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 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.MvPolynomial.Variables
#align_import data.mv_polynomial.supported from "leanprover-community/mathlib"@"2f5b500a507264de86d666a5f87ddb976e2d8de4"
... | Mathlib/Algebra/MvPolynomial/Supported.lean | 102 | 103 | theorem supported_univ : supported R (Set.univ : Set σ) = ⊤ := by |
simp [Algebra.eq_top_iff, mem_supported]
|
/-
Copyright (c) 2021 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Oliver Nash
-/
import Mathlib.Algebra.Ring.Divisibility.Lemmas
import Mathlib.Algebra.Lie.Nilpotent
import Mathlib.Algebra.Lie.Engel
import Mathlib.LinearAlgebra.Eigenspace.Triangularizab... | Mathlib/Algebra/Lie/Weights/Basic.lean | 306 | 313 | theorem exists_weightSpace_le_ker_of_isNoetherian [IsNoetherian R M] (χ : L → R) (x : L) :
∃ k : ℕ,
weightSpace M χ ≤ LinearMap.ker ((toEnd R L M x - algebraMap R _ (χ x)) ^ k) := by |
use (toEnd R L M x).maxGenEigenspaceIndex (χ x)
intro m hm
replace hm : m ∈ (toEnd R L M x).maxGenEigenspace (χ x) :=
weightSpace_le_weightSpaceOf M x χ hm
rwa [Module.End.maxGenEigenspace_eq] at hm
|
/-
Copyright (c) 2023 Richard M. Hill. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Richard M. Hill
-/
import Mathlib.RingTheory.PowerSeries.Trunc
import Mathlib.RingTheory.PowerSeries.Inverse
import Mathlib.RingTheory.Derivation.Basic
/-!
# Definitions
In this fil... | Mathlib/RingTheory/PowerSeries/Derivative.lean | 127 | 133 | theorem trunc_derivative' (f : R⟦X⟧) (n : ℕ) :
trunc (n-1) (d⁄dX R f) = Polynomial.derivative (trunc n f) := by |
cases n with
| zero =>
simp
| succ n =>
rw [succ_sub_one, trunc_derivative]
|
/-
Copyright (c) 2021 Johan Commelin. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johan Commelin
-/
import Mathlib.Algebra.Group.Equiv.TypeTags
import Mathlib.GroupTheory.FreeAbelianGroup
import Mathlib.GroupTheory.FreeGroup.IsFreeGroup
import Mathlib.LinearAlgebra.... | Mathlib/GroupTheory/FreeAbelianGroupFinsupp.lean | 165 | 166 | theorem support_of (x : X) : support (of x) = {x} := by |
rw [support, toFinsupp_of, Finsupp.support_single_ne_zero _ one_ne_zero]
|
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Bhavik Mehta
-/
import Mathlib.Analysis.Calculus.Deriv.Support
import Mathlib.Analysis.SpecialFunctions.Pow.Deriv
import Mathlib.MeasureTheory.Integral.FundThmCalcu... | Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean | 1,176 | 1,198 | theorem integrableOn_Ioi_comp_rpow_iff [NormedSpace ℝ E] (f : ℝ → E) {p : ℝ} (hp : p ≠ 0) :
IntegrableOn (fun x => (|p| * x ^ (p - 1)) • f (x ^ p)) (Ioi 0) ↔ IntegrableOn f (Ioi 0) := by |
let S := Ioi (0 : ℝ)
have a1 : ∀ x : ℝ, x ∈ S → HasDerivWithinAt (fun t : ℝ => t ^ p) (p * x ^ (p - 1)) S x :=
fun x hx => (hasDerivAt_rpow_const (Or.inl (mem_Ioi.mp hx).ne')).hasDerivWithinAt
have a2 : InjOn (fun x : ℝ => x ^ p) S := by
rcases lt_or_gt_of_ne hp with (h | h)
· apply StrictAntiOn.injO... |
/-
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.Init.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive... | Mathlib/Data/Option/Basic.lean | 240 | 250 | theorem pbind_eq_some {f : ∀ a : α, a ∈ x → Option β} {y : β} :
x.pbind f = some y ↔ ∃ (z : α) (H : z ∈ x), f z H = some y := by |
rcases x with (_|x)
· simp only [pbind, false_iff, not_exists]
intro z h
simp at h
· simp only [pbind]
refine ⟨fun h ↦ ⟨x, rfl, h⟩, ?_⟩
rintro ⟨z, H, hz⟩
simp only [mem_def, Option.some_inj] at H
simpa [H] using hz
|
/-
Copyright (c) 2018 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.RingTheory.Valuation.Basic
import Mathlib.NumberTheory.Padics.PadicNorm
import Mathlib.Analysis.Normed.Field.Basic
#align_import number_theory.padic... | Mathlib/NumberTheory/Padics/PadicNumbers.lean | 600 | 617 | theorem defn (f : PadicSeq p) {ε : ℚ} (hε : 0 < ε) :
∃ N, ∀ i ≥ N, padicNormE (Padic.mk f - f i : ℚ_[p]) < ε := by |
dsimp [padicNormE]
change ∃ N, ∀ i ≥ N, (f - const _ (f i)).norm < ε
by_contra! h
cases' cauchy₂ f hε with N hN
rcases h N with ⟨i, hi, hge⟩
have hne : ¬f - const (padicNorm p) (f i) ≈ 0 := fun h ↦ by
rw [PadicSeq.norm, dif_pos h] at hge
exact not_lt_of_ge hge hε
unfold PadicSeq.norm at hge; spli... |
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.CategoryTheory.Monoidal.Braided.Basic
import Mathlib.CategoryTheory.Monoidal.Discrete
import Mathlib.CategoryTheory.Monoidal.CoherenceLemmas
import Mat... | Mathlib/CategoryTheory/Monoidal/Mon_.lean | 421 | 432 | theorem Mon_tensor_mul_assoc (M N : Mon_ C) :
((tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ▷ (M.X ⊗ N.X)) ≫
tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =
(α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫
((M.X ⊗ N.X) ◁ (tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul))) ≫
... |
simp only [comp_whiskerRight_assoc, MonoidalCategory.whiskerLeft_comp_assoc]
slice_lhs 2 3 => rw [tensor_μ_natural_left]
slice_lhs 3 4 => rw [← tensor_comp, mul_assoc M, mul_assoc N, tensor_comp, tensor_comp]
slice_lhs 1 3 => rw [tensor_associativity]
slice_lhs 3 4 => rw [← tensor_μ_natural_right]
simp
|
/-
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 | 427 | 429 | theorem setToSimpleFunc_smul_left {m : MeasurableSpace α} (T : Set α → F →L[ℝ] F') (c : ℝ)
(f : α →ₛ F) : setToSimpleFunc (fun s => c • T s) f = c • setToSimpleFunc T f := by |
simp_rw [setToSimpleFunc, ContinuousLinearMap.smul_apply, smul_sum]
|
/-
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 | 346 | 406 | theorem lintegral_iSup {f : ℕ → α → ℝ≥0∞} (hf : ∀ n, Measurable (f n)) (h_mono : Monotone f) :
∫⁻ a, ⨆ n, f n a ∂μ = ⨆ n, ∫⁻ a, f n a ∂μ := by |
set c : ℝ≥0 → ℝ≥0∞ := (↑)
set F := fun a : α => ⨆ n, f n a
refine le_antisymm ?_ (iSup_lintegral_le _)
rw [lintegral_eq_nnreal]
refine iSup_le fun s => iSup_le fun hsf => ?_
refine ENNReal.le_of_forall_lt_one_mul_le fun a ha => ?_
rcases ENNReal.lt_iff_exists_coe.1 ha with ⟨r, rfl, _⟩
have ha : r < 1 :... |
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.Algebra.Category.MonCat.Limits
import Mathlib.Algebra.Category.GroupCat.ForgetCorepresentable
import Mathlib.Algebra.Category.GroupCat.Preadditive
impo... | Mathlib/Algebra/Category/GroupCat/Limits.lean | 550 | 553 | theorem kernelIsoKer_inv_comp_ι {G H : AddCommGroupCat.{u}} (f : G ⟶ H) :
(kernelIsoKer f).inv ≫ kernel.ι f = AddSubgroup.subtype f.ker := by |
ext
simp [kernelIsoKer]
|
/-
