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 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Yury Kudryashov, Sébastien Gouëzel, Rémy Degenne
-/
import Mathlib.MeasureTheory.Function.SimpleFuncDenseLp
#align_import measure_theory.integral.set_to_l1 from "leanprov... | Mathlib/MeasureTheory/Integral/SetToL1.lean | 1,101 | 1,107 | theorem setToL1_smul_left (hT : DominatedFinMeasAdditive μ T C) (c : ℝ) (f : α →₁[μ] E) :
setToL1 (hT.smul c) f = c • setToL1 hT f := by |
suffices setToL1 (hT.smul c) = c • setToL1 hT by rw [this, ContinuousLinearMap.smul_apply]
refine ContinuousLinearMap.extend_unique (setToL1SCLM α E μ (hT.smul c)) _ _ _ _ ?_
ext1 f
suffices c • setToL1 hT f = setToL1SCLM α E μ (hT.smul c) f by rw [← this]; congr
rw [setToL1_eq_setToL1SCLM, setToL1SCLM_smul_... |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Chris Hughes, Mario Carneiro
-/
import Mathlib.Tactic.FinCases
import Mathlib.Data.Nat.Choose.Sum
import Mathlib.LinearAlgebra.Finsupp
import Mathlib.Algebra.Field.IsField
#alig... | Mathlib/RingTheory/Ideal/Basic.lean | 521 | 522 | theorem span_singleton_mul_right_unit {a : α} (h2 : IsUnit a) (x : α) :
span ({x * a} : Set α) = span {x} := by | rw [mul_comm, span_singleton_mul_left_unit h2]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Michael Stoll
-/
import Mathlib.NumberTheory.LegendreSymbol.Basic
import Mathlib.NumberTheory.LegendreSymbol.QuadraticChar.GaussSum
#align_import number_theory.legendre_sy... | Mathlib/NumberTheory/LegendreSymbol/QuadraticReciprocity.lean | 89 | 96 | theorem exists_sq_eq_neg_two_iff : IsSquare (-2 : ZMod p) ↔ p % 8 = 1 ∨ p % 8 = 3 := by |
rw [FiniteField.isSquare_neg_two_iff, card p]
have h₁ := Prime.mod_two_eq_one_iff_ne_two.mpr hp
rw [← mod_mod_of_dvd p (by decide : 2 ∣ 8)] at h₁
have h₂ := mod_lt p (by norm_num : 0 < 8)
revert h₂ h₁
generalize p % 8 = m; clear! p
intros; interval_cases m <;> simp_all -- Porting note (#11043): was `deci... |
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker, Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Finsupp.Multiset
import Math... | Mathlib/RingTheory/UniqueFactorizationDomain.lean | 1,518 | 1,521 | theorem factors_mul (a b : Associates α) : (a * b).factors = a.factors + b.factors := by |
nontriviality α
refine eq_of_prod_eq_prod <| eq_of_factors_eq_factors ?_
rw [prod_add, factors_prod, factors_prod, factors_prod]
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers
-/
import Mathlib.Algebra.ModEq
import Mathlib.Algebra.Module.Defs
import Mathlib.Algebra.Order.Archimedean
import Mathlib.Algebra.Periodic
import Mathlib.Data.Int.SuccPred
... | Mathlib/Algebra/Order/ToIntervalMod.lean | 339 | 340 | theorem toIcoDiv_sub (a b : α) : toIcoDiv hp a (b - p) = toIcoDiv hp a b - 1 := by |
simpa only [one_zsmul] using toIcoDiv_sub_zsmul hp a b 1
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Michael Stoll
-/
import Mathlib.Data.Nat.Squarefree
import Mathlib.NumberTheory.Zsqrtd.QuadraticReciprocity
import Mathlib.Tactic.LinearCombination
#align_import number_th... | Mathlib/NumberTheory/SumTwoSquares.lean | 125 | 138 | theorem ZMod.isSquare_neg_one_iff' {n : ℕ} (hn : Squarefree n) :
IsSquare (-1 : ZMod n) ↔ ∀ {q : ℕ}, q ∣ n → q % 4 ≠ 3 := by |
have help : ∀ a b : ZMod 4, a ≠ 3 → b ≠ 3 → a * b ≠ 3 := by decide
rw [ZMod.isSquare_neg_one_iff hn]
refine ⟨?_, fun H q _ => H⟩
intro H
refine @induction_on_primes _ ?_ ?_ (fun p q hp hq hpq => ?_)
· exact fun _ => by norm_num
· exact fun _ => by norm_num
· replace hp := H hp (dvd_of_mul_right_dvd hpq... |
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Batteries.Control.ForInStep.Lemmas
import Batteries.Data.List.Basic
import Batteries.Ta... | .lake/packages/batteries/Batteries/Data/List/Lemmas.lean | 960 | 961 | theorem diff_cons_right (l₁ l₂ : List α) (a : α) : l₁.diff (a :: l₂) = (l₁.diff l₂).erase a := by |
apply Eq.symm; induction l₂ generalizing l₁ <;> simp [erase_comm, *]
|
/-
Copyright (c) 2022 Jujian Zhang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jujian Zhang, Scott Morrison, Joël Riou
-/
import Mathlib.CategoryTheory.Abelian.InjectiveResolution
import Mathlib.Algebra.Homology.Additive
import Mathlib.CategoryTheory.Abelian.Homolo... | Mathlib/CategoryTheory/Abelian/RightDerived.lean | 219 | 223 | theorem NatTrans.rightDerived_id (F : C ⥤ D) [F.Additive] (n : ℕ) :
NatTrans.rightDerived (𝟙 F) n = 𝟙 (F.rightDerived n) := by |
dsimp only [rightDerived]
simp only [rightDerivedToHomotopyCategory_id, whiskerRight_id']
rfl
|
/-
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
-/
import Mathlib.Algebra.Order.CauSeq.BigOperators
import Mathlib.Data.Complex.Abs
import Mathlib.Data.Complex.BigOperators
import Mathlib.Data.Na... | Mathlib/Data/Complex/Exponential.lean | 739 | 739 | theorem sin_sq : sin x ^ 2 = 1 - cos x ^ 2 := by | rw [← sin_sq_add_cos_sq x, add_sub_cancel_right]
|
/-
Copyright (c) 2018 Patrick Massot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Johannes Hölzl, Yaël Dillies
-/
import Mathlib.Analysis.Normed.Group.Seminorm
import Mathlib.Order.LiminfLimsup
import Mathlib.Topology.Instances.Rat
import Mathlib.Top... | Mathlib/Analysis/Normed/Group/Basic.lean | 777 | 777 | theorem mem_sphere_iff_norm' : b ∈ sphere a r ↔ ‖b / a‖ = r := by | simp [dist_eq_norm_div]
|
/-
Copyright (c) 2020 Aaron Anderson, Jalex Stark, Kyle Miller. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jalex Stark, Kyle Miller, Alena Gusakov, Hunter Monroe
-/
import Mathlib.Combinatorics.SimpleGraph.Init
import Mathlib.Data.Rel
import Mathlib... | Mathlib/Combinatorics/SimpleGraph/Basic.lean | 863 | 866 | theorem commonNeighbors_top_eq {v w : V} :
(⊤ : SimpleGraph V).commonNeighbors v w = Set.univ \ {v, w} := by |
ext u
simp [commonNeighbors, eq_comm, not_or]
|
/-
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.Interval
import Mathlib.Order.Interval.Set.Pi
import Mathlib.Tactic.TFAE
import Mathlib.Tactic.NormNum
im... | Mathlib/Topology/Order/Basic.lean | 615 | 617 | theorem countable_of_isolated_left' [SecondCountableTopology α] :
Set.Countable { x : α | ∃ y, y < x ∧ Ioo y x = ∅ } := by |
simpa only [← covBy_iff_Ioo_eq] using countable_setOf_covBy_left
|
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Yury Kudryashov
-/
import Mathlib.Analysis.Normed.Group.InfiniteSum
import Mathlib.Analysis.Normed.MulAction
import Mathlib.Topology.Algebra.Order.LiminfLimsup
import Mat... | Mathlib/Analysis/Asymptotics/Asymptotics.lean | 1,987 | 1,994 | theorem isBigOWith_of_eq_mul {u v : α → R} (φ : α → R) (hφ : ∀ᶠ x in l, ‖φ x‖ ≤ c)
(h : u =ᶠ[l] φ * v) :
IsBigOWith c l u v := by |
simp only [IsBigOWith_def]
refine h.symm.rw (fun x a => ‖a‖ ≤ c * ‖v x‖) (hφ.mono fun x hx => ?_)
simp only [Pi.mul_apply]
refine (norm_mul_le _ _).trans ?_
gcongr
|
/-
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.LinearAlgebra.AffineSpace.Independent
import Mathlib.LinearAlgebra.Basis
#align_import linear_algebra.affine_space.basis from "leanprover-community/mathlib"... | Mathlib/LinearAlgebra/AffineSpace/Basis.lean | 182 | 183 | theorem coord_apply [DecidableEq ι] (i j : ι) : b.coord i (b j) = if i = j then 1 else 0 := by |
rcases eq_or_ne i j with h | h <;> simp [h]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex
#align_import analysis.special_function... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean | 210 | 215 | theorem arccos_eq_arctan {x : ℝ} (h : 0 < x) : arccos x = arctan (√(1 - x ^ 2) / x) := by |
rw [arccos, eq_comm]
refine arctan_eq_of_tan_eq ?_ ⟨?_, ?_⟩
· rw_mod_cast [tan_pi_div_two_sub, tan_arcsin, inv_div]
· linarith only [arcsin_le_pi_div_two x, pi_pos]
· linarith only [arcsin_pos.2 h]
|
/-
Copyright (c) 2020 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Algebra.Group.Conj
import Mathlib.Algebra.Group.Pi.Lemmas
import Mathlib.Algebra.Group.Subsemigroup.Operations
import Mathlib.Algebra.Group.Submonoid.Operati... | Mathlib/Algebra/Group/Subgroup/Basic.lean | 3,121 | 3,124 | theorem codisjoint_subgroupOf_sup (H K : Subgroup G) :
Codisjoint (H.subgroupOf (H ⊔ K)) (K.subgroupOf (H ⊔ K)) := by |
rw [codisjoint_iff, sup_subgroupOf_eq, subgroupOf_self]
exacts [le_sup_left, le_sup_right]
|
/-
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.FieldTheory.Minpoly.Field
#align_import ring_theory.power_basis from "leanprover-community/mathlib"@"d1d69e99ed34c95266668af4e288fc1c598b9a7f"
/-!
