw [coeff_X_pow, if_neg, zero_mul]
rintro rfl
apply h2
rw [mem_antidiagonal, add_comm, add_right_cancel_iff] at h1
subst h1
rfl
· rw [add_comm]
exact fun h1 => (h1 (mem_antidiagonal.2 r... |
/-
Copyright (c) 2021 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.LinearAlgebra.TensorAlgebra.Basic
import Mathlib.LinearAlgebra.TensorPower
#align_import linear_algebra.tensor_algebra.to_tensor_power from "leanprover-comm... | Mathlib/LinearAlgebra/TensorAlgebra/ToTensorPower.lean | 107 | 110 | theorem ofDirectSum_comp_toDirectSum :
ofDirectSum.comp toDirectSum = AlgHom.id R (TensorAlgebra R M) := by |
ext
simp [DirectSum.lof_eq_of, tprod_apply]
|
/-
Copyright (c) 2021 Kalle Kytölä. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kalle Kytölä
-/
import Mathlib.MeasureTheory.Measure.ProbabilityMeasure
import Mathlib.MeasureTheory.Measure.Lebesgue.Basic
import Mathlib.MeasureTheory.Integral.Layercake
import Mathlib... | Mathlib/MeasureTheory/Measure/Portmanteau.lean | 281 | 311 | theorem FiniteMeasure.limsup_measure_closed_le_of_tendsto {Ω ι : Type*} {L : Filter ι}
[MeasurableSpace Ω] [TopologicalSpace Ω] [HasOuterApproxClosed Ω]
[OpensMeasurableSpace Ω] {μ : FiniteMeasure Ω}
{μs : ι → FiniteMeasure Ω} (μs_lim : Tendsto μs L (𝓝 μ)) {F : Set Ω} (F_closed : IsClosed F) :
(L.limsu... |
rcases L.eq_or_neBot with rfl | hne
· simp only [limsup_bot, bot_le]
apply ENNReal.le_of_forall_pos_le_add
intro ε ε_pos _
let fs := F_closed.apprSeq
have key₁ : Tendsto (fun n ↦ ∫⁻ ω, (fs n ω : ℝ≥0∞) ∂μ) atTop (𝓝 ((μ : Measure Ω) F)) :=
HasOuterApproxClosed.tendsto_lintegral_apprSeq F_closed (μ : Me... |
/-
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.Algebra.Order.Monoid.Unbundled.MinMax
import Mathlib.Algebra.Order.Monoid.WithTop
import Mathlib.Data.Finset.Image
import Mathlib.Data.Multiset.Fold
#... | Mathlib/Data/Finset/Fold.lean | 144 | 155 | theorem fold_ite' {g : α → β} (hb : op b b = b) (p : α → Prop) [DecidablePred p] :
Finset.fold op b (fun i => ite (p i) (f i) (g i)) s =
op (Finset.fold op b f (s.filter p)) (Finset.fold op b g (s.filter fun i => ¬p i)) := by |
classical
induction' s using Finset.induction_on with x s hx IH
· simp [hb]
· simp only [Finset.fold_insert hx]
split_ifs with h
· have : x ∉ Finset.filter p s := by simp [hx]
simp [Finset.filter_insert, h, Finset.fold_insert this, ha.assoc, IH]
· have : x ∉ Finset.filter (fun i... |
/-
Copyright (c) 2016 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad
-/
import Mathlib.Data.List.OfFn
import Mathlib.Data.List.Nodup
import Mathlib.Data.List.Infix
#align_import data.list.sort from "leanprover-community/mathlib"@"f694c7dea... | Mathlib/Data/List/Sort.lean | 172 | 174 | theorem sorted_ofFn_iff {r : α → α → Prop} : (ofFn f).Sorted r ↔ ((· < ·) ⇒ r) f f := by |
simp_rw [Sorted, pairwise_iff_get, get_ofFn, Relator.LiftFun]
exact Iff.symm (Fin.rightInverse_cast _).surjective.forall₂
|
/-
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 | 507 | 516 | theorem measure_iUnion_eq_iSup' {α ι : Type*} [MeasurableSpace α] {μ : Measure α}
[Countable ι] [Preorder ι] [IsDirected ι (· ≤ ·)]
{f : ι → Set α} : μ (⋃ i, f i) = ⨆ i, μ (Accumulate f i) := by |