Copyright (c) 2021 Heather Macbeth. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Heather Macbeth
-/
import Mathlib.Algebra.CharP.Invertible
import Mathlib.Analysis.NormedSpace.LinearIsometry
import Mathlib.Analysis.Normed.Group.AddTorsor
import Mathlib.Analysis.No... | Mathlib/Analysis/NormedSpace/AffineIsometry.lean | 389 | 392 | theorem linearIsometryEquiv_mk' (e : P₁ → P₂) (e' : V₁ ≃ₗᵢ[𝕜] V₂) (p h) :
(mk' e e' p h).linearIsometryEquiv = e' := by |
ext
rfl
|
/-
Copyright (c) 2021 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.Group.Commute.Defs
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.GroupWithZero.Defs
import Mathlib.Algebra.Order.Monoid.Unbundled.Bas... | Mathlib/Algebra/Regular/Basic.lean | 256 | 260 | theorem IsRightRegular.ne_zero [Nontrivial R] (ra : IsRightRegular a) : a ≠ 0 := by |
rintro rfl
rcases exists_pair_ne R with ⟨x, y, xy⟩
refine xy (ra (?_ : x * 0 = y * 0))
rw [mul_zero, mul_zero]
|
/-
Copyright (c) 2022 Damiano Testa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Damiano Testa
-/
import Mathlib.Algebra.MonoidAlgebra.Support
#align_import algebra.monoid_algebra.degree from "leanprover-community/mathlib"@"f694c7dead66f5d4c80f446c796a5aad14707f0e"... | Mathlib/Algebra/MonoidAlgebra/Degree.lean | 138 | 146 | theorem le_inf_support_list_prod (degt0 : 0 ≤ degt 0)
(degtm : ∀ a b, degt a + degt b ≤ degt (a + b)) (l : List R[A]) :
(l.map fun f : R[A] => f.support.inf degt).sum ≤ l.prod.support.inf degt := by |
refine OrderDual.ofDual_le_ofDual.mpr ?_
refine sup_support_list_prod_le ?_ ?_ l
· refine (OrderDual.ofDual_le_ofDual.mp ?_)
exact degt0
· refine (fun a b => OrderDual.ofDual_le_ofDual.mp ?_)
exact degtm a b
|
/-
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 | 1,638 | 1,639 | theorem dist_prod_same_left {x : α} {y₁ y₂ : β} : dist (x, y₁) (x, y₂) = dist y₁ y₂ := by |
simp [Prod.dist_eq, dist_nonneg]
|
/-
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.Data.Fin.VecNotation
import Mathlib.Logic.Embedding.Set
#align_import logic.equiv.fin from "leanprover-community/mathlib"@"bd835ef554f37ef9b804f0903089211f89cb3... | Mathlib/Logic/Equiv/Fin.lean | 376 | 379 | theorem finAddFlip_apply_mk_right {k : ℕ} (h₁ : m ≤ k) (h₂ : k < m + n) :
finAddFlip (⟨k, h₂⟩ : Fin (m + n)) = ⟨k - m, by omega⟩ := by |
convert @finAddFlip_apply_natAdd n ⟨k - m, by omega⟩ m
simp [Nat.add_sub_cancel' h₁]
|
/-
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.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Data.Finset.Fold
import Mathlib.Data.Finset.Option
import Mathlib.Data.Finset.Pi
import Mathlib.Data.... | Mathlib/Data/Finset/Lattice.lean | 926 | 929 | theorem _root_.map_finset_sup' [SemilatticeSup β] [FunLike F α β] [SupHomClass F α β]
(f : F) {s : Finset ι} (hs) (g : ι → α) :
f (s.sup' hs g) = s.sup' hs (f ∘ g) := by |
refine hs.cons_induction ?_ ?_ <;> intros <;> simp [*]
|
/-
Copyright (c) 2017 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Yury G. Kudryashov
-/
import Batteries.Data.Sum.Basic
import Batteries.Logic
/-!
# Disjoint union of types
Theorems about the definitions introduced in `Batteries.Da... | .lake/packages/batteries/Batteries/Data/Sum/Lemmas.lean | 206 | 211 | theorem Lex.mono (hr : ∀ a b, r₁ a b → r₂ a b) (hs : ∀ a b, s₁ a b → s₂ a b) (h : Lex r₁ s₁ x y) :
Lex r₂ s₂ x y := by |
cases h
· exact Lex.inl (hr _ _ ‹_›)
· exact Lex.inr (hs _ _ ‹_›)
· exact Lex.sep _ _
|
/-
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 | 757 | 757 | theorem algEquiv_symm_mk' (x : R) (y : M) : (algEquiv M S Q).symm (mk' Q x y) = mk' S x y := by | simp
|
/-
Copyright (c) 2017 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Mario Carneiro, Johannes Hölzl, Chris Hughes, Jens Wagemaker, Jon Eugster
-/
import Mathlib.Algebra.Group.Basic
import Mathlib.Algebra.Group.Commute.Defs
import Mathlib.Logic.Uni... | Mathlib/Algebra/Group/Units.lean | 603 | 604 | theorem divp_eq_divp_iff {x y : α} {ux uy : αˣ} : x /ₚ ux = y /ₚ uy ↔ x * uy = y * ux := by |
rw [divp_eq_iff_mul_eq, divp_mul_eq_mul_divp, divp_eq_iff_mul_eq]
|
/-
Copyright (c) 2020 Riccardo Brasca. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Riccardo Brasca
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Monic
#align_import data.polynomial.lifts from "leanprover-community/mathlib"@"63417... | Mathlib/Algebra/Polynomial/Lifts.lean | 166 | 202 | theorem mem_lifts_and_degree_eq {p : S[X]} (hlifts : p ∈ lifts f) :
∃ q : R[X], map f q = p ∧ q.degree = p.degree := by |
generalize hd : p.natDegree = d
revert hd p
induction' d using Nat.strong_induction_on with n hn
intros p hlifts hdeg
by_cases erase_zero : p.eraseLead = 0
· rw [← eraseLead_add_monomial_natDegree_leadingCoeff p, erase_zero, zero_add, leadingCoeff]
exact
monomial_mem_lifts_and_degree_eq
(... |
/-
Copyright (c) 2022 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Stoll
-/
import Mathlib.NumberTheory.LegendreSymbol.JacobiSymbol
#align_import number_theory.legendre_symbol.norm_num from "leanprover-community/mathlib"@"e2621d935895abe70071a... | Mathlib/Tactic/NormNum/LegendreSymbol.lean | 76 | 82 | theorem jacobiSymNat.zero_left (b : ℕ) (hb : Nat.beq (b / 2) 0 = false) : jacobiSymNat 0 b = 0 := by |
rw [jacobiSymNat, Nat.cast_zero, jacobiSym.zero_left ?_]
calc
1 < 2 * 1 := by decide
_ ≤ 2 * (b / 2) :=
Nat.mul_le_mul_left _ (Nat.succ_le.mpr (Nat.pos_of_ne_zero (Nat.ne_of_beq_eq_false hb)))
_ ≤ b := Nat.mul_div_le b 2
|
/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Tim Baumann, Stephen Morgan, Scott Morrison, Floris van Doorn
-/
import Mathlib.Tactic.CategoryTheory.Reassoc
#align_import category_theory.isomorphism from "leanprover-community/math... | Mathlib/CategoryTheory/Iso.lean | 642 | 643 | theorem map_hom_inv (F : C ⥤ D) {X Y : C} (f : X ⟶ Y) [IsIso f] :
F.map f ≫ F.map (inv f) = 𝟙 (F.obj X) := by | simp
|
/-
Copyright (c) 2022 Violeta Hernández Palacios. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Violeta Hernández Palacios
-/
import Mathlib.Order.SuccPred.Basic
import Mathlib.Order.BoundedOrder
#align_import order.succ_pred.limit from "leanprover-community/mathlib"... | Mathlib/Order/SuccPred/Limit.lean | 237 | 239 | theorem not_isSuccLimit_iff' : ¬IsSuccLimit a ↔ a ∈ range (@succ α _ _) := by |
simp_rw [isSuccLimit_iff_succ_ne, not_forall, not_ne_iff]
rfl
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Kevin Buzzard, Yury Kudryashov, Eric Wieser