# Power ... | Mathlib/RingTheory/PowerBasis.lean | 105 | 116 | theorem mem_span_pow {x y : S} {d : ℕ} (hd : d ≠ 0) :
y ∈ Submodule.span R (Set.range fun i : Fin d => x ^ (i : ℕ)) ↔
∃ f : R[X], f.natDegree < d ∧ y = aeval x f := by |
rw [mem_span_pow']
constructor <;>
· rintro ⟨f, h, hy⟩
refine ⟨f, ?_, hy⟩
by_cases hf : f = 0
· simp only [hf, natDegree_zero, degree_zero] at h ⊢
first | exact lt_of_le_of_ne (Nat.zero_le d) hd.symm | exact WithBot.bot_lt_coe d
simp_all only [degree_eq_natDegree hf]
· fir... |
/-
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.Geometry.Euclidean.Sphere.Basic
import Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional
import Mathlib.Tactic.DeriveFintype
#align_import geometry.eucl... | Mathlib/Geometry/Euclidean/Circumcenter.lean | 781 | 787 | theorem exists_circumradius_eq_of_cospherical {ps : Set P} {n : ℕ} [FiniteDimensional ℝ V]
(hd : finrank ℝ V = n) (hc : Cospherical ps) :
∃ r : ℝ, ∀ sx : Simplex ℝ P n, Set.range sx.points ⊆ ps → sx.circumradius = r := by |
haveI : Nonempty (⊤ : AffineSubspace ℝ P) := Set.univ.nonempty
rw [← finrank_top, ← direction_top ℝ V P] at hd
refine exists_circumradius_eq_of_cospherical_subset ?_ hd hc
exact Set.subset_univ _
|
/-
Copyright (c) 2015 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis
-/
import Mathlib.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Algebra.Ring.Parity
#align_import algebra.grou... | Mathlib/Algebra/Order/Ring/Basic.lean | 100 | 101 | theorem le_self_pow (ha : 1 ≤ a) (h : m ≠ 0) : a ≤ a ^ m := by |
simpa only [pow_one] using pow_le_pow_right ha <| Nat.pos_iff_ne_zero.2 h
|
/-
Copyright (c) 2016 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.Nat.Basic
/-! # Basic lemmas about natural numbers
The primary purpose ... | .lake/packages/batteries/Batteries/Data/Nat/Lemmas.lean | 81 | 86 | theorem recDiag_succ_zero {motive : Nat → Nat → Sort _} (zero_zero : motive 0 0)
(zero_succ : ∀ n, motive 0 n → motive 0 (n+1)) (succ_zero : ∀ m, motive m 0 → motive (m+1) 0)
(succ_succ : ∀ m n, motive m n → motive (m+1) (n+1)) (m) :
Nat.recDiag zero_zero zero_succ succ_zero succ_succ (m+1) 0
= succ_z... |
simp [Nat.recDiag]; cases m <;> rfl
|
/-
Copyright (c) 2014 Parikshit Khanna. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura, Floris van Doorn, Mario Carneiro
-/
import Batteries.Control.ForInStep.Lemmas
import Batteries.Data.List.Basic
import Batteries.Ta... | .lake/packages/batteries/Batteries/Data/List/Lemmas.lean | 1,382 | 1,382 | theorem not_mem_range_self {n : Nat} : n ∉ range n := by | simp
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Data.Nat.Lattice
import Mathlib.Logic.Denumerable
import Mathlib.Logic.Function.Iterate
import Mathlib.Order.Hom.Basic
import Mathlib.Data.Set.Subsing... | Mathlib/Order/OrderIsoNat.lean | 84 | 86 | theorem not_acc_of_decreasing_seq (f : ((· > ·) : ℕ → ℕ → Prop) ↪r r) (k : ℕ) : ¬Acc r (f k) := by |
rw [acc_iff_no_decreasing_seq, not_isEmpty_iff]
exact ⟨⟨f, k, rfl⟩⟩
|
/-
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 | 260 | 261 | theorem coeff_ne_zero_C {a : R} {n : ℕ} (h : n ≠ 0) : coeff R n (C R a) = 0 := by |
rw [coeff_C, if_neg h]
|
/-
Copyright (c) 2021 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Ashvni Narayanan
-/
import Mathlib.Algebra.Order.Group.TypeTags
import Mathlib.FieldTheory.RatFunc.Degree
import Mathlib.RingTheory.DedekindDomain.IntegralClosure
import Math... | Mathlib/NumberTheory/FunctionField.lean | 62 | 80 | theorem functionField_iff (Fqt : Type*) [Field Fqt] [Algebra Fq[X] Fqt]
[IsFractionRing Fq[X] Fqt] [Algebra (RatFunc Fq) F] [Algebra Fqt F] [Algebra Fq[X] F]
[IsScalarTower Fq[X] Fqt F] [IsScalarTower Fq[X] (RatFunc Fq) F] :
FunctionField Fq F ↔ FiniteDimensional Fqt F := by |
let e := IsLocalization.algEquiv Fq[X]⁰ (RatFunc Fq) Fqt
have : ∀ (c) (x : F), e c • x = c • x := by
intro c x
rw [Algebra.smul_def, Algebra.smul_def]
congr
refine congr_fun (f := fun c => algebraMap Fqt F (e c)) ?_ c -- Porting note: Added `(f := _)`
refine IsLocalization.ext (nonZeroDivisors ... |
/-
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.Polynomial.Expand
import Mathlib.Algebra.Polynomial.Splits
import Mathlib.Algebra.Squarefree.Basic
import Mathlib.FieldTheory.Minpoly.Field
import Mathli... | Mathlib/FieldTheory/Separable.lean | 380 | 403 | theorem exists_separable_of_irreducible {f : F[X]} (hf : Irreducible f) (hp : p ≠ 0) :
∃ (n : ℕ) (g : F[X]), g.Separable ∧ expand F (p ^ n) g = f := by |
replace hp : p.Prime := (CharP.char_is_prime_or_zero F p).resolve_right hp
induction' hn : f.natDegree using Nat.strong_induction_on with N ih generalizing f
rcases separable_or p hf with (h | ⟨h1, g, hg, hgf⟩)
· refine ⟨0, f, h, ?_⟩
rw [pow_zero, expand_one]
· cases' N with N
· rw [natDegree_eq_zero... |
/-
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 | 132 | 133 | theorem mem_ker_map (h : IsAdjoinRoot S f) {p} : p ∈ RingHom.ker h.map ↔ f ∣ p := by |
rw [h.ker_map, Ideal.mem_span_singleton]
|
/-
Copyright (c) 2019 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Analysis.Calculus.TangentCone
import Mathlib.Analysis.NormedSpace.OperatorNorm.Asymptotics
#align_import analysis.ca... | Mathlib/Analysis/Calculus/FDeriv/Basic.lean | 419 | 426 | theorem hasFDerivWithinAt_insert {y : E} :
HasFDerivWithinAt f f' (insert y s) x ↔ HasFDerivWithinAt f f' s x := by |
rcases eq_or_ne x y with (rfl | h)
· simp_rw [HasFDerivWithinAt, hasFDerivAtFilter_iff_isLittleO]
apply Asymptotics.isLittleO_insert
simp only [sub_self, map_zero]
refine ⟨fun h => h.mono <| subset_insert y s, fun hf => hf.mono_of_mem ?_⟩