have hd : Directed (· ⊆ ·) (Accumulate f) := by
intro i j
rcases directed_of (· ≤ ·) i j with ⟨k, rik, rjk⟩
exact ⟨k, biUnion_subset_biUnion_left fun l rli ↦ le_trans rli rik,
biUnion_subset_biUnion_left fun l rlj ↦ le_trans rlj rjk⟩
rw [← iUnion_accumulate]
exact measure_iUnion_eq_iSup hd
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Floris van Doorn
-/
import Mathlib.Data.Finsupp.Multiset
import Mathlib.Order.Bounded
import Mathlib.SetTheory.Cardinal.PartENat
import Mathlib.SetTheor... | Mathlib/SetTheory/Cardinal/Ordinal.lean | 616 | 624 | theorem mul_eq_max_of_aleph0_le_left {a b : Cardinal} (h : ℵ₀ ≤ a) (h' : b ≠ 0) :
a * b = max a b := by |
rcases le_or_lt ℵ₀ b with hb | hb
· exact mul_eq_max h hb
refine (mul_le_max_of_aleph0_le_left h).antisymm ?_
have : b ≤ a := hb.le.trans h
rw [max_eq_left this]
convert mul_le_mul_left' (one_le_iff_ne_zero.mpr h') a
rw [mul_one]
|
/-
Copyright (c) 2021 Julian Kuelshammer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Julian Kuelshammer
-/
import Mathlib.Data.ZMod.Quotient
import Mathlib.GroupTheory.NoncommPiCoprod
import Mathlib.GroupTheory.OrderOfElement
import Mathlib.Algebra.GCDMonoid.Finset... | Mathlib/GroupTheory/Exponent.lean | 628 | 633 | theorem MonoidHom.exponent_dvd {F M₁ M₂ : Type*} [Monoid M₁] [Monoid M₂]
[FunLike F M₁ M₂] [MonoidHomClass F M₁ M₂]
{f : F} (hf : Function.Surjective f) : exponent M₂ ∣ exponent M₁ := by |
refine Monoid.exponent_dvd_of_forall_pow_eq_one fun m₂ ↦ ?_
obtain ⟨m₁, rfl⟩ := hf m₂
rw [← map_pow, pow_exponent_eq_one, map_one]
|
/-
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.Data.Rat.Basic
import Batteries.Tactic.SeqFocus
/-! # Additional lemmas about the Rational Numbers -/
namespace Rat
theorem ext : {p q : Rat} → p.... | .lake/packages/batteries/Batteries/Data/Rat/Lemmas.lean | 215 | 217 | theorem mkRat_add_mkRat (n₁ n₂ : Int) {d₁ d₂} (z₁ : d₁ ≠ 0) (z₂ : d₂ ≠ 0) :
mkRat n₁ d₁ + mkRat n₂ d₂ = mkRat (n₁ * d₂ + n₂ * d₁) (d₁ * d₂) := by |
rw [← normalize_eq_mkRat z₁, ← normalize_eq_mkRat z₂, normalize_add_normalize, normalize_eq_mkRat]
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Algebra.Group.Nat
import Mathlib.Algebra.Order.Sub.Canonical
import Mathlib.Data.List.Perm
import Mathlib.Data.Set.List
import Mathlib.Init.Quot... | Mathlib/Data/Multiset/Basic.lean | 2,942 | 2,945 | theorem map_eq_map {f : α → β} (hf : Function.Injective f) {s t : Multiset α} :
s.map f = t.map f ↔ s = t := by |
rw [← rel_eq, ← rel_eq, rel_map]
simp only [hf.eq_iff]
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.Multilinear.Curry
#align_import analysis.calculus.formal_multilinear_series from "leanprover-community/mathlib"@"f2ce608671... | Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean | 258 | 260 | theorem order_eq_zero_iff' : p.order = 0 ↔ p = 0 ∨ p 0 ≠ 0 := by |
simpa [order, Nat.sInf_eq_zero, FormalMultilinearSeries.ext_iff, eq_empty_iff_forall_not_mem]
using or_comm
|
/-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Junyan Xu
-/
import Mathlib.Data.DFinsupp.Basic
#align_import data.dfinsupp.ne_locus from "leanprover-community/mathlib"@"f7fc89d5d5ff1db2d1242c7bb0e9062ce47ef47c"
/-!