-/
import Mathlib.Algebra.Algebra.Prod
import Mathlib.LinearAlgebra.Basic
import Mathlib.LinearAlgebra.Span
... | Mathlib/LinearAlgebra/Prod.lean | 632 | 636 | theorem fst_map_snd : (Submodule.fst R M M₂).map (LinearMap.snd R M M₂) = ⊥ := by |
-- Porting note (#10936): was `tidy`
rw [eq_bot_iff]; intro x
simp only [fst, comap_bot, mem_map, mem_ker, snd_apply, eq_comm, Prod.exists, exists_eq_left,
exists_const, mem_bot, imp_self]
|
/-
Copyright (c) 2021 Aaron Anderson, Jesse Michael Han, Floris van Doorn. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Aaron Anderson, Jesse Michael Han, Floris van Doorn
-/
import Mathlib.Data.Fin.VecNotation
import Mathlib.SetTheory.Cardinal.Basic
#align_import m... | Mathlib/ModelTheory/Basic.lean | 878 | 879 | theorem injective_toEmbedding : Function.Injective (toEmbedding : (M ≃[L] N) → M ↪[L] N) := by |
intro _ _ h; apply DFunLike.coe_injective; exact congr_arg (DFunLike.coe ∘ Embedding.toHom) h
|
/-
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.Degree.Definitions
import Mathlib.Algebra.Polynomial.Induction
#align_import data.polyn... | Mathlib/Algebra/Polynomial/Eval.lean | 844 | 844 | theorem map_id : p.map (RingHom.id _) = p := by | simp [Polynomial.ext_iff, coeff_map]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Johan Commelin, Mario Carneiro
-/
import Mathlib.Algebra.MvPolynomial.Degrees
#align_import data.mv_polynomial.variables from "leanprover-community/mathlib"@"2f5b500a5... | Mathlib/Algebra/MvPolynomial/Variables.lean | 108 | 112 | theorem vars_add_subset [DecidableEq σ] (p q : MvPolynomial σ R) :
(p + q).vars ⊆ p.vars ∪ q.vars := by |
intro x hx
simp only [vars_def, Finset.mem_union, Multiset.mem_toFinset] at hx ⊢
simpa using Multiset.mem_of_le (degrees_add _ _) hx
|
/-
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.Normed.Group.Hom
import Mathlib.Analysis.SpecialFunctions.Pow.Continuity
import Mathlib.Data.Set.Image
import Mathlib.MeasureTh... | Mathlib/MeasureTheory/Function/LpSpace.lean | 95 | 97 | theorem Memℒp.snorm_mk_lt_top {α E : Type*} [MeasurableSpace α] {μ : Measure α}
[NormedAddCommGroup E] {p : ℝ≥0∞} {f : α → E} (hfp : Memℒp f p μ) :
snorm (AEEqFun.mk f hfp.1) p μ < ∞ := by | simp [hfp.2]
|
/-
Copyright (c) 2019 Jean Lo. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jean Lo, Yaël Dillies, Moritz Doll
-/
import Mathlib.Data.Real.Pointwise
import Mathlib.Analysis.Convex.Function
import Mathlib.Analysis.LocallyConvex.Basic
import Mathlib.Data.Real.Sqrt
#al... | Mathlib/Analysis/Seminorm.lean | 693 | 693 | theorem mem_closedBall_zero : y ∈ closedBall p 0 r ↔ p y ≤ r := by | rw [mem_closedBall, sub_zero]
|
/-
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.Topology.Maps
import Mathlib.Topology.NhdsSet
#align_import topology.constructions from "leanprover-community/mathlib"... | Mathlib/Topology/Constructions.lean | 316 | 317 | theorem mem_nhds_iff {x : CofiniteTopology X} {s : Set (CofiniteTopology X)} :
s ∈ 𝓝 x ↔ x ∈ s ∧ sᶜ.Finite := by | simp [nhds_eq]
|
/-
Copyright (c) 2023 Xavier Roblot. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Xavier Roblot
-/
import Mathlib.Analysis.Normed.Field.Basic
import Mathlib.LinearAlgebra.Eigenspace.Basic
import Mathlib.LinearAlgebra.Determinant
/-!
# Gershgorin's circle theorem
Th... | Mathlib/LinearAlgebra/Matrix/Gershgorin.lean | 26 | 56 | theorem eigenvalue_mem_ball {μ : K} (hμ : Module.End.HasEigenvalue (Matrix.toLin' A) μ) :
∃ k, μ ∈ Metric.closedBall (A k k) (∑ j ∈ Finset.univ.erase k, ‖A k j‖) := by |
cases isEmpty_or_nonempty n
· exfalso
exact hμ Submodule.eq_bot_of_subsingleton
· obtain ⟨v, h_eg, h_nz⟩ := hμ.exists_hasEigenvector
obtain ⟨i, -, h_i⟩ := Finset.exists_mem_eq_sup' Finset.univ_nonempty (fun i => ‖v i‖)
have h_nz : v i ≠ 0 := by
contrapose! h_nz
ext j
rw [Pi.zero_app... |
/-
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 Mathlib.Data.Nat.Defs
import Mathlib.Data.Option.Basic
import Mathlib.Data.List.Defs
im... | Mathlib/Data/List/Basic.lean | 2,827 | 2,835 | theorem filterMap_congr {f g : α → Option β} {l : List α}
(h : ∀ x ∈ l, f x = g x) : l.filterMap f = l.filterMap g := by |
induction' l with a l ih <;> simp
simp [ih (fun x hx ↦ h x (List.mem_cons_of_mem a hx))]
cases' hfa : f a with b
· have : g a = none := Eq.symm (by simpa [hfa] using h a (by simp))
simp [this]
· have : g a = some b := Eq.symm (by simpa [hfa] using h a (by simp))
simp [this]
|
/-
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.CharP.Invertible
import Mathlib.Algebra.Order.Interval.Set.Group
import Mathlib.Analysis.Convex.Segment
import Mathlib.LinearAlgebra.AffineSpace.Fi... | Mathlib/Analysis/Convex/Between.lean | 910 | 913 | theorem sbtw_pointReflection_of_ne {x y : P} (h : x ≠ y) : Sbtw R y x (pointReflection R x y) := by |
refine ⟨wbtw_pointReflection _ _ _, h, ?_⟩
nth_rw 1 [← pointReflection_self R x]
exact (pointReflection_involutive R x).injective.ne h
|
/-
Copyright (c) 2024 Mitchell Lee. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mitchell Lee
-/
import Mathlib.GroupTheory.Coxeter.Length
import Mathlib.Data.ZMod.Parity
/-!
# Reflections, inversions, and inversion sequences
Throughout this file, `B` is a type and... | Mathlib/GroupTheory/Coxeter/Inversion.lean | 223 | 225 | theorem leftInvSeq_concat (ω : List B) (i : B) :
lis (ω.concat i) = (lis ω).concat ((π ω) * (s i) * (π ω)⁻¹) := by |
simp [leftInvSeq_eq_reverse_rightInvSeq_reverse, rightInvSeq]
|
/-
Copyright (c) 2022 Eric Wieser. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Eric Wieser
-/
import Mathlib.Data.Matrix.Notation
import Mathlib.Data.Matrix.Basic
import Mathlib.Data.Fin.Tuple.Reflection
#align_import data.matrix.reflection from "leanprover-communi... | Mathlib/Data/Matrix/Reflection.lean | 209 | 212 | theorem vecMulᵣ_eq [NonUnitalNonAssocSemiring α] (v : Fin l → α) (A : Matrix (Fin l) (Fin m) α) :
vecMulᵣ v A = v ᵥ* A := by |
simp [vecMulᵣ, Function.comp]
rfl
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Scott Morrison
-/
import Mathlib.Algebra.BigOperators.Finsupp
import Mathlib.Algebra.Module.Basic
import Mathlib.Algebra.Regular.SMul
import Mathlib.Data.Finset.Preimag... | Mathlib/Data/Finsupp/Basic.lean | 892 | 894 | theorem filter_eq_zero_iff : f.filter p = 0 ↔ ∀ x, p x → f x = 0 := by |
simp only [DFunLike.ext_iff, filter_eq_indicator, zero_apply, Set.indicator_apply_eq_zero,
Set.mem_setOf_eq]