simp_rw [nhdsWithin_insert_of_ne h, self_mem_nhdsWithin]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Johannes Hölzl
-/
import Mathlib.MeasureTheory.Constructions.BorelSpace.Order
#align_import measure_theory.function.simple_func from "leanprover-community/mathlib"@"bf... | Mathlib/MeasureTheory/Function/SimpleFunc.lean | 1,242 | 1,251 | theorem lintegral_lt_top {f : α →ₛ ℝ≥0∞} (hm : f.FinMeasSupp μ) (hf : ∀ᵐ a ∂μ, f a ≠ ∞) :
f.lintegral μ < ∞ := by |
refine sum_lt_top fun a ha => ?_
rcases eq_or_ne a ∞ with (rfl | ha)
· simp only [ae_iff, Ne, Classical.not_not] at hf
simp [Set.preimage, hf]
· by_cases ha0 : a = 0
· subst a
rwa [zero_mul]
· exact mul_ne_top ha (finMeasSupp_iff.1 hm _ ha0).ne
|
/-
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 | 148 | 151 | theorem associated_of_irreducible {a b : R} (ha : Irreducible a) (hb : Irreducible b) :
Associated a b := by |
rw [irreducible_iff_uniformizer] at ha hb
rw [← span_singleton_eq_span_singleton, ← ha, hb]
|
/-
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, Antoine Labelle
-/
import Mathlib.LinearAlgebra.Contraction
import Mathlib.LinearAlgebra.Matrix.Charpoly.Coeff
#align_import ... | Mathlib/LinearAlgebra/Trace.lean | 250 | 259 | theorem trace_comp_comm :
compr₂ (llcomp R M N M) (trace R M) = compr₂ (llcomp R N M N).flip (trace R N) := by |
apply
(compl₁₂_inj (show Surjective (dualTensorHom R N M) from (dualTensorHomEquiv R N M).surjective)
(show Surjective (dualTensorHom R M N) from (dualTensorHomEquiv R M N).surjective)).1
ext g m f n
simp only [AlgebraTensorModule.curry_apply, TensorProduct.curry_apply,
coe_restrictScalars, compl... |
/-
Copyright (c) 2019 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Group.Units.Equiv
import Mathlib.CategoryTheory.Endomorphism
#align_import category_theory.conj from "leanprover-community/mathlib"@"32253a1... | Mathlib/CategoryTheory/Conj.lean | 203 | 205 | theorem map_conjAut (F : C ⥤ D) {X Y : C} (α : X ≅ Y) (f : Aut X) :
F.mapIso (α.conjAut f) = (F.mapIso α).conjAut (F.mapIso f) := by |
ext; simp only [mapIso_hom, Iso.conjAut_hom, F.map_conj]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Order.Filter.SmallSets
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Compactness.Compact
import Mathlib.To... | Mathlib/Topology/UniformSpace/Basic.lean | 979 | 980 | theorem interior_mem_uniformity {s : Set (α × α)} (hs : s ∈ 𝓤 α) : interior s ∈ 𝓤 α := by |
rw [uniformity_eq_uniformity_interior]; exact mem_lift' hs
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura
-/
import Mathlib.Init.Algebra.Classes
import Mathlib.Init.Data.Ordering.Basic
#align_import init.data.ordering.lemmas from "leanprover-community/lean"@"4bd31... | Mathlib/Init/Data/Ordering/Lemmas.lean | 26 | 28 | theorem ite_eq_eq_distrib (c : Prop) [Decidable c] (a b : Ordering) :
((if c then a else b) = Ordering.eq) = if c then a = Ordering.eq else b = Ordering.eq := by |
by_cases c <;> simp [*]
|
/-
Copyright (c) 2020 Frédéric Dupuis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Frédéric Dupuis
-/
import Mathlib.Analysis.InnerProductSpace.Projection
import Mathlib.Analysis.NormedSpace.Dual
import Mathlib.Analysis.NormedSpace.Star.Basic
#align_import analysis... | Mathlib/Analysis/InnerProductSpace/Dual.lean | 94 | 99 | theorem ext_inner_right_basis {ι : Type*} {x y : E} (b : Basis ι 𝕜 E)
(h : ∀ i : ι, ⟪x, b i⟫ = ⟪y, b i⟫) : x = y := by |
refine ext_inner_left_basis b fun i => ?_
rw [← inner_conj_symm]
conv_rhs => rw [← inner_conj_symm]
exact congr_arg conj (h i)
|
/-
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.Sigma.Basic
import Mathlib.Algebra.Order.Ring.Nat
#align_import set_theory.lists from "leanprover-community/mathlib"@"497d1e06409995dd8ec95301fa8... | Mathlib/SetTheory/Lists.lean | 159 | 160 | theorem mem_cons {a y l} : a ∈ @cons α y l ↔ a ~ y ∨ a ∈ l := by |
simp [mem_def, or_and_right, exists_or]
|
/-
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.Group.Subgroup.Basic
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.PUnitInstances
import Mathlib.GroupTheory.Congr... | Mathlib/GroupTheory/Coprod/Basic.lean | 189 | 199 | theorem induction_on' {C : M ∗ N → Prop} (m : M ∗ N)
(one : C 1)
(inl_mul : ∀ m x, C x → C (inl m * x))
(inr_mul : ∀ n x, C x → C (inr n * x)) : C m := by |
rcases mk_surjective m with ⟨x, rfl⟩
induction x using FreeMonoid.recOn with
| h0 => exact one
| ih x xs ih =>
cases x with
| inl m => simpa using inl_mul m _ ih
| inr n => simpa using inr_mul n _ ih
|
/-
Copyright (c) 2023 Jz Pan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jz Pan
-/
import Mathlib.FieldTheory.SplittingField.Construction
import Mathlib.FieldTheory.IsAlgClosed.AlgebraicClosure
import Mathlib.FieldTheory.Separable
import Mathlib.FieldTheory.NormalC... | Mathlib/FieldTheory/SeparableDegree.lean | 479 | 488 | theorem natSepDegree_eq_one_iff_of_monic' (q : ℕ) [ExpChar F q] (hm : f.Monic)
(hi : Irreducible f) : f.natSepDegree = 1 ↔
∃ (n : ℕ) (y : F), f = expand F (q ^ n) (X - C y) := by |
refine ⟨fun h ↦ ?_, fun ⟨n, y, h⟩ ↦ ?_⟩
· obtain ⟨g, h1, n, rfl⟩ := hi.hasSeparableContraction q
have h2 : g.natDegree = 1 := by
rwa [natSepDegree_expand _ q, h1.natSepDegree_eq_natDegree] at h
rw [((monic_expand_iff <| expChar_pow_pos F q n).mp hm).eq_X_add_C h2]
exact ⟨n, -(g.coeff 0), by rw [m... |
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Yury Kudryashov
-/
import Mathlib.Data.ENNReal.Inv
#align_import data.real.ennreal from "leanprover-community/mathlib"@"c14c8fcde993801fca8946b0d80131a1a81d1520"
/-!