# Lo... | Mathlib/Data/DFinsupp/NeLocus.lean | 72 | 74 | theorem neLocus_zero_right : f.neLocus 0 = f.support := by |
ext
rw [mem_neLocus, mem_support_iff, coe_zero, Pi.zero_apply]
|
/-
Copyright (c) 2020 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne, Sébastien Gouëzel
-/
import Mathlib.Analysis.NormedSpace.IndicatorFunction
import Mathlib.MeasureTheory.Function.EssSup
import Mathlib.MeasureTheory.Function.AEEqFun
import... | Mathlib/MeasureTheory/Function/LpSeminorm/Basic.lean | 511 | 539 | theorem snorm_norm_rpow (f : α → F) (hq_pos : 0 < q) :
snorm (fun x => ‖f x‖ ^ q) p μ = snorm f (p * ENNReal.ofReal q) μ ^ q := by |
by_cases h0 : p = 0
· simp [h0, ENNReal.zero_rpow_of_pos hq_pos]
by_cases hp_top : p = ∞
· simp only [hp_top, snorm_exponent_top, ENNReal.top_mul', hq_pos.not_le, ENNReal.ofReal_eq_zero,
if_false, snorm_exponent_top, snormEssSup]
have h_rpow :
essSup (fun x : α => (‖‖f x‖ ^ q‖₊ : ℝ≥0∞)) μ =
... |
/-
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 | 74 | 74 | theorem rtake_nil : rtake ([] : List α) n = [] := by | simp [rtake]
|
/-
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.Reverse
import Mathlib.Algebra.Regular.SMul
#align_import data.polynomial.monic from "l... | Mathlib/Algebra/Polynomial/Monic.lean | 355 | 357 | theorem leadingCoeff_map' (p : R[X]) : leadingCoeff (p.map f) = f (leadingCoeff p) := by |
unfold leadingCoeff
rw [coeff_map, natDegree_map_eq_of_injective hf p]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Order.Filter.Basic
import Mathlib.Topology.Bases
import Mathlib.Data.Set.Accumulate
import Mathlib.Topology.Bornology.... | Mathlib/Topology/Compactness/Compact.lean | 134 | 139 | theorem isCompact_iff_ultrafilter_le_nhds :
IsCompact s ↔ ∀ f : Ultrafilter X, ↑f ≤ 𝓟 s → ∃ x ∈ s, ↑f ≤ 𝓝 x := by |
refine (forall_neBot_le_iff ?_).trans ?_
· rintro f g hle ⟨x, hxs, hxf⟩
exact ⟨x, hxs, hxf.mono hle⟩
· simp only [Ultrafilter.clusterPt_iff]
|
/-
Copyright (c) 2022 Junyan Xu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa, Junyan Xu
-/
import Mathlib.Algebra.Order.Group.PiLex
import Mathlib.Data.DFinsupp.Order
import Mathlib.Data.DFinsupp.NeLocus
import Mathlib.Order.WellFoundedSet
#align_impo... | Mathlib/Data/DFinsupp/Lex.lean | 61 | 64 | theorem lex_lt_of_lt [∀ i, PartialOrder (α i)] (r) [IsStrictOrder ι r] {x y : Π₀ i, α i}
(hlt : x < y) : Pi.Lex r (· < ·) x y := by |
simp_rw [Pi.Lex, le_antisymm_iff]
exact lex_lt_of_lt_of_preorder r hlt
|
/-
Copyright (c) 2020 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov, Patrick Massot
-/
import Mathlib.Order.Interval.Set.UnorderedInterval
import Mathlib.Algebra.Order.Interval.Set.Monoid
import Mathlib.Data.Set.Pointwise.Basic
i... | Mathlib/Data/Set/Pointwise/Interval.lean | 668 | 670 | theorem preimage_mul_const_Ioc_of_neg (a b : α) {c : α} (h : c < 0) :
(fun x => x * c) ⁻¹' Ioc a b = Ico (b / c) (a / c) := by |
simp [← Ioi_inter_Iic, ← Ici_inter_Iio, h, inter_comm]
|
/-
Copyright (c) 2020 Kevin Kappelmann. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kevin Kappelmann
-/
import Mathlib.Algebra.ContinuedFractions.Computation.CorrectnessTerminating
import Mathlib.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