|
/-
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.Convolution
import Mathlib.Analysis.SpecialFunctions.Trigonometric.EulerSineProd
import Mathlib.Analysis.SpecialFunctions.Gamma.BohrMollerup
i... | Mathlib/Analysis/SpecialFunctions/Gamma/Beta.lean | 93 | 102 | theorem betaIntegral_symm (u v : ℂ) : betaIntegral v u = betaIntegral u v := by |
rw [betaIntegral, betaIntegral]
have := intervalIntegral.integral_comp_mul_add (a := 0) (b := 1) (c := -1)
(fun x : ℝ => (x : ℂ) ^ (u - 1) * (1 - (x : ℂ)) ^ (v - 1)) neg_one_lt_zero.ne 1
rw [inv_neg, inv_one, neg_one_smul, ← intervalIntegral.integral_symm] at this
simp? at this says
simp only [neg_mul,... |
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.RingTheory.Ideal.IsPrimary
import Mathlib.RingTheory.Localization.AtPrime
import Mathlib.Order.Minimal
#align_import ring_theory.ideal.minimal_prime from "l... | Mathlib/RingTheory/Ideal/MinimalPrime.lean | 91 | 101 | theorem Ideal.sInf_minimalPrimes : sInf I.minimalPrimes = I.radical := by |
rw [I.radical_eq_sInf]
apply le_antisymm
· intro x hx
rw [Ideal.mem_sInf] at hx ⊢
rintro J ⟨e, hJ⟩
obtain ⟨p, hp, hp'⟩ := Ideal.exists_minimalPrimes_le e
exact hp' (hx hp)
· apply sInf_le_sInf _
intro I hI
exact hI.1.symm
|
/-
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.Algebra.BigOperators.Group.Finset
import Mathlib.Data.Fintype.Card
#align_import data.multiset.fintype from "leanprover-community/mathlib"@"e3d9ab8faa9dea8f... | Mathlib/Data/Multiset/Fintype.lean | 228 | 231 | theorem Multiset.map_univ {β : Type*} (m : Multiset α) (f : α → β) :
((Finset.univ : Finset m).val.map fun (x : m) ↦ f (x : α)) = m.map f := by |
erw [← Multiset.map_map]
rw [Multiset.map_univ_coe]
|
/-
Copyright (c) 2023 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Probability.Kernel.Composition
#align_import probability.kernel.invariance from "leanprover-community/mathlib"@"3b92d54a05ee592aa2c6181a4e76b1bb7cc45d0b"
/... | Mathlib/Probability/Kernel/Invariance.lean | 87 | 92 | theorem Invariant.comp [IsSFiniteKernel κ] (hκ : Invariant κ μ) (hη : Invariant η μ) :
Invariant (κ ∘ₖ η) μ := by |
cases' isEmpty_or_nonempty α with _ hα
· exact Subsingleton.elim _ _
· simp_rw [Invariant, ← comp_const_apply_eq_bind (κ ∘ₖ η) μ hα.some, comp_assoc, hη.comp_const,
hκ.comp_const, const_apply]
|
/-
Copyright (c) 2019 Amelia Livingston. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Amelia Livingston
-/
import Mathlib.Algebra.Group.Submonoid.Membership
import Mathlib.Algebra.Group.Units
import Mathlib.Algebra.Regular.Basic
import Mathlib.GroupTheory.Congruence.... | Mathlib/GroupTheory/MonoidLocalization.lean | 1,775 | 1,776 | theorem mulEquivOfQuotient_mk (x y) : mulEquivOfQuotient f (mk x y) = f.mk' x y := by |
rw [mk_eq_monoidOf_mk'_apply]; exact mulEquivOfQuotient_mk' _ _
|
/-
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 | 287 | 289 | theorem concat_append {u v w x : V} (p : G.Walk u v) (h : G.Adj v w) (q : G.Walk w x) :
(p.concat h).append q = p.append (cons h q) := by |
rw [concat_eq_append, ← append_assoc, cons_nil_append]
|
/-
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.Interval.Set.Basic
import Mathlib.Data.Set.Lattice
import Mathlib.Data.SetLike.Basic
#align_import order.interval from "leanprover-community/mathlib... | Mathlib/Order/Interval/Basic.lean | 759 | 760 | theorem coe_iInf [@DecidableRel α (· ≤ ·)] (f : ι → Interval α) :
↑(⨅ i, f i) = ⋂ i, (f i : Set α) := by | simp [iInf]
|
/-
Copyright (c) 2022 Rémy Degenne. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Rémy Degenne
-/
import Mathlib.Data.Countable.Basic
import Mathlib.Logic.Encodable.Basic
import Mathlib.Order.SuccPred.Basic
import Mathlib.Order.Interval.Finset.Defs
#align_import orde... | Mathlib/Order/SuccPred/LinearLocallyFinite.lean | 72 | 74 | theorem le_succFn (i : ι) : i ≤ succFn i := by |
rw [le_isGLB_iff (succFn_spec i), mem_lowerBounds]
exact fun x hx ↦ le_of_lt hx
|
/-
Copyright (c) 2019 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Yaël Dillies
-/
import Mathlib.Algebra.Module.BigOperators
import Mathlib.Data.Finset.NoncommProd
import Mathlib.Data.Fintype.Perm
import Mathlib.Data.Int.ModEq
import Mat... | Mathlib/GroupTheory/Perm/Cycle/Factors.lean | 530 | 531 | theorem cycleFactorsFinset_one : cycleFactorsFinset (1 : Perm α) = ∅ := by |
simp [cycleFactorsFinset_eq_empty_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.Topology.Compactness.SigmaCompact
import Mathlib.Topology.Connected.TotallyDisconnected
import Mathlib.Topology.Inseparable
#align_imp... | Mathlib/Topology/Separation.lean | 829 | 832 | theorem Dense.diff_finite [T1Space X] [∀ x : X, NeBot (𝓝[≠] x)] {s : Set X} (hs : Dense s)
{t : Set X} (ht : t.Finite) : Dense (s \ t) := by |
convert hs.diff_finset ht.toFinset
exact (Finite.coe_toFinset _).symm
|
/-
Copyright (c) 2022 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Analysis.Calculus.Deriv.Basic
import Mathlib.MeasureTheory.Constructions.BorelSpace.ContinuousLinearMap
import Mathlib.MeasureTheory.Covering.Bes... | Mathlib/MeasureTheory/Function/Jacobian.lean | 251 | 266 | theorem exists_partition_approximatesLinearOn_of_hasFDerivWithinAt [SecondCountableTopology F]
(f : E → F) (s : Set E) (f' : E → E →L[ℝ] F) (hf' : ∀ x ∈ s, HasFDerivWithinAt f (f' x) s x)
(r : (E →L[ℝ] F) → ℝ≥0) (rpos : ∀ A, r A ≠ 0) :
∃ (t : ℕ → Set E) (A : ℕ → E →L[ℝ] F),
Pairwise (Disjoint on t) ∧
... |
rcases exists_closed_cover_approximatesLinearOn_of_hasFDerivWithinAt f s f' hf' r rpos with
⟨t, A, t_closed, st, t_approx, ht⟩
refine
⟨disjointed t, A, disjoint_disjointed _,
MeasurableSet.disjointed fun n => (t_closed n).measurableSet, ?_, ?_, ht⟩
· rw [iUnion_disjointed]; exact st
· intro n; ex... |
/-
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.PurelyInseparable
import Mathlib.FieldTheory.PerfectClosure
/-!
# `IsPerfectClosure` predicate
This file contains `IsPerfectClosure` which asserts that ... | Mathlib/FieldTheory/IsPerfectClosure.lean | 84 | 85 | theorem pNilradical_eq_bot {R : Type*} [CommSemiring R] {p : ℕ} (hp : ¬ 1 < p) :
pNilradical R p = ⊥ := by | rw [pNilradical, if_neg hp]
|
/-
Copyright (c) 2018 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau
-/
import Mathlib.RingTheory.Ideal.Operations
#align_import ring_theory.ideal.operations from "leanprover-community/mathlib"@"e7f0ddbf65bd7181a85edb74b64bdc35ba4bdc74"
/-!