... | Mathlib/Data/ENNReal/Real.lean | 465 | 467 | theorem toReal_smul (r : ℝ≥0) (s : ℝ≥0∞) : (r • s).toReal = r • s.toReal := by |
rw [ENNReal.smul_def, smul_eq_mul, toReal_mul, coe_toReal]
rfl
|
/-
Copyright (c) 2015, 2017 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Robert Y. Lewis, Johannes Hölzl, Mario Carneiro, Sébastien Gouëzel
-/
import Mathlib.Topology.EMetricSpace.Basic
import Mathlib.Topology.Bornology.Constructions
imp... | Mathlib/Topology/MetricSpace/PseudoMetric.lean | 1,404 | 1,406 | theorem Real.Icc_eq_closedBall (x y : ℝ) : Icc x y = closedBall ((x + y) / 2) ((y - x) / 2) := by |
rw [Real.closedBall_eq_Icc, ← sub_div, add_comm, ← sub_add, add_sub_cancel_left, add_self_div_two,
← add_div, add_assoc, add_sub_cancel, add_self_div_two]
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jens Wagemaker, Aaron Anderson
-/
import Mathlib.Algebra.BigOperators.Associated
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Data.Finsupp.Multiset
import Math... | Mathlib/RingTheory/UniqueFactorizationDomain.lean | 1,636 | 1,640 | theorem mem_factors'_of_dvd {a p : α} (ha0 : a ≠ 0) (hp : Irreducible p) (hd : p ∣ a) :
Subtype.mk (Associates.mk p) (irreducible_mk.2 hp) ∈ factors' a := by |
obtain ⟨q, hq, hpq⟩ := exists_mem_factors_of_dvd ha0 hp hd
apply Multiset.mem_pmap.mpr; use q; use hq
exact Subtype.eq (Eq.symm (mk_eq_mk_iff_associated.mpr hpq))
|
/-
Copyright (c) 2023 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston, Joël Riou
-/
import Mathlib.Algebra.Homology.ShortComplex.ModuleCat
import Mathlib.RepresentationTheory.GroupCohomology.Basic
import Mathlib.RepresentationTheory... | Mathlib/RepresentationTheory/GroupCohomology/LowDegree.lean | 188 | 196 | theorem dTwo_comp_eq :
dTwo A ∘ₗ twoCochainsLequiv A = threeCochainsLequiv A ∘ₗ (inhomogeneousCochains A).d 2 3 := by |
ext x y
show A.ρ y.1 (x _) - x _ + x _ - x _ = _ + _
dsimp
rw [Fin.sum_univ_three]
simp only [sub_eq_add_neg, add_assoc, Fin.val_zero, zero_add, pow_one, neg_smul,
one_smul, Fin.val_one, Fin.val_two, pow_succ' (-1 : k) 2, neg_sq, Nat.one_add, one_pow, mul_one]
rcongr i <;> fin_cases i <;> rfl
|
/-
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.Calculus.ContDiff.Bounds
import Mathlib.Analysis.Calculus.IteratedDeriv.Defs
import Mathlib.Analysis.Calculus.LineDeriv.Basic
import Mathlib.Analysi... | Mathlib/Analysis/Distribution/SchwartzSpace.lean | 567 | 571 | theorem _root_.schwartz_withSeminorms : WithSeminorms (schwartzSeminormFamily 𝕜 E F) := by |
have A : WithSeminorms (schwartzSeminormFamily ℝ E F) := ⟨rfl⟩
rw [SeminormFamily.withSeminorms_iff_nhds_eq_iInf] at A ⊢
rw [A]
rfl
|
/-
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.LinearAlgebra.Matrix.Adjugate
import Mathlib.RingTheory.PolynomialAlgebra
#align_import linear_algebra.matrix.charpoly.basic from "leanprover-communit... | Mathlib/LinearAlgebra/Matrix/Charpoly/Basic.lean | 103 | 106 | theorem charpoly_reindex (e : n ≃ m)
(M : Matrix n n R) : (reindex e e M).charpoly = M.charpoly := by |
unfold Matrix.charpoly
rw [charmatrix_reindex, Matrix.det_reindex_self]
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Yaël Dillies
-/
import Mathlib.Order.Cover
import Mathlib.Order.Interval.Finset.Defs
#align_import data.finset.locally_finite from "leanprover-community/mathlib"@"442a... | Mathlib/Order/Interval/Finset/Basic.lean | 543 | 544 | theorem Icc_eq_singleton_iff : Icc a b = {c} ↔ a = c ∧ b = c := by |
rw [← coe_eq_singleton, coe_Icc, Set.Icc_eq_singleton_iff]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.Algebra.Algebra.Operations
import Mathlib.Data.Fintype.Lattice
import Mathlib.RingTheory.Coprime.Lemmas
#align_import ring_theory.ideal.operations from "leanpro... | Mathlib/RingTheory/Ideal/Operations.lean | 527 | 528 | theorem mem_span_singleton_mul {x y : R} {I : Ideal R} : x ∈ span {y} * I ↔ ∃ z ∈ I, y * z = x := by |
simp only [mul_comm, mem_mul_span_singleton]
|
/-
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 | 2,638 | 2,638 | theorem filter_eq_empty_iff : s.filter p = ∅ ↔ ∀ ⦃x⦄, x ∈ s → ¬p x := by | simp [Finset.ext_iff]
|
/-
Copyright (c) 2017 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro
-/
import Mathlib.Logic.Relation
import Mathlib.Data.Option.Basic
import Mathlib.Data.Seq.Seq
#align_import data.seq.wseq from "leanprover-community/mathlib"@"a7... | Mathlib/Data/Seq/WSeq.lean | 667 | 676 | theorem flatten_pure (s : WSeq α) : flatten (Computation.pure s) = s := by |
refine Seq.eq_of_bisim (fun s1 s2 => flatten (Computation.pure s2) = s1) ?_ rfl
intro s' s h
rw [← h]
simp only [Seq.BisimO, flatten, Seq.omap, pure_def, Seq.corec_eq, destruct_pure]
cases Seq.destruct s with
| none => simp
| some val =>
cases' val with o s'
simp
|
/-
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 | 345 | 348 | theorem partiallyWellOrderedOn_insert :
PartiallyWellOrderedOn (insert a s) r ↔ PartiallyWellOrderedOn s r := by |
simp only [← singleton_union, partiallyWellOrderedOn_union,
partiallyWellOrderedOn_singleton, true_and_iff]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.MeasureTheory.Measure.NullMeasurable
import Mathlib.MeasureTheory.MeasurableSpace.Basic
import Mathlib.Topology.Algebra.Order.LiminfLim... | Mathlib/MeasureTheory/Measure/MeasureSpace.lean | 329 | 330 | theorem union_ae_eq_right_iff_ae_subset : (s ∪ t : Set α) =ᵐ[μ] t ↔ s ≤ᵐ[μ] t := by |
rw [union_comm, union_ae_eq_left_iff_ae_subset]
|
/-
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 | 549 | 550 | theorem inner_re_zero_left (x : E) : re ⟪0, x⟫ = 0 := by |
simp only [inner_zero_left, AddMonoidHom.map_zero]
|
/-
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 | 279 | 294 | theorem Fix.rec_eq {α : Type _} (g : F α → α) (x : F (Fix F)) :
Fix.rec g (Fix.mk x) = g (Fix.rec g <$> x) := by |
have : recF g ∘ fixToW = Fix.rec g := by
apply funext
apply Quotient.ind
intro x
apply recF_eq_of_Wequiv
rw [fixToW]
apply Wrepr_equiv
conv =>
lhs
rw [Fix.rec, Fix.mk]
dsimp
cases' h : repr x with a f
rw [PFunctor.map_eq, recF_eq, ← PFunctor.map_eq, PFunctor.W.dest_mk, PFunc... |
/-
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 | 760 | 774 | theorem lintegral_indicator_le (f : α → ℝ≥0∞) (s : Set α) :
∫⁻ a, s.indicator f a ∂μ ≤ ∫⁻ a in s, f a ∂μ := by |
simp only [lintegral]
apply iSup_le (fun g ↦ (iSup_le (fun hg ↦ ?_)))
have : g ≤ f := hg.trans (indicator_le_self s f)
refine le_iSup_of_le g (le_iSup_of_le this (le_of_eq ?_))
rw [lintegral_restrict, SimpleFunc.lintegral]
congr with t
by_cases H : t = 0
· simp [H]
congr with x
simp only [mem_preim... |
/-
Copyright (c) 2022 Wrenna Robson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Wrenna Robson
-/
import Mathlib.Topology.MetricSpace.Basic
#align_import topology.metric_space.infsep from "leanprover-community/mathlib"@"5316314b553dcf8c6716541851517c1a9715e22b"
/-... | Mathlib/Topology/MetricSpace/Infsep.lean | 453 | 455 | theorem Nontrivial.infsep_eq_iInf (hs : s.Nontrivial) :
s.infsep = ⨅ d : s.offDiag, (uncurry dist) (d : α × α) := by |
classical rw [Set.infsep_eq_iInf, if_pos hs]
|
/-
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.FieldTheory.SeparableClosure
import Mathlib.Algebra.CharP.IntermediateField
/-!