impo... | Mathlib/Algebra/ContinuedFractions/Computation/Approximations.lean | 294 | 309 | theorem of_denom_mono : (of v).denominators n ≤ (of v).denominators (n + 1) := by |
let g := of v
cases' Decidable.em <| g.partialDenominators.TerminatedAt n with terminated not_terminated
· have : g.partialDenominators.get? n = none := by rwa [Stream'.Seq.TerminatedAt] at terminated
have : g.TerminatedAt n :=
terminatedAt_iff_part_denom_none.2 (by rwa [Stream'.Seq.TerminatedAt] at te... |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.ZPow
import Mathlib.Analysis.SpecialFunctions.Sqrt
import Mathlib.Analysis.SpecialFunctions.Log.Deriv
impo... | Mathlib/Analysis/Convex/SpecificFunctions/Deriv.lean | 168 | 171 | theorem strictConcaveOn_sin_Icc : StrictConcaveOn ℝ (Icc 0 π) sin := by |
apply strictConcaveOn_of_deriv2_neg (convex_Icc _ _) continuousOn_sin fun x hx => ?_
rw [interior_Icc] at hx
simp [sin_pos_of_mem_Ioo hx]
|
/-
Copyright (c) 2018 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Markus Himmel, Bhavik Mehta, Andrew Yang, Emily Riehl
-/
import Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks
import Mathlib.CategoryTheory.Limits.Shapes.BinaryPro... | Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean | 2,267 | 2,269 | theorem inl_inl_pushoutLeftPushoutInrIso_hom :
pushout.inl ≫ pushout.inl ≫ (pushoutLeftPushoutInrIso f g g').hom = pushout.inl := by |
rw [← Category.assoc, ← Iso.eq_comp_inv, inl_pushoutLeftPushoutInrIso_inv]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov
-/
import Mathlib.GroupTheory.QuotientGroup
import Mathlib.LinearAlgebra.Span
#align_import linear_algebra.quotient from... | Mathlib/LinearAlgebra/Quotient.lean | 257 | 259 | theorem mk_surjective : Function.Surjective (@mk _ _ _ _ _ p) := by |
rintro ⟨x⟩
exact ⟨x, rfl⟩
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Set.Function
import Mathlib.Logic.Equiv.Defs
import Mathlib.Tactic.Says
#align_import logic.equiv.set from "leanprover-... | Mathlib/Logic/Equiv/Set.lean | 503 | 506 | theorem image_symm_preimage {α β} {f : α → β} (hf : Injective f) (u s : Set α) :
(fun x => (Set.image f s hf).symm x : f '' s → α) ⁻¹' u = Subtype.val ⁻¹' (f '' u) := by |
ext ⟨b, a, has, rfl⟩
simp [hf.eq_iff]
|
/-
Copyright (c) 2021 Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller
-/
import Mathlib.Combinatorics.SimpleGraph.Subgraph
import Mathlib.Data.List.Rotate
#align_import combinatorics.simple_graph.connectivity from "leanprover-community/mathlib"... | Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | 574 | 575 | theorem support_reverse {u v : V} (p : G.Walk u v) : p.reverse.support = p.support.reverse := by |
induction p <;> simp [support_append, *]
|
/-
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 | 146 | 183 | theorem IsPushout.isVanKampen_inl {W E X Z : C} (c : BinaryCofan W E) [FinitaryExtensive C]
[HasPullbacks C] (hc : IsColimit c) (f : W ⟶ X) (h : X ⟶ Z) (i : c.pt ⟶ Z)
(H : IsPushout f c.inl h i) : H.IsVanKampen := by |
obtain ⟨hc₁⟩ := (is_coprod_iff_isPushout c hc H.1).mpr H
introv W' hf hg hh hi w
obtain ⟨hc₂⟩ := ((BinaryCofan.isVanKampen_iff _).mp (FinitaryExtensive.vanKampen c hc)
(BinaryCofan.mk _ pullback.fst) _ _ _ hg.w.symm pullback.condition.symm).mpr
⟨hg, IsPullback.of_hasPullback αY c.inr⟩
refine (is_coprod... |
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