# Map... | Mathlib/RingTheory/Ideal/Maps.lean | 891 | 895 | theorem eq_liftOfRightInverse (hf : Function.RightInverse f_inv f) (g : A →+* C)
(hg : RingHom.ker f ≤ RingHom.ker g) (h : B →+* C) (hh : h.comp f = g) :
h = f.liftOfRightInverse f_inv hf ⟨g, hg⟩ := by |
simp_rw [← hh]
exact ((f.liftOfRightInverse f_inv hf).apply_symm_apply _).symm
|
/-
Copyright (c) 2020 Bhavik Mehta. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Bhavik Mehta, Andrew Yang
-/
import Mathlib.CategoryTheory.Limits.Shapes.Equalizers
import Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts
import Mathlib.CategoryTheory.Limits.Shapes... | Mathlib/CategoryTheory/Limits/Constructions/Equalizers.lean | 51 | 54 | theorem pullbackFst_eq_pullback_snd (F : WalkingParallelPair ⥤ C) :
pullbackFst F = pullback.snd := by |
convert (eq_whisker pullback.condition Limits.prod.fst :
(_ : constructEqualizer F ⟶ F.obj WalkingParallelPair.zero) = _) <;> simp
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang, Joël Riou
-/
import Mathlib.CategoryTheory.Sites.Subsheaf
import Mathlib.CategoryTheory.Sites.CompatibleSheafification
import Mathlib.CategoryTheory.Sites.LocallyInjective
#... | Mathlib/CategoryTheory/Sites/LocallySurjective.lean | 113 | 116 | theorem isLocallySurjective_iff_whisker_forget {F G : Cᵒᵖ ⥤ A} (f : F ⟶ G) :
IsLocallySurjective J f ↔ IsLocallySurjective J (whiskerRight f (forget A)) := by |
simp only [isLocallySurjective_iff_imagePresheaf_sheafify_eq_top]
rfl
|
/-
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.Algebra.Order.Monoid.Unbundled.Pow
import Mathlib.Data.Finset.Fold
import Mathlib.Data.Finset.Option
import Mathlib.Data.Finset.Pi
import Mathlib.Data.... | Mathlib/Data/Finset/Lattice.lean | 2,017 | 2,019 | theorem iSup_eq_iSup_finset' (s : ι' → α) :
⨆ i, s i = ⨆ t : Finset (PLift ι'), ⨆ i ∈ t, s (PLift.down i) := by |
rw [← iSup_eq_iSup_finset, ← Equiv.plift.surjective.iSup_comp]; rfl
|
/-
Copyright (c) 2021 Yury Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury Kudryashov
-/
import Mathlib.Algebra.BigOperators.Option
import Mathlib.Analysis.BoxIntegral.Box.Basic
import Mathlib.Data.Set.Pairwise.Lattice
#align_import analysis.box_integr... | Mathlib/Analysis/BoxIntegral/Partition/Basic.lean | 533 | 541 | theorem restrict_biUnion (πi : ∀ J, Prepartition J) (hJ : J ∈ π) :
(π.biUnion πi).restrict J = πi J := by |
refine (eq_of_boxes_subset_iUnion_superset (fun J₁ h₁ => ?_) ?_).symm
· refine (mem_restrict _).2 ⟨J₁, π.mem_biUnion.2 ⟨J, hJ, h₁⟩, (inf_of_le_right ?_).symm⟩
exact WithBot.coe_le_coe.2 (le_of_mem _ h₁)
· simp only [iUnion_restrict, iUnion_biUnion, Set.subset_def, Set.mem_inter_iff, Set.mem_iUnion]
rintr... |
/-
Copyright (c) 2019 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Casper Putz, Anne Baanen
-/
import Mathlib.LinearAlgebra.FiniteDimensional
import Mathlib.LinearAlgebra.GeneralLinearGroup
import Mathlib.LinearAlgebra.... | Mathlib/LinearAlgebra/Determinant.lean | 96 | 99 | theorem det_conj_of_mul_eq_one [DecidableEq m] [DecidableEq n] {M : Matrix m n A}
{M' : Matrix n m A} {N : Matrix n n A} (hMM' : M * M' = 1) (hM'M : M' * M = 1) :
det (M * N * M') = det N := by |
rw [← det_comm' hM'M hMM', ← Matrix.mul_assoc, hM'M, Matrix.one_mul]
|
/-
Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Maximal
import Mathlib.Algebraic... | Mathlib/RingTheory/DedekindDomain/Ideal.lean | 188 | 196 | theorem mul_generator_self_inv {R₁ : Type*} [CommRing R₁] [Algebra R₁ K] [IsLocalization R₁⁰ K]
(I : FractionalIdeal R₁⁰ K) [Submodule.IsPrincipal (I : Submodule R₁ K)] (h : I ≠ 0) :
I * spanSingleton _ (generator (I : Submodule R₁ K))⁻¹ = 1 := by |
-- Rewrite only the `I` that appears alone.
conv_lhs => congr; rw [eq_spanSingleton_of_principal I]
rw [spanSingleton_mul_spanSingleton, mul_inv_cancel, spanSingleton_one]
intro generator_I_eq_zero
apply h
rw [eq_spanSingleton_of_principal I, generator_I_eq_zero, spanSingleton_zero]
|
/-
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.Exp
import Mathlib.Tactic.Positivity.Core
import Mathlib.Algeb... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean | 806 | 815 | theorem sin_pi_over_two_pow_succ (n : ℕ) :
sin (π / 2 ^ (n + 2)) = √(2 - sqrtTwoAddSeries 0 n) / 2 := by |
rw [eq_div_iff_mul_eq two_ne_zero, eq_comm, sqrt_eq_iff_sq_eq, mul_pow,
sin_sq_pi_over_two_pow_succ, sub_mul]
· congr <;> norm_num
· rw [sub_nonneg]
exact (sqrtTwoAddSeries_lt_two _).le
refine mul_nonneg (sin_nonneg_of_nonneg_of_le_pi ?_ ?_) zero_le_two
· positivity
· exact div_le_self pi_pos.le <|... |
/-
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 | 128 | 139 | theorem image_update_Icc (f : ∀ i, α i) (i : ι) (a b : α i) :
update f i '' Icc a b = Icc (update f i a) (update f i b) := by |
ext x
rw [← Set.pi_univ_Icc]
refine ⟨?_, fun h => ⟨x i, ?_, ?_⟩⟩
· rintro ⟨c, hc, rfl⟩
simpa [update_le_update_iff]
· simpa only [Function.update_same] using h i (mem_univ i)
· ext j
obtain rfl | hij := eq_or_ne i j
· exact Function.update_same _ _ _
· simpa only [Function.update_noteq hij.... |
/-
Copyright (c) 2017 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Patrick Massot, Scott Morrison, Mario Carneiro, Andrew Yang
-/
import Mathlib.Topology.Category.TopCat.Limits.Products
#align_import topology.category.Top.limits.pullbacks from "leanp... | Mathlib/Topology/Category/TopCat/Limits/Pullbacks.lean | 212 | 248 | theorem range_pullback_map {W X Y Z S T : TopCat} (f₁ : W ⟶ S) (f₂ : X ⟶ S) (g₁ : Y ⟶ T)
(g₂ : Z ⟶ T) (i₁ : W ⟶ Y) (i₂ : X ⟶ Z) (i₃ : S ⟶ T) [H₃ : Mono i₃] (eq₁ : f₁ ≫ i₃ = i₁ ≫ g₁)
(eq₂ : f₂ ≫ i₃ = i₂ ≫ g₂) :
Set.range (pullback.map f₁ f₂ g₁ g₂ i₁ i₂ i₃ eq₁ eq₂) =
(pullback.fst : pullback g₁ g₂ ⟶ _) ... |
ext
constructor
· rintro ⟨y, rfl⟩
simp only [Set.mem_inter_iff, Set.mem_preimage, Set.mem_range]
erw [← comp_apply, ← comp_apply] -- now `erw` after #13170
simp only [limit.lift_π, PullbackCone.mk_pt, PullbackCone.mk_π_app, comp_apply]
exact ⟨exists_apply_eq_apply _ _, exists_apply_eq_apply _ _⟩
... |
/-
Copyright (c) 2022 Adam Topaz. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Adam Topaz, Junyan Xu, Jack McKoen
-/
import Mathlib.RingTheory.Valuation.ValuationRing
import Mathlib.RingTheory.Localization.AsSubring
import Mathlib.Algebra.Ring.Subring.Pointwise
impor... | Mathlib/RingTheory/Valuation/ValuationSubring.lean | 204 | 211 | theorem valuation_eq_one_iff (a : A) : IsUnit a ↔ A.valuation a = 1 :=
⟨fun h => A.valuation_unit h.unit, fun h => by
have ha : (a : K) ≠ 0 := by |
intro c
rw [c, A.valuation.map_zero] at h
exact zero_ne_one h
have ha' : (a : K)⁻¹ ∈ A := by rw [← valuation_le_one_iff, map_inv₀, h, inv_one]
apply isUnit_of_mul_eq_one a ⟨a⁻¹, ha'⟩; ext; field_simp⟩
|
/-
Copyright (c) 2023 Kyle Miller, Rémi Bottinelli. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kyle Miller, Rémi Bottinelli
-/
import Mathlib.Combinatorics.SimpleGraph.Connectivity
/-!
# Connectivity of subgraphs and induced graphs
## Main definitions
* `SimpleG... | Mathlib/Combinatorics/SimpleGraph/Connectivity/Subgraph.lean | 64 | 69 | theorem singletonSubgraph_connected {v : V} : (G.singletonSubgraph v).Connected := by |
refine ⟨⟨?_⟩⟩
rintro ⟨a, ha⟩ ⟨b, hb⟩
simp only [singletonSubgraph_verts, Set.mem_singleton_iff] at ha hb
subst_vars
rfl
|
/-
Copyright (c) 2023 Scott Carnahan. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Carnahan
-/
import Mathlib.Algebra.Group.Defs
import Mathlib.Algebra.Group.Prod
import Mathlib.Data.PNat.Basic
import Mathlib.GroupTheory.GroupAction.Prod
/-!