# Purely inseparable extension and relative perfect closure
This file contains basic... | Mathlib/FieldTheory/PurelyInseparable.lean | 406 | 411 | theorem IsPurelyInseparable.tower_bot [Algebra E K] [IsScalarTower F E K]
[IsPurelyInseparable F K] : IsPurelyInseparable F E := by |
refine ⟨⟨fun x ↦ (isIntegral' F (algebraMap E K x)).tower_bot_of_field⟩, fun x h ↦ ?_⟩
rw [← minpoly.algebraMap_eq (algebraMap E K).injective] at h
obtain ⟨y, h⟩ := inseparable F _ h
exact ⟨y, (algebraMap E K).injective (h.symm ▸ (IsScalarTower.algebraMap_apply F E K y).symm)⟩
|
/-
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.Algebra.Ring.Int
import Mathlib.Data.Nat.Bitwise
import Mathlib.Data.Nat.Size
#align_import data.int.bitwise from "leanprover-community/mathlib"@"0743cc... | Mathlib/Data/Int/Bitwise.lean | 390 | 398 | theorem bitwise_bit (f : Bool → Bool → Bool) (a m b n) :
bitwise f (bit a m) (bit b n) = bit (f a b) (bitwise f m n) := by |
cases' m with m m <;> cases' n with n n <;>
simp [bitwise, ofNat_eq_coe, bit_coe_nat, natBitwise, Bool.not_false, Bool.not_eq_false',
bit_negSucc]
· by_cases h : f false false <;> simp (config := {decide := true}) [h]
· by_cases h : f false true <;> simp (config := {decide := true}) [h]
· by_cases h : f ... |
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Cone.Extension
import Mathlib.Analysis.NormedSpace.RCLike
import Mathlib.Analysis.NormedSpace.Extend
import Mathlib.... | Mathlib/Analysis/NormedSpace/HahnBanach/Extension.lean | 168 | 177 | theorem exists_dual_vector (x : E) (h : x ≠ 0) : ∃ g : E →L[𝕜] 𝕜, ‖g‖ = 1 ∧ g x = ‖x‖ := by |
let p : Submodule 𝕜 E := 𝕜 ∙ x
let f := (‖x‖ : 𝕜) • coord 𝕜 x h
obtain ⟨g, hg⟩ := exists_extension_norm_eq p f
refine ⟨g, ?_, ?_⟩
· rw [hg.2, coord_norm']
· calc
g x = g (⟨x, mem_span_singleton_self x⟩ : 𝕜 ∙ x) := by rw [coe_mk]
_ = ((‖x‖ : 𝕜) • coord 𝕜 x h) (⟨x, mem_span_singleton_self ... |
/-
Copyright (c) 2019 Zhouhang Zhou. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Zhouhang Zhou, Frédéric Dupuis, Heather Macbeth
-/
import Mathlib.Analysis.Convex.Basic
import Mathlib.Analysis.InnerProductSpace.Orthogonal
import Mathlib.Analysis.InnerProductSpace.Sy... | Mathlib/Analysis/InnerProductSpace/Projection.lean | 1,053 | 1,055 | theorem orthogonalProjection_add_orthogonalProjection_orthogonal [HasOrthogonalProjection K]
(w : E) : (orthogonalProjection K w : E) + (orthogonalProjection Kᗮ w : E) = w := by |
simp
|
/-
Copyright (c) 2015 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
import Mathlib.Data.Bool.Basic
import Mathlib.Data.Option.Defs
import Mathlib.Data.Prod.Basic
import Mathlib.Data.Sigma.Basic
import Mathlib... | Mathlib/Logic/Equiv/Basic.lean | 1,553 | 1,555 | theorem Perm.extendDomain_trans (e e' : Perm α') :
(e.extendDomain f).trans (e'.extendDomain f) = Perm.extendDomain (e.trans e') f := by |
simp [Perm.extendDomain, permCongr_trans]
|
/-
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.MeasureTheory.Group.GeometryOfNumbers
import Mathlib.MeasureTheory.Measure.Lebesgue.VolumeOfBalls
import Mathlib.NumberTheory.NumberField.CanonicalEmbedd... | Mathlib/NumberTheory/NumberField/CanonicalEmbedding/ConvexBody.lean | 349 | 360 | theorem convexBodySum_volume_eq_zero_of_le_zero {B} (hB : B ≤ 0) :
volume (convexBodySum K B) = 0 := by |
obtain hB | hB := lt_or_eq_of_le hB
· suffices convexBodySum K B = ∅ by rw [this, measure_empty]
ext x
refine ⟨fun hx => ?_, fun h => h.elim⟩
rw [Set.mem_setOf] at hx
linarith [convexBodySumFun_nonneg x]
· suffices convexBodySum K B = { 0 } by rw [this, measure_singleton]
ext
rw [convexBo... |
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Yury Kudryashov
-/
import Mathlib.Algebra.Algebra.Tower
#align_import algebra.algebra.restrict_scalars from "leanprover-community/mathlib"@"c310cfdc40da4d99a10a58c33a95360ef9e6e... | Mathlib/Algebra/Algebra/RestrictScalars.lean | 175 | 179 | theorem RestrictScalars.addEquiv_symm_map_smul_smul (r : R) (s : S) (x : M) :
(RestrictScalars.addEquiv R S M).symm ((r • s) • x) =
r • (RestrictScalars.addEquiv R S M).symm (s • x) := by |
rw [Algebra.smul_def, mul_smul]
rfl
|
/-
Copyright (c) 2018 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mitchell Rowett, Scott Morrison, Johan Commelin, Mario Carneiro,
Michael Howes
-/
import Mathlib.Algebra.Group.Subgroup.Basic
import Mathlib.Deprecated.Submonoid
#al... | Mathlib/Deprecated/Subgroup.lean | 302 | 308 | theorem normalizer_isSubgroup (s : Set G) : IsSubgroup (normalizer s) :=
{ one_mem := by | simp [normalizer]
mul_mem := fun {a b}
(ha : ∀ n, n ∈ s ↔ a * n * a⁻¹ ∈ s) (hb : ∀ n, n ∈ s ↔ b * n * b⁻¹ ∈ s) n =>
by rw [mul_inv_rev, ← mul_assoc, mul_assoc a, mul_assoc a, ← ha, ← hb]
inv_mem := fun {a} (ha : ∀ n, n ∈ s ↔ a * n * a⁻¹ ∈ s) n => by
rw [ha (a⁻¹ * n * a⁻¹⁻¹)]; simp [mul_assoc]... |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Eric Rodriguez
-/
import Mathlib.Algebra.Order.Field.Basic
import Mathlib.Data.Nat.Cast.Order
import Mathlib.Data.Nat.Choose.Basic
import Mathlib.Data.Nat.Cast.Order
#alig... | Mathlib/Data/Nat/Choose/Bounds.lean | 41 | 46 | theorem pow_le_choose (r n : ℕ) : ((n + 1 - r : ℕ) ^ r : α) / r ! ≤ n.choose r := by |
rw [div_le_iff']
· norm_cast
rw [← Nat.descFactorial_eq_factorial_mul_choose]
exact n.pow_sub_le_descFactorial r
exact mod_cast r.factorial_pos
|
/-
Copyright (c) 2021 Jakob von Raumer. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jakob von Raumer
-/
import Mathlib.CategoryTheory.Monoidal.Free.Coherence
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.Closed.Monoidal
import Mathlib.... | Mathlib/CategoryTheory/Monoidal/Rigid/Basic.lean | 450 | 458 | theorem tensorLeftHomEquiv_whiskerLeft_comp_evaluation {Y Z : C} [HasLeftDual Z] (f : Y ⟶ ᘁZ) :
(tensorLeftHomEquiv _ _ _ _) (Z ◁ f ≫ ε_ _ _) = f ≫ (ρ_ _).inv :=
calc
_ = 𝟙 _ ⊗≫ (η_ (ᘁZ) Z ▷ Y ≫ ((ᘁZ) ⊗ Z) ◁ f) ⊗≫ (ᘁZ) ◁ ε_ (ᘁZ) Z := by |
dsimp [tensorLeftHomEquiv]; coherence
_ = f ⊗≫ (η_ (ᘁZ) Z ▷ (ᘁZ) ⊗≫ (ᘁZ) ◁ ε_ (ᘁZ) Z) := by
rw [← whisker_exchange]; coherence
_ = _ := by
rw [evaluation_coevaluation'']; coherence
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.Group.Defs
#align_import algebra.invertible from "leanprover-community/mathlib"@"722b3b152ddd5e0cf21c0a29787c76596cb6b422"
/-!
# Invertible element... | Mathlib/Algebra/Group/Invertible/Defs.lean | 136 | 137 | theorem mul_invOf_mul_self_cancel [Monoid α] (a b : α) [Invertible b] : a * ⅟ b * b = a := by |
simp [mul_assoc]
|
/-
Copyright (c) 2020 Thomas Browning. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Thomas Browning
-/
import Mathlib.Algebra.GCDMonoid.Multiset
import Mathlib.Combinatorics.Enumerative.Partition
import Mathlib.Data.List.Rotate
import Mathlib.GroupTheory.Perm.Cycle.F... | Mathlib/GroupTheory/Perm/Cycle/Type.lean | 403 | 406 | theorem one_eq : vectorsProdEqOne G 1 = {Vector.nil.cons 1} := by |
simp_rw [Set.eq_singleton_iff_unique_mem, mem_iff, Vector.toList_singleton, List.prod_singleton,
Vector.head_cons, true_and]
exact fun v hv => v.cons_head_tail.symm.trans (congr_arg₂ Vector.cons hv v.tail.eq_nil)
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Sébastien Gouëzel, Yury Kudryashov
-/
import Mathlib.Dynamics.Ergodic.MeasurePreserving
import Mathlib.LinearAlgebra.Determinant
import Mathlib.LinearAlgebra.Matrix.Dia... | Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | 167 | 167 | theorem volume_Iic {a : ℝ} : volume (Iic a) = ∞ := by | rw [← measure_congr Iio_ae_eq_Iic]; simp
|
/-
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.SpecificLimits.Basic
import Mathlib.Topology.MetricSpace.IsometricSMul
#align_import topology.metric_space.hausdorff_distance from "lea... | Mathlib/Topology/MetricSpace/HausdorffDistance.lean | 889 | 890 | theorem hausdorffDist_closure : hausdorffDist (closure s) (closure t) = hausdorffDist s t := by |
simp [hausdorffDist]
|
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Reid Barton, Johan Commelin, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Equivalence
#align_import category_theory.adjunction.basic from "leanprover-community/mathlib"@"d101e93197bb5f6... | Mathlib/CategoryTheory/Adjunction/Basic.lean | 192 | 195 | theorem homEquiv_naturality_right_square (f : X' ⟶ X) (g : X ⟶ G.obj Y')
(h : X' ⟶ G.obj Y) (k : Y ⟶ Y') (w : f ≫ g = h ≫ G.map k) :
F.map f ≫ (adj.homEquiv X Y').symm g = (adj.homEquiv X' Y).symm h ≫ k := by |
rw [← homEquiv_naturality_left_symm, ← homEquiv_naturality_right_symm, w]
|
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth, Eric Wieser
-/
import Mathlib.Analysis.NormedSpace.PiLp
import Mathlib.Analysis.InnerProductSpace.PiL2
#align_import analysis.matrix from "leanprover-community/mathl... | Mathlib/Analysis/Matrix.lean | 94 | 95 | theorem nnnorm_le_iff {r : ℝ≥0} {A : Matrix m n α} : ‖A‖₊ ≤ r ↔ ∀ i j, ‖A i j‖₊ ≤ r := by |
simp_rw [nnnorm_def, pi_nnnorm_le_iff]
|
/-
Copyright (c) 2023 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov, Floris van Doorn
-/
import Mathlib.Order.Filter.AtTopBot
import Mathlib.Order.Filter.Subsingleton
/-!
# Functions that are eventually constant along a filter
In this... | Mathlib/Order/Filter/EventuallyConst.lean | 61 | 63 | theorem eventuallyConst_pred {p : α → Prop} :
EventuallyConst p l ↔ (∀ᶠ x in l, p x) ∨ (∀ᶠ x in l, ¬p x) := by |
simp [eventuallyConst_pred', or_comm, EventuallyEq]
|
/-
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.RingTheory.IntegrallyClosed
import Mathlib.RingTheory.Trace
import Mathlib.RingTheory.Norm
#align_import ring_theory.discriminant from "leanprover-c... | Mathlib/RingTheory/Discriminant.lean | 277 | 311 | theorem discr_mul_isIntegral_mem_adjoin [IsSeparable K L] [IsIntegrallyClosed R]
[IsFractionRing R K] {B : PowerBasis K L} (hint : IsIntegral R B.gen) {z : L}
(hz : IsIntegral R z) : discr K B.basis • z ∈ adjoin R ({B.gen} : Set L) := by |
have hinv : IsUnit (traceMatrix K B.basis).det := by
simpa [← discr_def] using discr_isUnit_of_basis _ B.basis
have H :
(traceMatrix K B.basis).det • (traceMatrix K B.basis) *ᵥ (B.basis.equivFun z) =
(traceMatrix K B.basis).det • fun i => trace K L (z * B.basis i) := by
congr; exact traceMatrix_o... |
/-
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 | 153 | 156 | theorem cons_copy {u v w v' w'} (h : G.Adj u v) (p : G.Walk v' w') (hv : v' = v) (hw : w' = w) :
Walk.cons h (p.copy hv hw) = (Walk.cons (hv ▸ h) p).copy rfl hw := by |
subst_vars
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, 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 | 1,522 | 1,523 | theorem insert_union (a : α) (s t : Finset α) : insert a s ∪ t = insert a (s ∪ t) := by |
simp only [insert_eq, union_assoc]
|
/-
Copyright (c) 2019 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin, Fabian Glöckle, Kyle Miller
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.FreeModule.Finite.Basic
import Mathlib.LinearAlgebra.FreeModu... | Mathlib/LinearAlgebra/Dual.lean | 1,540 | 1,543 | theorem dualPairing_eq (W : Subspace K V₁) :
W.dualPairing = W.quotAnnihilatorEquiv.toLinearMap := by |
ext
rfl
|
/-
Copyright (c) 2023 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Probability.Kernel.Basic
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.DominatedConvergence
#align_import pr... | Mathlib/Probability/Kernel/MeasurableIntegral.lean | 332 | 340 | theorem StronglyMeasurable.integral_kernel_prod_left'' {f : γ × β → E} (hf : StronglyMeasurable f) :
StronglyMeasurable fun y => ∫ x, f (x, y) ∂η (a, y) := by |
change
StronglyMeasurable
((fun y => ∫ x, (fun u : γ × α × β => f (u.1, u.2.2)) (x, y) ∂η y) ∘ fun x => (a, x))
apply StronglyMeasurable.comp_measurable _ (measurable_prod_mk_left (m := mα))
· have := MeasureTheory.StronglyMeasurable.integral_kernel_prod_left' (κ := η)
(hf.comp_measurable (measur... |
/-
Copyright (c) 2021 Alex Kontorovich and Heather Macbeth and Marc Masdeu. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Alex Kontorovich, Heather Macbeth, Marc Masdeu
-/
import Mathlib.Analysis.Complex.UpperHalfPlane.Basic
import Mathlib.LinearAlgebra.GeneralLinearG... | Mathlib/NumberTheory/Modular.lean | 376 | 377 | theorem normSq_S_smul_lt_one (h : 1 < normSq z) : normSq ↑(S • z) < 1 := by |
simpa [coe_S, num, denom] using (inv_lt_inv z.normSq_pos zero_lt_one).mpr h
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad
-/
import Mathlib.Algebra.Order.Ring.Defs
import Mathlib.Data.Set.Finite
#align_import order.filter.basic from "leanprover-community/mathlib"@"d4f691b9e5... | Mathlib/Order/Filter/Basic.lean | 2,336 | 2,339 | theorem _root_.Set.LeftInvOn.filter_map_Iic {f : α → β} {g : β → α} (hfg : LeftInvOn g f s) :
LeftInvOn (map g) (map f) (Iic <| 𝓟 s) := fun F (hF : F ≤ 𝓟 s) ↦ by
have : (g ∘ f) =ᶠ[𝓟 s] id := by | simpa only [eventuallyEq_principal] using hfg
rw [map_map, map_congr (this.filter_mono hF), map_id]
|
/-
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.LinearAlgebra.Matrix.Charpoly.Coeff
import Mathlib.LinearAlgebra.Matrix.ToLin
#align_import linear_algebra.matrix.charpoly.linear_map from "leanprover-commu... | Mathlib/LinearAlgebra/Matrix/Charpoly/LinearMap.lean | 114 | 121 | theorem Matrix.Represents.mul {A A' : Matrix ι ι R} {f f' : Module.End R M} (h : A.Represents b f)
(h' : Matrix.Represents b A' f') : (A * A').Represents b (f * f') := by |
delta Matrix.Represents PiToModule.fromMatrix
rw [LinearMap.comp_apply, AlgEquiv.toLinearMap_apply, _root_.map_mul]