# Typeclasses for... | Mathlib/Algebra/Group/PNatPowAssoc.lean | 88 | 92 | theorem ppow_eq_pow [Monoid M] [Pow M ℕ+] [PNatPowAssoc M] (x : M) (n : ℕ+) :
x ^ n = x ^ (n : ℕ) := by |
refine PNat.recOn n ?_ fun k hk ↦ ?_
· rw [ppow_one, PNat.one_coe, pow_one]
· rw [ppow_add, ppow_one, PNat.add_coe, pow_add, PNat.one_coe, pow_one, ← hk]
|
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Order.Interval.Finset.Basic
#align_import data.multiset.locally_finite from "leanprover-community/mathlib"@"59694bd07f0a39c5beccba34bd9f413a160782bf"
/-!... | Mathlib/Order/Interval/Multiset.lean | 202 | 202 | theorem Ioo_self : Ioo a a = 0 := by | rw [Ioo, Finset.Ioo_self, Finset.empty_val]
|
/-
Copyright (c) 2022 Antoine Chambert-Loir. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir
-/
import Mathlib.Algebra.Group.Subgroup.Actions
import Mathlib.GroupTheory.GroupAction.Basic
import Mathlib.GroupTheory.GroupAction.FixedPoints
#align_im... | Mathlib/GroupTheory/GroupAction/FixingSubgroup.lean | 129 | 131 | theorem mem_fixingSubgroup_compl_iff_movedBy_subset {s : Set α} {m : M} :
m ∈ fixingSubgroup M sᶜ ↔ (fixedBy α m)ᶜ ⊆ s := by |
rw [mem_fixingSubgroup_iff_subset_fixedBy, Set.compl_subset_comm]
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Jeremy Avigad, Yury Kudryashov, Patrick Massot
-/
import Mathlib.Algebra.BigOperators.Group.Finset
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Algebra.Order.... | Mathlib/Order/Filter/AtTopBot.lean | 1,695 | 1,698 | theorem atTop_Ioi_eq [SemilatticeSup α] (a : α) : atTop = comap ((↑) : Ioi a → α) atTop := by |
rcases isEmpty_or_nonempty (Ioi a) with h|⟨⟨b, hb⟩⟩
· exact Subsingleton.elim _ _
· rw [← map_val_atTop_of_Ici_subset (Ici_subset_Ioi.2 hb), comap_map Subtype.coe_injective]
|
/-
Copyright (c) 2023 Josha Dekker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Josha Dekker
-/
import Mathlib.Topology.Bases
import Mathlib.Order.Filter.CountableInter
import Mathlib.Topology.Compactness.SigmaCompact
/-!
# Lindelöf sets and Lindelöf spaces
## Mai... | Mathlib/Topology/Compactness/Lindelof.lean | 575 | 580 | theorem Filter.comap_coLindelof_le {f : X → Y} (hf : Continuous f) :
(Filter.coLindelof Y).comap f ≤ Filter.coLindelof X := by |
rw [(hasBasis_coLindelof.comap f).le_basis_iff hasBasis_coLindelof]
intro t ht
refine ⟨f '' t, ht.image hf, ?_⟩
simpa using t.subset_preimage_image f
|
/-
Copyright (c) 2022 Andrew Yang. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Andrew Yang
-/
import Mathlib.AlgebraicGeometry.AffineScheme
import Mathlib.AlgebraicGeometry.Pullbacks
import Mathlib.CategoryTheory.MorphismProperty.Limits
import Mathlib.Data.List.TFAE... | Mathlib/AlgebraicGeometry/Morphisms/Basic.lean | 337 | 370 | theorem AffineTargetMorphismProperty.IsLocal.targetAffineLocallyIsLocal
{P : AffineTargetMorphismProperty} (hP : P.IsLocal) :
PropertyIsLocalAtTarget (targetAffineLocally P) := by |
constructor
· exact targetAffineLocally_respectsIso hP.1
· intro X Y f U H V
rw [← P.toProperty_apply (i := V.2), hP.1.arrow_mk_iso_iff (morphismRestrictRestrict f _ _)]
convert H ⟨_, IsAffineOpen.imageIsOpenImmersion V.2 (Y.ofRestrict _)⟩
rw [← P.toProperty_apply]
· rintro X Y f 𝒰 h𝒰
-- Port... |
/-
Copyright (c) 2022 Kexing Ying. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kexing Ying
-/
import Mathlib.Order.Interval.Set.Monotone
import Mathlib.Probability.Process.HittingTime
import Mathlib.Probability.Martingale.Basic
import Mathlib.Tactic.AdaptationNote
... | Mathlib/Probability/Martingale/Upcrossing.lean | 301 | 306 | theorem upperCrossingTime_lt_bddAbove (hab : a < b) :
BddAbove {n | upperCrossingTime a b f N n ω < N} := by |
obtain ⟨k, hk⟩ := exists_upperCrossingTime_eq f N ω hab
refine ⟨k, fun n (hn : upperCrossingTime a b f N n ω < N) => ?_⟩
by_contra hn'
exact hn.ne (upperCrossingTime_stabilize (not_le.1 hn').le hk)
|
/-
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 | 1,030 | 1,034 | theorem iSup_dualAnnihilator_le_iInf {ι : Sort*} (U : ι → Submodule R M) :
⨆ i : ι, (U i).dualAnnihilator ≤ (⨅ i : ι, U i).dualAnnihilator := by |
rw [le_dualAnnihilator_iff_le_dualCoannihilator, dualCoannihilator_iSup_eq]
apply iInf_mono
exact fun i : ι => le_dualAnnihilator_dualCoannihilator (U i)
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes, Abhimanyu Pallavi Sudhir, Jean Lo, Calle Sönne, Benjamin Davidson
-/
import Mathlib.Algebra.QuadraticDiscriminant
import Mathlib.Analysis.Convex.SpecificFunctions.Deriv
imp... | Mathlib/Analysis/SpecialFunctions/Trigonometric/Complex.lean | 290 | 294 | theorem mul_le_sin {x : ℝ} (hx : 0 ≤ x) (hx' : x ≤ π / 2) : 2 / π * x ≤ sin x := by |
rw [← inv_div]
simpa [-inv_div, mul_inv_cancel_left₀ pi_div_two_pos.ne'] using @le_sin_mul ((π / 2)⁻¹ * x)
(mul_nonneg (inv_nonneg.2 pi_div_two_pos.le) hx)
(by rwa [← div_eq_inv_mul, div_le_one pi_div_two_pos])
|
/-
Copyright (c) 2017 Johannes Hölzl. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Mario Carneiro, Patrick Massot, Yury Kudryashov, Rémy Degenne
-/
import Mathlib.Order.Interval.Set.Basic
import Mathlib.Order.Hom.Set
#align_import data.set.intervals.... | Mathlib/Order/Interval/Set/OrderIso.lean | 48 | 49 | theorem preimage_Icc (e : α ≃o β) (a b : β) : e ⁻¹' Icc a b = Icc (e.symm a) (e.symm b) := by |
simp [← Ici_inter_Iic]
|
/-
Copyright (c) 2020 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis
-/
import Mathlib.Algebra.Field.Basic
import Mathlib.Algebra.Order.Field.Defs
import Mathlib.Data.Tree.Basic
import Mathlib.Logic.Basic
import Mathlib.Tactic.NormNum.C... | Mathlib/Tactic/CancelDenoms/Core.lean | 59 | 60 | theorem sub_subst {α} [Ring α] {n e1 e2 t1 t2 : α} (h1 : n * e1 = t1) (h2 : n * e2 = t2) :
n * (e1 - e2) = t1 - t2 := by | simp [left_distrib, *, sub_eq_add_neg]
|
/-
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.Init.Control.Combinators
import Mathlib.Data.Option.Defs
import Mathlib.Logic.IsEmpty
import Mathlib.Logic.Relator
import Mathlib.Util.CompileInductive... | Mathlib/Data/Option/Basic.lean | 175 | 177 | theorem map_pbind (f : β → γ) (x : Option α) (g : ∀ a, a ∈ x → Option β) :
Option.map f (x.pbind g) = x.pbind fun a H ↦ Option.map f (g a H) := by |
cases x <;> simp only [pbind, map_none']
|
/-
Copyright (c) 2023 Michael Stoll. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Michael Geißer, Michael Stoll