ext
dsimp [PiToModule.fromEnd]
rw [← h'.congr_fun, ← h.congr_fun]
rfl
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Yury Kudryashov
-/
import Mathlib.Data.Set.Image
import Mathlib.Order.SuccPred.Relation
import Mathlib.Topology.Clopen
import Mathlib.Topology.Irreducib... | Mathlib/Topology/Connected/Basic.lean | 641 | 652 | theorem IsPreconnected.subset_connectedComponentIn {x : α} {F : Set α} (hs : IsPreconnected s)
(hxs : x ∈ s) (hsF : s ⊆ F) : s ⊆ connectedComponentIn F x := by |
have : IsPreconnected (((↑) : F → α) ⁻¹' s) := by
refine inducing_subtype_val.isPreconnected_image.mp ?_
rwa [Subtype.image_preimage_coe, inter_eq_right.mpr hsF]
have h2xs : (⟨x, hsF hxs⟩ : F) ∈ (↑) ⁻¹' s := by
rw [mem_preimage]
exact hxs
have := this.subset_connectedComponent h2xs
rw [connecte... |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies, Violeta Hernández Palacios, Grayson Burton, Floris van Doorn
-/
import Mathlib.Order.Interval.Set.OrdConnected
import Mathlib.Order.Antisymmetrization
#align_import order.... | Mathlib/Order/Cover.lean | 421 | 422 | theorem CovBy.Ioc_eq (h : a ⋖ b) : Ioc a b = {b} := by |
rw [← Ioo_union_right h.lt, h.Ioo_eq, empty_union]
|
/-
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.NumberTheory.ClassNumber.AdmissibleAbs
import Mathlib.NumberTheory.ClassNumber.Finite
import Mathlib.NumberTheory.NumberField.Discriminant
#align_import num... | Mathlib/NumberTheory/NumberField/ClassNumber.lean | 77 | 92 | theorem _root_.RingOfIntegers.isPrincipalIdealRing_of_abs_discr_lt
(h : |discr K| < (2 * (π / 4) ^ NrComplexPlaces K *
((finrank ℚ K) ^ (finrank ℚ K) / (finrank ℚ K).factorial)) ^ 2) :
IsPrincipalIdealRing (𝓞 K) := by |
have : 0 < finrank ℚ K := finrank_pos -- Lean needs to know that for positivity to succeed
rw [← Real.sqrt_lt (by positivity) (by positivity), mul_assoc, ← inv_mul_lt_iff' (by positivity),
mul_inv, ← inv_pow, inv_div, inv_div, mul_assoc, Int.cast_abs] at h
rw [← classNumber_eq_one_iff, classNumber, Fintype.c... |
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Johan Commelin
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.TensorProduct.Tower
import Mathlib.RingTheory.Adjoin.Basic
import Mathlib.... | Mathlib/RingTheory/TensorProduct/Basic.lean | 1,014 | 1,015 | theorem productMap_eq_comp_map : productMap f g = (lmul' R).comp (TensorProduct.map f g) := by |
ext <;> 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.Algebra.Order.Group.Basic
import Mathlib.Algebra.Order.Ring.Basic
import Mathlib.Algebra.Star.Unitary
import Mathlib.Data.Nat.ModEq
import Mathlib.Numb... | Mathlib/NumberTheory/PellMatiyasevic.lean | 622 | 627 | theorem xn_modEq_x2n_sub {n j} (h : j ≤ 2 * n) : xn a1 (2 * n - j) + xn a1 j ≡ 0 [MOD xn a1 n] :=
(le_total j n).elim (xn_modEq_x2n_sub_lem a1) fun jn => by
have : 2 * n - j + j ≤ n + j := by |
rw [tsub_add_cancel_of_le h, two_mul]; exact Nat.add_le_add_left jn _
let t := xn_modEq_x2n_sub_lem a1 (Nat.le_of_add_le_add_right this)
rwa [tsub_tsub_cancel_of_le h, add_comm] at t
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen
-/
import Mathlib.Algebra.BigOperators.Fin
import Mathlib.Algebra.GeomSum
import Mathlib.LinearAlgebra.Matrix.Block
import Mathlib.LinearAlgebra.Matrix.Determinant.Basic
impor... | Mathlib/LinearAlgebra/Vandermonde.lean | 165 | 168 | theorem det_vandermonde_sub {n : ℕ} (v : Fin n → R) (a : R) :
(Matrix.vandermonde fun i ↦ v i - a).det = (Matrix.vandermonde v).det := by |
rw [← det_vandermonde_add v (- a)]
simp only [← sub_eq_add_neg]
|
/-
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.Data.Fin.Tuple.Basic
import Mathlib.Data.List.Range
#align_import data.fin.vec_notation from "leanprover-community/mathlib"@"2445c98ae4b87eabebdde552593519b... | Mathlib/Data/Fin/VecNotation.lean | 509 | 510 | theorem cons_sub_cons (x : α) (v : Fin n → α) (y : α) (w : Fin n → α) :
vecCons x v - vecCons y w = vecCons (x - y) (v - w) := by | simp
|
/-
Copyright (c) 2020 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.Algebra.Equiv
import Mathlib.LinearAlgebra.Dimension.StrongRankCondition
import Mathlib.LinearAlgebra.FreeModule.Basic
import Mathlib.LinearA... | Mathlib/Algebra/Quaternion.lean | 767 | 767 | theorem star_mul_eq_coe : star a * a = (star a * a).re := by | ext <;> simp <;> ring
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot
-/
import Mathlib.Order.Filter.SmallSets
import Mathlib.Tactic.Monotonicity
import Mathlib.Topology.Compactness.Compact
import Mathlib.To... | Mathlib/Topology/UniformSpace/Basic.lean | 575 | 580 | theorem uniformity_lift_le_swap {g : Set (α × α) → Filter β} {f : Filter β} (hg : Monotone g)
(h : ((𝓤 α).lift fun s => g (preimage Prod.swap s)) ≤ f) : (𝓤 α).lift g ≤ f :=
calc
(𝓤 α).lift g ≤ (Filter.map (@Prod.swap α α) <| 𝓤 α).lift g :=
lift_mono uniformity_le_symm le_rfl
_ ≤ _ := by | rw [map_lift_eq2 hg, image_swap_eq_preimage_swap]; exact h
|
/-
Copyright (c) 2023 David Loeffler. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler
-/
import Mathlib.Analysis.Complex.Circle
import Mathlib.MeasureTheory.Group.Integral
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.Meas... | Mathlib/Analysis/Fourier/FourierTransform.lean | 348 | 354 | theorem vector_fourierIntegral_eq_integral_exp_smul {V : Type*} [AddCommGroup V] [Module ℝ V]
[MeasurableSpace V] {W : Type*} [AddCommGroup W] [Module ℝ W] (L : V →ₗ[ℝ] W →ₗ[ℝ] ℝ)
(μ : Measure V) (f : V → E) (w : W) :
VectorFourier.fourierIntegral fourierChar μ L f w =
∫ v : V, Complex.exp (↑(-2 * π *... |
simp_rw [VectorFourier.fourierIntegral, Submonoid.smul_def, Real.fourierChar_apply, mul_neg,
neg_mul]
|
/-
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.Order.BoundedOrder
import Mathlib.Order.MinMax
import Mathlib.Algebra.NeZero
import Mathlib.Algebra.Or... | Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean | 347 | 352 | theorem min_mul_distrib (a b c : α) : min a (b * c) = min a (min a b * min a c) := by |
rcases le_total a b with hb | hb
· simp [hb, le_mul_right]
· rcases le_total a c with hc | hc
· simp [hc, le_mul_left]
· simp [hb, hc]
|
/-
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, Sébastien Gouëzel
-/
import Mathlib.Order.Interval.Set.Disjoint
import Mathlib.MeasureTheory.Integral.SetIntegral
import Mathlib.MeasureTheory.M... | Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | 782 | 784 | theorem integral_comp_mul_add (hc : c ≠ 0) (d) :
(∫ x in a..b, f (c * x + d)) = c⁻¹ • ∫ x in c * a + d..c * b + d, f x := by |
rw [← integral_comp_add_right, ← integral_comp_mul_left _ hc]
|
/-
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.Complex.Basic
import Mathlib.Analysis.NormedSpace.OperatorNorm.NormedSpace
import Mathlib.Data.Complex.Determinant
#align_import analys... | Mathlib/Analysis/Complex/OperatorNorm.lean | 50 | 54 | theorem imCLM_norm : ‖imCLM‖ = 1 :=
le_antisymm (LinearMap.mkContinuous_norm_le _ zero_le_one _) <|
calc
1 = ‖imCLM I‖ := by | simp
_ ≤ ‖imCLM‖ := unit_le_opNorm _ _ (by simp)
|
/-
Copyright (c) 2023 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Polynomial.Degree.Lemmas
/-!