-/
import Mathlib.Tactic.Qify
import Mathlib.Data.ZMod.Basic
import Mathlib.NumberTheory.DiophantineApproximation
import Mathlib.NumberTheory.Zsqrtd.Basic
... | Mathlib/NumberTheory/Pell.lean | 289 | 293 | theorem x_pow_pos {a : Solution₁ d} (hax : 0 < a.x) (n : ℕ) : 0 < (a ^ n).x := by |
induction' n with n ih
· simp only [Nat.zero_eq, pow_zero, x_one, zero_lt_one]
· rw [pow_succ]
exact x_mul_pos ih hax
|
/-
Copyright (c) 2024 Oliver Nash. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Antoine Chambert-Loir, Oliver Nash
-/
import Mathlib.Algebra.Polynomial.AlgebraMap
import Mathlib.Algebra.Polynomial.Identities
import Mathlib.RingTheory.Nilpotent.Lemmas
import Mathlib.R... | Mathlib/Dynamics/Newton.lean | 75 | 77 | theorem isFixedPt_newtonMap_of_isUnit_iff (h : IsUnit <| aeval x (derivative P)) :
IsFixedPt P.newtonMap x ↔ aeval x P = 0 := by |
rw [IsFixedPt, newtonMap_apply, sub_eq_self, Ring.inverse_mul_eq_iff_eq_mul _ _ _ h, mul_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.RingTheory.DedekindDomain.Ideal
#align_import number_theory.ramification_inertia from "leanprover-community/mathlib"@"039a089d2a4b93c761b234f3e5f5aeb752bac6... | Mathlib/NumberTheory/RamificationInertia.lean | 724 | 729 | theorem Factors.finrank_pow_ramificationIdx [p.IsMaximal]
(P : (factors (map (algebraMap R S) p)).toFinset) :
finrank (R ⧸ p) (S ⧸ (P : Ideal S) ^ ramificationIdx (algebraMap R S) p P) =
ramificationIdx (algebraMap R S) p P * inertiaDeg (algebraMap R S) p P := by |
rw [finrank_prime_pow_ramificationIdx, inertiaDeg_algebraMap]
exacts [NeZero.ne _, Factors.ne_bot p P]
|
/-
Copyright (c) 2018 Mario Carneiro. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Mario Carneiro, Chris Hughes
-/
import Mathlib.Algebra.GCDMonoid.Basic
import Mathlib.Algebra.EuclideanDomain.Basic
import Mathlib.RingTheory.Ideal.Basic
import Mathlib.RingTheory.Prin... | Mathlib/RingTheory/EuclideanDomain.lean | 42 | 47 | theorem left_div_gcd_ne_zero {p q : R} (hp : p ≠ 0) : p / GCDMonoid.gcd p q ≠ 0 := by |
obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_left p q
obtain ⟨pq0, r0⟩ : GCDMonoid.gcd p q ≠ 0 ∧ r ≠ 0 := mul_ne_zero_iff.mp (hr ▸ hp)
nth_rw 1 [hr]
rw [mul_comm, mul_div_cancel_right₀ _ pq0]
exact r0
|
/-
Copyright (c) 2022 Joseph Myers. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Joseph Myers, Heather Macbeth
-/
import Mathlib.Analysis.SpecialFunctions.Complex.Circle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic
#align_import geometry.euclidean.angle.or... | Mathlib/Geometry/Euclidean/Angle/Oriented/Rotation.lean | 297 | 298 | theorem eq_rotation_self_iff_angle_eq_zero {x : V} (hx : x ≠ 0) (θ : Real.Angle) :
x = o.rotation θ x ↔ θ = 0 := by | rw [← o.rotation_eq_self_iff_angle_eq_zero hx, eq_comm]
|
/-
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.RingTheory.WittVector.Basic
#align_import ring_theory.witt_vector.teichmuller from "leanprover-community/mathlib"@"c0a51cf2de54089d69301befc4c73bbc2f5... | Mathlib/RingTheory/WittVector/Teichmuller.lean | 116 | 117 | theorem teichmuller_zero : teichmuller p (0 : R) = 0 := by |
ext ⟨⟩ <;> · rw [zero_coeff]; rfl
|
/-
Copyright (c) 2019 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison, Bhavik Mehta
-/
import Mathlib.CategoryTheory.Comma.Over
import Mathlib.CategoryTheory.DiscreteCategory
import Mathlib.CategoryTheory.EpiMono
import Mathlib.CategoryThe... | Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | 919 | 921 | theorem coprod.map_id_comp {X Y Z W : C} (f : X ⟶ Y) (g : Y ⟶ Z) [HasBinaryCoproduct W X]
[HasBinaryCoproduct W Y] [HasBinaryCoproduct W Z] :
coprod.map (𝟙 W) (f ≫ g) = coprod.map (𝟙 W) f ≫ coprod.map (𝟙 W) g := by | simp
|
/-
Copyright (c) 2023 Yuma Mizuno. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yuma Mizuno
-/
import Mathlib.Tactic.CategoryTheory.Coherence
import Mathlib.CategoryTheory.Bicategory.Coherence
/-!
# Adjunctions in bicategories
For 1-morphisms `f : a ⟶ b` and `g : b... | Mathlib/CategoryTheory/Bicategory/Adjunction.lean | 232 | 247 | theorem adjointifyCounit_left_triangle (η : 𝟙 a ≅ f ≫ g) (ε : g ≫ f ≅ 𝟙 b) :
leftZigzagIso η (adjointifyCounit η ε) = λ_ f ≪≫ (ρ_ f).symm := by |
apply Iso.ext
dsimp [adjointifyCounit, bicategoricalIsoComp]
calc
_ = 𝟙 _ ⊗≫ (η.hom ▷ (f ≫ 𝟙 b) ≫ (f ≫ g) ◁ f ◁ ε.inv) ⊗≫
f ◁ g ◁ η.inv ▷ f ⊗≫ f ◁ ε.hom := by
simp [bicategoricalComp]; coherence
_ = 𝟙 _ ⊗≫ f ◁ ε.inv ⊗≫ (η.hom ▷ (f ≫ g) ≫ (f ≫ g) ◁ η.inv) ▷ f ⊗≫ f ◁ ε.hom := by
rw... |
/-
Copyright (c) 2021 Yaël Dillies. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yaël Dillies
-/
import Mathlib.Algebra.CharZero.Lemmas
import Mathlib.Order.Interval.Finset.Basic
#align_import data.int.interval from "leanprover-community/mathlib"@"1d29de43a5ba4662dd... | Mathlib/Data/Int/Interval.lean | 177 | 178 | theorem card_fintype_Ico_of_le (h : a ≤ b) : (Fintype.card (Set.Ico a b) : ℤ) = b - a := by |
rw [card_fintype_Ico, toNat_sub_of_le h]
|
/-
Copyright (c) 2020 Scott Morrison. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Scott Morrison
-/
import Mathlib.RepresentationTheory.Action.Limits
import Mathlib.RepresentationTheory.Action.Concrete
import Mathlib.CategoryTheory.Monoidal.FunctorCategory
import Ma... | Mathlib/RepresentationTheory/Action/Monoidal.lean | 413 | 415 | theorem mapAction_ε_inv_hom : (inv (F.mapAction G).ε).hom = inv F.ε := by |
rw [← cancel_mono F.ε, IsIso.inv_hom_id, ← F.mapAction_toLaxMonoidalFunctor_ε_hom G,
← Action.comp_hom, IsIso.inv_hom_id, Action.id_hom]
|
/-
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 | 212 | 214 | theorem ofReal_lt_ofReal_iff_of_nonneg {p q : ℝ} (hp : 0 ≤ p) :
ENNReal.ofReal p < ENNReal.ofReal q ↔ p < q := by |
rw [ENNReal.ofReal, ENNReal.ofReal, coe_lt_coe, Real.toNNReal_lt_toNNReal_iff_of_nonneg hp]
|
/-
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 | 117 | 119 | theorem map_mul : (I * J).map g = I.map g * J.map g := by |
simp only [mul_def]
exact coeToSubmodule_injective (Submodule.map_mul _ _ _)
|
/-
Copyright (c) 2023 Geoffrey Irving. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: David Loeffler, Geoffrey Irving
-/
import Mathlib.Analysis.Analytic.Composition
import Mathlib.Analysis.Analytic.Linear
import Mathlib.Analysis.NormedSpace.OperatorNorm.Mul
/-!