# `compute_degree` and `monicity`: tactics for explicit polynomials
This file defines two related tactics: `comp... | Mathlib/Tactic/ComputeDegree.lean | 157 | 165 | theorem degree_eq_of_le_of_coeff_ne_zero' {deg m o : WithBot ℕ} {c : R} {p : R[X]}
(h_deg_le : degree p ≤ m) (coeff_eq : coeff p (WithBot.unbot' 0 deg) = c)
(coeff_ne_zero : c ≠ 0) (deg_eq_deg : m = deg) (coeff_eq_deg : o = deg) :
degree p = deg := by |
subst coeff_eq coeff_eq_deg deg_eq_deg
rcases eq_or_ne m ⊥ with rfl|hh
· exact bot_unique h_deg_le
· obtain ⟨m, rfl⟩ := WithBot.ne_bot_iff_exists.mp hh
exact degree_eq_of_le_of_coeff_ne_zero ‹_› ‹_›
|
/-
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 | 614 | 626 | theorem perm_insertNth {α} (x : α) (l : List α) {n} (h : n ≤ l.length) :
insertNth n x l ~ x :: l := by |
induction l generalizing n with
| nil =>
cases n with
| zero => rfl
| succ => cases h
| cons _ _ ih =>
cases n with
| zero => simp [insertNth]
| succ =>
simp only [insertNth, modifyNthTail]
refine .trans (.cons _ (ih (Nat.le_of_succ_le_succ h))) (.swap ..)
|
/-
Copyright (c) 2021 Benjamin Davidson. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Benjamin Davidson
-/
import Mathlib.Algebra.Field.Opposite
import Mathlib.Algebra.Group.Subgroup.ZPowers
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Rin... | Mathlib/Algebra/Periodic.lean | 128 | 130 | theorem Periodic.const_inv_smul₀ [AddCommMonoid α] [DivisionSemiring γ] [Module γ α]
(h : Periodic f c) (a : γ) : Periodic (fun x => f (a⁻¹ • x)) (a • c) := by |
simpa only [inv_inv] using h.const_smul₀ a⁻¹
|
/-
Copyright (c) 2020 Anne Baanen. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.RingTheory.IntegralClosure
import Mathlib.RingTheory.FractionalIdeal.Basic
#align_import ring_theory.fractional_ideal from "leanprover... | Mathlib/RingTheory/FractionalIdeal/Operations.lean | 455 | 460 | theorem mul_one_div_le_one {I : FractionalIdeal R₁⁰ K} : I * (1 / I) ≤ 1 := by |
by_cases hI : I = 0
· rw [hI, div_zero, mul_zero]
exact zero_le 1
· rw [← coe_le_coe, coe_mul, coe_div hI, coe_one]
apply Submodule.mul_one_div_le_one
|
/-
Copyright (c) 2020 Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Floris van Doorn
-/
import Mathlib.MeasureTheory.Constructions.Prod.Basic
import Mathlib.MeasureTheory.Integral.DominatedConvergence
import Mathlib.MeasureTheory.Integral.SetIntegral... | Mathlib/MeasureTheory/Constructions/Prod/Integral.lean | 420 | 446 | theorem continuous_integral_integral :
Continuous fun f : α × β →₁[μ.prod ν] E => ∫ x, ∫ y, f (x, y) ∂ν ∂μ := by |
rw [continuous_iff_continuousAt]; intro g
refine
tendsto_integral_of_L1 _ (L1.integrable_coeFn g).integral_prod_left
(eventually_of_forall fun h => (L1.integrable_coeFn h).integral_prod_left) ?_
simp_rw [←
lintegral_fn_integral_sub (fun x => (‖x‖₊ : ℝ≥0∞)) (L1.integrable_coeFn _)
(L1.integrab... |
/-
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.PropInstances
#align_import order.heyting.basic from "leanprover-community/mathlib"@"9ac7c0c8c4d7a535ec3e5b34b8859aab9233b2f4"
/-!
# Heyting algebr... | Mathlib/Order/Heyting/Basic.lean | 513 | 514 | theorem sup_sdiff_cancel' (hab : a ≤ b) (hbc : b ≤ c) : b ⊔ c \ a = c := by |
rw [sup_sdiff_eq_sup hab, sup_of_le_right hbc]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro
-/
import Mathlib.Data.Finset.Update
import Mathlib.Data.Prod.TProd
import Mathlib.GroupTheory.Coset
import Mathlib.Logic.Equiv.Fin
import Mathlib.Measur... | Mathlib/MeasureTheory/MeasurableSpace/Basic.lean | 863 | 874 | theorem measurable_from_prod_countable' [Countable β]
{_ : MeasurableSpace γ} {f : α × β → γ} (hf : ∀ y, Measurable fun x => f (x, y))
(h'f : ∀ y y' x, y' ∈ measurableAtom y → f (x, y') = f (x, y)) :
Measurable f := fun s hs => by
have : f ⁻¹' s = ⋃ y, ((fun x => f (x, y)) ⁻¹' s) ×ˢ (measurableAtom y : Se... |
ext1 ⟨x, y⟩
simp only [mem_preimage, mem_iUnion, mem_prod]
refine ⟨fun h ↦ ⟨y, h, mem_measurableAtom_self y⟩, ?_⟩
rintro ⟨y', hy's, hy'⟩
rwa [h'f y' y x hy']
rw [this]
exact .iUnion (fun y ↦ (hf y hs).prod (.measurableAtom_of_countable y))
|
/-
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.Monoid.Defs
import Mathlib.Algebra.Order.Sub.Defs
import Mathlib.Util.AssertExists
#ali... | Mathlib/Algebra/Order/Group/Defs.lean | 887 | 888 | theorem one_lt_div' : 1 < a / b ↔ b < a := by |
rw [← mul_lt_mul_iff_right b, one_mul, div_eq_mul_inv, inv_mul_cancel_right]
|
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.Algebra.Group.Equiv.Basic
import Mathlib.Algebra.Group.Submonoid.Operations
import Mathlib.Data.Setoid.Basic
#align_import group_theory.congruen... | Mathlib/GroupTheory/Congruence/Basic.lean | 925 | 925 | theorem lift_comp_mk' (H : c ≤ ker f) : (c.lift f H).comp c.mk' = f := by | ext; rfl
|
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