# Var... | Mathlib/Analysis/Analytic/Constructions.lean | 305 | 313 | theorem Finset.analyticAt_sum {f : α → E → F} {c : E}
(N : Finset α) (h : ∀ n ∈ N, AnalyticAt 𝕜 (f n) c) :
AnalyticAt 𝕜 (fun z ↦ ∑ n ∈ N, f n z) c := by |
induction' N using Finset.induction with a B aB hB
· simp only [Finset.sum_empty]
exact analyticAt_const
· simp_rw [Finset.sum_insert aB]
simp only [Finset.mem_insert] at h
exact (h a (Or.inl rfl)).add (hB fun b m ↦ h b (Or.inr m))
|
/-
Copyright (c) 2022 Yury G. Kudryashov. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Yury G. Kudryashov
-/
import Mathlib.Analysis.Asymptotics.Asymptotics
import Mathlib.Analysis.NormedSpace.Basic
#align_import analysis.asymptotics.theta from "leanprover-community... | Mathlib/Analysis/Asymptotics/Theta.lean | 224 | 226 | theorem IsTheta.isBoundedUnder_le_iff (h : f' =Θ[l] g') :
IsBoundedUnder (· ≤ ·) l (norm ∘ f') ↔ IsBoundedUnder (· ≤ ·) l (norm ∘ g') := by |
simp only [← isBigO_const_of_ne (one_ne_zero' ℝ), h.isBigO_congr_left]
|
/-
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
-/
import Mathlib.Data.Sum.Order
import Mathlib.Order.InitialSeg
import Mathlib.SetTheory.Cardinal.Basic
import Mathlib.Tactic.PPWithUniv
#align_impor... | Mathlib/SetTheory/Ordinal/Basic.lean | 238 | 238 | theorem type_ne_zero_iff_nonempty [IsWellOrder α r] : type r ≠ 0 ↔ Nonempty α := by | simp
|
/-
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 | 1,415 | 1,418 | theorem linearIndependent_fin2 {f : Fin 2 → V} :
LinearIndependent K f ↔ f 1 ≠ 0 ∧ ∀ a : K, a • f 1 ≠ f 0 := by |
rw [linearIndependent_fin_succ, linearIndependent_unique_iff, range_unique, mem_span_singleton,
not_exists, show Fin.tail f default = f 1 by rw [← Fin.succ_zero_eq_one]; rfl]
|
/-
Copyright (c) 2018 Chris Hughes. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Chris Hughes
-/
import Mathlib.Algebra.Order.Ring.Nat
import Mathlib.Algebra.Order.Monoid.WithTop
#align_import data.nat.with_bot from "leanprover-community/mathlib"@"966e0cf0685c9cedf... | Mathlib/Data/Nat/WithBot.lean | 81 | 85 | theorem add_one_le_of_lt {n m : WithBot ℕ} (h : n < m) : n + 1 ≤ m := by |
cases n
· exact bot_le
cases m
exacts [(not_lt_bot h).elim, WithBot.coe_le_coe.2 (WithBot.coe_lt_coe.1 h)]
|
/-
Copyright (c) 2019 Reid Barton. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Johannes Hölzl, Patrick Massot, Sébastien Gouëzel, Zhouhang Zhou, Reid Barton
-/
import Mathlib.Logic.Equiv.Fin
import Mathlib.Topology.DenseEmbedding
import Mathlib.Topology.Support
impo... | Mathlib/Topology/Homeomorph.lean | 409 | 410 | theorem image_closure (h : X ≃ₜ Y) (s : Set X) : h '' closure s = closure (h '' s) := by |
rw [← preimage_symm, preimage_closure]
|
/-
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 | 1,185 | 1,189 | theorem mapRange_def [∀ (i) (x : β₁ i), Decidable (x ≠ 0)] {f : ∀ i, β₁ i → β₂ i}
{hf : ∀ i, f i 0 = 0} {g : Π₀ i, β₁ i} :
mapRange f hf g = mk g.support fun i => f i.1 (g i.1) := by |
ext i
by_cases h : g i ≠ 0 <;> simp at h <;> simp [h, hf]
|
/-
Copyright (c) 2014 Jeremy Avigad. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Jeremy Avigad, Mario Carneiro, Yaël Dillies
-/
import Mathlib.Logic.Function.Iterate
import Mathlib.Init.Data.Int.Order
import Mathlib.Order.Compare
import Mathlib.Order.Max
import Math... | Mathlib/Order/Monotone/Basic.lean | 261 | 263 | theorem strictMonoOn_dual_iff :
StrictMonoOn (toDual ∘ f ∘ ofDual : αᵒᵈ → βᵒᵈ) s ↔ StrictMonoOn f s := by |
rw [strictMonoOn_toDual_comp_iff, strictAntiOn_comp_ofDual_iff]
|
/-
Copyright (c) 2017 Robert Y. Lewis. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Robert Y. Lewis, Keeley Hoek
-/
import Mathlib.Algebra.NeZero
import Mathlib.Data.Nat.Defs
import Mathlib.Logic.Embedding.Basic
import Mathlib.Logic.Equiv.Set
import Mathlib.Tactic.Co... | Mathlib/Data/Fin/Basic.lean | 1,894 | 1,898 | theorem exists_eq_add_of_le {n : ℕ} {a b : Fin n} (h : a ≤ b) : ∃ k ≤ b, b = a + k := by |
obtain ⟨k, hk⟩ : ∃ k : ℕ, (b : ℕ) = a + k := Nat.exists_eq_add_of_le h
have hkb : k ≤ b := by omega
refine ⟨⟨k, hkb.trans_lt b.is_lt⟩, hkb, ?_⟩
simp [Fin.ext_iff, Fin.val_add, ← hk, Nat.mod_eq_of_lt b.is_lt]
|
/-
Copyright (c) 2019 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel, Floris van Doorn, Heather Macbeth
-/
import Mathlib.Topology.FiberBundle.Trivialization
import Mathlib.Topology.Order.LeftRightNhds
#align_import topology.fiber_... | Mathlib/Topology/FiberBundle/Basic.lean | 715 | 716 | theorem mk_mem_localTrivAt_source : (⟨b, a⟩ : Z.TotalSpace) ∈ (Z.localTrivAt b).source := by |
simp only [mfld_simps]
|
/-
Copyright (c) 2020 Sébastien Gouëzel. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Sébastien Gouëzel
-/
import Mathlib.Topology.Separation
import Mathlib.Topology.UniformSpace.Basic
import Mathlib.Topology.UniformSpace.Cauchy
#align_import topology.uniform_space.... | Mathlib/Topology/UniformSpace/UniformConvergence.lean | 273 | 280 | theorem TendstoUniformlyOnFilter.prod_map {ι' α' β' : Type*} [UniformSpace β'] {F' : ι' → α' → β'}
{f' : α' → β'} {q : Filter ι'} {q' : Filter α'} (h : TendstoUniformlyOnFilter F f p p')
(h' : TendstoUniformlyOnFilter F' f' q q') :
TendstoUniformlyOnFilter (fun i : ι × ι' => Prod.map (F i.1) (F' i.2)) (Prod... |
rw [tendstoUniformlyOnFilter_iff_tendsto] at h h' ⊢
rw [uniformity_prod_eq_comap_prod, tendsto_comap_iff, ← map_swap4_prod, tendsto_map'_iff]
convert h.prod_map h' -- seems to be faster than `exact` here
|
/-
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 | 267 | 270 | theorem add_ofNat' (m n) : Num.ofNat' (m + n) = Num.ofNat' m + Num.ofNat' n := by |
induction n
· simp only [Nat.add_zero, ofNat'_zero, add_zero]
· simp only [Nat.add_succ, Nat.add_zero, ofNat'_succ, add_one, add_succ, *]
|
/-
Copyright (c) 2021 Anatole Dedecker. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Anatole Dedecker, Eric Wieser
-/
import Mathlib.Analysis.Analytic.Basic
import Mathlib.Analysis.Complex.Basic
import Mathlib.Analysis.Normed.Field.InfiniteSum
import Mathlib.Data.Nat... | Mathlib/Analysis/NormedSpace/Exponential.lean | 591 | 597 | theorem _root_.Function.update_exp {ι : Type*} {𝔸 : ι → Type*} [Finite ι] [DecidableEq ι]
[∀ i, NormedRing (𝔸 i)] [∀ i, NormedAlgebra 𝕂 (𝔸 i)] [∀ i, CompleteSpace (𝔸 i)] (x : ∀ i, 𝔸 i)
(j : ι) (xj : 𝔸 j) :
Function.update (exp 𝕂 x) j (exp 𝕂 xj) = exp 𝕂 (Function.update x j xj) := by |
ext i
simp_rw [Pi.exp_def]
exact (Function.apply_update (fun i => exp 𝕂) x j xj i).symm
|
/-
Copyright (c) 2020 Kenny Lau. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenny Lau, Ken Lee, Chris Hughes
-/
import Mathlib.Algebra.BigOperators.Ring
import Mathlib.Data.Fintype.Basic
import Mathlib.Data.Int.GCD
import Mathlib.RingTheory.Coprime.Basic
#align_im... | Mathlib/RingTheory/Coprime/Lemmas.lean | 73 | 76 | theorem IsCoprime.prod_left_iff : IsCoprime (∏ i ∈ t, s i) x ↔ ∀ i ∈ t, IsCoprime (s i) x := by |
classical
refine Finset.induction_on t (iff_of_true isCoprime_one_left fun _ ↦ by simp) fun b t hbt ih ↦ ?_
rw [Finset.prod_insert hbt, IsCoprime.mul_left_iff, ih, Finset.forall_mem_insert]
|
/-
Copyright (c) 2020 Kenji Nakagawa. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Kenji Nakagawa, Anne Baanen, Filippo A. E. Nuccio
-/
import Mathlib.Algebra.Algebra.Subalgebra.Pointwise
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Maximal
import Mathlib.Algebraic... | Mathlib/RingTheory/DedekindDomain/Ideal.lean | 1,262 | 1,268 | theorem Ideal.le_of_pow_le_prime {I P : Ideal R} [hP : P.IsPrime] {n : ℕ} (h : I ^ n ≤ P) :
I ≤ P := by |
by_cases hP0 : P = ⊥
· simp only [hP0, le_bot_iff] at h ⊢
exact pow_eq_zero h
rw [← Ideal.dvd_iff_le] at h ⊢
exact ((Ideal.prime_iff_isPrime hP0).mpr hP).dvd_of_dvd_pow h
|
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